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Proof step 8 from proof restn-firstn-better (previous step)

x = a + y, n0 + m0 < # x, 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, y)) = firstn(n0, restn(m0, y)), x = []

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 9:
m0 ≤ # y, n0 + m0 ≤ # y, ∀ n0, x. n0 + m0 ≤ # x ∧ m0 ≤ # x → restn(m0, firstn(n0 + m0, x)) = firstn(n0, restn(m0, x)) ⊦ restn(m0, firstn(n0 + m0, y)) = firstn(n0, restn(m0, y))

Rule argument

Additional information

Used simplifier rules: ⊦ m < n ↔ ¬ n ≤ m; (ss)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m < n + 1 ↔ ¬ n < m; (s)% ⊦ # a ' = 0 +1; (s)% ⊦ #(x + y) = # x + # y; (s)%a: ⊦ (x + y) + z = x + y + z; from specification list% ⊦ a ' + x ≠ []; (s)% ⊦ a + x = a ' + x; (s)