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Proof step 18 from proof cyclic-def-shortest (previous step)

dups(refs), validrefs(str, refs, st[refs .first -- _type] .type, st) ⊦ ∃ refs0, r0. ¬ dups(refs0) ∧ r0 ∈ refs0 ∧ refs0 .first = refs .first ∧ validrefs(str, refs0 + r0 ', st[refs0 .first -- _type] .type, st)

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 17:
refs = r + refs1 ∧ refs ≠ [], dups(refs), validrefs(str, 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)

Rule argument

[speclemmaarg: refs ≠ [] ,r = refs .first ∧ refs0 = refs .rest ↔ refs = r + refs0 ,[refs, r, refs0] → [refs, r, refs1] ,T ,list-data ,references ,elim-list-c ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ ¬ dups([])], [CSIMPELIM : refs ≠ [] ⊦ r = refs .first ∧ refs0 = refs .rest ↔ refs = r + refs0])], [EXTRATERMS : (refs, r, refs1)]]