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Proof step 30 from proof add-preserves-reachable (previous step)

H[r1].nxt = H[ce.nxt].nxt, path(H[r1].nxt ' + x0, H), r1 ∈ H, ce.nxt ∈ H, reachable(H[r1].nxt, r1, H) → reachable(H[r1].nxt, ce.nxt, H) ∨ reachable(H[r1].nxt, (H[r1].nxt ' + x0).last, H[r1, ce]) ⊦ ce.nxt = null, r1 = null, reachable(r1, (H[r1].nxt ' + x0).last, H[r1, ce]), reachable(r1, ce.nxt, H), reachable(H[r1].nxt, (H[r1].nxt ' + x0).last, H)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 31:
⊦ reachable(r, r0, H) ↔ (∃ x. path(r ' + x, H) ∧ (r ' + x).last = r0)
Proof step 32:
H[r1].nxt = H[ce.nxt].nxt, path(H[r1].nxt ' + x0, H), ce.nxt ∈ H, r1 ∈ H, reachable(H[r1].nxt, r1, H) → reachable(H[r1].nxt, ce.nxt, H) ∨ reachable(H[r1].nxt, (H[r1].nxt ' + x0).last, H[r1, ce]) ⊦ ce.nxt = null, r1 = null, reachable(r1, ce.nxt, H), reachable(r1, (H[r1].nxt ' + x0).last, H[r1, ce]), ∃ x. path(H[r1].nxt ' + x, H) ∧ (H[r1].nxt ' + x).last = (H[r1].nxt ' + x0).last

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, (H[r1].nxt ' + x0).last, H[r1].nxt] flags: <all paths>

Additional information

Used terms: H, (H[r1].nxt ' + x0).last, H[r1].nxt