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Proof step 93 from proof lim-exists (previous step)

# σσ[0] = ⌜n⌝ , # σσ = ∞, σσ[m0] = σσ[0] + ⌜nf to ii(m0)⌝ , ii(0) = 0, n1 < ii(m0 + 1) + n, ii(m0) < ii(m0 + 1), n0 < ii(m0 + 1), n1 < n, mon(σσ), ii(m0) ≤ n0, ∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ (nf from ii(n)) to # lf(n) = lf(n), ∀ n. σσ[n + 1] = σσ[n] + ⌜lf(n)⌝ , ∀ m, n. m ≤ n → ii(m) ≤ ii(n) ⊦ ii(m0 + 1) + n = 0, ((σσ[0] + ⌜nf⌝ to n0 + n) + ⌜(nf to ii(m0 + 1)) from n0⌝ )[n1] = σσ[0][n1], ii(m0 + 1) = 0, n = 0, σσ = [], ∃ n. # σσ[n] = ∞

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: stream name: get-append-ls seq: ⊦ ⌜m⌝ < # σ → (σ + σ1)[m] = σ[m] subst: [m, σ1, σ] → [n1, ⌜(nf to ii(m0 + 1)) from n0⌝ , σσ[0] + ⌜nf⌝ to n0 + n] flags: <all paths>

Additional information

Used simplifier rules: ⊦ ¬ ⌜m⌝ < # σ → (σ + σ1)[m] = σ1[m - (# σ).n]; (s)% ⊦ #(σ + σ1) = # σ + # σ1; (s)% ⊦ # ⌜nf⌝ = ∞; (s)% ⊦ 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%append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream% ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ ¬ ∞ < n∞; (s)%¬ # σ < ⌜n⌝ ⊦ #(σ to n) = ⌜n⌝ ; (s)% ⊦ ⌜m⌝ ≠ ∞; (s)%n∞ ≠ ∞ ⊦ (⌜m⌝ + n∞).n = m + n∞.n; (s)% ⊦ ⌜m⌝ .n = m; (s)% ⊦ ⌜x⌝ [n] = x[n]; (s)% ⊦ m < n → ⌜m⌝ < ⌜n⌝ + n∞0; (s) Used terms: n1, ⌜(nf to ii(m0 + 1)) from n0⌝ , σσ[0] + ⌜nf⌝ to n0 + n