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Proof step 6 from proof restn-firstn-better (previous step)

m0 +1 ≤ # x, n0 + m0 +1 ≤ # x, ∀ n0, x. n0 + m0 ≤ # x ∧ m0 ≤ # x → restn(m0, firstn(n0 + m0, x)) = firstn(n0, restn(m0, x)) ⊦ restn(m0 +1, firstn(n0 + m0 +1, x)) = firstn(n0, restn(m0 +1, x))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 7:
m0 < # x, n0 + m0 < # x, ∀ n0, x. n0 + m0 ≤ # x ∧ m0 ≤ # x → restn(m0, firstn(n0 + m0, x)) = firstn(n0, restn(m0, x)) ⊦ restn(m0, firstn(n0 + m0, x.tail)) = firstn(n0, restn(m0, x.tail)), x = []

Rule argument

Additional information

Used simplifier rules: firstN-rec: ⊦ firstn(n + 1, x) = x.head + firstn(n, x.tail);%restN-rec: ⊦ restn(n + 1, x) = restn(n, x.tail);% ⊦ # x = 0 ↔ x = []; (s)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic% ⊦ m + 1 ≤ n ↔ m < n; (s)%a: ⊦ (x + y) + z = x + y + z; from specification list% ⊦ (a ' + x).tail = x; (s)%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ a + x = a ' + x; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m +1 = m + 1; (s)