l3lab/ntp-mathlib huggingface dataset:
theorem X_pow_dvd_iff {s : σ} {n : ℕ} {φ : MvPowerSeries σ R} :
    (X s : MvPowerSeries σ R) ^ n ∣ φ ↔ ∀ m : σ →₀ ℕ, m s < n → coeff R m φ = 0 := by
  classical
  constructor
  · rintro ⟨φ, rfl⟩ m h
    rw [coeff_mul, Finset.sum_eq_zero]
    rintro ⟨i, j⟩ hij
    rw [coeff_X_pow, if_neg, zero_mul]
    contrapose! h
    dsimp at h
    subst i
    rw [mem_antidiagonal] at hij
    rw [← hij, Finsupp.add_apply, Finsupp.single_eq_same]
    exact Nat.le_add_right n _
  · intro h
    refine' ⟨fun m => coeff R (m + single s n) φ, _⟩
    ext m
    by_cases H : m - single s n + single s n = m
    · rw [coeff_mul, Finset.sum_eq_single (single s n, m - single s n)]
      · rw [coeff_X_pow, if_pos rfl, one_mul]
        simpa using congr_arg (fun m : σ →₀ ℕ => coeff R m φ) H.symm
      · rintro ⟨i, j⟩ hij hne
        rw [mem_antidiagonal] at hij
        rw [coeff_X_pow]
        split_ifs with hi
        · exfalso
          apply hne
          rw [← hij, ← hi, Prod.mk.inj_iff]
          refine' ⟨rfl, _⟩
          ext t
          simp only [add_tsub_cancel_left, Finsupp.add_apply, Finsupp.tsub_apply]
        · exact zero_mul _
      · intro hni
        exfalso
        apply hni
        rwa [mem_antidiagonal, add_comm]
    · 

Proof state:
case neg
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
h : ∀ (m : σ →₀ ℕ), m s < n → (coeff R m) φ = 0
m : σ →₀ ℕ
H : ¬((m - fun₀ | s => n) + fun₀ | s => n) = m
⊢ (coeff R m) φ = (coeff R m) (X s ^ n * fun m => (coeff R (m + fun₀ | s => n)) φ)

Next tactic:
rw [h, coeff_mul, Finset.sum_eq_zero]
theorem X_pow_dvd_iff {s : σ} {n : ℕ} {φ : MvPowerSeries σ R} :
    (X s : MvPowerSeries σ R) ^ n ∣ φ ↔ ∀ m : σ →₀ ℕ, m s < n → coeff R m φ = 0 := by
  classical
  constructor
  · rintro ⟨φ, rfl⟩ m h
    rw [coeff_mul, Finset.sum_eq_zero]
    rintro ⟨i, j⟩ hij
    rw [coeff_X_pow, if_neg, zero_mul]
    contrapose! h
    dsimp at h
    subst i
    rw [mem_antidiagonal] at hij
    rw [← hij, Finsupp.add_apply, Finsupp.single_eq_same]
    exact Nat.le_add_right n _
  · intro h
    refine' ⟨fun m => coeff R (m + single s n) φ, _⟩
    ext m
    by_cases H : m - single s n + single s n = m
    · rw [coeff_mul, Finset.sum_eq_single (single s n, m - single s n)]
      · rw [coeff_X_pow, if_pos rfl, one_mul]
        simpa using congr_arg (fun m : σ →₀ ℕ => coeff R m φ) H.symm
      · rintro ⟨i, j⟩ hij hne
        rw [mem_antidiagonal] at hij
        rw [coeff_X_pow]
        split_ifs with hi
        · exfalso
          apply hne
          rw [← hij, ← hi, Prod.mk.inj_iff]
          refine' ⟨rfl, _⟩
          ext t
          simp only [add_tsub_cancel_left, Finsupp.add_apply, Finsupp.tsub_apply]
        · exact zero_mul _
      · intro hni
        exfalso
        apply hni
        rwa [mem_antidiagonal, add_comm]
    · rw [h, coeff_mul, Finset.sum_eq_zero]
      · 

Proof state:
case neg
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
h : ∀ (m : σ →₀ ℕ), m s < n → (coeff R m) φ = 0
m : σ →₀ ℕ
H : ¬((m - fun₀ | s => n) + fun₀ | s => n) = m
⊢ ∀ x ∈ antidiagonal m, ((coeff R x.1) (X s ^ n) * (coeff R x.2) fun m => (coeff R (m + fun₀ | s => n)) φ) = 0

Next tactic:
rintro ⟨i, j⟩ hij
theorem X_pow_dvd_iff {s : σ} {n : ℕ} {φ : MvPowerSeries σ R} :
    (X s : MvPowerSeries σ R) ^ n ∣ φ ↔ ∀ m : σ →₀ ℕ, m s < n → coeff R m φ = 0 := by
  classical
  constructor
  · rintro ⟨φ, rfl⟩ m h
    rw [coeff_mul, Finset.sum_eq_zero]
    rintro ⟨i, j⟩ hij
    rw [coeff_X_pow, if_neg, zero_mul]
    contrapose! h
    dsimp at h
    subst i
    rw [mem_antidiagonal] at hij
    rw [← hij, Finsupp.add_apply, Finsupp.single_eq_same]
    exact Nat.le_add_right n _
  · intro h
    refine' ⟨fun m => coeff R (m + single s n) φ, _⟩
    ext m
    by_cases H : m - single s n + single s n = m
    · rw [coeff_mul, Finset.sum_eq_single (single s n, m - single s n)]
      · rw [coeff_X_pow, if_pos rfl, one_mul]
        simpa using congr_arg (fun m : σ →₀ ℕ => coeff R m φ) H.symm
      · rintro ⟨i, j⟩ hij hne
        rw [mem_antidiagonal] at hij
        rw [coeff_X_pow]
        split_ifs with hi
        · exfalso
          apply hne
          rw [← hij, ← hi, Prod.mk.inj_iff]
          refine' ⟨rfl, _⟩
          ext t
          simp only [add_tsub_cancel_left, Finsupp.add_apply, Finsupp.tsub_apply]
        · exact zero_mul _
      · intro hni
        exfalso
        apply hni
        rwa [mem_antidiagonal, add_comm]
    · rw [h, coeff_mul, Finset.sum_eq_zero]
      · rintro ⟨i, j⟩ hij
        

Proof state:
case neg.mk
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
h : ∀ (m : σ →₀ ℕ), m s < n → (coeff R m) φ = 0
m : σ →₀ ℕ
H : ¬((m - fun₀ | s => n) + fun₀ | s => n) = m
i j : σ →₀ ℕ
hij : (i, j) ∈ antidiagonal m
⊢ ((coeff R (i, j).1) (X s ^ n) * (coeff R (i, j).2) fun m => (coeff R (m + fun₀ | s => n)) φ) = 0

Next tactic:
rw [mem_antidiagonal] at hij
theorem X_pow_dvd_iff {s : σ} {n : ℕ} {φ : MvPowerSeries σ R} :
    (X s : MvPowerSeries σ R) ^ n ∣ φ ↔ ∀ m : σ →₀ ℕ, m s < n → coeff R m φ = 0 := by
  classical
  constructor
  · rintro ⟨φ, rfl⟩ m h
    rw [coeff_mul, Finset.sum_eq_zero]
    rintro ⟨i, j⟩ hij
    rw [coeff_X_pow, if_neg, zero_mul]
    contrapose! h
    dsimp at h
    subst i
    rw [mem_antidiagonal] at hij
    rw [← hij, Finsupp.add_apply, Finsupp.single_eq_same]
    exact Nat.le_add_right n _
  · intro h
    refine' ⟨fun m => coeff R (m + single s n) φ, _⟩
    ext m
    by_cases H : m - single s n + single s n = m
    · rw [coeff_mul, Finset.sum_eq_single (single s n, m - single s n)]
      · rw [coeff_X_pow, if_pos rfl, one_mul]
        simpa using congr_arg (fun m : σ →₀ ℕ => coeff R m φ) H.symm
      · rintro ⟨i, j⟩ hij hne
        rw [mem_antidiagonal] at hij
        rw [coeff_X_pow]
        split_ifs with hi
        · exfalso
          apply hne
          rw [← hij, ← hi, Prod.mk.inj_iff]
          refine' ⟨rfl, _⟩
          ext t
          simp only [add_tsub_cancel_left, Finsupp.add_apply, Finsupp.tsub_apply]
        · exact zero_mul _
      · intro hni
        exfalso
        apply hni
        rwa [mem_antidiagonal, add_comm]
    · rw [h, coeff_mul, Finset.sum_eq_zero]
      · rintro ⟨i, j⟩ hij
        rw [mem_antidiagonal] at hij
        

Proof state:
case neg.mk
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
h : ∀ (m : σ →₀ ℕ), m s < n → (coeff R m) φ = 0
m : σ →₀ ℕ
H : ¬((m - fun₀ | s => n) + fun₀ | s => n) = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
⊢ ((coeff R (i, j).1) (X s ^ n) * (coeff R (i, j).2) fun m => (coeff R (m + fun₀ | s => n)) φ) = 0

Next tactic:
rw [coeff_X_pow]
theorem X_pow_dvd_iff {s : σ} {n : ℕ} {φ : MvPowerSeries σ R} :
    (X s : MvPowerSeries σ R) ^ n ∣ φ ↔ ∀ m : σ →₀ ℕ, m s < n → coeff R m φ = 0 := by
  classical
  constructor
  · rintro ⟨φ, rfl⟩ m h
    rw [coeff_mul, Finset.sum_eq_zero]
    rintro ⟨i, j⟩ hij
    rw [coeff_X_pow, if_neg, zero_mul]
    contrapose! h
    dsimp at h
    subst i
    rw [mem_antidiagonal] at hij
    rw [← hij, Finsupp.add_apply, Finsupp.single_eq_same]
    exact Nat.le_add_right n _
  · intro h
    refine' ⟨fun m => coeff R (m + single s n) φ, _⟩
    ext m
    by_cases H : m - single s n + single s n = m
    · rw [coeff_mul, Finset.sum_eq_single (single s n, m - single s n)]
      · rw [coeff_X_pow, if_pos rfl, one_mul]
        simpa using congr_arg (fun m : σ →₀ ℕ => coeff R m φ) H.symm
      · rintro ⟨i, j⟩ hij hne
        rw [mem_antidiagonal] at hij
        rw [coeff_X_pow]
        split_ifs with hi
        · exfalso
          apply hne
          rw [← hij, ← hi, Prod.mk.inj_iff]
          refine' ⟨rfl, _⟩
          ext t
          simp only [add_tsub_cancel_left, Finsupp.add_apply, Finsupp.tsub_apply]
        · exact zero_mul _
      · intro hni
        exfalso
        apply hni
        rwa [mem_antidiagonal, add_comm]
    · rw [h, coeff_mul, Finset.sum_eq_zero]
      · rintro ⟨i, j⟩ hij
        rw [mem_antidiagonal] at hij
        rw [coeff_X_pow]
        

Proof state:
case neg.mk
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
h : ∀ (m : σ →₀ ℕ), m s < n → (coeff R m) φ = 0
m : σ →₀ ℕ
H : ¬((m - fun₀ | s => n) + fun₀ | s => n) = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
⊢ ((if (i, j).1 = fun₀ | s => n then 1 else 0) * (coeff R (i, j).2) fun m => (coeff R (m + fun₀ | s => n)) φ) = 0

Next tactic:
split_ifs with hi
theorem X_pow_dvd_iff {s : σ} {n : ℕ} {φ : MvPowerSeries σ R} :
    (X s : MvPowerSeries σ R) ^ n ∣ φ ↔ ∀ m : σ →₀ ℕ, m s < n → coeff R m φ = 0 := by
  classical
  constructor
  · rintro ⟨φ, rfl⟩ m h
    rw [coeff_mul, Finset.sum_eq_zero]
    rintro ⟨i, j⟩ hij
    rw [coeff_X_pow, if_neg, zero_mul]
    contrapose! h
    dsimp at h
    subst i
    rw [mem_antidiagonal] at hij
    rw [← hij, Finsupp.add_apply, Finsupp.single_eq_same]
    exact Nat.le_add_right n _
  · intro h
    refine' ⟨fun m => coeff R (m + single s n) φ, _⟩
    ext m
    by_cases H : m - single s n + single s n = m
    · rw [coeff_mul, Finset.sum_eq_single (single s n, m - single s n)]
      · rw [coeff_X_pow, if_pos rfl, one_mul]
        simpa using congr_arg (fun m : σ →₀ ℕ => coeff R m φ) H.symm
      · rintro ⟨i, j⟩ hij hne
        rw [mem_antidiagonal] at hij
        rw [coeff_X_pow]
        split_ifs with hi
        · exfalso
          apply hne
          rw [← hij, ← hi, Prod.mk.inj_iff]
          refine' ⟨rfl, _⟩
          ext t
          simp only [add_tsub_cancel_left, Finsupp.add_apply, Finsupp.tsub_apply]
        · exact zero_mul _
      · intro hni
        exfalso
        apply hni
        rwa [mem_antidiagonal, add_comm]
    · rw [h, coeff_mul, Finset.sum_eq_zero]
      · rintro ⟨i, j⟩ hij
        rw [mem_antidiagonal] at hij
        rw [coeff_X_pow]
        split_ifs with hi
        · 

Proof state:
case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
h : ∀ (m : σ →₀ ℕ), m s < n → (coeff R m) φ = 0
m : σ →₀ ℕ
H : ¬((m - fun₀ | s => n) + fun₀ | s => n) = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
hi : (i, j).1 = fun₀ | s => n
⊢ (1 * (coeff R (i, j).2) fun m => (coeff R (m + fun₀ | s => n)) φ) = 0

Next tactic:
exfalso
theorem X_pow_dvd_iff {s : σ} {n : ℕ} {φ : MvPowerSeries σ R} :
    (X s : MvPowerSeries σ R) ^ n ∣ φ ↔ ∀ m : σ →₀ ℕ, m s < n → coeff R m φ = 0 := by
  classical
  constructor
  · rintro ⟨φ, rfl⟩ m h
    rw [coeff_mul, Finset.sum_eq_zero]
    rintro ⟨i, j⟩ hij
    rw [coeff_X_pow, if_neg, zero_mul]
    contrapose! h
    dsimp at h
    subst i
    rw [mem_antidiagonal] at hij
    rw [← hij, Finsupp.add_apply, Finsupp.single_eq_same]
    exact Nat.le_add_right n _
  · intro h
    refine' ⟨fun m => coeff R (m + single s n) φ, _⟩
    ext m
    by_cases H : m - single s n + single s n = m
    · rw [coeff_mul, Finset.sum_eq_single (single s n, m - single s n)]
      · rw [coeff_X_pow, if_pos rfl, one_mul]
        simpa using congr_arg (fun m : σ →₀ ℕ => coeff R m φ) H.symm
      · rintro ⟨i, j⟩ hij hne
        rw [mem_antidiagonal] at hij
        rw [coeff_X_pow]
        split_ifs with hi
        · exfalso
          apply hne
          rw [← hij, ← hi, Prod.mk.inj_iff]
          refine' ⟨rfl, _⟩
          ext t
          simp only [add_tsub_cancel_left, Finsupp.add_apply, Finsupp.tsub_apply]
        · exact zero_mul _
      · intro hni
        exfalso
        apply hni
        rwa [mem_antidiagonal, add_comm]
    · rw [h, coeff_mul, Finset.sum_eq_zero]
      · rintro ⟨i, j⟩ hij
        rw [mem_antidiagonal] at hij
        rw [coeff_X_pow]
        split_ifs with hi
        · exfalso
          

Proof state:
case pos.h
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
h : ∀ (m : σ →₀ ℕ), m s < n → (coeff R m) φ = 0
m : σ →₀ ℕ
H : ¬((m - fun₀ | s => n) + fun₀ | s => n) = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
hi : (i, j).1 = fun₀ | s => n
⊢ False

Next tactic:
apply H
theorem X_pow_dvd_iff {s : σ} {n : ℕ} {φ : MvPowerSeries σ R} :
    (X s : MvPowerSeries σ R) ^ n ∣ φ ↔ ∀ m : σ →₀ ℕ, m s < n → coeff R m φ = 0 := by
  classical
  constructor
  · rintro ⟨φ, rfl⟩ m h
    rw [coeff_mul, Finset.sum_eq_zero]
    rintro ⟨i, j⟩ hij
    rw [coeff_X_pow, if_neg, zero_mul]
    contrapose! h
    dsimp at h
    subst i
    rw [mem_antidiagonal] at hij
    rw [← hij, Finsupp.add_apply, Finsupp.single_eq_same]
    exact Nat.le_add_right n _
  · intro h
    refine' ⟨fun m => coeff R (m + single s n) φ, _⟩
    ext m
    by_cases H : m - single s n + single s n = m
    · rw [coeff_mul, Finset.sum_eq_single (single s n, m - single s n)]
      · rw [coeff_X_pow, if_pos rfl, one_mul]
        simpa using congr_arg (fun m : σ →₀ ℕ => coeff R m φ) H.symm
      · rintro ⟨i, j⟩ hij hne
        rw [mem_antidiagonal] at hij
        rw [coeff_X_pow]
        split_ifs with hi
        · exfalso
          apply hne
          rw [← hij, ← hi, Prod.mk.inj_iff]
          refine' ⟨rfl, _⟩
          ext t
          simp only [add_tsub_cancel_left, Finsupp.add_apply, Finsupp.tsub_apply]
        · exact zero_mul _
      · intro hni
        exfalso
        apply hni
        rwa [mem_antidiagonal, add_comm]
    · rw [h, coeff_mul, Finset.sum_eq_zero]
      · rintro ⟨i, j⟩ hij
        rw [mem_antidiagonal] at hij
        rw [coeff_X_pow]
        split_ifs with hi
        · exfalso
          apply H
          

Proof state:
case pos.h
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
h : ∀ (m : σ →₀ ℕ), m s < n → (coeff R m) φ = 0
m : σ →₀ ℕ
H : ¬((m - fun₀ | s => n) + fun₀ | s => n) = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
hi : (i, j).1 = fun₀ | s => n
⊢ ((m - fun₀ | s => n) + fun₀ | s => n) = m

Next tactic:
rw [← hij, hi]
theorem X_pow_dvd_iff {s : σ} {n : ℕ} {φ : MvPowerSeries σ R} :
    (X s : MvPowerSeries σ R) ^ n ∣ φ ↔ ∀ m : σ →₀ ℕ, m s < n → coeff R m φ = 0 := by
  classical
  constructor
  · rintro ⟨φ, rfl⟩ m h
    rw [coeff_mul, Finset.sum_eq_zero]
    rintro ⟨i, j⟩ hij
    rw [coeff_X_pow, if_neg, zero_mul]
    contrapose! h
    dsimp at h
    subst i
    rw [mem_antidiagonal] at hij
    rw [← hij, Finsupp.add_apply, Finsupp.single_eq_same]
    exact Nat.le_add_right n _
  · intro h
    refine' ⟨fun m => coeff R (m + single s n) φ, _⟩
    ext m
    by_cases H : m - single s n + single s n = m
    · rw [coeff_mul, Finset.sum_eq_single (single s n, m - single s n)]
      · rw [coeff_X_pow, if_pos rfl, one_mul]
        simpa using congr_arg (fun m : σ →₀ ℕ => coeff R m φ) H.symm
      · rintro ⟨i, j⟩ hij hne
        rw [mem_antidiagonal] at hij
        rw [coeff_X_pow]
        split_ifs with hi
        · exfalso
          apply hne
          rw [← hij, ← hi, Prod.mk.inj_iff]
          refine' ⟨rfl, _⟩
          ext t
          simp only [add_tsub_cancel_left, Finsupp.add_apply, Finsupp.tsub_apply]
        · exact zero_mul _
      · intro hni
        exfalso
        apply hni
        rwa [mem_antidiagonal, add_comm]
    · rw [h, coeff_mul, Finset.sum_eq_zero]
      · rintro ⟨i, j⟩ hij
        rw [mem_antidiagonal] at hij
        rw [coeff_X_pow]
        split_ifs with hi
        · exfalso
          apply H
          rw [← hij, hi]
          

Proof state:
case pos.h
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
h : ∀ (m : σ →₀ ℕ), m s < n → (coeff R m) φ = 0
m : σ →₀ ℕ
H : ¬((m - fun₀ | s => n) + fun₀ | s => n) = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
hi : (i, j).1 = fun₀ | s => n
⊢ (((fun₀ | s => n) + (i, j).2 - fun₀ | s => n) + fun₀ | s => n) = (fun₀ | s => n) + (i, j).2

Next tactic:
ext
theorem X_pow_dvd_iff {s : σ} {n : ℕ} {φ : MvPowerSeries σ R} :
    (X s : MvPowerSeries σ R) ^ n ∣ φ ↔ ∀ m : σ →₀ ℕ, m s < n → coeff R m φ = 0 := by
  classical
  constructor
  · rintro ⟨φ, rfl⟩ m h
    rw [coeff_mul, Finset.sum_eq_zero]
    rintro ⟨i, j⟩ hij
    rw [coeff_X_pow, if_neg, zero_mul]
    contrapose! h
    dsimp at h
    subst i
    rw [mem_antidiagonal] at hij
    rw [← hij, Finsupp.add_apply, Finsupp.single_eq_same]
    exact Nat.le_add_right n _
  · intro h
    refine' ⟨fun m => coeff R (m + single s n) φ, _⟩
    ext m
    by_cases H : m - single s n + single s n = m
    · rw [coeff_mul, Finset.sum_eq_single (single s n, m - single s n)]
      · rw [coeff_X_pow, if_pos rfl, one_mul]
        simpa using congr_arg (fun m : σ →₀ ℕ => coeff R m φ) H.symm
      · rintro ⟨i, j⟩ hij hne
        rw [mem_antidiagonal] at hij
        rw [coeff_X_pow]
        split_ifs with hi
        · exfalso
          apply hne
          rw [← hij, ← hi, Prod.mk.inj_iff]
          refine' ⟨rfl, _⟩
          ext t
          simp only [add_tsub_cancel_left, Finsupp.add_apply, Finsupp.tsub_apply]
        · exact zero_mul _
      · intro hni
        exfalso
        apply hni
        rwa [mem_antidiagonal, add_comm]
    · rw [h, coeff_mul, Finset.sum_eq_zero]
      · rintro ⟨i, j⟩ hij
        rw [mem_antidiagonal] at hij
        rw [coeff_X_pow]
        split_ifs with hi
        · exfalso
          apply H
          rw [← hij, hi]
          ext
          

Proof state:
case pos.h.h
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
h : ∀ (m : σ →₀ ℕ), m s < n → (coeff R m) φ = 0
m : σ →₀ ℕ
H : ¬((m - fun₀ | s => n) + fun₀ | s => n) = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
hi : (i, j).1 = fun₀ | s => n
a✝ : σ
⊢ (((fun₀ | s => n) + (i, j).2 - fun₀ | s => n) + fun₀ | s => n) a✝ = ((fun₀ | s => n) + (i, j).2) a✝

Next tactic:
rw [coe_add, coe_add, Pi.add_apply, Pi.add_apply, add_tsub_cancel_left, add_comm]
theorem X_pow_dvd_iff {s : σ} {n : ℕ} {φ : MvPowerSeries σ R} :
    (X s : MvPowerSeries σ R) ^ n ∣ φ ↔ ∀ m : σ →₀ ℕ, m s < n → coeff R m φ = 0 := by
  classical
  constructor
  · rintro ⟨φ, rfl⟩ m h
    rw [coeff_mul, Finset.sum_eq_zero]
    rintro ⟨i, j⟩ hij
    rw [coeff_X_pow, if_neg, zero_mul]
    contrapose! h
    dsimp at h
    subst i
    rw [mem_antidiagonal] at hij
    rw [← hij, Finsupp.add_apply, Finsupp.single_eq_same]
    exact Nat.le_add_right n _
  · intro h
    refine' ⟨fun m => coeff R (m + single s n) φ, _⟩
    ext m
    by_cases H : m - single s n + single s n = m
    · rw [coeff_mul, Finset.sum_eq_single (single s n, m - single s n)]
      · rw [coeff_X_pow, if_pos rfl, one_mul]
        simpa using congr_arg (fun m : σ →₀ ℕ => coeff R m φ) H.symm
      · rintro ⟨i, j⟩ hij hne
        rw [mem_antidiagonal] at hij
        rw [coeff_X_pow]
        split_ifs with hi
        · exfalso
          apply hne
          rw [← hij, ← hi, Prod.mk.inj_iff]
          refine' ⟨rfl, _⟩
          ext t
          simp only [add_tsub_cancel_left, Finsupp.add_apply, Finsupp.tsub_apply]
        · exact zero_mul _
      · intro hni
        exfalso
        apply hni
        rwa [mem_antidiagonal, add_comm]
    · rw [h, coeff_mul, Finset.sum_eq_zero]
      · rintro ⟨i, j⟩ hij
        rw [mem_antidiagonal] at hij
        rw [coeff_X_pow]
        split_ifs with hi
        · exfalso
          apply H
          rw [← hij, hi]
          ext
          rw [coe_add, coe_add, Pi.add_apply, Pi.add_apply, add_tsub_cancel_left, add_comm]
        · 

Proof state:
case neg
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
h : ∀ (m : σ →₀ ℕ), m s < n → (coeff R m) φ = 0
m : σ →₀ ℕ
H : ¬((m - fun₀ | s => n) + fun₀ | s => n) = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
hi : ¬(i, j).1 = fun₀ | s => n
⊢ (0 * (coeff R (i, j).2) fun m => (coeff R (m + fun₀ | s => n)) φ) = 0

Next tactic:
exact zero_mul _
theorem X_pow_dvd_iff {s : σ} {n : ℕ} {φ : MvPowerSeries σ R} :
    (X s : MvPowerSeries σ R) ^ n ∣ φ ↔ ∀ m : σ →₀ ℕ, m s < n → coeff R m φ = 0 := by
  classical
  constructor
  · rintro ⟨φ, rfl⟩ m h
    rw [coeff_mul, Finset.sum_eq_zero]
    rintro ⟨i, j⟩ hij
    rw [coeff_X_pow, if_neg, zero_mul]
    contrapose! h
    dsimp at h
    subst i
    rw [mem_antidiagonal] at hij
    rw [← hij, Finsupp.add_apply, Finsupp.single_eq_same]
    exact Nat.le_add_right n _
  · intro h
    refine' ⟨fun m => coeff R (m + single s n) φ, _⟩
    ext m
    by_cases H : m - single s n + single s n = m
    · rw [coeff_mul, Finset.sum_eq_single (single s n, m - single s n)]
      · rw [coeff_X_pow, if_pos rfl, one_mul]
        simpa using congr_arg (fun m : σ →₀ ℕ => coeff R m φ) H.symm
      · rintro ⟨i, j⟩ hij hne
        rw [mem_antidiagonal] at hij
        rw [coeff_X_pow]
        split_ifs with hi
        · exfalso
          apply hne
          rw [← hij, ← hi, Prod.mk.inj_iff]
          refine' ⟨rfl, _⟩
          ext t
          simp only [add_tsub_cancel_left, Finsupp.add_apply, Finsupp.tsub_apply]
        · exact zero_mul _
      · intro hni
        exfalso
        apply hni
        rwa [mem_antidiagonal, add_comm]
    · rw [h, coeff_mul, Finset.sum_eq_zero]
      · rintro ⟨i, j⟩ hij
        rw [mem_antidiagonal] at hij
        rw [coeff_X_pow]
        split_ifs with hi
        · exfalso
          apply H
          rw [← hij, hi]
          ext
          rw [coe_add, coe_add, Pi.add_apply, Pi.add_apply, add_tsub_cancel_left, add_comm]
        · exact zero_mul _
      · 

Proof state:
case neg.a
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
h : ∀ (m : σ →₀ ℕ), m s < n → (coeff R m) φ = 0
m : σ →₀ ℕ
H : ¬((m - fun₀ | s => n) + fun₀ | s => n) = m
⊢ m s < n

Next tactic:
contrapose! H
theorem X_pow_dvd_iff {s : σ} {n : ℕ} {φ : MvPowerSeries σ R} :
    (X s : MvPowerSeries σ R) ^ n ∣ φ ↔ ∀ m : σ →₀ ℕ, m s < n → coeff R m φ = 0 := by
  classical
  constructor
  · rintro ⟨φ, rfl⟩ m h
    rw [coeff_mul, Finset.sum_eq_zero]
    rintro ⟨i, j⟩ hij
    rw [coeff_X_pow, if_neg, zero_mul]
    contrapose! h
    dsimp at h
    subst i
    rw [mem_antidiagonal] at hij
    rw [← hij, Finsupp.add_apply, Finsupp.single_eq_same]
    exact Nat.le_add_right n _
  · intro h
    refine' ⟨fun m => coeff R (m + single s n) φ, _⟩
    ext m
    by_cases H : m - single s n + single s n = m
    · rw [coeff_mul, Finset.sum_eq_single (single s n, m - single s n)]
      · rw [coeff_X_pow, if_pos rfl, one_mul]
        simpa using congr_arg (fun m : σ →₀ ℕ => coeff R m φ) H.symm
      · rintro ⟨i, j⟩ hij hne
        rw [mem_antidiagonal] at hij
        rw [coeff_X_pow]
        split_ifs with hi
        · exfalso
          apply hne
          rw [← hij, ← hi, Prod.mk.inj_iff]
          refine' ⟨rfl, _⟩
          ext t
          simp only [add_tsub_cancel_left, Finsupp.add_apply, Finsupp.tsub_apply]
        · exact zero_mul _
      · intro hni
        exfalso
        apply hni
        rwa [mem_antidiagonal, add_comm]
    · rw [h, coeff_mul, Finset.sum_eq_zero]
      · rintro ⟨i, j⟩ hij
        rw [mem_antidiagonal] at hij
        rw [coeff_X_pow]
        split_ifs with hi
        · exfalso
          apply H
          rw [← hij, hi]
          ext
          rw [coe_add, coe_add, Pi.add_apply, Pi.add_apply, add_tsub_cancel_left, add_comm]
        · exact zero_mul _
      · contrapose! H
        

Proof state:
case neg.a
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
h : ∀ (m : σ →₀ ℕ), m s < n → (coeff R m) φ = 0
m : σ →₀ ℕ
H : n ≤ m s
⊢ ((m - fun₀ | s => n) + fun₀ | s => n) = m

Next tactic:
ext t