Theorem: algebra.grlex_add_le_add
Proof state before:
case inl.inl.inr.inr.inl
n : ℕ
a b c : Variables n order.GrLex
h✝¹ : Order.lex (Variables.mul a c) (Variables.mul b c)
h✝ : Order.lex a b
sum₁ : ℕ := Order.grlex.elem_sum (Variables.mul a c)
sum₂ : ℕ := Order.grlex.elem_sum (Variables.mul b c)
h₃ : Order.grlex.elem_sum a < Order.grlex.elem_sum b
h_le : True
h₁ : ¬Order.grlex.elem_sum a = Order.grlex.elem_sum b
h₂ : Order.grlex.elem_sum b ≤ Order.grlex.elem_sum a
h₄ : Order.grlex.elem_sum a ≤ Order.grlex.elem_sum b
⊢ False

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

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

case 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✝ : ¬Order.lex (Variables.mul a c) (Variables.mul b c)
⊢ let sum₁ := Order.grlex.elem_sum (Variables.mul a c);
  let sum₂ := Order.grlex.elem_sum (Variables.mul b c);
  if sum₁ < sum₂ then True else if sum₁ = sum₂ then False else False
Tactic:
have h₅ := Eq.symm (Nat.le_antisymm h₂ h₄)
Proof state after:
case inl.inl.inr.inr.inl
n : ℕ
a b c : Variables n order.GrLex
h✝¹ : Order.lex (Variables.mul a c) (Variables.mul b c)
h✝ : Order.lex a b
sum₁ : ℕ := Order.grlex.elem_sum (Variables.mul a c)
sum₂ : ℕ := Order.grlex.elem_sum (Variables.mul b c)
h₃ : Order.grlex.elem_sum a < Order.grlex.elem_sum b
h_le : True
h₁ : ¬Order.grlex.elem_sum a = Order.grlex.elem_sum b
h₂ : Order.grlex.elem_sum b ≤ Order.grlex.elem_sum a
h₄ : Order.grlex.elem_sum a ≤ Order.grlex.elem_sum b
h₅ : Order.grlex.elem_sum a = Order.grlex.elem_sum b
⊢ False

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

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

case 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✝ : ¬Order.lex (Variables.mul a c) (Variables.mul b c)
⊢ let sum₁ := Order.grlex.elem_sum (Variables.mul a c);
  let sum₂ := Order.grlex.elem_sum (Variables.mul b c);
  if sum₁ < sum₂ then True else if sum₁ = sum₂ then False else False