Definition. We define struct_b_semigroup 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)) of type set → prop.
In Proofgold the corresponding term root is 27f874... and object id is a1c8b8...
Theorem. (MetaCat_struct_b_semigroup)
MetaCat struct_b_semigroup Hom_struct_b struct_id struct_comp
In Proofgold the corresponding term root is 246240... and proposition id is 88d6d7...
Proof:
We prove the intermediate claim L1: ∀X, struct_b_semigroup 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_semigroup L1.
∎
Theorem. (MetaCat_struct_b_semigroup_Forgetful)
MetaFunctor struct_b_semigroup 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 11fb44... and proposition id is f4f7a0...
Proof:
We prove the intermediate claim L1: ∀X, struct_b_semigroup 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_semigroup L1.
∎
Proposition. (MetaCat_struct_b_semigroup_initial)
∃Y : set, ∃uniqa : set → set, initial_p struct_b_semigroup Hom_struct_b struct_id struct_comp Y uniqa
In Proofgold the corresponding term root is f48fd5... and proposition id is 28d3a5...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_semigroup_terminal)
∃Y : set, ∃uniqa : set → set, terminal_p struct_b_semigroup Hom_struct_b struct_id struct_comp Y uniqa
In Proofgold the corresponding term root is 4120b2... and proposition id is aa3f34...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_semigroup_coproduct_constr)
∃coprod : set → set → set, ∃i0 i1 : set → set → set, ∃copair : set → set → set → set → set → set, coproduct_constr_p struct_b_semigroup Hom_struct_b struct_id struct_comp coprod i0 i1 copair
In Proofgold the corresponding term root is 981b05... and proposition id is 3f944d...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_semigroup_product_constr)
∃prod : set → set → set, ∃pi0 pi1 : set → set → set, ∃pair : set → set → set → set → set → set, product_constr_p struct_b_semigroup Hom_struct_b struct_id struct_comp prod pi0 pi1 pair
In Proofgold the corresponding term root is d5b8a1... and proposition id is 183c50...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_semigroup_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_semigroup Hom_struct_b struct_id struct_comp quot canonmap fac
In Proofgold the corresponding term root is 1aa512... and proposition id is fede81...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_semigroup_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_semigroup Hom_struct_b struct_id struct_comp quot canonmap fac
In Proofgold the corresponding term root is b8cc50... and proposition id is 9347c5...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_semigroup_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_semigroup Hom_struct_b struct_id struct_comp po i0 i1 copair
In Proofgold the corresponding term root is 7aef08... and proposition id is 090905...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_semigroup_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_semigroup Hom_struct_b struct_id struct_comp pb pi0 pi1 pair
In Proofgold the corresponding term root is e1ac57... and proposition id is 61ba2e...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_semigroup_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_semigroup Hom_struct_b struct_id struct_comp prod pi0 pi1 pair exp a lm
In Proofgold the corresponding term root is 549129... and proposition id is c982c6...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_semigroup_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_semigroup Hom_struct_b struct_id struct_comp one uniqa Omega tru ch constr
In Proofgold the corresponding term root is d50205... and proposition id is bc0343...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_semigroup_nno)
∃one : set, ∃uniqa : set → set, ∃N : set, ∃zer suc : set, ∃rec : set → set → set → set, nno_p struct_b_semigroup Hom_struct_b struct_id struct_comp one uniqa N zer suc rec
In Proofgold the corresponding term root is 3c101a... and proposition id is 87fa7b...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_semigroup_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_semigroup 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 ab524a... and proposition id is 0887e4...
Proof:
The rest of the proof is missing.