ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Algebra/Polynomial/Lifts.lean
R : Type u
inst✝¹ : Semiring R
S : Type v
inst✝ : Semiring S
f : R →+* S
n : ℕ
hn : ∀ m < n, ∀ {p : S[X]}, p ∈ lifts f → p.natDegree = m → ∃ q, map f q = p ∧ q.degree = p.degree
p : S[X]
hlifts : p ∈ lifts f
hdeg : p.natDegree = n
erase_zero : ¬p.eraseLead = 0
\u22a2 ∃ q, map f q = p ∧ q.degree = p.degree
-- Mathlib/Algebra/Polynomial/Lifts.lean
R : Type u
inst✝¹ : Semiring R
S : Type v
inst✝ : Semiring S
f : R →+* S
n : ℕ
hn : ∀ m < n, ∀ {p : S[X]}, p ∈ lifts f → p.natDegree = m → ∃ q, map f q = p ∧ q.degree = p.degree
p : S[X]
hlifts : p ∈ lifts f
hdeg : p.natDegree = n
erase_zero : ¬p.eraseLead = 0
deg_erase : p.eraseLead.natDegree < p.natDegree
\u22a2 ∃ q, map f q = p ∧ q.degree = p.degree
-- Mathlib/Algebra/Polynomial/Lifts.lean
R : Type u
inst✝¹ : Semiring R
S : Type v
inst✝ : Semiring S
f : R →+* S
n : ℕ
hn : ∀ m < n, ∀ {p : S[X]}, p ∈ lifts f → p.natDegree = m → ∃ q, map f q = p ∧ q.degree = p.degree
p : S[X]
hlifts : p ∈ lifts f
hdeg : p.natDegree = n
erase_zero : ¬p.eraseLead = 0
deg_erase : p.eraseLead.natDegree < p.natDegree
pzero : p ≠ 0
\u22a2 ∃ q, map f q = p ∧ q.degree = p.degree
-- Mathlib/Algebra/Polynomial/Lifts.lean
R : Type u
inst✝¹ : Semiring R
S : Type v
inst✝ : Semiring S
f : R →+* S
n : ℕ
hn : ∀ m < n, ∀ {p : S[X]}, p ∈ lifts f → p.natDegree = m → ∃ q, map f q = p ∧ q.degree = p.degree
p : S[X]
hlifts : p ∈ lifts f
hdeg : p.natDegree = n
erase_zero : ¬p.eraseLead = 0
deg_erase : p.eraseLead.natDegree < p.natDegree
pzero : p ≠ 0
lead_zero : p.coeff p.natDegree ≠ 0
\u22a2 ∃ q, map f q = p ∧ q.degree = p.degree
-- Mathlib/Algebra/Polynomial/Lifts.lean
R : Type u
inst✝¹ : Semiring R
S : Type v
inst✝ : Semiring S
f : R →+* S
n : ℕ
hn : ∀ m < n, ∀ {p : S[X]}, p ∈ lifts f → p.natDegree = m → ∃ q, map f q = p ∧ q.degree = p.degree
p : S[X]
hlifts : p ∈ lifts f
hdeg : p.natDegree = n
erase_zero : ¬p.eraseLead = 0
deg_erase : p.eraseLead.natDegree < p.natDegree
pzero : p ≠ 0
lead_zero : p.coeff p.natDegree ≠ 0
lead : R[X]
hlead : map f lead = (monomial p.natDegree) (p.coeff p.natDegree) ∧ lead.degree = ((monomial p.natDegree) (p.coeff p.natDegree)).degree
\u22a2 ∃ q, map f q = p ∧ q.degree = p.degree
-- Mathlib/Algebra/Polynomial/Lifts.lean
R : Type u
inst✝¹ : Semiring R
S : Type v
inst✝ : Semiring S
f : R →+* S
n : ℕ
hn : ∀ m < n, ∀ {p : S[X]}, p ∈ lifts f → p.natDegree = m → ∃ q, map f q = p ∧ q.degree = p.degree
p : S[X]
hlifts : p ∈ lifts f
hdeg : p.natDegree = n
erase_zero : ¬p.eraseLead = 0
deg_erase : p.eraseLead.natDegree < p.natDegree
pzero : p ≠ 0
lead_zero : p.coeff p.natDegree ≠ 0
lead : R[X]
hlead : map f lead = (monomial p.natDegree) (p.coeff p.natDegree) ∧ lead.degree = ((monomial p.natDegree) (p.coeff p.natDegree)).degree
deg_lead : lead.degree = ↑p.natDegree
\u22a2 ∃ q, map f q = p ∧ q.degree = p.degree
-- Mathlib/Algebra/Polynomial/Lifts.lean
R : Type u
inst✝¹ : Semiring R
S : Type v
inst✝ : Semiring S
f : R →+* S
n : ℕ
hn : ∀ m < n, ∀ {p : S[X]}, p ∈ lifts f → p.natDegree = m → ∃ q, map f q = p ∧ q.degree = p.degree
p : S[X]
hlifts : p ∈ lifts f
hdeg : p.natDegree = n
erase_zero : ¬p.eraseLead = 0
deg_erase : p.eraseLead.natDegree < n
pzero : p ≠ 0
lead_zero : p.coeff p.natDegree ≠ 0
lead : R[X]
hlead : map f lead = (monomial p.natDegree) (p.coeff p.natDegree) ∧ lead.degree = ((monomial p.natDegree) (p.coeff p.natDegree)).degree
deg_lead : lead.degree = ↑p.natDegree
\u22a2 ∃ q, map f q = p ∧ q.degree = p.degree
-- Mathlib/Algebra/Polynomial/Lifts.lean
R : Type u
inst✝¹ : Semiring R
S : Type v
inst✝ : Semiring S
f : R →+* S
n : ℕ
hn : ∀ m < n, ∀ {p : S[X]}, p ∈ lifts f → p.natDegree = m → ∃ q, map f q = p ∧ q.degree = p.degree
p : S[X]
hlifts : p ∈ lifts f
hdeg : p.natDegree = n
erase_zero : ¬p.eraseLead = 0
deg_erase : p.eraseLead.natDegree < n
pzero : p ≠ 0
lead_zero : p.coeff p.natDegree ≠ 0
lead : R[X]
hlead : map f lead = (monomial p.natDegree) (p.coeff p.natDegree) ∧ lead.degree = ((monomial p.natDegree) (p.coeff p.natDegree)).degree
deg_lead : lead.degree = ↑p.natDegree
erase : R[X]
herase : map f erase = Polynomial.erase p.natDegree p ∧ erase.degree = (Polynomial.erase p.natDegree p).degree
\u22a2 ∃ q, map f q = p ∧ q.degree = p.degree
-- Mathlib/Algebra/Polynomial/Lifts.lean
R : Type u
inst✝¹ : Semiring R
S : Type v
inst✝ : Semiring S
f : R →+* S
n : ℕ
hn : ∀ m < n, ∀ {p : S[X]}, p ∈ lifts f → p.natDegree = m → ∃ q, map f q = p ∧ q.degree = p.degree
p : S[X]
hlifts : p ∈ lifts f
hdeg : p.natDegree = n
erase_zero : ¬p.eraseLead = 0
deg_erase : p.eraseLead.natDegree < n
pzero : p ≠ 0
lead_zero : p.coeff p.natDegree ≠ 0
lead : R[X]
hlead : map f lead = (monomial p.natDegree) (p.coeff p.natDegree) ∧ lead.degree = ((monomial p.natDegree) (p.coeff p.natDegree)).degree
deg_lead : lead.degree = ↑p.natDegree
erase : R[X]
herase : map f erase = Polynomial.erase p.natDegree p ∧ erase.degree = (Polynomial.erase p.natDegree p).degree
\u22a2 map f (erase + lead) = p ∧ (erase + lead).degree = p.degree
-- Mathlib/Algebra/Polynomial/Lifts.lean
R : Type u
inst✝¹ : Semiring R
S : Type v
inst✝ : Semiring S
f : R →+* S
n : ℕ
hn : ∀ m < n, ∀ {p : S[X]}, p ∈ lifts f → p.natDegree = m → ∃ q, map f q = p ∧ q.degree = p.degree
p : S[X]
hlifts : p ∈ lifts f
hdeg : p.natDegree = n
erase_zero : ¬p.eraseLead = 0
deg_erase : p.eraseLead.natDegree < n
pzero : p ≠ 0
lead_zero : p.coeff p.natDegree ≠ 0
lead : R[X]
hlead : map f lead = (monomial p.natDegree) (p.coeff p.natDegree) ∧ lead.degree = ((monomial p.natDegree) (p.coeff p.natDegree)).degree
deg_lead : lead.degree = ↑p.natDegree
erase : R[X]
herase : map f erase = Polynomial.erase p.natDegree p ∧ erase.degree = (Polynomial.erase p.natDegree p).degree
\u22a2 (erase + lead).degree = p.degree
-- Mathlib/Algebra/Polynomial/Lifts.lean
R : Type u
inst✝¹ : Semiring R
S : Type v
inst✝ : Semiring S
f : R →+* S
n : ℕ
hn : ∀ m < n, ∀ {p : S[X]}, p ∈ lifts f → p.natDegree = m → ∃ q, map f q = p ∧ q.degree = p.degree
p : S[X]
hlifts : p ∈ lifts f
hdeg : p.natDegree = n
erase_zero : ¬p.eraseLead = 0
deg_erase : p.eraseLead.natDegree < n
pzero : p ≠ 0
lead_zero : p.coeff p.natDegree ≠ 0
lead : R[X]
hlead : map f lead = (monomial p.natDegree) (p.coeff p.natDegree) ∧ lead.degree = ((monomial p.natDegree) (p.coeff p.natDegree)).degree
deg_lead : lead.degree = ↑p.natDegree
erase : R[X]
herase : map f erase = Polynomial.erase p.natDegree p ∧ erase.degree = (Polynomial.erase p.natDegree p).degree
\u22a2 (erase + lead).degree = lead.degree
-- Mathlib/Algebra/Polynomial/Lifts.lean
R : Type u
inst✝¹ : Semiring R
S : Type v
inst✝ : Semiring S
f : R →+* S
n : ℕ
hn : ∀ m < n, ∀ {p : S[X]}, p ∈ lifts f → p.natDegree = m → ∃ q, map f q = p ∧ q.degree = p.degree
p : S[X]
hlifts : p ∈ lifts f
hdeg : p.natDegree = n
erase_zero : ¬p.eraseLead = 0
deg_erase : p.eraseLead.natDegree < n
pzero : p ≠ 0
lead_zero : p.coeff p.natDegree ≠ 0
lead : R[X]
hlead : map f lead = (monomial p.natDegree) (p.coeff p.natDegree) ∧ lead.degree = ((monomial p.natDegree) (p.coeff p.natDegree)).degree
deg_lead : lead.degree = ↑p.natDegree
erase : R[X]
herase : map f erase = Polynomial.erase p.natDegree p ∧ erase.degree = (Polynomial.erase p.natDegree p).degree
\u22a2 erase.degree < lead.degree
-- Mathlib/Algebra/Polynomial/Lifts.lean
R : Type u
inst✝¹ : Semiring R
S : Type v
inst✝ : Semiring S
f : R →+* S
n : ℕ
hn : ∀ m < n, ∀ {p : S[X]}, p ∈ lifts f → p.natDegree = m → ∃ q, map f q = p ∧ q.degree = p.degree
p : S[X]
hlifts : p ∈ lifts f
hdeg : p.natDegree = n
erase_zero : ¬p.eraseLead = 0
deg_erase : p.eraseLead.natDegree < n
pzero : p ≠ 0
lead_zero : p.coeff p.natDegree ≠ 0
lead : R[X]
hlead : map f lead = (monomial p.natDegree) (p.coeff p.natDegree) ∧ lead.degree = ((monomial p.natDegree) (p.coeff p.natDegree)).degree
deg_lead : lead.degree = ↑p.natDegree
erase : R[X]
herase : map f erase = Polynomial.erase p.natDegree p ∧ erase.degree = (Polynomial.erase p.natDegree p).degree
\u22a2 (Polynomial.erase p.natDegree p).degree < p.degree
-- Mathlib/Data/List/Cycle.lean
α : Type u_1
s : Cycle α
\u22a2 s.reverse.Subsingleton ↔ s.Subsingleton
-- Mathlib/Topology/AlexandrovDiscrete.lean
ι : Sort u_1
κ : ι → Sort u_2
α : Type u_3
β : Type u_4
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
s✝ t : Set α
a x y : α
inst✝¹ : AlexandrovDiscrete α
inst✝ : AlexandrovDiscrete β
s : Set α
\u22a2 𝓟 (exterior s) = 𝓝ˢ s
-- Mathlib/RingTheory/PolynomialAlgebra.lean
R : Type u_1
A : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Algebra R A
n : Type w
inst✝¹ : DecidableEq n
inst✝ : Fintype n
\u22a2 matPolyEquiv (diagonal fun x => X) = X
-- Mathlib/Algebra/EuclideanDomain/Basic.lean
R : Type u
inst✝ : EuclideanDomain R
a b c : R
ha : a ≠ 0
h : a * b = c
\u22a2 b = c / a
-- Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean
R : Type u
inst✝¹ : CommRing R
W✝ : Affine R
F : Type u
inst✝ : Field F
W : Affine F
x₁ x₂ y₁ y₂ : F
hx : x₁ ≠ x₂
\u22a2 y₁ * (x₂ - W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂)) + y₂ * (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) - x₁) + W.negAddY x₁ x₂ y₁ (W.slope x₁ x₂ y₁ y₂) * (x₁ - x₂) = 0
-- Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean
R : Type u
inst✝¹ : CommRing R
W✝ : Affine R
F : Type u
inst✝ : Field F
W : Affine F
x₁ x₂ y₁ y₂ : F
hx : x₁ ≠ x₂
\u22a2 y₁ * (x₂ - (((y₁ - y₂) / (x₁ - x₂)) ^ 2 + W.a₁ * ((y₁ - y₂) / (x₁ - x₂)) - W.a₂ - x₁ - x₂)) + y₂ * (((y₁ - y₂) / (x₁ - x₂)) ^ 2 + W.a₁ * ((y₁ - y₂) / (x₁ - x₂)) - W.a₂ - x₁ - x₂ - x₁) + ((y₁ - y₂) / (x₁ - x₂) * (((y₁ - y₂) / (x₁ - x₂)) ^ 2 + W.a₁ * ((y₁ - y₂) / (x₁ - x₂)) - W.a₂ - x₁ - x₂ - x₁) + y₁) * (x₁ - x₂) = 0
-- Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean
R : Type u
inst✝¹ : CommRing R
W✝ : Affine R
F : Type u
inst✝ : Field F
W : Affine F
x₁ x₂ y₁ y₂ : F
hx : x₁ ≠ x₂
\u22a2 (y₁ * (x₂ * ((x₁ - x₂) ^ 2 * (x₁ - x₂)) - ((y₁ - y₂) ^ 2 * (x₁ - x₂) + W.a₁ * (y₁ - y₂) * (x₁ - x₂) ^ 2 - (x₁ - x₂) ^ 2 * (x₁ - x₂) * W.a₂ - (x₁ - x₂) ^ 2 * (x₁ - x₂) * x₁ - (x₁ - x₂) ^ 2 * (x₁ - x₂) * x₂)) + y₂ * ((y₁ - y₂) ^ 2 * (x₁ - x₂) + W.a₁ * (y₁ - y₂) * (x₁ - x₂) ^ 2 - (x₁ - x₂) ^ 2 * (x₁ - x₂) * W.a₂ - (x₁ - x₂) ^ 2 * (x₁ - x₂) * x₁ - (x₁ - x₂) ^ 2 * (x₁ - x₂) * x₂ - (x₁ - x₂) ^ 2 * (x₁ - x₂) * x₁)) * ((x₁ - x₂) * ((x₁ - x₂) ^ 2 * (x₁ - x₂))) + ((y₁ - y₂) * ((y₁ - y₂) ^ 2 * (x₁ - x₂) + W.a₁ * (y₁ - y₂) * (x₁ - x₂) ^ 2 - (x₁ - x₂) ^ 2 * (x₁ - x₂) * W.a₂ - (x₁ - x₂) ^ 2 * (x₁ - x₂) * x₁ - (x₁ - x₂) ^ 2 * (x₁ - x₂) * x₂ - (x₁ - x₂) ^ 2 * (x₁ - x₂) * x₁) + y₁ * ((x₁ - x₂) * ((x₁ - x₂) ^ 2 * (x₁ - x₂)))) * (x₁ - x₂) * ((x₁ - x₂) ^ 2 * (x₁ - x₂)) = 0
-- Mathlib/Data/Set/Image.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t✝ : Set α
f : α → β
t : Set β
x : β
x✝ : x ∈ range f ∩ t
y : α
h_eq : f y = x
hx : x ∈ t
\u22a2 y ∈ f ⁻¹' t
-- Mathlib/Data/Set/Image.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t✝ : Set α
f : α → β
t : Set β
x : β
x✝ : x ∈ range f ∩ t
y : α
h_eq : f y = x
hx : x ∈ t
\u22a2 x ∈ t
-- Mathlib/Data/Matrix/RowCol.lean
l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
R : Type u_5
α : Type v
β : Type w
ι : Type u_6
M : Matrix m n α
i : m
j : n
b : n → α
c : m → α
inst✝¹ : DecidableEq m
inst✝ : Star α
\u22a2 Mᴴ.updateColumn i (star b) = (M.updateRow i b)ᴴ
-- Mathlib/Data/Matrix/RowCol.lean
l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
R : Type u_5
α : Type v
β : Type w
ι : Type u_6
M : Matrix m n α
i : m
j : n
b : n → α
c : m → α
inst✝¹ : DecidableEq m
inst✝ : Star α
\u22a2 ((M.map star).updateRow i (star b))ᵀ = ((M.map star).updateRow i (star ∘ b))ᵀ
-- Mathlib/Dynamics/Ergodic/MeasurePreserving.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
μa : Measure α
μb : Measure β
μc : Measure γ
μd : Measure δ
μ : Measure α
f : α → α
s : Set α
hf : MeasurePreserving f μ μ
hs : NullMeasurableSet s μ
n : ℕ
ih : μ (s ∆ (f^[n] ⁻¹' s)) ≤ n • μ (s ∆ (f ⁻¹' s))
\u22a2 μ (s ∆ (f^[n + 1] ⁻¹' s)) ≤ (n + 1) • μ (s ∆ (f ⁻¹' s))
-- Mathlib/Dynamics/Ergodic/MeasurePreserving.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
μa : Measure α
μb : Measure β
μc : Measure γ
μd : Measure δ
μ : Measure α
f : α → α
s : Set α
hf : MeasurePreserving f μ μ
hs : NullMeasurableSet s μ
n : ℕ
ih : μ (s ∆ (f^[n] ⁻¹' s)) ≤ n • μ (s ∆ (f ⁻¹' s))
\u22a2 μ (s ∆ (f^[n + 1] ⁻¹' s)) ≤ n • μ (s ∆ (f ⁻¹' s)) + μ (s ∆ (f ⁻¹' s))
-- Mathlib/Dynamics/Ergodic/MeasurePreserving.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
μa : Measure α
μb : Measure β
μc : Measure γ
μd : Measure δ
μ : Measure α
f : α → α
s : Set α
hf : MeasurePreserving f μ μ
hs : NullMeasurableSet s μ
n : ℕ
ih : μ (s ∆ (f^[n] ⁻¹' s)) ≤ n • μ (s ∆ (f ⁻¹' s))
\u22a2 μ ((f^[n] ⁻¹' s) ∆ (f^[n + 1] ⁻¹' s)) ≤ μ (s ∆ (f ⁻¹' s))
-- Mathlib/Dynamics/Ergodic/MeasurePreserving.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
μa : Measure α
μb : Measure β
μc : Measure γ
μd : Measure δ
μ : Measure α
f : α → α
s : Set α
hf : MeasurePreserving f μ μ
n : ℕ
ih : μ (s ∆ (f^[n] ⁻¹' s)) ≤ n • μ (s ∆ (f ⁻¹' s))
hs : NullMeasurableSet (s ∆ (f ⁻¹' s)) μ
\u22a2 μ ((f^[n] ⁻¹' s) ∆ (f^[n + 1] ⁻¹' s)) ≤ μ (s ∆ (f ⁻¹' s))
-- Mathlib/Data/Matroid/Basic.lean
α : Type u_1
M : Matroid α
B B' I J D X : Set α
e f : α
hB : M.Base B
h : e ∈ M.E \ B
\u22a2 M.Dep (insert e B)
-- Mathlib/Data/Matroid/Basic.lean
α : Type u_1
M : Matroid α
B B' I J D X : Set α
e f : α
hB : M.Base B
h : e ∈ M.E \ B
\u22a2 ¬M.Indep (insert e B)
-- Mathlib/SetTheory/Cardinal/Ordinal.lean
a b : Cardinal.{u_1}
ha : ℵ₀ ≤ a
hb : ℵ₀ ≤ b
\u22a2 a ≤ a * b
-- Mathlib/SetTheory/Cardinal/Ordinal.lean
a b : Cardinal.{u_1}
ha : ℵ₀ ≤ a
hb : ℵ₀ ≤ b
\u22a2 b ≤ a * b
-- Mathlib/NumberTheory/MulChar/Basic.lean
R✝ : Type u_1
inst✝³ : CommMonoid R✝
R' : Type u_2
inst✝² : CommRing R'
R'' : Type u_3
inst✝¹ : CommRing R''
R : Type u_4
inst✝ : CommRing R
χ : MulChar R R'
f : R' →+* R''
a✝ : Rˣ
\u22a2 (χ.ringHomComp f)⁻¹ ↑a✝ = (χ⁻¹.ringHomComp f) ↑a✝
-- Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
\u22a2 ∑ i : Fin n, d * (2 * d + 1) ^ ↑i = ((2 * d + 1) ^ n - 1) / 2
-- Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean
α : Type u_1
β : Type u_2
n d k N : ℕ
x : Fin n → ℕ
\u22a2 (2 * d + 1) ^ n - 1 = (∑ i : Fin n, d * (2 * d + 1) ^ ↑i) * 2
-- Mathlib/Algebra/Group/Units.lean
α : Type u
M : Type u_1
N : Type u_2
inst✝ : DivisionMonoid α
a b c : α
ha : IsUnit a
hb : IsUnit b
\u22a2 IsUnit (a / b)
-- Mathlib/Algebra/Group/Units.lean
α : Type u
M : Type u_1
N : Type u_2
inst✝ : DivisionMonoid α
a b c : α
ha : IsUnit a
hb : IsUnit b
\u22a2 IsUnit (a * b⁻¹)
-- Mathlib/MeasureTheory/Measure/Stieltjes.lean
f g : StieltjesFunction
h : ∀ (x : ℝ), ↑f x = ↑g x
\u22a2 f = g
-- Mathlib/Data/Vector/Basic.lean
n : ℕ
α : Type u_1
β : Type u_2
γ : Type u_3
a✝ a b : α
i j : Fin (n + 1)
h : i ≤ j
l : List α
hl : l.length = n
\u22a2 insertNth b j.succ (insertNth a i ⟨l, hl⟩) = insertNth a i.castSucc (insertNth b j ⟨l, hl⟩)
-- Mathlib/Data/Vector/Basic.lean
n : ℕ
α : Type u_1
β : Type u_2
γ : Type u_3
a✝ a b : α
i j : Fin (n + 1)
h : i ≤ j
l : List α
hl : l.length = n
\u22a2 ↑(insertNth b j.succ (insertNth a i ⟨l, hl⟩)) = ↑(insertNth a i.castSucc (insertNth b j ⟨l, hl⟩))
-- Mathlib/Data/Vector/Basic.lean
n : ℕ
α : Type u_1
β : Type u_2
γ : Type u_3
a✝ a b : α
i j : Fin (n + 1)
h : i ≤ j
l : List α
hl : l.length = n
\u22a2 List.insertNth (↑j + 1) b (List.insertNth (↑i) a l) = List.insertNth (↑i) a (List.insertNth (↑j) b l)
-- Mathlib/Data/Vector/Basic.lean
n : ℕ
α : Type u_1
β : Type u_2
γ : Type u_3
a✝ a b : α
i j : Fin (n + 1)
h : i ≤ j
l : List α
hl : l.length = n
\u22a2 ↑i ≤ ↑j
-- Mathlib/Data/Vector/Basic.lean
n : ℕ
α : Type u_1
β : Type u_2
γ : Type u_3
a✝ a b : α
i j : Fin (n + 1)
h : i ≤ j
l : List α
hl : l.length = n
\u22a2 ↑j ≤ l.length
-- Mathlib/Algebra/Polynomial/FieldDivision.lean
R : Type u
S : Type v
k : Type y
A : Type z
a b : R
n✝ : ℕ
inst✝³ : Field R
p q : R[X]
inst✝² : CommRing S
inst✝¹ : IsDomain S
inst✝ : Algebra R S
n : ℕ
hn : n ≠ 0
\u22a2 (X ^ n).rootSet S = {0}
-- Mathlib/Algebra/Polynomial/FieldDivision.lean
R : Type u
S : Type v
k : Type y
A : Type z
a b : R
n✝ : ℕ
inst✝³ : Field R
p q : R[X]
inst✝² : CommRing S
inst✝¹ : IsDomain S
inst✝ : Algebra R S
n : ℕ
hn : n ≠ 0
\u22a2 1 ≠ 0
-- Mathlib/CategoryTheory/Limits/Opposites.lean
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
J : Type u₂
inst✝² : Category.{v₂, u₂} J
X✝ : Type v₂
X Y Z : C
f : X ⟶ Z
g : X ⟶ Y
inst✝¹ : HasPushout f g
inst✝ : HasPullback f.op g.op
\u22a2 (pushoutIsoUnopPullback f g).inv.op ≫ pullback.fst f.op g.op = (pushout.inl f g).op
-- Mathlib/CategoryTheory/Limits/Opposites.lean
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
J : Type u₂
inst✝² : Category.{v₂, u₂} J
X✝ : Type v₂
X Y Z : C
f : X ⟶ Z
g : X ⟶ Y
inst✝¹ : HasPushout f g
inst✝ : HasPullback f.op g.op
\u22a2 ((pushoutIsoUnopPullback f g).inv.op ≫ pullback.fst f.op g.op).unop = (pushout.inl f g).op.unop
-- Mathlib/CategoryTheory/Limits/Opposites.lean
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
J : Type u₂
inst✝² : Category.{v₂, u₂} J
X✝ : Type v₂
X Y Z : C
f : X ⟶ Z
g : X ⟶ Y
inst✝¹ : HasPushout f g
inst✝ : HasPullback f.op g.op
\u22a2 (pullback.fst f.op g.op).unop ≫ (pushoutIsoUnopPullback f g).inv = pushout.inl f g
-- Mathlib/SetTheory/Cardinal/Ordinal.lean
\u22a2 ℵ₀ < aleph 1
-- Mathlib/SetTheory/Cardinal/Ordinal.lean
\u22a2 ℵ₀ < succ ℵ₀
-- Mathlib/GroupTheory/HNNExtension.lean
G : Type u_1
inst✝² : Group G
A B : Subgroup G
φ : ↥A ≃* ↥B
H : Type u_2
inst✝¹ : Group H
M : Type u_3
inst✝ : Monoid M
a : ↥A
\u22a2 of ↑a * t⁻¹ = t⁻¹ * of ↑(φ a)
-- Mathlib/GroupTheory/HNNExtension.lean
G : Type u_1
inst✝² : Group G
A B : Subgroup G
φ : ↥A ≃* ↥B
H : Type u_2
inst✝¹ : Group H
M : Type u_3
inst✝ : Monoid M
a : ↥A
\u22a2 of ↑a * t⁻¹ = t⁻¹ * (t * of ↑a * t⁻¹)
-- Mathlib/Data/DFinsupp/Basic.lean
ι : Type u
γ : Type w
β : ι → Type v
β₁ : ι → Type v₁
β₂ : ι → Type v₂
inst✝¹ : DecidableEq ι
κ : Type u_1
inst✝ : (i : ι) → AddZeroClass (β i)
h : κ → ι
hh : Function.Injective h
f g : Π₀ (i : ι), β i
i✝ : κ
\u22a2 (comapDomain h hh (f + g)) i✝ = (comapDomain h hh f + comapDomain h hh g) i✝
-- Mathlib/LinearAlgebra/SesquilinearForm.lean
R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
R₃ : Type u_4
M : Type u_5
M₁ : Type u_6
M₂ : Type u_7
M₃ : Type u_8
Mₗ₁ : Type u_9
Mₗ₁' : Type u_10
Mₗ₂ : Type u_11
Mₗ₂' : Type u_12
K : Type u_13
K₁ : Type u_14
K₂ : Type u_15
V : Type u_16
V₁ : Type u_17
V₂ : Type u_18
n : Type u_19
inst✝⁷ : Field K
inst✝⁶ : AddCommGroup V
inst✝⁵ : Module K V
inst✝⁴ : Field K₁
inst✝³ : AddCommGroup V₁
inst✝² : Module K₁ V₁
inst✝¹ : AddCommGroup V₂
inst✝ : Module K V₂
J : K →+* K
J₁ J₁' : K₁ →+* K
B : V →ₗ[K] V →ₛₗ[J] V₂
x y : V
\u22a2 y ∈ (Submodule.span K {x}).orthogonalBilin B ↔ y ∈ ker (B x)
-- Mathlib/LinearAlgebra/SesquilinearForm.lean
R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
R₃ : Type u_4
M : Type u_5
M₁ : Type u_6
M₂ : Type u_7
M₃ : Type u_8
Mₗ₁ : Type u_9
Mₗ₁' : Type u_10
Mₗ₂ : Type u_11
Mₗ₂' : Type u_12
K : Type u_13
K₁ : Type u_14
K₂ : Type u_15
V : Type u_16
V₁ : Type u_17
V₂ : Type u_18
n : Type u_19
inst✝⁷ : Field K
inst✝⁶ : AddCommGroup V
inst✝⁵ : Module K V
inst✝⁴ : Field K₁
inst✝³ : AddCommGroup V₁
inst✝² : Module K₁ V₁
inst✝¹ : AddCommGroup V₂
inst✝ : Module K V₂
J : K →+* K
J₁ J₁' : K₁ →+* K
B : V →ₗ[K] V →ₛₗ[J] V₂
x y : V
\u22a2 (∀ (n : V), (∃ a, a • x = n) → B.IsOrtho n y) ↔ (B x) y = 0
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Basic.lean
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S✝ S : StarSubalgebra ℂ A
hS : IsClosed ↑S
x : A
h : IsUnit x
hxS : x ∈ S
\u22a2 ↑h.unit⁻¹ ∈ S
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Basic.lean
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S✝ S : StarSubalgebra ℂ A
hS : IsClosed ↑S
x : A
h : IsUnit x
hxS : x ∈ S
hx : IsUnit (star x * x)
\u22a2 ↑h.unit⁻¹ ∈ S
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Basic.lean
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S✝ S : StarSubalgebra ℂ A
hS : IsClosed ↑S
x : A
h : IsUnit x
hxS : x ∈ S
hx : IsUnit (star x * x)
\u22a2 ↑hx.unit⁻¹ ∈ S
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Basic.lean
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S✝ S : StarSubalgebra ℂ A
hS : IsClosed ↑S
x : A
h : IsUnit x
hxS : x ∈ S
hx : IsUnit (star x * x)
\u22a2 ↑hx.unit⁻¹ ∈ elementalStarAlgebra ℂ (star x * x)
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Basic.lean
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S✝ S : StarSubalgebra ℂ A
hS : IsClosed ↑S
x : A
h : IsUnit x
hxS : x ∈ S
hx : IsUnit (star x * x)
this : IsStarNormal (star x * x)
\u22a2 ↑hx.unit⁻¹ ∈ elementalStarAlgebra ℂ (star x * x)
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Basic.lean
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S✝ S : StarSubalgebra ℂ A
hS : IsClosed ↑S
x : A
h : IsUnit x
hxS : x ∈ S
hx : IsUnit (star x * x)
this : IsStarNormal (star x * x)
hx' : IsUnit ⟨star x * x, ⋯⟩
\u22a2 ↑hx.unit⁻¹ ∈ elementalStarAlgebra ℂ (star x * x)
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Basic.lean
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S✝ S : StarSubalgebra ℂ A
hS : IsClosed ↑S
x : A
h : IsUnit x
hxS : x ∈ S
hx : IsUnit (star x * x)
this : IsStarNormal (star x * x)
hx' : IsUnit ⟨star x * x, ⋯⟩
\u22a2 ↑hx.unit⁻¹ = ↑↑hx'.unit⁻¹
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Basic.lean
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S✝ S : StarSubalgebra ℂ A
hS : IsClosed ↑S
x : A
h : IsUnit x
hxS : x ∈ S
hx : IsUnit (star x * x)
this : IsStarNormal (star x * x)
hx' : IsUnit ⟨star x * x, ⋯⟩
\u22a2 ↑hx.unit * ↑↑hx'.unit⁻¹ = 1
-- Mathlib/LinearAlgebra/RootSystem/Defs.lean
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : RootPairing ι R M N
i j✝ j : ι
\u22a2 (P.reflection i) (P.root j) ∈ range ⇑P.root
-- Mathlib/Order/Hom/CompleteLattice.lean
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
δ : Type u_5
ι : Sort u_6
κ : ι → Sort u_7
inst✝⁴ : FunLike F α β
inst✝³ : InfSet α
inst✝² : InfSet β
inst✝¹ : InfSet γ
inst✝ : InfSet δ
g : sInfHom β γ
f₁ f₂ : sInfHom α β
hg : Injective ⇑g
h : g.comp f₁ = g.comp f₂
a : α
\u22a2 g (f₁ a) = g (f₂ a)
-- Mathlib/Algebra/Polynomial/Div.lean
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : Ring R
p q f g : R[X]
hg : g.Monic
a✝ : Nontrivial R
hfg hgf : ¬f /ₘ g = 0
\u22a2 (f /ₘ g).natDegree = f.natDegree - g.natDegree
-- Mathlib/Algebra/Polynomial/Div.lean
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : Ring R
p q f g : R[X]
hg : g.Monic
a✝ : Nontrivial R
hfg : ¬f /ₘ g = 0
hgf : ¬f.degree < g.degree
\u22a2 (f /ₘ g).natDegree = f.natDegree - g.natDegree
-- Mathlib/Algebra/Polynomial/Div.lean
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : Ring R
p q f g : R[X]
hg : g.Monic
a✝ : Nontrivial R
hfg : ¬f /ₘ g = 0
hgf : g.degree ≤ f.degree
\u22a2 (f /ₘ g).natDegree = f.natDegree - g.natDegree