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Proof step 53 from proof add-preserves-reachable (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 54:
⊦ reachable(r, r0, H) ↔ (∃ x. path(r ' + x, H) ∧ (r ' + x).last = r0)
Proof step 55:
∃ x. path(H[r].nxt ' + x, H) ∧ (H[r].nxt ' + x).last = ce.nxt, H[r1].nxt = H[ce.nxt].nxt, reachable(H[r].nxt, r1, H), (H[r].nxt ' + x0).last ∈ H, reachable(r, r1, H), ce.nxt ∈ H, path(H[r].nxt ' + x0, H), r ∈ H, r1 ∈ H ⊦ r = r1, r = null, ce.nxt = null, reachable(r, (H[r].nxt ' + x0).last, H[r1, ce]), ∃ x. path(r ' + x, H) ∧ (r ' + x).last = ce.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, ce.nxt, H[r].nxt] flags: rewrite, current

Additional information

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