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Proof step 165 from proof rem-stays-reachable-aux2 (previous step)

x1 + r2 ' + x2 = x + H[r1].nxt ' + r0 ' + x3, (r ' + x1).last = (r ' + x).last, (r1 ' + x0).last = (r ' + x).last, H[(r ' + x1).last].nxt = r2, H[(r ' + x).last].nxt = H[r1].nxt, path(r ' + x1, H), path(r ' + x1 + r2 ' + x2 + r1 ', H), path(r ' + x, H), path(r1 ' + x0, H), H[r1].nxt ∈ H, (r ' + x).last ∈ H, r1 ∈ H, r2 ∈ H ⊦ H[r1].nxt = null, r = r2, r1 = r, r2 = r1, r2 = null, dups(x0), dups(x1 + x2), dups(x1), dups(x2), r ∈ x2, r ∈ x1, r1 ∈ x0, r1 ∈ x, r2 ∈ x1, r2 ∈ x2, r1 ∈ x1, r1 ∈ x2

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 166:
(r1 ' + x0).last = (r ' + x).last, H[(r ' + x).last].nxt = H[r1].nxt, path(r1 ' + x0, H), path(r ' + x + H[r1].nxt ' + (r0 ' + x3) + r1 ', H), path(r ' + x, H), r1 ∈ H, (r ' + x).last ∈ H, H[r1].nxt ∈ H ⊦ H[r1].nxt = null, H[r1].nxt = r1, H[r1].nxt = r0, r = H[r1].nxt, r = r0, r1 = r, r1 = r0, dups(x0), dups(x + x3), dups(x), dups(x3), r1 ∈ x3, H[r1].nxt ∈ x3, H[r1].nxt ∈ x, r1 ∈ x, r1 ∈ x0, r ∈ x, r ∈ x3, r0 ∈ x3, r0 ∈ x

Rule argument

Additional information

Used simplifier rules: a: ⊦ ¬ dups(x0 + x1) → ¬ dups(x0) ∧ ¬ dups(x1); (forward) from specification list-dup[reflist]% ⊦ dups(x0 + r ' + x1) ↔ r ∈ x0 ∨ r ∈ x1 ∨ dups(x0 + x1); (ws)% ⊦ dups(r ' + x0) ↔ dups(x0) ∨ r ∈ x0; (ws)% ⊦ r ∈ r0 ' ↔ r = r0; (s)% ⊦ r ∈ x0 + x1 ↔ r ∈ x0 ∨ r ∈ x1; (s)%in-14: ⊦ r ' + x0 = x1 → r ∈ x1; (forward) from specification list[reflist]%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]%⊦ ¬ r ∈ x0 ∧ ¬ r ∈ x1 → (x0 + r ' + x1 = x2 + r ' + x3 ↔ x1 = x3 ∧ x0 = x2); (ws)