ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/NumberTheory/Padics/PadicNumbers.lean
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : f ≈ g
h : padicNorm p (↑f (stationaryPoint hf)) ≠ padicNorm p (↑g (stationaryPoint hg))
hlt : padicNorm p (↑g (stationaryPoint hg)) < padicNorm p (↑f (stationaryPoint hf))
hpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
N : ℕ
hN : ∀ j ≥ N, padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
i : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))
hi : N ≤ i
hN' : padicNorm p (↑(f - g) i) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
\u22a2 False
-- Mathlib/NumberTheory/Padics/PadicNumbers.lean
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : f ≈ g
hpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
N : ℕ
hlt : padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg)))) < padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg))))
h : padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) ≠ padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))
hN : ∀ j ≥ N, padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
i : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))
hi : N ≤ i
hN' : padicNorm p (↑(f - g) i) < padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) - padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))
\u22a2 False
-- Mathlib/NumberTheory/Padics/PadicNumbers.lean
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : f ≈ g
hpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
N : ℕ
hlt : padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg)))) < padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg))))
h : padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) ≠ padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))
hN : ∀ j ≥ N, padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
i : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))
hi : N ≤ i
hN' : padicNorm p (↑(f - g) i) < padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) - padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))
hpne : padicNorm p (↑f i) ≠ padicNorm p (-↑g i)
\u22a2 False
-- Mathlib/NumberTheory/Padics/PadicNumbers.lean
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : f ≈ g
hpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
N : ℕ
hlt : padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg)))) < padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg))))
h : padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) ≠ padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))
hN : ∀ j ≥ N, padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
i : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))
hi : N ≤ i
hN' : padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) < padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) - padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))
hpne : padicNorm p (↑f i) ≠ padicNorm p (-↑g i)
\u22a2 False
-- Mathlib/NumberTheory/Padics/PadicNumbers.lean
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : f ≈ g
hpn : 0 < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
N : ℕ
hlt : padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg)))) < padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg))))
h : padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) ≠ padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))
hN : ∀ j ≥ N, padicNorm p (↑(f - g) j) < padicNorm p (↑f (stationaryPoint hf)) - padicNorm p (↑g (stationaryPoint hg))
i : ℕ := max N (max (stationaryPoint hf) (stationaryPoint hg))
hi : N ≤ i
hN' : padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) < padicNorm p (↑f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) - padicNorm p (↑g (max N (max (stationaryPoint hf) (stationaryPoint hg))))
hpne : padicNorm p (↑f i) ≠ padicNorm p (-↑g i)
this : padicNorm p (↑f i) < padicNorm p (↑f i)
\u22a2 False
-- Mathlib/GroupTheory/Coprod/Basic.lean
M : Type u_1
N : Type u_2
P : Type u_3
inst✝³ : MulOneClass M
inst✝² : MulOneClass N
inst✝¹ : Monoid P
P' : Type u_4
inst✝ : Monoid P'
f : P →* P'
g₁ : M →* P
g₂ : N →* P
\u22a2 (f.comp (lift g₁ g₂)).comp inl = (lift (f.comp g₁) (f.comp g₂)).comp inl
-- Mathlib/GroupTheory/Coprod/Basic.lean
M : Type u_1
N : Type u_2
P : Type u_3
inst✝³ : MulOneClass M
inst✝² : MulOneClass N
inst✝¹ : Monoid P
P' : Type u_4
inst✝ : Monoid P'
f : P →* P'
g₁ : M →* P
g₂ : N →* P
\u22a2 (f.comp (lift g₁ g₂)).comp inr = (lift (f.comp g₁) (f.comp g₂)).comp inr
-- Mathlib/LinearAlgebra/PiTensorProduct.lean
ι : Type u_1
ι₂ : Type u_2
ι₃ : Type u_3
R : Type u_4
inst✝⁷ : CommSemiring R
R₁ : Type u_5
R₂ : Type u_6
s : ι → Type u_7
inst✝⁶ : (i : ι) → AddCommMonoid (s i)
inst✝⁵ : (i : ι) → Module R (s i)
M : Type u_8
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
E : Type u_9
inst✝² : AddCommMonoid E
inst✝¹ : Module R E
F : Type u_10
inst✝ : AddCommMonoid F
φ₁ φ₂ : (⨂[R] (i : ι), s i) →ₗ[R] E
H : φ₁.compMultilinearMap (tprod R) = φ₂.compMultilinearMap (tprod R)
\u22a2 φ₁ = φ₂
-- Mathlib/LinearAlgebra/PiTensorProduct.lean
ι : Type u_1
ι₂ : Type u_2
ι₃ : Type u_3
R : Type u_4
inst✝⁷ : CommSemiring R
R₁ : Type u_5
R₂ : Type u_6
s : ι → Type u_7
inst✝⁶ : (i : ι) → AddCommMonoid (s i)
inst✝⁵ : (i : ι) → Module R (s i)
M : Type u_8
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
E : Type u_9
inst✝² : AddCommMonoid E
inst✝¹ : Module R E
F : Type u_10
inst✝ : AddCommMonoid F
φ₁ φ₂ : (⨂[R] (i : ι), s i) →ₗ[R] E
H : φ₁.compMultilinearMap (tprod R) = φ₂.compMultilinearMap (tprod R)
\u22a2 ∀ (x : ⨂[R] (i : ι), s i), φ₁ x = φ₂ x
-- Mathlib/LinearAlgebra/PiTensorProduct.lean
ι : Type u_1
ι₂ : Type u_2
ι₃ : Type u_3
R : Type u_4
inst✝⁷ : CommSemiring R
R₁ : Type u_5
R₂ : Type u_6
s : ι → Type u_7
inst✝⁶ : (i : ι) → AddCommMonoid (s i)
inst✝⁵ : (i : ι) → Module R (s i)
M : Type u_8
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
E : Type u_9
inst✝² : AddCommMonoid E
inst✝¹ : Module R E
F : Type u_10
inst✝ : AddCommMonoid F
φ₁ φ₂ : (⨂[R] (i : ι), s i) →ₗ[R] E
H : φ₁.compMultilinearMap (tprod R) = φ₂.compMultilinearMap (tprod R)
z : ⨂[R] (i : ι), s i
\u22a2 ∀ (r : R) (f : (i : ι) → s i), φ₁ (tprodCoeff R r f) = φ₂ (tprodCoeff R r f)
-- Mathlib/Data/Nat/Bitwise.lean
i : ℕ
ih : ∀ (n : ℕ), n.testBit i = n.bits.getI i
n : ℕ
\u22a2 n.testBit (i + 1) = n.bits.getI (i + 1)
-- Mathlib/Data/Nat/Bitwise.lean
i : ℕ
ih : ∀ (n : ℕ), n.testBit i = n.bits.getI i
n : ℕ
\u22a2 (bit n.bodd n.div2).testBit (i + 1) = n.bits.getI (i + 1)
-- Mathlib/Data/Nat/Bitwise.lean
i : ℕ
ih : ∀ (n : ℕ), n.testBit i = n.bits.getI i
n : ℕ
\u22a2 n.bits.tail.getI i = n.bits.getI (i + 1)
-- Mathlib/CategoryTheory/Preadditive/Basic.lean
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
P Q R : C
f f' : P ⟶ Q
g g' : Q ⟶ R
inst✝ : IsIso f
\u22a2 f ≫ g = 0 ↔ g = 0
-- Mathlib/CategoryTheory/Limits/IsConnected.lean
C✝ : Type u
inst✝³ : Category.{v, u} C✝
C : Type u
D : Type u₂
inst✝² : Category.{v, u} C
inst✝¹ : Category.{v₂, u₂} D
F : C ⥤ D
inst✝ : F.Final
\u22a2 IsConnected C ↔ IsConnected D
-- Mathlib/CategoryTheory/Limits/IsConnected.lean
C✝ : Type u
inst✝³ : Category.{v, u} C✝
C : Type u
D : Type u₂
inst✝² : Category.{v, u} C
inst✝¹ : Category.{v₂, u₂} D
F : C ⥤ D
inst✝ : F.Final
\u22a2 Nonempty (colimit (constPUnitFunctor C) ≅ PUnit.{(max v u v₂ u₂) + 1}) ↔ Nonempty (colimit (constPUnitFunctor D) ≅ PUnit.{(max v u v₂ u₂) + 1})
-- Mathlib/Data/Finset/Pointwise/Interval.lean
α : Type u_1
inst✝⁵ : Mul α
inst✝⁴ : Preorder α
inst✝³ : DecidableEq α
inst✝² : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
inst✝¹ : CovariantClass α α (Function.swap HMul.hMul) LE.le
inst✝ : LocallyFiniteOrderTop α
a b : α
\u22a2 ↑(Ici a * Ici b) ⊆ ↑(Ici (a * b))
-- Mathlib/Topology/Algebra/Group/Basic.lean
G : Type w
H : Type x
α : Type u
β : Type v
inst✝² : TopologicalSpace α
inst✝¹ : Group α
inst✝ : ContinuousConstSMul αᵐᵒᵖ α
s t : Set α
hs : IsOpen s
\u22a2 IsOpen (s * t)
-- Mathlib/Topology/Algebra/Group/Basic.lean
G : Type w
H : Type x
α : Type u
β : Type v
inst✝² : TopologicalSpace α
inst✝¹ : Group α
inst✝ : ContinuousConstSMul αᵐᵒᵖ α
s t : Set α
hs : IsOpen s
\u22a2 IsOpen (⋃ a ∈ t, op a • s)
-- Mathlib/Analysis/InnerProductSpace/Calculus.lean
n : ℕ∞
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
y : E
hy : y ∈ ball 0 1
\u22a2 ContDiffWithinAt ℝ n (↑univUnitBall.symm) (ball 0 1) y
-- Mathlib/Analysis/InnerProductSpace/Calculus.lean
n : ℕ∞
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
y : E
hy : y ∈ ball 0 1
\u22a2 ContDiffAt ℝ n (↑univUnitBall.symm) y
-- Mathlib/Analysis/InnerProductSpace/Calculus.lean
n : ℕ∞
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
y : E
hy : y ∈ ball 0 1
\u22a2 ContDiffAt ℝ n (fun y => (√(1 - ‖y‖ ^ 2))⁻¹) y
-- Mathlib/Analysis/InnerProductSpace/Calculus.lean
n : ℕ∞
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
y : E
hy : y ∈ ball 0 1
h : 0 < 1 - ‖y‖ ^ 2
\u22a2 ContDiffAt ℝ n (fun y => (√(1 - ‖y‖ ^ 2))⁻¹) y
-- Mathlib/Analysis/InnerProductSpace/Calculus.lean
n : ℕ∞
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
y : E
hy : y ∈ ball 0 1
h : 0 < 1 - ‖y‖ ^ 2
\u22a2 ContDiffAt ℝ n (fun y => √(1 - ‖y‖ ^ 2)) y
-- Mathlib/Analysis/InnerProductSpace/Calculus.lean
n : ℕ∞
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
y : E
hy : y ∈ ball 0 1
h : 0 < 1 - ‖y‖ ^ 2
\u22a2 ContDiffAt ℝ n (fun y => 1 - ‖y‖ ^ 2) y
-- Mathlib/Data/Multiset/Lattice.lean
α✝ : Type u_1
inst✝³ : SemilatticeSup α✝
inst✝² : OrderBot α✝
inst✝¹ : DecidableEq α✝
α : Type u_2
inst✝ : DecidableEq α
m : Multiset (Multiset α)
\u22a2 m.sup.Nodup ↔ ∀ a ∈ m, a.Nodup
-- Mathlib/Data/Multiset/Lattice.lean
α✝ : Type u_1
inst✝³ : SemilatticeSup α✝
inst✝² : OrderBot α✝
inst✝¹ : DecidableEq α✝
α : Type u_2
inst✝ : DecidableEq α
\u22a2 (sup 0).Nodup ↔ ∀ a ∈ 0, a.Nodup
-- Mathlib/Data/Multiset/Lattice.lean
α✝ : Type u_1
inst✝³ : SemilatticeSup α✝
inst✝² : OrderBot α✝
inst✝¹ : DecidableEq α✝
α : Type u_2
inst✝ : DecidableEq α
a✝ : Multiset α
s✝ : Multiset (Multiset α)
h : s✝.sup.Nodup ↔ ∀ a ∈ s✝, a.Nodup
\u22a2 (a✝ ::ₘ s✝).sup.Nodup ↔ ∀ a ∈ a✝ ::ₘ s✝, a.Nodup
-- Mathlib/Algebra/Order/Module/Defs.lean
α : Type u_1
β : Type u_2
a a₁ a₂ : α
b b₁ b₂ : β
inst✝⁶ : Zero α
inst✝⁵ : Zero β
inst✝⁴ : SMulZeroClass α β
inst✝³ : Preorder α
inst✝² : Preorder β
inst✝¹ : PosSMulStrictMono α β
inst✝ : PosSMulReflectLT α β
ha : 0 < a
\u22a2 a • b < 0 ↔ b < 0
-- Mathlib/Topology/MetricSpace/Dilation.lean
α : Type u_1
β : Type u_2
γ : Type u_3
F : Type u_4
G : Type u_5
inst✝³ : PseudoMetricSpace α
inst✝² : PseudoMetricSpace β
inst✝¹ : FunLike F α β
inst✝ : DilationClass F α β
f : F
\u22a2 Metric.diam (range ⇑f) = ↑(ratio f) * Metric.diam univ
-- Mathlib/Algebra/Order/UpperLower.lean
α : Type u_1
inst✝ : OrderedCommGroup α
s t : Set α
a : α
ht : IsUpperSet t
\u22a2 IsLowerSet (s / t)
-- Mathlib/Algebra/Order/UpperLower.lean
α : Type u_1
inst✝ : OrderedCommGroup α
s t : Set α
a : α
ht : IsUpperSet t
\u22a2 IsLowerSet (s * t⁻¹)
-- Mathlib/Algebra/Module/Submodule/Range.lean
R : Type u_1
R₂ : Type u_2
R₃ : Type u_3
K : Type u_4
K₂ : Type u_5
M : Type u_6
M₂ : Type u_7
M₃ : Type u_8
V : Type u_9
V₂ : Type u_10
inst✝¹³ : Semiring R
inst✝¹² : Semiring R₂
inst✝¹¹ : Semiring R₃
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : AddCommMonoid M₂
inst✝⁸ : AddCommMonoid M₃
σ₁₂ : R →+* R₂
σ₂₃ : R₂ →+* R₃
σ₁₃ : R →+* R₃
inst✝⁷ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝⁶ : Module R M
inst✝⁵ : Module R₂ M₂
inst✝⁴ : Module R₃ M₃
σ₂₁ : R₂ →+* R
τ₁₂ : R →+* R₂
τ₂₃ : R₂ →+* R₃
τ₁₃ : R →+* R₃
inst✝³ : RingHomCompTriple τ₁₂ τ₂₃ τ₁₃
F : Type u_11
inst✝² : FunLike F M M₂
inst✝¹ : SemilinearMapClass F τ₁₂ M M₂
inst✝ : RingHomSurjective τ₁₂
f : M →ₛₗ[τ₁₂] M₂
\u22a2 range f = ⊥ ↔ f = 0
-- Mathlib/LinearAlgebra/CliffordAlgebra/Contraction.lean
R : Type u1
inst✝² : CommRing R
M : Type u2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q Q' Q'' : QuadraticForm R M
B B' : BilinForm R M
h : BilinMap.toQuadraticMap B = Q' - Q
h' : BilinMap.toQuadraticMap B' = Q'' - Q'
x : CliffordAlgebra Q
\u22a2 (changeForm ⋯) ((changeForm h) x) = x
-- Mathlib/LinearAlgebra/CliffordAlgebra/Contraction.lean
R : Type u1
inst✝² : CommRing R
M : Type u2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q Q' Q'' : QuadraticForm R M
B B' : BilinForm R M
h : BilinMap.toQuadraticMap B = Q' - Q
h' : BilinMap.toQuadraticMap B' = Q'' - Q'
x : CliffordAlgebra Q'
\u22a2 (changeForm h) ((changeForm ⋯) x) = x
-- Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
p p₀ p₁ p₂ p1 p2 : P
hp1p2 : p1 ≠ p2
\u22a2 ∠ p1 (midpoint ℝ p1 p2) p2 = π
-- Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
p p₀ p₁ p₂ p1 p2 : P
hp1p2 : p1 ≠ p2
\u22a2 InnerProductGeometry.angle (p1 -ᵥ p2) (p2 -ᵥ p1) = π
-- Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
p p₀ p₁ p₂ p1 p2 : P
hp1p2 : p1 ≠ p2
\u22a2 InnerProductGeometry.angle (p1 -ᵥ p2) (-(p1 -ᵥ p2)) = π
-- Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
p p₀ p₁ p₂ p1 p2 : P
hp1p2 : p1 ≠ p2
\u22a2 p1 -ᵥ p2 ≠ 0
-- Mathlib/Data/Complex/Exponential.lean
x✝ y x : ℝ
\u22a2 |sin x| = √(1 - cos x ^ 2)
-- Mathlib/Algebra/Homology/ShortComplex/Homology.lean
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
S S₁ S₂ S₃ S₄ : ShortComplex C
φ : S₁ ⟶ S₂
h₁✝ : S₁.HomologyData
h₂✝ : S₂.HomologyData
h₁ : S.LeftHomologyData
h₂ : S.RightHomologyData
\u22a2 h₁.f' ≫ h₂.liftH (h₁.i ≫ h₂.p) ⋯ = 0
-- Mathlib/Algebra/Homology/ShortComplex/Homology.lean
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
S S₁ S₂ S₃ S₄ : ShortComplex C
φ : S₁ ⟶ S₂
h₁✝ : S₁.HomologyData
h₂✝ : S₂.HomologyData
h₁ : S.LeftHomologyData
h₂ : S.RightHomologyData
\u22a2 leftRightHomologyComparison' h₁ h₂ = h₁.descH (h₂.liftH (h₁.i ≫ h₂.p) ⋯) ⋯
-- Mathlib/Order/Filter/Basic.lean
α : Type u
β : Type v
γ : Type w
δ : Type u_1
ι : Sort x
p : α → Prop
f : Filter α
hp : ∃ᶠ (x : α) in f, p x
\u22a2 ∃ x, p x
-- Mathlib/Order/Filter/Basic.lean
α : Type u
β : Type v
γ : Type w
δ : Type u_1
ι : Sort x
p : α → Prop
f : Filter α
hp : ∃ᶠ (x : α) in f, p x
H : ¬∃ x, p x
\u22a2 False
-- Mathlib/Order/Filter/Basic.lean
α : Type u
β : Type v
γ : Type w
δ : Type u_1
ι : Sort x
p : α → Prop
f : Filter α
hp : ∃ᶠ (x : α) in f, p x
H : ∀ᶠ (x : α) in f, ¬p x
\u22a2 False
-- Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean
ι : Type u_1
σ : Type u_2
R : Type u_3
A : Type u_4
inst✝⁵ : Semiring A
inst✝⁴ : DecidableEq ι
inst✝³ : AddMonoid ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
I : Ideal A
\u22a2 (homogeneousHull 𝒜 I).toIdeal = ⨆ i, span (⇑(GradedRing.proj 𝒜 i) '' ↑I)
-- Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean
ι : Type u_1
σ : Type u_2
R : Type u_3
A : Type u_4
inst✝⁵ : Semiring A
inst✝⁴ : DecidableEq ι
inst✝³ : AddMonoid ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
I : Ideal A
\u22a2 (homogeneousHull 𝒜 I).toIdeal = span (⋃ i, ⇑(GradedRing.proj 𝒜 i) '' ↑I)
-- Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean
ι : Type u_1
σ : Type u_2
R : Type u_3
A : Type u_4
inst✝⁵ : Semiring A
inst✝⁴ : DecidableEq ι
inst✝³ : AddMonoid ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
I : Ideal A
x✝ : A
\u22a2 x✝ ∈ {r | ∃ i x, ↑(((decompose 𝒜) ↑x) i) = r} ↔ x✝ ∈ ⋃ i, ⇑(GradedRing.proj 𝒜 i) '' ↑I
-- Mathlib/NumberTheory/Padics/Hensel.lean
p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
x y z : ℤ_[p]
hz : Polynomial.eval x F - Polynomial.eval y F = z * (x - y)
\u22a2 ‖z‖ ≤ 1
-- Mathlib/MeasureTheory/Constructions/HaarToSphere.lean
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
μ : Measure E
s : Set ↑(sphere 0 1)
hs : MeasurableSet s
\u22a2 μ.toSphere s = ↑(dim E) * μ (Ioo 0 1 • Subtype.val '' s)
-- Mathlib/Probability/Kernel/Disintegration/Density.lean
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
n : ℕ
a : α
x : γ
s s' : Set β
h : s ⊆ s'
h0 : ¬(ν a) (countablePartitionSet n x) = 0
\u22a2 ((κ a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x)).toReal ≤ ((κ a) (countablePartitionSet n x ×ˢ s') / (ν a) (countablePartitionSet n x)).toReal
-- Mathlib/Probability/Kernel/Disintegration/Density.lean
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
n : ℕ
a : α
x : γ
s s' : Set β
h : s ⊆ s'
h0 : ¬(ν a) (countablePartitionSet n x) = 0
h_ne_top : ∀ (s : Set β), (κ a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x) ≠ ⊤
\u22a2 ((κ a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x)).toReal ≤ ((κ a) (countablePartitionSet n x ×ˢ s') / (ν a) (countablePartitionSet n x)).toReal
-- Mathlib/Probability/Kernel/Disintegration/Density.lean
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
n : ℕ
a : α
x : γ
s s' : Set β
h : s ⊆ s'
h0 : ¬(ν a) (countablePartitionSet n x) = 0
h_ne_top : ∀ (s : Set β), (κ a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x) ≠ ⊤
\u22a2 (κ a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x) ≤ (κ a) (countablePartitionSet n x ×ˢ s') / (ν a) (countablePartitionSet n x)
-- Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
a b c d : ℝ
f g : ℝ → E
μ : Measure ℝ
hab : IntervalIntegrable f μ a b
hcd : IntervalIntegrable f μ c d
hac : IntervalIntegrable f μ a c
\u22a2 ∫ (x : ℝ) in a..b, f x ∂μ - ∫ (x : ℝ) in c..d, f x ∂μ = ∫ (x : ℝ) in a..c, f x ∂μ - ∫ (x : ℝ) in b..d, f x ∂μ
-- Mathlib/Algebra/Category/CoalgebraCat/ComonEquivalence.lean
R : Type u
inst✝¹² : CommRing R
M N P Q : Type u
inst✝¹¹ : AddCommGroup M
inst✝¹⁰ : AddCommGroup N
inst✝⁹ : AddCommGroup P
inst✝⁸ : AddCommGroup Q
inst✝⁷ : Module R M
inst✝⁶ : Module R N
inst✝⁵ : Module R P
inst✝⁴ : Module R Q
inst✝³ : Coalgebra R M
inst✝² : Coalgebra R N
inst✝¹ : Coalgebra R P
inst✝ : Coalgebra R Q
\u22a2 ((comonEquivalence R).symm.inverse.obj (of R M ⊗ of R N) ⊗ (comonEquivalence R).symm.inverse.obj (of R P)).comul = Coalgebra.comul
-- Mathlib/Algebra/Category/CoalgebraCat/ComonEquivalence.lean
R : Type u
inst✝¹² : CommRing R
M N P Q : Type u
inst✝¹¹ : AddCommGroup M
inst✝¹⁰ : AddCommGroup N
inst✝⁹ : AddCommGroup P
inst✝⁸ : AddCommGroup Q
inst✝⁷ : Module R M
inst✝⁶ : Module R N
inst✝⁵ : Module R P
inst✝⁴ : Module R Q
inst✝³ : Coalgebra R M
inst✝² : Coalgebra R N
inst✝¹ : Coalgebra R P
inst✝ : Coalgebra R Q
\u22a2 (tensor_μ (ModuleCat R)ᵒᵖ (Opposite.op ((ofComon R).obj (Comon_.Mon_OpOpToComon_obj' (ModuleCat R) (Comon_.Comon_ToMon_OpOp_obj' (ModuleCat R) (of R M).toComonObj ⊗ Comon_.Comon_ToMon_OpOp_obj' (ModuleCat R) (of R N).toComonObj))).toModuleCat, Opposite.op (ModuleCat.mk P)) (Opposite.op ((ofComon R).obj (Comon_.Mon_OpOpToComon_obj' (ModuleCat R) (Comon_.Comon_ToMon_OpOp_obj' (ModuleCat R) (of R M).toComonObj ⊗ Comon_.Comon_ToMon_OpOp_obj' (ModuleCat R) (of R N).toComonObj))).toModuleCat, Opposite.op (ModuleCat.mk P))).unop ∘ₗ map Coalgebra.comul Coalgebra.comul = ↑(tensorTensorTensorComm R (M ⊗[R] N) (M ⊗[R] N) P P) ∘ₗ map (↑(tensorTensorTensorComm R M M N N) ∘ₗ map Coalgebra.comul Coalgebra.comul) Coalgebra.comul
-- Mathlib/Algebra/Category/CoalgebraCat/ComonEquivalence.lean
R : Type u
inst✝¹² : CommRing R
M N P Q : Type u
inst✝¹¹ : AddCommGroup M
inst✝¹⁰ : AddCommGroup N
inst✝⁹ : AddCommGroup P
inst✝⁸ : AddCommGroup Q
inst✝⁷ : Module R M
inst✝⁶ : Module R N
inst✝⁵ : Module R P
inst✝⁴ : Module R Q
inst✝³ : Coalgebra R M
inst✝² : Coalgebra R N
inst✝¹ : Coalgebra R P
inst✝ : Coalgebra R Q
\u22a2 (tensor_μ (ModuleCat R)ᵒᵖ (Opposite.op ((ofComon R).obj (Comon_.Mon_OpOpToComon_obj' (ModuleCat R) (Comon_.Comon_ToMon_OpOp_obj' (ModuleCat R) (of R M).toComonObj ⊗ Comon_.Comon_ToMon_OpOp_obj' (ModuleCat R) (of R N).toComonObj))).toModuleCat, Opposite.op (ModuleCat.mk P)) (Opposite.op ((ofComon R).obj (Comon_.Mon_OpOpToComon_obj' (ModuleCat R) (Comon_.Comon_ToMon_OpOp_obj' (ModuleCat R) (of R M).toComonObj ⊗ Comon_.Comon_ToMon_OpOp_obj' (ModuleCat R) (of R N).toComonObj))).toModuleCat, Opposite.op (ModuleCat.mk P))).unop ∘ₗ map (Comon_.Mon_OpOpToComon_obj' (ModuleCat R) (Comon_.Comon_ToMon_OpOp_obj' (ModuleCat R) (of R M).toComonObj ⊗ Comon_.Comon_ToMon_OpOp_obj' (ModuleCat R) (of R N).toComonObj)).comul Coalgebra.comul = ↑(tensorTensorTensorComm R (M ⊗[R] N) (M ⊗[R] N) P P) ∘ₗ map (↑(tensorTensorTensorComm R M M N N) ∘ₗ map Coalgebra.comul Coalgebra.comul) Coalgebra.comul
-- Mathlib/Algebra/Category/CoalgebraCat/ComonEquivalence.lean
R : Type u
inst✝¹² : CommRing R
M N P Q : Type u
inst✝¹¹ : AddCommGroup M
inst✝¹⁰ : AddCommGroup N
inst✝⁹ : AddCommGroup P
inst✝⁸ : AddCommGroup Q
inst✝⁷ : Module R M
inst✝⁶ : Module R N
inst✝⁵ : Module R P
inst✝⁴ : Module R Q
inst✝³ : Coalgebra R M
inst✝² : Coalgebra R N
inst✝¹ : Coalgebra R P
inst✝ : Coalgebra R Q
\u22a2 (tensor_μ (ModuleCat R)ᵒᵖ (Opposite.op ((ofComon R).obj (Comon_.Mon_OpOpToComon_obj' (ModuleCat R) (Comon_.Comon_ToMon_OpOp_obj' (ModuleCat R) (of R M).toComonObj ⊗ Comon_.Comon_ToMon_OpOp_obj' (ModuleCat R) (of R N).toComonObj))).toModuleCat, Opposite.op (ModuleCat.mk P)) (Opposite.op ((ofComon R).obj (Comon_.Mon_OpOpToComon_obj' (ModuleCat R) (Comon_.Comon_ToMon_OpOp_obj' (ModuleCat R) (of R M).toComonObj ⊗ Comon_.Comon_ToMon_OpOp_obj' (ModuleCat R) (of R N).toComonObj))).toModuleCat, Opposite.op (ModuleCat.mk P))).unop ∘ₗ map ((tensor_μ (ModuleCat R)ᵒᵖ (Opposite.op (ModuleCat.mk M), Opposite.op (ModuleCat.mk N)) (Opposite.op (ModuleCat.mk M), Opposite.op (ModuleCat.mk N))).unop ∘ₗ map Coalgebra.comul Coalgebra.comul) Coalgebra.comul = ↑(tensorTensorTensorComm R (M ⊗[R] N) (M ⊗[R] N) P P) ∘ₗ map (↑(tensorTensorTensorComm R M M N N) ∘ₗ map Coalgebra.comul Coalgebra.comul) Coalgebra.comul