Merge remote-tracking branch 'Saizan/dev' into dev

src/Cat/Category/Product.agda

@@ -154,67 +154,40 @@ module Try0 {ℓa ℓb : Level} {ℂ : Category ℓa ℓb}
 
     open RawCategory raw
 
+    propEqs : ∀ {X' : Object}{Y' : Object} (let X , xa , xb = X') (let Y , ya , yb = Y')
+              → (xy : ℂ.Arrow X Y) → isProp (ℂ [ ya ∘ xy ] ≡ xa × ℂ [ yb ∘ xy ] ≡ xb)
+    propEqs xs = propSig (ℂ.arrowsAreSets _ _) (\ _ → ℂ.arrowsAreSets _ _)
+
     isAssocitaive : IsAssociative
-    isAssocitaive
-      {A , a0 , a1}
-      {B , b0 , b1}
-      {C , c0 , c1}
-      {D , d0 , d1}
-      {ff@(f , f0 , f1)}
-      {gg@(g , g0 , g1)}
-      {hh@(h , h0 , h1)}
-      i
-      = s0 i , rl i , rr i
+    isAssocitaive {A'@(A , a0 , a1)} {B , _} {C , c0 , c1} {D'@(D , d0 , d1)} {ff@(f , f0 , f1)} {gg@(g , g0 , g1)} {hh@(h , h0 , h1)} i
+      = s0 i , lemPropF propEqs s0 {proj₂ l} {proj₂ r} i
       where
       l = hh ∘ (gg ∘ ff)
       r = hh ∘ gg ∘ ff
       -- s0 : h ℂ.∘ (g ℂ.∘ f) ≡ h ℂ.∘ g ℂ.∘ f
       s0 : proj₁ l ≡ proj₁ r
       s0 = ℂ.isAssociative {f = f} {g} {h}
-      prop0 : ∀ a → isProp (ℂ [ d0 ∘ a ] ≡ a0)
-      prop0 a = ℂ.arrowsAreSets (ℂ [ d0 ∘ a ]) a0
-      rl : (λ i → (ℂ [ d0 ∘ s0 i ]) ≡ a0) [ proj₁ (proj₂ l) ≡ proj₁ (proj₂ r) ]
-      rl = lemPropF prop0 s0
-      prop1 : ∀ a → isProp ((ℂ [ d1 ∘ a ]) ≡ a1)
-      prop1 a = ℂ.arrowsAreSets _ _
-      rr : (λ i → (ℂ [ d1 ∘ s0 i ]) ≡ a1) [ proj₂ (proj₂ l) ≡ proj₂ (proj₂ r) ]
-      rr = lemPropF prop1 s0
+
 
     isIdentity : IsIdentity 𝟙
     isIdentity {AA@(A , a0 , a1)} {BB@(B , b0 , b1)} {f , f0 , f1} = leftIdentity , rightIdentity
       where
       leftIdentity : 𝟙 ∘ (f , f0 , f1) ≡ (f , f0 , f1)
-      leftIdentity i = l i , rl i , rr i
+      leftIdentity i = l i , lemPropF propEqs l {proj₂ L} {proj₂ R} i
         where
         L = 𝟙 ∘ (f , f0 , f1)
         R : Arrow AA BB
         R = f , f0 , f1
         l : proj₁ L ≡ proj₁ R
         l = ℂ.leftIdentity
-        prop0 : ∀ a → isProp ((ℂ [ b0 ∘ a ]) ≡ a0)
-        prop0 a = ℂ.arrowsAreSets _ _
-        rl : (λ i → (ℂ [ b0 ∘ l i ]) ≡ a0) [ proj₁ (proj₂ L) ≡ proj₁ (proj₂ R) ]
-        rl = lemPropF prop0 l
-        prop1 : ∀ a → isProp (ℂ [ b1 ∘ a ] ≡ a1)
-        prop1 _ = ℂ.arrowsAreSets _ _
-        rr : (λ i → (ℂ [ b1 ∘ l i ]) ≡ a1) [ proj₂ (proj₂ L) ≡ proj₂ (proj₂ R) ]
-        rr = lemPropF prop1 l
       rightIdentity : (f , f0 , f1) ∘ 𝟙 ≡ (f , f0 , f1)
-      rightIdentity i = l i , rl i , rr i
+      rightIdentity i = l i , lemPropF propEqs l {proj₂ L} {proj₂ R} i
         where
         L = (f , f0 , f1) ∘ 𝟙
         R : Arrow AA BB
         R = (f , f0 , f1)
         l : ℂ [ f ∘ ℂ.𝟙 ] ≡ f
         l = ℂ.rightIdentity
-        prop0 : ∀ a → isProp ((ℂ [ b0 ∘ a ]) ≡ a0)
-        prop0 _ = ℂ.arrowsAreSets _ _
-        rl : (λ i → (ℂ [ b0 ∘ l i ]) ≡ a0) [ proj₁ (proj₂ L) ≡ proj₁ (proj₂ R) ]
-        rl = lemPropF prop0 l
-        prop1 : ∀ a → isProp ((ℂ [ b1 ∘ a ]) ≡ a1)
-        prop1 _ = ℂ.arrowsAreSets _ _
-        rr : (λ i → (ℂ [ b1 ∘ l i ]) ≡ a1) [ proj₂ (proj₂ L) ≡ proj₂ (proj₂ R) ]
-        rr = lemPropF prop1 l
 
     arrowsAreSets : ArrowsAreSets
     arrowsAreSets {X , x0 , x1} {Y , y0 , y1} (f , f0 , f1) (g , g0 , g1) p q = {!!}