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
f : σ → τ
inst : TendstoCofinite f
inst_1 : DecidableEq τ
x : τ →₀ ℕ
hs :
  (↑((@Finset.antidiagonal _ Finsupp.instAddMonoid _ x).sup fun (q : (τ →₀ ℕ) × (τ →₀ ℕ)) ↦
          Finset.product {q}
            (Set.Finite.toFinset (s := mapDomain f ⁻¹' {q.1}) ⋯ ×ˢ
              Set.Finite.toFinset (s := mapDomain f ⁻¹' {q.2}) ⋯)) :
      Set (((τ →₀ ℕ) × (τ →₀ ℕ)) × (σ →₀ ℕ) × (σ →₀ ℕ))).Finite
⊢ Set.Finite (α := Prod.{u_2, u_1} ((τ →₀ ℕ) × (τ →₀ ℕ)) ((σ →₀ ℕ) × (σ →₀ ℕ)))
    {p : Prod.{u_2, u_1} ((τ →₀ ℕ) × (τ →₀ ℕ)) ((σ →₀ ℕ) × (σ →₀ ℕ)) |
      p.1 ∈ @Finset.antidiagonal _ Finsupp.instAddMonoid _ x ∧
        p.2 ∈
          Set.Finite.toFinset (s := mapDomain f ⁻¹' {p.1.1}) ⋯ ×ˢ Set.Finite.toFinset (s := mapDomain f ⁻¹' {p.1.2}) ⋯}
have ht : {p : Prod.{u_2, u_1} ((τ →₀ ℕ) × (τ →₀ ℕ)) ((σ →₀ ℕ) × (σ →₀ ℕ)) |
    p.1 ∈ @Finset.antidiagonal _ Finsupp.instAddMonoid _ x ∧
      p.2 ∈
        Set.Finite.toFinset (s := mapDomain f ⁻¹' {p.1.1}) sorry ×ˢ Set.Finite.toFinset (s := mapDomain f ⁻¹' {p.1.2}) sorry} ⊆
  (↑((@Finset.antidiagonal _ Finsupp.instAddMonoid _ x).sup fun (q : (τ →₀ ℕ) × (τ →₀ ℕ)) ↦
        Finset.product {q}
          (Set.Finite.toFinset (s := mapDomain f ⁻¹' {q.1}) sorry ×ˢ Set.Finite.toFinset (s := mapDomain f ⁻¹' {q.2}) sorry)) :
    Set (((τ →₀ ℕ) × (τ →₀ ℕ)) × (σ →₀ ℕ) × (σ →₀ ℕ)))
Tactic: apply Set.Finite.subset (s := ↑((Finset.antidiagonal x).sup (fun q ↦ Finset.product {q}
    ((TendstoCofinite.finite_preimage_singleton (Finsupp.mapDomain f) q.1).toFinset ×ˢ
      (TendstoCofinite.finite_preimage_singleton (Finsupp.mapDomain f) q.2).toFinset))))
-- 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
inst_1 : DecidableEq τ
x : τ →₀ ℕ
a : Prod.{u_2, u_1} ((τ →₀ ℕ) × (τ →₀ ℕ)) ((σ →₀ ℕ) × (σ →₀ ℕ))
⊢ Membership.mem (γ := Set (Prod.{u_2, u_1} ((τ →₀ ℕ) × (τ →₀ ℕ)) ((σ →₀ ℕ) × (σ →₀ ℕ))))
      {p : Prod.{u_2, u_1} ((τ →₀ ℕ) × (τ →₀ ℕ)) ((σ →₀ ℕ) × (σ →₀ ℕ)) |
        p.1 ∈ @Finset.antidiagonal _ Finsupp.instAddMonoid _ x ∧
          p.2 ∈
            Set.Finite.toFinset (s := mapDomain f ⁻¹' {p.1.1}) ⋯ ×ˢ
              Set.Finite.toFinset (s := mapDomain f ⁻¹' {p.1.2}) ⋯}
      a →
    a ∈
      (↑((@Finset.antidiagonal _ Finsupp.instAddMonoid _ x).sup fun (q : (τ →₀ ℕ) × (τ →₀ ℕ)) ↦
            Finset.product {q}
              (Set.Finite.toFinset (s := mapDomain f ⁻¹' {q.1}) ⋯ ×ˢ
                Set.Finite.toFinset (s := mapDomain f ⁻¹' {q.2}) ⋯)) :
        Set (((τ →₀ ℕ) × (τ →₀ ℕ)) × (σ →₀ ℕ) × (σ →₀ ℕ)))
have ht : a.1.1 + a.1.2 = x →
  mapDomain f a.2.1 = a.1.1 →
    mapDomain f a.2.2 = a.1.2 →
      ∃ (a_4 : σ →₀ ℕ) (b : σ →₀ ℕ),
        mapDomain f a_4 + mapDomain f b = x ∧ ((mapDomain f a_4, mapDomain f b), a_4, b) = a
Tactic: simp
-- 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
inst_1 : DecidableEq τ
x : τ →₀ ℕ
⊢ Set.Finite (α := Prod.{u_2, u_1} ((τ →₀ ℕ) × (τ →₀ ℕ)) ((σ →₀ ℕ) × (σ →₀ ℕ)))
    {p : Prod.{u_2, u_1} ((τ →₀ ℕ) × (τ →₀ ℕ)) ((σ →₀ ℕ) × (σ →₀ ℕ)) |
      p.1 ∈ @Finset.antidiagonal _ Finsupp.instAddMonoid _ x ∧
        p.2 ∈
          Set.Finite.toFinset (s := mapDomain f ⁻¹' {p.1.1}) ⋯ ×ˢ Set.Finite.toFinset (s := mapDomain f ⁻¹' {p.1.2}) ⋯}
have hs : (↑((@Finset.antidiagonal _ Finsupp.instAddMonoid _ x).sup fun (q : (τ →₀ ℕ) × (τ →₀ ℕ)) ↦
        Finset.product {q}
          (Set.Finite.toFinset (s := mapDomain f ⁻¹' {q.1}) sorry ×ˢ Set.Finite.toFinset (s := mapDomain f ⁻¹' {q.2}) sorry)) :
    Set (((τ →₀ ℕ) × (τ →₀ ℕ)) × (σ →₀ ℕ) × (σ →₀ ℕ))).Finite
Tactic: classical
  apply Set.Finite.subset (s := ↑((Finset.antidiagonal x).sup (fun q ↦ Finset.product {q}
    ((TendstoCofinite.finite_preimage_singleton (Finsupp.mapDomain f) q.1).toFinset ×ˢ
      (TendstoCofinite.finite_preimage_singleton (Finsupp.mapDomain f) q.2).toFinset))))
  · exact Finset.finite_toSet ..
  · intro; simp
    grind
-- 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
inst_1 : DecidableEq τ
x : τ →₀ ℕ
hs :
  (↑((@Finset.antidiagonal _ Finsupp.instAddMonoid _ x).sup fun (q : (τ →₀ ℕ) × (τ →₀ ℕ)) ↦
          Finset.product {q}
            (Set.Finite.toFinset (s := mapDomain f ⁻¹' {q.1}) ⋯ ×ˢ
              Set.Finite.toFinset (s := mapDomain f ⁻¹' {q.2}) ⋯)) :
      Set (((τ →₀ ℕ) × (τ →₀ ℕ)) × (σ →₀ ℕ) × (σ →₀ ℕ))).Finite
⊢ Set.Finite (α := Prod.{u_2, u_1} ((τ →₀ ℕ) × (τ →₀ ℕ)) ((σ →₀ ℕ) × (σ →₀ ℕ)))
    {p : Prod.{u_2, u_1} ((τ →₀ ℕ) × (τ →₀ ℕ)) ((σ →₀ ℕ) × (σ →₀ ℕ)) |
      p.1 ∈ @Finset.antidiagonal _ Finsupp.instAddMonoid _ x ∧
        p.2 ∈
          Set.Finite.toFinset (s := mapDomain f ⁻¹' {p.1.1}) ⋯ ×ˢ Set.Finite.toFinset (s := mapDomain f ⁻¹' {p.1.2}) ⋯}
have ht : {p : Prod.{u_2, u_1} ((τ →₀ ℕ) × (τ →₀ ℕ)) ((σ →₀ ℕ) × (σ →₀ ℕ)) |
    p.1 ∈ @Finset.antidiagonal _ Finsupp.instAddMonoid _ x ∧
      p.2 ∈
        Set.Finite.toFinset (s := mapDomain f ⁻¹' {p.1.1}) sorry ×ˢ Set.Finite.toFinset (s := mapDomain f ⁻¹' {p.1.2}) sorry} ⊆
  (↑((@Finset.antidiagonal _ Finsupp.instAddMonoid _ x).sup fun (q : (τ →₀ ℕ) × (τ →₀ ℕ)) ↦
        Finset.product {q}
          (Set.Finite.toFinset (s := mapDomain f ⁻¹' {q.1}) sorry ×ˢ Set.Finite.toFinset (s := mapDomain f ⁻¹' {q.2}) sorry)) :
    Set (((τ →₀ ℕ) × (τ →₀ ℕ)) × (σ →₀ ℕ) × (σ →₀ ℕ)))
Tactic: classical
  apply Set.Finite.subset (s := ↑((Finset.antidiagonal x).sup (fun q ↦ Finset.product {q}
    ((TendstoCofinite.finite_preimage_singleton (Finsupp.mapDomain f) q.1).toFinset ×ˢ
      (TendstoCofinite.finite_preimage_singleton (Finsupp.mapDomain f) q.2).toFinset))))
  · exact Finset.finite_toSet ..
  · intro; simp
    grind
-- 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
inst_1 : DecidableEq σ
inst_2 : DecidableEq τ
x : τ →₀ ℕ
⊢ Finset.image
      (fun (x : Prod.{u_2, u_1} ((τ →₀ ℕ) × (τ →₀ ℕ)) ((σ →₀ ℕ) × (σ →₀ ℕ))) ↦
        match (motive := Prod.{u_2, u_1} ((τ →₀ ℕ) × (τ →₀ ℕ)) ((σ →₀ ℕ) × (σ →₀ ℕ)) → (σ →₀ ℕ) × (σ →₀ ℕ) × (σ →₀ ℕ))
          x with
        | (fst, b) => (b.1 + b.2, b))
      (Set.Finite.toFinset (s :=
        {p : Prod.{u_2, u_1} ((τ →₀ ℕ) × (τ →₀ ℕ)) ((σ →₀ ℕ) × (σ →₀ ℕ)) |
          p.1 ∈ @Finset.antidiagonal _ Finsupp.instAddMonoid _ x ∧
            p.2 ∈
              Set.Finite.toFinset (s := mapDomain f ⁻¹' {p.1.1}) ⋯ ×ˢ
                Set.Finite.toFinset (s := mapDomain f ⁻¹' {p.1.2}) ⋯})
        ⋯) =
    Set.Finite.toFinset (s :=
      {p : (σ →₀ ℕ) × (σ →₀ ℕ) × (σ →₀ ℕ) |
        mapDomain f p.1 = x ∧ p.2 ∈ @Finset.antidiagonal _ Finsupp.instAddMonoid _ p.1})
      ⋯
have h : ∀ (fst fst_1 snd : σ →₀ ℕ),
  Membership.mem (γ := Finset ((σ →₀ ℕ) × (σ →₀ ℕ) × (σ →₀ ℕ)))
      (Finset.image
        (fun (x : Prod.{u_2, u_1} ((τ →₀ ℕ) × (τ →₀ ℕ)) ((σ →₀ ℕ) × (σ →₀ ℕ))) ↦
          match (motive := Prod.{u_2, u_1} ((τ →₀ ℕ) × (τ →₀ ℕ)) ((σ →₀ ℕ) × (σ →₀ ℕ)) → (σ →₀ ℕ) × (σ →₀ ℕ) × (σ →₀ ℕ))
            x with
          | (fst, b) => (b.1 + b.2, b))
        (Set.Finite.toFinset (s :=
          {p : Prod.{u_2, u_1} ((τ →₀ ℕ) × (τ →₀ ℕ)) ((σ →₀ ℕ) × (σ →₀ ℕ)) |
            p.1 ∈ @Finset.antidiagonal _ Finsupp.instAddMonoid _ x ∧
              p.2 ∈
                Set.Finite.toFinset (s := mapDomain f ⁻¹' {p.1.1}) sorry ×ˢ
                  Set.Finite.toFinset (s := mapDomain f ⁻¹' {p.1.2}) sorry})
          sorry))
      (fst, fst_1, snd) ↔
    (fst, fst_1, snd) ∈
      Set.Finite.toFinset (s :=
        {p : (σ →₀ ℕ) × (σ →₀ ℕ) × (σ →₀ ℕ) |
          mapDomain f p.1 = x ∧ p.2 ∈ @Finset.antidiagonal _ Finsupp.instAddMonoid _ p.1})
        sorry
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
inst_1 : DecidableEq σ
inst_2 : DecidableEq τ
x : τ →₀ ℕ
fst fst_1 snd : σ →₀ ℕ
⊢ Membership.mem (γ := Finset ((σ →₀ ℕ) × (σ →₀ ℕ) × (σ →₀ ℕ)))
      (Finset.image
        (fun (x : Prod.{u_2, u_1} ((τ →₀ ℕ) × (τ →₀ ℕ)) ((σ →₀ ℕ) × (σ →₀ ℕ))) ↦
          match (motive := Prod.{u_2, u_1} ((τ →₀ ℕ) × (τ →₀ ℕ)) ((σ →₀ ℕ) × (σ →₀ ℕ)) → (σ →₀ ℕ) × (σ →₀ ℕ) × (σ →₀ ℕ))
            x with
          | (fst, b) => (b.1 + b.2, b))
        (Set.Finite.toFinset (s :=
          {p : Prod.{u_2, u_1} ((τ →₀ ℕ) × (τ →₀ ℕ)) ((σ →₀ ℕ) × (σ →₀ ℕ)) |
            p.1 ∈ @Finset.antidiagonal _ Finsupp.instAddMonoid _ x ∧
              p.2 ∈
                Set.Finite.toFinset (s := mapDomain f ⁻¹' {p.1.1}) ⋯ ×ˢ
                  Set.Finite.toFinset (s := mapDomain f ⁻¹' {p.1.2}) ⋯})
          ⋯))
      (fst, fst_1, snd) ↔
    (fst, fst_1, snd) ∈
      Set.Finite.toFinset (s :=
        {p : (σ →₀ ℕ) × (σ →₀ ℕ) × (σ →₀ ℕ) |
          mapDomain f p.1 = x ∧ p.2 ∈ @Finset.antidiagonal _ Finsupp.instAddMonoid _ p.1})
        ⋯
have h : fst_1 + snd = fst → (mapDomain f fst_1 + mapDomain f snd = x ↔ mapDomain f fst = x)
Tactic: simp
-- Lean 4 proof step from Mathlib.RingTheory.MvPowerSeries.Rename (goal)
Goal:
σ : Type u_1
τ : Type u_2
R : Type u_4
f : σ → τ
inst : TendstoCofinite f
inst_1 : Semiring R
p q : MvPowerSeries σ R
⊢ renameFun f (p * q) = renameFun f p * renameFun f q
have h : ∀ (x : τ →₀ ℕ),
  Eq.{u_4 + 1} (α := R) ((coeff x : (a : MvPowerSeries τ R) → R) (renameFun f (p * q)) : R)
    ((coeff x : (a : MvPowerSeries τ R) → R) (renameFun f p * renameFun f q) : R)
Tactic: ext x
-- Lean 4 proof step from Mathlib.RingTheory.MvPowerSeries.Rename (goal)
Goal:
σ : Type u_1
τ : Type u_2
R : Type u_4
f : σ → τ
inst : TendstoCofinite f
inst_1 : Semiring R
p q : MvPowerSeries σ R
x : τ →₀ ℕ
⊢ Eq.{u_4 + 1} (α := R) ((coeff x : (a : MvPowerSeries τ R) → R) (renameFun f (p * q)) : R)
    ((coeff x : (a : MvPowerSeries τ R) → R) (renameFun f p * renameFun f q) : R)
have h : Eq.{u_4 + 1} (α := R)
  (∑ x ∈ Set.Finite.toFinset (s := mapDomain f ⁻¹' {x}) sorry,
    ∑ p_1 ∈ @antidiagonal _ Finsupp.instAddMonoid _ x,
      HMul.hMul (α := R) (β := R) ((coeff p_1.1 : (a : MvPowerSeries σ R) → R) p : R)
        ((coeff p_1.2 : (a : MvPowerSeries σ R) → R) q : R))
  (∑ x ∈ @antidiagonal _ Finsupp.instAddMonoid _ x,
    ∑ x ∈ Set.Finite.toFinset (s := mapDomain f ⁻¹' {x.1}) sorry ×ˢ Set.Finite.toFinset (s := mapDomain f ⁻¹' {x.2}) sorry,
      HMul.hMul (α := R) (β := R) ((coeff x.1 : (a : MvPowerSeries σ R) → R) p : R)
        ((coeff x.2 : (a : MvPowerSeries σ R) → R) q : R))
Tactic: simp only [coeff_renameFun, coeff_mul, sum_mul_sum, ← sum_product']
-- Lean 4 proof step from Mathlib.RingTheory.MvPowerSeries.Rename (goal)
Goal:
σ : Type u_1
τ : Type u_2
R : Type u_4
f : σ → τ
inst : TendstoCofinite f
inst_1 : Semiring R
p q : MvPowerSeries σ R
x : τ →₀ ℕ
x_1 : Prod.{u_2, u_1} ((τ →₀ ℕ) × (τ →₀ ℕ)) ((σ →₀ ℕ) × (σ →₀ ℕ))
⊢ x_1 ∈
      (↑(Set.Finite.toFinset (s :=
            {p : Prod.{u_2, u_1} ((τ →₀ ℕ) × (τ →₀ ℕ)) ((σ →₀ ℕ) × (σ →₀ ℕ)) |
              p.1 ∈ @antidiagonal _ Finsupp.instAddMonoid _ x ∧
                p.2 ∈
                  Set.Finite.toFinset (s := mapDomain f ⁻¹' {p.1.1}) ⋯ ×ˢ
                    Set.Finite.toFinset (s := mapDomain f ⁻¹' {p.1.2}) ⋯})
            ⋯) :
        Set (((τ →₀ ℕ) × (τ →₀ ℕ)) × (σ →₀ ℕ) × (σ →₀ ℕ))) →
    ∀ ⦃x₂ : Prod.{u_2, u_1} ((τ →₀ ℕ) × (τ →₀ ℕ)) ((σ →₀ ℕ) × (σ →₀ ℕ))⦄,
      x₂ ∈
          (↑(Set.Finite.toFinset (s :=
                {p : Prod.{u_2, u_1} ((τ →₀ ℕ) × (τ →₀ ℕ)) ((σ →₀ ℕ) × (σ →₀ ℕ)) |
                  p.1 ∈ @antidiagonal _ Finsupp.instAddMonoid _ x ∧
                    p.2 ∈
                      Set.Finite.toFinset (s := mapDomain f ⁻¹' {p.1.1}) ⋯ ×ˢ
                        Set.Finite.toFinset (s := mapDomain f ⁻¹' {p.1.2}) ⋯})
                ⋯) :
            Set (((τ →₀ ℕ) × (τ →₀ ℕ)) × (σ →₀ ℕ) × (σ →₀ ℕ))) →
        Eq (α := (σ →₀ ℕ) × (σ →₀ ℕ) × (σ →₀ ℕ))
            (match (motive :=
                Prod.{u_2, u_1} ((τ →₀ ℕ) × (τ →₀ ℕ)) ((σ →₀ ℕ) × (σ →₀ ℕ)) → (σ →₀ ℕ) × (σ →₀ ℕ) × (σ →₀ ℕ)) x_1 with
              | (fst, b) => (b.1 + b.2, b) :
              (σ →₀ ℕ) × (σ →₀ ℕ) × (σ →₀ ℕ))
            (match (motive :=
                Prod.{u_2, u_1} ((τ →₀ ℕ) × (τ →₀ ℕ)) ((σ →₀ ℕ) × (σ →₀ ℕ)) → (σ →₀ ℕ) × (σ →₀ ℕ) × (σ →₀ ℕ)) x₂ with
              | (fst, b) => (b.1 + b.2, b) :
              (σ →₀ ℕ) × (σ →₀ ℕ) × (σ →₀ ℕ)) →
          x_1 = x₂
have simp : x_1.1.1 + x_1.1.2 = x →
  mapDomain f x_1.2.1 = x_1.1.1 →
    mapDomain f x_1.2.2 = x_1.1.2 →
      ∀ (a b : τ →₀ ℕ) (a_1 b_1 : σ →₀ ℕ),
        a + b = x →
          mapDomain f a_1 = a →
            mapDomain f b_1 = b → x_1.2.1 + x_1.2.2 = a_1 + b_1 → x_1.2 = (a_1, b_1) → x_1 = ((a, b), a_1, b_1)
Tactic: simp
-- Lean 4 proof step from Mathlib.RingTheory.MvPowerSeries.Rename (goal)
Goal:
σ : Type u_1
τ : Type u_2
R : Type u_4
f : σ → τ
inst : TendstoCofinite f
inst_1 : Semiring R
p q : MvPowerSeries σ R
x : τ →₀ ℕ
⊢ Eq.{u_4 + 1} (α := R)
    (∑ x ∈ Set.Finite.toFinset (s := mapDomain f ⁻¹' {x}) ⋯,
      ∑ p_1 ∈ @antidiagonal _ Finsupp.instAddMonoid _ x,
        HMul.hMul (α := R) (β := R) ((coeff p_1.1 : (a : MvPowerSeries σ R) → R) p : R)
          ((coeff p_1.2 : (a : MvPowerSeries σ R) → R) q : R))
    (∑ x ∈ @antidiagonal _ Finsupp.instAddMonoid _ x,
      ∑ x ∈ Set.Finite.toFinset (s := mapDomain f ⁻¹' {x.1}) ⋯ ×ˢ Set.Finite.toFinset (s := mapDomain f ⁻¹' {x.2}) ⋯,
        HMul.hMul (α := R) (β := R) ((coeff x.1 : (a : MvPowerSeries σ R) → R) p : R)
          ((coeff x.2 : (a : MvPowerSeries σ R) → R) q : R))
have h : Eq.{u_4 + 1} (α := R)
  (∑
    p_1 ∈
      Set.Finite.toFinset (s :=
        {p : (σ →₀ ℕ) × (σ →₀ ℕ) × (σ →₀ ℕ) | mapDomain f p.1 = x ∧ p.2 ∈ @antidiagonal _ Finsupp.instAddMonoid _ p.1})
        sorry,
    HMul.hMul (α := R) (β := R) ((coeff p_1.2.1 : (a : MvPowerSeries σ R) → R) p : R)
      ((coeff p_1.2.2 : (a : MvPowerSeries σ R) → R) q : R))
  (∑ x ∈ @antidiagonal _ Finsupp.instAddMonoid _ x,
    ∑ x ∈ Set.Finite.toFinset (s := mapDomain f ⁻¹' {x.1}) sorry ×ˢ Set.Finite.toFinset (s := mapDomain f ⁻¹' {x.2}) sorry,
      HMul.hMul (α := R) (β := R) ((coeff x.1 : (a : MvPowerSeries σ R) → R) p : R)
        ((coeff x.2 : (a : MvPowerSeries σ R) → R) q : R))
Tactic: rw [← sum_finset_product' (renameFunAux f x).toFinset _ _ (by simp),
    ← sum_finset_product' (renameFunAux' f x).toFinset _ _ (by simp),
    ← renameFunAuxImage f x, sum_image fun _ ↦ by simp; grind]
-- Lean 4 proof step from Mathlib.RingTheory.MvPowerSeries.Rename (goal)
Goal:
σ : Type u_1
τ : Type u_2
R : Type u_4
f : σ → τ
inst : TendstoCofinite f
inst_1 : Semiring R
p q : MvPowerSeries σ R
x : τ →₀ ℕ
h :
  Eq.{u_4 + 1} (α := R)
    (∑
      p_1 ∈
        Set.Finite.toFinset (s :=
          {p : (σ →₀ ℕ) × (σ →₀ ℕ) × (σ →₀ ℕ) |
            mapDomain f p.1 = x ∧ p.2 ∈ @antidiagonal _ Finsupp.instAddMonoid _ p.1})
          ⋯,
      HMul.hMul (α := R) (β := R) ((coeff p_1.2.1 : (a : MvPowerSeries σ R) → R) p : R)
        ((coeff p_1.2.2 : (a : MvPowerSeries σ R) → R) q : R))
    (∑ x ∈ @antidiagonal _ Finsupp.instAddMonoid _ x,
      ∑ x ∈ Set.Finite.toFinset (s := mapDomain f ⁻¹' {x.1}) ⋯ ×ˢ Set.Finite.toFinset (s := mapDomain f ⁻¹' {x.2}) ⋯,
        HMul.hMul (α := R) (β := R) ((coeff x.1 : (a : MvPowerSeries σ R) → R) p : R)
          ((coeff x.2 : (a : MvPowerSeries σ R) → R) q : R))
⊢ Eq.{u_4 + 1} (α := R)
    (∑ x ∈ Set.Finite.toFinset (s := mapDomain f ⁻¹' {x}) ⋯,
      ∑ p_1 ∈ @antidiagonal _ Finsupp.instAddMonoid _ x,
        HMul.hMul (α := R) (β := R) ((coeff p_1.1 : (a : MvPowerSeries σ R) → R) p : R)
          ((coeff p_1.2 : (a : MvPowerSeries σ R) → R) q : R))
    (∑ x ∈ @antidiagonal _ Finsupp.instAddMonoid _ x,
      ∑ x ∈ Set.Finite.toFinset (s := mapDomain f ⁻¹' {x.1}) ⋯ ×ˢ Set.Finite.toFinset (s := mapDomain f ⁻¹' {x.2}) ⋯,
        HMul.hMul (α := R) (β := R) ((coeff x.1 : (a : MvPowerSeries σ R) → R) p : R)
          ((coeff x.2 : (a : MvPowerSeries σ R) → R) q : R))
have h₁ : Eq.{u_4 + 1} (α := R)
  (∑
    p_1 ∈
      Set.Finite.toFinset (s :=
        {p : (σ →₀ ℕ) × (σ →₀ ℕ) × (σ →₀ ℕ) | mapDomain f p.1 = x ∧ p.2 ∈ @antidiagonal _ Finsupp.instAddMonoid _ p.1})
        sorry,
    HMul.hMul (α := R) (β := R) ((coeff p_1.2.1 : (a : MvPowerSeries σ R) → R) p : R)
      ((coeff p_1.2.2 : (a : MvPowerSeries σ R) → R) q : R))
  (∑
    p_1 ∈
      Set.Finite.toFinset (s :=
        {p : Prod.{u_2, u_1} ((τ →₀ ℕ) × (τ →₀ ℕ)) ((σ →₀ ℕ) × (σ →₀ ℕ)) |
          p.1 ∈ @antidiagonal _ Finsupp.instAddMonoid _ x ∧
            p.2 ∈
              Set.Finite.toFinset (s := mapDomain f ⁻¹' {p.1.1}) sorry ×ˢ
                Set.Finite.toFinset (s := mapDomain f ⁻¹' {p.1.2}) sorry})
        sorry,
    HMul.hMul (α := R) (β := R) ((coeff p_1.2.1 : (a : MvPowerSeries σ R) → R) p : R)
      ((coeff p_1.2.2 : (a : MvPowerSeries σ R) → R) q : R))
Tactic: rw [← sum_finset_product' (renameFunAux f x).toFinset _ _ (by simp),
    ← sum_finset_product' (renameFunAux' f x).toFinset _ _ (by simp),
    ← renameFunAuxImage f x, sum_image fun _ ↦ by simp; grind]
-- Lean 4 proof step from Mathlib.RingTheory.MvPowerSeries.Rename (goal)
Goal:
σ : Type u_1
τ : Type u_2
R : Type u_4
f : σ → τ
inst : TendstoCofinite f
inst_1 : Semiring R
p q : MvPowerSeries σ R
⊢ renameFun f (p * q) = renameFun f p * renameFun f q
have h : ∀ (x : τ →₀ ℕ),
  Eq.{u_4 + 1} (α := R) ((coeff x : (a : MvPowerSeries τ R) → R) (renameFun f (p * q)) : R)
    ((coeff x : (a : MvPowerSeries τ R) → R) (renameFun f p * renameFun f q) : R)
Tactic: classical
  ext x
  simp only [coeff_renameFun, coeff_mul, sum_mul_sum, ← sum_product']
  rw [← sum_finset_product' (renameFunAux f x).toFinset _ _ (by simp),
    ← sum_finset_product' (renameFunAux' f x).toFinset _ _ (by simp),
    ← renameFunAuxImage f x, sum_image fun _ ↦ by simp; grind]
-- Lean 4 proof step from Mathlib.RingTheory.MvPowerSeries.Rename (goal)
Goal:
σ : Type u_1
τ : Type u_2
R : Type u_4
f : σ → τ
inst : TendstoCofinite f
inst_1 : Semiring R
p q : MvPowerSeries σ R
h :
  ∀ (x : τ →₀ ℕ),
    Eq.{u_4 + 1} (α := R) ((coeff x : (a : MvPowerSeries τ R) → R) (renameFun f (p * q)) : R)
      ((coeff x : (a : MvPowerSeries τ R) → R) (renameFun f p * renameFun f q) : R)
⊢ renameFun f (p * q) = renameFun f p * renameFun f q
have h₁ : ∀ (x : τ →₀ ℕ),
  Eq.{u_4 + 1} (α := R)
    (∑ x ∈ Set.Finite.toFinset (s := mapDomain f ⁻¹' {x}) sorry,
      ∑ p_1 ∈ @antidiagonal _ Finsupp.instAddMonoid _ x,
        HMul.hMul (α := R) (β := R) ((coeff p_1.1 : (a : MvPowerSeries σ R) → R) p : R)
          ((coeff p_1.2 : (a : MvPowerSeries σ R) → R) q : R))
    (∑ x ∈ @antidiagonal _ Finsupp.instAddMonoid _ x,
      ∑ x ∈ Set.Finite.toFinset (s := mapDomain f ⁻¹' {x.1}) sorry ×ˢ Set.Finite.toFinset (s := mapDomain f ⁻¹' {x.2}) sorry,
        HMul.hMul (α := R) (β := R) ((coeff x.1 : (a : MvPowerSeries σ R) → R) p : R)
          ((coeff x.2 : (a : MvPowerSeries σ R) → R) q : R))
Tactic: classical
  ext x
  simp only [coeff_renameFun, coeff_mul, sum_mul_sum, ← sum_product']
  rw [← sum_finset_product' (renameFunAux f x).toFinset _ _ (by simp),
    ← sum_finset_product' (renameFunAux' f x).toFinset _ _ (by simp),
    ← renameFunAuxImage f x, sum_image fun _ ↦ by simp; grind]
-- Lean 4 proof step from Mathlib.RingTheory.MvPowerSeries.Rename (goal)
Goal:
σ : Type u_1
τ : Type u_2
γ : Type u_3
R : Type u_4
S : Type u_5
f : σ → τ
g : τ → γ
inst : TendstoCofinite f
inst_1 : CommSemiring R
inst_2 : CommSemiring S
⊢ Eq (α := MvPowerSeries τ R) (renameFun f 0) 0
have h : ∀ (n : τ →₀ ℕ),
  Eq.{u_4 + 1} (α := R) ((coeff n : (a : MvPowerSeries τ R) → R) (renameFun f 0) : R)
    ((coeff n : (a : MvPowerSeries τ R) → R) 0 : R)
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.MvPowerSeries.Rename (goal)
Goal:
σ : Type u_1
τ : Type u_2
γ : Type u_3
R : Type u_4
S : Type u_5
f : σ → τ
g : τ → γ
inst : TendstoCofinite f
inst_1 : CommSemiring R
inst_2 : CommSemiring S
x x_1 : MvPowerSeries σ R
⊢ renameFun f (x + x_1) = renameFun f x + renameFun f x_1
have h : ∀ (n : τ →₀ ℕ),
  Eq.{u_4 + 1} (α := R) ((coeff n : (a : MvPowerSeries τ R) → R) (renameFun f (x + x_1)) : R)
    ((coeff n : (a : MvPowerSeries τ R) → R) (renameFun f x + renameFun f x_1) : 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
p : MvPowerSeries σ R
x : σ →₀ ℕ
⊢ Eq.{u_4 + 1} (α := R)
    ((coeff (embDomain e x) : (a : MvPowerSeries τ R) → R)
        ((rename (⇑e : (a : σ) → τ) : (a : MvPowerSeries σ R) → MvPowerSeries τ R) p : MvPowerSeries τ R) :
      R)
    ((coeff x : (a : MvPowerSeries σ R) → R) p : R)
have rw : ∀ (b : σ →₀ ℕ),
  Membership.mem (γ := Finset (σ →₀ ℕ))
      (Set.Finite.toFinset (s := Set.preimage (β := τ →₀ ℕ) (mapDomain (⇑e : (a : σ) → τ)) {embDomain e x}) sorry) b →
    b ≠ x → Eq.{u_4 + 1} (α := R) ((coeff b : (a : MvPowerSeries σ R) → R) p : R) (0 : R)
Tactic: rw [coeff_rename, Finset.sum_eq_single x _ (by simp [← embDomain_eq_mapDomain])]
-- Lean 4 proof step from Mathlib.RingTheory.MvPowerSeries.Rename (goal)
Goal:
σ : Type u_1
τ : Type u_2
γ : Type u_3
R : Type u_4
f : σ → τ
g : τ → γ
inst : TendstoCofinite f
inst_1 : CommSemiring R
inst_2 : TendstoCofinite g
p : MvPowerSeries σ R
⊢ Eq (α := MvPowerSeries γ R)
    ((rename g : (a : MvPowerSeries τ R) → MvPowerSeries γ R)
        ((rename f : (a : MvPowerSeries σ R) → MvPowerSeries τ R) p) :
      MvPowerSeries γ R)
    ((rename (g ∘ f) : (a : MvPowerSeries σ R) → MvPowerSeries γ R) p : MvPowerSeries γ R)
have h : ∀ (y : γ →₀ ℕ),
  Eq.{u_4 + 1} (α := R)
    ((coeff y : (a : MvPowerSeries γ R) → R)
        ((rename g : (a : MvPowerSeries τ R) → MvPowerSeries γ R)
            ((rename f : (a : MvPowerSeries σ R) → MvPowerSeries τ R) p) :
          MvPowerSeries γ R) :
      R)
    ((coeff y : (a : MvPowerSeries γ R) → R)
        ((rename (g ∘ f) : (a : MvPowerSeries σ R) → MvPowerSeries γ R) p : MvPowerSeries γ R) :
      R)
Tactic: ext y
-- Lean 4 proof step from Mathlib.RingTheory.MvPowerSeries.Rename (goal)
Goal:
σ : Type u_1
τ : Type u_2
γ : Type u_3
R : Type u_4
f : σ → τ
g : τ → γ
inst : TendstoCofinite f
inst_1 : CommSemiring R
inst_2 : TendstoCofinite g
p : MvPowerSeries σ R
y : γ →₀ ℕ
⊢ Eq.{u_4 + 1} (α := R)
    ((coeff y : (a : MvPowerSeries γ R) → R)
        ((rename g : (a : MvPowerSeries τ R) → MvPowerSeries γ R)
            ((rename f : (a : MvPowerSeries σ R) → MvPowerSeries τ R) p) :
          MvPowerSeries γ R) :
      R)
    ((coeff y : (a : MvPowerSeries γ R) → R)
        ((rename (g ∘ f) : (a : MvPowerSeries σ R) → MvPowerSeries γ R) p : MvPowerSeries γ R) :
      R)
have h : Eq.{u_4 + 1} (α := R)
  (∑ x ∈ Set.Finite.toFinset (s := mapDomain g ⁻¹' {y}) sorry,
    ∑ x ∈ Set.Finite.toFinset (s := mapDomain f ⁻¹' {x}) sorry, (coeff x : (a : MvPowerSeries σ R) → R) p)
  (∑ x ∈ Set.Finite.toFinset (s := mapDomain (g ∘ f) ⁻¹' {y}) sorry, (coeff x : (a : MvPowerSeries σ R) → R) p)
Tactic: simp only [coeff_rename]
-- Lean 4 proof step from Mathlib.RingTheory.MvPowerSeries.Rename (goal)
Goal:
σ : Type u_1
τ : Type u_2
γ : Type u_3
R : Type u_4
f : σ → τ
g : τ → γ
inst : TendstoCofinite f
inst_1 : CommSemiring R
inst_2 : TendstoCofinite g
p : MvPowerSeries σ R
y : γ →₀ ℕ
⊢ ∀ (p : Prod.{u_2, u_1} (τ →₀ ℕ) (σ →₀ ℕ)),
    Membership.mem (γ := Finset (Prod.{u_2, u_1} (τ →₀ ℕ) (σ →₀ ℕ)))
        (Finset.image (fun (u : σ →₀ ℕ) ↦ (mapDomain f u, u)) (Set.Finite.toFinset (s := mapDomain (g ∘ f) ⁻¹' {y}) ⋯))
        p ↔
      p.1 ∈ Set.Finite.toFinset (s := mapDomain g ⁻¹' {y}) ⋯ ∧ p.2 ∈ Set.Finite.toFinset (s := mapDomain f ⁻¹' {p.1}) ⋯
have simp : ∀ (a : τ →₀ ℕ) (b : σ →₀ ℕ), mapDomain f b = a → (mapDomain (g ∘ f) b = y ↔ mapDomain g a = y)
Tactic: simp