mathlib-initiative/mathlib-tactics huggingface dataset:
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
⊢ 0 (p ^ i * n) =
    Function.extend (fun x ↦ p ^ x) (fun x ↦ (PowerSeries.coeff x) (PowerSeries.subst (PowerSeries.X ^ k) f)) 0
        (p ^ i) *
      0 n

pp: 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

Tactic:
Pi.zero_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 =
    Function.extend (fun x ↦ p ^ x) (fun x ↦ (PowerSeries.coeff x) (PowerSeries.subst (PowerSeries.X ^ k) f)) 0
        (p ^ i) *
      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
n p k : ℕ
hp : Nat.Prime p
hk : 0 < k
i : ℕ
hmn : (p ^ i).Coprime n
hn : ¬∃ j, p ^ j = n
⊢ 0 =
    Function.extend (fun x ↦ p ^ x) (fun x ↦ (PowerSeries.coeff x) (PowerSeries.subst (PowerSeries.X ^ k) f)) 0
        (p ^ i) *
      0 n

pp: 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

Tactic:
Pi.zero_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 =
    Function.extend (fun x ↦ p ^ x) (fun x ↦ (PowerSeries.coeff x) (PowerSeries.subst (PowerSeries.X ^ k) f)) 0
        (p ^ i) *
      0

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.RingTheory.PowerSeries.Substitution

Goal state before:
pp: R : Type u_2
inst✝² : CommRing R
τ : Type u_3
S : Type u_4
inst✝¹ : CommRing S
inst✝ : Algebra R S
a : MvPowerSeries τ S
w : τ → ℕ
ha : HasSubst a
f : R⟦X⟧
i : Unit →₀ ℕ
⊢ ¬(MvPowerSeries.coeff i) f = 0 →
    f.order * MvPowerSeries.weightedOrder w a ≤
      (Finsupp.weight (Function.const Unit (MvPowerSeries.weightedOrder w a))) i

Tactic:
hi

Goal state after:
R : Type u_2
inst✝² : CommRing R
τ : Type u_3
S : Type u_4
inst✝¹ : CommRing S
inst✝ : Algebra R S
a : MvPowerSeries τ S
w : τ → ℕ
ha : HasSubst a
f : R⟦X⟧
i : Unit →₀ ℕ
hi : ¬(MvPowerSeries.coeff i) f = 0
⊢ f.order * MvPowerSeries.weightedOrder w a ≤ (Finsupp.weight (Function.const Unit (MvPowerSeries.weightedOrder w a))) i
Module: Mathlib.Analysis.Convex.EGauge

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹ : NNNorm 𝕜
ι : Type u_3
E : ι → Type u_4
inst✝ : (i : ι) → SMul 𝕜 (E i)
I : Set ι
i : ι
hi : i ∈ I
s : (i : ι) → Set (E i)
x✝ x : (i : ι) → E i
hx : x ∈ I.pi s
⊢ (Pi.evalMulActionHom i) x ∈ s i

Tactic:
exact hx i hi

Goal state after:
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
⊢ 0 (m * n) =
    0 m * Function.extend (fun x ↦ p ^ x) (fun x ↦ (PowerSeries.coeff x) (PowerSeries.subst (PowerSeries.X ^ k) f)) 0 n

pp: 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

Tactic:
Pi.zero_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 =
    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.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
⊢ 0 =
    0 m * Function.extend (fun x ↦ p ^ x) (fun x ↦ (PowerSeries.coeff x) (PowerSeries.subst (PowerSeries.X ^ k) f)) 0 n

pp: 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

Tactic:
Pi.zero_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 = 0 * 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.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: ι : Type u_1
R : Type u_2
inst✝ : CommSemiring R
f : ι → ArithmeticFunction R
this✝ : UniformSpace R := ⊥
this : IsUniformInducing DFunLike.coe
n : ℕ
s : Set ι
hs : s ∈ cofinite
hs' : ∀ y ∈ s, ∀ i ∈ Set.Iic n, (f y) i = 1 i
t : Finset ι := ⋯.toFinset
u v : Finset ι
hu : t ⊆ u
hv : t ⊆ v
i : ι
⊢ i ∈ u \ t → i ∈ s

Tactic:
hi

Goal state after:
ι : Type u_1
R : Type u_2
inst✝ : CommSemiring R
f : ι → ArithmeticFunction R
this✝ : UniformSpace R := ⊥
this : IsUniformInducing DFunLike.coe
n : ℕ
s : Set ι
hs : s ∈ cofinite
hs' : ∀ y ∈ s, ∀ i ∈ Set.Iic n, (f y) i = 1 i
t : Finset ι := ⋯.toFinset
u v : Finset ι
hu : t ⊆ u
hv : t ⊆ v
i : ι
hi : i ∈ u \ t
⊢ i ∈ s
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: ι : Type u_1
R : Type u_2
inst✝ : CommSemiring R
f : ι → ArithmeticFunction R
this✝ : UniformSpace R := ⊥
this : IsUniformInducing DFunLike.coe
n : ℕ
s : Set ι
hs : s ∈ cofinite
hs' : ∀ y ∈ s, ∀ i ∈ Set.Iic n, (f y) i = 1 i
t : Finset ι := ⋯.toFinset
u v : Finset ι
hv : t ⊆ v
hu : ∀ i ∈ u \ t, i ∈ s
i : ι
⊢ i ∈ v \ t → i ∈ s

Tactic:
hi

Goal state after:
ι : Type u_1
R : Type u_2
inst✝ : CommSemiring R
f : ι → ArithmeticFunction R
this✝ : UniformSpace R := ⊥
this : IsUniformInducing DFunLike.coe
n : ℕ
s : Set ι
hs : s ∈ cofinite
hs' : ∀ y ∈ s, ∀ i ∈ Set.Iic n, (f y) i = 1 i
t : Finset ι := ⋯.toFinset
u v : Finset ι
hv : t ⊆ v
hu : ∀ i ∈ u \ t, i ∈ s
i : ι
hi : i ∈ v \ t
⊢ i ∈ s
Module: Mathlib.Analysis.Convex.EGauge

Goal state before:
pp: case inl
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
inst✝² : NormedDivisionRing 𝕜
inst✝¹ : (i : ι) → AddCommGroup (E i)
inst✝ : (i : ι) → Module 𝕜 (E i)
U : (i : ι) → Set (E i)
x : (i : ι) → E i
r : ℝ≥0∞
c : ι → 𝕜
hr₀ : 0 < r
hI : ∅.Finite
hU : ∀ i ∈ ∅, Balanced 𝕜 (U i)
hI₀ : ∅ = univ ∨ (∃ i ∈ ∅, x i ≠ 0) ∨ (𝓝[≠] 0).NeBot
hr : ⨆ i ∈ ∅, egauge 𝕜 (U i) (x i) < r
hc : ∀ i ∈ ∅, x i ∈ c i • U i
hcr : ∀ i ∈ ∅, ‖c i‖ₑ < r
⊢ ∃ c₀, (c₀ ≠ 0 ∨ ∅ = univ) ∧ (∀ i ∈ ∅, ‖c i‖ ≤ ‖c₀‖) ∧ ‖c₀‖ₑ < r

pp: case inr
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
inst✝² : NormedDivisionRing 𝕜
inst✝¹ : (i : ι) → AddCommGroup (E i)
inst✝ : (i : ι) → Module 𝕜 (E i)
I : Set ι
hI : I.Finite
U : (i : ι) → Set (E i)
hU : ∀ i ∈ I, Balanced 𝕜 (U i)
x : (i : ι) → E i
hI₀ : I = univ ∨ (∃ i ∈ I, x i ≠ 0) ∨ (𝓝[≠] 0).NeBot
r : ℝ≥0∞
hr : ⨆ i ∈ I, egauge 𝕜 (U i) (x i) < r
c : ι → 𝕜
hc : ∀ i ∈ I, x i ∈ c i • U i
hcr : ∀ i ∈ I, ‖c i‖ₑ < r
hr₀ : 0 < r
hIne : I.Nonempty
⊢ ∃ c₀, (c₀ ≠ 0 ∨ I = univ) ∧ (∀ i ∈ I, ‖c i‖ ≤ ‖c₀‖) ∧ ‖c₀‖ₑ < r

Tactic:
· obtain hι | hbot : IsEmpty ι ∨ (𝓝[≠] (0 : 𝕜)).NeBot := by simpa [@eq_comm _ ∅] using hI₀
      · use 0
        simp [@eq_comm _ ∅, hι, hr₀]
      · rcases exists_enorm_lt 𝕜 hr₀.ne' with ⟨c₀, hc₀, hc₀r⟩
        exact ⟨c₀, .inl hc₀, by simp, hc₀r⟩

Goal state after:
case inr
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
inst✝² : NormedDivisionRing 𝕜
inst✝¹ : (i : ι) → AddCommGroup (E i)
inst✝ : (i : ι) → Module 𝕜 (E i)
I : Set ι
hI : I.Finite
U : (i : ι) → Set (E i)
hU : ∀ i ∈ I, Balanced 𝕜 (U i)
x : (i : ι) → E i
hI₀ : I = univ ∨ (∃ i ∈ I, x i ≠ 0) ∨ (𝓝[≠] 0).NeBot
r : ℝ≥0∞
hr : ⨆ i ∈ I, egauge 𝕜 (U i) (x i) < r
c : ι → 𝕜
hc : ∀ i ∈ I, x i ∈ c i • U i
hcr : ∀ i ∈ I, ‖c i‖ₑ < r
hr₀ : 0 < r
hIne : I.Nonempty
⊢ ∃ c₀, (c₀ ≠ 0 ∨ I = univ) ∧ (∀ i ∈ I, ‖c i‖ ≤ ‖c₀‖) ∧ ‖c₀‖ₑ < r
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
⊢ 0 (n + 1 + 1) = 1 (n + 1 + 1)

pp: 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

Tactic:
Pi.zero_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 = 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.Convex.EGauge

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
inst✝² : NormedDivisionRing 𝕜
inst✝¹ : (i : ι) → AddCommGroup (E i)
inst✝ : (i : ι) → Module 𝕜 (E i)
I : Set ι
hI : I.Finite
U : (i : ι) → Set (E i)
hU : ∀ i ∈ I, Balanced 𝕜 (U i)
x : (i : ι) → E i
hI₀ : I = univ ∨ (∃ i ∈ I, x i ≠ 0) ∨ (𝓝[≠] 0).NeBot
r : ℝ≥0∞
hr : ⨆ i ∈ I, egauge 𝕜 (U i) (x i) < r
c : ι → 𝕜
hc : ∀ i ∈ I, x i ∈ c i • U i
hcr : ∀ i ∈ I, ‖c i‖ₑ < r
c₀ : 𝕜
hc₀ : c₀ ≠ 0 ∨ I = univ
hc₀I : ∀ i ∈ I, ‖c i‖ ≤ ‖c₀‖
hc₀r : ‖c₀‖ₑ < r
i : ι
⊢ i ∈ I → x i ∈ (c₀ • U) i

Tactic:
hi

Goal state after:
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
inst✝² : NormedDivisionRing 𝕜
inst✝¹ : (i : ι) → AddCommGroup (E i)
inst✝ : (i : ι) → Module 𝕜 (E i)
I : Set ι
hI : I.Finite
U : (i : ι) → Set (E i)
hU : ∀ i ∈ I, Balanced 𝕜 (U i)
x : (i : ι) → E i
hI₀ : I = univ ∨ (∃ i ∈ I, x i ≠ 0) ∨ (𝓝[≠] 0).NeBot
r : ℝ≥0∞
hr : ⨆ i ∈ I, egauge 𝕜 (U i) (x i) < r
c : ι → 𝕜
hc : ∀ i ∈ I, x i ∈ c i • U i
hcr : ∀ i ∈ I, ‖c i‖ₑ < r
c₀ : 𝕜
hc₀ : c₀ ≠ 0 ∨ I = univ
hc₀I : ∀ i ∈ I, ‖c i‖ ≤ ‖c₀‖
hc₀r : ‖c₀‖ₑ < r
i : ι
hi : i ∈ I
⊢ x i ∈ (c₀ • U) i
Module: Mathlib.Analysis.Convex.EGauge

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
x : E
r : ℝ≥0
hc : 1 < ‖c‖
h₀ : r ≠ 0 ∨ ‖x‖ ≠ 0
this : NontriviallyNormedField 𝕜 := { toNormedField := inst✝², non_trivial := ⋯ }
hr : 0 < r
hx : ‖x‖ = 0
⊢ ‖x‖ₑ = 0

Tactic:
simpa [enorm, ← coe_nnnorm, NNReal.coe_eq_zero] using hx

Goal state after:
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✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : Module 𝕜 E
inst✝³ : AddCommGroup F
inst✝² : TopologicalSpace F
inst✝¹ : Module 𝕜 F
l : Filter α
f : α → E
g : α → F
inst✝ : ContinuousConstSMul 𝕜 F
ι : Sort u_7
p : ι → Prop
U : ι → Set E
hb : (𝓝 0).HasBasis p U
h : ∀ (i : ι), p i → ∃ C, ∃ V ∈ 𝓝 0, (fun x ↦ egauge 𝕜 (U i) (f x)) ≤ᶠ[l] fun x ↦ ↑C * egauge 𝕜 V (g x)
i : ι
⊢ p i → ∃ j ∈ 𝓝 0, ∀ᶠ (x : α) in l, egauge 𝕜 (U i) (f x) ≤ egauge 𝕜 (id j) (g x)

Tactic:
hi

Goal state after:
α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : Module 𝕜 E
inst✝³ : AddCommGroup F
inst✝² : TopologicalSpace F
inst✝¹ : Module 𝕜 F
l : Filter α
f : α → E
g : α → F
inst✝ : ContinuousConstSMul 𝕜 F
ι : Sort u_7
p : ι → Prop
U : ι → Set E
hb : (𝓝 0).HasBasis p U
h : ∀ (i : ι), p i → ∃ C, ∃ V ∈ 𝓝 0, (fun x ↦ egauge 𝕜 (U i) (f x)) ≤ᶠ[l] fun x ↦ ↑C * egauge 𝕜 V (g x)
i : ι
hi : p i
⊢ ∃ j ∈ 𝓝 0, ∀ᶠ (x : α) in l, egauge 𝕜 (U i) (f x) ≤ egauge 𝕜 (id j) (g x)
Module: Mathlib.Analysis.Calculus.Deriv.Basic

Goal state before:
pp: 𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
F : Type v
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 F
inst✝ : TopologicalSpace F
f : 𝕜 → F
x : 𝕜
s : Set 𝕜
h : ¬DifferentiableWithinAt 𝕜 f s x
⊢ derivWithin f s x = 0

Tactic:
by
  unfold derivWithin
  rw [fderivWithin_zero_of_not_differentiableWithinAt h]
  simp

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Basic

Goal state before:
pp: case e'_12
𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
f : 𝕜 → F
f' : F
x : 𝕜
σ σ' : 𝕜 →+* 𝕜
inst✝³ : RingHomIsometric σ
inst✝² : RingHomInvPair σ σ'
F' : Type u_1
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace 𝕜 F'
L : F →SL[σ] F'
hf : HasDerivAt f f' x
this : RingHomIsometric σ'
R : 𝕜 →SL[σ'] 𝕜 := ⋯
hR : ∀ (k : 𝕜), R k = σ' k
e_4✝ : inst✝⁶.toDivisionRing.toAddCommGroup = NonUnitalNormedRing.toNormedAddCommGroup.toAddCommGroup
e_5✝ : Semiring.toModule ≍ NormedField.toNormedSpace.toModule
⊢ toSpanSingleton 𝕜 (L f') = L ∘SL toSpanSingleton 𝕜 f' ∘SL R

pp: case convert_3
𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
f : 𝕜 → F
f' : F
x : 𝕜
σ σ' : 𝕜 →+* 𝕜
inst✝³ : RingHomIsometric σ
inst✝² : RingHomInvPair σ σ'
F' : Type u_1
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace 𝕜 F'
L : F →SL[σ] F'
hf : HasDerivAt f f' x
this : RingHomIsometric σ'
R : 𝕜 →SL[σ'] 𝕜 := { toLinearMap := σ'.toSemilinearMap, cont := ⋯ }
hR : ∀ (k : 𝕜), R k = σ' k
⊢ HasFDerivAt f (toSpanSingleton 𝕜 f') (R (σ x))

Tactic:
· ext
    simp [R]

Goal state after:
case convert_3
𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
f : 𝕜 → F
f' : F
x : 𝕜
σ σ' : 𝕜 →+* 𝕜
inst✝³ : RingHomIsometric σ
inst✝² : RingHomInvPair σ σ'
F' : Type u_1
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace 𝕜 F'
L : F →SL[σ] F'
hf : HasDerivAt f f' x
this : RingHomIsometric σ'
R : 𝕜 →SL[σ'] 𝕜 := { toLinearMap := σ'.toSemilinearMap, cont := ⋯ }
hR : ∀ (k : 𝕜), R k = σ' k
⊢ HasFDerivAt f (toSpanSingleton 𝕜 f') (R (σ x))
Module: Mathlib.Analysis.Calculus.FormalMultilinearSeries

Goal state before:
pp: 𝕜 : Type u
E : Type v
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
p : FormalMultilinearSeries 𝕜 𝕜 E
n : ℕ
⊢ p.coeff n = 0 ↔ p n = 0

Tactic:
rw [← mkPiRing_coeff_eq p, ContinuousMultilinearMap.mkPiRing_eq_zero_iff]

Goal state after:
Module: Mathlib.Analysis.Calculus.FormalMultilinearSeries

Goal state before:
pp: 𝕜 : Type u
E : Type v
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
p : FormalMultilinearSeries 𝕜 𝕜 E
n : ℕ
⊢ ‖p n‖ = ‖p.coeff n‖

Tactic:
rw [← mkPiRing_coeff_eq p, ContinuousMultilinearMap.norm_mkPiRing]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_3
F : Type u_4
inst✝¹⁵ : Semiring 𝕜
inst✝¹⁴ : AddCommMonoid E
inst✝¹³ : AddCommMonoid F
inst✝¹² : Module 𝕜 E
inst✝¹¹ : Module 𝕜 F
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : TopologicalSpace F
inst✝⁸ : ContinuousAdd E
inst✝⁷ : ContinuousAdd F
inst✝⁶ : ContinuousConstSMul 𝕜 E
inst✝⁵ : ContinuousConstSMul 𝕜 F
𝕜' : Type
inst✝⁴ : DivisionSemiring 𝕜'
inst✝³ : Module 𝕜' F
inst✝² : ContinuousConstSMul 𝕜' F
inst✝¹ : SMulCommClass 𝕜 𝕜' F
inst✝ : T2Space F
a : 𝕜'
f : FormalMultilinearSeries 𝕜 E F
⊢ a • f.sum = (a • f).sum

Tactic:
ext z

Goal state after:
𝕜 : Type u_1
E : Type u_3
F : Type u_4
inst✝¹⁵ : Semiring 𝕜
inst✝¹⁴ : AddCommMonoid E
inst✝¹³ : AddCommMonoid F
inst✝¹² : Module 𝕜 E
inst✝¹¹ : Module 𝕜 F
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : TopologicalSpace F
inst✝⁸ : ContinuousAdd E
inst✝⁷ : ContinuousAdd F
inst✝⁶ : ContinuousConstSMul 𝕜 E
inst✝⁵ : ContinuousConstSMul 𝕜 F
𝕜' : Type
inst✝⁴ : DivisionSemiring 𝕜'
inst✝³ : Module 𝕜' F
inst✝² : ContinuousConstSMul 𝕜' F
inst✝¹ : SMulCommClass 𝕜 𝕜' F
inst✝ : T2Space F
a : 𝕜'
f : FormalMultilinearSeries 𝕜 E F
z : E
⊢ (a • f.sum) z = (a • f).sum z
Module: Mathlib.Analysis.Analytic.ConvergenceRadius

Goal state before:
pp: case inr
𝕜 : Type u_1
E : Type u_3
F : Type u_4
G : Type u_5
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
p : FormalMultilinearSeries 𝕜 F G
u : E →ₗᵢ[𝕜] F
h : Nontrivial E
⊢ p.radius ≤ (p.compContinuousLinearMap u.toContinuousLinearMap).radius

Tactic:
simpa [u.norm_toContinuousLinearMap]
      using div_le_radius_compContinuousLinearMap p u.toContinuousLinearMap

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

Goal state before:
pp: case h.inr
𝕜 : Type u_1
E : Type u_3
F : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0
C : ℝ
h : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C
n : ℕ
hr : 0 < ↑r
⊢ ‖p (n + 1)‖ * ↑r ^ n ≤ C / ↑r

Tactic:
specialize h (n + 1)

Goal state after:
case h.inr
𝕜 : Type u_1
E : Type u_3
F : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0
C : ℝ
n : ℕ
h : ‖p (n + 1)‖ * ↑r ^ (n + 1) ≤ C
hr : 0 < ↑r
⊢ ‖p (n + 1)‖ * ↑r ^ n ≤ C / ↑r
Module: Mathlib.Analysis.Analytic.CPolynomialDef

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
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → p m = 0
k : ℕ
x : E
x✝ : Fin k → E
m : ℕ
hm : n - k ≤ m
⊢ n ≤ k + m

Tactic:
rw [add_comm]

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
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → p m = 0
k : ℕ
x : E
x✝ : Fin k → E
m : ℕ
hm : n - k ≤ m
⊢ n ≤ m + k
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
s : Set E
x : E
h' : g =ᶠ[𝓝[s] x] f
h'' : g x = f x
r : ℝ≥0∞
hr : HasFPowerSeriesWithinOnBall f p s x r
ε : ℝ≥0∞
εpos : ε > 0
hε : Metric.eball x ε ∩ s ⊆ {y | g y = f y}
r' : ℝ≥0∞ := min r ε
this : HasFPowerSeriesWithinOnBall f p s x r'
⊢ HasFPowerSeriesWithinOnBall g p s x r'

Tactic:
apply this.congr _ h''

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
s : Set E
x : E
h' : g =ᶠ[𝓝[s] x] f
h'' : g x = f x
r : ℝ≥0∞
hr : HasFPowerSeriesWithinOnBall f p s x r
ε : ℝ≥0∞
εpos : ε > 0
hε : Metric.eball x ε ∩ s ⊆ {y | g y = f y}
r' : ℝ≥0∞ := ⋯
this : HasFPowerSeriesWithinOnBall f p s x r'
⊢ EqOn g f (s ∩ Metric.eball x r')
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 : E → F
pf : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
hf : HasFPowerSeriesWithinOnBall f pf s x r
v : Fin 0 → E
v_eq : v = fun i ↦ 0
zero_mem : 0 ∈ Metric.eball 0 r
i : ℕ
⊢ i ≠ 0 → ((pf i) fun x ↦ 0) = 0

Tactic:
hi

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 : E → F
pf : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
hf : HasFPowerSeriesWithinOnBall f pf s x r
v : Fin 0 → E
v_eq : v = fun i ↦ 0
zero_mem : 0 ∈ Metric.eball 0 r
i : ℕ
hi : i ≠ 0
⊢ ((pf i) fun x ↦ 0) = 0
Module: Mathlib.Analysis.Analytic.CPolynomialDef

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 : E → F
x : E
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
r : ℝ≥0∞
hr : HasFiniteFPowerSeriesOnBall f p x n r
⊢ {x | CPolynomialAt 𝕜 f x} ∈ 𝓝 x

Tactic:
exact mem_of_superset (Metric.eball_mem_nhds _ hr.r_pos) fun y hy => hr.cpolynomialAt_of_mem hy

Goal state after:
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✝¹⁵ : CommRing 𝕜
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : AddCommGroup F
inst✝¹² : AddCommGroup G
inst✝¹¹ : Module 𝕜 E
inst✝¹⁰ : Module 𝕜 F
inst✝⁹ : Module 𝕜 G
inst✝⁸ : TopologicalSpace E
inst✝⁷ : TopologicalSpace F
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalAddGroup E
inst✝⁴ : ContinuousConstSMul 𝕜 E
inst✝³ : IsTopologicalAddGroup F
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜 G
n : ℕ
p : FormalMultilinearSeries 𝕜 F G
f : E →L[𝕜] F
c : Composition n
v : Fin n → E
⊢ (p.compContinuousLinearMap f).applyComposition c v = p.applyComposition c (⇑f ∘ v)

Tactic:
ext
  simp [applyComposition, Function.comp_def]

Goal state after:
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
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
hq : 0 < q.radius
hp : 0 < p.radius
rq rp : ℝ≥0
hrp : rp < 1 ∧ ↑rp < p.radius
hrq : rq < 1 ∧ ↑rq < q.radius
rp_pos : 0 < rp
rq_pos : 0 < rq
Cq : ℝ≥0
_hCq0 : Cq > 0
hCq : ∀ (n : ℕ), ‖q n‖₊ * rq ^ n ≤ Cq
Cp : ℝ≥0
hCp1 : Cp ≥ 1
hCp : ∀ (n : ℕ), ‖p n‖₊ * rp ^ n ≤ Cp
r0 : ℝ≥0 := (4 * Cp)⁻¹
r0_pos : 0 < r0
r : ℝ≥0 := rp * rq * r0
r_pos : 0 < r
I : ∀ (i : (n : ℕ) × Composition n), ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst ≤ Cq / 4 ^ i.fst
this : ∀ (n : ℕ), HasSum (fun c ↦ (4 ^ n)⁻¹) (2 ^ (n - 1) / 4 ^ n)
n : ℕ
⊢ 4 = 2 * 2

Tactic:
norm_num

Goal state after: