Proof step 1 from proof lastN-blastN (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  butlastn(n, x) + lastn(n, x) ≠ x, n ≤ # x ⊦

Rule argument

Additional information

Proof step 2 from proof lastN-blastN (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 3:

  butlastn(n, x) + lastn(n, x) ≠ x, n ≤ # x ⊦

Rule argument

Additional information

Proof step 3 from proof lastN-blastN (previous step) (back to tree)

User applied rule induction

And obtained the following new goals

• Proof step 4:

  butlastn(n, x) + lastn(n, x) ≠ x, n ≤ # x, ∀ x0, n. # x0 < # x → butlastn(n, x0) + lastn(n, x0) ≠ x0 ∧ n ≤ # x0 → false ⊦

Rule argument

Additional information

Used simplifier rules: <; (wellfounded) Used terms: # x

Proof step 4 from proof lastN-blastN (previous step) (back to tree)

User applied rule apply lemma

And obtained the following new goals

• Proof step 5:

  ⊦ x ≠ [] → (∃ y, a. x = y + a ')

• Proof step 6:

  butlastn(n, x) + lastn(n, x) ≠ x, n ≤ # x, x ≠ [] → (∃ y, a. x = y + a '), ∀ x0, n. # x0 < # x → butlastn(n, x0) + lastn(n, x0) ≠ x0 ∧ n ≤ # x0 → false ⊦

Rule argument

Additional information

Used terms: x

Proof step 5 from proof lastN-blastN (previous step) (back to tree)

A spec-lemma

Proof step 6 from proof lastN-blastN (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 7:

  butlastn(n, x) + lastn(n, x) ≠ x, n ≤ # x, x ≠ [] → x = y + a ', ∀ x0, n. # x0 < # x → butlastn(n, x0) + lastn(n, x0) ≠ x0 ∧ n ≤ # x0 → false ⊦

Rule argument

Additional information

Proof step 7 from proof lastN-blastN (previous step) (back to tree)

User applied rule cut formula

And obtained the following new goals

• Proof step 8:

  butlastn(n, x) + lastn(n, x) ≠ x, x ≠ [], n ≤ # x, x ≠ [] → x = y + a ', ∀ x0, n. # x0 < # x → butlastn(n, x0) + lastn(n, x0) ≠ x0 ∧ n ≤ # x0 → false ⊦

Rule argument

Additional information

Used simplifier rules: lastN-zero: ⊦ lastn(0, x) = []; ⊦ # [] = 0; (s) ⊦ m ≤ 0 ↔ m = 0; (s) ⊦ butlastn(0, x) = x; (s) ⊦ [] + x = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list Used formulas: x = []

Proof step 8 from proof lastN-blastN (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 9:

  butlastn(n, y + a ') + lastn(n, y + a ') ≠ y + a ', n ≤ # y + 1, ∀ x0, n. # x0 < #(y + a ') → butlastn(n, x0) + lastn(n, x0) ≠ x0 ∧ n ≤ # x0 → false ⊦

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ # a ' = 0 +1; (s) ⊦ #(x + y) = # x + # y; (s) ⊦ a ∈ a0 ' ↔ a = a0; (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) ⊦ x + a ' ≠ []; (s) in-15: ⊦ x + a ' = y → a ∈ y; (forward) from specification list

Proof step 9 from proof lastN-blastN (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 10:

  0 < n ⊦ lastn((* n), x + a ') = lastn(n -1, x) + a '

• Proof step 11:

  butlastn(n, y + a ') + lastn(n -1, y) + a ' ≠ y + a ', 0 < n, n ≤ # y + 1, ∀ x0, n. # x0 < #(y + a ') → butlastn(n, x0) + lastn(n, x0) ≠ x0 ∧ n ≤ # x0 → false ⊦

Rule argument

Additional information

Used simplifier rules: lastN-zero: ⊦ lastn(0, x) = []; ⊦ 0 < m ↔ m ≠ 0; (s) ⊦ butlastn(0, x) = x; (s) ⊦ x + [] = x; (s) ⊦ x + y = x + z ↔ y = z; (s) a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: a, y, n

Proof step 10 from proof lastN-blastN (previous step) (back to tree)

Lemma num-lastN

Proof step 11 from proof lastN-blastN (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 12:

  butlastn(n, y + a ') + lastn(n - 1, y) ≠ y, n ≠ 0, n ≤ # y + 1, ∀ x0, n. # x0 < #(y + a ') → butlastn(n, x0) + lastn(n, x0) ≠ x0 ∧ n ≤ # x0 → false ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ 0 < m ↔ m ≠ 0; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ x + y = z + y ↔ x = z; (s) ⊦ n -1 = n - 1; (s)

Proof step 12 from proof lastN-blastN (previous step) (back to tree)

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

• Proof step 13:

  n = m + 1, butlastn(n, y + a ') + lastn(m, y) ≠ y, n ≠ 0, n ≤ # y + 1, ∀ x0, n. # x0 < #(y + a ') → butlastn(n, x0) + lastn(n, x0) ≠ x0 ∧ n ≤ # x0 → false ⊦

Rule argument

Additional information

Used simplifier rules: n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1; (elim) Used terms: n, m

Proof step 13 from proof lastN-blastN (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 14:

  butlastn(m, y) + lastn(m, y) ≠ y, m ≤ # y, ∀ x0, n. # x0 < #(y + a ') → butlastn(n, x0) + lastn(n, x0) ≠ x0 ∧ n ≤ # x0 → false ⊦

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m + n ≤ m + n0 ↔ n ≤ n0; (s) ⊦ m + 1 ≠ 0; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ (x + a ').butlast = x; (s) ⊦ butlastn(n + 1, x) = butlastn(n, x.butlast); (s)

Proof step 14 from proof lastN-blastN (previous step) (back to tree)

User applied rule apply induction

And closed the goal!

Rule argument

Additional information

Used simplifier rules: ⊦ #(x + y) = # x + # y; (s) ⊦ # a ' = 0 +1; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ ¬ n < n; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: y, m