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-rw-r--r--theories/Relations/Relation_Operators.v132
1 files changed, 91 insertions, 41 deletions
diff --git a/theories/Relations/Relation_Operators.v b/theories/Relations/Relation_Operators.v
index 87cd1e6f..027a9e6c 100644
--- a/theories/Relations/Relation_Operators.v
+++ b/theories/Relations/Relation_Operators.v
@@ -6,68 +6,119 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: Relation_Operators.v 10681 2008-03-16 13:40:45Z msozeau $ i*)
+(*i $Id: Relation_Operators.v 11481 2008-10-20 19:23:51Z herbelin $ i*)
-(****************************************************************************)
-(* Bruno Barras, Cristina Cornes *)
-(* *)
-(* Some of these definitons were taken from : *)
-(* Constructing Recursion Operators in Type Theory *)
-(* L. Paulson JSC (1986) 2, 325-355 *)
-(****************************************************************************)
+(************************************************************************)
+(** * Bruno Barras, Cristina Cornes *)
+(** * *)
+(** * Some of these definitions were taken from : *)
+(** * Constructing Recursion Operators in Type Theory *)
+(** * L. Paulson JSC (1986) 2, 325-355 *)
+(************************************************************************)
Require Import Relation_Definitions.
Require Import List.
-(** Some operators to build relations *)
+(** * Some operators to build relations *)
+
+(** ** Transitive closure *)
Section Transitive_Closure.
Variable A : Type.
Variable R : relation A.
-
+
+ (** Definition by direct transitive closure *)
+
Inductive clos_trans (x: A) : A -> Prop :=
- | t_step : forall y:A, R x y -> clos_trans x y
- | t_trans :
- forall y z:A, clos_trans x y -> clos_trans y z -> clos_trans x z.
+ | t_step (y:A) : R x y -> clos_trans x y
+ | t_trans (y z:A) : clos_trans x y -> clos_trans y z -> clos_trans x z.
+
+ (** Alternative definition by transitive extension on the left *)
+
+ Inductive clos_trans_1n (x: A) : A -> Prop :=
+ | t1n_step (y:A) : R x y -> clos_trans_1n x y
+ | t1n_trans (y z:A) : R x y -> clos_trans_1n y z -> clos_trans_1n x z.
+
+ (** Alternative definition by transitive extension on the right *)
+
+ Inductive clos_trans_n1 (x: A) : A -> Prop :=
+ | tn1_step (y:A) : R x y -> clos_trans_n1 x y
+ | tn1_trans (y z:A) : R y z -> clos_trans_n1 x y -> clos_trans_n1 x z.
+
End Transitive_Closure.
+(** ** Reflexive-transitive closure *)
Section Reflexive_Transitive_Closure.
Variable A : Type.
Variable R : relation A.
- Inductive clos_refl_trans (x:A) : A -> Prop:=
- | rt_step : forall y:A, R x y -> clos_refl_trans x y
+ (** Definition by direct reflexive-transitive closure *)
+
+ Inductive clos_refl_trans (x:A) : A -> Prop :=
+ | rt_step (y:A) : R x y -> clos_refl_trans x y
| rt_refl : clos_refl_trans x x
- | rt_trans :
- forall y z:A,
+ | rt_trans (y z:A) :
clos_refl_trans x y -> clos_refl_trans y z -> clos_refl_trans x z.
+
+ (** Alternative definition by transitive extension on the left *)
+
+ Inductive clos_refl_trans_1n (x: A) : A -> Prop :=
+ | rt1n_refl : clos_refl_trans_1n x x
+ | rt1n_trans (y z:A) :
+ R x y -> clos_refl_trans_1n y z -> clos_refl_trans_1n x z.
+
+ (** Alternative definition by transitive extension on the right *)
+
+ Inductive clos_refl_trans_n1 (x: A) : A -> Prop :=
+ | rtn1_refl : clos_refl_trans_n1 x x
+ | rtn1_trans (y z:A) :
+ R y z -> clos_refl_trans_n1 x y -> clos_refl_trans_n1 x z.
+
End Reflexive_Transitive_Closure.
+(** ** Reflexive-symmetric-transitive closure *)
Section Reflexive_Symetric_Transitive_Closure.
Variable A : Type.
Variable R : relation A.
+ (** Definition by direct reflexive-symmetric-transitive closure *)
+
Inductive clos_refl_sym_trans : relation A :=
- | rst_step : forall x y:A, R x y -> clos_refl_sym_trans x y
- | rst_refl : forall x:A, clos_refl_sym_trans x x
- | rst_sym :
- forall x y:A, clos_refl_sym_trans x y -> clos_refl_sym_trans y x
- | rst_trans :
- forall x y z:A,
+ | rst_step (x y:A) : R x y -> clos_refl_sym_trans x y
+ | rst_refl (x:A) : clos_refl_sym_trans x x
+ | rst_sym (x y:A) : clos_refl_sym_trans x y -> clos_refl_sym_trans y x
+ | rst_trans (x y z:A) :
clos_refl_sym_trans x y ->
clos_refl_sym_trans y z -> clos_refl_sym_trans x z.
+
+ (** Alternative definition by symmetric-transitive extension on the left *)
+
+ Inductive clos_refl_sym_trans_1n (x: A) : A -> Prop :=
+ | rts1n_refl : clos_refl_sym_trans_1n x x
+ | rts1n_trans (y z:A) : R x y \/ R y x ->
+ clos_refl_sym_trans_1n y z -> clos_refl_sym_trans_1n x z.
+
+ (** Alternative definition by symmetric-transitive extension on the right *)
+
+ Inductive clos_refl_sym_trans_n1 (x: A) : A -> Prop :=
+ | rtsn1_refl : clos_refl_sym_trans_n1 x x
+ | rtsn1_trans (y z:A) : R y z \/ R z y ->
+ clos_refl_sym_trans_n1 x y -> clos_refl_sym_trans_n1 x z.
+
End Reflexive_Symetric_Transitive_Closure.
+(** ** Converse of a relation *)
-Section Transposee.
+Section Converse.
Variable A : Type.
Variable R : relation A.
Definition transp (x y:A) := R y x.
-End Transposee.
+End Converse.
+(** ** Union of relations *)
Section Union.
Variable A : Type.
@@ -76,6 +127,7 @@ Section Union.
Definition union (x y:A) := R1 x y \/ R2 x y.
End Union.
+(** ** Disjoint union of relations *)
Section Disjoint_Union.
Variables A B : Type.
@@ -83,16 +135,15 @@ Variable leA : A -> A -> Prop.
Variable leB : B -> B -> Prop.
Inductive le_AsB : A + B -> A + B -> Prop :=
- | le_aa : forall x y:A, leA x y -> le_AsB (inl _ x) (inl _ y)
- | le_ab : forall (x:A) (y:B), le_AsB (inl _ x) (inr _ y)
- | le_bb : forall x y:B, leB x y -> le_AsB (inr _ x) (inr _ y).
+ | le_aa (x y:A) : leA x y -> le_AsB (inl _ x) (inl _ y)
+ | le_ab (x:A) (y:B) : le_AsB (inl _ x) (inr _ y)
+ | le_bb (x y:B) : leB x y -> le_AsB (inr _ x) (inr _ y).
End Disjoint_Union.
-
+(** ** Lexicographic order on dependent pairs *)
Section Lexicographic_Product.
- (* Lexicographic order on dependent pairs *)
Variable A : Type.
Variable B : A -> Type.
@@ -106,8 +157,10 @@ Section Lexicographic_Product.
| right_lex :
forall (x:A) (y y':B x),
leB x y y' -> lexprod (existS B x y) (existS B x y').
+
End Lexicographic_Product.
+(** ** Product of relations *)
Section Symmetric_Product.
Variable A : Type.
@@ -123,16 +176,15 @@ Section Symmetric_Product.
End Symmetric_Product.
+(** ** Multiset of two relations *)
Section Swap.
Variable A : Type.
Variable R : A -> A -> Prop.
Inductive swapprod : A * A -> A * A -> Prop :=
- | sp_noswap : forall x x':A * A, symprod A A R R x x' -> swapprod x x'
- | sp_swap :
- forall (x y:A) (p:A * A),
- symprod A A R R (x, y) p -> swapprod (y, x) p.
+ | sp_noswap x y (p:A * A) : symprod A A R R (x, y) p -> swapprod (x, y) p
+ | sp_swap x y (p:A * A) : symprod A A R R (x, y) p -> swapprod (y, x) p.
End Swap.
@@ -144,16 +196,14 @@ Section Lexicographic_Exponentiation.
Let List := list A.
Inductive Ltl : List -> List -> Prop :=
- | Lt_nil : forall (a:A) (x:List), Ltl Nil (a :: x)
- | Lt_hd : forall a b:A, leA a b -> forall x y:list A, Ltl (a :: x) (b :: y)
- | Lt_tl : forall (a:A) (x y:List), Ltl x y -> Ltl (a :: x) (a :: y).
-
+ | Lt_nil (a:A) (x:List) : Ltl Nil (a :: x)
+ | Lt_hd (a b:A) : leA a b -> forall x y:list A, Ltl (a :: x) (b :: y)
+ | Lt_tl (a:A) (x y:List) : Ltl x y -> Ltl (a :: x) (a :: y).
Inductive Desc : List -> Prop :=
| d_nil : Desc Nil
- | d_one : forall x:A, Desc (x :: Nil)
- | d_conc :
- forall (x y:A) (l:List),
+ | d_one (x:A) : Desc (x :: Nil)
+ | d_conc (x y:A) (l:List) :
leA x y -> Desc (l ++ y :: Nil) -> Desc ((l ++ y :: Nil) ++ x :: Nil).
Definition Pow : Set := sig Desc.
generated by cgit v1.2.3 (git 2.39.1) at 2025-01-29 20:06:38 +0000