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Proof step 39 from proof rem-stays-reachable-aux2 (previous step)

r0 = H[r0].nxt, (r ' + x1).last = (r ' + x).last, (r0 ' + x0).last = (r ' + x).last, (r ' + x).last ∈ H, path(r0 ' + x0, H), r0 ∈ H, path(r ' + x + r0 ', H), path(r ' + x1, H) ⊦ r0 = r, r0 = null, dups(x), r ∈ x, dups(x1), r ∈ x1, dups(x0), r0 ∈ x0, r0 ∈ x

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 40:
⊦ x0 ≠ [] → (x1 + x0).last = x0.last
Proof step 41:
¬ x ≠ [], r0 = H[r0].nxt, (r ' + x1).last = (r ' + x).last, (r0 ' + x0).last = (r ' + x).last, (r ' + x).last ∈ H, path(r0 ' + x0, H), r0 ∈ H, path(r ' + x + r0 ', H), path(r ' + x1, H) ⊦ r0 = r, r0 = null, dups(x), r ∈ x, dups(x1), r ∈ x1, dups(x0), r0 ∈ x0, r0 ∈ x
Proof step 51:
x ≠ [], r0 = H[r0].nxt, (r ' + x1).last = x.last, (r0 ' + x0).last = x.last, x.last ∈ H, path(r0 ' + x0, H), r0 ∈ H, path(r ' + x + r0 ', H), path(r ' + x1, H) ⊦ r0 = r, r0 = null, dups(x), r ∈ x, dups(x1), r ∈ x1, dups(x0), r0 ∈ x0, r0 ∈ x

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 '