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changeset 1029:348b5c6d5670

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Tue, 30 Mar 2021 16:03:44 +0900
parents 28569574e3cf
children 76a7d5a8a4e0
files src/CCCSets.agda
diffstat 1 files changed, 17 insertions(+), 45 deletions(-) [+]
line wrap: on
line diff
--- a/src/CCCSets.agda	Tue Mar 30 15:20:08 2021 +0900
+++ b/src/CCCSets.agda	Tue Mar 30 16:03:44 2021 +0900
@@ -161,63 +161,35 @@
         ... | case1 t = true
         ... | case2 f = false
 
-        s2i : {a b : Obj Sets} (m : Hom Sets b a) → (mono : Mono Sets m ) →  (e : sequ a Bool (tchar m mono)  (λ _ → true )) → image m (equ e) 
+        b2i : {a b : Obj (Sets {c}) } (m : Hom Sets b a) →  (mono : Mono Sets m ) → (x : b) → image m (m x)
+        b2i m mono x = isImage x
+        i2b : {a b : Obj (Sets {c}) } (m : Hom Sets b a) →  (mono : Mono Sets m ) → {y : a} → image m y → b
+        i2b m mono (isImage x) = x
+        b2i-iso : {a b : Obj (Sets {c}) } (m : Hom Sets b a) →  (mono : Mono Sets m ) → (x : b) → i2b m mono (b2i m mono x) ≡ x
+        b2i-iso m mono x = refl
+        b2s : {a b : Obj (Sets {c}) } (m : Hom Sets b a) → (mono : Mono Sets m ) → (x : b) →  sequ a Bool (tchar m mono)  (λ _ → true )
+        b2s m mono x with tchar m mono (m x) | inspect (tchar m mono) (m x)
+        ... | true | record {eq = eq1} = elem (m x) eq1
+        ... | false | record { eq = eq1 } = {!!}
+        s2i  : {a b : Obj (Sets {c}) } (m : Hom Sets b a) → (mono : Mono Sets m ) → (e : sequ a Bool (tchar m mono)  (λ _ → true )) → image m (equ e)
         s2i {a} {b} m mono (elem y eq) with lem (image m y)
         ... | case1 im = im
-        i2s : {a b : Obj Sets} (m : Hom Sets b a) → (mono : Mono Sets m ) → {y : a} →  (i : image m y) → sequ a Bool (tchar m mono)  (λ _ → true )
-        i2s {a} {b} m mono {y} i with lem (image m y) | inspect (tchar m mono) y
-        ... | case1 (isImage x) | record { eq = eq1 } = elem (m x) eq1
-        ... | case2 n | record { eq = eq1 } = ⊥-elim (n i) 
-        open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) 
-        ii : {a b : Obj Sets} (m : Hom Sets b a) → (mono : Mono Sets m ) → {y : a} → (i : image m y) →  s2i m mono ( i2s m mono i ) ≅  i
-        ii {a} {b} m mono {y} i with lem (image m y) | inspect (tchar m mono) y
-        ... | case2 n | t =  ⊥-elim (n i)
-        ... | case1 (isImage x) |  record { eq = eq1 }  = {!!}
-        ss : {a b : Obj Sets} (m : Hom Sets b a) → (mono : Mono Sets m ) → (e : sequ a Bool (tchar m mono)  (λ _ → true )) → i2s m mono ( s2i m mono e ) ≡ e
-        ss = {!!}
-        tcharImg : {a b : Obj Sets} (m : Hom Sets b a) → (mono : Mono Sets m ) → (y : a) → tchar m mono y ≡ true → image m y
-        tcharImg  m mono y eq with lem (image m y)
-        ... | case1 t = t
-        tchar¬Img : {a b : Obj Sets} (m : Hom Sets b a) → (mono : Mono Sets m )  (y : a) → tchar m mono y ≡ false → ¬ image m y
-        tchar¬Img  m mono y eq im with lem (image m y) 
-        ... | case2 n  = n im
-        img-x : {a b : Obj (Sets {c}) } (m : Hom Sets b a) → {y : a} → image m y → b
-        img-x m {.(m x)} (isImage x) = x
-        m-img-x : {a b : Obj (Sets {c}) } (m : Hom Sets b a) → {y : a} → (t : image m y ) → m (img-x m t) ≡ y
-        m-img-x m (isImage x) = refl
-        img-cong : {a b : Obj (Sets {c}) } (m : Hom Sets b a) → (mono : Mono Sets m ) → (y y' : a) → y ≡ y' → (s : image m y ) (t : image m y') → s ≅ t
-        img-cong {a} {b} m mono .(m x) .(m x₁) eq (isImage x) (isImage x₁)
-            with cong (λ k → k ! ) ( Mono.isMono mono {One} (λ _ → x) (λ _ → x₁ ) ( extensionality Sets ( λ _ → eq )) )
-        ... | refl = HE.refl
-        img-x-cong : {a b : Obj (Sets {c}) } (m : Hom Sets b a) → (mono : Mono Sets m ) → (y y' : a) → y ≡ y' → (s : image m y ) →( t : image m y') → img-x m s ≡ img-x m t 
-        img-x-cong {a} {b} m mono .(m x) .(m x₁) eq (isImage x) (isImage x₁)
-            with cong (λ k → k ! ) ( Mono.isMono mono {One} (λ _ → x) (λ _ → x₁ ) ( extensionality Sets ( λ _ → eq )) )
-        ... | refl = refl
-        img-x-cong0 : {a b : Obj (Sets {c}) } (m : Hom Sets b a) → (mono : Mono Sets m ) → (y : a) → (s t : image m y ) → img-x m s ≡ img-x m t 
-        img-x-cong0 m mono y s t = img-x-cong m mono y y refl s t
+
+
         isol : {a b : Obj (Sets {c}) } (m : Hom Sets b a) → (mono : Mono Sets m ) → IsoL Sets m (λ (e : sequ a Bool (tchar m mono)  (λ _ → true )) → equ e )
         isol {a} {b} m mono  = record { iso-L = record { ≅→ = b→s ; ≅← = b←s ;
                iso→  =  extensionality Sets ( λ x → iso1 x )
              ; iso←  =  extensionality Sets ( λ x → iso2 x) } ; iso≈L = {!!} } where
           b→s : Hom Sets b (sequ a Bool (tchar m mono) (λ _ → true))
-          b→s x with tchar m mono (m x) | inspect (tchar m mono ) (m x)
-          ... | true | record { eq = eq1 } = elem (m x) eq1
-          b→s x  | false | record { eq = eq1 }  with tchar¬Img m mono (m x) eq1
-          ... | t = ⊥-elim (t (isImage x)) 
+          b→s x = {!!}
           b←s : Hom Sets (sequ a Bool (tchar m mono) (λ _ → true)) b
-          b←s (elem y eq) with tchar m mono y | inspect (tchar m mono ) y
-          ... | true | record { eq = eq1 } = img-x m (tcharImg m mono y eq1 )
-          i←s :  Hom Sets (sequ a Bool (tchar m mono) (λ _ → true)) (image m {!!}) 
-          i←s  (elem y eq) = {!!}
-          bs1 : (y : a) → (eq1 : tchar m mono y ≡ true ) →  m (b←s ( elem y eq1 )) ≡ y
-          bs1 y eq1 with tcharImg m mono y eq1
-          ... | isImage x = {!!}
+          b←s (elem y eq) = {!!}
           iso1 : (x : b) → b←s ( b→s x ) ≡  x
           iso1 x with  tchar m mono (m x) | inspect (tchar m mono ) (m x)
           ... | true | record { eq = eq1 }  = begin
-             b←s ( elem (m x) eq1 )  ≡⟨ cong (λ k → k ! ) (Mono.isMono mono {One} (λ _ → b←s ( elem (m x) eq1 ) ) (λ _ → x ) (cong (λ k _ → k ) (bs1 (m x) eq1 ))) ⟩
+             b←s ( elem (m x) eq1 )  ≡⟨ {!!} ⟩
              x ∎ where open ≡-Reasoning
-          iso1 x | false | record { eq = eq1 } = ⊥-elim ( tchar¬Img m mono (m x) eq1 (isImage x))
+          iso1 x | false | record { eq = eq1 } = {!!}
           iso2 : (x : sequ a Bool (tchar m mono) (λ _ → true) ) →  (Sets [ b→s o b←s ]) x ≡ id1 Sets (sequ a Bool (tchar m mono) (λ _ → true)) x
           iso2 (elem y eq) = {!!}
         imequ   : {a b : Obj Sets} (m : Hom Sets b a) (mono : Mono Sets m) → IsEqualizer Sets m (tchar m mono) (Sets [ (λ _ → true ) o CCC.○ sets a ])