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Proof step 62 from proof reachable-addnew-rev (previous step)

x1 = x + r0 ', path(r ' + x1 + H[r0].nxt ' + x0, H[H[r0].nxt, ce]), r ∈ H, (H[r0].nxt ' + x0).last ∈ H, path(r ' + x1, H) ⊦ H[r0].nxt = null, r0 = H[r0].nxt, x1 = [], reachable(ce.nxt, (H[r0].nxt ' + x0).last, H), H[r0].nxt ∈ H, r ∈ x1, r ∈ x0, H[r0].nxt ∈ x1, H[r0].nxt ∈ x0, dups(x1 + x0), reachable(r, (H[r0].nxt ' + x0).last, H), dups(x1), dups(x0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 63:
path(r ' + (x + r0 ') + H[r0].nxt ' + x0, H[H[r0].nxt, ce]), r ∈ H, (H[r0].nxt ' + x0).last ∈ H, path(r ' + x + r0 ', H) ⊦ H[r0].nxt = null, r0 = H[r0].nxt, r = r0, reachable(ce.nxt, (H[r0].nxt ' + x0).last, H), H[r0].nxt ∈ H, r ∈ x, r ∈ x0, H[r0].nxt ∈ x, H[r0].nxt ∈ x0, r0 ∈ x, r0 ∈ x0, dups(x + x0), reachable(r, (H[r0].nxt ' + x0).last, H), dups(x), dups(x0)

Rule argument

Additional information

Used simplifier rules: a: ⊦ ¬ dups(x0 + x1) → ¬ dups(x0) ∧ ¬ dups(x1); (forward) from specification list-dup[reflist]%l-03: ⊦ ¬ r ∈ x0 → (dups(x0 + r ') ↔ dups(x0)); from specification list-dup[reflist]%f-01: ⊦ dups(x0 + r ' + x1) ↔ r ∈ x0 ∨ r ∈ x1 ∨ dups(x0 + x1); from specification list-dup[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]