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Proof step 11 from proof cyclic-def-shortest (previous step)

validrefs(str, r ' + refs1, st[r -- _type] .type, st), ∀ st, r, str, refs. # refs < # refs1 → ( validrefs(str, r ' + refs, st[r -- _type] .type, st) → (∃ refs0, r0. ¬ dups(refs0) ∧ r0 ∈ refs0 ∧ refs0 .first = r ∧ validrefs(str, refs0 + r0 ', st[refs0 .first -- _type] .type, st)) ∨ ¬ r ∈ refs ∧ ¬ dups(refs)) ⊦ refs1 = [], ¬ r ∈ refs1 ∧ ¬ dups(refs1), ∃ refs0, r0. ¬ dups(refs0) ∧ r0 ∈ refs0 ∧ refs0 .first = r ∧ validrefs(str, refs0 + r0 ', st[refs0 .first -- _type] .type, st), ¬ r ∈ refs1 .butlast ∧ ¬ dups(refs1 .butlast) ∧ (refs1 .last = r ∨ refs1 .last ∈ refs1 .butlast)

User applied rule insert elim lemma

And obtained the following new goals

Proof step 10:
refs1 = refs2 + r1 ', validrefs(str, r ' + refs1, st[r -- _type] .type, st), ∀ st, r, str, refs. # refs < # refs1 → ( validrefs(str, r ' + refs, st[r -- _type] .type, st) → (∃ refs0, r0. ¬ dups(refs0) ∧ r0 ∈ refs0 ∧ refs0 .first = r ∧ validrefs(str, refs0 + r0 ', st[refs0 .first -- _type] .type, st)) ∨ ¬ r ∈ refs ∧ ¬ dups(refs)) ⊦ refs1 = [], ¬ r ∈ refs1 ∧ ¬ dups(refs1), ∃ refs0, r0. ¬ dups(refs0) ∧ r0 ∈ refs0 ∧ refs0 .first = r ∧ validrefs(str, refs0 + r0 ', st[refs0 .first -- _type] .type, st), ¬ r ∈ refs2 ∧ ¬ dups(refs2) ∧ (r1 = r ∨ r1 ∈ refs2)

Rule argument

[speclemmaarg: refs ≠ [] ,r = refs .last ∧ refs0 = refs .butlast ↔ refs = refs0 + r ' ,[refs, r, refs0] → [refs1, r1, refs2] ,T ,list-last ,references ,last-elim ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPELIM : refs ≠ [] ⊦ r = refs .last ∧ refs0 = refs .butlast ↔ refs = refs0 + r '])], [EXTRATERMS : (refs1, r1, refs2)]]