ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Topology/ExtremallyDisconnected.lean
X✝ : Type u
inst✝³ : TopologicalSpace X✝
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : ExtremallyDisconnected X
e : X ≃ₜ Y
U : Set Y
hU : IsOpen U

⊢ IsOpen (closure U)
-- Mathlib/Topology/ExtremallyDisconnected.lean
X✝ : Type u
inst✝³ : TopologicalSpace X✝
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : ExtremallyDisconnected X
e : X ≃ₜ Y
U : Set Y
hU : IsOpen U

⊢ IsOpen (closure (⇑e.symm '' U))
-- Mathlib/RingTheory/UniqueFactorizationDomain.lean
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a p : α
ha0 : a ≠ 0
hp : Irreducible p
hd : p ∣ a

⊢ Associates.mk p ∈ (Associates.mk a).factors
-- Mathlib/RingTheory/UniqueFactorizationDomain.lean
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a p : α
ha0 : a ≠ 0
hp : Irreducible p
hd : p ∣ a

⊢ Associates.mk p ∈ ↑(factors' a)
-- Mathlib/Data/Finset/Lattice.lean
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
ι : Type u_5
κ : Type u_6
inst✝ : SemilatticeSup α
s : Finset β
H : s.Nonempty
f : β → α
a : α

⊢ (s.sup' H fun x => a) = a
-- Mathlib/Data/Finset/Lattice.lean
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
ι : Type u_5
κ : Type u_6
inst✝ : SemilatticeSup α
s : Finset β
H : s.Nonempty
f : β → α
a : α

⊢ (s.sup' H fun x => a) ≤ a
-- Mathlib/Data/Finset/Lattice.lean
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
ι : Type u_5
κ : Type u_6
inst✝ : SemilatticeSup α
s : Finset β
H : s.Nonempty
f : β → α
a : α

⊢ a ≤ s.sup' H fun x => a
-- Mathlib/Analysis/Calculus/SmoothSeries.lean
α : Type u_1
β : Type u_2
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
g g' : α → 𝕜 → F
v : ℕ → α → ℝ
s : Set E
t : Set 𝕜
x₀ x✝ : E
y₀ y : 𝕜
N : ℕ∞
hf : ∀ (i : α), ContDiff 𝕜 N (f i)
hv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)
h'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i
k : ℕ
hk : ↑k ≤ N
x : E

⊢ iteratedFDeriv 𝕜 k (fun y => ∑' (n : α), f n y) x = ∑' (n : α), iteratedFDeriv 𝕜 k (f n) x
-- Mathlib/Data/ZMod/Basic.lean
m n✝ n : ℕ
a : ZMod (n + 1)
h : ¬a.val ≤ n.succ / 2

⊢ a.valMinAbs * 2 = ↑(n + 1) ↔ 2 * a.val = n + 1
-- Mathlib/Data/ZMod/Basic.lean
m n✝ n : ℕ
a : ZMod (n + 1)
h : ¬a.val ≤ n.succ / 2

⊢ ¬a.valMinAbs * 2 = ↑(n + 1)
-- Mathlib/Data/ZMod/Basic.lean
m n✝ n : ℕ
a : ZMod (n + 1)
h : ¬a.val ≤ n.succ / 2

⊢ 2 * a.val = n + 1 → a.val ≤ n.succ / 2
-- Mathlib/Data/Complex/Exponential.lean
x✝ y x : ℝ
hx : 0 ≤ x

⊢ 1 ≤ rexp x
-- Mathlib/Topology/UniformSpace/Cauchy.lean
α : Type u
β : Type v
uniformSpace : UniformSpace α
inst✝ : Preorder β
u : β → α
h : CauchySeq u

⊢ Tendsto (Prod.map u u) atTop (𝓤 α)
-- Mathlib/RingTheory/DedekindDomain/Ideal.lean
R : Type u_1
A : Type u_2
K : Type u_3
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing A
inst✝⁸ : Field K
R₁ : Type u_4
inst✝⁷ : CommRing R₁
inst✝⁶ : IsDomain R₁
inst✝⁵ : Algebra R₁ K
inst✝⁴ : IsFractionRing R₁ K
I✝ J : FractionalIdeal R₁⁰ K
K' : Type u_5
inst✝³ : Field K'
inst✝² : Algebra R₁ K'
inst✝¹ : IsFractionRing R₁ K'
I : FractionalIdeal R₁⁰ K
inst✝ : (↑I).IsPrincipal
h : I ≠ 0

⊢ (↑I⁻¹).IsPrincipal
-- Mathlib/RingTheory/DedekindDomain/Ideal.lean
R : Type u_1
A : Type u_2
K : Type u_3
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing A
inst✝⁸ : Field K
R₁ : Type u_4
inst✝⁷ : CommRing R₁
inst✝⁶ : IsDomain R₁
inst✝⁵ : Algebra R₁ K
inst✝⁴ : IsFractionRing R₁ K
I✝ J : FractionalIdeal R₁⁰ K
K' : Type u_5
inst✝³ : Field K'
inst✝² : Algebra R₁ K'
inst✝¹ : IsFractionRing R₁ K'
I : FractionalIdeal R₁⁰ K
inst✝ : (↑I).IsPrincipal
h : I ≠ 0

⊢ ∃ x, I⁻¹ = spanSingleton R₁⁰ x
-- Mathlib/RingTheory/DedekindDomain/Ideal.lean
R : Type u_1
A : Type u_2
K : Type u_3
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing A
inst✝⁸ : Field K
R₁ : Type u_4
inst✝⁷ : CommRing R₁
inst✝⁶ : IsDomain R₁
inst✝⁵ : Algebra R₁ K
inst✝⁴ : IsFractionRing R₁ K
I✝ J : FractionalIdeal R₁⁰ K
K' : Type u_5
inst✝³ : Field K'
inst✝² : Algebra R₁ K'
inst✝¹ : IsFractionRing R₁ K'
I : FractionalIdeal R₁⁰ K
inst✝ : (↑I).IsPrincipal
h : I ≠ 0

⊢ I⁻¹ = spanSingleton R₁⁰ (generator ↑I)⁻¹
-- Mathlib/RingTheory/DedekindDomain/Ideal.lean
R : Type u_1
A : Type u_2
K : Type u_3
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing A
inst✝⁸ : Field K
R₁ : Type u_4
inst✝⁷ : CommRing R₁
inst✝⁶ : IsDomain R₁
inst✝⁵ : Algebra R₁ K
inst✝⁴ : IsFractionRing R₁ K
I✝ J : FractionalIdeal R₁⁰ K
K' : Type u_5
inst✝³ : Field K'
inst✝² : Algebra R₁ K'
inst✝¹ : IsFractionRing R₁ K'
I : FractionalIdeal R₁⁰ K
inst✝ : (↑I).IsPrincipal
h : I ≠ 0
hI : I * spanSingleton R₁⁰ (generator ↑I)⁻¹ = 1

⊢ I⁻¹ = spanSingleton R₁⁰ (generator ↑I)⁻¹
-- Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean
a b : ℝ
h : a ≠ 0

⊢ ContinuousAt (fun x => a ^ x) b
-- Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean
a b : ℝ
h : a ≠ 0

⊢ ContinuousAt (fun x => (↑a ^ ↑x).re) b
-- Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean
a b : ℝ
h : a ≠ 0

⊢ ContinuousAt (fun x => ↑a ^ ↑x) b
-- Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean
a b : ℝ
h : a ≠ 0

⊢ ↑a ≠ 0
-- Mathlib/RingTheory/Trace/Basic.lean
R : Type u_1
S : Type u_2
T : Type u_3
inst✝¹⁸ : CommRing R
inst✝¹⁷ : CommRing S
inst✝¹⁶ : CommRing T
inst✝¹⁵ : Algebra R S
inst✝¹⁴ : Algebra R T
K : Type u_4
L : Type u_5
inst✝¹³ : Field K
inst✝¹² : Field L
inst✝¹¹ : Algebra K L
ι κ : Type w
inst✝¹⁰ : Fintype ι
F : Type u_6
inst✝⁹ : Field F
inst✝⁸ : Algebra R L
inst✝⁷ : Algebra L F
inst✝⁶ : Algebra R F
inst✝⁵ : IsScalarTower R L F
A : Type u_7
B : Type u_8
C : Type u_9
inst✝⁴ : CommRing A
inst✝³ : CommRing B
inst✝² : CommRing C
inst✝¹ : Algebra A B
inst✝ : Algebra A C
e : B ≃ₐ[A] C
x : B

⊢ (trace A C) (e x) = (trace A B) x
-- Mathlib/RingTheory/Trace/Basic.lean
R : Type u_1
S : Type u_2
T : Type u_3
inst✝¹⁸ : CommRing R
inst✝¹⁷ : CommRing S
inst✝¹⁶ : CommRing T
inst✝¹⁵ : Algebra R S
inst✝¹⁴ : Algebra R T
K : Type u_4
L : Type u_5
inst✝¹³ : Field K
inst✝¹² : Field L
inst✝¹¹ : Algebra K L
ι κ : Type w
inst✝¹⁰ : Fintype ι
F : Type u_6
inst✝⁹ : Field F
inst✝⁸ : Algebra R L
inst✝⁷ : Algebra L F
inst✝⁶ : Algebra R F
inst✝⁵ : IsScalarTower R L F
A : Type u_7
B : Type u_8
C : Type u_9
inst✝⁴ : CommRing A
inst✝³ : CommRing B
inst✝² : CommRing C
inst✝¹ : Algebra A B
inst✝ : Algebra A C
e : B ≃ₐ[A] C
x : B

⊢ (LinearMap.trace A C) ((lmul A C) (e x)) = (LinearMap.trace A C) (e.toLinearEquiv.conj ((lmul A B) x))
-- Mathlib/RingTheory/Trace/Basic.lean
R : Type u_1
S : Type u_2
T : Type u_3
inst✝¹⁸ : CommRing R
inst✝¹⁷ : CommRing S
inst✝¹⁶ : CommRing T
inst✝¹⁵ : Algebra R S
inst✝¹⁴ : Algebra R T
K : Type u_4
L : Type u_5
inst✝¹³ : Field K
inst✝¹² : Field L
inst✝¹¹ : Algebra K L
ι κ : Type w
inst✝¹⁰ : Fintype ι
F : Type u_6
inst✝⁹ : Field F
inst✝⁸ : Algebra R L
inst✝⁷ : Algebra L F
inst✝⁶ : Algebra R F
inst✝⁵ : IsScalarTower R L F
A : Type u_7
B : Type u_8
C : Type u_9
inst✝⁴ : CommRing A
inst✝³ : CommRing B
inst✝² : CommRing C
inst✝¹ : Algebra A B
inst✝ : Algebra A C
e : B ≃ₐ[A] C
x : B
x✝ : C

⊢ ((lmul A C) (e x)) x✝ = (e.toLinearEquiv.conj ((lmul A B) x)) x✝
-- Mathlib/Algebra/Homology/TotalComplex.lean
C : Type u_1
inst✝⁴ : Category.{u_5, u_1} C
inst✝³ : Preadditive C
I₁ : Type u_2
I₂ : Type u_3
I₁₂ : Type u_4
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K L M : HomologicalComplex₂ C c₁ c₂
φ : K ⟶ L
e : K ≅ L
ψ : L ⟶ M
c₁₂ : ComplexShape I₁₂
inst✝² : DecidableEq I₁₂
inst✝¹ : TotalComplexShape c₁ c₂ c₁₂
inst✝ : K.HasTotal c₁₂
i₁ : I₁
i₂ : I₂
i₁₂ : I₁₂
h : ¬c₂.Rel i₂ (c₂.next i₂)

⊢ c₁.ε₂ c₂ c₁₂ (i₁, i₂) • (K.X i₁).d i₂ (c₂.next i₂) ≫ K.toGradedObject.ιMapObjOrZero (c₁.π c₂ c₁₂) (i₁, c₂.next i₂) i₁₂ = 0
-- Mathlib/Control/Applicative.lean
F✝ : Type u → Type v
inst✝¹ : Applicative F✝
inst✝ : LawfulApplicative F✝
α β γ σ : Type u
F : Type u → Type u_1
F1 : Functor F
p1 : {α : Type u} → α → F α
s1 : {α β : Type u} → F (α → β) → (Unit → F α) → F β
sl1 : {α β : Type u} → F α → (Unit → F β) → F α
sr1 : {α β : Type u} → F α → (Unit → F β) → F β
F2 : Functor F
p2 : {α : Type u} → α → F α
s2 : {α β : Type u} → F (α → β) → (Unit → F α) → F β
sl2 : {α β : Type u} → F α → (Unit → F β) → F α
sr2 : {α β : Type u} → F α → (Unit → F β) → F β
L1 : LawfulApplicative F
L2 : LawfulApplicative F
H1 : ∀ {α : Type u} (x : α), pure x = pure x
H2 : ∀ {α β : Type u} (f : F (α → β)) (x : F α), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x

⊢ mk = mk
-- Mathlib/Control/Applicative.lean
F✝ : Type u → Type v
inst✝¹ : Applicative F✝
inst✝ : LawfulApplicative F✝
α β γ σ : Type u
F : Type u → Type u_1
F1 : Functor F
p1 : {α : Type u} → α → F α
s1 : {α β : Type u} → F (α → β) → (Unit → F α) → F β
sl1 : {α β : Type u} → F α → (Unit → F β) → F α
sr1 : {α β : Type u} → F α → (Unit → F β) → F β
F2 : Functor F
s2 : {α β : Type u} → F (α → β) → (Unit → F α) → F β
sl2 : {α β : Type u} → F α → (Unit → F β) → F α
sr2 : {α β : Type u} → F α → (Unit → F β) → F β
L1 : LawfulApplicative F
L2 : LawfulApplicative F
H1 : ∀ {α : Type u} (x : α), pure x = pure x
H2 : ∀ {α β : Type u} (f : F (α → β)) (x : F α), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x

⊢ mk = mk
-- Mathlib/Control/Applicative.lean
F✝ : Type u → Type v
inst✝¹ : Applicative F✝
inst✝ : LawfulApplicative F✝
α β γ σ : Type u
F : Type u → Type u_1
F1 : Functor F
p1 : {α : Type u} → α → F α
s1 : {α β : Type u} → F (α → β) → (Unit → F α) → F β
sl1 : {α β : Type u} → F α → (Unit → F β) → F α
sr1 : {α β : Type u} → F α → (Unit → F β) → F β
F2 : Functor F
sl2 : {α β : Type u} → F α → (Unit → F β) → F α
sr2 : {α β : Type u} → F α → (Unit → F β) → F β
L1 : LawfulApplicative F
L2 : LawfulApplicative F
H1 : ∀ {α : Type u} (x : α), pure x = pure x
H2 : ∀ {α β : Type u} (f : F (α → β)) (x : F α), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x

⊢ mk = mk
-- Mathlib/Control/Applicative.lean
F✝ : Type u → Type v
inst✝¹ : Applicative F✝
inst✝ : LawfulApplicative F✝
α β γ σ : Type u
F : Type u → Type u_1
F1 : Functor F
p1 : {α : Type u} → α → F α
s1 : {α β : Type u} → F (α → β) → (Unit → F α) → F β
sl1 : {α β : Type u} → F α → (Unit → F β) → F α
sr1 : {α β : Type u} → F α → (Unit → F β) → F β
F2 : Functor F
sl2 : {α β : Type u} → F α → (Unit → F β) → F α
sr2 : {α β : Type u} → F α → (Unit → F β) → F β
H1 : ∀ {α : Type u} (x : α), pure x = pure x
H2 : ∀ {α β : Type u} (f : F (α → β)) (x : F α), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x
toLawfulFunctor✝¹ : LawfulFunctor F
seqLeft_eq1 : ∀ {α β : Type u} (x : F α) (y : F β), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const β <$> x) fun x => y
seqRight_eq1 : ∀ {α β : Type u} (x : F α) (y : F β), (SeqRight.seqRight x fun x => y) = Seq.seq (const α id <$> x) fun x => y
pure_seq1 : ∀ {α β : Type u} (g : α → β) (x : F α), (Seq.seq (pure g) fun x_1 => x) = g <$> x
seq_pure✝¹ : ∀ {α β : Type u} (g : F (α → β)) (x : α), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g
seq_assoc✝¹ : ∀ {α β γ : Type u} (x : F α) (g : F (α → β)) (h : F (β → γ)), (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x
toLawfulFunctor✝ : LawfulFunctor F
seqLeft_eq2 : ∀ {α β : Type u} (x : F α) (y : F β), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const β <$> x) fun x => y
seqRight_eq2 : ∀ {α β : Type u} (x : F α) (y : F β), (SeqRight.seqRight x fun x => y) = Seq.seq (const α id <$> x) fun x => y
pure_seq2 : ∀ {α β : Type u} (g : α → β) (x : F α), (Seq.seq (pure g) fun x_1 => x) = g <$> x
seq_pure✝ : ∀ {α β : Type u} (g : F (α → β)) (x : α), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g
seq_assoc✝ : ∀ {α β γ : Type u} (x : F α) (g : F (α → β)) (h : F (β → γ)), (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x

⊢ mk = mk
-- Mathlib/Control/Applicative.lean
F✝ : Type u → Type v
inst✝¹ : Applicative F✝
inst✝ : LawfulApplicative F✝
α β γ σ : Type u
F : Type u → Type u_1
F1 : Functor F
p1 : {α : Type u} → α → F α
s1 : {α β : Type u} → F (α → β) → (Unit → F α) → F β
sl1 : {α β : Type u} → F α → (Unit → F β) → F α
sr1 : {α β : Type u} → F α → (Unit → F β) → F β
sl2 : {α β : Type u} → F α → (Unit → F β) → F α
sr2 : {α β : Type u} → F α → (Unit → F β) → F β
toLawfulFunctor✝¹ : LawfulFunctor F
seqLeft_eq1 : ∀ {α β : Type u} (x : F α) (y : F β), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const β <$> x) fun x => y
seqRight_eq1 : ∀ {α β : Type u} (x : F α) (y : F β), (SeqRight.seqRight x fun x => y) = Seq.seq (const α id <$> x) fun x => y
pure_seq1 : ∀ {α β : Type u} (g : α → β) (x : F α), (Seq.seq (pure g) fun x_1 => x) = g <$> x
seq_pure✝¹ : ∀ {α β : Type u} (g : F (α → β)) (x : α), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g
seq_assoc✝¹ : ∀ {α β γ : Type u} (x : F α) (g : F (α → β)) (h : F (β → γ)), (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x
H1 : ∀ {α : Type u} (x : α), pure x = pure x
H2 : ∀ {α β : Type u} (f : F (α → β)) (x : F α), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x
toLawfulFunctor✝ : LawfulFunctor F
seqLeft_eq2 : ∀ {α β : Type u} (x : F α) (y : F β), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const β <$> x) fun x => y
seqRight_eq2 : ∀ {α β : Type u} (x : F α) (y : F β), (SeqRight.seqRight x fun x => y) = Seq.seq (const α id <$> x) fun x => y
pure_seq2 : ∀ {α β : Type u} (g : α → β) (x : F α), (Seq.seq (pure g) fun x_1 => x) = g <$> x
seq_pure✝ : ∀ {α β : Type u} (g : F (α → β)) (x : α), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g
seq_assoc✝ : ∀ {α β γ : Type u} (x : F α) (g : F (α → β)) (h : F (β → γ)), (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x

⊢ mk = mk
-- Mathlib/Control/Applicative.lean
F✝ : Type u → Type v
inst✝¹ : Applicative F✝
inst✝ : LawfulApplicative F✝
α✝ β✝ γ σ : Type u
F : Type u → Type u_1
F1 : Functor F
p1 : {α : Type u} → α → F α
s1 : {α β : Type u} → F (α → β) → (Unit → F α) → F β
sl1 : {α β : Type u} → F α → (Unit → F β) → F α
sr1 : {α β : Type u} → F α → (Unit → F β) → F β
sl2 : {α β : Type u} → F α → (Unit → F β) → F α
sr2 : {α β : Type u} → F α → (Unit → F β) → F β
toLawfulFunctor✝¹ : LawfulFunctor F
seqLeft_eq1 : ∀ {α β : Type u} (x : F α) (y : F β), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const β <$> x) fun x => y
seqRight_eq1 : ∀ {α β : Type u} (x : F α) (y : F β), (SeqRight.seqRight x fun x => y) = Seq.seq (const α id <$> x) fun x => y
pure_seq1 : ∀ {α β : Type u} (g : α → β) (x : F α), (Seq.seq (pure g) fun x_1 => x) = g <$> x
seq_pure✝¹ : ∀ {α β : Type u} (g : F (α → β)) (x : α), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g
seq_assoc✝¹ : ∀ {α β γ : Type u} (x : F α) (g : F (α → β)) (h : F (β → γ)), (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x
H1 : ∀ {α : Type u} (x : α), pure x = pure x
H2 : ∀ {α β : Type u} (f : F (α → β)) (x : F α), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x
toLawfulFunctor✝ : LawfulFunctor F
seqLeft_eq2 : ∀ {α β : Type u} (x : F α) (y : F β), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const β <$> x) fun x => y
seqRight_eq2 : ∀ {α β : Type u} (x : F α) (y : F β), (SeqRight.seqRight x fun x => y) = Seq.seq (const α id <$> x) fun x => y
pure_seq2 : ∀ {α β : Type u} (g : α → β) (x : F α), (Seq.seq (pure g) fun x_1 => x) = g <$> x
seq_pure✝ : ∀ {α β : Type u} (g : F (α → β)) (x : α), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g
seq_assoc✝ : ∀ {α β γ : Type u} (x : F α) (g : F (α → β)) (h : F (β → γ)), (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x
α β : Type u
x : F α
y : Unit → F β

⊢ sl1 x y = sl2 x y
-- Mathlib/Control/Applicative.lean
F✝ : Type u → Type v
inst✝¹ : Applicative F✝
inst✝ : LawfulApplicative F✝
α✝ β✝ γ σ : Type u
F : Type u → Type u_1
F1 : Functor F
p1 : {α : Type u} → α → F α
s1 : {α β : Type u} → F (α → β) → (Unit → F α) → F β
sl1 : {α β : Type u} → F α → (Unit → F β) → F α
sr1 : {α β : Type u} → F α → (Unit → F β) → F β
sl2 : {α β : Type u} → F α → (Unit → F β) → F α
sr2 : {α β : Type u} → F α → (Unit → F β) → F β
toLawfulFunctor✝¹ : LawfulFunctor F
seqLeft_eq1 : ∀ {α β : Type u} (x : F α) (y : F β), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const β <$> x) fun x => y
seqRight_eq1 : ∀ {α β : Type u} (x : F α) (y : F β), (SeqRight.seqRight x fun x => y) = Seq.seq (const α id <$> x) fun x => y
pure_seq1 : ∀ {α β : Type u} (g : α → β) (x : F α), (Seq.seq (pure g) fun x_1 => x) = g <$> x
seq_pure✝¹ : ∀ {α β : Type u} (g : F (α → β)) (x : α), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g
seq_assoc✝¹ : ∀ {α β γ : Type u} (x : F α) (g : F (α → β)) (h : F (β → γ)), (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x
H1 : ∀ {α : Type u} (x : α), pure x = pure x
H2 : ∀ {α β : Type u} (f : F (α → β)) (x : F α), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x
toLawfulFunctor✝ : LawfulFunctor F
seqLeft_eq2 : ∀ {α β : Type u} (x : F α) (y : F β), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const β <$> x) fun x => y
seqRight_eq2 : ∀ {α β : Type u} (x : F α) (y : F β), (SeqRight.seqRight x fun x => y) = Seq.seq (const α id <$> x) fun x => y
pure_seq2 : ∀ {α β : Type u} (g : α → β) (x : F α), (Seq.seq (pure g) fun x_1 => x) = g <$> x
seq_pure✝ : ∀ {α β : Type u} (g : F (α → β)) (x : α), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g
seq_assoc✝ : ∀ {α β γ : Type u} (x : F α) (g : F (α → β)) (h : F (β → γ)), (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x
α β : Type u
x : F α
y : Unit → F β

⊢ sr1 x y = sr2 x y
-- Mathlib/Algebra/Polynomial/Lifts.lean
R : Type u
inst✝¹ : Ring R
S : Type v
inst✝ : Ring S
f : R →+* S
p : S[X]

⊢ p ∈ lifts f ↔ p ∈ liftsRing f
-- Mathlib/MeasureTheory/MeasurableSpace/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s✝ t✝ u : Set α
π : δ → Type u_6
inst✝² : MeasurableSpace α
inst✝¹ : (a : δ) → MeasurableSpace (π a)
inst✝ : MeasurableSpace γ
s : Set δ
t : (i : δ) → Set (π i)
hs : s.Countable
h : (s.pi t).Nonempty

⊢ MeasurableSet (s.pi t) ↔ ∀ i ∈ s, MeasurableSet (t i)
-- Mathlib/Data/List/Palindrome.lean
α : Type u_1
β : Type u_2
l✝¹ l✝ : List α
x : α
l : List α
y : α
hp : l.reverse = l → l.Palindrome
hr : (x :: (l ++ [y])).reverse = x :: (l ++ [y])

⊢ (x :: (l ++ [y])).Palindrome
-- Mathlib/Data/List/Palindrome.lean
α : Type u_1
β : Type u_2
l✝¹ l✝ : List α
x : α
l : List α
y : α
hp : l.reverse = l → l.Palindrome
hr : [y].reverse ++ l.reverse ++ [x] = x :: (l ++ [y])

⊢ (x :: (l ++ [y])).Palindrome
-- Mathlib/Data/List/Palindrome.lean
α : Type u_1
β : Type u_2
l✝¹ l✝ : List α
x : α
l : List α
y : α
hp : l.reverse = l → l.Palindrome
hr : [y].reverse ++ l.reverse ++ [x] = x :: (l ++ [y])

⊢ (x :: (l ++ [x])).Palindrome
-- Mathlib/Data/List/Palindrome.lean
α : Type u_1
β : Type u_2
l✝¹ l✝ : List α
x : α
l : List α
y : α
hp : l.reverse = l → l.Palindrome
hr : [y].reverse ++ l.reverse ++ [x] = x :: (l ++ [y])
this : l.Palindrome

⊢ (x :: (l ++ [x])).Palindrome
-- Mathlib/LinearAlgebra/AffineSpace/Basis.lean
ι : Type u_1
ι' : Type u_2
k : Type u_3
V : Type u_4
P : Type u_5
inst✝⁴ : AddCommGroup V
inst✝³ : AffineSpace V P
inst✝² : Ring k
inst✝¹ : Module k V
b : AffineBasis ι k P
s : Finset ι
i✝ j✝ : ι
e : ι ≃ ι'
inst✝ : DecidableEq ι
i j : ι

⊢ (b.coord i) (b j) = if i = j then 1 else 0
-- Mathlib/RingTheory/FractionalIdeal/Operations.lean
R : Type u_1
inst✝⁶ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁵ : CommRing P
inst✝⁴ : Algebra R P
loc : IsLocalization S P
P' : Type u_3
inst✝³ : CommRing P'
inst✝² : Algebra R P'
loc' : IsLocalization S P'
P'' : Type u_4
inst✝¹ : CommRing P''
inst✝ : Algebra R P''
loc'' : IsLocalization S P''
I✝ J : FractionalIdeal S P
g : P →ₐ[R] P'
inj : Function.Injective ⇑(algebraMap R P)
I : Ideal R
h : IsUnit ↑I

⊢ I.FG
-- Mathlib/RingTheory/FractionalIdeal/Operations.lean
R : Type u_1
inst✝⁶ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁵ : CommRing P
inst✝⁴ : Algebra R P
loc : IsLocalization S P
P' : Type u_3
inst✝³ : CommRing P'
inst✝² : Algebra R P'
loc' : IsLocalization S P'
P'' : Type u_4
inst✝¹ : CommRing P''
inst✝ : Algebra R P''
loc'' : IsLocalization S P''
I✝ J : FractionalIdeal S P
g : P →ₐ[R] P'
inj : Function.Injective ⇑(algebraMap R P)
I : Ideal R
h : IsUnit ↑I

⊢ (↑↑I).FG
-- 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✝¹ : CompleteLattice α
f✝ g s✝ t : ι → α
a b : α
inst✝ : CompleteLattice β
s : Set α
f : α → β
hf : Monotone f

⊢ ⨆ a ∈ s, f a ≤ f (sSup 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✝¹ : CompleteLattice α
f✝ g s✝ t : ι → α
a b : α
inst✝ : CompleteLattice β
s : Set α
f : α → β
hf : Monotone f

⊢ ⨆ a ∈ s, f a ≤ f (⨆ a ∈ s, a)
-- Mathlib/Combinatorics/Quiver/Covering.lean
U : Type u_1
inst✝⁵ : Quiver U
V : Type u_2
inst✝⁴ : Quiver V
φ : U ⥤q V
W : Type ?u.19860
inst✝³ : Quiver W
ψ : V ⥤q W
inst✝² : HasInvolutiveReverse U
inst✝¹ : HasInvolutiveReverse V
inst✝ : φ.MapReverse
u : U

⊢ Bijective (φ.costar u) ↔ Bijective (φ.star u)
-- Mathlib/Algebra/Order/Ring/Unbundled/Basic.lean
α : Type u
β : Type u_1
inst✝⁵ : Semiring α
inst✝⁴ : Preorder α
a b c d : α
inst✝³ : ExistsAddOfLE α
inst✝² : PosMulMono α
inst✝¹ : CovariantClass α α (swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝ : ContravariantClass α α (swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
ha : a ≤ 0
h : 1 ≤ b

⊢ a * b ≤ a
-- Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
x p : ℝ
hx_nonneg : 0 ≤ x
hp_nonneg : 0 ≤ p
hp0 : ¬p = 0
hx0 : x ≠ 0

⊢ ENNReal.ofReal x ^ p = ENNReal.ofReal (x ^ p)
-- Mathlib/MeasureTheory/Integral/DominatedConvergence.lean
ι✝ : Type u_1
𝕜 : Type u_2
E : Type u_3
F✝ : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
μ : Measure ℝ
ι : Type u_5
inst✝ : Countable ι
F : ι → ℝ → E
bound : ι → ℝ → ℝ
hF_meas : ∀ (n : ι), AEStronglyMeasurable (F n) (μ.restrict (Ι a b))
h_bound : ∀ (n : ι), ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι a b → ‖F n t‖ ≤ bound n t
bound_summable : ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι a b → Summable fun n => bound n t
bound_integrable : IntervalIntegrable (fun t => ∑' (n : ι), bound n t) μ a b
h_lim : ∀ᵐ (t : ℝ) ∂μ, t ∈ Ι a b → HasSum (fun n => F n t) (f t)

⊢ HasSum (fun n => ∫ (t : ℝ) in a..b, F n t ∂μ) (∫ (t : ℝ) in a..b, f t ∂μ)
-- Mathlib/MeasureTheory/Integral/DominatedConvergence.lean
ι✝ : Type u_1
𝕜 : Type u_2
E : Type u_3
F✝ : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
μ : Measure ℝ
ι : Type u_5
inst✝ : Countable ι
F : ι → ℝ → E
bound : ι → ℝ → ℝ
hF_meas : ∀ (n : ι), AEStronglyMeasurable (F n) (μ.restrict (Ι a b))
h_bound : ∀ (n : ι), ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), ‖F n x‖ ≤ bound n x
bound_summable : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), Summable fun n => bound n x
bound_integrable : IntegrableOn (fun t => ∑' (n : ι), bound n t) (Ι a b) μ
h_lim : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), HasSum (fun n => F n x) (f x)

⊢ HasSum (fun n => (if a ≤ b then 1 else -1) • ∫ (t : ℝ) in Ι a b, F n t ∂μ) ((if a ≤ b then 1 else -1) • ∫ (t : ℝ) in Ι a b, f t ∂μ)
-- Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean
α : Type u_1
ι : Type u_2
β : Type u_3
I : Set ι
hI : I.Nonempty
f : α → β
m : ι → OuterMeasure β

⊢ (map f) (⨅ i ∈ I, (comap f) (m i)) = ⨅ i ∈ I, (map f) ((comap f) (m i))
-- Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean
α : Type u_1
ι : Type u_2
β : Type u_3
I : Set ι
hI : I.Nonempty
f : α → β
m : ι → OuterMeasure β
this : Nonempty ↑I

⊢ (map f) (⨅ i ∈ I, (comap f) (m i)) = ⨅ i ∈ I, (map f) ((comap f) (m i))
-- Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean
α : Type u_1
ι : Type u_2
β : Type u_3
I : Set ι
hI : I.Nonempty
f : α → β
m : ι → OuterMeasure β
this : Nonempty ↑I

⊢ (map f) (⨅ i, (comap f) (m ↑i)) = ⨅ i, (map f) ((comap f) (m ↑i))
-- Mathlib/SetTheory/Surreal/Multiplication.lean
x x₁ x₂ x₃ x' y y₁✝ y₂✝ y₃ y' y₁ y₂ : PGame
h : ∀ (i : x₂.LeftMoves), IH3 x₁ (x₂.moveLeft i) x₂ y₁ y₂
hs : ∀ (i : (-x₁).LeftMoves), IH3 (-x₂) ((-x₁).moveLeft i) (-x₁) y₁ y₂
hl : x₁ < x₂

⊢ P3 x₁ x₂ y₁ y₂