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

Goal state before:
pp: case neg
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
f g : PowerSeries R
hq : 1 < q
k : ℕ
hs :
  Finset.map ({ toFun := fun k ↦ q ^ k, inj' := ⋯ }.prodMap { toFun := fun k ↦ q ^ k, inj' := ⋯ })
      (Finset.antidiagonal k) ⊆
    (q ^ k).divisorsAntidiagonal
a b : ℕ
hab : (a, b) ∈ (q ^ k).divisorsAntidiagonal
h :
  (a, b) ∉
    Finset.map ({ toFun := fun k ↦ q ^ k, inj' := ⋯ }.prodMap { toFun := fun k ↦ q ^ k, inj' := ⋯ })
      (Finset.antidiagonal k)
ha : ¬∃ i, q ^ i = a
⊢ 0 = 0

pp: case neg.hb
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
f g : PowerSeries R
hq : 1 < q
k : ℕ
hs :
  Finset.map ({ toFun := fun k ↦ q ^ k, inj' := ⋯ }.prodMap { toFun := fun k ↦ q ^ k, inj' := ⋯ })
      (Finset.antidiagonal k) ⊆
    (q ^ k).divisorsAntidiagonal
a b : ℕ
hab : (a, b) ∈ (q ^ k).divisorsAntidiagonal
h :
  (a, b) ∉
    Finset.map ({ toFun := fun k ↦ q ^ k, inj' := ⋯ }.prodMap { toFun := fun k ↦ q ^ k, inj' := ⋯ })
      (Finset.antidiagonal k)
ha : ¬∃ i, q ^ i = a
⊢ ¬∃ a_1, q ^ a_1 = (a, b).1

Tactic:
]

Goal state after:
case neg.hb
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
f g : PowerSeries R
hq : 1 < q
k : ℕ
hs :
  Finset.map ({ toFun := fun k ↦ q ^ k, inj' := ⋯ }.prodMap { toFun := fun k ↦ q ^ k, inj' := ⋯ })
      (Finset.antidiagonal k) ⊆
    (q ^ k).divisorsAntidiagonal
a b : ℕ
hab : (a, b) ∈ (q ^ k).divisorsAntidiagonal
h :
  (a, b) ∉
    Finset.map ({ toFun := fun k ↦ q ^ k, inj' := ⋯ }.prodMap { toFun := fun k ↦ q ^ k, inj' := ⋯ })
      (Finset.antidiagonal k)
ha : ¬∃ i, q ^ i = a
⊢ ¬∃ a_1, q ^ a_1 = (a, b).1
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝² : CommRing R
τ : Type u_4
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
a : σ → MvPowerSeries τ S
w : τ → ℕ
ha : HasSubst a
f : MvPowerSeries σ R
d : τ →₀ ℕ
hd : ↑((Finsupp.weight w) d) < ⨅ d, ⨅ (_ : (coeff d) f ≠ 0), (Finsupp.weight (weightedOrder w ∘ a)) d
⊢ 0 = 0

pp: σ : Type u_1
R : Type u_3
inst✝² : CommRing R
τ : Type u_4
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
a : σ → MvPowerSeries τ S
w : τ → ℕ
ha : HasSubst a
f : MvPowerSeries σ R
d : τ →₀ ℕ
hd : ↑((Finsupp.weight w) d) < ⨅ d, ⨅ (_ : (coeff d) f ≠ 0), (Finsupp.weight (weightedOrder w ∘ a)) d
⊢ ∀ (x : σ →₀ ℕ), (coeff x) f • (coeff d) (x.prod fun s e ↦ a s ^ e) = 0

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_3
inst✝² : CommRing R
τ : Type u_4
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
a : σ → MvPowerSeries τ S
w : τ → ℕ
ha : HasSubst a
f : MvPowerSeries σ R
d : τ →₀ ℕ
hd : ↑((Finsupp.weight w) d) < ⨅ d, ⨅ (_ : (coeff d) f ≠ 0), (Finsupp.weight (weightedOrder w ∘ a)) d
⊢ ∀ (x : σ →₀ ℕ), (coeff x) f • (coeff d) (x.prod fun s e ↦ a s ^ e) = 0
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: case neg
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
f g : PowerSeries R
n : ℕ
hq : 1 < q
hn : ¬∃ a, q ^ a = n
⊢ 0 = 0

pp: case neg
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
f g : PowerSeries R
n : ℕ
hq : 1 < q
hn : ¬∃ a, q ^ a = n
⊢ ∀ x ∈ n.divisorsAntidiagonal,
    Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) f) 0 x.1 *
        Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) g) 0 x.2 =
      0

Tactic:
]

Goal state after:
case neg
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
f g : PowerSeries R
n : ℕ
hq : 1 < q
hn : ¬∃ a, q ^ a = n
⊢ ∀ x ∈ n.divisorsAntidiagonal,
    Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) f) 0 x.1 *
        Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) g) 0 x.2 =
      0
Module: Mathlib.NumberTheory.LSeries.Basic

Goal state before:
pp: a : ℕ → ℂ
n : ℕ
h : 0 ≤ a n
x : ℝ
⊢ 0 ≤ if n = 0 then 0 else a n / ↑n ^ ↑x

Tactic:
]

Goal state after:
a : ℕ → ℂ
n : ℕ
h : 0 ≤ a n
x : ℝ
⊢ 0 ≤ if n = 0 then 0 else a n / ↑n ^ ↑x
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: case neg
σ : Type u_1
R : Type u_3
inst✝² : CommRing R
τ : Type u_4
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
a : σ → MvPowerSeries τ S
w : τ → ℕ
ha : HasSubst a
f : MvPowerSeries σ R
d : τ →₀ ℕ
hd : ↑((Finsupp.weight w) d) < ⨅ d, ⨅ (_ : (coeff d) f ≠ 0), (Finsupp.weight (weightedOrder w ∘ a)) d
x : σ →₀ ℕ
hfx : ¬(coeff x) f = 0
⊢ 0 = 0

pp: case neg
σ : Type u_1
R : Type u_3
inst✝² : CommRing R
τ : Type u_4
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
a : σ → MvPowerSeries τ S
w : τ → ℕ
ha : HasSubst a
f : MvPowerSeries σ R
d : τ →₀ ℕ
hd : ↑((Finsupp.weight w) d) < ⨅ d, ⨅ (_ : (coeff d) f ≠ 0), (Finsupp.weight (weightedOrder w ∘ a)) d
x : σ →₀ ℕ
hfx : ¬(coeff x) f = 0
⊢ ↑((Finsupp.weight w) d) < weightedOrder w (x.prod fun s e ↦ a s ^ e)

Tactic:
]

Goal state after:
case neg
σ : Type u_1
R : Type u_3
inst✝² : CommRing R
τ : Type u_4
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
a : σ → MvPowerSeries τ S
w : τ → ℕ
ha : HasSubst a
f : MvPowerSeries σ R
d : τ →₀ ℕ
hd : ↑((Finsupp.weight w) d) < ⨅ d, ⨅ (_ : (coeff d) f ≠ 0), (Finsupp.weight (weightedOrder w ∘ a)) d
x : σ →₀ ℕ
hfx : ¬(coeff x) f = 0
⊢ ↑((Finsupp.weight w) d) < weightedOrder w (x.prod fun s e ↦ a s ^ e)
Module: Mathlib.NumberTheory.LSeries.Basic

Goal state before:
pp: f g : ℕ → ℂ
s : ℂ
h : f =ᶠ[cofinite] g
hf : LSeriesSummable f s
⊢ LSeriesSummable g s

Tactic:
]

Goal state after:
f g : ℕ → ℂ
s : ℂ
h : f =ᶠ[cofinite] g
hf : LSeriesSummable f s
⊢ LSeriesSummable g s
Module: Mathlib.NumberTheory.LSeries.Basic

Goal state before:
pp: f g : ℕ → ℂ
s : ℂ
h : ∃ s ∈ cofinite, Set.EqOn f g s
hf : LSeriesSummable f s
⊢ ∃ s_1 ∈ cofinite, Set.EqOn (term f s) (term g s) s_1

Tactic:
]

Goal state after:
f g : ℕ → ℂ
s : ℂ
h : ∃ s ∈ cofinite, Set.EqOn f g s
hf : LSeriesSummable f s
⊢ ∃ s_1 ∈ cofinite, Set.EqOn (term f s) (term g s) s_1
Module: Mathlib.NumberTheory.LSeries.Basic

Goal state before:
pp: f g : ℕ → ℂ
s : ℂ
hf : LSeriesSummable f s
S : Set ℕ
hS : S ∈ cofinite
hS' : Set.EqOn f g S
n : ℕ
hn : n ∈ S ∧ ¬n = 0
⊢ term f s n = term g s n

Tactic:
]

Goal state after:
f g : ℕ → ℂ
s : ℂ
hf : LSeriesSummable f s
S : Set ℕ
hS : S ∈ cofinite
hS' : Set.EqOn f g S
n : ℕ
hn : n ∈ S ∧ ¬n = 0
⊢ term f s n = term g s n
Module: Mathlib.NumberTheory.LSeries.Basic

Goal state before:
pp: f : ℕ → ℂ
s s' : ℂ
h : s.re ≤ s'.re
hf : Summable fun x ↦ ‖term f s x‖
⊢ Summable fun x ↦ ‖term f s' x‖

Tactic:
]

Goal state after:
f : ℕ → ℂ
s s' : ℂ
h : s.re ≤ s'.re
hf : Summable fun x ↦ ‖term f s x‖
⊢ Summable fun x ↦ ‖term f s' x‖
Module: Mathlib.NumberTheory.LSeries.Basic

Goal state before:
pp: f : ℕ → ℂ
h : f 1 = 1
⊢ δ = δ

Tactic:
]

Goal state after:
Module: Mathlib.NumberTheory.LSeries.Basic

Goal state before:
pp: case h
f : ℕ → ℂ
s : ℂ
h : Summable fun x ↦ ‖term f s x‖
n : ℕ
hn₀ : n ≠ 0
hn : (∑' (n : ℕ), ‖term f s n‖) * ↑n ^ s.re < ‖f n‖
this : ‖f n‖ ≤ (∑' (i : ℕ), ‖term f s i‖) * ↑n ^ s.re
⊢ False

Tactic:
]

Goal state after:
case h
f : ℕ → ℂ
s : ℂ
h : Summable fun x ↦ ‖term f s x‖
n : ℕ
hn₀ : n ≠ 0
hn : (∑' (n : ℕ), ‖term f s n‖) * ↑n ^ s.re < ‖f n‖
this : ‖f n‖ ≤ (∑' (i : ℕ), ‖term f s i‖) * ↑n ^ s.re
⊢ False
Module: Mathlib.NumberTheory.LSeries.Basic

Goal state before:
pp: f : ℕ → ℂ
s : ℂ
h : LSeriesSummable f s
C : ℝ
hC : ∀ (n : ℕ), n ≠ 0 → ‖f n‖ ≤ C * ↑n ^ s.re
n : ℕ
hn : 1 ≤ n
⊢ ↑n ^ s.re = ↑n ^ s.re

Tactic:
]

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

Goal state before:
pp: case pos
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
f g : PowerSeries R
n : ℕ
hq : 1 < q
hn : ¬∃ a, q ^ a = n
i j : ℕ
hk : (q ^ i, q ^ j) ∈ n.divisorsAntidiagonal
hab : q ^ (i + j) = n
⊢ Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) f) 0 (q ^ i, q ^ j).1 *
      Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) g) 0 (q ^ i, q ^ j).2 =
    0

Tactic:
]

Goal state after:
case pos
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
f g : PowerSeries R
n : ℕ
hq : 1 < q
hn : ¬∃ a, q ^ a = n
i j : ℕ
hk : (q ^ i, q ^ j) ∈ n.divisorsAntidiagonal
hab : q ^ (i + j) = n
⊢ Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) f) 0 (q ^ i, q ^ j).1 *
      Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) g) 0 (q ^ i, q ^ j).2 =
    0
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
x : σ → ℕ
k : ℕ
ha : HasSubst a
hx : ∀ (i : σ), k ≤ x i
d : τ →₀ ℕ
hd : Finsupp.degree d < k
⊢ finsum ?m.148 = ∑ᶠ (d_1 : σ →₀ ℕ), (coeff d_1) f • (coeff d) (d_1.prod fun s e ↦ ↑((truncTotal (x s)) (a s)) ^ e)

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
x : σ → ℕ
k : ℕ
ha : HasSubst a
hx : ∀ (i : σ), k ≤ x i
d : τ →₀ ℕ
hd : Finsupp.degree d < k
⊢ ∀ (x : σ →₀ ℕ), (coeff x) f • (coeff d) (x.prod fun s e ↦ a s ^ e) = ?m.148 x

pp: 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
x : σ → ℕ
k : ℕ
ha : HasSubst a
hx : ∀ (i : σ), k ≤ x i
d : τ →₀ ℕ
hd : Finsupp.degree d < k
⊢ (σ →₀ ℕ) → S

pp: case pos.ha
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
x : σ → ℕ
k : ℕ
ha : HasSubst a
hx : ∀ (i : σ), k ≤ x i
d : τ →₀ ℕ
hd : Finsupp.degree d < k
⊢ HasSubst fun i ↦ ↑((truncTotal (x i)) (a i))

Tactic:
]

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
x : σ → ℕ
k : ℕ
ha : HasSubst a
hx : ∀ (i : σ), k ≤ x i
d : τ →₀ ℕ
hd : Finsupp.degree d < k
⊢ ∀ (x_1 : σ →₀ ℕ),
    (coeff x_1) f • (coeff d) (x_1.prod fun s e ↦ a s ^ e) =
      (coeff x_1) f • (coeff d) (x_1.prod fun s e ↦ ↑((truncTotal (x s)) (a s)) ^ e)

case pos.ha
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
x : σ → ℕ
k : ℕ
ha : HasSubst a
hx : ∀ (i : σ), k ≤ x i
d : τ →₀ ℕ
hd : Finsupp.degree d < k
⊢ HasSubst fun i ↦ ↑((truncTotal (x i)) (a i))
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: case neg
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
f g : PowerSeries R
n : ℕ
hq : 1 < q
hn : ¬∃ a, q ^ a = n
a b : ℕ
hk : (a, b) ∈ n.divisorsAntidiagonal
hab : (a, b).1 * (a, b).2 = n
ha : ∃ i, q ^ i = a
hb : ¬∃ j, q ^ j = b
⊢ 0 = 0

pp: case neg.hb
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
f g : PowerSeries R
n : ℕ
hq : 1 < q
hn : ¬∃ a, q ^ a = n
a b : ℕ
hk : (a, b) ∈ n.divisorsAntidiagonal
hab : (a, b).1 * (a, b).2 = n
ha : ∃ i, q ^ i = a
hb : ¬∃ j, q ^ j = b
⊢ ¬∃ a_1, q ^ a_1 = (a, b).2

Tactic:
]

Goal state after:
case neg.hb
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
f g : PowerSeries R
n : ℕ
hq : 1 < q
hn : ¬∃ a, q ^ a = n
a b : ℕ
hk : (a, b) ∈ n.divisorsAntidiagonal
hab : (a, b).1 * (a, b).2 = n
ha : ∃ i, q ^ i = a
hb : ¬∃ j, q ^ j = b
⊢ ¬∃ a_1, q ^ a_1 = (a, b).2
Module: Mathlib.RingTheory.PowerSeries.Substitution

Goal state before:
pp: τ : Type u_3
S : Type u_4
inst✝² : CommRing S
inst✝¹ : TopologicalSpace S
inst✝ : DiscreteTopology S
a : MvPowerSeries τ S
⊢ MvPowerSeries.HasEval (Function.const Unit a) ↔ MvPowerSeries.HasEval (Function.const Unit a)

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.PowerSeries.Substitution

Goal state before:
pp: τ : Type u_3
S : Type u_4
inst✝ : CommRing S
n : τ →₀ ℕ
hn : n ≠ 0
s : S
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.PowerSeries.Substitution

Goal state before:
pp: R : Type u_2
inst✝ : CommRing R
τ : Type u_3
⊢ MvPowerSeries.HasSubst (Function.const Unit 0)

Tactic:
]

Goal state after:
R : Type u_2
inst✝ : CommRing R
τ : Type u_3
⊢ MvPowerSeries.HasSubst (Function.const Unit 0)
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: case neg
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
f g : PowerSeries R
n : ℕ
hq : 1 < q
hn : ¬∃ a, q ^ a = n
a b : ℕ
hk : (a, b) ∈ n.divisorsAntidiagonal
hab : (a, b).1 * (a, b).2 = n
ha : ¬∃ i, q ^ i = a
⊢ 0 = 0

pp: case neg.hb
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
f g : PowerSeries R
n : ℕ
hq : 1 < q
hn : ¬∃ a, q ^ a = n
a b : ℕ
hk : (a, b) ∈ n.divisorsAntidiagonal
hab : (a, b).1 * (a, b).2 = n
ha : ¬∃ i, q ^ i = a
⊢ ¬∃ a_1, q ^ a_1 = (a, b).1

Tactic:
]

Goal state after:
case neg.hb
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
f g : PowerSeries R
n : ℕ
hq : 1 < q
hn : ¬∃ a, q ^ a = n
a b : ℕ
hk : (a, b) ∈ n.divisorsAntidiagonal
hab : (a, b).1 * (a, b).2 = n
ha : ¬∃ i, q ^ i = a
⊢ ¬∃ a_1, q ^ a_1 = (a, b).1
Module: Mathlib.RingTheory.PowerSeries.Substitution

Goal state before:
pp: A : Type u_1
inst✝ : CommRing A
f : A⟦X⟧
hf : HasSubst f
n k : ℕ
hk : MvPowerSeries.constantCoeff f ^ k = 0
n' : ℕ
hn' : n' ≤ n
m : ℕ
hm : k * (n + 1) + m ∈ Set.Ici (k * (n + 1))
⊢ constantCoeff f ^ k = 0

Tactic:
]

Goal state after:
A : Type u_1
inst✝ : CommRing A
f : A⟦X⟧
hf : HasSubst f
n k : ℕ
hk : MvPowerSeries.constantCoeff f ^ k = 0
n' : ℕ
hn' : n' ≤ n
m : ℕ
hm : k * (n + 1) + m ∈ Set.Ici (k * (n + 1))
⊢ constantCoeff f ^ k = 0
Module: Mathlib.RingTheory.PowerSeries.Substitution

Goal state before:
pp: A : Type u_1
inst✝ : CommRing A
f : A⟦X⟧
hf : HasSubst f
n k : ℕ
hk : MvPowerSeries.constantCoeff f ^ k = 0
n' : ℕ
hn' : n' ≤ n
m : ℕ
hm : k * (n + 1) + m ∈ Set.Ici (k * (n + 1))
⊢ n' < n + 1

Tactic:
]

Goal state after:
A : Type u_1
inst✝ : CommRing A
f : A⟦X⟧
hf : HasSubst f
n k : ℕ
hk : MvPowerSeries.constantCoeff f ^ k = 0
n' : ℕ
hn' : n' ≤ n
m : ℕ
hm : k * (n + 1) + m ∈ Set.Ici (k * (n + 1))
⊢ n' < n + 1
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
ha : HasSubst a
f g : R⟦X⟧
⊢ (substAlgHom ha) f + (substAlgHom ha) g = (substAlgHom ha) f + (substAlgHom ha) g

Tactic:
]

Goal state after:
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
ha : HasSubst a
f g : R⟦X⟧
⊢ (substAlgHom ha) f - (substAlgHom ha) g = (substAlgHom ha) f - (substAlgHom ha) g

Tactic:
]

Goal state after:
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
ha : HasSubst a
f : R⟦X⟧
n : ℕ
⊢ (substAlgHom ha) f ^ n = (substAlgHom ha) f ^ n

Tactic:
]

Goal state after:
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
ha : HasSubst a
f g : R⟦X⟧
⊢ (substAlgHom ha) f * (substAlgHom ha) g = (substAlgHom ha) f * (substAlgHom ha) g

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.PowerSeries.Substitution

Goal state before:
pp: A : Type u_1
inst✝⁶ : CommRing A
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Algebra A R
τ : Type u_3
S : Type u_4
inst✝³ : CommRing S
inst✝² : Algebra R S
a : MvPowerSeries τ S
inst✝¹ : Algebra A S
inst✝ : IsScalarTower A R S
ha : HasSubst a
r : A
f : R⟦X⟧
⊢ r • (substAlgHom ha) f = r • (substAlgHom ha) f

Tactic:
]

Goal state after:
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
ha : HasSubst a
f : R⟦X⟧
e : τ →₀ ℕ
⊢ (Function.support fun d ↦ (coeff d) f • (MvPowerSeries.coeff e) (a ^ d)).Finite

Tactic:
]

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
ha : HasSubst a
f : R⟦X⟧
e : τ →₀ ℕ
⊢ (Function.support fun d ↦ (coeff d) f • (MvPowerSeries.coeff e) (a ^ d)).Finite
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: case neg
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
x : R
n : ℕ
hq : 1 < q
hn : ¬∃ k, q ^ k = n
⊢ 0 = 0

pp: case neg
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
x : R
n : ℕ
hq : 1 < q
hn : ¬∃ k, q ^ k = n
⊢ n ≠ 1

Tactic:
]

Goal state after:
case neg
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
x : R
n : ℕ
hq : 1 < q
hn : ¬∃ k, q ^ k = n
⊢ n ≠ 1
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
ha : HasSubst a
f : R⟦X⟧
e : τ →₀ ℕ
⊢ Function.support
      ((fun d ↦ (coeff d) f • (MvPowerSeries.coeff e) (a ^ d)) ∘ ⇑(Finsupp.uniqueLinearEquiv ℕ ℕ ()).toEquiv) =
    Function.support fun d ↦ (MvPowerSeries.coeff d) f • (MvPowerSeries.coeff e) (d.prod fun s e ↦ a ^ e)

Tactic:
]

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
ha : HasSubst a
f : R⟦X⟧
e : τ →₀ ℕ
⊢ Function.support
      ((fun d ↦ (coeff d) f • (MvPowerSeries.coeff e) (a ^ d)) ∘ ⇑(Finsupp.uniqueLinearEquiv ℕ ℕ ()).toEquiv) =
    Function.support fun d ↦ (MvPowerSeries.coeff d) f • (MvPowerSeries.coeff e) (d.prod fun s e ↦ a ^ e)
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
q : ℕ
hq : 1 < q
f : PowerSeries R
k : ℕ
⊢ (PowerSeries.coeff k) f = (PowerSeries.coeff k) f

Tactic:
]

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

Goal state before:
pp: case pos
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
f : PowerSeries R
hq : 1 < q
⊢ PowerSeries.constantCoeff f = PowerSeries.constantCoeff f

Tactic:
]

Goal state after:
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
ha : HasSubst a
f : R⟦X⟧
e : τ →₀ ℕ
⊢ (fun d ↦ (coeff d) f • (MvPowerSeries.coeff e) (a ^ d)) =
    (fun d ↦ (MvPowerSeries.coeff d) f • (MvPowerSeries.coeff e) (d.prod fun s e ↦ a ^ e)) ∘
      ⇑(Finsupp.uniqueLinearEquiv ℕ ℕ ()).toEquiv.symm

Tactic:
]

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
ha : HasSubst a
f : R⟦X⟧
e : τ →₀ ℕ
⊢ (fun d ↦ (coeff d) f • (MvPowerSeries.coeff e) (a ^ d)) =
    (fun d ↦ (MvPowerSeries.coeff d) f • (MvPowerSeries.coeff e) (d.prod fun s e ↦ a ^ e)) ∘
      ⇑(Finsupp.uniqueLinearEquiv ℕ ℕ ()).toEquiv.symm
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)
⊢ ∑ 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:
]

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)
⊢ ∑ 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 pos
R : Type u_1
inst✝ : CommRing R
q k : ℕ
hk : k ≠ 0
f : PowerSeries R
hq : 1 < q
i : ℕ
⊢ ((ofPowerSeries (q ^ k)) f) (q ^ i) = if k ∣ i then (algebraMap R R) ((PowerSeries.coeff (i / k)) f) else 0

Tactic:
]

Goal state after:
case pos
R : Type u_1
inst✝ : CommRing R
q k : ℕ
hk : k ≠ 0
f : PowerSeries R
hq : 1 < q
i : ℕ
⊢ ((ofPowerSeries (q ^ k)) f) (q ^ i) = if k ∣ i then (algebraMap R R) ((PowerSeries.coeff (i / k)) f) else 0
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: case pos
R : Type u_1
inst✝ : CommRing R
q k : ℕ
hk : k ≠ 0
f : PowerSeries R
hq : 1 < q
j : ℕ
⊢ (PowerSeries.coeff j) f = (algebraMap R R) ((PowerSeries.coeff (k * j / k)) f)

Tactic:
]

Goal state after:
case pos
R : Type u_1
inst✝ : CommRing R
q k : ℕ
hk : k ≠ 0
f : PowerSeries R
hq : 1 < q
j : ℕ
⊢ (PowerSeries.coeff j) f = (algebraMap R R) ((PowerSeries.coeff (k * j / k)) f)
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
ha : HasSubst a
f : R⟦X⟧
e : τ →₀ ℕ
⊢ ∑ᶠ (i : ℕ),
      (MvPowerSeries.coeff ((Finsupp.uniqueLinearEquiv ℕ ℕ ()).toEquiv.symm i)) f •
        (MvPowerSeries.coeff e) (((Finsupp.uniqueLinearEquiv ℕ ℕ ()).toEquiv.symm i).prod fun s e ↦ a ^ e) =
    ∑ᶠ (d : ℕ), (coeff d) f • (MvPowerSeries.coeff e) (a ^ d)

Tactic:
]

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
ha : HasSubst a
f : R⟦X⟧
e : τ →₀ ℕ
⊢ ∑ᶠ (i : ℕ),
      (MvPowerSeries.coeff ((Finsupp.uniqueLinearEquiv ℕ ℕ ()).toEquiv.symm i)) f •
        (MvPowerSeries.coeff e) (((Finsupp.uniqueLinearEquiv ℕ ℕ ()).toEquiv.symm i).prod fun s e ↦ a ^ e) =
    ∑ᶠ (d : ℕ), (coeff d) f • (MvPowerSeries.coeff e) (a ^ d)
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: 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
n : σ →₀ ℕ
hc : Finsupp.degree d < Finsupp.degree n
⊢ (coeff d) (∏ a_1 ∈ n.support, a a_1 ^ n a_1) = 0

Tactic:
]

Goal state after:
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
n : σ →₀ ℕ
hc : Finsupp.degree d < Finsupp.degree n
⊢ (coeff d) (∏ a_1 ∈ n.support, a a_1 ^ n a_1) = 0
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
⊢ 0 = 0

pp: 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

Tactic:
]

Goal state after:
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