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Proof step 13 from proof reachable-nodups-def (previous step)

(H[r].nxt ' + x).last = (H[r].nxt ' + x0).last, path(H[r].nxt ' + x, H), path(H[r].nxt ' + x0, H), r ∈ H, ∀ r, H. path(r ' + x0, H) → (∃ x1. path(r ' + x1, H) ∧ (r ' + x1).last = (r ' + x0).last ∧ ¬ r ∈ x1 ∧ ¬ dups(x1)) ⊦ r = null, dups(x), H[r].nxt ∈ x, ∃ x. path(r ' + x, H) ∧ (r ' + x).last = (H[r].nxt ' + x0).last ∧ ¬ r ∈ x ∧ ¬ dups(x), path(r ' + H[r].nxt ' + x, H) ∧ (r ' + H[r].nxt ' + x).last = (H[r].nxt ' + x0).last ∧ ¬ r ∈ H[r].nxt ' + x ∧ ¬ dups(H[r].nxt ' + x)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 14:
(H[r].nxt ' + x).last = (H[r].nxt ' + x0).last, path(H[r].nxt ' + x0, H), path(H[r].nxt ' + x, H), r ∈ H, ∀ r, H. path(r ' + x0, H) → (∃ x1. path(r ' + x1, H) ∧ (r ' + x1).last = (r ' + x0).last ∧ ¬ r ∈ x1 ∧ ¬ dups(x1)) ⊦ r = null, dups(x), H[r].nxt ∈ x, ∃ x. path(r ' + x, H) ∧ (r ' + x).last = (H[r].nxt ' + x0).last ∧ ¬ r ∈ x ∧ ¬ dups(x), r ≠ H[r].nxt ∧ ¬ r ∈ x

Rule argument

Additional information

Used simplifier rules: path-cons: ⊦ path(r ' + r0 ' + x, H) ↔ r ∈ H ∧ r ≠ null ∧ H[r].nxt = r0 ∧ path(r0 ' + x, H);% ⊦ dups(r ' + x0) ↔ dups(x0) ∨ r ∈ x0; (ws)% ⊦ r ∈ r0 ' ↔ r = r0; (s)% ⊦ r ∈ x0 + x1 ↔ r ∈ x0 ∨ r ∈ x1; (s)% ⊦ (r ' + r0 ' + x0).last = (r0 ' + x0).last; (s)%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]