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Proof step 63 from proof infinite-append-butlast (previous step)

((n0 = 0 ⊃ 0;ii(n0) - 1) + # lf(n0) - 1) + 1 = ii(n0 + 1), ii(n0 + 1) = ii(n0) + # (n0 = 0 ⊃ lf(0);lf(n0).tail), ii(0) = 0, (σ from ii(n0)) to # (n0 = 0 ⊃ lf(0);lf(n0).tail) = ⌜(n0 = 0 ⊃ lf(0);lf(n0).tail)⌝ , ⌜(n0 = 0 ⊃ 0;ii(n0) - 1) + # lf(n0) - 1⌝ < # σ, (finp(σ) ⊃ # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → (m = 0 ⊃ lf(0);lf(m).tail) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n. n ≠ 0 → ii(n) ≠ 0, ∀ n. ii(n + 1) = ii(n) + # (n = 0 ⊃ lf(0);lf(n).tail) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # (n = 0 ⊃ lf(0);lf(n).tail) = ⌜(n = 0 ⊃ lf(0);lf(n).tail)⌝ , ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. lf(n).last = lf(n + 1).head ∧ lf(n) ≠ [], ∀ n. σ[ii(n + 1) - 1] = lf(n + 1).head ⊦ finp(σ), ∃ m0. (m0 = 0 ⊃ 0;ii(m0) - 1) ≤ n2 ∧ n2 < ii(m0 + 1) - 1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 64:
((n0 = 0 ⊃ 0;ii(n0) - 1) + # lf(n0) - 1) + 1 = ii(n0 + 1), ii(0) = 0, ii(n0 + 1) = ii(n0) + # (n0 = 0 ⊃ lf(0);lf(n0).tail), (nf from ii(n0)) to # (n0 = 0 ⊃ lf(0);lf(n0).tail) = (n0 = 0 ⊃ lf(0);lf(n0).tail), ∀ n. lf(n).last = lf(n + 1).head ∧ lf(n) ≠ [], ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. n ≠ 0 → ii(n) ≠ 0, ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n. nf(ii(n + 1) - 1) = lf(n + 1).head, ∀ n. ii(n + 1) = ii(n) + # (n = 0 ⊃ lf(0);lf(n).tail) ∧ (nf from ii(n)) to # (n = 0 ⊃ lf(0);lf(n).tail) = (n = 0 ⊃ lf(0);lf(n).tail) ⊦ ∃ m0. (m0 = 0 ⊃ 0;ii(m0) - 1) ≤ n2 ∧ n2 < ii(m0 + 1) - 1

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜nf⌝ [n] = nf(n); (s)% ⊦ ¬ finp(⌜nf⌝ ); (s)% ⊦ ⌜x⌝ = ⌜x0⌝ ↔ x = x0; (s)% ⊦ ⌜nf⌝ to n = ⌜nf to n⌝ ; (s)% ⊦ ⌜nf⌝ from n = ⌜nf from n⌝ ; (s)%stream-inf-forward: ⊦ ¬ finp(σ) → (∃ nf. ⌜nf⌝ = σ); (forward) from specification stream-basic% ⊦ ¬ ∞ < n∞; (s)% ⊦ ⌜m⌝ < ∞; (s)% ⊦ ¬ finp(σ) → # σ = ∞; (s)