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Proof step 90 from proof lim-exists (previous step)

# σσ = ∞, # σσ[0] = ⌜n⌝ , σσ[m0] = σσ[0] + ⌜nf to ii(m0)⌝ , ii(0) = 0, n2 < ii(m0 + 1), n0 < ii(m0 + 1), ii(m0) < ii(m0 + 1), mon(σσ), ii(m0) ≤ n0, ∀ 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) + n = 0, ⌜(nf to ii(m0 + 1)) from n0⌝ [n2 + n - (#(σσ[0] + ⌜nf⌝ to n0 + n)).n] = nf(n2), ii(m0 + 1) = 0, σσ = [], ⌜n2 + n⌝ < #(σσ[0] + ⌜nf⌝ to n0 + n), ∃ n. # σσ[n] = ∞

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 91:
# σσ[0] = ⌜n⌝ , # σσ = ∞, σσ[m0] = σσ[0] + ⌜nf to ii(m0)⌝ , ii(0) = 0, ii(m0) < ii(m0 + 1), n2 < ii(m0 + 1), n0 < ii(m0 + 1), mon(σσ), ii(m0) ≤ n0, n0 ≤ n2, ii(m0) ≤ n2, ∀ 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, ii(m0 + 1) = 0, nf(n0 + (n2 - n0)) = nf(n2), σσ = [], ∃ n. # σσ[n] = ∞

Rule argument

Additional information

Used simplifier rules: m < n ⊦ (nf to n)[m] = nf(m); (s)%m + m0 < # x ⊦ (x from m)[m0] = x[m + m0]; (s)% ⊦ m < n → (n ≤ m ↔ false); (s)% ⊦ n < m → (m - n) + n = m; (s)% ⊦ m ≠ n → (m ≤ n ↔ m < n); (s)% ⊦ m ≤ n → (n - m = 0 ↔ n = m); (s)% ⊦ m < m + n ↔ n ≠ 0; (s)%let: ⊦ n ≤ n0 ∧ n0 ≤ n1 → n ≤ n1; (forward) from specification nat%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat% ⊦ #(nf to n) = n; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s)% ⊦ ⌜n⌝ + n∞1 < ⌜n⌝ + n∞2 ↔ n∞1 < n∞2; (s)% ⊦ ⌜x⌝ [n] = x[n]; (s)% ⊦ m + n - n + n1 = m - n1; (s)% ⊦ ⌜m⌝ .n = m; (s)%n∞ ≠ ∞ ⊦ (⌜m⌝ + n∞).n = m + n∞.n; (s)% ⊦ ⌜m⌝ ≠ ∞; (s)%¬ # σ < ⌜n⌝ ⊦ #(σ to n) = ⌜n⌝ ; (s)% ⊦ ¬ ∞ < n∞; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s)%append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; 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% ⊦ n∞ + ∞ = ∞; (s)% ⊦ # ⌜nf⌝ = ∞; (s)% ⊦ #(σ + σ1) = # σ + # σ1; (s)