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Proof step 46 from proof acyclic-rem-not-reachable (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 47:
⊦ reachable(r, r0, H) ↔ (∃ x. path(r ' + x, H) ∧ (r ' + x).last = r0)
Proof step 48:
ce.nxt = H[H[r].nxt].nxt, path(ce.nxt ' + x + H[r].nxt ', H), r ∈ H, H[r].nxt ∈ H ⊦ H[r].nxt = ce.nxt, r = ce.nxt, r = null, r = H[r].nxt, H[r].nxt = null, dups(x), ce.nxt ∈ x, H[r].nxt ∈ x, r ∈ x, ∃ x. path(ce.nxt ' + x, H) ∧ (ce.nxt ' + x).last = H[r].nxt

Rule argument

spec: inst: name: reachable-def seq: ⊦ reachable(r, r0, H) ↔ (∃ x. path(r ' + x, H) ∧ (r ' + x).last = r0) subst: [H, r0, r] → [H, H[r].nxt, ce.nxt] flags: rewrite, current

Additional information

Used terms: H, H[r].nxt, ce.nxt