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Proof step 4 from proof put-in-02 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

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

Rule argument

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

Additional information

Used simplifier rules: put-zero: ⊦ (b ' + x)[0, a] = a + x;% ⊦ 0 ≤ m; (s)% ⊦ a + x = a ' + x; (s)%a: ⊦ (x + y) + z = x + y + z; from specification list% ⊦ a ∈ b ' + x ↔ ¬(a ≠ b ∧ ¬ a ∈ x); (ss)% ⊦ x + y = [] ↔ x = [] ∧ y = []; (as)% ⊦ a ' ≠ []; (s) Used terms: a, m, a0 ' + x0