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Proof step 45 from proof add-preserves-reachable-rev (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 46:
⊦ reachable(r, r0, H) ↔ (∃ x. path(r ' + x, H) ∧ (r ' + x).last = r0)
Proof step 47:
H[r1].nxt = H[ce.nxt].nxt, path(ce.nxt ' + x0, H[r1, ce]), (ce.nxt ' + x0).last ∈ H[r1, ce], r1 ∈ H, ce.nxt ∈ H ⊦ (ce.nxt ' + x0).last = ce.nxt, ce.nxt = null, r1 = null, reachable(r1, ce.nxt, H), reachable(r1, (ce.nxt ' + x0).last, H), ∃ x. path(r1 ' + x, H) ∧ (r1 ' + x).last = H[r1].nxt

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, r1] flags: <all paths>

Additional information

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