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Proof step 171 from proof rem-stays-reachable-aux2 (previous step)

(r1 ' + x0).last = (r ' + x).last, H[(r ' + x).last].nxt = H[r1].nxt, path(H[H[r1].nxt].nxt ' + x3 + r1 ', H), H[r1].nxt ∈ H, (r ' + x).last ∈ H, path(r ' + x, H), path(r ' + x + H[r1].nxt ' + (H[H[r1].nxt].nxt ' + x3) + r1 ', H), r1 ∈ H, path(r1 ' + x0, H) ⊦ H[r1].nxt = null, r = H[H[r1].nxt].nxt, H[r1].nxt = H[H[r1].nxt].nxt, r1 = H[H[r1].nxt].nxt, r1 = r, r = H[r1].nxt, H[r1].nxt = r1, dups(x3), dups(x), dups(x + x3), H[H[r1].nxt].nxt ∈ x, H[H[r1].nxt].nxt ∈ x3, r1 ∈ x, r1 ∈ x0, dups(x0), r ∈ x, r ∈ x3, H[r1].nxt ∈ x, H[r1].nxt ∈ x3, r1 ∈ x3

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 172:
⊦ x0 ≠ [] → (x1 + x0).last = x0.last
Proof step 173:
¬ x ≠ [], (r1 ' + x0).last = (r ' + x).last, H[(r ' + x).last].nxt = H[r1].nxt, path(H[H[r1].nxt].nxt ' + x3 + r1 ', H), H[r1].nxt ∈ H, (r ' + x).last ∈ H, path(r ' + x, H), path(r ' + x + H[r1].nxt ' + (H[H[r1].nxt].nxt ' + x3) + r1 ', H), r1 ∈ H, path(r1 ' + x0, H) ⊦ H[r1].nxt = null, r = H[H[r1].nxt].nxt, H[r1].nxt = H[H[r1].nxt].nxt, r1 = H[H[r1].nxt].nxt, r1 = r, r = H[r1].nxt, H[r1].nxt = r1, dups(x3), dups(x), dups(x + x3), H[H[r1].nxt].nxt ∈ x, H[H[r1].nxt].nxt ∈ x3, r1 ∈ x, r1 ∈ x0, dups(x0), r ∈ x, r ∈ x3, H[r1].nxt ∈ x, H[r1].nxt ∈ x3, r1 ∈ x3
Proof step 185:
x ≠ [], (r1 ' + x0).last = x.last, H[x.last].nxt = H[r1].nxt, path(H[H[r1].nxt].nxt ' + x3 + r1 ', H), H[r1].nxt ∈ H, x.last ∈ H, path(r ' + x, H), path(r ' + x + H[r1].nxt ' + (H[H[r1].nxt].nxt ' + x3) + r1 ', H), r1 ∈ H, path(r1 ' + x0, H) ⊦ H[r1].nxt = null, r = H[H[r1].nxt].nxt, H[r1].nxt = H[H[r1].nxt].nxt, r1 = H[H[r1].nxt].nxt, r1 = r, r = H[r1].nxt, H[r1].nxt = r1, dups(x3), dups(x), dups(x + x3), H[H[r1].nxt].nxt ∈ x, H[H[r1].nxt].nxt ∈ x3, r1 ∈ x, r1 ∈ x0, dups(x0), r ∈ x, r ∈ x3, H[r1].nxt ∈ x, H[r1].nxt ∈ x3, r1 ∈ x3

Rule argument

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

Additional information

Used terms: x, r '