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Proof step 40 from proof acyclic-rem-not-reachable (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: inst: name: path-modify seq: ⊦ ¬ r0 ∈ x → (path(x, H[r0, ce]) ↔ path(x, H)) subst: [ce, r0, H, x] → [ce, r, H, ce.nxt ' + x + H[r].nxt '] flags: rewrite, current

Additional information

Used simplifier rules: in-01: ⊦ r ≠ r0 → (r ∈ r0 ' + x0 ↔ r ∈ x0); from specification list[reflist]%in-append: ⊦ r ∈ x0 + x1 ↔ ¬ ¬(r ∈ x0 ∨ r ∈ x1); from specification list[reflist]%in-17: ⊦ r ∈ r0 ' ↔ r = r0; from specification list[reflist]%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist] Used terms: ce, r, H, ce.nxt ' + x + H[r].nxt '