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Proof step 50 from proof reachable-addnew-rev (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 51:
path(H[r].nxt ' + x0, H[H[r].nxt, ce]), r ∈ H, (H[r].nxt ' + x0).last ∈ H ⊦ H[r].nxt = null, r = H[r].nxt, r = null, dups(x0), reachable(r, (H[r].nxt ' + x0).last, H), reachable(ce.nxt, (H[r].nxt ' + x0).last, H), H[r].nxt ∈ H, r ∈ x0, H[r].nxt ∈ x0

Rule argument

Additional information

Used simplifier rules: path-one: ⊦ path(r ', H) ↔ r ∈ H ∧ r ≠ null;%path-cons: ⊦ path(r ' + r0 ' + x, H) ↔ r ∈ H ∧ r ≠ null ∧ H[r].nxt = r0 ∧ path(r0 ' + x, H);% ⊦ ¬ dups([]); (s)% ⊦ ¬ r ∈ []; (s)% ⊦ r ≠ r0 → H[r0, ce][r] = H[r]; (s)% ⊦ r ≠ r0 → (r ∈ H[r0, ce] ↔ r ∈ H); (s)% ⊦ [] + x0 = x0; (s)% ⊦ r '.last = r; (s)%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]% ⊦ x0 + [] = x0; (s)