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Proof step 86 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, ((σσ[0] + ⌜nf⌝ to n0 + n) + ⌜(nf to ii(m0 + 1)) from n0⌝ )[n2 + n] = nf(n2), ∃ n. # σσ[n] = ∞

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 87:
σσ[m0] = σσ[0] + ⌜nf to ii(m0)⌝ , # σσ[0] = ⌜n⌝ , ii(0) = 0, # σσ = ∞, n2 < ii(m0 + 1), n2 < n0, ii(m0) < ii(m0 + 1), ii(m0) ≤ n0, n0 < ii(m0 + 1), mon(σσ), n2 + 1 < ii(m0 + 1), ∀ 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) ⊦ n0 = 0, (σσ[0] + ⌜nf⌝ )[n2 + n] = nf(n2), ii(m0 + 1) = 0, σσ = [], ii(m0 + 1) + n = 0, ∃ n. # σσ[n] = ∞

Rule argument

Additional information

Used simplifier rules: succ: ⊦ m +1 = m + 1; from specification nat-basic2%trans-1: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification nat-basic1%lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic1%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic1%get-to: m < n, ¬ # σ < ⌜n⌝ ⊦ (σ to n)[m] = σ[m]; from specification stream%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%size-to: ¬ # σ < ⌜n⌝ ⊦ #(σ to n) = ⌜n⌝ ; from specification stream%less-inf1: ⊦ ¬ ∞ < n∞; from specification nat-inf%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%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%get-append-ls: ⊦ ⌜m⌝ < # σ → (σ + σ1)[m] = σ[m]; from specification stream