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Proof step 13 from proof sublist-move (previous step)

n + m ≤ 1 + # x0, n + m ≤ m1 + 1, m1 ≤ # x0, ∀ n0, m0, n, m. n0 + m0 ≤ # x0 ∧ n + m ≤ m0 → firstn(m, restn(n, firstn(m0, restn(n0, x0)))) = firstn(m, restn(n0 + n, x0)) ⊦ firstn(m, restn(n, a ' + firstn(m1, x0))) = firstn(m, restn(n, a ' + x0))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 14:
⊦ 0 < n → restn(n, x) = restn(n -1, x.tail)
Proof step 15:
m1 ≤ # x0, n + m ≤ m1 + 1, n + m ≤ 1 + # x0, ∀ n0, m0, n, m. n0 + m0 ≤ # x0 ∧ n + m ≤ m0 → firstn(m, restn(n, firstn(m0, restn(n0, x0)))) = firstn(m, restn(n0 + n, x0)) ⊦ firstn(m, restn(n, a ' + firstn(m1, x0))) = firstn(m, restn(n, a ' + x0)), 0 < n
Proof step 20:
0 < n, m1 ≤ # x0, n + m ≤ m1 + 1, n + m ≤ 1 + # x0, ∀ n0, m0, n, m. n0 + m0 ≤ # x0 ∧ n + m ≤ m0 → firstn(m, restn(n, firstn(m0, restn(n0, x0)))) = firstn(m, restn(n0 + n, x0)) ⊦ firstn(m, restn(n -1, (a ' + firstn(m1, x0)).tail)) = firstn(m, restn(n, a ' + x0))

Rule argument

spec: <toplevel> name: rest-rec seq: ⊦ 0 < n → restn(n, x) = restn(n -1, x.tail) subst: [x, n] → [a ' + firstn(m1, x0), n] flags: <all paths>

Additional information

Used terms: a ' + firstn(m1, x0), n