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Proof step 95 from proof lim-exists (previous step)

# σσ = ∞, ii(0) = 0, # σσ[0] = ⌜n⌝ , σσ[m0] = σσ[0] + ⌜nf to ii(m0)⌝ , n1 < ii(m0 + 1) + n, mon(σσ), n0 < ii(m0 + 1), ii(m0) ≤ n0, ii(m0) < ii(m0 + 1), n1 < n, ∀ 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, n = 0, (σσ[0] + ⌜nf⌝ )[n1] = σσ[0][n1], ∃ n. # σσ[n] = ∞

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: stream inst: name: get-append-ls seq: ⊦ ⌜m⌝ < # σ → (σ + σ1)[m] = σ[m] subst: [m, σ1, σ] → [n1, ⌜nf⌝ , σσ[0]] flags: rewrite

Additional information

Used simplifier rules: get-append-notls: ⊦ ¬ ⌜m⌝ < # σ → (σ + σ1)[m] = σ1[m - (# σ).n]; from specification stream%sel-inf: ⊦ ⌜nf⌝ [n] = nf(n); from specification stream%append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream%sel: ⊦ ⌜m⌝ .n = m; from specification nat-inf-basic%less-fin: ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; from specification nat-inf%lels: ⊦ ¬ m < n ↔ n ≤ m; from specification nat%ls-le-simp: ⊦ m < n → (n ≤ m ↔ false); from specification nat Used terms: n1, ⌜nf⌝ , σσ[0]