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Proof step 23 from proof rem-reachable-rev-weak (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 24:
⊦ reachable(r, r0, H) ↔ (∃ x. path(r ' + x, H) ∧ (r ' + x).last = r0)
Proof step 25:
(H[r].nxt ' + x).last = (H[r].nxt ' + x0).last, ce.nxt = H[H[r1].nxt].nxt, path(H[r].nxt ' + x0, H[r1, ce]), path(H[r].nxt ' + x, H), r ∈ H, r1 ∈ H, H[r1].nxt ∈ H ⊦ (H[r].nxt ' + x0).last = H[r1].nxt, H[r1].nxt = null, r = null, r = r1, r1 = null, ∃ x. path(r ' + x, H) ∧ (r ' + x).last = (H[r].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[r].nxt ' + x0).last, r] flags: <all paths>

Additional information

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