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Proof step 97 from proof rem-stays-reachable-aux2 (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 98:
H[r2].nxt ' + r0 ' + y = x + r2 ', ((r2 ' + x0).last ' + x).last = (r2 ' + x0).last, (r2 ' + x0).last ∈ H, path(r ' + x2 + (r2 ' + x0).last ' + x, H), path(r ' + (x2 + (r2 ' + x0).last ') + H[r2].nxt ', H), path(r ' + (x2 + (r2 ' + x0).last ' + x) + r2 ', H), H[r2].nxt ∈ H, r2 ∈ H, path(r2 ' + x0, H), H[r2].nxt ∈ x ∨ H[r2].nxt = r2, r2 = H[r2].nxt ∨ r2 = r0 ∨ r2 ∈ y ⊦ r2 = (r2 ' + x0).last, r = r2, H[r2].nxt = null, r = (r2 ' + x0).last, r2 ∈ x2, r2 ∈ x0, dups(x0), r ∈ x2, r ∈ x, r2 ∈ x, (r2 ' + x0).last ∈ x2, (r2 ' + x0).last ∈ x, dups(x2 + x), dups(x2), dups(x)

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]%lapp-03: ⊦ (r ' + x0 + r0 ' + x1).last = (r0 ' + x1).last; from specification list-last[reflist]%app-03: ⊦ x0 + x1 = x2 + x1 ↔ x0 = x2; from specification list[reflist]%app-06: ⊦ x0 + r ' = x1 + r0 ' ↔ x0 = x1 ∧ r = r0; from specification list[reflist]%in-14: ⊦ r ' + x0 = x1 → r ∈ x1; (forward) from specification list[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-15: ⊦ x0 + r ' = x1 → r ∈ x1; (forward) from specification list[reflist]%inex: ⊦ x0 + r ' = x1 + r0 ' + x2 ∧ r ≠ r0 → (∃ x3. x2 = x3 + r '); (forward) from specification list[reflist]%in-16: ⊦ x0 + r ' = x1 + r0 ' + x2 ∧ r ≠ r0 → r ∈ x2; (forward) from specification list[reflist]%in-17: ⊦ r ∈ r0 ' ↔ r = r0; 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]%e-02: ⊦ x0 + r ' ≠ []; from specification list[reflist]