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Proof step 52 from proof acyclic-rem-not-reachable (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 53:
ce.nxt = H[H[(r ' + x).last].nxt].nxt, (r ' + x).last ∈ H, path(r ' + x1 + H[(r ' + x).last].nxt ', H), path(r ' + x, H), H[(r ' + x).last].nxt ∈ H, ∀ r1. ¬ reachable(H[r1].nxt, r1, H) ⊦ H[(r ' + x).last].nxt = null, r = H[(r ' + x).last].nxt, (r ' + x).last = r, (r ' + x).last = H[(r ' + x).last].nxt, (r ' + x).last ∈ x1, r ∈ x1, H[(r ' + x).last].nxt ∈ x1, dups(x1), r ∈ x, dups(x)

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]%rec: ⊦ dups(r ' + x0) ↔ ¬ ¬(r ∈ x0 ∨ dups(x0)); from specification list-dup[reflist]