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Proof step 74 from proof acyclic-rem-not-reachable (previous step)

ce.nxt = H[H[r0].nxt].nxt, path(r ' + x1 + H[r0].nxt ' + (r2 ' + x2) + r0 ', H), path(r ' + x1 + H[r0].nxt ', H), r0 ∈ H, H[r0].nxt ∈ H, ∀ r1. ¬ reachable(H[r1].nxt, r1, H) ⊦ H[r0].nxt = r2, r0 = r2, r = r2, r0 = H[r0].nxt, r0 = r, r = H[r0].nxt, H[r0].nxt = null, dups(x2), dups(x1), dups(x1 + x2), r2 ∈ x2, r2 ∈ x1, H[r0].nxt ∈ x2, r0 ∈ x2, r ∈ x2, H[r0].nxt ∈ x1, r ∈ x1, r0 ∈ x1, reachable(ce.nxt, H[r0].nxt, H)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 75:
⊦ reachable(r, r0, H) ↔ (∃ x. path(r ' + x, H) ∧ (r ' + x).last = r0)
Proof step 76:
ce.nxt = H[H[r0].nxt].nxt, path(r ' + x1 + H[r0].nxt ' + (r2 ' + x2) + r0 ', H), path(r ' + x1 + H[r0].nxt ', H), r0 ∈ H, H[r0].nxt ∈ H, ∀ r1. ¬ reachable(H[r1].nxt, r1, H) ⊦ H[r0].nxt = r2, r0 = r2, r = r2, r0 = H[r0].nxt, r0 = r, r = H[r0].nxt, H[r0].nxt = null, dups(x2), dups(x1), dups(x1 + x2), r2 ∈ x2, r2 ∈ x1, H[r0].nxt ∈ x2, r0 ∈ x2, r ∈ x2, H[r0].nxt ∈ x1, r ∈ x1, r0 ∈ x1, ∃ x. path(ce.nxt ' + x, H) ∧ (ce.nxt ' + x).last = H[r0].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[r0].nxt, ce.nxt] flags: rewrite, current

Additional information

Used terms: H, H[r0].nxt, ce.nxt