ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Analysis/Fourier/FourierTransformDeriv.lean
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : NormedAddCommGroup W
inst✝ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f✝ f : V → E
K : ℕ∞
C : ℝ
hf : ContDiff ℝ K f
n k : ℕ
hk : ↑k ≤ K
v : V
hv : ∀ i ≤ k, ∀ j ≤ n, ‖v‖ ^ j * ‖iteratedFDeriv ℝ i f v‖ ≤ C
T : ContinuousMultilinearMap ℝ (fun i => W →L[ℝ] ℝ) (ContinuousMultilinearMap ℝ (fun i => W) ℝ) := (ContinuousMultilinearMap.mkPiAlgebra ℝ (Fin n) ℝ).compContinuousLinearMapLRight
I₁ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m ⇑T fun x => L v‖ ≤ ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m)
I₂ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m (⇑T ∘ ⇑(ContinuousLinearMap.pi fun x => L)) v‖ ≤ ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m) * ‖L‖ ^ m
I₃ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m (⇑T ∘ ⇑(ContinuousLinearMap.pi fun x => L)) v‖ ≤ ↑(n.descFactorial m) * ‖L‖ ^ n * ‖v‖ ^ (n - m)
A : ContDiff ℝ K fun y => T fun x => L y
\u22a2 ‖(-(2 * ↑π * I)) ^ n‖ * ‖iteratedFDeriv ℝ k (fun x => ((smulRightL ℝ (fun x => W) E) (T fun x_1 => L x, f x).1) (T fun x_1 => L x, f x).2) v‖ ≤ (2 * π) ^ n * (2 * ↑n + 2) ^ k * ‖L‖ ^ n * C
-- Mathlib/Analysis/Fourier/FourierTransformDeriv.lean
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : NormedAddCommGroup W
inst✝ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f✝ f : V → E
K : ℕ∞
C : ℝ
hf : ContDiff ℝ K f
n k : ℕ
hk : ↑k ≤ K
v : V
hv : ∀ i ≤ k, ∀ j ≤ n, ‖v‖ ^ j * ‖iteratedFDeriv ℝ i f v‖ ≤ C
T : ContinuousMultilinearMap ℝ (fun i => W →L[ℝ] ℝ) (ContinuousMultilinearMap ℝ (fun i => W) ℝ) := (ContinuousMultilinearMap.mkPiAlgebra ℝ (Fin n) ℝ).compContinuousLinearMapLRight
I₁ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m ⇑T fun x => L v‖ ≤ ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m)
I₂ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m (⇑T ∘ ⇑(ContinuousLinearMap.pi fun x => L)) v‖ ≤ ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m) * ‖L‖ ^ m
I₃ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m (⇑T ∘ ⇑(ContinuousLinearMap.pi fun x => L)) v‖ ≤ ↑(n.descFactorial m) * ‖L‖ ^ n * ‖v‖ ^ (n - m)
A : ContDiff ℝ K fun y => T fun x => L y
\u22a2 (2 * π) ^ n * ‖iteratedFDeriv ℝ k (fun x => ((smulRightL ℝ (fun x => W) E) (T fun x_1 => L x)) (f x)) v‖ ≤ (2 * π) ^ n * ((2 * ↑n + 2) ^ k * (‖L‖ ^ n * C))
-- Mathlib/Analysis/Fourier/FourierTransformDeriv.lean
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : NormedAddCommGroup W
inst✝ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f✝ f : V → E
K : ℕ∞
C : ℝ
hf : ContDiff ℝ K f
n k : ℕ
hk : ↑k ≤ K
v : V
hv : ∀ i ≤ k, ∀ j ≤ n, ‖v‖ ^ j * ‖iteratedFDeriv ℝ i f v‖ ≤ C
T : ContinuousMultilinearMap ℝ (fun i => W →L[ℝ] ℝ) (ContinuousMultilinearMap ℝ (fun i => W) ℝ) := (ContinuousMultilinearMap.mkPiAlgebra ℝ (Fin n) ℝ).compContinuousLinearMapLRight
I₁ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m ⇑T fun x => L v‖ ≤ ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m)
I₂ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m (⇑T ∘ ⇑(ContinuousLinearMap.pi fun x => L)) v‖ ≤ ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m) * ‖L‖ ^ m
I₃ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m (⇑T ∘ ⇑(ContinuousLinearMap.pi fun x => L)) v‖ ≤ ↑(n.descFactorial m) * ‖L‖ ^ n * ‖v‖ ^ (n - m)
A : ContDiff ℝ K fun y => T fun x => L y
\u22a2 ‖iteratedFDeriv ℝ k (fun x => ((smulRightL ℝ (fun x => W) E) (T fun x_1 => L x)) (f x)) v‖ ≤ (2 * ↑n + 2) ^ k * (‖L‖ ^ n * C)
-- Mathlib/Analysis/Fourier/FourierTransformDeriv.lean
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : NormedAddCommGroup W
inst✝ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f✝ f : V → E
K : ℕ∞
C : ℝ
hf : ContDiff ℝ K f
n k : ℕ
hk : ↑k ≤ K
v : V
hv : ∀ i ≤ k, ∀ j ≤ n, ‖v‖ ^ j * ‖iteratedFDeriv ℝ i f v‖ ≤ C
T : ContinuousMultilinearMap ℝ (fun i => W →L[ℝ] ℝ) (ContinuousMultilinearMap ℝ (fun i => W) ℝ) := (ContinuousMultilinearMap.mkPiAlgebra ℝ (Fin n) ℝ).compContinuousLinearMapLRight
I₁ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m ⇑T fun x => L v‖ ≤ ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m)
I₂ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m (⇑T ∘ ⇑(ContinuousLinearMap.pi fun x => L)) v‖ ≤ ↑(n.descFactorial m) * 1 * (‖L‖ * ‖v‖) ^ (n - m) * ‖L‖ ^ m
I₃ : ∀ (m : ℕ), ‖iteratedFDeriv ℝ m (⇑T ∘ ⇑(ContinuousLinearMap.pi fun x => L)) v‖ ≤ ↑(n.descFactorial m) * ‖L‖ ^ n * ‖v‖ ^ (n - m)
A : ContDiff ℝ K fun y => T fun x => L y
\u22a2 ∑ i ∈ Finset.range (k + 1), ↑(k.choose i) * ‖iteratedFDeriv ℝ i (fun y => T fun x => L y) v‖ * ‖iteratedFDeriv ℝ (k - i) f v‖ ≤ (2 * ↑n + 2) ^ k * (‖L‖ ^ n * C)
-- Mathlib/Data/Real/Basic.lean
x✝ y : ℝ
C : ℝ → Prop
h : ∀ (y : CauSeq ℚ abs), C (mk y)
x : Cauchy abs
\u22a2 C { cauchy := x }
-- Mathlib/Data/Real/Basic.lean
x✝ y : ℝ
C : ℝ → Prop
h : ∀ (y : CauSeq ℚ abs), C (mk y)
x : CauSeq ℚ abs
\u22a2 C { cauchy := Quot.mk Setoid.r x }
-- Mathlib/MeasureTheory/MeasurableSpace/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t✝ u : Set α
f g : α → β
m : MeasurableSpace α
mβ : MeasurableSpace β
inst✝ : MeasurableSingletonClass α
hf : Measurable f
h : {x | f x ≠ g x}.Countable
t : Set β
ht : MeasurableSet t
\u22a2 MeasurableSet (g ⁻¹' t)
-- Mathlib/MeasureTheory/MeasurableSpace/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t✝ u : Set α
f g : α → β
m : MeasurableSpace α
mβ : MeasurableSpace β
inst✝ : MeasurableSingletonClass α
hf : Measurable f
h : {x | f x ≠ g x}.Countable
t : Set β
ht : MeasurableSet t
this : g ⁻¹' t = g ⁻¹' t ∩ {x | f x = g x}ᶜ ∪ g ⁻¹' t ∩ {x | f x = g x}
\u22a2 MeasurableSet (g ⁻¹' t)
-- Mathlib/MeasureTheory/MeasurableSpace/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t✝ u : Set α
f g : α → β
m : MeasurableSpace α
mβ : MeasurableSpace β
inst✝ : MeasurableSingletonClass α
hf : Measurable f
h : {x | f x ≠ g x}.Countable
t : Set β
ht : MeasurableSet t
this : g ⁻¹' t = g ⁻¹' t ∩ {x | f x = g x}ᶜ ∪ g ⁻¹' t ∩ {x | f x = g x}
\u22a2 MeasurableSet (g ⁻¹' t ∩ {x | f x = g x}ᶜ ∪ g ⁻¹' t ∩ {x | f x = g x})
-- Mathlib/MeasureTheory/MeasurableSpace/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t✝ u : Set α
f g : α → β
m : MeasurableSpace α
mβ : MeasurableSpace β
inst✝ : MeasurableSingletonClass α
hf : Measurable f
h : {x | f x ≠ g x}.Countable
t : Set β
ht : MeasurableSet t
this : g ⁻¹' t = g ⁻¹' t ∩ {x | f x = g x}ᶜ ∪ g ⁻¹' t ∩ {x | f x = g x}
\u22a2 MeasurableSet (g ⁻¹' t ∩ {x | f x = g x})
-- Mathlib/MeasureTheory/MeasurableSpace/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t✝ u : Set α
f g : α → β
m : MeasurableSpace α
mβ : MeasurableSpace β
inst✝ : MeasurableSingletonClass α
hf : Measurable f
h : {x | f x ≠ g x}.Countable
t : Set β
ht : MeasurableSet t
this✝ : g ⁻¹' t = g ⁻¹' t ∩ {x | f x = g x}ᶜ ∪ g ⁻¹' t ∩ {x | f x = g x}
this : g ⁻¹' t ∩ {x | f x = g x} = f ⁻¹' t ∩ {x | f x = g x}
\u22a2 MeasurableSet (f ⁻¹' t ∩ {x | f x = g x})
-- Mathlib/LinearAlgebra/QuadraticForm/Prod.lean
ι : Type u_1
R : Type u_2
M₁ : Type u_3
M₂ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
P : Type u_7
Mᵢ : ι → Type u_8
Nᵢ : ι → Type u_9
inst✝⁸ : CommSemiring R
inst✝⁷ : (i : ι) → AddCommMonoid (Mᵢ i)
inst✝⁶ : (i : ι) → AddCommMonoid (Nᵢ i)
inst✝⁵ : AddCommMonoid P
inst✝⁴ : (i : ι) → Module R (Mᵢ i)
inst✝³ : (i : ι) → Module R (Nᵢ i)
inst✝² : Module R P
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
i : ι
Q : QuadraticMap R (Mᵢ i) P
m : (i : ι) → Mᵢ i
\u22a2 Q (m i) = (pi (Pi.single i Q)) m
-- Mathlib/LinearAlgebra/QuadraticForm/Prod.lean
ι : Type u_1
R : Type u_2
M₁ : Type u_3
M₂ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
P : Type u_7
Mᵢ : ι → Type u_8
Nᵢ : ι → Type u_9
inst✝⁸ : CommSemiring R
inst✝⁷ : (i : ι) → AddCommMonoid (Mᵢ i)
inst✝⁶ : (i : ι) → AddCommMonoid (Nᵢ i)
inst✝⁵ : AddCommMonoid P
inst✝⁴ : (i : ι) → Module R (Mᵢ i)
inst✝³ : (i : ι) → Module R (Nᵢ i)
inst✝² : Module R P
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
i : ι
Q : QuadraticMap R (Mᵢ i) P
m : (i : ι) → Mᵢ i
j : ι
hij : j ≠ i
\u22a2 (Pi.single i Q j) (m j) = 0
-- Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean
C : Type u_1
inst✝⁴ : Category.{u_2, u_1} C
inst✝³ : Preadditive C
K K' : ChainComplex C ℕ
f : K ⟶ K'
Δ✝ Δ'✝ Δ''✝ : SimplexCategory
inst✝² : HasFiniteCoproducts C
Δ Δ' : SimplexCategoryᵒᵖ
A : Splitting.IndexSet Δ
θ : Δ ⟶ Δ'
Δ'' : SimplexCategory
e : unop Δ' ⟶ Δ''
i : Δ'' ⟶ unop A.fst
inst✝¹ : Epi e
inst✝ : Mono i
fac : e ≫ i = θ.unop ≫ A.e
\u22a2 ((splitting K).ι (unop A.fst).len ≫ (obj K).map A.e.op) ≫ (obj K).map θ = Termwise.mapMono K i ≫ (splitting K).ι Δ''.len ≫ map K (Splitting.IndexSet.mk e).e.op
-- Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean
C : Type u_1
inst✝⁴ : Category.{u_2, u_1} C
inst✝³ : Preadditive C
K K' : ChainComplex C ℕ
f : K ⟶ K'
Δ✝ Δ'✝ Δ''✝ : SimplexCategory
inst✝² : HasFiniteCoproducts C
Δ Δ' : SimplexCategoryᵒᵖ
A : Splitting.IndexSet Δ
θ : Δ ⟶ Δ'
Δ'' : SimplexCategory
e : unop Δ' ⟶ Δ''
i : Δ'' ⟶ unop A.fst
inst✝¹ : Epi e
inst✝ : Mono i
fac : e ≫ i = θ.unop ≫ A.e
\u22a2 (splitting K).ι (unop A.fst).len ≫ (obj K).map (A.e.op ≫ θ) = Termwise.mapMono K i ≫ (splitting K).ι Δ''.len ≫ map K (Splitting.IndexSet.mk e).e.op
-- Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean
C : Type u_1
inst✝⁴ : Category.{u_2, u_1} C
inst✝³ : Preadditive C
K K' : ChainComplex C ℕ
f : K ⟶ K'
Δ✝ Δ'✝ Δ''✝ : SimplexCategory
inst✝² : HasFiniteCoproducts C
Δ Δ' : SimplexCategoryᵒᵖ
A : Splitting.IndexSet Δ
θ : Δ ⟶ Δ'
Δ'' : SimplexCategory
e : unop Δ' ⟶ Δ''
i : Δ'' ⟶ unop A.fst
inst✝¹ : Epi e
inst✝ : Mono i
fac : e ≫ i = θ.unop ≫ A.e
\u22a2 Sigma.ι (summand K A.fst) (Splitting.IndexSet.id A.fst) ≫ map K (A.e.op ≫ θ) = Termwise.mapMono K i ≫ Sigma.ι (summand K (op Δ'')) (Splitting.IndexSet.id (op Δ'')) ≫ map K (Splitting.IndexSet.mk e).e.op
-- Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
K L M : CochainComplex C ℤ
n : ℤ
γ γ₁ γ₂ : Cochain K L n
a n' : ℤ
hn' : n + a = n'
γ' : Cochain L M 0
\u22a2 (γ.comp γ' ⋯).leftShift a n' hn' = (γ.leftShift a n' hn').comp γ' ⋯
-- Mathlib/Order/Interval/Set/Pi.lean
ι : Type u_1
α : ι → Type u_2
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → PartialOrder (α i)
f : (i : ι) → α i
i : ι
a : α i
\u22a2 update f i '' Ico a (f i) = Ico (update f i a) f
-- Mathlib/Data/Complex/Module.lean
A : Type u_1
inst✝³ : AddCommGroup A
inst✝² : Module ℂ A
inst✝¹ : StarAddMonoid A
inst✝ : StarModule ℂ A
z : ℂ
a : A
\u22a2 ℜ (z • a) = z.re • ℜ a - z.im • ℑ a
-- Mathlib/Data/Complex/Module.lean
A : Type u_1
inst✝³ : AddCommGroup A
inst✝² : Module ℂ A
inst✝¹ : StarAddMonoid A
inst✝ : StarModule ℂ A
z : ℂ
a : A
this : ℜ (z • a) = ℜ ((↑z.re + ↑z.im * I) • a)
\u22a2 ℜ (z • a) = z.re • ℜ a - z.im • ℑ a
-- Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
inst✝⁴ : ConcreteCategory D
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst✝ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
P : Cᵒᵖ ⥤ D
hsep : ∀ (X : C) (S : J.Cover X) (x y : (forget D).obj (P.obj (op X))), (∀ (I : S.Arrow), (P.map I.f.op) x = (P.map I.f.op) y) → x = y
X : C
x y : (forget D).obj (P.obj (op X))
h : ((J.toPlus P).app (op X)) x = ((J.toPlus P).app (op X)) y
\u22a2 x = y
-- Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
inst✝⁴ : ConcreteCategory D
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst✝ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
P : Cᵒᵖ ⥤ D
hsep : ∀ (X : C) (S : J.Cover X) (x y : (forget D).obj (P.obj (op X))), (∀ (I : S.Arrow), (P.map I.f.op) x = (P.map I.f.op) y) → x = y
X : C
x y : (forget D).obj (P.obj (op X))
h : mk (Meq.mk ⊤ x) = mk (Meq.mk ⊤ y)
\u22a2 x = y
-- Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁵ : Category.{max v u, w} D
inst✝⁴ : ConcreteCategory D
inst✝³ : PreservesLimits (forget D)
inst✝² : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst✝ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
P : Cᵒᵖ ⥤ D
hsep : ∀ (X : C) (S : J.Cover X) (x y : (forget D).obj (P.obj (op X))), (∀ (I : S.Arrow), (P.map I.f.op) x = (P.map I.f.op) y) → x = y
X : C
x y : (forget D).obj (P.obj (op X))
h : ∃ W h1 h2, (Meq.mk ⊤ x).refine h1 = (Meq.mk ⊤ y).refine h2
\u22a2 x = y
-- Mathlib/FieldTheory/Adjoin.lean
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
P : IntermediateField F E → Prop
base : P ⊥
ih : ∀ (K : IntermediateField F E) (x : E), P K → P (restrictScalars F (↥K)⟮x⟯)
S : Finset E
\u22a2 P (adjoin F ↑S)
-- Mathlib/Order/Closure.lean
α : Type u_1
ι : Sort u_2
κ : ι → Sort u_3
β : Type u_4
inst✝ : SetLike α β
l : LowerAdjoint SetLike.coe
x y : α
\u22a2 l.toFun (↑(l.toFun ↑x) ∪ ↑(l.toFun ↑y)) = l.toFun (↑x ∪ ↑y)
-- Mathlib/Data/Real/Sqrt.lean
x y : ℝ
hx : 0 ≤ x
hy : 0 ≤ y
h : √x = y
\u22a2 y * y = x
-- Mathlib/Data/Real/Sqrt.lean
x y : ℝ
hx : 0 ≤ x
hy : 0 ≤ y
h : y * y = x
\u22a2 √x = y
-- Mathlib/Order/Interval/Finset/Basic.lean
ι : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : LocallyFiniteOrder α
a a₁ a₂ b b₁ b₂ c x : α
inst✝ : LocallyFiniteOrderTop α
\u22a2 Ioo a b ⊆ Ioi a
-- Mathlib/Analysis/Complex/Polynomial/Basic.lean
p : ℝ[X]
z : ℂ
h0 : (aeval z) p = 0
hz : z.im ≠ 0
\u22a2 X ^ 2 - C (2 * z.re) * X + C (‖z‖ ^ 2) ∣ p
-- Mathlib/Analysis/Complex/Polynomial/Basic.lean
p : ℝ[X]
z : ℂ
h0 : (aeval z) p = 0
hz : z.im ≠ 0
\u22a2 map (algebraMap ℝ ℂ) (X ^ 2 - C (2 * z.re) * X + C (‖z‖ ^ 2)) ∣ map (algebraMap ℝ ℂ) p
-- Mathlib/Analysis/Complex/Polynomial/Basic.lean
p : ℝ[X]
z : ℂ
h0 : (aeval z) p = 0
hz : z.im ≠ 0
\u22a2 map (algebraMap ℝ ℂ) (X ^ 2 - C (2 * z.re) * X + C (‖z‖ ^ 2)) = (X - C ((starRingEnd ℂ) z)) * (X - C z)
-- Mathlib/Algebra/RingQuot.lean
R : Type uR
inst✝³ : Semiring R
S : Type uS
inst✝² : CommSemiring S
T : Type uT
A : Type uA
inst✝¹ : Semiring A
inst✝ : Algebra S A
r✝ r : R → R → Prop
\u22a2 Function.Surjective ⇑(mkRingHom r)
-- Mathlib/Algebra/RingQuot.lean
R : Type uR
inst✝³ : Semiring R
S : Type uS
inst✝² : CommSemiring S
T : Type uT
A : Type uA
inst✝¹ : Semiring A
inst✝ : Algebra S A
r✝ r : R → R → Prop
\u22a2 Function.Surjective fun x => { toQuot := Quot.mk (Rel r) x }
-- Mathlib/Algebra/Order/Interval/Finset.lean
ι : Type u_1
α : Type u_2
inst✝³ : OrderedCancelAddCommMonoid α
inst✝² : ExistsAddOfLE α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b c : α
\u22a2 image (fun x => c + x) (Ioc a b) = Ioc (c + a) (c + b)
-- Mathlib/LinearAlgebra/TensorAlgebra/Basic.lean
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
m : M
\u22a2 (RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m) = (ι R) m
-- Mathlib/LinearAlgebra/TensorAlgebra/Basic.lean
R : Type u_1
inst✝² : CommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
m : M
\u22a2 (RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m) = { toFun := fun m => (RingQuot.mkAlgHom R (Rel R M)) (FreeAlgebra.ι R m), map_add' := ⋯, map_smul' := ⋯ } m
-- Mathlib/Data/Finset/Pi.lean
α : Type u_1
β✝ : α → Type u
δ : α → Sort v
inst✝ : DecidableEq α
s✝ : Finset α
t : (a : α) → Finset (β✝ a)
β : Type u_2
s : Finset α
f : α → β
\u22a2 (s.pi fun a => {f a}) = {fun a x => f a}
-- Mathlib/Data/Finset/Pi.lean
α : Type u_1
β✝ : α → Type u
δ : α → Sort v
inst✝ : DecidableEq α
s✝ : Finset α
t : (a : α) → Finset (β✝ a)
β : Type u_2
s : Finset α
f : α → β
\u22a2 ((fun a x => f a) ∈ s.pi fun a => {f a}) ∧ ∀ x ∈ s.pi fun a => {f a}, x = fun a x => f a
-- Mathlib/Data/Finset/Pi.lean
α : Type u_1
β✝ : α → Type u
δ : α → Sort v
inst✝ : DecidableEq α
s✝ : Finset α
t : (a : α) → Finset (β✝ a)
β : Type u_2
s : Finset α
f : α → β
a : (a : α) → a ∈ s → β
ha : a ∈ s.pi fun a => {f a}
i : α
hi : i ∈ s
\u22a2 a i hi = f i
-- Mathlib/Data/Finset/Pi.lean
α : Type u_1
β✝ : α → Type u
δ : α → Sort v
inst✝ : DecidableEq α
s✝ : Finset α
t : (a : α) → Finset (β✝ a)
β : Type u_2
s : Finset α
f : α → β
a : (a : α) → a ∈ s → β
ha : ∀ (a_1 : α) (h : a_1 ∈ s), a a_1 h ∈ {f a_1}
i : α
hi : i ∈ s
\u22a2 a i hi = f i
-- Mathlib/AlgebraicGeometry/Morphisms/Basic.lean
P : MorphismProperty Scheme
Q : AffineTargetMorphismProperty
inst✝¹ : HasAffineProperty P Q
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
hP' : Q.StableUnderBaseChange
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
inst✝ : IsAffine S
H : Q g
\u22a2 P (pullback.fst f g)
-- Mathlib/AlgebraicGeometry/Morphisms/Basic.lean
P : MorphismProperty Scheme
Q : AffineTargetMorphismProperty
inst✝¹ : HasAffineProperty P Q
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
hP' : Q.StableUnderBaseChange
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
inst✝ : IsAffine S
H : Q g
this : Q.IsLocal := isLocal_affineProperty P
\u22a2 P (pullback.fst f g)
-- Mathlib/AlgebraicGeometry/Morphisms/Basic.lean
P : MorphismProperty Scheme
Q : AffineTargetMorphismProperty
inst✝¹ : HasAffineProperty P Q
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
hP' : Q.StableUnderBaseChange
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
inst✝ : IsAffine S
H : Q g
this : Q.IsLocal := isLocal_affineProperty P
\u22a2 ∀ (i : (X.affineCover.pullbackCover (pullback.fst f g)).J), Q (X.affineCover.pullbackHom (pullback.fst f g) i)
-- Mathlib/AlgebraicGeometry/Morphisms/Basic.lean
P : MorphismProperty Scheme
Q : AffineTargetMorphismProperty
inst✝¹ : HasAffineProperty P Q
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
hP' : Q.StableUnderBaseChange
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
inst✝ : IsAffine S
H : Q g
this✝ : Q.IsLocal := isLocal_affineProperty P
i : (X.affineCover.pullbackCover (pullback.fst f g)).J
e : pullback (pullback.fst f g) (X.affineCover.map i) ≅ pullback (X.affineCover.map i ≫ f) g := pullbackSymmetry (pullback.fst f g) (X.affineCover.map i) ≪≫ pullbackRightPullbackFstIso f g (X.affineCover.map i)
this : e.hom ≫ pullback.fst (X.affineCover.map i ≫ f) g = X.affineCover.pullbackHom (pullback.fst f g) i
\u22a2 Q (X.affineCover.pullbackHom (pullback.fst f g) i)
-- Mathlib/AlgebraicGeometry/Morphisms/Basic.lean
P : MorphismProperty Scheme
Q : AffineTargetMorphismProperty
inst✝¹ : HasAffineProperty P Q
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
hP' : Q.StableUnderBaseChange
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
inst✝ : IsAffine S
H : Q g
this✝ : Q.IsLocal := isLocal_affineProperty P
i : (X.affineCover.pullbackCover (pullback.fst f g)).J
e : pullback (pullback.fst f g) (X.affineCover.map i) ≅ pullback (X.affineCover.map i ≫ f) g := pullbackSymmetry (pullback.fst f g) (X.affineCover.map i) ≪≫ pullbackRightPullbackFstIso f g (X.affineCover.map i)
this : e.hom ≫ pullback.fst (X.affineCover.map i ≫ f) g = X.affineCover.pullbackHom (pullback.fst f g) i
\u22a2 Q (pullback.fst (X.affineCover.map i ≫ f) g)
-- Mathlib/AlgebraicGeometry/Morphisms/Basic.lean
P : MorphismProperty Scheme
Q : AffineTargetMorphismProperty
inst✝¹ : HasAffineProperty P Q
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
hP' : Q.StableUnderBaseChange
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
inst✝ : IsAffine S
H : Q g
this✝ : Q.IsLocal := isLocal_affineProperty P
i : (X.affineCover.pullbackCover (pullback.fst f g)).J
e : pullback (pullback.fst f g) (X.affineCover.map i) ≅ pullback (X.affineCover.map i ≫ f) g := pullbackSymmetry (pullback.fst f g) (X.affineCover.map i) ≪≫ pullbackRightPullbackFstIso f g (X.affineCover.map i)
this : e.hom ≫ pullback.fst (X.affineCover.map i ≫ f) g = X.affineCover.pullbackHom (pullback.fst f g) i
\u22a2 Q g
-- Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
h : 3 ≠ 0
\u22a2 W.Ψ₃.leadingCoeff = 3
-- Mathlib/MeasureTheory/Measure/Stieltjes.lean
f✝ : StieltjesFunction
c : ℝ≥0
f : StieltjesFunction
\u22a2 (c • f).measure = c • f.measure
-- Mathlib/MeasureTheory/Measure/Stieltjes.lean
f✝ : StieltjesFunction
c : ℝ≥0
f : StieltjesFunction
a b : ℝ
x✝ : a < b
\u22a2 (c • f).measure (Ioc a b) = (c • f.measure) (Ioc a b)
-- Mathlib/MeasureTheory/Measure/Stieltjes.lean
f✝ : StieltjesFunction
c : ℝ≥0
f : StieltjesFunction
a b : ℝ
x✝ : a < b
\u22a2 ofReal (↑(c • f) b - ↑(c • f) a) = c • ofReal (↑f b - ↑f a)
-- Mathlib/MeasureTheory/Measure/Stieltjes.lean
f✝ : StieltjesFunction
c : ℝ≥0
f : StieltjesFunction
a b : ℝ
x✝ : a < b
\u22a2 ofReal (↑c * ↑f b - ↑c * ↑f a) = c • ofReal (↑f b - ↑f a)
-- Mathlib/MeasureTheory/Measure/Stieltjes.lean
f✝ : StieltjesFunction
c : ℝ≥0
f : StieltjesFunction
a b : ℝ
x✝ : a < b
\u22a2 ↑c • ofReal (↑f b - ↑f a) = c • ofReal (↑f b - ↑f a)
-- Mathlib/Data/Nat/Defs.lean
a b c d m n k : ℕ
p q : ℕ → Prop
hb : 0 < b
\u22a2 a / b ≤ c ↔ a ≤ b * c + (b - 1)
-- Mathlib/Data/Nat/Defs.lean
a b c d m n k : ℕ
p q : ℕ → Prop
hb : 0 < b
\u22a2 a < b * c + b ↔ a ≤ b * c + (b - 1)
-- Mathlib/AlgebraicGeometry/AffineScheme.lean
X : Scheme
inst✝ : IsAffine X
\u22a2 IsTopologicalBasis (SetLike.coe '' Set.range X.basicOpen)
-- Mathlib/AlgebraicGeometry/AffineScheme.lean
X : Scheme
inst✝ : IsAffine X
\u22a2 SetLike.coe '' Set.range X.basicOpen = Set.preimage ⇑(TopCat.homeoOfIso (Scheme.forgetToTop.mapIso X.isoSpec)) '' (SetLike.coe '' Set.range PrimeSpectrum.basicOpen)
-- Mathlib/AlgebraicGeometry/AffineScheme.lean
X : Scheme
inst✝ : IsAffine X
x✝ : Set ↑↑X.toPresheafedSpace
\u22a2 x✝ ∈ SetLike.coe '' Set.range X.basicOpen ↔ x✝ ∈ Set.preimage ⇑(TopCat.homeoOfIso (Scheme.forgetToTop.mapIso X.isoSpec)) '' (SetLike.coe '' Set.range PrimeSpectrum.basicOpen)
-- Mathlib/AlgebraicGeometry/AffineScheme.lean
X : Scheme
inst✝ : IsAffine X
x✝ : Set ↑↑X.toPresheafedSpace
\u22a2 (∃ x ∈ Set.range X.basicOpen, ↑x = x✝) ↔ ∃ x, (∃ x_1 ∈ Set.range PrimeSpectrum.basicOpen, ↑x_1 = x) ∧ ⇑(TopCat.homeoOfIso (Scheme.forgetToTop.mapIso X.isoSpec)) ⁻¹' x = x✝
-- Mathlib/Analysis/InnerProductSpace/OfNorm.lean
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
hI : ¬I = 0
x y : E
\u22a2 inner_ 𝕜 (I • x) y = (starRingEnd 𝕜) I * inner_ 𝕜 x y
-- Mathlib/Analysis/InnerProductSpace/OfNorm.lean
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
hI : ¬I = 0
x y : E
hI' : -I * I = 1
\u22a2 inner_ 𝕜 (I • x) y = (starRingEnd 𝕜) I * inner_ 𝕜 x y
-- Mathlib/Analysis/InnerProductSpace/OfNorm.lean
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
hI : ¬I = 0
x y : E
hI' : -I * I = 1
\u22a2 4⁻¹ * (𝓚 ‖I • x + y‖ * 𝓚 ‖I • x + y‖ - 𝓚 ‖I • x - y‖ * 𝓚 ‖I • x - y‖ + I * 𝓚 ‖I • I • x + y‖ * 𝓚 ‖I • I • x + y‖ - I * 𝓚 ‖I • I • x - y‖ * 𝓚 ‖I • I • x - y‖) = 4⁻¹ * (-I * (𝓚 ‖x + y‖ * 𝓚 ‖x + y‖ - 𝓚 ‖x - y‖ * 𝓚 ‖x - y‖ + I * 𝓚 ‖I • x + y‖ * 𝓚 ‖I • x + y‖ - I * 𝓚 ‖I • x - y‖ * 𝓚 ‖I • x - y‖))
-- Mathlib/Analysis/InnerProductSpace/OfNorm.lean
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
hI : ¬I = 0
x y : E
hI' : -I * I = 1
\u22a2 𝓚 ‖I • x + y‖ * 𝓚 ‖I • x + y‖ - 𝓚 ‖I • x - y‖ * 𝓚 ‖I • x - y‖ + I * 𝓚 ‖I • I • x + y‖ * 𝓚 ‖I • I • x + y‖ - I * 𝓚 ‖I • I • x - y‖ * 𝓚 ‖I • I • x - y‖ = -I * (𝓚 ‖x + y‖ * 𝓚 ‖x + y‖ - 𝓚 ‖x - y‖ * 𝓚 ‖x - y‖ + I * 𝓚 ‖I • x + y‖ * 𝓚 ‖I • x + y‖ - I * 𝓚 ‖I • x - y‖ * 𝓚 ‖I • x - y‖)
-- Mathlib/Analysis/InnerProductSpace/OfNorm.lean
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
hI : ¬I = 0
x y : E
hI' : -I * I = 1
\u22a2 𝓚 ‖I • x + y‖ * 𝓚 ‖I • x + y‖ - 𝓚 ‖I • x - y‖ * 𝓚 ‖I • x - y‖ + I * 𝓚 ‖-x + y‖ * 𝓚 ‖-x + y‖ - I * 𝓚 ‖-x - y‖ * 𝓚 ‖-x - y‖ = -I * (𝓚 ‖x + y‖ * 𝓚 ‖x + y‖ - 𝓚 ‖x - y‖ * 𝓚 ‖x - y‖ + I * 𝓚 ‖I • x + y‖ * 𝓚 ‖I • x + y‖ - I * 𝓚 ‖I • x - y‖ * 𝓚 ‖I • x - y‖)