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Proof step 47 from proof infinite-append-prefixnextnli (previous step)

ii(0) = 0, lf(0) = [], # x = ii(n), lf(n + 1) = nextnli(lf(n), n), nextnli(x, n) = x + nextnli0(x, n), ii(n + 1) = ii(n) + # nextnli0(lf(n), n), (x from ii(n)) to # nextnli0(lf(n), n) = nextnli0(lf(n), n), ii(n + 1) ≤ ii(n), n ≤ n → nextnli0(lf(n), n) = [] ∧ ii(n) = ii(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. lf(n + 1) = nextnli(lf(n), n), ∀ n0. lf(n0) = x to ii(n0) ⊦ nextnli(x, n) = x

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 48:
nextnli(x, n) = x + nextnli0(x, n), lf(n + 1) = nextnli(lf(n), n), # x = ii(n), lf(0) = [], ii(0) = 0, nextnli0(lf(n), n) = [], ii(n + 1) = ii(n), ∀ n0. lf(n0) = x to ii(n0), ∀ n. lf(n + 1) = nextnli(lf(n), n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n) ⊦ nextnli(x, n) = x

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic2%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic2%add-base: ⊦ n + 0 = n; from specification nat-basic2%to-zero-l: ⊦ x to 0 = []; from specification stream%size-01: ⊦ # [] = 0; from specification list-data%ler: ⊦ m ≤ m; from specification nat