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Proof step 71 from proof infinite-append-prefixnextli-weak (previous step)

# lf(n0) = ii(n0), #((σ from ii(n0)) to # nextli0(lf(n0))) = ⌜# nextli0(lf(n0))⌝ , # σ = ⌜ii(n)⌝ , ii(0) = 0, lf(0) = [], n3 < # nextli0(lf(n0)), finp(σ), ∀ n1. n1 < # nextli0(lf(n0)) → ((σ from ii(n0)) to # nextli0(lf(n0)))[n1] = nextli0(lf(n0))[n1], ∀ n. ii(n + 1) = ii(n) + # nextli0(lf(n)) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # nextli0(lf(n)) = ⌜nextli0(lf(n))⌝ , ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. lf(n + 1) = nextli(lf(n)), ∀ x. nextli(x) = x + nextli0(x), ∀ n2. n2 < ii(n0) → lf(n0)[n2] = σ[n2], ∀ m. n ≤ m → nextli0(lf(m)) = [] ∧ ii(m) = ii(n) ⊦ nextli0(lf(n0))[n3] = σ[n3 + ii(n0)], nextli0(lf(n0)) = [], # σ < ⌜ii(n0)⌝ + ⌜# nextli0(lf(n0))⌝

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: m + m0 < # x ⊦ (x from m)[m0] = x[m + m0]; (s)% ⊦ m < n → (n ≤ m ↔ false); (s)% ⊦ m + n ≤ m + n0 ↔ n ≤ n0; (s)%let: ⊦ n ≤ n0 ∧ n0 ≤ n1 → n ≤ n1; (forward) from specification nat%¬ # x < n, m < n ⊦ (x to n)[m] = x[m]; (s)% ⊦ ⌜x⌝ = ⌜x0⌝ ↔ x = x0; (s)% ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s)%¬ # x < n ⊦ #(x to n) = n; (s)% ⊦ m ≤ n → (n < m ↔ false); (s)%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat%n0 ≤ n ⊦ n - n0 < m ↔ n < m + n0; (s)% ⊦ m ≤ n ↔ ¬ n < m; (ss)%¬ # x < n ⊦ #(x from n) = # x - n; (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)% ⊦ m < n → n ≠ 0; (s)%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (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% ⊦ finp(⌜x⌝ ); (s)% ⊦ ⌜x⌝ [n] = x[n]; (s)% ⊦ ⌜x⌝ to n = ⌜x to n⌝ ; (s)% ⊦ ⌜x⌝ from n = ⌜x from n⌝ ; (s)% ⊦ ⌜m⌝ = ⌜m0⌝ ↔ m = m0; (s)% ⊦ # ⌜x⌝ = ⌜# x⌝ ; (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)% ⊦ ⌜n⌝ < ⌜m⌝ + ⌜n0⌝ ↔ n < m + n0; (s)%stream-fin-forward: ⊦ finp(σ) → (∃ x. ⌜x⌝ = σ); (forward) from specification stream-basic