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Proof step 3 from proof ocs-same-length-mklist (previous step)

∀ a. #oc(a, x0) = # x0 ↔ x0 = mklist(a, # x0) ⊦ #oc(a, a0 ') + #oc(a, x0) = 1 + # x0 ↔ a0 ' + x0 = mklist(a, 1 + # x0)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 4:
⊦ m ≠ 0 → mklist(a, m) = a + mklist(a, m - 1)
Proof step 5:
∀ a. #oc(a, x0) = # x0 ↔ x0 = mklist(a, # x0) ⊦ 1 + # x0 = 0, #oc(a, a0 ') + #oc(a, x0) = 1 + # x0 ↔ a0 ' + x0 = a + mklist(a, 1 + # x0 - 1)

Rule argument

spec: list-last name: mk-rec seq: ⊦ m ≠ 0 → mklist(a, m) = a + mklist(a, m - 1) subst: [m, a] → [1 + # x0, a] flags: <all paths>

Additional information

Used simplifier rules: ⊦ m + 1 ≠ 0; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic Used terms: 1 + # x0, a