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Proof step 43 from proof rem-stays-reachable-aux2 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 44:
⊦ x0 ≠ [] → (x1 + x0).last = x0.last
Proof step 45:
(H[r].nxt ' + x0).last = r, (r ' + x1).last = r, H[r].nxt = H[H[r].nxt].nxt, x1 = [], path(H[r].nxt ' + x0, H), path(r ' + x1, H), r ∈ H, H[r].nxt ∈ H ⊦ H[r].nxt = null, H[r].nxt = r, r = null, dups(x1), dups(x0), r ∈ x1, H[r].nxt ∈ x0

Rule argument

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

Additional information

Used simplifier rules: ⊦ x0.last ∈ x0 ↔ x0 ≠ []; (s) Used terms: x1, r '