ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/RingTheory/LocalRing/RingHom/Basic.lean
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : LocalRing R
inst✝² : CommSemiring S
inst✝¹ : Nontrivial S
f : R →+* S
inst✝ : IsLocalRingHom f
hf : Function.Surjective ⇑f
a b : R
hab : IsUnit (f a + f b)
\u22a2 IsUnit (f a) ∨ IsUnit (f b)
-- Mathlib/RingTheory/LocalRing/RingHom/Basic.lean
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : LocalRing R
inst✝² : CommSemiring S
inst✝¹ : Nontrivial S
f : R →+* S
inst✝ : IsLocalRingHom f
hf : Function.Surjective ⇑f
a b : R
hab : IsUnit (f (a + b))
\u22a2 IsUnit (f a) ∨ IsUnit (f b)
-- Mathlib/Geometry/Manifold/BumpFunction.lean
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
c : M
f : SmoothBumpFunction I c
x : M
inst✝¹ : T2Space M
inst✝ : SmoothManifoldWithCorners I M
\u22a2 Smooth I 𝓘(ℝ, ℝ) ↑f
-- Mathlib/Geometry/Manifold/BumpFunction.lean
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
c : M
f : SmoothBumpFunction I c
x✝ : M
inst✝¹ : T2Space M
inst✝ : SmoothManifoldWithCorners I M
x : M
hx : x ∈ tsupport ↑f
\u22a2 ContMDiffAt I 𝓘(ℝ, ℝ) ⊤ (↑f) x
-- Mathlib/Geometry/Manifold/BumpFunction.lean
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
c : M
f : SmoothBumpFunction I c
x✝ : M
inst✝¹ : T2Space M
inst✝ : SmoothManifoldWithCorners I M
x : M
hx : x ∈ tsupport ↑f
this : x ∈ (chartAt H c).source
\u22a2 ContMDiffAt I 𝓘(ℝ, ℝ) ⊤ (↑f) x
-- Mathlib/Geometry/Manifold/BumpFunction.lean
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
c : M
f : SmoothBumpFunction I c
x✝ : M
inst✝¹ : T2Space M
inst✝ : SmoothManifoldWithCorners I M
x : M
hx : x ∈ tsupport ↑f
this : x ∈ (chartAt H c).source
\u22a2 ContMDiffAt I 𝓘(ℝ, ℝ) ⊤ (↑f.toContDiffBump ∘ ↑(extChartAt I c)) x
-- Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean
R : Type u
inst✝¹ : CommRing R
W' : Jacobian R
F : Type v
inst✝ : Field F
W : Jacobian F
P Q : Fin 3 → R
hP : W'.Equation P
hPz : P z = 0
\u22a2 W'.addXYZ P Q = (P x * Q z) • Q
-- Mathlib/Algebra/Order/Field/Basic.lean
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝ : LinearOrderedField α
a b c d : α
n : ℤ
ha : a < 0
hb : b < 0
\u22a2 a < 1 / b ↔ b < 1 / a
-- Mathlib/Algebra/Module/Submodule/Ker.lean
R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
R₃ : Type u_4
K : Type u_5
M : Type u_6
M₁ : Type u_7
M₂ : Type u_8
M₃ : Type u_9
V : Type u_10
V₂ : Type u_11
inst✝⁷ : Semiring R
inst✝⁶ : Semiring R₂
inst✝⁵ : AddCommMonoid M
inst✝⁴ : AddCommMonoid M₂
inst✝³ : Module R M
inst✝² : Module R₂ M₂
p✝ p'✝ : Submodule R M
q : Submodule R₂ M₂
τ₁₂ : R →+* R₂
F : Type u_12
inst✝¹ : FunLike F M M₂
inst✝ : SemilinearMapClass F τ₁₂ M M₂
p p' : Submodule R M
h : p ≤ p'
\u22a2 ker (inclusion h) = ⊥
-- Mathlib/CategoryTheory/Abelian/Opposite.lean
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
A B : Cᵒᵖ
g : A ⟶ B
\u22a2 (imageUnopOp g).hom ≫ image.ι g = (factorThruImage g.unop).op
-- Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean
C : Type u_1
inst✝³ : Category.{u_3, u_1} C
inst✝² : HasZeroMorphisms C
ι : Type u_2
c : ComplexShape ι
K : HomologicalComplex C c
i j k : ι
hi : c.prev j = i
hk : c.next j = k
inst✝¹ : K.HasHomology j
inst✝ : (K.sc' i j k).HasHomology
\u22a2 K.pOpcycles j ≫ ShortComplex.opcyclesMap ((natIsoSc' C c i j k hi hk).hom.app K) = (K.sc' i j k).pOpcycles
-- Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean
C : Type u_1
inst✝³ : Category.{u_3, u_1} C
inst✝² : HasZeroMorphisms C
ι : Type u_2
c : ComplexShape ι
K : HomologicalComplex C c
i j k : ι
hi : c.prev j = i
hk : c.next j = k
inst✝¹ : K.HasHomology j
inst✝ : (K.sc' i j k).HasHomology
\u22a2 ((natIsoSc' C c i j k hi hk).hom.app K).τ₂ ≫ (K.sc' i j k).pOpcycles = (K.sc' i j k).pOpcycles
-- Mathlib/GroupTheory/Perm/Cycle/Factors.lean
ι : Type u_1
α : Type u_2
β : Type u_3
f✝ g : Perm α
x y : α
inst✝² : DecidableRel f✝.SameCycle
inst✝¹ : DecidableRel g.SameCycle
f : Perm α
inst✝ : DecidableRel f.SameCycle
hx : f x ≠ x
\u22a2 (f.cycleOf x) x ≠ x
-- Mathlib/GroupTheory/Perm/Cycle/Factors.lean
ι : Type u_1
α : Type u_2
β : Type u_3
f✝ g : Perm α
x y✝ : α
inst✝² : DecidableRel f✝.SameCycle
inst✝¹ : DecidableRel g.SameCycle
f : Perm α
inst✝ : DecidableRel f.SameCycle
hx : f x ≠ x
this : (f.cycleOf x) x ≠ x
y : α
h : (f.cycleOf x) y ≠ y
hxy : f.SameCycle x y
i : ℤ
hi : (f ^ i) x = y
\u22a2 (f.cycleOf x ^ i) x = y
-- Mathlib/GroupTheory/Perm/Cycle/Factors.lean
ι : Type u_1
α : Type u_2
β : Type u_3
f✝ g : Perm α
x y✝ : α
inst✝² : DecidableRel f✝.SameCycle
inst✝¹ : DecidableRel g.SameCycle
f : Perm α
inst✝ : DecidableRel f.SameCycle
hx : f x ≠ x
this : (f.cycleOf x) x ≠ x
y : α
h : (f.cycleOf x) y ≠ y
hxy : ¬f.SameCycle x y
\u22a2 (f.cycleOf x).SameCycle x y
-- Mathlib/GroupTheory/Perm/Cycle/Factors.lean
ι : Type u_1
α : Type u_2
β : Type u_3
f✝ g : Perm α
x y✝ : α
inst✝² : DecidableRel f✝.SameCycle
inst✝¹ : DecidableRel g.SameCycle
f : Perm α
inst✝ : DecidableRel f.SameCycle
hx : f x ≠ x
this : (f.cycleOf x) x ≠ x
y : α
h : y ≠ y
hxy : ¬f.SameCycle x y
\u22a2 (f.cycleOf x).SameCycle x y
-- Mathlib/LinearAlgebra/Coevaluation.lean
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
\u22a2 LinearMap.rTensor (Module.Dual K V) (contractLeft K V) ∘ₗ ↑(TensorProduct.assoc K (Module.Dual K V) V (Module.Dual K V)).symm ∘ₗ LinearMap.lTensor (Module.Dual K V) (coevaluation K V) = ↑(TensorProduct.lid K (Module.Dual K V)).symm ∘ₗ ↑(TensorProduct.rid K (Module.Dual K V))
-- Mathlib/LinearAlgebra/Coevaluation.lean
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
\u22a2 (mk K (Module.Dual K V) K).compr₂ (LinearMap.rTensor (Module.Dual K V) (contractLeft K V) ∘ₗ ↑(TensorProduct.assoc K (Module.Dual K V) V (Module.Dual K V)).symm ∘ₗ LinearMap.lTensor (Module.Dual K V) (coevaluation K V)) = (mk K (Module.Dual K V) K).compr₂ (↑(TensorProduct.lid K (Module.Dual K V)).symm ∘ₗ ↑(TensorProduct.rid K (Module.Dual K V)))
-- Mathlib/LinearAlgebra/Coevaluation.lean
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
\u22a2 ∀ (i : ↑(Basis.ofVectorSpaceIndex K V)), ((mk K (Module.Dual K V) K).compr₂ (LinearMap.rTensor (Module.Dual K V) (contractLeft K V) ∘ₗ ↑(TensorProduct.assoc K (Module.Dual K V) V (Module.Dual K V)).symm ∘ₗ LinearMap.lTensor (Module.Dual K V) (coevaluation K V))) ((Basis.ofVectorSpace K V).dualBasis i) = ((mk K (Module.Dual K V) K).compr₂ (↑(TensorProduct.lid K (Module.Dual K V)).symm ∘ₗ ↑(TensorProduct.rid K (Module.Dual K V)))) ((Basis.ofVectorSpace K V).dualBasis i)
-- Mathlib/LinearAlgebra/Coevaluation.lean
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
j : ↑(Basis.ofVectorSpaceIndex K V)
\u22a2 ((mk K (Module.Dual K V) K).compr₂ (LinearMap.rTensor (Module.Dual K V) (contractLeft K V) ∘ₗ ↑(TensorProduct.assoc K (Module.Dual K V) V (Module.Dual K V)).symm ∘ₗ LinearMap.lTensor (Module.Dual K V) (coevaluation K V))) ((Basis.ofVectorSpace K V).dualBasis j) = ((mk K (Module.Dual K V) K).compr₂ (↑(TensorProduct.lid K (Module.Dual K V)).symm ∘ₗ ↑(TensorProduct.rid K (Module.Dual K V)))) ((Basis.ofVectorSpace K V).dualBasis j)
-- Mathlib/LinearAlgebra/Coevaluation.lean
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
j : ↑(Basis.ofVectorSpaceIndex K V)
\u22a2 (((mk K (Module.Dual K V) K).compr₂ (LinearMap.rTensor (Module.Dual K V) (contractLeft K V) ∘ₗ ↑(TensorProduct.assoc K (Module.Dual K V) V (Module.Dual K V)).symm ∘ₗ LinearMap.lTensor (Module.Dual K V) (coevaluation K V))) ((Basis.ofVectorSpace K V).dualBasis j)) 1 = (((mk K (Module.Dual K V) K).compr₂ (↑(TensorProduct.lid K (Module.Dual K V)).symm ∘ₗ ↑(TensorProduct.rid K (Module.Dual K V)))) ((Basis.ofVectorSpace K V).dualBasis j)) 1
-- Mathlib/LinearAlgebra/Coevaluation.lean
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
j : ↑(Basis.ofVectorSpaceIndex K V)
\u22a2 (LinearMap.rTensor (Module.Dual K V) (contractLeft K V) ∘ₗ ↑(TensorProduct.assoc K (Module.Dual K V) V (Module.Dual K V)).symm ∘ₗ LinearMap.lTensor (Module.Dual K V) (coevaluation K V)) ((Basis.ofVectorSpace K V).dualBasis j ⊗ₜ[K] 1) = (↑(TensorProduct.lid K (Module.Dual K V)).symm ∘ₗ ↑(TensorProduct.rid K (Module.Dual K V))) ((Basis.ofVectorSpace K V).dualBasis j ⊗ₜ[K] 1)
-- Mathlib/LinearAlgebra/Coevaluation.lean
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
j : ↑(Basis.ofVectorSpaceIndex K V)
\u22a2 (LinearMap.rTensor (Module.Dual K V) (contractLeft K V)) ((TensorProduct.assoc K (Module.Dual K V) V (Module.Dual K V)).symm ((LinearMap.lTensor (Module.Dual K V) (coevaluation K V)) ((Basis.ofVectorSpace K V).dualBasis j ⊗ₜ[K] 1))) = (TensorProduct.lid K (Module.Dual K V)).symm ((TensorProduct.rid K (Module.Dual K V)) ((Basis.ofVectorSpace K V).dualBasis j ⊗ₜ[K] 1))
-- Mathlib/LinearAlgebra/Coevaluation.lean
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
j : ↑(Basis.ofVectorSpaceIndex K V)
\u22a2 (LinearMap.rTensor (Module.Dual K V) (contractLeft K V)) ((TensorProduct.assoc K (Module.Dual K V) V (Module.Dual K V)).symm ((LinearMap.lTensor (Module.Dual K V) (coevaluation K V)) ((Basis.ofVectorSpace K V).dualBasis j ⊗ₜ[K] 1))) = 1 ⊗ₜ[K] (Basis.ofVectorSpace K V).dualBasis j
-- Mathlib/LinearAlgebra/Coevaluation.lean
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
j : ↑(Basis.ofVectorSpaceIndex K V)
\u22a2 (LinearMap.rTensor (Module.Dual K V) (contractLeft K V)) ((TensorProduct.assoc K (Module.Dual K V) V (Module.Dual K V)).symm ((Basis.ofVectorSpace K V).dualBasis j ⊗ₜ[K] ∑ i : ↑(Basis.ofVectorSpaceIndex K V), (Basis.ofVectorSpace K V) i ⊗ₜ[K] (Basis.ofVectorSpace K V).coord i)) = 1 ⊗ₜ[K] (Basis.ofVectorSpace K V).dualBasis j
-- Mathlib/LinearAlgebra/Coevaluation.lean
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
j : ↑(Basis.ofVectorSpaceIndex K V)
\u22a2 (LinearMap.rTensor (Module.Dual K V) (contractLeft K V)) (∑ x : ↑(Basis.ofVectorSpaceIndex K V), (TensorProduct.assoc K (Module.Dual K V) V (Module.Dual K V)).symm ((Basis.ofVectorSpace K V).dualBasis j ⊗ₜ[K] (Basis.ofVectorSpace K V) x ⊗ₜ[K] (Basis.ofVectorSpace K V).coord x)) = 1 ⊗ₜ[K] (Basis.ofVectorSpace K V).dualBasis j
-- Mathlib/LinearAlgebra/Coevaluation.lean
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
j : ↑(Basis.ofVectorSpaceIndex K V)
\u22a2 (LinearMap.rTensor (Module.Dual K V) (contractLeft K V)) (∑ x : ↑(Basis.ofVectorSpaceIndex K V), ((Basis.ofVectorSpace K V).dualBasis j ⊗ₜ[K] (Basis.ofVectorSpace K V) x) ⊗ₜ[K] (Basis.ofVectorSpace K V).coord x) = 1 ⊗ₜ[K] (Basis.ofVectorSpace K V).dualBasis j
-- Mathlib/LinearAlgebra/Coevaluation.lean
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
j : ↑(Basis.ofVectorSpaceIndex K V)
\u22a2 ∑ x : ↑(Basis.ofVectorSpaceIndex K V), (LinearMap.rTensor (Module.Dual K V) (contractLeft K V)) (((Basis.ofVectorSpace K V).dualBasis j ⊗ₜ[K] (Basis.ofVectorSpace K V) x) ⊗ₜ[K] (Basis.ofVectorSpace K V).coord x) = 1 ⊗ₜ[K] (Basis.ofVectorSpace K V).dualBasis j
-- Mathlib/LinearAlgebra/Coevaluation.lean
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
j : ↑(Basis.ofVectorSpaceIndex K V)
\u22a2 ∑ x : ↑(Basis.ofVectorSpaceIndex K V), ((Basis.ofVectorSpace K V).dualBasis j) ((Basis.ofVectorSpace K V) x) ⊗ₜ[K] (Basis.ofVectorSpace K V).coord x = 1 ⊗ₜ[K] (Basis.ofVectorSpace K V).dualBasis j
-- Mathlib/LinearAlgebra/Coevaluation.lean
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
j : ↑(Basis.ofVectorSpaceIndex K V)
\u22a2 (∑ x : ↑(Basis.ofVectorSpaceIndex K V), if x = j then 1 ⊗ₜ[K] (Basis.ofVectorSpace K V).coord x else 0) = 1 ⊗ₜ[K] (Basis.ofVectorSpace K V).coord j
-- Mathlib/LinearAlgebra/Coevaluation.lean
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
j : ↑(Basis.ofVectorSpaceIndex K V)
\u22a2 (if j ∈ Finset.univ then 1 ⊗ₜ[K] (Basis.ofVectorSpace K V).coord j else 0) = 1 ⊗ₜ[K] (Basis.ofVectorSpace K V).coord j
-- Mathlib/Combinatorics/Hindman.lean
M : Type u_1
inst✝ : CommMonoid M
a : Stream' M
s : Finset ℕ
hs : s.Nonempty
\u22a2 ∏ i ∈ s, a.get i ∈ FP a
-- Mathlib/Combinatorics/Hindman.lean
M : Type u_1
inst✝ : CommMonoid M
a : Stream' M
s : Finset ℕ
hs : s.Nonempty
\u22a2 ∏ i ∈ s, a.get i ∈ FP (Stream'.drop (s.min' hs) a)
-- Mathlib/Combinatorics/Hindman.lean
M : Type u_1
inst✝ : CommMonoid M
a : Stream' M
s : Finset ℕ
ih : ∀ t ⊂ s, ∀ (hs : t.Nonempty), ∏ i ∈ t, a.get i ∈ FP (Stream'.drop (t.min' hs) a)
hs : s.Nonempty
\u22a2 ∏ i ∈ s, a.get i ∈ FP (Stream'.drop (s.min' hs) a)
-- Mathlib/Combinatorics/Hindman.lean
M : Type u_1
inst✝ : CommMonoid M
a : Stream' M
s : Finset ℕ
ih : ∀ t ⊂ s, ∀ (hs : t.Nonempty), ∏ i ∈ t, a.get i ∈ FP (Stream'.drop (t.min' hs) a)
hs : s.Nonempty
\u22a2 (Stream'.drop (s.min' hs) a).head * ∏ x ∈ s.erase (s.min' hs), a.get x ∈ FP (Stream'.drop (s.min' hs) a)
-- Mathlib/Combinatorics/Hindman.lean
M : Type u_1
inst✝ : CommMonoid M
a : Stream' M
s : Finset ℕ
ih : ∀ t ⊂ s, ∀ (hs : t.Nonempty), ∏ i ∈ t, a.get i ∈ FP (Stream'.drop (t.min' hs) a)
hs : s.Nonempty
h : s.erase (s.min' hs) = ∅
\u22a2 (Stream'.drop (s.min' hs) a).head * ∏ x ∈ s.erase (s.min' hs), a.get x ∈ FP (Stream'.drop (s.min' hs) a)
-- Mathlib/Combinatorics/Hindman.lean
M : Type u_1
inst✝ : CommMonoid M
a : Stream' M
s : Finset ℕ
ih : ∀ t ⊂ s, ∀ (hs : t.Nonempty), ∏ i ∈ t, a.get i ∈ FP (Stream'.drop (t.min' hs) a)
hs : s.Nonempty
h : (s.erase (s.min' hs)).Nonempty
\u22a2 (Stream'.drop (s.min' hs) a).head * ∏ x ∈ s.erase (s.min' hs), a.get x ∈ FP (Stream'.drop (s.min' hs) a)
-- Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
a x✝ z x : ℂ
\u22a2 ↑x⁻¹.arg = -↑x.arg
-- Mathlib/Analysis/Normed/Group/Basic.lean
𝓕 : Type u_1
𝕜 : Type u_2
α : Type u_3
ι : Type u_4
κ : Type u_5
E : Type u_6
F : Type u_7
G : Type u_8
inst✝¹ : SeminormedCommGroup E
inst✝ : SeminormedCommGroup F
a✝ a₁ a₂ b✝ b₁ b₂ : E
r✝ r₁ r₂ : ℝ
a b : E
r : ℝ
c : E
\u22a2 c ∈ (fun x => b * x) ⁻¹' sphere a r ↔ c ∈ sphere (a / b) r
-- Mathlib/FieldTheory/SeparableDegree.lean
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
K : Type w
inst✝¹ : Field K
inst✝ : Algebra F K
f : F[X]
h : Splits (algebraMap F E) f
\u22a2 f.natSepDegree = (f.aroots E).toFinset.card
-- Mathlib/CategoryTheory/Abelian/LeftDerived.lean
C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{u_2, u_1} D
inst✝³ : Abelian C
inst✝² : HasProjectiveResolutions C
inst✝¹ : Abelian D
X Y : C
f : X ⟶ Y
P : ProjectiveResolution X
Q : ProjectiveResolution Y
φ : P.complex ⟶ Q.complex
comm : φ.f 0 ≫ Q.π.f 0 = P.π.f 0 ≫ f
F : C ⥤ D
inst✝ : F.Additive
n : ℕ
\u22a2 (HomotopyCategory.homologyFunctor D (ComplexShape.down ℕ) n).map (F.leftDerivedToHomotopyCategory.map f) ≫ (HomotopyCategory.homologyFunctor D (ComplexShape.down ℕ) n).map (Q.isoLeftDerivedToHomotopyCategoryObj F).hom ≫ (HomotopyCategory.homologyFunctorFactors D (ComplexShape.down ℕ) n).hom.app ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj Q.complex) = ((HomotopyCategory.homologyFunctor D (ComplexShape.down ℕ) n).map (P.isoLeftDerivedToHomotopyCategoryObj F).hom ≫ (HomotopyCategory.homologyFunctorFactors D (ComplexShape.down ℕ) n).hom.app ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj P.complex)) ≫ HomologicalComplex.homologyMap ((F.mapHomologicalComplex (ComplexShape.down ℕ)).map φ) n
-- Mathlib/CategoryTheory/Abelian/LeftDerived.lean
C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{u_2, u_1} D
inst✝³ : Abelian C
inst✝² : HasProjectiveResolutions C
inst✝¹ : Abelian D
X Y : C
f : X ⟶ Y
P : ProjectiveResolution X
Q : ProjectiveResolution Y
φ : P.complex ⟶ Q.complex
comm : φ.f 0 ≫ Q.π.f 0 = P.π.f 0 ≫ f
F : C ⥤ D
inst✝ : F.Additive
n : ℕ
\u22a2 (HomotopyCategory.homologyFunctor D (ComplexShape.down ℕ) n).map (P.isoLeftDerivedToHomotopyCategoryObj F).hom ≫ (HomotopyCategory.homologyFunctor D (ComplexShape.down ℕ) n).map ((F.mapHomologicalComplex (ComplexShape.down ℕ) ⋙ HomotopyCategory.quotient D (ComplexShape.down ℕ)).map φ) ≫ (HomotopyCategory.homologyFunctorFactors D (ComplexShape.down ℕ) n).hom.app ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj Q.complex) = (HomotopyCategory.homologyFunctor D (ComplexShape.down ℕ) n).map (P.isoLeftDerivedToHomotopyCategoryObj F).hom ≫ (HomotopyCategory.homologyFunctorFactors D (ComplexShape.down ℕ) n).hom.app ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj P.complex) ≫ HomologicalComplex.homologyMap ((F.mapHomologicalComplex (ComplexShape.down ℕ)).map φ) n
-- Mathlib/CategoryTheory/Abelian/LeftDerived.lean
C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{u_2, u_1} D
inst✝³ : Abelian C
inst✝² : HasProjectiveResolutions C
inst✝¹ : Abelian D
X Y : C
f : X ⟶ Y
P : ProjectiveResolution X
Q : ProjectiveResolution Y
φ : P.complex ⟶ Q.complex
comm : φ.f 0 ≫ Q.π.f 0 = P.π.f 0 ≫ f
F : C ⥤ D
inst✝ : F.Additive
n : ℕ
\u22a2 (HomotopyCategory.homologyFunctor D (ComplexShape.down ℕ) n).map (P.isoLeftDerivedToHomotopyCategoryObj F).hom ≫ (HomotopyCategory.homologyFunctorFactors D (ComplexShape.down ℕ) n).hom.app ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj P.complex) ≫ (HomologicalComplex.homologyFunctor D (ComplexShape.down ℕ) n).map ((F.mapHomologicalComplex (ComplexShape.down ℕ)).map φ) = (HomotopyCategory.homologyFunctor D (ComplexShape.down ℕ) n).map (P.isoLeftDerivedToHomotopyCategoryObj F).hom ≫ (HomotopyCategory.homologyFunctorFactors D (ComplexShape.down ℕ) n).hom.app ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj P.complex) ≫ HomologicalComplex.homologyMap ((F.mapHomologicalComplex (ComplexShape.down ℕ)).map φ) n
-- Mathlib/Analysis/InnerProductSpace/Basic.lean
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : _root_.RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x : F
r : ℝ
hx : x ≠ 0
hr : 0 < r
\u22a2 ⟪x, r • x⟫_ℝ / (‖x‖ * ‖r • x‖) = 1
-- Mathlib/Analysis/InnerProductSpace/Basic.lean
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : _root_.RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x : F
r : ℝ
hx : x ≠ 0
hr : 0 < r
\u22a2 r * (‖x‖ * ‖x‖) ≠ 0
-- Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
x✝ y z : ℝ
n : ℕ
x ε : ℝ
hx : 0 ≤ x
hε : 0 < ε
h : 0 < x
\u22a2 log x ≤ x ^ ε / ε
-- Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
x✝ y z : ℝ
n : ℕ
x ε : ℝ
hx : 0 ≤ x
hε : 0 < ε
h : 0 < x
\u22a2 ε * log x ≤ x ^ ε
-- Mathlib/Geometry/Manifold/Sheaf/Smooth.lean
𝕜 : Type u_1
inst✝²¹ : NontriviallyNormedField 𝕜
EM : Type u_2
inst✝²⁰ : NormedAddCommGroup EM
inst✝¹⁹ : NormedSpace 𝕜 EM
HM : Type u_3
inst✝¹⁸ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
E : Type u_4
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : NormedSpace 𝕜 E
H : Type u_5
inst✝¹⁵ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
H' : Type u_6
inst✝¹⁴ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E H'
M : Type u
inst✝¹³ : TopologicalSpace M
inst✝¹² : ChartedSpace HM M
N G A A' R : Type u
inst✝¹¹ : TopologicalSpace N
inst✝¹⁰ : ChartedSpace H N
inst✝⁹ : TopologicalSpace G
inst✝⁸ : ChartedSpace H G
inst✝⁷ : TopologicalSpace A
inst✝⁶ : ChartedSpace H A
inst✝⁵ : TopologicalSpace A'
inst✝⁴ : ChartedSpace H' A'
inst✝³ : TopologicalSpace R
inst✝² : ChartedSpace H R
inst✝¹ : CommRing R
inst✝ : SmoothRing I R
x : M
r : R
\u22a2 ∃ a, (eval IM I M R x) a = r
-- Mathlib/Geometry/Manifold/Sheaf/Smooth.lean
𝕜 : Type u_1
inst✝²¹ : NontriviallyNormedField 𝕜
EM : Type u_2
inst✝²⁰ : NormedAddCommGroup EM
inst✝¹⁹ : NormedSpace 𝕜 EM
HM : Type u_3
inst✝¹⁸ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
E : Type u_4
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : NormedSpace 𝕜 E
H : Type u_5
inst✝¹⁵ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
H' : Type u_6
inst✝¹⁴ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E H'
M : Type u
inst✝¹³ : TopologicalSpace M
inst✝¹² : ChartedSpace HM M
N G A A' R : Type u
inst✝¹¹ : TopologicalSpace N
inst✝¹⁰ : ChartedSpace H N
inst✝⁹ : TopologicalSpace G
inst✝⁸ : ChartedSpace H G
inst✝⁷ : TopologicalSpace A
inst✝⁶ : ChartedSpace H A
inst✝⁵ : TopologicalSpace A'
inst✝⁴ : ChartedSpace H' A'
inst✝³ : TopologicalSpace R
inst✝² : ChartedSpace H R
inst✝¹ : CommRing R
inst✝ : SmoothRing I R
x : M
y : (smoothSheaf IM I M R).presheaf.stalk x
\u22a2 ∃ a, (eval IM I M R x) a = smoothSheaf.eval IM I R x y
-- Mathlib/Geometry/Manifold/Sheaf/Smooth.lean
𝕜 : Type u_1
inst✝²¹ : NontriviallyNormedField 𝕜
EM : Type u_2
inst✝²⁰ : NormedAddCommGroup EM
inst✝¹⁹ : NormedSpace 𝕜 EM
HM : Type u_3
inst✝¹⁸ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
E : Type u_4
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : NormedSpace 𝕜 E
H : Type u_5
inst✝¹⁵ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
H' : Type u_6
inst✝¹⁴ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E H'
M : Type u
inst✝¹³ : TopologicalSpace M
inst✝¹² : ChartedSpace HM M
N G A A' R : Type u
inst✝¹¹ : TopologicalSpace N
inst✝¹⁰ : ChartedSpace H N
inst✝⁹ : TopologicalSpace G
inst✝⁸ : ChartedSpace H G
inst✝⁷ : TopologicalSpace A
inst✝⁶ : ChartedSpace H A
inst✝⁵ : TopologicalSpace A'
inst✝⁴ : ChartedSpace H' A'
inst✝³ : TopologicalSpace R
inst✝² : ChartedSpace H R
inst✝¹ : CommRing R
inst✝ : SmoothRing I R
x : M
y : (smoothSheaf IM I M R).presheaf.stalk x
\u22a2 (eval IM I M R x) ((forgetStalk IM I M R x).inv y) = smoothSheaf.eval IM I R x y