ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Data/Set/Pointwise/Interval.lean
α : Type u_1
inst✝ : OrderedAddCommGroup α
a b c : α

⊢ (fun x => a + x) '' Iio b = Iio (a + b)
-- Mathlib/Data/Set/Sups.lean
F : Type u_1
α : Type u_2
β : Type u_3
inst✝ : SemilatticeInf α
s t : Set α
a : α

⊢ a ∈ ↑(lowerClosure (s ⊼ t)) ↔ a ∈ ↑(lowerClosure s ⊓ lowerClosure t)
-- Mathlib/Data/Set/Sups.lean
F : Type u_1
α : Type u_2
β : Type u_3
inst✝ : SemilatticeInf α
s t : Set α
a : α

⊢ (∃ a_1, (∃ a ∈ s, ∃ b ∈ t, a ⊓ b = a_1) ∧ a ≤ a_1) ↔ a ∈ lowerClosure s ⊓ lowerClosure t
-- Mathlib/NumberTheory/Padics/PadicVal.lean
p : ℕ
hp : Fact (Nat.Prime p)
a : ℤ

⊢ multiplicity.Finite (↑p) a ↔ a ≠ 0
-- Mathlib/LinearAlgebra/Pi.lean
R : Type u
K : Type u'
M : Type v
V : Type v'
M₂ : Type w
V₂ : Type w'
M₃ : Type y
V₃ : Type y'
M₄ : Type z
ι : Type x
ι' : Type x'
inst✝⁶ : Semiring R
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : Module R M₂
inst✝³ : AddCommMonoid M₃
inst✝² : Module R M₃
φ : ι → Type i
inst✝¹ : (i : ι) → AddCommMonoid (φ i)
inst✝ : (i : ι) → Module R (φ i)
f : (i : ι) → M₂ →ₗ[R] φ i
c : M₂

⊢ c ∈ ker (pi f) ↔ c ∈ ⨅ i, ker (f i)
-- Mathlib/Data/Real/Sqrt.lean
x y : ℝ

⊢ √x ≤ 1 ↔ x ≤ 1
-- Mathlib/GroupTheory/Submonoid/Inverses.lean
M : Type u_1
inst✝ : CommMonoid M
S : Submonoid M
hS : S ≤ IsUnit.submonoid M
x : ↥S

⊢ ↑((S.leftInvEquiv hS).symm x) * ↑x = 1
-- Mathlib/GroupTheory/Submonoid/Inverses.lean
M : Type u_1
inst✝ : CommMonoid M
S : Submonoid M
hS : S ≤ IsUnit.submonoid M
x : ↥S

⊢ x = (S.leftInvEquiv hS) ((S.leftInvEquiv hS).symm x)
-- Mathlib/Algebra/Order/Field/Basic.lean
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝ : LinearOrderedSemifield α
a b c d e : α
m n : ℤ
ha : 0 < a
hb : 0 < b

⊢ a ≤ 1 / b ↔ b ≤ 1 / a
-- Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
x : ℝ
hx : 0 ≤ x
y : ℝ
z : ℂ

⊢ ↑x ^ (↑y * z) = ↑(x ^ y) ^ z
-- Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
x : ℝ
hx : 0 ≤ x
y : ℝ
z : ℂ

⊢ -π < (log ↑x * ↑y).im
-- Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
x : ℝ
hx : 0 ≤ x
y : ℝ
z : ℂ

⊢ (log ↑x * ↑y).im ≤ π
-- Mathlib/CategoryTheory/Dialectica/Monoidal.lean
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasFiniteProducts C
inst✝ : HasPullbacks C
X₁ X₂ Y₁ Y₂ : Dial C
f : X₁ ⟶ X₂
g : Y₁ ⟶ Y₂

⊢ (Subobject.pullback π(π₁, π(prod.map π₁ π₁ ≫ f.F, prod.map π₂ π₂ ≫ g.F))).obj (X₁.tensorObj Y₁).rel ≤ (Subobject.pullback (prod.map (prod.map f.f g.f) (𝟙 (X₂.tensorObj Y₂).tgt))).obj (X₂.tensorObj Y₂).rel
-- Mathlib/CategoryTheory/Dialectica/Monoidal.lean
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasFiniteProducts C
inst✝ : HasPullbacks C
X₁ X₂ Y₁ Y₂ : Dial C
f : X₁ ⟶ X₂
g : Y₁ ⟶ Y₂

⊢ (Subobject.pullback π(π₁, π(prod.map π₁ π₁ ≫ f.F, prod.map π₂ π₂ ≫ g.F))).obj ((Subobject.pullback (prod.map π₁ π₁)).obj X₁.rel) ⊓ (Subobject.pullback π(π₁, π(prod.map π₁ π₁ ≫ f.F, prod.map π₂ π₂ ≫ g.F))).obj ((Subobject.pullback (prod.map π₂ π₂)).obj Y₁.rel) ≤ (Subobject.pullback (prod.map (prod.map f.f g.f) (𝟙 (X₂.tgt ⨯ Y₂.tgt)))).obj ((Subobject.pullback (prod.map π₁ π₁)).obj X₂.rel) ⊓ (Subobject.pullback (prod.map (prod.map f.f g.f) (𝟙 (X₂.tgt ⨯ Y₂.tgt)))).obj ((Subobject.pullback (prod.map π₂ π₂)).obj Y₂.rel)
-- Mathlib/CategoryTheory/Dialectica/Monoidal.lean
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasFiniteProducts C
inst✝ : HasPullbacks C
X₁ X₂ Y₁ Y₂ : Dial C
f : X₁ ⟶ X₂
g : Y₁ ⟶ Y₂

⊢ (Subobject.pullback (π(π₁, π(prod.map π₁ π₁ ≫ f.F, prod.map π₂ π₂ ≫ g.F)) ≫ prod.map π₁ π₁)).obj X₁.rel ≤ (Subobject.pullback (prod.map (prod.map f.f g.f) (𝟙 (X₂.tgt ⨯ Y₂.tgt)) ≫ prod.map π₁ π₁)).obj X₂.rel
-- Mathlib/CategoryTheory/Dialectica/Monoidal.lean
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasFiniteProducts C
inst✝ : HasPullbacks C
X₁ X₂ Y₁ Y₂ : Dial C
f : X₁ ⟶ X₂
g : Y₁ ⟶ Y₂

⊢ (Subobject.pullback (π(π₁, π(prod.map π₁ π₁ ≫ f.F, prod.map π₂ π₂ ≫ g.F)) ≫ prod.map π₂ π₂)).obj Y₁.rel ≤ (Subobject.pullback (prod.map (prod.map f.f g.f) (𝟙 (X₂.tgt ⨯ Y₂.tgt)) ≫ prod.map π₂ π₂)).obj Y₂.rel
-- Mathlib/Analysis/NormedSpace/Pointwise.lean
𝕜 : Type u_1
E : Type u_2
inst✝³ : NormedField 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace ℝ E
x y z : E
δ ε r : ℝ
hr : 0 < r

⊢ r • ball 0 1 = ball 0 r
-- Mathlib/Algebra/Order/CauSeq/Completion.lean
α : Type u_1
inst✝² : LinearOrderedField α
β : Type u_2
inst✝¹ : DivisionRing β
abv : β → α
inst✝ : IsAbsoluteValue abv
x : β

⊢ const abv x⁻¹ = if h : (const abv x).LimZero then 0 else (const abv x).inv h
-- Mathlib/SetTheory/Ordinal/Basic.lean
α : Type u
β : Type u_1
γ : Type u_2
r✝ : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
r : α → α → Prop
inst✝ : IsWellOrder α r
h0 : 0 < type r
a : α

⊢ ¬r a (enum r 0 h0)
-- Mathlib/SetTheory/Ordinal/Basic.lean
α : Type u
β : Type u_1
γ : Type u_2
r✝ : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
r : α → α → Prop
inst✝ : IsWellOrder α r
h0 : 0 < type r
a : α

⊢ 0 ≤ typein r a
-- Mathlib/CategoryTheory/Bicategory/Basic.lean
B : Type u
inst✝ : Bicategory B
a b c d e : B
f : a ⟶ b
g : b ⟶ c

⊢ f ◁ (λ_ g).inv ≫ (α_ f (𝟙 b) g).inv = (ρ_ f).inv ▷ g
-- Mathlib/Geometry/Euclidean/Circumcenter.lean
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P n
fs : Finset (Fin (n + 1))
m : ℕ
h : fs.card = m + 1

⊢ ∃ r, ∀ (i : Fin (m + 1)), dist ((s.face h).points i) s.circumcenter = r
-- Mathlib/Geometry/Euclidean/Circumcenter.lean
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P n
fs : Finset (Fin (n + 1))
m : ℕ
h : fs.card = m + 1

⊢ ∀ (i : Fin (m + 1)), dist ((s.face h).points i) s.circumcenter = s.circumradius
-- Mathlib/RepresentationTheory/Action/Basic.lean
V : Type (u + 1)
inst✝ : LargeCategory V
G : MonCat
M N : Action V G
f : M.V ≅ N.V
comm : autoParam (∀ (g : ↑G), M.ρ g ≫ f.hom = f.hom ≫ N.ρ g) _auto✝
g : ↑G

⊢ N.ρ g ≫ f.inv = f.inv ≫ M.ρ g
-- Mathlib/RepresentationTheory/Action/Basic.lean
V : Type (u + 1)
inst✝ : LargeCategory V
G : MonCat
M N : Action V G
f : M.V ≅ N.V
comm : autoParam (∀ (g : ↑G), M.ρ g ≫ f.hom = f.hom ≫ N.ρ g) _auto✝
g : ↑G
w : (M.ρ g ≫ f.hom) ≫ f.inv = (f.hom ≫ N.ρ g) ≫ f.inv

⊢ N.ρ g ≫ f.inv = f.inv ≫ M.ρ g
-- Mathlib/Algebra/BigOperators/Group/Finset.lean
ι : Type u_1
κ : Type u_2
α : Type u_3
β : Type u_4
γ : Type u_5
s✝ s₁ s₂ : Finset α
a : α
f✝ g : α → β
inst✝¹ : CommMonoid β
inst✝ : DecidableEq α
s t : Finset α
f : α → β

⊢ (∏ i ∈ s, if i ∈ t then f i else 1) = ∏ i ∈ s ∩ t, f i
-- Mathlib/Algebra/Polynomial/Basic.lean
R : Type u
a✝ b : R
m n : ℕ
inst✝ : Semiring R
p q : R[X]
a : R[ℕ]

⊢ { toFinsupp := a } = 0 ↔ a = 0
-- Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedAddCommGroup G
s : Set α
f g : α → E
inst✝ : DecidablePred fun x => x ∈ s
hs : MeasurableSet s

⊢ eLpNormEssSup (s.piecewise f g) μ = max (eLpNormEssSup f (μ.restrict s)) (eLpNormEssSup g (μ.restrict sᶜ))
-- Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedAddCommGroup G
s : Set α
f g : α → E
inst✝ : DecidablePred fun x => x ∈ s
hs : MeasurableSet s

⊢ essSup (fun x => ↑‖s.piecewise f g x‖₊) μ = essSup (s.piecewise (fun x => ↑‖f x‖₊) fun x => ↑‖g x‖₊) μ
-- Mathlib/Analysis/CstarAlgebra/Basic.lean
𝕜 : Type u_1
E : Type u_2
α : Type u_3
inst✝² : NonUnitalNormedRing E
inst✝¹ : StarRing E
inst✝ : CstarRing E
x : E

⊢ x * x⋆ ≠ 0 ↔ x ≠ 0
-- Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Basic.lean
C : Type u
inst✝ : Category.{v, u} C
X Y : C
𝒯 : LimitCone (Functor.empty C)
ℬ : (X Y : C) → LimitCone (pair X Y)
X₁ X₂ X₃ Y₁ Y₂ Y₃ : C
f₁ : X₁ ⟶ Y₁
f₂ : X₂ ⟶ Y₂
f₃ : X₃ ⟶ Y₃

⊢ (ℬ (tensorObj ℬ Y₁ Y₂) Y₃).isLimit.lift (BinaryFan.mk (BinaryFan.fst (ℬ (tensorObj ℬ X₁ X₂) X₃).cone ≫ (ℬ Y₁ Y₂).isLimit.lift (BinaryFan.mk (BinaryFan.fst (ℬ X₁ X₂).cone ≫ f₁) (BinaryFan.snd (ℬ X₁ X₂).cone ≫ f₂))) (BinaryFan.snd (ℬ (tensorObj ℬ X₁ X₂) X₃).cone ≫ f₃)) ≫ (BinaryFan.associatorOfLimitCone ℬ Y₁ Y₂ Y₃).hom = (BinaryFan.associatorOfLimitCone ℬ X₁ X₂ X₃).hom ≫ (ℬ Y₁ (tensorObj ℬ Y₂ Y₃)).isLimit.lift (BinaryFan.mk (BinaryFan.fst (ℬ X₁ (ℬ X₂ X₃).cone.pt).cone ≫ f₁) (BinaryFan.snd (ℬ X₁ (ℬ X₂ X₃).cone.pt).cone ≫ (ℬ Y₂ Y₃).isLimit.lift (BinaryFan.mk (BinaryFan.fst (ℬ X₂ X₃).cone ≫ f₂) (BinaryFan.snd (ℬ X₂ X₃).cone ≫ f₃))))
-- Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Basic.lean
C : Type u
inst✝ : Category.{v, u} C
X Y : C
𝒯 : LimitCone (Functor.empty C)
ℬ : (X Y : C) → LimitCone (pair X Y)
X₁ X₂ X₃ Y₁ Y₂ Y₃ : C
f₁ : X₁ ⟶ Y₁
f₂ : X₂ ⟶ Y₂
f₃ : X₃ ⟶ Y₃

⊢ ∀ (j : Discrete WalkingPair), ((ℬ (tensorObj ℬ Y₁ Y₂) Y₃).isLimit.lift (BinaryFan.mk (BinaryFan.fst (ℬ (tensorObj ℬ X₁ X₂) X₃).cone ≫ (ℬ Y₁ Y₂).isLimit.lift (BinaryFan.mk (BinaryFan.fst (ℬ X₁ X₂).cone ≫ f₁) (BinaryFan.snd (ℬ X₁ X₂).cone ≫ f₂))) (BinaryFan.snd (ℬ (tensorObj ℬ X₁ X₂) X₃).cone ≫ f₃)) ≫ (BinaryFan.associatorOfLimitCone ℬ Y₁ Y₂ Y₃).hom) ≫ (ℬ Y₁ (ℬ Y₂ Y₃).cone.pt).cone.π.app j = ((BinaryFan.associatorOfLimitCone ℬ X₁ X₂ X₃).hom ≫ (ℬ Y₁ (tensorObj ℬ Y₂ Y₃)).isLimit.lift (BinaryFan.mk (BinaryFan.fst (ℬ X₁ (ℬ X₂ X₃).cone.pt).cone ≫ f₁) (BinaryFan.snd (ℬ X₁ (ℬ X₂ X₃).cone.pt).cone ≫ (ℬ Y₂ Y₃).isLimit.lift (BinaryFan.mk (BinaryFan.fst (ℬ X₂ X₃).cone ≫ f₂) (BinaryFan.snd (ℬ X₂ X₃).cone ≫ f₃))))) ≫ (ℬ Y₁ (ℬ Y₂ Y₃).cone.pt).cone.π.app j
-- Mathlib/ModelTheory/DirectLimit.lean
L : Language
ι : Type v
inst✝⁴ : Preorder ι
G : ι → Type w
inst✝³ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝² : IsDirected ι fun x x_1 => x ≤ x_1
inst✝¹ : DirectedSystem G fun i j h => ⇑(f i j h)
inst✝ : Nonempty ι
S : L.Substructure (DirectLimit G f)
S_fg : S.FG
A : Finset (DirectLimit G f)
A_closure : (Substructure.closure L).toFun ↑A = S

⊢ ∃ i T, Substructure.map (of L ι G f i).toHom T = S
-- Mathlib/ModelTheory/DirectLimit.lean
L : Language
ι : Type v
inst✝⁴ : Preorder ι
G : ι → Type w
inst✝³ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝² : IsDirected ι fun x x_1 => x ≤ x_1
inst✝¹ : DirectedSystem G fun i j h => ⇑(f i j h)
inst✝ : Nonempty ι
S : L.Substructure (DirectLimit G f)
S_fg : S.FG
A : Finset (DirectLimit G f)
A_closure : (Substructure.closure L).toFun ↑A = S
i : ι
y : { x // x ∈ A } → G i
eq_y : (fun a => ↑a) = fun a => ⟦Structure.Sigma.mk f i (y a)⟧

⊢ ∃ i T, Substructure.map (of L ι G f i).toHom T = S
-- Mathlib/ModelTheory/DirectLimit.lean
L : Language
ι : Type v
inst✝⁴ : Preorder ι
G : ι → Type w
inst✝³ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝² : IsDirected ι fun x x_1 => x ≤ x_1
inst✝¹ : DirectedSystem G fun i j h => ⇑(f i j h)
inst✝ : Nonempty ι
S : L.Substructure (DirectLimit G f)
S_fg : S.FG
A : Finset (DirectLimit G f)
A_closure : (Substructure.closure L).toFun ↑A = S
i : ι
y : { x // x ∈ A } → G i
eq_y : (fun a => ↑a) = fun a => ⟦Structure.Sigma.mk f i (y a)⟧

⊢ ∃ T, Substructure.map (of L ι G f i).toHom T = S
-- Mathlib/ModelTheory/DirectLimit.lean
L : Language
ι : Type v
inst✝⁴ : Preorder ι
G : ι → Type w
inst✝³ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝² : IsDirected ι fun x x_1 => x ≤ x_1
inst✝¹ : DirectedSystem G fun i j h => ⇑(f i j h)
inst✝ : Nonempty ι
S : L.Substructure (DirectLimit G f)
S_fg : S.FG
A : Finset (DirectLimit G f)
A_closure : (Substructure.closure L).toFun ↑A = S
i : ι
y : { x // x ∈ A } → G i
eq_y : (fun a => ↑a) = fun a => ⟦Structure.Sigma.mk f i (y a)⟧

⊢ Substructure.map (of L ι G f i).toHom ((Substructure.closure L).toFun (range y)) = S
-- Mathlib/ModelTheory/DirectLimit.lean
L : Language
ι : Type v
inst✝⁴ : Preorder ι
G : ι → Type w
inst✝³ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝² : IsDirected ι fun x x_1 => x ≤ x_1
inst✝¹ : DirectedSystem G fun i j h => ⇑(f i j h)
inst✝ : Nonempty ι
S : L.Substructure (DirectLimit G f)
S_fg : S.FG
A : Finset (DirectLimit G f)
A_closure : (Substructure.closure L).toFun ↑A = S
i : ι
y : { x // x ∈ A } → G i
eq_y : (fun a => ↑a) = fun a => ⟦Structure.Sigma.mk f i (y a)⟧

⊢ (Substructure.closure L).toFun (⇑(of L ι G f i).toHom '' range y) = S
-- Mathlib/ModelTheory/DirectLimit.lean
L : Language
ι : Type v
inst✝⁴ : Preorder ι
G : ι → Type w
inst✝³ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝² : IsDirected ι fun x x_1 => x ≤ x_1
inst✝¹ : DirectedSystem G fun i j h => ⇑(f i j h)
inst✝ : Nonempty ι
S : L.Substructure (DirectLimit G f)
S_fg : S.FG
A : Finset (DirectLimit G f)
A_closure : (Substructure.closure L).toFun ↑A = S
i : ι
y : { x // x ∈ A } → G i
eq_y : (fun a => ↑a) = fun a => ⟦Structure.Sigma.mk f i (y a)⟧

⊢ (Substructure.closure L).toFun ((fun a => ⟦Structure.Sigma.mk f i a⟧) '' range y) = S
-- Mathlib/Data/ENNReal/Real.lean
ι : Sort u_1
f g : ι → ℝ≥0∞
a b c d : ℝ≥0∞
r p q : ℝ≥0
hf : ∀ (i : ι), f i ≠ ⊤

⊢ (iInf f).toReal = ⨅ i, (f i).toReal
-- Mathlib/MeasureTheory/Integral/IntegrableOn.lean
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
f g : α → E
s t : Set α
μ ν : Measure α

⊢ IntegrableOn f univ μ ↔ Integrable f μ
-- Mathlib/Algebra/Order/Field/Power.lean
α : Type u_1
inst✝ : LinearOrderedField α
a b c d : α
k : ℤ
h : k + k ≠ 0

⊢ 0 < a ^ (k + k) ↔ a ≠ 0
-- Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean
K : Type u
V : Type v
inst✝⁵ : DivisionRing K
inst✝⁴ : AddCommGroup V
inst✝³ : Module K V
V₂ : Type v'
inst✝² : AddCommGroup V₂
inst✝¹ : Module K V₂
inst✝ : FiniteDimensional K V
S : Submodule K V
h : finrank K ↥S = finrank K V

⊢ S = ⊤
-- Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean
K : Type u
V : Type v
inst✝⁵ : DivisionRing K
inst✝⁴ : AddCommGroup V
inst✝³ : Module K V
V₂ : Type v'
inst✝² : AddCommGroup V₂
inst✝¹ : Module K V₂
inst✝ : FiniteDimensional K V
S : Submodule K V
h : finrank K ↥S = finrank K V
this : IsNoetherian K V

⊢ S = ⊤
-- Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean
K : Type u
V : Type v
inst✝⁵ : DivisionRing K
inst✝⁴ : AddCommGroup V
inst✝³ : Module K V
V₂ : Type v'
inst✝² : AddCommGroup V₂
inst✝¹ : Module K V₂
inst✝ : FiniteDimensional K V
S : Submodule K V
h : finrank K ↥S = finrank K V
this : IsNoetherian K V
bS : Basis (↑(Basis.ofVectorSpaceIndex K ↥S)) K ↥S := Basis.ofVectorSpace K ↥S
bS_eq : bS = Basis.ofVectorSpace K ↥S

⊢ S = ⊤
-- Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean
K : Type u
V : Type v
inst✝⁵ : DivisionRing K
inst✝⁴ : AddCommGroup V
inst✝³ : Module K V
V₂ : Type v'
inst✝² : AddCommGroup V₂
inst✝¹ : Module K V₂
inst✝ : FiniteDimensional K V
S : Submodule K V
h : finrank K ↥S = finrank K V
this✝ : IsNoetherian K V
bS : Basis (↑(Basis.ofVectorSpaceIndex K ↥S)) K ↥S := Basis.ofVectorSpace K ↥S
bS_eq : bS = Basis.ofVectorSpace K ↥S
this : LinearIndependent (ι := { x // x ∈ Subtype.val '' Basis.ofVectorSpaceIndex K ↥S }) K Subtype.val

⊢ S = ⊤
-- Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean
K : Type u
V : Type v
inst✝⁵ : DivisionRing K
inst✝⁴ : AddCommGroup V
inst✝³ : Module K V
V₂ : Type v'
inst✝² : AddCommGroup V₂
inst✝¹ : Module K V₂
inst✝ : FiniteDimensional K V
S : Submodule K V
h : finrank K ↥S = finrank K V
this✝ : IsNoetherian K V
bS : Basis (↑(Basis.ofVectorSpaceIndex K ↥S)) K ↥S := Basis.ofVectorSpace K ↥S
bS_eq : bS = Basis.ofVectorSpace K ↥S
this : LinearIndependent (ι := { x // x ∈ Subtype.val '' Basis.ofVectorSpaceIndex K ↥S }) K Subtype.val
b : Basis (↑(this.extend ⋯)) K V := Basis.extend this
b_eq : b = Basis.extend this

⊢ S = ⊤
-- Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean
K : Type u
V : Type v
inst✝⁵ : DivisionRing K
inst✝⁴ : AddCommGroup V
inst✝³ : Module K V
V₂ : Type v'
inst✝² : AddCommGroup V₂
inst✝¹ : Module K V₂
inst✝ : FiniteDimensional K V
S : Submodule K V
h : finrank K ↥S = finrank K V
this✝ : IsNoetherian K V
bS : Basis (↑(Basis.ofVectorSpaceIndex K ↥S)) K ↥S := Basis.ofVectorSpace K ↥S
bS_eq : bS = Basis.ofVectorSpace K ↥S
this : LinearIndependent (ι := { x // x ∈ Subtype.val '' Basis.ofVectorSpaceIndex K ↥S }) K Subtype.val
b : Basis (↑(this.extend ⋯)) K V := Basis.extend this
b_eq : b = Basis.extend this
i1 : Fintype ↑(this.extend ⋯) := ⋯.fintype

⊢ S = ⊤
-- Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean
K : Type u
V : Type v
inst✝⁵ : DivisionRing K
inst✝⁴ : AddCommGroup V
inst✝³ : Module K V
V₂ : Type v'
inst✝² : AddCommGroup V₂
inst✝¹ : Module K V₂
inst✝ : FiniteDimensional K V
S : Submodule K V
h : finrank K ↥S = finrank K V
this✝ : IsNoetherian K V
bS : Basis (↑(Basis.ofVectorSpaceIndex K ↥S)) K ↥S := Basis.ofVectorSpace K ↥S
bS_eq : bS = Basis.ofVectorSpace K ↥S
this : LinearIndependent (ι := { x // x ∈ Subtype.val '' Basis.ofVectorSpaceIndex K ↥S }) K Subtype.val
b : Basis (↑(this.extend ⋯)) K V := Basis.extend this
b_eq : b = Basis.extend this
i1 : Fintype ↑(this.extend ⋯) := ⋯.fintype
i2 : Fintype ↑(Subtype.val '' Basis.ofVectorSpaceIndex K ↥S) := ⋯.fintype

⊢ S = ⊤
-- Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean
K : Type u
V : Type v
inst✝⁵ : DivisionRing K
inst✝⁴ : AddCommGroup V
inst✝³ : Module K V
V₂ : Type v'
inst✝² : AddCommGroup V₂
inst✝¹ : Module K V₂
inst✝ : FiniteDimensional K V
S : Submodule K V
h : finrank K ↥S = finrank K V
this✝ : IsNoetherian K V
bS : Basis (↑(Basis.ofVectorSpaceIndex K ↥S)) K ↥S := Basis.ofVectorSpace K ↥S
bS_eq : bS = Basis.ofVectorSpace K ↥S
this : LinearIndependent (ι := { x // x ∈ Subtype.val '' Basis.ofVectorSpaceIndex K ↥S }) K Subtype.val
b : Basis (↑(this.extend ⋯)) K V := Basis.extend this
b_eq : b = Basis.extend this
i1 : Fintype ↑(this.extend ⋯) := ⋯.fintype
i2 : Fintype ↑(Subtype.val '' Basis.ofVectorSpaceIndex K ↥S) := ⋯.fintype
i3 : Fintype ↑(Basis.ofVectorSpaceIndex K ↥S) := ⋯.fintype

⊢ S = ⊤
-- Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean
K : Type u
V : Type v
inst✝⁵ : DivisionRing K
inst✝⁴ : AddCommGroup V
inst✝³ : Module K V
V₂ : Type v'
inst✝² : AddCommGroup V₂
inst✝¹ : Module K V₂
inst✝ : FiniteDimensional K V
S : Submodule K V
h : finrank K ↥S = finrank K V
this✝¹ : IsNoetherian K V
bS : Basis (↑(Basis.ofVectorSpaceIndex K ↥S)) K ↥S := Basis.ofVectorSpace K ↥S
bS_eq : bS = Basis.ofVectorSpace K ↥S
this✝ : LinearIndependent (ι := { x // x ∈ Subtype.val '' Basis.ofVectorSpaceIndex K ↥S }) K Subtype.val
b : Basis (↑(this✝.extend ⋯)) K V := Basis.extend this✝
b_eq : b = Basis.extend this✝
i1 : Fintype ↑(this✝.extend ⋯) := ⋯.fintype
i2 : Fintype ↑(Subtype.val '' Basis.ofVectorSpaceIndex K ↥S) := ⋯.fintype
i3 : Fintype ↑(Basis.ofVectorSpaceIndex K ↥S) := ⋯.fintype
this : Subtype.val '' Basis.ofVectorSpaceIndex K ↥S = this✝.extend ⋯

⊢ S = ⊤
-- Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean
K : Type u
V : Type v
inst✝⁵ : DivisionRing K
inst✝⁴ : AddCommGroup V
inst✝³ : Module K V
V₂ : Type v'
inst✝² : AddCommGroup V₂
inst✝¹ : Module K V₂
inst✝ : FiniteDimensional K V
S : Submodule K V
h : finrank K ↥S = finrank K V
this✝¹ : IsNoetherian K V
bS : Basis (↑(Basis.ofVectorSpaceIndex K ↥S)) K ↥S := Basis.ofVectorSpace K ↥S
bS_eq : bS = Basis.ofVectorSpace K ↥S
this✝ : LinearIndependent (ι := { x // x ∈ Subtype.val '' Basis.ofVectorSpaceIndex K ↥S }) K Subtype.val
b : Basis (↑(this✝.extend ⋯)) K V := Basis.extend this✝
b_eq : b = Basis.extend this✝
i1 : Fintype ↑(this✝.extend ⋯) := ⋯.fintype
i2 : Fintype ↑(Subtype.val '' Basis.ofVectorSpaceIndex K ↥S) := ⋯.fintype
i3 : Fintype ↑(Basis.ofVectorSpaceIndex K ↥S) := ⋯.fintype
this : Subtype.val '' Basis.ofVectorSpaceIndex K ↥S = this✝.extend ⋯

⊢ S = Submodule.map S.subtype (span K (Basis.ofVectorSpaceIndex K ↥S))
-- Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean
K : Type u
V : Type v
inst✝⁵ : DivisionRing K
inst✝⁴ : AddCommGroup V
inst✝³ : Module K V
V₂ : Type v'
inst✝² : AddCommGroup V₂
inst✝¹ : Module K V₂
inst✝ : FiniteDimensional K V
S : Submodule K V
h : finrank K ↥S = finrank K V
this✝² : IsNoetherian K V
bS : Basis (↑(Basis.ofVectorSpaceIndex K ↥S)) K ↥S := Basis.ofVectorSpace K ↥S
bS_eq : bS = Basis.ofVectorSpace K ↥S
this✝¹ : LinearIndependent (ι := { x // x ∈ Subtype.val '' Basis.ofVectorSpaceIndex K ↥S }) K Subtype.val
b : Basis (↑(this✝¹.extend ⋯)) K V := Basis.extend this✝¹
b_eq : b = Basis.extend this✝¹
i1 : Fintype ↑(this✝¹.extend ⋯) := ⋯.fintype
i2 : Fintype ↑(Subtype.val '' Basis.ofVectorSpaceIndex K ↥S) := ⋯.fintype
i3 : Fintype ↑(Basis.ofVectorSpaceIndex K ↥S) := ⋯.fintype
this✝ : Subtype.val '' Basis.ofVectorSpaceIndex K ↥S = this✝¹.extend ⋯
this : span K (Set.range ⇑bS) = ⊤

⊢ S = Submodule.map S.subtype (span K (Basis.ofVectorSpaceIndex K ↥S))
-- Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean
K : Type u
V : Type v
inst✝⁵ : DivisionRing K
inst✝⁴ : AddCommGroup V
inst✝³ : Module K V
V₂ : Type v'
inst✝² : AddCommGroup V₂
inst✝¹ : Module K V₂
inst✝ : FiniteDimensional K V
S : Submodule K V
h : finrank K ↥S = finrank K V
this✝² : IsNoetherian K V
bS : Basis (↑(Basis.ofVectorSpaceIndex K ↥S)) K ↥S := Basis.ofVectorSpace K ↥S
bS_eq : bS = Basis.ofVectorSpace K ↥S
this✝¹ : LinearIndependent (ι := { x // x ∈ Subtype.val '' Basis.ofVectorSpaceIndex K ↥S }) K Subtype.val
b : Basis (↑(this✝¹.extend ⋯)) K V := Basis.extend this✝¹
b_eq : b = Basis.extend this✝¹
i1 : Fintype ↑(this✝¹.extend ⋯) := ⋯.fintype
i2 : Fintype ↑(Subtype.val '' Basis.ofVectorSpaceIndex K ↥S) := ⋯.fintype
i3 : Fintype ↑(Basis.ofVectorSpaceIndex K ↥S) := ⋯.fintype
this✝ : Subtype.val '' Basis.ofVectorSpaceIndex K ↥S = this✝¹.extend ⋯
this : span K (Basis.ofVectorSpaceIndex K ↥S) = ⊤

⊢ S = Submodule.map S.subtype (span K (Basis.ofVectorSpaceIndex K ↥S))
-- Mathlib/Analysis/InnerProductSpace/Orientation.lean
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
ι : Type u_2
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
ne : Nonempty ι
e f : OrthonormalBasis ι ℝ E
x : Orientation ℝ E ι

⊢ (e.adjustToOrientation x).toBasis.orientation = x
-- Mathlib/Analysis/InnerProductSpace/Orientation.lean
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
ι : Type u_2
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
ne : Nonempty ι
e f : OrthonormalBasis ι ℝ E
x : Orientation ℝ E ι

⊢ (e.toBasis.adjustToOrientation x).orientation = x
-- Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝⁴ : Countable ι
m : MeasurableSpace α
μ ν : Measure α
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace γ
f✝ g✝ : α → β
E : Type u_5
inst✝¹ : TopologicalSpace E
inst✝ : MetrizableSpace E
f g : α → E
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ

⊢ NullMeasurableSet {x | f x = g x} μ
-- Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝⁴ : Countable ι
m : MeasurableSpace α
μ ν : Measure α
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace γ
f✝ g✝ : α → β
E : Type u_5
inst✝¹ : TopologicalSpace E
inst✝ : MetrizableSpace E
f g : α → E
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ

⊢ {x | AEStronglyMeasurable.mk f hf x = AEStronglyMeasurable.mk g hg x} =ᶠ[ae μ] {x | f x = g x}
-- Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝⁴ : Countable ι
m : MeasurableSpace α
μ ν : Measure α
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace γ
f✝ g✝ : α → β
E : Type u_5
inst✝¹ : TopologicalSpace E
inst✝ : MetrizableSpace E
f g : α → E
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
x : α
hfx : f x = AEStronglyMeasurable.mk f hf x
hgx : g x = AEStronglyMeasurable.mk g hg x

⊢ {x | AEStronglyMeasurable.mk f hf x = AEStronglyMeasurable.mk g hg x} x = {x | f x = g x} x
-- Mathlib/MeasureTheory/Group/Measure.lean
𝕜 : Type u_1
G : Type u_2
H : Type u_3
inst✝⁶ : MeasurableSpace G
inst✝⁵ : MeasurableSpace H
inst✝⁴ : Group G
inst✝³ : TopologicalSpace G
μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : TopologicalGroup G
inst✝ : BorelSpace G
g : G

⊢ μ {g} = μ {1}
-- Mathlib/MeasureTheory/Group/Measure.lean
𝕜 : Type u_1
G : Type u_2
H : Type u_3
inst✝⁶ : MeasurableSpace G
inst✝⁵ : MeasurableSpace H
inst✝⁴ : Group G
inst✝³ : TopologicalSpace G
μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : TopologicalGroup G
inst✝ : BorelSpace G
g : G

⊢ {g} = (fun h => g⁻¹ * h) ⁻¹' {1}