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Proof step 14 from proof rmpos-in (previous step)

x0[n - 1] ∈ (x0[n - 1] ' + x0) -p n, n ≤ # x0, ∀ n0, a1. n0 < # x0 → (a1 ∈ x0 -p n0 ↔ a1 ≠ x0[n0] ∧ a1 ∈ x0) ⊦ n = 0, x0 = [], dups(x0), x0[n - 1] ∈ x0

Heuristic elimination applied rule insert elim lemma

And closed the goal!

Rule argument

spec: nat name: elim-pred-c seq: n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1 subst: [n, m] → [n, m]

Additional information

Used simplifier rules: delpos-rec: ⊦ (a ' + x) -p (n + 1) = a + (x -p n);%get-in: ⊦ m < # x → (x[m] ∈ x ↔ true);% ⊦ m + 1 ≠ 0; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m + 1 ≤ n ↔ m < n; (s)% ⊦ a + x = a ' + x; (s)%a: ⊦ (x + y) + z = x + y + z; from specification list% ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s)% ⊦ a ∈ b ' ↔ a = b; (s)%n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1; (elim) Used terms: n, m