Definition of categories.

Proof that basic pairs form a category.

diff --git a/helm/software/matita/library/formal_topology/basic_pairs.ma b/helm/software/matita/library/formal_topology/basic_pairs.ma
new file mode 100644 (file)
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+++ b/helm/software/matita/library/formal_topology/basic_pairs.ma
@@ -0,0 +1,183 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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".
+include "logic/cprop_connectives.ma".
+include "formal_topology/categories.ma".
+
+record basic_pair: Type ≝
+ { carr1: Type;
+   carr2: Type;
+   concr: Ω \sup carr1;
+   form: Ω \sup carr2;
+   rel: binary_relation ?? concr form
+ }.
+
+notation "x ⊩ y" with precedence 45 for @{'Vdash2 $x $y}.
+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);
+   commute: concr_rel ∘ ⊩ = ⊩ ∘ form_rel
+ }.
+
+notation "hvbox (r \sub \c)"  with precedence 90 for @{'concr_rel $r}.
+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;
+ constructor 1;
+  [ apply (λr,r'. r \sub\c ∘ ⊩ = r' \sub\c ∘ ⊩);
+  | simplify;
+    intros;
+    apply refl_equal_relations;
+  | simplify;
+    intros;
+    apply sym_equal_relations;
+    assumption
+  | simplify;
+    intros;
+    apply (trans_equal_relations ??????? H);
+    assumption
+  ]      
+qed.
+
+definition relation_pair_setoid: basic_pair → basic_pair → setoid.
+ intros;
+ constructor 1;
+  [ apply (relation_pair b b1)
+  | apply relation_pair_equality
+  ]
+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'.
+ intros 7 (o1 o2 r r' H c1 f2);
+ split; intro;
+  [ lapply (fi ?? (commute ?? r c1 f2) H1) as H2;
+    lapply (if ?? (H c1 f2) H2) as H3;
+    apply (if ?? (commute ?? r' c1 f2) H3);
+  | lapply (fi ?? (commute ?? r' c1 f2) H1) as H2;
+    lapply (fi ?? (H c1 f2) H2) as H3;
+    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 ]]]
+qed.
+
+definition relation_pair_composition:
+ ∀o1,o2,o3. binary_morphism (relation_pair_setoid o1 o2) (relation_pair_setoid o2 o3) (relation_pair_setoid o1 o3).
+ intros;
+ constructor 1;
+  [ intros (r r1);
+    constructor 1;
+     [ apply (r \sub\c ∘ r1 \sub\c) 
+     | 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
+        ]]
+  | 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]]
+qed.
+
+definition BP: category.
+ constructor 1;
+  [ apply basic_pair
+  | apply relation_pair_setoid
+  | apply id
+  | apply relation_pair_composition
+  | 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 
+  | 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]]
+  | 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]]
+  ]
+qed.

diff --git a/helm/software/matita/library/formal_topology/categories.ma b/helm/software/matita/library/formal_topology/categories.ma
new file mode 100644 (file)
index 0000000..7adcb27
--- /dev/null
+++ b/helm/software/matita/library/formal_topology/categories.ma
@@ -0,0 +1,47 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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