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
S : Type u_4
V : Type u_5
inst : AddCommMonoid Γ
inst_1 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
inst_4 : IsCancelAdd R
inst_5 : IsCancelMulZero R
x : R⟦Γ⟧
hx : x ≠ 0
y z : R⟦Γ⟧
this : AddCancelCommMonoid R := { toAddCommMonoid := inst_3.toAddCommMonoid, toIsLeftCancelAdd := ⋯ }
hyz : y ≠ z
⊢ y * x ≠ z * x
have simp : ∀ (hyz : ∃ (x : Γ), ¬y.coeff x = z.coeff x), ∃ (x_1 : Γ), ¬(y * x).coeff x_1 = (z * x).coeff x_1
Tactic: simp only [ne_eq, ← coeff_inj, funext_iff, not_forall] at ⊢ hyz
Cleared hypotheses: hyz
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (hypothesis)
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 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
inst_4 : IsCancelAdd R
inst_5 : IsCancelMulZero R
x : R⟦Γ⟧
hx : x ≠ 0
y z : R⟦Γ⟧
this : AddCancelCommMonoid R := { toAddCommMonoid := inst_3.toAddCommMonoid, toIsLeftCancelAdd := ⋯ }
hyz : ∃ (x : Γ), ¬y.coeff x = z.coeff x
⊢ ∃ (x_1 : Γ), ¬(y * x).coeff x_1 = (z * x).coeff x_1
have this₁ : Set.IsWF.{u_1} (α := Γ) {a : Γ | y.coeff a ≠ z.coeff a}
Tactic: have : Set.IsWF {a | y.coeff a ≠ z.coeff a} :=
      .mono (y.isWF_support.union z.isWF_support) (by intro; simp; grind)
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.Multiplication (hypothesis)
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 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
inst_4 : IsCancelAdd R
inst_5 : IsCancelMulZero R
x : R⟦Γ⟧
hx : x ≠ 0
y z : R⟦Γ⟧
this_1 : AddCancelCommMonoid R := { toAddCommMonoid := inst_3.toAddCommMonoid, toIsLeftCancelAdd := ⋯ }
hyz : ∃ (x : Γ), ¬y.coeff x = z.coeff x
this : Set.IsWF.{u_1} (α := Γ) {a : Γ | y.coeff a ≠ z.coeff a}
a : Γ := this.min (s := {a : Γ | y.coeff a ≠ z.coeff a}) hyz
⊢ ∃ (x_1 : Γ), ¬(y * x).coeff x_1 = (z * x).coeff x_1
have ha : y.coeff a ≠ z.coeff a
Tactic: have ha : y.coeff a ≠ z.coeff a := this.min_mem hyz
-- 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 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
inst_4 : IsCancelAdd R
inst_5 : IsCancelMulZero R
x : R⟦Γ⟧
hx : x ≠ 0
y z : R⟦Γ⟧
this_1 : AddCancelCommMonoid R := { toAddCommMonoid := inst_3.toAddCommMonoid, toIsLeftCancelAdd := ⋯ }
hyz : ∃ (x : Γ), ¬y.coeff x = z.coeff x
this : Set.IsWF.{u_1} (α := Γ) {a : Γ | y.coeff a ≠ z.coeff a}
a : Γ := this.min (s := {a : Γ | y.coeff a ≠ z.coeff a}) hyz
ha : y.coeff a ≠ z.coeff a
⊢ ∃ (x_1 : Γ), ¬(y * x).coeff x_1 = (z * x).coeff x_1
have refine : ¬(y * x).coeff (a + x.order) = (z * x).coeff (a + x.order)
Tactic: refine ⟨a + x.order, ?_⟩
-- 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 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
inst_4 : IsCancelAdd R
inst_5 : IsCancelMulZero R
x : R⟦Γ⟧
hx : x ≠ 0
y z : R⟦Γ⟧
this_1 : AddCancelCommMonoid R := { toAddCommMonoid := inst_3.toAddCommMonoid, toIsLeftCancelAdd := ⋯ }
hyz : ∃ (x : Γ), ¬y.coeff x = z.coeff x
this : Set.IsWF.{u_1} (α := Γ) {a : Γ | y.coeff a ≠ z.coeff a}
a : Γ := this.min (s := {a : Γ | y.coeff a ≠ z.coeff a}) hyz
ha : y.coeff a ≠ z.coeff a
⊢ ¬(y * x).coeff (a + x.order) = (z * x).coeff (a + x.order)
have hi : (a, x.order) ∈
  addAntidiagonal (s := y.support) (t := x.support) sorry sorry (a + x.order) ∪
    addAntidiagonal (s := z.support) (t := x.support) sorry sorry (a + x.order)
Tactic: rwa [coeff_mul, coeff_mul, sum_subset subset_union_left,
      sum_subset (s₁ := addAntidiagonal _ _ _) subset_union_right,
      sum_eq_sum_iff_single (i := (a, x.order)), mul_left_inj' (coeff_order_eq_zero.not.2 hx)]
-- 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 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
inst_4 : IsCancelAdd R
inst_5 : IsCancelMulZero R
x : R⟦Γ⟧
hx : x ≠ 0
y z : R⟦Γ⟧
this_1 : AddCancelCommMonoid R := { toAddCommMonoid := inst_3.toAddCommMonoid, toIsLeftCancelAdd := ⋯ }
hyz : ∃ (x : Γ), ¬y.coeff x = z.coeff x
this : Set.IsWF.{u_1} (α := Γ) {a : Γ | y.coeff a ≠ z.coeff a}
a : Γ := this.min (s := {a : Γ | y.coeff a ≠ z.coeff a}) hyz
ha : y.coeff a ≠ z.coeff a
hi :
  (a, x.order) ∈
    addAntidiagonal (s := y.support) (t := x.support) ⋯ ⋯ (a + x.order) ∪
      addAntidiagonal (s := z.support) (t := x.support) ⋯ ⋯ (a + x.order)
⊢ ¬(y * x).coeff (a + x.order) = (z * x).coeff (a + x.order)
have hfg : ∀
  j ∈
    addAntidiagonal (s := y.support) (t := x.support) sorry sorry (a + x.order) ∪
      addAntidiagonal (s := z.support) (t := x.support) sorry sorry (a + x.order),
  j ≠ (a, x.order) → y.coeff j.1 * x.coeff j.2 = z.coeff j.1 * x.coeff j.2
Tactic: rwa [coeff_mul, coeff_mul, sum_subset subset_union_left,
      sum_subset (s₁ := addAntidiagonal _ _ _) subset_union_right,
      sum_eq_sum_iff_single (i := (a, x.order)), mul_left_inj' (coeff_order_eq_zero.not.2 hx)]
-- 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 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
inst_4 : IsCancelAdd R
inst_5 : IsCancelMulZero R
x : R⟦Γ⟧
hx : x ≠ 0
y z : R⟦Γ⟧
this_1 : AddCancelCommMonoid R := { toAddCommMonoid := inst_3.toAddCommMonoid, toIsLeftCancelAdd := ⋯ }
hyz : ∃ (x : Γ), ¬y.coeff x = z.coeff x
this : Set.IsWF.{u_1} (α := Γ) {a : Γ | y.coeff a ≠ z.coeff a}
a : Γ := this.min (s := {a : Γ | y.coeff a ≠ z.coeff a}) hyz
ha : y.coeff a ≠ z.coeff a
hi :
  (a, x.order) ∈
    addAntidiagonal (s := y.support) (t := x.support) ⋯ ⋯ (a + x.order) ∪
      addAntidiagonal (s := z.support) (t := x.support) ⋯ ⋯ (a + x.order)
hfg :
  ∀
    j ∈
      addAntidiagonal (s := y.support) (t := x.support) ⋯ ⋯ (a + x.order) ∪
        addAntidiagonal (s := z.support) (t := x.support) ⋯ ⋯ (a + x.order),
    j ≠ (a, x.order) → y.coeff j.1 * x.coeff j.2 = z.coeff j.1 * x.coeff j.2
⊢ ¬(y * x).coeff (a + x.order) = (z * x).coeff (a + x.order)
have rwa : ∀
  x_1 ∈
    addAntidiagonal (s := y.support) (t := x.support) sorry sorry (a + x.order) ∪
      addAntidiagonal (s := z.support) (t := x.support) sorry sorry (a + x.order),
  x_1 ∉ addAntidiagonal (s := z.support) (t := x.support) sorry sorry (a + x.order) → z.coeff x_1.1 * x.coeff x_1.2 = 0
Tactic: rwa [coeff_mul, coeff_mul, sum_subset subset_union_left,
      sum_subset (s₁ := addAntidiagonal _ _ _) subset_union_right,
      sum_eq_sum_iff_single (i := (a, x.order)), mul_left_inj' (coeff_order_eq_zero.not.2 hx)]
-- 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 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
inst_4 : IsCancelAdd R
inst_5 : IsCancelMulZero R
x : R⟦Γ⟧
hx : x ≠ 0
y z : R⟦Γ⟧
this_1 : AddCancelCommMonoid R := { toAddCommMonoid := inst_3.toAddCommMonoid, toIsLeftCancelAdd := ⋯ }
hyz : ∃ (x : Γ), ¬y.coeff x = z.coeff x
this : Set.IsWF.{u_1} (α := Γ) {a : Γ | y.coeff a ≠ z.coeff a}
a : Γ := this.min (s := {a : Γ | y.coeff a ≠ z.coeff a}) hyz
ha : y.coeff a ≠ z.coeff a
hi :
  (a, x.order) ∈
    addAntidiagonal (s := y.support) (t := x.support) ⋯ ⋯ (a + x.order) ∪
      addAntidiagonal (s := z.support) (t := x.support) ⋯ ⋯ (a + x.order)
hfg :
  ∀
    j ∈
      addAntidiagonal (s := y.support) (t := x.support) ⋯ ⋯ (a + x.order) ∪
        addAntidiagonal (s := z.support) (t := x.support) ⋯ ⋯ (a + x.order),
    j ≠ (a, x.order) → y.coeff j.1 * x.coeff j.2 = z.coeff j.1 * x.coeff j.2
rwa :
  ∀
    x_1 ∈
      addAntidiagonal (s := y.support) (t := x.support) ⋯ ⋯ (a + x.order) ∪
        addAntidiagonal (s := z.support) (t := x.support) ⋯ ⋯ (a + x.order),
    x_1 ∉ addAntidiagonal (s := z.support) (t := x.support) ⋯ ⋯ (a + x.order) → z.coeff x_1.1 * x.coeff x_1.2 = 0
⊢ ¬(y * x).coeff (a + x.order) = (z * x).coeff (a + x.order)
have rwa₁ : ∀
  x_1 ∈
    addAntidiagonal (s := y.support) (t := x.support) sorry sorry (a + x.order) ∪
      addAntidiagonal (s := z.support) (t := x.support) sorry sorry (a + x.order),
  x_1 ∉ addAntidiagonal (s := y.support) (t := x.support) sorry sorry (a + x.order) → y.coeff x_1.1 * x.coeff x_1.2 = 0
Tactic: rwa [coeff_mul, coeff_mul, sum_subset subset_union_left,
      sum_subset (s₁ := addAntidiagonal _ _ _) subset_union_right,
      sum_eq_sum_iff_single (i := (a, x.order)), mul_left_inj' (coeff_order_eq_zero.not.2 hx)]
-- 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 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
inst_4 : IsCancelAdd R
inst_5 : IsCancelMulZero R
x : R⟦Γ⟧
hx : x ≠ 0
y z : R⟦Γ⟧
this_1 : AddCancelCommMonoid R := { toAddCommMonoid := inst_3.toAddCommMonoid, toIsLeftCancelAdd := ⋯ }
hyz : ∃ (x : Γ), ¬y.coeff x = z.coeff x
this : Set.IsWF.{u_1} (α := Γ) {a : Γ | y.coeff a ≠ z.coeff a}
a : Γ := this.min (s := {a : Γ | y.coeff a ≠ z.coeff a}) hyz
ha : y.coeff a ≠ z.coeff a
⊢ (a, x.order) ∈
    addAntidiagonal (s := y.support) (t := x.support) ⋯ ⋯ (a + x.order) ∪
      addAntidiagonal (s := z.support) (t := x.support) ⋯ ⋯ (a + x.order)
have hi : ¬y.coeff a = 0 ∨ ¬z.coeff a = 0
Tactic: simp [hx]
-- 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 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
inst_4 : IsCancelAdd R
inst_5 : IsCancelMulZero R
x : R⟦Γ⟧
hx : x ≠ 0
y z : R⟦Γ⟧
this_1 : AddCancelCommMonoid R := { toAddCommMonoid := inst_3.toAddCommMonoid, toIsLeftCancelAdd := ⋯ }
hyz : ∃ (x : Γ), ¬y.coeff x = z.coeff x
this : Set.IsWF.{u_1} (α := Γ) {a : Γ | y.coeff a ≠ z.coeff a}
a : Γ := this.min (s := {a : Γ | y.coeff a ≠ z.coeff a}) hyz
ha : y.coeff a ≠ z.coeff a
⊢ ∀
    j ∈
      addAntidiagonal (s := y.support) (t := x.support) ⋯ ⋯ (a + x.order) ∪
        addAntidiagonal (s := z.support) (t := x.support) ⋯ ⋯ (a + x.order),
    j ≠ (a, x.order) → y.coeff j.1 * x.coeff j.2 = z.coeff j.1 * x.coeff j.2
have hfg : ∀ (a_1 b : Γ),
  ¬y.coeff a_1 = 0 ∨ ¬z.coeff a_1 = 0 →
    ¬x.coeff b = 0 → a_1 + b = a + x.order → (a_1 = a → ¬b = x.order) → y.coeff a_1 = z.coeff a_1
Tactic: simp +contextual only [mem_union, mem_addAntidiagonal, mul_eq_mul_right_iff, Prod.mk.injEq,
        ne_eq, ← or_and_right, or_false, and_imp, Prod.forall, mem_support, not_and]
-- 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 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
inst_4 : IsCancelAdd R
inst_5 : IsCancelMulZero R
x : R⟦Γ⟧
hx : x ≠ 0
y z : R⟦Γ⟧
this_1 : AddCancelCommMonoid R := { toAddCommMonoid := inst_3.toAddCommMonoid, toIsLeftCancelAdd := ⋯ }
hyz : ∃ (x : Γ), ¬y.coeff x = z.coeff x
this : Set.IsWF.{u_1} (α := Γ) {a : Γ | y.coeff a ≠ z.coeff a}
a : Γ := this.min (s := {a : Γ | y.coeff a ≠ z.coeff a}) hyz
ha : y.coeff a ≠ z.coeff a
b c : Γ
hxb : ¬x.coeff c = 0
hbc : b + c = a + x.order
hbc' : b = a → ¬c = x.order
⊢ y.coeff b = z.coeff b
have hfg : ∀ (hbc' : y.coeff b ≠ z.coeff b), b = a ∧ c = x.order
Tactic: contrapose! hbc'
Cleared hypotheses: hbc'
-- 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 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
inst_4 : IsCancelAdd R
inst_5 : IsCancelMulZero R
x : R⟦Γ⟧
hx : x ≠ 0
y z : R⟦Γ⟧
this_1 : AddCancelCommMonoid R := { toAddCommMonoid := inst_3.toAddCommMonoid, toIsLeftCancelAdd := ⋯ }
hyz : ∃ (x : Γ), ¬y.coeff x = z.coeff x
this : Set.IsWF.{u_1} (α := Γ) {a : Γ | y.coeff a ≠ z.coeff a}
a : Γ := this.min (s := {a : Γ | y.coeff a ≠ z.coeff a}) hyz
ha : y.coeff a ≠ z.coeff a
b c : Γ
hxb : ¬x.coeff c = 0
hbc : b + c = a + x.order
hbc' : y.coeff b ≠ z.coeff b
⊢ b = a ∧ c = x.order
have hfg : a = b ∧ c = x.order
Tactic: rwa [eq_comm, eq_comm (a := c), ← add_eq_add_iff_eq_and_eq
        (Set.IsWF.min_le _ _ hbc') (order_le_of_coeff_ne_zero hxb), eq_comm]
-- 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 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
inst_4 : IsCancelAdd R
inst_5 : IsCancelMulZero R
x : R⟦Γ⟧
hx : x ≠ 0
y z : R⟦Γ⟧
this_1 : AddCancelCommMonoid R := { toAddCommMonoid := inst_3.toAddCommMonoid, toIsLeftCancelAdd := ⋯ }
hyz : ∃ (x : Γ), ¬y.coeff x = z.coeff x
this : Set.IsWF.{u_1} (α := Γ) {a : Γ | y.coeff a ≠ z.coeff a}
a : Γ := this.min (s := {a : Γ | y.coeff a ≠ z.coeff a}) hyz
ha : y.coeff a ≠ z.coeff a
b c : Γ
hxb : ¬x.coeff c = 0
hbc : b + c = a + x.order
hbc' : y.coeff b ≠ z.coeff b
hfg : a = b ∧ c = x.order
⊢ b = a ∧ c = x.order
have hfg₁ : a = b ∧ x.order = c
Tactic: rwa [eq_comm, eq_comm (a := c), ← add_eq_add_iff_eq_and_eq
        (Set.IsWF.min_le _ _ hbc') (order_le_of_coeff_ne_zero hxb), eq_comm]
-- 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 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
inst_4 : IsCancelAdd R
inst_5 : IsCancelMulZero R
x : R⟦Γ⟧
hx : x ≠ 0
y z : R⟦Γ⟧
this_1 : AddCancelCommMonoid R := { toAddCommMonoid := inst_3.toAddCommMonoid, toIsLeftCancelAdd := ⋯ }
hyz : ∃ (x : Γ), ¬y.coeff x = z.coeff x
this : Set.IsWF.{u_1} (α := Γ) {a : Γ | y.coeff a ≠ z.coeff a}
a : Γ := this.min (s := {a : Γ | y.coeff a ≠ z.coeff a}) hyz
ha : y.coeff a ≠ z.coeff a
b c : Γ
hxb : ¬x.coeff c = 0
hbc : b + c = a + x.order
hbc' : y.coeff b ≠ z.coeff b
hfg : a = b ∧ c = x.order
hfg₁ : a = b ∧ x.order = c
⊢ b = a ∧ c = x.order
have hfg₂ : this.min (s := {a : Γ | y.coeff a ≠ z.coeff a}) hyz + x.order = b + c
Tactic: rwa [eq_comm, eq_comm (a := c), ← add_eq_add_iff_eq_and_eq
        (Set.IsWF.min_le _ _ hbc') (order_le_of_coeff_ne_zero hxb), eq_comm]
-- 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 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
inst_4 : NoZeroDivisors R
x_1 y_1 x y : R⟦Γ⟧
hxy : x * y = 0
⊢ x = 0 ∨ y = 0
have contrapose! : ∀ (hxy : x ≠ 0 ∧ y ≠ 0), x * y ≠ 0
Tactic: contrapose! hxy
Cleared hypotheses: hxy
-- 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 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
inst_4 : NoZeroDivisors R
x_1 y_1 x y : R⟦Γ⟧
hxy : x ≠ 0 ∧ y ≠ 0
⊢ x * y ≠ 0
have simp : ∃ (x_2 : Γ), ¬(x * y).coeff x_2 = coeff 0 x_2
Tactic: simp only [ne_eq, HahnSeries.ext_iff, funext_iff, not_forall]
-- 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
inst_4 : NoZeroDivisors R
x y : R⟦Γ⟧
hx : x ≠ 0
hy : y ≠ 0
⊢ (x * y).order = x.order + y.order
have a : (x * y).order ≤ x.order + y.order
Tactic: apply le_antisymm
-- 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
inst_4 : NoZeroDivisors R
x y : R⟦Γ⟧
hx : x ≠ 0
hy : y ≠ 0
a : (x * y).order ≤ x.order + y.order
⊢ (x * y).order = x.order + y.order
have a₁ : x.order + y.order ≤ (x * y).order
Tactic: apply le_antisymm
-- 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
inst_4 : NoZeroDivisors R
x y : R⟦Γ⟧
hx : x ≠ 0
hy : y ≠ 0
⊢ (x * y).order ≤ x.order + y.order
have a : (x * y).coeff (x.order + y.order) ≠ 0
Tactic: apply order_le_of_coeff_ne_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 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : NonUnitalNonAssocSemiring R
inst_4 : NoZeroDivisors R
x y : R⟦Γ⟧
hx : x ≠ 0
hy : y ≠ 0
⊢ x.order + y.order ≤ (x * y).order
have a : Set.IsWF.min (s := x.support + y.support) sorry sorry ≤ sorry.min (s := (x * y).support) sorry
Tactic: rw [order_of_ne hx, order_of_ne hy, order_of_ne (mul_ne_zero hx hy), ← Set.IsWF.min_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 : LinearOrder Γ
inst_2 : IsOrderedCancelAddMonoid Γ
inst_3 : Semiring R
inst_4 : NoZeroDivisors R
x : R⟦Γ⟧
n : ℕ
⊢ (x ^ (n + 1)).order = (n + 1) • x.order
have inl : (HPow.hPow (α := R⟦Γ⟧) 0 (n + 1)).order = HSMul.hSMul (β := Γ) (n + 1) (order.{u_1, u_3} (R := R) 0)
Tactic: obtain rfl | hx := eq_or_ne x 0
Cleared hypotheses: x
-- 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 : Semiring R
inst_4 : NoZeroDivisors R
x : R⟦Γ⟧
n : ℕ
inl : (HPow.hPow (α := R⟦Γ⟧) 0 (n + 1)).order = HSMul.hSMul (β := Γ) (n + 1) (order.{u_1, u_3} (R := R) 0)
⊢ (x ^ (n + 1)).order = (n + 1) • x.order
have inr : ∀ (hx : x ≠ 0), (x ^ (n + 1)).order = (n + 1) • x.order
Tactic: obtain rfl | hx := eq_or_ne x 0
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.PowerSeries (goal)
Goal:
Γ : Type u_1
R : Type u_2
inst : Semiring R
f : R⟦ℕ⟧
⊢ (fun (f : PowerSeries R) ↦
        ({ coeff := fun (n : ℕ) ↦ (PowerSeries.coeff n : (a : PowerSeries R) → R) f, isPWO_support' := ⋯ } : R⟦ℕ⟧))
      ((fun (f : R⟦ℕ⟧) ↦ PowerSeries.mk f.coeff) f) =
    f
have coeff : ∀ (x : ℕ),
  coeff
      ((fun (f : PowerSeries R) ↦
          ({ coeff := fun (n : ℕ) ↦ (PowerSeries.coeff n : (a : PowerSeries R) → R) f, isPWO_support' := sorry } : R⟦ℕ⟧))
        ((fun (f : R⟦ℕ⟧) ↦ PowerSeries.mk f.coeff) f))
      x =
    f.coeff x
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.PowerSeries (goal)
Goal:
Γ : Type u_1
R : Type u_2
inst : Semiring R
f : PowerSeries R
⊢ (fun (f : R⟦ℕ⟧) ↦ PowerSeries.mk f.coeff)
      ((fun (f : PowerSeries R) ↦
          ({ coeff := fun (n : ℕ) ↦ (PowerSeries.coeff n : (a : PowerSeries R) → R) f, isPWO_support' := ⋯ } : R⟦ℕ⟧))
        f) =
    f
have h : ∀ (n : ℕ),
  Eq (α := R)
    ((PowerSeries.coeff n : (a : PowerSeries R) → R)
        ((fun (f : R⟦ℕ⟧) ↦ PowerSeries.mk f.coeff)
          ((fun (f : PowerSeries R) ↦
              ({ coeff := fun (n : ℕ) ↦ (PowerSeries.coeff n : (a : PowerSeries R) → R) f, isPWO_support' := sorry } :
                R⟦ℕ⟧))
            f)) :
      R)
    ((PowerSeries.coeff n : (a : PowerSeries R) → R) f : R)
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.PowerSeries (goal)
Goal:
Γ : Type u_1
R : Type u_2
inst : Semiring R
f g : R⟦ℕ⟧
⊢ PowerSeries.mk (f * g).coeff = PowerSeries.mk f.coeff * PowerSeries.mk g.coeff
have h : ∀ (n : ℕ),
  Eq (α := R) ((PowerSeries.coeff n : (a : PowerSeries R) → R) (PowerSeries.mk (f * g).coeff) : R)
    ((PowerSeries.coeff n : (a : PowerSeries R) → R) (PowerSeries.mk f.coeff * PowerSeries.mk g.coeff) : R)
Tactic: ext n
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.PowerSeries (goal)
Goal:
Γ : Type u_1
R : Type u_2
inst : Semiring R
f g : R⟦ℕ⟧
n : ℕ
⊢ Eq (α := R) ((PowerSeries.coeff n : (a : PowerSeries R) → R) (PowerSeries.mk (f * g).coeff) : R)
    ((PowerSeries.coeff n : (a : PowerSeries R) → R) (PowerSeries.mk f.coeff * PowerSeries.mk g.coeff) : R)
have h : Eq.{u_2 + 1} (α := R) (∑ ij ∈ addAntidiagonal (s := f.support) (t := g.support) sorry sorry n, f.coeff ij.1 * g.coeff ij.2)
  (∑ x ∈ @antidiagonal _ Nat.instAddMonoid _ n, f.coeff x.1 * g.coeff x.2)
Tactic: simp only [PowerSeries.coeff_mul, PowerSeries.coeff_mk, coeff_mul]
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.PowerSeries (goal)
Goal:
Γ : Type u_1
R : Type u_2
inst : Semiring R
f g : R⟦ℕ⟧
n : ℕ
⊢ Eq.{u_2 + 1} (α := R) (∑ ij ∈ addAntidiagonal (s := f.support) (t := g.support) ⋯ ⋯ n, f.coeff ij.1 * g.coeff ij.2)
    (∑ x ∈ @antidiagonal _ Nat.instAddMonoid _ n, f.coeff x.1 * g.coeff x.2)
have h : {x ∈ addAntidiagonal (s := f.support) (t := g.support) sorry sorry n | f.coeff x.1 * g.coeff x.2 ≠ 0} =
  {x ∈ @antidiagonal _ Nat.instAddMonoid _ n | f.coeff x.1 * g.coeff x.2 ≠ 0}
Tactic: refine (sum_filter_ne_zero _).symm.trans <| (sum_congr ?_ fun _ _ ↦ rfl).trans <|
      sum_filter_ne_zero _
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.PowerSeries (goal)
Goal:
Γ : Type u_1
R : Type u_2
inst : Semiring R
f g : R⟦ℕ⟧
n : ℕ
⊢ {x ∈ addAntidiagonal (s := f.support) (t := g.support) ⋯ ⋯ n | f.coeff x.1 * g.coeff x.2 ≠ 0} =
    {x ∈ @antidiagonal _ Nat.instAddMonoid _ n | f.coeff x.1 * g.coeff x.2 ≠ 0}
have h : ∀ (m : ℕ × ℕ),
  m ∈ {x ∈ addAntidiagonal (s := f.support) (t := g.support) sorry sorry n | f.coeff x.1 * g.coeff x.2 ≠ 0} ↔
    m ∈ {x ∈ @antidiagonal _ Nat.instAddMonoid _ n | f.coeff x.1 * g.coeff x.2 ≠ 0}
Tactic: ext m
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.PowerSeries (goal)
Goal:
Γ : Type u_1
R : Type u_2
inst : Semiring R
f g : R⟦ℕ⟧
n : ℕ
m : ℕ × ℕ
⊢ m ∈ {x ∈ addAntidiagonal (s := f.support) (t := g.support) ⋯ ⋯ n | f.coeff x.1 * g.coeff x.2 ≠ 0} ↔
    m ∈ {x ∈ @antidiagonal _ Nat.instAddMonoid _ n | f.coeff x.1 * g.coeff x.2 ≠ 0}
have h : f.coeff m.1 * g.coeff m.2 ≠ 0 → (f.coeff m.1 ≠ 0 ∧ g.coeff m.2 ≠ 0 ∧ m.1 + m.2 = n ↔ m.1 + m.2 = n)
Tactic: simp only [mem_antidiagonal, mem_addAntidiagonal, and_congr_left_iff, mem_filter,
      mem_support]
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.PowerSeries (goal)
Goal:
Γ : Type u_1
R : Type u_2
inst : Semiring R
f g : R⟦ℕ⟧
n : ℕ
m : ℕ × ℕ
h : f.coeff m.1 * g.coeff m.2 ≠ 0
⊢ f.coeff m.1 ≠ 0 ∧ g.coeff m.2 ≠ 0 ∧ m.1 + m.2 = n ↔ m.1 + m.2 = n
have h₁ : g.coeff m.2 ≠ 0 ∧ m.1 + m.2 = n ↔ m.1 + m.2 = n
Tactic: rw [and_iff_right (left_ne_zero_of_mul h), and_iff_right (right_ne_zero_of_mul h)]
-- Lean 4 proof step from Mathlib.RingTheory.HahnSeries.PowerSeries (goal)
Goal:
Γ : Type u_1
R : Type u_2
inst : Semiring R
f g : R⟦ℕ⟧
n : ℕ
m : ℕ × ℕ
h : f.coeff m.1 * g.coeff m.2 ≠ 0
h₁ : g.coeff m.2 ≠ 0 ∧ m.1 + m.2 = n ↔ m.1 + m.2 = n
⊢ f.coeff m.1 ≠ 0 ∧ g.coeff m.2 ≠ 0 ∧ m.1 + m.2 = n ↔ m.1 + m.2 = n
have h₂ : m.1 + m.2 = n ↔ m.1 + m.2 = n
Tactic: rw [and_iff_right (left_ne_zero_of_mul h), and_iff_right (right_ne_zero_of_mul h)]