ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Algebra/MonoidAlgebra/Degree.lean
R : Type u_1
R' : Type u_2
A : Type u_3
T : Type u_4
B : Type u_5
ι : Type u_6
inst✝⁷ : Semiring R
inst✝⁶ : Ring R'
inst✝⁵ : AddZeroClass A
inst✝⁴ : SemilatticeSup B
inst✝³ : Add B
inst✝² : OrderBot B
D : A → B
p q : R[A]
hadd : ∀ (a1 a2 : A), D (a1 + a2) = D a1 + D a2
inst✝¹ : CovariantClass B B (fun x x_1 => x + x_1) fun x x_1 => x < x_1
inst✝ : CovariantClass B B (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1
hD : Function.Injective D
ap aq : A
hp : supDegree D p ≤ D ap
hq : supDegree D q ≤ D aq
\u22a2 (p * q) (ap + aq) = p ap * q aq
-- Mathlib/Algebra/MonoidAlgebra/Degree.lean
R : Type u_1
R' : Type u_2
A : Type u_3
T : Type u_4
B : Type u_5
ι : Type u_6
inst✝⁷ : Semiring R
inst✝⁶ : Ring R'
inst✝⁵ : AddZeroClass A
inst✝⁴ : SemilatticeSup B
inst✝³ : Add B
inst✝² : OrderBot B
D : A → B
p q : R[A]
hadd : ∀ (a1 a2 : A), D (a1 + a2) = D a1 + D a2
inst✝¹ : CovariantClass B B (fun x x_1 => x + x_1) fun x x_1 => x < x_1
inst✝ : CovariantClass B B (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1
hD : Function.Injective D
ap aq : A
hp : supDegree D p ≤ D ap
hq : supDegree D q ≤ D aq
\u22a2 (∑ x ∈ p.support, ∑ x_1 ∈ q.support, if x + x_1 = ap + aq then p x * q x_1 else 0) = p ap * q aq
-- Mathlib/Algebra/MonoidAlgebra/Degree.lean
R : Type u_1
R' : Type u_2
A : Type u_3
T : Type u_4
B : Type u_5
ι : Type u_6
inst✝⁷ : Semiring R
inst✝⁶ : Ring R'
inst✝⁵ : AddZeroClass A
inst✝⁴ : SemilatticeSup B
inst✝³ : Add B
inst✝² : OrderBot B
D : A → B
p q : R[A]
hadd : ∀ (a1 a2 : A), D (a1 + a2) = D a1 + D a2
inst✝¹ : CovariantClass B B (fun x x_1 => x + x_1) fun x x_1 => x < x_1
inst✝ : CovariantClass B B (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1
hD : Function.Injective D
ap aq : A
hp : supDegree D p ≤ D ap
hq : supDegree D q ≤ D aq
\u22a2 ∀ b ∈ q.support, b ≠ aq → (if ap + b = ap + aq then p ap * q b else 0) = 0
-- Mathlib/Algebra/MonoidAlgebra/Degree.lean
R : Type u_1
R' : Type u_2
A : Type u_3
T : Type u_4
B : Type u_5
ι : Type u_6
inst✝⁷ : Semiring R
inst✝⁶ : Ring R'
inst✝⁵ : AddZeroClass A
inst✝⁴ : SemilatticeSup B
inst✝³ : Add B
inst✝² : OrderBot B
D : A → B
p q : R[A]
hadd : ∀ (a1 a2 : A), D (a1 + a2) = D a1 + D a2
inst✝¹ : CovariantClass B B (fun x x_1 => x + x_1) fun x x_1 => x < x_1
inst✝ : CovariantClass B B (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1
hD : Function.Injective D
ap aq : A
hp : supDegree D p ≤ D ap
hq : supDegree D q ≤ D aq
\u22a2 aq ∉ q.support → (if ap + aq = ap + aq then p ap * q aq else 0) = 0
-- Mathlib/Algebra/MonoidAlgebra/Degree.lean
R : Type u_1
R' : Type u_2
A : Type u_3
T : Type u_4
B : Type u_5
ι : Type u_6
inst✝⁷ : Semiring R
inst✝⁶ : Ring R'
inst✝⁵ : AddZeroClass A
inst✝⁴ : SemilatticeSup B
inst✝³ : Add B
inst✝² : OrderBot B
D : A → B
p q : R[A]
hadd : ∀ (a1 a2 : A), D (a1 + a2) = D a1 + D a2
inst✝¹ : CovariantClass B B (fun x x_1 => x + x_1) fun x x_1 => x < x_1
inst✝ : CovariantClass B B (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1
hD : Function.Injective D
ap aq : A
hp : supDegree D p ≤ D ap
hq : supDegree D q ≤ D aq
\u22a2 ∀ b ∈ p.support, b ≠ ap → (∑ x ∈ q.support, if b + x = ap + aq then p b * q x else 0) = 0
-- Mathlib/Algebra/MonoidAlgebra/Degree.lean
R : Type u_1
R' : Type u_2
A : Type u_3
T : Type u_4
B : Type u_5
ι : Type u_6
inst✝⁷ : Semiring R
inst✝⁶ : Ring R'
inst✝⁵ : AddZeroClass A
inst✝⁴ : SemilatticeSup B
inst✝³ : Add B
inst✝² : OrderBot B
D : A → B
p q : R[A]
hadd : ∀ (a1 a2 : A), D (a1 + a2) = D a1 + D a2
inst✝¹ : CovariantClass B B (fun x x_1 => x + x_1) fun x x_1 => x < x_1
inst✝ : CovariantClass B B (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1
hD : Function.Injective D
ap aq : A
hp : supDegree D p ≤ D ap
hq : supDegree D q ≤ D aq
\u22a2 ap ∉ p.support → (∑ x ∈ q.support, if ap + x = ap + aq then p ap * q x else 0) = 0
-- Mathlib/Order/LiminfLimsup.lean
α : Type u_1
β : Type u_2
γ✝ : Type u_3
ι : Type u_4
ι' : Type u_5
γ : Type u_6
inst✝¹ : ConditionallyCompleteLattice β
inst✝ : ConditionallyCompleteLattice γ
f : Filter α
u : α → β
g : β ≃o γ
hu : autoParam (IsBoundedUnder (fun x x_1 => x ≤ x_1) f u) _auto✝
hu_co : autoParam (IsCoboundedUnder (fun x x_1 => x ≤ x_1) f u) _auto✝
hgu : autoParam (IsBoundedUnder (fun x x_1 => x ≤ x_1) f fun x => g (u x)) _auto✝
hgu_co : autoParam (IsCoboundedUnder (fun x x_1 => x ≤ x_1) f fun x => g (u x)) _auto✝
\u22a2 g (limsup u f) = limsup (fun x => g (u x)) f
-- Mathlib/Order/LiminfLimsup.lean
α : Type u_1
β : Type u_2
γ✝ : Type u_3
ι : Type u_4
ι' : Type u_5
γ : Type u_6
inst✝¹ : ConditionallyCompleteLattice β
inst✝ : ConditionallyCompleteLattice γ
f : Filter α
u : α → β
g : β ≃o γ
hu : autoParam (IsBoundedUnder (fun x x_1 => x ≤ x_1) f u) _auto✝
hu_co : autoParam (IsCoboundedUnder (fun x x_1 => x ≤ x_1) f u) _auto✝
hgu : autoParam (IsBoundedUnder (fun x x_1 => x ≤ x_1) f fun x => g (u x)) _auto✝
hgu_co : autoParam (IsCoboundedUnder (fun x x_1 => x ≤ x_1) f fun x => g (u x)) _auto✝
\u22a2 limsup (fun x => g (u x)) f ≤ g (limsup u f)
-- Mathlib/Order/LiminfLimsup.lean
α : Type u_1
β : Type u_2
γ✝ : Type u_3
ι : Type u_4
ι' : Type u_5
γ : Type u_6
inst✝¹ : ConditionallyCompleteLattice β
inst✝ : ConditionallyCompleteLattice γ
f : Filter α
u : α → β
g : β ≃o γ
hu : autoParam (IsBoundedUnder (fun x x_1 => x ≤ x_1) f u) _auto✝
hu_co : autoParam (IsCoboundedUnder (fun x x_1 => x ≤ x_1) f u) _auto✝
hgu : autoParam (IsBoundedUnder (fun x x_1 => x ≤ x_1) f fun x => g (u x)) _auto✝
hgu_co : autoParam (IsCoboundedUnder (fun x x_1 => x ≤ x_1) f fun x => g (u x)) _auto✝
\u22a2 g (g.symm (limsup (fun x => g (u x)) f)) ≤ g (limsup u f)
-- Mathlib/Order/LiminfLimsup.lean
α : Type u_1
β : Type u_2
γ✝ : Type u_3
ι : Type u_4
ι' : Type u_5
γ : Type u_6
inst✝¹ : ConditionallyCompleteLattice β
inst✝ : ConditionallyCompleteLattice γ
f : Filter α
u : α → β
g : β ≃o γ
hu : autoParam (IsBoundedUnder (fun x x_1 => x ≤ x_1) f u) _auto✝
hu_co : autoParam (IsCoboundedUnder (fun x x_1 => x ≤ x_1) f u) _auto✝
hgu : autoParam (IsBoundedUnder (fun x x_1 => x ≤ x_1) f fun x => g (u x)) _auto✝
hgu_co : autoParam (IsCoboundedUnder (fun x x_1 => x ≤ x_1) f fun x => g (u x)) _auto✝
\u22a2 g.symm (limsup (fun x => g (u x)) f) ≤ limsup u f
-- Mathlib/Order/LiminfLimsup.lean
α : Type u_1
β : Type u_2
γ✝ : Type u_3
ι : Type u_4
ι' : Type u_5
γ : Type u_6
inst✝¹ : ConditionallyCompleteLattice β
inst✝ : ConditionallyCompleteLattice γ
f : Filter α
u : α → β
g : β ≃o γ
hu : autoParam (IsBoundedUnder (fun x x_1 => x ≤ x_1) f u) _auto✝
hu_co : autoParam (IsCoboundedUnder (fun x x_1 => x ≤ x_1) f u) _auto✝
hgu : autoParam (IsBoundedUnder (fun x x_1 => x ≤ x_1) f fun x => g (u x)) _auto✝
hgu_co : autoParam (IsCoboundedUnder (fun x x_1 => x ≤ x_1) f fun x => g (u x)) _auto✝
hf : u = fun i => g.symm (g (u i))
\u22a2 g.symm (limsup (fun x => g (u x)) f) ≤ limsup u f
-- Mathlib/Order/LiminfLimsup.lean
α : Type u_1
β : Type u_2
γ✝ : Type u_3
ι : Type u_4
ι' : Type u_5
γ : Type u_6
inst✝¹ : ConditionallyCompleteLattice β
inst✝ : ConditionallyCompleteLattice γ
f : Filter α
u : α → β
g : β ≃o γ
hu : autoParam (IsBoundedUnder (fun x x_1 => x ≤ x_1) f u) _auto✝
hu_co : autoParam (IsCoboundedUnder (fun x x_1 => x ≤ x_1) f u) _auto✝
hgu : autoParam (IsBoundedUnder (fun x x_1 => x ≤ x_1) f fun x => g (u x)) _auto✝
hgu_co : autoParam (IsCoboundedUnder (fun x x_1 => x ≤ x_1) f fun x => g (u x)) _auto✝
hf : u = fun i => g.symm (g (u i))
\u22a2 g.symm (limsup (fun x => g (u x)) f) ≤ limsup (fun i => g.symm (g (u i))) f
-- Mathlib/Order/LiminfLimsup.lean
α : Type u_1
β : Type u_2
γ✝ : Type u_3
ι : Type u_4
ι' : Type u_5
γ : Type u_6
inst✝¹ : ConditionallyCompleteLattice β
inst✝ : ConditionallyCompleteLattice γ
f : Filter α
u : α → β
g : β ≃o γ
hu : autoParam (IsBoundedUnder (fun x x_1 => x ≤ x_1) f u) _auto✝
hu_co : autoParam (IsCoboundedUnder (fun x x_1 => x ≤ x_1) f u) _auto✝
hgu : autoParam (IsBoundedUnder (fun x x_1 => x ≤ x_1) f fun x => g (u x)) _auto✝
hgu_co : autoParam (IsCoboundedUnder (fun x x_1 => x ≤ x_1) f fun x => g (u x)) _auto✝
hf : u = fun i => g.symm (g (u i))
\u22a2 IsBoundedUnder (fun x x_1 => x ≤ x_1) f fun x => g.symm (g (u x))
-- Mathlib/Order/LiminfLimsup.lean
α : Type u_1
β : Type u_2
γ✝ : Type u_3
ι : Type u_4
ι' : Type u_5
γ : Type u_6
inst✝¹ : ConditionallyCompleteLattice β
inst✝ : ConditionallyCompleteLattice γ
f : Filter α
u : α → β
g : β ≃o γ
hu : autoParam (IsBoundedUnder (fun x x_1 => x ≤ x_1) f u) _auto✝
hu_co : autoParam (IsCoboundedUnder (fun x x_1 => x ≤ x_1) f u) _auto✝
hgu : autoParam (IsBoundedUnder (fun x x_1 => x ≤ x_1) f fun x => g (u x)) _auto✝
hgu_co : autoParam (IsCoboundedUnder (fun x x_1 => x ≤ x_1) f fun x => g (u x)) _auto✝
hf : u = fun i => g.symm (g (u i))
\u22a2 IsBoundedUnder (fun x x_1 => x ≤ x_1) f fun x => u x
-- 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⁻¹ ↔ b ≤ a
-- Mathlib/Order/Filter/Cocardinal.lean
ι α β : Type u
c : Cardinal.{u}
hreg : c.IsRegular
l : Filter α
s : Set α
x✝ : ∃ i, {s | #↑s < c} i ∧ iᶜ ⊆ s
_t : Set α
htf : {s | #↑s < c} _t
hts : _tᶜ ⊆ s
\u22a2 s ∈ cocardinal α hreg
-- Mathlib/Order/Filter/Cocardinal.lean
ι α β : Type u
c : Cardinal.{u}
hreg : c.IsRegular
l : Filter α
s : Set α
x✝ : ∃ i, {s | #↑s < c} i ∧ iᶜ ⊆ s
_t : Set α
htf : {s | #↑s < c} _t
hts : _tᶜ ⊆ s
this : #↑sᶜ < c
\u22a2 s ∈ cocardinal α hreg
-- Mathlib/Data/Matrix/Composition.lean
I : Type u_1
J : Type u_2
K : Type u_3
L : Type u_4
R : Type u_5
inst✝⁴ : Semiring R
inst✝³ : Fintype I
inst✝² : Fintype J
inst✝¹ : DecidableEq I
inst✝ : DecidableEq J
x✝¹ x✝ : Matrix I I (Matrix J J R)
i✝ j✝ : I × J
\u22a2 __spread✝⁻⁰.toFun (x✝¹ * x✝) i✝ j✝ = (__spread✝⁻⁰.toFun x✝¹ * __spread✝⁻⁰.toFun x✝) i✝ j✝
-- Mathlib/GroupTheory/Submonoid/Inverses.lean
M : Type u_1
inst✝ : CommMonoid M
S : Submonoid M
a : ↥S.leftInv
b : M
\u22a2 ↑(S.fromLeftInv a) = b ↔ ↑a * b = 1
-- Mathlib/MeasureTheory/Function/LpSpace.lean
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
\u22a2 ‖0‖₊ = 0
-- Mathlib/MeasureTheory/Function/LpSpace.lean
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
\u22a2 (eLpNorm (↑↑0) p μ).toNNReal = 0
-- Mathlib/MeasureTheory/Function/LpSpace.lean
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
\u22a2 (eLpNorm (↑0) p μ).toNNReal = 0
-- Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean
P : MorphismProperty Scheme
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝ : HasRingHomProperty P Q
X✝ Y✝ Z✝ : Scheme
f✝ : X✝ ⟶ Y✝
g✝ : Y✝ ⟶ Z✝
H : ∀ {R S T : Type u} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] (f : R →+* S) (g : S →+* T), Q (g.comp f) → Q g
Z : Scheme
hZ : IsAffine Z
Y : Scheme
g : Y ⟶ Z
hY✝ : IsAffine Y
X : Scheme
f : X ⟶ Y
h : P (f ≫ g)
hY : IsAffine X
\u22a2 P f
-- Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean
P : MorphismProperty Scheme
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝ : HasRingHomProperty P Q
X✝ Y✝ Z✝ : Scheme
f✝ : X✝ ⟶ Y✝
g✝ : Y✝ ⟶ Z✝
H : ∀ {R S T : Type u} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] (f : R →+* S) (g : S →+* T), Q (g.comp f) → Q g
Z : Scheme
hZ : IsAffine Z
Y : Scheme
g : Y ⟶ Z
hY✝ : IsAffine Y
X : Scheme
f : X ⟶ Y
h : Q (Scheme.Hom.app (f ≫ g) ⊤)
hY : IsAffine X
\u22a2 Q (Scheme.Hom.app f ⊤)
-- Mathlib/Topology/Connected/PathConnected.lean
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x✝ y z : X
ι : Type u_3
F : Set X
p : ι → Prop
s : X → ι → Set X
h : ∀ (x : X), (𝓝 x).HasBasis p (s x)
h' : ∀ (x : X) (i : ι), p i → IsPathConnected (s x i)
x : X
\u22a2 (𝓝 x).HasBasis (fun s => s ∈ 𝓝 x ∧ IsPathConnected s) id
-- Mathlib/Topology/Connected/PathConnected.lean
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x✝ y z : X
ι : Type u_3
F : Set X
p : ι → Prop
s : X → ι → Set X
h : ∀ (x : X), (𝓝 x).HasBasis p (s x)
h' : ∀ (x : X) (i : ι), p i → IsPathConnected (s x i)
x : X
\u22a2 ∀ (i : ι), p i → ∃ i', (i' ∈ 𝓝 x ∧ IsPathConnected i') ∧ id i' ⊆ s x i
-- Mathlib/Topology/Connected/PathConnected.lean
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x✝ y z : X
ι : Type u_3
F : Set X
p : ι → Prop
s : X → ι → Set X
h : ∀ (x : X), (𝓝 x).HasBasis p (s x)
h' : ∀ (x : X) (i : ι), p i → IsPathConnected (s x i)
x : X
\u22a2 ∀ (i' : Set X), i' ∈ 𝓝 x ∧ IsPathConnected i' → ∃ i, p i ∧ s x i ⊆ id i'
-- Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean
a : ℤ
b : ℕ
\u22a2 {0, 1, -1} = ↑(MonoidHom.mrange ↑SignType.castHom)
-- Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean
a : ℤ
b : ℕ
\u22a2 {0, -1, 1} = ↑(MonoidHom.mrange ↑SignType.castHom)
-- Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean
a : ℤ
b : ℕ
x✝ : ℤ
ha' : x✝ ∈ List.pmap (fun p pp => legendreSym p a) b.primeFactorsList ⋯
\u22a2 x✝ ∈ (MonoidHom.mrange ↑SignType.castHom).copy {0, 1, -1} ⋯
-- Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean
a : ℤ
b p : ℕ
hp : p ∈ b.primeFactorsList
ha' : legendreSym p a ∈ List.pmap (fun p pp => legendreSym p a) b.primeFactorsList ⋯
\u22a2 legendreSym p a ∈ (MonoidHom.mrange ↑SignType.castHom).copy {0, 1, -1} ⋯
-- Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean
a : ℤ
b p : ℕ
hp : p ∈ b.primeFactorsList
ha' : legendreSym p a ∈ List.pmap (fun p pp => legendreSym p a) b.primeFactorsList ⋯
this : Fact (Nat.Prime p)
\u22a2 legendreSym p a ∈ (MonoidHom.mrange ↑SignType.castHom).copy {0, 1, -1} ⋯
-- Mathlib/Topology/Homotopy/HomotopyGroup.lean
N : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace X
x : X
inst✝ : DecidableEq N
i j : N
h : i ≠ j
a b c d : ↑(Ω^ N X x)
y✝ : N → ↑I
\u22a2 (transAt i (transAt j a b) (transAt j c d)) y✝ = (transAt j (transAt i a c) (transAt i b d)) y✝
-- Mathlib/Topology/Homotopy/HomotopyGroup.lean
N : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace X
x : X
inst✝ : DecidableEq N
i j : N
h : i ≠ j
a b c d : ↑(Ω^ N X x)
y✝ : N → ↑I
\u22a2 (if ↑(y✝ i) ≤ 1 / 2 then if ↑(y✝ j) ≤ 1 / 2 then a (Function.update (Function.update y✝ i (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ i)))) j (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ j)))) else b (Function.update (Function.update y✝ i (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ i)))) j (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ j) - 1))) else if ↑(y✝ j) ≤ 1 / 2 then c (Function.update (Function.update y✝ i (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ i) - 1))) j (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ j)))) else d (Function.update (Function.update y✝ i (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ i) - 1))) j (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ j) - 1)))) = if ↑(y✝ j) ≤ 1 / 2 then if ↑(y✝ i) ≤ 1 / 2 then a (Function.update (Function.update y✝ j (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ j)))) i (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ i)))) else c (Function.update (Function.update y✝ j (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ j)))) i (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ i) - 1))) else if ↑(y✝ i) ≤ 1 / 2 then b (Function.update (Function.update y✝ j (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ j) - 1))) i (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ i)))) else d (Function.update (Function.update y✝ j (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ j) - 1))) i (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ i) - 1)))
-- Mathlib/Data/ENNReal/Inv.lean
a b c d : ℝ≥0∞
r p q : ℝ≥0
\u22a2 a * b / c = a / c * b
-- Mathlib/Algebra/Star/NonUnitalSubalgebra.lean
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : StarRing R
inst✝¹⁴ : NonUnitalSemiring A
inst✝¹³ : StarRing A
inst✝¹² : Module R A
inst✝¹¹ : IsScalarTower R A A
inst✝¹⁰ : SMulCommClass R A A
inst✝⁹ : StarModule R A
inst✝⁸ : NonUnitalSemiring B
inst✝⁷ : StarRing B
inst✝⁶ : Module R B
inst✝⁵ : IsScalarTower R B B
inst✝⁴ : SMulCommClass R B B
inst✝³ : StarModule R B
inst✝² : FunLike F A B
inst✝¹ : NonUnitalAlgHomClass F R A B
inst✝ : NonUnitalStarAlgHomClass F R A B
s : Set A
S : NonUnitalStarSubalgebra R A
\u22a2 adjoin R s ≤ S ↔ s ≤ ↑S
-- Mathlib/Algebra/Star/NonUnitalSubalgebra.lean
F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : StarRing R
inst✝¹⁴ : NonUnitalSemiring A
inst✝¹³ : StarRing A
inst✝¹² : Module R A
inst✝¹¹ : IsScalarTower R A A
inst✝¹⁰ : SMulCommClass R A A
inst✝⁹ : StarModule R A
inst✝⁸ : NonUnitalSemiring B
inst✝⁷ : StarRing B
inst✝⁶ : Module R B
inst✝⁵ : IsScalarTower R B B
inst✝⁴ : SMulCommClass R B B
inst✝³ : StarModule R B
inst✝² : FunLike F A B
inst✝¹ : NonUnitalAlgHomClass F R A B
inst✝ : NonUnitalStarAlgHomClass F R A B
s : Set A
S : NonUnitalStarSubalgebra R A
\u22a2 s ∪ star s ⊆ ↑S ↔ s ≤ ↑S
-- Mathlib/Topology/Bornology/Absorbs.lean
G₀ : Type u_1
α : Type u_2
inst✝⁵ : GroupWithZero G₀
inst✝⁴ : Bornology G₀
inst✝³ : MulAction G₀ α
s✝ t u : Set α
S : Set (Set α)
inst✝² : (cobounded G₀).NeBot
E : Type u_3
inst✝¹ : AddMonoid E
inst✝ : DistribMulAction G₀ E
s : Set E
\u22a2 Absorbs G₀ s 0 ↔ 0 ∈ s
-- Mathlib/CategoryTheory/Sites/LocallySurjective.lean
C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝¹ : Category.{v', u'} A
inst✝ : ConcreteCategory A
F₁ F₂ F₃ : Sheaf J A
φ : F₁ ⟶ F₂
ψ : F₂ ⟶ F₃
\u22a2 Presheaf.IsLocallySurjective J ((sheafToPresheaf J A).map φ) ↔ IsLocallySurjective φ
-- Mathlib/Algebra/Polynomial/RingDivision.lean
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
p q : R[X]
h : IsUnit p
a✝ : Nontrivial R
\u22a2 p.natDegree = 0
-- Mathlib/Algebra/Polynomial/RingDivision.lean
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
p q✝ : R[X]
h : IsUnit p
a✝ : Nontrivial R
q : R[X]
hq : p * q = 1
\u22a2 p.natDegree = 0
-- Mathlib/Algebra/Polynomial/RingDivision.lean
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
p q✝ : R[X]
h : IsUnit p
a✝ : Nontrivial R
q : R[X]
hq : p * q = 1
this : (p * q).natDegree = p.natDegree + q.natDegree
\u22a2 p.natDegree = 0
-- Mathlib/Algebra/Polynomial/RingDivision.lean
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
p q✝ : R[X]
h : IsUnit p
a✝ : Nontrivial R
q : R[X]
hq : p * q = 1
this : p.natDegree = 0 ∧ q.natDegree = 0
\u22a2 p.natDegree = 0
-- Mathlib/CategoryTheory/Functor/Flat.lean
C D : Type u₁
inst✝⁹ : SmallCategory C
inst✝⁸ : SmallCategory D
E : Type u₂
inst✝⁷ : Category.{u₁, u₂} E
inst✝⁶ : ConcreteCategory E
inst✝⁵ : HasLimits E
inst✝⁴ : HasColimits E
inst✝³ : ReflectsLimits (forget E)
inst✝² : PreservesFilteredColimits (forget E)
inst✝¹ : PreservesLimits (forget E)
inst✝ : HasFiniteLimits C
F : C ⥤ D
H : RepresentablyFlat F.op.lan
this✝ : PreservesFiniteLimits F.op.lan
this : PreservesFiniteLimits F
\u22a2 RepresentablyFlat F
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Basic.lean
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
h : IsUnit a
a✝ : Nontrivial A
\u22a2 IsUnit ⟨a, ⋯⟩
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Basic.lean
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
h : IsUnit a
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
\u22a2 IsUnit a'
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Basic.lean
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
h : IsUnit a
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
\u22a2 IsUnit (star a' * a')
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Basic.lean
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
\u22a2 IsUnit (star a' * a')
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Basic.lean
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
\u22a2 IsUnit (star a' * a')
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Basic.lean
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ := (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
\u22a2 ‖star a' * a' - ↑u‖ < ‖↑u⁻¹‖⁻¹
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Basic.lean
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ := (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
\u22a2 ‖star a' * a' - (algebraMap ℂ ↥(elementalStarAlgebra ℂ a)) ↑‖star a * a‖‖ < ‖star a * a‖
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Basic.lean
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ := (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
h₂ : ∀ z ∈ spectrum ℂ ((algebraMap ℂ A) ↑‖star a * a‖ - star a * a), ‖z‖₊ < ‖star a * a‖₊
\u22a2 ‖star a' * a' - (algebraMap ℂ ↥(elementalStarAlgebra ℂ a)) ↑‖star a * a‖‖ < ‖star a * a‖
-- Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean
ι : Sort u_1
R : Type u_2
A : Type u_3
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set Aᵐᵒᵖ
\u22a2 (Algebra.adjoin R s).unop = Algebra.adjoin R (MulOpposite.op ⁻¹' s)
-- Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean
ι : Sort u_1
R : Type u_2
A : Type u_3
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set Aᵐᵒᵖ
\u22a2 (Algebra.adjoin R s).unop.toSubsemiring = (Algebra.adjoin R (MulOpposite.op ⁻¹' s)).toSubsemiring
-- Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean
ι : Sort u_1
R : Type u_2
A : Type u_3
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set Aᵐᵒᵖ
\u22a2 Subsemiring.closure (MulOpposite.op ⁻¹' Set.range ⇑(algebraMap R Aᵐᵒᵖ) ∪ MulOpposite.op ⁻¹' s) = Subsemiring.closure (Set.range ⇑(algebraMap R A) ∪ MulOpposite.op ⁻¹' s)
-- Mathlib/Analysis/Fourier/PoissonSummation.lean
f : C(ℝ, ℂ)
h_norm : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight ↑n))‖
h_sum : Summable fun n => 𝓕 ⇑f ↑n
x : ℝ
\u22a2 ∑' (n : ℤ), f (x + ↑n) = ∑' (n : ℤ), 𝓕 ⇑f ↑n * (fourier n) ↑x