Proof step 1 from proof nextl-nextj-lemma-butlast (conclusion) (back to tree)

User applied rule cut formula

And obtained the following new goals

• Proof step 2:

  ∀ m. ∃ lf, jj. lf(0) = a ' ∧ jj(0) = n0 ∧ (∀ n. n ≤ m → lf(n + 1) = nextl(jj(n), lf(n).last) ∧ jj(n + 1) = nextj(jj(n), lf(n).last)) ⊦ ∃ lf, jj. lf(0) = a ' ∧ jj(0) = n0 ∧ (∀ n. lf(n + 1) = nextl(jj(n), lf(n).last) ∧ jj(n + 1) = nextj(jj(n), lf(n).last))

• Proof step 42:

  ⊦ ∀ m. ∃ lf, jj. lf(0) = a ' ∧ jj(0) = n0 ∧ (∀ n. n ≤ m → lf(n + 1) = nextl(jj(n), lf(n).last) ∧ jj(n + 1) = nextj(jj(n), lf(n).last)), ∃ lf, jj. lf(0) = a ' ∧ jj(0) = n0 ∧ (∀ n. lf(n + 1) = nextl(jj(n), lf(n).last) ∧ jj(n + 1) = nextj(jj(n), lf(n).last))

Rule argument

Additional information

Used formulas: ∀ m. ∃ lf, jj. lf(0) = a ' ∧ jj(0) = n0 ∧ (∀ n. n ≤ m → lf(n + 1) = nextl(jj(n), lf(n).last) ∧ jj(n + 1) = nextj(jj(n), lf(n).last))

Proof step 2 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 3:

  ∀ m. ∃ lf, jj. lf(0) = a ' ∧ jj(0) = n0 ∧ (∀ n. n ≤ m → lf(n + 1) = nextl(jj(n), lf(n).last) ∧ jj(n + 1) = nextj(jj(n), lf(n).last)) ⊦ ∃ lf, jj. lf(0) = a ' ∧ jj(0) = n0 ∧ (∀ n. lf(n + 1) = nextl(jj(n), lf(n).last) ∧ jj(n + 1) = nextj(jj(n), lf(n).last))

Rule argument

Additional information

Proof step 3 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

User applied rule choice left

And obtained the following new goals

• Proof step 4:

  ∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = n0 ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last)) ⊦ ∃ lf, jj. lf(0) = a ' ∧ jj(0) = n0 ∧ (∀ n. lf(n + 1) = nextl(jj(n), lf(n).last) ∧ jj(n + 1) = nextj(jj(n), lf(n).last))

Rule argument

Additional information

Proof step 4 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

User applied rule cut formula

And obtained the following new goals

• Proof step 5:

  ∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = n0 ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last)), ∀ m. ∀ n. m ≤ n → lfgen(n)(m) = lfgen(m)(m) ∧ iigen(n)(m) = iigen(m)(m) ⊦ ∃ lf, jj. lf(0) = a ' ∧ jj(0) = n0 ∧ (∀ n. lf(n + 1) = nextl(jj(n), lf(n).last) ∧ jj(n + 1) = nextj(jj(n), lf(n).last))

• Proof step 15:

  ∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = n0 ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last)) ⊦ ∀ m. ∀ n. m ≤ n → lfgen(n)(m) = lfgen(m)(m) ∧ iigen(n)(m) = iigen(m)(m), ∃ lf, jj. lf(0) = a ' ∧ jj(0) = n0 ∧ (∀ n. lf(n + 1) = nextl(jj(n), lf(n).last) ∧ jj(n + 1) = nextj(jj(n), lf(n).last))

Rule argument

Additional information

Used formulas: ∀ m. ∀ n. m ≤ n → lfgen(n)(m) = lfgen(m)(m) ∧ iigen(n)(m) = iigen(m)(m)

Proof step 5 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 6:

  ∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = n0 ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last)), ∀ m, n. m ≤ n → lfgen(n)(m) = lfgen(m)(m) ∧ iigen(n)(m) = iigen(m)(m) ⊦ ∃ lf, jj. lf(0) = a ' ∧ jj(0) = n0 ∧ (∀ n. lf(n + 1) = nextl(jj(n), lf(n).last) ∧ jj(n + 1) = nextj(jj(n), lf(n).last))

Rule argument

Additional information

Proof step 6 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

User applied rule exists right

And obtained the following new goals

• Proof step 7:

  ∀ m, n. m ≤ n → lfgen(n)(m) = lfgen(m)(m) ∧ iigen(n)(m) = iigen(m)(m), ∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = n0 ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last)) ⊦ lfgen(0)(0) = a ' ∧ iigen(0)(0) = n0 ∧ (∀ n. lfgen(n + 1)(n + 1) = nextl(iigen(n)(n), lfgen(n)(n).last) ∧ iigen(n + 1)(n + 1) = nextj(iigen(n)(n), lfgen(n)(n).last))

Rule argument

Additional information

Used terms: λ n. lfgen(n)(n), λ n. iigen(n)(n)

Proof step 7 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 8:

  ∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = n0 ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last)), ∀ m, n. m ≤ n → lfgen(n)(m) = lfgen(m)(m) ∧ iigen(n)(m) = iigen(m)(m) ⊦ lfgen(0)(0) = a ' ∧ iigen(0)(0) = n0 ∧ lfgen(n + 1)(n + 1) = nextl(iigen(n)(n), lfgen(n)(n).last) ∧ iigen(n + 1)(n + 1) = nextj(iigen(n)(n), lfgen(n)(n).last)

Rule argument

Additional information

Proof step 8 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 9:

  iigen(0)(0) = n0, lfgen(0)(0) = a ', ∀ m, n. m ≤ n → lfgen(n)(m) = lfgen(m)(m) ∧ iigen(n)(m) = iigen(m)(m), ∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = n0 ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last)), ∀ n. n ≤ 0 → lfgen(0)(n + 1) = nextl(iigen(0)(n), lfgen(0)(n).last) ∧ iigen(0)(n + 1) = nextj(iigen(0)(n), lfgen(0)(n).last) ⊦ lfgen(0)(0) = a ' ∧ iigen(0)(0) = n0 ∧ lfgen(n + 1)(n + 1) = nextl(iigen(n)(n), lfgen(n)(n).last) ∧ iigen(n + 1)(n + 1) = nextj(iigen(n)(n), lfgen(n)(n).last)

Rule argument

Additional information

Used terms: 0

Proof step 9 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 10:

  iigen(0)(1) = nextj(iigen(0)(0), a), lfgen(0)(0) = a ', lfgen(0)(1) = nextl(iigen(0)(0), a), ∀ m, n. m ≤ n → lfgen(n)(m) = lfgen(m)(m) ∧ iigen(n)(m) = iigen(m)(m), ∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = iigen(0)(0) ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last)) ⊦ lfgen(n + 1)(n + 1) = nextl(iigen(n)(n), lfgen(n)(n).last) ∧ iigen(n + 1)(n + 1) = nextj(iigen(n)(n), lfgen(n)(n).last)

Rule argument

Additional information

Used simplifier rules: ⊦ a '.last = a; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m ≤ 0 ↔ m = 0; (s)

Proof step 10 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 11:

  iigen(0)(1) = nextj(iigen(0)(0), a), iigen(n + 1)(0) = iigen(0)(0), lfgen(0)(1) = nextl(iigen(0)(0), a), lfgen(0)(0) = a ', lfgen(n + 1)(0) = a ', ∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = iigen(0)(0) ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last)), ∀ m, n. m ≤ n → lfgen(n)(m) = lfgen(m)(m) ∧ iigen(n)(m) = iigen(m)(m), ∀ n0. n0 ≤ n + 1 → lfgen(n + 1)(n0 + 1) = nextl(iigen(n + 1)(n0), lfgen(n + 1)(n0).last) ∧ iigen(n + 1)(n0 + 1) = nextj(iigen(n + 1)(n0), lfgen(n + 1)(n0).last) ⊦ lfgen(n + 1)(n + 1) = nextl(iigen(n)(n), lfgen(n)(n).last) ∧ iigen(n + 1)(n + 1) = nextj(iigen(n)(n), lfgen(n)(n).last)

Rule argument

Additional information

Used terms: n + 1

Proof step 11 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 12:

  iigen(n + 1)(0) = iigen(0)(0), iigen(0)(1) = nextj(iigen(0)(0), a), lfgen(n + 1)(0) = a ', lfgen(0)(0) = a ', lfgen(0)(1) = nextl(iigen(0)(0), a), ∀ n0. n0 ≤ n + 1 → lfgen(n + 1)(n0 + 1) = nextl(iigen(n + 1)(n0), lfgen(n + 1)(n0).last) ∧ iigen(n + 1)(n0 + 1) = nextj(iigen(n + 1)(n0), lfgen(n + 1)(n0).last), ∀ m, n. m ≤ n → lfgen(n)(m) = lfgen(m)(m) ∧ iigen(n)(m) = iigen(m)(m), ∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = iigen(0)(0) ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last)) ⊦ lfgen(n + 1)(n + 1) = nextl(iigen(n)(n), lfgen(n)(n).last) ∧ iigen(n + 1)(n + 1) = nextj(iigen(n)(n), lfgen(n)(n).last)

Rule argument

Additional information

Proof step 12 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 13:

  iigen(0)(1) = nextj(iigen(0)(0), a), iigen(n + 1)(0) = iigen(0)(0), lfgen(0)(1) = nextl(iigen(0)(0), a), lfgen(0)(0) = a ', lfgen(n + 1)(0) = a ', n ≤ n + 1 → lfgen(n + 1)(n + 1) = nextl(iigen(n + 1)(n), lfgen(n + 1)(n).last) ∧ iigen(n + 1)(n + 1) = nextj(iigen(n + 1)(n), lfgen(n + 1)(n).last), ∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = iigen(0)(0) ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last)), ∀ m, n. m ≤ n → lfgen(n)(m) = lfgen(m)(m) ∧ iigen(n)(m) = iigen(m)(m) ⊦ lfgen(n + 1)(n + 1) = nextl(iigen(n)(n), lfgen(n)(n).last) ∧ iigen(n + 1)(n + 1) = nextj(iigen(n)(n), lfgen(n)(n).last)

Rule argument

Additional information

Used terms: n

Proof step 13 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 14:

  iigen(n + 1)(0) = iigen(0)(0), iigen(0)(1) = nextj(iigen(0)(0), a), iigen(n + 1)(n + 1) = nextj(iigen(n + 1)(n), lfgen(n + 1)(n).last), lfgen(n + 1)(0) = a ', lfgen(0)(0) = a ', lfgen(0)(1) = nextl(iigen(0)(0), a), lfgen(n + 1)(n + 1) = nextl(iigen(n + 1)(n), lfgen(n + 1)(n).last), ∀ m, n. m ≤ n → lfgen(n)(m) = lfgen(m)(m) ∧ iigen(n)(m) = iigen(m)(m), ∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = iigen(0)(0) ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last)) ⊦ lfgen(n + 1)(n + 1) = nextl(iigen(n)(n), lfgen(n)(n).last) ∧ iigen(n + 1)(n + 1) = nextj(iigen(n)(n), lfgen(n)(n).last)

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 ≤ m + n; (s)

Proof step 14 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

User applied rule all left

And closed the goal!

Rule argument

Additional information

Used simplifier rules: ⊦ m ≤ m + n; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: n, n + 1

Proof step 15 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

User applied rule all right

And obtained the following new goals

• Proof step 16:

  ∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = n0 ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last)) ⊦ ∀ n. m ≤ n → lfgen(n)(m) = lfgen(m)(m) ∧ iigen(n)(m) = iigen(m)(m), ∃ lf, jj. lf(0) = a ' ∧ jj(0) = n0 ∧ (∀ n. lf(n + 1) = nextl(jj(n), lf(n).last) ∧ jj(n + 1) = nextj(jj(n), lf(n).last))

Rule argument

Additional information

Proof step 16 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

User applied rule weakening

And obtained the following new goals

• Proof step 17:

  ∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = n0 ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last)) ⊦ ∀ n. m ≤ n → lfgen(n)(m) = lfgen(m)(m) ∧ iigen(n)(m) = iigen(m)(m)

Rule argument

Additional information

Proof step 17 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

User applied rule structural induction

And obtained the following new goals

• Proof step 18:

  ∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = n0 ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last)) ⊦ ∀ n. 0 ≤ n → lfgen(n)(0) = lfgen(0)(0) ∧ iigen(n)(0) = iigen(0)(0)

• Proof step 22:

  ∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = n0 ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last)), ∀ lfgen, a, iigen, n0, nextl, nextj. (∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = n0 ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last))) → (∀ n. m0 ≤ n → lfgen(n)(m0) = lfgen(m0)(m0) ∧ iigen(n)(m0) = iigen(m0)(m0)) ⊦ ∀ n. m0 +1 ≤ n → lfgen(n)(m0 +1) = lfgen(m0 +1)(m0 +1) ∧ iigen(n)(m0 +1) = iigen(m0 +1)(m0 +1)

Rule argument

Additional information

Used simplifier rules: nat freely generated by 0, +1;;

Proof step 18 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 19:

  ∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = n0 ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last)) ⊦ lfgen(n)(0) = lfgen(0)(0) ∧ iigen(n)(0) = iigen(0)(0)

Rule argument

Additional information

Used simplifier rules: ⊦ 0 ≤ m; (s)

Proof step 19 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 20:

  iigen(n)(0) = n0, lfgen(n)(0) = a ', ∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = n0 ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last)), ∀ n0. n0 ≤ n → lfgen(n)(n0 + 1) = nextl(iigen(n)(n0), lfgen(n)(n0).last) ∧ iigen(n)(n0 + 1) = nextj(iigen(n)(n0), lfgen(n)(n0).last) ⊦ lfgen(n)(0) = lfgen(0)(0) ∧ iigen(n)(0) = iigen(0)(0)

Rule argument

Additional information

Used terms: n

Proof step 20 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 21:

  lfgen(n)(0) = a ', ∀ n0. n0 ≤ n → lfgen(n)(n0 + 1) = nextl(iigen(n)(n0), lfgen(n)(n0).last) ∧ iigen(n)(n0 + 1) = nextj(iigen(n)(n0), lfgen(n)(n0).last), ∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = iigen(n)(0) ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last)) ⊦ a ' = lfgen(0)(0) ∧ iigen(n)(0) = iigen(0)(0)

Rule argument

Additional information

Proof step 21 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

User applied rule all left

And closed the goal!

Rule argument

Additional information

Used simplifier rules: ⊦ m ≤ 0 ↔ m = 0; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: 0

Proof step 22 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 23:

  m0 < n, ∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = n0 ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last)), ∀ lfgen, a, iigen, n0, nextl, nextj. (∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = n0 ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last))) → (∀ n. m0 ≤ n → lfgen(n)(m0) = lfgen(m0)(m0) ∧ iigen(n)(m0) = iigen(m0)(m0)) ⊦ lfgen(n)(m0 + 1) = lfgen(m0 + 1)(m0 + 1) ∧ iigen(n)(m0 + 1) = iigen(m0 + 1)(m0 + 1)

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ m +1 = m + 1; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m + 1 ≤ n ↔ m < n; (s)

Proof step 23 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

User applied rule cut formula

And obtained the following new goals

• Proof step 24:

  n ≠ m0 + 1, m0 < n, ∀ lfgen, a, iigen, n0, nextl, nextj. (∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = n0 ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last))) → (∀ n. m0 ≤ n → lfgen(n)(m0) = lfgen(m0)(m0) ∧ iigen(n)(m0) = iigen(m0)(m0)), ∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = n0 ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last)) ⊦ lfgen(n)(m0 + 1) = lfgen(m0 + 1)(m0 + 1) ∧ iigen(n)(m0 + 1) = iigen(m0 + 1)(m0 + 1)

Rule argument

Additional information

Used simplifier rules: ⊦ 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 formulas: n = m0 + 1

Proof step 24 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 25:

  n ≠ m0 + 1, m0 < n, m0 + 1 < n, ∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = n0 ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last)), ∀ lfgen, a, iigen, n0, nextl, nextj. (∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = n0 ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last))) → (∀ n. m0 ≤ n → lfgen(n)(m0) = lfgen(m0)(m0) ∧ iigen(n)(m0) = iigen(m0)(m0)) ⊦ lfgen(n)(m0 + 1) = lfgen(m0 + 1)(m0 + 1) ∧ iigen(n)(m0 + 1) = iigen(m0 + 1)(m0 + 1)

Rule argument

Additional information

Used simplifier rules: succ-03: ⊦ m < n ∧ m + 1 ≠ n → m + 1 < n; (forward) from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic

Proof step 25 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 26:

  n ≠ m0 + 1, m0 + 1 < n, m0 < n, (∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = n0 ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last))) → (∀ n. m0 ≤ n → lfgen(n)(m0) = lfgen(m0)(m0) ∧ iigen(n)(m0) = iigen(m0)(m0)), ∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = n0 ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last)) ⊦ lfgen(n)(m0 + 1) = lfgen(m0 + 1)(m0 + 1) ∧ iigen(n)(m0 + 1) = iigen(m0 + 1)(m0 + 1)

Rule argument

Additional information

Used terms: lfgen, a, iigen, n0, nextl, nextj

Proof step 26 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

User applied rule case distinction

And obtained the following new goals

• Proof step 27:

  n ≠ m0 + 1, m0 < n, m0 + 1 < n, ∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = n0 ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last)) ⊦ ∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = n0 ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last)), lfgen(n)(m0 + 1) = lfgen(m0 + 1)(m0 + 1) ∧ iigen(n)(m0 + 1) = iigen(m0 + 1)(m0 + 1)

• Proof step 31:

  n ≠ m0 + 1, m0 < n, m0 + 1 < n, ∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = n0 ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last)), ∀ n. m0 ≤ n → lfgen(n)(m0) = lfgen(m0)(m0) ∧ iigen(n)(m0) = iigen(m0)(m0) ⊦ lfgen(n)(m0 + 1) = lfgen(m0 + 1)(m0 + 1) ∧ iigen(n)(m0 + 1) = iigen(m0 + 1)(m0 + 1)

Rule argument

Additional information

Proof step 27 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 28:

  n ≠ m0 + 1, m0 + 1 < n, m0 < n, ∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = n0 ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last)) ⊦ lfgen(n)(m0 + 1) = lfgen(m0 + 1)(m0 + 1) ∧ iigen(n)(m0 + 1) = iigen(m0 + 1)(m0 + 1), lfgen(m)(0) = a ' ∧ iigen(m)(0) = n0 ∧ (n1 ≤ m → lfgen(m)(n1 + 1) = nextl(iigen(m)(n1), lfgen(m)(n1).last) ∧ iigen(m)(n1 + 1) = nextj(iigen(m)(n1), lfgen(m)(n1).last))

Rule argument

Additional information

Proof step 28 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 29:

  iigen(m)(0) = n0, lfgen(m)(0) = a ', n ≠ m0 + 1, m0 < n, m0 + 1 < n, ∀ n. n ≤ m → lfgen(m)(n + 1) = nextl(iigen(m)(n), lfgen(m)(n).last) ∧ iigen(m)(n + 1) = nextj(iigen(m)(n), lfgen(m)(n).last) ⊦ lfgen(m)(0) = a ' ∧ iigen(m)(0) = n0 ∧ (n1 ≤ m → lfgen(m)(n1 + 1) = nextl(iigen(m)(n1), lfgen(m)(n1).last) ∧ iigen(m)(n1 + 1) = nextj(iigen(m)(n1), lfgen(m)(n1).last)), lfgen(n)(m0 + 1) = lfgen(m0 + 1)(m0 + 1) ∧ iigen(n)(m0 + 1) = iigen(m0 + 1)(m0 + 1)

Rule argument

Additional information

Used terms: m

Proof step 29 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 30:

  lfgen(m)(0) = a ', n ≠ m0 + 1, m0 + 1 < n, m0 < n, n1 ≤ m, ∀ n. n ≤ m → lfgen(m)(n + 1) = nextl(iigen(m)(n), lfgen(m)(n).last) ∧ iigen(m)(n + 1) = nextj(iigen(m)(n), lfgen(m)(n).last) ⊦ lfgen(n)(m0 + 1) = lfgen(m0 + 1)(m0 + 1) ∧ iigen(n)(m0 + 1) = iigen(m0 + 1)(m0 + 1), lfgen(m)(n1 + 1) = nextl(iigen(m)(n1), lfgen(m)(n1).last) ∧ iigen(m)(n1 + 1) = nextj(iigen(m)(n1), lfgen(m)(n1).last)

Rule argument

Additional information

Proof step 30 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

User applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: n1

Proof step 31 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 32:

  iigen(n)(0) = n0, lfgen(n)(0) = a ', n ≠ m0 + 1, m0 + 1 < n, m0 < n, ∀ n. m0 ≤ n → lfgen(n)(m0) = lfgen(m0)(m0) ∧ iigen(n)(m0) = iigen(m0)(m0), ∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = n0 ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last)), ∀ n0. n0 ≤ n → lfgen(n)(n0 + 1) = nextl(iigen(n)(n0), lfgen(n)(n0).last) ∧ iigen(n)(n0 + 1) = nextj(iigen(n)(n0), lfgen(n)(n0).last) ⊦ lfgen(n)(m0 + 1) = lfgen(m0 + 1)(m0 + 1) ∧ iigen(n)(m0 + 1) = iigen(m0 + 1)(m0 + 1)

Rule argument

Additional information

Used terms: n

Proof step 32 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 33:

  lfgen(n)(0) = a ', n ≠ m0 + 1, m0 < n, m0 + 1 < n, ∀ n0. n0 ≤ n → lfgen(n)(n0 + 1) = nextl(iigen(n)(n0), lfgen(n)(n0).last) ∧ iigen(n)(n0 + 1) = nextj(iigen(n)(n0), lfgen(n)(n0).last), ∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = iigen(n)(0) ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last)), ∀ n. m0 ≤ n → lfgen(n)(m0) = lfgen(m0)(m0) ∧ iigen(n)(m0) = iigen(m0)(m0) ⊦ lfgen(n)(m0 + 1) = lfgen(m0 + 1)(m0 + 1) ∧ iigen(n)(m0 + 1) = iigen(m0 + 1)(m0 + 1)

Rule argument

Additional information

Proof step 33 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 34:

  lfgen(n)(0) = a ', n ≠ m0 + 1, m0 + 1 < n, m0 < n, m0 ≤ n → lfgen(n)(m0 + 1) = nextl(iigen(n)(m0), lfgen(n)(m0).last) ∧ iigen(n)(m0 + 1) = nextj(iigen(n)(m0), lfgen(n)(m0).last), ∀ n. m0 ≤ n → lfgen(n)(m0) = lfgen(m0)(m0) ∧ iigen(n)(m0) = iigen(m0)(m0), ∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = iigen(n)(0) ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last)), ∀ n0. n0 ≤ n → lfgen(n)(n0 + 1) = nextl(iigen(n)(n0), lfgen(n)(n0).last) ∧ iigen(n)(n0 + 1) = nextj(iigen(n)(n0), lfgen(n)(n0).last) ⊦ lfgen(n)(m0 + 1) = lfgen(m0 + 1)(m0 + 1) ∧ iigen(n)(m0 + 1) = iigen(m0 + 1)(m0 + 1)

Rule argument

Additional information

Used terms: m0

Proof step 34 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 35:

  iigen(n)(m0 + 1) = nextj(iigen(n)(m0), lfgen(n)(m0).last), lfgen(n)(0) = a ', lfgen(n)(m0 + 1) = nextl(iigen(n)(m0), lfgen(n)(m0).last), n ≠ m0 + 1, m0 < n, m0 + 1 < n, ∀ n0. n0 ≤ n → lfgen(n)(n0 + 1) = nextl(iigen(n)(n0), lfgen(n)(n0).last) ∧ iigen(n)(n0 + 1) = nextj(iigen(n)(n0), lfgen(n)(n0).last), ∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = iigen(n)(0) ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last)), ∀ n. m0 ≤ n → lfgen(n)(m0) = lfgen(m0)(m0) ∧ iigen(n)(m0) = iigen(m0)(m0) ⊦ lfgen(n)(m0 + 1) = lfgen(m0 + 1)(m0 + 1) ∧ iigen(n)(m0 + 1) = iigen(m0 + 1)(m0 + 1)

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → m ≤ n; (s)

Proof step 35 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 36:

  iigen(n)(m0 + 1) = nextj(iigen(n)(m0), lfgen(n)(m0).last), iigen(m0 + 1)(0) = iigen(n)(0), lfgen(n)(m0 + 1) = nextl(iigen(n)(m0), lfgen(n)(m0).last), lfgen(n)(0) = a ', lfgen(m0 + 1)(0) = a ', n ≠ m0 + 1, m0 + 1 < n, m0 < n, ∀ n. m0 ≤ n → lfgen(n)(m0) = lfgen(m0)(m0) ∧ iigen(n)(m0) = iigen(m0)(m0), ∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = iigen(n)(0) ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last)), ∀ n0. n0 ≤ n → lfgen(n)(n0 + 1) = nextl(iigen(n)(n0), lfgen(n)(n0).last) ∧ iigen(n)(n0 + 1) = nextj(iigen(n)(n0), lfgen(n)(n0).last), ∀ n. n ≤ m0 + 1 → lfgen(m0 + 1)(n + 1) = nextl(iigen(m0 + 1)(n), lfgen(m0 + 1)(n).last) ∧ iigen(m0 + 1)(n + 1) = nextj(iigen(m0 + 1)(n), lfgen(m0 + 1)(n).last) ⊦ lfgen(n)(m0 + 1) = lfgen(m0 + 1)(m0 + 1) ∧ iigen(n)(m0 + 1) = iigen(m0 + 1)(m0 + 1)

Rule argument

Additional information

Used terms: m0 + 1

Proof step 36 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 37:

  iigen(m0 + 1)(0) = iigen(n)(0), iigen(n)(m0 + 1) = nextj(iigen(n)(m0), lfgen(n)(m0).last), lfgen(m0 + 1)(0) = a ', lfgen(n)(0) = a ', lfgen(n)(m0 + 1) = nextl(iigen(n)(m0), lfgen(n)(m0).last), n ≠ m0 + 1, m0 < n, m0 + 1 < n, ∀ n. n ≤ m0 + 1 → lfgen(m0 + 1)(n + 1) = nextl(iigen(m0 + 1)(n), lfgen(m0 + 1)(n).last) ∧ iigen(m0 + 1)(n + 1) = nextj(iigen(m0 + 1)(n), lfgen(m0 + 1)(n).last), ∀ n0. n0 ≤ n → lfgen(n)(n0 + 1) = nextl(iigen(n)(n0), lfgen(n)(n0).last) ∧ iigen(n)(n0 + 1) = nextj(iigen(n)(n0), lfgen(n)(n0).last), ∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = iigen(n)(0) ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last)), ∀ n. m0 ≤ n → lfgen(n)(m0) = lfgen(m0)(m0) ∧ iigen(n)(m0) = iigen(m0)(m0) ⊦ lfgen(n)(m0 + 1) = lfgen(m0 + 1)(m0 + 1) ∧ iigen(n)(m0 + 1) = iigen(m0 + 1)(m0 + 1)

Rule argument

Additional information

Proof step 37 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 38:

  iigen(n)(m0 + 1) = nextj(iigen(n)(m0), lfgen(n)(m0).last), iigen(m0 + 1)(0) = iigen(n)(0), lfgen(n)(m0 + 1) = nextl(iigen(n)(m0), lfgen(n)(m0).last), lfgen(n)(0) = a ', lfgen(m0 + 1)(0) = a ', n ≠ m0 + 1, m0 + 1 < n, m0 < n, m0 ≤ m0 + 1 → lfgen(m0 + 1)(m0 + 1) = nextl(iigen(m0 + 1)(m0), lfgen(m0 + 1)(m0).last) ∧ iigen(m0 + 1)(m0 + 1) = nextj(iigen(m0 + 1)(m0), lfgen(m0 + 1)(m0).last), ∀ n. m0 ≤ n → lfgen(n)(m0) = lfgen(m0)(m0) ∧ iigen(n)(m0) = iigen(m0)(m0), ∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = iigen(n)(0) ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last)), ∀ n0. n0 ≤ n → lfgen(n)(n0 + 1) = nextl(iigen(n)(n0), lfgen(n)(n0).last) ∧ iigen(n)(n0 + 1) = nextj(iigen(n)(n0), lfgen(n)(n0).last), ∀ n. n ≤ m0 + 1 → lfgen(m0 + 1)(n + 1) = nextl(iigen(m0 + 1)(n), lfgen(m0 + 1)(n).last) ∧ iigen(m0 + 1)(n + 1) = nextj(iigen(m0 + 1)(n), lfgen(m0 + 1)(n).last) ⊦ lfgen(n)(m0 + 1) = lfgen(m0 + 1)(m0 + 1) ∧ iigen(n)(m0 + 1) = iigen(m0 + 1)(m0 + 1)

Rule argument

Additional information

Used terms: m0

Proof step 38 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 39:

  iigen(m0 + 1)(0) = iigen(n)(0), iigen(n)(m0 + 1) = nextj(iigen(n)(m0), lfgen(n)(m0).last), iigen(m0 + 1)(m0 + 1) = nextj(iigen(m0 + 1)(m0), lfgen(m0 + 1)(m0).last), lfgen(m0 + 1)(0) = a ', lfgen(n)(0) = a ', lfgen(n)(m0 + 1) = nextl(iigen(n)(m0), lfgen(n)(m0).last), lfgen(m0 + 1)(m0 + 1) = nextl(iigen(m0 + 1)(m0), lfgen(m0 + 1)(m0).last), n ≠ m0 + 1, m0 < n, m0 + 1 < n, ∀ n. n ≤ m0 + 1 → lfgen(m0 + 1)(n + 1) = nextl(iigen(m0 + 1)(n), lfgen(m0 + 1)(n).last) ∧ iigen(m0 + 1)(n + 1) = nextj(iigen(m0 + 1)(n), lfgen(m0 + 1)(n).last), ∀ n0. n0 ≤ n → lfgen(n)(n0 + 1) = nextl(iigen(n)(n0), lfgen(n)(n0).last) ∧ iigen(n)(n0 + 1) = nextj(iigen(n)(n0), lfgen(n)(n0).last), ∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = iigen(n)(0) ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last)), ∀ n. m0 ≤ n → lfgen(n)(m0) = lfgen(m0)(m0) ∧ iigen(n)(m0) = iigen(m0)(m0) ⊦ lfgen(n)(m0 + 1) = lfgen(m0 + 1)(m0 + 1) ∧ iigen(n)(m0 + 1) = iigen(m0 + 1)(m0 + 1)

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 ≤ m + n; (s)

Proof step 39 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 40:

  iigen(m0 + 1)(m0 + 1) = nextj(iigen(m0 + 1)(m0), lfgen(m0 + 1)(m0).last), iigen(n)(m0 + 1) = nextj(iigen(n)(m0), lfgen(n)(m0).last), iigen(m0 + 1)(0) = iigen(n)(0), lfgen(m0 + 1)(m0 + 1) = nextl(iigen(m0 + 1)(m0), lfgen(m0 + 1)(m0).last), lfgen(n)(m0 + 1) = nextl(iigen(n)(m0), lfgen(n)(m0).last), lfgen(n)(0) = a ', lfgen(m0 + 1)(0) = a ', n ≠ m0 + 1, m0 + 1 < n, m0 < n, m0 ≤ m0 +1 → lfgen(m0 +1)(m0) = lfgen(m0)(m0) ∧ iigen(m0 +1)(m0) = iigen(m0)(m0), ∀ n. m0 ≤ n → lfgen(n)(m0) = lfgen(m0)(m0) ∧ iigen(n)(m0) = iigen(m0)(m0), ∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = iigen(n)(0) ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last)), ∀ n0. n0 ≤ n → lfgen(n)(n0 + 1) = nextl(iigen(n)(n0), lfgen(n)(n0).last) ∧ iigen(n)(n0 + 1) = nextj(iigen(n)(n0), lfgen(n)(n0).last), ∀ n. n ≤ m0 + 1 → lfgen(m0 + 1)(n + 1) = nextl(iigen(m0 + 1)(n), lfgen(m0 + 1)(n).last) ∧ iigen(m0 + 1)(n + 1) = nextj(iigen(m0 + 1)(n), lfgen(m0 + 1)(n).last) ⊦ lfgen(n)(m0 + 1) = lfgen(m0 + 1)(m0 + 1) ∧ iigen(n)(m0 + 1) = iigen(m0 + 1)(m0 + 1)

Rule argument

Additional information

Used terms: m0 +1

Proof step 40 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 41:

  iigen(m0 + 1)(0) = iigen(n)(0), iigen(n)(m0 + 1) = nextj(iigen(n)(m0), lfgen(n)(m0).last), iigen(m0 + 1)(m0 + 1) = nextj(iigen(m0)(m0), lfgen(m0)(m0).last), iigen(m0 + 1)(m0) = iigen(m0)(m0), lfgen(m0 + 1)(0) = a ', lfgen(n)(0) = a ', lfgen(n)(m0 + 1) = nextl(iigen(n)(m0), lfgen(n)(m0).last), lfgen(m0 + 1)(m0 + 1) = nextl(iigen(m0)(m0), lfgen(m0)(m0).last), lfgen(m0 + 1)(m0) = lfgen(m0)(m0), n ≠ m0 + 1, m0 < n, m0 + 1 < n, ∀ n. n ≤ m0 + 1 → lfgen(m0 + 1)(n + 1) = nextl(iigen(m0 + 1)(n), lfgen(m0 + 1)(n).last) ∧ iigen(m0 + 1)(n + 1) = nextj(iigen(m0 + 1)(n), lfgen(m0 + 1)(n).last), ∀ n0. n0 ≤ n → lfgen(n)(n0 + 1) = nextl(iigen(n)(n0), lfgen(n)(n0).last) ∧ iigen(n)(n0 + 1) = nextj(iigen(n)(n0), lfgen(n)(n0).last), ∀ m0. lfgen(m0)(0) = a ' ∧ iigen(m0)(0) = iigen(n)(0) ∧ (∀ n. n ≤ m0 → lfgen(m0)(n + 1) = nextl(iigen(m0)(n), lfgen(m0)(n).last) ∧ iigen(m0)(n + 1) = nextj(iigen(m0)(n), lfgen(m0)(n).last)), ∀ n. m0 ≤ n → lfgen(n)(m0) = lfgen(m0)(m0) ∧ iigen(n)(m0) = iigen(m0)(m0) ⊦ lfgen(n)(m0 + 1) = lfgen(m0 + 1)(m0 + 1) ∧ iigen(n)(m0 + 1) = iigen(m0 + 1)(m0 + 1)

Rule argument

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m ≤ m + n; (s)

Proof step 41 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

User applied rule all left

And closed the goal!

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → m ≤ n; (s) Used terms: n

Proof step 42 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 43:

  ⊦ ∃ lf, jj. lf(0) = a ' ∧ jj(0) = n0 ∧ (∀ n. lf(n + 1) = nextl(jj(n), lf(n).last) ∧ jj(n + 1) = nextj(jj(n), lf(n).last)), ∃ lf, jj. lf(0) = a ' ∧ jj(0) = n0 ∧ (∀ n. n ≤ m → lf(n + 1) = nextl(jj(n), lf(n).last) ∧ jj(n + 1) = nextj(jj(n), lf(n).last))

Rule argument

Additional information

Proof step 43 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

User applied rule weakening

And obtained the following new goals

• Proof step 44:

  ⊦ ∃ lf, jj. lf(0) = a ' ∧ jj(0) = n0 ∧ (∀ n. n ≤ m → lf(n + 1) = nextl(jj(n), lf(n).last) ∧ jj(n + 1) = nextj(jj(n), lf(n).last))

Rule argument

Additional information

Proof step 44 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

User applied rule structural induction

And obtained the following new goals

• Proof step 45:

  ⊦ ∃ lf, jj. lf(0) = a ' ∧ jj(0) = n0 ∧ (∀ n. n ≤ 0 → lf(n + 1) = nextl(jj(n), lf(n).last) ∧ jj(n + 1) = nextj(jj(n), lf(n).last))

• Proof step 47:

  ∀ a, n0, nextl, nextj. ∃ lf, jj. lf(0) = a ' ∧ jj(0) = n0 ∧ (∀ n. n ≤ m0 → lf(n + 1) = nextl(jj(n), lf(n).last) ∧ jj(n + 1) = nextj(jj(n), lf(n).last)) ⊦ ∃ lf, jj. lf(0) = a ' ∧ jj(0) = n0 ∧ (∀ n. n ≤ m0 +1 → lf(n + 1) = nextl(jj(n), lf(n).last) ∧ jj(n + 1) = nextj(jj(n), lf(n).last))

Rule argument

Additional information

Used simplifier rules: nat freely generated by 0, +1;;

Proof step 45 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 46:

  ⊦ ∃ lf, jj. lf(0) = a ' ∧ jj(0) = n0 ∧ lf(1) = nextl(jj(0), lf(0).last) ∧ jj(1) = nextj(jj(0), lf(0).last)

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 ≤ 0 ↔ m = 0; (s)

Proof step 46 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

User applied rule exists right

And closed the goal!

Rule argument

Additional information

Used simplifier rules: ⊦ a ' = a0 ' ↔ a = a0; (s) ⊦ a '.last = a; (s) Used terms: λ n. (n = 0 ⊃ a ';nextl(n0, a)), λ n. (n = 0 ⊃ n0;nextj(n0, a))

Proof step 47 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 48:

  ∀ a, n0, nextl, nextj. ∃ lf, jj. lf(0) = a ' ∧ jj(0) = n0 ∧ (∀ n. n ≤ m0 → lf(n + 1) = nextl(jj(n), lf(n).last) ∧ jj(n + 1) = nextj(jj(n), lf(n).last)) ⊦ ∃ lf, jj. lf(0) = a ' ∧ jj(0) = n0 ∧ (∀ n. n ≤ m0 + 1 → lf(n + 1) = nextl(jj(n), lf(n).last) ∧ jj(n + 1) = nextj(jj(n), lf(n).last))

Rule argument

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (s)

Proof step 48 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 49:

  ∃ lf, jj. lf(0) = a ' ∧ jj(0) = n0 ∧ (∀ n. n ≤ m0 → lf(n + 1) = nextl(jj(n), lf(n).last) ∧ jj(n + 1) = nextj(jj(n), lf(n).last)) ⊦ ∃ lf, jj. lf(0) = a ' ∧ jj(0) = n0 ∧ (∀ n. n ≤ m0 + 1 → lf(n + 1) = nextl(jj(n), lf(n).last) ∧ jj(n + 1) = nextj(jj(n), lf(n).last))

Rule argument

Additional information

Used terms: a, n0, nextl, nextj

Proof step 49 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 50:

  lf(0) = a ', ∀ n. n ≤ m0 → lf(n + 1) = nextl(jj(n), lf(n).last) ∧ jj(n + 1) = nextj(jj(n), lf(n).last) ⊦ ∃ lf, jj0. lf(0) = a ' ∧ jj0(0) = jj(0) ∧ (∀ n. n ≤ m0 + 1 → lf(n + 1) = nextl(jj0(n), lf(n).last) ∧ jj0(n + 1) = nextj(jj0(n), lf(n).last))

Rule argument

Additional information

Proof step 50 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

User applied rule exists right

And obtained the following new goals

• Proof step 51:

  lf(0) = a ', ∀ n. n ≤ m0 → lf(n + 1) = nextl(jj(n), lf(n).last) ∧ jj(n + 1) = nextj(jj(n), lf(n).last) ⊦ (0 = m0 + 2 ⊃ nextl(jj(m0 + 1), lf(m0 + 1).last);lf(0)) = a ' ∧ (0 = m0 + 2 ⊃ nextj(jj(m0 + 1), lf(m0 + 1).last);jj(0)) = jj(0) ∧ (∀ n. n ≤ m0 + 1 → (n + 1 = m0 + 2 ⊃ nextl(jj(m0 + 1), lf(m0 + 1).last);lf(n + 1)) = nextl((n = m0 + 2 ⊃ nextj(jj(m0 + 1), lf(m0 + 1).last);jj(n)), (n = m0 + 2 ⊃ nextl(jj(m0 + 1), lf(m0 + 1).last);lf(n)).last) ∧ (n + 1 = m0 + 2 ⊃ nextj(jj(m0 + 1), lf(m0 + 1).last);jj(n + 1)) = nextj((n = m0 + 2 ⊃ nextj(jj(m0 + 1), lf(m0 + 1).last);jj(n)), (n = m0 + 2 ⊃ nextl(jj(m0 + 1), lf(m0 + 1).last);lf(n)).last))

Rule argument

Additional information

Used terms: λ n0. (n0 = m0 + 2 ⊃ nextl(jj(m0 + 1), lf(m0 + 1).last);lf(n0)), λ n0. (n0 = m0 + 2 ⊃ nextj(jj(m0 + 1), lf(m0 + 1).last);jj(n0))

Proof step 51 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 52:

  lf(0) = a ', n ≤ m0 + 1, ∀ n. n ≤ m0 → lf(n + 1) = nextl(jj(n), lf(n).last) ∧ jj(n + 1) = nextj(jj(n), lf(n).last) ⊦ (n = m0 + 1 ⊃ nextl(jj(m0 + 1), lf(m0 + 1).last);lf(n + 1)) = nextl((n = m0 + 2 ⊃ nextj(jj(m0 + 1), lf(m0 + 1).last);jj(n)), (n = m0 + 2 ⊃ nextl(jj(m0 + 1), lf(m0 + 1).last);lf(n)).last) ∧ (n = m0 + 1 ⊃ nextj(jj(m0 + 1), lf(m0 + 1).last);jj(n + 1)) = nextj((n = m0 + 2 ⊃ nextj(jj(m0 + 1), lf(m0 + 1).last);jj(n)), (n = m0 + 2 ⊃ nextl(jj(m0 + 1), lf(m0 + 1).last);lf(n)).last)

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 n < n0 ⊦ m + (* n) = m0 + (* n0) ↔ m = m0 + (n0 - n); (s) n < n0 ⊦ (* n) ≠ m0 + (* n0); (s)

Proof step 52 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

User applied rule cut formula

And obtained the following new goals

• Proof step 53:

  lf(0) = a ', n ≠ m0 + 1, n ≤ m0 + 1, ∀ n. n ≤ m0 → lf(n + 1) = nextl(jj(n), lf(n).last) ∧ jj(n + 1) = nextj(jj(n), lf(n).last) ⊦ (n = m0 + 1 ⊃ nextl(jj(m0 + 1), lf(m0 + 1).last);lf(n + 1)) = nextl((n = m0 + 2 ⊃ nextj(jj(m0 + 1), lf(m0 + 1).last);jj(n)), (n = m0 + 2 ⊃ nextl(jj(m0 + 1), lf(m0 + 1).last);lf(n)).last) ∧ (n = m0 + 1 ⊃ nextj(jj(m0 + 1), lf(m0 + 1).last);jj(n + 1)) = nextj((n = m0 + 2 ⊃ nextj(jj(m0 + 1), lf(m0 + 1).last);jj(n)), (n = m0 + 2 ⊃ nextl(jj(m0 + 1), lf(m0 + 1).last);lf(n)).last)

Rule argument

Additional information

Used simplifier rules: ⊦ m + n ≤ m + n0 ↔ n ≤ n0; (s) ⊦ m + n = m + n0 ↔ n = n0; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used formulas: n = m0 + 1

Proof step 53 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 54:

  lf(0) = a ', n ≠ m0 + 1, n ≤ m0, ∀ n. n ≤ m0 → lf(n + 1) = nextl(jj(n), lf(n).last) ∧ jj(n + 1) = nextj(jj(n), lf(n).last) ⊦ lf(n + 1) = nextl((n = m0 + 2 ⊃ nextj(jj(m0 + 1), lf(m0 + 1).last);jj(n)), (n = m0 + 2 ⊃ nextl(jj(m0 + 1), lf(m0 + 1).last);lf(n)).last) ∧ jj(n + 1) = nextj((n = m0 + 2 ⊃ nextj(jj(m0 + 1), lf(m0 + 1).last);jj(n)), (n = m0 + 2 ⊃ nextl(jj(m0 + 1), lf(m0 + 1).last);lf(n)).last)

Rule argument

Additional information

Used simplifier rules: ⊦ m < n ↔ ¬ n ≤ m; (ss) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ m ≠ n → (m ≤ n ↔ m < n); (s)

Proof step 54 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 55:

  lf(0) = a ', n ≠ m0 + 1, n ≤ m0, n ≤ m0 → lf(n + 1) = nextl(jj(n), lf(n).last) ∧ jj(n + 1) = nextj(jj(n), lf(n).last) ⊦ lf(n + 1) = nextl((n = m0 + 2 ⊃ nextj(jj(m0 + 1), lf(m0 + 1).last);jj(n)), (n = m0 + 2 ⊃ nextl(jj(m0 + 1), lf(m0 + 1).last);lf(n)).last) ∧ jj(n + 1) = nextj((n = m0 + 2 ⊃ nextj(jj(m0 + 1), lf(m0 + 1).last);jj(n)), (n = m0 + 2 ⊃ nextl(jj(m0 + 1), lf(m0 + 1).last);lf(n)).last)

Rule argument

Additional information

Used terms: n

Proof step 55 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 56:

  jj(n + 1) = nextj(jj(n), lf(n).last), lf(0) = a ', lf(n + 1) = nextl(jj(n), lf(n).last), n ≠ m0 + 1, n ≤ m0 ⊦ lf(n + 1) = nextl((n = m0 + 2 ⊃ nextj(jj(m0 + 1), lf(m0 + 1).last);jj(n)), (n = m0 + 2 ⊃ nextl(jj(m0 + 1), lf(m0 + 1).last);lf(n)).last) ∧ jj(n + 1) = nextj((n = m0 + 2 ⊃ nextj(jj(m0 + 1), lf(m0 + 1).last);jj(n)), (n = m0 + 2 ⊃ nextl(jj(m0 + 1), lf(m0 + 1).last);lf(n)).last)

Rule argument

Additional information

Proof step 56 from proof nextl-nextj-lemma-butlast (previous step) (back to tree)

User applied rule cut formula

And closed the goal!

Rule argument

Additional information

Used simplifier rules: ⊦ m + n = m + n0 ↔ n = n0; (s) ⊦ m + m0 ≤ m ↔ m0 = 0; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used formulas: n = m0 + 2