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Proof step 6 from proof rem-reachable (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

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

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);%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]