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Proof step 81 from proof infinite-append-prefixnextli (previous step)

¬ n0 ≤ m, nextli0(lf(n0)) = [], ii(n0) = ii(n), nextli0(lf(n)) = [], ii(n + 1) = ii(n), ii(0) = 0, lf(0) = [], # lf(n) = ii(n), nextli(lf(n)) = lf(n), n ≤ n0, ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. lf(n + 1) = nextli(lf(n)), ∀ x. nextli(x) = x + nextli0(x), ∀ m. n ≤ m → nextli0(lf(m)) = [] ∧ ii(m) = ii(n), ∀ n0. lf(n0) = lf(n) to ii(n0), ∀ n0. ii(n0 + 1) = ii(n0) + # nextli0(lf(n0)) ∧ ¬ ii(n) < ii(n0 + 1) ∧ (lf(n) from ii(n0)) to # nextli0(lf(n0)) = nextli0(lf(n0)), ∀ m. nextli0(lf(m)) = [] ↔ n0 ≤ m, ∀ n. nextli0(lf(n)) = [] → (∀ m. n < m → lf(m) = lf(n) ∧ nextli0(lf(m)) = [] ∧ ii(m) = ii(n)) ⊦ ii(m) < ii(m + 1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 82:
nextli0(lf(n0)) = [], ii(n0) = ii(n), nextli0(lf(n)) = [], ii(n + 1) = ii(n), ii(0) = 0, lf(0) = [], # lf(n) = ii(n), nextli(lf(n)) = lf(n), n ≤ n0, ii(m + 1) ≤ ii(m), m < n0, ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. lf(n + 1) = nextli(lf(n)), ∀ x. nextli(x) = x + nextli0(x), ∀ m. n ≤ m → nextli0(lf(m)) = [] ∧ ii(m) = ii(n), ∀ n0. lf(n0) = lf(n) to ii(n0), ∀ n0. ii(n0 + 1) = ii(n0) + # nextli0(lf(n0)) ∧ ¬ ii(n) < ii(n0 + 1) ∧ (lf(n) from ii(n0)) to # nextli0(lf(n0)) = nextli0(lf(n0)), ∀ m. nextli0(lf(m)) = [] ↔ n0 ≤ m, ∀ n. nextli0(lf(n)) = [] → (∀ m. n < m → lf(m) = lf(n) ∧ nextli0(lf(m)) = [] ∧ ii(m) = ii(n)) ⊦ n0 = 0

Rule argument

Additional information

Used simplifier rules: f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic1%lels-01: ⊦ ¬ m ≤ n ↔ n < m; from specification nat%lels: ⊦ ¬ m < n ↔ n ≤ m; from specification nat