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Proof step 123 from proof rem-stays-reachable-aux2 (previous step)

(r0 ' + x0).last = (r ' + x1).last, H[H[r0].nxt].nxt = r0, x0 = [], path(r ' + x1, H), path(r0 ' + x0, H), path(r ' + x1 + H[r0].nxt ', H), H[r0].nxt ∈ H, (r ' + x1).last ∈ H, r0 ∈ H ⊦ H[r0].nxt = r0, H[r0].nxt = null, r = H[r0].nxt, r0 = r, r0 = null, dups(x0), dups(x1), r0 ∈ x0, r ∈ x1, r0 ∈ x1, H[r0].nxt ∈ x1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 124:
H[H[(r ' + x1).last].nxt].nxt = (r ' + x1).last, path(r ' + x1, H), path(r ' + x1 + H[(r ' + x1).last].nxt ', H), H[(r ' + x1).last].nxt ∈ H, (r ' + x1).last ∈ H ⊦ (r ' + x1).last = r, (r ' + x1).last = null, H[(r ' + x1).last].nxt = (r ' + x1).last, H[(r ' + x1).last].nxt = null, r = H[(r ' + x1).last].nxt, dups(x1), r ∈ x1, (r ' + x1).last ∈ x1, H[(r ' + x1).last].nxt ∈ x1

Rule argument

Additional information

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