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Proof step 2 from proof acyclic-rem-not-reachable (previous step)

ce.nxt = H[H[r0].nxt].nxt, reachable(r, H[r0].nxt, H[r0, ce]), reachable(r, r0, H), H[r0].nxt ∈ H, H[r0].nxt ∈ H[r0, ce], r0 ∈ H, ∀ r1. ¬ reachable(H[r1].nxt, r1, H) ⊦ H[r0].nxt = null

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 3:
⊦ reachable(r, r0, H) ↔ (∃ x. path(r ' + x, H) ∧ (r ' + x).last = r0 ∧ ¬ dups(r ' + x))
Proof step 4:
ce.nxt = H[H[r0].nxt].nxt, ∃ x. path(r ' + x, H[r0, ce]) ∧ (r ' + x).last = H[r0].nxt ∧ ¬ dups(r ' + x), reachable(r, r0, H), H[r0].nxt ∈ H, H[r0].nxt ∈ H[r0, ce], r0 ∈ H, ∀ r1. ¬ reachable(H[r1].nxt, r1, H) ⊦ H[r0].nxt = null

Rule argument

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

Additional information

Used terms: H[r0, ce], H[r0].nxt, r