diff --git "a/random/random_our/test.jsonl" "b/random/random_our/test.jsonl"
new file mode 100644--- /dev/null
+++ "b/random/random_our/test.jsonl"
@@ -0,0 +1,4046 @@
+{"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": "((μ.restrict t).restrict s) u = (μ.restrict s) u"}, "premise": [32215, 32223, 133448], "module": ["Mathlib/MeasureTheory/Measure/Restrict.lean"]}
+{"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, 32223, 32215], "module": ["Mathlib/MeasureTheory/Measure/Restrict.lean"]}
+{"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], "module": ["Mathlib/Data/List/Nodup.lean"]}
+{"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], "module": ["Mathlib/Probability/Kernel/Basic.lean"]}
+{"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": "IsBigOWith c (map k l) f g ↔ IsBigOWith c l (f ∘ k) (g ∘ k)"}, "premise": [43347, 16164], "module": ["Mathlib/Analysis/Asymptotics/Asymptotics.lean"]}
+{"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": [43347, 16164], "module": ["Mathlib/Analysis/Asymptotics/Asymptotics.lean"]}
+{"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], "module": ["Mathlib/Analysis/Calculus/Deriv/Comp.lean"]}
+{"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, 106115], "module": ["Mathlib/Analysis/BoxIntegral/Basic.lean"]}
+{"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": [1674, 106115, 36534], "module": ["Mathlib/Analysis/BoxIntegral/Basic.lean"]}
+{"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", "ε₁ : ℝ", "ε₁0 : 0 < ε₁", "hε₁ : 2 * μ.toBoxAdditive I * ε₁ < 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": [106115], "module": ["Mathlib/Analysis/BoxIntegral/Basic.lean"]}
+{"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, 106115], "module": ["Mathlib/Analysis/BoxIntegral/Basic.lean"]}
+{"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": [14273, 106115, 34344, 42680, 34363], "module": ["Mathlib/Analysis/BoxIntegral/Basic.lean"]}
+{"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": [143385, 1674, 106115, 14284], "module": ["Mathlib/Analysis/BoxIntegral/Basic.lean"]}
+{"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"], "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], "module": ["Mathlib/Analysis/BoxIntegral/Basic.lean"]}
+{"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, 30620], "module": ["Mathlib/Analysis/BoxIntegral/Basic.lean"]}
+{"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": [143493, 1673, 27599, 11251, 27511, 16024, 2106, 30620], "module": ["Mathlib/Analysis/BoxIntegral/Basic.lean"]}
+{"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": [30620, 119727], "module": ["Mathlib/Analysis/BoxIntegral/Basic.lean"]}
+{"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": [64866, 133635, 55373, 58062, 119727, 34358], "module": ["Mathlib/Analysis/BoxIntegral/Basic.lean"]}
+{"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": [64866, 133635, 40867, 1674, 55373, 58062, 34358, 143385, 2106, 2107], "module": ["Mathlib/Analysis/BoxIntegral/Basic.lean"]}
+{"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": [40867, 40869, 1674, 143385, 2106, 2107], "module": ["Mathlib/Analysis/BoxIntegral/Basic.lean"]}
+{"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": [40869, 106102], "module": ["Mathlib/Analysis/BoxIntegral/Basic.lean"]}
+{"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": [108341, 33981, 106102, 36525], "module": ["Mathlib/Analysis/BoxIntegral/Basic.lean"]}
+{"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": [31842, 36525, 14288, 108341, 34358, 27511, 34363, 33981], "module": ["Mathlib/Analysis/BoxIntegral/Basic.lean"]}
+{"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 < ⊤"], "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], "module": ["Mathlib/Analysis/BoxIntegral/Basic.lean"]}
+{"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, 1673, 14288, 18791, 27511, 34168], "module": ["Mathlib/Analysis/BoxIntegral/Basic.lean"]}
+{"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": [18791, 1673, 139088, 14288, 27511, 34168], "module": ["Mathlib/Analysis/BoxIntegral/Basic.lean"]}
+{"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": [18791, 1673, 1674, 14288, 135251, 27511, 34168], "module": ["Mathlib/Analysis/BoxIntegral/Basic.lean"]}
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+{"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": [111329, 14273, 126952, 103917, 42671], "module": ["Mathlib/Analysis/BoxIntegral/Basic.lean"]}
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+{"state": {"context": ["E : Type u_1", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : CompleteSpace E", "H : Type u_2", "inst✝³ : TopologicalSpace H", "I : ModelWithCorners ℝ E H", "M : Type u_3", "inst✝² : TopologicalSpace M", "inst✝¹ : ChartedSpace H M", "inst✝ : SmoothManifoldWithCorners I M", "γ γ' : ℝ → M", "v : (x : M) → TangentSpace I x", "s s' : Set ℝ", "t₀ a : ℝ", "ha : a ≠ 0", "hγ : IsIntegralCurveOn (γ ∘ fun x => x * a) (a • v) {t | t * a ∈ s}"], "goal": "v = a⁻¹ • a • v"}, "premise": [68649], "module": ["Mathlib/Geometry/Manifold/IntegralCurve.lean"]}
+{"state": {"context": ["E : Type u_1", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : CompleteSpace E", "H : Type u_2", "inst✝³ : TopologicalSpace H", "I : ModelWithCorners ℝ E H", "M : Type u_3", "inst✝² : TopologicalSpace M", "inst✝¹ : ChartedSpace H M", "inst✝ : SmoothManifoldWithCorners I M", "γ γ' : ℝ → M", "v : (x : M) → TangentSpace I x", "s s' : Set ℝ", "t₀ a : ℝ", "ha : a ≠ 0", "hγ : IsIntegralCurveOn (γ ∘ fun x => x * a) (a • v) {t | t * a ∈ s}"], "goal": "s = {t | t * a⁻¹ ∈ {t | t * a ∈ s}}"}, "premise": [68649], "module": ["Mathlib/Geometry/Manifold/IntegralCurve.lean"]}
+{"state": {"context": ["C : Type u_1", "inst✝ : Category.{u_2, u_1} C", "P : Karoubi C"], "goal": "((toKaroubi C).map P.p).f = (P.decompId_p ≫ P.decompId_i).f"}, "premise": [97095, 97104, 97111, 97119, 97120], "module": ["Mathlib/CategoryTheory/Idempotents/Karoubi.lean"]}
+{"state": {"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "D : Type u₂", "inst✝¹ : Category.{v₂, u₂} D", "inst✝ : MonoidalCategory D", "A : Comon_ (C ⥤ D)", "X✝ Y✝ : C", "f : X✝ ⟶ Y✝"], "goal": "A.X.map f ≫ A.counit.app Y✝ = A.counit.app X✝"}, "premise": [97888, 98915, 96174], "module": ["Mathlib/CategoryTheory/Monoidal/Internal/FunctorCategory.lean"]}
+{"state": {"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "D : Type u₂", "inst✝¹ : Category.{v₂, u₂} D", "inst✝ : MonoidalCategory D", "A : Comon_ (C ⥤ D)", "X✝ Y✝ : C", "f : X✝ ⟶ Y✝"], "goal": "A.counit.app X✝ ≫ 𝟙 (𝟙_ D) = A.counit.app X✝"}, "premise": [97888, 98915, 96174], "module": ["Mathlib/CategoryTheory/Monoidal/Internal/FunctorCategory.lean"]}
+{"state": {"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "D : Type u₂", "inst✝¹ : Category.{v₂, u₂} D", "inst✝ : MonoidalCategory D", "A : Comon_ (C ⥤ D)", "X✝ Y✝ : C", "f : X✝ ⟶ Y✝"], "goal": "A.X.map f ≫ A.comul.app Y✝ = A.comul.app X✝ ≫ (A.X.map f ⊗ A.X.map f)"}, "premise": [97888, 98917], "module": ["Mathlib/CategoryTheory/Monoidal/Internal/FunctorCategory.lean"]}
+{"state": {"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "D : Type u₂", "inst✝¹ : Category.{v₂, u₂} D", "inst✝ : MonoidalCategory D", "A : Comon_ (C ⥤ D)", "X : C"], "goal": "A.X.map (𝟙 X) = 𝟙 (A.X.obj X)"}, "premise": [99920], "module": ["Mathlib/CategoryTheory/Monoidal/Internal/FunctorCategory.lean"]}
+{"state": {"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "D : Type u₂", "inst✝¹ : Category.{v₂, u₂} D", "inst✝ : MonoidalCategory D", "A : Comon_ (C ⥤ D)", "X✝ Y✝ Z✝ : C", "f : X✝ ⟶ Y✝", "g : Y✝ ⟶ Z✝"], "goal": "A.X.map (f ≫ g) = A.X.map f ≫ A.X.map g"}, "premise": [99919], "module": ["Mathlib/CategoryTheory/Monoidal/Internal/FunctorCategory.lean"]}
+{"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, 96173, 99219, 99222, 107105], "module": ["Mathlib/CategoryTheory/Monoidal/Mon_.lean"]}
+{"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": [107105, 100101, 107078, 107079, 107082, 96173, 99219, 100117, 99222, 99225], "module": ["Mathlib/CategoryTheory/Monoidal/Mon_.lean"]}
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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": "(((((((((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": [99259, 96173, 88742], "module": ["Mathlib/CategoryTheory/Monoidal/Mon_.lean"]}
+{"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": [96173, 88742, 96175], "module": ["Mathlib/CategoryTheory/Monoidal/Mon_.lean"]}
+{"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": [99264, 96173, 96175], "module": ["Mathlib/CategoryTheory/Monoidal/Mon_.lean"]}
+{"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": [99264, 99211, 96173], "module": ["Mathlib/CategoryTheory/Monoidal/Mon_.lean"]}
+{"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": [99211, 96173], "module": ["Mathlib/CategoryTheory/Monoidal/Mon_.lean"]}
+{"state": {"context": ["x y : ℝ"], "goal": "↑(tanh x) = ↑(sinh x / cosh x)"}, "premise": [149118], "module": ["Mathlib/Data/Complex/Exponential.lean"]}
+{"state": {"context": ["n m k : ℕ"], "goal": "n &&& m &&& k = n &&& (m &&& k)"}, "premise": [1241, 1242, 140199], "module": ["Mathlib/Data/Nat/Bitwise.lean"]}
+{"state": {"context": ["a b x : PGame"], "goal": "-x.birthday.toPGame ≤ x"}, "premise": [48114, 48116, 50300], "module": ["Mathlib/SetTheory/Game/Birthday.lean"]}
+{"state": {"context": ["E : Type u_1", "ι : Type u_2", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : NormedSpace ℝ E", "b : Basis ι ℝ E", "inst✝² : Finite ι", "inst✝¹ : MeasurableSpace E", "inst✝ : OpensMeasurableSpace E", "μ : Measure E"], "goal": "IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑b)).toAddSubgroup) (fundamentalDomain b) μ"}, "premise": [141384, 29169, 32981, 111682, 111687], "module": ["Mathlib/Algebra/Module/Zlattice/Basic.lean"]}
+{"state": {"context": ["E : Type u_1", "ι : Type u_2", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : NormedSpace ℝ E", "b : Basis ι ℝ E", "inst✝² : Finite ι", "inst✝¹ : MeasurableSpace E", "inst✝ : OpensMeasurableSpace E", "μ : Measure E", "val✝ : Fintype ι"], "goal": "IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑b)).toAddSubgroup) (fundamentalDomain b) μ"}, "premise": [111682, 111687, 141384, 29169, 32981], "module": ["Mathlib/Algebra/Module/Zlattice/Basic.lean"]}
+{"state": {"context": ["ι : Type u_1", "l : Filter ι", "E : Type u_2", "inst✝⁴ : NormedAddCommGroup E", "𝕜 : Type u_3", "inst✝³ : RCLike 𝕜", "inst✝² : NormedSpace 𝕜 E", "G : Type u_4", "inst✝¹ : NormedAddCommGroup G", "inst✝ : NormedSpace 𝕜 G", "f : ι → E → G", "g : E → G", "f' : ι → E → E →L[𝕜] G", "g' : E → E →L[𝕜] G", "x : E", "hf' : UniformCauchySeqOnFilter f' l (𝓝 x)", "hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2", "hfg : Cauchy (map (fun n => f n x) l)", "this : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E"], "goal": "UniformCauchySeqOnFilter f l (𝓝 x)"}, "premise": [42852, 2106, 2107, 60153, 67192, 119729], "module": ["Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean"]}
+{"state": {"context": ["ι : Type u_1", "l : Filter ι", "E : Type u_2", "inst✝⁴ : NormedAddCommGroup E", "𝕜 : Type u_3", "inst✝³ : RCLike 𝕜", "inst✝² : NormedSpace 𝕜 E", "G : Type u_4", "inst✝¹ : NormedAddCommGroup G", "inst✝ : NormedSpace 𝕜 G", "f : ι → E → G", "g : E → G", "f' : ι → E → E →L[𝕜] G", "g' : E → E →L[𝕜] G", "x : E", "hf' : TendstoUniformlyOnFilter (fun n z => f' n.1 z - f' n.2 z) 0 (l ×ˢ l) (𝓝 x)", "hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2", "hfg : Cauchy (map (fun n => f n x) l)", "this : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E"], "goal": "TendstoUniformlyOnFilter (fun n z => f n.1 z - f n.2 z) 0 (l ×ˢ l) (𝓝 x)"}, "premise": [42852, 119729, 67192, 60153, 2106, 2107], "module": ["Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean"]}
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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", "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 t₀ b)"}, "premise": [135720], "module": ["Mathlib/Analysis/ODE/Gronwall.lean"]}
+{"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, 2100], "module": ["Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean"]}
+{"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], "module": ["Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean"]}
+{"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], "module": ["Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean"]}
+{"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"], "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], "module": ["Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean"]}
+{"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": [99600, 99593, 2100, 99919], "module": ["Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean"]}
+{"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": [99600, 99593, 99919], "module": ["Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean"]}
+{"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": [99600, 99593, 89631, 99919], "module": ["Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean"]}
+{"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], "module": ["Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean"]}
+{"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 ⋙ 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], "module": ["Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean"]}
+{"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, 2100, 2101, 113673], "module": ["Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean"]}
+{"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": [99600, 113673, 2100, 2101], "module": ["Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean"]}
+{"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, 2107], "module": ["Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean"]}
+{"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, 75175], "module": ["Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean"]}
+{"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": [128914, 2107, 102934], "module": ["Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean"]}
+{"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": [128914, 2107, 102934], "module": ["Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean"]}
+{"state": {"context": ["x : EReal", "h : x < 0"], "goal": "⊤ * x = ⊥"}, "premise": [147093, 147252], "module": ["Mathlib/Data/Real/EReal.lean"]}
+{"state": {"context": ["x : EReal", "h : x < 0"], "goal": "x * ⊤ = ⊥"}, "premise": [147252, 147093], "module": ["Mathlib/Data/Real/EReal.lean"]}
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+{"state": {"context": ["R : Type uR", "inst✝⁶ : Semiring R", "S : Type uS", "inst✝⁵ : CommSemiring S", "T : Type uT", "A : Type uA", "inst✝⁴ : Semiring A", "inst✝³ : Algebra S A", "r : R → R → Prop", "inst✝² : Semiring T", "B : Type u₄", "inst✝¹ : Semiring B", "inst✝ : Algebra S B", "s : A → A → Prop", "f : { f // ∀ ⦃x y : A⦄, s x y → f x = f y }", "x✝ : A"], "goal": "↑((fun F => ⟨F.comp (mkAlgHom S s), ⋯⟩) ((fun f' => preLiftAlgHom S ⋯) f)) x✝ = ↑f x✝"}, "premise": [1670, 117118, 117125, 121043, 121074, 121547, 121549, 121551, 125003, 125014, 125019], "module": ["Mathlib/Algebra/RingQuot.lean"]}
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+{"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": "∑ x_1 ∈ p.support, f (p x_1) * x ^ x_1 = (liftNC ↑f ⇑((powersHom S) x)) p"}, "premise": [101221, 102824], "module": ["Mathlib/Algebra/Polynomial/Eval.lean"]}
+{"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, 15903, 16166, 16327], "module": ["Mathlib/Order/Filter/Basic.lean"]}
+{"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": [14273, 16229, 16166, 16327, 16013, 15903], "module": ["Mathlib/Order/Filter/Basic.lean"]}
+{"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": [131585, 16166, 16327, 15889, 15903], "module": ["Mathlib/Order/Filter/Basic.lean"]}
+{"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], "module": ["Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean"]}
+{"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, 142613], "module": ["Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean"]}
+{"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": [77058, 142613], "module": ["Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean"]}
+{"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": [74756, 78760, 75181, 1713, 102995, 142613], "module": ["Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean"]}
+{"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], "module": ["Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean"]}
+{"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": "(LinearMap.llcomp R (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ) (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ) (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)) mA ∘ₗ mA = (((LinearMap.llcomp R (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ) (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) (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)) LinearMap.lflip) ((LinearMap.llcomp R (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ) (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ) (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)) mA.flip ∘ₗ mA)).flip"}, "premise": [128589], "module": ["Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean"]}
+{"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, 110044, 118422, 118909, 136152], "module": ["Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean"]}
+{"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": [116934, 117033, 109741, 118422, 119703, 136152, 81818, 85979, 110044, 118909], "module": ["Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean"]}
+{"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 ����", "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 + (ixb * iza + 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 + (ixb * iya + ixb * 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], "module": ["Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean"]}
+{"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, 2100, 81869], "module": ["Mathlib/RepresentationTheory/Character.lean"]}
+{"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": [23732, 81869, 2100], "module": ["Mathlib/RepresentationTheory/Character.lean"]}
+{"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], "module": ["Mathlib/Analysis/Convex/Combination.lean"]}
+{"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], "module": ["Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean"]}
+{"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], "module": ["Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean"]}
+{"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], "module": ["Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean"]}
+{"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], "module": ["Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean"]}
+{"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": "(ρ_ (__src✝.obj (X₁, X₂))).hom = __src✝.obj (X₁, X₂) ◁ (λ_ (𝟙_ C)).inv ≫ tensor_μ C (X₁, X₂) (𝟙_ (C × C)) ≫ __src✝.map (ρ_ (X₁, X₂)).hom"}, "premise": [107142], "module": ["Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean"]}
+{"state": {"context": ["R : Type u_1", "inst✝¹ : LinearOrderedSemifield R", "inst✝ : FloorSemiring R", "b : ℕ", "hb : 1 < b", "z : ℤ"], "goal": "log b (↑b ^ z) = z"}, "premise": [3407], "module": ["Mathlib/Data/Int/Log.lean"]}
+{"state": {"context": ["R : Type u_1", "inst✝¹ : LinearOrderedSemifield R", "inst✝ : FloorSemiring R", "b : ℕ", "hb : 1 < b", "n : ℕ"], "goal": "log b (↑b ^ ↑n) = ↑n"}, "premise": [3407], "module": ["Mathlib/Data/Int/Log.lean"]}
+{"state": {"context": ["R : Type u_1", "inst✝¹ : LinearOrderedSemifield R", "inst✝ : FloorSemiring R", "b : ℕ", "hb : 1 < b", "n : ℕ"], "goal": "log b (↑b ^ (-↑n)) = -↑n"}, "premise": [3407], "module": ["Mathlib/Data/Int/Log.lean"]}
+{"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 : Semiring R", "inst✝³ : AddCommMonoid M", "inst✝² : AddCommMonoid M₂", "inst✝¹ : Module R M", "inst✝ : Module R M₂", "p : Submodule R M", "q : Submodule R M₂", "m : M"], "goal": "(m, 0) ∈ fst R M M₂"}, "premise": [84783, 109977, 110007], "module": ["Mathlib/LinearAlgebra/Prod.lean"]}
+{"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 : Semiring R", "inst✝³ : AddCommMonoid M", "inst✝² : AddCommMonoid M₂", "inst✝¹ : Module R M", "inst✝ : Module R M₂", "p : Submodule R M", "q : Submodule R M₂"], "goal": "∀ (x y : ↥(fst R M M₂)), (fun x => (↑x).1) (x + y) = (fun x => (↑x).1) x + (fun x => (↑x).1) y"}, "premise": [1199, 2029, 109628, 119463, 120171, 137123, 137296], "module": ["Mathlib/LinearAlgebra/Prod.lean"]}
+{"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 : Semiring R", "inst✝³ : AddCommMonoid M", "inst✝² : AddCommMonoid M₂", "inst✝¹ : Module R M", "inst✝ : Module R M₂", "p : Submodule R M", "q : Submodule R M₂"], "goal": "∀ (m : R) (x : ↥(fst R M M₂)), { toFun := fun x => (↑x).1, map_add' := ⋯ }.toFun (m • x) = (RingHom.id R) m • { toFun := fun x => (↑x).1, map_add' := ⋯ }.toFun x"}, "premise": [1199, 2029, 7143, 118593, 121577, 137123, 137296], "module": ["Mathlib/LinearAlgebra/Prod.lean"]}
+{"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 : Semiring R", "inst✝³ : AddCommMonoid M", "inst✝² : AddCommMonoid M₂", "inst✝¹ : Module R M", "inst✝ : Module R M₂", "p : Submodule R M", "q : Submodule R M₂", "x : M", "y : M₂", "hy : (x, y) ∈ fst R M M₂"], "goal": "(fun m => ⟨(m, 0), ⋯⟩) ({ toFun := fun x => (↑x).1, map_add' := ⋯, map_smul' := ⋯ }.toFun ⟨(x, y), hy⟩) = ⟨(x, y), hy⟩"}, "premise": [84783, 109977, 110007, 1180, 2100], "module": ["Mathlib/LinearAlgebra/Prod.lean"]}
+{"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 : Semiring R", "inst✝³ : AddCommMonoid M", "inst✝² : AddCommMonoid M₂", "inst✝¹ : Module R M", "inst✝ : Module R M₂", "p : Submodule R M", "q : Submodule R M₂", "x : M", "y : M₂", "hy✝ : (x, y) ∈ fst R M M₂", "hy : y = 0"], "goal": "(fun m => ⟨(m, 0), ⋯⟩) ({ toFun := fun x => (↑x).1, map_add' := ⋯, map_smul' := ⋯ }.toFun ⟨(x, y), hy✝⟩) = ⟨(x, y), hy✝⟩"}, "premise": [84783, 2100, 110007, 109977, 1180], "module": ["Mathlib/LinearAlgebra/Prod.lean"]}
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+{"state": {"context": ["α✝ : Type u_1", "inst✝¹ : OrderedCancelCommMonoid α✝", "s✝ t✝ : Set α✝", "hs✝ : s✝.IsPWO", "ht✝ : t✝.IsPWO", "a✝ : α✝", "u : Set α✝", "hu : u.IsPWO", "x : α✝ × α✝", "α : Type u_2", "inst✝ : LinearOrderedCancelCommMonoid α", "s t : Set α", "hs : s.IsWF", "ht : t.IsWF", "hns : s.Nonempty", "hnt : t.Nonempty", "a b : α"], "goal": "a ∈ s ∧ b ∈ t ∧ a * b = hs.min hns * ht.min hnt ↔ a = hs.min hns ∧ b = ht.min hnt"}, "premise": [136272, 137313, 138737], "module": ["Mathlib/Data/Finset/MulAntidiagonal.lean"]}
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+{"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, 105289, 105657], "module": ["Mathlib/Analysis/SumOverResidueClass.lean"]}
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+{"state": {"context": ["C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "D : Type u₂", "inst✝ : Category.{v₂, u₂} D", "G : C", "hG : IsDetector G", "A✝ B✝ : C", "f : A✝ ⟶ B✝", "hf : IsIso ((coyoneda.obj (op G)).map f)", "h : G ⟶ B✝"], "goal": "∃! h', h' ≫ f = h"}, "premise": [1674, 97999, 98000], "module": ["Mathlib/CategoryTheory/Generator.lean"]}
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+{"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": "∀ (x : R ⧸ I), x ∈ ker (Quotient.lift I f H) ↔ x ∈ map (Quotient.mk I) (ker f)"}, "premise": [80339], "module": ["Mathlib/RingTheory/Ideal/QuotientOperations.lean"]}
+{"state": {"context": ["n✝ n p : ℕ", "hp : Prime p"], "goal": "p ∣ (n + 1)! ↔ p ≤ n + 1"}, "premise": [144341, 144416, 2110, 2141, 2143, 2146, 3523, 14302], "module": ["Mathlib/Data/Nat/Prime/Basic.lean"]}
+{"state": {"context": ["n✝ n p : ℕ", "hp : Prime p"], "goal": "p ∣ n + 1 ∨ p ≤ n ↔ p ≤ n + 1"}, "premise": [144416, 2146, 3523, 144341, 14302, 2141, 2110, 2143], "module": ["Mathlib/Data/Nat/Prime/Basic.lean"]}
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+{"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], "module": ["Mathlib/Topology/MetricSpace/Isometry.lean"]}
+{"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], "module": ["Mathlib/CategoryTheory/Idempotents/KaroubiKaroubi.lean"]}
+{"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], "module": ["Mathlib/CategoryTheory/Idempotents/KaroubiKaroubi.lean"]}
+{"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], "module": ["Mathlib/RingTheory/Polynomial/Quotient.lean"]}
+{"state": {"context": ["R : Type u_1", "σ : Type u_2", "inst✝ : CommRing R", "r : R", "I : Ideal R", "f : MvPolynomial σ (R ⧸ I)"], "goal": "∀ (a : R ⧸ I), (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)) (C a)) = C a"}, "premise": [112219], "module": ["Mathlib/RingTheory/Polynomial/Quotient.lean"]}
+{"state": {"context": ["R : Type u_1", "σ : Type u_2", "inst✝ : CommRing R", "r : R", "I : Ideal R", "f : MvPolynomial σ (R ⧸ I)"], "goal": "∀ (p q : MvPolynomial σ (R ⧸ I)), (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)) p) = p → (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)) q) = q → (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)) (p + q)) = p + q"}, "premise": [112219], "module": ["Mathlib/RingTheory/Polynomial/Quotient.lean"]}
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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"], "goal": "∃ n e, Smooth I 𝓘(ℝ, EuclideanSpace ℝ (Fin n)) e ∧ Injective e ∧ ∀ (x : M), Injective ⇑(mfderiv I 𝓘(ℝ, EuclideanSpace ℝ (Fin n)) e x)"}, "premise": [141384], "module": ["Mathlib/Geometry/Manifold/WhitneyEmbedding.lean"]}
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+{"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, 118863, 119730, 81621, 81690], "module": ["Mathlib/RingTheory/OreLocalization/Basic.lean"]}
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+{"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": "∫ (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, 38967], "module": ["Mathlib/Analysis/SpecialFunctions/Integrals.lean"]}
+{"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", "H : ∫ (x : ℝ) in a..b, cos x ^ (n + 1) * cos x = cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a - ∫ (x : ℝ) in a..b, -↑(n + 1) * sin x * cos x ^ n * sin 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, 26326, 26326], "module": ["Mathlib/Analysis/SpecialFunctions/Integrals.lean"]}
+{"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"}, "premise": [27349, 26326], "module": ["Mathlib/Analysis/SpecialFunctions/Integrals.lean"]}
+{"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 cos MeasureTheory.volume a b"}, "premise": [27349, 26326], "module": ["Mathlib/Analysis/SpecialFunctions/Integrals.lean"]}
+{"state": {"context": ["m n : ℕ"], "goal": "(subNatNat m n).bodd = xor m.bodd n.bodd"}, "premise": [2483, 129210, 144078], "module": ["Mathlib/Data/Int/Bitwise.lean"]}
+{"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, 104749, 104871], "module": ["Mathlib/Algebra/Order/Module/Defs.lean"]}
+{"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": [104871, 104749, 110032, 105657, 119769], "module": ["Mathlib/Algebra/Order/Module/Defs.lean"]}
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+{"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 ≫ X.arrow = g ≫ X.arrow"}, "premise": [96191], "module": ["Mathlib/CategoryTheory/Subterminal.lean"]}
+{"state": {"context": ["f : ℕ → ℂ", "s : ℂ", "hs : abscissaOfAbsConv f < ↑s.re"], "goal": "LSeriesSummable f s"}, "premise": [19262, 70141, 131585, 131592, 147085], "module": ["Mathlib/NumberTheory/LSeries/Convergence.lean"]}
+{"state": {"context": ["f : ℕ → ℂ", "s : ℂ", "hs : ∃ a, LSeriesSummable f ↑a ∧ a < s.re"], "goal": "LSeriesSummable f s"}, "premise": [19262, 70141, 131585, 131592, 147085], "module": ["Mathlib/NumberTheory/LSeries/Convergence.lean"]}
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+{"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": [31461, 143757, 118863, 120658, 27956, 133462, 32215], "module": ["Mathlib/Probability/ConditionalProbability.lean"]}
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+{"state": {"context": ["α : Type u_1", "E : α → Type u_2", "q : ℝ≥0∞", "inst✝ : (i : α) → NormedAddCommGroup (E i)", "i : α", "hp : ⊤ ≠ 0", "f : ↥(lp E ⊤)"], "goal": "‖↑f i‖ ≤ ‖f‖"}, "premise": [70039, 143376], "module": ["Mathlib/Analysis/Normed/Lp/lpSpace.lean"]}
+{"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, 70039], "module": ["Mathlib/Analysis/Normed/Lp/lpSpace.lean"]}
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+{"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": [42680, 40018, 45196, 40087], "module": ["Mathlib/Analysis/Normed/Lp/lpSpace.lean"]}
+{"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": [63268, 45195, 45196, 40087, 42680], "module": ["Mathlib/Analysis/Normed/Lp/lpSpace.lean"]}
+{"state": {"context": ["θ : Angle"], "goal": "(θ - ↑(π / 2)).cos = θ.sin"}, "premise": [38265, 38585], "module": ["Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean"]}
+{"state": {"context": ["x✝ : ℝ"], "goal": "(↑x✝ - ↑(π / 2)).cos = (↑x✝).sin"}, "premise": [38265, 38585], "module": ["Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean"]}
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+{"state": {"context": ["α : Type u_1", "inst✝¹ : DecidableEq α", "inst✝ : CommGroup α", "e : α", "x : Finset α × Finset α"], "goal": "mulETransformLeft e⁻¹ x = (mulETransformRight e x.swap).swap"}, "premise": [100276, 118887], "module": ["Mathlib/Combinatorics/Additive/ETransform.lean"]}
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+{"state": {"context": ["ι : Type u_1", "α : Type u", "β : Type v", "γ : Type w", "l₁ l₂ : List α", "op : α → α → α", "ha : Std.Associative op", "hc : Std.Commutative op", "a : α", "l : List α", "a₁ a₂ : α"], "goal": "(l <*> op a₁ (op a₂ a)) = op a₁ ((a :: l) <*> a₂)"}, "premise": [2584], "module": ["Mathlib/Data/List/Basic.lean"]}
+{"state": {"context": ["α : Type u_1", "α' : Type u_2", "β : Type u_3", "β' : Type u_4", "γ : Type u_5", "E : Type u_6", "inst✝⁸ : MeasurableSpace α", "inst✝⁷ : MeasurableSpace α'", "inst✝⁶ : MeasurableSpace β", "inst✝⁵ : MeasurableSpace β'", "inst✝⁴ : MeasurableSpace γ", "μ μ' : Measure α", "ν ν' : Measure β", "τ : Measure γ", "inst✝³ : NormedAddCommGroup E", "inst✝² : SFinite ν", "inst✝¹ : SFinite μ", "inst✝ : NeZero ν", "s : Set α", "h : NullMeasurableSet (Prod.fst ⁻¹' s) (μ.prod ν)"], "goal": "NullMeasurableSet (s ×ˢ ?m.137705) (μ.prod ?m.137701)"}, "premise": [133916], "module": ["Mathlib/MeasureTheory/Constructions/Prod/Basic.lean"]}
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+{"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], "module": ["Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean"]}
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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": [28986, 34867], "module": ["Mathlib/MeasureTheory/Measure/LogLikelihoodRatio.lean"]}
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+{"state": {"context": ["α : Type u", "β : Type v", "ι : Sort w", "γ : Type x", "s✝ t : Set α", "inst✝² : Preorder α", "inst✝¹ : IsDirected α fun x x_1 => x ≤ x_1", "inst✝ : Nonempty α", "s : Set α", "I : Set β", "S : β → Set α", "H : I.Finite"], "goal": "BddAbove (⋃ i ∈ ∅, S i) ↔ ∀ i ∈ ∅, BddAbove (S i)"}, "premise": [17927, 133385, 135376], "module": ["Mathlib/Data/Set/Finite.lean"]}
+{"state": {"context": ["α : Type u", "β : Type v", "ι : Sort w", "γ : Type x", "s✝¹ t : Set α", "inst✝² : Preorder α", "inst✝¹ : IsDirected α fun x x_1 => x ≤ x_1", "inst✝ : Nonempty α", "s : Set α", "I : Set β", "S : β → Set α", "H : I.Finite", "a✝ : β", "s✝ : Set β", "x✝¹ : a✝ ∉ s✝", "x✝ : s✝.Finite", "hs : BddAbove (⋃ i ∈ s✝, S i) ↔ ∀ i ∈ s✝, BddAbove (S i)"], "goal": "BddAbove (⋃ i ∈ insert a✝ s✝, S i) ↔ ∀ i ∈ insert a✝ s✝, BddAbove (S i)"}, "premise": [17841, 133510, 135383], "module": ["Mathlib/Data/Set/Finite.lean"]}
+{"state": {"context": ["n : ℕ"], "goal": "0 < φ n ↔ 0 < n"}, "premise": [103552], "module": ["Mathlib/Data/Nat/Totient.lean"]}
+{"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], "module": ["Mathlib/LinearAlgebra/Basis.lean"]}
+{"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, 4618, 143302], "module": ["Mathlib/RingTheory/Perfection.lean"]}
+{"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, 1739, 143302], "module": ["Mathlib/RingTheory/Perfection.lean"]}
+{"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", "hfn : (coeff (ModP K v O hv p) p (Nat.find h + k)) 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)) f) ^ p ^ (Nat.find h + k)"}, "premise": [4618, 143302], "module": ["Mathlib/RingTheory/Perfection.lean"]}
+{"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, 2100], "module": ["Mathlib/RingTheory/Perfection.lean"]}
+{"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, 80150], "module": ["Mathlib/RingTheory/Perfection.lean"]}
+{"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": [80040, 80002, 2100, 121592], "module": ["Mathlib/RingTheory/Perfection.lean"]}
+{"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": [80032, 119745, 80002, 80040, 121592, 119761, 75416], "module": ["Mathlib/RingTheory/Perfection.lean"]}
+{"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": "v ((algebraMap O K) x) ^ (p * p ^ (Nat.find h + k)) = v ((algebraMap O K) x) ^ (p * p ^ (Nat.find h).add k)"}, "premise": [75416, 80002, 80032, 80040, 119745, 119761, 121592], "module": ["Mathlib/RingTheory/Perfection.lean"]}
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+{"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✝¹² : CommSemiring R", "inst✝¹¹ : AddCommMonoid M", "inst✝¹⁰ : Module R M", "R₂ : Type u_7", "M₂ : Type u_8", "inst✝⁹ : CommRing R₂", "inst✝⁸ : AddCommGroup M₂", "inst✝⁷ : Module R₂ M₂", "e : Basis ι R M", "v : ι' → M", "i : ι", "j : ι'", "N : Type u_9", "inst✝⁶ : AddCommMonoid N", "inst✝⁵ : 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✝⁴ : Fintype ι'", "inst✝³ : Finite κ", "inst✝² : Fintype ι", "inst✝¹ : DecidableEq ι", "inst✝ : DecidableEq ι'", "b₁ : Basis ι R M", "b₂ : Basis ι' R M", "b₃ : Basis κ R N", "A : Matrix κ ι R"], "goal": "A * b₁.toMatrix ⇑b₂ = (toMatrix b₂ b₃) ((toLin b₁ b₃) A)"}, "premise": [141384, 85032], "module": ["Mathlib/LinearAlgebra/Matrix/Basis.lean"]}
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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], "module": ["Mathlib/CategoryTheory/Abelian/Basic.lean"]}
+{"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": "IsColimit (CokernelCofork.ofπ 0 ⋯)"}, "premise": [94315], "module": ["Mathlib/CategoryTheory/Abelian/Basic.lean"]}
+{"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, 115834, 117880, 119769, 120670], "module": ["Mathlib/Analysis/Convex/Slope.lean"]}
+{"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": [104742, 35799, 117880, 119769, 115834, 120670], "module": ["Mathlib/Analysis/Convex/Slope.lean"]}
+{"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], "module": ["Mathlib/Analysis/Convex/Slope.lean"]}
+{"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, 56662], "module": ["Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean"]}
+{"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": [67763, 56660, 56662, 12602, 12573], "module": ["Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean"]}
+{"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": "∀ (x : H) (i : Set E), i ∈ 𝓝 (↑I x) ∧ IsCompact i → IsCompact (↑I.symm '' (i ∩ range ↑I))"}, "premise": [56662], "module": ["Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean"]}
+{"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], "module": ["Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean"]}
+{"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], "module": ["Mathlib/RingTheory/PowerSeries/Basic.lean"]}
+{"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": "(coeff R ?m.10640) φ = (coeff R ?m.10640) ψ"}, "premise": [79089], "module": ["Mathlib/RingTheory/PowerSeries/Basic.lean"]}
+{"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": "n () = ?m.10640"}, "premise": [79089], "module": ["Mathlib/RingTheory/PowerSeries/Basic.lean"]}
+{"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": "ℕ"}, "premise": [79089], "module": ["Mathlib/RingTheory/PowerSeries/Basic.lean"]}
+{"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], "module": ["Mathlib/Algebra/Group/Submonoid/Operations.lean"]}
+{"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], "module": ["Mathlib/CategoryTheory/Monoidal/Bimod.lean"]}
+{"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 ({ X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := ⋯, left_assoc := ⋯, actRight := TensorBimod.actRight M N, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ }.actRight ▷ P.X) ((α_ { X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := ⋯, left_assoc := ⋯, actRight := TensorBimod.actRight M N, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ }.X Y.X P.X).hom ≫ { X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := ⋯, left_assoc := ⋯, actRight := TensorBimod.actRight M N, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ }.X ◁ P.actLeft) ≫ (whiskerRight (M.whiskerLeft f) P).hom = coequalizer.π ({ X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := ⋯, left_assoc := ⋯, actRight := TensorBimod.actRight M N, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ }.actRight ▷ P.X) ((α_ { X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := ⋯, left_assoc := ⋯, actRight := TensorBimod.actRight M N, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ }.X Y.X P.X).hom ≫ { X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := ⋯, left_assoc := ⋯, actRight := TensorBimod.actRight M N, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ }.X ◁ P.actLeft) ≫ ((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], "module": ["Mathlib/CategoryTheory/Monoidal/Bimod.lean"]}
+{"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], "module": ["Mathlib/CategoryTheory/Monoidal/Bimod.lean"]}
+{"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) ≫ 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], "module": ["Mathlib/CategoryTheory/Monoidal/Bimod.lean"]}
+{"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], "module": ["Mathlib/CategoryTheory/Monoidal/Bimod.lean"]}
+{"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 = (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], "module": ["Mathlib/CategoryTheory/Monoidal/Bimod.lean"]}
+{"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, 99324], "module": ["Mathlib/CategoryTheory/Monoidal/Bimod.lean"]}
+{"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": [1673, 99324, 96190], "module": ["Mathlib/CategoryTheory/Monoidal/Bimod.lean"]}
+{"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": [96173, 94736, 93393, 99222, 99324], "module": ["Mathlib/CategoryTheory/Monoidal/Bimod.lean"]}
+{"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": [106541, 96173, 94736, 93393, 99222], "module": ["Mathlib/CategoryTheory/Monoidal/Bimod.lean"]}
+{"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": [94736, 93393, 106541, 96173], "module": ["Mathlib/CategoryTheory/Monoidal/Bimod.lean"]}
+{"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": [94736, 93393, 99219, 96173], "module": ["Mathlib/CategoryTheory/Monoidal/Bimod.lean"]}
+{"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) ≫ AssociatorBimod.inv M N' P"}, "premise": [93393, 94736, 96173, 99219], "module": ["Mathlib/CategoryTheory/Monoidal/Bimod.lean"]}
+{"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], "module": ["Mathlib/CategoryTheory/Monoidal/Bimod.lean"]}
+{"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) ≫ AssociatorBimod.invAux M N' P"}, "premise": [94804, 96173], "module": ["Mathlib/CategoryTheory/Monoidal/Bimod.lean"]}
+{"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, 96173, 106539], "module": ["Mathlib/CategoryTheory/Monoidal/Bimod.lean"]}
+{"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": [106539, 99219, 96173], "module": ["Mathlib/CategoryTheory/Monoidal/Bimod.lean"]}
+{"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": [106539, 96173, 99262], "module": ["Mathlib/CategoryTheory/Monoidal/Bimod.lean"]}
+{"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, 99262], "module": ["Mathlib/CategoryTheory/Monoidal/Bimod.lean"]}
+{"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], "module": ["Mathlib/CategoryTheory/Monoidal/Bimod.lean"]}
+{"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], "module": ["Mathlib/Algebra/BigOperators/Finprod.lean"]}
+{"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], "module": ["Mathlib/Logic/Equiv/Defs.lean"]}
+{"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, 3154, 14286, 22500, 23823], "module": ["Mathlib/NumberTheory/PellMatiyasevic.lean"]}
+{"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": [22500, 14286, 23823, 3154, 23773], "module": ["Mathlib/NumberTheory/PellMatiyasevic.lean"]}
+{"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], "module": ["Mathlib/Algebra/Group/Basic.lean"]}
+{"state": {"context": ["S : Set ℝ", "a : ℝ", "hS : ∀ x ∈ S, x ≤ a", "ha : 0 ≤ a"], "goal": "sSup S ≤ a"}, "premise": [133383, 16842, 146815, 16842, 146815], "module": ["Mathlib/Data/Real/Archimedean.lean"]}
+{"state": {"context": ["a : ℝ", "ha : 0 ≤ a", "hS : ∀ x ∈ ∅, x ≤ a"], "goal": "sSup ∅ ≤ a"}, "premise": [16842, 146815, 133383], "module": ["Mathlib/Data/Real/Archimedean.lean"]}
+{"state": {"context": ["S : Set ℝ", "a : ℝ", "hS : ∀ x ∈ S, x ≤ a", "ha : 0 ≤ a", "hS₂ : S.Nonempty"], "goal": "sSup S ≤ a"}, "premise": [16842, 146815, 133383], "module": ["Mathlib/Data/Real/Archimedean.lean"]}
+{"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], "module": ["Mathlib/Algebra/Ring/Parity.lean"]}
+{"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, 118076], "module": ["Mathlib/Analysis/Normed/Ring/Units.lean"]}
+{"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": [43397, 16105, 43722, 16090, 118076], "module": ["Mathlib/Analysis/Normed/Ring/Units.lean"]}
+{"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": [43424, 43587, 43724, 43411, 118076], "module": ["Mathlib/Analysis/Normed/Ring/Units.lean"]}
+{"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": [43424, 43587, 43724, 119730, 43411, 43455], "module": ["Mathlib/Analysis/Normed/Ring/Units.lean"]}
+{"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, 43594, 43563, 119730, 43455], "module": ["Mathlib/Analysis/Normed/Ring/Units.lean"]}
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+{"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"], "goal": "P.diagonal f"}, "premise": [126493], "module": ["Mathlib/AlgebraicGeometry/Morphisms/Constructors.lean"]}
+{"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], "module": ["Mathlib/AlgebraicGeometry/Morphisms/Constructors.lean"]}
+{"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": "∀ (i : (𝒱.pullbackCover (pullback.diagonal f)).J), Q (𝒱.pullbackHom (pullback.diagonal f) i)"}, "premise": [126501], "module": ["Mathlib/AlgebraicGeometry/Morphisms/Constructors.lean"]}
+{"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], "module": ["Mathlib/AlgebraicGeometry/Morphisms/Constructors.lean"]}
+{"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": "pullback.diagonal f ≫ 𝟙 (pullback.diagonalObj f) = 𝟙 X ≫ pullback.diagonal f"}, "premise": [1673, 126483], "module": ["Mathlib/AlgebraicGeometry/Morphisms/Constructors.lean"]}
+{"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": "pullback.map ((𝒰' i).map j ≫ pullback.snd f (𝒰.map i)) ((𝒰' i).map k ≫ pullback.snd f (𝒰.map i)) f f ((𝒰' i).map j ≫ pullback.fst f (𝒰.map i)) ((𝒰' i).map k ≫ pullback.fst f (𝒰.map i)) (𝒰.map i) ⋯ ⋯ ≫ 𝟙 (pullback.diagonalObj f) = 𝟙 (pullback ((𝒰' i).map j ≫ pullback.snd f (𝒰.map i)) ((𝒰' i).map k ≫ pullback.snd f (𝒰.map i))) ≫ pullback.map ((𝒰' i).map j ≫ pullback.snd f (𝒰.map i)) ((𝒰' i).map k ≫ pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i)) ((𝒰' i).map j) ((𝒰' i).map k) (𝟙 (𝒰.obj i)) ⋯ ⋯ ≫ pullback.map (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i)) f f (pullback.fst f (𝒰.map i)) (pullback.fst f (𝒰.map i)) (𝒰.map i) ⋯ ⋯"}, "premise": [1673, 126483], "module": ["Mathlib/AlgebraicGeometry/Morphisms/Constructors.lean"]}
+{"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 (((pullbackDiagonalMapIso f (𝒰.map i) ((𝒰' i).map j) ((𝒰' i).map k)).inv ≫ pullback.map (pullback.diagonal f) (pullback.map ((𝒰' i).map j ≫ pullback.snd f (𝒰.map i)) ((𝒰' i).map k ≫ pullback.snd f (𝒰.map i)) f f ((𝒰' i).map j ≫ pullback.fst f (𝒰.map i)) ((𝒰' i).map k ≫ pullback.fst f (𝒰.map i)) (𝒰.map i) ⋯ ⋯) (pullback.diagonal f) (pullback.map ((𝒰' i).map j ≫ pullback.snd f (𝒰.map i)) ((𝒰' i).map k ≫ pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i)) ((𝒰' i).map j) ((𝒰' i).map k) (𝟙 (𝒰.obj i)) ⋯ ⋯ ≫ pullback.map (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i)) f f (pullback.fst f (𝒰.map i)) (pullback.fst f (𝒰.map i)) (𝒰.map i) ⋯ ⋯) (𝟙 X) (𝟙 (pullback ((𝒰' i).map j ≫ pullback.snd f (𝒰.map i)) ((𝒰' i).map k ≫ pullback.snd f (𝒰.map i)))) (𝟙 (pullback.diagonalObj f)) ?mk.mk.convert_1 ?mk.mk.convert_2) ≫ pullback.snd (pullback.diagonal f) (pullback.map ((𝒰' i).map j ≫ pullback.snd f (𝒰.map i)) ((𝒰' i).map k ≫ pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i)) ((𝒰' i).map j) ((𝒰' i).map k) (𝟙 (𝒰.obj i)) ⋯ ⋯ ≫ pullback.map (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i)) f f (pullback.fst f (𝒰.map i)) (pullback.fst f (𝒰.map i)) (𝒰.map i) ⋯ ⋯))"}, "premise": [1673, 126483], "module": ["Mathlib/AlgebraicGeometry/Morphisms/Constructors.lean"]}
+{"state": {"context": ["X : Scheme", "r : ↑Γ(X, ⊤)"], "goal": "adjunction.unit.app X ⁻¹ᵁ basicOpen r = X.basicOpen r"}, "premise": [126625, 126614], "module": ["Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean"]}
+{"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": [126625, 126614], "module": ["Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean"]}
+{"state": {"context": ["X : Scheme", "r : ↑Γ(X, ⊤)"], "goal": "((𝟭 Scheme).obj X).basicOpen ((Scheme.Hom.app (adjunction.unit.app X) ⊤) ((Scheme.ΓSpecIso Γ(X, ⊤)).inv r)) = X.basicOpen r"}, "premise": [126614], "module": ["Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean"]}
+{"state": {"context": ["X : Scheme", "r : ↑Γ(X, ⊤)"], "goal": "(Scheme.Hom.app (adjunction.unit.app X) ⊤) ((Scheme.ΓSpecIso Γ(X, ⊤)).inv r) = r"}, "premise": [129600], "module": ["Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean"]}
+{"state": {"context": ["X : Scheme", "r : ↑Γ(X, ⊤)"], "goal": "(Scheme.ΓSpecIso Γ(X, ⊤)).hom ((Scheme.ΓSpecIso Γ(X, ⊤)).inv r) = r"}, "premise": [129600], "module": ["Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean"]}
+{"state": {"context": ["q : ℚ", "x y : ℝ", "h : Irrational (x ^ 0)"], "goal": "Irrational x"}, "premise": [119739, 148036], "module": ["Mathlib/Data/Real/Irrational.lean"]}
+{"state": {"context": ["q : ℚ", "x y : ℝ", "h : Irrational 1"], "goal": "Irrational x"}, "premise": [119739, 148036], "module": ["Mathlib/Data/Real/Irrational.lean"]}
+{"state": {"context": ["q : ℚ", "x y : ℝ", "n : ℕ", "h : Irrational (x ^ (n + 1))"], "goal": "Irrational x"}, "premise": [119742, 2110, 145727], "module": ["Mathlib/Data/Real/Irrational.lean"]}
+{"state": {"context": ["q : ℚ", "x y : ℝ", "n : ℕ", "h : Irrational (x ^ n * x)"], "goal": "Irrational x"}, "premise": [2110, 119742, 145727], "module": ["Mathlib/Data/Real/Irrational.lean"]}
+{"state": {"context": ["q : ℚ", "x y : ℝ", "h : Irrational x"], "goal": "Irrational (↑q - x)"}, "premise": [119789, 145702, 145714], "module": ["Mathlib/Data/Real/Irrational.lean"]}
+{"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", "F G K L : CochainComplex C ℤ", "n m n₁ n₂ n₁₂ : ℤ", "z₁ : Cochain F G n₁", "z₂ z₂' : Cochain G K n₂", "h : n₁ + n₂ = n₁₂", "p q : ℤ", "hpq : p + n₁₂ = q"], "goal": "(z₁.comp (z₂ + z₂') h).v p q hpq = (z₁.comp z₂ h + z₁.comp z₂' h).v p q hpq"}, "premise": [91599, 114432, 114452], "module": ["Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean"]}
+{"state": {"context": ["X : Type u", "Y : Type v", "ι : Sort w", "α : Type u_1", "β : Type u_2", "x : X", "s s₁ s₂ t : Set X", "p p₁ p₂ : X → Prop", "inst✝¹ : TopologicalSpace X", "f : Filter X", "H : f ≤ 𝓝 x", "inst✝ : f.NeBot"], "goal": "ClusterPt x f"}, "premise": [1674, 14569], "module": ["Mathlib/Topology/Basic.lean"]}
+{"state": {"context": ["C : Type u_1", "inst✝⁵ : Category.{u_2, u_1} C", "X Y Z : C", "inst✝⁴ : HasPullbacks C", "S T : C", "f : X ⟶ T", "g : Y ⟶ T", "i : T ⟶ S", "inst✝³ : HasPullback i i", "inst✝² : HasPullback f g", "inst✝¹ : HasPullback (f ≫ i) (g ≫ i)", "inst✝ : HasPullback (diagonal i) (map (f ≫ i) (g ≫ i) i i f g (𝟙 S) ⋯ ⋯)"], "goal": "IsPullback (fst f g ≫ f) (map f g (f ≫ i) (g ≫ i) (𝟙 X) (𝟙 Y) i ⋯ ⋯) (diagonal i) (map (f ≫ i) (g ≫ i) i i f g (𝟙 S) ⋯ ⋯)"}, "premise": [94102], "module": ["Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean"]}
+{"state": {"context": ["C : Type u_1", "inst✝⁵ : Category.{u_2, u_1} C", "X Y Z : C", "inst✝⁴ : HasPullbacks C", "S T : C", "f : X ⟶ T", "g : Y ⟶ T", "i : T ⟶ S", "inst✝³ : HasPullback i i", "inst✝² : HasPullback f g", "inst✝¹ : HasPullback (f ≫ i) (g ≫ i)", "inst✝ : HasPullback (diagonal i) (map (f ≫ i) (g ≫ i) i i f g (𝟙 S) ⋯ ⋯)"], "goal": "(pullbackDiagonalMapIdIso f g i).symm.hom ≫ fst (diagonal i) (map (f ≫ i) (g ≫ i) i i f g (𝟙 S) ⋯ ⋯) = fst f g ≫ f"}, "premise": [94102], "module": ["Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean"]}
+{"state": {"context": ["C : Type u_1", "inst✝⁵ : Category.{u_2, u_1} C", "X Y Z : C", "inst✝⁴ : HasPullbacks C", "S T : C", "f : X ⟶ T", "g : Y ⟶ T", "i : T ⟶ S", "inst✝³ : HasPullback i i", "inst✝² : HasPullback f g", "inst✝¹ : HasPullback (f ≫ i) (g ≫ i)", "inst✝ : HasPullback (diagonal i) (map (f ≫ i) (g ≫ i) i i f g (𝟙 S) ⋯ ⋯)"], "goal": "(pullbackDiagonalMapIdIso f g i).symm.hom ≫ snd (diagonal i) (map (f ≫ i) (g ≫ i) i i f g (𝟙 S) ⋯ ⋯) = map f g (f ≫ i) (g ≫ i) (𝟙 X) (𝟙 Y) i ⋯ ⋯"}, "premise": [94102], "module": ["Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean"]}
+{"state": {"context": ["C : Type u_1", "inst✝⁵ : Category.{u_2, u_1} C", "X Y Z : C", "inst✝⁴ : HasPullbacks C", "S T : C", "f : X ⟶ T", "g : Y ⟶ T", "i : T ⟶ S", "inst✝³ : HasPullback i i", "inst✝² : HasPullback f g", "inst✝¹ : HasPullback (f ≫ i) (g ≫ i)", "inst✝ : HasPullback (diagonal i) (map (f ≫ i) (g ≫ i) i i f g (𝟙 S) ⋯ ⋯)"], "goal": "CommSq (fst f g ≫ f) (map f g (f ≫ i) (g ≫ i) (𝟙 X) (𝟙 Y) i ⋯ ⋯) (diagonal i) (map (f ≫ i) (g ≫ i) i i f g (𝟙 S) ⋯ ⋯)"}, "premise": [94102], "module": ["Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean"]}
+{"state": {"context": ["G : Type u_1", "H : Type u_2", "inst✝¹ : Mul G", "inst✝ : Mul H", "A B : Finset G", "a0 b0 : G", "f : G ↪ H", "mul : ∀ (x y : G), f (x * y) = f x * f y"], "goal": "UniqueMul (Finset.map f A) (Finset.map f B) (f a0) (f b0) ↔ UniqueMul A B a0 b0"}, "premise": [1713, 70654, 122369, 137416], "module": ["Mathlib/Algebra/Group/UniqueProds.lean"]}
+{"state": {"context": ["R : Type u_1", "inst✝² : Monoid R", "S : Submonoid R", "inst✝¹ : OreSet S", "X : Type u_2", "inst✝ : MulAction R X", "r : X", "s t : ↥S"], "goal": "(1 /ₒ t) • (r /ₒ s) = r /ₒ (s * t)"}, "premise": [81641], "module": ["Mathlib/RingTheory/OreLocalization/Basic.lean"]}
+{"state": {"context": ["R : Type u", "S : Type v", "a b : R", "m n✝ : ℕ", "ι : Type y", "inst✝ : Ring R", "p : R[X]", "n : ℕ", "H : p.degree ≤ ↑n"], "goal": "(X ^ (n + 1) - p).Monic"}, "premise": [101729, 102168, 119789], "module": ["Mathlib/Algebra/Polynomial/Monic.lean"]}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : DecidableEq α", "inst✝ : Group α", "e : α", "x : Finset α × Finset α"], "goal": "(mulETransformLeft e x).1 * (mulETransformLeft e x).2 ⊆ x.1 * x.2"}, "premise": [138682, 138865, 141655, 118922, 141965], "module": ["Mathlib/Combinatorics/Additive/ETransform.lean"]}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : DecidableEq α", "inst✝ : Group α", "e : α", "x : Finset α × Finset α"], "goal": "op e • x.1 * e⁻¹ • x.2 ⊆ x.1 * x.2"}, "premise": [118922, 141965, 138865, 141655, 138682], "module": ["Mathlib/Combinatorics/Additive/ETransform.lean"]}
+{"state": {"context": ["α : Type u", "β : α → Type v", "inst✝ : DecidableEq α", "a : α", "b : β a", "s✝ : Finmap β", "s : AList β", "h : a ∉ ⟦s⟧"], "goal": "(insert a b ⟦s⟧).entries = ⟨a, b⟩ ::ₘ ⟦s⟧.entries"}, "premise": [1673, 1681, 129438, 129439, 140477], "module": ["Mathlib/Data/Finmap.lean"]}
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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", "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": "Integrable ?pos.bound✝ μ"}, "premise": [27173], "module": ["Mathlib/Analysis/Calculus/ParametricIntegral.lean"]}
+{"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": "∀ᵐ (a : α) ∂μ, Tendsto (fun n => ‖n - x₀‖⁻¹ • (F n a - F x₀ a - (F' a) (n - x₀))) (𝓝 x₀) (𝓝 (‖x₀ - x₀‖⁻¹ • (F x₀ a - F x₀ a - (F' a) (x₀ - x₀))))"}, "premise": [27173], "module": ["Mathlib/Analysis/Calculus/ParametricIntegral.lean"]}
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+{"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], "module": ["Mathlib/CategoryTheory/Subobject/Lattice.lean"]}
+{"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": "underlying.obj f ⟶ underlying.obj (sSup s)"}, "premise": [89263], "module": ["Mathlib/CategoryTheory/Subobject/Lattice.lean"]}
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+{"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, 43665], "module": ["Mathlib/Computability/AkraBazzi/AkraBazzi.lean"]}
+{"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, 76716], "module": ["Mathlib/Computability/AkraBazzi/AkraBazzi.lean"]}
+{"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": [43665, 76771], "module": ["Mathlib/Computability/AkraBazzi/AkraBazzi.lean"]}
+{"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": [76769, 76770, 76771, 15491, 15493, 131585, 15495, 34120, 1674, 76714, 76749, 76750, 76783, 15504, 15889, 15506], "module": ["Mathlib/Computability/AkraBazzi/AkraBazzi.lean"]}
+{"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": [76769, 76770, 15491, 131585, 15493, 15495, 34120, 1674, 76714, 76749, 76750, 76783, 15504, 15889, 15506, 76784], "module": ["Mathlib/Computability/AkraBazzi/AkraBazzi.lean"]}
+{"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"], "goal": "∃ c > 0, ∀ n ≥ n₀, c * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖"}, "premise": [76784], "module": ["Mathlib/Computability/AkraBazzi/AkraBazzi.lean"]}
+{"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], "module": ["Mathlib/Computability/AkraBazzi/AkraBazzi.lean"]}
+{"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"], "goal": "∃ c > 0, ∀ n ≥ n₀, c * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖"}, "premise": [2107, 19591, 19714, 76724, 76768, 139848], "module": ["Mathlib/Computability/AkraBazzi/AkraBazzi.lean"]}
+{"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], "module": ["Mathlib/Computability/AkraBazzi/AkraBazzi.lean"]}
+{"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"], "goal": "C * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖"}, "premise": [2107, 14271, 19591, 19691, 106022, 139802], "module": ["Mathlib/Computability/AkraBazzi/AkraBazzi.lean"]}
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+{"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], "module": ["Mathlib/NumberTheory/Cyclotomic/Basic.lean"]}
+{"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": "{1} = ∅ ∪ {1}"}, "premise": [1673, 24203, 24210], "module": ["Mathlib/NumberTheory/Cyclotomic/Basic.lean"]}
+{"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, 14272, 73596], "module": ["Mathlib/Computability/TuringMachine.lean"]}
+{"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": [73592, 73593, 73596, 14272], "module": ["Mathlib/Computability/TuringMachine.lean"]}
+{"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": [14272, 73595, 73596], "module": ["Mathlib/Computability/TuringMachine.lean"]}
+{"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, 73595], "module": ["Mathlib/Computability/TuringMachine.lean"]}
+{"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": [73538, 14277, 73513, 73516, 73585, 73588, 4985, 5278], "module": ["Mathlib/Computability/TuringMachine.lean"]}
+{"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)))))"], "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], "module": ["Mathlib/Computability/TuringMachine.lean"]}
+{"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, 73504, 73590], "module": ["Mathlib/Computability/TuringMachine.lean"]}
+{"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": [73536, 73504, 73590, 70456, 73530, 70459, 73597], "module": ["Mathlib/Computability/TuringMachine.lean"]}
+{"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) (bif true then TM1.stepAux (trNormal q) (stVar v (S k) o) (Tape.mk' ∅ (addBottom T')) else TM1.stepAux (move Dir.left (goto fun x x => ret q)) (stVar v (S k) o) (Tape.mk' ∅ (addBottom T'))) c"}, "premise": [73504, 73590], "module": ["Mathlib/Computability/TuringMachine.lean"]}
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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 := ⋯ }"], "goal": "∃ x, orderOf x = p"}, "premise": [70720, 103585, 103586, 144294, 7819, 70028, 2100, 4566, 119742], "module": ["Mathlib/GroupTheory/Perm/Cycle/Type.lean"]}
+{"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 : ∀ (k : ℕ) (v : ↑(vectorsProdEqOne G p)), (σ ^ k) v = f k v"], "goal": "∃ x, orderOf x = p"}, "premise": [70720, 70727, 7819, 7820, 2100, 4566, 119742, 119743], "module": ["Mathlib/GroupTheory/Perm/Cycle/Type.lean"]}
+{"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"], "goal": "∃ x, orderOf x = p"}, "premise": [70727, 124839, 119756, 7820, 2101, 119743], "module": ["Mathlib/GroupTheory/Perm/Cycle/Type.lean"]}
+{"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, ⋯⟩"], "goal": "∃ x, orderOf x = p"}, "premise": [137127, 124839, 130377, 119756, 2101], "module": ["Mathlib/GroupTheory/Perm/Cycle/Type.lean"]}
+{"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"], "goal": "∃ x, orderOf x = p"}, "premise": [130377, 9397, 137127], "module": ["Mathlib/GroupTheory/Perm/Cycle/Type.lean"]}
+{"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": [130401, 137128, 1673, 9397, 2011, 8415], "module": ["Mathlib/GroupTheory/Perm/Cycle/Type.lean"]}
+{"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", "g : G", "hg : ↑↑v = List.replicate (↑↑v).length g"], "goal": "g ^ p = 1"}, "premise": [1673, 2011, 8415, 130401, 137128], "module": ["Mathlib/GroupTheory/Perm/Cycle/Type.lean"]}
+{"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", "g : G", "hg : ↑↑v = List.replicate (↑↑v).length g", "hg' : g = 1"], "goal": "v = v₀"}, "premise": [1673, 2011, 8415, 130401, 137128], "module": ["Mathlib/GroupTheory/Perm/Cycle/Type.lean"]}
+{"state": {"context": ["θ : Angle"], "goal": "(2 • θ).toReal = 2 * θ.toReal - 2 * π ↔ π / 2 < θ.toReal"}, "premise": [38345, 20162, 38270, 38369, 119749, 122222], "module": ["Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean"]}
+{"state": {"context": ["θ : Angle"], "goal": "(2 • ↑θ.toReal).toReal = 2 * θ.toReal - 2 * π ↔ π / 2 < θ.toReal"}, "premise": [38369, 20162, 119749, 38345, 122222, 38270], "module": ["Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean"]}
+{"state": {"context": ["θ : Angle"], "goal": "π < 2 * θ.toReal ∧ 2 * θ.toReal ≤ 3 * π ↔ π / 2 < θ.toReal"}, "premise": [38369, 20162, 119749, 1673, 38347, 122222, 106030, 38514, 103768, 38270], "module": ["Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean"]}
+{"state": {"context": ["n✝ n : ℕ", "h1 : 1 < n", "h : ∀ (m : ℕ), m < n → m ≠ 0 → n.Coprime m"], "goal": "Prime n"}, "premise": [1674, 144299, 11234, 108656], "module": ["Mathlib/Data/Nat/Prime/Defs.lean"]}
+{"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": [108656, 1674, 144299, 11234], "module": ["Mathlib/Data/Nat/Prime/Defs.lean"]}
+{"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": [108656, 433, 11234, 395], "module": ["Mathlib/Data/Nat/Prime/Defs.lean"]}
+{"state": {"context": ["p n : ℕ+", "A : Type w", "B : Type z", "K : Type u", "L : Type v", "C : Type w", "inst✝⁷ : CommRing A", "inst✝⁶ : CommRing B", "inst✝⁵ : Algebra A B", "inst✝⁴ : IsCyclotomicExtension {n} A B", "inst✝³ : Field L", "ζ : L", "hζ✝ : IsPrimitiveRoot ζ ↑n", "inst✝² : Field K", "inst✝¹ : Algebra K L", "k : ℕ", "hζ : IsPrimitiveRoot ζ (2 ^ (k + 1))", "inst✝ : IsCyclotomicExtension {2 ^ (k + 1)} K L", "hirr : Irreducible (cyclotomic (2 ^ (k + 1)) K)"], "goal": "(Algebra.norm K) (ζ ^ 2 ^ k - 1) = (-2) ^ 2 ^ k"}, "premise": [2140, 78754, 108281, 108889, 113020, 2140, 3723, 3906, 4737, 14271, 103765, 119743], "module": ["Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean"]}
+{"state": {"context": ["p n : ℕ+", "A : Type w", "B : Type z", "K : Type u", "L : Type v", "C : Type w", "inst✝⁷ : CommRing A", "inst✝⁶ : CommRing B", "inst✝⁵ : Algebra A B", "inst✝⁴ : IsCyclotomicExtension {n} A B", "inst✝³ : Field L", "ζ : L", "hζ✝ : IsPrimitiveRoot ζ ↑n", "inst✝² : Field K", "inst✝¹ : Algebra K L", "k : ℕ", "hζ : IsPrimitiveRoot ζ (2 ^ (k + 1))", "inst✝ : IsCyclotomicExtension {2 ^ (k + 1)} K L", "hirr : Irreducible (cyclotomic (2 ^ (k + 1)) K)", "this : IsPrimitiveRoot (ζ ^ 2 ^ k) (2 ^ (k + 1) / 2 ^ k)"], "goal": "(Algebra.norm K) (ζ ^ 2 ^ k - 1) = (-2) ^ 2 ^ k"}, "premise": [4737, 78754, 3906, 3723, 119743, 108281, 103765, 113020, 108889, 2140, 14271], "module": ["Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean"]}
+{"state": {"context": ["p n : ℕ+", "A : Type w", "B : Type z", "K : Type u", "L : Type v", "C : Type w", "inst✝⁷ : CommRing A", "inst✝⁶ : CommRing B", "inst✝⁵ : Algebra A B", "inst✝⁴ : IsCyclotomicExtension {n} A B", "inst✝³ : Field L", "ζ : L", "hζ✝ : IsPrimitiveRoot ζ ↑n", "inst✝² : Field K", "inst✝¹ : Algebra K L", "k : ℕ", "hζ : IsPrimitiveRoot ζ (2 ^ (k + 1))", "inst✝ : IsCyclotomicExtension {2 ^ (k + 1)} K L", "hirr : Irreducible (cyclotomic (2 ^ (k + 1)) K)", "this : IsPrimitiveRoot (ζ ^ 2 ^ k) 2"], "goal": "(Algebra.norm K) (ζ ^ 2 ^ k - 1) = (-2) ^ 2 ^ k"}, "premise": [4737, 3906, 117094, 142664, 3723, 103765, 119743, 2140, 14271], "module": ["Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean"]}
+{"state": {"context": ["p n : ℕ+", "A : Type w", "B : Type z", "K : Type u", "L : Type v", "C : Type w", "inst✝⁷ : CommRing A", "inst✝⁶ : CommRing B", "inst✝⁵ : Algebra A B", "inst✝⁴ : IsCyclotomicExtension {n} A B", "inst✝³ : Field L", "ζ : L", "hζ✝ : IsPrimitiveRoot ζ ↑n", "inst✝² : Field K", "inst✝¹ : Algebra K L", "k : ℕ", "hζ : IsPrimitiveRoot ζ (2 ^ (k + 1))", "inst✝ : IsCyclotomicExtension {2 ^ (k + 1)} K L", "hirr : Irreducible (cyclotomic (2 ^ (k + 1)) K)", "this : IsPrimitiveRoot (ζ ^ 2 ^ k) 2", "H : -1 - 1 = (algebraMap K L) (-2)"], "goal": "(Algebra.norm K) (ζ ^ 2 ^ k - 1) = (-2) ^ 2 ^ k"}, "premise": [117094, 142664], "module": ["Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean"]}
+{"state": {"context": ["p n : ℕ+", "A : Type w", "B : Type z", "K : Type u", "L : Type v", "C : Type w", "inst✝⁷ : CommRing A", "inst✝⁶ : CommRing B", "inst✝⁵ : Algebra A B", "inst✝⁴ : IsCyclotomicExtension {n} A B", "inst✝³ : Field L", "ζ : L", "hζ✝ : IsPrimitiveRoot ζ ↑n", "inst✝² : Field K", "inst✝¹ : Algebra K L", "k : ℕ", "hζ : IsPrimitiveRoot ζ (2 ^ (k + 1))", "inst✝ : IsCyclotomicExtension {2 ^ (k + 1)} K L", "this : IsPrimitiveRoot (ζ ^ 2 ^ k) 2", "H : -1 - 1 = (algebraMap K L) (-2)", "hirr : Irreducible (cyclotomic (↑(2 ^ (k + 1))) K)"], "goal": "(Algebra.norm K) (ζ ^ 2 ^ k - 1) = (-2) ^ 2 ^ k"}, "premise": [2136, 3715, 25228, 77910, 78773, 103416, 141238, 144303, 145335], "module": ["Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean"]}
+{"state": {"context": ["p n : ℕ+", "A : Type w", "B : Type z", "K : Type u", "L : Type v", "C : Type w", "inst✝⁷ : CommRing A", "inst✝⁶ : CommRing B", "inst✝⁵ : Algebra A B", "inst✝⁴ : IsCyclotomicExtension {n} A B", "inst✝³ : Field L", "ζ : L", "hζ✝ : IsPrimitiveRoot ζ ↑n", "inst✝² : Field K", "inst✝¹ : Algebra K L", "k : ℕ", "hζ : IsPrimitiveRoot ζ (2 ^ (k + 1))", "inst✝ : IsCyclotomicExtension {2 ^ (k + 1)} K L", "this : IsPrimitiveRoot (ζ ^ 2 ^ k) 2", "H : -1 - 1 = (algebraMap K L) (-2)", "hirr : Irreducible (cyclotomic (↑(2 ^ (k + 1))) K)"], "goal": "(-2) ^ (2 ^ k * (2 - 1)) = (-2) ^ 2 ^ k"}, "premise": [2136, 3715, 25228, 77910, 78773, 103416, 141238, 144303, 145335], "module": ["Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean"]}
+{"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✝² : CommMonoid M", "inst✝¹ : CommMonoid N", "f : α → M", "s : Set α", "inst✝ : Fintype ↑s"], "goal": "s ∩ mulSupport f = ↑s.toFinset ∩ mulSupport f"}, "premise": [140898], "module": ["Mathlib/Algebra/BigOperators/Finprod.lean"]}
+{"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", "s : Set ι", "inst✝ : DecidablePred fun x => x ∈ s", "t : (i : ι) → Set (α i)", "x✝ : (i : ι) → α i"], "goal": "(x✝ ∈ univ.pi fun i => if i ∈ s then t i else univ) ↔ x✝ ∈ s.pi t"}, "premise": [131608, 2015], "module": ["Mathlib/Data/Set/Prod.lean"]}
+{"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", "s : Set ι", "inst✝ : DecidablePred fun x => x ∈ s", "t : (i : ι) → Set (α i)", "x✝ : (i : ι) → α i"], "goal": "(∀ (i : ι), x✝ i ∈ if i ∈ s then t i else univ) ↔ x✝ ∈ s.pi t"}, "premise": [131608, 2015], "module": ["Mathlib/Data/Set/Prod.lean"]}
+{"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", "s : Set ι", "inst✝ : DecidablePred fun x => x ∈ s", "t : (i : ι) → Set (α i)", "x✝ : (i : ι) → α i", "i : ι"], "goal": "(x✝ i ∈ if i ∈ s then t i else univ) ↔ i ∈ s → x✝ i ∈ t i"}, "premise": [2015], "module": ["Mathlib/Data/Set/Prod.lean"]}
+{"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"], "goal": "o.oangle y x = -o.oangle x y"}, "premise": [34610, 40357], "module": ["Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean"]}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "N : Type u_3", "P : Type u_4", "inst✝⁸ : CommRing R", "inst✝⁷ : AddCommGroup M", "inst✝⁶ : AddCommGroup N", "inst✝⁵ : AddCommGroup P", "inst✝⁴ : Module R M", "inst✝³ : Module R N", "inst✝² : Module R P", "f : M →ₗ[R] N", "g : N →ₗ[R] P", "Q : Type u_5", "inst✝¹ : AddCommGroup Q", "inst✝ : Module R Q", "hfg : Exact ⇑f ⇑g", "hg : Surjective ⇑g"], "goal": "inverse Q hfg hg ∘ₗ lTensor Q g = (range (lTensor Q f)).mkQ"}, "premise": [85680], "module": ["Mathlib/LinearAlgebra/TensorProduct/RightExactness.lean"]}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝⁶ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace 𝕜 E", "F : Type u_3", "inst✝³ : NormedAddCommGroup F", "inst✝² : NormedSpace 𝕜 F", "G : Type u_4", "inst✝¹ : NormedAddCommGroup G", "inst✝ : NormedSpace ℝ G", "x y : E", "s t : Set E", "h : UniqueDiffWithinAt 𝕜 s x", "st : 𝓝[s] x ≤ 𝓝[t] x"], "goal": "UniqueDiffWithinAt 𝕜 t x"}, "premise": [45662, 55574], "module": ["Mathlib/Analysis/Calculus/TangentCone.lean"]}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝⁶ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace 𝕜 E", "F : Type u_3", "inst✝³ : NormedAddCommGroup F", "inst✝² : NormedSpace 𝕜 F", "G : Type u_4", "inst✝¹ : NormedAddCommGroup G", "inst✝ : NormedSpace ℝ G", "x y : E", "s t : Set E", "st : 𝓝[s] x ≤ 𝓝[t] x", "h : Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 s x)) ∧ x ∈ closure s"], "goal": "Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 t x)) ∧ x ∈ closure t"}, "premise": [45662, 55574], "module": ["Mathlib/Analysis/Calculus/TangentCone.lean"]}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝⁶ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace 𝕜 E", "F : Type u_3", "inst✝³ : NormedAddCommGroup F", "inst✝² : NormedSpace 𝕜 F", "G : Type u_4", "inst✝¹ : NormedAddCommGroup G", "inst✝ : NormedSpace ℝ G", "x y : E", "s t : Set E", "st : 𝓝[s] x ≤ 𝓝[t] x", "h : Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 s x)) ∧ (𝓝[s] x).NeBot"], "goal": "Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 t x)) ∧ (𝓝[t] x).NeBot"}, "premise": [55457, 45669, 15923, 55574, 2106, 2107, 86685], "module": ["Mathlib/Analysis/Calculus/TangentCone.lean"]}
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+{"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": [14273, 103882, 19309, 103545, 59002, 102621, 33470], "module": ["Mathlib/Analysis/PSeries.lean"]}
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+{"state": {"context": ["m n a✝ b✝ c d a b : ℤ", "hb : 0 < b", "this : a % b < |b|"], "goal": "a % b < b"}, "premise": [105284, 14284, 104495], "module": ["Mathlib/Data/Int/ModEq.lean"]}
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+{"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", "a x : α", "a✝ : List α", "h : {x | x ∈ a ::ₘ ⟦a✝⟧}.Pairwise fun x y => op x y = op y x", "h' : {x | x ∈ ⟦a✝⟧}.Pairwise fun x y => op x y = op y x"], "goal": "noncommFold op (a ::ₘ ⟦a✝⟧) h x = op a (noncommFold op ⟦a✝⟧ h' x)"}, "premise": [1830], "module": ["Mathlib/Data/Finset/NoncommProd.lean"]}
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+{"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], "module": ["Mathlib/Analysis/SpecialFunctions/Complex/Analytic.lean"]}
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+{"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": [69361, 2100, 38393, 1690, 119708], "module": ["Mathlib/Geometry/Euclidean/Angle/Sphere.lean"]}