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Proof step 6 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) ⊦ a ∈ (a0 ' + x0)[m, (a0 ' + x0)[n]][n, (a0 ' + x0)[m]] ↔ a ∈ a0 ' + x0, a0 ' + x0 ≠ [] ∧ n ≠ 0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 7:
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][0, (a0 ' + x0)[m]] ↔ a ∈ a0 ' + x0

Rule argument

Additional information

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