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Proof step 69 from proof acyclic-rem-not-reachable (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 70:
⊦ reachable(r, r0, H) ↔ (∃ x. path(r ' + x, H) ∧ (r ' + x).last = r0)
Proof step 71:
H[H[ce.nxt].nxt].nxt = ce.nxt, path(r ' + x1 + H[ce.nxt].nxt ', H), ce.nxt ∈ H, H[ce.nxt].nxt ∈ H ⊦ ce.nxt = null, ce.nxt = H[ce.nxt].nxt, ce.nxt = r, H[ce.nxt].nxt = null, r = H[ce.nxt].nxt, dups(x1), ce.nxt ∈ x1, r ∈ x1, H[ce.nxt].nxt ∈ x1, ∃ x. path(H[ce.nxt].nxt ' + x, H) ∧ (H[ce.nxt].nxt ' + x).last = ce.nxt

Rule argument

spec: <toplevel> 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[ce.nxt].nxt] flags: <all paths>

Additional information

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