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Proof step 11 from proof reachable-modbehind (previous step)

path(r ' + (r0 + x0), H), reachable((r ' + (r0 + x0)).last, r1, H), r1 ∈ H, (r ' + (r0 + x0)).last ∈ H, ∀ r, H, r1, ce. path(r ' + x0, H) ∧ reachable((r ' + x0).last, r1, H) ∧ (r ' + x0).last ∈ H ∧ r1 ∈ H → path(r ' + x0, H[r1, ce]) ∨ reachable(r1, (r ' + x0).last, H) ⊦ path(r ' + (r0 + x0), H[r1, ce]), reachable(r1, (r ' + (r0 + x0)).last, H)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 12:
path(H[r].nxt ' + x0, H), reachable((H[r].nxt ' + x0).last, r1, H), r ∈ H, r1 ∈ H, (H[r].nxt ' + x0).last ∈ H, r ∈ H[r1, ce], ∀ r2, H0, r3, ce0. path(r2 ' + x0, H0) ∧ reachable((r2 ' + x0).last, r3, H0) → path(r2 ' + x0, H0[r3, ce0]) ∨ reachable(r3, (r2 ' + x0).last, H0) ⊦ r = null, reachable(r1, (H[r].nxt ' + x0).last, H), path(H[r1, ce][r].nxt ' + x0, H[r1, ce]) ∧ H[r1, ce][r].nxt = H[r].nxt

Rule argument

Additional information

Used simplifier rules: path-cons: ⊦ path(r ' + r0 ' + x, H) ↔ r ∈ H ∧ r ≠ null ∧ H[r].nxt = r0 ∧ path(r0 ' + x, H);%path-last: ⊦ path(r ' + x, H) → ((r ' + x).last ∈ H ↔ true);%reachable-notnullinH: ⊦ reachable(r, r1, H) → (r1 ∈ H ↔ true);% ⊦ r ∈ H[r0, ce] ↔ r = r0 ∨ r ∈ H; (ws)% ⊦ (r ' + r0 ' + x0).last = (r0 ' + x0).last; (s)%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]% ⊦ r + x0 = r ' + x0; (s)