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Proof step 54 from proof infinite-append-prefixnextli (previous step)

lf(0) = [], ii(0) = 0, nextli0(lf(n0)) = [], ii(n0 + 1) = ii(n0), n0 ≤ m0, (finp(σ) ⊃ # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → nextli0(lf(m)) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ x. nextli(x) = x + nextli0(x), ∀ n. lf(n + 1) = nextli(lf(n)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # nextli0(lf(n)) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # nextli0(lf(n)) = ⌜nextli0(lf(n))⌝ , ∀ n0. lf(n0) = (σ to ii(n0)).l, n0 < m0 → m0 = 0 ∨ ii(m0) = ii(n0) ∧ lf(m0) = lf(n0) ⊦ n0 < m0, # σ = ⌜ii(n0)⌝ , # σ < ⌜ii(n0)⌝ , ii(m0 + 1) = ii(n0) ∧ lf(m0 + 1) = lf(n0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 55:
ii(m0 + 1) = ii(m0), nextli0(lf(m0)) = [], ii(0) = 0, lf(0) = [], ∀ n0. lf(n0) = (σ to ii(n0)).l, ∀ 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), (finp(σ) ⊃ # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → nextli0(lf(m)) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)) ⊦ # σ = ⌜ii(m0)⌝ , lf(m0 + 1) = lf(m0), # σ < ⌜ii(m0)⌝

Rule argument

Additional information

Used simplifier rules: ref-1: ⊦ ¬ n < n; from specification nat-basic1%ler: ⊦ m ≤ m; from specification nat%anti: ⊦ m ≤ n → (n ≤ m ↔ m = n); from specification nat%lels: ⊦ ¬ m < n ↔ n ≤ m; from specification nat