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Proof step 16 from proof rem-reachable (previous step)

ce.nxt = H[H[r1].nxt].nxt, reachable(H[r1].nxt, (H[r1].nxt ' + x0).last, H[r1, ce]), path(H[r1].nxt ' + x0, H), H[r1].nxt ∈ H, r1 ∈ H, (H[r1].nxt ' + x0).last ∈ H, (H[r1].nxt ' + x0).last ∈ H[r1, ce] ⊦ H[r1].nxt = null, (H[r1].nxt ' + x0).last = H[r1].nxt, r1 = null, reachable(r1, (H[r1].nxt ' + x0).last, H[r1, ce]), reachable(ce.nxt, (H[r1].nxt ' + x0).last, H[r1, ce])

User applied rule insert lemma

And obtained the following new goals

Proof step 17:
⊦ reachable(r, r0, H) ∧ r ≠ r0 → (¬ reachable(H[r].nxt, r0, H) ↔ false)
Proof step 18:
reachable(H[r1].nxt, (H[r1].nxt ' + x0).last, H[r1, ce]) ∧ H[r1].nxt ≠ (H[r1].nxt ' + x0).last → (¬ reachable(H[r1, ce][H[r1].nxt].nxt, (H[r1].nxt ' + x0).last, H[r1, ce]) ↔ false), ce.nxt = H[H[r1].nxt].nxt, reachable(H[r1].nxt, (H[r1].nxt ' + x0).last, H[r1, ce]), path(H[r1].nxt ' + x0, H), H[r1].nxt ∈ H, r1 ∈ H, (H[r1].nxt ' + x0).last ∈ H, (H[r1].nxt ' + x0).last ∈ H[r1, ce] ⊦ H[r1].nxt = null, (H[r1].nxt ' + x0).last = H[r1].nxt, r1 = null, reachable(r1, (H[r1].nxt ' + x0).last, H[r1, ce]), reachable(ce.nxt, (H[r1].nxt ' + x0).last, H[r1, ce])

Rule argument

spec: inst: name: fromnext-reachable seq: ⊦ reachable(r, r0, H) ∧ r ≠ r0 → (¬ reachable(H[r].nxt, r0, H) ↔ false) subst: [r, r0, H] → [H[r1].nxt, (H[r1].nxt ' + x0).last, H[r1, ce]] flags: current, []

Additional information

Used terms: H[r1].nxt, (H[r1].nxt ' + x0).last, H[r1, ce]