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(bounded_nats
(every_nonempty_set_has_least 0
(every_nonempty_set_has_least-1 nil 3314637118
("" (skolem-typepred)
(("" (use "non_empty_bounded_below_has_least")
((""
(expand* "non_empty_bounded_below?" "nonempty?" "empty?"
"member")
(("" (skolem!)
(("" (lemma "non_empty_finite_bounded_below")
(("" (inst - "{t: T | x!1(t) AND t <= x!2}")
(("1" (expand "non_empty_bounded_below?")
(("1" (flatten)
(("1"
(expand* "bounded_below?" "lower_bound?"
"restrict")
(("1" (skolem!)
(("1" (inst + "t!1")
(("1" (skolem!)
(("1" (inst-cp - "x!2")
(("1" (inst - "r!1")
(("1" (assert) nil nil)) nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2" (split)
(("1" (expand "is_finite")
(("1"
(inst + "1 + x!2"
"LAMBDA (n: ({t: T | x!1(t) AND t <= x!2})): n")
(("1" (expand "injective?")
(("1" (skosimp) nil nil)) nil))
nil))
nil)
("2" (expand* "empty?" "member")
(("2" (inst?) (("2" (assert) nil nil)) nil)) nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
((non_empty_bounded_below_has_least judgement-tcc nil
bounded_integers nil)
(x!1 skolem-const-decl "(nonempty?[T])" bounded_nats nil)
(non_empty_bounded_below? const-decl "bool" non_empty_bounded_sets
nil)
(x!2 skolem-const-decl "T" bounded_nats nil)
(<= const-decl "bool" reals nil)
(is_finite const-decl "bool" finite_sets nil)
(AND const-decl "[bool, bool -> bool]" booleans nil)
(finite_set type-eq-decl nil finite_sets nil)
(non_empty_finite_set type-eq-decl nil finite_sets nil)
(r!1 skolem-const-decl "(x!1)" bounded_nats nil)
(real_le_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(bounded_below? const-decl "bool" bounded_orders nil)
(restrict const-decl "R" restrict nil)
(lower_bound? const-decl "bool" bounded_orders nil)
(reflexive_restrict application-judgement "(reflexive?[S])"
restrict_order_props nil)
(antisymmetric_restrict application-judgement "(antisymmetric?[S])"
restrict_order_props nil)
(transitive_restrict application-judgement "(transitive?[S])"
restrict_order_props nil)
(preorder_restrict application-judgement "(preorder?[S])"
restrict_order_props nil)
(partial_order_restrict application-judgement "(partial_order?[S])"
restrict_order_props nil)
(dichotomous_restrict application-judgement "(dichotomous?[S])"
restrict_order_props nil)
(total_order_restrict application-judgement "(total_order?[S])"
restrict_order_props nil)
(injective? const-decl "bool" functions nil)
(numfield nonempty-type-eq-decl nil number_fields nil)
(+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
(< const-decl "bool" reals nil)
(below type-eq-decl nil nat_types nil)
(posint_plus_nnint_is_posint application-judgement "posint"
integers nil)
(non_empty_finite_bounded_below judgement-tcc nil
non_empty_bounded_sets nil)
(empty? const-decl "bool" sets nil)
(member const-decl "bool" sets nil)
(nonempty? const-decl "bool" sets nil)
(set type-eq-decl nil sets nil)
(T formal-subtype-decl nil bounded_nats nil)
(T_pred const-decl "[nat -> boolean]" bounded_nats nil)
(nat nonempty-type-eq-decl nil naturalnumbers nil)
(>= const-decl "bool" reals nil)
(int nonempty-type-eq-decl nil integers nil)
(integer_pred const-decl "[rational -> boolean]" integers nil)
(rational nonempty-type-from-decl nil rationals nil)
(rational_pred const-decl "[real -> boolean]" rationals nil)
(real nonempty-type-from-decl nil reals nil)
(real_pred const-decl "[number_field -> boolean]" reals nil)
(number_field nonempty-type-from-decl nil number_fields nil)
(number_field_pred const-decl "[number -> boolean]" number_fields
nil)
(number nonempty-type-decl nil numbers nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(bool nonempty-type-eq-decl nil booleans nil)
(boolean nonempty-type-decl nil booleans nil))
nil)))
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