awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- 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
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
have coeff : ∀
  (hgh :
    ¬Eq.{u_4 + 1} (α := S) ((f : (a : R) → S) (x.coeff gh.1) : S) (0 : S) ∧
      ¬Eq.{u_4 + 1} (α := S) ((f : (a : R) → S) (y.coeff gh.2) : S) (0 : S) ∧ gh.1 + gh.2 = x_1),
  ¬x.coeff gh.1 = 0 ∧ ¬y.coeff gh.2 = 0
Tactic: simp_all only [mem_addAntidiagonal, mem_support, map_coeff, ne_eq, and_true]
Cleared hypotheses: hgh
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (hypothesis)
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
gh : Γ × Γ
hgh : gh ∈ addAntidiagonal (s := x.support) (t := y.support) ⋯ ⋯ x_1
hz : gh ∉ 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 gh.1) : S) ((f : (a : R) → S) (y.coeff gh.2) : S) = 0
have hgh₁ : ¬x.coeff gh.1 = 0 ∧ ¬y.coeff gh.2 = 0 ∧ gh.1 + gh.2 = x_1
Tactic: simp_all only [mem_addAntidiagonal, mem_support, ne_eq, map_coeff, and_true,
      not_and, not_not]
Cleared hypotheses: hgh, hz
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (hypothesis)
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
gh : Γ × Γ
hgh : gh ∈ addAntidiagonal (s := x.support) (t := y.support) ⋯ ⋯ x_1
hz : gh ∉ addAntidiagonal (s := support (x.map f)) (t := support (y.map f)) ⋯ ⋯ x_1
hgh₁ : ¬x.coeff gh.1 = 0 ∧ ¬y.coeff gh.2 = 0 ∧ gh.1 + gh.2 = 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
have hz₁ : ¬Eq.{u_4 + 1} (α := S) ((f : (a : R) → S) (x.coeff gh.1) : S) (0 : S) →
  Eq.{u_4 + 1} (α := S) ((f : (a : R) → S) (y.coeff gh.2) : S) (0 : S)
Tactic: simp_all only [mem_addAntidiagonal, mem_support, ne_eq, map_coeff, and_true,
      not_and, not_not]
Cleared hypotheses: 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
gh : Γ × Γ
hgh : ¬x.coeff gh.1 = 0 ∧ ¬y.coeff gh.2 = 0 ∧ gh.1 + gh.2 = x_1
hz :
  ¬Eq.{u_4 + 1} (α := S) ((f : (a : R) → S) (x.coeff gh.1) : S) (0 : S) →
    Eq.{u_4 + 1} (α := S) ((f : (a : R) → S) (y.coeff gh.2) : S) (0 : S)
⊢ 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
have pos : ∀ (h : Eq.{u_4 + 1} (α := S) ((f : (a : R) → S) (x.coeff gh.1) : S) (0 : S)),
  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: by_cases h : f (x.coeff gh.1) = 0
-- 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
gh : Γ × Γ
hgh : ¬x.coeff gh.1 = 0 ∧ ¬y.coeff gh.2 = 0 ∧ gh.1 + gh.2 = x_1
hz :
  ¬Eq.{u_4 + 1} (α := S) ((f : (a : R) → S) (x.coeff gh.1) : S) (0 : S) →
    Eq.{u_4 + 1} (α := S) ((f : (a : R) → S) (y.coeff gh.2) : S) (0 : S)
pos :
  ∀ (h : Eq.{u_4 + 1} (α := S) ((f : (a : R) → S) (x.coeff gh.1) : S) (0 : S)),
    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
⊢ 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
have neg : ∀ (h : ¬Eq.{u_4 + 1} (α := S) ((f : (a : R) → S) (x.coeff gh.1) : S) (0 : S)),
  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: by_cases h : f (x.coeff gh.1) = 0
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
Γ : Type u_1
Γ' : Type u_2
R : Type u_3
S : Type u_4
V : Type u_5
inst : AddCommMonoid Γ
inst_1 : PartialOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
x y z : R⟦Γ⟧
⊢ x * (y + z) = x * y + x * z
have simp : (Equiv.symm (HahnModule.of R) : (a : HahnModule Γ R R) → R⟦Γ⟧)
    (HSMul.hSMul (β := HahnModule Γ R R) x
      ((HahnModule.of R : (a : R⟦Γ⟧) → HahnModule Γ R R) (y + z) : HahnModule Γ R R)) =
  HAdd.hAdd (α := R⟦Γ⟧) (β := R⟦Γ⟧)
    ((Equiv.symm (HahnModule.of R) : (a : HahnModule Γ R R) → R⟦Γ⟧)
        (HSMul.hSMul (β := HahnModule Γ R R) x
          ((HahnModule.of R : (a : R⟦Γ⟧) → HahnModule Γ R R) y : HahnModule Γ R R)) :
      R⟦Γ⟧)
    ((Equiv.symm (HahnModule.of R) : (a : HahnModule Γ R R) → R⟦Γ⟧)
        (HSMul.hSMul (β := HahnModule Γ R R) x
          ((HahnModule.of R : (a : R⟦Γ⟧) → HahnModule Γ R R) z : HahnModule Γ R R)) :
      R⟦Γ⟧)
Tactic: simp only [← of_symm_smul_of_eq_mul]
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
Γ : Type u_1
Γ' : Type u_2
R : Type u_3
S : Type u_4
V : Type u_5
inst : AddCommMonoid Γ
inst_1 : PartialOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
x y z : R⟦Γ⟧
⊢ (x + y) * z = x * z + y * z
have simp : (Equiv.symm (HahnModule.of R) : (a : HahnModule Γ R R) → R⟦Γ⟧)
    (HSMul.hSMul (β := HahnModule Γ R R) (x + y)
      ((HahnModule.of R : (a : R⟦Γ⟧) → HahnModule Γ R R) z : HahnModule Γ R R)) =
  HAdd.hAdd (α := R⟦Γ⟧) (β := R⟦Γ⟧)
    ((Equiv.symm (HahnModule.of R) : (a : HahnModule Γ R R) → R⟦Γ⟧)
        (HSMul.hSMul (β := HahnModule Γ R R) x
          ((HahnModule.of R : (a : R⟦Γ⟧) → HahnModule Γ R R) z : HahnModule Γ R R)) :
      R⟦Γ⟧)
    ((Equiv.symm (HahnModule.of R) : (a : HahnModule Γ R R) → R⟦Γ⟧)
        (HSMul.hSMul (β := HahnModule Γ R R) y
          ((HahnModule.of R : (a : R⟦Γ⟧) → HahnModule Γ R R) z : HahnModule Γ R R)) :
      R⟦Γ⟧)
Tactic: simp only [← of_symm_smul_of_eq_mul]
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
Γ : Type u_1
Γ' : Type u_2
R : Type u_3
S : Type u_4
V : Type u_5
inst : AddCommMonoid Γ
inst_1 : PartialOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
x y z : R⟦Γ⟧
⊢ (Equiv.symm (HahnModule.of R) : (a : HahnModule Γ R R) → R⟦Γ⟧)
      (HSMul.hSMul (β := HahnModule Γ R R) (x + y)
        ((HahnModule.of R : (a : R⟦Γ⟧) → HahnModule Γ R R) z : HahnModule Γ R R)) =
    HAdd.hAdd (α := R⟦Γ⟧) (β := R⟦Γ⟧)
      ((Equiv.symm (HahnModule.of R) : (a : HahnModule Γ R R) → R⟦Γ⟧)
          (HSMul.hSMul (β := HahnModule Γ R R) x
            ((HahnModule.of R : (a : R⟦Γ⟧) → HahnModule Γ R R) z : HahnModule Γ R R)) :
        R⟦Γ⟧)
      ((Equiv.symm (HahnModule.of R) : (a : HahnModule Γ R R) → R⟦Γ⟧)
          (HSMul.hSMul (β := HahnModule Γ R R) y
            ((HahnModule.of R : (a : R⟦Γ⟧) → HahnModule Γ R R) z : HahnModule Γ R R)) :
        R⟦Γ⟧)
have refine : ∀ (r s u : R), (r + s) • u = r • u + s • u
Tactic: refine HahnModule.add_smul ?_
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
Γ : Type u_1
Γ' : Type u_2
R : Type u_3
S : Type u_4
V : Type u_5
inst : AddCommMonoid Γ
inst_1 : PartialOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
x y z : R⟦Γ⟧
⊢ ∀ (r s u : R), (r + s) • u = r • u + s • u
have simp : ∀ (r s u : R), (r + s) * u = r * u + s * u
Tactic: simp only [smul_eq_mul]
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
Γ : Type u_1
R : Type u_3
inst : AddCommMonoid Γ
inst_1 : PartialOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
r : R
x : R⟦Γ⟧
a b : Γ
⊢ (HMul.hMul (α := R⟦Γ⟧) ((single b : (a : R) → R⟦Γ⟧) r : R⟦Γ⟧) x).coeff (a + b) = r * x.coeff a
have rw : coeff
    ((Equiv.symm (HahnModule.of R) : (a : HahnModule Γ R R) → R⟦Γ⟧)
      (HSMul.hSMul (α := R⟦Γ⟧) (β := HahnModule Γ R R) ((single b : (a : R) → R⟦Γ⟧) r : R⟦Γ⟧)
        ((HahnModule.of R : (a : R⟦Γ⟧) → HahnModule Γ R R) x : HahnModule Γ R R)))
    (b +ᵥ a) =
  r * x.coeff a
Tactic: rw [← of_symm_smul_of_eq_mul, add_comm, ← vadd_eq_add]
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
Γ : Type u_1
R : Type u_3
inst : AddCommMonoid Γ
inst_1 : PartialOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
r : R
x : R⟦Γ⟧
a b : Γ
⊢ (HMul.hMul (β := R⟦Γ⟧) x ((single b : (a : R) → R⟦Γ⟧) r : R⟦Γ⟧)).coeff (a + b) = x.coeff a * r
have pos : ∀ (hr : r = 0), (HMul.hMul (β := R⟦Γ⟧) x ((single b : (a : R) → R⟦Γ⟧) r : R⟦Γ⟧)).coeff (a + b) = x.coeff a * r
Tactic: by_cases hr : r = 0
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
Γ : Type u_1
R : Type u_3
inst : AddCommMonoid Γ
inst_1 : PartialOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
r : R
x : R⟦Γ⟧
a b : Γ
pos : ∀ (hr : r = 0), (HMul.hMul (β := R⟦Γ⟧) x ((single b : (a : R) → R⟦Γ⟧) r : R⟦Γ⟧)).coeff (a + b) = x.coeff a * r
⊢ (HMul.hMul (β := R⟦Γ⟧) x ((single b : (a : R) → R⟦Γ⟧) r : R⟦Γ⟧)).coeff (a + b) = x.coeff a * r
have neg : ∀ (hr : ¬r = 0), (HMul.hMul (β := R⟦Γ⟧) x ((single b : (a : R) → R⟦Γ⟧) r : R⟦Γ⟧)).coeff (a + b) = x.coeff a * r
Tactic: by_cases hr : r = 0
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
Γ : Type u_1
R : Type u_3
inst : AddCommMonoid Γ
inst_1 : PartialOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
r : R
x : R⟦Γ⟧
a b : Γ
hr : ¬r = 0
⊢ (HMul.hMul (β := R⟦Γ⟧) x ((single b : (a : R) → R⟦Γ⟧) r : R⟦Γ⟧)).coeff (a + b) = x.coeff a * r
have neg : ∑ x_1 ∈ addAntidiagonal (s := x.support) (t := {b}) sorry sorry (a + b),
    x.coeff x_1.1 * coeff ((single b : (a : R) → R⟦Γ⟧) r) x_1.2 =
  x.coeff a * r
Tactic: simp only [hr, coeff_mul, support_single_of_ne, Ne, not_false_iff]
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
Γ : Type u_1
R : Type u_3
inst : AddCommMonoid Γ
inst_1 : PartialOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
r : R
x : R⟦Γ⟧
a b : Γ
hr : ¬r = 0
⊢ ∑ x_1 ∈ addAntidiagonal (s := x.support) (t := {b}) ⋯ ⋯ (a + b),
      x.coeff x_1.1 * coeff ((single b : (a : R) → R⟦Γ⟧) r) x_1.2 =
    x.coeff a * r
have pos : ∀ (hx : x.coeff a = 0),
  ∑ x_1 ∈ addAntidiagonal (s := x.support) (t := {b}) sorry sorry (a + b),
      x.coeff x_1.1 * coeff ((single b : (a : R) → R⟦Γ⟧) r) x_1.2 =
    x.coeff a * r
Tactic: by_cases hx : x.coeff a = 0
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
Γ : Type u_1
R : Type u_3
inst : AddCommMonoid Γ
inst_1 : PartialOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
r : R
x : R⟦Γ⟧
a b : Γ
hr : ¬r = 0
pos :
  ∀ (hx : x.coeff a = 0),
    ∑ x_1 ∈ addAntidiagonal (s := x.support) (t := {b}) ⋯ ⋯ (a + b),
        x.coeff x_1.1 * coeff ((single b : (a : R) → R⟦Γ⟧) r) x_1.2 =
      x.coeff a * r
⊢ ∑ x_1 ∈ addAntidiagonal (s := x.support) (t := {b}) ⋯ ⋯ (a + b),
      x.coeff x_1.1 * coeff ((single b : (a : R) → R⟦Γ⟧) r) x_1.2 =
    x.coeff a * r
have neg : ∀ (hx : ¬x.coeff a = 0),
  ∑ x_1 ∈ addAntidiagonal (s := x.support) (t := {b}) sorry sorry (a + b),
      x.coeff x_1.1 * coeff ((single b : (a : R) → R⟦Γ⟧) r) x_1.2 =
    x.coeff a * r
Tactic: by_cases hx : x.coeff a = 0
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
Γ : Type u_1
R : Type u_3
inst : AddCommMonoid Γ
inst_1 : PartialOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
r : R
x : R⟦Γ⟧
a b : Γ
hr : ¬r = 0
hx : x.coeff a = 0
⊢ ∑ x_1 ∈ addAntidiagonal (s := x.support) (t := {b}) ⋯ ⋯ (a + b),
      x.coeff x_1.1 * coeff ((single b : (a : R) → R⟦Γ⟧) r) x_1.2 =
    x.coeff a * r
have pos : Eq.{u_3 + 1} (α := R)
  (∑ x_1 ∈ addAntidiagonal (s := x.support) (t := {b}) sorry sorry (a + b),
    x.coeff x_1.1 * coeff ((single b : (a : R) → R⟦Γ⟧) r) x_1.2)
  0
Tactic: simp only [hx, zero_mul]
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
Γ : Type u_1
R : Type u_3
inst : AddCommMonoid Γ
inst_1 : PartialOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
r : R
x : R⟦Γ⟧
a b : Γ
hr : ¬r = 0
hx : x.coeff a = 0
⊢ Eq.{u_3 + 1} (α := R)
    (∑ x_1 ∈ addAntidiagonal (s := x.support) (t := {b}) ⋯ ⋯ (a + b),
      x.coeff x_1.1 * coeff ((single b : (a : R) → R⟦Γ⟧) r) x_1.2)
    0
have rw : addAntidiagonal (s := x.support) (t := {b}) sorry sorry (a + b) = ∅
Tactic: rw [sum_congr _ fun _ _ => rfl, sum_empty]
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
Γ : Type u_1
R : Type u_3
inst : AddCommMonoid Γ
inst_1 : PartialOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
r : R
x : R⟦Γ⟧
a b : Γ
hr : ¬r = 0
hx : x.coeff a = 0
⊢ addAntidiagonal (s := x.support) (t := {b}) ⋯ ⋯ (a + b) = ∅
have h : ∀ (a1 a2 : Γ),
  (a1, a2) ∈ addAntidiagonal (s := x.support) (t := {b}) sorry sorry (a + b) ↔ Membership.mem (γ := Finset (Γ × Γ)) ∅ (a1, a2)
Tactic: ext ⟨a1, a2⟩
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
Γ : Type u_1
R : Type u_3
inst : AddCommMonoid Γ
inst_1 : PartialOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
r : R
x : R⟦Γ⟧
a b : Γ
hr : ¬r = 0
hx : x.coeff a = 0
a1 a2 : Γ
⊢ (a1, a2) ∈ addAntidiagonal (s := x.support) (t := {b}) ⋯ ⋯ (a + b) ↔ Membership.mem (γ := Finset (Γ × Γ)) ∅ (a1, a2)
have h : a1 ∈ x.support → a2 = b → ¬a1 + a2 = a + b
Tactic: simp only [notMem_empty, not_and, Set.mem_singleton_iff,
      mem_addAntidiagonal, iff_false]
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
Γ : Type u_1
R : Type u_3
inst : AddCommMonoid Γ
inst_1 : PartialOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
r : R
x : R⟦Γ⟧
a : Γ
hr : ¬r = 0
hx : x.coeff a = 0
a1 a2 : Γ
h2 : a1 ∈ x.support
h1 : a1 + a2 = a + a2
⊢ False
have h : ∀ (hx : x.coeff a1 = 0) (a2 : Γ) (h2 : a1 ∈ x.support) (h1 : a1 + a2 = a + a2), False
Tactic: rw [← add_right_cancel h1] at hx
Cleared hypotheses: hx, a2, h2, h1
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
Γ : Type u_1
R : Type u_3
inst : AddCommMonoid Γ
inst_1 : PartialOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
r : R
x : R⟦Γ⟧
a b : Γ
hr : ¬r = 0
hx : ¬x.coeff a = 0
⊢ ∑ x_1 ∈ addAntidiagonal (s := x.support) (t := {b}) ⋯ ⋯ (a + b),
      x.coeff x_1.1 * coeff ((single b : (a : R) → R⟦Γ⟧) r) x_1.2 =
    x.coeff a * r
have trans : Eq.{u_3 + 1} (α := R)
  (∑ x_1 ∈ addAntidiagonal (s := x.support) (t := {b}) sorry sorry (a + b),
    x.coeff x_1.1 * coeff ((single b : (a : R) → R⟦Γ⟧) r) x_1.2)
  (∑ ij ∈ {(a, b)}, x.coeff ij.1 * coeff ((single b : (a : R) → R⟦Γ⟧) r) ij.2)
Tactic: trans ∑ ij ∈ {(a, b)}, x.coeff ij.fst * (single b r).coeff ij.snd
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
Γ : Type u_1
R : Type u_3
inst : AddCommMonoid Γ
inst_1 : PartialOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
r : R
x : R⟦Γ⟧
a b : Γ
hr : ¬r = 0
hx : ¬x.coeff a = 0
trans :
  Eq.{u_3 + 1} (α := R)
    (∑ x_1 ∈ addAntidiagonal (s := x.support) (t := {b}) ⋯ ⋯ (a + b),
      x.coeff x_1.1 * coeff ((single b : (a : R) → R⟦Γ⟧) r) x_1.2)
    (∑ ij ∈ {(a, b)}, x.coeff ij.1 * coeff ((single b : (a : R) → R⟦Γ⟧) r) ij.2)
⊢ ∑ x_1 ∈ addAntidiagonal (s := x.support) (t := {b}) ⋯ ⋯ (a + b),
      x.coeff x_1.1 * coeff ((single b : (a : R) → R⟦Γ⟧) r) x_1.2 =
    x.coeff a * r
have trans₁ : ∑ ij ∈ {(a, b)}, x.coeff ij.1 * coeff ((single b : (a : R) → R⟦Γ⟧) r) ij.2 = x.coeff a * r
Tactic: trans ∑ ij ∈ {(a, b)}, x.coeff ij.fst * (single b r).coeff ij.snd
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
Γ : Type u_1
R : Type u_3
inst : AddCommMonoid Γ
inst_1 : PartialOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
r : R
x : R⟦Γ⟧
a b : Γ
hr : ¬r = 0
hx : ¬x.coeff a = 0
⊢ Eq.{u_3 + 1} (α := R)
    (∑ x_1 ∈ addAntidiagonal (s := x.support) (t := {b}) ⋯ ⋯ (a + b),
      x.coeff x_1.1 * coeff ((single b : (a : R) → R⟦Γ⟧) r) x_1.2)
    (∑ ij ∈ {(a, b)}, x.coeff ij.1 * coeff ((single b : (a : R) → R⟦Γ⟧) r) ij.2)
have apply : addAntidiagonal (s := x.support) (t := {b}) sorry sorry (a + b) = {(a, b)}
Tactic: apply sum_congr _ fun _ _ => rfl
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
Γ : Type u_1
R : Type u_3
inst : AddCommMonoid Γ
inst_1 : PartialOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
r : R
x : R⟦Γ⟧
a b : Γ
hr : ¬r = 0
hx : ¬x.coeff a = 0
⊢ addAntidiagonal (s := x.support) (t := {b}) ⋯ ⋯ (a + b) = {(a, b)}
have h : ∀ (a1 a2 : Γ),
  (a1, a2) ∈ addAntidiagonal (s := x.support) (t := {b}) sorry sorry (a + b) ↔
    Membership.mem (γ := Finset (Γ × Γ)) {(a, b)} (a1, a2)
Tactic: ext ⟨a1, a2⟩
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
Γ : Type u_1
R : Type u_3
inst : AddCommMonoid Γ
inst_1 : PartialOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
r : R
x : R⟦Γ⟧
a b : Γ
hr : ¬r = 0
hx : ¬x.coeff a = 0
a1 a2 : Γ
⊢ (a1, a2) ∈ addAntidiagonal (s := x.support) (t := {b}) ⋯ ⋯ (a + b) ↔
    Membership.mem (γ := Finset (Γ × Γ)) {(a, b)} (a1, a2)
have h : a1 ∈ x.support ∧ a2 = b ∧ a1 + a2 = a + b ↔ a1 = a ∧ a2 = b
Tactic: simp only [Set.mem_singleton_iff, Prod.mk_inj, mem_addAntidiagonal, mem_singleton]
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
Γ : Type u_1
R : Type u_3
inst : AddCommMonoid Γ
inst_1 : PartialOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
r : R
x : R⟦Γ⟧
a b : Γ
hr : ¬r = 0
hx : ¬x.coeff a = 0
a1 a2 : Γ
⊢ a1 ∈ x.support ∧ a2 = b ∧ a1 + a2 = a + b ↔ a1 = a ∧ a2 = b
have h : a1 ∈ x.support ∧ a2 = b ∧ a1 + a2 = a + b → a1 = a ∧ a2 = b
Tactic: constructor
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
Γ : Type u_1
R : Type u_3
inst : AddCommMonoid Γ
inst_1 : PartialOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
r : R
x : R⟦Γ⟧
a b : Γ
hr : ¬r = 0
hx : ¬x.coeff a = 0
a1 a2 : Γ
h : a1 ∈ x.support ∧ a2 = b ∧ a1 + a2 = a + b → a1 = a ∧ a2 = b
⊢ a1 ∈ x.support ∧ a2 = b ∧ a1 + a2 = a + b ↔ a1 = a ∧ a2 = b
have h₁ : a1 = a ∧ a2 = b → a1 ∈ x.support ∧ a2 = b ∧ a1 + a2 = a + b
Tactic: constructor
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
Γ : Type u_1
R : Type u_3
inst : AddCommMonoid Γ
inst_1 : PartialOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
r : R
x : R⟦Γ⟧
a : Γ
⊢ (HMul.hMul (β := R⟦Γ⟧) x ((single 0 : (a : R) → R⟦Γ⟧) r : R⟦Γ⟧)).coeff a = x.coeff a * r
have rw : (HMul.hMul (β := R⟦Γ⟧) x ((single 0 : (a : R) → R⟦Γ⟧) r : R⟦Γ⟧)).coeff (a + 0) = x.coeff (a + 0) * r
Tactic: rw [← add_zero a, coeff_mul_single_add, add_zero]
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
Γ : Type u_1
R : Type u_3
inst : AddCommMonoid Γ
inst_1 : PartialOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
r : R
x : R⟦Γ⟧
a : Γ
rw : (HMul.hMul (β := R⟦Γ⟧) x ((single 0 : (a : R) → R⟦Γ⟧) r : R⟦Γ⟧)).coeff (a + 0) = x.coeff (a + 0) * r
⊢ (HMul.hMul (β := R⟦Γ⟧) x ((single 0 : (a : R) → R⟦Γ⟧) r : R⟦Γ⟧)).coeff a = x.coeff a * r
have rw₁ : x.coeff a * r = x.coeff (a + 0) * r
Tactic: rw [← add_zero a, coeff_mul_single_add, add_zero]
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
Γ : Type u_1
R : Type u_3
inst : AddCommMonoid Γ
inst_1 : PartialOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
r : R
x : R⟦Γ⟧
a : Γ
rw : (HMul.hMul (β := R⟦Γ⟧) x ((single 0 : (a : R) → R⟦Γ⟧) r : R⟦Γ⟧)).coeff (a + 0) = x.coeff (a + 0) * r
rw₁ : x.coeff a * r = x.coeff (a + 0) * r
⊢ (HMul.hMul (β := R⟦Γ⟧) x ((single 0 : (a : R) → R⟦Γ⟧) r : R⟦Γ⟧)).coeff a = x.coeff a * r
have rw₂ : x.coeff a * r = x.coeff a * r
Tactic: rw [← add_zero a, coeff_mul_single_add, add_zero]
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
Γ : Type u_1
R : Type u_3
inst : AddCommMonoid Γ
inst_1 : PartialOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
r : R
x : R⟦Γ⟧
a : Γ
⊢ (HMul.hMul (α := R⟦Γ⟧) ((single 0 : (a : R) → R⟦Γ⟧) r : R⟦Γ⟧) x).coeff a = r * x.coeff a
have rw : (HMul.hMul (α := R⟦Γ⟧) ((single 0 : (a : R) → R⟦Γ⟧) r : R⟦Γ⟧) x).coeff (a + 0) = r * x.coeff (a + 0)
Tactic: rw [← add_zero a, coeff_single_mul_add, add_zero]
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
Γ : Type u_1
R : Type u_3
inst : AddCommMonoid Γ
inst_1 : PartialOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
r : R
x : R⟦Γ⟧
a : Γ
rw : (HMul.hMul (α := R⟦Γ⟧) ((single 0 : (a : R) → R⟦Γ⟧) r : R⟦Γ⟧) x).coeff (a + 0) = r * x.coeff (a + 0)
⊢ (HMul.hMul (α := R⟦Γ⟧) ((single 0 : (a : R) → R⟦Γ⟧) r : R⟦Γ⟧) x).coeff a = r * x.coeff a
have rw₁ : r * x.coeff a = r * x.coeff (a + 0)
Tactic: rw [← add_zero a, coeff_single_mul_add, add_zero]
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
Γ : Type u_1
R : Type u_3
inst : AddCommMonoid Γ
inst_1 : PartialOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
r : R
x : R⟦Γ⟧
a : Γ
rw : (HMul.hMul (α := R⟦Γ⟧) ((single 0 : (a : R) → R⟦Γ⟧) r : R⟦Γ⟧) x).coeff (a + 0) = r * x.coeff (a + 0)
rw₁ : r * x.coeff a = r * x.coeff (a + 0)
⊢ (HMul.hMul (α := R⟦Γ⟧) ((single 0 : (a : R) → R⟦Γ⟧) r : R⟦Γ⟧) x).coeff a = r * x.coeff a
have rw₂ : r * x.coeff a = r * x.coeff a
Tactic: rw [← add_zero a, coeff_single_mul_add, add_zero]
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
Γ : Type u_1
R : Type u_3
inst : AddCommMonoid Γ
inst_1 : PartialOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : Semiring R
r : R
x : R⟦Γ⟧
⊢ HMul.hMul (α := R⟦Γ⟧) ((single 0 : (a : R) → R⟦Γ⟧) r : R⟦Γ⟧) x = r • x
have coeff : ∀ (x_1 : Γ), (HMul.hMul (α := R⟦Γ⟧) ((single 0 : (a : R) → R⟦Γ⟧) r : R⟦Γ⟧) x).coeff x_1 = (r • x).coeff x_1
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
Γ : Type u_1
Γ' : Type u_2
R : Type u_3
S : Type u_4
V : Type u_5
inst : AddCommMonoid Γ
inst_1 : PartialOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
x : R⟦Γ⟧
⊢ 0 * x = 0
have coeff : ∀ (x_1 : Γ), (0 * x).coeff x_1 = coeff 0 x_1
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
Γ : Type u_1
Γ' : Type u_2
R : Type u_3
S : Type u_4
V : Type u_5
inst : AddCommMonoid Γ
inst_1 : PartialOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
x : R⟦Γ⟧
⊢ x * 0 = 0
have coeff : ∀ (x_1 : Γ), (x * 0).coeff x_1 = coeff 0 x_1
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (goal)
Goal:
Γ : Type u_1
R : Type u_3
inst : AddCommMonoid Γ
inst_1 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
x y : R⟦Γ⟧
⊢ (x * y).coeff (x.order + y.order) = x.leadingCoeff * y.leadingCoeff
have simp : coeff
    ((Equiv.symm (HahnModule.of R) : (a : HahnModule Γ R R) → R⟦Γ⟧)
      (HSMul.hSMul (β := HahnModule Γ R R) x ((HahnModule.of R : (a : R⟦Γ⟧) → HahnModule Γ R R) y : HahnModule Γ R R)))
    (x.order + y.order) =
  x.leadingCoeff * y.leadingCoeff
Tactic: simp only [← of_symm_smul_of_eq_mul]