Mercurial > hg > Members > kono > Proof > category
diff CCCGraph.agda @ 817:177162990879
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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Sun, 28 Apr 2019 12:37:34 +0900 |
parents | CCChom.agda@4e48b331020a |
children | bc244fc604c8 |
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/CCCGraph.agda Sun Apr 28 12:37:34 2019 +0900 @@ -0,0 +1,237 @@ +open import Level +open import Category +module CCCgraph 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 import CCC + +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 + +open import Category.Sets + +-- Sets is a CCC + +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 -- formula + atom : (vertex G) → Objs G + ⊤ : Objs G + _∧_ : Objs G → Objs G → Objs G + _<=_ : Objs G → Objs G → Objs G + + data Arrow (G : Graph ) : Objs G → Objs G → Set where --- proof + arrow : {a b : vertex G} → (edge G) a b → Arrow G (atom a) (atom b) + ○ : (a : Objs G ) → Arrow G a ⊤ + π : {a b : Objs G } → Arrow G ( a ∧ b ) a + π' : {a b : Objs G } → Arrow G ( a ∧ b ) b + ε : {a b : Objs G } → Arrow G ((a <= b) ∧ b ) a + <_,_> : {a b c : Objs G } → Arrow G c a → Arrow G c b → Arrow G c (a ∧ b) + _* : {a b c : Objs G } → Arrow G (c ∧ b ) a → Arrow G c ( a <= b ) + + record SM {v : Level} : Set (suc v) where + field + graph : Graph {v} + 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 : Objs (graph 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 + amap : {G : SM} { a b : Objs (graph G) } → Arrow (graph G) a b → fobj {G} a → fobj {G} b + amap {G} (arrow x) = smap G x + amap (○ a) _ = OneObj + amap π ( x , _) = x + amap π'( _ , x) = x + amap ε ( f , x ) = f x + amap < f , g > x = (amap f x , amap g x) + amap (f *) x = λ y → amap f ( x , y ) + fmap : {G : SM} { a b : Objs (graph G) } → Hom (DX G) a b → fobj {G} a → fobj {G} b + fmap {G} {a} (id a) = λ z → z + fmap {G} (next x f ) = Sets [ amap {G} x o fmap f ] + + -- CS is a map from Positive logic to Sets + -- Sets is CCC, so we have a cartesian closed category generated by a graph + -- as a sub category of Sets + + 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} {b} {c} {f} {next {b} {d} {c} x g} z = adistr (distr {a} {b} {d} {f} {g} z ) x where + adistr : fmap (g ++ f) z ≡ fmap g (fmap f z) → + ( x : Arrow (graph G) d c ) → fmap ( next x (g ++ f) ) z ≡ fmap ( next x g ) (fmap f z ) + adistr eq x = cong ( λ k → amap x k ) eq + distr {a} {b} {b} {f} {id b} z = refl + 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 ) + +--- record CCCObj { c₁ c₂ ℓ : Level} : Set (suc (c₁ ⊔ c₂ ⊔ ℓ)) where +--- field +--- cat : Category c₁ c₂ ℓ +--- ccc : CCC cat +--- +--- open CCCObj +--- +--- record CCCMap {c₁ c₂ ℓ : Level} (A B : CCCObj {c₁} {c₂} {ℓ} ) : Set (suc (c₁ ⊔ c₂ ⊔ ℓ )) where +--- field +--- cmap : Functor (cat A ) (cat B ) +--- ccf : {A B : CCCObj {c₁} {c₂} {ℓ} } → CCC (cat A) → CCC (cat B) +--- +--- Cart : {c₁ c₂ ℓ : Level} → Category (suc (c₁ ⊔ c₂ ⊔ ℓ)) (suc (c₁ ⊔ c₂ ⊔ ℓ))(suc (c₁ ⊔ c₂ ⊔ ℓ)) +--- Cart {c₁} {c₂} {ℓ} = record { +--- Obj = CCCObj {c₁} {c₂} {ℓ} +--- ; Hom = CCCMap +--- ; _o_ = {!!} +--- ; _≈_ = {!!} +--- ; Id = {!!} +--- ; isCategory = record { +--- identityL = {!!} +--- ; identityR = {!!} +--- ; o-resp-≈ = {!!} +--- ; associative = {!!} +--- }} +--- +--- open import discrete +--- open Graph +--- +--- record Gmap {v : Level} (x y : Graph {v} ) : Set (suc v) where +--- vmap : vertex x → vertex y +--- emap : {a b : vertex x} → edge x a b → edge y (vmap a) (vmap b) +--- +--- Grp : {v : Level} → Category (suc v) v v +--- Grp {v} = record { +--- Obj = Graph {v} +--- ; Hom = {!!} +--- ; _o_ = {!!} +--- ; _≈_ = {!!} +--- ; Id = {!!} +--- ; isCategory = record { +--- identityL = {!!} +--- ; identityR = {!!} +--- ; o-resp-≈ = {!!} +--- ; associative = {!!} +--- }} +--- +--- UX : {c₁ c₂ ℓ : Level} → Functor (Cart {c₁} {c₂} {ℓ} ) (Grp {c₁}) +--- UX = {!!} +--- +--- open ccc-from-graph +--- +--- sm : {v : Level } → Graph {v} → SM {v} +--- SM.graph (sm g) = g +--- SM.sobj (sm g) = {!!} +--- SM.smap (sm g) = {!!} +--- +--- HX : {v : Level } ( x : Obj (Grp {v}) ) → Obj (Grp {v}) +--- HX {v} x = {!!} -- FObj UX ( record { cat = Sets {v} ; ccc = sets } ) +--- +--- +--- +---