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Proof step 89 from proof lim-exists (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: get-infto: m < n ⊦ (nf to n)[m] = nf(m); from specification stream%get-from-01: m + m0 < # x ⊦ (x from m)[m0] = x[m + m0]; from specification stream%ls-le-simp: ⊦ m < n → (n ≤ m ↔ false); from specification nat%prop-07: ⊦ n < m → (m - n) + n = m; from specification nat%lels-02: ⊦ m ≠ n → (m ≤ n ↔ m < n); from specification nat%zero-02: ⊦ m ≤ n → (n - m = 0 ↔ n = m); from specification nat%l-01: ⊦ m < m + n ↔ n ≠ 0; from specification nat-basic2%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%size-inf-to: ⊦ #(nf to n) = n; from specification stream%lels: ⊦ ¬ m < n ↔ n ≤ m; from specification nat%l: ⊦ m + n < m + n0 ↔ n < n0; from specification nat-basic2%ls-natinf-to-nat-2-2: ⊦ ⌜n⌝ + ⌜m⌝ < ⌜n0⌝ + ⌜n1⌝ ↔ n + m < n0 + n1; from specification nat-inf%sel-fin: ⊦ ⌜x⌝ [n] = x[n]; from specification stream%cancel-01: ⊦ m + n - n + n1 = m - n1; from specification nat%sel: ⊦ ⌜m⌝ .n = m; from specification nat-inf-basic%prime-add-fin: n∞ ≠ ∞ ⊦ (⌜m⌝ + n∞).n = m + n∞.n; from specification nat-inf%disj: ⊦ ⌜m⌝ ≠ ∞; from specification nat-inf-basic%size-to: ¬ # σ < ⌜n⌝ ⊦ #(σ to n) = ⌜n⌝ ; from specification stream%less-inf1: ⊦ ¬ ∞ < n∞; from specification nat-inf%c: ⊦ m + n = n + m; from specification nat-basic2%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic2%add-fin: ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; from specification nat-inf%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%add-all-inf: ⊦ n∞ + ∞ = ∞; from specification nat-inf%size-inf: ⊦ # ⌜nf⌝ = ∞; from specification stream%size-append: ⊦ #(σ + σ1) = # σ + # σ1; from specification stream