Proof step 1 from proof inv-ls-succ (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  ⌜n⌝ + ⌜1⌝ < leninv(nσ), stutf(nσ), ¬ inv(nσ)[n] < inv(nσ)[n + 1] ⊦

Rule argument

Additional information

Proof step 2 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 3:

  ⌜n⌝ + ⌜1⌝ < leninv(nσ), stutf(nσ), inv(nσ)[n + 1] ≤ inv(nσ)[n] ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ m < n ↔ ¬ n ≤ m; (ss)

Proof step 3 from proof inv-ls-succ (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 4:

  ⊦ stutf(nσ) → inv(nσ) = (leninv(nσ) = ∞ ⊃ ⌜λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n⌝ ;⌜λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n⌝ to leninv(nσ).n)

• Proof step 5:

  ⌜n⌝ + ⌜1⌝ < leninv(nσ), stutf(nσ), (leninv(nσ) = ∞ ⊃ ⌜λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n⌝ ;⌜λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n⌝ to leninv(nσ).n)[n + 1] ≤ (leninv(nσ) = ∞ ⊃ ⌜λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n⌝ ;⌜λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n⌝ to leninv(nσ).n)[n] ⊦

Rule argument

Additional information

Used terms: nσ

Proof step 4 from proof inv-ls-succ (previous step) (back to tree)

Lemma inv-def

Proof step 5 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 6:

  ⌜n⌝ + ⌜1⌝ < leninv(nσ), stutf(nσ), (leninv(nσ) = ∞ ⊃ ⌜λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n⌝ ;⌜(λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to leninv(nσ).n⌝ )[n + 1] ≤ (leninv(nσ) = ∞ ⊃ ⌜λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n⌝ ;⌜(λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to leninv(nσ).n⌝ )[n] ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜nf⌝ to n = ⌜nf to n⌝ ; (s)

Proof step 6 from proof inv-ls-succ (previous step) (back to tree)

User applied rule cut formula

And obtained the following new goals

• Proof step 7:

  max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n + 1) = ∞, ⌜n⌝ + ⌜1⌝ < leninv(nσ), stutf(nσ), (leninv(nσ) = ∞ ⊃ ⌜λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n⌝ ;⌜(λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to leninv(nσ).n⌝ )[n + 1] ≤ (leninv(nσ) = ∞ ⊃ ⌜λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n⌝ ;⌜(λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to leninv(nσ).n⌝ )[n] ⊦

• Proof step 99:

  max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n + 1) ≠ ∞, ⌜n⌝ + ⌜1⌝ < leninv(nσ), stutf(nσ), (leninv(nσ) = ∞ ⊃ ⌜λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n⌝ ;⌜(λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to leninv(nσ).n⌝ )[n + 1] ≤ (leninv(nσ) = ∞ ⊃ ⌜λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n⌝ ;⌜(λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to leninv(nσ).n⌝ )[n] ⊦

Rule argument

Additional information

Used formulas: max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n + 1) = ∞

Proof step 7 from proof inv-ls-succ (previous step) (back to tree)

User applied rule case distinction

And obtained the following new goals

• Proof step 8:

  leninv(nσ) = ∞, max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n + 1) = ∞, ⌜n⌝ + ⌜1⌝ < leninv(nσ), stutf(nσ), ⌜λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n⌝ [n + 1] ≤ (leninv(nσ) = ∞ ⊃ ⌜λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n⌝ ;⌜(λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to leninv(nσ).n⌝ )[n] ⊦

• Proof step 44:

  max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n + 1) = ∞, leninv(nσ) ≠ ∞, ⌜n⌝ + ⌜1⌝ < leninv(nσ), stutf(nσ), ⌜(λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to leninv(nσ).n⌝ [n + 1] ≤ (leninv(nσ) = ∞ ⊃ ⌜λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n⌝ ;⌜(λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to leninv(nσ).n⌝ )[n] ⊦

Rule argument

Additional information

Proof step 8 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 9:

  leninv(nσ) = ∞, max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n + 1) = ∞, stutf(nσ), ∞.n ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n ⊦

Rule argument

Additional information

Used simplifier rules: add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ⌜n⌝ ≠ ∞; (s) ⊦ ⌜m⌝ + n∞ = ∞ ↔ n∞ = ∞; (s) ⊦ n∞ < ∞ ↔ n∞ ≠ ∞; (s) ⊦ ⌜nf⌝ [n] = nf(n); (s)

Proof step 9 from proof inv-ls-succ (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 10:

  ⊦ max∞(np) = ∞ ↔ (∀ n. ∃ n0. n < n0 ∧ np(n0)) ∨ (∀ n. ¬ np(n))

• Proof step 11:

  leninv(nσ) = ∞, stutf(nσ), ∞.n ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, (∀ n0. ∃ n1. n0 < n1 ∧ ⌜n1⌝ < # nσ ∧ nσ[n1] = n + 1) ∨ (∀ n0. ¬(⌜n0⌝ < # nσ ∧ nσ[n0] = n + 1)) ⊦

Rule argument

Additional information

Used terms: λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n + 1

Proof step 10 from proof inv-ls-succ (previous step) (back to tree)

Lemma maxinf-inf

Proof step 11 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 12:

  leninv(nσ) = ∞, stutf(nσ), ∞.n ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, (∀ n0. ∃ n1. n0 < n1 ∧ ⌜n1⌝ < # nσ ∧ nσ[n1] = n + 1) ∨ (∀ n0. ¬ ⌜n0⌝ < # nσ ∨ nσ[n0] ≠ n + 1) ⊦

Rule argument

Additional information

Proof step 12 from proof inv-ls-succ (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 13:

  ⊦ stutf(nσ) → (leninv(nσ) = ∞ ↔ # nσ = ∞ ∧ (∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0]))

• Proof step 14:

  # nσ = ∞, stutf(nσ), ∞.n ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, ∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0], (∀ n0. ∃ n1. n0 < n1 ∧ ⌜n1⌝ < # nσ ∧ nσ[n1] = n + 1) ∨ (∀ n0. ¬ ⌜n0⌝ < # nσ ∨ nσ[n0] ≠ n + 1) ⊦

Rule argument

Additional information

Used terms: nσ

Proof step 13 from proof inv-ls-succ (previous step) (back to tree)

Lemma leninv-inf

Proof step 14 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 15:

  # nσ = ∞, stutf(nσ), ∞.n ≤ max∞(λ n0. nσ[n0] = n).n, ∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0], (∀ n0. ∃ n1. n0 < n1 ∧ nσ[n1] = n + 1) ∨ (∀ n0. nσ[n0] ≠ n + 1) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜m⌝ < ∞; (s)

Proof step 15 from proof inv-ls-succ (previous step) (back to tree)

User applied rule case distinction

And obtained the following new goals

• Proof step 16:

  # nσ = ∞, stutf(nσ), ∞.n ≤ max∞(λ n0. nσ[n0] = n).n, ∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0], ∀ n0. ∃ n1. n0 < n1 ∧ nσ[n1] = n + 1 ⊦

• Proof step 24:

  # nσ = ∞, stutf(nσ), ∞.n ≤ max∞(λ n0. nσ[n0] = n).n, ∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0], ∀ n0. nσ[n0] ≠ n + 1 ⊦

Rule argument

Additional information

Proof step 16 from proof inv-ls-succ (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 17:

  # nσ = ∞, stutf(nσ), ∞.n ≤ max∞(λ n0. nσ[n0] = n).n, ∀ n0. ∃ n1. n0 < n1 ∧ nσ[n1] = n + 1, ∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0], ∃ n1. 0 < n1 ∧ nσ[n1] = n + 1 ⊦

Rule argument

Additional information

Used terms: 0

Proof step 17 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 18:

  # nσ = ∞, nσ[n1] = n + 1, n1 ≠ 0, stutf(nσ), ∞.n ≤ max∞(λ n0. nσ[n0] = n).n, ∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0], ∀ n0. ∃ n1. n0 < n1 ∧ nσ[n1] = n + 1 ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ 0 < m ↔ m ≠ 0; (s)

Proof step 18 from proof inv-ls-succ (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 19:

  # nσ = ∞, nσ[n1] = n + 1, n1 ≠ 0, stutf(nσ), ∞.n ≤ max∞(λ n0. nσ[n0] = n).n, ∀ n0. ∃ n1. n0 < n1 ∧ nσ[n1] = n + 1, ∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0], ∃ m0. n1 < m0 ∧ nσ[n1] < nσ[m0] ⊦

Rule argument

Additional information

Used terms: n1

Proof step 19 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 20:

  # nσ = ∞, nσ[n1] = n + 1, n1 ≠ 0, n + 1 < nσ[n0], n1 < n0, stutf(nσ), ∞.n ≤ max∞(λ n0. nσ[n0] = n).n, ∀ n0. ∃ n1. n0 < n1 ∧ nσ[n1] = n + 1, ∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0] ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic

Proof step 20 from proof inv-ls-succ (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 21:

  # nσ = ∞, nσ[n1] = n + 1, n1 ≠ 0, n1 < n0, n + 1 < nσ[n0], stutf(nσ), ∞.n ≤ max∞(λ n0. nσ[n0] = n).n, ∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0], ∀ n0. ∃ n1. n0 < n1 ∧ nσ[n1] = n + 1, ∃ n1. n0 < n1 ∧ nσ[n1] = n + 1 ⊦

Rule argument

Additional information

Used terms: n0

Proof step 21 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 22:

  # nσ = ∞, nσ[n1] = n + 1, nσ[n2] = n + 1, n1 ≠ 0, n1 < n2, n1 < n0, n + 1 < nσ[n0], n0 < n2, n1 + 1 < n2, stutf(nσ), ∞.n ≤ max∞(λ n0. nσ[n0] = n).n, ∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0], ∀ n0. ∃ n1. n0 < n1 ∧ nσ[n1] = n + 1 ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (s) ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic trans-nat: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification nat-basic lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic

Proof step 22 from proof inv-ls-succ (previous step) (back to tree)

User applied rule apply lemma

And obtained the following new goals

• Proof step 23:

  ⊦ mon≤(nσ) ∧ m ≤ n ∧ ⌜n⌝ < # nσ → nσ[m] ≤ nσ[n]

Rule argument

Additional information

Used simplifier rules: stutf-monle: ⊦ stutf(nσ) → (mon≤(nσ) ↔ true); ⊦ m < n → m ≤ n; (s) ⊦ ⌜m⌝ < ∞; (s) ⊦ m < n → (n ≤ m ↔ false); (s) Used terms: nσ, n0, n2

Proof step 23 from proof inv-ls-succ (previous step) (back to tree)

Lemma monle-leq

Proof step 24 from proof inv-ls-succ (previous step) (back to tree)

User applied rule weakening

And obtained the following new goals

• Proof step 25:

  # nσ = ∞, stutf(nσ), ∀ n0. nσ[n0] ≠ n + 1, ∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0] ⊦

Rule argument

Additional information

Proof step 25 from proof inv-ls-succ (previous step) (back to tree)

User applied rule induction

And obtained the following new goals

• Proof step 26:

  # nσ = ∞, stutf(nσ), ∀ n0. nσ[n0] ≠ n + 1, ∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0], ∀ n1, nσ. n1 < n → # nσ = ∞ ∧ stutf(nσ) ∧ (∀ n0. nσ[n0] ≠ n1 + 1) ∧ (∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0]) → false ⊦

Rule argument

Additional information

Used simplifier rules: <; (wellfounded) Used terms: n

Proof step 26 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 27:

  # nσ = ∞, stutf(nσ), ∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0], ∀ n0. nσ[n0] ≠ n + 1, ∀ n1, nσ. n1 < n → # nσ = ∞ ∧ stutf(nσ) ∧ (∀ n0. nσ[n0] ≠ n1 + 1) ∧ (∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0]) → false ⊦

Rule argument

Additional information

Proof step 27 from proof inv-ls-succ (previous step) (back to tree)

User applied rule constructor cut

And obtained the following new goals

• Proof step 28:

  # nσ = ∞, n = 0, stutf(nσ), ∀ n0. nσ[n0] ≠ n + 1, ∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0], ∀ n1, nσ. n1 < n → # nσ = ∞ ∧ stutf(nσ) ∧ (∀ n0. nσ[n0] ≠ n1 + 1) ∧ (∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0]) → false ⊦

• Proof step 35:

  # nσ = ∞, n = n2 +1, stutf(nσ), ∀ n0. nσ[n0] ≠ n + 1, ∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0], ∀ n1, nσ. n1 < n → # nσ = ∞ ∧ stutf(nσ) ∧ (∀ n0. nσ[n0] ≠ n1 + 1) ∧ (∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0]) → false ⊦

Rule argument

Additional information

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

Proof step 28 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 29:

  # nσ = ∞, stutf(nσ), ∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0], ∀ n0. nσ[n0] ≠ 1, ∀ n1, nσ. n1 < 0 → # nσ = ∞ ∧ stutf(nσ) ∧ (∀ n0. nσ[n0] ≠ n1 + 1) ∧ (∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0]) → false ⊦

Rule argument

Additional information

Proof step 29 from proof inv-ls-succ (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 30:

  # nσ = ∞, stutf(nσ), ∀ n0. nσ[n0] ≠ 1, ∃ m0. 0 < m0 ∧ nσ[0] < nσ[m0], ∀ n1, nσ. n1 < 0 → # nσ = ∞ ∧ stutf(nσ) ∧ (∀ n0. nσ[n0] ≠ n1 + 1) ∧ (∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0]) → false ⊦

Rule argument

Additional information

Used terms: 0

Proof step 30 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 31:

  # nσ = ∞, nσ[n] ≠ 0, n ≠ 0, stutf(nσ), ∀ n0. nσ[n0] ≠ 1, ∀ n1, nσ. n1 < 0 → # nσ = ∞ ∧ stutf(nσ) ∧ (∀ n0. nσ[n0] ≠ n1 + 1) ∧ (∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0]) → false ⊦

Rule argument

Additional information

Used simplifier rules: stutf-zero: ⊦ stutf(nσ) → nσ[0] = 0; ⊦ 0 < m ↔ m ≠ 0; (s)

Proof step 31 from proof inv-ls-succ (previous step) (back to tree)

User applied rule apply lemma

And obtained the following new goals

• Proof step 32:

  ⊦ stutf(nσ) ∧ ⌜m⌝ < # nσ ∧ n ≤ nσ[m] → (∃ m0. m0 ≤ m ∧ nσ[m0] = n)

• Proof step 33:

  # nσ = ∞, nσ[n] ≠ 0, n ≠ 0, stutf(nσ), stutf(nσ) ∧ ⌜n⌝ < # nσ ∧ 1 ≤ nσ[n] → (∃ m0. m0 ≤ n ∧ nσ[m0] = 1), ∀ n0. nσ[n0] ≠ 1, ∀ n1, nσ. n1 < 0 → # nσ = ∞ ∧ stutf(nσ) ∧ (∀ n0. nσ[n0] ≠ n1 + 1) ∧ (∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0]) → false ⊦

Rule argument

Additional information

Used terms: nσ, n, 1

Proof step 32 from proof inv-ls-succ (previous step) (back to tree)

Lemma stutf-alltaken

Proof step 33 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 34:

  # nσ = ∞, nσ[n0] = 1, nσ[n] ≠ 0, n ≠ 0, stutf(nσ), 1 ≤ nσ[n], n0 ≤ n, ∀ n0. nσ[n0] ≠ 1, ∀ n1, nσ. n1 < 0 → # nσ = ∞ ∧ stutf(nσ) ∧ (∀ n0. nσ[n0] ≠ n1 + 1) ∧ (∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0]) → false ⊦

Rule argument

Additional information

Used simplifier rules: not-zero: 0 < n ⊦ (* n) ≤ m → m ≠ 0; (forward) from specification nat ⊦ m < 1 ↔ m = 0; (s) ⊦ m ≤ n ↔ ¬ n < m; (ss) ⊦ ⌜m⌝ < ∞; (s)

Proof step 34 from proof inv-ls-succ (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: n0

Proof step 35 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 36:

  # nσ = ∞, stutf(nσ), ∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0], ∀ n0. nσ[n0] ≠ (n2 + 1) + 1, ∀ n1, nσ. n1 < n2 +1 → # nσ = ∞ ∧ stutf(nσ) ∧ (∀ n0. nσ[n0] ≠ n1 + 1) ∧ (∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0]) → false ⊦

Rule argument

Additional information

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

Proof step 36 from proof inv-ls-succ (previous step) (back to tree)

User applied rule apply induction

And obtained the following new goals

• Proof step 37:

  # nσ = ∞, stutf(nσ), ∀ n0. nσ[n0] ≠ (n2 + 1) + 1, ∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0], ∀ n1, nσ. n1 < n2 +1 → # nσ = ∞ ∧ stutf(nσ) ∧ (∀ n0. nσ[n0] ≠ n1 + 1) ∧ (∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0]) → false ⊦ # nσ = ∞ ∧ stutf(nσ) ∧ (∀ n0. nσ[n0] ≠ n2 + 1) ∧ (∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0])

Rule argument

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (s) ⊦ 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: n2, nσ

Proof step 37 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 38:

  # nσ = ∞, nσ[n0] = n2 + 1, stutf(nσ), ∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0], ∀ n0. nσ[n0] ≠ (n2 + 1) + 1, ∀ n1, nσ. n1 < n2 +1 → # nσ = ∞ ∧ stutf(nσ) ∧ (∀ n0. nσ[n0] ≠ n1 + 1) ∧ (∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0]) → false ⊦

Rule argument

Additional information

Proof step 38 from proof inv-ls-succ (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 39:

  # nσ = ∞, nσ[n0] = n2 + 1, stutf(nσ), ∀ n0. nσ[n0] ≠ (n2 + 1) + 1, ∃ m0. n0 < m0 ∧ nσ[n0] < nσ[m0], ∀ n1, nσ. n1 < n2 +1 → # nσ = ∞ ∧ stutf(nσ) ∧ (∀ n0. nσ[n0] ≠ n1 + 1) ∧ (∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0]) → false ⊦

Rule argument

Additional information

Used terms: n0

Proof step 39 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 40:

  # nσ = ∞, nσ[n0] = n2 + 1, n2 + 1 < nσ[n], n0 < n, stutf(nσ), ∀ n0. nσ[n0] ≠ (n2 + 1) + 1, ∀ n1, nσ. n1 < n2 +1 → # nσ = ∞ ∧ stutf(nσ) ∧ (∀ n0. nσ[n0] ≠ n1 + 1) ∧ (∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0]) → false ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic

Proof step 40 from proof inv-ls-succ (previous step) (back to tree)

User applied rule apply lemma

And obtained the following new goals

• Proof step 41:

  ⊦ stutf(nσ) ∧ ⌜m⌝ < # nσ ∧ n ≤ nσ[m] → (∃ m0. m0 ≤ m ∧ nσ[m0] = n)

• Proof step 42:

  # nσ = ∞, nσ[n0] = n2 + 1, n0 < n, n2 + 1 < nσ[n], stutf(nσ), stutf(nσ) ∧ ⌜n⌝ < # nσ ∧ 2 + n2 ≤ nσ[n] → (∃ m0. m0 ≤ n ∧ nσ[m0] = 2 + n2), ∀ n0. nσ[n0] ≠ (n2 + 1) + 1, ∀ n1, nσ. n1 < n2 +1 → # nσ = ∞ ∧ stutf(nσ) ∧ (∀ n0. nσ[n0] ≠ n1 + 1) ∧ (∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0]) → false ⊦

Rule argument

Additional information

Used terms: nσ, n, 2 + n2

Proof step 41 from proof inv-ls-succ (previous step) (back to tree)

Lemma stutf-alltaken

Proof step 42 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 43:

  # nσ = ∞, nσ[n0] = n2 + 1, nσ[n1] = 2 + n2, n0 < n, n2 + 1 < nσ[n], stutf(nσ), 2 + n2 ≤ nσ[n], n1 ≤ n, ∀ n1. nσ[n1] ≠ 2 + n2, ∀ n1, nσ. n1 < n2 +1 → # nσ = ∞ ∧ stutf(nσ) ∧ (∀ n0. nσ[n0] ≠ n1 + 1) ∧ (∀ m. ∃ m0. m < m0 ∧ nσ[m] < nσ[m0]) → 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 ⊦ ¬ n < n; (s) ⊦ m + n < m + n0 ↔ n < n0; (s) ⊦ m +1 = m + 1; (s) n < n0 ⊦ (* n) ≠ m0 + (* n0); (s) trans-nat: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification nat-basic lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ m ≤ n ↔ ¬ n < m; (ss) ⊦ ⌜m⌝ < ∞; (s)

Proof step 43 from proof inv-ls-succ (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: n1

Proof step 44 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 45:

  max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n + 1) = ∞, leninv(nσ) ≠ ∞, ⌜n⌝ + ⌜1⌝ < leninv(nσ), stutf(nσ), ((λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to leninv(nσ).n)[n + 1] ≤ ((λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to leninv(nσ).n)[n] ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜x⌝ [n] = x[n]; (s)

Proof step 45 from proof inv-ls-succ (previous step) (back to tree)

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

• Proof step 46:

  leninv(nσ) = ⌜n1⌝ , max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n + 1) = ∞, leninv(nσ) ≠ ∞, ⌜n⌝ + ⌜1⌝ < leninv(nσ), stutf(nσ), ((λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to n1)[n + 1] ≤ ((λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to n1)[n] ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ n∞ ≠ ∞ → (n = n∞.n ↔ n∞ = ⌜n⌝ ); (elim) Used terms: leninv(nσ), n1

Proof step 46 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 47:

  leninv(nσ) = ⌜n1⌝ , max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n + 1) = ∞, n + 1 < n1, stutf(nσ), ∞.n ≤ ((λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to n1)[n] ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic m < n ⊦ (nf to n)[m] = nf(m); (s) ⊦ m ≤ n → (n < m ↔ false); (s) ⊦ m < n ↔ ¬ n ≤ m; (ss) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ⌜n⌝ + ⌜m⌝ < ⌜n0⌝ ↔ n + m < n0; (s) ⊦ ⌜n⌝ ≠ ∞; (s)

Proof step 47 from proof inv-ls-succ (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 48:

  ⊦ leninv(nσ) = max∞(λ n. n ≠ 0 → (∃ m. ⌜m⌝ < # nσ ∧ nσ[m] = n - 1 ∧ ¬(# nσ = ∞ ∧ (∀ m0. m < m0 → nσ[m0] = n - 1))))

• Proof step 49:

  max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n + 1) = ∞, n + 1 < n1, stutf(nσ), ∞.n ≤ ((λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to n1)[n], max∞(λ n. n ≠ 0 → (∃ m. ⌜m⌝ < # nσ ∧ nσ[m] = n - 1 ∧ ¬(# nσ = ∞ ∧ (∀ m0. m < m0 → nσ[m0] = n - 1)))) = ⌜n1⌝ ⊦

Rule argument

Additional information

Used terms: nσ

Proof step 48 from proof inv-ls-succ (previous step) (back to tree)

Lemma leninv-def

Proof step 49 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 50:

  max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n + 1) = ∞, n + 1 < n1, stutf(nσ), ∞.n ≤ ((λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to n1)[n], max∞(λ n0. n0 ≠ 0 → (∃ n2. ⌜n2⌝ < # nσ ∧ nσ[n2] = n0 - 1 ∧ (# nσ ≠ ∞ ∨ ¬(∀ m0. n2 < m0 → nσ[m0] = n0 - 1)))) = ⌜n1⌝ ⊦

Rule argument

Additional information

Proof step 50 from proof inv-ls-succ (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 51:

  ⊦ max∞(np) = ⌜n⌝ ↔ np(n) ∧ (∀ m. np(m) → ¬ n < m)

• Proof step 52:

  max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n + 1) = ∞, n + 1 < n1, stutf(nσ), ∞.n ≤ ((λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to n1)[n], n1 ≠ 0 → (∃ m. ⌜m⌝ < # nσ ∧ nσ[m] = n1 - 1 ∧ (# nσ ≠ ∞ ∨ ¬(∀ m0. m < m0 → nσ[m0] = n1 - 1))), ∀ m. (m ≠ 0 → (∃ m0. ⌜m0⌝ < # nσ ∧ nσ[m0] = m - 1 ∧ (# nσ ≠ ∞ ∨ ¬(∀ m1. m0 < m1 → nσ[m1] = m - 1)))) → ¬ n1 < m ⊦

Rule argument

Additional information

Used terms: n1, λ n0. n0 ≠ 0 → (∃ m. ⌜m⌝ < # nσ ∧ nσ[m] = n0 - 1 ∧ (# nσ ≠ ∞ ∨ ¬(∀ m0. m < m0 → nσ[m0] = n0 - 1)))

Proof step 51 from proof inv-ls-succ (previous step) (back to tree)

Lemma maxinf-fin

Proof step 52 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 53:

  nσ[n0] = n1 - 1, max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n + 1) = ∞, n + 1 < n1, ⌜n0⌝ < # nσ, stutf(nσ), ∞.n ≤ ((λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to n1)[n], ∀ m. (∃ m0. ⌜m0⌝ < # nσ ∧ nσ[m0] = m - 1 ∧ (# nσ ≠ ∞ ∨ ¬(∀ m1. m0 < m1 → nσ[m1] = m - 1))) → ¬ n1 < m, # nσ ≠ ∞ ∨ ¬(n0 < n2 → nσ[n2] = n1 - 1) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s)

Proof step 53 from proof inv-ls-succ (previous step) (back to tree)

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

• Proof step 54:

  nσ[n0] = n3, max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n + 1) = ∞, n1 = n3 + 1, n1 ≠ 0, ⌜n0⌝ < # nσ, n + 1 < n1, stutf(nσ), ∞.n ≤ ((λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to n1)[n], ∀ m. (∃ m0. ⌜m0⌝ < # nσ ∧ nσ[m0] = m - 1 ∧ (# nσ ≠ ∞ ∨ ¬(∀ m1. m0 < m1 → nσ[m1] = m - 1))) → ¬ n1 < m, # nσ ≠ ∞ ∨ ¬(n0 < n2 → nσ[n2] = n3) ⊦

Rule argument

Additional information

Used simplifier rules: n0 < n ⊦ ¬ m + (* n) < (* n0); (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1; (elim) Used terms: n1, m2

Proof step 54 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 55:

  max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n + 1) = ∞, n < nσ[n0], ⌜n0⌝ < # nσ, stutf(nσ), ∞.n ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, ∀ m. (∃ m0. ⌜m0⌝ < # nσ ∧ nσ[m0] = m - 1 ∧ (# nσ ≠ ∞ ∨ ¬(∀ m1. m0 < m1 → nσ[m1] = m - 1))) → ¬ nσ[n0] + 1 < m, # nσ ≠ ∞ ∨ ¬(n0 < n2 → nσ[n2] = nσ[n0]) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (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 ⊦ m + n < m + n0 ↔ n < n0; (s) m < n ⊦ (nf to n)[m] = nf(m); (s) ⊦ m < n → ¬ n < m; (s) ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ m + 1 ≠ 0; (s)

Proof step 55 from proof inv-ls-succ (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 56:

  ⊦ max∞(np) = ∞ ↔ (∀ n. ∃ n0. n < n0 ∧ np(n0)) ∨ (∀ n. ¬ np(n))

• Proof step 57:

  ⌜n0⌝ < # nσ, n < nσ[n0], stutf(nσ), ∞.n ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, ∀ m. (∃ m0. ⌜m0⌝ < # nσ ∧ nσ[m0] = m - 1 ∧ (# nσ ≠ ∞ ∨ ¬(∀ m1. m0 < m1 → nσ[m1] = m - 1))) → ¬ nσ[n0] + 1 < m, # nσ ≠ ∞ ∨ ¬(n0 < n2 → nσ[n2] = nσ[n0]), (∀ n0. ∃ n1. n0 < n1 ∧ ⌜n1⌝ < # nσ ∧ nσ[n1] = n + 1) ∨ (∀ n0. ¬(⌜n0⌝ < # nσ ∧ nσ[n0] = n + 1)) ⊦

Rule argument

Additional information

Used terms: λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n + 1

Proof step 56 from proof inv-ls-succ (previous step) (back to tree)

Lemma maxinf-inf

Proof step 57 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 58:

  n < nσ[n0], ⌜n0⌝ < # nσ, stutf(nσ), ∞.n ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, ∀ m. (∃ m0. ⌜m0⌝ < # nσ ∧ nσ[m0] = m - 1 ∧ (# nσ ≠ ∞ ∨ ¬(∀ m1. m0 < m1 → nσ[m1] = m - 1))) → ¬ nσ[n0] + 1 < m, # nσ ≠ ∞ ∨ ¬(n0 < n2 → nσ[n2] = nσ[n0]), (∀ n0. ∃ n1. n0 < n1 ∧ ⌜n1⌝ < # nσ ∧ nσ[n1] = n + 1) ∨ (∀ n1. ¬ ⌜n1⌝ < # nσ ∨ nσ[n1] ≠ n + 1) ⊦

Rule argument

Additional information

Proof step 58 from proof inv-ls-succ (previous step) (back to tree)

User applied rule case distinction

And obtained the following new goals

• Proof step 59:

  ⌜n0⌝ < # nσ, n < nσ[n0], stutf(nσ), ∞.n ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, ∀ m. (∃ m0. ⌜m0⌝ < # nσ ∧ nσ[m0] = m - 1 ∧ (# nσ ≠ ∞ ∨ ¬(∀ m1. m0 < m1 → nσ[m1] = m - 1))) → ¬ nσ[n0] + 1 < m, ∀ n0. ∃ n1. n0 < n1 ∧ ⌜n1⌝ < # nσ ∧ nσ[n1] = n + 1, # nσ ≠ ∞ ∨ ¬(n0 < n2 → nσ[n2] = nσ[n0]) ⊦

• Proof step 82:

  ⌜n0⌝ < # nσ, n < nσ[n0], stutf(nσ), ∞.n ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, ∀ m. (∃ m0. ⌜m0⌝ < # nσ ∧ nσ[m0] = m - 1 ∧ (# nσ ≠ ∞ ∨ ¬(∀ m1. m0 < m1 → nσ[m1] = m - 1))) → ¬ nσ[n0] + 1 < m, ∀ n1. ¬ ⌜n1⌝ < # nσ ∨ nσ[n1] ≠ n + 1, # nσ ≠ ∞ ∨ ¬(n0 < n2 → nσ[n2] = nσ[n0]) ⊦

Rule argument

Additional information

Proof step 59 from proof inv-ls-succ (previous step) (back to tree)

User applied rule case distinction

And obtained the following new goals

• Proof step 60:

  # nσ ≠ ∞, n < nσ[n0], ⌜n0⌝ < # nσ, stutf(nσ), ∞.n ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, ∀ n0. ∃ n1. n0 < n1 ∧ ⌜n1⌝ < # nσ ∧ nσ[n1] = n + 1, ∀ m. (∃ m0. ⌜m0⌝ < # nσ ∧ nσ[m0] = m - 1 ∧ (# nσ ≠ ∞ ∨ ¬(∀ m1. m0 < m1 → nσ[m1] = m - 1))) → ¬ nσ[n0] + 1 < m ⊦

• Proof step 65:

  nσ[n2] ≠ nσ[n0], n0 < n2, n < nσ[n0], ⌜n0⌝ < # nσ, stutf(nσ), ∞.n ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, ∀ n0. ∃ n1. n0 < n1 ∧ ⌜n1⌝ < # nσ ∧ nσ[n1] = n + 1, ∀ m. (∃ m0. ⌜m0⌝ < # nσ ∧ nσ[m0] = m - 1 ∧ (# nσ ≠ ∞ ∨ ¬(∀ m1. m0 < m1 → nσ[m1] = m - 1))) → ¬ nσ[n0] + 1 < m ⊦

Rule argument

Additional information

Proof step 60 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 61:

  # nσ ≠ ∞, ⌜n0⌝ < # nσ, n < nσ[n0], stutf(nσ), ∞.n ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, ∀ n0. ∃ n1. n0 < n1 ∧ ⌜n1⌝ < # nσ ∧ nσ[n1] = n + 1, ∀ m. (∃ m0. ⌜m0⌝ < # nσ ∧ nσ[m0] = m - 1) → ¬ nσ[n0] + 1 < m ⊦

Rule argument

Additional information

Proof step 61 from proof inv-ls-succ (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 62:

  ⊦ n∞ = ∞ ↔ ¬(∃ n. n∞ = ⌜n⌝ )

• Proof step 63:

  n < nσ[n0], ⌜n0⌝ < # nσ, stutf(nσ), ∞.n ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, ∀ m. (∃ m0. ⌜m0⌝ < # nσ ∧ nσ[m0] = m - 1) → ¬ nσ[n0] + 1 < m, ∀ n0. ∃ n1. n0 < n1 ∧ ⌜n1⌝ < # nσ ∧ nσ[n1] = n + 1, ∃ n. # nσ = ⌜n⌝ ⊦

Rule argument

Additional information

Used terms: # nσ

Proof step 62 from proof inv-ls-succ (previous step) (back to tree)

A spec-lemma

Proof step 63 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 64:

  # nσ = ⌜n1⌝ , n0 < n1, n < nσ[n0], stutf(nσ), ∞.n ≤ max∞(λ n2. n2 < n1 ∧ nσ[n2] = n).n, ∀ m. (∃ m0. m0 < n1 ∧ nσ[m0] = m - 1) → ¬ nσ[n0] + 1 < m, ∀ n0. ∃ n1. n0 < n1 ∧ ⌜n1⌝ < # nσ ∧ nσ[n1] = n + 1 ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s)

Proof step 64 from proof inv-ls-succ (previous step) (back to tree)

User applied rule all left

And closed the goal!

Rule argument

Additional information

Used simplifier rules: eqle-one: ⊦ m = n + 1 ∧ n < n0 → m ≤ n0; (forward) from specification nat a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic trans-nat: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification nat-basic f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ m < n → n ≠ 0; (s) ⊦ m +1 = m + 1; (s) ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) ⊦ m < n → ¬ n < m; (s) Used terms: n1

Proof step 65 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 66:

  nσ[n2] ≠ nσ[n0], n0 < n2, n < nσ[n0], ⌜n0⌝ < # nσ, stutf(nσ), ∞.n ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, ∀ n0. ∃ n1. n0 < n1 ∧ ⌜n1⌝ < # nσ ∧ nσ[n1] = n + 1, ∀ m. (∃ m0. ⌜m0⌝ < # nσ ∧ nσ[m0] = m - 1 ∧ (# nσ ≠ ∞ ∨ ¬(∀ m1. m0 < m1 → nσ[m1] = m - 1))) → ¬ nσ[n0] + 1 < m ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic

Proof step 66 from proof inv-ls-succ (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 67:

  nσ[n2] ≠ nσ[n0], ⌜n0⌝ < # nσ, n < nσ[n0], n0 < n2, stutf(nσ), ∞.n ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, ∀ m. (∃ m0. ⌜m0⌝ < # nσ ∧ nσ[m0] = m - 1 ∧ (# nσ ≠ ∞ ∨ ¬(∀ m1. m0 < m1 → nσ[m1] = m - 1))) → ¬ nσ[n0] + 1 < m, ∀ n0. ∃ n1. n0 < n1 ∧ ⌜n1⌝ < # nσ ∧ nσ[n1] = n + 1, ∃ n1. n0 < n1 ∧ ⌜n1⌝ < # nσ ∧ nσ[n1] = n + 1 ⊦

Rule argument

Additional information

Used terms: n0

Proof step 67 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 68:

  nσ[n1] = n + 1, nσ[n2] ≠ nσ[n0], n0 < n1, ⌜n1⌝ < # nσ, n0 < n2, ⌜n0⌝ < # nσ, n < nσ[n0], stutf(nσ), ∞.n ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, ∀ n0. ∃ n1. n0 < n1 ∧ ⌜n1⌝ < # nσ ∧ nσ[n1] = n + 1, ∀ m. (∃ m0. ⌜m0⌝ < # nσ ∧ nσ[m0] = m - 1 ∧ (# nσ ≠ ∞ ∨ ¬(∀ m1. m0 < m1 → nσ[m1] = m - 1))) → ¬ nσ[n0] + 1 < m ⊦

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 + 1 ≤ n ↔ m < n; (s) ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic eqle-one: ⊦ m = n + 1 ∧ n < n0 → m ≤ n0; (forward) from specification nat

Proof step 68 from proof inv-ls-succ (previous step) (back to tree)

User applied rule apply lemma

And obtained the following new goals

• Proof step 69:

  ⊦ mon≤(nσ) ∧ m ≤ n ∧ ⌜n⌝ < # nσ → nσ[m] ≤ nσ[n]

• Proof step 70:

  nσ[n1] = n + 1, nσ[n2] ≠ nσ[n0], n < nσ[n0], ⌜n0⌝ < # nσ, n0 < n2, ⌜n1⌝ < # nσ, n0 < n1, stutf(nσ), ∞.n ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, mon≤(nσ) ∧ n0 ≤ n1 ∧ ⌜n1⌝ < # nσ → nσ[n0] ≤ nσ[n1], ∀ m. (∃ m0. ⌜m0⌝ < # nσ ∧ nσ[m0] = m - 1 ∧ (# nσ ≠ ∞ ∨ ¬(∀ m1. m0 < m1 → nσ[m1] = m - 1))) → ¬ nσ[n0] + 1 < m, ∀ n0. ∃ n1. n0 < n1 ∧ ⌜n1⌝ < # nσ ∧ nσ[n1] = n + 1 ⊦

Rule argument

Additional information

Used terms: nσ, n0, n1

Proof step 69 from proof inv-ls-succ (previous step) (back to tree)

Lemma monle-leq

Proof step 70 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 71:

  nσ[n1] = n + 1, nσ[n0] = n + 1, nσ[n2] ≠ n + 1, ⌜n0⌝ < # nσ, n0 < n2, ⌜n1⌝ < # nσ, n0 < n1, stutf(nσ), ∞.n ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, ∀ m. (∃ m0. ⌜m0⌝ < # nσ ∧ nσ[m0] = m - 1 ∧ (# nσ ≠ ∞ ∨ ¬(∀ m1. m0 < m1 → nσ[m1] = m - 1))) → ¬ 2 + n < m, ∀ n0. ∃ n1. n0 < n1 ∧ ⌜n1⌝ < # nσ ∧ nσ[n1] = n + 1 ⊦

Rule argument

Additional information

Used simplifier rules: stutf-monle: ⊦ stutf(nσ) → (mon≤(nσ) ↔ true); ⊦ m ≤ m; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ ¬ n < n; (s) ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ m + 1 ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic eqle-one: ⊦ m = n + 1 ∧ n < n0 → m ≤ n0; (forward) from specification nat ⊦ m < n → (n ≤ m + 1 ↔ n = m + 1); (s) fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat le-one: ⊦ m ≤ n + 1 ∧ n < n0 → m ≤ n0; (forward) from specification nat ⊦ m < n → m ≤ n; (s)

Proof step 71 from proof inv-ls-succ (previous step) (back to tree)

User applied rule apply lemma

And obtained the following new goals

• Proof step 72:

  ⊦ mon≤(nσ) ∧ m ≤ n ∧ ⌜n⌝ < # nσ → nσ[m] ≤ nσ[n]

• Proof step 73:

  nσ[n0] = n + 1, nσ[n1] = n + 1, nσ[n2] ≠ n + 1, n0 < n1, ⌜n1⌝ < # nσ, n0 < n2, ⌜n0⌝ < # nσ, stutf(nσ), ∞.n ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, mon≤(nσ) ∧ n0 ≤ n2 ∧ ⌜n2⌝ < # nσ → nσ[n0] ≤ nσ[n2], ∀ n0. ∃ n1. n0 < n1 ∧ ⌜n1⌝ < # nσ ∧ nσ[n1] = n + 1, ∀ m. (∃ m0. ⌜m0⌝ < # nσ ∧ nσ[m0] = m - 1 ∧ (# nσ ≠ ∞ ∨ ¬(∀ m1. m0 < m1 → nσ[m1] = m - 1))) → ¬ 2 + n < m ⊦

Rule argument

Additional information

Used terms: nσ, n0, n2

Proof step 72 from proof inv-ls-succ (previous step) (back to tree)

Lemma monle-leq

Proof step 73 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 74:

  nσ[n0] = n + 1, nσ[n1] = n + 1, nσ[n2] ≠ n + 1, n0 < n1, ⌜n1⌝ < # nσ, n0 < n2, ⌜n0⌝ < # nσ, stutf(nσ), ∞.n ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, ⌜n2⌝ < # nσ → n < nσ[n2], ∀ n0. ∃ n1. n0 < n1 ∧ ⌜n1⌝ < # nσ ∧ nσ[n1] = n + 1, ∀ m. (∃ m0. ⌜m0⌝ < # nσ ∧ nσ[m0] = m - 1 ∧ (# nσ ≠ ∞ ∨ ¬(∀ m1. m0 < m1 → nσ[m1] = m - 1))) → ¬ 2 + n < m ⊦

Rule argument

Additional information

Used simplifier rules: stutf-monle: ⊦ stutf(nσ) → (mon≤(nσ) ↔ true); 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) ⊦ m < n → m ≤ n; (s)

Proof step 74 from proof inv-ls-succ (previous step) (back to tree)

User applied rule cut formula

And obtained the following new goals

• Proof step 75:

  nσ[n1] = n + 1, nσ[n0] = n + 1, nσ[n2] ≠ n + 1, ⌜n0⌝ < # nσ, n0 < n2, ⌜n1⌝ < # nσ, n0 < n1, ⌜n2⌝ < # nσ, stutf(nσ), ∞.n ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, ⌜n2⌝ < # nσ → n < nσ[n2], ∀ m. (∃ m0. ⌜m0⌝ < # nσ ∧ nσ[m0] = m - 1 ∧ (# nσ ≠ ∞ ∨ ¬(∀ m1. m0 < m1 → nσ[m1] = m - 1))) → ¬ 2 + n < m, ∀ n0. ∃ n1. n0 < n1 ∧ ⌜n1⌝ < # nσ ∧ nσ[n1] = n + 1 ⊦

• Proof step 80:

  nσ[n1] = n + 1, nσ[n0] = n + 1, nσ[n2] ≠ n + 1, ⌜n0⌝ < # nσ, n0 < n2, ⌜n1⌝ < # nσ, n0 < n1, stutf(nσ), ∞.n ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, ¬ ⌜n2⌝ < # nσ, ⌜n2⌝ < # nσ → n < nσ[n2], ∀ m. (∃ m0. ⌜m0⌝ < # nσ ∧ nσ[m0] = m - 1 ∧ (# nσ ≠ ∞ ∨ ¬(∀ m1. m0 < m1 → nσ[m1] = m - 1))) → ¬ 2 + n < m, ∀ n0. ∃ n1. n0 < n1 ∧ ⌜n1⌝ < # nσ ∧ nσ[n1] = n + 1 ⊦

Rule argument

Additional information

Used formulas: ⌜n2⌝ < # nσ

Proof step 75 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 76:

  nσ[n0] = n + 1, nσ[n1] = n + 1, nσ[n2] ≠ n + 1, ⌜n2⌝ < # nσ, n0 < n1, ⌜n1⌝ < # nσ, n0 < n2, ⌜n0⌝ < # nσ, n + 1 < nσ[n2], n < nσ[n2], stutf(nσ), ∞.n ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, ∀ n0. ∃ n1. n0 < n1 ∧ ⌜n1⌝ < # nσ ∧ nσ[n1] = n + 1, ∀ m. (∃ m0. ⌜m0⌝ < # nσ ∧ nσ[m0] = m - 1 ∧ (# nσ ≠ ∞ ∨ ¬(∀ m1. m0 < m1 → nσ[m1] = m - 1))) → ¬ 2 + n < m ⊦

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 + 1 ≤ n ↔ m < n; (s) ⊦ m < n → n ≠ 0; (s) succ-03: ⊦ m < n ∧ m + 1 ≠ n → m + 1 < n; (forward) from specification nat-basic f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic eqle-one: ⊦ m = n + 1 ∧ n < n0 → m ≤ n0; (forward) from specification nat

Proof step 76 from proof inv-ls-succ (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 77:

  nσ[n1] = n + 1, nσ[n0] = n + 1, nσ[n2] ≠ n + 1, n < nσ[n2], n + 1 < nσ[n2], ⌜n0⌝ < # nσ, n0 < n2, ⌜n1⌝ < # nσ, n0 < n1, ⌜n2⌝ < # nσ, stutf(nσ), ∞.n ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, ∀ m. (∃ m0. ⌜m0⌝ < # nσ ∧ nσ[m0] = m - 1 ∧ (# nσ ≠ ∞ ∨ ¬(∀ m1. m0 < m1 → nσ[m1] = m - 1))) → ¬ 2 + n < m, ∃ n1. n2 < n1 ∧ ⌜n1⌝ < # nσ ∧ nσ[n1] = n + 1 ⊦

Rule argument

Additional information

Used terms: n2

Proof step 77 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 78:

  nσ[n3] = n + 1, nσ[n0] = n + 1, nσ[n1] = n + 1, nσ[n2] ≠ n + 1, n0 + 1 < n3, n2 < n3, ⌜n3⌝ < # nσ, ⌜n2⌝ < # nσ, n0 < n1, ⌜n1⌝ < # nσ, n0 < n2, ⌜n0⌝ < # nσ, n + 1 < nσ[n2], n < nσ[n2], n0 < n3, stutf(nσ), ∞.n ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, ∀ m. (∃ m0. ⌜m0⌝ < # nσ ∧ nσ[m0] = m - 1 ∧ (# nσ ≠ ∞ ∨ ¬(∀ m1. m0 < m1 → nσ[m1] = m - 1))) → ¬ 2 + n < m ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → m ≤ n; (s) ⊦ m +1 = m + 1; (s) ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic trans-nat: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification nat-basic lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (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 eqle-one: ⊦ m = n + 1 ∧ n < n0 → m ≤ n0; (forward) from specification nat

Proof step 78 from proof inv-ls-succ (previous step) (back to tree)

User applied rule apply lemma

And obtained the following new goals

• Proof step 79:

  ⊦ mon≤(nσ) ∧ m ≤ n ∧ ⌜n⌝ < # nσ → nσ[m] ≤ nσ[n]

Rule argument

Additional information

Used simplifier rules: stutf-monle: ⊦ stutf(nσ) → (mon≤(nσ) ↔ true); le-one: ⊦ m ≤ n + 1 ∧ n < n0 → m ≤ n0; (forward) from specification nat a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat ⊦ m < n → m ≤ n; (s) ⊦ ¬ n < n; (s) Used terms: nσ, n2, n3

Proof step 79 from proof inv-ls-succ (previous step) (back to tree)

Lemma monle-leq

Proof step 80 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 81:

  nσ[n0] = n + 1, nσ[n1] = n + 1, nσ[n2] ≠ n + 1, n1 < n2, n0 < n1, ⌜n1⌝ < # nσ, n0 < n2, ⌜n0⌝ < # nσ, n0 + 1 < n2, stutf(nσ), ∞.n ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, ¬ ⌜n2⌝ < # nσ, ∀ n0. ∃ n1. n0 < n1 ∧ ⌜n1⌝ < # nσ ∧ nσ[n1] = n + 1, ∀ m. (∃ m0. ⌜m0⌝ < # nσ ∧ nσ[m0] = m - 1 ∧ (# nσ ≠ ∞ ∨ ¬(∀ m1. m0 < m1 → nσ[m1] = m - 1))) → ¬ 2 + n < m ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (s) trans-nat: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification nat-basic lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) lsle-trans: ⊦ n∞ < n∞1 ∧ ¬ n∞2 < n∞1 → n∞ < n∞2; (forward) from specification nat-inf

Proof step 81 from proof inv-ls-succ (previous step) (back to tree)

User applied rule all left

And closed the goal!

Rule argument

Additional information

Used simplifier rules: lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic trans-nat: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification nat-basic lsle-trans: ⊦ n∞ < n∞1 ∧ ¬ n∞2 < n∞1 → n∞ < n∞2; (forward) from specification nat-inf f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ m < n → n ≠ 0; (s) ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) ⊦ m < n → ¬ n < m; (s) Used terms: n2

Proof step 82 from proof inv-ls-succ (previous step) (back to tree)

User applied rule case distinction

And obtained the following new goals

• Proof step 83:

  # nσ ≠ ∞, n < nσ[n0], ⌜n0⌝ < # nσ, stutf(nσ), ∞.n ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, ∀ n1. ¬ ⌜n1⌝ < # nσ ∨ nσ[n1] ≠ n + 1, ∀ m. (∃ m0. ⌜m0⌝ < # nσ ∧ nσ[m0] = m - 1 ∧ (# nσ ≠ ∞ ∨ ¬(∀ m1. m0 < m1 → nσ[m1] = m - 1))) → ¬ nσ[n0] + 1 < m ⊦

• Proof step 94:

  nσ[n2] ≠ nσ[n0], n0 < n2, n < nσ[n0], ⌜n0⌝ < # nσ, stutf(nσ), ∞.n ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, ∀ n1. ¬ ⌜n1⌝ < # nσ ∨ nσ[n1] ≠ n + 1, ∀ m. (∃ m0. ⌜m0⌝ < # nσ ∧ nσ[m0] = m - 1 ∧ (# nσ ≠ ∞ ∨ ¬(∀ m1. m0 < m1 → nσ[m1] = m - 1))) → ¬ nσ[n0] + 1 < m ⊦

Rule argument

Additional information

Proof step 83 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 84:

  # nσ ≠ ∞, ⌜n0⌝ < # nσ, n < nσ[n0], stutf(nσ), ∞.n ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, ∀ n1. ¬ ⌜n1⌝ < # nσ ∨ nσ[n1] ≠ n + 1, ∀ m. (∃ m0. ⌜m0⌝ < # nσ ∧ nσ[m0] = m - 1) → ¬ nσ[n0] + 1 < m ⊦

Rule argument

Additional information

Proof step 84 from proof inv-ls-succ (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 85:

  ⊦ n∞ = ∞ ↔ ¬(∃ n. n∞ = ⌜n⌝ )

• Proof step 86:

  n < nσ[n0], ⌜n0⌝ < # nσ, stutf(nσ), ∞.n ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, ∀ m. (∃ m0. ⌜m0⌝ < # nσ ∧ nσ[m0] = m - 1) → ¬ nσ[n0] + 1 < m, ∀ n1. ¬ ⌜n1⌝ < # nσ ∨ nσ[n1] ≠ n + 1, ∃ n. # nσ = ⌜n⌝ ⊦

Rule argument

Additional information

Used terms: # nσ

Proof step 85 from proof inv-ls-succ (previous step) (back to tree)

A spec-lemma

Proof step 86 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 87:

  # nσ = ⌜n1⌝ , n0 < n1, n < nσ[n0], stutf(nσ), ∞.n ≤ max∞(λ n2. n2 < n1 ∧ nσ[n2] = n).n, ∀ m. (∃ m0. m0 < n1 ∧ nσ[m0] = m - 1) → ¬ nσ[n0] + 1 < m, ∀ n1. ¬ ⌜n1⌝ < # nσ ∨ nσ[n1] ≠ n + 1 ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s)

Proof step 87 from proof inv-ls-succ (previous step) (back to tree)

User applied rule apply lemma

And obtained the following new goals

• Proof step 88:

  ⊦ stutf(nσ) ∧ ⌜m⌝ < # nσ ∧ n ≤ nσ[m] → (∃ m0. m0 ≤ m ∧ nσ[m0] = n)

• Proof step 89:

  # nσ = ⌜n1⌝ , n < nσ[n0], n0 < n1, stutf(nσ), ∞.n ≤ max∞(λ n2. n2 < n1 ∧ nσ[n2] = n).n, stutf(nσ) ∧ ⌜n0⌝ < # nσ ∧ n + 1 ≤ nσ[n0] → (∃ m0. m0 ≤ n0 ∧ nσ[m0] = n + 1), ∀ n1. ¬ ⌜n1⌝ < # nσ ∨ nσ[n1] ≠ n + 1, ∀ m. (∃ m0. m0 < n1 ∧ nσ[m0] = m - 1) → ¬ nσ[n0] + 1 < m ⊦

Rule argument

Additional information

Used terms: nσ, n0, n + 1

Proof step 88 from proof inv-ls-succ (previous step) (back to tree)

Lemma stutf-alltaken

Proof step 89 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 90:

  # nσ = ⌜n1⌝ , nσ[n2] = n + 1, n0 < n1, n2 < n1, n < nσ[n0], stutf(nσ), ∞.n ≤ max∞(λ n2. n2 < n1 ∧ nσ[n2] = n).n, n2 ≤ n0, ∀ n1. ¬ ⌜n1⌝ < # nσ ∨ nσ[n1] ≠ n + 1, ∀ m. (∃ m0. m0 < n1 ∧ nσ[m0] = m - 1) → ¬ nσ[n0] + 1 < m ⊦

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 + 1 ≤ n ↔ m < n; (s) fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat eqle-one: ⊦ m = n + 1 ∧ n < n0 → m ≤ n0; (forward) from specification nat ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s)

Proof step 90 from proof inv-ls-succ (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 91:

  # nσ = ⌜n1⌝ , nσ[n2] = n + 1, n < nσ[n0], n2 < n1, n0 < n1, stutf(nσ), n2 ≤ n0, ∞.n ≤ max∞(λ n2. n2 < n1 ∧ nσ[n2] = n).n, (∃ m0. m0 < n1 ∧ nσ[m0] = n1 - 1) → ¬ nσ[n0] + 1 < n1, ∀ n1. ¬ ⌜n1⌝ < # nσ ∨ nσ[n1] ≠ n + 1 ⊦

Rule argument

Additional information

Used terms: n1

Proof step 91 from proof inv-ls-succ (previous step) (back to tree)

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

• Proof step 92:

  # nσ = ⌜n1⌝ , nσ[n2] = n + 1, n1 = n3 + 1, n1 ≠ 0, n0 < n1, n2 < n1, n < nσ[n0], stutf(nσ), ∞.n ≤ max∞(λ n2. n2 < n1 ∧ nσ[n2] = n).n, n2 ≤ n0, (∃ m0. m0 < n1 ∧ nσ[m0] = n3) → ¬ nσ[n0] + 1 < n1, ∀ n1. ¬ ⌜n1⌝ < # nσ ∨ nσ[n1] ≠ n + 1 ⊦

Rule argument

Additional information

Used simplifier rules: stutf-nonempty: ⊦ stutf(nσ) → (nσ = [] ↔ false); ⊦ # σ = ⌜0⌝ ↔ σ = []; (s) n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1; (elim) Used terms: n1, m

Proof step 92 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 93:

  # nσ = ⌜n3⌝ + ⌜1⌝ , nσ[n2] = n + 1, n < nσ[n0], stutf(nσ), n2 ≤ n0, ∞.n ≤ max∞(λ n1. ¬ n3 < n1 ∧ nσ[n1] = n).n, n0 ≤ n3, n2 ≤ n3, (∃ m0. ¬ nσ[m0] < m0 ∧ nσ[m0] = n3) → ¬ nσ[n0] < n3, ∀ n1. n3 < n1 ∨ nσ[n1] ≠ n + 1 ⊦

Rule argument

Additional information

Used simplifier rules: let: ⊦ n ≤ n0 ∧ n0 ≤ n1 → n ≤ n1; (forward) from specification nat 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 < n + 1 ↔ ¬ n < m; (s) ⊦ n∞ = ⌜m⌝ + ⌜n0⌝ → (⌜n⌝ < n∞ ↔ n < m + n0); (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ m + 1 ≠ 0; (s) ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s)

Proof step 93 from proof inv-ls-succ (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used simplifier rules: ⊦ m ≤ n → (n < m ↔ false); (s) Used terms: n2

Proof step 94 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 95:

  nσ[n2] ≠ nσ[n0], n0 < n2, n < nσ[n0], ⌜n0⌝ < # nσ, stutf(nσ), ∞.n ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, ∀ n1. ¬ ⌜n1⌝ < # nσ ∨ nσ[n1] ≠ n + 1, ∀ m. (∃ m0. ⌜m0⌝ < # nσ ∧ nσ[m0] = m - 1 ∧ (# nσ ≠ ∞ ∨ ¬(∀ m1. m0 < m1 → nσ[m1] = m - 1))) → ¬ nσ[n0] + 1 < m ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic

Proof step 95 from proof inv-ls-succ (previous step) (back to tree)

User applied rule apply lemma

And obtained the following new goals

• Proof step 96:

  ⊦ stutf(nσ) ∧ ⌜m⌝ < # nσ ∧ n ≤ nσ[m] → (∃ m0. m0 ≤ m ∧ nσ[m0] = n)

• Proof step 97:

  nσ[n2] ≠ nσ[n0], ⌜n0⌝ < # nσ, n < nσ[n0], n0 < n2, stutf(nσ), ∞.n ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, stutf(nσ) ∧ ⌜n0⌝ < # nσ ∧ n + 1 ≤ nσ[n0] → (∃ m0. m0 ≤ n0 ∧ nσ[m0] = n + 1), ∀ m. (∃ m0. ⌜m0⌝ < # nσ ∧ nσ[m0] = m - 1 ∧ (# nσ ≠ ∞ ∨ ¬(∀ m1. m0 < m1 → nσ[m1] = m - 1))) → ¬ nσ[n0] + 1 < m, ∀ n1. ¬ ⌜n1⌝ < # nσ ∨ nσ[n1] ≠ n + 1 ⊦

Rule argument

Additional information

Used terms: nσ, n0, n + 1

Proof step 96 from proof inv-ls-succ (previous step) (back to tree)

Lemma stutf-alltaken

Proof step 97 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 98:

  nσ[n1] = n + 1, nσ[n2] ≠ nσ[n0], n0 < n2, ⌜n0⌝ < # nσ, n1 < n2, n < nσ[n0], stutf(nσ), ∞.n ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, n1 ≤ n0, ∀ n1. ¬ ⌜n1⌝ < # nσ ∨ nσ[n1] ≠ n + 1, ∀ m. (∃ m0. ⌜m0⌝ < # nσ ∧ nσ[m0] = m - 1 ∧ (# nσ ≠ ∞ ∨ ¬(∀ m1. m0 < m1 → nσ[m1] = m - 1))) → ¬ nσ[n0] + 1 < m ⊦

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 + 1 ≤ n ↔ m < n; (s) fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat eqle-one: ⊦ m = n + 1 ∧ n < n0 → m ≤ n0; (forward) from specification nat

Proof step 98 from proof inv-ls-succ (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used simplifier rules: lsle-trans: ⊦ n∞ < n∞1 ∧ ¬ n∞2 < n∞1 → n∞ < n∞2; (forward) from specification nat-inf ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) ⊦ m ≤ n → (n < m ↔ false); (s) Used terms: n1

Proof step 99 from proof inv-ls-succ (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 100:

  ⊦ n∞ = ∞ ↔ ¬(∃ n. n∞ = ⌜n⌝ )

• Proof step 101:

  ⌜n⌝ + ⌜1⌝ < leninv(nσ), stutf(nσ), (leninv(nσ) = ∞ ⊃ ⌜λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n⌝ ;⌜(λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to leninv(nσ).n⌝ )[n + 1] ≤ (leninv(nσ) = ∞ ⊃ ⌜λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n⌝ ;⌜(λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to leninv(nσ).n⌝ )[n], ∃ n0. max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n + 1) = ⌜n0⌝ ⊦

Rule argument

Additional information

Used terms: max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n + 1)

Proof step 100 from proof inv-ls-succ (previous step) (back to tree)

A spec-lemma

Proof step 101 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 102:

  max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n + 1) = ⌜n0⌝ , ⌜n⌝ + ⌜1⌝ < leninv(nσ), stutf(nσ), (leninv(nσ) = ∞ ⊃ ⌜λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n⌝ ;⌜(λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to leninv(nσ).n⌝ )[n + 1] ≤ (leninv(nσ) = ∞ ⊃ ⌜λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n⌝ ;⌜(λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to leninv(nσ).n⌝ )[n] ⊦

Rule argument

Additional information

Proof step 102 from proof inv-ls-succ (previous step) (back to tree)

User applied rule case distinction

And obtained the following new goals

• Proof step 103:

  leninv(nσ) = ∞, max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n + 1) = ⌜n0⌝ , ⌜n⌝ + ⌜1⌝ < leninv(nσ), stutf(nσ), ⌜λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n⌝ [n + 1] ≤ (leninv(nσ) = ∞ ⊃ ⌜λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n⌝ ;⌜(λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to leninv(nσ).n⌝ )[n] ⊦

• Proof step 127:

  max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n + 1) = ⌜n0⌝ , leninv(nσ) ≠ ∞, ⌜n⌝ + ⌜1⌝ < leninv(nσ), stutf(nσ), ⌜(λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to leninv(nσ).n⌝ [n + 1] ≤ (leninv(nσ) = ∞ ⊃ ⌜λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n⌝ ;⌜(λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to leninv(nσ).n⌝ )[n] ⊦

Rule argument

Additional information

Proof step 103 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 104:

  leninv(nσ) = ∞, max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n + 1) = ⌜n0⌝ , stutf(nσ), n0 ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n ⊦

Rule argument

Additional information

Used simplifier rules: add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ⌜n⌝ ≠ ∞; (s) ⊦ ⌜m⌝ + n∞ = ∞ ↔ n∞ = ∞; (s) ⊦ n∞ < ∞ ↔ n∞ ≠ ∞; (s) ⊦ ⌜n⌝ .n = n; (s) ⊦ ⌜nf⌝ [n] = nf(n); (s)

Proof step 104 from proof inv-ls-succ (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 105:

  ⊦ max∞(np) = ⌜n⌝ ↔ np(n) ∧ (∀ m. np(m) → ¬ n < m)

• Proof step 106:

  nσ[n0] = n + 1, leninv(nσ) = ∞, ⌜n0⌝ < # nσ, stutf(nσ), n0 ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, ∀ m. ⌜m⌝ < # nσ ∧ nσ[m] = n + 1 → ¬ n0 < m ⊦

Rule argument

Additional information

Used terms: n0, λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n + 1

Proof step 105 from proof inv-ls-succ (previous step) (back to tree)

Lemma maxinf-fin

Proof step 106 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 107:

  nσ[n0] = n + 1, leninv(nσ) = ∞, ⌜n0⌝ < # nσ, stutf(nσ), n0 ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, ∀ m. ⌜m⌝ < # nσ ∧ nσ[m] = n + 1 → ¬ n0 < m ⊦

Rule argument

Additional information

Proof step 107 from proof inv-ls-succ (previous step) (back to tree)

User applied rule cut formula

And obtained the following new goals

• Proof step 108:

  nσ[n0] = n + 1, leninv(nσ) = ∞, max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n) = ∞, ⌜n0⌝ < # nσ, stutf(nσ), n0 ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, ∀ m. ⌜m⌝ < # nσ ∧ nσ[m] = n + 1 → ¬ n0 < m ⊦

• Proof step 120:

  nσ[n0] = n + 1, leninv(nσ) = ∞, max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n) ≠ ∞, ⌜n0⌝ < # nσ, stutf(nσ), n0 ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, ∀ m. ⌜m⌝ < # nσ ∧ nσ[m] = n + 1 → ¬ n0 < m ⊦

Rule argument

Additional information

Used formulas: max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n) = ∞

Proof step 108 from proof inv-ls-succ (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 109:

  ⊦ max∞(np) = ∞ ↔ (∀ n. ∃ n0. n < n0 ∧ np(n0)) ∨ (∀ n. ¬ np(n))

• Proof step 110:

  nσ[n0] = n + 1, leninv(nσ) = ∞, ⌜n0⌝ < # nσ, stutf(nσ), n0 ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, ∀ m. ⌜m⌝ < # nσ ∧ nσ[m] = n + 1 → ¬ n0 < m, (∀ n0. ∃ n1. n0 < n1 ∧ ⌜n1⌝ < # nσ ∧ nσ[n1] = n) ∨ (∀ n0. ¬(⌜n0⌝ < # nσ ∧ nσ[n0] = n)) ⊦

Rule argument

Additional information

Used terms: λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n

Proof step 109 from proof inv-ls-succ (previous step) (back to tree)

Lemma maxinf-inf

Proof step 110 from proof inv-ls-succ (previous step) (back to tree)

User applied rule case distinction

And obtained the following new goals

• Proof step 111:

  nσ[n0] = n + 1, leninv(nσ) = ∞, ⌜n0⌝ < # nσ, stutf(nσ), n0 ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, ∀ m. ⌜m⌝ < # nσ ∧ nσ[m] = n + 1 → ¬ n0 < m, ∀ n0. ∃ n1. n0 < n1 ∧ ⌜n1⌝ < # nσ ∧ nσ[n1] = n ⊦

• Proof step 115:

  nσ[n0] = n + 1, leninv(nσ) = ∞, ⌜n0⌝ < # nσ, stutf(nσ), n0 ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, ∀ m. ⌜m⌝ < # nσ ∧ nσ[m] = n + 1 → ¬ n0 < m, ∀ n0. ¬(⌜n0⌝ < # nσ ∧ nσ[n0] = n) ⊦

Rule argument

Additional information

Proof step 111 from proof inv-ls-succ (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 112:

  nσ[n0] = n + 1, leninv(nσ) = ∞, ⌜n0⌝ < # nσ, stutf(nσ), n0 ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, ∀ n0. ∃ n1. n0 < n1 ∧ ⌜n1⌝ < # nσ ∧ nσ[n1] = n, ∀ m. ⌜m⌝ < # nσ ∧ nσ[m] = n + 1 → ¬ n0 < m, ∃ n1. n0 < n1 ∧ ⌜n1⌝ < # nσ ∧ nσ[n1] = n ⊦

Rule argument

Additional information

Used terms: n0

Proof step 112 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 113:

  nσ[n0] = nσ[n1] + 1, leninv(nσ) = ∞, ⌜n0⌝ < # nσ, ⌜n1⌝ < # nσ, n0 < n1, stutf(nσ), n0 ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = nσ[n1]).n, ∀ n0. ∃ n2. n0 < n2 ∧ ⌜n2⌝ < # nσ ∧ nσ[n2] = nσ[n1], ∀ m. ⌜m⌝ < # nσ ∧ nσ[m] = nσ[n0] → ¬ n0 < m ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic

Proof step 113 from proof inv-ls-succ (previous step) (back to tree)

User applied rule apply lemma

And obtained the following new goals

• Proof step 114:

  ⊦ mon≤(nσ) ∧ m ≤ n ∧ ⌜n⌝ < # nσ → nσ[m] ≤ nσ[n]

Rule argument

Additional information

Used simplifier rules: stutf-monle: ⊦ stutf(nσ) → (mon≤(nσ) ↔ true); ⊦ m < n → m ≤ n; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ m = n + 1 → ¬ m ≤ n; (s) Used terms: nσ, n0, n1

Proof step 114 from proof inv-ls-succ (previous step) (back to tree)

Lemma monle-leq

Proof step 115 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 116:

  nσ[n0] = n + 1, leninv(nσ) = ∞, ⌜n0⌝ < # nσ, stutf(nσ), n0 ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, ∀ m. ⌜m⌝ < # nσ ∧ nσ[m] = n + 1 → ¬ n0 < m, ∀ n1. ¬ ⌜n1⌝ < # nσ ∨ nσ[n1] ≠ n ⊦

Rule argument

Additional information

Proof step 116 from proof inv-ls-succ (previous step) (back to tree)

User applied rule apply lemma

And obtained the following new goals

• Proof step 117:

  ⊦ stutf(nσ) ∧ ⌜m⌝ < # nσ ∧ n ≤ nσ[m] → (∃ m0. m0 ≤ m ∧ nσ[m0] = n)

• Proof step 118:

  nσ[n0] = n + 1, leninv(nσ) = ∞, ⌜n0⌝ < # nσ, stutf(nσ), n0 ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n, stutf(nσ) ∧ ⌜n0⌝ < # nσ ∧ n ≤ nσ[n0] → (∃ m0. m0 ≤ n0 ∧ nσ[m0] = n), ∀ n1. ¬ ⌜n1⌝ < # nσ ∨ nσ[n1] ≠ n, ∀ m. ⌜m⌝ < # nσ ∧ nσ[m] = n + 1 → ¬ n0 < m ⊦

Rule argument

Additional information

Used terms: nσ, n0, n

Proof step 117 from proof inv-ls-succ (previous step) (back to tree)

Lemma stutf-alltaken

Proof step 118 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 119:

  nσ[n0] = nσ[n1] + 1, leninv(nσ) = ∞, ⌜n0⌝ < # nσ, stutf(nσ), n1 ≤ n0, n0 ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = nσ[n1]).n, n1 ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = nσ[n1]).n, ∀ m. ⌜m⌝ < # nσ ∧ nσ[m] = nσ[n0] → ¬ n0 < m, ∀ n0. ¬ ⌜n0⌝ < # nσ ∨ nσ[n0] ≠ nσ[n1] ⊦

Rule argument

Additional information

Used simplifier rules: let: ⊦ n ≤ n0 ∧ n0 ≤ n1 → n ≤ n1; (forward) from specification nat 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 119 from proof inv-ls-succ (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used simplifier rules: lsle-trans: ⊦ n∞ < n∞1 ∧ ¬ n∞2 < n∞1 → n∞ < n∞2; (forward) from specification nat-inf ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) ⊦ m ≤ n → (n < m ↔ false); (s) Used terms: n1

Proof step 120 from proof inv-ls-succ (previous step) (back to tree)

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

• Proof step 121:

  nσ[n0] = n + 1, leninv(nσ) = ∞, max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n) = ⌜n1⌝ , max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n) ≠ ∞, ⌜n0⌝ < # nσ, stutf(nσ), n0 ≤ n1, ∀ m. ⌜m⌝ < # nσ ∧ nσ[m] = n + 1 → ¬ n0 < m ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ n∞ ≠ ∞ → (n = n∞.n ↔ n∞ = ⌜n⌝ ); (elim) Used terms: max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n), n1

Proof step 121 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 122:

  nσ[n0] = n + 1, leninv(nσ) = ∞, max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n) = ⌜n1⌝ , ⌜n0⌝ < # nσ, stutf(nσ), n0 ≤ n1, ∀ m. ⌜m⌝ < # nσ ∧ nσ[m] = n + 1 → ¬ n0 < m ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜n⌝ ≠ ∞; (s)

Proof step 122 from proof inv-ls-succ (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 123:

  ⊦ max∞(np) = ⌜n⌝ ↔ np(n) ∧ (∀ m. np(m) → ¬ n < m)

• Proof step 124:

  nσ[n1] = n, nσ[n0] = n + 1, leninv(nσ) = ∞, ⌜n0⌝ < # nσ, ⌜n1⌝ < # nσ, stutf(nσ), n0 ≤ n1, ∀ m. ⌜m⌝ < # nσ ∧ nσ[m] = n + 1 → ¬ n0 < m, ∀ m. ⌜m⌝ < # nσ ∧ nσ[m] = n → ¬ n1 < m ⊦

Rule argument

Additional information

Used terms: n1, λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n

Proof step 123 from proof inv-ls-succ (previous step) (back to tree)

Lemma maxinf-fin

Proof step 124 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 125:

  nσ[n0] = nσ[n1] + 1, leninv(nσ) = ∞, ⌜n1⌝ < # nσ, ⌜n0⌝ < # nσ, stutf(nσ), n0 ≤ n1, ∀ m. ⌜m⌝ < # nσ ∧ nσ[m] = nσ[n1] → ¬ n1 < m, ∀ m. ⌜m⌝ < # nσ ∧ nσ[m] = nσ[n0] → ¬ n0 < m ⊦

Rule argument

Additional information

Proof step 125 from proof inv-ls-succ (previous step) (back to tree)

User applied rule apply lemma

And obtained the following new goals

• Proof step 126:

  ⊦ mon≤(nσ) ∧ m ≤ n ∧ ⌜n⌝ < # nσ → nσ[m] ≤ nσ[n]

Rule argument

Additional information

Used simplifier rules: stutf-monle: ⊦ stutf(nσ) → (mon≤(nσ) ↔ true); a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ m = n + 1 → ¬ m ≤ n; (s) Used terms: nσ, n0, n1

Proof step 126 from proof inv-ls-succ (previous step) (back to tree)

Lemma monle-leq

Proof step 127 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 128:

  max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n + 1) = ⌜n0⌝ , leninv(nσ) ≠ ∞, ⌜n⌝ + ⌜1⌝ < leninv(nσ), stutf(nσ), ((λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to leninv(nσ).n)[n + 1] ≤ ((λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to leninv(nσ).n)[n] ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜x⌝ [n] = x[n]; (s)

Proof step 128 from proof inv-ls-succ (previous step) (back to tree)

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

• Proof step 129:

  leninv(nσ) = ⌜n1⌝ , max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n + 1) = ⌜n0⌝ , leninv(nσ) ≠ ∞, ⌜n⌝ + ⌜1⌝ < leninv(nσ), stutf(nσ), ((λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to n1)[n + 1] ≤ ((λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to n1)[n] ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ n∞ ≠ ∞ → (n = n∞.n ↔ n∞ = ⌜n⌝ ); (elim) Used terms: leninv(nσ), n1

Proof step 129 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 130:

  leninv(nσ) = ⌜n1⌝ , max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n + 1) = ⌜n0⌝ , n + 1 < n1, stutf(nσ), n0 ≤ ((λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to n1)[n] ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜n⌝ .n = n; (s) ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic m < n ⊦ (nf to n)[m] = nf(m); (s) ⊦ m ≤ n → (n < m ↔ false); (s) ⊦ m < n ↔ ¬ n ≤ m; (ss) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ⌜n⌝ + ⌜m⌝ < ⌜n0⌝ ↔ n + m < n0; (s) ⊦ ⌜n⌝ ≠ ∞; (s)

Proof step 130 from proof inv-ls-succ (previous step) (back to tree)

User applied rule cut formula

And obtained the following new goals

• Proof step 131:

  leninv(nσ) = ⌜n1⌝ , max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n + 1) = ⌜n0⌝ , max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n) = ∞, n + 1 < n1, stutf(nσ), n0 ≤ ((λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to n1)[n] ⊦

• Proof step 147:

  leninv(nσ) = ⌜n1⌝ , max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n + 1) = ⌜n0⌝ , max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n) ≠ ∞, n + 1 < n1, stutf(nσ), n0 ≤ ((λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to n1)[n] ⊦

Rule argument

Additional information

Used formulas: max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n) = ∞

Proof step 131 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 132:

  leninv(nσ) = ⌜n1⌝ , max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n) = ∞, max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n + 1) = ⌜n0⌝ , n + 1 < n1, stutf(nσ), n0 ≤ ((λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to n1)[n] ⊦

Rule argument

Additional information

Proof step 132 from proof inv-ls-succ (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 133:

  ⊦ max∞(np) = ∞ ↔ (∀ n. ∃ n0. n < n0 ∧ np(n0)) ∨ (∀ n. ¬ np(n))

• Proof step 134:

  leninv(nσ) = ⌜n1⌝ , max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n + 1) = ⌜n0⌝ , n + 1 < n1, stutf(nσ), n0 ≤ ((λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to n1)[n], (∀ n0. ∃ n1. n0 < n1 ∧ ⌜n1⌝ < # nσ ∧ nσ[n1] = n) ∨ (∀ n0. ¬(⌜n0⌝ < # nσ ∧ nσ[n0] = n)) ⊦

Rule argument

Additional information

Used terms: λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n

Proof step 133 from proof inv-ls-succ (previous step) (back to tree)

Lemma maxinf-inf

Proof step 134 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 135:

  leninv(nσ) = ⌜n1⌝ , max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n + 1) = ⌜n0⌝ , n + 1 < n1, stutf(nσ), n0 ≤ ((λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to n1)[n], (∀ n0. ∃ n1. n0 < n1 ∧ ⌜n1⌝ < # nσ ∧ nσ[n1] = n) ∨ (∀ n2. ¬ ⌜n2⌝ < # nσ ∨ nσ[n2] ≠ n) ⊦

Rule argument

Additional information

Proof step 135 from proof inv-ls-succ (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 136:

  ⊦ max∞(np) = ⌜n⌝ ↔ np(n) ∧ (∀ m. np(m) → ¬ n < m)

• Proof step 137:

  nσ[n0] = n + 1, leninv(nσ) = ⌜n1⌝ , n + 1 < n1, ⌜n0⌝ < # nσ, stutf(nσ), n0 ≤ ((λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to n1)[n], ∀ m. ⌜m⌝ < # nσ ∧ nσ[m] = n + 1 → ¬ n0 < m, (∀ n0. ∃ n1. n0 < n1 ∧ ⌜n1⌝ < # nσ ∧ nσ[n1] = n) ∨ (∀ n2. ¬ ⌜n2⌝ < # nσ ∨ nσ[n2] ≠ n) ⊦

Rule argument

Additional information

Used terms: n0, λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n + 1

Proof step 136 from proof inv-ls-succ (previous step) (back to tree)

Lemma maxinf-fin

Proof step 137 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 138:

  nσ[n0] = n + 1, leninv(nσ) = ⌜n1⌝ , ⌜n0⌝ < # nσ, n + 1 < n1, stutf(nσ), n0 ≤ ((λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to n1)[n], ∀ m. ⌜m⌝ < # nσ ∧ nσ[m] = n + 1 → ¬ n0 < m, (∀ n0. ∃ n1. n0 < n1 ∧ ⌜n1⌝ < # nσ ∧ nσ[n1] = n) ∨ (∀ n2. ¬ ⌜n2⌝ < # nσ ∨ nσ[n2] ≠ n) ⊦

Rule argument

Additional information

Proof step 138 from proof inv-ls-succ (previous step) (back to tree)

User applied rule case distinction

And obtained the following new goals

• Proof step 139:

  nσ[n0] = n + 1, leninv(nσ) = ⌜n1⌝ , n + 1 < n1, ⌜n0⌝ < # nσ, stutf(nσ), n0 ≤ ((λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to n1)[n], ∀ m. ⌜m⌝ < # nσ ∧ nσ[m] = n + 1 → ¬ n0 < m, ∀ n0. ∃ n1. n0 < n1 ∧ ⌜n1⌝ < # nσ ∧ nσ[n1] = n ⊦

• Proof step 143:

  nσ[n0] = n + 1, leninv(nσ) = ⌜n1⌝ , n + 1 < n1, ⌜n0⌝ < # nσ, stutf(nσ), n0 ≤ ((λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to n1)[n], ∀ m. ⌜m⌝ < # nσ ∧ nσ[m] = n + 1 → ¬ n0 < m, ∀ n2. ¬ ⌜n2⌝ < # nσ ∨ nσ[n2] ≠ n ⊦

Rule argument

Additional information

Proof step 139 from proof inv-ls-succ (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 140:

  nσ[n0] = n + 1, leninv(nσ) = ⌜n1⌝ , ⌜n0⌝ < # nσ, n + 1 < n1, stutf(nσ), n0 ≤ ((λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to n1)[n], ∀ n0. ∃ n1. n0 < n1 ∧ ⌜n1⌝ < # nσ ∧ nσ[n1] = n, ∀ m. ⌜m⌝ < # nσ ∧ nσ[m] = n + 1 → ¬ n0 < m, ∃ n1. n0 < n1 ∧ ⌜n1⌝ < # nσ ∧ nσ[n1] = n ⊦

Rule argument

Additional information

Used terms: n0

Proof step 140 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 141:

  nσ[n0] = nσ[n2] + 1, leninv(nσ) = ⌜n1⌝ , ⌜n0⌝ < # nσ, nσ[n0] < n1, ⌜n2⌝ < # nσ, n0 < n2, stutf(nσ), n0 ≤ ((λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to n1)[nσ[n2]], ∀ n0. ∃ n1. n0 < n1 ∧ ⌜n1⌝ < # nσ ∧ nσ[n1] = nσ[n2], ∀ m. ⌜m⌝ < # nσ ∧ nσ[m] = nσ[n0] → ¬ n0 < m ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic

Proof step 141 from proof inv-ls-succ (previous step) (back to tree)

User applied rule apply lemma

And obtained the following new goals

• Proof step 142:

  ⊦ mon≤(nσ) ∧ m ≤ n ∧ ⌜n⌝ < # nσ → nσ[m] ≤ nσ[n]

Rule argument

Additional information

Used simplifier rules: stutf-monle: ⊦ stutf(nσ) → (mon≤(nσ) ↔ true); ⊦ m < n → m ≤ n; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ m = n + 1 → ¬ m ≤ n; (s) Used terms: nσ, n0, n2

Proof step 142 from proof inv-ls-succ (previous step) (back to tree)

Lemma monle-leq

Proof step 143 from proof inv-ls-succ (previous step) (back to tree)

User applied rule apply lemma

And obtained the following new goals

• Proof step 144:

  ⊦ stutf(nσ) ∧ ⌜m⌝ < # nσ ∧ n ≤ nσ[m] → (∃ m0. m0 ≤ m ∧ nσ[m0] = n)

• Proof step 145:

  nσ[n0] = n + 1, leninv(nσ) = ⌜n1⌝ , ⌜n0⌝ < # nσ, n + 1 < n1, stutf(nσ), n0 ≤ ((λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to n1)[n], stutf(nσ) ∧ ⌜n0⌝ < # nσ ∧ n ≤ nσ[n0] → (∃ m0. m0 ≤ n0 ∧ nσ[m0] = n), ∀ n2. ¬ ⌜n2⌝ < # nσ ∨ nσ[n2] ≠ n, ∀ m. ⌜m⌝ < # nσ ∧ nσ[m] = n + 1 → ¬ n0 < m ⊦

Rule argument

Additional information

Used terms: nσ, n0, n

Proof step 144 from proof inv-ls-succ (previous step) (back to tree)

Lemma stutf-alltaken

Proof step 145 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 146:

  nσ[n0] = nσ[n2] + 1, leninv(nσ) = ⌜n1⌝ , ⌜n0⌝ < # nσ, nσ[n0] < n1, stutf(nσ), n2 ≤ ((λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to n1)[nσ[n2]], n0 ≤ ((λ m. max∞(λ n. ⌜n⌝ < # nσ ∧ nσ[n] = m).n) to n1)[nσ[n2]], n2 ≤ n0, ∀ n0. ¬ ⌜n0⌝ < # nσ ∨ nσ[n0] ≠ nσ[n2], ∀ m. ⌜m⌝ < # nσ ∧ nσ[m] = nσ[n0] → ¬ n0 < m ⊦

Rule argument

Additional information

Used simplifier rules: let: ⊦ n ≤ n0 ∧ n0 ≤ n1 → n ≤ n1; (forward) from specification nat lf-02: ⊦ m < n ∧ m = n0 → n0 < 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 ⊦ m ≤ m + n; (s)

Proof step 146 from proof inv-ls-succ (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used simplifier rules: lsle-trans: ⊦ n∞ < n∞1 ∧ ¬ n∞2 < n∞1 → n∞ < n∞2; (forward) from specification nat-inf ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) ⊦ m ≤ n → (n < m ↔ false); (s) Used terms: n2

Proof step 147 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 148:

  leninv(nσ) = ⌜n1⌝ , max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n + 1) = ⌜n0⌝ , max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n) ≠ ∞, n + 1 < n1, stutf(nσ), n0 ≤ max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n).n ⊦

Rule argument

Additional information

Used simplifier rules: m < n ⊦ (nf to n)[m] = nf(m); (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ ¬ m + n < m; (s) fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat ⊦ m < n ↔ ¬ n ≤ m; (ss)

Proof step 148 from proof inv-ls-succ (previous step) (back to tree)

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

• Proof step 149:

  leninv(nσ) = ⌜n1⌝ , max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n + 1) = ⌜n0⌝ , max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n) = ⌜n2⌝ , max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n) ≠ ∞, n + 1 < n1, stutf(nσ), n0 ≤ n2 ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ n∞ ≠ ∞ → (n = n∞.n ↔ n∞ = ⌜n⌝ ); (elim) Used terms: max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n), n2

Proof step 149 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 150:

  leninv(nσ) = ⌜n1⌝ , max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n + 1) = ⌜n0⌝ , max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n) = ⌜n2⌝ , n + 1 < n1, stutf(nσ), n0 ≤ n2 ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜n⌝ ≠ ∞; (s)

Proof step 150 from proof inv-ls-succ (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 151:

  ⊦ max∞(np) = ⌜n⌝ ↔ np(n) ∧ (∀ m. np(m) → ¬ n < m)

• Proof step 152:

  nσ[n0] = n + 1, leninv(nσ) = ⌜n1⌝ , max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n) = ⌜n2⌝ , n + 1 < n1, ⌜n0⌝ < # nσ, stutf(nσ), n0 ≤ n2, ∀ m. ⌜m⌝ < # nσ ∧ nσ[m] = n + 1 → ¬ n0 < m ⊦

Rule argument

Additional information

Used terms: n0, λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n + 1

Proof step 151 from proof inv-ls-succ (previous step) (back to tree)

Lemma maxinf-fin

Proof step 152 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 153:

  nσ[n0] = n + 1, leninv(nσ) = ⌜n1⌝ , max∞(λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n) = ⌜n2⌝ , ⌜n0⌝ < # nσ, n + 1 < n1, stutf(nσ), n0 ≤ n2, ∀ m. ⌜m⌝ < # nσ ∧ nσ[m] = n + 1 → ¬ n0 < m ⊦

Rule argument

Additional information

Proof step 153 from proof inv-ls-succ (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 154:

  ⊦ max∞(np) = ⌜n⌝ ↔ np(n) ∧ (∀ m. np(m) → ¬ n < m)

• Proof step 155:

  nσ[n2] = n, nσ[n0] = n + 1, leninv(nσ) = ⌜n1⌝ , n + 1 < n1, ⌜n0⌝ < # nσ, ⌜n2⌝ < # nσ, stutf(nσ), n0 ≤ n2, ∀ m. ⌜m⌝ < # nσ ∧ nσ[m] = n + 1 → ¬ n0 < m, ∀ m. ⌜m⌝ < # nσ ∧ nσ[m] = n → ¬ n2 < m ⊦

Rule argument

Additional information

Used terms: n2, λ n0. ⌜n0⌝ < # nσ ∧ nσ[n0] = n

Proof step 154 from proof inv-ls-succ (previous step) (back to tree)

Lemma maxinf-fin

Proof step 155 from proof inv-ls-succ (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 156:

  nσ[n0] = nσ[n2] + 1, leninv(nσ) = ⌜n1⌝ , nσ[n0] < n1, ⌜n0⌝ < # nσ, ⌜n2⌝ < # nσ, stutf(nσ), n0 ≤ n2, ∀ m. ⌜m⌝ < # nσ ∧ nσ[m] = nσ[n0] → ¬ n0 < m, ∀ m. ⌜m⌝ < # nσ ∧ nσ[m] = nσ[n2] → ¬ n2 < m ⊦

Rule argument

Additional information

Used simplifier rules: lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic

Proof step 156 from proof inv-ls-succ (previous step) (back to tree)

User applied rule apply lemma

And obtained the following new goals

• Proof step 157:

  ⊦ mon≤(nσ) ∧ m ≤ n ∧ ⌜n⌝ < # nσ → nσ[m] ≤ nσ[n]

Rule argument

Additional information

Used simplifier rules: stutf-monle: ⊦ stutf(nσ) → (mon≤(nσ) ↔ true); a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ m = n + 1 → ¬ m ≤ n; (s) Used terms: nσ, n0, n2

Proof step 157 from proof inv-ls-succ (previous step) (back to tree)

Lemma monle-leq