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Proof step 82 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), 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] + ⌜nf to ii(m0 + 1)⌝ )[n1], ii(m0 + 1) = 0, σσ = [], ⌜n1⌝ < # σσ[0], ∃ n. # σσ[n] = ∞

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 83:
# 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), mon(σσ), ii(m0) ≤ n0, 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[0] + ⌜nf⌝ to n0 + n) + ⌜(nf to ii(m0 + 1)) from n0⌝ )[n1] = nf(n1 - n), ii(m0 + 1) = 0, σσ = [], ∃ n. # σσ[n] = ∞

Rule argument

Additional information

Used simplifier rules: m < n ⊦ (nf to n)[m] = nf(m); (s)%n0 ≤ n ⊦ n - n0 < m ↔ n < m + n0; (s)% ⊦ m < n → (n ≤ m ↔ false); (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s)% ⊦ m ≤ n ↔ ¬ n < m; (ss)% ⊦ ⌜m⌝ .n = m; (s)%append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream% ⊦ ⌜x⌝ [n] = x[n]; (s)% ⊦ ¬ ⌜m⌝ < # σ → (σ + σ1)[m] = σ1[m - (# σ).n]; (s)