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Proof step 78 from proof lim-exists (previous step)

# 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), ∀ 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⌝ ) = #(σσ[0] + ⌜nf to ii(m0 + 1)⌝ ) ∧ (∀ n1. ⌜n1⌝ < #((σσ[0] + ⌜nf⌝ to n0 + n) + ⌜(nf to ii(m0 + 1)) from n0⌝ ) → ((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 79:
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] = ∞

Rule argument

Additional information

Used simplifier rules: eq-natinf-to-nat-2-1: ⊦ ⌜n⌝ + ⌜m⌝ = ⌜n0⌝ ↔ n + m = n0; from specification nat-inf%add-cancel-01: ⊦ ⌜n⌝ + n∞1 = ⌜n⌝ + n∞2 ↔ n∞1 = n∞2; from specification nat-inf%prop-07: ⊦ n < m → (m - n) + n = m; from specification nat%ls-natinf-to-nat-1-3: ⊦ ⌜n1⌝ < ⌜n0⌝ + ⌜n⌝ + ⌜m⌝ ↔ n1 < n0 + n + m; from specification nat-inf%inf-size-to: ⊦ # ⌜nf to n⌝ = ⌜n⌝ ; from specification stream%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%length-from: ¬ # x < n ⊦ #(x from n) = # x - n; from specification stream%ls-02: ⊦ m < n → ¬ n < m; from specification nat-basic1%size-inf-to: ⊦ #(nf to n) = n; from specification stream%size-fin: ⊦ # ⌜x⌝ = ⌜# x⌝ ; from specification stream%size-append: ⊦ #(σ + σ1) = # σ + # σ1; from specification stream