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Proof step 111 from proof rem-stays-reachable-aux2 (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 112:
x1 + r2 ' + x2 = x + H[r0].nxt ', (r0 ' + x0).last = (r ' + x).last, (r ' + x1).last = (r ' + x).last, path(r0 ' + x0, H), path(r ' + x1 + r2 ' + x2, H), path(r ' + x1 + r2 ' + x2 + r0 ', H), path(r ' + x1, H), H[r0].nxt ∈ H, r0 ∈ H, (r ' + x).last ∈ H, H[r0].nxt ∈ x1 ∨ H[r0].nxt = r2 ∨ H[r0].nxt ∈ x2 ⊦ H[r0].nxt = null, r = r2, r0 = r, r2 = r0, dups(x2), dups(x1), dups(x1 + x2), dups(x0), r0 ∈ x2, r0 ∈ x1, r2 ∈ x2, r2 ∈ x1, r0 ∈ x, r0 ∈ x0, r ∈ x1, r ∈ x2

Rule argument

Additional information

Used simplifier rules: ⊦ r ∈ r0 ' ↔ r = r0; (s)% ⊦ r ∈ x0 + x1 ↔ r ∈ x0 ∨ r ∈ x1; (s)%in-15: ⊦ x0 + r ' = x1 → r ∈ x1; (forward) from specification list[reflist]% ⊦ r '.butlast = []; (s)% ⊦ r '.last = r; (s)%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]% ⊦ x0 + [] = x0; (s)