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Proof step 29 from proof monls-mon-implies-monls (previous step)

⌜n⌝ + ⌜1⌝ < # lim(nσf), ((lim(nσf) to n + 2) + nσ)[n] < lim(nσf)[n + 1], ((lim(nσf) to n + 2) + nσ)[n] < lim(nσf)[n], mon(nσf), lim(nσf)[n + 1] ≤ lim(nσf)[n], ∀ n. nσf(n) ⊑ lim(nσf) ⊦ # lim(nσf) < ⌜n⌝ + ⌜2⌝

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, inst: natstream name: get-append-ls seq: ⊦ ⌜m⌝ < # nσ → (nσ + nσ0)[m] = nσ[m] subst: [m, nσ0, nσ] → [n, nσ, lim(nσf) to n + 2] flags: <all paths>

Additional information

Used simplifier rules: ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%¬ # nσ < ⌜n⌝ ⊦ #(nσ to n) = ⌜n⌝ ; (s)% ⊦ ⌜n⌝ < ⌜m⌝ + ⌜n0⌝ ↔ n < m + n0; (s)% ⊦ m < m + n ↔ n ≠ 0; (s)%add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf%add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf%m < n, ¬ # nσ < ⌜n⌝ ⊦ (nσ to n)[m] = nσ[m]; (s)% ⊦ m ≤ n → (n < m ↔ false); (s)% ⊦ ¬ ⌜m⌝ < # nσ → (nσ + nσ0)[m] = nσ0[m - (# nσ).n]; (s)% ⊦ ⌜m⌝ ≠ ∞; (s)%n∞ ≠ ∞ ⊦ (⌜m⌝ + n∞).n = m + n∞.n; (s)% ⊦ ⌜m⌝ .n = m; (s)%append-assoc: ⊦ (nσ + nσ0) + nσ1 = nσ + nσ0 + nσ1; from specification stream[natstream]% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m + m0 ≤ m ↔ m0 = 0; (s) Used terms: n, nσ, lim(nσf) to n + 2