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Proof step 81 from proof lim-exists (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 82:
# σσ = ∞, ii(0) = 0, # σσ[0] = ⌜n⌝ , σσ[m0] = σσ[0] + ⌜nf to ii(m0)⌝ , n1 < ii(m0 + 1) + n, mon(σσ), n0 < ii(m0 + 1), ii(m0) ≤ n0, ii(m0) < ii(m0 + 1), n ≤ n1, ∀ 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 = [], ii(m0 + 1) = 0, ((σσ[0] + ⌜nf⌝ to n0 + n) + ⌜(nf to ii(m0 + 1)) from n0⌝ )[n1] = nf(n1 - n), ∃ n. # σσ[n] = ∞

Rule argument

Additional information

Used simplifier rules: get-infto: m < n ⊦ (nf to n)[m] = nf(m); from specification stream%ls-01: n0 ≤ n ⊦ n - n0 < m ↔ n < m + n0; from specification nat%ls-le-simp: ⊦ m < n → (n ≤ m ↔ false); from specification nat%lels: ⊦ ¬ m < n ↔ n ≤ m; from specification nat%less-fin: ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; from specification nat-inf%lels-01: ⊦ ¬ m ≤ n ↔ n < m; from specification nat%sel: ⊦ ⌜m⌝ .n = m; from specification nat-inf-basic%append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream%sel-fin: ⊦ ⌜x⌝ [n] = x[n]; from specification stream%get-append-notls: ⊦ ¬ ⌜m⌝ < # σ → (σ + σ1)[m] = σ1[m - (# σ).n]; from specification stream