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Proof step 35 from proof lim-exists (previous step)

# x = ii(n), # σσ = ∞, ii(0) = 0, # σσ[n] = ⌜n0⌝ , mon(σσ), ∀ 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), n ≤ n1 → σσ[n1] = σσ[n], ∀ m. n ≤ m → σσ[m] = σσ[n] ⊦ ∃ σ1. σσ[n1] + σ1 = σσ[n], σσ = [], ∃ n. # σσ[n] = ∞

User applied rule insert lemma

And obtained the following new goals

Rule argument

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

Additional information

Used simplifier rules: prefix-def: ⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0);%less-inf2: ⊦ ⌜m⌝ < ∞; from specification nat-inf%lels-01: ⊦ ¬ m ≤ n ↔ n < m; from specification nat%lels-03: ⊦ m < n → m ≤ n; from specification nat%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic1%appendend-equal: ⊦ σ + σ0 = σ ↔ ¬ ¬(σ0 = [] ∨ # σ = ∞); from specification stream%append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream Used terms: σσ, n1, n