Theorem: algebra.lex_add_le_add
Proof state before:
case h_2.inl
n : ℕ
v₁ v₂ v₃ : Variables n order.Lex
m : ℕ
a b c : Variables m order.Lex
x : ℕ
xs : List ℕ
p₁ : List.length (x :: xs) = m
y : ℕ
ys : List ℕ
p₂ : List.length (y :: ys) = m
z : ℕ
zs : List ℕ
p₃ : List.length (z :: zs) = m
h : Order.lex_impl { val := x :: xs, property := p₁ } { val := y :: ys, property := p₂ }
v₁✝ v₂✝ : Vector ℕ m
x✝ : ℕ
tail✝¹ : List ℕ
property✝¹ : List.length (x✝ :: tail✝¹) = m
y✝ : ℕ
tail✝ : List ℕ
property✝ : List.length (y✝ :: tail✝) = m
heq✝¹ :
  Variables.mul { val := x :: xs, property := p₁ } { val := z :: zs, property := p₃ } =
    { val := x✝ :: tail✝¹, property := property✝¹ }
heq✝ :
  Variables.mul { val := y :: ys, property := p₂ } { val := z :: zs, property := p₃ } =
    { val := y✝ :: tail✝, property := property✝ }
h✝ : x✝ = y✝
⊢ Order.lex_impl (tail { val := x :: xs, property := p₁ }) (tail { val := y :: ys, property := p₂ })

case h_2.inr
n : ℕ
v₁ v₂ v₃ : Variables n order.Lex
m : ℕ
a b c : Variables m order.Lex
x : ℕ
xs : List ℕ
p₁ : List.length (x :: xs) = m
y : ℕ
ys : List ℕ
p₂ : List.length (y :: ys) = m
z : ℕ
zs : List ℕ
p₃ : List.length (z :: zs) = m
h : Order.lex_impl { val := x :: xs, property := p₁ } { val := y :: ys, property := p₂ }
v₁✝ v₂✝ : Vector ℕ m
x✝ : ℕ
tail✝¹ : List ℕ
property✝¹ : List.length (x✝ :: tail✝¹) = m
y✝ : ℕ
tail✝ : List ℕ
property✝ : List.length (y✝ :: tail✝) = m
heq✝¹ :
  Variables.mul { val := x :: xs, property := p₁ } { val := z :: zs, property := p₃ } =
    { val := x✝ :: tail✝¹, property := property✝¹ }
heq✝ :
  Variables.mul { val := y :: ys, property := p₂ } { val := z :: zs, property := p₃ } =
    { val := y✝ :: tail✝, property := property✝ }
h✝ : ¬x✝ = y✝
⊢ x✝ ≤ y✝
Tactic:
rw [Order.lex_impl] at h
Proof state after:
case h_2.inl
n : ℕ
v₁ v₂ v₃ : Variables n order.Lex
m : ℕ
a b c : Variables m order.Lex
x : ℕ
xs : List ℕ
p₁ : List.length (x :: xs) = m
y : ℕ
ys : List ℕ
p₂ : List.length (y :: ys) = m
z : ℕ
zs : List ℕ
p₃ : List.length (z :: zs) = m
h :
  match { val := x :: xs, property := p₁ }, { val := y :: ys, property := p₂ } with
  | { val := [], property := property }, { val := [], property := property_1 } => True
  | { val := x_1 :: tail, property := property }, { val := y_1 :: tail_1, property := property_1 } =>
    if x_1 = y_1 then
      Order.lex_impl (Vector.tail { val := x :: xs, property := p₁ }) (Vector.tail { val := y :: ys, property := p₂ })
    else x_1 ≤ y_1
v₁✝ v₂✝ : Vector ℕ m
x✝ : ℕ
tail✝¹ : List ℕ
property✝¹ : List.length (x✝ :: tail✝¹) = m
y✝ : ℕ
tail✝ : List ℕ
property✝ : List.length (y✝ :: tail✝) = m
heq✝¹ :
  Variables.mul { val := x :: xs, property := p₁ } { val := z :: zs, property := p₃ } =
    { val := x✝ :: tail✝¹, property := property✝¹ }
heq✝ :
  Variables.mul { val := y :: ys, property := p₂ } { val := z :: zs, property := p₃ } =
    { val := y✝ :: tail✝, property := property✝ }
h✝ : x✝ = y✝
⊢ Order.lex_impl (tail { val := x :: xs, property := p₁ }) (tail { val := y :: ys, property := p₂ })

case h_2.inr
n : ℕ
v₁ v₂ v₃ : Variables n order.Lex
m : ℕ
a b c : Variables m order.Lex
x : ℕ
xs : List ℕ
p₁ : List.length (x :: xs) = m
y : ℕ
ys : List ℕ
p₂ : List.length (y :: ys) = m
z : ℕ
zs : List ℕ
p₃ : List.length (z :: zs) = m
h : Order.lex_impl { val := x :: xs, property := p₁ } { val := y :: ys, property := p₂ }
v₁✝ v₂✝ : Vector ℕ m
x✝ : ℕ
tail✝¹ : List ℕ
property✝¹ : List.length (x✝ :: tail✝¹) = m
y✝ : ℕ
tail✝ : List ℕ
property✝ : List.length (y✝ :: tail✝) = m
heq✝¹ :
  Variables.mul { val := x :: xs, property := p₁ } { val := z :: zs, property := p₃ } =
    { val := x✝ :: tail✝¹, property := property✝¹ }
heq✝ :
  Variables.mul { val := y :: ys, property := p₂ } { val := z :: zs, property := p₃ } =
    { val := y✝ :: tail✝, property := property✝ }
h✝ : ¬x✝ = y✝
⊢ x✝ ≤ y✝