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Proof step 58 from proof rem-stays-reachable-aux (previous step)

ce.nxt = H[H[(r ' + x).last].nxt].nxt, path(r ' + x0 + H[(r ' + x).last].nxt ', H), path(r ' + x, H), H[(r ' + x).last].nxt ∈ H, (r ' + x).last ∈ H ⊦ H[(r ' + x).last].nxt = ce.nxt, H[(r ' + x).last].nxt = null, (r ' + x).last = r, r ∈ x, dups(x), (r ' + x).last ∈ x0, path(r ' + x0 + H[(r ' + x).last].nxt ', H[(r ' + x).last, ce])

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 59:
⊦ ¬ 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, (r ' + x).last, H, r ' + x0 + H[(r ' + x).last].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 ' + x).last, H, r ' + x0 + H[(r ' + x).last].nxt '