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Proof step 3 from proof first-append (previous step)

∀ x, y. m0 ≤ # x → firstn(m0, x + y) = firstn(m0, x), m0 +1 ≤ # x ⊦ firstn(m0 +1, x + y) = firstn(m0 +1, x)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 4:
m0 < # x, ∀ x, y. m0 ≤ # x → firstn(m0, x + y) = firstn(m0, x) ⊦ x = [], firstn(m0, x.rest + y) = firstn(m0, x.rest)

Rule argument

Additional information

Used simplifier rules: firstN-rec: ⊦ firstn(n + 1, x) = x.first + firstn(n, x.rest);%app-08: ⊦ a ' + x = a ' + y ↔ x = y; from specification list%rest-01: ⊦ x ≠ [] → (x + y).rest = x.rest + y; from specification list%a: ⊦ (x + y) + z = x + y + z; from specification list%first-01: ⊦ x ≠ [] → (x + y).first = x.first; from specification list%l: ⊦ # x = 0 ↔ x = []; from specification list-data%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic1%succ-01: ⊦ m + 1 ≤ n ↔ m < n; from specification nat%c: ⊦ m + n = n + m; from specification nat-basic2%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic2%c: ⊦ a + x = a ' + x; from specification list%succ: ⊦ m +1 = m + 1; from specification nat-basic2