ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Algebra/Order/Ring/Unbundled/Basic.lean
α : Type u
β : Type u_1
inst✝⁴ : Semiring α
inst✝³ : Preorder α
a b c d : α
inst✝² : ZeroLEOneClass α
inst✝¹ : PosMulMono α
inst✝ : MulPosMono α
ha₀ : 0 ≤ a
ha₁ : a ≤ 1
n✝ m✝ : ℕ
h : n✝.le m✝
\u22a2 a * a ^ m✝ ≤ a ^ n✝
-- Mathlib/Data/Set/Finite.lean
α : Type u
β : Type v
ι : Sort w
γ : Type x
s t : Set α
\u22a2 univ.Infinite ↔ Infinite α
-- Mathlib/Algebra/Polynomial/Basic.lean
R : Type u
a✝ b : R
m n : ℕ
inst✝ : Semiring R
p q a : R[X]
\u22a2 a.toFinsupp = 1 ↔ a = 1
-- Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m : ℤ
γ : Cochain F G n
p q : ℤ
hpq : p + n = q
\u22a2 γ.v p q hpq ≫ (HomologicalComplex.XIsoOfEq G ⋯).hom = γ.v p q ⋯
-- Mathlib/RingTheory/Perfection.lean
K : Type u₁
inst✝² : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝¹ : CommRing O
inst✝ : Algebra O K
hv : v.Integers O
p : ℕ
hp : Fact (Nat.Prime p)
hvp : Fact (v ↑p ≠ 1)
x : O
\u22a2 v ↑p < v ((algebraMap O K) x) ↔ (Ideal.Quotient.mk (Ideal.span {↑p})) x ≠ 0
-- Mathlib/GroupTheory/Torsion.lean
G : Type u_1
H : Type u_2
inst✝ : AddGroup G
\u22a2 AddMonoid.IsTorsionFree G ↔ NoZeroSMulDivisors ℤ G
-- Mathlib/GroupTheory/Torsion.lean
G : Type u_1
H : Type u_2
inst✝ : AddGroup G
\u22a2 (∀ (y : ℤ) (x : G), x ≠ 0 → y ≠ 0 → ¬y • x = 0) ↔ ∀ {c : ℤ} {x : G}, c • x = 0 → c = 0 ∨ x = 0
-- Mathlib/SetTheory/Game/Ordinal.lean
a b : Ordinal.{u_1}
h : a ≤ b
\u22a2 a.toPGame ≤ b.toPGame
-- Mathlib/SetTheory/Game/Ordinal.lean
a b : Ordinal.{u_1}
h : a ≤ b
i : a.toPGame.LeftMoves
\u22a2 a.toPGame.moveLeft i ⧏ b.toPGame
-- Mathlib/SetTheory/Game/Ordinal.lean
a b : Ordinal.{u_1}
h : a ≤ b
i : a.toPGame.LeftMoves
\u22a2 (↑(toLeftMovesToPGame.symm i)).toPGame ⧏ b.toPGame
-- Mathlib/Analysis/SpecialFunctions/Log/Base.lean
b✝ x y a b : ℝ
h₁ : a ≠ 0
h₂ : b ≠ 0
c : ℝ
\u22a2 logb (a * b) c = ((logb a c)⁻¹ + (logb b c)⁻¹)⁻¹
-- Mathlib/MeasureTheory/Constructions/Pi.lean
ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝³ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝² : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝¹ : ∀ (i : ι), SigmaFinite (μ i)
inst✝ : (i : ι) → MetricSpace (α i)
x : (i : ι) → α i
r : ℝ
hr : 0 < r
\u22a2 (Measure.pi μ) (Metric.ball x r) = ∏ i : ι, (μ i) (Metric.ball (x i) r)
-- Mathlib/Order/LiminfLimsup.lean
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Type u_5
α : Type u_6
f : Filter α
inst✝² : f.NeBot
R : Type u_7
inst✝¹ : Preorder R
κ : Type u_8
inst✝ : AddCommMonoid R
r : R → R → Prop
hr : ∀ (v₁ v₂ : α → R), IsBoundedUnder r f v₁ → IsBoundedUnder r f v₂ → IsBoundedUnder r f (v₁ + v₂)
hr₀ : r 0 0
u : κ → α → R
s : Finset κ
h : ∀ k ∈ s, IsBoundedUnder r f (u k)
\u22a2 IsBoundedUnder r f (∑ k ∈ s, u k)
-- Mathlib/Order/LiminfLimsup.lean
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Type u_5
α : Type u_6
f : Filter α
inst✝² : f.NeBot
R : Type u_7
inst✝¹ : Preorder R
κ : Type u_8
inst✝ : AddCommMonoid R
r : R → R → Prop
hr : ∀ (v₁ v₂ : α → R), IsBoundedUnder r f v₁ → IsBoundedUnder r f v₂ → IsBoundedUnder r f (v₁ + v₂)
hr₀ : r 0 0
u : κ → α → R
h : ∀ k ∈ ∅, IsBoundedUnder r f (u k)
\u22a2 IsBoundedUnder r f (∑ k ∈ ∅, u k)
-- Mathlib/Order/LiminfLimsup.lean
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Type u_5
α : Type u_6
f : Filter α
inst✝² : f.NeBot
R : Type u_7
inst✝¹ : Preorder R
κ : Type u_8
inst✝ : AddCommMonoid R
r : R → R → Prop
hr : ∀ (v₁ v₂ : α → R), IsBoundedUnder r f v₁ → IsBoundedUnder r f v₂ → IsBoundedUnder r f (v₁ + v₂)
hr₀ : r 0 0
u : κ → α → R
a✝¹ : κ
s✝ : Finset κ
h✝ : a✝¹ ∉ s✝
a✝ : (∀ k ∈ s✝, IsBoundedUnder r f (u k)) → IsBoundedUnder r f (∑ k ∈ s✝, u k)
h : ∀ k ∈ Finset.cons a✝¹ s✝ h✝, IsBoundedUnder r f (u k)
\u22a2 IsBoundedUnder r f (∑ k ∈ Finset.cons a✝¹ s✝ h✝, u k)
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/ComplexDeriv.lean
x : ℂ
hx : cos x = 0
\u22a2 Tendsto (fun x => abs (tan x)) (𝓝[≠] x) atTop
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/ComplexDeriv.lean
x : ℂ
hx : cos x = 0
\u22a2 Tendsto (fun x => ‖sin x‖ / ‖cos x‖) (𝓝[≠] x) atTop
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/ComplexDeriv.lean
x : ℂ
hx : cos x = 0
A : sin x ≠ 0
\u22a2 Tendsto (fun x => ‖sin x‖ / ‖cos x‖) (𝓝[≠] x) atTop
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/ComplexDeriv.lean
x : ℂ
hx : cos x = 0
A : sin x ≠ 0
B : Tendsto cos (𝓝[≠] x) (𝓝[≠] 0)
\u22a2 Tendsto (fun x => ‖sin x‖ / ‖cos x‖) (𝓝[≠] x) atTop
-- Mathlib/MeasureTheory/Constructions/Prod/Basic.lean
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁸ : MeasurableSpace α
inst✝⁷ : MeasurableSpace α'
inst✝⁶ : MeasurableSpace β
inst✝⁵ : MeasurableSpace β'
inst✝⁴ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝³ : NormedAddCommGroup E
inst✝² : SFinite ν
inst✝¹ : SFinite μ
inst✝ : SFinite τ
\u22a2 map (⇑MeasurableEquiv.prodAssoc) ((μ.prod ν).prod τ) = μ.prod (ν.prod τ)
-- Mathlib/MeasureTheory/Constructions/Prod/Basic.lean
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁸ : MeasurableSpace α
inst✝⁷ : MeasurableSpace α'
inst✝⁶ : MeasurableSpace β
inst✝⁵ : MeasurableSpace β'
inst✝⁴ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝³ : NormedAddCommGroup E
inst✝² : SFinite ν
inst✝¹ : SFinite μ
inst✝ : SFinite τ
this : (sum fun p => (sFiniteSeq μ p.1).prod ((sFiniteSeq ν p.2.1).prod (sFiniteSeq τ p.2.2))) = sum fun p => (sFiniteSeq μ p.1.1).prod ((sFiniteSeq ν p.1.2).prod (sFiniteSeq τ p.2))
\u22a2 map (⇑MeasurableEquiv.prodAssoc) ((μ.prod ν).prod τ) = μ.prod (ν.prod τ)
-- Mathlib/MeasureTheory/Constructions/Prod/Basic.lean
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁸ : MeasurableSpace α
inst✝⁷ : MeasurableSpace α'
inst✝⁶ : MeasurableSpace β
inst✝⁵ : MeasurableSpace β'
inst✝⁴ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝³ : NormedAddCommGroup E
inst✝² : SFinite ν
inst✝¹ : SFinite μ
inst✝ : SFinite τ
this : (sum fun p => (sFiniteSeq μ p.1).prod ((sFiniteSeq ν p.2.1).prod (sFiniteSeq τ p.2.2))) = sum fun p => (sFiniteSeq μ p.1.1).prod ((sFiniteSeq ν p.1.2).prod (sFiniteSeq τ p.2))
\u22a2 (sum fun i => map (⇑MeasurableEquiv.prodAssoc) (((sFiniteSeq μ i.1.1).prod (sFiniteSeq ν i.1.2)).prod (sFiniteSeq τ i.2))) = sum fun p => (sFiniteSeq μ p.1.1).prod ((sFiniteSeq ν p.1.2).prod (sFiniteSeq τ p.2))
-- Mathlib/MeasureTheory/Constructions/Prod/Basic.lean
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁸ : MeasurableSpace α
inst✝⁷ : MeasurableSpace α'
inst✝⁶ : MeasurableSpace β
inst✝⁵ : MeasurableSpace β'
inst✝⁴ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝³ : NormedAddCommGroup E
inst✝² : SFinite ν
inst✝¹ : SFinite μ
inst✝ : SFinite τ
this : (sum fun p => (sFiniteSeq μ p.1).prod ((sFiniteSeq ν p.2.1).prod (sFiniteSeq τ p.2.2))) = sum fun p => (sFiniteSeq μ p.1.1).prod ((sFiniteSeq ν p.1.2).prod (sFiniteSeq τ p.2))
i : (ℕ × ℕ) × ℕ
\u22a2 map (⇑MeasurableEquiv.prodAssoc) (((sFiniteSeq μ i.1.1).prod (sFiniteSeq ν i.1.2)).prod (sFiniteSeq τ i.2)) = (sFiniteSeq μ i.1.1).prod ((sFiniteSeq ν i.1.2).prod (sFiniteSeq τ i.2))
-- Mathlib/MeasureTheory/Constructions/Prod/Basic.lean
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁸ : MeasurableSpace α
inst✝⁷ : MeasurableSpace α'
inst✝⁶ : MeasurableSpace β
inst✝⁵ : MeasurableSpace β'
inst✝⁴ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝³ : NormedAddCommGroup E
inst✝² : SFinite ν
inst✝¹ : SFinite μ
inst✝ : SFinite τ
this : (sum fun p => (sFiniteSeq μ p.1).prod ((sFiniteSeq ν p.2.1).prod (sFiniteSeq τ p.2.2))) = sum fun p => (sFiniteSeq μ p.1.1).prod ((sFiniteSeq ν p.1.2).prod (sFiniteSeq τ p.2))
i : (ℕ × ℕ) × ℕ
\u22a2 ∀ s ∈ {s | MeasurableSet s}, ∀ t ∈ image2 (fun x x_1 => x ×ˢ x_1) {s | MeasurableSet s} {t | MeasurableSet t}, (map (⇑MeasurableEquiv.prodAssoc) (((sFiniteSeq μ i.1.1).prod (sFiniteSeq ν i.1.2)).prod (sFiniteSeq τ i.2))) (s ×ˢ t) = (sFiniteSeq μ i.1.1) s * ((sFiniteSeq ν i.1.2).prod (sFiniteSeq τ i.2)) t
-- Mathlib/MeasureTheory/Constructions/Prod/Basic.lean
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁸ : MeasurableSpace α
inst✝⁷ : MeasurableSpace α'
inst✝⁶ : MeasurableSpace β
inst✝⁵ : MeasurableSpace β'
inst✝⁴ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝³ : NormedAddCommGroup E
inst✝² : SFinite ν
inst✝¹ : SFinite μ
inst✝ : SFinite τ
this : (sum fun p => (sFiniteSeq μ p.1).prod ((sFiniteSeq ν p.2.1).prod (sFiniteSeq τ p.2.2))) = sum fun p => (sFiniteSeq μ p.1.1).prod ((sFiniteSeq ν p.1.2).prod (sFiniteSeq τ p.2))
i : (ℕ × ℕ) × ℕ
s : Set α
hs : s ∈ {s | MeasurableSet s}
t : Set β
ht : t ∈ {s | MeasurableSet s}
u : Set γ
hu : u ∈ {t | MeasurableSet t}
\u22a2 (map (⇑MeasurableEquiv.prodAssoc) (((sFiniteSeq μ i.1.1).prod (sFiniteSeq ν i.1.2)).prod (sFiniteSeq τ i.2))) (s ×ˢ (fun x x_1 => x ×ˢ x_1) t u) = (sFiniteSeq μ i.1.1) s * ((sFiniteSeq ν i.1.2).prod (sFiniteSeq τ i.2)) ((fun x x_1 => x ×ˢ x_1) t u)
-- Mathlib/MeasureTheory/Constructions/Prod/Basic.lean
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁸ : MeasurableSpace α
inst✝⁷ : MeasurableSpace α'
inst✝⁶ : MeasurableSpace β
inst✝⁵ : MeasurableSpace β'
inst✝⁴ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝³ : NormedAddCommGroup E
inst✝² : SFinite ν
inst✝¹ : SFinite μ
inst✝ : SFinite τ
this : (sum fun p => (sFiniteSeq μ p.1).prod ((sFiniteSeq ν p.2.1).prod (sFiniteSeq τ p.2.2))) = sum fun p => (sFiniteSeq μ p.1.1).prod ((sFiniteSeq ν p.1.2).prod (sFiniteSeq τ p.2))
i : (ℕ × ℕ) × ℕ
s : Set α
hs : MeasurableSet s
t : Set β
ht : MeasurableSet t
u : Set γ
hu : MeasurableSet u
\u22a2 (map (⇑MeasurableEquiv.prodAssoc) (((sFiniteSeq μ i.1.1).prod (sFiniteSeq ν i.1.2)).prod (sFiniteSeq τ i.2))) (s ×ˢ (fun x x_1 => x ×ˢ x_1) t u) = (sFiniteSeq μ i.1.1) s * ((sFiniteSeq ν i.1.2).prod (sFiniteSeq τ i.2)) ((fun x x_1 => x ×ˢ x_1) t u)
-- Mathlib/NumberTheory/Dioph.lean
α β γ : Type u
S S' : Set (α → ℕ)
d : Dioph S
H : ∀ (v : α → ℕ), v ∈ S ↔ v ∈ S'
\u22a2 Dioph S'
-- Mathlib/MeasureTheory/Integral/Bochner.lean
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedAddCommGroup F
m : MeasurableSpace α
μ : Measure α
inst✝⁸ : NormedField 𝕜
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : SMulCommClass ℝ 𝕜 E
F' : Type u_5
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace ℝ F'
E' : Type u_6
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace ℝ E'
inst✝ : NormedSpace 𝕜 E'
f : ↥(simpleFunc ℝ 1 μ)
eq : ∀ (a : α), ↑(toSimpleFunc f).posPart a = max (↑(toSimpleFunc f) a) 0
\u22a2 ↑(toSimpleFunc (posPart f)) =ᶠ[ae μ] ↑(toSimpleFunc f).posPart
-- Mathlib/MeasureTheory/Integral/Bochner.lean
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedAddCommGroup F
m : MeasurableSpace α
μ : Measure α
inst✝⁸ : NormedField 𝕜
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : SMulCommClass ℝ 𝕜 E
F' : Type u_5
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace ℝ F'
E' : Type u_6
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace ℝ E'
inst✝ : NormedSpace 𝕜 E'
f : ↥(simpleFunc ℝ 1 μ)
eq : ∀ (a : α), ↑(toSimpleFunc f).posPart a = max (↑(toSimpleFunc f) a) 0
ae_eq : ∀ᵐ (a : α) ∂μ, ↑(toSimpleFunc (posPart f)) a = max (↑(toSimpleFunc f) a) 0
\u22a2 ↑(toSimpleFunc (posPart f)) =ᶠ[ae μ] ↑(toSimpleFunc f).posPart
-- Mathlib/MeasureTheory/Integral/Bochner.lean
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedAddCommGroup F
m : MeasurableSpace α
μ : Measure α
inst✝⁸ : NormedField 𝕜
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : SMulCommClass ℝ 𝕜 E
F' : Type u_5
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace ℝ F'
E' : Type u_6
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace ℝ E'
inst✝ : NormedSpace 𝕜 E'
f : ↥(simpleFunc ℝ 1 μ)
eq : ∀ (a : α), ↑(toSimpleFunc f).posPart a = max (↑(toSimpleFunc f) a) 0
ae_eq : ∀ᵐ (a : α) ∂μ, ↑(toSimpleFunc (posPart f)) a = max (↑(toSimpleFunc f) a) 0
a : α
h : ↑(toSimpleFunc (posPart f)) a = max (↑(toSimpleFunc f) a) 0
\u22a2 ↑(toSimpleFunc (posPart f)) a = ↑(toSimpleFunc f).posPart a
-- Mathlib/SetTheory/Cardinal/Ordinal.lean
α β : Type u
β' : Type v
inst✝ : Infinite α
leq : lift.{v, u} #α = lift.{u, v} #β'
\u22a2 #(α ≃ β') = 2 ^ lift.{v, u} #α
-- Mathlib/SetTheory/Cardinal/Ordinal.lean
α β : Type u
β' : Type v
inst✝ : Infinite α
leq : lift.{v, u} #α = lift.{u, v} #β'
\u22a2 ↑2 ^ lift.{v, u} #α = 2 ^ lift.{v, u} #α
-- Mathlib/Topology/EMetricSpace/Basic.lean
α : Type u
β : Type v
X : Type u_1
inst✝ : PseudoEMetricSpace α
x y z : α
ε ε₁ ε₂ : ℝ≥0∞
s t : Set α
\u22a2 diam {x, y, z} = max (max (edist x y) (edist x z)) (edist y z)
-- Mathlib/RingTheory/Polynomial/ScaleRoots.lean
R : Type u_1
S : Type u_2
A : Type u_3
K : Type u_4
inst✝¹ : Semiring R
inst✝ : Semiring S
p : R[X]
s : R
this : (a : Prop) → Decidable a
hp : ¬p = 0
\u22a2 (p.scaleRoots s).degree = p.degree
-- Mathlib/RingTheory/Polynomial/ScaleRoots.lean
R : Type u_1
S : Type u_2
A : Type u_3
K : Type u_4
inst✝¹ : Semiring R
inst✝ : Semiring S
p : R[X]
s : R
this : (a : Prop) → Decidable a
hp : ¬p = 0
\u22a2 (p.scaleRoots s).coeff p.natDegree ≠ 0
-- Mathlib/RingTheory/Polynomial/ScaleRoots.lean
R : Type u_1
S : Type u_2
A : Type u_3
K : Type u_4
inst✝¹ : Semiring R
inst✝ : Semiring S
p : R[X]
s : R
this : (a : Prop) → Decidable a
hp : ¬p = 0
\u22a2 p.leadingCoeff ≠ 0
-- Mathlib/RingTheory/Polynomial/ScaleRoots.lean
R : Type u_1
S : Type u_2
A : Type u_3
K : Type u_4
inst✝¹ : Semiring R
inst✝ : Semiring S
p : R[X]
s : R
this : (a : Prop) → Decidable a
hp : ¬p = 0
h : p.leadingCoeff = 0
\u22a2 False
-- Mathlib/RingTheory/Polynomial/ScaleRoots.lean
R : Type u_1
S : Type u_2
A : Type u_3
K : Type u_4
inst✝¹ : Semiring R
inst✝ : Semiring S
p : R[X]
s : R
this✝ : (a : Prop) → Decidable a
hp : ¬p = 0
h : p.leadingCoeff = 0
this : p = 0
\u22a2 False
-- Mathlib/RingTheory/Polynomial/Quotient.lean
R : Type u_1
inst✝ : CommRing R
I : Ideal R[X]
x : ↥((Quotient.mk I).comp C).range
hx : C x ∈ map (mapRingHom ((Quotient.mk I).comp C).rangeRestrict) I
i : R →+* ↥((Quotient.mk I).comp C).range := ((Quotient.mk I).comp C).rangeRestrict
\u22a2 x = 0
-- Mathlib/RingTheory/Polynomial/Quotient.lean
R : Type u_1
inst✝ : CommRing R
I : Ideal R[X]
x : ↥((Quotient.mk I).comp C).range
hx : C x ∈ map (mapRingHom ((Quotient.mk I).comp C).rangeRestrict) I
i : R →+* ↥((Quotient.mk I).comp C).range := ((Quotient.mk I).comp C).rangeRestrict
hi' : RingHom.ker (mapRingHom i) ≤ I
\u22a2 x = 0
-- Mathlib/RingTheory/Polynomial/Quotient.lean
R : Type u_1
inst✝ : CommRing R
I : Ideal R[X]
i : R →+* ↥((Quotient.mk I).comp C).range := ((Quotient.mk I).comp C).rangeRestrict
hi' : RingHom.ker (mapRingHom i) ≤ I
x : R[X] ⧸ I
hx' : x ∈ ((Quotient.mk I).comp C).range
hx : C ⟨x, hx'⟩ ∈ map (mapRingHom ((Quotient.mk I).comp C).rangeRestrict) I
\u22a2 ⟨x, hx'⟩ = 0
-- Mathlib/RingTheory/Polynomial/Quotient.lean
R : Type u_1
inst✝ : CommRing R
I : Ideal R[X]
i : R →+* ↥((Quotient.mk I).comp C).range := ((Quotient.mk I).comp C).rangeRestrict
hi' : RingHom.ker (mapRingHom i) ≤ I
y : R
hx' : ((Quotient.mk I).comp C) y ∈ ((Quotient.mk I).comp C).range
hx : C ⟨((Quotient.mk I).comp C) y, hx'⟩ ∈ map (mapRingHom ((Quotient.mk I).comp C).rangeRestrict) I
\u22a2 ⟨((Quotient.mk I).comp C) y, hx'⟩ = 0
-- Mathlib/RingTheory/Polynomial/Quotient.lean
R : Type u_1
inst✝ : CommRing R
I : Ideal R[X]
i : R →+* ↥((Quotient.mk I).comp C).range := ((Quotient.mk I).comp C).rangeRestrict
hi' : RingHom.ker (mapRingHom i) ≤ I
y : R
hx' : ((Quotient.mk I).comp C) y ∈ ((Quotient.mk I).comp C).range
hx : C ⟨((Quotient.mk I).comp C) y, hx'⟩ ∈ map (mapRingHom ((Quotient.mk I).comp C).rangeRestrict) I
\u22a2 ↑⟨((Quotient.mk I).comp C) y, hx'⟩ = ↑0
-- Mathlib/RingTheory/Polynomial/Quotient.lean
R : Type u_1
inst✝ : CommRing R
I : Ideal R[X]
i : R →+* ↥((Quotient.mk I).comp C).range := ((Quotient.mk I).comp C).rangeRestrict
hi' : RingHom.ker (mapRingHom i) ≤ I
y : R
hx' : ((Quotient.mk I).comp C) y ∈ ((Quotient.mk I).comp C).range
hx : C ⟨((Quotient.mk I).comp C) y, hx'⟩ ∈ map (mapRingHom ((Quotient.mk I).comp C).rangeRestrict) I
\u22a2 C y ∈ I
-- Mathlib/RingTheory/Polynomial/Quotient.lean
R : Type u_1
inst✝ : CommRing R
I : Ideal R[X]
i : R →+* ↥((Quotient.mk I).comp C).range := ((Quotient.mk I).comp C).rangeRestrict
hi' : RingHom.ker (mapRingHom i) ≤ I
y : R
hx' : ((Quotient.mk I).comp C) y ∈ ((Quotient.mk I).comp C).range
hx : C ⟨((Quotient.mk I).comp C) y, hx'⟩ ∈ map (mapRingHom ((Quotient.mk I).comp C).rangeRestrict) I
\u22a2 C (i y) ∈ map (mapRingHom i) I
-- Mathlib/Algebra/MvPolynomial/Rename.lean
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
p : MvPolynomial σ R
f : σ → τ
inst✝ : DecidableEq τ
h : Injective f
\u22a2 ((rename f) p).support = Finset.image (Finsupp.mapDomain f) p.support
-- Mathlib/Algebra/MvPolynomial/Rename.lean
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
p : MvPolynomial σ R
f : σ → τ
inst✝ : DecidableEq τ
h : Injective f
\u22a2 support (Finsupp.mapDomain (Finsupp.mapDomain f) p) = Finset.image (Finsupp.mapDomain f) p.support
-- Mathlib/AlgebraicTopology/DoldKan/NCompGamma.lean
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
P : Karoubi (SimplicialObject C)
n : ℕ
\u22a2 (N₂Γ₂.inv.app (N₂.obj P) ≫ N₂.map (Γ₂N₂.natTrans.app P)).f.f n = (𝟙 (N₂.obj P)).f.f n
-- Mathlib/AlgebraicTopology/DoldKan/NCompGamma.lean
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
P : Karoubi (SimplicialObject C)
n : ℕ
eq₁ : (N₂Γ₂.inv.app (N₂.obj P)).f.f n = PInfty.f n ≫ P.p.app (op [n]) ≫ ((Γ₀.splitting (N₂.obj P).X).cofan (op [n])).inj (Splitting.IndexSet.id (op [n]))
\u22a2 (N₂Γ₂.inv.app (N₂.obj P) ≫ N₂.map (Γ₂N₂.natTrans.app P)).f.f n = (𝟙 (N₂.obj P)).f.f n
-- Mathlib/AlgebraicTopology/DoldKan/NCompGamma.lean
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
P : Karoubi (SimplicialObject C)
n : ℕ
eq₁ : (N₂Γ₂.inv.app (N₂.obj P)).f.f n = PInfty.f n ≫ P.p.app (op [n]) ≫ ((Γ₀.splitting (N₂.obj P).X).cofan (op [n])).inj (Splitting.IndexSet.id (op [n]))
eq₂ : ((Γ₀.splitting (N₂.obj P).X).cofan (op [n])).inj (Splitting.IndexSet.id (op [n])) ≫ (N₂.map (Γ₂N₂.natTrans.app P)).f.f n = PInfty.f n ≫ P.p.app (op [n])
\u22a2 (N₂Γ₂.inv.app (N₂.obj P) ≫ N₂.map (Γ₂N₂.natTrans.app P)).f.f n = (𝟙 (N₂.obj P)).f.f n
-- Mathlib/Data/Fin/Tuple/Sort.lean
n : ℕ
α : Type u_1
inst✝ : LinearOrder α
f : Fin n → α
σ : Equiv.Perm (Fin n)
h : f ∘ ⇑σ ≠ f ∘ ⇑(sort f)
\u22a2 ∃ i j, i < j ∧ (f ∘ ⇑σ) j < (f ∘ ⇑σ) i
-- Mathlib/Data/Fin/Tuple/Sort.lean
n : ℕ
α : Type u_1
inst✝ : LinearOrder α
f : Fin n → α
σ : Equiv.Perm (Fin n)
h : ∀ (i j : Fin n), i < j → (f ∘ ⇑σ) i ≤ (f ∘ ⇑σ) j
\u22a2 f ∘ ⇑σ = f ∘ ⇑(sort f)
-- Mathlib/CategoryTheory/Iso.lean
C : Type u
inst✝ : Category.{v, u} C
X Y Z : C
α : X ≅ Y
f : Z ⟶ Y
g : Z ⟶ X
H : f ≫ α.inv = g
\u22a2 f = g ≫ α.hom
-- Mathlib/Algebra/Homology/SingleHomology.lean
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
j : ι
A B : C
f : A ⟶ B
\u22a2 (singleObjCyclesSelfIso c j A).inv ≫ cyclesMap ((single C c j).map f) j = f ≫ (singleObjCyclesSelfIso c j B).inv
-- Mathlib/NumberTheory/Divisors.lean
n : ℕ
x : ℕ × ℕ
\u22a2 x.swap ∈ n.divisorsAntidiagonal ↔ x ∈ n.divisorsAntidiagonal
-- Mathlib/Combinatorics/SimpleGraph/Subgraph.lean
ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' G'' : G.Subgraph
s s' : Set V
\u22a2 G'.induce s' ≤ G'.induce (s ∪ s')
-- Mathlib/CategoryTheory/Comma/Over.lean
T : Type u₁
inst✝ : Category.{v₁, u₁} T
X : T
A B : Under X
f : A ⟶ B
\u22a2 A.hom ≫ f.right = B.hom
-- Mathlib/CategoryTheory/Comma/Over.lean
T : Type u₁
inst✝ : Category.{v₁, u₁} T
X : T
A B : Under X
f : A ⟶ B
this : (Functor.fromPUnit X).map f.left ≫ B.hom = A.hom ≫ (𝟭 T).map f.right
\u22a2 A.hom ≫ f.right = B.hom
-- Mathlib/Algebra/Order/ToIntervalMod.lean
α : Type u_1
inst✝ : LinearOrderedAddCommGroup α
hα : Archimedean α
p : α
hp : 0 < p
a✝ b✝ c✝ : α
n : ℤ
a b c : α
\u22a2 toIcoMod hp a (b + c) = toIcoMod hp (a - c) b + c
-- Mathlib/Algebra/Order/Group/Unbundled/Basic.lean
α✝ : Type u
inst✝⁵ : Group α✝
inst✝⁴ : LinearOrder α✝
inst✝³ : CovariantClass α✝ α✝ (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
a✝ b✝ c : α✝
α : Type u_1
inst✝² : CommGroup α
inst✝¹ : LinearOrder α
inst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
a b : α
\u22a2 a / b ≤ b / a ↔ a ≤ b
-- Mathlib/Algebra/Order/Group/Unbundled/Basic.lean
α✝ : Type u
inst✝⁵ : Group α✝
inst✝⁴ : LinearOrder α✝
inst✝³ : CovariantClass α✝ α✝ (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
a✝ b✝ c : α✝
α : Type u_1
inst✝² : CommGroup α
inst✝¹ : LinearOrder α
inst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
a b : α
\u22a2 a / b ≤ a⁻¹ * b ↔ a ≤ b
-- Mathlib/CategoryTheory/Galois/Basic.lean
C : Type u₁
inst✝² : Category.{u₂, u₁} C
inst✝¹ : PreGaloisCategory C
F : C ⥤ FintypeCat
inst✝ : FiberFunctor F
X A B : C
f : A ⟶ X
g : B ⟶ X
a : ↑(F.obj A)
b : ↑(F.obj B)
h : F.map f a = F.map g b
\u22a2 ((Types.pullbackIsoPullback (F.map f) (F.map g)).inv ≫ (PreservesPullback.iso (F ⋙ FintypeCat.incl) f g).inv ≫ (F ⋙ FintypeCat.incl).map (pullback.fst f g)) ⟨(a, b), h⟩ = a
-- Mathlib/CategoryTheory/Comma/Arrow.lean
T : Type u
inst✝ : Category.{v, u} T
f g : Arrow T
sq : f ⟶ g
e : f ≅ g
\u22a2 e.hom.right ≫ e.inv.right = 𝟙 f.right
-- Mathlib/MeasureTheory/Measure/Haar/Unique.lean
G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : Group G
inst✝⁷ : TopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
inst✝⁴ : LocallyCompactSpace G
μ' μ : Measure G
inst✝³ : IsProbabilityMeasure μ
inst✝² : IsProbabilityMeasure μ'
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ'.IsHaarMeasure
\u22a2 μ' = μ
-- Mathlib/MeasureTheory/Measure/Haar/Unique.lean
G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : Group G
inst✝⁷ : TopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
inst✝⁴ : LocallyCompactSpace G
μ' μ : Measure G
inst✝³ : IsProbabilityMeasure μ
inst✝² : IsProbabilityMeasure μ'
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ'.IsHaarMeasure
this : CompactSpace G
\u22a2 μ' = μ
-- Mathlib/MeasureTheory/Measure/Haar/Unique.lean
G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : Group G
inst✝⁷ : TopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
inst✝⁴ : LocallyCompactSpace G
μ' μ : Measure G
inst✝³ : IsProbabilityMeasure μ
inst✝² : IsProbabilityMeasure μ'
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ'.IsHaarMeasure
this : CompactSpace G
A : ∀ (s : Set G), μ' s = μ'.haarScalarFactor μ • μ s
\u22a2 μ' = μ
-- Mathlib/MeasureTheory/Measure/Haar/Unique.lean
G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : Group G
inst✝⁷ : TopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
inst✝⁴ : LocallyCompactSpace G
μ' μ : Measure G
inst✝³ : IsProbabilityMeasure μ
inst✝² : IsProbabilityMeasure μ'
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ'.IsHaarMeasure
this : CompactSpace G
A : ∀ (s : Set G), μ' s = μ'.haarScalarFactor μ • μ s
Z : μ' univ = μ'.haarScalarFactor μ • μ univ
\u22a2 μ' = μ
-- Mathlib/MeasureTheory/Measure/Haar/Unique.lean
G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : Group G
inst✝⁷ : TopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
inst✝⁴ : LocallyCompactSpace G
μ' μ : Measure G
inst✝³ : IsProbabilityMeasure μ
inst✝² : IsProbabilityMeasure μ'
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ'.IsHaarMeasure
this : CompactSpace G
A : ∀ (s : Set G), μ' s = μ'.haarScalarFactor μ • μ s
Z : 1 = μ'.haarScalarFactor μ
\u22a2 μ' = μ
-- Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean
α : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
f : α → ℚ → ℝ
hf : IsMeasurableRatCDF f
a : α
x : ℝ
\u22a2 ↑(hf.stieltjesFunction a) x ≤ 1
-- Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean
α : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
f : α → ℚ → ℝ
hf : IsMeasurableRatCDF f
a : α
x : ℝ
r : ℚ
hrx : x < ↑r
\u22a2 ↑(hf.stieltjesFunction a) x ≤ 1