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Proof step 77 from proof acyclic-rem-not-reachable (previous step)

path(H[r0].nxt ' + (r2 ' + x2) + r0 ', H), ce.nxt = H[H[r0].nxt].nxt, path(r ' + x1 + H[r0].nxt ' + (r2 ' + x2) + r0 ', H), path(r ' + x1 + H[r0].nxt ', H), r0 ∈ H, H[r0].nxt ∈ H, ∀ r1. ¬ reachable(H[r1].nxt, r1, H) ⊦ H[r0].nxt = r2, r0 = r2, r = r2, r0 = H[r0].nxt, r0 = r, r = H[r0].nxt, H[r0].nxt = null, dups(x2), dups(x1), dups(x1 + x2), r2 ∈ x2, r2 ∈ x1, H[r0].nxt ∈ x2, r0 ∈ x2, r ∈ x2, H[r0].nxt ∈ x1, r ∈ x1, r0 ∈ x1, ∃ x. path(ce.nxt ' + x, H) ∧ (ce.nxt ' + x).last = H[r0].nxt

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 78:
ce.nxt = H[H[r0].nxt].nxt, path(ce.nxt ' + x2 + r0 ', H), H[r0].nxt ∈ H, r0 ∈ H, path(r ' + x1 + H[r0].nxt ', H), path(r ' + x1 + H[r0].nxt ' + (ce.nxt ' + x2) + r0 ', H), ∀ r1. ¬ reachable(H[r1].nxt, r1, H) ⊦ H[r0].nxt = null, r = H[r0].nxt, r0 = r, r0 = H[r0].nxt, r = ce.nxt, r0 = ce.nxt, H[r0].nxt = ce.nxt, r0 ∈ x1, r ∈ x1, H[r0].nxt ∈ x1, r ∈ x2, r0 ∈ x2, H[r0].nxt ∈ x2, ce.nxt ∈ x1, ce.nxt ∈ x2, dups(x1 + x2), dups(x1), dups(x2), ∃ x. path(ce.nxt ' + x, H) ∧ (ce.nxt ' + x).last = H[r0].nxt

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));%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]