awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.HEval (goal)
Goal:
Γ : Type u_1
R : Type u_3
V : Type u_4
inst : AddCommMonoid Γ
inst_1 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : CommRing R
inst_4 : CommRing V
inst_5 : Algebra R V
x : V⟦Γ⟧
hx : ¬0 < x.orderTop
f : PowerSeries R
n : ℕ
g : Γ
⊢ Eq.{u_4 + 1} (α := V) (HahnSeries.coeff ((powerSeriesFamily x f : (a : ℕ) → V⟦Γ⟧) n) g)
    (HahnSeries.coeff ((powerSeriesFamily 0 f : (a : ℕ) → V⟦Γ⟧) n) g)
have h : Eq (α := V) (HahnSeries.coeff ((powerSeriesFamily x f : (a : ℕ) → V⟦Γ⟧) 0) g : V)
  (HahnSeries.coeff ((powerSeriesFamily 0 f : (a : ℕ) → V⟦Γ⟧) 0) g : V)
Tactic: obtain rfl | hn := eq_or_ne n 0
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.HEval (goal)
Goal:
Γ : Type u_1
R : Type u_3
V : Type u_4
inst : AddCommMonoid Γ
inst_1 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : CommRing R
inst_4 : CommRing V
inst_5 : Algebra R V
x : V⟦Γ⟧
hx : ¬0 < x.orderTop
f : PowerSeries R
n : ℕ
g : Γ
h :
  Eq (α := V) (HahnSeries.coeff ((powerSeriesFamily x f : (a : ℕ) → V⟦Γ⟧) 0) g : V)
    (HahnSeries.coeff ((powerSeriesFamily 0 f : (a : ℕ) → V⟦Γ⟧) 0) g : V)
⊢ Eq.{u_4 + 1} (α := V) (HahnSeries.coeff ((powerSeriesFamily x f : (a : ℕ) → V⟦Γ⟧) n) g)
    (HahnSeries.coeff ((powerSeriesFamily 0 f : (a : ℕ) → V⟦Γ⟧) n) g)
have h₁ : ∀ (hn : n ≠ 0),
  Eq.{u_4 + 1} (α := V) (HahnSeries.coeff ((powerSeriesFamily x f : (a : ℕ) → V⟦Γ⟧) n) g)
    (HahnSeries.coeff ((powerSeriesFamily 0 f : (a : ℕ) → V⟦Γ⟧) n) g)
Tactic: obtain rfl | hn := eq_or_ne n 0
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.HEval (goal)
Goal:
Γ : Type u_1
R : Type u_3
V : Type u_4
inst : AddCommMonoid Γ
inst_1 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : CommRing R
inst_4 : CommRing V
inst_5 : Algebra R V
f : PowerSeries R
⊢ hsum (powerSeriesFamily 0 f) =
    HSMul.hSMul (α := R) (β := V⟦Γ⟧) ((PowerSeries.constantCoeff : (a : PowerSeries R) → R) f : R) 1
have coeff : ∀ (g : Γ),
  HahnSeries.coeff (hsum (powerSeriesFamily 0 f)) g =
    (HSMul.hSMul (α := R) (β := V⟦Γ⟧) ((PowerSeries.constantCoeff : (a : PowerSeries R) → R) f : R) 1).coeff g
Tactic: ext g
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.HEval (goal)
Goal:
Γ : Type u_1
R : Type u_3
V : Type u_4
inst : AddCommMonoid Γ
inst_1 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : CommRing R
inst_4 : CommRing V
inst_5 : Algebra R V
f : PowerSeries R
g : Γ
⊢ HahnSeries.coeff (hsum (powerSeriesFamily 0 f)) g =
    (HSMul.hSMul (α := R) (β := V⟦Γ⟧) ((PowerSeries.constantCoeff : (a : PowerSeries R) → R) f : R) 1).coeff g
have pos : ∀ (hg : g = 0),
  HahnSeries.coeff (hsum (powerSeriesFamily 0 f)) g =
    (HSMul.hSMul (α := R) (β := V⟦Γ⟧) ((PowerSeries.constantCoeff : (a : PowerSeries R) → R) f : R) 1).coeff g
Tactic: by_cases hg : g = 0
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.HEval (goal)
Goal:
Γ : Type u_1
R : Type u_3
V : Type u_4
inst : AddCommMonoid Γ
inst_1 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : CommRing R
inst_4 : CommRing V
inst_5 : Algebra R V
f : PowerSeries R
g : Γ
pos :
  ∀ (hg : g = 0),
    HahnSeries.coeff (hsum (powerSeriesFamily 0 f)) g =
      (HSMul.hSMul (α := R) (β := V⟦Γ⟧) ((PowerSeries.constantCoeff : (a : PowerSeries R) → R) f : R) 1).coeff g
⊢ HahnSeries.coeff (hsum (powerSeriesFamily 0 f)) g =
    (HSMul.hSMul (α := R) (β := V⟦Γ⟧) ((PowerSeries.constantCoeff : (a : PowerSeries R) → R) f : R) 1).coeff g
have neg : ∀ (hg : ¬g = 0),
  HahnSeries.coeff (hsum (powerSeriesFamily 0 f)) g =
    (HSMul.hSMul (α := R) (β := V⟦Γ⟧) ((PowerSeries.constantCoeff : (a : PowerSeries R) → R) f : R) 1).coeff g
Tactic: by_cases hg : g = 0
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.HEval (goal)
Goal:
Γ : Type u_1
R : Type u_3
V : Type u_4
inst : AddCommMonoid Γ
inst_1 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : CommRing R
inst_4 : CommRing V
inst_5 : Algebra R V
f : PowerSeries R
g : Γ
hg : g = 0
⊢ HahnSeries.coeff (hsum (powerSeriesFamily 0 f)) g =
    (HSMul.hSMul (α := R) (β := V⟦Γ⟧) ((PowerSeries.constantCoeff : (a : PowerSeries R) → R) f : R) 1).coeff g
have pos : ∑ᶠ (i : ℕ), HahnSeries.coeff.{u_1, u_4} (Γ := Γ) ((powerSeriesFamily 0 f : (a : ℕ) → V⟦Γ⟧) i) 0 =
  (HSMul.hSMul (α := R) (β := V⟦Γ⟧) ((PowerSeries.constantCoeff : (a : PowerSeries R) → R) f : R) 1).coeff 0
Tactic: simp only [hg, coeff_hsum]
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.HEval (goal)
Goal:
Γ : Type u_1
R : Type u_3
V : Type u_4
inst : AddCommMonoid Γ
inst_1 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : CommRing R
inst_4 : CommRing V
inst_5 : Algebra R V
f : PowerSeries R
g : Γ
hg : g = 0
⊢ ∑ᶠ (i : ℕ), HahnSeries.coeff.{u_1, u_4} (Γ := Γ) ((powerSeriesFamily 0 f : (a : ℕ) → V⟦Γ⟧) i) 0 =
    (HSMul.hSMul (α := R) (β := V⟦Γ⟧) ((PowerSeries.constantCoeff : (a : PowerSeries R) → R) f : R) 1).coeff 0
have pos : HahnSeries.coeff.{u_1, u_4} (Γ := Γ) ((powerSeriesFamily 0 f : (a : ℕ) → V⟦Γ⟧) 0) 0 =
  (HSMul.hSMul (α := R) (β := V⟦Γ⟧) ((PowerSeries.constantCoeff : (a : PowerSeries R) → R) f : R) 1).coeff 0
Tactic: rw [finsum_eq_single _ 0 (fun n hn ↦ by simp [hn])]
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.HEval (goal)
Goal:
Γ : Type u_1
R : Type u_3
V : Type u_4
inst : AddCommMonoid Γ
inst_1 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : CommRing R
inst_4 : CommRing V
inst_5 : Algebra R V
f : PowerSeries R
g : Γ
hg : ¬g = 0
n : ℕ
⊢ Eq.{u_4 + 1} (α := V) (HahnSeries.coeff ((powerSeriesFamily 0 f : (a : ℕ) → V⟦Γ⟧) n) g) 0
have pos : ∀ (hn : n = 0), Eq.{u_4 + 1} (α := V) (HahnSeries.coeff ((powerSeriesFamily 0 f : (a : ℕ) → V⟦Γ⟧) n) g) 0
Tactic: by_cases hn : n = 0
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.HEval (goal)
Goal:
Γ : Type u_1
R : Type u_3
V : Type u_4
inst : AddCommMonoid Γ
inst_1 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : CommRing R
inst_4 : CommRing V
inst_5 : Algebra R V
f : PowerSeries R
g : Γ
hg : ¬g = 0
n : ℕ
pos : ∀ (hn : n = 0), Eq.{u_4 + 1} (α := V) (HahnSeries.coeff ((powerSeriesFamily 0 f : (a : ℕ) → V⟦Γ⟧) n) g) 0
⊢ Eq.{u_4 + 1} (α := V) (HahnSeries.coeff ((powerSeriesFamily 0 f : (a : ℕ) → V⟦Γ⟧) n) g) 0
have neg : ∀ (hn : ¬n = 0), Eq.{u_4 + 1} (α := V) (HahnSeries.coeff ((powerSeriesFamily 0 f : (a : ℕ) → V⟦Γ⟧) n) g) 0
Tactic: by_cases hn : n = 0
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.HEval (goal)
Goal:
Γ : Type u_1
R : Type u_3
V : Type u_4
inst : AddCommMonoid Γ
inst_1 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : CommRing R
inst_4 : CommRing V
inst_5 : Algebra R V
f : PowerSeries R
g : Γ
hg : ¬g = 0
⊢ HahnSeries.coeff (hsum (powerSeriesFamily 0 f)) g =
    (HSMul.hSMul (α := R) (β := V⟦Γ⟧) ((PowerSeries.constantCoeff : (a : PowerSeries R) → R) f : R) 1).coeff g
have neg : 0 = (HSMul.hSMul (α := R) (β := V⟦Γ⟧) ((PowerSeries.constantCoeff : (a : PowerSeries R) → R) f : R) 1).coeff g
Tactic: rw [coeff_hsum, finsum_eq_zero_of_forall_eq_zero
      fun n ↦ (by by_cases hn : n = 0 <;> simp [hg, hn])]
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.HEval (goal)
Goal:
Γ : Type u_1
R : Type u_3
V : Type u_4
inst : AddCommMonoid Γ
inst_1 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : CommRing R
inst_4 : CommRing V
inst_5 : Algebra R V
x : V⟦Γ⟧
f g : PowerSeries R
⊢ powerSeriesFamily x (f + g) = powerSeriesFamily x f + powerSeriesFamily x g
have h : ∀ (n : ℕ),
  Eq (α := V⟦Γ⟧) ((powerSeriesFamily x (f + g) : (a : ℕ) → V⟦Γ⟧) n : V⟦Γ⟧)
    ((powerSeriesFamily x f + powerSeriesFamily x g : (a : ℕ) → V⟦Γ⟧) n : V⟦Γ⟧)
Tactic: ext1 n
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.HEval (goal)
Goal:
Γ : Type u_1
R : Type u_3
V : Type u_4
inst : AddCommMonoid Γ
inst_1 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : CommRing R
inst_4 : CommRing V
inst_5 : Algebra R V
x : V⟦Γ⟧
f g : PowerSeries R
n : ℕ
⊢ Eq (α := V⟦Γ⟧) ((powerSeriesFamily x (f + g) : (a : ℕ) → V⟦Γ⟧) n : V⟦Γ⟧)
    ((powerSeriesFamily x f + powerSeriesFamily x g : (a : ℕ) → V⟦Γ⟧) n : V⟦Γ⟧)
have pos : ∀ (hx : 0 < x.orderTop),
  Eq (α := V⟦Γ⟧) ((powerSeriesFamily x (f + g) : (a : ℕ) → V⟦Γ⟧) n : V⟦Γ⟧)
    ((powerSeriesFamily x f + powerSeriesFamily x g : (a : ℕ) → V⟦Γ⟧) n : V⟦Γ⟧)
Tactic: by_cases hx : 0 < x.orderTop
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.HEval (goal)
Goal:
Γ : Type u_1
R : Type u_3
V : Type u_4
inst : AddCommMonoid Γ
inst_1 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : CommRing R
inst_4 : CommRing V
inst_5 : Algebra R V
x : V⟦Γ⟧
f g : PowerSeries R
n : ℕ
pos :
  ∀ (hx : 0 < x.orderTop),
    Eq (α := V⟦Γ⟧) ((powerSeriesFamily x (f + g) : (a : ℕ) → V⟦Γ⟧) n : V⟦Γ⟧)
      ((powerSeriesFamily x f + powerSeriesFamily x g : (a : ℕ) → V⟦Γ⟧) n : V⟦Γ⟧)
⊢ Eq (α := V⟦Γ⟧) ((powerSeriesFamily x (f + g) : (a : ℕ) → V⟦Γ⟧) n : V⟦Γ⟧)
    ((powerSeriesFamily x f + powerSeriesFamily x g : (a : ℕ) → V⟦Γ⟧) n : V⟦Γ⟧)
have neg : ∀ (hx : ¬0 < x.orderTop),
  Eq (α := V⟦Γ⟧) ((powerSeriesFamily x (f + g) : (a : ℕ) → V⟦Γ⟧) n : V⟦Γ⟧)
    ((powerSeriesFamily x f + powerSeriesFamily x g : (a : ℕ) → V⟦Γ⟧) n : V⟦Γ⟧)
Tactic: by_cases hx : 0 < x.orderTop
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.HEval (goal)
Goal:
Γ : Type u_1
R : Type u_3
V : Type u_4
inst : AddCommMonoid Γ
inst_1 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : CommRing R
inst_4 : CommRing V
inst_5 : Algebra R V
x : V⟦Γ⟧
f : PowerSeries R
r : R
⊢ powerSeriesFamily x (r • f) =
    HSMul.hSMul (α := R⟦Γ⟧) ((HahnSeries.single 0 : (a : R) → R⟦Γ⟧) r : R⟦Γ⟧) (powerSeriesFamily x f)
have h : ∀ (n : ℕ),
  Eq (α := V⟦Γ⟧) ((powerSeriesFamily x (r • f) : (a : ℕ) → V⟦Γ⟧) n : V⟦Γ⟧)
    ((HSMul.hSMul (α := R⟦Γ⟧) ((HahnSeries.single 0 : (a : R) → R⟦Γ⟧) r : R⟦Γ⟧) (powerSeriesFamily x f) :
          (a : ℕ) → V⟦Γ⟧)
        n :
      V⟦Γ⟧)
Tactic: ext1 n
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.HEval (goal)
Goal:
Γ : Type u_1
R : Type u_3
V : Type u_4
inst : AddCommMonoid Γ
inst_1 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : CommRing R
inst_4 : CommRing V
inst_5 : Algebra R V
x : V⟦Γ⟧
a b : PowerSeries R
g : Γ
⊢ ((powerSeriesFamily x (a * b)).coeff g).support ⊆
    image (fun (i : ℕ × ℕ) ↦ i.1 + i.2) (((powerSeriesFamily x a).mul (powerSeriesFamily x b)).coeff g).support
have pos : ∀ (h : 0 < x.orderTop),
  ((powerSeriesFamily x (a * b)).coeff g).support ⊆
    image (fun (i : ℕ × ℕ) ↦ i.1 + i.2) (((powerSeriesFamily x a).mul (powerSeriesFamily x b)).coeff g).support
Tactic: by_cases h : 0 < x.orderTop
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.HEval (goal)
Goal:
Γ : Type u_1
R : Type u_3
V : Type u_4
inst : AddCommMonoid Γ
inst_1 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : CommRing R
inst_4 : CommRing V
inst_5 : Algebra R V
x : V⟦Γ⟧
a b : PowerSeries R
g : Γ
pos :
  ∀ (h : 0 < x.orderTop),
    ((powerSeriesFamily x (a * b)).coeff g).support ⊆
      image (fun (i : ℕ × ℕ) ↦ i.1 + i.2) (((powerSeriesFamily x a).mul (powerSeriesFamily x b)).coeff g).support
⊢ ((powerSeriesFamily x (a * b)).coeff g).support ⊆
    image (fun (i : ℕ × ℕ) ↦ i.1 + i.2) (((powerSeriesFamily x a).mul (powerSeriesFamily x b)).coeff g).support
have neg : ∀ (h : ¬0 < x.orderTop),
  ((powerSeriesFamily x (a * b)).coeff g).support ⊆
    image (fun (i : ℕ × ℕ) ↦ i.1 + i.2) (((powerSeriesFamily x a).mul (powerSeriesFamily x b)).coeff g).support
Tactic: by_cases h : 0 < x.orderTop
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.HEval (goal)
Goal:
Γ : Type u_1
R : Type u_3
V : Type u_4
inst : AddCommMonoid Γ
inst_1 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : CommRing R
inst_4 : CommRing V
inst_5 : Algebra R V
x : V⟦Γ⟧
a b : PowerSeries R
g : Γ
h : 0 < x.orderTop
⊢ ((powerSeriesFamily x (a * b)).coeff g).support ⊆
    image (fun (i : ℕ × ℕ) ↦ i.1 + i.2) (((powerSeriesFamily x a).mul (powerSeriesFamily x b)).coeff g).support
have pos : ∀ (x_1 : ℕ),
  Ne.{u_4 + 1} (α := V) (HahnSeries.coeff ((powerSeriesFamily x (a * b) : (a : ℕ) → V⟦Γ⟧) x_1) g) 0 →
    x_1 ∈
      (↑(image (fun (i : ℕ × ℕ) ↦ i.1 + i.2)
            (Set.Finite.toFinset (s :=
              Function.support.{0, u_4} (M := V) fun (a_2 : ℕ × ℕ) ↦
                HahnSeries.coeff (((powerSeriesFamily x a).mul (powerSeriesFamily x b) : (a : ℕ × ℕ) → V⟦Γ⟧) a_2) g)
              sorry)) :
        Set ℕ)
Tactic: simp only [coeff_support, Set.Finite.toFinset_subset, support_subset_iff]
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.HEval (goal)
Goal:
Γ : Type u_1
R : Type u_3
V : Type u_4
inst : AddCommMonoid Γ
inst_1 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : CommRing R
inst_4 : CommRing V
inst_5 : Algebra R V
x : V⟦Γ⟧
a b : PowerSeries R
g : Γ
h : 0 < x.orderTop
n : ℕ
hn : Ne.{u_4 + 1} (α := V) (HahnSeries.coeff ((powerSeriesFamily x (a * b) : (a : ℕ) → V⟦Γ⟧) n) g) 0
⊢ ∃ c ∈ @antidiagonal _ Nat.instAddMonoid _ n,
    HSMul.hSMul (α := R) ((PowerSeries.coeff c.1 : (a : PowerSeries R) → R) a : R)
        (HSMul.hSMul (α := R) (β := V) ((PowerSeries.coeff c.2 : (a : PowerSeries R) → R) b : R)
          (HahnSeries.coeff ((powers x : (a : ℕ) → V⟦Γ⟧) n) g)) ≠
      0
have refine : Ne.{u_4 + 1} (α := V)
  (∑ x_1 ∈ @antidiagonal _ Nat.instAddMonoid _ n,
    HSMul.hSMul (α := R) ((PowerSeries.coeff x_1.1 : (a : PowerSeries R) → R) a : R)
      (HSMul.hSMul (α := R) (β := V) ((PowerSeries.coeff x_1.2 : (a : PowerSeries R) → R) b : R)
        (HahnSeries.coeff ((powers x : (a : ℕ) → V⟦Γ⟧) n) g)))
  0
Tactic: refine exists_ne_zero_of_sum_ne_zero ?_
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.HEval (hypothesis)
Goal:
Γ : Type u_1
R : Type u_3
V : Type u_4
inst : AddCommMonoid Γ
inst_1 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : CommRing R
inst_4 : CommRing V
inst_5 : Algebra R V
x : V⟦Γ⟧
a b : PowerSeries R
g : Γ
h : 0 < x.orderTop
n : ℕ
hn : Ne.{u_4 + 1} (α := V) (HahnSeries.coeff ((powerSeriesFamily x (a * b) : (a : ℕ) → V⟦Γ⟧) n) g) 0
⊢ n ∈
    (↑(image (fun (i : ℕ × ℕ) ↦ i.1 + i.2)
          (Set.Finite.toFinset (s :=
            Function.support.{0, u_4} (M := V) fun (a_1 : ℕ × ℕ) ↦
              HahnSeries.coeff (((powerSeriesFamily x a).mul (powerSeriesFamily x b) : (a : ℕ × ℕ) → V⟦Γ⟧) a_1) g)
            ⋯)) :
      Set ℕ)
have he : ∃ c ∈ @antidiagonal _ Nat.instAddMonoid _ n,
  HSMul.hSMul (α := R) ((PowerSeries.coeff c.1 : (a : PowerSeries R) → R) a : R)
      (HSMul.hSMul (α := R) (β := V) ((PowerSeries.coeff c.2 : (a : PowerSeries R) → R) b : R)
        (HahnSeries.coeff ((powers x : (a : ℕ) → V⟦Γ⟧) n) g)) ≠
    0
Tactic: have he : ∃ c ∈ antidiagonal n, (PowerSeries.coeff c.1) a • (PowerSeries.coeff c.2) b •
        ((powers x) n).coeff g ≠ 0 := by
      refine exists_ne_zero_of_sum_ne_zero ?_
      simpa [PowerSeries.coeff_mul, sum_smul, mul_smul, h] using hn
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.HEval (hypothesis)
Goal:
Γ : Type u_1
R : Type u_3
V : Type u_4
inst : AddCommMonoid Γ
inst_1 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : CommRing R
inst_4 : CommRing V
inst_5 : Algebra R V
x : V⟦Γ⟧
a b : PowerSeries R
g : Γ
h : 0 < x.orderTop
n : ℕ
hn : Ne.{u_4 + 1} (α := V) (HahnSeries.coeff ((powerSeriesFamily x (a * b) : (a : ℕ) → V⟦Γ⟧) n) g) 0
he :
  ∃ c ∈ @antidiagonal _ Nat.instAddMonoid _ n,
    HSMul.hSMul (α := R) ((PowerSeries.coeff c.1 : (a : PowerSeries R) → R) a : R)
        (HSMul.hSMul (α := R) (β := V) ((PowerSeries.coeff c.2 : (a : PowerSeries R) → R) b : R)
          (HahnSeries.coeff ((powers x : (a : ℕ) → V⟦Γ⟧) n) g)) ≠
      0
⊢ n ∈
    (↑(image (fun (i : ℕ × ℕ) ↦ i.1 + i.2)
          (Set.Finite.toFinset (s :=
            Function.support.{0, u_4} (M := V) fun (a_1 : ℕ × ℕ) ↦
              HahnSeries.coeff (((powerSeriesFamily x a).mul (powerSeriesFamily x b) : (a : ℕ × ℕ) → V⟦Γ⟧) a_1) g)
            ⋯)) :
      Set ℕ)
have he₁ : ∃ (c : ℕ × ℕ),
  c.1 + c.2 = n ∧
    HSMul.hSMul (α := R) ((PowerSeries.coeff c.1 : (a : PowerSeries R) → R) a : R)
        (HSMul.hSMul (α := R) ((PowerSeries.coeff c.2 : (a : PowerSeries R) → R) b : R) ((x ^ n).coeff g)) ≠
      0
Tactic: simp only [powers_of_orderTop_pos h, mem_antidiagonal] at he
Cleared hypotheses: he
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.HEval (goal)
Goal:
Γ : Type u_1
R : Type u_3
V : Type u_4
inst : AddCommMonoid Γ
inst_1 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : CommRing R
inst_4 : CommRing V
inst_5 : Algebra R V
x : V⟦Γ⟧
a b : PowerSeries R
g : Γ
h : 0 < x.orderTop
n : ℕ
hn : Ne.{u_4 + 1} (α := V) (HahnSeries.coeff ((powerSeriesFamily x (a * b) : (a : ℕ) → V⟦Γ⟧) n) g) 0
he :
  ∃ (c : ℕ × ℕ),
    c.1 + c.2 = n ∧
      HSMul.hSMul (α := R) ((PowerSeries.coeff c.1 : (a : PowerSeries R) → R) a : R)
          (HSMul.hSMul (α := R) ((PowerSeries.coeff c.2 : (a : PowerSeries R) → R) b : R) ((x ^ n).coeff g)) ≠
        0
⊢ n ∈
    (↑(image (fun (i : ℕ × ℕ) ↦ i.1 + i.2)
          (Set.Finite.toFinset (s :=
            Function.support.{0, u_4} (M := V) fun (a_1 : ℕ × ℕ) ↦
              HahnSeries.coeff (((powerSeriesFamily x a).mul (powerSeriesFamily x b) : (a : ℕ × ℕ) → V⟦Γ⟧) a_1) g)
            ⋯)) :
      Set ℕ)
have pos : ∀ (c : ℕ × ℕ) (hcn : c.1 + c.2 = n)
  (hc :
    HSMul.hSMul (α := R) ((PowerSeries.coeff c.1 : (a : PowerSeries R) → R) a : R)
        (HSMul.hSMul (α := R) ((PowerSeries.coeff c.2 : (a : PowerSeries R) → R) b : R) ((x ^ n).coeff g)) ≠
      0),
  n ∈
    (↑(image (fun (i : ℕ × ℕ) ↦ i.1 + i.2)
          (Set.Finite.toFinset (s :=
            Function.support.{0, u_4} (M := V) fun (a_1 : ℕ × ℕ) ↦
              HahnSeries.coeff (((powerSeriesFamily x a).mul (powerSeriesFamily x b) : (a : ℕ × ℕ) → V⟦Γ⟧) a_1) g)
            sorry)) :
      Set ℕ)
Tactic: obtain ⟨c, hcn, hc⟩ := he
Cleared hypotheses: he
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.HEval (goal)
Goal:
Γ : Type u_1
R : Type u_3
V : Type u_4
inst : AddCommMonoid Γ
inst_1 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : CommRing R
inst_4 : CommRing V
inst_5 : Algebra R V
x : V⟦Γ⟧
a b : PowerSeries R
g : Γ
h : 0 < x.orderTop
n : ℕ
hn : Ne.{u_4 + 1} (α := V) (HahnSeries.coeff ((powerSeriesFamily x (a * b) : (a : ℕ) → V⟦Γ⟧) n) g) 0
c : ℕ × ℕ
hcn : c.1 + c.2 = n
hc :
  HSMul.hSMul (α := R) ((PowerSeries.coeff c.1 : (a : PowerSeries R) → R) a : R)
      (HSMul.hSMul (α := R) ((PowerSeries.coeff c.2 : (a : PowerSeries R) → R) b : R) ((x ^ n).coeff g)) ≠
    0
⊢ n ∈
    (↑(image (fun (i : ℕ × ℕ) ↦ i.1 + i.2)
          (Set.Finite.toFinset (s :=
            Function.support.{0, u_4} (M := V) fun (a_1 : ℕ × ℕ) ↦
              HahnSeries.coeff (((powerSeriesFamily x a).mul (powerSeriesFamily x b) : (a : ℕ × ℕ) → V⟦Γ⟧) a_1) g)
            ⋯)) :
      Set ℕ)
have pos : ∃ (x_1 : ℕ × ℕ),
  Membership.mem (γ := Set (ℕ × ℕ))
      (Function.support.{0, u_4} (M := V) fun (a_1 : ℕ × ℕ) ↦
        HahnSeries.coeff (((powerSeriesFamily x a).mul (powerSeriesFamily x b) : (a : ℕ × ℕ) → V⟦Γ⟧) a_1) g)
      x_1 ∧
    x_1.1 + x_1.2 = n
Tactic: simp only [coe_image, Set.Finite.coe_toFinset, Set.mem_image]
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.HEval (goal)
Goal:
Γ : Type u_1
R : Type u_3
V : Type u_4
inst : AddCommMonoid Γ
inst_1 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : CommRing R
inst_4 : CommRing V
inst_5 : Algebra R V
x : V⟦Γ⟧
a b : PowerSeries R
g : Γ
h : 0 < x.orderTop
n : ℕ
hn : Ne.{u_4 + 1} (α := V) (HahnSeries.coeff ((powerSeriesFamily x (a * b) : (a : ℕ) → V⟦Γ⟧) n) g) 0
c : ℕ × ℕ
hcn : c.1 + c.2 = n
hc :
  HSMul.hSMul (α := R) ((PowerSeries.coeff c.1 : (a : PowerSeries R) → R) a : R)
      (HSMul.hSMul (α := R) ((PowerSeries.coeff c.2 : (a : PowerSeries R) → R) b : R) ((x ^ n).coeff g)) ≠
    0
⊢ ∃ (x_1 : ℕ × ℕ),
    Membership.mem (γ := Set (ℕ × ℕ))
        (Function.support.{0, u_4} (M := V) fun (a_1 : ℕ × ℕ) ↦
          HahnSeries.coeff (((powerSeriesFamily x a).mul (powerSeriesFamily x b) : (a : ℕ × ℕ) → V⟦Γ⟧) a_1) g)
        x_1 ∧
      x_1.1 + x_1.2 = n
have h₁ : Membership.mem (γ := Set (ℕ × ℕ))
    (Function.support.{0, u_4} (M := V) fun (a_1 : ℕ × ℕ) ↦
      HahnSeries.coeff (((powerSeriesFamily x a).mul (powerSeriesFamily x b) : (a : ℕ × ℕ) → V⟦Γ⟧) a_1) g)
    c ∧
  c.1 + c.2 = n
Tactic: use c
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.HEval (goal)
Goal:
Γ : Type u_1
R : Type u_3
V : Type u_4
inst : AddCommMonoid Γ
inst_1 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : CommRing R
inst_4 : CommRing V
inst_5 : Algebra R V
x : V⟦Γ⟧
a b : PowerSeries R
g : Γ
h : 0 < x.orderTop
n : ℕ
hn : Ne.{u_4 + 1} (α := V) (HahnSeries.coeff ((powerSeriesFamily x (a * b) : (a : ℕ) → V⟦Γ⟧) n) g) 0
c : ℕ × ℕ
hcn : c.1 + c.2 = n
hc :
  HSMul.hSMul (α := R) ((PowerSeries.coeff c.1 : (a : PowerSeries R) → R) a : R)
      (HSMul.hSMul (α := R) ((PowerSeries.coeff c.2 : (a : PowerSeries R) → R) b : R) ((x ^ n).coeff g)) ≠
    0
⊢ Membership.mem (γ := Set (ℕ × ℕ))
      (Function.support.{0, u_4} (M := V) fun (a_1 : ℕ × ℕ) ↦
        HahnSeries.coeff (((powerSeriesFamily x a).mul (powerSeriesFamily x b) : (a : ℕ × ℕ) → V⟦Γ⟧) a_1) g)
      c ∧
    c.1 + c.2 = n
have h₁ : (HSMul.hSMul (α := R) (β := V⟦Γ⟧) ((PowerSeries.coeff c.1 : (a : PowerSeries R) → R) a : R)
          ((powers x : (a : ℕ) → V⟦Γ⟧) c.1 : V⟦Γ⟧) *
        HSMul.hSMul (α := R) (β := V⟦Γ⟧) ((PowerSeries.coeff c.2 : (a : PowerSeries R) → R) b : R)
          ((powers x : (a : ℕ) → V⟦Γ⟧) c.2 : V⟦Γ⟧)).coeff
    g ≠
  0
Tactic: simp only [mul_toFun, smulFamily_toFun, Function.mem_support, hcn,
      and_true]
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.HEval (goal)
Goal:
Γ : Type u_1
R : Type u_3
V : Type u_4
inst : AddCommMonoid Γ
inst_1 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : CommRing R
inst_4 : CommRing V
inst_5 : Algebra R V
x : V⟦Γ⟧
a b : PowerSeries R
g : Γ
h : 0 < x.orderTop
n : ℕ
hn : Ne.{u_4 + 1} (α := V) (HahnSeries.coeff ((powerSeriesFamily x (a * b) : (a : ℕ) → V⟦Γ⟧) n) g) 0
c : ℕ × ℕ
hcn : c.1 + c.2 = n
hc :
  HSMul.hSMul (α := R) ((PowerSeries.coeff c.1 : (a : PowerSeries R) → R) a : R)
      (HSMul.hSMul (α := R) ((PowerSeries.coeff c.2 : (a : PowerSeries R) → R) b : R) ((x ^ n).coeff g)) ≠
    0
⊢ (HSMul.hSMul (α := R) (β := V⟦Γ⟧) ((PowerSeries.coeff c.1 : (a : PowerSeries R) → R) a : R)
            ((powers x : (a : ℕ) → V⟦Γ⟧) c.1 : V⟦Γ⟧) *
          HSMul.hSMul (α := R) (β := V⟦Γ⟧) ((PowerSeries.coeff c.2 : (a : PowerSeries R) → R) b : R)
            ((powers x : (a : ℕ) → V⟦Γ⟧) c.2 : V⟦Γ⟧)).coeff
      g ≠
    0
have h₁ : (HSMul.hSMul (α := R) ((PowerSeries.coeff c.1 : (a : PowerSeries R) → R) a : R)
        (HSMul.hSMul (α := R) ((PowerSeries.coeff c.2 : (a : PowerSeries R) → R) b : R) (x ^ n))).coeff
    g ≠
  0
Tactic: rw [powers_of_orderTop_pos h c.1, powers_of_orderTop_pos h c.2, Algebra.smul_mul_assoc,
      Algebra.mul_smul_comm, ← pow_add, hcn]
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.HEval (goal)
Goal:
Γ : Type u_1
R : Type u_3
V : Type u_4
inst : AddCommMonoid Γ
inst_1 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : CommRing R
inst_4 : CommRing V
inst_5 : Algebra R V
x : V⟦Γ⟧
a b : PowerSeries R
g : Γ
h : ¬0 < x.orderTop
⊢ ((powerSeriesFamily x (a * b)).coeff g).support ⊆
    image (fun (i : ℕ × ℕ) ↦ i.1 + i.2) (((powerSeriesFamily x a).mul (powerSeriesFamily x b)).coeff g).support
have neg : ∀ (x_1 : ℕ),
  Ne.{u_4 + 1} (α := V) (HahnSeries.coeff ((powerSeriesFamily x (a * b) : (a : ℕ) → V⟦Γ⟧) x_1) g) 0 →
    x_1 ∈
      (↑(image (fun (i : ℕ × ℕ) ↦ i.1 + i.2)
            (Set.Finite.toFinset (s :=
              Function.support.{0, u_4} (M := V) fun (a_2 : ℕ × ℕ) ↦
                HahnSeries.coeff (((powerSeriesFamily x a).mul (powerSeriesFamily x b) : (a : ℕ × ℕ) → V⟦Γ⟧) a_2) g)
              sorry)) :
        Set ℕ)
Tactic: simp only [coeff_support, Set.Finite.toFinset_subset, support_subset_iff]