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Proof step 14 from proof reachable-modbehind (previous step)

(H[r1].nxt ' + x0).last ∈ H, reachable((H[r1].nxt ' + x0).last, r1, H), path(H[r1].nxt ' + x0, H), r1 ∈ H, ∀ r2, H0, r3, ce0. path(r2 ' + x0, H0) ∧ reachable((r2 ' + x0).last, r3, H0) → reachable(r3, (r2 ' + x0).last, H0) ∨ path(r2 ' + x0, H0[r3, ce0]) ⊦ r1 = null, reachable(r1, (H[r1].nxt ' + x0).last, H), path(ce.nxt ' + x0, H[r1, ce]) ∧ ce.nxt = H[r1].nxt

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 15:
⊦ reachable(r, r0, H) ↔ (∃ x. path(r ' + x, H) ∧ (r ' + x).last = r0)
Proof step 16:
(H[r1].nxt ' + x0).last ∈ H, reachable((H[r1].nxt ' + x0).last, r1, H), path(H[r1].nxt ' + x0, H), r1 ∈ H, ∀ r2, H0, r3, ce0. path(r2 ' + x0, H0) ∧ reachable((r2 ' + x0).last, r3, H0) → reachable(r3, (r2 ' + x0).last, H0) ∨ path(r2 ' + x0, H0[r3, ce0]) ⊦ r1 = null, ∃ x. path(r1 ' + x, H) ∧ (r1 ' + x).last = (H[r1].nxt ' + x0).last, path(ce.nxt ' + x0, H[r1, ce]) ∧ ce.nxt = H[r1].nxt

Rule argument

spec: inst: name: reachable-def seq: ⊦ reachable(r, r0, H) ↔ (∃ x. path(r ' + x, H) ∧ (r ' + x).last = r0) subst: [H, r0, r] → [H, (H[r1].nxt ' + x0).last, r1] flags: rewrite, current

Additional information

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