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

# x = ii(n), ii(0) = 0, ii(n + 1) = ii(n) + # nextnli0(lf(n), n), lf(n + 1) = nextnli(lf(n), n), lf(0) = [], nextnli(x, n) = x + nextnli0(x, 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:
# x = ii(n), ii(0) = 0, ii(n + 1) = ii(n), lf(n + 1) = nextnli(lf(n), n), lf(0) = [], nextnli(x, n) = x + nextnli0(x, n), nextnli0(lf(n), 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

Rule argument

Additional information

Used simplifier rules: ⊦ x to 0 = []; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ n + 0 = n; (s)% ⊦ # [] = 0; (s)% ⊦ m ≤ m; (s)