Mercurial > hg > Members > kono > Proof > category
view SetsCompleteness.agda @ 589:6584db867bd0
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sun, 14 May 2017 09:55:10 +0900 |
| parents | 11b5eeb4a9e7 |
| children | 2c5d8c3c9086 |
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open import Category -- https://github.com/konn/category-agda open import Level open import Category.Sets renaming ( _o_ to _*_ ) module SetsCompleteness where open import cat-utility open import Relation.Binary.Core open import Function import Relation.Binary.PropositionalEquality -- Extensionality a b = {A : Set a} {B : A → Set b} {f g : (x : A) → B x} → (∀ x → f x ≡ g x) → f ≡ g → ( λ x → f x ≡ λ x → g x ) postulate extensionality : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) → Relation.Binary.PropositionalEquality.Extensionality c₂ c₂ ≡cong = Relation.Binary.PropositionalEquality.cong ≈-to-≡ : { c₂ : Level } {a b : Obj (Sets { c₂})} {f g : Hom Sets a b} → Sets [ f ≈ g ] → (x : a ) → f x ≡ g x ≈-to-≡ refl x = refl record Σ {a} (A : Set a) (B : Set a) : Set a where constructor _,_ field proj₁ : A proj₂ : B open Σ public SetsProduct : { c₂ : Level} → CreateProduct ( Sets { c₂} ) SetsProduct { c₂ } = record { product = λ a b → Σ a b ; π1 = λ a b → λ ab → (proj₁ ab) ; π2 = λ a b → λ ab → (proj₂ ab) ; isProduct = λ a b → record { _×_ = λ f g x → record { proj₁ = f x ; proj₂ = g x } -- ( f x , g x ) ; π1fxg=f = refl ; π2fxg=g = refl ; uniqueness = refl ; ×-cong = λ {c} {f} {f'} {g} {g'} f=f g=g → prod-cong a b f=f g=g } } where prod-cong : ( a b : Obj (Sets {c₂}) ) {c : Obj (Sets {c₂}) } {f f' : Hom Sets c a } {g g' : Hom Sets c b } → Sets [ f ≈ f' ] → Sets [ g ≈ g' ] → Sets [ (λ x → f x , g x) ≈ (λ x → f' x , g' x) ] prod-cong a b {c} {f} {.f} {g} {.g} refl refl = refl record sproduct {a} (I : Set a) ( Product : I → Set a ) : Set a where field proj : ( i : I ) → Product i open sproduct iproduct1 : { c₂ : Level} → (I : Obj (Sets { c₂}) ) (fi : I → Obj Sets ) {q : Obj Sets} → ((i : I) → Hom Sets q (fi i)) → Hom Sets q (sproduct I fi) iproduct1 I fi {q} qi x = record { proj = λ i → (qi i) x } ipcx : { c₂ : Level} → (I : Obj (Sets { c₂})) (fi : I → Obj Sets ) {q : Obj Sets} {qi qi' : (i : I) → Hom Sets q (fi i)} → ((i : I) → Sets [ qi i ≈ qi' i ]) → (x : q) → iproduct1 I fi qi x ≡ iproduct1 I fi qi' x ipcx I fi {q} {qi} {qi'} qi=qi x = begin record { proj = λ i → (qi i) x } ≡⟨ ≡cong ( λ qi → record { proj = qi } ) ( extensionality Sets (λ i → ≈-to-≡ (qi=qi i) x )) ⟩ record { proj = λ i → (qi' i) x } ∎ where open import Relation.Binary.PropositionalEquality open ≡-Reasoning ip-cong : { c₂ : Level} → (I : Obj (Sets { c₂}) ) (fi : I → Obj Sets ) {q : Obj Sets} {qi qi' : (i : I) → Hom Sets q (fi i)} → ((i : I) → Sets [ qi i ≈ qi' i ]) → Sets [ iproduct1 I fi qi ≈ iproduct1 I fi qi' ] ip-cong I fi {q} {qi} {qi'} qi=qi = extensionality Sets ( ipcx I fi qi=qi ) SetsIProduct : { c₂ : Level} → (I : Obj Sets) (fi : I → Obj Sets ) → IProduct ( Sets { c₂} ) I SetsIProduct I fi = record { ai = fi ; iprod = sproduct I fi ; pi = λ i prod → proj prod i ; isIProduct = record { iproduct = iproduct1 I fi ; pif=q = pif=q ; ip-uniqueness = ip-uniqueness ; ip-cong = ip-cong I fi } } where pif=q : {q : Obj Sets} (qi : (i : I) → Hom Sets q (fi i)) {i : I} → Sets [ Sets [ (λ prod → proj prod i) o iproduct1 I fi qi ] ≈ qi i ] pif=q {q} qi {i} = refl ip-uniqueness : {q : Obj Sets} {h : Hom Sets q (sproduct I fi)} → Sets [ iproduct1 I fi (λ i → Sets [ (λ prod → proj prod i) o h ]) ≈ h ] ip-uniqueness = refl -- -- e f -- c -------→ a ---------→ b f ( f' -- ^ . ---------→ -- | . g -- |k . -- | . h --y : d -- cf. https://github.com/danr/Agda-projects/blob/master/Category-Theory/Equalizer.agda data sequ {c : Level} (A B : Set c) ( f g : A → B ) : Set c where elem : (x : A ) → (eq : f x ≡ g x) → sequ A B f g equ : { c₂ : Level} {a b : Obj (Sets {c₂}) } { f g : Hom (Sets {c₂}) a b } → ( sequ a b f g ) → a equ (elem x eq) = x fe=ge0 : { c₂ : Level} {a b : Obj (Sets {c₂}) } { f g : Hom (Sets {c₂}) a b } → (x : sequ a b f g) → (Sets [ f o (λ e → equ e) ]) x ≡ (Sets [ g o (λ e → equ e) ]) x fe=ge0 (elem x eq ) = eq irr : { c₂ : Level} {d : Set c₂ } { x y : d } ( eq eq' : x ≡ y ) → eq ≡ eq' irr refl refl = refl elm-cong : { c₂ : Level} → {a b : Obj (Sets {c₂}) } {f g : Hom (Sets {c₂}) a b} → (x y : sequ a b f g) → equ x ≡ equ y → x ≡ y elm-cong ( elem x eq ) (elem .x eq' ) refl = ≡cong ( λ ee → elem x ee ) ( irr eq eq' ) fe=ge : { c₂ : Level} → {a b : Obj (Sets {c₂}) } {f g : Hom (Sets {c₂}) a b} → Sets [ Sets [ f o (λ e → equ {_} {a} {b} {f} {g} e ) ] ≈ Sets [ g o (λ e → equ e ) ] ] fe=ge = extensionality Sets (fe=ge0 ) k : { c₂ : Level} → {a b : Obj (Sets {c₂}) } {f g : Hom (Sets {c₂}) a b} → {d : Obj Sets} (h : Hom Sets d a) → Sets [ Sets [ f o h ] ≈ Sets [ g o h ] ] → Hom Sets d (sequ a b f g) k {_} {_} {_} {_} {_} {d} h eq = λ x → elem (h x) ( ≈-to-≡ eq x ) ek=h : { c₂ : Level} → {a b : Obj (Sets {c₂}) } {f g : Hom (Sets {c₂}) a b} → {d : Obj Sets} {h : Hom Sets d a} {eq : Sets [ Sets [ f o h ] ≈ Sets [ g o h ] ]} → Sets [ Sets [ (λ e → equ {_} {a} {b} {f} {g} e ) o k h eq ] ≈ h ] ek=h {_} {_} {_} {_} {_} {d} {h} {eq} = refl open sequ -- equalizer-c = sequ a b f g -- ; equalizer = λ e → equ e SetsIsEqualizer : { c₂ : Level} → (a b : Obj (Sets {c₂}) ) (f g : Hom (Sets {c₂}) a b) → IsEqualizer Sets (λ e → equ e )f g SetsIsEqualizer {c₂} a b f g = record { fe=ge = fe=ge { c₂ } {a} {b} {f} {g} ; k = λ {d} h eq → k { c₂ } {a} {b} {f} {g} {d} h eq ; ek=h = λ {d} {h} {eq} → ek=h {c₂} {a} {b} {f} {g} {d} {h} {eq} ; uniqueness = uniqueness } where injection : { c₂ : Level } {a b : Obj (Sets { c₂})} (f : Hom Sets a b) → Set c₂ injection f = ∀ x y → f x ≡ f y → x ≡ y lemma5 : {d : Obj Sets} {h : Hom Sets d a} {fh=gh : Sets [ Sets [ f o h ] ≈ Sets [ g o h ] ]} {k' : Hom Sets d (sequ a b f g)} → Sets [ Sets [ (λ e → equ e) o k' ] ≈ h ] → (x : d ) → equ {_} {a} {b} {f} {g} (k h fh=gh x) ≡ equ (k' x) lemma5 refl x = refl -- somehow this is not equal to ≈-to-≡ uniqueness : {d : Obj Sets} {h : Hom Sets d a} {fh=gh : Sets [ Sets [ f o h ] ≈ Sets [ g o h ] ]} {k' : Hom Sets d (sequ a b f g)} → Sets [ Sets [ (λ e → equ e) o k' ] ≈ h ] → Sets [ k h fh=gh ≈ k' ] uniqueness {d} {h} {fh=gh} {k'} ek'=h = extensionality Sets ( λ ( x : d ) → begin k h fh=gh x ≡⟨ elm-cong ( k h fh=gh x) ( k' x ) (lemma5 {d} {h} {fh=gh} {k'} ek'=h x ) ⟩ k' x ∎ ) where open import Relation.Binary.PropositionalEquality open ≡-Reasoning open Functor ---- -- C is locally small i.e. Hom C i j is a set c₁ -- record Small { c₁ c₂ ℓ : Level} ( C : Category c₁ c₂ ℓ ) ( I : Set c₁ ) : Set (suc (c₁ ⊔ c₂ ⊔ ℓ )) where field hom→ : {i j : Obj C } → Hom C i j → I → I hom← : {i j : Obj C } → ( f : I → I ) → Hom C i j hom-iso : {i j : Obj C } → { f : Hom C i j } → hom← ( hom→ f ) ≡ f -- ≈-≡ : {a b : Obj C } { x y : Hom C a b } → (x≈y : C [ x ≈ y ] ) → x ≡ y open Small ΓObj : { c₁ c₂ ℓ : Level} { C : Category c₁ c₂ ℓ } { I : Set c₁ } ( s : Small C I ) ( Γ : Functor C ( Sets { c₁} )) (i : Obj C ) → Set c₁ ΓObj s Γ i = FObj Γ i ΓMap : { c₁ c₂ ℓ : Level} { C : Category c₁ c₂ ℓ } { I : Set c₁ } ( s : Small C I ) ( Γ : Functor C ( Sets { c₁} )) {i j : Obj C } → ( f : I → I ) → ΓObj s Γ i → ΓObj s Γ j ΓMap s Γ {i} {j} f = FMap Γ ( hom← s f ) slid : { c₁ : Level} { I : Set c₁ } → I → I slid x = x record slim { c₂ } { I OC : Set c₂ } ( sobj : OC → Set c₂ ) ( smap : { i j : OC } → (f : I → I ) → sobj i → sobj j ) : Set c₂ where field slequ : (i j : OC) (f : I → I ) → sequ (sproduct OC sobj) (sobj j) (λ x → smap f (proj x i) ) (λ x → proj x j ) slprod : {i : OC } → sproduct OC sobj slprod {i} = equ ( slequ i i slid ) open slim open import HomReasoning open NTrans cone-equ : { c₁ c₂ ℓ : Level} ( C : Category c₁ c₂ ℓ ) ( I : Set c₁ ) ( s : Small C I ) ( Γ : Functor C (Sets {c₁} ) ) → (a b : Obj C) (f : I → I ) → (se : slim (ΓObj s Γ) (ΓMap s Γ) ) → sequ (slim (ΓObj s Γ) (ΓMap s Γ)) (FObj Γ b) (λ x → FMap Γ (hom← s f) (proj (slprod x {a}) a) ) (λ x → proj (slprod x {b}) b ) cone-equ C I s Γ a b f se = elem se {!!} -- (( fe=ge0 (slequ se a b f )) ) cone-equ' : { c₁ c₂ ℓ : Level} ( C : Category c₁ c₂ ℓ ) ( I : Set c₁ ) ( s : Small C I ) ( Γ : Functor C (Sets {c₁} ) ) → (a b : Obj C) (f : I → I ) → (se : slim (ΓObj s Γ) (ΓMap s Γ) ) → FMap Γ (hom← s f) (proj (equ (slequ se a b f )) a) ≡ proj (equ (slequ se a b f)) b cone-equ' C I s Γ a b f se = fe=ge0 (slequ se a b f ) Cone : { c₁ c₂ ℓ : Level} ( C : Category c₁ c₂ ℓ ) ( I : Set c₁ ) ( s : Small C I ) ( Γ : Functor C (Sets {c₁} ) ) → NTrans C Sets (K Sets C ( slim (ΓObj s Γ) (ΓMap s Γ))) Γ Cone C I s Γ = record { TMap = λ i → λ se → proj ( equ {_} { sproduct (Obj C) (ΓObj s Γ)} {FObj Γ i} {λ x → proj x i} {λ x → proj x i} (elem (slprod se) refl )) i ; isNTrans = record { commute = commute1 } } where commute1 : {a b : Obj C} {f : Hom C a b} → Sets [ Sets [ FMap Γ f o (λ se → proj (slprod se) a) ] ≈ Sets [ (λ se → proj (slprod se) b) o FMap (K Sets C ( slim (ΓObj s Γ) (ΓMap s Γ) )) f ] ] commute1 {a} {b} {f} = extensionality Sets ( λ se → begin (Sets [ FMap Γ f o (λ se → proj (slprod se {a}) a) ]) se ≡⟨⟩ FMap Γ f (proj (slprod se) a ) ≡⟨ ≡cong ( λ g → FMap Γ g (proj (slprod se) a)) (sym ( hom-iso s ) ) ⟩ FMap Γ (hom← s ( hom→ s f)) (proj (slprod se {a}) a) ≡⟨ fe=ge0 (cone-equ C I s Γ a b ( hom→ s f) se ) ⟩ proj (slprod se {b}) b ≡⟨⟩ (Sets [ (λ se → proj (slprod se {b}) b) o FMap (K Sets C ( slim (ΓObj s Γ) (ΓMap s Γ) )) f ]) se ∎ ) where open import Relation.Binary.PropositionalEquality open ≡-Reasoning SetsLimit : { c₁ c₂ ℓ : Level} ( C : Category c₁ c₂ ℓ ) ( I : Set c₁ ) ( small : Small C I ) ( Γ : Functor C (Sets {c₁} ) ) → Limit Sets C Γ SetsLimit { c₂} C I s Γ = record { a0 = slim (ΓObj s Γ) (ΓMap s Γ) ; t0 = Cone C I s Γ ; isLimit = record { limit = limit1 ; t0f=t = λ {a t i } → {!!} ; limit-uniqueness = λ {a t i } → {!!} } } where -- ( e : Obj C → sproduct (Obj C) sobj ) -- sequ (Obj C) (ΓObj s Γ j) (λ x₁ → ΓMap s Γ f (proj (e x₁) i)) (λ x₁ → proj (e x₁) j) limit1 : (a : Obj Sets) → NTrans C Sets (K Sets C a) Γ → Hom Sets a (slim (ΓObj s Γ) (ΓMap s Γ)) limit1 a t = λ x → record { slequ = λ i j f → k (λ p → iproduct1 (Obj C) (ΓObj s Γ) (TMap t) x) ( begin (λ x₁ → ΓMap s Γ f (proj x₁ i)) o (λ p → iproduct1 (Obj C) (ΓObj s Γ) (TMap t) x) ≈⟨ car (nat t) ⟩ (λ x₁ → proj x₁ j) o (λ p → iproduct1 (Obj C) (ΓObj s Γ) (TMap t) x) ∎ ) x } where open ≈-Reasoning (Sets { c₂})