commit by user andrea

[helm.git] / weblib / basics / relations.ma

diff --git a/weblib/basics/relations.ma b/weblib/basics/relations.ma
index b31db23d7ffffa2e8e6888311194c70ce73b3029..753ed32332d9e51f4a09f306ac9b929c103e96d3 100644 (file)
--- a/weblib/basics/relations.ma
+++ b/weblib/basics/relations.ma
@@ -15,27 +15,27 @@ include "basics/logic.ma".
 definition relation : Type[0] → Type[0]
 ≝ λA.A→A→Prop. 
 
-definition reflexive: ∀A.∀R :relation A.Prop
+definition reflexive: ∀A.∀R :\ 5a href="cic:/matita/basics/relations/relation.def(1)"\ 6relation\ 5/a\ 6 A.Prop
 ≝ λA.λR.∀x:A.R x x.
 
-definition symmetric: ∀A.∀R: relation A.Prop
+definition symmetric: ∀A.∀R: \ 5a href="cic:/matita/basics/relations/relation.def(1)"\ 6relation\ 5/a\ 6 A.Prop
 ≝ λA.λR.∀x,y:A.R x y → R y x.
 
-definition transitive: ∀A.∀R:relation A.Prop
+definition transitive: ∀A.∀R:\ 5a href="cic:/matita/basics/relations/relation.def(1)"\ 6relation\ 5/a\ 6 A.Prop
 ≝ λA.λR.∀x,y,z:A.R x y → R y z → R x z.
 
-definition irreflexive: ∀A.∀R:relation A.Prop
-≝ λA.λR.∀x:A.¬(R x x).
+definition irreflexive: ∀A.∀R:\ 5a href="cic:/matita/basics/relations/relation.def(1)"\ 6relation\ 5/a\ 6 A.Prop
+≝ λA.λR.∀x:A.\ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6(R x x).
 
-definition cotransitive: ∀A.∀R:relation A.Prop
-≝ λA.λR.∀x,y:A.R x y → ∀z:A. R x z ∨ R z y.
+definition cotransitive: ∀A.∀R:\ 5a href="cic:/matita/basics/relations/relation.def(1)"\ 6relation\ 5/a\ 6 A.Prop
+≝ λA.λR.∀x,y:A.R x y → ∀z:A. R x z \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 R z y.
 
-definition tight_apart: ∀A.∀eq,ap:relation A.Prop
-≝ λA.λeq,ap.∀x,y:A. (¬(ap x y) → eq x y) ∧
-(eq x y → ¬(ap x y)).
+definition tight_apart: ∀A.∀eq,ap:\ 5a href="cic:/matita/basics/relations/relation.def(1)"\ 6relation\ 5/a\ 6 A.Prop
+≝ λA.λeq,ap.∀x,y:A. (\ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6(ap x y) → eq x y) \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6
+(eq x y → \ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6(ap x y)).
 
-definition antisymmetric: ∀A.∀R:relation A.Prop
-≝ λA.λR.∀x,y:A. R x y → ¬(R y x).
+definition antisymmetric: ∀A.∀R:\ 5a href="cic:/matita/basics/relations/relation.def(1)"\ 6relation\ 5/a\ 6 A.Prop
+≝ λA.λR.∀x,y:A. R x y → \ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6(R y x).
 
 (********** functions **********)
 
@@ -45,19 +45,19 @@ definition compose ≝
 interpretation "function composition" 'compose f g = (compose ? ? ? f g).
 
 definition injective: ∀A,B:Type[0].∀ f:A→B.Prop
-≝ λA,B.λf.∀x,y:A.f x = f y → x=y.
+≝ λA,B.λf.∀x,y:A.f x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 f y → x\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6y.
 
 definition surjective: ∀A,B:Type[0].∀f:A→B.Prop
-≝λA,B.λf.∀z:B.∃x:A.z = f x.
+≝λA,B.λf.∀z:B.\ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6x:A\ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6.\ 5/a\ 6z \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 f x.
 
 definition commutative: ∀A:Type[0].∀f:A→A→A.Prop 
-≝ λA.λf.∀x,y.f x y = f y x.
+≝ λA.λf.∀x,y.f x y \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 f y x.
 
 definition commutative2: ∀A,B:Type[0].∀f:A→A→B.Prop
-≝ λA,B.λf.∀x,y.f x y = f y x.
+≝ λA,B.λf.∀x,y.f x y \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 f y x.
 
 definition associative: ∀A:Type[0].∀f:A→A→A.Prop
-≝ λA.λf.∀x,y,z.f (f x y) z = f x (f y z).
+≝ λA.λf.∀x,y,z.f (f x y) z \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 f x (f y z).
 
 (* functions and relations *)
 definition monotonic : ∀A:Type[0].∀R:A→A→Prop.
@@ -66,25 +66,25 @@ definition monotonic : ∀A:Type[0].∀R:A→A→Prop.
 
 (* functions and functions *)
 definition distributive: ∀A:Type[0].∀f,g:A→A→A.Prop
-≝ λA.λf,g.∀x,y,z:A. f x (g y z) = g (f x y) (f x z).
+≝ λA.λf,g.∀x,y,z:A. f x (g y z) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 g (f x y) (f x z).
 
 definition distributive2: ∀A,B:Type[0].∀f:A→B→B.∀g:B→B→B.Prop
-≝ λA,B.λf,g.∀x:A.∀y,z:B. f x (g y z) = g (f x y) (f x z).
+≝ λA,B.λf,g.∀x:A.∀y,z:B. f x (g y z) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 g (f x y) (f x z).
 
 lemma injective_compose : ∀A,B,C,f,g.
-injective A B f → injective B C g → injective A C (λx.g (f x)).
-/3/; qed.
+\ 5a href="cic:/matita/basics/relations/injective.def(1)"\ 6injective\ 5/a\ 6 A B f → \ 5a href="cic:/matita/basics/relations/injective.def(1)"\ 6injective\ 5/a\ 6 B C g → \ 5a href="cic:/matita/basics/relations/injective.def(1)"\ 6injective\ 5/a\ 6 A C (λx.g (f x)).
+/\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/; qed.
 
 (* extensional equality *)
 
 definition exteqP: ∀A:Type[0].∀P,Q:A→Prop.Prop ≝
-λA.λP,Q.∀a.iff (P a) (Q a).
+λA.λP,Q.∀a.\ 5a href="cic:/matita/basics/logic/iff.def(1)"\ 6iff\ 5/a\ 6 (P a) (Q a).
 
 definition exteqR: ∀A,B:Type[0].∀R,S:A→B→Prop.Prop ≝
-λA,B.λR,S.∀a.∀b.iff (R a b) (S a b).
+λA,B.λR,S.∀a.∀b.\ 5a href="cic:/matita/basics/logic/iff.def(1)"\ 6iff\ 5/a\ 6 (R a b) (S a b).
 
 definition exteqF: ∀A,B:Type[0].∀f,g:A→B.Prop ≝
-λA,B.λf,g.∀a.f a = g a.
+λA,B.λf,g.∀a.f a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 g a.
 
 notation " x \eqP y " non associative with precedence 45 
 for @{'eqP A $x $y}.
@@ -102,5 +102,4 @@ notation " f \eqF g " non associative with precedence 45
 for @{'eqF ? ? f g}.
 
 interpretation "functional extentional equality" 
-'eqF A B f g = (exteqF A B f g).
-
+'eqF A B f g = (exteqF A B f g).
\ No newline at end of file