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Proof step 77 from proof rem-stays-reachable-aux2 (previous step)

x1 = x2 + r1 ' + y0, x2 = (x + H[x2.last].nxt ') + r0 ' + y, (x2.last ' + x0).last = (r ' + x).last, (r ' + x1).last = (r ' + x).last, path(r ' + x1, H), path(r ' + x + H[x2.last].nxt ', H), path(r ' + x2, H), H[x2.last].nxt ∈ H, x2.last ∈ H, path(x2.last ' + x0, H), (r ' + x).last ∈ H ⊦ H[x2.last].nxt = null, x2.last = r, x2 = [], x2.last ∈ x, x2.last ∈ x0, dups(x0), r ∈ x1, dups(x1), r ∈ x2, dups(x2)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 78:
x2 = (x + H[x2.last].nxt ') + r0 ' + y, (x2.last ' + x0).last = (r ' + x).last, (r1 ' + y0).last = (r ' + x).last, path(r ' + x2 + r1 ' + y0, H), path(r ' + x + H[x2.last].nxt ', H), path(r ' + x2, H), H[x2.last].nxt ∈ H, x2.last ∈ H, path(x2.last ' + x0, H), (r ' + x).last ∈ H ⊦ H[x2.last].nxt = null, x2.last = r, x2 = [], r = r1, r ∈ x2, x2.last ∈ x, x2.last ∈ x0, dups(x0), r ∈ y0, r1 ∈ x2, r1 ∈ y0, dups(x2 + y0), dups(x2), dups(y0)

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]%in-rec: ⊦ r ∈ r0 ' + x0 ↔ ¬ ¬(r = r0 ∨ r ∈ x0); from specification list[reflist]%in-append: ⊦ r ∈ x0 + x1 ↔ ¬ ¬(r ∈ x0 ∨ r ∈ x1); from specification list[reflist]%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]%lapp-01: ⊦ (x0 + r ' + x1).last = (r ' + x1).last; from specification list-last[reflist]%lthree: ⊦ x0 ≠ [] → (x1 + x0 + x2).last = (x0 + x2).last; from specification list-last[reflist]