Theorem. (MetaCat_struct_b_b_e_rng)
MetaCat Rng Hom_struct_b_b_e struct_id struct_comp
In Proofgold the corresponding term root is 4f67cc... and proposition id is ea1d4c...
Proof:
We prove the intermediate claim L1: ∀X, Rng X → struct_b_b_e 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_b_e_gen Rng L1.
∎
Theorem. (MetaCat_struct_b_b_e_rng_Forgetful)
MetaFunctor Rng Hom_struct_b_b_e 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 f29160... and proposition id is e4ca27...
Proof:
We prove the intermediate claim L1: ∀X, Rng X → struct_b_b_e 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_b_e_Forgetful_gen Rng L1.
∎
Proposition. (MetaCat_struct_b_b_e_rng_initial)
∃Y : set, ∃uniqa : set → set, initial_p Rng Hom_struct_b_b_e struct_id struct_comp Y uniqa
In Proofgold the corresponding term root is ca7926... and proposition id is 929dd6...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_b_e_rng_terminal)
∃Y : set, ∃uniqa : set → set, terminal_p Rng Hom_struct_b_b_e struct_id struct_comp Y uniqa
In Proofgold the corresponding term root is 2fcb57... and proposition id is c549ea...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_b_e_rng_coproduct_constr)
∃coprod : set → set → set, ∃i0 i1 : set → set → set, ∃copair : set → set → set → set → set → set, coproduct_constr_p Rng Hom_struct_b_b_e struct_id struct_comp coprod i0 i1 copair
In Proofgold the corresponding term root is 9a09c5... and proposition id is ecb0cb...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_b_e_rng_product_constr)
∃prod : set → set → set, ∃pi0 pi1 : set → set → set, ∃pair : set → set → set → set → set → set, product_constr_p Rng Hom_struct_b_b_e struct_id struct_comp prod pi0 pi1 pair
In Proofgold the corresponding term root is 1a30d7... and proposition id is 073ed1...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_b_e_rng_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 Rng Hom_struct_b_b_e struct_id struct_comp quot canonmap fac
In Proofgold the corresponding term root is 9f6852... and proposition id is 2c44bb...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_b_e_rng_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 Rng Hom_struct_b_b_e struct_id struct_comp quot canonmap fac
In Proofgold the corresponding term root is 26883a... and proposition id is efe8ef...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_b_e_rng_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 Rng Hom_struct_b_b_e struct_id struct_comp po i0 i1 copair
In Proofgold the corresponding term root is 30f9bf... and proposition id is e728e1...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_b_e_rng_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 Rng Hom_struct_b_b_e struct_id struct_comp pb pi0 pi1 pair
In Proofgold the corresponding term root is 04d485... and proposition id is 4dc81a...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_b_e_rng_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 Rng Hom_struct_b_b_e struct_id struct_comp prod pi0 pi1 pair exp a lm
In Proofgold the corresponding term root is 4cb422... and proposition id is 6a0142...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_b_e_rng_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 Rng Hom_struct_b_b_e struct_id struct_comp one uniqa Omega tru ch constr
In Proofgold the corresponding term root is 69f320... and proposition id is 7bf95c...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_b_e_rng_nno)
∃one : set, ∃uniqa : set → set, ∃N : set, ∃zer suc : set, ∃rec : set → set → set → set, nno_p Rng Hom_struct_b_b_e struct_id struct_comp one uniqa N zer suc rec
In Proofgold the corresponding term root is 5bd842... and proposition id is 2332ca...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_b_e_rng_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)) Rng Hom_struct_b_b_e struct_id struct_comp F0 F1 (λX ⇒ X 0) (λX Y f ⇒ f) eta eps
In Proofgold the corresponding term root is 19beba... and proposition id is 08a752...
Proof:
The rest of the proof is missing.