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Proof step 4 from proof put-sw-in (previous step)

m ≤ # x0, n ≤ # x0, ∀ m, n, a. m < # x0 ∧ n < # x0 → (a ∈ x0[m, x0[n]][n, x0[m]] ↔ a ∈ x0) ⊦ a ∈ (a0 ' + x0)[m, (a0 ' + x0)[n]][n, (a0 ' + x0)[m]] ↔ a ∈ a0 ' + x0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 5:
⊦ x ≠ [] ∧ n ≠ 0 → x[n] = x.tail[n - 1]
Proof step 6:
n ≤ # x0, m ≤ # x0, ∀ m, n, a. m < # x0 ∧ n < # x0 → (a ∈ x0[m, x0[n]][n, x0[m]] ↔ a ∈ x0) ⊦ a ∈ (a0 ' + x0)[m, (a0 ' + x0)[n]][n, (a0 ' + x0)[m]] ↔ a ∈ a0 ' + x0, a0 ' + x0 ≠ [] ∧ n ≠ 0
Proof step 11:
n ≤ # x0, m ≤ # x0, ∀ m, n, a. m < # x0 ∧ n < # x0 → (a ∈ x0[m, x0[n]][n, x0[m]] ↔ a ∈ x0) ⊦ a0 ' + x0 = [], a ∈ (a0 ' + x0)[m, (a0 ' + x0).tail[n - 1]][n, (a0 ' + x0)[m]] ↔ a ∈ a0 ' + x0, n = 0

Rule argument

spec: <toplevel> name: get-rec seq: ⊦ x ≠ [] ∧ n ≠ 0 → x[n] = x.tail[n - 1] subst: [n, x] → [n, a0 ' + x0] flags: <all paths>

Additional information

Used terms: n, a0 ' + x0