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Proof step 10 from proof rem-stays-reachable-aux (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 11:
⊦ x0 ≠ [] → (x1 + x0).last = x0.last
Proof step 12:
¬ x ≠ [], ce.nxt = H[(r ' + x0).last].nxt, (r ' + x0).last = H[(r ' + x).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, r ∈ x, dups(x), r ∈ x0, dups(x0), (r ' + x0).last ≠ (r ' + x).last ∧ r = (r ' + x0).last, ∃ 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
Proof step 24:
x ≠ [], ce.nxt = H[(r ' + x0).last].nxt, (r ' + x0).last = H[x.last].nxt, path(r ' + x0, H[x.last, ce]), path(r ' + x, H), (r ' + x0).last ∈ H, x.last ∈ H, (r ' + x0).last ∈ H[x.last, ce] ⊦ (r ' + x0).last = ce.nxt, (r ' + x0).last = null, r ∈ x, dups(x), r ∈ x0, dups(x0), (r ' + x0).last ≠ x.last ∧ r = (r ' + x0).last, ∃ 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

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 '