ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
context: α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Type u_5
inst✝³ : Nonempty β
inst✝² : LinearOrder β
inst✝¹ : Preorder γ
inst✝ : NoMaxOrder γ
g : β → γ
f : α → β
l : Filter α
hg : Monotone g
hg' : Tendsto g atTop atTop
c : γ
hc : ∀ᶠ (x : γ) in map (g ∘ f) l, (fun x x_1 => x ≤ x_1) x c
goal: IsBoundedUnder (fun x x_1 => x ≤ x_1) l f
context: α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Type u_5
inst✝³ : Nonempty β
inst✝² : LinearOrder β
inst✝¹ : Preorder γ
inst✝ : NoMaxOrder γ
g : β → γ
f : α → β
l : Filter α
hg : Monotone g
hg' : Tendsto g atTop atTop
c : γ
hc : ∀ᶠ (a : α) in l, (fun x x_1 => x ≤ x_1) ((g ∘ f) a) c
goal: IsBoundedUnder (fun x x_1 => x ≤ x_1) l f
context: α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Type u_5
inst✝³ : Nonempty β
inst✝² : LinearOrder β
inst✝¹ : Preorder γ
inst✝ : NoMaxOrder γ
g : β → γ
f : α → β
l : Filter α
hg : Monotone g
hg' : Tendsto g atTop atTop
c : γ
hc : ∀ᶠ (a : α) in l, (fun x x_1 => x ≤ x_1) ((g ∘ f) a) c
b : β
hb : ∀ a ≥ b, c < g a
goal: IsBoundedUnder (fun x x_1 => x ≤ x_1) l f
context: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
s : ℕ → Set α := disjointed (spanningSets μ)
goal: ∃ g, (∀ (x : α), 0 < g x) ∧ Measurable g ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ε
context: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
s : ℕ → Set α := disjointed (spanningSets μ)
this : ∀ (n : ℕ), μ (s n) < ⊤
goal: ∃ g, (∀ (x : α), 0 < g x) ∧ Measurable g ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ε
context: α : Type u_1
β : Type u_2
γ : Type u_3
δ✝ : Type u_4
m m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
s : ℕ → Set α := disjointed (spanningSets μ)
this : ∀ (n : ℕ), μ (s n) < ⊤
δ : ℕ → ℝ≥0
δpos : ∀ (i : ℕ), 0 < δ i
δsum : ∑' (i : ℕ), μ (s i) * ↑(δ i) < ε
N : α → ℕ := spanningSetsIndex μ
goal: ∃ g, (∀ (x : α), 0 < g x) ∧ Measurable g ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ε
context: α : Type u_1
β : Type u_2
γ : Type u_3
δ✝ : Type u_4
m m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
s : ℕ → Set α := disjointed (spanningSets μ)
this : ∀ (n : ℕ), μ (s n) < ⊤
δ : ℕ → ℝ≥0
δpos : ∀ (i : ℕ), 0 < δ i
δsum : ∑' (i : ℕ), μ (s i) * ↑(δ i) < ε
N : α → ℕ := spanningSetsIndex μ
hN_meas : Measurable N
goal: ∃ g, (∀ (x : α), 0 < g x) ∧ Measurable g ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ε
context: α : Type u_1
β : Type u_2
γ : Type u_3
δ✝ : Type u_4
m m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
s : ℕ → Set α := disjointed (spanningSets μ)
this : ∀ (n : ℕ), μ (s n) < ⊤
δ : ℕ → ℝ≥0
δpos : ∀ (i : ℕ), 0 < δ i
δsum : ∑' (i : ℕ), μ (s i) * ↑(δ i) < ε
N : α → ℕ := spanningSetsIndex μ
hN_meas : Measurable N
hNs : ∀ (n : ℕ), N ⁻¹' {n} = s n
goal: ∃ g, (∀ (x : α), 0 < g x) ∧ Measurable g ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ε
context: α : Type u_1
β : Type u_2
γ : Type u_3
δ✝ : Type u_4
m m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
s : ℕ → Set α := disjointed (spanningSets μ)
this : ∀ (n : ℕ), μ (s n) < ⊤
δ : ℕ → ℝ≥0
δpos : ∀ (i : ℕ), 0 < δ i
δsum : ∑' (i : ℕ), μ (s i) * ↑(δ i) < ε
N : α → ℕ := spanningSetsIndex μ
hN_meas : Measurable N
hNs : ∀ (n : ℕ), N ⁻¹' {n} = s n
goal: ∫⁻ (x : α), ↑((δ ∘ N) x) ∂μ < ε
context: α : Type u_1
β : Type u_2
γ : Type u_3
δ✝ : Type u_4
m m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
s : ℕ → Set α := disjointed (spanningSets μ)
this : ∀ (n : ℕ), μ (s n) < ⊤
δ : ℕ → ℝ≥0
δpos : ∀ (i : ℕ), 0 < δ i
δsum : ∑' (i : ℕ), μ (s i) * ↑(δ i) < ε
N : α → ℕ := spanningSetsIndex μ
hN_meas : Measurable N
hNs : ∀ (n : ℕ), N ⁻¹' {n} = s n
goal: ∫⁻ (a : ℕ), ↑(δ a) ∂Measure.map N μ < ε
context: α : Type u_1
β : Type u_2
G : Type u_3
M : Type u_4
inst✝ : DivInvOneMonoid G
a : G
goal: a / 1 = a
context: R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
goal: T R (-2) = 2 * X ^ 2 - 1
context: V : Type u
V' : Type v
V'' : Type w
G : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
u v w : V
p : G.Walk u v
h : G.Adj v w
goal: (p.concat h).support = p.support.concat w
context: Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f✝ : ℕ → Ω → E
ℱ : Filtration ℕ m0
n✝ : ℕ
inst✝ : SigmaFiniteFiltration μ ℱ
f g : ℕ → Ω → E
hf : Martingale f ℱ μ
hg : Adapted ℱ fun n => g (n + 1)
hg0 : g 0 = 0
hgint : ∀ (n : ℕ), Integrable (g n) μ
n : ℕ
goal: predictablePart (f + g) ℱ μ n =ᶠ[ae μ] g n
context: Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f✝ : ℕ → Ω → E
ℱ : Filtration ℕ m0
n✝ : ℕ
inst✝ : SigmaFiniteFiltration μ ℱ
f g : ℕ → Ω → E
hf : Martingale f ℱ μ
hg : Adapted ℱ fun n => g (n + 1)
hg0 : g 0 = 0
hgint : ∀ (n : ℕ), Integrable (g n) μ
n : ℕ
ω : Ω
hω : martingalePart (f + g) ℱ μ n ω = f n ω
goal: predictablePart (f + g) ℱ μ n ω = g n ω
context: Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f✝ : ℕ → Ω → E
ℱ : Filtration ℕ m0
n✝ : ℕ
inst✝ : SigmaFiniteFiltration μ ℱ
f g : ℕ → Ω → E
hf : Martingale f ℱ μ
hg : Adapted ℱ fun n => g (n + 1)
hg0 : g 0 = 0
hgint : ∀ (n : ℕ), Integrable (g n) μ
n : ℕ
ω : Ω
hω : martingalePart (f + g) ℱ μ n ω = f n ω
goal: f n ω + predictablePart (f + g) ℱ μ n ω = f n ω + g n ω
context: Ω : Type u_1
E : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f✝ : ℕ → Ω → E
ℱ : Filtration ℕ m0
n✝ : ℕ
inst✝ : SigmaFiniteFiltration μ ℱ
f g : ℕ → Ω → E
hf : Martingale f ℱ μ
hg : Adapted ℱ fun n => g (n + 1)
hg0 : g 0 = 0
hgint : ∀ (n : ℕ), Integrable (g n) μ
n : ℕ
ω : Ω
hω : martingalePart (f + g) ℱ μ n ω = f n ω
goal: f n ω + predictablePart (f + g) ℱ μ n ω = (martingalePart (f + g) ℱ μ + predictablePart (f + g) ℱ μ) n ω
context: β : Type u_1
G : Type u_2
M : Type u_3
inst✝³ : Monoid M
inst✝² : Preorder M
inst✝¹ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
x : M
inst✝ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1
a : M
n m : ℕ
ha : 1 < a
h : n < m
goal: a ^ n < a ^ m
context: β : Type u_1
G : Type u_2
M : Type u_3
inst✝³ : Monoid M
inst✝² : Preorder M
inst✝¹ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
x : M
inst✝ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1
a : M
n : ℕ
ha : 1 < a
k : ℕ
goal: a ^ n < a ^ (n.succ + k)
context: β : Type u_1
G : Type u_2
M : Type u_3
inst✝³ : Monoid M
inst✝² : Preorder M
inst✝¹ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
x : M
inst✝ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1
a : M
n : ℕ
ha : 1 < a
k : ℕ
goal: a ^ n < a ^ n * a ^ (k + 1)
context: C : Type u_1
D : Type u_2
E : Type u_3
inst✝² : Category.{?u.20707, u_1} C
inst✝¹ : Category.{?u.20711, u_2} D
inst✝ : Category.{?u.20715, u_3} E
F : D ⥤ E
G : E ⥤ D
adj : F ⊣ G
x : D ⥤ C
x✝ : E
goal: (𝟙 (x.obj (G.obj x✝)) ≫ x.map (adj.unit.app (G.obj x✝)) ≫ 𝟙 (x.obj (G.obj (F.obj (G.obj x✝))))) ≫ 𝟙 (x.obj (G.obj (F.obj (G.obj x✝)))) ≫ 𝟙 (x.obj (G.obj (F.obj (G.obj x✝)))) ≫ x.map (G.map (adj.counit.app x✝)) ≫ 𝟙 (x.obj (G.obj x✝)) = 𝟙 (x.obj (G.obj x✝))
context: C : Type u_1
D : Type u_2
E : Type u_3
inst✝² : Category.{?u.20707, u_1} C
inst✝¹ : Category.{?u.20711, u_2} D
inst✝ : Category.{?u.20715, u_3} E
F : D ⥤ E
G : E ⥤ D
adj : F ⊣ G
x : E ⥤ C
x✝ : D
goal: (𝟙 (x.obj (F.obj x✝)) ≫ x.map (F.map (adj.unit.app x✝)) ≫ 𝟙 (x.obj (F.obj (G.obj (F.obj x✝))))) ≫ 𝟙 (x.obj (F.obj (G.obj (F.obj x✝)))) ≫ 𝟙 (x.obj (F.obj (G.obj (F.obj x✝)))) ≫ x.map (adj.counit.app (F.obj x✝)) ≫ 𝟙 (x.obj (F.obj x✝)) = 𝟙 (x.obj (F.obj x✝))
context: R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
n m : σ
s : σ →₀ ℕ
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S₁
p q : MvPolynomial σ R
S : Subalgebra R (MvPolynomial σ R) := Algebra.adjoin R (range X)
goal: S = ⊤
context: R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
n m : σ
s : σ →₀ ℕ
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S₁
p✝ q : MvPolynomial σ R
S : Subalgebra R (MvPolynomial σ R) := Algebra.adjoin R (range X)
p : MvPolynomial σ R
goal: p ∈ S
context: α : Type u_1
n : Nat
a' : α
l : List α
h : n < l.length
goal: ∃ l₁ a l₂, l = l₁ ++ a :: l₂ ∧ l₁.length = n ∧ l.set n a' = l₁ ++ a' :: l₂
context: α : Type u_1
n : Nat
a' : α
l : List α
h : n < l.length
goal: ∃ l₁ a l₂, l = l₁ ++ a :: l₂ ∧ l₁.length = n ∧ modifyNth (fun x => a') n l = l₁ ++ a' :: l₂
context: k : ℕ
hk : k ≠ 0
goal: ∫ (x : ℝ) in 0 ..1, bernoulliFun k x = 0
context: k : ℕ
hk : k ≠ 0
goal: bernoulliFun (k + 1) 1 / (↑k + 1) - bernoulliFun (k + 1) 0 / (↑k + 1) = 0
context: R✝ : Type u
S : Type v
T : Type w
inst✝³ : NonAssocSemiring R✝
M : Submonoid R✝
inst✝² : NonAssocSemiring S
inst✝¹ : NonAssocSemiring T
R : Type u_1
inst✝ : Semiring R
s : Set R
goal: centralizer s.centralizer.centralizer = centralizer s
context: R✝ : Type u
S : Type v
T : Type w
inst✝³ : NonAssocSemiring R✝
M : Submonoid R✝
inst✝² : NonAssocSemiring S
inst✝¹ : NonAssocSemiring T
R : Type u_1
inst✝ : Semiring R
s : Set R
goal: ↑(centralizer s.centralizer.centralizer) = ↑(centralizer s)
context: M : Type u_1
inst✝ : Mul M
c d : Con M
goal: { toFun := op, invFun := unop, left_inv := ⋯, right_inv := ⋯ } c ≤ { toFun := op, invFun := unop, left_inv := ⋯, right_inv := ⋯ } d ↔ c ≤ d
context: 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 b : Ordinal.{u_4}
l : a.IsLimit
h : b < a
goal: b + 0 < a
context: 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 b : Ordinal.{u_4}
l : a.IsLimit
h✝ : b < a
c : Ordinal.{u_4}
h : c < a - b
goal: succ c < a - b
context: 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 b : Ordinal.{u_4}
l : a.IsLimit
h✝ : b < a
c : Ordinal.{u_4}
h : c < a - b
goal: succ (b + c) < a
context: R✝ : Type u
A✝ : Type v
inst✝⁹ : CommSemiring R✝
inst✝⁸ : Ring A✝
inst✝⁷ : Algebra R✝ A✝
S : Type u_1
R : Type u_2
A : Type u_3
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring S
inst✝⁴ : Ring A
inst✝³ : Algebra R S
inst✝² : Algebra R A
inst✝¹ : Algebra S A
inst✝ : IsScalarTower R S A
a : A
r : R
goal: (algebraMap R S) r ∈ spectrum S a ↔ r ∈ σ a
context: α✝ α : Type u_1
inst✝¹ : Nonempty α
inst✝ : DecidableEq α
s : Finset α
f : α → ℤ
n : ℕ
h : ∑ i ∈ s, (f i).natAbs ≤ n
goal: ∃ β x sgn g, (∀ (b : β), g b ∉ s → sgn b = 0) ∧ Fintype.card β = n ∧ ∀ a ∈ s, (∑ i : β, if g i = a then ↑(sgn i) else 0) = f a
context: E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atBot] fun x => |x| ^ (-b)
R S : ℝ
goal: (fun x => ‖ContinuousMap.restrict (Icc (x + R) (x + S)) f‖) =O[atBot] fun x => |x| ^ (-b)
context: E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atBot] fun x => |x| ^ (-b)
R S : ℝ
h1 : ⇑(f.comp { toFun := fun a => -a, continuous_toFun := ⋯ }) =O[atTop] fun x => |x| ^ (-b)
goal: (fun x => ‖ContinuousMap.restrict (Icc (x + R) (x + S)) f‖) =O[atBot] fun x => |x| ^ (-b)
context: E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atBot] fun x => |x| ^ (-b)
R S : ℝ
h1 : ⇑(f.comp { toFun := fun a => -a, continuous_toFun := ⋯ }) =O[atTop] fun x => |x| ^ (-b)
h2 : ((fun x => ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖) ∘ Neg.neg) =O[atBot] ((fun x => |x| ^ (-b)) ∘ Neg.neg)
goal: (fun x => ‖ContinuousMap.restrict (Icc (x + R) (x + S)) f‖) =O[atBot] fun x => |x| ^ (-b)
context: E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atBot] fun x => |x| ^ (-b)
R S : ℝ
h1 : ⇑(f.comp { toFun := fun a => -a, continuous_toFun := ⋯ }) =O[atTop] fun x => |x| ^ (-b)
h2 : ((fun x => ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖) ∘ Neg.neg) =O[atBot] fun x => |x| ^ (-b)
this : (fun x => |x| ^ (-b)) ∘ Neg.neg = fun x => |x| ^ (-b)
goal: (fun x => ‖ContinuousMap.restrict (Icc (x + R) (x + S)) f‖) =O[atBot] fun x => |x| ^ (-b)
context: E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atBot] fun x => |x| ^ (-b)
R S : ℝ
h1 : ⇑(f.comp { toFun := fun a => -a, continuous_toFun := ⋯ }) =O[atTop] fun x => |x| ^ (-b)
h2 : ((fun x => ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖) ∘ Neg.neg) =O[atBot] fun x => |x| ^ (-b)
this : (fun x => |x| ^ (-b)) ∘ Neg.neg = fun x => |x| ^ (-b)
x : ℝ
goal: ‖‖ContinuousMap.restrict (Icc (x + R) (x + S)) f‖‖ ≤ ‖((fun x => ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖) ∘ Neg.neg) x‖
context: E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atBot] fun x => |x| ^ (-b)
R S : ℝ
h1 : ⇑(f.comp { toFun := fun a => -a, continuous_toFun := ⋯ }) =O[atTop] fun x => |x| ^ (-b)
h2 : ((fun x => ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖) ∘ Neg.neg) =O[atBot] fun x => |x| ^ (-b)
this : (fun x => |x| ^ (-b)) ∘ Neg.neg = fun x => |x| ^ (-b)
x✝ x : ℝ
hx : x ∈ Icc (x✝ + R) (x✝ + S)
goal: ‖(ContinuousMap.restrict (Icc (x✝ + R) (x✝ + S)) f) ⟨x, hx⟩‖ ≤ ‖ContinuousMap.restrict (Icc (-x✝ + -S) (-x✝ + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖
context: E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atBot] fun x => |x| ^ (-b)
R S : ℝ
h1 : ⇑(f.comp { toFun := fun a => -a, continuous_toFun := ⋯ }) =O[atTop] fun x => |x| ^ (-b)
h2 : ((fun x => ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖) ∘ Neg.neg) =O[atBot] fun x => |x| ^ (-b)
this : (fun x => |x| ^ (-b)) ∘ Neg.neg = fun x => |x| ^ (-b)
x✝ x : ℝ
hx : x ∈ Icc (x✝ + R) (x✝ + S)
goal: ‖f x‖ ≤ ‖ContinuousMap.restrict (Icc (-x✝ + -S) (-x✝ + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖
context: ι : Type u_1
α : Type u
β : Type v
γ : Type w
l₁ l₂ : List α
a : α
l : List α
x : α
n : ℕ
goal: l.length ≤ (insertNth n x l).length
context: K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
inst✝² : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
S : Solution hζ
S' : Solution' hζ
inst✝¹ : DecidableRel fun a b => a ∣ b
inst✝ : DecidableEq (𝓞 K)
goal: S.Solution'_descent.multiplicity = S.multiplicity - 1
context: K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
inst✝² : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
S : Solution hζ
S' : Solution' hζ
inst✝¹ : DecidableRel fun a b => a ∣ b
inst✝ : DecidableEq (𝓞 K)
k : 𝓞 K
hk : λ ^ (S.multiplicity - 1) * FermatLastTheoremForThreeGen.Solution.X S = λ ^ (S.multiplicity - 1 + 1) * k
goal: λ ∣ FermatLastTheoremForThreeGen.Solution.X S
context: K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
inst✝² : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
S : Solution hζ
S' : Solution' hζ
inst✝¹ : DecidableRel fun a b => a ∣ b
inst✝ : DecidableEq (𝓞 K)
k : 𝓞 K
hk : λ ^ (S.multiplicity - 1) * FermatLastTheoremForThreeGen.Solution.X S = λ ^ (S.multiplicity - 1) * (λ * k)
goal: λ ∣ FermatLastTheoremForThreeGen.Solution.X S
context: C : Type u
inst✝³ : Category.{v, u} C
X✝ Y Z : C
inst✝² : HasZeroMorphisms C
f : X✝ ⟶ Y
inst✝¹ : HasKernel f
inst✝ : HasCokernels C
X A B : C
g : B ⟶ X
hg : Mono g
i : A ≅ B
hf : Mono (i.hom ≫ g)
goal: (fun A f x => Subobject.mk (cokernel.π f).op) A (i.hom ≫ g) hf = (fun A f x => Subobject.mk (cokernel.π f).op) B g hg
context: C : Type u
inst✝³ : Category.{v, u} C
X✝ Y Z : C
inst✝² : HasZeroMorphisms C
f✝ : X✝ ⟶ Y
inst✝¹ : HasKernel f✝
inst✝ : HasCokernels C
X A B : C
f : A ⟶ X
g : B ⟶ X
hf : Mono f
hg : Mono g
h : Subobject.mk f ≤ Subobject.mk g
goal: Subobject.lift (fun A f x => Subobject.mk (cokernel.π f).op) ⋯ (Subobject.mk f) ≤ Subobject.lift (fun A f x => Subobject.mk (cokernel.π f).op) ⋯ (Subobject.mk g)
context: C : Type u
inst✝³ : Category.{v, u} C
X✝ Y Z : C
inst✝² : HasZeroMorphisms C
f✝ : X✝ ⟶ Y
inst✝¹ : HasKernel f✝
inst✝ : HasCokernels C
X A B : C
f : A ⟶ X
g : B ⟶ X
hf : Mono f
hg : Mono g
h : Subobject.mk f ≤ Subobject.mk g
goal: Subobject.mk (cokernel.π f).op ≤ Subobject.mk (cokernel.π g).op
context: α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
inst✝ : CompleteLattice β
S : Set (Set α)
f : α → β
goal: ⨅ x ∈ ⋃₀ S, f x = ⨅ s ∈ S, ⨅ x ∈ s, f x
context: f : ℝ → ℝ
x : ℝ
n : ℕ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hx : 0 < x
hn : n ≠ 0
goal: f 1 + (x * log ↑n + log ↑n ! - ∑ m ∈ Finset.range (n + 1), log (x + ↑m)) ≤ f x
context: f : ℝ → ℝ
x : ℝ
n : ℕ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hx : 0 < x
hn : n ≠ 0
goal: f 1 + (x * log ↑n + log ↑n !) ≤ f (↑(n + 1) + x)
context: J : Type v
inst✝ : SmallCategory J
F : J ⥤ Cat
X✝ Y : limit (F ⋙ objects)
X : J
goal: { obj := fun j => limit.π (F ⋙ objects) j X✝ ⟶ limit.π (F ⋙ objects) j Y, map := fun {X Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.map (𝟙 X) = 𝟙 ({ obj := fun j => limit.π (F ⋙ objects) j X✝ ⟶ limit.π (F ⋙ objects) j Y, map := fun {X Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.obj X)
context: J : Type v
inst✝ : SmallCategory J
F : J ⥤ Cat
X✝ Y : limit (F ⋙ objects)
X : J
f : { obj := fun j => limit.π (F ⋙ objects) j X✝ ⟶ limit.π (F ⋙ objects) j Y, map := fun {X Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.obj X
this : Category.{v, v} (objects.obj (F.obj X)) := inferInstance
goal: { obj := fun j => limit.π (F ⋙ objects) j X✝ ⟶ limit.π (F ⋙ objects) j Y, map := fun {X Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.map (𝟙 X) f = 𝟙 ({ obj := fun j => limit.π (F ⋙ objects) j X✝ ⟶ limit.π (F ⋙ objects) j Y, map := fun {X Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.obj X) f
context: J : Type v
inst✝ : SmallCategory J
F : J ⥤ Cat
X Y : limit (F ⋙ objects)
x✝¹ x✝ Z : J
f : x✝¹ ⟶ x✝
g : x✝ ⟶ Z
goal: { obj := fun j => limit.π (F ⋙ objects) j X ⟶ limit.π (F ⋙ objects) j Y, map := fun {X_1 Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.map (f ≫ g) = { obj := fun j => limit.π (F ⋙ objects) j X ⟶ limit.π (F ⋙ objects) j Y, map := fun {X_1 Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.map f ≫ { obj := fun j => limit.π (F ⋙ objects) j X ⟶ limit.π (F ⋙ objects) j Y, map := fun {X_1 Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.map g
context: J : Type v
inst✝ : SmallCategory J
F : J ⥤ Cat
X Y : limit (F ⋙ objects)
x✝¹ x✝ Z : J
f : x✝¹ ⟶ x✝
g : x✝ ⟶ Z
h : { obj := fun j => limit.π (F ⋙ objects) j X ⟶ limit.π (F ⋙ objects) j Y, map := fun {X_1 Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.obj x✝¹
this : Category.{v, v} (objects.obj (F.obj Z)) := inferInstance
goal: { obj := fun j => limit.π (F ⋙ objects) j X ⟶ limit.π (F ⋙ objects) j Y, map := fun {X_1 Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.map (f ≫ g) h = ({ obj := fun j => limit.π (F ⋙ objects) j X ⟶ limit.π (F ⋙ objects) j Y, map := fun {X_1 Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.map f ≫ { obj := fun j => limit.π (F ⋙ objects) j X ⟶ limit.π (F ⋙ objects) j Y, map := fun {X_1 Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.map g) h
context: α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
M✝ : Type u_5
M' : Type u_6
N : Type u_7
P : Type u_8
G : Type u_9
H : Type u_10
R : Type u_11
S : Type u_12
inst✝¹ : Zero M✝
α : Type u_13
M : Type u_14
inst✝ : Zero M
f g : α →₀ M
h : f.graph = g.graph
goal: f = g
context: k : Type u_1
V : Type u_2
P : Type u_3
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
p₁ p₂ : P
s : Set P
h : affineSpan k s = ⊤
goal: s.Nonempty
context: k : Type u_1
V : Type u_2
P : Type u_3
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
p₁ p₂ : P
h : affineSpan k ∅ = ⊤
goal: False
context: k : Type u_1
V : Type u_2
P : Type u_3
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
p₁ p₂ : P
h : ⊥ = ⊤
goal: False
context: α : Type ua
β : Type ub
γ : Type uc
δ : Type ud
ι : Sort u_1
inst✝ : UniformSpace α
p : ι → Prop
s : ι → Set (α × α)
h : (𝓤 α).HasBasis p s
x : α
goal: (𝓝 x).HasBasis p fun i => {y | (y, x) ∈ s i}
context: α : Type ua
β : Type ub
γ : Type uc
δ : Type ud
ι : Sort u_1
inst✝ : UniformSpace α
p : ι → Prop
s : ι → Set (α × α)
x : α
h : (comap Prod.swap (𝓤 α)).HasBasis p fun i => Prod.swap ⁻¹' s i
goal: (𝓝 x).HasBasis p fun i => {y | (y, x) ∈ s i}
context: α : Type ua
β : Type ub
γ : Type uc
δ : Type ud
ι : Sort u_1
inst✝ : UniformSpace α
p : ι → Prop
s : ι → Set (α × α)
x : α
h : (𝓤 α).HasBasis p fun i => Prod.swap ⁻¹' s i
goal: (𝓝 x).HasBasis p fun i => {y | (y, x) ∈ s i}
context: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ : Finite α
inst✝ : Finite β
this✝ : Fintype α
this : Fintype β
goal: Nat.card α = Nat.card β ↔ Nonempty (α ≃ β)
context: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁴ : MeasurableSpace α
μ ν✝ : Measure α
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace γ
inst✝¹ : TopologicalSpace δ
inst✝ : MeasurableSpace β
ν : Measure β
f : α → β
g : β →ₘ[ν] γ
hf : QuasiMeasurePreserving f μ ν
goal: ↑(g.compQuasiMeasurePreserving f hf) =ᶠ[ae μ] ↑g ∘ f
context: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁴ : MeasurableSpace α
μ ν✝ : Measure α
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace γ
inst✝¹ : TopologicalSpace δ
inst✝ : MeasurableSpace β
ν : Measure β
f : α → β
g : β →ₘ[ν] γ
hf : QuasiMeasurePreserving f μ ν
goal: ↑(mk (↑g ∘ f) ⋯) =ᶠ[ae μ] ↑g ∘ f
context: ι : Type u_1
M : ι → Type u_2
inst✝² : (i : ι) → Monoid (M i)
N : Type u_3
inst✝¹ : Monoid N
inst✝ : DecidableEq ι
i : ι
x : M i
goal: (lift (Pi.mulSingle i (MonoidHom.id (M i)))) (of x) = x
context: l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
R : Type u_5
inst✝⁴ : Fintype n
inst✝³ : Fintype o
inst✝² : CommRing R
inst✝¹ : StrongRankCondition R
inst✝ : DecidableEq n
goal: rank 1 = Fintype.card n
context: α : Type u
L L₁ L₂ L₃ L₄ : List (α × Bool)
inst✝ : DecidableEq α
x y : FreeGroup α
goal: (x * y).norm = (mk (x.toWord ++ y.toWord)).norm
context: α : Type u
β : Type v
γ : Type w
ι : Type x
inst✝¹ : UniformSpace β
F : ι → α → β
f : α → β
s s' : Set α
x : α
p : Filter ι
p' : Filter α
g : ι → α
inst✝ : TopologicalSpace α
h : ContinuousAt f x
hg : Tendsto g p (𝓝 x)
hunif : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝 x, ∀ᶠ (n : ι) in p, ∀ y ∈ t, (f y, F n y) ∈ u
goal: Tendsto (fun n => F n (g n)) p (𝓝 (f x))
context: α : Type u
β : Type v
γ : Type w
ι : Type x
inst✝¹ : UniformSpace β
F : ι → α → β
f : α → β
s s' : Set α
x : α
p : Filter ι
p' : Filter α
g : ι → α
inst✝ : TopologicalSpace α
h : ContinuousWithinAt f univ x
hg : Tendsto g p (𝓝 x)
hunif : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝 x, ∀ᶠ (n : ι) in p, ∀ y ∈ t, (f y, F n y) ∈ u
goal: Tendsto (fun n => F n (g n)) p (𝓝 (f x))
context: α : Type u
β : Type v
γ : Type w
ι : Type x
inst✝¹ : UniformSpace β
F : ι → α → β
f : α → β
s s' : Set α
x : α
p : Filter ι
p' : Filter α
g : ι → α
inst✝ : TopologicalSpace α
h : ContinuousWithinAt f univ x
hg : Tendsto g p (𝓝[univ] x)
hunif : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝[univ] x, ∀ᶠ (n : ι) in p, ∀ y ∈ t, (f y, F n y) ∈ u
goal: Tendsto (fun n => F n (g n)) p (𝓝 (f x))
context: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Mul α
s s₁ s₂ t t₁ t₂ : Finset α
h : 0 < Eₘ[s, t]
H : ¬(s.Nonempty ∧ t.Nonempty)
goal: False
context: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Mul α
s s₁ s₂ t t₁ t₂ : Finset α
h : 0 < Eₘ[s, t]
H : s = ∅ ∨ t = ∅
goal: False
context: M : Type u_1
A : Type u_2
B : Type u_3
inst✝ : MulOneClass M
ι : Sort u_4
hι : Nonempty ι
S : ι → Submonoid M
hS : Directed (fun x x_1 => x ≤ x_1) S
x : M
goal: x ∈ ⨆ i, S i ↔ ∃ i, x ∈ S i
context: M : Type u_1
A : Type u_2
B : Type u_3
inst✝ : MulOneClass M
ι : Sort u_4
hι : Nonempty ι
S : ι → Submonoid M
hS : Directed (fun x x_1 => x ≤ x_1) S
x : M
goal: x ∈ ⨆ i, S i → ∃ i, x ∈ S i
context: M : Type u_1
A : Type u_2
B : Type u_3
inst✝ : MulOneClass M
ι : Sort u_4
hι : Nonempty ι
S : ι → Submonoid M
hS : Directed (fun x x_1 => x ≤ x_1) S
x : M
goal: x ∈ closure (⋃ i, ↑(S i)) → ∃ i, x ∈ S i