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Proof step 2 from proof rem-stays-reachable-aux (previous step)

ce.nxt = H[H[r0].nxt].nxt, reachable(r, r0, H), H[r0].nxt ∈ H, r0 ∈ H ⊦ H[r0].nxt = ce.nxt, H[r0].nxt = null, reachable(r, H[r0].nxt, H[r0, ce]) ↔ H[r0].nxt ≠ r0 ∧ r = H[r0].nxt ∨ (∃ x. path(r ' + x + H[r0].nxt ', H) ∧ r0 ≠ r ∧ ¬ r0 ∈ x) ∨ (∃ x. path(H[r0].nxt ' + x + H[r0].nxt ', H) ∧ ¬ r0 ∈ x)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 3:
⊦ reachable(r, r0, H) ↔ (∃ x. path(r ' + x, H) ∧ (r ' + x).last = r0 ∧ ¬ dups(r ' + x))
Proof step 4:
ce.nxt = H[H[r0].nxt].nxt, ∃ x. path(r ' + x, H) ∧ (r ' + x).last = r0 ∧ ¬ dups(r ' + x), H[r0].nxt ∈ H, r0 ∈ H ⊦ H[r0].nxt = ce.nxt, H[r0].nxt = null, reachable(r, H[r0].nxt, H[r0, ce]) ↔ H[r0].nxt ≠ r0 ∧ r = H[r0].nxt ∨ (∃ x. path(r ' + x + H[r0].nxt ', H) ∧ r0 ≠ r ∧ ¬ r0 ∈ x) ∨ (∃ x. path(H[r0].nxt ' + x + H[r0].nxt ', H) ∧ ¬ r0 ∈ x)

Rule argument

spec: inst: 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, r0, r] flags: rewrite, current

Additional information

Used terms: H, r0, r