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Proof step 35 from proof infinite-append-prefixnextnli (previous step)

# (σ to ii(n0 + 1)).l = # lf(n0 + 1), ii(0) = 0, lf(n0 + 1) = lf(n0) + nextnli0(lf(n0), n0), lf(0) = [], lf(n0) = (σ to ii(n0)).l, ∀ x, n. nextnli(x, n) = x + nextnli0(x, n), ∀ n. lf(n + 1) = nextnli(lf(n), n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # nextnli0(lf(n), n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # nextnli0(lf(n), n) = ⌜nextnli0(lf(n), n)⌝ , ∀ n0. lf(n0) = (σ to ii(n0)).l, ∀ n. n < # lf(n0 + 1) → (σ to ii(n0 + 1)).l[n] = lf(n0 + 1)[n], ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1) ⊦ finp(σ), #(σ to ii(n0 + 1)) = ⌜# lf(n0 + 1)⌝ ∧ (⌜n1⌝ < #(σ to ii(n0 + 1)) → (σ to ii(n0 + 1))[n1] = lf(n0 + 1)[n1])

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 36:
ii(0) = 0, ii(n0 + 1) = # lf(n0 + 1), lf(0) = [], lf(n0 + 1) = lf(n0) + nextnli0(lf(n0), n0), lf(n0) = nf to ii(n0), n1 < ii(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. lf(n + 1) = nextnli(lf(n), n), ∀ x, n. nextnli(x, n) = x + nextnli0(x, n), ∀ n. lf(n) = nf to ii(n), ∀ n. n < ii(n0 + 1) → nf(n) = lf(n0 + 1)[n], ∀ n. ii(n + 1) = ii(n) + # nextnli0(lf(n), n) ∧ (nf from ii(n)) to # nextnli0(lf(n), n) = nextnli0(lf(n), n) ⊦ nf(n1) = lf(n0 + 1)[n1]

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜x⌝ = ⌜x0⌝ ↔ x = x0; (s)% ⊦ ⌜nf⌝ from n = ⌜nf from n⌝ ; (s)% ⊦ ¬ finp(⌜nf⌝ ); (s)% ⊦ ⌜nf⌝ [n] = nf(n); (s)% ⊦ ⌜x⌝ .l = x; (s)% ⊦ ⌜nf⌝ to n = ⌜nf to n⌝ ; (s)% ⊦ m < n → n ≠ 0; (s)%m < n, ¬ # σ < ⌜n⌝ ⊦ (σ to n)[m] = σ[m]; (s)% ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s)% ⊦ ⌜m⌝ = ⌜m0⌝ ↔ m = m0; (s)%¬ # σ < ⌜n⌝ ⊦ #(σ to n) = ⌜n⌝ ; (s)%m < n, ¬ # σ < ⌜n⌝ ⊦ (σ to n).l[m] = σ[m]; (s)% ⊦ m ≤ n → (n < m ↔ false); (s)% ⊦ # x = 0 ↔ x = []; (s)%stream-inf-forward: ⊦ ¬ finp(σ) → (∃ nf. ⌜nf⌝ = σ); (forward) from specification stream-basic%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat% ⊦ m < n ↔ ¬ n ≤ m; (ss)%¬ # σ < ⌜n⌝ ⊦ # (σ to n).l = n; (s)% ⊦ ¬ ∞ < n∞; (s)% ⊦ ¬ finp(σ) → # σ = ∞; (s)