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Proof step 16 from proof reachable-nodups-def (previous step)

x.last = (H[r].nxt ' + x0).last, path(H[r].nxt ' + x0, H), r ∈ H, path(H[r].nxt ' + x, H), r ∈ x, ∀ r, H. path(r ' + x0, H) → (∃ x1. path(r ' + x1, H) ∧ (r ' + x1).last = (r ' + x0).last ∧ ¬ r ∈ x1 ∧ ¬ dups(x1)) ⊦ r = null, r = H[r].nxt, dups(x), H[r].nxt ∈ x, ∃ x. path(r ' + x, H) ∧ (r ' + x).last = (H[r].nxt ' + x0).last ∧ ¬ r ∈ x ∧ ¬ dups(x)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 17:
⊦ r ∈ x0 ↔ (∃ x1, x2. x0 = x1 + r ' + x2 ∧ ¬ r ∈ x2)
Proof step 18:
x.last = (H[r].nxt ' + x0).last, path(H[r].nxt ' + x0, H), r ∈ H, path(H[r].nxt ' + x, H), ∃ x1, x2. x = x1 + r ' + x2 ∧ ¬ r ∈ x2, ∀ r, H. path(r ' + x0, H) → (∃ x1. path(r ' + x1, H) ∧ (r ' + x1).last = (r ' + x0).last ∧ ¬ r ∈ x1 ∧ ¬ dups(x1)) ⊦ r = null, r = H[r].nxt, dups(x), H[r].nxt ∈ x, ∃ x. path(r ' + x, H) ∧ (r ' + x).last = (H[r].nxt ' + x0).last ∧ ¬ r ∈ x ∧ ¬ dups(x)

Rule argument

spec: list inst: reflist name: lastoc-rew seq: ⊦ r ∈ x0 ↔ (∃ x1, x2. x0 = x1 + r ' + x2 ∧ ¬ r ∈ x2) subst: [x0, r] → [x, r] flags: rewrite

Additional information

Used terms: x, r