remove PreCategory

diff --git a/src/PreCategory.v b/src/PreCategory.v
deleted file mode 100644 (file)
index 7418261..0000000
--- a/src/PreCategory.v
+++ /dev/null
@@ -1,93 +0,0 @@
-(*********************************************************************************************************************************)
-(* SemiCategory:                                                                                                                 *)
-(*                                                                                                                               *)
-(*   Same as a category, but without identity maps.   See                                                                        *)
-(*                                                                                                                               *)
-(*       http://ncatlab.org/nlab/show/semicategory                                                                               *)
-(*                                                                                                                               *)
-(*********************************************************************************************************************************)
-
-Generalizable All Variables.
-Require Import Preamble.
-Require Import General.
-
-Class SemiCategory (Ob:Type)(Hom:Ob->Ob->Type) :=
-{ semi_hom              := Hom
-; semi_ob               := Ob
-; semi_comp             :  forall  {a}{b}{c}, Hom a b -> Hom b c -> Hom a c
-; semi_eqv              :  forall  a b,   (Hom a b) -> (Hom a b) -> Prop
-; semi_eqv_equivalence  :  forall  a b,   Equivalence (semi_eqv a b)
-; semi_comp_respects    :  forall  a b c, Proper (semi_eqv a b ==> semi_eqv b c ==> semi_eqv a c) (@semi_comp _ _ _)
-; semi_associativity    :
-       forall `(f:Hom a b)`(g:Hom b c)`(h:Hom c d), semi_eqv _ _ (semi_comp (semi_comp f g) h) (semi_comp f (semi_comp g h))
-}.
-Coercion semi_ob : SemiCategory >-> Sortclass.
-
-Notation "a ~->   b" := (@semi_hom  _ _ _       a b) (at level 70).
-Notation "f ~-~   g" := (@semi_eqv  _ _ _ _ _   f g) (at level 54).
-Notation "f >>->> g" := (@semi_comp _ _ _ _ _ _ f g) (at level 45).
-
-Add Parametric Relation (Ob:Type)(Hom:Ob->Ob->Type)(C:SemiCategory Ob Hom)(a b:Ob) : (semi_hom a b) (semi_eqv a b)
-  reflexivity proved by  (@Equivalence_Reflexive  _ _ (semi_eqv_equivalence a b))
-  symmetry proved by     (@Equivalence_Symmetric  _ _ (semi_eqv_equivalence a b))
-  transitivity proved by (@Equivalence_Transitive _ _ (semi_eqv_equivalence a b))
-  as parametric_relation_semi_eqv.
-  Add Parametric Morphism `(c:SemiCategory Ob Hom)(a b c:Ob) : (@semi_comp _ _ _ a b c)
-  with signature (semi_eqv _ _ ==> semi_eqv _ _ ==> semi_eqv _ _) as parametric_morphism_semi_comp.
-    intros.
-    apply semi_comp_respects; auto.
-  Defined.
-
-Class SemiFunctor
-`( c1                 : SemiCategory                           )
-`( c2                 : SemiCategory                           )
-( fobj                : c1 -> c2                               ) :=
-{ semifunctor_fobj         := fobj
-; semi_fmor                : forall {a b}, (a~->b) -> (fobj a)~->(fobj b)
-; semi_fmor_respects       : forall a b (f f':a~->b),   (f ~-~ f') ->        (semi_fmor f ~-~ semi_fmor f')
-; semi_fmor_preserves_comp : forall `(f:a~->b)`(g:b~->c), (semi_fmor f) >>->> (semi_fmor g) ~-~ semi_fmor (f >>->> g)
-}.
-Implicit Arguments semi_fmor [[Ob][Hom][c1][Ob0][Hom0][c2][fobj][a][b]].
-
-  (* register "fmor" so we can rewrite through it *)
-  Implicit Arguments semi_fmor                [ Ob Hom Ob0 Hom0 c1 c2 fobj a b ].
-  Implicit Arguments semi_fmor_respects       [ Ob Hom Ob0 Hom0 c1 c2 fobj a b ].
-  Implicit Arguments semi_fmor_preserves_comp [ Ob Hom Ob0 Hom0 c1 c2 fobj a b c ].
-  Notation "F \- f" := (semi_fmor F f) (at level 20)  : category_scope.
-  Add Parametric Morphism `(C1:SemiCategory)`(C2:SemiCategory)
-    (Fobj:C1->C2)
-    (F:SemiFunctor C1 C2 Fobj)
-    (a b:C1)
-  : (@semi_fmor _ _ C1 _ _ C2 Fobj F a b)
-  with signature ((@semi_eqv C1 _ C1 a b) ==> (@semi_eqv C2 _ C2 (Fobj a) (Fobj b))) as parametric_morphism_semi_fmor'.
-  intros; apply (@semi_fmor_respects _ _ C1 _ _ C2 Fobj F a b x y); auto.
-  Defined.
-  Coercion semifunctor_fobj : SemiFunctor >-> Funclass.
-
-Definition semifunctor_comp
-  `(C1:SemiCategory)
-  `(C2:SemiCategory)
-  `(C3:SemiCategory)
-  `(F:@SemiFunctor _ _ C1 _ _ C2 Fobj)`(G:@SemiFunctor _ _ C2 _ _ C3 Gobj) : SemiFunctor C1 C3 (Gobj ○ Fobj).
-  intros. apply (Build_SemiFunctor _ _ _ _ _ _ _ (fun a b m => semi_fmor G (semi_fmor F m))).
-  intros.
-  setoid_rewrite H.
-  reflexivity.
-  intros.
-  setoid_rewrite semi_fmor_preserves_comp; auto.
-  setoid_rewrite semi_fmor_preserves_comp; auto.
-  reflexivity.
-  Defined.
-Notation "f >>>–>>> g" := (@semifunctor_comp _ _ _ _ _ _ _ _ _ _ f _ g) (at level 20)  : category_scope.
-
-Class IsomorphicSemiCategories `(C:SemiCategory)`(D:SemiCategory) :=
-{ isc_f_obj    : C -> D
-; isc_g_obj    : D -> C
-; isc_f        : SemiFunctor C D isc_f_obj
-; isc_g        : SemiFunctor D C isc_g_obj
-; isc_forward  : forall a b (f:a~->b), semi_fmor isc_f  (semi_fmor isc_g  f) ~-~ f
-}.
-; isc_backward : isc_g >>>> isc_f ~~~~ functor_id D
-}.
-
-