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Proof step 6 from proof acyclic-rem-not-reachable (previous step)

(r ' + x).last = H[r0].nxt, ce.nxt = H[H[r0].nxt].nxt, path(r ' + x, H[r0, ce]), reachable(r, r0, H), H[r0].nxt ∈ H, r0 ∈ H, H[r0].nxt ∈ H[r0, ce], ∀ r1. ¬ reachable(H[r1].nxt, r1, H) ⊦ H[r0].nxt = null, dups(x), r ∈ x

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 7:
⊦ x0 ≠ [] → (x1 + x0).last = x0.last
Proof step 8:
(r ' + x).last = H[r0].nxt, ce.nxt = H[H[r0].nxt].nxt, x = [], path(r ' + x, H[r0, ce]), reachable(r, r0, H), H[r0].nxt ∈ H[r0, ce], r0 ∈ H, H[r0].nxt ∈ H, ∀ r1. ¬ reachable(H[r1].nxt, r1, H) ⊦ H[r0].nxt = null, dups(x), r ∈ x
Proof step 10:
x.last = H[r0].nxt, ce.nxt = H[H[r0].nxt].nxt, path(r ' + x, H[r0, ce]), reachable(r, r0, H), H[r0].nxt ∈ H[r0, ce], r0 ∈ H, H[r0].nxt ∈ H, ∀ r1. ¬ reachable(H[r1].nxt, r1, H) ⊦ H[r0].nxt = null, x = [], dups(x), r ∈ x

Rule argument

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

Additional information

Used terms: x, r '