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Proof step 13 from proof removed-stays-reachable (previous step)

reachable(ce.nxt, H[r0].nxt, H) ∧ ¬ reachable(ce.nxt, r0, H) ↔ (∃ x. path(H[r0].nxt ' + x + H[r0].nxt ', H) ∧ ¬ r0 ∈ x), ce.nxt = H[H[r0].nxt].nxt, r0 ∈ H, H[r0].nxt ∈ H, reachable(r, r0, H), H[r0].nxt = r0 ∨ r ≠ H[r0].nxt ⊦ H[r0].nxt = null, H[r0].nxt = ce.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) ↔ reachable(ce.nxt, H[r0].nxt, H) ∧ ¬ reachable(ce.nxt, r0, H) ∨ (∃ r1. reachable(r, r1, H) ∧ H[r1].nxt = H[r0].nxt ∧ ¬ reachable(r0, r1, H))

User applied rule insert lemma

And obtained the following new goals

Rule argument

spec: inst: name: rem-stays-reachable-aux2 seq: reachable(r, r0, H), H[r0].nxt ≠ null, H[r0].nxt ∈ H ⊦ (∃ r1. reachable(r, r1, H) ∧ H[r0].nxt = H[r1].nxt ∧ ¬ reachable(r0, r1, H)) ↔ (∃ x. path(r ' + x + H[r0].nxt ', H) ∧ r0 ≠ r ∧ ¬ r0 ∈ x) subst: [r, r0, H] → [r, r0, H] flags: current, []

Additional information

Used terms: r, r0, H