ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean
α : Type u_1
𝕜 : Type u_2
inst✝¹ : LinearOrderedField 𝕜
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : 𝕜
s t : Finset α
a b : α
s' : Finset α
hs' : s' ⊆ s
t' : Finset α
ht' : t' ⊆ t
hs : ↑s.card * 1 ≤ ↑s'.card
ht : ↑t.card * 1 ≤ ↑t'.card
\u22a2 |↑(G.edgeDensity s' t') - ↑(G.edgeDensity s t)| < 1
-- Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean
α : Type u_1
𝕜 : Type u_2
inst✝¹ : LinearOrderedField 𝕜
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : 𝕜
s t : Finset α
a b : α
s' : Finset α
hs' : s' ⊆ s
t' : Finset α
ht' : t' ⊆ t
hs : ↑s.card ≤ ↑s'.card
ht : ↑t.card ≤ ↑t'.card
\u22a2 |↑(G.edgeDensity s' t') - ↑(G.edgeDensity s t)| < 1
-- Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean
α : Type u_1
𝕜 : Type u_2
inst✝¹ : LinearOrderedField 𝕜
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : 𝕜
s t : Finset α
a b : α
s' : Finset α
hs' : s' ⊆ s
t' : Finset α
ht' : t' ⊆ t
hs : ↑s.card ≤ ↑s'.card
ht : ↑t.card ≤ ↑t'.card
\u22a2 0 < 1
-- Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean
R✝ : Type u
S : Type v
T : Type w
inst✝³ : NonUnitalNonAssocSemiring R✝
M : Subsemigroup R✝
inst✝² : NonUnitalNonAssocSemiring S
inst✝¹ : NonUnitalNonAssocSemiring T
R : Type u_1
inst✝ : NonUnitalSemiring R
z : R
\u22a2 z ∈ center R ↔ ∀ (g : R), g * z = z * g
-- Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean
R✝ : Type u
S : Type v
T : Type w
inst✝³ : NonUnitalNonAssocSemiring R✝
M : Subsemigroup R✝
inst✝² : NonUnitalNonAssocSemiring S
inst✝¹ : NonUnitalNonAssocSemiring T
R : Type u_1
inst✝ : NonUnitalSemiring R
z : R
\u22a2 z ∈ center R ↔ z ∈ Set.center R
-- Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean
R : Type u
inst✝¹ : CommRing R
W' : Jacobian R
F : Type v
inst✝ : Field F
W : Jacobian F
P : Fin 3 → R
hPz : P z = 0
\u22a2 W'.dblZ P = 0
-- Mathlib/SetTheory/Cardinal/Divisibility.lean
a✝ b✝ : Cardinal.{u}
n m : ℕ
a x : Cardinal.{u_1}
b0 : x ≠ 0
b : Cardinal.{u_1}
hab : x = a * b
\u22a2 a ≤ x
-- Mathlib/RingTheory/Ideal/Over.lean
R : Type u_1
inst✝² : CommRing R
S : Type u_2
inst✝¹ : CommRing S
f✝ : R →+* S
I✝ J : Ideal S
inst✝ : Algebra R S
f : R →+* S
I : Ideal R
hf₁ : RingHom.ker f ≤ I
hf₂ : f.IsIntegral
\u22a2 map f I = ⊤ → I = ⊤
-- Mathlib/RingTheory/Ideal/Over.lean
R : Type u_1
inst✝² : CommRing R
S : Type u_2
inst✝¹ : CommRing S
f✝ : R →+* S
I✝ J : Ideal S
inst✝ : Algebra R S
f : R →+* S
I : Ideal R
hf₁ : RingHom.ker f ≤ I
hf₂ : f.IsIntegral
\u22a2 ¬I = ⊤ → ¬map f I = ⊤
-- Mathlib/RingTheory/Ideal/Over.lean
R : Type u_1
inst✝² : CommRing R
S : Type u_2
inst✝¹ : CommRing S
f✝ : R →+* S
I✝ J : Ideal S
inst✝ : Algebra R S
f : R →+* S
I : Ideal R
hf₁ : RingHom.ker f ≤ I
hf₂ : f.IsIntegral
h : ¬I = ⊤
\u22a2 ¬map f I = ⊤
-- Mathlib/RingTheory/Ideal/Over.lean
R : Type u_1
inst✝² : CommRing R
S : Type u_2
inst✝¹ : CommRing S
f✝ : R →+* S
I✝ J : Ideal S
inst✝ : Algebra R S
f : R →+* S
I : Ideal R
hf₁ : RingHom.ker f ≤ I
hf₂ : f.IsIntegral
h : ¬I = ⊤
m : Ideal R
left✝ : m.IsMaximal
hm : I ≤ m
\u22a2 ¬map f I = ⊤
-- Mathlib/RingTheory/Ideal/Over.lean
R : Type u_1
inst✝² : CommRing R
S : Type u_2
inst✝¹ : CommRing S
f✝ : R →+* S
I✝ J : Ideal S
inst✝ : Algebra R S
f : R →+* S
I : Ideal R
hf₁ : RingHom.ker f ≤ I
hf₂ : f.IsIntegral
h : ¬I = ⊤
m : Ideal R
left✝ : m.IsMaximal
hm : I ≤ m
x✝ : Algebra R S := f.toAlgebra
this : Algebra.IsIntegral R S
\u22a2 ¬map f I = ⊤
-- Mathlib/RingTheory/Ideal/Over.lean
R : Type u_1
inst✝² : CommRing R
S : Type u_2
inst✝¹ : CommRing S
f✝ : R →+* S
I✝ J : Ideal S
inst✝ : Algebra R S
f : R →+* S
I : Ideal R
hf₁ : RingHom.ker f ≤ I
hf₂ : f.IsIntegral
h : ¬I = ⊤
x✝ : Algebra R S := f.toAlgebra
this : Algebra.IsIntegral R S
m' : Ideal S
left✝¹ : m'.IsMaximal
left✝ : (comap (algebraMap R S) m').IsMaximal
hm : I ≤ comap (algebraMap R S) m'
\u22a2 ¬map f I = ⊤
-- Mathlib/RingTheory/Ideal/Over.lean
R : Type u_1
inst✝² : CommRing R
S : Type u_2
inst✝¹ : CommRing S
f✝ : R →+* S
I✝ J : Ideal S
inst✝ : Algebra R S
f : R →+* S
I : Ideal R
hf₁ : RingHom.ker f ≤ I
hf₂ : f.IsIntegral
h : ¬I = ⊤
x✝ : Algebra R S := f.toAlgebra
this : Algebra.IsIntegral R S
m' : Ideal S
left✝¹ : m'.IsMaximal
left✝ : (comap (algebraMap R S) m').IsMaximal
hm : map (algebraMap R S) I ≤ m'
\u22a2 ¬map f I = ⊤
-- Mathlib/Analysis/Analytic/Constructions.lean
α : Type u_1
𝕜 : Type u_2
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedSpace 𝕜 G
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : NormedSpace 𝕜 H
𝕝 : Type u_7
inst✝³ : NontriviallyNormedField 𝕝
inst✝² : NormedAlgebra 𝕜 𝕝
A : Type u_8
inst✝¹ : NormedRing A
inst✝ : NormedAlgebra 𝕜 A
z : 𝕝
m : z ∈ {z | z ≠ 0}
\u22a2 AnalyticAt 𝕜 (fun z => z⁻¹) z
-- Mathlib/Data/Rat/Defs.lean
q a b c : ℚ
n d : ℤ
\u22a2 ↑n / ↑d = n /. d
-- Mathlib/MeasureTheory/Integral/Asymptotics.lean
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f : α → E
g : α → F
a b : α
μ : Measure α
l : Filter α
inst✝² : TopologicalSpace α
inst✝¹ : SecondCountableTopology α
inst✝ : (cocompact α).IsMeasurablyGenerated
hf : LocallyIntegrable f μ
ho : f =O[cocompact α] g
hg : IntegrableAtFilter g (cocompact α) μ
\u22a2 Integrable f μ
-- Mathlib/MeasureTheory/Integral/Asymptotics.lean
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f : α → E
g : α → F
a b : α
μ : Measure α
l : Filter α
inst✝² : TopologicalSpace α
inst✝¹ : SecondCountableTopology α
inst✝ : (cocompact α).IsMeasurablyGenerated
hf : LocallyIntegrable f μ
ho : f =O[cocompact α] g
hg : IntegrableAtFilter g (cocompact α) μ
\u22a2 StronglyMeasurableAtFilter f (cocompact α) μ
-- Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
q : ℝ
μ : Measure α
f✝ g : α → E
ι : Type u_3
s : Finset ι
f : ι → α → E
hf : ∀ i ∈ s, Memℒp (f i) p μ
\u22a2 Memℒp (∑ i ∈ s, f i) p μ
-- Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
q : ℝ
μ : Measure α
f✝ g : α → E
ι : Type u_3
s : Finset ι
f : ι → α → E
hf : ∀ i ∈ s, Memℒp (f i) p μ
\u22a2 ∑ i ∈ s, f i = fun a => ∑ i ∈ s, f i a
-- Mathlib/CategoryTheory/Pi/Basic.lean
I : Type w₀
J✝ : Type w₁
C : I → Type u₁
inst✝ : (i : I) → Category.{v₁, u₁} (C i)
J : Type w₁
K : Type w₂
f : K → J
g : J → I
X Y : (i : I) → C i
f' : X ⟶ Y
\u22a2 (({ obj := fun f i => f (g i), map := fun {X Y} α i => α (g i), map_id := ⋯, map_comp := ⋯ } ⋙ { obj := fun f_1 i => f_1 (f i), map := fun {X Y} α i => α (f i), map_id := ⋯, map_comp := ⋯ }).map f' ≫ fun b => 𝟙 (Y (g (f b)))) = (fun b => 𝟙 (X (g (f b)))) ≫ fun i => f' (g (f i))
-- Mathlib/CategoryTheory/Pi/Basic.lean
I : Type w₀
J✝ : Type w₁
C : I → Type u₁
inst✝ : (i : I) → Category.{v₁, u₁} (C i)
J : Type w₁
K : Type w₂
f : K → J
g : J → I
X Y : (i : I) → C i
f' : X ⟶ Y
x✝ : K
\u22a2 (({ obj := fun f i => f (g i), map := fun {X Y} α i => α (g i), map_id := ⋯, map_comp := ⋯ } ⋙ { obj := fun f_1 i => f_1 (f i), map := fun {X Y} α i => α (f i), map_id := ⋯, map_comp := ⋯ }).map f' ≫ fun b => 𝟙 (Y (g (f b)))) x✝ = ((fun b => 𝟙 (X (g (f b)))) ≫ fun i => f' (g (f i))) x✝
-- Mathlib/CategoryTheory/Pi/Basic.lean
I : Type w₀
J✝ : Type w₁
C : I → Type u₁
inst✝ : (i : I) → Category.{v₁, u₁} (C i)
J : Type w₁
K : Type w₂
f : K → J
g : J → I
X Y : (i : I) → C i
f' : X ⟶ Y
\u22a2 ((fun i => f' (g (f i))) ≫ fun b => 𝟙 (Y (g (f b)))) = (fun b => 𝟙 (X (g (f b)))) ≫ ({ obj := fun f i => f (g i), map := fun {X Y} α i => α (g i), map_id := ⋯, map_comp := ⋯ } ⋙ { obj := fun f_1 i => f_1 (f i), map := fun {X Y} α i => α (f i), map_id := ⋯, map_comp := ⋯ }).map f'
-- Mathlib/CategoryTheory/Pi/Basic.lean
I : Type w₀
J✝ : Type w₁
C : I → Type u₁
inst✝ : (i : I) → Category.{v₁, u₁} (C i)
J : Type w₁
K : Type w₂
f : K → J
g : J → I
X Y : (i : I) → C i
f' : X ⟶ Y
x✝ : K
\u22a2 ((fun i => f' (g (f i))) ≫ fun b => 𝟙 (Y (g (f b)))) x✝ = ((fun b => 𝟙 (X (g (f b)))) ≫ ({ obj := fun f i => f (g i), map := fun {X Y} α i => α (g i), map_id := ⋯, map_comp := ⋯ } ⋙ { obj := fun f_1 i => f_1 (f i), map := fun {X Y} α i => α (f i), map_id := ⋯, map_comp := ⋯ }).map f') x✝
-- Mathlib/CategoryTheory/Limits/Shapes/Pullback/Assoc.lean
C : Type u
inst✝⁶ : Category.{v, u} C
W X Y Z X₁ X₂ X₃ Y₁ Y₂ : C
f₁ : X₁ ⟶ Y₁
f₂ : X₂ ⟶ Y₁
f₃ : X₂ ⟶ Y₂
f₄ : X₃ ⟶ Y₂
inst✝⁵ : HasPullback f₁ f₂
inst✝⁴ : HasPullback f₃ f₄
inst✝³ : HasPullback (sorryAx (?m.176514 ⟶ X₂) true ≫ f₃) f₄
inst✝² : HasPullback f₁ (sorryAx (?m.181965 ⟶ X₂) true ≫ f₂)
inst✝¹ : HasPullback (pullback.snd f₁ f₂ ≫ f₃) f₄
inst✝ : HasPullback f₁ (pullback.fst f₃ f₄ ≫ f₂)
\u22a2 (pullbackAssoc f₁ f₂ f₃ f₄).inv ≫ pullback.fst (pullback.snd f₁ f₂ ≫ f₃) f₄ ≫ pullback.snd f₁ f₂ = pullback.snd f₁ (pullback.fst f₃ f₄ ≫ f₂) ≫ pullback.fst f₃ f₄
-- Mathlib/FieldTheory/PerfectClosure.lean
K : Type u
inst✝² : CommRing K
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP K p
n : ℕ
ih : mk K p (n, 0) = 0
this : R K p (n, 0) (n + 1, (frobenius K p) 0)
\u22a2 R K p (n, 0) (n + 1, 0)
-- .lake/packages/batteries/Batteries/Data/List/Perm.lean
α : Type u_1
inst✝ : DecidableEq α
l₁ l₂ : List α
h : l₁ <+~ l₂
x : α
hx : count x l₁ < count x l₂
\u22a2 x :: l₁ <+~ l₂
-- .lake/packages/batteries/Batteries/Data/List/Perm.lean
α : Type u_1
inst✝ : DecidableEq α
l₁ l₂ : List α
h : ∀ (x : α), x ∈ l₁ → count x l₁ ≤ count x l₂
x : α
hx : count x l₁ < count x l₂
\u22a2 ∀ (x_1 : α), x_1 ∈ x :: l₁ → count x_1 (x :: l₁) ≤ count x_1 l₂
-- .lake/packages/batteries/Batteries/Data/List/Perm.lean
α : Type u_1
inst✝ : DecidableEq α
l₁ l₂ : List α
h : ∀ (x : α), x ∈ l₁ → count x l₁ ≤ count x l₂
x : α
hx : count x l₁ < count x l₂
y : α
hy : y ∈ x :: l₁
\u22a2 count y (x :: l₁) ≤ count y l₂
-- Mathlib/Algebra/Algebra/Operations.lean
ι✝ : Sort uι
R : Type u
inst✝² : CommSemiring R
A : Type v
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
M N : Submodule R A
m n : A
ι : Type u_1
s : Finset ι
x : ι → A
\u22a2 ∏ i ∈ s, span R {x i} = span R {∏ i ∈ s, x i}
-- 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✝³ : SeminormedGroup E
inst✝² : SeminormedGroup F
inst✝¹ : SeminormedGroup G
s : Set E
a a₁ a₂ b b₁ b₂ : E
r r₁ r₂ : ℝ
inst✝ : FunLike 𝓕 E F
x : E
\u22a2 x ∈ closure {1} ↔ ‖x‖ = 0
-- Mathlib/MeasureTheory/Constructions/Prod/Integral.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✝¹² : MeasurableSpace α
inst✝¹¹ : MeasurableSpace α'
inst✝¹⁰ : MeasurableSpace β
inst✝⁹ : MeasurableSpace β'
inst✝⁸ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : SFinite ν
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : SFinite μ
E' : Type u_7
inst✝³ : NormedAddCommGroup E'
inst✝² : NormedSpace ℝ E'
𝕜 : Type u_8
inst✝¹ : _root_.RCLike 𝕜
inst✝ : NormedSpace 𝕜 E
f : α → 𝕜
g : β → E
hE : CompleteSpace E
h : ¬Integrable (fun z => f z.1 • g z.2) (μ.prod ν)
\u22a2 ∫ (z : α × β), f z.1 • g z.2 ∂μ.prod ν = (∫ (x : α), f x ∂μ) • ∫ (y : β), g y ∂ν
-- Mathlib/MeasureTheory/Constructions/Prod/Integral.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✝¹² : MeasurableSpace α
inst✝¹¹ : MeasurableSpace α'
inst✝¹⁰ : MeasurableSpace β
inst✝⁹ : MeasurableSpace β'
inst✝⁸ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : SFinite ν
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : SFinite μ
E' : Type u_7
inst✝³ : NormedAddCommGroup E'
inst✝² : NormedSpace ℝ E'
𝕜 : Type u_8
inst✝¹ : _root_.RCLike 𝕜
inst✝ : NormedSpace 𝕜 E
f : α → 𝕜
g : β → E
hE : CompleteSpace E
h : ¬Integrable (fun z => f z.1 • g z.2) (μ.prod ν)
H : ¬Integrable f μ ∨ ¬Integrable g ν
\u22a2 ∫ (z : α × β), f z.1 • g z.2 ∂μ.prod ν = (∫ (x : α), f x ∂μ) • ∫ (y : β), g y ∂ν
-- Mathlib/SetTheory/Ordinal/NaturalOps.lean
a b c d : Ordinal.{u}
\u22a2 a ⨳ b ≤ c ↔ ∀ a' < a, ∀ b' < b, a' ⨳ b ♯ a ⨳ b' < c ♯ a' ⨳ b'
-- Mathlib/SetTheory/Ordinal/NaturalOps.lean
a b c d : Ordinal.{u}
\u22a2 ¬a ⨳ b ≤ c ↔ ¬∀ a' < a, ∀ b' < b, a' ⨳ b ♯ a ⨳ b' < c ♯ a' ⨳ b'
-- Mathlib/SetTheory/Ordinal/FixedPoint.lean
o : Ordinal.{u}
f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v}
H : ∀ (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi)
a : Ordinal.{max u v}
\u22a2 o.nfpBFamily f a ∈ ⋂ i, ⋂ (hi : i < o), fixedPoints (f i hi)
-- Mathlib/SetTheory/Ordinal/FixedPoint.lean
o : Ordinal.{u}
f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v}
H : ∀ (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi)
a : Ordinal.{max u v}
\u22a2 ∀ (i : Ordinal.{u}) (j : i < o), o.nfpBFamily f a ∈ fixedPoints (f i j)
-- Mathlib/RingTheory/WittVector/WittPolynomial.lean
p : ℕ
R : Type u_1
inst✝² : CommRing R
inst✝¹ : DecidableEq R
inst✝ : Invertible ↑p
n : ℕ
\u22a2 xInTermsOfW p R n * C (↑p ^ n) = X n - ∑ i ∈ range n, C (↑p ^ i) * xInTermsOfW p R i ^ p ^ (n - i)
-- Mathlib/MeasureTheory/Constructions/Polish/Basic.lean
α : Type u_1
ι : Type u_2
X : Type u_3
Y : Type u_4
inst✝⁴ : TopologicalSpace X
inst✝³ : PolishSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : T0Space Y
inst✝ : SecondCountableTopology Y
f : X → Y
hf : Continuous f
hsurj : Surjective f
this✝¹ : MeasurableSpace X := borel X
this✝ : BorelSpace X
\u22a2 MeasurableSpace.map f (borel X) = borel Y
-- Mathlib/Order/Interval/Set/OrderIso.lean
α : Type u_1
β : Type u_2
inst✝¹ : Preorder α
inst✝ : Preorder β
e : α ≃o β
a : α
\u22a2 ⇑e '' Iic a = Iic (e a)
-- Mathlib/CategoryTheory/Sites/Subsheaf.lean
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F F' F'' : Cᵒᵖ ⥤ Type w
G G' : Subpresheaf F
h : G ≤ G'
hF : Presieve.IsSheaf J F
hG' : Presieve.IsSheaf J G'.toPresheaf
U : Cᵒᵖ
x : F.obj U
hx : x ∈ (sheafify J G).obj U
\u22a2 x ∈ G'.obj U
-- Mathlib/CategoryTheory/Sites/Subsheaf.lean
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F F' F'' : Cᵒᵖ ⥤ Type w
G G' : Subpresheaf F
h : G ≤ G'
hF : Presieve.IsSheaf J F
hG' : Presieve.IsSheaf J G'.toPresheaf
U : Cᵒᵖ
x : F.obj U
hx : x ∈ (sheafify J G).obj U
\u22a2 x = ↑((G.sheafifyLift (homOfLe h) hG').app U ⟨x, hx⟩)
-- Mathlib/CategoryTheory/Sites/Subsheaf.lean
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F F' F'' : Cᵒᵖ ⥤ Type w
G G' : Subpresheaf F
h : G ≤ G'
hF : Presieve.IsSheaf J F
hG' : Presieve.IsSheaf J G'.toPresheaf
U : Cᵒᵖ
x : F.obj U
hx : x ∈ (sheafify J G).obj U
\u22a2 ∀ ⦃Y : C⦄ ⦃f : Y ⟶ unop U⦄, (G.sieveOfSection x).arrows f → F.map f.op x = F.map f.op ↑((G.sheafifyLift (homOfLe h) hG').app U ⟨x, hx⟩)
-- Mathlib/CategoryTheory/Sites/Subsheaf.lean
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F F' F'' : Cᵒᵖ ⥤ Type w
G G' : Subpresheaf F
h : G ≤ G'
hF : Presieve.IsSheaf J F
hG' : Presieve.IsSheaf J G'.toPresheaf
U : Cᵒᵖ
x : F.obj U
hx : x ∈ (sheafify J G).obj U
V : C
i : V ⟶ unop U
hi : (G.sieveOfSection x).arrows i
\u22a2 F.map i.op x = F.map i.op ↑((G.sheafifyLift (homOfLe h) hG').app U ⟨x, hx⟩)
-- Mathlib/CategoryTheory/Sites/Subsheaf.lean
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F F' F'' : Cᵒᵖ ⥤ Type w
G G' : Subpresheaf F
h : G ≤ G'
hF : Presieve.IsSheaf J F
hG' : Presieve.IsSheaf J G'.toPresheaf
U : Cᵒᵖ
x : F.obj U
hx : x ∈ (sheafify J G).obj U
V : C
i : V ⟶ unop U
hi : (G.sieveOfSection x).arrows i
this : (fun f => ↑(f.app (op V) ⟨F.map i.op x, hi⟩)) (homOfLe ⋯ ≫ G.sheafifyLift (homOfLe h) hG') = (fun f => ↑(f.app (op V) ⟨F.map i.op x, hi⟩)) (homOfLe h)
\u22a2 F.map i.op x = F.map i.op ↑((G.sheafifyLift (homOfLe h) hG').app U ⟨x, hx⟩)
-- Mathlib/CategoryTheory/Sites/Subsheaf.lean
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F F' F'' : Cᵒᵖ ⥤ Type w
G G' : Subpresheaf F
h : G ≤ G'
hF : Presieve.IsSheaf J F
hG' : Presieve.IsSheaf J G'.toPresheaf
U : Cᵒᵖ
x : F.obj U
hx : x ∈ (sheafify J G).obj U
V : C
i : V ⟶ unop U
hi : (G.sieveOfSection x).arrows i
this : (fun f => ↑(f.app (op V) ⟨F.map i.op x, hi⟩)) (homOfLe ⋯ ≫ G.sheafifyLift (homOfLe h) hG') = (fun f => ↑(f.app (op V) ⟨F.map i.op x, hi⟩)) (homOfLe h)
\u22a2 F.map i.op ↑((G.sheafifyLift (homOfLe h) hG').app U ⟨x, hx⟩) = (fun f => ↑(f.app (op V) ⟨F.map i.op x, hi⟩)) (homOfLe ⋯ ≫ G.sheafifyLift (homOfLe h) hG')
-- Mathlib/CategoryTheory/Sites/Subsheaf.lean
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F F' F'' : Cᵒᵖ ⥤ Type w
G G' : Subpresheaf F
h : G ≤ G'
hF : Presieve.IsSheaf J F
hG' : Presieve.IsSheaf J G'.toPresheaf
U : Cᵒᵖ
x : F.obj U
hx : x ∈ (sheafify J G).obj U
V : C
i : V ⟶ unop U
hi : (G.sieveOfSection x).arrows i
this : (fun f => ↑(f.app (op V) ⟨F.map i.op x, hi⟩)) (homOfLe ⋯ ≫ G.sheafifyLift (homOfLe h) hG') = (fun f => ↑(f.app (op V) ⟨F.map i.op x, hi⟩)) (homOfLe h)
\u22a2 ↑((G.sheafifyLift (homOfLe h) hG').app (op V) ((sheafify J G).toPresheaf.map i.op ⟨x, hx⟩)) = (fun f => ↑(f.app (op V) ⟨F.map i.op x, hi⟩)) (homOfLe ⋯ ≫ G.sheafifyLift (homOfLe h) hG')
-- Mathlib/Data/Nat/Defs.lean
a✝ b c d m n k : ℕ
p q : ℕ → Prop
hdn : d ∣ n
a : ℕ
\u22a2 a < n / d ↔ d * a < n
-- Mathlib/Data/Nat/Defs.lean
a✝ b c m n k : ℕ
p q : ℕ → Prop
a : ℕ
hdn : 0 ∣ n
\u22a2 a < n / 0 ↔ 0 * a < n
-- Mathlib/Data/Nat/Defs.lean
a✝ b c d m n k : ℕ
p q : ℕ → Prop
hdn : d ∣ n
a : ℕ
hd : d > 0
\u22a2 a < n / d ↔ d * a < n
-- Mathlib/Data/Matroid/Basic.lean
α : Type u_1
M : Matroid α
E : Set α
hE : E.Finite
f : Matroid α → Set α × Set (Set α) := fun M => (M.E, {B | M.Base B})
\u22a2 {M | M.E ⊆ E}.Finite
-- Mathlib/Data/Matroid/Basic.lean
α : Type u_1
M : Matroid α
E : Set α
hE : E.Finite
f : Matroid α → Set α × Set (Set α) := fun M => (M.E, {B | M.Base B})
hf : Function.Injective f
\u22a2 {M | M.E ⊆ E}.Finite
-- Mathlib/Data/Matroid/Basic.lean
α : Type u_1
M : Matroid α
E : Set α
hE : E.Finite
f : Matroid α → Set α × Set (Set α) := fun M => (M.E, {B | M.Base B})
hf : Function.Injective f
\u22a2 (f '' {M | M.E ⊆ E}).Finite
-- Mathlib/Data/Matroid/Basic.lean
α : Type u_1
M : Matroid α
E : Set α
hE : E.Finite
f : Matroid α → Set α × Set (Set α) := fun M => (M.E, {B | M.Base B})
hf : Function.Injective f
\u22a2 f '' {M | M.E ⊆ E} ⊆ {b | b ⊆ E} ×ˢ {b | b ⊆ {b | b ⊆ E}}
-- Mathlib/Data/Matroid/Basic.lean
α : Type u_1
M✝ : Matroid α
E : Set α
hE✝ : E.Finite
f : Matroid α → Set α × Set (Set α) := fun M => (M.E, {B | M.Base B})
hf : Function.Injective f
M : Matroid α
hE : M.E ⊆ E
\u22a2 f M ∈ {b | b ⊆ E} ×ˢ {b | b ⊆ {b | b ⊆ E}}
-- Mathlib/Data/Matroid/Basic.lean
α : Type u_1
M✝ : Matroid α
E : Set α
hE✝ : E.Finite
f : Matroid α → Set α × Set (Set α) := fun M => (M.E, {B | M.Base B})
hf : Function.Injective f
M : Matroid α
hE : M.E ⊆ E
\u22a2 M.E ⊆ E ∧ ∀ (a : Set α), M.Base a → a ⊆ E
-- Mathlib/Data/Complex/Exponential.lean
x✝ y x : ℝ
\u22a2 rexp x < 1 ↔ x < 0
-- Mathlib/RingTheory/FractionalIdeal/Norm.lean
R : Type u_1
inst✝¹¹ : CommRing R
inst✝¹⁰ : IsDedekindDomain R
inst✝⁹ : Module.Free ℤ R
inst✝⁸ : Module.Finite ℤ R
K : Type u_2
inst✝⁷ : CommRing K
inst✝⁶ : Algebra R K
inst✝⁵ : IsFractionRing R K
inst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid R ℤ⁰) K
inst✝³ : Algebra ℚ K
inst✝² : NoZeroDivisors K
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
b : Basis ι ℤ R
I : FractionalIdeal R⁰ K
bI : Basis ι ℤ ↥↑I
\u22a2 |(Basis.localizationLocalization ℚ ℤ⁰ K b).det (Subtype.val ∘ ⇑bI)| = absNorm I
-- Mathlib/RingTheory/FractionalIdeal/Norm.lean
R : Type u_1
inst✝¹¹ : CommRing R
inst✝¹⁰ : IsDedekindDomain R
inst✝⁹ : Module.Free ℤ R
inst✝⁸ : Module.Finite ℤ R
K : Type u_2
inst✝⁷ : CommRing K
inst✝⁶ : Algebra R K
inst✝⁵ : IsFractionRing R K
inst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid R ℤ⁰) K
inst✝³ : Algebra ℚ K
inst✝² : NoZeroDivisors K
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
b : Basis ι ℤ R
I : FractionalIdeal R⁰ K
bI : Basis ι ℤ ↥↑I
this : Nontrivial K
\u22a2 |(Basis.localizationLocalization ℚ ℤ⁰ K b).det (Subtype.val ∘ ⇑bI)| = absNorm I
-- Mathlib/RingTheory/FractionalIdeal/Norm.lean
R : Type u_1
inst✝¹¹ : CommRing R
inst✝¹⁰ : IsDedekindDomain R
inst✝⁹ : Module.Free ℤ R
inst✝⁸ : Module.Finite ℤ R
K : Type u_2
inst✝⁷ : CommRing K
inst✝⁶ : Algebra R K
inst✝⁵ : IsFractionRing R K
inst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid R ℤ⁰) K
inst✝³ : Algebra ℚ K
inst✝² : NoZeroDivisors K
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
b : Basis ι ℤ R
I : FractionalIdeal R⁰ K
bI : Basis ι ℤ ↥↑I
this : Nontrivial K
b₀ : Basis ι ℚ K := Basis.localizationLocalization ℚ ℤ⁰ K b
\u22a2 |(Basis.localizationLocalization ℚ ℤ⁰ K b).det (Subtype.val ∘ ⇑bI)| = absNorm I
-- Mathlib/RingTheory/FractionalIdeal/Norm.lean
R : Type u_1
inst✝¹¹ : CommRing R
inst✝¹⁰ : IsDedekindDomain R
inst✝⁹ : Module.Free ℤ R
inst✝⁸ : Module.Finite ℤ R
K : Type u_2
inst✝⁷ : CommRing K
inst✝⁶ : Algebra R K
inst✝⁵ : IsFractionRing R K
inst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid R ℤ⁰) K
inst✝³ : Algebra ℚ K
inst✝² : NoZeroDivisors K
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
b : Basis ι ℤ R
I : FractionalIdeal R⁰ K
bI : Basis ι ℤ ↥↑I
this : Nontrivial K
b₀ : Basis ι ℚ K := Basis.localizationLocalization ℚ ℤ⁰ K b
bI.num : Basis ι ℤ ↥I.num := bI.map (LinearEquiv.restrictScalars ℤ (equivNum ⋯))
\u22a2 |(Basis.localizationLocalization ℚ ℤ⁰ K b).det (Subtype.val ∘ ⇑bI)| = absNorm I
-- Mathlib/RingTheory/FractionalIdeal/Norm.lean
R : Type u_1
inst✝¹¹ : CommRing R
inst✝¹⁰ : IsDedekindDomain R
inst✝⁹ : Module.Free ℤ R
inst✝⁸ : Module.Finite ℤ R
K : Type u_2
inst✝⁷ : CommRing K
inst✝⁶ : Algebra R K
inst✝⁵ : IsFractionRing R K
inst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid R ℤ⁰) K
inst✝³ : Algebra ℚ K
inst✝² : NoZeroDivisors K
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
b : Basis ι ℤ R
I : FractionalIdeal R⁰ K
bI : Basis ι ℤ ↥↑I
this : Nontrivial K
b₀ : Basis ι ℚ K := Basis.localizationLocalization ℚ ℤ⁰ K b
bI.num : Basis ι ℤ ↥I.num := bI.map (LinearEquiv.restrictScalars ℤ (equivNum ⋯))
\u22a2 |((Basis.localizationLocalization ℚ ℤ⁰ K b).toMatrix (Subtype.val ∘ ⇑bI)).det| = |↑(b.toMatrix (Subtype.val ∘ ⇑bI.num)).det| / |↑((Algebra.norm ℤ) ↑I.den)|
-- Mathlib/RingTheory/FractionalIdeal/Norm.lean
R : Type u_1
inst✝¹¹ : CommRing R
inst✝¹⁰ : IsDedekindDomain R
inst✝⁹ : Module.Free ℤ R
inst✝⁸ : Module.Finite ℤ R
K : Type u_2
inst✝⁷ : CommRing K
inst✝⁶ : Algebra R K
inst✝⁵ : IsFractionRing R K
inst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid R ℤ⁰) K
inst✝³ : Algebra ℚ K
inst✝² : NoZeroDivisors K
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
b : Basis ι ℤ R
I : FractionalIdeal R⁰ K
bI : Basis ι ℤ ↥↑I
this : Nontrivial K
b₀ : Basis ι ℚ K := Basis.localizationLocalization ℚ ℤ⁰ K b
bI.num : Basis ι ℤ ↥I.num := bI.map (LinearEquiv.restrictScalars ℤ (equivNum ⋯))
\u22a2 |((Basis.localizationLocalization ℚ ℤ⁰ K b).toMatrix (Subtype.val ∘ ⇑bI)).det| = |(algebraMap ℤ ℚ) (b.toMatrix (Subtype.val ∘ ⇑bI.num)).det| / |↑((Algebra.norm ℤ) ↑I.den)|
-- Mathlib/RingTheory/FractionalIdeal/Norm.lean
R : Type u_1
inst✝¹¹ : CommRing R
inst✝¹⁰ : IsDedekindDomain R
inst✝⁹ : Module.Free ℤ R
inst✝⁸ : Module.Finite ℤ R
K : Type u_2
inst✝⁷ : CommRing K
inst✝⁶ : Algebra R K
inst✝⁵ : IsFractionRing R K
inst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid R ℤ⁰) K
inst✝³ : Algebra ℚ K
inst✝² : NoZeroDivisors K
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
b : Basis ι ℤ R
I : FractionalIdeal R⁰ K
bI : Basis ι ℤ ↥↑I
this : Nontrivial K
b₀ : Basis ι ℚ K := Basis.localizationLocalization ℚ ℤ⁰ K b
bI.num : Basis ι ℤ ↥I.num := bI.map (LinearEquiv.restrictScalars ℤ (equivNum ⋯))
\u22a2 |((Basis.localizationLocalization ℚ ℤ⁰ K b).toMatrix (Subtype.val ∘ ⇑bI)).det| = |(b₀.toMatrix ((algebraMap R K) ↑I.den • Subtype.val ∘ ⇑bI)).det| / |↑((Algebra.norm ℤ) ↑I.den)|
-- Mathlib/Data/Nat/Factors.lean
a b : ℕ
ha : a ≠ 0
hb : b ≠ 0
\u22a2 (a * b).primeFactorsList ~ a.primeFactorsList ++ b.primeFactorsList
-- Mathlib/Data/Nat/Factors.lean
a b : ℕ
ha : a ≠ 0
hb : b ≠ 0
\u22a2 (a.primeFactorsList ++ b.primeFactorsList).prod = a * b
-- Mathlib/Data/Nat/Factors.lean
a b : ℕ
ha : a ≠ 0
hb : b ≠ 0
\u22a2 ∀ p ∈ a.primeFactorsList ++ b.primeFactorsList, Prime p
-- 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)
\u22a2 ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p 0) 1) ^ p ^ 0 = 1