awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.MvPowerSeries.Rename (goal)
Goal:
σ : Type u_1
τ : Type u_2
R : Type u_4
inst : CommSemiring R
e : σ ↪ τ
i : σ
⊢ Eq (α := MvPowerSeries σ R)
    ((killCompl e : (a : MvPowerSeries τ R) → MvPowerSeries σ R) (X ((e : (a : σ) → τ) i)) : MvPowerSeries σ R)
    (X i : MvPowerSeries σ R)
have h : ∀ (n : σ →₀ ℕ),
  Eq.{u_4 + 1} (α := R)
    ((coeff n : (a : MvPowerSeries σ R) → R)
        ((killCompl e : (a : MvPowerSeries τ R) → MvPowerSeries σ R) (X ((e : (a : σ) → τ) i)) : MvPowerSeries σ R) :
      R)
    ((coeff n : (a : MvPowerSeries σ R) → R) (X i) : R)
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.MvPowerSeries.Rename (goal)
Goal:
σ : Type u_1
τ : Type u_2
R : Type u_4
inst : CommSemiring R
e : σ ↪ τ
i : σ
⊢ Eq (α := MvPowerSeries σ R)
    ((killCompl e : (a : MvPowerSeries τ R) → MvPowerSeries σ R) (X ((e : (a : σ) → τ) i)) : MvPowerSeries σ R)
    (X i : MvPowerSeries σ R)
have h : ∀ (n : σ →₀ ℕ),
  Eq.{u_4 + 1} (α := R)
    ((coeff n : (a : MvPowerSeries σ R) → R)
        ((killCompl e : (a : MvPowerSeries τ R) → MvPowerSeries σ R) (X ((e : (a : σ) → τ) i)) : MvPowerSeries σ R) :
      R)
    ((coeff n : (a : MvPowerSeries σ R) → R) (X i) : R)
Tactic: classical
  ext; simp [coeff_X, coeff_killCompl, ← embDomain_single]
-- Lean 4 proof step from Mathlib.RingTheory.MvPowerSeries.Rename (goal)
Goal:
σ : Type u_1
τ : Type u_2
R : Type u_4
inst : CommSemiring R
e : σ ↪ τ
t : τ
h : t ∉ Set.range.{u_2, u_1 + 1} (ι := σ) (⇑e : (a : σ) → τ)
⊢ single t 1 ∉ Set.range (embDomain e)
have rwa : ¬(↑(support (M := ℕ) (single t 1)) : Set τ) ⊆ Set.range.{u_2, u_1 + 1} (ι := σ) (⇑e : (a : σ) → τ)
Tactic: rwa [mem_range_embDomain_iff, support_single _ (by simp), Finset.coe_singleton,
      Set.singleton_subset_iff]
-- Lean 4 proof step from Mathlib.RingTheory.MvPowerSeries.Rename (goal)
Goal:
σ : Type u_1
τ : Type u_2
R : Type u_4
inst : CommSemiring R
e : σ ↪ τ
t : τ
h : t ∉ Set.range.{u_2, u_1 + 1} (ι := σ) (⇑e : (a : σ) → τ)
rwa : ¬(↑(support (M := ℕ) (single t 1)) : Set τ) ⊆ Set.range.{u_2, u_1 + 1} (ι := σ) (⇑e : (a : σ) → τ)
⊢ single t 1 ∉ Set.range (embDomain e)
have rwa₁ : ¬(↑{t} : Set τ) ⊆ Set.range.{u_2, u_1 + 1} (ι := σ) (⇑e : (a : σ) → τ)
Tactic: rwa [mem_range_embDomain_iff, support_single _ (by simp), Finset.coe_singleton,
      Set.singleton_subset_iff]
-- Lean 4 proof step from Mathlib.RingTheory.MvPowerSeries.Rename (goal)
Goal:
σ : Type u_1
τ : Type u_2
R : Type u_4
inst : CommSemiring R
e : σ ↪ τ
t : τ
h : t ∉ Set.range.{u_2, u_1 + 1} (ι := σ) (⇑e : (a : σ) → τ)
rwa : ¬(↑(support (M := ℕ) (single t 1)) : Set τ) ⊆ Set.range.{u_2, u_1 + 1} (ι := σ) (⇑e : (a : σ) → τ)
rwa₁ : ¬(↑{t} : Set τ) ⊆ Set.range.{u_2, u_1 + 1} (ι := σ) (⇑e : (a : σ) → τ)
⊢ single t 1 ∉ Set.range (embDomain e)
have rwa₂ : ¬Subset (α := Set τ) {t} (Set.range.{u_2, u_1 + 1} (ι := σ) (⇑e : (a : σ) → τ))
Tactic: rwa [mem_range_embDomain_iff, support_single _ (by simp), Finset.coe_singleton,
      Set.singleton_subset_iff]
-- Lean 4 proof step from Mathlib.RingTheory.MvPowerSeries.Rename (hypothesis)
Goal:
σ : Type u_1
τ : Type u_2
R : Type u_4
inst : CommSemiring R
e : σ ↪ τ
t : τ
h : t ∉ Set.range.{u_2, u_1 + 1} (ι := σ) (⇑e : (a : σ) → τ)
⊢ Eq (α := MvPowerSeries σ R) ((killCompl e : (a : MvPowerSeries τ R) → MvPowerSeries σ R) (X t) : MvPowerSeries σ R)
    (0 : MvPowerSeries σ R)
have h₁ : single t 1 ∉ Set.range (embDomain e)
Tactic: replace h : single t 1 ∉ Set.range (embDomain e) := by
    rwa [mem_range_embDomain_iff, support_single _ (by simp), Finset.coe_singleton,
      Set.singleton_subset_iff]
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.RingTheory.MvPowerSeries.Rename (goal)
Goal:
σ : Type u_1
τ : Type u_2
R : Type u_4
inst : CommSemiring R
e : σ ↪ τ
⊢ Eq (α := MvPowerSeries σ R →ₐ[R] MvPowerSeries σ R) (AlgHom.comp (killCompl e) (rename (⇑e : (a : σ) → τ)))
    (AlgHom.id R (MvPowerSeries σ R))
have H : ∀ (x : MvPowerSeries σ R) (n : σ →₀ ℕ),
  Eq.{u_4 + 1} (α := R)
    ((coeff n : (a : MvPowerSeries σ R) → R)
        ((AlgHom.comp (killCompl e) (rename (⇑e : (a : σ) → τ)) : (a : MvPowerSeries σ R) → MvPowerSeries σ R) x :
          MvPowerSeries σ R) :
      R)
    ((coeff n : (a : MvPowerSeries σ R) → R)
        ((AlgHom.id R (MvPowerSeries σ R) : (a : MvPowerSeries σ R) → MvPowerSeries σ R) x : MvPowerSeries σ R) :
      R)
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.MvPowerSeries.Rename (goal)
Goal:
σ : Type u_1
τ : Type u_2
R : Type u_4
S : Type u_5
inst : CommSemiring R
inst_1 : CommSemiring S
e : σ ↪ τ
φ : R →+* S
p : MvPowerSeries τ R
⊢ Eq (α := MvPowerSeries σ S)
    ((killCompl e : (a : MvPowerSeries τ S) → MvPowerSeries σ S)
        ((map φ : (a : MvPowerSeries τ R) → MvPowerSeries τ S) p) :
      MvPowerSeries σ S)
    ((map φ : (a : MvPowerSeries σ R) → MvPowerSeries σ S)
        ((killCompl e : (a : MvPowerSeries τ R) → MvPowerSeries σ R) p) :
      MvPowerSeries σ S)
have h : ∀ (n : σ →₀ ℕ),
  Eq.{u_5 + 1} (α := S)
    ((coeff n : (a : MvPowerSeries σ S) → S)
        ((killCompl e : (a : MvPowerSeries τ S) → MvPowerSeries σ S)
            ((map φ : (a : MvPowerSeries τ R) → MvPowerSeries τ S) p) :
          MvPowerSeries σ S) :
      S)
    ((coeff n : (a : MvPowerSeries σ S) → S)
        ((map φ : (a : MvPowerSeries σ R) → MvPowerSeries σ S)
            ((killCompl e : (a : MvPowerSeries τ R) → MvPowerSeries σ R) p) :
          MvPowerSeries σ S) :
      S)
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.MvPowerSeries.Rename (goal)
Goal:
σ : Type u_1
τ : Type u_2
f : σ → τ
inst : TendstoCofinite f
R : Type u_6
inst_1 : CommRing R
d : τ →₀ ℕ
x : σ
⊢ Membership.mem (γ := Set σ)
      {s : σ | Ne.{u_6 + 1} (α := R) ((coeff d : (a : MvPowerSeries τ R) → R) ((X ∘ f) s) : R) (0 : R)} x →
    Membership.mem (γ := Set σ) (⋃ i ∈ (↑d.support : Set τ), f ⁻¹' {i}) x
have contrapose : x ∉ ⋃ i ∈ (↑d.support : Set τ), f ⁻¹' {i} →
  x ∉ {s : σ | Ne.{u_6 + 1} (α := R) ((coeff d : (a : MvPowerSeries τ R) → R) ((X ∘ f) s) : R) (0 : R)}
Tactic: contrapose
-- Lean 4 proof step from Mathlib.RingTheory.MvPowerSeries.Rename (goal)
Goal:
σ : Type u_1
τ : Type u_2
f : σ → τ
inst : TendstoCofinite f
R : Type u_6
inst_1 : CommRing R
p : MvPowerSeries σ R
⊢ Eq (α := MvPowerSeries τ R) ((rename f : (a : MvPowerSeries σ R) → MvPowerSeries τ R) p : MvPowerSeries τ R)
    (subst (X ∘ f) p : MvPowerSeries τ R)
have h : ∀ (n : τ →₀ ℕ),
  Eq.{u_6 + 1} (α := R)
    ((coeff n : (a : MvPowerSeries τ R) → R)
        ((rename f : (a : MvPowerSeries σ R) → MvPowerSeries τ R) p : MvPowerSeries τ R) :
      R)
    ((coeff n : (a : MvPowerSeries τ R) → R) (subst (X ∘ f) p) : R)
Tactic: ext n
-- Lean 4 proof step from Mathlib.RingTheory.MvPowerSeries.Rename (goal)
Goal:
σ : Type u_1
τ : Type u_2
f : σ → τ
inst : TendstoCofinite f
R : Type u_6
inst_1 : CommRing R
p : MvPowerSeries σ R
n : τ →₀ ℕ
⊢ Eq.{u_6 + 1} (α := R)
    ((coeff n : (a : MvPowerSeries τ R) → R)
        ((rename f : (a : MvPowerSeries σ R) → MvPowerSeries τ R) p : MvPowerSeries τ R) :
      R)
    ((coeff n : (a : MvPowerSeries τ R) → R) (subst (X ∘ f) p) : R)
have h : Eq.{u_6 + 1} (α := R)
  (∑ x ∈ Set.Finite.toFinset (s := mapDomain f ⁻¹' {n}) sorry, (coeff x : (a : MvPowerSeries σ R) → R) p)
  (∑
    i ∈
      Set.Finite.toFinset (s :=
        Function.support.{u_1, u_6} (M := R) fun (d : σ →₀ ℕ) ↦
          HSMul.hSMul (α := R) (β := R) ((coeff d : (a : MvPowerSeries σ R) → R) p : R)
            ((coeff n : (a : MvPowerSeries τ R) → R)
                (d.prod fun (s : σ) (e : ℕ) ↦ HPow.hPow (α := MvPowerSeries τ R) ((X ∘ f) s) e) :
              R))
        sorry,
    HSMul.hSMul (α := R) (β := R) ((coeff i : (a : MvPowerSeries σ R) → R) p : R)
      ((coeff n : (a : MvPowerSeries τ R) → R)
          (i.prod fun (s : σ) (e : ℕ) ↦ HPow.hPow (α := MvPowerSeries τ R) ((X ∘ f) s) e) :
        R))
Tactic: rw [coeff_rename, coeff_subst (HasSubst.X_comp _) p n, finsum_eq_sum _
    (coeff_subst_finite (HasSubst.X_comp _) p n)]
-- Lean 4 proof step from Mathlib.RingTheory.MvPowerSeries.Rename (hypothesis)
Goal:
σ : Type u_1
τ : Type u_2
f : σ → τ
inst : TendstoCofinite f
R : Type u_6
inst_1 : CommRing R
p : MvPowerSeries σ R
n : τ →₀ ℕ
d : σ →₀ ℕ
hd :
  HMul.hMul (α := R) (β := R) ((coeff d : (a : MvPowerSeries σ R) → R) p : R)
      ((coeff n : (a : MvPowerSeries τ R) → R)
          (d.prod fun (s : σ) (e : ℕ) ↦ HPow.hPow (α := MvPowerSeries τ R) (X (f s)) e) :
        R) ≠
    0
⊢ mapDomain f d = n
have hd₁ : HMul.hMul (α := R) (β := R) ((coeff d : (a : MvPowerSeries σ R) → R) p : R)
    ((coeff n : (a : MvPowerSeries τ R) → R) ((monomial (mapDomain f d) : (a : R) → MvPowerSeries τ R) 1) : R) ≠
  0
Tactic: simp_rw [← monomial_mapDomain_apply_one] at hd
Cleared hypotheses: hd
-- Lean 4 proof step from Mathlib.RingTheory.MvPowerSeries.Rename (hypothesis)
Goal:
σ : Type u_1
τ : Type u_2
f : σ → τ
inst : TendstoCofinite f
R : Type u_6
inst_1 : CommRing R
p : MvPowerSeries σ R
n : τ →₀ ℕ
⊢ Eq.{u_6 + 1} (α := R)
    (∑ x ∈ Set.Finite.toFinset (s := mapDomain f ⁻¹' {n}) ⋯, (coeff x : (a : MvPowerSeries σ R) → R) p)
    (∑
      i ∈
        Set.Finite.toFinset (s :=
          Function.support.{u_1, u_6} (M := R) fun (d : σ →₀ ℕ) ↦
            HSMul.hSMul (α := R) (β := R) ((coeff d : (a : MvPowerSeries σ R) → R) p : R)
              ((coeff n : (a : MvPowerSeries τ R) → R)
                  (d.prod fun (s : σ) (e : ℕ) ↦ HPow.hPow (α := MvPowerSeries τ R) ((X ∘ f) s) e) :
                R))
          ⋯,
      HSMul.hSMul (α := R) (β := R) ((coeff i : (a : MvPowerSeries σ R) → R) p : R)
        ((coeff n : (a : MvPowerSeries τ R) → R)
            (i.prod fun (s : σ) (e : ℕ) ↦ HPow.hPow (α := MvPowerSeries τ R) ((X ∘ f) s) e) :
          R))
have this : ∀ (d : σ →₀ ℕ)
  (hd :
    HMul.hMul (α := R) (β := R) ((coeff d : (a : MvPowerSeries σ R) → R) p : R)
        ((coeff n : (a : MvPowerSeries τ R) → R)
            (d.prod fun (s : σ) (e : ℕ) ↦ HPow.hPow (α := MvPowerSeries τ R) (X (f s)) e) :
          R) ≠
      0),
  mapDomain f d = n
Tactic: have (d : σ →₀ ℕ) (hd : (coeff d) p * (coeff n) (d.prod fun s e ↦ X (f s) ^ e) ≠ 0) :
      mapDomain f d = n := by
    simp_rw [← monomial_mapDomain_apply_one] at hd
    exact (eq_of_coeff_monomial_ne_zero (right_ne_zero_of_mul hd)).symm
-- Lean 4 proof step from Mathlib.RingTheory.MvPowerSeries.Rename (goal)
Goal:
σ : Type u_1
τ : Type u_2
f : σ → τ
inst : TendstoCofinite f
R : Type u_6
inst_1 : CommRing R
p : MvPowerSeries σ R
n : τ →₀ ℕ
this :
  ∀ (d : σ →₀ ℕ)
    (hd :
      HMul.hMul (α := R) (β := R) ((coeff d : (a : MvPowerSeries σ R) → R) p : R)
          ((coeff n : (a : MvPowerSeries τ R) → R)
              (d.prod fun (s : σ) (e : ℕ) ↦ HPow.hPow (α := MvPowerSeries τ R) (X (f s)) e) :
            R) ≠
        0),
    mapDomain f d = n
⊢ Eq.{u_6 + 1} (α := R)
    (∑ x ∈ Set.Finite.toFinset (s := mapDomain f ⁻¹' {n}) ⋯, (coeff x : (a : MvPowerSeries σ R) → R) p)
    (∑
      i ∈
        Set.Finite.toFinset (s :=
          Function.support.{u_1, u_6} (M := R) fun (d : σ →₀ ℕ) ↦
            HSMul.hSMul (α := R) (β := R) ((coeff d : (a : MvPowerSeries σ R) → R) p : R)
              ((coeff n : (a : MvPowerSeries τ R) → R)
                  (d.prod fun (s : σ) (e : ℕ) ↦ HPow.hPow (α := MvPowerSeries τ R) ((X ∘ f) s) e) :
                R))
          ⋯,
      HSMul.hSMul (α := R) (β := R) ((coeff i : (a : MvPowerSeries σ R) → R) p : R)
        ((coeff n : (a : MvPowerSeries τ R) → R)
            (i.prod fun (s : σ) (e : ℕ) ↦ HPow.hPow (α := MvPowerSeries τ R) ((X ∘ f) s) e) :
          R))
have h : Set.Finite.toFinset (s :=
    Function.support.{u_1, u_6} (M := R) fun (d : σ →₀ ℕ) ↦
      HSMul.hSMul (α := R) (β := R) ((coeff d : (a : MvPowerSeries σ R) → R) p : R)
        ((coeff n : (a : MvPowerSeries τ R) → R)
            (d.prod fun (s : σ) (e : ℕ) ↦ HPow.hPow (α := MvPowerSeries τ R) ((X ∘ f) s) e) :
          R))
    sorry ⊆
  Set.Finite.toFinset (s := mapDomain f ⁻¹' {n}) sorry
Tactic: refine (Finset.sum_subset_zero_on_sdiff ?_ ?_ (fun x hx => ?_)).symm
-- Lean 4 proof step from Mathlib.RingTheory.MvPowerSeries.Rename (goal)
Goal:
σ : Type u_1
τ : Type u_2
f : σ → τ
inst : TendstoCofinite f
R : Type u_6
inst_1 : CommRing R
p : MvPowerSeries σ R
n : τ →₀ ℕ
this :
  ∀ (d : σ →₀ ℕ)
    (hd :
      HMul.hMul (α := R) (β := R) ((coeff d : (a : MvPowerSeries σ R) → R) p : R)
          ((coeff n : (a : MvPowerSeries τ R) → R)
              (d.prod fun (s : σ) (e : ℕ) ↦ HPow.hPow (α := MvPowerSeries τ R) (X (f s)) e) :
            R) ≠
        0),
    mapDomain f d = n
h :
  Set.Finite.toFinset (s :=
      Function.support.{u_1, u_6} (M := R) fun (d : σ →₀ ℕ) ↦
        HSMul.hSMul (α := R) (β := R) ((coeff d : (a : MvPowerSeries σ R) → R) p : R)
          ((coeff n : (a : MvPowerSeries τ R) → R)
              (d.prod fun (s : σ) (e : ℕ) ↦ HPow.hPow (α := MvPowerSeries τ R) ((X ∘ f) s) e) :
            R))
      ⋯ ⊆
    Set.Finite.toFinset (s := mapDomain f ⁻¹' {n}) ⋯
⊢ Eq.{u_6 + 1} (α := R)
    (∑ x ∈ Set.Finite.toFinset (s := mapDomain f ⁻¹' {n}) ⋯, (coeff x : (a : MvPowerSeries σ R) → R) p)
    (∑
      i ∈
        Set.Finite.toFinset (s :=
          Function.support.{u_1, u_6} (M := R) fun (d : σ →₀ ℕ) ↦
            HSMul.hSMul (α := R) (β := R) ((coeff d : (a : MvPowerSeries σ R) → R) p : R)
              ((coeff n : (a : MvPowerSeries τ R) → R)
                  (d.prod fun (s : σ) (e : ℕ) ↦ HPow.hPow (α := MvPowerSeries τ R) ((X ∘ f) s) e) :
                R))
          ⋯,
      HSMul.hSMul (α := R) (β := R) ((coeff i : (a : MvPowerSeries σ R) → R) p : R)
        ((coeff n : (a : MvPowerSeries τ R) → R)
            (i.prod fun (s : σ) (e : ℕ) ↦ HPow.hPow (α := MvPowerSeries τ R) ((X ∘ f) s) e) :
          R))
have h₁ : ∀
  x ∈
    Set.Finite.toFinset (s := mapDomain f ⁻¹' {n}) sorry \
      Set.Finite.toFinset (s :=
        Function.support.{u_1, u_6} (M := R) fun (d : σ →₀ ℕ) ↦
          HSMul.hSMul (α := R) (β := R) ((coeff d : (a : MvPowerSeries σ R) → R) p : R)
            ((coeff n : (a : MvPowerSeries τ R) → R)
                (d.prod fun (s : σ) (e : ℕ) ↦ HPow.hPow (α := MvPowerSeries τ R) ((X ∘ f) s) e) :
              R))
        sorry,
  Eq.{u_6 + 1} (α := R) ((coeff x : (a : MvPowerSeries σ R) → R) p : R) (0 : R)
Tactic: refine (Finset.sum_subset_zero_on_sdiff ?_ ?_ (fun x hx => ?_)).symm
-- Lean 4 proof step from Mathlib.RingTheory.MvPowerSeries.Rename (goal)
Goal:
σ : Type u_1
τ : Type u_2
f : σ → τ
inst : TendstoCofinite f
R : Type u_6
inst_1 : CommRing R
p : MvPowerSeries σ R
n : τ →₀ ℕ
this :
  ∀ (d : σ →₀ ℕ)
    (hd :
      HMul.hMul (α := R) (β := R) ((coeff d : (a : MvPowerSeries σ R) → R) p : R)
          ((coeff n : (a : MvPowerSeries τ R) → R)
              (d.prod fun (s : σ) (e : ℕ) ↦ HPow.hPow (α := MvPowerSeries τ R) (X (f s)) e) :
            R) ≠
        0),
    mapDomain f d = n
h :
  Set.Finite.toFinset (s :=
      Function.support.{u_1, u_6} (M := R) fun (d : σ →₀ ℕ) ↦
        HSMul.hSMul (α := R) (β := R) ((coeff d : (a : MvPowerSeries σ R) → R) p : R)
          ((coeff n : (a : MvPowerSeries τ R) → R)
              (d.prod fun (s : σ) (e : ℕ) ↦ HPow.hPow (α := MvPowerSeries τ R) ((X ∘ f) s) e) :
            R))
      ⋯ ⊆
    Set.Finite.toFinset (s := mapDomain f ⁻¹' {n}) ⋯
h₁ :
  ∀
    x ∈
      Set.Finite.toFinset (s := mapDomain f ⁻¹' {n}) ⋯ \
        Set.Finite.toFinset (s :=
          Function.support.{u_1, u_6} (M := R) fun (d : σ →₀ ℕ) ↦
            HSMul.hSMul (α := R) (β := R) ((coeff d : (a : MvPowerSeries σ R) → R) p : R)
              ((coeff n : (a : MvPowerSeries τ R) → R)
                  (d.prod fun (s : σ) (e : ℕ) ↦ HPow.hPow (α := MvPowerSeries τ R) ((X ∘ f) s) e) :
                R))
          ⋯,
    Eq.{u_6 + 1} (α := R) ((coeff x : (a : MvPowerSeries σ R) → R) p : R) (0 : R)
⊢ Eq.{u_6 + 1} (α := R)
    (∑ x ∈ Set.Finite.toFinset (s := mapDomain f ⁻¹' {n}) ⋯, (coeff x : (a : MvPowerSeries σ R) → R) p)
    (∑
      i ∈
        Set.Finite.toFinset (s :=
          Function.support.{u_1, u_6} (M := R) fun (d : σ →₀ ℕ) ↦
            HSMul.hSMul (α := R) (β := R) ((coeff d : (a : MvPowerSeries σ R) → R) p : R)
              ((coeff n : (a : MvPowerSeries τ R) → R)
                  (d.prod fun (s : σ) (e : ℕ) ↦ HPow.hPow (α := MvPowerSeries τ R) ((X ∘ f) s) e) :
                R))
          ⋯,
      HSMul.hSMul (α := R) (β := R) ((coeff i : (a : MvPowerSeries σ R) → R) p : R)
        ((coeff n : (a : MvPowerSeries τ R) → R)
            (i.prod fun (s : σ) (e : ℕ) ↦ HPow.hPow (α := MvPowerSeries τ R) ((X ∘ f) s) e) :
          R))
have h₂ : ∀ (x : σ →₀ ℕ)
  (hx :
    x ∈
      Set.Finite.toFinset (s :=
        Function.support.{u_1, u_6} (M := R) fun (d : σ →₀ ℕ) ↦
          HSMul.hSMul (α := R) (β := R) ((coeff d : (a : MvPowerSeries σ R) → R) p : R)
            ((coeff n : (a : MvPowerSeries τ R) → R)
                (d.prod fun (s : σ) (e : ℕ) ↦ HPow.hPow (α := MvPowerSeries τ R) ((X ∘ f) s) e) :
              R))
        sorry),
  HSMul.hSMul (α := R) (β := R) ((coeff x : (a : MvPowerSeries σ R) → R) p : R)
      ((coeff n : (a : MvPowerSeries τ R) → R)
          (x.prod fun (s : σ) (e : ℕ) ↦ HPow.hPow (α := MvPowerSeries τ R) ((X ∘ f) s) e) :
        R) =
    (coeff x : (a : MvPowerSeries σ R) → R) p
Tactic: refine (Finset.sum_subset_zero_on_sdiff ?_ ?_ (fun x hx => ?_)).symm
-- Lean 4 proof step from Mathlib.RingTheory.MvPowerSeries.Rename (hypothesis)
Goal:
σ : Type u_1
τ : Type u_2
f : σ → τ
inst : TendstoCofinite f
R : Type u_6
inst_1 : CommRing R
p : MvPowerSeries σ R
n : τ →₀ ℕ
this :
  ∀ (d : σ →₀ ℕ)
    (hd :
      HMul.hMul (α := R) (β := R) ((coeff d : (a : MvPowerSeries σ R) → R) p : R)
          ((coeff n : (a : MvPowerSeries τ R) → R)
              (d.prod fun (s : σ) (e : ℕ) ↦ HPow.hPow (α := MvPowerSeries τ R) (X (f s)) e) :
            R) ≠
        0),
    mapDomain f d = n
x : σ →₀ ℕ
hx :
  x ∈
    Set.Finite.toFinset (s :=
      Function.support.{u_1, u_6} (M := R) fun (d : σ →₀ ℕ) ↦
        HSMul.hSMul (α := R) (β := R) ((coeff d : (a : MvPowerSeries σ R) → R) p : R)
          ((coeff n : (a : MvPowerSeries τ R) → R)
              (d.prod fun (s : σ) (e : ℕ) ↦ HPow.hPow (α := MvPowerSeries τ R) ((X ∘ f) s) e) :
            R))
      ⋯
⊢ HSMul.hSMul (α := R) (β := R) ((coeff x : (a : MvPowerSeries σ R) → R) p : R)
      ((coeff n : (a : MvPowerSeries τ R) → R)
          (x.prod fun (s : σ) (e : ℕ) ↦ HPow.hPow (α := MvPowerSeries τ R) ((X ∘ f) s) e) :
        R) =
    (coeff x : (a : MvPowerSeries σ R) → R) p
have hx₁ : Membership.mem (γ := Set (σ →₀ ℕ))
  (Function.support.{u_1, u_6} (M := R) fun (d : σ →₀ ℕ) ↦
    HSMul.hSMul (α := R) (β := R) ((coeff d : (a : MvPowerSeries σ R) → R) p : R)
      ((coeff n : (a : MvPowerSeries τ R) → R)
          (d.prod fun (s : σ) (e : ℕ) ↦ HPow.hPow (α := MvPowerSeries τ R) ((X ∘ f) s) e) :
        R))
  x
Tactic: simp only [Set.Finite.mem_toFinset] at hx
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.RingTheory.MvPowerSeries.Rename (goal)
Goal:
σ : Type u_1
τ : Type u_2
f : σ → τ
inst : TendstoCofinite f
R : Type u_6
inst_1 : CommRing R
p : MvPowerSeries σ R
⊢ Eq (α := MvPowerSeries τ R) ((rename f : (a : MvPowerSeries σ R) → MvPowerSeries τ R) p : MvPowerSeries τ R)
    (subst (X ∘ f) p : MvPowerSeries τ R)
have h : ∀ (n : τ →₀ ℕ),
  Eq.{u_6 + 1} (α := R)
    ((coeff n : (a : MvPowerSeries τ R) → R)
        ((rename f : (a : MvPowerSeries σ R) → MvPowerSeries τ R) p : MvPowerSeries τ R) :
      R)
    ((coeff n : (a : MvPowerSeries τ R) → R) (subst (X ∘ f) p) : R)
Tactic: classical
  ext n
  rw [coeff_rename, coeff_subst (HasSubst.X_comp _) p n, finsum_eq_sum _
    (coeff_subst_finite (HasSubst.X_comp _) p n)]
  have (d : σ →₀ ℕ) (hd : (coeff d) p * (coeff n) (d.prod fun s e ↦ X (f s) ^ e) ≠ 0) :
      mapDomain f d = n := by
    simp_rw [← monomial_mapDomain_apply_one] at hd
    exact (eq_of_coeff_monomial_ne_zero (right_ne_zero_of_mul hd)).symm
  refine (Finset.sum_subset_zero_on_sdiff ?_ ?_ (fun x hx => ?_)).symm
  · exact Set.Finite.toFinset_mono this
  · simp +contextual [← monomial_mapDomain_apply_one]
  · simp only [Set.Finite.mem_toFinset] at hx
    simp [← this _ hx, ← monomial_mapDomain_apply_one, coeff_monomial_same]
-- Lean 4 proof step from Mathlib.RingTheory.MvPowerSeries.Rename (goal)
Goal:
σ : Type u_1
τ : Type u_2
f : σ → τ
inst : TendstoCofinite f
R : Type u_6
inst_1 : CommRing R
p : MvPowerSeries σ R
h :
  ∀ (n : τ →₀ ℕ),
    Eq.{u_6 + 1} (α := R)
      ((coeff n : (a : MvPowerSeries τ R) → R)
          ((rename f : (a : MvPowerSeries σ R) → MvPowerSeries τ R) p : MvPowerSeries τ R) :
        R)
      ((coeff n : (a : MvPowerSeries τ R) → R) (subst (X ∘ f) p) : R)
⊢ Eq (α := MvPowerSeries τ R) ((rename f : (a : MvPowerSeries σ R) → MvPowerSeries τ R) p : MvPowerSeries τ R)
    (subst (X ∘ f) p : MvPowerSeries τ R)
have h₁ : ∀ (n : τ →₀ ℕ),
  Eq.{u_6 + 1} (α := R)
    (∑ x ∈ Set.Finite.toFinset (s := mapDomain f ⁻¹' {n}) sorry, (coeff x : (a : MvPowerSeries σ R) → R) p)
    (∑
      i ∈
        Set.Finite.toFinset (s :=
          Function.support.{u_1, u_6} (M := R) fun (d : σ →₀ ℕ) ↦
            HSMul.hSMul (α := R) (β := R) ((coeff d : (a : MvPowerSeries σ R) → R) p : R)
              ((coeff n : (a : MvPowerSeries τ R) → R)
                  (d.prod fun (s : σ) (e : ℕ) ↦ HPow.hPow (α := MvPowerSeries τ R) ((X ∘ f) s) e) :
                R))
          sorry,
      HSMul.hSMul (α := R) (β := R) ((coeff i : (a : MvPowerSeries σ R) → R) p : R)
        ((coeff n : (a : MvPowerSeries τ R) → R)
            (i.prod fun (s : σ) (e : ℕ) ↦ HPow.hPow (α := MvPowerSeries τ R) ((X ∘ f) s) e) :
          R))
Tactic: classical
  ext n
  rw [coeff_rename, coeff_subst (HasSubst.X_comp _) p n, finsum_eq_sum _
    (coeff_subst_finite (HasSubst.X_comp _) p n)]
  have (d : σ →₀ ℕ) (hd : (coeff d) p * (coeff n) (d.prod fun s e ↦ X (f s) ^ e) ≠ 0) :
      mapDomain f d = n := by
    simp_rw [← monomial_mapDomain_apply_one] at hd
    exact (eq_of_coeff_monomial_ne_zero (right_ne_zero_of_mul hd)).symm
  refine (Finset.sum_subset_zero_on_sdiff ?_ ?_ (fun x hx => ?_)).symm
  · exact Set.Finite.toFinset_mono this
  · simp +contextual [← monomial_mapDomain_apply_one]
  · simp only [Set.Finite.mem_toFinset] at hx
    simp [← this _ hx, ← monomial_mapDomain_apply_one, coeff_monomial_same]
-- Lean 4 proof step from Mathlib.RingTheory.MvPowerSeries.Substitution (goal)
Goal:
σ : Type u_1
τ : Type u_4
S : Type u_5
inst : CommRing S
a : σ → MvPowerSeries τ S
inst_1 : TopologicalSpace S
inst_2 : DiscreteTopology S
⊢ HasSubst a ↔ HasEval a
have simp_rw : ((∀ (s : σ), IsNilpotent.{u_5} (R := S) ((constantCoeff : (a : MvPowerSeries τ S) → S) (a s) : S)) ∧
    ∀ (d : τ →₀ ℕ),
      Set.Finite.{u_1} (α := σ)
        {s : σ | Ne.{u_5 + 1} (α := S) ((coeff d : (a : MvPowerSeries τ S) → S) (a s) : S) (0 : S)}) ↔
  HasEval a
Tactic: simp_rw [hasSubst_def, hasEval_def, coeff_zero_iff,
    isTopologicallyNilpotent_iff_constantCoeff_isNilpotent]
-- Lean 4 proof step from Mathlib.RingTheory.MvPowerSeries.Substitution (goal)
Goal:
σ : Type u_1
τ : Type u_4
S : Type u_5
inst : CommRing S
a : σ → MvPowerSeries τ S
inst_1 : TopologicalSpace S
inst_2 : DiscreteTopology S
simp_rw :
  ((∀ (s : σ), IsNilpotent.{u_5} (R := S) ((constantCoeff : (a : MvPowerSeries τ S) → S) (a s) : S)) ∧
      ∀ (d : τ →₀ ℕ),
        Set.Finite.{u_1} (α := σ)
          {s : σ | Ne.{u_5 + 1} (α := S) ((coeff d : (a : MvPowerSeries τ S) → S) (a s) : S) (0 : S)}) ↔
    HasEval a
⊢ HasSubst a ↔ HasEval a
have simp_rw₁ : ((∀ (s : σ), IsNilpotent.{u_5} (R := S) ((constantCoeff : (a : MvPowerSeries τ S) → S) (a s) : S)) ∧
    ∀ (d : τ →₀ ℕ),
      Set.Finite.{u_1} (α := σ)
        {s : σ | Ne.{u_5 + 1} (α := S) ((coeff d : (a : MvPowerSeries τ S) → S) (a s) : S) (0 : S)}) ↔
  (∀ (s : σ), IsTopologicallyNilpotent (a s)) ∧ Filter.Tendsto a Filter.cofinite (nhds 0)
Tactic: simp_rw [hasSubst_def, hasEval_def, coeff_zero_iff,
    isTopologicallyNilpotent_iff_constantCoeff_isNilpotent]