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Proof step 9 from proof rem-stays-reachable-aux (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 10:
ce.nxt = H[(r ' + x0).last].nxt, (r ' + x0).last = H[(r ' + x).last].nxt, path(r ' + x0, H[(r ' + x).last, ce]), path(r ' + x, H), (r ' + x0).last ∈ H, (r ' + x).last ∈ H, (r ' + x0).last ∈ H[(r ' + x).last, ce] ⊦ (r ' + x0).last = ce.nxt, (r ' + x0).last = null, r ∈ x, dups(x), r ∈ x0, dups(x0), (r ' + x0).last ≠ (r ' + x).last ∧ r = (r ' + x0).last, ∃ x0. path(r ' + x0 + H[(r ' + x).last].nxt ', H) ∧ (r ' + x).last ≠ r ∧ ¬ (r ' + x).last ∈ x0, ∃ x0. path(H[(r ' + x).last].nxt ' + x0 + H[(r ' + x).last].nxt ', H) ∧ ¬ (r ' + x).last ∈ x0

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]