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Proof step 80 from proof acyclic-rem-not-reachable (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 81:
⊦ path(r ' + x + r0 ', H) ↔ path(r ' + x, H) ∧ H[(r ' + x).last].nxt = r0 ∧ r0 ∈ H ∧ r0 ≠ null

Rule argument

spec: inst: name: path-split-last seq: ⊦ path(r ' + x + r0 ', H) ↔ path(r ' + x, H) ∧ H[(r ' + x).last].nxt = r0 ∧ r0 ∈ H ∧ r0 ≠ null subst: [H, r0, x, r] → [H, H[r0].nxt, x2 + r0 ', ce.nxt] flags: rewrite, current

Additional information

Used simplifier rules: path-postfix: ⊦ path(x + r ' + y, H) → (¬ path(r ' + y, H) ↔ false);%last-01: ⊦ (r ' + x0 + r0 ').last = r0; from specification list-last[reflist]%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist] Used terms: H, H[r0].nxt, x2 + r0 ', ce.nxt