ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
Theorem: contMDiffAt_iff_target_of_mem_source
File: Mathlib/Geometry/Manifold/ContMDiff/Defs.lean

Proof steps:
state_before: 𝕜 : Type u_1
inst✝³⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³⁶ : NormedAddCommGroup E
inst✝³⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝³⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³³ : TopologicalSpace M
inst✝³² : ChartedSpace H M
inst✝³¹ : SmoothManifoldWithCorners I M
E' : Type u_5
inst✝³⁰ : NormedAddCommGroup E'
inst✝²⁹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝²⁸ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝²⁷ : TopologicalSpace M'
inst✝²⁶ : ChartedSpace H' M'
inst✝²⁵ : SmoothManifoldWithCorners I' M'
E'' : Type u_8
inst✝²⁴ : NormedAddCommGroup E''
inst✝²³ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝²² : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝²¹ : TopologicalSpace M''
inst✝²⁰ : ChartedSpace H'' M''
F : Type u_11
inst✝¹⁹ : NormedAddCommGroup F
inst✝¹⁸ : NormedSpace 𝕜 F
G : Type u_12
inst✝¹⁷ : TopologicalSpace G
J : ModelWithCorners 𝕜 F G
N : Type u_13
inst✝¹⁶ : TopologicalSpace N
inst✝¹⁵ : ChartedSpace G N
inst✝¹⁴ : SmoothManifoldWithCorners J N
F' : Type u_14
inst✝¹³ : NormedAddCommGroup F'
inst✝¹² : NormedSpace 𝕜 F'
G' : Type u_15
inst✝¹¹ : TopologicalSpace G'
J' : ModelWithCorners 𝕜 F' G'
N' : Type u_16
inst✝¹⁰ : TopologicalSpace N'
inst✝⁹ : ChartedSpace G' N'
inst✝⁸ : SmoothManifoldWithCorners J' N'
F₁ : Type u_17
inst✝⁷ : NormedAddCommGroup F₁
inst✝⁶ : NormedSpace 𝕜 F₁
F₂ : Type u_18
inst✝⁵ : NormedAddCommGroup F₂
inst✝⁴ : NormedSpace 𝕜 F₂
F₃ : Type u_19
inst✝³ : NormedAddCommGroup F₃
inst✝² : NormedSpace 𝕜 F₃
F₄ : Type u_20
inst✝¹ : NormedAddCommGroup F₄
inst✝ : NormedSpace 𝕜 F₄
e : PartialHomeomorph M H
e' : PartialHomeomorph M' H'
f f₁ : M → M'
s s₁ t : Set M
x✝ : M
m n : ℕ∞
x : M
y : M'
hy : f x ∈ (chartAt H' y).source
⊢ ContMDiffAt I I' n f x ↔ ContinuousAt f x ∧ ContMDiffAt I 𝓘(𝕜, E') n (↑(extChartAt I' y) ∘ f) x
state_after: no goals
tactic: rw [ContMDiffAt, contMDiffWithinAt_iff_target_of_mem_source hy, continuousWithinAt_univ,
    ContMDiffAt]
Theorem: Finset.forall_mem_eraseNone
File: Mathlib/Data/Finset/Option.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
s : Finset (Option α)
p : Option α → Prop
⊢ (∀ a ∈ eraseNone s, p (some a)) ↔ ∀ (a : α), some a ∈ s → p (some a)
state_after: no goals
tactic: simp [Option.forall]
Theorem: AkraBazziRecurrence.eventually_one_sub_smoothingFn_nonneg
File: Mathlib/Computability/AkraBazzi/AkraBazzi.lean

Proof steps:
state_before: α : Type u_1
inst✝¹ : Fintype α
inst✝ : Nonempty α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
R : AkraBazziRecurrence T g a b r
⊢ ∀ᶠ (n : ℕ) in atTop, 0 ≤ 1 - ε ↑n
state_after: case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : Nonempty α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
R : AkraBazziRecurrence T g a b r
n : ℕ
hn : 0 < 1 - ε ↑n
⊢ 0 ≤ 1 - ε ↑n
tactic: filter_upwards [eventually_one_sub_smoothingFn_pos] with n hn
state_before: case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : Nonempty α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
R : AkraBazziRecurrence T g a b r
n : ℕ
hn : 0 < 1 - ε ↑n
⊢ 0 ≤ 1 - ε ↑n
state_after: no goals
tactic: exact le_of_lt hn
Theorem: HNNExtension.NormalWord.t_pow_smul_eq_unitsSMul
File: Mathlib/GroupTheory/HNNExtension.lean

Proof steps:
state_before: G : Type u_1
inst✝³ : Group G
A B : Subgroup G
φ : ↥A ≃* ↥B
H : Type u_2
inst✝² : Group H
M : Type u_3
inst✝¹ : Monoid M
d : TransversalPair G A B
inst✝ : DecidableEq G
u : ℤˣ
w : NormalWord d
⊢ t ^ ↑u • w = unitsSMul φ u w
state_after: no goals
tactic: rcases Int.units_eq_one_or u with (rfl | rfl) <;>
    simp [instHSMul, SMul.smul, MulAction.toEndHom, Equiv.Perm.inv_def]
Theorem: isClosed_setOf_map_neg
File: Mathlib/Topology/Algebra/Group/Basic.lean

Proof steps:
state_before: G : Type w
H : Type x
α : Type u
β : Type v
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Inv G
inst✝⁶ : ContinuousInv G
inst✝⁵ : TopologicalSpace α
f : α → G
s : Set α
x : α
ι : Type u_1
G₁ : Type u_2
G₂ : Type u_3
inst✝⁴ : TopologicalSpace G₂
inst✝³ : T2Space G₂
inst✝² : Inv G₁
inst✝¹ : Inv G₂
inst✝ : ContinuousInv G₂
⊢ IsClosed {f | ∀ (x : G₁), f x⁻¹ = (f x)⁻¹}
state_after: G : Type w
H : Type x
α : Type u
β : Type v
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Inv G
inst✝⁶ : ContinuousInv G
inst✝⁵ : TopologicalSpace α
f : α → G
s : Set α
x : α
ι : Type u_1
G₁ : Type u_2
G₂ : Type u_3
inst✝⁴ : TopologicalSpace G₂
inst✝³ : T2Space G₂
inst✝² : Inv G₁
inst✝¹ : Inv G₂
inst✝ : ContinuousInv G₂
⊢ IsClosed (⋂ i, {x | x i⁻¹ = (x i)⁻¹})
tactic: simp only [setOf_forall]
state_before: G : Type w
H : Type x
α : Type u
β : Type v
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Inv G
inst✝⁶ : ContinuousInv G
inst✝⁵ : TopologicalSpace α
f : α → G
s : Set α
x : α
ι : Type u_1
G₁ : Type u_2
G₂ : Type u_3
inst✝⁴ : TopologicalSpace G₂
inst✝³ : T2Space G₂
inst✝² : Inv G₁
inst✝¹ : Inv G₂
inst✝ : ContinuousInv G₂
⊢ IsClosed (⋂ i, {x | x i⁻¹ = (x i)⁻¹})
state_after: no goals
tactic: exact isClosed_iInter fun i => isClosed_eq (continuous_apply _) (continuous_apply _).inv
Theorem: IsMulFreimanIso.comp
File: Mathlib/Combinatorics/Additive/FreimanHom.lean

Proof steps:
state_before: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : CommMonoid α
inst✝¹ : CommMonoid β
inst✝ : CommMonoid γ
A A₁ A₂ : Set α
B B₁ B₂ : Set β
C : Set γ
f f₁ f₂ : α → β
g : β → γ
m n : ℕ
hg : IsMulFreimanIso n B C g
hf : IsMulFreimanIso n A B f
s t : Multiset α
hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
hs : card s = n
ht : card t = n
⊢ (map (g ∘ f) s).prod = (map (g ∘ f) t).prod ↔ s.prod = t.prod
state_after: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : CommMonoid α
inst✝¹ : CommMonoid β
inst✝ : CommMonoid γ
A A₁ A₂ : Set α
B B₁ B₂ : Set β
C : Set γ
f f₁ f₂ : α → β
g : β → γ
m n : ℕ
hg : IsMulFreimanIso n B C g
hf : IsMulFreimanIso n A B f
s t : Multiset α
hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
hs : card s = n
ht : card t = n
⊢ (map g (map f s)).prod = (map g (map f t)).prod ↔ s.prod = t.prod
tactic: rw [← map_map, ← map_map]
state_before: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : CommMonoid α
inst✝¹ : CommMonoid β
inst✝ : CommMonoid γ
A A₁ A₂ : Set α
B B₁ B₂ : Set β
C : Set γ
f f₁ f₂ : α → β
g : β → γ
m n : ℕ
hg : IsMulFreimanIso n B C g
hf : IsMulFreimanIso n A B f
s t : Multiset α
hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
hs : card s = n
ht : card t = n
⊢ (map g (map f s)).prod = (map g (map f t)).prod ↔ s.prod = t.prod
state_after: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : CommMonoid α
inst✝¹ : CommMonoid β
inst✝ : CommMonoid γ
A A₁ A₂ : Set α
B B₁ B₂ : Set β
C : Set γ
f f₁ f₂ : α → β
g : β → γ
m n : ℕ
hg : IsMulFreimanIso n B C g
hf : IsMulFreimanIso n A B f
s t : Multiset α
hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
hs : card s = n
ht : card t = n
⊢ ∀ ⦃x : β⦄, x ∈ map f s → x ∈ B

F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : CommMonoid α
inst✝¹ : CommMonoid β
inst✝ : CommMonoid γ
A A₁ A₂ : Set α
B B₁ B₂ : Set β
C : Set γ
f f₁ f₂ : α → β
g : β → γ
m n : ℕ
hg : IsMulFreimanIso n B C g
hf : IsMulFreimanIso n A B f
s t : Multiset α
hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
hs : card s = n
ht : card t = n
⊢ ∀ ⦃x : β⦄, x ∈ map f t → x ∈ B
tactic: rw [hg.map_prod_eq_map_prod _ _ (by rwa [card_map]) (by rwa [card_map]),
      hf.map_prod_eq_map_prod hsA htA hs ht]
Theorem: Filter.Tendsto.prod_map
File: Mathlib/Order/Filter/Prod.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ✝ : Type u_4
ι : Sort u_5
s : Set α
t : Set β
f✝ : Filter α
g✝ : Filter β
δ : Type u_6
f : α → γ
g : β → δ
a : Filter α
b : Filter β
c : Filter γ
d : Filter δ
hf : Tendsto f a c
hg : Tendsto g b d
⊢ Tendsto (Prod.map f g) (a ×ˢ b) (c ×ˢ d)
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ✝ : Type u_4
ι : Sort u_5
s : Set α
t : Set β
f✝ : Filter α
g✝ : Filter β
δ : Type u_6
f : α → γ
g : β → δ
a : Filter α
b : Filter β
c : Filter γ
d : Filter δ
hf : Tendsto f a c
hg : Tendsto g b d
⊢ map f a ×ˢ map g b ≤ c ×ˢ d
tactic: erw [Tendsto, ← prod_map_map_eq]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ✝ : Type u_4
ι : Sort u_5
s : Set α
t : Set β
f✝ : Filter α
g✝ : Filter β
δ : Type u_6
f : α → γ
g : β → δ
a : Filter α
b : Filter β
c : Filter γ
d : Filter δ
hf : Tendsto f a c
hg : Tendsto g b d
⊢ map f a ×ˢ map g b ≤ c ×ˢ d
state_after: no goals
tactic: exact Filter.prod_mono hf hg
Theorem: Polynomial.eraseLead_zero
File: Mathlib/Algebra/Polynomial/EraseLead.lean

Proof steps:
state_before: R : Type u_1
inst✝ : Semiring R
f : R[X]
⊢ eraseLead 0 = 0
state_after: no goals
tactic: simp only [eraseLead, erase_zero]
Theorem: MvPolynomial.coeffs_one
File: Mathlib/Algebra/MvPolynomial/Basic.lean

Proof steps:
state_before: 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
⊢ coeffs 1 ⊆ {1}
state_after: no goals
tactic: classical
    rw [coeffs, Finset.image_subset_iff]
    simp_all [coeff_one]
Theorem: Quiver.Path.comp_assoc
File: Mathlib/Combinatorics/Quiver/Path.lean

Proof steps:
state_before: V : Type u
inst✝ : Quiver V
a✝¹ b✝¹ c✝ d a b c x✝ : V
p : Path a b
q : Path b c
b✝ : V
r : Path c b✝
a✝ : b✝ ⟶ x✝
⊢ (p.comp q).comp (r.cons a✝) = p.comp (q.comp (r.cons a✝))
state_after: no goals
tactic: rw [comp_cons, comp_cons, comp_cons, comp_assoc p q r]
Theorem: ENNReal.tsum_fiberwise
File: Mathlib/Topology/Instances/ENNReal.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
a b c d : ℝ≥0∞
r p q : ℝ≥0
x y z ε ε₁ ε₂ : ℝ≥0∞
s : Set ℝ≥0∞
f✝ g✝ : α → ℝ≥0∞
f : β → ℝ≥0∞
g : β → γ
⊢ ∑' (x : γ) (b : ↑(g ⁻¹' {x})), f ↑b = ∑' (i : β), f i
state_after: case ha
α : Type u_1
β : Type u_2
γ : Type u_3
a b c d : ℝ≥0∞
r p q : ℝ≥0
x y z ε ε₁ ε₂ : ℝ≥0∞
s : Set ℝ≥0∞
f✝ g✝ : α → ℝ≥0∞
f : β → ℝ≥0∞
g : β → γ
⊢ HasSum (fun b => ∑' (b_1 : ↑(g ⁻¹' {b})), f ↑b_1) (∑' (i : β), f i)
tactic: apply HasSum.tsum_eq
state_before: case ha
α : Type u_1
β : Type u_2
γ : Type u_3
a b c d : ℝ≥0∞
r p q : ℝ≥0
x y z ε ε₁ ε₂ : ℝ≥0∞
s : Set ℝ≥0∞
f✝ g✝ : α → ℝ≥0∞
f : β → ℝ≥0∞
g : β → γ
⊢ HasSum (fun b => ∑' (b_1 : ↑(g ⁻¹' {b})), f ↑b_1) (∑' (i : β), f i)
state_after: case ha
α : Type u_1
β : Type u_2
γ : Type u_3
a b c d : ℝ≥0∞
r p q : ℝ≥0
x y z ε ε₁ ε₂ : ℝ≥0∞
s : Set ℝ≥0∞
f✝ g✝ : α → ℝ≥0∞
f : β → ℝ≥0∞
g : β → γ
equiv : (y : γ) × { x // g x = y } ≃ β := Equiv.sigmaFiberEquiv g
⊢ HasSum (fun b => ∑' (b_1 : ↑(g ⁻¹' {b})), f ↑b_1) (∑' (i : β), f i)
tactic: let equiv := Equiv.sigmaFiberEquiv g
state_before: case ha
α : Type u_1
β : Type u_2
γ : Type u_3
a b c d : ℝ≥0∞
r p q : ℝ≥0
x y z ε ε₁ ε₂ : ℝ≥0∞
s : Set ℝ≥0∞
f✝ g✝ : α → ℝ≥0∞
f : β → ℝ≥0∞
g : β → γ
equiv : (y : γ) × { x // g x = y } ≃ β := Equiv.sigmaFiberEquiv g
⊢ HasSum (fun b => ∑' (b_1 : ↑(g ⁻¹' {b})), f ↑b_1) (∑' (i : β), f i)
state_after: case ha
α : Type u_1
β : Type u_2
γ : Type u_3
a b c d : ℝ≥0∞
r p q : ℝ≥0
x y z ε ε₁ ε₂ : ℝ≥0∞
s : Set ℝ≥0∞
f✝ g✝ : α → ℝ≥0∞
f : β → ℝ≥0∞
g : β → γ
equiv : (y : γ) × { x // g x = y } ≃ β := Equiv.sigmaFiberEquiv g
⊢ ∀ (b : γ), HasSum (fun c => (f ∘ ⇑equiv) ⟨b, c⟩) (∑' (b_1 : ↑(g ⁻¹' {b})), f ↑b_1)
tactic: apply (equiv.hasSum_iff.mpr ENNReal.summable.hasSum).sigma
state_before: case ha
α : Type u_1
β : Type u_2
γ : Type u_3
a b c d : ℝ≥0∞
r p q : ℝ≥0
x y z ε ε₁ ε₂ : ℝ≥0∞
s : Set ℝ≥0∞
f✝ g✝ : α → ℝ≥0∞
f : β → ℝ≥0∞
g : β → γ
equiv : (y : γ) × { x // g x = y } ≃ β := Equiv.sigmaFiberEquiv g
⊢ ∀ (b : γ), HasSum (fun c => (f ∘ ⇑equiv) ⟨b, c⟩) (∑' (b_1 : ↑(g ⁻¹' {b})), f ↑b_1)
state_after: no goals
tactic: exact fun _ ↦ ENNReal.summable.hasSum_iff.mpr rfl
Theorem: HomologicalComplex.unopEquivalence_counitIso
File: Mathlib/Algebra/Homology/Opposite.lean

Proof steps:
state_before: ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.66755, u_2} V
c : ComplexShape ι
inst✝ : HasZeroMorphisms V
X : (HomologicalComplex Vᵒᵖ c)ᵒᵖ
⊢ (unopFunctor V c).map ((unopUnitIso V c).hom.app X) ≫ (unopCounitIso V c).hom.app ((unopFunctor V c).obj X) =
    𝟙 ((unopFunctor V c).obj X)
state_after: case h
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.66755, u_2} V
c : ComplexShape ι
inst✝ : HasZeroMorphisms V
X : (HomologicalComplex Vᵒᵖ c)ᵒᵖ
i✝ : ι
⊢ ((unopFunctor V c).map ((unopUnitIso V c).hom.app X) ≫ (unopCounitIso V c).hom.app ((unopFunctor V c).obj X)).f i✝ =
    (𝟙 ((unopFunctor V c).obj X)).f i✝
tactic: ext
state_before: case h
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.66755, u_2} V
c : ComplexShape ι
inst✝ : HasZeroMorphisms V
X : (HomologicalComplex Vᵒᵖ c)ᵒᵖ
i✝ : ι
⊢ ((unopFunctor V c).map ((unopUnitIso V c).hom.app X) ≫ (unopCounitIso V c).hom.app ((unopFunctor V c).obj X)).f i✝ =
    (𝟙 ((unopFunctor V c).obj X)).f i✝
state_after: case h
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.66755, u_2} V
c : ComplexShape ι
inst✝ : HasZeroMorphisms V
X : (HomologicalComplex Vᵒᵖ c)ᵒᵖ
i✝ : ι
⊢ ((unopFunctor V c).map ((unopUnitIso V c).hom.app X)).f i✝ ≫
      ((unopCounitIso V c).hom.app ((unopFunctor V c).obj X)).f i✝ =
    (𝟙 ((unopFunctor V c).obj X)).f i✝
tactic: simp only [opUnitIso, opCounitIso, NatIso.ofComponents_hom_app, Iso.op_hom, comp_f,
      opFunctor_map_f, Quiver.Hom.unop_op, Hom.isoOfComponents_hom_f]
state_before: case h
ι : Type u_1
V : Type u_2
inst✝¹ : Category.{?u.66755, u_2} V
c : ComplexShape ι
inst✝ : HasZeroMorphisms V
X : (HomologicalComplex Vᵒᵖ c)ᵒᵖ
i✝ : ι
⊢ ((unopFunctor V c).map ((unopUnitIso V c).hom.app X)).f i✝ ≫
      ((unopCounitIso V c).hom.app ((unopFunctor V c).obj X)).f i✝ =
    (𝟙 ((unopFunctor V c).obj X)).f i✝
state_after: no goals
tactic: exact Category.comp_id _
Theorem: ENNReal.mul_sSup
File: Mathlib/Topology/Instances/ENNReal.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
a✝ b c d : ℝ≥0∞
r p q : ℝ≥0
x y z ε ε₁ ε₂ : ℝ≥0∞
s✝ s : Set ℝ≥0∞
a : ℝ≥0∞
⊢ a * sSup s = ⨆ i ∈ s, a * i
state_after: no goals
tactic: simp only [sSup_eq_iSup, mul_iSup]
Theorem: List.Pi.mem_enum
File: Mathlib/Data/FinEnum.lean

Proof steps:
state_before: α : Type u_1
inst✝¹ : FinEnum α
β : α → Type u_2
inst✝ : (a : α) → FinEnum (β a)
f : (a : α) → β a
⊢ f ∈ enum β
state_after: no goals
tactic: simpa [Pi.enum] using ⟨fun a _ => f a, mem_pi_toList _ _, rfl⟩
Theorem: Finset.mem_nonMemberSubfamily
File: Mathlib/Combinatorics/SetFamily/Compression/Down.lean

Proof steps:
state_before: α : Type u_1
inst✝ : DecidableEq α
𝒜 ℬ : Finset (Finset α)
s : Finset α
a : α
⊢ s ∈ nonMemberSubfamily a 𝒜 ↔ s ∈ 𝒜 ∧ a ∉ s
state_after: no goals
tactic: simp [nonMemberSubfamily]
Theorem: CochainComplex.shiftShortComplexFunctorIso_zero_add_hom_app
File: Mathlib/Algebra/Homology/HomotopyCategory/ShiftSequence.lean

Proof steps:
state_before: C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
a : ℤ
K : CochainComplex C ℤ
⊢ (shiftShortComplexFunctorIso C 0 a a ⋯).hom.app K =
    (shortComplexFunctor C (up ℤ) a).map ((shiftFunctorZero (CochainComplex C ℤ) ℤ).hom.app K)
state_after: no goals
tactic: ext <;> dsimp <;> simp [one_smul, shiftFunctorZero_hom_app_f]
Theorem: Polynomial.isCoprime_X_sub_C_of_isUnit_sub
File: Mathlib/Algebra/Polynomial/RingDivision.lean

Proof steps:
state_before: R✝ : Type u
S : Type v
T : Type w
a✝ b✝ : R✝
n : ℕ
inst✝² : CommRing R✝
inst✝¹ : IsDomain R✝
p q : R✝[X]
R : Type u_1
inst✝ : CommRing R
a b : R
h : IsUnit (a - b)
⊢ -C ↑h.unit⁻¹ * (X - C a) + C ↑h.unit⁻¹ * (X - C b) = 1
state_after: R✝ : Type u
S : Type v
T : Type w
a✝ b✝ : R✝
n : ℕ
inst✝² : CommRing R✝
inst✝¹ : IsDomain R✝
p q : R✝[X]
R : Type u_1
inst✝ : CommRing R
a b : R
h : IsUnit (a - b)
⊢ C (↑h.unit⁻¹ * (a - b)) = 1
tactic: rw [neg_mul_comm, ← left_distrib, neg_add_eq_sub, sub_sub_sub_cancel_left, ← C_sub, ← C_mul]
state_before: R✝ : Type u
S : Type v
T : Type w
a✝ b✝ : R✝
n : ℕ
inst✝² : CommRing R✝
inst✝¹ : IsDomain R✝
p q : R✝[X]
R : Type u_1
inst✝ : CommRing R
a b : R
h : IsUnit (a - b)
⊢ C (↑h.unit⁻¹ * (a - b)) = 1
state_after: R✝ : Type u
S : Type v
T : Type w
a✝ b✝ : R✝
n : ℕ
inst✝² : CommRing R✝
inst✝¹ : IsDomain R✝
p q : R✝[X]
R : Type u_1
inst✝ : CommRing R
a b : R
h : IsUnit (a - b)
⊢ C (↑h.unit⁻¹ * (a - b)) = C 1
tactic: rw [← C_1]
state_before: R✝ : Type u
S : Type v
T : Type w
a✝ b✝ : R✝
n : ℕ
inst✝² : CommRing R✝
inst✝¹ : IsDomain R✝
p q : R✝[X]
R : Type u_1
inst✝ : CommRing R
a b : R
h : IsUnit (a - b)
⊢ C (↑h.unit⁻¹ * (a - b)) = C 1
state_after: case h.e_6.h
R✝ : Type u
S : Type v
T : Type w
a✝ b✝ : R✝
n : ℕ
inst✝² : CommRing R✝
inst✝¹ : IsDomain R✝
p q : R✝[X]
R : Type u_1
inst✝ : CommRing R
a b : R
h : IsUnit (a - b)
⊢ ↑h.unit⁻¹ * (a - b) = 1
tactic: congr
state_before: case h.e_6.h
R✝ : Type u
S : Type v
T : Type w
a✝ b✝ : R✝
n : ℕ
inst✝² : CommRing R✝
inst✝¹ : IsDomain R✝
p q : R✝[X]
R : Type u_1
inst✝ : CommRing R
a b : R
h : IsUnit (a - b)
⊢ ↑h.unit⁻¹ * (a - b) = 1
state_after: no goals
tactic: exact h.val_inv_mul
Theorem: Asymptotics.superpolynomialDecay_iff_superpolynomialDecay_abs
File: Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
l : Filter α
k f g g' : α → β
inst✝² : TopologicalSpace β
inst✝¹ : LinearOrderedCommRing β
inst✝ : OrderTopology β
⊢ (∀ (n : ℕ), Tendsto (fun a => |k a ^ n * f a|) l (𝓝 0)) ↔ SuperpolynomialDecay l (fun a => |k a|) fun a => |f a|
state_after: no goals
tactic: simp_rw [SuperpolynomialDecay, abs_mul, abs_pow]
Theorem: SimpleGraph.nonuniformWitness_subset
File: Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean

Proof steps:
state_before: α : Type u_1
𝕜 : Type u_2
inst✝¹ : LinearOrderedField 𝕜
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : 𝕜
s t : Finset α
a b : α
h : ¬G.IsUniform ε s t
⊢ G.nonuniformWitness ε s t ⊆ s
state_after: α : Type u_1
𝕜 : Type u_2
inst✝¹ : LinearOrderedField 𝕜
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : 𝕜
s t : Finset α
a b : α
h : ¬G.IsUniform ε s t
⊢ (if WellOrderingRel s t then (G.nonuniformWitnesses ε s t).1 else (G.nonuniformWitnesses ε t s).2) ⊆ s
tactic: unfold nonuniformWitness
state_before: α : Type u_1
𝕜 : Type u_2
inst✝¹ : LinearOrderedField 𝕜
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : 𝕜
s t : Finset α
a b : α
h : ¬G.IsUniform ε s t
⊢ (if WellOrderingRel s t then (G.nonuniformWitnesses ε s t).1 else (G.nonuniformWitnesses ε t s).2) ⊆ s
state_after: case pos
α : Type u_1
𝕜 : Type u_2
inst✝¹ : LinearOrderedField 𝕜
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : 𝕜
s t : Finset α
a b : α
h : ¬G.IsUniform ε s t
h✝ : WellOrderingRel s t
⊢ (G.nonuniformWitnesses ε s t).1 ⊆ s

case neg
α : Type u_1
𝕜 : Type u_2
inst✝¹ : LinearOrderedField 𝕜
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : 𝕜
s t : Finset α
a b : α
h : ¬G.IsUniform ε s t
h✝ : ¬WellOrderingRel s t
⊢ (G.nonuniformWitnesses ε t s).2 ⊆ s
tactic: split_ifs
Theorem: AddAction.zsmul_vadd_eq_iff_period_dvd
File: Mathlib/Dynamics/PeriodicPts.lean

Proof steps:
state_before: α : Type v
G : Type u
inst✝³ : Group G
inst✝² : MulAction G α
M : Type u
inst✝¹ : Monoid M
inst✝ : MulAction M α
j : ℤ
g : G
a : α
⊢ g ^ j • a = a ↔ ↑(period g a) ∣ j
state_after: case ofNat
α : Type v
G : Type u
inst✝³ : Group G
inst✝² : MulAction G α
M : Type u
inst✝¹ : Monoid M
inst✝ : MulAction M α
g : G
a : α
n : ℕ
⊢ g ^ Int.ofNat n • a = a ↔ ↑(period g a) ∣ Int.ofNat n

case negSucc
α : Type v
G : Type u
inst✝³ : Group G
inst✝² : MulAction G α
M : Type u
inst✝¹ : Monoid M
inst✝ : MulAction M α
g : G
a : α
n : ℕ
⊢ g ^ Int.negSucc n • a = a ↔ ↑(period g a) ∣ Int.negSucc n
tactic: rcases j with n | n
Theorem: MeasureTheory.continuous_diracProba
File: Mathlib/MeasureTheory/Measure/DiracProba.lean

Proof steps:
state_before: X : Type u_1
inst✝² : MeasurableSpace X
inst✝¹ : TopologicalSpace X
inst✝ : OpensMeasurableSpace X
⊢ Continuous fun x => diracProba x
state_after: X : Type u_1
inst✝² : MeasurableSpace X
inst✝¹ : TopologicalSpace X
inst✝ : OpensMeasurableSpace X
⊢ ∀ (x : X), ContinuousAt (fun x => diracProba x) x
tactic: rw [continuous_iff_continuousAt]
state_before: X : Type u_1
inst✝² : MeasurableSpace X
inst✝¹ : TopologicalSpace X
inst✝ : OpensMeasurableSpace X
⊢ ∀ (x : X), ContinuousAt (fun x => diracProba x) x
state_after: X : Type u_1
inst✝² : MeasurableSpace X
inst✝¹ : TopologicalSpace X
inst✝ : OpensMeasurableSpace X
x : X
f : X →ᵇ ℝ≥0
⊢ Tendsto (fun i => ∫⁻ (ω : X), ↑(f ω) ∂↑(diracProba i)) (𝓝 x) (𝓝 (∫⁻ (ω : X), ↑(f ω) ∂↑(diracProba x)))
tactic: apply fun x ↦ ProbabilityMeasure.tendsto_iff_forall_lintegral_tendsto.mpr fun f ↦ ?_
state_before: X : Type u_1
inst✝² : MeasurableSpace X
inst✝¹ : TopologicalSpace X
inst✝ : OpensMeasurableSpace X
x : X
f : X →ᵇ ℝ≥0
⊢ Tendsto (fun i => ∫⁻ (ω : X), ↑(f ω) ∂↑(diracProba i)) (𝓝 x) (𝓝 (∫⁻ (ω : X), ↑(f ω) ∂↑(diracProba x)))
state_after: X : Type u_1
inst✝² : MeasurableSpace X
inst✝¹ : TopologicalSpace X
inst✝ : OpensMeasurableSpace X
x : X
f : X →ᵇ ℝ≥0
f_mble : Measurable fun X_1 => ↑(f X_1)
⊢ Tendsto (fun i => ∫⁻ (ω : X), ↑(f ω) ∂↑(diracProba i)) (𝓝 x) (𝓝 (∫⁻ (ω : X), ↑(f ω) ∂↑(diracProba x)))
tactic: have f_mble : Measurable (fun X ↦ (f X : ℝ≥0∞)) :=
    measurable_coe_nnreal_ennreal_iff.mpr f.continuous.measurable
state_before: X : Type u_1
inst✝² : MeasurableSpace X
inst✝¹ : TopologicalSpace X
inst✝ : OpensMeasurableSpace X
x : X
f : X →ᵇ ℝ≥0
f_mble : Measurable fun X_1 => ↑(f X_1)
⊢ Tendsto (fun i => ∫⁻ (ω : X), ↑(f ω) ∂↑(diracProba i)) (𝓝 x) (𝓝 (∫⁻ (ω : X), ↑(f ω) ∂↑(diracProba x)))
state_after: X : Type u_1
inst✝² : MeasurableSpace X
inst✝¹ : TopologicalSpace X
inst✝ : OpensMeasurableSpace X
x : X
f : X →ᵇ ℝ≥0
f_mble : Measurable fun X_1 => ↑(f X_1)
⊢ Tendsto (fun i => ↑(f i)) (𝓝 x) (𝓝 ↑(f x))
tactic: simp only [diracProba, ProbabilityMeasure.coe_mk, lintegral_dirac' _ f_mble]
state_before: X : Type u_1
inst✝² : MeasurableSpace X
inst✝¹ : TopologicalSpace X
inst✝ : OpensMeasurableSpace X
x : X
f : X →ᵇ ℝ≥0
f_mble : Measurable fun X_1 => ↑(f X_1)
⊢ Tendsto (fun i => ↑(f i)) (𝓝 x) (𝓝 ↑(f x))
state_after: no goals
tactic: exact (ENNReal.continuous_coe.comp f.continuous).continuousAt
Theorem: ENNReal.log_rpow
File: Mathlib/Analysis/SpecialFunctions/Log/ENNRealLog.lean

Proof steps:
state_before: x : ℝ≥0∞
y : ℝ
⊢ (x ^ y).log = ↑y * x.log
state_after: case inl
x : ℝ≥0∞
y : ℝ
y_neg : y < 0
⊢ (x ^ y).log = ↑y * x.log

case inr.inl
x : ℝ≥0∞
⊢ (x ^ 0).log = ↑0 * x.log

case inr.inr
x : ℝ≥0∞
y : ℝ
y_pos : 0 < y
⊢ (x ^ y).log = ↑y * x.log
tactic: rcases lt_trichotomy y 0 with (y_neg | rfl | y_pos)
Theorem: TopologicalSpace.Closeds.coe_sup
File: Mathlib/Topology/Sets/Closeds.lean

Proof steps:
state_before: ι : Type u_1
α : Type u_2
β : Type u_3
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
s t : Closeds α
⊢ ↑(s ⊔ t) = ↑s ∪ ↑t
state_after: no goals
tactic: rfl