ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean
σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm : ↑k * esymm σ R k = (-1) ^ (k + 1) * (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 + ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides : (fun x => x - (-1) ^ (k + 1) * ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2) (↑k * esymm σ R k) = (fun x => x - (-1) ^ (k + 1) * ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2) ((-1) ^ (k + 1) * (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 + ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2))
\u22a2 psum σ R k = (-1) ^ (k + 1) * ↑k * esymm σ R k - ∑ x ∈ filter (fun a => a.1 < k ∧ 0 < a.1) (antidiagonal k), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2
-- Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean
σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm : ↑k * esymm σ R k = (-1) ^ (k + 1) * (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 + ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides : ↑k * esymm σ R k - (-1) ^ (k + 1) * ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 = (-1) ^ (k + 1) * ∑ a ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
\u22a2 psum σ R k = (-1) ^ (k + 1) * ↑k * esymm σ R k - ∑ x ∈ filter (fun a => a.1 < k ∧ 0 < a.1) (antidiagonal k), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2
-- Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean
σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm : ↑k * esymm σ R k = (-1) ^ (k + 1) * (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 + ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides✝ : ↑k * esymm σ R k - (-1) ^ (k + 1) * ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 = (-1) ^ (k + 1) * ∑ a ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
sub_both_sides : (fun x => (-1) ^ (k + 1) * x) (↑k * esymm σ R k - (-1) ^ (k + 1) * ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2) = (fun x => (-1) ^ (k + 1) * x) ((-1) ^ (k + 1) * ∑ a ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2)
\u22a2 psum σ R k = (-1) ^ (k + 1) * ↑k * esymm σ R k - ∑ x ∈ filter (fun a => a.1 < k ∧ 0 < a.1) (antidiagonal k), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2
-- Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean
σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm : ↑k * esymm σ R k = (-1) ^ (k + 1) * (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 + ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides✝ : ↑k * esymm σ R k - (-1) ^ (k + 1) * ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 = (-1) ^ (k + 1) * ∑ a ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
sub_both_sides : (-1) ^ (k + 1) * ↑k * esymm σ R k - ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 = ∑ a ∈ filter (fun a => a.1 < k ∧ ¬0 < a.1) (antidiagonal k), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
\u22a2 psum σ R k = (-1) ^ (k + 1) * ↑k * esymm σ R k - ∑ x ∈ filter (fun a => a.1 < k ∧ 0 < a.1) (antidiagonal k), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2
-- Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean
σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm : ↑k * esymm σ R k = (-1) ^ (k + 1) * (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 + ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides✝ : ↑k * esymm σ R k - (-1) ^ (k + 1) * ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 = (-1) ^ (k + 1) * ∑ a ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
sub_both_sides : (-1) ^ (k + 1) * ↑k * esymm σ R k - ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 = ∑ a ∈ filter (fun a => a.1 < k ∧ ¬0 < a.1) (antidiagonal k), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
this : filter (fun a => a.1 < k ∧ ¬0 < a.1) (antidiagonal k) = {(0, k)}
\u22a2 psum σ R k = (-1) ^ (k + 1) * ↑k * esymm σ R k - ∑ x ∈ filter (fun a => a.1 < k ∧ 0 < a.1) (antidiagonal k), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2
-- Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean
σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm : ↑k * esymm σ R k = (-1) ^ (k + 1) * (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 + ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides✝ : ↑k * esymm σ R k - (-1) ^ (k + 1) * ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 = (-1) ^ (k + 1) * ∑ a ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
sub_both_sides : (-1) ^ (k + 1) * ↑k * esymm σ R k - ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 = (-1) ^ (0, k).1 * esymm σ R (0, k).1 * psum σ R (0, k).2
this : filter (fun a => a.1 < k ∧ ¬0 < a.1) (antidiagonal k) = {(0, k)}
\u22a2 psum σ R k = (-1) ^ (k + 1) * ↑k * esymm σ R k - ∑ x ∈ filter (fun a => a.1 < k ∧ 0 < a.1) (antidiagonal k), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2
-- Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean
σ : Type u_1
inst✝¹ : Fintype σ
R : Type u_2
inst✝ : CommRing R
k : ℕ
h : 0 < k
hesymm : ↑k * esymm σ R k = (-1) ^ (k + 1) * (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 + ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)
sub_both_sides✝ : ↑k * esymm σ R k - (-1) ^ (k + 1) * ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 = (-1) ^ (k + 1) * ∑ a ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2
this : filter (fun a => a.1 < k ∧ ¬0 < a.1) (antidiagonal k) = {(0, k)}
sub_both_sides : (-1) ^ (k + 1) * ↑k * esymm σ R k - ∑ x ∈ filter (fun a => a.1 < k ∧ 0 < a.1) (antidiagonal k), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 = psum σ R k
\u22a2 psum σ R k = (-1) ^ (k + 1) * ↑k * esymm σ R k - ∑ x ∈ filter (fun a => a.1 < k ∧ 0 < a.1) (antidiagonal k), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2
-- Mathlib/RingTheory/GradedAlgebra/Basic.lean
ι : Type u_1
R : Type u_2
A : Type u_3
σ : Type u_4
inst✝⁷ : DecidableEq ι
inst✝⁶ : AddMonoid ι
inst✝⁵ : CommSemiring R
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
a : ⨁ (i : ι), ↥(𝒜 i)
i : ι
\u22a2 (proj 𝒜 i) ((decompose 𝒜).symm a) = (decompose 𝒜).symm ((of (fun i => ↥(𝒜 i)) i) (a i))
-- Mathlib/Order/Interval/Set/Basic.lean
α : Type u_1
β : Type u_2
inst✝¹ : Preorder α
inst✝ : Preorder β
a₁ a₂ : α
b₁ b₂ : β
x : α
y : β
\u22a2 (x, y) ∈ Icc a₁ a₂ ×ˢ Icc b₁ b₂ ↔ (x, y) ∈ Icc (a₁, b₁) (a₂, b₂)
-- Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
E : Type u_1
f : ℝ → E
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
g : ℝ → E
a b : ℝ
hb : 0 < b
\u22a2 ∫ (x : ℝ) in Ioi a, g (x * b) = b⁻¹ • ∫ (x : ℝ) in Ioi (a * b), g x
-- Mathlib/MeasureTheory/Measure/WithDensityFinite.lean
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
\u22a2 μ.toFiniteAux = 0 ↔ μ = 0
-- Mathlib/MeasureTheory/Measure/WithDensityFinite.lean
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
h : μ.toFiniteAux = 0
\u22a2 μ = 0
-- Mathlib/MeasureTheory/Measure/WithDensityFinite.lean
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
h : μ.toFiniteAux = 0
s : Set α
hs : MeasurableSet s
\u22a2 μ s = 0 s
-- Mathlib/MeasureTheory/Measure/WithDensityFinite.lean
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
h : ∀ (s : Set α), MeasurableSet s → μ.toFiniteAux s = 0 s
s : Set α
hs : MeasurableSet s
\u22a2 μ s = 0 s
-- Mathlib/MeasureTheory/Measure/WithDensityFinite.lean
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
s : Set α
hs : MeasurableSet s
h : μ.toFiniteAux s = 0 s
\u22a2 μ s = 0 s
-- Mathlib/MeasureTheory/Measure/WithDensityFinite.lean
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
s : Set α
hs : MeasurableSet s
h : ∀ (i : ℕ), 2 ^ (i + 1) * (sFiniteSeq μ i) Set.univ = ⊤ ∨ (sFiniteSeq μ i) s = 0
\u22a2 μ s = 0 s
-- Mathlib/MeasureTheory/Measure/WithDensityFinite.lean
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
s : Set α
hs : MeasurableSet s
h : ∀ (i : ℕ), 2 ^ (i + 1) * (sFiniteSeq μ i) Set.univ = ⊤ ∨ (sFiniteSeq μ i) s = 0
\u22a2 ∑' (i : ℕ), (sFiniteSeq μ i) s = 0 s
-- Mathlib/MeasureTheory/Measure/WithDensityFinite.lean
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
s : Set α
hs : MeasurableSet s
h : ∀ (i : ℕ), 2 ^ (i + 1) * (sFiniteSeq μ i) Set.univ = ⊤ ∨ (sFiniteSeq μ i) s = 0
\u22a2 ∀ (i : ℕ), (sFiniteSeq μ i) s = 0
-- Mathlib/MeasureTheory/Measure/WithDensityFinite.lean
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
s : Set α
hs : MeasurableSet s
n : ℕ
h : 2 ^ (n + 1) * (sFiniteSeq μ n) Set.univ = ⊤ ∨ (sFiniteSeq μ n) s = 0
\u22a2 (sFiniteSeq μ n) s = 0
-- Mathlib/Algebra/GradedMonoid.lean
ι : Type u_1
R : Type u_2
S : Type u_3
inst✝³ : SetLike S R
inst✝² : Monoid R
inst✝¹ : AddMonoid ι
A : ι → S
inst✝ : GradedMonoid A
n : ℕ
r : R
i : ι
h : r ∈ A i
\u22a2 r ^ n ∈ A (n • i)
-- Mathlib/Dynamics/Ergodic/Conservative.lean
ι : Type u_1
α : Type u_2
inst✝ : MeasurableSpace α
f : α → α
s : Set α
μ : Measure α
hf : Conservative f μ
hsm : NullMeasurableSet s μ
hs₀ : μ s ≠ 0
\u22a2 ∃ x ∈ s, ∃ m, m ≠ 0 ∧ f^[m] x ∈ s
-- Mathlib/Dynamics/Ergodic/Conservative.lean
ι : Type u_1
α : Type u_2
inst✝ : MeasurableSpace α
f : α → α
s : Set α
μ : Measure α
hf : Conservative f μ
hsm : NullMeasurableSet s μ
hs₀ : μ s ≠ 0
t : Set α
hsub : t ⊆ s
htm : MeasurableSet t
hts : t =ᶠ[ae μ] s
\u22a2 ∃ x ∈ s, ∃ m, m ≠ 0 ∧ f^[m] x ∈ s
-- Mathlib/Dynamics/Ergodic/Conservative.lean
ι : Type u_1
α : Type u_2
inst✝ : MeasurableSpace α
f : α → α
s : Set α
μ : Measure α
hf : Conservative f μ
hsm : NullMeasurableSet s μ
hs₀ : μ s ≠ 0
t : Set α
hsub : t ⊆ s
htm : MeasurableSet t
hts : t =ᶠ[ae μ] s
x : α
hxt : x ∈ t
m : ℕ
hm₀ : m ≠ 0
hmt : f^[m] x ∈ t
\u22a2 ∃ x ∈ s, ∃ m, m ≠ 0 ∧ f^[m] x ∈ s
-- Mathlib/Data/Finset/Prod.lean
α : Type u_1
β : Type u_2
γ : Type u_3
s✝ s' : Finset α
t✝ t' : Finset β
a : α
b : β
inst✝ : DecidableEq (α × β)
s : Finset α
t : Finset β
x✝ : α × β
x : α
y : β
\u22a2 (x, y) ∈ s ×ˢ t ↔ (x, y) ∈ t.biUnion fun b => image (fun a => (a, b)) s
-- Mathlib/Data/Finset/Sups.lean
F : Type u_1
α : Type u_2
β : Type u_3
inst✝⁴ : DecidableEq α
inst✝³ : DecidableEq β
inst✝² : DistribLattice α
inst✝¹ : OrderBot α
inst✝ : DecidableRel Disjoint
s t u v : Finset α
\u22a2 s ○ t ○ (u ○ v) = s ○ u ○ (t ○ v)
-- Mathlib/Data/List/Lattice.lean
α : Type u_1
l l₁✝ l₂✝ : List α
p : α → Prop
a✝ : α
inst✝ : DecidableEq α
a : α
l₁ l₂ t : List α
s : t <+ l₁
e : t ++ l₂ = l₁ ∪ l₂
h : a ∈ l₁ ∪ l₂
\u22a2 t ++ l₂ = a :: l₁ ∪ l₂
-- Mathlib/Data/List/Lattice.lean
α : Type u_1
l l₁✝ l₂✝ : List α
p : α → Prop
a✝ : α
inst✝ : DecidableEq α
a : α
l₁ l₂ t : List α
s : t <+ l₁
e : t ++ l₂ = l₁ ∪ l₂
h : ¬a ∈ l₁ ∪ l₂
\u22a2 a :: t ++ l₂ = a :: l₁ ∪ l₂
-- Mathlib/NumberTheory/ArithmeticFunction.lean
R : Type u_1
\u22a2 μ.IsMultiplicative
-- Mathlib/NumberTheory/ArithmeticFunction.lean
R : Type u_1
\u22a2 μ 1 = 1 ∧ ∀ {m n : ℕ}, m ≠ 0 → n ≠ 0 → m.Coprime n → μ (m * n) = μ m * μ n
-- Mathlib/NumberTheory/ArithmeticFunction.lean
R : Type u_1
n m : ℕ
hn : n ≠ 0
hm : m ≠ 0
hnm : n.Coprime m
\u22a2 μ (n * m) = μ n * μ m
-- 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]
hp : p.Monic
hq : q.Monic
a✝ : Nontrivial R
\u22a2 (p * q).natDegree = p.natDegree + q.natDegree
-- 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]
hp : p.Monic
hq : q.Monic
a✝ : Nontrivial R
\u22a2 p.leadingCoeff * q.leadingCoeff ≠ 0
-- Mathlib/NumberTheory/ArithmeticFunction.lean
R : Type u_1
inst✝ : Semiring R
f : ArithmeticFunction R
k x : ℕ
\u22a2 (f.ppow (k + 1)) x = (f.pmul (f.ppow k)) x
-- Mathlib/NumberTheory/ArithmeticFunction.lean
R : Type u_1
inst✝ : Semiring R
f : ArithmeticFunction R
k x : ℕ
\u22a2 f x * f x ^ k = (f.pmul (f.ppow k)) x
-- Mathlib/Analysis/InnerProductSpace/TwoDim.lean
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x y : E
\u22a2 |(o.areaForm x) y| ≤ ‖x‖ * ‖y‖
-- Mathlib/Algebra/Module/LocalizedModule.lean
R : Type u
inst✝⁵ : CommSemiring R
S : Submonoid R
M : Type v
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
T : Type u_1
inst✝² : CommSemiring T
inst✝¹ : Algebra R T
inst✝ : IsLocalization S T
x y : T
m : M
s : ↥S
\u22a2 (x * y) • mk m s = x • y • mk m s
-- Mathlib/Algebra/Module/LocalizedModule.lean
R : Type u
inst✝⁵ : CommSemiring R
S : Submonoid R
M : Type v
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
T : Type u_1
inst✝² : CommSemiring T
inst✝¹ : Algebra R T
inst✝ : IsLocalization S T
x y : T
m : M
s : ↥S
\u22a2 (IsLocalization.mk' T (IsLocalization.sec S x).1 (IsLocalization.sec S x).2 * IsLocalization.mk' T (IsLocalization.sec S y).1 (IsLocalization.sec S y).2) • mk m s = IsLocalization.mk' T (IsLocalization.sec S x).1 (IsLocalization.sec S x).2 • IsLocalization.mk' T (IsLocalization.sec S y).1 (IsLocalization.sec S y).2 • mk m s
-- Mathlib/MeasureTheory/Constructions/Cylinders.lean
ι : Type u_1
α : ι → Type u_2
s : Finset ι
\u22a2 cylinder s ∅ = ∅
-- 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✝ : CompleteSemilatticeInf α
s✝ t : Set α
a b : α
s : ι → α
\u22a2 iInf s ≤ a ↔ ∀ (b : α), (∀ (i : ι), b ≤ s i) → b ≤ a
-- Mathlib/MeasureTheory/Integral/Average.lean
α : Type u_1
E : Type u_2
F : Type u_3
m0 : MeasurableSpace α
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : CompleteSpace F
μ ν : Measure α
s t : Set α
f✝ g : α → E
inst✝ : IsFiniteMeasure μ
f : α → E
\u22a2 (μ univ).toReal • ⨍ (x : α), f x ∂μ = ∫ (x : α), f x ∂μ
-- Mathlib/MeasureTheory/Integral/Average.lean
α : Type u_1
E : Type u_2
F : Type u_3
m0 : MeasurableSpace α
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : CompleteSpace F
μ ν : Measure α
s t : Set α
f✝ g : α → E
inst✝ : IsFiniteMeasure μ
f : α → E
hμ : μ = 0
\u22a2 (μ univ).toReal • ⨍ (x : α), f x ∂μ = ∫ (x : α), f x ∂μ
-- Mathlib/MeasureTheory/Integral/Average.lean
α : Type u_1
E : Type u_2
F : Type u_3
m0 : MeasurableSpace α
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : CompleteSpace F
μ ν : Measure α
s t : Set α
f✝ g : α → E
inst✝ : IsFiniteMeasure μ
f : α → E
hμ : μ ≠ 0
\u22a2 (μ univ).toReal • ⨍ (x : α), f x ∂μ = ∫ (x : α), f x ∂μ
-- Mathlib/GroupTheory/Complement.lean
G : Type u_1
inst✝¹ : Group G
H K : Subgroup G
S T : Set G
α : Type u_2
inst✝ : MulAction G α
a : α
h1 : ∀ (h : ↥H), h • a = a → h = 1
h2 : ∀ (g : G), ∃ h, h • g • a = a
\u22a2 H.IsComplement' (MulAction.stabilizer G a)
-- Mathlib/GroupTheory/Complement.lean
G : Type u_1
inst✝¹ : Group G
H K : Subgroup G
S T : Set G
α : Type u_2
inst✝ : MulAction G α
a : α
h1 : ∀ (h : ↥H), h • a = a → h = 1
h2 : ∀ (g : G), ∃ h, h • g • a = a
g : G
\u22a2 ∃! x, ↑x.1 * ↑x.2 = g
-- Mathlib/GroupTheory/Complement.lean
G : Type u_1
inst✝¹ : Group G
H K : Subgroup G
S T : Set G
α : Type u_2
inst✝ : MulAction G α
a : α
h1 : ∀ (h : ↥H), h • a = a → h = 1
h2 : ∀ (g : G), ∃ h, h • g • a = a
g : G
h : ↥H
hh : h • g • a = a
\u22a2 ∃! x, ↑x.1 * ↑x.2 = g
-- Mathlib/GroupTheory/Complement.lean
G : Type u_1
inst✝¹ : Group G
H K : Subgroup G
S T : Set G
α : Type u_2
inst✝ : MulAction G α
a : α
h1 : ∀ (h : ↥H), h • a = a → h = 1
h2 : ∀ (g : G), ∃ h, h • g • a = a
g : G
h : ↥H
hh : h • g • a = a
hh' : (↑h * g) • a = a
\u22a2 ∃! x, ↑x.1 * ↑x.2 = g
-- Mathlib/GroupTheory/Complement.lean
G : Type u_1
inst✝¹ : Group G
H K : Subgroup G
S T : Set G
α : Type u_2
inst✝ : MulAction G α
a : α
h1 : ∀ (h : ↥H), h • a = a → h = 1
h2 : ∀ (g : G), ∃ h, h • g • a = a
g : G
h : ↥H
hh : h • g • a = a
hh' : (↑h * g) • a = a
\u22a2 ∀ (y : ↑↑H × ↑↑(MulAction.stabilizer G a)), (fun x => ↑x.1 * ↑x.2 = g) y → y = (h⁻¹, ⟨↑h * g, hh'⟩)
-- Mathlib/GroupTheory/Complement.lean
G : Type u_1
inst✝¹ : Group G
H K : Subgroup G
S T : Set G
α : Type u_2
inst✝ : MulAction G α
a : α
h1 : ∀ (h : ↥H), h • a = a → h = 1
h2 : ∀ (g : G), ∃ h, h • g • a = a
h : ↥H
h' : ↑↑H
g : G
hg : g • a = a
hh : h • (↑(h', ⟨g, hg⟩).1 * ↑(h', ⟨g, hg⟩).2) • a = a
hh' : (↑h * (↑(h', ⟨g, hg⟩).1 * ↑(h', ⟨g, hg⟩).2)) • a = a
\u22a2 (h', ⟨g, hg⟩) = (h⁻¹, ⟨↑h * (↑(h', ⟨g, hg⟩).1 * ↑(h', ⟨g, hg⟩).2), hh'⟩)
-- Mathlib/GroupTheory/Complement.lean
G : Type u_1
inst✝¹ : Group G
H K : Subgroup G
S T : Set G
α : Type u_2
inst✝ : MulAction G α
a : α
h2 : ∀ (g : G), ∃ h, h • g • a = a
h : ↥H
h' : ↑↑H
g : G
hg : g • a = a
hh : h • (↑(h', ⟨g, hg⟩).1 * ↑(h', ⟨g, hg⟩).2) • a = a
hh' : (↑h * (↑(h', ⟨g, hg⟩).1 * ↑(h', ⟨g, hg⟩).2)) • a = a
h1 : h * h' = 1
\u22a2 (h', ⟨g, hg⟩) = (h⁻¹, ⟨↑h * (↑(h', ⟨g, hg⟩).1 * ↑(h', ⟨g, hg⟩).2), hh'⟩)
-- Mathlib/GroupTheory/Complement.lean
G : Type u_1
inst✝¹ : Group G
H K : Subgroup G
S T : Set G
α : Type u_2
inst✝ : MulAction G α
a : α
h2 : ∀ (g : G), ∃ h, h • g • a = a
h : ↥H
h' : ↑↑H
g : G
hg : g • a = a
hh : h • (↑(h', ⟨g, hg⟩).1 * ↑(h', ⟨g, hg⟩).2) • a = a
hh' : (↑h * (↑(h', ⟨g, hg⟩).1 * ↑(h', ⟨g, hg⟩).2)) • a = a
h1 : h * h' = 1
\u22a2 ↑(h', ⟨g, hg⟩).2 = ↑(h⁻¹, ⟨↑h * (↑(h', ⟨g, hg⟩).1 * ↑(h', ⟨g, hg⟩).2), hh'⟩).2
-- Mathlib/Order/Filter/Archimedean.lean
α : Type u_1
R : Type u_2
inst✝¹ : StrictOrderedRing R
inst✝ : Archimedean R
p : R → Prop
h : ∀ᶠ (x : R) in atBot, p x
\u22a2 ∀ᶠ (n : ℤ) in atBot, p ↑n
-- Mathlib/Order/Filter/Archimedean.lean
α : Type u_1
R : Type u_2
inst✝¹ : StrictOrderedRing R
inst✝ : Archimedean R
p : R → Prop
h : ∀ᶠ (x : R) in atBot, p x
\u22a2 ∀ᶠ (n : ℤ) in Filter.comap Int.cast atBot, p ↑n
-- .lake/packages/batteries/Batteries/Data/String/Lemmas.lean
p : Char → Bool
s : String
\u22a2 s.find p = { byteIdx := utf8Len (List.takeWhile (fun x => !p x) s.data) }
-- Mathlib/Topology/MetricSpace/Thickening.lean
ι : Sort u_1
α : Type u
β : Type v
inst✝¹ : PseudoEMetricSpace α
δ✝ : ℝ
s : Set α
x✝ : α
X : Type u
inst✝ : PseudoMetricSpace X
δ : ℝ
E : Set X
x : X
\u22a2 x ∈ thickening δ E ↔ x ∈ ⋃ x ∈ E, ball x δ
-- Mathlib/Topology/MetricSpace/Thickening.lean
ι : Sort u_1
α : Type u
β : Type v
inst✝¹ : PseudoEMetricSpace α
δ✝ : ℝ
s : Set α
x✝ : α
X : Type u
inst✝ : PseudoMetricSpace X
δ : ℝ
E : Set X
x : X
\u22a2 x ∈ thickening δ E ↔ ∃ i ∈ E, x ∈ ball i δ
-- Mathlib/Data/PNat/Xgcd.lean
u✝ u : XgcdType
h : u.r ≠ 0
\u22a2 (if x : u.r = 0 then u.finish else u.step.reduce.flip) = u.step.reduce.flip
-- Mathlib/Logic/Encodable/Lattice.lean
α : Type u_1
β : Type u_2
inst✝ : Encodable β
f : β → Set α
C : Set α → Prop
H0 : C ∅
H1 : ∀ (b : β), C (f b)
n : ℕ
\u22a2 C (⋃ b ∈ none, f b)
-- Mathlib/Logic/Encodable/Lattice.lean
α : Type u_1
β : Type u_2
inst✝ : Encodable β
f : β → Set α
C : Set α → Prop
H0 : C ∅
H1 : ∀ (b : β), C (f b)
n : ℕ
\u22a2 C ∅
-- Mathlib/Logic/Encodable/Lattice.lean
α : Type u_1
β : Type u_2
inst✝ : Encodable β
f : β → Set α
C : Set α → Prop
H0 : C ∅
H1 : ∀ (b : β), C (f b)
n : ℕ
b : β
\u22a2 ⋃ b_1 ∈ some b, f b_1 = f b
-- Mathlib/Data/Matroid/Restrict.lean
α : Type u_1
M : Matroid α
R I J✝ X Y B J : Set α
e : α
hI : M.Basis' I X
hJ : M.Basis' J X
\u22a2 I.encard = J.encard
-- Mathlib/Data/Matroid/Restrict.lean
α : Type u_1
M : Matroid α
R I J✝ X Y B J : Set α
e : α
hI : (M ↾ X).Base I
hJ : (M ↾ X).Base J
\u22a2 I.encard = J.encard
-- Mathlib/Algebra/Tropical/BigOperators.lean
R : Type u_1
S : Type u_2
inst✝ : AddCommMonoid R
s : Multiset R
\u22a2 ∀ (a : List R), trop (sum ⟦a⟧) = (map trop ⟦a⟧).prod
-- Mathlib/Algebra/Order/Ring/Rat.lean
a b c p q : ℚ
\u22a2 p ≤ q ↔ p.num * ↑q.den ≤ q.num * ↑p.den
-- Mathlib/Algebra/Order/Ring/Rat.lean
a b c p q : ℚ
\u22a2 p.num /. ↑p.den ≤ q.num /. ↑q.den ↔ (p.num /. ↑p.den).num * ↑(q.num /. ↑q.den).den ≤ (q.num /. ↑q.den).num * ↑(p.num /. ↑p.den).den
-- Mathlib/Algebra/Order/Ring/Rat.lean
a b c p q : ℚ
\u22a2 p.num /. ↑p.den ≤ q.num /. ↑q.den ↔ p.num * ↑q.den ≤ q.num * ↑p.den
-- Mathlib/CategoryTheory/Monoidal/Preadditive.lean
C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : MonoidalCategory C
inst✝² : MonoidalPreadditive C
inst✝¹ : HasFiniteBiproducts C
J : Type
inst✝ : Fintype J
X Y : C
f : J → C
g h : X ⊗ ⨁ f ⟶ Y
w : ∀ (j : J), X ◁ biproduct.ι f j ≫ g = X ◁ biproduct.ι f j ≫ h
\u22a2 g = h
-- Mathlib/CategoryTheory/Monoidal/Preadditive.lean
C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : MonoidalCategory C
inst✝² : MonoidalPreadditive C
inst✝¹ : HasFiniteBiproducts C
J : Type
inst✝ : Fintype J
X Y : C
f : J → C
g h : X ⊗ ⨁ f ⟶ Y
w : ∀ (j : J), X ◁ biproduct.ι f j ≫ g = X ◁ biproduct.ι f j ≫ h
\u22a2 (leftDistributor X f).inv ≫ g = (leftDistributor X f).inv ≫ h
-- Mathlib/CategoryTheory/Monoidal/Preadditive.lean
C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : MonoidalCategory C
inst✝² : MonoidalPreadditive C
inst✝¹ : HasFiniteBiproducts C
J : Type
inst✝ : Fintype J
X Y : C
f : J → C
g h : X ⊗ ⨁ f ⟶ Y
w : ∀ (j : J), X ◁ biproduct.ι f j ≫ g = X ◁ biproduct.ι f j ≫ h
j✝ : J
\u22a2 biproduct.ι (fun j => X ⊗ f j) j✝ ≫ (leftDistributor X f).inv ≫ g = biproduct.ι (fun j => X ⊗ f j) j✝ ≫ (leftDistributor X f).inv ≫ h
-- Mathlib/CategoryTheory/Monoidal/Preadditive.lean
C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : MonoidalCategory C
inst✝² : MonoidalPreadditive C
inst✝¹ : HasFiniteBiproducts C
J : Type
inst✝ : Fintype J
X Y : C
f : J → C
g h : X ⊗ ⨁ f ⟶ Y
w : ∀ (j : J), X ◁ biproduct.ι f j ≫ g = X ◁ biproduct.ι f j ≫ h
j✝ : J
\u22a2 X ◁ biproduct.ι f j✝ ≫ g = X ◁ biproduct.ι f j✝ ≫ h
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
θ : Angle
\u22a2 ↑θ.toReal = θ
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
x✝ : ℝ
\u22a2 ↑(↑x✝).toReal = ↑x✝
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝¹ : MeasurableSpace Ω
inst✝ : PseudoEMetricSpace Ω
μ ν : Measure Ω
\u22a2 levyProkhorovEDist μ ν = levyProkhorovEDist ν μ