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Proof step 48 from proof rem-stays-reachable-aux1 (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 49:
path(H[H[(r1 ' + x).last].nxt].nxt ' + x0 + H[(r1 ' + x).last].nxt ', H), H[(r1 ' + x).last].nxt ∈ H, (r1 ' + x).last ∈ H, path(H[H[(r1 ' + x).last].nxt].nxt ' + x0 + H[(r1 ' + x).last].nxt ' + r1 ' + x, H), (r1 ' + x).last = r1 ∨ (r1 ' + x).last ∈ x ⊦ (r1 ' + x).last = H[(r1 ' + x).last].nxt, (r1 ' + x).last = H[H[(r1 ' + x).last].nxt].nxt, H[(r1 ' + x).last].nxt = null, H[(r1 ' + x).last].nxt = H[H[(r1 ' + x).last].nxt].nxt, dups((r1 ' + x).butlast), dups(x0), H[H[(r1 ' + x).last].nxt].nxt ∈ (r1 ' + x).butlast, H[H[(r1 ' + x).last].nxt].nxt ∈ x0, (r1 ' + x).last ∈ (r1 ' + x).butlast, dups(x0 + (r1 ' + x).butlast), H[(r1 ' + x).last].nxt ∈ (r1 ' + x).butlast, H[(r1 ' + x).last].nxt ∈ x0, (r1 ' + x).last ∈ x0

Rule argument

Additional information

Used simplifier rules: butlast-last: x0 ≠ [] ⊦ x0.butlast + x0.last ' = x0; from specification list-last[reflist]%e-01: ⊦ r ' + x0 ≠ []; from specification list[reflist]%a: ⊦ ¬ dups(x0 + x1) → ¬ dups(x0) ∧ ¬ dups(x1); (forward) from specification list-dup[reflist]%in-01: ⊦ r ≠ r0 → (r ∈ r0 ' + x0 ↔ r ∈ x0); from specification list[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]%f-01: ⊦ dups(x0 + r ' + x1) ↔ r ∈ x0 ∨ r ∈ x1 ∨ dups(x0 + x1); from specification list-dup[reflist]%btwo: ⊦ (r ' + r0 ' + x0).butlast = r ' + (r0 ' + x0).butlast; from specification list-last[reflist]%bin: ⊦ (x0 + r ' + x1).butlast = x0 + (r ' + x1).butlast; from specification list-last[reflist]%e-03: ⊦ x0 + r ' + x1 ≠ []; from specification list[reflist]%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]%l-05: ⊦ (r ' + r0 ' + x0).last = (r0 ' + x0).last; from specification list-last[reflist]%lapp-01: ⊦ (x0 + r ' + x1).last = (r ' + x1).last; from specification list-last[reflist]