l3lab/ntp-mathlib huggingface dataset:
theorem single_zero_mul_coeff [NonUnitalNonAssocSemiring R] {r : R} {x : HahnSeries Γ R} {a : Γ} :
    ((single 0 r : HahnSeries Γ R) * x).coeff a = r * x.coeff a := by
  

Proof state:
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : NonUnitalNonAssocSemiring R
r : R
x : HahnSeries Γ R
a : Γ
⊢ coeff ((single 0) r * x) a = r * coeff x a

Next tactic:
rw [← add_zero a, single_mul_coeff_add, add_zero]
@[simp]
theorem single_zero_mul_eq_smul [Semiring R] {r : R} {x : HahnSeries Γ R} :
    single 0 r * x = r • x := by
  

Proof state:
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
r : R
x : HahnSeries Γ R
⊢ (single 0) r * x = r • x

Next tactic:
ext
@[simp]
theorem single_zero_mul_eq_smul [Semiring R] {r : R} {x : HahnSeries Γ R} :
    single 0 r * x = r • x := by
  ext
  

Proof state:
case coeff.h
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : Semiring R
r : R
x : HahnSeries Γ R
x✝ : Γ
⊢ coeff ((single 0) r * x) x✝ = coeff (r • x) x✝

Next tactic:
exact single_zero_mul_coeff
theorem support_mul_subset_add_support [NonUnitalNonAssocSemiring R] {x y : HahnSeries Γ R} :
    support (x * y) ⊆ support x + support y := by
  

Proof state:
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : NonUnitalNonAssocSemiring R
x y : HahnSeries Γ R
⊢ support (x * y) ⊆ support x + support y

Next tactic:
apply Set.Subset.trans (fun x hx => _) support_addAntidiagonal_subset_add
theorem support_mul_subset_add_support [NonUnitalNonAssocSemiring R] {x y : HahnSeries Γ R} :
    support (x * y) ⊆ support x + support y := by
  apply Set.Subset.trans (fun x hx => _) support_addAntidiagonal_subset_add
  · 

Proof state:
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : NonUnitalNonAssocSemiring R
x y : HahnSeries Γ R
⊢ Set.IsPwo (support x)

Next tactic:
exact x.isPwo_support
theorem support_mul_subset_add_support [NonUnitalNonAssocSemiring R] {x y : HahnSeries Γ R} :
    support (x * y) ⊆ support x + support y := by
  apply Set.Subset.trans (fun x hx => _) support_addAntidiagonal_subset_add
  · exact x.isPwo_support
  · 

Proof state:
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : NonUnitalNonAssocSemiring R
x y : HahnSeries Γ R
⊢ Set.IsPwo (support y)

Next tactic:
exact y.isPwo_support
theorem support_mul_subset_add_support [NonUnitalNonAssocSemiring R] {x y : HahnSeries Γ R} :
    support (x * y) ⊆ support x + support y := by
  apply Set.Subset.trans (fun x hx => _) support_addAntidiagonal_subset_add
  · exact x.isPwo_support
  · exact y.isPwo_support
  

Proof state:
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : NonUnitalNonAssocSemiring R
x y : HahnSeries Γ R
⊢ ∀ x_1 ∈ support (x * y),
    x_1 ∈ {a | Finset.Nonempty (addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) a)}

Next tactic:
intro x hx
theorem support_mul_subset_add_support [NonUnitalNonAssocSemiring R] {x y : HahnSeries Γ R} :
    support (x * y) ⊆ support x + support y := by
  apply Set.Subset.trans (fun x hx => _) support_addAntidiagonal_subset_add
  · exact x.isPwo_support
  · exact y.isPwo_support
  intro x hx
  

Proof state:
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : NonUnitalNonAssocSemiring R
x✝ y : HahnSeries Γ R
x : Γ
hx : x ∈ support (x✝ * y)
⊢ x ∈ {a | Finset.Nonempty (addAntidiagonal (_ : Set.IsPwo (support x✝)) (_ : Set.IsPwo (support y)) a)}

Next tactic:
contrapose! hx
theorem support_mul_subset_add_support [NonUnitalNonAssocSemiring R] {x y : HahnSeries Γ R} :
    support (x * y) ⊆ support x + support y := by
  apply Set.Subset.trans (fun x hx => _) support_addAntidiagonal_subset_add
  · exact x.isPwo_support
  · exact y.isPwo_support
  intro x hx
  contrapose! hx
  

Proof state:
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : NonUnitalNonAssocSemiring R
x✝ y : HahnSeries Γ R
x : Γ
hx : x ∉ {a | Finset.Nonempty (addAntidiagonal (_ : Set.IsPwo (support x✝)) (_ : Set.IsPwo (support y)) a)}
⊢ x ∉ support (x✝ * y)

Next tactic:
simp only [not_nonempty_iff_eq_empty, Ne.def, Set.mem_setOf_eq] at hx
theorem support_mul_subset_add_support [NonUnitalNonAssocSemiring R] {x y : HahnSeries Γ R} :
    support (x * y) ⊆ support x + support y := by
  apply Set.Subset.trans (fun x hx => _) support_addAntidiagonal_subset_add
  · exact x.isPwo_support
  · exact y.isPwo_support
  intro x hx
  contrapose! hx
  simp only [not_nonempty_iff_eq_empty, Ne.def, Set.mem_setOf_eq] at hx
  

Proof state:
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : NonUnitalNonAssocSemiring R
x✝ y : HahnSeries Γ R
x : Γ
hx : addAntidiagonal (_ : Set.IsPwo (support x✝)) (_ : Set.IsPwo (support y)) x = ∅
⊢ x ∉ support (x✝ * y)

Next tactic:
simp [hx, mul_coeff]
theorem mul_coeff_order_add_order {Γ} [LinearOrderedCancelAddCommMonoid Γ]
    [NonUnitalNonAssocSemiring R] (x y : HahnSeries Γ R) :
    (x * y).coeff (x.order + y.order) = x.coeff x.order * y.coeff y.order := by
  

Proof state:
Γ✝ : Type u_1
R : Type u_2
inst✝² : OrderedCancelAddCommMonoid Γ✝
Γ : Type u_3
inst✝¹ : LinearOrderedCancelAddCommMonoid Γ
inst✝ : NonUnitalNonAssocSemiring R
x y : HahnSeries Γ R
⊢ coeff (x * y) (order x + order y) = coeff x (order x) * coeff y (order y)

Next tactic:
by_cases hx : x = 0
theorem mul_coeff_order_add_order {Γ} [LinearOrderedCancelAddCommMonoid Γ]
    [NonUnitalNonAssocSemiring R] (x y : HahnSeries Γ R) :
    (x * y).coeff (x.order + y.order) = x.coeff x.order * y.coeff y.order := by
  by_cases hx : x = 0; · 

Proof state:
case pos
Γ✝ : Type u_1
R : Type u_2
inst✝² : OrderedCancelAddCommMonoid Γ✝
Γ : Type u_3
inst✝¹ : LinearOrderedCancelAddCommMonoid Γ
inst✝ : NonUnitalNonAssocSemiring R
x y : HahnSeries Γ R
hx : x = 0
⊢ coeff (x * y) (order x + order y) = coeff x (order x) * coeff y (order y)

Next tactic:
simp [hx, mul_coeff]
theorem mul_coeff_order_add_order {Γ} [LinearOrderedCancelAddCommMonoid Γ]
    [NonUnitalNonAssocSemiring R] (x y : HahnSeries Γ R) :
    (x * y).coeff (x.order + y.order) = x.coeff x.order * y.coeff y.order := by
  by_cases hx : x = 0; · simp [hx, mul_coeff]
  

Proof state:
case neg
Γ✝ : Type u_1
R : Type u_2
inst✝² : OrderedCancelAddCommMonoid Γ✝
Γ : Type u_3
inst✝¹ : LinearOrderedCancelAddCommMonoid Γ
inst✝ : NonUnitalNonAssocSemiring R
x y : HahnSeries Γ R
hx : ¬x = 0
⊢ coeff (x * y) (order x + order y) = coeff x (order x) * coeff y (order y)

Next tactic:
by_cases hy : y = 0
theorem mul_coeff_order_add_order {Γ} [LinearOrderedCancelAddCommMonoid Γ]
    [NonUnitalNonAssocSemiring R] (x y : HahnSeries Γ R) :
    (x * y).coeff (x.order + y.order) = x.coeff x.order * y.coeff y.order := by
  by_cases hx : x = 0; · simp [hx, mul_coeff]
  by_cases hy : y = 0; · 

Proof state:
case pos
Γ✝ : Type u_1
R : Type u_2
inst✝² : OrderedCancelAddCommMonoid Γ✝
Γ : Type u_3
inst✝¹ : LinearOrderedCancelAddCommMonoid Γ
inst✝ : NonUnitalNonAssocSemiring R
x y : HahnSeries Γ R
hx : ¬x = 0
hy : y = 0
⊢ coeff (x * y) (order x + order y) = coeff x (order x) * coeff y (order y)

Next tactic:
simp [hy, mul_coeff]
theorem mul_coeff_order_add_order {Γ} [LinearOrderedCancelAddCommMonoid Γ]
    [NonUnitalNonAssocSemiring R] (x y : HahnSeries Γ R) :
    (x * y).coeff (x.order + y.order) = x.coeff x.order * y.coeff y.order := by
  by_cases hx : x = 0; · simp [hx, mul_coeff]
  by_cases hy : y = 0; · simp [hy, mul_coeff]
  

Proof state:
case neg
Γ✝ : Type u_1
R : Type u_2
inst✝² : OrderedCancelAddCommMonoid Γ✝
Γ : Type u_3
inst✝¹ : LinearOrderedCancelAddCommMonoid Γ
inst✝ : NonUnitalNonAssocSemiring R
x y : HahnSeries Γ R
hx : ¬x = 0
hy : ¬y = 0
⊢ coeff (x * y) (order x + order y) = coeff x (order x) * coeff y (order y)

Next tactic:
rw [order_of_ne hx, order_of_ne hy, mul_coeff, Finset.addAntidiagonal_min_add_min,
    Finset.sum_singleton]
private theorem mul_assoc' [NonUnitalSemiring R] (x y z : HahnSeries Γ R) :
    x * y * z = x * (y * z) := by
  

Proof state:
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : NonUnitalSemiring R
x y z : HahnSeries Γ R
⊢ x * y * z = x * (y * z)

Next tactic:
ext b
private theorem mul_assoc' [NonUnitalSemiring R] (x y z : HahnSeries Γ R) :
    x * y * z = x * (y * z) := by
  ext b
  

Proof state:
case coeff.h
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : NonUnitalSemiring R
x y z : HahnSeries Γ R
b : Γ
⊢ coeff (x * y * z) b = coeff (x * (y * z)) b

Next tactic:
rw [mul_coeff_left' (x.isPwo_support.add y.isPwo_support) support_mul_subset_add_support,
    mul_coeff_right' (y.isPwo_support.add z.isPwo_support) support_mul_subset_add_support]
private theorem mul_assoc' [NonUnitalSemiring R] (x y z : HahnSeries Γ R) :
    x * y * z = x * (y * z) := by
  ext b
  rw [mul_coeff_left' (x.isPwo_support.add y.isPwo_support) support_mul_subset_add_support,
    mul_coeff_right' (y.isPwo_support.add z.isPwo_support) support_mul_subset_add_support]
  

Proof state:
case coeff.h
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : NonUnitalSemiring R
x y z : HahnSeries Γ R
b : Γ
⊢ ∑ ij in addAntidiagonal (_ : Set.IsPwo (support x + support y)) (_ : Set.IsPwo (support z)) b,
      coeff (x * y) ij.1 * coeff z ij.2 =
    ∑ ij in addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y + support z)) b,
      coeff x ij.1 * coeff (y * z) ij.2

Next tactic:
simp only [mul_coeff, add_coeff, sum_mul, mul_sum, sum_sigma']
private theorem mul_assoc' [NonUnitalSemiring R] (x y z : HahnSeries Γ R) :
    x * y * z = x * (y * z) := by
  ext b
  rw [mul_coeff_left' (x.isPwo_support.add y.isPwo_support) support_mul_subset_add_support,
    mul_coeff_right' (y.isPwo_support.add z.isPwo_support) support_mul_subset_add_support]
  simp only [mul_coeff, add_coeff, sum_mul, mul_sum, sum_sigma']
  

Proof state:
case coeff.h
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : NonUnitalSemiring R
x y z : HahnSeries Γ R
b : Γ
⊢ ∑ x_1 in
      Finset.sigma (addAntidiagonal (_ : Set.IsPwo (support x + support y)) (_ : Set.IsPwo (support z)) b) fun a =>
        addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) a.1,
      coeff x x_1.snd.1 * coeff y x_1.snd.2 * coeff z x_1.fst.2 =
    ∑ x_1 in
      Finset.sigma (addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y + support z)) b) fun a =>
        addAntidiagonal (_ : Set.IsPwo (support y)) (_ : Set.IsPwo (support z)) a.2,
      coeff x x_1.fst.1 * (coeff y x_1.snd.1 * coeff z x_1.snd.2)

Next tactic:
refine' sum_bij_ne_zero (fun a _ _ => ⟨⟨a.2.1, a.2.2 + a.1.2⟩, ⟨a.2.2, a.1.2⟩⟩) _ _ _ _
private theorem mul_assoc' [NonUnitalSemiring R] (x y z : HahnSeries Γ R) :
    x * y * z = x * (y * z) := by
  ext b
  rw [mul_coeff_left' (x.isPwo_support.add y.isPwo_support) support_mul_subset_add_support,
    mul_coeff_right' (y.isPwo_support.add z.isPwo_support) support_mul_subset_add_support]
  simp only [mul_coeff, add_coeff, sum_mul, mul_sum, sum_sigma']
  refine' sum_bij_ne_zero (fun a _ _ => ⟨⟨a.2.1, a.2.2 + a.1.2⟩, ⟨a.2.2, a.1.2⟩⟩) _ _ _ _
  · 

Proof state:
case coeff.h.refine'_1
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : NonUnitalSemiring R
x y z : HahnSeries Γ R
b : Γ
⊢ ∀ (a : (_ : Γ × Γ) × Γ × Γ)
    (h₁ :
      a ∈
        Finset.sigma (addAntidiagonal (_ : Set.IsPwo (support x + support y)) (_ : Set.IsPwo (support z)) b) fun a =>
          addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) a.1)
    (h₂ : coeff x a.snd.1 * coeff y a.snd.2 * coeff z a.fst.2 ≠ 0),
    (fun a x_1 x => { fst := (a.snd.1, a.snd.2 + a.fst.2), snd := (a.snd.2, a.fst.2) }) a h₁ h₂ ∈
      Finset.sigma (addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y + support z)) b) fun a =>
        addAntidiagonal (_ : Set.IsPwo (support y)) (_ : Set.IsPwo (support z)) a.2

Next tactic:
rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩ H1 H2
private theorem mul_assoc' [NonUnitalSemiring R] (x y z : HahnSeries Γ R) :
    x * y * z = x * (y * z) := by
  ext b
  rw [mul_coeff_left' (x.isPwo_support.add y.isPwo_support) support_mul_subset_add_support,
    mul_coeff_right' (y.isPwo_support.add z.isPwo_support) support_mul_subset_add_support]
  simp only [mul_coeff, add_coeff, sum_mul, mul_sum, sum_sigma']
  refine' sum_bij_ne_zero (fun a _ _ => ⟨⟨a.2.1, a.2.2 + a.1.2⟩, ⟨a.2.2, a.1.2⟩⟩) _ _ _ _
  · rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩ H1 H2
    

Proof state:
case coeff.h.refine'_1.mk.mk.mk
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : NonUnitalSemiring R
x y z : HahnSeries Γ R
b i j k l : Γ
H1 :
  { fst := (i, j), snd := (k, l) } ∈
    Finset.sigma (addAntidiagonal (_ : Set.IsPwo (support x + support y)) (_ : Set.IsPwo (support z)) b) fun a =>
      addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) a.1
H2 :
  coeff x { fst := (i, j), snd := (k, l) }.snd.1 * coeff y { fst := (i, j), snd := (k, l) }.snd.2 *
      coeff z { fst := (i, j), snd := (k, l) }.fst.2 ≠
    0
⊢ (fun a x_1 x => { fst := (a.snd.1, a.snd.2 + a.fst.2), snd := (a.snd.2, a.fst.2) }) { fst := (i, j), snd := (k, l) }
      H1 H2 ∈
    Finset.sigma (addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y + support z)) b) fun a =>
      addAntidiagonal (_ : Set.IsPwo (support y)) (_ : Set.IsPwo (support z)) a.2

Next tactic:
simp only [and_true_iff, Set.image2_add, eq_self_iff_true, mem_addAntidiagonal, Ne.def,
      Set.image_prod, mem_sigma, Set.mem_setOf_eq] at H1 H2 ⊢
private theorem mul_assoc' [NonUnitalSemiring R] (x y z : HahnSeries Γ R) :
    x * y * z = x * (y * z) := by
  ext b
  rw [mul_coeff_left' (x.isPwo_support.add y.isPwo_support) support_mul_subset_add_support,
    mul_coeff_right' (y.isPwo_support.add z.isPwo_support) support_mul_subset_add_support]
  simp only [mul_coeff, add_coeff, sum_mul, mul_sum, sum_sigma']
  refine' sum_bij_ne_zero (fun a _ _ => ⟨⟨a.2.1, a.2.2 + a.1.2⟩, ⟨a.2.2, a.1.2⟩⟩) _ _ _ _
  · rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩ H1 H2
    simp only [and_true_iff, Set.image2_add, eq_self_iff_true, mem_addAntidiagonal, Ne.def,
      Set.image_prod, mem_sigma, Set.mem_setOf_eq] at H1 H2 ⊢
    

Proof state:
case coeff.h.refine'_1.mk.mk.mk
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : NonUnitalSemiring R
x y z : HahnSeries Γ R
b i j k l : Γ
H2 : ¬coeff x k * coeff y l * coeff z j = 0
H1 : (i ∈ support x + support y ∧ j ∈ support z ∧ i + j = b) ∧ k ∈ support x ∧ l ∈ support y ∧ k + l = i
⊢ (k ∈ support x ∧ l + j ∈ support y + support z ∧ k + (l + j) = b) ∧ l ∈ support y ∧ j ∈ support z

Next tactic:
obtain ⟨⟨H3, nz, rfl⟩, nx, ny, rfl⟩ := H1
private theorem mul_assoc' [NonUnitalSemiring R] (x y z : HahnSeries Γ R) :
    x * y * z = x * (y * z) := by
  ext b
  rw [mul_coeff_left' (x.isPwo_support.add y.isPwo_support) support_mul_subset_add_support,
    mul_coeff_right' (y.isPwo_support.add z.isPwo_support) support_mul_subset_add_support]
  simp only [mul_coeff, add_coeff, sum_mul, mul_sum, sum_sigma']
  refine' sum_bij_ne_zero (fun a _ _ => ⟨⟨a.2.1, a.2.2 + a.1.2⟩, ⟨a.2.2, a.1.2⟩⟩) _ _ _ _
  · rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩ H1 H2
    simp only [and_true_iff, Set.image2_add, eq_self_iff_true, mem_addAntidiagonal, Ne.def,
      Set.image_prod, mem_sigma, Set.mem_setOf_eq] at H1 H2 ⊢
    obtain ⟨⟨H3, nz, rfl⟩, nx, ny, rfl⟩ := H1
    

Proof state:
case coeff.h.refine'_1.mk.mk.mk.intro.intro.intro.intro.intro
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : NonUnitalSemiring R
x y z : HahnSeries Γ R
j k l : Γ
H2 : ¬coeff x k * coeff y l * coeff z j = 0
nz : j ∈ support z
nx : k ∈ support x
ny : l ∈ support y
H3 : k + l ∈ support x + support y
⊢ (k ∈ support x ∧ l + j ∈ support y + support z ∧ k + (l + j) = k + l + j) ∧ l ∈ support y ∧ j ∈ support z

Next tactic:
exact ⟨⟨nx, Set.add_mem_add ny nz, (add_assoc _ _ _).symm⟩, ny, nz⟩
private theorem mul_assoc' [NonUnitalSemiring R] (x y z : HahnSeries Γ R) :
    x * y * z = x * (y * z) := by
  ext b
  rw [mul_coeff_left' (x.isPwo_support.add y.isPwo_support) support_mul_subset_add_support,
    mul_coeff_right' (y.isPwo_support.add z.isPwo_support) support_mul_subset_add_support]
  simp only [mul_coeff, add_coeff, sum_mul, mul_sum, sum_sigma']
  refine' sum_bij_ne_zero (fun a _ _ => ⟨⟨a.2.1, a.2.2 + a.1.2⟩, ⟨a.2.2, a.1.2⟩⟩) _ _ _ _
  · rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩ H1 H2
    simp only [and_true_iff, Set.image2_add, eq_self_iff_true, mem_addAntidiagonal, Ne.def,
      Set.image_prod, mem_sigma, Set.mem_setOf_eq] at H1 H2 ⊢
    obtain ⟨⟨H3, nz, rfl⟩, nx, ny, rfl⟩ := H1
    exact ⟨⟨nx, Set.add_mem_add ny nz, (add_assoc _ _ _).symm⟩, ny, nz⟩
  · 

Proof state:
case coeff.h.refine'_2
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : NonUnitalSemiring R
x y z : HahnSeries Γ R
b : Γ
⊢ ∀ (a₁ a₂ : (_ : Γ × Γ) × Γ × Γ)
    (h₁₁ :
      a₁ ∈
        Finset.sigma (addAntidiagonal (_ : Set.IsPwo (support x + support y)) (_ : Set.IsPwo (support z)) b) fun a =>
          addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) a.1)
    (h₁₂ : coeff x a₁.snd.1 * coeff y a₁.snd.2 * coeff z a₁.fst.2 ≠ 0)
    (h₂₁ :
      a₂ ∈
        Finset.sigma (addAntidiagonal (_ : Set.IsPwo (support x + support y)) (_ : Set.IsPwo (support z)) b) fun a =>
          addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) a.1)
    (h₂₂ : coeff x a₂.snd.1 * coeff y a₂.snd.2 * coeff z a₂.fst.2 ≠ 0),
    (fun a x_1 x => { fst := (a.snd.1, a.snd.2 + a.fst.2), snd := (a.snd.2, a.fst.2) }) a₁ h₁₁ h₁₂ =
        (fun a x_1 x => { fst := (a.snd.1, a.snd.2 + a.fst.2), snd := (a.snd.2, a.fst.2) }) a₂ h₂₁ h₂₂ →
      a₁ = a₂

Next tactic:
rintro ⟨⟨i1, j1⟩, k1, l1⟩ ⟨⟨i2, j2⟩, k2, l2⟩ H1 H2 H3 H4 H5
private theorem mul_assoc' [NonUnitalSemiring R] (x y z : HahnSeries Γ R) :
    x * y * z = x * (y * z) := by
  ext b
  rw [mul_coeff_left' (x.isPwo_support.add y.isPwo_support) support_mul_subset_add_support,
    mul_coeff_right' (y.isPwo_support.add z.isPwo_support) support_mul_subset_add_support]
  simp only [mul_coeff, add_coeff, sum_mul, mul_sum, sum_sigma']
  refine' sum_bij_ne_zero (fun a _ _ => ⟨⟨a.2.1, a.2.2 + a.1.2⟩, ⟨a.2.2, a.1.2⟩⟩) _ _ _ _
  · rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩ H1 H2
    simp only [and_true_iff, Set.image2_add, eq_self_iff_true, mem_addAntidiagonal, Ne.def,
      Set.image_prod, mem_sigma, Set.mem_setOf_eq] at H1 H2 ⊢
    obtain ⟨⟨H3, nz, rfl⟩, nx, ny, rfl⟩ := H1
    exact ⟨⟨nx, Set.add_mem_add ny nz, (add_assoc _ _ _).symm⟩, ny, nz⟩
  · rintro ⟨⟨i1, j1⟩, k1, l1⟩ ⟨⟨i2, j2⟩, k2, l2⟩ H1 H2 H3 H4 H5
    

Proof state:
case coeff.h.refine'_2.mk.mk.mk.mk.mk.mk
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : NonUnitalSemiring R
x y z : HahnSeries Γ R
b i1 j1 k1 l1 i2 j2 k2 l2 : Γ
H1 :
  { fst := (i1, j1), snd := (k1, l1) } ∈
    Finset.sigma (addAntidiagonal (_ : Set.IsPwo (support x + support y)) (_ : Set.IsPwo (support z)) b) fun a =>
      addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) a.1
H2 :
  coeff x { fst := (i1, j1), snd := (k1, l1) }.snd.1 * coeff y { fst := (i1, j1), snd := (k1, l1) }.snd.2 *
      coeff z { fst := (i1, j1), snd := (k1, l1) }.fst.2 ≠
    0
H3 :
  { fst := (i2, j2), snd := (k2, l2) } ∈
    Finset.sigma (addAntidiagonal (_ : Set.IsPwo (support x + support y)) (_ : Set.IsPwo (support z)) b) fun a =>
      addAntidiagonal (_ : Set.IsPwo (support x)) (_ : Set.IsPwo (support y)) a.1
H4 :
  coeff x { fst := (i2, j2), snd := (k2, l2) }.snd.1 * coeff y { fst := (i2, j2), snd := (k2, l2) }.snd.2 *
      coeff z { fst := (i2, j2), snd := (k2, l2) }.fst.2 ≠
    0
H5 :
  (fun a x_1 x => { fst := (a.snd.1, a.snd.2 + a.fst.2), snd := (a.snd.2, a.fst.2) })
      { fst := (i1, j1), snd := (k1, l1) } H1 H2 =
    (fun a x_1 x => { fst := (a.snd.1, a.snd.2 + a.fst.2), snd := (a.snd.2, a.fst.2) })
      { fst := (i2, j2), snd := (k2, l2) } H3 H4
⊢ { fst := (i1, j1), snd := (k1, l1) } = { fst := (i2, j2), snd := (k2, l2) }

Next tactic:
simp only [Set.image2_add, Prod.mk.inj_iff, mem_addAntidiagonal, Ne.def, Set.image_prod,
      mem_sigma, Set.mem_setOf_eq, heq_iff_eq] at H1 H3 H5
private theorem mul_assoc' [NonUnitalSemiring R] (x y z : HahnSeries Γ R) :
    x * y * z = x * (y * z) := by
  ext b
  rw [mul_coeff_left' (x.isPwo_support.add y.isPwo_support) support_mul_subset_add_support,
    mul_coeff_right' (y.isPwo_support.add z.isPwo_support) support_mul_subset_add_support]
  simp only [mul_coeff, add_coeff, sum_mul, mul_sum, sum_sigma']
  refine' sum_bij_ne_zero (fun a _ _ => ⟨⟨a.2.1, a.2.2 + a.1.2⟩, ⟨a.2.2, a.1.2⟩⟩) _ _ _ _
  · rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩ H1 H2
    simp only [and_true_iff, Set.image2_add, eq_self_iff_true, mem_addAntidiagonal, Ne.def,
      Set.image_prod, mem_sigma, Set.mem_setOf_eq] at H1 H2 ⊢
    obtain ⟨⟨H3, nz, rfl⟩, nx, ny, rfl⟩ := H1
    exact ⟨⟨nx, Set.add_mem_add ny nz, (add_assoc _ _ _).symm⟩, ny, nz⟩
  · rintro ⟨⟨i1, j1⟩, k1, l1⟩ ⟨⟨i2, j2⟩, k2, l2⟩ H1 H2 H3 H4 H5
    simp only [Set.image2_add, Prod.mk.inj_iff, mem_addAntidiagonal, Ne.def, Set.image_prod,
      mem_sigma, Set.mem_setOf_eq, heq_iff_eq] at H1 H3 H5
    

Proof state:
case coeff.h.refine'_2.mk.mk.mk.mk.mk.mk
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : NonUnitalSemiring R
x y z : HahnSeries Γ R
b i1 j1 k1 l1 i2 j2 k2 l2 : Γ
H2 :
  coeff x { fst := (i1, j1), snd := (k1, l1) }.snd.1 * coeff y { fst := (i1, j1), snd := (k1, l1) }.snd.2 *
      coeff z { fst := (i1, j1), snd := (k1, l1) }.fst.2 ≠
    0
H4 :
  coeff x { fst := (i2, j2), snd := (k2, l2) }.snd.1 * coeff y { fst := (i2, j2), snd := (k2, l2) }.snd.2 *
      coeff z { fst := (i2, j2), snd := (k2, l2) }.fst.2 ≠
    0
H5 : { fst := (k1, l1 + j1), snd := (l1, j1) } = { fst := (k2, l2 + j2), snd := (l2, j2) }
H1 : (i1 ∈ support x + support y ∧ j1 ∈ support z ∧ i1 + j1 = b) ∧ k1 ∈ support x ∧ l1 ∈ support y ∧ k1 + l1 = i1
H3 : (i2 ∈ support x + support y ∧ j2 ∈ support z ∧ i2 + j2 = b) ∧ k2 ∈ support x ∧ l2 ∈ support y ∧ k2 + l2 = i2
⊢ { fst := (i1, j1), snd := (k1, l1) } = { fst := (i2, j2), snd := (k2, l2) }

Next tactic:
obtain (⟨⟨rfl, _⟩, rfl, rfl⟩ : (k1 = k2 ∧ l1 + j1 = l2 + j2) ∧ l1 = l2 ∧ j1 = j2) :=
      by simpa using H5
private theorem mul_assoc' [NonUnitalSemiring R] (x y z : HahnSeries Γ R) :
    x * y * z = x * (y * z) := by
  ext b
  rw [mul_coeff_left' (x.isPwo_support.add y.isPwo_support) support_mul_subset_add_support,
    mul_coeff_right' (y.isPwo_support.add z.isPwo_support) support_mul_subset_add_support]
  simp only [mul_coeff, add_coeff, sum_mul, mul_sum, sum_sigma']
  refine' sum_bij_ne_zero (fun a _ _ => ⟨⟨a.2.1, a.2.2 + a.1.2⟩, ⟨a.2.2, a.1.2⟩⟩) _ _ _ _
  · rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩ H1 H2
    simp only [and_true_iff, Set.image2_add, eq_self_iff_true, mem_addAntidiagonal, Ne.def,
      Set.image_prod, mem_sigma, Set.mem_setOf_eq] at H1 H2 ⊢
    obtain ⟨⟨H3, nz, rfl⟩, nx, ny, rfl⟩ := H1
    exact ⟨⟨nx, Set.add_mem_add ny nz, (add_assoc _ _ _).symm⟩, ny, nz⟩
  · rintro ⟨⟨i1, j1⟩, k1, l1⟩ ⟨⟨i2, j2⟩, k2, l2⟩ H1 H2 H3 H4 H5
    simp only [Set.image2_add, Prod.mk.inj_iff, mem_addAntidiagonal, Ne.def, Set.image_prod,
      mem_sigma, Set.mem_setOf_eq, heq_iff_eq] at H1 H3 H5
    obtain (⟨⟨rfl, _⟩, rfl, rfl⟩ : (k1 = k2 ∧ l1 + j1 = l2 + j2) ∧ l1 = l2 ∧ j1 = j2) :=
      by 

Proof state:
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : NonUnitalSemiring R
x y z : HahnSeries Γ R
b i1 j1 k1 l1 i2 j2 k2 l2 : Γ
H2 :
  coeff x { fst := (i1, j1), snd := (k1, l1) }.snd.1 * coeff y { fst := (i1, j1), snd := (k1, l1) }.snd.2 *
      coeff z { fst := (i1, j1), snd := (k1, l1) }.fst.2 ≠
    0
H4 :
  coeff x { fst := (i2, j2), snd := (k2, l2) }.snd.1 * coeff y { fst := (i2, j2), snd := (k2, l2) }.snd.2 *
      coeff z { fst := (i2, j2), snd := (k2, l2) }.fst.2 ≠
    0
H5 : { fst := (k1, l1 + j1), snd := (l1, j1) } = { fst := (k2, l2 + j2), snd := (l2, j2) }
H1 : (i1 ∈ support x + support y ∧ j1 ∈ support z ∧ i1 + j1 = b) ∧ k1 ∈ support x ∧ l1 ∈ support y ∧ k1 + l1 = i1
H3 : (i2 ∈ support x + support y ∧ j2 ∈ support z ∧ i2 + j2 = b) ∧ k2 ∈ support x ∧ l2 ∈ support y ∧ k2 + l2 = i2
⊢ (k1 = k2 ∧ l1 + j1 = l2 + j2) ∧ l1 = l2 ∧ j1 = j2

Next tactic:
simpa using H5