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Proof step 128 from proof rem-stays-reachable-aux2 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 129:
⊦ x0 ≠ [] → (x1 + x0).last = x0.last
Proof step 130:
¬ x1 ≠ [], H[H[r0].nxt].nxt = r0, path(r ' + x1 + H[r0].nxt ', H), H[r0].nxt ∈ H, r0 ∈ H, path(r0 ' + x + (r ' + x1).last ', H), (r ' + x1).last ∈ H, path(r ' + x1, H) ⊦ H[r0].nxt = null, r0 = r, r = H[r0].nxt, H[r0].nxt = r0, r0 = null, r0 = (r ' + x1).last, r ∈ x1, H[r0].nxt ∈ x1, r0 ∈ x1, dups(x1), r0 ∈ x, (r ' + x1).last ∈ x, dups(x)
Proof step 134:
x1 ≠ [], H[H[r0].nxt].nxt = r0, path(r ' + x1 + H[r0].nxt ', H), H[r0].nxt ∈ H, r0 ∈ H, path(r0 ' + x + x1.last ', H), x1.last ∈ H, path(r ' + x1, H) ⊦ H[r0].nxt = null, r0 = r, r = H[r0].nxt, H[r0].nxt = r0, r0 = null, r0 = x1.last, r ∈ x1, H[r0].nxt ∈ x1, r0 ∈ x1, dups(x1), r0 ∈ x, x1.last ∈ x, dups(x)

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 terms: x1, r '