Definition. We define struct_r_ord to be λX ⇒ struct_r X ∧ unpack_r_o X (λX' r ⇒ (∀x ∈ X', ¬ r x x) ∧ (∀x y ∈ X', r x y ∨ r y x) ∧ (∀x y z ∈ X', r x y → r y z → r x z)) of type set → prop.
In Proofgold the corresponding term root is 8d0d9c... and object id is 90aeaa...
Theorem. (MetaCat_struct_r_ord)
MetaCat struct_r_ord Hom_struct_r struct_id struct_comp
In Proofgold the corresponding term root is cd9a3a... and proposition id is 86ad88...
Proof:
We prove the intermediate claim L1: ∀X, struct_r_ord X → struct_r X.
Let X be given.
Assume HX.
Apply HX to the current goal.
Assume H _.
An exact proof term for the current goal is H.
An exact proof term for the current goal is MetaCat_struct_r_gen struct_r_ord L1.
∎
Theorem. (MetaCat_struct_r_ord_Forgetful)
MetaFunctor struct_r_ord Hom_struct_r struct_id struct_comp (λ_ ⇒ True) SetHom (λX ⇒ lam_id X) (λX Y Z f g ⇒ (lam_comp X f g)) (λX ⇒ X 0) (λX Y f ⇒ f)
In Proofgold the corresponding term root is 3e3cab... and proposition id is a0fbbb...
Proof:
We prove the intermediate claim L1: ∀X, struct_r_ord X → struct_r X.
Let X be given.
Assume HX.
Apply HX to the current goal.
Assume H _.
An exact proof term for the current goal is H.
An exact proof term for the current goal is MetaCat_struct_r_Forgetful_gen struct_r_ord L1.
∎
Proposition. (MetaCat_struct_r_ord_initial)
∃Y : set, ∃uniqa : set → set, initial_p struct_r_ord Hom_struct_r struct_id struct_comp Y uniqa
In Proofgold the corresponding term root is db4d4d... and proposition id is adce2c...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_r_ord_terminal)
∃Y : set, ∃uniqa : set → set, terminal_p struct_r_ord Hom_struct_r struct_id struct_comp Y uniqa
In Proofgold the corresponding term root is 36fa76... and proposition id is 18ea90...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_r_ord_coproduct_constr)
∃coprod : set → set → set, ∃i0 i1 : set → set → set, ∃copair : set → set → set → set → set → set, coproduct_constr_p struct_r_ord Hom_struct_r struct_id struct_comp coprod i0 i1 copair
In Proofgold the corresponding term root is cc3240... and proposition id is ebfe13...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_r_ord_product_constr)
∃prod : set → set → set, ∃pi0 pi1 : set → set → set, ∃pair : set → set → set → set → set → set, product_constr_p struct_r_ord Hom_struct_r struct_id struct_comp prod pi0 pi1 pair
In Proofgold the corresponding term root is e6c1f2... and proposition id is e28a37...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_r_ord_coequalizer_constr)
∃quot : set → set → set → set → set, ∃canonmap : set → set → set → set → set, ∃fac : set → set → set → set → set → set → set, coequalizer_constr_p struct_r_ord Hom_struct_r struct_id struct_comp quot canonmap fac
In Proofgold the corresponding term root is 955017... and proposition id is df0393...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_r_ord_equalizer_constr)
∃quot : set → set → set → set → set, ∃canonmap : set → set → set → set → set, ∃fac : set → set → set → set → set → set → set, equalizer_constr_p struct_r_ord Hom_struct_r struct_id struct_comp quot canonmap fac
In Proofgold the corresponding term root is c31488... and proposition id is 788eea...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_r_ord_pushout_constr)
∃po : set → set → set → set → set → set, ∃i0 : set → set → set → set → set → set, ∃i1 : set → set → set → set → set → set, ∃copair : set → set → set → set → set → set → set → set → set, pushout_constr_p struct_r_ord Hom_struct_r struct_id struct_comp po i0 i1 copair
In Proofgold the corresponding term root is a62226... and proposition id is e5c69f...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_r_ord_pullback_constr)
∃pb : set → set → set → set → set → set, ∃pi0 : set → set → set → set → set → set, ∃pi1 : set → set → set → set → set → set, ∃pair : set → set → set → set → set → set → set → set → set, pullback_constr_p struct_r_ord Hom_struct_r struct_id struct_comp pb pi0 pi1 pair
In Proofgold the corresponding term root is 8a70de... and proposition id is 95eddd...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_r_ord_product_exponent)
∃prod : set → set → set, ∃pi0 pi1 : set → set → set, ∃pair : set → set → set → set → set → set, ∃exp : set → set → set, ∃a : set → set → set, ∃lm : set → set → set → set → set, product_exponent_constr_p struct_r_ord Hom_struct_r struct_id struct_comp prod pi0 pi1 pair exp a lm
In Proofgold the corresponding term root is 3db465... and proposition id is d432e9...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_r_ord_subobject_classifier)
∃one : set, ∃uniqa : set → set, ∃Omega : set, ∃tru : set, ∃ch : set → set → set → set, ∃constr : set → set → set → set → set → set → set, subobject_classifier_p struct_r_ord Hom_struct_r struct_id struct_comp one uniqa Omega tru ch constr
In Proofgold the corresponding term root is 0443f5... and proposition id is 568d69...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_r_ord_nno)
∃one : set, ∃uniqa : set → set, ∃N : set, ∃zer suc : set, ∃rec : set → set → set → set, nno_p struct_r_ord Hom_struct_r struct_id struct_comp one uniqa N zer suc rec
In Proofgold the corresponding term root is 33e346... and proposition id is 1a5273...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_r_ord_left_adjoint_forgetful)
∃F0 : set → set, ∃F1 : set → set → set → set, ∃eta eps : set → set, MetaAdjunction_strict (λ_ ⇒ True) SetHom (λX ⇒ (lam_id X)) (λX Y Z f g ⇒ (lam_comp X f g)) struct_r_ord Hom_struct_r struct_id struct_comp F0 F1 (λX ⇒ X 0) (λX Y f ⇒ f) eta eps
In Proofgold the corresponding term root is 0abf3e... and proposition id is 820006...
Proof:
The rest of the proof is missing.