From: Claudio Sacerdoti Coen <claudio.sacerdoticoen@unibo.it>
Date: Mon, 25 Aug 2008 13:17:48 +0000 (+0000)
Subject: New categories REL and BP.
X-Git-Tag: make_still_working~4841
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=b195056adc77e652f59ec0b46afe277b150e12c8;p=helm.git

New categories REL and BP.
Nice naive infrastructure for setoid rewriting.
---

diff --git a/helm/software/matita/library/datatypes/categories.ma b/helm/software/matita/library/datatypes/categories.ma
new file mode 100644
index 000000000..0eb9b6818
--- /dev/null
+++ b/helm/software/matita/library/datatypes/categories.ma
@@ -0,0 +1,77 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "logic/cprop_connectives.ma".
+
+record equivalence_relation (A:Type) : Type ≝
+ { eq_rel:2> A → A → CProp;
+   refl: reflexive ? eq_rel;
+   sym: symmetric ? eq_rel;
+   trans: transitive ? eq_rel
+ }.
+
+record setoid : Type ≝
+ { carr:> Type;
+   eq: equivalence_relation carr
+ }.
+
+interpretation "setoid eq" 'eq x y = (eq_rel _ (eq _) x y).
+notation ".= r" with precedence 50 for @{trans ????? $r}.
+interpretation "setoid symmetry" 'invert r = (sym ____ r).
+
+record binary_morphism (A,B,C: setoid) : Type ≝
+ { fun:2> A → B → C;
+   prop: ∀a,a',b,b'. eq ? a a' → eq ? b b' → eq ? (fun a b) (fun a' b')
+ }.
+
+notation "# r" with precedence 60 for @{prop ???????? (refl ???) $r}.
+notation "r #" with precedence 60 for @{prop ???????? $r (refl ???)}.
+
+definition CPROP: setoid.
+ constructor 1;
+  [ apply CProp
+  | constructor 1;
+     [ apply Iff
+     | intros 1; split; intro; assumption
+     | intros 3; cases H; split; assumption
+     | intros 5; cases H; cases H1; split; intro;
+        [ apply (H4 (H2 H6)) | apply (H3 (H5 H6))]]]
+qed.
+
+definition eq_morphism: ∀S:setoid. binary_morphism S S CPROP.
+ intro;
+ constructor 1;
+  [ apply (eq_rel ? (eq S))
+  | intros; split; intro;
+     [ apply (.= H \sup -1);
+       apply (.= H2);
+       assumption
+     | apply (.= H);
+       apply (.= H2);
+       apply (H1 \sup -1)]]
+qed.
+
+record category : Type ≝
+ { objs:> Type;
+   arrows: objs → objs → setoid;
+   id: ∀o:objs. arrows o o;
+   comp: ∀o1,o2,o3. binary_morphism (arrows o1 o2) (arrows o2 o3) (arrows o1 o3);
+   comp_assoc: ∀o1,o2,o3,o4. ∀a12,a23,a34.
+    comp o1 o3 o4 (comp o1 o2 o3 a12 a23) a34 = comp o1 o2 o4 a12 (comp o2 o3 o4 a23 a34);
+   id_neutral_left: ∀o1,o2. ∀a: arrows o1 o2. comp ??? (id o1) a = a;
+   id_neutral_right: ∀o1,o2. ∀a: arrows o1 o2. comp ??? a (id o2) = a
+ }.
+
+interpretation "category composition" 'compose x y = (fun ___ (comp ____) x y).
+notation "'ASSOC'" with precedence 90 for @{comp_assoc ????????}.
\ No newline at end of file
diff --git a/helm/software/matita/library/datatypes/subsets.ma b/helm/software/matita/library/datatypes/subsets.ma
index 96ec347c7..26b449d74 100644
--- a/helm/software/matita/library/datatypes/subsets.ma
+++ b/helm/software/matita/library/datatypes/subsets.ma
@@ -13,6 +13,7 @@
 (**************************************************************************)
 
 include "logic/cprop_connectives.ma".
+include "datatypes/categories.ma".
 
 record powerset (A: Type) : Type ≝ { char: A → CProp }.
 
@@ -40,103 +41,8 @@ definition union ≝ λA:Type.λU,V:Ω \sup A.{a | a ∈ U ∨ a ∈ V}.
 
 interpretation "union" 'union U V = (union _ U V).
 
-record ssigma (A:Type) (S: powerset A) : Type ≝
- { witness:> A;
-   proof:> witness ∈ S
- }.
-
-coercion ssigma.
-
-record binary_relation (A,B: Type) (U: Ω \sup A) (V: Ω \sup B) : Type ≝
- { satisfy:2> U → V → CProp }.
-
-(*notation < "hvbox (x  (\circ term 19 r \frac \nbsp \circ) y)"  with precedence 45 for @{'satisfy $r $x $y}.*)
-notation < "hvbox (x \nbsp \natur term 90 r \nbsp y)"  with precedence 45 for @{'satisfy $r $x $y}.
-notation > "hvbox (x \natur term 90 r y)"  with precedence 45 for @{'satisfy $r $x $y}.
-interpretation "relation applied" 'satisfy r x y = (satisfy ____ r x y).
-
-definition composition:
- ∀A,B,C.∀U1: Ω \sup A.∀U2: Ω \sup B.∀U3: Ω \sup C.
-  binary_relation ?? U1 U2 → binary_relation ?? U2 U3 →
-   binary_relation ?? U1 U3.
- intros (A B C U1 U2 U3 R12 R23);
- constructor 1;
- intros (s1 s3);
- apply (∃s2. s1 ♮R12 s2 ∧ s2 ♮R23 s3);
-qed.
-
-interpretation "binary relation composition" 'compose x y = (composition ______ x y).
-
-definition equal_relations ≝
- λA,B,U,V.λr,r': binary_relation A B U V.
-  ∀x,y. r x y ↔ r' x y.
-
-interpretation "equal relation" 'eq x y = (equal_relations ____ x y).
-
-lemma refl_equal_relations: ∀A,B,U,V. reflexive ? (equal_relations A B U V).
- intros 5; intros 2; split; intro; assumption.
-qed.
-
-lemma sym_equal_relations: ∀A,B,U,V. symmetric ? (equal_relations A B U V).
- intros 7; intros 2; split; intro;
-  [ apply (fi ?? (H ??)) | apply (if ?? (H ??))] assumption.
-qed.
-
-lemma trans_equal_relations: ∀A,B,U,V. transitive ? (equal_relations A B U V).
- intros 9; intros 2; split; intro;
-  [ apply (if ?? (H1 ??)) | apply (fi ?? (H ??)) ]
-  [ apply (if ?? (H ??)) | apply (fi ?? (H1 ??)) ]
-  assumption.
-qed.
-
-lemma equal_morphism:
- ∀A,B,U,V.∀r1,r1',r2,r2':binary_relation A B U V.
-  r1' = r1 → r2 = r2' → r1 = r2 → r1' = r2'.
- intros 13;
- split; intro;
-  [ apply (if ?? (H1 ??));
-    apply (if ?? (H2 ??));
-    apply (if ?? (H ??));
-    assumption
-  | apply (fi ?? (H ??));
-    apply (fi ?? (H2 ??));
-    apply (fi ?? (H1 ??));
-    assumption
-  ]
-qed.
-
-lemma associative_composition:
- ∀A,B,C,D.∀U1,U2,U3,U4.
-  ∀r1:binary_relation A B U1 U2.
-   ∀r2:binary_relation B C U2 U3.
-    ∀r3:binary_relation C D U3 U4.
-     (r1 ∘ r2) ∘ r3 = r1 ∘ (r2 ∘ r3).
- intros 13;
- split; intro;
- cases H; clear H; cases H1; clear H1;
- [cases H; clear H | cases H2; clear H2]
- cases H1; clear H1;
- exists; try assumption;
- split; try assumption;
- exists; try assumption;
- split; assumption.
-qed.
-
-lemma composition_morphism:
- ∀A,B,C.∀U1,U2,U3.
-  ∀r1,r1':binary_relation A B U1 U2.
-   ∀r2,r2':binary_relation B C U2 U3.
-    r1 = r1' → r2 = r2' → r1 ∘ r2 = r1' ∘ r2'.
- intros 14; split; intro;
- cases H2; clear H2; cases H3; clear H3;
- [ lapply (if ?? (H x w) H2) | lapply (fi ?? (H x w) H2) ]
- [ lapply (if ?? (H1 w y) H4)| lapply (fi ?? (H1 w y) H4) ]
- exists; try assumption;
- split; assumption.
-qed.
-
 include "logic/equality.ma".
 
 definition singleton ≝ λA:Type.λa:A.{b | a=b}.
 
-interpretation "singleton" 'singl a = (singleton _ a).
+interpretation "singleton" 'singl a = (singleton _ a).
\ No newline at end of file
diff --git a/helm/software/matita/library/formal_topology/basic_pairs.ma b/helm/software/matita/library/formal_topology/basic_pairs.ma
index 5cc67a250..7b1f97559 100644
--- a/helm/software/matita/library/formal_topology/basic_pairs.ma
+++ b/helm/software/matita/library/formal_topology/basic_pairs.ma
@@ -12,16 +12,13 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "datatypes/subsets.ma".
-include "logic/cprop_connectives.ma".
-include "formal_topology/categories.ma".
+include "formal_topology/relations.ma".
+include "datatypes/categories.ma".
 
 record basic_pair: Type ≝
- { carr1: Type;
-   carr2: Type;
-   concr: Ω \sup carr1;
-   form: Ω \sup carr2;
-   rel: binary_relation ?? concr form
+ { concr: REL;
+   form: REL;
+   rel: arrows ? concr form
  }.
 
 notation "x ⊩ y" with precedence 45 for @{'Vdash2 $x $y}.
@@ -30,11 +27,9 @@ notation "⊩" with precedence 60 for @{'Vdash}.
 interpretation "basic pair relation" 'Vdash2 x y = (rel _ x y).
 interpretation "basic pair relation (non applied)" 'Vdash = (rel _).
 
-alias symbol "eq" = "equal relation".
-alias symbol "compose" = "binary relation composition".
 record relation_pair (BP1,BP2: basic_pair): Type ≝
- { concr_rel: binary_relation ?? (concr BP1) (concr BP2);
-   form_rel: binary_relation ?? (form BP1) (form BP2);
+ { concr_rel: arrows ? (concr BP1) (concr BP2);
+   form_rel: arrows ? (form BP1) (form BP2);
    commute: concr_rel ∘ ⊩ = ⊩ ∘ form_rel
  }.
 
@@ -44,7 +39,6 @@ notation "hvbox (r \sub \f)"  with precedence 90 for @{'form_rel $r}.
 interpretation "concrete relation" 'concr_rel r = (concr_rel __ r). 
 interpretation "formal relation" 'form_rel r = (form_rel __ r). 
 
-
 definition relation_pair_equality:
  ∀o1,o2. equivalence_relation (relation_pair o1 o2).
  intros;
@@ -52,15 +46,13 @@ definition relation_pair_equality:
   [ apply (λr,r'. r \sub\c ∘ ⊩ = r' \sub\c ∘ ⊩);
   | simplify;
     intros;
-    apply refl_equal_relations;
+    apply refl;
   | simplify;
-    intros;
-    apply sym_equal_relations;
-    assumption
+    intros 2;
+    apply sym;
   | simplify;
-    intros;
-    apply (trans_equal_relations ??????? H);
-    assumption
+    intros 3;
+    apply trans;
   ]      
 qed.
 
@@ -72,11 +64,7 @@ definition relation_pair_setoid: basic_pair → basic_pair → setoid.
   ]
 qed.
 
-definition eq' ≝
- λo1,o2.λr,r':relation_pair o1 o2.⊩ ∘ r \sub\f = ⊩ ∘ r' \sub\f.
-
-alias symbol "eq" = "setoid eq".
-lemma eq_to_eq': ∀o1,o2.∀r,r': relation_pair_setoid o1 o2. r=r' → eq' ?? r r'.
+lemma eq_to_eq': ∀o1,o2.∀r,r': relation_pair_setoid o1 o2. r=r' → ⊩ \circ r \sub\f = ⊩ \circ r'\sub\f.
  intros 7 (o1 o2 r r' H c1 f2);
  split; intro;
   [ lapply (fi ?? (commute ?? r c1 f2) H1) as H2;
@@ -87,27 +75,15 @@ lemma eq_to_eq': ∀o1,o2.∀r,r': relation_pair_setoid o1 o2. r=r' → eq' ?? r
     apply (if ?? (commute ?? r c1 f2) H3);
   ]
 qed.
-   
 
 definition id: ∀o:basic_pair. relation_pair o o.
  intro;
  constructor 1;
-  [1,2: constructor 1;
-    intros;
-    apply (s=s1)
-  | simplify; intros;
-    split;
-    intro;
-    cases H;
-    cases H1; clear H H1;
-     [ exists [ apply y ]
-       split
-        [ rewrite > H2; assumption
-        | reflexivity ]
-     | exists [ apply x ]
-       split
-        [2: rewrite < H3; assumption
-        | reflexivity ]]]
+  [1,2: apply id;
+  | lapply (id_neutral_left ? (concr o) ? (⊩)) as H;
+    lapply (id_neutral_right ?? (form o) (⊩)) as H1;
+    apply (.= H);
+    apply (H1 \sup -1);]
 qed.
 
 definition relation_pair_composition:
@@ -120,37 +96,25 @@ definition relation_pair_composition:
      | apply (r \sub\f ∘ r1 \sub\f)
      | lapply (commute ?? r) as H;
        lapply (commute ?? r1) as H1;
-       apply (equal_morphism ???? (r\sub\c ∘ (r1\sub\c ∘ ⊩)) ? ((⊩ ∘ r\sub\f) ∘ r1\sub\f));
-        [1,2: apply associative_composition]
-       apply (equal_morphism ???? (r\sub\c ∘ (⊩ ∘ r1\sub\f)) ? ((r\sub\c ∘ ⊩) ∘ r1\sub\f));
-        [1,2: apply composition_morphism; first [assumption | apply refl_equal_relations]
-        | apply sym_equal_relations;
-          apply associative_composition
-        ]]
+       apply (.= ASSOC);
+       apply (.= #H1);
+       apply (.= ASSOC\sup -1);
+       apply (.= H#);
+       apply comp_assoc]
   | intros;
-    alias symbol "eq" = "equal relation".
-    change with (a\sub\c ∘ b\sub\c ∘ ⊩ = a'\sub\c ∘ b'\sub\c ∘ ⊩);
-    apply (equal_morphism ???? (a\sub\c ∘ (b\sub\c ∘ ⊩)) ? (a'\sub\c ∘ (b'\sub\c ∘ ⊩)));
-     [ apply associative_composition
-     | apply sym_equal_relations; apply associative_composition]
-    apply (equal_morphism ???? (a\sub\c ∘ (b'\sub\c ∘ ⊩)) ? (a' \sub \c∘(b' \sub \c∘⊩)));
-     [2: apply refl_equal_relations;
-     |1: apply composition_morphism;
-          [ apply refl_equal_relations
-          | assumption]]
-    apply (equal_morphism ???? (a\sub\c ∘ (⊩ ∘ b'\sub\f)) ? (a'\sub\c ∘ (⊩ ∘ b'\sub\f)));
-     [1,2: apply composition_morphism;
-       [1,3: apply refl_equal_relations
-       | apply (commute ?? b');
-       | apply sym_equal_relations; apply (commute ?? b');]]
-    apply (equal_morphism ???? ((a\sub\c ∘ ⊩) ∘ b'\sub\f) ? ((a'\sub\c ∘ ⊩) ∘ b'\sub\f));
-     [2: apply associative_composition
-     |1: apply sym_equal_relations; apply associative_composition]
-    apply composition_morphism;
-     [ assumption
-     | apply refl_equal_relations]]
+    change with (a\sub\c ∘ b\sub\c ∘ ⊩ = a'\sub\c ∘ b'\sub\c ∘ ⊩);  
+    change in H with (a \sub\c ∘ ⊩ = a' \sub\c ∘ ⊩);
+    change in H1 with (b \sub\c ∘ ⊩ = b' \sub\c ∘ ⊩);
+    apply (.= ASSOC);
+    apply (.= #H1);
+    apply (.= #(commute ?? b'));
+    apply (.= ASSOC \sup -1);
+    apply (.= H#);
+    apply (.= ASSOC);
+    apply (.= #(commute ?? b')\sup -1);
+    apply (ASSOC \sup -1)]
 qed.
-
+    
 definition BP: category.
  constructor 1;
   [ apply basic_pair
@@ -160,24 +124,12 @@ definition BP: category.
   | intros;
     change with (a12\sub\c ∘ a23\sub\c ∘ a34\sub\c ∘ ⊩ =
                  (a12\sub\c ∘ (a23\sub\c ∘ a34\sub\c) ∘ ⊩));
-    apply composition_morphism;
-     [2: apply refl_equal_relations]
-    apply associative_composition 
+    apply (ASSOC#);
   | intros;
     change with ((id o1)\sub\c ∘ a\sub\c ∘ ⊩ = a\sub\c ∘ ⊩);
-    apply composition_morphism;
-     [2: apply refl_equal_relations]
-    intros 2; unfold id; simplify;
-    split; intro;
-     [ cases H; cases H1; rewrite > H2; assumption
-     | exists; [assumption] split; [reflexivity| assumption]]
+    apply ((id_neutral_left ????)#);
   | intros;
     change with (a\sub\c ∘ (id o2)\sub\c ∘ ⊩ = a\sub\c ∘ ⊩);
-    apply composition_morphism;
-     [2: apply refl_equal_relations]
-    intros 2; unfold id; simplify;
-    split; intro;
-     [ cases H; cases H1; rewrite < H3; assumption
-     | exists; [assumption] split; [assumption|reflexivity]]
+    apply ((id_neutral_right ????)#);
   ]
-qed.
+qed.
\ No newline at end of file
diff --git a/helm/software/matita/library/formal_topology/categories.ma b/helm/software/matita/library/formal_topology/categories.ma
deleted file mode 100644
index 7adcb279f..000000000
--- a/helm/software/matita/library/formal_topology/categories.ma
+++ /dev/null
@@ -1,47 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-include "logic/cprop_connectives.ma".
-
-record equivalence_relation (A:Type) : Type ≝
- { eq_rel:2> A → A → CProp;
-   refl: reflexive ? eq_rel;
-   sym: symmetric ? eq_rel;
-   trans: transitive ? eq_rel
- }.
-
-record setoid : Type ≝
- { carr:> Type;
-   eq: equivalence_relation carr
- }.
-
-interpretation "setoid eq" 'eq x y = (eq_rel _ (eq _) x y).
-
-record binary_morphism (A,B,C: setoid) : Type ≝
- { fun:2> A → B → C;
-   prop: ∀a,a',b,b'. eq ? a a' → eq ? b b' → eq ? (fun a b) (fun a' b')
- }.
-
-record category : Type ≝
- { objs: Type;
-   arrows: objs → objs → setoid;
-   id: ∀o:objs. arrows o o;
-   comp: ∀o1,o2,o3. binary_morphism (arrows o1 o2) (arrows o2 o3) (arrows o1 o3);
-   comp_assoc: ∀o1,o2,o3,o4. ∀a12,a23,a34.
-    comp o1 o3 o4 (comp o1 o2 o3 a12 a23) a34 = comp o1 o2 o4 a12 (comp o2 o3 o4 a23 a34);
-   id_neutral_left: ∀o1,o2. ∀a: arrows o1 o2. comp ??? (id o1) a = a;
-   id_neutral_right: ∀o1,o2. ∀a: arrows o1 o2. comp ??? a (id o2) = a
- }.
-
-interpretation "category composition" 'compose x y = (comp ____ x y).
\ No newline at end of file
diff --git a/helm/software/matita/library/formal_topology/relations.ma b/helm/software/matita/library/formal_topology/relations.ma
new file mode 100644
index 000000000..2386a34d4
--- /dev/null
+++ b/helm/software/matita/library/formal_topology/relations.ma
@@ -0,0 +1,130 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "datatypes/subsets.ma".
+
+record ssigma (A:Type) (S: powerset A) : Type ≝
+ { witness:> A;
+   proof:> witness ∈ S
+ }.
+
+coercion ssigma.
+
+record binary_relation (A,B: Type) (U: Ω \sup A) (V: Ω \sup B) : Type ≝
+ { satisfy:2> U → V → CProp }.
+
+notation < "hvbox (x \nbsp \natur term 90 r \nbsp y)"  with precedence 45 for @{'satisfy $r $x $y}.
+notation > "hvbox (x \natur term 90 r y)"  with precedence 45 for @{'satisfy $r $x $y}.
+interpretation "relation applied" 'satisfy r x y = (satisfy ____ r x y).
+
+definition composition:
+ ∀A,B,C.∀U1: Ω \sup A.∀U2: Ω \sup B.∀U3: Ω \sup C.
+  binary_relation ?? U1 U2 → binary_relation ?? U2 U3 →
+   binary_relation ?? U1 U3.
+ intros (A B C U1 U2 U3 R12 R23);
+ constructor 1;
+ intros (s1 s3);
+ apply (∃s2. s1 ♮R12 s2 ∧ s2 ♮R23 s3);
+qed.
+
+interpretation "binary relation composition" 'compose x y = (composition ______ x y).
+
+definition equal_relations ≝
+ λA,B,U,V.λr,r': binary_relation A B U V.
+  ∀x,y. r x y ↔ r' x y.
+
+interpretation "equal relation" 'eq x y = (equal_relations ____ x y).
+
+lemma refl_equal_relations: ∀A,B,U,V. reflexive ? (equal_relations A B U V).
+ intros 5; intros 2; split; intro; assumption.
+qed.
+
+lemma sym_equal_relations: ∀A,B,U,V. symmetric ? (equal_relations A B U V).
+ intros 7; intros 2; split; intro;
+  [ apply (fi ?? (H ??)) | apply (if ?? (H ??))] assumption.
+qed.
+
+lemma trans_equal_relations: ∀A,B,U,V. transitive ? (equal_relations A B U V).
+ intros 9; intros 2; split; intro;
+  [ apply (if ?? (H1 ??)) | apply (fi ?? (H ??)) ]
+  [ apply (if ?? (H ??)) | apply (fi ?? (H1 ??)) ]
+  assumption.
+qed.
+
+lemma associative_composition:
+ ∀A,B,C,D.∀U1,U2,U3,U4.
+  ∀r1:binary_relation A B U1 U2.
+   ∀r2:binary_relation B C U2 U3.
+    ∀r3:binary_relation C D U3 U4.
+     (r1 ∘ r2) ∘ r3 = r1 ∘ (r2 ∘ r3).
+ intros 13;
+ split; intro;
+ cases H; clear H; cases H1; clear H1;
+ [cases H; clear H | cases H2; clear H2]
+ cases H1; clear H1;
+ exists; try assumption;
+ split; try assumption;
+ exists; try assumption;
+ split; assumption.
+qed.
+
+lemma composition_morphism:
+ ∀A,B,C.∀U1,U2,U3.
+  ∀r1,r1':binary_relation A B U1 U2.
+   ∀r2,r2':binary_relation B C U2 U3.
+    r1 = r1' → r2 = r2' → r1 ∘ r2 = r1' ∘ r2'.
+ intros 14; split; intro;
+ cases H2; clear H2; cases H3; clear H3;
+ [ lapply (if ?? (H x w) H2) | lapply (fi ?? (H x w) H2) ]
+ [ lapply (if ?? (H1 w y) H4)| lapply (fi ?? (H1 w y) H4) ]
+ exists; try assumption;
+ split; assumption.
+qed.
+
+definition binary_relation_setoid: ∀A,B. Ω \sup A → Ω \sup B → setoid.
+ intros (A B U V);
+ constructor 1;
+  [ apply (binary_relation ?? U V)
+  | constructor 1;
+     [ apply equal_relations
+     | apply refl_equal_relations
+     | apply sym_equal_relations
+     | apply trans_equal_relations
+     ]]
+qed.
+
+record sigma (A:Type) (P: A → Type) : Type ≝
+ { s_witness:> A;
+   s_proof:> P s_witness
+ }.
+
+interpretation "sigma" 'sigma \eta.x = (sigma _ x).
+
+definition REL: category.
+ constructor 1;
+  [ apply (ΣA:Type.Ω \sup A)
+  | intros; apply (binary_relation_setoid ?? (s_proof ?? s) (s_proof ?? s1))
+  | intros; constructor 1; intros; apply (s=s1) 
+  | intros; constructor 1;
+     [ apply composition
+     | apply composition_morphism
+     ]
+  | intros; unfold mk_binary_morphism; simplify;
+    apply associative_composition
+  |6,7: intros 5; simplify; split; intro;
+     [1,3: cases H; clear H; cases H1; clear H1;
+       [ rewrite > H | rewrite < H2 ]
+       assumption
+     |*: exists; try assumption; split; first [ reflexivity | assumption ]]]
+qed.
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