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Proof step 23 from proof reachable-addnew-rev (previous step)

path(r ' + x1, H), r ∈ H ⊦ (r ' + x1).last = H[(r ' + x1).last].nxt, H[(r ' + x1).last].nxt = null, dups(x1), reachable(r, (r ' + x1).last, H), H[(r ' + x1).last].nxt ∈ x1, H[(r ' + x1).last].nxt ∈ H, r ∈ x1

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:
path(r ' + x1, H), r ∈ H ⊦ (r ' + x1).last = H[(r ' + x1).last].nxt, H[(r ' + x1).last].nxt = null, dups(x1), r ∈ x1, H[(r ' + x1).last].nxt ∈ H, H[(r ' + x1).last].nxt ∈ x1, ∃ x. path(r ' + x, H) ∧ (r ' + x).last = (r ' + x1).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, (r ' + x1).last, r] flags: <all paths>

Additional information

Used terms: H, (r ' + x1).last, r