ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/AlgebraicGeometry/Morphisms/Constructors.lean
P : MorphismProperty Scheme
Q : AffineTargetMorphismProperty
inst✝² : HasAffineProperty P Q
X Y : Scheme
f : X ⟶ Y
𝒰 : Y.OpenCover
inst✝¹ : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)
𝒰' : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover
inst✝ : ∀ (i : 𝒰.J) (j : (𝒰' i).J), IsAffine ((𝒰' i).obj j)
h𝒰' : ∀ (i : 𝒰.J) (j k : (𝒰' i).J), Q (pullback.mapDesc ((𝒰' i).map j) ((𝒰' i).map k) (𝒰.pullbackHom f i))
this : Q.IsLocal := isLocal_affineProperty P
𝒱 : (pullback f f).OpenCover := (Scheme.Pullback.openCoverOfBase 𝒰 f f).bind fun i => Scheme.Pullback.openCoverOfLeftRight (𝒰' i) (𝒰' i) (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i))
i1 : ∀ (i : 𝒱.J), IsAffine (𝒱.obj i)

⊢ ∀ (i : (𝒱.pullbackCover (pullback.diagonal f)).J), Q (𝒱.pullbackHom (pullback.diagonal f) i)
-- Mathlib/AlgebraicGeometry/Morphisms/Constructors.lean
P : MorphismProperty Scheme
Q : AffineTargetMorphismProperty
inst✝² : HasAffineProperty P Q
X Y : Scheme
f : X ⟶ Y
𝒰 : Y.OpenCover
inst✝¹ : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)
𝒰' : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover
inst✝ : ∀ (i : 𝒰.J) (j : (𝒰' i).J), IsAffine ((𝒰' i).obj j)
h𝒰' : ∀ (i : 𝒰.J) (j k : (𝒰' i).J), Q (pullback.mapDesc ((𝒰' i).map j) ((𝒰' i).map k) (𝒰.pullbackHom f i))
this : Q.IsLocal := isLocal_affineProperty P
𝒱 : (pullback f f).OpenCover := (Scheme.Pullback.openCoverOfBase 𝒰 f f).bind fun i => Scheme.Pullback.openCoverOfLeftRight (𝒰' i) (𝒰' i) (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i))
i1 : ∀ (i : 𝒱.J), IsAffine (𝒱.obj i)
i : (Scheme.Pullback.openCoverOfBase 𝒰 f f).J
j k : (𝒰' i).J

⊢ Q (((Scheme.Pullback.openCoverOfBase 𝒰 f f).bind fun i => Scheme.Pullback.openCoverOfLeftRight (𝒰' i) (𝒰' i) (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i))).pullbackHom (pullback.diagonal f) ⟨i, (j, k)⟩)
-- Mathlib/AlgebraicGeometry/Morphisms/Constructors.lean
P : MorphismProperty Scheme
Q : AffineTargetMorphismProperty
inst✝² : HasAffineProperty P Q
X Y : Scheme
f : X ⟶ Y
𝒰 : Y.OpenCover
inst✝¹ : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)
𝒰' : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover
inst✝ : ∀ (i : 𝒰.J) (j : (𝒰' i).J), IsAffine ((𝒰' i).obj j)
h𝒰' : ∀ (i : 𝒰.J) (j k : (𝒰' i).J), Q (pullback.mapDesc ((𝒰' i).map j) ((𝒰' i).map k) (𝒰.pullbackHom f i))
this : Q.IsLocal := isLocal_affineProperty P
𝒱 : (pullback f f).OpenCover := (Scheme.Pullback.openCoverOfBase 𝒰 f f).bind fun i => Scheme.Pullback.openCoverOfLeftRight (𝒰' i) (𝒰' i) (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i))
i1 : ∀ (i : 𝒱.J), IsAffine (𝒱.obj i)
i : (Scheme.Pullback.openCoverOfBase 𝒰 f f).J
j k : (𝒰' i).J

⊢ pullback.diagonal f ≫ 𝟙 (pullback.diagonalObj f) = 𝟙 X ≫ pullback.diagonal f
-- Mathlib/AlgebraicGeometry/Morphisms/Constructors.lean
P : MorphismProperty Scheme
Q : AffineTargetMorphismProperty
inst✝² : HasAffineProperty P Q
X Y : Scheme
f : X ⟶ Y
𝒰 : Y.OpenCover
inst✝¹ : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)
𝒰' : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover
inst✝ : ∀ (i : 𝒰.J) (j : (𝒰' i).J), IsAffine ((𝒰' i).obj j)
h𝒰' : ∀ (i : 𝒰.J) (j k : (𝒰' i).J), Q (pullback.mapDesc ((𝒰' i).map j) ((𝒰' i).map k) (𝒰.pullbackHom f i))
this : Q.IsLocal := isLocal_affineProperty P
𝒱 : (pullback f f).OpenCover := (Scheme.Pullback.openCoverOfBase 𝒰 f f).bind fun i => Scheme.Pullback.openCoverOfLeftRight (𝒰' i) (𝒰' i) (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i))
i1 : ∀ (i : 𝒱.J), IsAffine (𝒱.obj i)
i : (Scheme.Pullback.openCoverOfBase 𝒰 f f).J
j k : (𝒰' i).J

⊢ pullback.map ((𝒰' i).map j ≫ pullback.snd f (𝒰.map i)) ((𝒰' i).map k ≫ pullback.snd f (𝒰.map i)) f f ((𝒰' i).map j ≫ pullback.fst f (𝒰.map i)) ((𝒰' i).map k ≫ pullback.fst f (𝒰.map i)) (𝒰.map i) ⋯ ⋯ ≫ 𝟙 (pullback.diagonalObj f) = 𝟙 (pullback ((𝒰' i).map j ≫ pullback.snd f (𝒰.map i)) ((𝒰' i).map k ≫ pullback.snd f (𝒰.map i))) ≫ pullback.map ((𝒰' i).map j ≫ pullback.snd f (𝒰.map i)) ((𝒰' i).map k ≫ pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i)) ((𝒰' i).map j) ((𝒰' i).map k) (𝟙 (𝒰.obj i)) ⋯ ⋯ ≫ pullback.map (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i)) f f (pullback.fst f (𝒰.map i)) (pullback.fst f (𝒰.map i)) (𝒰.map i) ⋯ ⋯
-- Mathlib/AlgebraicGeometry/Morphisms/Constructors.lean
P : MorphismProperty Scheme
Q : AffineTargetMorphismProperty
inst✝² : HasAffineProperty P Q
X Y : Scheme
f : X ⟶ Y
𝒰 : Y.OpenCover
inst✝¹ : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)
𝒰' : (i : 𝒰.J) → (pullback f (𝒰.map i)).OpenCover
inst✝ : ∀ (i : 𝒰.J) (j : (𝒰' i).J), IsAffine ((𝒰' i).obj j)
h𝒰' : ∀ (i : 𝒰.J) (j k : (𝒰' i).J), Q (pullback.mapDesc ((𝒰' i).map j) ((𝒰' i).map k) (𝒰.pullbackHom f i))
this : Q.IsLocal := isLocal_affineProperty P
𝒱 : (pullback f f).OpenCover := (Scheme.Pullback.openCoverOfBase 𝒰 f f).bind fun i => Scheme.Pullback.openCoverOfLeftRight (𝒰' i) (𝒰' i) (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i))
i1 : ∀ (i : 𝒱.J), IsAffine (𝒱.obj i)
i : (Scheme.Pullback.openCoverOfBase 𝒰 f f).J
j k : (𝒰' i).J

⊢ Q (((pullbackDiagonalMapIso f (𝒰.map i) ((𝒰' i).map j) ((𝒰' i).map k)).inv ≫ pullback.map (pullback.diagonal f) (pullback.map ((𝒰' i).map j ≫ pullback.snd f (𝒰.map i)) ((𝒰' i).map k ≫ pullback.snd f (𝒰.map i)) f f ((𝒰' i).map j ≫ pullback.fst f (𝒰.map i)) ((𝒰' i).map k ≫ pullback.fst f (𝒰.map i)) (𝒰.map i) ⋯ ⋯) (pullback.diagonal f) (pullback.map ((𝒰' i).map j ≫ pullback.snd f (𝒰.map i)) ((𝒰' i).map k ≫ pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i)) ((𝒰' i).map j) ((𝒰' i).map k) (𝟙 (𝒰.obj i)) ⋯ ⋯ ≫ pullback.map (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i)) f f (pullback.fst f (𝒰.map i)) (pullback.fst f (𝒰.map i)) (𝒰.map i) ⋯ ⋯) (𝟙 X) (𝟙 (pullback ((𝒰' i).map j ≫ pullback.snd f (𝒰.map i)) ((𝒰' i).map k ≫ pullback.snd f (𝒰.map i)))) (𝟙 (pullback.diagonalObj f)) ?mk.mk.convert_1 ?mk.mk.convert_2) ≫ pullback.snd (pullback.diagonal f) (pullback.map ((𝒰' i).map j ≫ pullback.snd f (𝒰.map i)) ((𝒰' i).map k ≫ pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i)) ((𝒰' i).map j) ((𝒰' i).map k) (𝟙 (𝒰.obj i)) ⋯ ⋯ ≫ pullback.map (pullback.snd f (𝒰.map i)) (pullback.snd f (𝒰.map i)) f f (pullback.fst f (𝒰.map i)) (pullback.fst f (𝒰.map i)) (𝒰.map i) ⋯ ⋯))
-- Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean
X : Scheme
r : ↑Γ(X, ⊤)

⊢ adjunction.unit.app X ⁻¹ᵁ basicOpen r = X.basicOpen r
-- Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean
X : Scheme
r : ↑Γ(X, ⊤)

⊢ adjunction.unit.app X ⁻¹ᵁ (Spec Γ(X, ⊤)).basicOpen ((Scheme.ΓSpecIso Γ(X, ⊤)).inv r) = X.basicOpen r
-- Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean
X : Scheme
r : ↑Γ(X, ⊤)

⊢ ((𝟭 Scheme).obj X).basicOpen ((Scheme.Hom.app (adjunction.unit.app X) ⊤) ((Scheme.ΓSpecIso Γ(X, ⊤)).inv r)) = X.basicOpen r
-- Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean
X : Scheme
r : ↑Γ(X, ⊤)

⊢ (Scheme.Hom.app (adjunction.unit.app X) ⊤) ((Scheme.ΓSpecIso Γ(X, ⊤)).inv r) = r
-- Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean
X : Scheme
r : ↑Γ(X, ⊤)

⊢ (Scheme.ΓSpecIso Γ(X, ⊤)).hom ((Scheme.ΓSpecIso Γ(X, ⊤)).inv r) = r
-- Mathlib/Data/Real/Irrational.lean
q : ℚ
x y : ℝ
h : Irrational (x ^ 0)

⊢ Irrational x
-- Mathlib/Data/Real/Irrational.lean
q : ℚ
x y : ℝ
h : Irrational 1

⊢ Irrational x
-- Mathlib/Data/Real/Irrational.lean
q : ℚ
x y : ℝ
n : ℕ
h : Irrational (x ^ (n + 1))

⊢ Irrational x
-- Mathlib/Data/Real/Irrational.lean
q : ℚ
x y : ℝ
n : ℕ
h : Irrational (x ^ n * x)

⊢ Irrational x
-- Mathlib/Data/Real/Irrational.lean
q : ℚ
x y : ℝ
h : Irrational x

⊢ Irrational (↑q - x)
-- 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 n₁ n₂ n₁₂ : ℤ
z₁ : Cochain F G n₁
z₂ z₂' : Cochain G K n₂
h : n₁ + n₂ = n₁₂
p q : ℤ
hpq : p + n₁₂ = q

⊢ (z₁.comp (z₂ + z₂') h).v p q hpq = (z₁.comp z₂ h + z₁.comp z₂' h).v p q hpq
-- Mathlib/Topology/Basic.lean
X : Type u
Y : Type v
ι : Sort w
α : Type u_1
β : Type u_2
x : X
s s₁ s₂ t : Set X
p p₁ p₂ : X → Prop
inst✝¹ : TopologicalSpace X
f : Filter X
H : f ≤ 𝓝 x
inst✝ : f.NeBot

⊢ ClusterPt x f
-- Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean
C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
X Y Z : C
inst✝⁴ : HasPullbacks C
S T : C
f : X ⟶ T
g : Y ⟶ T
i : T ⟶ S
inst✝³ : HasPullback i i
inst✝² : HasPullback f g
inst✝¹ : HasPullback (f ≫ i) (g ≫ i)
inst✝ : HasPullback (diagonal i) (map (f ≫ i) (g ≫ i) i i f g (𝟙 S) ⋯ ⋯)

⊢ IsPullback (fst f g ≫ f) (map f g (f ≫ i) (g ≫ i) (𝟙 X) (𝟙 Y) i ⋯ ⋯) (diagonal i) (map (f ≫ i) (g ≫ i) i i f g (𝟙 S) ⋯ ⋯)
-- Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean
C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
X Y Z : C
inst✝⁴ : HasPullbacks C
S T : C
f : X ⟶ T
g : Y ⟶ T
i : T ⟶ S
inst✝³ : HasPullback i i
inst✝² : HasPullback f g
inst✝¹ : HasPullback (f ≫ i) (g ≫ i)
inst✝ : HasPullback (diagonal i) (map (f ≫ i) (g ≫ i) i i f g (𝟙 S) ⋯ ⋯)

⊢ (pullbackDiagonalMapIdIso f g i).symm.hom ≫ fst (diagonal i) (map (f ≫ i) (g ≫ i) i i f g (𝟙 S) ⋯ ⋯) = fst f g ≫ f
-- Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean
C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
X Y Z : C
inst✝⁴ : HasPullbacks C
S T : C
f : X ⟶ T
g : Y ⟶ T
i : T ⟶ S
inst✝³ : HasPullback i i
inst✝² : HasPullback f g
inst✝¹ : HasPullback (f ≫ i) (g ≫ i)
inst✝ : HasPullback (diagonal i) (map (f ≫ i) (g ≫ i) i i f g (𝟙 S) ⋯ ⋯)

⊢ (pullbackDiagonalMapIdIso f g i).symm.hom ≫ snd (diagonal i) (map (f ≫ i) (g ≫ i) i i f g (𝟙 S) ⋯ ⋯) = map f g (f ≫ i) (g ≫ i) (𝟙 X) (𝟙 Y) i ⋯ ⋯
-- Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean
C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
X Y Z : C
inst✝⁴ : HasPullbacks C
S T : C
f : X ⟶ T
g : Y ⟶ T
i : T ⟶ S
inst✝³ : HasPullback i i
inst✝² : HasPullback f g
inst✝¹ : HasPullback (f ≫ i) (g ≫ i)
inst✝ : HasPullback (diagonal i) (map (f ≫ i) (g ≫ i) i i f g (𝟙 S) ⋯ ⋯)

⊢ CommSq (fst f g ≫ f) (map f g (f ≫ i) (g ≫ i) (𝟙 X) (𝟙 Y) i ⋯ ⋯) (diagonal i) (map (f ≫ i) (g ≫ i) i i f g (𝟙 S) ⋯ ⋯)
-- Mathlib/Algebra/Group/UniqueProds.lean
G : Type u_1
H : Type u_2
inst✝¹ : Mul G
inst✝ : Mul H
A B : Finset G
a0 b0 : G
f : G ↪ H
mul : ∀ (x y : G), f (x * y) = f x * f y

⊢ UniqueMul (Finset.map f A) (Finset.map f B) (f a0) (f b0) ↔ UniqueMul A B a0 b0
-- Mathlib/RingTheory/OreLocalization/Basic.lean
R : Type u_1
inst✝² : Monoid R
S : Submonoid R
inst✝¹ : OreSet S
X : Type u_2
inst✝ : MulAction R X
r : X
s t : ↥S

⊢ (1 /ₒ t) • (r /ₒ s) = r /ₒ (s * t)
-- Mathlib/Algebra/Polynomial/Monic.lean
R : Type u
S : Type v
a b : R
m n✝ : ℕ
ι : Type y
inst✝ : Ring R
p : R[X]
n : ℕ
H : p.degree ≤ ↑n

⊢ (X ^ (n + 1) - p).Monic
-- Mathlib/Combinatorics/Additive/ETransform.lean
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Group α
e : α
x : Finset α × Finset α

⊢ (mulETransformLeft e x).1 * (mulETransformLeft e x).2 ⊆ x.1 * x.2
-- Mathlib/Combinatorics/Additive/ETransform.lean
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Group α
e : α
x : Finset α × Finset α

⊢ op e • x.1 * e⁻¹ • x.2 ⊆ x.1 * x.2
-- Mathlib/Data/Finmap.lean
α : Type u
β : α → Type v
inst✝ : DecidableEq α
a : α
b : β a
s✝ : Finmap β
s : AList β
h : a ∉ ⟦s⟧

⊢ (insert a b ⟦s⟧).entries = ⟨a, b⟩ ::ₘ ⟦s⟧.entries
-- Mathlib/Probability/ProbabilityMassFunction/Monad.lean
α : Type u_1
β : Type u_2
γ : Type u_3
p✝ : PMF α
f : α → PMF β
g : β → PMF γ
p : PMF α
q : PMF β
x : β

⊢ (p.bind fun x => q) x = q x
-- Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
x✝ y✝ z : ℝ
n✝ : ℕ
x : ℝ
hx : x ≠ 0
y : ℝ
n : ℕ

⊢ x ^ (y - ↑n) = x ^ y / x ^ n
-- Mathlib/Probability/ProbabilityMassFunction/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : Countable α
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSingletonClass α
p : PMF α
μ : Measure α
inst✝ : IsProbabilityMeasure μ
x : α

⊢ p.toMeasure.toPMF x = p x
-- Mathlib/CategoryTheory/Dialectica/Monoidal.lean
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasFiniteProducts C
inst✝ : HasPullbacks C
X : Dial C

⊢ (tensorUnit.tensorObj X).rel = (Subobject.pullback (prod.map (prod.leftUnitor X.src).hom (prod.leftUnitor X.tgt).hom)).obj X.rel
-- Mathlib/MeasureTheory/Measure/WithDensityFinite.lean
α : Type u_1
mα : MeasurableSpace α
μ✝ μ : Measure α
inst✝ : SFinite μ

⊢ μ.toFinite ≪ μ
-- Mathlib/MeasureTheory/Measure/WithDensityFinite.lean
α : Type u_1
mα : MeasurableSpace α
μ✝ μ : Measure α
inst✝ : SFinite μ

⊢ μ.toFinite ≪ Measure.sum (sFiniteSeq μ)
-- Mathlib/MeasureTheory/Measure/WithDensityFinite.lean
α : Type u_1
mα : MeasurableSpace α
μ✝ μ : Measure α
inst✝ : SFinite μ
s : Set α
hs : MeasurableSet s
hs0 : (Measure.sum (sFiniteSeq μ)) s = 0

⊢ μ.toFinite s = 0
-- Mathlib/MeasureTheory/Measure/WithDensityFinite.lean
α : Type u_1
mα : MeasurableSpace α
μ✝ μ : Measure α
inst✝ : SFinite μ
s : Set α
hs : MeasurableSet s
hs0 : ∀ (i : ℕ), (sFiniteSeq μ i) s = 0

⊢ μ.toFinite s = 0
-- Mathlib/Data/Matrix/Reflection.lean
l m✝ n✝ : ℕ
α : Type u_1
β : Type u_2
m n : ℕ
A : Matrix (Fin m) (Fin n) α

⊢ A.etaExpand = A
-- Mathlib/Data/Matrix/Reflection.lean
l m✝ n✝ : ℕ
α : Type u_1
β : Type u_2
m n : ℕ
A : Matrix (Fin m) (Fin n) α

⊢ ((Equiv.refl (Fin m → Fin n → α)) fun i j => A i j) = A
-- Mathlib/LinearAlgebra/Ray.lean
R : Type u_1
inst✝⁵ : StrictOrderedCommSemiring R
M : Type u_2
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
N : Type u_3
inst✝² : AddCommMonoid N
inst✝¹ : Module R N
ι : Type u_4
inst✝ : DecidableEq ι
u : Rˣ
hu : 0 < ↑u
v : Ray R M

⊢ u • v = v
-- Mathlib/LinearAlgebra/Ray.lean
R : Type u_1
inst✝⁵ : StrictOrderedCommSemiring R
M : Type u_2
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
N : Type u_3
inst✝² : AddCommMonoid N
inst✝¹ : Module R N
ι : Type u_4
inst✝ : DecidableEq ι
u : Rˣ
hu : 0 < ↑u
v✝ : M
hv✝ : v✝ ≠ 0

⊢ u • rayOfNeZero R v✝ hv✝ = rayOfNeZero R v✝ hv✝
-- Mathlib/LinearAlgebra/Ray.lean
R : Type u_1
inst✝⁵ : StrictOrderedCommSemiring R
M : Type u_2
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
N : Type u_3
inst✝² : AddCommMonoid N
inst✝¹ : Module R N
ι : Type u_4
inst✝ : DecidableEq ι
u : Rˣ
hu : 0 < ↑u
v✝ : M
hv✝ : v✝ ≠ 0

⊢ SameRay R (u • v✝) v✝
-- Mathlib/GroupTheory/Archimedean.lean
G : Type u_1
inst✝¹ : LinearOrderedAddCommGroup G
inst✝ : Archimedean G
H : AddSubgroup G
hbot : H ≠ ⊥
a : G
h₀ : 0 < a
hd : Disjoint (↑H) (Ioo 0 a)
hex : ∀ g > 0, ∃ n, g ∈ Ioc (n • a) ((n + 1) • a)

⊢ ∃ b, IsLeast {g | g ∈ H ∧ 0 < g} b
-- Mathlib/GroupTheory/Archimedean.lean
G : Type u_1
inst✝¹ : LinearOrderedAddCommGroup G
inst✝ : Archimedean G
H : AddSubgroup G
hbot : H ≠ ⊥
a : G
h₀ : 0 < a
hd : Disjoint (↑H) (Ioo 0 a)
hex : ∀ g > 0, ∃ n, g ∈ Ioc (n • a) ((n + 1) • a)
this : ∃ n, (↑H ∩ Ioc (n • a) ((n + 1) • a)).Nonempty

⊢ ∃ b, IsLeast {g | g ∈ H ∧ 0 < g} b
-- Mathlib/GroupTheory/Archimedean.lean
G : Type u_1
inst✝¹ : LinearOrderedAddCommGroup G
inst✝ : Archimedean G
H : AddSubgroup G
hbot : H ≠ ⊥
a : G
h₀ : 0 < a
hd : Disjoint (↑H) (Ioo 0 a)
hex : ∀ g > 0, ∃ n, g ∈ Ioc (n • a) ((n + 1) • a)
this : ∃ n, (↑H ∩ Ioc (n • a) ((n + 1) • a)).Nonempty
n : ℕ
hmin : ∀ m < n, ¬(↑H ∩ Ioc (m • a) ((m + 1) • a)).Nonempty
x : G
hxH : x ∈ ↑H
hnx : n • a < x
hxn : x ≤ (n + 1) • a

⊢ ∃ b, IsLeast {g | g ∈ H ∧ 0 < g} b
-- Mathlib/GroupTheory/Archimedean.lean
G : Type u_1
inst✝¹ : LinearOrderedAddCommGroup G
inst✝ : Archimedean G
H : AddSubgroup G
hbot : H ≠ ⊥
a : G
h₀ : 0 < a
hd : Disjoint (↑H) (Ioo 0 a)
hex : ∀ g > 0, ∃ n, g ∈ Ioc (n • a) ((n + 1) • a)
this : ∃ n, (↑H ∩ Ioc (n • a) ((n + 1) • a)).Nonempty
n : ℕ
hmin : ∀ m < n, ¬(↑H ∩ Ioc (m • a) ((m + 1) • a)).Nonempty
x : G
hxH : x ∈ ↑H
hnx : n • a < x
hxn : x ≤ (n + 1) • a
hxmin : ¬∃ b, IsLeast {g | g ∈ H ∧ 0 < g} b

⊢ False
-- Mathlib/GroupTheory/Archimedean.lean
G : Type u_1
inst✝¹ : LinearOrderedAddCommGroup G
inst✝ : Archimedean G
H : AddSubgroup G
hbot : H ≠ ⊥
a : G
h₀ : 0 < a
hd : Disjoint (↑H) (Ioo 0 a)
hex : ∀ g > 0, ∃ n, g ∈ Ioc (n • a) ((n + 1) • a)
this : ∃ n, (↑H ∩ Ioc (n • a) ((n + 1) • a)).Nonempty
n : ℕ
hmin : ∀ m < n, ¬(↑H ∩ Ioc (m • a) ((m + 1) • a)).Nonempty
x : G
hxH : x ∈ ↑H
hnx : n • a < x
hxn : x ≤ (n + 1) • a
hxmin : ∀ (x : G), x ∈ H ∧ 0 < x → ∃ x_1, ∃ (_ : x_1 ∈ H ∧ 0 < x_1), x_1 < x

⊢ False
-- Mathlib/GroupTheory/Archimedean.lean
G : Type u_1
inst✝¹ : LinearOrderedAddCommGroup G
inst✝ : Archimedean G
H : AddSubgroup G
hbot : H ≠ ⊥
a : G
h₀ : 0 < a
hd : Disjoint (↑H) (Ioo 0 a)
hex : ∀ g > 0, ∃ n, g ∈ Ioc (n • a) ((n + 1) • a)
this : ∃ n, (↑H ∩ Ioc (n • a) ((n + 1) • a)).Nonempty
n : ℕ
hmin : ∀ m < n, ¬(↑H ∩ Ioc (m • a) ((m + 1) • a)).Nonempty
x : G
hxH : x ∈ ↑H
hnx : n • a < x
hxn : x ≤ (n + 1) • a
hxmin : ∀ (x : G), x ∈ H ∧ 0 < x → ∃ x_1, ∃ (_ : x_1 ∈ H ∧ 0 < x_1), x_1 < x
y : G
hxy : y < x
hyH : y ∈ H
hy₀ : 0 < y

⊢ False
-- Mathlib/GroupTheory/Archimedean.lean
G : Type u_1
inst✝¹ : LinearOrderedAddCommGroup G
inst✝ : Archimedean G
H : AddSubgroup G
hbot : H ≠ ⊥
a : G
h₀ : 0 < a
hd : Disjoint (↑H) (Ioo 0 a)
hex : ∀ g > 0, ∃ n, g ∈ Ioc (n • a) ((n + 1) • a)
this : ∃ n, (↑H ∩ Ioc (n • a) ((n + 1) • a)).Nonempty
n : ℕ
hmin : ∀ m < n, ¬(↑H ∩ Ioc (m • a) ((m + 1) • a)).Nonempty
x : G
hxH : x ∈ ↑H
hnx : n • a < x
hxn : x ≤ (n + 1) • a
hxmin : ∀ (x : G), x ∈ H ∧ 0 < x → ∃ x_1, ∃ (_ : x_1 ∈ H ∧ 0 < x_1), x_1 < x
y : G
hxy : y < x
hyH : y ∈ H
hy₀ : 0 < y
m : ℕ
hm : y ∈ Ioc (m • a) ((m + 1) • a)

⊢ False
-- Mathlib/GroupTheory/Archimedean.lean
G : Type u_1
inst✝¹ : LinearOrderedAddCommGroup G
inst✝ : Archimedean G
H : AddSubgroup G
hbot : H ≠ ⊥
a : G
h₀ : 0 < a
hd : Disjoint (↑H) (Ioo 0 a)
hex : ∀ g > 0, ∃ n, g ∈ Ioc (n • a) ((n + 1) • a)
this : ∃ n, (↑H ∩ Ioc (n • a) ((n + 1) • a)).Nonempty
n : ℕ
hmin : ∀ m < n, ¬(↑H ∩ Ioc (m • a) ((m + 1) • a)).Nonempty
x : G
hxH : x ∈ ↑H
hnx : n • a < x
hxn : x ≤ (n + 1) • a
hxmin : ∀ (x : G), x ∈ H ∧ 0 < x → ∃ x_1, ∃ (_ : x_1 ∈ H ∧ 0 < x_1), x_1 < x
y : G
hxy : y < x
hyH : y ∈ H
hy₀ : 0 < y
m : ℕ
hm : y ∈ Ioc (m • a) ((m + 1) • a)
hmn : m < n

⊢ False
-- Mathlib/GroupTheory/Archimedean.lean
G : Type u_1
inst✝¹ : LinearOrderedAddCommGroup G
inst✝ : Archimedean G
H : AddSubgroup G
hbot : H ≠ ⊥
a : G
h₀ : 0 < a
hd : Disjoint (↑H) (Ioo 0 a)
hex : ∀ g > 0, ∃ n, g ∈ Ioc (n • a) ((n + 1) • a)
this : ∃ n, (↑H ∩ Ioc (n • a) ((n + 1) • a)).Nonempty
n : ℕ
hmin : ∀ m < n, ¬(↑H ∩ Ioc (m • a) ((m + 1) • a)).Nonempty
x : G
hxH : x ∈ ↑H
hnx : n • a < x
hxn : x ≤ (n + 1) • a
hxmin : ∀ (x : G), x ∈ H ∧ 0 < x → ∃ x_1, ∃ (_ : x_1 ∈ H ∧ 0 < x_1), x_1 < x
y : G
hxy : y < x
hyH : y ∈ H
hy₀ : 0 < y
m : ℕ
hm : y ∈ Ioc (m • a) ((m + 1) • a)
hnm : n ≤ m

⊢ False
-- Mathlib/Topology/Category/Profinite/Nobeling.lean
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
s : Finset I
x y : ↑(π C fun x => x ∈ s)
h : y = x

⊢ (List.map (⇑(LocallyConstant.evalMonoidHom y)) (factors C s x)).prod = 1
-- Mathlib/Topology/Category/Profinite/Nobeling.lean
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
s : Finset I
x y : ↑(π C fun x => x ∈ s)
h : y = x

⊢ ∀ x_1 ∈ List.map (⇑(LocallyConstant.evalMonoidHom y)) (factors C s x), x_1 = 1
-- Mathlib/Topology/Category/Profinite/Nobeling.lean
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
s : Finset I
x y : ↑(π C fun x => x ∈ s)
h : y = x

⊢ ∀ (x_1 : ℤ), (∃ a, (∃ a_1 ∈ Finset.sort (fun x x_2 => x ≥ x_2) s, (if ↑x a_1 = true then e (π C fun x => x ∈ s) a_1 else 1 - e (π C fun x => x ∈ s) a_1) = a) ∧ ((Pi.evalMonoidHom (fun a => ℤ) x).comp LocallyConstant.coeFnMonoidHom) a = x_1) → x_1 = 1
-- Mathlib/Testing/SlimCheck/Gen.lean
lo hi a : ℕ
h : lo.succ ≤ a ∧ a ≤ hi

⊢ a.pred.succ ≤ hi
-- Mathlib/Testing/SlimCheck/Gen.lean
lo hi a : ℕ
h : lo.succ ≤ a ∧ a ≤ hi

⊢ a ≤ hi
-- Mathlib/Testing/SlimCheck/Gen.lean
lo hi a : ℕ
h : lo.succ ≤ a ∧ a ≤ hi

⊢ 0 < a
-- Mathlib/Algebra/BigOperators/Group/Multiset.lean
F : Type u_1
ι : Type u_2
α : Type u_3
β : Type u_4
γ : Type u_5
inst✝² : CommMonoid α
inst✝¹ : CommMonoid β
s t : Multiset α
a : α
m : Multiset ι
f g : ι → α
inst✝ : DecidableEq α
h : a ∈ s

⊢ a * (s.erase a).prod = s.prod
-- Mathlib/Analysis/Normed/Group/SemiNormedGrp/Kernels.lean
X Y Z : SemiNormedGrp
f : X ⟶ Y
g : Y ⟶ Z
cond : f ≫ g = 0
hg : NormedAddGroupHom.NormNoninc g

⊢ NormedAddGroupHom.NormNoninc (explicitCokernelDesc cond)
-- Mathlib/Analysis/Normed/Group/SemiNormedGrp/Kernels.lean
X Y Z : SemiNormedGrp
f : X ⟶ Y
g : Y ⟶ Z
cond : f ≫ g = 0
hg : NormedAddGroupHom.NormNoninc g

⊢ ‖explicitCokernelDesc cond‖ ≤ 1
-- Mathlib/Analysis/Normed/Group/SemiNormedGrp/Kernels.lean
X Y Z : SemiNormedGrp
f : X ⟶ Y
g : Y ⟶ Z
cond : f ≫ g = 0
hg : NormedAddGroupHom.NormNoninc g

⊢ ‖explicitCokernelDesc cond‖ ≤ ↑1
-- Mathlib/Order/CompleteLattice.lean
α : Type u_1
β : Type u_2
β₂ : Type u_3
γ : Type u_4
ι : Sort u_5
ι' : Sort u_6
κ : ι → Sort u_7
κ' : ι' → Sort u_8
inst✝ : CompleteLattice α
f✝ g s t : ι → α
a b : α
ι₁ : Sort u_9
ι₂ : Sort u_10
κ₁ : ι₁ → Sort u_11
κ₂ : ι₂ → Sort u_12
f : (i₁ : ι₁) → κ₁ i₁ → (i₂ : ι₂) → κ₂ i₂ → α

⊢ ⨆ i₁, ⨆ j₁, ⨆ i₂, ⨆ j₂, f i₁ j₁ i₂ j₂ = ⨆ i₂, ⨆ j₂, ⨆ i₁, ⨆ j₁, f i₁ j₁ i₂ j₂
-- Mathlib/Analysis/RCLike/Basic.lean
K : Type u_1
E : Type u_2
inst✝ : RCLike K
z : K

⊢ z + (starRingEnd K) z = ↑(re z) + ↑(im z) * I + (↑(re z) - ↑(im z) * I)
-- Mathlib/Analysis/RCLike/Basic.lean
K : Type u_1
E : Type u_2
inst✝ : RCLike K
z : K

⊢ ↑(re z) + ↑(im z) * I + (↑(re z) - ↑(im z) * I) = 2 * ↑(re z)
-- Mathlib/Algebra/Polynomial/Monic.lean
R : Type u
S : Type v
a b : R
m n✝ : ℕ
ι : Type y
inst✝ : Semiring R
p q r : R[X]
n : ℕ
H1 : p.degree ≤ ↑n
H2 : p.coeff n = 1
H : ¬p.degree < ↑n

⊢ p.Monic
-- Mathlib/CategoryTheory/Sites/Continuous.lean
C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁴ : Category.{v₂, u₂} D
E : Type u₃
inst✝³ : Category.{v₃, u₃} E
F : C ⥤ D
A : Type u
inst✝² : Category.{t, u} A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝¹ : F.PreservesOneHypercovers J K
inst✝ : J.IsGeneratedByOneHypercovers
P : Dᵒᵖ ⥤ A
hP : Presheaf.IsSheaf K P

⊢ Presheaf.IsSheaf J (F.op ⋙ P)
-- Mathlib/CategoryTheory/Sites/Continuous.lean
C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁴ : Category.{v₂, u₂} D
E : Type u₃
inst✝³ : Category.{v₃, u₃} E
F : C ⥤ D
A : Type u
inst✝² : Category.{t, u} A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝¹ : F.PreservesOneHypercovers J K
inst✝ : J.IsGeneratedByOneHypercovers
P : Dᵒᵖ ⥤ A
hP : Presheaf.IsSheaf K P

⊢ ∀ ⦃X : C⦄ (E : J.OneHypercover X), Nonempty (IsLimit (E.multifork (F.op ⋙ P)))
-- Mathlib/Order/Ideal.lean
P : Type u_1
inst✝² : LE P
inst✝¹ : IsDirected P fun x x_1 => x ≤ x_1
inst✝ : Nonempty P
I : Ideal P
hI : IsCoatom I
x✝ : Ideal P
hJ : I < x✝

⊢ ↑x✝ = univ
-- Mathlib/MeasureTheory/Measure/Tilted.lean
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable (fun x => rexp (f x)) μ

⊢ (μ.tilted f).tilted (-f) = (μ Set.univ)⁻¹ • μ
-- Mathlib/MeasureTheory/Measure/Tilted.lean
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable (fun x => rexp (f x)) μ

⊢ μ.tilted (f + -f) = (μ Set.univ)⁻¹ • μ
-- Mathlib/Analysis/Complex/AbsMax.lean
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
U : Set E
c : E
hc : IsPreconnected U
ho : IsOpen U
hd : DifferentiableOn ℂ f U
hcU : c ∈ U
hm : IsMaxOn (norm ∘ f) U c
V : Set E := U ∩ {z | IsMaxOn (norm ∘ f) U z}

⊢ EqOn (norm ∘ f) (Function.const E ‖f c‖) U
-- Mathlib/Analysis/Complex/AbsMax.lean
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
U : Set E
c : E
hc : IsPreconnected U
ho : IsOpen U
hd : DifferentiableOn ℂ f U
hcU : c ∈ U
hm : IsMaxOn (norm ∘ f) U c
V : Set E := U ∩ {z | IsMaxOn (norm ∘ f) U z}
hV : ∀ x ∈ V, ‖f x‖ = ‖f c‖

⊢ EqOn (norm ∘ f) (Function.const E ‖f c‖) U
-- Mathlib/Analysis/Complex/AbsMax.lean
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
U : Set E
c : E
hc : IsPreconnected U
ho : IsOpen U
hd : DifferentiableOn ℂ f U
hcU : c ∈ U
hm : IsMaxOn (norm ∘ f) U c
V : Set E := U ∩ {z | IsMaxOn (norm ∘ f) U z}
hV : ∀ x ∈ V, ‖f x‖ = ‖f c‖

⊢ U ⊆ V
-- Mathlib/Analysis/Complex/AbsMax.lean
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
U : Set E
c : E
hc : IsPreconnected U
ho : IsOpen U
hd : DifferentiableOn ℂ f U
hcU : c ∈ U
hm : IsMaxOn (norm ∘ f) U c
V : Set E := U ∩ {z | IsMaxOn (norm ∘ f) U z}
hV : ∀ x ∈ V, ‖f x‖ = ‖f c‖
hVo : IsOpen V

⊢ U ⊆ V