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Proof step 51 from proof acyclic-rem-not-reachable (previous step)

ce.nxt = H[H[r0].nxt].nxt, path(r ' + x1 + H[r0].nxt ', H), reachable(r, r0, H), r0 ∈ H, H[r0].nxt ∈ H, ∀ r1. ¬ reachable(H[r1].nxt, r1, H) ⊦ H[r0].nxt = null, r = H[r0].nxt, r0 = r, r0 = H[r0].nxt, dups(x1), H[r0].nxt ∈ x1, r ∈ x1, r0 ∈ x1

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 52:
⊦ reachable(r, r0, H) ↔ (∃ x. path(r ' + x, H) ∧ (r ' + x).last = r0 ∧ ¬ dups(r ' + x))
Proof step 53:
ce.nxt = H[H[r0].nxt].nxt, path(r ' + x1 + H[r0].nxt ', H), H[r0].nxt ∈ H, r0 ∈ H, ∀ r1. ¬ reachable(H[r1].nxt, r1, H), ∃ x. path(r ' + x, H) ∧ (r ' + x).last = r0 ∧ ¬ dups(r ' + x) ⊦ H[r0].nxt = null, r = H[r0].nxt, r0 = H[r0].nxt, r0 = r, dups(x1), r0 ∈ x1, r ∈ x1, H[r0].nxt ∈ x1

Rule argument

spec: <toplevel> 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, r] flags: <all paths>

Additional information

Used terms: H, r0, r