minor fixes

[helm.git] / helm / software / matita / contribs / formal_topology / overlap / cprop_connectives.ma

diff --git a/helm/software/matita/contribs/formal_topology/overlap/cprop_connectives.ma b/helm/software/matita/contribs/formal_topology/overlap/cprop_connectives.ma
index 36616b2f8fc74dc10a8f53170abe94b4b1915229..13bb825ee9edc3ecca39ff0cecf3f90112cd40f2 100644 (file)
--- a/helm/software/matita/contribs/formal_topology/overlap/cprop_connectives.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/cprop_connectives.ma
@@ -14,23 +14,35 @@
 
 include "logic/connectives.ma".
 
-definition Type3 : Type := Type.
+definition Type4 : Type := Type.
+definition Type3 : Type4 := Type.
 definition Type2 : Type3 := Type.
 definition Type1 : Type2 := Type.
 definition Type0 : Type1 := Type.
 
-definition Type_OF_Type0: Type0 → Type := λx.x.
-definition Type_OF_Type1: Type1 → Type := λx.x.
-definition Type_OF_Type2: Type2 → Type := λx.x.
-definition Type_OF_Type3: Type3 → Type := λx.x.
-coercion Type_OF_Type0.
-coercion Type_OF_Type1.
-coercion Type_OF_Type2.
-coercion Type_OF_Type3.
+definition Type_of_Type0: Type0 → Type := λx.x.
+definition Type_of_Type1: Type1 → Type := λx.x.
+definition Type_of_Type2: Type2 → Type := λx.x.
+definition Type_of_Type3: Type3 → Type := λx.x.
+definition Type_of_Type4: Type4 → Type := λx.x.
+coercion Type_of_Type0.
+coercion Type_of_Type1.
+coercion Type_of_Type2.
+coercion Type_of_Type3.
+coercion Type_of_Type4.
 
 definition CProp0 : Type1 := Type0.
 definition CProp1 : Type2 := Type1.
 definition CProp2 : Type3 := Type2.
+definition CProp3 : Type4 := Type3.
+definition CProp_of_CProp0: CProp0 → CProp ≝ λx.x.
+definition CProp_of_CProp1: CProp1 → CProp ≝ λx.x.
+definition CProp_of_CProp2: CProp2 → CProp ≝ λx.x.
+definition CProp_of_CProp3: CProp3 → CProp ≝ λx.x.
+coercion CProp_of_CProp0.
+coercion CProp_of_CProp1.
+coercion CProp_of_CProp2.
+coercion CProp_of_CProp3.
 
 inductive Or (A,B:CProp0) : CProp0 ≝
  | Left : A → Or A B
@@ -86,18 +98,28 @@ record Iff1 (A,B:CProp1) : CProp1 ≝
    fi1: B → A
  }.
  
-interpretation "logical iff" 'iff x y = (Iff x y).
-
 notation "hvbox(a break ⇔ b)" right associative with precedence 25 for @{'iff1 $a $b}.
+interpretation "logical iff" 'iff x y = (Iff x y).
 interpretation "logical iff type1" 'iff1 x y = (Iff1 x y).
 
+inductive exT22 (A:Type2) (P:A→CProp2) : CProp2 ≝
+  ex_introT22: ∀w:A. P w → exT22 A P.
+  
+interpretation "CProp2 exists" 'exists \eta.x = (exT22 _ x).
+
+definition pi1exT22 ≝ λA,P.λx:exT22 A P.match x with [ex_introT22 x _ ⇒ x].
+definition pi2exT22 ≝ 
+  λA,P.λx:exT22 A P.match x return λx.P (pi1exT22 ?? x) with [ex_introT22 _ p ⇒ p].
+  
+interpretation "exT22 \fst" 'pi1 = (pi1exT22 _ _).
+interpretation "exT22 \snd" 'pi2 = (pi2exT22 _ _).
+interpretation "exT22 \fst a" 'pi1a x = (pi1exT22 _ _ x).
+interpretation "exT22 \snd a" 'pi2a x = (pi2exT22 _ _ x).
+interpretation "exT22 \fst b" 'pi1b x y = (pi1exT22 _ _ x y).
+interpretation "exT22 \snd b" 'pi2b x y = (pi2exT22 _ _ x y).
+
 inductive exT (A:Type0) (P:A→CProp0) : CProp0 ≝
   ex_introT: ∀w:A. P w → exT A P.
-  
-notation "\ll term 19 a, break term 19 b \gg" 
-with precedence 90 for @{'dependent_pair $a $b}.
-interpretation "dependent pair" 'dependent_pair a b = 
-  (ex_introT _ _ a b).
 
 interpretation "CProp exists" 'exists \eta.x = (exT _ x).
 
@@ -112,11 +134,11 @@ definition pi2exT ≝
   λA,P.λx:exT A P.match x return λx.P (pi1exT ?? x) with [ex_introT _ p ⇒ p].
 
 interpretation "exT \fst" 'pi1 = (pi1exT _ _).
-interpretation "exT \fst" 'pi1a x = (pi1exT _ _ x).
-interpretation "exT \fst" 'pi1b x y = (pi1exT _ _ x y).
+interpretation "exT \fst a" 'pi1a x = (pi1exT _ _ x).
+interpretation "exT \fst b" 'pi1b x y = (pi1exT _ _ x y).
 interpretation "exT \snd" 'pi2 = (pi2exT _ _).
-interpretation "exT \snd" 'pi2a x = (pi2exT _ _ x).
-interpretation "exT \snd" 'pi2b x y = (pi2exT _ _ x y).
+interpretation "exT \snd a" 'pi2a x = (pi2exT _ _ x).
+interpretation "exT \snd b" 'pi2b x y = (pi2exT _ _ x y).
 
 inductive exT23 (A:Type0) (P:A→CProp0) (Q:A→CProp0) (R:A→A→CProp0) : CProp0 ≝
   ex_introT23: ∀w,p:A. P w → Q p → R w p → exT23 A P Q R.
@@ -128,28 +150,43 @@ definition pi2exT23 ≝
 
 interpretation "exT2 \fst" 'pi1 = (pi1exT23 _ _ _ _).
 interpretation "exT2 \snd" 'pi2 = (pi2exT23 _ _ _ _).   
-interpretation "exT2 \fst" 'pi1a x = (pi1exT23 _ _ _ _ x).
-interpretation "exT2 \snd" 'pi2a x = (pi2exT23 _ _ _ _ x).
-interpretation "exT2 \fst" 'pi1b x y = (pi1exT23 _ _ _ _ x y).
-interpretation "exT2 \snd" 'pi2b x y = (pi2exT23 _ _ _ _ x y).
+interpretation "exT2 \fst a" 'pi1a x = (pi1exT23 _ _ _ _ x).
+interpretation "exT2 \snd a" 'pi2a x = (pi2exT23 _ _ _ _ x).
+interpretation "exT2 \fst b" 'pi1b x y = (pi1exT23 _ _ _ _ x y).
+interpretation "exT2 \snd b" 'pi2b x y = (pi2exT23 _ _ _ _ x y).
 
 inductive exT2 (A:Type0) (P,Q:A→CProp0) : CProp0 ≝
   ex_introT2: ∀w:A. P w → Q w → exT2 A P Q.
 
+
 definition Not : CProp0 → Prop ≝ λx:CProp.x → False.
 
 interpretation "constructive not" 'not x = (Not x).
   
-definition cotransitive ≝
+definition cotransitive: ∀C:Type0. ∀lt:C→C→CProp0.CProp0 ≝
  λC:Type0.λlt:C→C→CProp0.∀x,y,z:C. lt x y → lt x z ∨ lt z y. 
 
-definition coreflexive ≝ λC:Type0.λlt:C→C→CProp0. ∀x:C. ¬ (lt x x).
+definition coreflexive: ∀C:Type0. ∀lt:C→C→CProp0.CProp0 ≝
+ λC:Type0.λlt:C→C→CProp0. ∀x:C. ¬ (lt x x).
 
-definition symmetric ≝ λC:Type0.λlt:C→C→CProp0. ∀x,y:C.lt x y → lt y x.
+definition symmetric: ∀C:Type0. ∀lt:C→C→CProp0.CProp0 ≝
+ λC:Type0.λlt:C→C→CProp0. ∀x,y:C.lt x y → lt y x.
 
-definition antisymmetric ≝ λA:Type0.λR:A→A→CProp0.λeq:A→A→Prop.∀x:A.∀y:A.R x y→R y x→eq x y.
+definition antisymmetric: ∀A:Type0. ∀R:A→A→CProp0. ∀eq:A→A→Prop.CProp0 ≝
+ λA:Type0.λR:A→A→CProp0.λeq:A→A→Prop.∀x:A.∀y:A.R x y→R y x→eq x y.
 
-definition reflexive ≝ λA:Type0.λR:A→A→CProp0.∀x:A.R x x.
+definition reflexive: ∀C:Type0. ∀lt:C→C→CProp0.CProp0 ≝ λA:Type0.λR:A→A→CProp0.∀x:A.R x x.
 
-definition transitive ≝ λA:Type0.λR:A→A→CProp0.∀x,y,z:A.R x y → R y z → R x z.
- 
+definition transitive: ∀C:Type0. ∀lt:C→C→CProp0.CProp0 ≝ λA:Type0.λR:A→A→CProp0.∀x,y,z:A.R x y → R y z → R x z.
+
+definition reflexive1: ∀A:Type1.∀R:A→A→CProp1.CProp1 ≝ λA:Type1.λR:A→A→CProp1.∀x:A.R x x.
+definition symmetric1: ∀A:Type1.∀R:A→A→CProp1.CProp1 ≝ λC:Type1.λlt:C→C→CProp1. ∀x,y:C.lt x y → lt y x.
+definition transitive1: ∀A:Type1.∀R:A→A→CProp1.CProp1 ≝ λA:Type1.λR:A→A→CProp1.∀x,y,z:A.R x y → R y z → R x z.
+
+definition reflexive2: ∀A:Type2.∀R:A→A→CProp2.CProp2 ≝ λA:Type2.λR:A→A→CProp2.∀x:A.R x x.
+definition symmetric2: ∀A:Type2.∀R:A→A→CProp2.CProp2 ≝ λC:Type2.λlt:C→C→CProp2. ∀x,y:C.lt x y → lt y x.
+definition transitive2: ∀A:Type2.∀R:A→A→CProp2.CProp2 ≝ λA:Type2.λR:A→A→CProp2.∀x,y,z:A.R x y → R y z → R x z.
+
+definition reflexive3: ∀A:Type3.∀R:A→A→CProp3.CProp3 ≝ λA:Type3.λR:A→A→CProp3.∀x:A.R x x.
+definition symmetric3: ∀A:Type3.∀R:A→A→CProp3.CProp3 ≝ λC:Type3.λlt:C→C→CProp3. ∀x,y:C.lt x y → lt y x.
+definition transitive3: ∀A:Type3.∀R:A→A→CProp3.CProp3 ≝ λA:Type3.λR:A→A→CProp3.∀x,y,z:A.R x y → R y z → R x z.