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Proof step 93 from proof reachable-addnew-rev (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 94:
⊦ reachable(r, r0, H) ↔ (∃ x. path(r ' + x, H) ∧ (r ' + x).last = r0)
Proof step 95:
reachable(ce.nxt, r1, H), ∃ x. path(ce.nxt ' + x, H[H[r0].nxt, ce]) ∧ (ce.nxt ' + x).last = r1, reachable(r, r0, H[H[r0].nxt, ce]), reachable(r, r0, H), r ∈ H, r0 ∈ H, r0 ∈ H[H[r0].nxt, ce], r1 ∈ H[H[r0].nxt, ce], r1 ∈ H ⊦ H[r0].nxt = null, reachable(r, r1, H[H[r0].nxt, ce]), reachable(r, r1, H), H[r0].nxt ∈ H, reachable(r0, r1, H[H[r0].nxt, ce])

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[H[r0].nxt, ce], r1, ce.nxt] flags: rewrite, current

Additional information

Used terms: H[H[r0].nxt, ce], r1, ce.nxt