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Proof step 89 from proof reachable-addnew-rev (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 90:
⊦ reachable(r, r0, H) ↔ (∃ x. path(r ' + x, H) ∧ (r ' + x).last = r0)
Proof step 91:
r0 ∈ H[H[r0].nxt, ce], r0 ∈ H, r ∈ H, reachable(r, r0, H), reachable(r, r0, H[H[r0].nxt, ce]), reachable(ce.nxt, H[r0].nxt, H[H[r0].nxt, ce]) ⊦ H[r0].nxt = null, ∃ x. path(r0 ' + x, H[H[r0].nxt, ce]) ∧ (r0 ' + x).last = H[r0].nxt, H[r0].nxt ∈ H, reachable(r, H[r0].nxt, 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], H[r0].nxt, r0] flags: rewrite, current

Additional information

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