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Proof step 53 from proof add-preserves-reachable-rev (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 54:
⊦ reachable(r, r0, H) ↔ (∃ x. path(r ' + x, H) ∧ (r ' + x).last = r0)
Proof step 55:
H[r1].nxt = H[ce.nxt].nxt, r1 ∈ H, (H[r1].nxt ' + x).last ∈ H[r1, ce], path(H[r1].nxt ' + x, H[r1, ce]), ce.nxt ∈ H ⊦ (H[r1].nxt ' + x).last = ce.nxt, r1 = null, ce.nxt = null, ce.nxt = r1, ∃ x0. path(r1 ' + x0, H) ∧ (r1 ' + x0).last = (H[r1].nxt ' + x).last, reachable(r1, ce.nxt, H), H[r1].nxt ∈ H ∧ H[r1].nxt ≠ null

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, (H[r1].nxt ' + x).last, r1] flags: rewrite, current

Additional information

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