The full dataset viewer is not available (click to read why). Only showing a preview of the rows.
The dataset generation failed because of a cast error
Error code: DatasetGenerationCastError
Exception: DatasetGenerationCastError
Message: An error occurred while generating the dataset
All the data files must have the same columns, but at some point there are 1 new columns ({'all_premises'})
This happened while the json dataset builder was generating data using
hf://datasets/ruc-ai4math/mathlib_handler_benchmark_410/random/random_our/train_expand_premise.jsonl (at revision d7eebe1a607c49926b283b5cc129ed8314915c16)
Please either edit the data files to have matching columns, or separate them into different configurations (see docs at https://hf.co/docs/hub/datasets-manual-configuration#multiple-configurations)
Traceback: Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1870, in _prepare_split_single
writer.write_table(table)
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/arrow_writer.py", line 622, in write_table
pa_table = table_cast(pa_table, self._schema)
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/table.py", line 2292, in table_cast
return cast_table_to_schema(table, schema)
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/table.py", line 2240, in cast_table_to_schema
raise CastError(
datasets.table.CastError: Couldn't cast
state: struct<context: list<item: string>, goal: string>
child 0, context: list<item: string>
child 0, item: string
child 1, goal: string
premise: int64
module: list<item: string>
child 0, item: string
all_premises: list<item: int64>
child 0, item: int64
to
{'state': {'context': Sequence(feature=Value(dtype='string', id=None), length=-1, id=None), 'goal': Value(dtype='string', id=None)}, 'premise': Sequence(feature=Value(dtype='int64', id=None), length=-1, id=None), 'module': Sequence(feature=Value(dtype='string', id=None), length=-1, id=None)}
because column names don't match
During handling of the above exception, another exception occurred:
Traceback (most recent call last):
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1438, in compute_config_parquet_and_info_response
parquet_operations = convert_to_parquet(builder)
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1050, in convert_to_parquet
builder.download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 924, in download_and_prepare
self._download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1000, in _download_and_prepare
self._prepare_split(split_generator, **prepare_split_kwargs)
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1741, in _prepare_split
for job_id, done, content in self._prepare_split_single(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1872, in _prepare_split_single
raise DatasetGenerationCastError.from_cast_error(
datasets.exceptions.DatasetGenerationCastError: An error occurred while generating the dataset
All the data files must have the same columns, but at some point there are 1 new columns ({'all_premises'})
This happened while the json dataset builder was generating data using
hf://datasets/ruc-ai4math/mathlib_handler_benchmark_410/random/random_our/train_expand_premise.jsonl (at revision d7eebe1a607c49926b283b5cc129ed8314915c16)
Please either edit the data files to have matching columns, or separate them into different configurations (see docs at https://hf.co/docs/hub/datasets-manual-configuration#multiple-configurations)Need help to make the dataset viewer work? Make sure to review how to configure the dataset viewer, and open a discussion for direct support.
state
dict | premise
sequence | module
sequence |
|---|---|---|
{
"context": [
"α : Type u_1",
"inst✝ : LinearOrder α",
"s t : Set α",
"x✝ y z x : α"
],
"goal": "toDual x ∈ (⇑ofDual ⁻¹' s).ordConnectedComponent (toDual x✝) ↔ toDual x ∈ ⇑ofDual ⁻¹' s.ordConnectedComponent x✝"
} | [
17175,
18528,
1713
] | [
"Mathlib/Order/Interval/Set/OrdConnectedComponent.lean"
] |
{
"context": [
"α : Type u_1",
"inst✝ : LinearOrder α",
"s t : Set α",
"x✝ y z x : α"
],
"goal": "⇑ofDual ⁻¹' [[x✝, x]] ⊆ ⇑ofDual ⁻¹' s ↔ toDual x ∈ ⇑ofDual ⁻¹' s.ordConnectedComponent x✝"
} | [
18528,
1713,
17175
] | [
"Mathlib/Order/Interval/Set/OrdConnectedComponent.lean"
] |
{
"context": [
"α : Type u_1",
"G G' : SimpleGraph α",
"inst✝² : DecidableRel G.Adj",
"ε : ℝ",
"s t u : Finset α",
"inst✝¹ : DecidableEq α",
"inst✝ : Fintype α",
"P : Finpartition univ",
"dst : 2 * ε ≤ ↑(G.edgeDensity s t)",
"ust : G.IsUniform ε s t",
"hst : Disjoint s t",
"dsu : 2 * ε ≤ ↑(G.edgeDensity s u)",
"usu : G.IsUniform ε s u",
"hsu : Disjoint s u",
"dtu : 2 * ε ≤ ↑(G.edgeDensity t u)",
"utu : G.IsUniform ε t u",
"htu : Disjoint t u"
],
"goal": "(1 - 2 * ε) * ε ^ 3 * ↑s.card * ↑t.card * ↑u.card ≤ ↑(G.cliqueFinset 3).card"
} | [
52912,
142597
] | [
"Mathlib/Combinatorics/SimpleGraph/Triangle/Counting.lean"
] |
{
"context": [
"α : Type u_1",
"G G' : SimpleGraph α",
"inst✝² : DecidableRel G.Adj",
"ε : ℝ",
"s t u : Finset α",
"inst✝¹ : DecidableEq α",
"inst✝ : Fintype α",
"P : Finpartition univ",
"dst : 2 * ε ≤ ↑(G.edgeDensity s t)",
"ust : G.IsUniform ε s t",
"hst : Disjoint s t",
"dsu : 2 * ε ≤ ↑(G.edgeDensity s u)",
"usu : G.IsUniform ε s u",
"hsu : Disjoint s u",
"dtu : 2 * ε ≤ ↑(G.edgeDensity t u)",
"utu : G.IsUniform ε t u",
"htu : Disjoint t u"
],
"goal": "↑(filter (fun x => match x with | (a, b, c) => G.Adj a b ∧ G.Adj a c ∧ G.Adj b c) (s ×ˢ t ×ˢ u)).card ≤ ↑(G.cliqueFinset 3).card"
} | [
52912,
142597
] | [
"Mathlib/Combinatorics/SimpleGraph/Triangle/Counting.lean"
] |
{
"context": [
"α : Type u_1",
"G G' : SimpleGraph α",
"inst✝² : DecidableRel G.Adj",
"ε : ℝ",
"s t u : Finset α",
"inst✝¹ : DecidableEq α",
"inst✝ : Fintype α",
"P : Finpartition univ",
"dst : 2 * ε ≤ ↑(G.edgeDensity s t)",
"ust : G.IsUniform ε s t",
"hst : Disjoint s t",
"dsu : 2 * ε ≤ ↑(G.edgeDensity s u)",
"usu : G.IsUniform ε s u",
"hsu : Disjoint s u",
"dtu : 2 * ε ≤ ↑(G.edgeDensity t u)",
"utu : G.IsUniform ε t u",
"htu : Disjoint t u"
],
"goal": "(filter (fun x => match x with | (a, b, c) => G.Adj a b ∧ G.Adj a c ∧ G.Adj b c) (s ×ˢ t ×ˢ u)).card ≤ (G.cliqueFinset 3).card"
} | [
137677,
142597
] | [
"Mathlib/Combinatorics/SimpleGraph/Triangle/Counting.lean"
] |
{
"context": [
"α : Type u_1",
"G G' : SimpleGraph α",
"inst✝² : DecidableRel G.Adj",
"ε : ℝ",
"s t u : Finset α",
"inst✝¹ : DecidableEq α",
"inst✝ : Fintype α",
"P : Finpartition univ",
"dst : 2 * ε ≤ ↑(G.edgeDensity s t)",
"ust : G.IsUniform ε s t",
"hst : Disjoint s t",
"dsu : 2 * ε ≤ ↑(G.edgeDensity s u)",
"usu : G.IsUniform ε s u",
"hsu : Disjoint s u",
"dtu : 2 * ε ≤ ↑(G.edgeDensity t u)",
"utu : G.IsUniform ε t u",
"htu : Disjoint t u"
],
"goal": "∀ a ∈ filter (fun x => match x with | (a, b, c) => G.Adj a b ∧ G.Adj a c ∧ G.Adj b c) (s ×ˢ t ×ˢ u), (fun x => match x with | (x, y, z) => {x, y, z}) a ∈ G.cliqueFinset 3"
} | [
137677
] | [
"Mathlib/Combinatorics/SimpleGraph/Triangle/Counting.lean"
] |
{
"context": [
"α : Type u_1",
"G G' : SimpleGraph α",
"inst✝² : DecidableRel G.Adj",
"ε : ℝ",
"s t u : Finset α",
"inst✝¹ : DecidableEq α",
"inst✝ : Fintype α",
"P : Finpartition univ",
"dst : 2 * ε ≤ ↑(G.edgeDensity s t)",
"ust : G.IsUniform ε s t",
"hst : Disjoint s t",
"dsu : 2 * ε ≤ ↑(G.edgeDensity s u)",
"usu : G.IsUniform ε s u",
"hsu : Disjoint s u",
"dtu : 2 * ε ≤ ↑(G.edgeDensity t u)",
"utu : G.IsUniform ε t u",
"htu : Disjoint t u"
],
"goal": "Set.InjOn (fun x => match x with | (x, y, z) => {x, y, z}) ↑(filter (fun x => match x with | (a, b, c) => G.Adj a b ∧ G.Adj a c ∧ G.Adj b c) (s ×ˢ t ×ˢ u))"
} | [
137677
] | [
"Mathlib/Combinatorics/SimpleGraph/Triangle/Counting.lean"
] |
{
"context": [
"α : Type u_1",
"G G' : SimpleGraph α",
"inst✝² : DecidableRel G.Adj",
"ε : ℝ",
"s t✝ u : Finset α",
"inst✝¹ : DecidableEq α",
"inst✝ : Fintype α",
"P : Finpartition univ",
"dst : 2 * ε ≤ ↑(G.edgeDensity s t✝)",
"ust : G.IsUniform ε s t✝",
"hst : Disjoint s t✝",
"dsu : 2 * ε ≤ ↑(G.edgeDensity s u)",
"usu : G.IsUniform ε s u",
"hsu : Disjoint s u",
"dtu : 2 * ε ≤ ↑(G.edgeDensity t✝ u)",
"utu : G.IsUniform ε t✝ u",
"htu : Disjoint t✝ u",
"x₁ y₁ z₁ : α",
"h₁ : (x₁, y₁, z₁) ∈ ↑(filter (fun x => match x with | (a, b, c) => G.Adj a b ∧ G.Adj a c ∧ G.Adj b c) (s ×ˢ t✝ ×ˢ u))",
"x₂ y₂ z₂ : α",
"h₂ : (x₂, y₂, z₂) ∈ ↑(filter (fun x => match x with | (a, b, c) => G.Adj a b ∧ G.Adj a c ∧ G.Adj b c) (s ×ˢ t✝ ×ˢ u))",
"t : (fun x => match x with | (x, y, z) => {x, y, z}) (x₁, y₁, z₁) = (fun x => match x with | (x, y, z) => {x, y, z}) (x₂, y₂, z₂)"
],
"goal": "(x₁, y₁, z₁) = (x₂, y₂, z₂)"
} | [
136829,
138668,
139089,
1101,
1674,
52913
] | [
"Mathlib/Combinatorics/SimpleGraph/Triangle/Counting.lean"
] |
{
"context": [
"α : Type u_1",
"G G' : SimpleGraph α",
"inst✝² : DecidableRel G.Adj",
"ε : ℝ",
"s t✝ u : Finset α",
"inst✝¹ : DecidableEq α",
"inst✝ : Fintype α",
"P : Finpartition univ",
"dst : 2 * ε ≤ ↑(G.edgeDensity s t✝)",
"ust : G.IsUniform ε s t✝",
"hst : Disjoint s t✝",
"dsu : 2 * ε ≤ ↑(G.edgeDensity s u)",
"usu : G.IsUniform ε s u",
"hsu : Disjoint s u",
"dtu : 2 * ε ≤ ↑(G.edgeDensity t✝ u)",
"utu : G.IsUniform ε t✝ u",
"htu : Disjoint t✝ u",
"x₁ y₁ z₁ x₂ y₂ z₂ : α",
"t : (fun x => match x with | (x, y, z) => {x, y, z}) (x₁, y₁, z₁) = (fun x => match x with | (x, y, z) => {x, y, z}) (x₂, y₂, z₂)",
"h₁ : (x₁ ∈ s ∧ y₁ ∈ t✝ ∧ z₁ ∈ u) ∧ G.Adj x₁ y₁ ∧ G.Adj x₁ z₁ ∧ G.Adj y₁ z₁",
"h₂ : (x₂ ∈ s ∧ y₂ ∈ t✝ ∧ z₂ ∈ u) ∧ G.Adj x₂ y₂ ∧ G.Adj x₂ z₂ ∧ G.Adj y₂ z₂"
],
"goal": "(x₁, y₁, z₁) = (x₂, y₂, z₂)"
} | [
1674,
138668,
1101,
139089,
52913,
136829
] | [
"Mathlib/Combinatorics/SimpleGraph/Triangle/Counting.lean"
] |
{
"context": [
"R : Type u_1",
"A : Type u_2",
"p : A → Prop",
"inst✝¹⁸ : OrderedCommRing R",
"inst✝¹⁷ : Nontrivial R",
"inst✝¹⁶ : StarRing R",
"inst✝¹⁵ : StarOrderedRing R",
"inst✝¹⁴ : MetricSpace R",
"inst✝¹³ : TopologicalRing R",
"inst✝¹² : ContinuousStar R",
"inst✝¹¹ : ∀ (α : Type ?u.1282777) [inst : Zero α] [inst_1 : TopologicalSpace α], StarOrderedRing C(α, R)₀",
"inst✝¹⁰ : TopologicalSpace A",
"inst✝⁹ : NonUnitalRing A",
"inst✝⁸ : StarRing A",
"inst✝⁷ : PartialOrder A",
"inst✝⁶ : StarOrderedRing A",
"inst✝⁵ : Module R A",
"inst✝⁴ : IsScalarTower R A A",
"inst✝³ : SMulCommClass R A A",
"inst✝² : StarModule R A",
"inst✝¹ : NonUnitalContinuousFunctionalCalculus R p",
"inst✝ : NonnegSpectrumClass R A",
"f g : R → R",
"a : A",
"hf : autoParam (ContinuousOn f (σₙ R a)) _auto✝",
"hg : autoParam (ContinuousOn g (σₙ R a)) _auto✝",
"hf0 : autoParam (f 0 = 0) _auto✝",
"hg0 : autoParam (g 0 = 0) _auto✝",
"ha : autoParam (p a) _auto✝"
],
"goal": "cfcₙ f a ≤ cfcₙ g a ↔ ∀ x ∈ σₙ R a, f x ≤ g x"
} | [
38025,
38072,
62597
] | [
"Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean"
] |
{
"context": [
"R : Type u_1",
"A : Type u_2",
"p : A → Prop",
"inst✝¹⁸ : OrderedCommRing R",
"inst✝¹⁷ : Nontrivial R",
"inst✝¹⁶ : StarRing R",
"inst✝¹⁵ : StarOrderedRing R",
"inst✝¹⁴ : MetricSpace R",
"inst✝¹³ : TopologicalRing R",
"inst✝¹² : ContinuousStar R",
"inst✝¹¹ : ∀ (α : Type u_1) [inst : Zero α] [inst_1 : TopologicalSpace α], StarOrderedRing C(α, R)₀",
"inst✝¹⁰ : TopologicalSpace A",
"inst✝⁹ : NonUnitalRing A",
"inst✝⁸ : StarRing A",
"inst✝⁷ : PartialOrder A",
"inst✝⁶ : StarOrderedRing A",
"inst✝⁵ : Module R A",
"inst✝⁴ : IsScalarTower R A A",
"inst✝³ : SMulCommClass R A A",
"inst✝² : StarModule R A",
"inst✝¹ : NonUnitalContinuousFunctionalCalculus R p",
"inst✝ : NonnegSpectrumClass R A",
"f g : R → R",
"a : A",
"hf : autoParam (ContinuousOn f (σₙ R a)) _auto✝",
"hg : autoParam (ContinuousOn g (σₙ R a)) _auto✝",
"hf0 : autoParam (f 0 = 0) _auto✝",
"hg0 : autoParam (g 0 = 0) _auto✝",
"ha : autoParam (p a) _auto✝"
],
"goal": "(∀ (x : ↑(σₙ R a)), { toFun := (σₙ R a).restrict f, continuous_toFun := ⋯, map_zero' := ⋯ } x ≤ { toFun := (σₙ R a).restrict g, continuous_toFun := ⋯, map_zero' := ⋯ } x) ↔ ∀ x ∈ σₙ R a, f x ≤ g x"
} | [
38025,
38072,
62597
] | [
"Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean"
] |
{
"context": [
"α : Type u_1",
"β : Type u_2",
"γ : Type u_3",
"ι : Type u_4",
"ι' : Type u_5",
"κ : Sort u_6",
"r p q : α → α → Prop",
"s t : Set ι",
"f : ι → Set α",
"h : (s ∪ t).PairwiseDisjoint f"
],
"goal": "(⋃ i ∈ s, f i) \\ ⋃ i ∈ t, f i = ⋃ i ∈ s \\ t, f i"
} | [
1674,
2107,
131589,
135248,
135354,
135582,
135215
] | [
"Mathlib/Data/Set/Pairwise/Lattice.lean"
] |
{
"context": [
"α : Type u_1",
"β : Type u_2",
"γ : Type u_3",
"ι : Type u_4",
"ι' : Type u_5",
"κ : Sort u_6",
"r p q : α → α → Prop",
"s t : Set ι",
"f : ι → Set α",
"h : (s ∪ t).PairwiseDisjoint f",
"i : ι",
"hi : i ∈ s \\ t",
"a : α",
"ha : a ∈ f i"
],
"goal": "a ∉ ⋃ x ∈ t, f x"
} | [
131589,
1674,
135215,
135248,
135354,
2107,
135582
] | [
"Mathlib/Data/Set/Pairwise/Lattice.lean"
] |
{
"context": [
"α : Type u_1",
"β : Type u_2",
"γ : Type u_3",
"ι : Type u_4",
"ι' : Type u_5",
"κ : Sort u_6",
"r p q : α → α → Prop",
"s t : Set ι",
"f : ι → Set α",
"h : (s ∪ t).PairwiseDisjoint f",
"i : ι",
"hi : i ∈ s \\ t",
"a : α",
"ha : a ∈ f i"
],
"goal": "¬∃ i, ∃ (_ : i ∈ t), a ∈ f i"
} | [
135215
] | [
"Mathlib/Data/Set/Pairwise/Lattice.lean"
] |
{
"context": [
"α : Type u_1",
"β : Type u_2",
"γ : Type u_3",
"ι : Type u_4",
"ι' : Type u_5",
"κ : Sort u_6",
"r p q : α → α → Prop",
"s t : Set ι",
"f : ι → Set α",
"h : (s ∪ t).PairwiseDisjoint f",
"i : ι",
"hi : i ∈ s \\ t",
"a : α",
"ha : a ∈ f i",
"j : ι",
"hj : j ∈ t",
"haj : a ∈ f j"
],
"goal": "False"
} | [
35,
1690,
2106,
2107,
13484
] | [
"Mathlib/Data/Set/Pairwise/Lattice.lean"
] |
{
"context": [
"ι : Type u_1",
"α : Type u_2",
"β : Type u_3",
"inst✝² : Preorder α",
"inst✝¹ : LocallyFiniteOrder α",
"inst✝ : Preorder β",
"f : α → β"
],
"goal": "StrictMono f ↔ ∀ (a b : α), a ⋖ b → f a < f b"
} | [
16610,
70476
] | [
"Mathlib/Order/Interval/Finset/Basic.lean"
] |
{
"context": [
"ι : Type u_1",
"α : Type u_2",
"β : Type u_3",
"inst✝² : Preorder α",
"inst✝¹ : LocallyFiniteOrder α",
"inst✝ : Preorder β",
"f : α → β",
"h : ∀ (a b : α), a ⋖ b → f a < f b",
"a b : α",
"hab : a < b"
],
"goal": "f a < f b"
} | [
16610,
70476
] | [
"Mathlib/Order/Interval/Finset/Basic.lean"
] |
{
"context": [
"ι : Type u_1",
"α : Type u_2",
"β : Type u_3",
"inst✝² : Preorder α",
"inst✝¹ : LocallyFiniteOrder α",
"inst✝ : Preorder β",
"f : α → β",
"h : ∀ (a b : α), a ⋖ b → f a < f b",
"a b : α",
"hab : a < b",
"this : TransGen CovBy a b → TransGen LT.lt (f a) (f b)"
],
"goal": "f a < f b"
} | [
70473,
14281,
20706,
70476
] | [
"Mathlib/Order/Interval/Finset/Basic.lean"
] |
{
"context": [
"ι : Type u_1",
"α : Type u_2",
"β : Type u_3",
"inst✝² : Preorder α",
"inst✝¹ : LocallyFiniteOrder α",
"inst✝ : Preorder β",
"f : α → β",
"h : ∀ (a b : α), a ⋖ b → f a < f b",
"a b : α",
"hab : a < b",
"this : a < b → f a < f b"
],
"goal": "f a < f b"
} | [
14281,
20706,
70473
] | [
"Mathlib/Order/Interval/Finset/Basic.lean"
] |
{
"context": [
"𝕜 : Type u_1",
"inst✝⁴ : NontriviallyNormedField 𝕜",
"F : Type u_2",
"inst✝³ : NormedAddCommGroup F",
"inst✝² : NormedSpace 𝕜 F",
"E : Type u_3",
"inst✝¹ : NormedAddCommGroup E",
"inst✝ : NormedSpace 𝕜 E",
"f✝ f₀ f₁ : E → F",
"f' : F",
"s t : Set E",
"x v : E",
"L : E →L[𝕜] F",
"f : E → F",
"x₀ : E",
"C : ℝ",
"hC₀ : 0 ≤ C",
"hlip : ∀ᶠ (x : E) in 𝓝 x₀, ‖f x - f x₀‖ ≤ C * ‖x - x₀‖"
],
"goal": "‖lineDeriv 𝕜 f x₀ v‖ ≤ C * ‖v‖"
} | [
44446
] | [
"Mathlib/Analysis/Calculus/LineDeriv/Basic.lean"
] |
{
"context": [
"𝕜 : Type u_1",
"inst✝⁴ : NontriviallyNormedField 𝕜",
"F : Type u_2",
"inst✝³ : NormedAddCommGroup F",
"inst✝² : NormedSpace 𝕜 F",
"E : Type u_3",
"inst✝¹ : NormedAddCommGroup E",
"inst✝ : NormedSpace 𝕜 E",
"f✝ f₀ f₁ : E → F",
"f' : F",
"s t : Set E",
"x v : E",
"L : E →L[𝕜] F",
"f : E → F",
"x₀ : E",
"C : ℝ",
"hC₀ : 0 ≤ C",
"hlip : ∀ᶠ (x : E) in 𝓝 x₀, ‖f x - f x₀‖ ≤ C * ‖x - x₀‖"
],
"goal": "∀ᶠ (x : 𝕜) in 𝓝 0, ‖f (x₀ + x • v) - f (x₀ + 0 • v)‖ ≤ C * ‖v‖ * ‖x - 0‖"
} | [
44446
] | [
"Mathlib/Analysis/Calculus/LineDeriv/Basic.lean"
] |
{
"context": [
"𝕜 : Type u_1",
"inst✝⁴ : NontriviallyNormedField 𝕜",
"F : Type u_2",
"inst✝³ : NormedAddCommGroup F",
"inst✝² : NormedSpace 𝕜 F",
"E : Type u_3",
"inst✝¹ : NormedAddCommGroup E",
"inst✝ : NormedSpace 𝕜 E",
"f✝ f₀ f₁ : E → F",
"f' : F",
"s t : Set E",
"x v : E",
"L : E →L[𝕜] F",
"f : E → F",
"x₀ : E",
"C : ℝ",
"hC₀ : 0 ≤ C",
"hlip : ∀ᶠ (x : E) in 𝓝 x₀, ‖f x - f x₀‖ ≤ C * ‖x - x₀‖",
"A : Continuous fun t => x₀ + t • v",
"this : ∀ᶠ (x : E) in 𝓝 (x₀ + 0 • v), ‖f x - f x₀‖ ≤ C * ‖x - x₀‖"
],
"goal": "∀ᶠ (x : 𝕜) in 𝓝 0, ‖f (x₀ + x • v) - f (x₀ + 0 • v)‖ ≤ C * ‖v‖ * ‖x - 0‖"
} | [
15889,
55623,
55638,
131585,
41371,
118076,
119707,
131585,
134071
] | [
"Mathlib/Analysis/Calculus/LineDeriv/Basic.lean"
] |
{
"context": [
"𝕜 : Type u_1",
"inst✝⁴ : NontriviallyNormedField 𝕜",
"F : Type u_2",
"inst✝³ : NormedAddCommGroup F",
"inst✝² : NormedSpace 𝕜 F",
"E : Type u_3",
"inst✝¹ : NormedAddCommGroup E",
"inst✝ : NormedSpace 𝕜 E",
"f✝ f₀ f₁ : E → F",
"f' : F",
"s t✝ : Set E",
"x v : E",
"L : E →L[𝕜] F",
"f : E → F",
"x₀ : E",
"C : ℝ",
"hC₀ : 0 ≤ C",
"hlip : ∀ᶠ (x : E) in 𝓝 x₀, ‖f x - f x₀‖ ≤ C * ‖x - x₀‖",
"A : Continuous fun t => x₀ + t • v",
"this : ∀ᶠ (x : E) in 𝓝 (x₀ + 0 • v), ‖f x - f x₀‖ ≤ C * ‖x - x₀‖",
"t : 𝕜",
"ht : t ∈ (fun t => x₀ + t • v) ⁻¹' {x | ‖f x - f x₀‖ ≤ C * ‖x - x₀‖}"
],
"goal": "‖f (x₀ + t • v) - f (x₀ + 0 • v)‖ ≤ C * ‖v‖ * ‖t - 0‖"
} | [
131585,
119707,
55623,
15889,
55638,
134071,
41371,
118076
] | [
"Mathlib/Analysis/Calculus/LineDeriv/Basic.lean"
] |
{
"context": [
"𝕜 : Type u_1",
"inst✝⁴ : NontriviallyNormedField 𝕜",
"F : Type u_2",
"inst✝³ : NormedAddCommGroup F",
"inst✝² : NormedSpace 𝕜 F",
"E : Type u_3",
"inst✝¹ : NormedAddCommGroup E",
"inst✝ : NormedSpace 𝕜 E",
"f✝ f₀ f₁ : E → F",
"f' : F",
"s t✝ : Set E",
"x v : E",
"L : E →L[𝕜] F",
"f : E → F",
"x₀ : E",
"C : ℝ",
"hC₀ : 0 ≤ C",
"hlip : ∀ᶠ (x : E) in 𝓝 x₀, ‖f x - f x₀‖ ≤ C * ‖x - x₀‖",
"A : Continuous fun t => x₀ + t • v",
"this : ∀ᶠ (x : E) in 𝓝 (x₀ + 0 • v), ‖f x - f x₀‖ ≤ C * ‖x - x₀‖",
"t : 𝕜",
"ht : ‖f (x₀ + t • v) - f x₀‖ ≤ C * (‖v‖ * ‖t‖)"
],
"goal": "‖f (x₀ + t • v) - f (x₀ + 0 • v)‖ ≤ C * ‖v‖ * ‖t - 0‖"
} | [
131585,
119703,
41371,
134071,
119707,
118076
] | [
"Mathlib/Analysis/Calculus/LineDeriv/Basic.lean"
] |
{
"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 : ∀ t ∈ Ioo a b, HasDerivAt f (v t (f t)) t ∧ f t ∈ s t",
"hg : ∀ t ∈ Ioo a b, HasDerivAt g (v t (g t)) t ∧ g t ∈ s t",
"heq : f t₀ = g t₀",
"t' : ℝ",
"ht' : t' ∈ Ioo a b"
],
"goal": "f t' = g t'"
} | [
14316
] | [
"Mathlib/Analysis/ODE/Gronwall.lean"
] |
{
"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 : ∀ t ∈ Ioo a b, HasDerivAt f (v t (f t)) t ∧ f t ∈ s t",
"hg : ∀ t ∈ Ioo a b, HasDerivAt g (v t (g t)) t ∧ g t ∈ s t",
"heq : f t₀ = g t₀",
"t' : ℝ",
"ht' : t' ∈ Ioo a b",
"h : t' < t₀"
],
"goal": "f t' = g t'"
} | [
14316
] | [
"Mathlib/Analysis/ODE/Gronwall.lean"
] |
{
"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 : ∀ t ∈ Ioo a b, HasDerivAt f (v t (f t)) t ∧ f t ∈ s t",
"hg : ∀ t ∈ Ioo a b, HasDerivAt g (v t (g t)) t ∧ g t ∈ s t",
"heq : f t₀ = g t₀",
"t' : ℝ",
"ht' : t' ∈ Ioo a b",
"h : t₀ ≤ t'"
],
"goal": "f t' = g t'"
} | [
14316
] | [
"Mathlib/Analysis/ODE/Gronwall.lean"
] |
{
"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",
"Z₁ Z₂ X₁ X₂ Y₁ Y₂ : C",
"f₁ : X₁ ⟶ Y₁",
"f₂ : X₂ ⟶ Y₂"
],
"goal": "(Z₁ ⊗ Z₂) ◁ (f₁ ⊗ f₂) ≫ tensor_μ C (Z₁, Z₂) (Y₁, Y₂) = tensor_μ C (Z₁, Z₂) (X₁, X₂) ≫ (Z₁ ◁ f₁ ⊗ Z₂ ◁ f₂)"
} | [
107138
] | [
"Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean"
] |
{
"context": [
"C : Type u_1",
"inst✝⁸ : Category.{u_3, u_1} C",
"inst✝⁷ : Preadditive C",
"𝕜 : Type u_2",
"inst✝⁶ : Field 𝕜",
"inst✝⁵ : IsAlgClosed 𝕜",
"inst✝⁴ : Linear 𝕜 C",
"inst✝³ : HasKernels C",
"X Y : C",
"inst✝² : FiniteDimensional 𝕜 (X ⟶ X)",
"inst✝¹ : Simple X",
"inst✝ : Simple Y"
],
"goal": "finrank 𝕜 (X ⟶ Y) ≤ 1"
} | [
71959
] | [
"Mathlib/CategoryTheory/Preadditive/Schur.lean"
] |
{
"context": [
"C : Type u_1",
"inst✝⁸ : Category.{u_3, u_1} C",
"inst✝⁷ : Preadditive C",
"𝕜 : Type u_2",
"inst✝⁶ : Field 𝕜",
"inst✝⁵ : IsAlgClosed 𝕜",
"inst✝⁴ : Linear 𝕜 C",
"inst✝³ : HasKernels C",
"X Y : C",
"inst✝² : FiniteDimensional 𝕜 (X ⟶ X)",
"inst✝¹ : Simple X",
"inst✝ : Simple Y",
"h : Subsingleton (X ⟶ Y)"
],
"goal": "finrank 𝕜 (X ⟶ Y) ≤ 1"
} | [
71959
] | [
"Mathlib/CategoryTheory/Preadditive/Schur.lean"
] |
{
"context": [
"C : Type u_1",
"inst✝⁸ : Category.{u_3, u_1} C",
"inst✝⁷ : Preadditive C",
"𝕜 : Type u_2",
"inst✝⁶ : Field 𝕜",
"inst✝⁵ : IsAlgClosed 𝕜",
"inst✝⁴ : Linear 𝕜 C",
"inst✝³ : HasKernels C",
"X Y : C",
"inst✝² : FiniteDimensional 𝕜 (X ⟶ X)",
"inst✝¹ : Simple X",
"inst✝ : Simple Y",
"h : Nontrivial (X ⟶ Y)"
],
"goal": "finrank 𝕜 (X ⟶ Y) ≤ 1"
} | [
71959
] | [
"Mathlib/CategoryTheory/Preadditive/Schur.lean"
] |
{
"context": [
"n : ℕ",
"t : ℝ",
"ht' : t ≤ ↑n"
],
"goal": "(1 - t / ↑n) ^ n ≤ rexp (-t)"
} | [
70039,
106248,
149344
] | [
"Mathlib/Data/Complex/Exponential.lean"
] |
{
"context": [
"t : ℝ",
"ht' : t ≤ ↑0"
],
"goal": "(1 - t / ↑0) ^ 0 ≤ rexp (-t)"
} | [
70039,
106248,
149344
] | [
"Mathlib/Data/Complex/Exponential.lean"
] |
{
"context": [
"n : ℕ",
"t : ℝ",
"ht' : t ≤ ↑n",
"hn : n ≠ 0"
],
"goal": "(1 - t / ↑n) ^ n ≤ rexp (-t)"
} | [
106248,
149344,
70039
] | [
"Mathlib/Data/Complex/Exponential.lean"
] |
{
"context": [
"n : ℕ",
"t : ℝ",
"ht' : t ≤ ↑n",
"hn : n ≠ 0"
],
"goal": "rexp (-t) = rexp (-(t / ↑n)) ^ n"
} | [
106248,
149344
] | [
"Mathlib/Data/Complex/Exponential.lean"
] |
{
"context": [
"n : ℕ",
"t : ℝ",
"ht' : t ≤ ↑n",
"hn : n ≠ 0"
],
"goal": "0 ≤ 1 - t / ↑n"
} | [
106248,
149344
] | [
"Mathlib/Data/Complex/Exponential.lean"
] |
{
"context": [
"R : Type u_1",
"M : Type u_2",
"inst✝² : CommRing R",
"inst✝¹ : AddCommGroup M",
"inst✝ : Module R M",
"Q : QuadraticForm R M",
"x : CliffordAlgebra Q"
],
"goal": "↑((toEven Q) (reverse (involute x))) = reverse ↑((toEven Q) x)"
} | [
81956,
81967,
81968,
81970,
82298,
82718,
82721,
82723,
82725,
117079,
117080,
117094,
121060,
122047,
122048,
122051,
122054
] | [
"Mathlib/LinearAlgebra/CliffordAlgebra/EvenEquiv.lean"
] |
{
"context": [
"α : Type u_1",
"E : Type u_2",
"F : Type u_3",
"m0 : MeasurableSpace α",
"inst✝⁵ : NormedAddCommGroup E",
"inst✝⁴ : NormedSpace ℝ E",
"inst✝³ : CompleteSpace E",
"inst✝² : NormedAddCommGroup F",
"inst✝¹ : NormedSpace ℝ F",
"inst✝ : CompleteSpace F",
"μ ν : Measure α",
"s t : Set α",
"f✝ g✝ f g : α → ℝ≥0∞",
"h : f =ᶠ[ae μ] g"
],
"goal": "⨍⁻ (x : α), f x ∂μ = ⨍⁻ (x : α), g x ∂μ"
} | [
26860,
30260
] | [
"Mathlib/MeasureTheory/Integral/Average.lean"
] |
{
"context": [
"C : Type u",
"inst✝² : Category.{v, u} C",
"inst✝¹ : HasZeroMorphisms C",
"S S₁ S₂ S₃ S₄ : ShortComplex C",
"φ : S₁ ⟶ S₂",
"h₁ : S₁.HomologyData",
"h₂ : S₂.HomologyData",
"A : C",
"inst✝ : S.HasHomology",
"h : S.LeftHomologyData"
],
"goal": "(S.leftHomologyπ ≫ S.leftHomologyIso.hom) ≫ (S.leftHomologyIso.symm ≪≫ h.leftHomologyIso).hom = h.cyclesIso.hom ≫ h.π"
} | [
88745,
88756,
96173,
115100
] | [
"Mathlib/Algebra/Homology/ShortComplex/Homology.lean"
] |
{
"context": [
"ι : Sort u_1",
"V : Type u",
"W : Type v",
"G : SimpleGraph V",
"G' : SimpleGraph W",
"v w : V",
"hvw : G.Adj v w"
],
"goal": "G.subgraphOfAdj ⋯ = G.subgraphOfAdj hvw"
} | [
1723,
1726
] | [
"Mathlib/Combinatorics/SimpleGraph/Subgraph.lean"
] |
{
"context": [
"α : Type u",
"β : Type v",
"inst✝ : Finite α",
"s : Set α",
"f : ↑s ↪ β",
"h : Nonempty (α ≃ β)"
],
"goal": "∃ g, ∀ (x : ↑s), g ↑x = f x"
} | [
49534
] | [
"Mathlib/SetTheory/Cardinal/Ordinal.lean"
] |
{
"context": [
"α : Type u",
"β : Type v",
"inst✝ : Finite α",
"s : Set α",
"f : ↑s ↪ β",
"h : Nonempty (α ≃ β)"
],
"goal": "Nonempty (↑sᶜ ≃ ↑(range ⇑f)ᶜ)"
} | [
49534
] | [
"Mathlib/SetTheory/Cardinal/Ordinal.lean"
] |
{
"context": [
"α : Type u",
"β : Type v",
"inst✝ : Finite α",
"s : Set α",
"f : ↑s ↪ β",
"h : Nonempty (α ≃ β)",
"g : α ≃ β"
],
"goal": "Nonempty (↑sᶜ ≃ ↑(range ⇑f)ᶜ)"
} | [
48597,
48597,
49531
] | [
"Mathlib/SetTheory/Cardinal/Ordinal.lean"
] |
{
"context": [
"α : Type u",
"β : Type v",
"inst✝ : Finite α",
"s : Set α",
"f : ↑s ↪ β",
"h : lift.{max u v, u} #α = lift.{max u v, v} #β",
"g : α ≃ β"
],
"goal": "Nonempty (↑sᶜ ≃ ↑(range ⇑f)ᶜ)"
} | [
49531,
48597
] | [
"Mathlib/SetTheory/Cardinal/Ordinal.lean"
] |
{
"context": [
"α : Type u",
"β : Type v",
"inst✝ : Finite α",
"s : Set α",
"f : ↑s ↪ β",
"h : lift.{max u v, u} #α = lift.{max u v, v} #β",
"g : α ≃ β"
],
"goal": "lift.{max u v, u} #↑s = lift.{max u v, v} #↑(range ⇑f)"
} | [
49531,
48844,
48597
] | [
"Mathlib/SetTheory/Cardinal/Ordinal.lean"
] |
{
"context": [
"α : Type u",
"β : Type v",
"inst✝ : Finite α",
"s : Set α",
"f : ↑s ↪ β",
"h : lift.{max u v, u} #α = lift.{max u v, v} #β",
"g : α ≃ β"
],
"goal": "Injective ⇑f"
} | [
48844,
70654
] | [
"Mathlib/SetTheory/Cardinal/Ordinal.lean"
] |
{
"context": [],
"goal": "@zipWithLeft' = @zipWithLeft'TR"
} | [
1838
] | [
".lake/packages/batteries/Batteries/Data/List/Basic.lean"
] |
{
"context": [
"α : Type u_3",
"β : Type u_2",
"γ : Type u_1",
"f : α → Option β → γ",
"as : List α",
"bs : List β"
],
"goal": "zipWithLeft' f as bs = zipWithLeft'TR f as bs"
} | [
1838
] | [
".lake/packages/batteries/Batteries/Data/List/Basic.lean"
] |
{
"context": [
"α : Type u_3",
"β : Type u_2",
"γ : Type u_1",
"f : α → Option β → γ",
"as : List α",
"bs : List β",
"acc : Array γ",
"head✝ : α",
"tail✝ : List α"
],
"goal": "zipWithLeft'TR.go f (head✝ :: tail✝) [] acc = match zipWithLeft' f (head✝ :: tail✝) [] with | (l, r) => (acc.toList ++ l, r)"
} | [
5846
] | [
".lake/packages/batteries/Batteries/Data/List/Basic.lean"
] |
{
"context": [
"α : Type u_1",
"β : Type u_2",
"G : Type u_3",
"M : Type u_4",
"inst✝ : CommGroup G",
"a✝ b✝ c d a b : G"
],
"goal": "a * (b / a) = b"
} | [
118077,
117806
] | [
"Mathlib/Algebra/Group/Basic.lean"
] |
{
"context": [
"ι : Type u_1",
"X : Type u_2",
"Y : Type u_3",
"inst✝⁵ : EMetricSpace X",
"inst✝⁴ : EMetricSpace Y",
"inst✝³ : MeasurableSpace X",
"inst✝² : BorelSpace X",
"inst✝¹ : MeasurableSpace Y",
"inst✝ : BorelSpace Y"
],
"goal": "μH[1] = volume"
} | [
26594,
30824,
30825,
88683,
141373,
143125
] | [
"Mathlib/MeasureTheory/Measure/Hausdorff.lean"
] |
{
"context": [
"α : Type u_1",
"f g h : Perm α",
"x : α",
"hfx : f x = x",
"n : ℕ"
],
"goal": "(f ^ Int.negSucc n) x = x"
} | [
7831,
8652,
119787
] | [
"Mathlib/GroupTheory/Perm/Support.lean"
] |
{
"context": [
"𝕜 : Type u_1",
"inst✝¹⁰ : NontriviallyNormedField 𝕜",
"D : Type uD",
"inst✝⁹ : NormedAddCommGroup D",
"inst✝⁸ : NormedSpace 𝕜 D",
"E : Type uE",
"inst✝⁷ : NormedAddCommGroup E",
"inst✝⁶ : NormedSpace 𝕜 E",
"F : Type uF",
"inst✝⁵ : NormedAddCommGroup F",
"inst✝⁴ : NormedSpace 𝕜 F",
"G : Type uG",
"inst✝³ : NormedAddCommGroup G",
"inst✝² : NormedSpace 𝕜 G",
"X : Type u_2",
"inst✝¹ : NormedAddCommGroup X",
"inst✝ : NormedSpace 𝕜 X",
"s s₁ t u : Set E",
"f✝ f₁ : E → F",
"g : F → G",
"x x₀ : E",
"c : F",
"b : E × F → G",
"m n : ℕ∞",
"p : E → FormalMultilinearSeries 𝕜 E F",
"f : E → F",
"hf : ContDiff 𝕜 n f",
"hmn : m + 1 ≤ n"
],
"goal": "ContDiff 𝕜 m fun p => (fderiv 𝕜 f p.1) p.2"
} | [
48434,
46361,
133914
] | [
"Mathlib/Analysis/Calculus/ContDiff/Basic.lean"
] |
{
"context": [
"𝕜 : Type u_1",
"inst✝¹⁰ : NontriviallyNormedField 𝕜",
"D : Type uD",
"inst✝⁹ : NormedAddCommGroup D",
"inst✝⁸ : NormedSpace 𝕜 D",
"E : Type uE",
"inst✝⁷ : NormedAddCommGroup E",
"inst✝⁶ : NormedSpace 𝕜 E",
"F : Type uF",
"inst✝⁵ : NormedAddCommGroup F",
"inst✝⁴ : NormedSpace 𝕜 F",
"G : Type uG",
"inst✝³ : NormedAddCommGroup G",
"inst✝² : NormedSpace 𝕜 G",
"X : Type u_2",
"inst✝¹ : NormedAddCommGroup X",
"inst✝ : NormedSpace 𝕜 X",
"s s₁ t u : Set E",
"f✝ f₁ : E → F",
"g : F → G",
"x x₀ : E",
"c : F",
"b : E × F → G",
"m n : ℕ∞",
"p : E → FormalMultilinearSeries 𝕜 E F",
"f : E → F",
"hf : ContDiffOn 𝕜 n f univ",
"hmn : m + 1 ≤ n"
],
"goal": "ContDiffOn 𝕜 m (fun p => (fderiv 𝕜 f p.1) p.2) univ"
} | [
46361,
48434,
133914
] | [
"Mathlib/Analysis/Calculus/ContDiff/Basic.lean"
] |
{
"context": [
"𝕜 : Type u_1",
"inst✝¹⁰ : NontriviallyNormedField 𝕜",
"D : Type uD",
"inst✝⁹ : NormedAddCommGroup D",
"inst✝⁸ : NormedSpace 𝕜 D",
"E : Type uE",
"inst✝⁷ : NormedAddCommGroup E",
"inst✝⁶ : NormedSpace 𝕜 E",
"F : Type uF",
"inst✝⁵ : NormedAddCommGroup F",
"inst✝⁴ : NormedSpace 𝕜 F",
"G : Type uG",
"inst✝³ : NormedAddCommGroup G",
"inst✝² : NormedSpace 𝕜 G",
"X : Type u_2",
"inst✝¹ : NormedAddCommGroup X",
"inst✝ : NormedSpace 𝕜 X",
"s s₁ t u : Set E",
"f✝ f₁ : E → F",
"g : F → G",
"x x₀ : E",
"c : F",
"b : E × F → G",
"m n : ℕ∞",
"p : E → FormalMultilinearSeries 𝕜 E F",
"f : E → F",
"hf : ContDiffOn 𝕜 n f univ",
"hmn : m + 1 ≤ n"
],
"goal": "ContDiffOn 𝕜 m (fun p => (fderivWithin 𝕜 f univ p.1) p.2) (univ ×ˢ univ)"
} | [
46361,
133914,
51651,
45679
] | [
"Mathlib/Analysis/Calculus/ContDiff/Basic.lean"
] |
{
"context": [
"n : ℕ",
"c : Char",
"l : List Char"
],
"goal": "{ data := l }.IsSuffix (leftpad n c { data := l })"
} | [
1455
] | [
"Mathlib/Data/String/Lemmas.lean"
] |
{
"context": [
"X : Type u_1",
"Y : Type u_2",
"Z : Type u_3",
"inst✝² : PseudoEMetricSpace X",
"inst✝¹ : PseudoEMetricSpace Y",
"inst✝ : PseudoEMetricSpace Z",
"e : X ≃ᵢ Y"
],
"goal": "ratio e.toDilationEquiv = 1"
} | [
60798,
60825,
61569
] | [
"Mathlib/Topology/MetricSpace/DilationEquiv.lean"
] |
{
"context": [
"n : ℕ"
],
"goal": "χ₈ ↑n = χ₈ ↑(n % 8)"
} | [
138369
] | [
"Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean"
] |
{
"context": [
"α : Type u_1",
"β : Type u_2",
"γ : Type u_3",
"ι : Sort u_4",
"ι' : Sort u_5",
"ι₂ : Sort u_6",
"κ : ι → Sort u_7",
"κ₁ : ι → Sort u_8",
"κ₂ : ι → Sort u_9",
"κ' : ι' → Sort u_10",
"P : ι → α → Prop",
"x✝ : α"
],
"goal": "x✝ ∈ ⋂ i, {x | P i x} ↔ x✝ ∈ {x | ∀ (i : ι), P i x}"
} | [
16574
] | [
"Mathlib/Data/Set/Lattice.lean"
] |
{
"context": [
"α : Type u",
"β : Type v",
"γ : Type w",
"δ : Type u_1",
"ι : Sort x",
"f : Filter α",
"p : α → Prop",
"q : Prop"
],
"goal": "(∀ᶠ (x : α) in f, p x → q) ↔ (∃ᶠ (x : α) in f, p x) → q"
} | [
16036,
16061,
70070
] | [
"Mathlib/Order/Filter/Basic.lean"
] |
{
"context": [
"C : Type u",
"inst✝ : Category.{v, u} C",
"X✝ Y✝ X Y Z : C",
"sXY : BinaryFan X Y",
"P : IsLimit sXY",
"sYZ : BinaryFan Y Z",
"Q : IsLimit sYZ",
"s : BinaryFan sXY.pt Z",
"R : IsLimit s",
"t : Cone (pair X sYZ.pt)"
],
"goal": "R.lift (BinaryFan.assocInv P t) ≫ Q.lift (BinaryFan.mk (s.fst ≫ sXY.snd) s.snd) = BinaryFan.snd t"
} | [
94261
] | [
"Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Basic.lean"
] |
{
"context": [
"C : Type u",
"inst✝ : Category.{v, u} C",
"X✝ Y✝ X Y Z : C",
"sXY : BinaryFan X Y",
"P : IsLimit sXY",
"sYZ : BinaryFan Y Z",
"Q : IsLimit sYZ",
"s : BinaryFan sXY.pt Z",
"R : IsLimit s",
"t : Cone (pair X sYZ.pt)"
],
"goal": "∀ (j : Discrete WalkingPair), (R.lift (BinaryFan.assocInv P t) ≫ Q.lift (BinaryFan.mk (s.fst ≫ sXY.snd) s.snd)) ≫ sYZ.π.app j = BinaryFan.snd t ≫ sYZ.π.app j"
} | [
94261
] | [
"Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Basic.lean"
] |
{
"context": [
"C : Type u",
"inst✝ : Category.{v, u} C",
"X✝ Y✝ X Y Z : C",
"sXY : BinaryFan X Y",
"P : IsLimit sXY",
"sYZ : BinaryFan Y Z",
"Q : IsLimit sYZ",
"s : BinaryFan sXY.pt Z",
"R : IsLimit s",
"t : Cone (pair X sYZ.pt)",
"m : t.pt ⟶ (BinaryFan.assoc Q s).pt",
"w : ∀ (j : Discrete WalkingPair), m ≫ (BinaryFan.assoc Q s).π.app j = t.π.app j"
],
"goal": "m = (fun t => R.lift (BinaryFan.assocInv P t)) t"
} | [
94242
] | [
"Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Basic.lean"
] |
{
"context": [
"C : Type u",
"inst✝ : Category.{v, u} C",
"X✝ Y✝ X Y Z : C",
"sXY : BinaryFan X Y",
"P : IsLimit sXY",
"sYZ : BinaryFan Y Z",
"Q : IsLimit sYZ",
"s : BinaryFan sXY.pt Z",
"R : IsLimit s",
"t : Cone (pair X sYZ.pt)",
"m : t.pt ⟶ (BinaryFan.assoc Q s).pt",
"w : ∀ (j : Discrete WalkingPair), m ≫ (BinaryFan.assoc Q s).π.app j = t.π.app j",
"h : (∀ (j : Discrete WalkingPair), m ≫ s.π.app j = (BinaryFan.assocInv P t).π.app j) → m = R.lift (BinaryFan.assocInv P t)"
],
"goal": "m = (fun t => R.lift (BinaryFan.assocInv P t)) t"
} | [
94242
] | [
"Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Basic.lean"
] |
{
"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✝ : Finite G",
"p : ℕ",
"hp : Fact (Nat.Prime p)",
"hdvd : p ∣ Nat.card G",
"this : Fintype G"
],
"goal": "∃ x, orderOf x = p"
} | [
9409,
47564
] | [
"Mathlib/GroupTheory/Perm/Cycle/Type.lean"
] |
{
"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✝ : Finite G",
"p : ℕ",
"hp : Fact (Nat.Prime p)",
"this : Fintype G",
"hdvd : p ∣ Fintype.card G"
],
"goal": "∃ x, orderOf x = p"
} | [
9409,
47564
] | [
"Mathlib/GroupTheory/Perm/Cycle/Type.lean"
] |
{
"context": [
"K : Type u_1",
"inst✝¹ : Field K",
"inst✝ : NumberField K",
"B✝ B : ℝ",
"hB : B ≤ 0"
],
"goal": "volume (convexBodySum K B) = 0"
} | [
14302
] | [
"Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean"
] |
{
"context": [
"K : Type u_1",
"inst✝¹ : Field K",
"inst✝ : NumberField K",
"B✝ B : ℝ",
"hB✝ : B ≤ 0",
"hB : B < 0"
],
"goal": "volume (convexBodySum K B) = 0"
} | [
14302
] | [
"Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean"
] |
{
"context": [
"K : Type u_1",
"inst✝¹ : Field K",
"inst✝ : NumberField K",
"B✝ B : ℝ",
"hB✝ : B ≤ 0",
"hB : B = 0"
],
"goal": "volume (convexBodySum K B) = 0"
} | [
14302
] | [
"Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean"
] |
{
"context": [
"G : Type u_1",
"H : Type u_2",
"α : Type u_3",
"β : Type u_4",
"E : Type u_5",
"inst✝⁶ : Group G",
"inst✝⁵ : MulAction G α",
"inst✝⁴ : MeasurableSpace α",
"μ : Measure α",
"inst✝³ : Countable G",
"inst✝² : MeasurableSpace G",
"s t : Set α",
"inst✝¹ : SMulInvariantMeasure G α μ",
"inst✝ : MeasurableSMul G α",
"fund_dom_s : IsFundamentalDomain G s μ",
"fund_dom_t : IsFundamentalDomain G t μ",
"U : Set (Quotient α_mod_G)",
"meas_U : MeasurableSet U"
],
"goal": "(Measure.map (Quotient.mk α_mod_G) (μ.restrict s)) U = (Measure.map (Quotient.mk α_mod_G) (μ.restrict t)) U"
} | [
33097,
33035
] | [
"Mathlib/MeasureTheory/Group/FundamentalDomain.lean"
] |
{
"context": [
"G : Type u_1",
"H : Type u_2",
"α : Type u_3",
"β : Type u_4",
"E : Type u_5",
"inst✝⁶ : Group G",
"inst✝⁵ : MulAction G α",
"inst✝⁴ : MeasurableSpace α",
"μ : Measure α",
"inst✝³ : Countable G",
"inst✝² : MeasurableSpace G",
"s t : Set α",
"inst✝¹ : SMulInvariantMeasure G α μ",
"inst✝ : MeasurableSMul G α",
"fund_dom_s : IsFundamentalDomain G s μ",
"fund_dom_t : IsFundamentalDomain G t μ",
"U : Set (Quotient α_mod_G)",
"meas_U : MeasurableSet U"
],
"goal": "μ (Quotient.mk α_mod_G ⁻¹' U ∩ s) = μ (Quotient.mk α_mod_G ⁻¹' U ∩ t)"
} | [
33097,
33035
] | [
"Mathlib/MeasureTheory/Group/FundamentalDomain.lean"
] |
{
"context": [
"G : Type u_1",
"H : Type u_2",
"α : Type u_3",
"β : Type u_4",
"E : Type u_5",
"inst✝⁶ : Group G",
"inst✝⁵ : MulAction G α",
"inst✝⁴ : MeasurableSpace α",
"μ : Measure α",
"inst✝³ : Countable G",
"inst✝² : MeasurableSpace G",
"s t : Set α",
"inst✝¹ : SMulInvariantMeasure G α μ",
"inst✝ : MeasurableSMul G α",
"fund_dom_s : IsFundamentalDomain G s μ",
"fund_dom_t : IsFundamentalDomain G t μ",
"U : Set (Quotient α_mod_G)",
"meas_U : MeasurableSet U"
],
"goal": "MeasurableSet (Quotient.mk α_mod_G ⁻¹' U)"
} | [
33035
] | [
"Mathlib/MeasureTheory/Group/FundamentalDomain.lean"
] |
{
"context": [
"G : Type u_1",
"H : Type u_2",
"α : Type u_3",
"β : Type u_4",
"E : Type u_5",
"inst✝⁶ : Group G",
"inst✝⁵ : MulAction G α",
"inst✝⁴ : MeasurableSpace α",
"μ : Measure α",
"inst✝³ : Countable G",
"inst✝² : MeasurableSpace G",
"s t : Set α",
"inst✝¹ : SMulInvariantMeasure G α μ",
"inst✝ : MeasurableSMul G α",
"fund_dom_s : IsFundamentalDomain G s μ",
"fund_dom_t : IsFundamentalDomain G t μ",
"U : Set (Quotient α_mod_G)",
"meas_U : MeasurableSet U"
],
"goal": "∀ (g : G), (fun x => g • x) ⁻¹' (Quotient.mk α_mod_G ⁻¹' U) = Quotient.mk α_mod_G ⁻¹' U"
} | [
33035
] | [
"Mathlib/MeasureTheory/Group/FundamentalDomain.lean"
] |
{
"context": [
"R : Type u_1",
"inst✝¹⁶ : CommSemiring R",
"R' : Type u_2",
"inst✝¹⁵ : Monoid R'",
"R'' : Type u_3",
"inst✝¹⁴ : Semiring R''",
"M : Type u_4",
"N : Type u_5",
"P : Type u_6",
"Q : Type u_7",
"S : Type u_8",
"T : Type u_9",
"inst✝¹³ : AddCommMonoid M",
"inst✝¹² : AddCommMonoid N",
"inst✝¹¹ : AddCommMonoid P",
"inst✝¹⁰ : AddCommMonoid Q",
"inst✝⁹ : AddCommMonoid S",
"inst✝⁸ : AddCommMonoid T",
"inst✝⁷ : Module R M",
"inst✝⁶ : Module R N",
"inst✝⁵ : Module R P",
"inst✝⁴ : Module R Q",
"inst✝³ : Module R S",
"inst✝² : Module R T",
"inst✝¹ : DistribMulAction R' M",
"inst✝ : Module R'' M",
"g✝ : P →ₗ[R] Q",
"f✝ : N →ₗ[R] P",
"f : M →ₗ[R] P",
"g : N →ₗ[R] Q"
],
"goal": "lTensor P g ∘ₗ rTensor N f = map f g"
} | [
86898,
109760,
109761
] | [
"Mathlib/LinearAlgebra/TensorProduct/Basic.lean"
] |
{
"context": [
"𝕜 : Type u_1",
"inst✝¹¹ : NontriviallyNormedField 𝕜",
"E : Type u_2",
"inst✝¹⁰ : NormedAddCommGroup E",
"inst✝⁹ : NormedSpace 𝕜 E",
"H : Type u_3",
"inst✝⁸ : TopologicalSpace H",
"I : ModelWithCorners 𝕜 E H",
"M : Type u_4",
"inst✝⁷ : TopologicalSpace M",
"inst✝⁶ : ChartedSpace H M",
"E' : Type u_5",
"inst✝⁵ : NormedAddCommGroup E'",
"inst✝⁴ : NormedSpace 𝕜 E'",
"H' : Type u_6",
"inst✝³ : TopologicalSpace H'",
"N : Type u_7",
"inst✝² : TopologicalSpace N",
"inst✝¹ : ChartedSpace H' N",
"J : ModelWithCorners 𝕜 E' H'",
"inst✝ : SmoothManifoldWithCorners J N",
"x : M",
"y : N",
"p : M × N"
],
"goal": "p ∈ (I.prod J).interior (M × N) ↔ p ∈ I.interior M ×ˢ J.interior N"
} | [
66448,
67778,
133949
] | [
"Mathlib/Geometry/Manifold/InteriorBoundary.lean"
] |
{
"context": [
"𝕜 : Type u_1",
"inst✝¹¹ : NontriviallyNormedField 𝕜",
"E : Type u_2",
"inst✝¹⁰ : NormedAddCommGroup E",
"inst✝⁹ : NormedSpace 𝕜 E",
"H : Type u_3",
"inst✝⁸ : TopologicalSpace H",
"I : ModelWithCorners 𝕜 E H",
"M : Type u_4",
"inst✝⁷ : TopologicalSpace M",
"inst✝⁶ : ChartedSpace H M",
"E' : Type u_5",
"inst✝⁵ : NormedAddCommGroup E'",
"inst✝⁴ : NormedSpace 𝕜 E'",
"H' : Type u_6",
"inst✝³ : TopologicalSpace H'",
"N : Type u_7",
"inst✝² : TopologicalSpace N",
"inst✝¹ : ChartedSpace H' N",
"J : ModelWithCorners 𝕜 E' H'",
"inst✝ : SmoothManifoldWithCorners J N",
"x : M",
"y : N",
"p : M × N",
"aux : interior (range ↑I) ×ˢ interior (range ↑J) = interior (range ↑(I.prod J))"
],
"goal": "p ∈ (I.prod J).interior (M × N) ↔ p ∈ I.interior M ×ˢ J.interior N"
} | [
66448,
67778,
133949
] | [
"Mathlib/Geometry/Manifold/InteriorBoundary.lean"
] |
{
"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)",
"p₁ p₂ p₃ p₄ : P",
"h : Wbtw ℝ p₁ p₂ p₃",
"hne : p₂ ≠ p₃"
],
"goal": "(∡ p₂ p₄ p₃).sign = (∡ p₁ p₄ p₃).sign"
} | [
1690,
38396,
70344,
70410
] | [
"Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean"
] |
{
"context": [
"ι α β : Type u",
"c : Cardinal.{u}",
"l : Filter α",
"inst✝ : CardinalInterFilter l c",
"s : ι → Set α",
"hic : #ι < c"
],
"goal": "⋂ i, s i ∈ l ↔ ∀ (i : ι), s i ∈ l"
} | [
135425,
1715,
12656,
14288,
48841
] | [
"Mathlib/Order/Filter/CardinalInter.lean"
] |
{
"context": [
"ι α β : Type u",
"c : Cardinal.{u}",
"l : Filter α",
"inst✝ : CardinalInterFilter l c",
"s : ι → Set α",
"hic : #ι < c"
],
"goal": "⋂₀ range s ∈ l ↔ ∀ (i : ι), s i ∈ l"
} | [
135425,
48841,
12656,
14288,
1715
] | [
"Mathlib/Order/Filter/CardinalInter.lean"
] |
{
"context": [
"ι α β : Type u",
"c : Cardinal.{u}",
"l : Filter α",
"inst✝ : CardinalInterFilter l c",
"s : ι → Set α",
"hic : #ι < c"
],
"goal": "(∀ s_1 ∈ range s, s_1 ∈ l) ↔ ∀ (i : ι), s i ∈ l"
} | [
48841,
12656,
14288,
1715,
134168
] | [
"Mathlib/Order/Filter/CardinalInter.lean"
] |
{
"context": [
"α : Type u",
"a : α",
"inst✝² : Group α",
"inst✝¹ : DecidableEq α",
"inst✝ : Fintype α",
"hn : ∀ (n : ℕ), 0 < n → (filter (fun a => a ^ n = 1) univ).card ≤ n",
"d : ℕ",
"hd : d ∣ Fintype.card α",
"c : ℕ := Fintype.card α"
],
"goal": "(filter (fun a => orderOf a = d) univ).card = φ d"
} | [
141401,
1674,
7980
] | [
"Mathlib/GroupTheory/SpecificGroups/Cyclic.lean"
] |
{
"context": [
"α : Type u",
"a : α",
"inst✝² : Group α",
"inst✝¹ : DecidableEq α",
"inst✝ : Fintype α",
"hn : ∀ (n : ℕ), 0 < n → (filter (fun a => a ^ n = 1) univ).card ≤ n",
"d : ℕ",
"hd : d ∣ Fintype.card α",
"c : ℕ := Fintype.card α",
"hc0 : 0 < c"
],
"goal": "(filter (fun a => orderOf a = d) univ).card = φ d"
} | [
141401,
1674,
7980
] | [
"Mathlib/GroupTheory/SpecificGroups/Cyclic.lean"
] |
{
"context": [
"α : Type u",
"a : α",
"inst✝² : Group α",
"inst✝¹ : DecidableEq α",
"inst✝ : Fintype α",
"hn : ∀ (n : ℕ), 0 < n → (filter (fun a => a ^ n = 1) univ).card ≤ n",
"d : ℕ",
"hd : d ∣ Fintype.card α",
"c : ℕ := Fintype.card α",
"hc0 : 0 < c"
],
"goal": "0 < (filter (fun a => orderOf a = d) univ).card"
} | [
7980,
1734
] | [
"Mathlib/GroupTheory/SpecificGroups/Cyclic.lean"
] |
{
"context": [
"α : Type u",
"a : α",
"inst✝² : Group α",
"inst✝¹ : DecidableEq α",
"inst✝ : Fintype α",
"hn : ∀ (n : ℕ), 0 < n → (filter (fun a => a ^ n = 1) univ).card ≤ n",
"d : ℕ",
"hd : d ∣ Fintype.card α",
"c : ℕ := Fintype.card α",
"hc0 : 0 < c",
"h0 : ¬0 < (filter (fun a => orderOf a = d) univ).card"
],
"goal": "False"
} | [
137616,
14323,
3806,
1734
] | [
"Mathlib/GroupTheory/SpecificGroups/Cyclic.lean"
] |
{
"context": [
"α : Type u",
"a : α",
"inst✝² : Group α",
"inst✝¹ : DecidableEq α",
"inst✝ : Fintype α",
"hn : ∀ (n : ℕ), 0 < n → (filter (fun a => a ^ n = 1) univ).card ≤ n",
"d : ℕ",
"hd : d ∣ Fintype.card α",
"c : ℕ := Fintype.card α",
"hc0 : 0 < c",
"h0 : filter (fun a => orderOf a = d) univ = ∅"
],
"goal": "False"
} | [
137616,
14323,
3806,
14279
] | [
"Mathlib/GroupTheory/SpecificGroups/Cyclic.lean"
] |
{
"context": [
"α : Type u",
"a : α",
"inst✝² : Group α",
"inst✝¹ : DecidableEq α",
"inst✝ : Fintype α",
"hn : ∀ (n : ℕ), 0 < n → (filter (fun a => a ^ n = 1) univ).card ≤ n",
"d : ℕ",
"hd : d ∣ Fintype.card α",
"c : ℕ := Fintype.card α",
"hc0 : 0 < c",
"h0 : filter (fun a => orderOf a = d) univ = ∅"
],
"goal": "c < c"
} | [
14279
] | [
"Mathlib/GroupTheory/SpecificGroups/Cyclic.lean"
] |
{
"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",
"inst✝ : DecidableEq ι",
"hi : i ∈ s",
"f : (j : ι) → α j",
"a : α i",
"t : (j : ι) → α j → Set (β j)"
],
"goal": "(s.pi fun j => t j (update f i a j)) = ({i} ∪ s \\ {i}).pi fun j => t j (update f i a j)"
} | [
1674,
133417,
133525,
133678
] | [
"Mathlib/Data/Set/Prod.lean"
] |
{
"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",
"inst✝ : DecidableEq ι",
"hi : i ∈ s",
"f : (j : ι) → α j",
"a : α i",
"t : (j : ι) → α j → Set (β j)"
],
"goal": "(({i} ∪ s \\ {i}).pi fun j => t j (update f i a j)) = {x | x i ∈ t i a} ∩ (s \\ {i}).pi fun j => t j (f j)"
} | [
71464,
134029,
134033,
134036
] | [
"Mathlib/Data/Set/Prod.lean"
] |
{
"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",
"inst✝ : DecidableEq ι",
"hi : i ∈ s",
"f : (j : ι) → α j",
"a : α i",
"t : (j : ι) → α j → Set (β j)"
],
"goal": "i ∉ s \\ {i}"
} | [
71464,
134029,
134033,
134036
] | [
"Mathlib/Data/Set/Prod.lean"
] |
{
"context": [
"C : Type u_1",
"inst✝ : Category.{u_2, u_1} C",
"A✝ B✝ B'✝ X✝ Y✝ Y' : C",
"i✝ : A✝ ⟶ B✝",
"i'✝ : B✝ ⟶ B'✝",
"p✝ : X✝ ⟶ Y✝",
"p' : Y✝ ⟶ Y'",
"A B A' B' X Y : C",
"i : A ⟶ B",
"i' : A' ⟶ B'",
"e : Arrow.mk i ≅ Arrow.mk i'",
"p : X ⟶ Y",
"a✝ : HasLiftingProperty i p"
],
"goal": "HasLiftingProperty i' p"
} | [
96641
] | [
"Mathlib/CategoryTheory/LiftingProperties/Basic.lean"
] |
{
"context": [
"C : Type u_1",
"inst✝ : Category.{u_2, u_1} C",
"A✝ B✝ B'✝ X✝ Y✝ Y' : C",
"i✝ : A✝ ⟶ B✝",
"i'✝ : B✝ ⟶ B'✝",
"p✝ : X✝ ⟶ Y✝",
"p' : Y✝ ⟶ Y'",
"A B A' B' X Y : C",
"i : A ⟶ B",
"i' : A' ⟶ B'",
"e : Arrow.mk i ≅ Arrow.mk i'",
"p : X ⟶ Y",
"a✝ : HasLiftingProperty i' p"
],
"goal": "HasLiftingProperty i p"
} | [
96641
] | [
"Mathlib/CategoryTheory/LiftingProperties/Basic.lean"
] |
{
"context": [
"V : Type u_1",
"P : Type u_2",
"inst✝³ : NormedAddCommGroup V",
"inst✝² : InnerProductSpace ℝ V",
"inst✝¹ : MetricSpace P",
"inst✝ : NormedAddTorsor V P",
"s : Set P",
"hs : Cospherical s",
"p : Fin 3 → P",
"hps : Set.range p ⊆ s",
"hpi : Function.Injective p"
],
"goal": "AffineIndependent ℝ p"
} | [
83636
] | [
"Mathlib/Geometry/Euclidean/Sphere/Basic.lean"
] |
{
"context": [
"V : Type u_1",
"P : Type u_2",
"inst✝³ : NormedAddCommGroup V",
"inst✝² : InnerProductSpace ℝ V",
"inst✝¹ : MetricSpace P",
"inst✝ : NormedAddTorsor V P",
"s : Set P",
"hs : Cospherical s",
"p : Fin 3 → P",
"hps : Set.range p ⊆ s",
"hpi : Function.Injective p"
],
"goal": "¬Collinear ℝ (Set.range p)"
} | [
83636
] | [
"Mathlib/Geometry/Euclidean/Sphere/Basic.lean"
] |
{
"context": [
"V : Type u_1",
"P : Type u_2",
"inst✝³ : NormedAddCommGroup V",
"inst✝² : InnerProductSpace ℝ V",
"inst✝¹ : MetricSpace P",
"inst✝ : NormedAddTorsor V P",
"s : Set P",
"hs : Cospherical s",
"p : Fin 3 → P",
"hps : Set.range p ⊆ s",
"hpi : Function.Injective p",
"hc : Collinear ℝ (Set.range p)"
],
"goal": "False"
} | [
83633,
131596
] | [
"Mathlib/Geometry/Euclidean/Sphere/Basic.lean"
] |
{
"context": [
"V : Type u_1",
"P : Type u_2",
"inst✝³ : NormedAddCommGroup V",
"inst✝² : InnerProductSpace ℝ V",
"inst✝¹ : MetricSpace P",
"inst✝ : NormedAddTorsor V P",
"s : Set P",
"hs : Cospherical s",
"p : Fin 3 → P",
"hps : Set.range p ⊆ s",
"hpi : Function.Injective p",
"hc : ∃ v, ∀ p_1 ∈ Set.range p, ∃ r, p_1 = r • v +ᵥ p 0"
],
"goal": "False"
} | [
83633,
131596
] | [
"Mathlib/Geometry/Euclidean/Sphere/Basic.lean"
] |
{
"context": [
"V : Type u_1",
"P : Type u_2",
"inst✝³ : NormedAddCommGroup V",
"inst✝² : InnerProductSpace ℝ V",
"inst✝¹ : MetricSpace P",
"inst✝ : NormedAddTorsor V P",
"s : Set P",
"hs : Cospherical s",
"p : Fin 3 → P",
"hps : Set.range p ⊆ s",
"hpi : Function.Injective p",
"v : V",
"hv : ∀ p_1 ∈ Set.range p, ∃ r, p_1 = r • v +ᵥ p 0"
],
"goal": "False"
} | [
134168
] | [
"Mathlib/Geometry/Euclidean/Sphere/Basic.lean"
] |
{
"context": [
"V : Type u_1",
"P : Type u_2",
"inst✝³ : NormedAddCommGroup V",
"inst✝² : InnerProductSpace ℝ V",
"inst✝¹ : MetricSpace P",
"inst✝ : NormedAddTorsor V P",
"s : Set P",
"hs : Cospherical s",
"p : Fin 3 → P",
"hps : Set.range p ⊆ s",
"hpi : Function.Injective p",
"v : V",
"hv : ∀ (i : Fin 3), ∃ r, p i = r • v +ᵥ p 0"
],
"goal": "False"
} | [
134168
] | [
"Mathlib/Geometry/Euclidean/Sphere/Basic.lean"
] |
{
"context": [
"V : Type u_1",
"P : Type u_2",
"inst✝³ : NormedAddCommGroup V",
"inst✝² : InnerProductSpace ℝ V",
"inst✝¹ : MetricSpace P",
"inst✝ : NormedAddTorsor V P",
"s : Set P",
"p : Fin 3 → P",
"hps : Set.range p ⊆ s",
"hpi : Function.Injective p",
"v : V",
"hv : ∀ (i : Fin 3), ∃ r, p i = r • v +ᵥ p 0",
"hv0 : v ≠ 0",
"c : P",
"r : ℝ",
"hs : ∀ p ∈ s, dist p c = r"
],
"goal": "False"
} | [
131596,
133308
] | [
"Mathlib/Geometry/Euclidean/Sphere/Basic.lean"
] |
{
"context": [
"V : Type u_1",
"P : Type u_2",
"inst✝³ : NormedAddCommGroup V",
"inst✝² : InnerProductSpace ℝ V",
"inst✝¹ : MetricSpace P",
"inst✝ : NormedAddTorsor V P",
"s : Set P",
"p : Fin 3 → P",
"hps : Set.range p ⊆ s",
"hpi : Function.Injective p",
"v : V",
"hv : ∀ (i : Fin 3), ∃ r, p i = r • v +ᵥ p 0",
"hv0 : v ≠ 0",
"c : P",
"r : ℝ",
"hs : ∀ p ∈ s, dist p c = r",
"hs' : ∀ (i : Fin 3), dist (p i) c = r"
],
"goal": "False"
} | [
131596,
133308
] | [
"Mathlib/Geometry/Euclidean/Sphere/Basic.lean"
] |
{
"context": [
"V : Type u_1",
"P : Type u_2",
"inst✝³ : NormedAddCommGroup V",
"inst✝² : InnerProductSpace ℝ V",
"inst✝¹ : MetricSpace P",
"inst✝ : NormedAddTorsor V P",
"s : Set P",
"p : Fin 3 → P",
"hps : Set.range p ⊆ s",
"hpi : Function.Injective p",
"v : V",
"hv0 : v ≠ 0",
"c : P",
"r : ℝ",
"hs : ∀ p ∈ s, dist p c = r",
"hs' : ∀ (i : Fin 3), dist (p i) c = r",
"f : Fin 3 → ℝ",
"hf : ∀ (i : Fin 3), p i = f i • v +ᵥ p 0",
"hsd : ∀ (i : Fin 3), dist (f i • v +ᵥ p 0) c = r"
],
"goal": "False"
} | [
1717,
110053,
115870,
115877
] | [
"Mathlib/Geometry/Euclidean/Sphere/Basic.lean"
] |
End of preview.
Subsets and Splits