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Proof step 175 from proof infinite-append-prefixnextli (previous step)

ii(0) = 0, lf(0) = [], # lf(n0) = # (σ to ii(n0)).l ∧ (∀ n. n < # lf(n0) → lf(n0)[n] = (σ to ii(n0)).l[n]), lf(n0 + 1) = lf(n0) + nextli0(lf(n0)), (σ from ii(n0)) to # nextli0(lf(n0)) = ⌜nextli0(lf(n0))⌝ , ∀ 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), (finp(σ) ⊃ # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → nextli0(lf(m)) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)) ⊦ # σ < ⌜ii(n0)⌝ + ⌜# nextli0(lf(n0))⌝ , # lf(n0) = ii(n0) ∧ (n1 < # lf(n0) + # nextli0(lf(n0)) → lf(n0 + 1)[n1] = σ[n1])

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 176:
lf(n0 + 1) = lf(n0) + nextli0(lf(n0)), (σ from ii(n0)) to # nextli0(lf(n0)) = ⌜nextli0(lf(n0))⌝ , ii(0) = 0, lf(0) = [], # lf(n0) = ii(n0), n1 < ii(n0) + # nextli0(lf(n0)), ∀ 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), (finp(σ) ⊃ # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → nextli0(lf(m)) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n2. n2 < ii(n0) → lf(n0)[n2] = σ[n2] ⊦ lf(n0 + 1)[n1] = σ[n1], ii(n0) + # nextli0(lf(n0)) = 0, # σ < ⌜ii(n0)⌝ + ⌜# nextli0(lf(n0))⌝

Rule argument

Additional information

Used simplifier rules: get-to-02: m < n, ¬ # σ < ⌜n⌝ ⊦ (σ to n).l[m] = σ[m]; from specification stream%le-ls-simp: ⊦ m ≤ n → (n < m ↔ false); from specification nat%l: ⊦ # x = 0 ↔ x = []; from specification list-data%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic1%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic1%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat%lels: ⊦ ¬ m < n ↔ n ≤ m; from specification nat%size-to-02: ¬ # σ < ⌜n⌝ ⊦ # (σ to n).l = n; from specification stream%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%c: ⊦ m + n = n + m; from specification nat-basic2%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic2%l-02: ⊦ ¬ m + n < m; from specification nat-basic2%ls-natinf-to-nat-2-1: ⊦ ⌜n⌝ + ⌜m⌝ < ⌜n0⌝ ↔ n + m < n0; from specification nat-inf%lels-trans: ⊦ ¬ n∞1 < n∞ ∧ n∞1 < n∞2 → n∞ < n∞2; (forward) from specification nat-inf