contradiction
case inr.inr.inl.inr.inl
n : ℕ
a b c : Variables n order.GrLex
h_lex₁ : ¬Order.lex (Variables.mul a c) (Variables.mul b c)
sum₁ : ℕ := Order.grlex.elem_sum (Variables.mul a c)
sum₂ : ℕ := Order.grlex.elem_sum (Variables.mul b c)
h✝² : Order.grlex.elem_sum b ≤ Order.grlex.elem_sum a
h✝¹ : Order.grlex.elem_sum a = Order.grlex.elem_sum b
h✝ : Order.grlex.elem_sum b ≤ Order.grlex.elem_sum a
h_lex₂ : Order.lex a b
h_le : True
hr : Order.lex (Variables.mul a c) (Variables.mul b c)
⊢ False

case inr.inr.inl.inr.inr
n : ℕ
a b c : Variables n order.GrLex
h✝⁴ : ¬Order.lex (Variables.mul a c) (Variables.mul b c)
sum₁ : ℕ := Order.grlex.elem_sum (Variables.mul a c)
sum₂ : ℕ := Order.grlex.elem_sum (Variables.mul b c)
h✝³ : Order.grlex.elem_sum b ≤ Order.grlex.elem_sum a
h✝² : Order.grlex.elem_sum a = Order.grlex.elem_sum b
h✝¹ : Order.grlex.elem_sum b ≤ Order.grlex.elem_sum a
h✝ : ¬Order.lex a b
h_le : False
⊢ False

case inr.inr.inr
n : ℕ
a b c : Variables n order.GrLex
h_le :
  let sum₁ := Order.grlex.elem_sum a;
  let sum₂ := Order.grlex.elem_sum b;
  if sum₁ < sum₂ then True else if sum₁ = sum₂ then if Order.lex a b then True else False else False
h✝² : ¬Order.lex (Variables.mul a c) (Variables.mul b c)
sum₁ : ℕ := Order.grlex.elem_sum (Variables.mul a c)
sum₂ : ℕ := Order.grlex.elem_sum (Variables.mul b c)
h✝¹ : ¬sum₁ < sum₂
h✝ : ¬sum₁ = sum₂
⊢ False
case inr.inr.inl.inr.inr
n : ℕ
a b c : Variables n order.GrLex
h✝⁴ : ¬Order.lex (Variables.mul a c) (Variables.mul b c)
sum₁ : ℕ := Order.grlex.elem_sum (Variables.mul a c)
sum₂ : ℕ := Order.grlex.elem_sum (Variables.mul b c)
h✝³ : Order.grlex.elem_sum b ≤ Order.grlex.elem_sum a
h✝² : Order.grlex.elem_sum a = Order.grlex.elem_sum b
h✝¹ : Order.grlex.elem_sum b ≤ Order.grlex.elem_sum a
h✝ : ¬Order.lex a b
h_le : False
⊢ False

case inr.inr.inr
n : ℕ
a b c : Variables n order.GrLex
h_le :
  let sum₁ := Order.grlex.elem_sum a;
  let sum₂ := Order.grlex.elem_sum b;
  if sum₁ < sum₂ then True else if sum₁ = sum₂ then if Order.lex a b then True else False else False
h✝² : ¬Order.lex (Variables.mul a c) (Variables.mul b c)
sum₁ : ℕ := Order.grlex.elem_sum (Variables.mul a c)
sum₂ : ℕ := Order.grlex.elem_sum (Variables.mul b c)
h✝¹ : ¬sum₁ < sum₂
h✝ : ¬sum₁ = sum₂
⊢ False