mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Star.MonoidHom

Goal state before:
pp: F : Type u_1
A : Type u_2
B : Type u_3
C : Type u_4
D : Type u_5
inst✝⁷ : Mul A
inst✝⁶ : Mul B
inst✝⁵ : Mul C
inst✝⁴ : Mul D
inst✝³ : Star A
inst✝² : Star B
inst✝¹ : Star C
inst✝ : Star D
f g : A ≃⋆* B
h : f.toFun = g.toFun
⊢ f.invFun = g.invFun → f = g

Tactic:
cases f

Goal state after:
case mk
F : Type u_1
A : Type u_2
B : Type u_3
C : Type u_4
D : Type u_5
inst✝⁷ : Mul A
inst✝⁶ : Mul B
inst✝⁵ : Mul C
inst✝⁴ : Mul D
inst✝³ : Star A
inst✝² : Star B
inst✝¹ : Star C
inst✝ : Star D
g : A ≃⋆* B
toMulEquiv✝ : A ≃* B
map_star'✝ : ∀ (a : A), toMulEquiv✝.toFun (star a) = star (toMulEquiv✝.toFun a)
h : { toMulEquiv := toMulEquiv✝, map_star' := map_star'✝ }.toFun = g.toFun
⊢ { toMulEquiv := toMulEquiv✝, map_star' := map_star'✝ }.invFun = g.invFun →
    { toMulEquiv := toMulEquiv✝, map_star' := map_star'✝ } = g
Module: Mathlib.Analysis.Normed.Group.Hom

Goal state before:
pp: V : Type u_1
V₁ : Type u_2
V₂ : Type u_3
V₃ : Type u_4
inst✝³ : SeminormedAddCommGroup V
inst✝² : SeminormedAddCommGroup V₁
inst✝¹ : SeminormedAddCommGroup V₂
inst✝ : SeminormedAddCommGroup V₃
f✝ g✝ f g : NormedAddGroupHom V₁ V₂
h : f.toFun = g.toFun
⊢ f = g

Tactic:
cases f

Goal state after:
case mk
V : Type u_1
V₁ : Type u_2
V₂ : Type u_3
V₃ : Type u_4
inst✝³ : SeminormedAddCommGroup V
inst✝² : SeminormedAddCommGroup V₁
inst✝¹ : SeminormedAddCommGroup V₂
inst✝ : SeminormedAddCommGroup V₃
f g✝ g : NormedAddGroupHom V₁ V₂
toFun✝ : V₁ → V₂
map_add'✝ : ∀ (v₁ v₂ : V₁), toFun✝ (v₁ + v₂) = toFun✝ v₁ + toFun✝ v₂
bound'✝ : ∃ C, ∀ (v : V₁), ‖toFun✝ v‖ ≤ C * ‖v‖
h : { toFun := toFun✝, map_add' := map_add'✝, bound' := bound'✝ }.toFun = g.toFun
⊢ { toFun := toFun✝, map_add' := map_add'✝, bound' := bound'✝ } = g
Module: Mathlib.Analysis.Normed.Group.Hom

Goal state before:
pp: V₁ : Type u_2
V₂ : Type u_3
inst✝¹ : SeminormedAddCommGroup V₁
inst✝ : SeminormedAddCommGroup V₂
f g : NormedAddGroupHom V₁ V₂
H : ⇑f = ⇑g
⊢ f = g

Tactic:
cases f

Goal state after:
case mk
V₁ : Type u_2
V₂ : Type u_3
inst✝¹ : SeminormedAddCommGroup V₁
inst✝ : SeminormedAddCommGroup V₂
g : NormedAddGroupHom V₁ V₂
toFun✝ : V₁ → V₂
map_add'✝ : ∀ (v₁ v₂ : V₁), toFun✝ (v₁ + v₂) = toFun✝ v₁ + toFun✝ v₂
bound'✝ : ∃ C, ∀ (v : V₁), ‖toFun✝ v‖ ≤ C * ‖v‖
H : ⇑{ toFun := toFun✝, map_add' := map_add'✝, bound' := bound'✝ } = ⇑g
⊢ { toFun := toFun✝, map_add' := map_add'✝, bound' := bound'✝ } = g
Module: Mathlib.Analysis.Normed.Group.Hom

Goal state before:
pp: V₁ : Type u_2
V₂ : Type u_3
inst✝¹ : SeminormedAddCommGroup V₁
inst✝ : SeminormedAddCommGroup V₂
f : NormedAddGroupHom V₁ V₂
K : AddSubgroup V₂
C : ℝ
h : f.SurjectiveOnWith K C
⊢ ∃ C' > 0, f.SurjectiveOnWith K C'

Tactic:
refine ⟨|C| + 1, ?_, ?_⟩

Goal state after:
case refine_1
V₁ : Type u_2
V₂ : Type u_3
inst✝¹ : SeminormedAddCommGroup V₁
inst✝ : SeminormedAddCommGroup V₂
f : NormedAddGroupHom V₁ V₂
K : AddSubgroup V₂
C : ℝ
h : f.SurjectiveOnWith K C
⊢ |C| + 1 > 0

case refine_2
V₁ : Type u_2
V₂ : Type u_3
inst✝¹ : SeminormedAddCommGroup V₁
inst✝ : SeminormedAddCommGroup V₂
f : NormedAddGroupHom V₁ V₂
K : AddSubgroup V₂
C : ℝ
h : f.SurjectiveOnWith K C
⊢ f.SurjectiveOnWith K (|C| + 1)
Module: Mathlib.Analysis.Normed.Group.Hom

Goal state before:
pp: V₁ : Type u_2
V₂ : Type u_3
inst✝¹ : SeminormedAddCommGroup V₁
inst✝ : SeminormedAddCommGroup V₂
f : NormedAddGroupHom V₁ V₂
x y : V₁
⊢ ‖f x - f y‖ ≤ ↑⟨‖f‖, ⋯⟩ * ‖x - y‖

Tactic:
← map_sub

Goal state after:
V₁ : Type u_2
V₂ : Type u_3
inst✝¹ : SeminormedAddCommGroup V₁
inst✝ : SeminormedAddCommGroup V₂
f : NormedAddGroupHom V₁ V₂
x y : V₁
⊢ ‖f (x - y)‖ ≤ ↑⟨‖f‖, ⋯⟩ * ‖x - y‖
Module: Mathlib.Analysis.Normed.Group.Hom

Goal state before:
pp: V₁ : Type u_3
V₂ : Type u_4
V₃ : Type u_5
inst✝⁵ : SeminormedAddCommGroup V₁
inst✝⁴ : SeminormedAddCommGroup V₂
inst✝³ : SeminormedAddCommGroup V₃
W₁ : Type u_6
W₂ : Type u_7
W₃ : Type u_8
inst✝² : SeminormedAddCommGroup W₁
inst✝¹ : SeminormedAddCommGroup W₂
inst✝ : SeminormedAddCommGroup W₃
f₁ : NormedAddGroupHom V₁ W₁
f₂ : NormedAddGroupHom V₂ W₂
f₃ : NormedAddGroupHom V₃ W₃
φ : NormedAddGroupHom V₁ V₂
ψ : NormedAddGroupHom W₁ W₂
φ' : NormedAddGroupHom V₂ V₃
ψ' : NormedAddGroupHom W₂ W₃
hf : ψ.comp f₁ = f₂.comp φ
hf' : ψ'.comp f₂ = f₃.comp φ'
⊢ (ψ'.comp ψ).comp f₁ = f₃.comp (φ'.comp φ)

Tactic:
by
  rw [comp_assoc, hf, ← comp_assoc, hf', comp_assoc]

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.ContinuousLinearMap

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
inst✝⁴ : SeminormedRing 𝕜
inst✝³ : Ring 𝕜₂
inst✝² : SeminormedAddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
f : 𝕜 →ₗ[𝕜] E
x : 𝕜
⊢ ‖f x‖ ≤ ‖f 1‖ * ‖x‖

Tactic:
by
    conv_lhs => rw [← mul_one x]
    rw [← smul_eq_mul, f.map_smul, mul_comm]; exact norm_smul_le _ _

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.Module

Goal state before:
pp: case neg.refine_1
ι : Type u_5
R : Type u_7
R₂ : Type u_8
M : Type u_9
M₂ : Type u_10
inst✝¹¹ : Semiring R
inst✝¹⁰ : Semiring R₂
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
inst✝⁷ : AddCommMonoid M₂
inst✝⁶ : Module R₂ M₂
inst✝⁵ : TopologicalSpace M
inst✝⁴ : TopologicalSpace M₂
σ : R →+* R₂
σ' : R₂ →+* R
inst✝³ : RingHomInvPair σ σ'
inst✝² : RingHomInvPair σ' σ
L : SummationFilter ι
inst✝¹ : T2Space M
inst✝ : T2Space M₂
f : ι → M
e : M ≃SL[σ] M₂
y : M₂
hf : ¬Summable f L
hf' : ¬Summable (fun z ↦ e (f z)) L
⊢ 0 = y → 0 = e.symm y

Tactic:
rintro rfl
      simp

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.Module

Goal state before:
pp: ι : Type u_5
R : Type u_7
R₂ : Type u_8
M : Type u_9
M₂ : Type u_10
inst✝¹¹ : Semiring R
inst✝¹⁰ : Semiring R₂
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
inst✝⁷ : AddCommMonoid M₂
inst✝⁶ : Module R₂ M₂
inst✝⁵ : TopologicalSpace M
inst✝⁴ : TopologicalSpace M₂
σ : R →+* R₂
σ' : R₂ →+* R
inst✝³ : RingHomInvPair σ σ'
inst✝² : RingHomInvPair σ' σ
L : SummationFilter ι
inst✝¹ : T2Space M
inst✝ : T2Space M₂
f : ι → M
e : M ≃SL[σ] M₂
⊢ ∑'[L] (z : ι), f z = e.symm (e (∑'[L] (z : ι), f z))

Tactic:
rw [e.symm_apply_apply _]

Goal state after:
Module: Mathlib.Analysis.Complex.Basic

Goal state before:
pp: ι : Type u_2
a b : ℝ
f : ι → ℂ
s : Finset ι
ha₀ : 0 ≤ a
ha : ↑a = ∑ i ∈ s, f i
hb : b = ∑ i ∈ s, ‖f i‖
⊢ a ≤ b

Tactic:
← Complex.norm_of_nonneg ha₀,

Goal state after:
ι : Type u_2
a b : ℝ
f : ι → ℂ
s : Finset ι
ha₀ : 0 ≤ a
ha : ↑a = ∑ i ∈ s, f i
hb : b = ∑ i ∈ s, ‖f i‖
⊢ ‖↑a‖ ≤ b
Module: Mathlib.LinearAlgebra.Complex.Module

Goal state before:
pp: case e_a
A : Type u_1
inst✝¹ : Ring A
inst✝ : Algebra ℝ A
I' : A
hf : I' * I' = -1
x₁ y₁ x₂ y₂ : ℝ
⊢ (algebraMap ℝ A) (x₁ * x₂ - y₁ * y₂) = (algebraMap ℝ A) (x₁ * x₂) - (algebraMap ℝ A) (y₁ * y₂)

Tactic:
← map_sub

Goal state after:
case e_a
A : Type u_1
inst✝¹ : Ring A
inst✝ : Algebra ℝ A
I' : A
hf : I' * I' = -1
x₁ y₁ x₂ y₂ : ℝ
⊢ (algebraMap ℝ A) (x₁ * x₂ - y₁ * y₂) = (algebraMap ℝ A) (x₁ * x₂ - y₁ * y₂)
Module: Mathlib.Analysis.Complex.Basic

Goal state before:
pp: z : ℂ
E : Type u_1
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
𝕜 : Type u_2
inst✝ : RCLike 𝕜
h : RCLike.im RCLike.I = 1
x y : 𝕜
⊢ ↑(RCLike.re (x + y)) + ↑(RCLike.im (x + y)) * I =
    ↑(RCLike.re x) + ↑(RCLike.im x) * I + (↑(RCLike.re y) + ↑(RCLike.im y) * I)

Tactic:
simp only [map_add, ofReal_add]

Goal state after:
z : ℂ
E : Type u_1
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
𝕜 : Type u_2
inst✝ : RCLike 𝕜
h : RCLike.im RCLike.I = 1
x y : 𝕜
⊢ ↑(RCLike.re x) + ↑(RCLike.re y) + (↑(RCLike.im x) + ↑(RCLike.im y)) * I =
    ↑(RCLike.re x) + ↑(RCLike.im x) * I + (↑(RCLike.re y) + ↑(RCLike.im y) * I)
Module: Mathlib.LinearAlgebra.Complex.Module

Goal state before:
pp: A : Type u_1
inst✝³ : AddCommGroup A
inst✝² : Module ℂ A
inst✝¹ : StarAddMonoid A
inst✝ : StarModule ℂ A
x : A
hx : IsSelfAdjoint x
⊢ (skewAdjoint.negISMul ∘ₗ skewAdjointPart ℝ) x = 0

Tactic:
LinearMap.comp_apply,

Goal state after:
A : Type u_1
inst✝³ : AddCommGroup A
inst✝² : Module ℂ A
inst✝¹ : StarAddMonoid A
inst✝ : StarModule ℂ A
x : A
hx : IsSelfAdjoint x
⊢ skewAdjoint.negISMul ((skewAdjointPart ℝ) x) = 0
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝ : RCLike K
z : ℝ
⊢ (algebraMap ℝ ℝ) ((AddMonoidHom.id ℝ) z) + (algebraMap ℝ ℝ) (0 z) * 0 = z

Tactic:
by
    simp only [add_zero, mul_zero, Algebra.algebraMap_self, RingHom.id_apply, AddMonoidHom.id_apply]

Goal state after:
Module: Mathlib.Topology.OpenPartialHomeomorph.Continuity

Goal state before:
pp: X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : OpenPartialHomeomorph X Y
x : X
p : Y → Prop
s : Set X
hx : x ∈ e.source
⊢ (∀ᶠ (y : Y) in 𝓝[↑e.symm ⁻¹' s] ↑e x, p y) ↔ ∀ᶠ (x : X) in 𝓝[s] x, p (↑e x)

Tactic:
refine Iff.trans ?_ eventually_map

Goal state after:
X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : OpenPartialHomeomorph X Y
x : X
p : Y → Prop
s : Set X
hx : x ∈ e.source
⊢ (∀ᶠ (y : Y) in 𝓝[↑e.symm ⁻¹' s] ↑e x, p y) ↔ ∀ᶠ (b : Y) in map (↑e) (𝓝[s] x), p b
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
x y : K
⊢ ‖re x - re y‖ₑ ≤ ‖x - y‖ₑ

Tactic:
calc ‖re x - re y‖ₑ
  _ = ‖re (x - y)‖ₑ := by rw [map_sub re x y]
  _ ≤ ‖x - y‖ₑ := by rw [enorm_le_iff_norm_le]; exact norm_re_le_norm (x - y)

Goal state after:
Module: Mathlib.Analysis.Asymptotics.Defs

Goal state before:
pp: α : Type u_1
E : Type u_3
F : Type u_4
inst✝¹ : Norm E
inst✝ : Norm F
f : α → E
g : α → F
l : Filter α
⊢ f =O[l] g ↔ ∃ c, IsBigOWith c l f g

Tactic:
rw [IsBigO_def]

Goal state after:
Module: Mathlib.Analysis.Asymptotics.Defs

Goal state before:
pp: α : Type u_1
E : Type u_3
F : Type u_4
inst✝¹ : Norm E
inst✝ : Norm F
c : ℝ
f : α → E
g : α → F
l : Filter α
h : IsBigOWith c l f g
⊢ f =O[l] g

Tactic:
rw [IsBigO_def]

Goal state after:
α : Type u_1
E : Type u_3
F : Type u_4
inst✝¹ : Norm E
inst✝ : Norm F
c : ℝ
f : α → E
g : α → F
l : Filter α
h : IsBigOWith c l f g
⊢ ∃ c, IsBigOWith c l f g
Module: Mathlib.Analysis.Asymptotics.Defs

Goal state before:
pp: α : Type u_1
E : Type u_3
F : Type u_4
inst✝¹ : Norm E
inst✝ : Norm F
l : Filter α
f₁ f₂ : α → E
g₁ g₂ : α → F
hf : f₁ =ᶠ[l] f₂
hg : g₁ =ᶠ[l] g₂
⊢ f₁ =o[l] g₁ ↔ f₂ =o[l] g₂

Tactic:
simp only [IsLittleO_def]

Goal state after:
α : Type u_1
E : Type u_3
F : Type u_4
inst✝¹ : Norm E
inst✝ : Norm F
l : Filter α
f₁ f₂ : α → E
g₁ g₂ : α → F
hf : f₁ =ᶠ[l] f₂
hg : g₁ =ᶠ[l] g₂
⊢ (∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l f₁ g₁) ↔ ∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l f₂ g₂
Module: Mathlib.Analysis.Asymptotics.Defs

Goal state before:
pp: α : Type u_1
E : Type u_3
F' : Type u_7
inst✝¹ : Norm E
inst✝ : SeminormedAddCommGroup F'
f : α → E
g' : α → F'
l : Filter α
⊢ (f =o[l] fun x ↦ ‖g' x‖) ↔ f =o[l] g'

Tactic:
simp only [IsLittleO_def]

Goal state after:
α : Type u_1
E : Type u_3
F' : Type u_7
inst✝¹ : Norm E
inst✝ : SeminormedAddCommGroup F'
f : α → E
g' : α → F'
l : Filter α
⊢ (∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l f fun x ↦ ‖g' x‖) ↔ ∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l f g'
Module: Mathlib.Analysis.Asymptotics.Defs

Goal state before:
pp: α : Type u_1
F : Type u_4
E' : Type u_6
inst✝¹ : Norm F
inst✝ : SeminormedAddCommGroup E'
g : α → F
f' : α → E'
l : Filter α
⊢ (fun x ↦ ‖f' x‖) =o[l] g ↔ f' =o[l] g

Tactic:
simp only [IsLittleO_def]

Goal state after:
α : Type u_1
F : Type u_4
E' : Type u_6
inst✝¹ : Norm F
inst✝ : SeminormedAddCommGroup E'
g : α → F
f' : α → E'
l : Filter α
⊢ (∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l (fun x ↦ ‖f' x‖) g) ↔ ∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l f' g
Module: Mathlib.Analysis.Asymptotics.Defs

Goal state before:
pp: α : Type u_1
E : Type u_3
F' : Type u_7
inst✝¹ : Norm E
inst✝ : SeminormedAddCommGroup F'
f : α → E
g' : α → F'
l : Filter α
⊢ (f =o[l] fun x ↦ -g' x) ↔ f =o[l] g'

Tactic:
simp only [IsLittleO_def]

Goal state after:
α : Type u_1
E : Type u_3
F' : Type u_7
inst✝¹ : Norm E
inst✝ : SeminormedAddCommGroup F'
f : α → E
g' : α → F'
l : Filter α
⊢ (∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l f fun x ↦ -g' x) ↔ ∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l f g'
Module: Mathlib.Analysis.Asymptotics.Defs

Goal state before:
pp: α : Type u_1
F : Type u_4
E' : Type u_6
inst✝¹ : Norm F
inst✝ : SeminormedAddCommGroup E'
g : α → F
f' : α → E'
l : Filter α
⊢ (fun x ↦ -f' x) =o[l] g ↔ f' =o[l] g

Tactic:
simp only [IsLittleO_def]

Goal state after:
α : Type u_1
F : Type u_4
E' : Type u_6
inst✝¹ : Norm F
inst✝ : SeminormedAddCommGroup E'
g : α → F
f' : α → E'
l : Filter α
⊢ (∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l (fun x ↦ -f' x) g) ↔ ∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l f' g
Module: Mathlib.Analysis.Asymptotics.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝¹ : Norm E
inst✝ : Norm F
f : α → E
g : α → F
l : Filter α
l' : Filter β
⊢ (∃ c, ∀ᶠ (x : α) in l, ‖f x‖ ≤ c * ‖g x‖) →
    ∃ c pa,
      (∀ᶠ (x : β) in l', pa x) ∧
        ∃ pb,
          (∀ᶠ (y : α) in l, pb y) ∧
            ∀ {x : β}, pa x → ∀ {y : α}, pb y → ‖(f ∘ Prod.snd) (x, y)‖ ≤ c * ‖(g ∘ Prod.snd) (x, y)‖

Tactic:
exact fun ⟨c, hc⟩ ↦ ⟨c, fun _ ↦ True, eventually_true l', _, hc, fun _ ↦ id⟩

Goal state after:
Module: Mathlib.Analysis.Asymptotics.Lemmas

Goal state before:
pp: α : Type u_1
E'' : Type u_9
F'' : Type u_10
inst✝¹ : NormedAddCommGroup E''
inst✝ : NormedAddCommGroup F''
f'' : α → E''
l : Filter α
c : F''
h : f'' =O[l] fun _x ↦ c
hc : c = 0
⊢ f'' =O[l] fun _x ↦ 0

Tactic:
rwa [← hc]

Goal state after:
Module: Mathlib.Analysis.Asymptotics.Lemmas

Goal state before:
pp: α : Type u_1
l : Filter α
f : α → ℝ
hf : 1 ≤ᶠ[l] f
m n : ℕ
h : n ≤ m
⊢ (f ^ n) =O[l] (f ^ m)

Tactic:
rw [IsBigO_def]

Goal state after:
α : Type u_1
l : Filter α
f : α → ℝ
hf : 1 ≤ᶠ[l] f
m n : ℕ
h : n ≤ m
⊢ ∃ c, IsBigOWith c l (f ^ n) (f ^ m)
Module: Mathlib.Analysis.Asymptotics.Lemmas

Goal state before:
pp: case mpr
α : Type u_1
𝕜 : Type u_15
inst✝ : NormedDivisionRing 𝕜
l : Filter α
u v : α → 𝕜
⊢ (∃ φ, Tendsto φ l (𝓝 0) ∧ u =ᶠ[l] φ * v) → u =o[l] v

Tactic:
simp only [IsLittleO_def]

Goal state after:
case mpr
α : Type u_1
𝕜 : Type u_15
inst✝ : NormedDivisionRing 𝕜
l : Filter α
u v : α → 𝕜
⊢ (∃ φ, Tendsto φ l (𝓝 0) ∧ u =ᶠ[l] φ * v) → ∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l u v
Module: Mathlib.Algebra.Order.CauSeq.BigOperators

Goal state before:
pp: case h₁
α : Type u_1
β : Type u_2
inst✝⁴ : Field α
inst✝³ : LinearOrder α
inst✝² : IsStrictOrderedRing α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : ℕ → β
ha : IsCauSeq abs fun m ↦ ∑ n ∈ range m, abv (f n)
hb : IsCauSeq abv fun m ↦ ∑ n ∈ range m, g n
ε : α
ε0 : 0 < ε
P : α
hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P
Q : α
hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q
hP0 : 0 < P
hPε0 : 0 < ε / (2 * P)
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, abv (∑ n ∈ range j, g n - ∑ n ∈ range k, g n) < ε / (2 * P)
hQε0 : 0 < ε / (4 * Q)
M : ℕ
hM : ∀ j ≥ M, ∀ k ≥ M, |∑ n ∈ range j, abv (f n) - ∑ n ∈ range k, abv (f n)| < ε / (4 * Q)
K : ℕ
hK : K ≥ 2 * (max N M + 1)
h₁ : ∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k) = ∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n
h₂ : (fun i ↦ ∑ k ∈ range (K - i), f i * g k) = fun i ↦ f i * ∑ k ∈ range (K - i), g k
h₃ :
  ∑ i ∈ range K, f i * ∑ k ∈ range (K - i), g k =
    ∑ i ∈ range K, f i * (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k) + ∑ i ∈ range K, f i * ∑ k ∈ range K, g k
two_mul_two : 4 = 2 * 2
hQ0 : Q ≠ 0
h2Q0 : 2 * Q ≠ 0
hε : ε / (2 * P) * P + ε / (4 * Q) * (2 * Q) = ε
hNMK : max N M + 1 < K
hKN : N < K
m : ℕ
hmJ : m ∈ range (max N M + 1)
⊢ m ≤ max N M + 1

Tactic:
exact (mem_range.1 hmJ).le

Goal state after:
Module: Mathlib.Analysis.Asymptotics.Lemmas

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
E : Type u_3
inst✝¹ : Norm E
F : Type u_4
inst✝ : Norm F
e : OpenPartialHomeomorph α β
b : β
hb : b ∈ e.target
f : β → E
g : β → F
⊢ f =o[𝓝 b] g ↔ (f ∘ ↑e) =o[𝓝 (↑e.symm b)] (g ∘ ↑e)

Tactic:
simp only [IsLittleO_def]

Goal state after:
α : Type u_1
β : Type u_2
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
E : Type u_3
inst✝¹ : Norm E
F : Type u_4
inst✝ : Norm F
e : OpenPartialHomeomorph α β
b : β
hb : b ∈ e.target
f : β → E
g : β → F
⊢ (∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c (𝓝 b) f g) ↔ ∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c (𝓝 (↑e.symm b)) (f ∘ ↑e) (g ∘ ↑e)
Module: Mathlib.Analysis.Asymptotics.Lemmas

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
E : Type u_3
inst✝¹ : Norm E
F : Type u_4
inst✝ : Norm F
e : α ≃ₜ β
b : β
f : β → E
g : β → F
⊢ f =o[𝓝 b] g ↔ (f ∘ ⇑e) =o[𝓝 (e.symm b)] (g ∘ ⇑e)

Tactic:
simp only [IsLittleO_def]

Goal state after:
α : Type u_1
β : Type u_2
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
E : Type u_3
inst✝¹ : Norm E
F : Type u_4
inst✝ : Norm F
e : α ≃ₜ β
b : β
f : β → E
g : β → F
⊢ (∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c (𝓝 b) f g) ↔ ∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c (𝓝 (e.symm b)) (f ∘ ⇑e) (g ∘ ⇑e)
Module: Mathlib.Algebra.Order.CauSeq.BigOperators

Goal state before:
pp: case inr.succ
α : Type u_1
β : Type u_2
inst✝⁵ : Field α
inst✝⁴ : LinearOrder α
inst✝³ : IsStrictOrderedRing α
inst✝² : Ring β
abv : β → α
inst✝¹ : IsAbsoluteValue abv
inst✝ : Archimedean α
f : ℕ → β
r : α
hr0 : 0 ≤ r
hr1 : r < 1
har1 : |r| < 1
hr : 0 < r
k : ℕ
ih :
  ∀ (n : ℕ),
    (∀ (m : ℕ), n ≤ m → abv (f m.succ) ≤ r * abv (f m)) →
      ∀ (m : ℕ), n.succ ≤ m → m = k + n.succ → abv (f m) ≤ abv (f n.succ) * r⁻¹ ^ n.succ * r ^ m
n : ℕ
h : ∀ (m : ℕ), n ≤ m → abv (f m.succ) ≤ r * abv (f m)
m : ℕ
hmn : n.succ ≤ m
hk : m = k + 1 + n.succ
kn : k + n.succ ≥ n.succ
⊢ abv (f (k + n.succ).succ) ≤ abv (f n.succ) * r⁻¹ ^ n.succ * r ^ (k + n.succ).succ

Tactic:
pow_succ r,

Goal state after:
case inr.succ
α : Type u_1
β : Type u_2
inst✝⁵ : Field α
inst✝⁴ : LinearOrder α
inst✝³ : IsStrictOrderedRing α
inst✝² : Ring β
abv : β → α
inst✝¹ : IsAbsoluteValue abv
inst✝ : Archimedean α
f : ℕ → β
r : α
hr0 : 0 ≤ r
hr1 : r < 1
har1 : |r| < 1
hr : 0 < r
k : ℕ
ih :
  ∀ (n : ℕ),
    (∀ (m : ℕ), n ≤ m → abv (f m.succ) ≤ r * abv (f m)) →
      ∀ (m : ℕ), n.succ ≤ m → m = k + n.succ → abv (f m) ≤ abv (f n.succ) * r⁻¹ ^ n.succ * r ^ m
n : ℕ
h : ∀ (m : ℕ), n ≤ m → abv (f m.succ) ≤ r * abv (f m)
m : ℕ
hmn : n.succ ≤ m
hk : m = k + 1 + n.succ
kn : k + n.succ ≥ n.succ
⊢ abv (f (k + n.succ).succ) ≤ abv (f n.succ) * r⁻¹ ^ n.succ * (r ^ (k + n.succ) * r)
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic

Goal state before:
pp: x : ℝ
hx : x ∈ Ioo (-(π / 2)) (π / 2)
⊢ x + π / 2 < π

Tactic:
by linarith [hx.2]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic

Goal state before:
pp: x : ℝ
hx : x ∈ Icc (-(π / 2)) (π / 2)
⊢ x + π / 2 ≤ π

Tactic:
by linarith [hx.2]

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Quotient

Goal state before:
pp: M : Type u_1
inst✝ : SeminormedCommGroup M
S : Subgroup M
this : ∀ (ε : ℝ), mk '' ball 1 ε = {x | ‖x‖ < ε}
⊢ (Filter.map mk (𝓝 1)).HasBasis (fun ε ↦ 0 < ε) fun x ↦ mk '' ball 1 x

Tactic:
exact .map _ Metric.nhds_basis_ball

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Quotient

Goal state before:
pp: M : Type u_1
inst✝ : SeminormedCommGroup M
S T : Subgroup M
x : M ⧸ S
m : M
r ε : ℝ
⊢ Isometry ⇑quotientBot

Tactic:
rw [MonoidHomClass.isometry_iff_norm]

Goal state after:
M : Type u_1
inst✝ : SeminormedCommGroup M
S T : Subgroup M
x : M ⧸ S
m : M
r ε : ℝ
⊢ ∀ (x : M ⧸ ⊥), ‖quotientBot x‖ = ‖x‖
Module: Mathlib.Analysis.Normed.Group.Quotient

Goal state before:
pp: M : Type u_1
inst✝ : SeminormedCommGroup M
S T : Subgroup M
x : M ⧸ S
m : M
r ε : ℝ
h : S ≤ T
⊢ Isometry ⇑(quotientQuotientEquivQuotient S T h)

Tactic:
rw [MonoidHomClass.isometry_iff_norm]

Goal state after:
M : Type u_1
inst✝ : SeminormedCommGroup M
S T : Subgroup M
x : M ⧸ S
m : M
r ε : ℝ
h : S ≤ T
⊢ ∀ (x : (M ⧸ S) ⧸ Subgroup.map (mk' S) T), ‖(quotientQuotientEquivQuotient S T h) x‖ = ‖x‖
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic

Goal state before:
pp: ⊢ (range cos).Infinite

Tactic:
by
  rw [Real.range_cos]
  exact Icc_infinite (by simp)

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic

Goal state before:
pp: n : ℕ
⊢ sqrtTwoAddSeries 0 (n + 1) < 2

Tactic:
refine lt_of_lt_of_le ?_ (sqrt_sq zero_lt_two.le).le
    rw [sqrtTwoAddSeries, sqrt_lt_sqrt_iff, ← lt_sub_iff_add_lt']
    · refine (sqrtTwoAddSeries_lt_two n).trans_le ?_
      norm_num
    · exact add_nonneg zero_le_two (sqrtTwoAddSeries_zero_nonneg n)

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Ball.Pointwise

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : NormedDivisionRing 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : NormSMulClass 𝕜 E
c : 𝕜
hc : c ≠ 0
s : Set E
x : E
this : Function.Surjective fun x ↦ c • x
⊢ ⨅ y ∈ c • s, edist (c • x) y = ⨅ y ∈ s, ‖c‖₊ • edist x y

Tactic:
refine (this.iInf_congr _ fun y => ?_).symm

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝³ : NormedDivisionRing 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : NormSMulClass 𝕜 E
c : 𝕜
hc : c ≠ 0
s : Set E
x : E
this : Function.Surjective fun x ↦ c • x
y : E
⊢ ⨅ (_ : c • y ∈ c • s), edist (c • x) (c • y) = ⨅ (_ : y ∈ s), ‖c‖₊ • edist x y
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic

Goal state before:
pp: c : ℝ := cos (π / 5)
⊢ 4 * (c * c) + -2 * c + -1 = 0

Tactic:
rw [← sq, neg_mul, ← sub_eq_add_neg, ← sub_eq_add_neg]

Goal state after:
c : ℝ := cos (π / 5)
⊢ 4 * c ^ 2 - 2 * c - 1 = 0
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic

Goal state before:
pp: ⊢ tan (π / 6) = 1 / √3

Tactic:
rw [tan_eq_sin_div_cos, sin_pi_div_six, cos_pi_div_six]

Goal state after:
⊢ 1 / 2 / (√3 / 2) = 1 / √3
Module: Mathlib.Analysis.Normed.Module.Ball.Pointwise

Goal state before:
pp: case inl
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
δ : ℝ
s : Set E
x : E
hδ : δ ≤ 0
⊢ infEDist x (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s) = infEDist x s - ENNReal.ofReal δ

Tactic:
infEDist_closure,

Goal state after:
case inl
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
δ : ℝ
s : Set E
x : E
hδ : δ ≤ 0
⊢ infEDist x s = infEDist x s - ENNReal.ofReal δ
Module: Mathlib.Analysis.Normed.Module.Ball.Pointwise

Goal state before:
pp: case inr
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
δ : ℝ
s : Set E
x : E
hδ : 0 < δ
⊢ infEDist x (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (thickening δ s)) =
    infEDist x s - ENNReal.ofReal δ

Tactic:
infEDist_closure,

Goal state after:
case inr
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
δ : ℝ
s : Set E
x : E
hδ : 0 < δ
⊢ infEDist x (thickening δ s) = infEDist x s - ENNReal.ofReal δ
Module: Mathlib.Topology.CompactOpen

Goal state before:
pp: X : Type u_2
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : C(X, Y)
s : Set C(X, Y)
⊢ s ∈ 𝓝 f ↔
    ∃ i,
      (i.Finite ∧ ∀ (K : Set X) (U : Set Y), (K, U) ∈ i → IsCompact K ∧ IsOpen[inst✝] U ∧ MapsTo (⇑f) K U) ∧
        ⋂ KU ∈ i, {g | MapsTo (⇑g) KU.1 KU.2} ⊆ s

Tactic:
simp_rw [nhds_compactOpen, iInf_comm.{_, 0, _ + 1}, iInf_prod', iInf_and']

Goal state after:
X : Type u_2
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : C(X, Y)
s : Set C(X, Y)
⊢ s ∈ ⨅ x, ⨅ (_ : IsCompact x.1 ∧ IsOpen[inst✝] x.2 ∧ MapsTo (⇑f) x.1 x.2), 𝓟 {g | MapsTo (⇑g) x.1 x.2} ↔
    ∃ i,
      (i.Finite ∧ ∀ (K : Set X) (U : Set Y), (K, U) ∈ i → IsCompact K ∧ IsOpen[inst✝] U ∧ MapsTo (⇑f) K U) ∧
        ⋂ KU ∈ i, {g | MapsTo (⇑g) KU.1 KU.2} ⊆ s
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: α : Type u_1
inst✝ : SeminormedAddCommGroup α
r C : ℝ
f : ℕ → α
hr : r < 1
hf : ∀ (n : ℕ), ‖f n‖ ≤ C * r ^ n
a : α
ha : HasSum f a
n : ℕ
⊢ ‖∑ x ∈ Finset.range n, f x - a‖ ≤ C * r ^ n / (1 - r)

Tactic:
rw [← dist_eq_norm]

Goal state after:
α : Type u_1
inst✝ : SeminormedAddCommGroup α
r C : ℝ
f : ℕ → α
hr : r < 1
hf : ∀ (n : ℕ), ‖f n‖ ≤ C * r ^ n
a : α
ha : HasSum f a
n : ℕ
⊢ dist (∑ x ∈ Finset.range n, f x) a ≤ C * r ^ n / (1 - r)