mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: case neg
R : Type u_1
inst✝ : CommRing R
q k : ℕ
hk : k ≠ 0
f : PowerSeries R
hq : 1 < q
i : ℕ
hn : ¬k ∣ i
⊢ Function.extend (fun x ↦ (q ^ k) ^ x) (fun x ↦ (PowerSeries.coeff x) f) 0 (q ^ i) = 0

Tactic:
Function.extend_apply',

Goal state after:
case neg
R : Type u_1
inst✝ : CommRing R
q k : ℕ
hk : k ≠ 0
f : PowerSeries R
hq : 1 < q
i : ℕ
hn : ¬k ∣ i
⊢ 0 (q ^ i) = 0

case neg.hb
R : Type u_1
inst✝ : CommRing R
q k : ℕ
hk : k ≠ 0
f : PowerSeries R
hq : 1 < q
i : ℕ
hn : ¬k ∣ i
⊢ ¬∃ a, (q ^ k) ^ a = q ^ i
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: case pos
σ : Type u_1
R : Type u_3
inst✝³ : CommRing R
τ : Type u_4
S : Type u_5
inst✝² : CommRing S
inst✝¹ : Algebra R S
a : σ → MvPowerSeries τ S
f : MvPowerSeries σ R
inst✝ : Finite τ
k : ℕ
ha : HasSubst a
ha₁ : ∀ (i : σ), constantCoeff (a i) = 0
d : τ →₀ ℕ
hd : Finsupp.degree d < k
h1 : Function.HasFiniteSupport fun d_1 ↦ (coeff d_1) f • (coeff d) (d_1.prod fun s e ↦ a s ^ e)
h2 :
  Function.HasFiniteSupport fun d_1 ↦
    (coeff d_1) (∑ i ∈ range k, (homogeneousComponent i) f) • (coeff d) (d_1.prod fun s e ↦ a s ^ e)
this : Set.Finite.toFinset h2 ⊆ Set.Finite.toFinset h1
⊢ ∑ i ∈ Set.Finite.toFinset h1, (coeff i) f • (coeff d) (i.prod fun s e ↦ a s ^ e) =
    ∑ i ∈ Set.Finite.toFinset h2,
      (coeff i) (∑ i ∈ range k, (homogeneousComponent i) f) • (coeff d) (i.prod fun s e ↦ a s ^ e)

Tactic:
have aux {n : σ →₀ ℕ} : coeff d (n.prod fun s e ↦ a s ^ e) ≠ 0 → n.degree ≤ d.degree := by
      contrapose!
      intro hc
      rw [Finsupp.prod]
      refine coeff_of_lt_order (lt_of_lt_of_le (Nat.cast_lt.mpr hc)
        (.trans ?_ (le_order_prod _ n.support)))
      exact_mod_cast sum_le_sum fun i hi => le_order_pow_of_constantCoeff_eq_zero _ (ha₁ i)

Goal state after:
case pos
σ : Type u_1
R : Type u_3
inst✝³ : CommRing R
τ : Type u_4
S : Type u_5
inst✝² : CommRing S
inst✝¹ : Algebra R S
a : σ → MvPowerSeries τ S
f : MvPowerSeries σ R
inst✝ : Finite τ
k : ℕ
ha : HasSubst a
ha₁ : ∀ (i : σ), constantCoeff (a i) = 0
d : τ →₀ ℕ
hd : Finsupp.degree d < k
h1 : Function.HasFiniteSupport fun d_1 ↦ (coeff d_1) f • (coeff d) (d_1.prod fun s e ↦ a s ^ e)
h2 :
  Function.HasFiniteSupport fun d_1 ↦
    (coeff d_1) (∑ i ∈ range k, (homogeneousComponent i) f) • (coeff d) (d_1.prod fun s e ↦ a s ^ e)
this : Set.Finite.toFinset h2 ⊆ Set.Finite.toFinset h1
aux : ∀ {n : σ →₀ ℕ}, (coeff d) (n.prod fun s e ↦ a s ^ e) ≠ 0 → Finsupp.degree n ≤ Finsupp.degree d
⊢ ∑ i ∈ Set.Finite.toFinset h1, (coeff i) f • (coeff d) (i.prod fun s e ↦ a s ^ e) =
    ∑ i ∈ Set.Finite.toFinset h2,
      (coeff i) (∑ i ∈ range k, (homogeneousComponent i) f) • (coeff d) (i.prod fun s e ↦ a s ^ e)
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: case neg
R : Type u_1
inst✝ : CommRing R
q k : ℕ
hk : k ≠ 0
f : PowerSeries R
hq : 1 < q
n : ℕ
hn : ¬∃ i, q ^ i = n
⊢ Function.extend (fun x ↦ (q ^ k) ^ x) (fun x ↦ (PowerSeries.coeff x) f) 0 n =
    Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) (PowerSeries.subst (PowerSeries.X ^ k) f)) 0 n

Tactic:
Function.extend_apply',

Goal state after:
case neg
R : Type u_1
inst✝ : CommRing R
q k : ℕ
hk : k ≠ 0
f : PowerSeries R
hq : 1 < q
n : ℕ
hn : ¬∃ i, q ^ i = n
⊢ 0 n = Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) (PowerSeries.subst (PowerSeries.X ^ k) f)) 0 n

case neg.hb
R : Type u_1
inst✝ : CommRing R
q k : ℕ
hk : k ≠ 0
f : PowerSeries R
hq : 1 < q
n : ℕ
hn : ¬∃ i, q ^ i = n
⊢ ¬∃ a, (q ^ k) ^ a = n
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: case neg
R : Type u_1
inst✝ : CommRing R
f : PowerSeries R
hf : PowerSeries.constantCoeff f = 1
n p k : ℕ
hp : Nat.Prime p
hk : 0 < k
i : ℕ
hmn : (p ^ i).Coprime n
hn : ¬∃ j, p ^ j = n
⊢ Function.extend (fun x ↦ p ^ x) (fun x ↦ (PowerSeries.coeff x) (PowerSeries.subst (PowerSeries.X ^ k) f)) 0
      (p ^ i * n) =
    Function.extend (fun x ↦ p ^ x) (fun x ↦ (PowerSeries.coeff x) (PowerSeries.subst (PowerSeries.X ^ k) f)) 0
        (p ^ i) *
      Function.extend (fun x ↦ p ^ x) (fun x ↦ (PowerSeries.coeff x) (PowerSeries.subst (PowerSeries.X ^ k) f)) 0 n

Tactic:
Function.extend_apply',

Goal state after:
case neg
R : Type u_1
inst✝ : CommRing R
f : PowerSeries R
hf : PowerSeries.constantCoeff f = 1
n p k : ℕ
hp : Nat.Prime p
hk : 0 < k
i : ℕ
hmn : (p ^ i).Coprime n
hn : ¬∃ j, p ^ j = n
⊢ 0 (p ^ i * n) =
    Function.extend (fun x ↦ p ^ x) (fun x ↦ (PowerSeries.coeff x) (PowerSeries.subst (PowerSeries.X ^ k) f)) 0
        (p ^ i) *
      Function.extend (fun x ↦ p ^ x) (fun x ↦ (PowerSeries.coeff x) (PowerSeries.subst (PowerSeries.X ^ k) f)) 0 n

case neg.hb
R : Type u_1
inst✝ : CommRing R
f : PowerSeries R
hf : PowerSeries.constantCoeff f = 1
n p k : ℕ
hp : Nat.Prime p
hk : 0 < k
i : ℕ
hmn : (p ^ i).Coprime n
hn : ¬∃ j, p ^ j = n
⊢ ¬∃ a, p ^ a = p ^ i * n
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: case neg
R : Type u_1
inst✝ : CommRing R
f : PowerSeries R
hf : PowerSeries.constantCoeff f = 1
m n : ℕ
hmn : m.Coprime n
p k : ℕ
hp : Nat.Prime p
hk : 0 < k
hm : ¬∃ i, p ^ i = m
⊢ Function.extend (fun x ↦ p ^ x) (fun x ↦ (PowerSeries.coeff x) (PowerSeries.subst (PowerSeries.X ^ k) f)) 0 (m * n) =
    Function.extend (fun x ↦ p ^ x) (fun x ↦ (PowerSeries.coeff x) (PowerSeries.subst (PowerSeries.X ^ k) f)) 0 m *
      Function.extend (fun x ↦ p ^ x) (fun x ↦ (PowerSeries.coeff x) (PowerSeries.subst (PowerSeries.X ^ k) f)) 0 n

Tactic:
Function.extend_apply',

Goal state after:
case neg
R : Type u_1
inst✝ : CommRing R
f : PowerSeries R
hf : PowerSeries.constantCoeff f = 1
m n : ℕ
hmn : m.Coprime n
p k : ℕ
hp : Nat.Prime p
hk : 0 < k
hm : ¬∃ i, p ^ i = m
⊢ 0 (m * n) =
    Function.extend (fun x ↦ p ^ x) (fun x ↦ (PowerSeries.coeff x) (PowerSeries.subst (PowerSeries.X ^ k) f)) 0 m *
      Function.extend (fun x ↦ p ^ x) (fun x ↦ (PowerSeries.coeff x) (PowerSeries.subst (PowerSeries.X ^ k) f)) 0 n

case neg.hb
R : Type u_1
inst✝ : CommRing R
f : PowerSeries R
hf : PowerSeries.constantCoeff f = 1
m n : ℕ
hmn : m.Coprime n
p k : ℕ
hp : Nat.Prime p
hk : 0 < k
hm : ¬∃ i, p ^ i = m
⊢ ¬∃ a, p ^ a = m * n
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝ : CommRing R
f : MvPowerSeries σ R
⊢ (rescaleAlgHom 1) f = (AlgHom.id R (MvPowerSeries σ R)) f

Tactic:
simp [rescaleAlgHom, subst_self]

Goal state after:
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: case succ.succ
ι : Type u_1
R : Type u_2
inst✝ : CommSemiring R
q : ι → ℕ
hq : Northcott q
f : ι → PowerSeries R
hf : ∀ (i : ι), PowerSeries.constantCoeff (f i) = 1
n✝ : ℕ
i : ι
n : ℕ
hi : n + 1 + 1 + 1 ≤ q i
hqi : 1 < q i
⊢ Function.extend (fun x ↦ q i ^ x) (fun x ↦ (PowerSeries.coeff x) (f i)) 0 (n + 1 + 1) = 1 (n + 1 + 1)

Tactic:
Function.extend_apply',

Goal state after:
case succ.succ
ι : Type u_1
R : Type u_2
inst✝ : CommSemiring R
q : ι → ℕ
hq : Northcott q
f : ι → PowerSeries R
hf : ∀ (i : ι), PowerSeries.constantCoeff (f i) = 1
n✝ : ℕ
i : ι
n : ℕ
hi : n + 1 + 1 + 1 ≤ q i
hqi : 1 < q i
⊢ 0 (n + 1 + 1) = 1 (n + 1 + 1)

case succ.succ.hb
ι : Type u_1
R : Type u_2
inst✝ : CommSemiring R
q : ι → ℕ
hq : Northcott q
f : ι → PowerSeries R
hf : ∀ (i : ι), PowerSeries.constantCoeff (f i) = 1
n✝ : ℕ
i : ι
n : ℕ
hi : n + 1 + 1 + 1 ≤ q i
hqi : 1 < q i
⊢ ¬∃ a, q i ^ a = n + 1 + 1
Module: Mathlib.Analysis.Calculus.TangentCone.Defs

Goal state before:
pp: case refine_1
R : Type u
E : Type v
inst✝² : AddCommGroup E
inst✝¹ : SMul R E
inst✝ : TopologicalSpace E
s : Set E
x y : E
h : (comap (fun p ↦ p.1 • p.2) (𝓝 y) ⊓ ⊤ ×ˢ 𝓝[(fun x_1 ↦ x + x_1) ⁻¹' s] 0).NeBot
⊢ Tendsto Prod.snd (comap (fun p ↦ p.1 • p.2) (𝓝 y) ⊓ ⊤ ×ˢ 𝓝[(fun x_1 ↦ x + x_1) ⁻¹' s] 0) (𝓝 0)

Tactic:
refine (tendsto_snd (f := ⊤)).mono_left <| inf_le_right.trans <| ?_

Goal state after:
case refine_1
R : Type u
E : Type v
inst✝² : AddCommGroup E
inst✝¹ : SMul R E
inst✝ : TopologicalSpace E
s : Set E
x y : E
h : (comap (fun p ↦ p.1 • p.2) (𝓝 y) ⊓ ⊤ ×ˢ 𝓝[(fun x_1 ↦ x + x_1) ⁻¹' s] 0).NeBot
⊢ ⊤ ×ˢ 𝓝[(fun x_1 ↦ x + x_1) ⁻¹' s] 0 ≤ ⊤ ×ˢ 𝓝 0
Module: Mathlib.RingTheory.PowerSeries.Substitution

Goal state before:
pp: R : Type u_2
inst✝ : CommRing R
f : R⟦X⟧
e : Fin 2 →₀ ℕ
he : (single 0 (e 0), single 1 (e 1)) ∉ antidiagonal e
⊢ single 0 (e 0) + single 1 (e 1) = e

Tactic:
ext i

Goal state after:
R : Type u_2
inst✝ : CommRing R
f : R⟦X⟧
e : Fin 2 →₀ ℕ
he : (single 0 (e 0), single 1 (e 1)) ∉ antidiagonal e
i : Fin 2
⊢ (single 0 (e 0) + single 1 (e 1)) i = e i
Module: Mathlib.Analysis.Asymptotics.TVS

Goal state before:
pp: α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : TopologicalSpace E
inst✝³ : Module 𝕜 E
inst✝² : AddCommGroup F
inst✝¹ : TopologicalSpace F
inst✝ : Module 𝕜 F
l : Filter α
f : α → E
g : E →ₗ[𝕜] F
hg : comap (⇑g) (𝓝 0) ≤ 𝓝 0
⊢ ∀ U ∈ 𝓝 0, ∃ V ∈ 𝓝 0, (fun x ↦ egauge 𝕜 U (f x)) ≤ᶠ[l] fun x ↦ egauge 𝕜 V ((⇑g ∘ f) x)

Tactic:
intro U

Goal state after:
α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : TopologicalSpace E
inst✝³ : Module 𝕜 E
inst✝² : AddCommGroup F
inst✝¹ : TopologicalSpace F
inst✝ : Module 𝕜 F
l : Filter α
f : α → E
g : E →ₗ[𝕜] F
hg : comap (⇑g) (𝓝 0) ≤ 𝓝 0
U : Set E
⊢ U ∈ 𝓝 0 → ∃ V ∈ 𝓝 0, (fun x ↦ egauge 𝕜 U (f x)) ≤ᶠ[l] fun x ↦ egauge 𝕜 V ((⇑g ∘ f) x)
Module: Mathlib.Analysis.Calculus.FormalMultilinearSeries

Goal state before:
pp: case e_6
𝕜 : Type u
E : Type v
F : Type w
inst✝¹⁰ : Semiring 𝕜
inst✝⁹ : AddCommMonoid E
inst✝⁸ : Module 𝕜 E
inst✝⁷ : TopologicalSpace E
inst✝⁶ : ContinuousAdd E
inst✝⁵ : ContinuousConstSMul 𝕜 E
inst✝⁴ : AddCommMonoid F
inst✝³ : Module 𝕜 F
inst✝² : TopologicalSpace F
inst✝¹ : ContinuousAdd F
inst✝ : ContinuousConstSMul 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
m : ℕ
v w : Fin m → E
h2 : ∀ (i : ℕ) (him hin : i < m), v ⟨i, him⟩ = w ⟨i, hin⟩
i : ℕ
hi : i < m
⊢ v ⟨i, hi⟩ = w ⟨i, hi⟩

Tactic:
exact h2 i hi hi

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

Goal state before:
pp: α : Type u_1
𝕜 : Type u_3
E : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : AddCommGroup E
inst✝² : TopologicalSpace E
inst✝¹ : Module 𝕜 E
l : Filter α
inst✝ : ContinuousSMul 𝕜 E
f : α → 𝕜
g : α → E
h : g =o[𝕜; l] f
⊢ ∀ U ∈ 𝓝 0, ∃ V ∈ 𝓝 0, ∀ (ε : ℝ≥0), ε ≠ 0 → (fun x ↦ egauge 𝕜 U ((f x)⁻¹ • g x)) ≤ᶠ[l] fun x ↦ ↑ε * egauge 𝕜 V (1 x)

Tactic:
intro U

Goal state after:
α : Type u_1
𝕜 : Type u_3
E : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : AddCommGroup E
inst✝² : TopologicalSpace E
inst✝¹ : Module 𝕜 E
l : Filter α
inst✝ : ContinuousSMul 𝕜 E
f : α → 𝕜
g : α → E
h : g =o[𝕜; l] f
U : Set E
⊢ U ∈ 𝓝 0 → ∃ V ∈ 𝓝 0, ∀ (ε : ℝ≥0), ε ≠ 0 → (fun x ↦ egauge 𝕜 U ((f x)⁻¹ • g x)) ≤ᶠ[l] fun x ↦ ↑ε * egauge 𝕜 V (1 x)
Module: Mathlib.Analysis.Normed.Module.Multilinear.Curry

Goal state before:
pp: 𝕜 : Type u
n : ℕ
Ei : Fin n.succ → Type wEi
G : Type wG
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : (i : Fin n.succ) → NormedAddCommGroup (Ei i)
inst✝² : (i : Fin n.succ) → NormedSpace 𝕜 (Ei i)
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : Ei 0 →L[𝕜] Ei i.succ [×n]→L[𝕜] G
m : Ei 0
x : (i : Fin n) → Ei i.succ
⊢ (f.uncurryLeft.curryLeft m) x = (f m) x

Tactic:
ContinuousMultilinearMap.curryLeft_apply,

Goal state after:
𝕜 : Type u
n : ℕ
Ei : Fin n.succ → Type wEi
G : Type wG
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : (i : Fin n.succ) → NormedAddCommGroup (Ei i)
inst✝² : (i : Fin n.succ) → NormedSpace 𝕜 (Ei i)
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : Ei 0 →L[𝕜] Ei i.succ [×n]→L[𝕜] G
m : Ei 0
x : (i : Fin n) → Ei i.succ
⊢ f.uncurryLeft (Fin.cons m x) = (f m) x
Module: Mathlib.Analysis.Normed.Module.Multilinear.Curry

Goal state before:
pp: 𝕜 : Type u
ι : Type v
ι' : Type v'
n : ℕ
E : ι → Type wE
Ei : Fin n.succ → Type wEi
G✝ : Type wG
G' : Type wG'
inst✝¹⁴ : Fintype ι
inst✝¹³ : Fintype ι'
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)
inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁹ : (i : Fin n.succ) → NormedAddCommGroup (Ei i)
inst✝⁸ : (i : Fin n.succ) → NormedSpace 𝕜 (Ei i)
inst✝⁷ : NormedAddCommGroup G✝
inst✝⁶ : NormedSpace 𝕜 G✝
inst✝⁵ : NormedAddCommGroup G'
inst✝⁴ : NormedSpace 𝕜 G'
F : Type u_1
G : Type u_2
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
B : G →L[𝕜] ContinuousMultilinearMap 𝕜 E F
inst : DecidableEq (Option ι)
v : Option ι → G × ((i : ι) → E i)
j✝ : Option ι
x y : G × ((i : ι) → E i)
j : ι
z : G × ((i : ι) → E i)
⊢ (fun i ↦ (update v (some j) z (some i)).2 i) = update (fun i ↦ (v (some i)).2 i) j (z.2 j)

Tactic:
ext i

Goal state after:
𝕜 : Type u
ι : Type v
ι' : Type v'
n : ℕ
E : ι → Type wE
Ei : Fin n.succ → Type wEi
G✝ : Type wG
G' : Type wG'
inst✝¹⁴ : Fintype ι
inst✝¹³ : Fintype ι'
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)
inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁹ : (i : Fin n.succ) → NormedAddCommGroup (Ei i)
inst✝⁸ : (i : Fin n.succ) → NormedSpace 𝕜 (Ei i)
inst✝⁷ : NormedAddCommGroup G✝
inst✝⁶ : NormedSpace 𝕜 G✝
inst✝⁵ : NormedAddCommGroup G'
inst✝⁴ : NormedSpace 𝕜 G'
F : Type u_1
G : Type u_2
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
B : G →L[𝕜] ContinuousMultilinearMap 𝕜 E F
inst : DecidableEq (Option ι)
v : Option ι → G × ((i : ι) → E i)
j✝ : Option ι
x y : G × ((i : ι) → E i)
j : ι
z : G × ((i : ι) → E i)
i : ι
⊢ (update v (some j) z (some i)).2 i = update (fun i ↦ (v (some i)).2 i) j (z.2 j) i
Module: Mathlib.Analysis.Normed.Module.Multilinear.Curry

Goal state before:
pp: 𝕜 : Type u
ι : Type v
ι' : Type v'
n : ℕ
E : ι → Type wE
Ei : Fin n.succ → Type wEi
G✝ : Type wG
G' : Type wG'
inst✝¹⁴ : Fintype ι
inst✝¹³ : Fintype ι'
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)
inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁹ : (i : Fin n.succ) → NormedAddCommGroup (Ei i)
inst✝⁸ : (i : Fin n.succ) → NormedSpace 𝕜 (Ei i)
inst✝⁷ : NormedAddCommGroup G✝
inst✝⁶ : NormedSpace 𝕜 G✝
inst✝⁵ : NormedAddCommGroup G'
inst✝⁴ : NormedSpace 𝕜 G'
F : Type u_1
G : Type u_2
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
B : G →L[𝕜] ContinuousMultilinearMap 𝕜 E F
inst : DecidableEq (Option ι)
v : Option ι → G × ((i : ι) → E i)
j✝ : Option ι
c : 𝕜
x : G × ((i : ι) → E i)
j : ι
z : G × ((i : ι) → E i)
⊢ (fun i ↦ (update v (some j) z (some i)).2 i) = update (fun i ↦ (v (some i)).2 i) j (z.2 j)

Tactic:
ext i

Goal state after:
𝕜 : Type u
ι : Type v
ι' : Type v'
n : ℕ
E : ι → Type wE
Ei : Fin n.succ → Type wEi
G✝ : Type wG
G' : Type wG'
inst✝¹⁴ : Fintype ι
inst✝¹³ : Fintype ι'
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)
inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁹ : (i : Fin n.succ) → NormedAddCommGroup (Ei i)
inst✝⁸ : (i : Fin n.succ) → NormedSpace 𝕜 (Ei i)
inst✝⁷ : NormedAddCommGroup G✝
inst✝⁶ : NormedSpace 𝕜 G✝
inst✝⁵ : NormedAddCommGroup G'
inst✝⁴ : NormedSpace 𝕜 G'
F : Type u_1
G : Type u_2
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
B : G →L[𝕜] ContinuousMultilinearMap 𝕜 E F
inst : DecidableEq (Option ι)
v : Option ι → G × ((i : ι) → E i)
j✝ : Option ι
c : 𝕜
x : G × ((i : ι) → E i)
j : ι
z : G × ((i : ι) → E i)
i : ι
⊢ (update v (some j) z (some i)).2 i = update (fun i ↦ (v (some i)).2 i) j (z.2 j) i
Module: Mathlib.Analysis.Normed.Module.Multilinear.Curry

Goal state before:
pp: 𝕜 : Type u
ι : Type v
ι' : Type v'
n : ℕ
E : ι → Type wE
Ei : Fin n.succ → Type wEi
G✝ : Type wG
G' : Type wG'
inst✝¹⁴ : Fintype ι
inst✝¹³ : Fintype ι'
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)
inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁹ : (i : Fin n.succ) → NormedAddCommGroup (Ei i)
inst✝⁸ : (i : Fin n.succ) → NormedSpace 𝕜 (Ei i)
inst✝⁷ : NormedAddCommGroup G✝
inst✝⁶ : NormedSpace 𝕜 G✝
inst✝⁵ : NormedAddCommGroup G'
inst✝⁴ : NormedSpace 𝕜 G'
F : Type u_1
G : Type u_2
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
B : G →L[𝕜] ContinuousMultilinearMap 𝕜 E F
b : Option ι → G × ((i : ι) → E i)
⊢ ‖B (b none).1‖ * ∏ i, ‖(b (some i)).2 i‖ ≤ ‖B‖ * (‖b none‖ * ∏ i, ‖b (some i)‖)

Tactic:
rw [← mul_assoc]

Goal state after:
𝕜 : Type u
ι : Type v
ι' : Type v'
n : ℕ
E : ι → Type wE
Ei : Fin n.succ → Type wEi
G✝ : Type wG
G' : Type wG'
inst✝¹⁴ : Fintype ι
inst✝¹³ : Fintype ι'
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)
inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁹ : (i : Fin n.succ) → NormedAddCommGroup (Ei i)
inst✝⁸ : (i : Fin n.succ) → NormedSpace 𝕜 (Ei i)
inst✝⁷ : NormedAddCommGroup G✝
inst✝⁶ : NormedSpace 𝕜 G✝
inst✝⁵ : NormedAddCommGroup G'
inst✝⁴ : NormedSpace 𝕜 G'
F : Type u_1
G : Type u_2
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
B : G →L[𝕜] ContinuousMultilinearMap 𝕜 E F
b : Option ι → G × ((i : ι) → E i)
⊢ ‖B (b none).1‖ * ∏ i, ‖(b (some i)).2 i‖ ≤ ‖B‖ * ‖b none‖ * ∏ i, ‖b (some i)‖
Module: Mathlib.Analysis.Analytic.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f g : E → F
p : FormalMultilinearSeries 𝕜 E F
x : E
r : ℝ≥0∞
hf : HasFPowerSeriesOnBall f p x r
hg : EqOn f g (Metric.eball x r)
y : E
hy : y ∈ Metric.eball 0 r
⊢ HasSum (fun n ↦ (p n) fun x ↦ y) (g (x + y))

Tactic:
convert hf.hasSum hy

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f g : E → F
p : FormalMultilinearSeries 𝕜 E F
x : E
r : ℝ≥0∞
hf : HasFPowerSeriesOnBall f p x r
hg : EqOn f g (Metric.eball x r)
y : E
hy : y ∈ Metric.eball 0 r
⊢ g (x + y) = f (x + y)
Module: Mathlib.Analysis.Analytic.Composition

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
n : ℕ
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
c : Composition n
⊢ ‖q.compAlongComposition p c‖₊ ≤ ‖q c.length‖₊ * ∏ i, ‖p (c.blocksFun i)‖₊

Tactic:
← NNReal.coe_le_coe

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
n : ℕ
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
c : Composition n
⊢ ↑‖q.compAlongComposition p c‖₊ ≤ ↑(‖q c.length‖₊ * ∏ i, ‖p (c.blocksFun i)‖₊)
Module: Mathlib.Analysis.Analytic.CPolynomial

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
ι : Type u_5
Em : ι → Type u_6
inst✝² : (i : ι) → NormedAddCommGroup (Em i)
inst✝¹ : (i : ι) → NormedSpace 𝕜 (Em i)
inst✝ : Fintype ι
f : ContinuousMultilinearMap 𝕜 Em F
y : (i : ι) → Em i
x✝ : y ∈ Metric.eball 0 ∞
⊢ ∀ b ∈ Finset.range (Fintype.card ι + 1), b ≠ Fintype.card ι → ((f.toFormalMultilinearSeries b) fun x ↦ y) = 0

Tactic:
intro m _ ne; rw [toFormalMultilinearSeries, dif_neg ne.symm]; rfl

Goal state after:
Module: Mathlib.Analysis.Analytic.Inverse

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i
n✝ n : ℕ
v : Fin (n + 2) → E
A : univ = {c | c.length < n + 2}.toFinset ∪ {Composition.ones (n + 2)}
B : Disjoint {c | c.length < n + 2}.toFinset {Composition.ones (n + 2)}
C :
  ((p.leftInv i x (Composition.ones (n + 2)).length) fun j ↦ (p 1) fun x ↦ v (Fin.castLE ⋯ j)) =
    (p.leftInv i x (n + 2)) fun j ↦ (p 1) fun x ↦ v j
⊢ (∑ i_1,
      (ContinuousMultilinearMap.compAlongComposition (p.compContinuousLinearMap ↑i.symm) (↑i_1)
          (p.leftInv i x (↑i_1).length))
        fun j ↦ (p 1) fun x ↦ v j) =
    ∑ c ∈ {c | c.length < n + 2}.toFinset, (p.leftInv i x c.length) (p.applyComposition c v)

Tactic:
convert!
        (sum_toFinset_eq_subtype (fun c : Composition (n + 2) => c.length < n + 2)
              (fun c : Composition (n + 2) =>
                (ContinuousMultilinearMap.compAlongComposition
                    (p.compContinuousLinearMap (i.symm : F →L[𝕜] E)) c (p.leftInv i x c.length))
                  fun j : Fin (n + 2) => p 1 fun _ : Fin 1 => v j)).symm.trans
          _

Goal state after:
case convert_2
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i
n✝ n : ℕ
v : Fin (n + 2) → E
A : univ = {c | c.length < n + 2}.toFinset ∪ {Composition.ones (n + 2)}
B : Disjoint {c | c.length < n + 2}.toFinset {Composition.ones (n + 2)}
C :
  ((p.leftInv i x (Composition.ones (n + 2)).length) fun j ↦ (p 1) fun x ↦ v (Fin.castLE ⋯ j)) =
    (p.leftInv i x (n + 2)) fun j ↦ (p 1) fun x ↦ v j
⊢ (∑ a ∈ {x | x.length < n + 2}.toFinset,
      (ContinuousMultilinearMap.compAlongComposition (p.compContinuousLinearMap ↑i.symm) a (p.leftInv i x a.length))
        fun j ↦ (p 1) fun x ↦ v j) =
    ∑ c ∈ {c | c.length < n + 2}.toFinset, (p.leftInv i x c.length) (p.applyComposition c v)
Module: Mathlib.Analysis.Analytic.Inverse

Goal state before:
pp: case e_6
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i
n : ℕ
v : Fin 0 → F
⊢ (fun _i ↦ 0) = ![]

Tactic:
ext i

Goal state after:
case e_6
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i✝ : E ≃L[𝕜] F
x : E
h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i✝
n : ℕ
v : Fin 0 → F
i : Fin 0
⊢ 0 = ![] i
Module: Mathlib.Analysis.Analytic.Constructions

Goal state before:
pp: case a.left
𝕜 : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_3
F : Type u_4
G : Type u_5
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p : FormalMultilinearSeries 𝕜 E F
q : FormalMultilinearSeries 𝕜 E G
r : ℝ≥0
hr : ↑r < (p.prod q).radius
this : (fun n ↦ ‖p.prod q n‖ * ↑r ^ n) =o[atTop] fun x ↦ 1
n : ℕ
⊢ ‖p n‖ ≤ max ‖p n‖ ‖q n‖

pp: case a.right
𝕜 : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_3
F : Type u_4
G : Type u_5
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p : FormalMultilinearSeries 𝕜 E F
q : FormalMultilinearSeries 𝕜 E G
r : ℝ≥0
hr : ↑r < (p.prod q).radius
this : (fun n ↦ ‖p.prod q n‖ * ↑r ^ n) =o[atTop] fun x ↦ 1
n : ℕ
⊢ ‖q n‖ ≤ max ‖p n‖ ‖q n‖

Tactic:
· apply le_max_left

Goal state after:
case a.right
𝕜 : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_3
F : Type u_4
G : Type u_5
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p : FormalMultilinearSeries 𝕜 E F
q : FormalMultilinearSeries 𝕜 E G
r : ℝ≥0
hr : ↑r < (p.prod q).radius
this : (fun n ↦ ‖p.prod q n‖ * ↑r ^ n) =o[atTop] fun x ↦ 1
n : ℕ
⊢ ‖q n‖ ≤ max ‖p n‖ ‖q n‖