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Proof step 107 from proof infinite-append-prefixnextli (previous step)

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

User applied rule induction

And obtained the following new goals

Proof step 108:
ii(0) = 0 ∧ lf(0) = [] ∧ # x = ii(n) ∧ (∀ n. nextli0(lf(n)) = [] → (∀ m. n < m → lf(m) = lf(n) ∧ nextli0(lf(m)) = [] ∧ ii(m) = ii(n))) ∧ (∀ 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) = x to ii(n0)) ∧ (∀ n0. ii(n0 + 1) = ii(n0) + # nextli0(lf(n0)) ∧ ¬ ii(n) < ii(n0 + 1) ∧ (x from ii(n0)) to # nextli0(lf(n0)) = nextli0(lf(n0))), ∀ n0, ii, lf, x, nextli0, nextli. n0 < n → ii(0) = 0 ∧ lf(0) = [] ∧ # x = ii(n0) ∧ (∀ n. nextli0(lf(n)) = [] → (∀ m. n < m → lf(m) = lf(n) ∧ nextli0(lf(m)) = [] ∧ ii(m) = ii(n))) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (∀ n. lf(n + 1) = nextli(lf(n))) ∧ (∀ x. nextli(x) = x + nextli0(x)) ∧ (∀ m. n0 ≤ m → nextli0(lf(m)) = [] ∧ ii(m) = ii(n0)) ∧ (∀ n0. lf(n0) = x to ii(n0)) ∧ (∀ n1. ii(n1 + 1) = ii(n1) + # nextli0(lf(n1)) ∧ ¬ ii(n0) < ii(n1 + 1) ∧ (x from ii(n1)) to # nextli0(lf(n1)) = nextli0(lf(n1))) → (∃ n0. ∀ m. nextli0(lf(m)) = [] ↔ n0 ≤ m) ⊦ ∃ n0. ∀ m. nextli0(lf(m)) = [] ↔ n0 ≤ m

Rule argument

indvar: n wfpred: <

Additional information

Used simplifier rules: <; (wellfounded) Used terms: n