Definition. We define struct_r_per to be λX ⇒ struct_r X ∧ unpack_r_o X (λX' r ⇒ (∀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 3a8be9... and object id is 2f640c...
Theorem. (MetaCat_struct_r_per)
MetaCat struct_r_per Hom_struct_r struct_id struct_comp
In Proofgold the corresponding term root is a9109a... and proposition id is 259fb4...
Proof:
We prove the intermediate claim L1: ∀X, struct_r_per 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_per L1.
∎
Theorem. (MetaCat_struct_r_per_Forgetful)
MetaFunctor struct_r_per 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 fc754c... and proposition id is 1b780a...
Proof:
We prove the intermediate claim L1: ∀X, struct_r_per 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_per L1.
∎
Proposition. (MetaCat_struct_r_per_initial)
∃Y : set, ∃uniqa : set → set, initial_p struct_r_per Hom_struct_r struct_id struct_comp Y uniqa
In Proofgold the corresponding term root is 5aca84... and proposition id is 743bb0...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_r_per_terminal)
∃Y : set, ∃uniqa : set → set, terminal_p struct_r_per Hom_struct_r struct_id struct_comp Y uniqa
In Proofgold the corresponding term root is 825c02... and proposition id is bf5537...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_r_per_coproduct_constr)
∃coprod : set → set → set, ∃i0 i1 : set → set → set, ∃copair : set → set → set → set → set → set, coproduct_constr_p struct_r_per Hom_struct_r struct_id struct_comp coprod i0 i1 copair
In Proofgold the corresponding term root is aabb17... and proposition id is 5de9f8...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_r_per_product_constr)
∃prod : set → set → set, ∃pi0 pi1 : set → set → set, ∃pair : set → set → set → set → set → set, product_constr_p struct_r_per Hom_struct_r struct_id struct_comp prod pi0 pi1 pair
In Proofgold the corresponding term root is 7424d1... and proposition id is b370d6...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_r_per_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_per Hom_struct_r struct_id struct_comp quot canonmap fac
In Proofgold the corresponding term root is 62c8f4... and proposition id is d1bd9e...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_r_per_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_per Hom_struct_r struct_id struct_comp quot canonmap fac
In Proofgold the corresponding term root is da50e7... and proposition id is 57f509...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_r_per_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_per Hom_struct_r struct_id struct_comp po i0 i1 copair
In Proofgold the corresponding term root is bd83f4... and proposition id is e89213...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_r_per_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_per Hom_struct_r struct_id struct_comp pb pi0 pi1 pair
In Proofgold the corresponding term root is 9f0f80... and proposition id is 440674...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_r_per_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_per Hom_struct_r struct_id struct_comp prod pi0 pi1 pair exp a lm
In Proofgold the corresponding term root is 88a055... and proposition id is 97ee86...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_r_per_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_per Hom_struct_r struct_id struct_comp one uniqa Omega tru ch constr
In Proofgold the corresponding term root is ad8528... and proposition id is 91d7b9...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_r_per_nno)
∃one : set, ∃uniqa : set → set, ∃N : set, ∃zer suc : set, ∃rec : set → set → set → set, nno_p struct_r_per Hom_struct_r struct_id struct_comp one uniqa N zer suc rec
In Proofgold the corresponding term root is 6745ff... and proposition id is 7d1326...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_r_per_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_per 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 94e18c... and proposition id is 8dcfe2...
Proof:
The rest of the proof is missing.