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Proof step 13 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)

User applied rule exists right

And obtained the following new goals

Proof step 12:
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), ¬ dups(r ' + refs1 .butlast) ∧ refs1 .last ∈ r ' + refs1 .butlast ∧ (r ' + refs1 .butlast) .first = r ∧ validrefs(str, (r ' + refs1 .butlast) + refs1 .last ', st[(r ' + refs1 .butlast) .first -- _type] .type, st), ∃ refs0, r0. ¬ dups(refs0) ∧ r0 ∈ refs0 ∧ refs0 .first = r ∧ validrefs(str, refs0 + r0 ', st[refs0 .first -- _type] .type, st)

Rule argument

[EXRARG : pos: right 3, [extquanttermlist: (r ' + refs1 .butlast, refs1 .last) abort:F , discard:F , computed:F ]]

Additional information

[[CONCRETESIMPRULES : ()], [EXTRATERMS : (r ' + refs1 .butlast, refs1 .last)]]