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Proof step 36 from proof lim-exists (previous step)

# σσ[n] = ⌜n0⌝ , # x = ii(n), # σσ = ∞, ii(0) = 0, mon(σσ), n ≤ n1 → σσ[n1] = σσ[n], ∀ m. n ≤ m → σσ[m] = σσ[n], ∀ n0. ii(n0 + 1) = ii(n0) + # lf(n0) ∧ ¬ ii(n) < ii(n0 + 1) ∧ (x from ii(n0)) to # lf(n0) = lf(n0), ∀ n. σσ[n + 1] = σσ[n] + ⌜lf(n)⌝ , ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n) ⊦ σσ = [], ∃ σ1. σσ[n1] + σ1 = σσ[n], ∃ n. # σσ[n] = ∞

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel> name: mon-lem seq: mon(σσ), m ≤ n, ⌜n⌝ < # σσ ⊦ σσ[m] ⊑ σσ[n] subst: [σσ, m, n] → [σσ, n1, n]

Additional information

Used simplifier rules: prefix-def: ⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0);% ⊦ ⌜m⌝ < ∞; (s)% ⊦ m ≤ n ↔ ¬ n < m; (ss)% ⊦ m < n → m ≤ n; (s)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic% ⊦ σ0 + σ = σ0 ↔ σ = [] ∨ ¬ finp(σ0); (s)%append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream Used terms: σσ, n1, n