ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean
C : Type u
inst✝ : Category.{v, u} C
D : GlueData C
i j k : D.J
S : Set ↑↑(D.V (i, j))
eq₁ : (PreservesPullback.iso (forget C) (D.f i j) (D.f i k)).hom ≫ pullback.fst ((forget C).map (D.f i j)) ((forget C).map (D.f i k)) = (pullback.fst (D.f i j) (D.f i k)).base
eq₂ : (PreservesPullback.iso (forget C) (D.f i j) (D.f i k)).hom ≫ pullback.snd ((forget C).map (D.f i j)) ((forget C).map (D.f i k)) = (pullback.snd (D.f i j) (D.f i k)).base

⊢ ⇑(pullback.snd ((forget C).map (D.f i j)) ((forget C).map (D.f i k))) '' (⇑(PreservesPullback.iso (forget C) (D.f i j) (D.f i k)).hom '' (⇑((PreservesPullback.iso (forget C) (D.f i j) (D.f i k)).hom ≫ pullback.fst ((forget C).map (D.f i j)) ((forget C).map (D.f i k))) ⁻¹' S)) = ⇑(D.f i k).base ⁻¹' (⇑(D.f i j).base '' S)
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean
C : Type u
inst✝ : Category.{v, u} C
D : GlueData C
i j k : D.J
S : Set ↑↑(D.V (i, j))
eq₁ : (PreservesPullback.iso (forget C) (D.f i j) (D.f i k)).hom ≫ pullback.fst ((forget C).map (D.f i j)) ((forget C).map (D.f i k)) = (pullback.fst (D.f i j) (D.f i k)).base
eq₂ : (PreservesPullback.iso (forget C) (D.f i j) (D.f i k)).hom ≫ pullback.snd ((forget C).map (D.f i j)) ((forget C).map (D.f i k)) = (pullback.snd (D.f i j) (D.f i k)).base

⊢ ⇑(pullback.snd ((forget C).map (D.f i j)) ((forget C).map (D.f i k))) '' (⇑(PreservesPullback.iso (forget C) (D.f i j) (D.f i k)).hom '' (⇑(pullback.fst ((forget C).map (D.f i j)) ((forget C).map (D.f i k))) ∘ ⇑(PreservesPullback.iso (forget C) (D.f i j) (D.f i k)).hom ⁻¹' S)) = ⇑(D.f i k).base ⁻¹' (⇑(D.f i j).base '' S)
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean
C : Type u
inst✝ : Category.{v, u} C
D : GlueData C
i j k : D.J
S : Set ↑↑(D.V (i, j))
eq₁ : (PreservesPullback.iso (forget C) (D.f i j) (D.f i k)).hom ≫ pullback.fst ((forget C).map (D.f i j)) ((forget C).map (D.f i k)) = (pullback.fst (D.f i j) (D.f i k)).base
eq₂ : (PreservesPullback.iso (forget C) (D.f i j) (D.f i k)).hom ≫ pullback.snd ((forget C).map (D.f i j)) ((forget C).map (D.f i k)) = (pullback.snd (D.f i j) (D.f i k)).base

⊢ ⇑((forget C).map (D.f i k)) ⁻¹' (⇑((forget C).map (D.f i j)) '' S) = ⇑(D.f i k).base ⁻¹' (⇑(D.f i j).base '' S)
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean
C : Type u
inst✝ : Category.{v, u} C
D : GlueData C
i j k : D.J
S : Set ↑↑(D.V (i, j))
eq₁ : (PreservesPullback.iso (forget C) (D.f i j) (D.f i k)).hom ≫ pullback.fst ((forget C).map (D.f i j)) ((forget C).map (D.f i k)) = (pullback.fst (D.f i j) (D.f i k)).base
eq₂ : (PreservesPullback.iso (forget C) (D.f i j) (D.f i k)).hom ≫ pullback.snd ((forget C).map (D.f i j)) ((forget C).map (D.f i k)) = (pullback.snd (D.f i j) (D.f i k)).base

⊢ Function.Surjective ⇑(PreservesPullback.iso (forget C) (D.f i j) (D.f i k)).hom
-- Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean
C : Type u
inst✝ : Category.{v, u} C
D : GlueData C
i j k : D.J
S : Set ↑↑(D.V (i, j))
eq₁ : (PreservesPullback.iso (forget C) (D.f i j) (D.f i k)).hom ≫ pullback.fst ((forget C).map (D.f i j)) ((forget C).map (D.f i k)) = (pullback.fst (D.f i j) (D.f i k)).base
eq₂ : (PreservesPullback.iso (forget C) (D.f i j) (D.f i k)).hom ≫ pullback.snd ((forget C).map (D.f i j)) ((forget C).map (D.f i k)) = (pullback.snd (D.f i j) (D.f i k)).base

⊢ Epi (PreservesPullback.iso (forget C) (D.f i j) (D.f i k)).hom
-- Mathlib/Algebra/MvPolynomial/Variables.lean
R : Type u
S : Type v
σ : Type u_1
τ : Type u_2
r : R
e : ℕ
n m : σ
s : σ →₀ ℕ
inst✝² : CommSemiring R
p q : MvPolynomial σ R
inst✝¹ : CommSemiring S
f : R →+* S
inst✝ : DecidableEq σ
i : σ

⊢ i ∈ p.vars ↔ i ∈ p.support.biUnion Finsupp.support
-- Mathlib/Analysis/InnerProductSpace/l2Space.lean
ι : Type u_1
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
cplt : CompleteSpace E
G : ι → Type u_4
inst✝¹ : (i : ι) → NormedAddCommGroup (G i)
inst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
F : ι → Submodule 𝕜 E
hV : IsHilbertSum 𝕜 G V
W₀ : Π₀ (i : ι), G i

⊢ hV.linearIsometryEquiv.symm (W₀.sum (lp.single 2)) = W₀.sum fun i => ⇑(V i)
-- Mathlib/Algebra/Algebra/Spectrum.lean
F : Type u_1
R : Type u_2
A : Type u_3
B : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : Ring A
inst✝⁵ : Algebra R A
inst✝⁴ : Ring B
inst✝³ : Algebra R B
inst✝² : FunLike F A R
inst✝¹ : AlgHomClass F R A R
inst✝ : Nontrivial R
φ : F
a : A

⊢ φ a ∈ σ a
-- Mathlib/Algebra/Algebra/Spectrum.lean
F : Type u_1
R : Type u_2
A : Type u_3
B : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : Ring A
inst✝⁵ : Algebra R A
inst✝⁴ : Ring B
inst✝³ : Algebra R B
inst✝² : FunLike F A R
inst✝¹ : AlgHomClass F R A R
inst✝ : Nontrivial R
φ : F
a : A
h : ↑ₐ (φ a) - a ∈ RingHom.ker ↑φ

⊢ φ a ∈ σ a
-- Mathlib/Analysis/Normed/Lp/PiLp.lean
p : ℝ≥0∞
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
inst✝⁷ : Fact (1 ≤ p)
inst✝⁶ : Fintype ι
inst✝⁵ : Semiring 𝕜
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (α i)
inst✝³ : (i : ι) → SeminormedAddCommGroup (β i)
inst✝² : (i : ι) → Module 𝕜 (α i)
inst✝¹ : (i : ι) → Module 𝕜 (β i)
c : 𝕜
inst✝ : DecidableEq ι
i : ι
b₁ b₂ : β i

⊢ nndist ((WithLp.equiv p ((i : ι) → β i)).symm (Pi.single i b₁)) ((WithLp.equiv p ((i : ι) → β i)).symm (Pi.single i b₂)) = nndist b₁ b₂
-- Mathlib/Topology/Separation.lean
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : RegularSpace X
x✝ : X
s : Set X
B : Set (Set X)
hB : IsTopologicalBasis B
x : X

⊢ (𝓝 x).HasBasis (fun s => x ∈ s ∧ s ∈ B) closure
-- Mathlib/RingTheory/Valuation/ValuationRing.lean
A : Type u
inst✝⁵ : CommRing A
K : Type v
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsDomain A
inst✝¹ : ValuationRing A
inst✝ : IsFractionRing A K
x✝ : K

⊢ x✝ ∈ (valuation A K).integer ↔ x✝ ∈ (algebraMap A K).range
-- Mathlib/Topology/Order/ScottTopology.lean
α : Type u_1
β : Type u_2
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : IsScott α
s : Set α
a : α

⊢ IsOpen s ↔ IsUpperSet s ∧ DirSupInacc s
-- Mathlib/Topology/Order/ScottTopology.lean
α : Type u_1
β : Type u_2
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : IsScott α
s : Set α
a : α

⊢ IsUpperSet s ∧ IsOpen s ↔ IsUpperSet s ∧ DirSupInacc s
-- Mathlib/Topology/Order/ScottTopology.lean
α : Type u_1
β : Type u_2
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : IsScott α
s : Set α
a : α
h : IsUpperSet s
h' : DirSupInacc s
d : Set α
d₁ : d.Nonempty
d₂ : DirectedOn (fun x x_1 => x ≤ x_1) d
x✝ : α
d₃ : IsLUB d x✝
ha : x✝ ∈ s

⊢ ∃ b ∈ d, Ici b ∩ d ⊆ s
-- Mathlib/Topology/Order/ScottTopology.lean
α : Type u_1
β : Type u_2
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : IsScott α
s : Set α
a : α
h : IsUpperSet s
h' : DirSupInacc s
d : Set α
d₁ : d.Nonempty
d₂ : DirectedOn (fun x x_1 => x ≤ x_1) d
x✝ : α
d₃ : IsLUB d x✝
ha : x✝ ∈ s
b : α
hbd : b ∈ d
hbu : b ∈ s

⊢ ∃ b ∈ d, Ici b ∩ d ⊆ s
-- Mathlib/Data/Finset/Piecewise.lean
ι : Type u_1
π : ι → Sort u_2
s : Finset ι
f g : (i : ι) → π i
inst✝¹ : (j : ι) → Decidable (j ∈ s)
inst✝ : DecidableEq ι
i : ι
hi : i ∉ s
v : π i

⊢ update (s.piecewise f g) i v = s.piecewise f (update g i v)
-- Mathlib/Data/Finset/Piecewise.lean
ι : Type u_1
π : ι → Sort u_2
s : Finset ι
f g : (i : ι) → π i
inst✝¹ : (j : ι) → Decidable (j ∈ s)
inst✝ : DecidableEq ι
i : ι
hi : i ∉ s
v : π i

⊢ s.piecewise (update f i v) (update g i v) = s.piecewise f (update g i v)
-- Mathlib/Data/Finset/Piecewise.lean
ι : Type u_1
π : ι → Sort u_2
s : Finset ι
f g : (i : ι) → π i
inst✝¹ : (j : ι) → Decidable (j ∈ s)
inst✝ : DecidableEq ι
i : ι
hi : i ∉ s
v : π i
j : ι
hj : j ∈ s

⊢ j ≠ i
-- Mathlib/Order/BoundedOrder.lean
α✝ : Type u
β : Type v
γ : Type u_1
δ : Type u_2
α : Type u_3
hA : PartialOrder α
A : OrderBot α
hB : PartialOrder α
B : OrderBot α
H : ∀ (x y : α), x ≤ y ↔ x ≤ y

⊢ ⊥ = ⊥
-- Mathlib/Order/BoundedOrder.lean
α✝ : Type u
β : Type v
γ : Type u_1
δ : Type u_2
α : Type u_3
hA : PartialOrder α
A B : OrderBot α
H : ∀ (x y : α), x ≤ y ↔ x ≤ y

⊢ ⊥ = ⊥
-- Mathlib/Order/BoundedOrder.lean
α✝ : Type u
β : Type v
γ : Type u_1
δ : Type u_2
α : Type u_3
hA : PartialOrder α
A B : OrderBot α
H : ∀ (x y : α), x ≤ y ↔ x ≤ y

⊢ ⊥ ≤ ⊥
-- Mathlib/Algebra/Star/SelfAdjoint.lean
R : Type u_1
A : Type u_2
inst✝¹ : AddGroup R
inst✝ : StarAddMonoid R
x y : R
hx : IsSelfAdjoint x
hy : IsSelfAdjoint y

⊢ IsSelfAdjoint (x - y)
-- Mathlib/Data/ZMod/Basic.lean
m n✝ n : ℕ
inst✝ : NeZero n
a : ZMod n

⊢ ↑a.val = a.valMinAbs + ↑(if a.val ≤ n / 2 then 0 else n)
-- Mathlib/Data/ZMod/Basic.lean
m n✝ n : ℕ
inst✝ : NeZero n
a : ZMod n

⊢ ↑a.val = (if a.val ≤ n / 2 then ↑a.val else ↑a.val - ↑n) + ↑(if a.val ≤ n / 2 then 0 else n)
-- Mathlib/AlgebraicTopology/SimplicialObject.lean
C : Type u
inst✝ : Category.{v, u} C
X : CosimplicialObject C
n : ℕ
i : Fin (n + 2)
j : Fin (n + 3)
H : i.castSucc < j
hj : j = 0

⊢ False
-- Mathlib/AlgebraicTopology/SimplicialObject.lean
C : Type u
inst✝ : Category.{v, u} C
X : CosimplicialObject C
n : ℕ
i : Fin (n + 2)
j : Fin (n + 3)
H : i.castSucc < j

⊢ X.map (SimplexCategory.δ i) ≫ X.map (SimplexCategory.δ j) = X.map (SimplexCategory.δ (j.pred ⋯)) ≫ X.map (SimplexCategory.δ i.castSucc)
-- Mathlib/GroupTheory/Complement.lean
G : Type u_1
inst✝ : Group G
H K : Subgroup G
S T : Set G
hST : IsComplement S T
hHT : IsComplement (↑H) T
hSK : IsComplement S ↑K
g : G
h1 : 1 ∈ S
hg : g ∈ T

⊢ ↑(hST.equiv g).1 = ↑⟨1, h1⟩
-- Mathlib/Algebra/DirectSum/Module.lean
R : Type u
inst✝² : Semiring R
ι : Type v
dec_ι : DecidableEq ι
M : Type u_1
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
A : ι → Submodule R M
h : IsInternal A
i : ι
x : ↥(A i)

⊢ ((LinearEquiv.ofBijective (coeLinearMap A) h).symm ↑x) i = x
-- Mathlib/CategoryTheory/Limits/Types.lean
J : Type v
inst✝¹ : Category.{w, v} J
F✝ : J ⥤ Type u
inst✝ : HasColimit F✝
α β : Type u
f : α ⟶ β
F : ℕᵒᵖ ⥤ Type u
c : Cone F
hc : IsLimit c
hF : ∀ (n : ℕ), Function.Surjective (F.map (homOfLE ⋯).op)
a : F.obj (Opposite.op 0)
n m : ℕ
h : m ≤ n

⊢ F.map (Opposite.op { down := { down := h } }) ((fun x => match x with | Opposite.op n => CategoryTheory.Limits.Types.limitOfSurjectionsSurjective.preimage hF a n) (Opposite.op n)) = (fun x => match x with | Opposite.op n => CategoryTheory.Limits.Types.limitOfSurjectionsSurjective.preimage hF a n) (Opposite.op m)
-- Mathlib/Data/List/EditDistance/Defs.lean
α : Type u_1
β : Type u_2
δ : Type u_3
inst✝¹ : AddZeroClass δ
inst✝ : Min δ
C : Cost α β δ
x : α
xs : List α
y : β
ys : List β
w : 0 < (suffixLevenshtein C (x :: xs) (y :: ys)).val.length

⊢ (suffixLevenshtein C (x :: xs) (y :: ys)).val[0] = match suffixLevenshtein C xs (y :: ys) with | ⟨dx, property⟩ => match suffixLevenshtein C (x :: xs) ys with | ⟨dy, property_1⟩ => match suffixLevenshtein C xs ys with | ⟨dxy, property_2⟩ => min (C.delete x + dx[0]) (min (C.insert y + dy[0]) (C.substitute x y + dxy[0]))
-- Mathlib/Data/List/EditDistance/Defs.lean
α : Type u_1
β : Type u_2
δ : Type u_3
inst✝¹ : AddZeroClass δ
inst✝ : Min δ
C : Cost α β δ
x : α
xs : List α
y : β
ys : List β
w : 0 < (suffixLevenshtein C (x :: xs) (y :: ys)).val.length

⊢ (impl C (x :: xs) y (suffixLevenshtein C (x :: xs) ys)).val[0] = match suffixLevenshtein C xs (y :: ys) with | ⟨dx, property⟩ => match suffixLevenshtein C (x :: xs) ys with | ⟨dy, property_1⟩ => match suffixLevenshtein C xs ys with | ⟨dxy, property_2⟩ => min (C.delete x + dx[0]) (min (C.insert y + dy[0]) (C.substitute x y + dxy[0]))
-- Mathlib/Data/List/EditDistance/Defs.lean
α : Type u_1
β : Type u_2
δ : Type u_3
inst✝¹ : AddZeroClass δ
inst✝ : Min δ
C : Cost α β δ
x : α
xs : List α
y : β
ys : List β
w : 0 < (suffixLevenshtein C (x :: xs) (y :: ys)).val.length

⊢ (impl C (x :: xs) y ⟨levenshtein C (x :: xs) ys :: (suffixLevenshtein C xs ys).val, ⋯⟩).val[0] = min (C.delete x + (suffixLevenshtein C xs (y :: ys)).val[0]) (min (C.insert y + (levenshtein C (x :: xs) ys :: (suffixLevenshtein C xs ys).val)[0]) (C.substitute x y + (suffixLevenshtein C xs ys).val[0]))
-- Mathlib/Data/List/EditDistance/Defs.lean
α : Type u_1
β : Type u_2
δ : Type u_3
inst✝¹ : AddZeroClass δ
inst✝ : Min δ
C : Cost α β δ
x : α
xs : List α
y : β
ys : List β
w : 0 < (suffixLevenshtein C (x :: xs) (y :: ys)).val.length

⊢ (match impl C xs y ⟨(suffixLevenshtein C xs ys).val, ?w'⟩ with | ⟨r, w⟩ => min (C.delete x + r[0]) (min (C.insert y + levenshtein C (x :: xs) ys) (C.substitute x y + (suffixLevenshtein C xs ys).val[0]))) = min (C.delete x + (suffixLevenshtein C xs (y :: ys)).val[0]) (min (C.insert y + (levenshtein C (x :: xs) ys :: (suffixLevenshtein C xs ys).val)[0]) (C.substitute x y + (suffixLevenshtein C xs ys).val[0]))
-- Mathlib/Data/List/EditDistance/Defs.lean
α : Type u_1
β : Type u_2
δ : Type u_3
inst✝¹ : AddZeroClass δ
inst✝ : Min δ
C : Cost α β δ
x : α
xs : List α
y : β
ys : List β
w : 0 < (suffixLevenshtein C (x :: xs) (y :: ys)).val.length

⊢ 0 < (suffixLevenshtein C xs ys).val.length
-- Mathlib/Order/Interval/Finset/Basic.lean
ι : Type u_1
α : Type u_2
inst✝¹ : Preorder α
inst✝ : LocallyFiniteOrderBot α
a : α

⊢ Iio a ⊆ Iic a
-- Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R₀ : Cᵒᵖ ⥤ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.val
inst✝³ : Presheaf.IsLocallyInjective J α
inst✝² : Presheaf.IsLocallySurjective J α
M₀ : PresheafOfModules R₀
A : Sheaf J AddCommGrp
φ : M₀.presheaf ⟶ A.val
inst✝¹ : Presheaf.IsLocallyInjective J φ
inst✝ : Presheaf.IsLocallySurjective J φ
X Y : Cᵒᵖ
π : X ⟶ Y
r r' : ↑(R.val.obj X)
m m' : ↑(A.val.obj X)
S : Sieve (Opposite.unop Y) := Presheaf.imageSieve α ((R.val.map π) r) ⊓ Presheaf.imageSieve φ ((A.val.map π) m)

⊢ (A.val.map π) (smul α φ r m) = smul α φ ((R.val.map π) r) ((A.val.map π) m)
-- Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R₀ : Cᵒᵖ ⥤ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.val
inst✝³ : Presheaf.IsLocallyInjective J α
inst✝² : Presheaf.IsLocallySurjective J α
M₀ : PresheafOfModules R₀
A : Sheaf J AddCommGrp
φ : M₀.presheaf ⟶ A.val
inst✝¹ : Presheaf.IsLocallyInjective J φ
inst✝ : Presheaf.IsLocallySurjective J φ
X Y : Cᵒᵖ
π : X ⟶ Y
r r' : ↑(R.val.obj X)
m m' : ↑(A.val.obj X)
S : Sieve (Opposite.unop Y) := Presheaf.imageSieve α ((R.val.map π) r) ⊓ Presheaf.imageSieve φ ((A.val.map π) m)
hS : S ∈ J.sieves (Opposite.unop Y)

⊢ (A.val.map π) (smul α φ r m) = smul α φ ((R.val.map π) r) ((A.val.map π) m)
-- Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R₀ : Cᵒᵖ ⥤ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.val
inst✝³ : Presheaf.IsLocallyInjective J α
inst✝² : Presheaf.IsLocallySurjective J α
M₀ : PresheafOfModules R₀
A : Sheaf J AddCommGrp
φ : M₀.presheaf ⟶ A.val
inst✝¹ : Presheaf.IsLocallyInjective J φ
inst✝ : Presheaf.IsLocallySurjective J φ
X Y : Cᵒᵖ
π : X ⟶ Y
r r' : ↑(R.val.obj X)
m m' : ↑(A.val.obj X)
S : Sieve (Opposite.unop Y) := Presheaf.imageSieve α ((R.val.map π) r) ⊓ Presheaf.imageSieve φ ((A.val.map π) m)
hS : S ∈ J.sieves (Opposite.unop Y)

⊢ ∀ (Y_1 : C) (f : Y_1 ⟶ Opposite.unop Y), S.arrows f → (A.val.map f.op) ((A.val.map π) (smul α φ r m)) = (A.val.map f.op) (smul α φ ((R.val.map π) r) ((A.val.map π) m))
-- Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R₀ : Cᵒᵖ ⥤ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.val
inst✝³ : Presheaf.IsLocallyInjective J α
inst✝² : Presheaf.IsLocallySurjective J α
M₀ : PresheafOfModules R₀
A : Sheaf J AddCommGrp
φ : M₀.presheaf ⟶ A.val
inst✝¹ : Presheaf.IsLocallyInjective J φ
inst✝ : Presheaf.IsLocallySurjective J φ
X Y✝ : Cᵒᵖ
π : X ⟶ Y✝
r r' : ↑(R.val.obj X)
m m' : ↑(A.val.obj X)
S : Sieve (Opposite.unop Y✝) := Presheaf.imageSieve α ((R.val.map π) r) ⊓ Presheaf.imageSieve φ ((A.val.map π) m)
hS : S ∈ J.sieves (Opposite.unop Y✝)
Y : C
f : Y ⟶ Opposite.unop Y✝
r₀ : (forget RingCat).obj (R₀.obj (Opposite.op Y))
hr₀ : (α.app (Opposite.op Y)) r₀ = (R.val.map f.op) ((R.val.map π) r)
m₀ : (forget AddCommGrp).obj (M₀.presheaf.obj (Opposite.op Y))
hm₀ : (φ.app (Opposite.op Y)) m₀ = (A.val.map f.op) ((A.val.map π) m)

⊢ (A.val.map f.op) ((A.val.map π) (smul α φ r m)) = (A.val.map f.op) (smul α φ ((R.val.map π) r) ((A.val.map π) m))
-- Mathlib/RingTheory/Polynomial/Hermite/Gaussian.lean
n : ℕ
x : ℝ

⊢ deriv^[n] (fun y => Real.exp (-(y ^ 2 / 2))) x = (-1) ^ n * (aeval x) (hermite n) * Real.exp (-(x ^ 2 / 2))
-- Mathlib/RingTheory/Polynomial/Hermite/Gaussian.lean
n : ℕ
x : ℝ

⊢ deriv^[n] (fun y => Real.exp (-(y ^ 2 / 2))) x = (-1) ^ n * ((aeval x) (hermite n) * Real.exp (-(x ^ 2 / 2)))
-- Mathlib/ModelTheory/Satisfiability.lean
L : Language
T : L.Theory
α : Type w
n : ℕ
φ ψ θ : L.BoundedFormula α n
h1 : T.SemanticallyEquivalent φ ψ
h2 : T.SemanticallyEquivalent ψ θ
M : T.ModelType
v : α → ↑M
xs : Fin n → ↑M
h1' : (φ ⇔ ψ).Realize v xs
h2' : (ψ ⇔ θ).Realize v xs

⊢ (φ ⇔ θ).Realize v xs
-- Mathlib/ModelTheory/Satisfiability.lean
L : Language
T : L.Theory
α : Type w
n : ℕ
φ ψ θ : L.BoundedFormula α n
h1 : T.SemanticallyEquivalent φ ψ
h2 : T.SemanticallyEquivalent ψ θ
M : T.ModelType
v : α → ↑M
xs : Fin n → ↑M
h1' : φ.Realize v xs ↔ ψ.Realize v xs
h2' : ψ.Realize v xs ↔ θ.Realize v xs

⊢ φ.Realize v xs ↔ θ.Realize v xs
-- Mathlib/Dynamics/PeriodicPts.lean
α : Type v
G : Type u
inst✝³ : Group G
inst✝² : MulAction G α
M : Type u
inst✝¹ : Monoid M
inst✝ : MulAction M α
n : ℕ
m : M
a : α

⊢ m ^ (n % period m a) • a = m ^ n • a
-- Mathlib/Analysis/Calculus/ParametricIntegral.lean
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' : H → α → H →L[𝕜] E
ε_pos : 0 < ε
hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) μ
hF_int : Integrable (F x₀) μ
hF'_meas : AEStronglyMeasurable (F' x₀) μ
h_bound : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F' x a‖ ≤ bound a
bound_integrable : Integrable bound μ
h_diff : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, HasFDerivAt (fun x => F x a) (F' x a) x
this : NormedSpace ℝ H := NormedSpace.restrictScalars ℝ 𝕜 H

⊢ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' x₀ a ∂μ) x₀
-- Mathlib/Analysis/Calculus/ParametricIntegral.lean
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' : H → α → H →L[𝕜] E
ε_pos : 0 < ε
hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) μ
hF_int : Integrable (F x₀) μ
hF'_meas : AEStronglyMeasurable (F' x₀) μ
h_bound : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F' x a‖ ≤ bound a
bound_integrable : Integrable bound μ
h_diff : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, HasFDerivAt (fun x => F x a) (F' x a) x
this : NormedSpace ℝ H := NormedSpace.restrictScalars ℝ 𝕜 H
x₀_in : x₀ ∈ ball x₀ ε

⊢ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' x₀ a ∂μ) x₀
-- Mathlib/Analysis/Calculus/ParametricIntegral.lean
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' : H → α → H →L[𝕜] E
ε_pos : 0 < ε
hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) μ
hF_int : Integrable (F x₀) μ
hF'_meas : AEStronglyMeasurable (F' x₀) μ
h_bound : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F' x a‖ ≤ bound a
bound_integrable : Integrable bound μ
h_diff : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, HasFDerivAt (fun x => F x a) (F' x a) x
this✝ : NormedSpace ℝ H := NormedSpace.restrictScalars ℝ 𝕜 H
x₀_in : x₀ ∈ ball x₀ ε
diff_x₀ : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' x₀ a) x₀
this : ∀ᵐ (a : α) ∂μ, LipschitzOnWith (Real.nnabs (bound a)) (fun x => F x a) (ball x₀ ε)

⊢ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' x₀ a ∂μ) x₀
-- Mathlib/Data/List/Permutation.lean
α : Type u_1
β : Type u_2
t : α
ts ys l : List α

⊢ l ∈ (permutationsAux2 t ts [] ys fun x => [] ++ x).2 ↔ ∃ l₁ l₂, l₂ ≠ [] ∧ ys = l₁ ++ l₂ ∧ l = l₁ ++ t :: l₂ ++ ts
-- Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
α : Type u
a : α
inst✝² : Group α
inst✝¹ : IsCyclic α
inst✝ : Fintype α
d : ℕ
hd : d ∣ Fintype.card α

⊢ (filter (fun a => orderOf a = d) univ).card = φ d
-- Mathlib/Algebra/Polynomial/Basic.lean
R : Type u
a b : R
m n✝ : ℕ
inst✝ : Semiring R
p q : R[X]
k n : ℕ
r : R

⊢ X ^ k * (monomial n) r = (monomial (n + k)) r
-- Mathlib/Analysis/InnerProductSpace/Projection.lean
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : _root_.RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : InnerProductSpace ℝ F
K : Submodule 𝕜 E
x y : E
hK : Dense ↑K
h : ∀ (v : ↥K), ⟪↑v, x⟫_𝕜 = 0
v : ↥K

⊢ ⟪↑v, x⟫_𝕜 = ⟪↑v, 0⟫_𝕜
-- Mathlib/Combinatorics/SetFamily/FourFunctions.lean
α : Type u_1
β : Type u_2
inst✝³ : DistribLattice α
inst✝² : LinearOrderedCommSemiring β
inst✝¹ : ExistsAddOfLE β
f f₁ f₂ f₃ f₄ g μ : α → β
inst✝ : DecidableEq α
s t : Finset α

⊢ s.card * t.card ≤ (s ⊼ t).card * (s ⊻ t).card
-- Mathlib/Algebra/Lie/TraceForm.lean
R : Type u_1
K : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : LieRingModule L M
inst✝² : LieModule R L M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
x y z : L

⊢ ((traceForm R L M) ⁅x, y⁆) z = -((traceForm R L M) y) ⁅x, z⁆
-- Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean
C : Type u
inst✝ : Category.{v, u} C
I : MulticospanIndex C
K : Multifork I
b : I.R

⊢ K.π.app (WalkingMulticospan.right b) = K.ι (I.fstTo b) ≫ I.fst b
-- Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean
C : Type u
inst✝ : Category.{v, u} C
I : MulticospanIndex C
K : Multifork I
b : I.R

⊢ K.π.app (WalkingMulticospan.left (I.fstTo b)) ≫ I.multicospan.map (WalkingMulticospan.Hom.fst b) = K.ι (I.fstTo b) ≫ I.fst b
-- Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean
C : Type u_1
inst✝¹ : Category.{?u.121497, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
h' : Homotopy φ₃ φ₄

⊢ (φ₁ + φ₃).τ₁ = S₁.f ≫ (h.h₁ + h'.h₁) + (h.h₀ + h'.h₀) + (φ₂ + φ₄).τ₁
-- Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean
C : Type u_1
inst✝¹ : Category.{?u.121497, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
h' : Homotopy φ₃ φ₄

⊢ S₁.f ≫ h.h₁ + h.h₀ + φ₂.τ₁ + (S₁.f ≫ h'.h₁ + h'.h₀ + φ₄.τ₁) = S₁.f ≫ h.h₁ + S₁.f ≫ h'.h₁ + (h.h₀ + h'.h₀) + (φ₂.τ₁ + φ₄.τ₁)
-- Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean
C : Type u_1
inst✝¹ : Category.{?u.121497, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
h' : Homotopy φ₃ φ₄

⊢ (φ₁ + φ₃).τ₂ = S₁.g ≫ (h.h₂ + h'.h₂) + (h.h₁ + h'.h₁) ≫ S₂.f + (φ₂ + φ₄).τ₂
-- Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean
C : Type u_1
inst✝¹ : Category.{?u.121497, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
h' : Homotopy φ₃ φ₄

⊢ S₁.g ≫ h.h₂ + h.h₁ ≫ S₂.f + φ₂.τ₂ + (S₁.g ≫ h'.h₂ + h'.h₁ ≫ S₂.f + φ₄.τ₂) = S₁.g ≫ h.h₂ + S₁.g ≫ h'.h₂ + (h.h₁ ≫ S₂.f + h'.h₁ ≫ S₂.f) + (φ₂.τ₂ + φ₄.τ₂)
-- Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean
C : Type u_1
inst✝¹ : Category.{?u.121497, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
h' : Homotopy φ₃ φ₄

⊢ (φ₁ + φ₃).τ₃ = h.h₃ + h'.h₃ + (h.h₂ + h'.h₂) ≫ S₂.g + (φ₂ + φ₄).τ₃
-- Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean
C : Type u_1
inst✝¹ : Category.{?u.121497, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
h' : Homotopy φ₃ φ₄

⊢ h.h₃ + h.h₂ ≫ S₂.g + φ₂.τ₃ + (h'.h₃ + h'.h₂ ≫ S₂.g + φ₄.τ₃) = h.h₃ + h'.h₃ + (h.h₂ ≫ S₂.g + h'.h₂ ≫ S₂.g) + (φ₂.τ₃ + φ₄.τ₃)
-- Mathlib/RingTheory/Perfection.lean
R : Type u₁
inst✝¹ : CommSemiring R
p : ℕ
hp : Fact (Nat.Prime p)
inst✝ : CharP R p
f : Ring.Perfection R p
n : ℕ

⊢ (coeff R p (n + 1)) ((frobenius (Ring.Perfection R p) p) f) = (coeff R p n) f