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Proof step 90 from proof rem-stays-reachable-aux (previous step)

ce.nxt = H[H[r0].nxt].nxt, path(H[r0].nxt ' + x1 + H[r0].nxt ', H), path(r ' + x2 + r0 ', H), H[r0].nxt ∈ H, r0 ∈ H ⊦ path(H[r0].nxt ' + x1 + H[r0].nxt ', H[r0, ce]), H[r0].nxt = ce.nxt, H[r0].nxt = null, r = r0, r0 ∈ x1, r ∈ x2, r0 ∈ x2, dups(x2), ∃ x0. path(r ' + x0, H[r0, ce]) ∧ (r ' + x0).last = H[r0].nxt

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 91:
⊦ ¬ r0 ∈ x → (path(x, H[r0, ce]) ↔ path(x, H))

Rule argument

spec: inst: name: path-modify seq: ⊦ ¬ r0 ∈ x → (path(x, H[r0, ce]) ↔ path(x, H)) subst: [ce, r0, H, x] → [ce, r0, H, H[r0].nxt ' + x1 + H[r0].nxt '] flags: rewrite, current

Additional information

Used simplifier rules: in-rec: ⊦ r ∈ r0 ' + x0 ↔ ¬ ¬(r = r0 ∨ 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, r0, H, H[r0].nxt ' + x1 + H[r0].nxt '