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Proof step 11 from proof put-sw-in (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 12:
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, x0[n - 1]][n, (a0 ' + x0)[m]] ↔ a ∈ a0 ' + x0, n = 0

Rule argument

Additional information

Used simplifier rules: ⊦ a ' + x ≠ []; (s)%a: ⊦ (x + y) + z = x + y + z; from specification list% ⊦ (a ' + x).tail = x; (s)