ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean
X : LocallyRingedSpace
r : ↑(Γ.obj (op X))
x : ↑X.toTopCat
\u22a2 (toOpen (↑(Γ.obj (op X))) (basicOpen 1) ≫ (structureSheaf ↑(Γ.obj (op X))).presheaf.germ ⟨X.toΓSpecFun x, ⋯⟩ ≫ (stalkFunctor CommRingCat (X.toΓSpecSheafedSpace.base x)).map X.toΓSpecSheafedSpace.c) ≫ stalkPushforward CommRingCat X.toΓSpecSheafedSpace.base X.presheaf x = X.presheaf.Γgerm x
-- Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean
X : LocallyRingedSpace
r : ↑(Γ.obj (op X))
x : ↑X.toTopCat
\u22a2 (toOpen (↑(Γ.obj (op X))) (basicOpen 1) ≫ X.toΓSpecSheafedSpace.c.app (op (basicOpen 1)) ≫ ((pushforward CommRingCat X.toΓSpecSheafedSpace.base).obj X.presheaf).germ ⟨X.toΓSpecFun x, ⋯⟩) ≫ stalkPushforward CommRingCat X.toΓSpecSheafedSpace.base X.presheaf x = X.presheaf.Γgerm x
-- Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean
X : LocallyRingedSpace
r : ↑(Γ.obj (op X))
x : ↑X.toTopCat
\u22a2 (X.toToΓSpecMapBasicOpen 1 ≫ ((pushforward CommRingCat X.toΓSpecSheafedSpace.base).obj X.presheaf).germ ⟨X.toΓSpecFun x, ⋯⟩) ≫ stalkPushforward CommRingCat X.toΓSpecSheafedSpace.base X.presheaf x = X.presheaf.germ ⟨x, ⋯⟩
-- Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean
X : LocallyRingedSpace
r : ↑(Γ.obj (op X))
x : ↑X.toTopCat
\u22a2 (X.toToΓSpecMapBasicOpen 1 ≫ ((pushforward CommRingCat X.toΓSpecSheafedSpace.base).obj X.presheaf).germ ⟨X.toΓSpecFun x, ⋯⟩) ≫ stalkPushforward CommRingCat X.toΓSpecSheafedSpace.base X.presheaf x = ((pushforward CommRingCat X.toΓSpecBase).obj X.presheaf).germ ⟨X.toΓSpecBase ↑⟨x, ⋯⟩, ⋯⟩ ≫ stalkPushforward CommRingCat X.toΓSpecBase X.presheaf ↑⟨x, ⋯⟩
-- Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean
X : LocallyRingedSpace
r : ↑(Γ.obj (op X))
x : ↑X.toTopCat
\u22a2 X.toToΓSpecMapBasicOpen 1 ≫ ((pushforward CommRingCat X.toΓSpecSheafedSpace.base).obj X.presheaf).germ ⟨X.toΓSpecFun x, ⋯⟩ = ((pushforward CommRingCat X.toΓSpecBase).obj X.presheaf).germ ⟨X.toΓSpecBase ↑⟨x, ⋯⟩, ⋯⟩
-- Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean
X : LocallyRingedSpace
r : ↑(Γ.obj (op X))
x : ↑X.toTopCat
\u22a2 ((pushforward CommRingCat X.toΓSpecBase).obj X.presheaf).map ⋯.hom.op ≫ ((pushforward CommRingCat X.toΓSpecSheafedSpace.base).obj X.presheaf).germ ⟨X.toΓSpecFun x, ⋯⟩ = ((pushforward CommRingCat X.toΓSpecBase).obj X.presheaf).germ ⟨X.toΓSpecBase ↑⟨x, ⋯⟩, ⋯⟩
-- Mathlib/Algebra/Module/LinearMap/Polynomial.lean
R : Type u_1
M₁ : Type u_2
M₂ : Type u_3
M₃ : Type u_4
ι₁ : Type u_5
ι₂ : Type u_6
ι₃ : Type u_7
inst✝¹¹ : CommRing R
inst✝¹⁰ : AddCommGroup M₁
inst✝⁹ : AddCommGroup M₂
inst✝⁸ : AddCommGroup M₃
inst✝⁷ : Module R M₁
inst✝⁶ : Module R M₂
inst✝⁵ : Module R M₃
inst✝⁴ : Fintype ι₁
inst✝³ : Fintype ι₂
inst✝² : Finite ι₃
inst✝¹ : DecidableEq ι₁
inst✝ : DecidableEq ι₂
b₁ : Basis ι₁ R M₁
b₂ : Basis ι₂ R M₂
b₃ : Basis ι₃ R M₃
g : M₂ →ₗ[R] M₃
f : M₁ →ₗ[R] M₂
i : ι₃
\u22a2 toMvPolynomial b₁ b₃ (g ∘ₗ f) i = (bind₁ (toMvPolynomial b₁ b₂ f)) (toMvPolynomial b₂ b₃ g i)
-- Mathlib/Algebra/Module/LinearMap/Polynomial.lean
R : Type u_1
M₁ : Type u_2
M₂ : Type u_3
M₃ : Type u_4
ι₁ : Type u_5
ι₂ : Type u_6
ι₃ : Type u_7
inst✝¹¹ : CommRing R
inst✝¹⁰ : AddCommGroup M₁
inst✝⁹ : AddCommGroup M₂
inst✝⁸ : AddCommGroup M₃
inst✝⁷ : Module R M₁
inst✝⁶ : Module R M₂
inst✝⁵ : Module R M₃
inst✝⁴ : Fintype ι₁
inst✝³ : Fintype ι₂
inst✝² : Finite ι₃
inst✝¹ : DecidableEq ι₁
inst✝ : DecidableEq ι₂
b₁ : Basis ι₁ R M₁
b₂ : Basis ι₂ R M₂
b₃ : Basis ι₃ R M₃
g : M₂ →ₗ[R] M₃
f : M₁ →ₗ[R] M₂
i : ι₃
\u22a2 (bind₁ ((toMatrix b₁ b₂) f).toMvPolynomial) (((toMatrix b₂ b₃) g).toMvPolynomial i) = (bind₁ (toMvPolynomial b₁ b₂ f)) (((toMatrix b₂ b₃) g).toMvPolynomial i)
-- Mathlib/CategoryTheory/Sites/SheafHom.lean
C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝ : Category.{v', u'} A
F G : Cᵒᵖ ⥤ A
X : C
φ : { obj := fun X => (Over.forget (unop X)).op ⋙ F ⟶ (Over.forget (unop X)).op ⋙ G, map := fun {X Y} f => whiskerLeft (Over.map f.unop).op }.obj (op X)
Y : Over (unop (op X))
\u22a2 ({ obj := fun X => (Over.forget (unop X)).op ⋙ F ⟶ (Over.forget (unop X)).op ⋙ G, map := fun {X Y} f => whiskerLeft (Over.map f.unop).op }.map (𝟙 (op X)) φ).app (op Y) = (𝟙 ({ obj := fun X => (Over.forget (unop X)).op ⋙ F ⟶ (Over.forget (unop X)).op ⋙ G, map := fun {X Y} f => whiskerLeft (Over.map f.unop).op }.obj (op X)) φ).app (op Y)
-- Mathlib/CategoryTheory/Sites/SheafHom.lean
C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝ : Category.{v', u'} A
F G : Cᵒᵖ ⥤ A
X Y Z : C
f : Y ⟶ X
g : Z ⟶ Y
φ : { obj := fun X => (Over.forget (unop X)).op ⋙ F ⟶ (Over.forget (unop X)).op ⋙ G, map := fun {X Y} f => whiskerLeft (Over.map f.unop).op }.obj (op X)
W : Over (unop (op Z))
\u22a2 ({ obj := fun X => (Over.forget (unop X)).op ⋙ F ⟶ (Over.forget (unop X)).op ⋙ G, map := fun {X Y} f => whiskerLeft (Over.map f.unop).op }.map (op f ≫ op g) φ).app (op W) = (({ obj := fun X => (Over.forget (unop X)).op ⋙ F ⟶ (Over.forget (unop X)).op ⋙ G, map := fun {X Y} f => whiskerLeft (Over.map f.unop).op }.map (op f) ≫ { obj := fun X => (Over.forget (unop X)).op ⋙ F ⟶ (Over.forget (unop X)).op ⋙ G, map := fun {X Y} f => whiskerLeft (Over.map f.unop).op }.map (op g)) φ).app (op W)
-- Mathlib/Algebra/Polynomial/Laurent.lean
R : Type u_1
inst✝¹ : Semiring R
inst✝ : Nontrivial R
n : ℤ
\u22a2 (T n).degree = ↑n
-- Mathlib/Algebra/Polynomial/Laurent.lean
R : Type u_1
inst✝¹ : Semiring R
inst✝ : Nontrivial R
n : ℤ
\u22a2 (C 1 * T n).degree = ↑n
-- Mathlib/Data/Finset/Pointwise.lean
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝³ : DecidableEq β
inst✝² : GroupWithZero α
inst✝¹ : MulAction α β
s✝ t : Finset β
a✝ : α
b : β
inst✝ : DecidableEq α
a : α
s : Finset α
\u22a2 (op a • s)⁻¹ = a⁻¹ • s⁻¹
-- Mathlib/Data/Finset/Pointwise.lean
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝³ : DecidableEq β
inst✝² : GroupWithZero α
inst✝¹ : MulAction α β
s✝ t : Finset β
a : α
b : β
inst✝ : DecidableEq α
s : Finset α
\u22a2 (op 0 • s)⁻¹ = 0⁻¹ • s⁻¹
-- Mathlib/Data/Finset/Pointwise.lean
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝³ : DecidableEq β
inst✝² : GroupWithZero α
inst✝¹ : MulAction α β
s✝ t : Finset β
a✝ : α
b : β
inst✝ : DecidableEq α
a : α
s : Finset α
ha : a ≠ 0
\u22a2 (op a • s)⁻¹ = a⁻¹ • s⁻¹
-- Mathlib/Analysis/SpecialFunctions/Log/Basic.lean
x y : ℝ
hx : 0 < x
\u22a2 log x = expOrderIso.symm ⟨x, hx⟩
-- Mathlib/Analysis/SpecialFunctions/Log/Basic.lean
x y : ℝ
hx : 0 < x
\u22a2 expOrderIso.symm ⟨|x|, ⋯⟩ = expOrderIso.symm ⟨x, hx⟩
-- 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
F : C ⥤ D
inst✝ : F.Additive
n : ℕ
X Y : C
f : X ⟶ Y
P : ProjectiveResolution X
Q : ProjectiveResolution Y
g : P.complex ⟶ Q.complex
w : g ≫ Q.π = P.π ≫ (ChainComplex.single₀ C).map f
\u22a2 (F.leftDerived n).map f = (P.isoLeftDerivedObj F n).hom ≫ (F.mapHomologicalComplex (ComplexShape.down ℕ) ⋙ HomologicalComplex.homologyFunctor D (ComplexShape.down ℕ) n).map g ≫ (Q.isoLeftDerivedObj F n).inv
-- 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
F : C ⥤ D
inst✝ : F.Additive
n : ℕ
X Y : C
f : X ⟶ Y
P : ProjectiveResolution X
Q : ProjectiveResolution Y
g : P.complex ⟶ Q.complex
w : g ≫ Q.π = P.π ≫ (ChainComplex.single₀ C).map f
\u22a2 g.f 0 ≫ Q.π.f 0 = P.π.f 0 ≫ f
-- Mathlib/MeasureTheory/PiSystem.lean
α : Type u_1
d : DynkinSystem α
s₁ s₂ : Set α
h₁ : d.Has s₁
h₂ : d.Has s₂
h : s₂ ⊆ s₁
\u22a2 d.Has (s₁ \ s₂)
-- Mathlib/MeasureTheory/PiSystem.lean
α : Type u_1
d : DynkinSystem α
s₁ s₂ : Set α
h₁ : d.Has s₁
h₂ : d.Has s₂
h : s₂ ⊆ s₁
\u22a2 d.Has (s₁ \ s₂)ᶜ
-- Mathlib/MeasureTheory/PiSystem.lean
α : Type u_1
d : DynkinSystem α
s₁ s₂ : Set α
h₁ : d.Has s₁
h₂ : d.Has s₂
h : s₂ ⊆ s₁
\u22a2 d.Has (s₁ᶜ ∪ s₂)
-- Mathlib/Order/Filter/CountableSeparatingOn.lean
α : Type u_1
p : Set α → Prop
s₀ : Set α
hp : p s₀
t : Set α
inst✝ : HasCountableSeparatingOn α p t
\u22a2 ∃ S, (∀ (n : ℕ), p (S n)) ∧ ∀ x ∈ t, ∀ y ∈ t, (∀ (n : ℕ), x ∈ S n ↔ y ∈ S n) → x = y
-- Mathlib/Order/Filter/CountableSeparatingOn.lean
α : Type u_1
p : Set α → Prop
s₀ : Set α
hp : p s₀
t : Set α
inst✝ : HasCountableSeparatingOn α p t
S : Set (Set α)
hSne : S.Nonempty
hSc : S.Countable
hS : (∀ s ∈ S, p s) ∧ ∀ x ∈ t, ∀ y ∈ t, (∀ s ∈ S, x ∈ s ↔ y ∈ s) → x = y
\u22a2 ∃ S, (∀ (n : ℕ), p (S n)) ∧ ∀ x ∈ t, ∀ y ∈ t, (∀ (n : ℕ), x ∈ S n ↔ y ∈ S n) → x = y
-- Mathlib/Order/Filter/CountableSeparatingOn.lean
α : Type u_1
p : Set α → Prop
s₀ : Set α
hp : p s₀
t : Set α
inst✝ : HasCountableSeparatingOn α p t
S : ℕ → Set α
hSne : (range S).Nonempty
hSc : (range S).Countable
hS : (∀ s ∈ range S, p s) ∧ ∀ x ∈ t, ∀ y ∈ t, (∀ s ∈ range S, x ∈ s ↔ y ∈ s) → x = y
\u22a2 ∃ S, (∀ (n : ℕ), p (S n)) ∧ ∀ x ∈ t, ∀ y ∈ t, (∀ (n : ℕ), x ∈ S n ↔ y ∈ S n) → x = y
-- Mathlib/Order/Filter/CountableSeparatingOn.lean
α : Type u_1
p : Set α → Prop
s₀ : Set α
hp : p s₀
t : Set α
inst✝ : HasCountableSeparatingOn α p t
S : ℕ → Set α
hSne : (range S).Nonempty
hSc : (range S).Countable
hS : (∀ s ∈ range S, p s) ∧ ∀ x ∈ t, ∀ y ∈ t, (∀ s ∈ range S, x ∈ s ↔ y ∈ s) → x = y
\u22a2 (∀ (n : ℕ), p (S n)) ∧ ∀ x ∈ t, ∀ y ∈ t, (∀ (n : ℕ), x ∈ S n ↔ y ∈ S n) → x = y
-- Mathlib/Topology/List.lean
α : Type u_1
β : Type u_2
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : Monoid α
inst✝ : ContinuousMul α
x : α
l : List α
ih : Tendsto prod (𝓝 l) (𝓝 l.prod)
\u22a2 Tendsto prod (𝓝 (x :: l)) (𝓝 (x :: l).prod)
-- Mathlib/Topology/List.lean
α : Type u_1
β : Type u_2
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : Monoid α
inst✝ : ContinuousMul α
x : α
l : List α
ih : Tendsto prod (𝓝 l) (𝓝 l.prod)
\u22a2 Tendsto (fun p => p.1 * p.2.prod) (𝓝 x ×ˢ 𝓝 l) (𝓝 (x * l.prod))
-- Mathlib/Topology/List.lean
α : Type u_1
β : Type u_2
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : Monoid α
inst✝ : ContinuousMul α
x : α
l : List α
ih : Tendsto prod (𝓝 l) (𝓝 l.prod)
this : ContinuousAt (fun p => p.1 * p.2) (x, l.prod)
\u22a2 Tendsto (fun p => p.1 * p.2.prod) (𝓝 x ×ˢ 𝓝 l) (𝓝 (x * l.prod))
-- Mathlib/Topology/List.lean
α : Type u_1
β : Type u_2
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : Monoid α
inst✝ : ContinuousMul α
x : α
l : List α
ih : Tendsto prod (𝓝 l) (𝓝 l.prod)
this : Tendsto (fun p => p.1 * p.2) (𝓝 x ×ˢ 𝓝 l.prod) (𝓝 ((x, l.prod).1 * (x, l.prod).2))
\u22a2 Tendsto (fun p => p.1 * p.2.prod) (𝓝 x ×ˢ 𝓝 l) (𝓝 (x * l.prod))
-- Mathlib/Data/Real/EReal.lean
x y : EReal
\u22a2 sign (x * y) = sign x * sign y
-- Mathlib/Algebra/Lie/Submodule.lean
R : Type u
L : Type v
L' : Type w₂
M : Type w
M' : Type w₁
inst✝¹² : CommRing R
inst✝¹¹ : LieRing L
inst✝¹⁰ : LieAlgebra R L
inst✝⁹ : LieRing L'
inst✝⁸ : LieAlgebra R L'
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : LieRingModule L M
inst✝⁴ : LieModule R L M
inst✝³ : AddCommGroup M'
inst✝² : Module R M'
inst✝¹ : LieRingModule L M'
inst✝ : LieModule R L M'
f : M →ₗ⁅R,L⁆ M'
N✝ N₂ : LieSubmodule R L M
N' : LieSubmodule R L M'
e : M ≃ₗ⁅R,L⁆ M'
N : LieSubmodule R L M'
m✝ : M'
\u22a2 m✝ ∈ map e.toLieModuleHom (comap e.toLieModuleHom N) ↔ m✝ ∈ N
-- Mathlib/RingTheory/WittVector/Defs.lean
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
\u22a2 wittSub p 0 = X (0, 0) - X (1, 0)
-- Mathlib/RingTheory/WittVector/Defs.lean
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
\u22a2 (map (Int.castRingHom ℚ)) (wittSub p 0) = (map (Int.castRingHom ℚ)) (X (0, 0) - X (1, 0))
-- Mathlib/Data/Bool/Set.lean
α : Type u_1
f : Bool → α
\u22a2 range f = {f false, f true}
-- Mathlib/Analysis/SpecialFunctions/Log/ENNRealLog.lean
y : EReal
\u22a2 ∃ a, a.log = y
-- Mathlib/Analysis/SpecialFunctions/Log/ENNRealLog.lean
y : EReal
y_bot : y = ⊥
\u22a2 ∃ a, a.log = y
-- Mathlib/Analysis/SpecialFunctions/Log/ENNRealLog.lean
y : EReal
y_nbot : ⊥ < y
\u22a2 ∃ a, a.log = y
-- Mathlib/Analysis/SpecialFunctions/Log/ENNRealLog.lean
y : EReal
y_nbot : ⊥ < y
y_top : y = ⊤
\u22a2 ∃ a, a.log = y
-- Mathlib/Analysis/SpecialFunctions/Log/ENNRealLog.lean
y : EReal
y_nbot : ⊥ < y
y_ntop : y < ⊤
\u22a2 ∃ a, a.log = y
-- Mathlib/Analysis/SpecialFunctions/Log/ENNRealLog.lean
y : EReal
y_nbot : ⊥ < y
y_ntop : y < ⊤
\u22a2 (ENNReal.ofReal (Real.exp y.toReal)).log = y
-- Mathlib/Analysis/SpecialFunctions/Log/ENNRealLog.lean
y : EReal
y_nbot : ⊥ < y
y_ntop : y < ⊤
exp_y_pos : ¬Real.exp y.toReal ≤ 0
\u22a2 (ENNReal.ofReal (Real.exp y.toReal)).log = y
-- Mathlib/Analysis/SpecialFunctions/Log/ENNRealLog.lean
y : EReal
y_nbot : ⊥ < y
y_ntop : y < ⊤
exp_y_pos : ¬Real.exp y.toReal ≤ 0
\u22a2 ↑y.toReal = y
-- Mathlib/Logic/Equiv/PartialEquiv.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
e✝ : PartialEquiv α β
e' : PartialEquiv β γ
e : PartialEquiv α β
s : Set α
h : e.source ⊆ s
\u22a2 (e.restr s).source = e.source
-- .lake/packages/batteries/Batteries/Data/String/Lemmas.lean
l m r : List Char
s : Substring
h : ValidFor l m r s
n : Nat
\u22a2 ValidFor (l ++ List.take n m) (List.drop n m) r (s.drop n)
-- .lake/packages/batteries/Batteries/Data/String/Lemmas.lean
l m r : List Char
s : Substring
h : ValidFor l m r s
n : Nat
this : { str := s.str, startPos := s.startPos, stopPos := s.stopPos }.nextn n { byteIdx := utf8Len [] } = { byteIdx := utf8Len [] + utf8Len (List.take n m) }
\u22a2 ValidFor (l ++ List.take n m) (List.drop n m) r (s.drop n)
-- .lake/packages/batteries/Batteries/Data/String/Lemmas.lean
l m r : List Char
s : Substring
h : ValidFor l m r s
n : Nat
this : { str := s.str, startPos := s.startPos, stopPos := s.stopPos }.nextn n 0 = { byteIdx := utf8Len (List.take n m) }
\u22a2 ValidFor (l ++ List.take n m) (List.drop n m) r (s.drop n)
-- .lake/packages/batteries/Batteries/Data/String/Lemmas.lean
l m r : List Char
s : Substring
h : ValidFor l m r s
n : Nat
this : { str := s.str, startPos := s.startPos, stopPos := s.stopPos }.nextn n 0 = { byteIdx := utf8Len (List.take n m) }
\u22a2 ValidFor (l ++ List.take n m) (List.drop n m) r { str := s.str, startPos := s.startPos + { byteIdx := utf8Len (List.take n m) }, stopPos := s.stopPos }
-- .lake/packages/batteries/Batteries/Data/String/Lemmas.lean
l m r : List Char
s : Substring
h : ValidFor l m r s
n : Nat
this : { str := s.str, startPos := s.startPos, stopPos := s.stopPos }.nextn n 0 = { byteIdx := utf8Len (List.take n m) }
\u22a2 ValidFor (l ++ List.take n m) (List.drop n m) r { str := { data := l ++ (m ++ r) }, startPos := { byteIdx := utf8Len l } + { byteIdx := utf8Len (List.take n m) }, stopPos := { byteIdx := utf8Len l + utf8Len m } }
-- .lake/packages/batteries/Batteries/Data/String/Lemmas.lean
l m r : List Char
s : Substring
n : Nat
h : ValidFor l (List.take n m ++ List.drop n m) r s
this : { str := s.str, startPos := s.startPos, stopPos := s.stopPos }.nextn n 0 = { byteIdx := utf8Len (List.take n m) }
\u22a2 ValidFor (l ++ List.take n m) (List.drop n m) r { str := { data := l ++ (m ++ r) }, startPos := { byteIdx := utf8Len l } + { byteIdx := utf8Len (List.take n m) }, stopPos := { byteIdx := utf8Len l + utf8Len m } }
-- .lake/packages/batteries/Batteries/Data/String/Lemmas.lean
l m r : List Char
s : Substring
n : Nat
h : ValidFor l (List.take n m ++ List.drop n m) r s
this : { str := s.str, startPos := s.startPos, stopPos := s.stopPos }.nextn n 0 = { byteIdx := utf8Len (List.take n m) }
\u22a2 { str := { data := l ++ (m ++ r) }, startPos := { byteIdx := utf8Len l } + { byteIdx := utf8Len (List.take n m) }, stopPos := { byteIdx := utf8Len l + utf8Len m } }.stopPos.byteIdx = utf8Len (l ++ List.take n m) + utf8Len (List.drop n m)
-- .lake/packages/batteries/Batteries/Data/String/Lemmas.lean
l m r : List Char
s : Substring
n : Nat
h : ValidFor l (List.take n m ++ List.drop n m) r s
this : { str := s.str, startPos := s.startPos, stopPos := s.stopPos }.nextn n 0 = { byteIdx := utf8Len (List.take n m) }
\u22a2 { str := { data := l ++ (List.take n m ++ List.drop n m ++ r) }, startPos := { byteIdx := utf8Len l } + { byteIdx := utf8Len (List.take n (List.take n m ++ List.drop n m)) }, stopPos := { byteIdx := utf8Len l + utf8Len (List.take n m ++ List.drop n m) } }.stopPos.byteIdx = utf8Len (l ++ List.take n m) + utf8Len (List.drop n m)
-- Mathlib/Data/Set/Card.lean
α : Type u_1
β : Type u_2
s t : Set α
a b : α
h : a ∈ s
\u22a2 (s \ {a}).encard = s.encard - 1
-- Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
θ₁ θ₂ : Real.Angle
x✝ : V
\u22a2 ((o.rotation θ₁).trans (o.rotation θ₂)) x✝ = (o.rotation (θ₂ + θ₁)) x✝
-- Mathlib/MeasureTheory/Measure/Portmanteau.lean
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : Measure Ω
μs : ℕ → Measure Ω
f : Ω → ℝ
f_cont : Continuous f
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → μ G ≤ liminf (fun i => (μs i) G) atTop
\u22a2 ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μ ≤ liminf (fun i => ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs i) atTop
-- Mathlib/MeasureTheory/Measure/Portmanteau.lean
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : Measure Ω
μs : ℕ → Measure Ω
f : Ω → ℝ
f_cont : Continuous f
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → μ G ≤ liminf (fun i => (μs i) G) atTop
\u22a2 ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} ≤ liminf (fun i => ∫⁻ (t : ℝ) in Ioi 0, (μs i) {a | t < f a}) atTop
-- Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝³ : Countable ι
m : MeasurableSpace α
μ ν✝ : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f g : α → β
inst✝ : PseudoMetrizableSpace β
ν : Measure α
\u22a2 AEStronglyMeasurable f (μ + ν) ↔ AEStronglyMeasurable f μ ∧ AEStronglyMeasurable f ν
-- Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝³ : Countable ι
m : MeasurableSpace α
μ ν✝ : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
f g : α → β
inst✝ : PseudoMetrizableSpace β
ν : Measure α
\u22a2 AEStronglyMeasurable f (bif true then μ else ν) ∧ AEStronglyMeasurable f (bif false then μ else ν) ↔ AEStronglyMeasurable f μ ∧ AEStronglyMeasurable f ν
-- Mathlib/AlgebraicGeometry/Morphisms/Basic.lean
P : MorphismProperty Scheme
inst✝ : P.RespectsIso
restrict : ∀ {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens), P f → P (f ∣_ U)
of_sSup_eq_top : ∀ {X Y : Scheme} (f : X ⟶ Y) {ι : Type u} (U : ι → Y.Opens), iSup U = ⊤ → (∀ (i : ι), P (f ∣_ U i)) → P f
\u22a2 IsLocalAtTarget P
-- Mathlib/AlgebraicGeometry/Morphisms/Basic.lean
P : MorphismProperty Scheme
inst✝ : P.RespectsIso
restrict : ∀ {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens), P f → P (f ∣_ U)
of_sSup_eq_top : ∀ {X Y : Scheme} (f : X ⟶ Y) {ι : Type u} (U : ι → Y.Opens), iSup U = ⊤ → (∀ (i : ι), P (f ∣_ U i)) → P f
X Y : Scheme
f : X ⟶ Y
𝒰 : Y.OpenCover
\u22a2 P f → ∀ (i : (𝒰.pullbackCover f).J), P (𝒰.pullbackHom f i)
-- Mathlib/AlgebraicGeometry/Morphisms/Basic.lean
P : MorphismProperty Scheme
inst✝ : P.RespectsIso
restrict : ∀ {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens), P f → P (f ∣_ U)
of_sSup_eq_top : ∀ {X Y : Scheme} (f : X ⟶ Y) {ι : Type u} (U : ι → Y.Opens), iSup U = ⊤ → (∀ (i : ι), P (f ∣_ U i)) → P f
X Y : Scheme
f : X ⟶ Y
𝒰 : Y.OpenCover
H : ∀ (i : (𝒰.pullbackCover f).J), P (𝒰.pullbackHom f i)
\u22a2 ∀ (i : 𝒰.J), P (f ∣_ Scheme.Hom.opensRange (𝒰.map i))
-- Mathlib/Probability/Kernel/Composition.lean
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
mγ : MeasurableSpace γ
s : Set (β × γ)
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a : α
f : β → γ → ℝ≥0∞
hf : Measurable (Function.uncurry f)
F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f)
\u22a2 ∫⁻ (bc : β × γ), f bc.1 bc.2 ∂(κ ⊗ₖ η) a = ∫⁻ (b : β), ∫⁻ (c : γ), f b c ∂η (a, b) ∂κ a
-- Mathlib/Probability/Kernel/Composition.lean
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
mγ : MeasurableSpace γ
s : Set (β × γ)
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a : α
f : β → γ → ℝ≥0∞
hf : Measurable (Function.uncurry f)
F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f)
h : ∀ (a : β × γ), ⨆ n, ↑(F n) a = Function.uncurry f a
\u22a2 ∫⁻ (bc : β × γ), f bc.1 bc.2 ∂(κ ⊗ₖ η) a = ∫⁻ (b : β), ∫⁻ (c : γ), f b c ∂η (a, b) ∂κ a
-- Mathlib/Probability/Kernel/Composition.lean
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
mγ : MeasurableSpace γ
s : Set (β × γ)
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a : α
f : β → γ → ℝ≥0∞
hf : Measurable (Function.uncurry f)
F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f)
h : ∀ (a : β) (b : γ), ⨆ n, ↑(F n) (a, b) = f a b
\u22a2 ∫⁻ (bc : β × γ), f bc.1 bc.2 ∂(κ ⊗ₖ η) a = ∫⁻ (b : β), ∫⁻ (c : γ), f b c ∂η (a, b) ∂κ a
-- Mathlib/Probability/Kernel/Composition.lean
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
mγ : MeasurableSpace γ
s : Set (β × γ)
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a : α
f : β → γ → ℝ≥0∞
hf : Measurable (Function.uncurry f)
F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f)
h : ∀ (a : β) (b : γ), ⨆ n, ↑(F n) (a, b) = f a b
\u22a2 ∫⁻ (bc : β × γ), ⨆ n, ↑(F n) (bc.1, bc.2) ∂(κ ⊗ₖ η) a = ∫⁻ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂η (a, b) ∂κ a
-- Mathlib/Probability/Kernel/Composition.lean
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
mγ : MeasurableSpace γ
s : Set (β × γ)
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a : α
f : β → γ → ℝ≥0∞
hf : Measurable (Function.uncurry f)
F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f)
h : ∀ (a : β) (b : γ), ⨆ n, ↑(F n) (a, b) = f a b
h_mono : Monotone F
\u22a2 ∫⁻ (bc : β × γ), ⨆ n, ↑(F n) (bc.1, bc.2) ∂(κ ⊗ₖ η) a = ∫⁻ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂η (a, b) ∂κ a
-- Mathlib/Probability/Kernel/Composition.lean
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
mγ : MeasurableSpace γ
s : Set (β × γ)
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a : α
f : β → γ → ℝ≥0∞
hf : Measurable (Function.uncurry f)
F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f)
h : ∀ (a : β) (b : γ), ⨆ n, ↑(F n) (a, b) = f a b
h_mono : Monotone F
\u22a2 ⨆ n, ∫⁻ (a : β × γ), ↑(F n) a ∂(κ ⊗ₖ η) a = ∫⁻ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂η (a, b) ∂κ a
-- Mathlib/Probability/Kernel/Composition.lean
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
mγ : MeasurableSpace γ
s : Set (β × γ)
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a : α
f : β → γ → ℝ≥0∞
hf : Measurable (Function.uncurry f)
F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f)
h : ∀ (a : β) (b : γ), ⨆ n, ↑(F n) (a, b) = f a b
h_mono : Monotone F
this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂η (a, b)
\u22a2 ⨆ n, ∫⁻ (a : β × γ), ↑(F n) a ∂(κ ⊗ₖ η) a = ∫⁻ (b : β), ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂η (a, b) ∂κ a