l3lab/ntp-mathlib huggingface dataset:
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 simpa using H5
    

Proof state:
case coeff.h.refine'_2.mk.mk.mk.mk.mk.mk.intro.intro.intro
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 : 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
H1 : (i1 ∈ support x + support y ∧ j1 ∈ support z ∧ i1 + j1 = b) ∧ k1 ∈ support x ∧ l1 ∈ support y ∧ k1 + l1 = i1
right✝ : l1 + j1 = l1 + j1
H4 :
  coeff x { fst := (i2, j1), snd := (k1, l1) }.snd.1 * coeff y { fst := (i2, j1), snd := (k1, l1) }.snd.2 *
      coeff z { fst := (i2, j1), snd := (k1, l1) }.fst.2 ≠
    0
H5 : { fst := (k1, l1 + j1), snd := (l1, j1) } = { fst := (k1, l1 + j1), snd := (l1, j1) }
H3 : (i2 ∈ support x + support y ∧ j1 ∈ support z ∧ i2 + j1 = b) ∧ k1 ∈ support x ∧ l1 ∈ support y ∧ k1 + l1 = i2
⊢ { fst := (i1, j1), snd := (k1, l1) } = { fst := (i2, j1), snd := (k1, l1) }

Next tactic:
simp only [and_true_iff, Prod.mk.inj_iff, eq_self_iff_true, heq_iff_eq, ← H1.2.2.2, ← H3.2.2.2]
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⟩⟩) _ _ _ _
  · 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 simpa using H5
    simp only [and_true_iff, Prod.mk.inj_iff, eq_self_iff_true, heq_iff_eq, ← H1.2.2.2, ← H3.2.2.2]
  · 

Proof state:
case coeff.h.refine'_3
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : NonUnitalSemiring R
x y z : HahnSeries Γ R
b : Γ
⊢ ∀
    b_1 ∈
      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 b_1.fst.1 * (coeff y b_1.snd.1 * coeff z b_1.snd.2) ≠ 0 →
      ∃ a,
        ∃ (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),
          b_1 = (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

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
    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 simpa using H5
    simp only [and_true_iff, Prod.mk.inj_iff, eq_self_iff_true, heq_iff_eq, ← H1.2.2.2, ← H3.2.2.2]
  · rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩ H1 H2
    

Proof state:
case coeff.h.refine'_3.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)) (_ : Set.IsPwo (support y + support z)) b) fun a =>
      addAntidiagonal (_ : Set.IsPwo (support y)) (_ : Set.IsPwo (support z)) a.2
H2 :
  coeff x { fst := (i, j), snd := (k, l) }.fst.1 *
      (coeff y { fst := (i, j), snd := (k, l) }.snd.1 * coeff z { fst := (i, j), snd := (k, l) }.snd.2) ≠
    0
⊢ ∃ a,
    ∃ (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),
      { fst := (i, j), snd := (k, l) } =
        (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

Next tactic:
simp only [exists_prop, Set.image2_add, Prod.mk.inj_iff, mem_addAntidiagonal, Sigma.exists,
      Ne.def, Set.image_prod, mem_sigma, Set.mem_setOf_eq, heq_iff_eq, Prod.exists] 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 ⊢
    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 simpa using H5
    simp only [and_true_iff, Prod.mk.inj_iff, eq_self_iff_true, heq_iff_eq, ← H1.2.2.2, ← H3.2.2.2]
  · rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩ H1 H2
    simp only [exists_prop, Set.image2_add, Prod.mk.inj_iff, mem_addAntidiagonal, Sigma.exists,
      Ne.def, Set.image_prod, mem_sigma, Set.mem_setOf_eq, heq_iff_eq, Prod.exists] at H1 H2 ⊢
    

Proof state:
case coeff.h.refine'_3.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 i * (coeff y k * coeff z l) = 0
H1 : (i ∈ support x ∧ j ∈ support y + support z ∧ i + j = b) ∧ k ∈ support y ∧ l ∈ support z ∧ k + l = j
⊢ ∃ a b_1 a_1 b_2,
    ((a ∈ support x + support y ∧ b_1 ∈ support z ∧ a + b_1 = b) ∧ a_1 ∈ support x ∧ b_2 ∈ support y ∧ a_1 + b_2 = a) ∧
      ¬coeff x a_1 * coeff y b_2 * coeff z b_1 = 0 ∧
        { fst := (i, j), snd := (k, l) } = { fst := (a_1, b_2 + b_1), snd := (b_2, b_1) }

Next tactic:
obtain ⟨⟨nx, H, rfl⟩, ny, nz, 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
    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 simpa using H5
    simp only [and_true_iff, Prod.mk.inj_iff, eq_self_iff_true, heq_iff_eq, ← H1.2.2.2, ← H3.2.2.2]
  · rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩ H1 H2
    simp only [exists_prop, Set.image2_add, Prod.mk.inj_iff, mem_addAntidiagonal, Sigma.exists,
      Ne.def, Set.image_prod, mem_sigma, Set.mem_setOf_eq, heq_iff_eq, Prod.exists] at H1 H2 ⊢
    obtain ⟨⟨nx, H, rfl⟩, ny, nz, rfl⟩ := H1
    

Proof state:
case coeff.h.refine'_3.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
i k l : Γ
H2 : ¬coeff x i * (coeff y k * coeff z l) = 0
nx : i ∈ support x
ny : k ∈ support y
nz : l ∈ support z
H : k + l ∈ support y + support z
⊢ ∃ a b a_1 b_1,
    ((a ∈ support x + support y ∧ b ∈ support z ∧ a + b = i + (k + l)) ∧
        a_1 ∈ support x ∧ b_1 ∈ support y ∧ a_1 + b_1 = a) ∧
      ¬coeff x a_1 * coeff y b_1 * coeff z b = 0 ∧
        { fst := (i, k + l), snd := (k, l) } = { fst := (a_1, b_1 + b), snd := (b_1, b) }

Next tactic:
exact
      ⟨i + k, l, i, k, ⟨⟨Set.add_mem_add nx ny, nz, add_assoc _ _ _⟩ , nx, ny, rfl⟩,
        fun h => H2 <| by rw [← h, mul_assoc], rfl⟩
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 simpa using H5
    simp only [and_true_iff, Prod.mk.inj_iff, eq_self_iff_true, heq_iff_eq, ← H1.2.2.2, ← H3.2.2.2]
  · rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩ H1 H2
    simp only [exists_prop, Set.image2_add, Prod.mk.inj_iff, mem_addAntidiagonal, Sigma.exists,
      Ne.def, Set.image_prod, mem_sigma, Set.mem_setOf_eq, heq_iff_eq, Prod.exists] at H1 H2 ⊢
    obtain ⟨⟨nx, H, rfl⟩, ny, nz, rfl⟩ := H1
    exact
      ⟨i + k, l, i, k, ⟨⟨Set.add_mem_add nx ny, nz, add_assoc _ _ _⟩ , nx, ny, rfl⟩,
        fun h => H2 <| by 

Proof state:
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : NonUnitalSemiring R
x y z : HahnSeries Γ R
i k l : Γ
H2 : ¬coeff x i * (coeff y k * coeff z l) = 0
nx : i ∈ support x
ny : k ∈ support y
nz : l ∈ support z
H : k + l ∈ support y + support z
h : coeff x i * coeff y k * coeff z l = 0
⊢ coeff x i * (coeff y k * coeff z l) = 0

Next tactic:
rw [← h, mul_assoc]
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 simpa using H5
    simp only [and_true_iff, Prod.mk.inj_iff, eq_self_iff_true, heq_iff_eq, ← H1.2.2.2, ← H3.2.2.2]
  · rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩ H1 H2
    simp only [exists_prop, Set.image2_add, Prod.mk.inj_iff, mem_addAntidiagonal, Sigma.exists,
      Ne.def, Set.image_prod, mem_sigma, Set.mem_setOf_eq, heq_iff_eq, Prod.exists] at H1 H2 ⊢
    obtain ⟨⟨nx, H, rfl⟩, ny, nz, rfl⟩ := H1
    exact
      ⟨i + k, l, i, k, ⟨⟨Set.add_mem_add nx ny, nz, add_assoc _ _ _⟩ , nx, ny, rfl⟩,
        fun h => H2 <| by rw [← h, mul_assoc], rfl⟩
  · 

Proof state:
case coeff.h.refine'_4
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),
    coeff x a.snd.1 * coeff y a.snd.2 * coeff z a.fst.2 =
      coeff x ((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).fst.1 *
        (coeff y ((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).snd.1 *
          coeff z ((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).snd.2)

Next tactic:
rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩ _ _
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 simpa using H5
    simp only [and_true_iff, Prod.mk.inj_iff, eq_self_iff_true, heq_iff_eq, ← H1.2.2.2, ← H3.2.2.2]
  · rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩ H1 H2
    simp only [exists_prop, Set.image2_add, Prod.mk.inj_iff, mem_addAntidiagonal, Sigma.exists,
      Ne.def, Set.image_prod, mem_sigma, Set.mem_setOf_eq, heq_iff_eq, Prod.exists] at H1 H2 ⊢
    obtain ⟨⟨nx, H, rfl⟩, ny, nz, rfl⟩ := H1
    exact
      ⟨i + k, l, i, k, ⟨⟨Set.add_mem_add nx ny, nz, add_assoc _ _ _⟩ , nx, ny, rfl⟩,
        fun h => H2 <| by rw [← h, mul_assoc], rfl⟩
  · rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩ _ _
    

Proof state:
case coeff.h.refine'_4.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
⊢ 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 =
    coeff x
        ((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✝).fst.1 *
      (coeff y
          ((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✝).snd.1 *
        coeff z
          ((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✝).snd.2)

Next tactic:
simp [mul_assoc]
instance [NonUnitalNonAssocSemiring R] : NonUnitalNonAssocSemiring (HahnSeries Γ R) :=
  { inferInstanceAs (AddCommMonoid (HahnSeries Γ R)),
    inferInstanceAs (Distrib (HahnSeries Γ R)) with
    zero_mul := fun _ => by
      

Proof state:
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : NonUnitalNonAssocSemiring R
src✝¹ : AddCommMonoid (HahnSeries Γ R) := inferInstanceAs (AddCommMonoid (HahnSeries Γ R))
src✝ : Distrib (HahnSeries Γ R) := inferInstanceAs (Distrib (HahnSeries Γ R))
x✝ : HahnSeries Γ R
⊢ 0 * x✝ = 0

Next tactic:
ext
instance [NonUnitalNonAssocSemiring R] : NonUnitalNonAssocSemiring (HahnSeries Γ R) :=
  { inferInstanceAs (AddCommMonoid (HahnSeries Γ R)),
    inferInstanceAs (Distrib (HahnSeries Γ R)) with
    zero_mul := fun _ => by
      ext
      

Proof state:
case coeff.h
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : NonUnitalNonAssocSemiring R
src✝¹ : AddCommMonoid (HahnSeries Γ R) := inferInstanceAs (AddCommMonoid (HahnSeries Γ R))
src✝ : Distrib (HahnSeries Γ R) := inferInstanceAs (Distrib (HahnSeries Γ R))
x✝¹ : HahnSeries Γ R
x✝ : Γ
⊢ coeff (0 * x✝¹) x✝ = coeff 0 x✝

Next tactic:
simp [mul_coeff]
instance [NonUnitalNonAssocSemiring R] : NonUnitalNonAssocSemiring (HahnSeries Γ R) :=
  { inferInstanceAs (AddCommMonoid (HahnSeries Γ R)),
    inferInstanceAs (Distrib (HahnSeries Γ R)) with
    zero_mul := fun _ => by
      ext
      simp [mul_coeff]
    mul_zero := fun _ => by
      

Proof state:
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : NonUnitalNonAssocSemiring R
src✝¹ : AddCommMonoid (HahnSeries Γ R) := inferInstanceAs (AddCommMonoid (HahnSeries Γ R))
src✝ : Distrib (HahnSeries Γ R) := inferInstanceAs (Distrib (HahnSeries Γ R))
x✝ : HahnSeries Γ R
⊢ x✝ * 0 = 0

Next tactic:
ext
instance [NonUnitalNonAssocSemiring R] : NonUnitalNonAssocSemiring (HahnSeries Γ R) :=
  { inferInstanceAs (AddCommMonoid (HahnSeries Γ R)),
    inferInstanceAs (Distrib (HahnSeries Γ R)) with
    zero_mul := fun _ => by
      ext
      simp [mul_coeff]
    mul_zero := fun _ => by
      ext
      

Proof state:
case coeff.h
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : NonUnitalNonAssocSemiring R
src✝¹ : AddCommMonoid (HahnSeries Γ R) := inferInstanceAs (AddCommMonoid (HahnSeries Γ R))
src✝ : Distrib (HahnSeries Γ R) := inferInstanceAs (Distrib (HahnSeries Γ R))
x✝¹ : HahnSeries Γ R
x✝ : Γ
⊢ coeff (x✝¹ * 0) x✝ = coeff 0 x✝

Next tactic:
simp [mul_coeff]
instance [NonAssocSemiring R] : NonAssocSemiring (HahnSeries Γ R) :=
  { AddMonoidWithOne.unary,
    inferInstanceAs (NonUnitalNonAssocSemiring (HahnSeries Γ R)) with
    one_mul := fun x => by
      

Proof state:
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : NonAssocSemiring R
src✝¹ : AddMonoidWithOne (HahnSeries Γ R) := AddMonoidWithOne.unary
src✝ : NonUnitalNonAssocSemiring (HahnSeries Γ R) := inferInstanceAs (NonUnitalNonAssocSemiring (HahnSeries Γ R))
x : HahnSeries Γ R
⊢ 1 * x = x

Next tactic:
ext
instance [NonAssocSemiring R] : NonAssocSemiring (HahnSeries Γ R) :=
  { AddMonoidWithOne.unary,
    inferInstanceAs (NonUnitalNonAssocSemiring (HahnSeries Γ R)) with
    one_mul := fun x => by
      ext
      

Proof state:
case coeff.h
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : NonAssocSemiring R
src✝¹ : AddMonoidWithOne (HahnSeries Γ R) := AddMonoidWithOne.unary
src✝ : NonUnitalNonAssocSemiring (HahnSeries Γ R) := inferInstanceAs (NonUnitalNonAssocSemiring (HahnSeries Γ R))
x : HahnSeries Γ R
x✝ : Γ
⊢ coeff (1 * x) x✝ = coeff x x✝

Next tactic:
exact single_zero_mul_coeff.trans (one_mul _)
instance [NonAssocSemiring R] : NonAssocSemiring (HahnSeries Γ R) :=
  { AddMonoidWithOne.unary,
    inferInstanceAs (NonUnitalNonAssocSemiring (HahnSeries Γ R)) with
    one_mul := fun x => by
      ext
      exact single_zero_mul_coeff.trans (one_mul _)
    mul_one := fun x => by
      

Proof state:
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : NonAssocSemiring R
src✝¹ : AddMonoidWithOne (HahnSeries Γ R) := AddMonoidWithOne.unary
src✝ : NonUnitalNonAssocSemiring (HahnSeries Γ R) := inferInstanceAs (NonUnitalNonAssocSemiring (HahnSeries Γ R))
x : HahnSeries Γ R
⊢ x * 1 = x

Next tactic:
ext
instance [NonAssocSemiring R] : NonAssocSemiring (HahnSeries Γ R) :=
  { AddMonoidWithOne.unary,
    inferInstanceAs (NonUnitalNonAssocSemiring (HahnSeries Γ R)) with
    one_mul := fun x => by
      ext
      exact single_zero_mul_coeff.trans (one_mul _)
    mul_one := fun x => by
      ext
      

Proof state:
case coeff.h
Γ : Type u_1
R : Type u_2
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : NonAssocSemiring R
src✝¹ : AddMonoidWithOne (HahnSeries Γ R) := AddMonoidWithOne.unary
src✝ : NonUnitalNonAssocSemiring (HahnSeries Γ R) := inferInstanceAs (NonUnitalNonAssocSemiring (HahnSeries Γ R))
x : HahnSeries Γ R
x✝ : Γ
⊢ coeff (x * 1) x✝ = coeff x x✝

Next tactic:
exact mul_single_zero_coeff.trans (mul_one _)
instance [NonUnitalCommSemiring R] : NonUnitalCommSemiring (HahnSeries Γ R) :=
  { inferInstanceAs (NonUnitalSemiring (HahnSeries Γ R)) with
    mul_comm := fun x y => by
      

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

Next tactic:
ext