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Proof step 75 from proof diag-theorem-butlast (previous step)

# nextl(0, a) = ii(1) + 1, ii(n2) + # lf(n2 + 1) = ii(n2 + 1) + 1, (σ from ii(n2))[0] = lf(n2).last, ii(0) = 0, ii(n0 + 1) = ii(n0) + # lf(n0 + 1) - 1, lf(n0 + 1) = nextl(n0, lf(n0).last), lf(0) = a ', lf(1) = nextl(0, a), lf(n2 + 1) = nextl(n2, lf(n2).last), lf(n2 + 1) = lf(n2).last ' + y0, nextl(0, a) = a ' + y, σ to ii(1) + 1 = ⌜nextl(0, a)⌝ , (σ from ii(n0)) to # lf(n0 + 1) = ⌜lf(n0 + 1)⌝ , ⌜ii(n2 + 1)⌝ < # σ, ⌜ii(1)⌝ < # σ, ⌜ii(n0 + 1)⌝ < # σ, a ∈ nextl(0, a), lf(n2).last ∈ lf(n2 + 1), (finp(σ) ⊃ # σ = ⌜ii(n)⌝ + ⌜1⌝ ∧ (∀ m. n ≤ m → lf(m + 1).tail = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n, a. nextl(n, a) ≠ [] ∧ nextl(n, a).head = a, ∀ n. lf(n + 1) = nextl(n, lf(n).last) ∧ lf(n) ≠ [], ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # lf(n + 1) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ (σ from ii(n)) to # lf(n + 1) = ⌜lf(n + 1)⌝ ⊦ σ[ii(n2)] = lf(n2).last, lf(n2 + 1) = [], lf(n0) = [], lf(n2) = [], nextl(0, a) = [], σ = [], #(σ from ii(n2)) < ⌜# lf(n2 + 1)⌝ , (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ + ⌜1⌝ ∧ (∀ m. n ≤ m → nextl(m, σ[ii(m)]) = σ[ii(m)] ' ∧ ii(n) = ii(m));∃ m. ii(m) ≤ n1 ∧ n1 < ii(m + 1)) ∧ (σ from ii(n0)) to # nextl(n0, σ[ii(n0)]) = ⌜nextl(n0, σ[ii(n0)])⌝ ∧ ii(n0 + 1) = ii(n0) + # nextl(n0, σ[ii(n0)]) - 1 ∧ σ[0] = a

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: stream name: get-from seq: ⊦ ⌜n⌝ + ⌜m⌝ < # σ → (σ from n)[m] = σ[n + m] subst: [m, n, σ] → [0, ii(n2), σ] flags: <all paths>

Additional information

Used simplifier rules: ⊦ n + 0 = n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ n∞ + ⌜0⌝ = n∞; (s)%add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf%add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf% ⊦ ¬ finp(σ) → # σ = ∞; (s)% ⊦ ⌜m⌝ < ∞; (s)% ⊦ n∞ = ⌜m⌝ + ⌜n0⌝ → (⌜n⌝ < n∞ ↔ n < m + n0); (s)% ⊦ m < n + 1 ↔ ¬ n < m; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ ⌜n⌝ + n∞1 = ∞ ↔ n∞1 = ∞; (s)% ⊦ ⌜m⌝ ≠ ∞; (s)%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic% ⊦ m < n → n ≠ 0; (s)%eqle-one: ⊦ m = n + 1 ∧ n < n0 → m ≤ n0; (forward) from specification nat%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic% ⊦ m + m0 ≤ m ↔ m0 = 0; (s)% ⊦ # x = 0 ↔ x = []; (s) Used terms: 0, ii(n2), σ