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Proof step 7 from proof put-at (previous step)

∀ m, a, n. m < # x0 → x0[m, a][n] = x0[n] ∨ m = n ⊦ a0 ' + x0 = [], 0 = n, (a ' + x0)[n] = (a0 ' + x0)[n]

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 8:
⊦ x ≠ [] ∧ n ≠ 0 → x[n] = x.tail[n - 1]
Proof step 9:
∀ m, a, n. m < # x0 → x0[m, a][n] = x0[n] ∨ m = n ⊦ a0 ' + x0 = [], a ' + x0 = [], (a ' + x0).tail[n - 1] = (a0 ' + x0)[n], n = 0

Rule argument

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

Additional information

Used simplifier rules: ⊦ x + y = [] ↔ x = [] ∧ y = []; (as)% ⊦ a ' ≠ []; (s)%a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: n, a ' + x0