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Proof step 2 from proof diffone-nodupsget (previous step)

a ∈ x, n + 1 < # x ⊦ (x \ a ')[n] = (n < pos(a, x) ⊃ x[n];x[n + 1]), x = [], dups(x)

Heuristic structural induction applied rule structural induction

And obtained the following new goals

Proof step 3:
∀ a, n. a ∈ x0 ∧ n + 1 < # x0 → (x0 \ a ')[n] = (n < pos(a, x0) ⊃ x0[n];x0[n + 1]) ∨ x0 = [] ∨ dups(x0), a ∈ a0 + x0, n + 1 < #(a0 + x0) ⊦ ((a0 + x0) \ a ')[n] = (n < pos(a, a0 + x0) ⊃ (a0 + x0)[n];(a0 + x0)[n + 1]), a0 + x0 = [], dups(a0 + x0)

Rule argument

term: x

Additional information

Used simplifier rules: list freely generated by [], +;;