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Proof step 66 from proof infinite-append-prefixnextli-weak (previous step)

# lf(n0) = ii(n0), ii(0) = 0, lf(0) = [], n1 = n3 + ii(n0), (σ from ii(n0)) to # nextli0(lf(n0)) = ⌜nextli0(lf(n0))⌝ , n1 < ii(n0) + # nextli0(lf(n0)), ii(n0) ≤ n1, (finp(σ) ⊃ # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → nextli0(lf(m)) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n. ii(n + 1) = ii(n) + # nextli0(lf(n)) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # nextli0(lf(n)) = ⌜nextli0(lf(n))⌝ , ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. lf(n + 1) = nextli(lf(n)), ∀ x. nextli(x) = x + nextli0(x), ∀ n2. n2 < ii(n0) → lf(n0)[n2] = σ[n2] ⊦ nextli0(lf(n0))[n3] = σ[n1], nextli0(lf(n0)) = [], # σ < ⌜ii(n0)⌝ + ⌜# nextli0(lf(n0))⌝

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 67:
# lf(n0) = ii(n0), ii(0) = 0, lf(0) = [], (σ from ii(n0)) to # nextli0(lf(n0)) = ⌜nextli0(lf(n0))⌝ , n3 < # nextli0(lf(n0)), (finp(σ) ⊃ # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → nextli0(lf(m)) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n. ii(n + 1) = ii(n) + # nextli0(lf(n)) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # nextli0(lf(n)) = ⌜nextli0(lf(n))⌝ , ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. lf(n + 1) = nextli(lf(n)), ∀ x. nextli(x) = x + nextli0(x), ∀ n2. n2 < ii(n0) → lf(n0)[n2] = σ[n2] ⊦ nextli0(lf(n0))[n3] = σ[n3 + ii(n0)], nextli0(lf(n0)) = [], # σ < ⌜ii(n0)⌝ + ⌜# nextli0(lf(n0))⌝

Rule argument

Additional information

Used simplifier rules: ⊦ # x = 0 ↔ x = []; (s)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic% ⊦ m + n < m + n0 ↔ n < n0; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m ≤ m + n; (s)