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Proof step 164 from proof rem-stays-reachable-aux2 (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 165:
H[(r ' + x).last].nxt = H[r1].nxt, (r1 ' + x0).last = (r ' + x).last, path(r1 ' + x0, H), r1 ∈ H, path(r ' + x + H[r1].nxt ' + (r0 ' + x3) + r1 ', H), path(r ' + x, H), (r ' + x).last ∈ H, H[r1].nxt ∈ H ⊦ H[r1].nxt = null, H[r1].nxt = r1, r = H[r1].nxt, r1 = r, r1 = r0, H[r1].nxt = r0, r = r0, r1 ∈ x3, H[r1].nxt ∈ x3, H[r1].nxt ∈ x, r ∈ x3, r ∈ x, dups(x0), r1 ∈ x0, r1 ∈ x, r0 ∈ x3, r0 ∈ x, dups(x + x3), dups(x), dups(x3)

Rule argument

Additional information

Used simplifier rules: a: ⊦ ¬ dups(x0 + x1) → ¬ dups(x0) ∧ ¬ dups(x1); (forward) from specification list-dup[reflist]%f-01: ⊦ dups(x0 + r ' + x1) ↔ r ∈ x0 ∨ r ∈ x1 ∨ dups(x0 + x1); from specification list-dup[reflist]%rec: ⊦ dups(r ' + x0) ↔ ¬ ¬(r ∈ x0 ∨ dups(x0)); from specification list-dup[reflist]%in-06: ⊦ ¬ r ∈ r0 ' + x0 ↔ r ≠ r0 ∧ ¬ r ∈ x0; from specification list[reflist]%in-rec: ⊦ r ∈ r0 ' + x0 ↔ ¬ ¬(r = r0 ∨ r ∈ x0); from specification list[reflist]%in-14: ⊦ r ' + x0 = x1 → r ∈ x1; (forward) from specification list[reflist]%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]%inapp: ⊦ ¬ r ∈ x0 ∧ ¬ r ∈ x1 → (x0 + r ' + x1 = x2 + r ' + x3 ↔ x1 = x3 ∧ x0 = x2); from specification list[reflist]