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Proof step 18 from proof reachable-nodups-def (previous step)

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)), ∃ x1, x2. x = x1 + r ' + x2 ∧ ¬ r ∈ x2 ⊦ r = null, r = H[r].nxt, dups(x), H[r].nxt ∈ x, ∃ x. path(r ' + x, H) ∧ (r ' + x).last = (H[r].nxt ' + x0).last ∧ ¬ r ∈ x ∧ ¬ dups(x)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 19:
(r ' + x1).last = (H[r].nxt ' + x0).last, path(H[r].nxt ' + x0, H), path(H[r].nxt ' + x2 + r ' + x1, 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 = H[r].nxt, r = null, dups(x2 + x1), dups(x2), dups(x1), r ∈ x1, H[r].nxt ∈ x2, H[r].nxt ∈ x1, r ∈ x2, ∃ x. path(r ' + x, H) ∧ (r ' + x).last = (r ' + x1).last ∧ ¬ r ∈ x ∧ ¬ dups(x)

Rule argument

Additional information

Used simplifier rules: a: ⊦ ¬ dups(x0 + x1) → ¬ dups(x0) ∧ ¬ dups(x1); (forward) from specification list-dup[reflist]% ⊦ dups(x0 + r ' + x1) ↔ r ∈ x0 ∨ r ∈ x1 ∨ dups(x0 + x1); (ws)% ⊦ r ≠ r0 → (r ∈ r0 ' + x0 ↔ r ∈ x0); (s)% ⊦ r ∈ x0 + x1 ↔ r ∈ x0 ∨ r ∈ x1; (s)%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]% ⊦ (x0 + r ' + x1).last = (r ' + x1).last; (s)