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Proof step 3 from proof rem-reachable (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 4:
⊦ reachable(r, r0, H) ↔ (∃ x. path(r ' + x, H) ∧ (r ' + x).last = r0)
Proof step 5:
ce.nxt = H[H[r1].nxt].nxt, H[r1].nxt ∈ H, r1 ∈ H, ∃ x. path(r ' + x, H) ∧ (r ' + x).last = r0 ⊦ H[r1].nxt = null, r0 = H[r1].nxt, r1 = null, reachable(r, r0, H[r1, ce])

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, r0, r] flags: <all paths>

Additional information

Used terms: H, r0, r