Definition. We define struct_b_monoid to be λX ⇒ struct_b X ∧ unpack_b_o X (λX' op ⇒ (∀x y z ∈ X', op (op x y) z = op x (op y z)) ∧ (∃e ∈ X', ∀x ∈ X', op x e = x ∧ op e x = x)) of type set → prop.
In Proofgold the corresponding term root is 8d1ba8... and object id is 63485b...
Theorem. (MetaCat_struct_b_monoid)
MetaCat struct_b_monoid Hom_struct_b struct_id struct_comp
In Proofgold the corresponding term root is eea6d4... and proposition id is 0e62dc...
Proof:
We prove the intermediate claim L1: ∀X, struct_b_monoid X → struct_b 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_b_gen struct_b_monoid L1.
∎
Theorem. (MetaCat_struct_b_monoid_Forgetful)
MetaFunctor struct_b_monoid Hom_struct_b 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 39b6fb... and proposition id is 82499d...
Proof:
We prove the intermediate claim L1: ∀X, struct_b_monoid X → struct_b 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_b_Forgetful_gen struct_b_monoid L1.
∎
Proposition. (MetaCat_struct_b_monoid_initial)
∃Y : set, ∃uniqa : set → set, initial_p struct_b_monoid Hom_struct_b struct_id struct_comp Y uniqa
In Proofgold the corresponding term root is f7091c... and proposition id is 430617...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_monoid_terminal)
∃Y : set, ∃uniqa : set → set, terminal_p struct_b_monoid Hom_struct_b struct_id struct_comp Y uniqa
In Proofgold the corresponding term root is 14622a... and proposition id is e4b9ec...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_monoid_coproduct_constr)
∃coprod : set → set → set, ∃i0 i1 : set → set → set, ∃copair : set → set → set → set → set → set, coproduct_constr_p struct_b_monoid Hom_struct_b struct_id struct_comp coprod i0 i1 copair
In Proofgold the corresponding term root is 3ac946... and proposition id is 9d52d9...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_monoid_product_constr)
∃prod : set → set → set, ∃pi0 pi1 : set → set → set, ∃pair : set → set → set → set → set → set, product_constr_p struct_b_monoid Hom_struct_b struct_id struct_comp prod pi0 pi1 pair
In Proofgold the corresponding term root is ba0d7e... and proposition id is 64746d...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_monoid_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_b_monoid Hom_struct_b struct_id struct_comp quot canonmap fac
In Proofgold the corresponding term root is 224058... and proposition id is 092479...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_monoid_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_b_monoid Hom_struct_b struct_id struct_comp quot canonmap fac
In Proofgold the corresponding term root is 7e8304... and proposition id is 1451d1...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_monoid_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_b_monoid Hom_struct_b struct_id struct_comp po i0 i1 copair
In Proofgold the corresponding term root is 75fa41... and proposition id is a8dbb6...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_monoid_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_b_monoid Hom_struct_b struct_id struct_comp pb pi0 pi1 pair
In Proofgold the corresponding term root is e9abb0... and proposition id is f9bf8f...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_monoid_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_b_monoid Hom_struct_b struct_id struct_comp prod pi0 pi1 pair exp a lm
In Proofgold the corresponding term root is 197806... and proposition id is 65df6c...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_monoid_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_b_monoid Hom_struct_b struct_id struct_comp one uniqa Omega tru ch constr
In Proofgold the corresponding term root is f09c3d... and proposition id is d0e051...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_monoid_nno)
∃one : set, ∃uniqa : set → set, ∃N : set, ∃zer suc : set, ∃rec : set → set → set → set, nno_p struct_b_monoid Hom_struct_b struct_id struct_comp one uniqa N zer suc rec
In Proofgold the corresponding term root is 86189b... and proposition id is fd9cba...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_monoid_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_b_monoid Hom_struct_b struct_id struct_comp F0 F1 (λX ⇒ X 0) (λX Y f ⇒ f) eta eps
In Proofgold the corresponding term root is 5204ac... and proposition id is 7f3dca...
Proof:
The rest of the proof is missing.