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Proof step 146 from proof rem-stays-reachable-aux2 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 147:
⊦ x0 ≠ [] → (x1 + x0).last = x0.last
Proof step 148:
(r ' + x1).last = (r2 ' + x).last, (r0 ' + x0).last = (r2 ' + x).last, x = [], path(r ' + x1, H), path(r ' + x1 + r2 ' + (x + H[r0].nxt ') + r0 ', H), path(r ' + x1 + r2 ' + x + H[r0].nxt ', H), path(r0 ' + x0, H), (r2 ' + x).last ∈ H, r0 ∈ H, H[r0].nxt ∈ H ⊦ H[r0].nxt = null, r = H[r0].nxt, r = r2, r0 = H[r0].nxt, r0 = r, r2 = H[r0].nxt, r2 = r0, dups(x), dups(x1), dups(x1 + x), dups(x0), H[r0].nxt ∈ x1, H[r0].nxt ∈ x, r0 ∈ x, r2 ∈ x1, r2 ∈ x, r0 ∈ x1, r ∈ x, r ∈ x1, r0 ∈ x0
Proof step 151:
(r ' + x1).last = x.last, (r0 ' + x0).last = x.last, path(r ' + x1, H), path(r ' + x1 + r2 ' + (x + H[r0].nxt ') + r0 ', H), path(r ' + x1 + r2 ' + x + H[r0].nxt ', H), path(r0 ' + x0, H), x.last ∈ H, r0 ∈ H, H[r0].nxt ∈ H ⊦ H[r0].nxt = null, r = H[r0].nxt, r = r2, r0 = H[r0].nxt, r0 = r, r2 = H[r0].nxt, r2 = r0, x = [], dups(x), dups(x1), dups(x1 + x), dups(x0), H[r0].nxt ∈ x1, H[r0].nxt ∈ x, r0 ∈ x, r2 ∈ x1, r2 ∈ x, r0 ∈ x1, r ∈ x, r ∈ x1, r0 ∈ x0

Rule argument

spec: list-last, inst: reflist name: lapp seq: ⊦ x0 ≠ [] → (x1 + x0).last = x0.last subst: [x0, x1] → [x, r2 '] flags: <all paths>

Additional information

Used terms: x, r2 '