awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
Γ : Type u_1
Γ' : Type u_2
R : Type u_3
V : Type u_5
inst : PartialOrder Γ
inst_1 : PartialOrder Γ'
inst_2 : VAdd Γ Γ'
inst_3 : IsOrderedCancelVAdd Γ Γ'
inst_4 : AddCommMonoid V
inst_5 : MulZeroClass R
inst_6 : SMulWithZero R V
x_1 : R⟦Γ⟧
y : HahnModule Γ' R V
x : Γ'
hx :
  Membership.mem (γ := Set Γ')
    (HahnSeries.support.{u_2, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) (x_1 • y))) x
⊢ (VAddAntidiagonal (s := x_1.support) (t :=
      HahnSeries.support.{u_2, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) y)) x ⋯).Nonempty
have contrapose! : ∀
  (hx :
    VAddAntidiagonal (s := x_1.support) (t :=
        HahnSeries.support.{u_2, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) y)) x sorry =
      ∅),
  x ∉ HahnSeries.support.{u_2, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) (x_1 • y))
Tactic: contrapose! hx
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
R : Type u_3
V : Type u_5
inst : AddCommMonoid V
Γ : Type u_6
Γ' : Type u_7
inst_1 : LinearOrder Γ
inst_2 : LinearOrder Γ'
inst_3 : VAdd Γ Γ'
inst_4 : IsOrderedCancelVAdd Γ Γ'
inst_5 : MulZeroClass R
inst_6 : SMulWithZero R V
x : R⟦Γ⟧
inst_7 : VAdd.{u_6, u_7} (WithTop Γ) (WithTop Γ')
y : HahnModule Γ' R V
h : ∀ (γ : Γ) (γ' : Γ'), (↑(γ +ᵥ γ') : WithTop Γ') = (↑γ : WithTop Γ) +ᵥ (↑γ' : WithTop Γ')
⊢ HVAdd.hVAdd (β := WithTop Γ') x.orderTop
      (orderTop.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) y)) ≤
    orderTop.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) (x • y))
have pos : ∀ (hx : x = 0),
  HVAdd.hVAdd (β := WithTop Γ') x.orderTop
      (orderTop.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) y)) ≤
    orderTop.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) (x • y))
Tactic: by_cases hx : x = 0
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
R : Type u_3
V : Type u_5
inst : AddCommMonoid V
Γ : Type u_6
Γ' : Type u_7
inst_1 : LinearOrder Γ
inst_2 : LinearOrder Γ'
inst_3 : VAdd Γ Γ'
inst_4 : IsOrderedCancelVAdd Γ Γ'
inst_5 : MulZeroClass R
inst_6 : SMulWithZero R V
x : R⟦Γ⟧
inst_7 : VAdd.{u_6, u_7} (WithTop Γ) (WithTop Γ')
y : HahnModule Γ' R V
h : ∀ (γ : Γ) (γ' : Γ'), (↑(γ +ᵥ γ') : WithTop Γ') = (↑γ : WithTop Γ) +ᵥ (↑γ' : WithTop Γ')
pos :
  ∀ (hx : x = 0),
    HVAdd.hVAdd (β := WithTop Γ') x.orderTop
        (orderTop.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) y)) ≤
      orderTop.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) (x • y))
⊢ HVAdd.hVAdd (β := WithTop Γ') x.orderTop
      (orderTop.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) y)) ≤
    orderTop.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) (x • y))
have neg : ∀ (hx : ¬x = 0),
  HVAdd.hVAdd (β := WithTop Γ') x.orderTop
      (orderTop.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) y)) ≤
    orderTop.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) (x • y))
Tactic: by_cases hx : x = 0
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
R : Type u_3
V : Type u_5
inst : AddCommMonoid V
Γ : Type u_6
Γ' : Type u_7
inst_1 : LinearOrder Γ
inst_2 : LinearOrder Γ'
inst_3 : VAdd Γ Γ'
inst_4 : IsOrderedCancelVAdd Γ Γ'
inst_5 : MulZeroClass R
inst_6 : SMulWithZero R V
x : R⟦Γ⟧
inst_7 : VAdd.{u_6, u_7} (WithTop Γ) (WithTop Γ')
y : HahnModule Γ' R V
h : ∀ (γ : Γ) (γ' : Γ'), (↑(γ +ᵥ γ') : WithTop Γ') = (↑γ : WithTop Γ) +ᵥ (↑γ' : WithTop Γ')
hx : ¬x = 0
⊢ HVAdd.hVAdd (β := WithTop Γ') x.orderTop
      (orderTop.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) y)) ≤
    orderTop.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) (x • y))
have pos : ∀ (hy : y = 0),
  HVAdd.hVAdd (β := WithTop Γ') x.orderTop
      (orderTop.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) y)) ≤
    orderTop.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) (x • y))
Tactic: by_cases hy : y = 0
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
R : Type u_3
V : Type u_5
inst : AddCommMonoid V
Γ : Type u_6
Γ' : Type u_7
inst_1 : LinearOrder Γ
inst_2 : LinearOrder Γ'
inst_3 : VAdd Γ Γ'
inst_4 : IsOrderedCancelVAdd Γ Γ'
inst_5 : MulZeroClass R
inst_6 : SMulWithZero R V
x : R⟦Γ⟧
inst_7 : VAdd.{u_6, u_7} (WithTop Γ) (WithTop Γ')
y : HahnModule Γ' R V
h : ∀ (γ : Γ) (γ' : Γ'), (↑(γ +ᵥ γ') : WithTop Γ') = (↑γ : WithTop Γ) +ᵥ (↑γ' : WithTop Γ')
hx : ¬x = 0
pos :
  ∀ (hy : y = 0),
    HVAdd.hVAdd (β := WithTop Γ') x.orderTop
        (orderTop.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) y)) ≤
      orderTop.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) (x • y))
⊢ HVAdd.hVAdd (β := WithTop Γ') x.orderTop
      (orderTop.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) y)) ≤
    orderTop.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) (x • y))
have neg : ∀ (hy : ¬y = 0),
  HVAdd.hVAdd (β := WithTop Γ') x.orderTop
      (orderTop.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) y)) ≤
    orderTop.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) (x • y))
Tactic: by_cases hy : y = 0
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (hypothesis)
Goal:
R : Type u_3
V : Type u_5
inst : AddCommMonoid V
Γ : Type u_6
Γ' : Type u_7
inst_1 : LinearOrder Γ
inst_2 : LinearOrder Γ'
inst_3 : VAdd Γ Γ'
inst_4 : IsOrderedCancelVAdd Γ Γ'
inst_5 : MulZeroClass R
inst_6 : SMulWithZero R V
x : R⟦Γ⟧
inst_7 : VAdd.{u_6, u_7} (WithTop Γ) (WithTop Γ')
y : HahnModule Γ' R V
h : ∀ (γ : Γ) (γ' : Γ'), (↑(γ +ᵥ γ') : WithTop Γ') = (↑γ : WithTop Γ) +ᵥ (↑γ' : WithTop Γ')
hx : ¬x = 0
hy : ¬y = 0
⊢ HVAdd.hVAdd (β := WithTop Γ') x.orderTop
      (orderTop.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) y)) ≤
    orderTop.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) (x • y))
have hhy : Ne (α := V⟦Γ'⟧) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) y : V⟦Γ'⟧) (0 : V⟦Γ'⟧)
Tactic: have hhy : ((of R).symm y) ≠ 0 := hy
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
R : Type u_3
V : Type u_5
inst : AddCommMonoid V
Γ : Type u_6
Γ' : Type u_7
inst_1 : LinearOrder Γ
inst_2 : LinearOrder Γ'
inst_3 : VAdd Γ Γ'
inst_4 : IsOrderedCancelVAdd Γ Γ'
inst_5 : MulZeroClass R
inst_6 : SMulWithZero R V
x : R⟦Γ⟧
inst_7 : VAdd.{u_6, u_7} (WithTop Γ) (WithTop Γ')
y : HahnModule Γ' R V
h : ∀ (γ : Γ) (γ' : Γ'), (↑(γ +ᵥ γ') : WithTop Γ') = (↑γ : WithTop Γ) +ᵥ (↑γ' : WithTop Γ')
hx : ¬x = 0
hy : ¬y = 0
hhy : Ne (α := V⟦Γ'⟧) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) y : V⟦Γ'⟧) (0 : V⟦Γ'⟧)
⊢ HVAdd.hVAdd (β := WithTop Γ') x.orderTop
      (orderTop.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) y)) ≤
    orderTop.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) (x • y))
have neg : (↑(Set.IsWF.min (s :=
        HVAdd.hVAdd (β := Set Γ') x.support
          (HahnSeries.support.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) y)))
        sorry sorry) :
    WithTop Γ') ≤
  orderTop.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) (x • y))
Tactic: rw [HahnSeries.orderTop_of_ne_zero hx, HahnSeries.orderTop_of_ne_zero hhy, ← h,
      ← Set.IsWF.min_vadd]
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
R : Type u_3
V : Type u_5
inst : AddCommMonoid V
Γ : Type u_6
Γ' : Type u_7
inst_1 : LinearOrder Γ
inst_2 : LinearOrder Γ'
inst_3 : VAdd Γ Γ'
inst_4 : IsOrderedCancelVAdd Γ Γ'
inst_5 : MulZeroClass R
inst_6 : SMulWithZero R V
x : R⟦Γ⟧
inst_7 : VAdd.{u_6, u_7} (WithTop Γ) (WithTop Γ')
y : HahnModule Γ' R V
h : ∀ (γ : Γ) (γ' : Γ'), (↑(γ +ᵥ γ') : WithTop Γ') = (↑γ : WithTop Γ) +ᵥ (↑γ' : WithTop Γ')
hx : ¬x = 0
hy : ¬y = 0
hhy : Ne (α := V⟦Γ'⟧) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) y : V⟦Γ'⟧) (0 : V⟦Γ'⟧)
⊢ (↑(Set.IsWF.min (s :=
          HVAdd.hVAdd (β := Set Γ') x.support
            (HahnSeries.support.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) y)))
          ⋯ ⋯) :
      WithTop Γ') ≤
    orderTop.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) (x • y))
have pos : ∀ (hxy : Eq (α := V⟦Γ'⟧) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) (x • y) : V⟦Γ'⟧) (0 : V⟦Γ'⟧)),
  (↑(Set.IsWF.min (s :=
          HVAdd.hVAdd (β := Set Γ') x.support
            (HahnSeries.support.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) y)))
          sorry sorry) :
      WithTop Γ') ≤
    orderTop.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) (x • y))
Tactic: by_cases hxy : (of R).symm (x • y) = 0
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
R : Type u_3
V : Type u_5
inst : AddCommMonoid V
Γ : Type u_6
Γ' : Type u_7
inst_1 : LinearOrder Γ
inst_2 : LinearOrder Γ'
inst_3 : VAdd Γ Γ'
inst_4 : IsOrderedCancelVAdd Γ Γ'
inst_5 : MulZeroClass R
inst_6 : SMulWithZero R V
x : R⟦Γ⟧
inst_7 : VAdd.{u_6, u_7} (WithTop Γ) (WithTop Γ')
y : HahnModule Γ' R V
h : ∀ (γ : Γ) (γ' : Γ'), (↑(γ +ᵥ γ') : WithTop Γ') = (↑γ : WithTop Γ) +ᵥ (↑γ' : WithTop Γ')
hx : ¬x = 0
hy : ¬y = 0
hhy : Ne (α := V⟦Γ'⟧) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) y : V⟦Γ'⟧) (0 : V⟦Γ'⟧)
pos :
  ∀ (hxy : Eq (α := V⟦Γ'⟧) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) (x • y) : V⟦Γ'⟧) (0 : V⟦Γ'⟧)),
    (↑(Set.IsWF.min (s :=
            HVAdd.hVAdd (β := Set Γ') x.support
              (HahnSeries.support.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) y)))
            ⋯ ⋯) :
        WithTop Γ') ≤
      orderTop.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) (x • y))
⊢ (↑(Set.IsWF.min (s :=
          HVAdd.hVAdd (β := Set Γ') x.support
            (HahnSeries.support.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) y)))
          ⋯ ⋯) :
      WithTop Γ') ≤
    orderTop.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) (x • y))
have neg : ∀ (hxy : ¬Eq (α := V⟦Γ'⟧) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) (x • y) : V⟦Γ'⟧) (0 : V⟦Γ'⟧)),
  (↑(Set.IsWF.min (s :=
          HVAdd.hVAdd (β := Set Γ') x.support
            (HahnSeries.support.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) y)))
          sorry sorry) :
      WithTop Γ') ≤
    orderTop.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) (x • y))
Tactic: by_cases hxy : (of R).symm (x • y) = 0
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
R : Type u_3
V : Type u_5
inst : AddCommMonoid V
Γ : Type u_6
Γ' : Type u_7
inst_1 : LinearOrder Γ
inst_2 : LinearOrder Γ'
inst_3 : VAdd Γ Γ'
inst_4 : IsOrderedCancelVAdd Γ Γ'
inst_5 : MulZeroClass R
inst_6 : SMulWithZero R V
x : R⟦Γ⟧
inst_7 : VAdd.{u_6, u_7} (WithTop Γ) (WithTop Γ')
y : HahnModule Γ' R V
h : ∀ (γ : Γ) (γ' : Γ'), (↑(γ +ᵥ γ') : WithTop Γ') = (↑γ : WithTop Γ) +ᵥ (↑γ' : WithTop Γ')
hx : ¬x = 0
hy : ¬y = 0
hhy : Ne (α := V⟦Γ'⟧) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) y : V⟦Γ'⟧) (0 : V⟦Γ'⟧)
hxy : Eq (α := V⟦Γ'⟧) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) (x • y) : V⟦Γ'⟧) (0 : V⟦Γ'⟧)
⊢ (↑(Set.IsWF.min (s :=
          HVAdd.hVAdd (β := Set Γ') x.support
            (HahnSeries.support.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) y)))
          ⋯ ⋯) :
      WithTop Γ') ≤
    orderTop.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) (x • y))
have pos : (↑(Set.IsWF.min (s :=
        HVAdd.hVAdd (β := Set Γ') x.support
          (HahnSeries.support.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) y)))
        sorry sorry) :
    WithTop Γ') ≤
  ⊤
Tactic: rw [hxy, HahnSeries.orderTop_zero]
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
R : Type u_3
V : Type u_5
inst : AddCommMonoid V
Γ : Type u_6
Γ' : Type u_7
inst_1 : LinearOrder Γ
inst_2 : LinearOrder Γ'
inst_3 : VAdd Γ Γ'
inst_4 : IsOrderedCancelVAdd Γ Γ'
inst_5 : MulZeroClass R
inst_6 : SMulWithZero R V
x : R⟦Γ⟧
inst_7 : VAdd.{u_6, u_7} (WithTop Γ) (WithTop Γ')
y : HahnModule Γ' R V
h : ∀ (γ : Γ) (γ' : Γ'), (↑(γ +ᵥ γ') : WithTop Γ') = (↑γ : WithTop Γ) +ᵥ (↑γ' : WithTop Γ')
hx : ¬x = 0
hy : ¬y = 0
hhy : Ne (α := V⟦Γ'⟧) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) y : V⟦Γ'⟧) (0 : V⟦Γ'⟧)
hxy : ¬Eq (α := V⟦Γ'⟧) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) (x • y) : V⟦Γ'⟧) (0 : V⟦Γ'⟧)
⊢ (↑(Set.IsWF.min (s :=
          HVAdd.hVAdd (β := Set Γ') x.support
            (HahnSeries.support.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) y)))
          ⋯ ⋯) :
      WithTop Γ') ≤
    orderTop.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) (x • y))
have neg : Set.IsWF.min (s :=
    HVAdd.hVAdd (β := Set Γ') x.support
      (HahnSeries.support.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) y)))
    sorry sorry ≤
  Set.IsWF.min (s :=
    HahnSeries.support.{u_7, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ' R V) → V⟦Γ'⟧) (x • y))) sorry sorry
Tactic: rw [HahnSeries.orderTop_of_ne_zero hxy, WithTop.coe_le_coe]
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
R : Type u_3
V : Type u_5
inst : AddCommMonoid V
Γ : Type u_6
inst_1 : AddCommMonoid Γ
inst_2 : LinearOrder Γ
inst_3 : IsOrderedCancelAddMonoid Γ
inst_4 : Zero R
inst_5 : SMulWithZero R V
x : R⟦Γ⟧
y : HahnModule Γ R V
⊢ coeff ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) (x • y))
      (x.order + order.{u_6, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) y)) =
    HSMul.hSMul (β := V) x.leadingCoeff
      (leadingCoeff.{u_6, u_5} (Γ := Γ) ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) y))
have pos : ∀ (hx : x = 0),
  coeff ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) (x • y))
      (x.order + order.{u_6, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) y)) =
    HSMul.hSMul (β := V) x.leadingCoeff
      (leadingCoeff.{u_6, u_5} (Γ := Γ) ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) y))
Tactic: by_cases hx : x = (0 : R⟦Γ⟧)
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
R : Type u_3
V : Type u_5
inst : AddCommMonoid V
Γ : Type u_6
inst_1 : AddCommMonoid Γ
inst_2 : LinearOrder Γ
inst_3 : IsOrderedCancelAddMonoid Γ
inst_4 : Zero R
inst_5 : SMulWithZero R V
x : R⟦Γ⟧
y : HahnModule Γ R V
pos :
  ∀ (hx : x = 0),
    coeff ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) (x • y))
        (x.order + order.{u_6, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) y)) =
      HSMul.hSMul (β := V) x.leadingCoeff
        (leadingCoeff.{u_6, u_5} (Γ := Γ) ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) y))
⊢ coeff ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) (x • y))
      (x.order + order.{u_6, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) y)) =
    HSMul.hSMul (β := V) x.leadingCoeff
      (leadingCoeff.{u_6, u_5} (Γ := Γ) ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) y))
have neg : ∀ (hx : ¬x = 0),
  coeff ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) (x • y))
      (x.order + order.{u_6, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) y)) =
    HSMul.hSMul (β := V) x.leadingCoeff
      (leadingCoeff.{u_6, u_5} (Γ := Γ) ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) y))
Tactic: by_cases hx : x = (0 : R⟦Γ⟧)
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
R : Type u_3
V : Type u_5
inst : AddCommMonoid V
Γ : Type u_6
inst_1 : AddCommMonoid Γ
inst_2 : LinearOrder Γ
inst_3 : IsOrderedCancelAddMonoid Γ
inst_4 : Zero R
inst_5 : SMulWithZero R V
x : R⟦Γ⟧
y : HahnModule Γ R V
hx : ¬x = 0
⊢ coeff ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) (x • y))
      (x.order + order.{u_6, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) y)) =
    HSMul.hSMul (β := V) x.leadingCoeff
      (leadingCoeff.{u_6, u_5} (Γ := Γ) ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) y))
have pos : ∀ (hy : Eq (α := V⟦Γ⟧) ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) y : V⟦Γ⟧) (0 : V⟦Γ⟧)),
  coeff ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) (x • y))
      (x.order + order.{u_6, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) y)) =
    HSMul.hSMul (β := V) x.leadingCoeff
      (leadingCoeff.{u_6, u_5} (Γ := Γ) ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) y))
Tactic: by_cases hy : (of R).symm y = 0
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
R : Type u_3
V : Type u_5
inst : AddCommMonoid V
Γ : Type u_6
inst_1 : AddCommMonoid Γ
inst_2 : LinearOrder Γ
inst_3 : IsOrderedCancelAddMonoid Γ
inst_4 : Zero R
inst_5 : SMulWithZero R V
x : R⟦Γ⟧
y : HahnModule Γ R V
hx : ¬x = 0
pos :
  ∀ (hy : Eq (α := V⟦Γ⟧) ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) y : V⟦Γ⟧) (0 : V⟦Γ⟧)),
    coeff ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) (x • y))
        (x.order + order.{u_6, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) y)) =
      HSMul.hSMul (β := V) x.leadingCoeff
        (leadingCoeff.{u_6, u_5} (Γ := Γ) ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) y))
⊢ coeff ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) (x • y))
      (x.order + order.{u_6, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) y)) =
    HSMul.hSMul (β := V) x.leadingCoeff
      (leadingCoeff.{u_6, u_5} (Γ := Γ) ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) y))
have neg : ∀ (hy : ¬Eq (α := V⟦Γ⟧) ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) y : V⟦Γ⟧) (0 : V⟦Γ⟧)),
  coeff ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) (x • y))
      (x.order + order.{u_6, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) y)) =
    HSMul.hSMul (β := V) x.leadingCoeff
      (leadingCoeff.{u_6, u_5} (Γ := Γ) ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) y))
Tactic: by_cases hy : (of R).symm y = 0
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
R : Type u_3
V : Type u_5
inst : AddCommMonoid V
Γ : Type u_6
inst_1 : AddCommMonoid Γ
inst_2 : LinearOrder Γ
inst_3 : IsOrderedCancelAddMonoid Γ
inst_4 : Zero R
inst_5 : SMulWithZero R V
x : R⟦Γ⟧
y : HahnModule Γ R V
hx : ¬x = 0
hy : ¬Eq (α := V⟦Γ⟧) ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) y : V⟦Γ⟧) (0 : V⟦Γ⟧)
⊢ coeff ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) (x • y))
      (x.order + order.{u_6, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) y)) =
    HSMul.hSMul (β := V) x.leadingCoeff
      (leadingCoeff.{u_6, u_5} (Γ := Γ) ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) y))
have neg : HSMul.hSMul (β := V)
    (x.coeff
      (sorry.min (s := x.support) sorry,
          Set.IsWF.min (s :=
            HahnSeries.support.{u_6, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) y)) sorry sorry).1)
    (coeff ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) y)
      (sorry.min (s := x.support) sorry,
          Set.IsWF.min (s :=
            HahnSeries.support.{u_6, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) y)) sorry sorry).2) =
  HSMul.hSMul (β := V) (x.coeff (x.orderTop.untop sorry))
    (coeff ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) y)
      (WithTop.untop (orderTop.{u_6, u_5} (R := V) ((Equiv.symm (of R) : (a : HahnModule Γ R V) → V⟦Γ⟧) y)) sorry))
Tactic: rw [HahnSeries.order_of_ne hx, HahnSeries.order_of_ne hy, coeff_smul,
    HahnSeries.leadingCoeff_of_ne_zero hx, HahnSeries.leadingCoeff_of_ne_zero hy, ← vadd_eq_add,
    Finset.vaddAntidiagonal_min_vadd_min, Finset.sum_singleton]
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
Γ : Type u_1
R : Type u_3
S : Type u_4
inst : AddCommMonoid Γ
inst_1 : PartialOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
inst_4 : NonUnitalNonAssocSemiring S
f : R →ₙ+* S
x y : R⟦Γ⟧
⊢ (x * y).map f = HMul.hMul (α := S⟦Γ⟧) (x.map f) (y.map f)
have coeff : ∀ (x_1 : Γ), coeff ((x * y).map f) x_1 = (HMul.hMul (α := S⟦Γ⟧) (x.map f) (y.map f)).coeff x_1
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
Γ : Type u_1
R : Type u_3
S : Type u_4
inst : AddCommMonoid Γ
inst_1 : PartialOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
inst_4 : NonUnitalNonAssocSemiring S
f : R →ₙ+* S
x y : R⟦Γ⟧
x_1 : Γ
⊢ coeff ((x * y).map f) x_1 = (HMul.hMul (α := S⟦Γ⟧) (x.map f) (y.map f)).coeff x_1
have coeff : Eq.{u_4 + 1} (α := S)
  (∑ x_2 ∈ addAntidiagonal (s := x.support) (t := y.support) sorry sorry x_1,
    HMul.hMul (α := S) (β := S) ((f : (a : R) → S) (x.coeff x_2.1) : S) ((f : (a : R) → S) (y.coeff x_2.2) : S))
  (∑ x_2 ∈ addAntidiagonal (s := support (x.map f)) (t := support (y.map f)) sorry sorry x_1,
    HMul.hMul (α := S) (β := S) ((f : (a : R) → S) (x.coeff x_2.1) : S) ((f : (a : R) → S) (y.coeff x_2.2) : S))
Tactic: simp only [map_coeff, coeff_mul, map_sum, map_mul]
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
Γ : Type u_1
R : Type u_3
S : Type u_4
inst : AddCommMonoid Γ
inst_1 : PartialOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
inst_4 : NonUnitalNonAssocSemiring S
f : R →ₙ+* S
x y : R⟦Γ⟧
x_1 : Γ
⊢ Eq.{u_4 + 1} (α := S)
    (∑ x_2 ∈ addAntidiagonal (s := x.support) (t := y.support) ⋯ ⋯ x_1,
      HMul.hMul (α := S) (β := S) ((f : (a : R) → S) (x.coeff x_2.1) : S) ((f : (a : R) → S) (y.coeff x_2.2) : S))
    (∑ x_2 ∈ addAntidiagonal (s := support (x.map f)) (t := support (y.map f)) ⋯ ⋯ x_1,
      HMul.hMul (α := S) (β := S) ((f : (a : R) → S) (x.coeff x_2.1) : S) ((f : (a : R) → S) (y.coeff x_2.2) : S))
have coeff : ∀ (gh : Γ × Γ) (hgh : gh ∈ addAntidiagonal (s := support (x.map f)) (t := support (y.map f)) sorry sorry x_1),
  gh ∈ addAntidiagonal (s := x.support) (t := y.support) sorry sorry x_1
Tactic: refine Eq.symm (sum_subset (fun gh hgh => ?_) (fun gh hgh hz => ?_))
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
Γ : Type u_1
R : Type u_3
S : Type u_4
inst : AddCommMonoid Γ
inst_1 : PartialOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
inst_4 : NonUnitalNonAssocSemiring S
f : R →ₙ+* S
x y : R⟦Γ⟧
x_1 : Γ
coeff :
  ∀ (gh : Γ × Γ) (hgh : gh ∈ addAntidiagonal (s := support (x.map f)) (t := support (y.map f)) ⋯ ⋯ x_1),
    gh ∈ addAntidiagonal (s := x.support) (t := y.support) ⋯ ⋯ x_1
⊢ Eq.{u_4 + 1} (α := S)
    (∑ x_2 ∈ addAntidiagonal (s := x.support) (t := y.support) ⋯ ⋯ x_1,
      HMul.hMul (α := S) (β := S) ((f : (a : R) → S) (x.coeff x_2.1) : S) ((f : (a : R) → S) (y.coeff x_2.2) : S))
    (∑ x_2 ∈ addAntidiagonal (s := support (x.map f)) (t := support (y.map f)) ⋯ ⋯ x_1,
      HMul.hMul (α := S) (β := S) ((f : (a : R) → S) (x.coeff x_2.1) : S) ((f : (a : R) → S) (y.coeff x_2.2) : S))
have coeff₁ : ∀ (gh : Γ × Γ) (hgh : gh ∈ addAntidiagonal (s := x.support) (t := y.support) sorry sorry x_1)
  (hz : gh ∉ addAntidiagonal (s := support (x.map f)) (t := support (y.map f)) sorry sorry x_1),
  HMul.hMul (α := S) (β := S) ((f : (a : R) → S) (x.coeff gh.1) : S) ((f : (a : R) → S) (y.coeff gh.2) : S) = 0
Tactic: refine Eq.symm (sum_subset (fun gh hgh => ?_) (fun gh hgh hz => ?_))