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Proof step 5 from proof get-in (previous step)

m ≤ # x0, ∀ m. m < # x0 → x0 = [] ∨ x0[m] ∈ x0 ⊦ (a ' + x0)[m] = a, (a ' + x0)[m] ∈ x0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 6:
⊦ x ≠ [] ∧ n ≠ 0 → x[n] = x.tail[n - 1]
Proof step 7:
m ≤ # x0, ∀ m. m < # x0 → x0 = [] ∨ x0[m] ∈ x0 ⊦ a ' + x0 = [], (a ' + x0).tail[m - 1] = a, m = 0, (a ' + x0).tail[m - 1] ∈ x0

Rule argument

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

Additional information

Used simplifier rules: get-zero-01: ⊦ (a ' + x)[0] = a;% ⊦ x + y = [] ↔ x = [] ∧ y = []; (as)% ⊦ a ' ≠ []; (s)%a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: m, a ' + x0