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Proof step 3 from proof add-preserves-reachable-rev (previous step)

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

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

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[r1, ce], r0, r] flags: <all paths>

Additional information

Used terms: H[r1, ce], r0, r