internlm/Lean-Github huggingface dataset:
Theorem: algebra.lex_le_trans
Proof state before:
case h_2.inr.inr.h_2.inr
n : ℕ
v₁ v₂ v₃ : Variables n order.Lex
h₁ : Order.lex v₁ v₂
h₂ : Order.lex v₂ v₃
m : ℕ
a b c : Variables m order.Lex
x : ℕ
xs : List ℕ
p : List.length (x :: xs) = m
y : ℕ
ys : List ℕ
q : List.length (y :: ys) = m
z : ℕ
zs : List ℕ
l : List.length (z :: zs) = m
v₁✝¹ v₂✝¹ : Vector ℕ m
x✝¹ : ℕ
tail✝³ : List ℕ
property✝³ : List.length (x✝¹ :: tail✝³) = m
y✝¹ : ℕ
tail✝² : List ℕ
property✝² : List.length (y✝¹ :: tail✝²) = m
heq✝³ : { val := x :: xs, property := p } = { val := x✝¹ :: tail✝³, property := property✝³ }
heq✝² : { val := y :: ys, property := q } = { val := y✝¹ :: tail✝², property := property✝² }
h✝² : ¬x = z
h✝¹ : ¬x✝¹ = y✝¹
ab : 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 : { val := y :: ys, property := q } = { val := x✝ :: tail✝¹, property := property✝¹ }
heq✝ : { val := z :: zs, property := l } = { val := y✝ :: tail✝, property := property✝ }
h✝ : ¬x✝ = y✝
bc : x✝ ≤ y✝
⊢ x ≤ z
Tactic:
simp at *
Proof state after:
case h_2.inr.inr.h_2.inr
n : ℕ
v₁ v₂ v₃ : Variables n order.Lex
h₁ : Order.lex v₁ v₂
h₂ : Order.lex v₂ v₃
m : ℕ
a b c : Variables m order.Lex
x : ℕ
xs : List ℕ
p : List.length (x :: xs) = m
y : ℕ
ys : List ℕ
q : List.length (y :: ys) = m
z : ℕ
zs : List ℕ
l : List.length (z :: zs) = m
v₁✝¹ v₂✝¹ : Vector ℕ m
x✝¹ : ℕ
tail✝³ : List ℕ
property✝³ : List.length (x✝¹ :: tail✝³) = m
y✝¹ : ℕ
tail✝² : List ℕ
property✝² : List.length (y✝¹ :: tail✝²) = m
h✝² : ¬x = z
h✝¹ : ¬x✝¹ = y✝¹
ab : 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
h✝ : ¬x✝ = y✝
bc : x✝ ≤ y✝
heq✝³ : x = x✝¹ ∧ xs = tail✝³
heq✝² : y = y✝¹ ∧ ys = tail✝²
heq✝¹ : y = x✝ ∧ ys = tail✝¹
heq✝ : z = y✝ ∧ zs = tail✝
⊢ x ≤ z
Theorem: algebra.lex_le_trans
Proof state before:
case h_2.inr.inr.h_2.inr
n : ℕ
v₁ v₂ v₃ : Variables n order.Lex
h₁ : Order.lex v₁ v₂
h₂ : Order.lex v₂ v₃
m : ℕ
a b c : Variables m order.Lex
x : ℕ
xs : List ℕ
p : List.length (x :: xs) = m
y : ℕ
ys : List ℕ
q : List.length (y :: ys) = m
z : ℕ
zs : List ℕ
l : List.length (z :: zs) = m
v₁✝¹ v₂✝¹ : Vector ℕ m
x✝¹ : ℕ
tail✝³ : List ℕ
property✝³ : List.length (x✝¹ :: tail✝³) = m
y✝¹ : ℕ
tail✝² : List ℕ
property✝² : List.length (y✝¹ :: tail✝²) = m
h✝² : ¬x = z
h✝¹ : ¬x✝¹ = y✝¹
ab : 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
h✝ : ¬x✝ = y✝
bc : x✝ ≤ y✝
heq✝³ : x = x✝¹ ∧ xs = tail✝³
heq✝² : y = y✝¹ ∧ ys = tail✝²
heq✝¹ : y = x✝ ∧ ys = tail✝¹
heq✝ : z = y✝ ∧ zs = tail✝
⊢ x ≤ z
Tactic:
rename_i neq₁ neq₂ _ _ _ _ _ _ _ _ _ heq₁ heq₂ heq₃ heq₄
Proof state after:
case h_2.inr.inr.h_2.inr
n : ℕ
v₁ v₂ v₃ : Variables n order.Lex
h₁ : Order.lex v₁ v₂
h₂ : Order.lex v₂ v₃
m : ℕ
a b c : Variables m order.Lex
x : ℕ
xs : List ℕ
p : List.length (x :: xs) = m
y : ℕ
ys : List ℕ
q : List.length (y :: ys) = m
z : ℕ
zs : List ℕ
l : List.length (z :: zs) = m
v₁✝¹ v₂✝¹ : Vector ℕ m
x✝¹ : ℕ
tail✝³ : List ℕ
property✝³ : List.length (x✝¹ :: tail✝³) = m
y✝¹ : ℕ
tail✝² : List ℕ
property✝² : List.length (y✝¹ :: tail✝²) = m
neq₁ : ¬x = z
neq₂ : ¬x✝¹ = y✝¹
ab : 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
h✝ : ¬x✝ = y✝
bc : x✝ ≤ y✝
heq₁ : x = x✝¹ ∧ xs = tail✝³
heq₂ : y = y✝¹ ∧ ys = tail✝²
heq₃ : y = x✝ ∧ ys = tail✝¹
heq₄ : z = y✝ ∧ zs = tail✝
⊢ x ≤ z
Theorem: algebra.lex_le_trans
Proof state before:
case h_2.inr.inr.h_2.inr
n : ℕ
v₁ v₂ v₃ : Variables n order.Lex
h₁ : Order.lex v₁ v₂
h₂ : Order.lex v₂ v₃
m : ℕ
a b c : Variables m order.Lex
x : ℕ
xs : List ℕ
p : List.length (x :: xs) = m
y : ℕ
ys : List ℕ
q : List.length (y :: ys) = m
z : ℕ
zs : List ℕ
l : List.length (z :: zs) = m
v₁✝¹ v₂✝¹ : Vector ℕ m
x✝¹ : ℕ
tail✝³ : List ℕ
property✝³ : List.length (x✝¹ :: tail✝³) = m
y✝¹ : ℕ
tail✝² : List ℕ
property✝² : List.length (y✝¹ :: tail✝²) = m
neq₁ : ¬x = z
neq₂ : ¬x✝¹ = y✝¹
ab : 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
h✝ : ¬x✝ = y✝
bc : x✝ ≤ y✝
heq₁ : x = x✝¹ ∧ xs = tail✝³
heq₂ : y = y✝¹ ∧ ys = tail✝²
heq₃ : y = x✝ ∧ ys = tail✝¹
heq₄ : z = y✝ ∧ zs = tail✝
⊢ x ≤ z
Tactic:
have x_cross_one_eq_x := Eq.symm (heq₁.left)
Proof state after:
case h_2.inr.inr.h_2.inr
n : ℕ
v₁ v₂ v₃ : Variables n order.Lex
h₁ : Order.lex v₁ v₂
h₂ : Order.lex v₂ v₃
m : ℕ
a b c : Variables m order.Lex
x : ℕ
xs : List ℕ
p : List.length (x :: xs) = m
y : ℕ
ys : List ℕ
q : List.length (y :: ys) = m
z : ℕ
zs : List ℕ
l : List.length (z :: zs) = m
v₁✝¹ v₂✝¹ : Vector ℕ m
x✝¹ : ℕ
tail✝³ : List ℕ
property✝³ : List.length (x✝¹ :: tail✝³) = m
y✝¹ : ℕ
tail✝² : List ℕ
property✝² : List.length (y✝¹ :: tail✝²) = m
neq₁ : ¬x = z
neq₂ : ¬x✝¹ = y✝¹
ab : 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
h✝ : ¬x✝ = y✝
bc : x✝ ≤ y✝
heq₁ : x = x✝¹ ∧ xs = tail✝³
heq₂ : y = y✝¹ ∧ ys = tail✝²
heq₃ : y = x✝ ∧ ys = tail✝¹
heq₄ : z = y✝ ∧ zs = tail✝
x_cross_one_eq_x : x✝¹ = x
⊢ x ≤ z
Theorem: algebra.lex_le_trans
Proof state before:
case h_2.inr.inr.h_2.inr
n : ℕ
v₁ v₂ v₃ : Variables n order.Lex
h₁ : Order.lex v₁ v₂
h₂ : Order.lex v₂ v₃
m : ℕ
a b c : Variables m order.Lex
x : ℕ
xs : List ℕ
p : List.length (x :: xs) = m
y : ℕ
ys : List ℕ
q : List.length (y :: ys) = m
z : ℕ
zs : List ℕ
l : List.length (z :: zs) = m
v₁✝¹ v₂✝¹ : Vector ℕ m
x✝¹ : ℕ
tail✝³ : List ℕ
property✝³ : List.length (x✝¹ :: tail✝³) = m
y✝¹ : ℕ
tail✝² : List ℕ
property✝² : List.length (y✝¹ :: tail✝²) = m
neq₁ : ¬x = z
neq₂ : ¬x✝¹ = y✝¹
ab : 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
h✝ : ¬x✝ = y✝
bc : x✝ ≤ y✝
heq₁ : x = x✝¹ ∧ xs = tail✝³
heq₂ : y = y✝¹ ∧ ys = tail✝²
heq₃ : y = x✝ ∧ ys = tail✝¹
heq₄ : z = y✝ ∧ zs = tail✝
x_cross_one_eq_x : x✝¹ = x
⊢ x ≤ z
Tactic:
have y_cross_one_eq_y := Eq.symm (heq₂.left)
Proof state after:
case h_2.inr.inr.h_2.inr
n : ℕ
v₁ v₂ v₃ : Variables n order.Lex
h₁ : Order.lex v₁ v₂
h₂ : Order.lex v₂ v₃
m : ℕ
a b c : Variables m order.Lex
x : ℕ
xs : List ℕ
p : List.length (x :: xs) = m
y : ℕ
ys : List ℕ
q : List.length (y :: ys) = m
z : ℕ
zs : List ℕ
l : List.length (z :: zs) = m
v₁✝¹ v₂✝¹ : Vector ℕ m
x✝¹ : ℕ
tail✝³ : List ℕ
property✝³ : List.length (x✝¹ :: tail✝³) = m
y✝¹ : ℕ
tail✝² : List ℕ
property✝² : List.length (y✝¹ :: tail✝²) = m
neq₁ : ¬x = z
neq₂ : ¬x✝¹ = y✝¹
ab : 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
h✝ : ¬x✝ = y✝
bc : x✝ ≤ y✝
heq₁ : x = x✝¹ ∧ xs = tail✝³
heq₂ : y = y✝¹ ∧ ys = tail✝²
heq₃ : y = x✝ ∧ ys = tail✝¹
heq₄ : z = y✝ ∧ zs = tail✝
x_cross_one_eq_x : x✝¹ = x
y_cross_one_eq_y : y✝¹ = y
⊢ x ≤ z
Theorem: algebra.lex_le_trans
Proof state before:
case h_2.inr.inr.h_2.inr
n : ℕ
v₁ v₂ v₃ : Variables n order.Lex
h₁ : Order.lex v₁ v₂
h₂ : Order.lex v₂ v₃
m : ℕ
a b c : Variables m order.Lex
x : ℕ
xs : List ℕ
p : List.length (x :: xs) = m
y : ℕ
ys : List ℕ
q : List.length (y :: ys) = m
z : ℕ
zs : List ℕ
l : List.length (z :: zs) = m
v₁✝¹ v₂✝¹ : Vector ℕ m
x✝¹ : ℕ
tail✝³ : List ℕ
property✝³ : List.length (x✝¹ :: tail✝³) = m
y✝¹ : ℕ
tail✝² : List ℕ
property✝² : List.length (y✝¹ :: tail✝²) = m
neq₁ : ¬x = z
neq₂ : ¬x✝¹ = y✝¹
ab : 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
h✝ : ¬x✝ = y✝
bc : x✝ ≤ y✝
heq₁ : x = x✝¹ ∧ xs = tail✝³
heq₂ : y = y✝¹ ∧ ys = tail✝²
heq₃ : y = x✝ ∧ ys = tail✝¹
heq₄ : z = y✝ ∧ zs = tail✝
x_cross_one_eq_x : x✝¹ = x
y_cross_one_eq_y : y✝¹ = y
⊢ x ≤ z
Tactic:
have x_cross_eq_x     := Eq.symm (heq₃.left)
Proof state after:
case h_2.inr.inr.h_2.inr
n : ℕ
v₁ v₂ v₃ : Variables n order.Lex
h₁ : Order.lex v₁ v₂
h₂ : Order.lex v₂ v₃
m : ℕ
a b c : Variables m order.Lex
x : ℕ
xs : List ℕ
p : List.length (x :: xs) = m
y : ℕ
ys : List ℕ
q : List.length (y :: ys) = m
z : ℕ
zs : List ℕ
l : List.length (z :: zs) = m
v₁✝¹ v₂✝¹ : Vector ℕ m
x✝¹ : ℕ
tail✝³ : List ℕ
property✝³ : List.length (x✝¹ :: tail✝³) = m
y✝¹ : ℕ
tail✝² : List ℕ
property✝² : List.length (y✝¹ :: tail✝²) = m
neq₁ : ¬x = z
neq₂ : ¬x✝¹ = y✝¹
ab : 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
h✝ : ¬x✝ = y✝
bc : x✝ ≤ y✝
heq₁ : x = x✝¹ ∧ xs = tail✝³
heq₂ : y = y✝¹ ∧ ys = tail✝²
heq₃ : y = x✝ ∧ ys = tail✝¹
heq₄ : z = y✝ ∧ zs = tail✝
x_cross_one_eq_x : x✝¹ = x
y_cross_one_eq_y : y✝¹ = y
x_cross_eq_x : x✝ = y
⊢ x ≤ z
Theorem: algebra.lex_le_trans
Proof state before:
case h_2.inr.inr.h_2.inr
n : ℕ
v₁ v₂ v₃ : Variables n order.Lex
h₁ : Order.lex v₁ v₂
h₂ : Order.lex v₂ v₃
m : ℕ
a b c : Variables m order.Lex
x : ℕ
xs : List ℕ
p : List.length (x :: xs) = m
y : ℕ
ys : List ℕ
q : List.length (y :: ys) = m
z : ℕ
zs : List ℕ
l : List.length (z :: zs) = m
v₁✝¹ v₂✝¹ : Vector ℕ m
x✝¹ : ℕ
tail✝³ : List ℕ
property✝³ : List.length (x✝¹ :: tail✝³) = m
y✝¹ : ℕ
tail✝² : List ℕ
property✝² : List.length (y✝¹ :: tail✝²) = m
neq₁ : ¬x = z
neq₂ : ¬x✝¹ = y✝¹
ab : 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
h✝ : ¬x✝ = y✝
bc : x✝ ≤ y✝
heq₁ : x = x✝¹ ∧ xs = tail✝³
heq₂ : y = y✝¹ ∧ ys = tail✝²
heq₃ : y = x✝ ∧ ys = tail✝¹
heq₄ : z = y✝ ∧ zs = tail✝
x_cross_one_eq_x : x✝¹ = x
y_cross_one_eq_y : y✝¹ = y
x_cross_eq_x : x✝ = y
⊢ x ≤ z
Tactic:
have y_cross_eq_x     := Eq.symm (heq₄.left)
Proof state after:
case h_2.inr.inr.h_2.inr
n : ℕ
v₁ v₂ v₃ : Variables n order.Lex
h₁ : Order.lex v₁ v₂
h₂ : Order.lex v₂ v₃
m : ℕ
a b c : Variables m order.Lex
x : ℕ
xs : List ℕ
p : List.length (x :: xs) = m
y : ℕ
ys : List ℕ
q : List.length (y :: ys) = m
z : ℕ
zs : List ℕ
l : List.length (z :: zs) = m
v₁✝¹ v₂✝¹ : Vector ℕ m
x✝¹ : ℕ
tail✝³ : List ℕ
property✝³ : List.length (x✝¹ :: tail✝³) = m
y✝¹ : ℕ
tail✝² : List ℕ
property✝² : List.length (y✝¹ :: tail✝²) = m
neq₁ : ¬x = z
neq₂ : ¬x✝¹ = y✝¹
ab : 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
h✝ : ¬x✝ = y✝
bc : x✝ ≤ y✝
heq₁ : x = x✝¹ ∧ xs = tail✝³
heq₂ : y = y✝¹ ∧ ys = tail✝²
heq₃ : y = x✝ ∧ ys = tail✝¹
heq₄ : z = y✝ ∧ zs = tail✝
x_cross_one_eq_x : x✝¹ = x
y_cross_one_eq_y : y✝¹ = y
x_cross_eq_x : x✝ = y
y_cross_eq_x : y✝ = z
⊢ x ≤ z
Theorem: algebra.lex_le_antisymm
Proof state before:
n : ℕ
⊢ ∀ (a b : Variables n order.Lex), Order.lex a b → Order.lex b a → a = b
Tactic:
intros v₁ v₂ h₁ h₂
Proof state after:
n : ℕ
v₁ v₂ : Variables n order.Lex
h₁ : Order.lex v₁ v₂
h₂ : Order.lex v₂ v₁
⊢ v₁ = v₂
Theorem: algebra.lex_le_antisymm
Proof state before:
n : ℕ
v₁ v₂ : Variables n order.Lex
h₁ : Order.lex v₁ v₂
h₂ : Order.lex v₂ v₁
⊢ v₁ = v₂
Tactic:
exact aux n v₁ v₂ h₁ h₂
Proof state after:
no goals
Theorem: algebra.lex_le_antisymm
Proof state before:
n : ℕ
v₁ v₂ : Variables n order.Lex
h₁ : Order.lex v₁ v₂
h₂ : Order.lex v₂ v₁
m : ℕ
a b : Vector ℕ m
ab : Order.lex_impl a b
ba : Order.lex_impl b a
⊢ a = b
Tactic:
match a, b with
  | ⟨[], p⟩, ⟨[], q⟩       => simp 
  | ⟨x::xs, p⟩, ⟨y::ys, q⟩ => rw [Order.lex_impl] at ab ba
                              simp at *
                              split at ab
                              split at ba
                              rename_i heq₁ heq₂
                              simp [Vector]
                              constructor
                              exact heq₁ 
                              have eq := aux (m-1) (tail ⟨x::xs, p⟩) (tail ⟨y::ys, q⟩) ab ba
                              simp [tail, Vector] at eq
                              exact eq
                              rename_i eq neq
                              have eq_symm := Eq.symm eq
                              contradiction
                              split at ba
                              rename_i neq eq
                              have eq_symm := Eq.symm eq
                              contradiction
                              rename_i neq₁ neq₂
                              have eq := Nat.le_antisymm ab ba
                              contradiction
Proof state after:
no goals
Theorem: algebra.lex_le_antisymm
Proof state before:
n : ℕ
v₁ v₂ : Variables n order.Lex
h₁ : Order.lex v₁ v₂
h₂ : Order.lex v₂ v₁
m : ℕ
a b : Vector ℕ m
p q : List.length [] = m
ab : Order.lex_impl { val := [], property := p } { val := [], property := q }
ba : Order.lex_impl { val := [], property := q } { val := [], property := p }
⊢ { val := [], property := p } = { val := [], property := q }
Tactic:
simp
Proof state after:
no goals
Theorem: algebra.lex_le_antisymm
Proof state before:
n : ℕ
v₁ v₂ : Variables n order.Lex
h₁ : Order.lex v₁ v₂
h₂ : Order.lex v₂ v₁
m : ℕ
a b : Vector ℕ m
x : ℕ
xs : List ℕ
p : List.length (x :: xs) = m
y : ℕ
ys : List ℕ
q : List.length (y :: ys) = m
ab : Order.lex_impl { val := x :: xs, property := p } { val := y :: ys, property := q }
ba : Order.lex_impl { val := y :: ys, property := q } { val := x :: xs, property := p }
⊢ { val := x :: xs, property := p } = { val := y :: ys, property := q }
Tactic:
rw [Order.lex_impl] at ab ba
Proof state after:
n : ℕ
v₁ v₂ : Variables n order.Lex
h₁ : Order.lex v₁ v₂
h₂ : Order.lex v₂ v₁
m : ℕ
a b : Vector ℕ m
x : ℕ
xs : List ℕ
p : List.length (x :: xs) = m
y : ℕ
ys : List ℕ
q : List.length (y :: ys) = m
ab :
  match { val := x :: xs, property := p }, { val := y :: ys, property := q } 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 := p }) (Vector.tail { val := y :: ys, property := q })
    else x_1 ≤ y_1
ba :
  match { val := y :: ys, property := q }, { val := x :: xs, property := p } 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 := y :: ys, property := q }) (Vector.tail { val := x :: xs, property := p })
    else x_1 ≤ y_1
⊢ { val := x :: xs, property := p } = { val := y :: ys, property := q }
Theorem: algebra.lex_le_antisymm
Proof state before:
n : ℕ
v₁ v₂ : Variables n order.Lex
h₁ : Order.lex v₁ v₂
h₂ : Order.lex v₂ v₁
m : ℕ
a b : Vector ℕ m
x : ℕ
xs : List ℕ
p : List.length (x :: xs) = m
y : ℕ
ys : List ℕ
q : List.length (y :: ys) = m
ab :
  match { val := x :: xs, property := p }, { val := y :: ys, property := q } 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 := p }) (Vector.tail { val := y :: ys, property := q })
    else x_1 ≤ y_1
ba :
  match { val := y :: ys, property := q }, { val := x :: xs, property := p } 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 := y :: ys, property := q }) (Vector.tail { val := x :: xs, property := p })
    else x_1 ≤ y_1
⊢ { val := x :: xs, property := p } = { val := y :: ys, property := q }
Tactic:
simp at *
Proof state after:
n : ℕ
v₁ v₂ : Variables n order.Lex
h₁ : Order.lex v₁ v₂
h₂ : Order.lex v₂ v₁
m : ℕ
a b : Vector ℕ m
x : ℕ
xs : List ℕ
p : List.length (x :: xs) = m
y : ℕ
ys : List ℕ
q : List.length (y :: ys) = m
ab :
  if x = y then Order.lex_impl (tail { val := x :: xs, property := p }) (tail { val := y :: ys, property := q })
  else x ≤ y
ba :
  if y = x then Order.lex_impl (tail { val := y :: ys, property := q }) (tail { val := x :: xs, property := p })
  else y ≤ x
⊢ { val := x :: xs, property := p } = { val := y :: ys, property := q }
Theorem: algebra.lex_le_antisymm
Proof state before:
n : ℕ
v₁ v₂ : Variables n order.Lex
h₁ : Order.lex v₁ v₂
h₂ : Order.lex v₂ v₁
m : ℕ
a b : Vector ℕ m
x : ℕ
xs : List ℕ
p : List.length (x :: xs) = m
y : ℕ
ys : List ℕ
q : List.length (y :: ys) = m
ab :
  if x = y then Order.lex_impl (tail { val := x :: xs, property := p }) (tail { val := y :: ys, property := q })
  else x ≤ y
ba :
  if y = x then Order.lex_impl (tail { val := y :: ys, property := q }) (tail { val := x :: xs, property := p })
  else y ≤ x
⊢ { val := x :: xs, property := p } = { val := y :: ys, property := q }
Tactic:
split at ab
Proof state after:
case inl
n : ℕ
v₁ v₂ : Variables n order.Lex
h₁ : Order.lex v₁ v₂
h₂ : Order.lex v₂ v₁
m : ℕ
a b : Vector ℕ m
x : ℕ
xs : List ℕ
p : List.length (x :: xs) = m
y : ℕ
ys : List ℕ
q : List.length (y :: ys) = m
ba :
  if y = x then Order.lex_impl (tail { val := y :: ys, property := q }) (tail { val := x :: xs, property := p })
  else y ≤ x
h✝ : x = y
ab : Order.lex_impl (tail { val := x :: xs, property := p }) (tail { val := y :: ys, property := q })
⊢ { val := x :: xs, property := p } = { val := y :: ys, property := q }

case inr
n : ℕ
v₁ v₂ : Variables n order.Lex
h₁ : Order.lex v₁ v₂
h₂ : Order.lex v₂ v₁
m : ℕ
a b : Vector ℕ m
x : ℕ
xs : List ℕ
p : List.length (x :: xs) = m
y : ℕ
ys : List ℕ
q : List.length (y :: ys) = m
ba :
  if y = x then Order.lex_impl (tail { val := y :: ys, property := q }) (tail { val := x :: xs, property := p })
  else y ≤ x
h✝ : ¬x = y
ab : x ≤ y
⊢ { val := x :: xs, property := p } = { val := y :: ys, property := q }
Theorem: algebra.lex_le_antisymm
Proof state before:
case inl
n : ℕ
v₁ v₂ : Variables n order.Lex
h₁ : Order.lex v₁ v₂
h₂ : Order.lex v₂ v₁
m : ℕ
a b : Vector ℕ m
x : ℕ
xs : List ℕ
p : List.length (x :: xs) = m
y : ℕ
ys : List ℕ
q : List.length (y :: ys) = m
ba :
  if y = x then Order.lex_impl (tail { val := y :: ys, property := q }) (tail { val := x :: xs, property := p })
  else y ≤ x
h✝ : x = y
ab : Order.lex_impl (tail { val := x :: xs, property := p }) (tail { val := y :: ys, property := q })
⊢ { val := x :: xs, property := p } = { val := y :: ys, property := q }

case inr
n : ℕ
v₁ v₂ : Variables n order.Lex
h₁ : Order.lex v₁ v₂
h₂ : Order.lex v₂ v₁
m : ℕ
a b : Vector ℕ m
x : ℕ
xs : List ℕ
p : List.length (x :: xs) = m
y : ℕ
ys : List ℕ
q : List.length (y :: ys) = m
ba :
  if y = x then Order.lex_impl (tail { val := y :: ys, property := q }) (tail { val := x :: xs, property := p })
  else y ≤ x
h✝ : ¬x = y
ab : x ≤ y
⊢ { val := x :: xs, property := p } = { val := y :: ys, property := q }
Tactic:
split at ba
Proof state after:
case inl.inl
n : ℕ
v₁ v₂ : Variables n order.Lex
h₁ : Order.lex v₁ v₂
h₂ : Order.lex v₂ v₁
m : ℕ
a b : Vector ℕ m
x : ℕ
xs : List ℕ
p : List.length (x :: xs) = m
y : ℕ
ys : List ℕ
q : List.length (y :: ys) = m
h✝¹ : x = y
ab : Order.lex_impl (tail { val := x :: xs, property := p }) (tail { val := y :: ys, property := q })
h✝ : y = x
ba : Order.lex_impl (tail { val := y :: ys, property := q }) (tail { val := x :: xs, property := p })
⊢ { val := x :: xs, property := p } = { val := y :: ys, property := q }

case inl.inr
n : ℕ
v₁ v₂ : Variables n order.Lex
h₁ : Order.lex v₁ v₂
h₂ : Order.lex v₂ v₁
m : ℕ
a b : Vector ℕ m
x : ℕ
xs : List ℕ
p : List.length (x :: xs) = m
y : ℕ
ys : List ℕ
q : List.length (y :: ys) = m
h✝¹ : x = y
ab : Order.lex_impl (tail { val := x :: xs, property := p }) (tail { val := y :: ys, property := q })
h✝ : ¬y = x
ba : y ≤ x
⊢ { val := x :: xs, property := p } = { val := y :: ys, property := q }

case inr
n : ℕ
v₁ v₂ : Variables n order.Lex
h₁ : Order.lex v₁ v₂
h₂ : Order.lex v₂ v₁
m : ℕ
a b : Vector ℕ m
x : ℕ
xs : List ℕ
p : List.length (x :: xs) = m
y : ℕ
ys : List ℕ
q : List.length (y :: ys) = m
ba :
  if y = x then Order.lex_impl (tail { val := y :: ys, property := q }) (tail { val := x :: xs, property := p })
  else y ≤ x
h✝ : ¬x = y
ab : x ≤ y
⊢ { val := x :: xs, property := p } = { val := y :: ys, property := q }
Theorem: algebra.lex_le_antisymm
Proof state before:
case inl.inl
n : ℕ
v₁ v₂ : Variables n order.Lex
h₁ : Order.lex v₁ v₂
h₂ : Order.lex v₂ v₁
m : ℕ
a b : Vector ℕ m
x : ℕ
xs : List ℕ
p : List.length (x :: xs) = m
y : ℕ
ys : List ℕ
q : List.length (y :: ys) = m
h✝¹ : x = y
ab : Order.lex_impl (tail { val := x :: xs, property := p }) (tail { val := y :: ys, property := q })
h✝ : y = x
ba : Order.lex_impl (tail { val := y :: ys, property := q }) (tail { val := x :: xs, property := p })
⊢ { val := x :: xs, property := p } = { val := y :: ys, property := q }

case inl.inr
n : ℕ
v₁ v₂ : Variables n order.Lex
h₁ : Order.lex v₁ v₂
h₂ : Order.lex v₂ v₁
m : ℕ
a b : Vector ℕ m
x : ℕ
xs : List ℕ
p : List.length (x :: xs) = m
y : ℕ
ys : List ℕ
q : List.length (y :: ys) = m
h✝¹ : x = y
ab : Order.lex_impl (tail { val := x :: xs, property := p }) (tail { val := y :: ys, property := q })
h✝ : ¬y = x
ba : y ≤ x
⊢ { val := x :: xs, property := p } = { val := y :: ys, property := q }

case inr
n : ℕ
v₁ v₂ : Variables n order.Lex
h₁ : Order.lex v₁ v₂
h₂ : Order.lex v₂ v₁
m : ℕ
a b : Vector ℕ m
x : ℕ
xs : List ℕ
p : List.length (x :: xs) = m
y : ℕ
ys : List ℕ
q : List.length (y :: ys) = m
ba :
  if y = x then Order.lex_impl (tail { val := y :: ys, property := q }) (tail { val := x :: xs, property := p })
  else y ≤ x
h✝ : ¬x = y
ab : x ≤ y
⊢ { val := x :: xs, property := p } = { val := y :: ys, property := q }
Tactic:
rename_i heq₁ heq₂
Proof state after:
case inl.inl
n : ℕ
v₁ v₂ : Variables n order.Lex
h₁ : Order.lex v₁ v₂
h₂ : Order.lex v₂ v₁
m : ℕ
a b : Vector ℕ m
x : ℕ
xs : List ℕ
p : List.length (x :: xs) = m
y : ℕ
ys : List ℕ
q : List.length (y :: ys) = m
heq₁ : x = y
ab : Order.lex_impl (tail { val := x :: xs, property := p }) (tail { val := y :: ys, property := q })
heq₂ : y = x
ba : Order.lex_impl (tail { val := y :: ys, property := q }) (tail { val := x :: xs, property := p })
⊢ { val := x :: xs, property := p } = { val := y :: ys, property := q }

case inl.inr
n : ℕ
v₁ v₂ : Variables n order.Lex
h₁ : Order.lex v₁ v₂
h₂ : Order.lex v₂ v₁
m : ℕ
a b : Vector ℕ m
x : ℕ
xs : List ℕ
p : List.length (x :: xs) = m
y : ℕ
ys : List ℕ
q : List.length (y :: ys) = m
h✝¹ : x = y
ab : Order.lex_impl (tail { val := x :: xs, property := p }) (tail { val := y :: ys, property := q })
h✝ : ¬y = x
ba : y ≤ x
⊢ { val := x :: xs, property := p } = { val := y :: ys, property := q }

case inr
n : ℕ
v₁ v₂ : Variables n order.Lex
h₁ : Order.lex v₁ v₂
h₂ : Order.lex v₂ v₁
m : ℕ
a b : Vector ℕ m
x : ℕ
xs : List ℕ
p : List.length (x :: xs) = m
y : ℕ
ys : List ℕ
q : List.length (y :: ys) = m
ba :
  if y = x then Order.lex_impl (tail { val := y :: ys, property := q }) (tail { val := x :: xs, property := p })
  else y ≤ x
h✝ : ¬x = y
ab : x ≤ y
⊢ { val := x :: xs, property := p } = { val := y :: ys, property := q }