Beginning of Section MetaCat
Variable Obj : set → prop
Variable Hom : set → set → set → prop
Variable id : set → set
Variable comp : set → set → set → set → set → set
Definition. We define idT to be ∀X : set, Obj X → Hom X X (id X) of type prop.
In Proofgold the corresponding term root is 5b5295... and object id is 6e71b7...
Definition. We define compT to be ∀X Y Z : set, ∀f g : set, Obj X → Obj Y → Obj Z → Hom X Y f → Hom Y Z g → Hom X Z (comp X Y Z g f) of type prop.
In Proofgold the corresponding term root is fd3b02... and object id is b651ec...
Definition. We define idL to be ∀X Y : set, ∀f : set, Obj X → Obj Y → Hom X Y f → comp X X Y f (id X) = f of type prop.
In Proofgold the corresponding term root is 8bf949... and object id is 274215...
Definition. We define idR to be ∀X Y : set, ∀f : set, Obj X → Obj Y → Hom X Y f → comp X Y Y (id Y) f = f of type prop.
In Proofgold the corresponding term root is e0f11c... and object id is 9fe6a5...
Definition. We define compAssoc to be ∀X Y Z W : set, ∀f g h : set, Obj X → Obj Y → Obj Z → Obj W → Hom X Y f → Hom Y Z g → Hom Z W h → comp X Y W (comp Y Z W h g) f = comp X Z W h (comp X Y Z g f) of type prop.
In Proofgold the corresponding term root is 6fdbfa... and object id is 67897c...
Definition. We define MetaCat to be (idT ∧ compT) ∧ (idL ∧ idR) ∧ compAssoc of type prop.
In Proofgold the corresponding term root is fd9743... and object id is 1fb9aa...
Theorem. (MetaCat_I)
idT → compT → (∀X Y : set, ∀f : set, Obj X → Obj Y → Hom X Y f → comp X X Y f (id X) = f) → (∀X Y : set, ∀f : set, Obj X → Obj Y → Hom X Y f → comp X Y Y (id Y) f = f) → (∀X Y Z W : set, ∀f g h : set, Obj X → Obj Y → Obj Z → Obj W → Hom X Y f → Hom Y Z g → Hom Z W h → comp X Y W (comp Y Z W h g) f = comp X Z W h (comp X Y Z g f)) → MetaCat
In Proofgold the corresponding term root is fc5379... and proposition id is 047c95...
Proof:
Assume H1 H2 H3 H4 H5.
We will prove (idT ∧ compT) ∧ (idL ∧ idR) ∧ compAssoc.
Apply and3I to the current goal.
Apply andI to the current goal.
An exact proof term for the current goal is H1.
An exact proof term for the current goal is H2.
Apply andI to the current goal.
An exact proof term for the current goal is H3.
An exact proof term for the current goal is H4.
An exact proof term for the current goal is H5.
∎
Theorem. (MetaCat_E)
MetaCat → ∀p : prop, (idT → compT → (∀X Y : set, ∀f : set, Obj X → Obj Y → Hom X Y f → comp X X Y f (id X) = f) → (∀X Y : set, ∀f : set, Obj X → Obj Y → Hom X Y f → comp X Y Y (id Y) f = f) → (∀X Y Z W : set, ∀f g h : set, Obj X → Obj Y → Obj Z → Obj W → Hom X Y f → Hom Y Z g → Hom Z W h → comp X Y W (comp Y Z W h g) f = comp X Z W h (comp X Y Z g f)) → p) → p
In Proofgold the corresponding term root is d158dd... and proposition id is 7da4b2...
Proof:
Assume HC.
Let p be given.
Assume Hp.
Apply HC to the current goal.
Assume H14 H5.
Apply H14 to the current goal.
Assume H12 H34.
Apply H12 to the current goal.
Assume H1 H2.
Apply H34 to the current goal.
Assume H3 H4.
An exact proof term for the current goal is Hp H1 H2 H3 H4 H5.
∎
End of Section MetaCat
Beginning of Section MetaCatOp
Variable Obj : set → prop
Variable Hom : set → set → set → prop
Variable id : set → set
Variable comp : set → set → set → set → set → set
Theorem. (MetaCatOp)
MetaCat Obj Hom id comp → MetaCat Obj (λX Y ⇒ Hom Y X) id (λX Y Z f g ⇒ comp Z Y X g f)
In Proofgold the corresponding term root is e74d2a... and proposition id is 6f34fb...
Proof:
Assume HC.
Apply MetaCat_E Obj Hom id comp HC to the current goal.
Assume HC1: ∀X : set, Obj X → Hom X X (id X).
Assume HC2: ∀X Y Z : set, ∀f g : set, Obj X → Obj Y → Obj Z → Hom X Y f → Hom Y Z g → Hom X Z (comp X Y Z g f).
Assume HC3: ∀X Y : set, ∀f : set, Obj X → Obj Y → Hom X Y f → comp X X Y f (id X) = f.
Assume HC4: ∀X Y : set, ∀f : set, Obj X → Obj Y → Hom X Y f → comp X Y Y (id Y) f = f.
Assume HC5: ∀X Y Z W : set, ∀f g h : set, Obj X → Obj Y → Obj Z → Obj W → Hom X Y f → Hom Y Z g → Hom Z W h → comp X Y W (comp Y Z W h g) f = comp X Z W h (comp X Y Z g f).
Apply MetaCat_I to the current goal.
An exact proof term for the current goal is HC1.
Let X, Y, Z, f and g be given.
Assume HX HY HZ Hf Hg.
An exact proof term for the current goal is HC2 Z Y X g f HZ HY HX Hg Hf.
Let X, Y and f be given.
Assume HX HY Hf.
An exact proof term for the current goal is HC4 Y X f HY HX Hf.
Let X, Y and f be given.
Assume HX HY Hf.
An exact proof term for the current goal is HC3 Y X f HY HX Hf.
Let X, Y, Z, W, f, g and h be given.
Assume HX HY HZ HW Hf Hg Hh.
We will prove comp W Y X f (comp W Z Y g h) = comp W Z X (comp Z Y X f g) h.
Use symmetry.
An exact proof term for the current goal is HC5 W Z Y X h g f HW HZ HY HX Hh Hg Hf.
∎
End of Section MetaCatOp
Beginning of Section LimsCoLims
Variable Obj : set → prop
Variable Hom : set → set → set → prop
Variable id : set → set
Variable comp : set → set → set → set → set → set
Definition. We define monic to be λX Y f ⇒ Obj X ∧ Obj Y ∧ Hom X Y f ∧ ∀Z : set, Obj Z → ∀g h : set, Hom Z X g → Hom Z X h → comp Z X Y f g = comp Z X Y f h → g = h of type set → set → set → prop.
In Proofgold the corresponding term root is bb42af... and object id is 4204ac...
Definition. We define terminal_p to be λY h ⇒ Obj Y ∧ ∀X : set, Obj X → Hom X Y (h X) ∧ ∀h' : set, Hom X Y h' → h' = h X of type set → (set → set) → prop.
In Proofgold the corresponding term root is fdae77... and object id is fc25a5...
Definition. We define initial_p to be λY h ⇒ Obj Y ∧ ∀X : set, Obj X → Hom Y X (h X) ∧ ∀h' : set, Hom Y X h' → h' = h X of type set → (set → set) → prop.
In Proofgold the corresponding term root is a7ed72... and object id is f206b9...
Definition. We define product_p to be λX Y Z pi0 pi1 pair ⇒ Obj X ∧ Obj Y ∧ Obj Z ∧ Hom Z X pi0 ∧ Hom Z Y pi1 ∧ ∀W : set, Obj W → ∀h k : set, Hom W X h → Hom W Y k → Hom W Z (pair W h k) ∧ comp W Z X pi0 (pair W h k) = h ∧ comp W Z Y pi1 (pair W h k) = k ∧ ∀u : set, Hom W Z u → comp W Z X pi0 u = h → comp W Z Y pi1 u = k → u = pair W h k of type set → set → set → set → set → (set → set → set → set) → prop.
In Proofgold the corresponding term root is 493d09... and object id is c575e7...
Definition. We define product_constr_p to be λprod pi0 pi1 pair ⇒ ∀X Y : set, Obj X → Obj Y → product_p X Y (prod X Y) (pi0 X Y) (pi1 X Y) (pair X Y) of type (set → set → set) → (set → set → set) → (set → set → set) → (set → set → set → set → set → set) → prop.
In Proofgold the corresponding term root is 8627a9... and object id is a34635...
Definition. We define coproduct_p to be λX Y Z i0 i1 comb ⇒ Obj X ∧ Obj Y ∧ Obj Z ∧ Hom X Z i0 ∧ Hom Y Z i1 ∧ ∀W : set, Obj W → ∀h k : set, Hom X W h → Hom Y W k → Hom Z W (comb W h k) ∧ comp X Z W (comb W h k) i0 = h ∧ comp Y Z W (comb W h k) i1 = k ∧ ∀hk : set, Hom Z W hk → comp X Z W hk i0 = h → comp Y Z W hk i1 = k → hk = comb W h k of type set → set → set → set → set → (set → set → set → set) → prop.
In Proofgold the corresponding term root is 1aca76... and object id is 545d6c...
Definition. We define coproduct_constr_p to be λcoprod i0 i1 copair ⇒ ∀X Y : set, Obj X → Obj Y → coproduct_p X Y (coprod X Y) (i0 X Y) (i1 X Y) (copair X Y) of type (set → set → set) → (set → set → set) → (set → set → set) → (set → set → set → set → set → set) → prop.
In Proofgold the corresponding term root is 9283ca... and object id is a3944e...
Definition. We define equalizer_p to be λX Y f g Q q fac ⇒ Obj X ∧ Obj Y ∧ Hom X Y f ∧ Hom X Y g ∧ Obj Q ∧ Hom Q X q ∧ comp Q X Y f q = comp Q X Y g q ∧ ∀W : set, Obj W → ∀h : set, Hom W X h → comp W X Y f h = comp W X Y g h → Hom W Q (fac W h) ∧ comp W Q X q (fac W h) = h ∧ ∀u : set, Hom W Q u → comp W Q X q u = h → u = fac W h of type set → set → set → set → set → set → (set → set → set) → prop.
In Proofgold the corresponding term root is e54193... and object id is 067f6a...
Definition. We define equalizer_constr_p to be λquot canonmap fac ⇒ ∀X Y : set, Obj X → Obj Y → ∀f g : set, Hom X Y f → Hom X Y g → equalizer_p X Y f g (quot X Y f g) (canonmap X Y f g) (fac X Y f g) of type (set → set → set → set → set) → (set → set → set → set → set) → (set → set → set → set → set → set → set) → prop.
In Proofgold the corresponding term root is 57caf1... and object id is 1c082c...
Definition. We define coequalizer_p to be λX Y f g Q q fac ⇒ Obj X ∧ Obj Y ∧ Hom X Y f ∧ Hom X Y g ∧ Obj Q ∧ Hom Y Q q ∧ comp X Y Q q f = comp X Y Q q g ∧ ∀W : set, Obj W → ∀h : set, Hom Y W h → comp X Y W h f = comp X Y W h g → Hom Q W (fac W h) ∧ comp Y Q W (fac W h) q = h ∧ ∀u : set, Hom Q W u → comp Y Q W u q = h → u = fac W h of type set → set → set → set → set → set → (set → set → set) → prop.
In Proofgold the corresponding term root is 7f834d... and object id is dfe3bc...
Definition. We define coequalizer_constr_p to be λquot canonmap fac ⇒ ∀X Y : set, Obj X → Obj Y → ∀f g : set, Hom X Y f → Hom X Y g → coequalizer_p X Y f g (quot X Y f g) (canonmap X Y f g) (fac X Y f g) of type (set → set → set → set → set) → (set → set → set → set → set) → (set → set → set → set → set → set → set) → prop.
In Proofgold the corresponding term root is 1d8fdb... and object id is bfc97c...
Definition. We define pullback_p to be λX Y Z f g P pi0 pi1 pair ⇒ Obj X ∧ Obj Y ∧ Obj Z ∧ Hom X Z f ∧ Hom Y Z g ∧ Obj P ∧ Hom P X pi0 ∧ Hom P Y pi1 ∧ comp P X Z f pi0 = comp P Y Z g pi1 ∧ ∀W : set, Obj W → ∀h : set, Hom W X h → ∀k : set, Hom W Y k → comp W X Z f h = comp W Y Z g k → Hom W P (pair W h k) ∧ comp W P X pi0 (pair W h k) = h ∧ comp W P Y pi1 (pair W h k) = k ∧ ∀u : set, Hom W P u → comp W P X pi0 u = h → comp W P Y pi1 u = k → u = pair W h k of type set → set → set → set → set → set → set → set → (set → set → set → set) → prop.
In Proofgold the corresponding term root is aa77e5... and object id is 86c867...
Definition. We define pullback_constr_p to be λpb pi0 pi1 pair ⇒ ∀X Y Z : set, Obj X → Obj Y → Obj Z → ∀f g : set, Hom X Z f → Hom Y Z g → pullback_p X Y Z f g (pb X Y Z f g) (pi0 X Y Z f g) (pi1 X Y Z f g) (pair X Y Z f g) of type (set → set → set → set → set → set) → (set → set → set → set → set → set) → (set → set → set → set → set → set) → (set → set → set → set → set → set → set → set → set) → prop.
In Proofgold the corresponding term root is c0b415... and object id is 39fb29...
Definition. We define pushout_p to be λX Y Z f g P i0 i1 copair ⇒ Obj X ∧ Obj Y ∧ Obj Z ∧ Hom Z X f ∧ Hom Z Y g ∧ Obj P ∧ Hom X P i0 ∧ Hom Y P i1 ∧ comp Z X P i0 f = comp Z Y P i1 g ∧ ∀W : set, Obj W → ∀h : set, Hom X W h → ∀k : set, Hom Y W k → comp Z X W h f = comp Z Y W k g → Hom P W (copair W h k) ∧ comp X P W (copair W h k) i0 = h ∧ comp Y P W (copair W h k) i1 = k ∧ ∀u : set, Hom P W u → comp X P W u i0 = h → comp Y P W u i1 = k → u = copair W h k of type set → set → set → set → set → set → set → set → (set → set → set → set) → prop.
In Proofgold the corresponding term root is a2ad18... and object id is 062995...
Definition. We define pushout_constr_p to be λpo i0 i1 copair ⇒ ∀X Y Z : set, Obj X → Obj Y → Obj Z → ∀f g : set, Hom Z X f → Hom Z Y g → pushout_p X Y Z f g (po X Y Z f g) (i0 X Y Z f g) (i1 X Y Z f g) (copair X Y Z f g) of type (set → set → set → set → set → set) → (set → set → set → set → set → set) → (set → set → set → set → set → set) → (set → set → set → set → set → set → set → set → set) → prop.
In Proofgold the corresponding term root is 6b9515... and object id is 455559...
Definition. We define exponent_p to be λprod pi0 pi1 pair X Y Z a lm ⇒ Obj X ∧ Obj Y ∧ Obj Z ∧ Hom (prod Z X) Y a ∧ ∀W : set, Obj W → ∀f : set, Hom (prod W X) Y f → Hom W Z (lm W f) ∧ comp (prod W X) (prod Z X) Y a (pair Z X (prod W X) (comp (prod W X) W Z (lm W f) (pi0 W X)) (pi1 W X)) = f ∧ ∀g : set, Hom W Z g → comp (prod W X) (prod Z X) Y a (pair Z X (prod W X) (comp (prod W X) W Z g (pi0 W X)) (pi1 W X)) = f → g = lm W f of type (set → set → set) → (set → set → set) → (set → set → set) → (set → set → set → set → set → set) → set → set → set → set → (set → set → set) → prop.
In Proofgold the corresponding term root is 7f9d1d... and object id is 900ac5...
Definition. We define product_exponent_constr_p to be λprod pi0 pi1 pair exp a lm ⇒ product_constr_p prod pi0 pi1 pair ∧ ∀X Y : set, Obj X → Obj Y → exponent_p prod pi0 pi1 pair X Y (exp X Y) (a X Y) (lm X Y) of type (set → set → set) → (set → set → set) → (set → set → set) → (set → set → set → set → set → set) → (set → set → set) → (set → set → set) → (set → set → set → set → set) → prop.
In Proofgold the corresponding term root is b618df... and object id is fbdca4...
Definition. We define subobject_classifier_p to be λone uniqa Omega tru ch constr_p ⇒ terminal_p one uniqa ∧ Obj Omega ∧ Hom one Omega tru ∧ ∀X Y : set, ∀m : set, monic X Y m → Hom Y Omega (ch X Y m) ∧ pullback_p one Y Omega tru (ch X Y m) X (uniqa X) m (constr_p X Y m) of type set → (set → set) → set → set → (set → set → set → set) → (set → set → set → set → set → set → set) → prop.
In Proofgold the corresponding term root is 23398d... and object id is b2f9d5...
Definition. We define nno_p to be λone uniqa N zer suc rec ⇒ terminal_p one uniqa ∧ Obj N ∧ Hom one N zer ∧ Hom N N suc ∧ ∀X : set, ∀x : set, ∀f : set, Obj X → Hom one X x → Hom X X f → Hom N X (rec X x f) ∧ comp one N X (rec X x f) zer = x ∧ comp N N X (rec X x f) suc = comp N X X f (rec X x f) ∧ ∀u : set, Hom N X u → comp one N X u zer = x → comp N N X u suc = comp N X X f u → u = rec X x f of type set → (set → set) → set → set → set → (set → set → set → set) → prop.
In Proofgold the corresponding term root is 037a74... and object id is b05ec9...
End of Section LimsCoLims
Beginning of Section LimsCoLims2
Variable Obj : set → prop
Variable Hom : set → set → set → prop
Variable id : set → set
Variable comp : set → set → set → set → set → set
Theorem. (product_coproduct_Op)
∀X Y Z : set, ∀pi0 pi1 : set, ∀pair : set → set → set → set, product_p Obj Hom id comp X Y Z pi0 pi1 pair → coproduct_p Obj (λX Y ⇒ Hom Y X) id (λX Y Z f g ⇒ comp Z Y X g f) X Y Z pi0 pi1 pair
In Proofgold the corresponding term root is 38cf5b... and proposition id is 9dd4bb...
Proof:
Let X, Y, Z, pi0, pi1 and pair be given.
Assume H1.
An exact proof term for the current goal is H1.
∎
Theorem. (product_coproduct_constr_Op)
∀prod : set → set → set, ∀pi0 pi1 : set → set → set, ∀pair : set → set → set → set → set → set, product_constr_p Obj Hom id comp prod pi0 pi1 pair → coproduct_constr_p Obj (λX Y ⇒ Hom Y X) id (λX Y Z f g ⇒ comp Z Y X g f) prod pi0 pi1 pair
In Proofgold the corresponding term root is 5f1c74... and proposition id is 31c777...
Proof:
Let prod, pi0, pi1 and pair be given.
Assume H1.
An exact proof term for the current goal is H1.
∎
Theorem. (coproduct_product_Op)
∀X Y Z : set, ∀i0 i1 : set, ∀copair : set → set → set → set, coproduct_p Obj Hom id comp X Y Z i0 i1 copair → product_p Obj (λX Y ⇒ Hom Y X) id (λX Y Z f g ⇒ comp Z Y X g f) X Y Z i0 i1 copair
In Proofgold the corresponding term root is 84a847... and proposition id is df89e7...
Proof:
Let X, Y, Z, i0, i1 and copair be given.
Assume H1.
An exact proof term for the current goal is H1.
∎
Theorem. (coproduct_product_constr_Op)
∀coprod : set → set → set, ∀i0 i1 : set → set → set, ∀copair : set → set → set → set → set → set, coproduct_constr_p Obj Hom id comp coprod i0 i1 copair → product_constr_p Obj (λX Y ⇒ Hom Y X) id (λX Y Z f g ⇒ comp Z Y X g f) coprod i0 i1 copair
In Proofgold the corresponding term root is 3e498f... and proposition id is 2bad67...
Proof:
Let coprod, i0, i1 and copair be given.
Assume H1.
An exact proof term for the current goal is H1.
∎
Theorem. (equalizer_coequalizer_Op)
∀X Y : set, ∀f g : set, ∀Q : set, ∀q : set, ∀fac : set → set → set, equalizer_p Obj Hom id comp X Y f g Q q fac → coequalizer_p Obj (λX Y ⇒ Hom Y X) id (λX Y Z f g ⇒ comp Z Y X g f) Y X f g Q q fac
In Proofgold the corresponding term root is 6fb40a... and proposition id is 434dd4...
Proof:
Let X, Y, f, g, Q, q and fac be given.
Assume HE.
Apply HE to the current goal.
Assume HE.
Apply HE to the current goal.
Assume HE.
Apply HE to the current goal.
Assume HE.
Apply HE to the current goal.
Assume HE.
Apply HE to the current goal.
Assume HE.
Apply HE to the current goal.
Assume HE.
Apply HE to the current goal.
Assume HE1: Obj X.
Assume HE2: Obj Y.
Assume HE3: Hom X Y f.
Assume HE4: Hom X Y g.
Assume HE5: Obj Q.
Assume HE6: Hom Q X q.
Assume HE7: comp Q X Y f q = comp Q X Y g q.
Assume HE8: ∀W : set, Obj W → ∀h : set, Hom W X h → comp W X Y f h = comp W X Y g h → Hom W Q (fac W h) ∧ comp W Q X q (fac W h) = h ∧ ∀u : set, Hom W Q u → comp W Q X q u = h → u = fac W h.
We will prove Obj Y ∧ Obj X ∧ Hom X Y f ∧ Hom X Y g ∧ Obj Q ∧ Hom Q X q ∧ comp Q X Y f q = comp Q X Y g q ∧ ∀W : set, Obj W → ∀h : set, Hom W X h → comp W X Y f h = comp W X Y g h → Hom W Q (fac W h) ∧ comp W Q X q (fac W h) = h ∧ ∀u : set, Hom W Q u → comp W Q X q u = h → u = fac W h.
Apply and8I to the current goal.
An exact proof term for the current goal is HE2.
An exact proof term for the current goal is HE1.
An exact proof term for the current goal is HE3.
An exact proof term for the current goal is HE4.
An exact proof term for the current goal is HE5.
An exact proof term for the current goal is HE6.
An exact proof term for the current goal is HE7.
An exact proof term for the current goal is HE8.
∎
Theorem. (equalizer_coequalizer_constr_Op)
∀quot : set → set → set → set → set, ∀canonmap : set → set → set → set → set, ∀fac : set → set → set → set → set → set → set, equalizer_constr_p Obj Hom id comp quot canonmap fac → coequalizer_constr_p Obj (λX Y ⇒ Hom Y X) id (λX Y Z f g ⇒ comp Z Y X g f) (λX Y f g ⇒ quot Y X f g) (λX Y f g ⇒ canonmap Y X f g) (λX Y f g ⇒ fac Y X f g)
In Proofgold the corresponding term root is ebf9a6... and proposition id is 71f1f9...
Proof:
Let quot, canonmap and fac be given.
Assume HE.
Let X and Y be given.
Assume HX HY.
Let f and g be given.
Assume Hf Hg.
An exact proof term for the current goal is equalizer_coequalizer_Op Y X f g (quot Y X f g) (canonmap Y X f g) (fac Y X f g) (HE Y X HY HX f g Hf Hg).
∎
Theorem. (coequalizer_equalizer_Op)
∀X Y : set, ∀f g : set, ∀Q : set, ∀q : set, ∀fac : set → set → set, coequalizer_p Obj Hom id comp X Y f g Q q fac → equalizer_p Obj (λX Y ⇒ Hom Y X) id (λX Y Z f g ⇒ comp Z Y X g f) Y X f g Q q fac
In Proofgold the corresponding term root is 23000b... and proposition id is ed80d2...
Proof:
Let X, Y, f, g, Q, q and fac be given.
Assume HE.
Apply HE to the current goal.
Assume HE.
Apply HE to the current goal.
Assume HE.
Apply HE to the current goal.
Assume HE.
Apply HE to the current goal.
Assume HE.
Apply HE to the current goal.
Assume HE.
Apply HE to the current goal.
Assume HE.
Apply HE to the current goal.
Assume HE1: Obj X.
Assume HE2: Obj Y.
Assume HE3: Hom X Y f.
Assume HE4: Hom X Y g.
Assume HE5: Obj Q.
Assume HE6: Hom Y Q q.
Assume HE7: comp X Y Q q f = comp X Y Q q g.
Assume HE8: ∀W : set, Obj W → ∀h : set, Hom Y W h → comp X Y W h f = comp X Y W h g → Hom Q W (fac W h) ∧ comp Y Q W (fac W h) q = h ∧ ∀u : set, Hom Q W u → comp Y Q W u q = h → u = fac W h.
We will prove Obj Y ∧ Obj X ∧ Hom X Y f ∧ Hom X Y g ∧ Obj Q ∧ Hom Y Q q ∧ comp X Y Q q f = comp X Y Q q g ∧ ∀W : set, Obj W → ∀h : set, Hom Y W h → comp X Y W h f = comp X Y W h g → Hom Q W (fac W h) ∧ comp Y Q W (fac W h) q = h ∧ ∀u : set, Hom Q W u → comp Y Q W u q = h → u = fac W h.
Apply and8I to the current goal.
An exact proof term for the current goal is HE2.
An exact proof term for the current goal is HE1.
An exact proof term for the current goal is HE3.
An exact proof term for the current goal is HE4.
An exact proof term for the current goal is HE5.
An exact proof term for the current goal is HE6.
An exact proof term for the current goal is HE7.
An exact proof term for the current goal is HE8.
∎
Theorem. (coequalizer_equalizer_constr_Op)
∀quot : set → set → set → set → set, ∀canonmap : set → set → set → set → set, ∀fac : set → set → set → set → set → set → set, coequalizer_constr_p Obj Hom id comp quot canonmap fac → equalizer_constr_p Obj (λX Y ⇒ Hom Y X) id (λX Y Z f g ⇒ comp Z Y X g f) (λX Y f g ⇒ quot Y X f g) (λX Y f g ⇒ canonmap Y X f g) (λX Y f g ⇒ fac Y X f g)
In Proofgold the corresponding term root is 20c134... and proposition id is 466a77...
Proof:
Let quot, canonmap and fac be given.
Assume HE.
Let X and Y be given.
Assume HX HY.
Let f and g be given.
Assume Hf Hg.
An exact proof term for the current goal is coequalizer_equalizer_Op Y X f g (quot Y X f g) (canonmap Y X f g) (fac Y X f g) (HE Y X HY HX f g Hf Hg).
∎
Theorem. (pullback_pushout_Op)
∀X Y Z : set, ∀f g : set, ∀P : set, ∀pi0 pi1 : set, ∀pair : set → set → set → set, pullback_p Obj Hom id comp X Y Z f g P pi0 pi1 pair → pushout_p Obj (λX Y ⇒ Hom Y X) id (λX Y Z f g ⇒ comp Z Y X g f) X Y Z f g P pi0 pi1 pair
In Proofgold the corresponding term root is a1e515... and proposition id is 038557...
Proof:
Let X, Y, Z, f, g, P, pi0, pi1 and pair be given.
Assume HP.
An exact proof term for the current goal is HP.
∎
Theorem. (pullback_pushout_constr_Op)
∀pb : set → set → set → set → set → set, ∀pi0 pi1 : set → set → set → set → set → set, ∀pair : set → set → set → set → set → set → set → set → set, pullback_constr_p Obj Hom id comp pb pi0 pi1 pair → pushout_constr_p Obj (λX Y ⇒ Hom Y X) id (λX Y Z f g ⇒ comp Z Y X g f) pb pi0 pi1 pair
In Proofgold the corresponding term root is c883c5... and proposition id is 9b6aee...
Proof:
Let pb, pi0, pi1 and pair be given.
Assume HP.
An exact proof term for the current goal is HP.
∎
Theorem. (pushout_pullback_Op)
∀X Y Z : set, ∀f g : set, ∀P : set, ∀i0 i1 : set, ∀copair : set → set → set → set, pushout_p Obj Hom id comp X Y Z f g P i0 i1 copair → pullback_p Obj (λX Y ⇒ Hom Y X) id (λX Y Z f g ⇒ comp Z Y X g f) X Y Z f g P i0 i1 copair
In Proofgold the corresponding term root is ce8d4a... and proposition id is 4e78ad...
Proof:
Let X, Y, Z, f, g, P, i0, i1 and copair be given.
Assume HP.
An exact proof term for the current goal is HP.
∎
Theorem. (pushout_pullback_constr_Op)
∀po : set → set → set → set → set → set, ∀i0 i1 : set → set → set → set → set → set, ∀copair : set → set → set → set → set → set → set → set → set, pushout_constr_p Obj Hom id comp po i0 i1 copair → pullback_constr_p Obj (λX Y ⇒ Hom Y X) id (λX Y Z f g ⇒ comp Z Y X g f) po i0 i1 copair
In Proofgold the corresponding term root is 340ba6... and proposition id is d2ca56...
Proof:
Let po, i0, i1 and copair be given.
Assume HP.
An exact proof term for the current goal is HP.
∎
Theorem. (product_equalizer_pullback_constr)
MetaCat Obj Hom id comp → ∀quot : set → set → set → set → set, ∀canonmap : set → set → set → set → set, ∀fac : set → set → set → set → set → set → set, equalizer_constr_p Obj Hom id comp quot canonmap fac → ∀prod : set → set → set, ∀pi0 : set → set → set, ∀pi1 : set → set → set, ∀pair : set → set → set → set → set → set, product_constr_p Obj Hom id comp prod pi0 pi1 pair → pullback_constr_p Obj Hom id comp (λX Y Z f g ⇒ quot (prod X Y) Z (comp (prod X Y) X Z f (pi0 X Y)) (comp (prod X Y) Y Z g (pi1 X Y))) (λX Y Z f g ⇒ comp (quot (prod X Y) Z (comp (prod X Y) X Z f (pi0 X Y)) (comp (prod X Y) Y Z g (pi1 X Y))) (prod X Y) X (pi0 X Y) (canonmap (prod X Y) Z (comp (prod X Y) X Z f (pi0 X Y)) (comp (prod X Y) Y Z g (pi1 X Y)))) (λX Y Z f g ⇒ comp (quot (prod X Y) Z (comp (prod X Y) X Z f (pi0 X Y)) (comp (prod X Y) Y Z g (pi1 X Y))) (prod X Y) Y (pi1 X Y) (canonmap (prod X Y) Z (comp (prod X Y) X Z f (pi0 X Y)) (comp (prod X Y) Y Z g (pi1 X Y)))) (λX Y Z f g W h k ⇒ fac (prod X Y) Z (comp (prod X Y) X Z f (pi0 X Y)) (comp (prod X Y) Y Z g (pi1 X Y)) W (pair X Y W h k))
In Proofgold the corresponding term root is 1c512a... and proposition id is 1abd16...
Proof:
Assume HC.
Apply MetaCat_E Obj Hom id comp HC to the current goal.
Assume HC1: ∀X : set, Obj X → Hom X X (id X).
Assume HC2: ∀X Y Z : set, ∀f g : set, Obj X → Obj Y → Obj Z → Hom X Y f → Hom Y Z g → Hom X Z (comp X Y Z g f).
Assume HC3: ∀X Y : set, ∀f : set, Obj X → Obj Y → Hom X Y f → comp X X Y f (id X) = f.
Assume HC4: ∀X Y : set, ∀f : set, Obj X → Obj Y → Hom X Y f → comp X Y Y (id Y) f = f.
Assume HC5: ∀X Y Z W : set, ∀f g h : set, Obj X → Obj Y → Obj Z → Obj W → Hom X Y f → Hom Y Z g → Hom Z W h → comp X Y W (comp Y Z W h g) f = comp X Z W h (comp X Y Z g f).
Let quot, canonmap and fac be given.
Assume HE: equalizer_constr_p Obj Hom id comp quot canonmap fac.
Let prod, pi0, pi1 and pair be given.
Assume HP: product_constr_p Obj Hom id comp prod pi0 pi1 pair.
Let X, Y and Z be given.
Assume HX: Obj X.
Assume HY: Obj Y.
Assume HZ: Obj Z.
Let f and g be given.
Assume Hf: Hom X Z f.
Assume Hg: Hom Y Z g.
Set pi0f to be the term comp (prod X Y) X Z f (pi0 X Y).
Set pi1g to be the term comp (prod X Y) Y Z g (pi1 X Y).
Set P to be the term quot (prod X Y) Z pi0f pi1g.
Set pi0P to be the term comp P (prod X Y) X (pi0 X Y) (canonmap (prod X Y) Z pi0f pi1g).
Set pi1P to be the term comp P (prod X Y) Y (pi1 X Y) (canonmap (prod X Y) Z pi0f pi1g).
Set pairP to be the term (λW : set ⇒ λh k : set ⇒ fac (prod X Y) Z pi0f pi1g W (pair X Y W h k)).
Apply HP X Y HX HY to the current goal.
Assume HP1.
Apply HP1 to the current goal.
Assume HP1.
Apply HP1 to the current goal.
Assume HP1.
Apply HP1 to the current goal.
Assume HP1.
Apply HP1 to the current goal.
Assume _ _.
Assume HP1: Obj (prod X Y).
Assume HP2: Hom (prod X Y) X (pi0 X Y).
Assume HP3: Hom (prod X Y) Y (pi1 X Y).
Assume HP4: ∀W : set, Obj W → ∀h k : set, Hom W X h → Hom W Y k → Hom W (prod X Y) (pair X Y W h k) ∧ comp W (prod X Y) X (pi0 X Y) (pair X Y W h k) = h ∧ comp W (prod X Y) Y (pi1 X Y) (pair X Y W h k) = k ∧ ∀u : set, Hom W (prod X Y) u → comp W (prod X Y) X (pi0 X Y) u = h → comp W (prod X Y) Y (pi1 X Y) u = k → u = pair X Y W h k.
We prove the intermediate claim Lpi0f: Hom (prod X Y) Z pi0f.
An exact proof term for the current goal is HC2 (prod X Y) X Z (pi0 X Y) f HP1 HX HZ HP2 Hf.
We prove the intermediate claim Lpi1g: Hom (prod X Y) Z pi1g.
An exact proof term for the current goal is HC2 (prod X Y) Y Z (pi1 X Y) g HP1 HY HZ HP3 Hg.
Apply HE (prod X Y) Z HP1 HZ pi0f pi1g Lpi0f Lpi1g to the current goal.
Assume HE1.
Apply HE1 to the current goal.
Assume HE1.
Apply HE1 to the current goal.
Assume HE1.
Apply HE1 to the current goal.
Assume HE1.
Apply HE1 to the current goal.
Assume HE1.
Apply HE1 to the current goal.
Assume HE1.
Apply HE1 to the current goal.
Assume _ _ _ _.
Assume HE1: Obj P.
Assume HE2: Hom P (prod X Y) (canonmap (prod X Y) Z pi0f pi1g).
Assume HE3: comp P (prod X Y) Z pi0f (canonmap (prod X Y) Z pi0f pi1g) = comp P (prod X Y) Z pi1g (canonmap (prod X Y) Z pi0f pi1g).
Assume HE4: ∀W : set, Obj W → ∀h : set, Hom W (prod X Y) h → comp W (prod X Y) Z pi0f h = comp W (prod X Y) Z pi1g h → Hom W P (fac (prod X Y) Z pi0f pi1g W h) ∧ comp W P (prod X Y) (canonmap (prod X Y) Z pi0f pi1g) (fac (prod X Y) Z pi0f pi1g W h) = h ∧ ∀u : set, Hom W P u → comp W P (prod X Y) (canonmap (prod X Y) Z pi0f pi1g) u = h → u = fac (prod X Y) Z pi0f pi1g W h.
We prove the intermediate claim Lpi0P: Hom P X pi0P.
An exact proof term for the current goal is HC2 P (prod X Y) X (canonmap (prod X Y) Z pi0f pi1g) (pi0 X Y) HE1 HP1 HX HE2 HP2.
We prove the intermediate claim Lpi1P: Hom P Y pi1P.
An exact proof term for the current goal is HC2 P (prod X Y) Y (canonmap (prod X Y) Z pi0f pi1g) (pi1 X Y) HE1 HP1 HY HE2 HP3.
We will prove pullback_p Obj Hom id comp X Y Z f g P pi0P pi1P pairP.
We will prove Obj X ∧ Obj Y ∧ Obj Z ∧ Hom X Z f ∧ Hom Y Z g ∧ Obj P ∧ Hom P X pi0P ∧ Hom P Y pi1P ∧ comp P X Z f pi0P = comp P Y Z g pi1P ∧ ∀W : set, Obj W → ∀h : set, Hom W X h → ∀k : set, Hom W Y k → comp W X Z f h = comp W Y Z g k → Hom W P (pairP W h k) ∧ comp W P X pi0P (pairP W h k) = h ∧ comp W P Y pi1P (pairP W h k) = k ∧ ∀u : set, Hom W P u → comp W P X pi0P u = h → comp W P Y pi1P u = k → u = pairP W h k.
Apply and10I to the current goal.
An exact proof term for the current goal is HX.
An exact proof term for the current goal is HY.
An exact proof term for the current goal is HZ.
An exact proof term for the current goal is Hf.
An exact proof term for the current goal is Hg.
We will prove Obj P.
An exact proof term for the current goal is HE1.
We will prove Hom P X pi0P.
An exact proof term for the current goal is Lpi0P.
We will prove Hom P Y pi1P.
An exact proof term for the current goal is Lpi1P.
We will prove comp P X Z f pi0P = comp P Y Z g pi1P.
We will prove comp P X Z f (comp P (prod X Y) X (pi0 X Y) (canonmap (prod X Y) Z pi0f pi1g)) = comp P Y Z g (comp P (prod X Y) Y (pi1 X Y) (canonmap (prod X Y) Z pi0f pi1g)).
Use transitivity with comp P (prod X Y) Z pi0f (canonmap (prod X Y) Z pi0f pi1g), and comp P (prod X Y) Z pi1g (canonmap (prod X Y) Z pi0f pi1g).
Use symmetry.
Apply HC5 to the current goal.
An exact proof term for the current goal is HE1.
An exact proof term for the current goal is HP1.
An exact proof term for the current goal is HX.
An exact proof term for the current goal is HZ.
An exact proof term for the current goal is HE2.
An exact proof term for the current goal is HP2.
An exact proof term for the current goal is Hf.
We will prove comp P (prod X Y) Z pi0f (canonmap (prod X Y) Z pi0f pi1g) = comp P (prod X Y) Z pi1g (canonmap (prod X Y) Z pi0f pi1g).
An exact proof term for the current goal is HE3.
Apply HC5 to the current goal.
An exact proof term for the current goal is HE1.
An exact proof term for the current goal is HP1.
An exact proof term for the current goal is HY.
An exact proof term for the current goal is HZ.
An exact proof term for the current goal is HE2.
An exact proof term for the current goal is HP3.
An exact proof term for the current goal is Hg.
Let W be given.
Assume HW: Obj W.
Let h be given.
Assume Hh: Hom W X h.
Let k be given.
Assume Hk: Hom W Y k.
Assume Hhk: comp W X Z f h = comp W Y Z g k.
Apply HP4 W HW h k Hh Hk to the current goal.
Assume HP4a.
Apply HP4a to the current goal.
Assume HP4a.
Apply HP4a to the current goal.
Assume HP4a: Hom W (prod X Y) (pair X Y W h k).
Assume HP4b: comp W (prod X Y) X (pi0 X Y) (pair X Y W h k) = h.
Assume HP4c: comp W (prod X Y) Y (pi1 X Y) (pair X Y W h k) = k.
Assume HP4d: ∀u : set, Hom W (prod X Y) u → comp W (prod X Y) X (pi0 X Y) u = h → comp W (prod X Y) Y (pi1 X Y) u = k → u = pair X Y W h k.
We prove the intermediate claim LP1: comp W (prod X Y) Z pi0f (pair X Y W h k) = comp W (prod X Y) Z pi1g (pair X Y W h k).
Use transitivity with comp W X Z f (comp W (prod X Y) X (pi0 X Y) (pair X Y W h k)), and comp W Y Z g (comp W (prod X Y) Y (pi1 X Y) (pair X Y W h k)).
An exact proof term for the current goal is HC5 W (prod X Y) X Z (pair X Y W h k) (pi0 X Y) f HW HP1 HX HZ HP4a HP2 Hf.
We will prove comp W X Z f (comp W (prod X Y) X (pi0 X Y) (pair X Y W h k)) = comp W Y Z g (comp W (prod X Y) Y (pi1 X Y) (pair X Y W h k)).
rewrite the current goal using HP4b (from left to right).
rewrite the current goal using HP4c (from left to right).
We will prove comp W X Z f h = comp W Y Z g k.
An exact proof term for the current goal is Hhk.
Use symmetry.
An exact proof term for the current goal is HC5 W (prod X Y) Y Z (pair X Y W h k) (pi1 X Y) g HW HP1 HY HZ HP4a HP3 Hg.
Apply HE4 W HW (pair X Y W h k) HP4a LP1 to the current goal.
Assume HE4a.
Apply HE4a to the current goal.
Assume HE4a: Hom W P (fac (prod X Y) Z pi0f pi1g W (pair X Y W h k)).
Assume HE4b: comp W P (prod X Y) (canonmap (prod X Y) Z pi0f pi1g) (fac (prod X Y) Z pi0f pi1g W (pair X Y W h k)) = pair X Y W h k.
Assume HE4c: ∀u : set, Hom W P u → comp W P (prod X Y) (canonmap (prod X Y) Z pi0f pi1g) u = pair X Y W h k → u = fac (prod X Y) Z pi0f pi1g W (pair X Y W h k).
We prove the intermediate claim Lhk: Hom W P (pairP W h k).
An exact proof term for the current goal is HE4a.
Apply and4I to the current goal.
We will prove Hom W P (pairP W h k).
An exact proof term for the current goal is Lhk.
We will prove comp W P X pi0P (pairP W h k) = h.
Use transitivity with and comp W (prod X Y) X (pi0 X Y) (comp W P (prod X Y) (canonmap (prod X Y) Z pi0f pi1g) (fac (prod X Y) Z pi0f pi1g W (pair X Y W h k))).
An exact proof term for the current goal is HC5 W P (prod X Y) X (pairP W h k) (canonmap (prod X Y) Z pi0f pi1g) (pi0 X Y) HW HE1 HP1 HX HE4a HE2 HP2.
rewrite the current goal using HE4b (from left to right).
We will prove comp W (prod X Y) X (pi0 X Y) (pair X Y W h k) = h.
An exact proof term for the current goal is HP4b.
We will prove comp W P Y pi1P (pairP W h k) = k.
Use transitivity with and comp W (prod X Y) Y (pi1 X Y) (comp W P (prod X Y) (canonmap (prod X Y) Z pi0f pi1g) (fac (prod X Y) Z pi0f pi1g W (pair X Y W h k))).
An exact proof term for the current goal is HC5 W P (prod X Y) Y (pairP W h k) (canonmap (prod X Y) Z pi0f pi1g) (pi1 X Y) HW HE1 HP1 HY HE4a HE2 HP3.
rewrite the current goal using HE4b (from left to right).
We will prove comp W (prod X Y) Y (pi1 X Y) (pair X Y W h k) = k.
An exact proof term for the current goal is HP4c.
Let u be given.
Assume Hu: Hom W P u.
Assume Huh: comp W P X pi0P u = h.
Assume Huk: comp W P Y pi1P u = k.
We will prove u = pairP W h k.
We will prove u = fac (prod X Y) Z pi0f pi1g W (pair X Y W h k).
Apply HE4c u Hu to the current goal.
We will prove comp W P (prod X Y) (canonmap (prod X Y) Z pi0f pi1g) u = pair X Y W h k.
Apply HP4d (comp W P (prod X Y) (canonmap (prod X Y) Z pi0f pi1g) u) to the current goal.
We will prove Hom W (prod X Y) (comp W P (prod X Y) (canonmap (prod X Y) Z pi0f pi1g) u).
An exact proof term for the current goal is HC2 W P (prod X Y) u (canonmap (prod X Y) Z pi0f pi1g) HW HE1 HP1 Hu HE2.
We will prove comp W (prod X Y) X (pi0 X Y) (comp W P (prod X Y) (canonmap (prod X Y) Z pi0f pi1g) u) = h.
rewrite the current goal using HC5 W P (prod X Y) X u (canonmap (prod X Y) Z pi0f pi1g) (pi0 X Y) HW HE1 HP1 HX Hu HE2 HP2 (from right to left).
We will prove comp W P X (comp P (prod X Y) X (pi0 X Y) (canonmap (prod X Y) Z pi0f pi1g)) u = h.
An exact proof term for the current goal is Huh.
We will prove comp W (prod X Y) Y (pi1 X Y) (comp W P (prod X Y) (canonmap (prod X Y) Z pi0f pi1g) u) = k.
rewrite the current goal using HC5 W P (prod X Y) Y u (canonmap (prod X Y) Z pi0f pi1g) (pi1 X Y) HW HE1 HP1 HY Hu HE2 HP3 (from right to left).
We will prove comp W P Y (comp P (prod X Y) Y (pi1 X Y) (canonmap (prod X Y) Z pi0f pi1g)) u = k.
An exact proof term for the current goal is Huk.
∎
Theorem. (product_equalizer_pullback_constr_ex)
MetaCat Obj Hom id comp → (∃quot : set → set → set → set → set, ∃canonmap : set → set → set → set → set, ∃fac : set → set → set → set → set → set → set, equalizer_constr_p Obj Hom id comp quot canonmap fac) → (∃prod : set → set → set, ∃pi0 : set → set → set, ∃pi1 : set → set → set, ∃pair : set → set → set → set → set → set, product_constr_p Obj Hom id comp prod pi0 pi1 pair) → ∃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 Obj Hom id comp pb pi0 pi1 pair
In Proofgold the corresponding term root is 9a59ec... and proposition id is ed2b0e...
Proof:
Assume HC HE HP.
Apply HE to the current goal.
Let quot be given.
Assume HE.
Apply HE to the current goal.
Let canonmap be given.
Assume HE.
Apply HE to the current goal.
Let fac be given.
Assume HE: equalizer_constr_p Obj Hom id comp quot canonmap fac.
Apply HP to the current goal.
Let prod be given.
Assume HP.
Apply HP to the current goal.
Let pi0 be given.
Assume HP.
Apply HP to the current goal.
Let pi1 be given.
Assume HP.
Apply HP to the current goal.
Let pair be given.
Assume HP: product_constr_p Obj Hom id comp prod pi0 pi1 pair.
We use (λX Y Z f g ⇒ quot (prod X Y) Z (comp (prod X Y) X Z f (pi0 X Y)) (comp (prod X Y) Y Z g (pi1 X Y))) to witness the existential quantifier.
We use (λX Y Z f g ⇒ comp (quot (prod X Y) Z (comp (prod X Y) X Z f (pi0 X Y)) (comp (prod X Y) Y Z g (pi1 X Y))) (prod X Y) X (pi0 X Y) (canonmap (prod X Y) Z (comp (prod X Y) X Z f (pi0 X Y)) (comp (prod X Y) Y Z g (pi1 X Y)))) to witness the existential quantifier.
We use (λX Y Z f g ⇒ comp (quot (prod X Y) Z (comp (prod X Y) X Z f (pi0 X Y)) (comp (prod X Y) Y Z g (pi1 X Y))) (prod X Y) Y (pi1 X Y) (canonmap (prod X Y) Z (comp (prod X Y) X Z f (pi0 X Y)) (comp (prod X Y) Y Z g (pi1 X Y)))) to witness the existential quantifier.
We use (λX Y Z f g W h k ⇒ fac (prod X Y) Z (comp (prod X Y) X Z f (pi0 X Y)) (comp (prod X Y) Y Z g (pi1 X Y)) W (pair X Y W h k)) to witness the existential quantifier.
An exact proof term for the current goal is product_equalizer_pullback_constr HC quot canonmap fac HE prod pi0 pi1 pair HP.
∎
End of Section LimsCoLims2
Beginning of Section LimsCoLims3
Variable Obj : set → prop
Variable Hom : set → set → set → prop
Variable id : set → set
Variable comp : set → set → set → set → set → set
Theorem. (coproduct_coequalizer_pushout_constr_ex)
MetaCat Obj Hom id comp → (∃quot : set → set → set → set → set, ∃canonmap : set → set → set → set → set, ∃fac : set → set → set → set → set → set → set, coequalizer_constr_p Obj Hom id comp quot canonmap fac) → (∃coprod : set → set → set, ∃i0 : set → set → set, ∃i1 : set → set → set, ∃copair : set → set → set → set → set → set, coproduct_constr_p Obj Hom id comp coprod i0 i1 copair) → ∃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 Obj Hom id comp po i0 i1 copair
In Proofgold the corresponding term root is 70eb6d... and proposition id is b0aadc...
Proof:
Assume HC HE HP.
We prove the intermediate claim LC: MetaCat Obj (λX Y ⇒ Hom Y X) id (λX Y Z f g ⇒ comp Z Y X g f).
An exact proof term for the current goal is MetaCatOp Obj Hom id comp HC.
We will prove ∃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 Obj (λX Y ⇒ Hom Y X) id (λX Y Z f g ⇒ comp Z Y X g f) pb pi0 pi1 pair.
Apply product_equalizer_pullback_constr_ex Obj (λX Y ⇒ Hom Y X) id (λX Y Z f g ⇒ comp Z Y X g f) LC to the current goal.
Apply HE to the current goal.
Let quot be given.
Assume HE.
Apply HE to the current goal.
Let canonmap be given.
Assume HE.
Apply HE to the current goal.
Let fac be given.
Assume HE: coequalizer_constr_p Obj Hom id comp quot canonmap fac.
We use (λX Y : set ⇒ quot Y X) to witness the existential quantifier.
We use (λX Y : set ⇒ canonmap Y X) to witness the existential quantifier.
We use (λX Y : set ⇒ fac Y X) to witness the existential quantifier.
An exact proof term for the current goal is coequalizer_equalizer_constr_Op Obj Hom id comp quot canonmap fac HE.
An exact proof term for the current goal is HP.
∎
End of Section LimsCoLims3
Definition. We define SetHom to be λX Y f ⇒ f ∈ YX of type set → set → set → prop.
In Proofgold the corresponding term root is de8fdf... and object id is af2f63...
Theorem. (MetaCatSet_initial_gen)
∀Obj : set → prop, Obj 0 → ∃Y : set, ∃uniqa : set → set, initial_p Obj SetHom (λX ⇒ (λx ∈ X ⇒ x)) (λX Y Z f g ⇒ (λx ∈ X ⇒ f (g x))) Y uniqa
In Proofgold the corresponding term root is 31dcb8... and proposition id is fa807d...
Proof:
Let Obj be given.
Assume HO.
We use 0 to witness the existential quantifier.
We use (λX ⇒ (λx ∈ 0 ⇒ 0)) to witness the existential quantifier.
We will prove Obj 0 ∧ ∀X : set, Obj X → SetHom 0 X (λx ∈ 0 ⇒ 0) ∧ ∀h' : set, SetHom 0 X h' → h' = (λx ∈ 0 ⇒ 0).
Apply andI to the current goal.
An exact proof term for the current goal is HO.
Let X be given.
Assume HX.
Apply andI to the current goal.
We will prove SetHom 0 X (λx ∈ 0 ⇒ 0).
We will prove (λx ∈ 0 ⇒ 0) ∈ ∏x ∈ 0, X.
Apply lam_Pi to the current goal.
Let x be given.
Assume Hx: x ∈ 0.
We will prove False.
An exact proof term for the current goal is EmptyE x Hx.
Let h be given.
Assume Hh: h ∈ ∏x ∈ 0, X.
We will prove h = (λx ∈ 0 ⇒ 0).
Use transitivity with and (λx ∈ 0 ⇒ h x).
Use symmetry.
An exact proof term for the current goal is Pi_eta 0 (λ_ ⇒ X) h Hh.
Apply lam_ext to the current goal.
Let x be given.
Assume Hx: x ∈ 0.
We will prove False.
An exact proof term for the current goal is EmptyE x Hx.
∎
Theorem. (MetaCatSet_initial)
∃Y : set, ∃uniqa : set → set, initial_p (λ_ ⇒ True) SetHom (λX ⇒ (λx ∈ X ⇒ x)) (λX Y Z f g ⇒ (λx ∈ X ⇒ f (g x))) Y uniqa
In Proofgold the corresponding term root is 648295... and proposition id is 80cab9...
Proof:
An exact proof term for the current goal is MetaCatSet_initial_gen (λ_ ⇒ True) TrueI.
∎
Theorem. (MetaCatSet_terminal_gen)
∀Obj : set → prop, Obj 1 → ∃Y : set, ∃uniqa : set → set, terminal_p Obj SetHom (λX ⇒ (λx ∈ X ⇒ x)) (λX Y Z f g ⇒ (λx ∈ X ⇒ f (g x))) Y uniqa
In Proofgold the corresponding term root is b6fb54... and proposition id is 4cd1b7...
Proof:
Let Obj be given.
Assume HO.
We use 1 to witness the existential quantifier.
We use (λX ⇒ (λx ∈ X ⇒ 0)) to witness the existential quantifier.
We will prove Obj 1 ∧ ∀X : set, Obj X → SetHom X 1 (λx ∈ X ⇒ 0) ∧ ∀h' : set, SetHom X 1 h' → h' = (λx ∈ X ⇒ 0).
Apply andI to the current goal.
An exact proof term for the current goal is HO.
Let X be given.
Assume HX.
Apply andI to the current goal.
We will prove SetHom X 1 (λx ∈ X ⇒ 0).
We will prove (λx ∈ X ⇒ 0) ∈ ∏x ∈ X, 1.
Apply lam_Pi to the current goal.
Let x be given.
Assume Hx: x ∈ X.
An exact proof term for the current goal is In_0_1.
Let h be given.
Assume Hh: h ∈ ∏x ∈ X, 1.
We will prove h = (λx ∈ X ⇒ 0).
Use transitivity with and (λx ∈ X ⇒ h x).
Use symmetry.
An exact proof term for the current goal is Pi_eta X (λ_ ⇒ 1) h Hh.
Apply lam_ext to the current goal.
Let x be given.
Assume Hx: x ∈ X.
We will prove h x = 0.
Apply SingE 0 (h x) to the current goal.
We will prove h x ∈ {0}.
rewrite the current goal using eq_1_Sing0 (from right to left).
We will prove h x ∈ 1.
An exact proof term for the current goal is ap_Pi X (λ_ ⇒ 1) h x Hh Hx.
∎
Theorem. (MetaCatSet_terminal)
∃Y : set, ∃uniqa : set → set, terminal_p (λ_ ⇒ True) SetHom (λX ⇒ (λx ∈ X ⇒ x)) (λX Y Z f g ⇒ (λx ∈ X ⇒ f (g x))) Y uniqa
In Proofgold the corresponding term root is bf7783... and proposition id is 370ea7...
Proof:
An exact proof term for the current goal is MetaCatSet_terminal_gen (λ_ ⇒ True) TrueI.
∎
Theorem. (MetaCatSet_coproduct_gen)
∀Obj : set → prop, (∀X, Obj X → ∀Y, Obj Y → Obj (setsum X Y)) → ∃coprod : set → set → set, ∃i0 i1 : set → set → set, ∃copair : set → set → set → set → set → set, coproduct_constr_p Obj SetHom (λX ⇒ (λx ∈ X ⇒ x)) (λX Y Z f g ⇒ (λx ∈ X ⇒ f (g x))) coprod i0 i1 copair
In Proofgold the corresponding term root is 7e8001... and proposition id is 3c078b...
Proof:
Let Obj be given.
Assume HO: ∀X, Obj X → ∀Y, Obj Y → Obj (setsum X Y).
We use setsum to witness the existential quantifier.
We use (λX Y ⇒ λx ∈ X ⇒ Inj0 x) to witness the existential quantifier.
We use (λX Y ⇒ λy ∈ Y ⇒ Inj1 y) to witness the existential quantifier.
We use (λX Y W h k ⇒ (λz ∈ X + Y ⇒ combine_funcs X Y (λx ⇒ h x) (λy ⇒ k y) z)) to witness the existential quantifier.
Let X and Y be given.
Assume HX: Obj X.
Assume HY: Obj Y.
Set Z to be the term X + Y.
Set i0 to be the term (λx ∈ X ⇒ Inj0 x).
Set i1 to be the term (λy ∈ Y ⇒ Inj1 y).
Set comb to be the term λW : set ⇒ λh k : set ⇒ (λz ∈ X + Y ⇒ combine_funcs X Y (λx ⇒ h x) (λy ⇒ k y) z).
We will prove Obj X ∧ Obj Y ∧ Obj Z ∧ SetHom X Z i0 ∧ SetHom Y Z i1 ∧ ∀W, Obj W → ∀h k : set, SetHom X W h → SetHom Y W k → SetHom Z W (comb W h k) ∧ (λx ∈ X ⇒ comb W h k (i0 x)) = h ∧ (λy ∈ Y ⇒ comb W h k (i1 y)) = k ∧ ∀hk : set, SetHom Z W hk → (λx ∈ X ⇒ hk (i0 x)) = h → (λy ∈ Y ⇒ hk (i1 y)) = k → hk = comb W h k.
Apply and6I to the current goal.
An exact proof term for the current goal is HX.
An exact proof term for the current goal is HY.
An exact proof term for the current goal is HO X HX Y HY.
We will prove SetHom X Z i0.
We will prove i0 ∈ ∏x ∈ X, Z.
Apply lam_Pi to the current goal.
Let x be given.
Assume Hx.
We will prove Inj0 x ∈ X + Y.
An exact proof term for the current goal is Inj0_setsum X Y x Hx.
We will prove SetHom Y Z i1.
We will prove i1 ∈ ∏y ∈ Y, Z.
Apply lam_Pi to the current goal.
Let y be given.
Assume Hy.
We will prove Inj1 y ∈ X + Y.
An exact proof term for the current goal is Inj1_setsum X Y y Hy.
Let W be given.
Assume HW: Obj W.
Let h and k be given.
Assume Hh: SetHom X W h.
Assume Hk: SetHom Y W k.
We will prove SetHom Z W (comb W h k) ∧ (λx ∈ X ⇒ comb W h k (i0 x)) = h ∧ (λy ∈ Y ⇒ comb W h k (i1 y)) = k ∧ ∀hk : set, SetHom Z W hk → (λx ∈ X ⇒ hk (i0 x)) = h → (λy ∈ Y ⇒ hk (i1 y)) = k → hk = comb W h k.
Apply and4I to the current goal.
We will prove SetHom Z W (comb W h k).
We will prove (λz ∈ X + Y ⇒ combine_funcs X Y (λx ⇒ h x) (λy ⇒ k y) z) ∈ ∏z ∈ Z, W.
Apply lam_Pi to the current goal.
Let z be given.
Assume Hz: z ∈ X + Y.
We will prove (combine_funcs X Y (λx ⇒ h x) (λy ⇒ k y) z) ∈ W.
Apply setsum_Inj_inv X Y z Hz to the current goal.
Assume H1.
Apply H1 to the current goal.
Let x be given.
Assume H2.
Apply H2 to the current goal.
Assume Hx: x ∈ X.
Assume H2: z = Inj0 x.
rewrite the current goal using H2 (from left to right).
We will prove (combine_funcs X Y (λx ⇒ h x) (λy ⇒ k y) (Inj0 x)) ∈ W.
rewrite the current goal using combine_funcs_eq1 X Y (λx ⇒ h x) (λy ⇒ k y) x (from left to right).
We will prove h x ∈ W.
An exact proof term for the current goal is ap_Pi X (λ_ ⇒ W) h x Hh Hx.
Assume H1.
Apply H1 to the current goal.
Let y be given.
Assume H2.
Apply H2 to the current goal.
Assume Hy: y ∈ Y.
Assume H2: z = Inj1 y.
rewrite the current goal using H2 (from left to right).
We will prove (combine_funcs X Y (λx ⇒ h x) (λy ⇒ k y) (Inj1 y)) ∈ W.
rewrite the current goal using combine_funcs_eq2 X Y (λx ⇒ h x) (λy ⇒ k y) y (from left to right).
We will prove k y ∈ W.
An exact proof term for the current goal is ap_Pi Y (λ_ ⇒ W) k y Hk Hy.
We will prove (λx ∈ X ⇒ comb W h k (i0 x)) = h.
Use transitivity with and (λx ∈ X ⇒ h x).
Apply lam_ext to the current goal.
Let x be given.
Assume Hx.
We will prove comb W h k (i0 x) = h x.
We will prove (λz ∈ X + Y ⇒ combine_funcs X Y (λx ⇒ h x) (λy ⇒ k y) z) ((λx ∈ X ⇒ Inj0 x) x) = h x.
rewrite the current goal using beta X Inj0 x Hx (from left to right).
We will prove (λz ∈ X + Y ⇒ combine_funcs X Y (λx ⇒ h x) (λy ⇒ k y) z) (Inj0 x) = h x.
rewrite the current goal using beta (X + Y) (λz ⇒ combine_funcs X Y (λx ⇒ h x) (λy ⇒ k y) z) (Inj0 x) (Inj0_setsum X Y x Hx) (from left to right).
Apply combine_funcs_eq1 to the current goal.
An exact proof term for the current goal is Pi_eta X (λ_ ⇒ W) h Hh.
We will prove (λy ∈ Y ⇒ comb W h k (i1 y)) = k.
Use transitivity with and (λy ∈ Y ⇒ k y).
Apply lam_ext to the current goal.
Let y be given.
Assume Hy.
We will prove comb W h k (i1 y) = k y.
We will prove (λz ∈ X + Y ⇒ combine_funcs X Y (λx ⇒ h x) (λy ⇒ k y) z) ((λy ∈ Y ⇒ Inj1 y) y) = k y.
rewrite the current goal using beta Y Inj1 y Hy (from left to right).
We will prove (λz ∈ X + Y ⇒ combine_funcs X Y (λx ⇒ h x) (λy ⇒ k y) z) (Inj1 y) = k y.
rewrite the current goal using beta (X + Y) (λz ⇒ combine_funcs X Y (λx ⇒ h x) (λy ⇒ k y) z) (Inj1 y) (Inj1_setsum X Y y Hy) (from left to right).
Apply combine_funcs_eq2 to the current goal.
An exact proof term for the current goal is Pi_eta Y (λ_ ⇒ W) k Hk.
Let hk be given.
Assume Hhk: SetHom Z W hk.
Assume Hhkh: (λx ∈ X ⇒ hk (i0 x)) = h.
Assume Hhkk: (λy ∈ Y ⇒ hk (i1 y)) = k.
We will prove hk = comb W h k.
Use transitivity with and (λz ∈ Z ⇒ hk z).
Use symmetry.
An exact proof term for the current goal is Pi_eta Z (λ_ ⇒ W) hk Hhk.
We will prove (λz ∈ Z ⇒ hk z) = (λz ∈ X + Y ⇒ combine_funcs X Y (λx ⇒ h x) (λy ⇒ k y) z).
Apply lam_ext to the current goal.
Let z be given.
Assume Hz.
We will prove hk z = combine_funcs X Y (λx ⇒ h x) (λy ⇒ k y) z.
Apply setsum_Inj_inv X Y z Hz to the current goal.
Assume H1.
Apply H1 to the current goal.
Let x be given.
Assume H2.
Apply H2 to the current goal.
Assume Hx: x ∈ X.
Assume H2: z = Inj0 x.
rewrite the current goal using H2 (from left to right).
We will prove hk (Inj0 x) = combine_funcs X Y (λx ⇒ h x) (λy ⇒ k y) (Inj0 x).
rewrite the current goal using combine_funcs_eq1 (from left to right).
We will prove hk (Inj0 x) = h x.
rewrite the current goal using Hhkh (from right to left).
rewrite the current goal using beta X (λx ⇒ hk (i0 x)) x Hx (from left to right).
We will prove hk (Inj0 x) = hk (i0 x).
Use f_equal.
We will prove Inj0 x = (λx ∈ X ⇒ Inj0 x) x.
rewrite the current goal using beta X Inj0 x Hx (from left to right).
Use reflexivity.
Assume H1.
Apply H1 to the current goal.
Let y be given.
Assume H2.
Apply H2 to the current goal.
Assume Hy: y ∈ Y.
Assume H2: z = Inj1 y.
rewrite the current goal using H2 (from left to right).
We will prove hk (Inj1 y) = combine_funcs X Y (λx ⇒ h x) (λy ⇒ k y) (Inj1 y).
rewrite the current goal using combine_funcs_eq2 (from left to right).
We will prove hk (Inj1 y) = k y.
rewrite the current goal using Hhkk (from right to left).
rewrite the current goal using beta Y (λy ⇒ hk (i1 y)) y Hy (from left to right).
We will prove hk (Inj1 y) = hk (i1 y).
Use f_equal.
We will prove Inj1 y = (λy ∈ Y ⇒ Inj1 y) y.
rewrite the current goal using beta Y Inj1 y Hy (from left to right).
Use reflexivity.
∎
Theorem. (MetaCatSet_coproduct)
∃coprod : set → set → set, ∃i0 i1 : set → set → set, ∃copair : set → set → set → set → set → set, coproduct_constr_p (λ_ ⇒ True) SetHom (λX ⇒ λx ∈ X ⇒ x) (λX Y Z f g ⇒ (λx ∈ X ⇒ f (g x))) coprod i0 i1 copair
In Proofgold the corresponding term root is 2c4f22... and proposition id is c28eaa...
Proof:
We prove the intermediate claim L1: ∀X, True → ∀Y, True → True.
An exact proof term for the current goal is (λ_ _ _ _ ⇒ TrueI).
An exact proof term for the current goal is MetaCatSet_coproduct_gen (λ_ ⇒ True) L1.
∎
Theorem. (MetaCatSet_product_gen)
∀Obj : set → prop, (∀X, Obj X → ∀Y, Obj Y → Obj (setprod X Y)) → ∃prod : set → set → set, ∃pi0 pi1 : set → set → set, ∃pair : set → set → set → set → set → set, product_constr_p Obj SetHom (λX ⇒ (λx ∈ X ⇒ x)) (λX Y Z f g ⇒ (λx ∈ X ⇒ f (g x))) prod pi0 pi1 pair
In Proofgold the corresponding term root is 2d427e... and proposition id is 21e789...
Proof:
Let Obj be given.
Assume HO: ∀X, Obj X → ∀Y, Obj Y → Obj (X ⨯ Y).
We use setprod to witness the existential quantifier.
We use (λX Y ⇒ (λz ∈ X ⨯ Y ⇒ z 0)) to witness the existential quantifier.
We use (λX Y ⇒ (λz ∈ X ⨯ Y ⇒ z 1)) to witness the existential quantifier.
We use (λX Y W h k ⇒ (λw ∈ W ⇒ (h w,k w))) to witness the existential quantifier.
Let X and Y be given.
Assume HX: Obj X.
Assume HY: Obj Y.
Set Z to be the term X ⨯ Y.
Set pi0 to be the term (λz ∈ Z ⇒ z 0).
Set pi1 to be the term (λz ∈ Z ⇒ z 1).
Set pair to be the term λW : set ⇒ λh k : set ⇒ (λw ∈ W ⇒ (h w,k w)).
We will prove product_p Obj SetHom (λX ⇒ (λx ∈ X ⇒ x)) (λX Y Z f g ⇒ (λx ∈ X ⇒ f (g x))) X Y Z pi0 pi1 pair.
We will prove Obj X ∧ Obj Y ∧ Obj Z ∧ SetHom Z X pi0 ∧ SetHom Z Y pi1 ∧ ∀W, Obj W → ∀h k : set, SetHom W X h → SetHom W Y k → SetHom W Z (pair W h k) ∧ (λw ∈ W ⇒ pi0 (pair W h k w)) = h ∧ (λw ∈ W ⇒ pi1 (pair W h k w)) = k ∧ ∀u : set, SetHom W Z u → (λw ∈ W ⇒ pi0 (u w)) = h → (λw ∈ W ⇒ pi1 (u w)) = k → u = pair W h k.
Apply and6I to the current goal.
An exact proof term for the current goal is HX.
An exact proof term for the current goal is HY.
An exact proof term for the current goal is HO X HX Y HY.
We will prove SetHom Z X pi0.
We will prove pi0 ∈ ∏z ∈ Z, X.
Apply lam_Pi to the current goal.
An exact proof term for the current goal is ap0_Sigma X (λ_ ⇒ Y).
We will prove SetHom Z Y pi1.
We will prove pi1 ∈ ∏z ∈ Z, Y.
Apply lam_Pi to the current goal.
An exact proof term for the current goal is ap1_Sigma X (λ_ ⇒ Y).
Let W be given.
Assume HW: Obj W.
Let h and k be given.
Assume Hh: SetHom W X h.
Assume Hk: SetHom W Y k.
We prove the intermediate claim L1: ∀w ∈ W, (h w,k w) ∈ Z.
Let w be given.
Assume Hw: w ∈ W.
We will prove (h w,k w) ∈ X ⨯ Y.
Apply tuple_2_setprod to the current goal.
An exact proof term for the current goal is ap_Pi W (λ_ ⇒ X) h w Hh Hw.
An exact proof term for the current goal is ap_Pi W (λ_ ⇒ Y) k w Hk Hw.
We prove the intermediate claim L2: (λw ∈ W ⇒ (h w,k w)) ∈ ∏w ∈ W, Z.
We will prove (λw ∈ W ⇒ (h w,k w)) ∈ ∏w ∈ W, Z.
Apply lam_Pi to the current goal.
An exact proof term for the current goal is L1.
We will prove SetHom W Z (pair W h k) ∧ (λw ∈ W ⇒ pi0 (pair W h k w)) = h ∧ (λw ∈ W ⇒ pi1 (pair W h k w)) = k ∧ ∀u : set, SetHom W Z u → (λw ∈ W ⇒ pi0 (u w)) = h → (λw ∈ W ⇒ pi1 (u w)) = k → u = pair W h k.
Apply and4I to the current goal.
We will prove SetHom W Z (pair W h k).
An exact proof term for the current goal is L2.
We will prove (λw ∈ W ⇒ pi0 (pair W h k w)) = h.
Use transitivity with and (λw ∈ W ⇒ h w).
Apply lam_ext to the current goal.
Let w be given.
Assume Hw.
We will prove pi0 ((λw ∈ W ⇒ (h w,k w)) w) = h w.
rewrite the current goal using beta W (λw ⇒ (h w,k w)) w Hw (from left to right).
We will prove (λz ∈ Z ⇒ z 0) (h w,k w) = h w.
rewrite the current goal using beta Z (λz ⇒ z 0) (h w,k w) (L1 w Hw) (from left to right).
We will prove (h w,k w) 0 = h w.
Apply tuple_2_0_eq to the current goal.
We will prove (λw ∈ W ⇒ h w) = h.
An exact proof term for the current goal is Pi_eta W (λ_ ⇒ X) h Hh.
We will prove (λw ∈ W ⇒ pi1 (pair W h k w)) = k.
Use transitivity with and (λw ∈ W ⇒ k w).
Apply lam_ext to the current goal.
Let w be given.
Assume Hw.
We will prove pi1 ((λw ∈ W ⇒ (h w,k w)) w) = k w.
rewrite the current goal using beta W (λw ⇒ (h w,k w)) w Hw (from left to right).
We will prove (λz ∈ Z ⇒ z 1) (h w,k w) = k w.
rewrite the current goal using beta Z (λz ⇒ z 1) (h w,k w) (L1 w Hw) (from left to right).
We will prove (h w,k w) 1 = k w.
Apply tuple_2_1_eq to the current goal.
We will prove (λw ∈ W ⇒ k w) = k.
An exact proof term for the current goal is Pi_eta W (λ_ ⇒ Y) k Hk.
Let u be given.
Assume Hu: SetHom W Z u.
Assume Huh: (λw ∈ W ⇒ pi0 (u w)) = h.
Assume Huk: (λw ∈ W ⇒ pi1 (u w)) = k.
We will prove u = pair W h k.
Use transitivity with and (λw ∈ W ⇒ u w).
Use symmetry.
An exact proof term for the current goal is Pi_eta W (λ_ ⇒ Z) u Hu.
We will prove (λw ∈ W ⇒ u w) = (λw ∈ W ⇒ (h w,k w)).
Apply lam_ext to the current goal.
Let w be given.
Assume Hw.
We will prove u w = (h w,k w).
We prove the intermediate claim L3: u w ∈ Z.
An exact proof term for the current goal is ap_Pi W (λ_ ⇒ Z) u w Hu Hw.
rewrite the current goal using tuple_Sigma_eta X (λ_ ⇒ Y) (u w) L3 (from right to left).
We will prove (u w 0,u w 1) = (h w,k w).
We prove the intermediate claim L4: u w 0 = h w.
rewrite the current goal using Huh (from right to left).
We will prove u w 0 = (λw ∈ W ⇒ pi0 (u w)) w.
Use transitivity with and pi0 (u w).
We will prove u w 0 = (λz ∈ Z ⇒ z 0) (u w).
Use symmetry.
An exact proof term for the current goal is beta Z (λz ⇒ z 0) (u w) L3.
Use symmetry.
An exact proof term for the current goal is beta W (λw ⇒ pi0 (u w)) w Hw.
We prove the intermediate claim L5: u w 1 = k w.
rewrite the current goal using Huk (from right to left).
We will prove u w 1 = (λw ∈ W ⇒ pi1 (u w)) w.
Use transitivity with and pi1 (u w).
We will prove u w 1 = (λz ∈ Z ⇒ z 1) (u w).
Use symmetry.
An exact proof term for the current goal is beta Z (λz ⇒ z 1) (u w) L3.
Use symmetry.
An exact proof term for the current goal is beta W (λw ⇒ pi1 (u w)) w Hw.
rewrite the current goal using L4 (from left to right).
rewrite the current goal using L5 (from left to right).
Use reflexivity.
∎
Theorem. (MetaCatSet_product)
∃prod : set → set → set, ∃pi0 pi1 : set → set → set, ∃pair : set → set → set → set → set → set, product_constr_p (λ_ ⇒ True) SetHom (λX ⇒ (λx ∈ X ⇒ x)) (λX Y Z f g ⇒ (λx ∈ X ⇒ f (g x))) prod pi0 pi1 pair
In Proofgold the corresponding term root is 8c56c9... and proposition id is 14d110...
Proof:
We prove the intermediate claim L1: ∀X, True → ∀Y, True → True.
An exact proof term for the current goal is (λ_ _ _ _ ⇒ TrueI).
An exact proof term for the current goal is MetaCatSet_product_gen (λ_ ⇒ True) L1.
∎
Theorem. (UnivOf_Subq_closed)
∀N, ∀X ∈ UnivOf N, ∀Q ⊆ X, Q ∈ UnivOf N
In Proofgold the corresponding term root is 1bcb03... and proposition id is 6aa34e...
Proof:
Let N and X be given.
Assume HX: X ∈ UnivOf N.
Let Q be given.
Assume HQ: Q ⊆ X.
We prove the intermediate claim LZ: ZF_closed (UnivOf N).
An exact proof term for the current goal is UnivOf_ZF_closed N.
We prove the intermediate claim LQ: Q ∈ 𝒫 X.
An exact proof term for the current goal is PowerI X Q HQ.
An exact proof term for the current goal is UnivOf_TransSet N (𝒫 X) (ZF_Power_closed (UnivOf N) LZ X HX) Q LQ.
∎
Definition. We define famunion_closed to be λU : set ⇒ ∀X ∈ U, ∀F : set → set, (∀x ∈ X, F x ∈ U) → famunion X F ∈ U of type set → prop.
In Proofgold the corresponding term root is caf77a... and object id is f8891c...
Theorem. (Union_Repl_famunion_closed)
∀U : set, Union_closed U → Repl_closed U → famunion_closed U
In Proofgold the corresponding term root is 35501a... and proposition id is 345f8f...
Proof:
Let U be given.
Assume HU: Union_closed U.
Assume HR: Repl_closed U.
Let X be given.
Assume HX: X ∈ U.
Let F be given.
Assume HF: ∀x ∈ X, F x ∈ U.
We will prove famunion X F ∈ U.
We will prove ⋃ {F x|x ∈ X} ∈ U.
Apply HU to the current goal.
We will prove {F x|x ∈ X} ∈ U.
An exact proof term for the current goal is HR X HX F HF.
∎
Theorem. (ZF_closed_0)
∀U X, TransSet U → ZF_closed U → X ∈ U → 0 ∈ U
In Proofgold the corresponding term root is e747e5... and proposition id is 1037dc...
Proof:
Let U and X be given.
Assume H1 H2 H3.
Apply ZF_closed_E U H2 to the current goal.
Assume HU HP HR.
We prove the intermediate claim L1: 𝒫 X ∈ U.
Apply HP to the current goal.
An exact proof term for the current goal is H3.
Apply H1 (𝒫 X) L1 0 to the current goal.
We will prove 0 ∈ 𝒫 X.
Apply Empty_In_Power to the current goal.
∎
Theorem. (ZF_Inj1_closed)
∀U, TransSet U → ZF_closed U → ∀X ∈ U, Inj1 X ∈ U
In Proofgold the corresponding term root is 617147... and proposition id is 7eedbc...
Proof:
Let U be given.
Assume HUT: TransSet U.
Assume HU: ZF_closed U.
Apply In_ind to the current goal.
Let X be given.
Assume IH: ∀x ∈ X, x ∈ U → Inj1 x ∈ U.
Assume HX: X ∈ U.
rewrite the current goal using Inj1_eq (from left to right).
We will prove ({Empty} ∪ {Inj1 x|x ∈ X}) ∈ U.
Apply ZF_binunion_closed U HU to the current goal.
Apply ZF_Sing_closed U HU to the current goal.
Apply HUT (𝒫 X) (ZF_Power_closed U HU X HX) Empty (Empty_In_Power X) to the current goal.
Apply ZF_Repl_closed U HU X HX to the current goal.
Let x be given.
Assume Hx: x ∈ X.
Apply IH x Hx to the current goal.
We will prove x ∈ U.
An exact proof term for the current goal is HUT X HX x Hx.
∎
Theorem. (ZF_Inj0_closed)
∀U, TransSet U → ZF_closed U → ∀X ∈ U, Inj0 X ∈ U
In Proofgold the corresponding term root is 219267... and proposition id is 1045a1...
Proof:
Let U be given.
Assume HUT: TransSet U.
Assume HU: ZF_closed U.
Let X be given.
Assume HX: X ∈ U.
We will prove {Inj1 x|x ∈ X} ∈ U.
Apply ZF_Repl_closed U HU X HX to the current goal.
Let x be given.
Assume Hx: x ∈ X.
Apply ZF_Inj1_closed U HUT HU x to the current goal.
We will prove x ∈ U.
An exact proof term for the current goal is HUT X HX x Hx.
∎
Theorem. (ZF_setsum_closed)
∀U, TransSet U → ZF_closed U → ∀X Y ∈ U, (X + Y) ∈ U
In Proofgold the corresponding term root is 527cc5... and proposition id is 32cb89...
Proof:
Let U be given.
Assume HUT: TransSet U.
Assume HU: ZF_closed U.
Let X be given.
Assume HX: X ∈ U.
Let Y be given.
Assume HY: Y ∈ U.
We will prove ({Inj0 x|x ∈ X} ∪ {Inj1 y|y ∈ Y}) ∈ U.
Apply ZF_binunion_closed U HU to the current goal.
We will prove {Inj0 x|x ∈ X} ∈ U.
Apply ZF_Repl_closed U HU X HX Inj0 to the current goal.
Let x be given.
Assume Hx: x ∈ X.
We will prove Inj0 x ∈ U.
Apply ZF_Inj0_closed U HUT HU x to the current goal.
We will prove x ∈ U.
An exact proof term for the current goal is HUT X HX x Hx.
Apply ZF_Repl_closed U HU Y HY Inj1 to the current goal.
Let y be given.
Assume Hy: y ∈ Y.
We will prove Inj1 y ∈ U.
Apply ZF_Inj1_closed U HUT HU y to the current goal.
We will prove y ∈ U.
An exact proof term for the current goal is HUT Y HY y Hy.
∎
Theorem. (ZF_Sigma_closed)
∀U, TransSet U → ZF_closed U → ∀X ∈ U, ∀Y : set → set, (∀x ∈ X, Y x ∈ U) → (∑x ∈ X, Y x) ∈ U
In Proofgold the corresponding term root is d9ae82... and proposition id is c59b39...
Proof:
Let U be given.
Assume HUT: TransSet U.
Assume HU: ZF_closed U.
Apply ZF_closed_E U HU to the current goal.
Assume HUU HUP HUR.
Let X be given.
Assume HX: X ∈ U.
Let Y be given.
Assume HY: ∀x ∈ X, Y x ∈ U.
We will prove (⋃x ∈ X{setsum x y|y ∈ Y x}) ∈ U.
We prove the intermediate claim L1: famunion_closed U.
Apply Union_Repl_famunion_closed to the current goal.
An exact proof term for the current goal is HUU.
An exact proof term for the current goal is HUR.
Apply L1 X HX (λx ⇒ {setsum x y|y ∈ Y x}) to the current goal.
Let x be given.
Assume Hx: x ∈ X.
We will prove {setsum x y|y ∈ Y x} ∈ U.
Apply HUR (Y x) (HY x Hx) (λy ⇒ setsum x y) to the current goal.
Let y be given.
Assume Hy: y ∈ Y x.
We will prove setsum x y ∈ U.
Apply ZF_setsum_closed U HUT HU to the current goal.
We will prove x ∈ U.
An exact proof term for the current goal is HUT X HX x Hx.
We will prove y ∈ U.
An exact proof term for the current goal is HUT (Y x) (HY x Hx) y Hy.
∎
Theorem. (ZF_setprod_closed)
∀U, TransSet U → ZF_closed U → ∀X Y ∈ U, (X ⨯ Y) ∈ U
In Proofgold the corresponding term root is 168889... and proposition id is ecfb59...
Proof:
Let U be given.
Assume HUT HU.
Let X be given.
Assume HX.
Let Y be given.
Assume HY.
An exact proof term for the current goal is ZF_Sigma_closed U HUT HU X HX (λ_ ⇒ Y) (λ_ _ ⇒ HY).
∎
Theorem. (ZF_Pi_closed)
∀U, TransSet U → ZF_closed U → ∀X ∈ U, ∀Y : set → set, (∀x ∈ X, Y x ∈ U) → (∏x ∈ X, Y x) ∈ U
In Proofgold the corresponding term root is 632304... and proposition id is 43ba53...
Proof:
Let U be given.
Assume HUT: TransSet U.
Assume HU: ZF_closed U.
Apply ZF_closed_E U HU to the current goal.
Assume HUU HUP HUR.
Let X be given.
Assume HX: X ∈ U.
Let Y be given.
Assume HY: ∀x ∈ X, Y x ∈ U.
We will prove (∏x ∈ X, Y x) ∈ U.
We prove the intermediate claim L1: (∏x ∈ X, Y x) ∈ 𝒫 (𝒫 (∑x ∈ X, ⋃ (Y x))).
An exact proof term for the current goal is Sep_In_Power (𝒫 (∑x ∈ X, ⋃ (Y x))) (λf ⇒ ∀x ∈ X, f x ∈ Y x).
We prove the intermediate claim L2: (𝒫 (𝒫 (∑x ∈ X, ⋃ (Y x)))) ∈ U.
Apply HUP to the current goal.
Apply HUP to the current goal.
Apply ZF_Sigma_closed U HUT HU X HX (λx ⇒ ⋃ (Y x)) to the current goal.
Let x be given.
Assume Hx: x ∈ X.
We will prove ⋃ (Y x) ∈ U.
Apply HUU to the current goal.
An exact proof term for the current goal is HY x Hx.
An exact proof term for the current goal is HUT (𝒫 (𝒫 (∑x ∈ X, ⋃ (Y x)))) L2 (∏x ∈ X, Y x) L1.
∎
Theorem. (ZF_setexp_closed)
∀U, TransSet U → ZF_closed U → ∀X Y ∈ U, (YX) ∈ U
In Proofgold the corresponding term root is 248526... and proposition id is 331185...
Proof:
Let U be given.
Assume HUT HU.
Let X be given.
Assume HX.
Let Y be given.
Assume HY.
An exact proof term for the current goal is ZF_Pi_closed U HUT HU X HX (λ_ ⇒ Y) (λx Hx ⇒ HY).
∎
Theorem. (MetaCatHFSet_initial)
∃Y : set, ∃uniqa : set → set, initial_p (λX ⇒ X ∈ UnivOf Empty) SetHom (λX ⇒ (λx ∈ X ⇒ x)) (λX Y Z f g ⇒ (λx ∈ X ⇒ f (g x))) Y uniqa
In Proofgold the corresponding term root is 0741fe... and proposition id is 6694e8...
Proof:
An exact proof term for the current goal is MetaCatSet_initial_gen (λX ⇒ X ∈ UnivOf Empty) (UnivOf_In Empty).
∎
Theorem. (MetaCatSmallSet_initial)
∃Y : set, ∃uniqa : set → set, initial_p (λX ⇒ X ∈ UnivOf (UnivOf Empty)) SetHom (λX ⇒ (λx ∈ X ⇒ x)) (λX Y Z f g ⇒ (λx ∈ X ⇒ f (g x))) Y uniqa
In Proofgold the corresponding term root is 964635... and proposition id is 4e15ce...
Proof:
We prove the intermediate claim L1: 0 ∈ UnivOf (UnivOf Empty).
An exact proof term for the current goal is ZF_closed_0 (UnivOf (UnivOf Empty)) (UnivOf Empty) (UnivOf_TransSet (UnivOf Empty)) (UnivOf_ZF_closed (UnivOf Empty)) (UnivOf_In (UnivOf Empty)).
An exact proof term for the current goal is MetaCatSet_initial_gen (λX ⇒ X ∈ UnivOf (UnivOf Empty)) L1.
∎
Theorem. (MetaCatHFSet_terminal)
∃Y : set, ∃uniqa : set → set, terminal_p (λX ⇒ X ∈ UnivOf Empty) SetHom (λX ⇒ (λx ∈ X ⇒ x)) (λX Y Z f g ⇒ (λx ∈ X ⇒ f (g x))) Y uniqa
In Proofgold the corresponding term root is 703b92... and proposition id is d36462...
Proof:
We prove the intermediate claim L1: 1 ∈ UnivOf Empty.
Apply ZF_ordsucc_closed to the current goal.
An exact proof term for the current goal is UnivOf_ZF_closed Empty.
An exact proof term for the current goal is ZF_closed_0 (UnivOf Empty) Empty (UnivOf_TransSet Empty) (UnivOf_ZF_closed Empty) (UnivOf_In Empty).
An exact proof term for the current goal is MetaCatSet_terminal_gen (λX ⇒ X ∈ UnivOf Empty) L1.
∎
Theorem. (MetaCatSmallSet_terminal)
∃Y : set, ∃uniqa : set → set, terminal_p (λX ⇒ X ∈ UnivOf (UnivOf Empty)) SetHom (λX ⇒ (λx ∈ X ⇒ x)) (λX Y Z f g ⇒ (λx ∈ X ⇒ f (g x))) Y uniqa
In Proofgold the corresponding term root is 1f7930... and proposition id is b5da2c...
Proof:
We prove the intermediate claim L1: 1 ∈ UnivOf (UnivOf Empty).
Apply ZF_ordsucc_closed to the current goal.
An exact proof term for the current goal is UnivOf_ZF_closed (UnivOf Empty).
An exact proof term for the current goal is ZF_closed_0 (UnivOf (UnivOf Empty)) (UnivOf Empty) (UnivOf_TransSet (UnivOf Empty)) (UnivOf_ZF_closed (UnivOf Empty)) (UnivOf_In (UnivOf Empty)).
An exact proof term for the current goal is MetaCatSet_terminal_gen (λX ⇒ X ∈ UnivOf (UnivOf Empty)) L1.
∎
Theorem. (MetaCatHFSet_coproduct)
∃coprod : set → set → set, ∃i0 i1 : set → set → set, ∃copair : set → set → set → set → set → set, coproduct_constr_p (λX ⇒ X ∈ UnivOf Empty) SetHom (λX ⇒ λx ∈ X ⇒ x) (λX Y Z f g ⇒ (λx ∈ X ⇒ f (g x))) coprod i0 i1 copair
In Proofgold the corresponding term root is df899b... and proposition id is 5604f1...
Proof:
We prove the intermediate claim L1: ∀X Y ∈ UnivOf Empty, X + Y ∈ UnivOf Empty.
An exact proof term for the current goal is ZF_setsum_closed (UnivOf Empty) (UnivOf_TransSet Empty) (UnivOf_ZF_closed Empty).
An exact proof term for the current goal is MetaCatSet_coproduct_gen (λX ⇒ X ∈ UnivOf Empty) L1.
∎
Theorem. (MetaCatSmallSet_coproduct)
∃coprod : set → set → set, ∃i0 i1 : set → set → set, ∃copair : set → set → set → set → set → set, coproduct_constr_p (λX ⇒ X ∈ UnivOf (UnivOf Empty)) SetHom (λX ⇒ (λx ∈ X ⇒ x)) (λX Y Z f g ⇒ (λx ∈ X ⇒ f (g x))) coprod i0 i1 copair
In Proofgold the corresponding term root is 15e47c... and proposition id is 002eee...
Proof:
We prove the intermediate claim L1: ∀X Y ∈ UnivOf (UnivOf Empty), X + Y ∈ UnivOf (UnivOf Empty).
An exact proof term for the current goal is ZF_setsum_closed (UnivOf (UnivOf Empty)) (UnivOf_TransSet (UnivOf Empty)) (UnivOf_ZF_closed (UnivOf Empty)).
An exact proof term for the current goal is MetaCatSet_coproduct_gen (λX ⇒ X ∈ UnivOf (UnivOf Empty)) L1.
∎
Theorem. (MetaCatHFSet_product)
∃prod : set → set → set, ∃pi0 pi1 : set → set → set, ∃pair : set → set → set → set → set → set, product_constr_p (λX ⇒ X ∈ UnivOf Empty) SetHom (λX ⇒ (λx ∈ X ⇒ x)) (λX Y Z f g ⇒ (λx ∈ X ⇒ f (g x))) prod pi0 pi1 pair
In Proofgold the corresponding term root is 2bef5b... and proposition id is 8ae2a3...
Proof:
We prove the intermediate claim L1: ∀X Y ∈ UnivOf Empty, X ⨯ Y ∈ UnivOf Empty.
An exact proof term for the current goal is ZF_setprod_closed (UnivOf Empty) (UnivOf_TransSet Empty) (UnivOf_ZF_closed Empty).
An exact proof term for the current goal is MetaCatSet_product_gen (λX ⇒ X ∈ UnivOf Empty) L1.
∎
Theorem. (MetaCatSmallSet_product)
∃prod : set → set → set, ∃pi0 pi1 : set → set → set, ∃pair : set → set → set → set → set → set, product_constr_p (λX ⇒ X ∈ UnivOf (UnivOf Empty)) SetHom (λX ⇒ (λx ∈ X ⇒ x)) (λX Y Z f g ⇒ (λx ∈ X ⇒ f (g x))) prod pi0 pi1 pair
In Proofgold the corresponding term root is 2285d1... and proposition id is 4281ba...
Proof:
We prove the intermediate claim L1: ∀X Y ∈ UnivOf (UnivOf Empty), X ⨯ Y ∈ UnivOf (UnivOf Empty).
An exact proof term for the current goal is ZF_setprod_closed (UnivOf (UnivOf Empty)) (UnivOf_TransSet (UnivOf Empty)) (UnivOf_ZF_closed (UnivOf Empty)).
An exact proof term for the current goal is MetaCatSet_product_gen (λX ⇒ X ∈ UnivOf (UnivOf Empty)) L1.
∎
Beginning of Section MetaFunctor
Variable Obj : set → prop
Variable Hom : set → set → set → prop
Variable id : set → set
Variable comp : set → set → set → set → set → set
Variable Obj' : set → prop
Variable Hom' : set → set → set → prop
Variable id' : set → set
Variable comp' : set → set → set → set → set → set
Variable F0 : set → set
Variable F1 : set → set → set → set
Definition. We define MetaFunctor to be (∀X, Obj X → Obj' (F0 X)) ∧ (∀X Y f, Obj X → Obj Y → Hom X Y f → Hom' (F0 X) (F0 Y) (F1 X Y f)) ∧ (∀X, Obj X → F1 X X (id X) = id' (F0 X)) ∧ (∀X Y Z f g, Obj X → Obj Y → Obj Z → Hom X Y f → Hom Y Z g → F1 X Z (comp X Y Z g f) = comp' (F0 X) (F0 Y) (F0 Z) (F1 Y Z g) (F1 X Y f)) of type prop.
In Proofgold the corresponding term root is f35b67... and object id is c80743...
Theorem. (MetaFunctorI)
(∀X, Obj X → Obj' (F0 X)) → (∀X Y f, Obj X → Obj Y → Hom X Y f → Hom' (F0 X) (F0 Y) (F1 X Y f)) → (∀X, Obj X → F1 X X (id X) = id' (F0 X)) → (∀X Y Z f g, Obj X → Obj Y → Obj Z → Hom X Y f → Hom Y Z g → F1 X Z (comp X Y Z g f) = comp' (F0 X) (F0 Y) (F0 Z) (F1 Y Z g) (F1 X Y f)) → MetaFunctor
In Proofgold the corresponding term root is 795e29... and proposition id is 2cb62d...
Proof:
An exact proof term for the current goal is and4I (∀X, Obj X → Obj' (F0 X)) (∀X Y f, Obj X → Obj Y → Hom X Y f → Hom' (F0 X) (F0 Y) (F1 X Y f)) (∀X, Obj X → F1 X X (id X) = id' (F0 X)) (∀X Y Z f g, Obj X → Obj Y → Obj Z → Hom X Y f → Hom Y Z g → F1 X Z (comp X Y Z g f) = comp' (F0 X) (F0 Y) (F0 Z) (F1 Y Z g) (F1 X Y f)).
∎
Theorem. (MetaFunctorE)
MetaFunctor → ∀p : prop, ((∀X, Obj X → Obj' (F0 X)) → (∀X Y f, Obj X → Obj Y → Hom X Y f → Hom' (F0 X) (F0 Y) (F1 X Y f)) → (∀X, Obj X → F1 X X (id X) = id' (F0 X)) → (∀X Y Z f g, Obj X → Obj Y → Obj Z → Hom X Y f → Hom Y Z g → F1 X Z (comp X Y Z g f) = comp' (F0 X) (F0 Y) (F0 Z) (F1 Y Z g) (F1 X Y f)) → p) → p
In Proofgold the corresponding term root is d4b40e... and proposition id is 973e25...
Proof:
An exact proof term for the current goal is and4E (∀X, Obj X → Obj' (F0 X)) (∀X Y f, Obj X → Obj Y → Hom X Y f → Hom' (F0 X) (F0 Y) (F1 X Y f)) (∀X, Obj X → F1 X X (id X) = id' (F0 X)) (∀X Y Z f g, Obj X → Obj Y → Obj Z → Hom X Y f → Hom Y Z g → F1 X Z (comp X Y Z g f) = comp' (F0 X) (F0 Y) (F0 Z) (F1 Y Z g) (F1 X Y f)).
∎
Definition. We define MetaFunctor_strict to be MetaCat Obj Hom id comp ∧ MetaCat Obj' Hom' id' comp' ∧ MetaFunctor of type prop.
In Proofgold the corresponding term root is 8085cf... and object id is 070aeb...
Theorem. (MetaFunctor_strict_I)
MetaCat Obj Hom id comp → MetaCat Obj' Hom' id' comp' → MetaFunctor → MetaFunctor_strict
In Proofgold the corresponding term root is 3d0579... and proposition id is 5cbb4a...
Proof:
An exact proof term for the current goal is and3I (MetaCat Obj Hom id comp) (MetaCat Obj' Hom' id' comp') MetaFunctor.
∎
Theorem. (MetaFunctor_strict_E)
MetaFunctor_strict → ∀p : prop, (MetaCat Obj Hom id comp → MetaCat Obj' Hom' id' comp' → MetaFunctor → p) → p
In Proofgold the corresponding term root is b9f4ec... and proposition id is 953054...
Proof:
An exact proof term for the current goal is and3E (MetaCat Obj Hom id comp) (MetaCat Obj' Hom' id' comp') MetaFunctor.
∎
End of Section MetaFunctor
Beginning of Section IdFunctor
Variable Obj : set → prop
Variable Hom : set → set → set → prop
Variable id : set → set
Variable comp : set → set → set → set → set → set
Theorem. (MetaCat_IdFunctor)
MetaFunctor Obj Hom id comp Obj Hom id comp (λX ⇒ X) (λX Y f ⇒ f)
In Proofgold the corresponding term root is ee743b... and proposition id is b7eb16...
Proof:
Apply MetaFunctorI to the current goal.
Let X be given.
Assume HX.
An exact proof term for the current goal is HX.
Let X, Y and f be given.
Assume HX HY Hf.
An exact proof term for the current goal is Hf.
Let X be given.
Assume HX.
Use reflexivity.
Let X, Y, Z, f and g be given.
Assume HX HY HZ Hf Hg.
Use reflexivity.
∎
Theorem. (MetaCat_IdFunctor_strict)
MetaCat Obj Hom id comp → MetaFunctor_strict Obj Hom id comp Obj Hom id comp (λX ⇒ X) (λX Y f ⇒ f)
In Proofgold the corresponding term root is 73c677... and proposition id is 2447d9...
Proof:
Assume HC.
Apply MetaFunctor_strict_I to the current goal.
An exact proof term for the current goal is HC.
An exact proof term for the current goal is HC.
An exact proof term for the current goal is MetaCat_IdFunctor.
∎
End of Section IdFunctor
Beginning of Section CompFunctors
Variable Obj : set → prop
Variable Hom : set → set → set → prop
Variable id : set → set
Variable comp : set → set → set → set → set → set
Variable Obj' : set → prop
Variable Hom' : set → set → set → prop
Variable id' : set → set
Variable comp' : set → set → set → set → set → set
Variable Obj'' : set → prop
Variable Hom'' : set → set → set → prop
Variable id'' : set → set
Variable comp'' : set → set → set → set → set → set
Variable F0 : set → set
Variable F1 : set → set → set → set
Variable G0 : set → set
Variable G1 : set → set → set → set
Theorem. (MetaCat_CompFunctors)
MetaFunctor Obj Hom id comp Obj' Hom' id' comp' F0 F1 → MetaFunctor Obj' Hom' id' comp' Obj'' Hom'' id'' comp'' G0 G1 → MetaFunctor Obj Hom id comp Obj'' Hom'' id'' comp'' (λX ⇒ G0 (F0 X)) (λX Y f ⇒ G1 (F0 X) (F0 Y) (F1 X Y f))
In Proofgold the corresponding term root is 830254... and proposition id is d7211d...
Proof:
Assume HF HG.
Apply MetaFunctorE Obj Hom id comp Obj' Hom' id' comp' F0 F1 HF to the current goal.
Assume HF1 HF2 HF3 HF4.
Apply MetaFunctorE Obj' Hom' id' comp' Obj'' Hom'' id'' comp'' G0 G1 HG to the current goal.
Assume HG1 HG2 HG3 HG4.
Apply MetaFunctorI to the current goal.
Let X be given.
Assume HX.
We will prove Obj'' (G0 (F0 X)).
Apply HG1 to the current goal.
Apply HF1 to the current goal.
An exact proof term for the current goal is HX.
Let X, Y and f be given.
Assume HX HY Hf.
We will prove Hom'' (G0 (F0 X)) (G0 (F0 Y)) (G1 (F0 X) (F0 Y) (F1 X Y f)).
Apply HG2 to the current goal.
Apply HF1 to the current goal.
An exact proof term for the current goal is HX.
Apply HF1 to the current goal.
An exact proof term for the current goal is HY.
Apply HF2 to the current goal.
An exact proof term for the current goal is HX.
An exact proof term for the current goal is HY.
An exact proof term for the current goal is Hf.
Let X be given.
Assume HX.
We will prove G1 (F0 X) (F0 X) (F1 X X (id X)) = id'' (G0 (F0 X)).
Use transitivity with and G1 (F0 X) (F0 X) (id' (F0 X)).
Use f_equal.
An exact proof term for the current goal is HF3 X HX.
An exact proof term for the current goal is HG3 (F0 X) (HF1 X HX).
Let X, Y, Z, f and g be given.
Assume HX HY HZ Hf Hg.
We will prove G1 (F0 X) (F0 Z) (F1 X Z (comp X Y Z g f)) = comp'' (G0 (F0 X)) (G0 (F0 Y)) (G0 (F0 Z)) (G1 (F0 Y) (F0 Z) (F1 Y Z g)) (G1 (F0 X) (F0 Y) (F1 X Y f)).
Use transitivity with and G1 (F0 X) (F0 Z) (comp' (F0 X) (F0 Y) (F0 Z) (F1 Y Z g) (F1 X Y f)).
Use f_equal.
An exact proof term for the current goal is HF4 X Y Z f g HX HY HZ Hf Hg.
Apply HG4 (F0 X) (F0 Y) (F0 Z) (F1 X Y f) (F1 Y Z g) (HF1 X HX) (HF1 Y HY) (HF1 Z HZ) to the current goal.
An exact proof term for the current goal is HF2 X Y f HX HY Hf.
An exact proof term for the current goal is HF2 Y Z g HY HZ Hg.
∎
Theorem. (MetaCat_CompFunctors_strict)
MetaFunctor_strict Obj Hom id comp Obj' Hom' id' comp' F0 F1 → MetaFunctor_strict Obj' Hom' id' comp' Obj'' Hom'' id'' comp'' G0 G1 → MetaFunctor_strict Obj Hom id comp Obj'' Hom'' id'' comp'' (λX ⇒ G0 (F0 X)) (λX Y f ⇒ G1 (F0 X) (F0 Y) (F1 X Y f))
In Proofgold the corresponding term root is 7ad5e2... and proposition id is 1eb285...
Proof:
Assume HF HG.
Apply MetaFunctor_strict_E Obj Hom id comp Obj' Hom' id' comp' F0 F1 HF to the current goal.
Assume HC _ HF.
Apply MetaFunctor_strict_E Obj' Hom' id' comp' Obj'' Hom'' id'' comp'' G0 G1 HG to the current goal.
Assume _ HC'' HG.
Apply MetaFunctor_strict_I to the current goal.
An exact proof term for the current goal is HC.
An exact proof term for the current goal is HC''.
Apply MetaCat_CompFunctors to the current goal.
An exact proof term for the current goal is HF.
An exact proof term for the current goal is HG.
∎
End of Section CompFunctors
Beginning of Section MetaNatTrans
Variable Obj : set → prop
Variable Hom : set → set → set → prop
Variable id : set → set
Variable comp : set → set → set → set → set → set
Variable Obj' : set → prop
Variable Hom' : set → set → set → prop
Variable id' : set → set
Variable comp' : set → set → set → set → set → set
Variable F0 : set → set
Variable F1 : set → set → set → set
Variable G0 : set → set
Variable G1 : set → set → set → set
Variable eta : set → set
Definition. We define MetaNatTrans to be (∀X, Obj X → Hom' (F0 X) (G0 X) (eta X)) ∧ (∀X Y f, Obj X → Obj Y → Hom X Y f → comp' (F0 X) (G0 X) (G0 Y) (G1 X Y f) (eta X) = comp' (F0 X) (F0 Y) (G0 Y) (eta Y) (F1 X Y f)) of type prop.
In Proofgold the corresponding term root is cf0ad1... and object id is 08be86...
Theorem. (MetaNatTransI)
(∀X, Obj X → Hom' (F0 X) (G0 X) (eta X)) → (∀X Y f, Obj X → Obj Y → Hom X Y f → comp' (F0 X) (G0 X) (G0 Y) (G1 X Y f) (eta X) = comp' (F0 X) (F0 Y) (G0 Y) (eta Y) (F1 X Y f)) → MetaNatTrans
In Proofgold the corresponding term root is 8286b8... and proposition id is c1d681...
Proof:
An exact proof term for the current goal is andI (∀X, Obj X → Hom' (F0 X) (G0 X) (eta X)) (∀X Y f, Obj X → Obj Y → Hom X Y f → comp' (F0 X) (G0 X) (G0 Y) (G1 X Y f) (eta X) = comp' (F0 X) (F0 Y) (G0 Y) (eta Y) (F1 X Y f)).
∎
Theorem. (MetaNatTransE)
MetaNatTrans → ∀p : prop, ((∀X, Obj X → Hom' (F0 X) (G0 X) (eta X)) → (∀X Y f, Obj X → Obj Y → Hom X Y f → comp' (F0 X) (G0 X) (G0 Y) (G1 X Y f) (eta X) = comp' (F0 X) (F0 Y) (G0 Y) (eta Y) (F1 X Y f)) → p) → p
In Proofgold the corresponding term root is 60c5d3... and proposition id is aa53ac...
Proof:
Assume H.
An exact proof term for the current goal is H.
∎
Definition. We define MetaNatTrans_strict to be MetaCat Obj Hom id comp ∧ MetaCat Obj' Hom' id' comp' ∧ MetaFunctor Obj Hom id comp Obj' Hom' id' comp' F0 F1 ∧ MetaFunctor Obj Hom id comp Obj' Hom' id' comp' G0 G1 ∧ MetaNatTrans of type prop.
In Proofgold the corresponding term root is 1b6ee9... and object id is bd2de2...
Theorem. (MetaNatTrans_strict_I)
MetaCat Obj Hom id comp → MetaCat Obj' Hom' id' comp' → MetaFunctor Obj Hom id comp Obj' Hom' id' comp' F0 F1 → MetaFunctor Obj Hom id comp Obj' Hom' id' comp' G0 G1 → MetaNatTrans → MetaNatTrans_strict
In Proofgold the corresponding term root is fea59f... and proposition id is 59a372...
Proof:
An exact proof term for the current goal is and5I (MetaCat Obj Hom id comp) (MetaCat Obj' Hom' id' comp') (MetaFunctor Obj Hom id comp Obj' Hom' id' comp' F0 F1) (MetaFunctor Obj Hom id comp Obj' Hom' id' comp' G0 G1) MetaNatTrans.
∎
Theorem. (MetaNatTrans_strict_E)
MetaNatTrans_strict → ∀p : prop, (MetaCat Obj Hom id comp → MetaCat Obj' Hom' id' comp' → MetaFunctor Obj Hom id comp Obj' Hom' id' comp' F0 F1 → MetaFunctor Obj Hom id comp Obj' Hom' id' comp' G0 G1 → MetaNatTrans → p) → p
In Proofgold the corresponding term root is db0322... and proposition id is b8f26b...
Proof:
Assume H.
An exact proof term for the current goal is and5E (MetaCat Obj Hom id comp) (MetaCat Obj' Hom' id' comp') (MetaFunctor Obj Hom id comp Obj' Hom' id' comp' F0 F1) (MetaFunctor Obj Hom id comp Obj' Hom' id' comp' G0 G1) MetaNatTrans H.
∎
End of Section MetaNatTrans
Beginning of Section CompFunctorNatTrans
Variable Obj : set → prop
Variable Hom : set → set → set → prop
Variable id : set → set
Variable comp : set → set → set → set → set → set
Variable Obj' : set → prop
Variable Hom' : set → set → set → prop
Variable id' : set → set
Variable comp' : set → set → set → set → set → set
Variable Obj'' : set → prop
Variable Hom'' : set → set → set → prop
Variable id'' : set → set
Variable comp'' : set → set → set → set → set → set
Variable F0 : set → set
Variable F1 : set → set → set → set
Variable G0 : set → set
Variable G1 : set → set → set → set
Variable H0 : set → set
Variable H1 : set → set → set → set
Variable eta : set → set
Theorem. (MetaCat_CompFunctorNatTrans)
MetaFunctor Obj Hom id comp Obj' Hom' id' comp' F0 F1 → MetaFunctor Obj Hom id comp Obj' Hom' id' comp' G0 G1 → MetaNatTrans Obj Hom id comp Obj' Hom' id' comp' F0 F1 G0 G1 eta → MetaFunctor Obj' Hom' id' comp' Obj'' Hom'' id'' comp'' H0 H1 → MetaNatTrans Obj Hom id comp Obj'' Hom'' id'' comp'' (λX ⇒ H0 (F0 X)) (λX Y f ⇒ H1 (F0 X) (F0 Y) (F1 X Y f)) (λX ⇒ H0 (G0 X)) (λX Y f ⇒ H1 (G0 X) (G0 Y) (G1 X Y f)) (λX ⇒ H1 (F0 X) (G0 X) (eta X))
In Proofgold the corresponding term root is e1978b... and proposition id is d7aeb3...
Proof:
Assume HF HG Heta HH.
Apply MetaFunctorE Obj Hom id comp Obj' Hom' id' comp' F0 F1 HF to the current goal.
Assume HF1 HF2 HF3 HF4.
Apply MetaFunctorE Obj Hom id comp Obj' Hom' id' comp' G0 G1 HG to the current goal.
Assume HG1 HG2 HG3 HG4.
Apply MetaNatTransE Obj Hom id comp Obj' Hom' id' comp' F0 F1 G0 G1 eta Heta to the current goal.
Assume Heta1 Heta2.
Apply MetaFunctorE Obj' Hom' id' comp' Obj'' Hom'' id'' comp'' H0 H1 HH to the current goal.
Assume HH1 HH2 HH3 HH4.
Apply MetaNatTransI to the current goal.
Let X be given.
Assume HX.
We will prove Hom'' (H0 (F0 X)) (H0 (G0 X)) (H1 (F0 X) (G0 X) (eta X)).
Apply HH2 to the current goal.
We will prove Obj' (F0 X).
Apply HF1 to the current goal.
An exact proof term for the current goal is HX.
We will prove Obj' (G0 X).
Apply HG1 to the current goal.
An exact proof term for the current goal is HX.
We will prove Hom' (F0 X) (G0 X) (eta X).
Apply Heta1 to the current goal.
An exact proof term for the current goal is HX.
Let X, Y and f be given.
Assume HX HY Hf.
We will prove comp'' (H0 (F0 X)) (H0 (G0 X)) (H0 (G0 Y)) (H1 (G0 X) (G0 Y) (G1 X Y f)) (H1 (F0 X) (G0 X) (eta X)) = comp'' (H0 (F0 X)) (H0 (F0 Y)) (H0 (G0 Y)) (H1 (F0 Y) (G0 Y) (eta Y)) (H1 (F0 X) (F0 Y) (F1 X Y f)).
Use transitivity with H1 (F0 X) (G0 Y) (comp' (F0 X) (G0 X) (G0 Y) (G1 X Y f) (eta X)), and H1 (F0 X) (G0 Y) (comp' (F0 X) (F0 Y) (G0 Y) (eta Y) (F1 X Y f)).
Use symmetry.
Apply HH4 to the current goal.
Apply HF1 to the current goal.
An exact proof term for the current goal is HX.
Apply HG1 to the current goal.
An exact proof term for the current goal is HX.
Apply HG1 to the current goal.
An exact proof term for the current goal is HY.
An exact proof term for the current goal is Heta1 X HX.
An exact proof term for the current goal is HG2 X Y f HX HY Hf.
Use f_equal.
An exact proof term for the current goal is Heta2 X Y f HX HY Hf.
Apply HH4 to the current goal.
Apply HF1 to the current goal.
An exact proof term for the current goal is HX.
Apply HF1 to the current goal.
An exact proof term for the current goal is HY.
Apply HG1 to the current goal.
An exact proof term for the current goal is HY.
An exact proof term for the current goal is HF2 X Y f HX HY Hf.
An exact proof term for the current goal is Heta1 Y HY.
∎
End of Section CompFunctorNatTrans
Beginning of Section CompNatTransFunctor
Variable Obj : set → prop
Variable Hom : set → set → set → prop
Variable id : set → set
Variable comp : set → set → set → set → set → set
Variable Obj' : set → prop
Variable Hom' : set → set → set → prop
Variable id' : set → set
Variable comp' : set → set → set → set → set → set
Variable Obj'' : set → prop
Variable Hom'' : set → set → set → prop
Variable id'' : set → set
Variable comp'' : set → set → set → set → set → set
Variable F0 : set → set
Variable F1 : set → set → set → set
Variable G0 : set → set
Variable G1 : set → set → set → set
Variable H0 : set → set
Variable H1 : set → set → set → set
Variable eta : set → set
Theorem. (MetaCat_CompNatTransFunctor)
MetaNatTrans Obj' Hom' id' comp' Obj'' Hom'' id'' comp'' F0 F1 G0 G1 eta → MetaFunctor Obj Hom id comp Obj' Hom' id' comp' H0 H1 → MetaNatTrans Obj Hom id comp Obj'' Hom'' id'' comp'' (λX ⇒ F0 (H0 X)) (λX Y f ⇒ F1 (H0 X) (H0 Y) (H1 X Y f)) (λX ⇒ G0 (H0 X)) (λX Y f ⇒ G1 (H0 X) (H0 Y) (H1 X Y f)) (λX ⇒ eta (H0 X))
In Proofgold the corresponding term root is b7392d... and proposition id is b8a6b7...
Proof:
Assume Heta HH.
Apply MetaNatTransE Obj' Hom' id' comp' Obj'' Hom'' id'' comp'' F0 F1 G0 G1 eta Heta to the current goal.
Assume Heta1 Heta2.
Apply MetaFunctorE Obj Hom id comp Obj' Hom' id' comp' H0 H1 HH to the current goal.
Assume HH1 HH2 HH3 HH4.
Apply MetaNatTransI to the current goal.
Let X be given.
Assume HX.
We will prove Hom'' (F0 (H0 X)) (G0 (H0 X)) (eta (H0 X)).
Apply Heta1 to the current goal.
We will prove Obj' (H0 X).
Apply HH1 to the current goal.
An exact proof term for the current goal is HX.
Let X, Y and f be given.
Assume HX HY Hf.
We will prove comp'' (F0 (H0 X)) (G0 (H0 X)) (G0 (H0 Y)) (G1 (H0 X) (H0 Y) (H1 X Y f)) (eta (H0 X)) = comp'' (F0 (H0 X)) (F0 (H0 Y)) (G0 (H0 Y)) (eta (H0 Y)) (F1 (H0 X) (H0 Y) (H1 X Y f)).
Apply Heta2 (H0 X) (H0 Y) (H1 X Y f) to the current goal.
Apply HH1 to the current goal.
An exact proof term for the current goal is HX.
Apply HH1 to the current goal.
An exact proof term for the current goal is HY.
Apply HH2 to the current goal.
An exact proof term for the current goal is HX.
An exact proof term for the current goal is HY.
An exact proof term for the current goal is Hf.
∎
End of Section CompNatTransFunctor
Beginning of Section MetaMonad
Variable Obj : set → prop
Variable Hom : set → set → set → prop
Variable id : set → set
Variable comp : set → set → set → set → set → set
Variable T0 : set → set
Variable T1 : set → set → set → set
Variable eta : set → set
Variable mu : set → set
Definition. We define MetaMonad to be (∀X, Obj X → comp (T0 (T0 (T0 X))) (T0 (T0 X)) (T0 X) (mu X) (T1 (T0 (T0 X)) (T0 X) (mu X)) = comp (T0 (T0 (T0 X))) (T0 (T0 X)) (T0 X) (mu X) (mu (T0 X))) ∧ (∀X, Obj X → comp (T0 X) (T0 (T0 X)) (T0 X) (mu X) (eta (T0 X)) = id (T0 X)) ∧ (∀X, Obj X → comp (T0 X) (T0 (T0 X)) (T0 X) (mu X) (T1 X (T0 X) (eta X)) = id (T0 X)) of type prop.
In Proofgold the corresponding term root is bb95d4... and object id is 7c9f95...
Definition. We define MetaMonad_strict to be MetaNatTrans_strict Obj Hom id comp Obj Hom id comp (λX ⇒ X) (λX Y f ⇒ f) T0 T1 eta ∧ MetaNatTrans_strict Obj Hom id comp Obj Hom id comp (λX ⇒ T0 (T0 X)) (λX Y f ⇒ T1 (T0 X) (T0 Y) (T1 X Y f)) T0 T1 mu ∧ MetaMonad of type prop.
In Proofgold the corresponding term root is 8e3f5c... and object id is ce28f9...
Theorem. (MetaMonadI)
(∀X, Obj X → comp (T0 (T0 (T0 X))) (T0 (T0 X)) (T0 X) (mu X) (T1 (T0 (T0 X)) (T0 X) (mu X)) = comp (T0 (T0 (T0 X))) (T0 (T0 X)) (T0 X) (mu X) (mu (T0 X))) → (∀X, Obj X → comp (T0 X) (T0 (T0 X)) (T0 X) (mu X) (eta (T0 X)) = id (T0 X)) → (∀X, Obj X → comp (T0 X) (T0 (T0 X)) (T0 X) (mu X) (T1 X (T0 X) (eta X)) = id (T0 X)) → MetaMonad
In Proofgold the corresponding term root is 7671e5... and proposition id is 46096c...
Proof:
An exact proof term for the current goal is and3I (∀X, Obj X → comp (T0 (T0 (T0 X))) (T0 (T0 X)) (T0 X) (mu X) (T1 (T0 (T0 X)) (T0 X) (mu X)) = comp (T0 (T0 (T0 X))) (T0 (T0 X)) (T0 X) (mu X) (mu (T0 X))) (∀X, Obj X → comp (T0 X) (T0 (T0 X)) (T0 X) (mu X) (eta (T0 X)) = id (T0 X)) (∀X, Obj X → comp (T0 X) (T0 (T0 X)) (T0 X) (mu X) (T1 X (T0 X) (eta X)) = id (T0 X)).
∎
Theorem. (MetaMonadE)
MetaMonad → (∀p : prop, ((∀X, Obj X → comp (T0 (T0 (T0 X))) (T0 (T0 X)) (T0 X) (mu X) (T1 (T0 (T0 X)) (T0 X) (mu X)) = comp (T0 (T0 (T0 X))) (T0 (T0 X)) (T0 X) (mu X) (mu (T0 X))) → (∀X, Obj X → comp (T0 X) (T0 (T0 X)) (T0 X) (mu X) (eta (T0 X)) = id (T0 X)) → (∀X, Obj X → comp (T0 X) (T0 (T0 X)) (T0 X) (mu X) (T1 X (T0 X) (eta X)) = id (T0 X)) → p) → p)
In Proofgold the corresponding term root is 40a66b... and proposition id is f4ba21...
Proof:
An exact proof term for the current goal is and3E (∀X, Obj X → comp (T0 (T0 (T0 X))) (T0 (T0 X)) (T0 X) (mu X) (T1 (T0 (T0 X)) (T0 X) (mu X)) = comp (T0 (T0 (T0 X))) (T0 (T0 X)) (T0 X) (mu X) (mu (T0 X))) (∀X, Obj X → comp (T0 X) (T0 (T0 X)) (T0 X) (mu X) (eta (T0 X)) = id (T0 X)) (∀X, Obj X → comp (T0 X) (T0 (T0 X)) (T0 X) (mu X) (T1 X (T0 X) (eta X)) = id (T0 X)).
∎
Theorem. (MetaMonad_strict_I)
MetaNatTrans_strict Obj Hom id comp Obj Hom id comp (λX ⇒ X) (λX Y f ⇒ f) T0 T1 eta → MetaNatTrans_strict Obj Hom id comp Obj Hom id comp (λX ⇒ T0 (T0 X)) (λX Y f ⇒ T1 (T0 X) (T0 Y) (T1 X Y f)) T0 T1 mu → MetaMonad → MetaMonad_strict
In Proofgold the corresponding term root is 73fd1c... and proposition id is 163090...
Proof:
An exact proof term for the current goal is and3I (MetaNatTrans_strict Obj Hom id comp Obj Hom id comp (λX ⇒ X) (λX Y f ⇒ f) T0 T1 eta) (MetaNatTrans_strict Obj Hom id comp Obj Hom id comp (λX ⇒ T0 (T0 X)) (λX Y f ⇒ T1 (T0 X) (T0 Y) (T1 X Y f)) T0 T1 mu) MetaMonad.
∎
Theorem. (MetaMonad_strict_E)
MetaMonad_strict → ∀p : prop, (MetaNatTrans_strict Obj Hom id comp Obj Hom id comp (λX ⇒ X) (λX Y f ⇒ f) T0 T1 eta → MetaNatTrans_strict Obj Hom id comp Obj Hom id comp (λX ⇒ T0 (T0 X)) (λX Y f ⇒ T1 (T0 X) (T0 Y) (T1 X Y f)) T0 T1 mu → MetaMonad → p) → p
In Proofgold the corresponding term root is 84fe03... and proposition id is 80e837...
Proof:
An exact proof term for the current goal is and3E (MetaNatTrans_strict Obj Hom id comp Obj Hom id comp (λX ⇒ X) (λX Y f ⇒ f) T0 T1 eta) (MetaNatTrans_strict Obj Hom id comp Obj Hom id comp (λX ⇒ T0 (T0 X)) (λX Y f ⇒ T1 (T0 X) (T0 Y) (T1 X Y f)) T0 T1 mu) MetaMonad.
∎
End of Section MetaMonad
Beginning of Section MetaAdjunction
Variable Obj : set → prop
Variable Hom : set → set → set → prop
Variable id : set → set
Variable comp : set → set → set → set → set → set
Variable Obj' : set → prop
Variable Hom' : set → set → set → prop
Variable id' : set → set
Variable comp' : set → set → set → set → set → set
Variable F0 : set → set
Variable F1 : set → set → set → set
Variable G0 : set → set
Variable G1 : set → set → set → set
Variable eta : set → set
Variable eps : set → set
Definition. We define MetaAdjunction to be (∀X, Obj X → comp' (F0 X) (F0 (G0 (F0 X))) (F0 X) (eps (F0 X)) (F1 X (G0 (F0 X)) (eta X)) = id' (F0 X)) ∧ (∀Y, Obj' Y → comp (G0 Y) (G0 (F0 (G0 Y))) (G0 Y) (G1 (F0 (G0 Y)) Y (eps Y)) (eta (G0 Y)) = id (G0 Y)) of type prop.
In Proofgold the corresponding term root is e8b6c4... and object id is bda632...
Definition. We define MetaAdjunction_strict to be MetaFunctor_strict Obj Hom id comp Obj' Hom' id' comp' F0 F1 ∧ MetaFunctor Obj' Hom' id' comp' Obj Hom id comp G0 G1 ∧ MetaNatTrans Obj Hom id comp Obj Hom id comp (λX ⇒ X) (λX Y f ⇒ f) (λX ⇒ G0 (F0 X)) (λX Y f ⇒ G1 (F0 X) (F0 Y) (F1 X Y f)) eta ∧ MetaNatTrans Obj' Hom' id' comp' Obj' Hom' id' comp' (λY ⇒ F0 (G0 Y)) (λX Y g ⇒ F1 (G0 X) (G0 Y) (G1 X Y g)) (λY ⇒ Y) (λX Y g ⇒ g) eps ∧ MetaAdjunction of type prop.
In Proofgold the corresponding term root is c8ed7d... and object id is 3ab943...
Theorem. (MetaAdjunctionI)
(∀X, Obj X → comp' (F0 X) (F0 (G0 (F0 X))) (F0 X) (eps (F0 X)) (F1 X (G0 (F0 X)) (eta X)) = id' (F0 X)) → (∀Y, Obj' Y → comp (G0 Y) (G0 (F0 (G0 Y))) (G0 Y) (G1 (F0 (G0 Y)) Y (eps Y)) (eta (G0 Y)) = id (G0 Y)) → MetaAdjunction
In Proofgold the corresponding term root is 1ed388... and proposition id is fd4945...
Proof:
An exact proof term for the current goal is andI (∀X, Obj X → comp' (F0 X) (F0 (G0 (F0 X))) (F0 X) (eps (F0 X)) (F1 X (G0 (F0 X)) (eta X)) = id' (F0 X)) (∀Y, Obj' Y → comp (G0 Y) (G0 (F0 (G0 Y))) (G0 Y) (G1 (F0 (G0 Y)) Y (eps Y)) (eta (G0 Y)) = id (G0 Y)).
∎
Theorem. (MetaAdjunctionE)
MetaAdjunction → ∀p : prop, ((∀X, Obj X → comp' (F0 X) (F0 (G0 (F0 X))) (F0 X) (eps (F0 X)) (F1 X (G0 (F0 X)) (eta X)) = id' (F0 X)) → (∀Y, Obj' Y → comp (G0 Y) (G0 (F0 (G0 Y))) (G0 Y) (G1 (F0 (G0 Y)) Y (eps Y)) (eta (G0 Y)) = id (G0 Y)) → p) → p
In Proofgold the corresponding term root is 67ed42... and proposition id is e6292b...
Proof:
An exact proof term for the current goal is (λH ⇒ H).
∎
Theorem. (MetaAdjunction_strict_I)
MetaFunctor_strict Obj Hom id comp Obj' Hom' id' comp' F0 F1 → MetaFunctor Obj' Hom' id' comp' Obj Hom id comp G0 G1 → MetaNatTrans Obj Hom id comp Obj Hom id comp (λX ⇒ X) (λX Y f ⇒ f) (λX ⇒ G0 (F0 X)) (λX Y f ⇒ G1 (F0 X) (F0 Y) (F1 X Y f)) eta → MetaNatTrans Obj' Hom' id' comp' Obj' Hom' id' comp' (λY ⇒ F0 (G0 Y)) (λX Y g ⇒ F1 (G0 X) (G0 Y) (G1 X Y g)) (λY ⇒ Y) (λX Y g ⇒ g) eps → MetaAdjunction → MetaAdjunction_strict
In Proofgold the corresponding term root is 712520... and proposition id is d6aa5e...
Proof:
An exact proof term for the current goal is and5I (MetaFunctor_strict Obj Hom id comp Obj' Hom' id' comp' F0 F1) (MetaFunctor Obj' Hom' id' comp' Obj Hom id comp G0 G1) (MetaNatTrans Obj Hom id comp Obj Hom id comp (λX ⇒ X) (λX Y f ⇒ f) (λX ⇒ G0 (F0 X)) (λX Y f ⇒ G1 (F0 X) (F0 Y) (F1 X Y f)) eta) (MetaNatTrans Obj' Hom' id' comp' Obj' Hom' id' comp' (λY ⇒ F0 (G0 Y)) (λX Y g ⇒ F1 (G0 X) (G0 Y) (G1 X Y g)) (λY ⇒ Y) (λX Y g ⇒ g) eps) MetaAdjunction.
∎
Theorem. (MetaAdjunction_strict_E)
MetaAdjunction_strict → ∀p : prop, (MetaFunctor_strict Obj Hom id comp Obj' Hom' id' comp' F0 F1 → MetaFunctor Obj' Hom' id' comp' Obj Hom id comp G0 G1 → MetaNatTrans Obj Hom id comp Obj Hom id comp (λX ⇒ X) (λX Y f ⇒ f) (λX ⇒ G0 (F0 X)) (λX Y f ⇒ G1 (F0 X) (F0 Y) (F1 X Y f)) eta → MetaNatTrans Obj' Hom' id' comp' Obj' Hom' id' comp' (λY ⇒ F0 (G0 Y)) (λX Y g ⇒ F1 (G0 X) (G0 Y) (G1 X Y g)) (λY ⇒ Y) (λX Y g ⇒ g) eps → MetaAdjunction → p) → p
In Proofgold the corresponding term root is 9a5dd9... and proposition id is 296719...
Proof:
An exact proof term for the current goal is and5E (MetaFunctor_strict Obj Hom id comp Obj' Hom' id' comp' F0 F1) (MetaFunctor Obj' Hom' id' comp' Obj Hom id comp G0 G1) (MetaNatTrans Obj Hom id comp Obj Hom id comp (λX ⇒ X) (λX Y f ⇒ f) (λX ⇒ G0 (F0 X)) (λX Y f ⇒ G1 (F0 X) (F0 Y) (F1 X Y f)) eta) (MetaNatTrans Obj' Hom' id' comp' Obj' Hom' id' comp' (λY ⇒ F0 (G0 Y)) (λX Y g ⇒ F1 (G0 X) (G0 Y) (G1 X Y g)) (λY ⇒ Y) (λX Y g ⇒ g) eps) MetaAdjunction.
∎
Theorem. (MetaAdjunctionMonad)
MetaFunctor Obj Hom id comp Obj' Hom' id' comp' F0 F1 → MetaFunctor Obj' Hom' id' comp' Obj Hom id comp G0 G1 → MetaNatTrans Obj Hom id comp Obj Hom id comp (λX ⇒ X) (λX Y f ⇒ f) (λX ⇒ G0 (F0 X)) (λX Y f ⇒ G1 (F0 X) (F0 Y) (F1 X Y f)) eta → MetaNatTrans Obj' Hom' id' comp' Obj' Hom' id' comp' (λY ⇒ F0 (G0 Y)) (λX Y g ⇒ F1 (G0 X) (G0 Y) (G1 X Y g)) (λY ⇒ Y) (λX Y g ⇒ g) eps → MetaAdjunction → MetaMonad Obj Hom id comp (λX ⇒ G0 (F0 X)) (λX Y f ⇒ G1 (F0 X) (F0 Y) (F1 X Y f)) eta (λX ⇒ G1 (F0 (G0 (F0 X))) (F0 X) (eps (F0 X)))
In Proofgold the corresponding term root is 19ca69... and proposition id is db40a9...
Proof:
Assume HF.
Apply MetaFunctorE Obj Hom id comp Obj' Hom' id' comp' F0 F1 HF to the current goal.
Assume HF1 HF2 HF3 HF4.
Assume HG.
Apply MetaFunctorE Obj' Hom' id' comp' Obj Hom id comp G0 G1 HG to the current goal.
Assume HG1 HG2 HG3 HG4.
Assume Heta.
Apply MetaNatTransE Obj Hom id comp Obj Hom id comp (λX ⇒ X) (λX Y f ⇒ f) (λX ⇒ G0 (F0 X)) (λX Y f ⇒ G1 (F0 X) (F0 Y) (F1 X Y f)) eta Heta to the current goal.
Assume Heta1 Heta2.
Assume Heps.
Apply MetaNatTransE Obj' Hom' id' comp' Obj' Hom' id' comp' (λY ⇒ F0 (G0 Y)) (λX Y g ⇒ F1 (G0 X) (G0 Y) (G1 X Y g)) (λY ⇒ Y) (λX Y g ⇒ g) eps Heps to the current goal.
Assume Heps1: ∀X, Obj' X → Hom' (F0 (G0 X)) X (eps X).
Assume Heps2: ∀X Y f, Obj' X → Obj' Y → Hom' X Y f → comp' (F0 (G0 X)) X Y f (eps X) = comp' (F0 (G0 X)) (F0 (G0 Y)) Y (eps Y) (F1 (G0 X) (G0 Y) (G1 X Y f)).
Assume HA.
Apply MetaAdjunctionE HA to the current goal.
Assume HA1 HA2.
Set T0 to be the term λX ⇒ G0 (F0 X) of type set → set.
Set T1 to be the term λX Y f ⇒ G1 (F0 X) (F0 Y) (F1 X Y f) of type set → set → set → set.
Set mu to be the term λX ⇒ G1 (F0 (G0 (F0 X))) (F0 X) (eps (F0 X)) of type set → set.
We prove the intermediate claim LT: MetaFunctor Obj Hom id comp Obj Hom id comp T0 T1.
An exact proof term for the current goal is MetaCat_CompFunctors Obj Hom id comp Obj' Hom' id' comp' Obj Hom id comp F0 F1 G0 G1 HF HG.
Apply MetaFunctorE Obj Hom id comp Obj Hom id comp T0 T1 LT to the current goal.
Assume HT01 HT02 HT03 HT04.
Apply MetaMonadI to the current goal.
We will prove ∀X, Obj X → comp (T0 (T0 (T0 X))) (T0 (T0 X)) (T0 X) (mu X) (T1 (T0 (T0 X)) (T0 X) (mu X)) = comp (T0 (T0 (T0 X))) (T0 (T0 X)) (T0 X) (mu X) (mu (T0 X)).
Let X be given.
Assume HX.
We prove the intermediate claim LFX: Obj' (F0 X).
Apply HF1 to the current goal.
An exact proof term for the current goal is HX.
We prove the intermediate claim LTX: Obj (T0 X).
An exact proof term for the current goal is HT01 X HX.
We prove the intermediate claim LFTX: Obj' (F0 (T0 X)).
Apply HF1 to the current goal.
An exact proof term for the current goal is LTX.
We prove the intermediate claim LTTX: Obj (T0 (T0 X)).
An exact proof term for the current goal is HT01 (T0 X) LTX.
We prove the intermediate claim LFTTX: Obj' (F0 (T0 (T0 X))).
Apply HF1 to the current goal.
An exact proof term for the current goal is LTTX.
We prove the intermediate claim LepsFX: Hom' (F0 (T0 X)) (F0 X) (eps (F0 X)).
An exact proof term for the current goal is Heps1 (F0 X) LFX.
We prove the intermediate claim LmuX: Hom (T0 (T0 X)) (T0 X) (mu X).
An exact proof term for the current goal is HG2 (F0 (T0 X)) (F0 X) (eps (F0 X)) LFTX LFX LepsFX.
We prove the intermediate claim LFmuX: Hom' (F0 (T0 (T0 X))) (F0 (T0 X)) (F1 (T0 (T0 X)) (T0 X) (mu X)).
An exact proof term for the current goal is HF2 (T0 (T0 X)) (T0 X) (mu X) LTTX LTX LmuX.
We will prove comp (T0 (T0 (T0 X))) (T0 (T0 X)) (T0 X) (mu X) (T1 (T0 (T0 X)) (T0 X) (mu X)) = comp (T0 (T0 (T0 X))) (T0 (T0 X)) (T0 X) (mu X) (mu (T0 X)).
Use transitivity with G1 (F0 (T0 (T0 X))) (F0 X) (comp' (F0 (T0 (T0 X))) (F0 (T0 X)) (F0 X) (eps (F0 X)) (F1 (T0 (T0 X)) (T0 X) (mu X))), and G1 (F0 (T0 (T0 X))) (F0 X) (comp' (F0 (T0 (T0 X))) (F0 (T0 X)) (F0 X) (eps (F0 X)) (eps (F0 (T0 X)))).
We will prove comp (T0 (T0 (T0 X))) (T0 (T0 X)) (T0 X) (mu X) (T1 (T0 (T0 X)) (T0 X) (mu X)) = G1 (F0 (T0 (T0 X))) (F0 X) (comp' (F0 (T0 (T0 X))) (F0 (T0 X)) (F0 X) (eps (F0 X)) (F1 (T0 (T0 X)) (T0 X) (mu X))).
Use symmetry.
An exact proof term for the current goal is HG4 (F0 (T0 (T0 X))) (F0 (T0 X)) (F0 X) (F1 (T0 (T0 X)) (T0 X) (mu X)) (eps (F0 X)) LFTTX LFTX LFX LFmuX LepsFX.
We will prove G1 (F0 (T0 (T0 X))) (F0 X) (comp' (F0 (T0 (T0 X))) (F0 (T0 X)) (F0 X) (eps (F0 X)) (F1 (T0 (T0 X)) (T0 X) (mu X))) = G1 (F0 (T0 (T0 X))) (F0 X) (comp' (F0 (T0 (T0 X))) (F0 (T0 X)) (F0 X) (eps (F0 X)) (eps (F0 (T0 X)))).
Use f_equal.
Use symmetry.
We will prove comp' (F0 (T0 (T0 X))) (F0 (T0 X)) (F0 X) (eps (F0 X)) (eps (F0 (T0 X))) = comp' (F0 (T0 (T0 X))) (F0 (T0 X)) (F0 X) (eps (F0 X)) (F1 (T0 (T0 X)) (T0 X) (mu X)).
An exact proof term for the current goal is Heps2 (F0 (T0 X)) (F0 X) (eps (F0 X)) LFTX LFX LepsFX.
We will prove G1 (F0 (T0 (T0 X))) (F0 X) (comp' (F0 (T0 (T0 X))) (F0 (T0 X)) (F0 X) (eps (F0 X)) (eps (F0 (T0 X)))) = comp (T0 (T0 (T0 X))) (T0 (T0 X)) (T0 X) (mu X) (mu (T0 X)).
An exact proof term for the current goal is HG4 (F0 (T0 (T0 X))) (F0 (T0 X)) (F0 X) (eps (F0 (T0 X))) (eps (F0 X)) LFTTX LFTX LFX (Heps1 (F0 (T0 X)) LFTX) LepsFX.
We will prove ∀X, Obj X → comp (T0 X) (T0 (T0 X)) (T0 X) (mu X) (eta (T0 X)) = id (T0 X).
Let X be given.
Assume HX.
We will prove comp (G0 (F0 X)) (G0 (F0 (G0 (F0 X)))) (G0 (F0 X)) (G1 (F0 (G0 (F0 X))) (F0 X) (eps (F0 X))) (eta (G0 (F0 X))) = id (G0 (F0 X)).
An exact proof term for the current goal is HA2 (F0 X) (HF1 X HX).
We will prove ∀X, Obj X → comp (T0 X) (T0 (T0 X)) (T0 X) (mu X) (T1 X (T0 X) (eta X)) = id (T0 X).
Let X be given.
Assume HX.
We will prove comp (G0 (F0 X)) (G0 (F0 (T0 X))) (G0 (F0 X)) (G1 (F0 (T0 X)) (F0 X) (eps (F0 X))) (G1 (F0 X) (F0 (T0 X)) (F1 X (T0 X) (eta X))) = id (T0 X).
We prove the intermediate claim LFX: Obj' (F0 X).
Apply HF1 to the current goal.
An exact proof term for the current goal is HX.
We prove the intermediate claim LTX: Obj (T0 X).
An exact proof term for the current goal is HT01 X HX.
We prove the intermediate claim LFTX: Obj' (F0 (T0 X)).
Apply HF1 to the current goal.
An exact proof term for the current goal is LTX.
Use transitivity with G1 (F0 X) (F0 X) (comp' (F0 X) (F0 (T0 X)) (F0 X) (eps (F0 X)) (F1 X (T0 X) (eta X))), and G1 (F0 X) (F0 X) (id' (F0 X)).
We will prove comp (G0 (F0 X)) (G0 (F0 (T0 X))) (G0 (F0 X)) (G1 (F0 (T0 X)) (F0 X) (eps (F0 X))) (G1 (F0 X) (F0 (T0 X)) (F1 X (T0 X) (eta X))) = G1 (F0 X) (F0 X) (comp' (F0 X) (F0 (T0 X)) (F0 X) (eps (F0 X)) (F1 X (T0 X) (eta X))).
Use symmetry.
Apply HG4 (F0 X) (F0 (T0 X)) (F0 X) (F1 X (T0 X) (eta X)) (eps (F0 X)) to the current goal.
An exact proof term for the current goal is LFX.
An exact proof term for the current goal is LFTX.
An exact proof term for the current goal is LFX.
We will prove Hom' (F0 X) (F0 (T0 X)) (F1 X (T0 X) (eta X)).
An exact proof term for the current goal is HF2 X (T0 X) (eta X) HX LTX (Heta1 X HX).
We will prove Hom' (F0 (G0 (F0 X))) (F0 X) (eps (F0 X)).
An exact proof term for the current goal is Heps1 (F0 X) (HF1 X HX).
We will prove G1 (F0 X) (F0 X) (comp' (F0 X) (F0 (T0 X)) (F0 X) (eps (F0 X)) (F1 X (T0 X) (eta X))) = G1 (F0 X) (F0 X) (id' (F0 X)).
Use f_equal.
An exact proof term for the current goal is HA1 X HX.
We will prove G1 (F0 X) (F0 X) (id' (F0 X)) = id (T0 X).
An exact proof term for the current goal is HG3 (F0 X) LFX.
∎
Theorem. (MetaAdjunctionMonad_strict)
MetaAdjunction_strict → MetaMonad_strict Obj Hom id comp (λX ⇒ G0 (F0 X)) (λX Y f ⇒ G1 (F0 X) (F0 Y) (F1 X Y f)) eta (λX ⇒ G1 (F0 (G0 (F0 X))) (F0 X) (eps (F0 X)))
In Proofgold the corresponding term root is 8f71ab... and proposition id is 07be67...
Proof:
Assume HA.
Apply MetaAdjunction_strict_E HA to the current goal.
Assume HA1 HG Heta Heps HA5.
Apply MetaFunctor_strict_E Obj Hom id comp Obj' Hom' id' comp' F0 F1 HA1 to the current goal.
Assume HC HC' HF.
Set T0 to be the term λX ⇒ G0 (F0 X) of type set → set.
Set T1 to be the term λX Y f ⇒ G1 (F0 X) (F0 Y) (F1 X Y f) of type set → set → set → set.
Set mu to be the term λX ⇒ G1 (F0 (G0 (F0 X))) (F0 X) (eps (F0 X)) of type set → set.
We prove the intermediate claim LT: MetaFunctor Obj Hom id comp Obj Hom id comp T0 T1.
An exact proof term for the current goal is MetaCat_CompFunctors Obj Hom id comp Obj' Hom' id' comp' Obj Hom id comp F0 F1 G0 G1 HF HG.
We prove the intermediate claim LTT: MetaFunctor Obj Hom id comp Obj Hom id comp (λX ⇒ T0 (T0 X)) (λX Y f ⇒ T1 (T0 X) (T0 Y) (T1 X Y f)).
An exact proof term for the current goal is MetaCat_CompFunctors Obj Hom id comp Obj Hom id comp Obj Hom id comp T0 T1 T0 T1 LT LT.
We prove the intermediate claim LFT: MetaFunctor Obj Hom id comp Obj' Hom' id' comp' (λX ⇒ F0 (T0 X)) (λX Y f ⇒ F1 (T0 X) (T0 Y) (T1 X Y f)).
An exact proof term for the current goal is MetaCat_CompFunctors Obj Hom id comp Obj Hom id comp Obj' Hom' id' comp' T0 T1 F0 F1 LT HF.
We prove the intermediate claim LepsF: MetaNatTrans Obj Hom id comp Obj' Hom' id' comp' (λX ⇒ F0 (G0 (F0 X))) (λX Y f ⇒ F1 (G0 (F0 X)) (G0 (F0 Y)) (G1 (F0 X) (F0 Y) (F1 X Y f))) F0 F1 (λX ⇒ eps (F0 X)).
An exact proof term for the current goal is MetaCat_CompNatTransFunctor Obj Hom id comp Obj' Hom' id' comp' Obj' Hom' id' comp' (λX ⇒ F0 (G0 X)) (λX Y f ⇒ F1 (G0 X) (G0 Y) (G1 X Y f)) (λX ⇒ X) (λX Y f ⇒ f) F0 F1 eps Heps HF.
We prove the intermediate claim Lmu: MetaNatTrans Obj Hom id comp Obj Hom id comp (λY ⇒ T0 (T0 Y)) (λX Y g ⇒ T1 (T0 X) (T0 Y) (T1 X Y g)) T0 T1 mu.
An exact proof term for the current goal is MetaCat_CompFunctorNatTrans Obj Hom id comp Obj' Hom' id' comp' Obj Hom id comp (λX ⇒ F0 (G0 (F0 X))) (λX Y f ⇒ F1 (G0 (F0 X)) (G0 (F0 Y)) (G1 (F0 X) (F0 Y) (F1 X Y f))) F0 F1 G0 G1 (λX ⇒ eps (F0 X)) LFT HF LepsF HG.
Apply MetaMonad_strict_I to the current goal.
Apply MetaNatTrans_strict_I to the current goal.
An exact proof term for the current goal is HC.
An exact proof term for the current goal is HC.
An exact proof term for the current goal is MetaCat_IdFunctor Obj Hom id comp.
An exact proof term for the current goal is LT.
We will prove MetaNatTrans Obj Hom id comp Obj Hom id comp (λX ⇒ X) (λX Y f ⇒ f) (λX ⇒ G0 (F0 X)) (λX Y f ⇒ G1 (F0 X) (F0 Y) (F1 X Y f)) eta.
An exact proof term for the current goal is Heta.
Apply MetaNatTrans_strict_I to the current goal.
An exact proof term for the current goal is HC.
An exact proof term for the current goal is HC.
An exact proof term for the current goal is LTT.
An exact proof term for the current goal is LT.
We will prove MetaNatTrans Obj Hom id comp Obj Hom id comp (λY ⇒ T0 (T0 Y)) (λX Y g ⇒ T1 (T0 X) (T0 Y) (T1 X Y g)) T0 T1 mu.
An exact proof term for the current goal is Lmu.
An exact proof term for the current goal is MetaAdjunctionMonad HF HG Heta Heps HA5.
∎
End of Section MetaAdjunction
Beginning of Section MetaCatConcrete
Variable Obj : set → prop
Variable U : set → set
Variable Hom : set → set → set → prop
Theorem. (MetaCatConcrete)
(∀X Y f, Obj X → Obj Y → Hom X Y f → f ∈ U YU X) → (∀X, Obj X → Hom X X (lam_id (U X))) → (∀X Y Z f g, Obj X → Obj Y → Obj Z → Hom X Y f → Hom Y Z g → Hom X Z (lam_comp (U X) g f)) → MetaCat Obj Hom (λX ⇒ lam_id (U X)) (λX Y Z g f ⇒ lam_comp (U X) g f)
In Proofgold the corresponding term root is 1db157... and proposition id is 081b4e...
Proof:
Assume H1 H2 H3.
Apply MetaCat_I to the current goal.
An exact proof term for the current goal is H2.
An exact proof term for the current goal is H3.
Let X, Y and f be given.
Assume HX HY Hf.
We will prove lam_comp (U X) f (lam_id (U X)) = f.
An exact proof term for the current goal is lam_comp_id_R (U X) (U Y) f (H1 X Y f HX HY Hf).
Let X, Y and f be given.
Assume HX HY Hf.
We will prove lam_comp (U X) (lam_id (U Y)) f = f.
An exact proof term for the current goal is lam_comp_id_L (U X) (U Y) f (H1 X Y f HX HY Hf).
Let X, Y, Z, W, f, g and h be given.
Assume HX HY HZ HW Hf Hg Hh.
Use symmetry.
An exact proof term for the current goal is lam_comp_assoc (U X) (U Y) f (H1 X Y f HX HY Hf) g h.
∎
Theorem. (MetaCatConcreteForgetful)
(∀X Y f, Obj X → Obj Y → Hom X Y f → f ∈ U YU X) → MetaFunctor Obj Hom (λX ⇒ lam_id (U X)) (λX Y Z f g ⇒ (lam_comp (U X) f g)) (λ_ ⇒ True) SetHom (λX ⇒ lam_id X) (λX Y Z f g ⇒ (lam_comp X f g)) U (λX Y f ⇒ f)
In Proofgold the corresponding term root is cb7abf... and proposition id is b606c5...
Proof:
Assume H1.
Set F0 to be the term U.
Set F1 to be the term λX Y f : set ⇒ f.
Apply MetaFunctorI to the current goal.
Let X be given.
Assume _.
An exact proof term for the current goal is TrueI.
An exact proof term for the current goal is H1.
Let X be given.
Assume _.
Use reflexivity.
Let X, Y, Z, f and g be given.
Assume _ _ _ _ _.
Use reflexivity.
∎
End of Section MetaCatConcrete
Theorem. (MetaCatSet)
MetaCat (λ_ ⇒ True) SetHom (λX ⇒ lam_id X) (λX Y Z g f ⇒ lam_comp X g f)
In Proofgold the corresponding term root is dcf573... and proposition id is e41257...
Proof:
Apply MetaCatConcrete (λ_ ⇒ True) (λX ⇒ X) SetHom to the current goal.
Let X, Y and f be given.
Assume HX HY Hf.
An exact proof term for the current goal is Hf.
Let X be given.
Assume HX.
An exact proof term for the current goal is lam_id_exp_In X.
Let X, Y, Z, f and g be given.
Assume HX HY HZ Hf Hg.
An exact proof term for the current goal is lam_comp_exp_In X Y Z f Hf g Hg.
∎
Theorem. (MetaCatHFSet)
MetaCat (λX ⇒ X ∈ UnivOf Empty) SetHom (λX ⇒ lam_id X) (λX Y Z f g ⇒ (lam_comp X f g))
In Proofgold the corresponding term root is 3e43f7... and proposition id is 2fb6ad...
Proof:
Apply MetaCatConcrete (λX ⇒ X ∈ UnivOf Empty) (λX ⇒ X) SetHom to the current goal.
Let X, Y and f be given.
Assume HX HY Hf.
An exact proof term for the current goal is Hf.
Let X be given.
Assume HX.
An exact proof term for the current goal is lam_id_exp_In X.
Let X, Y, Z, f and g be given.
Assume HX HY HZ Hf Hg.
An exact proof term for the current goal is lam_comp_exp_In X Y Z f Hf g Hg.
∎
Theorem. (MetaCatSmallSet)
MetaCat (λX ⇒ X ∈ UnivOf (UnivOf Empty)) SetHom (λX ⇒ lam_id X) (λX Y Z f g ⇒ (lam_comp X f g))
In Proofgold the corresponding term root is ea08c8... and proposition id is 689787...
Proof:
Apply MetaCatConcrete (λX ⇒ X ∈ UnivOf (UnivOf Empty)) (λX ⇒ X) SetHom to the current goal.
Let X, Y and f be given.
Assume HX HY Hf.
An exact proof term for the current goal is Hf.
Let X be given.
Assume HX.
An exact proof term for the current goal is lam_id_exp_In X.
Let X, Y, Z, f and g be given.
Assume HX HY HZ Hf Hg.
An exact proof term for the current goal is lam_comp_exp_In X Y Z f Hf g Hg.
∎
Theorem. (MetaCatConcreteForgetful_strict)
∀Obj : set → prop, ∀U : set → set, ∀Hom : set → set → set → prop, (∀X Y f, Obj X → Obj Y → Hom X Y f → f ∈ U YU X) → (∀X, Obj X → Hom X X (lam_id (U X))) → (∀X Y Z f g, Obj X → Obj Y → Obj Z → Hom X Y f → Hom Y Z g → Hom X Z (lam_comp (U X) g f)) → MetaFunctor_strict Obj Hom (λX ⇒ lam_id (U X)) (λX Y Z f g ⇒ (lam_comp (U X) f g)) (λ_ ⇒ True) SetHom (λX ⇒ lam_id X) (λX Y Z f g ⇒ (lam_comp X f g)) U (λX Y f ⇒ f)
In Proofgold the corresponding term root is b158c7... and proposition id is ec1a3b...
Proof:
Let Obj, U and Hom be given.
Assume H1 H2 H3.
Apply MetaFunctor_strict_I to the current goal.
An exact proof term for the current goal is MetaCatConcrete Obj U Hom H1 H2 H3.
An exact proof term for the current goal is MetaCatSet.
An exact proof term for the current goal is MetaCatConcreteForgetful Obj U Hom H1.
∎
Definition. We define Hom_struct_e to be λX Y f ⇒ unpack_e_o X (λX' eX ⇒ unpack_e_o Y (λY' eY ⇒ f ∈ Y'X' ∧ f eX = eY)) of type set → set → set → prop.
In Proofgold the corresponding term root is d6f688... and object id is 0f5046...
Definition. We define Hom_struct_u to be λX Y f ⇒ unpack_u_o X (λX' uX ⇒ unpack_u_o Y (λY' uY ⇒ f ∈ Y'X' ∧ ∀x ∈ X', f (uX x) = uY (f x))) of type set → set → set → prop.
In Proofgold the corresponding term root is 957911... and object id is 44b620...
Definition. We define Hom_struct_b to be λX Y f ⇒ unpack_b_o X (λX' opX ⇒ unpack_b_o Y (λY' opY ⇒ f ∈ Y'X' ∧ ∀x y ∈ X', f (opX x y) = opY (f x) (f y))) of type set → set → set → prop.
In Proofgold the corresponding term root is e81578... and object id is 9b88c8...
Definition. We define Hom_struct_p to be λX Y f ⇒ unpack_p_o X (λX' pX ⇒ unpack_p_o Y (λY' pY ⇒ f ∈ Y'X' ∧ ∀x ∈ X', pX x → pY (f x))) of type set → set → set → prop.
In Proofgold the corresponding term root is 5d0201... and object id is 8f474b...
Definition. We define Hom_struct_r to be λX Y f ⇒ unpack_r_o X (λX' rX ⇒ unpack_r_o Y (λY' rY ⇒ f ∈ Y'X' ∧ ∀x y ∈ X', rX x y → rY (f x) (f y))) of type set → set → set → prop.
In Proofgold the corresponding term root is acf0f0... and object id is 3f6068...
Definition. We define Hom_struct_c to be λX Y f ⇒ unpack_c_o X (λX' CX ⇒ unpack_c_o Y (λY' CY ⇒ f ∈ Y'X' ∧ ∀U : set → prop, (∀y, U y → y ∈ Y') → CY U → CX (λx ⇒ x ∈ X' ∧ U (f x)))) of type set → set → set → prop.
In Proofgold the corresponding term root is 9dfc2a... and object id is 4ebd3f...
Definition. We define Hom_struct_b_b_e to be λX Y f ⇒ unpack_b_b_e_o X (λX' opX op2X eX ⇒ unpack_b_b_e_o Y (λY' opY op2Y eY ⇒ f ∈ Y'X' ∧ (∀x y ∈ X', f (opX x y) = opY (f x) (f y)) ∧ (∀x y ∈ X', f (op2X x y) = op2Y (f x) (f y)) ∧ f eX = eY)) of type set → set → set → prop.
In Proofgold the corresponding term root is 070862... and object id is 26f87b...
Definition. We define Hom_struct_b_b_e_e to be λX Y f ⇒ unpack_b_b_e_e_o X (λX' opX op2X eX e2X ⇒ unpack_b_b_e_e_o Y (λY' opY op2Y eY e2Y ⇒ f ∈ Y'X' ∧ (∀x y ∈ X', f (opX x y) = opY (f x) (f y)) ∧ (∀x y ∈ X', f (op2X x y) = op2Y (f x) (f y)) ∧ f eX = eY ∧ f e2X = e2Y)) of type set → set → set → prop.
In Proofgold the corresponding term root is a11d43... and object id is a8f5c0...
Definition. We define Hom_struct_b_b_r_e_e to be λX Y f ⇒ unpack_b_b_r_e_e_o X (λX' opX op2X rX eX e2X ⇒ unpack_b_b_r_e_e_o Y (λY' opY op2Y rY eY e2Y ⇒ f ∈ Y'X' ∧ (∀x y ∈ X', f (opX x y) = opY (f x) (f y)) ∧ (∀x y ∈ X', f (op2X x y) = op2Y (f x) (f y)) ∧ (∀x y ∈ X', rX x y → rY (f x) (f y)) ∧ f eX = eY ∧ f e2X = e2Y)) of type set → set → set → prop.
In Proofgold the corresponding term root is d75651... and object id is a6f30d...
Theorem. (Hom_struct_e_pack)
∀X Y eX eY f, (Hom_struct_e (pack_e X eX) (pack_e Y eY) f) = (f ∈ YX ∧ f eX = eY)
In Proofgold the corresponding term root is 266cf3... and proposition id is f65a35...
Proof:
Let X, Y, eX, eY and f be given.
Set Phi to be the term λX eX Y eY ⇒ f ∈ YX ∧ f eX = eY of type set → set → set → set → prop.
Set PhiX to be the term λX eX ⇒ unpack_e_o (pack_e Y eY) (λY' eY ⇒ Phi X eX Y' eY) of type set → set → prop.
We will prove unpack_e_o (pack_e X eX) PhiX = Phi X eX Y eY.
Use transitivity with and PhiX X eX.
An exact proof term for the current goal is unpack_e_o_eq PhiX X eX.
We will prove unpack_e_o (pack_e Y eY) (Phi X eX) = Phi X eX Y eY.
An exact proof term for the current goal is unpack_e_o_eq (Phi X eX) Y eY.
∎
Theorem. (Hom_struct_u_pack)
∀X Y, ∀opX opY : set → set, ∀f, (Hom_struct_u (pack_u X opX) (pack_u Y opY) f) = (f ∈ YX ∧ (∀x ∈ X, f (opX x) = opY (f x)))
In Proofgold the corresponding term root is c0506b... and proposition id is 66c4c4...
Proof:
Let X, Y, opX, opY and f be given.
Set Phi to be the term λX opX Y opY ⇒ f ∈ YX ∧ (∀x ∈ X, f (opX x) = opY (f x)) of type set → (set → set) → set → (set → set) → prop.
Set PhiX to be the term λX opX ⇒ unpack_u_o (pack_u Y opY) (λY' opY ⇒ Phi X opX Y' opY) of type set → (set → set) → prop.
We will prove unpack_u_o (pack_u X opX) PhiX = Phi X opX Y opY.
We prove the intermediate claim L1: ∀opX : set → set, ∀opY' : set → set, (∀x ∈ Y, opY x = opY' x) → Phi X opX Y opY' = Phi X opX Y opY.
Let opX be given.
Let opY' be given.
Assume HopY'.
Apply prop_ext_2 to the current goal.
Assume H.
Apply H to the current goal.
Assume H1 H2.
We prove the intermediate claim Lf: ∀x ∈ X, f x ∈ Y.
Let x be given.
An exact proof term for the current goal is ap_Pi X (λ_ ⇒ Y) f x H1.
We will prove f ∈ YX ∧ (∀x ∈ X, f (opX x) = opY (f x)).
Apply andI to the current goal.
An exact proof term for the current goal is H1.
Let x be given.
Assume Hx.
rewrite the current goal using HopY' (f x) (Lf x Hx) (from left to right).
An exact proof term for the current goal is H2 x Hx.
Assume H.
Apply H to the current goal.
Assume H1 H2.
We prove the intermediate claim Lf: ∀x ∈ X, f x ∈ Y.
Let x be given.
An exact proof term for the current goal is ap_Pi X (λ_ ⇒ Y) f x H1.
We will prove f ∈ YX ∧ (∀x ∈ X, f (opX x) = opY' (f x)).
Apply andI to the current goal.
An exact proof term for the current goal is H1.
Let x be given.
Assume Hx.
rewrite the current goal using HopY' (f x) (Lf x Hx) (from right to left).
An exact proof term for the current goal is H2 x Hx.
We prove the intermediate claim L2: ∀opX' : set → set, (∀x ∈ X, opX x = opX' x) → PhiX X opX' = PhiX X opX.
Let opX' be given.
Assume HopX'.
We will prove PhiX X opX' = PhiX X opX.
We will prove unpack_u_o (pack_u Y opY) (λY' opY ⇒ Phi X opX' Y' opY) = unpack_u_o (pack_u Y opY) (λY' opY ⇒ Phi X opX Y' opY).
Use transitivity with Phi X opX' Y opY, and Phi X opX Y opY.
We will prove unpack_u_o (pack_u Y opY) (Phi X opX') = Phi X opX' Y opY.
An exact proof term for the current goal is unpack_u_o_eq (Phi X opX') Y opY (L1 opX').
We will prove Phi X opX' Y opY = Phi X opX Y opY.
Apply prop_ext_2 to the current goal.
Assume H.
Apply H to the current goal.
Assume H1 H2.
We will prove f ∈ YX ∧ (∀x ∈ X, f (opX x) = opY (f x)).
Apply andI to the current goal.
An exact proof term for the current goal is H1.
Let x be given.
Assume Hx.
rewrite the current goal using HopX' x Hx (from left to right).
An exact proof term for the current goal is H2 x Hx.
Assume H.
Apply H to the current goal.
Assume H1 H2.
We will prove f ∈ YX ∧ (∀x ∈ X, f (opX' x) = opY (f x)).
Apply andI to the current goal.
An exact proof term for the current goal is H1.
Let x be given.
Assume Hx.
rewrite the current goal using HopX' x Hx (from right to left).
An exact proof term for the current goal is H2 x Hx.
Use symmetry.
An exact proof term for the current goal is unpack_u_o_eq (Phi X opX) Y opY (L1 opX).
Use transitivity with and PhiX X opX.
An exact proof term for the current goal is unpack_u_o_eq PhiX X opX L2.
We will prove unpack_u_o (pack_u Y opY) (Phi X opX) = Phi X opX Y opY.
An exact proof term for the current goal is unpack_u_o_eq (Phi X opX) Y opY (L1 opX).
∎
Theorem. (Hom_struct_b_pack)
∀X Y, ∀opX opY : set → set → set, ∀f, (Hom_struct_b (pack_b X opX) (pack_b Y opY) f) = (f ∈ YX ∧ (∀x y ∈ X, f (opX x y) = opY (f x) (f y)))
In Proofgold the corresponding term root is 7ee20a... and proposition id is 2cd8d7...
Proof:
Let X, Y, opX, opY and f be given.
Set Phi to be the term λX opX Y opY ⇒ f ∈ YX ∧ (∀x y ∈ X, f (opX x y) = opY (f x) (f y)) of type set → (set → set → set) → set → (set → set → set) → prop.
Set PhiX to be the term λX opX ⇒ unpack_b_o (pack_b Y opY) (λY' opY ⇒ Phi X opX Y' opY) of type set → (set → set → set) → prop.
We will prove unpack_b_o (pack_b X opX) PhiX = Phi X opX Y opY.
We prove the intermediate claim L1: ∀opX : set → set → set, ∀opY' : set → set → set, (∀x y ∈ Y, opY x y = opY' x y) → Phi X opX Y opY' = Phi X opX Y opY.
Let opX be given.
Let opY' be given.
Assume HopY'.
Apply prop_ext_2 to the current goal.
Assume H.
Apply H to the current goal.
Assume H1 H2.
We prove the intermediate claim Lf: ∀x ∈ X, f x ∈ Y.
Let x be given.
An exact proof term for the current goal is ap_Pi X (λ_ ⇒ Y) f x H1.
We will prove f ∈ YX ∧ (∀x y ∈ X, f (opX x y) = opY (f x) (f y)).
Apply andI to the current goal.
An exact proof term for the current goal is H1.
Let x be given.
Assume Hx.
Let y be given.
Assume Hy.
rewrite the current goal using HopY' (f x) (Lf x Hx) (f y) (Lf y Hy) (from left to right).
An exact proof term for the current goal is H2 x Hx y Hy.
Assume H.
Apply H to the current goal.
Assume H1 H2.
We prove the intermediate claim Lf: ∀x ∈ X, f x ∈ Y.
Let x be given.
An exact proof term for the current goal is ap_Pi X (λ_ ⇒ Y) f x H1.
We will prove f ∈ YX ∧ (∀x y ∈ X, f (opX x y) = opY' (f x) (f y)).
Apply andI to the current goal.
An exact proof term for the current goal is H1.
Let x be given.
Assume Hx.
Let y be given.
Assume Hy.
rewrite the current goal using HopY' (f x) (Lf x Hx) (f y) (Lf y Hy) (from right to left).
An exact proof term for the current goal is H2 x Hx y Hy.
We prove the intermediate claim L2: ∀opX' : set → set → set, (∀x y ∈ X, opX x y = opX' x y) → PhiX X opX' = PhiX X opX.
Let opX' be given.
Assume HopX'.
We will prove PhiX X opX' = PhiX X opX.
We will prove unpack_b_o (pack_b Y opY) (λY' opY ⇒ Phi X opX' Y' opY) = unpack_b_o (pack_b Y opY) (λY' opY ⇒ Phi X opX Y' opY).
Use transitivity with Phi X opX' Y opY, and Phi X opX Y opY.
We will prove unpack_b_o (pack_b Y opY) (Phi X opX') = Phi X opX' Y opY.
An exact proof term for the current goal is unpack_b_o_eq (Phi X opX') Y opY (L1 opX').
We will prove Phi X opX' Y opY = Phi X opX Y opY.
Apply prop_ext_2 to the current goal.
Assume H.
Apply H to the current goal.
Assume H1 H2.
We will prove f ∈ YX ∧ (∀x y ∈ X, f (opX x y) = opY (f x) (f y)).
Apply andI to the current goal.
An exact proof term for the current goal is H1.
Let x be given.
Assume Hx.
Let y be given.
Assume Hy.
rewrite the current goal using HopX' x Hx y Hy (from left to right).
An exact proof term for the current goal is H2 x Hx y Hy.
Assume H.
Apply H to the current goal.
Assume H1 H2.
We will prove f ∈ YX ∧ (∀x y ∈ X, f (opX' x y) = opY (f x) (f y)).
Apply andI to the current goal.
An exact proof term for the current goal is H1.
Let x be given.
Assume Hx.
Let y be given.
Assume Hy.
rewrite the current goal using HopX' x Hx y Hy (from right to left).
An exact proof term for the current goal is H2 x Hx y Hy.
Use symmetry.
An exact proof term for the current goal is unpack_b_o_eq (Phi X opX) Y opY (L1 opX).
Use transitivity with and PhiX X opX.
An exact proof term for the current goal is unpack_b_o_eq PhiX X opX L2.
We will prove unpack_b_o (pack_b Y opY) (Phi X opX) = Phi X opX Y opY.
An exact proof term for the current goal is unpack_b_o_eq (Phi X opX) Y opY (L1 opX).
∎
Theorem. (Hom_struct_p_pack)
∀X Y, ∀pX pY : set → prop, ∀f, (Hom_struct_p (pack_p X pX) (pack_p Y pY) f) = (f ∈ YX ∧ (∀x ∈ X, pX x → pY (f x)))
In Proofgold the corresponding term root is 63c01b... and proposition id is 55fb5e...
Proof:
Let X, Y, pX, pY and f be given.
Set Phi to be the term λX pX Y pY ⇒ f ∈ YX ∧ (∀x ∈ X, pX x → pY (f x)) of type set → (set → prop) → set → (set → prop) → prop.
Set PhiX to be the term λX pX ⇒ unpack_p_o (pack_p Y pY) (λY' pY ⇒ Phi X pX Y' pY) of type set → (set → prop) → prop.
We will prove unpack_p_o (pack_p X pX) PhiX = Phi X pX Y pY.
We prove the intermediate claim L1: ∀pX : set → prop, ∀pY' : set → prop, (∀x ∈ Y, pY x ↔ pY' x) → Phi X pX Y pY' = Phi X pX Y pY.
Let pX be given.
Let pY' be given.
Assume HpY'.
Apply prop_ext_2 to the current goal.
Assume H.
Apply H to the current goal.
Assume H1 H2.
We prove the intermediate claim Lf: ∀x ∈ X, f x ∈ Y.
Let x be given.
An exact proof term for the current goal is ap_Pi X (λ_ ⇒ Y) f x H1.
We will prove f ∈ YX ∧ (∀x ∈ X, pX x → pY (f x)).
Apply andI to the current goal.
An exact proof term for the current goal is H1.
Let x be given.
Assume Hx H7.
Apply iffER (pY (f x)) (pY' (f x)) (HpY' (f x) (Lf x Hx)) to the current goal.
An exact proof term for the current goal is H2 x Hx H7.
Assume H.
Apply H to the current goal.
Assume H1 H2.
We prove the intermediate claim Lf: ∀x ∈ X, f x ∈ Y.
Let x be given.
An exact proof term for the current goal is ap_Pi X (λ_ ⇒ Y) f x H1.
We will prove f ∈ YX ∧ (∀x ∈ X, pX x → pY' (f x)).
Apply andI to the current goal.
An exact proof term for the current goal is H1.
Let x be given.
Assume Hx H7.
Apply iffEL (pY (f x)) (pY' (f x)) (HpY' (f x) (Lf x Hx)) to the current goal.
An exact proof term for the current goal is H2 x Hx H7.
We prove the intermediate claim L2: ∀pX' : set → prop, (∀x ∈ X, pX x ↔ pX' x) → PhiX X pX' = PhiX X pX.
Let pX' be given.
Assume HpX'.
We will prove PhiX X pX' = PhiX X pX.
We will prove unpack_p_o (pack_p Y pY) (λY' pY ⇒ Phi X pX' Y' pY) = unpack_p_o (pack_p Y pY) (λY' pY ⇒ Phi X pX Y' pY).
Use transitivity with Phi X pX' Y pY, and Phi X pX Y pY.
We will prove unpack_p_o (pack_p Y pY) (Phi X pX') = Phi X pX' Y pY.
An exact proof term for the current goal is unpack_p_o_eq (Phi X pX') Y pY (L1 pX').
We will prove Phi X pX' Y pY = Phi X pX Y pY.
Apply prop_ext_2 to the current goal.
Assume H.
Apply H to the current goal.
Assume H1 H2.
We will prove f ∈ YX ∧ (∀x ∈ X, pX x → pY (f x)).
Apply andI to the current goal.
An exact proof term for the current goal is H1.
Let x be given.
Assume Hx H7.
Apply H2 x Hx to the current goal.
Apply iffEL (pX x) (pX' x) (HpX' x Hx) to the current goal.
An exact proof term for the current goal is H7.
Assume H.
Apply H to the current goal.
Assume H1 H2.
We will prove f ∈ YX ∧ (∀x ∈ X, pX' x → pY (f x)).
Apply andI to the current goal.
An exact proof term for the current goal is H1.
Let x be given.
Assume Hx H7.
Apply H2 x Hx to the current goal.
Apply iffER (pX x) (pX' x) (HpX' x Hx) to the current goal.
An exact proof term for the current goal is H7.
Use symmetry.
An exact proof term for the current goal is unpack_p_o_eq (Phi X pX) Y pY (L1 pX).
Use transitivity with and PhiX X pX.
An exact proof term for the current goal is unpack_p_o_eq PhiX X pX L2.
We will prove unpack_p_o (pack_p Y pY) (Phi X pX) = Phi X pX Y pY.
An exact proof term for the current goal is unpack_p_o_eq (Phi X pX) Y pY (L1 pX).
∎
Theorem. (Hom_struct_r_pack)
∀X Y, ∀rX rY : set → set → prop, ∀f, (Hom_struct_r (pack_r X rX) (pack_r Y rY) f) = (f ∈ YX ∧ (∀x y ∈ X, rX x y → rY (f x) (f y)))
In Proofgold the corresponding term root is 4e4867... and proposition id is c84abf...
Proof:
Let X, Y, rX, rY and f be given.
Set Phi to be the term λX rX Y rY ⇒ f ∈ YX ∧ (∀x y ∈ X, rX x y → rY (f x) (f y)) of type set → (set → set → prop) → set → (set → set → prop) → prop.
Set PhiX to be the term λX rX ⇒ unpack_r_o (pack_r Y rY) (λY' rY ⇒ Phi X rX Y' rY) of type set → (set → set → prop) → prop.
We will prove unpack_r_o (pack_r X rX) PhiX = Phi X rX Y rY.
We prove the intermediate claim L1: ∀rX : set → set → prop, ∀rY' : set → set → prop, (∀x y ∈ Y, rY x y ↔ rY' x y) → Phi X rX Y rY' = Phi X rX Y rY.
Let rX be given.
Let rY' be given.
Assume HrY'.
Apply prop_ext_2 to the current goal.
Assume H.
Apply H to the current goal.
Assume H1 H2.
We prove the intermediate claim Lf: ∀x ∈ X, f x ∈ Y.
Let x be given.
An exact proof term for the current goal is ap_Pi X (λ_ ⇒ Y) f x H1.
We will prove f ∈ YX ∧ (∀x y ∈ X, rX x y → rY (f x) (f y)).
Apply andI to the current goal.
An exact proof term for the current goal is H1.
Let x be given.
Assume Hx.
Let y be given.
Assume Hy H7.
Apply iffER (rY (f x) (f y)) (rY' (f x) (f y)) (HrY' (f x) (Lf x Hx) (f y) (Lf y Hy)) to the current goal.
An exact proof term for the current goal is H2 x Hx y Hy H7.
Assume H.
Apply H to the current goal.
Assume H1 H2.
We prove the intermediate claim Lf: ∀x ∈ X, f x ∈ Y.
Let x be given.
An exact proof term for the current goal is ap_Pi X (λ_ ⇒ Y) f x H1.
We will prove f ∈ YX ∧ (∀x y ∈ X, rX x y → rY' (f x) (f y)).
Apply andI to the current goal.
An exact proof term for the current goal is H1.
Let x be given.
Assume Hx.
Let y be given.
Assume Hy H7.
Apply iffEL (rY (f x) (f y)) (rY' (f x) (f y)) (HrY' (f x) (Lf x Hx) (f y) (Lf y Hy)) to the current goal.
An exact proof term for the current goal is H2 x Hx y Hy H7.
We prove the intermediate claim L2: ∀rX' : set → set → prop, (∀x y ∈ X, rX x y ↔ rX' x y) → PhiX X rX' = PhiX X rX.
Let rX' be given.
Assume HrX'.
We will prove PhiX X rX' = PhiX X rX.
We will prove unpack_r_o (pack_r Y rY) (λY' rY ⇒ Phi X rX' Y' rY) = unpack_r_o (pack_r Y rY) (λY' rY ⇒ Phi X rX Y' rY).
Use transitivity with Phi X rX' Y rY, and Phi X rX Y rY.
We will prove unpack_r_o (pack_r Y rY) (Phi X rX') = Phi X rX' Y rY.
An exact proof term for the current goal is unpack_r_o_eq (Phi X rX') Y rY (L1 rX').
We will prove Phi X rX' Y rY = Phi X rX Y rY.
Apply prop_ext_2 to the current goal.
Assume H.
Apply H to the current goal.
Assume H1 H2.
We will prove f ∈ YX ∧ (∀x y ∈ X, rX x y → rY (f x) (f y)).
Apply andI to the current goal.
An exact proof term for the current goal is H1.
Let x be given.
Assume Hx.
Let y be given.
Assume Hy H7.
Apply H2 x Hx y Hy to the current goal.
Apply iffEL (rX x y) (rX' x y) (HrX' x Hx y Hy) to the current goal.
An exact proof term for the current goal is H7.
Assume H.
Apply H to the current goal.
Assume H1 H2.
We will prove f ∈ YX ∧ (∀x y ∈ X, rX' x y → rY (f x) (f y)).
Apply andI to the current goal.
An exact proof term for the current goal is H1.
Let x be given.
Assume Hx.
Let y be given.
Assume Hy H7.
Apply H2 x Hx y Hy to the current goal.
Apply iffER (rX x y) (rX' x y) (HrX' x Hx y Hy) to the current goal.
An exact proof term for the current goal is H7.
Use symmetry.
An exact proof term for the current goal is unpack_r_o_eq (Phi X rX) Y rY (L1 rX).
Use transitivity with and PhiX X rX.
An exact proof term for the current goal is unpack_r_o_eq PhiX X rX L2.
We will prove unpack_r_o (pack_r Y rY) (Phi X rX) = Phi X rX Y rY.
An exact proof term for the current goal is unpack_r_o_eq (Phi X rX) Y rY (L1 rX).
∎
Theorem. (Hom_struct_c_pack)
∀X Y, ∀CX CY : (set → prop) → prop, ∀f, (Hom_struct_c (pack_c X CX) (pack_c Y CY) f) = (f ∈ YX ∧ (∀U : set → prop, (∀y, U y → y ∈ Y) → CY U → CX (λx ⇒ x ∈ X ∧ U (f x))))
In Proofgold the corresponding term root is 147946... and proposition id is 5059f2...
Proof:
Let X, Y, CX, CY and f be given.
Set Phi to be the term λX CX Y CY ⇒ f ∈ YX ∧ (∀U : set → prop, (∀y, U y → y ∈ Y) → CY U → CX (λx ⇒ x ∈ X ∧ U (f x))) of type set → ((set → prop) → prop) → set → ((set → prop) → prop) → prop.
Set PhiX to be the term λX CX ⇒ unpack_c_o (pack_c Y CY) (λY' CY ⇒ Phi X CX Y' CY) of type set → ((set → prop) → prop) → prop.
We will prove unpack_c_o (pack_c X CX) PhiX = Phi X CX Y CY.
We prove the intermediate claim L1: ∀CX : (set → prop) → prop, ∀CY' : (set → prop) → prop, (∀U : set → prop, (∀y, U y → y ∈ Y) → (CY U ↔ CY' U)) → Phi X CX Y CY' = Phi X CX Y CY.
Let CX be given.
Let CY' be given.
Assume HCY'.
Apply prop_ext_2 to the current goal.
Assume H.
Apply H to the current goal.
Assume H1 H2.
We will prove f ∈ YX ∧ (∀U : set → prop, (∀y, U y → y ∈ Y) → CY U → CX (λx ⇒ x ∈ X ∧ U (f x))).
Apply andI to the current goal.
An exact proof term for the current goal is H1.
Let U be given.
Assume HU1 HU2.
We will prove CX (λx ⇒ x ∈ X ∧ U (f x)).
Apply H2 to the current goal.
An exact proof term for the current goal is HU1.
We will prove CY' U.
An exact proof term for the current goal is iffEL (CY U) (CY' U) (HCY' U HU1) HU2.
Assume H.
Apply H to the current goal.
Assume H1 H2.
We will prove f ∈ YX ∧ (∀U : set → prop, (∀y, U y → y ∈ Y) → CY' U → CX (λx ⇒ x ∈ X ∧ U (f x))).
Apply andI to the current goal.
An exact proof term for the current goal is H1.
Let U be given.
Assume HU1 HU2.
We will prove CX (λx ⇒ x ∈ X ∧ U (f x)).
Apply H2 to the current goal.
An exact proof term for the current goal is HU1.
We will prove CY U.
An exact proof term for the current goal is iffER (CY U) (CY' U) (HCY' U HU1) HU2.
We prove the intermediate claim L2: ∀CX' : (set → prop) → prop, (∀U : set → prop, (∀x, U x → x ∈ X) → (CX U ↔ CX' U)) → PhiX X CX' = PhiX X CX.
Let CX' be given.
Assume HCX'.
We will prove PhiX X CX' = PhiX X CX.
We will prove unpack_c_o (pack_c Y CY) (λY' CY ⇒ Phi X CX' Y' CY) = unpack_c_o (pack_c Y CY) (λY' CY ⇒ Phi X CX Y' CY).
Use transitivity with Phi X CX' Y CY, and Phi X CX Y CY.
We will prove unpack_c_o (pack_c Y CY) (Phi X CX') = Phi X CX' Y CY.
An exact proof term for the current goal is unpack_c_o_eq (Phi X CX') Y CY (L1 CX').
We will prove Phi X CX' Y CY = Phi X CX Y CY.
Apply prop_ext_2 to the current goal.
Assume H.
Apply H to the current goal.
Assume H1: f ∈ YX.
Assume H2: ∀U : set → prop, (∀y, U y → y ∈ Y) → CY U → CX' (λx ⇒ x ∈ X ∧ U (f x)).
We will prove f ∈ YX ∧ (∀U : set → prop, (∀y, U y → y ∈ Y) → CY U → CX (λx ⇒ x ∈ X ∧ U (f x))).
Apply andI to the current goal.
An exact proof term for the current goal is H1.
Let U be given.
Assume HU1: ∀y, U y → y ∈ Y.
Assume HU2: CY U.
We prove the intermediate claim LU: ∀x, x ∈ X ∧ U (f x) → x ∈ X.
Let x be given.
An exact proof term for the current goal is andEL (x ∈ X) (U (f x)).
Apply iffER (CX (λx ⇒ x ∈ X ∧ U (f x))) (CX' (λx ⇒ x ∈ X ∧ U (f x))) (HCX' (λx ⇒ x ∈ X ∧ U (f x)) LU) to the current goal.
An exact proof term for the current goal is H2 U HU1 HU2.
Assume H.
Apply H to the current goal.
Assume H1 H2.
We will prove f ∈ YX ∧ (∀U : set → prop, (∀y, U y → y ∈ Y) → CY U → CX' (λx ⇒ x ∈ X ∧ U (f x))).
Apply andI to the current goal.
An exact proof term for the current goal is H1.
Let U be given.
Assume HU1: ∀y, U y → y ∈ Y.
Assume HU2: CY U.
We prove the intermediate claim LU: ∀x, x ∈ X ∧ U (f x) → x ∈ X.
Let x be given.
An exact proof term for the current goal is andEL (x ∈ X) (U (f x)).
Apply iffEL (CX (λx ⇒ x ∈ X ∧ U (f x))) (CX' (λx ⇒ x ∈ X ∧ U (f x))) (HCX' (λx ⇒ x ∈ X ∧ U (f x)) LU) to the current goal.
An exact proof term for the current goal is H2 U HU1 HU2.
Use symmetry.
An exact proof term for the current goal is unpack_c_o_eq (Phi X CX) Y CY (L1 CX).
Use transitivity with and PhiX X CX.
An exact proof term for the current goal is unpack_c_o_eq PhiX X CX L2.
We will prove unpack_c_o (pack_c Y CY) (Phi X CX) = Phi X CX Y CY.
An exact proof term for the current goal is unpack_c_o_eq (Phi X CX) Y CY (L1 CX).
∎
Theorem. (Hom_struct_b_b_e_pack)
∀X Y, ∀opX op2X opY op2Y : set → set → set, ∀eX eY f, (Hom_struct_b_b_e (pack_b_b_e X opX op2X eX) (pack_b_b_e Y opY op2Y eY) f) = (f ∈ YX ∧ (∀x y ∈ X, f (opX x y) = opY (f x) (f y)) ∧ (∀x y ∈ X, f (op2X x y) = op2Y (f x) (f y)) ∧ f eX = eY)
In Proofgold the corresponding term root is 10267d... and proposition id is 024cda...
Proof:
Let X, Y, opX, op2X, opY, op2Y, eX, eY and f be given.
Set Phi to be the term λX opX op2X eX Y opY op2Y eY ⇒ f ∈ YX ∧ (∀x y ∈ X, f (opX x y) = opY (f x) (f y)) ∧ (∀x y ∈ X, f (op2X x y) = op2Y (f x) (f y)) ∧ f eX = eY of type set → (set → set → set) → (set → set → set) → set → set → (set → set → set) → (set → set → set) → set → prop.
Set PhiX to be the term λX opX op2X eX ⇒ unpack_b_b_e_o (pack_b_b_e Y opY op2Y eY) (λY' opY op2Y eY ⇒ Phi X opX op2X eX Y' opY op2Y eY) of type set → (set → set → set) → (set → set → set) → set → prop.
We will prove unpack_b_b_e_o (pack_b_b_e X opX op2X eX) PhiX = Phi X opX op2X eX Y opY op2Y eY.
We prove the intermediate claim L1: ∀opX op2X : set → set → set, ∀opY' : set → set → set, (∀x y ∈ Y, opY x y = opY' x y) → ∀op2Y' : set → set → set, (∀x y ∈ Y, op2Y x y = op2Y' x y) → Phi X opX op2X eX Y opY' op2Y' eY = Phi X opX op2X eX Y opY op2Y eY.
Let opX and op2X be given.
Let opY' be given.
Assume HopY'.
Let op2Y' be given.
Assume Hop2Y'.
Apply prop_ext_2 to the current goal.
Assume H.
Apply and4E (f ∈ YX) (∀x y ∈ X, f (opX x y) = opY' (f x) (f y)) (∀x y ∈ X, f (op2X x y) = op2Y' (f x) (f y)) (f eX = eY) H to the current goal.
Assume H1 H2 H3 H4.
We prove the intermediate claim Lf: ∀x ∈ X, f x ∈ Y.
Let x be given.
An exact proof term for the current goal is ap_Pi X (λ_ ⇒ Y) f x H1.
We will prove f ∈ YX ∧ (∀x y ∈ X, f (opX x y) = opY (f x) (f y)) ∧ (∀x y ∈ X, f (op2X x y) = op2Y (f x) (f y)) ∧ f eX = eY.
Apply and4I to the current goal.
An exact proof term for the current goal is H1.
Let x be given.
Assume Hx.
Let y be given.
Assume Hy.
rewrite the current goal using HopY' (f x) (Lf x Hx) (f y) (Lf y Hy) (from left to right).
An exact proof term for the current goal is H2 x Hx y Hy.
Let x be given.
Assume Hx.
Let y be given.
Assume Hy.
rewrite the current goal using Hop2Y' (f x) (Lf x Hx) (f y) (Lf y Hy) (from left to right).
An exact proof term for the current goal is H3 x Hx y Hy.
An exact proof term for the current goal is H4.
Assume H.
Apply and4E (f ∈ YX) (∀x y ∈ X, f (opX x y) = opY (f x) (f y)) (∀x y ∈ X, f (op2X x y) = op2Y (f x) (f y)) (f eX = eY) H to the current goal.
Assume H1 H2 H3 H4.
We prove the intermediate claim Lf: ∀x ∈ X, f x ∈ Y.
Let x be given.
An exact proof term for the current goal is ap_Pi X (λ_ ⇒ Y) f x H1.
We will prove f ∈ YX ∧ (∀x y ∈ X, f (opX x y) = opY' (f x) (f y)) ∧ (∀x y ∈ X, f (op2X x y) = op2Y' (f x) (f y)) ∧ f eX = eY.
Apply and4I to the current goal.
An exact proof term for the current goal is H1.
Let x be given.
Assume Hx.
Let y be given.
Assume Hy.
rewrite the current goal using HopY' (f x) (Lf x Hx) (f y) (Lf y Hy) (from right to left).
An exact proof term for the current goal is H2 x Hx y Hy.
Let x be given.
Assume Hx.
Let y be given.
Assume Hy.
rewrite the current goal using Hop2Y' (f x) (Lf x Hx) (f y) (Lf y Hy) (from right to left).
An exact proof term for the current goal is H3 x Hx y Hy.
An exact proof term for the current goal is H4.
We prove the intermediate claim L2: ∀opX' : set → set → set, (∀x y ∈ X, opX x y = opX' x y) → ∀op2X' : set → set → set, (∀x y ∈ X, op2X x y = op2X' x y) → PhiX X opX' op2X' eX = PhiX X opX op2X eX.
Let opX' be given.
Assume HopX'.
Let op2X' be given.
Assume Hop2X'.
We will prove PhiX X opX' op2X' eX = PhiX X opX op2X eX.
We will prove unpack_b_b_e_o (pack_b_b_e Y opY op2Y eY) (λY' opY op2Y eY ⇒ Phi X opX' op2X' eX Y' opY op2Y eY) = unpack_b_b_e_o (pack_b_b_e Y opY op2Y eY) (λY' opY op2Y eY ⇒ Phi X opX op2X eX Y' opY op2Y eY).
Use transitivity with Phi X opX' op2X' eX Y opY op2Y eY, and Phi X opX op2X eX Y opY op2Y eY.
We will prove unpack_b_b_e_o (pack_b_b_e Y opY op2Y eY) (Phi X opX' op2X' eX) = Phi X opX' op2X' eX Y opY op2Y eY.
An exact proof term for the current goal is unpack_b_b_e_o_eq (Phi X opX' op2X' eX) Y opY op2Y eY (L1 opX' op2X').
We will prove Phi X opX' op2X' eX Y opY op2Y eY = Phi X opX op2X eX Y opY op2Y eY.
Apply prop_ext_2 to the current goal.
Assume H.
Apply and4E (f ∈ YX) (∀x y ∈ X, f (opX' x y) = opY (f x) (f y)) (∀x y ∈ X, f (op2X' x y) = op2Y (f x) (f y)) (f eX = eY) H to the current goal.
Assume H1 H2 H3 H4.
We will prove f ∈ YX ∧ (∀x y ∈ X, f (opX x y) = opY (f x) (f y)) ∧ (∀x y ∈ X, f (op2X x y) = op2Y (f x) (f y)) ∧ f eX = eY.
Apply and4I to the current goal.
An exact proof term for the current goal is H1.
Let x be given.
Assume Hx.
Let y be given.
Assume Hy.
rewrite the current goal using HopX' x Hx y Hy (from left to right).
An exact proof term for the current goal is H2 x Hx y Hy.
Let x be given.
Assume Hx.
Let y be given.
Assume Hy.
rewrite the current goal using Hop2X' x Hx y Hy (from left to right).
An exact proof term for the current goal is H3 x Hx y Hy.
An exact proof term for the current goal is H4.
Assume H.
Apply and4E (f ∈ YX) (∀x y ∈ X, f (opX x y) = opY (f x) (f y)) (∀x y ∈ X, f (op2X x y) = op2Y (f x) (f y)) (f eX = eY) H to the current goal.
Assume H1 H2 H3 H4.
We will prove f ∈ YX ∧ (∀x y ∈ X, f (opX' x y) = opY (f x) (f y)) ∧ (∀x y ∈ X, f (op2X' x y) = op2Y (f x) (f y)) ∧ f eX = eY.
Apply and4I to the current goal.
An exact proof term for the current goal is H1.
Let x be given.
Assume Hx.
Let y be given.
Assume Hy.
rewrite the current goal using HopX' x Hx y Hy (from right to left).
An exact proof term for the current goal is H2 x Hx y Hy.
Let x be given.
Assume Hx.
Let y be given.
Assume Hy.
rewrite the current goal using Hop2X' x Hx y Hy (from right to left).
An exact proof term for the current goal is H3 x Hx y Hy.
An exact proof term for the current goal is H4.
Use symmetry.
An exact proof term for the current goal is unpack_b_b_e_o_eq (Phi X opX op2X eX) Y opY op2Y eY (L1 opX op2X).
Use transitivity with and PhiX X opX op2X eX.
An exact proof term for the current goal is unpack_b_b_e_o_eq PhiX X opX op2X eX L2.
We will prove unpack_b_b_e_o (pack_b_b_e Y opY op2Y eY) (Phi X opX op2X eX) = Phi X opX op2X eX Y opY op2Y eY.
An exact proof term for the current goal is unpack_b_b_e_o_eq (Phi X opX op2X eX) Y opY op2Y eY (L1 opX op2X).
∎
Theorem. (Hom_struct_b_b_e_e_pack)
∀X Y, ∀opX op2X opY op2Y : set → set → set, ∀eX e2X eY e2Y f, (Hom_struct_b_b_e_e (pack_b_b_e_e X opX op2X eX e2X) (pack_b_b_e_e Y opY op2Y eY e2Y) f) = (f ∈ YX ∧ (∀x y ∈ X, f (opX x y) = opY (f x) (f y)) ∧ (∀x y ∈ X, f (op2X x y) = op2Y (f x) (f y)) ∧ f eX = eY ∧ f e2X = e2Y)
In Proofgold the corresponding term root is c390c3... and proposition id is 4ba5a3...
Proof:
Let X, Y, opX, op2X, opY, op2Y, eX, e2X, eY, e2Y and f be given.
Set Phi to be the term λX opX op2X eX e2X Y opY op2Y eY e2Y ⇒ f ∈ YX ∧ (∀x y ∈ X, f (opX x y) = opY (f x) (f y)) ∧ (∀x y ∈ X, f (op2X x y) = op2Y (f x) (f y)) ∧ f eX = eY ∧ f e2X = e2Y of type set → (set → set → set) → (set → set → set) → set → set → set → (set → set → set) → (set → set → set) → set → set → prop.
Set PhiX to be the term λX opX op2X eX e2X ⇒ unpack_b_b_e_e_o (pack_b_b_e_e Y opY op2Y eY e2Y) (λY' opY op2Y eY e2Y ⇒ Phi X opX op2X eX e2X Y' opY op2Y eY e2Y) of type set → (set → set → set) → (set → set → set) → set → set → prop.
We will prove unpack_b_b_e_e_o (pack_b_b_e_e X opX op2X eX e2X) PhiX = Phi X opX op2X eX e2X Y opY op2Y eY e2Y.
We prove the intermediate claim L1: ∀opX op2X : set → set → set, ∀opY' : set → set → set, (∀x y ∈ Y, opY x y = opY' x y) → ∀op2Y' : set → set → set, (∀x y ∈ Y, op2Y x y = op2Y' x y) → Phi X opX op2X eX e2X Y opY' op2Y' eY e2Y = Phi X opX op2X eX e2X Y opY op2Y eY e2Y.
Let opX and op2X be given.
Let opY' be given.
Assume HopY'.
Let op2Y' be given.
Assume Hop2Y'.
Apply prop_ext_2 to the current goal.
Assume H.
Apply and5E (f ∈ YX) (∀x y ∈ X, f (opX x y) = opY' (f x) (f y)) (∀x y ∈ X, f (op2X x y) = op2Y' (f x) (f y)) (f eX = eY) (f e2X = e2Y) H to the current goal.
Assume H1 H2 H3 H4 H5.
We prove the intermediate claim Lf: ∀x ∈ X, f x ∈ Y.
Let x be given.
An exact proof term for the current goal is ap_Pi X (λ_ ⇒ Y) f x H1.
We will prove f ∈ YX ∧ (∀x y ∈ X, f (opX x y) = opY (f x) (f y)) ∧ (∀x y ∈ X, f (op2X x y) = op2Y (f x) (f y)) ∧ f eX = eY ∧ f e2X = e2Y.
Apply and5I to the current goal.
An exact proof term for the current goal is H1.
Let x be given.
Assume Hx.
Let y be given.
Assume Hy.
rewrite the current goal using HopY' (f x) (Lf x Hx) (f y) (Lf y Hy) (from left to right).
An exact proof term for the current goal is H2 x Hx y Hy.
Let x be given.
Assume Hx.
Let y be given.
Assume Hy.
rewrite the current goal using Hop2Y' (f x) (Lf x Hx) (f y) (Lf y Hy) (from left to right).
An exact proof term for the current goal is H3 x Hx y Hy.
An exact proof term for the current goal is H4.
An exact proof term for the current goal is H5.
Assume H.
Apply and5E (f ∈ YX) (∀x y ∈ X, f (opX x y) = opY (f x) (f y)) (∀x y ∈ X, f (op2X x y) = op2Y (f x) (f y)) (f eX = eY) (f e2X = e2Y) H to the current goal.
Assume H1 H2 H3 H4 H5.
We prove the intermediate claim Lf: ∀x ∈ X, f x ∈ Y.
Let x be given.
An exact proof term for the current goal is ap_Pi X (λ_ ⇒ Y) f x H1.
We will prove f ∈ YX ∧ (∀x y ∈ X, f (opX x y) = opY' (f x) (f y)) ∧ (∀x y ∈ X, f (op2X x y) = op2Y' (f x) (f y)) ∧ f eX = eY ∧ f e2X = e2Y.
Apply and5I to the current goal.
An exact proof term for the current goal is H1.
Let x be given.
Assume Hx.
Let y be given.
Assume Hy.
rewrite the current goal using HopY' (f x) (Lf x Hx) (f y) (Lf y Hy) (from right to left).
An exact proof term for the current goal is H2 x Hx y Hy.
Let x be given.
Assume Hx.
Let y be given.
Assume Hy.
rewrite the current goal using Hop2Y' (f x) (Lf x Hx) (f y) (Lf y Hy) (from right to left).
An exact proof term for the current goal is H3 x Hx y Hy.
An exact proof term for the current goal is H4.
An exact proof term for the current goal is H5.
We prove the intermediate claim L2: ∀opX' : set → set → set, (∀x y ∈ X, opX x y = opX' x y) → ∀op2X' : set → set → set, (∀x y ∈ X, op2X x y = op2X' x y) → PhiX X opX' op2X' eX e2X = PhiX X opX op2X eX e2X.
Let opX' be given.
Assume HopX'.
Let op2X' be given.
Assume Hop2X'.
We will prove PhiX X opX' op2X' eX e2X = PhiX X opX op2X eX e2X.
We will prove unpack_b_b_e_e_o (pack_b_b_e_e Y opY op2Y eY e2Y) (λY' opY op2Y eY e2Y ⇒ Phi X opX' op2X' eX e2X Y' opY op2Y eY e2Y) = unpack_b_b_e_e_o (pack_b_b_e_e Y opY op2Y eY e2Y) (λY' opY op2Y eY e2Y ⇒ Phi X opX op2X eX e2X Y' opY op2Y eY e2Y).
Use transitivity with Phi X opX' op2X' eX e2X Y opY op2Y eY e2Y, and Phi X opX op2X eX e2X Y opY op2Y eY e2Y.
We will prove unpack_b_b_e_e_o (pack_b_b_e_e Y opY op2Y eY e2Y) (Phi X opX' op2X' eX e2X) = Phi X opX' op2X' eX e2X Y opY op2Y eY e2Y.
An exact proof term for the current goal is unpack_b_b_e_e_o_eq (Phi X opX' op2X' eX e2X) Y opY op2Y eY e2Y (L1 opX' op2X').
We will prove Phi X opX' op2X' eX e2X Y opY op2Y eY e2Y = Phi X opX op2X eX e2X Y opY op2Y eY e2Y.
Apply prop_ext_2 to the current goal.
Assume H.
Apply and5E (f ∈ YX) (∀x y ∈ X, f (opX' x y) = opY (f x) (f y)) (∀x y ∈ X, f (op2X' x y) = op2Y (f x) (f y)) (f eX = eY) (f e2X = e2Y) H to the current goal.
Assume H1 H2 H3 H4 H5.
We will prove f ∈ YX ∧ (∀x y ∈ X, f (opX x y) = opY (f x) (f y)) ∧ (∀x y ∈ X, f (op2X x y) = op2Y (f x) (f y)) ∧ f eX = eY ∧ f e2X = e2Y.
Apply and5I to the current goal.
An exact proof term for the current goal is H1.
Let x be given.
Assume Hx.
Let y be given.
Assume Hy.
rewrite the current goal using HopX' x Hx y Hy (from left to right).
An exact proof term for the current goal is H2 x Hx y Hy.
Let x be given.
Assume Hx.
Let y be given.
Assume Hy.
rewrite the current goal using Hop2X' x Hx y Hy (from left to right).
An exact proof term for the current goal is H3 x Hx y Hy.
An exact proof term for the current goal is H4.
An exact proof term for the current goal is H5.
Assume H.
Apply and5E (f ∈ YX) (∀x y ∈ X, f (opX x y) = opY (f x) (f y)) (∀x y ∈ X, f (op2X x y) = op2Y (f x) (f y)) (f eX = eY) (f e2X = e2Y) H to the current goal.
Assume H1 H2 H3 H4 H5.
We will prove f ∈ YX ∧ (∀x y ∈ X, f (opX' x y) = opY (f x) (f y)) ∧ (∀x y ∈ X, f (op2X' x y) = op2Y (f x) (f y)) ∧ f eX = eY ∧ f e2X = e2Y.
Apply and5I to the current goal.
An exact proof term for the current goal is H1.
Let x be given.
Assume Hx.
Let y be given.
Assume Hy.
rewrite the current goal using HopX' x Hx y Hy (from right to left).
An exact proof term for the current goal is H2 x Hx y Hy.
Let x be given.
Assume Hx.
Let y be given.
Assume Hy.
rewrite the current goal using Hop2X' x Hx y Hy (from right to left).
An exact proof term for the current goal is H3 x Hx y Hy.
An exact proof term for the current goal is H4.
An exact proof term for the current goal is H5.
Use symmetry.
An exact proof term for the current goal is unpack_b_b_e_e_o_eq (Phi X opX op2X eX e2X) Y opY op2Y eY e2Y (L1 opX op2X).
Use transitivity with and PhiX X opX op2X eX e2X.
An exact proof term for the current goal is unpack_b_b_e_e_o_eq PhiX X opX op2X eX e2X L2.
We will prove unpack_b_b_e_e_o (pack_b_b_e_e Y opY op2Y eY e2Y) (Phi X opX op2X eX e2X) = Phi X opX op2X eX e2X Y opY op2Y eY e2Y.
An exact proof term for the current goal is unpack_b_b_e_e_o_eq (Phi X opX op2X eX e2X) Y opY op2Y eY e2Y (L1 opX op2X).
∎
Theorem. (Hom_struct_b_b_r_e_e_pack)
∀X Y, ∀opX op2X opY op2Y : set → set → set, ∀rX rY : set → set → prop, ∀eX e2X eY e2Y f, (Hom_struct_b_b_r_e_e (pack_b_b_r_e_e X opX op2X rX eX e2X) (pack_b_b_r_e_e Y opY op2Y rY eY e2Y) f) = (f ∈ YX ∧ (∀x y ∈ X, f (opX x y) = opY (f x) (f y)) ∧ (∀x y ∈ X, f (op2X x y) = op2Y (f x) (f y)) ∧ (∀x y ∈ X, rX x y → rY (f x) (f y)) ∧ f eX = eY ∧ f e2X = e2Y)
In Proofgold the corresponding term root is 3b7da8... and proposition id is 3e8fcc...
Proof:
Let X, Y, opX, op2X, opY, op2Y, rX, rY, eX, e2X, eY, e2Y and f be given.
Set Phi to be the term λX opX op2X rX eX e2X Y opY op2Y rY eY e2Y ⇒ f ∈ YX ∧ (∀x y ∈ X, f (opX x y) = opY (f x) (f y)) ∧ (∀x y ∈ X, f (op2X x y) = op2Y (f x) (f y)) ∧ (∀x y ∈ X, rX x y → rY (f x) (f y)) ∧ f eX = eY ∧ f e2X = e2Y of type set → (set → set → set) → (set → set → set) → (set → set → prop) → set → set → set → (set → set → set) → (set → set → set) → (set → set → prop) → set → set → prop.
Set PhiX to be the term λX opX op2X rX eX e2X ⇒ unpack_b_b_r_e_e_o (pack_b_b_r_e_e Y opY op2Y rY eY e2Y) (λY' opY op2Y rY eY e2Y ⇒ Phi X opX op2X rX eX e2X Y' opY op2Y rY eY e2Y) of type set → (set → set → set) → (set → set → set) → (set → set → prop) → set → set → prop.
We will prove unpack_b_b_r_e_e_o (pack_b_b_r_e_e X opX op2X rX eX e2X) PhiX = Phi X opX op2X rX eX e2X Y opY op2Y rY eY e2Y.
We prove the intermediate claim L1: ∀opX op2X : set → set → set, ∀rX : set → set → prop, ∀opY' : set → set → set, (∀x y ∈ Y, opY x y = opY' x y) → ∀op2Y' : set → set → set, (∀x y ∈ Y, op2Y x y = op2Y' x y) → ∀rY' : set → set → prop, (∀x y ∈ Y, rY x y ↔ rY' x y) → Phi X opX op2X rX eX e2X Y opY' op2Y' rY' eY e2Y = Phi X opX op2X rX eX e2X Y opY op2Y rY eY e2Y.
Let opX, op2X and rX be given.
Let opY' be given.
Assume HopY'.
Let op2Y' be given.
Assume Hop2Y'.
Let rY' be given.
Assume HrY'.
Apply prop_ext_2 to the current goal.
Assume H.
Apply and6E (f ∈ YX) (∀x y ∈ X, f (opX x y) = opY' (f x) (f y)) (∀x y ∈ X, f (op2X x y) = op2Y' (f x) (f y)) (∀x y ∈ X, rX x y → rY' (f x) (f y)) (f eX = eY) (f e2X = e2Y) H to the current goal.
Assume H1 H2 H3 H4 H5 H6.
We prove the intermediate claim Lf: ∀x ∈ X, f x ∈ Y.
Let x be given.
An exact proof term for the current goal is ap_Pi X (λ_ ⇒ Y) f x H1.
We will prove f ∈ YX ∧ (∀x y ∈ X, f (opX x y) = opY (f x) (f y)) ∧ (∀x y ∈ X, f (op2X x y) = op2Y (f x) (f y)) ∧ (∀x y ∈ X, rX x y → rY (f x) (f y)) ∧ f eX = eY ∧ f e2X = e2Y.
Apply and6I to the current goal.
An exact proof term for the current goal is H1.
Let x be given.
Assume Hx.
Let y be given.
Assume Hy.
rewrite the current goal using HopY' (f x) (Lf x Hx) (f y) (Lf y Hy) (from left to right).
An exact proof term for the current goal is H2 x Hx y Hy.
Let x be given.
Assume Hx.
Let y be given.
Assume Hy.
rewrite the current goal using Hop2Y' (f x) (Lf x Hx) (f y) (Lf y Hy) (from left to right).
An exact proof term for the current goal is H3 x Hx y Hy.
Let x be given.
Assume Hx.
Let y be given.
Assume Hy H7.
Apply iffER (rY (f x) (f y)) (rY' (f x) (f y)) (HrY' (f x) (Lf x Hx) (f y) (Lf y Hy)) to the current goal.
An exact proof term for the current goal is H4 x Hx y Hy H7.
An exact proof term for the current goal is H5.
An exact proof term for the current goal is H6.
Assume H.
Apply and6E (f ∈ YX) (∀x y ∈ X, f (opX x y) = opY (f x) (f y)) (∀x y ∈ X, f (op2X x y) = op2Y (f x) (f y)) (∀x y ∈ X, rX x y → rY (f x) (f y)) (f eX = eY) (f e2X = e2Y) H to the current goal.
Assume H1 H2 H3 H4 H5 H6.
We prove the intermediate claim Lf: ∀x ∈ X, f x ∈ Y.
Let x be given.
An exact proof term for the current goal is ap_Pi X (λ_ ⇒ Y) f x H1.
We will prove f ∈ YX ∧ (∀x y ∈ X, f (opX x y) = opY' (f x) (f y)) ∧ (∀x y ∈ X, f (op2X x y) = op2Y' (f x) (f y)) ∧ (∀x y ∈ X, rX x y → rY' (f x) (f y)) ∧ f eX = eY ∧ f e2X = e2Y.
Apply and6I to the current goal.
An exact proof term for the current goal is H1.
Let x be given.
Assume Hx.
Let y be given.
Assume Hy.
rewrite the current goal using HopY' (f x) (Lf x Hx) (f y) (Lf y Hy) (from right to left).
An exact proof term for the current goal is H2 x Hx y Hy.
Let x be given.
Assume Hx.
Let y be given.
Assume Hy.
rewrite the current goal using Hop2Y' (f x) (Lf x Hx) (f y) (Lf y Hy) (from right to left).
An exact proof term for the current goal is H3 x Hx y Hy.
Let x be given.
Assume Hx.
Let y be given.
Assume Hy H7.
Apply iffEL (rY (f x) (f y)) (rY' (f x) (f y)) (HrY' (f x) (Lf x Hx) (f y) (Lf y Hy)) to the current goal.
An exact proof term for the current goal is H4 x Hx y Hy H7.
An exact proof term for the current goal is H5.
An exact proof term for the current goal is H6.
We prove the intermediate claim L2: ∀opX' : set → set → set, (∀x y ∈ X, opX x y = opX' x y) → ∀op2X' : set → set → set, (∀x y ∈ X, op2X x y = op2X' x y) → ∀rX' : set → set → prop, (∀x y ∈ X, rX x y ↔ rX' x y) → PhiX X opX' op2X' rX' eX e2X = PhiX X opX op2X rX eX e2X.
Let opX' be given.
Assume HopX'.
Let op2X' be given.
Assume Hop2X'.
Let rX' be given.
Assume HrX'.
We will prove PhiX X opX' op2X' rX' eX e2X = PhiX X opX op2X rX eX e2X.
We will prove unpack_b_b_r_e_e_o (pack_b_b_r_e_e Y opY op2Y rY eY e2Y) (λY' opY op2Y rY eY e2Y ⇒ Phi X opX' op2X' rX' eX e2X Y' opY op2Y rY eY e2Y) = unpack_b_b_r_e_e_o (pack_b_b_r_e_e Y opY op2Y rY eY e2Y) (λY' opY op2Y rY eY e2Y ⇒ Phi X opX op2X rX eX e2X Y' opY op2Y rY eY e2Y).
Use transitivity with Phi X opX' op2X' rX' eX e2X Y opY op2Y rY eY e2Y, and Phi X opX op2X rX eX e2X Y opY op2Y rY eY e2Y.
We will prove unpack_b_b_r_e_e_o (pack_b_b_r_e_e Y opY op2Y rY eY e2Y) (Phi X opX' op2X' rX' eX e2X) = Phi X opX' op2X' rX' eX e2X Y opY op2Y rY eY e2Y.
An exact proof term for the current goal is unpack_b_b_r_e_e_o_eq (Phi X opX' op2X' rX' eX e2X) Y opY op2Y rY eY e2Y (L1 opX' op2X' rX').
We will prove Phi X opX' op2X' rX' eX e2X Y opY op2Y rY eY e2Y = Phi X opX op2X rX eX e2X Y opY op2Y rY eY e2Y.
Apply prop_ext_2 to the current goal.
Assume H.
Apply and6E (f ∈ YX) (∀x y ∈ X, f (opX' x y) = opY (f x) (f y)) (∀x y ∈ X, f (op2X' x y) = op2Y (f x) (f y)) (∀x y ∈ X, rX' x y → rY (f x) (f y)) (f eX = eY) (f e2X = e2Y) H to the current goal.
Assume H1 H2 H3 H4 H5 H6.
We will prove f ∈ YX ∧ (∀x y ∈ X, f (opX x y) = opY (f x) (f y)) ∧ (∀x y ∈ X, f (op2X x y) = op2Y (f x) (f y)) ∧ (∀x y ∈ X, rX x y → rY (f x) (f y)) ∧ f eX = eY ∧ f e2X = e2Y.
Apply and6I to the current goal.
An exact proof term for the current goal is H1.
Let x be given.
Assume Hx.
Let y be given.
Assume Hy.
rewrite the current goal using HopX' x Hx y Hy (from left to right).
An exact proof term for the current goal is H2 x Hx y Hy.
Let x be given.
Assume Hx.
Let y be given.
Assume Hy.
rewrite the current goal using Hop2X' x Hx y Hy (from left to right).
An exact proof term for the current goal is H3 x Hx y Hy.
Let x be given.
Assume Hx.
Let y be given.
Assume Hy H7.
Apply H4 x Hx y Hy to the current goal.
Apply iffEL (rX x y) (rX' x y) (HrX' x Hx y Hy) to the current goal.
An exact proof term for the current goal is H7.
An exact proof term for the current goal is H5.
An exact proof term for the current goal is H6.
Assume H.
Apply and6E (f ∈ YX) (∀x y ∈ X, f (opX x y) = opY (f x) (f y)) (∀x y ∈ X, f (op2X x y) = op2Y (f x) (f y)) (∀x y ∈ X, rX x y → rY (f x) (f y)) (f eX = eY) (f e2X = e2Y) H to the current goal.
Assume H1 H2 H3 H4 H5 H6.
We will prove f ∈ YX ∧ (∀x y ∈ X, f (opX' x y) = opY (f x) (f y)) ∧ (∀x y ∈ X, f (op2X' x y) = op2Y (f x) (f y)) ∧ (∀x y ∈ X, rX' x y → rY (f x) (f y)) ∧ f eX = eY ∧ f e2X = e2Y.
Apply and6I to the current goal.
An exact proof term for the current goal is H1.
Let x be given.
Assume Hx.
Let y be given.
Assume Hy.
rewrite the current goal using HopX' x Hx y Hy (from right to left).
An exact proof term for the current goal is H2 x Hx y Hy.
Let x be given.
Assume Hx.
Let y be given.
Assume Hy.
rewrite the current goal using Hop2X' x Hx y Hy (from right to left).
An exact proof term for the current goal is H3 x Hx y Hy.
Let x be given.
Assume Hx.
Let y be given.
Assume Hy H7.
Apply H4 x Hx y Hy to the current goal.
Apply iffER (rX x y) (rX' x y) (HrX' x Hx y Hy) to the current goal.
An exact proof term for the current goal is H7.
An exact proof term for the current goal is H5.
An exact proof term for the current goal is H6.
Use symmetry.
An exact proof term for the current goal is unpack_b_b_r_e_e_o_eq (Phi X opX op2X rX eX e2X) Y opY op2Y rY eY e2Y (L1 opX op2X rX).
Use transitivity with and PhiX X opX op2X rX eX e2X.
An exact proof term for the current goal is unpack_b_b_r_e_e_o_eq PhiX X opX op2X rX eX e2X L2.
We will prove unpack_b_b_r_e_e_o (pack_b_b_r_e_e Y opY op2Y rY eY e2Y) (Phi X opX op2X rX eX e2X) = Phi X opX op2X rX eX e2X Y opY op2Y rY eY e2Y.
An exact proof term for the current goal is unpack_b_b_r_e_e_o_eq (Phi X opX op2X rX eX e2X) Y opY op2Y rY eY e2Y (L1 opX op2X rX).
∎
Beginning of Section MetaCatStruct
Variable Obj : set → prop
Theorem. (MetaCat_struct_e_gen)
(∀X, Obj X → struct_e X) → MetaCat Obj Hom_struct_e (λX ⇒ lam_id (X 0)) (λX Y Z g f ⇒ lam_comp (X 0) g f)
In Proofgold the corresponding term root is f2377b... and proposition id is 1c0e13...
Proof:
Assume HO.
Apply MetaCatConcrete Obj (λX ⇒ X 0) Hom_struct_e to the current goal.
Let X, Y and f be given.
Assume HX HY.
Apply HO X HX (λu ⇒ Hom_struct_e u Y f → f ∈ Y 0u 0) to the current goal.
Let X' and eX be given.
Assume HeX.
Apply HO Y HY (λu ⇒ Hom_struct_e (pack_e X' eX) u f → f ∈ u 0pack_e X' eX 0) to the current goal.
Let Y' and eY be given.
Assume HeY.
We will prove Hom_struct_e (pack_e X' eX) (pack_e Y' eY) f → f ∈ pack_e Y' eY 0pack_e X' eX 0.
rewrite the current goal using Hom_struct_e_pack (from left to right).
Assume Hf: f ∈ Y'X' ∧ f eX = eY.
Apply Hf to the current goal.
Assume Hf1 Hf2.
rewrite the current goal using pack_e_0_eq2 (from right to left).
rewrite the current goal using pack_e_0_eq2 (from right to left).
An exact proof term for the current goal is Hf1.
Let X be given.
Assume HX.
Apply HO X HX (λu ⇒ Hom_struct_e u u (lam_id (u 0))) to the current goal.
Let X' and eX be given.
Assume HeX.
We will prove Hom_struct_e (pack_e X' eX) (pack_e X' eX) (lam_id (pack_e X' eX 0)).
rewrite the current goal using pack_e_0_eq2 (from right to left).
rewrite the current goal using Hom_struct_e_pack (from left to right).
We will prove lam_id X' ∈ X'X' ∧ lam_id X' eX = eX.
Apply andI to the current goal.
An exact proof term for the current goal is lam_id_exp_In X'.
We will prove (λx ∈ X' ⇒ x) eX = eX.
An exact proof term for the current goal is beta X' (λx ⇒ x) eX HeX.
Let X, Y, Z, f and g be given.
Assume HX HY HZ.
Apply HO X HX (λu ⇒ Hom_struct_e u Y f → Hom_struct_e Y Z g → Hom_struct_e u Z (lam_comp (u 0) g f)) to the current goal.
Let X' and eX be given.
Assume HeX.
We will prove Hom_struct_e (pack_e X' eX) Y f → Hom_struct_e Y Z g → Hom_struct_e (pack_e X' eX) Z (lam_comp (pack_e X' eX 0) g f).
rewrite the current goal using pack_e_0_eq2 (from right to left).
We will prove Hom_struct_e (pack_e X' eX) Y f → Hom_struct_e Y Z g → Hom_struct_e (pack_e X' eX) Z (lam_comp X' g f).
Apply HO Y HY (λu ⇒ Hom_struct_e (pack_e X' eX) u f → Hom_struct_e u Z g → Hom_struct_e (pack_e X' eX) Z (lam_comp X' g f)) to the current goal.
Let Y' and eY be given.
Assume HeY.
We will prove Hom_struct_e (pack_e X' eX) (pack_e Y' eY) f → Hom_struct_e (pack_e Y' eY) Z g → Hom_struct_e (pack_e X' eX) Z (lam_comp X' g f).
rewrite the current goal using Hom_struct_e_pack (from left to right).
Assume Hf: f ∈ Y'X' ∧ f eX = eY.
Apply HO Z HZ (λu ⇒ Hom_struct_e (pack_e Y' eY) u g → Hom_struct_e (pack_e X' eX) u (lam_comp X' g f)) to the current goal.
Let Z' and eZ be given.
Assume HeZ.
We will prove Hom_struct_e (pack_e Y' eY) (pack_e Z' eZ) g → Hom_struct_e (pack_e X' eX) (pack_e Z' eZ) (lam_comp X' g f).
rewrite the current goal using Hom_struct_e_pack (from left to right).
rewrite the current goal using Hom_struct_e_pack (from left to right).
Assume Hg: g ∈ Z'Y' ∧ g eY = eZ.
Apply Hf to the current goal.
Assume Hf1 Hf2.
Apply Hg to the current goal.
Assume Hg1 Hg2.
We will prove lam_comp X' g f ∈ Z'X' ∧ lam_comp X' g f eX = eZ.
Apply andI to the current goal.
An exact proof term for the current goal is lam_comp_exp_In X' Y' Z' f Hf1 g Hg1.
We will prove (λx ∈ X' ⇒ g (f x)) eX = eZ.
rewrite the current goal using beta X' (λx ⇒ g (f x)) eX HeX (from left to right).
We will prove g (f eX) = eZ.
rewrite the current goal using Hf2 (from left to right).
An exact proof term for the current goal is Hg2.
∎
Theorem. (MetaCat_struct_e_Forgetful_gen)
(∀X, Obj X → struct_e X) → MetaFunctor Obj Hom_struct_e (λX ⇒ lam_id (X 0)) (λX Y Z g f ⇒ lam_comp (X 0) g f) (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 7938f4... and proposition id is 0f4ef6...
Proof:
Assume HO.
We prove the intermediate claim L1: ∀X Y f, Obj X → Obj Y → Hom_struct_e X Y f → f ∈ Y 0X 0.
Let X, Y and f be given.
Assume HX HY.
Apply HO X HX (λu ⇒ Hom_struct_e u Y f → f ∈ Y 0u 0) to the current goal.
Let X' and eX be given.
Assume HeX.
Apply HO Y HY (λu ⇒ Hom_struct_e (pack_e X' eX) u f → f ∈ u 0pack_e X' eX 0) to the current goal.
Let Y' and eY be given.
Assume HeY.
We will prove Hom_struct_e (pack_e X' eX) (pack_e Y' eY) f → f ∈ pack_e Y' eY 0pack_e X' eX 0.
rewrite the current goal using Hom_struct_e_pack (from left to right).
Assume Hf.
rewrite the current goal using pack_e_0_eq2 (from right to left).
rewrite the current goal using pack_e_0_eq2 (from right to left).
Apply Hf to the current goal.
Assume Hf1 _.
An exact proof term for the current goal is Hf1.
An exact proof term for the current goal is MetaCatConcreteForgetful Obj (λX ⇒ X 0) Hom_struct_e L1.
∎
Theorem. (MetaCat_struct_p_gen)
(∀X, Obj X → struct_p X) → MetaCat Obj Hom_struct_p (λX ⇒ lam_id (X 0)) (λX Y Z g f ⇒ lam_comp (X 0) g f)
In Proofgold the corresponding term root is 8e7946... and proposition id is 5387a7...
Proof:
Assume HO.
Apply MetaCatConcrete Obj (λX ⇒ X 0) Hom_struct_p to the current goal.
Let X, Y and f be given.
Assume HX HY.
Apply HO X HX (λu ⇒ Hom_struct_p u Y f → f ∈ Y 0u 0) to the current goal.
Let X' and pX be given.
Apply HO Y HY (λu ⇒ Hom_struct_p (pack_p X' pX) u f → f ∈ u 0pack_p X' pX 0) to the current goal.
Let Y' and pY be given.
We will prove Hom_struct_p (pack_p X' pX) (pack_p Y' pY) f → f ∈ pack_p Y' pY 0pack_p X' pX 0.
rewrite the current goal using Hom_struct_p_pack (from left to right).
Assume Hf: f ∈ Y'X' ∧ (∀x ∈ X', pX x → pY (f x)).
Apply Hf to the current goal.
Assume Hf1 Hf2.
rewrite the current goal using pack_p_0_eq2 (from right to left).
rewrite the current goal using pack_p_0_eq2 (from right to left).
An exact proof term for the current goal is Hf1.
Let X be given.
Assume HX.
Apply HO X HX (λu ⇒ Hom_struct_p u u (lam_id (u 0))) to the current goal.
Let X' and pX be given.
We will prove Hom_struct_p (pack_p X' pX) (pack_p X' pX) (lam_id (pack_p X' pX 0)).
rewrite the current goal using pack_p_0_eq2 (from right to left).
rewrite the current goal using Hom_struct_p_pack (from left to right).
We will prove lam_id X' ∈ X'X' ∧ (∀x ∈ X', pX x → pX (lam_id X' x)).
Apply andI to the current goal.
An exact proof term for the current goal is lam_id_exp_In X'.
Let x be given.
Assume Hx Hx2.
We will prove pX ((λx ∈ X' ⇒ x) x).
rewrite the current goal using beta X' (λx ⇒ x) x Hx (from left to right).
An exact proof term for the current goal is Hx2.
Let X, Y, Z, f and g be given.
Assume HX HY HZ.
Apply HO X HX (λu ⇒ Hom_struct_p u Y f → Hom_struct_p Y Z g → Hom_struct_p u Z (lam_comp (u 0) g f)) to the current goal.
Let X' and pX be given.
We will prove Hom_struct_p (pack_p X' pX) Y f → Hom_struct_p Y Z g → Hom_struct_p (pack_p X' pX) Z (lam_comp (pack_p X' pX 0) g f).
rewrite the current goal using pack_p_0_eq2 (from right to left).
We will prove Hom_struct_p (pack_p X' pX) Y f → Hom_struct_p Y Z g → Hom_struct_p (pack_p X' pX) Z (lam_comp X' g f).
Apply HO Y HY (λu ⇒ Hom_struct_p (pack_p X' pX) u f → Hom_struct_p u Z g → Hom_struct_p (pack_p X' pX) Z (lam_comp X' g f)) to the current goal.
Let Y' and pY be given.
We will prove Hom_struct_p (pack_p X' pX) (pack_p Y' pY) f → Hom_struct_p (pack_p Y' pY) Z g → Hom_struct_p (pack_p X' pX) Z (lam_comp X' g f).
rewrite the current goal using Hom_struct_p_pack (from left to right).
Assume Hf: f ∈ Y'X' ∧ (∀x ∈ X', pX x → pY (f x)).
Apply HO Z HZ (λu ⇒ Hom_struct_p (pack_p Y' pY) u g → Hom_struct_p (pack_p X' pX) u (lam_comp X' g f)) to the current goal.
Let Z' and pZ be given.
We will prove Hom_struct_p (pack_p Y' pY) (pack_p Z' pZ) g → Hom_struct_p (pack_p X' pX) (pack_p Z' pZ) (lam_comp X' g f).
rewrite the current goal using Hom_struct_p_pack (from left to right).
rewrite the current goal using Hom_struct_p_pack (from left to right).
Assume Hg: g ∈ Z'Y' ∧ (∀y ∈ Y', pY y → pZ (g y)).
Apply Hf to the current goal.
Assume Hf1 Hf2.
Apply Hg to the current goal.
Assume Hg1 Hg2.
We will prove lam_comp X' g f ∈ Z'X' ∧ (∀x ∈ X', pX x → pZ (lam_comp X' g f x)).
Apply andI to the current goal.
An exact proof term for the current goal is lam_comp_exp_In X' Y' Z' f Hf1 g Hg1.
Let x be given.
Assume Hx Hx2.
We will prove pZ ((λx ∈ X' ⇒ g (f x)) x).
rewrite the current goal using beta X' (λx ⇒ g (f x)) x Hx (from left to right).
We will prove pZ (g (f x)).
Apply Hg2 (f x) (ap_Pi X' (λ_ ⇒ Y') f x Hf1 Hx) to the current goal.
Apply Hf2 x Hx to the current goal.
An exact proof term for the current goal is Hx2.
∎
Theorem. (MetaCat_struct_p_Forgetful_gen)
(∀X, Obj X → struct_p X) → MetaFunctor Obj Hom_struct_p (λX ⇒ lam_id (X 0)) (λX Y Z g f ⇒ lam_comp (X 0) g f) (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 59e3f3... and proposition id is 57ba03...
Proof:
Assume HO.
We prove the intermediate claim L1: ∀X Y f, Obj X → Obj Y → Hom_struct_p X Y f → f ∈ Y 0X 0.
Let X, Y and f be given.
Assume HX HY.
Apply HO X HX (λu ⇒ Hom_struct_p u Y f → f ∈ Y 0u 0) to the current goal.
Let X' and pX be given.
Apply HO Y HY (λu ⇒ Hom_struct_p (pack_p X' pX) u f → f ∈ u 0pack_p X' pX 0) to the current goal.
Let Y' and pY be given.
We will prove Hom_struct_p (pack_p X' pX) (pack_p Y' pY) f → f ∈ pack_p Y' pY 0pack_p X' pX 0.
rewrite the current goal using Hom_struct_p_pack (from left to right).
Assume Hf.
rewrite the current goal using pack_p_0_eq2 (from right to left).
rewrite the current goal using pack_p_0_eq2 (from right to left).
Apply Hf to the current goal.
Assume Hf1 _.
An exact proof term for the current goal is Hf1.
An exact proof term for the current goal is MetaCatConcreteForgetful Obj (λX ⇒ X 0) Hom_struct_p L1.
∎
Theorem. (MetaCat_struct_r_gen)
(∀X, Obj X → struct_r X) → MetaCat Obj Hom_struct_r (λX ⇒ lam_id (X 0)) (λX Y Z g f ⇒ lam_comp (X 0) g f)
In Proofgold the corresponding term root is cbfb7b... and proposition id is 626580...
Proof:
Assume HO.
Apply MetaCatConcrete Obj (λX ⇒ X 0) Hom_struct_r to the current goal.
Let X, Y and f be given.
Assume HX HY.
Apply HO X HX (λu ⇒ Hom_struct_r u Y f → f ∈ Y 0u 0) to the current goal.
Let X' and rX be given.
Apply HO Y HY (λu ⇒ Hom_struct_r (pack_r X' rX) u f → f ∈ u 0pack_r X' rX 0) to the current goal.
Let Y' and rY be given.
We will prove Hom_struct_r (pack_r X' rX) (pack_r Y' rY) f → f ∈ pack_r Y' rY 0pack_r X' rX 0.
rewrite the current goal using Hom_struct_r_pack (from left to right).
Assume Hf: f ∈ Y'X' ∧ (∀x x' ∈ X', rX x x' → rY (f x) (f x')).
Apply Hf to the current goal.
Assume Hf1 Hf2.
rewrite the current goal using pack_r_0_eq2 (from right to left).
rewrite the current goal using pack_r_0_eq2 (from right to left).
An exact proof term for the current goal is Hf1.
Let X be given.
Assume HX.
Apply HO X HX (λu ⇒ Hom_struct_r u u (lam_id (u 0))) to the current goal.
Let X' and rX be given.
We will prove Hom_struct_r (pack_r X' rX) (pack_r X' rX) (lam_id (pack_r X' rX 0)).
rewrite the current goal using pack_r_0_eq2 (from right to left).
rewrite the current goal using Hom_struct_r_pack (from left to right).
We will prove lam_id X' ∈ X'X' ∧ (∀x x' ∈ X', rX x x' → rX (lam_id X' x) (lam_id X' x')).
Apply andI to the current goal.
An exact proof term for the current goal is lam_id_exp_In X'.
Let x be given.
Assume Hx.
Let x' be given.
Assume Hx' Hx2.
We will prove rX ((λx ∈ X' ⇒ x) x) ((λx ∈ X' ⇒ x) x').
rewrite the current goal using beta X' (λx ⇒ x) x Hx (from left to right).
rewrite the current goal using beta X' (λx ⇒ x) x' Hx' (from left to right).
An exact proof term for the current goal is Hx2.
Let X, Y, Z, f and g be given.
Assume HX HY HZ.
Apply HO X HX (λu ⇒ Hom_struct_r u Y f → Hom_struct_r Y Z g → Hom_struct_r u Z (lam_comp (u 0) g f)) to the current goal.
Let X' and rX be given.
We will prove Hom_struct_r (pack_r X' rX) Y f → Hom_struct_r Y Z g → Hom_struct_r (pack_r X' rX) Z (lam_comp (pack_r X' rX 0) g f).
rewrite the current goal using pack_r_0_eq2 (from right to left).
We will prove Hom_struct_r (pack_r X' rX) Y f → Hom_struct_r Y Z g → Hom_struct_r (pack_r X' rX) Z (lam_comp X' g f).
Apply HO Y HY (λu ⇒ Hom_struct_r (pack_r X' rX) u f → Hom_struct_r u Z g → Hom_struct_r (pack_r X' rX) Z (lam_comp X' g f)) to the current goal.
Let Y' and rY be given.
We will prove Hom_struct_r (pack_r X' rX) (pack_r Y' rY) f → Hom_struct_r (pack_r Y' rY) Z g → Hom_struct_r (pack_r X' rX) Z (lam_comp X' g f).
rewrite the current goal using Hom_struct_r_pack (from left to right).
Assume Hf: f ∈ Y'X' ∧ (∀x x' ∈ X', rX x x' → rY (f x) (f x')).
Apply HO Z HZ (λu ⇒ Hom_struct_r (pack_r Y' rY) u g → Hom_struct_r (pack_r X' rX) u (lam_comp X' g f)) to the current goal.
Let Z' and rZ be given.
We will prove Hom_struct_r (pack_r Y' rY) (pack_r Z' rZ) g → Hom_struct_r (pack_r X' rX) (pack_r Z' rZ) (lam_comp X' g f).
rewrite the current goal using Hom_struct_r_pack (from left to right).
rewrite the current goal using Hom_struct_r_pack (from left to right).
Assume Hg: g ∈ Z'Y' ∧ (∀y y' ∈ Y', rY y y' → rZ (g y) (g y')).
Apply Hf to the current goal.
Assume Hf1 Hf2.
Apply Hg to the current goal.
Assume Hg1 Hg2.
We will prove lam_comp X' g f ∈ Z'X' ∧ (∀x x' ∈ X', rX x x' → rZ (lam_comp X' g f x) (lam_comp X' g f x')).
Apply andI to the current goal.
An exact proof term for the current goal is lam_comp_exp_In X' Y' Z' f Hf1 g Hg1.
Let x be given.
Assume Hx.
Let x' be given.
Assume Hx' Hx2.
We will prove rZ ((λx ∈ X' ⇒ g (f x)) x) ((λx ∈ X' ⇒ g (f x)) x').
rewrite the current goal using beta X' (λx ⇒ g (f x)) x Hx (from left to right).
rewrite the current goal using beta X' (λx ⇒ g (f x)) x' Hx' (from left to right).
We will prove rZ (g (f x)) (g (f x')).
Apply Hg2 (f x) (ap_Pi X' (λ_ ⇒ Y') f x Hf1 Hx) (f x') (ap_Pi X' (λ_ ⇒ Y') f x' Hf1 Hx') to the current goal.
Apply Hf2 x Hx x' Hx' to the current goal.
An exact proof term for the current goal is Hx2.
∎
Theorem. (MetaCat_struct_r_Forgetful_gen)
(∀X, Obj X → struct_r X) → MetaFunctor Obj Hom_struct_r (λX ⇒ lam_id (X 0)) (λX Y Z g f ⇒ lam_comp (X 0) g f) (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 8faf0f... and proposition id is 45945b...
Proof:
Assume HO.
We prove the intermediate claim L1: ∀X Y f, Obj X → Obj Y → Hom_struct_r X Y f → f ∈ Y 0X 0.
Let X, Y and f be given.
Assume HX HY.
Apply HO X HX (λu ⇒ Hom_struct_r u Y f → f ∈ Y 0u 0) to the current goal.
Let X' and rX be given.
Apply HO Y HY (λu ⇒ Hom_struct_r (pack_r X' rX) u f → f ∈ u 0pack_r X' rX 0) to the current goal.
Let Y' and rY be given.
We will prove Hom_struct_r (pack_r X' rX) (pack_r Y' rY) f → f ∈ pack_r Y' rY 0pack_r X' rX 0.
rewrite the current goal using Hom_struct_r_pack (from left to right).
Assume Hf.
rewrite the current goal using pack_r_0_eq2 (from right to left).
rewrite the current goal using pack_r_0_eq2 (from right to left).
Apply Hf to the current goal.
Assume Hf1 _.
An exact proof term for the current goal is Hf1.
An exact proof term for the current goal is MetaCatConcreteForgetful Obj (λX ⇒ X 0) Hom_struct_r L1.
∎
Theorem. (MetaCat_struct_u_gen)
(∀X, Obj X → struct_u X) → MetaCat Obj Hom_struct_u (λX ⇒ lam_id (X 0)) (λX Y Z g f ⇒ lam_comp (X 0) g f)
In Proofgold the corresponding term root is 137f84... and proposition id is 7ce95d...
Proof:
Assume HO.
Apply MetaCatConcrete Obj (λX ⇒ X 0) Hom_struct_u to the current goal.
Let X, Y and f be given.
Assume HX HY.
Apply HO X HX (λu ⇒ Hom_struct_u u Y f → f ∈ Y 0u 0) to the current goal.
Let X' and uX be given.
Assume HuX.
Apply HO Y HY (λu ⇒ Hom_struct_u (pack_u X' uX) u f → f ∈ u 0pack_u X' uX 0) to the current goal.
Let Y' and uY be given.
Assume HuY.
We will prove Hom_struct_u (pack_u X' uX) (pack_u Y' uY) f → f ∈ pack_u Y' uY 0pack_u X' uX 0.
rewrite the current goal using Hom_struct_u_pack (from left to right).
Assume Hf: f ∈ Y'X' ∧ (∀x ∈ X', f (uX x) = uY (f x)).
Apply Hf to the current goal.
Assume Hf1 Hf2.
rewrite the current goal using pack_u_0_eq2 (from right to left).
rewrite the current goal using pack_u_0_eq2 (from right to left).
An exact proof term for the current goal is Hf1.
Let X be given.
Assume HX.
Apply HO X HX (λu ⇒ Hom_struct_u u u (lam_id (u 0))) to the current goal.
Let X' and uX be given.
Assume HuX.
We will prove Hom_struct_u (pack_u X' uX) (pack_u X' uX) (lam_id (pack_u X' uX 0)).
rewrite the current goal using pack_u_0_eq2 (from right to left).
rewrite the current goal using Hom_struct_u_pack (from left to right).
We will prove lam_id X' ∈ X'X' ∧ (∀x ∈ X', lam_id X' (uX x) = uX (lam_id X' x)).
Apply andI to the current goal.
An exact proof term for the current goal is lam_id_exp_In X'.
Let x be given.
Assume Hx.
We will prove lam_id X' (uX x) = uX ((λx ∈ X' ⇒ x) x).
rewrite the current goal using beta X' (λx ⇒ x) x Hx (from left to right).
An exact proof term for the current goal is beta X' (λx ⇒ x) (uX x) (HuX x Hx).
Let X, Y, Z, f and g be given.
Assume HX HY HZ.
Apply HO X HX (λu ⇒ Hom_struct_u u Y f → Hom_struct_u Y Z g → Hom_struct_u u Z (lam_comp (u 0) g f)) to the current goal.
Let X' and uX be given.
Assume HuX.
We will prove Hom_struct_u (pack_u X' uX) Y f → Hom_struct_u Y Z g → Hom_struct_u (pack_u X' uX) Z (lam_comp (pack_u X' uX 0) g f).
rewrite the current goal using pack_u_0_eq2 (from right to left).
We will prove Hom_struct_u (pack_u X' uX) Y f → Hom_struct_u Y Z g → Hom_struct_u (pack_u X' uX) Z (lam_comp X' g f).
Apply HO Y HY (λu ⇒ Hom_struct_u (pack_u X' uX) u f → Hom_struct_u u Z g → Hom_struct_u (pack_u X' uX) Z (lam_comp X' g f)) to the current goal.
Let Y' and uY be given.
Assume HuY.
We will prove Hom_struct_u (pack_u X' uX) (pack_u Y' uY) f → Hom_struct_u (pack_u Y' uY) Z g → Hom_struct_u (pack_u X' uX) Z (lam_comp X' g f).
rewrite the current goal using Hom_struct_u_pack (from left to right).
Assume Hf: f ∈ Y'X' ∧ (∀x ∈ X', f (uX x) = uY (f x)).
Apply HO Z HZ (λu ⇒ Hom_struct_u (pack_u Y' uY) u g → Hom_struct_u (pack_u X' uX) u (lam_comp X' g f)) to the current goal.
Let Z' and uZ be given.
Assume HuZ.
We will prove Hom_struct_u (pack_u Y' uY) (pack_u Z' uZ) g → Hom_struct_u (pack_u X' uX) (pack_u Z' uZ) (lam_comp X' g f).
rewrite the current goal using Hom_struct_u_pack (from left to right).
rewrite the current goal using Hom_struct_u_pack (from left to right).
Assume Hg: g ∈ Z'Y' ∧ (∀y ∈ Y', g (uY y) = uZ (g y)).
Apply Hf to the current goal.
Assume Hf1 Hf2.
Apply Hg to the current goal.
Assume Hg1 Hg2.
We will prove lam_comp X' g f ∈ Z'X' ∧ (∀x ∈ X', lam_comp X' g f (uX x) = uZ (lam_comp X' g f x)).
Apply andI to the current goal.
An exact proof term for the current goal is lam_comp_exp_In X' Y' Z' f Hf1 g Hg1.
Let x be given.
Assume Hx.
We will prove lam_comp X' g f (uX x) = uZ ((λx ∈ X' ⇒ g (f x)) x).
rewrite the current goal using beta X' (λx ⇒ g (f x)) x Hx (from left to right).
We will prove (λx ∈ X' ⇒ g (f x)) (uX x) = uZ (g (f x)).
rewrite the current goal using beta X' (λx ⇒ g (f x)) (uX x) (HuX x Hx) (from left to right).
We will prove g (f (uX x)) = uZ (g (f x)).
rewrite the current goal using Hg2 (f x) (ap_Pi X' (λ_ ⇒ Y') f x Hf1 Hx) (from right to left).
We will prove g (f (uX x)) = g (uY (f x)).
Use f_equal.
An exact proof term for the current goal is Hf2 x Hx.
∎
Theorem. (MetaCat_struct_u_Forgetful_gen)
(∀X, Obj X → struct_u X) → MetaFunctor Obj Hom_struct_u (λX ⇒ lam_id (X 0)) (λX Y Z g f ⇒ lam_comp (X 0) g f) (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 6809a8... and proposition id is 6eadb3...
Proof:
Assume HO.
We prove the intermediate claim L1: ∀X Y f, Obj X → Obj Y → Hom_struct_u X Y f → f ∈ Y 0X 0.
Let X, Y and f be given.
Assume HX HY.
Apply HO X HX (λu ⇒ Hom_struct_u u Y f → f ∈ Y 0u 0) to the current goal.
Let X' and uX be given.
Assume HuX.
Apply HO Y HY (λu ⇒ Hom_struct_u (pack_u X' uX) u f → f ∈ u 0pack_u X' uX 0) to the current goal.
Let Y' and uY be given.
Assume HuY.
We will prove Hom_struct_u (pack_u X' uX) (pack_u Y' uY) f → f ∈ pack_u Y' uY 0pack_u X' uX 0.
rewrite the current goal using Hom_struct_u_pack (from left to right).
Assume Hf.
rewrite the current goal using pack_u_0_eq2 (from right to left).
rewrite the current goal using pack_u_0_eq2 (from right to left).
Apply Hf to the current goal.
Assume Hf1 _.
An exact proof term for the current goal is Hf1.
An exact proof term for the current goal is MetaCatConcreteForgetful Obj (λX ⇒ X 0) Hom_struct_u L1.
∎
Theorem. (MetaCat_struct_b_gen)
(∀X, Obj X → struct_b X) → MetaCat Obj Hom_struct_b (λX ⇒ lam_id (X 0)) (λX Y Z g f ⇒ lam_comp (X 0) g f)
In Proofgold the corresponding term root is be2503... and proposition id is 125f16...
Proof:
Assume HO.
Apply MetaCatConcrete Obj (λX ⇒ X 0) Hom_struct_b to the current goal.
Let X, Y and f be given.
Assume HX HY.
Apply HO X HX (λu ⇒ Hom_struct_b u Y f → f ∈ Y 0u 0) to the current goal.
Let X' and bX be given.
Assume HbX.
Apply HO Y HY (λu ⇒ Hom_struct_b (pack_b X' bX) u f → f ∈ u 0pack_b X' bX 0) to the current goal.
Let Y' and bY be given.
Assume HbY.
We will prove Hom_struct_b (pack_b X' bX) (pack_b Y' bY) f → f ∈ pack_b Y' bY 0pack_b X' bX 0.
rewrite the current goal using Hom_struct_b_pack (from left to right).
Assume Hf: f ∈ Y'X' ∧ (∀x x' ∈ X', f (bX x x') = bY (f x) (f x')).
Apply Hf to the current goal.
Assume Hf1 Hf2.
rewrite the current goal using pack_b_0_eq2 (from right to left).
rewrite the current goal using pack_b_0_eq2 (from right to left).
An exact proof term for the current goal is Hf1.
Let X be given.
Assume HX.
Apply HO X HX (λu ⇒ Hom_struct_b u u (lam_id (u 0))) to the current goal.
Let X' and bX be given.
Assume HbX.
We will prove Hom_struct_b (pack_b X' bX) (pack_b X' bX) (lam_id (pack_b X' bX 0)).
rewrite the current goal using pack_b_0_eq2 (from right to left).
rewrite the current goal using Hom_struct_b_pack (from left to right).
We will prove lam_id X' ∈ X'X' ∧ (∀x x' ∈ X', lam_id X' (bX x x') = bX (lam_id X' x) (lam_id X' x')).
Apply andI to the current goal.
An exact proof term for the current goal is lam_id_exp_In X'.
Let x be given.
Assume Hx.
Let x' be given.
Assume Hx'.
We will prove lam_id X' (bX x x') = bX ((λx ∈ X' ⇒ x) x) ((λx ∈ X' ⇒ x) x').
rewrite the current goal using beta X' (λx ⇒ x) x Hx (from left to right).
rewrite the current goal using beta X' (λx ⇒ x) x' Hx' (from left to right).
An exact proof term for the current goal is beta X' (λx ⇒ x) (bX x x') (HbX x Hx x' Hx').
Let X, Y, Z, f and g be given.
Assume HX HY HZ.
Apply HO X HX (λu ⇒ Hom_struct_b u Y f → Hom_struct_b Y Z g → Hom_struct_b u Z (lam_comp (u 0) g f)) to the current goal.
Let X' and bX be given.
Assume HbX.
We will prove Hom_struct_b (pack_b X' bX) Y f → Hom_struct_b Y Z g → Hom_struct_b (pack_b X' bX) Z (lam_comp (pack_b X' bX 0) g f).
rewrite the current goal using pack_b_0_eq2 (from right to left).
We will prove Hom_struct_b (pack_b X' bX) Y f → Hom_struct_b Y Z g → Hom_struct_b (pack_b X' bX) Z (lam_comp X' g f).
Apply HO Y HY (λu ⇒ Hom_struct_b (pack_b X' bX) u f → Hom_struct_b u Z g → Hom_struct_b (pack_b X' bX) Z (lam_comp X' g f)) to the current goal.
Let Y' and bY be given.
Assume HbY.
We will prove Hom_struct_b (pack_b X' bX) (pack_b Y' bY) f → Hom_struct_b (pack_b Y' bY) Z g → Hom_struct_b (pack_b X' bX) Z (lam_comp X' g f).
rewrite the current goal using Hom_struct_b_pack (from left to right).
Assume Hf: f ∈ Y'X' ∧ (∀x x' ∈ X', f (bX x x') = bY (f x) (f x')).
Apply HO Z HZ (λu ⇒ Hom_struct_b (pack_b Y' bY) u g → Hom_struct_b (pack_b X' bX) u (lam_comp X' g f)) to the current goal.
Let Z' and bZ be given.
Assume HbZ.
We will prove Hom_struct_b (pack_b Y' bY) (pack_b Z' bZ) g → Hom_struct_b (pack_b X' bX) (pack_b Z' bZ) (lam_comp X' g f).
rewrite the current goal using Hom_struct_b_pack (from left to right).
rewrite the current goal using Hom_struct_b_pack (from left to right).
Assume Hg: g ∈ Z'Y' ∧ (∀y y' ∈ Y', g (bY y y') = bZ (g y) (g y')).
Apply Hf to the current goal.
Assume Hf1 Hf2.
Apply Hg to the current goal.
Assume Hg1 Hg2.
We will prove lam_comp X' g f ∈ Z'X' ∧ (∀x x' ∈ X', lam_comp X' g f (bX x x') = bZ (lam_comp X' g f x) (lam_comp X' g f x')).
Apply andI to the current goal.
An exact proof term for the current goal is lam_comp_exp_In X' Y' Z' f Hf1 g Hg1.
Let x be given.
Assume Hx.
Let x' be given.
Assume Hx'.
We will prove lam_comp X' g f (bX x x') = bZ ((λx ∈ X' ⇒ g (f x)) x) ((λx ∈ X' ⇒ g (f x)) x').
rewrite the current goal using beta X' (λx ⇒ g (f x)) x Hx (from left to right).
rewrite the current goal using beta X' (λx ⇒ g (f x)) x' Hx' (from left to right).
We will prove (λx ∈ X' ⇒ g (f x)) (bX x x') = bZ (g (f x)) (g (f x')).
rewrite the current goal using beta X' (λx ⇒ g (f x)) (bX x x') (HbX x Hx x' Hx') (from left to right).
We will prove g (f (bX x x')) = bZ (g (f x)) (g (f x')).
rewrite the current goal using Hg2 (f x) (ap_Pi X' (λ_ ⇒ Y') f x Hf1 Hx) (f x') (ap_Pi X' (λ_ ⇒ Y') f x' Hf1 Hx') (from right to left).
We will prove g (f (bX x x')) = g (bY (f x) (f x')).
Use f_equal.
An exact proof term for the current goal is Hf2 x Hx x' Hx'.
∎
Theorem. (MetaCat_struct_b_Forgetful_gen)
(∀X, Obj X → struct_b X) → MetaFunctor Obj Hom_struct_b (λX ⇒ lam_id (X 0)) (λX Y Z g f ⇒ lam_comp (X 0) g f) (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 5ef6db... and proposition id is 79957c...
Proof:
Assume HO.
We prove the intermediate claim L1: ∀X Y f, Obj X → Obj Y → Hom_struct_b X Y f → f ∈ Y 0X 0.
Let X, Y and f be given.
Assume HX HY.
Apply HO X HX (λu ⇒ Hom_struct_b u Y f → f ∈ Y 0u 0) to the current goal.
Let X' and bX be given.
Assume HbX.
Apply HO Y HY (λu ⇒ Hom_struct_b (pack_b X' bX) u f → f ∈ u 0pack_b X' bX 0) to the current goal.
Let Y' and bY be given.
Assume HbY.
We will prove Hom_struct_b (pack_b X' bX) (pack_b Y' bY) f → f ∈ pack_b Y' bY 0pack_b X' bX 0.
rewrite the current goal using Hom_struct_b_pack (from left to right).
Assume Hf.
rewrite the current goal using pack_b_0_eq2 (from right to left).
rewrite the current goal using pack_b_0_eq2 (from right to left).
Apply Hf to the current goal.
Assume Hf1 _.
An exact proof term for the current goal is Hf1.
An exact proof term for the current goal is MetaCatConcreteForgetful Obj (λX ⇒ X 0) Hom_struct_b L1.
∎
Theorem. (MetaCat_struct_c_gen)
(∀X, Obj X → struct_c X) → MetaCat Obj Hom_struct_c (λX ⇒ lam_id (X 0)) (λX Y Z g f ⇒ lam_comp (X 0) g f)
In Proofgold the corresponding term root is a1cc13... and proposition id is dd75cc...
Proof:
Assume HO.
Apply MetaCatConcrete Obj (λX ⇒ X 0) Hom_struct_c to the current goal.
Let X, Y and f be given.
Assume HX HY.
Apply HO X HX (λu ⇒ Hom_struct_c u Y f → f ∈ Y 0u 0) to the current goal.
Let X' and CX be given.
Apply HO Y HY (λu ⇒ Hom_struct_c (pack_c X' CX) u f → f ∈ u 0pack_c X' CX 0) to the current goal.
Let Y' and CY be given.
We will prove Hom_struct_c (pack_c X' CX) (pack_c Y' CY) f → f ∈ pack_c Y' CY 0pack_c X' CX 0.
rewrite the current goal using Hom_struct_c_pack (from left to right).
Assume Hf: f ∈ Y'X' ∧ (∀U : set → prop, (∀y, U y → y ∈ Y') → CY U → CX (λx ⇒ x ∈ X' ∧ U (f x))).
Apply Hf to the current goal.
Assume Hf1 Hf2.
rewrite the current goal using pack_c_0_eq2 (from right to left).
rewrite the current goal using pack_c_0_eq2 (from right to left).
An exact proof term for the current goal is Hf1.
Let X be given.
Assume HX.
Apply HO X HX (λu ⇒ Hom_struct_c u u (lam_id (u 0))) to the current goal.
Let X' and CX be given.
We will prove Hom_struct_c (pack_c X' CX) (pack_c X' CX) (lam_id (pack_c X' CX 0)).
rewrite the current goal using pack_c_0_eq2 (from right to left).
rewrite the current goal using Hom_struct_c_pack (from left to right).
We will prove lam_id X' ∈ X'X' ∧ (∀U : set → prop, (∀y, U y → y ∈ X') → CX U → CX (λx ⇒ x ∈ X' ∧ U (lam_id X' x))).
Apply andI to the current goal.
An exact proof term for the current goal is lam_id_exp_In X'.
Let U be given.
Assume HU1 HU2.
We prove the intermediate claim L1: U = (λx ⇒ x ∈ X' ∧ U (lam_id X' x)).
Apply pred_ext_2 to the current goal.
Let x be given.
Assume Hx.
We will prove x ∈ X' ∧ U (lam_id X' x).
Apply andI to the current goal.
An exact proof term for the current goal is HU1 x Hx.
We will prove U ((λx ∈ X' ⇒ x) x).
rewrite the current goal using beta X' (λx ⇒ x) x (HU1 x Hx) (from left to right).
An exact proof term for the current goal is Hx.
Let x be given.
Assume Hx.
Apply Hx to the current goal.
Assume Hx1.
We will prove U ((λx ∈ X' ⇒ x) x) → U x.
rewrite the current goal using beta X' (λx ⇒ x) x Hx1 (from left to right).
Assume Hx2.
An exact proof term for the current goal is Hx2.
rewrite the current goal using L1 (from right to left).
An exact proof term for the current goal is HU2.
Let X, Y, Z, f and g be given.
Assume HX HY HZ.
Apply HO X HX (λu ⇒ Hom_struct_c u Y f → Hom_struct_c Y Z g → Hom_struct_c u Z (lam_comp (u 0) g f)) to the current goal.
Let X' and CX be given.
We will prove Hom_struct_c (pack_c X' CX) Y f → Hom_struct_c Y Z g → Hom_struct_c (pack_c X' CX) Z (lam_comp (pack_c X' CX 0) g f).
rewrite the current goal using pack_c_0_eq2 (from right to left).
We will prove Hom_struct_c (pack_c X' CX) Y f → Hom_struct_c Y Z g → Hom_struct_c (pack_c X' CX) Z (lam_comp X' g f).
Apply HO Y HY (λu ⇒ Hom_struct_c (pack_c X' CX) u f → Hom_struct_c u Z g → Hom_struct_c (pack_c X' CX) Z (lam_comp X' g f)) to the current goal.
Let Y' and CY be given.
We will prove Hom_struct_c (pack_c X' CX) (pack_c Y' CY) f → Hom_struct_c (pack_c Y' CY) Z g → Hom_struct_c (pack_c X' CX) Z (lam_comp X' g f).
rewrite the current goal using Hom_struct_c_pack (from left to right).
Assume Hf: f ∈ Y'X' ∧ (∀U : set → prop, (∀y, U y → y ∈ Y') → CY U → CX (λx ⇒ x ∈ X' ∧ U (f x))).
Apply HO Z HZ (λu ⇒ Hom_struct_c (pack_c Y' CY) u g → Hom_struct_c (pack_c X' CX) u (lam_comp X' g f)) to the current goal.
Let Z' and CZ be given.
We will prove Hom_struct_c (pack_c Y' CY) (pack_c Z' CZ) g → Hom_struct_c (pack_c X' CX) (pack_c Z' CZ) (lam_comp X' g f).
rewrite the current goal using Hom_struct_c_pack (from left to right).
rewrite the current goal using Hom_struct_c_pack (from left to right).
Assume Hg: g ∈ Z'Y' ∧ (∀U : set → prop, (∀z, U z → z ∈ Z') → CZ U → CY (λy ⇒ y ∈ Y' ∧ U (g y))).
Apply Hf to the current goal.
Assume Hf1 Hf2.
Apply Hg to the current goal.
Assume Hg1 Hg2.
We will prove lam_comp X' g f ∈ Z'X' ∧ (∀U : set → prop, (∀z, U z → z ∈ Z') → CZ U → CX (λx ⇒ x ∈ X' ∧ U (lam_comp X' g f x))).
Apply andI to the current goal.
An exact proof term for the current goal is lam_comp_exp_In X' Y' Z' f Hf1 g Hg1.
Let U be given.
Assume HU1: ∀z, U z → z ∈ Z'.
Assume HU2: CZ U.
We prove the intermediate claim L2: CY (λy ⇒ y ∈ Y' ∧ U (g y)).
An exact proof term for the current goal is Hg2 U HU1 HU2.
We prove the intermediate claim L3: ∀y, y ∈ Y' ∧ U (g y) → y ∈ Y'.
Let y be given.
An exact proof term for the current goal is andEL (y ∈ Y') (U (g y)).
We prove the intermediate claim L4: CX (λx ⇒ x ∈ X' ∧ (f x ∈ Y' ∧ U (g (f x)))).
An exact proof term for the current goal is Hf2 (λy ⇒ y ∈ Y' ∧ U (g y)) L3 L2.
We prove the intermediate claim L5: (λx : set ⇒ x ∈ X' ∧ (f x ∈ Y' ∧ U (g (f x)))) = (λx : set ⇒ x ∈ X' ∧ U (lam_comp X' g f x)).
Apply pred_ext_2 to the current goal.
Let x be given.
Assume Hx.
Apply Hx to the current goal.
Assume Hx1 Hx23.
Apply Hx23 to the current goal.
Assume Hx2 Hx3.
We will prove x ∈ X' ∧ U (lam_comp X' g f x).
Apply andI to the current goal.
An exact proof term for the current goal is Hx1.
We will prove U ((λx ∈ X' ⇒ g (f x)) x).
rewrite the current goal using beta X' (λx ⇒ g (f x)) x Hx1 (from left to right).
An exact proof term for the current goal is Hx3.
Let x be given.
Assume Hx.
Apply Hx to the current goal.
Assume Hx1.
We will prove U ((λx ∈ X' ⇒ g (f x)) x) → x ∈ X' ∧ (f x ∈ Y' ∧ U (g (f x))).
rewrite the current goal using beta X' (λx ⇒ g (f x)) x Hx1 (from left to right).
Assume Hx2.
We will prove x ∈ X' ∧ (f x ∈ Y' ∧ U (g (f x))).
Apply andI to the current goal.
An exact proof term for the current goal is Hx1.
Apply andI to the current goal.
An exact proof term for the current goal is ap_Pi X' (λ_ ⇒ Y') f x Hf1 Hx1.
An exact proof term for the current goal is Hx2.
rewrite the current goal using L5 (from right to left).
An exact proof term for the current goal is L4.
∎
Theorem. (MetaCat_struct_c_Forgetful_gen)
(∀X, Obj X → struct_c X) → MetaFunctor Obj Hom_struct_c (λX ⇒ lam_id (X 0)) (λX Y Z g f ⇒ lam_comp (X 0) g f) (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 c43bbf... and proposition id is 58a40d...
Proof:
Assume HO.
We prove the intermediate claim L1: ∀X Y f, Obj X → Obj Y → Hom_struct_c X Y f → f ∈ Y 0X 0.
Let X, Y and f be given.
Assume HX HY.
Apply HO X HX (λu ⇒ Hom_struct_c u Y f → f ∈ Y 0u 0) to the current goal.
Let X' and CX be given.
Apply HO Y HY (λu ⇒ Hom_struct_c (pack_c X' CX) u f → f ∈ u 0pack_c X' CX 0) to the current goal.
Let Y' and CY be given.
We will prove Hom_struct_c (pack_c X' CX) (pack_c Y' CY) f → f ∈ pack_c Y' CY 0pack_c X' CX 0.
rewrite the current goal using Hom_struct_c_pack (from left to right).
Assume Hf.
rewrite the current goal using pack_c_0_eq2 (from right to left).
rewrite the current goal using pack_c_0_eq2 (from right to left).
Apply Hf to the current goal.
Assume Hf1 _.
An exact proof term for the current goal is Hf1.
An exact proof term for the current goal is MetaCatConcreteForgetful Obj (λX ⇒ X 0) Hom_struct_c L1.
∎
Theorem. (MetaCat_struct_b_b_e_gen)
(∀X, Obj X → struct_b_b_e X) → MetaCat Obj Hom_struct_b_b_e (λX ⇒ lam_id (X 0)) (λX Y Z g f ⇒ lam_comp (X 0) g f)
In Proofgold the corresponding term root is c7aff8... and proposition id is 5f5c51...
Proof:
Assume HO.
Apply MetaCatConcrete Obj (λX ⇒ X 0) Hom_struct_b_b_e to the current goal.
Let X, Y and f be given.
Assume HX HY.
Apply HO X HX (λu ⇒ Hom_struct_b_b_e u Y f → f ∈ Y 0u 0) to the current goal.
Let X' and bX be given.
Assume HbX.
Let b2X be given.
Assume Hb2X.
Let eX be given.
Assume HeX.
Apply HO Y HY (λu ⇒ Hom_struct_b_b_e (pack_b_b_e X' bX b2X eX) u f → f ∈ u 0pack_b_b_e X' bX b2X eX 0) to the current goal.
Let Y' and bY be given.
Assume HbY.
Let b2Y be given.
Assume Hb2Y.
Let eY be given.
Assume HeY.
We will prove Hom_struct_b_b_e (pack_b_b_e X' bX b2X eX) (pack_b_b_e Y' bY b2Y eY) f → f ∈ pack_b_b_e Y' bY b2Y eY 0pack_b_b_e X' bX b2X eX 0.
rewrite the current goal using Hom_struct_b_b_e_pack (from left to right).
Assume Hf: f ∈ Y'X' ∧ (∀x x' ∈ X', f (bX x x') = bY (f x) (f x')) ∧ (∀x x' ∈ X', f (b2X x x') = b2Y (f x) (f x')) ∧ f eX = eY.
Apply and4E (f ∈ Y'X') (∀x x' ∈ X', f (bX x x') = bY (f x) (f x')) (∀x x' ∈ X', f (b2X x x') = b2Y (f x) (f x')) (f eX = eY) Hf to the current goal.
Assume Hf1: f ∈ Y'X'.
Assume Hf2: ∀x x' ∈ X', f (bX x x') = bY (f x) (f x').
Assume Hf3: ∀x x' ∈ X', f (b2X x x') = b2Y (f x) (f x').
Assume Hf4: f eX = eY.
We will prove f ∈ pack_b_b_e Y' bY b2Y eY 0pack_b_b_e X' bX b2X eX 0.
rewrite the current goal using pack_b_b_e_0_eq2 (from right to left).
rewrite the current goal using pack_b_b_e_0_eq2 (from right to left).
We will prove f ∈ Y'X'.
An exact proof term for the current goal is Hf1.
Let X be given.
Assume HX.
Apply HO X HX (λu ⇒ Hom_struct_b_b_e u u (lam_id (u 0))) to the current goal.
Let X' and bX be given.
Assume HbX.
Let b2X be given.
Assume Hb2X.
Let eX be given.
Assume HeX.
We will prove Hom_struct_b_b_e (pack_b_b_e X' bX b2X eX) (pack_b_b_e X' bX b2X eX) (lam_id (pack_b_b_e X' bX b2X eX 0)).
rewrite the current goal using pack_b_b_e_0_eq2 (from right to left).
rewrite the current goal using Hom_struct_b_b_e_pack (from left to right).
We will prove lam_id X' ∈ X'X' ∧ (∀x x' ∈ X', lam_id X' (bX x x') = bX (lam_id X' x) (lam_id X' x')) ∧ (∀x x' ∈ X', lam_id X' (b2X x x') = b2X (lam_id X' x) (lam_id X' x')) ∧ lam_id X' eX = eX.
Apply and4I to the current goal.
An exact proof term for the current goal is lam_id_exp_In X'.
Let x be given.
Assume Hx.
Let x' be given.
Assume Hx'.
We will prove lam_id X' (bX x x') = bX ((λx ∈ X' ⇒ x) x) ((λx ∈ X' ⇒ x) x').
rewrite the current goal using beta X' (λx ⇒ x) x Hx (from left to right).
rewrite the current goal using beta X' (λx ⇒ x) x' Hx' (from left to right).
An exact proof term for the current goal is beta X' (λx ⇒ x) (bX x x') (HbX x Hx x' Hx').
Let x be given.
Assume Hx.
Let x' be given.
Assume Hx'.
We will prove lam_id X' (b2X x x') = b2X ((λx ∈ X' ⇒ x) x) ((λx ∈ X' ⇒ x) x').
rewrite the current goal using beta X' (λx ⇒ x) x Hx (from left to right).
rewrite the current goal using beta X' (λx ⇒ x) x' Hx' (from left to right).
An exact proof term for the current goal is beta X' (λx ⇒ x) (b2X x x') (Hb2X x Hx x' Hx').
We will prove (λx ∈ X' ⇒ x) eX = eX.
An exact proof term for the current goal is beta X' (λx ⇒ x) eX HeX.
Let X, Y, Z, f and g be given.
Assume HX HY HZ.
Apply HO X HX (λu ⇒ Hom_struct_b_b_e u Y f → Hom_struct_b_b_e Y Z g → Hom_struct_b_b_e u Z (lam_comp (u 0) g f)) to the current goal.
Let X' and bX be given.
Assume HbX.
Let b2X be given.
Assume Hb2X.
Let eX be given.
Assume HeX.
We will prove Hom_struct_b_b_e (pack_b_b_e X' bX b2X eX) Y f → Hom_struct_b_b_e Y Z g → Hom_struct_b_b_e (pack_b_b_e X' bX b2X eX) Z (lam_comp (pack_b_b_e X' bX b2X eX 0) g f).
rewrite the current goal using pack_b_b_e_0_eq2 (from right to left).
We will prove Hom_struct_b_b_e (pack_b_b_e X' bX b2X eX) Y f → Hom_struct_b_b_e Y Z g → Hom_struct_b_b_e (pack_b_b_e X' bX b2X eX) Z (lam_comp X' g f).
Apply HO Y HY (λu ⇒ Hom_struct_b_b_e (pack_b_b_e X' bX b2X eX) u f → Hom_struct_b_b_e u Z g → Hom_struct_b_b_e (pack_b_b_e X' bX b2X eX) Z (lam_comp X' g f)) to the current goal.
Let Y' and bY be given.
Assume HbY.
Let b2Y be given.
Assume Hb2Y.
Let eY be given.
Assume HeY.
We will prove Hom_struct_b_b_e (pack_b_b_e X' bX b2X eX) (pack_b_b_e Y' bY b2Y eY) f → Hom_struct_b_b_e (pack_b_b_e Y' bY b2Y eY) Z g → Hom_struct_b_b_e (pack_b_b_e X' bX b2X eX) Z (lam_comp X' g f).
rewrite the current goal using Hom_struct_b_b_e_pack (from left to right).
Assume Hf: f ∈ Y'X' ∧ (∀x x' ∈ X', f (bX x x') = bY (f x) (f x')) ∧ (∀x x' ∈ X', f (b2X x x') = b2Y (f x) (f x')) ∧ f eX = eY.
Apply HO Z HZ (λu ⇒ Hom_struct_b_b_e (pack_b_b_e Y' bY b2Y eY) u g → Hom_struct_b_b_e (pack_b_b_e X' bX b2X eX) u (lam_comp X' g f)) to the current goal.
Let Z' and bZ be given.
Assume HbZ.
Let b2Z be given.
Assume Hb2Z.
Let eZ be given.
Assume HeZ.
We will prove Hom_struct_b_b_e (pack_b_b_e Y' bY b2Y eY) (pack_b_b_e Z' bZ b2Z eZ) g → Hom_struct_b_b_e (pack_b_b_e X' bX b2X eX) (pack_b_b_e Z' bZ b2Z eZ) (lam_comp X' g f).
rewrite the current goal using Hom_struct_b_b_e_pack (from left to right).
rewrite the current goal using Hom_struct_b_b_e_pack (from left to right).
Assume Hg: g ∈ Z'Y' ∧ (∀y y' ∈ Y', g (bY y y') = bZ (g y) (g y')) ∧ (∀y y' ∈ Y', g (b2Y y y') = b2Z (g y) (g y')) ∧ g eY = eZ.
Apply and4E (f ∈ Y'X') (∀x x' ∈ X', f (bX x x') = bY (f x) (f x')) (∀x x' ∈ X', f (b2X x x') = b2Y (f x) (f x')) (f eX = eY) Hf to the current goal.
Assume Hf1: f ∈ Y'X'.
Assume Hf2: ∀x x' ∈ X', f (bX x x') = bY (f x) (f x').
Assume Hf3: ∀x x' ∈ X', f (b2X x x') = b2Y (f x) (f x').
Assume Hf4: f eX = eY.
Apply and4E (g ∈ Z'Y') (∀y y' ∈ Y', g (bY y y') = bZ (g y) (g y')) (∀y y' ∈ Y', g (b2Y y y') = b2Z (g y) (g y')) (g eY = eZ) Hg to the current goal.
Assume Hg1: g ∈ Z'Y'.
Assume Hg2: ∀y y' ∈ Y', g (bY y y') = bZ (g y) (g y').
Assume Hg3: ∀y y' ∈ Y', g (b2Y y y') = b2Z (g y) (g y').
Assume Hg4: g eY = eZ.
We will prove lam_comp X' g f ∈ Z'X' ∧ (∀x x' ∈ X', lam_comp X' g f (bX x x') = bZ (lam_comp X' g f x) (lam_comp X' g f x')) ∧ (∀x x' ∈ X', lam_comp X' g f (b2X x x') = b2Z (lam_comp X' g f x) (lam_comp X' g f x')) ∧ lam_comp X' g f eX = eZ.
Apply and4I to the current goal.
An exact proof term for the current goal is lam_comp_exp_In X' Y' Z' f Hf1 g Hg1.
Let x be given.
Assume Hx.
Let x' be given.
Assume Hx'.
We will prove lam_comp X' g f (bX x x') = bZ ((λx ∈ X' ⇒ g (f x)) x) ((λx ∈ X' ⇒ g (f x)) x').
rewrite the current goal using beta X' (λx ⇒ g (f x)) x Hx (from left to right).
rewrite the current goal using beta X' (λx ⇒ g (f x)) x' Hx' (from left to right).
We will prove (λx ∈ X' ⇒ g (f x)) (bX x x') = bZ (g (f x)) (g (f x')).
rewrite the current goal using beta X' (λx ⇒ g (f x)) (bX x x') (HbX x Hx x' Hx') (from left to right).
We will prove g (f (bX x x')) = bZ (g (f x)) (g (f x')).
rewrite the current goal using Hg2 (f x) (ap_Pi X' (λ_ ⇒ Y') f x Hf1 Hx) (f x') (ap_Pi X' (λ_ ⇒ Y') f x' Hf1 Hx') (from right to left).
We will prove g (f (bX x x')) = g (bY (f x) (f x')).
Use f_equal.
An exact proof term for the current goal is Hf2 x Hx x' Hx'.
Let x be given.
Assume Hx.
Let x' be given.
Assume Hx'.
We will prove lam_comp X' g f (b2X x x') = b2Z ((λx ∈ X' ⇒ g (f x)) x) ((λx ∈ X' ⇒ g (f x)) x').
rewrite the current goal using beta X' (λx ⇒ g (f x)) x Hx (from left to right).
rewrite the current goal using beta X' (λx ⇒ g (f x)) x' Hx' (from left to right).
We will prove (λx ∈ X' ⇒ g (f x)) (b2X x x') = b2Z (g (f x)) (g (f x')).
rewrite the current goal using beta X' (λx ⇒ g (f x)) (b2X x x') (Hb2X x Hx x' Hx') (from left to right).
We will prove g (f (b2X x x')) = b2Z (g (f x)) (g (f x')).
rewrite the current goal using Hg3 (f x) (ap_Pi X' (λ_ ⇒ Y') f x Hf1 Hx) (f x') (ap_Pi X' (λ_ ⇒ Y') f x' Hf1 Hx') (from right to left).
We will prove g (f (b2X x x')) = g (b2Y (f x) (f x')).
Use f_equal.
An exact proof term for the current goal is Hf3 x Hx x' Hx'.
We will prove (λx ∈ X' ⇒ g (f x)) eX = eZ.
rewrite the current goal using beta X' (λx ⇒ g (f x)) eX HeX (from left to right).
We will prove g (f eX) = eZ.
rewrite the current goal using Hf4 (from left to right).
An exact proof term for the current goal is Hg4.
∎
Theorem. (MetaCat_struct_b_b_e_Forgetful_gen)
(∀X, Obj X → struct_b_b_e X) → MetaFunctor Obj Hom_struct_b_b_e (λX ⇒ lam_id (X 0)) (λX Y Z g f ⇒ lam_comp (X 0) g f) (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 f8aa78... and proposition id is 9c6b91...
Proof:
Assume HO.
We prove the intermediate claim L1: ∀X Y f, Obj X → Obj Y → Hom_struct_b_b_e X Y f → f ∈ Y 0X 0.
Let X, Y and f be given.
Assume HX HY.
Apply HO X HX (λu ⇒ Hom_struct_b_b_e u Y f → f ∈ Y 0u 0) to the current goal.
Let X' and bX be given.
Assume HbX.
Let b2X be given.
Assume Hb2X.
Let eX be given.
Assume HeX.
Apply HO Y HY (λu ⇒ Hom_struct_b_b_e (pack_b_b_e X' bX b2X eX) u f → f ∈ u 0pack_b_b_e X' bX b2X eX 0) to the current goal.
Let Y' and bY be given.
Assume HbY.
Let b2Y be given.
Assume Hb2Y.
Let eY be given.
Assume HeY.
We will prove Hom_struct_b_b_e (pack_b_b_e X' bX b2X eX) (pack_b_b_e Y' bY b2Y eY) f → f ∈ pack_b_b_e Y' bY b2Y eY 0pack_b_b_e X' bX b2X eX 0.
rewrite the current goal using Hom_struct_b_b_e_pack (from left to right).
Assume Hf.
rewrite the current goal using pack_b_b_e_0_eq2 (from right to left).
rewrite the current goal using pack_b_b_e_0_eq2 (from right to left).
Apply Hf to the current goal.
Assume Hf _.
Apply Hf to the current goal.
Assume Hf _.
Apply Hf to the current goal.
Assume Hf1 _.
An exact proof term for the current goal is Hf1.
An exact proof term for the current goal is MetaCatConcreteForgetful Obj (λX ⇒ X 0) Hom_struct_b_b_e L1.
∎
Theorem. (MetaCat_struct_b_b_e_e_gen)
(∀X, Obj X → struct_b_b_e_e X) → MetaCat Obj Hom_struct_b_b_e_e (λX ⇒ lam_id (X 0)) (λX Y Z g f ⇒ lam_comp (X 0) g f)
In Proofgold the corresponding term root is d57683... and proposition id is 936d91...
Proof:
Assume HO.
Apply MetaCatConcrete Obj (λX ⇒ X 0) Hom_struct_b_b_e_e to the current goal.
Let X, Y and f be given.
Assume HX HY.
Apply HO X HX (λu ⇒ Hom_struct_b_b_e_e u Y f → f ∈ Y 0u 0) to the current goal.
Let X' and bX be given.
Assume HbX.
Let b2X be given.
Assume Hb2X.
Let eX be given.
Assume HeX.
Let e2X be given.
Assume He2X.
Apply HO Y HY (λu ⇒ Hom_struct_b_b_e_e (pack_b_b_e_e X' bX b2X eX e2X) u f → f ∈ u 0pack_b_b_e_e X' bX b2X eX e2X 0) to the current goal.
Let Y' and bY be given.
Assume HbY.
Let b2Y be given.
Assume Hb2Y.
Let eY be given.
Assume HeY.
Let e2Y be given.
Assume He2Y.
We will prove Hom_struct_b_b_e_e (pack_b_b_e_e X' bX b2X eX e2X) (pack_b_b_e_e Y' bY b2Y eY e2Y) f → f ∈ pack_b_b_e_e Y' bY b2Y eY e2Y 0pack_b_b_e_e X' bX b2X eX e2X 0.
rewrite the current goal using Hom_struct_b_b_e_e_pack (from left to right).
Assume Hf: f ∈ Y'X' ∧ (∀x x' ∈ X', f (bX x x') = bY (f x) (f x')) ∧ (∀x x' ∈ X', f (b2X x x') = b2Y (f x) (f x')) ∧ f eX = eY ∧ f e2X = e2Y.
Apply and5E (f ∈ Y'X') (∀x x' ∈ X', f (bX x x') = bY (f x) (f x')) (∀x x' ∈ X', f (b2X x x') = b2Y (f x) (f x')) (f eX = eY) (f e2X = e2Y) Hf to the current goal.
Assume Hf1: f ∈ Y'X'.
Assume Hf2: ∀x x' ∈ X', f (bX x x') = bY (f x) (f x').
Assume Hf3: ∀x x' ∈ X', f (b2X x x') = b2Y (f x) (f x').
Assume Hf4: f eX = eY.
Assume Hf5: f e2X = e2Y.
We will prove f ∈ pack_b_b_e_e Y' bY b2Y eY e2Y 0pack_b_b_e_e X' bX b2X eX e2X 0.
rewrite the current goal using pack_b_b_e_e_0_eq2 (from right to left).
rewrite the current goal using pack_b_b_e_e_0_eq2 (from right to left).
We will prove f ∈ Y'X'.
An exact proof term for the current goal is Hf1.
Let X be given.
Assume HX.
Apply HO X HX (λu ⇒ Hom_struct_b_b_e_e u u (lam_id (u 0))) to the current goal.
Let X' and bX be given.
Assume HbX.
Let b2X be given.
Assume Hb2X.
Let eX be given.
Assume HeX.
Let e2X be given.
Assume He2X.
We will prove Hom_struct_b_b_e_e (pack_b_b_e_e X' bX b2X eX e2X) (pack_b_b_e_e X' bX b2X eX e2X) (lam_id (pack_b_b_e_e X' bX b2X eX e2X 0)).
rewrite the current goal using pack_b_b_e_e_0_eq2 (from right to left).
rewrite the current goal using Hom_struct_b_b_e_e_pack (from left to right).
We will prove lam_id X' ∈ X'X' ∧ (∀x x' ∈ X', lam_id X' (bX x x') = bX (lam_id X' x) (lam_id X' x')) ∧ (∀x x' ∈ X', lam_id X' (b2X x x') = b2X (lam_id X' x) (lam_id X' x')) ∧ lam_id X' eX = eX ∧ lam_id X' e2X = e2X.
Apply and5I to the current goal.
An exact proof term for the current goal is lam_id_exp_In X'.
Let x be given.
Assume Hx.
Let x' be given.
Assume Hx'.
We will prove lam_id X' (bX x x') = bX ((λx ∈ X' ⇒ x) x) ((λx ∈ X' ⇒ x) x').
rewrite the current goal using beta X' (λx ⇒ x) x Hx (from left to right).
rewrite the current goal using beta X' (λx ⇒ x) x' Hx' (from left to right).
An exact proof term for the current goal is beta X' (λx ⇒ x) (bX x x') (HbX x Hx x' Hx').
Let x be given.
Assume Hx.
Let x' be given.
Assume Hx'.
We will prove lam_id X' (b2X x x') = b2X ((λx ∈ X' ⇒ x) x) ((λx ∈ X' ⇒ x) x').
rewrite the current goal using beta X' (λx ⇒ x) x Hx (from left to right).
rewrite the current goal using beta X' (λx ⇒ x) x' Hx' (from left to right).
An exact proof term for the current goal is beta X' (λx ⇒ x) (b2X x x') (Hb2X x Hx x' Hx').
We will prove (λx ∈ X' ⇒ x) eX = eX.
An exact proof term for the current goal is beta X' (λx ⇒ x) eX HeX.
We will prove (λx ∈ X' ⇒ x) e2X = e2X.
An exact proof term for the current goal is beta X' (λx ⇒ x) e2X He2X.
Let X, Y, Z, f and g be given.
Assume HX HY HZ.
Apply HO X HX (λu ⇒ Hom_struct_b_b_e_e u Y f → Hom_struct_b_b_e_e Y Z g → Hom_struct_b_b_e_e u Z (lam_comp (u 0) g f)) to the current goal.
Let X' and bX be given.
Assume HbX.
Let b2X be given.
Assume Hb2X.
Let eX be given.
Assume HeX.
Let e2X be given.
Assume He2X.
We will prove Hom_struct_b_b_e_e (pack_b_b_e_e X' bX b2X eX e2X) Y f → Hom_struct_b_b_e_e Y Z g → Hom_struct_b_b_e_e (pack_b_b_e_e X' bX b2X eX e2X) Z (lam_comp (pack_b_b_e_e X' bX b2X eX e2X 0) g f).
rewrite the current goal using pack_b_b_e_e_0_eq2 (from right to left).
We will prove Hom_struct_b_b_e_e (pack_b_b_e_e X' bX b2X eX e2X) Y f → Hom_struct_b_b_e_e Y Z g → Hom_struct_b_b_e_e (pack_b_b_e_e X' bX b2X eX e2X) Z (lam_comp X' g f).
Apply HO Y HY (λu ⇒ Hom_struct_b_b_e_e (pack_b_b_e_e X' bX b2X eX e2X) u f → Hom_struct_b_b_e_e u Z g → Hom_struct_b_b_e_e (pack_b_b_e_e X' bX b2X eX e2X) Z (lam_comp X' g f)) to the current goal.
Let Y' and bY be given.
Assume HbY.
Let b2Y be given.
Assume Hb2Y.
Let eY be given.
Assume HeY.
Let e2Y be given.
Assume He2Y.
We will prove Hom_struct_b_b_e_e (pack_b_b_e_e X' bX b2X eX e2X) (pack_b_b_e_e Y' bY b2Y eY e2Y) f → Hom_struct_b_b_e_e (pack_b_b_e_e Y' bY b2Y eY e2Y) Z g → Hom_struct_b_b_e_e (pack_b_b_e_e X' bX b2X eX e2X) Z (lam_comp X' g f).
rewrite the current goal using Hom_struct_b_b_e_e_pack (from left to right).
Assume Hf: f ∈ Y'X' ∧ (∀x x' ∈ X', f (bX x x') = bY (f x) (f x')) ∧ (∀x x' ∈ X', f (b2X x x') = b2Y (f x) (f x')) ∧ f eX = eY ∧ f e2X = e2Y.
Apply HO Z HZ (λu ⇒ Hom_struct_b_b_e_e (pack_b_b_e_e Y' bY b2Y eY e2Y) u g → Hom_struct_b_b_e_e (pack_b_b_e_e X' bX b2X eX e2X) u (lam_comp X' g f)) to the current goal.
Let Z' and bZ be given.
Assume HbZ.
Let b2Z be given.
Assume Hb2Z.
Let eZ be given.
Assume HeZ.
Let e2Z be given.
Assume He2Z.
We will prove Hom_struct_b_b_e_e (pack_b_b_e_e Y' bY b2Y eY e2Y) (pack_b_b_e_e Z' bZ b2Z eZ e2Z) g → Hom_struct_b_b_e_e (pack_b_b_e_e X' bX b2X eX e2X) (pack_b_b_e_e Z' bZ b2Z eZ e2Z) (lam_comp X' g f).
rewrite the current goal using Hom_struct_b_b_e_e_pack (from left to right).
rewrite the current goal using Hom_struct_b_b_e_e_pack (from left to right).
Assume Hg: g ∈ Z'Y' ∧ (∀y y' ∈ Y', g (bY y y') = bZ (g y) (g y')) ∧ (∀y y' ∈ Y', g (b2Y y y') = b2Z (g y) (g y')) ∧ g eY = eZ ∧ g e2Y = e2Z.
Apply and5E (f ∈ Y'X') (∀x x' ∈ X', f (bX x x') = bY (f x) (f x')) (∀x x' ∈ X', f (b2X x x') = b2Y (f x) (f x')) (f eX = eY) (f e2X = e2Y) Hf to the current goal.
Assume Hf1: f ∈ Y'X'.
Assume Hf2: ∀x x' ∈ X', f (bX x x') = bY (f x) (f x').
Assume Hf3: ∀x x' ∈ X', f (b2X x x') = b2Y (f x) (f x').
Assume Hf4: f eX = eY.
Assume Hf5: f e2X = e2Y.
Apply and5E (g ∈ Z'Y') (∀y y' ∈ Y', g (bY y y') = bZ (g y) (g y')) (∀y y' ∈ Y', g (b2Y y y') = b2Z (g y) (g y')) (g eY = eZ) (g e2Y = e2Z) Hg to the current goal.
Assume Hg1: g ∈ Z'Y'.
Assume Hg2: ∀y y' ∈ Y', g (bY y y') = bZ (g y) (g y').
Assume Hg3: ∀y y' ∈ Y', g (b2Y y y') = b2Z (g y) (g y').
Assume Hg4: g eY = eZ.
Assume Hg5: g e2Y = e2Z.
We will prove lam_comp X' g f ∈ Z'X' ∧ (∀x x' ∈ X', lam_comp X' g f (bX x x') = bZ (lam_comp X' g f x) (lam_comp X' g f x')) ∧ (∀x x' ∈ X', lam_comp X' g f (b2X x x') = b2Z (lam_comp X' g f x) (lam_comp X' g f x')) ∧ lam_comp X' g f eX = eZ ∧ lam_comp X' g f e2X = e2Z.
Apply and5I to the current goal.
An exact proof term for the current goal is lam_comp_exp_In X' Y' Z' f Hf1 g Hg1.
Let x be given.
Assume Hx.
Let x' be given.
Assume Hx'.
We will prove lam_comp X' g f (bX x x') = bZ ((λx ∈ X' ⇒ g (f x)) x) ((λx ∈ X' ⇒ g (f x)) x').
rewrite the current goal using beta X' (λx ⇒ g (f x)) x Hx (from left to right).
rewrite the current goal using beta X' (λx ⇒ g (f x)) x' Hx' (from left to right).
We will prove (λx ∈ X' ⇒ g (f x)) (bX x x') = bZ (g (f x)) (g (f x')).
rewrite the current goal using beta X' (λx ⇒ g (f x)) (bX x x') (HbX x Hx x' Hx') (from left to right).
We will prove g (f (bX x x')) = bZ (g (f x)) (g (f x')).
rewrite the current goal using Hg2 (f x) (ap_Pi X' (λ_ ⇒ Y') f x Hf1 Hx) (f x') (ap_Pi X' (λ_ ⇒ Y') f x' Hf1 Hx') (from right to left).
We will prove g (f (bX x x')) = g (bY (f x) (f x')).
Use f_equal.
An exact proof term for the current goal is Hf2 x Hx x' Hx'.
Let x be given.
Assume Hx.
Let x' be given.
Assume Hx'.
We will prove lam_comp X' g f (b2X x x') = b2Z ((λx ∈ X' ⇒ g (f x)) x) ((λx ∈ X' ⇒ g (f x)) x').
rewrite the current goal using beta X' (λx ⇒ g (f x)) x Hx (from left to right).
rewrite the current goal using beta X' (λx ⇒ g (f x)) x' Hx' (from left to right).
We will prove (λx ∈ X' ⇒ g (f x)) (b2X x x') = b2Z (g (f x)) (g (f x')).
rewrite the current goal using beta X' (λx ⇒ g (f x)) (b2X x x') (Hb2X x Hx x' Hx') (from left to right).
We will prove g (f (b2X x x')) = b2Z (g (f x)) (g (f x')).
rewrite the current goal using Hg3 (f x) (ap_Pi X' (λ_ ⇒ Y') f x Hf1 Hx) (f x') (ap_Pi X' (λ_ ⇒ Y') f x' Hf1 Hx') (from right to left).
We will prove g (f (b2X x x')) = g (b2Y (f x) (f x')).
Use f_equal.
An exact proof term for the current goal is Hf3 x Hx x' Hx'.
We will prove (λx ∈ X' ⇒ g (f x)) eX = eZ.
rewrite the current goal using beta X' (λx ⇒ g (f x)) eX HeX (from left to right).
We will prove g (f eX) = eZ.
rewrite the current goal using Hf4 (from left to right).
An exact proof term for the current goal is Hg4.
We will prove (λx ∈ X' ⇒ g (f x)) e2X = e2Z.
rewrite the current goal using beta X' (λx ⇒ g (f x)) e2X He2X (from left to right).
We will prove g (f e2X) = e2Z.
rewrite the current goal using Hf5 (from left to right).
An exact proof term for the current goal is Hg5.
∎
Theorem. (MetaCat_struct_b_b_e_e_Forgetful_gen)
(∀X, Obj X → struct_b_b_e_e X) → MetaFunctor Obj Hom_struct_b_b_e_e (λX ⇒ lam_id (X 0)) (λX Y Z g f ⇒ lam_comp (X 0) g f) (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 45b449... and proposition id is 72690d...
Proof:
Assume HO.
We prove the intermediate claim L1: ∀X Y f, Obj X → Obj Y → Hom_struct_b_b_e_e X Y f → f ∈ Y 0X 0.
Let X, Y and f be given.
Assume HX HY.
Apply HO X HX (λu ⇒ Hom_struct_b_b_e_e u Y f → f ∈ Y 0u 0) to the current goal.
Let X' and bX be given.
Assume HbX.
Let b2X be given.
Assume Hb2X.
Let eX be given.
Assume HeX.
Let e2X be given.
Assume He2X.
Apply HO Y HY (λu ⇒ Hom_struct_b_b_e_e (pack_b_b_e_e X' bX b2X eX e2X) u f → f ∈ u 0pack_b_b_e_e X' bX b2X eX e2X 0) to the current goal.
Let Y' and bY be given.
Assume HbY.
Let b2Y be given.
Assume Hb2Y.
Let eY be given.
Assume HeY.
Let e2Y be given.
Assume He2Y.
We will prove Hom_struct_b_b_e_e (pack_b_b_e_e X' bX b2X eX e2X) (pack_b_b_e_e Y' bY b2Y eY e2Y) f → f ∈ pack_b_b_e_e Y' bY b2Y eY e2Y 0pack_b_b_e_e X' bX b2X eX e2X 0.
rewrite the current goal using Hom_struct_b_b_e_e_pack (from left to right).
Assume Hf.
rewrite the current goal using pack_b_b_e_e_0_eq2 (from right to left).
rewrite the current goal using pack_b_b_e_e_0_eq2 (from right to left).
Apply Hf to the current goal.
Assume Hf _.
Apply Hf to the current goal.
Assume Hf _.
Apply Hf to the current goal.
Assume Hf _.
Apply Hf to the current goal.
Assume Hf1 _.
An exact proof term for the current goal is Hf1.
An exact proof term for the current goal is MetaCatConcreteForgetful Obj (λX ⇒ X 0) Hom_struct_b_b_e_e L1.
∎
Theorem. (MetaCat_struct_b_b_r_e_e_gen)
(∀X, Obj X → struct_b_b_r_e_e X) → MetaCat Obj Hom_struct_b_b_r_e_e (λX ⇒ lam_id (X 0)) (λX Y Z g f ⇒ lam_comp (X 0) g f)
In Proofgold the corresponding term root is d69e51... and proposition id is dc6cf0...
Proof:
Assume HO.
Apply MetaCatConcrete Obj (λX ⇒ X 0) Hom_struct_b_b_r_e_e to the current goal.
Let X, Y and f be given.
Assume HX HY.
Apply HO X HX (λu ⇒ Hom_struct_b_b_r_e_e u Y f → f ∈ Y 0u 0) to the current goal.
Let X' and bX be given.
Assume HbX.
Let b2X be given.
Assume Hb2X.
Let rX and eX be given.
Assume HeX.
Let e2X be given.
Assume He2X.
Apply HO Y HY (λu ⇒ Hom_struct_b_b_r_e_e (pack_b_b_r_e_e X' bX b2X rX eX e2X) u f → f ∈ u 0pack_b_b_r_e_e X' bX b2X rX eX e2X 0) to the current goal.
Let Y' and bY be given.
Assume HbY.
Let b2Y be given.
Assume Hb2Y.
Let rY and eY be given.
Assume HeY.
Let e2Y be given.
Assume He2Y.
We will prove Hom_struct_b_b_r_e_e (pack_b_b_r_e_e X' bX b2X rX eX e2X) (pack_b_b_r_e_e Y' bY b2Y rY eY e2Y) f → f ∈ pack_b_b_r_e_e Y' bY b2Y rY eY e2Y 0pack_b_b_r_e_e X' bX b2X rX eX e2X 0.
rewrite the current goal using Hom_struct_b_b_r_e_e_pack (from left to right).
Assume Hf: f ∈ Y'X' ∧ (∀x x' ∈ X', f (bX x x') = bY (f x) (f x')) ∧ (∀x x' ∈ X', f (b2X x x') = b2Y (f x) (f x')) ∧ (∀x x' ∈ X', rX x x' → rY (f x) (f x')) ∧ f eX = eY ∧ f e2X = e2Y.
Apply and6E (f ∈ Y'X') (∀x x' ∈ X', f (bX x x') = bY (f x) (f x')) (∀x x' ∈ X', f (b2X x x') = b2Y (f x) (f x')) (∀x x' ∈ X', rX x x' → rY (f x) (f x')) (f eX = eY) (f e2X = e2Y) Hf to the current goal.
Assume Hf1: f ∈ Y'X'.
Assume Hf2: ∀x x' ∈ X', f (bX x x') = bY (f x) (f x').
Assume Hf3: ∀x x' ∈ X', f (b2X x x') = b2Y (f x) (f x').
Assume Hf4: ∀x x' ∈ X', rX x x' → rY (f x) (f x').
Assume Hf5: f eX = eY.
Assume Hf6: f e2X = e2Y.
We will prove f ∈ pack_b_b_r_e_e Y' bY b2Y rY eY e2Y 0pack_b_b_r_e_e X' bX b2X rX eX e2X 0.
rewrite the current goal using pack_b_b_r_e_e_0_eq2 (from right to left).
rewrite the current goal using pack_b_b_r_e_e_0_eq2 (from right to left).
We will prove f ∈ Y'X'.
An exact proof term for the current goal is Hf1.
Let X be given.
Assume HX.
Apply HO X HX (λu ⇒ Hom_struct_b_b_r_e_e u u (lam_id (u 0))) to the current goal.
Let X' and bX be given.
Assume HbX.
Let b2X be given.
Assume Hb2X.
Let rX and eX be given.
Assume HeX.
Let e2X be given.
Assume He2X.
We will prove Hom_struct_b_b_r_e_e (pack_b_b_r_e_e X' bX b2X rX eX e2X) (pack_b_b_r_e_e X' bX b2X rX eX e2X) (lam_id (pack_b_b_r_e_e X' bX b2X rX eX e2X 0)).
rewrite the current goal using pack_b_b_r_e_e_0_eq2 (from right to left).
rewrite the current goal using Hom_struct_b_b_r_e_e_pack (from left to right).
We will prove lam_id X' ∈ X'X' ∧ (∀x x' ∈ X', lam_id X' (bX x x') = bX (lam_id X' x) (lam_id X' x')) ∧ (∀x x' ∈ X', lam_id X' (b2X x x') = b2X (lam_id X' x) (lam_id X' x')) ∧ (∀x x' ∈ X', rX x x' → rX (lam_id X' x) (lam_id X' x')) ∧ lam_id X' eX = eX ∧ lam_id X' e2X = e2X.
Apply and6I to the current goal.
An exact proof term for the current goal is lam_id_exp_In X'.
Let x be given.
Assume Hx.
Let x' be given.
Assume Hx'.
We will prove lam_id X' (bX x x') = bX ((λx ∈ X' ⇒ x) x) ((λx ∈ X' ⇒ x) x').
rewrite the current goal using beta X' (λx ⇒ x) x Hx (from left to right).
rewrite the current goal using beta X' (λx ⇒ x) x' Hx' (from left to right).
An exact proof term for the current goal is beta X' (λx ⇒ x) (bX x x') (HbX x Hx x' Hx').
Let x be given.
Assume Hx.
Let x' be given.
Assume Hx'.
We will prove lam_id X' (b2X x x') = b2X ((λx ∈ X' ⇒ x) x) ((λx ∈ X' ⇒ x) x').
rewrite the current goal using beta X' (λx ⇒ x) x Hx (from left to right).
rewrite the current goal using beta X' (λx ⇒ x) x' Hx' (from left to right).
An exact proof term for the current goal is beta X' (λx ⇒ x) (b2X x x') (Hb2X x Hx x' Hx').
Let x be given.
Assume Hx.
Let x' be given.
Assume Hx' Hxx'.
We will prove rX ((λx ∈ X' ⇒ x) x) ((λx ∈ X' ⇒ x) x').
rewrite the current goal using beta X' (λx ⇒ x) x Hx (from left to right).
rewrite the current goal using beta X' (λx ⇒ x) x' Hx' (from left to right).
An exact proof term for the current goal is Hxx'.
We will prove (λx ∈ X' ⇒ x) eX = eX.
An exact proof term for the current goal is beta X' (λx ⇒ x) eX HeX.
We will prove (λx ∈ X' ⇒ x) e2X = e2X.
An exact proof term for the current goal is beta X' (λx ⇒ x) e2X He2X.
Let X, Y, Z, f and g be given.
Assume HX HY HZ.
Apply HO X HX (λu ⇒ Hom_struct_b_b_r_e_e u Y f → Hom_struct_b_b_r_e_e Y Z g → Hom_struct_b_b_r_e_e u Z (lam_comp (u 0) g f)) to the current goal.
Let X' and bX be given.
Assume HbX.
Let b2X be given.
Assume Hb2X.
Let rX and eX be given.
Assume HeX.
Let e2X be given.
Assume He2X.
We will prove Hom_struct_b_b_r_e_e (pack_b_b_r_e_e X' bX b2X rX eX e2X) Y f → Hom_struct_b_b_r_e_e Y Z g → Hom_struct_b_b_r_e_e (pack_b_b_r_e_e X' bX b2X rX eX e2X) Z (lam_comp (pack_b_b_r_e_e X' bX b2X rX eX e2X 0) g f).
rewrite the current goal using pack_b_b_r_e_e_0_eq2 (from right to left).
We will prove Hom_struct_b_b_r_e_e (pack_b_b_r_e_e X' bX b2X rX eX e2X) Y f → Hom_struct_b_b_r_e_e Y Z g → Hom_struct_b_b_r_e_e (pack_b_b_r_e_e X' bX b2X rX eX e2X) Z (lam_comp X' g f).
Apply HO Y HY (λu ⇒ Hom_struct_b_b_r_e_e (pack_b_b_r_e_e X' bX b2X rX eX e2X) u f → Hom_struct_b_b_r_e_e u Z g → Hom_struct_b_b_r_e_e (pack_b_b_r_e_e X' bX b2X rX eX e2X) Z (lam_comp X' g f)) to the current goal.
Let Y' and bY be given.
Assume HbY.
Let b2Y be given.
Assume Hb2Y.
Let rY and eY be given.
Assume HeY.
Let e2Y be given.
Assume He2Y.
We will prove Hom_struct_b_b_r_e_e (pack_b_b_r_e_e X' bX b2X rX eX e2X) (pack_b_b_r_e_e Y' bY b2Y rY eY e2Y) f → Hom_struct_b_b_r_e_e (pack_b_b_r_e_e Y' bY b2Y rY eY e2Y) Z g → Hom_struct_b_b_r_e_e (pack_b_b_r_e_e X' bX b2X rX eX e2X) Z (lam_comp X' g f).
rewrite the current goal using Hom_struct_b_b_r_e_e_pack (from left to right).
Assume Hf: f ∈ Y'X' ∧ (∀x x' ∈ X', f (bX x x') = bY (f x) (f x')) ∧ (∀x x' ∈ X', f (b2X x x') = b2Y (f x) (f x')) ∧ (∀x x' ∈ X', rX x x' → rY (f x) (f x')) ∧ f eX = eY ∧ f e2X = e2Y.
Apply HO Z HZ (λu ⇒ Hom_struct_b_b_r_e_e (pack_b_b_r_e_e Y' bY b2Y rY eY e2Y) u g → Hom_struct_b_b_r_e_e (pack_b_b_r_e_e X' bX b2X rX eX e2X) u (lam_comp X' g f)) to the current goal.
Let Z' and bZ be given.
Assume HbZ.
Let b2Z be given.
Assume Hb2Z.
Let rZ and eZ be given.
Assume HeZ.
Let e2Z be given.
Assume He2Z.
We will prove Hom_struct_b_b_r_e_e (pack_b_b_r_e_e Y' bY b2Y rY eY e2Y) (pack_b_b_r_e_e Z' bZ b2Z rZ eZ e2Z) g → Hom_struct_b_b_r_e_e (pack_b_b_r_e_e X' bX b2X rX eX e2X) (pack_b_b_r_e_e Z' bZ b2Z rZ eZ e2Z) (lam_comp X' g f).
rewrite the current goal using Hom_struct_b_b_r_e_e_pack (from left to right).
rewrite the current goal using Hom_struct_b_b_r_e_e_pack (from left to right).
Assume Hg: g ∈ Z'Y' ∧ (∀y y' ∈ Y', g (bY y y') = bZ (g y) (g y')) ∧ (∀y y' ∈ Y', g (b2Y y y') = b2Z (g y) (g y')) ∧ (∀y y' ∈ Y', rY y y' → rZ (g y) (g y')) ∧ g eY = eZ ∧ g e2Y = e2Z.
Apply and6E (f ∈ Y'X') (∀x x' ∈ X', f (bX x x') = bY (f x) (f x')) (∀x x' ∈ X', f (b2X x x') = b2Y (f x) (f x')) (∀x x' ∈ X', rX x x' → rY (f x) (f x')) (f eX = eY) (f e2X = e2Y) Hf to the current goal.
Assume Hf1: f ∈ Y'X'.
Assume Hf2: ∀x x' ∈ X', f (bX x x') = bY (f x) (f x').
Assume Hf3: ∀x x' ∈ X', f (b2X x x') = b2Y (f x) (f x').
Assume Hf4: ∀x x' ∈ X', rX x x' → rY (f x) (f x').
Assume Hf5: f eX = eY.
Assume Hf6: f e2X = e2Y.
Apply and6E (g ∈ Z'Y') (∀y y' ∈ Y', g (bY y y') = bZ (g y) (g y')) (∀y y' ∈ Y', g (b2Y y y') = b2Z (g y) (g y')) (∀y y' ∈ Y', rY y y' → rZ (g y) (g y')) (g eY = eZ) (g e2Y = e2Z) Hg to the current goal.
Assume Hg1: g ∈ Z'Y'.
Assume Hg2: ∀y y' ∈ Y', g (bY y y') = bZ (g y) (g y').
Assume Hg3: ∀y y' ∈ Y', g (b2Y y y') = b2Z (g y) (g y').
Assume Hg4: ∀y y' ∈ Y', rY y y' → rZ (g y) (g y').
Assume Hg5: g eY = eZ.
Assume Hg6: g e2Y = e2Z.
We will prove lam_comp X' g f ∈ Z'X' ∧ (∀x x' ∈ X', lam_comp X' g f (bX x x') = bZ (lam_comp X' g f x) (lam_comp X' g f x')) ∧ (∀x x' ∈ X', lam_comp X' g f (b2X x x') = b2Z (lam_comp X' g f x) (lam_comp X' g f x')) ∧ (∀x x' ∈ X', rX x x' → rZ (lam_comp X' g f x) (lam_comp X' g f x')) ∧ lam_comp X' g f eX = eZ ∧ lam_comp X' g f e2X = e2Z.
Apply and6I to the current goal.
An exact proof term for the current goal is lam_comp_exp_In X' Y' Z' f Hf1 g Hg1.
Let x be given.
Assume Hx.
Let x' be given.
Assume Hx'.
We will prove lam_comp X' g f (bX x x') = bZ ((λx ∈ X' ⇒ g (f x)) x) ((λx ∈ X' ⇒ g (f x)) x').
rewrite the current goal using beta X' (λx ⇒ g (f x)) x Hx (from left to right).
rewrite the current goal using beta X' (λx ⇒ g (f x)) x' Hx' (from left to right).
We will prove (λx ∈ X' ⇒ g (f x)) (bX x x') = bZ (g (f x)) (g (f x')).
rewrite the current goal using beta X' (λx ⇒ g (f x)) (bX x x') (HbX x Hx x' Hx') (from left to right).
We will prove g (f (bX x x')) = bZ (g (f x)) (g (f x')).
rewrite the current goal using Hg2 (f x) (ap_Pi X' (λ_ ⇒ Y') f x Hf1 Hx) (f x') (ap_Pi X' (λ_ ⇒ Y') f x' Hf1 Hx') (from right to left).
We will prove g (f (bX x x')) = g (bY (f x) (f x')).
Use f_equal.
An exact proof term for the current goal is Hf2 x Hx x' Hx'.
Let x be given.
Assume Hx.
Let x' be given.
Assume Hx'.
We will prove lam_comp X' g f (b2X x x') = b2Z ((λx ∈ X' ⇒ g (f x)) x) ((λx ∈ X' ⇒ g (f x)) x').
rewrite the current goal using beta X' (λx ⇒ g (f x)) x Hx (from left to right).
rewrite the current goal using beta X' (λx ⇒ g (f x)) x' Hx' (from left to right).
We will prove (λx ∈ X' ⇒ g (f x)) (b2X x x') = b2Z (g (f x)) (g (f x')).
rewrite the current goal using beta X' (λx ⇒ g (f x)) (b2X x x') (Hb2X x Hx x' Hx') (from left to right).
We will prove g (f (b2X x x')) = b2Z (g (f x)) (g (f x')).
rewrite the current goal using Hg3 (f x) (ap_Pi X' (λ_ ⇒ Y') f x Hf1 Hx) (f x') (ap_Pi X' (λ_ ⇒ Y') f x' Hf1 Hx') (from right to left).
We will prove g (f (b2X x x')) = g (b2Y (f x) (f x')).
Use f_equal.
An exact proof term for the current goal is Hf3 x Hx x' Hx'.
Let x be given.
Assume Hx.
Let x' be given.
Assume Hx' Hxx'.
We will prove rZ (lam_comp X' g f x) (lam_comp X' g f x').
We will prove rZ ((λx ∈ X' ⇒ g (f x)) x) ((λx ∈ X' ⇒ g (f x)) x').
rewrite the current goal using beta X' (λx ⇒ g (f x)) x Hx (from left to right).
rewrite the current goal using beta X' (λx ⇒ g (f x)) x' Hx' (from left to right).
We will prove rZ (g (f x)) (g (f x')).
Apply Hg4 (f x) (ap_Pi X' (λ_ ⇒ Y') f x Hf1 Hx) (f x') (ap_Pi X' (λ_ ⇒ Y') f x' Hf1 Hx') to the current goal.
An exact proof term for the current goal is Hf4 x Hx x' Hx' Hxx'.
We will prove (λx ∈ X' ⇒ g (f x)) eX = eZ.
rewrite the current goal using beta X' (λx ⇒ g (f x)) eX HeX (from left to right).
We will prove g (f eX) = eZ.
rewrite the current goal using Hf5 (from left to right).
An exact proof term for the current goal is Hg5.
We will prove (λx ∈ X' ⇒ g (f x)) e2X = e2Z.
rewrite the current goal using beta X' (λx ⇒ g (f x)) e2X He2X (from left to right).
We will prove g (f e2X) = e2Z.
rewrite the current goal using Hf6 (from left to right).
An exact proof term for the current goal is Hg6.
∎
Theorem. (MetaCat_struct_b_b_r_e_e_Forgetful_gen)
(∀X, Obj X → struct_b_b_r_e_e X) → MetaFunctor Obj Hom_struct_b_b_r_e_e (λX ⇒ lam_id (X 0)) (λX Y Z g f ⇒ lam_comp (X 0) g f) (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 de2d15... and proposition id is 49006a...
Proof:
Assume HO.
We prove the intermediate claim L1: ∀X Y f, Obj X → Obj Y → Hom_struct_b_b_r_e_e X Y f → f ∈ Y 0X 0.
Let X, Y and f be given.
Assume HX HY.
Apply HO X HX (λu ⇒ Hom_struct_b_b_r_e_e u Y f → f ∈ Y 0u 0) to the current goal.
Let X' and bX be given.
Assume HbX.
Let b2X be given.
Assume Hb2X.
Let rX and eX be given.
Assume HeX.
Let e2X be given.
Assume He2X.
Apply HO Y HY (λu ⇒ Hom_struct_b_b_r_e_e (pack_b_b_r_e_e X' bX b2X rX eX e2X) u f → f ∈ u 0pack_b_b_r_e_e X' bX b2X rX eX e2X 0) to the current goal.
Let Y' and bY be given.
Assume HbY.
Let b2Y be given.
Assume Hb2Y.
Let rY and eY be given.
Assume HeY.
Let e2Y be given.
Assume He2Y.
We will prove Hom_struct_b_b_r_e_e (pack_b_b_r_e_e X' bX b2X rX eX e2X) (pack_b_b_r_e_e Y' bY b2Y rY eY e2Y) f → f ∈ pack_b_b_r_e_e Y' bY b2Y rY eY e2Y 0pack_b_b_r_e_e X' bX b2X rX eX e2X 0.
rewrite the current goal using Hom_struct_b_b_r_e_e_pack (from left to right).
Assume Hf.
rewrite the current goal using pack_b_b_r_e_e_0_eq2 (from right to left).
rewrite the current goal using pack_b_b_r_e_e_0_eq2 (from right to left).
Apply Hf to the current goal.
Assume Hf _.
Apply Hf to the current goal.
Assume Hf _.
Apply Hf to the current goal.
Assume Hf _.
Apply Hf to the current goal.
Assume Hf _.
Apply Hf to the current goal.
Assume Hf1 _.
An exact proof term for the current goal is Hf1.
An exact proof term for the current goal is MetaCatConcreteForgetful Obj (λX ⇒ X 0) Hom_struct_b_b_r_e_e L1.
∎
End of Section MetaCatStruct