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Proof step 87 from proof lim-exists (previous step)

σσ[m0] = σσ[0] + ⌜nf to ii(m0)⌝ , # σσ[0] = ⌜n⌝ , ii(0) = 0, # σσ = ∞, n2 < ii(m0 + 1), n2 < n0, ii(m0) < ii(m0 + 1), ii(m0) ≤ n0, n0 < ii(m0 + 1), mon(σσ), n2 + 1 < ii(m0 + 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) ⊦ n0 = 0, (σσ[0] + ⌜nf⌝ )[n2 + n] = nf(n2), ii(m0 + 1) = 0, σσ = [], ii(m0 + 1) + n = 0, ∃ n. # σσ[n] = ∞

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: stream inst: name: get-append-notls seq: ⊦ ¬ ⌜m⌝ < # σ → (σ + σ1)[m] = σ1[m - (# σ).n] subst: [m, σ1, σ] → [n2 + n, ⌜nf⌝ , σσ[0]] flags: rewrite

Additional information

Used simplifier rules: sel-inf: ⊦ ⌜nf⌝ [n] = nf(n); from specification stream%add-fin: ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; from specification nat-inf%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic2%c: ⊦ m + n = n + m; from specification nat-basic2%sel: ⊦ ⌜m⌝ .n = m; from specification nat-inf-basic%eq-01: ⊦ m + n - m = n; from specification nat%get-append-ls: ⊦ ⌜m⌝ < # σ → (σ + σ1)[m] = σ[m]; from specification stream%append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream%ls-natinf-to-nat-2-1: ⊦ ⌜n⌝ + ⌜m⌝ < ⌜n0⌝ ↔ n + m < n0; from specification nat-inf%l-02: ⊦ ¬ m + n < m; from specification nat-basic2%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 Used terms: n2 + n, ⌜nf⌝ , σσ[0]