Mercurial > hg > Members > kono > Proof > category
view CCChom.agda @ 811:2de0358ea10a
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sat, 27 Apr 2019 07:09:35 +0900 |
| parents | 084dc5e170f8 |
| children | 4ff300e1e98c |
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open import Level open import Category module CCChom where open import HomReasoning open import cat-utility open import Data.Product renaming (_×_ to _/\_ ) hiding ( <_,_> ) open import Category.Constructions.Product open import Relation.Binary.PropositionalEquality hiding ( [_] ) open Functor -- ccc-1 : Hom A a 1 ≅ {*} -- ccc-2 : Hom A c (a × b) ≅ (Hom A c a ) × ( Hom A c b ) -- ccc-3 : Hom A a (c ^ b) ≅ Hom A (a × b) c data One : Set where OneObj : One -- () in Haskell ( or any one object set ) OneCat : Category Level.zero Level.zero Level.zero OneCat = record { Obj = One ; Hom = λ a b → One ; _o_ = λ{a} {b} {c} x y → OneObj ; _≈_ = λ x y → x ≡ y ; Id = λ{a} → OneObj ; isCategory = record { isEquivalence = record {refl = refl ; trans = trans ; sym = sym } ; identityL = λ{a b f} → lemma {a} {b} {f} ; identityR = λ{a b f} → lemma {a} {b} {f} ; o-resp-≈ = λ{a b c f g h i} _ _ → refl ; associative = λ{a b c d f g h } → refl } } where lemma : {a b : One } → { f : One } → OneObj ≡ f lemma {a} {b} {f} with f ... | OneObj = refl record IsoS {c₁ c₂ ℓ c₁' c₂' ℓ' : Level} (A : Category c₁ c₂ ℓ) (B : Category c₁' c₂' ℓ') (a b : Obj A) ( a' b' : Obj B ) : Set ( c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁' ⊔ c₂' ⊔ ℓ' ) where field ≅→ : Hom A a b → Hom B a' b' ≅← : Hom B a' b' → Hom A a b iso→ : {f : Hom B a' b' } → B [ ≅→ ( ≅← f) ≈ f ] iso← : {f : Hom A a b } → A [ ≅← ( ≅→ f) ≈ f ] cong→ : {f g : Hom A a b } → A [ f ≈ g ] → B [ ≅→ f ≈ ≅→ g ] cong← : {f g : Hom B a' b'} → B [ f ≈ g ] → A [ ≅← f ≈ ≅← g ] record IsCCChom {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) (１ : Obj A) ( _*_ : Obj A → Obj A → Obj A ) ( _^_ : Obj A → Obj A → Obj A ) : Set ( c₁ ⊔ c₂ ⊔ ℓ ) where field ccc-1 : {a : Obj A} {b c : Obj OneCat} → -- Hom A a １ ≅ {*} IsoS A OneCat a １ b c ccc-2 : {a b c : Obj A} → -- Hom A c ( a * b ) ≅ ( Hom A c a ) * ( Hom A c b ) IsoS A (A × A) c (a * b) (c , c ) (a , b ) ccc-3 : {a b c : Obj A} → -- Hom A a ( c ^ b ) ≅ Hom A ( a * b ) c IsoS A A a (c ^ b) (a * b) c nat-2 : {a b c : Obj A} → {f : Hom A (b * c) (b * c) } → {g : Hom A a (b * c) } → (A × A) [ (A × A) [ IsoS.≅→ ccc-2 f o (g , g) ] ≈ IsoS.≅→ ccc-2 ( A [ f o g ] ) ] nat-3 : {a b c : Obj A} → { k : Hom A c (a ^ b ) } → A [ A [ IsoS.≅→ (ccc-3) (id1 A (a ^ b)) o (IsoS.≅← (ccc-2 ) (A [ k o (proj₁ ( IsoS.≅→ ccc-2 (id1 A (c * b)))) ] , (proj₂ ( IsoS.≅→ ccc-2 (id1 A (c * b) ))))) ] ≈ IsoS.≅→ (ccc-3 ) k ] open import CCC record CCChom {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) : Set ( c₁ ⊔ c₂ ⊔ ℓ ) where field one : Obj A _*_ : Obj A → Obj A → Obj A _^_ : Obj A → Obj A → Obj A isCCChom : IsCCChom A one _*_ _^_ open import HomReasoning CCC→hom : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) ( c : CCC A ) → CCChom A CCC→hom A c = record { one = CCC.１ c ; _*_ = CCC._∧_ c ; _^_ = CCC._<=_ c ; isCCChom = record { ccc-1 = λ {a} {b} {c'} → record { ≅→ = c101 ; ≅← = c102 ; iso→ = c103 {a} {b} {c'} ; iso← = c104 ; cong← = c105 ; cong→ = c106 } ; ccc-2 = record { ≅→ = c201 ; ≅← = c202 ; iso→ = c203 ; iso← = c204 ; cong← = c205; cong→ = c206 } ; ccc-3 = record { ≅→ = c301 ; ≅← = c302 ; iso→ = c303 ; iso← = c304 ; cong← = c305 ; cong→ = c306 } ; nat-2 = nat-2 ; nat-3 = nat-3 } } module CCC→HOM where c101 : {a : Obj A} → Hom A a (CCC.１ c) → Hom OneCat OneObj OneObj c101 _ = OneObj c102 : {a : Obj A} → Hom OneCat OneObj OneObj → Hom A a (CCC.１ c) c102 {a} OneObj = CCC.○ c a c103 : {a : Obj A } {b c : Obj OneCat} {f : Hom OneCat b b } → _[_≈_] OneCat {b} {c} ( c101 {a} (c102 {a} f) ) f c103 {a} {OneObj} {OneObj} {OneObj} = refl c104 : {a : Obj A} → {f : Hom A a (CCC.１ c)} → A [ (c102 ( c101 f )) ≈ f ] c104 {a} {f} = let open ≈-Reasoning A in HomReasoning.≈-Reasoning.sym A (IsCCC.e2 (CCC.isCCC c) ) c201 : { c₁ a b : Obj A} → Hom A c₁ ((c CCC.∧ a) b) → Hom (A × A) (c₁ , c₁) (a , b) c201 f = ( A [ CCC.π c o f ] , A [ CCC.π' c o f ] ) c202 : { c₁ a b : Obj A} → Hom (A × A) (c₁ , c₁) (a , b) → Hom A c₁ ((c CCC.∧ a) b) c202 (f , g ) = CCC.<_,_> c f g c203 : { c₁ a b : Obj A} → {f : Hom (A × A) (c₁ , c₁) (a , b)} → (A × A) [ (c201 ( c202 f )) ≈ f ] c203 = ( IsCCC.e3a (CCC.isCCC c) , IsCCC.e3b (CCC.isCCC c)) c204 : { c₁ a b : Obj A} → {f : Hom A c₁ ((c CCC.∧ a) b)} → A [ (c202 ( c201 f )) ≈ f ] c204 = IsCCC.e3c (CCC.isCCC c) c301 : { d a b : Obj A} → Hom A a ((c CCC.<= d) b) → Hom A ((c CCC.∧ a) b) d -- a -> d <= b -> (a ∧ b ) -> d c301 {d} {a} {b} f = A [ CCC.ε c o CCC.<_,_> c ( A [ f o CCC.π c ] ) ( CCC.π' c ) ] c302 : { d a b : Obj A} → Hom A ((c CCC.∧ a) b) d → Hom A a ((c CCC.<= d) b) c302 f = CCC._* c f c303 : { c₁ a b : Obj A} → {f : Hom A ((c CCC.∧ a) b) c₁} → A [ (c301 ( c302 f )) ≈ f ] c303 = IsCCC.e4a (CCC.isCCC c) c304 : { c₁ a b : Obj A} → {f : Hom A a ((c CCC.<= c₁) b)} → A [ (c302 ( c301 f )) ≈ f ] c304 = IsCCC.e4b (CCC.isCCC c) c105 : {a : Obj A } {f g : Hom OneCat OneObj OneObj} → _[_≈_] OneCat {OneObj} {OneObj} f g → A [ c102 {a} f ≈ c102 {a} g ] c105 refl = let open ≈-Reasoning A in refl-hom c106 : { a : Obj A } {f g : Hom A a (CCC.１ c)} → A [ f ≈ g ] → _[_≈_] OneCat {OneObj} {OneObj} OneObj OneObj c106 _ = refl c205 : { a b c₁ : Obj A } {f g : Hom (A × A) (c₁ , c₁) (a , b)} → (A × A) [ f ≈ g ] → A [ c202 f ≈ c202 g ] c205 f=g = IsCCC.π-cong (CCC.isCCC c ) (proj₁ f=g ) (proj₂ f=g ) c206 : { a b c₁ : Obj A } {f g : Hom A c₁ ((c CCC.∧ a) b)} → A [ f ≈ g ] → (A × A) [ c201 f ≈ c201 g ] c206 {a} {b} {c₁} {f} {g} f=g = ( begin CCC.π c o f ≈⟨ cdr f=g ⟩ CCC.π c o g ∎ ) , ( begin CCC.π' c o f ≈⟨ cdr f=g ⟩ CCC.π' c o g ∎ ) where open ≈-Reasoning A c305 : { a b c₁ : Obj A } {f g : Hom A ((c CCC.∧ a) b) c₁} → A [ f ≈ g ] → A [ (c CCC.*) f ≈ (c CCC.*) g ] c305 f=g = IsCCC.*-cong (CCC.isCCC c ) f=g c306 : { a b c₁ : Obj A } {f g : Hom A a ((c CCC.<= c₁) b)} → A [ f ≈ g ] → A [ c301 f ≈ c301 g ] c306 {a} {b} {c₁} {f} {g} f=g = begin CCC.ε c o CCC.<_,_> c ( f o CCC.π c ) ( CCC.π' c ) ≈⟨ cdr ( IsCCC.π-cong (CCC.isCCC c ) (car f=g ) refl-hom) ⟩ CCC.ε c o CCC.<_,_> c ( g o CCC.π c ) ( CCC.π' c ) ∎ where open ≈-Reasoning A nat-2 : {a b : Obj A} {c = c₁ : Obj A} {f : Hom A ((c CCC.∧ b) c₁) ((c CCC.∧ b) c₁)} {g : Hom A a ((c CCC.∧ b) c₁)} → (A × A) [ (A × A) [ c201 f o g , g ] ≈ c201 (A [ f o g ]) ] nat-2 {a} {b} {c₁} {f} {g} = ( begin ( CCC.π c o f) o g ≈↑⟨ assoc ⟩ ( CCC.π c ) o (f o g) ∎ ) , (sym-hom assoc) where open ≈-Reasoning A nat-3 : {a b : Obj A} {c = c₁ : Obj A} {k : Hom A c₁ ((c CCC.<= a) b)} → A [ A [ c301 (id1 A ((c CCC.<= a) b)) o c202 (A [ k o proj₁ (c201 (id1 A ((c CCC.∧ c₁) b))) ] , proj₂ (c201 (id1 A ((c CCC.∧ c₁) b)))) ] ≈ c301 k ] nat-3 {a} {b} { c₁} {k} = begin c301 (id1 A ((c CCC.<= a) b)) o c202 ( k o proj₁ (c201 (id1 A ((c CCC.∧ c₁) b))) , proj₂ (c201 (id1 A ((c CCC.∧ c₁) b)))) ≈⟨⟩ ( CCC.ε c o CCC.<_,_> c ((id1 A (CCC._<=_ c a b )) o CCC.π c) (CCC.π' c)) o (CCC.<_,_> c (k o ( CCC.π c o (id1 A (CCC._∧_ c c₁ b )))) ( CCC.π' c o (id1 A (CCC._∧_ c c₁ b)))) ≈↑⟨ assoc ⟩ (CCC.ε c) o (( CCC.<_,_> c ((id1 A (CCC._<=_ c a b )) o CCC.π c) (CCC.π' c)) o (CCC.<_,_> c (k o ( CCC.π c o (id1 A (CCC._∧_ c c₁ b )))) ( CCC.π' c o (id1 A (CCC._∧_ c c₁ b))))) ≈⟨ cdr (car (IsCCC.π-cong (CCC.isCCC c ) idL refl-hom ) ) ⟩ (CCC.ε c) o ( CCC.<_,_> c (CCC.π c) (CCC.π' c) o (CCC.<_,_> c (k o ( CCC.π c o (id1 A (CCC._∧_ c c₁ b )))) ( CCC.π' c o (id1 A (CCC._∧_ c c₁ b))))) ≈⟨ cdr (car (IsCCC.π-id (CCC.isCCC c))) ⟩ (CCC.ε c) o ( id1 A (CCC._∧_ c ((c CCC.<= a) b) b ) o (CCC.<_,_> c (k o ( CCC.π c o (id1 A (CCC._∧_ c c₁ b )))) ( CCC.π' c o (id1 A (CCC._∧_ c c₁ b))))) ≈⟨ cdr ( cdr ( IsCCC.π-cong (CCC.isCCC c) (cdr idR) idR )) ⟩ (CCC.ε c) o ( id1 A (CCC._∧_ c ((c CCC.<= a) b) b ) o (CCC.<_,_> c (k o ( CCC.π c )) ( CCC.π' c ))) ≈⟨ cdr idL ⟩ (CCC.ε c) o (CCC.<_,_> c ( k o (CCC.π c) ) (CCC.π' c)) ≈⟨⟩ c301 k ∎ where open ≈-Reasoning A U_b : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) → ( ccc : CCC A ) → (b : Obj A) → Functor A A FObj (U_b A ccc b) = λ a → (CCC._<=_ ccc a b ) FMap (U_b A ccc b) = λ f → CCC._* ccc ( A [ f o CCC.ε ccc ] ) isFunctor (U_b A ccc b) = isF where _<=_ = CCC._<=_ ccc _∧_ = CCC._∧_ ccc <_,_> = CCC.<_,_> ccc _* = CCC._* ccc ε = CCC.ε ccc π = CCC.π ccc π' = CCC.π' ccc isc = CCC.isCCC ccc *-cong = IsCCC.*-cong (CCC.isCCC ccc) π-cong = IsCCC.π-cong (CCC.isCCC ccc) isF : IsFunctor A A ( λ a → (a <= b)) ( λ f → CCC._* ccc ( A [ f o ε ] ) ) IsFunctor.≈-cong isF f≈g = IsCCC.*-cong (CCC.isCCC ccc) ( car f≈g ) where open ≈-Reasoning A -- e4b : {a b c : Obj A} → { k : Hom A c (a <= b ) } → A [ ( A [ ε o < A [ k o π ] , π' > ] ) * ≈ k ] IsFunctor.identity isF {a} = begin (id1 A a o ε ) * ≈⟨ IsCCC.*-cong (CCC.isCCC ccc) ( begin id1 A a o CCC.ε ccc ≈⟨ idL ⟩ ε ≈↑⟨ idR ⟩ ε o id1 A ( ( a <= b ) ∧ b ) ≈↑⟨ cdr ( IsCCC.π-id isc) ⟩ ε o ( < π , π' > ) ≈↑⟨ cdr ( π-cong idL refl-hom) ⟩ ε o ( < id1 A (a <= b) o π , π' > ) ∎ ) ⟩ ( ε o ( < id1 A ( a <= b) o π , π' > ) ) * ≈⟨ IsCCC.e4b (CCC.isCCC ccc) ⟩ id1 A (a <= b) ∎ where open ≈-Reasoning A IsFunctor.distr isF {x} {y} {z} {f} {g} = begin ( ( g o f ) o ε ) * ≈↑⟨ *-cong assoc ⟩ ( g o (f o ε ) ) * ≈↑⟨ *-cong ( cdr (IsCCC.e4a isc) ) ⟩ ( g o ( ε o ( < ( ( f o ε ) * ) o π , π' > ) )) * ≈⟨ *-cong assoc ⟩ ( ( g o ε ) o ( < ( ( f o ε ) * ) o π , π' > ) ) * ≈↑⟨ IsCCC.distr-* isc ⟩ ( g o ε ) * o ( f o ε ) * ∎ where open ≈-Reasoning A F_b : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) → ( ccc : CCC A ) → (b : Obj A) → Functor A A FObj (F_b A ccc b) = λ a → ( CCC._∧_ ccc a b ) FMap (F_b A ccc b) = λ f → ( CCC.<_,_> ccc (A [ f o CCC.π ccc ] ) ( CCC.π' ccc) ) isFunctor (F_b A ccc b) = isF where _∧_ = CCC._∧_ ccc <_,_> = CCC.<_,_> ccc π = CCC.π ccc π' = CCC.π' ccc isc = CCC.isCCC ccc π-cong = IsCCC.π-cong (CCC.isCCC ccc) isF : IsFunctor A A ( λ a → (a ∧ b)) ( λ f → < ( A [ f o π ] ) , π' > ) IsFunctor.≈-cong isF f≈g = π-cong ( car f≈g ) refl-hom where open ≈-Reasoning A IsFunctor.identity isF {a} = trans-hom (π-cong idL refl-hom ) (IsCCC.π-id isc) where open ≈-Reasoning A IsFunctor.distr isF {x} {y} {z} {f} {g} = begin < ( g o f ) o π , π' > ≈↑⟨ π-cong assoc refl-hom ⟩ < g o ( f o π ) , π' > ≈↑⟨ π-cong (cdr (IsCCC.e3a isc )) refl-hom ⟩ < g o ( π o < ( f o π ) , π' > ) , π' > ≈⟨ π-cong assoc ( sym-hom (IsCCC.e3b isc )) ⟩ < ( g o π ) o < ( f o π ) , π' > , π' o < ( f o π ) , π' > > ≈↑⟨ IsCCC.distr-π isc ⟩ < ( g o π ) , π' > o < ( f o π ) , π' > ∎ where open ≈-Reasoning A CCCtoAdj : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) ( ccc : CCC A ) → (b : Obj A ) → coUniversalMapping A A (F_b A ccc b) CCCtoAdj A ccc b = record { U = λ a → a <= b ; ε = ε' ; _*' = solution ; iscoUniversalMapping = record { couniversalMapping = couniversalMapping ; couniquness = couniquness } } where _<=_ = CCC._<=_ ccc <_,_> = CCC.<_,_> ccc _* = CCC._* ccc ε = CCC.ε ccc π = CCC.π ccc π' = CCC.π' ccc isc = CCC.isCCC ccc *-cong = IsCCC.*-cong (CCC.isCCC ccc) ε' : (a : Obj A) → Hom A (FObj (F_b A ccc b) (a <= b)) a ε' a = ε solution : { b' : Obj A} {a : Obj A} → Hom A (FObj (F_b A ccc b) a) b' → Hom A a (b' <= b) solution f = f * couniversalMapping : {b = b₁ : Obj A} {a : Obj A} {f : Hom A (FObj (F_b A ccc b) a) b₁} → A [ A [ ε' b₁ o FMap (F_b A ccc b) (solution f) ] ≈ f ] couniversalMapping {c} {a} {f} = IsCCC.e4a isc couniquness : {b = b₁ : Obj A} {a : Obj A} {f : Hom A (FObj (F_b A ccc b) a) b₁} {g : Hom A a (b₁ <= b)} → A [ A [ ε' b₁ o FMap (F_b A ccc b) g ] ≈ f ] → A [ solution f ≈ g ] couniquness {c} {a} {f} {g} eq = begin f * ≈↑⟨ *-cong eq ⟩ ( ε o FMap (F_b A ccc b) g ) * ≈⟨⟩ ( ε o < ( g o π ) , π' > ) * ≈⟨ IsCCC.e4b isc ⟩ g ∎ where open ≈-Reasoning A c^b=b<=c : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) ( ccc : CCC A ) → {a b c : Obj A} → -- Hom A １ ( c ^ b ) ≅ Hom A b c IsoS A A (CCC.１ ccc ) (CCC._<=_ ccc c b) b c c^b=b<=c A ccc {a} {b} {c} = record { ≅→ = λ f → A [ CCC.ε ccc o CCC.<_,_> ccc ( A [ f o CCC.○ ccc b ] ) ( id1 A b ) ] --- g ’ (g : 1 → b ^ a) of ; ≅← = λ f → CCC._* ccc ( A [ f o CCC.π' ccc ] ) --- ┌ f ┐ name of (f : b ^a → 1 ) ; iso→ = iso→ ; iso← = iso← ; cong→ = cong* ; cong← = cong2 } where cc = IsCCChom.ccc-3 ( CCChom.isCCChom (CCC→hom A ccc ) ) -- e4a : {a b c : Obj A} → { k : Hom A (c /\ b) a } → A [ A [ ε o ( <,> ( A [ (k *) o π ] ) π') ] ≈ k ] iso→ : {f : Hom A b c} → A [ (A Category.o CCC.ε ccc) (CCC.< ccc , (A Category.o (ccc CCC.*) ((A Category.o f) (CCC.π' ccc))) (CCC.○ ccc b) > (Category.Category.Id A)) ≈ f ] iso→ {f} = begin CCC.ε ccc o (CCC.<_,_> ccc (CCC._* ccc ( f o (CCC.π' ccc)) o (CCC.○ ccc b)) (id1 A b ) ) ≈↑⟨ cdr ( IsCCC.π-cong ( CCC.isCCC ccc ) refl-hom (IsCCC.e3b (CCC.isCCC ccc) ) ) ⟩ CCC.ε ccc o ( CCC.<_,_> ccc (CCC._* ccc (f o CCC.π' ccc) o CCC.○ ccc b) ((CCC.π' ccc) o CCC.<_,_> ccc (CCC.○ ccc b) (id1 A b) ) ) ≈↑⟨ cdr ( IsCCC.π-cong ( CCC.isCCC ccc ) (cdr (IsCCC.e3a (CCC.isCCC ccc))) refl-hom ) ⟩ CCC.ε ccc o ( CCC.<_,_> ccc (CCC._* ccc (f o CCC.π' ccc) o ( CCC.π ccc o CCC.<_,_> ccc (CCC.○ ccc b) (id1 A b) ) ) ((CCC.π' ccc) o CCC.<_,_> ccc (CCC.○ ccc b) (id1 A b) ) ) ≈⟨ cdr ( IsCCC.π-cong ( CCC.isCCC ccc ) assoc refl-hom ) ⟩ CCC.ε ccc o ( CCC.<_,_> ccc ((CCC._* ccc (f o CCC.π' ccc) o CCC.π ccc ) o CCC.<_,_> ccc (CCC.○ ccc b) (id1 A b) ) ((CCC.π' ccc) o CCC.<_,_> ccc (CCC.○ ccc b) (id1 A b) ) ) ≈↑⟨ cdr ( IsCCC.distr-π ( CCC.isCCC ccc ) ) ⟩ CCC.ε ccc o ( CCC.<_,_> ccc (CCC._* ccc (f o CCC.π' ccc) o CCC.π ccc ) (CCC.π' ccc) o CCC.<_,_> ccc (CCC.○ ccc b) (id1 A b) ) ≈⟨ assoc ⟩ ( CCC.ε ccc o CCC.<_,_> ccc (CCC._* ccc (f o CCC.π' ccc) o CCC.π ccc ) (CCC.π' ccc) ) o CCC.<_,_> ccc (CCC.○ ccc b) (id1 A b) ≈⟨ car ( IsCCC.e4a (CCC.isCCC ccc) ) ⟩ ( f o CCC.π' ccc ) o CCC.<_,_> ccc (CCC.○ ccc b) (id1 A b) ≈↑⟨ assoc ⟩ f o ( CCC.π' ccc o CCC.<_,_> ccc (CCC.○ ccc b) (id1 A b) ) ≈⟨ cdr (IsCCC.e3b (CCC.isCCC ccc)) ⟩ f o id1 A b ≈⟨ idR ⟩ f ∎ where open ≈-Reasoning A lemma : {f : Hom A (CCC.１ ccc) ((ccc CCC.<= c) b)} → A [ A [ A [ f o (CCC.○ ccc b) ] o (CCC.π' ccc) ] ≈ A [ f o (CCC.π ccc) ] ] lemma {f} = begin ( f o (CCC.○ ccc b) ) o (CCC.π' ccc) ≈↑⟨ assoc ⟩ f o ( (CCC.○ ccc b) o (CCC.π' ccc) ) ≈⟨ cdr ( IsCCC.e2 (CCC.isCCC ccc) ) ⟩ f o (CCC.○ ccc ( CCC._∧_ ccc (CCC.１ ccc) b ) ) ≈↑⟨ cdr ( IsCCC.e2 (CCC.isCCC ccc) ) ⟩ f o ( (CCC.○ ccc) (CCC.１ ccc) o (CCC.π ccc) ) ≈↑⟨ cdr ( car ( IsCCC.e2 (CCC.isCCC ccc) )) ⟩ f o ( id1 A (CCC.１ ccc) o (CCC.π ccc) ) ≈⟨ cdr (idL) ⟩ f o (CCC.π ccc) ∎ where open ≈-Reasoning A -- e4b : {a b c : Obj A} → { k : Hom A c (a <= b ) } → A [ ( A [ ε o ( <,> ( A [ k o π ] ) π' ) ] ) * ≈ k ] iso← : {f : Hom A (CCC.１ ccc) ((ccc CCC.<= c) b)} → A [ (ccc CCC.*) ((A Category.o (A Category.o CCC.ε ccc) (CCC.< ccc , (A Category.o f) (CCC.○ ccc b) > (Category.Category.Id A))) (CCC.π' ccc)) ≈ f ] iso← {f} = begin CCC._* ccc (( CCC.ε ccc o ( CCC.<_,_> ccc ( f o (CCC.○ ccc b) ) (id1 A b ))) o (CCC.π' ccc)) ≈↑⟨ IsCCC.*-cong ( CCC.isCCC ccc ) assoc ⟩ CCC._* ccc ( CCC.ε ccc o (( CCC.<_,_> ccc ( f o (CCC.○ ccc b) ) (id1 A b )) o (CCC.π' ccc))) ≈⟨ IsCCC.*-cong ( CCC.isCCC ccc ) (cdr ( IsCCC.distr-π ( CCC.isCCC ccc ) ) ) ⟩ CCC._* ccc ( CCC.ε ccc o CCC.<_,_> ccc ( (f o (CCC.○ ccc b)) o CCC.π' ccc ) (id1 A b o CCC.π' ccc) ) ≈⟨ IsCCC.*-cong ( CCC.isCCC ccc ) (cdr ( IsCCC.π-cong ( CCC.isCCC ccc ) lemma idL )) ⟩ CCC._* ccc ( CCC.ε ccc o CCC.<_,_> ccc ( f o CCC.π ccc ) (CCC.π' ccc) ) ≈⟨ IsCCC.e4b (CCC.isCCC ccc) ⟩ f ∎ where open ≈-Reasoning A cong* : {f g : Hom A (CCC.１ ccc) ((ccc CCC.<= c) b)} → A [ f ≈ g ] → A [ (A Category.o CCC.ε ccc) (CCC.< ccc , (A Category.o f) (CCC.○ ccc b) > (Category.Category.Id A)) ≈ (A Category.o CCC.ε ccc) (CCC.< ccc , (A Category.o g) (CCC.○ ccc b) > (Category.Category.Id A)) ] cong* {f} {g} f≈g = begin CCC.ε ccc o ( CCC.<_,_> ccc ( f o ( CCC.○ ccc b )) (id1 A b )) ≈⟨ cdr (IsCCC.π-cong ( CCC.isCCC ccc ) (car f≈g) refl-hom ) ⟩ CCC.ε ccc o ( CCC.<_,_> ccc ( g o ( CCC.○ ccc b )) (id1 A b )) ∎ where open ≈-Reasoning A cong2 : {f g : Hom A b c} → A [ f ≈ g ] → A [ (ccc CCC.*) ((A Category.o f) (CCC.π' ccc)) ≈ (ccc CCC.*) ((A Category.o g) (CCC.π' ccc)) ] cong2 {f} {g} f≈g = begin CCC._* ccc ( f o (CCC.π' ccc) ) ≈⟨ IsCCC.*-cong ( CCC.isCCC ccc ) (car f≈g ) ⟩ CCC._* ccc ( g o (CCC.π' ccc) ) ∎ where open ≈-Reasoning A open CCChom open IsCCChom open IsoS hom→CCC : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) ( h : CCChom A ) → CCC A hom→CCC A h = record { １ = １ ; ○ = ○ ; _∧_ = _∧_ ; <_,_> = <,> ; π = π ; π' = π' ; _<=_ = _<=_ ; _* = _* ; ε = ε ; isCCC = isCCC } where １ : Obj A １ = one h ○ : (a : Obj A ) → Hom A a １ ○ a = ≅← ( ccc-1 (isCCChom h ) {_} {OneObj} {OneObj} ) OneObj _∧_ : Obj A → Obj A → Obj A _∧_ a b = _*_ h a b <,> : {a b c : Obj A } → Hom A c a → Hom A c b → Hom A c ( a ∧ b) <,> f g = ≅← ( ccc-2 (isCCChom h ) ) ( f , g ) π : {a b : Obj A } → Hom A (a ∧ b) a π {a} {b} = proj₁ ( ≅→ ( ccc-2 (isCCChom h ) ) (id1 A (_*_ h a b) )) π' : {a b : Obj A } → Hom A (a ∧ b) b π' {a} {b} = proj₂ ( ≅→ ( ccc-2 (isCCChom h ) ) (id1 A (_*_ h a b) )) _<=_ : (a b : Obj A ) → Obj A _<=_ = _^_ h _* : {a b c : Obj A } → Hom A (a ∧ b) c → Hom A a (c <= b) _* = ≅← ( ccc-3 (isCCChom h ) ) ε : {a b : Obj A } → Hom A ((a <= b ) ∧ b) a ε {a} {b} = ≅→ ( ccc-3 (isCCChom h ) {_^_ h a b} {b} ) (id1 A ( _^_ h a b )) isCCC : CCC.IsCCC A １ ○ _∧_ <,> π π' _<=_ _* ε isCCC = record { e2 = e2 ; e3a = e3a ; e3b = e3b ; e3c = e3c ; π-cong = π-cong ; e4a = e4a ; e4b = e4b ; *-cong = *-cong } where e20 : ∀ ( f : Hom OneCat OneObj OneObj ) → _[_≈_] OneCat {OneObj} {OneObj} f OneObj e20 OneObj = refl e2 : {a : Obj A} → ∀ { f : Hom A a １ } → A [ f ≈ ○ a ] e2 {a} {f} = begin f ≈↑⟨ iso← ( ccc-1 (isCCChom h )) ⟩ ≅← ( ccc-1 (isCCChom h ) {a} {OneObj} {OneObj}) ( ≅→ ( ccc-1 (isCCChom h ) {a} {OneObj} {OneObj} ) f ) ≈⟨ ≡-cong {Level.zero} {Level.zero} {Level.zero} {OneCat} {OneObj} {OneObj} ( λ y → ≅← ( ccc-1 (isCCChom h ) {a} {OneObj} {OneObj} ) y ) (e20 ( ≅→ ( ccc-1 (isCCChom h ) {a} {OneObj} {OneObj} ) f) ) ⟩ ≅← ( ccc-1 (isCCChom h ) {a} {OneObj} {OneObj} ) OneObj ≈⟨⟩ ○ a ∎ where open ≈-Reasoning A -- -- g id -- a -------------> b * c ------> b * c -- -- a -------------> b * c ------> b -- a -------------> b * c ------> c -- cong-proj₁ : {a b c d : Obj A} → { f g : Hom (A × A) ( a , b ) ( c , d ) } → (A × A) [ f ≈ g ] → A [ proj₁ f ≈ proj₁ g ] cong-proj₁ eq = proj₁ eq cong-proj₂ : {a b c d : Obj A} → { f g : Hom (A × A) ( a , b ) ( c , d ) } → (A × A) [ f ≈ g ] → A [ proj₂ f ≈ proj₂ g ] cong-proj₂ eq = proj₂ eq e3a : {a b c : Obj A} → { f : Hom A c a }{ g : Hom A c b } → A [ A [ π o <,> f g ] ≈ f ] e3a {a} {b} {c} {f} {g} = begin π o <,> f g ≈⟨⟩ proj₁ (≅→ (ccc-2 (isCCChom h)) (id1 A (_*_ h a b) )) o (≅← (ccc-2 (isCCChom h)) (f , g)) ≈⟨ cong-proj₁ (nat-2 (isCCChom h)) ⟩ proj₁ (≅→ (ccc-2 (isCCChom h)) (( id1 A ( _*_ h a b )) o ( ≅← (ccc-2 (isCCChom h)) (f , g) ) )) ≈⟨ cong-proj₁ ( cong→ (ccc-2 (isCCChom h)) idL ) ⟩ proj₁ (≅→ (ccc-2 (isCCChom h)) ( ≅← (ccc-2 (isCCChom h)) (f , g) )) ≈⟨ cong-proj₁ ( iso→ (ccc-2 (isCCChom h))) ⟩ proj₁ ( f , g ) ≈⟨⟩ f ∎ where open ≈-Reasoning A e3b : {a b c : Obj A} → { f : Hom A c a }{ g : Hom A c b } → A [ A [ π' o <,> f g ] ≈ g ] e3b {a} {b} {c} {f} {g} = begin π' o <,> f g ≈⟨⟩ proj₂ (≅→ (ccc-2 (isCCChom h)) (id1 A (_*_ h a b) )) o (≅← (ccc-2 (isCCChom h)) (f , g)) ≈⟨ cong-proj₂ (nat-2 (isCCChom h)) ⟩ proj₂ (≅→ (ccc-2 (isCCChom h)) (( id1 A ( _*_ h a b )) o ( ≅← (ccc-2 (isCCChom h)) (f , g) ) )) ≈⟨ cong-proj₂ ( cong→ (ccc-2 (isCCChom h)) idL ) ⟩ proj₂ (≅→ (ccc-2 (isCCChom h)) ( ≅← (ccc-2 (isCCChom h)) (f , g) )) ≈⟨ cong-proj₂ ( iso→ (ccc-2 (isCCChom h))) ⟩ proj₂ ( f , g ) ≈⟨⟩ g ∎ where open ≈-Reasoning A e3c : {a b c : Obj A} → { h : Hom A c (a ∧ b) } → A [ <,> ( A [ π o h ] ) ( A [ π' o h ] ) ≈ h ] e3c {a} {b} {c} {f} = begin <,> ( π o f ) ( π' o f ) ≈⟨⟩ ≅← (ccc-2 (isCCChom h)) ( ( proj₁ (≅→ (ccc-2 (isCCChom h)) (id1 A (_*_ h a b) ))) o f , ( proj₂ (≅→ (ccc-2 (isCCChom h)) (id1 A (_*_ h a b)))) o f ) ≈⟨⟩ ≅← (ccc-2 (isCCChom h)) (_[_o_] (A × A) (≅→ (ccc-2 (isCCChom h)) (id1 A (_*_ h a b) )) (f , f ) ) ≈⟨ cong← (ccc-2 (isCCChom h)) (nat-2 (isCCChom h)) ⟩ ≅← (ccc-2 (isCCChom h)) (≅→ (ccc-2 (isCCChom h)) (id1 A (_*_ h a b) o f )) ≈⟨ cong← (ccc-2 (isCCChom h)) (cong→ (ccc-2 (isCCChom h)) idL ) ⟩ ≅← (ccc-2 (isCCChom h)) (≅→ (ccc-2 (isCCChom h)) f ) ≈⟨ iso← (ccc-2 (isCCChom h)) ⟩ f ∎ where open ≈-Reasoning A π-cong : {a b c : Obj A} → { f f' : Hom A c a }{ g g' : Hom A c b } → A [ f ≈ f' ] → A [ g ≈ g' ] → A [ <,> f g ≈ <,> f' g' ] π-cong {a} {b} {c} {f} {f'} {g} {g'} eq1 eq2 = begin <,> f g ≈⟨⟩ ≅← (ccc-2 (isCCChom h)) (f , g) ≈⟨ cong← (ccc-2 (isCCChom h)) ( eq1 , eq2 ) ⟩ ≅← (ccc-2 (isCCChom h)) (f' , g') ≈⟨⟩ <,> f' g' ∎ where open ≈-Reasoning A e4a : {a b c : Obj A} → { k : Hom A (c ∧ b) a } → A [ A [ ε o ( <,> ( A [ (k *) o π ] ) π') ] ≈ k ] e4a {a} {b} {c} {k} = begin ε o ( <,> ((k *) o π ) π' ) ≈⟨⟩ ≅→ (ccc-3 (isCCChom h)) (id1 A (_^_ h a b)) o (≅← (ccc-2 (isCCChom h)) ((( ≅← (ccc-3 (isCCChom h)) k) o π ) , π')) ≈⟨ nat-3 (isCCChom h) ⟩ ≅→ (ccc-3 (isCCChom h)) (≅← (ccc-3 (isCCChom h)) k) ≈⟨ iso→ (ccc-3 (isCCChom h)) ⟩ k ∎ where open ≈-Reasoning A e4b : {a b c : Obj A} → { k : Hom A c (a <= b ) } → A [ ( A [ ε o ( <,> ( A [ k o π ] ) π' ) ] ) * ≈ k ] e4b {a} {b} {c} {k} = begin ( ε o ( <,> ( k o π ) π' ) ) * ≈⟨⟩ ≅← (ccc-3 (isCCChom h)) ( ≅→ ( ccc-3 (isCCChom h ) {_^_ h a b} {b} ) (id1 A ( _^_ h a b )) o (≅← (ccc-2 (isCCChom h)) ( k o π , π'))) ≈⟨ cong← (ccc-3 (isCCChom h)) (nat-3 (isCCChom h)) ⟩ ≅← (ccc-3 (isCCChom h)) (≅→ (ccc-3 (isCCChom h)) k) ≈⟨ iso← (ccc-3 (isCCChom h)) ⟩ k ∎ where open ≈-Reasoning A *-cong : {a b c : Obj A} {f f' : Hom A (a ∧ b) c} → A [ f ≈ f' ] → A [ f * ≈ f' * ] *-cong eq = cong← ( ccc-3 (isCCChom h )) eq open import Category.Sets postulate extensionality : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) → Relation.Binary.PropositionalEquality.Extensionality c₂ c₂ data One' {l : Level} : Set l where OneObj' : One' -- () in Haskell ( or any one object set ) sets : {l : Level } → CCC (Sets {l}) sets {l} = record { １ = One' ; ○ = λ _ → λ _ → OneObj' ; _∧_ = _∧_ ; <_,_> = <,> ; π = π ; π' = π' ; _<=_ = _<=_ ; _* = _* ; ε = ε ; isCCC = isCCC } where １ : Obj Sets １ = One' ○ : (a : Obj Sets ) → Hom Sets a １ ○ a = λ _ → OneObj' _∧_ : Obj Sets → Obj Sets → Obj Sets _∧_ a b = a /\ b <,> : {a b c : Obj Sets } → Hom Sets c a → Hom Sets c b → Hom Sets c ( a ∧ b) <,> f g = λ x → ( f x , g x ) π : {a b : Obj Sets } → Hom Sets (a ∧ b) a π {a} {b} = proj₁ π' : {a b : Obj Sets } → Hom Sets (a ∧ b) b π' {a} {b} = proj₂ _<=_ : (a b : Obj Sets ) → Obj Sets a <= b = b → a _* : {a b c : Obj Sets } → Hom Sets (a ∧ b) c → Hom Sets a (c <= b) f * = λ x → λ y → f ( x , y ) ε : {a b : Obj Sets } → Hom Sets ((a <= b ) ∧ b) a ε {a} {b} = λ x → ( proj₁ x ) ( proj₂ x ) isCCC : CCC.IsCCC Sets １ ○ _∧_ <,> π π' _<=_ _* ε isCCC = record { e2 = e2 ; e3a = λ {a} {b} {c} {f} {g} → e3a {a} {b} {c} {f} {g} ; e3b = λ {a} {b} {c} {f} {g} → e3b {a} {b} {c} {f} {g} ; e3c = e3c ; π-cong = π-cong ; e4a = e4a ; e4b = e4b ; *-cong = *-cong } where e2 : {a : Obj Sets} {f : Hom Sets a １} → Sets [ f ≈ ○ a ] e2 {a} {f} = extensionality Sets ( λ x → e20 x ) where e20 : (x : a ) → f x ≡ ○ a x e20 x with f x e20 x | OneObj' = refl e3a : {a b c : Obj Sets} {f : Hom Sets c a} {g : Hom Sets c b} → Sets [ ( Sets [ π o ( <,> f g) ] ) ≈ f ] e3a = refl e3b : {a b c : Obj Sets} {f : Hom Sets c a} {g : Hom Sets c b} → Sets [ Sets [ π' o ( <,> f g ) ] ≈ g ] e3b = refl e3c : {a b c : Obj Sets} {h : Hom Sets c (a ∧ b)} → Sets [ <,> (Sets [ π o h ]) (Sets [ π' o h ]) ≈ h ] e3c = refl π-cong : {a b c : Obj Sets} {f f' : Hom Sets c a} {g g' : Hom Sets c b} → Sets [ f ≈ f' ] → Sets [ g ≈ g' ] → Sets [ <,> f g ≈ <,> f' g' ] π-cong refl refl = refl e4a : {a b c : Obj Sets} {h : Hom Sets (c ∧ b) a} → Sets [ Sets [ ε o <,> (Sets [ h * o π ]) π' ] ≈ h ] e4a = refl e4b : {a b c : Obj Sets} {k : Hom Sets c (a <= b)} → Sets [ (Sets [ ε o <,> (Sets [ k o π ]) π' ]) * ≈ k ] e4b = refl *-cong : {a b c : Obj Sets} {f f' : Hom Sets (a ∧ b) c} → Sets [ f ≈ f' ] → Sets [ f * ≈ f' * ] *-cong refl = refl module ccc-from-graph where open import Relation.Binary.PropositionalEquality renaming ( cong to ≡-cong ) hiding ( [_] ) open import discrete open graphtocat open Graph data Objs {G : Graph } : Set where atom : (vertex G) → Objs ⊤ : Objs _∧_ : Objs {G} → Objs {G} → Objs _<=_ : Objs {G} → Objs {G} → Objs data Arrow {G : Graph } : Objs {G} → Objs {G} → Set where arrow : {a b : vertex G} → (edge G) a b → Arrow (atom a) (atom b) ○ : (a : Objs ) → Arrow a ⊤ π : {a b : Objs } → Arrow ( a ∧ b ) a π' : {a b : Objs } → Arrow ( a ∧ b ) b ε : {a b : Objs } → Arrow ((a <= b) ∧ b ) a -- <_,_> : {a b c : Objs } → Arrow c a → Arrow c b → Arrow c (a ∧ b) -- _* : {a b c : Objs } → Arrow (c ∧ b ) a → Arrow c ( a <= b ) record SM : Set (suc Level.zero) where field graph : Graph sobj : vertex graph → Set smap : { a b : vertex graph } → edge graph a b → sobj a → sobj b open SM -- positive intutionistic calculus PL : (G : SM) → Graph PL G = record { vertex = Objs {graph G} ; edge = Arrow {graph G}} DX : (G : SM) → Category Level.zero Level.zero Level.zero DX G = GraphtoCat (PL G) -- open import Category.Sets -- postulate extensionality : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) → Relation.Binary.PropositionalEquality.Extensionality c₂ c₂ fobj : {G : SM} ( a : Obj (DX G) ) → Set fobj {G} (atom x) = sobj G x fobj {G} (a ∧ b) = (fobj {G} a ) /\ (fobj {G} b ) fobj {G} (a <= b) = fobj {G} b → fobj {G} a fobj ⊤ = One' fmap : {G : SM} { a b : Obj (DX G) } → Hom (DX G) a b → fobj {G} a → fobj {G} b fmap {G} {a} {a} (id a) = λ z → z fmap {G} {a} {(atom b)} (next {a} {c} (arrow x) f) = λ (z : fobj a) → smap G x ( fmap f z ) fmap {G} {a} {⊤} (next (○ b) f) = λ _ → OneObj' fmap {G} {a} {b} (next {a} {.b ∧ y} {b} π f) = λ z → proj₁ ( fmap {G} {a} {b ∧ y} f z ) fmap {G} {a} {b} (next {.a} {x ∧ .b} {.b} π' f) = λ z → proj₂ ( fmap f z ) where fmap {G} {a} {b} (next {.a} {(x <= y) ∧ y} {.b} ε f) = λ z → ( proj₁ (fmap f z))( proj₂ (fmap f z)) CS : (G : SM ) → Functor (DX G) (Sets {Level.zero}) FObj (CS G) a = fobj a FMap (CS G) {a} {b} f = fmap {G} {a} {b} f isFunctor (CS G) = isf where _++_ = Category._o_ (DX G) ++idR = IsCategory.identityR ( Category.isCategory ( DX G ) ) distr : {a b c : Obj (DX G)} { f : Hom (DX G) a b } { g : Hom (DX G) b c } → (z : fobj {G} a ) → fmap (g ++ f) z ≡ fmap g (fmap f z) distr {a} {.a} {.a} {id a} {id .a} z = refl distr {a} {.a} {c} {id a} {next x g} z = begin fmap (DX G [ next x g o Chain.id a ]) z ≡⟨⟩ fmap ( next x ( g ++ (Chain.id a))) z ≡⟨ cong ( λ k → fmap ( next x k ) z ) ++idR ⟩ fmap ( next x g ) z ≡⟨ refl ⟩ ( Sets [ fmap (next x g) o fmap (Chain.id a) ] ) z ∎ where open ≡-Reasoning distr {a} {b} {b} {f} {id b} z = refl distr {a} {b} {atom z} {f} {next {.b} {atom y} {atom z} (arrow x) g} w = begin fmap (DX G [ next (arrow x) g o f ]) w ≡⟨⟩ fmap ( next (arrow x) (g ++ f ) ) w ≡⟨⟩ smap G x ( fmap (g ++ f) w ) ≡⟨ cong ( λ k → smap G x k ) (distr {a} {b} {atom y} {f} {g} w) ⟩ smap G x (fmap g (fmap f w)) ≡⟨⟩ fmap (next (arrow x) g) ( fmap f w ) ≡⟨⟩ (Sets [ fmap (next (arrow x) g) o fmap f ]) w ∎ where open ≡-Reasoning distr {a} {b} {.⊤} {f} {next (○ a₁) g} z = begin fmap (DX G [ next (○ a₁) g o f ]) z ≡⟨⟩ (Sets [ fmap (next (○ a₁) g) o fmap f ]) z ∎ where open ≡-Reasoning distr {a} {b} {c} {f} {next {.b} {.c ∧ x} {.c} π g} z = begin fmap (DX G [ next π g o f ]) z ≡⟨⟩ fmap {G} {a} {c} (next {PL G} {a} {c ∧ x} {c} π ( g ++ f)) z ≡⟨ {!!} ⟩ proj₁ ( fmap {G} {a} {c ∧ x} (g ++ f ) z ) ≡⟨ cong ( λ k → proj₁ k ) ( distr {a} {b} {(c ∧ x)} {f} {g} z ) ⟩ proj₁ ( fmap g ( fmap f z )) ≡⟨ {!!} ⟩ (Sets [ fmap (next π g) o fmap f ]) z ∎ where open ≡-Reasoning distr {a} {b} {c} {f} {next π' g} z = begin fmap (DX G [ next π' g o f ]) z ≡⟨⟩ fmap (next π' (g ++ f)) z ≡⟨ {!!} ⟩ proj₂ (fmap (g ++ f) z) ≡⟨ {!!} ⟩ ( Sets [ fmap (next π' g) o fmap f ] ) z ∎ where open ≡-Reasoning distr {a} {b} {c} {f} {next {b} {(c <= x) ∧ x} {c} ε g} z = begin fmap (DX G [ next ε g o f ] ) z ≡⟨⟩ fmap (next ε (g ++ f)) z ≡⟨ {!!} ⟩ proj₁ (fmap (g ++ f) z) ( proj₂ (fmap (g ++ f) z) ) ≡⟨ cong ( λ k → proj₁ k ( proj₂ k)) (distr {a} {b} {(c <= x) ∧ x} {f} {g} z) ⟩ proj₁ (fmap g (fmap f z)) ( proj₂ (fmap g (fmap f z))) ≡⟨ {!!} ⟩ fmap (next ε g) (fmap f z) ≡⟨⟩ ( Sets [ fmap (next ε g) o fmap f ] ) z ∎ where open ≡-Reasoning isf : IsFunctor (DX G) Sets fobj fmap IsFunctor.identity isf = extensionality Sets ( λ x → refl ) IsFunctor.≈-cong isf refl = refl IsFunctor.distr isf {a} {b} {c} {g} {f} = extensionality Sets ( λ z → distr {a} {b} {c} {g} {f} z ) <_,_> : { G : SM } → {a b c : Objs {graph G}} → Arrow c a → Arrow c b → Hom Sets (FObj (CS G) c ) (FObj (CS G) (a ∧ b) ) <_,_> {G} {a} {b} {c} f g = λ z → ( (FMap (CS G) ( next f (id c))) z , FMap (CS G) (next g (id c)) z ) _* : { G : SM } → {a b c : Objs {graph G}} → Arrow (c ∧ b ) a → Hom Sets (FObj (CS G) c) (FObj (CS G) ( a <= b )) _* {G} {a} {b} {c} f = λ z y → FMap (CS G) ( next f (id (c ∧ b) ) ) ( z , y )