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

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 72:
# lf(n0) = ii(n0), lf(0) = [], ii(0) = 0, # x = ii(n), # nextli0(lf(n0)) + ii(n0) ≤ ii(n), n3 < # nextli0(lf(n0)), ∀ m. n ≤ m → nextli0(lf(m)) = [] ∧ ii(m) = ii(n), ∀ x. nextli(x) = x + nextli0(x), ∀ n. lf(n + 1) = nextli(lf(n)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n1. n1 < ii(n0) → lf(n0)[n1] = x[n1], ∀ n1. ii(n1 + 1) = ii(n1) + # nextli0(lf(n1)) ∧ ¬ ii(n) < ii(n1 + 1) ∧ (x from ii(n1)) to # nextli0(lf(n1)) = nextli0(lf(n1)), ∀ n1. n1 < # nextli0(lf(n0)) → x[ii(n0) + n1] = nextli0(lf(n0))[n1] ⊦ ii(n0) + # nextli0(lf(n0)) = 0, nextli0(lf(n0)) = [], nextli0(lf(n0))[n3] = x[n3 + ii(n0)]

Rule argument

Additional information

Used simplifier rules: get-from-01: m + m0 < # x ⊦ (x from m)[m0] = x[m + m0]; from specification stream%ls-le-simp: ⊦ m < n → (n ≤ m ↔ false); from specification nat%lep-02: ⊦ m + n ≤ m + n0 ↔ n ≤ n0; from specification nat%let: ⊦ n ≤ n0 ∧ n0 ≤ n1 → n ≤ n1; (forward) from specification nat%get-finto: ¬ # x < n, m < n ⊦ (x to n)[m] = x[m]; from specification stream%inj-02: ⊦ ⌜x⌝ = ⌜x0⌝ ↔ x = x0; from specification stream-basic%less-fin: ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; from specification nat-inf%length-to: ¬ # x < n ⊦ #(x to n) = n; from specification stream%le-ls-simp: ⊦ m ≤ n → (n < m ↔ false); from specification nat%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat%ls-01: n0 ≤ n ⊦ n - n0 < m ↔ n < m + n0; from specification nat%lels-01: ⊦ ¬ m ≤ n ↔ n < m; from specification nat%length-from: ¬ # x < n ⊦ #(x from n) = # x - n; from specification stream%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%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic1%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (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%test-02: ⊦ finp(⌜x⌝ ); from specification stream-basic%sel-fin: ⊦ ⌜x⌝ [n] = x[n]; from specification stream%to-fin: ⊦ ⌜x⌝ to n = ⌜x to n⌝ ; from specification stream%from-fin: ⊦ ⌜x⌝ from n = ⌜x from n⌝ ; from specification stream%inj: ⊦ ⌜m⌝ = ⌜m0⌝ ↔ m = m0; from specification nat-inf-basic%size-fin: ⊦ # ⌜x⌝ = ⌜# x⌝ ; 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%lels: ⊦ ¬ m < n ↔ n ≤ m; from specification nat%ls-natinf-to-nat-1-2: ⊦ ⌜n⌝ < ⌜m⌝ + ⌜n0⌝ ↔ n < m + n0; from specification nat-inf%stream-fin-forward: ⊦ finp(σ) → (∃ x. ⌜x⌝ = σ); (forward) from specification stream-basic