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Proof step 2 from proof add-preserves-reachable (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 3:
⊦ reachable(r, r0, H) ↔ (∃ x. path(r ' + x, H) ∧ (r ' + x).last = r0)
Proof step 4:
H[r1].nxt = H[ce.nxt].nxt, ∃ x. path(r ' + x, H) ∧ (r ' + x).last = r0, ce.nxt ∈ H, reachable(r, r1, H), r0 ∈ H, r1 ∈ H ⊦ ce.nxt = null, reachable(r, r0, H[r1, ce]), reachable(r, ce.nxt, H)

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, r0, r] flags: rewrite, current

Additional information

Used terms: H, r0, r