Theorem: MvPowerSeries.X_pow_dvd_iff
File: Mathlib/RingTheory/MvPowerSeries/Basic.lean

tactic: constructor
state_before: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
⊢ X s ^ n ∣ φ ↔ ∀ (m : σ →₀ ℕ), m s < n → (coeff R m) φ = 0
state_after: case mp
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
⊢ X s ^ n ∣ φ → ∀ (m : σ →₀ ℕ), m s < n → (coeff R m) φ = 0

case mpr
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
⊢ (∀ (m : σ →₀ ℕ), m s < n → (coeff R m) φ = 0) → X s ^ n ∣ φ

tactic: rintro ⟨φ, rfl⟩ m h
state_before: case mp
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
⊢ X s ^ n ∣ φ → ∀ (m : σ →₀ ℕ), m s < n → (coeff R m) φ = 0
state_after: case mp.intro
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
m : σ →₀ ℕ
h : m s < n
⊢ (coeff R m) (X s ^ n * φ) = 0

tactic: rw [coeff_mul, Finset.sum_eq_zero]
state_before: case mp.intro
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
m : σ →₀ ℕ
h : m s < n
⊢ (coeff R m) (X s ^ n * φ) = 0
state_after: case mp.intro
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
m : σ →₀ ℕ
h : m s < n
⊢ ∀ x ∈ antidiagonal m, (coeff R x.1) (X s ^ n) * (coeff R x.2) φ = 0

tactic: rintro ⟨i, j⟩ hij
state_before: case mp.intro
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
m : σ →₀ ℕ
h : m s < n
⊢ ∀ x ∈ antidiagonal m, (coeff R x.1) (X s ^ n) * (coeff R x.2) φ = 0
state_after: case mp.intro.mk
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
m : σ →₀ ℕ
h : m s < n
i j : σ →₀ ℕ
hij : (i, j) ∈ antidiagonal m
⊢ (coeff R (i, j).1) (X s ^ n) * (coeff R (i, j).2) φ = 0

tactic: rw [coeff_X_pow, if_neg, zero_mul]
state_before: case mp.intro.mk
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
m : σ →₀ ℕ
h : m s < n
i j : σ →₀ ℕ
hij : (i, j) ∈ antidiagonal m
⊢ (coeff R (i, j).1) (X s ^ n) * (coeff R (i, j).2) φ = 0
state_after: case mp.intro.mk.hnc
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
m : σ →₀ ℕ
h : m s < n
i j : σ →₀ ℕ
hij : (i, j) ∈ antidiagonal m
⊢ ¬(i, j).1 = single s n

tactic: contrapose! h
state_before: case mp.intro.mk.hnc
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
m : σ →₀ ℕ
h : m s < n
i j : σ →₀ ℕ
hij : (i, j) ∈ antidiagonal m
⊢ ¬(i, j).1 = single s n
state_after: case mp.intro.mk.hnc
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
m i j : σ →₀ ℕ
hij : (i, j) ∈ antidiagonal m
h : (i, j).1 = single s n
⊢ n ≤ m s

tactic: dsimp at h
state_before: case mp.intro.mk.hnc
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
m i j : σ →₀ ℕ
hij : (i, j) ∈ antidiagonal m
h : (i, j).1 = single s n
⊢ n ≤ m s
state_after: case mp.intro.mk.hnc
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
m i j : σ →₀ ℕ
hij : (i, j) ∈ antidiagonal m
h : i = single s n
⊢ n ≤ m s

tactic: subst i
state_before: case mp.intro.mk.hnc
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
m i j : σ →₀ ℕ
hij : (i, j) ∈ antidiagonal m
h : i = single s n
⊢ n ≤ m s
state_after: case mp.intro.mk.hnc
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
m j : σ →₀ ℕ
hij : (single s n, j) ∈ antidiagonal m
⊢ n ≤ m s

tactic: rw [mem_antidiagonal] at hij
state_before: case mp.intro.mk.hnc
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
m j : σ →₀ ℕ
hij : (single s n, j) ∈ antidiagonal m
⊢ n ≤ m s
state_after: case mp.intro.mk.hnc
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
m j : σ →₀ ℕ
hij : (single s n, j).1 + (single s n, j).2 = m
⊢ n ≤ m s

tactic: rw [← hij, Finsupp.add_apply, Finsupp.single_eq_same]
state_before: case mp.intro.mk.hnc
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
m j : σ →₀ ℕ
hij : (single s n, j).1 + (single s n, j).2 = m
⊢ n ≤ m s
state_after: case mp.intro.mk.hnc
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
m j : σ →₀ ℕ
hij : (single s n, j).1 + (single s n, j).2 = m
⊢ n ≤ n + (single s n, j).2 s

tactic: exact Nat.le_add_right n _
state_before: case mp.intro.mk.hnc
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
m j : σ →₀ ℕ
hij : (single s n, j).1 + (single s n, j).2 = m
⊢ n ≤ n + (single s n, j).2 s
state_after: no goals

tactic: intro h
state_before: case mpr
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
⊢ (∀ (m : σ →₀ ℕ), m s < n → (coeff R m) φ = 0) → X s ^ n ∣ φ
state_after: case mpr
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
⊢ X s ^ n ∣ φ

tactic: refine ⟨fun m => coeff R (m + single s n) φ, ?_⟩
state_before: case mpr
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
⊢ X s ^ n ∣ φ
state_after: case mpr
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
⊢ φ = X s ^ n * fun m => (coeff R (m + single s n)) φ

tactic: ext m
state_before: case mpr
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
⊢ φ = X s ^ n * fun m => (coeff R (m + single s n)) φ
state_after: case mpr.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
⊢ (coeff R m) φ = (coeff R m) (X s ^ n * fun m => (coeff R (m + single s n)) φ)

tactic: by_cases H : m - single s n + single s n = m
state_before: case mpr.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
⊢ (coeff R m) φ = (coeff R m) (X s ^ n * fun m => (coeff R (m + single s n)) φ)
state_after: 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 - single s n + single s n = m
⊢ (coeff R m) φ = (coeff R m) (X s ^ n * fun m => (coeff R (m + single s n)) φ)

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 - single s n + single s n = m
⊢ (coeff R m) φ = (coeff R m) (X s ^ n * fun m => (coeff R (m + single s n)) φ)

tactic: rw [coeff_mul, Finset.sum_eq_single (single s n, m - single s n)]
state_before: 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 - single s n + single s n = m
⊢ (coeff R m) φ = (coeff R m) (X s ^ n * fun m => (coeff R (m + single s n)) φ)
state_after: 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 - single s n + single s n = m
⊢ (coeff R m) φ =
    (coeff R (single s n, m - single s n).1) (X s ^ n) *
      (coeff R (single s n, m - single s n).2) fun m => (coeff R (m + single s n)) φ

case pos.h₀
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 - single s n + single s n = m
⊢ ∀ b ∈ antidiagonal m,
    b ≠ (single s n, m - single s n) →
      ((coeff R b.1) (X s ^ n) * (coeff R b.2) fun m => (coeff R (m + single s n)) φ) = 0

case pos.h₁
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 - single s n + single s n = m
⊢ (single s n, m - single s n) ∉ antidiagonal m →
    ((coeff R (single s n, m - single s n).1) (X s ^ n) *
        (coeff R (single s n, m - single s n).2) fun m => (coeff R (m + single s n)) φ) =
      0

tactic: rw [coeff_X_pow, if_pos rfl, one_mul]
state_before: 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 - single s n + single s n = m
⊢ (coeff R m) φ =
    (coeff R (single s n, m - single s n).1) (X s ^ n) *
      (coeff R (single s n, m - single s n).2) fun m => (coeff R (m + single s n)) φ
state_after: 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 - single s n + single s n = m
⊢ (coeff R m) φ = (coeff R (single s n, m - single s n).2) fun m => (coeff R (m + single s n)) φ

tactic: simpa using congr_arg (fun m : σ →₀ ℕ => coeff R m φ) H.symm
state_before: 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 - single s n + single s n = m
⊢ (coeff R m) φ = (coeff R (single s n, m - single s n).2) fun m => (coeff R (m + single s n)) φ
state_after: no goals

tactic: rintro ⟨i, j⟩ hij hne
state_before: case pos.h₀
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 - single s n + single s n = m
⊢ ∀ b ∈ antidiagonal m,
    b ≠ (single s n, m - single s n) →
      ((coeff R b.1) (X s ^ n) * (coeff R b.2) fun m => (coeff R (m + single s n)) φ) = 0
state_after: case pos.h₀.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 - single s n + single s n = m
i j : σ →₀ ℕ
hij : (i, j) ∈ antidiagonal m
hne : (i, j) ≠ (single s n, m - single s n)
⊢ ((coeff R (i, j).1) (X s ^ n) * (coeff R (i, j).2) fun m => (coeff R (m + single s n)) φ) = 0

tactic: rw [mem_antidiagonal] at hij
state_before: case pos.h₀.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 - single s n + single s n = m
i j : σ →₀ ℕ
hij : (i, j) ∈ antidiagonal m
hne : (i, j) ≠ (single s n, m - single s n)
⊢ ((coeff R (i, j).1) (X s ^ n) * (coeff R (i, j).2) fun m => (coeff R (m + single s n)) φ) = 0
state_after: case pos.h₀.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 - single s n + single s n = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
hne : (i, j) ≠ (single s n, m - single s n)
⊢ ((coeff R (i, j).1) (X s ^ n) * (coeff R (i, j).2) fun m => (coeff R (m + single s n)) φ) = 0

tactic: rw [coeff_X_pow]
state_before: case pos.h₀.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 - single s n + single s n = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
hne : (i, j) ≠ (single s n, m - single s n)
⊢ ((coeff R (i, j).1) (X s ^ n) * (coeff R (i, j).2) fun m => (coeff R (m + single s n)) φ) = 0
state_after: case pos.h₀.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 - single s n + single s n = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
hne : (i, j) ≠ (single s n, m - single s n)
⊢ ((if (i, j).1 = single s n then 1 else 0) * (coeff R (i, j).2) fun m => (coeff R (m + single s n)) φ) = 0

tactic: split_ifs with hi
state_before: case pos.h₀.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 - single s n + single s n = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
hne : (i, j) ≠ (single s n, m - single s n)
⊢ ((if (i, j).1 = single s n then 1 else 0) * (coeff R (i, j).2) fun m => (coeff R (m + single s n)) φ) = 0
state_after: 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 - single s n + single s n = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
hne : (i, j) ≠ (single s n, m - single s n)
hi : (i, j).1 = single s n
⊢ (1 * (coeff R (i, j).2) fun m => (coeff R (m + single s n)) φ) = 0

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 - single s n + single s n = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
hne : (i, j) ≠ (single s n, m - single s n)
hi : ¬(i, j).1 = single s n
⊢ (0 * (coeff R (i, j).2) fun m => (coeff R (m + single s n)) φ) = 0

tactic: exfalso
state_before: 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 - single s n + single s n = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
hne : (i, j) ≠ (single s n, m - single s n)
hi : (i, j).1 = single s n
⊢ (1 * (coeff R (i, j).2) fun m => (coeff R (m + single s n)) φ) = 0
state_after: 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 - single s n + single s n = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
hne : (i, j) ≠ (single s n, m - single s n)
hi : (i, j).1 = single s n
⊢ False

tactic: apply hne
state_before: 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 - single s n + single s n = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
hne : (i, j) ≠ (single s n, m - single s n)
hi : (i, j).1 = single s n
⊢ False
state_after: 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 - single s n + single s n = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
hne : (i, j) ≠ (single s n, m - single s n)
hi : (i, j).1 = single s n
⊢ (i, j) = (single s n, m - single s n)

tactic: rw [← hij, ← hi, Prod.mk.inj_iff]
state_before: 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 - single s n + single s n = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
hne : (i, j) ≠ (single s n, m - single s n)
hi : (i, j).1 = single s n
⊢ (i, j) = (single s n, m - single s n)
state_after: 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 - single s n + single s n = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
hne : (i, j) ≠ (single s n, m - single s n)
hi : (i, j).1 = single s n
⊢ i = (i, j).1 ∧ j = (i, j).1 + (i, j).2 - (i, j).1

tactic: refine ⟨rfl, ?_⟩
state_before: 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 - single s n + single s n = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
hne : (i, j) ≠ (single s n, m - single s n)
hi : (i, j).1 = single s n
⊢ i = (i, j).1 ∧ j = (i, j).1 + (i, j).2 - (i, j).1
state_after: 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 - single s n + single s n = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
hne : (i, j) ≠ (single s n, m - single s n)
hi : (i, j).1 = single s n
⊢ j = (i, j).1 + (i, j).2 - (i, j).1

tactic: ext t
state_before: 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 - single s n + single s n = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
hne : (i, j) ≠ (single s n, m - single s n)
hi : (i, j).1 = single s n
⊢ j = (i, j).1 + (i, j).2 - (i, j).1
state_after: 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 - single s n + single s n = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
hne : (i, j) ≠ (single s n, m - single s n)
hi : (i, j).1 = single s n
t : σ
⊢ j t = ((i, j).1 + (i, j).2 - (i, j).1) t

tactic: simp only [add_tsub_cancel_left, Finsupp.add_apply, Finsupp.tsub_apply]
state_before: 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 - single s n + single s n = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
hne : (i, j) ≠ (single s n, m - single s n)
hi : (i, j).1 = single s n
t : σ
⊢ j t = ((i, j).1 + (i, j).2 - (i, j).1) t
state_after: no goals

tactic: exact zero_mul _
state_before: 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 - single s n + single s n = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
hne : (i, j) ≠ (single s n, m - single s n)
hi : ¬(i, j).1 = single s n
⊢ (0 * (coeff R (i, j).2) fun m => (coeff R (m + single s n)) φ) = 0
state_after: no goals

tactic: intro hni
state_before: case pos.h₁
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 - single s n + single s n = m
⊢ (single s n, m - single s n) ∉ antidiagonal m →
    ((coeff R (single s n, m - single s n).1) (X s ^ n) *
        (coeff R (single s n, m - single s n).2) fun m => (coeff R (m + single s n)) φ) =
      0
state_after: case pos.h₁
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 - single s n + single s n = m
hni : (single s n, m - single s n) ∉ antidiagonal m
⊢ ((coeff R (single s n, m - single s n).1) (X s ^ n) *
      (coeff R (single s n, m - single s n).2) fun m => (coeff R (m + single s n)) φ) =
    0

tactic: exfalso
state_before: case pos.h₁
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 - single s n + single s n = m
hni : (single s n, m - single s n) ∉ antidiagonal m
⊢ ((coeff R (single s n, m - single s n).1) (X s ^ n) *
      (coeff R (single s n, m - single s n).2) fun m => (coeff R (m + single s n)) φ) =
    0
state_after: case pos.h₁
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 - single s n + single s n = m
hni : (single s n, m - single s n) ∉ antidiagonal m
⊢ False

tactic: apply hni
state_before: case pos.h₁
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 - single s n + single s n = m
hni : (single s n, m - single s n) ∉ antidiagonal m
⊢ False
state_after: case pos.h₁
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 - single s n + single s n = m
hni : (single s n, m - single s n) ∉ antidiagonal m
⊢ (single s n, m - single s n) ∈ antidiagonal m

tactic: rwa [mem_antidiagonal, add_comm]
state_before: case pos.h₁
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 - single s n + single s n = m
hni : (single s n, m - single s n) ∉ antidiagonal m
⊢ (single s n, m - single s n) ∈ antidiagonal m
state_after: no goals

tactic: rw [h, coeff_mul, Finset.sum_eq_zero]
state_before: 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 - single s n + single s n = m
⊢ (coeff R m) φ = (coeff R m) (X s ^ n * fun m => (coeff R (m + single s n)) φ)
state_after: 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 - single s n + single s n = m
⊢ ∀ x ∈ antidiagonal m, ((coeff R x.1) (X s ^ n) * (coeff R x.2) fun m => (coeff R (m + single s n)) φ) = 0

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 - single s n + single s n = m
⊢ m s < n

tactic: rintro ⟨i, j⟩ hij
state_before: 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 - single s n + single s n = m
⊢ ∀ x ∈ antidiagonal m, ((coeff R x.1) (X s ^ n) * (coeff R x.2) fun m => (coeff R (m + single s n)) φ) = 0
state_after: 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 - single s n + single 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 + single s n)) φ) = 0

tactic: rw [mem_antidiagonal] at hij
state_before: 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 - single s n + single 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 + single s n)) φ) = 0
state_after: 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 - single s n + single 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 + single s n)) φ) = 0

tactic: rw [coeff_X_pow]
state_before: 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 - single s n + single 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 + single s n)) φ) = 0
state_after: 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 - single s n + single s n = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
⊢ ((if (i, j).1 = single s n then 1 else 0) * (coeff R (i, j).2) fun m => (coeff R (m + single s n)) φ) = 0

tactic: split_ifs with hi
state_before: 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 - single s n + single s n = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
⊢ ((if (i, j).1 = single s n then 1 else 0) * (coeff R (i, j).2) fun m => (coeff R (m + single s n)) φ) = 0
state_after: 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 - single s n + single s n = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
hi : (i, j).1 = single s n
⊢ (1 * (coeff R (i, j).2) fun m => (coeff R (m + single s n)) φ) = 0

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 - single s n + single s n = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
hi : ¬(i, j).1 = single s n
⊢ (0 * (coeff R (i, j).2) fun m => (coeff R (m + single s n)) φ) = 0

tactic: exfalso
state_before: 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 - single s n + single s n = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
hi : (i, j).1 = single s n
⊢ (1 * (coeff R (i, j).2) fun m => (coeff R (m + single s n)) φ) = 0
state_after: 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 - single s n + single s n = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
hi : (i, j).1 = single s n
⊢ False

tactic: apply H
state_before: 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 - single s n + single s n = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
hi : (i, j).1 = single s n
⊢ False
state_after: 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 - single s n + single s n = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
hi : (i, j).1 = single s n
⊢ m - single s n + single s n = m

tactic: rw [← hij, hi]
state_before: 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 - single s n + single s n = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
hi : (i, j).1 = single s n
⊢ m - single s n + single s n = m
state_after: 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 - single s n + single s n = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
hi : (i, j).1 = single s n
⊢ single s n + (i, j).2 - single s n + single s n = single s n + (i, j).2

tactic: ext
state_before: 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 - single s n + single s n = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
hi : (i, j).1 = single s n
⊢ single s n + (i, j).2 - single s n + single s n = single s n + (i, j).2
state_after: 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 - single s n + single s n = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
hi : (i, j).1 = single s n
a✝ : σ
⊢ (single s n + (i, j).2 - single s n + single s n) a✝ = (single s n + (i, j).2) a✝

tactic: rw [coe_add, coe_add, Pi.add_apply, Pi.add_apply, add_tsub_cancel_left, add_comm]
state_before: 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 - single s n + single s n = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
hi : (i, j).1 = single s n
a✝ : σ
⊢ (single s n + (i, j).2 - single s n + single s n) a✝ = (single s n + (i, j).2) a✝
state_after: no goals

tactic: exact zero_mul _
state_before: 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 - single s n + single s n = m
i j : σ →₀ ℕ
hij : (i, j).1 + (i, j).2 = m
hi : ¬(i, j).1 = single s n
⊢ (0 * (coeff R (i, j).2) fun m => (coeff R (m + single s n)) φ) = 0
state_after: no goals

tactic: contrapose! H
state_before: 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 - single s n + single s n = m
⊢ m s < n
state_after: 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 - single s n + single s n = m

tactic: ext t
state_before: 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 - single s n + single s n = m
state_after: case neg.a.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 : n ≤ m s
t : σ
⊢ (m - single s n + single s n) t = m t

tactic: by_cases hst : s = t
state_before: case neg.a.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 : n ≤ m s
t : σ
⊢ (m - single s n + single s n) t = m t
state_after: 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 : n ≤ m s
t : σ
hst : s = t
⊢ (m - single s n + single s n) t = m t

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 : n ≤ m s
t : σ
hst : ¬s = t
⊢ (m - single s n + single s n) t = m t

tactic: subst t
state_before: 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 : n ≤ m s
t : σ
hst : s = t
⊢ (m - single s n + single s n) t = m t
state_after: 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 : n ≤ m s
⊢ (m - single s n + single s n) s = m s

tactic: simpa using tsub_add_cancel_of_le H
state_before: 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 : n ≤ m s
⊢ (m - single s n + single s n) s = m s
state_after: no goals

tactic: simp [Finsupp.single_apply, hst]
state_before: 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 : n ≤ m s
t : σ
hst : ¬s = t
⊢ (m - single s n + single s n) t = m t
state_after: no goals