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Proof step 82 from proof rem-stays-reachable-aux2 (previous step)

(r ' + x1).last = (r ' + x).last, x1 + r2 ' + x3 = x + H[(r2 ' + x3).last].nxt ' + r0 ' + y, ((r2 ' + x3).last ' + x0).last = (r ' + x).last, (r ' + x).last ∈ H, path(r ' + x1, H), path(r ' + x + H[(r2 ' + x3).last].nxt ', H), path(r ' + x1 + r2 ' + x3, H), H[(r2 ' + x3).last].nxt ∈ H, (r2 ' + x3).last ∈ H, path((r2 ' + x3).last ' + x0, H) ⊦ H[(r2 ' + x3).last].nxt = null, (r2 ' + x3).last = r, r = r2, (r2 ' + x3).last ∈ x, (r2 ' + x3).last ∈ x0, dups(x0), r ∈ x1, r ∈ x3, r2 ∈ x1, r2 ∈ x3, dups(x1 + x3), dups(x1), dups(x3)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 83:
⊦ x0 ≠ [] → (x1 + x0).last = x0.last
Proof step 84:
¬ x3 ≠ [], (r ' + x1).last = (r ' + x).last, x1 + r2 ' + x3 = x + H[(r2 ' + x3).last].nxt ' + r0 ' + y, ((r2 ' + x3).last ' + x0).last = (r ' + x).last, (r ' + x).last ∈ H, path(r ' + x1, H), path(r ' + x + H[(r2 ' + x3).last].nxt ', H), path(r ' + x1 + r2 ' + x3, H), H[(r2 ' + x3).last].nxt ∈ H, (r2 ' + x3).last ∈ H, path((r2 ' + x3).last ' + x0, H) ⊦ H[(r2 ' + x3).last].nxt = null, (r2 ' + x3).last = r, r = r2, (r2 ' + x3).last ∈ x, (r2 ' + x3).last ∈ x0, dups(x0), r ∈ x1, r ∈ x3, r2 ∈ x1, r2 ∈ x3, dups(x1 + x3), dups(x1), dups(x3)
Proof step 102:
x3 ≠ [], (r ' + x1).last = (r ' + x).last, x1 + r2 ' + x3 = x + H[x3.last].nxt ' + r0 ' + y, (x3.last ' + x0).last = (r ' + x).last, (r ' + x).last ∈ H, path(r ' + x1, H), path(r ' + x + H[x3.last].nxt ', H), path(r ' + x1 + r2 ' + x3, H), H[x3.last].nxt ∈ H, x3.last ∈ H, path(x3.last ' + x0, H) ⊦ H[x3.last].nxt = null, x3.last = r, r = r2, x3.last ∈ x, x3.last ∈ x0, dups(x0), r ∈ x1, r ∈ x3, r2 ∈ x1, r2 ∈ x3, dups(x1 + x3), dups(x1), dups(x3)

Rule argument

spec: list-last inst: reflist name: lapp seq: ⊦ x0 ≠ [] → (x1 + x0).last = x0.last subst: [x0, x1] → [x3, r2 '] flags: rewrite

Additional information

Used terms: x3, r2 '