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Proof step 148 from proof rem-stays-reachable-aux2 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: list-last inst: reflist name: lapp seq: ⊦ x0 ≠ [] → (x1 + x0).last = x0.last subst: [x0, x1] → [x1, r '] flags: rewrite

Additional information

Used simplifier rules: lin-01: ⊦ x0.last ∈ x0 ↔ x0 ≠ []; from specification list-last[reflist]%app: ⊦ x0 + [] = x0; from specification list[reflist]%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]%l: ⊦ r '.last = r; from specification list-last[reflist] Used terms: x1, r '