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Proof step 12 from proof add-preserves-reachable (previous step)

H[r1].nxt = H[ce.nxt].nxt, path(H[r1].nxt ' + x0, H), r1 ∈ H, ce.nxt ∈ H, reachable(H[r1].nxt, r1, H) → reachable(H[r1].nxt, ce.nxt, H) ∨ reachable(H[r1].nxt, (H[r1].nxt ' + x0).last, H[r1, ce]) ⊦ ce.nxt = null, r1 = null, reachable(r1, (H[r1].nxt ' + x0).last, H[r1, ce]), reachable(r1, ce.nxt, H)

User applied rule insert lemma

And obtained the following new goals

Proof step 13:
⊦ ¬ reachable(r, r0, H) ∧ reachable(r, r1, H) ∧ r0 ≠ null ∧ r0 ∈ H → (reachable(r, r1, H[r0, ce]) ↔ true)
Proof step 14:
H[r1].nxt = H[ce.nxt].nxt, path(H[r1].nxt ' + x0, H), ce.nxt ∈ H, r1 ∈ H, reachable(H[r1].nxt, r1, H) → reachable(H[r1].nxt, ce.nxt, H) ∨ reachable(H[r1].nxt, (H[r1].nxt ' + x0).last, H[r1, ce]), ¬ reachable(H[ce.nxt].nxt, r1, H) ∧ reachable(H[ce.nxt].nxt, (H[ce.nxt].nxt ' + x0).last, H) ∧ r1 ≠ null ∧ r1 ∈ H → (reachable(H[ce.nxt].nxt, (H[ce.nxt].nxt ' + x0).last, H[r1, ce]) ↔ true) ⊦ ce.nxt = null, r1 = null, reachable(r1, ce.nxt, H), reachable(r1, (H[r1].nxt ' + x0).last, H[r1, ce])

Rule argument

spec: <toplevel> name: modnotreachable-reachable-weak seq: ⊦ ¬ reachable(r, r0, H) ∧ reachable(r, r1, H) ∧ r0 ≠ null ∧ r0 ∈ H → (reachable(r, r1, H[r0, ce]) ↔ true) subst: [r, r0, H, r1, ce] → [H[ce.nxt].nxt, r1, H, (H[ce.nxt].nxt ' + x0).last, ce]

Additional information

Used terms: H[ce.nxt].nxt, r1, H, (H[ce.nxt].nxt ' + x0).last, ce