ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Data/Int/SuccPred.lean
a b : ℕ

⊢ ↑a ⋖ ↑b ↔ a ⋖ b
-- Mathlib/Data/Int/SuccPred.lean
a b : ℕ

⊢ ↑a + 1 = ↑b ↔ a + 1 = b
-- Mathlib/Analysis/SpecialFunctions/PolarCoord.lean
r θ : ℝ
hr : r ∈ Ioi 0
hθ : θ ∈ Ioo (-π) π

⊢ (fun p => (p.1 * cos p.2, p.1 * sin p.2)) (r, θ) ∈ {q | 0 < q.1} ∪ {q | q.2 ≠ 0}
-- Mathlib/Analysis/SpecialFunctions/PolarCoord.lean
r : ℝ
hr : r ∈ Ioi 0
hθ : 0 ∈ Ioo (-π) π

⊢ (fun p => (p.1 * cos p.2, p.1 * sin p.2)) (r, 0) ∈ {q | 0 < q.1} ∪ {q | q.2 ≠ 0}
-- Mathlib/Analysis/SpecialFunctions/PolarCoord.lean
r θ : ℝ
hr : r ∈ Ioi 0
hθ : θ ∈ Ioo (-π) π
h'θ : θ ≠ 0

⊢ (fun p => (p.1 * cos p.2, p.1 * sin p.2)) (r, θ) ∈ {q | 0 < q.1} ∪ {q | q.2 ≠ 0}
-- Mathlib/Analysis/SpecialFunctions/PolarCoord.lean
x y : ℝ
hxy : (x, y) ∈ {q | 0 < q.1} ∪ {q | q.2 ≠ 0}

⊢ (fun q => (√(q.1 ^ 2 + q.2 ^ 2), (Complex.equivRealProd.symm q).arg)) (x, y) ∈ Ioi 0 ×ˢ Ioo (-π) π
-- Mathlib/Analysis/SpecialFunctions/PolarCoord.lean
x y : ℝ
hxy : (x, y) ∈ {q | 0 < q.1} ∪ {q | q.2 ≠ 0}

⊢ 0 < x ^ 2 + y ^ 2 ∧ (0 ≤ (Complex.equivRealProd.symm (x, y)).re ∨ (Complex.equivRealProd.symm (x, y)).im ≠ 0)
-- Mathlib/Analysis/SpecialFunctions/PolarCoord.lean
r θ : ℝ
hr : r ∈ Ioi 0
hθ : θ ∈ Ioo (-π) π

⊢ (fun q => (√(q.1 ^ 2 + q.2 ^ 2), (Complex.equivRealProd.symm q).arg)) ((fun p => (p.1 * cos p.2, p.1 * sin p.2)) (r, θ)) = (r, θ)
-- Mathlib/Analysis/SpecialFunctions/PolarCoord.lean
r θ : ℝ
hr : r ∈ Ioi 0
hθ : θ ∈ Ioo (-π) π

⊢ √((r * cos θ) ^ 2 + (r * sin θ) ^ 2) = r ∧ (Complex.equivRealProd.symm (r * cos θ, r * sin θ)).arg = θ
-- Mathlib/Analysis/SpecialFunctions/PolarCoord.lean
x y : ℝ
a✝ : (x, y) ∈ {q | 0 < q.1} ∪ {q | q.2 ≠ 0}

⊢ (fun p => (p.1 * cos p.2, p.1 * sin p.2)) ((fun q => (√(q.1 ^ 2 + q.2 ^ 2), (Complex.equivRealProd.symm q).arg)) (x, y)) = (x, y)
-- Mathlib/Analysis/SpecialFunctions/PolarCoord.lean
x y : ℝ
a✝ : (x, y) ∈ {q | 0 < q.1} ∪ {q | q.2 ≠ 0}
A : √(x ^ 2 + y ^ 2) = Complex.abs (↑x + ↑y * Complex.I)

⊢ (fun p => (p.1 * cos p.2, p.1 * sin p.2)) ((fun q => (√(q.1 ^ 2 + q.2 ^ 2), (Complex.equivRealProd.symm q).arg)) (x, y)) = (x, y)
-- Mathlib/Analysis/SpecialFunctions/PolarCoord.lean
x y : ℝ
a✝ : (x, y) ∈ {q | 0 < q.1} ∪ {q | q.2 ≠ 0}
A : √(x ^ 2 + y ^ 2) = Complex.abs (↑x + ↑y * Complex.I)
Z : ↑(Complex.abs (↑x + ↑y * Complex.I)) * (Complex.cos ↑(↑x + ↑y * Complex.I).arg + Complex.sin ↑(↑x + ↑y * Complex.I).arg * Complex.I) = ↑x + ↑y * Complex.I

⊢ (fun p => (p.1 * cos p.2, p.1 * sin p.2)) ((fun q => (√(q.1 ^ 2 + q.2 ^ 2), (Complex.equivRealProd.symm q).arg)) (x, y)) = (x, y)
-- Mathlib/Analysis/SpecialFunctions/PolarCoord.lean
x y : ℝ
a✝ : (x, y) ∈ {q | 0 < q.1} ∪ {q | q.2 ≠ 0}
A : √(x ^ 2 + y ^ 2) = Complex.abs (↑x + ↑y * Complex.I)
Z : ↑(Complex.abs (↑x + ↑y * Complex.I) * cos (↑x + ↑y * Complex.I).arg) + ↑(Complex.abs (↑x + ↑y * Complex.I) * sin (↑x + ↑y * Complex.I).arg) * Complex.I = ↑x + ↑y * Complex.I

⊢ (fun p => (p.1 * cos p.2, p.1 * sin p.2)) ((fun q => (√(q.1 ^ 2 + q.2 ^ 2), (Complex.equivRealProd.symm q).arg)) (x, y)) = (x, y)
-- Mathlib/Analysis/SpecialFunctions/PolarCoord.lean

⊢ ContinuousOn ↑{ toFun := fun q => (√(q.1 ^ 2 + q.2 ^ 2), (Complex.equivRealProd.symm q).arg), invFun := fun p => (p.1 * cos p.2, p.1 * sin p.2), source := {q | 0 < q.1} ∪ {q | q.2 ≠ 0}, target := Ioi 0 ×ˢ Ioo (-π) π, map_source' := ⋯, map_target' := ⋯, left_inv' := ⋯, right_inv' := ⋯ } { toFun := fun q => (√(q.1 ^ 2 + q.2 ^ 2), (Complex.equivRealProd.symm q).arg), invFun := fun p => (p.1 * cos p.2, p.1 * sin p.2), source := {q | 0 < q.1} ∪ {q | q.2 ≠ 0}, target := Ioi 0 ×ˢ Ioo (-π) π, map_source' := ⋯, map_target' := ⋯, left_inv' := ⋯, right_inv' := ⋯ }.source
-- Mathlib/Analysis/SpecialFunctions/PolarCoord.lean

⊢ ContinuousOn (fun x => (Complex.equivRealProd.symm x).arg) { toFun := fun q => (√(q.1 ^ 2 + q.2 ^ 2), (Complex.equivRealProd.symm q).arg), invFun := fun p => (p.1 * cos p.2, p.1 * sin p.2), source := {q | 0 < q.1} ∪ {q | q.2 ≠ 0}, target := Ioi 0 ×ˢ Ioo (-π) π, map_source' := ⋯, map_target' := ⋯, left_inv' := ⋯, right_inv' := ⋯ }.source
-- Mathlib/Analysis/SpecialFunctions/PolarCoord.lean
A : MapsTo (⇑Complex.equivRealProd.symm) ({q | 0 < q.1} ∪ {q | q.2 ≠ 0}) Complex.slitPlane

⊢ ContinuousOn (fun x => (Complex.equivRealProd.symm x).arg) { toFun := fun q => (√(q.1 ^ 2 + q.2 ^ 2), (Complex.equivRealProd.symm q).arg), invFun := fun p => (p.1 * cos p.2, p.1 * sin p.2), source := {q | 0 < q.1} ∪ {q | q.2 ≠ 0}, target := Ioi 0 ×ˢ Ioo (-π) π, map_source' := ⋯, map_target' := ⋯, left_inv' := ⋯, right_inv' := ⋯ }.source
-- Mathlib/Analysis/SpecialFunctions/PolarCoord.lean
A : MapsTo (⇑Complex.equivRealProd.symm) ({q | 0 < q.1} ∪ {q | q.2 ≠ 0}) Complex.slitPlane
z : ℂ
hz : z ∈ Complex.slitPlane

⊢ ContinuousWithinAt Complex.arg Complex.slitPlane z
-- Mathlib/Analysis/SpecialFunctions/PolarCoord.lean
A : MapsTo (⇑Complex.equivRealProd.symm) ({q | 0 < q.1} ∪ {q | q.2 ≠ 0}) Complex.slitPlane

⊢ ContinuousOn ⇑Complex.equivRealProd.symm { toFun := fun q => (√(q.1 ^ 2 + q.2 ^ 2), (Complex.equivRealProd.symm q).arg), invFun := fun p => (p.1 * cos p.2, p.1 * sin p.2), source := {q | 0 < q.1} ∪ {q | q.2 ≠ 0}, target := Ioi 0 ×ˢ Ioo (-π) π, map_source' := ⋯, map_target' := ⋯, left_inv' := ⋯, right_inv' := ⋯ }.source
-- Mathlib/Analysis/Convex/Normed.lean
ι : Type u_1
E : Type u_2
P : Type u_3
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor E P
s✝ t s : Set E

⊢ EMetric.diam ((convexHull ℝ) s) = EMetric.diam s
-- Mathlib/Analysis/Convex/Normed.lean
ι : Type u_1
E : Type u_2
P : Type u_3
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor E P
s✝ t s : Set E
x : E
hx : x ∈ (convexHull ℝ) s
y : E
hy : y ∈ (convexHull ℝ) s

⊢ edist x y ≤ EMetric.diam s
-- Mathlib/Analysis/Convex/Normed.lean
ι : Type u_1
E : Type u_2
P : Type u_3
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor E P
s✝ t s : Set E
x : E
hx : x ∈ (convexHull ℝ) s
y : E
hy : y ∈ (convexHull ℝ) s
x' : E
hx' : x' ∈ s
y' : E
hy' : y' ∈ s
H : dist x y ≤ dist x' y'

⊢ edist x y ≤ EMetric.diam s
-- Mathlib/Analysis/Convex/Normed.lean
ι : Type u_1
E : Type u_2
P : Type u_3
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor E P
s✝ t s : Set E
x : E
hx : x ∈ (convexHull ℝ) s
y : E
hy : y ∈ (convexHull ℝ) s
x' : E
hx' : x' ∈ s
y' : E
hy' : y' ∈ s
H : dist x y ≤ dist x' y'

⊢ ENNReal.ofReal (dist x y) ≤ EMetric.diam s
-- Mathlib/Analysis/Convex/Normed.lean
ι : Type u_1
E : Type u_2
P : Type u_3
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor E P
s✝ t s : Set E
x : E
hx : x ∈ (convexHull ℝ) s
y : E
hy : y ∈ (convexHull ℝ) s
x' : E
hx' : x' ∈ s
y' : E
hy' : y' ∈ s
H : dist x y ≤ dist x' y'

⊢ ENNReal.ofReal (dist x' y') ≤ EMetric.diam s
-- Mathlib/Analysis/Convex/Normed.lean
ι : Type u_1
E : Type u_2
P : Type u_3
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor E P
s✝ t s : Set E
x : E
hx : x ∈ (convexHull ℝ) s
y : E
hy : y ∈ (convexHull ℝ) s
x' : E
hx' : x' ∈ s
y' : E
hy' : y' ∈ s
H : dist x y ≤ dist x' y'

⊢ edist x' y' ≤ EMetric.diam s
-- Mathlib/Algebra/Order/BigOperators/Group/Finset.lean
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G✝ : Type u_6
k : Type u_7
R : Type u_8
G : Type u_9
inst✝ : LinearOrderedAddCommGroup G
f : ι → G
s : Finset ι
hf : ∀ i ∈ s, 0 ≤ f i

⊢ |∑ i ∈ s, f i| = ∑ i ∈ s, f i
-- Mathlib/RingTheory/FractionalIdeal/Operations.lean
R : Type u_1
inst✝⁸ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁷ : CommRing P
inst✝⁶ : Algebra R P
loc : IsLocalization S P
K : Type u_3
K' : Type u_4
inst✝⁵ : Field K
inst✝⁴ : Field K'
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : Algebra R K'
inst✝ : IsFractionRing R K'
I✝ J : FractionalIdeal R⁰ K
h : K →ₐ[R] K'
I : Ideal R

⊢ ↑I = 1 ↔ I = 1
-- Mathlib/MeasureTheory/Function/L1Space.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝² : MeasurableSpace δ
inst✝¹ : NormedAddCommGroup β
inst✝ : NormedAddCommGroup γ
ι : Type u_5
s : Finset ι
f : ι → α → β
hf : ∀ i ∈ s, Integrable (f i) μ

⊢ Integrable (fun a => ∑ i ∈ s, f i a) μ
-- Mathlib/GroupTheory/Perm/Sign.lean
α : Type u
inst✝ : DecidableEq α
β : Type v
n : ℕ
f : Perm (Fin n)
a₁ a₂ : Fin n
ha : ⟨a₁, a₂⟩ ∈ ↑(finPairsLT n)
b₁ b₂ : Fin n
hb : ⟨b₁, b₂⟩ ∈ ↑(finPairsLT n)
h : (if f a₂ < f a₁ then ⟨f a₁, f a₂⟩ else ⟨f a₂, f a₁⟩) = if f b₂ < f b₁ then ⟨f b₁, f b₂⟩ else ⟨f b₂, f b₁⟩

⊢ ⟨a₁, a₂⟩ = ⟨b₁, b₂⟩
-- Mathlib/GroupTheory/Perm/Sign.lean
α : Type u
inst✝ : DecidableEq α
β : Type v
n : ℕ
f : Perm (Fin n)
a₁ a₂ : Fin n
ha : ⟨a₁, a₂⟩.snd < ⟨a₁, a₂⟩.fst
b₁ b₂ : Fin n
hb : ⟨b₁, b₂⟩.snd < ⟨b₁, b₂⟩.fst
h : (if f a₂ < f a₁ then ⟨f a₁, f a₂⟩ else ⟨f a₂, f a₁⟩) = if f b₂ < f b₁ then ⟨f b₁, f b₂⟩ else ⟨f b₂, f b₁⟩

⊢ ⟨a₁, a₂⟩ = ⟨b₁, b₂⟩
-- Mathlib/GroupTheory/Perm/Sign.lean
α : Type u
inst✝ : DecidableEq α
β : Type v
n : ℕ
f : Perm (Fin n)
a₁ a₂ : Fin n
ha : ⟨a₁, a₂⟩.snd < ⟨a₁, a₂⟩.fst
b₁ b₂ : Fin n
hb : ⟨b₁, b₂⟩.snd < ⟨b₁, b₂⟩.fst
h : (if f a₂ < f a₁ then ⟨f a₁, f a₂⟩ else ⟨f a₂, f a₁⟩) = if f b₂ < f b₁ then ⟨f b₁, f b₂⟩ else ⟨f b₂, f b₁⟩
this : ¬b₁ < b₂

⊢ ⟨a₁, a₂⟩ = ⟨b₁, b₂⟩
-- Mathlib/GroupTheory/Perm/Sign.lean
α : Type u
inst✝ : DecidableEq α
β : Type v
n : ℕ
f : Perm (Fin n)
a₁ a₂ b₁ b₂ : Fin n
ha : b₁ < b₂
hb : b₂ < b₁
this : b₂ ≤ b₁
h✝¹ : f b₁ < f b₂
h✝ : f b₁ ≤ f b₂
h : a₁ = b₂ ∧ a₂ = b₁

⊢ b₂ = b₁ ∧ b₁ = b₂
-- Mathlib/GroupTheory/Perm/Sign.lean
α : Type u
inst✝ : DecidableEq α
β : Type v
n : ℕ
f : Perm (Fin n)
a₁ a₂ b₁ b₂ : Fin n
ha : b₁ < b₂
hb : b₂ < b₁
this : b₂ ≤ b₁
h✝¹ : f b₂ ≤ f b₁
h✝ : f b₂ < f b₁
h : a₂ = b₁ ∧ a₁ = b₂

⊢ b₂ = b₁ ∧ b₁ = b₂
-- Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean

⊢ doublingGamma 1 = 1
-- .lake/packages/batteries/Batteries/Data/List/Lemmas.lean
α✝ : Type u_1
l₁ l₂ : List α✝
x✝ : l₁.reverse <:+ l₂.reverse
r : List α✝
e : r ++ l₁.reverse = l₂.reverse

⊢ l₁ ++ r.reverse = l₂
-- .lake/packages/batteries/Batteries/Data/List/Lemmas.lean
α✝ : Type u_1
l₁ l₂ : List α✝
x✝ : l₁ <+: l₂
r : List α✝
e : l₁ ++ r = l₂

⊢ r.reverse ++ l₁.reverse = l₂.reverse
-- Mathlib/RepresentationTheory/Action/Monoidal.lean
V : Type (u + 1)
inst✝¹ : LargeCategory V
G✝ : MonCat
G : Type u
inst✝ : Group G
X : Action (Type u) (MonCat.of G)
g : G

⊢ ((leftRegular G ⊗ X).ρ g ≫ fun g => (g.1, X.ρ g.1⁻¹ g.2)) = (fun g => (g.1, X.ρ g.1⁻¹ g.2)) ≫ (leftRegular G ⊗ { V := X.V, ρ := 1 }).ρ g
-- Mathlib/RepresentationTheory/Action/Monoidal.lean
V : Type (u + 1)
inst✝¹ : LargeCategory V
G✝ : MonCat
G : Type u
inst✝ : Group G
X : Action (Type u) (MonCat.of G)
g x₁ : G
x₂ : X.V

⊢ ((leftRegular G ⊗ X).ρ g ≫ fun g => (g.1, X.ρ g.1⁻¹ g.2)) (x₁, x₂) = ((fun g => (g.1, X.ρ g.1⁻¹ g.2)) ≫ (leftRegular G ⊗ { V := X.V, ρ := 1 }).ρ g) (x₁, x₂)
-- Mathlib/RepresentationTheory/Action/Monoidal.lean
V : Type (u + 1)
inst✝¹ : LargeCategory V
G✝ : MonCat
G : Type u
inst✝ : Group G
X : Action (Type u) (MonCat.of G)
g x₁ : G
x₂ : X.V

⊢ (((leftRegular G ⊗ X).ρ g ≫ fun g => (g.1, X.ρ g.1⁻¹ g.2)) (x₁, x₂)).2 = (((fun g => (g.1, X.ρ g.1⁻¹ g.2)) ≫ (leftRegular G ⊗ { V := X.V, ρ := 1 }).ρ g) (x₁, x₂)).2
-- Mathlib/RepresentationTheory/Action/Monoidal.lean
V : Type (u + 1)
inst✝¹ : LargeCategory V
G✝ : MonCat
G : Type u
inst✝ : Group G
X : Action (Type u) (MonCat.of G)
g x₁ : G
x₂ : X.V

⊢ (X.ρ (g * x₁)⁻¹ * X.ρ g) x₂ = X.ρ (x₁, x₂).1⁻¹ (x₁, x₂).2
-- Mathlib/RepresentationTheory/Action/Monoidal.lean
V : Type (u + 1)
inst✝¹ : LargeCategory V
G✝ : MonCat
G : Type u
inst✝ : Group G
X : Action (Type u) (MonCat.of G)
g : G

⊢ ((leftRegular G ⊗ { V := X.V, ρ := 1 }).ρ g ≫ fun g => (g.1, X.ρ g.1 g.2)) = (fun g => (g.1, X.ρ g.1 g.2)) ≫ (leftRegular G ⊗ X).ρ g
-- Mathlib/RepresentationTheory/Action/Monoidal.lean
V : Type (u + 1)
inst✝¹ : LargeCategory V
G✝ : MonCat
G : Type u
inst✝ : Group G
X : Action (Type u) (MonCat.of G)
g x₁ : G
x₂ : X.V

⊢ ((leftRegular G ⊗ { V := X.V, ρ := 1 }).ρ g ≫ fun g => (g.1, X.ρ g.1 g.2)) (x₁, x₂) = ((fun g => (g.1, X.ρ g.1 g.2)) ≫ (leftRegular G ⊗ X).ρ g) (x₁, x₂)
-- Mathlib/RepresentationTheory/Action/Monoidal.lean
V : Type (u + 1)
inst✝¹ : LargeCategory V
G✝ : MonCat
G : Type u
inst✝ : Group G
X : Action (Type u) (MonCat.of G)
g x₁ : G
x₂ : X.V

⊢ (((leftRegular G ⊗ { V := X.V, ρ := 1 }).ρ g ≫ fun g => (g.1, X.ρ g.1 g.2)) (x₁, x₂)).2 = (((fun g => (g.1, X.ρ g.1 g.2)) ≫ (leftRegular G ⊗ X).ρ g) (x₁, x₂)).2
-- Mathlib/RepresentationTheory/Action/Monoidal.lean
V : Type (u + 1)
inst✝¹ : LargeCategory V
G✝ : MonCat
G : Type u
inst✝ : Group G
X : Action (Type u) (MonCat.of G)
g x₁ : G
x₂ : X.V

⊢ ((((leftRegular G).ρ g ⊗ { V := X.V, ρ := 1 }.ρ g) ≫ fun g => (g.1, X.ρ g.1 g.2)) (x₁, x₂)).2 = (((fun g => (g.1, X.ρ g.1 g.2)) ≫ ((leftRegular G).ρ g ⊗ X.ρ g)) (x₁, x₂)).2
-- Mathlib/RepresentationTheory/Action/Monoidal.lean
V : Type (u + 1)
inst✝¹ : LargeCategory V
G✝ : MonCat
G : Type u
inst✝ : Group G
X : Action (Type u) (MonCat.of G)
g x₁ : G
x₂ : X.V

⊢ X.ρ ((leftRegular G).ρ g x₁) x₂ = X.ρ g (X.ρ x₁ x₂)
-- Mathlib/RepresentationTheory/Action/Monoidal.lean
V : Type (u + 1)
inst✝¹ : LargeCategory V
G✝ : MonCat
G : Type u
inst✝ : Group G
X : Action (Type u) (MonCat.of G)
g x₁ : G
x₂ : X.V

⊢ X.ρ (g * x₁) x₂ = X.ρ g (X.ρ x₁ x₂)
-- Mathlib/RepresentationTheory/Action/Monoidal.lean
V : Type (u + 1)
inst✝¹ : LargeCategory V
G✝ : MonCat
G : Type u
inst✝ : Group G
X : Action (Type u) (MonCat.of G)
g x₁ : G
x₂ : X.V

⊢ (X.ρ g * X.ρ x₁) x₂ = X.ρ g (X.ρ x₁ x₂)
-- Mathlib/RepresentationTheory/Action/Monoidal.lean
V : Type (u + 1)
inst✝¹ : LargeCategory V
G✝ : MonCat
G : Type u
inst✝ : Group G
X : Action (Type u) (MonCat.of G)

⊢ ({ hom := fun g => (g.1, X.ρ g.1⁻¹ g.2), comm := ⋯ } ≫ { hom := fun g => (g.1, X.ρ g.1 g.2), comm := ⋯ }).hom = (𝟙 (leftRegular G ⊗ X)).hom
-- Mathlib/RepresentationTheory/Action/Monoidal.lean
V : Type (u + 1)
inst✝¹ : LargeCategory V
G✝ : MonCat
G : Type u
inst✝ : Group G
X : Action (Type u) (MonCat.of G)
x : (leftRegular G ⊗ X).V

⊢ ({ hom := fun g => (g.1, X.ρ g.1⁻¹ g.2), comm := ⋯ } ≫ { hom := fun g => (g.1, X.ρ g.1 g.2), comm := ⋯ }).hom x = (𝟙 (leftRegular G ⊗ X)).hom x
-- Mathlib/RepresentationTheory/Action/Monoidal.lean
V : Type (u + 1)
inst✝¹ : LargeCategory V
G✝ : MonCat
G : Type u
inst✝ : Group G
X : Action (Type u) (MonCat.of G)
x : (leftRegular G ⊗ X).V

⊢ (({ hom := fun g => (g.1, X.ρ g.1⁻¹ g.2), comm := ⋯ } ≫ { hom := fun g => (g.1, X.ρ g.1 g.2), comm := ⋯ }).hom x).2 = ((𝟙 (leftRegular G ⊗ X)).hom x).2
-- Mathlib/RepresentationTheory/Action/Monoidal.lean
V : Type (u + 1)
inst✝¹ : LargeCategory V
G✝ : MonCat
G : Type u
inst✝ : Group G
X : Action (Type u) (MonCat.of G)
x : (leftRegular G ⊗ X).V

⊢ (X.ρ x.1 * X.ρ x.1⁻¹) x.2 = x.2
-- Mathlib/RepresentationTheory/Action/Monoidal.lean
V : Type (u + 1)
inst✝¹ : LargeCategory V
G✝ : MonCat
G : Type u
inst✝ : Group G
X : Action (Type u) (MonCat.of G)

⊢ ({ hom := fun g => (g.1, X.ρ g.1 g.2), comm := ⋯ } ≫ { hom := fun g => (g.1, X.ρ g.1⁻¹ g.2), comm := ⋯ }).hom = (𝟙 (leftRegular G ⊗ { V := X.V, ρ := 1 })).hom
-- Mathlib/RepresentationTheory/Action/Monoidal.lean
V : Type (u + 1)
inst✝¹ : LargeCategory V
G✝ : MonCat
G : Type u
inst✝ : Group G
X : Action (Type u) (MonCat.of G)
x : (leftRegular G ⊗ { V := X.V, ρ := 1 }).V

⊢ ({ hom := fun g => (g.1, X.ρ g.1 g.2), comm := ⋯ } ≫ { hom := fun g => (g.1, X.ρ g.1⁻¹ g.2), comm := ⋯ }).hom x = (𝟙 (leftRegular G ⊗ { V := X.V, ρ := 1 })).hom x
-- Mathlib/RepresentationTheory/Action/Monoidal.lean
V : Type (u + 1)
inst✝¹ : LargeCategory V
G✝ : MonCat
G : Type u
inst✝ : Group G
X : Action (Type u) (MonCat.of G)
x : (leftRegular G ⊗ { V := X.V, ρ := 1 }).V

⊢ (({ hom := fun g => (g.1, X.ρ g.1 g.2), comm := ⋯ } ≫ { hom := fun g => (g.1, X.ρ g.1⁻¹ g.2), comm := ⋯ }).hom x).2 = ((𝟙 (leftRegular G ⊗ { V := X.V, ρ := 1 })).hom x).2
-- Mathlib/RepresentationTheory/Action/Monoidal.lean
V : Type (u + 1)
inst✝¹ : LargeCategory V
G✝ : MonCat
G : Type u
inst✝ : Group G
X : Action (Type u) (MonCat.of G)
x : (leftRegular G ⊗ { V := X.V, ρ := 1 }).V

⊢ (X.ρ x.1⁻¹ * X.ρ x.1) x.2 = x.2
-- Mathlib/Analysis/Convex/Deriv.lean
S : Set ℝ
f : ℝ → ℝ
x y f' : ℝ
hfc : StrictConcaveOn ℝ S f
hfd : ∀ x ∈ S, DifferentiableAt ℝ f x

⊢ StrictAntiOn (deriv f) S
-- Mathlib/Algebra/Algebra/Subalgebra/Pointwise.lean
R : Type u_1
A : Type u_2
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
S : Subalgebra R A

⊢ toSubmodule S * toSubmodule S = toSubmodule S
-- Mathlib/Algebra/Algebra/Subalgebra/Pointwise.lean
R : Type u_1
A : Type u_2
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
S : Subalgebra R A

⊢ toSubmodule S * toSubmodule S ≤ toSubmodule S
-- Mathlib/Algebra/Algebra/Subalgebra/Pointwise.lean
R : Type u_1
A : Type u_2
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
S : Subalgebra R A

⊢ toSubmodule S ≤ toSubmodule S * toSubmodule S
-- Mathlib/Algebra/Order/Group/Abs.lean
α : Type u_1
inst✝ : LinearOrderedAddCommGroup α
a✝ b✝ c d a b : α

⊢ |a| = |a - b + b|
-- Mathlib/Topology/EMetricSpace/Basic.lean
α : Type u
β : Type v
X : Type u_1
inst✝ : PseudoEMetricSpace α
x y z : α
ε ε₁ ε₂ : ℝ≥0∞
s✝ t s : Set α
hs : ∀ ε > 0, ∃ t, t.Countable ∧ s ⊆ ⋃ x ∈ t, closedBall x ε

⊢ ∃ t ⊆ s, t.Countable ∧ s ⊆ closure t
-- Mathlib/Topology/EMetricSpace/Basic.lean
α : Type u
β : Type v
X : Type u_1
inst✝ : PseudoEMetricSpace α
x y z : α
ε ε₁ ε₂ : ℝ≥0∞
s t : Set α
hs : ∀ ε > 0, ∃ t, t.Countable ∧ ∅ ⊆ ⋃ x ∈ t, closedBall x ε

⊢ ∃ t ⊆ ∅, t.Countable ∧ ∅ ⊆ closure t
-- Mathlib/Topology/EMetricSpace/Basic.lean
α : Type u
β : Type v
X : Type u_1
inst✝ : PseudoEMetricSpace α
x y z : α
ε ε₁ ε₂ : ℝ≥0∞
s✝ t s : Set α
hs : ∀ ε > 0, ∃ t, t.Countable ∧ s ⊆ ⋃ x ∈ t, closedBall x ε
x₀ : α
hx₀ : x₀ ∈ s

⊢ ∃ t ⊆ s, t.Countable ∧ s ⊆ closure t
-- Mathlib/Topology/EMetricSpace/Basic.lean
α : Type u
β : Type v
X : Type u_1
inst✝ : PseudoEMetricSpace α
x y z : α
ε ε₁ ε₂ : ℝ≥0∞
s✝ t s : Set α
hs : ∀ ε > 0, ∃ t, t.Countable ∧ s ⊆ ⋃ x ∈ t, closedBall x ε
x₀ : α
hx₀ : x₀ ∈ s
T : ℕ → Set α
hTc : ∀ (n : ℕ), (T n).Countable
hsT : ∀ (n : ℕ), s ⊆ ⋃ x ∈ T n, closedBall x (↑n)⁻¹
f : ℝ≥0∞ → α → α
hfs : ∀ (r : ℝ≥0∞) (x : α), f r x ∈ s
hf : ∀ (r : ℝ≥0∞) (x : α), closedBall x r ∩ s ⊆ closedBall (f r x) (r * 2)

⊢ ∃ t ⊆ s, t.Countable ∧ s ⊆ closure t
-- Mathlib/Topology/EMetricSpace/Basic.lean
α : Type u
β : Type v
X : Type u_1
inst✝ : PseudoEMetricSpace α
x y z : α
ε ε₁ ε₂ : ℝ≥0∞
s✝ t s : Set α
hs : ∀ ε > 0, ∃ t, t.Countable ∧ s ⊆ ⋃ x ∈ t, closedBall x ε
x₀ : α
hx₀ : x₀ ∈ s
T : ℕ → Set α
hTc : ∀ (n : ℕ), (T n).Countable
hsT : ∀ (n : ℕ), s ⊆ ⋃ x ∈ T n, closedBall x (↑n)⁻¹
f : ℝ≥0∞ → α → α
hfs : ∀ (r : ℝ≥0∞) (x : α), f r x ∈ s
hf : ∀ (r : ℝ≥0∞) (x : α), closedBall x r ∩ s ⊆ closedBall (f r x) (r * 2)

⊢ s ⊆ closure (⋃ n, f (↑n)⁻¹ '' T n)
-- Mathlib/Topology/EMetricSpace/Basic.lean
α : Type u
β : Type v
X : Type u_1
inst✝ : PseudoEMetricSpace α
x✝ y z : α
ε✝ ε₁ ε₂ : ℝ≥0∞
s✝ t s : Set α
hs : ∀ ε > 0, ∃ t, t.Countable ∧ s ⊆ ⋃ x ∈ t, closedBall x ε
x₀ : α
hx₀ : x₀ ∈ s
T : ℕ → Set α
hTc : ∀ (n : ℕ), (T n).Countable
hsT : ∀ (n : ℕ), s ⊆ ⋃ x ∈ T n, closedBall x (↑n)⁻¹
f : ℝ≥0∞ → α → α
hfs : ∀ (r : ℝ≥0∞) (x : α), f r x ∈ s
hf : ∀ (r : ℝ≥0∞) (x : α), closedBall x r ∩ s ⊆ closedBall (f r x) (r * 2)
x : α
hx : x ∈ s
ε : ℝ≥0∞
ε0 : ε > 0

⊢ ∃ y ∈ ⋃ n, f (↑n)⁻¹ '' T n, edist x y < ε
-- Mathlib/Topology/EMetricSpace/Basic.lean
α : Type u
β : Type v
X : Type u_1
inst✝ : PseudoEMetricSpace α
x✝ y z : α
ε✝ ε₁ ε₂ : ℝ≥0∞
s✝ t s : Set α
hs : ∀ ε > 0, ∃ t, t.Countable ∧ s ⊆ ⋃ x ∈ t, closedBall x ε
x₀ : α
hx₀ : x₀ ∈ s
T : ℕ → Set α
hTc : ∀ (n : ℕ), (T n).Countable
hsT : ∀ (n : ℕ), s ⊆ ⋃ x ∈ T n, closedBall x (↑n)⁻¹
f : ℝ≥0∞ → α → α
hfs : ∀ (r : ℝ≥0∞) (x : α), f r x ∈ s
hf : ∀ (r : ℝ≥0∞) (x : α), closedBall x r ∩ s ⊆ closedBall (f r x) (r * 2)
x : α
hx : x ∈ s
ε : ℝ≥0∞
ε0 : ε > 0
n : ℕ
hn : (↑n)⁻¹ < ε / 2

⊢ ∃ y ∈ ⋃ n, f (↑n)⁻¹ '' T n, edist x y < ε
-- Mathlib/Topology/EMetricSpace/Basic.lean
α : Type u
β : Type v
X : Type u_1
inst✝ : PseudoEMetricSpace α
x✝ y✝ z : α
ε✝ ε₁ ε₂ : ℝ≥0∞
s✝ t s : Set α
hs : ∀ ε > 0, ∃ t, t.Countable ∧ s ⊆ ⋃ x ∈ t, closedBall x ε
x₀ : α
hx₀ : x₀ ∈ s
T : ℕ → Set α
hTc : ∀ (n : ℕ), (T n).Countable
hsT : ∀ (n : ℕ), s ⊆ ⋃ x ∈ T n, closedBall x (↑n)⁻¹
f : ℝ≥0∞ → α → α
hfs : ∀ (r : ℝ≥0∞) (x : α), f r x ∈ s
hf : ∀ (r : ℝ≥0∞) (x : α), closedBall x r ∩ s ⊆ closedBall (f r x) (r * 2)
x : α
hx : x ∈ s
ε : ℝ≥0∞
ε0 : ε > 0
n : ℕ
hn : (↑n)⁻¹ < ε / 2
y : α
hyn : y ∈ T n
hyx : x ∈ closedBall y (↑n)⁻¹

⊢ ∃ y ∈ ⋃ n, f (↑n)⁻¹ '' T n, edist x y < ε
-- Mathlib/Topology/EMetricSpace/Basic.lean
α : Type u
β : Type v
X : Type u_1
inst✝ : PseudoEMetricSpace α
x✝ y✝ z : α
ε✝ ε₁ ε₂ : ℝ≥0∞
s✝ t s : Set α
hs : ∀ ε > 0, ∃ t, t.Countable ∧ s ⊆ ⋃ x ∈ t, closedBall x ε
x₀ : α
hx₀ : x₀ ∈ s
T : ℕ → Set α
hTc : ∀ (n : ℕ), (T n).Countable
hsT : ∀ (n : ℕ), s ⊆ ⋃ x ∈ T n, closedBall x (↑n)⁻¹
f : ℝ≥0∞ → α → α
hfs : ∀ (r : ℝ≥0∞) (x : α), f r x ∈ s
hf : ∀ (r : ℝ≥0∞) (x : α), closedBall x r ∩ s ⊆ closedBall (f r x) (r * 2)
x : α
hx : x ∈ s
ε : ℝ≥0∞
ε0 : ε > 0
n : ℕ
hn : (↑n)⁻¹ < ε / 2
y : α
hyn : y ∈ T n
hyx : x ∈ closedBall y (↑n)⁻¹

⊢ edist x (f (↑n)⁻¹ y) < ε
-- Mathlib/MeasureTheory/Integral/SetIntegral.lean
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s t : Set X
μ ν : Measure X
l l' : Filter X
ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0

⊢ ∫ (x : X) in s ∪ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ
-- Mathlib/MeasureTheory/Integral/SetIntegral.lean
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s t : Set X
μ ν : Measure X
l l' : Filter X
ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0
ht : IntegrableOn f t μ

⊢ ∫ (x : X) in s ∪ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ
-- Mathlib/MeasureTheory/Integral/SetIntegral.lean
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s t : Set X
μ ν : Measure X
l l' : Filter X
ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0
ht : IntegrableOn f t μ
H : IntegrableOn f (s ∪ t) μ

⊢ ∫ (x : X) in s ∪ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ
-- Mathlib/MeasureTheory/Integral/SetIntegral.lean
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s t : Set X
μ ν : Measure X
l l' : Filter X
ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0
ht : IntegrableOn f t μ
H : IntegrableOn f (s ∪ t) μ
f' : X → E := AEStronglyMeasurable.mk f ⋯

⊢ ∫ (x : X) in s ∪ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ
-- Mathlib/NumberTheory/WellApproximable.lean
A : Type u_1
inst✝ : SeminormedCommGroup A
a : A
m n : ℕ
δ : ℝ
hn : 0 < n
han : orderOf a ^ 2 ∣ n

⊢ a • approxOrderOf A n δ = approxOrderOf A n δ
-- Mathlib/NumberTheory/WellApproximable.lean
A : Type u_1
inst✝ : SeminormedCommGroup A
a : A
m n : ℕ
δ : ℝ
hn : 0 < n
han : orderOf a ^ 2 ∣ n

⊢ ⋃ i, ⋃ (_ : orderOf i = n), ball (a * i) δ = ⋃ x, ⋃ (_ : orderOf x = n), ball x δ
-- Mathlib/NumberTheory/WellApproximable.lean
A : Type u_1
inst✝ : SeminormedCommGroup A
a : A
m n : ℕ
δ : ℝ
hn : 0 < n
han : ∀ {b : A}, orderOf b = n → orderOf (a * b) = n

⊢ ⋃ i, ⋃ (_ : orderOf i = n), ball (a * i) δ = ⋃ x, ⋃ (_ : orderOf x = n), ball x δ
-- Mathlib/NumberTheory/WellApproximable.lean
A : Type u_1
inst✝ : SeminormedCommGroup A
a : A
m n : ℕ
δ : ℝ
hn : 0 < n
han : ∀ {b : A}, orderOf b = n → orderOf (a * b) = n
f : ↑{b | orderOf b = n} → ↑{b | orderOf b = n} := fun b => ⟨a * ↑b, ⋯⟩

⊢ ⋃ i, ⋃ (_ : orderOf i = n), ball (a * i) δ = ⋃ x, ⋃ (_ : orderOf x = n), ball x δ
-- Mathlib/NumberTheory/WellApproximable.lean
A : Type u_1
inst✝ : SeminormedCommGroup A
a : A
m n : ℕ
δ : ℝ
hn : 0 < n
han : ∀ {b : A}, orderOf b = n → orderOf (a * b) = n
f : ↑{b | orderOf b = n} → ↑{b | orderOf b = n} := fun b => ⟨a * ↑b, ⋯⟩
hf : Surjective f

⊢ ⋃ i, ⋃ (_ : orderOf i = n), ball (a * i) δ = ⋃ x, ⋃ (_ : orderOf x = n), ball x δ