ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Analysis/Analytic/CPolynomial.lean
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f g : E → F
p✝ pf pg : FormalMultilinearSeries 𝕜 E F
x : E
r r' : ℝ≥0∞
n✝ m : ℕ
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → p m = 0

⊢ E
-- Mathlib/Order/Filter/AtTopBot.lean
ι : Type u_1
ι' : Type u_2
α : Type u_3
β : Type u_4
γ : Type u_5
inst✝¹ : SemilatticeSup α
inst✝ : Nonempty α
p : α × α → Prop

⊢ (∀ᶠ (x : α × α) in atTop, p x) ↔ ∃ a, ∀ k ≥ a, ∀ l ≥ a, p (k, l)
-- Mathlib/SetTheory/Cardinal/Ordinal.lean
a b : Cardinal.{u_1}
h : b ≠ 0

⊢ a = 1 * a
-- Mathlib/Data/Seq/Computation.lean
α : Type u
β : Type v
γ : Type w
a : α
n : ℕ

⊢ ((pure a).thinkN n).Results a n
-- Mathlib/Data/Seq/Computation.lean
α : Type u
β : Type v
γ : Type w
a : α
n : ℕ
this : ((pure a).thinkN n).Results a (0 + n)

⊢ ((pure a).thinkN n).Results a n
-- Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
x y z : ℝ
hx : 0 ≤ x
hy : y ≠ 0

⊢ x ^ y = 0 ↔ x = 0
-- Mathlib/CategoryTheory/Limits/Lattice.lean
α : Type u
J : Type w
inst✝³ : SmallCategory J
inst✝² : FinCategory J
inst✝¹ : SemilatticeSup α
inst✝ : OrderBot α
x y z : α
f : z ⟶ x
g : z ⟶ y

⊢ colimit (span f g) = Finset.univ.sup (span f g).obj
-- Mathlib/CategoryTheory/Limits/Lattice.lean
α : Type u
J : Type w
inst✝³ : SmallCategory J
inst✝² : FinCategory J
inst✝¹ : SemilatticeSup α
inst✝ : OrderBot α
x y z : α
f : z ⟶ x
g : z ⟶ y

⊢ z ⊔ (x ⊔ (y ⊔ ⊥)) = z ⊔ (x ⊔ y)
-- Mathlib/Data/ENat/Basic.lean
m n : ℕ∞
x✝ : ℕ

⊢ ↑(↑x✝).toNat = ↑x✝ ↔ ↑x✝ ≠ ⊤
-- Mathlib/Algebra/MonoidAlgebra/Support.lean
k : Type u₁
G : Type u₂
inst✝² : Semiring k
inst✝¹ : Mul G
inst✝ : DecidableEq G
a b : MonoidAlgebra k G

⊢ (a * b).support ⊆ a.support * b.support
-- Mathlib/Algebra/MonoidAlgebra/Support.lean
k : Type u₁
G : Type u₂
inst✝² : Semiring k
inst✝¹ : Mul G
inst✝ : DecidableEq G
a b : MonoidAlgebra k G

⊢ (sum a fun a₁ b₁ => sum b fun a₂ b₂ => single (a₁ * a₂) (b₁ * b₂)).support ⊆ a.support * b.support
-- Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean
a : ℝ
s : ℂ
hs : 1 < s.re

⊢ HasSum (fun n => ↑(SignType.sign (↑n + a)) / ↑|↑n + a| ^ s / 2) (hurwitzZetaOdd (↑a) s)
-- Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean
a : ℝ
s : ℂ
hs : 1 < s.re
n : ℤ

⊢ ↑(SignType.sign (↑n + a)) / ↑|↑n + a| ^ s / 2 = (s + 1).Gammaℝ * ↑(SignType.sign (↑n + a)) / ↑|↑n + a| ^ s / 2 / (s + 1).Gammaℝ
-- Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean
a : ℝ
s : ℂ
hs : 1 < s.re
n : ℤ
this : 0 < (s + 1).re

⊢ ↑(SignType.sign (↑n + a)) / ↑|↑n + a| ^ s / 2 = (s + 1).Gammaℝ * ↑(SignType.sign (↑n + a)) / ↑|↑n + a| ^ s / 2 / (s + 1).Gammaℝ
-- Mathlib/Algebra/Field/Basic.lean
α : Type u_1
β : Type u_2
K : Type u_3
inst✝ : DivisionSemiring α
a b c d : α
hab : Commute a b
ha : a ≠ 0
hb : b ≠ 0

⊢ 1 / a + 1 / b = (a + b) / (a * b)
-- Mathlib/Data/ENNReal/Real.lean
a b c d : ℝ≥0∞
r p✝ q✝ : ℝ≥0
p q : ℝ
hp : 0 ≤ p

⊢ ENNReal.ofReal (p * q) = ENNReal.ofReal p * ENNReal.ofReal q
-- Mathlib/SetTheory/Cardinal/ToNat.lean
α : Type u
c✝ d : Cardinal.{u}
c : Cardinal.{v}

⊢ toNat (lift.{u, v} c) = toNat c
-- Mathlib/Data/Matroid/Map.lean
α : Type u_1
β : Type u_2
f : α → β
E I s : Set α
M : Matroid α
N✝ N : Matroid β

⊢ ∀ I ⊆ (N.comap id).E, (N.comap id).Indep I ↔ N.Indep I
-- Mathlib/Order/Bounded.lean
α : Type u_1
r : α → α → Prop
s t : Set α
inst✝¹ : LinearOrder α
inst✝ : NoMaxOrder α
a : α

⊢ Unbounded (fun x x_1 => x < x_1) (s ∩ {b | a < b}) ↔ Unbounded (fun x x_1 => x < x_1) s
-- Mathlib/AlgebraicGeometry/Morphisms/Basic.lean
P : MorphismProperty Scheme
Q : AffineTargetMorphismProperty
inst✝ : HasAffineProperty P Q
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
hP' : Q.StableUnderBaseChange
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
H : P g

⊢ P (pullback.fst f g)
-- Mathlib/AlgebraicGeometry/Morphisms/Basic.lean
P : MorphismProperty Scheme
Q : AffineTargetMorphismProperty
inst✝ : HasAffineProperty P Q
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
hP' : Q.StableUnderBaseChange
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
H : P g

⊢ ∀ (i : ((S.affineCover.pullbackCover f).pullbackCover (pullback.fst f g)).J), P ((S.affineCover.pullbackCover f).pullbackHom (pullback.fst f g) i)
-- Mathlib/AlgebraicGeometry/Morphisms/Basic.lean
P : MorphismProperty Scheme
Q : AffineTargetMorphismProperty
inst✝ : HasAffineProperty P Q
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
hP' : Q.StableUnderBaseChange
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
H : P g
i : ((S.affineCover.pullbackCover f).pullbackCover (pullback.fst f g)).J

⊢ P ((S.affineCover.pullbackCover f).pullbackHom (pullback.fst f g) i)
-- Mathlib/AlgebraicGeometry/Morphisms/Basic.lean
P : MorphismProperty Scheme
Q : AffineTargetMorphismProperty
inst✝ : HasAffineProperty P Q
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
hP' : Q.StableUnderBaseChange
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
H : P g
i : ((S.affineCover.pullbackCover f).pullbackCover (pullback.fst f g)).J
e : pullback (pullback.fst f g) ((S.affineCover.pullbackCover f).map i) ≅ pullback (pullback.snd f (S.affineCover.map i)) (pullback.fst (S.affineCover.map i) g) := pullbackSymmetry (pullback.fst f g) ((S.affineCover.pullbackCover f).map i) ≪≫ pullbackRightPullbackFstIso f g ((S.affineCover.pullbackCover f).map i) ≪≫ asIso (pullback.map ((S.affineCover.pullbackCover f).map i ≫ f) g (pullback.snd f (S.affineCover.map i) ≫ S.affineCover.map i) g (𝟙 ((S.affineCover.pullbackCover f).obj i)) (𝟙 Y) (𝟙 S) ⋯ ⋯) ≪≫ (pullbackRightPullbackFstIso (S.affineCover.map i) g (pullback.snd f (S.affineCover.map i))).symm

⊢ P ((S.affineCover.pullbackCover f).pullbackHom (pullback.fst f g) i)
-- Mathlib/AlgebraicGeometry/Morphisms/Basic.lean
P : MorphismProperty Scheme
Q : AffineTargetMorphismProperty
inst✝ : HasAffineProperty P Q
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
hP' : Q.StableUnderBaseChange
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
H : P g
i : ((S.affineCover.pullbackCover f).pullbackCover (pullback.fst f g)).J
e : pullback (pullback.fst f g) ((S.affineCover.pullbackCover f).map i) ≅ pullback (pullback.snd f (S.affineCover.map i)) (pullback.fst (S.affineCover.map i) g) := pullbackSymmetry (pullback.fst f g) ((S.affineCover.pullbackCover f).map i) ≪≫ pullbackRightPullbackFstIso f g ((S.affineCover.pullbackCover f).map i) ≪≫ asIso (pullback.map ((S.affineCover.pullbackCover f).map i ≫ f) g (pullback.snd f (S.affineCover.map i) ≫ S.affineCover.map i) g (𝟙 ((S.affineCover.pullbackCover f).obj i)) (𝟙 Y) (𝟙 S) ⋯ ⋯) ≪≫ (pullbackRightPullbackFstIso (S.affineCover.map i) g (pullback.snd f (S.affineCover.map i))).symm
this : e.hom ≫ pullback.fst (pullback.snd f (S.affineCover.map i)) (pullback.fst (S.affineCover.map i) g) = (S.affineCover.pullbackCover f).pullbackHom (pullback.fst f g) i

⊢ P ((S.affineCover.pullbackCover f).pullbackHom (pullback.fst f g) i)
-- Mathlib/AlgebraicGeometry/Morphisms/Basic.lean
P : MorphismProperty Scheme
Q : AffineTargetMorphismProperty
inst✝ : HasAffineProperty P Q
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
hP' : Q.StableUnderBaseChange
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
H : P g
i : ((S.affineCover.pullbackCover f).pullbackCover (pullback.fst f g)).J
e : pullback (pullback.fst f g) ((S.affineCover.pullbackCover f).map i) ≅ pullback (pullback.snd f (S.affineCover.map i)) (pullback.fst (S.affineCover.map i) g) := pullbackSymmetry (pullback.fst f g) ((S.affineCover.pullbackCover f).map i) ≪≫ pullbackRightPullbackFstIso f g ((S.affineCover.pullbackCover f).map i) ≪≫ asIso (pullback.map ((S.affineCover.pullbackCover f).map i ≫ f) g (pullback.snd f (S.affineCover.map i) ≫ S.affineCover.map i) g (𝟙 ((S.affineCover.pullbackCover f).obj i)) (𝟙 Y) (𝟙 S) ⋯ ⋯) ≪≫ (pullbackRightPullbackFstIso (S.affineCover.map i) g (pullback.snd f (S.affineCover.map i))).symm
this : e.hom ≫ pullback.fst (pullback.snd f (S.affineCover.map i)) (pullback.fst (S.affineCover.map i) g) = (S.affineCover.pullbackCover f).pullbackHom (pullback.fst f g) i

⊢ P (pullback.fst (pullback.snd f (S.affineCover.map i)) (pullback.fst (S.affineCover.map i) g))
-- Mathlib/AlgebraicGeometry/Morphisms/Basic.lean
P : MorphismProperty Scheme
Q : AffineTargetMorphismProperty
inst✝ : HasAffineProperty P Q
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
hP' : Q.StableUnderBaseChange
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
H : P g
i : ((S.affineCover.pullbackCover f).pullbackCover (pullback.fst f g)).J
e : pullback (pullback.fst f g) ((S.affineCover.pullbackCover f).map i) ≅ pullback (pullback.snd f (S.affineCover.map i)) (pullback.fst (S.affineCover.map i) g) := pullbackSymmetry (pullback.fst f g) ((S.affineCover.pullbackCover f).map i) ≪≫ pullbackRightPullbackFstIso f g ((S.affineCover.pullbackCover f).map i) ≪≫ asIso (pullback.map ((S.affineCover.pullbackCover f).map i ≫ f) g (pullback.snd f (S.affineCover.map i) ≫ S.affineCover.map i) g (𝟙 ((S.affineCover.pullbackCover f).obj i)) (𝟙 Y) (𝟙 S) ⋯ ⋯) ≪≫ (pullbackRightPullbackFstIso (S.affineCover.map i) g (pullback.snd f (S.affineCover.map i))).symm
this : e.hom ≫ pullback.fst (pullback.snd f (S.affineCover.map i)) (pullback.fst (S.affineCover.map i) g) = (S.affineCover.pullbackCover f).pullbackHom (pullback.fst f g) i

⊢ Q (pullback.fst (S.affineCover.map i) g)
-- Mathlib/AlgebraicGeometry/Morphisms/Basic.lean
P : MorphismProperty Scheme
Q : AffineTargetMorphismProperty
inst✝ : HasAffineProperty P Q
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
hP' : Q.StableUnderBaseChange
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
H : P g
i : ((S.affineCover.pullbackCover f).pullbackCover (pullback.fst f g)).J
e : pullback (pullback.fst f g) ((S.affineCover.pullbackCover f).map i) ≅ pullback (pullback.snd f (S.affineCover.map i)) (pullback.fst (S.affineCover.map i) g) := pullbackSymmetry (pullback.fst f g) ((S.affineCover.pullbackCover f).map i) ≪≫ pullbackRightPullbackFstIso f g ((S.affineCover.pullbackCover f).map i) ≪≫ asIso (pullback.map ((S.affineCover.pullbackCover f).map i ≫ f) g (pullback.snd f (S.affineCover.map i) ≫ S.affineCover.map i) g (𝟙 ((S.affineCover.pullbackCover f).obj i)) (𝟙 Y) (𝟙 S) ⋯ ⋯) ≪≫ (pullbackRightPullbackFstIso (S.affineCover.map i) g (pullback.snd f (S.affineCover.map i))).symm
this✝ : e.hom ≫ pullback.fst (pullback.snd f (S.affineCover.map i)) (pullback.fst (S.affineCover.map i) g) = (S.affineCover.pullbackCover f).pullbackHom (pullback.fst f g) i
this : Q.IsLocal := isLocal_affineProperty P

⊢ Q (pullback.fst (S.affineCover.map i) g)
-- Mathlib/AlgebraicGeometry/Morphisms/Basic.lean
P : MorphismProperty Scheme
Q : AffineTargetMorphismProperty
inst✝ : HasAffineProperty P Q
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
hP' : Q.StableUnderBaseChange
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
H : P g
i : ((S.affineCover.pullbackCover f).pullbackCover (pullback.fst f g)).J
e : pullback (pullback.fst f g) ((S.affineCover.pullbackCover f).map i) ≅ pullback (pullback.snd f (S.affineCover.map i)) (pullback.fst (S.affineCover.map i) g) := pullbackSymmetry (pullback.fst f g) ((S.affineCover.pullbackCover f).map i) ≪≫ pullbackRightPullbackFstIso f g ((S.affineCover.pullbackCover f).map i) ≪≫ asIso (pullback.map ((S.affineCover.pullbackCover f).map i ≫ f) g (pullback.snd f (S.affineCover.map i) ≫ S.affineCover.map i) g (𝟙 ((S.affineCover.pullbackCover f).obj i)) (𝟙 Y) (𝟙 S) ⋯ ⋯) ≪≫ (pullbackRightPullbackFstIso (S.affineCover.map i) g (pullback.snd f (S.affineCover.map i))).symm
this✝ : e.hom ≫ pullback.fst (pullback.snd f (S.affineCover.map i)) (pullback.fst (S.affineCover.map i) g) = (S.affineCover.pullbackCover f).pullbackHom (pullback.fst f g) i
this : Q.IsLocal := isLocal_affineProperty P

⊢ Q (pullback.snd g (S.affineCover.map i))
-- .lake/packages/batteries/Batteries/Data/List/Count.lean
α : Type u_1
inst✝ : DecidableEq α
head : α
tail : List α
a : α
h : head :: tail ≠ []

⊢ count a (head :: tail).tail = count a (head :: tail) - if a = (head :: tail).head h then 1 else 0
-- Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Submodule ℝ E
hs : s ≠ ⊤

⊢ μ ↑s = 0
-- Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Submodule ℝ E
hs : s ≠ ⊤
x : E
hx : x ∉ s

⊢ μ ↑s = 0
-- Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Submodule ℝ E
hs : s ≠ ⊤
x : E
hx : x ∉ s
c : ℝ
cpos : 0 < c
cone : c < 1

⊢ μ ↑s = 0
-- Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Submodule ℝ E
hs : s ≠ ⊤
x : E
hx : x ∉ s
c : ℝ
cpos : 0 < c
cone : c < 1
A : Bornology.IsBounded (range fun n => c ^ n • x)

⊢ μ ↑s = 0
-- Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Submodule ℝ E
hs : s ≠ ⊤
x : E
hx : x ∉ s
c : ℝ
cpos : 0 < c
cone : c < 1
A : Bornology.IsBounded (range fun n => c ^ n • x)

⊢ Pairwise (Disjoint on fun n => {c ^ n • x} + ↑s)
-- Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Submodule ℝ E
hs : s ≠ ⊤
x : E
hx : x ∉ s
c : ℝ
cpos : 0 < c
cone : c < 1
A : Bornology.IsBounded (range fun n => c ^ n • x)
m n : ℕ
hmn : m ≠ n

⊢ (Disjoint on fun n => {c ^ n • x} + ↑s) m n
-- Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Submodule ℝ E
hs : s ≠ ⊤
x : E
hx : x ∉ s
c : ℝ
cpos : 0 < c
cone : c < 1
A : Bornology.IsBounded (range fun n => c ^ n • x)
m n : ℕ
hmn : m ≠ n

⊢ ∀ ⦃a : E⦄, -(c ^ m • x) + a ∈ s → -(c ^ n • x) + a ∉ s
-- Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Submodule ℝ E
hs : s ≠ ⊤
x : E
hx : x ∉ s
c : ℝ
cpos : 0 < c
cone : c < 1
A : Bornology.IsBounded (range fun n => c ^ n • x)
m n : ℕ
hmn : m ≠ n
y : E
hym : -(c ^ m • x) + y ∈ s
hyn : -(c ^ n • x) + y ∈ s

⊢ False
-- Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Submodule ℝ E
hs : s ≠ ⊤
x : E
hx : x ∉ s
c : ℝ
cpos : 0 < c
cone : c < 1
A✝ : Bornology.IsBounded (range fun n => c ^ n • x)
m n : ℕ
hmn : m ≠ n
y : E
hym : -(c ^ m • x) + y ∈ s
hyn : -(c ^ n • x) + y ∈ s
A : (c ^ n - c ^ m) • x ∈ s

⊢ False
-- Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Submodule ℝ E
hs : s ≠ ⊤
x : E
hx : x ∉ s
c : ℝ
cpos : 0 < c
cone : c < 1
A✝ : Bornology.IsBounded (range fun n => c ^ n • x)
m n : ℕ
hmn : m ≠ n
y : E
hym : -(c ^ m • x) + y ∈ s
hyn : -(c ^ n • x) + y ∈ s
A : (c ^ n - c ^ m) • x ∈ s
H : c ^ n - c ^ m ≠ 0

⊢ False
-- Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Submodule ℝ E
hs : s ≠ ⊤
x : E
hx : x ∉ s
c : ℝ
cpos : 0 < c
cone : c < 1
A✝ : Bornology.IsBounded (range fun n => c ^ n • x)
m n : ℕ
hmn : m ≠ n
y : E
hym : -(c ^ m • x) + y ∈ s
hyn : -(c ^ n • x) + y ∈ s
A : (c ^ n - c ^ m) • x ∈ s
H : c ^ n - c ^ m ≠ 0
this : x ∈ s

⊢ False
-- Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean
x : ℂ

⊢ x ^ (-1) = x⁻¹
-- Mathlib/AlgebraicTopology/SimplicialSet.lean
n : ℕ
i : Fin (n + 2)
S : SSet
σ₁ σ₂ : Λ[n + 1, i] ⟶ S
h : ∀ (j : Fin (n + 2)) (h : j ≠ i), σ₁.app (op [n]) (face i j h) = σ₂.app (op [n]) (face i j h)

⊢ σ₁.app = σ₂.app
-- Mathlib/AlgebraicTopology/SimplicialSet.lean
n : ℕ
i : Fin (n + 2)
S : SSet
σ₁ σ₂ : Λ[n + 1, i] ⟶ S
h : ∀ (j : Fin (n + 2)) (h : j ≠ i), σ₁.app (op [n]) (face i j h) = σ₂.app (op [n]) (face i j h)

⊢ ∀ (x : SimplexCategoryᵒᵖ), σ₁.app x = σ₂.app x
-- Mathlib/AlgebraicTopology/SimplicialSet.lean
n : ℕ
i : Fin (n + 2)
S : SSet
σ₁ σ₂ : Λ[n + 1, i] ⟶ S
h : ∀ (j : Fin (n + 2)) (h : j ≠ i), σ₁.app (op [n]) (face i j h) = σ₂.app (op [n]) (face i j h)
m : ℕ
f : Λ[n + 1, i] _[m]

⊢ σ₁.app (op [m]) f = σ₂.app (op [m]) f
-- Mathlib/AlgebraicTopology/SimplicialSet.lean
n : ℕ
i : Fin (n + 2)
S : SSet
σ₁ σ₂ : Λ[n + 1, i] ⟶ S
h : ∀ (j : Fin (n + 2)) (h : j ≠ i), σ₁.app (op [n]) (face i j h) = σ₂.app (op [n]) (face i j h)
m : ℕ
f : Λ[n + 1, i] _[m]
f' : unop (op [m]) ⟶ [n + 1]
hf : (standardSimplex.objEquiv [n + 1] (op [m])).symm f' = ↑f

⊢ σ₁.app (op [m]) f = σ₂.app (op [m]) f
-- Mathlib/AlgebraicTopology/SimplicialSet.lean
n : ℕ
i : Fin (n + 2)
S : SSet
σ₁ σ₂ : Λ[n + 1, i] ⟶ S
h : ∀ (j : Fin (n + 2)) (h : j ≠ i), σ₁.app (op [n]) (face i j h) = σ₂.app (op [n]) (face i j h)
m : ℕ
f : Λ[n + 1, i] _[m]
f' : unop (op [m]) ⟶ [n + 1]
hf : (standardSimplex.objEquiv [n + 1] (op [m])).symm f' = ↑f
j : Fin (n + 2)
hji : ¬j = i
hfj : ∀ (k : Fin ((unop (op [m])).len + 1)), (Hom.toOrderHom f') k ≠ j

⊢ σ₁.app (op [m]) f = σ₂.app (op [m]) f
-- Mathlib/AlgebraicTopology/SimplicialSet.lean
n : ℕ
i : Fin (n + 2)
S : SSet
σ₁ σ₂ : Λ[n + 1, i] ⟶ S
h : ∀ (j : Fin (n + 2)) (h : j ≠ i), σ₁.app (op [n]) (face i j h) = σ₂.app (op [n]) (face i j h)
m : ℕ
f : Λ[n + 1, i] _[m]
f' : unop (op [m]) ⟶ [n + 1]
hf : (standardSimplex.objEquiv [n + 1] (op [m])).symm f' = ↑f
j : Fin (n + 2)
hji : ¬j = i
hfj : ∀ (k : Fin ((unop (op [m])).len + 1)), (Hom.toOrderHom f') k ≠ j
H : f = Λ[n + 1, i].map (factor_δ f' j).op (face i j hji)

⊢ σ₁.app (op [m]) f = σ₂.app (op [m]) f
-- Mathlib/AlgebraicTopology/SimplicialSet.lean
n : ℕ
i : Fin (n + 2)
S : SSet
σ₁ σ₂ : Λ[n + 1, i] ⟶ S
h : ∀ (j : Fin (n + 2)) (h : j ≠ i), σ₁.app (op [n]) (face i j h) = σ₂.app (op [n]) (face i j h)
m : ℕ
f : Λ[n + 1, i] _[m]
f' : unop (op [m]) ⟶ [n + 1]
hf : (standardSimplex.objEquiv [n + 1] (op [m])).symm f' = ↑f
j : Fin (n + 2)
hji : ¬j = i
hfj : ∀ (k : Fin ((unop (op [m])).len + 1)), (Hom.toOrderHom f') k ≠ j
H : f = Λ[n + 1, i].map (factor_δ f' j).op (face i j hji)
H₁ : (Λ[n + 1, i].map (factor_δ f' j).op ≫ σ₁.app (op [m])) (face i j hji) = (σ₁.app (op [n]) ≫ S.map (factor_δ f' j).op) (face i j hji)

⊢ σ₁.app (op [m]) f = σ₂.app (op [m]) f
-- Mathlib/AlgebraicTopology/SimplicialSet.lean
n : ℕ
i : Fin (n + 2)
S : SSet
σ₁ σ₂ : Λ[n + 1, i] ⟶ S
h : ∀ (j : Fin (n + 2)) (h : j ≠ i), σ₁.app (op [n]) (face i j h) = σ₂.app (op [n]) (face i j h)
m : ℕ
f : Λ[n + 1, i] _[m]
f' : unop (op [m]) ⟶ [n + 1]
hf : (standardSimplex.objEquiv [n + 1] (op [m])).symm f' = ↑f
j : Fin (n + 2)
hji : ¬j = i
hfj : ∀ (k : Fin ((unop (op [m])).len + 1)), (Hom.toOrderHom f') k ≠ j
H : f = Λ[n + 1, i].map (factor_δ f' j).op (face i j hji)
H₁ : (Λ[n + 1, i].map (factor_δ f' j).op ≫ σ₁.app (op [m])) (face i j hji) = (σ₁.app (op [n]) ≫ S.map (factor_δ f' j).op) (face i j hji)
H₂ : (Λ[n + 1, i].map (factor_δ f' j).op ≫ σ₂.app (op [m])) (face i j hji) = (σ₂.app (op [n]) ≫ S.map (factor_δ f' j).op) (face i j hji)

⊢ σ₁.app (op [m]) f = σ₂.app (op [m]) f
-- Mathlib/NumberTheory/Padics/RingHoms.lean
p : ℕ
hp_prime : Fact (Nat.Prime p)
r : ℚ
x : ℤ_[p]

⊢ x - (toZMod x).cast ∈ maximalIdeal ℤ_[p]
-- Mathlib/NumberTheory/Padics/RingHoms.lean
p : ℕ
hp_prime : Fact (Nat.Prime p)
r : ℚ
x : ℤ_[p]

⊢ (toZMod x).cast = ↑x.zmodRepr
-- Mathlib/NumberTheory/Padics/RingHoms.lean
p : ℕ
hp_prime : Fact (Nat.Prime p)
r : ℚ
x : ℤ_[p]

⊢ (↑x.zmodRepr).cast = ↑x.zmodRepr
-- Mathlib/NumberTheory/Padics/RingHoms.lean
r : ℚ
p' : ℕ
hp_prime : Fact (Nat.Prime (0 + p' + 1))
x : ℤ_[0 + p' + 1]

⊢ (↑x.zmodRepr).cast = ↑x.zmodRepr
-- Mathlib/NumberTheory/Padics/RingHoms.lean
r : ℚ
p' : ℕ
hp_prime : Fact (Nat.Prime (0 + p' + 1))
x : ℤ_[0 + p' + 1]

⊢ ↑(↑x.zmodRepr).val = ↑x.zmodRepr
-- Mathlib/NumberTheory/Padics/RingHoms.lean
r : ℚ
p' : ℕ
hp_prime : Fact (Nat.Prime (0 + p' + 1))
x : ℤ_[0 + p' + 1]

⊢ x.zmodRepr % (p' + 1) = x.zmodRepr
-- Mathlib/NumberTheory/Padics/RingHoms.lean
r : ℚ
p' : ℕ
hp_prime : Fact (Nat.Prime (0 + p' + 1))
x : ℤ_[0 + p' + 1]

⊢ x.zmodRepr < p' + 1
-- Mathlib/AlgebraicGeometry/Morphisms/OpenImmersion.lean
X Y Z : Scheme
f : X ⟶ Y
g : Y ⟶ Z
x✝² x✝¹ : Scheme
x✝ : x✝² ⟶ x✝¹

⊢ IsOpenImmersion x✝ ↔ ((topologically fun {α β} [TopologicalSpace α] [TopologicalSpace β] => OpenEmbedding) ⊓ stalkwise fun {R S} [CommRing R] [CommRing S] f => Function.Bijective ⇑f) x✝
-- Mathlib/Analysis/Normed/Group/Basic.lean
𝓕 : Type u_1
𝕜 : Type u_2
α : Type u_3
ι : Type u_4
κ : Type u_5
E : Type u_6
F : Type u_7
G : Type u_8
inst✝² : SeminormedGroup E
inst✝¹ : SeminormedGroup F
inst✝ : SeminormedGroup G
s : Set E
a✝ a₁ a₂ b✝ b₁ b₂ : E
r r₁ r₂ : ℝ
a b c : E

⊢ ‖a / c‖ ≤ ‖a / b‖ + ‖b / c‖
-- Mathlib/Data/ENNReal/Inv.lean
a b c d : ℝ≥0∞
r p q : ℝ≥0
h0 : a ≠ 0
hI : a ≠ ⊤

⊢ a * (b / a) = b
-- Mathlib/Data/Int/ModEq.lean
m n a b c d : ℤ

⊢ a ≡ b [ZMOD -n] ↔ a ≡ b [ZMOD n]
-- Mathlib/Data/Complex/Exponential.lean
x : ℝ
h1 : 0 ≤ x
h2 : x < 1

⊢ rexp x ≤ 1 / (1 - x)
-- Mathlib/Data/Complex/Exponential.lean
h1 : 0 ≤ 0
h2 : 0 < 1

⊢ rexp 0 ≤ 1 / (1 - 0)
-- Mathlib/Data/Complex/Exponential.lean
x : ℝ
h1✝ : 0 ≤ x
h2 : x < 1
h1 : 0 < x

⊢ rexp x ≤ 1 / (1 - x)
-- Mathlib/NumberTheory/NumberField/Discriminant.lean
K : Type u_1
inst✝³ : Field K
inst✝² : NumberField K
A : Type u_2
inst✝¹ : Field A
inst✝ : CharZero A
N : ℕ
hK : |discr K| ≤ ↑N
D : ℕ := rankOfDiscrBdd N
B : ℝ≥0 := boundOfDiscBdd N
C : ℕ := ⌈max B 1 ^ D * ↑(D.choose (D / 2))⌉₊

⊢ {K | {w | w.IsReal}.Nonempty ∧ |discr ↥↑K| ≤ ↑N}.Finite
-- Mathlib/NumberTheory/NumberField/Discriminant.lean
K✝ : Type u_1
inst✝³ : Field K✝
inst✝² : NumberField K✝
A : Type u_2
inst✝¹ : Field A
inst✝ : CharZero A
N : ℕ
hK : |discr K✝| ≤ ↑N
D : ℕ := rankOfDiscrBdd N
B : ℝ≥0 := boundOfDiscBdd N
C : ℕ := ⌈max B 1 ^ D * ↑(D.choose (D / 2))⌉₊
x✝¹ : { F // FiniteDimensional ℚ ↥F }
K : IntermediateField ℚ A
hK₀ : FiniteDimensional ℚ ↥K
x✝ : ⟨K, hK₀⟩ ∈ {K | {w | w.IsReal}.Nonempty ∧ |discr ↥↑K| ≤ ↑N}
hK₁ : {w | w.IsReal}.Nonempty
hK₂ : |discr ↥↑⟨K, hK₀⟩| ≤ ↑N

⊢ ∃ x ∈ ⋃ f, ⋃ (_ : f.natDegree ≤ D ∧ ∀ (i : ℕ), f.coeff i ∈ Set.Icc (-↑C) ↑C), ↑(Polynomial.map (algebraMap ℤ A) f).roots.toFinset, ↑⟨K, hK₀⟩ = ℚ⟮x⟯
-- Mathlib/NumberTheory/NumberField/Discriminant.lean
K✝ : Type u_1
inst✝³ : Field K✝
inst✝² : NumberField K✝
A : Type u_2
inst✝¹ : Field A
inst✝ : CharZero A
N : ℕ
hK : |discr K✝| ≤ ↑N
D : ℕ := rankOfDiscrBdd N
B : ℝ≥0 := boundOfDiscBdd N
C : ℕ := ⌈max B 1 ^ D * ↑(D.choose (D / 2))⌉₊
x✝¹ : { F // FiniteDimensional ℚ ↥F }
K : IntermediateField ℚ A
hK₀ : FiniteDimensional ℚ ↥K
x✝ : ⟨K, hK₀⟩ ∈ {K | {w | w.IsReal}.Nonempty ∧ |discr ↥↑K| ≤ ↑N}
hK₁ : {w | w.IsReal}.Nonempty
hK₂ : |discr ↥↑⟨K, hK₀⟩| ≤ ↑N

⊢ ∃ x, (∃ i, ∃ (_ : i.natDegree ≤ D ∧ ∀ (i_1 : ℕ), i.coeff i_1 ∈ Set.Icc (-↑C) ↑C), x ∈ ↑(Polynomial.map (algebraMap ℤ A) i).roots.toFinset) ∧ K = ℚ⟮x⟯
-- Mathlib/NumberTheory/NumberField/Discriminant.lean
K✝ : Type u_1
inst✝³ : Field K✝
inst✝² : NumberField K✝
A : Type u_2
inst✝¹ : Field A
inst✝ : CharZero A
N : ℕ
hK : |discr K✝| ≤ ↑N
D : ℕ := rankOfDiscrBdd N
B : ℝ≥0 := boundOfDiscBdd N
C : ℕ := ⌈max B 1 ^ D * ↑(D.choose (D / 2))⌉₊
x✝¹ : { F // FiniteDimensional ℚ ↥F }
K : IntermediateField ℚ A
hK₀ : FiniteDimensional ℚ ↥K
x✝ : ⟨K, hK₀⟩ ∈ {K | {w | w.IsReal}.Nonempty ∧ |discr ↥↑K| ≤ ↑N}
hK₂ : |discr ↥↑⟨K, hK₀⟩| ≤ ↑N
this : NumberField ↥K
w₀ : InfinitePlace ↥↑⟨K, hK₀⟩
hw₀ : w₀ ∈ {w | w.IsReal}

⊢ minkowskiBound (↥K) 1 < ↑(convexBodyLTFactor ↥K) * ↑B