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Proof step 2 from proof put-sw-in (previous step)

m < # x, n < # x ⊦ a ∈ x[m, x[n]][n, x[m]] ↔ a ∈ x

User applied rule structural induction

And obtained the following new goals

Proof step 3:
n < #(a0 + x0), m < #(a0 + 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

Rule argument

indvar: x

Additional information

Used simplifier rules: list freely generated by [], +;;% ⊦ ¬ a ∈ []; (s)% ⊦ # [] = 0; (s)% ⊦ ¬ n < 0; (s)