-- Mathlib/RingTheory/MvPowerSeries/LexOrder.lean
theorem le_lexOrder_mul (φ ψ : MvPowerSeries σ R) :
    lexOrder φ + lexOrder ψ ≤ lexOrder (φ * ψ)

Proof state:
case inr
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
d : σ →₀ ℕ
hd : ↑(toLex d) < φ.lexOrder + ψ.lexOrder
u v : σ →₀ ℕ
h : u + v = d
hv : ↑(toLex v) < ψ.lexOrder
⊢ (coeff R u) φ * (coeff R v) ψ = 0

Tactic:
rw [coeff_eq_zero_of_lt_lexOrder hv, mul_zero]

Result:
no goals

Proof state:
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
d : σ →₀ ℕ
hd : ↑(toLex d) < φ.lexOrder + ψ.lexOrder
⊢ (coeff R d) (φ * ψ) = 0

Tactic:
rw [coeff_mul]

Result:
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
d : σ →₀ ℕ
hd : ↑(toLex d) < φ.lexOrder + ψ.lexOrder
⊢ ∑ p ∈ Finset.antidiagonal d, (coeff R p.1) φ * (coeff R p.2) ψ = 0

Proof state:
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
⊢ ∀ (d : σ →₀ ℕ), ↑(toLex d) < φ.lexOrder + ψ.lexOrder → (coeff R d) (φ * ψ) = 0

Tactic:
intro d hd

Result:
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
d : σ →₀ ℕ
hd : ↑(toLex d) < φ.lexOrder + ψ.lexOrder
⊢ (coeff R d) (φ * ψ) = 0

Proof state:
case h.mk
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
d : σ →₀ ℕ
hd : ¬φ.lexOrder + ψ.lexOrder ≤ ↑(toLex d)
u v : σ →₀ ℕ
h : u + v = d
hu : φ.lexOrder ≤ ↑(toLex u)
hv : ψ.lexOrder ≤ ↑(toLex v)
⊢ φ.lexOrder + ψ.lexOrder ≤ ↑(toLex d)

Tactic:
simp only [← h, toLex_add, WithTop.coe_add, add_le_add hu hv]

Result:
no goals

Proof state:
case h.mk
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
d : σ →₀ ℕ
hd : ¬φ.lexOrder + ψ.lexOrder ≤ ↑(toLex d)
u v : σ →₀ ℕ
h : u + v = d
hu : φ.lexOrder ≤ ↑(toLex u)
hv : ψ.lexOrder ≤ ↑(toLex v)
⊢ False

Tactic:
apply hd

Result:
case h.mk
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
d : σ →₀ ℕ
hd : ¬φ.lexOrder + ψ.lexOrder ≤ ↑(toLex d)
u v : σ →₀ ℕ
h : u + v = d
hu : φ.lexOrder ≤ ↑(toLex u)
hv : ψ.lexOrder ≤ ↑(toLex v)
⊢ φ.lexOrder + ψ.lexOrder ≤ ↑(toLex d)

Proof state:
case h
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
d : σ →₀ ℕ
hd : ↑(toLex d) < φ.lexOrder + ψ.lexOrder
⊢ ∀ x ∈ Finset.antidiagonal d, (coeff R x.1) φ * (coeff R x.2) ψ = 0

Tactic:
rintro ⟨u, v⟩ h

Result:
case h.mk
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
d : σ →₀ ℕ
hd : ↑(toLex d) < φ.lexOrder + ψ.lexOrder
u v : σ →₀ ℕ
h : (u, v) ∈ Finset.antidiagonal d
⊢ (coeff R (u, v).1) φ * (coeff R (u, v).2) ψ = 0

Proof state:
case h.mk
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
d : σ →₀ ℕ
hd : ↑(toLex d) < φ.lexOrder + ψ.lexOrder
u v : σ →₀ ℕ
h : (u, v) ∈ Finset.antidiagonal d
⊢ (coeff R (u, v).1) φ * (coeff R (u, v).2) ψ = 0

Tactic:
simp only [Finset.mem_antidiagonal] at h

Result:
case h.mk
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
d : σ →₀ ℕ
hd : ↑(toLex d) < φ.lexOrder + ψ.lexOrder
u v : σ →₀ ℕ
h : u + v = d
⊢ (coeff R (u, v).1) φ * (coeff R (u, v).2) ψ = 0

Proof state:
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
d : σ →₀ ℕ
hd : ↑(toLex d) < φ.lexOrder + ψ.lexOrder
u v : σ →₀ ℕ
h : u + v = d
this : ↑(toLex u) < φ.lexOrder ∨ ↑(toLex v) < ψ.lexOrder
⊢ (coeff R u) φ * (coeff R v) ψ = 0

Tactic:
rcases this with (hu | hv)

Result:
case inl
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
d : σ →₀ ℕ
hd : ↑(toLex d) < φ.lexOrder + ψ.lexOrder
u v : σ →₀ ℕ
h : u + v = d
hu : ↑(toLex u) < φ.lexOrder
⊢ (coeff R u) φ * (coeff R v) ψ = 0

case inr
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
d : σ →₀ ℕ
hd : ↑(toLex d) < φ.lexOrder + ψ.lexOrder
u v : σ →₀ ℕ
h : u + v = d
hv : ↑(toLex v) < ψ.lexOrder
⊢ (coeff R u) φ * (coeff R v) ψ = 0

Proof state:
case h.mk
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
d : σ →₀ ℕ
hd : ↑(toLex d) < φ.lexOrder + ψ.lexOrder
u v : σ →₀ ℕ
h : u + v = d
⊢ ↑(toLex u) < φ.lexOrder ∨ ↑(toLex v) < ψ.lexOrder

Tactic:
rw [or_iff_not_imp_left, not_lt, ← not_le]

Result:
case h.mk
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
d : σ →₀ ℕ
hd : ↑(toLex d) < φ.lexOrder + ψ.lexOrder
u v : σ →₀ ℕ
h : u + v = d
⊢ φ.lexOrder ≤ ↑(toLex u) → ¬ψ.lexOrder ≤ ↑(toLex v)

Proof state:
case h.mk
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
d : σ →₀ ℕ
hd : ↑(toLex d) < φ.lexOrder + ψ.lexOrder
u v : σ →₀ ℕ
h : u + v = d
⊢ φ.lexOrder ≤ ↑(toLex u) → ¬ψ.lexOrder ≤ ↑(toLex v)

Tactic:
intro hu hv

Result:
case h.mk
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
d : σ →₀ ℕ
hd : ↑(toLex d) < φ.lexOrder + ψ.lexOrder
u v : σ →₀ ℕ
h : u + v = d
hu : φ.lexOrder ≤ ↑(toLex u)
hv : ψ.lexOrder ≤ ↑(toLex v)
⊢ False

Proof state:
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
⊢ φ.lexOrder + ψ.lexOrder ≤ (φ * ψ).lexOrder

Tactic:
rw [le_lexOrder_iff]

Result:
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
⊢ ∀ (d : σ →₀ ℕ), ↑(toLex d) < φ.lexOrder + ψ.lexOrder → (coeff R d) (φ * ψ) = 0

Proof state:
case h.mk
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
d : σ →₀ ℕ
hd : ↑(toLex d) < φ.lexOrder + ψ.lexOrder
u v : σ →₀ ℕ
h : u + v = d
hu : φ.lexOrder ≤ ↑(toLex u)
hv : ψ.lexOrder ≤ ↑(toLex v)
⊢ False

Tactic:
rw [← not_le] at hd

Result:
case h.mk
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
d : σ →₀ ℕ
hd : ¬φ.lexOrder + ψ.lexOrder ≤ ↑(toLex d)
u v : σ →₀ ℕ
h : u + v = d
hu : φ.lexOrder ≤ ↑(toLex u)
hv : ψ.lexOrder ≤ ↑(toLex v)
⊢ False

Proof state:
case inl
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
d : σ →₀ ℕ
hd : ↑(toLex d) < φ.lexOrder + ψ.lexOrder
u v : σ →₀ ℕ
h : u + v = d
hu : ↑(toLex u) < φ.lexOrder
⊢ (coeff R u) φ * (coeff R v) ψ = 0

Tactic:
rw [coeff_eq_zero_of_lt_lexOrder hu, zero_mul]

Result:
no goals

Proof state:
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
d : σ →₀ ℕ
hd : ↑(toLex d) < φ.lexOrder + ψ.lexOrder
⊢ ∑ p ∈ Finset.antidiagonal d, (coeff R p.1) φ * (coeff R p.2) ψ = 0

Tactic:
apply Finset.sum_eq_zero

Result:
case h
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
d : σ →₀ ℕ
hd : ↑(toLex d) < φ.lexOrder + ψ.lexOrder
⊢ ∀ x ∈ Finset.antidiagonal d, (coeff R x.1) φ * (coeff R x.2) ψ = 0