Theorem: algebra.lex_add_le_add
Proof state before:
case h_2.inl.h_2.inr.refl.refl
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
v₁✝¹ v₂✝¹ v₁✝ v₂✝ : Vector ℕ m
x✝ : ℕ
tail✝¹ : List ℕ
property✝² : List.length (x✝ :: tail✝¹) = m
y✝ : ℕ
tail✝ : List ℕ
eq : List.length (y✝ :: tail✝) = m
neq : ¬x✝ = y✝
h : x✝ ≤ y✝
heq₁ : x = x✝ ∧ xs = tail✝¹
heq₂ : y = y✝ ∧ ys = tail✝
eq₁ : x = x✝
eq₂ : y = y✝
property✝¹ : List.length ((x + z) :: List.zipWith (fun x y => x + y) xs zs) = m
sum_eq : x + z = y + z
property✝ : List.length ((y + z) :: List.zipWith (fun x y => x + y) ys zs) = m
⊢ 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:
simp [eq₁, eq₂] at sum_eq
Proof state after:
case h_2.inl.h_2.inr.refl.refl
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
v₁✝¹ v₂✝¹ v₁✝ v₂✝ : Vector ℕ m
x✝ : ℕ
tail✝¹ : List ℕ
property✝² : List.length (x✝ :: tail✝¹) = m
y✝ : ℕ
tail✝ : List ℕ
eq : List.length (y✝ :: tail✝) = m
neq : ¬x✝ = y✝
h : x✝ ≤ y✝
heq₁ : x = x✝ ∧ xs = tail✝¹
heq₂ : y = y✝ ∧ ys = tail✝
eq₁ : x = x✝
eq₂ : y = y✝
property✝¹ : List.length ((x + z) :: List.zipWith (fun x y => x + y) xs zs) = m
property✝ : List.length ((y + z) :: List.zipWith (fun x y => x + y) ys zs) = m
sum_eq : 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✝