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Proof step 53 from proof lim-exists (previous step)

∀ n. σσ[n] = σσ[0] + ⌜nf to ii(n)⌝ , ii(0) = 0, # σσ = ∞, mon(σσ), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 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) ⊦ σσ = [], ∃ n. # σσ[n] = ∞, (∀ n. ∃ σ1. σσ[n] + σ1 = σσ[0] + ⌜nf⌝ ) ∧ (∀ m. ¬ #(σσ[0] + ⌜nf⌝ ) < ⌜m⌝ → (∃ n, σ1. (σσ[0] + ⌜nf⌝ to m) + σ1 = σσ[n]))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 54:
ii(0) = 0, # σσ = ∞, mon(σσ), ∀ n. σσ[n] = σσ[0] + ⌜nf to ii(n)⌝ , ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 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) ⊦ σσ = [], ∃ n. # σσ[n] = ∞, (∃ σ1. σσ[n] + σ1 = σσ[0] + ⌜nf⌝ ) ∧ (∃ n, σ1. (σσ[0] + ⌜nf⌝ to m) + σ1 = σσ[n])

Rule argument

Additional information

Used simplifier rules: less-inf1: ⊦ ¬ ∞ < 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