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Proof step 27 from proof rem-stays-reachable-aux2 (previous step)

(r0 ' + x0).last = (r ' + x).last, path(r0 ' + x0, H), path(r ' + x + H[r0].nxt ', H), reachable(r, (r ' + x).last, H), reachable(r, r0, H), H[r0].nxt ∈ H, r0 ∈ H, (r ' + x).last ∈ H ⊦ H[r0].nxt = null, r0 = r, dups(x0), r0 ∈ x, r0 ∈ x0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 28:
⊦ reachable(r, r0, H) ↔ (∃ x. path(r ' + x, H) ∧ (r ' + x).last = r0 ∧ ¬ dups(r ' + x))
Proof step 29:
(r0 ' + x0).last = (r ' + x).last, path(r ' + x + H[r0].nxt ', H), path(r0 ' + x0, H), reachable(r, r0, H), (r ' + x).last ∈ H, r0 ∈ H, H[r0].nxt ∈ H, ∃ x0. path(r ' + x0, H) ∧ (r ' + x0).last = (r ' + x).last ∧ ¬ dups(r ' + x0) ⊦ H[r0].nxt = null, r0 = r, dups(x0), r0 ∈ x0, r0 ∈ x

Rule argument

spec: <toplevel> name: reachable-nodups-def seq: ⊦ reachable(r, r0, H) ↔ (∃ x. path(r ' + x, H) ∧ (r ' + x).last = r0 ∧ ¬ dups(r ' + x)) subst: [H, r0, r] → [H, (r ' + x).last, r] flags: <all paths>

Additional information

Used terms: H, (r ' + x).last, r