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Proof step 73 from proof infinite-append-prefixnextnli (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 74:
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)⌝ , (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)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n) ⊦ # σ < ⌜ii(n0)⌝ + ⌜# nextnli0(lf(n0), n0)⌝ , # lf(n0) = ii(n0) ∧ (n1 < # lf(n0) + # nextnli0(lf(n0), n0) → (lf(n0) + nextnli0(lf(n0), n0))[n1] = (σ to ii(n0) + # nextnli0(lf(n0), n0)).l[n1])

Rule argument

Additional information

Used simplifier rules: ⊦ m + n = m + n0 ↔ n = n0; (s)%¬ # σ < ⌜n⌝ ⊦ # (σ to n).l = n; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s)%a: ⊦ (x + y) + z = x + y + z; from specification list% ⊦ #(x + y) = # x + # y; (s)