ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Computability/PartrecCode.lean
x✝ : Unit
p n : ℕ
this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))
c : Code
k' : ℕ
k : ℕ := k' + 1
nk : n ≤ k'
hg : ∀ {k' : ℕ} {c' : Code} {n : ℕ}, Nat.pair k' (encode c') < Nat.pair k (encode c) → Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range (Nat.pair k (encode c)))) (k', c') n = evaln k' c' n
\u22a2 rec (some 0) (some n.succ) (some (unpair n).1) (some (unpair n).2) (fun cf cg x x => do let x ← Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range (Nat.pair (k' + 1) (encode c)))) (k' + 1, cf) n let y ← Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range (Nat.pair (k' + 1) (encode c)))) (k' + 1, cg) n some (Nat.pair x y)) (fun cf cg x x => do let x ← Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range (Nat.pair (k' + 1) (encode c)))) (k' + 1, cg) n Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range (Nat.pair (k' + 1) (encode c)))) (k' + 1, cf) x) (fun cf cg x x => Nat.rec (Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range (Nat.pair (k' + 1) (encode c)))) (k' + 1, cf) (unpair n).1) (fun n_1 n_ih => do let i ← Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range (Nat.pair (k' + 1) (encode c)))) (k', c) (Nat.pair (unpair n).1 n_1) Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range (Nat.pair (k' + 1) (encode c)))) (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n_1 i))) (unpair n).2) (fun cf x => do let x ← Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range (Nat.pair (k' + 1) (encode c)))) (k' + 1, cf) n Nat.rec (some (unpair n).2) (fun n_1 n_ih => Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range (Nat.pair (k' + 1) (encode c)))) (k', c) (Nat.pair (unpair n).1 ((unpair n).2 + 1))) x) c = evaln (k' + 1) c n
-- Mathlib/Computability/PartrecCode.lean
this : Primrec₂ fun x n => let a := ofNat (ℕ × Code) n; List.map (evaln a.1 a.2) (List.range a.1)
x✝ : (ℕ × Code) × ℕ
k : ℕ
c : Code
n : ℕ
\u22a2 (((let a := ofNat (ℕ × Code) (encode ((k, c), n).1); List.map (evaln a.1 a.2) (List.range a.1)).get? ((k, c), n).2).bind fun b => (((k, c), n), b).2) = evaln ((k, c), n).1.1 ((k, c), n).1.2 ((k, c), n).2
-- Mathlib/Data/Set/Image.lean
α : Type u_1
s : Set α
\u22a2 range val = s
-- Mathlib/Data/Set/Image.lean
α : Type u_1
s : Set α
\u22a2 val '' univ = s
-- Mathlib/MeasureTheory/Integral/Average.lean
α : Type u_1
E : Type u_2
F : Type u_3
m0 : MeasurableSpace α
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
μ ν : Measure α
s✝ t : Set α
f✝ g f : α → E
s : Set α
h : μ s ≠ ⊤
\u22a2 (μ s).toReal • ⨍ (x : α) in s, f x ∂μ = ∫ (x : α) in s, f x ∂μ
-- Mathlib/MeasureTheory/Integral/Average.lean
α : Type u_1
E : Type u_2
F : Type u_3
m0 : MeasurableSpace α
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
μ ν : Measure α
s✝ t : Set α
f✝ g f : α → E
s : Set α
h : μ s ≠ ⊤
this : Fact (μ s < ⊤)
\u22a2 (μ s).toReal • ⨍ (x : α) in s, f x ∂μ = ∫ (x : α) in s, f x ∂μ
-- Mathlib/RingTheory/PowerSeries/Derivative.lean
R : Type u_1
inst✝ : CommSemiring R
f : R⟦X⟧
n d : ℕ
\u22a2 (trunc n f.derivativeFun).coeff d = (derivative (trunc (n + 1) f)).coeff d
-- Mathlib/RingTheory/PowerSeries/Derivative.lean
R : Type u_1
inst✝ : CommSemiring R
f : R⟦X⟧
n d : ℕ
\u22a2 (if d < n then (coeff R d) f.derivativeFun else 0) = (derivative (trunc (n + 1) f)).coeff d
-- Mathlib/SetTheory/Ordinal/Arithmetic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
a : Ordinal.{u_4}
\u22a2 0 - a = 0
-- Mathlib/SetTheory/Ordinal/Arithmetic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
a : Ordinal.{u_4}
\u22a2 0 - a ≤ 0
-- Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean
x : ℝ≥0
y : ℝ
h✝¹ : x ≠ 0 ∨ 0 ≤ y
h✝ : 0 ≤ y
h : 0 = y
\u22a2 ContinuousAt (fun z => z ^ 0) x
-- Mathlib/SetTheory/Ordinal/CantorNormalForm.lean
C : Ordinal.{u_2} → Sort u_1
b : Ordinal.{u_2}
H0 : C 0
H : (o : Ordinal.{u_2}) → o ≠ 0 → C (o % b ^ log b o) → C o
\u22a2 b.CNFRec H0 H 0 = H0
-- Mathlib/SetTheory/Ordinal/CantorNormalForm.lean
C : Ordinal.{u_2} → Sort u_1
b : Ordinal.{u_2}
H0 : C 0
H : (o : Ordinal.{u_2}) → o ≠ 0 → C (o % b ^ log b o) → C o
\u22a2 ⋯.mpr H0 = H0
-- Mathlib/RingTheory/FinitePresentation.lean
R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : CommRing B
inst✝¹ : Algebra R B
inst✝ : IsNoetherianRing R
h : FiniteType R A
\u22a2 FinitePresentation R A
-- Mathlib/RingTheory/FinitePresentation.lean
R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : CommRing B
inst✝¹ : Algebra R B
inst✝ : IsNoetherianRing R
h : FiniteType R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] A
hf : Surjective ⇑f
\u22a2 FinitePresentation R A
-- Mathlib/RingTheory/FinitePresentation.lean
R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : CommRing B
inst✝¹ : Algebra R B
inst✝ : IsNoetherianRing R
h : FiniteType R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] A
hf : Surjective ⇑f
hnoet : IsNoetherianRing (MvPolynomial (Fin n) R)
\u22a2 (RingHom.ker f.toRingHom).FG
-- Mathlib/RingTheory/FinitePresentation.lean
R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : CommRing B
inst✝¹ : Algebra R B
inst✝ : IsNoetherianRing R
h : FiniteType R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] A
hf : Surjective ⇑f
hnoet : IsNoetherian (MvPolynomial (Fin n) R) (MvPolynomial (Fin n) R)
\u22a2 (RingHom.ker f.toRingHom).FG
-- Mathlib/RingTheory/FinitePresentation.lean
R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : CommRing B
inst✝¹ : Algebra R B
inst✝ : IsNoetherianRing R
h : FiniteType R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] A
hf : Surjective ⇑f
hnoet : IsNoetherian (MvPolynomial (Fin n) R) (MvPolynomial (Fin n) R)
this : Module (MvPolynomial (Fin n) R) (MvPolynomial (Fin n) R) := Semiring.toModule
\u22a2 (RingHom.ker f.toRingHom).FG
-- Mathlib/RingTheory/FinitePresentation.lean
R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : CommRing B
inst✝¹ : Algebra R B
inst✝ : IsNoetherianRing R
h : FiniteType R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] A
hf : Surjective ⇑f
hnoet : IsNoetherian (MvPolynomial (Fin n) R) (MvPolynomial (Fin n) R)
this✝ : Module (MvPolynomial (Fin n) R) (MvPolynomial (Fin n) R) := Semiring.toModule
this : Submodule.FG (RingHom.ker f.toRingHom)
\u22a2 (RingHom.ker f.toRingHom).FG
-- Mathlib/Algebra/Group/Invertible/Defs.lean
α : Type u
inst✝¹ : Monoid α
a b : α
inst✝ : Invertible b
\u22a2 a * b * ⅟b = a
-- Mathlib/AlgebraicGeometry/EllipticCurve/Group.lean
R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
W✝ : Affine R
f : R →+* S
F : Type u
inst✝ : Field F
W : Affine F
\u22a2 zero + zero = 0 ↔ zero = -zero
-- Mathlib/AlgebraicGeometry/EllipticCurve/Group.lean
R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
W✝ : Affine R
f : R →+* S
F : Type u
inst✝ : Field F
W : Affine F
x₂ y₂ : F
h✝ : W.Nonsingular x₂ y₂
\u22a2 zero + some h✝ = 0 ↔ zero = -some h✝
-- Mathlib/AlgebraicGeometry/EllipticCurve/Group.lean
R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
W✝ : Affine R
f : R →+* S
F : Type u
inst✝ : Field F
W : Affine F
x₁ y₁ : F
h✝ : W.Nonsingular x₁ y₁
\u22a2 some h✝ + zero = 0 ↔ some h✝ = -zero
-- Mathlib/AlgebraicGeometry/EllipticCurve/Group.lean
R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
W✝ : Affine R
f : R →+* S
F : Type u
inst✝ : Field F
W : Affine F
x₁ y₁ : F
h✝¹ : W.Nonsingular x₁ y₁
x₂ y₂ : F
h✝ : W.Nonsingular x₂ y₂
\u22a2 some h✝¹ + some h✝ = 0 ↔ some h✝¹ = -some h✝
-- Mathlib/CategoryTheory/Shift/ShiftSequence.lean
C : Type u_1
A : Type u_2
inst✝⁶ : Category.{u_6, u_1} C
inst✝⁵ : Category.{u_5, u_2} A
F : C ⥤ A
M : Type u_3
inst✝⁴ : AddMonoid M
inst✝³ : HasShift C M
G : Type u_4
inst✝² : AddGroup G
inst✝¹ : HasShift C G
inst✝ : F.ShiftSequence M
a : M
X : C
\u22a2 (F.shiftIso 0 a a ⋯).hom.app X = (F.shift a).map ((shiftFunctorZero C M).hom.app X)
-- Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean
k✝ : Type u_1
V1 : Type u_2
P1 : Type u_3
V2 : Type u_4
P2 : Type u_5
V3 : Type u_6
P3 : Type u_7
V4 : Type u_8
P4 : Type u_9
inst✝¹³ : Ring k✝
inst✝¹² : AddCommGroup V1
inst✝¹¹ : Module k✝ V1
inst✝¹⁰ : AffineSpace V1 P1
inst✝⁹ : AddCommGroup V2
inst✝⁸ : Module k✝ V2
inst✝⁷ : AffineSpace V2 P2
inst✝⁶ : AddCommGroup V3
inst✝⁵ : Module k✝ V3
inst✝⁴ : AffineSpace V3 P3
inst✝³ : AddCommGroup V4
inst✝² : Module k✝ V4
inst✝¹ : AffineSpace V4 P4
k : Type u_10
inst✝ : LinearOrderedField k
f : k →ᵃ[k] k
a b : k
\u22a2 ⇑f '' Set.uIcc a b = Set.uIcc (f a) (f b)
-- Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean
k✝ : Type u_1
V1 : Type u_2
P1 : Type u_3
V2 : Type u_4
P2 : Type u_5
V3 : Type u_6
P3 : Type u_7
V4 : Type u_8
P4 : Type u_9
inst✝¹³ : Ring k✝
inst✝¹² : AddCommGroup V1
inst✝¹¹ : Module k✝ V1
inst✝¹⁰ : AffineSpace V1 P1
inst✝⁹ : AddCommGroup V2
inst✝⁸ : Module k✝ V2
inst✝⁷ : AffineSpace V2 P2
inst✝⁶ : AddCommGroup V3
inst✝⁵ : Module k✝ V3
inst✝⁴ : AffineSpace V3 P3
inst✝³ : AddCommGroup V4
inst✝² : Module k✝ V4
inst✝¹ : AffineSpace V4 P4
k : Type u_10
inst✝ : LinearOrderedField k
f : k →ᵃ[k] k
a b : k
this : ⇑f = (fun x => x + f 0) ∘ fun x => x * (f 1 - f 0)
\u22a2 ⇑f '' Set.uIcc a b = Set.uIcc (f a) (f b)
-- Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean
k✝ : Type u_1
V1 : Type u_2
P1 : Type u_3
V2 : Type u_4
P2 : Type u_5
V3 : Type u_6
P3 : Type u_7
V4 : Type u_8
P4 : Type u_9
inst✝¹³ : Ring k✝
inst✝¹² : AddCommGroup V1
inst✝¹¹ : Module k✝ V1
inst✝¹⁰ : AffineSpace V1 P1
inst✝⁹ : AddCommGroup V2
inst✝⁸ : Module k✝ V2
inst✝⁷ : AffineSpace V2 P2
inst✝⁶ : AddCommGroup V3
inst✝⁵ : Module k✝ V3
inst✝⁴ : AffineSpace V3 P3
inst✝³ : AddCommGroup V4
inst✝² : Module k✝ V4
inst✝¹ : AffineSpace V4 P4
k : Type u_10
inst✝ : LinearOrderedField k
f : k →ᵃ[k] k
a b : k
this : ⇑f = (fun x => x + f 0) ∘ fun x => x * (f 1 - f 0)
\u22a2 (fun x => x + f 0) '' ((fun x => x * (f 1 - f 0)) '' Set.uIcc a b) = Set.uIcc (((fun x => x + f 0) ∘ fun x => x * (f 1 - f 0)) a) (((fun x => x + f 0) ∘ fun x => x * (f 1 - f 0)) b)
-- Mathlib/Data/List/Nodup.lean
α : Type u
β : Type v
l✝ l₁ l₂ : List α
r : α → α → Prop
a b : α
l : List α
\u22a2 l.Nodup ↔ ∀ (i j : ℕ), i < j → j < l.length → l[i]? ≠ l[j]?
-- Mathlib/Data/List/Nodup.lean
α : Type u
β : Type v
l✝ l₁ l₂ : List α
r : α → α → Prop
a b : α
l : List α
\u22a2 (∀ (i j : ℕ) (_hi : i < l.length) (_hj : j < l.length), i < j → l[i] ≠ l[j]) ↔ ∀ (i j : ℕ), i < j → j < l.length → l[i]? ≠ l[j]?
-- Mathlib/SetTheory/Ordinal/Exponential.lean
a b c : Ordinal.{u_1}
h : b ≤ c
\u22a2 a ^ c = a ^ b * a ^ (c - b)
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
x : ℝ
h : sin x = 0
h₃ : x - ↑⌊x / π⌋ * π > 0
\u22a2 0 = sin (x - ↑⌊x / π⌋ * π)
-- Mathlib/Algebra/Polynomial/Degree/Definitions.lean
R : Type u
S : Type v
a b c d : R
n m : ℕ
inst✝ : Semiring R
p✝ q r p : R[X]
\u22a2 ∑ i ∈ p.support, (monomial i) (p.coeff i) = ∑ i ∈ p.support, C (p.coeff i) * X ^ i
-- Mathlib/Analysis/MeanInequalities.lean
ι : Type u
s : Finset ι
f g : ι → ℝ≥0
A B : ℝ≥0
p q : ℝ
hpq : p.IsConjExponent q
hf : HasSum (fun i => f i ^ p) (A ^ p)
hg : HasSum (fun i => g i ^ q) (B ^ q)
\u22a2 ∃ C ≤ A * B, HasSum (fun i => f i * g i) C
-- Mathlib/Analysis/MeanInequalities.lean
ι : Type u
s : Finset ι
f g : ι → ℝ≥0
A B : ℝ≥0
p q : ℝ
hpq : p.IsConjExponent q
hf : HasSum (fun i => f i ^ p) (A ^ p)
hg : HasSum (fun i => g i ^ q) (B ^ q)
H₁ : Summable fun i => f i * g i
H₂ : ∑' (i : ι), f i * g i ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)
\u22a2 ∃ C ≤ A * B, HasSum (fun i => f i * g i) C
-- Mathlib/Analysis/MeanInequalities.lean
ι : Type u
s : Finset ι
f g : ι → ℝ≥0
A B : ℝ≥0
p q : ℝ
hpq : p.IsConjExponent q
hf : HasSum (fun i => f i ^ p) (A ^ p)
hg : HasSum (fun i => g i ^ q) (B ^ q)
H₁ : Summable fun i => f i * g i
H₂ : ∑' (i : ι), f i * g i ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)
hA : A = (∑' (i : ι), f i ^ p) ^ (1 / p)
\u22a2 ∃ C ≤ A * B, HasSum (fun i => f i * g i) C
-- Mathlib/Analysis/MeanInequalities.lean
ι : Type u
s : Finset ι
f g : ι → ℝ≥0
A B : ℝ≥0
p q : ℝ
hpq : p.IsConjExponent q
hf : HasSum (fun i => f i ^ p) (A ^ p)
hg : HasSum (fun i => g i ^ q) (B ^ q)
H₁ : Summable fun i => f i * g i
H₂ : ∑' (i : ι), f i * g i ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)
hA : A = (∑' (i : ι), f i ^ p) ^ (1 / p)
hB : B = (∑' (i : ι), g i ^ q) ^ (1 / q)
\u22a2 ∃ C ≤ A * B, HasSum (fun i => f i * g i) C
-- Mathlib/SetTheory/Game/Basic.lean
x✝ y✝ z✝ : PGame
xl xr : Type u_1
xL : xl → PGame
xR : xr → PGame
yl yr : Type u_1
yL : yl → PGame
yR : yr → PGame
zl zr : Type u_1
zL : zl → PGame
zR : zr → PGame
x : PGame := mk xl xr xL xR
y : PGame := mk yl yr yL yR
z : PGame := mk zl zr zL zR
\u22a2 ⟦mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR⟧ = ⟦mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR)⟧
-- Mathlib/SetTheory/Game/Basic.lean
x✝ y✝ z✝ : PGame
xl xr : Type u_1
xL : xl → PGame
xR : xr → PGame
yl yr : Type u_1
yL : yl → PGame
yR : yr → PGame
zl zr : Type u_1
zL : zl → PGame
zR : zr → PGame
x : PGame := mk xl xr xL xR
y : PGame := mk yl yr yL yR
z : PGame := mk zl zr zL zR
\u22a2 (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR).LeftMoves ≃ (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR)).LeftMoves
-- Mathlib/SetTheory/Game/Basic.lean
x✝ y✝ z✝ : PGame
xl xr : Type u_1
xL : xl → PGame
xR : xr → PGame
yl yr : Type u_1
yL : yl → PGame
yR : yr → PGame
zl zr : Type u_1
zL : zl → PGame
zR : zr → PGame
x : PGame := mk xl xr xL xR
y : PGame := mk yl yr yL yR
z : PGame := mk zl zr zL zR
\u22a2 (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR).RightMoves ≃ (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR)).RightMoves
-- Mathlib/SetTheory/Game/Basic.lean
x✝ y✝ z✝ : PGame
xl xr : Type u_1
xL : xl → PGame
xR : xr → PGame
yl yr : Type u_1
yL : yl → PGame
yR : yr → PGame
zl zr : Type u_1
zL : zl → PGame
zR : zr → PGame
x : PGame := mk xl xr xL xR
y : PGame := mk yl yr yL yR
z : PGame := mk zl zr zL zR
\u22a2 ∀ (i : (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR).LeftMoves), ⟦(mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR).moveLeft i⟧ = ⟦(mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR)).moveLeft (?refine_1 i)⟧
-- Mathlib/SetTheory/Game/Basic.lean
x✝ y✝ z✝ : PGame
xl xr : Type u_1
xL : xl → PGame
xR : xr → PGame
yl yr : Type u_1
yL : yl → PGame
yR : yr → PGame
zl zr : Type u_1
zL : zl → PGame
zR : zr → PGame
x : PGame := mk xl xr xL xR
y : PGame := mk yl yr yL yR
z : PGame := mk zl zr zL zR
\u22a2 ∀ (j : (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR).RightMoves), ⟦(mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR).moveRight j⟧ = ⟦(mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR)).moveRight (?refine_2 j)⟧
-- Mathlib/RingTheory/HahnSeries/Multiplication.lean
Γ : Type u_1
Γ' : Type u_2
R : Type u_3
V : Type u_4
inst✝² : PartialOrder Γ
inst✝¹ : Zero V
inst✝ : SMul R V
x y : HahnModule Γ R V
\u22a2 ((of R).symm x).coeff = ((of R).symm y).coeff ↔ x = y
-- Mathlib/RingTheory/Adjoin/Basic.lean
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝² : CommSemiring R
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
s t : Set A
\u22a2 adjoin R (s ∪ t) = Subalgebra.restrictScalars R (adjoin (↥(adjoin R s)) t)
-- Mathlib/Algebra/Order/Ring/Abs.lean
α : Type u_1
R : Type u_2
inst✝ : LinearOrderedRing R
a b : R
n : ℕ
\u22a2 a ^ n = -1 ↔ a = -1 ∧ Odd n
-- Mathlib/AlgebraicTopology/DoldKan/Homotopies.lean
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
q n m : ℕ
hnq : n < q
hnm : c.Rel m n
\u22a2 0 ≫ eqToHom ⋯ = 0
-- Mathlib/MeasureTheory/Measure/WithDensity.lean
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
g : α → ℝ≥0∞
\u22a2 μ.withDensity (f + g) = μ.withDensity f + μ.withDensity g
-- Mathlib/MeasureTheory/Measure/WithDensity.lean
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
g : α → ℝ≥0∞
s : Set α
hs : MeasurableSet s
\u22a2 (μ.withDensity (f + g)) s = (μ.withDensity f + μ.withDensity g) s
-- Mathlib/MeasureTheory/Measure/WithDensity.lean
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
g : α → ℝ≥0∞
s : Set α
hs : MeasurableSet s
\u22a2 ∫⁻ (a : α) in s, (f + g) a ∂μ = ∫⁻ (a : α) in s, f a + g a ∂μ
-- Mathlib/MeasureTheory/Measure/Typeclasses.lean
α : Type u_1
β✝ : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝⁴ : MeasurableSpace β✝
μ ν ν₁ ν₂ : Measure α
s t : Set α
β : Type u_5
inst✝³ : TopologicalSpace β
inst✝² : T1Space β
inst✝¹ : SecondCountableTopology β
inst✝ : Nonempty β
f : α → β
h : ∀ (b : β), ∃ᵐ (x : α) ∂μ, f x ≠ b
m : OuterMeasure β := (OuterMeasure.map f) μ.toOuterMeasure
\u22a2 ∃ a b, a ≠ b ∧ (∀ s ∈ 𝓝 a, 0 < μ (f ⁻¹' s)) ∧ ∀ t ∈ 𝓝 b, 0 < μ (f ⁻¹' t)
-- Mathlib/MeasureTheory/Measure/Typeclasses.lean
α : Type u_1
β✝ : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝⁴ : MeasurableSpace β✝
μ ν ν₁ ν₂ : Measure α
s t : Set α
β : Type u_5
inst✝³ : TopologicalSpace β
inst✝² : T1Space β
inst✝¹ : SecondCountableTopology β
inst✝ : Nonempty β
f : α → β
m : OuterMeasure β := (OuterMeasure.map f) μ.toOuterMeasure
h : ∀ (b : β), m {b}ᶜ ≠ 0
\u22a2 ∃ a b, a ≠ b ∧ (∀ s ∈ 𝓝 a, 0 < μ (f ⁻¹' s)) ∧ ∀ t ∈ 𝓝 b, 0 < μ (f ⁻¹' t)
-- Mathlib/MeasureTheory/Measure/Typeclasses.lean
α : Type u_1
β✝ : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝⁴ : MeasurableSpace β✝
μ ν ν₁ ν₂ : Measure α
s t : Set α
β : Type u_5
inst✝³ : TopologicalSpace β
inst✝² : T1Space β
inst✝¹ : SecondCountableTopology β
inst✝ : Nonempty β
f : α → β
m : OuterMeasure β := (OuterMeasure.map f) μ.toOuterMeasure
h : ∀ (b : β), m {b}ᶜ ≠ 0
inhabited_h : Inhabited β
\u22a2 ∃ a b, a ≠ b ∧ (∀ s ∈ 𝓝 a, 0 < μ (f ⁻¹' s)) ∧ ∀ t ∈ 𝓝 b, 0 < μ (f ⁻¹' t)
-- Mathlib/MeasureTheory/Measure/Typeclasses.lean
α : Type u_1
β✝ : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝⁴ : MeasurableSpace β✝
μ ν ν₁ ν₂ : Measure α
s t : Set α
β : Type u_5
inst✝³ : TopologicalSpace β
inst✝² : T1Space β
inst✝¹ : SecondCountableTopology β
inst✝ : Nonempty β
f : α → β
m : OuterMeasure β := (OuterMeasure.map f) μ.toOuterMeasure
h : ∀ (b : β), m {b}ᶜ ≠ 0
inhabited_h : Inhabited β
this : m univ ≠ 0
\u22a2 ∃ a b, a ≠ b ∧ (∀ s ∈ 𝓝 a, 0 < μ (f ⁻¹' s)) ∧ ∀ t ∈ 𝓝 b, 0 < μ (f ⁻¹' t)
-- Mathlib/MeasureTheory/Measure/Typeclasses.lean
α : Type u_1
β✝ : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝⁴ : MeasurableSpace β✝
μ ν ν₁ ν₂ : Measure α
s t : Set α
β : Type u_5
inst✝³ : TopologicalSpace β
inst✝² : T1Space β
inst✝¹ : SecondCountableTopology β
inst✝ : Nonempty β
f : α → β
m : OuterMeasure β := (OuterMeasure.map f) μ.toOuterMeasure
h : ∀ (b : β), m {b}ᶜ ≠ 0
inhabited_h : Inhabited β
this : m univ ≠ 0
b : β
hb : ∀ t ∈ 𝓝[univ] b, 0 < m t
\u22a2 ∃ a b, a ≠ b ∧ (∀ s ∈ 𝓝 a, 0 < μ (f ⁻¹' s)) ∧ ∀ t ∈ 𝓝 b, 0 < μ (f ⁻¹' t)
-- Mathlib/MeasureTheory/Measure/Typeclasses.lean
α : Type u_1
β✝ : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝⁴ : MeasurableSpace β✝
μ ν ν₁ ν₂ : Measure α
s t : Set α
β : Type u_5
inst✝³ : TopologicalSpace β
inst✝² : T1Space β
inst✝¹ : SecondCountableTopology β
inst✝ : Nonempty β
f : α → β
m : OuterMeasure β := (OuterMeasure.map f) μ.toOuterMeasure
h : ∀ (b : β), m {b}ᶜ ≠ 0
inhabited_h : Inhabited β
this : m univ ≠ 0
b : β
hb : ∀ t ∈ 𝓝 b, 0 < m t
\u22a2 ∃ a b, a ≠ b ∧ (∀ s ∈ 𝓝 a, 0 < μ (f ⁻¹' s)) ∧ ∀ t ∈ 𝓝 b, 0 < μ (f ⁻¹' t)
-- Mathlib/MeasureTheory/Measure/Typeclasses.lean
α : Type u_1
β✝ : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝⁴ : MeasurableSpace β✝
μ ν ν₁ ν₂ : Measure α
s t : Set α
β : Type u_5
inst✝³ : TopologicalSpace β
inst✝² : T1Space β
inst✝¹ : SecondCountableTopology β
inst✝ : Nonempty β
f : α → β
m : OuterMeasure β := (OuterMeasure.map f) μ.toOuterMeasure
h : ∀ (b : β), m {b}ᶜ ≠ 0
inhabited_h : Inhabited β
this : m univ ≠ 0
b : β
hb : ∀ t ∈ 𝓝 b, 0 < m t
a : β
hab : a ≠ b
ha : ∀ t ∈ 𝓝[{b}ᶜ] a, 0 < m t
\u22a2 ∃ a b, a ≠ b ∧ (∀ s ∈ 𝓝 a, 0 < μ (f ⁻¹' s)) ∧ ∀ t ∈ 𝓝 b, 0 < μ (f ⁻¹' t)
-- Mathlib/MeasureTheory/Measure/Typeclasses.lean
α : Type u_1
β✝ : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝⁴ : MeasurableSpace β✝
μ ν ν₁ ν₂ : Measure α
s t : Set α
β : Type u_5
inst✝³ : TopologicalSpace β
inst✝² : T1Space β
inst✝¹ : SecondCountableTopology β
inst✝ : Nonempty β
f : α → β
m : OuterMeasure β := (OuterMeasure.map f) μ.toOuterMeasure
h : ∀ (b : β), m {b}ᶜ ≠ 0
inhabited_h : Inhabited β
this : m univ ≠ 0
b : β
hb : ∀ t ∈ 𝓝 b, 0 < m t
a : β
hab : a ≠ b
ha : ∀ t ∈ 𝓝 a, 0 < m t
\u22a2 ∃ a b, a ≠ b ∧ (∀ s ∈ 𝓝 a, 0 < μ (f ⁻¹' s)) ∧ ∀ t ∈ 𝓝 b, 0 < μ (f ⁻¹' t)
-- Mathlib/Analysis/MeanInequalitiesPow.lean
ι : Type u
s : Finset ι
w₁ w₂ z₁ z₂ : ℝ≥0∞
hw' : w₁ + w₂ = 1
p : ℝ
hp : 1 ≤ p
\u22a2 (w₁ * z₁ + w₂ * z₂) ^ p ≤ w₁ * z₁ ^ p + w₂ * z₂ ^ p
-- Mathlib/Analysis/MeanInequalitiesPow.lean
ι : Type u
s : Finset ι
w₁ w₂ z₁ z₂ : ℝ≥0∞
hw' : w₁ + w₂ = 1
p : ℝ
hp : 1 ≤ p
h : (∑ i : Fin (Nat.succ 0).succ, ![w₁, w₂] i * ![z₁, z₂] i) ^ p ≤ ∑ i : Fin (Nat.succ 0).succ, ![w₁, w₂] i * ![z₁, z₂] i ^ p
\u22a2 (w₁ * z₁ + w₂ * z₂) ^ p ≤ w₁ * z₁ ^ p + w₂ * z₂ ^ p
-- Mathlib/Analysis/MeanInequalitiesPow.lean
ι : Type u
s : Finset ι
w₁ w₂ z₁ z₂ : ℝ≥0∞
hw' : w₁ + w₂ = 1
p : ℝ
hp : 1 ≤ p
\u22a2 ∑ i : Fin (Nat.succ 0).succ, ![w₁, w₂] i = 1
-- Mathlib/GroupTheory/PushoutI.lean
ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Monoid (G i)
inst✝ : Monoid H
φ : (i : ι) → H →* G i
i : ι
x : H
\u22a2 ((of i).comp (φ i)) x = (base φ) x
-- Mathlib/GroupTheory/PushoutI.lean
ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Monoid (G i)
inst✝ : Monoid H
φ : (i : ι) → H →* G i
i : ι
x : H
\u22a2 (con φ) ((inl.comp CoprodI.of) ((φ i) x)) (inr x)
-- Mathlib/GroupTheory/PushoutI.lean
ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Monoid (G i)
inst✝ : Monoid H
φ : (i : ι) → H →* G i
i : ι
x : H
\u22a2 ∃ i_1 x', (inl.comp CoprodI.of) ((φ i) x) = inl (CoprodI.of ((φ i_1) x')) ∧ inr x = inr x'
-- Mathlib/GroupTheory/PushoutI.lean
ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Monoid (G i)
inst✝ : Monoid H
φ : (i : ι) → H →* G i
i : ι
x : H
\u22a2 ∃ i_1 x', inl (CoprodI.of ((φ i) x)) = inl (CoprodI.of ((φ i_1) x')) ∧ inr x = inr x'
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
x : ℂ
\u22a2 cos (x + ↑π / 2) = -sin x
-- Mathlib/CategoryTheory/Abelian/Exact.lean
C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
inst✝⁴ : Abelian C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
inst✝² : Abelian D
F : C ⥤ D
inst✝¹ : F.PreservesZeroMorphisms
inst✝ : F.Faithful
S : ShortComplex C
hS : (S.map F).Exact
\u22a2 S.Exact
-- Mathlib/CategoryTheory/Abelian/Exact.lean
C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
inst✝⁴ : Abelian C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
inst✝² : Abelian D
F : C ⥤ D
inst✝¹ : F.PreservesZeroMorphisms
inst✝ : F.Faithful
S : ShortComplex C
hS : kernel.ι (S.map F).g ≫ cokernel.π (S.map F).f = 0
\u22a2 kernel.ι S.g ≫ cokernel.π S.f = 0