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Proof step 129 from proof rem-stays-reachable-aux2 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 130:
⊦ x0 ≠ [] → (x1 + x0).last = x0.last
Proof step 131:
H[H[r0].nxt].nxt = r0, x1 = [], path(r ' + x1, H), path(r0 ' + x + (r ' + x1).last ', H), path(r ' + x1 + H[r0].nxt ', H), H[r0].nxt ∈ H, (r ' + x1).last ∈ H, r0 ∈ H ⊦ H[r0].nxt = r0, H[r0].nxt = null, r = H[r0].nxt, r0 = (r ' + x1).last, r0 = r, r0 = null, dups(x), dups(x1), (r ' + x1).last ∈ x, r0 ∈ x, r ∈ x1, r0 ∈ x1, H[r0].nxt ∈ x1
Proof step 135:
H[H[r0].nxt].nxt = r0, path(r ' + x1, H), path(r0 ' + x + x1.last ', H), path(r ' + x1 + H[r0].nxt ', H), H[r0].nxt ∈ H, x1.last ∈ H, r0 ∈ H ⊦ H[r0].nxt = r0, H[r0].nxt = null, r = H[r0].nxt, r0 = x1.last, r0 = r, r0 = null, x1 = [], dups(x), dups(x1), x1.last ∈ x, r0 ∈ x, r ∈ x1, r0 ∈ x1, H[r0].nxt ∈ x1

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 terms: x1, r '