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Proof step 3 from proof cyclic-def-shortest (previous step)

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

User applied rule apply induction

And closed the goal!

Rule argument

[substlist-arg: [st, r, str, refs] → [st, r, str, refs2]]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ # (refs + refs0) = # refs + # refs0], [CSIMPSEQ : ⊦ # r ' = 0 +1], [CSIMPSEQ : ⊦ (refs + refs0) + refs1 = refs + refs0 + refs1], [CSIMPSEQ : ⊦ m < n + 1 ↔ ¬ n < m], [CSIMPSEQ : ⊦ ¬ n < n], [CSIMPSEQ : ⊦ (m + n0) + n1 = m + n0 + n1], [CSIMPSEQ : ⊦ m + n = n + m])], [EXTRATERMS : (st, r, str, refs2)]]