From: Claudio Sacerdoti Coen <claudio.sacerdoticoen@unibo.it>
Date: Sat, 23 Aug 2008 19:12:21 +0000 (+0000)
Subject: Some notation moved to core_notation.
X-Git-Tag: make_still_working~4851
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=42c44d828983e4ea2d115eba20a8020b62108384;p=helm.git

Some notation moved to core_notation.
A few new things in datatypes/subsets and related stuff.
---

diff --git a/helm/software/matita/core_notation.moo b/helm/software/matita/core_notation.moo
index 1ea2f922b..9891129af 100644
--- a/helm/software/matita/core_notation.moo
+++ b/helm/software/matita/core_notation.moo
@@ -137,6 +137,15 @@ notation "hvbox(\lnot a)"
   non associative with precedence 40
 for @{ 'not $a }.
 
+notation > "hvbox(a break \liff b)" 
+  left associative with precedence 25
+for @{ 'iff $a $b }.
+
+notation "hvbox(a break \leftrightarrow b)" 
+  left associative with precedence 25
+for @{ 'iff $a $b }.
+
+
 notation "hvbox(\Omega \sup term 90 A)" non associative with precedence 70
 for @{ 'powerset $A }.
 
diff --git a/helm/software/matita/library/datatypes/subsets.ma b/helm/software/matita/library/datatypes/subsets.ma
index f9fae64a0..96ec347c7 100644
--- a/helm/software/matita/library/datatypes/subsets.ma
+++ b/helm/software/matita/library/datatypes/subsets.ma
@@ -40,6 +40,101 @@ 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}.
diff --git a/helm/software/matita/library/logic/coimplication.ma b/helm/software/matita/library/logic/coimplication.ma
index a8fc4a232..67f04b903 100644
--- a/helm/software/matita/library/logic/coimplication.ma
+++ b/helm/software/matita/library/logic/coimplication.ma
@@ -17,16 +17,7 @@ include "logic/connectives.ma".
 definition Iff : Prop \to Prop \to Prop \def
    \lambda A,B. (A \to B) \land (B \to A).
    
- (*CSC: the URI must disappear: there is a bug now *)
-interpretation "logical iff" 'iff x y = (cic:/matita/logic/coimplication/Iff.con x y).
-
-notation > "hvbox(a break \liff b)" 
-  left associative with precedence 25
-for @{ 'iff $a $b }.
-
-notation < "hvbox(a break \leftrightarrow b)" 
-  left associative with precedence 25
-for @{ 'iff $a $b }.
+interpretation "logical iff" 'iff x y = (Iff x y).
 
 theorem iff_intro: \forall A,B. (A \to B) \to (B \to A) \to (A \liff B).
  unfold Iff. intros. split; intros; autobatch.
diff --git a/helm/software/matita/library/logic/cprop_connectives.ma b/helm/software/matita/library/logic/cprop_connectives.ma
index 3b43ef399..c0bbdda18 100644
--- a/helm/software/matita/library/logic/cprop_connectives.ma
+++ b/helm/software/matita/library/logic/cprop_connectives.ma
@@ -58,6 +58,13 @@ notation < "hvbox(a break ∧ b break ∧ c break ∧ d)" with precedence 35 for
  
 interpretation "constructive quaternary and" 'and4 x y z t = (And4 x y z t).
 
+record Iff (A,B:CProp) : CProp ≝
+ { if: A → B;
+   fi: B → A
+ }.
+
+interpretation "logical iff" 'iff x y = (Iff x y).
+
 inductive exT (A:Type) (P:A→CProp) : CProp ≝
   ex_introT: ∀w:A. P w → exT A P.