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Proof step 122 from proof rem-stays-reachable-aux2 (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 123:
H[H[(r ' + x1).last].nxt].nxt = (r ' + x1).last, path(r ' + x1, H), (r ' + x1).last ∈ H, H[(r ' + x1).last].nxt ∈ H, path(r ' + x1 + H[(r ' + x1).last].nxt ', H) ⊦ (r ' + x1).last = null, H[(r ' + x1).last].nxt = (r ' + x1).last, r = H[(r ' + x1).last].nxt, (r ' + x1).last = r, H[(r ' + x1).last].nxt = null, (r ' + x1).last ∈ x1, H[(r ' + x1).last].nxt ∈ x1, r ∈ x1, dups(x1)

Rule argument

Additional information

Used simplifier rules: path-one: ⊦ path(r ', H) ↔ ¬ ¬(r ∈ H ∧ r ≠ null);%e-01: ⊦ ¬ dups([]); from specification list-dup[reflist]%in-empty: ⊦ ¬ r ∈ []; from specification list[reflist]%l: ⊦ r '.last = r; from specification list-last[reflist]%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]%app: ⊦ x0 + [] = x0; from specification list[reflist]