ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean
R : Type u_1
A : Type u_2
B : Type u_3
S : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing A
inst✝⁸ : Ring B
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R B
f✝ : R →+* S
M : Type u_5
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Module A M
inst✝¹ : IsScalarTower R A M
inst✝ : NoZeroSMulDivisors A M
N : Submodule R M
hN : N ≠ ⊥
hN' : N.FG
x : A
hx : ∀ n ∈ N, x • n ∈ N
A' : Subalgebra R A := { carrier := {x | ∀ n ∈ N, x • n ∈ N}, mul_mem' := ⋯, one_mem' := ⋯, add_mem' := ⋯, zero_mem' := ⋯, algebraMap_mem' := ⋯ }
f : ↥A' →ₐ[R] Module.End R ↥N := AlgHom.ofLinearMap { toFun := fun x => (DistribMulAction.toLinearMap R M x).restrict ⋯, map_add' := ⋯, map_smul' := ⋯ } ⋯ ⋯
a : M
ha₁ : a ∈ N
ha₂ : a ≠ 0
this✝ : Function.Injective ⇑f
this : Module.Finite R ↥N
\u22a2 IsIntegral R (f ⟨x, hx⟩)
-- Mathlib/Analysis/Calculus/LagrangeMultipliers.lean
E : Type u_1
F : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : CompleteSpace F
f✝ : E → F
φ : E → ℝ
x₀ : E
f'✝ : E →L[ℝ] F
φ' : E →L[ℝ] ℝ
ι : Type u_3
inst✝ : Fintype ι
f : ι → E → ℝ
f' : ι → E →L[ℝ] ℝ
hextr : IsLocalExtrOn φ {x | ∀ (i : ι), f i x = f i x₀} x₀
hf' : ∀ (i : ι), HasStrictFDerivAt (f i) (f' i) x₀
hφ' : HasStrictFDerivAt φ φ' x₀
this : DecidableEq ι := Classical.decEq ι
\u22a2 ∃ Λ Λ₀, (Λ, Λ₀) ≠ 0 ∧ ∑ i : ι, Λ i • f' i + Λ₀ • φ' = 0
-- Mathlib/Analysis/Calculus/LagrangeMultipliers.lean
E : Type u_1
F : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : CompleteSpace F
f✝ : E → F
φ : E → ℝ
x₀ : E
f'✝ : E →L[ℝ] F
φ' : E →L[ℝ] ℝ
ι : Type u_3
inst✝ : Fintype ι
f : ι → E → ℝ
f' : ι → E →L[ℝ] ℝ
hf' : ∀ (i : ι), HasStrictFDerivAt (f i) (f' i) x₀
hφ' : HasStrictFDerivAt φ φ' x₀
this : DecidableEq ι := Classical.decEq ι
hextr : IsLocalExtrOn φ {x | (fun i => f i x) = fun i => f i x₀} x₀
\u22a2 ∃ Λ Λ₀, (Λ, Λ₀) ≠ 0 ∧ ∑ i : ι, Λ i • f' i + Λ₀ • φ' = 0
-- Mathlib/Analysis/Calculus/LagrangeMultipliers.lean
E : Type u_1
F : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : CompleteSpace F
f✝ : E → F
φ : E → ℝ
x₀ : E
f'✝ : E →L[ℝ] F
φ' : E →L[ℝ] ℝ
ι : Type u_3
inst✝ : Fintype ι
f : ι → E → ℝ
f' : ι → E →L[ℝ] ℝ
hf' : ∀ (i : ι), HasStrictFDerivAt (f i) (f' i) x₀
hφ' : HasStrictFDerivAt φ φ' x₀
this : DecidableEq ι := Classical.decEq ι
hextr : IsLocalExtrOn φ {x | (fun i => f i x) = fun i => f i x₀} x₀
Λ : Module.Dual ℝ (ι → ℝ)
Λ₀ : ℝ
h0 : (Λ, Λ₀) ≠ 0
hsum : ∀ (x : E), Λ ((ContinuousLinearMap.pi f') x) + Λ₀ • φ' x = 0
\u22a2 ∃ Λ Λ₀, (Λ, Λ₀) ≠ 0 ∧ ∑ i : ι, Λ i • f' i + Λ₀ • φ' = 0
-- Mathlib/Analysis/Calculus/LagrangeMultipliers.lean
E : Type u_1
F : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : CompleteSpace F
f✝ : E → F
φ : E → ℝ
x₀ : E
f'✝ : E →L[ℝ] F
φ' : E →L[ℝ] ℝ
ι : Type u_3
inst✝ : Fintype ι
f : ι → E → ℝ
f' : ι → E →L[ℝ] ℝ
hf' : ∀ (i : ι), HasStrictFDerivAt (f i) (f' i) x₀
hφ' : HasStrictFDerivAt φ φ' x₀
this : DecidableEq ι := Classical.decEq ι
hextr : IsLocalExtrOn φ {x | (fun i => f i x) = fun i => f i x₀} x₀
Λ₀ : ℝ
Λ : ι → ℝ
h0 : ((LinearEquiv.piRing ℝ ℝ ι ℝ).symm Λ, Λ₀) ≠ 0
hsum : ∀ (x : E), ((LinearEquiv.piRing ℝ ℝ ι ℝ).symm Λ) ((ContinuousLinearMap.pi f') x) + Λ₀ • φ' x = 0
\u22a2 ∃ Λ Λ₀, (Λ, Λ₀) ≠ 0 ∧ ∑ i : ι, Λ i • f' i + Λ₀ • φ' = 0
-- Mathlib/Algebra/Polynomial/Laurent.lean
R : Type u_1
inst✝¹ : Semiring R
inst✝ : DecidableEq R
n : ℤ
a : R
\u22a2 (C a * T n).degree = if a = 0 then ⊥ else ↑n
-- Mathlib/Data/List/Chain.lean
α : Type u
β : Type v
R r : α → α → Prop
l l₁ l₂ : List α
a b : α
\u22a2 Chain R a (l ++ [b]) ↔ Forall₂ R (a :: l) (l ++ [b])
-- 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
f g : ↥(Lp E p μ)
\u22a2 dist f g = (eLpNorm (↑↑f - ↑↑g) p μ).toReal
-- 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
f g : ↥(Lp E p μ)
\u22a2 (eLpNorm (↑↑(f - g)) p μ).toReal = (eLpNorm (↑↑f - ↑↑g) p μ).toReal
-- 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
f g : ↥(Lp E p μ)
\u22a2 eLpNorm (↑↑(f - g)) p μ = eLpNorm (↑↑f - ↑↑g) p μ
-- Mathlib/NumberTheory/DirichletCharacter/Basic.lean
R : Type u_1
inst✝ : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
\u22a2 χ.FactorsThrough 1 ↔ χ = 1
-- Mathlib/NumberTheory/DirichletCharacter/Basic.lean
R : Type u_1
inst✝ : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
x✝ : χ.FactorsThrough 1
w✝ : 1 ∣ n
χ₀ : DirichletCharacter R 1
hχ₀ : χ = (changeLevel w✝) χ₀
\u22a2 χ = 1
-- Mathlib/CategoryTheory/Abelian/ProjectiveResolution.lean
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
Y Z : C
f : Y ⟶ Z
P : ProjectiveResolution Y
Q : ProjectiveResolution Z
\u22a2 liftFOne f P Q ≫ Q.complex.d 1 0 = P.complex.d 1 0 ≫ liftFZero f P Q
-- Mathlib/Data/Set/Pointwise/Basic.lean
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝ : Monoid α
s t : Set α
a : α
m n✝ n : ℕ
hn : n ≠ 0
\u22a2 ∅ ^ n = ∅
-- Mathlib/Combinatorics/Enumerative/Composition.lean
n : ℕ
c : CompositionAsSet n
i : ℕ
IH : ∀ (h : i < c.boundaries.card), (take i c.blocks).sum = ↑(c.boundary ⟨i, h⟩)
h : i + 1 < c.boundaries.card
\u22a2 (take (i + 1) c.blocks).sum = ↑(c.boundary ⟨i + 1, h⟩)
-- Mathlib/Combinatorics/Enumerative/Composition.lean
n : ℕ
c : CompositionAsSet n
i : ℕ
IH : ∀ (h : i < c.boundaries.card), (take i c.blocks).sum = ↑(c.boundary ⟨i, h⟩)
h : i + 1 < c.boundaries.card
A : i < c.blocks.length
\u22a2 (take (i + 1) c.blocks).sum = ↑(c.boundary ⟨i + 1, h⟩)
-- Mathlib/Combinatorics/Enumerative/Composition.lean
n : ℕ
c : CompositionAsSet n
i : ℕ
IH : ∀ (h : i < c.boundaries.card), (take i c.blocks).sum = ↑(c.boundary ⟨i, h⟩)
h : i + 1 < c.boundaries.card
A : i < c.blocks.length
B : i < c.boundaries.card
\u22a2 (take (i + 1) c.blocks).sum = ↑(c.boundary ⟨i + 1, h⟩)
-- Mathlib/Combinatorics/Enumerative/Composition.lean
n : ℕ
c : CompositionAsSet n
i : ℕ
IH : ∀ (h : i < c.boundaries.card), (take i c.blocks).sum = ↑(c.boundary ⟨i, h⟩)
h : i + 1 < c.boundaries.card
A : i < c.blocks.length
B : i < c.boundaries.card
\u22a2 ↑(c.boundary ⟨i, B⟩) + c.blocks[i] = ↑(c.boundary ⟨i + 1, h⟩)
-- Mathlib/SetTheory/Cardinal/ENat.lean
x : Cardinal.{u_1}
n : ℕ
\u22a2 x.toENatAux ≤ ↑n ↔ x ≤ ↑n
-- Mathlib/Analysis/Normed/Group/Basic.lean
𝓕 : Type u_1
𝕜 : Type u_2
α : Type u_3
ι : Type u_4
κ : Type u_5
E : Type u_6
F : Type u_7
G : Type u_8
inst✝² : SeminormedGroup E
inst✝¹ : SeminormedGroup F
inst✝ : SeminormedGroup G
s : Set E
a✝ a₁ a₂ b b₁ b₂ : E
r r₁ r₂ : ℝ
a : E
\u22a2 ‖a⁻¹‖ = ‖a‖
-- Mathlib/NumberTheory/ModularForms/EisensteinSeries/MDifferentiable.lean
k : ℤ
a : Fin 2 → ℤ
\u22a2 DifferentiableOn ℂ (eisSummand k a ∘ ↑ofComplex) {z | 0 < z.im}
-- Mathlib/NumberTheory/ModularForms/EisensteinSeries/MDifferentiable.lean
k : ℤ
a : Fin 2 → ℤ
\u22a2 ∀ x ∈ {z | 0 < z.im}, (eisSummand k a ∘ ↑ofComplex) x = (↑(a 0) * x + ↑(a 1)) ^ (-k)
-- Mathlib/NumberTheory/ModularForms/EisensteinSeries/MDifferentiable.lean
k : ℤ
a : Fin 2 → ℤ
z : ℍ
\u22a2 (eisSummand k a ∘ ↑ofComplex) ↑z = (↑(a 0) * ↑z + ↑(a 1)) ^ (-k)
-- Mathlib/Data/PNat/Prime.lean
m n : ℕ+
k l : ℕ
h : m.Coprime n
\u22a2 (↑m ^ k).Coprime (↑n ^ l)
-- Mathlib/Data/PNat/Prime.lean
m n : ℕ+
k l : ℕ
h : (↑m).Coprime ↑n
\u22a2 (↑m ^ k).Coprime (↑n ^ l)
-- Mathlib/Data/PNat/Prime.lean
m n : ℕ+
k l : ℕ
h : (↑m).Coprime ↑n
\u22a2 (↑m).Coprime ↑n
-- Mathlib/NumberTheory/SmoothNumbers.lean
m : ℕ
\u22a2 m ∈ factoredNumbers ∅ ↔ m ∈ {1}
-- Mathlib/GroupTheory/GroupAction/FixedPoints.lean
α : Type u_1
G : Type u_2
inst✝³ : Group G
inst✝² : MulAction G α
M : Type u_3
inst✝¹ : Monoid M
inst✝ : MulAction M α
g h : G
a : α
\u22a2 a ∈ h • fixedBy α g ↔ a ∈ fixedBy α (h * g * h⁻¹)
-- Mathlib/Data/Nat/Defs.lean
a b c d m n✝ k : ℕ
p q : ℕ → Prop
x : ℕ
x✝ : ∃ n, n * n = x
n : ℕ
hn : n * n = x
\u22a2 x.sqrt * x.sqrt = x
-- Mathlib/Combinatorics/SimpleGraph/Density.lean
𝕜 : Type u_1
ι : Type u_2
κ : Type u_3
α : Type u_4
β : Type u_5
r : α → α → Prop
inst✝ : DecidableRel r
s s₁ s₂ t t₁ t₂ : Finset α
a b : α
hr : Symmetric r
x : α × α
\u22a2 x.swap ∈ interedges r s t ↔ x ∈ interedges r t s
-- Mathlib/Combinatorics/SimpleGraph/Density.lean
𝕜 : Type u_1
ι : Type u_2
κ : Type u_3
α : Type u_4
β : Type u_5
r : α → α → Prop
inst✝ : DecidableRel r
s s₁ s₂ t t₁ t₂ : Finset α
a b : α
hr : Symmetric r
x : α × α
\u22a2 x.swap.1 ∈ s ∧ x.swap.2 ∈ t ∧ r x.swap.2 x.swap.1 ↔ x.1 ∈ t ∧ x.2 ∈ s ∧ r x.1 x.2
-- Mathlib/Analysis/SpecificLimits/FloorPow.lean
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim : ∀ (a : ℝ), 1 < a → ∃ c, (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧ Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
\u22a2 Tendsto (fun n => u n / ↑n) atTop (𝓝 l)
-- Mathlib/Analysis/SpecificLimits/FloorPow.lean
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim : ∀ (a : ℝ), 1 < a → ∃ c, (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧ Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
\u22a2 Tendsto (fun n => u n / ↑n) atTop (𝓝 l)
-- Mathlib/Analysis/SpecificLimits/FloorPow.lean
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim : ∀ (a : ℝ), 1 < a → ∃ c, (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧ Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
\u22a2 Tendsto (fun n => u n / ↑n) atTop (𝓝 l)
-- Mathlib/Analysis/SpecificLimits/FloorPow.lean
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim : ∀ (a : ℝ), 1 < a → ∃ c, (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧ Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
\u22a2 Tendsto (fun n => u n / ↑n) atTop (𝓝 l)
-- Mathlib/Analysis/SpecificLimits/FloorPow.lean
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim : ∀ (a : ℝ), 1 < a → ∃ c, (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧ Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d < l
\u22a2 ∀ᶠ (b : ℕ) in atTop, d < u b / ↑b
-- Mathlib/Analysis/SpecificLimits/FloorPow.lean
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim : ∀ (a : ℝ), 1 < a → ∃ c, (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧ Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d > l
\u22a2 ∀ᶠ (b : ℕ) in atTop, u b / ↑b < d
-- Mathlib/RingTheory/WittVector/Truncated.lean
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
R : Type u_1
inst✝ : CommRing R
i : Fin n
\u22a2 coeff i (truncateFun n 0) = 0
-- Mathlib/Algebra/Order/Group/Abs.lean
α : Type u_1
inst✝ : LinearOrderedAddCommGroup α
a c d : α
\u22a2 |a - a| ≤ 0
-- Mathlib/Algebra/Module/Torsion.lean
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set R
a : R
h : torsionBySet R M s = ⊤
x : M
\u22a2 ∀ ⦃a : ↑s⦄, ↑a • x = 0
-- Mathlib/Algebra/Module/Torsion.lean
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set R
a : R
h : torsionBySet R M s = ⊤
x : M
\u22a2 x ∈ ⊤
-- Mathlib/Algebra/Group/Defs.lean
G : Type u_1
M : Type u_2
inst✝ : Monoid M
a✝ b c : M
m n : ℕ
a : M
\u22a2 a ^ 1 = a
-- Mathlib/Topology/Gluing.lean
D : GlueData
i : D.J
a : ↑(D.U i)
j : D.J
b : ↑(D.U j)
k : D.J
c : ↑(D.U k)
x : ↑(D.V (⟨i, a⟩.fst, ⟨j, b⟩.fst))
e₁ : (D.f ⟨i, a⟩.fst ⟨j, b⟩.fst) x = ⟨i, a⟩.snd
e₂ : (D.f ⟨j, b⟩.fst ⟨i, a⟩.fst) ((D.t ⟨i, a⟩.fst ⟨j, b⟩.fst) x) = ⟨j, b⟩.snd
y : ↑(D.V (⟨j, b⟩.fst, ⟨k, c⟩.fst))
e₃ : (D.f ⟨j, b⟩.fst ⟨k, c⟩.fst) y = ⟨j, b⟩.snd
e₄ : (D.f ⟨k, c⟩.fst ⟨j, b⟩.fst) ((D.t ⟨j, b⟩.fst ⟨k, c⟩.fst) y) = ⟨k, c⟩.snd
\u22a2 D.Rel ⟨i, a⟩ ⟨k, c⟩
-- Mathlib/Topology/Gluing.lean
D : GlueData
i : D.J
a : ↑(D.U i)
j : D.J
b : ↑(D.U j)
k : D.J
c : ↑(D.U k)
x : ↑(D.V (⟨i, a⟩.fst, ⟨j, b⟩.fst))
e₁ : (D.f ⟨i, a⟩.fst ⟨j, b⟩.fst) x = ⟨i, a⟩.snd
e₂ : (D.f ⟨j, b⟩.fst ⟨i, a⟩.fst) ((D.t ⟨i, a⟩.fst ⟨j, b⟩.fst) x) = ⟨j, b⟩.snd
y : ↑(D.V (⟨j, b⟩.fst, ⟨k, c⟩.fst))
e₃ : (D.f ⟨j, b⟩.fst ⟨k, c⟩.fst) y = ⟨j, b⟩.snd
e₄ : (D.f ⟨k, c⟩.fst ⟨j, b⟩.fst) ((D.t ⟨j, b⟩.fst ⟨k, c⟩.fst) y) = ⟨k, c⟩.snd
z : ↑(pullback (D.f j i) (D.f j k)) := (pullbackIsoProdSubtype (D.f j i) (D.f j k)).inv ⟨((D.t ⟨i, a⟩.fst ⟨j, b⟩.fst) x, y), ⋯⟩
\u22a2 D.Rel ⟨i, a⟩ ⟨k, c⟩
-- Mathlib/Topology/Gluing.lean
D : GlueData
i : D.J
a : ↑(D.U i)
j : D.J
b : ↑(D.U j)
k : D.J
c : ↑(D.U k)
x : ↑(D.V (⟨i, a⟩.fst, ⟨j, b⟩.fst))
e₁ : (D.f ⟨i, a⟩.fst ⟨j, b⟩.fst) x = ⟨i, a⟩.snd
e₂ : (D.f ⟨j, b⟩.fst ⟨i, a⟩.fst) ((D.t ⟨i, a⟩.fst ⟨j, b⟩.fst) x) = ⟨j, b⟩.snd
y : ↑(D.V (⟨j, b⟩.fst, ⟨k, c⟩.fst))
e₃ : (D.f ⟨j, b⟩.fst ⟨k, c⟩.fst) y = ⟨j, b⟩.snd
e₄ : (D.f ⟨k, c⟩.fst ⟨j, b⟩.fst) ((D.t ⟨j, b⟩.fst ⟨k, c⟩.fst) y) = ⟨k, c⟩.snd
z : ↑(pullback (D.f j i) (D.f j k)) := (pullbackIsoProdSubtype (D.f j i) (D.f j k)).inv ⟨((D.t ⟨i, a⟩.fst ⟨j, b⟩.fst) x, y), ⋯⟩
eq₁ : (D.t j i) ((pullback.fst (D.f j i) (D.f j k)) z) = x
\u22a2 D.Rel ⟨i, a⟩ ⟨k, c⟩
-- Mathlib/Topology/Gluing.lean
D : GlueData
i : D.J
a : ↑(D.U i)
j : D.J
b : ↑(D.U j)
k : D.J
c : ↑(D.U k)
x : ↑(D.V (⟨i, a⟩.fst, ⟨j, b⟩.fst))
e₁ : (D.f ⟨i, a⟩.fst ⟨j, b⟩.fst) x = ⟨i, a⟩.snd
e₂ : (D.f ⟨j, b⟩.fst ⟨i, a⟩.fst) ((D.t ⟨i, a⟩.fst ⟨j, b⟩.fst) x) = ⟨j, b⟩.snd
y : ↑(D.V (⟨j, b⟩.fst, ⟨k, c⟩.fst))
e₃ : (D.f ⟨j, b⟩.fst ⟨k, c⟩.fst) y = ⟨j, b⟩.snd
e₄ : (D.f ⟨k, c⟩.fst ⟨j, b⟩.fst) ((D.t ⟨j, b⟩.fst ⟨k, c⟩.fst) y) = ⟨k, c⟩.snd
z : ↑(pullback (D.f j i) (D.f j k)) := (pullbackIsoProdSubtype (D.f j i) (D.f j k)).inv ⟨((D.t ⟨i, a⟩.fst ⟨j, b⟩.fst) x, y), ⋯⟩
eq₁ : (D.t j i) ((pullback.fst (D.f j i) (D.f j k)) z) = x
eq₂ : (pullback.snd (D.f j i) (D.f j k)) z = y
\u22a2 D.Rel ⟨i, a⟩ ⟨k, c⟩
-- Mathlib/Topology/Gluing.lean
D : GlueData
i : D.J
a : ↑(D.U i)
j : D.J
b : ↑(D.U j)
k : D.J
c : ↑(D.U k)
x : ↑(D.V (⟨i, a⟩.fst, ⟨j, b⟩.fst))
e₁ : (D.f ⟨i, a⟩.fst ⟨j, b⟩.fst) x = ⟨i, a⟩.snd
e₂ : (D.f ⟨j, b⟩.fst ⟨i, a⟩.fst) ((D.t ⟨i, a⟩.fst ⟨j, b⟩.fst) x) = ⟨j, b⟩.snd
y : ↑(D.V (⟨j, b⟩.fst, ⟨k, c⟩.fst))
e₃ : (D.f ⟨j, b⟩.fst ⟨k, c⟩.fst) y = ⟨j, b⟩.snd
e₄ : (D.f ⟨k, c⟩.fst ⟨j, b⟩.fst) ((D.t ⟨j, b⟩.fst ⟨k, c⟩.fst) y) = ⟨k, c⟩.snd
z : ↑(pullback (D.f j i) (D.f j k))
eq₁ : (D.t j i) ((pullback.fst (D.f j i) (D.f j k)) z) = x
eq₂ : (pullback.snd (D.f j i) (D.f j k)) z = y
\u22a2 ∃ x, (D.f ⟨i, a⟩.fst ⟨k, c⟩.fst) x = ⟨i, a⟩.snd ∧ (D.f ⟨k, c⟩.fst ⟨i, a⟩.fst) ((D.t ⟨i, a⟩.fst ⟨k, c⟩.fst) x) = ⟨k, c⟩.snd
-- Mathlib/Topology/Gluing.lean
D : GlueData
i : D.J
a : ↑(D.U i)
j : D.J
b : ↑(D.U j)
k : D.J
c : ↑(D.U k)
x : ↑(D.V (⟨i, a⟩.fst, ⟨j, b⟩.fst))
e₁ : (D.f ⟨i, a⟩.fst ⟨j, b⟩.fst) x = ⟨i, a⟩.snd
e₂ : (D.f ⟨j, b⟩.fst ⟨i, a⟩.fst) ((D.t ⟨i, a⟩.fst ⟨j, b⟩.fst) x) = ⟨j, b⟩.snd
y : ↑(D.V (⟨j, b⟩.fst, ⟨k, c⟩.fst))
e₃ : (D.f ⟨j, b⟩.fst ⟨k, c⟩.fst) y = ⟨j, b⟩.snd
e₄ : (D.f ⟨k, c⟩.fst ⟨j, b⟩.fst) ((D.t ⟨j, b⟩.fst ⟨k, c⟩.fst) y) = ⟨k, c⟩.snd
z : ↑(pullback (D.f j i) (D.f j k))
eq₁ : (D.t j i) ((pullback.fst (D.f j i) (D.f j k)) z) = x
eq₂ : (pullback.snd (D.f j i) (D.f j k)) z = y
\u22a2 (D.f ⟨i, a⟩.fst ⟨k, c⟩.fst) ((pullback.fst (D.f i k) (D.f i j)) ((D.t' j i k) z)) = ⟨i, a⟩.snd ∧ (D.f ⟨k, c⟩.fst ⟨i, a⟩.fst) ((D.t ⟨i, a⟩.fst ⟨k, c⟩.fst) ((pullback.fst (D.f i k) (D.f i j)) ((D.t' j i k) z))) = ⟨k, c⟩.snd
-- Mathlib/Topology/Gluing.lean
D : GlueData
i j k : D.J
z : ↑(pullback (D.f j i) (D.f j k))
e₂ : (D.f j i) ((D.t i j) ((D.t j i) ((pullback.fst (D.f j i) (D.f j k)) z))) = (D.f j k) ((pullback.snd (D.f j i) (D.f j k)) z)
\u22a2 (D.f i k) ((pullback.fst (D.f i k) (D.f i j)) ((D.t' j i k) z)) = (D.f i j) ((D.t j i) ((pullback.fst (D.f j i) (D.f j k)) z)) ∧ (D.f k i) ((D.t i k) ((pullback.fst (D.f i k) (D.f i j)) ((D.t' j i k) z))) = (D.f k j) ((D.t j k) ((pullback.snd (D.f j i) (D.f j k)) z))
-- Mathlib/Topology/Gluing.lean
D : GlueData
i j k : D.J
z : ↑(pullback (D.f j i) (D.f j k))
e₂ : (D.f j i) ((D.t i j) ((D.t j i) ((pullback.fst (D.f j i) (D.f j k)) z))) = (D.f j k) ((pullback.snd (D.f j i) (D.f j k)) z)
h₁ : D.t' j i k ≫ pullback.fst (D.f i k) (D.f i j) ≫ D.f i k = pullback.fst (D.f j i) (D.f j k) ≫ D.t j i ≫ D.f i j
\u22a2 (D.f i k) ((pullback.fst (D.f i k) (D.f i j)) ((D.t' j i k) z)) = (D.f i j) ((D.t j i) ((pullback.fst (D.f j i) (D.f j k)) z)) ∧ (D.f k i) ((D.t i k) ((pullback.fst (D.f i k) (D.f i j)) ((D.t' j i k) z))) = (D.f k j) ((D.t j k) ((pullback.snd (D.f j i) (D.f j k)) z))
-- Mathlib/Topology/Gluing.lean
D : GlueData
i j k : D.J
z : ↑(pullback (D.f j i) (D.f j k))
e₂ : (D.f j i) ((D.t i j) ((D.t j i) ((pullback.fst (D.f j i) (D.f j k)) z))) = (D.f j k) ((pullback.snd (D.f j i) (D.f j k)) z)
h₁ : D.t' j i k ≫ pullback.fst (D.f i k) (D.f i j) ≫ D.f i k = pullback.fst (D.f j i) (D.f j k) ≫ D.t j i ≫ D.f i j
h₂ : D.t' j i k ≫ pullback.fst (D.f i k) (D.f i j) ≫ D.t i k ≫ D.f k i = pullback.snd (D.f j i) (D.f j k) ≫ D.t j k ≫ D.f k j
\u22a2 (D.f i k) ((pullback.fst (D.f i k) (D.f i j)) ((D.t' j i k) z)) = (D.f i j) ((D.t j i) ((pullback.fst (D.f j i) (D.f j k)) z)) ∧ (D.f k i) ((D.t i k) ((pullback.fst (D.f i k) (D.f i j)) ((D.t' j i k) z))) = (D.f k j) ((D.t j k) ((pullback.snd (D.f j i) (D.f j k)) z))
-- Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s₁ : AffineSubspace k P
v₁₂ v₂₃ : V
\u22a2 map (↑(constVAdd k P v₂₃)) (map (↑(constVAdd k P v₁₂)) s₁) = map (↑(constVAdd k P (v₂₃ + v₁₂))) s₁
-- Mathlib/Logic/Function/Basic.lean
α : Sort u_1
β : Sort u_2
γ : Sort u_3
f : α → β
g : α → γ
e' : β → γ
b : β
hb : ¬∃ a, f a = b
\u22a2 extend f g e' b = e' b
-- Mathlib/Algebra/BigOperators/Intervals.lean
α✝ : Type u_1
M : Type u_2
inst✝¹ : CommMonoid M
s₂ s₁ s : Finset α✝
a : α✝
g f✝ : α✝ → M
α : Type u_3
inst✝ : CommGroup α
f : ℕ → α
n m : ℕ
hnm : n ≤ m
\u22a2 (∏ k ∈ range m, f k) / ∏ k ∈ range n, f k = ∏ k ∈ filter (fun k => n ≤ k) (range m), f k
-- Mathlib/Algebra/BigOperators/Intervals.lean
α✝ : Type u_1
M : Type u_2
inst✝¹ : CommMonoid M
s₂ s₁ s : Finset α✝
a : α✝
g f✝ : α✝ → M
α : Type u_3
inst✝ : CommGroup α
f : ℕ → α
n m : ℕ
hnm : n ≤ m
\u22a2 ∏ k ∈ Ico n m, f k = ∏ k ∈ filter (fun k => n ≤ k) (range m), f k
-- Mathlib/Algebra/BigOperators/Intervals.lean
α✝ : Type u_1
M : Type u_2
inst✝¹ : CommMonoid M
s₂ s₁ s : Finset α✝
a : α✝
g f✝ : α✝ → M
α : Type u_3
inst✝ : CommGroup α
f : ℕ → α
n m : ℕ
hnm : n ≤ m
\u22a2 Ico n m = filter (fun k => n ≤ k) (range m)
-- Mathlib/Algebra/BigOperators/Intervals.lean
α✝ : Type u_1
M : Type u_2
inst✝¹ : CommMonoid M
s₂ s₁ s : Finset α✝
a : α✝
g f✝ : α✝ → M
α : Type u_3
inst✝ : CommGroup α
f : ℕ → α
n m : ℕ
hnm : n ≤ m
\u22a2 ∀ (a : ℕ), a ∈ Ico n m ↔ a ∈ filter (fun k => n ≤ k) (range m)
-- Mathlib/Algebra/BigOperators/Intervals.lean
α✝ : Type u_1
M : Type u_2
inst✝¹ : CommMonoid M
s₂ s₁ s : Finset α✝
a : α✝
g f✝ : α✝ → M
α : Type u_3
inst✝ : CommGroup α
f : ℕ → α
n m : ℕ
hnm : n ≤ m
\u22a2 ∀ (a : ℕ), n ≤ a ∧ a < m ↔ a < m ∧ n ≤ a
-- Mathlib/LinearAlgebra/Dual.lean
R : Type uR
M : Type uM
K : Type uK
V : Type uV
ι : Type uι
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq ι
b : Basis ι R M
inst✝ : _root_.Finite ι
\u22a2 range b.toDual = ⊤
-- Mathlib/LinearAlgebra/Dual.lean
R : Type uR
M : Type uM
K : Type uK
V : Type uV
ι : Type uι
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq ι
b : Basis ι R M
inst✝ : _root_.Finite ι
f : Dual R M
\u22a2 f ∈ range b.toDual
-- Mathlib/LinearAlgebra/Dual.lean
R : Type uR
M : Type uM
K : Type uK
V : Type uV
ι : Type uι
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq ι
b : Basis ι R M
inst✝ : _root_.Finite ι
f : Dual R M
lin_comb : ι →₀ R := Finsupp.equivFunOnFinite.symm fun i => f (b i)
\u22a2 f ∈ range b.toDual
-- Mathlib/LinearAlgebra/Dual.lean
R : Type uR
M : Type uM
K : Type uK
V : Type uV
ι : Type uι
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq ι
b : Basis ι R M
inst✝ : _root_.Finite ι
f : Dual R M
lin_comb : ι →₀ R := Finsupp.equivFunOnFinite.symm fun i => f (b i)
i : ι
\u22a2 (b.toDual ((Finsupp.total ι M R ⇑b) lin_comb)) (b i) = f (b i)
-- Mathlib/LinearAlgebra/Dual.lean
R : Type uR
M : Type uM
K : Type uK
V : Type uV
ι : Type uι
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : DecidableEq ι
b : Basis ι R M
inst✝ : _root_.Finite ι
f : Dual R M
lin_comb : ι →₀ R := Finsupp.equivFunOnFinite.symm fun i => f (b i)
i : ι
\u22a2 lin_comb i = f (b i)
-- Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
h : S.LeftHomologyData
inst✝ : S.HasLeftHomology
\u22a2 h.cyclesIso.inv ≫ S.iCycles = h.i
-- Mathlib/CategoryTheory/Limits/Preserves/Shapes/Zero.lean
C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁶ : Category.{v₂, u₂} D
E : Type u₃
inst✝⁵ : Category.{v₃, u₃} E
inst✝⁴ : HasZeroObject C
inst✝³ : HasZeroObject D
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroMorphisms D
F : C ⥤ D
inst✝ : F.PreservesZeroMorphisms
\u22a2 0 ≫ 0 = 𝟙 (F.obj 0)
-- Mathlib/CategoryTheory/Limits/Preserves/Shapes/Zero.lean
C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁶ : Category.{v₂, u₂} D
E : Type u₃
inst✝⁵ : Category.{v₃, u₃} E
inst✝⁴ : HasZeroObject C
inst✝³ : HasZeroObject D
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroMorphisms D
F : C ⥤ D
inst✝ : F.PreservesZeroMorphisms
\u22a2 0 ≫ 0 = 𝟙 0
-- Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean
K : Type u_1
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
v : K
n : ℕ
h : fract v ≠ 0
\u22a2 (of v).s.get? 0 = some { a := 1, b := ↑⌊(fract v)⁻¹⌋ }
-- Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean
K : Type u_1
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
v : K
n : ℕ
h : fract v ≠ 0
\u22a2 (match some (IntFractPair.of (IntFractPair.of v).fr⁻¹), some ∘ fun p => { a := 1, b := ↑p.b } with | none, x => none | some a, f => f a) = some { a := 1, b := ↑⌊(fract v)⁻¹⌋ }
-- Mathlib/MeasureTheory/Measure/Haar/Basic.lean
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : (interior U).Nonempty
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ K₂.carrier
\u22a2 ↑(index (↑K₁) U) / ↑(index (↑K₀) U) ≤ ↑(index (↑K₂) U) / ↑(index (↑K₀) U)
-- Mathlib/MeasureTheory/Measure/Haar/Basic.lean
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : (interior U).Nonempty
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ K₂.carrier
\u22a2 ↑(index (↑K₁) U) ≤ ↑(index (↑K₂) U)
-- Mathlib/MeasureTheory/Measure/Haar/Basic.lean
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : (interior U).Nonempty
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ K₂.carrier
\u22a2 0 < ↑(index (↑K₀) U)
-- Mathlib/SetTheory/Cardinal/Cofinality.lean
α : Type u_1
r : α → α → Prop
ι : Type (max u_2 u_3)
f : ι → Ordinal.{max u_2 u_3}
c : Cardinal.{max u_2 u_3}
hc : c.IsRegular
hι : #ι < c
\u22a2 #ι < c.ord.cof