diff --git "a/random/random_our/test_increase.jsonl" "b/random/random_our/test_increase.jsonl"
new file mode 100644--- /dev/null
+++ "b/random/random_our/test_increase.jsonl"
@@ -0,0 +1,3172 @@
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+{"state": [{"context": ["R : Type u_1", "α : Type u_2", "β : Type u_3", "δ : Type u_4", "γ : Type u_5", "ι : Type u_6", "m0 : MeasurableSpace α", "inst✝¹ : MeasurableSpace β", "inst✝ : MeasurableSpace γ", "μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α", "s s' t : Set α", "h : s ⊆ t", "u : Set α", "hu : MeasurableSet u"], "goal": "u ∩ s ⊆ t"}], "premise": [133448], "state_str": "case h.h\nR : Type u_1\nα : Type u_2\nβ : Type u_3\nδ : Type u_4\nγ : Type u_5\nι : Type u_6\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nh : s ⊆ t\nu : Set α\nhu : MeasurableSet u\n⊢ u ∩ s ⊆ t"}
+{"state": [{"context": ["α : Type u", "β : Type v", "l✝ l₁ l₂ : List α", "r : α → α → Prop", "a✝ b a : α", "l : List α", "IH : (∀ (a : α), ¬[a, a] <+ l) → l.Nodup", "h : ∀ (a_1 : α), ¬[a_1, a_1] <+ a :: l"], "goal": "(a :: l).Nodup"}], "premise": [1312, 1674, 129717, 132445, 132447], "state_str": "case cons\nα : Type u\nβ : Type v\nl✝ l₁ l₂ : List α\nr : α → α → Prop\na✝ b a : α\nl : List α\nIH : (∀ (a : α), ¬[a, a] <+ l) → l.Nodup\nh : ∀ (a_1 : α), ¬[a_1, a_1] <+ a :: l\n⊢ (a :: l).Nodup"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "κ✝ : Kernel α β", "s t✝ : Set β", "κ : Kernel α β", "hs : MeasurableSet s", "a : α", "f : β → ℝ≥0∞", "t : Set β"], "goal": "∫⁻ (b : β) in t, f b ∂(κ.restrict hs) a = ∫⁻ (b : β) in t ∩ s, f b ∂κ a"}], "premise": [32236, 72667], "state_str": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ✝ : Kernel α β\ns t✝ : Set β\nκ : Kernel α β\nhs : MeasurableSet s\na : α\nf : β → ℝ≥0∞\nt : Set β\n⊢ ∫⁻ (b : β) in t, f b ∂(κ.restrict hs) a = ∫⁻ (b : β) in t ∩ s, f b ∂κ a"}
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+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "E : Type u_3", "F : Type u_4", "G : Type u_5", "E' : Type u_6", "F' : Type u_7", "G' : Type u_8", "E'' : Type u_9", "F'' : Type u_10", "G'' : Type u_11", "E''' : Type u_12", "R : Type u_13", "R' : Type u_14", "𝕜 : Type u_15", "𝕜' : Type u_16", "inst✝¹³ : Norm E", "inst✝¹² : Norm F", "inst✝¹¹ : Norm G", "inst✝¹⁰ : SeminormedAddCommGroup E'", "inst✝⁹ : SeminormedAddCommGroup F'", "inst✝⁸ : SeminormedAddCommGroup G'", "inst✝⁷ : NormedAddCommGroup E''", "inst✝⁶ : NormedAddCommGroup F''", "inst✝⁵ : NormedAddCommGroup G''", "inst✝⁴ : SeminormedRing R", "inst✝³ : SeminormedAddGroup E'''", "inst✝² : SeminormedRing R'", "inst✝¹ : NormedDivisionRing 𝕜", "inst✝ : NormedDivisionRing 𝕜'", "c c' c₁ c₂ : ℝ", "f : α → E", "g : α → F", "k✝ : α → G", "f' : α → E'", "g' : α → F'", "k' : α → G'", "f'' : α → E''", "g'' : α → F''", "k'' : α → G''", "l✝ l' : Filter α", "k : β → α", "l : Filter β"], "goal": "(∀ᶠ (x : α) in map k l, ‖f x‖ ≤ c * ‖g x‖) ↔ ∀ᶠ (x : β) in l, ‖(f ∘ k) x‖ ≤ c * ‖(g ∘ k) x‖"}], "premise": [16164], "state_str": "α : Type u_1\nβ : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n𝕜 : Type u_15\n𝕜' : Type u_16\ninst✝¹³ : Norm E\ninst✝¹² : Norm F\ninst✝¹¹ : Norm G\ninst✝¹⁰ : SeminormedAddCommGroup E'\ninst✝⁹ : SeminormedAddCommGroup F'\ninst✝⁸ : SeminormedAddCommGroup G'\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedAddCommGroup F''\ninst✝⁵ : NormedAddCommGroup G''\ninst✝⁴ : SeminormedRing R\ninst✝³ : SeminormedAddGroup E'''\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedDivisionRing 𝕜\ninst✝ : NormedDivisionRing 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk✝ : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl✝ l' : Filter α\nk : β → α\nl : Filter β\n⊢ (∀ᶠ (x : α) in map k l, ‖f x‖ ≤ c * ‖g x‖) ↔ ∀ᶠ (x : β) in l, ‖(f ∘ k) x‖ ≤ c * ‖(g ∘ k) x‖"}
+{"state": [{"context": ["𝕜 : Type u", "inst✝⁸ : NontriviallyNormedField 𝕜", "F : Type v", "inst✝⁷ : NormedAddCommGroup F", "inst✝⁶ : NormedSpace 𝕜 F", "E : Type w", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace 𝕜 E", "f f₀ f₁ g : 𝕜 → F", "f' f₀' f₁' g' : F", "x : 𝕜", "s t : Set 𝕜", "L L₁ L₂ : Filter 𝕜", "𝕜' : Type u_1", "inst✝³ : NontriviallyNormedField 𝕜'", "inst✝² : NormedAlgebra 𝕜 𝕜'", "inst✝¹ : NormedSpace 𝕜' F", "inst✝ : IsScalarTower 𝕜 𝕜' F", "s' t' : Set 𝕜'", "h : 𝕜 → 𝕜'", "h₁ : 𝕜 → 𝕜", "h₂ : 𝕜' → 𝕜'", "h' h₂' : 𝕜'", "h₁' : 𝕜", "g₁ : 𝕜' → F", "g₁' : F", "L' : Filter 𝕜'", "y : 𝕜'", "hg : HasStrictDerivAt g₁ g₁' (h x)", "hh : HasStrictDerivAt h h' x"], "goal": "HasStrictDerivAt (g₁ ∘ h) (h' • g₁') x"}], "premise": [44335, 44942, 45128], "state_str": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nE : Type w\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\n𝕜' : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜'\ninst✝² : NormedAlgebra 𝕜 𝕜'\ninst✝¹ : NormedSpace 𝕜' F\ninst✝ : IsScalarTower 𝕜 𝕜' F\ns' t' : Set 𝕜'\nh : 𝕜 → 𝕜'\nh₁ : 𝕜 → 𝕜\nh₂ : 𝕜' → 𝕜'\nh' h₂' : 𝕜'\nh₁' : 𝕜\ng₁ : 𝕜' → F\ng₁' : F\nL' : Filter 𝕜'\ny : 𝕜'\nhg : HasStrictDerivAt g₁ g₁' (h x)\nhh : HasStrictDerivAt h h' x\n⊢ HasStrictDerivAt (g₁ ∘ h) (h' • g₁') x"}
+{"state": [{"context": ["ι : Type u", "E : Type v", "F : Type w", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "I✝ J : Box ι", "π : TaggedPrepartition I✝", "inst✝² : Fintype ι", "l : IntegrationParams", "f✝ g : (ι → ℝ) → E", "vol : ι →ᵇᵃ[⊤] E →L[ℝ] F", "y y' : F", "inst✝¹ : CompleteSpace E", "I : Box ι", "f : (ι → ℝ) → E", "hb : ∃ C, ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C", "μ : Measure (ι → ℝ)", "inst✝ : IsLocallyFiniteMeasure μ", "hc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x"], "goal": "Integrable I l f μ.toBoxAdditive.toSMul"}], "premise": [1674, 36534], "state_str": "ι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nhb : ∃ C, ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\n⊢ Integrable I l f μ.toBoxAdditive.toSMul"}
+{"state": [{"context": ["ι : Type u", "E : Type v", "F : Type w", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "I✝ J : Box ι", "π : TaggedPrepartition I✝", "inst✝² : Fintype ι", "l : IntegrationParams", "f✝ g : (ι → ℝ) → E", "vol : ι →ᵇᵃ[⊤] E →L[ℝ] F", "y y' : F", "inst✝¹ : CompleteSpace E", "I : Box ι", "f : (ι → ℝ) → E", "hb : ∃ C, ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C", "μ : Measure (ι → ℝ)", "inst✝ : IsLocallyFiniteMeasure μ", "hc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x", "ε : ℝ", "ε0 : ε > 0"], "goal": "∃ r, (∀ (c : ℝ≥0), l.RCond (r c)) ∧ ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I), l.MemBaseSet I c₁ (r c₁) π₁ → π₁.IsPartition → l.MemBaseSet I c₂ (r c₂) π₂ → π₂.IsPartition → dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε"}], "premise": [106115], "state_str": "ι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nhb : ∃ C, ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\n⊢ ∃ r,\n    (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),\n        l.MemBaseSet I c₁ (r c₁) π₁ →\n          π₁.IsPartition →\n            l.MemBaseSet I c₂ (r c₂) π₂ →\n              π₂.IsPartition →\n                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε"}
+{"state": [{"context": ["ι : Type u", "E : Type v", "F : Type w", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "I✝ J : Box ι", "π : TaggedPrepartition I✝", "inst✝² : Fintype ι", "l : IntegrationParams", "f✝ g : (ι → ℝ) → E", "vol : ι →ᵇᵃ[⊤] E →L[ℝ] F", "y y' : F", "inst✝¹ : CompleteSpace E", "I : Box ι", "f : (ι → ℝ) → E", "μ : Measure (ι → ℝ)", "inst✝ : IsLocallyFiniteMeasure μ", "hc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x", "ε : ℝ", "ε0 : ε > 0", "ε₁ : ℝ", "ε₁0 : 0 < ε₁", "hε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε", "C : ℝ", "hC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C"], "goal": "∃ r, (∀ (c : ℝ≥0), l.RCond (r c)) ∧ ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I), l.MemBaseSet I c₁ (r c₁) π₁ → π₁.IsPartition → l.MemBaseSet I c₂ (r c₂) π₂ → π₂.IsPartition → dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε"}], "premise": [14273, 34344, 34363, 42680], "state_str": "case intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\n⊢ ∃ r,\n    (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),\n        l.MemBaseSet I c₁ (r c₁) π₁ →\n          π₁.IsPartition →\n            l.MemBaseSet I c₂ (r c₂) π₂ →\n              π₂.IsPartition →\n                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε"}
+{"state": [{"context": ["ι : Type u", "E : Type v", "F : Type w", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "I✝ J : Box ι", "π : TaggedPrepartition I✝", "inst✝² : Fintype ι", "l : IntegrationParams", "f✝ g : (ι → ℝ) → E", "vol : ι →ᵇᵃ[⊤] E →L[ℝ] F", "y y' : F", "inst✝¹ : CompleteSpace E", "I : Box ι", "f : (ι → ℝ) → E", "μ : Measure (ι → ℝ)", "inst✝ : IsLocallyFiniteMeasure μ", "hc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x", "ε : ℝ", "ε0 : ε > 0", "ε₁ : ℝ", "ε₁0 : 0 < ε₁", "hε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε", "C : ℝ", "hC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C", "C0 : 0 ≤ C"], "goal": "∃ r, (∀ (c : ℝ≥0), l.RCond (r c)) ∧ ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I), l.MemBaseSet I c₁ (r c₁) π₁ → π₁.IsPartition → l.MemBaseSet I c₂ (r c₂) π₂ → π₂.IsPartition → dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε"}], "premise": [106115], "state_str": "case intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\n⊢ ∃ r,\n    (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),\n        l.MemBaseSet I c₁ (r c₁) π₁ →\n          π₁.IsPartition →\n            l.MemBaseSet I c₂ (r c₂) π₂ →\n              π₂.IsPartition →\n                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε"}
+{"state": [{"context": ["ι : Type u", "E : Type v", "F : Type w", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "I✝ J : Box ι", "π : TaggedPrepartition I✝", "inst✝² : Fintype ι", "l : IntegrationParams", "f✝ g : (ι → ℝ) → E", "vol : ι →ᵇᵃ[⊤] E →L[ℝ] F", "y y' : F", "inst✝¹ : CompleteSpace E", "I : Box ι", "f : (ι → ℝ) → E", "μ : Measure (ι → ℝ)", "inst✝ : IsLocallyFiniteMeasure μ", "hc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x", "ε : ℝ", "ε0 : ε > 0", "ε₁ : ℝ", "ε₁0 : 0 < ε₁", "hε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε", "C : ℝ", "hC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C", "C0 : 0 ≤ C", "ε₂ : ℝ", "ε₂0 : 0 < ε₂", "hε₂ : 4 * C * ε₂ < ε"], "goal": "∃ r, (∀ (c : ℝ≥0), l.RCond (r c)) ∧ ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I), l.MemBaseSet I c₁ (r c₁) π₁ → π₁.IsPartition → l.MemBaseSet I c₂ (r c₂) π₂ → π₂.IsPartition → dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε"}], "premise": [1674, 14284, 143385], "state_str": "case intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\n⊢ ∃ r,\n    (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),\n        l.MemBaseSet I c₁ (r c₁) π₁ →\n          π₁.IsPartition →\n            l.MemBaseSet I c₂ (r c₂) π₂ →\n              π₂.IsPartition →\n                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε"}
+{"state": [{"context": ["ι : Type u", "E : Type v", "F : Type w", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "I✝ J : Box ι", "π : TaggedPrepartition I✝", "inst✝² : Fintype ι", "l : IntegrationParams", "f✝ g : (ι → ℝ) → E", "vol : ι →ᵇᵃ[⊤] E →L[ℝ] F", "y y' : F", "inst✝¹ : CompleteSpace E", "I : Box ι", "f : (ι → ℝ) → E", "μ : Measure (ι → ℝ)", "inst✝ : IsLocallyFiniteMeasure μ", "hc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x", "ε : ℝ", "ε0 : ε > 0", "ε₁ : ℝ", "ε₁0 : 0 < ε₁", "hε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε", "C : ℝ", "hC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C", "C0 : 0 ≤ C", "ε₂ : ℝ", "ε₂0 : 0 < ε₂", "hε₂ : 4 * C * ε₂ < ε", "ε₂0' : ENNReal.ofReal ε₂ ≠ 0", "D : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}", "μ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)"], "goal": "∃ r, (∀ (c : ℝ≥0), l.RCond (r c)) ∧ ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I), l.MemBaseSet I c₁ (r c₁) π₁ → π₁.IsPartition → l.MemBaseSet I c₂ (r c₂) π₂ → π₂.IsPartition → dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε"}], "premise": [1673, 2106, 11251, 16024, 27511, 27599, 143493], "state_str": "case intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\n⊢ ∃ r,\n    (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),\n        l.MemBaseSet I c₁ (r c₁) π₁ →\n          π₁.IsPartition →\n            l.MemBaseSet I c₂ (r c₂) π₂ →\n              π₂.IsPartition →\n                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε"}
+{"state": [{"context": ["ι : Type u", "E : Type v", "F : Type w", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "I✝ J : Box ι", "π : TaggedPrepartition I✝", "inst✝² : Fintype ι", "l : IntegrationParams", "f✝ g : (ι → ℝ) → E", "vol : ι →ᵇᵃ[⊤] E →L[ℝ] F", "y y' : F", "inst✝¹ : CompleteSpace E", "I : Box ι", "f : (ι → ℝ) → E", "μ : Measure (ι → ℝ)", "inst✝ : IsLocallyFiniteMeasure μ", "hc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x", "ε : ℝ", "ε0 : ε > 0", "ε₁ : ℝ", "ε₁0 : 0 < ε₁", "hε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε", "C : ℝ", "hC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C", "C0 : 0 ≤ C", "ε₂ : ℝ", "ε₂0 : 0 < ε₂", "hε₂ : 4 * C * ε₂ < ε", "ε₂0' : ENNReal.ofReal ε₂ ≠ 0", "D : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}", "μ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)", "μ'D : μ' D = 0"], "goal": "∃ r, (∀ (c : ℝ≥0), l.RCond (r c)) ∧ ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I), l.MemBaseSet I c₁ (r c₁) π₁ → π₁.IsPartition → l.MemBaseSet I c₂ (r c₂) π₂ → π₂.IsPartition → dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε"}], "premise": [30620], "state_str": "case intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\n⊢ ∃ r,\n    (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),\n        l.MemBaseSet I c₁ (r c₁) π₁ →\n          π₁.IsPartition →\n            l.MemBaseSet I c₂ (r c₂) π₂ →\n              π₂.IsPartition →\n                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε"}
+{"state": [{"context": ["ι : Type u", "E : Type v", "F : Type w", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "I✝ J : Box ι", "π : TaggedPrepartition I✝", "inst✝² : Fintype ι", "l : IntegrationParams", "f✝ g : (ι → ℝ) → E", "vol : ι →ᵇᵃ[⊤] E →L[ℝ] F", "y y' : F", "inst✝¹ : CompleteSpace E", "I : Box ι", "f : (ι → ℝ) → E", "μ : Measure (ι → ℝ)", "inst✝ : IsLocallyFiniteMeasure μ", "hc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x", "ε : ℝ", "ε0 : ε > 0", "ε₁ : ℝ", "ε₁0 : 0 < ε₁", "hε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε", "C : ℝ", "hC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C", "C0 : 0 ≤ C", "ε₂ : ℝ", "ε₂0 : 0 < ε₂", "hε₂ : 4 * C * ε₂ < ε", "ε₂0' : ENNReal.ofReal ε₂ ≠ 0", "D : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}", "μ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)", "μ'D : μ' D = 0", "U : Set (ι → ℝ)", "UD : U ⊇ D", "Uopen : IsOpen U", "hU : μ' U < μ' D + ENNReal.ofReal ε₂"], "goal": "∃ r, (∀ (c : ℝ≥0), l.RCond (r c)) ∧ ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I), l.MemBaseSet I c₁ (r c₁) π₁ → π₁.IsPartition → l.MemBaseSet I c₂ (r c₂) π₂ → π₂.IsPartition → dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε"}], "premise": [119727], "state_str": "case intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < μ' D + ENNReal.ofReal ε₂\n⊢ ∃ r,\n    (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),\n        l.MemBaseSet I c₁ (r c₁) π₁ →\n          π₁.IsPartition →\n            l.MemBaseSet I c₂ (r c₂) π₂ →\n              π₂.IsPartition →\n                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε"}
+{"state": [{"context": ["ι : Type u", "E : Type v", "F : Type w", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "I✝ J : Box ι", "π : TaggedPrepartition I✝", "inst✝² : Fintype ι", "l : IntegrationParams", "f✝ g : (ι → ℝ) → E", "vol : ι →ᵇᵃ[⊤] E →L[ℝ] F", "y y' : F", "inst✝¹ : CompleteSpace E", "I : Box ι", "f : (ι → ℝ) → E", "μ : Measure (ι → ℝ)", "inst✝ : IsLocallyFiniteMeasure μ", "hc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x", "ε : ℝ", "ε0 : ε > 0", "ε₁ : ℝ", "ε₁0 : 0 < ε₁", "hε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε", "C : ℝ", "hC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C", "C0 : 0 ≤ C", "ε₂ : ℝ", "ε₂0 : 0 < ε₂", "hε₂ : 4 * C * ε₂ < ε", "ε₂0' : ENNReal.ofReal ε₂ ≠ 0", "D : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}", "μ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)", "μ'D : μ' D = 0", "U : Set (ι → ℝ)", "UD : U ⊇ D", "Uopen : IsOpen U", "hU : μ' U < ENNReal.ofReal ε₂"], "goal": "∃ r, (∀ (c : ℝ≥0), l.RCond (r c)) ∧ ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I), l.MemBaseSet I c₁ (r c₁) π₁ → π₁.IsPartition → l.MemBaseSet I c₂ (r c₂) π₂ → π₂.IsPartition → dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε"}], "premise": [34358, 55373, 58062, 64866, 133635], "state_str": "case intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\n⊢ ∃ r,\n    (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),\n        l.MemBaseSet I c₁ (r c₁) π₁ →\n          π₁.IsPartition →\n            l.MemBaseSet I c₂ (r c₂) π₂ →\n              π₂.IsPartition →\n                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε"}
+{"state": [{"context": ["ι : Type u", "E : Type v", "F : Type w", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "I✝ J : Box ι", "π : TaggedPrepartition I✝", "inst✝² : Fintype ι", "l : IntegrationParams", "f✝ g : (ι → ℝ) → E", "vol : ι →ᵇᵃ[⊤] E →L[ℝ] F", "y y' : F", "inst✝¹ : CompleteSpace E", "I : Box ι", "f : (ι → ℝ) → E", "μ : Measure (ι → ℝ)", "inst✝ : IsLocallyFiniteMeasure μ", "hc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x", "ε : ℝ", "ε0 : ε > 0", "ε₁ : ℝ", "ε₁0 : 0 < ε₁", "hε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε", "C : ℝ", "hC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C", "C0 : 0 ≤ C", "ε₂ : ℝ", "ε₂0 : 0 < ε₂", "hε₂ : 4 * C * ε₂ < ε", "ε₂0' : ENNReal.ofReal ε₂ ≠ 0", "D : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}", "μ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)", "μ'D : μ' D = 0", "U : Set (ι → ℝ)", "UD : U ⊇ D", "Uopen : IsOpen U", "hU : μ' U < ENNReal.ofReal ε₂", "comp : IsCompact (Box.Icc I \\ U)"], "goal": "∃ r, (∀ (c : ℝ≥0), l.RCond (r c)) ∧ ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I), l.MemBaseSet I c₁ (r c₁) π₁ → π₁.IsPartition → l.MemBaseSet I c₂ (r c₂) π₂ → π₂.IsPartition → dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε"}], "premise": [1674, 2106, 2107, 40867, 143385], "state_str": "case intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\ncomp : IsCompact (Box.Icc I \\ U)\n⊢ ∃ r,\n    (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),\n        l.MemBaseSet I c₁ (r c₁) π₁ →\n          π₁.IsPartition →\n            l.MemBaseSet I c₂ (r c₂) π₂ →\n              π₂.IsPartition →\n                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε"}
+{"state": [{"context": ["ι : Type u", "E : Type v", "F : Type w", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "I✝ J : Box ι", "π : TaggedPrepartition I✝", "inst✝² : Fintype ι", "l : IntegrationParams", "f✝ g : (ι → ℝ) → E", "vol : ι →ᵇᵃ[⊤] E →L[ℝ] F", "y y' : F", "inst✝¹ : CompleteSpace E", "I : Box ι", "f : (ι → ℝ) → E", "μ : Measure (ι → ℝ)", "inst✝ : IsLocallyFiniteMeasure μ", "hc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x", "ε : ℝ", "ε0 : ε > 0", "ε₁ : ℝ", "ε₁0 : 0 < ε₁", "hε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε", "C : ℝ", "hC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C", "C0 : 0 ≤ C", "ε₂ : ℝ", "ε₂0 : 0 < ε₂", "hε₂ : 4 * C * ε₂ < ε", "ε₂0' : ENNReal.ofReal ε₂ ≠ 0", "D : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}", "μ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)", "μ'D : μ' D = 0", "U : Set (ι → ℝ)", "UD : U ⊇ D", "Uopen : IsOpen U", "hU : μ' U < ENNReal.ofReal ε₂", "comp : IsCompact (Box.Icc I \\ U)", "this : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁"], "goal": "∃ r, (∀ (c : ℝ≥0), l.RCond (r c)) ∧ ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I), l.MemBaseSet I c₁ (r c₁) π₁ → π₁.IsPartition → l.MemBaseSet I c₂ (r c₂) π₂ → π₂.IsPartition → dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε"}], "premise": [40869], "state_str": "case intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\ncomp : IsCompact (Box.Icc I \\ U)\nthis : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁\n⊢ ∃ r,\n    (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),\n        l.MemBaseSet I c₁ (r c₁) π₁ →\n          π₁.IsPartition →\n            l.MemBaseSet I c₂ (r c₂) π₂ →\n              π₂.IsPartition →\n                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε"}
+{"state": [{"context": ["ι : Type u", "E : Type v", "F : Type w", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "I✝ J : Box ι", "π : TaggedPrepartition I✝", "inst✝² : Fintype ι", "l : IntegrationParams", "f✝ g : (ι → ℝ) → E", "vol : ι →ᵇᵃ[⊤] E →L[ℝ] F", "y y' : F", "inst✝¹ : CompleteSpace E", "I : Box ι", "f : (ι → ℝ) → E", "μ : Measure (ι → ℝ)", "inst✝ : IsLocallyFiniteMeasure μ", "hc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x", "ε : ℝ", "ε0 : ε > 0", "ε₁ : ℝ", "ε₁0 : 0 < ε₁", "hε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε", "C : ℝ", "hC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C", "C0 : 0 ≤ C", "ε₂ : ℝ", "ε₂0 : 0 < ε₂", "hε₂ : 4 * C * ε₂ < ε", "ε₂0' : ENNReal.ofReal ε₂ ≠ 0", "D : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}", "μ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)", "μ'D : μ' D = 0", "U : Set (ι → ℝ)", "UD : U ⊇ D", "Uopen : IsOpen U", "hU : μ' U < ENNReal.ofReal ε₂", "comp : IsCompact (Box.Icc I \\ U)", "this : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁", "r : ℝ", "r0 : r > 0", "hr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁"], "goal": "∃ r, (∀ (c : ℝ≥0), l.RCond (r c)) ∧ ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I), l.MemBaseSet I c₁ (r c₁) π₁ → π₁.IsPartition → l.MemBaseSet I c₂ (r c₂) π₂ → π₂.IsPartition → dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε"}], "premise": [106102], "state_str": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\ncomp : IsCompact (Box.Icc I \\ U)\nthis : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁\nr : ℝ\nr0 : r > 0\nhr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁\n⊢ ∃ r,\n    (∀ (c : ℝ≥0), l.RCond (r c)) ∧\n      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),\n        l.MemBaseSet I c₁ (r c₁) π₁ →\n          π₁.IsPartition →\n            l.MemBaseSet I c₂ (r c₂) π₂ →\n              π₂.IsPartition →\n                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε"}
+{"state": [{"context": ["ι : Type u", "E : Type v", "F : Type w", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "I✝ J : Box ι", "π : TaggedPrepartition I✝", "inst✝² : Fintype ι", "l : IntegrationParams", "f✝ g : (ι → ℝ) → E", "vol : ι →ᵇᵃ[⊤] E →L[ℝ] F", "y y' : F", "inst✝¹ : CompleteSpace E", "I : Box ι", "f : (ι → ℝ) → E", "μ : Measure (ι → ℝ)", "inst✝ : IsLocallyFiniteMeasure μ", "hc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x", "ε : ℝ", "ε0 : ε > 0", "ε₁ : ℝ", "ε₁0 : 0 < ε₁", "hε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε", "C : ℝ", "hC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C", "C0 : 0 ≤ C", "ε₂ : ℝ", "ε₂0 : 0 < ε₂", "hε₂ : 4 * C * ε₂ < ε", "ε₂0' : ENNReal.ofReal ε₂ ≠ 0", "D : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}", "μ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)", "μ'D : μ' D = 0", "U : Set (ι → ℝ)", "UD : U ⊇ D", "Uopen : IsOpen U", "hU : μ' U < ENNReal.ofReal ε₂", "comp : IsCompact (Box.Icc I \\ U)", "this : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁", "r : ℝ", "r0 : r > 0", "hr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁", "c₁ c₂ : ℝ≥0", "π₁ π₂ : TaggedPrepartition I", "h₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁", "h₁p : π₁.IsPartition", "h₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂", "h₂p : π₂.IsPartition"], "goal": "dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε"}], "premise": [33981, 36525, 108341], "state_str": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\ncomp : IsCompact (Box.Icc I \\ U)\nthis : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁\nr : ℝ\nr0 : r > 0\nhr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁\nc₁ c₂ : ℝ≥0\nπ₁ π₂ : TaggedPrepartition I\nh₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁\nh₁p : π₁.IsPartition\nh₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂\nh₂p : π₂.IsPartition\n⊢ dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε"}
+{"state": [{"context": ["ι : Type u", "E : Type v", "F : Type w", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "I✝ J : Box ι", "π : TaggedPrepartition I✝", "inst✝² : Fintype ι", "l : IntegrationParams", "f✝ g : (ι → ℝ) → E", "vol : ι →ᵇᵃ[⊤] E →L[ℝ] F", "y y' : F", "inst✝¹ : CompleteSpace E", "I : Box ι", "f : (ι → ℝ) → E", "μ : Measure (ι → ℝ)", "inst✝ : IsLocallyFiniteMeasure μ", "hc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x", "ε : ℝ", "ε0 : ε > 0", "ε₁ : ℝ", "ε₁0 : 0 < ε₁", "hε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε", "C : ℝ", "hC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C", "C0 : 0 ≤ C", "ε₂ : ℝ", "ε₂0 : 0 < ε₂", "hε₂ : 4 * C * ε₂ < ε", "ε₂0' : ENNReal.ofReal ε₂ ≠ 0", "D : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}", "μ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)", "μ'D : μ' D = 0", "U : Set (ι → ℝ)", "UD : U ⊇ D", "Uopen : IsOpen U", "hU : μ' U < ENNReal.ofReal ε₂", "comp : IsCompact (Box.Icc I \\ U)", "this : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁", "r : ℝ", "r0 : r > 0", "hr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁", "c₁ c₂ : ℝ≥0", "π₁ π₂ : TaggedPrepartition I", "h₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁", "h₁p : π₁.IsPartition", "h₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂", "h₂p : π₂.IsPartition"], "goal": "‖∑ x ∈ (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes, μ.toBoxAdditive x • (f ((π₁.infPrepartition π₂.toPrepartition).tag x) - f ((π₂.infPrepartition π₁.toPrepartition).tag x))‖ ≤ ε"}], "premise": [14288, 27511, 31842, 34358, 34363], "state_str": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\ncomp : IsCompact (Box.Icc I \\ U)\nthis : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁\nr : ℝ\nr0 : r > 0\nhr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁\nc₁ c₂ : ℝ≥0\nπ₁ π₂ : TaggedPrepartition I\nh₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁\nh₁p : π₁.IsPartition\nh₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂\nh₂p : π₂.IsPartition\n⊢ ‖∑ x ∈ (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes,\n        μ.toBoxAdditive x •\n          (f ((π₁.infPrepartition π₂.toPrepartition).tag x) - f ((π₂.infPrepartition π₁.toPrepartition).tag x))‖ ≤\n    ε"}
+{"state": [{"context": ["ι : Type u", "E : Type v", "F : Type w", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "I✝ J : Box ι", "π : TaggedPrepartition I✝", "inst✝² : Fintype ι", "l : IntegrationParams", "f✝ g : (ι → ℝ) → E", "vol : ι →ᵇᵃ[⊤] E →L[ℝ] F", "y y' : F", "inst✝¹ : CompleteSpace E", "I : Box ι", "f : (ι → ℝ) → E", "μ : Measure (ι → ℝ)", "inst✝ : IsLocallyFiniteMeasure μ", "hc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x", "ε : ℝ", "ε0 : ε > 0", "ε₁ : ℝ", "ε₁0 : 0 < ε₁", "hε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε", "C : ℝ", "hC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C", "C0 : 0 ≤ C", "ε₂ : ℝ", "ε₂0 : 0 < ε₂", "hε₂ : 4 * C * ε₂ < ε", "ε₂0' : ENNReal.ofReal ε₂ ≠ 0", "D : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}", "μ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)", "μ'D : μ' D = 0", "U : Set (ι → ℝ)", "UD : U ⊇ D", "Uopen : IsOpen U", "hU : μ' U < ENNReal.ofReal ε₂", "comp : IsCompact (Box.Icc I \\ U)", "this : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁", "r : ℝ", "r0 : r > 0", "hr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁", "c₁ c₂ : ℝ≥0", "π₁ π₂ : TaggedPrepartition I", "h₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁", "h₁p : π₁.IsPartition", "h₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂", "h₂p : π₂.IsPartition", "μI : μ ↑I < ⊤", "t₁ : Box ι → ι → ℝ := fun J => (π₁.infPrepartition π₂.toPrepartition).tag J", "t₂ : Box ι → ι → ℝ := fun J => (π₂.infPrepartition π₁.toPrepartition).tag J", "B : Finset (Box ι) := (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes", "B' : Finset (Box ι) := Finset.filter (fun J => ↑J ⊆ U) B"], "goal": "‖∑ x ∈ (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes, μ.toBoxAdditive x • (f ((π₁.infPrepartition π₂.toPrepartition).tag x) - f ((π₂.infPrepartition π₁.toPrepartition).tag x))‖ ≤ ε"}], "premise": [139088], "state_str": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\ncomp : IsCompact (Box.Icc I \\ U)\nthis : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁\nr : ℝ\nr0 : r > 0\nhr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁\nc₁ c₂ : ℝ≥0\nπ₁ π₂ : TaggedPrepartition I\nh₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁\nh₁p : π₁.IsPartition\nh₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂\nh₂p : π₂.IsPartition\nμI : μ ↑I < ⊤\nt₁ : Box ι → ι → ℝ := fun J => (π₁.infPrepartition π₂.toPrepartition).tag J\nt₂ : Box ι → ι → ℝ := fun J => (π₂.infPrepartition π₁.toPrepartition).tag J\nB : Finset (Box ι) := (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes\nB' : Finset (Box ι) := Finset.filter (fun J => ↑J ⊆ U) B\n⊢ ‖∑ x ∈ (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes,\n        μ.toBoxAdditive x •\n          (f ((π₁.infPrepartition π₂.toPrepartition).tag x) - f ((π₂.infPrepartition π₁.toPrepartition).tag x))‖ ≤\n    ε"}
+{"state": [{"context": ["ι : Type u", "E : Type v", "F : Type w", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "I✝ J : Box ι", "π : TaggedPrepartition I✝", "inst✝² : Fintype ι", "l : IntegrationParams", "f✝ g : (ι → ℝ) → E", "vol : ι →ᵇᵃ[⊤] E →L[ℝ] F", "y y' : F", "inst✝¹ : CompleteSpace E", "I : Box ι", "f : (ι → ℝ) → E", "μ : Measure (ι → ℝ)", "inst✝ : IsLocallyFiniteMeasure μ", "hc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x", "ε : ℝ", "ε0 : ε > 0", "ε₁ : ℝ", "ε₁0 : 0 < ε₁", "hε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε", "C : ℝ", "hC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C", "C0 : 0 ≤ C", "ε₂ : ℝ", "ε₂0 : 0 < ε₂", "hε₂ : 4 * C * ε₂ < ε", "ε₂0' : ENNReal.ofReal ε₂ ≠ 0", "D : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}", "μ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)", "μ'D : μ' D = 0", "U : Set (ι → ℝ)", "UD : U ⊇ D", "Uopen : IsOpen U", "hU : μ' U < ENNReal.ofReal ε₂", "comp : IsCompact (Box.Icc I \\ U)", "this : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁", "r : ℝ", "r0 : r > 0", "hr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁", "c₁ c₂ : ℝ≥0", "π₁ π₂ : TaggedPrepartition I", "h₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁", "h₁p : π₁.IsPartition", "h₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂", "h₂p : π₂.IsPartition", "μI : μ ↑I < ⊤", "t₁ : Box ι → ι → ℝ := fun J => (π₁.infPrepartition π₂.toPrepartition).tag J", "t₂ : Box ι → ι → ℝ := fun J => (π₂.infPrepartition π₁.toPrepartition).tag J", "B : Finset (Box ι) := (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes", "B' : Finset (Box ι) := Finset.filter (fun J => ↑J ⊆ U) B", "hB' : B' ⊆ B"], "goal": "‖∑ x ∈ (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes, μ.toBoxAdditive x • (f ((π₁.infPrepartition π₂.toPrepartition).tag x) - f ((π₂.infPrepartition π₁.toPrepartition).tag x))‖ ≤ ε"}], "premise": [1673, 14288, 18791, 27511, 34168], "state_str": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\ncomp : IsCompact (Box.Icc I \\ U)\nthis : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁\nr : ℝ\nr0 : r > 0\nhr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁\nc₁ c₂ : ℝ≥0\nπ₁ π₂ : TaggedPrepartition I\nh₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁\nh₁p : π₁.IsPartition\nh₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂\nh₂p : π₂.IsPartition\nμI : μ ↑I < ⊤\nt₁ : Box ι → ι → ℝ := fun J => (π₁.infPrepartition π₂.toPrepartition).tag J\nt₂ : Box ι → ι → ℝ := fun J => (π₂.infPrepartition π₁.toPrepartition).tag J\nB : Finset (Box ι) := (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes\nB' : Finset (Box ι) := Finset.filter (fun J => ↑J ⊆ U) B\nhB' : B' ⊆ B\n⊢ ‖∑ x ∈ (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes,\n        μ.toBoxAdditive x •\n          (f ((π₁.infPrepartition π₂.toPrepartition).tag x) - f ((π₂.infPrepartition π₁.toPrepartition).tag x))‖ ≤\n    ε"}
+{"state": [{"context": ["ι : Type u", "E : Type v", "F : Type w", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "I✝ J : Box ι", "π : TaggedPrepartition I✝", "inst✝² : Fintype ι", "l : IntegrationParams", "f✝ g : (ι → ℝ) → E", "vol : ι →ᵇᵃ[⊤] E →L[ℝ] F", "y y' : F", "inst✝¹ : CompleteSpace E", "I : Box ι", "f : (ι → ℝ) → E", "μ : Measure (ι → ℝ)", "inst✝ : IsLocallyFiniteMeasure μ", "hc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x", "ε : ℝ", "ε0 : ε > 0", "ε₁ : ℝ", "ε₁0 : 0 < ε₁", "hε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε", "C : ℝ", "hC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C", "C0 : 0 ≤ C", "ε₂ : ℝ", "ε₂0 : 0 < ε₂", "hε₂ : 4 * C * ε₂ < ε", "ε₂0' : ENNReal.ofReal ε₂ ≠ 0", "D : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}", "μ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)", "μ'D : μ' D = 0", "U : Set (ι → ℝ)", "UD : U ⊇ D", "Uopen : IsOpen U", "hU : μ' U < ENNReal.ofReal ε₂", "comp : IsCompact (Box.Icc I \\ U)", "this : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁", "r : ℝ", "r0 : r > 0", "hr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁", "c₁ c₂ : ℝ≥0", "π₁ π₂ : TaggedPrepartition I", "h₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁", "h₁p : π₁.IsPartition", "h₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂", "h₂p : π₂.IsPartition", "μI : μ ↑I < ⊤", "t₁ : Box ι → ι → ℝ := fun J => (π₁.infPrepartition π₂.toPrepartition).tag J", "t₂ : Box ι → ι → ℝ := fun J => (π₂.infPrepartition π₁.toPrepartition).tag J", "B : Finset (Box ι) := (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes", "B' : Finset (Box ι) := Finset.filter (fun J => ↑J ⊆ U) B", "hB' : B' ⊆ B", "μJ_ne_top : ∀ J ∈ B, μ ↑J ≠ ⊤"], "goal": "‖∑ x ∈ (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes, μ.toBoxAdditive x • (f ((π₁.infPrepartition π₂.toPrepartition).tag x) - f ((π₂.infPrepartition π₁.toPrepartition).tag x))‖ ≤ ε"}], "premise": [1674, 34168, 135251], "state_str": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\ncomp : IsCompact (Box.Icc I \\ U)\nthis : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁\nr : ℝ\nr0 : r > 0\nhr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁\nc₁ c₂ : ℝ≥0\nπ₁ π₂ : TaggedPrepartition I\nh₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁\nh₁p : π₁.IsPartition\nh₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂\nh₂p : π₂.IsPartition\nμI : μ ↑I < ⊤\nt₁ : Box ι → ι → ℝ := fun J => (π₁.infPrepartition π₂.toPrepartition).tag J\nt₂ : Box ι → ι → ℝ := fun J => (π₂.infPrepartition π₁.toPrepartition).tag J\nB : Finset (Box ι) := (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes\nB' : Finset (Box ι) := Finset.filter (fun J => ↑J ⊆ U) B\nhB' : B' ⊆ B\nμJ_ne_top : ∀ J ∈ B, μ ↑J ≠ ⊤\n⊢ ‖∑ x ∈ (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes,\n        μ.toBoxAdditive x •\n          (f ((π₁.infPrepartition π₂.toPrepartition).tag x) - f ((π₂.infPrepartition π₁.toPrepartition).tag x))‖ ≤\n    ε"}
+{"state": [{"context": ["ι : Type u", "E : Type v", "F : Type w", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "I✝ J : Box ι", "π : TaggedPrepartition I✝", "inst✝² : Fintype ι", "l : IntegrationParams", "f✝ g : (ι → ℝ) → E", "vol : ι →ᵇᵃ[⊤] E →L[ℝ] F", "y y' : F", "inst✝¹ : CompleteSpace E", "I : Box ι", "f : (ι → ℝ) → E", "μ : Measure (ι → ℝ)", "inst✝ : IsLocallyFiniteMeasure μ", "hc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x", "ε : ℝ", "ε0 : ε > 0", "ε₁ : ℝ", "ε₁0 : 0 < ε₁", "hε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε", "C : ℝ", "hC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C", "C0 : 0 ≤ C", "ε₂ : ℝ", "ε₂0 : 0 < ε₂", "hε₂ : 4 * C * ε₂ < ε", "ε₂0' : ENNReal.ofReal ε₂ ≠ 0", "D : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}", "μ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)", "μ'D : μ' D = 0", "U : Set (ι → ℝ)", "UD : U ⊇ D", "Uopen : IsOpen U", "hU : μ' U < ENNReal.ofReal ε₂", "comp : IsCompact (Box.Icc I \\ U)", "this : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁", "r : ℝ", "r0 : r > 0", "hr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁", "c₁ c₂ : ℝ≥0", "π₁ π₂ : TaggedPrepartition I", "h₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁", "h₁p : π₁.IsPartition", "h₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂", "h₂p : π₂.IsPartition", "μI : μ ↑I < ⊤", "t₁ : Box ι → ι → ℝ := fun J => (π₁.infPrepartition π₂.toPrepartition).tag J", "t₂ : Box ι → ι → ℝ := fun J => (π₂.infPrepartition π₁.toPrepartition).tag J", "B : Finset (Box ι) := (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes", "B' : Finset (Box ι) := Finset.filter (fun J => ↑J ⊆ U) B", "hB' : B' ⊆ B", "μJ_ne_top : ∀ J ∈ B, μ ↑J ≠ ⊤", "un : ∀ S ⊆ B, ⋃ J ∈ S, ↑J ⊆ ↑I"], "goal": "‖∑ x ∈ (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes, μ.toBoxAdditive x • (f ((π₁.infPrepartition π₂.toPrepartition).tag x) - f ((π₂.infPrepartition π₁.toPrepartition).tag x))‖ ≤ ε"}], "premise": [111329, 126952], "state_str": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\ncomp : IsCompact (Box.Icc I \\ U)\nthis : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁\nr : ℝ\nr0 : r > 0\nhr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁\nc₁ c₂ : ℝ≥0\nπ₁ π₂ : TaggedPrepartition I\nh₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁\nh₁p : π₁.IsPartition\nh₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂\nh₂p : π₂.IsPartition\nμI : μ ↑I < ⊤\nt₁ : Box ι → ι → ℝ := fun J => (π₁.infPrepartition π₂.toPrepartition).tag J\nt₂ : Box ι → ι → ℝ := fun J => (π₂.infPrepartition π₁.toPrepartition).tag J\nB : Finset (Box ι) := (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes\nB' : Finset (Box ι) := Finset.filter (fun J => ↑J ⊆ U) B\nhB' : B' ⊆ B\nμJ_ne_top : ∀ J ∈ B, μ ↑J ≠ ⊤\nun : ∀ S ⊆ B, ⋃ J ∈ S, ↑J ⊆ ↑I\n⊢ ‖∑ x ∈ (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes,\n        μ.toBoxAdditive x •\n          (f ((π₁.infPrepartition π₂.toPrepartition).tag x) - f ((π₂.infPrepartition π₁.toPrepartition).tag x))‖ ≤\n    ε"}
+{"state": [{"context": ["ι : Type u", "E : Type v", "F : Type w", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "I✝ J : Box ι", "π : TaggedPrepartition I✝", "inst✝² : Fintype ι", "l : IntegrationParams", "f✝ g : (ι → ℝ) → E", "vol : ι →ᵇᵃ[⊤] E →L[ℝ] F", "y y' : F", "inst✝¹ : CompleteSpace E", "I : Box ι", "f : (ι → ℝ) → E", "μ : Measure (ι → ℝ)", "inst✝ : IsLocallyFiniteMeasure μ", "hc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x", "ε : ℝ", "ε0 : ε > 0", "ε₁ : ℝ", "ε₁0 : 0 < ε₁", "hε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε", "C : ℝ", "hC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C", "C0 : 0 ≤ C", "ε₂ : ℝ", "ε₂0 : 0 < ε₂", "hε₂ : 4 * C * ε₂ < ε", "ε₂0' : ENNReal.ofReal ε₂ ≠ 0", "D : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}", "μ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)", "μ'D : μ' D = 0", "U : Set (ι → ℝ)", "UD : U ⊇ D", "Uopen : IsOpen U", "hU : μ' U < ENNReal.ofReal ε₂", "comp : IsCompact (Box.Icc I \\ U)", "this : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁", "r : ℝ", "r0 : r > 0", "hr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁", "c₁ c₂ : ℝ≥0", "π₁ π₂ : TaggedPrepartition I", "h₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁", "h₁p : π₁.IsPartition", "h₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂", "h₂p : π₂.IsPartition", "μI : μ ↑I < ⊤", "t₁ : Box ι → ι → ℝ := fun J => (π₁.infPrepartition π₂.toPrepartition).tag J", "t₂ : Box ι → ι → ℝ := fun J => (π₂.infPrepartition π₁.toPrepartition).tag J", "B : Finset (Box ι) := (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes", "B' : Finset (Box ι) := Finset.filter (fun J => ↑J ⊆ U) B", "hB' : B' ⊆ B", "μJ_ne_top : ∀ J ∈ B, μ ↑J ≠ ⊤", "un : ∀ S ⊆ B, ⋃ J ∈ S, ↑J ⊆ ↑I"], "goal": "‖∑ x ∈ B \\ B', μ.toBoxAdditive x • (f ((π₁.infPrepartition π₂.toPrepartition).tag x) - f ((π₂.infPrepartition π₁.toPrepartition).tag x)) + ∑ x ∈ B', μ.toBoxAdditive x • (f ((π₁.infPrepartition π₂.toPrepartition).tag x) - f ((π₂.infPrepartition π₁.toPrepartition).tag x))‖ ≤ ε / 2 + ε / 2"}], "premise": [14273, 42671, 103917], "state_str": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nι : Type u\nE : Type v\nF : Type w\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nI✝ J : Box ι\nπ : TaggedPrepartition I✝\ninst✝² : Fintype ι\nl : IntegrationParams\nf✝ g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\ninst✝¹ : CompleteSpace E\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nhε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε\nC : ℝ\nhC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nC0 : 0 ≤ C\nε₂ : ℝ\nε₂0 : 0 < ε₂\nhε₂ : 4 * C * ε₂ < ε\nε₂0' : ENNReal.ofReal ε₂ ≠ 0\nD : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}\nμ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)\nμ'D : μ' D = 0\nU : Set (ι → ℝ)\nUD : U ⊇ D\nUopen : IsOpen U\nhU : μ' U < ENNReal.ofReal ε₂\ncomp : IsCompact (Box.Icc I \\ U)\nthis : ∀ x ∈ Box.Icc I \\ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁\nr : ℝ\nr0 : r > 0\nhr : ∀ x ∈ Box.Icc I \\ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁\nc₁ c₂ : ℝ≥0\nπ₁ π₂ : TaggedPrepartition I\nh₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁\nh₁p : π₁.IsPartition\nh₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂\nh₂p : π₂.IsPartition\nμI : μ ↑I < ⊤\nt₁ : Box ι → ι → ℝ := fun J => (π₁.infPrepartition π₂.toPrepartition).tag J\nt₂ : Box ι → ι → ℝ := fun J => (π₂.infPrepartition π₁.toPrepartition).tag J\nB : Finset (Box ι) := (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes\nB' : Finset (Box ι) := Finset.filter (fun J => ↑J ⊆ U) B\nhB' : B' ⊆ B\nμJ_ne_top : ∀ J ∈ B, μ ↑J ≠ ⊤\nun : ∀ S ⊆ B, ⋃ J ∈ S, ↑J ⊆ ↑I\n⊢ ‖∑ x ∈ B \\ B',\n          μ.toBoxAdditive x •\n            (f ((π₁.infPrepartition π₂.toPrepartition).tag x) - f ((π₂.infPrepartition π₁.toPrepartition).tag x)) +\n        ∑ x ∈ B',\n          μ.toBoxAdditive x •\n            (f ((π₁.infPrepartition π₂.toPrepartition).tag x) - f ((π₂.infPrepartition π₁.toPrepartition).tag x))‖ ≤\n    ε / 2 + ε / 2"}
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+{"state": [{"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "inst✝¹ : MonoidalCategory C", "inst✝ : SymmetricCategory C", "X Y : Mon_ C"], "goal": "(X ⊗ Y).mul ≫ (β_ X.X Y.X).hom = ((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫ (Y ⊗ X).mul"}], "premise": [96173, 99219, 99222, 99225, 100101, 100117, 107078, 107079, 107082], "state_str": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : SymmetricCategory C\nX Y : Mon_ C\n⊢ (X ⊗ Y).mul ≫ (β_ X.X Y.X).hom = ((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫ (Y ⊗ X).mul"}
+{"state": [{"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "inst✝¹ : MonoidalCategory C", "inst✝ : SymmetricCategory C", "X Y : Mon_ C"], "goal": "(α_ X.X Y.X (X.X ⊗ Y.X)).hom ≫ X.X ◁ (α_ Y.X X.X Y.X).inv ≫ X.X ◁ (β_ Y.X X.X).hom ▷ Y.X ≫ X.X ◁ (β_ X.X Y.X).hom ▷ Y.X ≫ (α_ X.X (Y.X ⊗ X.X) Y.X).inv ≫ (α_ X.X Y.X X.X).inv ▷ Y.X ≫ (β_ X.X Y.X).hom ▷ X.X ▷ Y.X ≫ (α_ (Y.X ⊗ X.X) X.X Y.X).hom ≫ (α_ Y.X X.X (X.X ⊗ Y.X)).hom ≫ Y.X ◁ X.X ◁ (β_ X.X Y.X).hom ≫ Y.X ◁ (α_ X.X Y.X X.X).inv ≫ Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫ Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul) = ((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫ (α_ Y.X X.X (Y.X ⊗ X.X)).hom ≫ Y.X ◁ (α_ X.X Y.X X.X).inv ≫ Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫ Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul)"}], "premise": [96173, 99219, 99222, 107105], "state_str": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : SymmetricCategory C\nX Y : Mon_ C\n⊢ (α_ X.X Y.X (X.X ⊗ Y.X)).hom ≫\n      X.X ◁ (α_ Y.X X.X Y.X).inv ≫\n        X.X ◁ (β_ Y.X X.X).hom ▷ Y.X ≫\n          X.X ◁ (β_ X.X Y.X).hom ▷ Y.X ≫\n            (α_ X.X (Y.X ⊗ X.X) Y.X).inv ≫\n              (α_ X.X Y.X X.X).inv ▷ Y.X ≫\n                (β_ X.X Y.X).hom ▷ X.X ▷ Y.X ≫\n                  (α_ (Y.X ⊗ X.X) X.X Y.X).hom ≫\n                    (α_ Y.X X.X (X.X ⊗ Y.X)).hom ≫\n                      Y.X ◁ X.X ◁ (β_ X.X Y.X).hom ≫\n                        Y.X ◁ (α_ X.X Y.X X.X).inv ≫\n                          Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫\n                            Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul) =\n    ((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫\n      (α_ Y.X X.X (Y.X ⊗ X.X)).hom ≫\n        Y.X ◁ (α_ X.X Y.X X.X).inv ≫\n          Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫ Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul)"}
+{"state": [{"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "inst✝¹ : MonoidalCategory C", "inst✝ : SymmetricCategory C", "X Y : Mon_ C"], "goal": "(α_ X.X Y.X (X.X ⊗ Y.X)).hom ≫ X.X ◁ (α_ Y.X X.X Y.X).inv ≫ ((((((((((X.X ◁ 𝟙 (Y.X ⊗ X.X) ▷ Y.X ≫ (α_ X.X (Y.X ⊗ X.X) Y.X).inv) ≫ (α_ X.X Y.X X.X).inv ▷ Y.X) ≫ (β_ X.X Y.X).hom ▷ X.X ▷ Y.X) ≫ (α_ (Y.X ⊗ X.X) X.X Y.X).hom) ≫ (α_ Y.X X.X (X.X ⊗ Y.X)).hom) ≫ Y.X ◁ X.X ◁ (β_ X.X Y.X).hom) ≫ Y.X ◁ (α_ X.X Y.X X.X).inv) ≫ Y.X ◁ (β_ X.X Y.X).hom ▷ X.X) ≫ Y.X ◁ (α_ Y.X X.X X.X).hom) ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv) ≫ (Y.mul ⊗ X.mul) = ((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫ (α_ Y.X X.X (Y.X ⊗ X.X)).hom ≫ Y.X ◁ (α_ X.X Y.X X.X).inv ≫ Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫ Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul)"}], "premise": [96173, 96175, 99212, 99216], "state_str": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : SymmetricCategory C\nX Y : Mon_ C\n⊢ (α_ X.X Y.X (X.X ⊗ Y.X)).hom ≫\n      X.X ◁ (α_ Y.X X.X Y.X).inv ≫\n        ((((((((((X.X ◁ 𝟙 (Y.X ⊗ X.X) ▷ Y.X ≫ (α_ X.X (Y.X ⊗ X.X) Y.X).inv) ≫ (α_ X.X Y.X X.X).inv ▷ Y.X) ≫\n                          (β_ X.X Y.X).hom ▷ X.X ▷ Y.X) ≫\n                        (α_ (Y.X ⊗ X.X) X.X Y.X).hom) ≫\n                      (α_ Y.X X.X (X.X ⊗ Y.X)).hom) ≫\n                    Y.X ◁ X.X ◁ (β_ X.X Y.X).hom) ≫\n                  Y.X ◁ (α_ X.X Y.X X.X).inv) ≫\n                Y.X ◁ (β_ X.X Y.X).hom ▷ X.X) ≫\n              Y.X ◁ (α_ Y.X X.X X.X).hom) ≫\n            (α_ Y.X Y.X (X.X ⊗ X.X)).inv) ≫\n          (Y.mul ⊗ X.mul) =\n    ((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫\n      (α_ Y.X X.X (Y.X ⊗ X.X)).hom ≫\n        Y.X ◁ (α_ X.X Y.X X.X).inv ≫\n          Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫ Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul)"}
+{"state": [{"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "inst✝¹ : MonoidalCategory C", "inst✝ : SymmetricCategory C", "X Y : Mon_ C"], "goal": "(α_ (X.X ⊗ Y.X) X.X Y.X).inv ≫ (β_ X.X Y.X).hom ▷ X.X ▷ Y.X ≫ (α_ (Y.X ⊗ X.X) X.X Y.X).hom ≫ (α_ Y.X X.X (X.X ⊗ Y.X)).hom ≫ Y.X ◁ X.X ◁ (β_ X.X Y.X).hom ≫ Y.X ◁ (α_ X.X Y.X X.X).inv ≫ Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫ Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul) = ((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫ (α_ Y.X X.X (Y.X ⊗ X.X)).hom ≫ Y.X ◁ (α_ X.X Y.X X.X).inv ≫ Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫ Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul)"}], "premise": [96173, 99259], "state_str": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : SymmetricCategory C\nX Y : Mon_ C\n⊢ (α_ (X.X ⊗ Y.X) X.X Y.X).inv ≫\n      (β_ X.X Y.X).hom ▷ X.X ▷ Y.X ≫\n        (α_ (Y.X ⊗ X.X) X.X Y.X).hom ≫\n          (α_ Y.X X.X (X.X ⊗ Y.X)).hom ≫\n            Y.X ◁ X.X ◁ (β_ X.X Y.X).hom ≫\n              Y.X ◁ (α_ X.X Y.X X.X).inv ≫\n                Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫\n                  Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul) =\n    ((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫\n      (α_ Y.X X.X (Y.X ⊗ X.X)).hom ≫\n        Y.X ◁ (α_ X.X Y.X X.X).inv ≫\n          Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫ Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul)"}
+{"state": [{"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "inst✝¹ : MonoidalCategory C", "inst✝ : SymmetricCategory C", "X Y : Mon_ C"], "goal": "(((((((((β_ X.X Y.X).hom ▷ (X.X ⊗ Y.X) ≫ (α_ (Y.X ⊗ X.X) X.X Y.X).inv) ≫ (α_ (Y.X ⊗ X.X) X.X Y.X).hom) ≫ (α_ Y.X X.X (X.X ⊗ Y.X)).hom) ≫ Y.X ◁ X.X ◁ (β_ X.X Y.X).hom) ≫ Y.X ◁ (α_ X.X Y.X X.X).inv) ≫ Y.X ◁ (β_ X.X Y.X).hom ▷ X.X) ≫ Y.X ◁ (α_ Y.X X.X X.X).hom) ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv) ≫ (Y.mul ⊗ X.mul) = ((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫ (α_ Y.X X.X (Y.X ⊗ X.X)).hom ≫ Y.X ◁ (α_ X.X Y.X X.X).inv ≫ Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫ Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul)"}], "premise": [88742, 96173], "state_str": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : SymmetricCategory C\nX Y : Mon_ C\n⊢ (((((((((β_ X.X Y.X).hom ▷ (X.X ⊗ Y.X) ≫ (α_ (Y.X ⊗ X.X) X.X Y.X).inv) ≫ (α_ (Y.X ⊗ X.X) X.X Y.X).hom) ≫\n                  (α_ Y.X X.X (X.X ⊗ Y.X)).hom) ≫\n                Y.X ◁ X.X ◁ (β_ X.X Y.X).hom) ≫\n              Y.X ◁ (α_ X.X Y.X X.X).inv) ≫\n            Y.X ◁ (β_ X.X Y.X).hom ▷ X.X) ≫\n          Y.X ◁ (α_ Y.X X.X X.X).hom) ≫\n        (α_ Y.X Y.X (X.X ⊗ X.X)).inv) ≫\n      (Y.mul ⊗ X.mul) =\n    ((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫\n      (α_ Y.X X.X (Y.X ⊗ X.X)).hom ≫\n        Y.X ◁ (α_ X.X Y.X X.X).inv ≫\n          Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫ Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul)"}
+{"state": [{"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "inst✝¹ : MonoidalCategory C", "inst✝ : SymmetricCategory C", "X Y : Mon_ C"], "goal": "(β_ X.X Y.X).hom ▷ (X.X ⊗ Y.X) ≫ ((((((𝟙 ((Y.X ⊗ X.X) ⊗ X.X ⊗ Y.X) ≫ (α_ Y.X X.X (X.X ⊗ Y.X)).hom) ≫ Y.X ◁ X.X ◁ (β_ X.X Y.X).hom) ≫ Y.X ◁ (α_ X.X Y.X X.X).inv) ≫ Y.X ◁ (β_ X.X Y.X).hom ▷ X.X) ≫ Y.X ◁ (α_ Y.X X.X X.X).hom) ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv) ≫ (Y.mul ⊗ X.mul) = ((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫ (α_ Y.X X.X (Y.X ⊗ X.X)).hom ≫ Y.X ◁ (α_ X.X Y.X X.X).inv ≫ Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫ Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul)"}], "premise": [96175], "state_str": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : SymmetricCategory C\nX Y : Mon_ C\n⊢ (β_ X.X Y.X).hom ▷ (X.X ⊗ Y.X) ≫\n      ((((((𝟙 ((Y.X ⊗ X.X) ⊗ X.X ⊗ Y.X) ≫ (α_ Y.X X.X (X.X ⊗ Y.X)).hom) ≫ Y.X ◁ X.X ◁ (β_ X.X Y.X).hom) ≫\n                Y.X ◁ (α_ X.X Y.X X.X).inv) ≫\n              Y.X ◁ (β_ X.X Y.X).hom ▷ X.X) ≫\n            Y.X ◁ (α_ Y.X X.X X.X).hom) ≫\n          (α_ Y.X Y.X (X.X ⊗ X.X)).inv) ≫\n        (Y.mul ⊗ X.mul) =\n    ((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫\n      (α_ Y.X X.X (Y.X ⊗ X.X)).hom ≫\n        Y.X ◁ (α_ X.X Y.X X.X).inv ≫\n          Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫ Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul)"}
+{"state": [{"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "inst✝¹ : MonoidalCategory C", "inst✝ : SymmetricCategory C", "X Y : Mon_ C"], "goal": "(β_ X.X Y.X).hom ▷ (X.X ⊗ Y.X) ≫ ((((((α_ Y.X X.X (X.X ⊗ Y.X)).hom ≫ Y.X ◁ X.X ◁ (β_ X.X Y.X).hom) ≫ Y.X ◁ (α_ X.X Y.X X.X).inv) ≫ Y.X ◁ (β_ X.X Y.X).hom ▷ X.X) ≫ Y.X ◁ (α_ Y.X X.X X.X).hom) ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv) ≫ (Y.mul ⊗ X.mul) = ((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫ (α_ Y.X X.X (Y.X ⊗ X.X)).hom ≫ Y.X ◁ (α_ X.X Y.X X.X).inv ≫ Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫ Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul)"}], "premise": [96173, 99264], "state_str": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : SymmetricCategory C\nX Y : Mon_ C\n⊢ (β_ X.X Y.X).hom ▷ (X.X ⊗ Y.X) ≫\n      ((((((α_ Y.X X.X (X.X ⊗ Y.X)).hom ≫ Y.X ◁ X.X ◁ (β_ X.X Y.X).hom) ≫ Y.X ◁ (α_ X.X Y.X X.X).inv) ≫\n              Y.X ◁ (β_ X.X Y.X).hom ▷ X.X) ≫\n            Y.X ◁ (α_ Y.X X.X X.X).hom) ≫\n          (α_ Y.X Y.X (X.X ⊗ X.X)).inv) ≫\n        (Y.mul ⊗ X.mul) =\n    ((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫\n      (α_ Y.X X.X (Y.X ⊗ X.X)).hom ≫\n        Y.X ◁ (α_ X.X Y.X X.X).inv ≫\n          Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫ Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul)"}
+{"state": [{"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "inst✝¹ : MonoidalCategory C", "inst✝ : SymmetricCategory C", "X Y : Mon_ C"], "goal": "(β_ X.X Y.X).hom ▷ (X.X ⊗ Y.X) ≫ ((((((Y.X ⊗ X.X) ◁ (β_ X.X Y.X).hom ≫ (α_ Y.X X.X (Y.X ⊗ X.X)).hom) ≫ Y.X ◁ (α_ X.X Y.X X.X).inv) ≫ Y.X ◁ (β_ X.X Y.X).hom ▷ X.X) ≫ Y.X ◁ (α_ Y.X X.X X.X).hom) ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv) ≫ (Y.mul ⊗ X.mul) = ((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫ (α_ Y.X X.X (Y.X ⊗ X.X)).hom ≫ Y.X ◁ (α_ X.X Y.X X.X).inv ≫ Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫ Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul)"}], "premise": [96173, 99211], "state_str": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : SymmetricCategory C\nX Y : Mon_ C\n⊢ (β_ X.X Y.X).hom ▷ (X.X ⊗ Y.X) ≫\n      ((((((Y.X ⊗ X.X) ◁ (β_ X.X Y.X).hom ≫ (α_ Y.X X.X (Y.X ⊗ X.X)).hom) ≫ Y.X ◁ (α_ X.X Y.X X.X).inv) ≫\n              Y.X ◁ (β_ X.X Y.X).hom ▷ X.X) ≫\n            Y.X ◁ (α_ Y.X X.X X.X).hom) ≫\n          (α_ Y.X Y.X (X.X ⊗ X.X)).inv) ≫\n        (Y.mul ⊗ X.mul) =\n    ((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫\n      (α_ Y.X X.X (Y.X ⊗ X.X)).hom ≫\n        Y.X ◁ (α_ X.X Y.X X.X).inv ≫\n          Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫ Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul)"}
+{"state": [{"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "inst✝¹ : MonoidalCategory C", "inst✝ : SymmetricCategory C", "X Y : Mon_ C"], "goal": "(((((((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫ (α_ Y.X X.X (Y.X ⊗ X.X)).hom) ≫ Y.X ◁ (α_ X.X Y.X X.X).inv) ≫ Y.X ◁ (β_ X.X Y.X).hom ▷ X.X) ≫ Y.X ◁ (α_ Y.X X.X X.X).hom) ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv) ≫ (Y.mul ⊗ X.mul) = ((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫ (α_ Y.X X.X (Y.X ⊗ X.X)).hom ≫ Y.X ◁ (α_ X.X Y.X X.X).inv ≫ Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫ Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul)"}], "premise": [96173], "state_str": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : SymmetricCategory C\nX Y : Mon_ C\n⊢ (((((((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫ (α_ Y.X X.X (Y.X ⊗ X.X)).hom) ≫ Y.X ◁ (α_ X.X Y.X X.X).inv) ≫\n            Y.X ◁ (β_ X.X Y.X).hom ▷ X.X) ≫\n          Y.X ◁ (α_ Y.X X.X X.X).hom) ≫\n        (α_ Y.X Y.X (X.X ⊗ X.X)).inv) ≫\n      (Y.mul ⊗ X.mul) =\n    ((β_ X.X Y.X).hom ⊗ (β_ X.X Y.X).hom) ≫\n      (α_ Y.X X.X (Y.X ⊗ X.X)).hom ≫\n        Y.X ◁ (α_ X.X Y.X X.X).inv ≫\n          Y.X ◁ (β_ X.X Y.X).hom ▷ X.X ≫ Y.X ◁ (α_ Y.X X.X X.X).hom ≫ (α_ Y.X Y.X (X.X ⊗ X.X)).inv ≫ (Y.mul ⊗ X.mul)"}
+{"state": [{"context": ["x y : ℝ"], "goal": "↑(tanh x) = ↑(sinh x / cosh x)"}], "premise": [149118], "state_str": "x y : ℝ\n⊢ ↑(tanh x) = ↑(sinh x / cosh x)"}
+{"state": [{"context": ["n m k : ℕ"], "goal": "n &&& m &&& k = n &&& (m &&& k)"}], "premise": [1241, 1242, 140199], "state_str": "n m k : ℕ\n⊢ n &&& m &&& k = n &&& (m &&& k)"}
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+{"state": [{"context": ["ι : Type u_1", "k : ℕ", "l : List ℕ"], "goal": "k.Coprime l.prod ↔ ∀ n ∈ l, k.Coprime n"}], "premise": [396, 143464], "state_str": "ι : Type u_1\nk : ℕ\nl : List ℕ\n⊢ k.Coprime l.prod ↔ ∀ n ∈ l, k.Coprime n"}
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+{"state": [{"context": ["M : Type u_1", "inst✝² : CommMonoid M", "S : Submonoid M", "N : Type u_2", "inst✝¹ : CommMonoid N", "P : Type u_3", "inst✝ : CommMonoid P", "f : S.LocalizationMap N", "z w : N", "a : M × ↥S", "ha : z * f.toMap ↑a.2 = f.toMap a.1"], "goal": "∃ z' w' d, z * f.toMap ↑d = f.toMap z' ∧ w * f.toMap ↑d = f.toMap w'"}], "premise": [9127], "state_str": "M : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type u_3\ninst✝ : CommMonoid P\nf : S.LocalizationMap N\nz w : N\na : M × ↥S\nha : z * f.toMap ↑a.2 = f.toMap a.1\n⊢ ∃ z' w' d, z * f.toMap ↑d = f.toMap z' ∧ w * f.toMap ↑d = f.toMap w'"}
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+{"state": [{"context": ["R : Type u_1", "α : Type u_2", "β : Type u_3", "δ : Type u_4", "γ : Type u_5", "ι : Type u_6", "inst✝ : MeasureSpace α", "s t : Set α", "hs : NullMeasurableSet s volume", "ht : MeasurableSet t"], "goal": "volume (Subtype.val ⁻¹' t) = volume (t ∩ s)"}], "premise": [28835, 29169, 31525, 32327, 32347, 134192, 134291, 137138], "state_str": "R : Type u_1\nα : Type u_2\nβ : Type u_3\nδ : Type u_4\nγ : Type u_5\nι : Type u_6\ninst✝ : MeasureSpace α\ns t : Set α\nhs : NullMeasurableSet s volume\nht : MeasurableSet t\n⊢ volume (Subtype.val ⁻¹' t) = volume (t ∩ s)"}
+{"state": [{"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "inst✝⁴ : CommMonoid α", "inst✝³ : CommMonoid β", "inst✝² : CommMonoid γ", "A A₁ A₂ : Set α", "B B₁ B₂ : Set β", "C : Set γ", "f✝ f₁ f₂ : α → β", "g : β → γ", "m n : ℕ", "inst✝¹ : EquivLike F α β", "inst✝ : MulEquivClass F α β", "f : F", "hfAB : BijOn (⇑f) A B", "s t : Multiset α", "x✝³ : ∀ ⦃x : α⦄, x ∈ s → x ∈ A", "x✝² : ∀ ⦃x : α⦄, x ∈ t → x ∈ A", "x✝¹ : card s = n", "x✝ : card t = n"], "goal": "(map (⇑f) s).prod = (map (⇑f) t).prod ↔ s.prod = t.prod"}], "premise": [1713, 124360, 128634], "state_str": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝⁴ : CommMonoid α\ninst✝³ : CommMonoid β\ninst✝² : CommMonoid γ\nA A₁ A₂ : Set α\nB B₁ B₂ : Set β\nC : Set γ\nf✝ f₁ f₂ : α → β\ng : β → γ\nm n : ℕ\ninst✝¹ : EquivLike F α β\ninst✝ : MulEquivClass F α β\nf : F\nhfAB : BijOn (⇑f) A B\ns t : Multiset α\nx✝³ : ∀ ⦃x : α⦄, x ∈ s → x ∈ A\nx✝² : ∀ ⦃x : α⦄, x ∈ t → x ∈ A\nx✝¹ : card s = n\nx✝ : card t = n\n⊢ (map (⇑f) s).prod = (map (⇑f) t).prod ↔ s.prod = t.prod"}
+{"state": [{"context": ["C : Type u₁", "inst✝⁸ : Category.{v₁, u₁} C", "inst✝⁷ : MonoidalCategory C", "inst✝⁶ : BraidedCategory C", "D : Type u₂", "inst✝⁵ : Category.{v₂, u₂} D", "inst✝⁴ : MonoidalCategory D", "inst✝³ : BraidedCategory D", "E : Type u₃", "inst✝² : Category.{v₃, u₃} E", "inst✝¹ : MonoidalCategory E", "inst✝ : BraidedCategory E", "X₁ X₂ : C"], "goal": "(λ_ X₁).hom ⊗ (λ_ X₂).hom = ((α_ (𝟙_ C) X₁ (𝟙_ C ⊗ X₂)).hom ≫ 𝟙_ C ◁ (α_ X₁ (𝟙_ C) X₂).inv ≫ 𝟙_ C ◁ (β_ X₁ (𝟙_ C)).hom ▷ X₂ ≫ 𝟙_ C ◁ (α_ (𝟙_ C) X₁ X₂).hom ≫ (α_ (𝟙_ C) (𝟙_ C) (X₁ ⊗ X₂)).inv) ≫ (λ_ (𝟙_ C)).hom ▷ (X₁ ⊗ X₂) ≫ (λ_ (X₁ ⊗ X₂)).hom"}], "premise": [96173, 99211, 99212, 99216, 99217, 99218, 99219, 99220, 99221, 99222, 99223, 99224, 99225, 99601, 99602, 99603, 99604, 99605, 99606, 99607, 99611, 99612], "state_str": "C : Type u₁\ninst✝⁸ : Category.{v₁, u₁} C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category.{v₃, u₃} E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ : C\n⊢ (λ_ X₁).hom ⊗ (λ_ X₂).hom =\n    ((α_ (𝟙_ C) X₁ (𝟙_ C ⊗ X₂)).hom ≫\n        𝟙_ C ◁ (α_ X₁ (𝟙_ C) X₂).inv ≫\n          𝟙_ C ◁ (β_ X₁ (𝟙_ C)).hom ▷ X₂ ≫ 𝟙_ C ◁ (α_ (𝟙_ C) X₁ X₂).hom ≫ (α_ (𝟙_ C) (𝟙_ C) (X₁ ⊗ X₂)).inv) ≫\n      (λ_ (𝟙_ C)).hom ▷ (X₁ ⊗ X₂) ≫ (λ_ (X₁ ⊗ X₂)).hom"}
+{"state": [{"context": ["C : Type u₁", "inst✝⁸ : Category.{v₁, u₁} C", "inst✝⁷ : MonoidalCategory C", "inst✝⁶ : BraidedCategory C", "D : Type u₂", "inst✝⁵ : Category.{v₂, u₂} D", "inst✝⁴ : MonoidalCategory D", "inst✝³ : BraidedCategory D", "E : Type u₃", "inst✝² : Category.{v₃, u₃} E", "inst✝¹ : MonoidalCategory E", "inst✝ : BraidedCategory E", "X₁ X₂ : C"], "goal": "(α_ (𝟙_ C) X₁ (𝟙_ C ⊗ X₂)).hom ≫ 𝟙_ C ◁ (α_ X₁ (𝟙_ C) X₂).inv ≫ (λ_ ((X₁ ⊗ 𝟙_ C) ⊗ X₂)).hom ≫ (ρ_ X₁).hom ▷ X₂ = ((α_ (𝟙_ C) X₁ (𝟙_ C ⊗ X₂)).hom ≫ 𝟙_ C ◁ (α_ X₁ (𝟙_ C) X₂).inv ≫ 𝟙_ C ◁ (β_ X₁ (𝟙_ C)).hom ▷ X₂ ≫ 𝟙_ C ◁ (α_ (𝟙_ C) X₁ X₂).hom ≫ (α_ (𝟙_ C) (𝟙_ C) (X₁ ⊗ X₂)).inv) ≫ (λ_ (𝟙_ C)).hom ▷ (X₁ ⊗ X₂) ≫ (λ_ (X₁ ⊗ X₂)).hom"}], "premise": [107095], "state_str": "C : Type u₁\ninst✝⁸ : Category.{v₁, u₁} C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category.{v₃, u₃} E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ : C\n⊢ (α_ (𝟙_ C) X₁ (𝟙_ C ⊗ X₂)).hom ≫ 𝟙_ C ◁ (α_ X₁ (𝟙_ C) X₂).inv ≫ (λ_ ((X₁ ⊗ 𝟙_ C) ⊗ X₂)).hom ≫ (ρ_ X₁).hom ▷ X₂ =\n    ((α_ (𝟙_ C) X₁ (𝟙_ C ⊗ X₂)).hom ≫\n        𝟙_ C ◁ (α_ X₁ (𝟙_ C) X₂).inv ≫\n          𝟙_ C ◁ (β_ X₁ (𝟙_ C)).hom ▷ X₂ ≫ 𝟙_ C ◁ (α_ (𝟙_ C) X₁ X₂).hom ≫ (α_ (𝟙_ C) (𝟙_ C) (X₁ ⊗ X₂)).inv) ≫\n      (λ_ (𝟙_ C)).hom ▷ (X₁ ⊗ X₂) ≫ (λ_ (X₁ ⊗ X₂)).hom"}
+{"state": [{"context": ["C : Type u₁", "inst✝⁸ : Category.{v₁, u₁} C", "inst✝⁷ : MonoidalCategory C", "inst✝⁶ : BraidedCategory C", "D : Type u₂", "inst✝⁵ : Category.{v₂, u₂} D", "inst✝⁴ : MonoidalCategory D", "inst✝³ : BraidedCategory D", "E : Type u₃", "inst✝² : Category.{v₃, u₃} E", "inst✝¹ : MonoidalCategory E", "inst✝ : BraidedCategory E", "X₁ X₂ : C"], "goal": "(α_ (𝟙_ C) X₁ (𝟙_ C ⊗ X₂)).hom ≫ 𝟙_ C ◁ (α_ X₁ (𝟙_ C) X₂).inv ≫ (λ_ ((X₁ ⊗ 𝟙_ C) ⊗ X₂)).hom ≫ ((β_ X₁ (𝟙_ C)).hom ≫ (λ_ X₁).hom) ▷ X₂ = ((α_ (𝟙_ C) X₁ (𝟙_ C ⊗ X₂)).hom ≫ 𝟙_ C ◁ (α_ X₁ (𝟙_ C) X₂).inv ≫ 𝟙_ C ◁ (β_ X₁ (𝟙_ C)).hom ▷ X₂ ≫ 𝟙_ C ◁ (α_ (𝟙_ C) X₁ X₂).hom ≫ (α_ (𝟙_ C) (𝟙_ C) (X₁ ⊗ X₂)).inv) ≫ (λ_ (𝟙_ C)).hom ▷ (X₁ ⊗ X₂) ≫ (λ_ (X₁ ⊗ X₂)).hom"}], "premise": [1673, 53720, 53750, 96173, 96190, 99211, 99212, 99216, 99217, 99218, 99219, 99220, 99221, 99222, 99223, 99224, 99225, 99601, 99602, 99603, 99604, 99605, 99606, 99607, 99611, 99612], "state_str": "C : Type u₁\ninst✝⁸ : Category.{v₁, u₁} C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category.{v₃, u₃} E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ : C\n⊢ (α_ (𝟙_ C) X₁ (𝟙_ C ⊗ X₂)).hom ≫\n      𝟙_ C ◁ (α_ X₁ (𝟙_ C) X₂).inv ≫ (λ_ ((X₁ ⊗ 𝟙_ C) ⊗ X₂)).hom ≫ ((β_ X₁ (𝟙_ C)).hom ≫ (λ_ X₁).hom) ▷ X₂ =\n    ((α_ (𝟙_ C) X₁ (𝟙_ C ⊗ X₂)).hom ≫\n        𝟙_ C ◁ (α_ X₁ (𝟙_ C) X₂).inv ≫\n          𝟙_ C ◁ (β_ X₁ (𝟙_ C)).hom ▷ X₂ ≫ 𝟙_ C ◁ (α_ (𝟙_ C) X₁ X₂).hom ≫ (α_ (𝟙_ C) (𝟙_ C) (X₁ ⊗ X₂)).inv) ≫\n      (λ_ (𝟙_ C)).hom ▷ (X₁ ��� X₂) ≫ (λ_ (X₁ ⊗ X₂)).hom"}
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+{"state": [{"context": ["a b c : EReal", "h : 0 < b", "h' : b ≠ ⊤"], "goal": "a / b ≤ b * (c / b) ↔ a ≤ b * c"}], "premise": [119707, 147320], "state_str": "a b c : EReal\nh : 0 < b\nh' : b ≠ ⊤\n⊢ a / b ≤ b * (c / b) ↔ a ≤ b * c"}
+{"state": [{"context": ["a b c : EReal", "h : 0 < b", "h' : b ≠ ⊤"], "goal": "a / b ≤ c * b / b ↔ a ≤ c * b"}], "premise": [19830, 147329], "state_str": "a b c : EReal\nh : 0 < b\nh' : b ≠ ⊤\n⊢ a / b ≤ c * b / b ↔ a ≤ c * b"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "s : α → Set β", "t : α → Set γ", "inst✝ : Preorder α"], "goal": "⋃ x, Accumulate s x = ⋃ x, s x"}], "premise": [133329], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ns : α → Set β\nt : α → Set γ\ninst✝ : Preorder α\n⊢ ⋃ x, Accumulate s x = ⋃ x, s x"}
+{"state": [{"context": ["𝕜 : Type u_1", "inst✝⁴ : NontriviallyNormedField 𝕜", "F : Type u_2", "inst✝³ : NormedAddCommGroup F", "inst✝² : NormedSpace 𝕜 F", "E : Type u_3", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace 𝕜 E", "f f₀ f₁ : E → F", "f' : F", "s t : Set E", "x v : E", "L : E →L[𝕜] F", "h : s ∈ 𝓝 x"], "goal": "lineDerivWithin 𝕜 f s x v = lineDeriv 𝕜 f x v"}], "premise": [44398], "state_str": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ : E → F\nf' : F\ns t : Set E\nx v : E\nL : E →L[𝕜] F\nh : s ∈ 𝓝 x\n⊢ lineDerivWithin 𝕜 f s x v = lineDeriv 𝕜 f x v"}
+{"state": [{"context": ["𝕜 : Type u_1", "inst✝⁴ : NontriviallyNormedField 𝕜", "F : Type u_2", "inst✝³ : NormedAddCommGroup F", "inst✝² : NormedSpace 𝕜 F", "E : Type u_3", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace 𝕜 E", "f f₀ f₁ : E → F", "f' : F", "s t : Set E", "x v : E", "L : E →L[𝕜] F", "h : s ∈ 𝓝 x"], "goal": "(fun t => x + t • v) ⁻¹' s ∈ 𝓝 0"}], "premise": [55623, 55638], "state_str": "case h\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ : E → F\nf' : F\ns t : Set E\nx v : E\nL : E →L[𝕜] F\nh : s ∈ 𝓝 x\n⊢ (fun t => x + t • v) ⁻¹' s ∈ 𝓝 0"}
+{"state": [{"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "D : Type u_1", "inst✝ : Category.{u_2, u_1} D", "Z : D", "F : C ⥤ D", "M : (x : C) → F.obj x ⟶ Z", "hM : ∀ (x y : C) (f : x ⟶ y), F.map f ≫ M y = M x", "x : C"], "goal": "(lift F M hM).map (starTerminal.from (incl.obj x)) ≫ (liftStar F M hM).hom = (inclLift F M hM).hom.app x ≫ M x"}], "premise": [96174, 96175], "state_str": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u_1\ninst✝ : Category.{u_2, u_1} D\nZ : D\nF : C ⥤ D\nM : (x : C) → F.obj x ⟶ Z\nhM : ∀ (x y : C) (f : x ⟶ y), F.map f ≫ M y = M x\nx : C\n⊢ (lift F M hM).map (starTerminal.from (incl.obj x)) ≫ (liftStar F M hM).hom = (inclLift F M hM).hom.app x ≫ M x"}
+{"state": [{"context": ["C : Type u₁", "inst✝³ : Category.{v₁, u₁} C", "inst✝² : MonoidalCategory C", "D : Type u₂", "inst✝¹ : Category.{v₂, u₂} D", "inst✝ : MonoidalCategory D", "F : LaxMonoidalFunctor C D", "X Y Z : C"], "goal": "F.obj X ◁ F.μ Y Z ≫ F.μ X (Y ⊗ Z) ≫ F.map (α_ X Y Z).inv = (α_ (F.obj X) (F.obj Y) (F.obj Z)).inv ≫ F.μ X Y ▷ F.obj Z ≫ F.μ (X ⊗ Y) Z"}], "premise": [88743, 88768, 96174, 99919, 99920], "state_str": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : MonoidalCategory D\nF : LaxMonoidalFunctor C D\nX Y Z : C\n⊢ F.obj X ◁ F.μ Y Z ≫ F.μ X (Y ⊗ Z) ≫ F.map (α_ X Y Z).inv =\n    (α_ (F.obj X) (F.obj Y) (F.obj Z)).inv ≫ F.μ X Y ▷ F.obj Z ≫ F.μ (X ⊗ Y) Z"}
+{"state": [{"context": ["α : Type u", "M : Type u_1", "N : Type u_2", "inst✝ : DivisionCommMonoid α", "a✝ b c✝ d : α", "hb : IsUnit b", "hd : IsUnit d", "a c : α", "h : a / b = c / d"], "goal": "a * d = c * b"}], "premise": [117906, 117910, 119730, 120562, 120579], "state_str": "α : Type u\nM : Type u_1\nN : Type u_2\ninst✝ : DivisionCommMonoid α\na✝ b c✝ d : α\nhb : IsUnit b\nhd : IsUnit d\na c : α\nh : a / b = c / d\n⊢ a * d = c * b"}
+{"state": [{"context": ["μ : ℝ", "v : ℝ≥0"], "goal": "Integrable (gaussianPDFReal μ v) ℙ"}], "premise": [72683], "state_str": "μ : ℝ\nv : ℝ≥0\n⊢ Integrable (gaussianPDFReal μ v) ℙ"}
+{"state": [{"context": ["μ : ℝ", "v : ��≥0", "hv : ¬v = 0", "g : ℝ → ℝ := fun x => (√(2 * π * ↑v))⁻¹ * rexp (-x ^ 2 / (2 * ↑v))"], "goal": "Integrable (fun x => (√(2 * π * ↑v))⁻¹ * rexp (-(x - μ) ^ 2 / (2 * ↑v))) ℙ"}], "premise": [1192, 1690, 14298, 28544, 39348, 103545, 103765, 106882, 108288, 108311, 108583, 119707, 119808, 122240, 146027, 146036, 146616, 146617, 146621, 146640, 146677, 149306], "state_str": "case neg\nμ : ℝ\nv : ℝ≥0\nhv : ¬v = 0\ng : ℝ → ℝ := fun x => (√(2 * π * ↑v))⁻¹ * rexp (-x ^ 2 / (2 * ↑v))\n⊢ Integrable (fun x => (√(2 * π * ↑v))⁻¹ * rexp (-(x - μ) ^ 2 / (2 * ↑v))) ℙ"}
+{"state": [{"context": ["μ : ℝ", "v : ℝ≥0", "hv : ¬v = 0", "g : ℝ → ℝ := fun x => (√(2 * π * ↑v))⁻¹ * rexp (-x ^ 2 / (2 * ↑v))", "hg : Integrable g ℙ"], "goal": "Integrable (fun x => (√(2 * π * ↑v))⁻¹ * rexp (-(x - μ) ^ 2 / (2 * ↑v))) ℙ"}], "premise": [30925], "state_str": "case neg\nμ : ℝ\nv : ℝ≥0\nhv : ¬v = 0\ng : ℝ → ℝ := fun x => (√(2 * π * ↑v))⁻¹ * rexp (-x ^ 2 / (2 * ↑v))\nhg : Integrable g ℙ\n⊢ Integrable (fun x => (√(2 * π * ↑v))⁻¹ * rexp (-(x - μ) ^ 2 / (2 * ↑v))) ℙ"}
+{"state": [{"context": ["C : Type u", "inst✝² : Category.{v, u} C", "inst✝¹ : HasFiniteProducts C", "inst✝ : HasPullbacks C", "X Y Z : Dial C"], "goal": "((X.tensorObj Y).tensorObj Z).rel = (Subobject.pullback (prod.map (prod.associator X.src Y.src Z.src).hom (prod.associator X.tgt Y.tgt Z.tgt).hom)).obj (X.tensorObj (Y.tensorObj Z)).rel"}], "premise": [14580, 89300, 90541], "state_str": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasFiniteProducts C\ninst✝ : HasPullbacks C\nX Y Z : Dial C\n⊢ ((X.tensorObj Y).tensorObj Z).rel =\n    (Subobject.pullback (prod.map (prod.associator X.src Y.src Z.src).hom (prod.associator X.tgt Y.tgt Z.tgt).hom)).obj\n      (X.tensorObj (Y.tensorObj Z)).rel"}
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+{"state": [{"context": ["Ω : Type u_1", "β : Type u_2", "ι : Type u_3", "m : MeasurableSpace Ω", "inst✝ : LinearOrder ι", "f : Filtration ι m", "τ π : Ω → ι", "hτ : IsStoppingTime f τ", "s : Set Ω", "i : ι", "h : MeasurableSet (s ∩ {ω | τ ω ≤ i} ∩ {ω | τ ω ≤ i})"], "goal": "MeasurableSet (s ∩ {ω | τ ω ≤ i})"}], "premise": [133440, 133444], "state_str": "Ω : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝ : LinearOrder ι\nf : Filtration ι m\nτ π : Ω → ι\nhτ : IsStoppingTime f τ\ns : Set Ω\ni : ι\nh : MeasurableSet (s ∩ {ω | τ ω ≤ i} ∩ {ω | τ ω ≤ i})\n⊢ MeasurableSet (s ∩ {ω | τ ω ≤ i})"}
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+{"state": [{"context": [], "goal": "@zipWithLeft = @zipWithLeftTR"}], "premise": [1838], "state_str": "⊢ @zipWithLeft = @zipWithLeftTR"}
+{"state": [{"context": ["α : Type u_3", "β : Type u_2", "γ : Type u_1", "f : α → Option β → γ", "as : List α", "bs : List β", "acc : Array γ", "head✝ : α", "tail✝ : List α"], "goal": "zipWithLeftTR.go f (head✝ :: tail✝) [] acc = acc.toList ++ zipWithLeft f (head✝ :: tail✝) []"}], "premise": [5846], "state_str": "α : Type u_3\nβ : Type u_2\nγ : Type u_1\nf : α → Option β → γ\nas : List α\nbs : List β\nacc : Array γ\nhead✝ : α\ntail✝ : List α\n⊢ zipWithLeftTR.go f (head✝ :: tail✝) [] acc = acc.toList ++ zipWithLeft f (head✝ :: tail✝) []"}
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+{"state": [{"context": ["ι : Type u_1", "inst✝ : Fintype ι", "a b : ι → ℝ", "h : a ≤ b"], "goal": "(volume (univ.pi fun i => Ico (a i) (b i))).toReal = ∏ i : ι, (b i - a i)"}], "premise": [1674, 30109, 105706, 143161, 143426], "state_str": "ι : Type u_1\ninst✝ : Fintype ι\na b : ι → ℝ\nh : a ≤ b\n⊢ (volume (univ.pi fun i => Ico (a i) (b i))).toReal = ∏ i : ι, (b i - a i)"}
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+{"state": [{"context": ["p : ℕ+", "k : ℕ", "K : Type u", "inst✝² : Field K", "inst✝¹ : CharZero K", "ζ : K", "hp : Fact (Nat.Prime ↑p)", "inst✝ : IsCyclotomicExtension {2 ^ (k + 1)} ℚ K", "hζ : IsPrimitiveRoot ζ ↑(2 ^ (k + 1))"], "goal": "Prime (hζ.toInteger - 1)"}], "premise": [24218], "state_str": "p : ℕ+\nk : ℕ\nK : Type u\ninst✝² : Field K\ninst✝¹ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {2 ^ (k + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ ↑(2 ^ (k + 1))\n⊢ Prime (hζ.toInteger - 1)"}
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+{"state": [{"context": ["p : ℕ+", "k : ℕ", "K : Type u", "inst✝² : Field K", "inst✝¹ : CharZero K", "ζ : K", "hp : Fact (Nat.Prime ↑p)", "inst✝ : IsCyclotomicExtension {2 ^ (k + 1)} ℚ K", "hζ : IsPrimitiveRoot ζ ↑(2 ^ (k + 1))", "this : NumberField K := numberField {2 ^ (k + 1)} ℚ K"], "goal": "Irreducible (Ideal.absNorm (Ideal.span {hζ.toInteger - 1}))"}], "premise": [75117, 81579, 144342, 144343], "state_str": "case refine_2\np : ℕ+\nk : ℕ\nK : Type u\ninst✝² : Field K\ninst✝¹ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {2 ^ (k + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ ↑(2 ^ (k + 1))\nthis : NumberField K := numberField {2 ^ (k + 1)} ℚ K\n⊢ Irreducible (Ideal.absNorm (Ideal.span {hζ.toInteger - 1}))"}
+{"state": [{"context": ["p : ℕ+", "K : Type u", "inst✝² : Field K", "inst✝¹ : CharZero K", "ζ : K", "hp : Fact (Nat.Prime ↑p)", "n✝ : ℕ", "inst✝ : IsCyclotomicExtension {2 ^ (n✝ + 1 + 1)} ℚ K", "hζ : IsPrimitiveRoot ζ ↑(2 ^ (n✝ + 1 + 1))", "this : NumberField K := numberField {2 ^ (n✝ + 1 + 1)} ℚ K"], "goal": "Prime ((Algebra.norm ℤ) (hζ.toInteger - 1))"}], "premise": [144261], "state_str": "case refine_2.succ\np : ℕ+\nK : Type u\ninst✝² : Field K\ninst✝¹ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\nn✝ : ℕ\ninst✝ : IsCyclotomicExtension {2 ^ (n✝ + 1 + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ ↑(2 ^ (n✝ + 1 + 1))\nthis : NumberField K := numberField {2 ^ (n✝ + 1 + 1)} ℚ K\n⊢ Prime ((Algebra.norm ℤ) (hζ.toInteger - 1))"}
+{"state": [{"context": ["p : ℕ+", "K : Type u", "inst✝² : Field K", "inst✝¹ : CharZero K", "ζ : K", "hp : Fact (Nat.Prime ↑p)", "n✝ : ℕ", "inst✝ : IsCyclotomicExtension {2 ^ (n✝ + 1 + 1)} ℚ K", "hζ : IsPrimitiveRoot ζ ↑(2 ^ (n✝ + 1 + 1))", "this : NumberField K := numberField {2 ^ (n✝ + 1 + 1)} ℚ K"], "goal": "(Algebra.norm ℤ) (hζ.toInteger - 1) = 2"}], "premise": [129080], "state_str": "case h.e'_3\np : ℕ+\nK : Type u\ninst✝² : Field K\ninst✝¹ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\nn✝ : ℕ\ninst✝ : IsCyclotomicExtension {2 ^ (n✝ + 1 + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ ↑(2 ^ (n✝ + 1 + 1))\nthis : NumberField K := numberField {2 ^ (n✝ + 1 + 1)} ℚ K\n⊢ (Algebra.norm ℤ) (hζ.toInteger - 1) = 2"}
+{"state": [{"context": ["p : ℕ+", "K : Type u", "inst✝² : Field K", "inst✝¹ : CharZero K", "ζ : K", "hp : Fact (Nat.Prime ↑p)", "n✝ : ℕ", "inst✝ : IsCyclotomicExtension {2 ^ (n✝ + 1 + 1)} ℚ K", "hζ : IsPrimitiveRoot ζ ↑(2 ^ (n✝ + 1 + 1))", "this : NumberField K := numberField {2 ^ (n✝ + 1 + 1)} ℚ K"], "goal": "(algebraMap ℤ ℚ) ((Algebra.norm ℤ) (hζ.toInteger - 1)) = (algebraMap ℤ ℚ) 2"}], "premise": [76899], "state_str": "case h.e'_3.a\np : ℕ+\nK : Type u\ninst✝² : Field K\ninst✝¹ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\nn✝ : ℕ\ninst✝ : IsCyclotomicExtension {2 ^ (n✝ + 1 + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ ↑(2 ^ (n✝ + 1 + 1))\nthis : NumberField K := numberField {2 ^ (n✝ + 1 + 1)} ℚ K\n⊢ (algebraMap ℤ ℚ) ((Algebra.norm ℤ) (hζ.toInteger - 1)) = (algebraMap ℤ ℚ) 2"}
+{"state": [{"context": ["p : ℕ+", "K : Type u", "inst✝² : Field K", "inst✝¹ : CharZero K", "ζ : K", "hp : Fact (Nat.Prime ↑p)", "n✝ : ℕ", "inst✝ : IsCyclotomicExtension {2 ^ (n✝ + 1 + 1)} ℚ K", "hζ : IsPrimitiveRoot ζ ↑(2 ^ (n✝ + 1 + 1))", "this : NumberField K := numberField {2 ^ (n✝ + 1 + 1)} ℚ K"], "goal": "(Algebra.norm ℚ) ((algebraMap (𝓞 K) K) (hζ.toInteger - 1)) = (algebraMap ℤ ℚ) 2"}], "premise": [121179, 121269, 121548, 122058, 122412, 141238, 142663], "state_str": "case h.e'_3.a\np : ℕ+\nK : Type u\ninst✝² : Field K\ninst✝¹ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\nn✝ : ℕ\ninst✝ : IsCyclotomicExtension {2 ^ (n✝ + 1 + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ ↑(2 ^ (n✝ + 1 + 1))\nthis : NumberField K := numberField {2 ^ (n✝ + 1 + 1)} ℚ K\n⊢ (Algebra.norm ℚ) ((algebraMap (𝓞 K) K) (hζ.toInteger - 1)) = (algebraMap ℤ ℚ) 2"}
+{"state": [{"context": ["p : ℕ+", "K : Type u", "inst✝² : Field K", "inst✝¹ : CharZero K", "ζ : K", "hp : Fact (Nat.Prime ↑p)", "n✝ : ℕ", "inst✝ : IsCyclotomicExtension {2 ^ (n✝ + 1 + 1)} ℚ K", "hζ : IsPrimitiveRoot ζ ↑(2 ^ (n✝ + 1 + 1))", "this : NumberField K := numberField {2 ^ (n✝ + 1 + 1)} ℚ K"], "goal": "(Algebra.norm ℚ) ((algebraMap (𝓞 K) K) (hζ.toInteger - 1)) = (Int.castRingHom ℚ) 2"}], "premise": [25245, 74768, 143114], "state_str": "case h.e'_3.a\np : ℕ+\nK : Type u\ninst✝² : Field K\ninst✝¹ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\nn✝ : ℕ\ninst✝ : IsCyclotomicExtension {2 ^ (n✝ + 1 + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ ↑(2 ^ (n✝ + 1 + 1))\nthis : NumberField K := numberField {2 ^ (n✝ + 1 + 1)} ℚ K\n⊢ (Algebra.norm ℚ) ((algebraMap (𝓞 K) K) (hζ.toInteger - 1)) = (Int.castRingHom ℚ) 2"}
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+{"state": [{"context": ["𝕜 : Type u_1", "inst✝⁴ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace 𝕜 E", "F : Type u_3", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace 𝕜 F", "p : FormalMultilinearSeries 𝕜 E F", "i : E ≃L[𝕜] F", "n : ℕ", "IH : ∀ m < n, p.removeZero.leftInv i m = p.leftInv i m"], "goal": "p.removeZero.leftInv i n = p.leftInv i n"}], "premise": [117792, 126920], "state_str": "case h.H\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\ni : E ≃L[𝕜] F\nn : ℕ\nIH : ∀ m < n, p.removeZero.leftInv i m = p.leftInv i m\n⊢ p.removeZero.leftInv i n = p.leftInv i n"}
+{"state": [{"context": ["ι : Type u_1", "α : Type u_2", "M : Type u_3", "N : Type u_4", "X : Type u_5", "inst✝³ : TopologicalSpace X", "inst✝² : TopologicalSpace M", "inst✝¹ : Monoid M", "inst✝ : ContinuousMul M", "f : ι → α → M", "x : Filter α", "a : ι → M", "x✝ : ∀ i ∈ [], Tendsto (f i) x (𝓝 (a i))"], "goal": "Tendsto (fun b => (List.map (fun c => f c b) []).prod) x (𝓝 (List.map a []).prod)"}], "premise": [55522], "state_str": "ι : Type u_1\nα : Type u_2\nM : Type u_3\nN : Type u_4\nX : Type u_5\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : Monoid M\ninst✝ : ContinuousMul M\nf : ι → α → M\nx : Filter α\na : ι → M\nx✝ : ∀ i ∈ [], Tendsto (f i) x (𝓝 (a i))\n⊢ Tendsto (fun b => (List.map (fun c => f c b) []).prod) x (𝓝 (List.map a []).prod)"}
+{"state": [{"context": ["ι : Type u_1", "α : Type u_2", "M : Type u_3", "N : Type u_4", "X : Type u_5", "inst✝³ : TopologicalSpace X", "inst✝² : TopologicalSpace M", "inst✝¹ : Monoid M", "inst✝ : ContinuousMul M", "f✝ : ι → α → M", "x : Filter α", "a : ι → M", "f : ι", "l : List ι", "h : ∀ i ∈ f :: l, Tendsto (f✝ i) x (𝓝 (a i))"], "goal": "Tendsto (fun b => (List.map (fun c => f✝ c b) (f :: l)).prod) x (𝓝 (List.map a (f :: l)).prod)"}], "premise": [2611, 124826], "state_str": "ι : Type u_1\nα : Type u_2\nM : Type u_3\nN : Type u_4\nX : Type u_5\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : Monoid M\ninst✝ : ContinuousMul M\nf✝ : ι → α → M\nx : Filter α\na : ι → M\nf : ι\nl : List ι\nh : ∀ i ∈ f :: l, Tendsto (f✝ i) x (𝓝 (a i))\n⊢ Tendsto (fun b => (List.map (fun c => f✝ c b) (f :: l)).prod) x (𝓝 (List.map a (f :: l)).prod)"}
+{"state": [{"context": ["ι : Type u_1", "α : Type u_2", "M : Type u_3", "N : Type u_4", "X : Type u_5", "inst✝³ : TopologicalSpace X", "inst✝² : TopologicalSpace M", "inst✝¹ : Monoid M", "inst✝ : ContinuousMul M", "f✝ : ι → α → M", "x : Filter α", "a : ι → M", "f : ι", "l : List ι", "h : ∀ i ∈ f :: l, Tendsto (f✝ i) x (𝓝 (a i))"], "goal": "Tendsto (fun b => f✝ f b * (List.map (fun c => f✝ c b) l).prod) x (𝓝 (a f * (List.map a l).prod))"}], "premise": [5022, 5023, 65038], "state_str": "ι : Type u_1\nα : Type u_2\nM : Type u_3\nN : Type u_4\nX : Type u_5\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : Monoid M\ninst✝ : ContinuousMul M\nf✝ : ι → α → M\nx : Filter α\na : ι → M\nf : ι\nl : List ι\nh : ∀ i ∈ f :: l, Tendsto (f✝ i) x (𝓝 (a i))\n⊢ Tendsto (fun b => f✝ f b * (List.map (fun c => f✝ c b) l).prod) x (𝓝 (a f * (List.map a l).prod))"}
+{"state": [{"context": ["F : Type u_1", "inst✝² : Field F", "E : Type u_2", "inst✝¹ : Field E", "inst✝ : Algebra F E", "S : Set E", "K : IntermediateField F E"], "goal": "lift ⊤ = K"}], "premise": [88474, 90789], "state_str": "F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : Set E\nK : IntermediateField F E\n⊢ lift ⊤ = K"}
+{"state": [{"context": ["K : Type u", "inst✝ : Field K", "A : ValuationSubring K", "a✝ b✝ : (PrimeSpectrum ↥A)ᵒᵈ", "h : A.primeSpectrumEquiv a✝ ≤ A.primeSpectrumEquiv b✝"], "goal": "a✝ ≤ b✝"}], "premise": [76007], "state_str": "K : Type u\ninst✝ : Field K\nA : ValuationSubring K\na✝ b✝ : (PrimeSpectrum ↥A)ᵒᵈ\nh : A.primeSpectrumEquiv a✝ ≤ A.primeSpectrumEquiv b✝\n⊢ a✝ ≤ b✝"}
+{"state": [{"context": ["K : Type u", "inst✝ : Field K", "A : ValuationSubring K", "a✝ b✝ : (PrimeSpectrum ↥A)ᵒᵈ", "h : A.primeSpectrumEquiv a✝ ≤ A.primeSpectrumEquiv b✝", "this : A.idealOfLE ↑(A.primeSpectrumEquiv b✝) ?refine_2 ≤ A.idealOfLE ↑(A.primeSpectrumEquiv a✝) ?refine_1"], "goal": "a✝ ≤ b✝"}, {"context": ["K : Type u", "inst✝ : Field K", "A : ValuationSubring K", "a✝ b✝ : (PrimeSpectrum ↥A)ᵒᵈ", "h : A.primeSpectrumEquiv a✝ ≤ A.primeSpectrumEquiv b✝"], "goal": "A ≤ ↑(A.primeSpectrumEquiv a✝)"}, {"context": ["K : Type u", "inst✝ : Field K", "A : ValuationSubring K", "a✝ b✝ : (PrimeSpectrum ↥A)ᵒᵈ", "h : A.primeSpectrumEquiv a✝ ≤ A.primeSpectrumEquiv b✝"], "goal": "A ≤ ↑(A.primeSpectrumEquiv b✝)"}], "premise": [76004], "state_str": "case refine_3\nK : Type u\ninst✝ : Field K\nA : ValuationSubring K\na✝ b✝ : (PrimeSpectrum ↥A)ᵒᵈ\nh : A.primeSpectrumEquiv a✝ ≤ A.primeSpectrumEquiv b✝\nthis : A.idealOfLE ↑(A.primeSpectrumEquiv b✝) ?refine_2 ≤ A.idealOfLE ↑(A.primeSpectrumEquiv a✝) ?refine_1\n⊢ a✝ ≤ b✝\n\ncase refine_1\nK : Type u\ninst✝ : Field K\nA : ValuationSubring K\na✝ b✝ : (PrimeSpectrum ↥A)ᵒᵈ\nh : A.primeSpectrumEquiv a✝ ≤ A.primeSpectrumEquiv b✝\n⊢ A ≤ ↑(A.primeSpectrumEquiv a✝)\n\ncase refine_2\nK : Type u\ninst✝ : Field K\nA : ValuationSubring K\na✝ b✝ : (PrimeSpectrum ↥A)ᵒᵈ\nh : A.primeSpectrumEquiv a✝ ≤ A.primeSpectrumEquiv b✝\n⊢ A ≤ ↑(A.primeSpectrumEquiv b✝)"}
+{"state": [{"context": ["K : Type u", "inst✝ : Field K", "A : ValuationSubring K", "a✝ b✝ : (PrimeSpectrum ↥A)ᵒᵈ", "h : A.primeSpectrumEquiv a✝ ≤ A.primeSpectrumEquiv b✝"], "goal": "A ≤ ↑(A.primeSpectrumEquiv a✝)"}, {"context": ["K : Type u", "inst✝ : Field K", "A : ValuationSubring K", "a✝ b✝ : (PrimeSpectrum ↥A)ᵒᵈ", "h : A.primeSpectrumEquiv a✝ ≤ A.primeSpectrumEquiv b✝"], "goal": "A ≤ ↑(A.primeSpectrumEquiv b✝)"}], "premise": [76002], "state_str": "case refine_1\nK : Type u\ninst✝ : Field K\nA : ValuationSubring K\na✝ b✝ : (PrimeSpectrum ↥A)ᵒᵈ\nh : A.primeSpectrumEquiv a✝ ≤ A.primeSpectrumEquiv b✝\n⊢ A ≤ ↑(A.primeSpectrumEquiv a✝)\n\ncase refine_2\nK : Type u\ninst✝ : Field K\nA : ValuationSubring K\na✝ b✝ : (PrimeSpectrum ↥A)ᵒᵈ\nh : A.primeSpectrumEquiv a✝ ≤ A.primeSpectrumEquiv b✝\n⊢ A ≤ ↑(A.primeSpectrumEquiv b✝)"}
+{"state": [{"context": ["K : Type u", "inst✝ : Field K", "A : ValuationSubring K", "a✝ b✝ : (PrimeSpectrum ↥A)ᵒᵈ", "h : a✝ ≤ b✝"], "goal": "__src✝ a✝ ≤ __src✝ b✝"}], "premise": [76006], "state_str": "K : Type u\ninst✝ : Field K\nA : ValuationSubring K\na✝ b✝ : (PrimeSpectrum ↥A)ᵒᵈ\nh : a✝ ≤ b✝\n⊢ __src✝ a✝ ≤ __src✝ b✝"}
+{"state": [{"context": ["R : Type u", "inst✝¹ : CommRing R", "W' : Jacobian R", "F : Type v", "inst✝ : Field F", "W : Jacobian F", "P : Fin 3 → R", "u : R"], "goal": "W'.dblU (u • P) = u ^ 4 * W'.dblU P"}], "premise": [145446, 145508], "state_str": "R : Type u\ninst✝¹ : CommRing R\nW' : Jacobian R\nF : Type v\ninst✝ : Field F\nW : Jacobian F\nP : Fin 3 → R\nu : R\n⊢ W'.dblU (u • P) = u ^ 4 * W'.dblU P"}
+{"state": [{"context": ["n : ℕ"], "goal": "Irrational √↑n ↔ ¬IsSquare n"}], "premise": [1713, 142636, 145681, 147900, 148033], "state_str": "n : ℕ\n⊢ Irrational √↑n ↔ ¬IsSquare n"}
+{"state": [{"context": ["R : Type u", "S : Type v", "T : Type w", "ι : Type y", "a b : R", "m n : ℕ", "inst✝¹ : Semiring R", "p q r : R[X]", "inst✝ : Semiring S", "f : R →+* S", "x : S"], "goal": "eval₂ f x X = x"}], "premise": [102824], "state_str": "R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\nx : S\n⊢ eval₂ f x X = x"}
+{"state": [{"context": ["R : Type u", "inst✝ : CommRing R", "x y : R", "h : IsCoprime x y", "z : R"], "goal": "IsCoprime x (z * x + y)"}], "premise": [119708], "state_str": "R : Type u\ninst✝ : CommRing R\nx y : R\nh : IsCoprime x y\nz : R\n⊢ IsCoprime x (z * x + y)"}
+{"state": [{"context": ["R : Type u", "inst✝ : CommRing R", "x y : R", "h : IsCoprime x y", "z : R"], "goal": "IsCoprime x (y + z * x)"}], "premise": [74115], "state_str": "R : Type u\ninst✝ : CommRing R\nx y : R\nh : IsCoprime x y\nz : R\n⊢ IsCoprime x (y + z * x)"}
+{"state": [{"context": ["C : Type u_1", "D : Type u_2", "inst✝³ : Category.{u_3, u_1} C", "inst✝² : Category.{?u.82092, u_2} D", "inst✝¹ : Preadditive C", "inst✝ : Preadditive D", "S : ShortComplex C", "s s' : S.Splitting", "h : s.r = s'.r"], "goal": "s = s'"}], "premise": [114606], "state_str": "C : Type u_1\nD : Type u_2\ninst✝³ : Category.{u_3, u_1} C\ninst✝² : Category.{?u.82092, u_2} D\ninst✝¹ : Preadditive C\ninst✝ : Preadditive D\nS : ShortComplex C\ns s' : S.Splitting\nh : s.r = s'.r\n⊢ s = s'"}
+{"state": [{"context": ["C : Type u_1", "D : Type u_2", "inst✝³ : Category.{u_3, u_1} C", "inst✝² : Category.{?u.82092, u_2} D", "inst✝¹ : Preadditive C", "inst✝ : Preadditive D", "S : ShortComplex C", "s s' : S.Splitting", "h : s.r = s'.r", "this : Epi S.g"], "goal": "s = s'"}], "premise": [114596], "state_str": "C : Type u_1\nD : Type u_2\ninst✝³ : Category.{u_3, u_1} C\ninst✝² : Category.{?u.82092, u_2} D\ninst✝¹ : Preadditive C\ninst✝ : Preadditive D\nS : ShortComplex C\ns s' : S.Splitting\nh : s.r = s'.r\nthis : Epi S.g\n⊢ s = s'"}
+{"state": [{"context": ["C : Type u_1", "D : Type u_2", "inst✝³ : Category.{u_3, u_1} C", "inst✝² : Category.{?u.82092, u_2} D", "inst✝¹ : Preadditive C", "inst✝ : Preadditive D", "S : ShortComplex C", "s s' : S.Splitting", "h : s.r = s'.r", "this : Epi S.g", "eq : s.r ≫ S.f + S.g ≫ s.s = 𝟙 S.X₂"], "goal": "s = s'"}], "premise": [96190, 114596, 117714], "state_str": "C : Type u_1\nD : Type u_2\ninst✝³ : Category.{u_3, u_1} C\ninst✝² : Category.{?u.82092, u_2} D\ninst✝¹ : Preadditive C\ninst✝ : Preadditive D\nS : ShortComplex C\ns s' : S.Splitting\nh : s.r = s'.r\nthis : Epi S.g\neq : s.r ≫ S.f + S.g ≫ s.s = 𝟙 S.X₂\n⊢ s = s'"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m : MeasurableSpace α", "μ ν : Measure α", "ι : Type u_5", "f : α → ℝ≥0∞", "h : ∫⁻ (x : α), f x ∂μ ≠ ⊤", "l : Filter ι", "s : ι → Set α", "hl : Tendsto (⇑μ ∘ s) l (𝓝 0)"], "goal": "Tendsto (fun i => ∫⁻ (x : α) in s i, f x ∂μ) l (𝓝 0)"}], "premise": [16372, 16380, 20159, 58904, 103552], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nι : Type u_5\nf : α → ℝ≥0∞\nh : ∫⁻ (x : α), f x ∂μ ≠ ⊤\nl : Filter ι\ns : ι → Set α\nhl : Tendsto (⇑μ ∘ s) l (𝓝 0)\n⊢ Tendsto (fun i => ∫⁻ (x : α) in s i, f x ∂μ) l (𝓝 0)"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m : MeasurableSpace α", "μ ν : Measure α", "ι : Type u_5", "f : α → ℝ≥0∞", "h : ∫⁻ (x : α), f x ∂μ ≠ ⊤", "l : Filter ι", "s : ι → Set α", "hl : ∀ (i : ℝ≥0∞), 0 < i → ∀ᶠ (a : ι) in l, (⇑μ ∘ s) a < i", "ε : ℝ≥0∞", "ε0 : 0 < ε"], "goal": "∀ᶠ (a : ι) in l, ∫⁻ (x : α) in s a, f x ∂μ < ε"}], "premise": [11234, 30271], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nι : Type u_5\nf : α → ℝ≥0∞\nh : ∫⁻ (x : α), f x ∂μ ≠ ⊤\nl : Filter ι\ns : ι → Set α\nhl : ∀ (i : ℝ≥0∞), 0 < i → ∀ᶠ (a : ι) in l, (⇑μ ∘ s) a < i\nε : ℝ≥0∞\nε0 : 0 < ε\n⊢ ∀ᶠ (a : ι) in l, ∫⁻ (x : α) in s a, f x ∂μ < ε"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ✝ : Type u_4", "m : MeasurableSpace α", "μ ν : Measure α", "ι : Type u_5", "f : α → ℝ≥0∞", "h : ∫⁻ (x : α), f x ∂μ ≠ ⊤", "l : Filter ι", "s : ι → Set α", "hl : ∀ (i : ℝ≥0∞), 0 < i → ∀ᶠ (a : ι) in l, (⇑μ ∘ s) a < i", "ε : ℝ≥0∞", "ε0 : 0 < ε", "δ : ℝ≥0∞", "δ0 : δ > 0", "hδ : ∀ (s : Set α), μ s < δ → ∫⁻ (x : α) in s, f x ∂μ < ε"], "goal": "∀ᶠ (a : ι) in l, ∫⁻ (x : α) in s a, f x ∂μ < ε"}], "premise": [16027], "state_str": "case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ✝ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nι : Type u_5\nf : α → ℝ≥0∞\nh : ∫⁻ (x : α), f x ∂μ ≠ ⊤\nl : Filter ι\ns : ι → Set α\nhl : ∀ (i : ℝ≥0∞), 0 < i → ∀ᶠ (a : ι) in l, (⇑μ ∘ s) a < i\nε : ℝ≥0∞\nε0 : 0 < ε\nδ : ℝ≥0∞\nδ0 : δ > 0\nhδ : ∀ (s : Set α), μ s < δ → ∫⁻ (x : α) in s, f x ∂μ < ε\n⊢ ∀ᶠ (a : ι) in l, ∫⁻ (x : α) in s a, f x ∂μ < ε"}
+{"state": [{"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝ : LinearOrderedSemifield α", "a b c d e : α", "m n : ℤ", "ha : 0 < a", "h : a < b"], "goal": "1 / b < 1 / a"}], "premise": [106028, 106089, 117815], "state_str": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : LinearOrderedSemifield α\na b c d e : α\nm n : ℤ\nha : 0 < a\nh : a < b\n⊢ 1 / b < 1 / a"}
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+{"state": [{"context": ["ι : Type u_1", "V : Type u", "inst✝¹ : Category.{v, u} V", "inst✝ : Preadditive V", "c : ComplexShape ι", "C D E : HomologicalComplex V c", "f✝ g : C ⟶ D", "h k : D ⟶ E", "i : ι", "P Q : CochainComplex V ℕ", "f : (i j : ℕ) → P.X i ⟶ Q.X j", "j : ℕ"], "goal": "P.d j ((ComplexShape.up ℕ).next j) ≫ f ((ComplexShape.up ℕ).next j) j = P.d j (j + 1) ≫ f (j + 1) j"}], "premise": [113846], "state_str": "ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf✝ g : C ⟶ D\nh k : D ⟶ E\ni : ι\nP Q : CochainComplex V ℕ\nf : (i j : ℕ) → P.X i ⟶ Q.X j\nj : ℕ\n⊢ P.d j ((ComplexShape.up ℕ).next j) ≫ f ((ComplexShape.up ℕ).next j) j = P.d j (j + 1) ≫ f (j + 1) j"}
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+{"state": [{"context": ["R : Type u", "inst✝¹ : CommRing R", "W' : Jacobian R", "F : Type v", "inst✝ : Field F", "W : Jacobian F", "P : Fin 3 → F", "n d : F", "hPz : P z ≠ 0", "hd : d ≠ 0"], "goal": "W.toAffine.addX (P x / P z ^ 2) (P x / P z ^ 2) (-n / (P z * d)) = (n ^ 2 - W.a₁ * n * P z * d - W.a₂ * P z ^ 2 * d ^ 2 - 2 * P x * d ^ 2) / (P z * d) ^ 2"}], "premise": [108269], "state_str": "R : Type u\ninst✝¹ : CommRing R\nW' : Jacobian R\nF : Type v\ninst✝ : Field F\nW : Jacobian F\nP : Fin 3 → F\nn d : F\nhPz : P z ≠ 0\nhd : d ≠ 0\n⊢ W.toAffine.addX (P x / P z ^ 2) (P x / P z ^ 2) (-n / (P z * d)) =\n    (n ^ 2 - W.a₁ * n * P z * d - W.a₂ * P z ^ 2 * d ^ 2 - 2 * P x * d ^ 2) / (P z * d) ^ 2"}
+{"state": [{"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝⁵ : _root_.RCLike 𝕜", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : NormedAddCommGroup F", "inst✝² : InnerProductSpace 𝕜 E", "inst✝¹ : InnerProductSpace ℝ F", "K : Submodule 𝕜 E", "inst✝ : HasOrthogonalProjection K", "v w : E", "hv : ¬v = 0"], "goal": "↑((orthogonalProjection (Submodule.span 𝕜 {v})) w) = (⟪v, w⟫_𝕜 / ↑(‖v‖ ^ 2)) • v"}], "premise": [1674, 14284, 42922], "state_str": "case neg\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : _root_.RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : HasOrthogonalProjection K\nv w : E\nhv : ¬v = 0\n⊢ ↑((orthogonalProjection (Submodule.span 𝕜 {v})) w) = (⟪v, w⟫_𝕜 / ↑(‖v‖ ^ 2)) • v"}
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+{"state": [{"context": ["G : Type u_1", "G' : Type u_2", "G'' : Type u_3", "inst✝³ : Group G", "inst✝² : Group G'", "inst✝¹ : Group G''", "A : Type u_4", "inst✝ : AddGroup A", "H K : Subgroup G", "k : Set G", "ι : Sort u_5", "p : ι → Subgroup G"], "goal": "⨆ i, p i = closure (⋃ i, ↑(p i))"}], "premise": [122733, 122743], "state_str": "G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst✝³ : Group G\ninst✝² : Group G'\ninst✝¹ : Group G''\nA : Type u_4\ninst✝ : AddGroup A\nH K : Subgroup G\nk : Set G\nι : Sort u_5\np : ι → Subgroup G\n⊢ ⨆ i, p i = closure (⋃ i, ↑(p i))"}
+{"state": [{"context": ["α : Sort u_1", "β : Sort u_2", "γ : Sort u_3", "f : α → β", "hf : Injective f", "e' : β → γ", "g₁ g₂ : α → γ", "hg : (fun g => extend f g e') g₁ = (fun g => extend f g e') g₂"], "goal": "g₁ = g₂"}], "premise": [1838], "state_str": "α : Sort u_1\nβ : Sort u_2\nγ : Sort u_3\nf : α → β\nhf : Injective f\ne' : β → γ\ng₁ g₂ : α → γ\nhg : (fun g => extend f g e') g₁ = (fun g => extend f g e') g₂\n⊢ g₁ = g₂"}
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+{"state": [{"context": ["m n k : ℕ"], "goal": "(m + k * n).Coprime n ↔ m.Coprime n"}], "premise": [1713, 143592], "state_str": "m n k : ℕ\n⊢ (m + k * n).Coprime n ↔ m.Coprime n"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "p : α → Prop", "inst✝ : DecidablePred p", "s : Finset α", "x : α"], "goal": "x ∈ map (Embedding.subtype p) (Finset.subtype p s) ↔ x ∈ filter p s"}], "premise": [1723, 1970], "state_str": "case a\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\np : α → Prop\ninst✝ : DecidablePred p\ns : Finset α\nx : α\n⊢ x ∈ map (Embedding.subtype p) (Finset.subtype p s) ↔ x ∈ filter p s"}
+{"state": [{"context": ["a b : ℕ+"], "goal": "(Ioo a b).card = ↑b - ↑a - 1"}], "premise": [19118], "state_str": "a b : ℕ+\n⊢ (Ioo a b).card = ↑b - ↑a - 1"}
+{"state": [{"context": ["a b : ℕ+"], "goal": "(Ioo a b).card = (Ioo ↑a ↑b).card"}], "premise": [19667], "state_str": "a b : ℕ+\n⊢ (Ioo a b).card = (Ioo ↑a ↑b).card"}
+{"state": [{"context": ["a b : ℕ+"], "goal": "(Ioo a b).card = (map (Embedding.subtype fun n => 0 < n) (Ioo a b)).card"}], "premise": [137660], "state_str": "a b : ℕ+\n⊢ (Ioo a b).card = (map (Embedding.subtype fun n => 0 < n) (Ioo a b)).card"}
+{"state": [{"context": ["p : ℕ", "hp : Fact (Nat.Prime p)", "m : ℕ"], "goal": "padicNorm p ↑m < 1 ↔ p ∣ m"}], "premise": [1713, 22580, 128750, 130033], "state_str": "p : ℕ\nhp : Fact (Nat.Prime p)\nm : ℕ\n⊢ padicNorm p ↑m < 1 ↔ p ∣ m"}
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+{"state": [{"context": ["𝕜 : Type u_1", "V : Type u_2", "V₁ : Type u_3", "V₁' : Type u_4", "V₂ : Type u_5", "V₃ : Type u_6", "V₄ : Type u_7", "P₁ : Type u_8", "P₁' : Type u_9", "P : Type u_10", "P₂ : Type u_11", "P₃ : Type u_12", "P₄ : Type u_13", "inst✝²⁵ : NormedField 𝕜", "inst✝²⁴ : SeminormedAddCommGroup V", "inst✝²³ : NormedSpace 𝕜 V", "inst✝²² : PseudoMetricSpace P", "inst✝²¹ : NormedAddTorsor V P", "inst✝²⁰ : SeminormedAddCommGroup V₁", "inst✝¹⁹ : NormedSpace 𝕜 V₁", "inst✝¹⁸ : PseudoMetricSpace P₁", "inst✝¹⁷ : NormedAddTorsor V₁ P₁", "inst✝¹⁶ : SeminormedAddCommGroup V₁'", "inst✝¹⁵ : NormedSpace 𝕜 V₁'", "inst✝¹⁴ : MetricSpace P₁'", "inst✝¹³ : NormedAddTorsor V₁' P₁'", "inst✝¹² : SeminormedAddCommGroup V₂", "inst✝¹¹ : NormedSpace 𝕜 V₂", "inst✝¹⁰ : PseudoMetricSpace P₂", "inst✝⁹ : NormedAddTorsor V₂ P₂", "inst✝⁸ : SeminormedAddCommGroup V₃", "inst✝⁷ : NormedSpace 𝕜 V₃", "inst✝⁶ : PseudoMetricSpace P₃", "inst✝⁵ : NormedAddTorsor V₃ P₃", "inst✝⁴ : SeminormedAddCommGroup V₄", "inst✝³ : NormedSpace 𝕜 V₄", "inst✝² : PseudoMetricSpace P₄", "inst✝¹ : NormedAddTorsor V₄ P₄", "e : P ≃ᵃⁱ[𝕜] P₂", "α : Type u_14", "inst✝ : TopologicalSpace α", "x y : P"], "goal": "dist ((pointReflection 𝕜 x) y) x = dist y x"}], "premise": [41069, 41093], "state_str": "𝕜 : Type u_1\nV : Type u_2\nV₁ : Type u_3\nV₁' : Type u_4\nV₂ : Type u_5\nV₃ : Type u_6\nV₄ : Type u_7\nP₁ : Type u_8\nP₁' : Type u_9\nP : Type u_10\nP₂ : Type u_11\nP₃ : Type u_12\nP₄ : Type u_13\ninst✝²⁵ : NormedField 𝕜\ninst✝²⁴ : SeminormedAddCommGroup V\ninst✝²³ : NormedSpace 𝕜 V\ninst✝²² : PseudoMetricSpace P\ninst✝²¹ : NormedAddTorsor V P\ninst✝²⁰ : SeminormedAddCommGroup V₁\ninst✝¹⁹ : NormedSpace 𝕜 V₁\ninst✝¹⁸ : PseudoMetricSpace P₁\ninst✝¹⁷ : NormedAddTorsor V₁ P₁\ninst✝¹⁶ : SeminormedAddCommGroup V₁'\ninst✝¹⁵ : NormedSpace 𝕜 V₁'\ninst✝¹⁴ : MetricSpace P₁'\ninst✝¹³ : NormedAddTorsor V₁' P₁'\ninst✝¹² : SeminormedAddCommGroup V₂\ninst✝¹¹ : NormedSpace 𝕜 V₂\ninst✝¹⁰ : PseudoMetricSpace P₂\ninst✝⁹ : NormedAddTorsor V₂ P₂\ninst✝⁸ : SeminormedAddCommGroup V₃\ninst✝⁷ : NormedSpace 𝕜 V₃\ninst✝⁶ : PseudoMetricSpace P₃\ninst✝⁵ : NormedAddTorsor V₃ P₃\ninst✝⁴ : SeminormedAddCommGroup V₄\ninst✝³ : NormedSpace 𝕜 V₄\ninst✝² : PseudoMetricSpace P₄\ninst✝¹ : NormedAddTorsor V₄ P₄\ne : P ≃ᵃⁱ[𝕜] P₂\nα : Type u_14\ninst✝ : TopologicalSpace α\nx y : P\n⊢ dist ((pointReflection 𝕜 x) y) x = dist y x"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "m : MeasurableSpace α", "s : SignedMeasure α", "i j : Set α", "hi : VectorMeasure.restrict 0 i ≤ VectorMeasure.restrict s i", "hi₁ : MeasurableSet i", "hj₁ : MeasurableSet j"], "goal": "(s.toMeasureOfZeroLE i hi₁ hi) j = ↑⟨↑s (i ∩ j), ⋯⟩"}], "premise": [29065, 32661, 133443], "state_str": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\ns : SignedMeasure α\ni j : Set α\nhi : VectorMeasure.restrict 0 i ≤ VectorMeasure.restrict s i\nhi₁ : MeasurableSet i\nhj₁ : MeasurableSet j\n⊢ (s.toMeasureOfZeroLE i hi₁ hi) j = ↑⟨↑s (i ∩ j), ⋯⟩"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : Preorder α", "inst✝ : SuccOrder α", "a b : α", "n m : ℕ", "h_eq : succ^[n] a = succ^[m] a", "h_ne : n ≠ m"], "goal": "IsMax (succ^[n] a)"}], "premise": [14308], "state_str": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : SuccOrder α\na b : α\nn m : ℕ\nh_eq : succ^[n] a = succ^[m] a\nh_ne : n ≠ m\n⊢ IsMax (succ^[n] a)"}
+{"state": [{"context": ["α : Type u", "β : Type v", "γ : Type w", "δ : Type u_1", "ι✝ : Sort x", "f✝ g : Filter α", "s t : Set α", "ι : Sort w", "inst✝ : Finite ι", "f : ι → Set α"], "goal": "⨅ i, 𝓟 (f i) = 𝓟 (⋂ i, f i)"}], "premise": [141384], "state_str": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type u_1\nι✝ : Sort x\nf✝ g : Filter α\ns t : Set α\nι : Sort w\ninst✝ : Finite ι\nf : ι → Set α\n⊢ ⨅ i, 𝓟 (f i) = 𝓟 (⋂ i, f i)"}
+{"state": [{"context": ["α : Type u", "β : Type v", "γ : Type w", "δ : Type u_1", "ι✝ : Sort x", "f✝ g : Filter α", "s t : Set α", "ι : Sort w", "inst✝ : Finite ι", "f : ι → Set α", "val✝ : Fintype (PLift ι)"], "goal": "⨅ i, 𝓟 (f i) = 𝓟 (⋂ i, f i)"}], "premise": [19292, 135236], "state_str": "case intro\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type u_1\nι✝ : Sort x\nf✝ g : Filter α\ns t : Set α\nι : Sort w\ninst✝ : Finite ι\nf : ι → Set α\nval✝ : Fintype (PLift ι)\n⊢ ⨅ i, 𝓟 (f i) = 𝓟 (⋂ i, f i)"}
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+{"state": [{"context": ["s b : ℝ", "hb : 0 < b"], "goal": "Tendsto (fun x => x ^ s * rexp (-b * x)) atTop (𝓝 0)"}], "premise": [16350, 39189, 62831], "state_str": "s b : ℝ\nhb : 0 < b\n⊢ Tendsto (fun x => x ^ s * rexp (-b * x)) atTop (𝓝 0)"}
+{"state": [{"context": ["s b : ℝ", "hb : 0 < b"], "goal": "(fun x => rexp (b * x) / x ^ s)⁻¹ =ᶠ[atTop] fun x => x ^ s * rexp (-b * x)"}], "premise": [117836, 119790, 131585, 149217], "state_str": "s b : ℝ\nhb : 0 < b\n⊢ (fun x => rexp (b * x) / x ^ s)⁻¹ =ᶠ[atTop] fun x => x ^ s * rexp (-b * x)"}
+{"state": [{"context": ["ι : Type u_1", "ι' : Type u_2", "c : ComplexShape ι", "c' : ComplexShape ι'", "C : Type u_3", "inst✝² : Category.{u_4, u_3} C", "inst✝¹ : HasZeroObject C", "inst✝ : HasZeroMorphisms C", "K : HomologicalComplex C c", "e : c.Embedding c'"], "goal": "IsZero (X K none)"}], "premise": [94063], "state_str": "ι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝² : Category.{u_4, u_3} C\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nK : HomologicalComplex C c\ne : c.Embedding c'\n⊢ IsZero (X K none)"}
+{"state": [{"context": ["R : Type u_1", "inst✝¹ : Ring R", "p q : R[X]", "inst✝ : NoZeroDivisors R", "a : R"], "goal": "(a • p).mirror = a • p.mirror"}], "premise": [101415, 101454, 101471], "state_str": "R : Type u_1\ninst✝¹ : Ring R\np q : R[X]\ninst✝ : NoZeroDivisors R\na : R\n⊢ (a • p).mirror = a • p.mirror"}
+{"state": [{"context": ["𝕜 : Type u", "inst✝⁸ : NontriviallyNormedField 𝕜", "F : Type v", "inst✝⁷ : NormedAddCommGroup F", "inst✝⁶ : NormedSpace 𝕜 F", "E : Type w", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace 𝕜 E", "G✝ : Type u_1", "inst���³ : NormedAddCommGroup G✝", "inst✝² : NormedSpace 𝕜 G✝", "f f₀ f₁ g : 𝕜 → F", "f' f₀' f₁' g' : F", "x : 𝕜", "s t : Set 𝕜", "L L₁ L₂ : Filter 𝕜", "G : Type u_2", "inst✝¹ : NormedAddCommGroup G", "inst✝ : NormedSpace 𝕜 G", "c : 𝕜 → F →L[𝕜] G", "c' : F →L[𝕜] G", "d : 𝕜 → E →L[𝕜] F", "d' : E →L[𝕜] F", "u : 𝕜 → F", "u' : F", "hc : HasDerivWithinAt c c' s x", "hd : HasDerivWithinAt d d' s x"], "goal": "HasDerivWithinAt (fun y => (c y).comp (d y)) (c'.comp (d x) + (c x).comp d') s x"}], "premise": [44330, 44331, 49227], "state_str": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nE : Type w\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nG✝ : Type u_1\ninst✝³ : NormedAddCommGroup G✝\ninst✝² : NormedSpace 𝕜 G✝\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\nG : Type u_2\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nc : 𝕜 → F →L[𝕜] G\nc' : F →L[𝕜] G\nd : 𝕜 → E →L[𝕜] F\nd' : E →L[𝕜] F\nu : 𝕜 → F\nu' : F\nhc : HasDerivWithinAt c c' s x\nhd : HasDerivWithinAt d d' s x\n⊢ HasDerivWithinAt (fun y => (c y).comp (d y)) (c'.comp (d x) + (c x).comp d') s x"}
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+{"state": [{"context": ["m n : ℕ", "α : Fin 2 → Type u", "s : Set (α 0)", "t : Set (α 1)", "f : (i : Fin 2) → α i"], "goal": "f ∈ (fun f => (f 0, f 1)) ⁻¹' s ×ˢ t ↔ f ∈ Set.univ.pi (cons s (cons t finZeroElim))"}], "premise": [4165], "state_str": "case h\nm n : ℕ\nα : Fin 2 → Type u\ns : Set (α 0)\nt : Set (α 1)\nf : (i : Fin 2) → α i\n⊢ f ∈ (fun f => (f 0, f 1)) ⁻¹' s ×ˢ t ↔ f ∈ Set.univ.pi (cons s (cons t finZeroElim))"}
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+{"state": [{"context": ["R✝ : Type u", "S : Type u_1", "σ✝ : Type v", "M : Type w", "inst✝⁶ : CommRing R✝", "inst✝⁵ : CommRing S", "inst✝⁴ : AddCommGroup M", "inst✝³ : Module R✝ M", "R : Type u", "σ : Type v", "inst✝² : CommSemiring R", "inst✝¹ : Finite σ", "inst✝ : NoZeroDivisors R", "val✝ : Fintype σ"], "goal": "NoZeroDivisors (MvPolynomial σ R)"}], "premise": [75837], "state_str": "case intro\nR✝ : Type u\nS : Type u_1\nσ✝ : Type v\nM : Type w\ninst✝⁶ : CommRing R��\ninst✝⁵ : CommRing S\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R✝ M\nR : Type u\nσ : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Finite σ\ninst✝ : NoZeroDivisors R\nval✝ : Fintype σ\n⊢ NoZeroDivisors (MvPolynomial σ R)"}
+{"state": [{"context": ["R✝ : Type u", "S : Type u_1", "σ✝ : Type v", "M : Type w", "inst✝⁶ : CommRing R✝", "inst✝⁵ : CommRing S", "inst✝⁴ : AddCommGroup M", "inst✝³ : Module R✝ M", "R : Type u", "σ : Type v", "inst✝² : CommSemiring R", "inst✝¹ : Finite σ", "inst✝ : NoZeroDivisors R", "val✝ : Fintype σ", "this : NoZeroDivisors (MvPolynomial (Fin (Fintype.card σ)) R)"], "goal": "NoZeroDivisors (MvPolynomial σ R)"}], "premise": [108592, 117063, 117080, 121684], "state_str": "case intro\nR✝ : Type u\nS : Type u_1\nσ✝ : Type v\nM : Type w\ninst✝⁶ : CommRing R✝\ninst✝⁵ : CommRing S\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R✝ M\nR : Type u\nσ : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Finite σ\ninst✝ : NoZeroDivisors R\nval✝ : Fintype σ\nthis : NoZeroDivisors (MvPolynomial (Fin (Fintype.card σ)) R)\n⊢ NoZeroDivisors (MvPolynomial σ R)"}
+{"state": [{"context": ["G : Type w", "H : Type x", "α : Type u", "β : Type v", "inst✝⁶ : TopologicalSpace G", "inst✝⁵ : Group G", "inst✝⁴ : TopologicalGroup G", "inst✝³ : TopologicalSpace α", "f : α → G", "s : Set α", "x : α", "inst✝² : TopologicalSpace H", "inst✝¹ : OrderedCommGroup H", "inst✝ : ContinuousInv H", "a : H"], "goal": "Tendsto (fun a => a⁻¹) (𝓟 (Ici a)) (𝓟 (Iic a⁻¹))"}], "premise": [16381], "state_str": "G : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : Group G\ninst✝⁴ : TopologicalGroup G\ninst✝³ : TopologicalSpace α\nf : α → G\ns : Set α\nx : α\ninst✝² : TopologicalSpace H\ninst✝¹ : OrderedCommGroup H\ninst✝ : ContinuousInv H\na : H\n⊢ Tendsto (fun a => a⁻¹) (𝓟 (Ici a)) (𝓟 (Iic a⁻¹))"}
+{"state": [{"context": ["ι : Sort u_1", "α : Type u_2", "β : Type u_3", "X : Type u_4", "Y : Type u_5", "l : Filter α", "p : ι → Prop", "s : ι → Set α", "h : l.HasBasis p s"], "goal": "(𝓝 l).HasBasis p fun i => {l' | s i ∈ l'}"}], "premise": [15950, 55686], "state_str": "ι : Sort u_1\nα : Type u_2\nβ : Type u_3\nX : Type u_4\nY : Type u_5\nl : Filter α\np : ι → Prop\ns : ι → Set α\nh : l.HasBasis p s\n⊢ (𝓝 l).HasBasis p fun i => {l' | s i ∈ l'}"}
+{"state": [{"context": ["α : Type u_1", "m : MeasurableSpace α", "μ ν : Measure α", "s : Set α", "inst✝ : IsFiniteMeasure ν", "h₁ : MeasurableSet s", "h₂ : ν ≤ μ", "measure_sub : Measure α := ofMeasurable (fun t x => μ t - ν t) ⋯ ⋯"], "goal": "(μ - ν) s = μ s - ν s"}], "premise": [29065, 31458, 103586, 119708, 120650], "state_str": "α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ns : Set α\ninst✝ : IsFiniteMeasure ν\nh₁ : MeasurableSet s\nh₂ : ν ≤ μ\nmeasure_sub : Measure α := ofMeasurable (fun t x => μ t - ν t) ⋯ ⋯\n⊢ (μ - ν) s = μ s - ν s"}
+{"state": [{"context": ["α : Type u_1", "m : MeasurableSpace α", "μ ν : Measure α", "s : Set α", "inst✝ : IsFiniteMeasure ν", "h₁ : MeasurableSet s", "h₂ : ν ≤ μ", "measure_sub : Measure α := ofMeasurable (fun t x => μ t - ν t) ⋯ ⋯", "h_measure_sub_add : ν + measure_sub = μ", "h_measure_sub_eq : μ - ν = measure_sub"], "goal": "measure_sub s = μ s - ν s"}], "premise": [29065], "state_str": "α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ns : Set α\ninst✝ : IsFiniteMeasure ν\nh₁ : MeasurableSet s\nh₂ : ν ≤ μ\nmeasure_sub : Measure α := ofMeasurable (fun t x => μ t - ν t) ⋯ ⋯\nh_measure_sub_add : ν + measure_sub = μ\nh_measure_sub_eq : μ - ν = measure_sub\n⊢ measure_sub s = μ s - ν s"}
+{"state": [{"context": ["ι : Type u_1", "𝕜 : Type u_2", "E : Type u_3", "F : Type u_4", "A : Type u_5", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℝ E", "f✝¹ : ℝ → E", "g' g✝ φ : ℝ → ℝ", "a b : ℝ", "inst✝ : CompleteSpace E", "f✝ f'✝ : ℝ → E", "f f' g : ℝ → ℝ", "h : ∀ x ∈ [[a, b]], HasDerivAt f (f' x) x", "h' : ContinuousOn f' [[a, b]]", "hg : ContinuousOn g (f '' [[a, b]])"], "goal": "∫ (x : ℝ) in a..b, (g ∘ f) x * f' x = ∫ (x : ℝ) in f a..f b, g x"}], "premise": [27352, 119707], "state_str": "ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nf✝¹ : ℝ → E\ng' g✝ φ : ℝ → ℝ\na b : ℝ\ninst✝ : CompleteSpace E\nf✝ f'✝ : ℝ → E\nf f' g : ℝ → ℝ\nh : ∀ x ∈ [[a, b]], HasDerivAt f (f' x) x\nh' : ContinuousOn f' [[a, b]]\nhg : ContinuousOn g (f '' [[a, b]])\n⊢ ∫ (x : ℝ) in a..b, (g ∘ f) x * f' x = ∫ (x : ℝ) in f a..f b, g x"}
+{"state": [{"context": ["ι : Type u_1", "κ : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "s✝ s₁ s₂ : Finset α", "a : α", "f✝ g : α → β", "inst✝² : CommMonoid β", "inst✝¹ : DecidableEq α", "s : Multiset α", "M : Type u_6", "inst✝ : CommMonoid M", "f : α → M"], "goal": "(Multiset.map f s).prod = ∏ m ∈ s.toFinset, f m ^ Multiset.count m s"}], "premise": [127768], "state_str": "ι : Type u_1\nκ : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝² : CommMonoid β\ninst✝¹ : DecidableEq α\ns : Multiset α\nM : Type u_6\ninst✝ : CommMonoid M\nf : α → M\n⊢ (Multiset.map f s).prod = ∏ m ∈ s.toFinset, f m ^ Multiset.count m s"}
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+{"state": [{"context": ["R : Type u_1", "M : Type u_2", "N : Type u_3", "inst✝⁴ : Semiring R", "inst✝³ : AddCommMonoid M", "inst✝² : Module R M", "inst✝¹ : AddCommMonoid N", "inst✝ : Module R N", "α : Type u_4", "f : M →ₗ[R] α →₀ R", "s : Surjective ⇑f"], "goal": "f ∘ₗ f.splittingOfFinsuppSurjective s = id"}], "premise": [85184, 109754, 148057], "state_str": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nα : Type u_4\nf : M →ₗ[R] α →₀ R\ns : Surjective ⇑f\n⊢ f ∘ₗ f.splittingOfFinsuppSurjective s = id"}
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+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "γ : Type u_4", "mγ : MeasurableSpace γ", "s : Set (β × γ)", "κ : Kernel α β", "inst✝¹ : IsSFiniteKernel κ", "η : Kernel (α × β) γ", "inst✝ : IsSFiniteKernel η", "a : α", "h : ((κ ⊗ₖ η) a) s = 0", "t : Set (β × γ)", "hst : s ⊆ t", "mt : MeasurableSet t", "ht : (fun b => (η (a, b)) (Prod.mk b ⁻¹' t)) =ᶠ[ae (κ a)] 0"], "goal": "(fun b => (η (a, b)) (Prod.mk b ⁻¹' s)) =ᶠ[ae (κ a)] 0"}], "premise": [16133], "state_str": "case intro.intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_4\nmγ : MeasurableSpace γ\ns : Set (β × γ)\nκ : Kernel α β\ninst✝¹ : IsSFiniteKernel κ\nη : Kernel (α × β) γ\ninst✝ : IsSFiniteKernel η\na : α\nh : ((κ ⊗ₖ η) a) s = 0\nt : Set (β × γ)\nhst : s ⊆ t\nmt : MeasurableSet t\nht : (fun b => (η (a, b)) (Prod.mk b ⁻¹' t)) =ᶠ[ae (κ a)] 0\n⊢ (fun b => (η (a, b)) (Prod.mk b ⁻¹' s)) =ᶠ[ae (κ a)] 0"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "γ : Type u_4", "mγ : MeasurableSpace γ", "s : Set (β × γ)", "κ : Kernel α β", "inst✝¹ : IsSFiniteKernel κ", "η : Kernel (α × β) γ", "inst✝ : IsSFiniteKernel η", "a : α", "h : ((κ ⊗ₖ η) a) s = 0", "t : Set (β × γ)", "hst : s ⊆ t", "mt : MeasurableSet t", "ht : (fun b => (η (a, b)) (Prod.mk b ⁻¹' t)) =ᶠ[ae (κ a)] 0"], "goal": "(fun b => (η (a, b)) (Prod.mk b ⁻¹' s)) ≤ᶠ[ae (κ a)] 0 ∧ 0 ≤ᶠ[ae (κ a)] fun b => (η (a, b)) (Prod.mk b ⁻¹' s)"}], "premise": [16021, 16131, 27559, 103545, 134062], "state_str": "case intro.intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_4\nmγ : MeasurableSpace γ\ns : Set (β × γ)\nκ : Kernel α β\ninst✝¹ : IsSFiniteKernel κ\nη : Kernel (α × β) γ\ninst✝ : IsSFiniteKernel η\na : α\nh : ((κ ⊗ₖ η) a) s = 0\nt : Set (β × γ)\nhst : s ⊆ t\nmt : MeasurableSet t\nht : (fun b => (η (a, b)) (Prod.mk b ⁻¹' t)) =ᶠ[ae (κ a)] 0\n⊢ (fun b => (η (a, b)) (Prod.mk b ⁻¹' s)) ≤ᶠ[ae (κ a)] 0 ∧ 0 ≤ᶠ[ae (κ a)] fun b => (η (a, b)) (Prod.mk b ⁻¹' s)"}
+{"state": [{"context": ["α : Type u_1", "inst✝ : LinearOrder α", "s t : Set α", "x y z : α"], "goal": "s.ordConnectedComponent x = ∅ ↔ x ∉ s"}], "premise": [1713, 17180, 133377], "state_str": "α : Type u_1\ninst✝ : LinearOrder α\ns t : Set α\nx y z : α\n⊢ s.ordConnectedComponent x = ∅ ↔ x ∉ s"}
+{"state": [{"context": ["R : Type u", "inst✝⁶ : CommSemiring R", "M : Type v", "inst✝⁵ : AddCommMonoid M", "inst✝⁴ : Module R M", "ι : Type w", "inst✝³ : DecidableEq ι", "inst✝² : Fintype ι", "κ : Type u_1", "inst✝¹ : DecidableEq κ", "inst✝ : Fintype κ", "b : Basis ι R M", "c : Basis κ R M", "f : M →ₗ[R] M"], "goal": "(trace R M) f = ((toMatrix b b) f).trace"}], "premise": [81843, 81844, 81845], "state_str": "R : Type u\ninst✝⁶ : CommSemiring R\nM : Type v\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nι : Type w\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\nκ : Type u_1\ninst✝¹ : DecidableEq κ\ninst✝ : Fintype κ\nb : Basis ι R M\nc : Basis κ R M\nf : M →ₗ[R] M\n⊢ (trace R M) f = ((toMatrix b b) f).trace"}
+{"state": [{"context": ["𝓕 : Type u_1", "𝕜 : Type u_2", "α : Type u_3", "ι : Type u_4", "κ : Type u_5", "E : Type u_6", "F : Type u_7", "G : Type u_8", "inst✝¹ : NormedGroup E", "inst✝ : NormedGroup F", "a b : E"], "goal": "comap norm (𝓝[>] 0) = 𝓝[≠] 1"}], "premise": [42800, 133594], "state_str": "𝓕 : Type u_1\n𝕜 : Type u_2\nα : Type u_3\nι : Type u_4\nκ : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst✝¹ : NormedGroup E\ninst✝ : NormedGroup F\na b : E\n⊢ comap norm (𝓝[>] 0) = 𝓝[≠] 1"}
+{"state": [{"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝⁴ : OrderedSemiring α", "inst✝³ : OrderedAddCommMonoid β", "inst✝² : MulActionWithZero α β", "inst✝¹ : CeilDiv α β", "a : α", "inst✝ : Nontrivial α", "b c : β"], "goal": "b ⌈/⌉ 1 ≤ c ↔ b ≤ c"}], "premise": [101703], "state_str": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝⁴ : OrderedSemiring α\ninst✝³ : OrderedAddCommMonoid β\ninst✝² : MulActionWithZero α β\ninst✝¹ : CeilDiv α β\na : α\ninst✝ : Nontrivial α\nb c : β\n⊢ b ⌈/⌉ 1 ≤ c ↔ b ≤ c"}
+{"state": [{"context": ["α : Type u_1", "inst✝ : LinearOrderedField α", "a✝ a b c : α", "h : c < 0"], "goal": "(fun x => x * c) ⁻¹' Icc a b = Icc (b / c) (a / c)"}], "premise": [20257, 133443], "state_str": "α : Type u_1\ninst✝ : LinearOrderedField α\na✝ a b c : α\nh : c < 0\n⊢ (fun x => x * c) ⁻¹' Icc a b = Icc (b / c) (a / c)"}
+{"state": [{"context": ["ι : Type u_1", "α : Type u", "β : Type v", "γ : Type w", "l₁ l₂ : List α", "p : α → Prop", "f : (a : α) → p a → β", "s t : List α", "hs : ∀ (a : α), a ∈ s → p a", "ht : ∀ (a : α), a ∈ t → p a", "hf : ∀ (a a' : α) (ha : p a) (ha' : p a'), f a ha = f a' ha' → a = a'", "h : s.Disjoint t"], "goal": "(pmap f s hs).Disjoint (pmap f t ht)"}], "premise": [132618], "state_str": "ι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\np : α → Prop\nf : (a : α) → p a → β\ns t : List α\nhs : ∀ (a : α), a ∈ s → p a\nht : ∀ (a : α), a ∈ t → p a\nhf : ∀ (a a' : α) (ha : p a) (ha' : p a'), f a ha = f a' ha' → a = a'\nh : s.Disjoint t\n⊢ (pmap f s hs).Disjoint (pmap f t ht)"}
+{"state": [{"context": ["ι : Type u_1", "α : Type u", "β : Type v", "γ : Type w", "l₁ l₂ : List α", "p : α → Prop", "f : (a : α) → p a → β", "s t : List α", "hs : ∀ (a : α), a ∈ s → p a", "ht : ∀ (a : α), a ∈ t → p a", "hf : ∀ (a a' : α) (ha : p a) (ha' : p a'), f a ha = f a' ha' → a = a'", "h : s.Disjoint t", "a : α", "ha : a ∈ s", "a' : α", "ha' : a' ∈ t", "ha'' : f a' ⋯ = f a ⋯"], "goal": "a ∈ t"}], "premise": [2100], "state_str": "case intro.intro.intro.intro\nι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\np : α → Prop\nf : (a : α) → p a → β\ns t : List α\nhs : ∀ (a : α), a ∈ s → p a\nht : ∀ (a : α), a ∈ t → p a\nhf : ∀ (a a' : α) (ha : p a) (ha' : p a'), f a ha = f a' ha' → a = a'\nh : s.Disjoint t\na : α\nha : a ∈ s\na' : α\nha' : a' ∈ t\nha'' : f a' ⋯ = f a ⋯\n⊢ a ∈ t"}
+{"state": [{"context": ["M : Type u_1", "N : Type u_2", "P : Type u_3", "inst✝² : MulOneClass M", "inst✝¹ : MulOneClass N", "inst✝ : MulOneClass P", "c : Con M", "x✝ y : M", "f✝ f : M →* P", "g : P → M", "hf : Function.RightInverse g ⇑f", "x : (ker f).Quotient"], "goal": "(kerLift f) ((toQuotient ∘ g) ((kerLift f) x)) = (kerLift f) x"}], "premise": [1670, 7319], "state_str": "M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝² : MulOneClass M\ninst✝¹ : MulOneClass N\ninst✝ : MulOneClass P\nc : Con M\nx✝ y : M\nf✝ f : M →* P\ng : P → M\nhf : Function.RightInverse g ⇑f\nx : (ker f).Quotient\n⊢ (kerLift f) ((toQuotient ∘ g) ((kerLift f) x)) = (kerLift f) x"}
+{"state": [{"context": ["𝕜 : Type u_1", "inst✝ : LinearOrderedField 𝕜", "p q : ℕ"], "goal": "(fun x => x ^ p / x ^ q) =ᶠ[atBot] fun x => x ^ (↑p - ↑q)"}], "premise": [15499, 16027], "state_str": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\np q : ℕ\n⊢ (fun x => x ^ p / x ^ q) =ᶠ[atBot] fun x => x ^ (↑p - ↑q)"}
+{"state": [{"context": ["𝕜 : Type u_1", "inst✝ : LinearOrderedField 𝕜", "p q : ℕ", "x : 𝕜", "hx : x < 0"], "goal": "(fun x => x ^ p / x ^ q) x = (fun x => x ^ (↑p - ↑q)) x"}], "premise": [108445], "state_str": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\np q : ℕ\nx : 𝕜\nhx : x < 0\n⊢ (fun x => x ^ p / x ^ q) x = (fun x => x ^ (↑p - ↑q)) x"}
+{"state": [{"context": ["C : Type u_1", "D : Type u_2", "E : Type u_3", "inst✝⁹ : Category.{u_4, u_1} C", "inst✝⁸ : Category.{u_5, u_2} D", "inst✝⁷ : Category.{u_6, u_3} E", "inst✝⁶ : Preadditive C", "inst✝⁵ : Preadditive D", "inst✝⁴ : Preadditive E", "F : C ⥤ D", "inst✝³ : F.Additive", "inst✝² : F.Full", "inst✝¹ : F.EssSurj", "G : D ⥤ E", "inst✝ : (F ⋙ G).Additive", "X Y : D", "f g : X ⟶ Y"], "goal": "G.map (f + g) = G.map f + G.map g"}], "premise": [100150], "state_str": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁹ : Category.{u_4, u_1} C\ninst✝⁸ : Category.{u_5, u_2} D\ninst✝⁷ : Category.{u_6, u_3} E\ninst✝⁶ : Preadditive C\ninst✝⁵ : Preadditive D\ninst✝⁴ : Preadditive E\nF : C ⥤ D\ninst✝³ : F.Additive\ninst✝² : F.Full\ninst✝¹ : F.EssSurj\nG : D ⥤ E\ninst✝ : (F ⋙ G).Additive\nX Y : D\nf g : X ⟶ Y\n⊢ G.map (f + g) = G.map f + G.map g"}
+{"state": [{"context": ["C : Type u_1", "D : Type u_2", "E : Type u_3", "inst✝⁹ : Category.{u_4, u_1} C", "inst✝⁸ : Category.{u_5, u_2} D", "inst✝⁷ : Category.{u_6, u_3} E", "inst✝⁶ : Preadditive C", "inst✝⁵ : Preadditive D", "inst✝⁴ : Preadditive E", "F : C ⥤ D", "inst✝³ : F.Additive", "inst✝² : F.Full", "inst✝¹ : F.EssSurj", "G : D ⥤ E", "inst✝ : (F ⋙ G).Additive", "X Y : D", "f g : X ⟶ Y", "f' : F.objPreimage X ⟶ F.objPreimage Y", "hf' : F.map f' = (F.objObjPreimageIso X).hom ≫ f ≫ (F.objObjPreimageIso Y).inv"], "goal": "G.map (f + g) = G.map f + G.map g"}], "premise": [100150], "state_str": "case intro\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁹ : Category.{u_4, u_1} C\ninst✝⁸ : Category.{u_5, u_2} D\ninst✝⁷ : Category.{u_6, u_3} E\ninst✝⁶ : Preadditive C\ninst✝⁵ : Preadditive D\ninst✝⁴ : Preadditive E\nF : C ⥤ D\ninst✝³ : F.Additive\ninst✝² : F.Full\ninst✝¹ : F.EssSurj\nG : D ⥤ E\ninst✝ : (F ⋙ G).Additive\nX Y : D\nf g : X ⟶ Y\nf' : F.objPreimage X ⟶ F.objPreimage Y\nhf' : F.map f' = (F.objObjPreimageIso X).hom ≫ f ≫ (F.objObjPreimageIso Y).inv\n⊢ G.map (f + g) = G.map f + G.map g"}
+{"state": [{"context": ["C : Type u_1", "D : Type u_2", "E : Type u_3", "inst✝⁹ : Category.{u_4, u_1} C", "inst✝⁸ : Category.{u_5, u_2} D", "inst✝⁷ : Category.{u_6, u_3} E", "inst✝⁶ : Preadditive C", "inst✝⁵ : Preadditive D", "inst✝⁴ : Preadditive E", "F : C ⥤ D", "inst✝³ : F.Additive", "inst✝² : F.Full", "inst✝¹ : F.EssSurj", "G : D ⥤ E", "inst✝ : (F ⋙ G).Additive", "X Y : D", "f g : X ⟶ Y", "f' : F.objPreimage X ⟶ F.objPreimage Y", "hf' : F.map f' = (F.objObjPreimageIso X).hom ≫ f ≫ (F.objObjPreimageIso Y).inv", "g' : F.objPreimage X ⟶ F.objPreimage Y", "hg' : F.map g' = (F.objObjPreimageIso X).hom ≫ g ≫ (F.objObjPreimageIso Y).inv"], "goal": "G.map (f + g) = G.map f + G.map g"}], "premise": [91598, 91599, 96190, 96191, 99919], "state_str": "case intro.intro\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁹ : Category.{u_4, u_1} C\ninst✝⁸ : Category.{u_5, u_2} D\ninst✝⁷ : Category.{u_6, u_3} E\ninst✝⁶ : Preadditive C\ninst✝⁵ : Preadditive D\ninst✝⁴ : Preadditive E\nF : C ⥤ D\ninst✝³ : F.Additive\ninst✝² : F.Full\ninst✝¹ : F.EssSurj\nG : D ⥤ E\ninst✝ : (F ⋙ G).Additive\nX Y : D\nf g : X ⟶ Y\nf' : F.objPreimage X ⟶ F.objPreimage Y\nhf' : F.map f' = (F.objObjPreimageIso X).hom ≫ f ≫ (F.objObjPreimageIso Y).inv\ng' : F.objPreimage X ⟶ F.objPreimage Y\nhg' : F.map g' = (F.objObjPreimageIso X).hom ≫ g ≫ (F.objObjPreimageIso Y).inv\n⊢ G.map (f + g) = G.map f + G.map g"}
+{"state": [{"context": ["C : Type u_1", "D : Type u_2", "E : Type u_3", "inst✝⁹ : Category.{u_4, u_1} C", "inst✝⁸ : Category.{u_5, u_2} D", "inst✝⁷ : Category.{u_6, u_3} E", "inst✝⁶ : Preadditive C", "inst✝⁵ : Preadditive D", "inst✝⁴ : Preadditive E", "F : C ⥤ D", "inst✝³ : F.Additive", "inst✝² : F.Full", "inst✝¹ : F.EssSurj", "G : D ⥤ E", "inst✝ : (F ⋙ G).Additive", "X Y : D", "f g : X ⟶ Y", "f' : F.objPreimage X ⟶ F.objPreimage Y", "hf' : F.map f' = (F.objObjPreimageIso X).hom ≫ f ≫ (F.objObjPreimageIso Y).inv", "g' : F.objPreimage X ⟶ F.objPreimage Y", "hg' : F.map g' = (F.objObjPreimageIso X).hom ≫ g ≫ (F.objObjPreimageIso Y).inv"], "goal": "G.map ((F.objObjPreimageIso X).hom ≫ f ≫ (F.objObjPreimageIso Y).inv + (F.objObjPreimageIso X).hom ≫ g ≫ (F.objObjPreimageIso Y).inv) = G.map ((F.objObjPreimageIso X).hom ≫ f ≫ (F.objObjPreimageIso Y).inv) + G.map ((F.objObjPreimageIso X).hom ≫ g ≫ (F.objObjPreimageIso Y).inv)"}], "premise": [91682], "state_str": "case intro.intro\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁹ : Category.{u_4, u_1} C\ninst✝⁸ : Category.{u_5, u_2} D\ninst✝⁷ : Category.{u_6, u_3} E\ninst✝⁶ : Preadditive C\ninst✝⁵ : Preadditive D\ninst✝⁴ : Preadditive E\nF : C ⥤ D\ninst✝³ : F.Additive\ninst✝² : F.Full\ninst✝¹ : F.EssSurj\nG : D ⥤ E\ninst✝ : (F ⋙ G).Additive\nX Y : D\nf g : X ⟶ Y\nf' : F.objPreimage X ⟶ F.objPreimage Y\nhf' : F.map f' = (F.objObjPreimageIso X).hom ≫ f ≫ (F.objObjPreimageIso Y).inv\ng' : F.objPreimage X ⟶ F.objPreimage Y\nhg' : F.map g' = (F.objObjPreimageIso X).hom ≫ g ≫ (F.objObjPreimageIso Y).inv\n⊢ G.map\n      ((F.objObjPreimageIso X).hom ≫ f ≫ (F.objObjPreimageIso Y).inv +\n        (F.objObjPreimageIso X).hom ≫ g ≫ (F.objObjPreimageIso Y).inv) =\n    G.map ((F.objObjPreimageIso X).hom ≫ f ≫ (F.objObjPreimageIso Y).inv) +\n      G.map ((F.objObjPreimageIso X).hom ≫ g ≫ (F.objObjPreimageIso Y).inv)"}
+{"state": [{"context": ["C : Type u_1", "D : Type u_2", "E : Type u_3", "inst✝⁹ : Category.{u_4, u_1} C", "inst✝⁸ : Category.{u_5, u_2} D", "inst✝⁷ : Category.{u_6, u_3} E", "inst✝⁶ : Preadditive C", "inst✝⁵ : Preadditive D", "inst✝⁴ : Preadditive E", "F : C ⥤ D", "inst✝³ : F.Additive", "inst✝² : F.Full", "inst✝¹ : F.EssSurj", "G : D ⥤ E", "inst✝ : (F ⋙ G).Additive", "X Y : D", "f g : X ⟶ Y", "f' : F.objPreimage X ⟶ F.objPreimage Y", "hf' : F.map f' = (F.objObjPreimageIso X).hom ≫ f ≫ (F.objObjPreimageIso Y).inv", "g' : F.objPreimage X ⟶ F.objPreimage Y", "hg' : F.map g' = (F.objObjPreimageIso X).hom ≫ g ≫ (F.objObjPreimageIso Y).inv"], "goal": "G.map (F.map f' + F.map g') = G.map (F.map (f' + g'))"}], "premise": [91682], "state_str": "case intro.intro\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁹ : Category.{u_4, u_1} C\ninst✝⁸ : Category.{u_5, u_2} D\ninst✝⁷ : Category.{u_6, u_3} E\ninst✝⁶ : Preadditive C\ninst✝⁵ : Preadditive D\ninst✝⁴ : Preadditive E\nF : C ⥤ D\ninst✝³ : F.Additive\ninst✝² : F.Full\ninst✝¹ : F.EssSurj\nG : D ⥤ E\ninst✝ : (F ⋙ G).Additive\nX Y : D\nf g : X ⟶ Y\nf' : F.objPreimage X ⟶ F.objPreimage Y\nhf' : F.map f' = (F.objObjPreimageIso X).hom ≫ f ≫ (F.objObjPreimageIso Y).inv\ng' : F.objPreimage X ⟶ F.objPreimage Y\nhg' : F.map g' = (F.objObjPreimageIso X).hom ≫ g ≫ (F.objObjPreimageIso Y).inv\n⊢ G.map (F.map f' + F.map g') = G.map (F.map (f' + g'))"}
+{"state": [{"context": ["C : Type u", "inst✝⁴ : Category.{v, u} C", "inst✝³ : HasZeroObject C", "inst✝² : HasShift C ℤ", "inst✝¹ : Preadditive C", "inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive", "hC : Pretriangulated C", "T : Triangle C", "H : T ∈ distinguishedTriangles"], "goal": "T.mor₁ ≫ T.mor₂ = 0"}], "premise": [99353, 99355], "state_str": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : Preadditive C\ninst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive\nhC : Pretriangulated C\nT : Triangle C\nH : T ∈ distinguishedTriangles\n⊢ T.mor₁ ≫ T.mor₂ = 0"}
+{"state": [{"context": ["C : Type u", "inst✝⁴ : Category.{v, u} C", "inst✝³ : HasZeroObject C", "inst✝² : HasShift C ℤ", "inst✝¹ : Preadditive C", "inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive", "hC : Pretriangulated C", "T : Triangle C", "H : T ∈ distinguishedTriangles", "c : (contractibleTriangle T.obj₁).obj₃ ⟶ T.obj₃", "hc : (contractibleTriangle T.obj₁).mor₂ ≫ c = T.mor₁ ≫ T.mor₂ ∧ (contractibleTriangle T.obj₁).mor₃ ≫ (shiftFunctor C 1).map (𝟙 T.obj₁) = c ≫ T.mor₃"], "goal": "T.mor₁ ≫ T.mor₂ = 0"}], "premise": [2100, 2107, 93605, 98124], "state_str": "case intro\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroObject C\ninst✝² : HasShift C ℤ\ninst✝¹ : Preadditive C\ninst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive\nhC : Pretriangulated C\nT : Triangle C\nH : T ∈ distinguishedTriangles\nc : (contractibleTriangle T.obj₁).obj₃ ⟶ T.obj₃\nhc :\n  (contractibleTriangle T.obj₁).mor₂ ≫ c = T.mor₁ ≫ T.mor₂ ∧\n    (contractibleTriangle T.obj₁).mor₃ ≫ (shiftFunctor C 1).map (𝟙 T.obj₁) = c ≫ T.mor₃\n⊢ T.mor₁ ≫ T.mor₂ = 0"}
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+{"state": [{"context": ["Γ : Type u_1", "inst✝¹ : Inhabited Γ", "Λ : Type u_2", "inst✝ : Inhabited Λ", "M : TM0.Machine Γ Λ", "x✝ : Cfg₀", "q : Λ", "T : Tape Γ", "q' : Λ", "s : TM0.Stmt Γ", "this : TM1.step (tr M) { l := some (Λ'.act s q'), var := (), Tape := T } = some { l := some (Λ'.normal q'), var := (), Tape := match s with | TM0.Stmt.move d => Tape.move d T | TM0.Stmt.write a => Tape.write a T }", "e : M q T.head = some (q', s)"], "goal": "Reaches₁ (TM1.step (tr M)) { l := some (Λ'.normal q), var := (), Tape := T } { l := match match M q' (match s with | TM0.Stmt.move d => Tape.move d T | TM0.Stmt.write a => Tape.write a T).head with | some val => true | none => false with | true => some (Λ'.normal q') | false => none, var := (), Tape := match s with | TM0.Stmt.move d => Tape.move d T | TM0.Stmt.write a => Tape.write a T }"}], "premise": [70459, 70461], "state_str": "case some.mk\nΓ : Type u_1\ninst✝¹ : Inhabited Γ\nΛ : Type u_2\ninst✝ : Inhabited Λ\nM : TM0.Machine Γ Λ\nx✝ : Cfg₀\nq : Λ\nT : Tape Γ\nq' : Λ\ns : TM0.Stmt Γ\nthis :\n  TM1.step (tr M) { l := some (Λ'.act s q'), var := (), Tape := T } =\n    some\n      { l := some (Λ'.normal q'), var := (),\n        Tape :=\n          match s with\n          | TM0.Stmt.move d => Tape.move d T\n          | TM0.Stmt.write a => Tape.write a T }\ne : M q T.head = some (q', s)\n⊢ Reaches₁ (TM1.step (tr M)) { l := some (Λ'.normal q), var := (), Tape := T }\n    {\n      l :=\n        match\n          match\n            M q'\n              (match s with\n                | TM0.Stmt.move d => Tape.move d T\n                | TM0.Stmt.write a => Tape.write a T).head with\n          | some val => true\n          | none => false with\n        | true => some (Λ'.normal q')\n        | false => none,\n      var := (),\n      Tape :=\n        match s with\n        | TM0.Stmt.move d => Tape.move d T\n        | TM0.Stmt.write a => Tape.write a T }"}
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+{"state": [{"context": ["G : PGame", "inst✝ : G.Impartial"], "goal": "(-G).grundyValue = G.grundyValue"}], "premise": [48146, 48153, 50303], "state_str": "G : PGame\ninst✝ : G.Impartial\n⊢ (-G).grundyValue = G.grundyValue"}
+{"state": [{"context": ["C : Type u_1", "D : Type u_2", "E : Type u_3", "inst✝¹⁰ : Category.{u_7, u_1} C", "inst✝⁹ : Category.{u_6, u_2} D", "inst✝⁸ : Category.{?u.40009, u_3} E", "F : C ⥤ D", "G : D ⥤ E", "A : Type u_4", "B : Type u_5", "inst✝⁷ : AddMonoid A", "inst✝⁶ : AddCommMonoid B", "inst✝⁵ : HasShift C A", "inst✝⁴ : HasShift D A", "inst✝³ : HasShift E A", "inst✝² : HasShift C B", "inst✝¹ : HasShift D B", "inst✝ : F.CommShift A", "a b : A"], "goal": "F.commShiftIso (a + b) = CommShift.isoAdd' ⋯ (F.commShiftIso a) (F.commShiftIso b)"}], "premise": [92619], "state_str": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝¹⁰ : Category.{u_7, u_1} C\ninst✝⁹ : Category.{u_6, u_2} D\ninst✝⁸ : Category.{?u.40009, u_3} E\nF : C ⥤ D\nG : D ⥤ E\nA : Type u_4\nB : Type u_5\ninst✝⁷ : AddMonoid A\ninst✝⁶ : AddCommMonoid B\ninst✝⁵ : HasShift C A\ninst✝⁴ : HasShift D A\ninst✝³ : HasShift E A\ninst✝² : HasShift C B\ninst✝¹ : HasShift D B\ninst✝ : F.CommShift A\na b : A\n⊢ F.commShiftIso (a + b) = CommShift.isoAdd' ⋯ (F.commShiftIso a) (F.commShiftIso b)"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4", "M✝ : Type u_5", "M' : Type u_6", "N : Type u_7", "P : Type u_8", "G : Type u_9", "H : Type u_10", "R : Type u_11", "S : Type u_12", "inst✝¹ : AddZeroClass M✝", "M : Type u_13", "inst✝ : AddZeroClass M", "d : M"], "goal": "AddEquiv.finsuppUnique.symm d = single () d"}], "premise": [148107, 148108], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nM✝ : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst✝¹ : AddZeroClass M✝\nM : Type u_13\ninst✝ : AddZeroClass M\nd : M\n⊢ AddEquiv.finsuppUnique.symm d = single () d"}
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+{"state": [{"context": ["A : Type u_1", "K : Type u_2", "R : Type u_3", "S : Type u_4", "inst✝⁹ : CommRing A", "inst✝⁸ : Field K", "inst✝⁷ : CommRing R", "inst✝⁶ : CommRing S", "M : Submonoid A", "inst✝⁵ : Algebra A S", "inst✝⁴ : IsLocalization M S", "inst✝³ : Algebra A K", "inst✝² : IsFractionRing A K", "inst✝¹ : IsDomain A", "inst✝ : UniqueFactorizationMonoid A", "p : A[X]", "x : K", "hr : (aeval x) p = 0"], "goal": "eval₂ (algebraMap A K) ((algebraMap A K) (num A x)) (p.scaleRoots ↑(den A x)) = 0"}], "premise": [74319], "state_str": "A : Type u_1\nK : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝⁹ : CommRing A\ninst✝⁸ : Field K\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\nM : Submonoid A\ninst✝⁵ : Algebra A S\ninst✝⁴ : IsLocalization M S\ninst✝³ : Algebra A K\ninst✝² : IsFractionRing A K\ninst✝¹ : IsDomain A\ninst✝ : UniqueFactorizationMonoid A\np : A[X]\nx : K\nhr : (aeval x) p = 0\n⊢ eval₂ (algebraMap A K) ((algebraMap A K) (num A x)) (p.scaleRoots ↑(den A x)) = 0"}
+{"state": [{"context": ["A : Type u_1", "K : Type u_2", "R : Type u_3", "S : Type u_4", "inst✝⁹ : CommRing A", "inst✝⁸ : Field K", "inst✝⁷ : CommRing R", "inst✝⁶ : CommRing S", "M : Submonoid A", "inst✝⁵ : Algebra A S", "inst✝⁴ : IsLocalization M S", "inst✝³ : Algebra A K", "inst✝² : IsFractionRing A K", "inst✝¹ : IsDomain A", "inst✝ : UniqueFactorizationMonoid A", "p : A[X]", "x : K", "hr : (aeval x) p = 0"], "goal": "(aeval (mk' K (num A x) (den A x))) p = 0"}], "premise": [76929], "state_str": "A : Type u_1\nK : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝⁹ : CommRing A\ninst✝⁸ : Field K\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\nM : Submonoid A\ninst✝⁵ : Algebra A S\ninst✝⁴ : IsLocalization M S\ninst✝³ : Algebra A K\ninst✝² : IsFractionRing A K\ninst✝¹ : IsDomain A\ninst✝ : UniqueFactorizationMonoid A\np : A[X]\nx : K\nhr : (aeval x) p = 0\n⊢ (aeval (mk' K (num A x) (den A x))) p = 0"}
+{"state": [{"context": ["ι : Type u_1", "α : ι → Type u_2", "β : ι → Type u_3", "s s₁ s₂ : Set ι", "t t₁ t₂ : (i : ι) → Set (α i)", "i : ι"], "goal": "(s.pi t).Nonempty ↔ ∀ (i : ι), ∃ x, i ∈ s → x ∈ t i"}], "premise": [1091], "state_str": "ι : Type u_1\nα : ι → Type u_2\nβ : ι → Type u_3\ns s₁ s₂ : Set ι\nt t₁ t₂ : (i : ι) → Set (α i)\ni : ι\n⊢ (s.pi t).Nonempty ↔ ∀ (i : ι), ∃ x, i ∈ s → x ∈ t i"}
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+{"state": [{"context": ["R : Type u_1", "inst✝² : CommRing R", "hR : HasUnitMulPowIrreducibleFactorization R", "inst✝¹ : IsDomain R", "inst✝ : UniqueFactorizationMonoid R", "h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q", "p : R", "hp : Irreducible p", "x : R", "hx : x ≠ 0", "fx : Multiset R", "hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x"], "goal": "Associated (p ^ Multiset.card fx) x"}], "premise": [2106], "state_str": "case intro.intro\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x\n⊢ Associated (p ^ Multiset.card fx) x"}
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+{"state": [{"context": ["R : Type u_1", "inst✝² : CommRing R", "hR : HasUnitMulPowIrreducibleFactorization R", "inst✝¹ : IsDomain R", "inst✝ : UniqueFactorizationMonoid R", "h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q", "p : R", "hp : Irreducible p", "x : R", "hx : x ≠ 0", "fx : Multiset R", "hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x", "H : Associates.mk fx.prod = Associates.mk x"], "goal": "Multiset.map Associates.mk fx = Multiset.replicate (Multiset.card fx) (Associates.mk p)"}], "premise": [137939], "state_str": "case intro.intro.e_a\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x\nH : Associates.mk fx.prod = Associates.mk x\n⊢ Multiset.map Associates.mk fx = Multiset.replicate (Multiset.card fx) (Associates.mk p)"}
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+{"state": [{"context": ["R : Type u_1", "inst✝² : CommRing R", "hR : HasUnitMulPowIrreducibleFactorization R", "inst✝¹ : IsDomain R", "inst✝ : UniqueFactorizationMonoid R", "h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q", "p : R", "hp : Irreducible p", "x : R", "hx : x ≠ 0", "fx : Multiset R", "hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x", "H : Associates.mk fx.prod = Associates.mk x", "q : R", "hq : q ∈ fx"], "goal": "Associated q p"}], "premise": [2107], "state_str": "case intro.intro.e_a\nR : Type u_1\ninst✝² : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\np : R\nhp : Irreducible p\nx : R\nhx : x ≠ 0\nfx : Multiset R\nhfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x\nH : Associates.mk fx.prod = Associates.mk x\nq : R\nhq : q ∈ fx\n⊢ Associated q p"}
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+{"state": [{"context": ["α : Type u_1", "inst✝ : LinearOrderedAddCommGroup α", "a✝ b✝ c✝ a b c : α"], "goal": "|max a c - max b c| ≤ |a - b|"}], "premise": [19708, 104518, 105293, 105294, 117981], "state_str": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\na✝ b✝ c✝ a b c : α\n⊢ |max a c - max b c| ≤ |a - b|"}
+{"state": [{"context": ["x y : ℝ", "hy : 0 < y"], "goal": "x ≤ log y ↔ rexp x ≤ y"}], "premise": [1713, 37865, 149304], "state_str": "x y : ℝ\nhy : 0 < y\n⊢ x ≤ log y ↔ rexp x ≤ y"}
+{"state": [{"context": ["x y z : ℝ", "n : ℕ", "hx : 0 ≤ x", "h : y - ↑n ≠ 0"], "goal": "x ^ (y - ↑n) = x ^ y / x ^ n"}], "premise": [40000, 40032], "state_str": "x y z : ℝ\nn : ℕ\nhx : 0 ≤ x\nh : y - ↑n ≠ 0\n⊢ x ^ (y - ↑n) = x ^ y / x ^ n"}
+{"state": [{"context": ["𝕜 : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝ : Fintype α", "s t : Finset α", "a b : α", "h : a ∉ s"], "goal": "(cons a s h).dens = s.dens + (↑(Fintype.card α))⁻¹"}], "premise": [115817], "state_str": "𝕜 : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : Fintype α\ns t : Finset α\na b : α\nh : a ∉ s\n⊢ (cons a s h).dens = s.dens + (↑(Fintype.card α))⁻¹"}
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+{"state": [{"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "M : Type u_4", "N : Type u_5", "P : Type u_6", "G : Type u_7", "inst✝ : CommMonoid M", "l₁ l₂ : List M", "h : l₁ <+ l₂", "l : List M", "hl : l₂ ~ l₁ ++ l"], "goal": "l₁.prod ∣ l₂.prod"}], "premise": [124830, 124899], "state_str": "case intro\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst✝ : CommMonoid M\nl₁ l₂ : List M\nh : l₁ <+ l₂\nl : List M\nhl : l₂ ~ l₁ ++ l\n⊢ l₁.prod ∣ l₂.prod"}
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+{"state": [{"context": ["α : Type u_1", "s t : Finset α", "a : α", "inst✝¹ : Fintype α", "inst✝ : DecidableEq α"], "goal": "IsCoatom s ↔ ∃ a, s = {a}ᶜ"}], "premise": [12068, 12069, 19018, 136918], "state_str": "α : Type u_1\ns t : Finset α\na : α\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\n⊢ IsCoatom s ↔ ∃ a, s = {a}ᶜ"}
+{"state": [{"context": ["ι : Type u_1", "σ : Type u_2", "R : Type u_3", "A : Type u_4", "inst✝⁵ : Semiring A", "inst✝⁴ : SetLike σ A", "inst✝³ : AddSubmonoidClass σ A", "𝒜 : ι → σ", "inst✝² : DecidableEq ι", "inst✝¹ : AddMonoid ι", "inst✝ : GradedRing 𝒜", "I✝ : Ideal A", "I : HomogeneousIdeal 𝒜"], "goal": "(Ideal.homogeneousCore 𝒜 I.toIdeal).toIdeal = I.toIdeal"}], "premise": [76558, 76572], "state_str": "case h\nι : Type u_1\nσ : Type u_2\nR : Type u_3\nA : Type u_4\ninst✝⁵ : Semiring A\ninst✝⁴ : SetLike σ A\ninst✝³ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝² : DecidableEq ι\ninst✝¹ : AddMonoid ι\ninst✝ : GradedRing 𝒜\nI✝ : Ideal A\nI : HomogeneousIdeal 𝒜\n⊢ (Ideal.homogeneousCore 𝒜 I.toIdeal).toIdeal = I.toIdeal"}
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+{"state": [{"context": ["α : Type u_1", "inst✝² : PseudoEMetricSpace α", "β : Type u_2", "inst✝¹ : One β", "inst✝ : TopologicalSpace β", "f : α → β", "E : Set α"], "goal": "Tendsto (fun δ => (cthickening δ E).mulIndicator f) (𝓝 0) (𝓝 ((closure E).mulIndicator f))"}], "premise": [66543], "state_str": "α : Type u_1\ninst✝² : PseudoEMetricSpace α\nβ : Type u_2\ninst✝¹ : One β\ninst✝ : TopologicalSpace β\nf : α → β\nE : Set α\n⊢ Tendsto (fun δ => (cthickening δ E).mulIndicator f) (𝓝 0) (𝓝 ((closure E).mulIndicator f))"}
+{"state": [{"context": ["α : Type u_1", "inst✝² : PseudoEMetricSpace α", "β : Type u_2", "inst✝¹ : One β", "inst✝ : TopologicalSpace β", "f : α → β", "E : Set α", "x : α"], "goal": "Tendsto (fun i => (cthickening i E).mulIndicator f x) (𝓝 0) (𝓝 ((closure E).mulIndicator f x))"}], "premise": [16349, 60782], "state_str": "α : Type u_1\ninst✝² : PseudoEMetricSpace α\nβ : Type u_2\ninst✝¹ : One β\ninst✝ : TopologicalSpace β\nf : α → β\nE : Set α\nx : α\n⊢ Tendsto (fun i => (cthickening i E).mulIndicator f x) (𝓝 0) (𝓝 ((closure E).mulIndicator f x))"}
+{"state": [{"context": ["α : Type u_1", "inst✝² : PseudoEMetricSpace α", "β : Type u_2", "inst✝¹ : One β", "inst✝ : TopologicalSpace β", "f : α → β", "E : Set α", "x : α"], "goal": "Tendsto (fun x_1 => (closure E).mulIndicator f x) (𝓝 0) (𝓝 ((closure E).mulIndicator f x))"}], "premise": [55522], "state_str": "α : Type u_1\ninst✝² : PseudoEMetricSpace α\nβ : Type u_2\ninst✝¹ : One β\ninst✝ : TopologicalSpace β\nf : α → β\nE : Set α\nx : α\n⊢ Tendsto (fun x_1 => (closure E).mulIndicator f x) (𝓝 0) (𝓝 ((closure E).mulIndicator f x))"}
+{"state": [{"context": ["L : Language", "L' : Language", "M : Type w", "N : Type u_1", "P : Type u_2", "inst✝² : L.Structure M", "inst✝¹ : L.Structure N", "inst✝ : L.Structure P", "α : Type u'", "β : Type v'", "γ : Type u_3", "n : ℕ", "φ ψ : L.Formula α", "v : α → M", "f : L.Functions n", "x : Fin n → M", "y : M"], "goal": "(graph f).Realize (cons y x) ↔ funMap f x = y"}], "premise": [25313, 143929, 143930], "state_str": "L : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst✝² : L.Structure M\ninst✝¹ : L.Structure N\ninst✝ : L.Structure P\nα : Type u'\nβ : Type v'\nγ : Type u_3\nn : ℕ\nφ ψ : L.Formula α\nv : α → M\nf : L.Functions n\nx : Fin n → M\ny : M\n⊢ (graph f).Realize (cons y x) ↔ funMap f x = y"}
+{"state": [{"context": ["L : Language", "L' : Language", "M : Type w", "N : Type u_1", "P : Type u_2", "inst✝² : L.Structure M", "inst✝¹ : L.Structure N", "inst✝ : L.Structure P", "α : Type u'", "β : Type v'", "γ : Type u_3", "n : ℕ", "φ ψ : L.Formula α", "v : α → M", "f : L.Functions n", "x : Fin n → M", "y : M"], "goal": "(y = funMap f fun i => x i) ↔ funMap f x = y"}], "premise": [1713, 1717], "state_str": "L : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst✝² : L.Structure M\ninst✝¹ : L.Structure N\ninst✝ : L.Structure P\nα : Type u'\nβ : Type v'\nγ : Type u_3\nn : ℕ\nφ ψ : L.Formula α\nv : α → M\nf : L.Functions n\nx : Fin n → M\ny : M\n⊢ (y = funMap f fun i => x i) ↔ funMap f x = y"}
+{"state": [{"context": ["p : ℕ → Prop", "hf : (setOf p).Infinite"], "goal": "Set.range (nth p) = setOf p"}], "premise": [144368], "state_str": "p : ℕ → Prop\nhf : (setOf p).Infinite\n⊢ Set.range (nth p) = setOf p"}
+{"state": [{"context": ["p : ℕ → Prop", "hf : (setOf p).Infinite", "this : Infinite ↑(setOf p)"], "goal": "Set.range (Subtype.val ∘ ⇑(Subtype.orderIsoOfNat (setOf p))) = setOf p"}], "premise": [70613], "state_str": "p : ℕ → Prop\nhf : (setOf p).Infinite\nthis : Infinite ↑(setOf p)\n⊢ Set.range (Subtype.val ∘ ⇑(Subtype.orderIsoOfNat (setOf p))) = setOf p"}
+{"state": [{"context": ["C₁ : Type u₁", "C₂ : Type u₂", "C₃ : Type u₃", "C₄ : Type u₄", "inst✝³ : Category.{v₁, u₁} C₁", "inst✝² : Category.{v₂, u₂} C₂", "inst✝¹ : Category.{v₃, u₃} C₃", "inst✝ : Category.{v₄, u₄} C₄", "T : C₁ ⥤ C₂", "L : C₁ ⥤ C₃", "R : C₂ ⥤ C₄", "B : C₃ ⥤ C₄", "w : TwoSquare T L R B", "X₂ : C₂", "X₃ : C₃", "g : R.obj X₂ ⟶ B.obj X₃", "f : w.StructuredArrowRightwards g"], "goal": "((w.structuredArrowDownwards X₂).obj (StructuredArrow.mk f.hom.left)).hom ≫ B.map f.right.hom = (StructuredArrow.mk g).hom"}], "premise": [98352], "state_str": "C₁ : Type u₁\nC₂ : Type u₂\nC₃ : Type u₃\nC₄ : Type u₄\ninst✝³ : Category.{v₁, u₁} C₁\ninst✝² : Category.{v₂, u₂} C₂\ninst✝¹ : Category.{v₃, u₃} C₃\ninst✝ : Category.{v₄, u₄} C₄\nT : C₁ ⥤ C₂\nL : C₁ ⥤ C₃\nR : C₂ ⥤ C₄\nB : C₃ ⥤ C₄\nw : TwoSquare T L R B\nX₂ : C₂\nX₃ : C₃\ng : R.obj X₂ ⟶ B.obj X₃\nf : w.StructuredArrowRightwards g\n⊢ ((w.structuredArrowDownwards X₂).obj (StructuredArrow.mk f.hom.left)).hom ≫ B.map f.right.hom =\n    (StructuredArrow.mk g).hom"}
+{"state": [{"context": ["C₁ : Type u₁", "C₂ : Type u₂", "C₃ : Type u₃", "C₄ : Type u₄", "inst✝³ : Category.{v₁, u₁} C₁", "inst✝² : Category.{v₂, u₂} C₂", "inst✝¹ : Category.{v₃, u₃} C₃", "inst✝ : Category.{v₄, u₄} C₄", "T : C₁ ⥤ C₂", "L : C₁ ⥤ C₃", "R : C₂ ⥤ C₄", "B : C₃ ⥤ C₄", "w : TwoSquare T L R B", "X₂ : C₂", "X₃ : C₃", "g : R.obj X₂ ⟶ B.obj X₃", "f₁ f₂ : w.StructuredArrowRightwards g", "φ : f₁ ⟶ f₂"], "goal": "f₁.hom.left ≫ T.map φ.right.left = f₂.hom.left"}], "premise": [98288], "state_str": "C₁ : Type u₁\nC₂ : Type u₂\nC₃ : Type u₃\nC₄ : Type u₄\ninst✝³ : Category.{v₁, u₁} C₁\ninst✝² : Category.{v₂, u₂} C₂\ninst✝¹ : Category.{v₃, u₃} C₃\ninst✝ : Category.{v₄, u₄} C₄\nT : C₁ ⥤ C₂\nL : C₁ ⥤ C₃\nR : C₂ ⥤ C₄\nB : C₃ ⥤ C₄\nw : TwoSquare T L R B\nX₂ : C₂\nX₃ : C₃\ng : R.obj X₂ ⟶ B.obj X₃\nf₁ f₂ : w.StructuredArrowRightwards g\nφ : f₁ ⟶ f₂\n⊢ f₁.hom.left ≫ T.map φ.right.left = f₂.hom.left"}
+{"state": [{"context": ["C₁ : Type u₁", "C₂ : Type u₂", "C₃ : Type u₃", "C₄ : Type u₄", "inst✝³ : Category.{v₁, u₁} C₁", "inst✝² : Category.{v₂, u₂} C₂", "inst✝¹ : Category.{v₃, u₃} C₃", "inst✝ : Category.{v₄, u₄} C₄", "T : C₁ ⥤ C₂", "L : C₁ ⥤ C₃", "R : C₂ ⥤ C₄", "B : C₃ ⥤ C₄", "w : TwoSquare T L R B", "X₂ : C₂", "X₃ : C₃", "g : R.obj X₂ ⟶ B.obj X₃", "f₁ f₂ : w.StructuredArrowRightwards g", "φ : f₁ ⟶ f₂"], "goal": "((w.structuredArrowDownwards X₂).map (StructuredArrow.homMk φ.right.left ⋯) ≫ ((fun f => CostructuredArrow.mk (StructuredArrow.homMk f.right.hom ⋯)) f₂).hom).right = ((fun f => CostructuredArrow.mk (StructuredArrow.homMk f.right.hom ⋯)) f₁).hom.right"}], "premise": [98352], "state_str": "case h\nC₁ : Type u₁\nC₂ : Type u₂\nC₃ : Type u₃\nC₄ : Type u₄\ninst✝³ : Category.{v₁, u₁} C₁\ninst✝² : Category.{v₂, u₂} C₂\ninst✝¹ : Category.{v₃, u₃} C₃\ninst✝ : Category.{v₄, u₄} C₄\nT : C₁ ⥤ C₂\nL : C₁ ⥤ C₃\nR : C₂ ⥤ C₄\nB : C₃ ⥤ C₄\nw : TwoSquare T L R B\nX₂ : C₂\nX₃ : C₃\ng : R.obj X₂ ⟶ B.obj X₃\nf₁ f₂ : w.StructuredArrowRightwards g\nφ : f₁ ⟶ f₂\n⊢ ((w.structuredArrowDownwards X₂).map (StructuredArrow.homMk φ.right.left ⋯) ≫\n        ((fun f => CostructuredArrow.mk (StructuredArrow.homMk f.right.hom ⋯)) f₂).hom).right =\n    ((fun f => CostructuredArrow.mk (StructuredArrow.homMk f.right.hom ⋯)) f₁).hom.right"}
+{"state": [{"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "D : Type u₂", "inst✝¹ : Category.{v₂, u₂} D", "F : C ⥤ D", "X Y Z : C", "f : Y ⟶ X", "E : Type u₃", "inst✝ : Category.{v₃, u₃} E", "G : D ⥤ E", "R : Presieve X"], "goal": "functorPushforward (F ⋙ G) R = functorPushforward G (functorPushforward F R)"}], "premise": [1838], "state_str": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y Z : C\nf : Y ⟶ X\nE : Type u₃\ninst✝ : Category.{v₃, u₃} E\nG : D ⥤ E\nR : Presieve X\n⊢ functorPushforward (F ⋙ G) R = functorPushforward G (functorPushforward F R)"}
+{"state": [{"context": ["C : Type u", "𝒞 : Category.{v, u} C", "inst✝ : MonoidalCategory C", "X Y Z Z' : C", "f : Z ⟶ Z'"], "goal": "(X ⊗ Y) ◁ f = (α_ X Y Z).hom ≫ X ◁ Y ◁ f ≫ (α_ X Y Z').inv"}], "premise": [99217, 99218], "state_str": "C : Type u\n𝒞 : Category.{v, u} C\ninst✝ : MonoidalCategory C\nX Y Z Z' : C\nf : Z ⟶ Z'\n⊢ (X ⊗ Y) ◁ f = (α_ X Y Z).hom ≫ X ◁ Y ◁ f ≫ (α_ X Y Z').inv"}
+{"state": [{"context": ["C : Type u", "𝒞 : Category.{v, u} C", "inst✝ : MonoidalCategory C", "X Y Z Z' : C", "f : Z ⟶ Z'"], "goal": "𝟙 (X ⊗ Y) ⊗ f = (α_ X Y Z).hom ≫ (𝟙 X ⊗ 𝟙 Y ⊗ f) ≫ (α_ X Y Z').inv"}], "premise": [96173, 99210], "state_str": "C : Type u\n𝒞 : Category.{v, u} C\ninst✝ : MonoidalCategory C\nX Y Z Z' : C\nf : Z ⟶ Z'\n⊢ 𝟙 (X ⊗ Y) ⊗ f = (α_ X Y Z).hom ≫ (𝟙 X ⊗ 𝟙 Y ⊗ f) ≫ (α_ X Y Z').inv"}
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+{"state": [{"context": ["α : Type u_1", "ι : Type u_2", "inst✝ : MetricSpace α", "t : Set ι", "x : ι → α", "r : ι → ℝ", "R : ℝ", "hr : ∀ a ∈ t, r a ≤ R", "τ : ℝ", "hτ : 3 < τ", "h✝ : t.Nonempty", "ht : ∃ a ∈ t, 0 ≤ r a", "t' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}", "u : Set ι", "ut' : u ⊆ t'", "u_disj : u.PairwiseDisjoint fun a => closedBall (x a) (r a)", "hu : ∀ a ∈ t', ∃ b ∈ u, (closedBall (x a) (r a) ∩ closedBall (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b", "A : ∀ a ∈ t', ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)"], "goal": "∃ u ⊆ t, (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧ ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)"}], "premise": [2107], "state_str": "case neg.intro.intro.intro\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ a ∈ t, r a ≤ R\nτ : ℝ\nhτ : 3 < τ\nh✝ : t.Nonempty\nht : ∃ a ∈ t, 0 ≤ r a\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : u.PairwiseDisjoint fun a => closedBall (x a) (r a)\nhu : ∀ a ∈ t', ∃ b ∈ u, (closedBall (x a) (r a) ∩ closedBall (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b\nA : ∀ a ∈ t', ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)\n⊢ ∃ u ⊆ t,\n    (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧\n      ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)"}
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+{"state": [{"context": ["R : Type u", "L : Type v", "L' : Type w₂", "inst✝⁴ : CommRing R", "inst✝³ : LieRing L", "inst✝² : LieAlgebra R L", "inst✝¹ : LieRing L'", "inst✝ : LieAlgebra R L'", "f : L →ₗ⁅R⁆ L'", "I : LieIdeal R L", "J : LieIdeal R L'", "I₁ I₂ : LieIdeal R L"], "goal": "⁅comap I.incl I₁, comap I.incl I₂⁆ = comap I.incl ⁅I ⊓ I₁, I ⊓ I₂⁆"}], "premise": [109410], "state_str": "R : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nI₁ I₂ : LieIdeal R L\n⊢ ⁅comap I.incl I₁, comap I.incl I₂⁆ = comap I.incl ⁅I ⊓ I₁, I ⊓ I₂⁆"}
+{"state": [{"context": ["R : Type u", "L : Type v", "L' : Type w₂", "inst✝⁴ : CommRing R", "inst✝³ : LieRing L", "inst✝² : LieAlgebra R L", "inst✝¹ : LieRing L'", "inst✝ : LieAlgebra R L'", "f : L →ₗ⁅R⁆ L'", "I : LieIdeal R L", "J : LieIdeal R L'", "I₁ I₂ : LieIdeal R L"], "goal": "⁅comap I.incl I₁, comap I.incl I₂⁆ = comap I.incl ⁅I.incl.idealRange ⊓ I₁, I.incl.idealRange ⊓ I₂⁆"}], "premise": [108507], "state_str": "R : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nI₁ I₂ : LieIdeal R L\n⊢ ⁅comap I.incl I₁, comap I.incl I₂⁆ = comap I.incl ⁅I.incl.idealRange ⊓ I₁, I.incl.idealRange ⊓ I₂⁆"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4", "ι' : Type u_5", "inst✝ : CompleteDistribLattice α", "f : Filter β", "p q : β → Prop", "u : β → α"], "goal": "(blimsup u f p ⊔ blimsup u f fun x => ¬p x) = limsup u f"}], "premise": [14767, 14868, 70036], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Type u_5\ninst✝ : CompleteDistribLattice α\nf : Filter β\np q : β → Prop\nu : β → α\n⊢ (blimsup u f p ⊔ blimsup u f fun x => ¬p x) = limsup u f"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝ : DecidableEq α", "l : List α"], "goal": "image l.get univ = l.toFinset"}], "premise": [140928], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : DecidableEq α\nl : List α\n⊢ image l.get univ = l.toFinset"}
+{"state": [{"context": ["ι : Type u_1", "α : Type u_2", "M : Type u_3", "N : Type u_4", "X : Type u_5", "inst✝³ : TopologicalSpace X", "inst✝² : TopologicalSpace M", "inst✝¹ : CommMonoid M", "inst✝ : ContinuousMul M", "f : ι → X → M", "hc : ∀ (i : ι), Continuous (f i)", "hf : LocallyFinite fun i => mulSupport (f i)"], "goal": "Continuous fun x => ∏ᶠ (i : ι), f i x"}], "premise": [1674, 55639], "state_str": "ι : Type u_1\nα : Type u_2\nM : Type u_3\nN : Type u_4\nX : Type u_5\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : CommMonoid M\ninst✝ : ContinuousMul M\nf : ι → X → M\nhc : ∀ (i : ι), Continuous (f i)\nhf : LocallyFinite fun i => mulSupport (f i)\n⊢ Continuous fun x => ∏ᶠ (i : ι), f i x"}
+{"state": [{"context": ["ι : Type u_1", "α : Type u_2", "M : Type u_3", "N : Type u_4", "X : Type u_5", "inst✝³ : TopologicalSpace X", "inst✝² : TopologicalSpace M", "inst✝¹ : CommMonoid M", "inst✝ : ContinuousMul M", "f : ι → X → M", "hc : ∀ (i : ι), Continuous (f i)", "hf : LocallyFinite fun i => mulSupport (f i)", "x : X"], "goal": "ContinuousAt (fun x => ∏ᶠ (i : ι), f i x) x"}], "premise": [65154], "state_str": "ι : Type u_1\nα : Type u_2\nM : Type u_3\nN : Type u_4\nX : Type u_5\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : CommMonoid M\ninst✝ : ContinuousMul M\nf : ι → X → M\nhc : ∀ (i : ι), Continuous (f i)\nhf : LocallyFinite fun i => mulSupport (f i)\nx : X\n⊢ ContinuousAt (fun x => ∏ᶠ (i : ι), f i x) x"}
+{"state": [{"context": ["ι : Type u_1", "α : Type u_2", "M : Type u_3", "N : Type u_4", "X : Type u_5", "inst✝³ : TopologicalSpace X", "inst✝² : TopologicalSpace M", "inst✝¹ : CommMonoid M", "inst✝ : ContinuousMul M", "f : ι → X → M", "hc : ∀ (i : ι), Continuous (f i)", "hf : LocallyFinite fun i => mulSupport (f i)", "x : X", "s : Finset ι", "hs : ∀ᶠ (y : X) in 𝓝 x, ∏ᶠ (i : ι), f i y = ∏ i ∈ s, f i y"], "goal": "ContinuousAt (fun x => ∏ᶠ (i : ι), f i x) x"}], "premise": [16092, 55622], "state_str": "case intro\nι : Type u_1\nα : Type u_2\nM : Type u_3\nN : Type u_4\nX : Type u_5\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : CommMonoid M\ninst✝ : ContinuousMul M\nf : ι → X → M\nhc : ∀ (i : ι), Continuous (f i)\nhf : LocallyFinite fun i => mulSupport (f i)\nx : X\ns : Finset ι\nhs : ∀ᶠ (y : X) in 𝓝 x, ∏ᶠ (i : ι), f i y = ∏ i ∈ s, f i y\n⊢ ContinuousAt (fun x => ∏ᶠ (i : ι), f i x) x"}
+{"state": [{"context": ["ι : Type u_1", "α : Type u_2", "M : Type u_3", "N : Type u_4", "X : Type u_5", "inst✝³ : TopologicalSpace X", "inst✝² : TopologicalSpace M", "inst✝¹ : CommMonoid M", "inst✝ : ContinuousMul M", "f : ι → X → M", "hc : ∀ (i : ι), Continuous (f i)", "hf : LocallyFinite fun i => mulSupport (f i)", "x : X", "s : Finset ι", "hs : ∀ᶠ (y : X) in 𝓝 x, ∏ᶠ (i : ι), f i y = ∏ i ∈ s, f i y"], "goal": "ContinuousAt (fun x => ∏ i ∈ s, f i x) x"}], "premise": [55638, 65141], "state_str": "case intro\nι : Type u_1\nα : Type u_2\nM : Type u_3\nN : Type u_4\nX : Type u_5\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : CommMonoid M\ninst✝ : ContinuousMul M\nf : ι → X → M\nhc : ∀ (i : ι), Continuous (f i)\nhf : LocallyFinite fun i => mulSupport (f i)\nx : X\ns : Finset ι\nhs : ∀ᶠ (y : X) in 𝓝 x, ∏ᶠ (i : ι), f i y = ∏ i ∈ s, f i y\n⊢ ContinuousAt (fun x => ∏ i ∈ s, f i x) x"}
+{"state": [{"context": ["x y : ℂ"], "goal": "sin x = sin y ↔ ∃ k, y = 2 * ↑k * ↑π + x ∨ y = (2 * ↑k + 1) * ↑π - x"}], "premise": [37407, 38733, 118013], "state_str": "x y : ℂ\n⊢ sin x = sin y ↔ ∃ k, y = 2 * ↑k * ↑π + x ∨ y = (2 * ↑k + 1) * ↑π - x"}
+{"state": [{"context": ["x y : ℂ"], "goal": "(∃ k, y = 2 * ↑k * ↑π + (x - ↑π / 2) + ↑π / 2 ∨ y = 2 * ↑k * ↑π - (x - ↑π / 2) + ↑π / 2) ↔ ∃ k, y = 2 * ↑k * ↑π + x ∨ y = (2 * ↑k + 1) * ↑π - x"}], "premise": [26, 1980, 2016], "state_str": "x y : ℂ\n⊢ (∃ k, y = 2 * ↑k * ↑π + (x - ↑π / 2) + ↑π / 2 ∨ y = 2 * ↑k * ↑π - (x - ↑π / 2) + ↑π / 2) ↔\n    ∃ k, y = 2 * ↑k * ↑π + x ∨ y = (2 * ↑k + 1) * ↑π - x"}
+{"state": [{"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "M : Type u_4", "N : Type u_5", "P : Type u_6", "G : Type u_7", "inst✝² : Monoid M", "inst✝¹ : Monoid N", "inst✝ : Monoid P", "l✝ l₁ l₂ : List M", "a : M", "l : List (List M)"], "goal": "l.join.prod = (map prod l).prod"}], "premise": [124826, 124830], "state_str": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : Monoid P\nl✝ l₁ l₂ : List M\na : M\nl : List (List M)\n⊢ l.join.prod = (map prod l).prod"}
+{"state": [{"context": ["E : Type u_1", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℝ E", "F : Type u_2", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace ℝ F", "v : ℝ → E → E", "s : ℝ → Set E", "K : ℝ≥0", "f g f' g' : ℝ → E", "a b t₀ εf εg δ : ℝ", "hv : ∀ (t : ℝ), LipschitzOnWith K (v t) (s t)", "ht : t₀ ∈ Ioo a b", "hf : ContinuousOn f (Icc a b)", "hf' : ∀ t ∈ Ioo a b, HasDerivAt f (v t (f t)) t", "hfs : ∀ t ∈ Ioo a b, f t ∈ s t", "hg : ContinuousOn g (Icc a b)", "hg' : ∀ t ∈ Ioo a b, HasDerivAt g (v t (g t)) t", "hgs : ∀ t ∈ Ioo a b, g t ∈ s t", "heq : f t₀ = g t₀"], "goal": "EqOn f g (Icc a b)"}], "premise": [2106, 2107, 14286, 20447], "state_str": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nv : ℝ → E → E\ns : ℝ → Set E\nK : ℝ≥0\nf g f' g' : ℝ → E\na b t₀ εf εg δ : ℝ\nhv : ∀ (t : ℝ), LipschitzOnWith K (v t) (s t)\nht : t₀ ∈ Ioo a b\nhf : ContinuousOn f (Icc a b)\nhf' : ∀ t ∈ Ioo a b, HasDerivAt f (v t (f t)) t\nhfs : ∀ t ∈ Ioo a b, f t ∈ s t\nhg : ContinuousOn g (Icc a b)\nhg' : ∀ t ∈ Ioo a b, HasDerivAt g (v t (g t)) t\nhgs : ∀ t ∈ Ioo a b, g t ∈ s t\nheq : f t₀ = g t₀\n⊢ EqOn f g (Icc a b)"}
+{"state": [{"context": ["E : Type u_1", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℝ E", "F : Type u_2", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace ℝ F", "v : ℝ → E → E", "s : ℝ → Set E", "K : ℝ≥0", "f g f' g' : ℝ → E", "a b t₀ εf εg δ : ℝ", "hv : ∀ (t : ℝ), LipschitzOnWith K (v t) (s t)", "ht : t₀ ∈ Ioo a b", "hf : ContinuousOn f (Icc a b)", "hf' : ∀ t ∈ Ioo a b, HasDerivAt f (v t (f t)) t", "hfs : ∀ t ∈ Ioo a b, f t ∈ s t", "hg : ContinuousOn g (Icc a b)", "hg' : ∀ t ∈ Ioo a b, HasDerivAt g (v t (g t)) t", "hgs : ∀ t ∈ Ioo a b, g t ∈ s t", "heq : f t₀ = g t₀"], "goal": "EqOn f g (Icc a t₀ ∪ Icc t₀ b)"}], "premise": [135720], "state_str": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nv : ℝ → E → E\ns : ℝ → Set E\nK : ℝ≥0\nf g f' g' : ℝ → E\na b t₀ εf εg δ : ℝ\nhv : ∀ (t : ℝ), LipschitzOnWith K (v t) (s t)\nht : t₀ ∈ Ioo a b\nhf : ContinuousOn f (Icc a b)\nhf' : ∀ t ∈ Ioo a b, HasDerivAt f (v t (f t)) t\nhfs : ∀ t ∈ Ioo a b, f t ∈ s t\nhg : ContinuousOn g (Icc a b)\nhg' : ∀ t ∈ Ioo a b, HasDerivAt g (v t (g t)) t\nhgs : ∀ t ∈ Ioo a b, g t ∈ s t\nheq : f t₀ = g t₀\n⊢ EqOn f g (Icc a t₀ ∪ Icc t₀ b)"}
+{"state": [{"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "J : GrothendieckTopology C", "R₀ R : Cᵒᵖ ⥤ RingCat", "α : R₀ ⟶ R", "inst✝¹ : Presheaf.IsLocallyInjective J α", "M₀ : PresheafOfModules R₀", "A : Cᵒᵖ ⥤ AddCommGrp", "φ : M₀.presheaf ⟶ A", "inst✝ : Presheaf.IsLocallyInjective J φ", "hA : Presheaf.IsSeparated J A", "X : C", "r : ↑(R.obj (Opposite.op X))", "m : ↑(A.obj (Opposite.op X))", "P : Presieve X", "r₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P", "m₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P", "hr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r", "hm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m", "Y₁ Y₂ Z : C", "g₁ : Z ⟶ Y₁", "g₂ : Z ⟶ Y₂", "f₁ : Y₁ ⟶ X", "f₂ : Y₂ ⟶ X", "h₁ : P f₁", "h₂ : P f₂", "fac : g₁ ≫ f₁ = g₂ ≫ f₂", "a₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁", "b₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁", "a₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂", "b₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂", "a₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁", "b₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁"], "goal": "(A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) = (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)"}], "premise": [2100], "state_str": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\na₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂\nb₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂\na₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁\nb₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁\n⊢ (A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) =\n    (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)"}
+{"state": [{"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "J : GrothendieckTopology C", "R₀ R : Cᵒᵖ ⥤ RingCat", "α : R₀ ⟶ R", "inst✝¹ : Presheaf.IsLocallyInjective J α", "M₀ : PresheafOfModules R₀", "A : Cᵒᵖ ⥤ AddCommGrp", "φ : M₀.presheaf ⟶ A", "inst✝ : Presheaf.IsLocallyInjective J φ", "hA : Presheaf.IsSeparated J A", "X : C", "r : ↑(R.obj (Opposite.op X))", "m : ↑(A.obj (Opposite.op X))", "P : Presieve X", "r₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P", "m₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P", "hr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r", "hm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m", "Y₁ Y₂ Z : C", "g₁ : Z ⟶ Y₁", "g₂ : Z ⟶ Y₂", "f₁ : Y₁ ⟶ X", "f₂ : Y₂ ⟶ X", "h₁ : P f₁", "h₂ : P f₂", "fac : g₁ ≫ f₁ = g₂ ≫ f₂", "a₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁", "b₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁", "a₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂", "b₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂", "a₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁", "b₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁", "ha₁ : (α.app (Opposite.op Y₁)) a₁ = (R.map f₁.op) r"], "goal": "(A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) = (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)"}], "premise": [2100], "state_str": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r�� f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\na₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂\nb₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂\na₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁\nb₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁\nha₁ : (α.app (Opposite.op Y₁)) a₁ = (R.map f₁.op) r\n⊢ (A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) =\n    (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)"}
+{"state": [{"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "J : GrothendieckTopology C", "R₀ R : Cᵒᵖ ⥤ RingCat", "α : R₀ ⟶ R", "inst✝¹ : Presheaf.IsLocallyInjective J α", "M₀ : PresheafOfModules R₀", "A : Cᵒᵖ ⥤ AddCommGrp", "φ : M₀.presheaf ⟶ A", "inst✝ : Presheaf.IsLocallyInjective J φ", "hA : Presheaf.IsSeparated J A", "X : C", "r : ↑(R.obj (Opposite.op X))", "m : ↑(A.obj (Opposite.op X))", "P : Presieve X", "r₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P", "m₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P", "hr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r", "hm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m", "Y₁ Y₂ Z : C", "g₁ : Z ⟶ Y₁", "g₂ : Z ⟶ Y₂", "f₁ : Y₁ ⟶ X", "f₂ : Y₂ ⟶ X", "h₁ : P f₁", "h₂ : P f₂", "fac : g₁ ≫ f₁ = g₂ ≫ f₂", "a₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁", "b₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁", "a₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂", "b₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂", "a₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁", "b₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁", "ha₁ : (α.app (Opposite.op Y₁)) a₁ = (R.map f₁.op) r", "ha₂ : (α.app (Opposite.op Y₂)) a₂ = (R.map f₂.op) r"], "goal": "(A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) = (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)"}], "premise": [2100], "state_str": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\na₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂\nb₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂\na₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁\nb₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁\nha₁ : (α.app (Opposite.op Y₁)) a₁ = (R.map f₁.op) r\nha₂ : (α.app (Opposite.op Y₂)) a₂ = (R.map f₂.op) r\n⊢ (A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) =\n    (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)"}
+{"state": [{"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "J : GrothendieckTopology C", "R₀ R : Cᵒᵖ ⥤ RingCat", "α : R₀ ⟶ R", "inst✝¹ : Presheaf.IsLocallyInjective J α", "M₀ : PresheafOfModules R₀", "A : Cᵒᵖ ⥤ AddCommGrp", "φ : M₀.presheaf ⟶ A", "inst✝ : Presheaf.IsLocallyInjective J φ", "hA : Presheaf.IsSeparated J A", "X : C", "r : ↑(R.obj (Opposite.op X))", "m : ↑(A.obj (Opposite.op X))", "P : Presieve X", "r₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P", "m₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P", "hr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r", "hm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m", "Y₁ Y₂ Z : C", "g₁ : Z ⟶ Y₁", "g₂ : Z ⟶ Y₂", "f₁ : Y₁ ⟶ X", "f₂ : Y₂ ⟶ X", "h₁ : P f₁", "h₂ : P f₂", "fac : g₁ ≫ f₁ = g₂ ≫ f₂", "a₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h���", "b₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁", "a₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂", "b₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂", "a₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁", "b₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁", "ha₁ : (α.app (Opposite.op Y₁)) a₁ = (R.map f₁.op) r", "ha₂ : (α.app (Opposite.op Y₂)) a₂ = (R.map f₂.op) r", "hb₁ : (φ.app (Opposite.op Y₁)) b₁ = (A.map f₁.op) m"], "goal": "(A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) = (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)"}], "premise": [2100], "state_str": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\na₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂\nb₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂\na₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁\nb₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁\nha₁ : (α.app (Opposite.op Y₁)) a₁ = (R.map f₁.op) r\nha₂ : (α.app (Opposite.op Y₂)) a₂ = (R.map f₂.op) r\nhb₁ : (φ.app (Opposite.op Y₁)) b₁ = (A.map f₁.op) m\n⊢ (A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) =\n    (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)"}
+{"state": [{"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "J : GrothendieckTopology C", "R₀ R : Cᵒᵖ ⥤ RingCat", "α : R₀ ⟶ R", "inst✝¹ : Presheaf.IsLocallyInjective J α", "M₀ : PresheafOfModules R₀", "A : Cᵒᵖ ⥤ AddCommGrp", "φ : M₀.presheaf ⟶ A", "inst✝ : Presheaf.IsLocallyInjective J φ", "hA : Presheaf.IsSeparated J A", "X : C", "r : ↑(R.obj (Opposite.op X))", "m : ↑(A.obj (Opposite.op X))", "P : Presieve X", "r₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P", "m₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P", "hr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r", "hm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m", "Y₁ Y₂ Z : C", "g₁ : Z ⟶ Y₁", "g₂ : Z ⟶ Y₂", "f₁ : Y₁ ⟶ X", "f₂ : Y₂ ⟶ X", "h₁ : P f₁", "h₂ : P f₂", "fac : g₁ ≫ f₁ = g₂ ≫ f₂", "a₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁", "b₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁", "a₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂", "b₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂", "a₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁", "b₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁", "ha₁ : (α.app (Opposite.op Y₁)) a₁ = (R.map f₁.op) r", "ha₂ : (α.app (Opposite.op Y₂)) a₂ = (R.map f₂.op) r", "hb₁ : (φ.app (Opposite.op Y₁)) b₁ = (A.map f₁.op) m", "hb₂ : (φ.app (Opposite.op Y₂)) b₂ = (A.map f₂.op) m"], "goal": "(A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) = (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)"}], "premise": [99593, 99600, 99919], "state_str": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\na₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂\nb₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂\na₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁\nb₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁\nha₁ : (α.app (Opposite.op Y₁)) a₁ = (R.map f₁.op) r\nha₂ : (α.app (Opposite.op Y₂)) a₂ = (R.map f₂.op) r\nhb₁ : (φ.app (Opposite.op Y₁)) b₁ = (A.map f₁.op) m\nhb₂ : (φ.app (Opposite.op Y₂)) b₂ = (A.map f₂.op) m\n⊢ (A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) =\n    (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)"}
+{"state": [{"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "J : GrothendieckTopology C", "R₀ R : Cᵒᵖ ⥤ RingCat", "α : R₀ ⟶ R", "inst✝¹ : Presheaf.IsLocallyInjective J α", "M₀ : PresheafOfModules R₀", "A : Cᵒᵖ ⥤ AddCommGrp", "φ : M₀.presheaf ⟶ A", "inst✝ : Presheaf.IsLocallyInjective J φ", "hA : Presheaf.IsSeparated J A", "X : C", "r : ↑(R.obj (Opposite.op X))", "m : ↑(A.obj (Opposite.op X))", "P : Presieve X", "r₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P", "m₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P", "hr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r", "hm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m", "Y₁ Y₂ Z : C", "g₁ : Z ⟶ Y₁", "g₂ : Z ⟶ Y₂", "f₁ : Y₁ ⟶ X", "f₂ : Y₂ ⟶ X", "h₁ : P f₁", "h₂ : P f₂", "fac : g₁ ≫ f₁ = g₂ ≫ f₂", "a₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁", "b₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁", "a₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂", "b₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂", "a₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁", "b₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁", "ha₁ : (α.app (Opposite.op Y₁)) a₁ = (R.map f₁.op) r", "ha₂ : (α.app (Opposite.op Y₂)) a₂ = (R.map f₂.op) r", "hb₁ : (φ.app (Opposite.op Y₁)) b₁ = (A.map f₁.op) m", "hb₂ : (φ.app (Opposite.op Y₂)) b₂ = (A.map f₂.op) m", "ha₀ : (α.app (Opposite.op Z)) a₀ = (R.map (f₁.op ≫ g₁.op)) r"], "goal": "(A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) = (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)"}], "premise": [99593, 99600, 99919], "state_str": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\na₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂\nb₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂\na₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁\nb₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁\nha₁ : (α.app (Opposite.op Y₁)) a₁ = (R.map f₁.op) r\nha₂ : (α.app (Opposite.op Y₂)) a₂ = (R.map f₂.op) r\nhb₁ : (φ.app (Opposite.op Y₁)) b₁ = (A.map f₁.op) m\nhb₂ : (φ.app (Opposite.op Y₂)) b₂ = (A.map f₂.op) m\nha₀ : (α.app (Opposite.op Z)) a₀ = (R.map (f₁.op ≫ g₁.op)) r\n⊢ (A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) =\n    (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)"}
+{"state": [{"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "J : GrothendieckTopology C", "R₀ R : Cᵒᵖ ⥤ RingCat", "α : R₀ ⟶ R", "inst✝¹ : Presheaf.IsLocallyInjective J α", "M₀ : PresheafOfModules R₀", "A : Cᵒᵖ ⥤ AddCommGrp", "φ : M₀.presheaf ⟶ A", "inst✝ : Presheaf.IsLocallyInjective J φ", "hA : Presheaf.IsSeparated J A", "X : C", "r : ↑(R.obj (Opposite.op X))", "m : ↑(A.obj (Opposite.op X))", "P : Presieve X", "r₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P", "m₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P", "hr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r", "hm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m", "Y₁ Y₂ Z : C", "g₁ : Z ⟶ Y₁", "g₂ : Z ⟶ Y₂", "f₁ : Y₁ ⟶ X", "f₂ : Y₂ ⟶ X", "h₁ : P f₁", "h₂ : P f₂", "fac : g₁ ≫ f₁ = g₂ ≫ f₂", "a₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁", "b₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁", "a₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂", "b₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂", "a₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁", "b₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁", "ha₁ : (α.app (Opposite.op Y₁)) a₁ = (R.map f₁.op) r", "ha₂ : (α.app (Opposite.op Y₂)) a₂ = (R.map f₂.op) r", "hb₁ : (φ.app (Opposite.op Y₁)) b₁ = (A.map f₁.op) m", "hb₂ : (φ.app (Opposite.op Y₂)) b₂ = (A.map f₂.op) m", "ha₀ : (α.app (Opposite.op Z)) a₀ = (R.map (f₁.op ≫ g₁.op)) r", "hb₀ : (φ.app (Opposite.op Z)) b₀ = (A.map (f₁.op ≫ g₁.op)) m"], "goal": "(A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) = (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)"}], "premise": [89631], "state_str": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\na₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂\nb₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂\na₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁\nb₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁\nha₁ : (α.app (Opposite.op Y₁)) a₁ = (R.map f₁.op) r\nha₂ : (α.app (Opposite.op Y₂)) a₂ = (R.map f₂.op) r\nhb₁ : (φ.app (Opposite.op Y₁)) b₁ = (A.map f₁.op) m\nhb₂ : (φ.app (Opposite.op Y₂)) b₂ = (A.map f₂.op) m\nha₀ : (α.app (Opposite.op Z)) a₀ = (R.map (f₁.op ≫ g₁.op)) r\nhb₀ : (φ.app (Opposite.op Z)) b₀ = (A.map (f₁.op ≫ g₁.op)) m\n⊢ (A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) =\n    (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)"}
+{"state": [{"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "J : GrothendieckTopology C", "R₀ R : Cᵒᵖ ⥤ RingCat", "α : R₀ ⟶ R", "inst✝¹ : Presheaf.IsLocallyInjective J α", "M₀ : PresheafOfModules R₀", "A : Cᵒᵖ ⥤ AddCommGrp", "φ : M₀.presheaf ⟶ A", "inst✝ : Presheaf.IsLocallyInjective J φ", "hA : Presheaf.IsSeparated J A", "X : C", "r : ↑(R.obj (Opposite.op X))", "m : ↑(A.obj (Opposite.op X))", "P : Presieve X", "r₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P", "m₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P", "hr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r", "hm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m", "Y₁ Y₂ Z : C", "g₁ : Z ⟶ Y₁", "g₂ : Z ⟶ Y₂", "f₁ : Y₁ ⟶ X", "f₂ : Y₂ ⟶ X", "h₁ : P f₁", "h₂ : P f₂", "fac : g₁ ≫ f₁ = g₂ ≫ f₂", "a₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁", "b₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁", "a₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂", "b₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂", "a₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁", "b₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁", "ha₁ : (α.app (Opposite.op Y₁)) a₁ = (R.map f₁.op) r", "ha₂ : (α.app (Opposite.op Y₂)) a₂ = (R.map f₂.op) r", "hb₁ : (φ.app (Opposite.op Y₁)) b₁ = (A.map f₁.op) m", "hb₂ : (φ.app (Opposite.op Y₂)) b₂ = (A.map f₂.op) m", "ha₀ : (α.app (Opposite.op Z)) a₀ = (R.map (f₁.op ≫ g₁.op)) r", "hb₀ : (φ.app (Opposite.op Z)) b₀ = (A.map (f₁.op ≫ g₁.op)) m", "ha₀' : (α.app (Opposite.op Z)) a₀ = (R.map (f₂.op ≫ g₂.op)) r"], "goal": "(A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) = (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)"}], "premise": [89631], "state_str": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\na₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂\nb₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂\na₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁\nb₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁\nha₁ : (α.app (Opposite.op Y₁)) a₁ = (R.map f₁.op) r\nha₂ : (α.app (Opposite.op Y₂)) a₂ = (R.map f₂.op) r\nhb₁ : (φ.app (Opposite.op Y₁)) b₁ = (A.map f₁.op) m\nhb₂ : (φ.app (Opposite.op Y₂)) b₂ = (A.map f₂.op) m\nha₀ : (α.app (Opposite.op Z)) a₀ = (R.map (f₁.op ≫ g₁.op)) r\nhb₀ : (φ.app (Opposite.op Z)) b₀ = (A.map (f₁.op ≫ g₁.op)) m\nha₀' : (α.app (Opposite.op Z)) a₀ = (R.map (f₂.op ≫ g₂.op)) r\n⊢ (A ⋙ forget AddCommGrp).map g₁.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₁ h₁) =\n    (A ⋙ forget AddCommGrp).map g₂.op ((r₀.smul m₀).map (whiskerRight φ (forget AddCommGrp)) f₂ h₂)"}
+{"state": [{"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "J : GrothendieckTopology C", "R₀ R : Cᵒᵖ ⥤ RingCat", "α : R₀ ⟶ R", "inst✝¹ : Presheaf.IsLocallyInjective J α", "M₀ : PresheafOfModules R₀", "A : Cᵒᵖ ⥤ AddCommGrp", "φ : M₀.presheaf ⟶ A", "inst✝ : Presheaf.IsLocallyInjective J φ", "hA : Presheaf.IsSeparated J A", "X : C", "r : ↑(R.obj (Opposite.op X))", "m : ↑(A.obj (Opposite.op X))", "P : Presieve X", "r₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P", "m₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P", "hr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r", "hm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m", "Y₁ Y₂ Z : C", "g₁ : Z ⟶ Y₁", "g₂ : Z ⟶ Y₂", "f₁ : Y₁ ⟶ X", "f₂ : Y₂ ⟶ X", "h₁ : P f₁", "h₂ : P f₂", "fac : g₁ ≫ f₁ = g₂ ≫ f₂", "a₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁", "b₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁", "a₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂", "b₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂", "a₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁", "b₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁", "ha₁ : (α.app (Opposite.op Y₁)) a₁ = (R.map f₁.op) r", "ha₂ : (α.app (Opposite.op Y₂)) a₂ = (R.map f₂.op) r", "hb₁ : (φ.app (Opposite.op Y₁)) b₁ = (A.map f₁.op) m", "hb₂ : (φ.app (Opposite.op Y₂)) b₂ = (A.map f₂.op) m", "ha₀ : (α.app (Opposite.op Z)) a₀ = (R.map (f₁.op ≫ g₁.op)) r", "hb₀ : (φ.app (Opposite.op Z)) b₀ = (A.map (f₁.op ≫ g₁.op)) m", "ha₀' : (α.app (Opposite.op Z)) a₀ = (R.map (f₂.op ≫ g₂.op)) r", "hb₀' : (φ.app (Opposite.op Z)) b₀ = (A.map (f₂.op ≫ g₂.op)) m"], "goal": "(A.map g₁.op) ((φ.app (Opposite.op Y₁)) (r₀.smul m₀ f₁ h₁)) = (A.map g₂.op) ((φ.app (Opposite.op Y₂)) (r₀.smul m₀ f₂ h₂))"}], "premise": [99600], "state_str": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\na₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂\nb₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂\na₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁\nb₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁\nha₁ : (α.app (Opposite.op Y₁)) a₁ = (R.map f₁.op) r\nha₂ : (α.app (Opposite.op Y₂)) a₂ = (R.map f₂.op) r\nhb₁ : (φ.app (Opposite.op Y₁)) b₁ = (A.map f₁.op) m\nhb₂ : (φ.app (Opposite.op Y₂)) b₂ = (A.map f₂.op) m\nha₀ : (α.app (Opposite.op Z)) a₀ = (R.map (f₁.op ≫ g₁.op)) r\nhb₀ : (φ.app (Opposite.op Z)) b₀ = (A.map (f₁.op ≫ g₁.op)) m\nha₀' : (α.app (Opposite.op Z)) a₀ = (R.map (f₂.op ≫ g₂.op)) r\nhb₀' : (φ.app (Opposite.op Z)) b₀ = (A.map (f₂.op ≫ g₂.op)) m\n⊢ (A.map g₁.op) ((φ.app (Opposite.op Y₁)) (r₀.smul m₀ f₁ h₁)) =\n    (A.map g₂.op) ((φ.app (Opposite.op Y₂)) (r₀.smul m₀ f₂ h₂))"}
+{"state": [{"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "J : GrothendieckTopology C", "R₀ R : Cᵒᵖ ⥤ RingCat", "α : R₀ ⟶ R", "inst✝¹ : Presheaf.IsLocallyInjective J α", "M₀ : PresheafOfModules R₀", "A : Cᵒᵖ ⥤ AddCommGrp", "φ : M₀.presheaf ⟶ A", "inst✝ : Presheaf.IsLocallyInjective J φ", "hA : Presheaf.IsSeparated J A", "X : C", "r : ↑(R.obj (Opposite.op X))", "m : ↑(A.obj (Opposite.op X))", "P : Presieve X", "r₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P", "m₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P", "hr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r", "hm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m", "Y₁ Y₂ Z : C", "g₁ : Z ⟶ Y₁", "g₂ : Z ⟶ Y₂", "f₁ : Y₁ ⟶ X", "f₂ : Y₂ ⟶ X", "h₁ : P f₁", "h₂ : P f₂", "fac : g₁ ≫ f₁ = g₂ ≫ f₂", "a₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁", "b₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁", "a₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂", "b₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂", "a₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁", "b₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁", "ha₁ : (α.app (Opposite.op Y₁)) a₁ = (R.map f₁.op) r", "ha₂ : (α.app (Opposite.op Y₂)) a₂ = (R.map f₂.op) r", "hb₁ : (φ.app (Opposite.op Y₁)) b₁ = (A.map f₁.op) m", "hb₂ : (φ.app (Opposite.op Y₂)) b₂ = (A.map f₂.op) m", "ha₀ : (α.app (Opposite.op Z)) a₀ = (R.map (f₁.op ≫ g₁.op)) r", "hb₀ : (φ.app (Opposite.op Z)) b₀ = (A.map (f₁.op ≫ g₁.op)) m", "ha₀' : (α.app (Opposite.op Z)) a₀ = (R.map (f₂.op ≫ g₂.op)) r", "hb₀' : (φ.app (Opposite.op Z)) b₀ = (A.map (f₂.op ≫ g₂.op)) m"], "goal": "(φ.app (Opposite.op Z)) ((M₀.presheaf.map g₁.op) (r₀.smul m₀ f₁ h₁)) = (φ.app (Opposite.op Z)) ((M₀.presheaf.map g₂.op) (r₀.smul m₀ f₂ h₂))"}], "premise": [2100, 2101, 113673], "state_str": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ R : Cᵒᵖ ⥤ RingCat\nα : R₀ ⟶ R\ninst✝¹ : Presheaf.IsLocallyInjective J α\nM₀ : PresheafOfModules R₀\nA : Cᵒᵖ ⥤ AddCommGrp\nφ : M₀.presheaf ⟶ A\ninst✝ : Presheaf.IsLocallyInjective J φ\nhA : Presheaf.IsSeparated J A\nX : C\nr : ↑(R.obj (Opposite.op X))\nm : ↑(A.obj (Opposite.op X))\nP : Presieve X\nr₀ : FamilyOfElements (R₀ ⋙ forget RingCat) P\nm₀ : FamilyOfElements (M₀.presheaf ⋙ forget AddCommGrp) P\nhr₀ : (r₀.map (whiskerRight α (forget RingCat))).IsAmalgamation r\nhm₀ : (m₀.map (whiskerRight φ (forget AddCommGrp))).IsAmalgamation m\nY₁ Y₂ Z : C\ng₁ : Z ⟶ Y₁\ng₂ : Z ⟶ Y₂\nf₁ : Y₁ ⟶ X\nf₂ : Y₂ ⟶ X\nh₁ : P f₁\nh₂ : P f₂\nfac : g₁ ≫ f₁ = g₂ ≫ f₂\na₁ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₁) := r₀ f₁ h₁\nb₁ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₁) := m₀ f₁ h₁\na₂ : (R₀ ⋙ forget RingCat).obj (Opposite.op Y₂) := r₀ f₂ h₂\nb₂ : (M₀.presheaf ⋙ forget AddCommGrp).obj (Opposite.op Y₂) := m₀ f₂ h₂\na₀ : ↑(R₀.obj (Opposite.op Z)) := (R₀.map g₁.op) a₁\nb₀ : ↑(M₀.obj (Opposite.op Z)) := (M₀.map g₁.op) b₁\nha₁ : (α.app (Opposite.op Y₁)) a₁ = (R.map f₁.op) r\nha₂ : (α.app (Opposite.op Y₂)) a₂ = (R.map f₂.op) r\nhb₁ : (φ.app (Opposite.op Y₁)) b₁ = (A.map f₁.op) m\nhb₂ : (φ.app (Opposite.op Y₂)) b₂ = (A.map f₂.op) m\nha₀ : (α.app (Opposite.op Z)) a₀ = (R.map (f₁.op ≫ g₁.op)) r\nhb₀ : (φ.app (Opposite.op Z)) b₀ = (A.map (f₁.op ≫ g₁.op)) m\nha₀' : (α.app (Opposite.op Z)) a₀ = (R.map (f₂.op ≫ g₂.op)) r\nhb₀' : (φ.app (Opposite.op Z)) b₀ = (A.map (f₂.op ≫ g₂.op)) m\n⊢ (φ.app (Opposite.op Z)) ((M₀.presheaf.map g₁.op) (r₀.smul m₀ f₁ h₁)) =\n    (φ.app (Opposite.op Z)) ((M₀.presheaf.map g₂.op) (r₀.smul m₀ f₂ h₂))"}
+{"state": [{"context": ["K✝ : Type u_1", "inst✝³ : Field K✝", "K : Type u_2", "inst✝² : CommRing K", "inst✝¹ : IsDomain K", "inst✝ : CharZero K", "ζ : K", "n : ℕ+", "h : IsPrimitiveRoot ζ ↑n"], "goal": "∃! P, map (Int.castRingHom K) P = cyclotomic' (↑n) K"}], "premise": [75175], "state_str": "K✝ : Type u_1\ninst✝³ : Field K✝\nK : Type u_2\ninst✝² : CommRing K\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\nζ : K\nn : ℕ+\nh : IsPrimitiveRoot ζ ↑n\n⊢ ∃! P, map (Int.castRingHom K) P = cyclotomic' (↑n) K"}
+{"state": [{"context": ["K✝ : Type u_1", "inst✝³ : Field K✝", "K : Type u_2", "inst✝² : CommRing K", "inst✝¹ : IsDomain K", "inst✝ : CharZero K", "ζ : K", "n : ℕ+", "h : IsPrimitiveRoot ζ ↑n", "P : ℤ[X]", "hP : map (Int.castRingHom K) P = cyclotomic' (↑n) K ∧ P.degree = (cyclotomic' (↑n) K).degree ∧ P.Monic"], "goal": "∃! P, map (Int.castRingHom K) P = cyclotomic' (↑n) K"}], "premise": [2107], "state_str": "case intro\nK✝ : Type u_1\ninst✝³ : Field K✝\nK : Type u_2\ninst✝² : CommRing K\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\nζ : K\nn : ℕ+\nh : IsPrimitiveRoot ζ ↑n\nP : ℤ[X]\nhP : map (Int.castRingHom K) P = cyclotomic' (↑n) K ∧ P.degree = (cyclotomic' (↑n) K).degree ∧ P.Monic\n⊢ ∃! P, map (Int.castRingHom K) P = cyclotomic' (↑n) K"}
+{"state": [{"context": ["K✝ : Type u_1", "inst✝³ : Field K✝", "K : Type u_2", "inst✝² : CommRing K", "inst✝¹ : IsDomain K", "inst✝ : CharZero K", "ζ : K", "n : ℕ+", "h : IsPrimitiveRoot ζ ↑n", "P : ℤ[X]", "hP : map (Int.castRingHom K) P = cyclotomic' (↑n) K ∧ P.degree = (cyclotomic' (↑n) K).degree ∧ P.Monic", "Q : ℤ[X]", "hQ : (fun P => map (Int.castRingHom K) P = cyclotomic' (↑n) K) Q"], "goal": "Q = P"}], "premise": [102934, 128914], "state_str": "case intro\nK✝ : Type u_1\ninst✝³ : Field K✝\nK : Type u_2\ninst✝² : CommRing K\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\nζ : K\nn : ℕ+\nh : IsPrimitiveRoot ζ ↑n\nP : ℤ[X]\nhP : map (Int.castRingHom K) P = cyclotomic' (↑n) K ∧ P.degree = (cyclotomic' (↑n) K).degree ∧ P.Monic\nQ : ℤ[X]\nhQ : (fun P => map (Int.castRingHom K) P = cyclotomic' (↑n) K) Q\n⊢ Q = P"}
+{"state": [{"context": ["K✝ : Type u_1", "inst✝³ : Field K✝", "K : Type u_2", "inst✝² : CommRing K", "inst✝¹ : IsDomain K", "inst✝ : CharZero K", "ζ : K", "n : ℕ+", "h : IsPrimitiveRoot ζ ↑n", "P : ℤ[X]", "hP : map (Int.castRingHom K) P = cyclotomic' (↑n) K ∧ P.degree = (cyclotomic' (↑n) K).degree ∧ P.Monic", "Q : ℤ[X]", "hQ : (fun P => map (Int.castRingHom K) P = cyclotomic' (↑n) K) Q"], "goal": "map (Int.castRingHom K) Q = map (Int.castRingHom K) P"}], "premise": [2107], "state_str": "case intro.a\nK✝ : Type u_1\ninst✝³ : Field K✝\nK : Type u_2\ninst✝² : CommRing K\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\nζ : K\nn : ℕ+\nh : IsPrimitiveRoot ζ ↑n\nP : ℤ[X]\nhP : map (Int.castRingHom K) P = cyclotomic' (↑n) K ∧ P.degree = (cyclotomic' (↑n) K).degree ∧ P.Monic\nQ : ℤ[X]\nhQ : (fun P => map (Int.castRingHom K) P = cyclotomic' (↑n) K) Q\n⊢ map (Int.castRingHom K) Q = map (Int.castRingHom K) P"}
+{"state": [{"context": ["x : EReal", "h : x < 0"], "goal": "⊤ * x = ⊥"}], "premise": [147093], "state_str": "x : EReal\nh : x < 0\n⊢ ⊤ * x = ⊥"}
+{"state": [{"context": ["x : EReal", "h : x < 0"], "goal": "x * ⊤ = ⊥"}], "premise": [147252], "state_str": "x : EReal\nh : x < 0\n⊢ x * ⊤ = ⊥"}
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+{"state": [{"context": ["V : Type u_1", "G✝ : SimpleGraph V", "e : Sym2 V", "inst✝³ : G✝.LocallyFinite", "inst✝² : Fintype V", "inst✝¹ : DecidableEq V", "G : SimpleGraph V", "inst✝ : DecidableRel G.Adj", "k : ℕ", "h : G.IsRegularOfDegree k", "v : V"], "goal": "Gᶜ.degree v = Fintype.card V - 1 - k"}], "premise": [52160], "state_str": "V : Type u_1\nG✝ : SimpleGraph V\ne : Sym2 V\ninst✝³ : G✝.LocallyFinite\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nk : ℕ\nh : G.IsRegularOfDegree k\nv : V\n⊢ Gᶜ.degree v = Fintype.card V - 1 - k"}
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+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "ι' : Sort u_5", "l l' : Filter α", "p : ι → Prop", "s : ι → Set α", "t : Set α", "i : ι", "p' : ι' → Prop", "s' : ι' → Set α", "i' : ι'", "f g : Filter α"], "goal": "(g ⊓ f).NeBot ↔ ∀ {p : α → Prop}, (∀ᶠ (x : α) in g, p x) → ∃ᶠ (x : α) in f, p x"}], "premise": [12594], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nι' : Sort u_5\nl l' : Filter α\np : ι → Prop\ns : ι → Set α\nt : Set α\ni : ι\np' : ι' → Prop\ns' : ι' → Set α\ni' : ι'\nf g : Filter α\n⊢ (g ⊓ f).NeBot ↔ ∀ {p : α → Prop}, (∀ᶠ (x : α) in g, p x) → ∃ᶠ (x : α) in f, p x"}
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+{"state": [{"context": ["a : UnitAddCircle", "r v : ℝ", "hv : 0 < v", "hv' : oddKernel a =O[atTop] fun x => rexp (-v * x)"], "goal": "(fun x => (ofReal' ∘ oddKernel a) x - 0) =O[atTop] fun x => x ^ r"}], "premise": [43455], "state_str": "a : UnitAddCircle\nr v : ℝ\nhv : 0 < v\nhv' : oddKernel a =O[atTop] fun x => rexp (-v * x)\n⊢ (fun x => (ofReal' ∘ oddKernel a) x - 0) =O[atTop] fun x => x ^ r"}
+{"state": [{"context": ["a : UnitAddCircle", "r v : ℝ", "hv : 0 < v", "hv' : (fun x => ‖oddKernel a x‖) =O[atTop] fun x => rexp (-v * x)"], "goal": "(fun x => ‖(ofReal' ∘ oddKernel a) x - 0‖) =O[atTop] fun x => x ^ r"}], "premise": [1671, 46181, 117816], "state_str": "a : UnitAddCircle\nr v : ℝ\nhv : 0 < v\nhv' : (fun x => ‖oddKernel a x‖) =O[atTop] fun x => rexp (-v * x)\n⊢ (fun x => ‖(ofReal' ∘ oddKernel a) x - 0‖) =O[atTop] fun x => x ^ r"}
+{"state": [{"context": ["a : UnitAddCircle", "r v : ℝ", "hv : 0 < v", "hv' : (fun x => ‖oddKernel a x‖) =O[atTop] fun x => rexp (-v * x)"], "goal": "(fun x => ‖oddKernel a x‖) =O[atTop] fun x => x ^ r"}], "premise": [39208, 43365, 43411], "state_str": "a : UnitAddCircle\nr v : ℝ\nhv : 0 < v\nhv' : (fun x => ‖oddKernel a x‖) =O[atTop] fun x => rexp (-v * x)\n⊢ (fun x => ‖oddKernel a x‖) =O[atTop] fun x => x ^ r"}
+{"state": [{"context": ["a : UnitAddCircle", "r : ℝ"], "goal": "(fun x => (ofReal' ∘ sinKernel a) x - 0) =O[atTop] fun x => x ^ r"}], "premise": [22987], "state_str": "a : UnitAddCircle\nr : ℝ\n⊢ (fun x => (ofReal' ∘ sinKernel a) x - 0) =O[atTop] fun x => x ^ r"}
+{"state": [{"context": ["a : UnitAddCircle", "r v : ℝ", "hv : 0 < v", "hv' : sinKernel a =O[atTop] fun x => rexp (-v * x)"], "goal": "(fun x => (ofReal' ∘ sinKernel a) x - 0) =O[atTop] fun x => x ^ r"}], "premise": [43455], "state_str": "a : UnitAddCircle\nr v : ℝ\nhv : 0 < v\nhv' : sinKernel a =O[atTop] fun x => rexp (-v * x)\n⊢ (fun x => (ofReal' ∘ sinKernel a) x - 0) =O[atTop] fun x => x ^ r"}
+{"state": [{"context": ["a : UnitAddCircle", "r v : ℝ", "hv : 0 < v", "hv' : (fun x => ‖sinKernel a x‖) =O[atTop] fun x => rexp (-v * x)"], "goal": "(fun x => ‖(ofReal' ∘ sinKernel a) x - 0‖) =O[atTop] fun x => x ^ r"}], "premise": [1671, 46181, 117816], "state_str": "a : UnitAddCircle\nr v : ℝ\nhv : 0 < v\nhv' : (fun x => ‖sinKernel a x‖) =O[atTop] fun x => rexp (-v * x)\n⊢ (fun x => ‖(ofReal' ∘ sinKernel a) x - 0‖) =O[atTop] fun x => x ^ r"}
+{"state": [{"context": ["a : UnitAddCircle", "r v : ℝ", "hv : 0 < v", "hv' : (fun x => ‖sinKernel a x‖) =O[atTop] fun x => rexp (-v * x)"], "goal": "(fun x => ‖sinKernel a x‖) =O[atTop] fun x => x ^ r"}], "premise": [39208, 43365, 43411], "state_str": "a : UnitAddCircle\nr v : ℝ\nhv : 0 < v\nhv' : (fun x => ‖sinKernel a x‖) =O[atTop] fun x => rexp (-v * x)\n⊢ (fun x => ‖sinKernel a x‖) =O[atTop] fun x => x ^ r"}
+{"state": [{"context": ["a : UnitAddCircle", "x : ℝ", "hx : x ∈ Ioi 0"], "goal": "(ofReal' ∘ oddKernel a) (1 / x) = (1 * ↑(x ^ (3 / 2))) • (ofReal' ∘ sinKernel a) x"}], "premise": [1670, 14286, 22982, 40068, 117810, 118863, 119728, 119770, 148304], "state_str": "a : UnitAddCircle\nx : ℝ\nhx : x ∈ Ioi 0\n⊢ (ofReal' ∘ oddKernel a) (1 / x) = (1 * ↑(x ^ (3 / 2))) • (ofReal' ∘ sinKernel a) x"}
+{"state": [{"context": ["α : Type u_1", "inst✝¹ : DistribLattice α", "inst✝ : BoundedOrder α", "a b : Complementeds α"], "goal": "Codisjoint ↑a ↑b ↔ Codisjoint a b"}], "premise": [1713, 13520, 13588, 13592, 13595], "state_str": "α : Type u_1\ninst✝¹ : DistribLattice α\ninst✝ : BoundedOrder α\na b : Complementeds α\n⊢ Codisjoint ↑a ↑b ↔ Codisjoint a b"}
+{"state": [{"context": ["R : Type u_1", "L : Type u_2", "M : Type u_3", "n : Type u_4", "ι : Type u_5", "ι' : Type u_6", "ιM : Type u_7", "inst✝¹² : CommRing R", "inst✝¹¹ : AddCommGroup L", "inst✝¹⁰ : Module R L", "inst✝⁹ : AddCommGroup M", "inst✝⁸ : Module R M", "φ : L →ₗ[R] Module.End R M", "inst✝⁷ : Fintype ι", "inst✝⁶ : Fintype ι'", "inst✝⁵ : Fintype ιM", "inst✝⁴ : DecidableEq ι", "inst✝³ : DecidableEq ι'", "inst✝² : DecidableEq ιM", "b : Basis ι R L", "bₘ : Basis ιM R M", "A : Type u_8", "inst✝¹ : CommRing A", "inst✝ : Algebra R A"], "goal": "Polynomial.map (↑(bind₁ (toMvPolynomial (basis A b) (basis A bₘ).end (tensorProduct R A M M ∘ₗ baseChange A φ)))) (charpoly.univ A ιM) = Polynomial.map (MvPolynomial.map (algebraMap R A)) (Polynomial.map (↑(bind₁ (toMvPolynomial b bₘ.end φ))) (charpoly.univ R ιM))"}], "premise": [84134], "state_str": "R : Type u_1\nL : Type u_2\nM : Type u_3\nn : Type u_4\nι : Type u_5\nι' : Type u_6\nιM : Type u_7\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup L\ninst✝¹⁰ : Module R L\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\nφ : L →ₗ[R] Module.End R M\ninst✝⁷ : Fintype ι\ninst✝⁶ : Fintype ι'\ninst✝⁵ : Fintype ιM\ninst✝⁴ : DecidableEq ι\ninst✝³ : DecidableEq ι'\ninst✝² : DecidableEq ιM\nb : Basis ι R L\nbₘ : Basis ιM R M\nA : Type u_8\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\n⊢ Polynomial.map (↑(bind₁ (toMvPolynomial (basis A b) (basis A bₘ).end (tensorProduct R A M M ∘ₗ baseChange A φ))))\n      (charpoly.univ A ιM) =\n    Polynomial.map (MvPolynomial.map (algebraMap R A))\n      (Polynomial.map (↑(bind₁ (toMvPolynomial b bₘ.end φ))) (charpoly.univ R ιM))"}
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+{"state": [{"context": ["a b : ℤ", "n : ℕ", "m : ℤ", "hm : m ≠ 0", "d : ℤ", "h2 : |m * d| < m"], "goal": "|m * d| < |m|"}], "premise": [14287, 105272], "state_str": "case inr.intro.h\na b : ℤ\nn : ℕ\nm : ℤ\nhm : m ≠ 0\nd : ℤ\nh2 : |m * d| < m\n⊢ |m * d| < |m|"}
+{"state": [{"context": ["G : Type u_1", "inst✝⁷ : MeasurableSpace G", "inst✝⁶ : Group G", "inst✝⁵ : MeasurableMul₂ G", "μ ν : Measure G", "inst✝⁴ : SFinite ν", "inst✝³ : SFinite μ", "s : Set G", "inst✝² : MeasurableInv G", "inst✝¹ : μ.IsMulLeftInvariant", "inst✝ : ν.IsMulLeftInvariant", "f : G → G → ℝ≥0∞", "hf : AEMeasurable (uncurry f) (μ.prod ν)"], "goal": "∫⁻ (x : G), ∫⁻ (y : G), f (y * x) x⁻¹ ∂ν ∂μ = ∫⁻ (x : G), ∫⁻ (y : G), f x y ∂ν ∂μ"}], "premise": [28856, 28857, 28861, 31188, 31233], "state_str": "G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SFinite ν\ninst✝³ : SFinite μ\ns : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : μ.IsMulLeftInvariant\ninst✝ : ν.IsMulLeftInvariant\nf : G → G → ℝ≥0∞\nhf : AEMeasurable (uncurry f) (μ.prod ν)\n⊢ ∫⁻ (x : G), ∫⁻ (y : G), f (y * x) x⁻¹ ∂ν ∂μ = ∫⁻ (x : G), ∫⁻ (y : G), f x y ∂ν ∂μ"}
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+{"state": [{"context": ["α : Type u_1", "inst✝² : PartialOrder α", "inst✝¹ : SuccOrder α", "a b : α", "C : α → Sort u_2", "inst✝ : WellFoundedLT α", "H_succ : (a : α) → ¬IsMax a → C a → C (succ a)", "H_lim : (a : α) → IsSuccLimit a → ((b : α) → b < a → C b) → C a", "ha : ¬IsMax a", "h : ¬IsSuccLimit (succ a)", "this : ∀ {b c : α} {hb : ¬IsMax b} {hc : ¬IsMax c} {x : (a : α) → C a} (h : b = c), ⋯ ▸ H_succ b hb (x b) = H_succ c hc (x c)", "x : { x // ¬IsMax x ∧ succ x = succ a } := Classical.indefiniteDescription (fun x => ¬IsMax x ∧ succ x = succ a) ⋯"], "goal": "(let x := Classical.indefiniteDescription (fun x => ¬IsMax x ∧ succ x = succ a) ⋯; ⋯ ▸ H_succ ↑x ⋯ ((fun y x => ⋯.fix (fun a IH => if h : IsSuccLimit a then H_lim a h IH else let x := Classical.indefiniteDescription (fun x => ¬IsMax x ∧ succ x = a) ⋯; ⋯ ▸ H_succ ↑x ⋯ (IH ↑x ⋯)) y) ↑x ⋯)) = H_succ a ha (SuccOrder.limitRecOn a H_succ H_lim)"}], "premise": [1673, 2106, 2107, 2115, 17402], "state_str": "α : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\na b : α\nC : α → Sort u_2\ninst✝ : WellFoundedLT α\nH_succ : (a : α) → ¬IsMax a → C a → C (succ a)\nH_lim : (a : α) → IsSuccLimit a → ((b : α) → b < a → C b) → C a\nha : ¬IsMax a\nh : ¬IsSuccLimit (succ a)\nthis :\n  ∀ {b c : α} {hb : ¬IsMax b} {hc : ¬IsMax c} {x : (a : α) → C a} (h : b = c), ⋯ ▸ H_succ b hb (x b) = H_succ c hc (x c)\nx : { x // ¬IsMax x ∧ succ x = succ a } := Classical.indefiniteDescription (fun x => ¬IsMax x ∧ succ x = succ a) ⋯\n⊢ (let x := Classical.indefiniteDescription (fun x => ¬IsMax x ∧ succ x = succ a) ⋯;\n    ⋯ ▸\n      H_succ ↑x ⋯\n        ((fun y x =>\n            ⋯.fix\n              (fun a IH =>\n                if h : IsSuccLimit a then H_lim a h IH\n                else\n                  let x := Classical.indefiniteDescription (fun x => ¬IsMax x ∧ succ x = a) ⋯;\n                  ⋯ ▸ H_succ ↑x ⋯ (IH ↑x ⋯))\n              y)\n          ↑x ⋯)) =\n    H_succ a ha (SuccOrder.limitRecOn a H_succ H_lim)"}
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+{"state": [{"context": ["R : Type u", "inst✝¹ : CommRing R", "W' : Jacobian R", "F : Type v", "inst✝ : Field F", "W : Jacobian F", "P Q : Fin 3 → F", "hP : W.Equation P", "hQ : W.Equation Q", "hPz : P z ≠ 0", "hQz : Q z ≠ 0", "hx : P x * Q z ^ 2 = Q x * P z ^ 2", "hy : P y * Q z ^ 3 ≠ W.negY Q * P z ^ 3"], "goal": "W.negDblY P / W.dblZ P ^ 3 = W.toAffine.negAddY (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (W.toAffine.slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3))"}], "premise": [1673, 117831, 145460, 145505, 145520, 145529], "state_str": "R : Type u\ninst✝¹ : CommRing R\nW' : Jacobian R\nF : Type v\ninst✝ : Field F\nW : Jacobian F\nP Q : Fin 3 → F\nhP : W.Equation P\nhQ : W.Equation Q\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhx : P x * Q z ^ 2 = Q x * P z ^ 2\nhy : P y * Q z ^ 3 ≠ W.negY Q * P z ^ 3\n⊢ W.negDblY P / W.dblZ P ^ 3 =\n    W.toAffine.negAddY (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3)\n      (W.toAffine.slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3))"}
+{"state": [{"context": ["ι : Sort u_1", "α : Type u_2", "inst✝² : CompleteLattice α", "f : ι → α", "inst✝¹ : CompleteLattice α", "inst✝ : IsCompactlyGenerated α", "a b✝ : α", "s : Set α", "b : α"], "goal": "sSup {c | CompleteLattice.IsCompactElement c ∧ c ≤ b} = b"}], "premise": [17150], "state_str": "ι : Sort u_1\nα : Type u_2\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b✝ : α\ns : Set α\nb : α\n⊢ sSup {c | CompleteLattice.IsCompactElement c ∧ c ≤ b} = b"}
+{"state": [{"context": ["ι : Sort u_1", "α : Type u_2", "inst✝² : CompleteLattice α", "f : ι → α", "inst✝¹ : CompleteLattice α", "inst✝ : IsCompactlyGenerated α", "a b : α", "s✝ s : Set α", "hs : ∀ x ∈ s, CompleteLattice.IsCompactElement x"], "goal": "sSup {c | CompleteLattice.IsCompactElement c ∧ c ≤ sSup s} = sSup s"}], "premise": [2106, 14296, 19202, 19203, 19207], "state_str": "case intro.intro\nι : Sort u_1\nα : Type u_2\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns✝ s : Set α\nhs : ∀ x ∈ s, CompleteLattice.IsCompactElement x\n⊢ sSup {c | CompleteLattice.IsCompactElement c ∧ c ≤ sSup s} = sSup s"}
+{"state": [{"context": ["k : Type u_1", "E : Type u_2", "PE : Type u_3", "inst✝³ : OrderedRing k", "inst✝² : OrderedAddCommGroup E", "inst✝¹ : Module k E", "inst✝ : OrderedSMul k E", "a a' b b' : E", "r r' : k", "hb : b ≤ b'", "hr : 0 ≤ r"], "goal": "(lineMap a b) r ≤ (lineMap a b') r"}], "premise": [84327], "state_str": "k : Type u_1\nE : Type u_2\nPE : Type u_3\ninst✝³ : OrderedRing k\ninst✝² : OrderedAddCommGroup E\ninst✝¹ : Module k E\ninst✝ : OrderedSMul k E\na a' b b' : E\nr r' : k\nhb : b ≤ b'\nhr : 0 ≤ r\n⊢ (lineMap a b) r ≤ (lineMap a b') r"}
+{"state": [{"context": ["k : Type u_1", "E : Type u_2", "PE : Type u_3", "inst✝³ : OrderedRing k", "inst✝² : OrderedAddCommGroup E", "inst✝¹ : Module k E", "inst✝ : OrderedSMul k E", "a a' b b' : E", "r r' : k", "hb : b ≤ b'", "hr : 0 ≤ r"], "goal": "(1 - r) • a + r • b ≤ (1 - r) • a + r • b'"}], "premise": [103882, 104871], "state_str": "k : Type u_1\nE : Type u_2\nPE : Type u_3\ninst✝³ : OrderedRing k\ninst✝² : OrderedAddCommGroup E\ninst✝¹ : Module k E\ninst✝ : OrderedSMul k E\na a' b b' : E\nr r' : k\nhb : b ≤ b'\nhr : 0 ≤ r\n⊢ (1 - r) • a + r • b ≤ (1 - r) • a + r • b'"}
+{"state": [{"context": ["ι : Sort u_1", "α : Type u", "β : Type v", "inst✝¹ : PseudoMetricSpace α", "inst✝ : PseudoMetricSpace β", "s t u : Set α", "x y : α", "Φ : α → β"], "goal": "infEdist x s = ⊤ ↔ s = ∅"}], "premise": [61811, 133383], "state_str": "ι : Sort u_1\nα : Type u\nβ : Type v\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\ns t u : Set α\nx y : α\nΦ : α → β\n⊢ infEdist x s = ⊤ ↔ s = ∅"}
+{"state": [{"context": ["R : Type u_1", "p n : ℕ", "hp : Fact (Nat.Prime p)", "inst✝¹ : Ring R", "inst✝ : CharP R p", "hn : p ∣ n", "this : Algebra (ZMod p) R := ZMod.algebra R p"], "goal": "cyclotomic (n * p) R = cyclotomic n R ^ p"}], "premise": [75181, 102948], "state_str": "R : Type u_1\np n : ℕ\nhp : Fact (Nat.Prime p)\ninst✝¹ : Ring R\ninst✝ : CharP R p\nhn : p ∣ n\nthis : Algebra (ZMod p) R := ZMod.algebra R p\n⊢ cyclotomic (n * p) R = cyclotomic n R ^ p"}
+{"state": [{"context": ["R : Type u_1", "p n : ℕ", "hp : Fact (Nat.Prime p)", "inst✝¹ : Ring R", "inst✝ : CharP R p", "hn : p ∣ n", "this : Algebra (ZMod p) R := ZMod.algebra R p"], "goal": "cyclotomic (n * p) (ZMod p) = cyclotomic n (ZMod p) ^ p"}], "premise": [70028, 74793, 75178, 75181, 88917, 101869], "state_str": "R : Type u_1\np n : ℕ\nhp : Fact (Nat.Prime p)\ninst✝¹ : Ring R\ninst✝ : CharP R p\nhn : p ∣ n\nthis : Algebra (ZMod p) R := ZMod.algebra R p\n⊢ cyclotomic (n * p) (ZMod p) = cyclotomic n (ZMod p) ^ p"}
+{"state": [{"context": ["u : Level", "R : Type u_1", "inst✝¹ : Ring R", "p : ℕ", "inst✝ : CharP R p", "k : ℤ"], "goal": "↑k = ↑(k % ↑p)"}], "premise": [4236, 124624], "state_str": "u : Level\nR : Type u_1\ninst✝¹ : Ring R\np : ℕ\ninst✝ : CharP R p\nk : ℤ\n⊢ ↑k = ↑(k % ↑p)"}
+{"state": [{"context": ["α : Type u_1", "inst✝¹ : LinearOrder α", "E : Type u_2", "inst✝ : PseudoEMetricSpace E", "f : α → E", "s t : Set α", "h : ∀ x ∈ s, ∀ y ∈ t, x ≤ y", "hs : ¬s = ∅"], "goal": "eVariationOn f s + eVariationOn f t ≤ eVariationOn f (s ∪ t)"}], "premise": [1674, 42098, 133378], "state_str": "case neg\nα : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns t : Set α\nh : ∀ x ∈ s, ∀ y ∈ t, x ≤ y\nhs : ¬s = ∅\n⊢ eVariationOn f s + eVariationOn f t ≤ eVariationOn f (s ∪ t)"}
+{"state": [{"context": ["α : Type u_1", "inst✝¹ : LinearOrder α", "E : Type u_2", "inst✝ : PseudoEMetricSpace E", "f : α → E", "s t : Set α", "h : ∀ x ∈ s, ∀ y ∈ t, x ≤ y", "hs : ¬s = ∅", "this : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }", "ht : ¬t = ∅"], "goal": "eVariationOn f s + eVariationOn f t ≤ eVariationOn f (s ∪ t)"}], "premise": [1674, 42098, 133378], "state_str": "case neg\nα : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns t : Set α\nh : ∀ x ∈ s, ∀ y ∈ t, x ≤ y\nhs : ¬s = ∅\nthis : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }\nht : ¬t = ∅\n⊢ eVariationOn f s + eVariationOn f t ≤ eVariationOn f (s ∪ t)"}
+{"state": [{"context": ["α : Type u_1", "inst���¹ : LinearOrder α", "E : Type u_2", "inst✝ : PseudoEMetricSpace E", "f : α → E", "s t : Set α", "h : ∀ x ∈ s, ∀ y ∈ t, x ≤ y", "hs : ¬s = ∅", "this✝ : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }", "ht : ¬t = ∅", "this : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ t }"], "goal": "eVariationOn f s + eVariationOn f t ≤ eVariationOn f (s ∪ t)"}], "premise": [58965], "state_str": "case neg\nα : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns t : Set α\nh : ∀ x ∈ s, ∀ y ∈ t, x ≤ y\nhs : ¬s = ∅\nthis✝ : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }\nht : ¬t = ∅\nthis : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ t }\n⊢ eVariationOn f s + eVariationOn f t ≤ eVariationOn f (s ∪ t)"}
+{"state": [{"context": ["α : Type u_1", "inst✝³ : Lattice α", "inst✝² : IsModularLattice α", "x y z : α", "β : Type u_2", "γ : Type u_3", "inst✝¹ : PartialOrder β", "inst✝ : Preorder γ", "h₁ : WellFounded fun x x_1 => x < x_1", "h₂ : WellFounded fun x x_1 => x < x_1", "K : α", "f₁ : β → α", "f₂ : α → β", "g₁ : γ → α", "g₂ : α → γ", "gci : GaloisCoinsertion f₁ f₂", "gi : GaloisInsertion g₂ g₁", "hf : ∀ (a : α), f₁ (f₂ a) = a ⊓ K", "hg : ∀ (a : α), g₁ (g₂ a) = a ⊔ K", "A B : α", "hAB : A < B"], "goal": "Prod.Lex (fun x x_1 => x < x_1) (fun x x_1 => x < x_1) (f₂ A, g₂ A) (f₂ B, g₂ B)"}], "premise": [968, 12842, 12863, 14274, 14297], "state_str": "α : Type u_1\ninst✝³ : Lattice α\ninst✝² : IsModularLattice α\nx y z : α\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : PartialOrder β\ninst✝ : Preorder γ\nh₁ : WellFounded fun x x_1 => x < x_1\nh₂ : WellFounded fun x x_1 => x < x_1\nK : α\nf₁ : β → α\nf₂ : α → β\ng₁ : γ → α\ng₂ : α → γ\ngci : GaloisCoinsertion f₁ f₂\ngi : GaloisInsertion g₂ g₁\nhf : ∀ (a : α), f₁ (f₂ a) = a ⊓ K\nhg : ∀ (a : α), g₁ (g₂ a) = a ⊔ K\nA B : α\nhAB : A < B\n⊢ Prod.Lex (fun x x_1 => x < x_1) (fun x x_1 => x < x_1) (f₂ A, g₂ A) (f₂ B, g₂ B)"}
+{"state": [{"context": ["α : Type u_1", "inst✝³ : Lattice α", "inst✝² : IsModularLattice α", "x y z : α", "β : Type u_2", "γ : Type u_3", "inst✝¹ : PartialOrder β", "inst✝ : Preorder γ", "h₁ : WellFounded fun x x_1 => x < x_1", "h₂ : WellFounded fun x x_1 => x < x_1", "K : α", "f₁ : β → α", "f₂ : α → β", "g₁ : γ → α", "g₂ : α → γ", "gci : GaloisCoinsertion f₁ f₂", "gi : GaloisInsertion g₂ g₁", "hf : ∀ (a : α), f₁ (f₂ a) = a ⊓ K", "hg : ∀ (a : α), g₁ (g₂ a) = a ⊔ K", "A B : α", "hAB : A < B"], "goal": "A ⊓ K ≤ B ⊓ K ∧ ¬B ⊓ K ≤ A ⊓ K ∨ (A ⊓ K ≤ B ⊓ K ∧ B ⊓ K ≤ A ⊓ K) ∧ A ⊔ K ≤ B ⊔ K ∧ ¬B ⊔ K ≤ A ⊔ K"}], "premise": [12842, 12863, 14274, 14297], "state_str": "α : Type u_1\ninst✝³ : Lattice α\ninst✝² : IsModularLattice α\nx y z : α\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : PartialOrder β\ninst✝ : Preorder γ\nh₁ : WellFounded fun x x_1 => x < x_1\nh₂ : WellFounded fun x x_1 => x < x_1\nK : α\nf₁ : β → α\nf₂ : α → β\ng₁ : γ → α\ng₂ : α → γ\ngci : GaloisCoinsertion f₁ f₂\ngi : GaloisInsertion g₂ g₁\nhf : ∀ (a : α), f₁ (f₂ a) = a ⊓ K\nhg : ∀ (a : α), g₁ (g₂ a) = a ⊔ K\nA B : α\nhAB : A < B\n⊢ A ⊓ K ≤ B ⊓ K ∧ ¬B ⊓ K ≤ A ⊓ K ∨ (A ⊓ K ≤ B ⊓ K ∧ B ⊓ K ≤ A ⊓ K) ∧ A ⊔ K ≤ B ⊔ K ∧ ¬B ⊔ K ≤ A ⊔ K"}
+{"state": [{"context": ["α : Type u_1", "inst✝³ : Lattice α", "inst✝² : IsModularLattice α", "x y z : α", "β : Type u_2", "γ : Type u_3", "inst✝¹ : PartialOrder β", "inst✝ : Preorder γ", "h₁ : WellFounded fun x x_1 => x < x_1", "h₂ : WellFounded fun x x_1 => x < x_1", "K : α", "f₁ : β → α", "f₂ : α → β", "g₁ : γ → α", "g₂ : α → γ", "gci : GaloisCoinsertion f₁ f₂", "gi : GaloisInsertion g₂ g₁", "hf : ∀ (a : α), f₁ (f₂ a) = a ⊓ K", "hg : ∀ (a : α), g₁ (g₂ a) = a ⊔ K", "A B : α", "hAB : A < B"], "goal": "A ⊓ K < B ⊓ K ∨ A ⊓ K = B ⊓ K ∧ A ⊔ K < B ⊔ K"}], "premise": [14286, 14302, 14576], "state_str": "α : Type u_1\ninst✝³ : Lattice α\ninst✝² : IsModularLattice α\nx y z : α\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : PartialOrder β\ninst✝ : Preorder γ\nh₁ : WellFounded fun x x_1 => x < x_1\nh₂ : WellFounded fun x x_1 => x < x_1\nK : α\nf₁ : β → α\nf₂ : α → β\ng₁ : γ → α\ng₂ : α → γ\ngci : GaloisCoinsertion f₁ f₂\ngi : GaloisInsertion g₂ g₁\nhf : ∀ (a : α), f₁ (f₂ a) = a ⊓ K\nhg : ∀ (a : α), g₁ (g₂ a) = a ⊔ K\nA B : α\nhAB : A < B\n⊢ A ⊓ K < B ⊓ K ∨ A ⊓ K = B ⊓ K ∧ A ⊔ K < B ⊔ K"}
+{"state": [{"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "X✝ : C", "inst✝ : HasPushouts C", "X Y : C", "f : X ⟶ Y", "x : Under X", "y : Under Y", "u : (pushout f).obj x ⟶ y"], "goal": "x.hom ≫ pushout.inl x.hom f ≫ u.right = ((map f).obj y).hom"}], "premise": [96772], "state_str": "C : Type u\ninst✝¹ : Category.{v, u} C\nX✝ : C\ninst✝ : HasPushouts C\nX Y : C\nf : X ⟶ Y\nx : Under X\ny : Under Y\nu : (pushout f).obj x ⟶ y\n⊢ x.hom ≫ pushout.inl x.hom f ≫ u.right = ((map f).obj y).hom"}
+{"state": [{"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "X✝ : C", "inst✝ : HasPushouts C", "X Y : C", "f : X ⟶ Y", "x : Under X", "y : Under Y", "u : (pushout f).obj x ⟶ y"], "goal": "x.hom ≫ pushout.inl x.hom f ≫ u.right = f ≫ y.hom"}], "premise": [96760], "state_str": "C : Type u\ninst✝¹ : Category.{v, u} C\nX✝ : C\ninst✝ : HasPushouts C\nX Y : C\nf : X ⟶ Y\nx : Under X\ny : Under Y\nu : (pushout f).obj x ⟶ y\n⊢ x.hom ≫ pushout.inl x.hom f ≫ u.right = f ≫ y.hom"}
+{"state": [{"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "X✝ : C", "inst✝ : HasPushouts C", "X Y : C", "f : X ⟶ Y", "x : Under X", "y : Under Y", "u : (pushout f).obj x ⟶ y"], "goal": "x.hom ≫ pushout.inl x.hom f ≫ u.right = f ≫ ((pushout f).obj x).hom ≫ u.right"}], "premise": [95896, 96761, 96762, 96771, 99922, 99936], "state_str": "C : Type u\ninst✝¹ : Category.{v, u} C\nX✝ : C\ninst✝ : HasPushouts C\nX Y : C\nf : X ⟶ Y\nx : Under X\ny : Under Y\nu : (pushout f).obj x ⟶ y\n⊢ x.hom ≫ pushout.inl x.hom f ≫ u.right = f ≫ ((pushout f).obj x).hom ≫ u.right"}
+{"state": [{"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "X✝ : C", "inst✝ : HasPushouts C", "X Y : C", "f : X ⟶ Y", "x : Under X", "y : Under Y", "u : (pushout f).obj x ⟶ y"], "goal": "x.hom ≫ pushout.inl x.hom f ≫ u.right = f ≫ pushout.inr x.hom f ≫ u.right"}], "premise": [93875, 96173], "state_str": "C : Type u\ninst✝¹ : Category.{v, u} C\nX✝ : C\ninst✝ : HasPushouts C\nX Y : C\nf : X ⟶ Y\nx : Under X\ny : Under Y\nu : (pushout f).obj x ⟶ y\n⊢ x.hom ≫ pushout.inl x.hom f ≫ u.right = f ≫ pushout.inr x.hom f ≫ u.right"}
+{"state": [{"context": ["R : Type u_1", "M : Type u_2", "inst✝² : CommRing R", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M", "Q : QuadraticForm R M", "motive : (x : CliffordAlgebra Q) → x ∈ evenOdd Q 0 → Prop", "algebraMap : ∀ (r : R), motive ((_root_.algebraMap R (CliffordAlgebra Q)) r) ⋯", "add : ∀ (x y : CliffordAlgebra Q) (hx : x ∈ evenOdd Q 0) (hy : y ∈ evenOdd Q 0), motive x hx → motive y hy → motive (x + y) ⋯", "ι_mul_ι_mul : ∀ (m₁ m₂ : M) (x : CliffordAlgebra Q) (hx : x ∈ evenOdd Q 0), motive x hx → motive ((ι Q) m₁ * (ι Q) m₂ * x) ⋯", "x : CliffordAlgebra Q", "hx : x ∈ evenOdd Q 0"], "goal": "motive x hx"}], "premise": [82278], "state_str": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nmotive : (x : CliffordAlgebra Q) → x ∈ evenOdd Q 0 → Prop\nalgebraMap : ∀ (r : R), motive ((_root_.algebraMap R (CliffordAlgebra Q)) r) ⋯\nadd :\n  ∀ (x y : CliffordAlgebra Q) (hx : x ∈ evenOdd Q 0) (hy : y ∈ evenOdd Q 0),\n    motive x hx → motive y hy → motive (x + y) ⋯\nι_mul_ι_mul :\n  ∀ (m₁ m₂ : M) (x : CliffordAlgebra Q) (hx : x ∈ evenOdd Q 0), motive x hx → motive ((ι Q) m₁ * (ι Q) m₂ * x) ⋯\nx : CliffordAlgebra Q\nhx : x ∈ evenOdd Q 0\n⊢ motive x hx"}
+{"state": [{"context": ["G : Type u", "inst✝ : Monoid G"], "goal": "exponent G = 1 ↔ Subsingleton G"}], "premise": [14296], "state_str": "G : Type u\ninst✝ : Monoid G\n⊢ exponent G = 1 ↔ Subsingleton G"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "G : Type u_3", "M : Type u_4", "inst✝ : Group G", "a✝ b✝ c d : G", "n : ℤ", "a b : G"], "goal": "a / (b * a) = b⁻¹"}], "premise": [117836, 117982], "state_str": "α : Type u_1\nβ : Type u_2\nG : Type u_3\nM : Type u_4\ninst✝ : Group G\na✝ b✝ c d : G\nn : ℤ\na b : G\n⊢ a / (b * a) = b⁻¹"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "G : Type u_3", "M : Type u_4", "inst✝ : DivisionCommMonoid α", "a b c d : α", "n : ℕ"], "goal": "(a * b) ^ ↑n = a ^ ↑n * b ^ ↑n"}], "premise": [117764, 119784], "state_str": "α : Type u_1\nβ : Type u_2\nG : Type u_3\nM : Type u_4\ninst✝ : DivisionCommMonoid α\na b c d : α\nn : ℕ\n⊢ (a * b) ^ ↑n = a ^ ↑n * b ^ ↑n"}
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+{"state": [{"context": ["𝕜 : Type u_1", "E : Type u_2", "inst✝³ : NormedField 𝕜", "inst✝² : SeminormedAddCommGroup E", "inst✝¹ : NormedSpace 𝕜 E", "inst✝ : NormedSpace ℝ E", "x✝ y z : E", "δ ε : ℝ", "hε : 0 ≤ ε", "hδ : 0 < δ", "s : Set E", "x : E"], "goal": "infEdist x s ≤ ENNReal.ofReal ε + ENNReal.ofReal δ → infEdist x s - ENNReal.ofReal δ ≤ ENNReal.ofReal ε"}], "premise": [1674, 103356], "state_str": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : NormedField 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace ℝ E\nx✝ y z : E\nδ ε : ℝ\nhε : 0 ≤ ε\nhδ : 0 < δ\ns : Set E\nx : E\n⊢ infEdist x s ≤ ENNReal.ofReal ε + ENNReal.ofReal δ → infEdist x s - ENNReal.ofReal δ ≤ ENNReal.ofReal ε"}
+{"state": [{"context": ["F : Type u_1", "inst✝⁴ : Field F", "E : Type u_2", "inst✝³ : Field E", "inst✝² : Algebra F E", "inst✝¹ : FiniteDimensional F E", "inst✝ : IsGalois F E"], "goal": "∃ p, p.Separable ∧ Polynomial.IsSplittingField F E p"}], "premise": [88192], "state_str": "F : Type u_1\ninst✝⁴ : Field F\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : FiniteDimensional F E\ninst✝ : IsGalois F E\n⊢ ∃ p, p.Separable ∧ Polynomial.IsSplittingField F E p"}
+{"state": [{"context": ["F : Type u_1", "inst✝⁴ : Field F", "E : Type u_2", "inst✝³ : Field E", "inst✝² : Algebra F E", "inst✝¹ : FiniteDimensional F E", "inst✝ : IsGalois F E", "α : E", "h1 : F⟮α⟯ = ⊤"], "goal": "∃ p, p.Separable ∧ Polynomial.IsSplittingField F E p"}], "premise": [1674, 2045, 89084, 89085], "state_str": "case intro\nF : Type u_1\ninst✝⁴ : Field F\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : FiniteDimensional F E\ninst✝ : IsGalois F E\nα : E\nh1 : F⟮α⟯ = ⊤\n⊢ ∃ p, p.Separable ∧ Polynomial.IsSplittingField F E p"}
+{"state": [{"context": ["F : Type u_1", "inst✝⁴ : Field F", "E : Type u_2", "inst✝³ : Field E", "inst✝² : Algebra F E", "inst✝¹ : FiniteDimensional F E", "inst✝ : IsGalois F E", "α : E", "h1 : F⟮α⟯ = ⊤"], "goal": "Algebra.adjoin F ((minpoly F α).rootSet E) = ⊤"}], "premise": [18789, 90763], "state_str": "case adjoin_rootSet'\nF : Type u_1\ninst✝⁴ : Field F\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : FiniteDimensional F E\ninst✝ : IsGalois F E\nα : E\nh1 : F⟮α⟯ = ⊤\n⊢ Algebra.adjoin F ((minpoly F α).rootSet E) = ⊤"}
+{"state": [{"context": ["F : Type u_1", "inst✝⁴ : Field F", "E : Type u_2", "inst✝³ : Field E", "inst✝² : Algebra F E", "inst✝¹ : FiniteDimensional F E", "inst✝ : IsGalois F E", "α : E", "h1 : F⟮α⟯ = ⊤"], "goal": "F⟮α⟯.toSubalgebra ≤ Algebra.adjoin F ((minpoly F α).rootSet E)"}], "premise": [89083, 90831], "state_str": "case adjoin_rootSet'\nF : Type u_1\ninst✝⁴ : Field F\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : FiniteDimensional F E\ninst✝ : IsGalois F E\nα : E\nh1 : F⟮α⟯ = ⊤\n⊢ F⟮α⟯.toSubalgebra ≤ Algebra.adjoin F ((minpoly F α).rootSet E)"}
+{"state": [{"context": ["F : Type u_1", "inst✝⁴ : Field F", "E : Type u_2", "inst✝³ : Field E", "inst✝² : Algebra F E", "inst✝¹ : FiniteDimensional F E", "inst✝ : IsGalois F E", "α : E", "h1 : F⟮α⟯ = ⊤"], "goal": "Algebra.adjoin F {α} ≤ Algebra.adjoin F ((minpoly F α).rootSet E)"}], "premise": [78310], "state_str": "case adjoin_rootSet'\nF : Type u_1\ninst✝⁴ : Field F\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : FiniteDimensional F E\ninst✝ : IsGalois F E\nα : E\nh1 : F⟮α⟯ = ⊤\n⊢ Algebra.adjoin F {α} ≤ Algebra.adjoin F ((minpoly F α).rootSet E)"}
+{"state": [{"context": ["F : Type u_1", "inst✝⁴ : Field F", "E : Type u_2", "inst✝³ : Field E", "inst✝² : Algebra F E", "inst✝¹ : FiniteDimensional F E", "inst✝ : IsGalois F E", "α : E", "h1 : F⟮α⟯ = ⊤"], "goal": "{α} ⊆ (minpoly F α).rootSet E"}], "premise": [102593, 133525], "state_str": "case adjoin_rootSet'.H\nF : Type u_1\ninst✝⁴ : Field F\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : FiniteDimensional F E\ninst✝ : IsGalois F E\nα : E\nh1 : F⟮α⟯ = ⊤\n⊢ {α} ⊆ (minpoly F α).rootSet E"}
+{"state": [{"context": ["F : Type u_1", "inst✝⁴ : Field F", "E : Type u_2", "inst✝³ : Field E", "inst✝² : Algebra F E", "inst✝¹ : FiniteDimensional F E", "inst✝ : IsGalois F E", "α : E", "h1 : F⟮α⟯ = ⊤"], "goal": "minpoly F α ≠ 0 ∧ (Polynomial.aeval α) (minpoly F α) = 0"}], "premise": [87690, 87695, 89083], "state_str": "case adjoin_rootSet'.H\nF : Type u_1\ninst✝⁴ : Field F\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : FiniteDimensional F E\ninst✝ : IsGalois F E\nα : E\nh1 : F⟮α⟯ = ⊤\n⊢ minpoly F α ≠ 0 ∧ (Polynomial.aeval α) (minpoly F α) = 0"}
+{"state": [{"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "m m0 : MeasurableSpace α", "p : ℝ≥0∞", "q : ℝ", "μ ν : Measure α", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedAddCommGroup G", "c : F", "hq_pos : 0 < q"], "goal": "eLpNorm' (fun x => c) q μ = ↑‖c‖₊ * μ Set.univ ^ (1 / q)"}], "premise": [30239, 39780], "state_str": "α : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nc : F\nhq_pos : 0 < q\n⊢ eLpNorm' (fun x => c) q μ = ↑‖c‖₊ * μ Set.univ ^ (1 / q)"}
+{"state": [{"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "m m0 : MeasurableSpace α", "p : ℝ≥0∞", "q : ℝ", "μ ν : Measure α", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedAddCommGroup G", "c : F", "hq_pos : 0 < q"], "goal": "(↑‖c‖₊ ^ q) ^ (1 / q) = ↑‖c‖₊"}], "premise": [39770], "state_str": "case e_a\nα : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nc : F\nhq_pos : 0 < q\n⊢ (↑‖c‖₊ ^ q) ^ (1 / q) = ↑‖c‖₊"}
+{"state": [{"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "m m0 : MeasurableSpace α", "p : ℝ≥0∞", "q : ℝ", "μ ν : Measure α", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedAddCommGroup G", "c : F", "hq_pos : 0 < q"], "goal": "↑‖c‖₊ ^ (q * (1 / q)) = ↑‖c‖₊"}], "premise": [39755], "state_str": "case e_a\nα : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nc : F\nhq_pos : 0 < q\n⊢ ↑‖c‖₊ ^ (q * (1 / q)) = ↑‖c‖₊"}
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+{"state": [{"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "inst✝⁷ : NontriviallyNormedField 𝕜", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace 𝕜 E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace 𝕜 F", "inst✝² : NormedAddCommGroup G", "inst✝¹ : NormedSpace 𝕜 G", "inst✝ : CompleteSpace F", "p : FormalMultilinearSeries 𝕜 E F", "x y : E", "r R : ℝ≥0", "h : ↑‖x‖₊ + ↑‖y‖₊ < p.radius", "radius_pos : 0 < p.radius", "x_mem_ball : x ∈ EMetric.ball 0 p.radius", "y_mem_ball : y ∈ EMetric.ball 0 (p.changeOrigin x).radius", "x_add_y_mem_ball : x + y ∈ EMetric.ball 0 p.radius", "f : (k : ℕ) × (l : ℕ) × { s // s.card = l } → F := fun s => ((p.changeOriginSeriesTerm s.fst s.snd.fst ↑s.snd.snd ⋯) fun x_1 => x) fun x => y", "hsf : Summable f", "hf : HasSum f ((p.changeOrigin x).sum y)"], "goal": "(p.changeOrigin x).sum y = p.sum (x + y)"}], "premise": [1673, 63037, 64097], "state_str": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\ninst✝ : CompleteSpace F\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\nh : ↑‖x‖₊ + ↑‖y‖₊ < p.radius\nradius_pos : 0 < p.radius\nx_mem_ball : x ∈ EMetric.ball 0 p.radius\ny_mem_ball : y ∈ EMetric.ball 0 (p.changeOrigin x).radius\nx_add_y_mem_ball : x + y ∈ EMetric.ball 0 p.radius\nf : (k : ℕ) × (l : ℕ) × { s // s.card = l } → F :=\n  fun s => ((p.changeOriginSeriesTerm s.fst s.snd.fst ↑s.snd.snd ⋯) fun x_1 => x) fun x => y\nhsf : Summable f\nhf : HasSum f ((p.changeOrigin x).sum y)\n⊢ (p.changeOrigin x).sum y = p.sum (x + y)"}
+{"state": [{"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "inst✝⁷ : NontriviallyNormedField 𝕜", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace 𝕜 E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace 𝕜 F", "inst✝² : NormedAddCommGroup G", "inst✝¹ : NormedSpace 𝕜 G", "inst✝ : CompleteSpace F", "p : FormalMultilinearSeries 𝕜 E F", "x y : E", "r R : ℝ≥0", "h : ↑‖x‖₊ + ↑‖y‖₊ < p.radius", "radius_pos : 0 < p.radius", "x_mem_ball : x ∈ EMetric.ball 0 p.radius", "y_mem_ball : y ∈ EMetric.ball 0 (p.changeOrigin x).radius", "x_add_y_mem_ball : x + y ∈ EMetric.ball 0 p.radius", "f : (k : ℕ) × (l : ℕ) × { s // s.card = l } → F := fun s => ((p.changeOriginSeriesTerm s.fst s.snd.fst ↑s.snd.snd ⋯) fun x_1 => x) fun x => y", "hsf : Summable f", "hf : HasSum f ((p.changeOrigin x).sum y)"], "goal": "HasSum (f ∘ ⇑changeOriginIndexEquiv.symm) (p.sum (x + y))"}], "premise": [1674, 35850, 63393, 64101], "state_str": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\ninst✝ : CompleteSpace F\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\nh : ↑‖x‖₊ + ↑‖y‖₊ < p.radius\nradius_pos : 0 < p.radius\nx_mem_ball : x ∈ EMetric.ball 0 p.radius\ny_mem_ball : y ∈ EMetric.ball 0 (p.changeOrigin x).radius\nx_add_y_mem_ball : x + y ∈ EMetric.ball 0 p.radius\nf : (k : ℕ) × (l : ℕ) × { s // s.card = l } → F :=\n  fun s => ((p.changeOriginSeriesTerm s.fst s.snd.fst ↑s.snd.snd ⋯) fun x_1 => x) fun x => y\nhsf : Summable f\nhf : HasSum f ((p.changeOrigin x).sum y)\n⊢ HasSum (f ∘ ⇑changeOriginIndexEquiv.symm) (p.sum (x + y))"}
+{"state": [{"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "inst✝⁷ : NontriviallyNormedField 𝕜", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace 𝕜 E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace 𝕜 F", "inst✝² : NormedAddCommGroup G", "inst✝¹ : NormedSpace 𝕜 G", "inst✝ : CompleteSpace F", "p : FormalMultilinearSeries 𝕜 E F", "x y : E", "r R : ℝ≥0", "h : ↑‖x‖₊ + ↑‖y‖₊ < p.radius", "radius_pos : 0 < p.radius", "x_mem_ball : x ∈ EMetric.ball 0 p.radius", "y_mem_ball : y ∈ EMetric.ball 0 (p.changeOrigin x).radius", "x_add_y_mem_ball : x + y ∈ EMetric.ball 0 p.radius", "f : (k : ℕ) × (l : ℕ) × { s // s.card = l } → F := fun s => ((p.changeOriginSeriesTerm s.fst s.snd.fst ↑s.snd.snd ⋯) fun x_1 => x) fun x => y", "hsf : Summable f", "hf : HasSum f ((p.changeOrigin x).sum y)", "n : ℕ"], "goal": "HasSum (fun c => (f ∘ ⇑changeOriginIndexEquiv.symm) ⟨n, c⟩) ((p n) fun x_1 => x + y)"}], "premise": [64577], "state_str": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\ninst✝ : CompleteSpace F\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\nh : ↑‖x‖₊ + ↑‖y‖₊ < p.radius\nradius_pos : 0 < p.radius\nx_mem_ball : x ∈ EMetric.ball 0 p.radius\ny_mem_ball : y ∈ EMetric.ball 0 (p.changeOrigin x).radius\nx_add_y_mem_ball : x + y ∈ EMetric.ball 0 p.radius\nf : (k : ℕ) × (l : ℕ) × { s // s.card = l } → F :=\n  fun s => ((p.changeOriginSeriesTerm s.fst s.snd.fst ↑s.snd.snd ⋯) fun x_1 => x) fun x => y\nhsf : Summable f\nhf : HasSum f ((p.changeOrigin x).sum y)\nn : ℕ\n⊢ HasSum (fun c => (f ∘ ⇑changeOriginIndexEquiv.symm) ⟨n, c⟩) ((p n) fun x_1 => x + y)"}
+{"state": [{"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "inst✝⁷ : NontriviallyNormedField 𝕜", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace 𝕜 E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace 𝕜 F", "inst✝² : NormedAddCommGroup G", "inst✝¹ : NormedSpace 𝕜 G", "inst✝ : CompleteSpace F", "p : FormalMultilinearSeries 𝕜 E F", "x y : E", "r R : ℝ≥0", "h : ↑‖x‖₊ + ↑‖y‖₊ < p.radius", "radius_pos : 0 < p.radius", "x_mem_ball : x ∈ EMetric.ball 0 p.radius", "y_mem_ball : y ∈ EMetric.ball 0 (p.changeOrigin x).radius", "x_add_y_mem_ball : x + y ∈ EMetric.ball 0 p.radius", "f : (k : ℕ) × (l : ℕ) × { s // s.card = l } → F := fun s => ((p.changeOriginSeriesTerm s.fst s.snd.fst ↑s.snd.snd ⋯) fun x_1 => x) fun x => y", "hsf : Summable f", "hf : HasSum f ((p.changeOrigin x).sum y)", "n : ℕ"], "goal": "HasSum (fun c => (f ∘ ⇑changeOriginIndexEquiv.symm) ⟨n, c⟩) (∑ s : Finset (Fin n), (p n) (s.piecewise (fun x_1 => x) fun x => y))"}], "premise": [35940], "state_str": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\ninst✝ : CompleteSpace F\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\nh : ↑‖x‖₊ + ↑‖y‖₊ < p.radius\nradius_pos : 0 < p.radius\nx_mem_ball : x ∈ EMetric.ball 0 p.radius\ny_mem_ball : y ∈ EMetric.ball 0 (p.changeOrigin x).radius\nx_add_y_mem_ball : x + y ∈ EMetric.ball 0 p.radius\nf : (k : ℕ) × (l : ℕ) × { s // s.card = l } → F :=\n  fun s => ((p.changeOriginSeriesTerm s.fst s.snd.fst ↑s.snd.snd ⋯) fun x_1 => x) fun x => y\nhsf : Summable f\nhf : HasSum f ((p.changeOrigin x).sum y)\nn : ℕ\n⊢ HasSum (fun c => (f ∘ ⇑changeOriginIndexEquiv.symm) ⟨n, c⟩)\n    (∑ s : Finset (Fin n), (p n) (s.piecewise (fun x_1 => x) fun x => y))"}
+{"state": [{"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "inst✝⁷ : NontriviallyNormedField 𝕜", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace 𝕜 E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace 𝕜 F", "inst✝² : NormedAddCommGroup G", "inst✝¹ : NormedSpace 𝕜 G", "inst✝ : CompleteSpace F", "p : FormalMultilinearSeries 𝕜 E F", "x y : E", "r R : ℝ≥0", "h : ↑‖x‖₊ + ↑‖y‖₊ < p.radius", "radius_pos : 0 < p.radius", "x_mem_ball : x ∈ EMetric.ball 0 p.radius", "y_mem_ball : y ∈ EMetric.ball 0 (p.changeOrigin x).radius", "x_add_y_mem_ball : x + y ∈ EMetric.ball 0 p.radius", "f : (k : ℕ) × (l : ℕ) × { s // s.card = l } → F := fun s => ((p.changeOriginSeriesTerm s.fst s.snd.fst ↑s.snd.snd ⋯) fun x_1 => x) fun x => y", "hsf : Summable f", "hf : HasSum f ((p.changeOrigin x).sum y)", "n : ℕ"], "goal": "HasSum (fun c => (f ∘ ⇑changeOriginIndexEquiv.symm) ⟨n, c⟩) (∑ x_1 : Finset (Fin n), ((p.changeOriginSeriesTerm (changeOriginIndexEquiv.symm ⟨n, x_1⟩).fst (changeOriginIndexEquiv.symm ⟨n, x_1⟩).snd.fst ↑(changeOriginIndexEquiv.symm ⟨n, x_1⟩).snd.snd ⋯) fun x_2 => x) fun x => y)"}], "premise": [63026], "state_str": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\ninst✝ : CompleteSpace F\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\nh : ↑‖x‖₊ + ↑‖y‖₊ < p.radius\nradius_pos : 0 < p.radius\nx_mem_ball : x ∈ EMetric.ball 0 p.radius\ny_mem_ball : y ∈ EMetric.ball 0 (p.changeOrigin x).radius\nx_add_y_mem_ball : x + y ∈ EMetric.ball 0 p.radius\nf : (k : ℕ) × (l : ℕ) × { s // s.card = l } → F :=\n  fun s => ((p.changeOriginSeriesTerm s.fst s.snd.fst ↑s.snd.snd ⋯) fun x_1 => x) fun x => y\nhsf : Summable f\nhf : HasSum f ((p.changeOrigin x).sum y)\nn : ℕ\n⊢ HasSum (fun c => (f ∘ ⇑changeOriginIndexEquiv.symm) ⟨n, c⟩)\n    (∑ x_1 : Finset (Fin n),\n      ((p.changeOriginSeriesTerm (changeOriginIndexEquiv.symm ⟨n, x_1⟩).fst\n            (changeOriginIndexEquiv.symm ⟨n, x_1⟩).snd.fst ↑(changeOriginIndexEquiv.symm ⟨n, x_1⟩).snd.snd ⋯)\n          fun x_2 => x)\n        fun x => y)"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "ι' : Sort u_5", "ι₂ : Sort u_6", "κ : ι → Sort u_7", "κ₁ : ι → Sort u_8", "κ₂ : ι → Sort u_9", "κ' : ι' → Sort u_10", "c : Set (Set α)"], "goal": "(⋂₀ c).Nonempty ↔ ∃ a, ∀ b ∈ c, a ∈ b"}], "premise": [133378, 135431], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nι' : Sort u_5\nι₂ : Sort u_6\nκ : ι → Sort u_7\nκ₁ : ι → Sort u_8\nκ₂ : ι → Sort u_9\nκ' : ι' → Sort u_10\nc : Set (Set α)\n⊢ (⋂₀ c).Nonempty ↔ ∃ a, ∀ b ∈ c, a ∈ b"}
+{"state": [{"context": ["R : Type u", "S : Type v", "T : Type w", "ι : Type y", "a b : R", "m n : ℕ", "inst✝² : Semiring R", "p✝ q r : R[X]", "inst✝¹ : Semiring S", "f✝ : R →+* S", "x✝ : S", "inst✝ : Semiring T", "f : R →+* S", "x : S", "p : R[ℕ]"], "goal": "eval₂ f x { toFinsupp := p } = (liftNC ↑f ⇑((powersHom S) x)) p"}], "premise": [101221, 102824], "state_str": "R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝² : Semiring R\np✝ q r : R[X]\ninst✝¹ : Semiring S\nf✝ : R →+* S\nx✝ : S\ninst✝ : Semiring T\nf : R →+* S\nx : S\np : R[ℕ]\n⊢ eval₂ f x { toFinsupp := p } = (liftNC ↑f ⇑((powersHom S) x)) p"}
+{"state": [{"context": ["α : Type u", "β : Type v", "γ : Type w", "δ : Type u_1", "ι : Sort x", "l f₁ f₂ : Filter α", "g₁ g₂ : α → Filter β", "hf : f₁ ≤ f₂", "hg : g₁ ≤ᶠ[f₁] g₂"], "goal": "f₁.bind g₁ ≤ f₂.bind g₂"}], "premise": [14273, 16013, 16229], "state_str": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type u_1\nι : Sort x\nl f₁ f₂ : Filter α\ng₁ g₂ : α → Filter β\nhf : f₁ ≤ f₂\nhg : g₁ ≤ᶠ[f₁] g₂\n⊢ f₁.bind g₁ ≤ f₂.bind g₂"}
+{"state": [{"context": ["α : Type u", "β : Type v", "γ : Type w", "δ : Type u_1", "ι : Sort x", "l f₁ f₂ : Filter α", "g₁ g₂ : α → Filter β", "hf : f₁ ≤ f₂", "hg : g₁ ≤ᶠ[f₁] g₂", "s : Set β", "hs : s ∈ (map g₂ f₁).join"], "goal": "s ∈ f₁.bind g₁"}], "premise": [15903, 16166, 16327], "state_str": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type u_1\nι : Sort x\nl f₁ f₂ : Filter α\ng₁ g₂ : α → Filter β\nhf : f₁ ≤ f₂\nhg : g₁ ≤ᶠ[f₁] g₂\ns : Set β\nhs : s ∈ (map g₂ f₁).join\n⊢ s ∈ f₁.bind g₁"}
+{"state": [{"context": ["α : Type u", "β : Type v", "γ : Type w", "δ : Type u_1", "ι : Sort x", "l f₁ f₂ : Filter α", "g₁ g₂ : α → Filter β", "hf : f₁ ≤ f₂", "hg : g₁ ≤ᶠ[f₁] g₂", "s : Set β", "hs : g₂ ⁻¹' {t | s ∈ t} ∈ f₁"], "goal": "{a | s ∈ g₁ a} ∈ f₁"}], "premise": [15889, 131585], "state_str": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type u_1\nι : Sort x\nl f₁ f₂ : Filter α\ng₁ g₂ : α → Filter β\nhf : f₁ ≤ f₂\nhg : g₁ ≤ᶠ[f₁] g₂\ns : Set β\nhs : g₂ ⁻¹' {t | s ∈ t} ∈ f₁\n⊢ {a | s ∈ g₁ a} ∈ f₁"}
+{"state": [{"context": ["ι : Type u_1", "α : Type u_2", "inst✝ : PseudoMetricSpace α", "x y : ℝ", "hx : x ∈ Icc 0 1", "hy : y ∈ Icc 0 1"], "goal": "dist x y ≤ 1"}], "premise": [59426, 117816], "state_str": "ι : Type u_1\nα : Type u_2\ninst✝ : PseudoMetricSpace α\nx y : ℝ\nhx : x ∈ Icc 0 1\nhy : y ∈ Icc 0 1\n⊢ dist x y ≤ 1"}
+{"state": [{"context": ["R : Type u_1", "inst✝² : CommRing R", "n : ℕ", "inst✝¹ : IsDomain R", "inst✝ : NeZero ↑n", "μ : R"], "goal": "(cyclotomic n R).IsRoot μ ↔ IsPrimitiveRoot μ n"}], "premise": [77058], "state_str": "R : Type u_1\ninst✝² : CommRing R\nn : ℕ\ninst✝¹ : IsDomain R\ninst✝ : NeZero ↑n\nμ : R\n⊢ (cyclotomic n R).IsRoot μ ↔ IsPrimitiveRoot μ n"}
+{"state": [{"context": ["R : Type u_1", "inst✝² : CommRing R", "n : ℕ", "inst✝¹ : IsDomain R", "inst✝ : NeZero ↑n", "μ : R", "hf : Function.Injective ⇑(algebraMap R (FractionRing R))"], "goal": "(cyclotomic n R).IsRoot μ ↔ IsPrimitiveRoot μ n"}], "premise": [142613], "state_str": "R : Type u_1\ninst✝² : CommRing R\nn : ℕ\ninst✝¹ : IsDomain R\ninst✝ : NeZero ↑n\nμ : R\nhf : Function.Injective ⇑(algebraMap R (FractionRing R))\n⊢ (cyclotomic n R).IsRoot μ ↔ IsPrimitiveRoot μ n"}
+{"state": [{"context": ["R : Type u_1", "inst✝² : CommRing R", "n : ℕ", "inst✝¹ : IsDomain R", "inst✝ : NeZero ↑n", "μ : R", "hf : Function.Injective ⇑(algebraMap R (FractionRing R))", "this : NeZero ↑n"], "goal": "(cyclotomic n R).IsRoot μ ↔ IsPrimitiveRoot μ n"}], "premise": [1713, 74756, 75181, 78760, 102995], "state_str": "R : Type u_1\ninst✝² : CommRing R\nn : ℕ\ninst✝¹ : IsDomain R\ninst✝ : NeZero ↑n\nμ : R\nhf : Function.Injective ⇑(algebraMap R (FractionRing R))\nthis : NeZero ↑n\n⊢ (cyclotomic n R).IsRoot μ ↔ IsPrimitiveRoot μ n"}
+{"state": [{"context": ["R : Type u_1", "ι : Type u_2", "A : Type u_3", "B : Type u_4", "inst✝¹¹ : CommSemiring ι", "inst✝¹⁰ : Module ι (Additive ℤˣ)", "inst✝⁹ : DecidableEq ι", "𝒜 : ι → Type u_5", "ℬ : ι → Type u_6", "inst✝⁸ : CommRing R", "inst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)", "inst✝⁶ : (i : ι) → AddCommGroup (ℬ i)", "inst✝⁵ : (i : ι) → Module R (𝒜 i)", "inst✝⁴ : (i : ι) → Module R (ℬ i)", "inst✝³ : DirectSum.GRing 𝒜", "inst✝² : DirectSum.GRing ℬ", "inst✝¹ : DirectSum.GAlgebra R 𝒜", "inst✝ : DirectSum.GAlgebra R ℬ", "x y z : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ", "mA : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ := gradedMul R 𝒜 ℬ"], "goal": "((gradedMul R 𝒜 ℬ) (((gradedMul R 𝒜 ℬ) x) y)) z = ((gradedMul R 𝒜 ℬ) x) (((gradedMul R 𝒜 ℬ) y) z)"}], "premise": [128589], "state_str": "R : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹¹ : CommSemiring ι\ninst✝¹⁰ : Module ι (Additive ℤˣ)\ninst✝⁹ : DecidableEq ι\n𝒜 : ι → Type u_5\nℬ : ι → Type u_6\ninst✝⁸ : CommRing R\ninst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)\ninst✝⁶ : (i : ι) → AddCommGroup (ℬ i)\ninst✝⁵ : (i : ι) → Module R (𝒜 i)\ninst✝⁴ : (i : ι) → Module R (ℬ i)\ninst✝³ : DirectSum.GRing 𝒜\ninst✝² : DirectSum.GRing ℬ\ninst✝¹ : DirectSum.GAlgebra R 𝒜\ninst✝ : DirectSum.GAlgebra R ℬ\nx y z : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ\nmA : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ :=\n  gradedMul R 𝒜 ℬ\n⊢ ((gradedMul R 𝒜 ℬ) (((gradedMul R 𝒜 ℬ) x) y)) z = ((gradedMul R 𝒜 ℬ) x) (((gradedMul R 𝒜 ℬ) y) z)"}
+{"state": [{"context": ["R : Type u_1", "ι : Type u_2", "A : Type u_3", "B : Type u_4", "inst✝¹¹ : CommSemiring ι", "inst✝¹⁰ : Module ι (Additive ℤˣ)", "inst✝⁹ : DecidableEq ι", "𝒜 : ι → Type u_5", "ℬ : ι → Type u_6", "inst✝⁸ : CommRing R", "inst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)", "inst✝⁶ : (i : ι) → AddCommGroup (ℬ i)", "inst✝⁵ : (i : ι) → Module R (𝒜 i)", "inst✝⁴ : (i : ι) → Module R (ℬ i)", "inst✝³ : DirectSum.GRing 𝒜", "inst✝² : DirectSum.GRing ℬ", "inst✝¹ : DirectSum.GAlgebra R 𝒜", "inst✝ : DirectSum.GAlgebra R ℬ", "x y z : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ", "mA : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ := gradedMul R 𝒜 ℬ", "ixa : ι", "xa : 𝒜 ixa", "ixb : ι", "xb : ℬ ixb", "iya : ι", "ya : 𝒜 iya", "iyb : ι", "yb : ℬ iyb", "iza : ι", "za : 𝒜 iza", "izb : ι", "zb : ℬ izb"], "goal": "((gradedMul R 𝒜 ℬ) (((gradedMul R 𝒜 ℬ) ((lof R ι 𝒜 ixa) xa ⊗ₜ[R] (lof R ι ℬ ixb) xb)) ((lof R ι 𝒜 iya) ya ⊗ₜ[R] (lof R ι ℬ iyb) yb))) ((lof R ι 𝒜 iza) za ⊗ₜ[R] (lof R ι ℬ izb) zb) = ((gradedMul R 𝒜 ℬ) ((lof R ι 𝒜 ixa) xa ⊗ₜ[R] (lof R ι ℬ ixb) xb)) (((gradedMul R 𝒜 ℬ) ((lof R ι 𝒜 iya) ya ⊗ₜ[R] (lof R ι ℬ iyb) yb)) ((lof R ι 𝒜 iza) za ⊗ₜ[R] (lof R ι ℬ izb) zb))"}], "premise": [81818, 85979, 109741, 110044, 116934, 117033, 118422, 119703], "state_str": "case a.H.h.H.h.a.H.h.H.h.a.H.h.H.h\nR : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹¹ : CommSemiring ι\ninst✝¹⁰ : Module ι (Additive ℤˣ)\ninst✝⁹ : DecidableEq ι\n𝒜 : ι → Type u_5\nℬ : ι → Type u_6\ninst✝⁸ : CommRing R\ninst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)\ninst✝⁶ : (i : ι) → AddCommGroup (ℬ i)\ninst✝⁵ : (i : ι) → Module R (𝒜 i)\ninst✝⁴ : (i : ι) → Module R (ℬ i)\ninst✝³ : DirectSum.GRing 𝒜\ninst✝² : DirectSum.GRing ℬ\ninst✝¹ : DirectSum.GAlgebra R 𝒜\ninst✝ : DirectSum.GAlgebra R ℬ\nx y z : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ\nmA : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ :=\n  gradedMul R 𝒜 ℬ\nixa : ι\nxa : 𝒜 ixa\nixb : ι\nxb : ℬ ixb\niya : ι\nya : 𝒜 iya\niyb : ι\nyb : ℬ iyb\niza : ι\nza : 𝒜 iza\nizb : ι\nzb : ℬ izb\n⊢ ((gradedMul R 𝒜 ℬ)\n        (((gradedMul R 𝒜 ℬ) ((lof R ι 𝒜 ixa) xa ⊗ₜ[R] (lof R ι ℬ ixb) xb))\n          ((lof R �� 𝒜 iya) ya ⊗ₜ[R] (lof R ι ℬ iyb) yb)))\n      ((lof R ι 𝒜 iza) za ⊗ₜ[R] (lof R ι ℬ izb) zb) =\n    ((gradedMul R 𝒜 ℬ) ((lof R ι 𝒜 ixa) xa ⊗ₜ[R] (lof R ι ℬ ixb) xb))\n      (((gradedMul R 𝒜 ℬ) ((lof R ι 𝒜 iya) ya ⊗ₜ[R] (lof R ι ℬ iyb) yb)) ((lof R ι 𝒜 iza) za ⊗ₜ[R] (lof R ι ℬ izb) zb))"}
+{"state": [{"context": ["R : Type u_1", "ι : Type u_2", "A : Type u_3", "B : Type u_4", "inst✝¹¹ : CommSemiring ι", "inst✝¹⁰ : Module ι (Additive ℤˣ)", "inst✝⁹ : DecidableEq ι", "𝒜 : ι → Type u_5", "ℬ : ι → Type u_6", "inst✝⁸ : CommRing R", "inst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)", "inst✝⁶ : (i : ι) → AddCommGroup (ℬ i)", "inst✝⁵ : (i : ι) → Module R (𝒜 i)", "inst✝⁴ : (i : ι) → Module R (ℬ i)", "inst✝³ : DirectSum.GRing 𝒜", "inst✝² : DirectSum.GRing ℬ", "inst✝¹ : DirectSum.GAlgebra R 𝒜", "inst✝ : DirectSum.GAlgebra R ℬ", "x y z : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ", "mA : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ := gradedMul R 𝒜 ℬ", "ixa : ι", "xa : 𝒜 ixa", "ixb : ι", "xb : ℬ ixb", "iya : ι", "ya : 𝒜 iya", "iyb : ι", "yb : ℬ iyb", "iza : ι", "za : 𝒜 iza", "izb : ι", "zb : ℬ izb"], "goal": "↑↑((-1) ^ (ixb * iya)) • (-1) ^ ((ixb + iyb) * iza) • ((lof R ι 𝒜 ixa) xa * ((lof R ι 𝒜 iya) ya * (lof R ι 𝒜 iza) za)) ⊗ₜ[R] ((lof R ι ℬ ixb) xb * ((lof R ι ℬ iyb) yb * (lof R ι ℬ izb) zb)) = ↑↑((-1) ^ (iyb * iza)) • (-1) ^ (ixb * (iya + iza)) • ((lof R ι 𝒜 ixa) xa * ((lof R ι 𝒜 iya) ya * (lof R ι 𝒜 iza) za)) ⊗ₜ[R] ((lof R ι ℬ ixb) xb * ((lof R ι ℬ iyb) yb * (lof R ι ℬ izb) zb))"}], "premise": [110044, 118422, 118909, 136152], "state_str": "case a.H.h.H.h.a.H.h.H.h.a.H.h.H.h\nR : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹¹ : CommSemiring ι\ninst✝¹⁰ : Module ι (Additive ℤˣ)\ninst✝⁹ : DecidableEq ι\n𝒜 : ι → Type u_5\nℬ : ι → Type u_6\ninst✝⁸ : CommRing R\ninst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)\ninst✝⁶ : (i : ι) → AddCommGroup (ℬ i)\ninst✝⁵ : (i : ι) → Module R (𝒜 i)\ninst✝⁴ : (i : ι) → Module R (ℬ i)\ninst✝³ : DirectSum.GRing 𝒜\ninst✝² : DirectSum.GRing ℬ\ninst✝¹ : DirectSum.GAlgebra R 𝒜\ninst✝ : DirectSum.GAlgebra R ℬ\nx y z : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ\nmA : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ :=\n  gradedMul R 𝒜 ℬ\nixa : ι\nxa : 𝒜 ixa\nixb : ι\nxb : ℬ ixb\niya : ι\nya : 𝒜 iya\niyb : ι\nyb : ℬ iyb\niza : ι\nza : 𝒜 iza\nizb : ι\nzb : ℬ izb\n⊢ ↑↑((-1) ^ (ixb * iya)) •\n      (-1) ^ ((ixb + iyb) * iza) •\n        ((lof R ι 𝒜 ixa) xa * ((lof R ι 𝒜 iya) ya * (lof R ι 𝒜 iza) za)) ⊗ₜ[R]\n          ((lof R ι ℬ ixb) xb * ((lof R ι ℬ iyb) yb * (lof R ι ℬ izb) zb)) =\n    ↑↑((-1) ^ (iyb * iza)) •\n      (-1) ^ (ixb * (iya + iza)) •\n        ((lof R ι 𝒜 ixa) xa * ((lof R ι 𝒜 iya) ya * (lof R ι 𝒜 iza) za)) ⊗ₜ[R]\n          ((lof R ι ℬ ixb) xb * ((lof R ι ℬ iyb) yb * (lof R ι ℬ izb) zb))"}
+{"state": [{"context": ["k : Type u", "inst✝¹ : Field k", "G : Type u", "inst✝ : Monoid G", "V W : FDRep k G", "i : V ≅ W", "g : G"], "goal": "V.character g = W.character g"}], "premise": [23732], "state_str": "case h\nk : Type u\ninst✝¹ : Field k\nG : Type u\ninst✝ : Monoid G\nV W : FDRep k G\ni : V ≅ W\ng : G\n⊢ V.character g = W.character g"}
+{"state": [{"context": ["k : Type u", "inst✝¹ : Field k", "G : Type u", "inst✝ : Monoid G", "V W : FDRep k G", "i : V ≅ W", "g : G"], "goal": "(trace k (CoeSort.coe V)) (V.ρ g) = (trace k (CoeSort.coe W)) ((isoToLinearEquiv i).conj (V.ρ g))"}], "premise": [2100, 81869], "state_str": "case h\nk : Type u\ninst✝¹ : Field k\nG : Type u\ninst✝ : Monoid G\nV W : FDRep k G\ni : V ≅ W\ng : G\n⊢ (trace k (CoeSort.coe V)) (V.ρ g) = (trace k (CoeSort.coe W)) ((isoToLinearEquiv i).conj (V.ρ g))"}
+{"state": [{"context": ["R : Type u_1", "R' : Type u_2", "E : Type u_3", "F : Type u_4", "ι : Type u_5", "ι' : Type u_6", "α : Type u_7", "inst✝⁸ : LinearOrderedField R", "inst✝⁷ : LinearOrderedField R'", "inst✝⁶ : AddCommGroup E", "inst✝⁵ : AddCommGroup F", "inst✝⁴ : LinearOrderedAddCommGroup α", "inst✝³ : Module R E", "inst✝² : Module R F", "inst✝¹ : Module R α", "inst✝ : OrderedSMul R α", "s✝ : Set E", "i j : ι", "c : R", "t✝ : Finset ι", "w : ι → R", "z : ι → E", "s t : Finset ι", "hw : ∑ i ∈ s, w i + ∑ i ∈ t, w i = 0", "hz : ∑ i ∈ s, w i • z i + ∑ i ∈ t, w i • z i = 0"], "goal": "s.centerMass w z = t.centerMass w z"}], "premise": [115837, 117821, 119813], "state_str": "R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nι : Type u_5\nι' : Type u_6\nα : Type u_7\ninst✝⁸ : LinearOrderedField R\ninst✝⁷ : LinearOrderedField R'\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns✝ : Set E\ni j : ι\nc : R\nt✝ : Finset ι\nw : ι → R\nz : ι → E\ns t : Finset ι\nhw : ∑ i ∈ s, w i + ∑ i ∈ t, w i = 0\nhz : ∑ i ∈ s, w i • z i + ∑ i ∈ t, w i • z i = 0\n⊢ s.centerMass w z = t.centerMass w z"}
+{"state": [{"context": ["C : Type u₁", "inst✝⁸ : Category.{v₁, u₁} C", "inst✝⁷ : MonoidalCategory C", "inst✝⁶ : BraidedCategory C", "D : Type u₂", "inst✝⁵ : Category.{v₂, u₂} D", "inst✝⁴ : MonoidalCategory D", "inst✝³ : BraidedCategory D", "E : Type u₃", "inst✝² : Category.{v₃, u₃} E", "inst✝¹ : MonoidalCategory E", "inst✝ : BraidedCategory E", "X✝ Y✝ : C × C", "f : X✝ ⟶ Y✝", "Z : C × C"], "goal": "__src✝.map f ▷ __src✝.obj Z ≫ tensor_μ C Y✝ Z = tensor_μ C X✝ Z ≫ __src✝.map (f ▷ Z)"}], "premise": [107139], "state_str": "C : Type u₁\ninst✝⁸ : Category.{v₁, u₁} C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category.{v₃, u₃} E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX✝ Y✝ : C × C\nf : X✝ ⟶ Y✝\nZ : C × C\n⊢ __src✝.map f ▷ __src✝.obj Z ≫ tensor_μ C Y✝ Z = tensor_μ C X✝ Z ≫ __src✝.map (f ▷ Z)"}
+{"state": [{"context": ["C : Type u₁", "inst✝⁸ : Category.{v₁, u₁} C", "inst✝⁷ : MonoidalCategory C", "inst✝⁶ : BraidedCategory C", "D : Type u₂", "inst✝⁵ : Category.{v₂, u₂} D", "inst✝⁴ : MonoidalCategory D", "inst✝³ : BraidedCategory D", "E : Type u₃", "inst✝² : Category.{v₃, u₃} E", "inst✝¹ : MonoidalCategory E", "inst✝ : BraidedCategory E", "X✝ Y✝ Z : C × C", "f : X✝ ⟶ Y✝"], "goal": "__src✝.obj Z ◁ __src✝.map f ≫ tensor_μ C Z Y✝ = tensor_μ C Z X✝ ≫ __src✝.map (Z ◁ f)"}], "premise": [107140], "state_str": "C : Type u₁\ninst✝⁸ : Category.{v₁, u₁} C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category.{v₃, u₃} E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX✝ Y✝ Z : C × C\nf : X✝ ⟶ Y✝\n⊢ __src✝.obj Z ◁ __src✝.map f ≫ tensor_μ C Z Y✝ = tensor_μ C Z X✝ ≫ __src✝.map (Z ◁ f)"}
+{"state": [{"context": ["C : Type u₁", "inst✝⁸ : Category.{v₁, u₁} C", "inst✝⁷ : MonoidalCategory C", "inst✝⁶ : BraidedCategory C", "D : Type u₂", "inst✝⁵ : Category.{v₂, u₂} D", "inst✝⁴ : MonoidalCategory D", "inst✝³ : BraidedCategory D", "E : Type u₃", "inst✝² : Category.{v₃, u₃} E", "inst✝¹ : MonoidalCategory E", "inst✝ : BraidedCategory E", "X Y Z : C × C"], "goal": "tensor_μ C X Y ▷ __src✝.obj Z ≫ tensor_μ C (X ⊗ Y) Z ≫ __src✝.map (α_ X Y Z).hom = (α_ (__src✝.obj X) (__src✝.obj Y) (__src✝.obj Z)).hom ≫ __src✝.obj X ◁ tensor_μ C Y Z ≫ tensor_μ C X (Y ⊗ Z)"}], "premise": [107143], "state_str": "C : Type u₁\ninst✝⁸ : Category.{v₁, u₁} C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category.{v₃, u₃} E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX Y Z : C × C\n⊢ tensor_μ C X Y ▷ __src✝.obj Z ≫ tensor_μ C (X ⊗ Y) Z ≫ __src✝.map (α_ X Y Z).hom =\n    (α_ (__src✝.obj X) (__src✝.obj Y) (__src✝.obj Z)).hom ≫ __src✝.obj X ◁ tensor_μ C Y Z ≫ tensor_μ C X (Y ⊗ Z)"}
+{"state": [{"context": ["C : Type u₁", "inst✝⁸ : Category.{v₁, u₁} C", "inst✝⁷ : MonoidalCategory C", "inst✝⁶ : BraidedCategory C", "D : Type u₂", "inst✝⁵ : Category.{v₂, u₂} D", "inst✝⁴ : MonoidalCategory D", "inst✝³ : BraidedCategory D", "E : Type u₃", "inst✝² : Category.{v₃, u₃} E", "inst✝¹ : MonoidalCategory E", "inst✝ : BraidedCategory E", "x✝ : C × C", "X₁ X₂ : C"], "goal": "(λ_ (__src✝.obj (X₁, X₂))).hom = (λ_ (𝟙_ C)).inv ▷ __src✝.obj (X₁, X₂) ≫ tensor_μ C (𝟙_ (C × C)) (X₁, X₂) ≫ __src✝.map (λ_ (X₁, X₂)).hom"}], "premise": [107141], "state_str": "C : Type u₁\ninst✝⁸ : Category.{v₁, u₁} C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category.{v₃, u₃} E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nx✝ : C × C\nX₁ X₂ : C\n⊢ (λ_ (__src✝.obj (X₁, X₂))).hom =\n    (λ_ (𝟙_ C)).inv ▷ __src✝.obj (X₁, X₂) ≫ tensor_μ C (𝟙_ (C × C)) (X₁, X₂) ≫ __src✝.map (λ_ (X₁, X₂)).hom"}
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+{"state": [{"context": ["α : Type u_1", "inst✝³ : TopologicalSpace α", "inst✝² : MeasurableSpace α", "inst✝¹ : BorelSpace α", "μ : Measure α", "inst✝ : μ.WeaklyRegular", "f : α → ℝ≥0", "fint : Integrable (fun x => ↑(f x)) μ", "ε : ℝ≥0", "εpos : 0 < ↑ε", "If : ∫⁻ (x : α), ↑(f x) ∂μ < ⊤"], "goal": "∃ g, (∀ (x : α), g x ≤ f x) ∧ UpperSemicontinuous g ∧ Integrable (fun x => ↑(g x)) μ ∧ ∫ (x : α), ↑(f x) ∂μ - ↑ε ≤ ∫ (x : α), ↑(g x) ∂μ"}], "premise": [11234, 26211], "state_str": "case intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nf : α → ℝ≥0\nfint : Integrable (fun x => ↑(f x)) μ\nε : ℝ≥0\nεpos : 0 < ↑ε\nIf : ∫⁻ (x : α), ↑(f x) ∂μ < ⊤\n⊢ ∃ g,\n    (∀ (x : α), g x ≤ f x) ∧\n      UpperSemicontinuous g ∧ Integrable (fun x => ↑(g x)) μ ∧ ∫ (x : α), ↑(f x) ∂μ - ↑ε ≤ ∫ (x : α), ↑(g x) ∂μ"}
+{"state": [{"context": ["�� : Type u_1", "inst✝³ : TopologicalSpace α", "inst✝² : MeasurableSpace α", "inst✝¹ : BorelSpace α", "μ : Measure α", "inst✝ : μ.WeaklyRegular", "f : α → ℝ≥0", "fint : Integrable (fun x => ↑(f x)) μ", "ε : ℝ≥0", "εpos : 0 < ↑ε", "If : ∫⁻ (x : α), ↑(f x) ∂μ < ⊤", "g : α → ℝ≥0", "gf : ∀ (x : α), g x ≤ f x", "gcont : UpperSemicontinuous g", "gint : ∫⁻ (x : α), ↑(f x) ∂μ ≤ ∫⁻ (x : α), ↑(g x) ∂μ + ↑ε"], "goal": "∃ g, (∀ (x : α), g x ≤ f x) ∧ UpperSemicontinuous g ∧ Integrable (fun x => ↑(g x)) μ ∧ ∫ (x : α), ↑(f x) ∂μ - ↑ε ≤ ∫ (x : α), ↑(g x) ∂μ"}], "premise": [14288, 30232], "state_str": "case intro.intro.intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nf : α → ℝ≥0\nfint : Integrable (fun x => ↑(f x)) μ\nε : ℝ≥0\nεpos : 0 < ↑ε\nIf : ∫⁻ (x : α), ↑(f x) ∂μ < ⊤\ng : α → ℝ≥0\ngf : ∀ (x : α), g x ≤ f x\ngcont : UpperSemicontinuous g\ngint : ∫⁻ (x : α), ↑(f x) ∂μ ≤ ∫⁻ (x : α), ↑(g x) ∂μ + ↑ε\n⊢ ∃ g,\n    (∀ (x : α), g x ≤ f x) ∧\n      UpperSemicontinuous g ∧ Integrable (fun x => ↑(g x)) μ ∧ ∫ (x : α), ↑(f x) ∂μ - ↑ε ≤ ∫ (x : α), ↑(g x) ∂μ"}
+{"state": [{"context": ["C₁ : Type u_1", "C₂ : Type u_2", "D₁ : Type u_3", "D₂ : Type u_4", "inst✝⁷ : Category.{u_8, u_1} C₁", "inst✝⁶ : Category.{u_7, u_2} C₂", "inst✝⁵ : Category.{u_6, u_3} D₁", "inst✝⁴ : Category.{u_5, u_4} D₂", "G : C₁ ⥤ C₂", "F : C₂ ⥤ C₁", "adj : G ⊣ F", "L₁ : C₁ ⥤ D₁", "W₁ : MorphismProperty C₁", "inst✝³ : L₁.IsLocalization W₁", "L₂ : C₂ ⥤ D₂", "W₂ : MorphismProperty C₂", "inst✝² : L₂.IsLocalization W₂", "G' : D₁ ⥤ D₂", "F' : D₂ ⥤ D₁", "inst✝¹ : CatCommSq G L₁ L₂ G'", "inst✝ : CatCommSq F L₂ L₁ F'", "X₂ : C₂"], "goal": "(adj.localization L₁ W₁ L₂ W₂ G' F').counit.app (L₂.obj X₂) = G'.map ((CatCommSq.iso F L₂ L₁ F').inv.app X₂) ≫ (CatCommSq.iso G L₁ L₂ G').inv.app (F.obj X₂) ≫ L₂.map (adj.counit.app X₂)"}], "premise": [92024], "state_str": "C₁ : Type u_1\nC₂ : Type u_2\nD₁ : Type u_3\nD₂ : Type u_4\ninst✝⁷ : Category.{u_8, u_1} C₁\ninst✝⁶ : Category.{u_7, u_2} C₂\ninst✝⁵ : Category.{u_6, u_3} D₁\ninst✝⁴ : Category.{u_5, u_4} D₂\nG : C₁ ⥤ C₂\nF : C₂ ⥤ C₁\nadj : G ⊣ F\nL₁ : C₁ ⥤ D₁\nW₁ : MorphismProperty C₁\ninst✝³ : L₁.IsLocalization W₁\nL₂ : C₂ ⥤ D₂\nW₂ : MorphismProperty C₂\ninst✝² : L₂.IsLocalization W₂\nG' : D₁ ⥤ D₂\nF' : D₂ ⥤ D₁\ninst✝¹ : CatCommSq G L₁ L₂ G'\ninst✝ : CatCommSq F L₂ L₁ F'\nX₂ : C₂\n⊢ (adj.localization L₁ W₁ L₂ W₂ G' F').counit.app (L₂.obj X₂) =\n    G'.map ((CatCommSq.iso F L₂ L₁ F').inv.app X₂) ≫\n      (CatCommSq.iso G L₁ L₂ G').inv.app (F.obj X₂) ≫ L₂.map (adj.counit.app X₂)"}
+{"state": [{"context": ["f : ℕ → ℝ", "hf : Antitone f", "n : ℕ", "hn : f n < 0", "hs : Summable f"], "goal": "False"}], "premise": [63917], "state_str": "f : ℕ → ℝ\nhf : Antitone f\nn : ℕ\nhn : f n < 0\nhs : Summable f\n⊢ False"}
+{"state": [{"context": ["f : ℕ → ℝ", "hf : Antitone f", "n : ℕ", "hn : f n < 0", "hs : Summable f", "this : Filter.Tendsto f Filter.atTop (nhds 0)"], "goal": "False"}], "premise": [42659, 42903, 61276], "state_str": "f : ℕ → ℝ\nhf : Antitone f\nn : ℕ\nhn : f n < 0\nhs : Summable f\nthis : Filter.Tendsto f Filter.atTop (nhds 0)\n⊢ False"}
+{"state": [{"context": ["f : ℕ → ℝ", "hf : Antitone f", "n : ℕ", "hn : f n < 0", "hs : Summable f", "this : ∀ ε > 0, ∃ N, ∀ n ≥ N, |f n| < ε"], "goal": "False"}], "premise": [105336], "state_str": "f : ℕ → ℝ\nhf : Antitone f\nn : ℕ\nhn : f n < 0\nhs : Summable f\nthis : ∀ ε > 0, ∃ N, ∀ n ≥ N, |f n| < ε\n⊢ False"}
+{"state": [{"context": ["f : ℕ → ℝ", "hf : Antitone f", "n : ℕ", "hn : f n < 0", "hs : Summable f", "this : ∀ ε > 0, ∃ N, ∀ n ≥ N, |f n| < ε", "N : ℕ", "hN : ∀ n_1 ≥ N, |f n_1| < |f n|"], "goal": "False"}], "premise": [3511], "state_str": "case intro\nf : ℕ → ℝ\nhf : Antitone f\nn : ℕ\nhn : f n < 0\nhs : Summable f\nthis : ∀ ε > 0, ∃ N, ∀ n ≥ N, |f n| < ε\nN : ℕ\nhN : ∀ n_1 ≥ N, |f n_1| < |f n|\n⊢ False"}
+{"state": [{"context": ["f : ℕ → ℝ", "hf : Antitone f", "n : ℕ", "hn : f n < 0", "hs : Summable f", "this : ∀ ε > 0, ∃ N, ∀ n ≥ N, |f n| < ε", "N : ℕ", "hN : |f (max n N)| < |f n|"], "goal": "False"}], "premise": [53688], "state_str": "case intro\nf : ℕ → ℝ\nhf : Antitone f\nn : ℕ\nhn : f n < 0\nhs : Summable f\nthis : ∀ ε > 0, ∃ N, ∀ n ≥ N, |f n| < ε\nN : ℕ\nhN : |f (max n N)| < |f n|\n⊢ False"}
+{"state": [{"context": ["f : ℕ → ℝ", "hf : Antitone f", "n : ℕ", "hn : f n < 0", "hs : Summable f", "this : ∀ ε > 0, ∃ N, ∀ n ≥ N, |f n| < ε", "N : ℕ"], "goal": "|f n| ≤ |f (max n N)|"}], "premise": [3510], "state_str": "case intro\nf : ℕ → ℝ\nhf : Antitone f\nn : ℕ\nhn : f n < 0\nhs : Summable f\nthis : ∀ ε > 0, ∃ N, ∀ n ≥ N, |f n| < ε\nN : ℕ\n⊢ |f n| ≤ |f (max n N)|"}
+{"state": [{"context": ["f : ℕ → ℝ", "hf : Antitone f", "n : ℕ", "hn : f n < 0", "hs : Summable f", "this : ∀ ε > 0, ∃ N, ∀ n ≥ N, |f n| < ε", "N : ℕ", "H : f (max n N) ≤ f n"], "goal": "|f n| ≤ |f (max n N)|"}], "premise": [105289, 105657], "state_str": "case intro\nf : ℕ → ℝ\nhf : Antitone f\nn : ℕ\nhn : f n < 0\nhs : Summable f\nthis : ∀ ε > 0, ∃ N, ∀ n ≥ N, |f n| < ε\nN : ℕ\nH : f (max n N) ≤ f n\n⊢ |f n| ≤ |f (max n N)|"}
+{"state": [{"context": ["C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "D : Type u₂", "inst✝ : Category.{v₂, u₂} D", "G : C"], "goal": "IsDetector G ↔ (coyoneda.obj (op G)).ReflectsIsomorphisms"}], "premise": [1674, 97999, 98000], "state_str": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nG : C\n⊢ IsDetector G ↔ (coyoneda.obj (op G)).ReflectsIsomorphisms"}
+{"state": [{"context": ["R : Type u", "S : Type v", "F : Type w", "inst✝¹ : CommRing R", "inst✝ : Semiring S", "I : Ideal R", "f : R →+* S", "H : I ≤ ker f"], "goal": "ker (Quotient.lift I f H) = map (Quotient.mk I) (ker f)"}], "premise": [80339], "state_str": "R : Type u\nS : Type v\nF : Type w\ninst✝¹ : CommRing R\ninst✝ : Semiring S\nI : Ideal R\nf : R →+* S\nH : I ≤ ker f\n⊢ ker (Quotient.lift I f H) = map (Quotient.mk I) (ker f)"}
+{"state": [{"context": ["n✝ n p : ℕ", "hp : Prime p"], "goal": "p ∣ (n + 1)! ↔ p ≤ n + 1"}], "premise": [144341, 144416], "state_str": "n✝ n p : ℕ\nhp : Prime p\n⊢ p ∣ (n + 1)! ↔ p ≤ n + 1"}
+{"state": [{"context": ["n✝ n p : ℕ", "hp : Prime p"], "goal": "p ∣ n + 1 ∨ p ≤ n ↔ p ≤ n + 1"}], "premise": [2110, 2141, 2143, 2146, 3523, 14302], "state_str": "n✝ n p : ℕ\nhp : Prime p\n⊢ p ∣ n + 1 ∨ p ≤ n ↔ p ≤ n + 1"}
+{"state": [{"context": ["M : Type u_1", "inst✝² : CommMonoid M", "S : Submonoid M", "N : Type u_2", "inst✝¹ : CommMonoid N", "P : Type u_3", "inst✝ : CommMonoid P", "f : S.LocalizationMap N", "x₁ x₂ : M", "y : ↥S"], "goal": "f.mk' x₂ y * f.toMap x₁ = f.mk' (x₁ * x₂) y"}], "premise": [9192, 119707], "state_str": "M : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type u_3\ninst✝ : CommMonoid P\nf : S.LocalizationMap N\nx₁ x₂ : M\ny : ↥S\n⊢ f.mk' x₂ y * f.toMap x₁ = f.mk' (x₁ * x₂) y"}
+{"state": [{"context": ["ι : Type u_1", "α : Type u", "β : Type v", "γ : Type w", "inst✝² : PseudoEMetricSpace α", "inst✝¹ : PseudoEMetricSpace β", "inst✝ : PseudoEMetricSpace γ", "h : α ≃ᵢ β", "s : Set β"], "goal": "EMetric.diam (⇑h ⁻¹' s) = EMetric.diam s"}], "premise": [59864, 59876], "state_str": "ι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\nh : α ≃ᵢ β\ns : Set β\n⊢ EMetric.diam (⇑h ⁻¹' s) = EMetric.diam s"}
+{"state": [{"context": ["C : Type u_1", "inst✝ : Category.{?u.1672, u_1} C", "P : Karoubi (Karoubi C)"], "goal": "P.p.f ≫ P.p.f = P.p.f"}], "premise": [97095, 97102], "state_str": "C : Type u_1\ninst✝ : Category.{?u.1672, u_1} C\nP : Karoubi (Karoubi C)\n⊢ P.p.f ≫ P.p.f = P.p.f"}
+{"state": [{"context": ["C : Type u_1", "inst✝ : Category.{?u.1672, u_1} C", "X✝ Y✝ : Karoubi (Karoubi C)", "f : X✝ ⟶ Y✝"], "goal": "f.f.f = ((fun P => { X := P.X.X, p := P.p.f, idem := ⋯ }) X✝).p ≫ f.f.f ≫ ((fun P => { X := P.X.X, p := P.p.f, idem := ⋯ }) Y✝).p"}], "premise": [97097, 97102], "state_str": "C : Type u_1\ninst✝ : Category.{?u.1672, u_1} C\nX✝ Y✝ : Karoubi (Karoubi C)\nf : X✝ ⟶ Y✝\n⊢ f.f.f =\n    ((fun P => { X := P.X.X, p := P.p.f, idem := ⋯ }) X✝).p ≫\n      f.f.f ≫ ((fun P => { X := P.X.X, p := P.p.f, idem := ⋯ }) Y✝).p"}
+{"state": [{"context": ["R : Type u_1", "σ : Type u_2", "inst✝ : CommRing R", "r : R", "I : Ideal R", "f : MvPolynomial σ (R ⧸ I)"], "goal": "(Ideal.Quotient.lift (Ideal.map C I) (eval₂Hom (C.comp (Ideal.Quotient.mk I)) X) ⋯) (eval₂ (Ideal.Quotient.lift I ((Ideal.Quotient.mk (Ideal.map C I)).comp C) ⋯) (fun i => (Ideal.Quotient.mk (Ideal.map C I)) (X i)) f) = f"}], "premise": [112219], "state_str": "R : Type u_1\nσ : Type u_2\ninst✝ : CommRing R\nr : R\nI : Ideal R\nf : MvPolynomial σ (R ⧸ I)\n⊢ (Ideal.Quotient.lift (Ideal.map C I) (eval₂Hom (C.comp (Ideal.Quotient.mk I)) X) ⋯)\n      (eval₂ (Ideal.Quotient.lift I ((Ideal.Quotient.mk (Ideal.map C I)).comp C) ⋯)\n        (fun i => (Ideal.Quotient.mk (Ideal.map C I)) (X i)) f) =\n    f"}
+{"state": [{"context": ["R : Type u_1", "A : Type u_2", "B : Type u_3", "inst✝⁷ : CommRing R", "inst✝⁶ : CommRing A", "inst✝⁵ : CommRing B", "inst✝⁴ : Algebra R B", "inst✝³ : Algebra A B", "inst✝² : IsIntegralClosure A R B", "inst✝¹ : Algebra R A", "inst✝ : IsScalarTower R A B", "x : R", "h : optParam (IsIntegral R ((algebraMap R B) x)) ⋯"], "goal": "(algebraMap A B) (mk' A ((algebraMap R B) x) h) = (algebraMap A B) ((algebraMap R A) x)"}], "premise": [81930, 121202], "state_str": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing A\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\ninst✝³ : Algebra A B\ninst✝² : IsIntegralClosure A R B\ninst✝¹ : Algebra R A\ninst✝ : IsScalarTower R A B\nx : R\nh : optParam (IsIntegral R ((algebraMap R B) x)) ⋯\n⊢ (algebraMap A B) (mk' A ((algebraMap R B) x) h) = (algebraMap A B) ((algebraMap R A) x)"}
+{"state": [{"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "inst✝² : Preadditive C", "R : Type u_1", "inst✝¹ : Ring R", "inst✝ : Linear R C", "K L M : CochainComplex C ℤ", "n✝ : ℤ", "γ✝ γ₁ γ₂ : Cochain K L n✝", "a n' : ℤ", "γ : Cochain ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K) L n'", "n : ℤ", "hn' : n + a = n'", "p q : ℤ", "hpq : p + n' = q"], "goal": "((γ.leftUnshift n hn').leftShift a n' hn').v p q hpq = γ.v p q hpq"}], "premise": [97904, 115392, 115394, 118909, 118910, 128841], "state_str": "case h\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nK L M : CochainComplex C ℤ\nn✝ : ℤ\nγ✝ γ₁ γ₂ : Cochain K L n✝\na n' : ℤ\nγ : Cochain ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K) L n'\nn : ℤ\nhn' : n + a = n'\np q : ℤ\nhpq : p + n' = q\n⊢ ((γ.leftUnshift n hn').leftShift a n' hn').v p q hpq = γ.v p q hpq"}
+{"state": [{"context": [], "goal": "size = 2 ^ System.Platform.numBits"}], "premise": [3596, 3734], "state_str": "⊢ size = 2 ^ System.Platform.numBits"}
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+{"state": [{"context": ["R✝ : Type u_1", "R : Type u", "inst✝² : CommRing R", "inst✝¹ : IsDomain R", "inst✝ : UniqueFactorizationMonoid R", "h₁ : ∃ p, Irreducible p", "h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q", "I : Ideal R", "I0 : ¬I = ⊥", "x : R", "hxI : x ∈ I", "hx0 : x ≠ 0", "p : R", "left✝ : Irreducible p", "H : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (p ^ n) x"], "goal": "Submodule.IsPrincipal I"}], "premise": [80415, 120413], "state_str": "case neg.intro.intro.intro.intro\nR✝ : Type u_1\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₁ : ∃ p, Irreducible p\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\nI : Ideal R\nI0 : ¬I = ⊥\nx : R\nhxI : x ∈ I\nhx0 : x ≠ 0\np : R\nleft✝ : Irreducible p\nH : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (p ^ n) x\n⊢ Submodule.IsPrincipal I"}
+{"state": [{"context": ["R✝ : Type u_1", "R : Type u", "inst✝² : CommRing R", "inst✝¹ : IsDomain R", "inst✝ : UniqueFactorizationMonoid R", "h₁ : ∃ p, Irreducible p", "h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q", "I : Ideal R", "I0 : ¬I = ⊥", "x : R", "hxI : x ∈ I", "hx0 : x ≠ 0", "p : R", "left✝ : Irreducible p", "H : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (p ^ n) x", "ex : ∃ n, p ^ n ∈ I"], "goal": "∃ a, I = Submodule.span R {a}"}], "premise": [1674, 2045], "state_str": "case neg.intro.intro.intro.intro.principal'\nR✝ : Type u_1\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₁ : ∃ p, Irreducible p\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\nI : Ideal R\nI0 : ¬I = ⊥\nx : R\nhxI : x ∈ I\nhx0 : x ≠ 0\np : R\nleft✝ : Irreducible p\nH : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (p ^ n) x\nex : ∃ n, p ^ n ∈ I\n⊢ ∃ a, I = Submodule.span R {a}"}
+{"state": [{"context": ["R✝ : Type u_1", "R : Type u", "inst✝² : CommRing R", "inst✝¹ : IsDomain R", "inst✝ : UniqueFactorizationMonoid R", "h₁ : ∃ p, Irreducible p", "h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q", "I : Ideal R", "I0 : ¬I = ⊥", "x : R", "hxI : x ∈ I", "hx0 : x ≠ 0", "p : R", "left✝ : Irreducible p", "H : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (p ^ n) x", "ex : ∃ n, p ^ n ∈ I"], "goal": "I = span {p ^ Nat.find ex}"}], "premise": [14296], "state_str": "case h\nR✝ : Type u_1\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₁ : ∃ p, Irreducible p\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\nI : Ideal R\nI0 : ¬I = ⊥\nx : R\nhxI : x ∈ I\nhx0 : x ≠ 0\np : R\nleft✝ : Irreducible p\nH : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (p ^ n) x\nex : ∃ n, p ^ n ∈ I\n⊢ I = span {p ^ Nat.find ex}"}
+{"state": [{"context": ["C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "X✝ Y Z : C", "D : Type u₂", "inst✝ : Category.{v₂, u₂} D", "B : C", "X : Subobject B"], "goal": "X.ofLE X ⋯ = 𝟙 (underlying.obj X)"}], "premise": [1673, 96191], "state_str": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX✝ Y Z : C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nB : C\nX : Subobject B\n⊢ X.ofLE X ⋯ = 𝟙 (underlying.obj X)"}
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+{"state": [{"context": ["𝕜 : Type u_1", "inst✝³⁷ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝³⁶ : NormedAddCommGroup E", "inst✝³⁵ : NormedSpace 𝕜 E", "H : Type u_3", "inst✝³⁴ : TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "inst✝³³ : TopologicalSpace M", "inst✝³² : ChartedSpace H M", "inst✝³¹ : SmoothManifoldWithCorners I M", "E' : Type u_5", "inst✝³⁰ : NormedAddCommGroup E'", "inst✝²⁹ : NormedSpace 𝕜 E'", "H' : Type u_6", "inst✝²⁸ : TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M' : Type u_7", "inst✝²⁷ : TopologicalSpace M'", "inst✝²⁶ : ChartedSpace H' M'", "inst✝²⁵ : SmoothManifoldWithCorners I' M'", "E'' : Type u_8", "inst✝²⁴ : NormedAddCommGroup E''", "inst✝²³ : NormedSpace 𝕜 E''", "H'' : Type u_9", "inst✝²² : TopologicalSpace H''", "I'' : ModelWithCorners 𝕜 E'' H''", "M'' : Type u_10", "inst✝²¹ : TopologicalSpace M''", "inst✝²⁰ : ChartedSpace H'' M''", "F : Type u_11", "inst✝¹⁹ : NormedAddCommGroup F", "inst✝¹⁸ : NormedSpace 𝕜 F", "G : Type u_12", "inst✝¹⁷ : TopologicalSpace G", "J : ModelWithCorners 𝕜 F G", "N : Type u_13", "inst✝¹⁶ : TopologicalSpace N", "inst✝¹⁵ : ChartedSpace G N", "inst✝¹⁴ : SmoothManifoldWithCorners J N", "F' : Type u_14", "inst✝¹³ : NormedAddCommGroup F'", "inst✝¹² : NormedSpace 𝕜 F'", "G' : Type u_15", "inst✝¹¹ : TopologicalSpace G'", "J' : ModelWithCorners 𝕜 F' G'", "N' : Type u_16", "inst✝¹⁰ : TopologicalSpace N'", "inst✝⁹ : ChartedSpace G' N'", "inst✝⁸ : SmoothManifoldWithCorners J' N'", "F₁ : Type u_17", "inst✝⁷ : NormedAddCommGroup F₁", "inst✝⁶ : NormedSpace 𝕜 F₁", "F₂ : Type u_18", "inst✝⁵ : NormedAddCommGroup F₂", "inst✝⁴ : NormedSpace 𝕜 F₂", "F₃ : Type u_19", "inst✝³ : NormedAddCommGroup F₃", "inst✝² : NormedSpace 𝕜 F₃", "F₄ : Type u_20", "inst✝¹ : NormedAddCommGroup F₄", "inst✝ : NormedSpace 𝕜 F₄", "e : PartialHomeomorph M H", "e' : PartialHomeomorph M' H'", "f f₁ : M → M'", "s s₁ t : Set M", "x✝ : M", "m n : ℕ∞", "x : M", "he : e ∈ maximalAtlas I M", "he' : e' ∈ maximalAtlas I' M'", "hs : s ⊆ e.source", "hx : x ∈ e.source", "hy : f x ∈ e'.source"], "goal": "ContinuousWithinAt f s x → (ContDiffWithinAt 𝕜 n (↑(e'.extend I') ∘ f ∘ ↑(e.extend I).symm) (↑(e.extend I).symm ⁻¹' s ∩ range ↑I) (↑(e.extend I) x) ↔ ContDiffWithinAt 𝕜 n (↑(e'.extend I') ∘ f ∘ ↑(e.extend I).symm) (↑(e.extend I) '' s) (↑(e.extend I) x))"}], "premise": [48335], "state_str": "𝕜 : Type u_1\ninst✝³⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³⁶ : NormedAddCommGroup E\ninst✝³⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝³⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝³³ : TopologicalSpace M\ninst✝³² : ChartedSpace H M\ninst✝³¹ : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst✝³⁰ : NormedAddCommGroup E'\ninst✝²⁹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝²⁸ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM' : Type u_7\ninst✝²⁷ : TopologicalSpace M'\ninst✝²⁶ : ChartedSpace H' M'\ninst✝²⁵ : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst✝²⁴ : NormedAddCommGroup E''\ninst✝²³ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝²² : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝²¹ : TopologicalSpace M''\ninst✝²⁰ : ChartedSpace H'' M''\nF : Type u_11\ninst✝¹⁹ : NormedAddCommGroup F\ninst✝¹⁸ : NormedSpace 𝕜 F\nG : Type u_12\ninst✝¹⁷ : TopologicalSpace G\nJ : ModelWithCorners 𝕜 F G\nN : Type u_13\ninst✝¹⁶ : TopologicalSpace N\ninst✝¹⁵ : ChartedSpace G N\ninst✝¹⁴ : SmoothManifoldWithCorners J N\nF' : Type u_14\ninst✝¹³ : NormedAddCommGroup F'\ninst✝¹² : NormedSpace 𝕜 F'\nG' : Type u_15\ninst✝¹¹ : TopologicalSpace G'\nJ' : ModelWithCorners 𝕜 F' G'\nN' : Type u_16\ninst✝¹⁰ : TopologicalSpace N'\ninst✝⁹ : ChartedSpace G' N'\ninst✝⁸ : SmoothManifoldWithCorners J' N'\nF₁ : Type u_17\ninst✝⁷ : NormedAddCommGroup F₁\ninst✝⁶ : NormedSpace 𝕜 F₁\nF₂ : Type u_18\ninst✝⁵ : NormedAddCommGroup F₂\ninst✝⁴ : NormedSpace 𝕜 F₂\nF₃ : Type u_19\ninst✝³ : NormedAddCommGroup F₃\ninst✝² : NormedSpace 𝕜 F₃\nF₄ : Type u_20\ninst✝¹ : NormedAddCommGroup F₄\ninst✝ : NormedSpace 𝕜 F₄\ne : PartialHomeomorph M H\ne' : PartialHomeomorph M' H'\nf f₁ : M → M'\ns s₁ t : Set M\nx✝ : M\nm n : ℕ∞\nx : M\nhe : e ∈ maximalAtlas I M\nhe' : e' ∈ maximalAtlas I' M'\nhs : s ⊆ e.source\nhx : x ∈ e.source\nhy : f x ∈ e'.source\n⊢ ContinuousWithinAt f s x →\n    (ContDiffWithinAt 𝕜 n (↑(e'.extend I') ∘ f ∘ ↑(e.extend I).symm) (↑(e.extend I).symm ⁻¹' s ∩ range ↑I)\n        (↑(e.extend I) x) ↔\n      ContDiffWithinAt 𝕜 n (↑(e'.extend I') ∘ f ∘ ↑(e.extend I).symm) (↑(e.extend I) '' s) (↑(e.extend I) x))"}
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+{"state": [{"context": ["C : Type u_1", "inst✝³ : Category.{u_2, u_1} C", "inst✝² : Preadditive C", "K L : HomologicalComplex₂ C (up ℤ) (up ℤ)", "f : K ⟶ L", "x y : ℤ", "inst✝¹ : K.HasTotal (up ℤ)", "inst✝ : L.HasTotal (up ℤ)", "n i₁ i₂ : ℤ", "h : i₁ + i₂ = n"], "goal": "((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) i₁ i₂ n h ≫ (total.map ((shiftFunctor₁ C x).map f) (up ℤ)).f n ≫ (L.totalShift₁Iso x).hom.f n = ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) i₁ i₂ n h ≫ (K.totalShift₁Iso x).hom.f n ≫ (total.map f (up ℤ)).f (n + x)"}], "premise": [116982], "state_str": "case h.h\nC : Type u_1\ninst✝³ : Category.{u_2, u_1} C\ninst✝² : Preadditive C\nK L : HomologicalComplex₂ C (up ℤ) (up ℤ)\nf : K ⟶ L\nx y : ℤ\ninst✝¹ : K.HasTotal (up ℤ)\ninst✝ : L.HasTotal (up ℤ)\nn i₁ i₂ : ℤ\nh : i₁ + i₂ = n\n⊢ ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) i₁ i₂ n h ≫\n      (total.map ((shiftFunctor₁ C x).map f) (up ℤ)).f n ≫ (L.totalShift₁Iso x).hom.f n =\n    ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) i₁ i₂ n h ≫ (K.totalShift₁Iso x).hom.f n ≫ (total.map f (up ℤ)).f (n + x)"}
+{"state": [{"context": ["C : Type u_1", "inst✝³ : Category.{u_2, u_1} C", "inst✝² : Preadditive C", "K L : HomologicalComplex₂ C (up ℤ) (up ℤ)", "f : K ⟶ L", "x y : ℤ", "inst✝¹ : K.HasTotal (up ℤ)", "inst✝ : L.HasTotal (up ℤ)", "n i₁ i₂ : ℤ", "h : i₁ + i₂ = n"], "goal": "(f.f (i₁ + x)).f i₂ ≫ 𝟙 ((L.X (i₁ + x)).X i₂) ≫ L.ιTotal (up ℤ) (i₁ + x) i₂ (n + x) ⋯ ≫ 𝟙 ((L.total (up ℤ)).X (n + x)) = 𝟙 ((K.X (i₁ + x)).X i₂) ≫ K.ιTotal (up ℤ) (i₁ + x) i₂ (n + x) ⋯ ≫ 𝟙 ((K.total (up ℤ)).X (n + x)) ≫ (total.map f (up ℤ)).f (n + x)"}], "premise": [96174, 96175, 115962], "state_str": "case h.h\nC : Type u_1\ninst✝³ : Category.{u_2, u_1} C\ninst✝² : Preadditive C\nK L : HomologicalComplex₂ C (up ℤ) (up ℤ)\nf : K ⟶ L\nx y : ℤ\ninst✝¹ : K.HasTotal (up ℤ)\ninst✝ : L.HasTotal (up ℤ)\nn i₁ i₂ : ℤ\nh : i₁ + i₂ = n\n⊢ (f.f (i₁ + x)).f i₂ ≫\n      𝟙 ((L.X (i₁ + x)).X i₂) ≫ L.ιTotal (up ℤ) (i₁ + x) i₂ (n + x) ⋯ ≫ 𝟙 ((L.total (up ℤ)).X (n + x)) =\n    𝟙 ((K.X (i₁ + x)).X i₂) ≫\n      K.ιTotal (up ℤ) (i₁ + x) i₂ (n + x) ⋯ ≫ 𝟙 ((K.total (up ℤ)).X (n + x)) ≫ (total.map f (up ℤ)).f (n + x)"}
+{"state": [{"context": ["K : Type u", "inst✝² : CommRing K", "p : ℕ", "inst✝¹ : Fact (Nat.Prime p)", "inst✝ : CharP K p", "n : ℕ", "x : K", "m : ℕ", "ih : mk K p (n, x) = mk K p (m + n, (⇑(frobenius K p))^[m] x)"], "goal": "mk K p (n, x) = mk K p (m + 1 + n, (⇑(frobenius K p))^[m + 1] x)"}], "premise": [3676, 71272], "state_str": "case succ\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nn : ℕ\nx : K\nm : ℕ\nih : mk K p (n, x) = mk K p (m + n, (⇑(frobenius K p))^[m] x)\n⊢ mk K p (n, x) = mk K p (m + 1 + n, (⇑(frobenius K p))^[m + 1] x)"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m : MeasurableSpace α", "μ ν : Measure α", "inst✝⁵ : MeasurableSpace δ", "inst✝⁴ : NormedAddCommGroup β", "inst✝³ : NormedAddCommGroup γ", "𝕜 : Type u_5", "inst✝² : NormedRing 𝕜", "inst✝¹ : MulActionWithZero 𝕜 β", "inst✝ : BoundedSMul 𝕜 β", "f : α → β", "c : 𝕜ˣ"], "goal": "HasFiniteIntegral f μ → HasFiniteIntegral (↑c • f) μ"}], "premise": [28445], "state_str": "case intro.mpr\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁵ : MeasurableSpace δ\ninst✝⁴ : NormedAddCommGroup β\ninst✝³ : NormedAddCommGroup γ\n𝕜 : Type u_5\ninst✝² : NormedRing 𝕜\ninst✝¹ : MulActionWithZero 𝕜 β\ninst✝ : BoundedSMul 𝕜 β\nf : α → β\nc : 𝕜ˣ\n⊢ HasFiniteIntegral f μ → HasFiniteIntegral (↑c • f) μ"}
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+{"state": [{"context": ["R : Type u_1", "inst✝¹³ : CommRing R", "inst✝¹² : IsDedekindDomain R", "S : Type u_2", "inst✝¹¹ : CommRing S", "inst✝¹⁰ : Algebra R S", "inst✝⁹ : Module.Free R S", "inst✝⁸ : Module.Finite R S", "p : Ideal R", "hp0 : p ≠ ⊥", "inst✝⁷ : p.IsPrime", "Sₚ : Type u_3", "inst✝⁶ : CommRing Sₚ", "inst✝⁵ : Algebra S Sₚ", "inst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ", "inst✝³ : Algebra R Sₚ", "inst✝² : IsScalarTower R S Sₚ", "inst✝¹ : IsDedekindDomain Sₚ", "inst✝ : IsDomain S", "this✝ : DecidableEq (Ideal Sₚ) := Classical.decEq (Ideal Sₚ)", "this : DecidablePred fun P => P.IsPrime := Classical.decPred fun P => P.IsPrime", "P : Ideal Sₚ"], "goal": "P ∈ Finset.filter (fun P => P.IsPrime) ({⊥} ∪ (normalizedFactors (map (algebraMap R Sₚ) p)).toFinset) ↔ P ∈ {I | I.IsPrime}"}], "premise": [133310, 138737, 138858, 139089, 139173], "state_str": "R : Type u_1\ninst✝¹³ : CommRing R\ninst✝¹² : IsDedekindDomain R\nS : Type u_2\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Algebra R S\ninst✝⁹ : Module.Free R S\ninst✝⁸ : Module.Finite R S\np : Ideal R\nhp0 : p ≠ ⊥\ninst✝⁷ : p.IsPrime\nSₚ : Type u_3\ninst✝⁶ : CommRing Sₚ\ninst✝⁵ : Algebra S Sₚ\ninst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ\ninst✝³ : Algebra R Sₚ\ninst✝² : IsScalarTower R S Sₚ\ninst✝¹ : IsDedekindDomain Sₚ\ninst✝ : IsDomain S\nthis✝ : DecidableEq (Ideal Sₚ) := Classical.decEq (Ideal Sₚ)\nthis : DecidablePred fun P => P.IsPrime := Classical.decPred fun P => P.IsPrime\nP : Ideal Sₚ\n⊢ P ∈ Finset.filter (fun P => P.IsPrime) ({⊥} ∪ (normalizedFactors (map (algebraMap R Sₚ) p)).toFinset) ↔\n    P ∈ {I | I.IsPrime}"}
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+{"state": [{"context": ["ι : Type u_1", "R : Type u_2", "R' : Type u_3", "E : Type u_4", "F : Type u_5", "G : Type u_6", "inst✝² : AddGroup E", "inst✝¹ : FunLike F E ℝ", "inst✝ : NonarchAddGroupSeminormClass F E", "f : F", "x y : E"], "goal": "f (x - y) ≤ max (f x) (f y)"}], "premise": [42399, 119789], "state_str": "ι : Type u_1\nR : Type u_2\nR' : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst✝² : AddGroup E\ninst✝¹ : FunLike F E ℝ\ninst✝ : NonarchAddGroupSeminormClass F E\nf : F\nx y : E\n⊢ f (x - y) ≤ max (f x) (f y)"}
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+{"state": [{"context": ["ι : Type uι", "E : Type uE", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : NormedSpace ℝ E", "inst✝⁶ : FiniteDimensional ℝ E", "H : Type uH", "inst✝⁵ : TopologicalSpace H", "I : ModelWithCorners ℝ E H", "M : Type uM", "inst✝⁴ : TopologicalSpace M", "inst✝³ : ChartedSpace H M", "inst✝² : SmoothManifoldWithCorners I M", "inst✝¹ : T2Space M", "hi : Fintype ι", "s : Set M", "f✝ : SmoothBumpCovering ι I M s", "inst✝ : Finite ι", "f : SmoothBumpCovering ι I M", "val✝ : Fintype ι", "F : Type := EuclideanSpace ℝ (Fin (finrank ℝ (ι → E × ℝ)))", "this✝ : IsNoetherian ℝ (E × ℝ) := IsNoetherian.iff_fg.mpr inferInstance", "this : FiniteDimensional ℝ (ι → E × ℝ) := IsNoetherian.iff_fg.mp inferInstance", "eEF : (ι → E × ℝ) ≃L[ℝ] F := ContinuousLinearEquiv.ofFinrankEq ⋯", "x : M"], "goal": "Injective ⇑(mfderiv I 𝓘(ℝ, EuclideanSpace ℝ (Fin (finrank ℝ (ι → E × ℝ)))) (⇑eEF ∘ ⇑f.embeddingPiTangent) x)"}], "premise": [45899, 66048, 67946, 68544, 68588], "state_str": "case intro\nι : Type uι\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\nH : Type uH\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\ninst✝² : SmoothManifoldWithCorners I M\ninst✝¹ : T2Space M\nhi : Fintype ι\ns : Set M\nf✝ : SmoothBumpCovering ι I M s\ninst✝ : Finite ι\nf : SmoothBumpCovering ι I M\nval✝ : Fintype ι\nF : Type := EuclideanSpace ℝ (Fin (finrank ℝ (ι → E × ℝ)))\nthis✝ : IsNoetherian ℝ (E × ℝ) := IsNoetherian.iff_fg.mpr inferInstance\nthis : FiniteDimensional ℝ (ι → E × ℝ) := IsNoetherian.iff_fg.mp inferInstance\neEF : (ι → E × ℝ) ≃L[ℝ] F := ContinuousLinearEquiv.ofFinrankEq ⋯\nx : M\n⊢ Injective ⇑(mfderiv I 𝓘(ℝ, EuclideanSpace ℝ (Fin (finrank ℝ (ι → E × ℝ)))) (⇑eEF ∘ ⇑f.embeddingPiTangent) x)"}
+{"state": [{"context": ["ι : Type uι", "E : Type uE", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : NormedSpace ℝ E", "inst✝⁶ : FiniteDimensional ℝ E", "H : Type uH", "inst✝⁵ : TopologicalSpace H", "I : ModelWithCorners ℝ E H", "M : Type uM", "inst✝⁴ : TopologicalSpace M", "inst✝³ : ChartedSpace H M", "inst✝² : SmoothManifoldWithCorners I M", "inst✝¹ : T2Space M", "hi : Fintype ι", "s : Set M", "f✝ : SmoothBumpCovering ι I M s", "inst✝ : Finite ι", "f : SmoothBumpCovering ι I M", "val✝ : Fintype ι", "F : Type := EuclideanSpace ℝ (Fin (finrank ℝ (ι → E × ℝ)))", "this✝ : IsNoetherian ℝ (E × ℝ) := IsNoetherian.iff_fg.mpr inferInstance", "this : FiniteDimensional ℝ (ι → E × ℝ) := IsNoetherian.iff_fg.mp inferInstance", "eEF : (ι → E × ℝ) ≃L[ℝ] F := ContinuousLinearEquiv.ofFinrankEq ⋯", "x : M"], "goal": "Injective ⇑((↑eEF).comp (mfderiv I 𝓘(ℝ, ι → E × ℝ) (⇑f.embeddingPiTangent) x))"}], "premise": [1680, 66062, 68937, 71025], "state_str": "case intro\nι : Type uι\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\nH : Type uH\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\ninst✝² : SmoothManifoldWithCorners I M\ninst✝¹ : T2Space M\nhi : Fintype ι\ns : Set M\nf✝ : SmoothBumpCovering ι I M s\ninst✝ : Finite ι\nf : SmoothBumpCovering ι I M\nval✝ : Fintype ι\nF : Type := EuclideanSpace ℝ (Fin (finrank ℝ (ι → E × ℝ)))\nthis✝ : IsNoetherian ℝ (E × ℝ) := IsNoetherian.iff_fg.mpr inferInstance\nthis : FiniteDimensional ℝ (ι → E × ℝ) := IsNoetherian.iff_fg.mp inferInstance\neEF : (ι → E × ℝ) ≃L[ℝ] F := ContinuousLinearEquiv.ofFinrankEq ⋯\nx : M\n⊢ Injective ⇑((↑eEF).comp (mfderiv I 𝓘(ℝ, ι → E × ℝ) (⇑f.embeddingPiTangent) x))"}
+{"state": [{"context": ["n : ℕ", "hpos : 0 < n", "p : ℕ", "hprime : Fact (Nat.Prime p)", "a : ℕ", "hroot : (cyclotomic n (ZMod p)).IsRoot ((Nat.castRingHom (ZMod p)) a)"], "goal": "a.Coprime p"}], "premise": [395], "state_str": "n : ℕ\nhpos : 0 < n\np : ℕ\nhprime : Fact (Nat.Prime p)\na : ℕ\nhroot : (cyclotomic n (ZMod p)).IsRoot ((Nat.castRingHom (ZMod p)) a)\n⊢ a.Coprime p"}
+{"state": [{"context": ["n : ℕ", "hpos : 0 < n", "p : ℕ", "hprime : Fact (Nat.Prime p)", "a : ℕ", "hroot : (cyclotomic n (ZMod p)).IsRoot ((Nat.castRingHom (ZMod p)) a)"], "goal": "p.Coprime a"}], "premise": [70028, 144340], "state_str": "case a\nn : ℕ\nhpos : 0 < n\np : ℕ\nhprime : Fact (Nat.Prime p)\na : ℕ\nhroot : (cyclotomic n (ZMod p)).IsRoot ((Nat.castRingHom (ZMod p)) a)\n⊢ p.Coprime a"}
+{"state": [{"context": ["n : ℕ", "hpos : 0 < n", "p : ℕ", "hprime : Fact (Nat.Prime p)", "a : ℕ", "hroot : (cyclotomic n (ZMod p)).IsRoot ((Nat.castRingHom (ZMod p)) a)", "h : p ∣ a"], "goal": "False"}], "premise": [1674, 138337], "state_str": "case a\nn : ℕ\nhpos : 0 < n\np : ℕ\nhprime : Fact (Nat.Prime p)\na : ℕ\nhroot : (cyclotomic n (ZMod p)).IsRoot ((Nat.castRingHom (ZMod p)) a)\nh : p ∣ a\n⊢ False"}
+{"state": [{"context": ["n : ℕ", "hpos : 0 < n", "p : ℕ", "hprime : Fact (Nat.Prime p)", "a : ℕ", "hroot : (cyclotomic n (ZMod p)).IsRoot ((Nat.castRingHom (ZMod p)) a)", "h : ↑a = 0"], "goal": "False"}], "premise": [102884, 102886, 142662], "state_str": "case a\nn : ℕ\nhpos : 0 < n\np : ℕ\nhprime : Fact (Nat.Prime p)\na : ℕ\nhroot : (cyclotomic n (ZMod p)).IsRoot ((Nat.castRingHom (ZMod p)) a)\nh : ↑a = 0\n⊢ False"}
+{"state": [{"context": ["n : ℕ", "hpos : 0 < n", "p : ℕ", "hprime : Fact (Nat.Prime p)", "a : ℕ", "hroot : (cyclotomic n (ZMod p)).coeff 0 = 0", "h : ↑a = 0", "hone : ¬n = 1"], "goal": "False"}], "premise": [1690, 3734, 14298, 75208], "state_str": "case neg\nn : ℕ\nhpos : 0 < n\np : ℕ\nhprime : Fact (Nat.Prime p)\na : ℕ\nhroot : (cyclotomic n (ZMod p)).coeff 0 = 0\nh : ↑a = 0\nhone : ¬n = 1\n⊢ False"}
+{"state": [{"context": ["n : ℕ", "hpos : 0 < n", "p : ℕ", "hprime : Fact (Nat.Prime p)", "a : ℕ", "hroot : 1 = 0", "h : ↑a = 0", "hone : ¬n = 1"], "goal": "False"}], "premise": [113018], "state_str": "case neg\nn : ℕ\nhpos : 0 < n\np : ℕ\nhprime : Fact (Nat.Prime p)\na : ℕ\nhroot : 1 = 0\nh : ↑a = 0\nhone : ¬n = 1\n⊢ False"}
+{"state": [{"context": ["R : Type u_1", "R' : Type u_2", "M : Type u_3", "X : Type u_4", "inst✝¹² : Monoid M", "S : Submonoid M", "inst✝¹¹ : OreSet S", "inst✝¹⁰ : MulAction M X", "inst✝⁹ : SMul R X", "inst✝⁸ : SMul R M", "inst✝⁷ : IsScalarTower R M M", "inst✝⁶ : IsScalarTower R M X", "inst✝⁵ : SMul R' X", "inst✝⁴ : SMul R' M", "inst✝³ : IsScalarTower R' M M", "inst✝² : IsScalarTower R' M X", "inst✝¹ : SMul R R'", "inst✝ : IsScalarTower R R' M", "r : M", "x : OreLocalization S X"], "goal": "(r /ₒ 1) • x = r • x"}], "premise": [81621], "state_str": "R : Type u_1\nR' : Type u_2\nM : Type u_3\nX : Type u_4\ninst✝¹² : Monoid M\nS : Submonoid M\ninst✝¹¹ : OreSet S\ninst✝¹⁰ : MulAction M X\ninst✝⁹ : SMul R X\ninst✝⁸ : SMul R M\ninst✝⁷ : IsScalarTower R M M\ninst✝⁶ : IsScalarTower R M X\ninst✝⁵ : SMul R' X\ninst✝⁴ : SMul R' M\ninst✝³ : IsScalarTower R' M M\ninst✝² : IsScalarTower R' M X\ninst✝¹ : SMul R R'\ninst✝ : IsScalarTower R R' M\nr : M\nx : OreLocalization S X\n⊢ (r /ₒ 1) • x = r • x"}
+{"state": [{"context": ["R : Type u_1", "R' : Type u_2", "M : Type u_3", "X : Type u_4", "inst✝¹² : Monoid M", "S : Submonoid M", "inst✝¹¹ : OreSet S", "inst✝¹⁰ : MulAction M X", "inst✝⁹ : SMul R X", "inst✝⁸ : SMul R M", "inst✝⁷ : IsScalarTower R M M", "inst✝⁶ : IsScalarTower R M X", "inst✝⁵ : SMul R' X", "inst✝⁴ : SMul R' M", "inst✝³ : IsScalarTower R' M M", "inst✝² : IsScalarTower R' M X", "inst✝¹ : SMul R R'", "inst✝ : IsScalarTower R R' M", "r : M", "r' : X", "s : ↥S"], "goal": "(r /ₒ 1) • (r' /ₒ s) = r • (r' /ₒ s)"}], "premise": [81637, 81690, 118863, 119730], "state_str": "case c\nR : Type u_1\nR' : Type u_2\nM : Type u_3\nX : Type u_4\ninst✝¹² : Monoid M\nS : Submonoid M\ninst✝¹¹ : OreSet S\ninst✝¹⁰ : MulAction M X\ninst✝⁹ : SMul R X\ninst✝⁸ : SMul R M\ninst✝⁷ : IsScalarTower R M M\ninst✝⁶ : IsScalarTower R M X\ninst✝⁵ : SMul R' X\ninst✝⁴ : SMul R' M\ninst✝³ : IsScalarTower R' M M\ninst✝² : IsScalarTower R' M X\ninst✝¹ : SMul R R'\ninst✝ : IsScalarTower R R' M\nr : M\nr' : X\ns : ↥S\n⊢ (r /ₒ 1) • (r' /ₒ s) = r • (r' /ₒ s)"}
+{"state": [{"context": ["a b : ℝ", "n : ℕ", "C : ℝ := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a", "h : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)"], "goal": "∫ (x : ℝ) in a..b, cos x ^ (n + 2) = (cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a + (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ n) - (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ (n + 2)"}], "premise": [38973, 44066, 117740, 122240, 122241], "state_str": "a b : ℝ\nn : ℕ\nC : ℝ := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a\nh : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)\n⊢ ∫ (x : ℝ) in a..b, cos x ^ (n + 2) =\n    (cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a + (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ n) -\n      (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ (n + 2)"}
+{"state": [{"context": ["a b : ℝ", "n : ℕ", "C : ℝ := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a", "h : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)", "hu : ∀ x ∈ [[a, b]], HasDerivAt (fun y => cos y ^ (n + 1)) (-↑(n + 1) * sin x * cos x ^ n) x"], "goal": "∫ (x : ℝ) in a..b, cos x ^ (n + 2) = (cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a + (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ n) - (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ (n + 2)"}], "premise": [38967], "state_str": "a b : ℝ\nn : ℕ\nC : ℝ := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a\nh : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)\nhu : ∀ x ∈ [[a, b]], HasDerivAt (fun y => cos y ^ (n + 1)) (-↑(n + 1) * sin x * cos x ^ n) x\n⊢ ∫ (x : ℝ) in a..b, cos x ^ (n + 2) =\n    (cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a + (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ n) -\n      (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ (n + 2)"}
+{"state": [{"context": ["a b : ℝ", "n : ℕ", "C : �� := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a", "h : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)", "hu : ∀ x ∈ [[a, b]], HasDerivAt (fun y => cos y ^ (n + 1)) (-↑(n + 1) * sin x * cos x ^ n) x", "hv : ∀ x ∈ [[a, b]], HasDerivAt sin (cos x) x"], "goal": "∫ (x : ℝ) in a..b, cos x ^ (n + 2) = (cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a + (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ n) - (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ (n + 2)"}], "premise": [27349], "state_str": "a b : ℝ\nn : ℕ\nC : ℝ := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a\nh : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)\nhu : ∀ x ∈ [[a, b]], HasDerivAt (fun y => cos y ^ (n + 1)) (-↑(n + 1) * sin x * cos x ^ n) x\nhv : ∀ x ∈ [[a, b]], HasDerivAt sin (cos x) x\n⊢ ∫ (x : ℝ) in a..b, cos x ^ (n + 2) =\n    (cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a + (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ n) -\n      (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ (n + 2)"}
+{"state": [{"context": ["a b : ℝ", "n : ℕ", "C : ℝ := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a", "h : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)", "hu : ∀ x ∈ [[a, b]], HasDerivAt (fun y => cos y ^ (n + 1)) (-↑(n + 1) * sin x * cos x ^ n) x", "hv : ∀ x ∈ [[a, b]], HasDerivAt sin (cos x) x"], "goal": "IntervalIntegrable (fun x => -↑(n + 1) * sin x * cos x ^ n) MeasureTheory.volume a b"}, {"context": ["a b : ℝ", "n : ℕ", "C : ℝ := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a", "h : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)", "hu : ∀ x ∈ [[a, b]], HasDerivAt (fun y => cos y ^ (n + 1)) (-↑(n + 1) * sin x * cos x ^ n) x", "hv : ∀ x ∈ [[a, b]], HasDerivAt sin (cos x) x"], "goal": "IntervalIntegrable cos MeasureTheory.volume a b"}], "premise": [26326], "state_str": "case refine_1\na b : ℝ\nn : ℕ\nC : ℝ := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a\nh : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)\nhu : ∀ x ∈ [[a, b]], HasDerivAt (fun y => cos y ^ (n + 1)) (-↑(n + 1) * sin x * cos x ^ n) x\nhv : ∀ x ∈ [[a, b]], HasDerivAt sin (cos x) x\n⊢ IntervalIntegrable (fun x => -↑(n + 1) * sin x * cos x ^ n) MeasureTheory.volume a b\n\ncase refine_2\na b : ℝ\nn : ℕ\nC : ℝ := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a\nh : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)\nhu : ∀ x ∈ [[a, b]], HasDerivAt (fun y => cos y ^ (n + 1)) (-↑(n + 1) * sin x * cos x ^ n) x\nhv : ∀ x ∈ [[a, b]], HasDerivAt sin (cos x) x\n⊢ IntervalIntegrable cos MeasureTheory.volume a b"}
+{"state": [{"context": ["m n : ℕ"], "goal": "(subNatNat m n).bodd = xor m.bodd n.bodd"}], "premise": [2483, 129210, 144078], "state_str": "m n : ℕ\n⊢ (subNatNat m n).bodd = xor m.bodd n.bodd"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "a a₁ a₂ : α", "b b₁ b₂ : β", "inst✝³ : OrderedRing α", "inst✝² : OrderedAddCommGroup β", "inst✝¹ : Module α β", "inst✝ : PosSMulMono α β", "h : b₁ ≤ b₂", "ha : a ≤ 0"], "goal": "a • b₂ ≤ a • b₁"}], "premise": [105657, 110032, 119769], "state_str": "α : Type u_1\nβ : Type u_2\na a₁ a₂ : α\nb b₁ b₂ : β\ninst✝³ : OrderedRing α\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : Module α β\ninst✝ : PosSMulMono α β\nh : b₁ ≤ b₂\nha : a ≤ 0\n⊢ a • b₂ ≤ a • b₁"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "a a₁ a₂ : α", "b b₁ b₂ : β", "inst✝³ : OrderedRing α", "inst✝² : OrderedAddCommGroup β", "inst✝¹ : Module α β", "inst✝ : PosSMulMono α β", "h : b₁ ≤ b₂", "ha : a ≤ 0"], "goal": "-a • b₁ ≤ -a • b₂"}], "premise": [104749, 104871], "state_str": "α : Type u_1\nβ : Type u_2\na a₁ a₂ : α\nb b₁ b₂ : β\ninst✝³ : OrderedRing α\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : Module α β\ninst✝ : PosSMulMono α β\nh : b₁ ≤ b₂\nha : a ≤ 0\n⊢ -a • b₁ ≤ -a • b₂"}
+{"state": [{"context": ["C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "A : C", "inst✝ : HasTerminal C", "X : MonoOver (⊤_ C)", "Z : C", "f g : Z ⟶ X.obj.left"], "goal": "f = g"}], "premise": [96191], "state_str": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nA : C\ninst✝ : HasTerminal C\nX : MonoOver (⊤_ C)\nZ : C\nf g : Z ⟶ X.obj.left\n⊢ f = g"}
+{"state": [{"context": ["f : ℕ → ℂ", "s : ℂ", "hs : abscissaOfAbsConv f < ↑s.re"], "goal": "LSeriesSummable f s"}], "premise": [19262, 70141, 131585, 131592, 147085], "state_str": "f : ℕ → ℂ\ns : ℂ\nhs : abscissaOfAbsConv f < ↑s.re\n⊢ LSeriesSummable f s"}
+{"state": [{"context": ["f : ℕ → ℂ", "s : ℂ", "y : ℝ", "hy : LSeriesSummable f ↑y", "hys : y < s.re"], "goal": "LSeriesSummable f s"}], "premise": [22362, 148280], "state_str": "case intro.intro\nf : ℕ → ℂ\ns : ℂ\ny : ℝ\nhy : LSeriesSummable f ↑y\nhys : y < s.re\n⊢ LSeriesSummable f s"}
+{"state": [{"context": ["α : Type u", "β : Type v", "γ : Type w", "inst✝⁴ : TopologicalSpace α", "inst✝³ : LinearOrder α", "inst✝² : OrderTop α", "inst✝¹ : OrderTopology α", "inst✝ : Nontrivial α"], "goal": "(𝓝 ⊤).HasBasis (fun a => a < ⊤) fun a => Ioi a"}], "premise": [2011, 18794, 71951], "state_str": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : LinearOrder α\ninst✝² : OrderTop α\ninst✝¹ : OrderTopology α\ninst✝ : Nontrivial α\n⊢ (𝓝 ⊤).HasBasis (fun a => a < ⊤) fun a => Ioi a"}
+{"state": [{"context": ["α : Type u", "β : Type v", "γ : Type w", "inst✝⁴ : TopologicalSpace α", "inst✝³ : LinearOrder α", "inst✝² : OrderTop α", "inst✝¹ : OrderTopology α", "inst✝ : Nontrivial α", "this : ∃ x, x < ⊤"], "goal": "(𝓝 ⊤).HasBasis (fun a => a < ⊤) fun a => Ioi a"}], "premise": [20336, 20338, 55931, 57165], "state_str": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : LinearOrder α\ninst✝² : OrderTop α\ninst✝¹ : OrderTopology α\ninst✝ : Nontrivial α\nthis : ∃ x, x < ⊤\n⊢ (𝓝 ⊤).HasBasis (fun a => a < ⊤) fun a => Ioi a"}
+{"state": [{"context": ["X : Type u_1", "inst✝² : NormedAddCommGroup X", "M : Type u_2", "inst✝¹ : Ring M", "inst✝ : Module M X", "P : M", "h : IsLprojection X P", "x : X"], "goal": "‖x‖ = ‖(1 - P) • x‖ + ‖(1 - (1 - P)) • x‖"}], "premise": [118068, 119708], "state_str": "X : Type u_1\ninst✝² : NormedAddCommGroup X\nM : Type u_2\ninst✝¹ : Ring M\ninst✝ : Module M X\nP : M\nh : IsLprojection X P\nx : X\n⊢ ‖x‖ = ‖(1 - P) • x‖ + ‖(1 - (1 - P)) • x‖"}
+{"state": [{"context": ["X : Type u_1", "inst✝² : NormedAddCommGroup X", "M : Type u_2", "inst✝¹ : Ring M", "inst✝ : Module M X", "P : M", "h : IsLprojection X P", "x : X"], "goal": "‖x‖ = ‖P • x‖ + ‖(1 - P) • x‖"}], "premise": [40389], "state_str": "X : Type u_1\ninst✝² : NormedAddCommGroup X\nM : Type u_2\ninst✝¹ : Ring M\ninst✝ : Module M X\nP : M\nh : IsLprojection X P\nx : X\n⊢ ‖x‖ = ‖P • x‖ + ‖(1 - P) • x‖"}
+{"state": [{"context": ["α : Type u", "r : α → α → Prop", "a : α", "l✝ : List α", "inst✝ : PartialOrder α", "l : List α", "h₁ : Sorted (fun x x_1 => x ≥ x_1) l", "h₂ : l.Nodup"], "goal": "Pairwise (fun x x_1 => x_1 ≠ x) l"}], "premise": [1691], "state_str": "α : Type u\nr : α → α → Prop\na : α\nl✝ : List α\ninst✝ : PartialOrder α\nl : List α\nh₁ : Sorted (fun x x_1 => x ≥ x_1) l\nh₂ : l.Nodup\n⊢ Pairwise (fun x x_1 => x_1 ≠ x) l"}
+{"state": [{"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "inst✝² : Ring k", "inst✝¹ : AddCommGroup V", "inst✝ : Module k V", "S : AffineSpace V P", "ι : Type u_4", "s : Finset ι", "ι₂ : Type u_5", "s₂ : Finset ι₂", "w₁ w₂ : ι → k", "p : ι → P"], "goal": "(affineCombination k s p) w₁ -ᵥ (affineCombination k s p) w₂ = (s.weightedVSub p) (w₁ - w₂)"}], "premise": [84200, 84279, 115868], "state_str": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\nι : Type u_4\ns : Finset ι\nι₂ : Type u_5\ns₂ : Finset ι₂\nw₁ w₂ : ι → k\np : ι → P\n⊢ (affineCombination k s p) w₁ -ᵥ (affineCombination k s p) w₂ = (s.weightedVSub p) (w₁ - w₂)"}
+{"state": [{"context": ["G : Type u_1", "inst✝⁷ : TopologicalSpace G", "inst✝⁶ : Group G", "inst✝⁵ : TopologicalGroup G", "inst✝⁴ : MeasurableSpace G", "inst✝³ : BorelSpace G", "inst✝² : LocallyCompactSpace G", "μ' μ : Measure G", "inst✝¹ : μ.IsHaarMeasure", "inst✝ : μ'.IsHaarMeasure", "s : Set G", "hs : MeasurableSet s", "h's : μ.IsEverywherePos s", "ν : Measure G := μ'.haarScalarFactor μ • μ"], "goal": "μ' s = ν s"}], "premise": [15884, 54090, 55409, 55410, 64785], "state_str": "G : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\n⊢ μ' s = ν s"}
+{"state": [{"context": ["G : Type u_1", "inst✝⁷ : TopologicalSpace G", "inst✝⁶ : Group G", "inst✝⁵ : TopologicalGroup G", "inst✝⁴ : MeasurableSpace G", "inst✝³ : BorelSpace G", "inst✝² : LocallyCompactSpace G", "μ' μ : Measure G", "inst✝¹ : μ.IsHaarMeasure", "inst✝ : μ'.IsHaarMeasure", "s : Set G", "hs : MeasurableSet s", "h's : μ.IsEverywherePos s", "ν : Measure G := μ'.haarScalarFactor μ • μ", "k : Set G", "k_comp : IsCompact k", "k_closed : IsClosed k", "k_mem : k ∈ 𝓝 1"], "goal": "μ' s = ν s"}], "premise": [55499], "state_str": "case intro.intro.intro\nG : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\nk : Set G\nk_comp : IsCompact k\nk_closed : IsClosed k\nk_mem : k ∈ 𝓝 1\n⊢ μ' s = ν s"}
+{"state": [{"context": ["G : Type u_1", "inst✝⁷ : TopologicalSpace G", "inst✝⁶ : Group G", "inst✝⁵ : TopologicalGroup G", "inst✝⁴ : MeasurableSpace G", "inst✝³ : BorelSpace G", "inst✝² : LocallyCompactSpace G", "μ' μ : Measure G", "inst✝¹ : μ.IsHaarMeasure", "inst✝ : μ'.IsHaarMeasure", "s : Set G", "hs : MeasurableSet s", "h's : μ.IsEverywherePos s", "ν : Measure G := μ'.haarScalarFactor μ • μ", "k : Set G", "k_comp : IsCompact k", "k_closed : IsClosed k", "k_mem : k ∈ 𝓝 1", "one_k : 1 ∈ k", "A : Set (Set G) := {t | t ⊆ s ∧ t.PairwiseDisjoint fun x => x • k}", "m : Set G", "m_max : ∀ a ∈ A, m ⊆ a → a = m", "mA : m ⊆ s ∧ m.PairwiseDisjoint fun x => x • k", "sm : s ⊆ ⋃ x ∈ m, x • (k * k⁻¹)"], "goal": "μ' s = ν s"}], "premise": [133383], "state_str": "case intro.intro.intro.intro.intro\nG : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\nk : Set G\nk_comp : IsCompact k\nk_closed : IsClosed k\nk_mem : k ∈ 𝓝 1\none_k : 1 ∈ k\nA : Set (Set G) := {t | t ⊆ s ∧ t.PairwiseDisjoint fun x => x • k}\nm : Set G\nm_max : ∀ a ∈ A, m ⊆ a → a = m\nmA : m ⊆ s ∧ m.PairwiseDisjoint fun x => x • k\nsm : s ⊆ ⋃ x ∈ m, x • (k * k⁻¹)\n⊢ μ' s = ν s"}
+{"state": [{"context": ["R : Type v", "inst✝¹ : NonAssocSemiring R", "inst✝ : StarRing R", "S T : StarSubsemiring R", "h : S.toSubsemiring = T.toSubsemiring", "x : R"], "goal": "x ∈ S ↔ x ∈ T"}], "premise": [1713, 110748], "state_str": "R : Type v\ninst✝¹ : NonAssocSemiring R\ninst✝ : StarRing R\nS T : StarSubsemiring R\nh : S.toSubsemiring = T.toSubsemiring\nx : R\n⊢ x ∈ S ↔ x ∈ T"}
+{"state": [{"context": ["α : Type u", "β : Type v", "γ : Type u_1", "inst✝² : OmegaCompletePartialOrder α", "inst✝¹ : OmegaCompletePartialOrder β", "inst✝ : OmegaCompletePartialOrder γ", "f : α →o β", "g : β →o γ", "c : Chain α"], "goal": "OrderHom.id (ωSup c) = ωSup (c.map OrderHom.id)"}], "premise": [10874], "state_str": "α : Type u\nβ : Type v\nγ : Type u_1\ninst✝² : OmegaCompletePartialOrder α\ninst✝¹ : OmegaCompletePartialOrder β\ninst✝ : OmegaCompletePartialOrder γ\nf : α →o β\ng : β →o γ\nc : Chain α\n⊢ OrderHom.id (ωSup c) = ωSup (c.map OrderHom.id)"}
+{"state": [{"context": ["R : Type u", "K : Type u'", "M : Type v", "V : Type v'", "M₂ : Type w", "V₂ : Type w'", "M₃ : Type y", "V₃ : Type y'", "M₄ : Type z", "ι : Type x", "M₅ : Type u_1", "M₆ : Type u_2", "inst✝⁴ : Ring R", "N : Type u_3", "inst✝³ : AddCommGroup M", "inst✝² : Module R M", "inst✝¹ : AddCommGroup N", "inst✝ : Module R N", "f : M × N →ₗ[R] M", "i : Injective ⇑f", "n : ℕ"], "goal": "Submodule.map (((f.tunnel' i n).fst.subtype ∘ₗ ↑(f.tunnel' i n).snd.symm) ∘ₗ f) (Submodule.snd R M N) ⊔ Submodule.map (((f.tunnel' i n).fst.subtype ∘ₗ ↑(f.tunnel' i n).snd.symm) ∘ₗ f) (Submodule.fst R M N) ≤ (f.tunnel' i n).fst"}], "premise": [14537, 84875, 110266, 110285], "state_str": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type u_1\nM₆ : Type u_2\ninst✝⁴ : Ring R\nN : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : M × N →ₗ[R] M\ni : Injective ⇑f\nn : ℕ\n⊢ Submodule.map (((f.tunnel' i n).fst.subtype ∘ₗ ↑(f.tunnel' i n).snd.symm) ∘ₗ f) (Submodule.snd R M N) ⊔\n      Submodule.map (((f.tunnel' i n).fst.subtype ∘ₗ ↑(f.tunnel' i n).snd.symm) ∘ₗ f) (Submodule.fst R M N) ≤\n    (f.tunnel' i n).fst"}
+{"state": [{"context": ["R : Type u", "K : Type u'", "M : Type v", "V : Type v'", "M₂ : Type w", "V₂ : Type w'", "M₃ : Type y", "V₃ : Type y'", "M₄ : Type z", "ι : Type x", "M₅ : Type u_1", "M₆ : Type u_2", "inst✝⁴ : Ring R", "N : Type u_3", "inst✝³ : AddCommGroup M", "inst✝² : Module R M", "inst✝¹ : AddCommGroup N", "inst✝ : Module R N", "f : M × N →ₗ[R] M", "i : Injective ⇑f", "n : ℕ"], "goal": "Submodule.map (f.tunnel' i n).fst.subtype (Submodule.map (↑(f.tunnel' i n).snd.symm) (Submodule.map f ⊤)) ≤ (f.tunnel' i n).fst"}], "premise": [109960], "state_str": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type u_1\nM₆ : Type u_2\ninst✝⁴ : Ring R\nN : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : M × N →ₗ[R] M\ni : Injective ⇑f\nn : ℕ\n⊢ Submodule.map (f.tunnel' i n).fst.subtype (Submodule.map (↑(f.tunnel' i n).snd.symm) (Submodule.map f ⊤)) ≤\n    (f.tunnel' i n).fst"}
+{"state": [{"context": ["R : Type u_1", "inst✝² : CommSemiring R", "M : Type u_2", "inst✝¹ : AddCommMonoid M", "inst✝ : Module R M", "x y : M"], "goal": "(fun m => (RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m)) (x + y) = (fun m => (RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m)) x + (fun m => (RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m)) y"}], "premise": [117079], "state_str": "R : Type u_1\ninst✝² : CommSemiring R\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx y : M\n⊢ (fun m => (RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m)) (x + y) =\n    (fun m => (RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m)) x +\n      (fun m => (RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m)) y"}
+{"state": [{"context": ["R : Type u_1", "inst✝² : CommSemiring R", "M : Type u_2", "inst✝¹ : AddCommMonoid M", "inst✝ : Module R M", "x y : M"], "goal": "(fun m => (RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m)) (x + y) = (RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R x + FreeAlgebra.ι R y)"}], "premise": [125016], "state_str": "R : Type u_1\ninst✝² : CommSemiring R\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx y : M\n⊢ (fun m => (RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m)) (x + y) =\n    (RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R x + FreeAlgebra.ι R y)"}
+{"state": [{"context": ["R : Type u_1", "inst✝² : CommSemiring R", "M : Type u_2", "inst✝¹ : AddCommMonoid M", "inst✝ : Module R M", "r : R", "x : M"], "goal": "{ toFun := fun m => (RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m), map_add' := ⋯ }.toFun (r • x) = (RingHom.id R) r • { toFun := fun m => (RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m), map_add' := ⋯ }.toFun x"}], "premise": [8242], "state_str": "R : Type u_1\ninst✝² : CommSemiring R\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nr : R\nx : M\n⊢ { toFun := fun m => (RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m), map_add' := ⋯ }.toFun (r • x) =\n    (RingHom.id R) r • { toFun := fun m => (RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m), map_add' := ⋯ }.toFun x"}
+{"state": [{"context": ["R : Type u_1", "inst✝² : CommSemiring R", "M : Type u_2", "inst✝¹ : AddCommMonoid M", "inst✝ : Module R M", "r : R", "x : M"], "goal": "{ toFun := fun m => (RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m), map_add' := ⋯ }.toFun (r • x) = (RingQuot.mkAlgHom R (Rel R M)) ((RingHom.id R) r • FreeAlgebra.ι R x)"}], "premise": [125016], "state_str": "R : Type u_1\ninst✝² : CommSemiring R\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nr : R\nx : M\n⊢ { toFun := fun m => (RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m), map_add' := ⋯ }.toFun (r • x) =\n    (RingQuot.mkAlgHom R (Rel R M)) ((RingHom.id R) r • FreeAlgebra.ι R x)"}
+{"state": [{"context": ["E : Type uE", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : FiniteDimensional ℝ E", "H : Type uH", "inst✝³ : TopologicalSpace H", "I : ModelWithCorners ℝ E H", "M : Type uM", "inst✝² : TopologicalSpace M", "inst✝¹ : ChartedSpace H M", "inst✝ : SmoothManifoldWithCorners I M", "c : M", "f : SmoothBumpFunction I c", "x✝ x : M", "hx : ↑f x = 1 x"], "goal": "1 x ≠ 0"}], "premise": [113018], "state_str": "E : Type uE\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : FiniteDimensional ℝ E\nH : Type uH\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx✝ x : M\nhx : ↑f x = 1 x\n⊢ 1 x ≠ 0"}
+{"state": [{"context": [], "goal": "sign 0 = 0"}], "premise": [38310, 146316], "state_str": "⊢ sign 0 = 0"}
+{"state": [{"context": ["α : Type u", "inst✝¹ : DecidableEq α", "M : Type u_1", "inst✝ : CommMonoid M", "n : α →₀ ℕ", "f : (α →₀ ℕ) → (α →₀ ℕ) → M"], "goal": "∀ (i : (α →₀ ℕ) × (α →₀ ℕ)), i ∈ antidiagonal n ↔ (Equiv.prodComm (α →₀ ℕ) (α →₀ ℕ)) i ∈ antidiagonal n"}], "premise": [119708], "state_str": "α : Type u\ninst✝¹ : DecidableEq α\nM : Type u_1\ninst✝ : CommMonoid M\nn : α →₀ ℕ\nf : (α →₀ ℕ) → (α →₀ ℕ) → M\n⊢ ∀ (i : (α →₀ ℕ) × (α →₀ ℕ)), i ∈ antidiagonal n ↔ (Equiv.prodComm (α →₀ ℕ) (α →₀ ℕ)) i ∈ antidiagonal n"}
+{"state": [{"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "M : Type u_4", "N : Type u_5", "P : Type u_6", "G : Type u_7", "inst✝⁴ : Monoid M", "inst✝³ : Monoid N", "inst✝² : Monoid P", "l✝ l₁ l₂ : List M", "a : M", "l : List M", "F : Type u_8", "inst✝¹ : FunLike F M N", "inst✝ : MonoidHomClass F M N", "f : F"], "goal": "(map (⇑f) l).prod = f l.prod"}], "premise": [5093, 117064], "state_str": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst✝⁴ : Monoid M\ninst✝³ : Monoid N\ninst✝² : Monoid P\nl✝ l₁ l₂ : List M\na : M\nl : List M\nF : Type u_8\ninst✝¹ : FunLike F M N\ninst✝ : MonoidHomClass F M N\nf : F\n⊢ (map (⇑f) l).prod = f l.prod"}
+{"state": [{"context": ["p : ℕ", "inst✝ : Fact (Nat.Prime p)", "hp2✝ : Fact (p % 2 = 1)", "a : ℕ", "hap : ↑a ≠ 0", "hp2 : ↑p = ↑1"], "goal": "↑(∑ x ∈ Ico 1 (p / 2).succ, a * x) = ↑(∑ x ∈ Ico 1 (p / 2).succ, (a * x % p + p * (a * x / p)))"}], "premise": [4476], "state_str": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp2✝ : Fact (p % 2 = 1)\na : ℕ\nhap : ↑a ≠ 0\nhp2 : ↑p = ↑1\n⊢ ↑(∑ x ∈ Ico 1 (p / 2).succ, a * x) = ↑(∑ x ∈ Ico 1 (p / 2).succ, (a * x % p + p * (a * x / p)))"}
+{"state": [{"context": ["p : ℕ", "inst✝ : Fact (Nat.Prime p)", "hp2✝ : Fact (p % 2 = 1)", "a : ℕ", "hap : ↑a ≠ 0", "hp2 : ↑p = ↑1"], "goal": "↑(∑ x ∈ Ico 1 (p / 2).succ, (a * x % p + p * (a * x / p))) = ↑(∑ x ∈ Ico 1 (p / 2).succ, (↑(a * x)).val) + ↑(∑ x ∈ Ico 1 (p / 2).succ, a * x / p)"}], "premise": [138278], "state_str": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp2✝ : Fact (p % 2 = 1)\na : ℕ\nhap : ↑a ≠ 0\nhp2 : ↑p = ↑1\n⊢ ↑(∑ x ∈ Ico 1 (p / 2).succ, (a * x % p + p * (a * x / p))) =\n    ↑(∑ x ∈ Ico 1 (p / 2).succ, (↑(a * x)).val) + ↑(∑ x ∈ Ico 1 (p / 2).succ, a * x / p)"}
+{"state": [{"context": ["p : ℕ", "inst✝ : Fact (Nat.Prime p)", "hp2✝ : Fact (p % 2 = 1)", "a : ℕ", "hap : ↑a ≠ 0", "hp2 : ↑p = ↑1"], "goal": "↑(∑ x ∈ Ico 1 (p / 2).succ, (a * x % p + p * (a * x / p))) = ↑(∑ x ∈ Ico 1 (p / 2).succ, a * x % p) + ↑(∑ x ∈ Ico 1 (p / 2).succ, a * x / p)"}], "premise": [125068, 125099, 127008, 142648, 143126], "state_str": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp2✝ : Fact (p % 2 = 1)\na : ℕ\nhap : ↑a ≠ 0\nhp2 : ↑p = ↑1\n⊢ ↑(∑ x ∈ Ico 1 (p / 2).succ, (a * x % p + p * (a * x / p))) =\n    ↑(∑ x ∈ Ico 1 (p / 2).succ, a * x % p) + ↑(∑ x ∈ Ico 1 (p / 2).succ, a * x / p)"}
+{"state": [{"context": ["p : ℕ", "inst✝ : Fact (Nat.Prime p)", "hp2✝ : Fact (p % 2 = 1)", "a : ℕ", "hap : ↑a ≠ 0", "hp2 : ↑p = ↑1"], "goal": "↑(∑ x ∈ Ico 1 (p / 2).succ, (↑(a * x)).val) = ∑ x ∈ Ico 1 (p / 2).succ, ↑((↑a * ↑x).valMinAbs + ↑(if (↑a * ↑x).val ≤ p / 2 then 0 else p))"}], "premise": [2100, 138415], "state_str": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp2✝ : Fact (p % 2 = 1)\na : ℕ\nhap : ↑a ≠ 0\nhp2 : ↑p = ↑1\n⊢ ↑(∑ x ∈ Ico 1 (p / 2).succ, (↑(a * x)).val) =\n    ∑ x ∈ Ico 1 (p / 2).succ, ↑((↑a * ↑x).valMinAbs + ↑(if (↑a * ↑x).val ≤ p / 2 then 0 else p))"}
+{"state": [{"context": ["p : ℕ", "inst✝ : Fact (Nat.Prime p)", "hp2✝ : Fact (p % 2 = 1)", "a : ℕ", "hap : ↑a ≠ 0", "hp2 : ↑p = ↑1"], "goal": "↑(∑ x ∈ Ico 1 (p / 2).succ, (↑(a * x)).val) = ∑ x ∈ Ico 1 (p / 2).succ, ↑↑(↑a * ↑x).val"}], "premise": [125099], "state_str": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp2✝ : Fact (p % 2 = 1)\na : ℕ\nhap : ↑a ≠ 0\nhp2 : ↑p = ↑1\n⊢ ↑(∑ x ∈ Ico 1 (p / 2).succ, (↑(a * x)).val) = ∑ x ∈ Ico 1 (p / 2).succ, ↑↑(↑a * ↑x).val"}
+{"state": [{"context": ["p : ℕ", "inst✝ : Fact (Nat.Prime p)", "hp2✝ : Fact (p % 2 = 1)", "a : ℕ", "hap : ↑a ≠ 0", "hp2 : ↑p = ↑1"], "goal": "∑ x ∈ Ico 1 (p / 2).succ, ↑((↑a * ↑x).valMinAbs + ↑(if (↑a * ↑x).val ≤ p / 2 then 0 else p)) = ↑(filter (fun x => p / 2 < (↑a * ↑x).val) (Ico 1 (p / 2).succ)).card + ↑(∑ x ∈ Ico 1 (p / 2).succ, (↑a * ↑x).valMinAbs.natAbs)"}], "premise": [119708, 125099, 127008, 127074], "state_str": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp2✝ : Fact (p % 2 = 1)\na : ℕ\nhap : ↑a ≠ 0\nhp2 : ↑p = ↑1\n⊢ ∑ x ∈ Ico 1 (p / 2).succ, ↑((↑a * ↑x).valMinAbs + ↑(if (↑a * ↑x).val ≤ p / 2 then 0 else p)) =\n    ↑(filter (fun x => p / 2 < (↑a * ↑x).val) (Ico 1 (p / 2).succ)).card +\n      ↑(∑ x ∈ Ico 1 (p / 2).succ, (↑a * ↑x).valMinAbs.natAbs)"}
+{"state": [{"context": ["p : ℕ", "inst✝ : Fact (Nat.Prime p)", "hp2✝ : Fact (p % 2 = 1)", "a : ℕ", "hap : ↑a ≠ 0", "hp2 : ↑p = ↑1"], "goal": "↑(filter (fun x => p / 2 < (↑a * ↑x).val) (Ico 1 (p / 2).succ)).card + ↑(∑ x ∈ Ico 1 (p / 2).succ, (↑a * ↑x).valMinAbs.natAbs) = ↑(filter (fun x => p / 2 < (↑a * ↑x).val) (Ico 1 (p / 2).succ)).card + ↑(∑ x ∈ Ico 1 (p / 2).succ, x)"}], "premise": [21926, 126881], "state_str": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp2✝ : Fact (p % 2 = 1)\na : ℕ\nhap : ↑a ≠ 0\nhp2 : ↑p = ↑1\n⊢ ↑(filter (fun x => p / 2 < (↑a * ↑x).val) (Ico 1 (p / 2).succ)).card +\n      ↑(∑ x ∈ Ico 1 (p / 2).succ, (↑a * ↑x).valMinAbs.natAbs) =\n    ↑(filter (fun x => p / 2 < (↑a * ↑x).val) (Ico 1 (p / 2).succ)).card + ↑(∑ x ∈ Ico 1 (p / 2).succ, x)"}
+{"state": [{"context": ["α : Type u", "inst✝ : LinearOrderedCommGroup α", "a b c : α"], "goal": "a⁻¹ ≤ a ↔ 1 ≤ a"}], "premise": [105596], "state_str": "α : Type u\ninst✝ : LinearOrderedCommGroup α\na b c : α\n⊢ a⁻¹ ≤ a ↔ 1 ≤ a"}
+{"state": [{"context": ["K : Type u_1", "inst✝¹ : Field K", "x : (𝓞 K)ˣ", "inst✝ : NumberField K"], "goal": "x ∈ torsion K ↔ ∀ (w : InfinitePlace K), w ((algebraMap (𝓞 K) K) ↑x) = 1"}], "premise": [6219, 23314], "state_str": "K : Type u_1\ninst✝¹ : Field K\nx : (𝓞 K)ˣ\ninst✝ : NumberField K\n⊢ x ∈ torsion K ↔ ∀ (w : InfinitePlace K), w ((algebraMap (𝓞 K) K) ↑x) = 1"}
+{"state": [{"context": ["K : Type u_1", "inst✝¹ : Field K", "x : (𝓞 K)ˣ", "inst✝ : NumberField K"], "goal": "IsOfFinOrder x ↔ ∀ (φ : K →+* ℂ), ‖φ ((algebraMap (𝓞 K) K) ↑x)‖ = 1"}], "premise": [1674, 8311, 8320, 43320], "state_str": "K : Type u_1\ninst✝¹ : Field K\nx : (𝓞 K)ˣ\ninst✝ : NumberField K\n⊢ IsOfFinOrder x ↔ ∀ (φ : K →+* ℂ), ‖φ ((algebraMap (𝓞 K) K) ↑x)‖ = 1"}
+{"state": [{"context": ["K : Type u_1", "inst✝¹ : Field K", "x : (𝓞 K)ˣ", "inst✝ : NumberField K", "h : ∀ (φ : K →+* ℂ), ‖φ ((algebraMap (𝓞 K) K) ↑x)‖ = 1"], "goal": "∃ n, 0 < n ∧ x ^ n = 1"}], "premise": [22030, 23293], "state_str": "K : Type u_1\ninst✝¹ : Field K\nx : (𝓞 K)ˣ\ninst✝ : NumberField K\nh : ∀ (φ : K →+* ℂ), ‖φ ((algebraMap (𝓞 K) K) ↑x)‖ = 1\n⊢ ∃ n, 0 < n ∧ x ^ n = 1"}
+{"state": [{"context": ["K : Type u_1", "inst✝¹ : Field K", "x : (𝓞 K)ˣ", "inst✝ : NumberField K", "h : ∀ (φ : K →+* ℂ), ‖φ ((algebraMap (𝓞 K) K) ↑x)‖ = 1", "n : ℕ", "hn : 0 < n", "hx : (algebraMap (𝓞 K) K) ↑x ^ n = 1"], "goal": "∃ n, 0 < n ∧ x ^ n = 1"}], "premise": [21863, 21865, 22014], "state_str": "case intro.intro\nK : Type u_1\ninst✝¹ : Field K\nx : (𝓞 K)ˣ\ninst✝ : NumberField K\nh : ∀ (φ : K →+* ℂ), ‖φ ((algebraMap (𝓞 K) K) ↑x)‖ = 1\nn : ℕ\nhn : 0 < n\nhx : (algebraMap (𝓞 K) K) ↑x ^ n = 1\n⊢ ∃ n, 0 < n ∧ x ^ n = 1"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "δ : Type u_4", "mδ : MeasurableSpace δ", "γ✝ : Type u_5", "mγ✝ : MeasurableSpace γ✝", "f✝ : β → γ✝", "g : γ✝ → α", "κ : Kernel α β", "γ : Type u_6", "mγ : MeasurableSpace γ", "f : β → γ", "hf : Measurable f"], "goal": "(κ.map (fun a => (a, f a)) ⋯).fst = κ"}], "premise": [28024, 74253, 74291], "state_str": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nδ : Type u_4\nmδ : MeasurableSpace δ\nγ✝ : Type u_5\nmγ✝ : MeasurableSpace γ✝\nf✝ : β → γ✝\ng : γ✝ → α\nκ : Kernel α β\nγ : Type u_6\nmγ : MeasurableSpace γ\nf : β → γ\nhf : Measurable f\n⊢ (κ.map (fun a => (a, f a)) ⋯).fst = κ"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "ι : Type u_5", "R : Type u_6", "R' : Type u_7", "m : MeasurableSpace α", "μ μ₁ μ₂ : Measure α", "s✝ s₁ s₂ t✝ : Set α", "s : ℕ → Set α", "t : Set α", "h : ∀ (n : ℕ), s n =ᶠ[ae μ] t"], "goal": "liminf s atTop =ᶠ[ae μ] t"}], "premise": [27622], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t✝ : Set α\ns : ℕ → Set α\nt : Set α\nh : ∀ (n : ℕ), s n =ᶠ[ae μ] t\n⊢ liminf s atTop =ᶠ[ae μ] t"}
+{"state": [{"context": ["α : Type u_1", "inst✝¹ : CancelCommMonoidWithZero α", "inst✝ : UniqueFactorizationMonoid α", "a : α", "p : Associates α", "hp : Irreducible p", "hz : a ≠ 0", "h_mem : ⟨p, hp⟩ ∈ factors' a", "this : DecidableEq (Associates α)"], "goal": "p ∣ Associates.mk a"}], "premise": [76189], "state_str": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\na : α\np : Associates α\nhp : Irreducible p\nhz : a ≠ 0\nh_mem : ⟨p, hp⟩ ∈ factors' a\nthis : DecidableEq (Associates α)\n⊢ p ∣ Associates.mk a"}
+{"state": [{"context": ["α : Type u_1", "inst✝¹ : CancelCommMonoidWithZero α", "inst✝ : UniqueFactorizationMonoid α", "a : α", "p : Associates α", "hp : Irreducible p", "hz : a ≠ 0", "h_mem : ⟨p, hp⟩ ∈ factors' a", "this : DecidableEq (Associates α)"], "goal": "p ∈ (Associates.mk a).factors"}], "premise": [76172], "state_str": "case hm\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\na : α\np : Associates α\nhp : Irreducible p\nhz : a ≠ 0\nh_mem : ⟨p, hp⟩ ∈ factors' a\nthis : DecidableEq (Associates α)\n⊢ p ∈ (Associates.mk a).factors"}
+{"state": [{"context": ["α : Type u_1", "inst✝¹ : CancelCommMonoidWithZero α", "inst✝ : UniqueFactorizationMonoid α", "a : α", "p : Associates α", "hp : Irreducible p", "hz : a ≠ 0", "h_mem : ⟨p, hp⟩ ∈ factors' a", "this : DecidableEq (Associates α)"], "goal": "p ∈ ↑(factors' a)"}], "premise": [1674, 76163], "state_str": "case hm\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\na : α\np : Associates α\nhp : Irreducible p\nhz : a ≠ 0\nh_mem : ⟨p, hp⟩ ∈ factors' a\nthis : DecidableEq (Associates α)\n⊢ p ∈ ↑(factors' a)"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "s t : Set α", "f : α → β", "n : ℕ", "hst : 2 * n < (s ∪ t).ncard"], "goal": "∃ r, n < r.ncard ∧ (r ⊆ s ∨ r ⊆ t)"}], "premise": [1690, 2134, 133419, 133420, 135018, 135040, 136394, 136406, 137708], "state_str": "α : Type u_1\nβ : Type u_2\ns t : Set α\nf : α → β\nn : ℕ\nhst : 2 * n < (s ∪ t).ncard\n⊢ ∃ r, n < r.ncard ∧ (r ⊆ s ∨ r ⊆ t)"}
+{"state": [{"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "inst✝¹ : DecidableEq α", "inst✝ : InvolutiveInv α", "s✝ : Finset α", "a : α", "s : Finset α"], "goal": "↑(s.preimage (fun x => x⁻¹) ⋯) = ↑s⁻¹"}], "premise": [131763, 136235, 141596], "state_str": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝¹ : DecidableEq α\ninst✝ : InvolutiveInv α\ns✝ : Finset α\na : α\ns : Finset α\n⊢ ↑(s.preimage (fun x => x⁻¹) ⋯) = ↑s⁻¹"}
+{"state": [{"context": ["C : Type u", "inst✝¹⁰ : Category.{v, u} C", "X Y : C", "D : Type u₂", "inst✝⁹ : Category.{w, u₂} D", "E : Type u₃", "inst✝⁸ : Category.{w', u₃} E", "F : C ⥤ D", "G : D ⥤ E", "A A' B B' : C", "inst✝⁷ : HasBinaryProduct A B", "inst✝⁶ : HasBinaryProduct A' B'", "inst✝⁵ : HasBinaryProduct (F.obj A) (F.obj B)", "inst✝⁴ : HasBinaryProduct (F.obj A') (F.obj B')", "inst✝³ : HasBinaryProduct (G.obj (F.obj A)) (G.obj (F.obj B))", "inst✝² : HasBinaryProduct ((F ⋙ G).obj A) ((F ⋙ G).obj B)", "f : A ⟶ A'", "g : B ⟶ B'", "inst✝¹ : IsIso (prodComparison F A B)", "inst✝ : IsIso (prodComparison F A' B')"], "goal": "inv (prodComparison F A B) ≫ F.map (prod.map f g) = prod.map (F.map f) (F.map g) ≫ inv (prodComparison F A' B')"}], "premise": [88799, 88802, 94580, 96173], "state_str": "C : Type u\ninst✝¹⁰ : Category.{v, u} C\nX Y : C\nD : Type u₂\ninst✝⁹ : Category.{w, u₂} D\nE : Type u₃\ninst✝⁸ : Category.{w', u₃} E\nF : C ⥤ D\nG : D ⥤ E\nA A' B B' : C\ninst✝⁷ : HasBinaryProduct A B\ninst✝⁶ : HasBinaryProduct A' B'\ninst✝⁵ : HasBinaryProduct (F.obj A) (F.obj B)\ninst✝⁴ : HasBinaryProduct (F.obj A') (F.obj B')\ninst✝³ : HasBinaryProduct (G.obj (F.obj A)) (G.obj (F.obj B))\ninst✝² : HasBinaryProduct ((F ⋙ G).obj A) ((F ⋙ G).obj B)\nf : A ⟶ A'\ng : B ⟶ B'\ninst✝¹ : IsIso (prodComparison F A B)\ninst✝ : IsIso (prodComparison F A' B')\n⊢ inv (prodComparison F A B) ≫ F.map (prod.map f g) = prod.map (F.map f) (F.map g) ≫ inv (prodComparison F A' B')"}
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+{"state": [{"context": ["K : Type u_1", "v : K", "n : ℕ", "inst✝¹ : LinearOrderedField K", "inst✝ : FloorRing K", "b : K", "nth_partDen_eq : (of v).partDens.get? n = some b", "not_terminatedAt_n : ¬(of v).TerminatedAt n"], "goal": "|v - (of v).convs n| ≤ 1 / (b * (of v).dens n * (of v).dens n)"}], "premise": [118589], "state_str": "K : Type u_1\nv : K\nn : ℕ\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nb : K\nnth_partDen_eq : (of v).partDens.get? n = some b\nnot_terminatedAt_n : ¬(of v).TerminatedAt n\n⊢ |v - (of v).convs n| ≤ 1 / (b * (of v).dens n * (of v).dens n)"}
+{"state": [{"context": ["K : Type u_1", "v : K", "n : ℕ", "inst✝¹ : LinearOrderedField K", "inst✝ : FloorRing K", "b : K", "nth_partDen_eq : (of v).partDens.get? n = some b", "not_terminatedAt_n : ¬(of v).TerminatedAt n"], "goal": "1 / ((of v).dens n * (of v).dens (n + 1)) ≤ 1 / (b * (of v).dens n * (of v).dens n)"}], "premise": [118584], "state_str": "K : Type u_1\nv : K\nn : ℕ\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nb : K\nnth_partDen_eq : (of v).partDens.get? n = some b\nnot_terminatedAt_n : ¬(of v).TerminatedAt n\n⊢ 1 / ((of v).dens n * (of v).dens (n + 1)) ≤ 1 / (b * (of v).dens n * (of v).dens n)"}
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+{"state": [{"context": ["R : Type u_1", "A : Type u_2", "inst✝³ : CommRing R", "inst✝² : CommRing A", "inst✝¹ : Algebra R A", "𝒜 : ℕ → Submodule R A", "inst✝ : GradedAlgebra 𝒜", "f : A", "x : ↑↑(Proj.restrict ⋯).toPresheafedSpace"], "goal": "awayToΓ 𝒜 f ≫ (Proj.restrict ⋯).presheaf.Γgerm x = HomogeneousLocalization.mapId 𝒜 ⋯ ≫ (Proj.stalkIso' 𝒜 ↑x).toCommRingCatIso.inv ≫ (Proj.restrictStalkIso ⋯ x).inv"}], "premise": [88770, 96173], "state_str": "R : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : A\nx : ↑↑(Proj.restrict ⋯).toPresheafedSpace\n⊢ awayToΓ 𝒜 f ≫ (Proj.restrict ⋯).presheaf.Γgerm x =\n    HomogeneousLocalization.mapId 𝒜 ⋯ ≫ (Proj.stalkIso' 𝒜 ↑x).toCommRingCatIso.inv ≫ (Proj.restrictStalkIso ⋯ x).inv"}
+{"state": [{"context": ["R : Type u_1", "A : Type u_2", "inst✝³ : CommRing R", "inst✝² : CommRing A", "inst✝¹ : Algebra R A", "𝒜 : ℕ → Submodule R A", "inst✝ : GradedAlgebra 𝒜", "f : A", "x : ↑↑(Proj.restrict ⋯).toPresheafedSpace"], "goal": "awayToSection 𝒜 f ≫ (structureSheaf 𝒜).val.map (homOfLE ⋯).op ≫ (Proj.restrict ⋯).presheaf.germ ⟨x, ⋯⟩ ≫ (Proj.restrictStalkIso ⋯ x).hom = HomogeneousLocalization.mapId 𝒜 ⋯ ≫ (Proj.stalkIso' 𝒜 ↑x).toCommRingCatIso.inv"}], "premise": [68476], "state_str": "R : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : A\nx : ↑↑(Proj.restrict ⋯).toPresheafedSpace\n⊢ awayToSection 𝒜 f ≫\n      (structureSheaf 𝒜).val.map (homOfLE ⋯).op ≫\n        (Proj.restrict ⋯).presheaf.germ ⟨x, ⋯⟩ ≫ (Proj.restrictStalkIso ⋯ x).hom =\n    HomogeneousLocalization.mapId 𝒜 ⋯ ≫ (Proj.stalkIso' 𝒜 ↑x).toCommRingCatIso.inv"}
+{"state": [{"context": ["R : Type u_1", "A : Type u_2", "inst✝³ : CommRing R", "inst✝² : CommRing A", "inst✝¹ : Algebra R A", "𝒜 : ℕ → Submodule R A", "inst✝ : GradedAlgebra 𝒜", "f : A", "x : ↑↑(Proj.restrict ⋯).toPresheafedSpace"], "goal": "awayToSection 𝒜 f ≫ (structureSheaf 𝒜).val.map (homOfLE ⋯).op ≫ Presheaf.germ (structureSheaf 𝒜).val ⟨(pbo f).inclusion x, ⋯⟩ = HomogeneousLocalization.mapId 𝒜 ⋯ ≫ (Proj.stalkIso' 𝒜 ↑x).toCommRingCatIso.inv"}], "premise": [64375, 136636], "state_str": "R : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : A\nx : ↑↑(Proj.restrict ⋯).toPresheafedSpace\n⊢ awayToSection 𝒜 f ≫\n      (structureSheaf 𝒜).val.map (homOfLE ⋯).op ≫ Presheaf.germ (structureSheaf 𝒜).val ⟨(pbo f).inclusion x, ⋯⟩ =\n    HomogeneousLocalization.mapId 𝒜 ⋯ ≫ (Proj.stalkIso' 𝒜 ↑x).toCommRingCatIso.inv"}
+{"state": [{"context": ["C : Type u_1", "inst✝² : Category.{u_2, u_1} C", "inst✝¹ : Preadditive C", "K L : HomologicalComplex₂ C (up ℤ) (up ℤ)", "f : K ⟶ L", "x y : ℤ", "inst✝ : K.HasTotal (up ℤ)", "a n b' : ℤ", "hb' : a + b' = n + y", "h : a + (b' - y) = n"], "goal": "(K.ιTotal (up ℤ) a b' (n + y) hb' ≫ 𝟙 ((K.total (up ℤ)).X (n + y)) ≫ K.totalDesc fun p q hpq => (p * y).negOnePow • eqToHom ⋯ ≫ ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) p (q - y) n ⋯) = (a * y).negOnePow • eqToHom ⋯ ≫ ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) a (b' - y) n h"}], "premise": [96175, 115951], "state_str": "C : Type u_1\ninst✝² : Category.{u_2, u_1} C\ninst✝¹ : Preadditive C\nK L : HomologicalComplex₂ C (up ℤ) (up ℤ)\nf : K ⟶ L\nx y : ℤ\ninst✝ : K.HasTotal (up ℤ)\na n b' : ℤ\nhb' : a + b' = n + y\nh : a + (b' - y) = n\n⊢ (K.ιTotal (up ℤ) a b' (n + y) hb' ≫\n      𝟙 ((K.total (up ℤ)).X (n + y)) ≫\n        K.totalDesc fun p q hpq =>\n          (p * y).negOnePow • eqToHom ⋯ ≫ ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) p (q - y) n ⋯) =\n    (a * y).negOnePow • eqToHom ⋯ ≫ ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) a (b' - y) n h"}
+{"state": [{"context": ["H : Type u_1", "M : Type u_2", "H' : Type u_3", "M' : Type u_4", "X : Type u_5", "inst✝⁶ : TopologicalSpace H", "inst✝⁵ : TopologicalSpace M", "inst✝⁴ : ChartedSpace H M", "inst✝³ : TopologicalSpace H'", "inst✝² : TopologicalSpace M'", "inst✝¹ : ChartedSpace H' M'", "inst✝ : TopologicalSpace X", "G : StructureGroupoid H", "G' : StructureGroupoid H'", "e e' : PartialHomeomorph M H", "f f' : PartialHomeomorph M' H'", "P : (H → H') → Set H → H → Prop", "g g' : M → M'", "s t : Set M", "x✝ : M", "Q✝ : (H → H) → Set H → H → Prop", "hG✝ : G.LocalInvariantProp G' P", "Q : (H → H) → Set H → H → Prop", "hG : G.LocalInvariantProp G Q", "hQ : ∀ (y : H), Q id univ y", "U : Opens M", "x : ↥U"], "goal": "LiftPropAt Q (id ∘ Subtype.val) x"}], "premise": [68175], "state_str": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns t : Set M\nx✝ : M\nQ✝ : (H → H) → Set H → H → Prop\nhG✝ : G.LocalInvariantProp G' P\nQ : (H → H) → Set H → H → Prop\nhG : G.LocalInvariantProp G Q\nhQ : ∀ (y : H), Q id univ y\nU : Opens M\nx : ↥U\n⊢ LiftPropAt Q (id ∘ Subtype.val) x"}
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+{"state": [{"context": ["p : ℕ", "hp : Fact (Nat.Prime p)", "n : ℕ", "R✝ : Type u_1", "inst✝¹ : CommRing R✝", "R : Type u_2", "inst✝ : Fintype R"], "goal": "Fintype.card (TruncatedWittVector p n R) = Fintype.card R ^ n"}], "premise": [140601, 141363], "state_str": "p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\nR✝ : Type u_1\ninst✝¹ : CommRing R✝\nR : Type u_2\ninst✝ : Fintype R\n⊢ Fintype.card (TruncatedWittVector p n R) = Fintype.card R ^ n"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "R : Type u_3", "n : Type u_4", "m : Type u_5", "inst✝ : MulZeroClass α", "A : Matrix m m α", "B : Matrix n n α", "hA : A.IsDiag", "hB : B.IsDiag", "a : m", "b : n", "c : m", "d : n", "h : (a, b) ≠ (c, d)"], "goal": "kroneckerMap (fun x x_1 => x * x_1) A B (a, b) (c, d) = 0"}], "premise": [70089, 137308], "state_str": "case mk.mk\nα : Type u_1\nβ : Type u_2\nR : Type u_3\nn : Type u_4\nm : Type u_5\ninst✝ : MulZeroClass α\nA : Matrix m m α\nB : Matrix n n α\nhA : A.IsDiag\nhB : B.IsDiag\na : m\nb : n\nc : m\nd : n\nh : (a, b) ≠ (c, d)\n⊢ kroneckerMap (fun x x_1 => x * x_1) A B (a, b) (c, d) = 0"}
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+{"state": [{"context": ["X : Type u_1", "Y : Type u_2", "Z : Type u_3", "inst✝¹ : PseudoMetricSpace X", "inst✝ : PseudoMetricSpace Y", "C r : ℝ≥0", "f : X → Y", "hf : HolderWith C r f", "x y : X", "d : ℝ≥0", "hd : dist x y ≤ ↑d"], "goal": "dist (f x) (f y) ≤ ↑C * ↑d ^ ↑r"}], "premise": [61137], "state_str": "case intro\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝¹ : PseudoMetricSpace X\ninst✝ : PseudoMetricSpace Y\nC r : ℝ≥0\nf : X → Y\nhf : HolderWith C r f\nx y : X\nd : ℝ≥0\nhd : dist x y ≤ ↑d\n⊢ dist (f x) (f y) ≤ ↑C * ↑d ^ ↑r"}
+{"state": [{"context": ["X : Type u_1", "Y : Type u_2", "Z : Type u_3", "inst✝¹ : PseudoMetricSpace X", "inst✝ : PseudoMetricSpace Y", "C r : ℝ≥0", "f : X → Y", "hf : HolderWith C r f", "x y : X", "d : ℝ≥0", "hd : nndist x y ≤ d"], "goal": "nndist (f x) (f y) ≤ C * d ^ ↑r"}], "premise": [60751], "state_str": "case intro\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝¹ : PseudoMetricSpace X\ninst�� : PseudoMetricSpace Y\nC r : ℝ≥0\nf : X → Y\nhf : HolderWith C r f\nx y : X\nd : ℝ≥0\nhd : nndist x y ≤ d\n⊢ nndist (f x) (f y) ≤ C * d ^ ↑r"}
+{"state": [{"context": ["R : Type u_1", "R₁ : Type u_2", "R₂ : Type u_3", "R₃ : Type u_4", "M : Type u_5", "M₁ : Type u_6", "M₂ : Type u_7", "M₃ : Type u_8", "ι : Type u_9", "inst✝¹¹ : Semiring R", "inst✝¹⁰ : Semiring R₂", "inst✝⁹ : Semiring R₃", "inst✝⁸ : AddCommMonoid M", "inst✝⁷ : AddCommMonoid M₁", "inst✝⁶ : AddCommMonoid M₂", "inst✝⁵ : AddCommMonoid M₃", "inst✝⁴ : Module R M", "inst✝³ : Module R M₁", "inst✝² : Module R₂ M₂", "inst✝¹ : Module R₃ M₃", "σ₁₂ : R →+* R₂", "σ₂₃ : R₂ →+* R₃", "σ₁₃ : R →+* R₃", "inst✝ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃", "f : M →ₛₗ[σ₁₂] M₂", "g✝ : M₂ →ₛₗ[σ₂₃] M₃", "N : Submodule R M", "g : Module.End R ↥N", "G : Module.End R M", "h : G ∘ₗ N.subtype = N.subtype ∘ₗ g", "k : ℕ", "hG : G ^ k = 0", "m : ↥N"], "goal": "↑((g ^ k) m) = ↑(0 m)"}], "premise": [109207, 109784, 109786], "state_str": "case h.a\nR : Type u_1\nR₁ : Type u_2\nR₂ : Type u_3\nR₃ : Type u_4\nM : Type u_5\nM₁ : Type u_6\nM₂ : Type u_7\nM₃ : Type u_8\nι : Type u_9\ninst✝¹¹ : Semiring R\ninst✝¹⁰ : Semiring R₂\ninst✝⁹ : Semiring R₃\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : AddCommMonoid M₂\ninst✝⁵ : AddCommMonoid M₃\ninst✝⁴ : Module R M\ninst✝³ : Module R M₁\ninst✝² : Module R₂ M₂\ninst✝¹ : Module R₃ M₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\ninst✝ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\nf : M →ₛₗ[σ₁₂] M₂\ng✝ : M₂ →ₛₗ[σ₂₃] M₃\nN : Submodule R M\ng : Module.End R ↥N\nG : Module.End R M\nh : G ∘ₗ N.subtype = N.subtype ∘ₗ g\nk : ℕ\nhG : G ^ k = 0\nm : ↥N\n⊢ ↑((g ^ k) m) = ↑(0 m)"}
+{"state": [{"context": ["a b : Rat", "g ad bd : Nat", "hg : g = a.den.gcd b.den", "had : ad = a.den / g", "hbd : bd = b.den / g", "den : Nat := ad * b.den", "num : Int := a.num * ↑bd + b.num * ↑ad"], "goal": "num.natAbs.gcd g = num.natAbs.gcd den"}], "premise": [3533, 3616], "state_str": "a b : Rat\ng ad bd : Nat\nhg : g = a.den.gcd b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * ↑bd + b.num * ↑ad\n⊢ num.natAbs.gcd g = num.natAbs.gcd den"}
+{"state": [{"context": ["a b : Rat", "g ad bd : Nat", "hg : g = a.den.gcd b.den", "had : ad = a.den / g", "hbd : bd = b.den / g", "den : Nat := ad * b.den", "num : Int := a.num * ↑bd + b.num * ↑ad", "ae : ad * g = a.den"], "goal": "num.natAbs.gcd g = num.natAbs.gcd den"}], "premise": [3533, 3617], "state_str": "a b : Rat\ng ad bd : Nat\nhg : g = a.den.gcd b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * ↑bd + b.num * ↑ad\nae : ad * g = a.den\n⊢ num.natAbs.gcd g = num.natAbs.gcd den"}
+{"state": [{"context": ["a b : Rat", "g ad bd : Nat", "hg : g = a.den.gcd b.den", "had : ad = a.den / g", "hbd : bd = b.den / g", "den : Nat := ad * b.den", "num : Int := a.num * ↑bd + b.num * ↑ad", "ae : ad * g = a.den", "be : bd * g = b.den"], "goal": "num.natAbs.gcd g = num.natAbs.gcd den"}], "premise": [3703], "state_str": "a b : Rat\ng ad bd : Nat\nhg : g = a.den.gcd b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * ↑bd + b.num * ↑ad\nae : ad * g = a.den\nbe : bd * g = b.den\n⊢ num.natAbs.gcd g = num.natAbs.gcd den"}
+{"state": [{"context": ["a b : Rat", "g ad bd : Nat", "hg : g = a.den.gcd b.den", "had : ad = a.den / g", "hbd : bd = b.den / g", "den : Nat := ad * b.den", "num : Int := a.num * ↑bd + b.num * ↑ad", "ae : ad * g = a.den", "be : bd * g = b.den", "hden : den = ad * bd * g"], "goal": "num.natAbs.gcd g = num.natAbs.gcd den"}], "premise": [401], "state_str": "a b : Rat\ng ad bd : Nat\nhg : g = a.den.gcd b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * ↑bd + b.num * ↑ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\n⊢ num.natAbs.gcd g = num.natAbs.gcd den"}
+{"state": [{"context": ["a b : Rat", "g ad bd : Nat", "hg : g = a.den.gcd b.den", "had : ad = a.den / g", "hbd : bd = b.den / g", "den : Nat := ad * b.den", "num : Int := a.num * ↑bd + b.num * ↑ad", "ae : ad * g = a.den", "be : bd * g = b.den", "hden : den = ad * bd * g"], "goal": "(ad * bd).Coprime num.natAbs"}], "premise": [403, 588, 3629], "state_str": "case H\na b : Rat\ng ad bd : Nat\nhg : g = a.den.gcd b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * ↑bd + b.num * ↑ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\n⊢ (ad * bd).Coprime num.natAbs"}
+{"state": [{"context": ["a b : Rat", "g ad bd : Nat", "hg : g = a.den.gcd b.den", "had : ad = a.den / g", "hbd : bd = b.den / g", "den : Nat := ad * b.den", "num : Int := a.num * ↑bd + b.num * ↑ad", "ae : ad * g = a.den", "be : bd * g = b.den", "hden : den = ad * bd * g", "cop : ad.Coprime bd"], "goal": "(ad * bd).Coprime num.natAbs"}], "premise": [3250, 3617, 4195, 4209, 4212], "state_str": "case H\na b : Rat\ng ad bd : Nat\nhg : g = a.den.gcd b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * ↑bd + b.num * ↑ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\ncop : ad.Coprime bd\n⊢ (ad * bd).Coprime num.natAbs"}
+{"state": [{"context": ["a b : Rat", "g ad bd : Nat", "hg : g = a.den.gcd b.den", "had : ad = a.den / g", "hbd : bd = b.den / g", "den : Nat := ad * b.den", "num : Int := a.num * ↑bd + b.num * ↑ad", "ae : ad * g = a.den", "be : bd * g = b.den", "hden : den = ad * bd * g", "cop : ad.Coprime bd", "H1 : ∀ (d : Nat), d.gcd num.natAbs ∣ a.num.natAbs * bd ↔ d.gcd num.natAbs ∣ b.num.natAbs * ad"], "goal": "(ad * bd).Coprime num.natAbs"}], "premise": [407], "state_str": "case H\na b : Rat\ng ad bd : Nat\nhg : g = a.den.gcd b.den\nhad : ad = a.den / g\nhbd : bd = b.den / g\nden : Nat := ad * b.den\nnum : Int := a.num * ↑bd + b.num * ↑ad\nae : ad * g = a.den\nbe : bd * g = b.den\nhden : den = ad * bd * g\ncop : ad.Coprime bd\nH1 : ∀ (d : Nat), d.gcd num.natAbs ∣ a.num.natAbs * bd ↔ d.gcd num.natAbs ∣ b.num.natAbs * ad\n⊢ (ad * bd).Coprime num.natAbs"}
+{"state": [{"context": ["R S : CommRingCat", "f : R ⟶ S", "p : PrimeSpectrum ↑S"], "goal": "toStalk (↑R) ((PrimeSpectrum.comap f) p) ≫ PresheafedSpace.Hom.stalkMap (Spec.sheafedSpaceMap f) p = f ≫ toStalk (↑S) p"}], "premise": [1680, 68614, 96173, 128400, 131474], "state_str": "R S : CommRingCat\nf : R ⟶ S\np : PrimeSpectrum ↑S\n⊢ toStalk (↑R) ((PrimeSpectrum.comap f) p) ≫ PresheafedSpace.Hom.stalkMap (Spec.sheafedSpaceMap f) p =\n    f ≫ toStalk (↑S) p"}
+{"state": [{"context": ["G : Type u_1", "G' : Type u_2", "inst✝¹ : Group G", "inst✝ : Group G'", "H K L : Subgroup G", "hHK : H ≤ K", "hKL : K ≤ L"], "goal": "H.relindex K * K.relindex L = H.relindex L"}], "premise": [6527], "state_str": "G : Type u_1\nG' : Type u_2\ninst✝¹ : Group G\ninst✝ : Group G'\nH K L : Subgroup G\nhHK : H ≤ K\nhKL : K ≤ L\n⊢ H.relindex K * K.relindex L = H.relindex L"}
+{"state": [{"context": ["G : Type u_1", "G' : Type u_2", "inst✝¹ : Group G", "inst✝ : Group G'", "H K L : Subgroup G", "hHK : H ≤ K", "hKL : K ≤ L"], "goal": "(H.subgroupOf L).relindex (K.subgroupOf L) * K.relindex L = H.relindex L"}], "premise": [6522], "state_str": "G : Type u_1\nG' : Type u_2\ninst✝¹ : Group G\ninst✝ : Group G'\nH K L : Subgroup G\nhHK : H ≤ K\nhKL : K ≤ L\n⊢ (H.subgroupOf L).relindex (K.subgroupOf L) * K.relindex L = H.relindex L"}
+{"state": [{"context": ["V : Type u_1", "inst✝¹ : NormedAddCommGroup V", "inst✝ : InnerProductSpace ℝ V", "x y : V", "h : ⟪x, y⟫_ℝ = 0", "h0 : ¬x = 0"], "goal": "Real.tan (angle x (x + y)) = ‖y‖ / ‖x‖"}], "premise": [37957, 69414], "state_str": "case neg\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : ⟪x, y⟫_ℝ = 0\nh0 : ¬x = 0\n⊢ Real.tan (angle x (x + y)) = ‖y‖ / ‖x‖"}
+{"state": [{"context": ["α : Type u", "inst✝² : Group α", "inst✝¹ : LT α", "inst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1", "a b c : α"], "goal": "a⁻¹ < 1 ↔ 1 < a"}], "premise": [1713, 103895, 119730, 119825], "state_str": "α : Type u\ninst✝² : Group α\ninst✝¹ : LT α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b c : α\n⊢ a⁻¹ < 1 ↔ 1 < a"}
+{"state": [{"context": ["p : ℕ", "G : Type u_1", "inst✝ : Group G"], "goal": "Nat.card ↥⊥ = p ^ ?m.2519"}], "premise": [118542, 119739], "state_str": "p : ℕ\nG : Type u_1\ninst✝ : Group G\n⊢ Nat.card ↥⊥ = p ^ ?m.2519"}
+{"state": [{"context": ["C : Type u₁", "inst✝³ : Category.{v₁, u₁} C", "D : Type u₂", "inst✝² : Category.{v₂, u₂} D", "G : C ⥤ D", "J : Type w", "inst✝¹ : Category.{w', w} J", "F : J ⥤ C", "inst✝ : PreservesLimit F G", "c₁ c₂ : Cone F", "t : IsLimit c₁"], "goal": "∀ (j : J), G.map (t.lift c₂) ≫ (G.mapCone c₁).π.app j = (G.mapCone c₂).π.app j"}], "premise": [99919], "state_str": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nG : C ⥤ D\nJ : Type w\ninst✝¹ : Category.{w', w} J\nF : J ⥤ C\ninst✝ : PreservesLimit F G\nc₁ c₂ : Cone F\nt : IsLimit c₁\n⊢ ∀ (j : J), G.map (t.lift c₂) ≫ (G.mapCone c₁).π.app j = (G.mapCone c₂).π.app j"}
+{"state": [{"context": ["α : Type u_1", "a : α", "m n : Nat", "l : List α", "h : m < l.length"], "goal": "(l.set m a).get? n = if m = n then some a else l.get? n"}], "premise": [5073], "state_str": "α : Type u_1\na : α\nm n : Nat\nl : List α\nh : m < l.length\n⊢ (l.set m a).get? n = if m = n then some a else l.get? n"}
+{"state": [{"context": ["α : Type u_1", "a : α", "m n : Nat", "l : List α", "h : m < l.length"], "goal": "(if m = n then if m < l.length then some a else none else l[n]?) = if m = n then some a else l[n]?"}], "premise": [4995], "state_str": "α : Type u_1\na : α\nm n : Nat\nl : List α\nh : m < l.length\n⊢ (if m = n then if m < l.length then some a else none else l[n]?) = if m = n then some a else l[n]?"}
+{"state": [{"context": ["K : Type u_1", "L : Type u_2", "inst✝² : Field K", "inst✝¹ : Field L", "inst✝ : Algebra K L", "x✝ : L ≃ₐ[K] L"], "goal": "x✝ ∈ ⊥.fixingSubgroup ↔ x✝ ∈ ⊤"}], "premise": [87986, 90759], "state_str": "case h\nK : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx✝ : L ≃ₐ[K] L\n⊢ x✝ ∈ ⊥.fixingSubgroup ↔ x✝ ∈ ⊤"}
+{"state": [{"context": ["A : Type u_1", "inst✝⁵ : NormedRing A", "inst✝⁴ : NormedAlgebra ℂ A", "inst✝³ : StarRing A", "inst✝² : ContinuousStar A", "inst✝¹ : CompleteSpace A", "inst✝ : StarModule ℂ A", "a b : ↥(selfAdjoint A)", "h : Commute ↑a ↑b"], "goal": "selfAdjoint.expUnitary a * selfAdjoint.expUnitary b = selfAdjoint.expUnitary b * selfAdjoint.expUnitary a"}], "premise": [34431, 118331, 119708], "state_str": "A : Type u_1\ninst✝⁵ : NormedRing A\ninst✝⁴ : NormedAlgebra ℂ A\ninst✝³ : StarRing A\ninst✝² : ContinuousStar A\ninst✝¹ : CompleteSpace A\ninst✝ : StarModule ℂ A\na b : ↥(selfAdjoint A)\nh : Commute ↑a ↑b\n⊢ selfAdjoint.expUnitary a * selfAdjoint.expUnitary b = selfAdjoint.expUnitary b * selfAdjoint.expUnitary a"}
+{"state": [{"context": ["ι : Type u", "α : ι → Type v", "i j : ι", "l✝ : List ι", "f : (i : ι) → α i", "inst✝ : DecidableEq ι", "l : List ι", "hnd : l.Nodup", "h : ∀ (i : ι), i ∈ l", "t : (i : ι) → Set (α i)", "h2 : {i | i ∈ l} = univ"], "goal": "TProd.elim' h ⁻¹' univ.pi t = Set.tprod l t"}], "premise": [134082, 137362], "state_str": "ι : Type u\nα : ι → Type v\ni j : ι\nl✝ : List ι\nf : (i : ι) → α i\ninst✝ : DecidableEq ι\nl : List ι\nhnd : l.Nodup\nh : ∀ (i : ι), i ∈ l\nt : (i : ι) → Set (α i)\nh2 : {i | i ∈ l} = univ\n⊢ TProd.elim' h ⁻¹' univ.pi t = Set.tprod l t"}
+{"state": [{"context": ["ι : Type u", "α : ι → Type v", "i j : ι", "l✝ : List ι", "f : (i : ι) → α i", "inst✝ : DecidableEq ι", "l : List ι", "hnd : l.Nodup", "h : ∀ (i : ι), i ∈ l", "t : (i : ι) → Set (α i)", "h2 : {i | i ∈ l} = univ"], "goal": "(fun x => TProd.mk l (TProd.elim' h x)) ⁻¹' Set.tprod l t = Set.tprod l t"}], "premise": [137361], "state_str": "ι : Type u\nα : ι → Type v\ni j : ι\nl✝ : List ι\nf : (i : ι) → α i\ninst✝ : DecidableEq ι\nl : List ι\nhnd : l.Nodup\nh : ∀ (i : ι), i ∈ l\nt : (i : ι) → Set (α i)\nh2 : {i | i ∈ l} = univ\n⊢ (fun x => TProd.mk l (TProd.elim' h x)) ⁻¹' Set.tprod l t = Set.tprod l t"}
+{"state": [{"context": ["n : ℕ", "x : ℚ"], "goal": "eval (1 + x) (bernoulli n) = eval x (bernoulli n) + ↑n * x ^ (n - 1)"}], "premise": [20720], "state_str": "n : ℕ\nx : ℚ\n⊢ eval (1 + x) (bernoulli n) = eval x (bernoulli n) + ↑n * x ^ (n - 1)"}
+{"state": [{"context": ["n : ℕ", "x : ℚ", "d : ℕ", "hd : ∀ m < d, eval (1 + x) (bernoulli m) = eval x (bernoulli m) + ↑m * x ^ (m - 1)", "nz : ↑d.succ ≠ 0"], "goal": "eval (1 + x) (bernoulli d) = eval x (bernoulli d) + ↑d * x ^ (d - 1)"}], "premise": [1673, 108567], "state_str": "n : ℕ\nx : ℚ\nd : ℕ\nhd : ∀ m < d, eval (1 + x) (bernoulli m) = eval x (bernoulli m) + ↑m * x ^ (m - 1)\nnz : ↑d.succ ≠ 0\n⊢ eval (1 + x) (bernoulli d) = eval x (bernoulli d) + ↑d * x ^ (d - 1)"}
+{"state": [{"context": ["n : ℕ", "x : ℚ", "d : ℕ", "hd : ∀ m < d, eval (1 + x) (bernoulli m) = eval x (bernoulli m) + ↑m * x ^ (m - 1)", "nz : ↑d.succ ≠ 0"], "goal": "↑d.succ * eval (1 + x) (bernoulli d) = ↑d.succ * (eval x (bernoulli d) + ↑d * x ^ (d - 1))"}], "premise": [22339, 102875, 102883, 103003, 118863], "state_str": "n : ℕ\nx : ℚ\nd : ℕ\nhd : ∀ m < d, eval (1 + x) (bernoulli m) = eval x (bernoulli m) + ↑m * x ^ (m - 1)\nnz : ↑d.succ ≠ 0\n⊢ ↑d.succ * eval (1 + x) (bernoulli d) = ↑d.succ * (eval x (bernoulli d) + ↑d * x ^ (d - 1))"}
+{"state": [{"context": ["n : ℕ", "x : ℚ", "d : ℕ", "hd : ∀ m < d, eval (1 + x) (bernoulli m) = eval x (bernoulli m) + ↑m * x ^ (m - 1)", "nz : ↑d.succ ≠ 0"], "goal": "eval (1 + x) ((monomial d) ↑d.succ) - ∑ x_1 ∈ range d, ↑((d + 1).choose x_1) • (eval x (bernoulli x_1) + ↑x_1 * x ^ (x_1 - 1)) = eval x ((monomial d) ↑d.succ - ∑ k ∈ range d, ↑((d + 1).choose k) • bernoulli k) + ↑d.succ * (↑d * x ^ (d - 1))"}], "premise": [102883, 103003], "state_str": "n : ℕ\nx : ℚ\nd : ℕ\nhd : ∀ m < d, eval (1 + x) (bernoulli m) = eval x (bernoulli m) + ↑m * x ^ (m - 1)\nnz : ↑d.succ ≠ 0\n⊢ eval (1 + x) ((monomial d) ↑d.succ) -\n      ∑ x_1 ∈ range d, ↑((d + 1).choose x_1) • (eval x (bernoulli x_1) + ↑x_1 * x ^ (x_1 - 1)) =\n    eval x ((monomial d) ↑d.succ - ∑ k ∈ range d, ↑((d + 1).choose k) • bernoulli k) + ↑d.succ * (↑d * x ^ (d - 1))"}
+{"state": [{"context": ["n : ℕ", "x : ℚ", "d : ℕ", "hd : ∀ m < d, eval (1 + x) (bernoulli m) = eval x (bernoulli m) + ↑m * x ^ (m - 1)", "nz : ↑d.succ ≠ 0"], "goal": "eval (1 + x) ((monomial d) ↑d.succ) - ∑ x_1 ∈ range d, ↑((d + 1).choose x_1) • (eval x (bernoulli x_1) + ↑x_1 * x ^ (x_1 - 1)) = eval x ((monomial d) ↑d.succ) - ∑ i ∈ range d, eval x (↑((d + 1).choose i) • bernoulli i) + ↑d.succ * (↑d * x ^ (d - 1))"}], "premise": [102875, 108334], "state_str": "n : ℕ\nx : ℚ\nd : ℕ\nhd : ∀ m < d, eval (1 + x) (bernoulli m) = eval x (bernoulli m) + ↑m * x ^ (m - 1)\nnz : ↑d.succ ≠ 0\n⊢ eval (1 + x) ((monomial d) ↑d.succ) -\n      ∑ x_1 ∈ range d, ↑((d + 1).choose x_1) • (eval x (bernoulli x_1) + ↑x_1 * x ^ (x_1 - 1)) =\n    eval x ((monomial d) ↑d.succ) - ∑ i ∈ range d, eval x (↑((d + 1).choose i) • bernoulli i) +\n      ↑d.succ * (↑d * x ^ (d - 1))"}
+{"state": [{"context": ["n : ℕ", "x : ℚ", "d : ℕ", "hd : ∀ m < d, eval (1 + x) (bernoulli m) = eval x (bernoulli m) + ↑m * x ^ (m - 1)", "nz : ↑d.succ ≠ 0"], "goal": "eval (1 + x) ((monomial d) ↑d.succ) - ∑ x_1 ∈ range d, (↑((d + 1).choose x_1) • eval x (bernoulli x_1) + ↑((d + 1).choose x_1) • (↑x_1 * x ^ (x_1 - 1))) = eval x ((monomial d) ↑d.succ) - ∑ x_1 ∈ range d, ↑((d + 1).choose x_1) • eval x (bernoulli x_1) + ↑d.succ * (↑d * x ^ (d - 1))"}], "premise": [117899, 118091, 118096, 127008], "state_str": "n : ℕ\nx : ℚ\nd : ℕ\nhd : ∀ m < d, eval (1 + x) (bernoulli m) = eval x (bernoulli m) + ↑m * x ^ (m - 1)\nnz : ↑d.succ ≠ 0\n⊢ eval (1 + x) ((monomial d) ↑d.succ) -\n      ∑ x_1 ∈ range d,\n        (↑((d + 1).choose x_1) • eval x (bernoulli x_1) + ↑((d + 1).choose x_1) • (↑x_1 * x ^ (x_1 - 1))) =\n    eval x ((monomial d) ↑d.succ) - ∑ x_1 ∈ range d, ↑((d + 1).choose x_1) • eval x (bernoulli x_1) +\n      ↑d.succ * (↑d * x ^ (d - 1))"}
+{"state": [{"context": ["n : ℕ", "x : ℚ", "d : ℕ", "hd : ∀ m < d, eval (1 + x) (bernoulli m) = eval x (bernoulli m) + ↑m * x ^ (m - 1)", "nz : ↑d.succ ≠ 0"], "goal": "eval (1 + x) ((monomial d) ↑d.succ) - eval x ((monomial d) ↑d.succ) = ∑ x_1 ∈ range d, ↑((d + 1).choose x_1) • (↑x_1 * x ^ (x_1 - 1)) + ↑((d + 1).choose d) * (↑d * x ^ (d - 1))"}], "premise": [102877, 118863, 127135, 143122], "state_str": "n : ℕ\nx : ℚ\nd : ℕ\nhd : ∀ m < d, eval (1 + x) (bernoulli m) = eval x (bernoulli m) + ↑m * x ^ (m - 1)\nnz : ↑d.succ ≠ 0\n⊢ eval (1 + x) ((monomial d) ↑d.succ) - eval x ((monomial d) ↑d.succ) =\n    ∑ x_1 ∈ range d, ↑((d + 1).choose x_1) • (↑x_1 * x ^ (x_1 - 1)) + ↑((d + 1).choose d) * (↑d * x ^ (d - 1))"}
+{"state": [{"context": ["n : ℕ", "x : ℚ", "d : ℕ", "hd : ∀ m < d, eval (1 + x) (bernoulli m) = eval x (bernoulli m) + ↑m * x ^ (m - 1)", "nz : ↑d.succ ≠ 0"], "goal": "∑ x_1 ∈ range (d + 1), ↑((d + 1).choose x_1) * (↑x_1 * x ^ (x_1 - 1)) = ∑ x_1 ∈ range (d + 1), ↑((d + 1).choose x_1) • (↑x_1 * x ^ (x_1 - 1))"}], "premise": [118863], "state_str": "n : ℕ\nx : ℚ\nd : ℕ\nhd : ∀ m < d, eval (1 + x) (bernoulli m) = eval x (bernoulli m) + ↑m * x ^ (m - 1)\nnz : ↑d.succ ≠ 0\n⊢ ∑ x_1 ∈ range (d + 1), ↑((d + 1).choose x_1) * (↑x_1 * x ^ (x_1 - 1)) =\n    ∑ x_1 ∈ range (d + 1), ↑((d + 1).choose x_1) • (↑x_1 * x ^ (x_1 - 1))"}
+{"state": [{"context": ["R : Type u_1", "S : Type u_2", "inst✝² : CommSemiring R", "inst✝¹ : CommRing S", "inst✝ : Algebra R S", "I : Ideal S", "P : ⦃S : Type u_2⦄ → [inst : CommRing S] → Ideal S → Prop", "h₁ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I : Ideal S), I ^ 2 = ⊥ → P I", "h₂ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I J : Ideal S), I ≤ J → P I → P (map (Quotient.mk I) J) → P J", "n : ℕ", "hI : I ^ n = ⊥"], "goal": "P I"}], "premise": [20720], "state_str": "case intro\nR : Type u_1\nS : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nI : Ideal S\nP : ⦃S : Type u_2⦄ → [inst : CommRing S] → Ideal S → Prop\nh₁ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I : Ideal S), I ^ 2 = ⊥ → P I\nh₂ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I J : Ideal S), I ≤ J → P I → P (map (Quotient.mk I) J) → P J\nn : ℕ\nhI : I ^ n = ⊥\n⊢ P I"}
+{"state": [{"context": ["R : Type u_1", "inst✝² : CommSemiring R", "P : ⦃S : Type u_2⦄ → [inst : CommRing S] → Ideal S → Prop", "h₁ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I : Ideal S), I ^ 2 = ⊥ → P I", "h₂ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I J : Ideal S), I ≤ J → P I → P (map (Quotient.mk I) J) → P J", "S : Type u_2", "inst✝¹ : CommRing S", "inst✝ : Algebra R S", "I : Ideal S", "hI' : ¬I = ⊥", "n : ℕ", "H : ∀ m < n + 1 + 1, ∀ {S : Type u_2} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = ⊥ → P I", "hI : I ^ (n + 1 + 1) = ⊥"], "goal": "P I"}], "premise": [82116, 113020], "state_str": "case neg.succ.succ\nR : Type u_1\ninst✝² : CommSemiring R\nP : ⦃S : Type u_2⦄ → [inst : CommRing S] → Ideal S → Prop\nh₁ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I : Ideal S), I ^ 2 = ⊥ → P I\nh₂ : ∀ ⦃S : Type u_2⦄ [inst : CommRing S] (I J : Ideal S), I ≤ J → P I → P (map (Quotient.mk I) J) → P J\nS : Type u_2\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nI : Ideal S\nhI' : ¬I = ⊥\nn : ℕ\nH : ∀ m < n + 1 + 1, ∀ {S : Type u_2} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = ⊥ → P I\nhI : I ^ (n + 1 + 1) = ⊥\n⊢ P I"}
+{"state": [{"context": ["C : Type u₁", "inst✝ : Category.{v, u₁} C", "X Y : Scheme", "f : X ⟶ Y", "U : Y.Opens", "V : (↑U).Opens"], "goal": "Hom.app (f ∣_ U) V = Hom.app f (U.ι ''ᵁ V) ≫ X.presheaf.map (eqToHom ⋯).op"}], "premise": [126574, 129868], "state_str": "C : Type u₁\ninst✝ : Category.{v, u₁} C\nX Y : Scheme\nf : X ⟶ Y\nU : Y.Opens\nV : (↑U).Opens\n⊢ Hom.app (f ∣_ U) V = Hom.app f (U.ι ''ᵁ V) ≫ X.presheaf.map (eqToHom ⋯).op"}
+{"state": [{"context": ["C : Type u₁", "inst✝ : Category.{v, u₁} C", "X Y : Scheme", "f : X ⟶ Y", "U : Y.Opens", "V : (↑U).Opens", "this : Hom.app (f ∣_ U ≫ U.ι) (U.ι ''ᵁ V) = Hom.app ((f ⁻¹ᵁ U).ι ≫ f) (U.ι ''ᵁ V) ≫ (↑(f ⁻¹ᵁ U)).presheaf.map (eqToHom ⋯).op"], "goal": "Hom.app (f ∣_ U) V = Hom.app f (U.ι ''ᵁ V) ≫ X.presheaf.map (eqToHom ⋯).op"}], "premise": [97751, 97752, 126553, 126559, 126573, 129829, 129830, 129832], "state_str": "C : Type u₁\ninst✝ : Category.{v, u₁} C\nX Y : Scheme\nf : X ⟶ Y\nU : Y.Opens\nV : (↑U).Opens\nthis :\n  Hom.app (f ∣_ U ≫ U.ι) (U.ι ''ᵁ V) = Hom.app ((f ⁻¹ᵁ U).ι ≫ f) (U.ι ''ᵁ V) ≫ (↑(f ⁻¹ᵁ U)).presheaf.map (eqToHom ⋯).op\n⊢ Hom.app (f ∣_ U) V = Hom.app f (U.ι ''ᵁ V) ≫ X.presheaf.map (eqToHom ⋯).op"}
+{"state": [{"context": ["C : Type u₁", "inst✝ : Category.{v, u₁} C", "X Y : Scheme", "f : X ⟶ Y", "U : Y.Opens", "V : (↑U).Opens", "this : Y.presheaf.map (homOfLE ⋯).op ≫ Hom.appLE (f ∣_ U) (U.ι ⁻¹ᵁ U.ι ''ᵁ V) ((f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V) ⋯ = (Hom.appLE f (U.ι ''ᵁ V) (f ⁻¹ᵁ U.ι ''ᵁ V) ⋯ ≫ X.presheaf.map (homOfLE ⋯).op) ≫ X.presheaf.map ((Hom.opensFunctor (f ⁻¹ᵁ U).ι).map (eqToHom ⋯)).op"], "goal": "Hom.app (f ∣_ U) V = Hom.app f (U.ι ''ᵁ V) ≫ X.presheaf.map (eqToHom ⋯).op"}], "premise": [55752, 134143, 137138], "state_str": "C : Type u₁\ninst✝ : Category.{v, u₁} C\nX Y : Scheme\nf : X ⟶ Y\nU : Y.Opens\nV : (↑U).Opens\nthis :\n  Y.presheaf.map (homOfLE ⋯).op ≫ Hom.appLE (f ∣_ U) (U.ι ⁻¹ᵁ U.ι ''ᵁ V) ((f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V) ⋯ =\n    (Hom.appLE f (U.ι ''ᵁ V) (f ⁻¹ᵁ U.ι ''ᵁ V) ⋯ ≫ X.presheaf.map (homOfLE ⋯).op) ≫\n      X.presheaf.map ((Hom.opensFunctor (f ⁻¹ᵁ U).ι).map (eqToHom ⋯)).op\n⊢ Hom.app (f ∣_ U) V = Hom.app f (U.ι ''ᵁ V) ≫ X.presheaf.map (eqToHom ⋯).op"}
+{"state": [{"context": ["C : Type u₁", "inst✝ : Category.{v, u₁} C", "X Y : Scheme", "f : X ⟶ Y", "U : Y.Opens", "V : (↑U).Opens", "this : Y.presheaf.map (homOfLE ⋯).op ≫ Hom.appLE (f ∣_ U) (U.ι ⁻¹ᵁ U.ι ''ᵁ V) ((f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V) ⋯ = (Hom.appLE f (U.ι ''ᵁ V) (f ⁻¹ᵁ U.ι ''ᵁ V) ⋯ ≫ X.presheaf.map (homOfLE ⋯).op) ≫ X.presheaf.map ((Hom.opensFunctor (f ⁻¹ᵁ U).ι).map (eqToHom ⋯)).op", "e : U.ι ⁻¹ᵁ U.ι ''ᵁ V = V", "e' : (f ∣_ U) ⁻¹ᵁ V = (f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V"], "goal": "Hom.app (f ∣_ U) V = Hom.app f (U.ι ''ᵁ V) ≫ X.presheaf.map (eqToHom ⋯).op"}], "premise": [97751, 126549, 126553, 129829], "state_str": "C : Type u₁\ninst✝ : Category.{v, u₁} C\nX Y : Scheme\nf : X ⟶ Y\nU : Y.Opens\nV : (↑U).Opens\nthis :\n  Y.presheaf.map (homOfLE ⋯).op ≫ Hom.appLE (f ∣_ U) (U.ι ⁻¹ᵁ U.ι ''ᵁ V) ((f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V) ⋯ =\n    (Hom.appLE f (U.ι ''ᵁ V) (f ⁻¹ᵁ U.ι ''ᵁ V) ⋯ ≫ X.presheaf.map (homOfLE ⋯).op) ≫\n      X.presheaf.map ((Hom.opensFunctor (f ⁻¹ᵁ U).ι).map (eqToHom ⋯)).op\ne : U.ι ⁻¹ᵁ U.ι ''ᵁ V = V\ne' : (f ∣_ U) ⁻¹ᵁ V = (f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V\n⊢ Hom.app (f ∣_ U) V = Hom.app f (U.ι ''ᵁ V) ≫ X.presheaf.map (eqToHom ⋯).op"}
+{"state": [{"context": ["C : Type u₁", "inst✝ : Category.{v, u₁} C", "X Y : Scheme", "f : X ⟶ Y", "U : Y.Opens", "V : (↑U).Opens", "this : Y.presheaf.map (homOfLE ⋯).op ≫ Hom.appLE (f ∣_ U) (U.ι ⁻¹ᵁ U.ι ''ᵁ V) ((f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V) ⋯ = (Hom.appLE f (U.ι ''ᵁ V) (f ⁻¹ᵁ U.ι ''ᵁ V) ⋯ ≫ X.presheaf.map (homOfLE ⋯).op) ≫ X.presheaf.map ((Hom.opensFunctor (f ⁻¹ᵁ U).ι).map (eqToHom ⋯)).op", "e : U.ι ⁻¹ᵁ U.ι ''ᵁ V = V", "e' : (f ∣_ U) ⁻¹ᵁ V = (f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V"], "goal": "Hom.appLE (f ∣_ U) V ((f ∣_ U) ⁻¹ᵁ V) ⋯ = Hom.appLE f (U.ι ''��� V) ((f ⁻¹ᵁ U).ι ''ᵁ (f ∣_ U) ⁻¹ᵁ V) ⋯"}], "premise": [126550, 126552], "state_str": "C : Type u₁\ninst✝ : Category.{v, u₁} C\nX Y : Scheme\nf : X ⟶ Y\nU : Y.Opens\nV : (↑U).Opens\nthis :\n  Y.presheaf.map (homOfLE ⋯).op ≫ Hom.appLE (f ∣_ U) (U.ι ⁻¹ᵁ U.ι ''ᵁ V) ((f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V) ⋯ =\n    (Hom.appLE f (U.ι ''ᵁ V) (f ⁻¹ᵁ U.ι ''ᵁ V) ⋯ ≫ X.presheaf.map (homOfLE ⋯).op) ≫\n      X.presheaf.map ((Hom.opensFunctor (f ⁻¹ᵁ U).ι).map (eqToHom ⋯)).op\ne : U.ι ⁻¹ᵁ U.ι ''ᵁ V = V\ne' : (f ∣_ U) ⁻¹ᵁ V = (f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V\n⊢ Hom.appLE (f ∣_ U) V ((f ∣_ U) ⁻¹ᵁ V) ⋯ = Hom.appLE f (U.ι ''ᵁ V) ((f ⁻¹ᵁ U).ι ''ᵁ (f ∣_ U) ⁻¹ᵁ V) ⋯"}
+{"state": [{"context": ["C : Type u₁", "inst✝ : Category.{v, u₁} C", "X Y : Scheme", "f : X ⟶ Y", "U : Y.Opens", "V : (↑U).Opens", "this : Y.presheaf.map (homOfLE ⋯).op ≫ Hom.appLE (f ∣_ U) (U.ι ⁻¹ᵁ U.ι ''ᵁ V) ((f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V) ⋯ = (Hom.appLE f (U.ι ''ᵁ V) (f ⁻¹ᵁ U.ι ''ᵁ V) ⋯ ≫ X.presheaf.map (homOfLE ⋯).op) ≫ X.presheaf.map ((Hom.opensFunctor (f ⁻¹ᵁ U).ι).map (eqToHom ⋯)).op", "e : U.ι ⁻¹ᵁ U.ι ''ᵁ V = V", "e' : (f ∣_ U) ⁻¹ᵁ V = (f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V"], "goal": "((↑U).presheaf.map (eqToHom e).op ≫ Hom.appLE (f ∣_ U) (U.ι ⁻¹ᵁ U.ι ''ᵁ V) ((f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V) ⋯) ≫ (↑(f ⁻¹ᵁ U)).presheaf.map (eqToHom e').op = Hom.appLE f (U.ι ''ᵁ V) ((f ⁻¹ᵁ U).ι ''ᵁ (f ∣_ U) ⁻¹ᵁ V) ⋯"}], "premise": [89627, 127358, 129829, 129830, 129875], "state_str": "C : Type u₁\ninst✝ : Category.{v, u₁} C\nX Y : Scheme\nf : X ⟶ Y\nU : Y.Opens\nV : (↑U).Opens\nthis :\n  Y.presheaf.map (homOfLE ⋯).op ≫ Hom.appLE (f ∣_ U) (U.ι ⁻¹ᵁ U.ι ''ᵁ V) ((f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V) ⋯ =\n    (Hom.appLE f (U.ι ''ᵁ V) (f ⁻¹ᵁ U.ι ''ᵁ V) ⋯ ≫ X.presheaf.map (homOfLE ⋯).op) ≫\n      X.presheaf.map ((Hom.opensFunctor (f ⁻¹ᵁ U).ι).map (eqToHom ⋯)).op\ne : U.ι ⁻¹ᵁ U.ι ''ᵁ V = V\ne' : (f ∣_ U) ⁻¹ᵁ V = (f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V\n⊢ ((↑U).presheaf.map (eqToHom e).op ≫ Hom.appLE (f ∣_ U) (U.ι ⁻¹ᵁ U.ι ''ᵁ V) ((f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V) ⋯) ≫\n      (↑(f ⁻¹ᵁ U)).presheaf.map (eqToHom e').op =\n    Hom.appLE f (U.ι ''ᵁ V) ((f ⁻¹ᵁ U).ι ''ᵁ (f ∣_ U) ⁻¹ᵁ V) ⋯"}
+{"state": [{"context": ["C : Type u₁", "inst✝ : Category.{v, u₁} C", "X Y : Scheme", "f : X ⟶ Y", "U : Y.Opens", "V : (↑U).Opens", "this : Y.presheaf.map (homOfLE ⋯).op ≫ Hom.appLE (f ∣_ U) (U.ι ⁻¹ᵁ U.ι ''ᵁ V) ((f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V) ⋯ = (Hom.appLE f (U.ι ''ᵁ V) (f ⁻¹ᵁ U.ι ''ᵁ V) ⋯ ≫ X.presheaf.map (homOfLE ⋯).op) ≫ X.presheaf.map ((Hom.opensFunctor (f ⁻¹ᵁ U).ι).map (eqToHom ⋯)).op", "e : U.ι ⁻¹ᵁ U.ι ''ᵁ V = V", "e' : (f ∣_ U) ⁻¹ᵁ V = (f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V"], "goal": "(Y.presheaf.map (homOfLE ⋯).op ≫ Hom.appLE (f ∣_ U) (U.ι ⁻¹ᵁ U.ι ''ᵁ V) ((f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V) ⋯) ≫ X.presheaf.map (homOfLE ⋯).op = Hom.appLE f (U.ι ''ᵁ V) ((f ⁻¹ᵁ U).ι ''ᵁ (f ∣_ U) ⁻¹ᵁ V) ⋯"}], "premise": [126549], "state_str": "C : Type u₁\ninst✝ : Category.{v, u₁} C\nX Y : Scheme\nf : X ⟶ Y\nU : Y.Opens\nV : (↑U).Opens\nthis :\n  Y.presheaf.map (homOfLE ⋯).op ≫ Hom.appLE (f ∣_ U) (U.ι ⁻¹ᵁ U.ι ''ᵁ V) ((f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V) ⋯ =\n    (Hom.appLE f (U.ι ''ᵁ V) (f ⁻¹ᵁ U.ι ''ᵁ V) ⋯ ≫ X.presheaf.map (homOfLE ⋯).op) ≫\n      X.presheaf.map ((Hom.opensFunctor (f ⁻¹ᵁ U).ι).map (eqToHom ⋯)).op\ne : U.ι ⁻¹ᵁ U.ι ''ᵁ V = V\ne' : (f ∣_ U) ⁻¹ᵁ V = (f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V\n⊢ (Y.presheaf.map (homOfLE ⋯).op ≫ Hom.appLE (f ∣_ U) (U.ι ⁻¹ᵁ U.ι ''ᵁ V) ((f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V) ⋯) ≫\n      X.presheaf.map (homOfLE ⋯).op =\n    Hom.appLE f (U.ι ''ᵁ V) ((f ⁻¹ᵁ U).ι ''ᵁ (f ∣_ U) ⁻¹ᵁ V) ⋯"}
+{"state": [{"context": ["Ω : Type u_1", "Ω' : Type u_2", "α : Type u_3", "m : MeasurableSpace Ω", "m' : MeasurableSpace Ω'", "μ : Measure Ω", "s t : Set Ω", "hcs : μ s ≠ 0", "hs : μ s ≠ ⊤"], "goal": "((μ s)⁻¹ • μ.restrict s) univ = 1"}], "premise": [27956, 31461, 32215, 118863, 120658, 133462], "state_str": "Ω : Type u_1\nΩ' : Type u_2\nα : Type u_3\nm : MeasurableSpace Ω\nm' : MeasurableSpace Ω'\nμ : Measure Ω\ns t : Set Ω\nhcs : μ s ≠ 0\nhs : μ s ≠ ⊤\n⊢ ((μ s)⁻¹ • μ.restrict s) univ = 1"}
+{"state": [{"context": ["Ω : Type u_1", "Ω' : Type u_2", "α : Type u_3", "m : MeasurableSpace Ω", "m' : MeasurableSpace Ω'", "μ : Measure Ω", "s t : Set Ω", "hcs : μ s ≠ 0", "hs : μ s ≠ ⊤"], "goal": "(μ s)⁻¹ * μ s = 1"}], "premise": [143757], "state_str": "Ω : Type u_1\nΩ' : Type u_2\nα : Type u_3\nm : MeasurableSpace Ω\nm' : MeasurableSpace Ω'\nμ : Measure Ω\ns t : Set Ω\nhcs : μ s ≠ 0\nhs : μ s ≠ ⊤\n⊢ (μ s)⁻¹ * μ s = 1"}
+{"state": [{"context": ["R : Type u_1", "inst✝¹ : AddCommMonoid R", "m : ℕ", "inst✝ : NeZero m", "f : ℕ → R", "n : ℕ"], "goal": "f n = (∑ a : ZMod m, {n | ↑n = a}.indicator f) n"}], "premise": [120852, 123837, 127097, 131585, 140822], "state_str": "case h\nR : Type u_1\ninst✝¹ : AddCommMonoid R\nm : ℕ\ninst✝ : NeZero m\nf : ℕ → R\nn : ℕ\n⊢ f n = (∑ a : ZMod m, {n | ↑n = a}.indicator f) n"}
+{"state": [{"context": ["α : Type u_1", "E : α → Type u_2", "p q : ℝ≥0∞", "inst✝ : (i : α) → NormedAddCommGroup (E i)", "hp : p ≠ 0", "f : ↥(lp E p)", "i : α"], "goal": "‖↑f i‖ ≤ ‖f‖"}], "premise": [70039], "state_str": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nhp : p ≠ 0\nf : ↥(lp E p)\ni : α\n⊢ ‖↑f i‖ ≤ ‖f‖"}
+{"state": [{"context": ["α : Type u_1", "E : α → Type u_2", "p q : ℝ≥0∞", "inst✝ : (i : α) → NormedAddCommGroup (E i)", "hp : p ≠ 0", "f : ↥(lp E p)", "i : α", "hp' : p ≠ ⊤"], "goal": "‖↑f i‖ ≤ ‖f‖"}], "premise": [143376], "state_str": "case inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nhp : p ≠ 0\nf : ↥(lp E p)\ni : α\nhp' : p ≠ ⊤\n⊢ ‖↑f i‖ ≤ ‖f‖"}
+{"state": [{"context": ["α : Type u_1", "E : α → Type u_2", "p q : ℝ≥0∞", "inst✝ : (i : α) → NormedAddCommGroup (E i)", "hp : p ≠ 0", "f : ↥(lp E p)", "i : α", "hp' : p ≠ ⊤", "hp'' : 0 < p.toReal"], "goal": "‖↑f i‖ ≤ ‖f‖"}], "premise": [40018, 42680], "state_str": "case inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nhp : p ≠ 0\nf : ↥(lp E p)\ni : α\nhp' : p ≠ ⊤\nhp'' : 0 < p.toReal\n⊢ ‖↑f i‖ ≤ ‖f‖"}
+{"state": [{"context": ["α : Type u_1", "E : α → Type u_2", "p q : ℝ≥0∞", "inst✝ : (i : α) → NormedAddCommGroup (E i)", "hp : p ≠ 0", "f : ↥(lp E p)", "i : α", "hp' : p ≠ ⊤", "hp'' : 0 < p.toReal", "this : ∀ (i : α), 0 ≤ ‖↑f i‖ ^ p.toReal"], "goal": "‖↑f i‖ ≤ ‖f‖"}], "premise": [40087, 42680, 45196], "state_str": "case inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nhp : p ≠ 0\nf : ↥(lp E p)\ni : α\nhp' : p ≠ ⊤\nhp'' : 0 < p.toReal\nthis : ∀ (i : α), 0 ≤ ‖↑f i‖ ^ p.toReal\n⊢ ‖↑f i‖ ≤ ‖f‖"}
+{"state": [{"context": ["α : Type u_1", "E : α → Type u_2", "p q : ℝ≥0∞", "inst✝ : (i : α) → NormedAddCommGroup (E i)", "hp : p ≠ 0", "f : ↥(lp E p)", "i : α", "hp' : p ≠ ⊤", "hp'' : 0 < p.toReal", "this : ∀ (i : α), 0 ≤ ‖↑f i‖ ^ p.toReal"], "goal": "‖↑f i‖ ^ p.toReal ≤ ‖f‖ ^ p.toReal"}], "premise": [45195, 63268], "state_str": "case inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nhp : p ≠ 0\nf : ↥(lp E p)\ni : α\nhp' : p ≠ ⊤\nhp'' : 0 < p.toReal\nthis : ∀ (i : α), 0 ≤ ‖↑f i‖ ^ p.toReal\n⊢ ‖↑f i‖ ^ p.toReal ≤ ‖f‖ ^ p.toReal"}
+{"state": [{"context": ["θ : Angle"], "goal": "(θ - ↑(π / 2)).cos = θ.sin"}], "premise": [38265], "state_str": "θ : Angle\n⊢ (θ - ↑(π / 2)).cos = θ.sin"}
+{"state": [{"context": ["x✝ : ℝ"], "goal": "(↑x✝ - ↑(π / 2)).cos = (↑x✝).sin"}], "premise": [38585], "state_str": "case h\nx✝ : ℝ\n⊢ (↑x✝ - ↑(π / 2)).cos = (↑x✝).sin"}
+{"state": [{"context": ["R : Type u_1", "M : Type u_2", "r : R", "x y : M", "inst✝² : Ring R", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M", "p p' : Submodule R M"], "goal": "mk x = 0 ↔ x ∈ p"}], "premise": [82369], "state_str": "R : Type u_1\nM : Type u_2\nr : R\nx y : M\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np p' : Submodule R M\n⊢ mk x = 0 ↔ x ∈ p"}
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+{"state": [{"context": ["R : Type u", "S : Type v", "T : Type w", "a b : R", "n : ℕ", "inst✝¹ : CommRing R", "inst✝ : IsDomain R", "p✝ q✝ p q : R[X]", "hpq : p ∣ q", "h₁ : q.natDegree ≤ p.natDegree", "h₂ : q.leadingCoeff ∣ p.leadingCoeff", "r : R[X]", "hr : q = p * r", "u : Rˣ", "hu : p.leadingCoeff * ↑u = q.leadingCoeff"], "goal": "q.natDegree ≤ (p * ↑((Units.map ↑C) u)).natDegree"}], "premise": [100829], "state_str": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np✝ q✝ p q : R[X]\nhpq : p ∣ q\nh₁ : q.natDegree ≤ p.natDegree\nh₂ : q.leadingCoeff ∣ p.leadingCoeff\nr : R[X]\nhr : q = p * r\nu : Rˣ\nhu : p.leadingCoeff * ↑u = q.leadingCoeff\n⊢ q.natDegree ≤ (p * ↑((Units.map ↑C) u)).natDegree"}
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+{"state": [{"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "inst✝² : CommMonoid α", "inst✝¹ : CommMonoid β", "inst✝ : CommMonoid γ", "A A₁ A₂ : Set α", "B B₁ B₂ : Set β", "C : Set γ", "f f₁ f₂ : α → β", "g : β → γ", "m n : ℕ", "hf : IsMulFreimanHom 2 A B f", "a b c d : α", "ha : a ∈ A", "hb : b ∈ A", "hc : c ∈ A", "hd : d ∈ A", "h : a * b = c * d"], "goal": "f a * f b = f c * f d"}], "premise": [124321], "state_str": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\ninst✝ : CommMonoid γ\nA A₁ A₂ : Set α\nB B₁ B₂ : Set β\nC : Set γ\nf f₁ f₂ : α → β\ng : β → γ\nm n : ℕ\nhf : IsMulFreimanHom 2 A B f\na b c d : α\nha : a ∈ A\nhb : b ∈ A\nhc : c ∈ A\nhd : d ∈ A\nh : a * b = c * d\n⊢ f a * f b = f c * f d"}
+{"state": [{"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "inst✝² : CommMonoid α", "inst✝¹ : CommMonoid β", "inst✝ : CommMonoid γ", "A A₁ A₂ : Set α", "B B₁ B₂ : Set β", "C : Set γ", "f f₁ f₂ : α → β", "g : β → γ", "m n : ℕ", "hf : IsMulFreimanHom 2 A B f", "a b c d : α", "ha : a ∈ A", "hb : b ∈ A", "hc : c ∈ A", "hd : d ∈ A", "h : {a, b}.prod = {c, d}.prod"], "goal": "{f a, f b}.prod = {f c, f d}.prod"}], "premise": [53292, 137912], "state_str": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : CommMonoid α\ninst✝¹ : CommMonoid β\ninst✝ : CommMonoid γ\nA A₁ A₂ : Set α\nB B₁ B₂ : Set β\nC : Set γ\nf f₁ f₂ : α → β\ng : β → γ\nm n : ℕ\nhf : IsMulFreimanHom 2 A B f\na b c d : α\nha : a ∈ A\nhb : b ∈ A\nhc : c ∈ A\nhd : d ∈ A\nh : {a, b}.prod = {c, d}.prod\n⊢ {f a, f b}.prod = {f c, f d}.prod"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "ι : Type u_5", "R : Type u_6", "R' : Type u_7", "m0 : MeasurableSpace α", "inst✝¹ : MeasurableSpace β", "inst✝ : MeasurableSpace γ", "μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α", "s✝ s' t : Set α", "f✝ : α → β", "f : ι → Measure α", "s : Set α", "hs : NullMeasurableSet s (sum f)"], "goal": "(sum f) s = ∑' (i : ι), (f i) s"}], "premise": [14296, 31533], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\nf✝ : α → β\nf : ι → Measure α\ns : Set α\nhs : NullMeasurableSet s (sum f)\n⊢ (sum f) s = ∑' (i : ι), (f i) s"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "ι : Type u_5", "R : Type u_6", "R' : Type u_7", "m0 : MeasurableSpace α", "inst✝¹ : MeasurableSpace β", "inst✝ : MeasurableSpace γ", "μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α", "s✝ s' t : Set α", "f✝ : α → β", "f : ι → Measure α", "s : Set α", "hs : NullMeasurableSet s (sum f)"], "goal": "(sum f) s ≤ ∑' (i : ι), (f i) s"}], "premise": [29194], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\nf✝ : α → β\nf : ι → Measure α\ns : Set α\nhs : NullMeasurableSet s (sum f)\n⊢ (sum f) s ≤ ∑' (i : ι), (f i) s"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "ι : Type u_5", "R : Type u_6", "R' : Type u_7", "m0 : MeasurableSpace α", "inst✝¹ : MeasurableSpace β", "inst✝ : MeasurableSpace γ", "μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α", "s✝ s' t✝ : Set α", "f✝ : α → β", "f : ι → Measure α", "s : Set α", "hs : NullMeasurableSet s (sum f)", "t : Set α", "ts : t ⊆ s", "t_meas : MeasurableSet t", "ht : t =ᶠ[ae (sum f)] s"], "goal": "(sum f) s ≤ ∑' (i : ι), (f i) s"}], "premise": [16092, 27559, 27639, 31534, 59000], "state_str": "case intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ : Set α\nf✝ : α → β\nf : ι → Measure α\ns : Set α\nhs : NullMeasurableSet s (sum f)\nt : Set α\nts : t ⊆ s\nt_meas : MeasurableSet t\nht : t =ᶠ[ae (sum f)] s\n⊢ (sum f) s ≤ ∑' (i : ι), (f i) s"}
+{"state": [{"context": ["x✝ y : ℂ", "x : ℝ"], "goal": "(cexp ↑x).im = 0"}], "premise": [148281, 149092], "state_str": "x✝ y : ℂ\nx : ℝ\n⊢ (cexp ↑x).im = 0"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "inst✝ : TopologicalSpace α", "S : Set (Set α)", "hS : S.Finite", "a : α"], "goal": "𝓝[⋃₀ S] a = ⨆ s ∈ S, 𝓝[s] a"}], "premise": [57198, 135449], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝ : TopologicalSpace α\nS : Set (Set α)\nhS : S.Finite\na : α\n⊢ 𝓝[⋃₀ S] a = ⨆ s ∈ S, 𝓝[s] a"}
+{"state": [{"context": ["V : Type u_1", "P : Type u_2", "inst✝³ : NormedAddCommGroup V", "inst✝² : InnerProductSpace ℝ V", "inst✝¹ : MetricSpace P", "inst✝ : NormedAddTorsor V P", "n : ℕ", "s : Simplex ℝ P (n + 2)", "i₁ i₂ : Fin (n + 3)"], "goal": "s.mongePoint ∈ s.mongePlane i₁ i₂"}], "premise": [33499, 73436, 84414, 84427, 84430, 84480], "state_str": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nn : ℕ\ns : Simplex ℝ P (n + 2)\ni₁ i₂ : Fin (n + 3)\n⊢ s.mongePoint ∈ s.mongePlane i₁ i₂"}
+{"state": [{"context": ["V : Type u_1", "P : Type u_2", "inst✝³ : NormedAddCommGroup V", "inst✝² : InnerProductSpace ℝ V", "inst✝¹ : MetricSpace P", "inst✝ : NormedAddTorsor V P", "n : ℕ", "s : Simplex ℝ P (n + 2)", "i₁ i₂ : Fin (n + 3)"], "goal": "(∀ u ∈ Submodule.span ℝ {s.points i₁ -ᵥ s.points i₂}, ⟪s.mongePoint -ᵥ centroid ℝ {i₁, i₂}ᶜ s.points, u⟫_ℝ = 0) ∧ s.mongePoint ∈ affineSpan ℝ (Set.range s.points)"}], "premise": [73428], "state_str": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nn : ℕ\ns : Simplex ℝ P (n + 2)\ni₁ i₂ : Fin (n + 3)\n⊢ (∀ u ∈ Submodule.span ℝ {s.points i₁ -ᵥ s.points i₂}, ⟪s.mongePoint -ᵥ centroid ℝ {i₁, i₂}ᶜ s.points, u⟫_ℝ = 0) ∧\n    s.mongePoint ∈ affineSpan ℝ (Set.range s.points)"}
+{"state": [{"context": ["V : Type u_1", "P : Type u_2", "inst✝³ : NormedAddCommGroup V", "inst✝² : InnerProductSpace ℝ V", "inst✝¹ : MetricSpace P", "inst✝ : NormedAddTorsor V P", "n : ℕ", "s : Simplex ℝ P (n + 2)", "i₁ i₂ : Fin (n + 3)", "v : V", "hv : v ∈ Submodule.span ℝ {s.points i₁ -ᵥ s.points i₂}"], "goal": "⟪s.mongePoint -ᵥ centroid ℝ {i₁, i₂}ᶜ s.points, v⟫_ℝ = 0"}], "premise": [1673, 86740], "state_str": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nn : ℕ\ns : Simplex ℝ P (n + 2)\ni₁ i₂ : Fin (n + 3)\nv : V\nhv : v ∈ Submodule.span ℝ {s.points i₁ -ᵥ s.points i₂}\n⊢ ⟪s.mongePoint -ᵥ centroid ℝ {i₁, i₂}ᶜ s.points, v⟫_ℝ = 0"}
+{"state": [{"context": ["V : Type u_1", "P : Type u_2", "inst✝³ : NormedAddCommGroup V", "inst✝² : InnerProductSpace ℝ V", "inst✝¹ : MetricSpace P", "inst✝ : NormedAddTorsor V P", "n : ℕ", "s : Simplex ℝ P (n + 2)", "i₁ i₂ : Fin (n + 3)", "r : ℝ", "hv : r • (s.points i₁ -ᵥ s.points i₂) ∈ Submodule.span ℝ {s.points i₁ -ᵥ s.points i₂}"], "goal": "⟪s.mongePoint -ᵥ centroid ℝ {i₁, i₂}ᶜ s.points, r • (s.points i₁ -ᵥ s.points i₂)⟫_ℝ = 0"}], "premise": [36761, 73435, 108558], "state_str": "case intro\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nn : ℕ\ns : Simplex ℝ P (n + 2)\ni₁ i₂ : Fin (n + 3)\nr : ℝ\nhv : r • (s.points i₁ -ᵥ s.points i₂) ∈ Submodule.span ℝ {s.points i₁ -ᵥ s.points i₂}\n⊢ ⟪s.mongePoint -ᵥ centroid ℝ {i₁, i₂}ᶜ s.points, r • (s.points i₁ -ᵥ s.points i₂)⟫_ℝ = 0"}
+{"state": [{"context": ["ι : Type u_1", "𝕜 : Type u_2", "E : Type u_3", "F : Type u_4", "A : Type u_5", "inst✝² : NormedAddCommGroup E", "inst✝¹ : CompleteSpace E", "inst✝ : NormedSpace ℝ E", "a✝ b✝ c d : ℝ", "f g : ℝ → E", "μ : Measure ℝ", "a b : ℝ", "ha : IntegrableOn f (Iic a) μ", "hb : IntegrableOn f (Iic b) μ", "hab : a ≤ b"], "goal": "∫ (x : ℝ) in Iic b, f x ∂μ - ∫ (x : ℝ) in Iic a, f x ∂μ = ∫ (x : ℝ) in a..b, f x ∂μ"}], "premise": [14272, 17669, 20409, 26334, 28266, 118074], "state_str": "ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b✝ c d : ℝ\nf g : ℝ → E\nμ : Measure ℝ\na b : ℝ\nha : IntegrableOn f (Iic a) μ\nhb : IntegrableOn f (Iic b) μ\nhab : a ≤ b\n⊢ ∫ (x : ℝ) in Iic b, f x ∂μ - ∫ (x : ℝ) in Iic a, f x ∂μ = ∫ (x : ℝ) in a..b, f x ∂μ"}
+{"state": [{"context": ["ι : Type u_1", "𝕜 : Type u_2", "E : Type u_3", "F : Type u_4", "A : Type u_5", "inst✝² : NormedAddCommGroup E", "inst✝¹ : CompleteSpace E", "inst✝ : NormedSpace ℝ E", "a✝ b✝ c d : ℝ", "f g : ℝ → E", "μ : Measure ℝ", "a b : ℝ", "ha : IntegrableOn f (Iic a) μ", "hb : IntegrableOn f (Iic b) μ", "hab : a ≤ b"], "goal": "MeasurableSet (Ioc a b)"}, {"context": ["ι : Type u_1", "𝕜 : Type u_2", "E : Type u_3", "F : Type u_4", "A : Type u_5", "inst✝² : NormedAddCommGroup E", "inst✝¹ : CompleteSpace E", "inst✝ : NormedSpace ℝ E", "a✝ b✝ c d : ℝ", "f g : ℝ → E", "μ : Measure ℝ", "a b : ℝ", "ha : IntegrableOn f (Iic a) μ", "hb : IntegrableOn f (Iic b) μ", "hab : a ≤ b"], "goal": "IntegrableOn f (Iic a) μ"}, {"context": ["ι : Type u_1", "𝕜 : Type u_2", "E : Type u_3", "F : Type u_4", "A : Type u_5", "inst✝² : NormedAddCommGroup E", "inst✝¹ : CompleteSpace E", "inst✝ : NormedSpace ℝ E", "a✝ b✝ c d : ℝ", "f g : ℝ → E", "μ : Measure ℝ", "a b : ℝ", "ha : IntegrableOn f (Iic a) μ", "hb : IntegrableOn f (Iic b) μ", "hab : a ≤ b"], "goal": "IntegrableOn f (Ioc a b) μ"}], "premise": [2106, 25580, 26449], "state_str": "case ht\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b✝ c d : ℝ\nf g : ℝ → E\nμ : Measure ℝ\na b : ℝ\nha : IntegrableOn f (Iic a) μ\nhb : IntegrableOn f (Iic b) μ\nhab : a ≤ b\n⊢ MeasurableSet (Ioc a b)\n\ncase hfs\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b✝ c d : ℝ\nf g : ℝ → E\nμ : Measure ℝ\na b : ℝ\nha : IntegrableOn f (Iic a) μ\nhb : IntegrableOn f (Iic b) μ\nhab : a ≤ b\n⊢ IntegrableOn f (Iic a) μ\n\ncase hft\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b✝ c d : ℝ\nf g : ℝ → E\nμ : Measure ℝ\na b : ℝ\nha : IntegrableOn f (Iic a) μ\nhb : IntegrableOn f (Iic b) μ\nhab : a ≤ b\n⊢ IntegrableOn f (Ioc a b) μ"}
+{"state": [{"context": ["ι : Type u_1", "α : Type u_2", "inst✝¹ : MeasurableSpace α", "μ : Measure α", "inst✝ : IsFiniteMeasure μ", "r : ℝ≥0∞", "s : ℕ → Set α", "hs : ∀ (n : ℕ), MeasurableSet (s n)", "hr₀ : r ≠ 0", "hr : ∀ (n : ℕ), r ≤ μ (s n)", "M : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}", "N : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)"], "goal": "∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)"}], "premise": [28724], "state_str": "ι : Type u_1\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)\n⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)"}
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+{"state": [{"context": ["ι : Type u_1", "α : Type u_2", "inst✝¹ : MeasurableSpace α", "μ : Measure α", "inst✝ : IsFiniteMeasure μ", "r : ℝ≥0∞", "s : ℕ → Set α", "hs : ∀ (n : ℕ), MeasurableSet (s n)", "hr₀ : r ≠ 0", "hr : ∀ (n : ℕ), r ≤ μ (s n)", "M : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}", "N : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)", "hN₀ : μ N = 0", "hN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \\ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)", "f : ℕ → α → ℝ≥0∞ := fun n => (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1", "hfapp : ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a"], "goal": "∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)"}], "premise": [28027, 28794, 28807, 31187, 31306], "state_str": "ι : Type u_1\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)\nhN₀ : μ N = 0\nhN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \\ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)\nf : ℕ → α → ℝ≥0∞ := fun n => (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1\nhfapp : ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a\n⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)"}
+{"state": [{"context": ["ι : Type u_1", "α : Type u_2", "inst✝¹ : MeasurableSpace α", "μ : Measure α", "inst✝ : IsFiniteMeasure μ", "r : ℝ≥0∞", "s : ℕ → Set α", "hs : ∀ (n : ℕ), MeasurableSet (s n)", "hr₀ : r ≠ 0", "hr : ∀ (n : ℕ), r ≤ μ (s n)", "M : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}", "N : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)", "hN₀ : μ N = 0", "hN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \\ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)", "f : ℕ → α → ℝ≥0∞ := fun n => (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1", "hfapp : ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a", "hf : ∀ (n : ℕ), Measurable (f n)"], "goal": "∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)"}], "premise": [1674, 3849, 14272, 105455, 106981, 111256, 113018, 119707, 137642, 142651, 143746, 143802], "state_str": "ι : Type u_1\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)\nhN₀ : μ N = 0\nhN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \\ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)\nf : ℕ → α → ℝ≥0∞ := fun n => (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1\nhfapp : ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a\nhf : ∀ (n : ℕ), Measurable (f n)\n⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)"}
+{"state": [{"context": ["ι : Type u_1", "α : Type u_2", "inst✝¹ : MeasurableSpace α", "μ : Measure α", "inst✝ : IsFiniteMeasure μ", "r : ℝ≥0∞", "s : ℕ → Set α", "hs : ∀ (n : ℕ), MeasurableSet (s n)", "hr₀ : r ≠ 0", "hr : ∀ (n : ℕ), r ≤ μ (s n)", "M : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}", "N : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)", "hN₀ : μ N = 0", "hN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \\ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)", "f : ℕ → α → ℝ≥0∞ := fun n => (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1", "hfapp : ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a", "hf : ∀ (n : ℕ), Measurable (f n)", "hf₁ : ∀ (n : ℕ), f n ≤ 1"], "goal": "∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)"}], "premise": [28794, 28807, 30291, 30292, 30313, 31309, 106984, 142650, 143746, 143801], "state_str": "ι : Type u_1\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)\nhN₀ : μ N = 0\nhN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \\ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)\nf : ℕ → α → ℝ≥0∞ := fun n => (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1\nhfapp : ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a\nhf : ∀ (n : ℕ), Measurable (f n)\nhf₁ : ∀ (n : ℕ), f n ≤ 1\n⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)"}
+{"state": [{"context": ["ι : Type u_1", "α : Type u_2", "inst✝¹ : MeasurableSpace α", "μ : Measure α", "inst✝ : IsFiniteMeasure μ", "r : ℝ≥0∞", "s : ℕ → Set α", "hs : ∀ (n : ℕ), MeasurableSet (s n)", "hr₀ : r ≠ 0", "hr : ∀ (n : ℕ), r ≤ μ (s n)", "M : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}", "N : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)", "hN₀ : μ N = 0", "hN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \\ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)", "f : ℕ → α → ℝ≥0∞ := fun n => (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1", "hfapp : ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a", "hf : ∀ (n : ℕ), Measurable (f n)", "hf₁ : ∀ (n : ℕ), f n ≤ 1", "hrf : ∀ (n : ℕ), r ≤ ∫⁻ (a : α), f n a ∂μ"], "goal": "∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)"}], "premise": [1673, 18816], "state_str": "ι : Type u_1\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)\nhN₀ : μ N = 0\nhN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \\ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)\nf : ℕ → α → ℝ≥0∞ := fun n => (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1\nhfapp : ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a\nhf : ∀ (n : ℕ), Measurable (f n)\nhf₁ : ∀ (n : ℕ), f n ≤ 1\nhrf : ∀ (n : ℕ), r ≤ ∫⁻ (a : α), f n a ∂μ\n⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)"}
+{"state": [{"context": ["ι : Type u_1", "α : Type u_2", "inst✝¹ : MeasurableSpace α", "μ : Measure α", "inst✝ : IsFiniteMeasure μ", "r : ℝ≥0∞", "s : ℕ → Set α", "hs : ∀ (n : ℕ), MeasurableSet (s n)", "hr₀ : r ≠ 0", "hr : ∀ (n : ℕ), r ≤ μ (s n)", "M : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}", "N : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)", "hN₀ : μ N = 0", "hN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \\ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)", "f : ℕ → α → ℝ≥0∞ := fun n => (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1", "hfapp : ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a", "hf : ∀ (n : ℕ), Measurable (f n)", "hf₁ : ∀ (n : ℕ), f n ≤ 1", "hrf : ∀ (n : ℕ), r ≤ ∫⁻ (a : α), f n a ∂μ", "hμ : μ ≠ 0", "this : ∫⁻ (x : α), limsup (fun x_1 => f x_1 x) atTop ∂μ ≤ μ univ"], "goal": "∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)"}], "premise": [18796, 26935, 31710], "state_str": "ι : Type u_1\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)\nhN₀ : μ N = 0\nhN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \\ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)\nf : ℕ → α → ℝ≥0∞ := fun n => (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1\nhfapp : ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a\nhf : ∀ (n : ℕ), Measurable (f n)\nhf₁ : ∀ (n : ℕ), f n ≤ 1\nhrf : ∀ (n : ℕ), r ≤ ∫⁻ (a : α), f n a ∂μ\nhμ : μ ≠ 0\nthis : ∫⁻ (x : α), limsup (fun x_1 => f x_1 x) atTop ∂μ ≤ μ univ\n⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)"}
+{"state": [{"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "f g : Perm α", "p : α → Prop", "x y z : α", "n : ℕ"], "goal": "f.SameCycle ((f ^ n) x) y ↔ f.SameCycle x y"}], "premise": [1713, 9950, 119784], "state_str": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\nf g : Perm α\np : α → Prop\nx y z : α\nn : ℕ\n⊢ f.SameCycle ((f ^ n) x) y ↔ f.SameCycle x y"}
+{"state": [{"context": ["R : Type u_1", "R₂ : Type u_2", "K : Type u_3", "M✝ : Type u_4", "M₂ : Type u_5", "V : Type u_6", "S : Type u_7", "inst✝⁸ : Semiring R", "inst✝⁷ : AddCommMonoid M✝", "inst✝⁶ : Module R M✝", "x : M✝", "p p' : Submodule R M✝", "inst✝⁵ : Semiring R₂", "σ₁₂ : R →+* R₂", "inst✝⁴ : AddCommMonoid M₂", "inst✝³ : Module R₂ M₂", "F : Type u_8", "inst✝² : FunLike F M✝ M₂", "inst✝¹ : SemilinearMapClass F σ₁₂ M✝ M₂", "s✝ t : Set M✝", "M : Type u_9", "inst✝ : AddCommGroup M", "s : AddSubgroup M"], "goal": "(span ℤ ↑s).toAddSubgroup = s"}], "premise": [86711, 122734], "state_str": "R : Type u_1\nR₂ : Type u_2\nK : Type u_3\nM✝ : Type u_4\nM₂ : Type u_5\nV : Type u_6\nS : Type u_7\ninst✝⁸ : Semiring R\ninst✝⁷ : AddCommMonoid M✝\ninst✝⁶ : Module R M✝\nx : M✝\np p' : Submodule R M✝\ninst✝⁵ : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : Module R₂ M₂\nF : Type u_8\ninst✝² : FunLike F M✝ M₂\ninst✝¹ : SemilinearMapClass F σ₁₂ M✝ M₂\ns✝ t : Set M✝\nM : Type u_9\ninst✝ : AddCommGroup M\ns : AddSubgroup M\n⊢ (span ℤ ↑s).toAddSubgroup = s"}
+{"state": [{"context": ["α : Type u_1", "ι : Sort u_2", "κ : ι → Sort u_3", "β : Type u_4", "inst✝ : SetLike α β", "l : LowerAdjoint SetLike.coe", "s : Set β", "x : β"], "goal": "x ∈ l.toFun s ↔ ∀ (S : α), s ⊆ ↑S → x ∈ S"}], "premise": [13385, 128379, 133525], "state_str": "α : Type u_1\nι : Sort u_2\nκ : ι → Sort u_3\nβ : Type u_4\ninst✝ : SetLike α β\nl : LowerAdjoint SetLike.coe\ns : Set β\nx : β\n⊢ x ∈ l.toFun s ↔ ∀ (S : α), s ⊆ ↑S → x ∈ S"}
+{"state": [{"context": ["α : Type u_1", "ι : Sort u_2", "κ : ι → Sort u_3", "β : Type u_4", "inst✝ : SetLike α β", "l : LowerAdjoint SetLike.coe", "s : Set β", "x : β"], "goal": "l.toFun {x} ≤ l.toFun s ↔ ∀ (S : α), l.toFun s ≤ S → l.toFun {x} ≤ S"}], "premise": [14272], "state_str": "α : Type u_1\nι : Sort u_2\nκ : ι → Sort u_3\nβ : Type u_4\ninst✝ : SetLike α β\nl : LowerAdjoint SetLike.coe\ns : Set β\nx : β\n⊢ l.toFun {x} ≤ l.toFun s ↔ ∀ (S : α), l.toFun s ≤ S → l.toFun {x} ≤ S"}
+{"state": [{"context": ["J : Type w", "C✝ : Type u", "inst✝² : Category.{v, u} C✝", "C : Type u", "inst✝¹ : Category.{v, u} C", "B : C", "objs : J → C", "arrows : (j : J) → B ⟶ objs j", "inst✝ : HasWidePushout B objs arrows", "j : J"], "goal": "arrows j ≫ ι arrows j = head arrows"}], "premise": [93390], "state_str": "J : Type w\nC✝ : Type u\ninst✝² : Category.{v, u} C✝\nC : Type u\ninst✝¹ : Category.{v, u} C\nB : C\nobjs : J → C\narrows : (j : J) → B ⟶ objs j\ninst✝ : HasWidePushout B objs arrows\nj : J\n⊢ arrows j ≫ ι arrows j = head arrows"}
+{"state": [{"context": ["P : Game → Prop", "α✝¹ β✝¹ : Type u_1", "a✝³ : α✝¹ → PGame", "a✝² : β✝¹ → PGame", "α✝ β✝ : Type u_1", "a✝¹ : α✝ → PGame", "a✝ : β✝ → PGame", "h : ∀ (k : (mk α✝¹ β✝¹ a✝³ a✝² * mk α✝ β✝ a✝¹ a✝).LeftMoves), P ⟦(mk α✝¹ β✝¹ a✝³ a✝² * mk α✝ β✝ a✝¹ a✝).moveLeft k⟧", "i : (-mk α✝¹ β✝¹ a✝³ a✝²).LeftMoves", "j : (-mk α✝ β✝ a✝¹ a✝).LeftMoves"], "goal": "⟦(-mk α✝¹ β✝¹ a✝³ a✝²).mulOption (-mk α✝ β✝ a✝¹ a✝) i j⟧ = ⟦(mk α✝¹ β✝¹ a✝³ a✝² * mk α✝ β✝ a✝¹ a✝).moveLeft (Sum.inr (i, j))⟧"}, {"context": ["P : Game → Prop", "α✝¹ β✝¹ : Type u_1", "a✝³ : α✝¹ → PGame", "a✝² : β✝¹ → PGame", "α✝ β✝ : Type u_1", "a✝¹ : α✝ → PGame", "a✝ : β✝ → PGame", "h : (∀ (i : (mk α✝¹ β✝¹ a✝³ a✝²).LeftMoves) (j : (mk α✝ β✝ a✝¹ a✝).LeftMoves), P ⟦(mk α✝¹ β✝¹ a✝³ a✝²).mulOption (mk α✝ β✝ a✝¹ a✝) i j⟧) ∧ ∀ (i : (-mk α✝¹ β✝¹ a✝³ a✝²).LeftMoves) (j : (-mk α✝ β✝ a✝¹ a✝).LeftMoves), P ⟦(-mk α✝¹ β✝¹ a✝³ a✝²).mulOption (-mk α✝ β✝ a✝¹ a✝) i j⟧", "i : β✝¹", "j : β✝"], "goal": "⟦(mk α✝¹ β✝¹ a✝³ a✝² * mk α✝ β✝ a✝¹ a✝).moveLeft (Sum.inr (i, j))⟧ = ⟦(-mk α✝¹ β✝¹ a✝³ a✝²).mulOption (-mk α✝ β✝ a✝¹ a✝) i j⟧"}], "premise": [50134, 50280, 53767, 53768, 53774, 53794], "state_str": "case h.e'_1\nP : Game → Prop\nα✝¹ β✝¹ : Type u_1\na✝³ : α✝¹ → PGame\na✝² : β✝¹ → PGame\nα✝ β✝ : Type u_1\na✝¹ : α✝ → PGame\na✝ : β✝ → PGame\nh : ∀ (k : (mk α✝¹ β✝¹ a✝³ a✝² * mk α✝ β✝ a✝¹ a✝).LeftMoves), P ⟦(mk α✝¹ β✝¹ a✝³ a✝² * mk α✝ β✝ a✝¹ a✝).moveLeft k⟧\ni : (-mk α✝¹ β✝¹ a✝³ a✝²).LeftMoves\nj : (-mk α✝ β✝ a✝¹ a✝).LeftMoves\n⊢ ⟦(-mk α✝¹ β✝¹ a✝³ a✝²).mulOption (-mk α✝ β✝ a✝¹ a✝) i j⟧ =\n    ⟦(mk α✝¹ β✝¹ a✝³ a✝² * mk α✝ β✝ a✝¹ a✝).moveLeft (Sum.inr (i, j))⟧\n\ncase h.e'_1\nP : Game → Prop\nα✝¹ β✝¹ : Type u_1\na✝³ : α✝¹ → PGame\na✝² : β✝¹ → PGame\nα✝ β✝ : Type u_1\na✝¹ : α✝ → PGame\na✝ : β✝ → PGame\nh :\n  (∀ (i : (mk α✝¹ β✝¹ a✝³ a✝²).LeftMoves) (j : (mk α✝ β✝ a✝¹ a✝).LeftMoves),\n      P ⟦(mk α✝¹ β✝¹ a✝³ a✝²).mulOption (mk α✝ β✝ a✝¹ a✝) i j⟧) ∧\n    ∀ (i : (-mk α✝¹ β✝¹ a✝³ a✝²).LeftMoves) (j : (-mk α✝ β✝ a✝¹ a✝).LeftMoves),\n      P ⟦(-mk α✝¹ β✝¹ a✝³ a✝²).mulOption (-mk α✝ β✝ a✝¹ a✝) i j⟧\ni : β✝¹\nj : β✝\n⊢ ⟦(mk α✝¹ β✝¹ a✝³ a✝² * mk α✝ β✝ a✝¹ a✝).moveLeft (Sum.inr (i, j))⟧ =\n    ⟦(-mk α✝¹ β✝¹ a✝³ a✝²).mulOption (-mk α✝ β✝ a✝¹ a✝) i j⟧"}
+{"state": [{"context": ["x : ℝ", "h : |x| < 1"], "goal": "HasSum (fun n => x ^ (n + 1) / (↑n + 1)) (-log (1 - x))"}], "premise": [63878], "state_str": "x : ℝ\nh : |x| < 1\n⊢ HasSum (fun n => x ^ (n + 1) / (↑n + 1)) (-log (1 - x))"}
+{"state": [{"context": ["x : ℝ", "h : |x| < 1"], "goal": "Summable fun n => x ^ (n + 1) / (↑n + 1)"}], "premise": [34079, 42178, 105294], "state_str": "x : ℝ\nh : |x| < 1\n⊢ Summable fun n => x ^ (n + 1) / (↑n + 1)"}
+{"state": [{"context": ["x : ℝ", "h : |x| < 1", "i : ℕ"], "goal": "‖x ^ (i + 1) / (↑i + 1)‖ ≤ |x| ^ i"}], "premise": [14286, 42903, 102701, 105283, 105294, 106204, 106302, 106802, 119742, 142593, 142636], "state_str": "x : ℝ\nh : |x| < 1\ni : ℕ\n⊢ ‖x ^ (i + 1) / (↑i + 1)‖ ≤ |x| ^ i"}
+{"state": [{"context": ["R : Type u_1", "M : Type u_2", "inst✝³ : Semiring R", "inst✝² : AddCommMonoid M", "inst✝¹ : Module R M", "G : Type u_3", "inst✝ : AddCommGroup G", "P : Submodule ℤ G", "x✝ : P.FG", "S : Finset G", "hS : span ℤ ↑S = P"], "goal": "AddSubgroup.closure ↑S = P.toAddSubgroup"}], "premise": [86711], "state_str": "R : Type u_1\nM : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nG : Type u_3\ninst✝ : AddCommGroup G\nP : Submodule ℤ G\nx✝ : P.FG\nS : Finset G\nhS : span ℤ ↑S = P\n⊢ AddSubgroup.closure ↑S = P.toAddSubgroup"}
+{"state": [{"context": ["R : Type u_1", "M : Type u_2", "inst✝³ : Semiring R", "inst✝² : AddCommMonoid M", "inst✝¹ : Module R M", "G : Type u_3", "inst✝ : AddCommGroup G", "P : Submodule ℤ G", "x✝ : P.toAddSubgroup.FG", "S : Finset G", "hS : AddSubgroup.closure ↑S = P.toAddSubgroup"], "goal": "span ℤ ↑S = P"}], "premise": [86711], "state_str": "R : Type u_1\nM : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nG : Type u_3\ninst✝ : AddCommGroup G\nP : Submodule ℤ G\nx✝ : P.toAddSubgroup.FG\nS : Finset G\nhS : AddSubgroup.closure ↑S = P.toAddSubgroup\n⊢ span ℤ ↑S = P"}
+{"state": [{"context": ["k : Type u", "inst✝ : Field k", "s : Finset (MonicIrreducible k)", "f : MonicIrreducible k", "hf : f ∈ s"], "goal": "(toSplittingField k s) (evalXSelf k f) = 0"}], "premise": [1739, 102959, 112176, 112372, 121047, 121061], "state_str": "k : Type u\ninst✝ : Field k\ns : Finset (MonicIrreducible k)\nf : MonicIrreducible k\nhf : f ∈ s\n⊢ (toSplittingField k s) (evalXSelf k f) = 0"}
+{"state": [{"context": ["k : Type u", "inst✝ : Field k", "s : Finset (MonicIrreducible k)", "f : MonicIrreducible k", "hf : f ∈ s"], "goal": "Polynomial.eval₂ (algebraMap k (∏ x ∈ s, ↑x).SplittingField) (rootOfSplits (algebraMap k (∏ x ∈ s, ↑x).SplittingField) ⋯ ⋯) ↑f = 0"}], "premise": [101142], "state_str": "k : Type u\ninst✝ : Field k\ns : Finset (MonicIrreducible k)\nf : MonicIrreducible k\nhf : f ∈ s\n⊢ Polynomial.eval₂ (algebraMap k (∏ x ∈ s, ↑x).SplittingField)\n      (rootOfSplits (algebraMap k (∏ x ∈ s, ↑x).SplittingField) ⋯ ⋯) ↑f =\n    0"}
+{"state": [{"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "inst✝² : Preadditive C", "inst✝¹ : HasShift C ℤ", "inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive", "T : Triangle C"], "goal": "(1 • (CategoryTheory.shiftFunctor C 0).map T.mor₃ ≫ (shiftFunctorComm C 1 0).hom.app T.obj₁) ≫ (CategoryTheory.shiftFunctor C 1).map ((CategoryTheory.shiftFunctorZero C ℤ).hom.app T.obj₁) = (CategoryTheory.shiftFunctorZero C ℤ).hom.app T.obj₃ ≫ T.mor₃"}], "premise": [92878, 96099, 96173, 96174, 97888, 99919, 99920, 99922, 99923, 118910], "state_str": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\ninst✝¹ : HasShift C ℤ\ninst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive\nT : Triangle C\n⊢ (1 • (CategoryTheory.shiftFunctor C 0).map T.mor₃ ≫ (shiftFunctorComm C 1 0).hom.app T.obj₁) ≫\n      (CategoryTheory.shiftFunctor C 1).map ((CategoryTheory.shiftFunctorZero C ℤ).hom.app T.obj₁) =\n    (CategoryTheory.shiftFunctorZero C ℤ).hom.app T.obj₃ ≫ T.mor₃"}
+{"state": [{"context": ["N k : ℕ"], "goal": "(N.roughNumbersUpTo k).card ≤ ∑ p ∈ N.succ.primesBelow \\ k.primesBelow, N / p"}], "premise": [21742], "state_str": "N k : ℕ\n⊢ (N.roughNumbersUpTo k).card ≤ ∑ p ∈ N.succ.primesBelow \\ k.primesBelow, N / p"}
+{"state": [{"context": ["N k : ℕ"], "goal": "((N.succ.primesBelow \\ k.primesBelow).biUnion fun p => Finset.filter (fun m => m ≠ 0 ∧ p ∣ m) (Finset.range N.succ)).card ≤ ∑ p ∈ N.succ.primesBelow \\ k.primesBelow, N / p"}], "premise": [106957, 127268, 144737], "state_str": "N k : ℕ\n⊢ ((N.succ.primesBelow \\ k.primesBelow).biUnion fun p =>\n        Finset.filter (fun m => m ≠ 0 ∧ p ∣ m) (Finset.range N.succ)).card ≤\n    ∑ p ∈ N.succ.primesBelow \\ k.primesBelow, N / p"}
+{"state": [{"context": ["R : Type u_1", "inst✝¹¹ : CommSemiring R", "l : Type u_2", "m : Type u_3", "n : Type u_4", "inst✝¹⁰ : Fintype n", "inst✝⁹ : Fintype m", "inst✝⁸ : DecidableEq n", "M₁ : Type u_5", "M₂ : Type u_6", "inst✝⁷ : AddCommMonoid M₁", "inst✝⁶ : AddCommMonoid M₂", "inst✝⁵ : Module R M₁", "inst✝⁴ : Module R M₂", "v₁ : Basis n R M₁", "v₂ : Basis m R M₂", "M₃ : Type u_7", "inst✝³ : AddCommMonoid M₃", "inst✝² : Module R M₃", "v₃ : Basis l R M₃", "inst✝¹ : Finite l", "inst✝ : DecidableEq m", "A : Matrix l m R", "B : Matrix m n R"], "goal": "(toLin v₁ v₃) (A * B) = (toLin v₂ v₃) A ∘ₗ (toLin v₁ v₂) B"}], "premise": [110557], "state_str": "R : Type u_1\ninst✝¹¹ : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst✝¹⁰ : Fintype n\ninst✝⁹ : Fintype m\ninst✝⁸ : DecidableEq n\nM₁ : Type u_5\nM₂ : Type u_6\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : AddCommMonoid M₂\ninst✝⁵ : Module R M₁\ninst✝⁴ : Module R M₂\nv₁ : Basis n R M₁\nv₂ : Basis m R M₂\nM₃ : Type u_7\ninst✝³ : AddCommMonoid M₃\ninst✝² : Module R M₃\nv₃ : Basis l R M₃\ninst✝¹ : Finite l\ninst✝ : DecidableEq m\nA : Matrix l m R\nB : Matrix m n R\n⊢ (toLin v₁ v₃) (A * B) = (toLin v₂ v₃) A ∘ₗ (toLin v₁ v₂) B"}
+{"state": [{"context": ["R : Type u_1", "inst✝¹¹ : CommSemiring R", "l : Type u_2", "m : Type u_3", "n : Type u_4", "inst✝¹⁰ : Fintype n", "inst✝⁹ : Fintype m", "inst✝⁸ : DecidableEq n", "M₁ : Type u_5", "M₂ : Type u_6", "inst✝⁷ : AddCommMonoid M₁", "inst✝⁶ : AddCommMonoid M₂", "inst✝⁵ : Module R M₁", "inst✝⁴ : Module R M₂", "v₁ : Basis n R M₁", "v₂ : Basis m R M₂", "M₃ : Type u_7", "inst✝³ : AddCommMonoid M₃", "inst✝² : Module R M₃", "v₃ : Basis l R M₃", "inst✝¹ : Finite l", "inst✝ : DecidableEq m", "A : Matrix l m R", "B : Matrix m n R", "this : DecidableEq l"], "goal": "(toMatrix v₁ v₃) ((toLin v₁ v₃) (A * B)) = (toMatrix v₁ v₃) ((toLin v₂ v₃) A ∘ₗ (toLin v₁ v₂) B)"}], "premise": [87085], "state_str": "case a\nR : Type u_1\ninst✝¹¹ : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst✝¹⁰ : Fintype n\ninst✝⁹ : Fintype m\ninst✝⁸ : DecidableEq n\nM₁ : Type u_5\nM₂ : Type u_6\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : AddCommMonoid M₂\ninst✝⁵ : Module R M₁\ninst✝⁴ : Module R M₂\nv₁ : Basis n R M₁\nv₂ : Basis m R M₂\nM₃ : Type u_7\ninst✝³ : AddCommMonoid M₃\ninst✝² : Module R M₃\nv₃ : Basis l R M₃\ninst✝¹ : Finite l\ninst✝ : DecidableEq m\nA : Matrix l m R\nB : Matrix m n R\nthis : DecidableEq l\n⊢ (toMatrix v₁ v₃) ((toLin v₁ v₃) (A * B)) = (toMatrix v₁ v₃) ((toLin v₂ v₃) A ∘ₗ (toLin v₁ v₂) B)"}
+{"state": [{"context": ["R : Type u_1", "inst✝¹¹ : CommSemiring R", "l : Type u_2", "m : Type u_3", "n : Type u_4", "inst✝¹⁰ : Fintype n", "inst✝⁹ : Fintype m", "inst✝⁸ : DecidableEq n", "M₁ : Type u_5", "M₂ : Type u_6", "inst✝⁷ : AddCommMonoid M₁", "inst✝⁶ : AddCommMonoid M₂", "inst✝⁵ : Module R M₁", "inst✝⁴ : Module R M₂", "v₁ : Basis n R M₁", "v₂ : Basis m R M₂", "M₃ : Type u_7", "inst✝³ : AddCommMonoid M₃", "inst✝² : Module R M₃", "v₃ : Basis l R M₃", "inst✝¹ : Finite l", "inst✝ : DecidableEq m", "A : Matrix l m R", "B : Matrix m n R", "this : DecidableEq l"], "goal": "(toMatrix v₁ v₃) ((toLin v₁ v₃) (A * B)) = (toMatrix v₂ v₃) ((toLin v₂ v₃) A) * (toMatrix v₁ v₂) ((toLin v₁ v₂) B)"}], "premise": [87069], "state_str": "case a\nR : Type u_1\ninst✝¹¹ : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst✝¹⁰ : Fintype n\ninst✝⁹ : Fintype m\ninst✝⁸ : DecidableEq n\nM₁ : Type u_5\nM₂ : Type u_6\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : AddCommMonoid M₂\ninst✝⁵ : Module R M₁\ninst✝⁴ : Module R M₂\nv₁ : Basis n R M₁\nv₂ : Basis m R M₂\nM₃ : Type u_7\ninst✝³ : AddCommMonoid M₃\ninst✝² : Module R M₃\nv₃ : Basis l R M₃\ninst✝¹ : Finite l\ninst✝ : DecidableEq m\nA : Matrix l m R\nB : Matrix m n R\nthis : DecidableEq l\n⊢ (toMatrix v₁ v₃) ((toLin v₁ v₃) (A * B)) = (toMatrix v₂ v₃) ((toLin v₂ v₃) A) * (toMatrix v₁ v₂) ((toLin v₁ v₂) B)"}
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+{"state": [{"context": ["R : Type u_1", "M : Type u_2", "inst✝² : CommRing R", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M", "Q : QuadraticForm R M", "n : ZMod 2", "motive : (x : CliffordAlgebra Q) → x ∈ evenOdd Q n → Prop", "range_ι_pow : ∀ (v : CliffordAlgebra Q) (h : v ∈ LinearMap.range (ι Q) ^ n.val), motive v ⋯", "add : ∀ (x y : CliffordAlgebra Q) (hx : x ∈ evenOdd Q n) (hy : y ∈ evenOdd Q n), motive x hx → motive y hy → motive (x + y) ⋯", "ι_mul_ι_mul : ∀ (m₁ m₂ : M) (x : CliffordAlgebra Q) (hx : x ∈ evenOdd Q n), motive x hx → motive ((ι Q) m₁ * (ι Q) m₂ * x) ⋯", "x : CliffordAlgebra Q", "hx : x ∈ evenOdd Q n", "k : ℕ", "xv : CliffordAlgebra Q"], "goal": "∀ (hx : xv ∈ LinearMap.range (ι Q) ^ (2 * k + n.val)), motive xv ⋯"}], "premise": [119758, 119761], "state_str": "case intro\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nn : ZMod 2\nmotive : (x : CliffordAlgebra Q) → x ∈ evenOdd Q n → Prop\nrange_ι_pow : ∀ (v : CliffordAlgebra Q) (h : v ∈ LinearMap.range (ι Q) ^ n.val), motive v ⋯\nadd :\n  ∀ (x y : CliffordAlgebra Q) (hx : x ∈ evenOdd Q n) (hy : y ∈ evenOdd Q n),\n    motive x hx → motive y hy → motive (x + y) ⋯\nι_mul_ι_mul :\n  ∀ (m₁ m₂ : M) (x : CliffordAlgebra Q) (hx : x ∈ evenOdd Q n), motive x hx → motive ((ι Q) m₁ * (ι Q) m₂ * x) ⋯\nx : CliffordAlgebra Q\nhx : x ∈ evenOdd Q n\nk : ℕ\nxv : CliffordAlgebra Q\n⊢ ∀ (hx : xv ∈ LinearMap.range (ι Q) ^ (2 * k + n.val)), motive xv ⋯"}
+{"state": [{"context": ["R : Type u_1", "M : Type u_2", "inst✝² : CommRing R", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M", "Q : QuadraticForm R M", "n : ZMod 2", "motive : (x : CliffordAlgebra Q) → x ∈ evenOdd Q n → Prop", "range_ι_pow : ∀ (v : CliffordAlgebra Q) (h : v ∈ LinearMap.range (ι Q) ^ n.val), motive v ⋯", "add : ∀ (x y : CliffordAlgebra Q) (hx : x ∈ evenOdd Q n) (hy : y ∈ evenOdd Q n), motive x hx → motive y hy → motive (x + y) ⋯", "ι_mul_ι_mul : ∀ (m₁ m₂ : M) (x : CliffordAlgebra Q) (hx : x ∈ evenOdd Q n), motive x hx → motive ((ι Q) m₁ * (ι Q) m₂ * x) ⋯", "x : CliffordAlgebra Q", "hx : x ∈ evenOdd Q n", "k : ℕ", "xv : CliffordAlgebra Q", "hxv : xv ∈ (LinearMap.range (ι Q) ^ 2) ^ k * LinearMap.range (ι Q) ^ n.val"], "goal": "motive xv ⋯"}], "premise": [109637, 109931, 119703, 119750, 121165, 122454, 122489], "state_str": "case intro\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nn : ZMod 2\nmotive : (x : CliffordAlgebra Q) → x ∈ evenOdd Q n → Prop\nrange_ι_pow : ∀ (v : CliffordAlgebra Q) (h : v ∈ LinearMap.range (ι Q) ^ n.val), motive v ⋯\nadd :\n  ∀ (x y : CliffordAlgebra Q) (hx : x ∈ evenOdd Q n) (hy : y ∈ evenOdd Q n),\n    motive x hx → motive y hy → motive (x + y) ⋯\nι_mul_ι_mul :\n  ∀ (m₁ m₂ : M) (x : CliffordAlgebra Q) (hx : x ∈ evenOdd Q n), motive x hx → motive ((ι Q) m₁ * (ι Q) m₂ * x) ⋯\nx : CliffordAlgebra Q\nhx : x ∈ evenOdd Q n\nk : ℕ\nxv : CliffordAlgebra Q\nhxv : xv ∈ (LinearMap.range (ι Q) ^ 2) ^ k * LinearMap.range (ι Q) ^ n.val\n⊢ motive xv ⋯"}
+{"state": [{"context": ["R : Type u_1", "inst✝¹² : CommSemiring R", "S : Submonoid R", "M : Type u_2", "M' : Type u_3", "M'' : Type u_4", "inst✝¹¹ : AddCommMonoid M", "inst✝¹⁰ : AddCommMonoid M'", "inst✝⁹ : AddCommMonoid M''", "A : Type u_5", "inst✝⁸ : CommSemiring A", "inst✝⁷ : Algebra R A", "inst✝⁶ : Module A M'", "inst✝⁵ : IsLocalization S A", "inst✝⁴ : Module R M", "inst✝³ : Module R M'", "inst✝² : Module R M''", "inst✝¹ : IsScalarTower R A M'", "f : M →ₗ[R] M'", "g : M →ₗ[R] M''", "inst✝ : IsLocalizedModule S f", "r : R", "x : LocalizedModule S M", "a : M", "b : ↥S"], "goal": "r • fromLocalizedModule' S f (LocalizedModule.mk a b) = fromLocalizedModule' S f (r • LocalizedModule.mk a b)"}], "premise": [113151, 113176], "state_str": "R : Type u_1\ninst✝¹² : CommSemiring R\nS : Submonoid R\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : AddCommMonoid M''\nA : Type u_5\ninst✝⁸ : CommSemiring A\ninst✝⁷ : Algebra R A\ninst✝⁶ : Module A M'\ninst✝⁵ : IsLocalization S A\ninst✝⁴ : Module R M\ninst✝³ : Module R M'\ninst✝² : Module R M''\ninst✝¹ : IsScalarTower R A M'\nf : M →ₗ[R] M'\ng : M →ₗ[R] M''\ninst✝ : IsLocalizedModule S f\nr : R\nx : LocalizedModule S M\na : M\nb : ↥S\n⊢ r • fromLocalizedModule' S f (LocalizedModule.mk a b) = fromLocalizedModule' S f (r • LocalizedModule.mk a b)"}
+{"state": [{"context": ["R : Type u_1", "inst✝¹² : CommSemiring R", "S : Submonoid R", "M : Type u_2", "M' : Type u_3", "M'' : Type u_4", "inst✝¹¹ : AddCommMonoid M", "inst✝¹⁰ : AddCommMonoid M'", "inst✝⁹ : AddCommMonoid M''", "A : Type u_5", "inst✝⁸ : CommSemiring A", "inst✝⁷ : Algebra R A", "inst✝⁶ : Module A M'", "inst✝⁵ : IsLocalization S A", "inst✝⁴ : Module R M", "inst✝³ : Module R M'", "inst✝² : Module R M''", "inst✝¹ : IsScalarTower R A M'", "f : M →ₗ[R] M'", "g : M →ₗ[R] M''", "inst✝ : IsLocalizedModule S f", "r : R", "x : LocalizedModule S M", "a : M", "b : ↥S"], "goal": "r • ↑⋯.unit⁻¹ (f a) = ↑⋯.unit⁻¹ (f (r • a))"}], "premise": [8242, 109741], "state_str": "R : Type u_1\ninst✝¹² : CommSemiring R\nS : Submonoid R\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : AddCommMonoid M''\nA : Type u_5\ninst✝⁸ : CommSemiring A\ninst✝⁷ : Algebra R A\ninst✝⁶ : Module A M'\ninst✝⁵ : IsLocalization S A\ninst✝⁴ : Module R M\ninst✝³ : Module R M'\ninst✝² : Module R M''\ninst✝¹ : IsScalarTower R A M'\nf : M →ₗ[R] M'\ng : M →ₗ[R] M''\ninst✝ : IsLocalizedModule S f\nr : R\nx : LocalizedModule S M\na : M\nb : ↥S\n⊢ r • ↑⋯.unit⁻¹ (f a) = ↑⋯.unit⁻¹ (f (r • a))"}
+{"state": [{"context": ["R : Type u_1", "p i : ℕ", "hp : Nat.Prime p"], "goal": "(σ 1) (p ^ i) = ∑ k ∈ range (i + 1), p ^ k"}], "premise": [23917], "state_str": "R : Type u_1\np i : ℕ\nhp : Nat.Prime p\n⊢ (σ 1) (p ^ i) = ∑ k ∈ range (i + 1), p ^ k"}
+{"state": [{"context": ["α : Type u_1", "s : Set α", "t : Set Bool", "x : α"], "goal": "x ∈ s.boolIndicator ⁻¹' t ↔ x ∈ (if true ∈ t then s else ∅) ∪ if false ∈ t then sᶜ else ∅"}], "premise": [131591], "state_str": "case h\nα : Type u_1\ns : Set α\nt : Set Bool\nx : α\n⊢ x ∈ s.boolIndicator ⁻¹' t ↔ x ∈ (if true ∈ t then s else ∅) ∪ if false ∈ t then sᶜ else ∅"}
+{"state": [{"context": ["G : Type u_1", "H : Type u_2", "A : Type u_3", "α : Type u_4", "β : Type u_5", "inst✝¹ : Group G", "inst✝ : Fintype G", "x : G", "n✝ : ℕ", "a : G", "n : ℕ"], "goal": "a ^ (n % Nat.card G) = a ^ n"}], "premise": [1717, 3534, 8352, 8513], "state_str": "G : Type u_1\nH : Type u_2\nA : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝¹ : Group G\ninst✝ : Fintype G\nx : G\nn✝ : ℕ\na : G\nn : ℕ\n⊢ a ^ (n % Nat.card G) = a ^ n"}
+{"state": [{"context": ["α : Type u", "β : Type v", "γ : Type w", "ι : Type x", "inst✝¹ : PseudoEMetricSpace α", "inst✝ : PseudoEMetricSpace β", "K : ℝ≥0", "s t : Set α", "f : α → β"], "goal": "LipschitzOnWith K f s ↔ LipschitzWith K (s.restrict f)"}], "premise": [58458, 133301], "state_str": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nK : ℝ≥0\ns t : Set α\nf : α → β\n⊢ LipschitzOnWith K f s ↔ LipschitzWith K (s.restrict f)"}
+{"state": [{"context": ["R : Type u", "inst✝ : CommRing R", "U : Opens ↑(PrimeSpectrum.Top R)", "f : (x : ↥U) → Localizations R ↑x", "r s : R", "h : ∀ (x : ↥U), s ∉ (↑x).asIdeal ∧ f x * (algebraMap R (Localizations R ↑x)) s = (algebraMap R (Localizations R ↑x)) r"], "goal": "∃ r s, ∀ (x : ↥U), ∃ (hs : s ∉ (↑x).asIdeal), f x = IsLocalization.mk' (Localization.AtPrime (↑x).asIdeal) r ⟨s, hs⟩"}], "premise": [1674, 2100, 2107, 77597], "state_str": "case intro.intro\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\nf : (x : ↥U) → Localizations R ↑x\nr s : R\nh : ∀ (x : ↥U), s ∉ (↑x).asIdeal ∧ f x * (algebraMap R (Localizations R ↑x)) s = (algebraMap R (Localizations R ↑x)) r\n⊢ ∃ r s, ∀ (x : ↥U), ∃ (hs : s ∉ (↑x).asIdeal), f x = IsLocalization.mk' (Localization.AtPrime (↑x).asIdeal) r ⟨s, hs⟩"}
+{"state": [{"context": ["R : Type u", "inst✝ : CommRing R", "U : Opens ↑(PrimeSpectrum.Top R)", "f : (x : ↥U) → Localizations R ↑x", "r s : R", "h : ∀ (x : ↥U), s ∉ (↑x).asIdeal ∧ f x * (algebraMap R (Localizations R ↑x)) s = (algebraMap R (Localizations R ↑x)) r", "x : ↥U"], "goal": "(algebraMap R (Localizations R ↑x)) r = f x * (algebraMap R (Localizations R ↑x)) ↑⟨s, ⋯⟩"}], "premise": [2100, 2106], "state_str": "case intro.intro\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\nf : (x : ↥U) → Localizations R ↑x\nr s : R\nh : ∀ (x : ↥U), s ∉ (↑x).asIdeal ∧ f x * (algebraMap R (Localizations R ↑x)) s = (algebraMap R (Localizations R ↑x)) r\nx : ↥U\n⊢ (algebraMap R (Localizations R ↑x)) r = f x * (algebraMap R (Localizations R ↑x)) ↑⟨s, ⋯⟩"}
+{"state": [{"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : DecidableEq α", "p✝ : Perm α", "x✝ : α", "p : Perm α", "x : α", "hx : ¬p x = x"], "goal": "(p.toList x).Nodup"}], "premise": [9447], "state_str": "case neg\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : ¬p x = x\n⊢ (p.toList x).Nodup"}
+{"state": [{"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : DecidableEq α", "p✝ : Perm α", "x✝ : α", "p : Perm α", "x : α", "hx : ¬p x = x", "hc : (p.cycleOf x).IsCycle"], "goal": "(p.toList x).Nodup"}], "premise": [129727], "state_str": "case neg\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : ¬p x = x\nhc : (p.cycleOf x).IsCycle\n⊢ (p.toList x).Nodup"}
+{"state": [{"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : DecidableEq α", "p✝ : Perm α", "x✝ : α", "p : Perm α", "x : α", "hx : ¬p x = x", "hc : (p.cycleOf x).IsCycle", "n m : ℕ", "hn : n < (p.toList x).length", "hm : m < (p.toList x).length"], "goal": "(p.toList x).nthLe n hn = (p.toList x).nthLe m hm → n = m"}], "premise": [8863, 9975], "state_str": "case neg\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : ¬p x = x\nhc : (p.cycleOf x).IsCycle\nn m : ℕ\nhn : n < (p.toList x).length\nhm : m < (p.toList x).length\n⊢ (p.toList x).nthLe n hn = (p.toList x).nthLe m hm → n = m"}
+{"state": [{"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : DecidableEq α", "p✝ : Perm α", "x✝ : α", "p : Perm α", "x : α", "hx : ¬p x = x", "hc : (p.cycleOf x).IsCycle", "n m : ℕ", "hn✝ : n < (p.toList x).length", "hn : n < orderOf (p.cycleOf x)", "hm✝ : m < (p.toList x).length", "hm : m < orderOf (p.cycleOf x)"], "goal": "(p.toList x).nthLe n hn✝ = (p.toList x).nthLe m hm✝ → n = m"}], "premise": [8669, 9439], "state_str": "case neg\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : ¬p x = x\nhc : (p.cycleOf x).IsCycle\nn m : ℕ\nhn✝ : n < (p.toList x).length\nhn : n < orderOf (p.cycleOf x)\nhm✝ : m < (p.toList x).length\nhm : m < orderOf (p.cycleOf x)\n⊢ (p.toList x).nthLe n hn✝ = (p.toList x).nthLe m hm✝ → n = m"}
+{"state": [{"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : DecidableEq α", "p✝ : Perm α", "x✝ : α", "p : Perm α", "x : α", "hx : x ∈ (p.cycleOf x).support", "hc : (p.cycleOf x).IsCycle", "n m : ℕ", "hn✝ : n < (p.toList x).length", "hn : n < orderOf (p.cycleOf x)", "hm✝ : m < (p.toList x).length", "hm : m < orderOf (p.cycleOf x)"], "goal": "(p.toList x).nthLe n hn✝ = (p.toList x).nthLe m hm✝ → n = m"}], "premise": [8869, 9431], "state_str": "case neg\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : x ∈ (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn m : ℕ\nhn✝ : n < (p.toList x).length\nhn : n < orderOf (p.cycleOf x)\nhm✝ : m < (p.toList x).length\nhm : m < orderOf (p.cycleOf x)\n⊢ (p.toList x).nthLe n hn✝ = (p.toList x).nthLe m hm✝ → n = m"}
+{"state": [{"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : DecidableEq α", "p✝ : Perm α", "x✝ : α", "p : Perm α", "x : α", "hx : x ∈ (p.cycleOf x).support", "hc : (p.cycleOf x).IsCycle", "n : ℕ", "hn✝ : n + 1 < (p.toList x).length", "hn : n + 1 < orderOf (p.cycleOf x)", "m : ℕ", "hm✝ : m + 1 < (p.toList x).length", "hm : m + 1 < orderOf (p.cycleOf x)", "h : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x"], "goal": "n + 1 = m + 1"}], "premise": [2136, 145226], "state_str": "case neg.succ.succ\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : x ∈ (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : ℕ\nhn✝ : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : ℕ\nhm✝ : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\nh : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x\n⊢ n + 1 = m + 1"}
+{"state": [{"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : DecidableEq α", "p✝ : Perm α", "x✝ : α", "p : Perm α", "x : α", "hx : x ∈ (p.cycleOf x).support", "hc : (p.cycleOf x).IsCycle", "n : ℕ", "hn✝ : n + 1 < (p.toList x).length", "hn : n + 1 < orderOf (p.cycleOf x)", "m : ℕ", "hm✝ : m + 1 < (p.toList x).length", "hm : m + 1 < orderOf (p.cycleOf x)", "h : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x", "hn' : ¬orderOf (p.cycleOf x) ∣ n.succ"], "goal": "n + 1 = m + 1"}], "premise": [2136, 145226], "state_str": "case neg.succ.succ\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : x ∈ (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : ℕ\nhn✝ : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : ℕ\nhm✝ : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\nh : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x\nhn' : ¬orderOf (p.cycleOf x) ∣ n.succ\n⊢ n + 1 = m + 1"}
+{"state": [{"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : DecidableEq α", "p✝ : Perm α", "x✝ : α", "p : Perm α", "x : α", "hx : x ∈ (p.cycleOf x).support", "hc : (p.cycleOf x).IsCycle", "n : ℕ", "hn✝ : n + 1 < (p.toList x).length", "hn : n + 1 < orderOf (p.cycleOf x)", "m : ℕ", "hm✝ : m + 1 < (p.toList x).length", "hm : m + 1 < orderOf (p.cycleOf x)", "h : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x", "hn' : ¬orderOf (p.cycleOf x) ∣ n.succ", "hm' : ¬orderOf (p.cycleOf x) ∣ m.succ"], "goal": "n + 1 = m + 1"}], "premise": [9986], "state_str": "case neg.succ.succ\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : x ∈ (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : ℕ\nhn✝ : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : ℕ\nhm✝ : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\nh : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x\nhn' : ¬orderOf (p.cycleOf x) ∣ n.succ\nhm' : ¬orderOf (p.cycleOf x) ∣ m.succ\n⊢ n + 1 = m + 1"}
+{"state": [{"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : DecidableEq α", "p✝ : Perm α", "x✝ : α", "p : Perm α", "x : α", "hx : x ∈ (p.cycleOf x).support", "hc : (p.cycleOf x).IsCycle", "n : ℕ", "hn✝ : n + 1 < (p.toList x).length", "hn : n + 1 < orderOf (p.cycleOf x)", "m : ℕ", "hm✝ : m + 1 < (p.toList x).length", "hm : m + 1 < orderOf (p.cycleOf x)", "h : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x", "hn' : (p.cycleOf x ^ n.succ).support = (p.cycleOf x).support", "hm' : (p.cycleOf x ^ m.succ).support = (p.cycleOf x).support"], "goal": "n + 1 = m + 1"}], "premise": [4467, 8424], "state_str": "case neg.succ.succ\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : x ∈ (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : ℕ\nhn✝ : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : ℕ\nhm✝ : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\nh : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x\nhn' : (p.cycleOf x ^ n.succ).support = (p.cycleOf x).support\nhm' : (p.cycleOf x ^ m.succ).support = (p.cycleOf x).support\n⊢ n + 1 = m + 1"}
+{"state": [{"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : DecidableEq α", "p✝ : Perm α", "x✝ : α", "p : Perm α", "x : α", "hx : x ∈ (p.cycleOf x).support", "hc : (p.cycleOf x).IsCycle", "n : ℕ", "hn✝ : n + 1 < (p.toList x).length", "hn : n + 1 < orderOf (p.cycleOf x)", "m : ℕ", "hm✝ : m + 1 < (p.toList x).length", "hm : m + 1 < orderOf (p.cycleOf x)", "h : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x", "hn' : (p.cycleOf x ^ n.succ).support = (p.cycleOf x).support", "hm' : (p.cycleOf x ^ m.succ).support = (p.cycleOf x).support"], "goal": "p.cycleOf x ^ (n + 1) = p.cycleOf x ^ (m + 1)"}], "premise": [8675], "state_str": "case neg.succ.succ\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np✝ : Perm α\nx✝ : α\np : Perm α\nx : α\nhx : x ∈ (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : ℕ\nhn✝ : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : ℕ\nhm✝ : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\nh : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x\nhn' : (p.cycleOf x ^ n.succ).support = (p.cycleOf x).support\nhm' : (p.cycleOf x ^ m.succ).support = (p.cycleOf x).support\n⊢ p.cycleOf x ^ (n + 1) = p.cycleOf x ^ (m + 1)"}
+{"state": [{"context": ["u✝ : Lean.Level", "arg : Q(Type u✝)", "sα✝ : Q(CommSemiring «$arg»)", "u : Lean.Level", "α : Q(Type u)", "sα : Q(CommSemiring «$α»)", "R✝ : Type u_1", "inst✝¹ : CommSemiring R✝", "a✝ a' a₁ a₂ a₃ b✝ b' b₁ b₂ b₃ c c₁ c₂ : R✝", "R : Type u_2", "inst✝ : Ring R", "a b : R"], "goal": "a - b = a + -b"}], "premise": [119789], "state_str": "u✝ : Lean.Level\narg : Q(Type u✝)\nsα✝ : Q(CommSemiring «$arg»)\nu : Lean.Level\nα : Q(Type u)\nsα : Q(CommSemiring «$α»)\nR✝ : Type u_1\ninst✝¹ : CommSemiring R✝\na✝ a' a₁ a₂ a₃ b✝ b' b₁ b₂ b₃ c c₁ c₂ : R✝\nR : Type u_2\ninst✝ : Ring R\na b : R\n⊢ a - b = a + -b"}
+{"state": [{"context": ["C : Type u", "A : Type u_1", "inst✝² : Category.{v, u} C", "inst✝¹ : AddCommMonoid A", "inst✝ : HasShift C A", "X✝ Y : C", "f : X✝ ⟶ Y", "m₁ m₂ m₃ : A", "X : C"], "goal": "(shiftFunctorComm C m₁ (m₂ + m₃)).hom.app X ≫ (shiftFunctor C m₁).map ((shiftFunctorAdd C m₂ m₃).hom.app X) = (shiftFunctorAdd C m₂ m₃).hom.app ((shiftFunctor C m₁).obj X) ≫ (shiftFunctor C m₃).map ((shiftFunctorComm C m₁ m₂).hom.app X) ≫ (shiftFunctorComm C m₁ m₃).hom.app ((shiftFunctor C m₂).obj X)"}], "premise": [96191], "state_str": "C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddCommMonoid A\ninst✝ : HasShift C A\nX✝ Y : C\nf : X✝ ⟶ Y\nm₁ m₂ m₃ : A\nX : C\n⊢ (shiftFunctorComm C m₁ (m₂ + m₃)).hom.app X ≫ (shiftFunctor C m₁).map ((shiftFunctorAdd C m₂ m₃).hom.app X) =\n    (shiftFunctorAdd C m₂ m₃).hom.app ((shiftFunctor C m₁).obj X) ≫\n      (shiftFunctor C m₃).map ((shiftFunctorComm C m₁ m₂).hom.app X) ≫\n        (shiftFunctorComm C m₁ m₃).hom.app ((shiftFunctor C m₂).obj X)"}
+{"state": [{"context": ["C : Type u", "A : Type u_1", "inst✝² : Category.{v, u} C", "inst✝¹ : AddCommMonoid A", "inst✝ : HasShift C A", "X✝ Y : C", "f : X✝ ⟶ Y", "m₁ m₂ m₃ : A", "X : C"], "goal": "(((shiftFunctorComm C m₁ (m₂ + m₃)).hom.app X ≫ (shiftFunctor C m₁).map ((shiftFunctorAdd C m₂ m₃).hom.app X)) ≫ (shiftFunctorComm C m₁ m₃).inv.app ((shiftFunctor C m₂).obj X)) ≫ (shiftFunctor C m₃).map ((shiftFunctorComm C m₁ m₂).inv.app X) = (((shiftFunctorAdd C m₂ m₃).hom.app ((shiftFunctor C m₁).obj X) ≫ (shiftFunctor C m₃).map ((shiftFunctorComm C m₁ m₂).hom.app X) ≫ (shiftFunctorComm C m₁ m₃).hom.app ((shiftFunctor C m₂).obj X)) ≫ (shiftFunctorComm C m₁ m₃).inv.app ((shiftFunctor C m₂).obj X)) ≫ (shiftFunctor C m₃).map ((shiftFunctorComm C m₁ m₂).inv.app X)"}], "premise": [96098, 96173], "state_str": "C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddCommMonoid A\ninst✝ : HasShift C A\nX✝ Y : C\nf : X✝ ⟶ Y\nm₁ m₂ m₃ : A\nX : C\n⊢ (((shiftFunctorComm C m₁ (m₂ + m₃)).hom.app X ≫ (shiftFunctor C m₁).map ((shiftFunctorAdd C m₂ m₃).hom.app X)) ≫\n        (shiftFunctorComm C m₁ m₃).inv.app ((shiftFunctor C m₂).obj X)) ≫\n      (shiftFunctor C m₃).map ((shiftFunctorComm C m₁ m₂).inv.app X) =\n    (((shiftFunctorAdd C m₂ m₃).hom.app ((shiftFunctor C m₁).obj X) ≫\n          (shiftFunctor C m₃).map ((shiftFunctorComm C m₁ m₂).hom.app X) ≫\n            (shiftFunctorComm C m₁ m₃).hom.app ((shiftFunctor C m₂).obj X)) ≫\n        (shiftFunctorComm C m₁ m₃).inv.app ((shiftFunctor C m₂).obj X)) ≫\n      (shiftFunctor C m₃).map ((shiftFunctorComm C m₁ m₂).inv.app X)"}
+{"state": [{"context": ["C : Type u", "A : Type u_1", "inst✝² : Category.{v, u} C", "inst✝¹ : AddCommMonoid A", "inst✝ : HasShift C A", "X✝ Y : C", "f : X✝ ⟶ Y", "m₁ m₂ m₃ : A", "X : C"], "goal": "(shiftFunctorComm C m₁ (m₂ + m₃)).hom.app X ≫ (shiftFunctor C m₁).map ((shiftFunctorAdd C m₂ m₃).hom.app X) ≫ (shiftFunctorComm C m₁ m₃).inv.app ((shiftFunctor C m₂).obj X) ≫ (shiftFunctor C m₃).map ((shiftFunctorComm C m₁ m₂).inv.app X) = (shiftFunctorAdd C m₂ m₃).hom.app ((shiftFunctor C m₁).obj X) ≫ (shiftFunctor C m₃).map ((shiftFunctorComm C m₁ m₂).hom.app X) ≫ 𝟙 ((shiftFunctor C m₃).obj ((shiftFunctor C m₁).obj ((shiftFunctor C m₂).obj X))) ≫ (shiftFunctor C m₃).map ((shiftFunctorComm C m₁ m₂).inv.app X)"}], "premise": [96098, 96175, 99919], "state_str": "C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddCommMonoid A\ninst✝ : HasShift C A\nX✝ Y : C\nf : X✝ ⟶ Y\nm₁ m₂ m₃ : A\nX : C\n⊢ (shiftFunctorComm C m₁ (m₂ + m₃)).hom.app X ≫\n      (shiftFunctor C m₁).map ((shiftFunctorAdd C m₂ m₃).hom.app X) ≫\n        (shiftFunctorComm C m₁ m₃).inv.app ((shiftFunctor C m₂).obj X) ≫\n          (shiftFunctor C m₃).map ((shiftFunctorComm C m₁ m₂).inv.app X) =\n    (shiftFunctorAdd C m₂ m₃).hom.app ((shiftFunctor C m₁).obj X) ≫\n      (shiftFunctor C m₃).map ((shiftFunctorComm C m₁ m₂).hom.app X) ≫\n        𝟙 ((shiftFunctor C m₃).obj ((shiftFunctor C m₁).obj ((shiftFunctor C m₂).obj X))) ≫\n          (shiftFunctor C m₃).map ((shiftFunctorComm C m₁ m₂).inv.app X)"}
+{"state": [{"context": ["C : Type u", "A : Type u_1", "inst✝² : Category.{v, u} C", "inst✝¹ : AddCommMonoid A", "inst✝ : HasShift C A", "X✝ Y : C", "f : X✝ ⟶ Y", "m₁ m₂ m₃ : A", "X : C"], "goal": "(shiftFunctorComm C m₁ (m₂ + m₃)).hom.app X ≫ (shiftFunctor C m₁).map ((shiftFunctorAdd C m₂ m₃).hom.app X) ≫ (shiftFunctorComm C m₁ m₃).inv.app ((shiftFunctor C m₂).obj X) ≫ (shiftFunctor C m₃).map ((shiftFunctorComm C m₁ m₂).inv.app X) = (shiftFunctorAdd C m₂ m₃).hom.app ((shiftFunctor C m₁).obj X) ≫ (shiftFunctor C m₃).map (𝟙 ((shiftFunctor C m₂).obj ((shiftFunctor C m₁).obj X)))"}], "premise": [92833, 92870, 96174, 99920], "state_str": "C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddCommMonoid A\ninst✝ : HasShift C A\nX✝ Y : C\nf : X✝ ⟶ Y\nm₁ m₂ m₃ : A\nX : C\n⊢ (shiftFunctorComm C m₁ (m₂ + m₃)).hom.app X ≫\n      (shiftFunctor C m₁).map ((shiftFunctorAdd C m₂ m₃).hom.app X) ≫\n        (shiftFunctorComm C m₁ m₃).inv.app ((shiftFunctor C m₂).obj X) ≫\n          (shiftFunctor C m₃).map ((shiftFunctorComm C m₁ m₂).inv.app X) =\n    (shiftFunctorAdd C m₂ m₃).hom.app ((shiftFunctor C m₁).obj X) ≫\n      (shiftFunctor C m₃).map (𝟙 ((shiftFunctor C m₂).obj ((shiftFunctor C m₁).obj X)))"}
+{"state": [{"context": ["C : Type u", "A : Type u_1", "inst✝² : Category.{v, u} C", "inst✝¹ : AddCommMonoid A", "inst✝ : HasShift C A", "X✝ Y : C", "f : X✝ ⟶ Y", "m₁ m₂ m₃ : A", "X : C"], "goal": "((shiftFunctorAdd' C m₁ (m₂ + m₃) (m₁ + (m₂ + m₃)) ⋯).inv.app X ≫ (shiftFunctorAdd' C (m₂ + m₃) m₁ (m₁ + (m₂ + m₃)) ⋯).hom.app X) ≫ (shiftFunctor C m₁).map ((shiftFunctorAdd' C m₂ m₃ (m₂ + m₃) ⋯).hom.app X) ≫ ((shiftFunctorAdd' C m₃ m₁ (m₁ + m₃) ⋯).inv.app ((shiftFunctor C m₂).obj X) ≫ (shiftFunctorAdd' C m₁ m₃ (m₁ + m₃) ⋯).hom.app ((shiftFunctor C m₂).obj X)) ≫ (shiftFunctor C m₃).map ((shiftFunctorAdd' C m₂ m₁ (m₁ + m₂) ⋯).inv.app X ≫ (shiftFunctorAdd' C m₁ m₂ (m₁ + m₂) ⋯).hom.app X) = (shiftFunctorAdd' C m₂ m₃ (m₂ + m₃) ⋯).hom.app ((shiftFunctor C m₁).obj X)"}], "premise": [92849, 96173, 99919, 119704, 119708], "state_str": "C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddCommMonoid A\ninst✝ : HasShift C A\nX✝ Y : C\nf : X✝ ⟶ Y\nm₁ m₂ m₃ : A\nX : C\n⊢ ((shiftFunctorAdd' C m₁ (m₂ + m₃) (m₁ + (m₂ + m₃)) ⋯).inv.app X ≫\n        (shiftFunctorAdd' C (m₂ + m₃) m₁ (m₁ + (m₂ + m₃)) ⋯).hom.app X) ≫\n      (shiftFunctor C m₁).map ((shiftFunctorAdd' C m₂ m₃ (m₂ + m₃) ⋯).hom.app X) ≫\n        ((shiftFunctorAdd' C m₃ m₁ (m₁ + m₃) ⋯).inv.app ((shiftFunctor C m₂).obj X) ≫\n            (shiftFunctorAdd' C m₁ m₃ (m₁ + m₃) ⋯).hom.app ((shiftFunctor C m₂).obj X)) ≫\n          (shiftFunctor C m₃).map\n            ((shiftFunctorAdd' C m₂ m₁ (m₁ + m₂) ⋯).inv.app X ≫ (shiftFunctorAdd' C m₁ m₂ (m₁ + m₂) ⋯).hom.app X) =\n    (shiftFunctorAdd' C m₂ m₃ (m₂ + m₃) ⋯).hom.app ((shiftFunctor C m₁).obj X)"}
+{"state": [{"context": ["G : Type u_1", "H : Type u_2", "A : Type u_3", "α : Type u_4", "β : Type u_5", "inst✝ : Group G", "x✝ y : G", "i : ℤ", "x : G"], "goal": "orderOf x⁻¹ = orderOf x"}], "premise": [8374], "state_str": "G : Type u_1\nH : Type u_2\nA : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝ : Group G\nx✝ y : G\ni : ℤ\nx : G\n⊢ orderOf x⁻¹ = orderOf x"}
+{"state": [{"context": ["α : Type u", "β : Type v", "φ : Ultrafilter α", "inst✝ : Preorder β", "x✝¹ : (↑φ).Germ β", "f : α → β", "x✝ : (↑φ).Germ β", "g : α → β"], "goal": "Quot.mk Setoid.r f < Quot.mk Setoid.r g ↔ LiftRel (fun x x_1 => x < x_1) (Quot.mk Setoid.r f) (Quot.mk Setoid.r g)"}], "premise": [11395], "state_str": "case h.mk.h.mk.a\nα : Type u\nβ : Type v\nφ : Ultrafilter α\ninst✝ : Preorder β\nx✝¹ : (↑φ).Germ β\nf : α → β\nx✝ : (↑φ).Germ β\ng : α → β\n⊢ Quot.mk Setoid.r f < Quot.mk Setoid.r g ↔ LiftRel (fun x x_1 => x < x_1) (Quot.mk Setoid.r f) (Quot.mk Setoid.r g)"}
+{"state": [{"context": ["R : Type u", "S : Type v", "a✝ b c d : R", "n m : ℕ", "inst✝ : Semiring R", "p✝ p q : R[X]", "ι : Type u_1", "a : R"], "goal": "(C a * X).leadingCoeff = a"}], "premise": [102222, 119743], "state_str": "R : Type u\nS : Type v\na✝ b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ p q : R[X]\nι : Type u_1\na : R\n⊢ (C a * X).leadingCoeff = a"}
+{"state": [{"context": ["α : Type u_1", "ι : Type u_2", "γ : Type u_3", "A : Type u_4", "B : Type u_5", "C : Type u_6", "inst✝⁵ : AddCommMonoid A", "inst✝⁴ : AddCommMonoid B", "inst✝³ : AddCommMonoid C", "t : ι → A → C", "h0 : ∀ (i : ι), t i 0 = 0", "h1 : ∀ (i : ι) (x y : A), t i (x + y) = t i x + t i y", "s : Finset α", "f✝ : α → ι →₀ A", "i : ι", "g✝ : ι →₀ A", "k : ι → A → γ → B", "x : γ", "β : Type u_7", "M : Type u_8", "M' : Type u_9", "N : Type u_10", "P : Type u_11", "G : Type u_12", "H : Type u_13", "R : Type u_14", "S : Type u_15", "inst✝² : DecidableEq β", "inst✝¹ : Zero M", "inst✝ : AddCommMonoid N", "f : α →₀ M", "g : α → M → β →₀ N"], "goal": "(f.sum g).support ⊆ f.support.biUnion fun a => (g a (f a)).support"}], "premise": [1673, 127131, 148063], "state_str": "α : Type u_1\nι : Type u_2\nγ : Type u_3\nA : Type u_4\nB : Type u_5\nC : Type u_6\ninst✝⁵ : AddCommMonoid A\ninst✝⁴ : AddCommMonoid B\ninst✝³ : AddCommMonoid C\nt : ι → A → C\nh0 : ∀ (i : ι), t i 0 = 0\nh1 : ∀ (i : ι) (x y : A), t i (x + y) = t i x + t i y\ns : Finset α\nf✝ : α → ι →₀ A\ni : ι\ng✝ : ι →₀ A\nk : ι → A → γ → B\nx : γ\nβ : Type u_7\nM : Type u_8\nM' : Type u_9\nN : Type u_10\nP : Type u_11\nG : Type u_12\nH : Type u_13\nR : Type u_14\nS : Type u_15\ninst✝² : DecidableEq β\ninst✝¹ : Zero M\ninst✝ : AddCommMonoid N\nf : α →₀ M\ng : α → M → β →₀ N\n⊢ (f.sum g).support ⊆ f.support.biUnion fun a => (g a (f a)).support"}
+{"state": [{"context": ["α : Type u_1", "ι : Type u_2", "γ : Type u_3", "A : Type u_4", "B : Type u_5", "C : Type u_6", "inst✝⁵ : AddCommMonoid A", "inst✝⁴ : AddCommMonoid B", "inst✝³ : AddCommMonoid C", "t : ι → A → C", "h0 : ∀ (i : ι), t i 0 = 0", "h1 : ∀ (i : ι) (x y : A), t i (x + y) = t i x + t i y", "s : Finset α", "f✝ : α → ι →₀ A", "i : ι", "g✝ : ι →₀ A", "k : ι → A → γ → B", "x : γ", "β : Type u_7", "M : Type u_8", "M' : Type u_9", "N : Type u_10", "P : Type u_11", "G : Type u_12", "H : Type u_13", "R : Type u_14", "S : Type u_15", "inst✝² : DecidableEq β", "inst✝¹ : Zero M", "inst✝ : AddCommMonoid N", "f : α →₀ M", "g : α → M → β →₀ N", "this : ∀ (c : β), (f.sum fun a b => (g a b) c) ≠ 0 → ∃ a, f a ≠ 0 ∧ ¬(g a (f a)) c = 0"], "goal": "(f.sum g).support ⊆ f.support.biUnion fun a => (g a (f a)).support"}], "premise": [2045, 125203, 136565, 138689, 148063], "state_str": "α : Type u_1\nι : Type u_2\nγ : Type u_3\nA : Type u_4\nB : Type u_5\nC : Type u_6\ninst✝⁵ : AddCommMonoid A\ninst✝⁴ : AddCommMonoid B\ninst✝³ : AddCommMonoid C\nt : ι → A → C\nh0 : ∀ (i : ι), t i 0 = 0\nh1 : ∀ (i : ι) (x y : A), t i (x + y) = t i x + t i y\ns : Finset α\nf✝ : α → ι →₀ A\ni : ι\ng✝ : ι →₀ A\nk : ι → A → γ → B\nx : γ\nβ : Type u_7\nM : Type u_8\nM' : Type u_9\nN : Type u_10\nP : Type u_11\nG : Type u_12\nH : Type u_13\nR : Type u_14\nS : Type u_15\ninst✝² : DecidableEq β\ninst✝¹ : Zero M\ninst✝ : AddCommMonoid N\nf : α →₀ M\ng : α → M → β →₀ N\nthis : ∀ (c : β), (f.sum fun a b => (g a b) c) ≠ 0 → ∃ a, f a ≠ 0 ∧ ¬(g a (f a)) c = 0\n⊢ (f.sum g).support ⊆ f.support.biUnion fun a => (g a (f a)).support"}
+{"state": [{"context": [], "goal": "(↑(π / 2)).sign = 1"}], "premise": [38302, 38580, 146326], "state_str": "⊢ (↑(π / 2)).sign = 1"}
+{"state": [{"context": ["α : Type u", "β : Type v", "γ : Type u_1", "δ : Type u_2", "inst✝¹ : LinearOrder α", "inst✝ : OrderBot α", "a b : α"], "goal": "min a b = ⊥ ↔ a = ⊥ ∨ b = ⊥"}], "premise": [14620, 14626, 18816], "state_str": "α : Type u\nβ : Type v\nγ : Type u_1\nδ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : OrderBot α\na b : α\n⊢ min a b = ⊥ ↔ a = ⊥ ∨ b = ⊥"}
+{"state": [{"context": ["x✝ y✝ z : ℝ", "n : ℕ", "x y : ℝ", "hx : 0 < x", "hz : 0 < x ^ y"], "goal": "y * log x = log (x ^ y)"}], "premise": [40070], "state_str": "x✝ y✝ z : ℝ\nn : ℕ\nx y : ℝ\nhx : 0 < x\nhz : 0 < x ^ y\n⊢ y * log x = log (x ^ y)"}
+{"state": [{"context": ["C : Type u_1", "inst✝³ : Category.{u_3, u_1} C", "ι : Type u_2", "c : ComplexShape ι", "inst✝² : Preadditive C", "inst✝¹ : CategoryWithHomology C", "inst✝ : (HomologicalComplex.quasiIso C c).HasLocalization", "X Y : HomotopyCategory C c", "f : X ⟶ Y"], "goal": "quasiIso C c f ↔ ∀ (n : ι), IsIso ((homologyFunctor C c n).map f)"}], "premise": [1713], "state_str": "C : Type u_1\ninst✝³ : Category.{u_3, u_1} C\nι : Type u_2\nc : ComplexShape ι\ninst✝² : Preadditive C\ninst✝¹ : CategoryWithHomology C\ninst✝ : (HomologicalComplex.quasiIso C c).HasLocalization\nX Y : HomotopyCategory C c\nf : X ⟶ Y\n⊢ quasiIso C c f ↔ ∀ (n : ι), IsIso ((homologyFunctor C c n).map f)"}
+{"state": [{"context": ["R : Type u", "S : Type v", "inst✝³ : CommRing R", "inst✝² : Ring S", "f : R[X]", "inst✝¹ : Algebra R S", "inst✝ : Nontrivial S", "h : IsAdjoinRootMonic S f"], "goal": "0 < f.natDegree"}], "premise": [87337], "state_str": "R : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : Ring S\nf : R[X]\ninst✝¹ : Algebra R S\ninst✝ : Nontrivial S\nh : IsAdjoinRootMonic S f\n⊢ 0 < f.natDegree"}
+{"state": [{"context": ["R : Type u", "S : Type v", "inst✝³ : CommRing R", "inst✝² : Ring S", "f : R[X]", "inst✝¹ : Algebra R S", "inst✝ : Nontrivial S", "h : IsAdjoinRootMonic S f", "i : ℕ", "hi : i < f.natDegree"], "goal": "0 < f.natDegree"}], "premise": [2134], "state_str": "case intro.mk\nR : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : Ring S\nf : R[X]\ninst✝¹ : Algebra R S\ninst✝ : Nontrivial S\nh : IsAdjoinRootMonic S f\ni : ℕ\nhi : i < f.natDegree\n⊢ 0 < f.natDegree"}
+{"state": [{"context": ["𝕜 : Type u_1", "V : Type u_2", "inst✝² : NormedField 𝕜", "inst✝¹ : AddCommGroup V", "inst✝ : Module 𝕜 V", "e : ENorm 𝕜 V"], "goal": "↑e 0 = 0"}], "premise": [40681, 115738], "state_str": "𝕜 : Type u_1\nV : Type u_2\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup V\ninst✝ : Module 𝕜 V\ne : ENorm 𝕜 V\n⊢ ↑e 0 = 0"}
+{"state": [{"context": ["α : Type u", "β : Type v", "l✝ l₁ l₂ : List α", "r : α → α → Prop", "a b : α", "inst✝ : DecidableEq α", "l : List α", "H : l.Nodup", "i : Fin l.length"], "goal": "indexOf (l.get i) l = ↑i"}], "premise": [129735], "state_str": "α : Type u\nβ : Type v\nl✝ l₁ l₂ : List α\nr : α → α → Prop\na b : α\ninst✝ : DecidableEq α\nl : List α\nH : l.Nodup\ni : Fin l.length\n⊢ indexOf (l.get i) l = ↑i"}
+{"state": [{"context": ["R : Type u_1", "α : Type u_2", "β : Type u_3", "δ : Type u_4", "γ : Type u_5", "ι : Type u_6", "m0 : MeasurableSpace α", "inst✝¹ : MeasurableSpace β", "inst✝ : MeasurableSpace γ", "μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α", "s s' t✝ : Set α", "h : μ s ≠ ⊤", "t : Set α", "ht : MeasurableSet t"], "goal": "(μ.restrict (toMeasurable μ s)) t = (μ.restrict s) t"}], "premise": [31452, 32215, 133443], "state_str": "R : Type u_1\nα : Type u_2\nβ : Type u_3\nδ : Type u_4\nγ : Type u_5\nι : Type u_6\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t✝ : Set α\nh : μ s ≠ ⊤\nt : Set α\nht : MeasurableSet t\n⊢ (μ.restrict (toMeasurable μ s)) t = (μ.restrict s) t"}
+{"state": [{"context": ["ι : Type u_1", "inst✝ : Fintype ι", "s : Set (ι → ℝ)", "x y : ι → ℝ", "δ : ℝ", "hs : IsLowerSet s"], "goal": "volume (frontier s) = 0"}], "premise": [25868, 27560, 29874, 55409], "state_str": "ι : Type u_1\ninst✝ : Fintype ι\ns : Set (ι → ℝ)\nx y : ι → ℝ\nδ : ℝ\nhs : IsLowerSet s\n⊢ volume (frontier s) = 0"}
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+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "m : Multiset α", "z : Sym2 α", "xs : List α"], "goal": "z ∈ Multiset.sym2 ⟦xs⟧ ↔ ∀ y ∈ z, y ∈ ⟦xs⟧"}], "premise": [134869], "state_str": "α : Type u_1\nβ : Type u_2\nm : Multiset α\nz : Sym2 α\nxs : List α\n⊢ z ∈ Multiset.sym2 ⟦xs⟧ ↔ ∀ y ∈ z, y ∈ ⟦xs⟧"}
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+{"state": [{"context": ["p : ℕ", "G : Type u_1", "inst✝² : Group G", "hG : IsPGroup p G", "hp : Fact (Nat.Prime p)", "α : Type u_2", "inst✝¹ : MulAction G α", "inst✝ : Finite α", "hpα : ¬p ∣ Nat.card α", "this : Finite α"], "goal": "Nat.card ↑(fixedPoints G α) ≠ 0"}], "premise": [53688], "state_str": "p : ℕ\nG : Type u_1\ninst✝² : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\nα : Type u_2\ninst✝¹ : MulAction G α\ninst✝ : Finite α\nhpα : ¬p ∣ Nat.card α\nthis : Finite α\n⊢ Nat.card ↑(fixedPoints G α) ≠ 0"}
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+{"state": [{"context": ["M✝ : Type u_1", "inst✝² : Monoid M✝", "s✝ : Set M✝", "A : Type u_2", "inst✝¹ : AddMonoid A", "t : Set A", "M : Type u_3", "inst✝ : CommMonoid M", "s : Set M", "hs : IsSubmonoid s", "m : Multiset M", "l : List M", "hl : ∀ a ∈ ⟦l⟧, a ∈ s"], "goal": "l.prod ∈ s"}], "premise": [125539], "state_str": "M✝ : Type u_1\ninst✝² : Monoid M✝\ns✝ : Set M✝\nA : Type u_2\ninst✝¹ : AddMonoid A\nt : Set A\nM : Type u_3\ninst✝ : CommMonoid M\ns : Set M\nhs : IsSubmonoid s\nm : Multiset M\nl : List M\nhl : ∀ a ∈ ⟦l⟧, a ∈ s\n⊢ l.prod ∈ s"}
+{"state": [{"context": ["α : Type u_1", "s✝¹ t✝ : Finset α", "n✝ : ?m.21109", "s✝ t : Finset α", "n : ℕ", "s a✝ : Finset α"], "goal": "a✝ ∈ powersetCard n s ↔ a✝ ∈ filter (fun x => x.card = n) s.powerset"}], "premise": [136723], "state_str": "case a\nα : Type u_1\ns✝¹ t✝ : Finset α\nn✝ : ?m.21109\ns✝ t : Finset α\nn : ℕ\ns a✝ : Finset α\n⊢ a✝ ∈ powersetCard n s ↔ a✝ ∈ filter (fun x => x.card = n) s.powerset"}
+{"state": [{"context": ["α : Type u_1", "β : Type v", "γ : Type u_2", "inst✝ : DecidableEq α", "x y z : Multiset α", "h : z ≤ y"], "goal": "x + y = x ∪ z ↔ y = z ∧ x.Disjoint y"}], "premise": [1723, 138261], "state_str": "α : Type u_1\nβ : Type v\nγ : Type u_2\ninst✝ : DecidableEq α\nx y z : Multiset α\nh : z ≤ y\n⊢ x + y = x ∪ z ↔ y = z ∧ x.Disjoint y"}
+{"state": [{"context": ["β : Type w", "α : Type w₂", "γ : Type w₃", "C : Type u", "inst✝¹ : Category.{v, u} C", "f : β → C", "inst✝ : HasCoproduct f", "c : Cofan f", "x✝ : Nonempty (IsColimit c)", "hc : IsColimit c", "this : IsIso (((coproductIsCoproduct f).coconePointUniqueUpToIso hc).hom ≫ hc.desc c)", "e_3✝ : ∐ f = (mk (∐ f) (Sigma.ι f)).pt", "b✝ : β"], "goal": "Sigma.ι f b✝ ≫ Sigma.desc c.inj = Sigma.ι f b✝ ≫ ((coproductIsCoproduct f).coconePointUniqueUpToIso hc).hom ≫ hc.desc c"}], "premise": [88752, 88826, 93389, 93391, 93392, 93678, 93679, 94280, 94281, 94286, 94292, 95603, 95630, 96174, 96175], "state_str": "case h.e'_5.h.h\nβ : Type w\nα : Type w₂\nγ : Type w₃\nC : Type u\ninst✝¹ : Category.{v, u} C\nf : β → C\ninst✝ : HasCoproduct f\nc : Cofan f\nx✝ : Nonempty (IsColimit c)\nhc : IsColimit c\nthis : IsIso (((coproductIsCoproduct f).coconePointUniqueUpToIso hc).hom ≫ hc.desc c)\ne_3✝ : ∐ f = (mk (∐ f) (Sigma.ι f)).pt\nb✝ : β\n⊢ Sigma.ι f b✝ ≫ Sigma.desc c.inj =\n    Sigma.ι f b✝ ≫ ((coproductIsCoproduct f).coconePointUniqueUpToIso hc).hom ≫ hc.desc c"}
+{"state": [{"context": ["F : Type u_1", "inst✝¹ : Field F", "ι : Type u_2", "inst✝ : DecidableEq ι", "s : Finset ι", "v : ι → F", "i j : ι", "hvs : Set.InjOn v ↑s", "hs : s.Nonempty"], "goal": "∑ j ∈ s, Lagrange.basis s v j = 1"}], "premise": [14288, 82764, 102216], "state_str": "F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns : Finset ι\nv : ι → F\ni j : ι\nhvs : Set.InjOn v ↑s\nhs : s.Nonempty\n⊢ ∑ j ∈ s, Lagrange.basis s v j = 1"}
+{"state": [{"context": ["α : Type u_1", "inst✝ : DecidableEq α", "s : Multiset α", "l₁ l₂ : List α"], "goal": "(↑l₁).ndinter ↑l₂ = ↑(l₁ ∩ l₂)"}], "premise": [5039, 137814, 137828, 138087], "state_str": "α : Type u_1\ninst✝ : DecidableEq α\ns : Multiset α\nl₁ l₂ : List α\n⊢ (↑l₁).ndinter ↑l₂ = ↑(l₁ ∩ l₂)"}
+{"state": [{"context": ["α : Type u_1", "inst✝ : DecidableEq α", "s : Multiset α", "l₁ l₂ : List α"], "goal": "List.filter (fun b => elem b l₂) l₁ ~ l₁ ∩ l₂"}], "premise": [762], "state_str": "α : Type u_1\ninst✝ : DecidableEq α\ns : Multiset α\nl₁ l₂ : List α\n⊢ List.filter (fun b => elem b l₂) l₁ ~ l₁ ∩ l₂"}
+{"state": [{"context": ["α : Type u_1", "inst✝² : Fintype α", "inst✝¹ : DecidableEq α", "P : Finpartition univ", "hP : P.IsEquipartition", "G : SimpleGraph α", "inst✝ : DecidableRel G.Adj", "ε : ℝ", "U : Finset α", "hU : U ∈ P.parts", "V : Finset α", "𝒜 : Finset (Finset α)", "s : Finset α", "h𝒜 : 𝒜 ⊆ (chunk hP G ε hU).parts", "hs : s ∈ 𝒜"], "goal": "↑𝒜.card * ↑s.card * (↑m / (↑m + 1)) ≤ ↑(𝒜.sup id).card"}], "premise": [51927, 106024, 117740, 117808], "state_str": "α : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nU : Finset α\nhU : U ∈ P.parts\nV : Finset α\n𝒜 : Finset (Finset α)\ns : Finset α\nh𝒜 : 𝒜 ⊆ (chunk hP G ε hU).parts\nhs : s ∈ 𝒜\n⊢ ↑𝒜.card * ↑s.card * (↑m / (↑m + 1)) ≤ ↑(𝒜.sup id).card"}
+{"state": [{"context": ["α : Type u_1", "inst✝² : Fintype α", "inst✝¹ : DecidableEq α", "P : Finpartition univ", "hP : P.IsEquipartition", "G : SimpleGraph α", "inst✝ : DecidableRel G.Adj", "ε : ℝ", "U : Finset α", "hU : U ∈ P.parts", "V : Finset α", "𝒜 : Finset (Finset α)", "s : Finset α", "h𝒜 : 𝒜 ⊆ (chunk hP G ε hU).parts", "hs : s ∈ 𝒜"], "goal": "↑𝒜.card * ↑m * ↑s.card ≤ ↑(𝒜.sup id).card * (↑m + 1)"}], "premise": [142636], "state_str": "α : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nU : Finset α\nhU : U ∈ P.parts\nV : Finset α\n𝒜 : Finset (Finset α)\ns : Finset α\nh𝒜 : 𝒜 ⊆ (chunk hP G ε hU).parts\nhs : s ∈ 𝒜\n⊢ ↑𝒜.card * ↑m * ↑s.card ≤ ↑(𝒜.sup id).card * (↑m + 1)"}
+{"state": [{"context": ["M : Type u_1", "N : Type u_2", "P : Type u_3", "inst✝³ : MulOneClass M", "inst✝² : MulOneClass N", "inst✝¹ : MulOneClass P", "S : Submonoid M", "A : Type u_4", "inst✝ : SetLike A M", "hA : SubmonoidClass A M", "S' : A", "s : Set M", "x✝¹ : { x // x ∈ ↑(closure s) }", "x : M", "hx : x ∈ ↑(closure s)", "x✝ : ⟨x, hx⟩ ∈ ⊤"], "goal": "⟨x, hx⟩ ∈ closure (Subtype.val ⁻¹' s)"}], "premise": [117649, 117662], "state_str": "M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Set M\nx✝¹ : { x // x ∈ ↑(closure s) }\nx : M\nhx : x ∈ ↑(closure s)\nx✝ : ⟨x, hx⟩ ∈ ⊤\n⊢ ⟨x, hx⟩ ∈ closure (Subtype.val ⁻¹' s)"}
+{"state": [{"context": ["α : Type u_1", "G G' : SimpleGraph α", "inst✝¹ : DecidableRel G.Adj", "ε : ℝ", "s t u : Finset α", "inst✝ : DecidableEq α", "hst : Disjoint s t", "hsu : Disjoint s u", "htu : Disjoint t u", "x₁ x₂ y₁ y₂ z₁ z₂ : α", "h : {x₁, y₁, z₁} = {x₂, y₂, z₂}", "hx₁ : x₁ ∈ s", "hx₂ : x₂ ∈ s", "hy₁ : y₁ ∈ t", "hy₂ : y₂ ∈ t", "hz₁ : z₁ ∈ u", "hz₂ : z₂ ∈ u"], "goal": "(x₁, y₁, z₁) = (x₂, y₂, z₂)"}], "premise": [2032, 2041, 138689, 138692, 138737, 138814], "state_str": "α : Type u_1\nG G' : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\ns t u : Finset α\ninst✝ : DecidableEq α\nhst : Disjoint s t\nhsu : Disjoint s u\nhtu : Disjoint t u\nx₁ x₂ y₁ y₂ z₁ z₂ : α\nh : {x₁, y₁, z₁} = {x₂, y₂, z₂}\nhx₁ : x₁ ∈ s\nhx₂ : x₂ ∈ s\nhy₁ : y₁ ∈ t\nhy₂ : y₂ ∈ t\nhz₁ : z₁ ∈ u\nhz₂ : z₂ ∈ u\n⊢ (x₁, y₁, z₁) = (x₂, y₂, z₂)"}
+{"state": [{"context": ["α : Type u_1", "G G' : SimpleGraph α", "inst✝¹ : DecidableRel G.Adj", "ε : ℝ", "s t u : Finset α", "inst✝ : DecidableEq α", "hst : Disjoint s t", "hsu : Disjoint s u", "htu : Disjoint t u", "x₁ x₂ y₁ y₂ z₁ z₂ : α", "hx₁ : x₁ ∈ s", "hx₂ : x₂ ∈ s", "hy₁ : y₁ ∈ t", "hy₂ : y₂ ∈ t", "hz₁ : z₁ ∈ u", "hz₂ : z₂ ∈ u", "h : ((x₁ = x₂ ∨ x₁ = y₂ ∨ x₁ = z₂) ∧ (y₁ = x₂ ∨ y₁ = y₂ ∨ y₁ = z₂) ∧ (z₁ = x₂ ∨ z₁ = y₂ ∨ z₁ = z₂)) ∧ (x₂ = x₁ ∨ x₂ = y₁ ∨ x₂ = z₁) ∧ (y₂ = x₁ ∨ y₂ = y₁ ∨ y₂ = z₁) ∧ (z₂ = x₁ ∨ z₂ = y₁ ∨ z₂ = z₁)"], "goal": "(x₁, y₁, z₁) = (x₂, y₂, z₂)"}], "premise": [138787], "state_str": "α : Type u_1\nG G' : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\ns t u : Finset α\ninst✝ : DecidableEq α\nhst : Disjoint s t\nhsu : Disjoint s u\nhtu : Disjoint t u\nx₁ x₂ y₁ y₂ z₁ z₂ : α\nhx₁ : x₁ ∈ s\nhx₂ : x₂ ∈ s\nhy₁ : y₁ ∈ t\nhy₂ : y₂ ∈ t\nhz₁ : z₁ ∈ u\nhz₂ : z₂ ∈ u\nh :\n  ((x₁ = x₂ ∨ x₁ = y₂ ∨ x₁ = z₂) ∧ (y₁ = x₂ ∨ y₁ = y₂ ∨ y₁ = z₂) ∧ (z₁ = x₂ ∨ z₁ = y₂ ∨ z₁ = z₂)) ∧\n    (x₂ = x₁ ∨ x₂ = y₁ ∨ x₂ = z₁) ∧ (y₂ = x₁ ∨ y₂ = y₁ ∨ y₂ = z₁) ∧ (z₂ = x₁ ∨ z₂ = y₁ ∨ z₂ = z₁)\n⊢ (x₁, y₁, z₁) = (x₂, y₂, z₂)"}
+{"state": [{"context": ["α : Type u_1", "G G' : SimpleGraph α", "inst✝¹ : DecidableRel G.Adj", "ε : ℝ", "s t u : Finset α", "inst✝ : DecidableEq α", "hst : ∀ ⦃a : α⦄, a ∈ s → a ∉ t", "hsu : ∀ ⦃a : α⦄, a ∈ s → a ∉ u", "htu : ∀ ⦃a : α⦄, a ∈ t → a ∉ u", "x₁ x₂ y₁ y₂ z₁ z₂ : α", "hx₁ : x₁ ∈ s", "hx₂ : x₂ ∈ s", "hy₁ : y₁ ∈ t", "hy₂ : y₂ ∈ t", "hz₁ : z₁ ∈ u", "hz₂ : z₂ ∈ u", "h : ((x₁ = x₂ ∨ x₁ = y₂ ∨ x₁ = z₂) ∧ (y₁ = x₂ ∨ y₁ = y₂ ∨ y₁ = z₂) ∧ (z₁ = x₂ ∨ z₁ = y₂ ∨ z₁ = z₂)) ∧ (x₂ = x₁ ∨ x₂ = y₁ ∨ x₂ = z₁) ∧ (y₂ = x₁ ∨ y₂ = y₁ ∨ y₂ = z₁) ∧ (z₂ = x₁ ∨ z₂ = y₁ ∨ z₂ = z₁)"], "goal": "(x₁, y₁, z₁) = (x₂, y₂, z₂)"}], "premise": [137308], "state_str": "α : Type u_1\nG G' : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\ns t u : Finset α\ninst✝ : DecidableEq α\nhst : ∀ ⦃a : α⦄, a ∈ s → a ∉ t\nhsu : ∀ ⦃a : α⦄, a ∈ s → a ∉ u\nhtu : ∀ ⦃a : α⦄, a ∈ t → a ∉ u\nx₁ x₂ y₁ y₂ z₁ z₂ : α\nhx₁ : x₁ ∈ s\nhx₂ : x₂ ∈ s\nhy₁ : y₁ ∈ t\nhy₂ : y₂ ∈ t\nhz₁ : z₁ ∈ u\nhz₂ : z₂ ∈ u\nh :\n  ((x₁ = x₂ ∨ x₁ = y₂ ∨ x₁ = z₂) ∧ (y₁ = x₂ ∨ y₁ = y₂ ∨ y₁ = z₂) ∧ (z₁ = x₂ ∨ z₁ = y₂ ∨ z₁ = z₂)) ∧\n    (x₂ = x₁ ∨ x₂ = y₁ ∨ x₂ = z₁) ∧ (y₂ = x₁ ∨ y₂ = y₁ ∨ y₂ = z₁) ∧ (z₂ = x₁ ∨ z₂ = y₁ ∨ z₂ = z₁)\n⊢ (x₁, y₁, z₁) = (x₂, y₂, z₂)"}
+{"state": [{"context": ["α : Type u_1", "G G' : SimpleGraph α", "inst✝¹ : DecidableRel G.Adj", "ε : ℝ", "s t u : Finset α", "inst✝ : DecidableEq α", "hst : ∀ ⦃a : α⦄, a ∈ s → a ∉ t", "hsu : ∀ ⦃a : α⦄, a ∈ s → a ∉ u", "htu : ∀ ⦃a : α⦄, a ∈ t → a ∉ u", "x₁ x₂ y₁ y₂ z₁ z₂ : α", "hx₁ : x₁ ∈ s", "hx₂ : x₂ ∈ s", "hy₁ : y₁ ∈ t", "hy₂ : y₂ ∈ t", "hz₁ : z₁ ∈ u", "hz₂ : z₂ ∈ u", "h : ((x₁ = x₂ ∨ x₁ = y₂ ∨ x₁ = z₂) ∧ (y₁ = x₂ ∨ y₁ = y₂ ∨ y₁ = z₂) ∧ (z₁ = x₂ ∨ z₁ = y₂ ∨ z₁ = z₂)) ∧ (x₂ = x₁ ∨ x₂ = y₁ ∨ x₂ = z₁) ∧ (y₂ = x₁ ∨ y₂ = y₁ ∨ y₂ = z₁) ∧ (z₂ = x₁ ∨ z₂ = y₁ ∨ z₂ = z₁)"], "goal": "x₁ = x₂ ∧ y₁ = y₂ ∧ z₁ = z₂"}], "premise": [1206, 1726, 1983], "state_str": "α : Type u_1\nG G' : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\ns t u : Finset α\ninst✝ : DecidableEq α\nhst : ∀ ⦃a : α⦄, a ∈ s → a ∉ t\nhsu : ∀ ⦃a : α⦄, a ∈ s → a ∉ u\nhtu : ∀ ⦃a : α⦄, a ∈ t → a ∉ u\nx₁ x₂ y₁ y₂ z₁ z₂ : α\nhx₁ : x₁ ∈ s\nhx₂ : x₂ ∈ s\nhy₁ : y₁ ∈ t\nhy₂ : y₂ ∈ t\nhz₁ : z₁ ∈ u\nhz₂ : z₂ ∈ u\nh :\n  ((x₁ = x₂ ∨ x₁ = y₂ ∨ x₁ = z₂) ∧ (y₁ = x₂ ∨ y₁ = y₂ ∨ y₁ = z₂) ∧ (z₁ = x₂ ∨ z₁ = y₂ ∨ z₁ = z₂)) ∧\n    (x₂ = x₁ ∨ x₂ = y₁ ∨ x₂ = z₁) ∧ (y₂ = x₁ ∨ y₂ = y₁ ∨ y₂ = z₁) ∧ (z₂ = x₁ ∨ z₂ = y₁ ∨ z₂ = z₁)\n⊢ x₁ = x₂ ∧ y₁ = y₂ ∧ z₁ = z₂"}
+{"state": [{"context": ["α : Type u_1", "inst✝⁵ : Mul α", "inst✝⁴ : PartialOrder α", "inst✝³ : DecidableEq α", "inst✝² : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1", "inst✝¹ : CovariantClass α α (Function.swap HMul.hMul) LT.lt", "inst✝ : LocallyFiniteOrder α", "a b c d : α"], "goal": "↑(Icc a b * Ico c d) ⊆ ↑(Ico (a * c) (b * d))"}], "premise": [134456], "state_str": "α : Type u_1\ninst✝⁵ : Mul α\ninst✝⁴ : PartialOrder α\ninst✝³ : DecidableEq α\ninst✝² : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst✝¹ : CovariantClass α α (Function.swap HMul.hMul) LT.lt\ninst✝ : LocallyFiniteOrder α\na b c d : α\n⊢ ↑(Icc a b * Ico c d) ⊆ ↑(Ico (a * c) (b * d))"}
+{"state": [{"context": ["C : Type u", "inst✝¹ : Groupoid C", "S : Subgroupoid C", "D : Type u_1", "inst✝ : Groupoid D", "φ : C ⥤ D", "hφ : Function.Injective φ.obj", "d : D"], "goal": "d ∈ (im φ hφ).objs ↔ ∃ c, φ.obj c = d"}], "premise": [1180, 91476, 91514], "state_str": "C : Type u\ninst✝¹ : Groupoid C\nS : Subgroupoid C\nD : Type u_1\ninst✝ : Groupoid D\nφ : C ⥤ D\nhφ : Function.Injective φ.obj\nd : D\n⊢ d ∈ (im φ hφ).objs ↔ ∃ c, φ.obj c = d"}
+{"state": [{"context": ["s : Finset ℕ", "m n✝ p n k : ℕ", "n0 : 0 < n", "i : ℕ", "e : k = 2 * i + 3", "ih : ∀ (m : ℕ), Prime m → m ∣ n → k ≤ m", "h : ¬n < k * k", "k2 : 2 ≤ k"], "goal": "n.MinSqFacProp (if k ∣ n then if k ∣ n / k then some k else (n / k).minSqFacAux (k + 2) else n.minSqFacAux (k + 2))"}], "premise": [14287], "state_str": "case neg\ns : Finset ℕ\nm n✝ p n k : ℕ\nn0 : 0 < n\ni : ℕ\ne : k = 2 * i + 3\nih : ∀ (m : ℕ), Prime m → m ∣ n → k ≤ m\nh : ¬n < k * k\nk2 : 2 ≤ k\n⊢ n.MinSqFacProp (if k ∣ n then if k ∣ n / k then some k else (n / k).minSqFacAux (k + 2) else n.minSqFacAux (k + 2))"}
+{"state": [{"context": ["s : Finset ℕ", "m n✝ p n k : ℕ", "n0 : 0 < n", "i : ℕ", "e : k = 2 * i + 3", "ih : ∀ (m : ℕ), Prime m → m ∣ n → k ≤ m", "h : ¬n < k * k", "k2 : 2 ≤ k", "k0 : 0 < k", "IH : ∀ (n' : ℕ), n' ∣ n → ¬k ∣ n' → n'.MinSqFacProp (n'.minSqFacAux (k + 2))"], "goal": "n.MinSqFacProp (if k ∣ n then if k ∣ n / k then some k else (n / k).minSqFacAux (k + 2) else n.minSqFacAux (k + 2))"}], "premise": [1674, 3770, 14296, 108875, 144323, 144324, 144327, 144330], "state_str": "case neg\ns : Finset ℕ\nm n✝ p n k : ℕ\nn0 : 0 < n\ni : ℕ\ne : k = 2 * i + 3\nih : ∀ (m : ℕ), Prime m → m ∣ n → k ≤ m\nh : ¬n < k * k\nk2 : 2 ≤ k\nk0 : 0 < k\nIH : ∀ (n' : ℕ), n' ∣ n → ¬k ∣ n' → n'.MinSqFacProp (n'.minSqFacAux (k + 2))\n⊢ n.MinSqFacProp (if k ∣ n then if k ∣ n / k then some k else (n / k).minSqFacAux (k + 2) else n.minSqFacAux (k + 2))"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4", "s s' : Set α", "x : α", "p : Filter ι", "g : ι → α", "inst✝ : UniformSpace β", "𝔖 : Set (Set α)"], "goal": "UniformContinuous fun f => (ofFun 𝔖) (UniformFun.toFun f)"}], "premise": [12537, 16372, 16380, 60256, 60290], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\n⊢ UniformContinuous fun f => (ofFun 𝔖) (UniformFun.toFun f)"}
+{"state": [{"context": ["ι : Type u_1", "ι' : Type u_2", "κ : Type u_3", "κ' : Type u_4", "R : Type u_5", "M : Type u_6", "inst✝¹³ : CommSemiring R", "inst✝¹² : AddCommMonoid M", "inst✝¹¹ : Module R M", "R₂ : Type u_7", "M₂ : Type u_8", "inst✝¹⁰ : CommRing R₂", "inst✝⁹ : AddCommGroup M₂", "inst✝⁸ : Module R₂ M₂", "e : Basis ι R M", "v : ι' → M", "i : ι", "j : ι'", "N : Type u_9", "inst✝⁷ : AddCommMonoid N", "inst✝⁶ : Module R N", "b : Basis ι R M", "b' : Basis ι' R M", "c : Basis κ R N", "c' : Basis κ' R N", "f : M →ₗ[R] N", "inst✝⁵ : Fintype ι'", "inst✝⁴ : Finite κ", "inst✝³ : Fintype ι", "inst✝² : Fintype κ'", "inst✝¹ : DecidableEq ι", "inst✝ : DecidableEq ι'"], "goal": "c.toMatrix ⇑c' * (toMatrix b' c') f * b'.toMatrix ⇑b = (toMatrix b c) f"}], "premise": [141384], "state_str": "ι : Type u_1\nι' : Type u_2\nκ : Type u_3\nκ' : Type u_4\nR : Type u_5\nM : Type u_6\ninst✝¹³ : CommSemiring R\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : Module R M\nR₂ : Type u_7\nM₂ : Type u_8\ninst✝¹⁰ : CommRing R₂\ninst✝⁹ : AddCommGroup M₂\ninst✝⁸ : Module R₂ M₂\ne : Basis ι R M\nv : ι' → M\ni : ι\nj : ι'\nN : Type u_9\ninst✝⁷ : AddCommMonoid N\ninst✝⁶ : Module R N\nb : Basis ι R M\nb' : Basis ι' R M\nc : Basis κ R N\nc' : Basis κ' R N\nf : M →ₗ[R] N\ninst✝⁵ : Fintype ι'\ninst✝⁴ : Finite κ\ninst✝³ : Fintype ι\ninst✝² : Fintype κ'\ninst✝¹ : DecidableEq ι\ninst✝ : DecidableEq ι'\n⊢ c.toMatrix ⇑c' * (toMatrix b' c') f * b'.toMatrix ⇑b = (toMatrix b c) f"}
+{"state": [{"context": ["ι : Type u_1", "ι' : Type u_2", "κ : Type u_3", "κ' : Type u_4", "R : Type u_5", "M : Type u_6", "inst✝¹³ : CommSemiring R", "inst✝¹² : AddCommMonoid M", "inst✝¹¹ : Module R M", "R₂ : Type u_7", "M₂ : Type u_8", "inst✝¹⁰ : CommRing R₂", "inst✝⁹ : AddCommGroup M₂", "inst✝⁸ : Module R₂ M₂", "e : Basis ι R M", "v : ι' → M", "i : ι", "j : ι'", "N : Type u_9", "inst✝⁷ : AddCommMonoid N", "inst✝⁶ : Module R N", "b : Basis ι R M", "b' : Basis ι' R M", "c : Basis κ R N", "c' : Basis κ' R N", "f : M →ₗ[R] N", "inst✝⁵ : Fintype ι'", "inst✝⁴ : Finite κ", "inst✝³ : Fintype ι", "inst✝² : Fintype κ'", "inst✝¹ : DecidableEq ι", "inst✝ : DecidableEq ι'", "val✝ : Fintype κ"], "goal": "c.toMatrix ⇑c' * (toMatrix b' c') f * b'.toMatrix ⇑b = (toMatrix b c) f"}], "premise": [85030, 85032], "state_str": "case intro\nι : Type u_1\nι' : Type u_2\nκ : Type u_3\nκ' : Type u_4\nR : Type u_5\nM : Type u_6\ninst✝¹³ : CommSemiring R\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : Module R M\nR₂ : Type u_7\nM₂ : Type u_8\ninst✝¹⁰ : CommRing R₂\ninst✝⁹ : AddCommGroup M₂\ninst✝⁸ : Module R₂ M₂\ne : Basis ι R M\nv : ι' → M\ni : ι\nj : ι'\nN : Type u_9\ninst✝⁷ : AddCommMonoid N\ninst✝⁶ : Module R N\nb : Basis ι R M\nb' : Basis ι' R M\nc : Basis κ R N\nc' : Basis κ' R N\nf : M →ₗ[R] N\ninst✝⁵ : Fintype ι'\ninst✝⁴ : Finite κ\ninst✝³ : Fintype ι\ninst✝² : Fintype κ'\ninst✝¹ : DecidableEq ι\ninst✝ : DecidableEq ι'\nval✝ : Fintype κ\n⊢ c.toMatrix ⇑c' * (toMatrix b' c') f * b'.toMatrix ⇑b = (toMatrix b c) f"}
+{"state": [{"context": ["α : Type u_1", "m0 : MeasurableSpace α", "μ : Measure α", "ι : Type u_2", "inst✝ : Countable ι", "f : ι → α → ℝ≥0∞", "h : ∀ (i : ι), Measurable (f i)", "s : Set α", "hs : MeasurableSet s"], "goal": "(μ.withDensity (∑' (n : ι), f n)) s = (sum fun n => μ.withDensity (f n)) s"}], "premise": [31314, 31534], "state_str": "case h\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nι : Type u_2\ninst✝ : Countable ι\nf : ι → α → ℝ≥0∞\nh : ∀ (i : ι), Measurable (f i)\ns : Set α\nhs : MeasurableSet s\n⊢ (μ.withDensity (∑' (n : ι), f n)) s = (sum fun n => μ.withDensity (f n)) s"}
+{"state": [{"context": ["α : Type u_1", "m0 : MeasurableSpace α", "μ : Measure α", "ι : Type u_2", "inst✝ : Countable ι", "f : ι → α → ℝ≥0∞", "h : ∀ (i : ι), Measurable (f i)", "s : Set α", "hs : MeasurableSet s"], "goal": "∫⁻ (x : α) in s, tsum (fun n => f n) x ∂μ = ∑' (i : ι), ∫⁻ (x : α) in s, f i x ∂μ"}], "premise": [29106, 30351], "state_str": "case h\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nι : Type u_2\ninst✝ : Countable ι\nf : ι → α → ℝ≥0∞\nh : ∀ (i : ι), Measurable (f i)\ns : Set α\nhs : MeasurableSet s\n⊢ ∫⁻ (x : α) in s, tsum (fun n => f n) x ∂μ = ∑' (i : ι), ∫⁻ (x : α) in s, f i x ∂μ"}
+{"state": [{"context": ["α : Type u_1", "m0 : MeasurableSpace α", "μ : Measure α", "ι : Type u_2", "inst✝ : Countable ι", "f : ι → α → ℝ≥0∞", "h : ∀ (i : ι), Measurable (f i)", "s : Set α", "hs : MeasurableSet s"], "goal": "∫⁻ (x : α) in s, tsum (fun n => f n) x ∂μ = ∫⁻ (a : α) in s, ∑' (i : ι), f i a ∂μ"}], "premise": [1674, 30261, 58990, 63417, 63418], "state_str": "case h\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nι : Type u_2\ninst✝ : Countable ι\nf : ι → α → ℝ≥0∞\nh : ∀ (i : ι), Measurable (f i)\ns : Set α\nhs : MeasurableSet s\n⊢ ∫⁻ (x : α) in s, tsum (fun n => f n) x ∂μ = ∫⁻ (a : α) in s, ∑' (i : ι), f i a ∂μ"}
+{"state": [{"context": ["R : Type u", "S : Type v", "σ : Type u_1", "τ : Type u_2", "r : R", "e : ℕ", "n m : σ", "s : σ →₀ ℕ", "inst✝¹ : CommSemiring R", "p✝ q✝ : MvPolynomial σ R", "inst✝ : DecidableEq σ", "p q : MvPolynomial σ R", "h : p.degrees.Disjoint q.degrees"], "goal": "(p + q).degrees = p.degrees ∪ q.degrees"}], "premise": [14296], "state_str": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np✝ q✝ : MvPolynomial σ R\ninst✝ : DecidableEq σ\np q : MvPolynomial σ R\nh : p.degrees.Disjoint q.degrees\n⊢ (p + q).degrees = p.degrees ∪ q.degrees"}
+{"state": [{"context": ["X : Type u_1", "Y : Type u_2", "Z : Type u_3", "inst✝² : PseudoEMetricSpace X", "inst✝¹ : PseudoEMetricSpace Y", "inst✝ : PseudoEMetricSpace Z", "C : ℝ≥0", "f : X → Y", "s : Set X"], "goal": "HolderOnWith C 1 f s ↔ LipschitzOnWith C f s"}], "premise": [39755, 146614], "state_str": "X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝² : PseudoEMetricSpace X\ninst✝¹ : PseudoEMetricSpace Y\ninst✝ : PseudoEMetricSpace Z\nC : ℝ≥0\nf : X → Y\ns : Set X\n⊢ HolderOnWith C 1 f s ↔ LipschitzOnWith C f s"}
+{"state": [{"context": ["f : ZFSet → ZFSet", "H : PSet.Definable 1 f", "x y : ZFSet", "h : y ∈ x", "z : ZFSet"], "goal": "Class.ToSet (fun x_1 => ↑(map f x) (x_1.pair z)) ↑y ↔ z = f y"}], "premise": [47911, 47934, 47936], "state_str": "f : ZFSet → ZFSet\nH : PSet.Definable 1 f\nx y : ZFSet\nh : y ∈ x\nz : ZFSet\n⊢ Class.ToSet (fun x_1 => ↑(map f x) (x_1.pair z)) ↑y ↔ z = f y"}
+{"state": [{"context": ["f : ZFSet → ZFSet", "H : PSet.Definable 1 f", "x y : ZFSet", "h : y ∈ x", "z : ZFSet"], "goal": "(∃ z_1 ∈ x, z_1.pair (f z_1) = y.pair z) ↔ z = f y"}], "premise": [47906], "state_str": "f : ZFSet → ZFSet\nH : PSet.Definable 1 f\nx y : ZFSet\nh : y ∈ x\nz : ZFSet\n⊢ (∃ z_1 ∈ x, z_1.pair (f z_1) = y.pair z) ↔ z = f y"}
+{"state": [{"context": ["α : Type u_1", "inst✝ : ConditionallyCompleteLattice α", "f : ℕ → α", "h : BddAbove (Set.range f)"], "goal": "⨆ n, (partialSups f) n = ⨆ n, f n"}], "premise": [14063, 16906, 16909], "state_str": "α : Type u_1\ninst✝ : ConditionallyCompleteLattice α\nf : ℕ → α\nh : BddAbove (Set.range f)\n⊢ ⨆ n, (partialSups f) n = ⨆ n, f n"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "f : α → β", "E I s : Set α", "M : Matroid α", "N : Matroid β"], "goal": "M.map id ⋯ = M"}], "premise": [139558], "state_str": "α : Type u_1\nβ : Type u_2\nf : α → β\nE I s : Set α\nM : Matroid α\nN : Matroid β\n⊢ M.map id ⋯ = M"}
+{"state": [{"context": ["C : Type u_1", "D : Type u_2", "inst✝³ : Category.{u_3, u_1} C", "inst✝² : Category.{?u.15006, u_2} D", "L : C ⥤ D", "W : MorphismProperty C", "inst✝¹ : L.IsLocalization W", "X Y : C", "φ : W.RightFraction₂ X Y", "inst✝ : W.HasLeftCalculusOfFractions"], "goal": "∃ ψ, φ.f ≫ ψ.s = φ.s ≫ ψ.f ∧ φ.f' ≫ ψ.s = φ.s ≫ ψ.f'"}], "premise": [92920], "state_str": "C : Type u_1\nD : Type u_2\ninst✝³ : Category.{u_3, u_1} C\ninst✝² : Category.{?u.15006, u_2} D\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\nX Y : C\nφ : W.RightFraction₂ X Y\ninst✝ : W.HasLeftCalculusOfFractions\n⊢ ∃ ψ, φ.f ≫ ψ.s = φ.s ≫ ψ.f ∧ φ.f' ≫ ψ.s = φ.s ≫ ψ.f'"}
+{"state": [{"context": ["C : Type u_1", "D : Type u_2", "inst✝³ : Category.{u_3, u_1} C", "inst✝² : Category.{?u.15006, u_2} D", "L : C ⥤ D", "W : MorphismProperty C", "inst✝¹ : L.IsLocalization W", "X Y : C", "φ : W.RightFraction₂ X Y", "inst✝ : W.HasLeftCalculusOfFractions", "ψ₁ : W.LeftFraction X Y", "hψ₁ : φ.fst.f ≫ ψ₁.s = φ.fst.s ≫ ψ₁.f"], "goal": "∃ ψ, φ.f ≫ ψ.s = φ.s ≫ ψ.f ∧ φ.f' ≫ ψ.s = φ.s ≫ ψ.f'"}], "premise": [92920], "state_str": "case intro\nC : Type u_1\nD : Type u_2\ninst✝³ : Category.{u_3, u_1} C\ninst✝² : Category.{?u.15006, u_2} D\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\nX Y : C\nφ : W.RightFraction₂ X Y\ninst✝ : W.HasLeftCalculusOfFractions\nψ₁ : W.LeftFraction X Y\nhψ₁ : φ.fst.f ≫ ψ₁.s = φ.fst.s ≫ ψ₁.f\n⊢ ∃ ψ, φ.f ≫ ψ.s = φ.s ≫ ψ.f ∧ φ.f' ≫ ψ.s = φ.s ≫ ψ.f'"}
+{"state": [{"context": ["C : Type u_1", "D : Type u_2", "inst✝³ : Category.{u_3, u_1} C", "inst✝² : Category.{?u.15006, u_2} D", "L : C ⥤ D", "W : MorphismProperty C", "inst✝¹ : L.IsLocalization W", "X Y : C", "φ : W.RightFraction₂ X Y", "inst✝ : W.HasLeftCalculusOfFractions", "ψ₁ : W.LeftFraction X Y", "hψ₁ : φ.fst.f ≫ ψ₁.s = φ.fst.s ≫ ψ₁.f", "ψ₂ : W.LeftFraction X Y", "hψ₂ : φ.snd.f ≫ ψ₂.s = φ.snd.s ≫ ψ₂.f"], "goal": "∃ ψ, φ.f ≫ ψ.s = φ.s ≫ ψ.f ∧ φ.f' ≫ ψ.s = φ.s ≫ ψ.f'"}], "premise": [92892, 92920], "state_str": "case intro.intro\nC : Type u_1\nD : Type u_2\ninst✝³ : Category.{u_3, u_1} C\ninst✝² : Category.{?u.15006, u_2} D\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\nX Y : C\nφ : W.RightFraction₂ X Y\ninst✝ : W.HasLeftCalculusOfFractions\nψ₁ : W.LeftFraction X Y\nhψ₁ : φ.fst.f ≫ ψ₁.s = φ.fst.s ≫ ψ₁.f\nψ₂ : W.LeftFraction X Y\nhψ₂ : φ.snd.f ≫ ψ₂.s = φ.snd.s ≫ ψ₂.f\n⊢ ∃ ψ, φ.f ≫ ψ.s = φ.s ≫ ψ.f ∧ φ.f' ≫ ψ.s = φ.s ≫ ψ.f'"}
+{"state": [{"context": ["C : Type u_1", "D : Type u_2", "inst✝³ : Category.{u_3, u_1} C", "inst✝² : Category.{?u.15006, u_2} D", "L : C ⥤ D", "W : MorphismProperty C", "inst✝¹ : L.IsLocalization W", "X Y : C", "φ : W.RightFraction₂ X Y", "inst✝ : W.HasLeftCalculusOfFractions", "ψ₁ : W.LeftFraction X Y", "hψ₁ : φ.f ≫ ψ₁.s = φ.s ≫ ψ₁.f", "ψ₂ : W.LeftFraction X Y", "hψ₂ : φ.f' ≫ ψ₂.s = φ.s ≫ ψ₂.f", "α : W.LeftFraction ψ₁.Y' ψ₂.Y'", "hα : ψ₂.s ≫ α.s = ψ₁.s ≫ α.f"], "goal": "∃ ψ, φ.f ≫ ψ.s = φ.s ≫ ψ.f ∧ φ.f' ≫ ψ.s = φ.s ≫ ψ.f'"}], "premise": [92892, 97269], "state_str": "case intro.intro.intro\nC : Type u_1\nD : Type u_2\ninst✝³ : Category.{u_3, u_1} C\ninst✝² : Category.{?u.15006, u_2} D\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\nX Y : C\nφ : W.RightFraction₂ X Y\ninst✝ : W.HasLeftCalculusOfFractions\nψ₁ : W.LeftFraction X Y\nhψ₁ : φ.f ≫ ψ₁.s = φ.s ≫ ψ₁.f\nψ₂ : W.LeftFraction X Y\nhψ₂ : φ.f' ≫ ψ₂.s = φ.s ≫ ψ₂.f\nα : W.LeftFraction ψ₁.Y' ψ₂.Y'\nhα : ψ₂.s ≫ α.s = ψ₁.s ≫ α.f\n⊢ ∃ ψ, φ.f ≫ ψ.s = φ.s ≫ ψ.f ∧ φ.f' ≫ ψ.s = φ.s ≫ ψ.f'"}
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+{"state": [{"context": ["𝕜 : Type u_1", "inst✝⁸ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace 𝕜 E", "F : Type u_3", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace 𝕜 F", "G : Type u_4", "inst✝³ : NormedAddCommGroup G", "inst✝² : NormedSpace 𝕜 G", "G' : Type u_5", "inst✝¹ : NormedAddCommGroup G'", "inst✝ : NormedSpace 𝕜 G'", "f f₀ f₁ g : E → F", "f' f₀' f₁' g' e : E →L[𝕜] F", "x : E", "s t : Set E", "L L₁ L₂ : Filter E", "b : E × F → G", "u : Set (E × F)", "h : IsBoundedBilinearMap 𝕜 b", "p : E × F"], "goal": "(fun p_1 => b p_1.1 - b p_1.2 - (h.deriv p) (p_1.1 - p_1.2)) =o[𝓝 (p, p)] fun p => p.1 - p.2"}], "premise": [43406, 66783], "state_str": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type u_5\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nb : E × F → G\nu : Set (E × F)\nh : IsBoundedBilinearMap 𝕜 b\np : E × F\n⊢ (fun p_1 => b p_1.1 - b p_1.2 - (h.deriv p) (p_1.1 - p_1.2)) =o[𝓝 (p, p)] fun p => p.1 - p.2"}
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+{"state": [{"context": ["R : Type u_1", "S : Type u_2", "inst✝² : Ring R", "inst✝¹ : Ring S", "f : R →+* S", "I : Type u_3", "e : I → R", "he✝¹ : OrthogonalIdempotents e", "inst✝ : Fintype I", "he✝ : CompleteOrthogonalIdempotents e", "h : ∀ x ∈ RingHom.ker f, IsNilpotent x", "he : ∀ (i : I), IsIdempotentElem (e i)", "he' : ∀ (i : I), IsMulCentral (e i)", "he'' : CompleteOrthogonalIdempotents (⇑f ∘ e)", "e' : I → R", "h₁ : CompleteOrthogonalIdempotents e'", "h₂ : ⇑f ∘ e' = ⇑f ∘ e"], "goal": "CompleteOrthogonalIdempotents e"}], "premise": [2100, 77172, 77198, 80704, 118004, 119304], "state_str": "case intro.intro\nR : Type u_1\nS : Type u_2\ninst✝² : Ring R\ninst✝¹ : Ring S\nf : R →+* S\nI : Type u_3\ne : I → R\nhe✝¹ : OrthogonalIdempotents e\ninst✝ : Fintype I\nhe✝ : CompleteOrthogonalIdempotents e\nh : ∀ x ∈ RingHom.ker f, IsNilpotent x\nhe : ∀ (i : I), IsIdempotentElem (e i)\nhe' : ∀ (i : I), IsMulCentral (e i)\nhe'' : CompleteOrthogonalIdempotents (⇑f ∘ e)\ne' : I → R\nh₁ : CompleteOrthogonalIdempotents e'\nh₂ : ⇑f ∘ e' = ⇑f ∘ e\n⊢ CompleteOrthogonalIdempotents e"}
+{"state": [{"context": ["α : Type u_1", "ι : Type u_2", "E : Type u_3", "m : MeasurableSpace α", "μ : Measure α", "inst✝ : MetricSpace E", "f : ℕ → α → E", "g : α → E", "hfg : TendstoInMeasure μ f atTop g", "n : ℕ"], "goal": "∃ N, ∀ m_1 ≥ N, μ {x | 2⁻¹ ^ n ≤ dist (f m_1 x) (g x)} ≤ 2⁻¹ ^ n"}], "premise": [40000, 103765, 104330, 106843], "state_str": "α : Type u_1\nι : Type u_2\nE : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nn : ℕ\n⊢ ∃ N, ∀ m_1 ≥ N, μ {x | 2⁻¹ ^ n ≤ dist (f m_1 x) (g x)} ≤ 2⁻¹ ^ n"}
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+{"state": [{"context": ["R : Type u", "K : Type u'", "M : Type v", "V : Type v'", "M₂ : Type w", "V₂ : Type w'", "M₃ : Type y", "V₃ : Type y'", "M₄ : Type z", "ι : Type x", "M₅ : Type u_1", "M₆ : Type u_2", "inst✝⁴ : Ring R", "N : Type u_3", "inst✝³ : AddCommGroup M", "inst✝² : Module R M", "inst✝¹ : AddCommGroup N", "inst✝ : Module R N", "f : M × N →ₗ[R] M", "i : Injective ⇑f", "n : ℕ"], "goal": "Disjoint (f.tailing i n) (OrderDual.ofDual ((f.tunnel i) (n + 1)))"}], "premise": [13483], "state_str": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type u_1\nM₆ : Type u_2\ninst✝⁴ : Ring R\nN : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : M × N →ₗ[R] M\ni : Injective ⇑f\nn : ℕ\n⊢ Disjoint (f.tailing i n) (OrderDual.ofDual ((f.tunnel i) (n + 1)))"}
+{"state": [{"context": ["R : Type u", "K : Type u'", "M : Type v", "V : Type v'", "M₂ : Type w", "V₂ : Type w'", "M₃ : Type y", "V₃ : Type y'", "M₄ : Type z", "ι : Type x", "M₅ : Type u_1", "M₆ : Type u_2", "inst✝⁴ : Ring R", "N : Type u_3", "inst✝³ : AddCommGroup M", "inst✝² : Module R M", "inst✝¹ : AddCommGroup N", "inst✝ : Module R N", "f : M × N →ₗ[R] M", "i : Injective ⇑f", "n : ℕ"], "goal": "Submodule.map (f.tunnelAux (f.tunnel' i n)) (Submodule.snd R M N) ⊓ Submodule.map (f.tunnelAux (f.tunnel' i n)) (Submodule.fst R M N) = ⊥"}], "premise": [14579, 84876, 84893, 110284, 110302, 110315], "state_str": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type u_1\nM₆ : Type u_2\ninst✝⁴ : Ring R\nN : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : M × N →ₗ[R] M\ni : Injective ⇑f\nn : ℕ\n⊢ Submodule.map (f.tunnelAux (f.tunnel' i n)) (Submodule.snd R M N) ⊓\n      Submodule.map (f.tunnelAux (f.tunnel' i n)) (Submodule.fst R M N) =\n    ⊥"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "G : Type u_3", "M : Type u_4", "inst✝ : Group G", "a b c d : G", "n : ℤ", "H : a / b = c / d"], "goal": "a = b ↔ c = d"}], "premise": [1713, 118005], "state_str": "α : Type u_1\nβ : Type u_2\nG : Type u_3\nM : Type u_4\ninst✝ : Group G\na b c d : G\nn : ℤ\nH : a / b = c / d\n⊢ a = b ↔ c = d"}
+{"state": [{"context": ["R : Type u_1", "inst✝⁴ : AddCommGroupWithOne R", "inst✝³ : PartialOrder R", "inst✝² : CovariantClass R R (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "inst✝¹ : ZeroLEOneClass R", "inst✝ : NeZero 1", "m n : ℤ", "h : m ≤ n"], "goal": "↑m ≤ ↑n"}], "premise": [105706], "state_str": "R : Type u_1\ninst✝⁴ : AddCommGroupWithOne R\ninst✝³ : PartialOrder R\ninst✝² : CovariantClass R R (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : ZeroLEOneClass R\ninst✝ : NeZero 1\nm n : ℤ\nh : m ≤ n\n⊢ ↑m ≤ ↑n"}
+{"state": [{"context": ["R : Type u_1", "inst✝⁴ : AddCommGroupWithOne R", "inst✝³ : PartialOrder R", "inst✝² : CovariantClass R R (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "inst✝¹ : ZeroLEOneClass R", "inst✝ : NeZero 1", "m n : ℤ", "k : ℕ", "hk : ↑k = n - m"], "goal": "↑m ≤ ↑n"}], "premise": [105706, 128750, 128757], "state_str": "case intro\nR : Type u_1\ninst✝⁴ : AddCommGroupWithOne R\ninst✝³ : PartialOrder R\ninst✝² : CovariantClass R R (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : ZeroLEOneClass R\ninst✝ : NeZero 1\nm n : ℤ\nk : ℕ\nhk : ↑k = n - m\n⊢ ↑m ≤ ↑n"}
+{"state": [{"context": ["R : Type u_1", "inst✝⁴ : AddCommGroupWithOne R", "inst✝³ : PartialOrder R", "inst✝² : CovariantClass R R (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "inst✝¹ : ZeroLEOneClass R", "inst✝ : NeZero 1", "m n : ℤ", "k : ℕ", "hk : ↑k = n - m"], "goal": "0 ≤ ↑k"}], "premise": [142591], "state_str": "case intro\nR : Type u_1\ninst✝⁴ : AddCommGroupWithOne R\ninst✝³ : PartialOrder R\ninst✝² : CovariantClass R R (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : ZeroLEOneClass R\ninst✝ : NeZero 1\nm n : ℤ\nk : ℕ\nhk : ↑k = n - m\n⊢ 0 ≤ ↑k"}
+{"state": [{"context": ["R : Type u", "inst✝⁵ : CommSemiring R", "S : Submonoid R", "M : Type v", "inst✝⁴ : AddCommMonoid M", "inst✝³ : Module R M", "T : Type u_1", "inst✝² : CommSemiring T", "inst✝¹ : Algebra R T", "inst✝ : IsLocalization S T", "s : ↥S", "p : LocalizedModule S M", "x✝² x✝¹ : M × ↥S", "a : M", "b : ↥S", "a' : M", "b' : ↥S", "x✝ : (a, b) ≈ (a', b')", "c : ↥S", "eq1 : c • (a', b').2 • (a, b).1 = c • (a, b).2 • (a', b').1"], "goal": "c • ((a', b').2 * s) • (a, b).1 = c • ((a, b).2 * s) • (a', b').1"}], "premise": [118866, 118875], "state_str": "R : Type u\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nT : Type u_1\ninst✝² : CommSemiring T\ninst✝¹ : Algebra R T\ninst✝ : IsLocalization S T\ns : ↥S\np : LocalizedModule S M\nx✝² x✝¹ : M × ↥S\na : M\nb : ↥S\na' : M\nb' : ↥S\nx✝ : (a, b) ≈ (a', b')\nc : ↥S\neq1 : c • (a', b').2 • (a, b).1 = c • (a, b).2 • (a', b').1\n⊢ c • ((a', b').2 * s) • (a, b).1 = c • ((a, b).2 * s) • (a', b').1"}
+{"state": [{"context": ["R : Type u", "inst✝⁵ : CommSemiring R", "S : Submonoid R", "M : Type v", "inst✝⁴ : AddCommMonoid M", "inst✝³ : Module R M", "T : Type u_1", "inst✝² : CommSemiring T", "inst✝¹ : Algebra R T", "inst✝ : IsLocalization S T", "s : ↥S", "x y : LocalizedModule S M"], "goal": "(fun p => p.liftOn (fun p => mk p.1 (p.2 * s)) ⋯) (x + y) = (fun p => p.liftOn (fun p => mk p.1 (p.2 * s)) ⋯) x + (fun p => p.liftOn (fun p => mk p.1 (p.2 * s)) ⋯) y"}], "premise": [113125], "state_str": "R : Type u\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nT : Type u_1\ninst✝² : CommSemiring T\ninst✝¹ : Algebra R T\ninst✝ : IsLocalization S T\ns : ↥S\nx y : LocalizedModule S M\n⊢ (fun p => p.liftOn (fun p => mk p.1 (p.2 * s)) ⋯) (x + y) =\n    (fun p => p.liftOn (fun p => mk p.1 (p.2 * s)) ⋯) x + (fun p => p.liftOn (fun p => mk p.1 (p.2 * s)) ⋯) y"}
+{"state": [{"context": ["R : Type u", "inst✝⁵ : CommSemiring R", "S : Submonoid R", "M : Type v", "inst✝⁴ : AddCommMonoid M", "inst✝³ : Module R M", "T : Type u_1", "inst✝² : CommSemiring T", "inst✝¹ : Algebra R T", "inst✝ : IsLocalization S T", "s : ↥S", "x y : LocalizedModule S M", "m₁ m₂ : M", "t₁ t₂ : ↥S"], "goal": "(fun p => p.liftOn (fun p => mk p.1 (p.2 * s)) ⋯) (mk m₁ t₁ + mk m₂ t₂) = (fun p => p.liftOn (fun p => mk p.1 (p.2 * s)) ⋯) (mk m₁ t₁) + (fun p => p.liftOn (fun p => mk p.1 (p.2 * s)) ⋯) (mk m₂ t₂)"}], "premise": [108334, 113126, 113130, 113148, 117739, 118866, 118875, 119703, 119707], "state_str": "R : Type u\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nT : Type u_1\ninst✝² : CommSemiring T\ninst✝¹ : Algebra R T\ninst✝ : IsLocalization S T\ns : ↥S\nx y : LocalizedModule S M\nm₁ m₂ : M\nt₁ t₂ : ↥S\n⊢ (fun p => p.liftOn (fun p => mk p.1 (p.2 * s)) ⋯) (mk m₁ t₁ + mk m₂ t₂) =\n    (fun p => p.liftOn (fun p => mk p.1 (p.2 * s)) ⋯) (mk m₁ t₁) +\n      (fun p => p.liftOn (fun p => mk p.1 (p.2 * s)) ⋯) (mk m₂ t₂)"}
+{"state": [{"context": ["R : Type u", "inst✝⁵ : CommSemiring R", "S : Submonoid R", "M : Type v", "inst✝⁴ : AddCommMonoid M", "inst✝³ : Module R M", "T : Type u_1", "inst✝² : CommSemiring T", "inst✝¹ : Algebra R T", "inst✝ : IsLocalization S T", "s : ↥S", "r : R", "x : LocalizedModule S M"], "goal": "{ toFun := fun p => p.liftOn (fun p => mk p.1 (p.2 * s)) ⋯, map_add' := ⋯ }.toFun (r • x) = (RingHom.id R) r • { toFun := fun p => p.liftOn (fun p => mk p.1 (p.2 * s)) ⋯, map_add' := ⋯ }.toFun x"}], "premise": [113124], "state_str": "R : Type u\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nT : Type u_1\ninst✝² : CommSemiring T\ninst✝¹ : Algebra R T\ninst✝ : IsLocalization S T\ns : ↥S\nr : R\nx : LocalizedModule S M\n⊢ { toFun := fun p => p.liftOn (fun p => mk p.1 (p.2 * s)) ⋯, map_add' := ⋯ }.toFun (r • x) =\n    (RingHom.id R) r • { toFun := fun p => p.liftOn (fun p => mk p.1 (p.2 * s)) ⋯, map_add' := ⋯ }.toFun x"}
+{"state": [{"context": ["R : Type u", "inst✝⁵ : CommSemiring R", "S : Submonoid R", "M : Type v", "inst✝⁴ : AddCommMonoid M", "inst✝³ : Module R M", "T : Type u_1", "inst✝² : CommSemiring T", "inst✝¹ : Algebra R T", "inst✝ : IsLocalization S T", "s : ↥S", "r : R", "x : LocalizedModule S M", "x✝¹ : M", "x✝ : ↥S"], "goal": "(mk ((IsLocalization.sec S (↑(algebraMap R (Localization S)).toMonoidWithZeroHom r)).1 • (x✝¹, x✝).1) ((IsLocalization.sec S (↑(algebraMap R (Localization S)).toMonoidWithZeroHom r)).2 * (x✝¹, x✝).2)).liftOn (fun p => mk p.1 (p.2 * s)) ⋯ = r • (mk x✝¹ x✝).liftOn (fun p => mk p.1 (p.2 * s)) ⋯"}], "premise": [113126, 113139, 119703], "state_str": "R : Type u\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nT : Type u_1\ninst✝² : CommSemiring T\ninst✝¹ : Algebra R T\ninst✝ : IsLocalization S T\ns : ↥S\nr : R\nx : LocalizedModule S M\nx✝¹ : M\nx✝ : ↥S\n⊢ (mk ((IsLocalization.sec S (↑(algebraMap R (Localization S)).toMonoidWithZeroHom r)).1 • (x✝¹, x✝).1)\n          ((IsLocalization.sec S (↑(algebraMap R (Localization S)).toMonoidWithZeroHom r)).2 * (x✝¹, x✝).2)).liftOn\n      (fun p => mk p.1 (p.2 * s)) ⋯ =\n    r • (mk x✝¹ x✝).liftOn (fun p => mk p.1 (p.2 * s)) ⋯"}
+{"state": [{"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "X Y Z : Scheme", "𝒰 : X.OpenCover", "f : X ⟶ Z", "g : Y ⟶ Z", "inst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g", "i j k : 𝒰.J"], "goal": "(fun i j k => t' 𝒰 f g i j k) i j k ≫ pullback.snd ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j k) ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j i) = pullback.fst ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i j) ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i k) ≫ (fun i j => t 𝒰 f g i j) i j"}], "premise": [93876], "state_str": "C : Type u\ninst✝¹ : Category.{v, u} C\nX Y Z : Scheme\n𝒰 : X.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g\ni j k : 𝒰.J\n⊢ (fun i j k => t' 𝒰 f g i j k) i j k ≫\n      pullback.snd ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j k)\n        ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j i) =\n    pullback.fst ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i j)\n        ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i k) ≫\n      (fun i j => t 𝒰 f g i j) i j"}
+{"state": [{"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "X Y Z : Scheme", "𝒰 : X.OpenCover", "f : X ⟶ Z", "g : Y ⟶ Z", "inst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g", "i j k : 𝒰.J"], "goal": "((fun i j k => t' 𝒰 f g i j k) i j k ≫ pullback.snd ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j k) ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j i)) ≫ pullback.fst (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i) = (pullback.fst ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i j) ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i k) ≫ (fun i j => t 𝒰 f g i j) i j) ≫ pullback.fst (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)"}, {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "X Y Z : Scheme", "𝒰 : X.OpenCover", "f : X ⟶ Z", "g : Y ⟶ Z", "inst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g", "i j k : 𝒰.J"], "goal": "((fun i j k => t' 𝒰 f g i j k) i j k ≫ pullback.snd ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j k) ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j i)) ≫ pullback.snd (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i) = (pullback.fst ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i j) ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i k) ≫ (fun i j => t 𝒰 f g i j) i j) ≫ pullback.snd (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)"}], "premise": [93876], "state_str": "case h₀\nC : Type u\ninst✝¹ : Category.{v, u} C\nX Y Z : Scheme\n𝒰 : X.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g\ni j k : 𝒰.J\n⊢ ((fun i j k => t' 𝒰 f g i j k) i j k ≫\n        pullback.snd ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j k)\n          ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j i)) ≫\n      pullback.fst (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i) =\n    (pullback.fst ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i j)\n          ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i k) ≫\n        (fun i j => t 𝒰 f g i j) i j) ≫\n      pullback.fst (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)\n\ncase h₁\nC : Type u\ninst✝¹ : Category.{v, u} C\nX Y Z : Scheme\n𝒰 : X.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g\ni j k : 𝒰.J\n⊢ ((fun i j k => t' 𝒰 f g i j k) i j k ≫\n        pullback.snd ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j k)\n          ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j i)) ≫\n      pullback.snd (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i) =\n    (pullback.fst ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i j)\n          ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i k) ≫\n        (fun i j => t 𝒰 f g i j) i j) ≫\n      pullback.snd (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)"}
+{"state": [{"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "X Y Z : Scheme", "𝒰 : X.OpenCover", "f : X ⟶ Z", "g : Y ⟶ Z", "inst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g", "i j k : 𝒰.J"], "goal": "(((fun i j k => t' 𝒰 f g i j k) i j k ≫ pullback.snd ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j k) ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j i)) ≫ pullback.fst (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)) ≫ pullback.fst (𝒰.map j ≫ f) g = ((pullback.fst ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i j) ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i k) ≫ (fun i j => t 𝒰 f g i j) i j) ≫ pullback.fst (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)) ≫ pullback.fst (𝒰.map j ≫ f) g"}, {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "X Y Z : Scheme", "𝒰 : X.OpenCover", "f : X ⟶ Z", "g : Y ⟶ Z", "inst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g", "i j k : 𝒰.J"], "goal": "(((fun i j k => t' 𝒰 f g i j k) i j k ≫ pullback.snd ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j k) ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j i)) ≫ pullback.fst (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)) ≫ pullback.snd (𝒰.map j ≫ f) g = ((pullback.fst ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i j) ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i k) ≫ (fun i j => t 𝒰 f g i j) i j) ≫ pullback.fst (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)) ≫ pullback.snd (𝒰.map j ≫ f) g"}, {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "X Y Z : Scheme", "𝒰 : X.OpenCover", "f : X ⟶ Z", "g : Y ⟶ Z", "inst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g", "i j k : 𝒰.J"], "goal": "((fun i j k => t' 𝒰 f g i j k) i j k ≫ pullback.snd ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j k) ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j i)) ≫ pullback.snd (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i) = (pullback.fst ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i j) ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i k) ≫ (fun i j => t 𝒰 f g i j) i j) ≫ pullback.snd (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)"}], "premise": [96173, 130190, 130191, 130192, 130197, 130198, 130199], "state_str": "case h₀.h₀\nC : Type u\ninst✝¹ : Category.{v, u} C\nX Y Z : Scheme\n𝒰 : X.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g\ni j k : 𝒰.J\n⊢ (((fun i j k => t' 𝒰 f g i j k) i j k ≫\n          pullback.snd ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j k)\n            ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j i)) ≫\n        pullback.fst (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)) ≫\n      pullback.fst (𝒰.map j ≫ f) g =\n    ((pullback.fst ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i j)\n            ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i k) ≫\n          (fun i j => t 𝒰 f g i j) i j) ≫\n        pullback.fst (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)) ≫\n      pullback.fst (𝒰.map j ≫ f) g\n\ncase h₀.h₁\nC : Type u\ninst✝¹ : Category.{v, u} C\nX Y Z : Scheme\n𝒰 : X.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g\ni j k : 𝒰.J\n⊢ (((fun i j k => t' 𝒰 f g i j k) i j k ≫\n          pullback.snd ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j k)\n            ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j i)) ≫\n        pullback.fst (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)) ≫\n      pullback.snd (𝒰.map j ≫ f) g =\n    ((pullback.fst ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i j)\n            ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i k) ≫\n          (fun i j => t 𝒰 f g i j) i j) ≫\n        pullback.fst (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)) ≫\n      pullback.snd (𝒰.map j ≫ f) g\n\ncase h₁\nC : Type u\ninst✝¹ : Category.{v, u} C\nX Y Z : Scheme\n𝒰 : X.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g\ni j k : 𝒰.J\n⊢ ((fun i j k => t' 𝒰 f g i j k) i j k ≫\n        pullback.snd ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j k)\n          ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j i)) ≫\n      pullback.snd (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i) =\n    (pullback.fst ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i j)\n          ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i k) ≫\n        (fun i j => t 𝒰 f g i j) i j) ≫\n      pullback.snd (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)"}
+{"state": [{"context": ["n : ℕ", "i a b : Fin (n + 1)", "hab : a ≤ b", "H : {i, a, b}.card ≤ n"], "goal": "Set.range ⇑(asOrderHom (standardSimplex.edge n a b hab)) ∪ {i} ≠ Set.univ"}], "premise": [1096, 1101, 1169, 1717, 1999, 2025, 4165, 47447, 47451, 53688, 70178, 130988, 131586, 131595, 133323, 133392, 133487, 133529, 137614, 141365], "state_str": "case range\nn : ℕ\ni a b : Fin (n + 1)\nhab : a ≤ b\nH : {i, a, b}.card ≤ n\n⊢ Set.range ⇑(asOrderHom (standardSimplex.edge n a b hab)) ∪ {i} ≠ Set.univ"}
+{"state": [{"context": ["G : Type u_1", "inst✝¹ : Group G", "ι : Type u_2", "hdec : DecidableEq ι", "hfin : Fintype ι", "H : ι → Type u_3", "inst✝ : (i : ι) → Group (H i)", "ϕ : (i : ι) → H i →* G", "f g : (i : ι) → H i", "hcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)", "hind : CompleteLattice.Independent fun i => (ϕ i).range", "hinj : ∀ (i : ι), Function.Injective ⇑(ϕ i)"], "goal": "Function.Injective ⇑(noncommPiCoprod ϕ hcomm)"}], "premise": [1673, 6982, 18818, 123039, 140822], "state_str": "G : Type u_1\ninst✝¹ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝ : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf g : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\nhind : CompleteLattice.Independent fun i => (ϕ i).range\nhinj : ∀ (i : ι), Function.Injective ⇑(ϕ i)\n⊢ Function.Injective ⇑(noncommPiCoprod ϕ hcomm)"}
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+{"state": [{"context": ["𝕜 : Type u", "inst✝⁵ : NontriviallyNormedField 𝕜", "F : Type v", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace 𝕜 F", "E : Type w", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace 𝕜 E", "f✝ f₀ f₁ g : 𝕜 → F", "f' f₀' f₁' g' : F", "x : 𝕜", "s✝ t✝ : Set 𝕜", "L L₁ L₂ : Filter 𝕜", "inst✝ : SeparableSpace 𝕜", "f : 𝕜 → F", "s t : Set 𝕜", "ts : t ⊆ s", "t_count : t.Countable", "ht : s ⊆ closure t", "this : s ⊆ closure (s ∩ t)"], "goal": "IsSeparable (closure ↑(Submodule.span 𝕜 (f '' t)))"}], "premise": [57753, 68726, 132748], "state_str": "𝕜 : Type u\ninst✝⁵ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\nE : Type w\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\nf✝ f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns✝ t✝ : Set 𝕜\nL L₁ L₂ : Filter 𝕜\ninst✝ : SeparableSpace 𝕜\nf : 𝕜 → F\ns t : Set 𝕜\nts : t ⊆ s\nt_count : t.Countable\nht : s ⊆ closure t\nthis : s ⊆ closure (s ∩ t)\n⊢ IsSeparable (closure ↑(Submodule.span 𝕜 (f '' t)))"}
+{"state": [{"context": [], "goal": "2 • ↑π = 0"}], "premise": [38273], "state_str": "⊢ 2 • ↑π = 0"}
+{"state": [{"context": ["Ω : Type u_1", "F : Type u_2", "m mΩ : MeasurableSpace Ω", "inst✝³ : StandardBorelSpace Ω", "inst✝² : Nonempty Ω", "μ : Measure Ω", "inst✝¹ inst✝ : IsFiniteMeasure μ", "hm : m ≤ mΩ", "s : Set Ω", "hs : MeasurableSet s"], "goal": "(fun ω => (((condexpKernel μ m) ω) s).toReal) =ᶠ[ae (μ.trim hm)] μ[s.indicator fun ω => 1|m]"}], "premise": [27927, 33747], "state_str": "Ω : Type u_1\nF : Type u_2\nm mΩ : MeasurableSpace Ω\ninst✝³ : StandardBorelSpace Ω\ninst✝² : Nonempty Ω\nμ : Measure Ω\ninst✝¹ inst✝ : IsFiniteMeasure μ\nhm : m ≤ mΩ\ns : Set Ω\nhs : MeasurableSet s\n⊢ (fun ω => (((condexpKernel μ m) ω) s).toReal) =ᶠ[ae (μ.trim hm)] μ[s.indicator fun ω => 1|m]"}
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+{"state": [{"context": ["p : ℕ+", "k : ℕ", "K : Type u", "inst✝² : Field K", "inst✝¹ : CharZero K", "ζ : K", "hp : Fact (Nat.Prime ↑p)", "inst✝ : IsCyclotomicExtension {p ^ k} ℚ K", "hζ : IsPrimitiveRoot ζ ↑(p ^ k)"], "goal": "∃ u n, Algebra.discr ℚ ⇑(IsPrimitiveRoot.subOnePowerBasis ℚ hζ).basis = ↑↑u * ↑↑p ^ n"}], "premise": [2100, 24040], "state_str": "p : ℕ+\nk : ℕ\nK : Type u\ninst✝² : Field K\ninst✝¹ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ k} ℚ K\nhζ : IsPrimitiveRoot ζ ↑(p ^ k)\n⊢ ∃ u n, Algebra.discr ℚ ⇑(IsPrimitiveRoot.subOnePowerBasis ℚ hζ).basis = ↑↑u * ↑↑p ^ n"}
+{"state": [{"context": ["p : ℕ+", "k : ℕ", "K : Type u", "inst✝² : Field K", "inst✝¹ : CharZero K", "ζ : K", "hp : Fact (Nat.Prime ↑p)", "inst✝ : IsCyclotomicExtension {p ^ k} ℚ K", "hζ : IsPrimitiveRoot ζ ↑(p ^ k)"], "goal": "∃ u n, Algebra.discr ℚ ⇑(IsPrimitiveRoot.powerBasis ℚ hζ).basis = ↑↑u * ↑↑p ^ n"}], "premise": [24044, 74768, 141094], "state_str": "p : ℕ+\nk : ℕ\nK : Type u\ninst✝² : Field K\ninst✝¹ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ k} ℚ K\nhζ : IsPrimitiveRoot ζ ↑(p ^ k)\n⊢ ∃ u n, Algebra.discr ℚ ⇑(IsPrimitiveRoot.powerBasis ℚ hζ).basis = ↑↑u * ↑↑p ^ n"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "m : OuterMeasure α", "f : α → β", "s : Set β"], "goal": "((map f) ⊤) s = ((restrict (range f)) ⊤) s"}], "premise": [27690, 27702, 27715, 134115, 134192], "state_str": "α : Type u_1\nβ : Type u_2\nm : OuterMeasure α\nf : α → β\ns : Set β\n⊢ ((map f) ⊤) s = ((restrict (range f)) ⊤) s"}
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+{"state": [{"context": ["α : Type u_1", "mα : MeasurableSpace α", "μ ν : Measure α", "f : α → ℝ", "inst✝¹ : IsFiniteMeasure μ", "inst✝ : μ.HaveLebesgueDecomposition ν", "hμν : μ ≪ ν", "c : ℝ≥0∞", "hc : c ≠ 0", "hc_ne_top : c ≠ ⊤"], "goal": "(fun x => log ((c • μ).rnDeriv ν x).toReal) =ᶠ[ae μ] fun x => log (μ.rnDeriv ν x).toReal + log c.toReal"}], "premise": [34867], "state_str": "α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : μ.HaveLebesgueDecomposition ν\nhμν : μ ≪ ν\nc : ℝ≥0∞\nhc : c ≠ 0\nhc_ne_top : c ≠ ⊤\n⊢ (fun x => log ((c • μ).rnDeriv ν x).toReal) =ᶠ[ae μ] fun x => log (μ.rnDeriv ν x).toReal + log c.toReal"}
+{"state": [{"context": ["α : Type u_1", "mα : MeasurableSpace α", "μ ν : Measure α", "f : α → ℝ", "inst✝¹ : IsFiniteMeasure μ", "inst✝ : μ.HaveLebesgueDecomposition ν", "hμν : μ ≪ ν", "c : ℝ≥0∞", "hc : c ≠ 0", "hc_ne_top : c ≠ ⊤", "h : (c • μ).rnDeriv ν =ᶠ[ae ν] c • μ.rnDeriv ν"], "goal": "(fun x => log ((c • μ).rnDeriv ν x).toReal) =ᶠ[ae μ] fun x => log (μ.rnDeriv ν x).toReal + log c.toReal"}], "premise": [15889, 32752, 34851, 131585], "state_str": "α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : μ.HaveLebesgueDecomposition ν\nhμν : μ ≪ ν\nc : ℝ≥0∞\nhc : c ≠ 0\nhc_ne_top : c ≠ ⊤\nh : (c • μ).rnDeriv ν =ᶠ[ae ν] c • μ.rnDeriv ν\n⊢ (fun x => log ((c • μ).rnDeriv ν x).toReal) =ᶠ[ae μ] fun x => log (μ.rnDeriv ν x).toReal + log c.toReal"}
+{"state": [{"context": ["α : Type u_1", "mα : MeasurableSpace α", "μ ν : Measure α", "f : α → ℝ", "inst✝¹ : IsFiniteMeasure μ", "inst✝ : μ.HaveLebesgueDecomposition ν", "hμν : μ ≪ ν", "c : ℝ≥0∞", "hc : c ≠ 0", "hc_ne_top : c ≠ ⊤", "h : (c • μ).rnDeriv ν =ᶠ[ae ν] c • μ.rnDeriv ν", "x : α", "hx_eq : (c • μ).rnDeriv ν x = (c • μ.rnDeriv ν) x", "hx_pos : 0 < μ.rnDeriv ν x", "hx_ne_top : μ.rnDeriv ν x < ⊤"], "goal": "log ((c • μ.rnDeriv ν) x).toReal = log (μ.rnDeriv ν x).toReal + log c.toReal"}], "premise": [118863, 120658, 143423], "state_str": "case h\nα : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : μ.HaveLebesgueDecomposition ν\nhμν : μ ≪ ν\nc : ℝ≥0∞\nhc : c ≠ 0\nhc_ne_top : c ≠ ⊤\nh : (c • μ).rnDeriv ν =ᶠ[ae ν] c • μ.rnDeriv ν\nx : α\nhx_eq : (c • μ).rnDeriv ν x = (c • μ.rnDeriv ν) x\nhx_pos : 0 < μ.rnDeriv ν x\nhx_ne_top : μ.rnDeriv ν x < ⊤\n⊢ log ((c • μ.rnDeriv ν) x).toReal = log (μ.rnDeriv ν x).toReal + log c.toReal"}
+{"state": [{"context": ["α : Type u_1", "mα : MeasurableSpace α", "μ ν : Measure α", "f : α → ℝ", "inst✝¹ : IsFiniteMeasure μ", "inst✝ : μ.HaveLebesgueDecomposition ν", "hμν : μ ≪ ν", "c : ℝ≥0∞", "hc : c ≠ 0", "hc_ne_top : c ≠ ⊤", "h : (c • μ).rnDeriv ν =ᶠ[ae ν] c • μ.rnDeriv ν", "x : α", "hx_eq : (c • μ).rnDeriv ν x = (c • μ.rnDeriv ν) x", "hx_pos : 0 < μ.rnDeriv ν x", "hx_ne_top : μ.rnDeriv ν x < ⊤"], "goal": "log (c.toReal * (μ.rnDeriv ν x).toReal) = log (μ.rnDeriv ν x).toReal + log c.toReal"}], "premise": [37879], "state_str": "case h\nα : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : μ.HaveLebesgueDecomposition ν\nhμν : μ ≪ ν\nc : ℝ≥0∞\nhc : c ≠ 0\nhc_ne_top : c ≠ ⊤\nh : (c • μ).rnDeriv ν =ᶠ[ae ν] c • μ.rnDeriv ν\nx : α\nhx_eq : (c • μ).rnDeriv ν x = (c • μ.rnDeriv ν) x\nhx_pos : 0 < μ.rnDeriv ν x\nhx_ne_top : μ.rnDeriv ν x < ⊤\n⊢ log (c.toReal * (μ.rnDeriv ν x).toReal) = log (μ.rnDeriv ν x).toReal + log c.toReal"}
+{"state": [{"context": ["R : Type u_1", "S : Type u_2", "inst✝⁶ : CommRing R", "inst✝⁵ : CommRing S", "inst✝⁴ : Algebra R S", "A B : Subalgebra R S", "inst✝³ : Module.Free R ↥A", "inst✝² : Module.Free R ↥B", "inst✝¹ : Module.Free ↥A ↥(Algebra.adjoin ↥A ↑B)", "inst✝ : Module.Free ↥B ↥(Algebra.adjoin ↥B ↑A)"], "goal": "finrank R ↥(A ⊔ B) = finrank R ↥B * finrank ↥B ↥(Algebra.adjoin ↥B ↑A)"}], "premise": [14537, 122431], "state_str": "R : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nA B : Subalgebra R S\ninst✝³ : Module.Free R ↥A\ninst✝² : Module.Free R ↥B\ninst✝¹ : Module.Free ↥A ↥(Algebra.adjoin ↥A ↑B)\ninst✝ : Module.Free ↥B ↥(Algebra.adjoin ↥B ↑A)\n⊢ finrank R ↥(A ⊔ B) = finrank R ↥B * finrank ↥B ↥(Algebra.adjoin ↥B ↑A)"}
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+{"state": [{"context": ["α✝ : Type u_1", "β : Type u_2", "ι : Type u_3", "ι' : Type u_4", "inst✝¹ : CompleteLattice α✝", "s✝ : Set α✝", "hs : SetIndependent s✝", "α : Type u_5", "inst✝ : CompleteLattice α", "s : Set α"], "goal": "SetIndependent s ↔ Independent Subtype.val"}], "premise": [19334, 133298], "state_str": "α✝ : Type u_1\nβ : Type u_2\nι : Type u_3\nι' : Type u_4\ninst✝¹ : CompleteLattice α✝\ns✝ : Set α✝\nhs : SetIndependent s✝\nα : Type u_5\ninst✝ : CompleteLattice α\ns : Set α\n⊢ SetIndependent s ↔ Independent Subtype.val"}
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+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "n : ℕ", "x : α", "xs : List α"], "goal": "enumFrom n (x :: xs) = (n, x) :: map (Prod.map Nat.succ id) (enumFrom n xs)"}], "premise": [2704, 3682, 130168], "state_str": "α : Type u_1\nβ : Type u_2\nn : ℕ\nx : α\nxs : List α\n⊢ enumFrom n (x :: xs) = (n, x) :: map (Prod.map Nat.succ id) (enumFrom n xs)"}
+{"state": [{"context": ["M : Type u_1", "inst✝² : CommMonoid M", "S : Submonoid M", "N : Type u_2", "inst✝¹ : CommMonoid N", "P : Type u_3", "inst✝ : CommMonoid P", "f : S.LocalizationMap N", "a₁ b₁ : M", "a₂ b₂ : ↥S"], "goal": "f.mk' a₁ a₂ = f.mk' b₁ b₂ ↔ (Localization.r S) (a₁, a₂) (b₁, b₂)"}], "premise": [1713, 9085, 9170], "state_str": "M : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type u_3\ninst✝ : CommMonoid P\nf : S.LocalizationMap N\na₁ b₁ : M\na₂ b₂ : ↥S\n⊢ f.mk' a₁ a₂ = f.mk' b₁ b₂ ↔ (Localization.r S) (a₁, a₂) (b₁, b₂)"}
+{"state": [{"context": ["𝕜₁ : Type u_1", "𝕜₂ : Type u_2", "inst✝⁸ : NormedField 𝕜₁", "inst✝⁷ : NormedField 𝕜₂", "σ : 𝕜₁ →+* 𝕜₂", "E : Type u_3", "F : Type u_4", "inst✝⁶ : AddCommGroup E", "inst✝⁵ : Module 𝕜₁ E", "inst✝⁴ : TopologicalSpace E", "inst✝³ : AddCommGroup F", "inst✝² : Module 𝕜₂ F", "inst✝¹ : UniformSpace F", "inst✝ : UniformAddGroup F", "𝔖 : Set (Set E)"], "goal": "TopologicalAddGroup.toUniformSpace F = inst✝¹"}], "premise": [67226], "state_str": "case e_t.h.e_3.h\n𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝⁸ : NormedField 𝕜₁\ninst✝⁷ : NormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nE : Type u_3\nF : Type u_4\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module 𝕜₁ E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : AddCommGroup F\ninst✝² : Module 𝕜₂ F\ninst✝¹ : UniformSpace F\ninst✝ : UniformAddGroup F\n𝔖 : Set (Set E)\n⊢ TopologicalAddGroup.toUniformSpace F = inst✝¹"}
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+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : Lattice α", "inst✝ : Group α", "a : α", "ha : 1 < a"], "goal": "1 < a⁺ᵐ"}], "premise": [1674, 105475], "state_str": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Lattice α\ninst✝ : Group α\na : α\nha : 1 < a\n⊢ 1 < a⁺ᵐ"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "s✝ s : Finset α"], "goal": "IsEmpty { x // x ∈ s } ↔ s = ∅"}], "premise": [133382], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ns✝ s : Finset α\n⊢ IsEmpty { x // x ∈ s } ↔ s = ∅"}
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+{"state": [{"context": ["K : Type u_1", "inst✝¹ : Field K", "inst✝ : NumberField K", "I✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ", "f : InfinitePlace K → ℝ≥0", "I : (FractionalIdeal (𝓞 K)⁰ K)ˣ", "h : minkowskiBound K I < volume (convexBodyLT K f)"], "goal": "∃ a ∈ ↑I, a ≠ 0 ∧ ∀ (w : InfinitePlace K), w a < ↑(f w)"}], "premise": [111688], "state_str": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nf : InfinitePlace K → ℝ≥0\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nh : minkowskiBound K I < volume (convexBodyLT K f)\n⊢ ∃ a ∈ ↑I, a ≠ 0 ∧ ∀ (w : InfinitePlace K), w a < ↑(f w)"}
+{"state": [{"context": ["K : Type u_1", "inst✝¹ : Field K", "inst✝ : NumberField K", "I✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ", "f : InfinitePlace K → ℝ≥0", "I : (FractionalIdeal (𝓞 K)⁰ K)ˣ", "h : minkowskiBound K I < volume (convexBodyLT K f)", "h_fund : IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup) (fundamentalDomain (fractionalIdealLatticeBasis K I)) volume", "this : Countable ↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup"], "goal": "∃ a ∈ ↑I, a ≠ 0 ∧ ∀ (w : InfinitePlace K), w a < ↑(f w)"}], "premise": [23692, 23693, 31957], "state_str": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nf : InfinitePlace K → ℝ≥0\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nh : minkowskiBound K I < volume (convexBodyLT K f)\nh_fund :\n  IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup)\n    (fundamentalDomain (fractionalIdealLatticeBasis K I)) volume\nthis : Countable ↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup\n⊢ ∃ a ∈ ↑I, a ≠ 0 ∧ ∀ (w : InfinitePlace K), w a < ↑(f w)"}
+{"state": [{"context": ["K : Type u_1", "inst✝¹ : Field K", "inst✝ : NumberField K", "I✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ", "f : InfinitePlace K → ℝ≥0", "I : (FractionalIdeal (𝓞 K)⁰ K)ˣ", "h : minkowskiBound K I < volume (convexBodyLT K f)", "h_fund : IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup) (fundamentalDomain (fractionalIdealLatticeBasis K I)) volume", "this : Countable ↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup", "x : ({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)", "hx : x ∈ (span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup", "h_nz : ⟨x, hx⟩ ≠ 0", "h_mem : ↑⟨x, hx⟩ ∈ convexBodyLT K f"], "goal": "∃ a ∈ ↑I, a ≠ 0 ∧ ∀ (w : InfinitePlace K), w a < ↑(f w)"}], "premise": [23238, 109654], "state_str": "case intro.mk.intro\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nf : InfinitePlace K → ℝ≥0\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nh : minkowskiBound K I < volume (convexBodyLT K f)\nh_fund :\n  IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup)\n    (fundamentalDomain (fractionalIdealLatticeBasis K I)) volume\nthis : Countable ↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup\nx : ({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)\nhx : x ∈ (span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup\nh_nz : ⟨x, hx⟩ ≠ 0\nh_mem : ↑⟨x, hx⟩ ∈ convexBodyLT K f\n⊢ ∃ a ∈ ↑I, a ≠ 0 ∧ ∀ (w : InfinitePlace K), w a < ↑(f w)"}
+{"state": [{"context": ["K : Type u_1", "inst✝¹ : Field K", "inst✝ : NumberField K", "I✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ", "f : InfinitePlace K → ℝ≥0", "I : (FractionalIdeal (𝓞 K)⁰ K)ˣ", "h : minkowskiBound K I < volume (convexBodyLT K f)", "h_fund : IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup) (fundamentalDomain (fractionalIdealLatticeBasis K I)) volume", "this : Countable ↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup", "a : K", "ha : a ∈ ↑↑I", "hx : (mixedEmbedding K) a ∈ (span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup", "h_nz : ⟨(mixedEmbedding K) a, hx⟩ ≠ 0", "h_mem : ↑⟨(mixedEmbedding K) a, hx⟩ ∈ convexBodyLT K f"], "goal": "∃ a ∈ ↑I, a ≠ 0 ∧ ∀ (w : InfinitePlace K), w a < ↑(f w)"}], "premise": [1673, 23691], "state_str": "case intro.mk.intro.intro.intro\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nf : InfinitePlace K → ℝ≥0\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nh : minkowskiBound K I < volume (convexBodyLT K f)\nh_fund :\n  IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup)\n    (fundamentalDomain (fractionalIdealLatticeBasis K I)) volume\nthis : Countable ↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup\na : K\nha : a ∈ ↑↑I\nhx : (mixedEmbedding K) a ∈ (span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup\nh_nz : ⟨(mixedEmbedding K) a, hx⟩ ≠ 0\nh_mem : ↑⟨(mixedEmbedding K) a, hx⟩ ∈ convexBodyLT K f\n⊢ ∃ a ∈ ↑I, a ≠ 0 ∧ ∀ (w : InfinitePlace K), w a < ↑(f w)"}
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+{"state": [{"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝¹ : LinearOrderedSemiring α", "inst✝ : FloorSemiring α", "a✝ : α", "n : ℕ", "a : α", "ha : 0 ≤ a", "x✝ : ℕ"], "goal": "x✝ ∈ Nat.cast ⁻¹' Ioi a ↔ x✝ ∈ Ioi ⌊a⌋₊"}], "premise": [105006], "state_str": "case h\nF : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : LinearOrderedSemiring α\ninst✝ : FloorSemiring α\na✝ : α\nn : ℕ\na : α\nha : 0 ≤ a\nx✝ : ℕ\n⊢ x✝ ∈ Nat.cast ⁻¹' Ioi a ↔ x✝ ∈ Ioi ⌊a⌋₊"}
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+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "ι : Sort u_5", "inst✝ : MeasurableSpace α", "μ✝ μ₁ μ₂ : Measure α", "s✝ s₁ s₂ t : Set α", "μ : Measure α", "s : Set α"], "goal": "MeasurableSet (toMeasurable μ s)"}], "premise": [29101], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Sort u_5\ninst✝ : MeasurableSpace α\nμ✝ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t : Set α\nμ : Measure α\ns : Set α\n⊢ MeasurableSet (toMeasurable μ s)"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "ι : Sort u_5", "inst✝ : MeasurableSpace α", "μ✝ μ₁ μ₂ : Measure α", "s✝ s₁ s₂ t : Set α", "μ : Measure α", "s : Set α", "hs : ∃ t ⊇ s, MeasurableSet t ∧ t =ᶠ[ae μ] s"], "goal": "MeasurableSet hs.choose"}, {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "ι : Sort u_5", "inst✝ : MeasurableSpace α", "μ✝ μ₁ μ₂ : Measure α", "s✝ s₁ s₂ t : Set α", "μ : Measure α", "s : Set α", "hs : ¬∃ t ⊇ s, MeasurableSet t ∧ t =ᶠ[ae μ] s", "h's : ∃ t ⊇ s, MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → μ (t ∩ u) = μ (s ∩ u)"], "goal": "MeasurableSet h's.choose"}, {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "ι : Sort u_5", "inst✝ : MeasurableSpace α", "μ✝ μ₁ μ₂ : Measure α", "s✝ s₁ s₂ t : Set α", "μ : Measure α", "s : Set α", "hs : ¬∃ t ⊇ s, MeasurableSet t ∧ t =ᶠ[ae μ] s", "h's : ¬∃ t ⊇ s, MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → μ (t ∩ u) = μ (s ∩ u)"], "goal": "MeasurableSet ⋯.choose"}], "premise": [1111, 2106, 2107, 29082], "state_str": "case pos\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Sort u_5\ninst✝ : MeasurableSpace α\nμ✝ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t : Set α\nμ : Measure α\ns : Set α\nhs : ∃ t ⊇ s, MeasurableSet t ∧ t =ᶠ[ae μ] s\n⊢ MeasurableSet hs.choose\n\ncase pos\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Sort u_5\ninst✝ : MeasurableSpace α\nμ✝ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t : Set α\nμ : Measure α\ns : Set α\nhs : ¬∃ t ⊇ s, MeasurableSet t ∧ t =ᶠ[ae μ] s\nh's : ∃ t ⊇ s, MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → μ (t ∩ u) = μ (s ∩ u)\n⊢ MeasurableSet h's.choose\n\ncase neg\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Sort u_5\ninst✝ : MeasurableSpace α\nμ✝ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t : Set α\nμ : Measure α\ns : Set α\nhs : ¬∃ t ⊇ s, MeasurableSet t ∧ t =ᶠ[ae μ] s\nh's : ¬∃ t ⊇ s, MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → μ (t ∩ u) = μ (s ∩ u)\n⊢ MeasurableSet ⋯.choose"}
+{"state": [{"context": ["α : Type u", "β : Type v", "X : Type u_1", "ι : Type u_2", "inst✝ : PseudoMetricSpace α", "x✝ y z : α", "δ ε ε₁ ε₂ : ℝ", "s✝ s : Set α", "x : α"], "goal": "⋃ n, s ∩ closedBall x ↑n = s"}], "premise": [61207, 133461, 135293], "state_str": "α : Type u\nβ : Type v\nX : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α\nx✝ y z : α\nδ ε ε₁ ε₂ : ℝ\ns✝ s : Set α\nx : α\n⊢ ⋃ n, s ∩ closedBall x ↑n = s"}
+{"state": [{"context": ["R : Type u_1", "inst✝² : CommRing R", "ι : Type u_2", "s : Finset ι", "v : ι → R", "inst✝¹ : Nontrivial R", "inst✝ : NoZeroDivisors R", "x : R", "hx : ∀ i ∈ s, x ≠ v i"], "goal": "eval x (nodal s v) ≠ 0"}], "premise": [102867, 102870, 102981, 103003, 118007, 123871], "state_str": "R : Type u_1\ninst✝² : CommRing R\nι : Type u_2\ns : Finset ι\nv : ι → R\ninst✝¹ : Nontrivial R\ninst✝ : NoZeroDivisors R\nx : R\nhx : ∀ i ∈ s, x ≠ v i\n⊢ eval x (nodal s v) ≠ 0"}
+{"state": [{"context": ["θ : Angle"], "goal": "θ ≠ -θ ↔ θ ≠ 0 ∧ θ ≠ ↑π"}], "premise": [1713, 1999, 38293], "state_str": "θ : Angle\n⊢ θ ≠ -θ ↔ θ ≠ 0 ∧ θ ≠ ↑π"}
+{"state": [{"context": ["a : ℍ"], "goal": "‖star a‖ = ‖a‖"}], "premise": [36830, 43811, 127722], "state_str": "a : ℍ\n⊢ ‖star a‖ = ‖a‖"}
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+{"state": [{"context": ["α : Type v", "G : Type u", "inst✝³ : Group G", "inst✝² : MulAction G α", "M : Type u", "inst✝¹ : Monoid M", "inst✝ : MulAction M α", "m : M", "a : α", "n : ℕ"], "goal": "IsPeriodicPt (fun x => m • x) n a ↔ m ^ n • a = a"}], "premise": [1713, 115682], "state_str": "α : Type v\nG : Type u\ninst✝³ : Group G\ninst✝² : MulAction G α\nM : Type u\ninst✝¹ : Monoid M\ninst✝ : MulAction M α\nm : M\na : α\nn : ℕ\n⊢ IsPeriodicPt (fun x => m • x) n a ↔ m ^ n • a = a"}
+{"state": [{"context": ["R : Type u_1", "inst✝¹ : Semiring R", "p : ℕ", "hp : Fact (Nat.Prime p)", "inst✝ : CharP R p", "x y : R", "h : Commute x y"], "goal": "(x + y) ^ p = x ^ p + y ^ p"}], "premise": [70028, 124510], "state_str": "R : Type u_1\ninst✝¹ : Semiring R\np : ℕ\nhp : Fact (Nat.Prime p)\ninst✝ : CharP R p\nx y : R\nh : Commute x y\n⊢ (x + y) ^ p = x ^ p + y ^ p"}
+{"state": [{"context": ["E : Type u_1", "inst✝ : NormedAddCommGroup E", "c w : ℂ", "R : ℝ"], "goal": "CircleIntegrable (fun z => (z - w)⁻¹) c R ↔ R = 0 ∨ w ∉ sphere c |R|"}], "premise": [25821, 119795], "state_str": "E : Type u_1\ninst✝ : NormedAddCommGroup E\nc w : ℂ\nR : ℝ\n⊢ CircleIntegrable (fun z => (z - w)⁻¹) c R ↔ R = 0 ∨ w ∉ sphere c |R|"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m : MeasurableSpace α", "μ ν : Measure α", "inst✝³ : MeasurableSpace δ", "inst✝² : NormedAddCommGroup β", "inst✝¹ : NormedAddCommGroup γ", "inst✝ : MeasurableSingletonClass α", "f : α → β"], "goal": "HasFiniteIntegral f Measure.count ↔ Summable fun x => ‖f x‖"}], "premise": [18791, 30390, 42755, 56440, 58991], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝³ : MeasurableSpace δ\ninst✝² : NormedAddCommGroup β\ninst✝¹ : NormedAddCommGroup γ\ninst✝ : MeasurableSingletonClass α\nf : α → β\n⊢ HasFiniteIntegral f Measure.count ↔ Summable fun x => ‖f x‖"}
+{"state": [{"context": ["V : Type u_1", "inst✝¹ : Category.{?u.28, u_1} V", "inst✝ : Abelian V", "X Y Z : V", "f : X ⟶ Y", "g : Y ⟶ Z", "w : f ≫ g = 0"], "goal": "g.op ≫ f.op = 0"}], "premise": [89631, 93608], "state_str": "V : Type u_1\ninst✝¹ : Category.{?u.28, u_1} V\ninst✝ : Abelian V\nX Y Z : V\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\n⊢ g.op ≫ f.op = 0"}
+{"state": [{"context": ["V : Type u_1", "inst✝¹ : Category.{?u.28, u_1} V", "inst✝ : Abelian V", "X Y Z : V", "f : X ⟶ Y", "g : Y ⟶ Z", "w : f ≫ g = 0"], "goal": "f ≫ factorThruImage g = 0"}], "premise": [93605, 94368, 96173, 96191], "state_str": "V : Type u_1\ninst✝¹ : Category.{?u.28, u_1} V\ninst✝ : Abelian V\nX Y Z : V\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\n⊢ f ≫ factorThruImage g = 0"}
+{"state": [{"context": ["V : Type u_1", "inst✝¹ : Category.{u_2, u_1} V", "inst✝ : Abelian V", "X Y Z : V", "f : X ⟶ Y", "g : Y ⟶ Z", "w : f ≫ g = 0"], "goal": "imageToKernel g.op f.op ⋯ ≫ (kernelSubobject f.op).arrow = ((imageSubobjectIso g.op ≪≫ (imageOpOp g).symm).hom ≫ (cokernel.desc f (factorThruImage g) ⋯).op ≫ (kernelSubobjectIso f.op ≪≫ kernelOpOp f).inv) ≫ (kernelSubobject f.op).arrow"}], "premise": [88745, 88755, 88756, 89321, 89340, 89631, 94851, 94883, 95118, 95131, 96173, 114276], "state_str": "case h\nV : Type u_1\ninst✝¹ : Category.{u_2, u_1} V\ninst✝ : Abelian V\nX Y Z : V\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\n⊢ imageToKernel g.op f.op ⋯ ≫ (kernelSubobject f.op).arrow =\n    ((imageSubobjectIso g.op ≪≫ (imageOpOp g).symm).hom ≫\n        (cokernel.desc f (factorThruImage g) ⋯).op ≫ (kernelSubobjectIso f.op ≪≫ kernelOpOp f).inv) ≫\n      (kernelSubobject f.op).arrow"}
+{"state": [{"context": ["xl xr : Type u", "x y : PGame"], "goal": "y ⧏ -x ↔ x ⧏ -y"}], "premise": [1713, 50296, 119769], "state_str": "xl xr : Type u\nx y : PGame\n⊢ y ⧏ -x ↔ x ⧏ -y"}
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+{"state": [{"context": ["z : ℝ", "h : 0 < z", "y : ℝ≥0∞", "x : ℝ≥0", "hxy : ↑x < y"], "goal": "(fun x => x ^ z) ↑x < (fun x => x ^ z) y"}], "premise": [70039], "state_str": "case intro\nz : ℝ\nh : 0 < z\ny : ℝ≥0∞\nx : ℝ≥0\nhxy : ↑x < y\n⊢ (fun x => x ^ z) ↑x < (fun x => x ^ z) y"}
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+{"state": [{"context": ["X : Type u_1", "Y : Type u_2", "Z : Type u_3", "inst✝² : TopologicalSpace X", "inst✝¹ : TopologicalSpace Y", "inst✝ : TopologicalSpace Z", "f✝² : X → Y", "s✝ : Set X", "x : X", "y : Y", "α : Type u_4", "ι : Type u_5", "f✝¹ : α → X", "g : X → Y", "f✝ : α → X", "s : Set X", "f : X → Y", "hf : Continuous f", "hs : Dense s"], "goal": "f '' closure s ⊆ closure (f '' s)"}], "premise": [55655], "state_str": "X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nf✝² : X → Y\ns✝ : Set X\nx : X\ny : Y\nα : Type u_4\nι : Type u_5\nf✝¹ : α → X\ng : X → Y\nf✝ : α → X\ns : Set X\nf : X → Y\nhf : Continuous f\nhs : Dense s\n⊢ f '' closure s ⊆ closure (f '' s)"}
+{"state": [{"context": ["K : Type u_1", "inst✝¹ : Field K", "inst✝ : NumberField K", "ι : Type u_2", "s : Finset ι", "α : ι → K"], "goal": "house (∑ i ∈ s, α i) ≤ ∑ i ∈ s, house (α i)"}], "premise": [126889], "state_str": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nι : Type u_2\ns : Finset ι\nα : ι → K\n⊢ house (∑ i ∈ s, α i) ≤ ∑ i ∈ s, house (α i)"}
+{"state": [{"context": ["K : Type u_1", "inst✝¹ : Field K", "inst✝ : NumberField K", "ι : Type u_2", "s : Finset ι", "α : ι → K"], "goal": "‖∑ x ∈ s, (canonicalEmbedding K) (α x)‖ ≤ ∑ x ∈ s, ‖(canonicalEmbedding K) (α x)‖"}], "premise": [42861], "state_str": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nι : Type u_2\ns : Finset ι\nα : ι → K\n⊢ ‖∑ x ∈ s, (canonicalEmbedding K) (α x)‖ ≤ ∑ x ∈ s, ‖(canonicalEmbedding K) (α x)‖"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "G : Type u_3", "M : Type u_4", "inst✝ : Group G", "a b c d : G", "n : ℤ", "H : a / b = c / d"], "goal": "a = b ↔ c = d"}], "premise": [1713, 118005], "state_str": "α : Type u_1\nβ : Type u_2\nG : Type u_3\nM : Type u_4\ninst✝ : Group G\na b c d : G\nn : ℤ\nH : a / b = c / d\n⊢ a = b ↔ c = d"}
+{"state": [{"context": ["R : Type u_1", "inst✝¹ : CommRing R", "ι : Type u_2", "s : Finset ι", "v : ι → R", "inst✝ : DecidableEq ι", "i : ι", "hi : i ∈ s"], "goal": "nodal s v = (X - C (v i)) * nodal (s.erase i) v"}], "premise": [127223], "state_str": "R : Type u_1\ninst✝¹ : CommRing R\nι : Type u_2\ns : Finset ι\nv : ι → R\ninst✝ : DecidableEq ι\ni : ι\nhi : i ∈ s\n⊢ nodal s v = (X - C (v i)) * nodal (s.erase i) v"}
+{"state": [{"context": ["X Y : Scheme", "f : X ⟶ Y"], "goal": "QuasiCompact f ↔ ∀ (U : Y.Opens), IsAffineOpen U → IsCompact ↑(f ⁻¹ᵁ U)"}], "premise": [128057], "state_str": "X Y : Scheme\nf : X ⟶ Y\n⊢ QuasiCompact f ↔ ∀ (U : Y.Opens), IsAffineOpen U → IsCompact ↑(f ⁻¹ᵁ U)"}
+{"state": [{"context": ["X Y : Scheme", "f : X ⟶ Y"], "goal": "(∀ (U : Set ↑↑Y.toPresheafedSpace), IsOpen U → IsCompact U → IsCompact (⇑f.val.base ⁻¹' U)) ↔ ∀ (U : Y.Opens), IsAffineOpen U → IsCompact ↑(f ⁻¹ᵁ U)"}], "premise": [55754, 128475], "state_str": "X Y : Scheme\nf : X ⟶ Y\n⊢ (∀ (U : Set ↑↑Y.toPresheafedSpace), IsOpen U → IsCompact U → IsCompact (⇑f.val.base ⁻¹' U)) ↔\n    ∀ (U : Y.Opens), IsAffineOpen U → IsCompact ↑(f ⁻¹ᵁ U)"}
+{"state": [{"context": ["X Y : Scheme", "f : X ⟶ Y", "H : ∀ (U : Y.Opens), IsAffineOpen U → IsCompact ↑(f ⁻¹ᵁ U)", "U : Set ↑↑Y.toPresheafedSpace", "hU : IsOpen U", "hU' : IsCompact U"], "goal": "IsCompact (⇑f.val.base ⁻¹' U)"}], "premise": [1673, 128059], "state_str": "X Y : Scheme\nf : X ⟶ Y\nH : ∀ (U : Y.Opens), IsAffineOpen U → IsCompact ↑(f ⁻¹ᵁ U)\nU : Set ↑↑Y.toPresheafedSpace\nhU : IsOpen U\nhU' : IsCompact U\n⊢ IsCompact (⇑f.val.base ⁻¹' U)"}
+{"state": [{"context": ["X Y : Scheme", "f : X ⟶ Y", "H : ∀ (U : Y.Opens), IsAffineOpen U → IsCompact ↑(f ⁻¹ᵁ U)", "S : Set ↑Y.affineOpens", "hS : S.Finite", "hU : IsOpen (⋃ i ∈ S, ↑↑i)", "hU' : IsCompact (⋃ i ∈ S, ↑↑i)"], "goal": "IsCompact (⇑f.val.base ⁻¹' ⋃ i ∈ S, ↑↑i)"}], "premise": [135513], "state_str": "case intro.intro\nX Y : Scheme\nf : X ⟶ Y\nH : ∀ (U : Y.Opens), IsAffineOpen U → IsCompact ↑(f ⁻¹ᵁ U)\nS : Set ↑Y.affineOpens\nhS : S.Finite\nhU : IsOpen (⋃ i ∈ S, ↑↑i)\nhU' : IsCompact (⋃ i ∈ S, ↑↑i)\n⊢ IsCompact (⇑f.val.base ⁻¹' ⋃ i ∈ S, ↑↑i)"}
+{"state": [{"context": ["X Y : Scheme", "f : X ⟶ Y", "H : ∀ (U : Y.Opens), IsAffineOpen U → IsCompact ↑(f ⁻¹ᵁ U)", "S : Set ↑Y.affineOpens", "hS : S.Finite", "hU : IsOpen (⋃ i ∈ S, ↑↑i)", "hU' : IsCompact (⋃ i ∈ S, ↑↑i)"], "goal": "IsCompact (⋃ i ∈ S, ⇑f.val.base ⁻¹' ↑↑i)"}], "premise": [58102, 137122], "state_str": "case intro.intro\nX Y : Scheme\nf : X ⟶ Y\nH : ∀ (U : Y.Opens), IsAffineOpen U → IsCompact ↑(f ⁻¹ᵁ U)\nS : Set ↑Y.affineOpens\nhS : S.Finite\nhU : IsOpen (⋃ i ∈ S, ↑↑i)\nhU' : IsCompact (⋃ i ∈ S, ↑↑i)\n⊢ IsCompact (⋃ i ∈ S, ⇑f.val.base ⁻¹' ↑↑i)"}
+{"state": [{"context": ["ι : Type u", "E : Type v", "F : Type w", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : NormedSpace ℝ E", "inst✝² : NormedAddCommGroup F", "inst✝¹ : NormedSpace ℝ F", "I J : Box ι", "π : TaggedPrepartition I", "inst✝ : Fintype ι", "l : IntegrationParams", "f g : (ι → ℝ) → E", "vol : ι →ᵇᵃ[⊤] E →L[ℝ] F", "y y' : F", "c c₁ c₂ : ℝ≥0", "ε ε₁ ε₂ : ℝ", "π₁ π₂ : TaggedPrepartition I", "h : Integrable I l f vol", "h₀ : 0 < ε", "hπ : l.MemBaseSet I c (h.convergenceR ε c) π", "hπp : π.IsPartition"], "goal": "dist (integralSum f vol π) (integral I l f vol) ≤ ε"}], "premise": [1739], "state_str": "ι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI J : Box ι\nπ : TaggedPrepartition I\ninst✝ : Fintype ι\nl : IntegrationParams\nf g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\nc c₁ c₂ : ℝ≥0\nε ε₁ ε₂ : ℝ\nπ₁ π₂ : TaggedPrepartition I\nh : Integrable I l f vol\nh₀ : 0 < ε\nhπ : l.MemBaseSet I c (h.convergenceR ε c) π\nhπp : π.IsPartition\n⊢ dist (integralSum f vol π) (integral I l f vol) ≤ ε"}
+{"state": [{"context": ["ι : Type u", "E : Type v", "F : Type w", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : NormedSpace ℝ E", "inst✝² : NormedAddCommGroup F", "inst✝¹ : NormedSpace ℝ F", "I J : Box ι", "π : TaggedPrepartition I", "inst✝ : Fintype ι", "l : IntegrationParams", "f g : (ι → ℝ) → E", "vol : ι →ᵇᵃ[⊤] E →L[ℝ] F", "y y' : F", "c c₁ c₂ : ℝ≥0", "ε ε₁ ε₂ : ℝ", "π₁ π₂ : TaggedPrepartition I", "h : Integrable I l f vol", "h₀ : 0 < ε", "hπ : l.MemBaseSet I c (⋯.choose c) π", "hπp : π.IsPartition"], "goal": "dist (integralSum f vol π) (integral I l f vol) ≤ ε"}], "premise": [1111, 1673, 2106, 36531, 36536], "state_str": "ι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI J : Box ι\nπ : TaggedPrepartition I\ninst✝ : Fintype ι\nl : IntegrationParams\nf g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\nc c₁ c₂ : ℝ≥0\nε ε₁ ε₂ : ℝ\nπ₁ π₂ : TaggedPrepartition I\nh : Integrable I l f vol\nh₀ : 0 < ε\nhπ : l.MemBaseSet I c (⋯.choose c) π\nhπp : π.IsPartition\n⊢ dist (integralSum f vol π) (integral I l f vol) ≤ ε"}
+{"state": [{"context": ["R : Type u_1", "K : Type u_2", "L : Type u_3", "M : Type u_4", "inst✝⁷ : CommRing R", "inst✝⁶ : Field K", "inst✝⁵ : LieRing L", "inst✝⁴ : LieAlgebra R L", "inst✝³ : LieAlgebra K L", "inst✝² : Module.Free R L", "inst✝¹ : Module.Finite R L", "inst✝ : IsKilling R L"], "goal": "(killingForm R L).Nondegenerate"}], "premise": [83279], "state_str": "R : Type u_1\nK : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : Field K\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : LieAlgebra K L\ninst✝² : Module.Free R L\ninst✝¹ : Module.Finite R L\ninst✝ : IsKilling R L\n⊢ (killingForm R L).Nondegenerate"}
+{"state": [{"context": ["α : Type u_1", "inst✝ : LinearOrderedAddCommGroup α", "a b c d : α", "h : [[c, d]] ⊆ [[a, b]]"], "goal": "|d - c| ≤ |b - a|"}], "premise": [105360], "state_str": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\na b c d : α\nh : [[c, d]] ⊆ [[a, b]]\n⊢ |d - c| ≤ |b - a|"}
+{"state": [{"context": ["α : Type u_1", "inst✝ : LinearOrderedAddCommGroup α", "a b c d : α", "h : [[c, d]] ⊆ [[a, b]]"], "goal": "max c d - min c d ≤ max a b - min a b"}], "premise": [18570], "state_str": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\na b c d : α\nh : [[c, d]] ⊆ [[a, b]]\n⊢ max c d - min c d ≤ max a b - min a b"}
+{"state": [{"context": ["α : Type u_1", "inst✝ : LinearOrderedAddCommGroup α", "a b c d : α", "h : min a b ≤ min c d ∧ max c d ≤ max a b"], "goal": "max c d - min c d ≤ max a b - min a b"}], "premise": [2106, 2107, 105733], "state_str": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\na b c d : α\nh : min a b ≤ min c d ∧ max c d ≤ max a b\n⊢ max c d - min c d ≤ max a b - min a b"}
+{"state": [{"context": ["K : Type u_1", "inst✝¹ : Field K", "inst✝ : NeZero 2", "a b c x : K", "ha : a ≠ 0", "s : K", "hs : discrim a b c = s * s"], "goal": "∃ x, a * x * x + b * x + c = 0"}], "premise": [1674, 2045], "state_str": "case intro\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NeZero 2\na b c x : K\nha : a ≠ 0\ns : K\nhs : discrim a b c = s * s\n⊢ ∃ x, a * x * x + b * x + c = 0"}
+{"state": [{"context": ["K : Type u_1", "inst✝¹ : Field K", "inst✝ : NeZero 2", "a b c x : K", "ha : a ≠ 0", "s : K", "hs : discrim a b c = s * s"], "goal": "a * ((-b + s) / (2 * a)) * ((-b + s) / (2 * a)) + b * ((-b + s) / (2 * a)) + c = 0"}], "premise": [101790], "state_str": "case h\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NeZero 2\na b c x : K\nha : a ≠ 0\ns : K\nhs : discrim a b c = s * s\n⊢ a * ((-b + s) / (2 * a)) * ((-b + s) / (2 * a)) + b * ((-b + s) / (2 * a)) + c = 0"}
+{"state": [{"context": ["G : Type u_1", "inst✝ : Group G", "ι : Type u_2", "s : Finset ι", "H : Subgroup G", "g : ι → G"], "goal": "⋃ i ∈ s, g i • ↑H = Set.univ ↔ Set.SurjOn (fun x => ↑(g x)) (↑s) Set.univ"}], "premise": [6873, 6918, 6923, 133393], "state_str": "G : Type u_1\ninst✝ : Group G\nι : Type u_2\ns : Finset ι\nH : Subgroup G\ng : ι → G\n⊢ ⋃ i ∈ s, g i • ↑H = Set.univ ↔ Set.SurjOn (fun x => ↑(g x)) (↑s) Set.univ"}
+{"state": [{"context": ["α : Type u", "β : α → Type v", "l✝ l₁ l₂ : List (Sigma β)", "inst✝ : DecidableEq α", "a : α", "l : List (Sigma β)", "h : a ∉ l.keys"], "goal": "kerase a l = l"}], "premise": [1999, 2106, 2107], "state_str": "α : Type u\nβ : α → Type v\nl✝ l₁ l₂ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nl : List (Sigma β)\nh : a ∉ l.keys\n⊢ kerase a l = l"}
+{"state": [{"context": ["α : Type u_1", "inst✝ : DecidableEq α", "𝒜✝ ℬ : Finset (Finset α)", "s : Finset α", "a✝ a : α", "𝒜 : Finset (Finset α)"], "goal": "(𝓓 a 𝒜).card = 𝒜.card"}], "premise": [135801, 137443, 137654, 137683, 138995], "state_str": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜✝ ℬ : Finset (Finset α)\ns : Finset α\na✝ a : α\n𝒜 : Finset (Finset α)\n⊢ (𝓓 a 𝒜).card = 𝒜.card"}
+{"state": [{"context": ["α : Type u_1", "inst✝ : DecidableEq α", "𝒜✝ ℬ : Finset (Finset α)", "s✝ : Finset α", "a✝ a : α", "𝒜 : Finset (Finset α)", "s : Finset α", "hs : s ∈ ↑(filter (fun a_1 => a_1.erase a ∉ 𝒜) 𝒜)"], "goal": "s ∈ {s | a ∈ s}"}], "premise": [138668, 139089], "state_str": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜✝ ℬ : Finset (Finset α)\ns✝ : Finset α\na✝ a : α\n𝒜 : Finset (Finset α)\ns : Finset α\nhs : s ∈ ↑(filter (fun a_1 => a_1.erase a ∉ 𝒜) 𝒜)\n⊢ s ∈ {s | a ∈ s}"}
+{"state": [{"context": ["α : Type u_1", "inst✝ : DecidableEq α", "𝒜✝ ℬ : Finset (Finset α)", "s✝ : Finset α", "a✝ a : α", "𝒜 : Finset (Finset α)", "s : Finset α", "hs : s ∈ 𝒜 ∧ s.erase a ∉ 𝒜"], "goal": "s ∈ {s | a ∈ s}"}], "premise": [35, 1673, 1690, 2106, 2107, 70047, 138968], "state_str": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜✝ ℬ : Finset (Finset α)\ns✝ : Finset α\na✝ a : α\n𝒜 : Finset (Finset α)\ns : Finset α\nhs : s ∈ 𝒜 ∧ s.erase a ∉ 𝒜\n⊢ s ∈ {s | a ∈ s}"}
+{"state": [{"context": ["α✝ : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "ι : Sort y", "s t u : Set α✝", "inst✝¹¹ : TopologicalSpace α✝", "inst✝¹⁰ : MeasurableSpace α✝", "inst✝⁹ : OpensMeasurableSpace α✝", "inst✝⁸ : MeasurableSpace δ", "inst✝⁷ : LinearOrder α✝", "inst✝⁶ : OrderClosedTopology α✝", "a b x : α✝", "α : Type u_5", "inst✝⁵ : TopologicalSpace α", "m : MeasurableSpace α", "inst✝⁴ : SecondCountableTopology α", "inst✝³ : LinearOrder α", "inst✝² : OrderTopology α", "inst✝¹ : BorelSpace α", "inst✝ : NoMinOrder α", "μ ν : Measure α", "hμ : ∀ ⦃a b : α⦄, a < b → μ (Ioc a b) ≠ ⊤", "h : ∀ ⦃a b : α⦄, a < b → μ (Ioc a b) = ν (Ioc a b)"], "goal": "μ = ν"}], "premise": [20181, 26467], "state_str": "α✝ : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Sort y\ns t u : Set α✝\ninst✝¹¹ : TopologicalSpace α✝\ninst✝¹⁰ : MeasurableSpace α✝\ninst✝⁹ : OpensMeasurableSpace α✝\ninst✝⁸ : MeasurableSpace δ\ninst✝⁷ : LinearOrder α✝\ninst✝⁶ : OrderClosedTopology α✝\na b x : α✝\nα : Type u_5\ninst✝⁵ : TopologicalSpace α\nm : MeasurableSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : BorelSpace α\ninst✝ : NoMinOrder α\nμ ν : Measure α\nhμ : ∀ ⦃a b : α⦄, a < b → μ (Ioc a b) ≠ ⊤\nh : ∀ ⦃a b : α⦄, a < b → μ (Ioc a b) = ν (Ioc a b)\n⊢ μ = ν"}
+{"state": [{"context": ["V : Type u_1", "P : Type u_2", "inst✝⁴ : NormedAddCommGroup V", "inst✝³ : NormedSpace ℝ V", "inst✝² : MetricSpace P", "inst✝¹ : NormedAddTorsor V P", "ι : Type u_3", "inst✝ : Fintype ι", "b : AffineBasis ι ℝ V"], "goal": "Finset.centroid ℝ Finset.univ ⇑b ∈ interior ((convexHull ℝ) (range ⇑b))"}], "premise": [83164], "state_str": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nι : Type u_3\ninst✝ : Fintype ι\nb : AffineBasis ι ℝ V\n⊢ Finset.centroid ℝ Finset.univ ⇑b ∈ interior ((convexHull ℝ) (range ⇑b))"}
+{"state": [{"context": ["V : Type u_1", "P : Type u_2", "inst✝⁴ : NormedAddCommGroup V", "inst✝³ : NormedSpace ℝ V", "inst✝² : MetricSpace P", "inst✝¹ : NormedAddTorsor V P", "ι : Type u_3", "inst✝ : Fintype ι", "b : AffineBasis ι ℝ V", "this : Nonempty ι"], "goal": "Finset.centroid ℝ Finset.univ ⇑b ∈ interior ((convexHull ℝ) (range ⇑b))"}], "premise": [40292, 70124, 83187, 104330, 131585, 137618, 140822, 140830, 142640], "state_str": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nι : Type u_3\ninst✝ : Fintype ι\nb : AffineBasis ι ℝ V\nthis : Nonempty ι\n⊢ Finset.centroid ℝ Finset.univ ⇑b ∈ interior ((convexHull ℝ) (range ⇑b))"}
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+{"state": [{"context": ["n : ℕ"], "goal": "0 < φ n ↔ 0 < n"}], "premise": [103552], "state_str": "n : ℕ\n⊢ 0 < φ n ↔ 0 < n"}
+{"state": [{"context": ["ι : Type u_1", "ι' : Type u_2", "R : Type u_3", "R₂ : Type u_4", "K : Type u_5", "M : Type u_6", "M' : Type u_7", "M'' : Type u_8", "V : Type u", "V' : Type u_9", "inst✝⁵ : Semiring R", "inst✝⁴ : AddCommMonoid M", "inst✝³ : Module R M", "inst✝² : AddCommMonoid M'", "inst✝¹ : Module R M'", "inst✝ : Finite ι", "e : M ≃ₗ[R] ι → R", "j : M", "x✝ : ι"], "goal": "(ofEquivFun e).equivFun j x✝ = e j x✝"}], "premise": [87366, 87372], "state_str": "case h.h\nι : Type u_1\nι' : Type u_2\nR : Type u_3\nR₂ : Type u_4\nK : Type u_5\nM : Type u_6\nM' : Type u_7\nM'' : Type u_8\nV : Type u\nV' : Type u_9\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M'\ninst✝ : Finite ι\ne : M ≃ₗ[R] ι → R\nj : M\nx✝ : ι\n⊢ (ofEquivFun e).equivFun j x✝ = e j x✝"}
+{"state": [{"context": ["K : Type u₁", "inst✝² : Field K", "v : Valuation K ℝ≥0", "O : Type u₂", "inst✝¹ : CommRing O", "inst✝ : Algebra O K", "hv : v.Integers O", "p : ℕ", "hp : Fact (Nat.Prime p)", "hvp : Fact (v ↑p ≠ 1)", "f : PreTilt K v O hv p", "n : ℕ", "hfn : (coeff (ModP K v O hv p) p n) f ≠ 0", "h : ∃ n, (coeff (ModP K v O hv p) p n) f ≠ 0"], "goal": "valAux K v O hv p f = ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p n) f) ^ p ^ n"}], "premise": [1739], "state_str": "K : Type u₁\ninst✝² : Field K\nv : Valuation K ℝ≥0\nO : Type u₂\ninst✝¹ : CommRing O\ninst✝ : Algebra O K\nhv : v.Integers O\np : ℕ\nhp : Fact (Nat.Prime p)\nhvp : Fact (v ↑p ≠ 1)\nf : PreTilt K v O hv p\nn : ℕ\nhfn : (coeff (ModP K v O hv p) p n) f ≠ 0\nh : ∃ n, (coeff (ModP K v O hv p) p n) f ≠ 0\n⊢ valAux K v O hv p f = ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p n) f) ^ p ^ n"}
+{"state": [{"context": ["K : Type u₁", "inst✝² : Field K", "v : Valuation K ℝ≥0", "O : Type u₂", "inst✝¹ : CommRing O", "inst✝ : Algebra O K", "hv : v.Integers O", "p : ℕ", "hp : Fact (Nat.Prime p)", "hvp : Fact (v ↑p ≠ 1)", "f : PreTilt K v O hv p", "n : ℕ", "hfn : (coeff (ModP K v O hv p) p n) f ≠ 0", "h : ∃ n, (coeff (ModP K v O hv p) p n) f ≠ 0"], "goal": "ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h = ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p n) f) ^ p ^ n"}], "premise": [4618, 143302], "state_str": "K : Type u₁\ninst✝² : Field K\nv : Valuation K ℝ≥0\nO : Type u₂\ninst✝¹ : CommRing O\ninst✝ : Algebra O K\nhv : v.Integers O\np : ℕ\nhp : Fact (Nat.Prime p)\nhvp : Fact (v ↑p ≠ 1)\nf : PreTilt K v O hv p\nn : ℕ\nhfn : (coeff (ModP K v O hv p) p n) f ≠ 0\nh : ∃ n, (coeff (ModP K v O hv p) p n) f ≠ 0\n⊢ ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h =\n    ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p n) f) ^ p ^ n"}
+{"state": [{"context": ["K : Type u₁", "inst✝² : Field K", "v : Valuation K ℝ≥0", "O : Type u₂", "inst✝¹ : CommRing O", "inst✝ : Algebra O K", "hv : v.Integers O", "p : ℕ", "hp : Fact (Nat.Prime p)", "hvp : Fact (v ↑p ≠ 1)", "f : PreTilt K v O hv p", "h : ∃ n, (coeff (ModP K v O hv p) p n) f ≠ 0", "k : ℕ", "ih : (coeff (ModP K v O hv p) p (Nat.find h + k)) f ≠ 0 → ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h = ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h + k)) f) ^ p ^ (Nat.find h + k)", "hfn : (coeff (ModP K v O hv p) p (Nat.find h + (k + 1))) f ≠ 0"], "goal": "ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h = ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h + (k + 1))) f) ^ p ^ (Nat.find h + (k + 1))"}], "premise": [80150], "state_str": "case intro.succ\nK : Type u₁\ninst✝² : Field K\nv : Valuation K ℝ≥0\nO : Type u₂\ninst✝¹ : CommRing O\ninst✝ : Algebra O K\nhv : v.Integers O\np : ℕ\nhp : Fact (Nat.Prime p)\nhvp : Fact (v ↑p ≠ 1)\nf : PreTilt K v O hv p\nh : ∃ n, (coeff (ModP K v O hv p) p n) f ≠ 0\nk : ℕ\nih :\n  (coeff (ModP K v O hv p) p (Nat.find h + k)) f ≠ 0 →\n    ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h =\n      ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h + k)) f) ^ p ^ (Nat.find h + k)\nhfn : (coeff (ModP K v O hv p) p (Nat.find h + (k + 1))) f ≠ 0\n⊢ ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h =\n    ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h + (k + 1))) f) ^ p ^ (Nat.find h + (k + 1))"}
+{"state": [{"context": ["K : Type u₁", "inst✝² : Field K", "v : Valuation K ℝ≥0", "O : Type u₂", "inst✝¹ : CommRing O", "inst✝ : Algebra O K", "hv : v.Integers O", "p : ℕ", "hp : Fact (Nat.Prime p)", "hvp : Fact (v ↑p ≠ 1)", "f : PreTilt K v O hv p", "h : ∃ n, (coeff (ModP K v O hv p) p n) f ≠ 0", "k : ℕ", "ih : (coeff (ModP K v O hv p) p (Nat.find h + k)) f ≠ 0 → ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h = ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h + k)) f) ^ p ^ (Nat.find h + k)", "hfn : (coeff (ModP K v O hv p) p (Nat.find h + (k + 1))) f ≠ 0", "x : O", "hx : (Ideal.Quotient.mk (Ideal.span {↑p})) x = (coeff (ModP K v O hv p) p (Nat.find h + k + 1)) f"], "goal": "ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h = ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h + (k + 1))) f) ^ p ^ (Nat.find h + (k + 1))"}], "premise": [2100], "state_str": "case intro.succ.intro\nK : Type u₁\ninst✝² : Field K\nv : Valuation K ℝ≥0\nO : Type u₂\ninst✝¹ : CommRing O\ninst✝ : Algebra O K\nhv : v.Integers O\np : ℕ\nhp : Fact (Nat.Prime p)\nhvp : Fact (v ↑p ≠ 1)\nf : PreTilt K v O hv p\nh : ∃ n, (coeff (ModP K v O hv p) p n) f ≠ 0\nk : ℕ\nih :\n  (coeff (ModP K v O hv p) p (Nat.find h + k)) f ≠ 0 →\n    ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h =\n      ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h + k)) f) ^ p ^ (Nat.find h + k)\nhfn : (coeff (ModP K v O hv p) p (Nat.find h + (k + 1))) f ≠ 0\nx : O\nhx : (Ideal.Quotient.mk (Ideal.span {↑p})) x = (coeff (ModP K v O hv p) p (Nat.find h + k + 1)) f\n⊢ ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h =\n    ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h + (k + 1))) f) ^ p ^ (Nat.find h + (k + 1))"}
+{"state": [{"context": ["K : Type u₁", "inst✝² : Field K", "v : Valuation K ℝ≥0", "O : Type u₂", "inst✝¹ : CommRing O", "inst✝ : Algebra O K", "hv : v.Integers O", "p : ℕ", "hp : Fact (Nat.Prime p)", "hvp : Fact (v ↑p ≠ 1)", "f : PreTilt K v O hv p", "h : ∃ n, (coeff (ModP K v O hv p) p n) f ≠ 0", "k : ℕ", "ih : (coeff (ModP K v O hv p) p (Nat.find h + k)) f ≠ 0 → ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h = ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h + k)) f) ^ p ^ (Nat.find h + k)", "hfn : (coeff (ModP K v O hv p) p (Nat.find h + (k + 1))) f ≠ 0", "x : O", "hx : (Ideal.Quotient.mk (Ideal.span {↑p})) x = (coeff (ModP K v O hv p) p (Nat.find h + k + 1)) f", "h1 : (Ideal.Quotient.mk (Ideal.span {↑p})) x ≠ 0"], "goal": "ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h = ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h + (k + 1))) f) ^ p ^ (Nat.find h + (k + 1))"}], "premise": [80002, 80040, 121592], "state_str": "case intro.succ.intro\nK : Type u₁\ninst✝² : Field K\nv : Valuation K ℝ≥0\nO : Type u₂\ninst✝¹ : CommRing O\ninst✝ : Algebra O K\nhv : v.Integers O\np : ℕ\nhp : Fact (Nat.Prime p)\nhvp : Fact (v ↑p ≠ 1)\nf : PreTilt K v O hv p\nh : ∃ n, (coeff (ModP K v O hv p) p n) f ≠ 0\nk : ℕ\nih :\n  (coeff (ModP K v O hv p) p (Nat.find h + k)) f ≠ 0 →\n    ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h =\n      ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h + k)) f) ^ p ^ (Nat.find h + k)\nhfn : (coeff (ModP K v O hv p) p (Nat.find h + (k + 1))) f ≠ 0\nx : O\nhx : (Ideal.Quotient.mk (Ideal.span {↑p})) x = (coeff (ModP K v O hv p) p (Nat.find h + k + 1)) f\nh1 : (Ideal.Quotient.mk (Ideal.span {↑p})) x ≠ 0\n⊢ ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h =\n    ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h + (k + 1))) f) ^ p ^ (Nat.find h + (k + 1))"}
+{"state": [{"context": ["K : Type u₁", "inst✝² : Field K", "v : Valuation K ℝ≥0", "O : Type u₂", "inst✝¹ : CommRing O", "inst✝ : Algebra O K", "hv : v.Integers O", "p : ℕ", "hp : Fact (Nat.Prime p)", "hvp : Fact (v ↑p ≠ 1)", "f : PreTilt K v O hv p", "h : ∃ n, (coeff (ModP K v O hv p) p n) f ≠ 0", "k : ℕ", "ih : (coeff (ModP K v O hv p) p (Nat.find h + k)) f ≠ 0 → ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h = ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h + k)) f) ^ p ^ (Nat.find h + k)", "hfn : (coeff (ModP K v O hv p) p (Nat.find h + (k + 1))) f ≠ 0", "x : O", "hx : (Ideal.Quotient.mk (Ideal.span {↑p})) x = (coeff (ModP K v O hv p) p (Nat.find h + k + 1)) f", "h1 : (Ideal.Quotient.mk (Ideal.span {↑p})) x ≠ 0", "h2 : (Ideal.Quotient.mk (Ideal.span {↑p})) (x ^ p) ≠ 0"], "goal": "ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h = ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h + (k + 1))) f) ^ p ^ (Nat.find h + (k + 1))"}], "premise": [75416, 80002, 80032, 80040, 119745, 119761, 121592], "state_str": "case intro.succ.intro\nK : Type u₁\ninst✝² : Field K\nv : Valuation K ℝ≥0\nO : Type u₂\ninst✝¹ : CommRing O\ninst✝ : Algebra O K\nhv : v.Integers O\np : ℕ\nhp : Fact (Nat.Prime p)\nhvp : Fact (v ↑p ≠ 1)\nf : PreTilt K v O hv p\nh : ∃ n, (coeff (ModP K v O hv p) p n) f ≠ 0\nk : ℕ\nih :\n  (coeff (ModP K v O hv p) p (Nat.find h + k)) f ≠ 0 →\n    ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h =\n      ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h + k)) f) ^ p ^ (Nat.find h + k)\nhfn : (coeff (ModP K v O hv p) p (Nat.find h + (k + 1))) f ≠ 0\nx : O\nhx : (Ideal.Quotient.mk (Ideal.span {↑p})) x = (coeff (ModP K v O hv p) p (Nat.find h + k + 1)) f\nh1 : (Ideal.Quotient.mk (Ideal.span {↑p})) x ≠ 0\nh2 : (Ideal.Quotient.mk (Ideal.span {↑p})) (x ^ p) ≠ 0\n⊢ ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h)) f) ^ p ^ Nat.find h =\n    ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (Nat.find h + (k + 1))) f) ^ p ^ (Nat.find h + (k + 1))"}
+{"state": [{"context": ["R : Type u", "inst✝² : Ring R", "ι : Type v", "dec_ι : DecidableEq ι", "M : Type u_1", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M", "A : ι → Submodule R M", "s : Set ι", "h : CompleteLattice.Independent fun i => A ↑i"], "goal": "IsInternal fun i => Submodule.comap (⨆ i ∈ s, A i).subtype (A ↑i)"}], "premise": [1674, 19386, 116972], "state_str": "R : Type u\ninst✝² : Ring R\nι : Type v\ndec_ι : DecidableEq ι\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nA : ι → Submodule R M\ns : Set ι\nh : CompleteLattice.Independent fun i => A ↑i\n⊢ IsInternal fun i => Submodule.comap (⨆ i ∈ s, A i).subtype (A ↑i)"}
+{"state": [{"context": ["R : Type u", "inst✝² : Ring R", "ι : Type v", "dec_ι : DecidableEq ι", "M : Type u_1", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M", "A : ι → Submodule R M", "s : Set ι", "h : CompleteLattice.Independent fun i => A ↑i", "p : Submodule R M := ⨆ i ∈ s, A i"], "goal": "CompleteLattice.Independent fun i => Submodule.comap (⨆ i ∈ s, A i).subtype (A ↑i)"}], "premise": [19399], "state_str": "R : Type u\ninst✝² : Ring R\nι : Type v\ndec_ι : DecidableEq ι\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nA : ι → Submodule R M\ns : Set ι\nh : CompleteLattice.Independent fun i => A ↑i\np : Submodule R M := ⨆ i ∈ s, A i\n⊢ CompleteLattice.Independent fun i => Submodule.comap (⨆ i ∈ s, A i).subtype (A ↑i)"}
+{"state": [{"context": ["R : Type u", "inst✝² : Ring R", "ι : Type v", "dec_ι : DecidableEq ι", "M : Type u_1", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M", "A : ι → Submodule R M", "s : Set ι", "h : CompleteLattice.Independent fun i => A ↑i", "p : Submodule R M := ⨆ i ∈ s, A i", "hp : ∀ i ∈ s, A i ≤ p", "e : Submodule R ↥p ≃o ↑(Set.Iic p) := p.mapIic"], "goal": "CompleteLattice.Independent fun i => Submodule.comap (⨆ i ∈ s, A i).subtype (A ↑i)"}], "premise": [2115, 15815, 15817], "state_str": "R : Type u\ninst✝² : Ring R\nι : Type v\ndec_ι : DecidableEq ι\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nA : ι → Submodule R M\ns : Set ι\nh : CompleteLattice.Independent fun i => A ↑i\np : Submodule R M := ⨆ i ∈ s, A i\nhp : ∀ i ∈ s, A i ≤ p\ne : Submodule R ↥p ≃o ↑(Set.Iic p) := p.mapIic\n⊢ CompleteLattice.Independent fun i => Submodule.comap (⨆ i ∈ s, A i).subtype (A ↑i)"}
+{"state": [{"context": ["R : Type u", "inst✝² : Ring R", "ι : Type v", "dec_ι : DecidableEq ι", "M : Type u_1", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M", "A : ι → Submodule R M", "s : Set ι", "h : CompleteLattice.Independent fun i => A ↑i", "p : Submodule R M := ⨆ i ∈ s, A i", "hp : ∀ i ∈ s, A i ≤ p", "e : Submodule R ↥p ≃o ↑(Set.Iic p) := p.mapIic", "i : ↑s", "m : M"], "goal": "m ∈ Submodule.map p.subtype (Submodule.comap p.subtype (A ↑i)) ↔ m ∈ ↑⟨A ↑i, ⋯⟩"}], "premise": [1713, 2115, 110316], "state_str": "case h.a.h\nR : Type u\ninst✝² : Ring R\nι : Type v\ndec_ι : DecidableEq ι\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nA : ι → Submodule R M\ns : Set ι\nh : CompleteLattice.Independent fun i => A ↑i\np : Submodule R M := ⨆ i ∈ s, A i\nhp : ∀ i ∈ s, A i ≤ p\ne : Submodule R ↥p ≃o ↑(Set.Iic p) := p.mapIic\ni : ↑s\nm : M\n⊢ m ∈ Submodule.map p.subtype (Submodule.comap p.subtype (A ↑i)) ↔ m ∈ ↑⟨A ↑i, ⋯⟩"}
+{"state": [{"context": ["R : Type uR", "S : Type uS", "A : Type uA", "B : Type uB", "C : Type uC", "D : Type uD", "E : Type uE", "F : Type uF", "inst✝⁸ : CommSemiring R", "inst✝⁷ : NonAssocSemiring A", "inst✝⁶ : Module R A", "inst✝⁵ : SMulCommClass R A A", "inst✝⁴ : IsScalarTower R A A", "inst✝³ : NonAssocSemiring B", "inst✝² : Module R B", "inst✝¹ : SMulCommClass R B B", "inst✝ : IsScalarTower R B B", "x : A ⊗[R] B"], "goal": "(mul x) (1 ⊗ₜ[R] 1) = x"}], "premise": [86834], "state_str": "R : Type uR\nS : Type uS\nA : Type uA\nB : Type uB\nC : Type uC\nD : Type uD\nE : Type uE\nF : Type uF\ninst✝⁸ : CommSemiring R\ninst✝⁷ : NonAssocSemiring A\ninst✝⁶ : Module R A\ninst✝⁵ : SMulCommClass R A A\ninst✝⁴ : IsScalarTower R A A\ninst✝³ : NonAssocSemiring B\ninst✝² : Module R B\ninst✝¹ : SMulCommClass R B B\ninst✝ : IsScalarTower R B B\nx : A ⊗[R] B\n⊢ (mul x) (1 ⊗ₜ[R] 1) = x"}
+{"state": [{"context": ["α : Type u", "β : Type v", "γ : Type w", "ι : Type x", "inst✝² : PseudoEMetricSpace α", "inst✝¹ : PseudoEMetricSpace β", "inst✝ : PseudoEMetricSpace γ", "K : ℝ≥0", "f : α → β", "x✝ y✝ : α", "r : ℝ≥0∞", "h : ∀ (x y : α), edist (f x) (f y) ≤ edist x y", "x y : α"], "goal": "edist (f x) (f y) ≤ ↑1 * edist x y"}], "premise": [119728, 143168], "state_str": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\nK : ℝ≥0\nf : α → β\nx✝ y✝ : α\nr : ℝ≥0∞\nh : ∀ (x y : α), edist (f x) (f y) ≤ edist x y\nx y : α\n⊢ edist (f x) (f y) ≤ ↑1 * edist x y"}
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+{"state": [{"context": ["V : Type u_1", "V' : Type u_2", "inst✝⁵ : NormedAddCommGroup V", "inst✝⁴ : NormedAddCommGroup V'", "inst✝³ : InnerProductSpace ℝ V", "inst✝² : InnerProductSpace ℝ V'", "inst✝¹ : Fact (finrank ℝ V = 2)", "inst✝ : Fact (finrank ℝ V' = 2)", "o : Orientation ℝ V (Fin 2)", "x y : V", "r : ℝ", "h : ¬(o.oangle x y = 0 ∨ o.oangle x y = ↑π)", "h' : ∀ (r' : ℝ), o.oangle x (r' • x + y) ≠ 0 ∧ o.oangle x (r' • x + y) ≠ ↑π", "s : Set (V × V) := (fun r' => (x, r' • x + y)) '' Set.univ", "hc : IsConnected s", "hf : ContinuousOn (fun z => o.oangle z.1 z.2) s", "hs : ∀ z ∈ s, o.oangle z.1 z.2 ≠ 0 ∧ o.oangle z.1 z.2 ≠ ↑π", "hx : (x, y) ∈ s", "hy : (x, r • x + y) ∈ s"], "goal": "(o.oangle x (r • x + y)).sign = (o.oangle x y).sign"}], "premise": [38419], "state_str": "case neg\nV : Type u_1\nV' : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : NormedAddCommGroup V'\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : InnerProductSpace ℝ V'\ninst✝¹ : Fact (finrank ℝ V = 2)\ninst✝ : Fact (finrank ℝ V' = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nr : ℝ\nh : ¬(o.oangle x y = 0 ∨ o.oangle x y = ↑π)\nh' : ∀ (r' : ℝ), o.oangle x (r' • x + y) ≠ 0 ∧ o.oangle x (r' • x + y) ≠ ↑π\ns : Set (V × V) := (fun r' => (x, r' • x + y)) '' Set.univ\nhc : IsConnected s\nhf : ContinuousOn (fun z => o.oangle z.1 z.2) s\nhs : ∀ z ∈ s, o.oangle z.1 z.2 ≠ 0 ∧ o.oangle z.1 z.2 ≠ ↑π\nhx : (x, y) ∈ s\nhy : (x, r • x + y) ∈ s\n⊢ (o.oangle x (r • x + y)).sign = (o.oangle x y).sign"}
+{"state": [{"context": ["ι : Type u_1", "α : Type u", "β : Type v", "γ : Type w", "l₁ l₂ l : List α", "i : Fin l.length"], "goal": "(l.get? ↑i).isSome = true"}], "premise": [4995], "state_str": "ι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ l : List α\ni : Fin l.length\n⊢ (l.get? ↑i).isSome = true"}
+{"state": [{"context": ["ι : Type u_1", "α : Type u", "β : Type v", "γ : Type w", "l₁ l₂ l : List α", "i : Fin l.length"], "goal": "l.get i = (l.get? ↑i).get ⋯"}], "premise": [1319], "state_str": "ι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ l : List α\ni : Fin l.length\n⊢ l.get i = (l.get? ↑i).get ⋯"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "κ : ι → Sort u_5", "inst✝ : LinearOrder α", "s t : Set α", "hs : IsUpperSet s", "ht : IsUpperSet t"], "goal": "s ⊆ t ∨ t ⊆ s"}], "premise": [1094], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nκ : ι → Sort u_5\ninst✝ : LinearOrder α\ns t : Set α\nhs : IsUpperSet s\nht : IsUpperSet t\n⊢ s ⊆ t ∨ t ⊆ s"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "κ : ι → Sort u_5", "inst✝ : LinearOrder α", "s t : Set α", "hs : IsUpperSet s", "ht : IsUpperSet t", "h : ¬s ⊆ t ∧ ¬t ⊆ s"], "goal": "False"}], "premise": [133334], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nκ : ι → Sort u_5\ninst✝ : LinearOrder α\ns t : Set α\nhs : IsUpperSet s\nht : IsUpperSet t\nh : ¬s ⊆ t ∧ ¬t ⊆ s\n⊢ False"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "κ : ι → Sort u_5", "inst✝ : LinearOrder α", "s t : Set α", "hs : IsUpperSet s", "ht : IsUpperSet t", "a : α", "has : a ∈ s", "hat : a ∉ t", "b : α", "hbt : b ∈ t", "hbs : b ∉ s"], "goal": "False"}], "premise": [14308], "state_str": "case intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nκ : ι → Sort u_5\ninst✝ : LinearOrder α\ns t : Set α\nhs : IsUpperSet s\nht : IsUpperSet t\na : α\nhas : a ∈ s\nhat : a ∉ t\nb : α\nhbt : b ∈ t\nhbs : b ∉ s\n⊢ False"}
+{"state": [{"context": ["l : Type u_1", "m : Type u_2", "n : Type u_3", "o : Type u_4", "m' : o → Type u_5", "n' : o → Type u_6", "R : Type u_7", "S : Type u_8", "α : Type v", "β : Type w", "γ : Type u_9", "inst✝² : DecidableEq n", "inst✝¹ : Zero α", "inst✝ : One α", "i j : n"], "goal": "1 i j = Pi.single i 1 j"}], "premise": [1717, 120695, 142182], "state_str": "l : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nm' : o → Type u_5\nn' : o → Type u_6\nR : Type u_7\nS : Type u_8\nα : Type v\nβ : Type w\nγ : Type u_9\ninst✝² : DecidableEq n\ninst✝¹ : Zero α\ninst✝ : One α\ni j : n\n⊢ 1 i j = Pi.single i 1 j"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "ι' : Sort u_5", "ι₂ : Sort u_6", "κ : ι → Sort u_7", "κ₁ : ι → Sort u_8", "κ₂ : ι → Sort u_9", "κ' : ι' → Sort u_10", "inst✝ : Nonempty ι", "s : ι → Set α", "t : Set β", "f : α → β", "H : ∀ (i : ι), SurjOn f (s i) t", "Hinj : InjOn f (⋃ i, s i)", "y : β", "hy : y ∈ t"], "goal": "y ∈ f '' ⋂ i, s i"}], "premise": [16574, 135485], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nι' : Sort u_5\nι₂ : Sort u_6\nκ : ι → Sort u_7\nκ₁ : ι → Sort u_8\nκ₂ : ι → Sort u_9\nκ' : ι' → Sort u_10\ninst✝ : Nonempty ι\ns : ι → Set α\nt : Set β\nf : α → β\nH : ∀ (i : ι), SurjOn f (s i) t\nHinj : InjOn f (⋃ i, s i)\ny : β\nhy : y ∈ t\n⊢ y ∈ f '' ⋂ i, s i"}
+{"state": [{"context": ["x✝ y z✝ : ℝ", "n : ℕ", "x z : ℝ", "hx : 1 < x", "hz : z < 0"], "goal": "x ^ z < 1"}], "premise": [40098], "state_str": "x✝ y z✝ : ℝ\nn : ℕ\nx z : ℝ\nhx : 1 < x\nhz : z < 0\n⊢ x ^ z < 1"}
+{"state": [{"context": ["x✝ y z✝ : ℝ", "n : ℕ", "x z : ℝ", "hx : 1 < x", "hz : z < 0"], "goal": "1 = x ^ 0"}], "premise": [2100, 40009], "state_str": "case h.e'_4\nx✝ y z✝ : ℝ\nn : ℕ\nx z : ℝ\nhx : 1 < x\nhz : z < 0\n⊢ 1 = x ^ 0"}
+{"state": [{"context": ["o : Ordinal.{u_1}", "ho : o ≠ 0"], "goal": "nim o ‖ 0"}], "premise": [47984, 48126, 50200], "state_str": "o : Ordinal.{u_1}\nho : o ≠ 0\n⊢ nim o ‖ 0"}
+{"state": [{"context": ["o : Ordinal.{u_1}", "ho : o ≠ 0"], "goal": "∃ j, (mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)).moveRight j ≤ 0"}], "premise": [49697], "state_str": "o : Ordinal.{u_1}\nho : o ≠ 0\n⊢ ∃ j,\n    (mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>\n            nim (typein (fun x x_1 => x < x_1) o₂)).moveRight\n        j ≤\n      0"}
+{"state": [{"context": ["s t : ℂ", "a : ℝ", "ha : 0 < a"], "goal": "∫ (x : ℝ) in 0 ..a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) = ↑a ^ (s + t - 1) * s.betaIntegral t"}], "premise": [1674, 11234, 148290], "state_str": "s t : ℂ\na : ℝ\nha : 0 < a\n⊢ ∫ (x : ℝ) in 0 ..a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) = ↑a ^ (s + t - 1) * s.betaIntegral t"}
+{"state": [{"context": ["s t : ℂ", "a : ℝ", "ha : 0 < a", "ha' : ↑a ≠ 0"], "goal": "∫ (x : ℝ) in 0 ..a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) = ↑a ^ (s + t - 1) * ∫ (x : ℝ) in 0 ..1, ↑x ^ (s - 1) * (1 - ↑x) ^ (t - 1)"}], "premise": [39266, 39268, 119703], "state_str": "s t : ℂ\na : ℝ\nha : 0 < a\nha' : ↑a ≠ 0\n⊢ ∫ (x : ℝ) in 0 ..a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) =\n    ↑a ^ (s + t - 1) * ∫ (x : ℝ) in 0 ..1, ↑x ^ (s - 1) * (1 - ↑x) ^ (t - 1)"}
+{"state": [{"context": ["s t : ℂ", "a : ℝ", "ha : 0 < a", "ha' : ↑a ≠ 0", "A : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))"], "goal": "∫ (x : ℝ) in 0 ..a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) = ↑a ^ (s + t - 1) * ∫ (x : ℝ) in 0 ..1, ↑x ^ (s - 1) * (1 - ↑x) ^ (t - 1)"}], "premise": [11234, 26361, 26375, 108302, 108408, 119703, 148328], "state_str": "s t : ℂ\na : ℝ\nha : 0 < a\nha' : ↑a ≠ 0\nA : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))\n⊢ ∫ (x : ℝ) in 0 ..a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) =\n    ↑a ^ (s + t - 1) * ∫ (x : ℝ) in 0 ..1, ↑x ^ (s - 1) * (1 - ↑x) ^ (t - 1)"}
+{"state": [{"context": ["s t : ℂ", "a : ℝ", "ha : 0 < a", "ha' : ↑a ≠ 0", "A : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))"], "goal": "∫ (x : ℝ) in 0 ..a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) = ∫ (x : ℝ) in 0 ..a, ↑a ^ (s - 1) * ↑a ^ (t - 1) * (↑(x / a) ^ (s - 1) * (1 - ↑(x / a)) ^ (t - 1))"}], "premise": [26334], "state_str": "s t : ℂ\na : ℝ\nha : 0 < a\nha' : ↑a ≠ 0\nA : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))\n⊢ ∫ (x : ℝ) in 0 ..a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) =\n    ∫ (x : ℝ) in 0 ..a, ↑a ^ (s - 1) * ↑a ^ (t - 1) * (↑(x / a) ^ (s - 1) * (1 - ↑(x / a)) ^ (t - 1))"}
+{"state": [{"context": ["s t : ℂ", "a : ℝ", "ha : 0 < a", "ha' : ↑a ≠ 0", "A : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))"], "goal": "∫ (x : ℝ) in Ioc 0 a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) ∂volume = ∫ (x : ℝ) in Ioc 0 a, ↑a ^ (s - 1) * ↑a ^ (t - 1) * (↑(x / a) ^ (s - 1) * (1 - ↑(x / a)) ^ (t - 1)) ∂volume"}], "premise": [26449, 28263], "state_str": "s t : ℂ\na : ℝ\nha : 0 < a\nha' : ↑a ≠ 0\nA : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))\n⊢ ∫ (x : ℝ) in Ioc 0 a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) ∂volume =\n    ∫ (x : ℝ) in Ioc 0 a, ↑a ^ (s - 1) * ↑a ^ (t - 1) * (↑(x / a) ^ (s - 1) * (1 - ↑(x / a)) ^ (t - 1)) ∂volume"}
+{"state": [{"context": ["s t : ℂ", "a : ℝ", "ha : 0 < a", "ha' : ↑a ≠ 0", "A : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))", "x : ℝ", "hx : x ∈ Ioc 0 a"], "goal": "↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) = ↑a ^ (s - 1) * ↑a ^ (t - 1) * (↑(x / a) ^ (s - 1) * (1 - ↑(x / a)) ^ (t - 1))"}], "premise": [117742], "state_str": "s t : ℂ\na : ℝ\nha : 0 < a\nha' : ↑a ≠ 0\nA : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))\nx : ℝ\nhx : x ∈ Ioc 0 a\n⊢ ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) = ↑a ^ (s - 1) * ↑a ^ (t - 1) * (↑(x / a) ^ (s - 1) * (1 - ↑(x / a)) ^ (t - 1))"}
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+{"state": [{"context": ["α : Type u_1", "m0 : MeasurableSpace α", "μ : Measure α", "f : α → ℝ≥0∞", "s : Set α", "t : Set α := toMeasurable (μ.withDensity f) s"], "goal": "∫⁻ (a : α) in s, f a ∂μ ≤ (μ.withDensity f) s"}], "premise": [2100, 29102, 29104, 29105, 30236, 31314], "state_str": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\ns : Set α\nt : Set α := toMeasurable (μ.withDensity f) s\n⊢ ∫⁻ (a : α) in s, f a ∂μ ≤ (μ.withDensity f) s"}
+{"state": [{"context": ["C : Type u", "inst✝² : Category.{v, u} C", "I : MultispanIndex C", "inst✝¹ : HasCoproduct I.left", "inst✝ : HasCoproduct I.right", "K : Cofork I.fstSigmaMap I.sndSigmaMap"], "goal": "((I.ofSigmaCoforkFunctor ⋙ I.toSigmaCoforkFunctor).obj K).π ≫ (Iso.refl ((I.ofSigmaCoforkFunctor ⋙ I.toSigmaCoforkFunctor).obj K).pt).hom = ((𝟭 (Cofork I.fstSigmaMap I.sndSigmaMap)).obj K).π"}], "premise": [93401], "state_str": "C : Type u\ninst✝² : Category.{v, u} C\nI : MultispanIndex C\ninst✝¹ : HasCoproduct I.left\ninst✝ : HasCoproduct I.right\nK : Cofork I.fstSigmaMap I.sndSigmaMap\n⊢ ((I.ofSigmaCoforkFunctor ⋙ I.toSigmaCoforkFunctor).obj K).π ≫\n      (Iso.refl ((I.ofSigmaCoforkFunctor ⋙ I.toSigmaCoforkFunctor).obj K).pt).hom =\n    ((𝟭 (Cofork I.fstSigmaMap I.sndSigmaMap)).obj K).π"}
+{"state": [{"context": ["C : Type u", "inst✝² : Category.{v, u} C", "I : MultispanIndex C", "inst✝¹ : HasCoproduct I.left", "inst✝ : HasCoproduct I.right", "K : Cofork I.fstSigmaMap I.sndSigmaMap", "j : I.R"], "goal": "colimit.ι (Discrete.functor I.right) { as := j } ≫ Sigma.desc (Multicofork.ofSigmaCofork I K).π ≫ 𝟙 K.pt = colimit.ι (Discrete.functor I.right) { as := j } ≫ K.π"}], "premise": [93392, 93679, 96174], "state_str": "case w.mk\nC : Type u\ninst✝² : Category.{v, u} C\nI : MultispanIndex C\ninst✝¹ : HasCoproduct I.left\ninst✝ : HasCoproduct I.right\nK : Cofork I.fstSigmaMap I.sndSigmaMap\nj : I.R\n⊢ colimit.ι (Discrete.functor I.right) { as := j } ≫ Sigma.desc (Multicofork.ofSigmaCofork I K).π ≫ 𝟙 K.pt =\n    colimit.ι (Discrete.functor I.right) { as := j } ≫ K.π"}
+{"state": [{"context": ["R : Type u", "inst✝¹² : CommRing R", "M N P Q : Type u", "inst✝¹¹ : AddCommGroup M", "inst✝¹⁰ : AddCommGroup N", "inst✝⁹ : AddCommGroup P", "inst✝⁸ : AddCommGroup Q", "inst✝⁷ : Module R M", "inst✝⁶ : Module R N", "inst✝⁵ : Module R P", "inst✝⁴ : Module R Q", "inst✝³ : Coalgebra R M", "inst✝² : Coalgebra R N", "inst✝¹ : Coalgebra R P", "inst✝ : Coalgebra R Q", "K L : CoalgebraCat R"], "goal": "Coalgebra.comul = ↑(TensorProduct.tensorTensorTensorComm R ↑K.toModuleCat ↑K.toModuleCat ↑L.toModuleCat ↑L.toModuleCat) ∘ₗ TensorProduct.map Coalgebra.comul Coalgebra.comul"}], "premise": [112661], "state_str": "R : Type u\ninst✝¹² : CommRing R\nM N P Q : Type u\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : AddCommGroup N\ninst✝⁹ : AddCommGroup P\ninst✝⁸ : AddCommGroup Q\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\ninst✝⁵ : Module R P\ninst✝⁴ : Module R Q\ninst✝³ : Coalgebra R M\ninst✝² : Coalgebra R N\ninst✝¹ : Coalgebra R P\ninst✝ : Coalgebra R Q\nK L : CoalgebraCat R\n⊢ Coalgebra.comul =\n    ↑(TensorProduct.tensorTensorTensorComm R ↑K.toModuleCat ↑K.toModuleCat ↑L.toModuleCat ↑L.toModuleCat) ∘ₗ\n      TensorProduct.map Coalgebra.comul Coalgebra.comul"}
+{"state": [{"context": ["R : Type u", "inst✝¹² : CommRing R", "M N P Q : Type u", "inst✝¹¹ : AddCommGroup M", "inst✝¹⁰ : AddCommGroup N", "inst✝⁹ : AddCommGroup P", "inst✝⁸ : AddCommGroup Q", "inst✝⁷ : Module R M", "inst✝⁶ : Module R N", "inst✝⁵ : Module R P", "inst✝⁴ : Module R Q", "inst✝³ : Coalgebra R M", "inst✝² : Coalgebra R N", "inst✝¹ : Coalgebra R P", "inst✝ : Coalgebra R Q", "K L : CoalgebraCat R"], "goal": "((comonEquivalence R).symm.inverse.obj K ⊗ (comonEquivalence R).symm.inverse.obj L).comul = ↑(TensorProduct.tensorTensorTensorComm R ↑K.toModuleCat ↑K.toModuleCat ↑L.toModuleCat ↑L.toModuleCat) ∘ₗ TensorProduct.map Coalgebra.comul Coalgebra.comul"}], "premise": [100101, 112717, 112960], "state_str": "R : Type u\ninst✝¹² : CommRing R\nM N P Q : Type u\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : AddCommGroup N\ninst✝⁹ : AddCommGroup P\ninst✝⁸ : AddCommGroup Q\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\ninst✝⁵ : Module R P\ninst✝⁴ : Module R Q\ninst✝³ : Coalgebra R M\ninst✝² : Coalgebra R N\ninst✝¹ : Coalgebra R P\ninst✝ : Coalgebra R Q\nK L : CoalgebraCat R\n⊢ ((comonEquivalence R).symm.inverse.obj K ⊗ (comonEquivalence R).symm.inverse.obj L).comul =\n    ↑(TensorProduct.tensorTensorTensorComm R ↑K.toModuleCat ↑K.toModuleCat ↑L.toModuleCat ↑L.toModuleCat) ∘ₗ\n      TensorProduct.map Coalgebra.comul Coalgebra.comul"}
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+{"state": [{"context": ["x x₁ x₂ y y₁ y₂ : PGame", "hx : x.Numeric", "hx₁ : x₁.Numeric", "hx₂ : x₂.Numeric", "hy : y.Numeric", "hy₁ : y₁.Numeric", "hy₂ : y₂.Numeric", "hp₁ : 0 < x₁", "hp₂ : 0 < x₂"], "goal": "⟦0⟧ < ⟦x₁ * x₂⟧"}], "premise": [48926, 49586], "state_str": "x x₁ x₂ y y₁ y₂ : PGame\nhx : x.Numeric\nhx₁ : x₁.Numeric\nhx₂ : x₂.Numeric\nhy : y.Numeric\nhy₁ : y₁.Numeric\nhy₂ : y₂.Numeric\nhp₁ : 0 < x₁\nhp₂ : 0 < x₂\n⊢ ⟦0⟧ < ⟦x₁ * x₂⟧"}
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+{"state": [{"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "inst✝² : Preadditive C", "X✝ Y✝ : C", "inst✝¹ : HasBinaryBiproduct X✝ Y✝", "X Y : C", "f : X ⟶ Y", "inst✝ : IsSplitMono f", "c : CokernelCofork f", "i : IsColimit c", "this : Epi (Cofork.π c) := epi_of_isColimit_cofork i"], "goal": "i.desc (Cofork.ofπ (Cofork.π (CokernelCofork.ofπ (𝟙 Y - retraction f ≫ f) ⋯)) ⋯) ≫ retraction f = 0"}], "premise": [93611], "state_str": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nX✝ Y✝ : C\ninst✝¹ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\nc : CokernelCofork f\ni : IsColimit c\nthis : Epi (Cofork.π c) := epi_of_isColimit_cofork i\n⊢ i.desc (Cofork.ofπ (Cofork.π (CokernelCofork.ofπ (𝟙 Y - retraction f ≫ f) ⋯)) ⋯) ≫ retraction f = 0"}
+{"state": [{"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "inst✝² : Preadditive C", "X✝ Y✝ : C", "inst✝¹ : HasBinaryBiproduct X✝ Y✝", "X Y : C", "f : X ⟶ Y", "inst✝ : IsSplitMono f", "c : CokernelCofork f", "i : IsColimit c", "this : Epi (Cofork.π c) := epi_of_isColimit_cofork i"], "goal": "Cofork.π c ≫ i.desc (Cofork.ofπ (Cofork.π (CokernelCofork.ofπ (𝟙 Y - retraction f ≫ f) ⋯)) ⋯) ≫ retraction f = 0"}], "premise": [88661, 91600, 91601, 94875, 96173, 96174, 117981], "state_str": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nX✝ Y✝ : C\ninst✝¹ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\nc : CokernelCofork f\ni : IsColimit c\nthis : Epi (Cofork.π c) := epi_of_isColimit_cofork i\n⊢ Cofork.π c ≫ i.desc (Cofork.ofπ (Cofork.π (CokernelCofork.ofπ (𝟙 Y - retraction f ≫ f) ⋯)) ⋯) ≫ retraction f = 0"}
+{"state": [{"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "inst✝² : Preadditive C", "X✝ Y✝ : C", "inst✝¹ : HasBinaryBiproduct X✝ Y✝", "X Y : C", "f : X ⟶ Y", "inst✝ : IsSplitMono f", "c : CokernelCofork f", "i : IsColimit c", "this : Epi (Cofork.π c) := epi_of_isColimit_cofork i"], "goal": "𝟙 Y ≫ retraction f = retraction f"}], "premise": [96175], "state_str": "case a\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nX✝ Y✝ : C\ninst✝¹ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\nc : CokernelCofork f\ni : IsColimit c\nthis : Epi (Cofork.π c) := epi_of_isColimit_cofork i\n⊢ 𝟙 Y ≫ retraction f = retraction f"}
+{"state": [{"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "inst✝² : Preadditive C", "X✝ Y✝ : C", "inst✝¹ : HasBinaryBiproduct X✝ Y✝", "X Y : C", "f : X ⟶ Y", "inst✝ : IsSplitMono f", "c : CokernelCofork f", "i : IsColimit c"], "goal": "(let c' := CokernelCofork.ofπ (Cofork.π c) ⋯; let i' := isCokernelEpiComp i (retraction f) ⋯; let i'' := isColimitCoforkOfCokernelCofork i'; (splitEpiOfIdempotentOfIsColimitCofork C ⋯ i'').section_) ≫ Cofork.π c = 𝟙 c.pt"}], "premise": [88658], "state_str": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nX✝ Y✝ : C\ninst✝¹ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nf : X ⟶ Y\ninst✝ : IsSplitMono f\nc : CokernelCofork f\ni : IsColimit c\n⊢ (let c' := CokernelCofork.ofπ (Cofork.π c) ⋯;\n      let i' := isCokernelEpiComp i (retraction f) ⋯;\n      let i'' := isColimitCoforkOfCokernelCofork i';\n      (splitEpiOfIdempotentOfIsColimitCofork C ⋯ i'').section_) ≫\n      Cofork.π c =\n    𝟙 c.pt"}
+{"state": [{"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "π : ι → Type u_4", "inst✝ : GeneralizedBooleanAlgebra α", "a b c d : α"], "goal": "a ∆ b ∆ c = a ∆ c ∆ b"}], "premise": [17207, 17278], "state_str": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\nπ : ι → Type u_4\ninst✝ : GeneralizedBooleanAlgebra α\na b c d : α\n⊢ a ∆ b ∆ c = a ∆ c ∆ b"}
+{"state": [{"context": ["E : Type u_1", "inst✝ : NormedAddCommGroup E", "a b : ℝ", "f : ℝ → ℝ", "s : ℝ", "hfc : LocallyIntegrableOn f (Ioi 0) volume", "hf_top : f =O[atTop] fun x => x ^ (-a)", "hs_top : s < a", "hf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)", "hs_bot : b < s"], "goal": "IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi 0) volume"}], "premise": [27429, 34455], "state_str": "E : Type u_1\ninst✝ : NormedAddCommGroup E\na b : ℝ\nf : ℝ → ℝ\ns : ℝ\nhfc : LocallyIntegrableOn f (Ioi 0) volume\nhf_top : f =O[atTop] fun x => x ^ (-a)\nhs_top : s < a\nhf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)\nhs_bot : b < s\n⊢ IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi 0) volume"}
+{"state": [{"context": ["E : Type u_1", "inst✝ : NormedAddCommGroup E", "a b : ℝ", "f : ℝ → ℝ", "s : ℝ", "hfc : LocallyIntegrableOn f (Ioi 0) volume", "hf_top : f =O[atTop] fun x => x ^ (-a)", "hs_top : s < a", "hf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)", "hs_bot : b < s", "c1 : ℝ", "hc1 : 0 < c1", "hc1' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume"], "goal": "IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi 0) volume"}], "premise": [27429, 34456], "state_str": "case intro.intro\nE : Type u_1\ninst✝ : NormedAddCommGroup E\na b : ℝ\nf : ℝ → ℝ\ns : ℝ\nhfc : LocallyIntegrableOn f (Ioi 0) volume\nhf_top : f =O[atTop] fun x => x ^ (-a)\nhs_top : s < a\nhf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)\nhs_bot : b < s\nc1 : ℝ\nhc1 : 0 < c1\nhc1' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume\n⊢ IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi 0) volume"}
+{"state": [{"context": ["E : Type u_1", "inst✝ : NormedAddCommGroup E", "a b : ℝ", "f : ℝ → ℝ", "s : ℝ", "hfc : LocallyIntegrableOn f (Ioi 0) volume", "hf_top : f =O[atTop] fun x => x ^ (-a)", "hs_top : s < a", "hf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)", "hs_bot : b < s", "c1 : ℝ", "hc1 : 0 < c1", "hc1' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume", "c2 : ℝ", "hc2 : 0 < c2", "hc2' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc 0 c2) volume"], "goal": "IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi 0) volume"}], "premise": [19690, 19694, 19701, 19714, 20393, 133412], "state_str": "case intro.intro.intro.intro\nE : Type u_1\ninst✝ : NormedAddCommGroup E\na b : ℝ\nf : ℝ → ℝ\ns : ℝ\nhfc : LocallyIntegrableOn f (Ioi 0) volume\nhf_top : f =O[atTop] fun x => x ^ (-a)\nhs_top : s < a\nhf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)\nhs_bot : b < s\nc1 : ℝ\nhc1 : 0 < c1\nhc1' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume\nc2 : ℝ\nhc2 : 0 < c2\nhc2' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc 0 c2) volume\n⊢ IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi 0) volume"}
+{"state": [{"context": ["E : Type u_1", "inst✝ : NormedAddCommGroup E", "a b : ℝ", "f : ℝ → ℝ", "s : ℝ", "hfc : LocallyIntegrableOn f (Ioi 0) volume", "hf_top : f =O[atTop] fun x => x ^ (-a)", "hs_top : s < a", "hf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)", "hs_bot : b < s", "c1 : ℝ", "hc1 : 0 < c1", "hc1' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume", "c2 : ℝ", "hc2 : 0 < c2", "hc2' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc 0 c2) volume", "this : Ioi 0 = Ioc 0 c2 ∪ Ioc c2 c1 ∪ Ioi c1"], "goal": "IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi 0) volume"}], "premise": [25595], "state_str": "case intro.intro.intro.intro\nE : Type u_1\ninst✝ : NormedAddCommGroup E\na b : ℝ\nf : ℝ → ℝ\ns : ℝ\nhfc : LocallyIntegrableOn f (Ioi 0) volume\nhf_top : f =O[atTop] fun x => x ^ (-a)\nhs_top : s < a\nhf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)\nhs_bot : b < s\nc1 : ℝ\nhc1 : 0 < c1\nhc1' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume\nc2 : ℝ\nhc2 : 0 < c2\nhc2' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc 0 c2) volume\nthis : Ioi 0 = Ioc 0 c2 ∪ Ioc c2 c1 ∪ Ioi c1\n⊢ IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi 0) volume"}
+{"state": [{"context": ["E : Type u_1", "inst✝ : NormedAddCommGroup E", "a b : ℝ", "f : ℝ → ℝ", "s : ℝ", "hfc : LocallyIntegrableOn f (Ioi 0) volume", "hf_top : f =O[atTop] fun x => x ^ (-a)", "hs_top : s < a", "hf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)", "hs_bot : b < s", "c1 : ℝ", "hc1 : 0 < c1", "hc1' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume", "c2 : ℝ", "hc2 : 0 < c2", "hc2' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc 0 c2) volume", "this : Ioi 0 = Ioc 0 c2 ∪ Ioc c2 c1 ∪ Ioi c1"], "goal": "(IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc 0 c2) volume ∧ IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc c2 c1) volume) ∧ IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume"}], "premise": [1673, 25660], "state_str": "case intro.intro.intro.intro\nE : Type u_1\ninst✝ : NormedAddCommGroup E\na b : ℝ\nf : ℝ → ℝ\ns : ℝ\nhfc : LocallyIntegrableOn f (Ioi 0) volume\nhf_top : f =O[atTop] fun x => x ^ (-a)\nhs_top : s < a\nhf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)\nhs_bot : b < s\nc1 : ℝ\nhc1 : 0 < c1\nhc1' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume\nc2 : ℝ\nhc2 : 0 < c2\nhc2' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc 0 c2) volume\nthis : Ioi 0 = Ioc 0 c2 ∪ Ioc c2 c1 ∪ Ioi c1\n⊢ (IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc 0 c2) volume ∧\n      IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc c2 c1) volume) ∧\n    IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume"}
+{"state": [{"context": ["E : Type u_1", "inst✝ : NormedAddCommGroup E", "a b : ℝ", "f : ℝ → ℝ", "s : ℝ", "hfc : LocallyIntegrableOn f (Ioi 0) volume", "hf_top : f =O[atTop] fun x => x ^ (-a)", "hs_top : s < a", "hf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)", "hs_bot : b < s", "c1 : ℝ", "hc1 : 0 < c1", "hc1' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume", "c2 : ℝ", "hc2 : 0 < c2", "hc2' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc 0 c2) volume", "this : Ioi 0 = Ioc 0 c2 ∪ Ioc c2 c1 ∪ Ioi c1"], "goal": "IntegrableOn (fun t => t ^ (s - 1) * f t) (Icc c2 c1) volume"}], "premise": [2107, 27426, 27492, 54886, 63094], "state_str": "case intro.intro.intro.intro\nE : Type u_1\ninst✝ : NormedAddCommGroup E\na b : ℝ\nf : ℝ → ℝ\ns : ℝ\nhfc : LocallyIntegrableOn f (Ioi 0) volume\nhf_top : f =O[atTop] fun x => x ^ (-a)\nhs_top : s < a\nhf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)\nhs_bot : b < s\nc1 : ℝ\nhc1 : 0 < c1\nhc1' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume\nc2 : ℝ\nhc2 : 0 < c2\nhc2' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc 0 c2) volume\nthis : Ioi 0 = Ioc 0 c2 ∪ Ioc c2 c1 ∪ Ioi c1\n⊢ IntegrableOn (fun t => t ^ (s - 1) * f t) (Icc c2 c1) volume"}
+{"state": [{"context": ["E : Type u_1", "inst✝ : NormedAddCommGroup E", "a b : ℝ", "f : ℝ → ℝ", "s : ℝ", "hfc : LocallyIntegrableOn f (Ioi 0) volume", "hf_top : f =O[atTop] fun x => x ^ (-a)", "hs_top : s < a", "hf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)", "hs_bot : b < s", "c1 : ℝ", "hc1 : 0 < c1", "hc1' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume", "c2 : ℝ", "hc2 : 0 < c2", "hc2' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc 0 c2) volume", "this : Ioi 0 = Ioc 0 c2 ∪ Ioc c2 c1 ∪ Ioi c1"], "goal": "ContinuousOn (fun t => t ^ (s - 1)) (Ioi 0)"}], "premise": [14284, 39566, 57315], "state_str": "case intro.intro.intro.intro\nE : Type u_1\ninst✝ : NormedAddCommGroup E\na b : ℝ\nf : ℝ → ℝ\ns : ℝ\nhfc : LocallyIntegrableOn f (Ioi 0) volume\nhf_top : f =O[atTop] fun x => x ^ (-a)\nhs_top : s < a\nhf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)\nhs_bot : b < s\nc1 : ℝ\nhc1 : 0 < c1\nhc1' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume\nc2 : ℝ\nhc2 : 0 < c2\nhc2' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc 0 c2) volume\nthis : Ioi 0 = Ioc 0 c2 ∪ Ioc c2 c1 ∪ Ioi c1\n⊢ ContinuousOn (fun t => t ^ (s - 1)) (Ioi 0)"}
+{"state": [{"context": ["R : Type u_1", "inst✝² : Monoid R", "S : Submonoid R", "inst✝¹ : OreSet S", "X : Type ?u.24237", "inst✝ : MulAction R X", "r r' r₁ r₂ : R", "s t : ↥S", "h : ↑t * r = ↑t * r'", "r₁' : R", "t' : ↥S", "hr₁ : ↑t' * r₁ = r₁' * ↑t"], "goal": "r₁ * r * r₂ /ₒ s = r₁ * r' * r₂ /ₒ s"}], "premise": [81627], "state_str": "case mk.mk\nR : Type u_1\ninst✝² : Monoid R\nS : Submonoid R\ninst✝¹ : OreSet S\nX : Type ?u.24237\ninst✝ : MulAction R X\nr r' r₁ r₂ : R\ns t : ↥S\nh : ↑t * r = ↑t * r'\nr₁' : R\nt' : ↥S\nhr₁ : ↑t' * r₁ = r₁' * ↑t\n⊢ r₁ * r * r₂ /ₒ s = r₁ * r' * r₂ /ₒ s"}
+{"state": [{"context": ["R : Type u_1", "inst✝² : Monoid R", "S : Submonoid R", "inst✝¹ : OreSet S", "X : Type ?u.24237", "inst✝ : MulAction R X", "r r' r₁ r₂ : R", "s t : ↥S", "h : ↑t * r = ↑t * r'", "r₁' : R", "t' : ↥S", "hr₁ : ↑t' * r₁ = r₁' * ↑t"], "goal": "t' • (r₁ * r * r₂) = t' • (r₁ * r' * r₂)"}], "premise": [119703], "state_str": "case mk.mk.e_r\nR : Type u_1\ninst✝² : Monoid R\nS : Submonoid R\ninst✝¹ : OreSet S\nX : Type ?u.24237\ninst✝ : MulAction R X\nr r' r₁ r₂ : R\ns t : ↥S\nh : ↑t * r = ↑t * r'\nr₁' : R\nt' : ↥S\nhr₁ : ↑t' * r₁ = r₁' * ↑t\n⊢ t' • (r₁ * r * r₂) = t' • (r₁ * r' * r₂)"}
+{"state": [{"context": ["𝒮 : Type u₁", "𝒳 : Type u₂", "inst✝² : Category.{v₁, u₂} 𝒳", "inst✝¹ : Category.{v₂, u₁} 𝒮", "p : 𝒳 ⥤ 𝒮", "R S : 𝒮", "a b : 𝒳", "f : R ⟶ S", "φ : a ⟶ b", "inst✝ : p.IsHomLift f φ"], "goal": "p.map φ ≫ eqToHom ⋯ = eqToHom ⋯ ≫ f"}], "premise": [92223, 96175, 97736], "state_str": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₂} 𝒳\ninst✝¹ : Category.{v₂, u₁} 𝒮\np : 𝒳 ⥤ 𝒮\nR S : 𝒮\na b : 𝒳\nf : R ⟶ S\nφ : a ⟶ b\ninst✝ : p.IsHomLift f φ\n⊢ p.map φ ≫ eqToHom ⋯ = eqToHom ⋯ ≫ f"}
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+{"state": [{"context": ["𝓕 : Type u_1", "𝕜 : Type u_2", "α : Type u_3", "ι : Type u_4", "κ : Type u_5", "E : Type u_6", "F : Type u_7", "G : Type u_8", "inst✝² : SeminormedGroup E", "inst✝¹ : SeminormedGroup F", "inst✝ : SeminormedGroup G", "s : Set E", "a a₁ a₂ b b₁ b₂ : E", "r r₁ r₂ : ℝ", "hr : r ≠ 0", "x : ↑(sphere 1 r)"], "goal": "‖↑x‖ ≠ 0"}], "premise": [42738], "state_str": "𝓕 : Type u_1\n𝕜 : Type u_2\nα : Type u_3\nι : Type u_4\nκ : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nhr : r ≠ 0\nx : ↑(sphere 1 r)\n⊢ ‖↑x‖ ≠ 0"}
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+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "inst✝⁵ : TopologicalSpace α", "ι : Type u_5", "π : ι → Type u_6", "inst✝⁴ : (i : ι) → TopologicalSpace (π i)", "inst✝³ : TopologicalSpace β", "inst✝² : TopologicalSpace γ", "inst✝¹ : TopologicalSpace δ", "inst✝ : DiscreteTopology α", "f : α × β → γ"], "goal": "(∀ (a : α) (b : β), 𝓝 (f (a, b)) ≤ map (f ∘ Prod.mk a) (𝓝 b)) ↔ ∀ (a : α) (x : β), 𝓝 (f (a, x)) ≤ map (fun x => f (a, x)) (𝓝 x)"}], "premise": [1713], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝⁵ : TopologicalSpace α\nι : Type u_5\nπ : ι → Type u_6\ninst✝⁴ : (i : ι) → TopologicalSpace (π i)\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\ninst✝ : DiscreteTopology α\nf : α × β → γ\n⊢ (∀ (a : α) (b : β), 𝓝 (f (a, b)) ≤ map (f ∘ Prod.mk a) (𝓝 b)) ↔\n    ∀ (a : α) (x : β), 𝓝 (f (a, x)) ≤ map (fun x => f (a, x)) (𝓝 x)"}
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+{"state": [{"context": ["R : Type u", "inst✝³ : NonUnitalSemiring R", "inst✝² : PartialOrder R", "inst✝¹ : StarRing R", "inst✝ : StarOrderedRing R", "a b : R", "hab : a < b", "c : R", "hc : IsRegular c"], "goal": "star c * a * c < star c * b * c"}], "premise": [11223, 71391, 107592, 107593, 111030, 111350], "state_str": "R : Type u\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\na b : R\nhab : a < b\nc : R\nhc : IsRegular c\n⊢ star c * a * c < star c * b * c"}
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+{"state": [{"context": ["R : Type u_1", "inst✝⁴ : CommRing R", "inst✝³ : IsDedekindDomain R", "K : Type u_2", "inst✝² : Field K", "inst✝¹ : Algebra R K", "inst✝ : IsFractionRing R K", "v : HeightOneSpectrum R", "r : R"], "goal": "v.valuation ((algebraMap R K) r) < 1 ↔ v.asIdeal ∣ Ideal.span {r}"}], "premise": [79215], "state_str": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nr : R\n⊢ v.valuation ((algebraMap R K) r) < 1 ↔ v.asIdeal ∣ Ideal.span {r}"}
+{"state": [{"context": ["R : Type u_1", "inst✝⁴ : CommRing R", "inst✝³ : IsDedekindDomain R", "K : Type u_2", "inst✝² : Field K", "inst✝¹ : Algebra R K", "inst✝ : IsFractionRing R K", "v : HeightOneSpectrum R", "r : R"], "goal": "v.intValuation r < 1 ↔ v.asIdeal ∣ Ideal.span {r}"}], "premise": [79202], "state_str": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nr : R\n⊢ v.intValuation r < 1 ↔ v.asIdeal ∣ Ideal.span {r}"}
+{"state": [{"context": ["K : Type u_1", "F : Type u_2", "R : Type u_3", "inst✝³ : DivisionRing K", "Γ₀ : Type u_4", "Γ'₀ : Type u_5", "Γ''₀ : Type u_6", "inst✝² : LinearOrderedCommMonoidWithZero Γ''₀", "inst✝¹ : Ring R", "inst✝ : LinearOrderedCommGroupWithZero Γ₀", "v : Valuation R Γ₀", "x y z : R", "h : v x < v y"], "goal": "v (x - y) = v y"}], "premise": [75424, 75431, 119789], "state_str": "K : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝³ : DivisionRing K\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx y z : R\nh : v x < v y\n⊢ v (x - y) = v y"}
+{"state": [{"context": ["K : Type u_1", "F : Type u_2", "R : Type u_3", "inst✝³ : DivisionRing K", "Γ₀ : Type u_4", "Γ'₀ : Type u_5", "Γ''₀ : Type u_6", "inst✝² : LinearOrderedCommMonoidWithZero Γ''₀", "inst✝¹ : Ring R", "inst✝ : LinearOrderedCommGroupWithZero Γ₀", "v : Valuation R Γ₀", "x y z : R", "h : v x < v y"], "goal": "v x < v (-y)"}], "premise": [75424], "state_str": "case h\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝³ : DivisionRing K\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx y z : R\nh : v x < v y\n⊢ v x < v (-y)"}
+{"state": [{"context": ["α : Type u_1", "inst✝ : AddCommGroup α", "p a a₁ a₂ b b₁ b₂ c : α", "n : ℕ", "z : ℤ", "x✝ : a₁ ≡ b₁ [PMOD p]", "m : ℤ", "hm : b₁ - a₁ = m • p"], "goal": "(∃ b, b₁ - b₂ - (a₁ - a₂) = (Equiv.subLeft m).symm.symm b • p) ↔ a₂ ≡ b₂ [PMOD p]"}], "premise": [110036, 117918], "state_str": "α : Type u_1\ninst✝ : AddCommGroup α\np a a₁ a₂ b b₁ b₂ c : α\nn : ℕ\nz : ℤ\nx✝ : a₁ ≡ b₁ [PMOD p]\nm : ℤ\nhm : b₁ - a₁ = m • p\n⊢ (∃ b, b₁ - b₂ - (a₁ - a₂) = (Equiv.subLeft m).symm.symm b • p) ↔ a₂ ≡ b₂ [PMOD p]"}
+{"state": [{"context": ["G : Type u_1", "inst✝ : Group G", "a✝ b✝ c a b : G"], "goal": "a * (a⁻¹ * b) = b"}], "premise": [119703, 119728, 119823], "state_str": "G : Type u_1\ninst✝ : Group G\na✝ b✝ c a b : G\n⊢ a * (a⁻¹ * b) = b"}
+{"state": [{"context": ["C : Type u_1", "D : Type u_2", "inst✝⁵ : Category.{u_3, u_1} C", "inst✝⁴ : Category.{?u.130873, u_2} D", "inst✝³ : Preadditive C", "inst✝² : Preadditive D", "S : ShortComplex C", "inst✝¹ : Balanced C", "hS : S.Exact", "h : S.LeftHomologyData", "inst✝ : Mono S.f"], "goal": "IsIso h.f'"}], "premise": [114569], "state_str": "C : Type u_1\nD : Type u_2\ninst✝⁵ : Category.{u_3, u_1} C\ninst✝⁴ : Category.{?u.130873, u_2} D\ninst✝³ : Preadditive C\ninst✝² : Preadditive D\nS : ShortComplex C\ninst✝¹ : Balanced C\nhS : S.Exact\nh : S.LeftHomologyData\ninst✝ : Mono S.f\n⊢ IsIso h.f'"}
+{"state": [{"context": ["C : Type u_1", "D : Type u_2", "inst✝⁵ : Category.{u_3, u_1} C", "inst✝⁴ : Category.{?u.130873, u_2} D", "inst✝³ : Preadditive C", "inst✝² : Preadditive D", "S : ShortComplex C", "inst✝¹ : Balanced C", "hS : S.Exact", "h : S.LeftHomologyData", "inst✝ : Mono S.f", "this : Epi h.f'"], "goal": "IsIso h.f'"}], "premise": [96197, 115008], "state_str": "C : Type u_1\nD : Type u_2\ninst✝⁵ : Category.{u_3, u_1} C\ninst✝⁴ : Category.{?u.130873, u_2} D\ninst✝³ : Preadditive C\ninst✝² : Preadditive D\nS : ShortComplex C\ninst✝¹ : Balanced C\nhS : S.Exact\nh : S.LeftHomologyData\ninst✝ : Mono S.f\nthis : Epi h.f'\n⊢ IsIso h.f'"}
+{"state": [{"context": ["C : Type u_1", "D : Type u_2", "inst✝⁵ : Category.{u_3, u_1} C", "inst✝⁴ : Category.{?u.130873, u_2} D", "inst✝³ : Preadditive C", "inst✝² : Preadditive D", "S : ShortComplex C", "inst✝¹ : Balanced C", "hS : S.Exact", "h : S.LeftHomologyData", "inst✝ : Mono S.f", "this✝ : Epi h.f'", "this : Mono h.f'"], "goal": "IsIso h.f'"}], "premise": [99006], "state_str": "C : Type u_1\nD : Type u_2\ninst✝⁵ : Category.{u_3, u_1} C\ninst✝⁴ : Category.{?u.130873, u_2} D\ninst✝³ : Preadditive C\ninst✝² : Preadditive D\nS : ShortComplex C\ninst✝¹ : Balanced C\nhS : S.Exact\nh : S.LeftHomologyData\ninst✝ : Mono S.f\nthis✝ : Epi h.f'\nthis : Mono h.f'\n⊢ IsIso h.f'"}
+{"state": [{"context": ["C : Type u_1", "inst✝² : Category.{u_2, u_1} C", "inst✝¹ : Preadditive C", "inst✝ : CategoryWithHomology C", "K L : CochainComplex C ℤ", "φ : K ⟶ L", "n : ℤ"], "goal": "QuasiIso ((CategoryTheory.shiftFunctor (HomologicalComplex C (up ℤ)) n).map φ) ↔ QuasiIso φ"}], "premise": [113822, 115540], "state_str": "C : Type u_1\ninst✝² : Category.{u_2, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : CategoryWithHomology C\nK L : CochainComplex C ℤ\nφ : K ⟶ L\nn : ℤ\n⊢ QuasiIso ((CategoryTheory.shiftFunctor (HomologicalComplex C (up ℤ)) n).map φ) ↔ QuasiIso φ"}
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+{"state": [{"context": ["F : Type u", "E : Type v", "inst✝⁵ : Field F", "inst✝⁴ : Field E", "inst✝³ : Algebra F E", "K : Type w", "inst✝² : Field K", "inst✝¹ : Algebra F K", "inst✝ : FiniteDimensional F E", "heq : finSepDegree F E = finrank F E", "x : E", "halg : IsAlgebraic F x", "h : finSepDegree F ↥F⟮x⟯ < finrank F ↥F⟮x⟯"], "goal": "False"}], "premise": [3971, 4677, 90367], "state_str": "F : Type u\nE : Type v\ninst✝⁵ : Field F\ninst✝⁴ : Field E\ninst✝³ : Algebra F E\nK : Type w\ninst✝² : Field K\ninst✝¹ : Algebra F K\ninst✝ : FiniteDimensional F E\nheq : finSepDegree F E = finrank F E\nx : E\nhalg : IsAlgebraic F x\nh : finSepDegree F ↥F⟮x⟯ < finrank F ↥F⟮x⟯\n⊢ False"}
+{"state": [{"context": ["F : Type u", "E : Type v", "inst✝⁵ : Field F", "inst✝⁴ : Field E", "inst✝³ : Algebra F E", "K : Type w", "inst✝² : Field K", "inst✝¹ : Algebra F K", "inst✝ : FiniteDimensional F E", "heq : finSepDegree F E = finrank F E", "x : E", "halg : IsAlgebraic F x", "h : finSepDegree F ↥F⟮x⟯ < finrank F ↥F⟮x⟯", "this : finSepDegree F ↥F⟮x⟯ * finSepDegree (↥F⟮x⟯) E < finrank F ↥F⟮x⟯ * finrank (↥F⟮x⟯) E"], "goal": "False"}], "premise": [85823, 90319], "state_str": "F : Type u\nE : Type v\ninst✝⁵ : Field F\ninst✝⁴ : Field E\ninst✝³ : Algebra F E\nK : Type w\ninst✝² : Field K\ninst✝¹ : Algebra F K\ninst✝ : FiniteDimensional F E\nheq : finSepDegree F E = finrank F E\nx : E\nhalg : IsAlgebraic F x\nh : finSepDegree F ↥F⟮x⟯ < finrank F ↥F⟮x⟯\nthis : finSepDegree F ↥F⟮x⟯ * finSepDegree (↥F⟮x⟯) E < finrank F ↥F⟮x⟯ * finrank (↥F⟮x⟯) E\n⊢ False"}
+{"state": [{"context": ["a b : ℤ"], "goal": "a = b ↔ ↑a = ↑b"}], "premise": [128913], "state_str": "a b : ℤ\n⊢ a = b ↔ ↑a = ↑b"}
+{"state": [{"context": ["R : Type u_1", "inst✝⁴ : CommRing R", "inst✝³ : IsArtinianRing R", "S : Submonoid R", "L : Type u_2", "inst✝² : CommRing L", "inst✝¹ : Algebra R L", "inst✝ : IsLocalization S L", "r' : L"], "goal": "∃ a, (algebraMap R L) a = r'"}], "premise": [77600], "state_str": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\ninst✝ : IsLocalization S L\nr' : L\n⊢ ∃ a, (algebraMap R L) a = r'"}
+{"state": [{"context": ["R : Type u_1", "inst✝⁴ : CommRing R", "inst✝³ : IsArtinianRing R", "S : Submonoid R", "L : Type u_2", "inst✝² : CommRing L", "inst✝¹ : Algebra R L", "inst✝ : IsLocalization S L", "r₁ : R", "s : ↥S"], "goal": "∃ a, (algebraMap R L) a = IsLocalization.mk' L r₁ s"}], "premise": [1674, 2045, 77573, 77591, 77601, 79863, 117080, 118863, 119466, 119703, 119730, 119742, 120533], "state_str": "case intro.intro\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\ninst✝ : IsLocalization S L\nr₁ : R\ns : ↥S\n⊢ ∃ a, (algebraMap R L) a = IsLocalization.mk' L r₁ s"}
+{"state": [{"context": ["R : Type u_1", "inst✝⁴ : CommRing R", "inst✝³ : IsArtinianRing R", "S : Submonoid R", "L : Type u_2", "inst✝² : CommRing L", "inst✝¹ : Algebra R L", "inst✝ : IsLocalization S L", "r₁ : R", "s : ↥S", "r₂ : R", "h : IsLocalization.mk' L 1 s = (algebraMap R L) r₂"], "goal": "∃ a, (algebraMap R L) a = IsLocalization.mk' L r₁ s"}], "premise": [77617, 117080], "state_str": "case intro.intro.intro\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\ninst✝ : IsLocalization S L\nr₁ : R\ns : ↥S\nr₂ : R\nh : IsLocalization.mk' L 1 s = (algebraMap R L) r₂\n⊢ ∃ a, (algebraMap R L) a = IsLocalization.mk' L r₁ s"}
+{"state": [{"context": ["R : Type u", "inst✝⁵ : CommSemiring R", "S : Submonoid R", "M : Type v", "inst✝⁴ : AddCommMonoid M", "inst✝³ : Module R M", "T : Type u_1", "inst✝² : CommSemiring T", "inst✝¹ : Algebra R T", "inst✝ : IsLocalization S T", "x y : M"], "goal": "(fun m => mk m 1) (x + y) = (fun m => mk m 1) x + (fun m => mk m 1) y"}], "premise": [113130], "state_str": "R : Type u\ninst✝⁵ : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nT : Type u_1\ninst✝² : CommSemiring T\ninst✝¹ : Algebra R T\ninst✝ : IsLocalization S T\nx y : M\n⊢ (fun m => mk m 1) (x + y) = (fun m => mk m 1) x + (fun m => mk m 1) y"}
+{"state": [{"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "π : ι → Type u_4", "inst✝¹ : Preorder α", "inst✝ : Nonempty β", "a b : α"], "goal": "const β a < const β b ↔ a < b"}], "premise": [11318, 14286], "state_str": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\nπ : ι → Type u_4\ninst✝¹ : Preorder α\ninst✝ : Nonempty β\na b : α\n⊢ const β a < const β b ↔ a < b"}
+{"state": [{"context": ["n : ℕ", "K : Type u_1", "inst✝² : CommRing K", "μ : K", "h : IsPrimitiveRoot μ n", "inst✝¹ : IsDomain K", "inst✝ : CharZero K", "p : ℕ", "hprime : Fact (Nat.Prime p)", "hdiv : ¬p ∣ n", "Q : ℤ[X] := minpoly ℤ (μ ^ p)"], "goal": "map (Int.castRingHom (ZMod p)) (minpoly ℤ μ) ∣ map (Int.castRingHom (ZMod p)) Q ^ p"}], "premise": [88917, 101869], "state_str": "n : ℕ\nK : Type u_1\ninst✝² : CommRing K\nμ : K\nh : IsPrimitiveRoot μ n\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\np : ℕ\nhprime : Fact (Nat.Prime p)\nhdiv : ¬p ∣ n\nQ : ℤ[X] := minpoly ℤ (μ ^ p)\n⊢ map (Int.castRingHom (ZMod p)) (minpoly ℤ μ) ∣ map (Int.castRingHom (ZMod p)) Q ^ p"}
+{"state": [{"context": ["n : ℕ", "K : Type u_1", "inst✝² : CommRing K", "μ : K", "h : IsPrimitiveRoot μ n", "inst✝¹ : IsDomain K", "inst✝ : CharZero K", "p : ℕ", "hprime : Fact (Nat.Prime p)", "hdiv : ¬p ∣ n", "Q : ℤ[X] := minpoly ℤ (μ ^ p)", "hfrob : map (Int.castRingHom (ZMod p)) Q ^ p = map (Int.castRingHom (ZMod p)) ((expand ℤ p) Q)"], "goal": "map (Int.castRingHom (ZMod p)) (minpoly ℤ μ) ∣ map (Int.castRingHom (ZMod p)) ((expand ℤ p) Q)"}], "premise": [121185], "state_str": "n : ℕ\nK : Type u_1\ninst✝² : CommRing K\nμ : K\nh : IsPrimitiveRoot μ n\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\np : ℕ\nhprime : Fact (Nat.Prime p)\nhdiv : ¬p ∣ n\nQ : ℤ[X] := minpoly ℤ (μ ^ p)\nhfrob : map (Int.castRingHom (ZMod p)) Q ^ p = map (Int.castRingHom (ZMod p)) ((expand ℤ p) Q)\n⊢ map (Int.castRingHom (ZMod p)) (minpoly ℤ μ) ∣ map (Int.castRingHom (ZMod p)) ((expand ℤ p) Q)"}
+{"state": [{"context": ["n : ℕ", "K : Type u_1", "inst✝² : CommRing K", "μ : K", "h : IsPrimitiveRoot μ n", "inst✝¹ : IsDomain K", "inst✝ : CharZero K", "p : ℕ", "hprime : Fact (Nat.Prime p)", "hdiv : ¬p ∣ n", "Q : ℤ[X] := minpoly ℤ (μ ^ p)", "hfrob : map (Int.castRingHom (ZMod p)) Q ^ p = map (Int.castRingHom (ZMod p)) ((expand ℤ p) Q)"], "goal": "minpoly ℤ μ ∣ (expand ℤ p) Q"}], "premise": [78115], "state_str": "n : ℕ\nK : Type u_1\ninst✝² : CommRing K\nμ : K\nh : IsPrimitiveRoot μ n\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\np : ℕ\nhprime : Fact (Nat.Prime p)\nhdiv : ¬p ∣ n\nQ : ℤ[X] := minpoly ℤ (μ ^ p)\nhfrob : map (Int.castRingHom (ZMod p)) Q ^ p = map (Int.castRingHom (ZMod p)) ((expand ℤ p) Q)\n⊢ minpoly ℤ μ ∣ (expand ℤ p) Q"}
+{"state": [{"context": ["α : Type u_1", "mα : MeasurableSpace α", "μ✝ : Measure α", "f : α → ℝ", "μ ν : Measure α", "inst✝¹ : SigmaFinite μ", "inst✝ : SigmaFinite ν", "hf : Integrable (fun x => rexp (f x)) ν"], "goal": "(fun x => (μ.rnDeriv (ν.tilted f) x).toReal) =ᶠ[ae ν] fun x => (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * (μ.rnDeriv ν x).toReal"}], "premise": [15889, 29903, 131585], "state_str": "α : Type u_1\nmα : MeasurableSpace α\nμ✝ : Measure α\nf : α → ℝ\nμ ν : Measure α\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite ν\nhf : Integrable (fun x => rexp (f x)) ν\n⊢ (fun x => (μ.rnDeriv (ν.tilted f) x).toReal) =ᶠ[ae ν] fun x =>\n    (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * (μ.rnDeriv ν x).toReal"}
+{"state": [{"context": ["α : Type u_1", "mα : MeasurableSpace α", "μ✝ : Measure α", "f : α → ℝ", "μ ν : Measure α", "inst✝¹ : SigmaFinite μ", "inst✝ : SigmaFinite ν", "hf : Integrable (fun x => rexp (f x)) ν", "x : α", "hx : μ.rnDeriv (ν.tilted f) x = ENNReal.ofReal (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * μ.rnDeriv ν x"], "goal": "(ENNReal.ofReal (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * μ.rnDeriv ν x).toReal = (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * (μ.rnDeriv ν x).toReal"}], "premise": [1842, 108287, 143198, 143423], "state_str": "case h\nα : Type u_1\nmα : MeasurableSpace α\nμ✝ : Measure α\nf : α → ℝ\nμ ν : Measure α\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite ν\nhf : Integrable (fun x => rexp (f x)) ν\nx : α\nhx : μ.rnDeriv (ν.tilted f) x = ENNReal.ofReal (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * μ.rnDeriv ν x\n⊢ (ENNReal.ofReal (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * μ.rnDeriv ν x).toReal =\n    (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * (μ.rnDeriv ν x).toReal"}
+{"state": [{"context": ["n✝ : ZFSet", "x : PSet", "x✝ : ⟦x⟧ ∈ omega", "n : ℕ", "h : x.Equiv (PSet.omega.Func { down := n })"], "goal": "insert (mk x) (mk x) = insert (mk (ofNat n)) (mk (ofNat n))"}], "premise": [47837], "state_str": "n✝ : ZFSet\nx : PSet\nx✝ : ⟦x⟧ ∈ omega\nn : ℕ\nh : x.Equiv (PSet.omega.Func { down := n })\n⊢ insert (mk x) (mk x) = insert (mk (ofNat n)) (mk (ofNat n))"}
+{"state": [{"context": ["α : Type u", "β : Type v", "γ : Type u_1", "f : Filter α", "s : Set α", "a : α", "hf : f.NeBot", "x : α"], "goal": "f = pure x ↔ {x} ∈ f"}], "premise": [1713, 11976, 16312], "state_str": "α : Type u\nβ : Type v\nγ : Type u_1\nf : Filter α\ns : Set α\na : α\nhf : f.NeBot\nx : α\n⊢ f = pure x ↔ {x} ∈ f"}
+{"state": [{"context": ["l : Type u_1", "m : Type u_2", "n : Type u_3", "o : Type u_4", "m' : o → Type u_5", "n' : o → Type u_6", "R : Type u_7", "S : Type u_8", "α : Type v", "β : Type w", "γ : Type u_9", "inst✝¹ : NonUnitalNonAssocSemiring α", "inst✝ : Fintype m", "A : Matrix m n α", "x y : m → α", "x✝ : n"], "goal": "((x + y) ᵥ* A) x✝ = (x ᵥ* A + y ᵥ* A) x✝"}], "premise": [142215], "state_str": "case h\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nm' : o → Type u_5\nn' : o → Type u_6\nR : Type u_7\nS : Type u_8\nα : Type v\nβ : Type w\nγ : Type u_9\ninst✝¹ : NonUnitalNonAssocSemiring α\ninst✝ : Fintype m\nA : Matrix m n α\nx y : m → α\nx✝ : n\n⊢ ((x + y) ᵥ* A) x✝ = (x ᵥ* A + y ᵥ* A) x✝"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "G H : SimpleGraph α", "s t✝ : Set α", "a b : α", "f : α ↪ β", "t : Set β", "ht : t.Nontrivial"], "goal": "(SimpleGraph.map f G).IsClique t ↔ ∃ s, G.IsClique s ∧ ⇑f '' s = t"}], "premise": [51301], "state_str": "α : Type u_1\nβ : Type u_2\nG H : SimpleGraph α\ns t✝ : Set α\na b : α\nf : α ↪ β\nt : Set β\nht : t.Nontrivial\n⊢ (SimpleGraph.map f G).IsClique t ↔ ∃ s, G.IsClique s ∧ ⇑f '' s = t"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "G H : SimpleGraph α", "s t✝ : Set α", "a b : α", "f : α ↪ β", "t : Set β", "ht : t.Nontrivial", "h : (SimpleGraph.map f G).IsClique t"], "goal": "⇑f '' (⇑f ⁻¹' t) = t"}], "premise": [134194], "state_str": "case refine_2\nα : Type u_1\nβ : Type u_2\nG H : SimpleGraph α\ns t✝ : Set α\na b : α\nf : α ↪ β\nt : Set β\nht : t.Nontrivial\nh : (SimpleGraph.map f G).IsClique t\n⊢ ⇑f '' (⇑f ⁻¹' t) = t"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "G H : SimpleGraph α", "s t✝ : Set α", "a b : α", "f : α ↪ β", "t : Set β", "ht : t.Nontrivial", "h : (SimpleGraph.map f G).IsClique t", "x : β", "hxt : x ∈ t"], "goal": "x ∈ Set.range ⇑f"}], "premise": [132292], "state_str": "case refine_2\nα : Type u_1\nβ : Type u_2\nG H : SimpleGraph α\ns t✝ : Set α\na b : α\nf : α ↪ β\nt : Set β\nht : t.Nontrivial\nh : (SimpleGraph.map f G).IsClique t\nx : β\nhxt : x ∈ t\n⊢ x ∈ Set.range ⇑f"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "G H : SimpleGraph α", "s t✝ : Set α", "a b : α", "f : α ↪ β", "t : Set β", "ht : t.Nontrivial", "h : (SimpleGraph.map f G).IsClique t", "u v : α", "hyt : f u ∈ t", "hxt : f v ∈ t", "hyne : f u ≠ f v"], "goal": "f v ∈ Set.range ⇑f"}], "premise": [131596], "state_str": "case refine_2.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nG H : SimpleGraph α\ns t✝ : Set α\na b : α\nf : α ↪ β\nt : Set β\nht : t.Nontrivial\nh : (SimpleGraph.map f G).IsClique t\nu v : α\nhyt : f u ∈ t\nhxt : f v ∈ t\nhyne : f u ≠ f v\n⊢ f v ∈ Set.range ⇑f"}
+{"state": [{"context": ["R : Type u_1", "F : Type u", "K : Type v", "L : Type w", "inst✝⁵ : CommRing R", "inst✝⁴ : Field K", "inst✝³ : Field L", "inst✝² : Field F", "i : K →+* L", "inst✝¹ : Algebra R K", "inst✝ : Algebra R L", "f : R[X]", "h : Splits (algebraMap R K) f", "e : K →ₐ[R] L"], "goal": "Splits (algebraMap R L) f"}], "premise": [121208], "state_str": "R : Type u_1\nF : Type u\nK : Type v\nL : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Field F\ni : K →+* L\ninst✝¹ : Algebra R K\ninst✝ : Algebra R L\nf : R[X]\nh : Splits (algebraMap R K) f\ne : K →ₐ[R] L\n⊢ Splits (algebraMap R L) f"}
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+{"state": [{"context": ["α : Type u_1", "E✝ : Type u_2", "F : Type u_3", "F' : Type u_4", "G : Type u_5", "𝕜 : Type u_6", "p : ℝ≥0∞", "inst✝¹² : NormedAddCommGroup E✝", "inst✝¹¹ : NormedSpace ℝ E✝", "inst✝¹⁰ : NormedAddCommGroup F", "inst✝⁹ : NormedSpace ℝ F", "inst✝⁸ : NormedAddCommGroup F'", "inst✝⁷ : NormedSpace ℝ F'", "inst✝⁶ : NormedAddCommGroup G", "m : MeasurableSpace α", "μ : Measure α", "inst✝⁵ : NormedField 𝕜", "inst✝⁴ : NormedSpace 𝕜 E✝", "E : Type u_7", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℝ E", "inst✝¹ : NormedSpace 𝕜 E", "inst✝ : NormedSpace 𝕜 F", "T : Set α → E →L[ℝ] F", "h_zero : ∀ (s : Set α), MeasurableSet s → μ s = 0 → T s = 0", "h_add : FinMeasAdditive μ T", "h_smul : ∀ (c : 𝕜) (s : Set α) (x : E), (T s) (c • x) = c • (T s) x", "c : 𝕜", "f : ↥(simpleFunc E 1 μ)"], "goal": "SimpleFunc.setToSimpleFunc T (toSimpleFunc (c • f)) = c • SimpleFunc.setToSimpleFunc T (toSimpleFunc f)"}], "premise": [30216, 32086], "state_str": "α : Type u_1\nE✝ : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝¹² : NormedAddCommGroup E✝\ninst✝¹¹ : NormedSpace ℝ E✝\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\ninst✝⁸ : NormedAddCommGroup F'\ninst✝⁷ : NormedSpace ℝ F'\ninst✝⁶ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝⁵ : NormedField 𝕜\ninst✝⁴ : NormedSpace 𝕜 E✝\nE : Type u_7\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace 𝕜 F\nT : Set α → E →L[ℝ] F\nh_zero : ∀ (s : Set α), MeasurableSet s → μ s = 0 → T s = 0\nh_add : FinMeasAdditive μ T\nh_smul : ∀ (c : 𝕜) (s : Set α) (x : E), (T s) (c • x) = c • (T s) x\nc : 𝕜\nf : ↥(simpleFunc E 1 μ)\n⊢ SimpleFunc.setToSimpleFunc T (toSimpleFunc (c • f)) = c • SimpleFunc.setToSimpleFunc T (toSimpleFunc f)"}
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+{"state": [{"context": ["M : Type u_1", "inst✝² : CommMonoid M", "S : Submonoid M", "N : Type u_2", "inst✝¹ : CommMonoid N", "P : Type u_3", "inst✝ : CommMonoid P", "f : S.LocalizationMap N", "x y : M", "c : ↥S", "u : Nˣ", "h : ↑u * f.toMap x = ↑u * f.toMap y", "hu : ↑u = f.toMap ↑c"], "goal": "f.toMap x = f.toMap y"}], "premise": [1673, 120417], "state_str": "M : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type u_3\ninst✝ : CommMonoid P\nf : S.LocalizationMap N\nx y : M\nc : ↥S\nu : Nˣ\nh : ↑u * f.toMap x = ↑u * f.toMap y\nhu : ↑u = f.toMap ↑c\n⊢ f.toMap x = f.toMap y"}
+{"state": [{"context": ["L : Language", "ι : Type v", "inst✝⁴ : Preorder ι", "G : ι → Type w", "inst✝³ : (i : ι) → L.Structure (G i)", "f : (i j : ι) → i ≤ j → G i ↪[L] G j", "inst✝² : IsDirected ι fun x x_1 => x ≤ x_1", "inst✝¹ : DirectedSystem G fun i j h => ⇑(f i j h)", "inst✝ : Nonempty ι", "n : ℕ", "R : L.Relations n", "i : ι", "x : Fin n → G i"], "goal": "(RelMap R fun a => ⟦Structure.Sigma.mk f i (x a)⟧) = RelMap R x"}], "premise": [24580], "state_str": "L : Language\nι : Type v\ninst✝⁴ : Preorder ι\nG : ι → Type w\ninst✝³ : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝² : IsDirected ι fun x x_1 => x ≤ x_1\ninst✝¹ : DirectedSystem G fun i j h => ⇑(f i j h)\ninst✝ : Nonempty ι\nn : ℕ\nR : L.Relations n\ni : ι\nx : Fin n → G i\n⊢ (RelMap R fun a => ⟦Structure.Sigma.mk f i (x a)⟧) = RelMap R x"}
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+{"state": [{"context": ["R : Type u_1", "l : Type u_2", "m : Type u_3", "n : Type u_4", "α : Type u_5", "β : Type u_6", "ι : Type u_7", "inst✝⁴ : Fintype l", "inst✝³ : Fintype m", "inst✝² : Fintype n", "inst✝¹ : Unique ι", "inst✝ : RCLike α", "A : Matrix l m α", "B : Matrix m n α", "i : l", "a✝¹ : i ∈ Finset.univ", "j : n", "a✝ : j ∈ Finset.univ"], "goal": "‖∑ j_1 : m, A i j_1 * B j_1 j‖₊ ^ 2 ≤ (∑ j : m, ‖A i j‖₊ ^ 2) * ∑ x : m, ‖B x j‖₊ ^ 2"}], "premise": [39672, 39678, 39691, 39704, 106103, 108459, 113020], "state_str": "case h₁.a.h.h\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\nα : Type u_5\nβ : Type u_6\nι : Type u_7\ninst✝⁴ : Fintype l\ninst✝³ : Fintype m\ninst✝² : Fintype n\ninst✝¹ : Unique ι\ninst✝ : RCLike α\nA : Matrix l m α\nB : Matrix m n α\ni : l\na✝¹ : i ∈ Finset.univ\nj : n\na✝ : j ∈ Finset.univ\n⊢ ‖∑ j_1 : m, A i j_1 * B j_1 j‖₊ ^ 2 ≤ (∑ j : m, ‖A i j‖₊ ^ 2) * ∑ x : m, ‖B x j‖₊ ^ 2"}
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+{"state": [{"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "H : Type u_5", "inst✝¹ : NormedAddCommGroup H", "inst✝ : InnerProductSpace ℝ H", "s t : Set H", "ι : Sort u_6", "f : ι → Set H", "x : H", "hx : ∀ (i : ι), x ∈ (f i).innerDualCone", "y : H", "hy : y ∈ ⋃ i, f i"], "goal": "0 ≤ ⟪y, x⟫_ℝ"}], "premise": [1673, 16573], "state_str": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℝ H\ns t : Set H\nι : Sort u_6\nf : ι → Set H\nx : H\nhx : ∀ (i : ι), x ∈ (f i).innerDualCone\ny : H\nhy : y ∈ ⋃ i, f i\n⊢ 0 ≤ ⟪y, x⟫_ℝ"}
+{"state": [{"context": ["𝓕 : Type u_1", "𝕜 : Type u_2", "α : Type u_3", "ι : Type u_4", "κ : Type u_5", "E : Type u_6", "F : Type u_7", "G : Type u_8", "inst✝² : SeminormedGroup E", "inst✝¹ : SeminormedGroup F", "inst✝ : SeminormedGroup G", "s : Set E", "a a₁ a₂ b b₁ b₂ : E", "r r₁ r₂ : ℝ", "x : E"], "goal": "(𝓝 x).HasBasis (fun ε => 0 < ε) fun ε => {y | ‖y / x‖ < ε}"}], "premise": [42710], "state_str": "𝓕 : Type u_1\n𝕜 : Type u_2\nα : Type u_3\nι : Type u_4\nκ : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nx : E\n⊢ (𝓝 x).HasBasis (fun ε => 0 < ε) fun ε => {y | ‖y / x‖ < ε}"}
+{"state": [{"context": ["𝓕 : Type u_1", "𝕜 : Type u_2", "α : Type u_3", "ι : Type u_4", "κ : Type u_5", "E : Type u_6", "F : Type u_7", "G : Type u_8", "inst✝² : SeminormedGroup E", "inst✝¹ : SeminormedGroup F", "inst✝ : SeminormedGroup G", "s : Set E", "a a₁ a₂ b b₁ b₂ : E", "r r₁ r₂ : ℝ", "x : E"], "goal": "(𝓝 x).HasBasis (fun ε => 0 < ε) fun ε => ball x ε"}], "premise": [61246], "state_str": "𝓕 : Type u_1\n𝕜 : Type u_2\nα : Type u_3\nι : Type u_4\nκ : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nx : E\n⊢ (𝓝 x).HasBasis (fun ε => 0 < ε) fun ε => ball x ε"}
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+{"state": [{"context": ["V : Type u_1", "E : Type u_2", "inst✝⁷ : NormedAddCommGroup V", "inst✝⁶ : InnerProductSpace ℝ V", "inst✝⁵ : MeasurableSpace V", "inst✝⁴ : BorelSpace V", "inst✝³ : FiniteDimensional ℝ V", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℂ E", "f : V → E", "inst✝ : CompleteSpace E", "hf : Integrable f volume", "h'f : Integrable (𝓕 f) volume", "v : V", "A : Tendsto (fun c => ∫ (w : V), cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ) • 𝓕 f w) atTop (𝓝 (𝓕⁻ (𝓕 f) v))", "B : Tendsto (fun c => ∫ (w : V), 𝓕 (fun w => cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ)) w • f w) atTop (𝓝 (𝓕⁻ (𝓕 f) v))"], "goal": "Tendsto (fun c => ∫ (w : V), ((↑π * ↑c) ^ (↑(finrank ℝ V) / 2) * cexp (-↑π ^ 2 * ↑c * ↑‖v - w‖ ^ 2)) • f w) atTop (𝓝 (𝓕⁻ (𝓕 f) v))"}], "premise": [16350], "state_str": "V : Type u_1\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MeasurableSpace V\ninst✝⁴ : BorelSpace V\ninst✝³ : FiniteDimensional ℝ V\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nf : V → E\ninst✝ : CompleteSpace E\nhf : Integrable f volume\nh'f : Integrable (𝓕 f) volume\nv : V\nA : Tendsto (fun c => ∫ (w : V), cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ) • 𝓕 f w) atTop (𝓝 (𝓕⁻ (𝓕 f) v))\nB :\n  Tendsto (fun c => ∫ (w : V), 𝓕 (fun w => cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ)) w • f w) atTop\n    (𝓝 (𝓕⁻ (𝓕 f) v))\n⊢ Tendsto (fun c => ∫ (w : V), ((↑π * ↑c) ^ (↑(finrank ℝ V) / 2) * cexp (-↑π ^ 2 * ↑c * ↑‖v - w‖ ^ 2)) • f w) atTop\n    (𝓝 (𝓕⁻ (𝓕 f) v))"}
+{"state": [{"context": ["V : Type u_1", "E : Type u_2", "inst✝⁷ : NormedAddCommGroup V", "inst✝⁶ : InnerProductSpace ℝ V", "inst✝⁵ : MeasurableSpace V", "inst✝⁴ : BorelSpace V", "inst✝³ : FiniteDimensional ℝ V", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℂ E", "f : V → E", "inst✝ : CompleteSpace E", "hf : Integrable f volume", "h'f : Integrable (𝓕 f) volume", "v : V", "A : Tendsto (fun c => ∫ (w : V), cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ) • 𝓕 f w) atTop (𝓝 (𝓕⁻ (𝓕 f) v))", "B : Tendsto (fun c => ∫ (w : V), 𝓕 (fun w => cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ)) w • f w) atTop (𝓝 (𝓕⁻ (𝓕 f) v))"], "goal": "(fun c => ∫ (w : V), 𝓕 (fun w => cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ)) w • f w) =ᶠ[atTop] fun c => ∫ (w : V), ((↑π * ↑c) ^ (↑(finrank ℝ V) / 2) * cexp (-↑π ^ 2 * ↑c * ↑‖v - w‖ ^ 2)) • f w"}], "premise": [15465, 15889, 131585], "state_str": "V : Type u_1\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MeasurableSpace V\ninst✝⁴ : BorelSpace V\ninst✝³ : FiniteDimensional ℝ V\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nf : V → E\ninst✝ : CompleteSpace E\nhf : Integrable f volume\nh'f : Integrable (𝓕 f) volume\nv : V\nA : Tendsto (fun c => ∫ (w : V), cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ) • 𝓕 f w) atTop (𝓝 (𝓕⁻ (𝓕 f) v))\nB :\n  Tendsto (fun c => ∫ (w : V), 𝓕 (fun w => cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ)) w • f w) atTop\n    (𝓝 (𝓕⁻ (𝓕 f) v))\n⊢ (fun c => ∫ (w : V), 𝓕 (fun w => cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ)) w • f w) =ᶠ[atTop] fun c =>\n    ∫ (w : V), ((↑π * ↑c) ^ (↑(finrank ℝ V) / 2) * cexp (-↑π ^ 2 * ↑c * ↑‖v - w‖ ^ 2)) • f w"}
+{"state": [{"context": ["V : Type u_1", "E : Type u_2", "inst✝⁷ : NormedAddCommGroup V", "inst✝⁶ : InnerProductSpace ℝ V", "inst✝⁵ : MeasurableSpace V", "inst✝⁴ : BorelSpace V", "inst✝³ : FiniteDimensional ℝ V", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℂ E", "f : V → E", "inst✝ : CompleteSpace E", "hf : Integrable f volume", "h'f : Integrable (𝓕 f) volume", "v : V", "A : Tendsto (fun c => ∫ (w : V), cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ) • 𝓕 f w) atTop (𝓝 (𝓕⁻ (𝓕 f) v))", "B : Tendsto (fun c => ∫ (w : V), 𝓕 (fun w => cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ)) w • f w) atTop (𝓝 (𝓕⁻ (𝓕 f) v))", "c : ℝ", "hc : 0 < c", "w : V"], "goal": "𝓕 (fun w => cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ)) w • f w = ((↑π * ↑c) ^ (↑(finrank ℝ V) / 2) * cexp (-↑π ^ 2 * ↑c * ↑‖v - w‖ ^ 2)) • f w"}], "premise": [40189], "state_str": "case h.e_f.h\nV : Type u_1\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MeasurableSpace V\ninst✝⁴ : BorelSpace V\ninst✝³ : FiniteDimensional ℝ V\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nf : V → E\ninst✝ : CompleteSpace E\nhf : Integrable f volume\nh'f : Integrable (𝓕 f) volume\nv : V\nA : Tendsto (fun c => ∫ (w : V), cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ) • 𝓕 f w) atTop (𝓝 (𝓕⁻ (𝓕 f) v))\nB :\n  Tendsto (fun c => ∫ (w : V), 𝓕 (fun w => cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ)) w • f w) atTop\n    (𝓝 (𝓕⁻ (𝓕 f) v))\nc : ℝ\nhc : 0 < c\nw : V\n⊢ 𝓕 (fun w => cexp (-↑c⁻¹ * ↑‖w‖ ^ 2 + 2 * ↑π * I * ↑⟪v, w⟫_ℝ)) w • f w =\n    ((↑π * ↑c) ^ (↑(finrank ℝ V) / 2) * cexp (-↑π ^ 2 * ↑c * ↑‖v - w‖ ^ 2)) • f w"}
+{"state": [{"context": ["G : Type w", "H : Type x", "α : Type u", "β : Type v", "inst✝² : TopologicalSpace G", "inst✝¹ : Group G", "inst✝ : TopologicalGroup G", "K V : Set G", "hK : IsCompact K", "hV : (interior V).Nonempty", "t : Finset G", "ht : K ⊆ ⋃ x ∈ t, interior ((fun x_1 => x * x_1) ⁻¹' V)"], "goal": "∃ t, K ⊆ ⋃ g ∈ t, (fun x => g * x) ⁻¹' V"}], "premise": [55381, 133327, 135265], "state_str": "case intro\nG : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : TopologicalGroup G\nK V : Set G\nhK : IsCompact K\nhV : (interior V).Nonempty\nt : Finset G\nht : K ⊆ ⋃ x ∈ t, interior ((fun x_1 => x * x_1) ⁻¹' V)\n⊢ ∃ t, K ⊆ ⋃ g ∈ t, (fun x => g * x) ⁻¹' V"}
+{"state": [{"context": ["R : Type u", "M : Type v", "inst✝³ : Ring R", "inst✝² : AddCommGroup M", "inst✝¹ : Module R M", "ι✝ : Type w", "ι' : Type w'", "inst✝ : StrongRankCondition R", "ι : Type w", "b : Finset ι", "h : Basis { x // x ∈ b } R M"], "goal": "finrank R M = b.card"}], "premise": [85910, 141428], "state_str": "R : Type u\nM : Type v\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι✝ : Type w\nι' : Type w'\ninst✝ : StrongRankCondition R\nι : Type w\nb : Finset ι\nh : Basis { x // x ∈ b } R M\n⊢ finrank R M = b.card"}
+{"state": [{"context": ["𝕜 : Type u_1", "inst✝² : RCLike 𝕜", "n : Type u_2", "inst✝¹ : Fintype n", "A : Matrix n n 𝕜", "inst✝ : DecidableEq n", "hA : A.IsHermitian", "j : n"], "goal": "A *ᵥ (WithLp.equiv 2 ((i : n) → (fun x => 𝕜) i)) (hA.eigenvectorBasis j) = hA.eigenvalues j • (WithLp.equiv 2 ((i : n) → (fun x => 𝕜) i)) (hA.eigenvectorBasis j)"}], "premise": [1673, 33144, 33401, 35283, 35329, 35360, 85174, 141363], "state_str": "𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nn : Type u_2\ninst✝¹ : Fintype n\nA : Matrix n n 𝕜\ninst✝ : DecidableEq n\nhA : A.IsHermitian\nj : n\n⊢ A *ᵥ (WithLp.equiv 2 ((i : n) → (fun x => 𝕜) i)) (hA.eigenvectorBasis j) =\n    hA.eigenvalues j • (WithLp.equiv 2 ((i : n) → (fun x => 𝕜) i)) (hA.eigenvectorBasis j)"}
+{"state": [{"context": ["R : Type u", "S : Type u'", "M : Type v", "N : Type v'", "inst✝¹⁷ : CommRing R", "inst✝¹⁶ : CommRing S", "inst✝¹⁵ : AddCommGroup M", "inst✝¹⁴ : AddCommGroup N", "inst✝¹³ : Module R M", "inst✝¹² : Module R N", "inst✝¹¹ : Algebra R S", "inst✝¹⁰ : Module S N", "inst✝⁹ : IsScalarTower R S N", "p : Submonoid R", "inst✝⁸ : IsLocalization p S", "f : M →ₗ[R] N", "inst✝⁷ : IsLocalizedModule p f", "hp : p ≤ R⁰", "M' : Submodule R M", "P : Type w", "inst✝⁶ : AddCommGroup P", "inst✝⁵ : Module R P", "Q : Type w'", "inst✝⁴ : AddCommGroup Q", "inst✝³ : Module R Q", "inst✝² : Module S Q", "inst✝¹ : IsScalarTower R S Q", "f' : P →ₗ[R] Q", "inst✝ : IsLocalizedModule p f'", "g : M →ₗ[R] P", "x : M", "hx : x ∈ ker g"], "goal": "f x ∈ ker ((IsLocalizedModule.map p f f') g)"}], "premise": [1673, 109977], "state_str": "R : Type u\nS : Type u'\nM : Type v\nN : Type v'\ninst✝¹⁷ : CommRing R\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : AddCommGroup M\ninst✝¹⁴ : AddCommGroup N\ninst✝¹³ : Module R M\ninst✝¹² : Module R N\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module S N\ninst✝⁹ : IsScalarTower R S N\np : Submonoid R\ninst✝⁸ : IsLocalization p S\nf : M →ₗ[R] N\ninst✝⁷ : IsLocalizedModule p f\nhp : p ≤ R⁰\nM' : Submodule R M\nP : Type w\ninst✝⁶ : AddCommGroup P\ninst✝⁵ : Module R P\nQ : Type w'\ninst✝⁴ : AddCommGroup Q\ninst✝³ : Module R Q\ninst✝² : Module S Q\ninst✝¹ : IsScalarTower R S Q\nf' : P →ₗ[R] Q\ninst✝ : IsLocalizedModule p f'\ng : M →ₗ[R] P\nx : M\nhx : x ∈ ker g\n⊢ f x ∈ ker ((IsLocalizedModule.map p f f') g)"}
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+{"state": [{"context": ["C : Type u_1", "inst✝⁴ : Category.{u_4, u_1} C", "J : GrothendieckTopology C", "A : Type u_2", "inst✝³ : Category.{u_3, u_2} A", "inst✝² : HasWeakSheafify J A", "inst✝¹ : (constantSheaf J A).Faithful", "inst✝ : (constantSheaf J A).Full", "t : C", "ht : IsTerminal t", "L : A ⥤ Sheaf J A", "adj : L ⊣ (sheafSections J A).obj (op t)", "F : Sheaf J A"], "goal": "IsDiscrete J ht F ↔ IsIso (adj.counit.app F)"}], "premise": [100177], "state_str": "C : Type u_1\ninst✝⁴ : Category.{u_4, u_1} C\nJ : GrothendieckTopology C\nA : Type u_2\ninst✝³ : Category.{u_3, u_2} A\ninst✝² : HasWeakSheafify J A\ninst✝¹ : (constantSheaf J A).Faithful\ninst✝ : (constantSheaf J A).Full\nt : C\nht : IsTerminal t\nL : A ⥤ Sheaf J A\nadj : L ⊣ (sheafSections J A).obj (op t)\nF : Sheaf J A\n⊢ IsDiscrete J ht F ↔ IsIso (adj.counit.app F)"}
+{"state": [{"context": ["C : Type u_1", "inst✝⁴ : Category.{u_4, u_1} C", "J : GrothendieckTopology C", "A : Type u_2", "inst✝³ : Category.{u_3, u_2} A", "inst✝² : HasWeakSheafify J A", "inst✝¹ : (constantSheaf J A).Faithful", "inst✝ : (constantSheaf J A).Full", "t : C", "ht : IsTerminal t", "L : A ⥤ Sheaf J A", "adj : L ⊣ (sheafSections J A).obj (op t)", "F : Sheaf J A", "this : L.Faithful"], "goal": "IsDiscrete J ht F ↔ IsIso (adj.counit.app F)"}], "premise": [100176], "state_str": "C : Type u_1\ninst✝⁴ : Category.{u_4, u_1} C\nJ : GrothendieckTopology C\nA : Type u_2\ninst✝³ : Category.{u_3, u_2} A\ninst✝² : HasWeakSheafify J A\ninst✝¹ : (constantSheaf J A).Faithful\ninst✝ : (constantSheaf J A).Full\nt : C\nht : IsTerminal t\nL : A ⥤ Sheaf J A\nadj : L ⊣ (sheafSections J A).obj (op t)\nF : Sheaf J A\nthis : L.Faithful\n⊢ IsDiscrete J ht F ↔ IsIso (adj.counit.app F)"}
+{"state": [{"context": ["C : Type u_1", "inst✝⁴ : Category.{u_4, u_1} C", "J : GrothendieckTopology C", "A : Type u_2", "inst✝³ : Category.{u_3, u_2} A", "inst✝² : HasWeakSheafify J A", "inst✝¹ : (constantSheaf J A).Faithful", "inst✝ : (constantSheaf J A).Full", "t : C", "ht : IsTerminal t", "L : A ⥤ Sheaf J A", "adj : L ⊣ (sheafSections J A).obj (op t)", "F : Sheaf J A", "this✝ : L.Faithful", "this : L.Full"], "goal": "IsDiscrete J ht F ↔ IsIso (adj.counit.app F)"}], "premise": [95374], "state_str": "C : Type u_1\ninst✝⁴ : Category.{u_4, u_1} C\nJ : GrothendieckTopology C\nA : Type u_2\ninst✝³ : Category.{u_3, u_2} A\ninst✝² : HasWeakSheafify J A\ninst✝¹ : (constantSheaf J A).Faithful\ninst✝ : (constantSheaf J A).Full\nt : C\nht : IsTerminal t\nL : A ⥤ Sheaf J A\nadj : L ⊣ (sheafSections J A).obj (op t)\nF : Sheaf J A\nthis✝ : L.Faithful\nthis : L.Full\n⊢ IsDiscrete J ht F ↔ IsIso (adj.counit.app F)"}
+{"state": [{"context": ["C : Type u_1", "inst✝⁴ : Category.{u_4, u_1} C", "J : GrothendieckTopology C", "A : Type u_2", "inst✝³ : Category.{u_3, u_2} A", "inst✝² : HasWeakSheafify J A", "inst✝¹ : (constantSheaf J A).Faithful", "inst✝ : (constantSheaf J A).Full", "t : C", "ht : IsTerminal t", "L : A ⥤ Sheaf J A", "adj : L ⊣ (sheafSections J A).obj (op t)", "F : Sheaf J A", "this✝ : L.Faithful", "this : L.Full"], "goal": "IsDiscrete J ht F ↔ F ∈ L.essImage"}], "premise": [91424], "state_str": "C : Type u_1\ninst✝⁴ : Category.{u_4, u_1} C\nJ : GrothendieckTopology C\nA : Type u_2\ninst✝³ : Category.{u_3, u_2} A\ninst✝² : HasWeakSheafify J A\ninst✝¹ : (constantSheaf J A).Faithful\ninst✝ : (constantSheaf J A).Full\nt : C\nht : IsTerminal t\nL : A ⥤ Sheaf J A\nadj : L ⊣ (sheafSections J A).obj (op t)\nF : Sheaf J A\nthis✝ : L.Faithful\nthis : L.Full\n⊢ IsDiscrete J ht F ↔ F ∈ L.essImage"}
+{"state": [{"context": ["B : Type u_1", "W : Type u_2", "inst✝ : Group W", "M : CoxeterMatrix B", "cs : CoxeterSystem M W", "w : W", "i : B"], "goal": "cs.IsRightDescent w⁻¹ i ↔ cs.IsLeftDescent w i"}], "premise": [1718, 8186], "state_str": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw : W\ni : B\n⊢ cs.IsRightDescent w⁻¹ i ↔ cs.IsLeftDescent w i"}
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+{"state": [{"context": ["K : Type u_1", "inst✝³ : Field K", "inst✝² : NumberField K", "inst✝¹ : IsCyclotomicExtension {3} ℚ K", "ζ : K", "hζ : IsPrimitiveRoot ζ ↑3", "S : Solution hζ", "S' : Solution' hζ", "inst✝ : DecidableRel fun a b => a ∣ b", "p : 𝓞 K", "hp : Prime p", "hpab : p ∣ 1 * S.a + 1 * S.b", "hpaηsqb : p ∣ S.a + ↑η ^ 2 * S.b"], "goal": "p ∣ λ"}], "premise": [119728], "state_str": "K : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\np : 𝓞 K\nhp : Prime p\nhpab : p ∣ 1 * S.a + 1 * S.b\nhpaηsqb : p ∣ S.a + ↑η ^ 2 * S.b\n⊢ p ∣ λ"}
+{"state": [{"context": ["K : Type u_1", "inst✝³ : Field K", "inst✝² : NumberField K", "inst✝¹ : IsCyclotomicExtension {3} ℚ K", "ζ : K", "hζ : IsPrimitiveRoot ζ ↑3", "S : Solution hζ", "S' : Solution' hζ", "inst✝ : DecidableRel fun a b => a ∣ b", "p : 𝓞 K", "hp : Prime p", "hpab : p ∣ 1 * S.a + 1 * S.b", "hpaηsqb : p ∣ 1 * S.a + ↑η ^ 2 * S.b"], "goal": "p ∣ λ"}], "premise": [122308], "state_str": "K : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\np : 𝓞 K\nhp : Prime p\nhpab : p ∣ 1 * S.a + 1 * S.b\nhpaηsqb : p ∣ 1 * S.a + ↑η ^ 2 * S.b\n⊢ p ∣ λ"}
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+{"state": [{"context": ["K : Type u_1", "inst✝³ : Field K", "inst✝² : NumberField K", "inst✝¹ : IsCyclotomicExtension {3} ℚ K", "ζ : K", "hζ : IsPrimitiveRoot ζ ↑3", "S : Solution hζ", "S' : Solution' hζ", "inst✝ : DecidableRel fun a b => a ∣ b", "p : 𝓞 K", "hp : Prime p", "hpab : p ∣ 1 * S.a + 1 * S.b", "hpaηsqb : p ∣ 1 * S.a + ↑η ^ 2 * S.b", "this : p ∣ -(↑η ^ 2 - 1)"], "goal": "p ∣ λ"}], "premise": [24753, 78735, 108877], "state_str": "K : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\np : 𝓞 K\nhp : Prime p\nhpab : p ∣ 1 * S.a + 1 * S.b\nhpaηsqb : p ∣ 1 * S.a + ↑η ^ 2 * S.b\nthis : p ∣ -(↑η ^ 2 - 1)\n⊢ p ∣ λ"}
+{"state": [{"context": ["K : Type u_1", "inst✝³ : Field K", "inst✝² : NumberField K", "inst✝¹ : IsCyclotomicExtension {3} ℚ K", "ζ : K", "hζ : IsPrimitiveRoot ζ ↑3", "S : Solution hζ", "S' : Solution' hζ", "inst✝ : DecidableRel fun a b => a ∣ b", "p : 𝓞 K", "hp : Prime p", "hpab : p ∣ 1 * S.a + 1 * S.b", "hpaηsqb : p ∣ 1 * S.a + ↑η ^ 2 * S.b", "this : p ∣ -(↑η ^ 2 - 1)"], "goal": "λ = -(↑η ^ 2 - 1) * ↑η"}], "premise": [24126, 24130, 117837, 119750, 122240], "state_str": "case h.e'_4\nK : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\np : 𝓞 K\nhp : Prime p\nhpab : p ∣ 1 * S.a + 1 * S.b\nhpaηsqb : p ∣ 1 * S.a + ↑η ^ 2 * S.b\nthis : p ∣ -(↑η ^ 2 - 1)\n⊢ λ = -(↑η ^ 2 - 1) * ↑η"}
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+{"state": [{"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "inst✝ : Abelian C", "P Q : C", "f : P ⟶ Q", "h : cokernel.π f = 0"], "goal": "Epi f"}], "premise": [94315], "state_str": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\nh : cokernel.π f = 0\n⊢ Epi f"}
+{"state": [{"context": ["𝕜 : Type u_1", "inst✝ : LinearOrderedField 𝕜", "s : Set 𝕜", "f : 𝕜 → 𝕜", "hf : StrictConcaveOn 𝕜 s f", "x y z : 𝕜", "hx : x ∈ s", "hz : z ∈ s", "hxy : x < y", "hyz : y < z"], "goal": "(f z - f y) / (z - y) < (f y - f x) / (y - x)"}], "premise": [35799, 104742], "state_str": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : StrictConcaveOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\n⊢ (f z - f y) / (z - y) < (f y - f x) / (y - x)"}
+{"state": [{"context": ["𝕜 : Type u_1", "inst✝ : LinearOrderedField 𝕜", "s : Set 𝕜", "f : 𝕜 → 𝕜", "hf : StrictConcaveOn 𝕜 s f", "x y z : 𝕜", "hx : x ∈ s", "hz : z ∈ s", "hxy : x < y", "hyz : y < z", "this : -(((-f) z - (-f) y) / (z - y)) < -(((-f) y - (-f) x) / (y - x))"], "goal": "(f z - f y) / (z - y) < (f y - f x) / (y - x)"}], "premise": [115834, 117880, 119769, 120670], "state_str": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : StrictConcaveOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nthis : -(((-f) z - (-f) y) / (z - y)) < -(((-f) y - (-f) x) / (y - x))\n⊢ (f z - f y) / (z - y) < (f y - f x) / (y - x)"}
+{"state": [{"context": ["𝕜 : Type u_1", "inst✝⁴ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace 𝕜 E", "H : Type u_3", "inst✝¹ : TopologicalSpace H", "I✝ : ModelWithCorners 𝕜 E H", "inst✝ : LocallyCompactSpace E", "I : ModelWithCorners 𝕜 E H"], "goal": "LocallyCompactSpace H"}], "premise": [12573, 12602, 56660, 67763], "state_str": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹ : TopologicalSpace H\nI✝ : ModelWithCorners 𝕜 E H\ninst✝ : LocallyCompactSpace E\nI : ModelWithCorners 𝕜 E H\n⊢ LocallyCompactSpace H"}
+{"state": [{"context": ["𝕜 : Type u_1", "inst✝⁴ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace 𝕜 E", "H : Type u_3", "inst✝¹ : TopologicalSpace H", "I✝ : ModelWithCorners 𝕜 E H", "inst✝ : LocallyCompactSpace E", "I : ModelWithCorners 𝕜 E H", "this : ∀ (x : H), (𝓝 x).HasBasis (fun s => s ∈ 𝓝 (↑I x) ∧ IsCompact s) fun s => ↑I.symm '' (s ∩ range ↑I)"], "goal": "LocallyCompactSpace H"}], "premise": [56662], "state_str": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹ : TopologicalSpace H\nI✝ : ModelWithCorners 𝕜 E H\ninst✝ : LocallyCompactSpace E\nI : ModelWithCorners 𝕜 E H\nthis : ∀ (x : H), (𝓝 x).HasBasis (fun s => s ∈ 𝓝 (↑I x) ∧ IsCompact s) fun s => ↑I.symm '' (s ∩ range ↑I)\n⊢ LocallyCompactSpace H"}
+{"state": [{"context": ["𝕜 : Type u_1", "inst✝⁴ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace 𝕜 E", "H : Type u_3", "inst✝¹ : TopologicalSpace H", "I✝ : ModelWithCorners 𝕜 E H", "inst✝ : LocallyCompactSpace E", "I : ModelWithCorners 𝕜 E H", "this : ∀ (x : H), (𝓝 x).HasBasis (fun s => s ∈ 𝓝 (↑I x) ∧ IsCompact s) fun s => ↑I.symm '' (s ∩ range ↑I)", "x : H", "s : Set E", "hsc : IsCompact s"], "goal": "IsCompact (↑I.symm '' (s ∩ range ↑I))"}], "premise": [58059, 58064, 67743, 67758], "state_str": "case intro\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹ : TopologicalSpace H\nI✝ : ModelWithCorners 𝕜 E H\ninst✝ : LocallyCompactSpace E\nI : ModelWithCorners 𝕜 E H\nthis : ∀ (x : H), (𝓝 x).HasBasis (fun s => s ∈ 𝓝 (↑I x) ∧ IsCompact s) fun s => ↑I.symm '' (s ∩ range ↑I)\nx : H\ns : Set E\nhsc : IsCompact s\n⊢ IsCompact (↑I.symm '' (s ∩ range ↑I))"}
+{"state": [{"context": ["R : Type u_1", "inst✝ : Semiring R", "φ ψ : R⟦X⟧", "h : ∀ (n : ℕ), (coeff R n) φ = (coeff R n) ψ", "n : Unit →₀ ℕ"], "goal": "(MvPowerSeries.coeff R n) φ = (MvPowerSeries.coeff R n) ψ"}], "premise": [79089], "state_str": "R : Type u_1\ninst✝ : Semiring R\nφ ψ : R⟦X⟧\nh : ∀ (n : ℕ), (coeff R n) φ = (coeff R n) ψ\nn : Unit →₀ ℕ\n⊢ (MvPowerSeries.coeff R n) φ = (MvPowerSeries.coeff R n) ψ"}
+{"state": [{"context": ["M : Type u_1", "N : Type u_2", "P : Type u_3", "inst✝⁴ : MulOneClass M", "inst✝³ : MulOneClass N", "inst✝² : MulOneClass P", "S : Submonoid M", "A : Type u_4", "inst✝¹ : SetLike A M", "hA : SubmonoidClass A M", "S' : A", "F : Type u_5", "inst✝ : FunLike F M N", "mc : MonoidHomClass F M N", "g : N →* P", "f : M →* N"], "goal": "map g (mrange f) = mrange (g.comp f)"}], "premise": [119378, 119525], "state_str": "M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝⁴ : MulOneClass M\ninst✝³ : MulOneClass N\ninst✝² : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst✝¹ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\nF : Type u_5\ninst✝ : FunLike F M N\nmc : MonoidHomClass F M N\ng : N →* P\nf : M →* N\n⊢ map g (mrange f) = mrange (g.comp f)"}
+{"state": [{"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "inst✝³ : MonoidalCategory C", "A B : Mon_ C", "M✝ : Bimod A B", "inst✝² : HasCoequalizers C", "inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)", "inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)", "W X Y Z : Mon_ C", "M : Bimod W X", "N N' : Bimod X Y", "f : N ⟶ N'", "P : Bimod Y Z"], "goal": "(whiskerRight (M.whiskerLeft f) P).hom = ((isoOfIso { hom := AssociatorBimod.hom M N P, inv := AssociatorBimod.inv M N P, hom_inv_id := ⋯, inv_hom_id := ⋯ } ⋯ ⋯).hom ≫ M.whiskerLeft (whiskerRight f P) ≫ (isoOfIso { hom := AssociatorBimod.hom M N' P, inv := AssociatorBimod.inv M N' P, hom_inv_id := ⋯, inv_hom_id := ⋯ } ⋯ ⋯).inv).hom"}], "premise": [94806], "state_str": "case h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ (whiskerRight (M.whiskerLeft f) P).hom =\n    ((isoOfIso { hom := AssociatorBimod.hom M N P, inv := AssociatorBimod.inv M N P, hom_inv_id := ⋯, inv_hom_id := ⋯ }\n            ⋯ ⋯).hom ≫\n        M.whiskerLeft (whiskerRight f P) ≫\n          (isoOfIso\n              { hom := AssociatorBimod.hom M N' P, inv := AssociatorBimod.inv M N' P, hom_inv_id := ⋯, inv_hom_id := ⋯ }\n              ⋯ ⋯).inv).hom"}
+{"state": [{"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "inst✝³ : MonoidalCategory C", "A B : Mon_ C", "M✝ : Bimod A B", "inst✝² : HasCoequalizers C", "inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)", "inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)", "W X Y Z : Mon_ C", "M : Bimod W X", "N N' : Bimod X Y", "f : N ⟶ N'", "P : Bimod Y Z"], "goal": "coequalizer.π (TensorBimod.actRight M N ▷ P.X) ((α_ (TensorBimod.X M N) Y.X P.X).hom ≫ TensorBimod.X M N ◁ P.actLeft) ≫ colimMap (parallelPairHom (TensorBimod.actRight M N ▷ P.X) ((α_ (TensorBimod.X M N) Y.X P.X).hom ≫ TensorBimod.X M N ◁ P.actLeft) (TensorBimod.actRight M N' ▷ P.X) ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft) (colimMap (parallelPairHom (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ((M.X ⊗ X.X) ◁ f.hom) (M.X ◁ f.hom) ⋯ ⋯) ▷ Y.X ▷ P.X) (colimMap (parallelPairHom (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ((M.X ⊗ X.X) ◁ f.hom) (M.X ◁ f.hom) ⋯ ⋯) ▷ P.X) ⋯ ⋯) = coequalizer.π (TensorBimod.actRight M N ▷ P.X) ((α_ (TensorBimod.X M N) Y.X P.X).hom ≫ TensorBimod.X M N ◁ P.actLeft) ≫ AssociatorBimod.hom M N P ≫ colimMap (parallelPairHom (M.actRight ▷ TensorBimod.X N P) ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P) (M.actRight ▷ TensorBimod.X N' P) ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P) ((M.X ⊗ X.X) ◁ colimMap (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X) ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯)) (M.X ◁ colimMap (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X) ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯)) ⋯ ⋯) ≫ AssociatorBimod.inv M N' P"}], "premise": [93393, 94736, 96173], "state_str": "case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ coequalizer.π (TensorBimod.actRight M N ▷ P.X)\n        ((α_ (TensorBimod.X M N) Y.X P.X).hom ≫ TensorBimod.X M N ◁ P.actLeft) ≫\n      colimMap\n        (parallelPairHom (TensorBimod.actRight M N ▷ P.X)\n          ((α_ (TensorBimod.X M N) Y.X P.X).hom ≫ TensorBimod.X M N ◁ P.actLeft) (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft)\n          (colimMap\n                (parallelPairHom (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) (M.actRight ▷ N'.X)\n                  ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ((M.X ⊗ X.X) ◁ f.hom) (M.X ◁ f.hom) ⋯ ⋯) ▷\n              Y.X ▷\n            P.X)\n          (colimMap\n              (parallelPairHom (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) (M.actRight ▷ N'.X)\n                ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ((M.X ⊗ X.X) ◁ f.hom) (M.X ◁ f.hom) ⋯ ⋯) ▷\n            P.X)\n          ⋯ ⋯) =\n    coequalizer.π (TensorBimod.actRight M N ▷ P.X)\n        ((α_ (TensorBimod.X M N) Y.X P.X).hom ≫ TensorBimod.X M N ◁ P.actLeft) ≫\n      AssociatorBimod.hom M N P ≫\n        colimMap\n            (parallelPairHom (M.actRight ▷ TensorBimod.X N P)\n              ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P) (M.actRight ▷ TensorBimod.X N' P)\n              ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P)\n              ((M.X ⊗ X.X) ◁\n                colimMap\n                  (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                    ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n              (M.X ◁\n                colimMap\n                  (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                    ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n              ⋯ ⋯) ≫\n          AssociatorBimod.inv M N' P"}
+{"state": [{"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "inst✝³ : MonoidalCategory C", "A B : Mon_ C", "M✝ : Bimod A B", "inst✝² : HasCoequalizers C", "inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)", "inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)", "W X Y Z : Mon_ C", "M : Bimod W X", "N N' : Bimod X Y", "f : N ⟶ N'", "P : Bimod Y Z"], "goal": "colimMap (parallelPairHom (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ((M.X ⊗ X.X) ◁ f.hom) (M.X ◁ f.hom) ⋯ ⋯) ▷ P.X ≫ colimit.ι (parallelPair (TensorBimod.actRight M N' ▷ P.X) ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft)) WalkingParallelPair.one = coequalizer.π (TensorBimod.actRight M N ▷ P.X) ((α_ (TensorBimod.X M N) Y.X P.X).hom ≫ TensorBimod.X M N ◁ P.actLeft) ≫ coequalizer.desc (AssociatorBimod.homAux M N P) ⋯ ≫ colimMap (parallelPairHom (M.actRight ▷ TensorBimod.X N P) ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P) (M.actRight ▷ TensorBimod.X N' P) ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P) ((M.X ⊗ X.X) ◁ colimMap (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X) ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯)) (M.X ◁ colimMap (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X) ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯)) ⋯ ⋯) ≫ AssociatorBimod.inv M N' P"}], "premise": [94804, 96173], "state_str": "case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ colimMap\n          (parallelPairHom (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) (M.actRight ▷ N'.X)\n            ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ((M.X ⊗ X.X) ◁ f.hom) (M.X ◁ f.hom) ⋯ ⋯) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    coequalizer.π (TensorBimod.actRight M N ▷ P.X)\n        ((α_ (TensorBimod.X M N) Y.X P.X).hom ≫ TensorBimod.X M N ◁ P.actLeft) ≫\n      coequalizer.desc (AssociatorBimod.homAux M N P) ⋯ ≫\n        colimMap\n            (parallelPairHom (M.actRight ▷ TensorBimod.X N P)\n              ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P) (M.actRight ▷ TensorBimod.X N' P)\n              ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P)\n              ((M.X ⊗ X.X) ◁\n                colimMap\n                  (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                    ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n              (M.X ◁\n                colimMap\n                  (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                    ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom �� Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n              ⋯ ⋯) ≫\n          AssociatorBimod.inv M N' P"}
+{"state": [{"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "inst✝³ : MonoidalCategory C", "A B : Mon_ C", "M✝ : Bimod A B", "inst✝² : HasCoequalizers C", "inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)", "inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)", "W X Y Z : Mon_ C", "M : Bimod W X", "N N' : Bimod X Y", "f : N ⟶ N'", "P : Bimod Y Z"], "goal": "colimMap (parallelPairHom (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ((M.X ⊗ X.X) ◁ f.hom) (M.X ◁ f.hom) ⋯ ⋯) ▷ P.X ≫ colimit.ι (parallelPair (TensorBimod.actRight M N' ▷ P.X) ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft)) WalkingParallelPair.one = (((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫ coequalizer.desc ((α_ M.X N.X P.X).hom ≫ M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫ coequalizer.π (M.actRight ▷ TensorBimod.X N P) ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P)) ⋯) ≫ colimMap (parallelPairHom (M.actRight ▷ TensorBimod.X N P) ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P) (M.actRight ▷ TensorBimod.X N' P) ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P) ((M.X ⊗ X.X) ◁ colimMap (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X) ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯)) (M.X ◁ colimMap (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X) ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯)) ⋯ ⋯)) ≫ AssociatorBimod.inv M N' P"}], "premise": [1673, 96190], "state_str": "case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ colimMap\n          (parallelPairHom (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) (M.actRight ▷ N'.X)\n            ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ((M.X ⊗ X.X) ◁ f.hom) (M.X ◁ f.hom) ⋯ ⋯) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    (((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫\n          coequalizer.desc\n            ((α_ M.X N.X P.X).hom ≫\n              M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫\n                coequalizer.π (M.actRight ▷ TensorBimod.X N P)\n                  ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P))\n            ⋯) ≫\n        colimMap\n          (parallelPairHom (M.actRight ▷ TensorBimod.X N P)\n            ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P) (M.actRight ▷ TensorBimod.X N' P)\n            ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P)\n            ((M.X ⊗ X.X) ◁\n              colimMap\n                (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                  ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n            (M.X ◁\n              colimMap\n                (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                  ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n            ⋯ ⋯)) ≫\n      AssociatorBimod.inv M N' P"}
+{"state": [{"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "inst✝³ : MonoidalCategory C", "A B : Mon_ C", "M✝ : Bimod A B", "inst✝² : HasCoequalizers C", "inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)", "inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)", "W X Y Z : Mon_ C", "M : Bimod W X", "N N' : Bimod X Y", "f : N ⟶ N'", "P : Bimod Y Z"], "goal": "(tensorRight P.X).map (coequalizer.π (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft)) ≫ colimMap (parallelPairHom (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ((M.X ⊗ X.X) ◁ f.hom) (M.X ◁ f.hom) ⋯ ⋯) ▷ P.X ≫ colimit.ι (parallelPair (TensorBimod.actRight M N' ▷ P.X) ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft)) WalkingParallelPair.one = (tensorRight P.X).map (coequalizer.π (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft)) ≫ (((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫ coequalizer.desc ((α_ M.X N.X P.X).hom ≫ M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫ coequalizer.π (M.actRight ▷ TensorBimod.X N P) ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P)) ⋯) ≫ colimMap (parallelPairHom (M.actRight ▷ TensorBimod.X N P) ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P) (M.actRight ▷ TensorBimod.X N' P) ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P) ((M.X ⊗ X.X) ◁ colimMap (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X) ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯)) (M.X ◁ colimMap (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X) ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯)) ⋯ ⋯)) ≫ AssociatorBimod.inv M N' P"}], "premise": [99324], "state_str": "case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ (tensorRight P.X).map (coequalizer.π (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft)) ≫\n      colimMap\n            (parallelPairHom (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) (M.actRight ▷ N'.X)\n              ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ((M.X ⊗ X.X) ◁ f.hom) (M.X ◁ f.hom) ⋯ ⋯) ▷\n          P.X ≫\n        colimit.ι\n          (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n            ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n          WalkingParallelPair.one =\n    (tensorRight P.X).map (coequalizer.π (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft)) ≫\n      (((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫\n            coequalizer.desc\n              ((α_ M.X N.X P.X).hom ≫\n                M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫\n                  coequalizer.π (M.actRight ▷ TensorBimod.X N P)\n                    ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P))\n              ⋯) ≫\n          colimMap\n            (parallelPairHom (M.actRight ▷ TensorBimod.X N P)\n              ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P) (M.actRight ▷ TensorBimod.X N' P)\n              ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P)\n              ((M.X ⊗ X.X) ◁\n                colimMap\n                  (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                    ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n              (M.X ◁\n                colimMap\n                  (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                    ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n              ⋯ ⋯)) ≫\n        AssociatorBimod.inv M N' P"}
+{"state": [{"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "inst✝³ : MonoidalCategory C", "A B : Mon_ C", "M✝ : Bimod A B", "inst✝² : HasCoequalizers C", "inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)", "inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)", "W X Y Z : Mon_ C", "M : Bimod W X", "N N' : Bimod X Y", "f : N ⟶ N'", "P : Bimod Y Z"], "goal": "coequalizer.π (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) ▷ P.X ≫ colimMap (parallelPairHom (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ((M.X ⊗ X.X) ◁ f.hom) (M.X ◁ f.hom) ⋯ ⋯) ▷ P.X ≫ colimit.ι (parallelPair (TensorBimod.actRight M N' ▷ P.X) ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft)) WalkingParallelPair.one = coequalizer.π (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) ▷ P.X ≫ (((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫ coequalizer.desc ((α_ M.X N.X P.X).hom ≫ M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫ coequalizer.π (M.actRight ▷ TensorBimod.X N P) ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P)) ⋯) ≫ colimMap (parallelPairHom (M.actRight ▷ TensorBimod.X N P) ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P) (M.actRight ▷ TensorBimod.X N' P) ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P) ((M.X ⊗ X.X) ◁ colimMap (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X) ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯)) (M.X ◁ colimMap (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X) ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯)) ⋯ ⋯)) ≫ AssociatorBimod.inv M N' P"}], "premise": [93393, 94736, 96173, 99222], "state_str": "case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ coequalizer.π (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) ▷ P.X ≫\n      colimMap\n            (parallelPairHom (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) (M.actRight ▷ N'.X)\n              ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ((M.X ⊗ X.X) ◁ f.hom) (M.X ◁ f.hom) ⋯ ⋯) ▷\n          P.X ≫\n        colimit.ι\n          (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n            ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n          WalkingParallelPair.one =\n    coequalizer.π (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) ▷ P.X ≫\n      (((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫\n            coequalizer.desc\n              ((α_ M.X N.X P.X).hom ≫\n                M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫\n                  coequalizer.π (M.actRight ▷ TensorBimod.X N P)\n                    ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P))\n              ⋯) ≫\n          colimMap\n            (parallelPairHom (M.actRight ▷ TensorBimod.X N P)\n              ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P) (M.actRight ▷ TensorBimod.X N' P)\n              ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P)\n              ((M.X ⊗ X.X) ◁\n                colimMap\n                  (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                    ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n              (M.X ◁\n                colimMap\n                  (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                    ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n              ⋯ ⋯)) ≫\n        AssociatorBimod.inv M N' P"}
+{"state": [{"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "inst✝³ : MonoidalCategory C", "A B : Mon_ C", "M✝ : Bimod A B", "inst✝² : HasCoequalizers C", "inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)", "inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)", "W X Y Z : Mon_ C", "M : Bimod W X", "N N' : Bimod X Y", "f : N ⟶ N'", "P : Bimod Y Z"], "goal": "(M.X ◁ f.hom ≫ colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft)) WalkingParallelPair.one) ▷ P.X ≫ colimit.ι (parallelPair (TensorBimod.actRight M N' ▷ P.X) ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft)) WalkingParallelPair.one = coequalizer.π (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) ▷ P.X ≫ (((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫ coequalizer.desc ((α_ M.X N.X P.X).hom ≫ M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫ coequalizer.π (M.actRight ▷ TensorBimod.X N P) ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P)) ⋯) ≫ colimMap (parallelPairHom (M.actRight ▷ TensorBimod.X N P) ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P) (M.actRight ▷ TensorBimod.X N' P) ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P) ((M.X ⊗ X.X) ◁ colimMap (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X) ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯)) (M.X ◁ colimMap (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X) ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯)) ⋯ ⋯)) ≫ AssociatorBimod.inv M N' P"}], "premise": [96173, 106541], "state_str": "case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ (M.X ◁ f.hom ≫\n          colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft))\n            WalkingParallelPair.one) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    coequalizer.π (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) ▷ P.X ≫\n      (((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫\n            coequalizer.desc\n              ((α_ M.X N.X P.X).hom ≫\n                M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫\n                  coequalizer.π (M.actRight ▷ TensorBimod.X N P)\n                    ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P))\n              ⋯) ≫\n          colimMap\n            (parallelPairHom (M.actRight ▷ TensorBimod.X N P)\n              ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P) (M.actRight ▷ TensorBimod.X N' P)\n              ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P)\n              ((M.X ⊗ X.X) ◁\n                colimMap\n                  (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                    ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n              (M.X ◁\n                colimMap\n                  (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                    ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n              ⋯ ⋯)) ≫\n        AssociatorBimod.inv M N' P"}
+{"state": [{"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "inst✝³ : MonoidalCategory C", "A B : Mon_ C", "M✝ : Bimod A B", "inst✝² : HasCoequalizers C", "inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)", "inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)", "W X Y Z : Mon_ C", "M : Bimod W X", "N N' : Bimod X Y", "f : N ⟶ N'", "P : Bimod Y Z"], "goal": "(M.X ◁ f.hom ≫ colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft)) WalkingParallelPair.one) ▷ P.X ≫ colimit.ι (parallelPair (TensorBimod.actRight M N' ▷ P.X) ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft)) WalkingParallelPair.one = (((α_ M.X N.X P.X).hom ≫ M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫ coequalizer.π (M.actRight ▷ TensorBimod.X N P) ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P)) ≫ colimMap (parallelPairHom (M.actRight ▷ TensorBimod.X N P) ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P) (M.actRight ▷ TensorBimod.X N' P) ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P) ((M.X ⊗ X.X) ◁ colimMap (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X) ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯)) (M.X ◁ colimMap (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X) ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯)) ⋯ ⋯)) ≫ AssociatorBimod.inv M N' P"}], "premise": [93393, 94736, 96173], "state_str": "case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ (M.X ◁ f.hom ≫\n          colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft))\n            WalkingParallelPair.one) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    (((α_ M.X N.X P.X).hom ≫\n          M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫\n            coequalizer.π (M.actRight ▷ TensorBimod.X N P)\n              ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P)) ≫\n        colimMap\n          (parallelPairHom (M.actRight ▷ TensorBimod.X N P)\n            ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P) (M.actRight ▷ TensorBimod.X N' P)\n            ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P)\n            ((M.X ⊗ X.X) ◁\n              colimMap\n                (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                  ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n            (M.X ◁\n              colimMap\n                (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                  ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))\n            ⋯ ⋯)) ≫\n      AssociatorBimod.inv M N' P"}
+{"state": [{"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "inst✝³ : MonoidalCategory C", "A B : Mon_ C", "M✝ : Bimod A B", "inst✝² : HasCoequalizers C", "inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)", "inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)", "W X Y Z : Mon_ C", "M : Bimod W X", "N N' : Bimod X Y", "f : N ⟶ N'", "P : Bimod Y Z"], "goal": "(M.X ◁ f.hom ≫ colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft)) WalkingParallelPair.one) ▷ P.X ≫ colimit.ι (parallelPair (TensorBimod.actRight M N' ▷ P.X) ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft)) WalkingParallelPair.one = (α_ M.X N.X P.X).hom ≫ M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫ (M.X ◁ colimMap (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X) ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯) ≫ colimit.ι (parallelPair (M.actRight ▷ TensorBimod.X N' P) ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P)) WalkingParallelPair.one) ≫ AssociatorBimod.inv M N' P"}], "premise": [93393, 94736, 96173, 99219], "state_str": "case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ (M.X ◁ f.hom ≫\n          colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft))\n            WalkingParallelPair.one) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    (α_ M.X N.X P.X).hom ≫\n      M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫\n        (M.X ◁\n              colimMap\n                (parallelPairHom (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)\n                  ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯) ≫\n            colimit.ι\n              (parallelPair (M.actRight ▷ TensorBimod.X N' P)\n                ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P))\n              WalkingParallelPair.one) ≫\n          AssociatorBimod.inv M N' P"}
+{"state": [{"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "inst✝³ : MonoidalCategory C", "A B : Mon_ C", "M✝ : Bimod A B", "inst✝² : HasCoequalizers C", "inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)", "inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)", "W X Y Z : Mon_ C", "M : Bimod W X", "N N' : Bimod X Y", "f : N ⟶ N'", "P : Bimod Y Z"], "goal": "(M.X ◁ f.hom ≫ colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft)) WalkingParallelPair.one) ▷ P.X ≫ colimit.ι (parallelPair (TensorBimod.actRight M N' ▷ P.X) ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft)) WalkingParallelPair.one = (α_ M.X N.X P.X).hom ≫ (M.X ◁ (f.hom ▷ P.X ≫ colimit.ι (parallelPair (N'.actRight ▷ P.X) ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft)) WalkingParallelPair.one) ≫ colimit.ι (parallelPair (M.actRight ▷ TensorBimod.X N' P) ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P)) WalkingParallelPair.one) ≫ coequalizer.desc (AssociatorBimod.invAux M N' P) ⋯"}], "premise": [94804, 96173], "state_str": "case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ (M.X ◁ f.hom ≫\n          colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft))\n            WalkingParallelPair.one) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    (α_ M.X N.X P.X).hom ≫\n      (M.X ◁\n            (f.hom ▷ P.X ≫\n              colimit.ι (parallelPair (N'.actRight ▷ P.X) ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft))\n                WalkingParallelPair.one) ≫\n          colimit.ι\n            (parallelPair (M.actRight ▷ TensorBimod.X N' P)\n              ((α_ M.X X.X (TensorBimod.X N' P)).hom ≫ M.X ◁ TensorBimod.actLeft N' P))\n            WalkingParallelPair.one) ≫\n        coequalizer.desc (AssociatorBimod.invAux M N' P) ⋯"}
+{"state": [{"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "inst✝³ : MonoidalCategory C", "A B : Mon_ C", "M✝ : Bimod A B", "inst✝² : HasCoequalizers C", "inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)", "inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)", "W X Y Z : Mon_ C", "M : Bimod W X", "N N' : Bimod X Y", "f : N ⟶ N'", "P : Bimod Y Z"], "goal": "(M.X ◁ f.hom ≫ colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft)) WalkingParallelPair.one) ▷ P.X ≫ colimit.ι (parallelPair (TensorBimod.actRight M N' ▷ P.X) ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft)) WalkingParallelPair.one = (α_ M.X N.X P.X).hom ≫ M.X ◁ (f.hom ▷ P.X ≫ colimit.ι (parallelPair (N'.actRight ▷ P.X) ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft)) WalkingParallelPair.one) ≫ (PreservesCoequalizer.iso (tensorLeft M.X) (N'.actRight ▷ P.X) ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft)).inv ≫ coequalizer.desc ((α_ M.X N'.X P.X).inv ≫ coequalizer.π (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ▷ P.X ≫ coequalizer.π (TensorBimod.actRight M N' ▷ P.X) ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft)) ⋯"}], "premise": [96173, 99219], "state_str": "case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ (M.X ◁ f.hom ≫\n          colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft))\n            WalkingParallelPair.one) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    (α_ M.X N.X P.X).hom ≫\n      M.X ◁\n          (f.hom ▷ P.X ≫\n            colimit.ι (parallelPair (N'.actRight ▷ P.X) ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft))\n              WalkingParallelPair.one) ≫\n        (PreservesCoequalizer.iso (tensorLeft M.X) (N'.actRight ▷ P.X) ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft)).inv ≫\n          coequalizer.desc\n            ((α_ M.X N'.X P.X).inv ≫\n              coequalizer.π (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ▷ P.X ≫\n                coequalizer.π (TensorBimod.actRight M N' ▷ P.X)\n                  ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n            ⋯"}
+{"state": [{"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "inst✝³ : MonoidalCategory C", "A B : Mon_ C", "M✝ : Bimod A B", "inst✝² : HasCoequalizers C", "inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)", "inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)", "W X Y Z : Mon_ C", "M : Bimod W X", "N N' : Bimod X Y", "f : N ⟶ N'", "P : Bimod Y Z"], "goal": "(M.X ◁ f.hom ≫ colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft)) WalkingParallelPair.one) ▷ P.X ≫ colimit.ι (parallelPair (TensorBimod.actRight M N' ▷ P.X) ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft)) WalkingParallelPair.one = (α_ M.X N.X P.X).hom ≫ ((M.X ◁ f.hom ▷ P.X ≫ M.X ◁ colimit.ι (parallelPair (N'.actRight ▷ P.X) ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft)) WalkingParallelPair.one) ≫ (PreservesCoequalizer.iso (tensorLeft M.X) (N'.actRight ▷ P.X) ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft)).inv) ≫ coequalizer.desc ((α_ M.X N'.X P.X).inv ≫ coequalizer.π (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ▷ P.X ≫ coequalizer.π (TensorBimod.actRight M N' ▷ P.X) ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft)) ⋯"}], "premise": [96173, 106539], "state_str": "case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ (M.X ◁ f.hom ≫\n          colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft))\n            WalkingParallelPair.one) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    (α_ M.X N.X P.X).hom ≫\n      ((M.X ◁ f.hom ▷ P.X ≫\n            M.X ◁\n              colimit.ι (parallelPair (N'.actRight ▷ P.X) ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft))\n                WalkingParallelPair.one) ≫\n          (PreservesCoequalizer.iso (tensorLeft M.X) (N'.actRight ▷ P.X)\n              ((α_ N'.X Y.X P.X).hom ≫ N'.X ◁ P.actLeft)).inv) ≫\n        coequalizer.desc\n          ((α_ M.X N'.X P.X).inv ≫\n            coequalizer.π (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ▷ P.X ≫\n              coequalizer.π (TensorBimod.actRight M N' ▷ P.X)\n                ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n          ⋯"}
+{"state": [{"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "inst✝³ : MonoidalCategory C", "A B : Mon_ C", "M✝ : Bimod A B", "inst✝² : HasCoequalizers C", "inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)", "inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)", "W X Y Z : Mon_ C", "M : Bimod W X", "N N' : Bimod X Y", "f : N ⟶ N'", "P : Bimod Y Z"], "goal": "(M.X ◁ f.hom ≫ colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft)) WalkingParallelPair.one) ▷ P.X ≫ colimit.ι (parallelPair (TensorBimod.actRight M N' ▷ P.X) ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft)) WalkingParallelPair.one = (α_ M.X N.X P.X).hom ≫ M.X ◁ f.hom ▷ P.X ≫ (α_ M.X N'.X P.X).inv ≫ coequalizer.π (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ▷ P.X ≫ coequalizer.π (TensorBimod.actRight M N' ▷ P.X) ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft)"}], "premise": [96173, 99262], "state_str": "case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ (M.X ◁ f.hom ≫\n          colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft))\n            WalkingParallelPair.one) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    (α_ M.X N.X P.X).hom ≫\n      M.X ◁ f.hom ▷ P.X ≫\n        (α_ M.X N'.X P.X).inv ≫\n          coequalizer.π (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ▷ P.X ≫\n            coequalizer.π (TensorBimod.actRight M N' ▷ P.X)\n              ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft)"}
+{"state": [{"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "inst✝³ : MonoidalCategory C", "A B : Mon_ C", "M✝ : Bimod A B", "inst✝² : HasCoequalizers C", "inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)", "inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)", "W X Y Z : Mon_ C", "M : Bimod W X", "N N' : Bimod X Y", "f : N ⟶ N'", "P : Bimod Y Z"], "goal": "(M.X ◁ f.hom ≫ colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft)) WalkingParallelPair.one) ▷ P.X ≫ colimit.ι (parallelPair (TensorBimod.actRight M N' ▷ P.X) ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft)) WalkingParallelPair.one = (α_ M.X N.X P.X).hom ≫ (((α_ M.X N.X P.X).inv ≫ (M.X ◁ f.hom) ▷ P.X) ≫ coequalizer.π (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ▷ P.X) ≫ coequalizer.π (TensorBimod.actRight M N' ▷ P.X) ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft)"}], "premise": [96173], "state_str": "case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ (M.X ◁ f.hom ≫\n          colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft))\n            WalkingParallelPair.one) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    (α_ M.X N.X P.X).hom ≫\n      (((α_ M.X N.X P.X).inv ≫ (M.X ◁ f.hom) ▷ P.X) ≫\n          coequalizer.π (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ▷ P.X) ≫\n        coequalizer.π (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft)"}
+{"state": [{"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "inst✝³ : MonoidalCategory C", "A B : Mon_ C", "M✝ : Bimod A B", "inst✝² : HasCoequalizers C", "inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)", "inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)", "W X Y Z : Mon_ C", "M : Bimod W X", "N N' : Bimod X Y", "f : N ⟶ N'", "P : Bimod Y Z"], "goal": "(M.X ◁ f.hom ≫ colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft)) WalkingParallelPair.one) ▷ P.X ≫ colimit.ι (parallelPair (TensorBimod.actRight M N' ▷ P.X) ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft)) WalkingParallelPair.one = ((M.X ◁ f.hom) ▷ P.X ≫ coequalizer.π (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ▷ P.X) ≫ coequalizer.π (TensorBimod.actRight M N' ▷ P.X) ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft)"}], "premise": [96173, 99222], "state_str": "case h.h\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM✝ : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N ⟶ N'\nP : Bimod Y Z\n⊢ (M.X ◁ f.hom ≫\n          colimit.ι (parallelPair (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft))\n            WalkingParallelPair.one) ▷\n        P.X ≫\n      colimit.ι\n        (parallelPair (TensorBimod.actRight M N' ▷ P.X)\n          ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft))\n        WalkingParallelPair.one =\n    ((M.X ◁ f.hom) ▷ P.X ≫ coequalizer.π (M.actRight ▷ N'.X) ((α_ M.X X.X N'.X).hom ≫ M.X ◁ N'.actLeft) ▷ P.X) ≫\n      coequalizer.π (TensorBimod.actRight M N' ▷ P.X)\n        ((α_ (TensorBimod.X M N') Y.X P.X).hom ≫ TensorBimod.X M N' ◁ P.actLeft)"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "G : Type u_4", "M : Type u_5", "N : Type u_6", "inst✝¹ : CommMonoid M", "inst✝ : CommMonoid N", "f g : α → M", "a b : α", "s t : Finset α"], "goal": "(∏ᶠ (i : α) (_ : i ∈ ↑s ∪ ↑t), f i) * ∏ᶠ (i : α) (_ : i ∈ ↑s ∩ ↑t), f i = (∏ᶠ (i : α) (_ : i ∈ ↑s), f i) * ∏ᶠ (i : α) (_ : i ∈ ↑t), f i"}], "premise": [125648, 126929, 138864, 138907], "state_str": "case intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g : α → M\na b : α\ns t : Finset α\n⊢ (∏ᶠ (i : α) (_ : i ∈ ↑s ∪ ↑t), f i) * ∏ᶠ (i : α) (_ : i ∈ ↑s ∩ ↑t), f i =\n    (∏ᶠ (i : α) (_ : i ∈ ↑s), f i) * ∏ᶠ (i : α) (_ : i ∈ ↑t), f i"}
+{"state": [{"context": ["α✝ : Sort u", "β : Sort v", "γ : Sort w", "α : Sort u_1", "a b : α"], "goal": "(Equiv.cast ⋯) a = b ↔ HEq a b"}], "premise": [70746], "state_str": "α✝ : Sort u\nβ : Sort v\nγ : Sort w\nα : Sort u_1\na b : α\n⊢ (Equiv.cast ⋯) a = b ↔ HEq a b"}
+{"state": [{"context": ["a : ℕ", "a1 : 1 < a", "b : ℤ√↑(Pell.d a1)", "b1 : 1 ≤ b", "hp : IsPell b", "n : ℕ", "h : b ≤ ↑n"], "goal": "↑n ≤ pellZd a1 n"}], "premise": [23773], "state_str": "a : ℕ\na1 : 1 < a\nb : ℤ√↑(Pell.d a1)\nb1 : 1 ≤ b\nhp : IsPell b\nn : ℕ\nh : b ≤ ↑n\n⊢ ↑n ≤ pellZd a1 n"}
+{"state": [{"context": ["a : ℕ", "a1 : 1 < a", "b : ℤ√↑(Pell.d a1)", "b1 : 1 ≤ b", "hp : IsPell b", "n : ℕ", "h : b ≤ ↑n"], "goal": "{ re := ↑n, im := 0 } ≤ pellZd a1 n"}], "premise": [3154, 14286, 22500, 23823], "state_str": "a : ℕ\na1 : 1 < a\nb : ℤ√↑(Pell.d a1)\nb1 : 1 ≤ b\nhp : IsPell b\nn : ℕ\nh : b ≤ ↑n\n⊢ { re := ↑n, im := 0 } ≤ pellZd a1 n"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "G : Type u_3", "M : Type u_4", "inst✝ : CommGroup G", "a b c d : G"], "goal": "a / b = c ↔ a = b * c"}], "premise": [1713, 118012, 119707], "state_str": "α : Type u_1\nβ : Type u_2\nG : Type u_3\nM : Type u_4\ninst✝ : CommGroup G\na b c d : G\n⊢ a / b = c ↔ a = b * c"}
+{"state": [{"context": ["G✝ : Type w", "H : Type x", "α : Type u", "β : Type v", "inst✝⁶ : TopologicalSpace G✝", "inst✝⁵ : Group G✝", "inst✝⁴ : TopologicalGroup G✝", "inst✝³ : TopologicalSpace α", "f : α → G✝", "s : Set α", "x : α", "G : Type u_1", "inst✝² : TopologicalSpace G", "inst✝¹ : Group G", "inst✝ : TopologicalGroup G", "g : G", "hg : g ∈ connectedComponent 1"], "goal": "g⁻¹ ∈ connectedComponent 1"}], "premise": [119801], "state_str": "G✝ : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝⁶ : TopologicalSpace G✝\ninst✝⁵ : Group G✝\ninst✝⁴ : TopologicalGroup G✝\ninst✝³ : TopologicalSpace α\nf : α → G✝\ns : Set α\nx : α\nG : Type u_1\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : TopologicalGroup G\ng : G\nhg : g ∈ connectedComponent 1\n⊢ g⁻¹ ∈ connectedComponent 1"}
+{"state": [{"context": ["G✝ : Type w", "H : Type x", "α : Type u", "β : Type v", "inst✝⁶ : TopologicalSpace G✝", "inst✝⁵ : Group G✝", "inst✝⁴ : TopologicalGroup G✝", "inst✝³ : TopologicalSpace α", "f : α → G✝", "s : Set α", "x : α", "G : Type u_1", "inst✝² : TopologicalSpace G", "inst✝¹ : Group G", "inst✝ : TopologicalGroup G", "g : G", "hg : g ∈ connectedComponent 1"], "goal": "g⁻¹ ∈ connectedComponent 1⁻¹"}], "premise": [1673, 65911, 66647, 131592], "state_str": "G✝ : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝⁶ : TopologicalSpace G✝\ninst✝⁵ : Group G✝\ninst✝⁴ : TopologicalGroup G✝\ninst✝³ : TopologicalSpace α\nf : α → G✝\ns : Set α\nx : α\nG : Type u_1\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : TopologicalGroup G\ng : G\nhg : g ∈ connectedComponent 1\n⊢ g⁻¹ ∈ connectedComponent 1⁻¹"}
+{"state": [{"context": ["S : Set ℝ", "a : ℝ", "hS : ∀ x ∈ S, x ≤ a", "ha : 0 ≤ a"], "goal": "sSup S ≤ a"}], "premise": [133383], "state_str": "S : Set ℝ\na : ℝ\nhS : ∀ x ∈ S, x ≤ a\nha : 0 ≤ a\n⊢ sSup S ≤ a"}
+{"state": [{"context": ["a : ℝ", "ha : 0 ≤ a", "hS : ∀ x ∈ ∅, x ≤ a"], "goal": "sSup ∅ ≤ a"}, {"context": ["S : Set ℝ", "a : ℝ", "hS : ∀ x ∈ S, x ≤ a", "ha : 0 ≤ a", "hS₂ : S.Nonempty"], "goal": "sSup S ≤ a"}], "premise": [16842, 146815], "state_str": "case inl\na : ℝ\nha : 0 ≤ a\nhS : ∀ x ∈ ∅, x ≤ a\n⊢ sSup ∅ ≤ a\n\ncase inr\nS : Set ℝ\na : ℝ\nhS : ∀ x ∈ S, x ≤ a\nha : 0 ≤ a\nhS₂ : S.Nonempty\n⊢ sSup S ≤ a"}
+{"state": [{"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "R : Type u_4", "m n : ℕ"], "goal": "Even (m / n) ↔ m % (2 * n) / n = 0"}], "premise": [1713, 3530, 3570, 119707, 121775], "state_str": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nR : Type u_4\nm n : ℕ\n⊢ Even (m / n) ↔ m % (2 * n) / n = 0"}
+{"state": [{"context": ["R : Type u_1", "inst✝¹ : NormedRing R", "inst✝ : CompleteSpace R", "x : Rˣ", "n : ℕ"], "goal": "(fun t => inverse (↑x + t) - (∑ i ∈ range n, (-↑x⁻¹ * t) ^ i) * ↑x⁻¹) =O[𝓝 0] fun t => ‖t‖ ^ n"}], "premise": [16090, 16105, 43397, 43722], "state_str": "R : Type u_1\ninst✝¹ : NormedRing R\ninst✝ : CompleteSpace R\nx : Rˣ\nn : ℕ\n⊢ (fun t => inverse (↑x + t) - (∑ i ∈ range n, (-↑x⁻¹ * t) ^ i) * ↑x⁻¹) =O[𝓝 0] fun t => ‖t‖ ^ n"}
+{"state": [{"context": ["R : Type u_1", "inst✝¹ : NormedRing R", "inst✝ : CompleteSpace R", "x : Rˣ", "n : ℕ"], "goal": "(fun x_1 => (∑ i ∈ range n, (-↑x⁻¹ * x_1) ^ i) * ↑x⁻¹ + (-↑x⁻¹ * x_1) ^ n * inverse (↑x + x_1) - (∑ i ∈ range n, (-↑x⁻¹ * x_1) ^ i) * ↑x⁻¹) =O[𝓝 0] fun t => ‖t‖ ^ n"}], "premise": [118076], "state_str": "R : Type u_1\ninst✝¹ : NormedRing R\ninst✝ : CompleteSpace R\nx : Rˣ\nn : ℕ\n⊢ (fun x_1 =>\n      (∑ i ∈ range n, (-↑x⁻¹ * x_1) ^ i) * ↑x⁻¹ + (-↑x⁻¹ * x_1) ^ n * inverse (↑x + x_1) -\n        (∑ i ∈ range n, (-↑x⁻¹ * x_1) ^ i) * ↑x⁻¹) =O[𝓝 0]\n    fun t => ‖t‖ ^ n"}
+{"state": [{"context": ["R : Type u_1", "inst✝¹ : NormedRing R", "inst✝ : CompleteSpace R", "x : Rˣ", "n : ℕ"], "goal": "(fun x_1 => (-↑x⁻¹ * x_1) ^ n * inverse (↑x + x_1)) =O[𝓝 0] fun t => ‖t‖ ^ n"}], "premise": [43411, 43424, 43587, 43724], "state_str": "R : Type u_1\ninst✝¹ : NormedRing R\ninst✝ : CompleteSpace R\nx : Rˣ\nn : ℕ\n⊢ (fun x_1 => (-↑x⁻¹ * x_1) ^ n * inverse (↑x + x_1)) =O[𝓝 0] fun t => ‖t‖ ^ n"}
+{"state": [{"context": ["R : Type u_1", "inst✝¹ : NormedRing R", "inst✝ : CompleteSpace R", "x : Rˣ", "n : ℕ"], "goal": "(fun x_1 => ‖(-↑x⁻¹ * x_1) ^ n‖ * 1) =O[𝓝 0] fun t => ‖t‖ ^ n"}], "premise": [43455, 119730], "state_str": "R : Type u_1\ninst✝¹ : NormedRing R\ninst✝ : CompleteSpace R\nx : Rˣ\nn : ℕ\n⊢ (fun x_1 => ‖(-↑x⁻¹ * x_1) ^ n‖ * 1) =O[𝓝 0] fun t => ‖t‖ ^ n"}
+{"state": [{"context": ["R : Type u_1", "inst✝¹ : NormedRing R", "inst✝ : CompleteSpace R", "x : Rˣ", "n : ℕ"], "goal": "(fun x_1 => (-↑x⁻¹ * x_1) ^ n) =O[𝓝 0] fun t => ‖t‖ ^ n"}], "premise": [43424, 43563, 43594], "state_str": "R : Type u_1\ninst✝¹ : NormedRing R\ninst✝ : CompleteSpace R\nx : Rˣ\nn : ℕ\n⊢ (fun x_1 => (-↑x⁻¹ * x_1) ^ n) =O[𝓝 0] fun t => ‖t‖ ^ n"}
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+{"state": [{"context": ["E : Type u_1", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℝ E", "F : Type u_2", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace ℝ F", "f g : ℝ → E", "x : ℝ", "f_diff : ∀ (y : ℝ), y ≠ x → HasDerivAt f (g y) y", "hf : ContinuousAt f x", "hg : ContinuousAt g x", "A : HasDerivWithinAt f (g x) (Ici x) x"], "goal": "HasDerivAt f (g x) x"}], "premise": [2100, 14283, 15884, 16350, 16369, 44367, 44379, 45602, 46351, 57179, 57310], "state_str": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf g : ℝ → E\nx : ℝ\nf_diff : ∀ (y : ℝ), y ≠ x → HasDerivAt f (g y) y\nhf : ContinuousAt f x\nhg : ContinuousAt g x\nA : HasDerivWithinAt f (g x) (Ici x) x\n⊢ HasDerivAt f (g x) x"}
+{"state": [{"context": ["E : Type u_1", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℝ E", "F : Type u_2", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace ℝ F", "f g : ℝ → E", "x : ℝ", "f_diff : ∀ (y : ℝ), y ≠ x → HasDerivAt f (g y) y", "hf : ContinuousAt f x", "hg : ContinuousAt g x", "A : HasDerivWithinAt f (g x) (Ici x) x", "B : HasDerivWithinAt f (g x) (Iic x) x"], "goal": "HasDerivAt f (g x) x"}], "premise": [44372], "state_str": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf g : ℝ → E\nx : ℝ\nf_diff : ∀ (y : ℝ), y ≠ x → HasDerivAt f (g y) y\nhf : ContinuousAt f x\nhg : ContinuousAt g x\nA : HasDerivWithinAt f (g x) (Ici x) x\nB : HasDerivWithinAt f (g x) (Iic x) x\n⊢ HasDerivAt f (g x) x"}
+{"state": [{"context": ["R : Type u_1", "S : Type u_2", "inst✝⁵ : CommRing R", "inst✝⁴ : CommRing S", "m : Type u_3", "n : Type u_4", "inst✝³ : DecidableEq m", "inst✝² : DecidableEq n", "inst✝¹ : Fintype m", "inst✝ : Fintype n", "M₁₁ : Matrix m m R", "M₁₂ : Matrix m n R", "M₂₁ : Matrix n m R", "M₂₂ M✝ : Matrix n n R", "i j : n", "M : Matrix n n R", "f : R →+* S"], "goal": "(M.map ⇑f).charpoly = Polynomial.map f M.charpoly"}], "premise": [84126, 86469, 102924, 142299], "state_str": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\nm : Type u_3\nn : Type u_4\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : Fintype m\ninst✝ : Fintype n\nM₁₁ : Matrix m m R\nM₁₂ : Matrix m n R\nM₂₁ : Matrix n m R\nM₂₂ M✝ : Matrix n n R\ni j : n\nM : Matrix n n R\nf : R →+* S\n⊢ (M.map ⇑f).charpoly = Polynomial.map f M.charpoly"}
+{"state": [{"context": ["F✝ : Type u → Type v", "α β γ : Type u", "inst✝¹ : Functor F✝", "inst✝ : LawfulFunctor F✝", "F : Type u_1 → Type u_2", "m : {α β : Type u_1} → (α → β) → F α → F β", "mc : {α β : Type u_1} → α → F β → F α", "m' : {α β : Type u_1} → (α → β) → F α → F β", "mc' : {α β : Type u_1} → α → F β → F α", "H1 : LawfulFunctor F", "H2 : LawfulFunctor F", "H : ∀ (α β : Type u_1) (f : α → β) (x : F α), f <$> x = f <$> x"], "goal": "{ map := m, mapConst := mc } = { map := m', mapConst := mc' }"}], "premise": [1838], "state_str": "F✝ : Type u → Type v\nα β γ : Type u\ninst✝¹ : Functor F✝\ninst✝ : LawfulFunctor F✝\nF : Type u_1 → Type u_2\nm : {α β : Type u_1} → (α → β) → F α → F β\nmc : {α β : Type u_1} → α → F β → F α\nm' : {α β : Type u_1} → (α → β) → F α → F β\nmc' : {α β : Type u_1} → α → F β → F α\nH1 : LawfulFunctor F\nH2 : LawfulFunctor F\nH : ∀ (α β : Type u_1) (f : α → β) (x : F α), f <$> x = f <$> x\n⊢ { map := m, mapConst := mc } = { map := m', mapConst := mc' }"}
+{"state": [{"context": ["F✝ : Type u → Type v", "α β γ : Type u", "inst✝¹ : Functor F✝", "inst✝ : LawfulFunctor F✝", "F : Type u_1 → Type u_2", "m : {α β : Type u_1} → (α → β) → F α → F β", "mc mc' : {α β : Type u_1} → α → F β → F α", "H1 : LawfulFunctor F", "H2 : LawfulFunctor F", "H : ∀ (α β : Type u_1) (f : α → β) (x : F α), f <$> x = f <$> x"], "goal": "mc = mc'"}], "premise": [1838], "state_str": "case refl.e_mapConst\nF✝ : Type u → Type v\nα β γ : Type u\ninst✝¹ : Functor F✝\ninst✝ : LawfulFunctor F✝\nF : Type u_1 → Type u_2\nm : {α β : Type u_1} → (α → β) → F α → F β\nmc mc' : {α β : Type u_1} → α → F β → F α\nH1 : LawfulFunctor F\nH2 : LawfulFunctor F\nH : ∀ (α β : Type u_1) (f : α → β) (x : F α), f <$> x = f <$> x\n⊢ mc = mc'"}
+{"state": [{"context": ["F✝ : Type u → Type v", "α✝ β✝ γ : Type u", "inst✝¹ : Functor F✝", "inst✝ : LawfulFunctor F✝", "F : Type u_1 → Type u_2", "m : {α β : Type u_1} → (α → β) → F α → F β", "mc mc' : {α β : Type u_1} → α → F β → F α", "H1 : LawfulFunctor F", "H2 : LawfulFunctor F", "H : ∀ (α β : Type u_1) (f : α → β) (x : F α), f <$> x = f <$> x", "α β : Type u_1"], "goal": "mc = mc'"}], "premise": [977], "state_str": "case refl.e_mapConst.h.h\nF✝ : Type u → Type v\nα✝ β✝ γ : Type u\ninst✝¹ : Functor F✝\ninst✝ : LawfulFunctor F✝\nF : Type u_1 → Type u_2\nm : {α β : Type u_1} → (α → β) → F α → F β\nmc mc' : {α β : Type u_1} → α → F β → F α\nH1 : LawfulFunctor F\nH2 : LawfulFunctor F\nH : ∀ (α β : Type u_1) (f : α → β) (x : F α), f <$> x = f <$> x\nα β : Type u_1\n⊢ mc = mc'"}
+{"state": [{"context": ["F✝ : Type u → Type v", "α✝ β✝ γ : Type u", "inst✝¹ : Functor F✝", "inst✝ : LawfulFunctor F✝", "F : Type u_1 → Type u_2", "m : {α β : Type u_1} → (α → β) → F α → F β", "mc mc' : {α β : Type u_1} → α → F β → F α", "H1 : LawfulFunctor F", "H2 : LawfulFunctor F", "H : ∀ (α β : Type u_1) (f : α → β) (x : F α), f <$> x = f <$> x", "α β : Type u_1", "E1 : ∀ {α β : Type u_1}, mapConst = map ∘ Function.const β"], "goal": "mc = mc'"}], "premise": [977], "state_str": "case refl.e_mapConst.h.h\nF✝ : Type u → Type v\nα✝ β✝ γ : Type u\ninst✝¹ : Functor F✝\ninst✝ : LawfulFunctor F✝\nF : Type u_1 → Type u_2\nm : {α β : Type u_1} → (α → β) → F α → F β\nmc mc' : {α β : Type u_1} → α → F β → F α\nH1 : LawfulFunctor F\nH2 : LawfulFunctor F\nH : ∀ (α β : Type u_1) (f : α → β) (x : F α), f <$> x = f <$> x\nα β : Type u_1\nE1 : ∀ {α β : Type u_1}, mapConst = map ∘ Function.const β\n⊢ mc = mc'"}
+{"state": [{"context": ["F✝ : Type u → Type v", "α✝ β✝ γ : Type u", "inst✝¹ : Functor F✝", "inst✝ : LawfulFunctor F✝", "F : Type u_1 → Type u_2", "m : {α β : Type u_1} → (α → β) → F α → F β", "mc mc' : {α β : Type u_1} → α → F β → F α", "H1 : LawfulFunctor F", "H2 : LawfulFunctor F", "H : ∀ (α β : Type u_1) (f : α → β) (x : F α), f <$> x = f <$> x", "α β : Type u_1", "E1 : ∀ {α β : Type u_1}, mapConst = map ∘ Function.const β", "E2 : ∀ {α β : Type u_1}, mapConst = map ∘ Function.const β"], "goal": "mc = mc'"}], "premise": [2100, 2101], "state_str": "case refl.e_mapConst.h.h\nF✝ : Type u → Type v\nα✝ β✝ γ : Type u\ninst✝¹ : Functor F✝\ninst✝ : LawfulFunctor F✝\nF : Type u_1 → Type u_2\nm : {α β : Type u_1} → (α → β) → F α → F β\nmc mc' : {α β : Type u_1} → α → F β → F α\nH1 : LawfulFunctor F\nH2 : LawfulFunctor F\nH : ∀ (α β : Type u_1) (f : α → β) (x : F α), f <$> x = f <$> x\nα β : Type u_1\nE1 : ∀ {α β : Type u_1}, mapConst = map ∘ Function.const β\nE2 : ∀ {α β : Type u_1}, mapConst = map ∘ Function.const β\n⊢ mc = mc'"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι✝ : Type u_4", "s s' : Set α", "x : α", "p✝ : Filter ι✝", "g : ι✝ → α", "inst✝¹ : UniformSpace β", "𝔖 : Set (Set α)", "ι : Type u_5", "ι' : Type u_6", "inst✝ : Nonempty ι", "t : ι → Set α", "p : ι' → Prop", "V : ι' → Set (β × β)", "ht : ∀ (i : ι), t i ∈ 𝔖", "hdir : Directed (fun x x_1 => x ⊆ x_1) t", "hex : ∀ s ∈ 𝔖, ∃ i, s ⊆ t i", "hb : (𝓤 β).HasBasis p V"], "goal": "(𝓤 (α →ᵤ[𝔖] β)).HasBasis (fun i => p i.2) fun i => UniformOnFun.gen 𝔖 (t i.1) (V i.2)"}], "premise": [1674, 133349, 134181, 134186], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι✝ : Type u_4\ns s' : Set α\nx : α\np✝ : Filter ι✝\ng : ι✝ → α\ninst✝¹ : UniformSpace β\n𝔖 : Set (Set α)\nι : Type u_5\nι' : Type u_6\ninst✝ : Nonempty ι\nt : ι → Set α\np : ι' → Prop\nV : ι' → Set (β × β)\nht : ∀ (i : ι), t i ∈ 𝔖\nhdir : Directed (fun x x_1 => x ⊆ x_1) t\nhex : ∀ s ∈ 𝔖, ∃ i, s ⊆ t i\nhb : (𝓤 β).HasBasis p V\n⊢ (𝓤 (α →ᵤ[𝔖] β)).HasBasis (fun i => p i.2) fun i => UniformOnFun.gen 𝔖 (t i.1) (V i.2)"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι✝ : Type u_4", "s s' : Set α", "x : α", "p✝ : Filter ι✝", "g : ι✝ → α", "inst✝¹ : UniformSpace β", "𝔖 : Set (Set α)", "ι : Type u_5", "ι' : Type u_6", "inst✝ : Nonempty ι", "t : ι → Set α", "p : ι' → Prop", "V : ι' → Set (β × β)", "ht : ∀ (i : ι), t i ∈ 𝔖", "hdir : Directed (fun x x_1 => x ⊆ x_1) t", "hex : ∀ s ∈ 𝔖, ∃ i, s ⊆ t i", "hb : (𝓤 β).HasBasis p V", "hne : 𝔖.Nonempty", "hd : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖"], "goal": "(𝓤 (α →ᵤ[𝔖] β)).HasBasis (fun i => p i.2) fun i => UniformOnFun.gen 𝔖 (t i.1) (V i.2)"}], "premise": [12534, 60281, 133326], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι✝ : Type u_4\ns s' : Set α\nx : α\np✝ : Filter ι✝\ng : ι✝ → α\ninst✝¹ : UniformSpace β\n𝔖 : Set (Set α)\nι : Type u_5\nι' : Type u_6\ninst✝ : Nonempty ι\nt : ι → Set α\np : ι' → Prop\nV : ι' → Set (β × β)\nht : ∀ (i : ι), t i ∈ 𝔖\nhdir : Directed (fun x x_1 => x ⊆ x_1) t\nhex : ∀ s ∈ 𝔖, ∃ i, s ⊆ t i\nhb : (𝓤 β).HasBasis p V\nhne : 𝔖.Nonempty\nhd : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖\n⊢ (𝓤 (α →ᵤ[𝔖] β)).HasBasis (fun i => p i.2) fun i => UniformOnFun.gen 𝔖 (t i.1) (V i.2)"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι✝ : Type u_4", "s✝ s' : Set α", "x : α", "p✝ : Filter ι✝", "g : ι✝ → α", "inst✝¹ : UniformSpace β", "𝔖 : Set (Set α)", "ι : Type u_5", "ι' : Type u_6", "inst✝ : Nonempty ι", "t : ι → Set α", "p : ι' → Prop", "V : ι' → Set (β × β)", "ht : ∀ (i : ι), t i ∈ 𝔖", "hdir : Directed (fun x x_1 => x ⊆ x_1) t", "hex : ∀ s ∈ 𝔖, ∃ i, s ⊆ t i", "hb : (𝓤 β).HasBasis p V", "hne : 𝔖.Nonempty", "hd : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖", "x✝¹ : Set α × ι'", "s : Set α", "i' : ι'", "x✝ : (s, i').1 ∈ 𝔖 ∧ p (s, i').2", "hs : (s, i').1 ∈ 𝔖", "hi' : p (s, i').2", "i : ι", "hi : s ⊆ t i"], "goal": "∃ i'_1, p i'_1.2 ∧ UniformOnFun.gen 𝔖 (t i'_1.1) (V i'_1.2) ⊆ UniformOnFun.gen 𝔖 (s, i').1 (V (s, i').2)"}], "premise": [60276, 133326], "state_str": "case intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι✝ : Type u_4\ns✝ s' : Set α\nx : α\np✝ : Filter ι✝\ng : ι✝ → α\ninst✝¹ : UniformSpace β\n𝔖 : Set (Set α)\nι : Type u_5\nι' : Type u_6\ninst✝ : Nonempty ι\nt : ι → Set α\np : ι' → Prop\nV : ι' → Set (β × β)\nht : ∀ (i : ι), t i ∈ 𝔖\nhdir : Directed (fun x x_1 => x ⊆ x_1) t\nhex : ∀ s ∈ 𝔖, ∃ i, s ⊆ t i\nhb : (𝓤 β).HasBasis p V\nhne : 𝔖.Nonempty\nhd : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖\nx✝¹ : Set α × ι'\ns : Set α\ni' : ι'\nx✝ : (s, i').1 ∈ 𝔖 ∧ p (s, i').2\nhs : (s, i').1 ∈ 𝔖\nhi' : p (s, i').2\ni : ι\nhi : s ⊆ t i\n⊢ ∃ i'_1, p i'_1.2 ∧ UniformOnFun.gen 𝔖 (t i'_1.1) (V i'_1.2) ⊆ UniformOnFun.gen 𝔖 (s, i').1 (V (s, i').2)"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4", "inst✝⁵ : Countable ι", "G : Type u_5", "p : ℝ≥0∞", "m m0 : MeasurableSpace α", "μ✝ : Measure α", "inst✝⁴ : SeminormedAddCommGroup G", "inst✝³ : MeasurableSpace G", "inst✝² : BorelSpace G", "inst✝¹ : SecondCountableTopology G", "f : α → G", "_m0 : MeasurableSpace α", "μ : Measure α", "inst✝ : SigmaFinite μ"], "goal": "AEFinStronglyMeasurable f μ ↔ AEMeasurable f μ"}], "premise": [29464], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ninst✝⁵ : Countable ι\nG : Type u_5\np : ℝ≥0∞\nm m0 : MeasurableSpace α\nμ✝ : Measure α\ninst✝⁴ : SeminormedAddCommGroup G\ninst✝³ : MeasurableSpace G\ninst✝² : BorelSpace G\ninst✝¹ : SecondCountableTopology G\nf : α → G\n_m0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\n⊢ AEFinStronglyMeasurable f μ ↔ AEMeasurable f μ"}
+{"state": [{"context": ["α✝ : Type u", "β : Type v", "ι : Type u_1", "π : ι → Type u_2", "inst✝² : TopologicalSpace α✝", "s t u v : Set α✝", "α : Type u_3", "inst✝¹ : TopologicalSpace α", "inst✝ : TotallySeparatedSpace α", "x y : α", "hxy : x ≠ y"], "goal": "∃ U, IsClopen U ∧ x ∈ U ∧ y ∈ Uᶜ"}], "premise": [65551, 131586], "state_str": "α✝ : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝² : TopologicalSpace α✝\ns t u v : Set α✝\nα : Type u_3\ninst✝¹ : TopologicalSpace α\ninst✝ : TotallySeparatedSpace α\nx y : α\nhxy : x ≠ y\n⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∈ Uᶜ"}
+{"state": [{"context": ["α✝ : Type u", "β : Type v", "ι : Type u_1", "π : ι → Type u_2", "inst✝² : TopologicalSpace α✝", "s t u v : Set α✝", "α : Type u_3", "inst✝¹ : TopologicalSpace α", "inst✝ : TotallySeparatedSpace α", "x y : α", "hxy : x ≠ y", "U V : Set α", "hU : IsOpen U", "hV : IsOpen V", "Ux : x ∈ U", "Vy : y ∈ V", "f : univ ⊆ U ∪ V", "disj : Disjoint U V"], "goal": "∃ U, IsClopen U ∧ x ∈ U ∧ y ∈ Uᶜ"}], "premise": [56686, 56700], "state_str": "case intro.intro.intro.intro.intro.intro.intro\nα✝ : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝² : TopologicalSpace α✝\ns t u v : Set α✝\nα : Type u_3\ninst✝¹ : TopologicalSpace α\ninst✝ : TotallySeparatedSpace α\nx y : α\nhxy : x ≠ y\nU V : Set α\nhU : IsOpen U\nhV : IsOpen V\nUx : x ∈ U\nVy : y ∈ V\nf : univ ⊆ U ∪ V\ndisj : Disjoint U V\n⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∈ Uᶜ"}
+{"state": [{"context": ["α✝ : Type u", "β : Type v", "ι : Type u_1", "π : ι → Type u_2", "inst✝² : TopologicalSpace α✝", "s t u v : Set α✝", "α : Type u_3", "inst✝¹ : TopologicalSpace α", "inst✝ : TotallySeparatedSpace α", "x y : α", "hxy : x ≠ y", "U V : Set α", "hU✝ : IsOpen U", "hV : IsOpen V", "Ux : x ∈ U", "Vy : y ∈ V", "f : univ ⊆ U ∪ V", "disj : Disjoint U V", "hU : IsClopen (univ ∩ U)"], "goal": "∃ U, IsClopen U ∧ x ∈ U ∧ y ∈ Uᶜ"}], "premise": [133462], "state_str": "case intro.intro.intro.intro.intro.intro.intro\nα✝ : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝² : TopologicalSpace α✝\ns t u v : Set α✝\nα : Type u_3\ninst✝¹ : TopologicalSpace α\ninst✝ : TotallySeparatedSpace α\nx y : α\nhxy : x ≠ y\nU V : Set α\nhU✝ : IsOpen U\nhV : IsOpen V\nUx : x ∈ U\nVy : y ∈ V\nf : univ ⊆ U ∪ V\ndisj : Disjoint U V\nhU : IsClopen (univ ∩ U)\n⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∈ Uᶜ"}
+{"state": [{"context": ["α✝ : Type u", "β : Type v", "ι : Type u_1", "π : ι → Type u_2", "inst✝² : TopologicalSpace α✝", "s t u v : Set α✝", "α : Type u_3", "inst✝¹ : TopologicalSpace α", "inst✝ : TotallySeparatedSpace α", "x y : α", "hxy : x ≠ y", "U V : Set α", "hU✝ : IsOpen U", "hV : IsOpen V", "Ux : x ∈ U", "Vy : y ∈ V", "f : univ ⊆ U ∪ V", "disj : Disjoint U V", "hU : IsClopen U"], "goal": "∃ U, IsClopen U ∧ x ∈ U ∧ y ∈ Uᶜ"}], "premise": [133618, 133622], "state_str": "case intro.intro.intro.intro.intro.intro.intro\nα✝ : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝² : TopologicalSpace α✝\ns t u v : Set α✝\nα : Type u_3\ninst✝¹ : TopologicalSpace α\ninst✝ : TotallySeparatedSpace α\nx y : α\nhxy : x ≠ y\nU V : Set α\nhU✝ : IsOpen U\nhV : IsOpen V\nUx : x ∈ U\nVy : y ∈ V\nf : univ ⊆ U ∪ V\ndisj : Disjoint U V\nhU : IsClopen U\n⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∈ Uᶜ"}
+{"state": [{"context": ["p : ℕ", "G : Type u_1", "inst✝⁴ : Group G", "hG : IsPGroup p G", "hp : Fact (Nat.Prime p)", "α : Type u_2", "inst✝³ : MulAction G α", "inst✝² : Finite α", "inst✝¹ : Nontrivial G", "inst✝ : Finite G"], "goal": "Nontrivial ↥(Subgroup.center G)"}], "premise": [1673, 1681, 2100, 2106, 2107, 6313, 6319, 6323, 7547, 14284, 108891, 122601, 137128], "state_str": "p : ℕ\nG : Type u_1\ninst✝⁴ : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\nα : Type u_2\ninst✝³ : MulAction G α\ninst✝² : Finite α\ninst✝¹ : Nontrivial G\ninst✝ : Finite G\n⊢ Nontrivial ↥(Subgroup.center G)"}
+{"state": [{"context": ["X : Type u", "Y : Type v", "Z : Type u_1", "W : Type u_2", "ε : Type u_3", "ζ : Type u_4", "inst✝² : TopologicalSpace X", "inst✝¹ : TopologicalSpace Y", "inst✝ : TopologicalSpace Z", "p : X → Prop", "f : X → Y", "hf : QuotientMap f", "s : Set Y", "hs : IsOpen s"], "goal": "QuotientMap (s.restrictPreimage f)"}], "premise": [1674, 56046, 56054], "state_str": "X : Type u\nY : Type v\nZ : Type u_1\nW : Type u_2\nε : Type u_3\nζ : Type u_4\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\np : X → Prop\nf : X → Y\nhf : QuotientMap f\ns : Set Y\nhs : IsOpen s\n⊢ QuotientMap (s.restrictPreimage f)"}
+{"state": [{"context": ["X : Type u", "Y : Type v", "Z : Type u_1", "W : Type u_2", "ε : Type u_3", "ζ : Type u_4", "inst✝² : TopologicalSpace X", "inst✝¹ : TopologicalSpace Y", "inst✝ : TopologicalSpace Z", "p : X → Prop", "f : X → Y", "hf : QuotientMap f", "s : Set Y", "hs : IsOpen s", "U : Set ↑s"], "goal": "IsOpen U ↔ IsOpen (s.restrictPreimage f ⁻¹' U)"}], "premise": [1713, 55616, 56053, 56055, 56094, 66500, 135788], "state_str": "X : Type u\nY : Type v\nZ : Type u_1\nW : Type u_2\nε : Type u_3\nζ : Type u_4\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\np : X → Prop\nf : X → Y\nhf : QuotientMap f\ns : Set Y\nhs : IsOpen s\nU : Set ↑s\n⊢ IsOpen U ↔ IsOpen (s.restrictPreimage f ⁻¹' U)"}
+{"state": [{"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝ : HeytingAlgebra α", "a b c : α"], "goal": "a ≤ aᶜ ↔ a = ⊥"}], "premise": [1713, 13474, 15128], "state_str": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : HeytingAlgebra α\na b c : α\n⊢ a ≤ aᶜ ↔ a = ⊥"}
+{"state": [{"context": ["ι : Type u_1", "X : Type u_2", "Y : Type u_3", "inst✝¹ : EMetricSpace X", "inst✝ : EMetricSpace Y", "m✝ : Set X → ℝ≥0∞", "r : ℝ≥0∞", "μ : OuterMeasure X", "s✝ : Set X", "m : Set X → ℝ≥0∞", "s : Set X"], "goal": "Tendsto (fun n => (pre m (↑n)⁻¹) s) atTop (𝓝 ((mkMetric' m) s))"}], "premise": [1674, 16355, 16368, 30773, 58957], "state_str": "ι : Type u_1\nX : Type u_2\nY : Type u_3\ninst✝¹ : EMetricSpace X\ninst✝ : EMetricSpace Y\nm✝ : Set X → ℝ≥0∞\nr : ℝ≥0∞\nμ : OuterMeasure X\ns✝ : Set X\nm : Set X → ℝ≥0∞\ns : Set X\n⊢ Tendsto (fun n => (pre m (↑n)⁻¹) s) atTop (𝓝 ((mkMetric' m) s))"}
+{"state": [{"context": ["ι : Type u_1", "X : Type u_2", "Y : Type u_3", "inst✝¹ : EMetricSpace X", "inst✝ : EMetricSpace Y", "m✝ : Set X → ℝ≥0∞", "r : ℝ≥0∞", "μ : OuterMeasure X", "s✝ : Set X", "m : Set X → ℝ≥0∞", "s : Set X"], "goal": "Tendsto (fun n => (↑n)⁻¹) atTop (𝓟 (Ioi 0))"}], "premise": [1674, 16021, 16380], "state_str": "ι : Type u_1\nX : Type u_2\nY : Type u_3\ninst✝¹ : EMetricSpace X\ninst✝ : EMetricSpace Y\nm✝ : Set X → ℝ≥0∞\nr : ℝ≥0∞\nμ : OuterMeasure X\ns✝ : Set X\nm : Set X → ℝ≥0∞\ns : Set X\n⊢ Tendsto (fun n => (↑n)⁻¹) atTop (𝓟 (Ioi 0))"}
+{"state": [{"context": ["F : Type u_1", "G : Type u_2", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "inst✝² : NormedAddCommGroup G", "inst✝¹ : NormedSpace ℝ G", "X : Type u_3", "Y : Type u_4", "Z : Type u_5", "inst✝ : TopologicalSpace X", "f g : X → Y", "A : Set X", "x : X", "P : (x : X) → (𝓝 x).Germ Y → Prop"], "goal": "(∀ (x : X), RestrictGermPredicate P A x ↑f) ↔ ∀ᶠ (x : X) in 𝓝ˢ A, P x ↑f"}], "premise": [56927], "state_str": "F : Type u_1\nG : Type u_2\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace ℝ G\nX : Type u_3\nY : Type u_4\nZ : Type u_5\ninst✝ : TopologicalSpace X\nf g : X → Y\nA : Set X\nx : X\nP : (x : X) → (𝓝 x).Germ Y → Prop\n⊢ (∀ (x : X), RestrictGermPredicate P A x ↑f) ↔ ∀ᶠ (x : X) in 𝓝ˢ A, P x ↑f"}
+{"state": [{"context": ["F : Type u_1", "G : Type u_2", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "inst✝² : NormedAddCommGroup G", "inst✝¹ : NormedSpace ℝ G", "X : Type u_3", "Y : Type u_4", "Z : Type u_5", "inst✝ : TopologicalSpace X", "f g : X → Y", "A : Set X", "x : X", "P : (x : X) → (𝓝 x).Germ Y → Prop"], "goal": "(∀ (x : X), RestrictGermPredicate P A x ↑f) ↔ ∀ x ∈ A, ∀ᶠ (y : X) in 𝓝 x, P y ↑f"}], "premise": [1713], "state_str": "F : Type u_1\nG : Type u_2\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace ℝ G\nX : Type u_3\nY : Type u_4\nZ : Type u_5\ninst✝ : TopologicalSpace X\nf g : X → Y\nA : Set X\nx : X\nP : (x : X) → (𝓝 x).Germ Y → Prop\n⊢ (∀ (x : X), RestrictGermPredicate P A x ↑f) ↔ ∀ x ∈ A, ∀ᶠ (y : X) in �� x, P y ↑f"}
+{"state": [{"context": ["α : Type u", "β : Type v", "γ : Type w", "inst✝¹⁰ : ConditionallyCompleteLinearOrder α", "inst✝⁹ : TopologicalSpace α", "inst✝⁸ : OrderTopology α", "inst✝⁷ : ConditionallyCompleteLinearOrder β", "inst✝⁶ : TopologicalSpace β", "inst✝⁵ : OrderTopology β", "inst✝⁴ : Nonempty γ", "inst✝³ : DenselyOrdered α", "a✝ b✝ : α", "δ : Type u_1", "inst✝² : LinearOrder δ", "inst✝¹ : TopologicalSpace δ", "inst✝ : OrderClosedTopology δ", "s : Set α", "hs : s.OrdConnected", "f : α → δ", "hf : ContinuousOn f s", "a b : α", "ha : a ∈ s", "hb : b ∈ s"], "goal": "SurjOn f s (uIcc (f a) (f b))"}], "premise": [14308, 56524], "state_str": "α : Type u\nβ : Type v\nγ : Type w\ninst✝¹⁰ : ConditionallyCompleteLinearOrder α\ninst✝⁹ : TopologicalSpace α\ninst✝⁸ : OrderTopology α\ninst✝⁷ : ConditionallyCompleteLinearOrder β\ninst✝⁶ : TopologicalSpace β\ninst✝⁵ : OrderTopology β\ninst✝⁴ : Nonempty γ\ninst✝³ : DenselyOrdered α\na✝ b✝ : α\nδ : Type u_1\ninst✝² : LinearOrder δ\ninst✝¹ : TopologicalSpace δ\ninst✝ : OrderClosedTopology δ\ns : Set α\nhs : s.OrdConnected\nf : α → δ\nhf : ContinuousOn f s\na b : α\nha : a ∈ s\nhb : b ∈ s\n⊢ SurjOn f s (uIcc (f a) (f b))"}
+{"state": [{"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "inst✝² : Ring k", "inst✝¹ : AddCommGroup V", "inst✝ : Module k V", "S : AffineSpace V P", "ι : Type u_4", "s : Finset ι", "ι₂ : Type u_5", "s₂ : Finset ι₂", "w : ι → k", "p : ι → P"], "goal": "(s.weightedVSub p) w = ∑ i ∈ s, w i • (p i -ᵥ Classical.choice ⋯)"}], "premise": [109864], "state_str": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\nι : Type u_4\ns : Finset ι\nι₂ : Type u_5\ns₂ : Finset ι₂\nw : ι → k\np : ι → P\n⊢ (s.weightedVSub p) w = ∑ i ∈ s, w i • (p i -ᵥ Classical.choice ⋯)"}
+{"state": [{"context": ["α✝ : Type u", "β : Type v", "α : Type u_1", "s : Set (Set α)", "t : Set α", "ht : GenerateOpen s t"], "goal": "generateFrom (insert t s) = generateFrom s"}], "premise": [14296, 57498, 57553, 133481], "state_str": "α✝ : Type u\nβ : Type v\nα : Type u_1\ns : Set (Set α)\nt : Set α\nht : GenerateOpen s t\n⊢ generateFrom (insert t s) = generateFrom s"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "R : Type u_3", "s t✝ u : Finset α", "f✝ : α → β", "n : ℕ", "t : Finset β", "f : α → β", "hf : Set.SurjOn f ↑s ↑t"], "goal": "t.card ≤ s.card"}], "premise": [137614, 137653], "state_str": "α : Type u_1\nβ : Type u_2\nR : Type u_3\ns t✝ u : Finset α\nf✝ : α → β\nn : ℕ\nt : Finset β\nf : α → β\nhf : Set.SurjOn f ↑s ↑t\n⊢ t.card ≤ s.card"}
+{"state": [{"context": ["P : MorphismProperty Scheme", "Q : AffineTargetMorphismProperty", "inst✝² : HasAffineProperty P Q", "X Y : Scheme", "f : X ⟶ Y", "𝒰 : Y.OpenCover", "inst✝¹ : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)", "𝒰' : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover", "inst✝ : ∀ (i : 𝒰.J) (j : (𝒰' i).J), IsAffine ((𝒰' i).obj j)", "h𝒰' : ∀ (i : 𝒰.J) (j k : (𝒰' i).J), Q (pullback.mapDesc ((𝒰' i).map j) ((𝒰' i).map k) (𝒰.pullbackHom f i))"], "goal": "P.diagonal f"}], "premise": [126493], "state_str": "P : MorphismProperty Scheme\nQ : AffineTargetMorphismProperty\ninst✝² : HasAffineProperty P Q\nX Y : Scheme\nf : X ⟶ Y\n𝒰 : Y.OpenCover\ninst✝¹ : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)\n𝒰' : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover\ninst✝ : ∀ (i : 𝒰.J) (j : (𝒰' i).J), IsAffine ((𝒰' i).obj j)\nh𝒰' : ∀ (i : 𝒰.J) (j k : (𝒰' i).J), Q (pullback.mapDesc ((𝒰' i).map j) ((𝒰' i).map k) (𝒰.pullbackHom f i))\n⊢ P.diagonal f"}
+{"state": [{"context": ["P : MorphismProperty Scheme", "Q : AffineTargetMorphismProperty", "inst✝² : HasAffineProperty P Q", "X Y : Scheme", "f : X ⟶ Y", "𝒰 : Y.OpenCover", "inst✝¹ : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)", "𝒰' : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover", "inst✝ : ∀ (i : 𝒰.J) (j : (𝒰' i).J), IsAffine ((𝒰' i).obj j)", "h𝒰' : ∀ (i : 𝒰.J) (j k : (𝒰' i).J), Q (pullback.mapDesc ((𝒰' i).map j) ((𝒰' i).map k) (𝒰.pullbackHom f i))", "this : Q.IsLocal := isLocal_affineProperty P", "𝒱 : (pullback f f).OpenCover := (Scheme.Pullback.openCoverOfBase 𝒰 f f).bind fun i => Scheme.Pullback.openCoverOfLeftRight (𝒰' i) (𝒰' i) (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i))", "i1 : ∀ (i : 𝒱.J), IsAffine (𝒱.obj i)"], "goal": "P.diagonal f"}], "premise": [126501], "state_str": "P : MorphismProperty Scheme\nQ : AffineTargetMorphismProperty\ninst✝² : HasAffineProperty P Q\nX Y : Scheme\nf : X ⟶ Y\n𝒰 : Y.OpenCover\ninst✝¹ : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)\n𝒰' : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover\ninst✝ : ∀ (i : 𝒰.J) (j : (𝒰' i).J), IsAffine ((𝒰' i).obj j)\nh𝒰' : ∀ (i : 𝒰.J) (j k : (𝒰' i).J), Q (pullback.mapDesc ((𝒰' i).map j) ((𝒰' i).map k) (𝒰.pullbackHom f i))\nthis : Q.IsLocal := isLocal_affineProperty P\n𝒱 : (pullback f f).OpenCover :=\n  (Scheme.Pullback.openCoverOfBase 𝒰 f f).bind fun i =>\n    Scheme.Pullback.openCoverOfLeftRight (𝒰' i) (𝒰' i) (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i))\ni1 : ∀ (i : 𝒱.J), IsAffine (𝒱.obj i)\n⊢ P.diagonal f"}
+{"state": [{"context": ["P : MorphismProperty Scheme", "Q : AffineTargetMorphismProperty", "inst✝² : HasAffineProperty P Q", "X Y : Scheme", "f : X ⟶ Y", "𝒰 : Y.OpenCover", "inst✝¹ : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)", "𝒰' : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover", "inst✝ : ∀ (i : 𝒰.J) (j : (𝒰' i).J), IsAffine ((𝒰' i).obj j)", "h𝒰' : ∀ (i : 𝒰.J) (j k : (𝒰' i).J), Q (pullback.mapDesc ((𝒰' i).map j) ((𝒰' i).map k) (𝒰.pullbackHom f i))", "this : Q.IsLocal := isLocal_affineProperty P", "𝒱 : (pullback f f).OpenCover := (Scheme.Pullback.openCoverOfBase 𝒰 f f).bind fun i => Scheme.Pullback.openCoverOfLeftRight (𝒰' i) (𝒰' i) (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i))", "i1 : ∀ (i : 𝒱.J), IsAffine (𝒱.obj i)", "i : (Scheme.Pullback.openCoverOfBase 𝒰 f f).J", "j k : (𝒰' i).J"], "goal": "Q (((Scheme.Pullback.openCoverOfBase 𝒰 f f).bind fun i => Scheme.Pullback.openCoverOfLeftRight (𝒰' i) (𝒰' i) (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i))).pullbackHom (pullback.diagonal f) ⟨i, (j, k)⟩)"}], "premise": [1673, 126483], "state_str": "case mk.mk\nP : MorphismProperty Scheme\nQ : AffineTargetMorphismProperty\ninst✝² : HasAffineProperty P Q\nX Y : Scheme\nf : X ⟶ Y\n𝒰 : Y.OpenCover\ninst✝¹ : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)\n𝒰' : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover\ninst✝ : ∀ (i : 𝒰.J) (j : (𝒰' i).J), IsAffine ((𝒰' i).obj j)\nh𝒰' : ∀ (i : 𝒰.J) (j k : (𝒰' i).J), Q (pullback.mapDesc ((𝒰' i).map j) ((𝒰' i).map k) (𝒰.pullbackHom f i))\nthis : Q.IsLocal := isLocal_affineProperty P\n𝒱 : (pullback f f).OpenCover :=\n  (Scheme.Pullback.openCoverOfBase 𝒰 f f).bind fun i =>\n    Scheme.Pullback.openCoverOfLeftRight (𝒰' i) (𝒰' i) (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i))\ni1 : ∀ (i : 𝒱.J), IsAffine (𝒱.obj i)\ni : (Scheme.Pullback.openCoverOfBase 𝒰 f f).J\nj k : (𝒰' i).J\n⊢ Q\n    (((Scheme.Pullback.openCoverOfBase 𝒰 f f).bind fun i =>\n          Scheme.Pullback.openCoverOfLeftRight (𝒰' i) (𝒰' i) (pullback.snd f (𝒰.map i))\n            (pullback.snd f (𝒰.map i))).pullbackHom\n      (pullback.diagonal f) ⟨i, (j, k)⟩)"}
+{"state": [{"context": ["X : Scheme", "r : ↑Γ(X, ⊤)"], "goal": "adjunction.unit.app X ⁻¹ᵁ basicOpen r = X.basicOpen r"}], "premise": [126625], "state_str": "X : Scheme\nr : ↑Γ(X, ⊤)\n⊢ adjunction.unit.app X ⁻¹ᵁ basicOpen r = X.basicOpen r"}
+{"state": [{"context": ["X : Scheme", "r : ↑Γ(X, ⊤)"], "goal": "adjunction.unit.app X ⁻¹ᵁ (Spec Γ(X, ⊤)).basicOpen ((Scheme.ΓSpecIso Γ(X, ⊤)).inv r) = X.basicOpen r"}], "premise": [126614], "state_str": "X : Scheme\nr : ↑Γ(X, ⊤)\n⊢ adjunction.unit.app X ⁻¹ᵁ (Spec Γ(X, ⊤)).basicOpen ((Scheme.ΓSpecIso Γ(X, ⊤)).inv r) = X.basicOpen r"}
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+{"state": [{"context": ["q : ℚ", "x y : ℝ", "h : Irrational (x ^ 0)"], "goal": "Irrational x"}], "premise": [119739], "state_str": "q : ℚ\nx y : ℝ\nh : Irrational (x ^ 0)\n⊢ Irrational x"}
+{"state": [{"context": ["q : ℚ", "x y : ℝ", "h : Irrational 1"], "goal": "Irrational x"}], "premise": [148036], "state_str": "q : ℚ\nx y : ℝ\nh : Irrational 1\n⊢ Irrational x"}
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+{"state": [{"context": ["q : ℚ", "x y : ℝ", "n : ℕ", "h : Irrational (x ^ n * x)"], "goal": "Irrational x"}], "premise": [2110, 145727], "state_str": "q : ℚ\nx y : ℝ\nn : ℕ\nh : Irrational (x ^ n * x)\n⊢ Irrational x"}
+{"state": [{"context": ["q : ℚ", "x y : ℝ", "h : Irrational x"], "goal": "Irrational (↑q - x)"}], "premise": [119789, 145702, 145714], "state_str": "q : ℚ\nx y : ℝ\nh : Irrational x\n⊢ Irrational (↑q - x)"}
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+{"state": [{"context": ["α : Type u_1", "mα : MeasurableSpace α", "μ✝ μ : Measure α", "inst✝ : SFinite μ", "s : Set α", "hs : MeasurableSet s", "hs0 : ∀ (i : ℕ), (sFiniteSeq μ i) s = 0"], "goal": "μ.toFinite s = 0"}], "premise": [30478, 30479], "state_str": "α : Type u_1\nmα : MeasurableSpace α\nμ✝ μ : Measure α\ninst✝ : SFinite μ\ns : Set α\nhs : MeasurableSet s\nhs0 : ∀ (i : ℕ), (sFiniteSeq μ i) s = 0\n⊢ μ.toFinite s = 0"}
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+{"state": [{"context": ["l m✝ n✝ : ℕ", "α : Type u_1", "β : Type u_2", "m n : ℕ", "A : Matrix (Fin m) (Fin n) α"], "goal": "((Equiv.refl (Fin m → Fin n → α)) fun i j => A i j) = A"}], "premise": [70748], "state_str": "l m✝ n✝ : ℕ\nα : Type u_1\nβ : Type u_2\nm n : ℕ\nA : Matrix (Fin m) (Fin n) α\n⊢ ((Equiv.refl (Fin m → Fin n → α)) fun i j => A i j) = A"}
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+{"state": [{"context": ["R : Type u_1", "inst✝⁵ : StrictOrderedCommSemiring R", "M : Type u_2", "inst✝⁴ : AddCommMonoid M", "inst✝³ : Module R M", "N : Type u_3", "inst✝² : AddCommMonoid N", "inst✝¹ : Module R N", "ι : Type u_4", "inst✝ : DecidableEq ι", "u : Rˣ", "hu : 0 < ↑u", "v✝ : M", "hv✝ : v✝ ≠ 0"], "goal": "SameRay R (u • v✝) v✝"}], "premise": [82998], "state_str": "case h\nR : Type u_1\ninst✝⁵ : StrictOrderedCommSemiring R\nM : Type u_2\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nN : Type u_3\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nι : Type u_4\ninst✝ : DecidableEq ι\nu : Rˣ\nhu : 0 < ↑u\nv✝ : M\nhv✝ : v✝ ≠ 0\n⊢ SameRay R (u • v✝) v✝"}
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+{"state": [{"context": ["K : Type u_1", "E : Type u_2", "inst✝ : RCLike K", "z : K"], "goal": "↑(re z) + ↑(im z) * I + (↑(re z) - ↑(im z) * I) = 2 * ↑(re z)"}], "premise": [118084, 122222], "state_str": "K : Type u_1\nE : Type u_2\ninst✝ : RCLike K\nz : K\n⊢ ↑(re z) + ↑(im z) * I + (↑(re z) - ↑(im z) * I) = 2 * ↑(re z)"}
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+{"state": [{"context": ["α : Type u_1", "inst✝⁶ : MeasurableSpace α", "μ : Measure α", "𝕜 : Type u_2", "inst✝⁵ : RCLike 𝕜", "E : Type u_3", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : NormedSpace ℝ E", "inst✝² : NormedSpace 𝕜 E", "H : Type u_4", "inst✝¹ : NormedAddCommGroup H", "inst✝ : NormedSpace 𝕜 H", "F : H → α → E", "x₀ : H", "bound : α → ℝ", "ε : ℝ", "F' : α → H →L[𝕜] E", "ε_pos : 0 < ε", "hF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ", "hF_int : Integrable (F x₀) μ", "hF'_meas : AEStronglyMeasurable F' μ", "h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖", "bound_integrable : Integrable bound μ", "h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀", "x₀_in : x₀ ∈ ball x₀ ε", "nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹", "b : α → ℝ := fun a => |bound a|"], "goal": "Integrable F' μ ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀"}], "premise": [28502], "state_str": "α : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\n⊢ Integrable F' μ ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀"}
+{"state": [{"context": ["α : Type u_1", "inst✝⁶ : MeasurableSpace α", "μ : Measure α", "𝕜 : Type u_2", "inst✝⁵ : RCLike 𝕜", "E : Type u_3", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : NormedSpace ℝ E", "inst✝² : NormedSpace 𝕜 E", "H : Type u_4", "inst✝¹ : NormedAddCommGroup H", "inst✝ : NormedSpace 𝕜 H", "F : H → α → E", "x₀ : H", "bound : α → ℝ", "ε : ℝ", "F' : α → H →L[𝕜] E", "ε_pos : 0 < ε", "hF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ", "hF_int : Integrable (F x₀) μ", "hF'_meas : AEStronglyMeasurable F' μ", "h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖", "bound_integrable : Integrable bound μ", "h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀", "x₀_in : x₀ ∈ ball x₀ ε", "nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹", "b : α → ℝ := fun a => |bound a|", "b_int : Integrable b μ"], "goal": "Integrable F' μ ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀"}], "premise": [105294], "state_str": "α : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\nb_int : Integrable b μ\n⊢ Integrable F' μ ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀"}
+{"state": [{"context": ["α : Type u_1", "inst✝⁶ : MeasurableSpace α", "μ : Measure α", "𝕜 : Type u_2", "inst✝⁵ : RCLike 𝕜", "E : Type u_3", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : NormedSpace ℝ E", "inst✝² : NormedSpace 𝕜 E", "H : Type u_4", "inst✝¹ : NormedAddCommGroup H", "inst✝ : NormedSpace 𝕜 H", "F : H → α → E", "x₀ : H", "bound : α → ℝ", "ε : ℝ", "F' : α → H →L[𝕜] E", "ε_pos : 0 < ε", "hF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ", "hF_int : Integrable (F x₀) μ", "hF'_meas : AEStronglyMeasurable F' μ", "h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖", "bound_integrable : Integrable bound μ", "h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀", "x₀_in : x₀ ∈ ball x₀ ε", "nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹", "b : α → ℝ := fun a => |bound a|", "b_int : Integrable b μ", "b_nonneg : ∀ (a : α), 0 ≤ b a"], "goal": "Integrable F' μ ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀"}], "premise": [16027, 42680, 102622, 105272], "state_str": "α : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\nb_int : Integrable b μ\nb_nonneg : ∀ (a : α), 0 ≤ b a\n⊢ Integrable F' μ ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀"}
+{"state": [{"context": ["α : Type u_1", "inst✝⁶ : MeasurableSpace α", "μ : Measure α", "𝕜 : Type u_2", "inst✝⁵ : RCLike 𝕜", "E : Type u_3", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : NormedSpace ℝ E", "inst✝² : NormedSpace 𝕜 E", "H : Type u_4", "inst✝¹ : NormedAddCommGroup H", "inst✝ : NormedSpace 𝕜 H", "F : H → α → E", "x₀ : H", "bound : α → ℝ", "ε : ℝ", "F' : α → H →L[𝕜] E", "ε_pos : 0 < ε", "hF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ", "hF_int : Integrable (F x₀) μ", "hF'_meas : AEStronglyMeasurable F' μ", "bound_integrable : Integrable bound μ", "h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀", "x₀_in : x₀ ∈ ball x₀ ε", "nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹", "b : α → ℝ := fun a => |bound a|", "b_int : Integrable b μ", "b_nonneg : ∀ (a : α), 0 ≤ b a", "h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖"], "goal": "Integrable F' μ ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀"}], "premise": [16027, 28502, 28510, 28544, 42665, 61157, 102621, 119707], "state_str": "α : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\nb_int : Integrable b μ\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\n⊢ Integrable F' μ ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀"}
+{"state": [{"context": ["α : Type u_1", "inst✝⁶ : MeasurableSpace α", "μ : Measure α", "𝕜 : Type u_2", "inst✝⁵ : RCLike 𝕜", "E : Type u_3", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : NormedSpace ℝ E", "inst✝² : NormedSpace 𝕜 E", "H : Type u_4", "inst✝¹ : NormedAddCommGroup H", "inst✝ : NormedSpace 𝕜 H", "F : H → α → E", "x₀ : H", "bound : α → ℝ", "ε : ℝ", "F' : α → H →L[𝕜] E", "ε_pos : 0 < ε", "hF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ", "hF_int : Integrable (F x₀) μ", "hF'_meas : AEStronglyMeasurable F' μ", "bound_integrable : Integrable bound μ", "h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀", "x₀_in : x₀ ∈ ball x₀ ε", "nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹", "b : α → ℝ := fun a => |bound a|", "b_int : Integrable b μ", "b_nonneg : ∀ (a : α), 0 ≤ b a", "h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖", "hF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ"], "goal": "Integrable F' μ ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀"}], "premise": [15884, 16019, 16027, 28454, 46304, 61259], "state_str": "α : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\nb_int : Integrable b μ\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ\n⊢ Integrable F' μ ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀"}
+{"state": [{"context": ["α : Type u_1", "inst✝⁶ : MeasurableSpace α", "μ : Measure α", "𝕜 : Type u_2", "inst✝⁵ : RCLike 𝕜", "E : Type u_3", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : NormedSpace ℝ E", "inst✝² : NormedSpace 𝕜 E", "H : Type u_4", "inst✝¹ : NormedAddCommGroup H", "inst✝ : NormedSpace 𝕜 H", "F : H → α → E", "x₀ : H", "bound : α → ℝ", "ε : ℝ", "F' : α → H →L[𝕜] E", "ε_pos : 0 < ε", "hF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ", "hF_int : Integrable (F x₀) μ", "hF'_meas : AEStronglyMeasurable F' μ", "bound_integrable : Integrable bound μ", "h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀", "x₀_in : x₀ ∈ ball x₀ ε", "nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹", "b : α → ℝ := fun a => |bound a|", "b_int : Integrable b μ", "b_nonneg : ∀ (a : α), 0 ≤ b a", "h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖", "hF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ", "hF'_int : Integrable F' μ", "hE : CompleteSpace E"], "goal": "HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀"}], "premise": [61259], "state_str": "case pos\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\nb_int : Integrable b μ\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ\nhF'_int : Integrable F' μ\nhE : CompleteSpace E\n⊢ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀"}
+{"state": [{"context": ["α : Type u_1", "inst✝⁶ : MeasurableSpace α", "μ : Measure α", "𝕜 : Type u_2", "inst✝⁵ : RCLike 𝕜", "E : Type u_3", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : NormedSpace ℝ E", "inst✝² : NormedSpace 𝕜 E", "H : Type u_4", "inst✝¹ : NormedAddCommGroup H", "inst✝ : NormedSpace 𝕜 H", "F : H → α → E", "x₀ : H", "bound : α → ℝ", "ε : ℝ", "F' : α → H →L[𝕜] E", "ε_pos : 0 < ε", "hF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ", "hF_int : Integrable (F x₀) μ", "hF'_meas : AEStronglyMeasurable F' μ", "bound_integrable : Integrable bound μ", "h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀", "x₀_in : x₀ ∈ ball x₀ ε", "nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹", "b : α → ℝ := fun a => |bound a|", "b_int : Integrable b μ", "b_nonneg : ∀ (a : α), 0 ≤ b a", "h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀���", "hF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ", "hF'_int : Integrable F' μ", "hE : CompleteSpace E", "h_ball : ball x₀ ε ∈ 𝓝 x₀"], "goal": "HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀"}], "premise": [15884, 28372, 28501, 28600, 33646, 33648, 40548, 61259, 131585], "state_str": "case pos\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\nb_int : Integrable b μ\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ\nhF'_int : Integrable F' μ\nhE : CompleteSpace E\nh_ball : ball x₀ ε ∈ 𝓝 x₀\n⊢ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀"}
+{"state": [{"context": ["α : Type u_1", "inst✝⁶ : MeasurableSpace α", "μ : Measure α", "𝕜 : Type u_2", "inst✝⁵ : RCLike 𝕜", "E : Type u_3", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : NormedSpace ℝ E", "inst✝² : NormedSpace 𝕜 E", "H : Type u_4", "inst✝¹ : NormedAddCommGroup H", "inst✝ : NormedSpace 𝕜 H", "F : H → α → E", "x₀ : H", "bound : α → ℝ", "ε : ℝ", "F' : α → H →L[𝕜] E", "ε_pos : 0 < ε", "hF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ", "hF_int : Integrable (F x₀) μ", "hF'_meas : AEStronglyMeasurable F' μ", "bound_integrable : Integrable bound μ", "h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀", "x₀_in : x₀ ∈ ball x₀ ε", "nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹", "b : α → ℝ := fun a => |bound a|", "b_int : Integrable b μ", "b_nonneg : ∀ (a : α), 0 ≤ b a", "h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖", "hF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ", "hF'_int : Integrable F' μ", "hE : CompleteSpace E", "h_ball : ball x₀ ε ∈ 𝓝 x₀", "this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - (∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - (F' a) (x - x₀)) ∂μ‖"], "goal": "HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀"}], "premise": [16349, 42797, 46302], "state_str": "case pos\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\nb_int : Integrable b μ\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ\nhF'_int : Integrable F' μ\nhE : CompleteSpace E\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n  ∀ᶠ (x : H) in 𝓝 x₀,\n    ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - (∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - (F' a) (x - x₀)) ∂μ‖\n⊢ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀"}
+{"state": [{"context": ["α : Type u_1", "inst✝⁶ : MeasurableSpace α", "μ : Measure α", "𝕜 : Type u_2", "inst✝⁵ : RCLike 𝕜", "E : Type u_3", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : NormedSpace ℝ E", "inst✝² : NormedSpace 𝕜 E", "H : Type u_4", "inst✝¹ : NormedAddCommGroup H", "inst✝ : NormedSpace 𝕜 H", "F : H → α → E", "x₀ : H", "bound : α → ℝ", "ε : ℝ", "F' : α → H →L[𝕜] E", "ε_pos : 0 < ε", "hF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ", "hF_int : Integrable (F x₀) μ", "hF'_meas : AEStronglyMeasurable F' μ", "bound_integrable : Integrable bound μ", "h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀", "x₀_in : x₀ ∈ ball x₀ ε", "nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹", "b : α → ℝ := fun a => |bound a|", "b_int : Integrable b μ", "b_nonneg : ∀ (a : α), 0 ≤ b a", "h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖", "hF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ", "hF'_int : Integrable F' μ", "hE : CompleteSpace E", "h_ball : ball x₀ ε ∈ 𝓝 x₀", "this : ∀ᶠ (x : H) in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - (∫ (a : α), F' a ∂μ) (x - x₀)‖ = ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - (F' a) (x - x₀)) ∂μ‖"], "goal": "Tendsto (fun x => ∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - (F' a) (x - x₀)) ∂μ) (𝓝 x₀) (𝓝 (∫ (a : α), ‖x₀ - x₀‖⁻¹ • (F x₀ a - F x₀ a - (F' a) (x₀ - x₀)) ∂μ))"}], "premise": [27173], "state_str": "case pos\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nF' : α → H →L[𝕜] E\nε_pos : 0 < ε\nhF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀) μ\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound μ\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => |bound a|\nb_int : Integrable b μ\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ\nhF'_int : Integrable F' μ\nhE : CompleteSpace E\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n  ∀ᶠ (x : H) in 𝓝 x₀,\n    ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - (∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - (F' a) (x - x₀)) ∂μ‖\n⊢ Tendsto (fun x => ∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - (F' a) (x - x₀)) ∂μ) (𝓝 x₀)\n    (𝓝 (∫ (a : α), ‖x₀ - x₀‖⁻¹ • (F x₀ a - F x₀ a - (F' a) (x₀ - x₀)) ∂μ))"}
+{"state": [{"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "X Y Z : C", "D : Type u₂", "inst✝³ : Category.{v₂, u₂} D", "inst✝² : WellPowered C", "inst✝¹ : HasCoproducts C", "inst✝ : HasImages C", "A : C", "s : Set (Subobject A)", "f : Subobject A", "hf : f ∈ s"], "goal": "f ≤ sSup s"}], "premise": [89263], "state_str": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nX Y Z : C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\ninst✝² : WellPowered C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nhf : f ∈ s\n⊢ f ≤ sSup s"}
+{"state": [{"context": ["A : Type u_1", "M : Type u_2", "inst✝¹ : AddGroup A", "inst✝ : DivisionMonoid M", "ψ : AddChar A M", "a : A"], "goal": "ψ (-a) = (ψ a)⁻¹"}], "premise": [119814], "state_str": "A : Type u_1\nM : Type u_2\ninst✝¹ : AddGroup A\ninst✝ : DivisionMonoid M\nψ : AddChar A M\na : A\n⊢ ψ (-a) = (ψ a)⁻¹"}
+{"state": [{"context": ["A : Type u_1", "M : Type u_2", "inst✝¹ : AddGroup A", "inst✝ : DivisionMonoid M", "ψ : AddChar A M", "a : A"], "goal": "ψ (-a) * ψ a = 1"}], "premise": [119817, 120318, 120319], "state_str": "case h\nA : Type u_1\nM : Type u_2\ninst✝¹ : AddGroup A\ninst✝ : DivisionMonoid M\nψ : AddChar A M\na : A\n⊢ ψ (-a) * ψ a = 1"}
+{"state": [{"context": ["α : Type u_1", "inst✝¹ : PartialOrder α", "inst✝ : SuccOrder α", "a b : α", "C : α → Sort u_2", "hs : (a : α) → ¬IsMax a → C (succ a)", "hl : (a : α) → IsSuccLimit a → C a", "hb : IsSuccLimit b"], "goal": "isSuccLimitRecOn b hs hl = hl b hb"}], "premise": [1739], "state_str": "α : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : SuccOrder α\na b : α\nC : α → Sort u_2\nhs : (a : α) → ¬IsMax a → C (succ a)\nhl : (a : α) → IsSuccLimit a → C a\nhb : IsSuccLimit b\n⊢ isSuccLimitRecOn b hs hl = hl b hb"}
+{"state": [{"context": ["M : Type u_1", "inst✝² : CommMonoid M", "S : Submonoid M", "N : Type u_2", "inst✝¹ : CommMonoid N", "P : Type u_3", "inst✝ : CommMonoid P", "f : S.LocalizationMap N", "x y : M", "c : ↥S", "h : f.toMap (x * ↑c) = f.toMap (y * ↑c)"], "goal": "f.toMap (↑c * x) = f.toMap (↑c * y)"}], "premise": [119707], "state_str": "M : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type u_3\ninst✝ : CommMonoid P\nf : S.LocalizationMap N\nx y : M\nc : ↥S\nh : f.toMap (x * ↑c) = f.toMap (y * ↑c)\n⊢ f.toMap (↑c * x) = f.toMap (↑c * y)"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m : MeasurableSpace α", "μ ν : Measure α", "inst✝² : MeasurableSpace δ", "inst✝¹ : NormedAddCommGroup β", "inst✝ : NormedAddCommGroup γ", "f : α → β", "c : ℝ≥0∞", "h₁ : c ≠ 0", "h₂ : c ≠ ⊤", "h : Integrable f (c • μ)"], "goal": "Integrable f μ"}], "premise": [1674, 28472, 118909, 118910, 143757, 143764], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → β\nc : ℝ≥0∞\nh₁ : c ≠ 0\nh₂ : c ≠ ⊤\nh : Integrable f (c • μ)\n⊢ Integrable f μ"}
+{"state": [{"context": ["R : Type u_1", "inst✝² : CommRing R", "inst✝¹ : IsDomain R", "inst✝ : NormalizedGCDMonoid R", "p : R[X]"], "goal": "p.content = 0 ↔ p = 0"}], "premise": [125056], "state_str": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np : R[X]\n⊢ p.content = 0 ↔ p = 0"}
+{"state": [{"context": ["ι : Type u_1", "R : Type u_2", "inst✝ : CommSemiring R", "t : Finset ι", "h : t.Nonempty", "I : ι → Ideal R", "this : DecidableEq ι"], "goal": "⨆ i ∈ t, ⨅ j ∈ t, ⨅ (_ : j ≠ i), I j = ⊤ ↔ (↑t).Pairwise fun i j => I i ⊔ I j = ⊤"}], "premise": [80343, 86663], "state_str": "ι : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nt : Finset ι\nh : t.Nonempty\nI : ι → Ideal R\nthis : DecidableEq ι\n⊢ ⨆ i ∈ t, ⨅ j ∈ t, ⨅ (_ : j ≠ i), I j = ⊤ ↔ (↑t).Pairwise fun i j => I i ⊔ I j = ⊤"}
+{"state": [{"context": ["ι : Type u_1", "R : Type u_2", "inst✝ : CommSemiring R", "t : Finset ι", "h : t.Nonempty", "I : ι → Ideal R", "this : DecidableEq ι"], "goal": "(∃ μ, ∑ i ∈ t, ↑(μ i) = 1) ↔ (↑t).Pairwise fun i j => I i ⊔ I j = ⊤"}], "premise": [138849], "state_str": "ι : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nt : Finset ι\nh : t.Nonempty\nI : ι → Ideal R\nthis : DecidableEq ι\n⊢ (∃ μ, ∑ i ∈ t, ↑(μ i) = 1) ↔ (↑t).Pairwise fun i j => I i ⊔ I j = ⊤"}
+{"state": [{"context": ["ι : Type u_1", "R : Type u_2", "inst✝ : CommSemiring R", "I : ι → Ideal R", "this : DecidableEq ι", "a : ι", "t : Finset ι", "hat : a ∉ t", "h : t.Nonempty", "ih : (∃ μ, ∑ i ∈ t, ↑(μ i) = 1) ↔ (↑t).Pairwise fun i j => I i ⊔ I j = ⊤"], "goal": "(∃ μ, ∑ i ∈ Finset.cons a t hat, ↑(μ i) = 1) ↔ (↑(Finset.cons a t hat)).Pairwise fun i j => I i ⊔ I j = ⊤"}], "premise": [2101, 14537, 133102, 138781], "state_str": "case refine_2\nι : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nI : ι → Ideal R\nthis : DecidableEq ι\na : ι\nt : Finset ι\nhat : a ∉ t\nh : t.Nonempty\nih : (∃ μ, ∑ i ∈ t, ↑(μ i) = 1) ↔ (↑t).Pairwise fun i j => I i ⊔ I j = ⊤\n⊢ (∃ μ, ∑ i ∈ Finset.cons a t hat, ↑(μ i) = 1) ↔ (↑(Finset.cons a t hat)).Pairwise fun i j => I i ⊔ I j = ⊤"}
+{"state": [{"context": ["k : Type u_1", "inst✝³ : Field k", "K : Type u_2", "inst✝² : Field K", "F : Type u_3", "inst✝¹ : Field F", "inst✝ : NumberField K"], "goal": "card { φ // ¬ComplexEmbedding.IsReal φ } = 2 * NrComplexPlaces K"}], "premise": [118863, 119730, 125068, 126990, 127179, 127247], "state_str": "k : Type u_1\ninst✝³ : Field k\nK : Type u_2\ninst✝² : Field K\nF : Type u_3\ninst✝¹ : Field F\ninst✝ : NumberField K\n⊢ card { φ // ¬ComplexEmbedding.IsReal φ } = 2 * NrComplexPlaces K"}
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+{"state": [{"context": ["xl xr : Type u", "x x' : PGame"], "goal": "x.insertRight x' ≤ x"}], "premise": [50295, 50362], "state_str": "xl xr : Type u\nx x' : PGame\n⊢ x.insertRight x' ≤ x"}
+{"state": [{"context": ["xl xr : Type u", "x x' : PGame"], "goal": "-x ≤ (-x).insertLeft (-x')"}], "premise": [50359], "state_str": "xl xr : Type u\nx x' : PGame\n⊢ -x ≤ (-x).insertLeft (-x')"}
+{"state": [{"context": ["R : Type u_1", "inst✝⁴ : CommSemiring R", "M : Submonoid R", "S : Type u_2", "inst✝³ : CommSemiring S", "inst✝² : Algebra R S", "P✝ : Type u_3", "inst✝¹ : CommSemiring P✝", "P : Ideal R", "hp : P.IsPrime", "inst✝ : IsLocalization.AtPrime S P", "this : _root_.Nontrivial S := Nontrivial S P", "x y : S", "hx : x ∈ nonunits S", "hy : y ∈ nonunits S", "hu : IsUnit (x + y)"], "goal": "False"}], "premise": [1673, 120505], "state_str": "R : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP✝ : Type u_3\ninst✝¹ : CommSemiring P✝\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nthis : _root_.Nontrivial S := Nontrivial S P\nx y : S\nhx : x ∈ nonunits S\nhy : y ∈ nonunits S\nhu : IsUnit (x + y)\n⊢ False"}
+{"state": [{"context": ["R : Type u_1", "inst✝⁴ : CommSemiring R", "M : Submonoid R", "S : Type u_2", "inst✝³ : CommSemiring S", "inst✝² : Algebra R S", "P✝ : Type u_3", "inst✝¹ : CommSemiring P✝", "P : Ideal R", "hp : P.IsPrime", "inst✝ : IsLocalization.AtPrime S P", "this : _root_.Nontrivial S := Nontrivial S P", "x y : S", "hx : x ∈ nonunits S", "hy : y ∈ nonunits S", "hu : IsUnit (x + y)", "z : S", "hxyz : (x + y) * z = 1"], "goal": "False"}], "premise": [1673, 1674, 70047, 77621, 120505], "state_str": "case intro\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP✝ : Type u_3\ninst✝¹ : CommSemiring P✝\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nthis : _root_.Nontrivial S := Nontrivial S P\nx y : S\nhx : x ∈ nonunits S\nhy : y ∈ nonunits S\nhu : IsUnit (x + y)\nz : S\nhxyz : (x + y) * z = 1\n⊢ False"}
+{"state": [{"context": ["R : Type u_1", "inst✝⁴ : CommSemiring R", "M : Submonoid R", "S : Type u_2", "inst✝³ : CommSemiring S", "inst✝² : Algebra R S", "P✝ : Type u_3", "inst✝¹ : CommSemiring P✝", "P : Ideal R", "hp : P.IsPrime", "inst✝ : IsLocalization.AtPrime S P", "this✝ : _root_.Nontrivial S := Nontrivial S P", "x y : S", "hx : x ∈ nonunits S", "hy : y ∈ nonunits S", "hu : IsUnit (x + y)", "z : S", "hxyz : (x + y) * z = 1", "this : ∀ {r : R} {s : ↥P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P"], "goal": "False"}], "premise": [77600], "state_str": "case intro\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP✝ : Type u_3\ninst✝¹ : CommSemiring P✝\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nthis✝ : _root_.Nontrivial S := Nontrivial S P\nx y : S\nhx : x ∈ nonunits S\nhy : y ∈ nonunits S\nhu : IsUnit (x + y)\nz : S\nhxyz : (x + y) * z = 1\nthis : ∀ {r : R} {s : ↥P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P\n⊢ False"}
+{"state": [{"context": ["R : Type u_1", "inst✝⁴ : CommSemiring R", "M : Submonoid R", "S : Type u_2", "inst✝³ : CommSemiring S", "inst✝² : Algebra R S", "P✝ : Type u_3", "inst✝¹ : CommSemiring P✝", "P : Ideal R", "hp : P.IsPrime", "inst✝ : IsLocalization.AtPrime S P", "this✝ : _root_.Nontrivial S := Nontrivial S P", "x y : S", "hx : x ∈ nonunits S", "hy : y ∈ nonunits S", "hu : IsUnit (x + y)", "z : S", "hxyz : (x + y) * z = 1", "this : ∀ {r : R} {s : ↥P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P", "rx : R", "sx : ↥P.primeCompl", "hrx : mk' S rx sx = x"], "goal": "False"}], "premise": [77600], "state_str": "case intro.intro.intro\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP✝ : Type u_3\ninst✝¹ : CommSemiring P✝\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nthis✝ : _root_.Nontrivial S := Nontrivial S P\nx y : S\nhx : x ∈ nonunits S\nhy : y ∈ nonunits S\nhu : IsUnit (x + y)\nz : S\nhxyz : (x + y) * z = 1\nthis : ∀ {r : R} {s : ↥P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P\nrx : R\nsx : ↥P.primeCompl\nhrx : mk' S rx sx = x\n⊢ False"}
+{"state": [{"context": ["R : Type u_1", "inst✝⁴ : CommSemiring R", "M : Submonoid R", "S : Type u_2", "inst✝³ : CommSemiring S", "inst✝² : Algebra R S", "P✝ : Type u_3", "inst✝¹ : CommSemiring P✝", "P : Ideal R", "hp : P.IsPrime", "inst✝ : IsLocalization.AtPrime S P", "this✝ : _root_.Nontrivial S := Nontrivial S P", "x y : S", "hx : x ∈ nonunits S", "hy : y ∈ nonunits S", "hu : IsUnit (x + y)", "z : S", "hxyz : (x + y) * z = 1", "this : ∀ {r : R} {s : ↥P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P", "rx : R", "sx : ↥P.primeCompl", "hrx : mk' S rx sx = x", "ry : R", "sy : ↥P.primeCompl", "hry : mk' S ry sy = y"], "goal": "False"}], "premise": [77600], "state_str": "case intro.intro.intro.intro.intro\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP✝ : Type u_3\ninst✝¹ : CommSemiring P✝\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nthis✝ : _root_.Nontrivial S := Nontrivial S P\nx y : S\nhx : x ∈ nonunits S\nhy : y ∈ nonunits S\nhu : IsUnit (x + y)\nz : S\nhxyz : (x + y) * z = 1\nthis : ∀ {r : R} {s : ↥P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P\nrx : R\nsx : ↥P.primeCompl\nhrx : mk' S rx sx = x\nry : R\nsy : ↥P.primeCompl\nhry : mk' S ry sy = y\n⊢ False"}
+{"state": [{"context": ["R : Type u_1", "inst✝⁴ : CommSemiring R", "M : Submonoid R", "S : Type u_2", "inst✝³ : CommSemiring S", "inst✝² : Algebra R S", "P✝ : Type u_3", "inst✝¹ : CommSemiring P✝", "P : Ideal R", "hp : P.IsPrime", "inst✝ : IsLocalization.AtPrime S P", "this✝ : _root_.Nontrivial S := Nontrivial S P", "x y : S", "hx : x ∈ nonunits S", "hy : y ∈ nonunits S", "hu : IsUnit (x + y)", "z : S", "hxyz : (x + y) * z = 1", "this : ∀ {r : R} {s : ↥P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P", "rx : R", "sx : ↥P.primeCompl", "hrx : mk' S rx sx = x", "ry : R", "sy : ↥P.primeCompl", "hry : mk' S ry sy = y", "rz : R", "sz : ↥P.primeCompl", "hrz : mk' S rz sz = z"], "goal": "False"}], "premise": [77590, 77613, 77628, 117620], "state_str": "case intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP✝ : Type u_3\ninst✝¹ : CommSemiring P✝\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nthis✝ : _root_.Nontrivial S := Nontrivial S P\nx y : S\nhx : x ∈ nonunits S\nhy : y ∈ nonunits S\nhu : IsUnit (x + y)\nz : S\nhxyz : (x + y) * z = 1\nthis : ∀ {r : R} {s : ↥P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P\nrx : R\nsx : ↥P.primeCompl\nhrx : mk' S rx sx = x\nry : R\nsy : ↥P.primeCompl\nhry : mk' S ry sy = y\nrz : R\nsz : ↥P.primeCompl\nhrz : mk' S rz sz = z\n⊢ False"}
+{"state": [{"context": ["R : Type u_1", "inst✝⁴ : CommSemiring R", "M : Submonoid R", "S : Type u_2", "inst✝³ : CommSemiring S", "inst✝² : Algebra R S", "P✝ : Type u_3", "inst✝¹ : CommSemiring P✝", "P : Ideal R", "hp : P.IsPrime", "inst✝ : IsLocalization.AtPrime S P", "this✝ : _root_.Nontrivial S := Nontrivial S P", "x y : S", "hu : IsUnit (x + y)", "z : S", "this : ∀ {r : R} {s : ↥P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P", "rx : R", "sx : ↥P.primeCompl", "hx : mk' S rx sx ∈ nonunits S", "hrx : mk' S rx sx = x", "ry : R", "sy : ↥P.primeCompl", "hy : mk' S ry sy ∈ nonunits S", "hry : mk' S ry sy = y", "rz : R", "sz : ↥P.primeCompl", "hxyz : mk' S ((rx * ↑sy + ry * ↑sx) * rz) (sx * sy * sz) = mk' S 1 ⟨1, ⋯⟩", "hrz : mk' S rz sz = z"], "goal": "False"}], "premise": [1673, 77604], "state_str": "case intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP✝ : Type u_3\ninst✝¹ : CommSemiring P✝\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nthis✝ : _root_.Nontrivial S := Nontrivial S P\nx y : S\nhu : IsUnit (x + y)\nz : S\nthis : ∀ {r : R} {s : ↥P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P\nrx : R\nsx : ↥P.primeCompl\nhx : mk' S rx sx ∈ nonunits S\nhrx : mk' S rx sx = x\nry : R\nsy : ↥P.primeCompl\nhy : mk' S ry sy ∈ nonunits S\nhry : mk' S ry sy = y\nrz : R\nsz : ↥P.primeCompl\nhxyz : mk' S ((rx * ↑sy + ry * ↑sx) * rz) (sx * sy * sz) = mk' S 1 ⟨1, ⋯⟩\nhrz : mk' S rz sz = z\n⊢ False"}
+{"state": [{"context": ["R : Type u_1", "inst✝⁴ : CommSemiring R", "M : Submonoid R", "S : Type u_2", "inst✝³ : CommSemiring S", "inst✝² : Algebra R S", "P✝ : Type u_3", "inst✝¹ : CommSemiring P✝", "P : Ideal R", "hp : P.IsPrime", "inst✝ : IsLocalization.AtPrime S P", "this✝ : _root_.Nontrivial S := Nontrivial S P", "x y : S", "hu : IsUnit (x + y)", "z : S", "this : ∀ {r : R} {s : ↥P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P", "rx : R", "sx : ↥P.primeCompl", "hx : mk' S rx sx ∈ nonunits S", "hrx : mk' S rx sx = x", "ry : R", "sy : ↥P.primeCompl", "hy : mk' S ry sy �� nonunits S", "hry : mk' S ry sy = y", "rz : R", "sz : ↥P.primeCompl", "hxyz : mk' S ((rx * ↑sy + ry * ↑sx) * rz) (sx * sy * sz) = mk' S 1 ⟨1, ⋯⟩", "hrz : mk' S rz sz = z", "t : ↥P.primeCompl", "ht : ↑t * (↑⟨1, ⋯⟩ * ((rx * ↑sy + ry * ↑sx) * rz)) = ↑t * (↑(sx * sy * sz) * 1)"], "goal": "False"}], "premise": [119464, 119728, 119730, 137134], "state_str": "case intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP✝ : Type u_3\ninst✝¹ : CommSemiring P✝\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nthis✝ : _root_.Nontrivial S := Nontrivial S P\nx y : S\nhu : IsUnit (x + y)\nz : S\nthis : ∀ {r : R} {s : ↥P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P\nrx : R\nsx : ↥P.primeCompl\nhx : mk' S rx sx ∈ nonunits S\nhrx : mk' S rx sx = x\nry : R\nsy : ↥P.primeCompl\nhy : mk' S ry sy ∈ nonunits S\nhry : mk' S ry sy = y\nrz : R\nsz : ↥P.primeCompl\nhxyz : mk' S ((rx * ↑sy + ry * ↑sx) * rz) (sx * sy * sz) = mk' S 1 ⟨1, ⋯⟩\nhrz : mk' S rz sz = z\nt : ↥P.primeCompl\nht : ↑t * (↑⟨1, ⋯⟩ * ((rx * ↑sy + ry * ↑sx) * rz)) = ↑t * (↑(sx * sy * sz) * 1)\n⊢ False"}
+{"state": [{"context": ["R : Type u_1", "inst✝⁴ : CommSemiring R", "M : Submonoid R", "S : Type u_2", "inst✝³ : CommSemiring S", "inst✝² : Algebra R S", "P✝ : Type u_3", "inst✝¹ : CommSemiring P✝", "P : Ideal R", "hp : P.IsPrime", "inst✝ : IsLocalization.AtPrime S P", "this✝ : _root_.Nontrivial S := Nontrivial S P", "x y : S", "hu : IsUnit (x + y)", "z : S", "this : ∀ {r : R} {s : ↥P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P", "rx : R", "sx : ↥P.primeCompl", "hx : mk' S rx sx ∈ nonunits S", "hrx : mk' S rx sx = x", "ry : R", "sy : ↥P.primeCompl", "hy : mk' S ry sy ∈ nonunits S", "hry : mk' S ry sy = y", "rz : R", "sz : ↥P.primeCompl", "hxyz : mk' S ((rx * ↑sy + ry * ↑sx) * rz) (sx * sy * sz) = mk' S 1 ⟨1, ⋯⟩", "hrz : mk' S rz sz = z", "t : ↥P.primeCompl", "ht : ↑t * ((rx * ↑sy + ry * ↑sx) * rz) = ↑t * (↑sx * ↑sy * ↑sz)"], "goal": "False"}], "premise": [1681, 2111, 2115, 20010, 80374], "state_str": "case intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP✝ : Type u_3\ninst✝¹ : CommSemiring P✝\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nthis✝ : _root_.Nontrivial S := Nontrivial S P\nx y : S\nhu : IsUnit (x + y)\nz : S\nthis : ∀ {r : R} {s : ↥P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P\nrx : R\nsx : ↥P.primeCompl\nhx : mk' S rx sx ∈ nonunits S\nhrx : mk' S rx sx = x\nry : R\nsy : ↥P.primeCompl\nhy : mk' S ry sy ∈ nonunits S\nhry : mk' S ry sy = y\nrz : R\nsz : ↥P.primeCompl\nhxyz : mk' S ((rx * ↑sy + ry * ↑sx) * rz) (sx * sy * sz) = mk' S 1 ⟨1, ⋯⟩\nhrz : mk' S rz sz = z\nt : ↥P.primeCompl\nht : ↑t * ((rx * ↑sy + ry * ↑sx) * rz) = ↑t * (↑sx * ↑sy * ↑sz)\n⊢ False"}
+{"state": [{"context": ["R : Type u_1", "inst✝⁴ : CommSemiring R", "M : Submonoid R", "S : Type u_2", "inst✝³ : CommSemiring S", "inst✝² : Algebra R S", "P✝ : Type u_3", "inst✝¹ : CommSemiring P✝", "P : Ideal R", "hp : P.IsPrime", "inst✝ : IsLocalization.AtPrime S P", "this✝ : _root_.Nontrivial S := Nontrivial S P", "x y : S", "hu : IsUnit (x + y)", "z : S", "this : ∀ {r : R} {s : ↥P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P", "rx : R", "sx : ↥P.primeCompl", "hx : mk' S rx sx ∈ nonunits S", "hrx : mk' S rx sx = x", "ry : R", "sy : ↥P.primeCompl", "hy : mk' S ry sy ∈ nonunits S", "hry : mk' S ry sy = y", "rz : R", "sz : ↥P.primeCompl", "hxyz : mk' S ((rx * ↑sy + ry * ↑sx) * rz) (sx * sy * sz) = mk' S 1 ⟨1, ⋯⟩", "hrz : mk' S rz sz = z", "t : ↥P.primeCompl", "ht : ↑t * ((rx * ↑sy + ry * ↑sx) * rz) = ↑t * (↑sx * ↑sy * ↑sz)"], "goal": "↑t * ((rx * ↑sy + ry * ↑sx) * rz) ∈ P"}], "premise": [80337, 80338, 80415], "state_str": "case intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP✝ : Type u_3\ninst✝¹ : CommSemiring P✝\nP : Ideal R\nhp : P.IsPrime\ninst✝ : IsLocalization.AtPrime S P\nthis✝ : _root_.Nontrivial S := Nontrivial S P\nx y : S\nhu : IsUnit (x + y)\nz : S\nthis : ∀ {r : R} {s : ↥P.primeCompl}, mk' S r s ∈ nonunits S → r ∈ P\nrx : R\nsx : ↥P.primeCompl\nhx : mk' S rx sx ∈ nonunits S\nhrx : mk' S rx sx = x\nry : R\nsy : ↥P.primeCompl\nhy : mk' S ry sy ∈ nonunits S\nhry : mk' S ry sy = y\nrz : R\nsz : ↥P.primeCompl\nhxyz : mk' S ((rx * ↑sy + ry * ↑sx) * rz) (sx * sy * sz) = mk' S 1 ⟨1, ⋯⟩\nhrz : mk' S rz sz = z\nt : ↥P.primeCompl\nht : ↑t * ((rx * ↑sy + ry * ↑sx) * rz) = ↑t * (↑sx * ↑sy * ↑sz)\n⊢ ↑t * ((rx * ↑sy + ry * ↑sx) * rz) ∈ P"}
+{"state": [{"context": ["C : Type u", "inst✝ : Category.{v, u} C", "F F₁ F₂ : C ⥤ Type w", "α : F₁ ⟶ F₂", "t₁ t₂ : F₁.Elements", "k : t₁ ⟶ t₂"], "goal": "F₂.map (↑k) ((fun t => ⟨t.fst, α.app t.fst t.snd⟩) t₁).snd = ((fun t => ⟨t.fst, α.app t.fst t.snd⟩) t₂).snd"}], "premise": [2100, 91300, 97666], "state_str": "C : Type u\ninst✝ : Category.{v, u} C\nF F₁ F₂ : C ⥤ Type w\nα : F₁ ⟶ F₂\nt₁ t₂ : F₁.Elements\nk : t₁ ⟶ t₂\n⊢ F₂.map (↑k) ((fun t => ⟨t.fst, α.app t.fst t.snd⟩) t₁).snd = ((fun t => ⟨t.fst, α.app t.fst t.snd⟩) t₂).snd"}
+{"state": [{"context": ["G : Type u_1", "H : Type u_2", "A : Type u_3", "α : Type u_4", "β : Type u_5", "inst✝¹ : Monoid G", "a b x y : G", "n m : ℕ", "inst✝ : Monoid H", "f : G →* H", "hf : Injective ⇑f"], "goal": "IsOfFinOrder (f x) ↔ IsOfFinOrder x"}], "premise": [1713, 8340, 8375], "state_str": "G : Type u_1\nH : Type u_2\nA : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝¹ : Monoid G\na b x y : G\nn m : ℕ\ninst✝ : Monoid H\nf : G →* H\nhf : Injective ⇑f\n⊢ IsOfFinOrder (f x) ↔ IsOfFinOrder x"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "δ' : Type u_5", "ι : Sort uι", "s t u : Set α", "mα : MeasurableSpace α", "inst✝¹ : MeasurableSpace β", "inst✝ : MeasurableSpace γ", "f : α → β", "g : β → γ", "hg : MeasurableEmbedding g"], "goal": "Measurable (g ∘ f) ↔ Measurable f"}], "premise": [28025, 28187], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns t u : Set α\nmα : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf : α → β\ng : β → γ\nhg : MeasurableEmbedding g\n⊢ Measurable (g ∘ f) ↔ Measurable f"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "δ' : Type u_5", "ι : Sort uι", "s t u : Set α", "mα : MeasurableSpace α", "inst✝¹ : MeasurableSpace β", "inst✝ : MeasurableSpace γ", "f : α → β", "g : β → γ", "hg : MeasurableEmbedding g", "H : Measurable (g ∘ f)"], "goal": "Measurable f"}], "premise": [28186, 71429, 134255], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns t u : Set α\nmα : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf : α → β\ng : β → γ\nhg : MeasurableEmbedding g\nH : Measurable (g ∘ f)\n⊢ Measurable f"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "δ' : Type u_5", "ι : Sort uι", "s t u : Set α", "mα : MeasurableSpace α", "inst✝¹ : MeasurableSpace β", "inst✝ : MeasurableSpace γ", "f : α → β", "g : β → γ", "hg : MeasurableEmbedding g", "H : Measurable (g ∘ f)"], "goal": "Measurable ((rangeSplitting g ∘ rangeFactorization g) ∘ f)"}], "premise": [28025, 28194, 28839], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns t u : Set α\nmα : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf : α → β\ng : β → γ\nhg : MeasurableEmbedding g\nH : Measurable (g ∘ f)\n⊢ Measurable ((rangeSplitting g ∘ rangeFactorization g) ∘ f)"}
+{"state": [{"context": ["R : Type u_1", "inst✝⁸ : CommRing R", "A : Type u_2", "inst✝⁷ : CommRing A", "inst✝⁶ : Algebra R A", "M : Type u_3", "inst✝⁵ : AddCommGroup M", "inst✝⁴ : Module A M", "inst✝³ : Module R M", "D✝ D1 D2 : Derivation R A M", "r : R", "a✝ b : A", "K : Type u_4", "inst✝² : Field K", "inst✝¹ : Module K M", "inst✝ : Algebra R K", "D : Derivation R K M", "a : K"], "goal": "D a⁻¹ = -a⁻¹ ^ 2 • D a"}], "premise": [70039], "state_str": "R : Type u_1\ninst✝⁸ : CommRing R\nA : Type u_2\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R A\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module A M\ninst✝³ : Module R M\nD✝ D1 D2 : Derivation R A M\nr : R\na✝ b : A\nK : Type u_4\ninst✝² : Field K\ninst✝¹ : Module K M\ninst✝ : Algebra R K\nD : Derivation R K M\na : K\n⊢ D a⁻¹ = -a⁻¹ ^ 2 • D a"}
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+{"state": [{"context": ["ι : Type u_1", "inst✝¹³ : Fintype ι", "A : ι → Type u_2", "inst✝¹² : (i : ι) → MeasurableSpace (A i)", "μ✝ : (i : ι) → Measure (A i)", "inst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)", "F : Type u_3", "inst✝¹⁰ : NormedAddCommGroup F", "inst✝⁹ : NormedSpace ℝ F", "E : Type u_4", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : NormedSpace ℝ E", "inst✝⁶ : MeasurableSpace E", "inst✝⁵ : BorelSpace E", "inst✝⁴ : FiniteDimensional ℝ E", "μ : Measure E", "inst✝³ : μ.IsAddHaarMeasure", "F' : Type u_5", "inst✝² : NormedAddCommGroup F'", "inst✝¹ : InnerProductSpace ℝ F'", "inst✝ : CompleteSpace F'", "u : E → F'", "hu : ContDiff ℝ 1 u", "h2u : HasCompactSupport u", "p p' : ℝ≥0", "hp✝ : 1 ≤ p", "hp'0 : ¬p' = 0", "n : ℕ := finrank ℝ E", "hn✝ : 0 < n", "hp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹", "n' : ℝ≥0 := (↑n).conjExponent", "h2p : ↑p < ↑n", "h0n : 2 ≤ n", "hn : (↑n).IsConjExponent n'", "h1n : 1 ≤ ↑n", "h2n : 0 < ↑n - 1", "hnp : 0 < ↑n - ↑p", "hp : 1 < p", "q : ℝ := (↑p).conjExponent", "hq : (↑p).IsConjExponent q", "h0p : p ≠ 0", "h1p : ↑p ≠ 1", "h3p : ↑p - 1 ≠ 0", "h0p' : p' ≠ 0", "h2q : 1 / ↑n' - 1 / q = 1 / ↑p'", "γ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩", "h0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)", "h1γ : 1 < ↑γ", "h2γ : γ * n' = p'", "h3γ : (↑γ - 1) * q = ↑p'", "h4γ : ↑γ ≠ 0", "h3u : ¬∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ = 0"], "goal": "eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ"}], "premise": [1674, 28616, 28624, 28626, 31053, 48447, 103552, 146600], "state_str": "case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\nγ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩\nh0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)\nh1γ : 1 < ↑γ\nh2γ : γ * n' = p'\nh3γ : (↑γ - 1) * q = ↑p'\nh4γ : ↑γ ≠ 0\nh3u : ¬∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ = 0\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ"}
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+{"state": [{"context": ["ι : Type u_1", "inst✝¹³ : Fintype ι", "A : ι → Type u_2", "inst✝¹² : (i : ι) → MeasurableSpace (A i)", "μ✝ : (i : ι) → Measure (A i)", "inst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)", "F : Type u_3", "inst✝¹⁰ : NormedAddCommGroup F", "inst✝⁹ : NormedSpace ℝ F", "E : Type u_4", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : NormedSpace ℝ E", "inst✝⁶ : MeasurableSpace E", "inst✝⁵ : BorelSpace E", "inst✝⁴ : FiniteDimensional ℝ E", "μ : Measure E", "inst✝³ : μ.IsAddHaarMeasure", "F' : Type u_5", "inst✝² : NormedAddCommGroup F'", "inst✝¹ : InnerProductSpace ℝ F'", "inst✝ : CompleteSpace F'", "u : E → F'", "hu : ContDiff ℝ 1 u", "h2u : HasCompactSupport u", "p p' : ℝ≥0", "hp✝ : 1 ≤ p", "hp'0 : ¬p' = 0", "n : ℕ := finrank ℝ E", "hn✝ : 0 < n", "hp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹", "n' : ℝ≥0 := (↑n).conjExponent", "h2p : ↑p < ↑n", "h0n : 2 ≤ n", "hn : (↑n).IsConjExponent n'", "h1n : 1 ≤ ↑n", "h2n : 0 < ↑n - 1", "hnp : 0 < ↑n - ↑p", "hp : 1 < p", "q : ℝ := (↑p).conjExponent", "hq : (↑p).IsConjExponent q", "h0p : p ≠ 0", "h1p : ↑p ≠ 1", "h3p : ↑p - 1 ≠ 0", "h0p' : p' ≠ 0", "h2q : 1 / ↑n' - 1 / q = 1 / ↑p'", "γ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩", "h0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)", "h1γ : 1 < ↑γ", "h2γ : γ * n' = p'", "h3γ : (↑γ - 1) * q = ↑p'", "h4γ : ↑γ ≠ 0", "h3u : ¬∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ = 0", "h4u : ∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ ≠ ⊤", "h5u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0"], "goal": "eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ"}], "premise": [39763, 101700, 104339, 145356, 145372], "state_str": "case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\nγ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩\nh0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)\nh1γ : 1 < ↑γ\nh2γ : γ * n' = p'\nh3γ : (↑γ - 1) * q = ↑p'\nh4γ : ↑γ ≠ 0\nh3u : ¬∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ = 0\nh4u : ∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ ≠ ⊤\nh5u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ"}
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+{"state": [{"context": ["ι : Type u_1", "inst✝¹³ : Fintype ι", "A : ι → Type u_2", "inst✝¹² : (i : ι) → MeasurableSpace (A i)", "μ✝ : (i : ι) → Measure (A i)", "inst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)", "F : Type u_3", "inst✝¹⁰ : NormedAddCommGroup F", "inst✝⁹ : NormedSpace ℝ F", "E : Type u_4", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : NormedSpace ℝ E", "inst✝⁶ : MeasurableSpace E", "inst✝⁵ : BorelSpace E", "inst✝⁴ : FiniteDimensional ℝ E", "μ : Measure E", "inst✝³ : μ.IsAddHaarMeasure", "F' : Type u_5", "inst✝² : NormedAddCommGroup F'", "inst✝¹ : InnerProductSpace ℝ F'", "inst✝ : CompleteSpace F'", "u : E → F'", "hu : ContDiff ℝ 1 u", "h2u : HasCompactSupport u", "p p' : ℝ≥0", "hp✝ : 1 ≤ p", "hp'0 : ¬p' = 0", "n : ℕ := finrank ℝ E", "hn✝ : 0 < n", "hp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹", "n' : ℝ≥0 := (↑n).conjExponent", "h2p : ↑p < ↑n", "h0n : 2 ≤ n", "hn : (↑n).IsConjExponent n'", "h1n : 1 ≤ ↑n", "h2n : 0 < ↑n - 1", "hnp : 0 < ↑n - ↑p", "hp : 1 < p", "q : ℝ := (↑p).conjExponent", "hq : (↑p).IsConjExponent q", "h0p : p ≠ 0", "h1p : ↑p ≠ 1", "h3p : ↑p - 1 ≠ 0", "h0p' : p' ≠ 0", "h2q : 1 / ↑n' - 1 / q = 1 / ↑p'", "γ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩", "h0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)", "h1γ : 1 < ↑γ", "h2γ : γ * n' = p'", "h3γ : (↑γ - 1) * q = ↑p'", "h4γ : ↑γ ≠ 0", "h3u : ¬∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ = 0", "h4u : ∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ ≠ ⊤", "h5u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0", "h6u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ ⊤", "h7u : Continuous u"], "goal": "eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ"}], "premise": [14272, 48447, 51647], "state_str": "case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\nγ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩\nh0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)\nh1γ : 1 < ↑γ\nh2γ : γ * n' = p'\nh3γ : (↑γ - 1) * q = ↑p'\nh4γ : ↑γ ≠ 0\nh3u : ¬∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ = 0\nh4u : ∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ ≠ ⊤\nh5u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0\nh6u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ ⊤\nh7u : Continuous u\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ"}
+{"state": [{"context": ["ι : Type u_1", "inst✝¹³ : Fintype ι", "A : ι → Type u_2", "inst✝¹² : (i : ι) → MeasurableSpace (A i)", "μ✝ : (i : ι) → Measure (A i)", "inst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)", "F : Type u_3", "inst✝¹⁰ : NormedAddCommGroup F", "inst✝⁹ : NormedSpace ℝ F", "E : Type u_4", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : NormedSpace ℝ E", "inst✝⁶ : MeasurableSpace E", "inst✝⁵ : BorelSpace E", "inst✝⁴ : FiniteDimensional ℝ E", "μ : Measure E", "inst✝³ : μ.IsAddHaarMeasure", "F' : Type u_5", "inst✝² : NormedAddCommGroup F'", "inst✝¹ : InnerProductSpace ℝ F'", "inst✝ : CompleteSpace F'", "u : E → F'", "hu : ContDiff ℝ 1 u", "h2u : HasCompactSupport u", "p p' : ℝ≥0", "hp✝ : 1 ≤ p", "hp'0 : ¬p' = 0", "n : ℕ := finrank ℝ E", "hn✝ : 0 < n", "hp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹", "n' : ℝ≥0 := (↑n).conjExponent", "h2p : ↑p < ↑n", "h0n : 2 ≤ n", "hn : (↑n).IsConjExponent n'", "h1n : 1 ≤ ↑n", "h2n : 0 < ↑n - 1", "hnp : 0 < ↑n - ↑p", "hp : 1 < p", "q : ℝ := (↑p).conjExponent", "hq : (↑p).IsConjExponent q", "h0p : p ≠ 0", "h1p : ↑p ≠ 1", "h3p : ↑p - 1 ≠ 0", "h0p' : p' ≠ 0", "h2q : 1 / ↑n' - 1 / q = 1 / ↑p'", "γ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩", "h0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)", "h1γ : 1 < ↑γ", "h2γ : γ * n' = p'", "h3γ : (↑γ - 1) * q = ↑p'", "h4γ : ↑γ ≠ 0", "h3u : ¬∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ = 0", "h4u : ∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ ≠ ⊤", "h5u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0", "h6u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ ⊤", "h7u : Continuous u", "h8u : Continuous (fderiv ℝ u)", "v : E → ℝ := fun x => ‖u x‖ ^ ↑γ"], "goal": "eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ"}], "premise": [33462], "state_str": "case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\nγ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩\nh0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)\nh1γ : 1 < ↑γ\nh2γ : γ * n' = p'\nh3γ : (↑γ - 1) * q = ↑p'\nh4γ : ↑γ ≠ 0\nh3u : ¬∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ = 0\nh4u : ∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ ≠ ⊤\nh5u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0\nh6u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ ⊤\nh7u : Continuous u\nh8u : Continuous (fderiv ℝ u)\nv : E → ℝ := fun x => ‖u x‖ ^ ↑γ\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ"}
+{"state": [{"context": ["ι : Type u_1", "inst✝¹³ : Fintype ι", "A : ι → Type u_2", "inst✝¹² : (i : ι) → MeasurableSpace (A i)", "μ✝ : (i : ι) → Measure (A i)", "inst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)", "F : Type u_3", "inst✝¹⁰ : NormedAddCommGroup F", "inst✝⁹ : NormedSpace ℝ F", "E : Type u_4", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : NormedSpace ℝ E", "inst✝⁶ : MeasurableSpace E", "inst✝⁵ : BorelSpace E", "inst✝⁴ : FiniteDimensional ℝ E", "μ : Measure E", "inst✝³ : μ.IsAddHaarMeasure", "F' : Type u_5", "inst✝² : NormedAddCommGroup F'", "inst✝¹ : InnerProductSpace ℝ F'", "inst✝ : CompleteSpace F'", "u : E → F'", "hu : ContDiff ℝ 1 u", "h2u : HasCompactSupport u", "p p' : ℝ≥0", "hp✝ : 1 ≤ p", "hp'0 : ¬p' = 0", "n : ℕ := finrank ℝ E", "hn✝ : 0 < n", "hp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹", "n' : ℝ≥0 := (↑n).conjExponent", "h2p : ↑p < ↑n", "h0n : 2 ≤ n", "hn : (↑n).IsConjExponent n'", "h1n : 1 ≤ ↑n", "h2n : 0 < ↑n - 1", "hnp : 0 < ↑n - ↑p", "hp : 1 < p", "q : ℝ := (↑p).conjExponent", "hq : (↑p).IsConjExponent q", "h0p : p ≠ 0", "h1p : ↑p ≠ 1", "h3p : ↑p - 1 ≠ 0", "h0p' : p' ≠ 0", "h2q : 1 / ↑n' - 1 / q = 1 / ↑p'", "γ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩", "h0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)", "h1γ : 1 < ↑γ", "h2γ : γ * n' = p'", "h3γ : (↑γ - 1) * q = ↑p'", "h4γ : ↑γ ≠ 0", "h3u : ¬∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ = 0", "h4u : ∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ ≠ ⊤", "h5u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0", "h6u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ ⊤", "h7u : Continuous u", "h8u : Continuous (fderiv ℝ u)", "v : E → ℝ := fun x => ‖u x‖ ^ ↑γ", "hv : ContDiff ℝ 1 v"], "goal": "eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ"}], "premise": [40033], "state_str": "case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\nγ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩\nh0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)\nh1γ : 1 < ↑γ\nh2γ : γ * n' = p'\nh3γ : (↑γ - 1) * q = ↑p'\nh4γ : ↑γ ≠ 0\nh3u : ¬∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ = 0\nh4u : ∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ ≠ ⊤\nh5u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0\nh6u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ ⊤\nh7u : Continuous u\nh8u : Continuous (fderiv ℝ u)\nv : E → ℝ := fun x => ‖u x‖ ^ ↑γ\nhv : ContDiff ℝ 1 v\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ"}
+{"state": [{"context": ["ι : Type u_1", "inst✝¹³ : Fintype ι", "A : ι → Type u_2", "inst✝¹² : (i : ι) → MeasurableSpace (A i)", "μ✝ : (i : ι) → Measure (A i)", "inst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)", "F : Type u_3", "inst✝¹⁰ : NormedAddCommGroup F", "inst✝⁹ : NormedSpace ℝ F", "E : Type u_4", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : NormedSpace ℝ E", "inst✝⁶ : MeasurableSpace E", "inst✝⁵ : BorelSpace E", "inst✝⁴ : FiniteDimensional ℝ E", "μ : Measure E", "inst✝³ : μ.IsAddHaarMeasure", "F' : Type u_5", "inst✝² : NormedAddCommGroup F'", "inst✝¹ : InnerProductSpace ℝ F'", "inst✝ : CompleteSpace F'", "u : E → F'", "hu : ContDiff ℝ 1 u", "h2u : HasCompactSupport u", "p p' : ℝ≥0", "hp✝ : 1 ≤ p", "hp'0 : ¬p' = 0", "n : ℕ := finrank ℝ E", "hn✝ : 0 < n", "hp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹", "n' : ℝ≥0 := (↑n).conjExponent", "h2p : ↑p < ↑n", "h0n : 2 ≤ n", "hn : (↑n).IsConjExponent n'", "h1n : 1 ≤ ↑n", "h2n : 0 < ↑n - 1", "hnp : 0 < ↑n - ↑p", "hp : 1 < p", "q : ℝ := (↑p).conjExponent", "hq : (↑p).IsConjExponent q", "h0p : p ≠ 0", "h1p : ↑p ≠ 1", "h3p : ↑p - 1 ≠ 0", "h0p' : p' ≠ 0", "h2q : 1 / ↑n' - 1 / q = 1 / ↑p'", "γ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩", "h0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)", "h1γ : 1 < ↑γ", "h2γ : γ * n' = p'", "h3γ : (↑γ - 1) * q = ↑p'", "h4γ : ↑γ ≠ 0", "h3u : ¬∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ = 0", "h4u : ∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ ≠ ⊤", "h5u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0", "h6u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ ⊤", "h7u : Continuous u", "h8u : Continuous (fderiv ℝ u)", "v : E → ℝ := fun x => ‖u x‖ ^ ↑γ", "hv : ContDiff ℝ 1 v", "h2v : HasCompactSupport v", "C : ℝ≥0 := eLpNormLESNormFDerivOneConst μ ↑n'", "this : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / ↑n') ≤ ↑C * ↑γ * (∫⁻ (x : E), ↑‖fderiv ℝ u x‖₊ ^ ↑p ∂μ) ^ (1 / ↑p) * (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q)"], "goal": "eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ"}], "premise": [1717, 12981, 28618, 39768, 108288, 143772, 146610, 146616, 146678], "state_str": "case neg\nι : Type u_1\ninst✝¹³ : Fintype ι\nA : ι → Type u_2\ninst✝¹² : (i : ι) → MeasurableSpace (A i)\nμ✝ : (i : ι) → Measure (A i)\ninst✝¹¹ : ∀ (i : ι), SigmaFinite (μ✝ i)\nF : Type u_3\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : InnerProductSpace ℝ F'\ninst✝ : CompleteSpace F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u : HasCompactSupport u\np p' : ℝ≥0\nhp✝ : 1 ≤ p\nhp'0 : ¬p' = 0\nn : ℕ := finrank ℝ E\nhn✝ : 0 < n\nhp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹\nn' : ℝ≥0 := (↑n).conjExponent\nh2p : ↑p < ↑n\nh0n : 2 ≤ n\nhn : (↑n).IsConjExponent n'\nh1n : 1 ≤ ↑n\nh2n : 0 < ↑n - 1\nhnp : 0 < ↑n - ↑p\nhp : 1 < p\nq : ℝ := (↑p).conjExponent\nhq : (↑p).IsConjExponent q\nh0p : p ≠ 0\nh1p : ↑p ≠ 1\nh3p : ↑p - 1 ≠ 0\nh0p' : p' ≠ 0\nh2q : 1 / ↑n' - 1 / q = 1 / ↑p'\nγ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩\nh0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)\nh1γ : 1 < ↑γ\nh2γ : γ * n' = p'\nh3γ : (↑γ - 1) * q = ↑p'\nh4γ : ↑γ ≠ 0\nh3u : ¬∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ = 0\nh4u : ∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ ≠ ⊤\nh5u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0\nh6u : (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q) ≠ ⊤\nh7u : Continuous u\nh8u : Continuous (fderiv ℝ u)\nv : E → ℝ := fun x => ‖u x‖ ^ ↑γ\nhv : ContDiff ℝ 1 v\nh2v : HasCompactSupport v\nC : ℝ≥0 := eLpNormLESNormFDerivOneConst μ ↑n'\nthis :\n  (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / ↑n') ≤\n    ↑C * ↑γ * (∫⁻ (x : E), ↑‖fderiv ℝ u x‖₊ ^ ↑p ∂μ) ^ (1 / ↑p) * (∫⁻ (x : E), ↑‖u x‖₊ ^ ↑p' ∂μ) ^ (1 / q)\n⊢ eLpNorm u (↑p') μ ≤ ↑(eLpNormLESNormFDerivOfEqInnerConst μ ↑p) * eLpNorm (fderiv ℝ u) (↑p) μ"}
+{"state": [{"context": ["V : Type u_1", "V' : Type u_2", "inst✝⁵ : NormedAddCommGroup V", "inst✝⁴ : NormedAddCommGroup V'", "inst✝³ : InnerProductSpace ℝ V", "inst✝² : InnerProductSpace ℝ V'", "inst✝¹ : Fact (finrank ℝ V = 2)", "inst✝ : Fact (finrank ℝ V' = 2)", "o : Orientation ℝ V (Fin 2)", "θ : Real.Angle", "x : V"], "goal": "-(o.rotation θ) x = (o.rotation (↑π + θ)) x"}], "premise": [69313, 69315], "state_str": "V : Type u_1\nV' : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : NormedAddCommGroup V'\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : InnerProductSpace ℝ V'\ninst✝¹ : Fact (finrank ℝ V = 2)\ninst✝ : Fact (finrank ℝ V' = 2)\no : Orientation ℝ V (Fin 2)\nθ : Real.Angle\nx : V\n⊢ -(o.rotation θ) x = (o.rotation (↑π + θ)) x"}
+{"state": [{"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "inst✝² : SMul αᵐᵒᵖ β", "inst✝¹ : SMul β γ", "inst✝ : SMul α γ", "a : α", "s : Set β", "t : Set γ", "h : ∀ (a : α) (b : β) (c : γ), (op a • b) • c = b • a • c", "x✝ : γ"], "goal": "x✝ ∈ (op a • s) • t ↔ x✝ ∈ s • a • t"}], "premise": [132793, 132862], "state_str": "case h\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : SMul αᵐᵒᵖ β\ninst✝¹ : SMul β γ\ninst✝ : SMul α γ\na : α\ns : Set β\nt : Set γ\nh : ∀ (a : α) (b : β) (c : γ), (op a • b) • c = b • a • c\nx✝ : γ\n⊢ x✝ ∈ (op a • s) • t ↔ x✝ ∈ s • a • t"}
+{"state": [{"context": ["𝕜 : Type u_1", "inst✝⁴ : RCLike 𝕜", "n : Type u_2", "inst✝³ : LinearOrder n", "inst✝² : IsWellOrder n fun x x_1 => x < x_1", "inst✝¹ : LocallyFiniteOrderBot n", "S : Matrix n n 𝕜", "inst✝ : Fintype n", "hS : S.PosDef"], "goal": "lowerInv hS = ((Pi.basisFun 𝕜 n).toMatrix ⇑(gramSchmidtBasis (Pi.basisFun 𝕜 n)))ᵀ"}], "premise": [85787], "state_str": "𝕜 : Type u_1\ninst✝⁴ : RCLike 𝕜\nn : Type u_2\ninst✝³ : LinearOrder n\ninst✝² : IsWellOrder n fun x x_1 => x < x_1\ninst✝¹ : LocallyFiniteOrderBot n\nS : Matrix n n 𝕜\ninst✝ : Fintype n\nhS : S.PosDef\n⊢ lowerInv hS = ((Pi.basisFun 𝕜 n).toMatrix ⇑(gramSchmidtBasis (Pi.basisFun 𝕜 n)))ᵀ"}
+{"state": [{"context": ["𝕜 : Type u_1", "inst✝⁴ : RCLike 𝕜", "n : Type u_2", "inst✝³ : LinearOrder n", "inst✝² : IsWellOrder n fun x x_1 => x < x_1", "inst✝¹ : LocallyFiniteOrderBot n", "S : Matrix n n 𝕜", "inst✝ : Fintype n", "hS : S.PosDef", "this : NormedAddCommGroup (n → 𝕜) := NormedAddCommGroup.ofMatrix ⋯"], "goal": "lowerInv hS = ((Pi.basisFun 𝕜 n).toMatrix ⇑(gramSchmidtBasis (Pi.basisFun 𝕜 n)))ᵀ"}], "premise": [85787], "state_str": "𝕜 : Type u_1\ninst✝⁴ : RCLike 𝕜\nn : Type u_2\ninst✝³ : LinearOrder n\ninst✝² : IsWellOrder n fun x x_1 => x < x_1\ninst✝¹ : LocallyFiniteOrderBot n\nS : Matrix n n 𝕜\ninst✝ : Fintype n\nhS : S.PosDef\nthis : NormedAddCommGroup (n → 𝕜) := NormedAddCommGroup.ofMatrix ⋯\n⊢ lowerInv hS = ((Pi.basisFun 𝕜 n).toMatrix ⇑(gramSchmidtBasis (Pi.basisFun 𝕜 n)))ᵀ"}
+{"state": [{"context": ["𝕜 : Type u_1", "inst✝⁴ : RCLike 𝕜", "n : Type u_2", "inst✝³ : LinearOrder n", "inst✝² : IsWellOrder n fun x x_1 => x < x_1", "inst✝¹ : LocallyFiniteOrderBot n", "S : Matrix n n 𝕜", "inst✝ : Fintype n", "hS : S.PosDef", "this✝ : NormedAddCommGroup (n → 𝕜) := NormedAddCommGroup.ofMatrix ⋯", "this : InnerProductSpace 𝕜 (n → 𝕜) := InnerProductSpace.ofMatrix ��", "i j : n"], "goal": "lowerInv hS i j = ((Pi.basisFun 𝕜 n).toMatrix ⇑(gramSchmidtBasis (Pi.basisFun 𝕜 n)))ᵀ i j"}], "premise": [33887, 85018], "state_str": "case a\n𝕜 : Type u_1\ninst✝⁴ : RCLike 𝕜\nn : Type u_2\ninst✝³ : LinearOrder n\ninst✝² : IsWellOrder n fun x x_1 => x < x_1\ninst✝¹ : LocallyFiniteOrderBot n\nS : Matrix n n 𝕜\ninst✝ : Fintype n\nhS : S.PosDef\nthis✝ : NormedAddCommGroup (n → 𝕜) := NormedAddCommGroup.ofMatrix ⋯\nthis : InnerProductSpace 𝕜 (n → 𝕜) := InnerProductSpace.ofMatrix ⋯\ni j : n\n⊢ lowerInv hS i j = ((Pi.basisFun 𝕜 n).toMatrix ⇑(gramSchmidtBasis (Pi.basisFun 𝕜 n)))ᵀ i j"}
+{"state": [{"context": ["R : Type u", "inst✝¹ : CommRing R", "W' : Jacobian R", "F : Type v", "inst✝ : Field F", "W : Jacobian F", "P Q : Fin 3 → F", "hP : W.Nonsingular P", "hQ : W.Nonsingular Q", "hPz : P z = 0", "hQz : Q z = 0"], "goal": "W.add P Q = P x ^ 2 • ![1, 1, 0]"}], "premise": [1737, 2107, 145491, 145536], "state_str": "R : Type u\ninst✝¹ : CommRing R\nW' : Jacobian R\nF : Type v\ninst✝ : Field F\nW : Jacobian F\nP Q : Fin 3 → F\nhP : W.Nonsingular P\nhQ : W.Nonsingular Q\nhPz : P z = 0\nhQz : Q z = 0\n⊢ W.add P Q = P x ^ 2 • ![1, 1, 0]"}
+{"state": [{"context": ["M : Type u_1", "α : Type u_2", "β : Type u_3", "Γ : Type u_4", "inst✝⁶ : Group Γ", "T : Type u_5", "inst✝⁵ : TopologicalSpace T", "inst✝⁴ : MulAction Γ T", "G₀ : Type u_6", "inst✝³ : GroupWithZero G₀", "inst✝² : MulAction G₀ α", "inst✝¹ : TopologicalSpace α", "inst✝ : ContinuousConstSMul G₀ α", "c : G₀", "s : Set α", "x : α", "hc : c ≠ 0"], "goal": "c • s ∈ 𝓝 (c • x) ↔ s ∈ 𝓝 x"}], "premise": [65017], "state_str": "M : Type u_1\nα : Type u_2\nβ : Type u_3\nΓ : Type u_4\ninst✝⁶ : Group Γ\nT : Type u_5\ninst✝⁵ : TopologicalSpace T\ninst✝⁴ : MulAction Γ T\nG₀ : Type u_6\ninst✝³ : GroupWithZero G₀\ninst✝² : MulAction G₀ α\ninst✝¹ : TopologicalSpace α\ninst✝ : ContinuousConstSMul G₀ α\nc : G₀\ns : Set α\nx : α\nhc : c ≠ 0\n⊢ c • s ∈ 𝓝 (c • x) ↔ s ∈ 𝓝 x"}
+{"state": [{"context": ["M : Type u_1", "α : Type u_2", "β : Type u_3", "Γ : Type u_4", "inst✝⁶ : Group Γ", "T : Type u_5", "inst✝⁵ : TopologicalSpace T", "inst✝⁴ : MulAction Γ T", "G₀ : Type u_6", "inst✝³ : GroupWithZero G₀", "inst✝² : MulAction G₀ α", "inst✝¹ : TopologicalSpace α", "inst✝ : ContinuousConstSMul G₀ α", "c : G₀", "s : Set α", "x : α", "hc : c ≠ 0", "h : c • s ∈ 𝓝 (c • x)"], "goal": "s ∈ 𝓝 x"}], "premise": [7382], "state_str": "M : Type u_1\nα : Type u_2\nβ : Type u_3\nΓ : Type u_4\ninst✝⁶ : Group Γ\nT : Type u_5\ninst✝⁵ : TopologicalSpace T\ninst✝⁴ : MulAction Γ T\nG₀ : Type u_6\ninst✝³ : GroupWithZero G₀\ninst✝² : MulAction G₀ α\ninst✝¹ : TopologicalSpace α\ninst✝ : ContinuousConstSMul G₀ α\nc : G₀\ns : Set α\nx : α\nhc : c ≠ 0\nh : c • s ∈ 𝓝 (c • x)\n⊢ s ∈ 𝓝 x"}
+{"state": [{"context": ["M : Type u_1", "α : Type u_2", "β : Type u_3", "Γ : Type u_4", "inst✝⁶ : Group Γ", "T : Type u_5", "inst✝⁵ : TopologicalSpace T", "inst✝⁴ : MulAction Γ T", "G₀ : Type u_6", "inst✝³ : GroupWithZero G₀", "inst✝² : MulAction G₀ α", "inst✝¹ : TopologicalSpace α", "inst✝ : ContinuousConstSMul G₀ α", "c : G₀", "s : Set α", "x : α", "hc : c ≠ 0", "h : c • s ∈ 𝓝 (c • x)"], "goal": "c⁻¹ • c • s ∈ 𝓝 (c⁻¹ • c • x)"}], "premise": [65017, 108555], "state_str": "M : Type u_1\nα : Type u_2\nβ : Type u_3\nΓ : Type u_4\ninst✝⁶ : Group Γ\nT : Type u_5\ninst✝⁵ : TopologicalSpace T\ninst✝⁴ : MulAction Γ T\nG₀ : Type u_6\ninst✝³ : GroupWithZero G₀\ninst✝² : MulAction G₀ α\ninst✝¹ : TopologicalSpace α\ninst✝ : ContinuousConstSMul G₀ α\nc : G₀\ns : Set α\nx : α\nhc : c ≠ 0\nh : c • s ∈ 𝓝 (c • x)\n⊢ c⁻¹ • c • s ∈ 𝓝 (c⁻¹ • c • x)"}
+{"state": [{"context": ["K : Type u_1", "inst✝³ : Field K", "inst✝² : NumberField K", "inst✝¹ : IsCyclotomicExtension {3} ℚ K", "ζ : K", "hζ : IsPrimitiveRoot ζ ↑3", "S : Solution hζ", "S' : Solution' hζ", "inst✝ : DecidableRel fun a b => a ∣ b"], "goal": "λ ^ (3 * S.multiplicity - 2) ∣ S.a + S.b"}], "premise": [79540], "state_str": "K : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\n⊢ λ ^ (3 * S.multiplicity - 2) ∣ S.a + S.b"}
+{"state": [{"context": ["K : Type u_1", "inst✝³ : Field K", "inst✝² : NumberField K", "inst✝¹ : IsCyclotomicExtension {3} ℚ K", "ζ : K", "hζ : IsPrimitiveRoot ζ ↑3", "S : Solution hζ", "S' : Solution' hζ", "inst✝ : DecidableRel fun a b => a ∣ b", "h : (λ ^ S.multiplicity) ^ 3 ∣ ↑S.u * S.c ^ 3"], "goal": "λ ^ (3 * S.multiplicity - 2) ∣ S.a + S.b"}], "premise": [24495, 24504, 24516, 24517, 119707, 119761], "state_str": "K : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\nh : (λ ^ S.multiplicity) ^ 3 ∣ ↑S.u * S.c ^ 3\n⊢ λ ^ (3 * S.multiplicity - 2) ∣ S.a + S.b"}
+{"state": [{"context": ["K : Type u_1", "inst✝³ : Field K", "inst✝² : NumberField K", "inst✝¹ : IsCyclotomicExtension {3} ℚ K", "ζ : K", "hζ : IsPrimitiveRoot ζ ↑3", "S : Solution hζ", "S' : Solution' hζ", "inst✝ : DecidableRel fun a b => a ∣ b", "h : λ ^ (3 * S.multiplicity) ∣ (S.a + S.b) * (λ * FermatLastTheoremForThreeGen.Solution.y S) * (λ * FermatLastTheoremForThreeGen.Solution.z S)"], "goal": "λ ^ (3 * S.multiplicity - 2) ∣ S.a + S.b"}], "premise": [24519, 24765, 125812], "state_str": "K : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\nh :\n  λ ^ (3 * S.multiplicity) ∣\n    (S.a + S.b) * (λ * FermatLastTheoremForThreeGen.Solution.y S) * (λ * FermatLastTheoremForThreeGen.Solution.z S)\n⊢ λ ^ (3 * S.multiplicity - 2) ∣ S.a + S.b"}
+{"state": [{"context": ["K : Type u_1", "inst✝³ : Field K", "inst✝² : NumberField K", "inst✝¹ : IsCyclotomicExtension {3} ℚ K", "ζ : K", "hζ : IsPrimitiveRoot ζ ↑3", "S : Solution hζ", "S' : Solution' hζ", "inst✝ : DecidableRel fun a b => a ∣ b", "h : λ ^ (3 * S.multiplicity) ∣ (S.a + S.b) * (λ * FermatLastTheoremForThreeGen.Solution.y S) * (λ * FermatLastTheoremForThreeGen.Solution.z S)"], "goal": "λ ^ (3 * S.multiplicity - 2) ∣ FermatLastTheoremForThreeGen.Solution.z S * (S.a + S.b)"}], "premise": [24518, 24765, 125812], "state_str": "K : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\nh :\n  λ ^ (3 * S.multiplicity) ∣\n    (S.a + S.b) * (λ * FermatLastTheoremForThreeGen.Solution.y S) * (λ * FermatLastTheoremForThreeGen.Solution.z S)\n⊢ λ ^ (3 * S.multiplicity - 2) ∣ FermatLastTheoremForThreeGen.Solution.z S * (S.a + S.b)"}
+{"state": [{"context": ["K : Type u_1", "inst✝³ : Field K", "inst✝² : NumberField K", "inst✝¹ : IsCyclotomicExtension {3} ℚ K", "ζ : K", "hζ : IsPrimitiveRoot ζ ↑3", "S : Solution hζ", "S' : Solution' hζ", "inst✝ : DecidableRel fun a b => a ∣ b", "h : λ ^ (3 * S.multiplicity) ∣ (S.a + S.b) * (λ * FermatLastTheoremForThreeGen.Solution.y S) * (λ * FermatLastTheoremForThreeGen.Solution.z S)"], "goal": "λ ^ (3 * S.multiplicity - 2) ∣ FermatLastTheoremForThreeGen.Solution.y S * (FermatLastTheoremForThreeGen.Solution.z S * (S.a + S.b))"}], "premise": [24503], "state_str": "K : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\nh :\n  λ ^ (3 * S.multiplicity) ∣\n    (S.a + S.b) * (λ * FermatLastTheoremForThreeGen.Solution.y S) * (λ * FermatLastTheoremForThreeGen.Solution.z S)\n⊢ λ ^ (3 * S.multiplicity - 2) ∣\n    FermatLastTheoremForThreeGen.Solution.y S * (FermatLastTheoremForThreeGen.Solution.z S * (S.a + S.b))"}
+{"state": [{"context": ["K : Type u_1", "inst✝³ : Field K", "inst✝² : NumberField K", "inst✝¹ : IsCyclotomicExtension {3} ℚ K", "ζ : K", "hζ : IsPrimitiveRoot ζ ↑3", "S : Solution hζ", "S' : Solution' hζ", "inst✝ : DecidableRel fun a b => a ∣ b", "h : λ ^ (3 * S.multiplicity) ∣ (S.a + S.b) * (λ * FermatLastTheoremForThreeGen.Solution.y S) * (λ * FermatLastTheoremForThreeGen.Solution.z S)", "this : 2 ≤ S.multiplicity"], "goal": "λ ^ (3 * S.multiplicity - 2) ∣ FermatLastTheoremForThreeGen.Solution.y S * (FermatLastTheoremForThreeGen.Solution.z S * (S.a + S.b))"}], "premise": [119742], "state_str": "K : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\nh :\n  λ ^ (3 * S.multiplicity) ∣\n    (S.a + S.b) * (λ * FermatLastTheoremForThreeGen.Solution.y S) * (λ * FermatLastTheoremForThreeGen.Solution.z S)\nthis : 2 ≤ S.multiplicity\n⊢ λ ^ (3 * S.multiplicity - 2) ∣\n    FermatLastTheoremForThreeGen.Solution.y S * (FermatLastTheoremForThreeGen.Solution.z S * (S.a + S.b))"}
+{"state": [{"context": ["K : Type u_1", "inst✝³ : Field K", "inst✝² : NumberField K", "inst✝¹ : IsCyclotomicExtension {3} ℚ K", "ζ : K", "hζ : IsPrimitiveRoot ζ ↑3", "S : Solution hζ", "S' : Solution' hζ", "inst✝ : DecidableRel fun a b => a ∣ b", "h : λ ^ (3 * S.multiplicity - 2) * λ * λ ∣ (S.a + S.b) * FermatLastTheoremForThreeGen.Solution.y S * FermatLastTheoremForThreeGen.Solution.z S * λ * λ", "this : 2 ≤ S.multiplicity"], "goal": "λ ^ (3 * S.multiplicity - 2) ∣ FermatLastTheoremForThreeGen.Solution.y S * (FermatLastTheoremForThreeGen.Solution.z S * (S.a + S.b))"}], "premise": [24765, 108659, 125797], "state_str": "K : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\ninst✝¹ : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝ : DecidableRel fun a b => a ∣ b\nh :\n  λ ^ (3 * S.multiplicity - 2) * λ * λ ∣\n    (S.a + S.b) * FermatLastTheoremForThreeGen.Solution.y S * FermatLastTheoremForThreeGen.Solution.z S * λ * λ\nthis : 2 ≤ S.multiplicity\n⊢ λ ^ (3 * S.multiplicity - 2) ∣\n    FermatLastTheoremForThreeGen.Solution.y S * (FermatLastTheoremForThreeGen.Solution.z S * (S.a + S.b))"}
+{"state": [{"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : Nonempty α", "T : ℕ → ℝ", "g : ℝ → ℝ", "a b : α → ℝ", "r : α → ℕ → ℕ", "R : AkraBazziRecurrence T g a b r", "b' : ℝ := b (min_bi b) / 2"], "goal": "(fun n => (1 + ε ↑n) * asympBound g a b n) =O[atTop] T"}], "premise": [76716], "state_str": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := b (min_bi b) / 2\n⊢ (fun n => (1 + ε ↑n) * asympBound g a b n) =O[atTop] T"}
+{"state": [{"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : Nonempty α", "T : ℕ → ℝ", "g : ℝ → ℝ", "a b : α → ℝ", "r : α → ℕ → ℕ", "R : AkraBazziRecurrence T g a b r", "b' : ℝ := b (min_bi b) / 2", "hb_pos : 0 < b'"], "goal": "(fun n => (1 + ε ↑n) * asympBound g a b n) =O[atTop] T"}], "premise": [43665], "state_str": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := b (min_bi b) / 2\nhb_pos : 0 < b'\n⊢ (fun n => (1 + ε ↑n) * asympBound g a b n) =O[atTop] T"}
+{"state": [{"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : Nonempty α", "T : ℕ → ℝ", "g : ℝ → ℝ", "a b : α → ℝ", "r : α → ℕ → ℕ", "R : AkraBazziRecurrence T g a b r", "b' : ℝ := b (min_bi b) / 2", "hb_pos : 0 < b'"], "goal": "∀ᶠ (n₀ : ℕ) in atTop, ∃ c > 0, ∀ n ≥ n₀, c * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖"}], "premise": [76771], "state_str": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := b (min_bi b) / 2\nhb_pos : 0 < b'\n⊢ ∀ᶠ (n₀ : ℕ) in atTop, ∃ c > 0, ∀ n ≥ n₀, c * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖"}
+{"state": [{"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : Nonempty α", "T : ℕ → ℝ", "g : ℝ → ℝ", "a b : α → ℝ", "r : α → ℕ → ℕ", "R : AkraBazziRecurrence T g a b r", "b' : ℝ := b (min_bi b) / 2", "hb_pos : 0 < b'", "c₁ : ℝ", "hc₁ : c₁ > 0", "h_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n"], "goal": "∀ᶠ (n₀ : ℕ) in atTop, ∃ c > 0, ∀ n ≥ n₀, c * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖"}], "premise": [1674, 15491, 15493, 15495, 15504, 15506, 15889, 34120, 76714, 76749, 76750, 76769, 76770, 76783, 131585], "state_str": "case intro.intro\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := b (min_bi b) / 2\nhb_pos : 0 < b'\nc₁ : ℝ\nhc₁ : c₁ > 0\nh_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n\n⊢ ∀ᶠ (n₀ : ℕ) in atTop, ∃ c > 0, ∀ n ≥ n₀, c * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖"}
+{"state": [{"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : Nonempty α", "T : ℕ → ℝ", "g : ℝ → ℝ", "a b : α → ℝ", "r : α → ℕ → ℕ", "R : AkraBazziRecurrence T g a b r", "b' : ℝ := b (min_bi b) / 2", "hb_pos : 0 < b'", "c₁ : ℝ", "hc₁ : c₁ > 0", "h_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n", "n₀ : ℕ", "n₀_ge_Rn₀ : R.n₀ ≤ n₀", "h_b_floor : 0 < ⌊b' * ↑n₀⌋₊", "h_smoothing_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < 1 + ε ↑y", "h_smoothing_pos' : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y", "h_asympBound_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < asympBound g a b y", "h_asympBound_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < asympBound g a b (r i y)", "h_asympBound_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < asympBound g a b y", "n₀_pos : 0 < n₀", "h_smoothing_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < 1 + ε ↑(r i y)", "bound2 : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b i ^ p a b * ↑y ^ p a b * (1 + ε ↑y) ≤ ↑(r i y) ^ p a b * (1 + ε ↑(r i y))", "h_smoothingFn_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y", "h_sumTransform : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), sumTransform (p a b) g (r i y) y ≤ c₁ * g ↑y", "h_bi_le_r : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b (min_bi b) / 2 * ↑y ≤ ↑(r i y)", "h_exp : ∀ (y : ℕ), n₀ ≤ y → ⌈rexp 1⌉₊ ≤ y"], "goal": "∃ c > 0, ∀ n ≥ n₀, c * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖"}], "premise": [76784], "state_str": "case h\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := b (min_bi b) / 2\nhb_pos : 0 < b'\nc₁ : ℝ\nhc₁ : c₁ > 0\nh_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n\nn₀ : ℕ\nn₀_ge_Rn₀ : R.n₀ ≤ n₀\nh_b_floor : 0 < ⌊b' * ↑n₀⌋₊\nh_smoothing_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < 1 + ε ↑y\nh_smoothing_pos' : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_asympBound_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < asympBound g a b y\nh_asympBound_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < asympBound g a b (r i y)\nh_asympBound_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < asympBound g a b y\nn₀_pos : 0 < n₀\nh_smoothing_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < 1 + ε ↑(r i y)\nbound2 : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b i ^ p a b * ↑y ^ p a b * (1 + ε ↑y) ≤ ↑(r i y) ^ p a b * (1 + ε ↑(r i y))\nh_smoothingFn_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_sumTransform : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), sumTransform (p a b) g (r i y) y ≤ c₁ * g ↑y\nh_bi_le_r : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b (min_bi b) / 2 * ↑y ≤ ↑(r i y)\nh_exp : ∀ (y : ℕ), n₀ ≤ y → ⌈rexp 1⌉₊ ≤ y\n⊢ ∃ c > 0, ∀ n ≥ n₀, c * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖"}
+{"state": [{"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : Nonempty α", "T : ℕ → ℝ", "g : ℝ → ℝ", "a b : α → ℝ", "r : α → ℕ → ℕ", "R : AkraBazziRecurrence T g a b r", "b' : ℝ := b (min_bi b) / 2", "hb_pos : 0 < b'", "c₁ : ℝ", "hc₁ : c₁ > 0", "h_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n", "n₀ : ℕ", "n₀_ge_Rn₀ : R.n₀ ≤ n₀", "h_b_floor : 0 < ⌊b' * ↑n₀⌋₊", "h_smoothing_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < 1 + ε ↑y", "h_smoothing_pos' : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y", "h_asympBound_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < asympBound g a b y", "h_asympBound_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < asympBound g a b (r i y)", "h_asympBound_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < asympBound g a b y", "n₀_pos : 0 < n₀", "h_smoothing_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < 1 + ε ↑(r i y)", "bound2 : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b i ^ p a b * ↑y ^ p a b * (1 + ε ↑y) ≤ ↑(r i y) ^ p a b * (1 + ε ↑(r i y))", "h_smoothingFn_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y", "h_sumTransform : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), sumTransform (p a b) g (r i y) y ≤ c₁ * g ↑y", "h_bi_le_r : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b (min_bi b) / 2 * ↑y ≤ ↑(r i y)", "h_exp : ∀ (y : ℕ), n₀ ≤ y → ⌈rexp 1⌉₊ ≤ y", "h_base_nonempty : (Ico ⌊b (min_bi b) / 2 * ↑n₀⌋₊ n₀).Nonempty", "base_min : ℝ := (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)", "base_min_def : base_min = (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)", "C : ℝ := min (2 * c₁)⁻¹ base_min"], "goal": "∃ c > 0, ∀ n ≥ n₀, c * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖"}], "premise": [2107, 19591, 19714, 76724, 76768, 139848], "state_str": "case h\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := b (min_bi b) / 2\nhb_pos : 0 < b'\nc₁ : ℝ\nhc₁ : c₁ > 0\nh_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n\nn₀ : ℕ\nn₀_ge_Rn₀ : R.n₀ ≤ n₀\nh_b_floor : 0 < ⌊b' * ↑n₀⌋₊\nh_smoothing_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < 1 + ε ↑y\nh_smoothing_pos' : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_asympBound_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < asympBound g a b y\nh_asympBound_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < asympBound g a b (r i y)\nh_asympBound_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < asympBound g a b y\nn₀_pos : 0 < n₀\nh_smoothing_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < 1 + ε ↑(r i y)\nbound2 : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b i ^ p a b * ↑y ^ p a b * (1 + ε ↑y) ≤ ↑(r i y) ^ p a b * (1 + ε ↑(r i y))\nh_smoothingFn_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_sumTransform : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), sumTransform (p a b) g (r i y) y ≤ c₁ * g ↑y\nh_bi_le_r : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b (min_bi b) / 2 * ↑y ≤ ↑(r i y)\nh_exp : ∀ (y : ℕ), n₀ ≤ y → ⌈rexp 1⌉₊ ≤ y\nh_base_nonempty : (Ico ⌊b (min_bi b) / 2 * ↑n₀⌋₊ n₀).Nonempty\nbase_min : ℝ := (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)\nbase_min_def : base_min = (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)\nC : ℝ := min (2 * c₁)⁻¹ base_min\n⊢ ∃ c > 0, ∀ n ≥ n₀, c * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖"}
+{"state": [{"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : Nonempty α", "T : ℕ → ℝ", "g : ℝ → ℝ", "a b : α → ℝ", "r : α → ℕ → ℕ", "R : AkraBazziRecurrence T g a b r", "b' : ℝ := b (min_bi b) / 2", "hb_pos : 0 < b'", "c₁ : ℝ", "hc₁ : c₁ > 0", "h_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n", "n₀ : ℕ", "n₀_ge_Rn₀ : R.n₀ ≤ n₀", "h_b_floor : 0 < ⌊b' * ↑n₀⌋₊", "h_smoothing_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < 1 + ε ↑y", "h_smoothing_pos' : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y", "h_asympBound_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < asympBound g a b y", "h_asympBound_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < asympBound g a b (r i y)", "h_asympBound_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < asympBound g a b y", "n₀_pos : 0 < n₀", "h_smoothing_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < 1 + ε ↑(r i y)", "bound2 : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b i ^ p a b * ↑y ^ p a b * (1 + ε ↑y) ≤ ↑(r i y) ^ p a b * (1 + ε ↑(r i y))", "h_smoothingFn_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y", "h_sumTransform : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), sumTransform (p a b) g (r i y) y ≤ c₁ * g ↑y", "h_bi_le_r : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b (min_bi b) / 2 * ↑y ≤ ↑(r i y)", "h_exp : ∀ (y : ℕ), n₀ ≤ y → ⌈rexp 1⌉₊ ≤ y", "h_base_nonempty : (Ico ⌊b (min_bi b) / 2 * ↑n₀⌋₊ n₀).Nonempty", "base_min : ℝ := (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)", "base_min_def : base_min = (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)", "C : ℝ := min (2 * c₁)⁻¹ base_min", "hC_pos : 0 < C", "n : ℕ", "hn : n ≥ n₀"], "goal": "C * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖"}], "premise": [2107, 14271, 19591, 19691, 106022, 139802], "state_str": "case h\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := b (min_bi b) / 2\nhb_pos : 0 < b'\nc₁ : ℝ\nhc₁ : c₁ > 0\nh_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n\nn₀ : ℕ\nn₀_ge_Rn₀ : R.n₀ ≤ n₀\nh_b_floor : 0 < ⌊b' * ↑n₀⌋₊\nh_smoothing_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < 1 + ε ↑y\nh_smoothing_pos' : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_asympBound_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < asympBound g a b y\nh_asympBound_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < asympBound g a b (r i y)\nh_asympBound_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < asympBound g a b y\nn₀_pos : 0 < n₀\nh_smoothing_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < 1 + ε ↑(r i y)\nbound2 : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b i ^ p a b * ↑y ^ p a b * (1 + ε ↑y) ≤ ↑(r i y) ^ p a b * (1 + ε ↑(r i y))\nh_smoothingFn_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_sumTransform : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), sumTransform (p a b) g (r i y) y ≤ c₁ * g ↑y\nh_bi_le_r : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b (min_bi b) / 2 * ↑y ≤ ↑(r i y)\nh_exp : ∀ (y : ℕ), n₀ ≤ y → ⌈rexp 1⌉₊ ≤ y\nh_base_nonempty : (Ico ⌊b (min_bi b) / 2 * ↑n₀⌋₊ n₀).Nonempty\nbase_min : ℝ := (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)\nbase_min_def : base_min = (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)\nC : ℝ := min (2 * c₁)⁻¹ base_min\nhC_pos : 0 < C\nn : ℕ\nhn : n ≥ n₀\n⊢ C * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖"}
+{"state": [{"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : Nonempty α", "T : ℕ → ℝ", "g : ℝ → ℝ", "a b : α → ℝ", "r : α → ℕ → ℕ", "R : AkraBazziRecurrence T g a b r", "b' : ℝ := b (min_bi b) / 2", "hb_pos : 0 < b'", "c₁ : ℝ", "hc₁ : c₁ > 0", "h_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n", "n₀ : ℕ", "n₀_ge_Rn₀ : R.n₀ ≤ n₀", "h_b_floor : 0 < ⌊b' * ↑n₀⌋₊", "h_smoothing_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < 1 + ε ↑y", "h_smoothing_pos' : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y", "h_asympBound_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < asympBound g a b y", "h_asympBound_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < asympBound g a b (r i y)", "h_asympBound_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < asympBound g a b y", "n₀_pos : 0 < n₀", "h_smoothing_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < 1 + ε ↑(r i y)", "bound2 : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b i ^ p a b * ↑y ^ p a b * (1 + ε ↑y) ≤ ↑(r i y) ^ p a b * (1 + ε ↑(r i y))", "h_smoothingFn_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y", "h_sumTransform : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), sumTransform (p a b) g (r i y) y ≤ c₁ * g ↑y", "h_bi_le_r : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b (min_bi b) / 2 * ↑y ≤ ↑(r i y)", "h_exp : ∀ (y : ℕ), n₀ ≤ y → ⌈rexp 1⌉₊ ≤ y", "h_base_nonempty : (Ico ⌊b (min_bi b) / 2 * ↑n₀⌋₊ n₀).Nonempty", "base_min : ℝ := (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)", "base_min_def : base_min = (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)", "C : ℝ := min (2 * c₁)⁻¹ base_min", "hC_pos : 0 < C", "n : ℕ", "hn : n ≥ n₀", "h_base : ∀ n ∈ Ico ⌊b' * ↑n₀⌋₊ n₀, C * ((1 + ε ↑n) * asympBound g a b n) ≤ T n", "h_asympBound_pos' : 0 < asympBound g a b n", "h_one_sub_smoothingFn_pos' : 0 < 1 + ε ↑n"], "goal": "C * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖"}], "premise": [42904, 76725], "state_str": "case h\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := b (min_bi b) / 2\nhb_pos : 0 < b'\nc₁ : ℝ\nhc₁ : c₁ > 0\nh_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n\nn₀ : ℕ\nn₀_ge_Rn₀ : R.n₀ ≤ n₀\nh_b_floor : 0 < ⌊b' * ↑n₀⌋₊\nh_smoothing_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < 1 + ε ↑y\nh_smoothing_pos' : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_asympBound_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < asympBound g a b y\nh_asympBound_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < asympBound g a b (r i y)\nh_asympBound_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < asympBound g a b y\nn₀_pos : 0 < n₀\nh_smoothing_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < 1 + ε ↑(r i y)\nbound2 : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b i ^ p a b * ↑y ^ p a b * (1 + ε ↑y) ≤ ↑(r i y) ^ p a b * (1 + ε ↑(r i y))\nh_smoothingFn_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y\nh_sumTransform : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), sumTransform (p a b) g (r i y) y ≤ c₁ * g ↑y\nh_bi_le_r : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b (min_bi b) / 2 * ↑y ≤ ↑(r i y)\nh_exp : ∀ (y : ℕ), n₀ ≤ y → ⌈rexp 1⌉₊ ≤ y\nh_base_nonempty : (Ico ⌊b (min_bi b) / 2 * ↑n₀⌋₊ n₀).Nonempty\nbase_min : ℝ := (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)\nbase_min_def : base_min = (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)\nC : ℝ := min (2 * c₁)⁻¹ base_min\nhC_pos : 0 < C\nn : ℕ\nhn : n ≥ n₀\nh_base : ∀ n ∈ Ico ⌊b' * ↑n₀⌋₊ n₀, C * ((1 + ε ↑n) * asympBound g a b n) ≤ T n\nh_asympBound_pos' : 0 < asympBound g a b n\nh_one_sub_smoothingFn_pos' : 0 < 1 + ε ↑n\n⊢ C * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖"}
+{"state": [{"context": ["F : Type u_1", "inst✝¹ : Field F", "ι : Type u_2", "inst✝ : DecidableEq ι", "s t : Finset ι", "i j : ι", "v r r' : ι → F"], "goal": "(interpolate {i} v) r = C (r i)"}], "premise": [82777, 82787, 119730, 126908], "state_str": "F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns t : Finset ι\ni j : ι\nv r r' : ι → F\n⊢ (interpolate {i} v) r = C (r i)"}
+{"state": [{"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝¹ : LinearOrderedSemiring α", "inst✝ : FloorSemiring α", "a : α", "n : ℕ", "ha : 0 ≤ a"], "goal": "a < ↑1 ↔ a < 1"}], "premise": [1713, 143125], "state_str": "F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : LinearOrderedSemiring α\ninst✝ : FloorSemiring α\na : α\nn : ℕ\nha : 0 ≤ a\n⊢ a < ↑1 ↔ a < 1"}
+{"state": [{"context": ["R : Type u_1", "inst✝⁷ : CommRing R", "inst✝⁶ : IsDedekindDomain R", "inst✝⁵ : Module.Free ℤ R", "inst✝⁴ : Module.Finite ℤ R", "K : Type u_2", "inst✝³ : CommRing K", "inst✝² : Algebra R K", "inst✝¹ : IsFractionRing R K", "inst✝ : NoZeroDivisors K", "I : FractionalIdeal R⁰ K"], "goal": "absNorm I = 0 ↔ I = 0"}], "premise": [75611, 75661, 109075], "state_str": "R : Type u_1\ninst✝⁷ : CommRing R\ninst✝⁶ : IsDedekindDomain R\ninst✝⁵ : Module.Free ℤ R\ninst✝⁴ : Module.Finite ℤ R\nK : Type u_2\ninst✝³ : CommRing K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : NoZeroDivisors K\nI : FractionalIdeal R⁰ K\n⊢ absNorm I = 0 ↔ I = 0"}
+{"state": [{"context": ["R : Type u_1", "inst✝⁷ : CommRing R", "inst✝⁶ : IsDedekindDomain R", "inst✝⁵ : Module.Free ℤ R", "inst✝⁴ : Module.Finite ℤ R", "K : Type u_2", "inst✝³ : CommRing K", "inst✝² : Algebra R K", "inst✝¹ : IsFractionRing R K", "inst✝ : NoZeroDivisors K", "I : FractionalIdeal R⁰ K", "h : absNorm I = 0"], "goal": "I.num = ⊥"}], "premise": [75608, 108406], "state_str": "R : Type u_1\ninst✝⁷ : CommRing R\ninst✝⁶ : IsDedekindDomain R\ninst✝⁵ : Module.Free ℤ R\ninst✝⁴ : Module.Finite ℤ R\nK : Type u_2\ninst✝³ : CommRing K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : NoZeroDivisors K\nI : FractionalIdeal R⁰ K\nh : absNorm I = 0\n⊢ I.num = ⊥"}
+{"state": [{"context": ["R : Type u_1", "inst✝⁷ : CommRing R", "inst✝⁶ : IsDedekindDomain R", "inst✝⁵ : Module.Free ℤ R", "inst✝⁴ : Module.Finite ℤ R", "K : Type u_2", "inst✝³ : CommRing K", "inst✝² : Algebra R K", "inst✝¹ : IsFractionRing R K", "inst✝ : NoZeroDivisors K", "I : FractionalIdeal R⁰ K", "h : ↑(Ideal.absNorm I.num) = 0 ∨ ↑|(Algebra.norm ℤ) ↑I.den| = 0"], "goal": "I.num = ⊥"}], "premise": [1673, 2112, 81583, 111255], "state_str": "R : Type u_1\ninst✝⁷ : CommRing R\ninst✝⁶ : IsDedekindDomain R\ninst✝⁵ : Module.Free ℤ R\ninst✝⁴ : Module.Finite ℤ R\nK : Type u_2\ninst✝³ : CommRing K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : NoZeroDivisors K\nI : FractionalIdeal R⁰ K\nh : ↑(Ideal.absNorm I.num) = 0 ∨ ↑|(Algebra.norm ℤ) ↑I.den| = 0\n⊢ I.num = ⊥"}
+{"state": [{"context": ["R : Type u_1", "inst✝⁷ : CommRing R", "inst✝⁶ : IsDedekindDomain R", "inst✝⁵ : Module.Free ℤ R", "inst✝⁴ : Module.Finite ℤ R", "K : Type u_2", "inst✝³ : CommRing K", "inst✝² : Algebra R K", "inst✝¹ : IsFractionRing R K", "inst✝ : NoZeroDivisors K", "I : FractionalIdeal R⁰ K", "h : ↑(Ideal.absNorm I.num) = 0 ∨ ↑|(Algebra.norm ℤ) ↑I.den| = 0"], "goal": "¬↑|(Algebra.norm ℤ) ↑I.den| = 0"}], "premise": [78556], "state_str": "R : Type u_1\ninst✝⁷ : CommRing R\ninst✝⁶ : IsDedekindDomain R\ninst✝⁵ : Module.Free ℤ R\ninst✝⁴ : Module.Finite ℤ R\nK : Type u_2\ninst✝³ : CommRing K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : NoZeroDivisors K\nI : FractionalIdeal R⁰ K\nh : ↑(Ideal.absNorm I.num) = 0 ∨ ↑|(Algebra.norm ℤ) ↑I.den| = 0\n⊢ ¬↑|(Algebra.norm ℤ) ↑I.den| = 0"}
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+{"state": [{"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "f✝ g : Perm α", "x✝ y : α", "inst✝³ : DecidableRel f✝.SameCycle", "inst✝² : DecidableRel g.SameCycle", "inst✝¹ : DecidableEq α", "inst✝ : Fintype α", "f : Perm α", "x a : α", "ha : a ∈ ↑(f.cycleOf x).support", "b : α", "hb : b ∈ ↑(f.cycleOf x).support"], "goal": "f.SameCycle a b"}], "premise": [9456, 138668], "state_str": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\nf✝ g : Perm α\nx✝ y : α\ninst✝³ : DecidableRel f✝.SameCycle\ninst✝² : DecidableRel g.SameCycle\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf : Perm α\nx a : α\nha : a ∈ ↑(f.cycleOf x).support\nb : α\nhb : b ∈ ↑(f.cycleOf x).support\n⊢ f.SameCycle a b"}
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+{"state": [{"context": ["G : Type u_1", "H : Type u_2", "inst✝ : CommMonoid G", "p : ℕ", "hp : Fact (Nat.Prime p)"], "goal": "orderOf 1 = p ^ 0"}], "premise": [8347, 119739], "state_str": "G : Type u_1\nH : Type u_2\ninst✝ : CommMonoid G\np : ℕ\nhp : Fact (Nat.Prime p)\n⊢ orderOf 1 = p ^ 0"}
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+{"state": [{"context": ["ι : Type u_1", "ι' : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "inst✝¹ : SemilatticeSup α", "inst✝ : Nonempty α", "F : Filter β", "u : α → β"], "goal": "(∀ {p : β → Prop}, (∀ᶠ (x : β) in F, p x) → ∀ (a : α), ∃ b ≥ a, p (u b)) ↔ ∀ U ∈ F, ∀ (N : α), ∃ n ≥ N, u n ∈ U"}], "premise": [1713], "state_str": "ι : Type u_1\nι' : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝¹ : SemilatticeSup α\ninst✝ : Nonempty α\nF : Filter β\nu : α → β\n⊢ (∀ {p : β → Prop}, (∀ᶠ (x : β) in F, p x) → ∀ (a : α), ∃ b ≥ a, p (u b)) ↔ ∀ U ∈ F, ∀ (N : α), ∃ n ≥ N, u n ∈ U"}
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+{"state": [{"context": ["R : Type u", "S : Type v", "T : Type w", "inst✝² : Ring R", "inst✝¹ : Ring S", "inst✝ : Ring T", "f : R →+* S", "s : Subring R", "h : ⇑f ⁻¹' {0} ⊆ ↑s"], "goal": "comap f (map f s) = s"}], "premise": [123620], "state_str": "R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\nf : R →+* S\ns : Subring R\nh : ⇑f ⁻¹' {0} ⊆ ↑s\n⊢ comap f (map f s) = s"}
+{"state": [{"context": ["R : Type u", "S : Type v", "T : Type w", "inst✝² : Ring R", "inst✝¹ : Ring S", "inst✝ : Ring T", "f : R →+* S", "s : Subring R", "h : ⇑f ⁻¹' {0} ⊆ ↑s"], "goal": "s = s ⊔ closure (⇑f ⁻¹' {0})"}], "premise": [14527, 123560], "state_str": "case h.e'_3\nR : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\nf : R →+* S\ns : Subring R\nh : ⇑f ⁻¹' {0} ⊆ ↑s\n⊢ s = s ⊔ closure (⇑f ⁻¹' {0})"}
+{"state": [{"context": ["α✝ : Type u_1", "β : Type u_2", "inst✝³ : MeasurableSpace α✝", "α : Type u_3", "inst✝² : MeasurableSpace α", "A : Set α", "ι : Type u_4", "L : Filter ι", "inst✝¹ : L.IsCountablyGenerated", "As : ι → Set α", "inst✝ : L.NeBot", "As_mble : ∀ (i : ι), MeasurableSet (As i)", "h_lim : ∀ (x : α), ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A"], "goal": "MeasurableSet A"}], "premise": [28808], "state_str": "α✝ : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α✝\nα : Type u_3\ninst✝² : MeasurableSpace α\nA : Set α\nι : Type u_4\nL : Filter ι\ninst✝¹ : L.IsCountablyGenerated\nAs : ι → Set α\ninst✝ : L.NeBot\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nh_lim : ∀ (x : α), ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A\n⊢ MeasurableSet A"}
+{"state": [{"context": ["α✝ : Type u_1", "β : Type u_2", "inst✝³ : MeasurableSpace α✝", "α : Type u_3", "inst✝² : MeasurableSpace α", "A : Set α", "ι : Type u_4", "L : Filter ι", "inst✝¹ : L.IsCountablyGenerated", "As : ι → Set α", "inst✝ : L.NeBot", "h_lim : ∀ (x : α), ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A", "As_mble : ∀ (i : ι), Measurable ((As i).indicator fun x => 1)"], "goal": "Measurable (A.indicator fun x => 1)"}], "premise": [1674, 25981, 59310], "state_str": "α✝ : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α✝\nα : Type u_3\ninst✝² : MeasurableSpace α\nA : Set α\nι : Type u_4\nL : Filter ι\ninst✝¹ : L.IsCountablyGenerated\nAs : ι → Set α\ninst✝ : L.NeBot\nh_lim : ∀ (x : α), ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A\nAs_mble : ∀ (i : ι), Measurable ((As i).indicator fun x => 1)\n⊢ Measurable (A.indicator fun x => 1)"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "inst✝³ : MulOneClass α", "inst✝² : LT α", "inst✝¹ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1", "inst✝ : ContravariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1", "a b : α"], "goal": "a < b * a ↔ 1 * a < b * a"}], "premise": [1713, 119728], "state_str": "α : Type u_1\nβ : Type u_2\ninst✝³ : MulOneClass α\ninst✝² : LT α\ninst✝¹ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst✝ : ContravariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b : α\n⊢ a < b * a ↔ 1 * a < b * a"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : LT α", "inst✝ : LT β", "s t : Set α", "l : List α", "a : α"], "goal": "[a] ∈ s.subchain ↔ a ∈ s"}], "premise": [15320], "state_str": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LT α\ninst✝ : LT β\ns t : Set α\nl : List α\na : α\n⊢ [a] ∈ s.subchain ↔ a ∈ s"}
+{"state": [{"context": ["α : Type u", "β : Type v", "γ : Type w", "f : α → β", "a : α", "l : List α"], "goal": "(map f (a :: l)).sublists = map (map f) (a :: l).sublists"}], "premise": [2611, 5247, 130926, 132380], "state_str": "α : Type u\nβ : Type v\nγ : Type w\nf : α → β\na : α\nl : List α\n⊢ (map f (a :: l)).sublists = map (map f) (a :: l).sublists"}
+{"state": [{"context": ["𝕜 : Type u", "𝕜' : Type u'", "E : Type v", "F : Type w", "G : Type x", "inst✝² : NontriviallyNormedField 𝕜", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace 𝕜 E", "p : FormalMultilinearSeries 𝕜 𝕜 E", "n✝ k n : ℕ"], "goal": "(fslope^[k] p).coeff n = p.coeff (n + k)"}], "premise": [119704], "state_str": "𝕜 : Type u\n𝕜' : Type u'\nE : Type v\nF : Type w\nG : Type x\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\np : FormalMultilinearSeries 𝕜 𝕜 E\nn✝ k n : ℕ\n⊢ (fslope^[k] p).coeff n = p.coeff (n + k)"}
+{"state": [{"context": ["Ω : Type u_1", "m : MeasurableSpace Ω", "X : Ω → ℝ", "μ : Measure Ω", "inst✝ : IsFiniteMeasure μ", "hX : Memℒp X 2 μ"], "goal": "ENNReal.ofReal (variance X μ) = evariance X μ"}], "premise": [143160], "state_str": "Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nhX : Memℒp X 2 μ\n⊢ ENNReal.ofReal (variance X μ) = evariance X μ"}
+{"state": [{"context": ["Ω : Type u_1", "m : MeasurableSpace Ω", "X : Ω → ℝ", "μ : Measure Ω", "inst✝ : IsFiniteMeasure μ", "hX : Memℒp X 2 μ"], "goal": "evariance X μ ≠ ⊤"}], "premise": [74172], "state_str": "Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nhX : Memℒp X 2 μ\n⊢ evariance X μ ≠ ⊤"}
+{"state": [{"context": ["E : Type u_1", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℂ E", "f g : ℝ → E", "s : ℂ", "hf : MellinConvergent f s", "hg : MellinConvergent g s"], "goal": "MellinConvergent (fun t => f t - g t) s"}], "premise": [28501, 108341], "state_str": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf g : ℝ → E\ns : ℂ\nhf : MellinConvergent f s\nhg : MellinConvergent g s\n⊢ MellinConvergent (fun t => f t - g t) s"}
+{"state": [{"context": ["E : Type u_1", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℂ E", "f g : ℝ → E", "s : ℂ", "hf : MellinConvergent f s", "hg : MellinConvergent g s"], "goal": "mellin (fun t => f t - g t) s = mellin f s - mellin g s"}], "premise": [33646, 108341], "state_str": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf g : ℝ → E\ns : ℂ\nhf : MellinConvergent f s\nhg : MellinConvergent g s\n⊢ mellin (fun t => f t - g t) s = mellin f s - mellin g s"}
+{"state": [{"context": ["A B : Grp", "f : A ⟶ B", "x : ↑B", "y : ↑(Set.range fun x => x • ↑(MonoidHom.range f))"], "goal": "x • ↑y ∈ Set.range fun x => x • ↑(MonoidHom.range f)"}], "premise": [2115], "state_str": "A B : Grp\nf : A ⟶ B\nx : ↑B\ny : ↑(Set.range fun x => x • ↑(MonoidHom.range f))\n⊢ x • ↑y ∈ Set.range fun x => x • ↑(MonoidHom.range f)"}
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+{"state": [{"context": ["n : ℕ", "α : TypeVec.{u_1} (n + 1)", "β : TypeVec.{u_2} (n + 1)", "f g : α ⟹ β", "h₀ : dropFun f = dropFun g", "h₁ : lastFun f = lastFun g"], "goal": "f = g"}], "premise": [1838], "state_str": "n : ℕ\nα : TypeVec.{u_1} (n + 1)\nβ : TypeVec.{u_2} (n + 1)\nf g : α ⟹ β\nh₀ : dropFun f = dropFun g\nh₁ : lastFun f = lastFun g\n⊢ f = g"}
+{"state": [{"context": ["n : ℕ+", "S T : Set ℕ+", "A : Type u", "B : Type v", "K : Type w", "L : Type z", "inst✝⁵ : CommRing A", "inst✝⁴ : CommRing B", "inst✝³ : Algebra A B", "inst✝² : Field K", "inst✝¹ : Field L", "inst✝ : Algebra K L", "h : ⊥ = ⊤"], "goal": "IsCyclotomicExtension {1} A B"}], "premise": [1673, 24203, 24210], "state_str": "n : ℕ+\nS T : Set ℕ+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra A B\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nh : ⊥ = ⊤\n⊢ IsCyclotomicExtension {1} A B"}
+{"state": [{"context": ["K : Type u_1", "inst✝² : DecidableEq K", "Γ : K → Type u_2", "Λ : Type u_3", "inst✝¹ : Inhabited Λ", "σ : Type u_4", "inst✝ : Inhabited σ", "M : Λ → Stmt₂", "q : Stmt₂", "v : σ", "T : ListBlank ((i : K) → Option (Γ i))", "k : K", "S : (k : K) → List (Γ k)", "hT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse", "o : StAct k", "IH : ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))}, (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) → ∃ b, TrCfg (TM2.stepAux q v S) b ∧ Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b"], "goal": "∃ b, TrCfg (TM2.stepAux (stRun o q) v S) b ∧ Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal (stRun o q)) v (Tape.mk' ∅ (addBottom T))) b"}], "premise": [73592, 73593], "state_str": "K : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_3\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nq : Stmt₂\nv : σ\nT : ListBlank ((i : K) → Option (Γ i))\nk : K\nS : (k : K) → List (Γ k)\nhT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse\no : StAct k\nIH :\n  ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))},\n    (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) →\n      ∃ b,\n        TrCfg (TM2.stepAux q v S) b ∧\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b\n⊢ ∃ b,\n    TrCfg (TM2.stepAux (stRun o q) v S) b ∧\n      Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal (stRun o q)) v (Tape.mk' ∅ (addBottom T))) b"}
+{"state": [{"context": ["K : Type u_1", "inst✝² : DecidableEq K", "Γ : K → Type u_2", "Λ : Type u_3", "inst✝¹ : Inhabited Λ", "σ : Type u_4", "inst✝ : Inhabited σ", "M : Λ → Stmt₂", "q : Stmt₂", "v : σ", "T : ListBlank ((i : K) → Option (Γ i))", "k : K", "S : (k : K) → List (Γ k)", "hT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse", "o : StAct k", "IH : ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))}, (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) → ∃ b, TrCfg (TM2.stepAux q v S) b ∧ Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b"], "goal": "∃ b, TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b ∧ Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' ∅ (addBottom T))) b"}], "premise": [14272, 73596], "state_str": "K : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_3\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nq : Stmt₂\nv : σ\nT : ListBlank ((i : K) → Option (Γ i))\nk : K\nS : (k : K) → List (Γ k)\nhT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse\no : StAct k\nIH :\n  ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))},\n    (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) →\n      ∃ b,\n        TrCfg (TM2.stepAux q v S) b ∧\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b\n⊢ ∃ b,\n    TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b ∧\n      Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' ∅ (addBottom T))) b"}
+{"state": [{"context": ["K : Type u_1", "inst✝² : DecidableEq K", "Γ : K → Type u_2", "Λ : Type u_3", "inst✝¹ : Inhabited Λ", "σ : Type u_4", "inst✝ : Inhabited σ", "M : Λ → Stmt₂", "q : Stmt₂", "v : σ", "T : ListBlank ((i : K) → Option (Γ i))", "k : K", "S : (k : K) → List (Γ k)", "hT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse", "o : StAct k", "IH : ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))}, (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) → ∃ b, TrCfg (TM2.stepAux q v S) b ∧ Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b", "hgo : Reaches₀ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) } { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)) }"], "goal": "∃ b, TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b ∧ Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' ∅ (addBottom T))) b"}], "premise": [73595], "state_str": "K : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_3\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nq : Stmt₂\nv : σ\nT : ListBlank ((i : K) → Option (Γ i))\nk : K\nS : (k : K) → List (Γ k)\nhT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse\no : StAct k\nIH :\n  ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))},\n    (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) →\n      ∃ b,\n        TrCfg (TM2.stepAux q v S) b ∧\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b\nhgo :\n  Reaches₀ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) }\n    { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)) }\n⊢ ∃ b,\n    TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b ∧\n      Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' ∅ (addBottom T))) b"}
+{"state": [{"context": ["K : Type u_1", "inst✝² : DecidableEq K", "Γ : K → Type u_2", "Λ : Type u_3", "inst✝¹ : Inhabited Λ", "σ : Type u_4", "inst✝ : Inhabited σ", "M : Λ → Stmt₂", "q : Stmt₂", "v : σ", "T : ListBlank ((i : K) → Option (Γ i))", "k : K", "S : (k : K) → List (Γ k)", "hT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse", "o : StAct k", "IH : ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))}, (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) → ∃ b, TrCfg (TM2.stepAux q v S) b ∧ Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b", "hgo : Reaches₀ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) } { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)) }", "T' : ListBlank ((k : K) → Option (Γ k))", "hT' : ∀ (k_1 : K), ListBlank.map (proj k_1) T' = ListBlank.mk (List.map some (update S k (stWrite ?m.379536 (S k) o) k_1)).reverse", "hrun : TM1.stepAux (trStAct ?m.379535 o) ?m.379536 ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T))) = TM1.stepAux ?m.379535 (stVar ?m.379536 (S k) o) ((Tape.move Dir.right)^[(update S k (stWrite ?m.379536 (S k) o) k).length] (Tape.mk' ∅ (addBottom T')))"], "goal": "∃ b, TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b ∧ Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' ∅ (addBottom T))) b"}], "premise": [73538], "state_str": "case intro.intro\nK : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_3\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nq : Stmt₂\nv : σ\nT : ListBlank ((i : K) → Option (Γ i))\nk : K\nS : (k : K) → List (Γ k)\nhT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse\no : StAct k\nIH :\n  ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))},\n    (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) →\n      ∃ b,\n        TrCfg (TM2.stepAux q v S) b ∧\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b\nhgo :\n  Reaches₀ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) }\n    { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)) }\nT' : ListBlank ((k : K) → Option (Γ k))\nhT' :\n  ∀ (k_1 : K),\n    ListBlank.map (proj k_1) T' = ListBlank.mk (List.map some (update S k (stWrite ?m.379536 (S k) o) k_1)).reverse\nhrun :\n  TM1.stepAux (trStAct ?m.379535 o) ?m.379536 ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T))) =\n    TM1.stepAux ?m.379535 (stVar ?m.379536 (S k) o)\n      ((Tape.move Dir.right)^[(update S k (stWrite ?m.379536 (S k) o) k).length] (Tape.mk' ∅ (addBottom T')))\n⊢ ∃ b,\n    TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b ∧\n      Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' ∅ (addBottom T))) b"}
+{"state": [{"context": ["K : Type u_1", "inst✝² : DecidableEq K", "Γ : K → Type u_2", "Λ : Type u_3", "inst✝¹ : Inhabited Λ", "σ : Type u_4", "inst✝ : Inhabited σ", "M : Λ → Stmt₂", "q : Stmt₂", "v : σ", "T : ListBlank ((i : K) → Option (Γ i))", "k : K", "S : (k : K) → List (Γ k)", "hT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse", "o : StAct k", "IH : ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))}, (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) → ∃ b, TrCfg (TM2.stepAux q v S) b ∧ Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b", "hgo : Reaches₀ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) } { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)) }", "T' : ListBlank ((k : K) → Option (Γ k))", "hT' : ∀ (k_1 : K), ListBlank.map (proj k_1) T' = ListBlank.mk (List.map some (update S k (stWrite ?m.379536 (S k) o) k_1)).reverse", "hrun : TM1.stepAux (trStAct ?m.379535 o) ?m.379536 ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T))) = TM1.stepAux ?m.379535 (stVar ?m.379536 (S k) o) ((Tape.move Dir.right)^[(update S k (stWrite ?m.379536 (S k) o) k).length] (Tape.mk' ∅ (addBottom T')))", "this : Reaches₁ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) } (TM1.stepAux (tr M (go k o q)) v ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T))))"], "goal": "∃ b, TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b ∧ Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' ∅ (addBottom T))) b"}], "premise": [4985, 5278, 14277, 73513, 73516, 73585, 73588], "state_str": "case intro.intro\nK : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_3\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nq : Stmt₂\nv : σ\nT : ListBlank ((i : K) → Option (Γ i))\nk : K\nS : (k : K) → List (Γ k)\nhT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse\no : StAct k\nIH :\n  ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))},\n    (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) →\n      ∃ b,\n        TrCfg (TM2.stepAux q v S) b ∧\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b\nhgo :\n  Reaches₀ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) }\n    { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)) }\nT' : ListBlank ((k : K) → Option (Γ k))\nhT' :\n  ∀ (k_1 : K),\n    ListBlank.map (proj k_1) T' = ListBlank.mk (List.map some (update S k (stWrite ?m.379536 (S k) o) k_1)).reverse\nhrun :\n  TM1.stepAux (trStAct ?m.379535 o) ?m.379536 ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T))) =\n    TM1.stepAux ?m.379535 (stVar ?m.379536 (S k) o)\n      ((Tape.move Dir.right)^[(update S k (stWrite ?m.379536 (S k) o) k).length] (Tape.mk' ∅ (addBottom T')))\nthis :\n  Reaches₁ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) }\n    (TM1.stepAux (tr M (go k o q)) v ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T))))\n⊢ ∃ b,\n    TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b ∧\n      Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' ∅ (addBottom T))) b"}
+{"state": [{"context": ["K : Type u_1", "inst✝² : DecidableEq K", "Γ : K → Type u_2", "Λ : Type u_3", "inst✝¹ : Inhabited Λ", "σ : Type u_4", "inst✝ : Inhabited σ", "M : Λ → Stmt₂", "q : Stmt₂", "v : σ", "T : ListBlank ((i : K) → Option (Γ i))", "k : K", "S : (k : K) → List (Γ k)", "hT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse", "o : StAct k", "IH : ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))}, (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) → ∃ b, TrCfg (TM2.stepAux q v S) b ∧ Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b", "hgo : Reaches₀ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) } { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)) }", "T' : ListBlank ((k : K) → Option (Γ k))", "hT' : ∀ (k_1 : K), ListBlank.map (proj k_1) T' = ListBlank.mk (List.map some (update S k (stWrite v (S k) o) k_1)).reverse", "hrun : TM1.stepAux (trStAct (goto fun x x => ret q) o) v ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T))) = TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o) ((Tape.move Dir.right)^[(update S k (stWrite v (S k) o) k).length] (Tape.mk' ∅ (addBottom T')))", "this : Reaches₁ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) } (match match none with | some val => false | none => true with | true => TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o) ((Tape.move Dir.right)^[(update S k (stWrite v (S k) o) k).length] (Tape.mk' ∅ (addBottom T'))) | false => TM1.stepAux (goto fun x x => go k o q) v (Tape.move Dir.right ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)))))", "c : TM1.Cfg Γ' Λ' σ", "gc : TrCfg (TM2.stepAux q ?m.383145 (update S k (stWrite v (S k) o))) c", "rc : Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) ?m.383145 (Tape.mk' ∅ (addBottom T'))) c"], "goal": "∃ b, TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b ∧ Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' ∅ (addBottom T))) b"}], "premise": [70456, 70459, 73530, 73536, 73597], "state_str": "case intro.intro.intro.intro\nK : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_3\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nq : Stmt₂\nv : σ\nT : ListBlank ((i : K) → Option (Γ i))\nk : K\nS : (k : K) → List (Γ k)\nhT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse\no : StAct k\nIH :\n  ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))},\n    (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) →\n      ∃ b,\n        TrCfg (TM2.stepAux q v S) b ∧\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b\nhgo :\n  Reaches₀ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) }\n    { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)) }\nT' : ListBlank ((k : K) → Option (Γ k))\nhT' :\n  ∀ (k_1 : K), ListBlank.map (proj k_1) T' = ListBlank.mk (List.map some (update S k (stWrite v (S k) o) k_1)).reverse\nhrun :\n  TM1.stepAux (trStAct (goto fun x x => ret q) o) v ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T))) =\n    TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n      ((Tape.move Dir.right)^[(update S k (stWrite v (S k) o) k).length] (Tape.mk' ∅ (addBottom T')))\nthis :\n  Reaches₁ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) }\n    (match\n      match none with\n      | some val => false\n      | none => true with\n    | true =>\n      TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n        ((Tape.move Dir.right)^[(update S k (stWrite v (S k) o) k).length] (Tape.mk' ∅ (addBottom T')))\n    | false =>\n      TM1.stepAux (goto fun x x => go k o q) v\n        (Tape.move Dir.right ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)))))\nc : TM1.Cfg Γ' Λ' σ\ngc : TrCfg (TM2.stepAux q ?m.383145 (update S k (stWrite v (S k) o))) c\nrc : Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) ?m.383145 (Tape.mk' ∅ (addBottom T'))) c\n⊢ ∃ b,\n    TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b ∧\n      Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' ∅ (addBottom T))) b"}
+{"state": [{"context": ["K : Type u_1", "inst✝² : DecidableEq K", "Γ : K → Type u_2", "Λ : Type u_3", "inst✝¹ : Inhabited Λ", "σ : Type u_4", "inst✝ : Inhabited σ", "M : Λ → Stmt₂", "q : Stmt₂", "v : σ", "T : ListBlank ((i : K) → Option (Γ i))", "k : K", "S : (k : K) → List (Γ k)", "hT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse", "o : StAct k", "IH : ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))}, (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) → ∃ b, TrCfg (TM2.stepAux q v S) b ∧ Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b", "hgo : Reaches₀ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) } { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)) }", "T' : ListBlank ((k : K) → Option (Γ k))", "hT' : ∀ (k_1 : K), ListBlank.map (proj k_1) T' = ListBlank.mk (List.map some (update S k (stWrite v (S k) o) k_1)).reverse", "hrun : TM1.stepAux (trStAct (goto fun x x => ret q) o) v ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T))) = TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o) ((Tape.move Dir.right)^[(update S k (stWrite v (S k) o) k).length] (Tape.mk' ∅ (addBottom T')))", "this : Reaches₁ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) } (match match none with | some val => false | none => true with | true => TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o) ((Tape.move Dir.right)^[(update S k (stWrite v (S k) o) k).length] (Tape.mk' ∅ (addBottom T'))) | false => TM1.stepAux (goto fun x x => go k o q) v (Tape.move Dir.right ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)))))", "c : TM1.Cfg Γ' Λ' σ", "gc : TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) c", "rc : Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) (stVar v (S k) o) (Tape.mk' ∅ (addBottom T'))) c"], "goal": "ReflTransGen (fun a b => b ∈ TM1.step (tr M) a) (TM1.stepAux (tr M (ret q)) (stVar v (S k) o) (Tape.mk' ∅ (addBottom T'))) c"}], "premise": [73504, 73590], "state_str": "case intro.intro.intro.intro\nK : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_3\ninst✝¹ : Inhabited Λ\nσ : Type u_4\ninst✝ : Inhabited σ\nM : Λ → Stmt₂\nq : Stmt₂\nv : σ\nT : ListBlank ((i : K) → Option (Γ i))\nk : K\nS : (k : K) → List (Γ k)\nhT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse\no : StAct k\nIH :\n  ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))},\n    (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) →\n      ∃ b,\n        TrCfg (TM2.stepAux q v S) b ∧\n          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b\nhgo :\n  Reaches₀ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) }\n    { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)) }\nT' : ListBlank ((k : K) → Option (Γ k))\nhT' :\n  ∀ (k_1 : K), ListBlank.map (proj k_1) T' = ListBlank.mk (List.map some (update S k (stWrite v (S k) o) k_1)).reverse\nhrun :\n  TM1.stepAux (trStAct (goto fun x x => ret q) o) v ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T))) =\n    TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n      ((Tape.move Dir.right)^[(update S k (stWrite v (S k) o) k).length] (Tape.mk' ∅ (addBottom T')))\nthis :\n  Reaches₁ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) }\n    (match\n      match none with\n      | some val => false\n      | none => true with\n    | true =>\n      TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o)\n        ((Tape.move Dir.right)^[(update S k (stWrite v (S k) o) k).length] (Tape.mk' ∅ (addBottom T')))\n    | false =>\n      TM1.stepAux (goto fun x x => go k o q) v\n        (Tape.move Dir.right ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)))))\nc : TM1.Cfg Γ' Λ' σ\ngc : TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) c\nrc : Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) (stVar v (S k) o) (Tape.mk' ∅ (addBottom T'))) c\n⊢ ReflTransGen (fun a b => b ∈ TM1.step (tr M) a)\n    (TM1.stepAux (tr M (ret q)) (stVar v (S k) o) (Tape.mk' ∅ (addBottom T'))) c"}
+{"state": [{"context": ["α : Type u_1", "ι : Type u_2", "β : ι → Type u_3", "inst✝² : Fintype ι", "inst✝¹ : (i : ι) → DecidableEq (β i)", "γ : ι → Type u_4", "inst✝ : (i : ι) → DecidableEq (γ i)", "x y z : (i : ι) → β i"], "goal": "hammingDist x y ≤ hammingDist x z + hammingDist y z"}], "premise": [20096], "state_str": "α : Type u_1\nι : Type u_2\nβ : ι → Type u_3\ninst✝² : Fintype ι\ninst✝¹ : (i : ι) → DecidableEq (β i)\nγ : ι → Type u_4\ninst✝ : (i : ι) → DecidableEq (γ i)\nx y z : (i : ι) → β i\n⊢ hammingDist x y ≤ hammingDist x z + hammingDist y z"}
+{"state": [{"context": ["α : Type u_1", "ι : Type u_2", "β : ι → Type u_3", "inst✝² : Fintype ι", "inst✝¹ : (i : ι) → DecidableEq (β i)", "γ : ι → Type u_4", "inst✝ : (i : ι) → DecidableEq (γ i)", "x y z : (i : ι) → β i"], "goal": "hammingDist x y ≤ hammingDist x z + hammingDist z y"}], "premise": [20097], "state_str": "α : Type u_1\nι : Type u_2\nβ : ι → Type u_3\ninst✝² : Fintype ι\ninst✝¹ : (i : ι) → DecidableEq (β i)\nγ : ι → Type u_4\ninst✝ : (i : ι) → DecidableEq (γ i)\nx y z : (i : ι) → β i\n⊢ hammingDist x y ≤ hammingDist x z + hammingDist z y"}
+{"state": [{"context": ["K R : Type v", "V M : Type w", "inst✝⁵ : CommRing R", "inst✝⁴ : AddCommGroup M", "inst✝³ : Module R M", "inst✝² : Field K", "inst✝¹ : AddCommGroup V", "inst✝ : Module K V", "f : End R M", "μ : R", "v : M", "hv : f.HasEigenvector μ v", "n : ℕ"], "goal": "(f ^ n) v = μ ^ n • v"}], "premise": [88219, 118909, 119742, 119745], "state_str": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End R M\nμ : R\nv : M\nhv : f.HasEigenvector μ v\nn : ℕ\n⊢ (f ^ n) v = μ ^ n • v"}
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+{"state": [{"context": ["Γ✝ : Type u_1", "Γ' : Type u_2", "R : Type u_3", "V : Type u_4", "inst✝³ : OrderedCancelAddCommMonoid Γ✝", "Γ : Type u_5", "inst✝² : LinearOrderedCancelAddCommMonoid Γ", "inst✝¹ : Semiring R", "inst✝ : NoZeroDivisors R", "x : HahnSeries Γ R", "h : ℕ", "IH : (x ^ h).order = h • x.order"], "goal": "(x ^ (h + 1)).order = (h + 1) • x.order"}], "premise": [70039], "state_str": "case succ\nΓ✝ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\ninst✝³ : OrderedCancelAddCommMonoid Γ✝\nΓ : Type u_5\ninst✝² : LinearOrderedCancelAddCommMonoid Γ\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\nx : HahnSeries Γ R\nh : ℕ\nIH : (x ^ h).order = h • x.order\n⊢ (x ^ (h + 1)).order = (h + 1) • x.order"}
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+{"state": [{"context": ["R : Type uR", "S : Type uS", "A : Type uA", "B : Type uB", "inst✝⁸ : CommSemiring R", "inst✝⁷ : CommSemiring S", "inst✝⁶ : Semiring A", "inst✝⁵ : Semiring B", "inst✝⁴ : Algebra R S", "inst✝³ : Algebra R A", "inst✝² : Algebra S A", "inst✝¹ : Algebra R B", "inst✝ : IsScalarTower R S A", "s t : Set A"], "goal": "Submonoid.closure s ≤ (adjoin R s).toSubmonoid"}], "premise": [1674, 78306, 117653], "state_str": "case a\nR : Type uR\nS : Type uS\nA : Type uA\nB : Type uB\ninst✝⁸ : CommSemiring R\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Semiring A\ninst✝⁵ : Semiring B\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : Algebra R B\ninst✝ : IsScalarTower R S A\ns t : Set A\n⊢ Submonoid.closure s ≤ (adjoin R s).toSubmonoid"}
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+{"state": [{"context": ["B : Type u₁", "inst✝² : Bicategory B", "C : Type u₂", "inst✝¹ : Bicategory C", "D : Type u₃", "inst✝ : Bicategory D", "F✝ : Pseudofunctor B C", "F : LaxFunctor B C", "F' : F.PseudoCore", "a✝ b✝ c✝ : B", "f✝ g✝ : a✝ ⟶ b✝", "η : f✝ ⟶ g✝", "h : b✝ ⟶ c✝"], "goal": "F.map₂ (η ▷ h) = ((fun {a b c} => F'.mapCompIso) f✝ h).hom ≫ F.map₂ η ▷ F.map h ≫ ((fun {a b c} => F'.mapCompIso) g✝ h).inv"}], "premise": [99888, 99902], "state_str": "B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory D\nF✝ : Pseudofunctor B C\nF : LaxFunctor B C\nF' : F.PseudoCore\na✝ b✝ c✝ : B\nf✝ g✝ : a✝ ⟶ b✝\nη : f✝ ⟶ g✝\nh : b✝ ⟶ c✝\n⊢ F.map₂ (η ▷ h) =\n    ((fun {a b c} => F'.mapCompIso) f✝ h).hom ≫ F.map₂ η ▷ F.map h ≫ ((fun {a b c} => F'.mapCompIso) g✝ h).inv"}
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+{"state": [{"context": ["α : Type u_1", "inst✝³ : Fintype α", "G✝ : Type u_2", "inst✝² : Group G✝", "n : ℕ", "G : Type u_3", "inst✝¹ : Group G", "inst✝ : Fintype G", "p : ℕ", "hp : Fact (Nat.Prime p)", "hdvd : p ∣ Fintype.card G", "hp' : p - 1 ≠ 0", "Scard : p ∣ Fintype.card ↑(vectorsProdEqOne G p)", "f : ℕ → ↑(vectorsProdEqOne G p) → ↑(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k", "hf1 : ∀ (v : ↑(vectorsProdEqOne G p)), f 0 v = v", "hf2 : ∀ (j k : ℕ) (v : ↑(vectorsProdEqOne G p)), f k (f j v) = f (j + k) v", "hf3 : ∀ (v : ↑(vectorsProdEqOne G p)), f p v = v", "σ : ↑(vectorsProdEqOne G p) ≃ ↑(vectorsProdEqOne G p) := { toFun := f 1, invFun := f (p - 1), left_inv := ⋯, right_inv := ⋯ }", "hσ : σ ^ p ^ 1 = 1", "v₀ : ↑(vectorsProdEqOne G p) := ⟨Vector.replicate p 1, ⋯⟩", "hv₀ : σ v₀ = v₀", "v : ↑(vectorsProdEqOne G p)", "hv1 : σ v = v", "hv2 : v ≠ v₀"], "goal": "∃ x, orderOf x = p"}], "premise": [1673, 2011, 8415, 130401, 137128], "state_str": "case intro.intro\nα : Type u_1\ninst✝³ : Fintype α\nG✝ : Type u_2\ninst✝² : Group G✝\nn : ℕ\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : Fintype G\np : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ∣ Fintype.card G\nhp' : p - 1 ≠ 0\nScard : p ∣ Fintype.card ↑(vectorsProdEqOne G p)\nf : ℕ → ↑(vectorsProdEqOne G p) → ↑(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k\nhf1 : ∀ (v : ↑(vectorsProdEqOne G p)), f 0 v = v\nhf2 : ∀ (j k : ℕ) (v : ↑(vectorsProdEqOne G p)), f k (f j v) = f (j + k) v\nhf3 : ∀ (v : ↑(vectorsProdEqOne G p)), f p v = v\nσ : ↑(vectorsProdEqOne G p) ≃ ↑(vectorsProdEqOne G p) :=\n  { toFun := f 1, invFun := f (p - 1), left_inv := ⋯, right_inv := ⋯ }\nhσ : σ ^ p ^ 1 = 1\nv₀ : ↑(vectorsProdEqOne G p) := ⟨Vector.replicate p 1, ⋯⟩\nhv₀ : σ v₀ = v₀\nv : ↑(vectorsProdEqOne G p)\nhv1 : σ v = v\nhv2 : v ≠ v₀\n⊢ ∃ x, orderOf x = p"}
+{"state": [{"context": ["θ : Angle"], "goal": "(2 • θ).toReal = 2 * θ.toReal - 2 * π ↔ π / 2 < θ.toReal"}], "premise": [38345], "state_str": "θ : Angle\n⊢ (2 • θ).toReal = 2 * θ.toReal - 2 * π ↔ π / 2 < θ.toReal"}
+{"state": [{"context": ["θ : Angle"], "goal": "(2 • ↑θ.toReal).toReal = 2 * θ.toReal - 2 * π ↔ π / 2 < θ.toReal"}], "premise": [20162, 38270, 38369, 119749, 122222], "state_str": "θ : Angle\n⊢ (2 • ↑θ.toReal).toReal = 2 * θ.toReal - 2 * π ↔ π / 2 < θ.toReal"}
+{"state": [{"context": ["θ : Angle"], "goal": "π < 2 * θ.toReal ∧ 2 * θ.toReal ≤ 3 * π ↔ π / 2 < θ.toReal"}], "premise": [1673, 38347, 38514, 103768, 106030], "state_str": "θ : Angle\n⊢ π < 2 * θ.toReal ∧ 2 * θ.toReal ≤ 3 * π ↔ π / 2 < θ.toReal"}
+{"state": [{"context": ["n✝ n : ℕ", "h1 : 1 < n", "h : ∀ (m : ℕ), m < n → m ≠ 0 → n.Coprime m"], "goal": "Prime n"}], "premise": [1674, 144299], "state_str": "n✝ n : ℕ\nh1 : 1 < n\nh : ∀ (m : ℕ), m < n → m ≠ 0 → n.Coprime m\n⊢ Prime n"}
+{"state": [{"context": ["n✝ n : ℕ", "h1 : 1 < n", "h : ∀ (m : ℕ), m < n → m ≠ 0 → n.Coprime m", "m : ℕ", "mlt : m < n", "mdvd : m ∣ n"], "goal": "m = 1"}], "premise": [11234, 108656], "state_str": "n✝ n : ℕ\nh1 : 1 < n\nh : ∀ (m : ℕ), m < n → m ≠ 0 → n.Coprime m\nm : ℕ\nmlt : m < n\nmdvd : m ∣ n\n⊢ m = 1"}
+{"state": [{"context": ["n✝ n : ℕ", "h1 : 1 < n", "h : ∀ (m : ℕ), m < n → m ≠ 0 → n.Coprime m", "m : ℕ", "mlt : m < n", "mdvd : m ∣ n", "hm : m ≠ 0"], "goal": "m = 1"}], "premise": [395, 433], "state_str": "n✝ n : ℕ\nh1 : 1 < n\nh : ∀ (m : ℕ), m < n → m ≠ 0 → n.Coprime m\nm : ℕ\nmlt : m < n\nmdvd : m ∣ n\nhm : m ≠ 0\n⊢ m = 1"}
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+{"state": [{"context": ["z : ℍ", "N : ℕ", "hn : 0 < N", "n : ℤ := ⌊z.re / ↑N⌋"], "goal": "∃ n, ModularGroup.T ^ (↑N * n) • z ∈ verticalStrip (↑N) z.im"}], "premise": [1674, 2045], "state_str": "z : ℍ\nN : ℕ\nhn : 0 < N\nn : ℤ := ⌊z.re / ↑N⌋\n⊢ ∃ n, ModularGroup.T ^ (↑N * n) • z ∈ verticalStrip (↑N) z.im"}
+{"state": [{"context": ["z : ℍ", "N : ℕ", "hn : 0 < N", "n : ℤ := ⌊z.re / ↑N⌋"], "goal": "ModularGroup.T ^ (↑N * -n) • z ∈ verticalStrip (↑N) z.im"}], "premise": [46751], "state_str": "case h\nz : ℍ\nN : ℕ\nhn : 0 < N\nn : ℤ := ⌊z.re / ↑N⌋\n⊢ ModularGroup.T ^ (↑N * -n) • z ∈ verticalStrip (↑N) z.im"}
+{"state": [{"context": ["z : ℍ", "N : ℕ", "hn : 0 < N", "n : ℤ := ⌊z.re / ↑N⌋"], "goal": "↑(↑N * -n) +ᵥ z ∈ verticalStrip (↑N) z.im"}], "premise": [14271, 46749, 121603, 122241, 128750, 128753], "state_str": "case h\nz : ℍ\nN : ℕ\nhn : 0 < N\nn : ℤ := ⌊z.re / ↑N⌋\n⊢ ↑(↑N * -n) +ᵥ z ∈ verticalStrip (↑N) z.im"}
+{"state": [{"context": ["z : ℍ", "N : ℕ", "hn : 0 < N", "n : ℤ := ⌊z.re / ↑N⌋"], "goal": "|(↑(↑N * -n) +ᵥ z).re| ≤ ↑N"}], "premise": [46748, 122240, 122241, 128750], "state_str": "case h\nz : ℍ\nN : ℕ\nhn : 0 < N\nn : ℤ := ⌊z.re / ↑N⌋\n⊢ |(↑(↑N * -n) +ᵥ z).re| ≤ ↑N"}
+{"state": [{"context": ["z : ℍ", "N : ℕ", "hn : 0 < N", "n : ℤ := ⌊z.re / ↑N⌋", "h : (↑(↑N * -n) +ᵥ z).re = ↑(-↑N * ⌊z.re / ↑N⌋) + z.re"], "goal": "|(↑(↑N * -n) +ᵥ z).re| ≤ ↑N"}], "premise": [119708], "state_str": "case h\nz : ℍ\nN : ℕ\nhn : 0 < N\nn : ℤ := ⌊z.re / ↑N⌋\nh : (↑(↑N * -n) +ᵥ z).re = ↑(-↑N * ⌊z.re / ↑N⌋) + z.re\n⊢ |(↑(↑N * -n) +ᵥ z).re| ≤ ↑N"}
+{"state": [{"context": ["z : ℍ", "N : ℕ", "hn : 0 < N", "n : ℤ := ⌊z.re / ↑N⌋", "h : (↑(↑N * -n) +ᵥ z).re = ↑(-↑N * ⌊z.re / ↑N⌋) + z.re"], "goal": "|z.re + ↑(-↑N * ⌊z.re / ↑N⌋)| ≤ ↑N"}], "premise": [121603, 122240, 128750, 128753], "state_str": "case h\nz : ℍ\nN : ℕ\nhn : 0 < N\nn : ℤ := ⌊z.re / ↑N⌋\nh : (↑(↑N * -n) +ᵥ z).re = ↑(-↑N * ⌊z.re / ↑N⌋) + z.re\n⊢ |z.re + ↑(-↑N * ⌊z.re / ↑N⌋)| ≤ ↑N"}
+{"state": [{"context": ["z : ℍ", "N : ℕ", "hn : 0 < N", "n : ℤ := ⌊z.re / ↑N⌋", "h : (↑(↑N * -n) +ᵥ z).re = ↑(-↑N * ⌊z.re / ↑N⌋) + z.re", "hnn : 0 < ↑N", "h2 : z.re + -(↑N * ↑n) = z.re - ↑n * ↑N"], "goal": "|z.re + -(↑N * ↑⌊z.re / ↑N⌋)| ≤ ↑N"}], "premise": [1674, 105170, 105406], "state_str": "case h\nz : ℍ\nN : ℕ\nhn : 0 < N\nn : ℤ := ⌊z.re / ↑N⌋\nh : (↑(↑N * -n) +ᵥ z).re = ↑(-↑N * ⌊z.re / ↑N⌋) + z.re\nhnn : 0 < ↑N\nh2 : z.re + -(↑N * ↑n) = z.re - ↑n * ↑N\n⊢ |z.re + -(↑N * ↑⌊z.re / ↑N⌋)| ≤ ↑N"}
+{"state": [{"context": ["z : ℍ", "N : ℕ", "hn : 0 < N", "n : ℤ := ⌊z.re / ↑N⌋", "h : (↑(↑N * -n) +ᵥ z).re = ↑(-↑N * ⌊z.re / ↑N⌋) + z.re", "hnn : 0 < ↑N", "h2 : z.re + -(↑N * ↑n) = z.re - ↑n * ↑N"], "goal": "z.re - ↑⌊z.re / ↑N⌋ * ↑N ≤ ↑N"}], "premise": [105171], "state_str": "case h\nz : ℍ\nN : ℕ\nhn : 0 < N\nn : ℤ := ⌊z.re / ↑N⌋\nh : (↑(↑N * -n) +ᵥ z).re = ↑(-↑N * ⌊z.re / ↑N⌋) + z.re\nhnn : 0 < ↑N\nh2 : z.re + -(↑N * ↑n) = z.re - ↑n * ↑N\n⊢ z.re - ↑⌊z.re / ↑N⌋ * ↑N ≤ ↑N"}
+{"state": [{"context": ["C : Type u", "inst✝² : Category.{v, u} C", "inst✝¹ : HasFiniteProducts C", "inst✝ : HasPullbacks C", "X Y : Dial C", "e₁ : X.src ≅ Y.src", "e₂ : X.tgt ≅ Y.tgt", "eq : X.rel = (Subobject.pullback (prod.map e₁.hom e₂.hom)).obj Y.rel"], "goal": "(Subobject.pullback π(π₁, π₂ ≫ e₂.inv)).obj X.rel ≤ (Subobject.pullback (prod.map e₁.hom (𝟙 Y.tgt))).obj Y.rel"}], "premise": [89300], "state_str": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasFiniteProducts C\ninst✝ : HasPullbacks C\nX Y : Dial C\ne₁ : X.src ≅ Y.src\ne₂ : X.tgt ≅ Y.tgt\neq : X.rel = (Subobject.pullback (prod.map e₁.hom e₂.hom)).obj Y.rel\n⊢ (Subobject.pullback π(π₁, π₂ ≫ e₂.inv)).obj X.rel ≤ (Subobject.pullback (prod.map e₁.hom (𝟙 Y.tgt))).obj Y.rel"}
+{"state": [{"context": ["C : Type u", "inst✝² : Category.{v, u} C", "inst✝¹ : HasFiniteProducts C", "inst✝ : HasPullbacks C", "X Y : Dial C", "e₁ : X.src ≅ Y.src", "e₂ : X.tgt ≅ Y.tgt", "eq : X.rel = (Subobject.pullback (prod.map e₁.hom e₂.hom)).obj Y.rel"], "goal": "(Subobject.pullback (π(π₁, π₂ ≫ e₂.inv) ≫ prod.map e₁.hom e₂.hom)).obj Y.rel ≤ (Subobject.pullback (prod.map e₁.hom (𝟙 Y.tgt))).obj Y.rel"}], "premise": [14277], "state_str": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasFiniteProducts C\ninst✝ : HasPullbacks C\nX Y : Dial C\ne₁ : X.src ≅ Y.src\ne₂ : X.tgt ≅ Y.tgt\neq : X.rel = (Subobject.pullback (prod.map e₁.hom e₂.hom)).obj Y.rel\n⊢ (Subobject.pullback (π(π₁, π₂ ≫ e₂.inv) ≫ prod.map e₁.hom e₂.hom)).obj Y.rel ≤\n    (Subobject.pullback (prod.map e₁.hom (𝟙 Y.tgt))).obj Y.rel"}
+{"state": [{"context": ["C : Type u", "inst✝² : Category.{v, u} C", "inst✝¹ : HasFiniteProducts C", "inst✝ : HasPullbacks C", "X Y : Dial C", "e₁ : X.src ≅ Y.src", "e₂ : X.tgt ≅ Y.tgt", "eq : X.rel = (Subobject.pullback (prod.map e₁.hom e₂.hom)).obj Y.rel"], "goal": "(Subobject.pullback π(π₁, π₂ ≫ e₂.hom)).obj Y.rel ≤ (Subobject.pullback (prod.map e₁.inv (𝟙 X.tgt))).obj X.rel"}], "premise": [89300], "state_str": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasFiniteProducts C\ninst✝ : HasPullbacks C\nX Y : Dial C\ne₁ : X.src ≅ Y.src\ne₂ : X.tgt ≅ Y.tgt\neq : X.rel = (Subobject.pullback (prod.map e₁.hom e₂.hom)).obj Y.rel\n⊢ (Subobject.pullback π(π₁, π₂ ≫ e₂.hom)).obj Y.rel ≤ (Subobject.pullback (prod.map e₁.inv (𝟙 X.tgt))).obj X.rel"}
+{"state": [{"context": ["C : Type u", "inst✝² : Category.{v, u} C", "inst✝¹ : HasFiniteProducts C", "inst✝ : HasPullbacks C", "X Y : Dial C", "e₁ : X.src ≅ Y.src", "e₂ : X.tgt ≅ Y.tgt", "eq : X.rel = (Subobject.pullback (prod.map e₁.hom e₂.hom)).obj Y.rel"], "goal": "(Subobject.pullback π(π₁, π₂ ≫ e₂.hom)).obj Y.rel ≤ (Subobject.pullback (prod.map e₁.inv (𝟙 X.tgt) ≫ prod.map e₁.hom e₂.hom)).obj Y.rel"}], "premise": [14277], "state_str": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasFiniteProducts C\ninst✝ : HasPullbacks C\nX Y : Dial C\ne₁ : X.src ≅ Y.src\ne₂ : X.tgt ≅ Y.tgt\neq : X.rel = (Subobject.pullback (prod.map e₁.hom e₂.hom)).obj Y.rel\n⊢ (Subobject.pullback π(π₁, π₂ ≫ e₂.hom)).obj Y.rel ≤\n    (Subobject.pullback (prod.map e₁.inv (𝟙 X.tgt) ≫ prod.map e₁.hom e₂.hom)).obj Y.rel"}
+{"state": [{"context": ["f : StieltjesFunction", "c : ℝ"], "goal": "MeasurableSet (Ioi c)"}], "premise": [27909], "state_str": "f : StieltjesFunction\nc : ℝ\n⊢ MeasurableSet (Ioi c)"}
+{"state": [{"context": ["f : StieltjesFunction", "c : ℝ", "t : Set ℝ"], "goal": "f.length (t ∩ Ioi c) + f.length (t \\ Ioi c) ≤ f.length t"}], "premise": [19310], "state_str": "f : StieltjesFunction\nc : ℝ\nt : Set ℝ\n⊢ f.length (t ∩ Ioi c) + f.length (t \\ Ioi c) ≤ f.length t"}
+{"state": [{"context": ["f : StieltjesFunction", "c : ℝ", "t : Set ℝ", "a b : ℝ", "h : t ⊆ Ioc a b"], "goal": "f.length (t ∩ Ioi c) + f.length (t \\ Ioi c) ≤ ofReal (↑f b - ↑f a)"}], "premise": [14273, 30567, 103917, 133464, 133650], "state_str": "f : StieltjesFunction\nc : ℝ\nt : Set ℝ\na b : ℝ\nh : t ⊆ Ioc a b\n⊢ f.length (t ∩ Ioi c) + f.length (t \\ Ioi c) ≤ ofReal (↑f b - ↑f a)"}
+{"state": [{"context": ["f : StieltjesFunction", "c : ℝ", "t : Set ℝ", "a b : ℝ", "h : t ⊆ Ioc a b"], "goal": "f.length (Ioc a b ∩ Ioi c) + f.length (Ioc a b \\ Ioi c) ≤ ofReal (↑f b - ↑f a)"}], "premise": [14308], "state_str": "f : StieltjesFunction\nc : ℝ\nt : Set ℝ\na b : ℝ\nh : t ⊆ Ioc a b\n⊢ f.length (Ioc a b ∩ Ioi c) + f.length (Ioc a b \\ Ioi c) ≤ ofReal (↑f b - ↑f a)"}
+{"state": [{"context": ["A : Type u_1", "inst✝³ : AddCommGroup A", "inst✝² : Module ℂ A", "inst✝¹ : StarAddMonoid A", "inst✝ : StarModule ℂ A", "a : ↥(skewAdjoint A)"], "goal": "-I • ↑a ∈ selfAdjoint A"}], "premise": [108340, 110032, 111020, 111611, 111625, 119769, 122529, 148351, 148358], "state_str": "A : Type u_1\ninst✝³ : AddCommGroup A\ninst✝² : Module ℂ A\ninst✝¹ : StarAddMonoid A\ninst✝ : StarModule ℂ A\na : ↥(skewAdjoint A)\n⊢ -I • ↑a ∈ selfAdjoint A"}
+{"state": [{"context": ["A : Type u_1", "inst✝³ : AddCommGroup A", "inst✝² : Module ℂ A", "inst✝¹ : StarAddMonoid A", "inst✝ : StarModule ℂ A", "a b : ↥(skewAdjoint A)"], "goal": "↑((fun a => ⟨-I • ↑a, ⋯⟩) (a + b)) = ↑((fun a => ⟨-I • ↑a, ⋯⟩) a + (fun a => ⟨-I • ↑a, ⋯⟩) b)"}], "premise": [108334, 116560, 122623], "state_str": "case a\nA : Type u_1\ninst✝³ : AddCommGroup A\ninst✝² : Module ℂ A\ninst✝¹ : StarAddMonoid A\ninst✝ : StarModule ℂ A\na b : ↥(skewAdjoint A)\n⊢ ↑((fun a => ⟨-I • ↑a, ⋯⟩) (a + b)) = ↑((fun a => ⟨-I • ↑a, ⋯⟩) a + (fun a => ⟨-I • ↑a, ⋯⟩) b)"}
+{"state": [{"context": ["A : Type u_1", "inst✝³ : AddCommGroup A", "inst✝² : Module ℂ A", "inst✝¹ : StarAddMonoid A", "inst✝ : StarModule ℂ A", "a : ℝ", "b : ↥(skewAdjoint A)"], "goal": "↑({ toFun := fun a => ⟨-I • ↑a, ⋯⟩, map_add' := ⋯ }.toFun (a • b)) = ↑((RingHom.id ℝ) a • { toFun := fun a => ⟨-I • ↑a, ⋯⟩, map_add' := ⋯ }.toFun b)"}], "premise": [108340, 110032, 111623, 111630, 117792, 121577, 122632], "state_str": "case a\nA : Type u_1\ninst✝³ : AddCommGroup A\ninst✝² : Module ℂ A\ninst✝¹ : StarAddMonoid A\ninst✝ : StarModule ℂ A\na : ℝ\nb : ↥(skewAdjoint A)\n⊢ ↑({ toFun := fun a => ⟨-I • ↑a, ⋯⟩, map_add' := ⋯ }.toFun (a • b)) =\n    ↑((RingHom.id ℝ) a • { toFun := fun a => ⟨-I • ↑a, ⋯⟩, map_add' := ⋯ }.toFun b)"}
+{"state": [{"context": ["A : Type u_1", "inst✝³ : AddCommGroup A", "inst✝² : Module ℂ A", "inst✝¹ : StarAddMonoid A", "inst✝ : StarModule ℂ A", "a : ℝ", "b : ↥(skewAdjoint A)"], "goal": "I • a • ↑b = a • I • ↑b"}], "premise": [118875], "state_str": "case a\nA : Type u_1\ninst✝³ : AddCommGroup A\ninst✝² : Module ℂ A\ninst✝¹ : StarAddMonoid A\ninst✝ : StarModule ℂ A\na : ℝ\nb : ↥(skewAdjoint A)\n⊢ I • a • ↑b = a • I • ↑b"}
+{"state": [{"context": ["R : Type u_1", "inst✝⁵ : CommSemiring R", "M : Submonoid R", "S : Type u_2", "inst✝⁴ : CommSemiring S", "inst✝³ : Algebra R S", "P : Type u_3", "inst✝² : CommSemiring P", "inst✝¹ : IsLocalization M S", "g✝ : R →+* P", "hg : ∀ (y : ↥M), IsUnit (g✝ ↑y)", "inst✝ : Algebra R P", "f g : S →ₐ[R] P"], "goal": "(↑f).comp (algebraMap R S) = (↑g).comp (algebraMap R S)"}], "premise": [121061], "state_str": "R : Type u_1\ninst✝⁵ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝⁴ : CommSemiring S\ninst✝³ : Algebra R S\nP : Type u_3\ninst✝² : CommSemiring P\ninst✝¹ : IsLocalization M S\ng✝ : R →+* P\nhg : ∀ (y : ↥M), IsUnit (g✝ ↑y)\ninst✝ : Algebra R P\nf g : S →ₐ[R] P\n⊢ (↑f).comp (algebraMap R S) = (↑g).comp (algebraMap R S)"}
+{"state": [{"context": ["𝕜 : Type u", "inst✝⁴ : NontriviallyNormedField 𝕜", "F : Type v", "inst✝³ : NormedAddCommGroup F", "inst✝² : NormedSpace 𝕜 F", "E : Type w", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace 𝕜 E", "f f₀ f₁ g : 𝕜 → F", "f' f₀' f₁' g' : F", "x : 𝕜", "s t : Set 𝕜", "L : Filter 𝕜", "ι : Type u_1", "u : Finset ι", "A : ι → 𝕜 → F", "A' : ι → F", "h : ∀ i ∈ u, HasDerivAtFilter (A i) (A' i) x L"], "goal": "HasDerivAtFilter (fun y => ∑ i ∈ u, A i y) (∑ i ∈ u, A' i) x L"}], "premise": [44327, 45814, 68790], "state_str": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nι : Type u_1\nu : Finset ι\nA : ι → 𝕜 → F\nA' : ι → F\nh : ∀ i ∈ u, HasDerivAtFilter (A i) (A' i) x L\n⊢ HasDerivAtFilter (fun y => ∑ i ∈ u, A i y) (∑ i ∈ u, A' i) x L"}
+{"state": [{"context": ["J : Type u₁", "inst✝² : Category.{v₁, u₁} J", "K : Type u₂", "inst✝¹ : Category.{v₂, u₂} K", "C : Type u₃", "inst✝ : Category.{v₃, u₃} C", "F✝ : J ⥤ C", "t✝ : Cone F✝", "F : J ⥤ C", "s : Cone F", "G : K ⥤ C", "t : Cone G", "P : IsLimit s", "Q : IsLimit t", "e : J ≌ K", "w : e.functor ⋙ G ≅ F", "w' : e.inverse ⋙ F ≅ G := (isoWhiskerLeft e.inverse w).symm ≪≫ e.invFunIdAssoc G"], "goal": "Q.lift ((Cones.equivalenceOfReindexing e.symm w').functor.obj s) ≫ P.lift ((Cones.equivalenceOfReindexing e w).functor.obj t) = 𝟙 s.pt"}], "premise": [94261], "state_str": "J : Type u₁\ninst✝² : Category.{v₁, u₁} J\nK : Type u₂\ninst✝¹ : Category.{v₂, u₂} K\nC : Type u₃\ninst✝ : Category.{v₃, u₃} C\nF✝ : J ⥤ C\nt✝ : Cone F✝\nF : J ⥤ C\ns : Cone F\nG : K ⥤ C\nt : Cone G\nP : IsLimit s\nQ : IsLimit t\ne : J ≌ K\nw : e.functor ⋙ G ≅ F\nw' : e.inverse ⋙ F ≅ G := (isoWhiskerLeft e.inverse w).symm ≪≫ e.invFunIdAssoc G\n⊢ Q.lift ((Cones.equivalenceOfReindexing e.symm w').functor.obj s) ≫\n      P.lift ((Cones.equivalenceOfReindexing e w).functor.obj t) =\n    𝟙 s.pt"}
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+{"state": [{"context": ["p : ℕ+", "k : ℕ", "K : Type u", "inst✝¹ : Field K", "inst✝ : CharZero K", "ζ : K", "hp : Fact (Nat.Prime ↑p)", "hcycl : IsCyclotomicExtension {p ^ k} ℚ K", "hζ : IsPrimitiveRoot ζ ↑(p ^ k)"], "goal": "(algebraMap (𝓞 K) K) (hζ.adjoinEquivRingOfIntegers ⟨ζ, ⋯⟩) = ↑hζ.toInteger"}], "premise": [24748, 81936], "state_str": "p : ℕ+\nk : ℕ\nK : Type u\ninst✝¹ : Field K\ninst✝ : CharZero K\nζ : K\nhp : Fact (Nat.Prime ↑p)\nhcycl : IsCyclotomicExtension {p ^ k} ℚ K\nhζ : IsPrimitiveRoot ζ ↑(p ^ k)\n⊢ (algebraMap (𝓞 K) K) (hζ.adjoinEquivRingOfIntegers ⟨ζ, ⋯⟩) = ↑hζ.toInteger"}
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+{"state": [{"context": ["K : Type u_1", "n : ℕ", "g : GenContFract K", "s : Stream'.Seq (Pair K)", "inst✝ : DivisionRing K"], "goal": "g.convs' (n + 1) = (g.squashGCF n).convs' n"}], "premise": [117559], "state_str": "K : Type u_1\nn : ℕ\ng : GenContFract K\ns : Stream'.Seq (Pair K)\ninst✝ : DivisionRing K\n⊢ g.convs' (n + 1) = (g.squashGCF n).convs' n"}
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+{"state": [{"context": ["p q r : ℕ+"], "goal": "sumInv {p, q, r} = (↑↑p)⁻¹ + (↑↑q)⁻¹ + (↑↑r)⁻¹"}], "premise": [119704, 119729, 124314, 124320, 137819, 137983, 137985], "state_str": "p q r : ℕ+\n⊢ sumInv {p, q, r} = (↑↑p)⁻¹ + (↑↑q)⁻¹ + (↑↑r)⁻¹"}
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+{"state": [{"context": ["𝕜 : Type u_1", "inst✝³ : RCLike 𝕜", "E : Type u_2", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace 𝕜 E", "inst✝ : InnerProductSpaceable E", "x y z : E"], "goal": "‖x + y + z‖ * ‖x + y + z‖ - ‖x + y - z‖ * ‖x + y - z‖ = ‖x + z‖ * ‖x + z‖ - ‖x - z‖ * ‖x - z‖ + (‖2 • z + y‖ * ‖2 • z + y‖ - ‖y - 2 • z‖ * ‖y - 2 • z‖) / 2"}], "premise": [34411, 34412], "state_str": "𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : InnerProductSpaceable E\nx y z : E\n⊢ ‖x + y + z‖ * ‖x + y + z‖ - ‖x + y - z‖ * ‖x + y - z‖ =\n    ‖x + z‖ * ‖x + z‖ - ‖x - z‖ * ‖x - z‖ + (‖2 • z + y‖ * ‖2 • z + y‖ - ‖y - 2 • z‖ * ‖y - 2 • z‖) / 2"}
+{"state": [{"context": ["C : Type u", "inst✝⁵ : Category.{v, u} C", "inst✝⁴ : HasShift C ℤ", "T₁ T₂ T₃ : Triangle C", "J : Type u_1", "T : J → Triangle C", "inst✝³ : HasProduct fun j => (T j).obj₁", "inst✝² : HasProduct fun j => (T j).obj₂", "inst✝¹ : HasProduct fun j => (T j).obj₃", "inst✝ : HasProduct fun j => (shiftFunctor C 1).obj (T j).obj₁", "T' : Triangle C", "φ : (j : J) → T' ⟶ T j"], "goal": "T'.mor₃ ≫ (shiftFunctor C 1).map (Pi.lift fun j => (φ j).hom₁) = (Pi.lift fun j => (φ j).hom₃) ≫ (Pi.map fun j => (T j).mor₃) ≫ inv (piComparison (shiftFunctor C 1) fun j => (T j).obj₁)"}], "premise": [88783, 96173, 96174, 96191], "state_str": "C : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : HasShift C ℤ\nT₁ T₂ T₃ : Triangle C\nJ : Type u_1\nT : J → Triangle C\ninst✝³ : HasProduct fun j => (T j).obj₁\ninst✝² : HasProduct fun j => (T j).obj₂\ninst✝¹ : HasProduct fun j => (T j).obj₃\ninst✝ : HasProduct fun j => (shiftFunctor C 1).obj (T j).obj₁\nT' : Triangle C\nφ : (j : J) → T' ⟶ T j\n⊢ T'.mor₃ ≫ (shiftFunctor C 1).map (Pi.lift fun j => (φ j).hom₁) =\n    (Pi.lift fun j => (φ j).hom₃) ≫\n      (Pi.map fun j => (T j).mor₃) ≫ inv (piComparison (shiftFunctor C 1) fun j => (T j).obj₁)"}
+{"state": [{"context": ["E : Type u_1", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "c : ℝ", "f : ℝ → E"], "goal": "∫ (x : ℝ) in Ioi c, f (-x) = ∫ (x : ℝ) in Iic (-c), f x"}], "premise": [29681, 119769], "state_str": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nc : ℝ\nf : ℝ → E\n⊢ ∫ (x : ℝ) in Ioi c, f (-x) = ∫ (x : ℝ) in Iic (-c), f x"}
+{"state": [{"context": ["E : Type u_1", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "c : ℝ", "f : ℝ → E"], "goal": "∫ (x : ℝ) in Iic (-c), f (- -x) = ∫ (x : ℝ) in Iic (- - -c), f x"}], "premise": [119769], "state_str": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nc : ℝ\nf : ℝ → E\n⊢ ∫ (x : ℝ) in Iic (-c), f (- -x) = ∫ (x : ℝ) in Iic (- - -c), f x"}
+{"state": [{"context": ["R : Type u_1", "A : Type u_2", "inst✝ : CommSemiring R", "f g : R[X]"], "goal": "{ toFun := ⇑derivative, map_add' := ⋯, map_smul' := ⋯ } (f * g) = f • { toFun := ⇑derivative, map_add' := ⋯, map_smul' := ⋯ } g + g • { toFun := ⇑derivative, map_add' := ⋯, map_smul' := ⋯ } f"}], "premise": [101508, 119707, 119708], "state_str": "R : Type u_1\nA : Type u_2\ninst✝ : CommSemiring R\nf g : R[X]\n⊢ { toFun := ⇑derivative, map_add' := ⋯, map_smul' := ⋯ } (f * g) =\n    f • { toFun := ⇑derivative, map_add' := ⋯, map_smul' := ⋯ } g +\n      g • { toFun := ⇑derivative, map_add' := ⋯, map_smul' := ⋯ } f"}
+{"state": [{"context": ["α : Type u_1", "inst✝ : LinearOrderedCommGroup α", "a✝ b✝ c✝ a b c : α"], "goal": "max (a / b) (a / c) = a / min b c"}], "premise": [103682, 104507, 119790], "state_str": "α : Type u_1\ninst✝ : LinearOrderedCommGroup α\na✝ b✝ c✝ a b c : α\n⊢ max (a / b) (a / c) = a / min b c"}
+{"state": [{"context": ["C : Type u", "inst✝² : Category.{v, u} C", "inst✝¹ : Abelian C", "inst✝ : HasExt C", "X : C", "n : ℕ", "X✝ Y✝ Z✝ : C", "f : X✝ ⟶ Y✝", "f' : Y✝ ⟶ Z✝", "α : ↑({ obj := fun Y => AddCommGrp.of (Ext X Y n), map := fun {X_1 Y} f => AddCommGrp.ofHom ((Ext.mk₀ f).postcomp X ⋯) }.obj X✝)"], "goal": "Ext.comp α (Ext.mk₀ (f ≫ f')) ⋯ = (Ext.comp α (Ext.mk₀ f) ⋯).comp (Ext.mk₀ f') ⋯"}], "premise": [113720], "state_str": "case w\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX : C\nn : ℕ\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\nf' : Y✝ ⟶ Z✝\nα :\n  ↑({ obj := fun Y => AddCommGrp.of (Ext X Y n),\n          map := fun {X_1 Y} f => AddCommGrp.ofHom ((Ext.mk₀ f).postcomp X ⋯) }.obj\n      X✝)\n⊢ Ext.comp α (Ext.mk₀ (f ≫ f')) ⋯ = (Ext.comp α (Ext.mk₀ f) ⋯).comp (Ext.mk₀ f') ⋯"}
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+{"state": [{"context": ["𝕜 : Type u_1", "inst✝⁹ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : NormedSpace 𝕜 E", "inst✝⁶ : CompleteSpace E", "F : Type u_3", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace 𝕜 F", "inst✝³ : CompleteSpace F", "G : Type u_4", "inst✝² : NormedAddCommGroup G", "inst✝¹ : NormedSpace 𝕜 G", "inst✝ : CompleteSpace G", "φ : ImplicitFunctionData 𝕜 E F G"], "goal": "map (Prod.fst ∘ φ.prodFun) (𝓝 φ.pt) = 𝓝 (φ.prodFun φ.pt).1"}], "premise": [16175, 44572, 45366, 66441], "state_str": "𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : CompleteSpace E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : CompleteSpace F\nG : Type u_4\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\ninst✝ : CompleteSpace G\nφ : ImplicitFunctionData 𝕜 E F G\n⊢ map (Prod.fst ∘ φ.prodFun) (𝓝 φ.pt) = 𝓝 (φ.prodFun φ.pt).1"}
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+{"state": [{"context": ["p : ℕ → Prop", "inst✝ : DecidablePred p", "n : ℕ"], "goal": "↑(count p n) ≤ Cardinal.mk ↑{k | p k}"}], "premise": [48619, 144586], "state_str": "p : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\n⊢ ↑(count p n) ≤ Cardinal.mk ↑{k | p k}"}
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+{"state": [{"context": ["R : Type u_1", "inst✝⁵ : CommRing R", "x : R", "B : Type u_2", "inst✝⁴ : CommRing B", "inst✝³ : Algebra R B", "inst✝² : IsLocalization.Away x B", "inst✝¹ : IsDomain R", "inst✝ : WfDvdMonoid R", "b : B", "hb : b ≠ 0", "hx : Irreducible x"], "goal": "∃ a n, ¬x ∣ a ∧ selfZPow x B n * (algebraMap R B) a = b"}], "premise": [77574], "state_str": "R : Type u_1\ninst✝⁵ : CommRing R\nx : R\nB : Type u_2\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R B\ninst✝² : IsLocalization.Away x B\ninst✝¹ : IsDomain R\ninst✝ : WfDvdMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\n⊢ ∃ a n, ¬x ∣ a ∧ selfZPow x B n * (algebraMap R B) a = b"}
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+{"state": [{"context": ["R : Type u_1", "inst✝⁵ : CommRing R", "x : R", "B : Type u_2", "inst✝⁴ : CommRing B", "inst✝³ : Algebra R B", "inst✝² : IsLocalization.Away x B", "inst✝¹ : IsDomain R", "inst✝ : WfDvdMonoid R", "b : B", "hb : b ≠ 0", "hx : Irreducible x", "a₀ : R", "y : ↥(Submonoid.powers x)", "d : ℕ", "hy : x ^ d = ↑y", "ha₀ : a₀ ≠ 0", "H : b * (algebraMap R B) (x ^ d) = (algebraMap R B) a₀"], "goal": "∃ a n, ¬x ∣ a ∧ selfZPow x B n * (algebraMap R B) a = b"}], "premise": [76076], "state_str": "case intro.mk.intro\nR : Type u_1\ninst✝⁵ : CommRing R\nx : R\nB : Type u_2\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R B\ninst✝² : IsLocalization.Away x B\ninst✝¹ : IsDomain R\ninst✝ : WfDvdMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : ↥(Submonoid.powers x)\nd : ℕ\nhy : x ^ d = ↑y\nha₀ : a₀ ≠ 0\nH : b * (algebraMap R B) (x ^ d) = (algebraMap R B) a₀\n⊢ ∃ a n, ¬x ∣ a ∧ selfZPow x B n * (algebraMap R B) a = b"}
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+{"state": [{"context": ["R : Type u_1", "inst✝⁵ : CommRing R", "x : R", "B : Type u_2", "inst✝⁴ : CommRing B", "inst✝³ : Algebra R B", "inst✝² : IsLocalization.Away x B", "inst✝¹ : IsDomain R", "inst✝ : WfDvdMonoid R", "b : B", "hb : b ≠ 0", "hx : Irreducible x", "a₀ : R", "y : ↥(Submonoid.powers x)", "d : ℕ", "hy : x ^ d = ↑y", "ha₀ : a₀ ≠ 0", "H : b * (algebraMap R B) (x ^ d) = (algebraMap R B) a₀", "m : ℕ", "a : R", "hyp1 : ¬x ∣ a", "hyp2 : a₀ = x ^ m * a"], "goal": "¬x ∣ a ∧ (algebraMap R B) x ^ m * mk' B a 1 = (algebraMap R B) x ^ m * (algebraMap R B) a"}], "premise": [77591], "state_str": "case intro.mk.intro.intro.intro.intro\nR : Type u_1\ninst✝⁵ : CommRing R\nx : R\nB : Type u_2\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R B\ninst✝² : IsLocalization.Away x B\ninst✝¹ : IsDomain R\ninst✝ : WfDvdMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\na₀ : R\ny : ↥(Submonoid.powers x)\nd : ℕ\nhy : x ^ d = ↑y\nha₀ : a₀ ≠ 0\nH : b * (algebraMap R B) (x ^ d) = (algebraMap R B) a₀\nm : ℕ\na : R\nhyp1 : ¬x ∣ a\nhyp2 : a₀ = x ^ m * a\n⊢ ¬x ∣ a ∧ (algebraMap R B) x ^ m * mk' B a 1 = (algebraMap R B) x ^ m * (algebraMap R B) a"}
+{"state": [{"context": ["ι : Type u_1", "ι₂ : Type u_2", "ι₃ : Type u_3", "R : Type u_4", "inst✝⁷ : CommSemiring R", "R₁ : Type u_5", "R₂ : Type u_6", "s : ι → Type u_7", "inst✝⁶ : (i : ι) → AddCommMonoid (s i)", "inst✝⁵ : (i : ι) → Module R (s i)", "M : Type u_8", "inst✝⁴ : AddCommMonoid M", "inst✝³ : Module R M", "E : Type u_9", "inst✝² : AddCommMonoid E", "inst✝¹ : Module R E", "F : Type u_10", "inst✝ : AddCommMonoid F", "e : ι ≃ ι₂", "x : ⨂[R] (i : ι₂), M"], "goal": "(reindex R (fun x => M) e).symm x = (reindex R (fun x => M) e.symm) x"}], "premise": [70760, 71902, 110228, 110229, 110506, 110555], "state_str": "case h\nι : Type u_1\nι₂ : Type u_2\nι₃ : Type u_3\nR : Type u_4\ninst✝⁷ : CommSemiring R\nR₁ : Type u_5\nR₂ : Type u_6\ns : ι → Type u_7\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type u_8\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_9\ninst✝² : AddCommMonoid E\ninst✝¹ : Module R E\nF : Type u_10\ninst✝ : AddCommMonoid F\ne : ι ≃ ι₂\nx : ⨂[R] (i : ι₂), M\n⊢ (reindex R (fun x => M) e).symm x = (reindex R (fun x => M) e.symm) x"}
+{"state": [{"context": ["X : Type u_1", "α : Type u_2", "ι : Sort u_3", "inst✝¹ : TopologicalSpace X", "inst✝ : BaireSpace X", "S : Set α", "f : α → Set X", "ho : ∀ s ∈ S, IsOpen (f s)", "hS : S.Countable", "hd : ∀ s ∈ S, Dense (f s)"], "goal": "Dense (⋂ s ∈ S, f s)"}], "premise": [135423], "state_str": "X : Type u_1\nα : Type u_2\nι : Sort u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : BaireSpace X\nS : Set α\nf : α → Set X\nho : ∀ s ∈ S, IsOpen (f s)\nhS : S.Countable\nhd : ∀ s ∈ S, Dense (f s)\n⊢ Dense (⋂ s ∈ S, f s)"}
+{"state": [{"context": ["X : Type u_1", "α : Type u_2", "ι : Sort u_3", "inst✝¹ : TopologicalSpace X", "inst✝ : BaireSpace X", "S : Set α", "f : α → Set X", "ho : ∀ s ∈ S, IsOpen (f s)", "hS : S.Countable", "hd : ∀ s ∈ S, Dense (f s)"], "goal": "Dense (⋂₀ (f '' S))"}], "premise": [65429, 132748, 134092], "state_str": "X : Type u_1\nα : Type u_2\nι : Sort u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : BaireSpace X\nS : Set α\nf : α → Set X\nho : ∀ s ∈ S, IsOpen (f s)\nhS : S.Countable\nhd : ∀ s ∈ S, Dense (f s)\n⊢ Dense (⋂₀ (f '' S))"}
+{"state": [{"context": ["n m k : ℕ"], "goal": "n.dist k ≤ n.dist m + m.dist k"}], "premise": [117738, 119704, 119708], "state_str": "n m k : ℕ\n⊢ n.dist k ≤ n.dist m + m.dist k"}
+{"state": [{"context": ["n m k : ℕ", "this : n.dist m + m.dist k = n - m + (m - k) + (k - m + (m - n))"], "goal": "n - k + (k - n) ≤ n - m + (m - k) + (k - m + (m - n))"}], "premise": [3788, 103372], "state_str": "n m k : ℕ\nthis : n.dist m + m.dist k = n - m + (m - k) + (k - m + (m - n))\n⊢ n - k + (k - n) ≤ n - m + (m - k) + (k - m + (m - n))"}
+{"state": [{"context": ["K : Type u_1", "inst✝¹ : Field K", "inst✝ : NumberField K", "c : ℝ"], "goal": "mixedEmbedding.norm (fun x => c, fun x => ↑c) = |c| ^ finrank ℚ K"}], "premise": [23218, 117064, 119730], "state_str": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nc : ℝ\n⊢ mixedEmbedding.norm (fun x => c, fun x => ↑c) = |c| ^ finrank ℚ K"}
+{"state": [{"context": ["R : Type u", "inst✝¹ : CommRing R", "W✝ : Affine R", "F : Type u", "inst✝ : Field F", "W : Affine F", "x₁ x₂ y₁ y₂ : F", "h₁ : W.Equation x₁ y₁", "h₂ : W.Equation x₂ y₂", "hxy : x₁ = x₂ → y₁ ≠ W.negY x₂ y₂"], "goal": "derivative (W.addPolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂)) = -((X - C x₁) * (X - C x₂) + (X - C x₁) * (X - C (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂))) + (X - C x₂) * (X - C (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂))))"}], "premise": [132204], "state_str": "R : Type u\ninst✝¹ : CommRing R\nW✝ : Affine R\nF : Type u\ninst✝ : Field F\nW : Affine F\nx₁ x₂ y₁ y₂ : F\nh₁ : W.Equation x₁ y₁\nh₂ : W.Equation x₂ y₂\nhxy : x₁ = x₂ → y₁ ≠ W.negY x₂ y₂\n⊢ derivative (W.addPolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂)) =\n    -((X - C x₁) * (X - C x₂) + (X - C x₁) * (X - C (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂))) +\n        (X - C x₂) * (X - C (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂))))"}
+{"state": [{"context": ["R : Type u", "inst✝¹ : CommRing R", "W✝ : Affine R", "F : Type u", "inst✝ : Field F", "W : Affine F", "x₁ x₂ y₁ y₂ : F", "h₁ : W.Equation x₁ y₁", "h₂ : W.Equation x₂ y₂", "hxy : x₁ = x₂ → y₁ ≠ W.negY x₂ y₂"], "goal": "derivative (-((X - C x₁) * (X - C x₂) * (X - C (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂))))) = -((X - C x₁) * (X - C x₂) + (X - C x₁) * (X - C (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂))) + (X - C x₂) * (X - C (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂))))"}], "premise": [101482, 101484, 101486, 101488, 101508, 101520, 101534, 101536], "state_str": "R : Type u\ninst✝¹ : CommRing R\nW✝ : Affine R\nF : Type u\ninst✝ : Field F\nW : Affine F\nx₁ x₂ y₁ y₂ : F\nh₁ : W.Equation x₁ y₁\nh₂ : W.Equation x₂ y₂\nhxy : x₁ = x₂ → y₁ ≠ W.negY x₂ y₂\n⊢ derivative (-((X - C x₁) * (X - C x₂) * (X - C (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂))))) =\n    -((X - C x₁) * (X - C x₂) + (X - C x₁) * (X - C (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂))) +\n        (X - C x₂) * (X - C (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂))))"}
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+{"state": [{"context": ["J : Type u₁", "inst✝³ : Category.{v₁, u₁} J", "K : Type u₂", "inst✝² : Category.{v₂, u₂} K", "C : Type u₃", "inst✝¹ : Category.{v₃, u₃} C", "D : Type u₄", "inst✝ : Category.{v₄, u₄} D", "F : J ⥤ C", "G : C ⥤ D", "X✝ Y✝ : Cone F", "f : X✝ ⟶ Y✝", "j : J"], "goal": "G.map f.hom ≫ ((fun A => { pt := G.obj A.pt, π := { app := fun j => G.map (A.π.app j), naturality := ⋯ } }) Y✝).π.app j = ((fun A => { pt := G.obj A.pt, π := { app := fun j => G.map (A.π.app j), naturality := ⋯ } }) X✝).π.app j"}], "premise": [95553], "state_str": "J : Type u₁\ninst✝³ : Category.{v₁, u₁} J\nK : Type u₂\ninst✝² : Category.{v₂, u₂} K\nC : Type u₃\ninst✝¹ : Category.{v₃, u₃} C\nD : Type u₄\ninst✝ : Category.{v₄, u₄} D\nF : J ⥤ C\nG : C ⥤ D\nX✝ Y✝ : Cone F\nf : X✝ ⟶ Y✝\nj : J\n⊢ G.map f.hom ≫\n      ((fun A => { pt := G.obj A.pt, π := { app := fun j => G.map (A.π.app j), naturality := ⋯ } }) Y✝).π.app j =\n    ((fun A => { pt := G.obj A.pt, π := { app := fun j => G.map (A.π.app j), naturality := ⋯ } }) X✝).π.app j"}
+{"state": [{"context": ["R : Type u_1", "inst✝³ : Semiring R", "p q : R[X]", "S : Type u_2", "inst✝² : AddCommMonoid S", "inst✝¹ : Pow S ℕ", "inst✝ : Module R S", "x : S"], "goal": "(p + q).smeval x = p.smeval x + q.smeval x"}], "premise": [101066], "state_str": "R : Type u_1\ninst✝³ : Semiring R\np q : R[X]\nS : Type u_2\ninst✝² : AddCommMonoid S\ninst✝¹ : Pow S ℕ\ninst✝ : Module R S\nx : S\n⊢ (p + q).smeval x = p.smeval x + q.smeval x"}
+{"state": [{"context": ["R : Type u_1", "inst✝³ : Semiring R", "p q : R[X]", "S : Type u_2", "inst✝² : AddCommMonoid S", "inst✝¹ : Pow S ℕ", "inst✝ : Module R S", "x : S"], "goal": "(p + q).sum (smul_pow x) = p.sum (smul_pow x) + q.sum (smul_pow x)"}], "premise": [101329], "state_str": "R : Type u_1\ninst✝³ : Semiring R\np q : R[X]\nS : Type u_2\ninst✝² : AddCommMonoid S\ninst✝¹ : Pow S ℕ\ninst✝ : Module R S\nx : S\n⊢ (p + q).sum (smul_pow x) = p.sum (smul_pow x) + q.sum (smul_pow x)"}
+{"state": [{"context": ["α : Type u", "β : Type v", "γ : Type w", "δ : Type u_1", "ι : Sort x", "f✝ g : Filter α", "s✝ t✝ : Set α", "f : β → Filter α", "s : Set β", "h : DirectedOn (f ⁻¹'o fun x x_1 => x ≥ x_1) s", "ne : s.Nonempty", "t : Set α"], "goal": "t ∈ (⨅ i ∈ s, f i).sets ↔ t ∈ ⋃ i ∈ s, (f i).sets"}], "premise": [15980], "state_str": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type u_1\nι : Sort x\nf✝ g : Filter α\ns✝ t✝ : Set α\nf : β → Filter α\ns : Set β\nh : DirectedOn (f ⁻¹'o fun x x_1 => x ≥ x_1) s\nne : s.Nonempty\nt : Set α\n⊢ t ∈ (⨅ i ∈ s, f i).sets ↔ t ∈ ⋃ i ∈ s, (f i).sets"}
+{"state": [{"context": ["R : Type u", "inst✝¹ : CommRing R", "E : EllipticCurve R", "A : Type v", "inst✝ : CommRing A", "φ : R →+* A"], "goal": "↑((Units.map ↑φ) E.Δ') = (E.map φ).Δ"}], "premise": [116758, 129504, 129529], "state_str": "R : Type u\ninst✝¹ : CommRing R\nE : EllipticCurve R\nA : Type v\ninst✝ : CommRing A\nφ : R →+* A\n⊢ ↑((Units.map ↑φ) E.Δ') = (E.map φ).Δ"}
+{"state": [{"context": ["u : ℕ → ℕ", "f : ℕ → ℝ≥0∞", "C : ℕ", "hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m", "h_pos : ∀ (n : ℕ), 0 < u n", "h_nonneg : ∀ (n : ℕ), 0 ≤ f n", "hu : Monotone u", "h_succ_diff : SuccDiffBounded C u"], "goal": "∑' (k : ℕ), (↑(u (k + 1)) - ↑(u k)) * f (u k) ≤ (↑(u 1) - ↑(u 0)) * f (u 0) + ↑C * ∑' (k : ℕ), f k"}], "premise": [2140, 15536, 16354, 59003], "state_str": "u : ℕ → ��\nf : ℕ → ℝ≥0∞\nC : ℕ\nhf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m\nh_pos : ∀ (n : ℕ), 0 < u n\nh_nonneg : ∀ (n : ℕ), 0 ≤ f n\nhu : Monotone u\nh_succ_diff : SuccDiffBounded C u\n⊢ ∑' (k : ℕ), (↑(u (k + 1)) - ↑(u k)) * f (u k) ≤ (↑(u 1) - ↑(u 0)) * f (u 0) + ↑C * ∑' (k : ℕ), f k"}
+{"state": [{"context": ["u : ℕ → ℕ", "f : ℕ → ℝ≥0∞", "C : ℕ", "hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m", "h_pos : ∀ (n : ℕ), 0 < u n", "h_nonneg : ∀ (n : ℕ), 0 ≤ f n", "hu : Monotone u", "h_succ_diff : SuccDiffBounded C u"], "goal": "⨆ i, ∑ a ∈ range i.succ, (↑(u (a + 1)) - ↑(u a)) * f (u a) ≤ (↑(u 1) - ↑(u 0)) * f (u 0) + ↑C * ∑' (k : ℕ), f k"}], "premise": [14273, 19309, 59002, 102621, 103882], "state_str": "u : ℕ → ℕ\nf : ℕ → ℝ≥0∞\nC : ℕ\nhf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m\nh_pos : ∀ (n : ℕ), 0 < u n\nh_nonneg : ∀ (n : ℕ), 0 ≤ f n\nhu : Monotone u\nh_succ_diff : SuccDiffBounded C u\n⊢ ⨆ i, ∑ a ∈ range i.succ, (↑(u (a + 1)) - ↑(u a)) * f (u a) ≤ (↑(u 1) - ↑(u 0)) * f (u 0) + ↑C * ∑' (k : ℕ), f k"}
+{"state": [{"context": ["u : ℕ → ℕ", "f : ℕ → ℝ≥0∞", "C : ℕ", "hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m", "h_pos : ∀ (n : ℕ), 0 < u n", "h_nonneg : ∀ (n : ℕ), 0 ≤ f n", "hu : Monotone u", "h_succ_diff : SuccDiffBounded C u", "n : ℕ"], "goal": "∑ a ∈ range n.succ, (↑(u (a + 1)) - ↑(u a)) * f (u a) ≤ (↑(u 1) - ↑(u 0)) * f (u 0) + ↑C * ∑ x ∈ Ico (u 0 + 1) (u n + 1), f x"}, {"context": ["u : ℕ → ℕ", "f : ℕ → ℝ≥0∞", "C : ℕ", "hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m", "h_pos : ∀ (n : ℕ), 0 < u n", "h_nonneg : ∀ (n : ℕ), 0 ≤ f n", "hu : Monotone u", "h_succ_diff : SuccDiffBounded C u", "n : ℕ"], "goal": "0 ≤ ↑C"}], "premise": [33470], "state_str": "case refine_1\nu : ℕ → ℕ\nf : ℕ → ℝ≥0∞\nC : ℕ\nhf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m\nh_pos : ∀ (n : ℕ), 0 < u n\nh_nonneg : ∀ (n : ℕ), 0 ≤ f n\nhu : Monotone u\nh_succ_diff : SuccDiffBounded C u\nn : ℕ\n⊢ ∑ a ∈ range n.succ, (↑(u (a + 1)) - ↑(u a)) * f (u a) ≤\n    (↑(u 1) - ↑(u 0)) * f (u 0) + ↑C * ∑ x ∈ Ico (u 0 + 1) (u n + 1), f x\n\ncase refine_2\nu : ℕ → ℕ\nf : ℕ → ℝ≥0∞\nC : ℕ\nhf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m\nh_pos : ∀ (n : ℕ), 0 < u n\nh_nonneg : ∀ (n : ℕ), 0 ≤ f n\nhu : Monotone u\nh_succ_diff : SuccDiffBounded C u\nn : ℕ\n⊢ 0 ≤ ↑C"}
+{"state": [{"context": ["u : ℕ → ℕ", "f : ℕ → ℝ≥0∞", "C : ℕ", "hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m", "h_pos : ∀ (n : ℕ), 0 < u n", "h_nonneg : ∀ (n : ℕ), 0 ≤ f n", "hu : Monotone u", "h_succ_diff : SuccDiffBounded C u", "n : ℕ"], "goal": "0 ≤ ↑C"}], "premise": [103545], "state_str": "case refine_2\nu : ℕ → ℕ\nf : ℕ → ℝ≥0∞\nC : ℕ\nhf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m\nh_pos : ∀ (n : ℕ), 0 < u n\nh_nonneg : ∀ (n : ℕ), 0 ≤ f n\nhu : Monotone u\nh_succ_diff : SuccDiffBounded C u\nn : ℕ\n⊢ 0 ≤ ↑C"}
+{"state": [{"context": ["ι : Type u_1", "X : Type u_2", "Y : Type u_3", "inst✝¹¹ : EMetricSpace X", "inst✝¹⁰ : EMetricSpace Y", "inst✝⁹ : MeasurableSpace X", "inst✝⁸ : BorelSpace X", "inst✝⁷ : MeasurableSpace Y", "inst✝⁶ : BorelSpace Y", "𝕜 : Type u_4", "E : Type u_5", "P : Type u_6", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace ℝ E", "inst✝³ : MeasurableSpace P", "inst✝² : MetricSpace P", "inst✝¹ : NormedAddTorsor E P", "inst✝ : BorelSpace P", "x y : P", "s : Set ℝ", "this✝¹ : MeasurableSpace E := borel E", "this✝ : BorelSpace E"], "goal": "μH[1] (⇑(IsometryEquiv.vaddConst x) '' ((fun x_1 => x_1 • (y -ᵥ x)) '' s)) = nndist x y • μH[1] s"}], "premise": [30818, 30829, 42507], "state_str": "ι : Type u_1\nX : Type u_2\nY : Type u_3\ninst✝¹¹ : EMetricSpace X\ninst✝¹⁰ : EMetricSpace Y\ninst✝⁹ : MeasurableSpace X\ninst✝⁸ : BorelSpace X\ninst✝⁷ : MeasurableSpace Y\ninst✝⁶ : BorelSpace Y\n𝕜 : Type u_4\nE : Type u_5\nP : Type u_6\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace P\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor E P\ninst✝ : BorelSpace P\nx y : P\ns : Set ℝ\nthis✝¹ : MeasurableSpace E := borel E\nthis✝ : BorelSpace E\n⊢ μH[1] (⇑(IsometryEquiv.vaddConst x) '' ((fun x_1 => x_1 • (y -ᵥ x)) '' s)) = nndist x y • μH[1] s"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "a a₁ a₂ : α", "b b₁ b₂ : β", "inst✝⁷ : Monoid α", "inst✝⁶ : Zero α", "inst✝⁵ : Zero β", "inst✝⁴ : MulAction α β", "inst✝³ : Preorder α", "inst✝² : Preorder β", "inst✝¹ : SMulPosStrictMono α β", "inst✝ : SMulPosReflectLT α β", "hb : 0 < b"], "goal": "a • b < b ↔ a • b < 1 • b"}], "premise": [1713, 118910], "state_str": "α : Type u_1\nβ : Type u_2\na a₁ a₂ : α\nb b₁ b₂ : β\ninst✝⁷ : Monoid α\ninst✝⁶ : Zero α\ninst✝⁵ : Zero β\ninst✝⁴ : MulAction α β\ninst✝³ : Preorder α\ninst✝² : Preorder β\ninst✝¹ : SMulPosStrictMono α β\ninst✝ : SMulPosReflectLT α β\nhb : 0 < b\n⊢ a • b < b ↔ a • b < 1 • b"}
+{"state": [{"context": ["m n a✝ b✝ c d a b : ℤ", "hb : 0 < b"], "goal": "a % b < b"}], "premise": [14284, 104495], "state_str": "m n a✝ b✝ c d a b : ℤ\nhb : 0 < b\n⊢ a % b < b"}
+{"state": [{"context": ["m n a✝ b✝ c d a b : ℤ", "hb : 0 < b", "this : a % b < |b|"], "goal": "a % b < b"}], "premise": [105284], "state_str": "m n a✝ b✝ c d a b : ℤ\nhb : 0 < b\nthis : a % b < |b|\n⊢ a % b < b"}
+{"state": [{"context": ["M : Type u_1", "inst✝³ : Monoid M", "inst✝² : MeasurableSpace M", "inst✝¹ : MeasurableMul₂ M", "μ : Measure M", "inst✝ : SFinite μ"], "goal": "map (fun x => x.1 * x.2) (μ.prod (dirac 1)) = μ"}], "premise": [27006, 31518], "state_str": "M : Type u_1\ninst✝³ : Monoid M\ninst✝² : MeasurableSpace M\ninst✝¹ : MeasurableMul₂ M\nμ : Measure M\ninst✝ : SFinite μ\n⊢ map (fun x => x.1 * x.2) (μ.prod (dirac 1)) = μ"}
+{"state": [{"context": ["F : Type u_1", "ι : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "f : α → β → β", "op : α → α → α", "assoc : Std.Associative op", "s : Multiset α", "a : α", "h : {x | x ∈ a ::ₘ s}.Pairwise fun x y => op x y = op y x", "h' : {x | x ∈ s}.Pairwise fun x y => op x y = op y x", "x : α"], "goal": "noncommFold op (a ::ₘ s) h x = op a (noncommFold op s h' x)"}], "premise": [1830], "state_str": "F : Type u_1\nι : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\nf : α → β → β\nop : α → α → α\nassoc : Std.Associative op\ns : Multiset α\na : α\nh : {x | x ∈ a ::ₘ s}.Pairwise fun x y => op x y = op y x\nh' : {x | x ∈ s}.Pairwise fun x y => op x y = op y x\nx : α\n⊢ noncommFold op (a ::ₘ s) h x = op a (noncommFold op s h' x)"}
+{"state": [{"context": ["E : Type", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℂ E", "f g : E → ℂ", "z : ℂ", "x : E", "s : Set E", "m : z ∈ slitPlane"], "goal": "AnalyticAt ℂ log z"}], "premise": [47144], "state_str": "E : Type\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf g : E → ℂ\nz : ℂ\nx : E\ns : Set E\nm : z ∈ slitPlane\n⊢ AnalyticAt ℂ log z"}
+{"state": [{"context": ["E : Type", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℂ E", "f g : E → ℂ", "z : ℂ", "x : E", "s : Set E", "m : z ∈ slitPlane"], "goal": "∀ᶠ (z : ℂ) in 𝓝 z, DifferentiableAt ℂ log z"}], "premise": [15889, 46276, 55504, 131585], "state_str": "E : Type\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf g : E → ℂ\nz : ℂ\nx : E\ns : Set E\nm : z ∈ slitPlane\n⊢ ∀ᶠ (z : ℂ) in 𝓝 z, DifferentiableAt ℂ log z"}
+{"state": [{"context": ["E : Type", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℂ E", "f g : E → ℂ", "z✝ : ℂ", "x : E", "s : Set E", "m✝ : z✝ ∈ slitPlane", "z : ℂ", "m : z ∈ slitPlane"], "goal": "DifferentiableAt ℂ log z"}], "premise": [39475, 46429], "state_str": "case h\nE : Type\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf g : E → ℂ\nz✝ : ℂ\nx : E\ns : Set E\nm✝ : z✝ ∈ slitPlane\nz : ℂ\nm : z ∈ slitPlane\n⊢ DifferentiableAt ℂ log z"}
+{"state": [{"context": ["V : Type u", "V' : Type v", "V'' : Type w", "G : SimpleGraph V", "G' : SimpleGraph V'", "G'' : SimpleGraph V''", "v w : V"], "goal": "G.IsBridge s(v, w) ↔ G.Adj v w ∧ ∀ ⦃u : V⦄ (p : G.Walk u u), p.IsCycle → s(v, w) ∉ p.edges"}], "premise": [1209, 51096], "state_str": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w : V\n⊢ G.IsBridge s(v, w) ↔ G.Adj v w ∧ ∀ ⦃u : V⦄ (p : G.Walk u u), p.IsCycle → s(v, w) ∉ p.edges"}
+{"state": [{"context": ["V : Type u", "V' : Type v", "V'' : Type w", "G : SimpleGraph V", "G' : SimpleGraph V'", "G'' : SimpleGraph V''", "v w : V", "h : G.Adj v w"], "goal": "¬(G \\ fromEdgeSet {s(v, w)}).Reachable v w ↔ ∀ ⦃u : V⦄ (p : G.Walk u u), p.IsCycle → s(v, w) ∉ p.edges"}], "premise": [70082], "state_str": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w : V\nh : G.Adj v w\n⊢ ¬(G \\ fromEdgeSet {s(v, w)}).Reachable v w ↔ ∀ ⦃u : V⦄ (p : G.Walk u u), p.IsCycle → s(v, w) ∉ p.edges"}
+{"state": [{"context": ["V : Type u", "V' : Type v", "V'' : Type w", "G : SimpleGraph V", "G' : SimpleGraph V'", "G'' : SimpleGraph V''", "v w : V", "h : G.Adj v w"], "goal": "(G \\ fromEdgeSet {s(v, w)}).Reachable v w ↔ ∃ u p, p.IsCycle ∧ s(v, w) ∈ p.edges"}], "premise": [51100], "state_str": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w : V\nh : G.Adj v w\n⊢ (G \\ fromEdgeSet {s(v, w)}).Reachable v w ↔ ∃ u p, p.IsCycle ∧ s(v, w) ∈ p.edges"}
+{"state": [{"context": ["V : Type u", "V' : Type v", "V'' : Type w", "G : SimpleGraph V", "G' : SimpleGraph V'", "G'' : SimpleGraph V''", "v w : V", "h : G.Adj v w"], "goal": "(G \\ fromEdgeSet {s(v, w)}).Reachable v w ↔ G.Adj v w ∧ (G \\ fromEdgeSet {s(v, w)}).Reachable v w"}], "premise": [20003], "state_str": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w : V\nh : G.Adj v w\n⊢ (G \\ fromEdgeSet {s(v, w)}).Reachable v w ↔ G.Adj v w ∧ (G \\ fromEdgeSet {s(v, w)}).Reachable v w"}
+{"state": [{"context": ["V : Type u_1", "P : Type u_2", "inst✝⁴ : NormedAddCommGroup V", "inst✝³ : InnerProductSpace ℝ V", "inst✝² : MetricSpace P", "inst✝¹ : NormedAddTorsor V P", "hd2 : Fact (finrank ℝ V = 2)", "inst✝ : Module.Oriented ℝ V (Fin 2)", "s : Sphere P", "p₁ p₂ p₃ : P", "hp₁ : p₁ ∈ s", "hp₂ : p₂ ∈ s", "hp₃ : p₃ ∈ s", "hp₁p₂ : p₁ ≠ p₂", "hp₁p₃ : p₁ ≠ p₃", "hp₂p₃ : p₂ ≠ p₃"], "goal": "((∡ p₁ p₂ p₃).tan⁻¹ / 2) • (o.rotation ↑(π / 2)) (p₃ -ᵥ p₁) +ᵥ midpoint ℝ p₁ p₃ = s.center"}], "premise": [69364], "state_str": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Module.Oriented ℝ V (Fin 2)\ns : Sphere P\np₁ p₂ p₃ : P\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₃ : p₃ ∈ s\nhp₁p₂ : p₁ ≠ p₂\nhp₁p₃ : p₁ ≠ p₃\nhp₂p₃ : p₂ ≠ p₃\n⊢ ((∡ p₁ p₂ p₃).tan⁻¹ / 2) • (o.rotation ↑(π / 2)) (p₃ -ᵥ p₁) +ᵥ midpoint ℝ p₁ p₃ = s.center"}
+{"state": [{"context": ["V : Type u_1", "P : Type u_2", "inst✝⁴ : NormedAddCommGroup V", "inst✝³ : InnerProductSpace ℝ V", "inst✝² : MetricSpace P", "inst✝¹ : NormedAddTorsor V P", "hd2 : Fact (finrank ℝ V = 2)", "inst✝ : Module.Oriented ℝ V (Fin 2)", "s : Sphere P", "p₁ p₂ p₃ : P", "hp₁ : p₁ ∈ s", "hp₂ : p₂ ∈ s", "hp₃ : p₃ ∈ s", "hp₁p₂ : p₁ ≠ p₂", "hp₁p₃ : p₁ ≠ p₃", "hp₂p₃ : p₂ ≠ p₃"], "goal": "(∡ p₁ p₂ p₃).tan⁻¹ = (∡ p₃ p₁ s.center).tan"}], "premise": [2100, 38393], "state_str": "case h.e'_2.h.e'_5.h.e'_5.h.e'_5\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Module.Oriented ℝ V (Fin 2)\ns : Sphere P\np₁ p₂ p₃ : P\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₃ : p₃ ∈ s\nhp₁p₂ : p₁ ≠ p₂\nhp₁p₃ : p₁ ≠ p₃\nhp₂p₃ : p₂ ≠ p₃\n⊢ (∡ p₁ p₂ p₃).tan⁻¹ = (∡ p₃ p₁ s.center).tan"}
+{"state": [{"context": ["V : Type u_1", "P : Type u_2", "inst✝⁴ : NormedAddCommGroup V", "inst✝³ : InnerProductSpace ℝ V", "inst✝² : MetricSpace P", "inst✝¹ : NormedAddTorsor V P", "hd2 : Fact (finrank ℝ V = 2)", "inst✝ : Module.Oriented ℝ V (Fin 2)", "s : Sphere P", "p₁ p₂ p₃ : P", "hp₁ : p₁ ∈ s", "hp₂ : p₂ ∈ s", "hp₃ : p₃ ∈ s", "hp₁p₂ : p₁ ≠ p₂", "hp₁p₃ : p₁ ≠ p₃", "hp₂p₃ : p₂ ≠ p₃"], "goal": "2 • ∡ p₁ p₂ p₃ + 2 • ∡ p₃ p₁ s.center = ↑π"}], "premise": [1690, 69361, 119708], "state_str": "case h.e'_2.h.e'_5.h.e'_5.h.e'_5.convert_3\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Module.Oriented ℝ V (Fin 2)\ns : Sphere P\np₁ p₂ p₃ : P\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₃ : p₃ ∈ s\nhp₁p₂ : p₁ ≠ p₂\nhp₁p₃ : p₁ ≠ p₃\nhp₂p₃ : p₂ ≠ p₃\n⊢ 2 • ∡ p₁ p₂ p₃ + 2 • ∡ p₃ p₁ s.center = ↑π"}