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Proof step 108 from proof rem-stays-reachable-aux2 (previous step)

x1 + r2 ' + x2 = (x + H[r1].nxt ' + r0 ' + y).butlast, (r ' + x1).last = (r ' + x).last, (r1 ' + x0).last = (r ' + x).last, r1 = (x + H[r1].nxt ' + r0 ' + y).last, path(r ' + x1, H), path(r ' + x1 + r2 ' + x2 + r1 ', H), path(r ' + x + H[r1].nxt ', H), path(r1 ' + x0, H), (r ' + x).last ∈ H, r1 ∈ H, H[r1].nxt ∈ H, r1 = H[r1].nxt ∨ r1 = r0 ∨ r1 ∈ y ⊦ x + H[r1].nxt ' + r0 ' + y = [], H[r1].nxt = null, r = r2, r1 = r, r2 = r1, 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 109:
x1 + r2 ' + x2 = x + H[(r0 ' + y).last].nxt ' + (r0 ' + y).butlast, (r ' + x1).last = (r ' + x).last, ((r0 ' + y).last ' + x0).last = (r ' + x).last, path(r ' + x1, H), path(r ' + x1 + r2 ' + x2 + (r0 ' + y).last ', H), path(r ' + x + H[(r0 ' + y).last].nxt ', H), path((r0 ' + y).last ' + x0, H), (r ' + x).last ∈ H, (r0 ' + y).last ∈ H, H[(r0 ' + y).last].nxt ∈ 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(x0), dups(x1 + x2), dups(x1), dups(x2), r ∈ x2, r ∈ x1, (r0 ' + y).last ∈ x0, (r0 ' + y).last ∈ x, r2 ∈ x1, r2 ∈ x2, (r0 ' + y).last ∈ x1, (r0 ' + y).last ∈ x2

Rule argument

Additional information

Used simplifier rules: ⊦ x0 + r ' + x1 ≠ []; (s)% ⊦ (x0 + r ' + x1).butlast = x0 + (r ' + x1).butlast; (s)%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]% ⊦ (x0 + r ' + x1).last = (r ' + x1).last; (s)