have le_contr := Nat.not_lt_of_le nleq
case inl.inr
n✝ : ℕ
v₁ v₂ : Variables n✝ order.GrLex
le : Order.grlex.elem_sum v₁ < Order.grlex.elem_sum v₂
h : True
neq : ¬algebra.Order.bgrlex.elem_sum v₁ = algebra.Order.bgrlex.elem_sum v₂
nleq : algebra.Order.bgrlex.elem_sum v₂ ≤ algebra.Order.bgrlex.elem_sum v₁
⊢ False

case inr
n✝ : ℕ
v₁ v₂ : Variables n✝ order.GrLex
h✝ : ¬Order.grlex.elem_sum v₁ < Order.grlex.elem_sum v₂
h : if Order.grlex.elem_sum v₁ = Order.grlex.elem_sum v₂ then if Order.lex v₁ v₂ then True else False else False
⊢ algebra.Order.bgrlex v₁ v₂ = true
case inl.inr
n✝ : ℕ
v₁ v₂ : Variables n✝ order.GrLex
le : Order.grlex.elem_sum v₁ < Order.grlex.elem_sum v₂
h : True
neq : ¬algebra.Order.bgrlex.elem_sum v₁ = algebra.Order.bgrlex.elem_sum v₂
nleq : algebra.Order.bgrlex.elem_sum v₂ ≤ algebra.Order.bgrlex.elem_sum v₁
le_contr : ¬algebra.Order.bgrlex.elem_sum v₁ < algebra.Order.bgrlex.elem_sum v₂
⊢ False

case inr
n✝ : ℕ
v₁ v₂ : Variables n✝ order.GrLex
h✝ : ¬Order.grlex.elem_sum v₁ < Order.grlex.elem_sum v₂
h : if Order.grlex.elem_sum v₁ = Order.grlex.elem_sum v₂ then if Order.lex v₁ v₂ then True else False else False
⊢ algebra.Order.bgrlex v₁ v₂ = true