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Proof step 16 from proof cyclic-def-shortest (previous step)

validrefs(str, r ' + refs1, 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 ∈ refs1 ∧ ¬ dups(refs1)

User applied rule induction

And obtained the following new goals

Proof step 15:
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)) ⊦ (∃ refs0, r0. ¬ dups(refs0) ∧ r0 ∈ refs0 ∧ refs0 .first = r ∧ validrefs(str, refs0 + r0 ', st[refs0 .first -- _type] .type, st)) ∨ ¬ r ∈ refs1 ∧ ¬ dups(refs1)

Rule argument

[INDHYPARG : [STANDARDINDUCTIONTYPE : ], true, false, # refs1, [] → [], <]

Additional information

[[CONCRETESIMPRULES : ([CSIMPLESSPRED : <])], [EXTRATERMS : (# refs1)]]