From 9f45bb0df10f8367bff0851da5fea3129a0512c0 Mon Sep 17 00:00:00 2001
From: krasimir <krasimir@chalmers.se>
Date: Mon, 15 Mar 2010 10:43:20 +0000
Subject: [PATCH] refactor Morphisms.gf and InitialAndTerminal.gf

---
 .../category-theory/InitialAndTerminal.gf     | 43 +++++++++++--------
 examples/category-theory/Morphisms.gf         | 33 +++++++++-----
 2 files changed, 49 insertions(+), 27 deletions(-)

diff --git a/examples/category-theory/InitialAndTerminal.gf b/examples/category-theory/InitialAndTerminal.gf
index 3a033dcd5..856fc1788 100644
--- a/examples/category-theory/InitialAndTerminal.gf
+++ b/examples/category-theory/InitialAndTerminal.gf
@@ -1,35 +1,44 @@
 abstract InitialAndTerminal = Morphisms ** {
 
-cat  Initial  (c : Category) ;
+cat  Initial  ({c} : Category) (El c) ;
 data initial  : ({c} : Category)
               -> (x : El c)
               -> ((y : El c) -> Arrow x y)
-              -> Initial c ;
-
-fun initEl : ({c} : Category)
-              -> Initial c
-              -> El c ;
-def initEl {c} (initial {c} x f) = x ;
+              -> Initial x ;
+              
+fun initAr :  ({c} : Category)
+           -> ({x} : El c)
+           -> Initial x
+           -> (y : El c)
+           -> Arrow x y ;
+def initAr {c} {x} (initial {c} x f) y = f y ;
 
 fun initials2iso :  ({c} : Category)
-                 -> ({x,y} : Initial c)
-                 -> Iso (initEl x) (initEl y) ;
+                 -> ({x,y} : El c)
+                 -> (ix : Initial x)
+                 -> (iy : Initial y)
+                 -> Iso (initAr ix y) (initAr iy x) ;
 -- def initials2iso = .. ;
 
-cat  Terminal (c : Category) ;
+
+cat  Terminal ({c} : Category) (El c) ;
 data terminal : ({c} : Category)
               -> (y : El c)
               -> ((x : El c) -> Arrow x y)
-              -> Terminal c ;
+              -> Terminal y ;
 
-fun termEl : ({c} : Category)
-             -> Terminal c
-             -> El c ;
-def termEl {c} (terminal {c} x f) = x ;
+fun terminalAr :  ({c} : Category)
+               -> (x : El c)
+               -> ({y} : El c)
+               -> Terminal y
+               -> Arrow x y ;
+def terminalAr {c} x {y} (terminal {c} y f) = f x ;
 
 fun terminals2iso :  ({c} : Category)
-                  -> ({x,y} : Terminal c)
-                  -> Iso (termEl x) (termEl y) ;
+                  -> ({x,y} : El c)
+                  -> (tx : Terminal x)
+                  -> (ty : Terminal y)
+                  -> Iso (terminalAr x ty) (terminalAr y tx) ;
 -- def terminals2iso = .. ;
 
 }
\ No newline at end of file
diff --git a/examples/category-theory/Morphisms.gf b/examples/category-theory/Morphisms.gf
index 1d7f7e05c..e5f21c925 100644
--- a/examples/category-theory/Morphisms.gf
+++ b/examples/category-theory/Morphisms.gf
@@ -1,6 +1,6 @@
 abstract Morphisms = Categories ** {
 
-cat Iso ({c} : Category) (x,y : El c) ;
+cat Iso ({c} : Category) ({x,y} : El c) (Arrow x y) (Arrow y x) ;
 
 data iso :  ({c} : Category)
          -> ({x,y} : El c)
@@ -8,12 +8,23 @@ data iso :  ({c} : Category)
          -> (g : Arrow y x)
          -> (EqAr (comp f g) (id y))
          -> (EqAr (comp g f) (id x))
-         -> Iso x y ;
+         -> Iso f g ;
+
+fun isoOp : ({c} : Category)
+         -> ({x,y} : El c)
+         -> ({f} : Arrow x y)
+         -> ({g} : Arrow y x)
+         -> Iso f g
+         -> Iso (opAr g) (opAr f) ;
+def isoOp {c} {x} {y} {f} {g} (iso {c} {x} {y} f g id_fg id_gf) =
+        iso {Op c} (opAr g) (opAr f) (eqOp id_fg) (eqOp id_gf) ;
 
 fun iso2mono :  ({c} : Category)
              -> ({x,y} : El c)
-             -> (Iso x y -> Mono x y) ;
-def iso2mono {c} {x} {y} (iso {c} {x} {y} f g id_fg id_gf) = 
+             -> ({f} : Arrow x y)
+             -> ({g} : Arrow y x)
+             -> (Iso f g -> Mono f) ;
+def iso2mono {c} {x} {y} {f} {g} (iso {c} {x} {y} f g id_fg id_gf) = 
         mono f (\h,m,eq_fh_fm -> 
                       eqSym (eqTran (eqIdR m)                                                                      --           h = m
                                     (eqTran (eqCompR id_gf m)                                                      --      id . m = h
@@ -26,8 +37,10 @@ def iso2mono {c} {x} {y} (iso {c} {x} {y} f g id_fg id_gf) =
 
 fun iso2epi  :  ({c} : Category)
              -> ({x,y} : El c)
-             -> (Iso x y -> Epi x y) ;
-def iso2epi {c} {x} {y} (iso {c} {x} {y} f g id_fg id_gf) =
+             -> ({f} : Arrow x y)
+             -> ({g} : Arrow y x)
+             -> (Iso f g -> Epi f) ;
+def iso2epi {c} {x} {y} {f} {g} (iso {c} {x} {y} f g id_fg id_gf) =
         epi {c} {x} {y} f (\{z},h,m,eq_hf_mf -> 
                       eqSym (eqTran (eqIdL m)                                                                     --           h = m
                                     (eqTran (eqCompL m id_fg)                                                     --      m . id = h
@@ -38,21 +51,21 @@ def iso2epi {c} {x} {y} (iso {c} {x} {y} f g id_fg id_gf) =
                                                                                    (eqCompR eq_hf_mf g))))))))) ; -- (h . f) . g = (m . f) . g
                                                                                                                   --  h . f      =  m . f
 
-cat Mono ({c} : Category) (x,y : El c) ;
+cat Mono ({c} : Category) ({x,y} : El c) (Arrow x y) ;
 
 data mono :  ({c} : Category)
           -> ({x,y} : El c)
           -> (f : Arrow x y)
           -> (({z} : El c) -> (h,m : Arrow z x) -> EqAr (comp f h) (comp f m) -> EqAr h m)
-          -> Mono x y ;
+          -> Mono f ;
 
 
-cat Epi ({c} : Category) (x,y : El c) ;
+cat Epi ({c} : Category) ({x,y} : El c) (Arrow x y) ;
 
 data epi  :  ({c} : Category)
           -> ({x,y} : El c)
           -> (f : Arrow x y)
           -> (({z} : El c) -> (h,m : Arrow y z) -> EqAr (comp h f) (comp m f) -> EqAr h m)
-          -> Epi x y ;
+          -> Epi f ;
 
 }
\ No newline at end of file