From: Claudio Sacerdoti Coen <claudio.sacerdoticoen@unibo.it>
Date: Wed, 30 Dec 2009 17:31:56 +0000 (+0000)
Subject: Porting of Sambin's stuff started.
X-Git-Tag: make_still_working~3156
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=a8285ad2e16e571100a666bc9178347d9e61dbe5;p=helm.git

Porting of Sambin's stuff started.
---

diff --git a/helm/software/matita/nlibrary/properties/relations2.ma b/helm/software/matita/nlibrary/properties/relations2.ma
new file mode 100644
index 000000000..67d09e938
--- /dev/null
+++ b/helm/software/matita/nlibrary/properties/relations2.ma
@@ -0,0 +1,26 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/pts.ma".
+
+ndefinition reflexive2 ≝ λT:Type[2].λP:T → T → CProp[2]. ∀x.P x x.
+ndefinition symmetric2 ≝ λT:Type[2].λP:T → T → CProp[2]. ∀x,y.P x y → P y x.
+ndefinition transitive2 ≝ λT:Type[2].λP:T → T → CProp[2]. ∀z,x,y. P x z → P z y → P x y.
+
+nrecord equivalence_relation2 (A:Type[2]) : Type[3] ≝
+ { eq_rel2:2> A → A → CProp[2];
+   refl2: reflexive2 ? eq_rel2;
+   sym2: symmetric2 ? eq_rel2;
+   trans2: transitive2 ? eq_rel2
+ }.
diff --git a/helm/software/matita/nlibrary/sets/categories.ma b/helm/software/matita/nlibrary/sets/categories.ma
new file mode 100644
index 000000000..80f3b4cf6
--- /dev/null
+++ b/helm/software/matita/nlibrary/sets/categories.ma
@@ -0,0 +1,98 @@
+
+include "sets/sets.ma".
+
+ndefinition binary_morph_setoid : setoid → setoid → setoid → setoid.
+#S1; #S2; #T; @ (binary_morphism S1 S2 T); @;
+##[ #f; #g; napply (∀x,y. f x y = g x y);
+##| #f; #x; #y; napply #;
+##| #f; #g; #H; #x; #y; napply ((H x y)^-1);
+##| #f; #g; #h; #H1; #H2; #x; #y; napply (trans … (H1 …) (H2 …)); ##]
+nqed.
+
+ndefinition unary_morph_setoid : setoid → setoid → setoid.
+#S1; #S2; @ (unary_morphism S1 S2); @;
+##[ #f; #g; napply (∀x. f x = g x);
+##| #f; #x; napply #;
+##| #f; #g; #H; #x; napply ((H x)^-1);
+##| #f; #g; #h; #H1; #H2; #x; napply (trans … (H1 …) (H2 …)); ##]
+nqed.
+
+nrecord category : Type[2] ≝
+ { objs:> Type[1];
+   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
+ }.
+
+notation "hvbox(A break ⇒ B)" right associative with precedence 50 for @{ 'arrows $A $B }.
+interpretation "arrows1" 'arrows A B = (unary_morphism1 A B).
+interpretation "arrows" 'arrows A B = (unary_morphism A B).
+
+notation > "𝐈𝐝 term 90 A" non associative with precedence 90 for @{ 'id $A }.
+notation < "mpadded width -90% (𝐈) 𝐝 \sub term 90 A" non associative with precedence 90 for @{ 'id $A }.
+
+interpretation "id" 'id A = (id ? A).
+
+ndefinition SETOID : category.
+@; 
+##[ napply setoid;
+##| napply unary_morph_setoid;
+##| #o; @ (λx.x); #a; #b; #H; napply H;
+##| #o1; #o2; #o3; @; 
+    ##[ #f; #g; @(λx.g (f x)); #a; #b; #H; napply (.= (††H)); napply #;
+    ##| #f; #g; #f'; #g'; #H1; #H2; nwhd; #x; napply (.= (H2 (f x)));
+        napply (.= (†(H1 x))); napply #; ##]
+##| #o1; #o2; #o3; #o4; #f; #g; #h; nwhd; #x; napply #;
+##|##6,7: #o1; #o2; #f; nwhd; #x; napply #; ##]
+nqed.
+
+unification hint 0 ≔ ;
+   R ≟ (mk_category setoid unary_morph_setoid (id SETOID) (comp SETOID)
+          (comp_assoc SETOID) (id_neutral_left SETOID)
+          (id_neutral_right SETOID))
+ (* -------------------------------------------------------------------- *) ⊢
+                              objs R ≡ setoid.
+                              
+ unification hint 0 ≔ x,y ;
+   R ≟ (mk_category setoid unary_morph_setoid (id SETOID) (comp SETOID)
+          (comp_assoc SETOID) (id_neutral_left SETOID)
+          (id_neutral_right SETOID))
+ (* -------------------------------------------------------------------- *) ⊢
+                  arrows R x y ≡ unary_morph_setoid x y. 
+                  
+unification hint 0 ≔ A,B ;               
+                  T ≟ (unary_morph_setoid A B)
+           (* ----------------------------------- *) ⊢              
+                  unary_morphism A B ≡ carr T. 
+                
+                
+ndefinition TYPE : setoid1.
+@ setoid; @; 
+##[ #T1; #T2; 
+    alias symbol "eq" = "setoid eq".
+    napply (∃f:T1 ⇒ T2.∃g:T2 ⇒ T1. (∀x.f (g x) = x) ∧ (∀y.g (f y) = y));
+##| #A; @ (𝐈𝐝 A); @ (𝐈𝐝 A); @; #x; napply #;
+##| #A; #B; *; #f; *; #g; *; #Hfg; #Hgf; @g; @f; @; nassumption; 
+##| #A; #B; #C; *; #f; *; #f'; *; #Hf; #Hf'; *; #g; *; #g'; *; #Hg; #Hg'; 
+    @; ##[ @(λx.g (f x)); #a; #b; #H; napply (.= (††H)); napply #;
+       ##| @; ##[ @(λx.f'(g' x)); #a; #b; #H; napply (.= (††H)); napply #; ##]
+    @; #x;
+    ##[ napply (.= (†(Hf …))); napply Hg;
+    ##| napply (.= (†(Hg' …))); napply Hf'; ##] ##] 
+nqed.
+
+unification hint 0 ≔ ;
+          R ≟ (mk_setoid1 setoid (eq1 TYPE))
+  (* -------------------------------------------- *) ⊢
+                 carr1 R ≡ setoid.
+   
+nrecord unary_morphism01 (A : setoid) (B: setoid1) : Type[1] ≝
+ { fun01:1> A → B;
+   prop01: ∀a,a'. eq ? a a' → eq1 ? (fun01 a) (fun01 a')
+ }.
+ 
+interpretation "prop01" 'prop1 c  = (prop01 ????? c).
diff --git a/helm/software/matita/nlibrary/sets/categories2.ma b/helm/software/matita/nlibrary/sets/categories2.ma
new file mode 100644
index 000000000..17d621431
--- /dev/null
+++ b/helm/software/matita/nlibrary/sets/categories2.ma
@@ -0,0 +1,104 @@
+
+include "sets/sets.ma".
+include "sets/setoids2.ma".
+include "sets/categories.ma".
+
+(*
+ndefinition binary_morph_setoid : setoid → setoid → setoid → setoid.
+#S1; #S2; #T; @ (binary_morphism S1 S2 T); @;
+##[ #f; #g; napply (∀x,y. f x y = g x y);
+##| #f; #x; #y; napply #;
+##| #f; #g; #H; #x; #y; napply ((H x y)^-1);
+##| #f; #g; #h; #H1; #H2; #x; #y; napply (trans … (H1 …) (H2 …)); ##]
+nqed.
+
+ndefinition unary_morph_setoid : setoid → setoid → setoid.
+#S1; #S2; @ (unary_morphism S1 S2); @;
+##[ #f; #g; napply (∀x. f x = g x);
+##| #f; #x; napply #;
+##| #f; #g; #H; #x; napply ((H x)^-1);
+##| #f; #g; #h; #H1; #H2; #x; napply (trans … (H1 …) (H2 …)); ##]
+nqed.
+
+nrecord category : Type[2] ≝
+ { objs:> Type[1];
+   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
+ }.
+*)
+
+notation "hvbox(A break ⇒ B)" right associative with precedence 50 for @{ 'arrows $A $B }.
+interpretation "arrows1" 'arrows A B = (unary_morphism1 A B).
+interpretation "arrows" 'arrows A B = (unary_morphism A B).
+
+(*
+notation > "𝐈𝐝 term 90 A" non associative with precedence 90 for @{ 'id $A }.
+notation < "mpadded width -90% (𝐈) 𝐝 \sub term 90 A" non associative with precedence 90 for @{ 'id $A }.
+
+interpretation "id" 'id A = (id ? A).
+
+ndefinition SETOID : category.
+@; 
+##[ napply setoid;
+##| napply unary_morph_setoid;
+##| #o; @ (λx.x); #a; #b; #H; napply H;
+##| #o1; #o2; #o3; @; 
+    ##[ #f; #g; @(λx.g (f x)); #a; #b; #H; napply (.= (††H)); napply #;
+    ##| #f; #g; #f'; #g'; #H1; #H2; nwhd; #x; napply (.= (H2 (f x)));
+        napply (.= (†(H1 x))); napply #; ##]
+##| #o1; #o2; #o3; #o4; #f; #g; #h; nwhd; #x; napply #;
+##|##6,7: #o1; #o2; #f; nwhd; #x; napply #; ##]
+nqed.
+
+unification hint 0 ≔ ;
+   R ≟ (mk_category setoid unary_morph_setoid (id SETOID) (comp SETOID)
+          (comp_assoc SETOID) (id_neutral_left SETOID)
+          (id_neutral_right SETOID))
+ (* -------------------------------------------------------------------- *) ⊢
+                              objs R ≡ setoid.
+                              
+ unification hint 0 ≔ x,y ;
+   R ≟ (mk_category setoid unary_morph_setoid (id SETOID) (comp SETOID)
+          (comp_assoc SETOID) (id_neutral_left SETOID)
+          (id_neutral_right SETOID))
+ (* -------------------------------------------------------------------- *) ⊢
+                  arrows R x y ≡ unary_morph_setoid x y. 
+                  
+unification hint 0 ≔ A,B ;               
+                  T ≟ (unary_morph_setoid A B)
+           (* ----------------------------------- *) ⊢              
+                  unary_morphism A B ≡ carr T. 
+                
+                
+ndefinition TYPE : setoid1.
+@ setoid; @; 
+##[ #T1; #T2; 
+    alias symbol "eq" = "setoid eq".
+    napply (∃f:T1 ⇒ T2.∃g:T2 ⇒ T1. (∀x.f (g x) = x) ∧ (∀y.g (f y) = y));
+##| #A; @ (𝐈𝐝 A); @ (𝐈𝐝 A); @; #x; napply #;
+##| #A; #B; *; #f; *; #g; *; #Hfg; #Hgf; @g; @f; @; nassumption; 
+##| #A; #B; #C; *; #f; *; #f'; *; #Hf; #Hf'; *; #g; *; #g'; *; #Hg; #Hg'; 
+    @; ##[ @(λx.g (f x)); #a; #b; #H; napply (.= (††H)); napply #;
+       ##| @; ##[ @(λx.f'(g' x)); #a; #b; #H; napply (.= (††H)); napply #; ##]
+    @; #x;
+    ##[ napply (.= (†(Hf …))); napply Hg;
+    ##| napply (.= (†(Hg' …))); napply Hf'; ##] ##] 
+nqed.
+
+unification hint 0 ≔ ;
+          R ≟ (mk_setoid1 setoid (eq1 TYPE))
+  (* -------------------------------------------- *) ⊢
+                 carr1 R ≡ setoid.
+   
+nrecord unary_morphism01 (A : setoid) (B: setoid1) : Type[1] ≝
+ { fun01:1> A → B;
+   prop01: ∀a,a'. eq ? a a' → eq1 ? (fun01 a) (fun01 a')
+ }.
+ 
+interpretation "prop01" 'prop1 c  = (prop01 ????? c).
+*)
diff --git a/helm/software/matita/nlibrary/sets/setoids2.ma b/helm/software/matita/nlibrary/sets/setoids2.ma
new file mode 100644
index 000000000..578503f6c
--- /dev/null
+++ b/helm/software/matita/nlibrary/sets/setoids2.ma
@@ -0,0 +1,69 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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 "properties/relations2.ma".
+include "sets/setoids1.ma".
+
+nrecord setoid2: Type[3] ≝
+ { carr2:> Type[2];
+   eq2: equivalence_relation2 carr2
+ }.
+
+ndefinition setoid2_of_setoid1: setoid1 → setoid2.
+ #s; @ (carr1 s); @ (carr1 s) (eq1 ?)
+  [ napply refl1
+  | napply sym1
+  | napply trans1]
+nqed.
+
+(*ncoercion setoid1_of_setoid : ∀s:setoid. setoid1 ≝ setoid1_of_setoid
+ on _s: setoid to setoid1.*)
+(*prefer coercion Type_OF_setoid.*)
+
+interpretation "setoid2 eq" 'eq t x y = (eq_rel2 ? (eq2 t) x y).
+interpretation "setoid1 eq" 'eq t x y = (eq_rel1 ? (eq1 t) x y).
+interpretation "setoid eq" 'eq t x y = (eq_rel ? (eq t) x y).
+
+(*
+notation > "hvbox(a break =_12 b)" non associative with precedence 45
+for @{ eq_rel2 (carr2 (setoid2_of_setoid1 ?)) (eq2 (setoid2_of_setoid1 ?)) $a $b }.
+*)
+notation > "hvbox(a break =_0 b)" non associative with precedence 45
+for @{ eq_rel ? (eq ?) $a $b }.
+notation > "hvbox(a break =_1 b)" non associative with precedence 45
+for @{ eq_rel1 ? (eq1 ?) $a $b }.
+notation > "hvbox(a break =_2 b)" non associative with precedence 45
+for @{ eq_rel2 ? (eq2 ?) $a $b }.
+
+interpretation "setoid2 symmetry" 'invert r = (sym2 ???? r).
+interpretation "setoid1 symmetry" 'invert r = (sym1 ???? r).
+interpretation "setoid symmetry" 'invert r = (sym ???? r).
+notation ".= r" with precedence 50 for @{'trans $r}.
+interpretation "trans2" 'trans r = (trans2 ????? r).
+interpretation "trans1" 'trans r = (trans1 ????? r).
+interpretation "trans" 'trans r = (trans ????? r).
+
+nrecord unary_morphism2 (A,B: setoid2) : Type[2] ≝
+ { fun12:1> A → B;
+   prop12: ∀a,a'. eq2 ? a a' → eq2 ? (fun12 a) (fun12 a')
+ }.
+
+nrecord binary_morphism2 (A,B,C:setoid2) : Type[2] ≝
+ { fun22:2> A → B → C;
+   prop22: ∀a,a',b,b'. eq2 ? a a' → eq2 ? b b' → eq2 ? (fun22 a b) (fun22 a' b')
+ }.
+
+interpretation "prop12" 'prop1 c = (prop12 ????? c).
+interpretation "prop22" 'prop2 l r = (prop22 ???????? l r).
+interpretation "refl2" 'refl = (refl2 ???).
diff --git a/helm/software/matita/nlibrary/topology/igft-setoid.ma b/helm/software/matita/nlibrary/topology/igft-setoid.ma
index 30f2d161e..7235b4ba0 100644
--- a/helm/software/matita/nlibrary/topology/igft-setoid.ma
+++ b/helm/software/matita/nlibrary/topology/igft-setoid.ma
@@ -200,7 +200,7 @@ nlet rec famU (A : nAx) (U : 𝛀^A) (x : Ord A) on x : 𝛀^A ≝
                 @; #w; #Hw; nwhd;
                 ncut (𝐈𝐦[𝐝 y (f i)] = 𝐈𝐦[𝐝 x i]);                    
                   
-    
+(*    
   
 notation < "term 90 U \sub (term 90 x)" non associative with precedence 50 for @{ 'famU $U $x }.
 notation > "U ⎽ term 90 x" non associative with precedence 50 for @{ 'famU $U $x }.
@@ -574,3 +574,4 @@ D*)
 [1]: http://upsilon.cc/~zack/research/publications/notation.pdf 
 
 D*)
+*)
\ No newline at end of file