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Proof step 11 from proof rem-stays-reachable-aux (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 12:
⊦ x0 ≠ [] → (x1 + x0).last = x0.last
Proof step 13:
(r ' + x0).last = H[(r ' + x).last].nxt, ce.nxt = H[(r ' + x0).last].nxt, x = [], path(r ' + x, H), path(r ' + x0, H[(r ' + x).last, ce]), (r ' + x0).last ∈ H[(r ' + x).last, ce], (r ' + x).last ∈ H, (r ' + x0).last ∈ H ⊦ (r ' + x0).last = null, (r ' + x0).last = ce.nxt, dups(x0), dups(x), r ∈ x0, r ∈ x, ∃ x0. path(r ' + x0 + H[(r ' + x).last].nxt ', H) ∧ (r ' + x).last ≠ r ∧ ¬ (r ' + x).last ∈ x0, ∃ x0. path(H[(r ' + x).last].nxt ' + x0 + H[(r ' + x).last].nxt ', H) ∧ ¬ (r ' + x).last ∈ x0, (r ' + x0).last ≠ (r ' + x).last ∧ r = (r ' + x0).last
Proof step 25:
(r ' + x0).last = H[x.last].nxt, ce.nxt = H[(r ' + x0).last].nxt, path(r ' + x, H), path(r ' + x0, H[x.last, ce]), (r ' + x0).last ∈ H[x.last, ce], x.last ∈ H, (r ' + x0).last ∈ H ⊦ (r ' + x0).last = null, (r ' + x0).last = ce.nxt, x = [], dups(x0), dups(x), r ∈ x0, r ∈ x, ∃ x0. path(r ' + x0 + H[x.last].nxt ', H) ∧ x.last ≠ r ∧ ¬ x.last ∈ x0, ∃ x0. path(H[x.last].nxt ' + x0 + H[x.last].nxt ', H) ∧ ¬ x.last ∈ x0, (r ' + x0).last ≠ x.last ∧ r = (r ' + x0).last

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 '