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Proof step 62 from proof rem-stays-reachable-aux (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 63:
⊦ x0 ≠ [] → (x1 + x0).last = x0.last
Proof step 64:
ce.nxt = H[H[(r ' + x).last].nxt].nxt, x = [], path(r ' + x, H), path(H[(r ' + x).last].nxt ' + x1 + H[(r ' + x).last].nxt ', H), H[(r ' + x).last].nxt ∈ H, (r ' + x).last ∈ H ⊦ H[(r ' + x).last].nxt = ce.nxt, H[(r ' + x).last].nxt = null, dups(x), r ∈ x, (r ' + x).last ∈ x1, ∃ x0. path(r ' + x0, H[(r ' + x).last, ce]) ∧ (r ' + x0).last = H[(r ' + x).last].nxt
Proof step 71:
ce.nxt = H[H[x.last].nxt].nxt, path(r ' + x, H), path(H[x.last].nxt ' + x1 + H[x.last].nxt ', H), H[x.last].nxt ∈ H, x.last ∈ H ⊦ H[x.last].nxt = ce.nxt, H[x.last].nxt = null, x = [], dups(x), r ∈ x, x.last ∈ x1, ∃ x0. path(r ' + x0, H[x.last, ce]) ∧ (r ' + x0).last = H[x.last].nxt

Rule argument

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

Additional information

Used terms: x, r '