ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Analysis/Normed/Lp/lpSpace.lean
α : Type u_1
E : α → Type u_2
p✝ q✝ : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
p q : ℝ≥0∞
hpq : p.toReal.IsConjExponent q.toReal
f : ↥(lp E p)
g : ↥(lp E q)
hf₁ : ∀ (i : α), 0 ≤ ‖↑f i‖
hg₁ : ∀ (i : α), 0 ≤ ‖↑g i‖
hf₂ : HasSum (fun i => ‖↑f i‖ ^ p.toReal) (‖f‖ ^ p.toReal)
hg₂ : HasSum (fun i => ‖↑g i‖ ^ q.toReal) (‖g‖ ^ q.toReal)
C : ℝ
hC' : C ≤ ‖f‖ * ‖g‖
hC : HasSum (fun i => ‖↑f i‖ * ‖↑g i‖) C
\u22a2 (Summable fun i => ‖↑f i‖ * ‖↑g i‖) ∧ ∑' (i : α), ‖↑f i‖ * ‖↑g i‖ ≤ ‖f‖ * ‖g‖
-- Mathlib/Analysis/Normed/Lp/lpSpace.lean
α : Type u_1
E : α → Type u_2
p✝ q✝ : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
p q : ℝ≥0∞
hpq : p.toReal.IsConjExponent q.toReal
f : ↥(lp E p)
g : ↥(lp E q)
hf₁ : ∀ (i : α), 0 ≤ ‖↑f i‖
hg₁ : ∀ (i : α), 0 ≤ ‖↑g i‖
hf₂ : HasSum (fun i => ‖↑f i‖ ^ p.toReal) (‖f‖ ^ p.toReal)
hg₂ : HasSum (fun i => ‖↑g i‖ ^ q.toReal) (‖g‖ ^ q.toReal)
C : ℝ
hC' : ∑' (b : α), ‖↑f b‖ * ‖↑g b‖ ≤ ‖f‖ * ‖g‖
hC : HasSum (fun i => ‖↑f i‖ * ‖↑g i‖) C
\u22a2 (Summable fun i => ‖↑f i‖ * ‖↑g i‖) ∧ ∑' (i : α), ‖↑f i‖ * ‖↑g i‖ ≤ ‖f‖ * ‖g‖
-- Mathlib/Topology/Instances/Rat.lean
ε : ℝ
ε0 : ε > 0
a✝ b✝ : ℚ
h : dist a✝ b✝ < ε
\u22a2 dist |a✝| |b✝| ≤ dist a✝ b✝
-- Mathlib/Data/List/Intervals.lean
n m : ℕ
h : n < m
\u22a2 Ico n m = n :: Ico (n + 1) m
-- Mathlib/Data/List/Intervals.lean
n m : ℕ
h : n < m
\u22a2 [n] ++ Ico n.succ m = n :: Ico (n + 1) m
-- Mathlib/Topology/Algebra/Field.lean
K : Type u_1
inst✝⁶ : DivisionRing K
inst✝⁵ : TopologicalSpace K
α : Type u_2
𝕜 : Type u_3
f g : α → 𝕜
S : Set α
inst✝⁴ : TopologicalSpace α
inst✝³ : TopologicalSpace 𝕜
inst✝² : T1Space 𝕜
inst✝¹ : Ring 𝕜
inst✝ : NoZeroDivisors 𝕜
hS : IsPreconnected S
hf : ContinuousOn f S
hsq : EqOn (f ^ 2) 1 S
this : DiscreteTopology ↑{1, -1}
\u22a2 EqOn f 1 S ∨ EqOn f (-1) S
-- Mathlib/Topology/Algebra/Field.lean
K : Type u_1
inst✝⁶ : DivisionRing K
inst✝⁵ : TopologicalSpace K
α : Type u_2
𝕜 : Type u_3
f g : α → 𝕜
S : Set α
inst✝⁴ : TopologicalSpace α
inst✝³ : TopologicalSpace 𝕜
inst✝² : T1Space 𝕜
inst✝¹ : Ring 𝕜
inst✝ : NoZeroDivisors 𝕜
hS : IsPreconnected S
hf : ContinuousOn f S
hsq : EqOn (f ^ 2) 1 S
this : DiscreteTopology ↑{1, -1}
hmaps : MapsTo f S {1, -1}
\u22a2 EqOn f 1 S ∨ EqOn f (-1) S
-- Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℕ
\u22a2 (W.preΨ' n).natDegree ≤ WeierstrassCurve.expDegree n ∧ (W.preΨ' n).coeff (WeierstrassCurve.expDegree n) = ↑(WeierstrassCurve.expCoeff n)
-- Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℕ
dm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n := fun {m n} {p q} => natDegree_mul_le_of_le
\u22a2 (W.preΨ' n).natDegree ≤ WeierstrassCurve.expDegree n ∧ (W.preΨ' n).coeff (WeierstrassCurve.expDegree n) = ↑(WeierstrassCurve.expCoeff n)
-- Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℕ
dm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n := fun {m n} {p q} => natDegree_mul_le_of_le
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} => natDegree_pow_le_of_le n
\u22a2 (W.preΨ' n).natDegree ≤ WeierstrassCurve.expDegree n ∧ (W.preΨ' n).coeff (WeierstrassCurve.expDegree n) = ↑(WeierstrassCurve.expCoeff n)
-- Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℕ
dm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n := fun {m n} {p q} => natDegree_mul_le_of_le
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} => natDegree_pow_le_of_le n
cm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).coeff (m + n) = p.coeff m * q.coeff n := fun {m n} {p q} => coeff_mul_of_natDegree_le
\u22a2 (W.preΨ' n).natDegree ≤ WeierstrassCurve.expDegree n ∧ (W.preΨ' n).coeff (WeierstrassCurve.expDegree n) = ↑(WeierstrassCurve.expCoeff n)
-- Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℕ
dm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n := fun {m n} {p q} => natDegree_mul_le_of_le
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} => natDegree_pow_le_of_le n
cm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).coeff (m + n) = p.coeff m * q.coeff n := fun {m n} {p q} => coeff_mul_of_natDegree_le
cp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ n → (p ^ m).coeff (m * n) = p.coeff n ^ m := fun {m n} {p} => coeff_pow_of_natDegree_le
\u22a2 (W.preΨ' n).natDegree ≤ WeierstrassCurve.expDegree n ∧ (W.preΨ' n).coeff (WeierstrassCurve.expDegree n) = ↑(WeierstrassCurve.expCoeff n)
-- Mathlib/Analysis/SpecificLimits/Normed.lean
α : Type u_1
β : Type u_2
ι : Type u_3
R : Type u_4
inst✝ : NormedRing R
k : ℕ
r : ℝ
hr : 1 < r
\u22a2 (fun n => ↑n ^ k) =o[atTop] fun n => r ^ n
-- Mathlib/Analysis/SpecificLimits/Normed.lean
α : Type u_1
β : Type u_2
ι : Type u_3
R : Type u_4
inst✝ : NormedRing R
k : ℕ
r : ℝ
hr : 1 < r
this : Tendsto (fun x => x ^ k) (𝓝[>] 1) (𝓝 1)
\u22a2 (fun n => ↑n ^ k) =o[atTop] fun n => r ^ n
-- Mathlib/Analysis/SpecificLimits/Normed.lean
α : Type u_1
β : Type u_2
ι : Type u_3
R : Type u_4
inst✝ : NormedRing R
k : ℕ
r : ℝ
hr : 1 < r
this : Tendsto (fun x => x ^ k) (𝓝[>] 1) (𝓝 1)
r' : ℝ
hr' : r' ^ k < r
h1 : 1 < r'
\u22a2 (fun n => ↑n ^ k) =o[atTop] fun n => r ^ n
-- Mathlib/Analysis/SpecificLimits/Normed.lean
α : Type u_1
β : Type u_2
ι : Type u_3
R : Type u_4
inst✝ : NormedRing R
k : ℕ
r : ℝ
hr : 1 < r
this : Tendsto (fun x => x ^ k) (𝓝[>] 1) (𝓝 1)
r' : ℝ
hr' : r' ^ k < r
h1 : 1 < r'
h0 : 0 ≤ r'
\u22a2 (fun n => ↑n ^ k) =o[atTop] fun n => r ^ n
-- Mathlib/Analysis/SpecificLimits/Normed.lean
α : Type u_1
β : Type u_2
ι : Type u_3
R : Type u_4
inst✝ : NormedRing R
k : ℕ
r : ℝ
hr : 1 < r
this : Tendsto (fun x => x ^ k) (𝓝[>] 1) (𝓝 1)
r' : ℝ
hr' : r' ^ k < r
h1 : 1 < r'
h0 : 0 ≤ r'
\u22a2 (fun n => ↑n ^ k) =O[atTop] fun n => (r' ^ k) ^ n
-- Mathlib/Analysis/SpecificLimits/Normed.lean
α : Type u_1
β : Type u_2
ι : Type u_3
R : Type u_4
inst✝ : NormedRing R
k : ℕ
r : ℝ
hr : 1 < r
this : Tendsto (fun x => x ^ k) (𝓝[>] 1) (𝓝 1)
r' : ℝ
hr' : r' ^ k < r
h1 : 1 < r'
h0 : 0 ≤ r'
\u22a2 (fun n => ↑n ^ k) =O[atTop] fun n => (r' ^ n) ^ k
-- Mathlib/Analysis/SpecificLimits/Normed.lean
α : Type u_1
β : Type u_2
ι : Type u_3
R : Type u_4
inst✝ : NormedRing R
k : ℕ
r : ℝ
hr : 1 < r
this : Tendsto (fun x => x ^ k) (𝓝[>] 1) (𝓝 1)
r' : ℝ
hr' : r' ^ k < r
h1 : 1 < r'
h0 : 0 ≤ r'
\u22a2 ∀ (n : ℕ), ‖↑n‖ ≤ (r' - 1)⁻¹ * ‖1‖ * ‖r' ^ n‖
-- Mathlib/Analysis/SpecificLimits/Normed.lean
α : Type u_1
β : Type u_2
ι : Type u_3
R : Type u_4
inst✝ : NormedRing R
k : ℕ
r : ℝ
hr : 1 < r
this : Tendsto (fun x => x ^ k) (𝓝[>] 1) (𝓝 1)
r' : ℝ
hr' : r' ^ k < r
h1 : 1 < r'
h0 : 0 ≤ r'
n : ℕ
\u22a2 ‖↑n‖ ≤ (r' - 1)⁻¹ * ‖1‖ * ‖r' ^ n‖
-- Mathlib/Analysis/SpecificLimits/Normed.lean
α : Type u_1
β : Type u_2
ι : Type u_3
R : Type u_4
inst✝ : NormedRing R
k : ℕ
r : ℝ
hr : 1 < r
this : Tendsto (fun x => x ^ k) (𝓝[>] 1) (𝓝 1)
r' : ℝ
hr' : r' ^ k < r
h1 : 1 < r'
h0 : 0 ≤ r'
n : ℕ
\u22a2 ‖↑n‖ ≤ (r' - 1)⁻¹ * ‖r' ^ n‖ * ‖1‖
-- Mathlib/Analysis/SpecificLimits/Normed.lean
α : Type u_1
β : Type u_2
ι : Type u_3
R : Type u_4
inst✝ : NormedRing R
k : ℕ
r : ℝ
hr : 1 < r
this : Tendsto (fun x => x ^ k) (𝓝[>] 1) (𝓝 1)
r' : ℝ
hr' : r' ^ k < r
h1 : 1 < r'
h0 : 0 ≤ r'
n : ℕ
\u22a2 ↑n ≤ (r' - 1)⁻¹ * ‖r' ^ n‖
-- Mathlib/Algebra/Homology/HomologicalComplexAbelian.lean
C : Type u_1
ι : Type u_2
c : ComplexShape ι
inst✝¹ : Category.{u_3, u_1} C
inst✝ : Abelian C
S : ShortComplex (HomologicalComplex C c)
hS : ∀ (i : ι), (S.map (eval C c i)).Exact
\u22a2 S.Exact
-- Mathlib/Algebra/Homology/HomologicalComplexAbelian.lean
C : Type u_1
ι : Type u_2
c : ComplexShape ι
inst✝¹ : Category.{u_3, u_1} C
inst✝ : Abelian C
S : ShortComplex (HomologicalComplex C c)
hS : ∀ (i : ι), IsZero (S.map (eval C c i)).homology
\u22a2 IsZero S.homology
-- Mathlib/Algebra/Homology/HomologicalComplexAbelian.lean
C : Type u_1
ι : Type u_2
c : ComplexShape ι
inst✝¹ : Category.{u_3, u_1} C
inst✝ : Abelian C
S : ShortComplex (HomologicalComplex C c)
hS : ∀ (i : ι), IsZero (S.map (eval C c i)).homology
\u22a2 𝟙 S.homology = 0
-- Mathlib/Algebra/Homology/HomologicalComplexAbelian.lean
C : Type u_1
ι : Type u_2
c : ComplexShape ι
inst✝¹ : Category.{u_3, u_1} C
inst✝ : Abelian C
S : ShortComplex (HomologicalComplex C c)
hS : ∀ (i : ι), IsZero (S.map (eval C c i)).homology
i : ι
\u22a2 (𝟙 S.homology).f i = Hom.f 0 i
-- Mathlib/Topology/Instances/ENNReal.lean
α : Type u_1
β : Type u_2
γ : Type u_3
f g : α → ℝ≥0
sf sg : ℝ≥0
i : α
h : ∀ (a : α), f a ≤ g a
hi : f i < g i
hf : HasSum f sf
hg : HasSum g sg
\u22a2 sf < sg
-- Mathlib/Topology/Instances/ENNReal.lean
α : Type u_1
β : Type u_2
γ : Type u_3
f g : α → ℝ≥0
sf sg : ℝ≥0
i : α
h : ∀ (a : α), f a ≤ g a
hi : f i < g i
hf : HasSum f sf
hg : HasSum g sg
A : ∀ (a : α), ↑(f a) ≤ ↑(g a)
\u22a2 sf < sg
-- Mathlib/Topology/Instances/ENNReal.lean
α : Type u_1
β : Type u_2
γ : Type u_3
f g : α → ℝ≥0
sf sg : ℝ≥0
i : α
h : ∀ (a : α), f a ≤ g a
hi : f i < g i
hf : HasSum f sf
hg : HasSum g sg
A : ∀ (a : α), ↑(f a) ≤ ↑(g a)
this : ↑sf < ↑sg
\u22a2 sf < sg
-- Mathlib/Algebra/Homology/ShortComplex/SnakeLemma.lean
C : Type u_1
inst✝¹ : Category.{?u.39934, u_1} C
inst✝ : Abelian C
S : SnakeInput C
\u22a2 S.v₁₂.τ₂ ≫ S.v₂₃.τ₂ = 0
-- Mathlib/Algebra/Homology/ShortComplex/ConcreteCategory.lean
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : ConcreteCategory C
inst✝³ : HasForget₂ C Ab
inst✝² : Abelian C
inst✝¹ : (forget₂ C Ab).Additive
inst✝ : (forget₂ C Ab).PreservesHomology
D : SnakeInput C
x₃ : (forget C).obj D.L₀.X₃
x₂ : (forget C).obj D.L₁.X₂
x₁ : (forget C).obj D.L₂.X₁
h₂ : D.L₁.g x₂ = D.v₀₁.τ₃ x₃
h₁ : D.L₂.f x₁ = D.v₁₂.τ₂ x₂
this : PreservesFiniteLimits (forget₂ C Ab)
\u22a2 D.δ x₃ = D.v₂₃.τ₁ x₁
-- Mathlib/Algebra/Homology/ShortComplex/ConcreteCategory.lean
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : ConcreteCategory C
inst✝³ : HasForget₂ C Ab
inst✝² : Abelian C
inst✝¹ : (forget₂ C Ab).Additive
inst✝ : (forget₂ C Ab).PreservesHomology
D : SnakeInput C
x₃ : (forget C).obj D.L₀.X₃
x₂ : (forget C).obj D.L₁.X₂
x₁ : (forget C).obj D.L₂.X₁
h₂ : D.L₁.g x₂ = D.v₀₁.τ₃ x₃
h₁ : D.L₂.f x₁ = D.v₁₂.τ₂ x₂
this✝ : PreservesFiniteLimits (forget₂ C Ab)
this : PreservesFiniteLimits (forget C)
\u22a2 D.δ x₃ = D.v₂₃.τ₁ x₁
-- Mathlib/Algebra/Homology/ShortComplex/ConcreteCategory.lean
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : ConcreteCategory C
inst✝³ : HasForget₂ C Ab
inst✝² : Abelian C
inst✝¹ : (forget₂ C Ab).Additive
inst✝ : (forget₂ C Ab).PreservesHomology
D : SnakeInput C
x₃ : (forget C).obj D.L₀.X₃
x₂ : (forget C).obj D.L₁.X₂
x₁ : (forget C).obj D.L₂.X₁
h₂ : D.L₁.g x₂ = D.v₀₁.τ₃ x₃
h₁ : D.L₂.f x₁ = D.v₁₂.τ₂ x₂
this✝ : PreservesFiniteLimits (forget₂ C Ab)
this : PreservesFiniteLimits (forget C)
eq : (forget C).map (pullback.snd D.L₁.g D.v₀₁.τ₃ ≫ D.δ) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂) = (forget C).map (D.φ₁ ≫ D.v₂₃.τ₁) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂)
\u22a2 D.δ x₃ = D.v₂₃.τ₁ x₁
-- Mathlib/Algebra/Homology/ShortComplex/ConcreteCategory.lean
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : ConcreteCategory C
inst✝³ : HasForget₂ C Ab
inst✝² : Abelian C
inst✝¹ : (forget₂ C Ab).Additive
inst✝ : (forget₂ C Ab).PreservesHomology
D : SnakeInput C
x₃ : (forget C).obj D.L₀.X₃
x₂ : (forget C).obj D.L₁.X₂
x₁ : (forget C).obj D.L₂.X₁
h₂ : D.L₁.g x₂ = D.v₀₁.τ₃ x₃
h₁ : D.L₂.f x₁ = D.v₁₂.τ₂ x₂
this✝ : PreservesFiniteLimits (forget₂ C Ab)
this : PreservesFiniteLimits (forget C)
eq : (forget C).map (pullback.snd D.L₁.g D.v₀₁.τ₃ ≫ D.δ) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂) = (forget C).map (D.φ₁ ≫ D.v₂₃.τ₁) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂)
eq₁ : (pullback.fst D.L₁.g D.v₀₁.τ₃) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂) = x₂
\u22a2 D.δ x₃ = D.v₂₃.τ₁ x₁
-- Mathlib/Algebra/Homology/ShortComplex/ConcreteCategory.lean
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : ConcreteCategory C
inst✝³ : HasForget₂ C Ab
inst✝² : Abelian C
inst✝¹ : (forget₂ C Ab).Additive
inst✝ : (forget₂ C Ab).PreservesHomology
D : SnakeInput C
x₃ : (forget C).obj D.L₀.X₃
x₂ : (forget C).obj D.L₁.X₂
x₁ : (forget C).obj D.L₂.X₁
h₂ : D.L₁.g x₂ = D.v₀₁.τ₃ x₃
h₁ : D.L₂.f x₁ = D.v₁₂.τ₂ x₂
this✝ : PreservesFiniteLimits (forget₂ C Ab)
this : PreservesFiniteLimits (forget C)
eq : (forget C).map (pullback.snd D.L₁.g D.v₀₁.τ₃ ≫ D.δ) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂) = (forget C).map (D.φ₁ ≫ D.v₂₃.τ₁) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂)
eq₁ : (pullback.fst D.L₁.g D.v₀₁.τ₃) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂) = x₂
eq₂ : (pullback.snd D.L₁.g D.v₀₁.τ₃) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂) = x₃
\u22a2 D.δ x₃ = D.v₂₃.τ₁ x₁
-- Mathlib/Algebra/Homology/ShortComplex/ConcreteCategory.lean
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : ConcreteCategory C
inst✝³ : HasForget₂ C Ab
inst✝² : Abelian C
inst✝¹ : (forget₂ C Ab).Additive
inst✝ : (forget₂ C Ab).PreservesHomology
D : SnakeInput C
x₃ : (forget C).obj D.L₀.X₃
x₂ : (forget C).obj D.L₁.X₂
x₁ : (forget C).obj D.L₂.X₁
h₂ : D.L₁.g x₂ = D.v₀₁.τ₃ x₃
h₁ : D.L₂.f x₁ = D.v₁₂.τ₂ x₂
this✝ : PreservesFiniteLimits (forget₂ C Ab)
this : PreservesFiniteLimits (forget C)
eq : (forget C).map (pullback.snd D.L₁.g D.v₀₁.τ₃ ≫ D.δ) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂) = (forget C).map (D.φ₁ ≫ D.v₂₃.τ₁) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂)
eq₁ : (forget C).map (pullback.fst D.L₁.g D.v₀₁.τ₃) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂) = x₂
eq₂ : (forget C).map (pullback.snd D.L₁.g D.v₀₁.τ₃) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂) = x₃
\u22a2 D.δ x₃ = D.v₂₃.τ₁ x₁
-- Mathlib/Algebra/Homology/ShortComplex/ConcreteCategory.lean
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : ConcreteCategory C
inst✝³ : HasForget₂ C Ab
inst✝² : Abelian C
inst✝¹ : (forget₂ C Ab).Additive
inst✝ : (forget₂ C Ab).PreservesHomology
D : SnakeInput C
x₃ : (forget C).obj D.L₀.X₃
x₂ : (forget C).obj D.L₁.X₂
x₁ : (forget C).obj D.L₂.X₁
h₂ : D.L₁.g x₂ = D.v₀₁.τ₃ x₃
h₁ : D.L₂.f x₁ = D.v₁₂.τ₂ x₂
this✝ : PreservesFiniteLimits (forget₂ C Ab)
this : PreservesFiniteLimits (forget C)
eq : (forget C).map D.δ ((forget C).map (pullback.snd D.L₁.g D.v₀₁.τ₃) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂)) = (forget C).map D.v₂₃.τ₁ ((forget C).map D.φ₁ (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂))
eq₁ : (forget C).map (pullback.fst D.L₁.g D.v₀₁.τ₃) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂) = x₂
eq₂ : (forget C).map (pullback.snd D.L₁.g D.v₀₁.τ₃) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂) = x₃
\u22a2 D.δ x₃ = D.v₂₃.τ₁ x₁
-- Mathlib/Algebra/Homology/ShortComplex/ConcreteCategory.lean
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : ConcreteCategory C
inst✝³ : HasForget₂ C Ab
inst✝² : Abelian C
inst✝¹ : (forget₂ C Ab).Additive
inst✝ : (forget₂ C Ab).PreservesHomology
D : SnakeInput C
x₃ : (forget C).obj D.L₀.X₃
x₂ : (forget C).obj D.L₁.X₂
x₁ : (forget C).obj D.L₂.X₁
h₂ : D.L₁.g x₂ = D.v₀₁.τ₃ x₃
h₁ : D.L₂.f x₁ = D.v₁₂.τ₂ x₂
this✝ : PreservesFiniteLimits (forget₂ C Ab)
this : PreservesFiniteLimits (forget C)
eq : (forget C).map D.δ x₃ = (forget C).map D.v₂₃.τ₁ ((forget C).map D.φ₁ (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂))
eq₁ : (forget C).map (pullback.fst D.L₁.g D.v₀₁.τ₃) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂) = x₂
eq₂ : (forget C).map (pullback.snd D.L₁.g D.v₀₁.τ₃) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂) = x₃
\u22a2 D.δ x₃ = D.v₂₃.τ₁ x₁
-- Mathlib/Algebra/Homology/ShortComplex/ConcreteCategory.lean
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : ConcreteCategory C
inst✝³ : HasForget₂ C Ab
inst✝² : Abelian C
inst✝¹ : (forget₂ C Ab).Additive
inst✝ : (forget₂ C Ab).PreservesHomology
D : SnakeInput C
x₃ : (forget C).obj D.L₀.X₃
x₂ : (forget C).obj D.L₁.X₂
x₁ : (forget C).obj D.L₂.X₁
h₂ : D.L₁.g x₂ = D.v₀₁.τ₃ x₃
h₁ : D.L₂.f x₁ = D.v₁₂.τ₂ x₂
this✝ : PreservesFiniteLimits (forget₂ C Ab)
this : PreservesFiniteLimits (forget C)
eq : (forget C).map D.δ x₃ = (forget C).map D.v₂₃.τ₁ ((forget C).map D.φ₁ (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂))
eq₁ : (forget C).map (pullback.fst D.L₁.g D.v₀₁.τ₃) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂) = x₂
eq₂ : (forget C).map (pullback.snd D.L₁.g D.v₀₁.τ₃) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂) = x₃
\u22a2 (forget C).map D.φ₁ (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂) = x₁
-- Mathlib/Algebra/Homology/ShortComplex/ConcreteCategory.lean
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : ConcreteCategory C
inst✝³ : HasForget₂ C Ab
inst✝² : Abelian C
inst✝¹ : (forget₂ C Ab).Additive
inst✝ : (forget₂ C Ab).PreservesHomology
D : SnakeInput C
x₃ : (forget C).obj D.L₀.X₃
x₂ : (forget C).obj D.L₁.X₂
x₁ : (forget C).obj D.L₂.X₁
h₂ : D.L₁.g x₂ = D.v₀₁.τ₃ x₃
h₁ : D.L₂.f x₁ = D.v₁₂.τ₂ x₂
this✝ : PreservesFiniteLimits (forget₂ C Ab)
this : PreservesFiniteLimits (forget C)
eq : (forget C).map D.δ x₃ = (forget C).map D.v₂₃.τ₁ ((forget C).map D.φ₁ (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂))
eq₁ : (forget C).map (pullback.fst D.L₁.g D.v₀₁.τ₃) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂) = x₂
eq₂ : (forget C).map (pullback.snd D.L₁.g D.v₀₁.τ₃) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂) = x₃
\u22a2 (forget C).map D.L₂.f ((forget C).map D.φ₁ (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂)) = (forget C).map D.L₂.f x₁
-- Mathlib/Algebra/Homology/ShortComplex/ConcreteCategory.lean
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : ConcreteCategory C
inst✝³ : HasForget₂ C Ab
inst✝² : Abelian C
inst✝¹ : (forget₂ C Ab).Additive
inst✝ : (forget₂ C Ab).PreservesHomology
D : SnakeInput C
x₃ : (forget C).obj D.L₀.X₃
x₂ : (forget C).obj D.L₁.X₂
x₁ : (forget C).obj D.L₂.X₁
h₂ : D.L₁.g x₂ = D.v₀₁.τ₃ x₃
h₁ : D.L₂.f x₁ = D.v₁₂.τ₂ x₂
this✝ : PreservesFiniteLimits (forget₂ C Ab)
this : PreservesFiniteLimits (forget C)
eq : (forget C).map D.δ x₃ = (forget C).map D.v₂₃.τ₁ ((forget C).map D.φ₁ (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂))
eq₁ : (forget C).map (pullback.fst D.L₁.g D.v₀₁.τ₃) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂) = x₂
eq₂ : (forget C).map (pullback.snd D.L₁.g D.v₀₁.τ₃) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂) = x₃
\u22a2 (forget C).map D.φ₂ (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂) = (forget C).map D.L₂.f x₁
-- Mathlib/Algebra/Homology/ShortComplex/ConcreteCategory.lean
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : ConcreteCategory C
inst✝³ : HasForget₂ C Ab
inst✝² : Abelian C
inst✝¹ : (forget₂ C Ab).Additive
inst✝ : (forget₂ C Ab).PreservesHomology
D : SnakeInput C
x₃ : (forget C).obj D.L₀.X₃
x₂ : (forget C).obj D.L₁.X₂
x₁ : (forget C).obj D.L₂.X₁
h₂ : D.L₁.g x₂ = D.v₀₁.τ₃ x₃
h₁ : D.L₂.f x₁ = D.v₁₂.τ₂ x₂
this✝ : PreservesFiniteLimits (forget₂ C Ab)
this : PreservesFiniteLimits (forget C)
eq : (forget C).map D.δ x₃ = (forget C).map D.v₂₃.τ₁ ((forget C).map D.φ₁ (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂))
eq₁ : (forget C).map (pullback.fst D.L₁.g D.v₀₁.τ₃) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂) = x₂
eq₂ : (forget C).map (pullback.snd D.L₁.g D.v₀₁.τ₃) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂) = x₃
\u22a2 (forget C).map (pullback.fst D.L₁.g D.v₀₁.τ₃ ≫ D.v₁₂.τ₂) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂) = (forget C).map D.L₂.f x₁
-- Mathlib/Algebra/Homology/ShortComplex/ConcreteCategory.lean
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : ConcreteCategory C
inst✝³ : HasForget₂ C Ab
inst✝² : Abelian C
inst✝¹ : (forget₂ C Ab).Additive
inst✝ : (forget₂ C Ab).PreservesHomology
D : SnakeInput C
x₃ : (forget C).obj D.L₀.X₃
x₂ : (forget C).obj D.L₁.X₂
x₁ : (forget C).obj D.L₂.X₁
h₂ : D.L₁.g x₂ = D.v₀₁.τ₃ x₃
h₁ : D.L₂.f x₁ = D.v₁₂.τ₂ x₂
this✝ : PreservesFiniteLimits (forget₂ C Ab)
this : PreservesFiniteLimits (forget C)
eq : (forget C).map D.δ x₃ = (forget C).map D.v₂₃.τ₁ ((forget C).map D.φ₁ (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂))
eq₁ : (forget C).map (pullback.fst D.L₁.g D.v₀₁.τ₃) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂) = x₂
eq₂ : (forget C).map (pullback.snd D.L₁.g D.v₀₁.τ₃) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂) = x₃
\u22a2 (forget C).map D.v₁₂.τ₂ x₂ = (forget C).map D.L₂.f x₁
-- Mathlib/MeasureTheory/Measure/Tilted.lean
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f✝ f : α → ℝ
g : α → ℝ≥0∞
\u22a2 ∫⁻ (x : α), g x ∂μ.tilted f = ∫⁻ (x : α), ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) * g x ∂μ
-- Mathlib/Topology/Sheaves/SheafCondition/Sites.lean
C : Type u
inst✝ : Category.{v, u} C
X Y : TopCat
f : X ⟶ Y
F : TopCat.Presheaf C Y
hf : IsOpenMap ⇑f
U : Opens ↑X
S : Sieve U
hU : S ∈ (Opens.grothendieckTopology ↑X).sieves U
x : ↑X
hx : x ∈ ↑U
V : Opens ↑X
i : V ⟶ U
hV : S.arrows i
hxV : x ∈ V
\u22a2 ∃ U_1 f_1, (Sieve.functorPushforward hf.functor S).arrows f_1 ∧ f x ∈ U_1
-- Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean
x : ℝ
hx : x ≠ 0
\u22a2 HasStrictDerivAt log x⁻¹ x
-- Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean
x : ℝ
hx✝ : x ≠ 0
hx : x < 0
\u22a2 HasStrictDerivAt log x⁻¹ x
-- Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean
x : ℝ
hx✝ : x ≠ 0
hx : 0 < x
\u22a2 HasStrictDerivAt log x⁻¹ x
-- Mathlib/Data/Int/Defs.lean
a b c d m n✝ : ℤ
n : ℕ
\u22a2 -[n+1].toNat = 0 ↔ -[n+1] ≤ 0
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
x : ℂ
\u22a2 HasStrictDerivAt cosh (sinh x) x
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
x : ℂ
\u22a2 HasStrictDerivAt cosh ((cexp x - cexp (-x)) * 2⁻¹) x
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
x : ℂ
\u22a2 (cexp x - cexp (-x)) * 2⁻¹ = (cexp x + cexp (-id x) * -1) * 2⁻¹
-- Mathlib/LinearAlgebra/Matrix/Adjugate.lean
m : Type u
n : Type v
α : Type w
inst✝⁵ : DecidableEq n
inst✝⁴ : Fintype n
inst✝³ : DecidableEq m
inst✝² : Fintype m
inst✝¹ : CommRing α
inst✝ : Nontrivial n
i j : n
\u22a2 adjugate 0 i j = 0 i j
-- Mathlib/LinearAlgebra/Matrix/Adjugate.lean
m : Type u
n : Type v
α : Type w
inst✝⁵ : DecidableEq n
inst✝⁴ : Fintype n
inst✝³ : DecidableEq m
inst✝² : Fintype m
inst✝¹ : CommRing α
inst✝ : Nontrivial n
i j j' : n
hj' : j' ≠ j
\u22a2 adjugate 0 i j = 0 i j
-- Mathlib/LinearAlgebra/Matrix/Adjugate.lean
m : Type u
n : Type v
α : Type w
inst✝⁵ : DecidableEq n
inst✝⁴ : Fintype n
inst✝³ : DecidableEq m
inst✝² : Fintype m
inst✝¹ : CommRing α
inst✝ : Nontrivial n
i j j' : n
hj' : j' ≠ j
\u22a2 ∀ (i_1 : n), 0ᵀ.updateColumn j (Pi.single i 1) i_1 j' = 0
-- Mathlib/LinearAlgebra/Matrix/Adjugate.lean
m : Type u
n : Type v
α : Type w
inst✝⁵ : DecidableEq n
inst✝⁴ : Fintype n
inst✝³ : DecidableEq m
inst✝² : Fintype m
inst✝¹ : CommRing α
inst✝ : Nontrivial n
i j j' : n
hj' : j' ≠ j
j'' : n
\u22a2 0ᵀ.updateColumn j (Pi.single i 1) j'' j' = 0
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean
x✝ y x : ℝ
\u22a2 π / 4 ≤ arcsin x ↔ √2 / 2 ≤ x
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean
x✝ y x : ℝ
\u22a2 π / 4 ∈ Ioc (-(π / 2)) (π / 2)
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean
x✝ y x : ℝ
this : 0 < π
\u22a2 π / 4 ∈ Ioc (-(π / 2)) (π / 2)
-- Mathlib/Computability/TuringMachine.lean
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
M : Λ → Stmt₁
S : Finset Λ
q₁ q₂ : Stmt₁
h₁ : q₁ ∈ stmts₁ q₂
\u22a2 some q₂ ∈ stmts M S → some q₁ ∈ stmts M S
-- Mathlib/Computability/TuringMachine.lean
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
M : Λ → Stmt₁
S : Finset Λ
q₁ q₂ : Stmt₁
h₁ : q₁ ∈ stmts₁ q₂
\u22a2 ∀ x ∈ S, q₂ ∈ stmts₁ (M x) → ∃ a ∈ S, q₁ ∈ stmts₁ (M a)
-- Mathlib/Topology/NoetherianSpace.lean
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : NoetherianSpace α
s : Set α
\u22a2 IsCompact s
-- Mathlib/Topology/NoetherianSpace.lean
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : NoetherianSpace α
s : Set α
ι✝ : Type u_1
U : ι✝ → Set α
hUo : ∀ (i : ι✝), IsOpen (U i)
hs : s ⊆ ⋃ i, U i
\u22a2 ∃ t, s ⊆ ⋃ i ∈ t, U i
-- Mathlib/Topology/NoetherianSpace.lean
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : NoetherianSpace α
s : Set α
ι✝ : Type u_1
U : ι✝ → Set α
hUo : ∀ (i : ι✝), IsOpen (U i)
hs : s ⊆ ⋃ i, U i
t : Finset ι✝
ht : ↑{ carrier := ⋃ i, U i, is_open' := ⋯ } ⊆ ⋃ i ∈ t, U i
\u22a2 ∃ t, s ⊆ ⋃ i ∈ t, U i