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Proof step 6 from proof add-preserves-reachable (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 7:
H[r1].nxt = H[ce.nxt].nxt, (H[r].nxt ' + x0).last ∈ H, reachable(r, r1, H), ce.nxt ∈ H, path(H[r].nxt ' + x0, H), r ∈ H, r1 ∈ H, ∀ H0, r2, ce0, r3. H0[r2].nxt = H0[ce0.nxt].nxt ∧ path(r3 ' + x0, H0) ∧ ce0.nxt ∈ H0 ∧ reachable(r3, r2, H0) → ce0.nxt = null ∨ reachable(r3, (r3 ' + x0).last, H0[r2, ce0]) ∨ reachable(r3, ce0.nxt, H0) ⊦ r = null, ce.nxt = null, reachable(r, (H[r].nxt ' + x0).last, H[r1, ce]), reachable(r, ce.nxt, H)

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 ↔ false);%reachable-notnullinH: ⊦ reachable(r, r1, H) → (¬ r1 ∈ H ↔ false);%l-05: ⊦ (r ' + r0 ' + x0).last = (r0 ' + x0).last; from specification list-last[reflist]%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]%c: ⊦ r + x0 = r ' + x0; from specification list[reflist]