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Proof step 67 from proof infinite-append-prefixnextnli (previous step)

ii(0) = 0, lf(0) = [], lf(n0) = (σ to ii(n0)).l, (σ from ii(n0)) to # nextnli0(lf(n0), n0) = ⌜nextnli0(lf(n0), n0)⌝ , lf(n0 + 1) = nextnli(lf(n0), n0), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), nextnli(lf(n0), n0) = lf(n0) + nextnli0(lf(n0), n0), (finp(σ) ⊃ # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → nextnli0(lf(m), m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)) ⊦ nextnli(lf(n0), n0) = (σ to ii(n0) + # nextnli0(lf(n0), n0)).l, # σ < ⌜ii(n0) + # nextnli0(lf(n0), n0)⌝

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 68:
ii(0) = 0, lf(0) = [], lf(n0) = (σ to ii(n0)).l, (σ from ii(n0)) to # nextnli0(lf(n0), n0) = ⌜nextnli0(lf(n0), n0)⌝ , lf(n0 + 1) = nextnli(lf(n0), n0), nextnli(lf(n0), n0) = lf(n0) + nextnli0(lf(n0), n0), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), (finp(σ) ⊃ # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → nextnli0(lf(m), m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)) ⊦ nextnli(lf(n0), n0) = (σ to ii(n0) + # nextnli0(lf(n0), n0)).l, # σ < ⌜ii(n0)⌝ + ⌜# nextnli0(lf(n0), n0)⌝

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic2%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic2%add-fin: ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; from specification nat-inf