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(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
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(** * Some operators on relations *)
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(** * Initial authors: Bruno Barras, Cristina Cornes *)
(** * *)
(** * Some of the initial definitions were taken from : *)
(** * Constructing Recursion Operators in Type Theory *)
(** * L. Paulson JSC (1986) 2, 325-355 *)
(** * *)
(** * Further extensions by Pierre Castéran *)
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Require Import Relation_Definitions.
(** ** 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 (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 closure *)
Section Reflexive_Closure.
Variable A : Type.
Variable R : relation A.
(** Definition by direct transitive closure *)
Inductive clos_refl (x: A) : A -> Prop :=
| r_step (y:A) : R x y -> clos_refl x y
| r_refl : clos_refl x x.
End Reflexive_Closure.
(** ** Reflexive-transitive closure *)
Section Reflexive_Transitive_Closure.
Variable A : Type.
Variable R : relation A.
(** 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 (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_Symmetric_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 (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 :=
| rst1n_refl : clos_refl_sym_trans_1n x x
| rst1n_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 :=
| rstn1_refl : clos_refl_sym_trans_n1 x x
| rstn1_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_Symmetric_Transitive_Closure.
(** ** Converse of a relation *)
Section Converse.
Variable A : Type.
Variable R : relation A.
Definition transp (x y:A) := R y x.
End Converse.
(** ** Union of relations *)
Section Union.
Variable A : Type.
Variables R1 R2 : relation A.
Definition union (x y:A) := R1 x y \/ R2 x y.
End Union.
(** ** Disjoint union of relations *)
Section Disjoint_Union.
Variables A B : Type.
Variable leA : A -> A -> Prop.
Variable leB : B -> B -> Prop.
Inductive le_AsB : A + B -> A + B -> Prop :=
| 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.
Variable A : Type.
Variable B : A -> Type.
Variable leA : A -> A -> Prop.
Variable leB : forall x:A, B x -> B x -> Prop.
Inductive lexprod : sigT B -> sigT B -> Prop :=
| left_lex :
forall (x x':A) (y:B x) (y':B x'),
leA x x' -> lexprod (existT B x y) (existT B x' y')
| right_lex :
forall (x:A) (y y':B x),
leB x y y' -> lexprod (existT B x y) (existT B x y').
End Lexicographic_Product.
(** ** Product of relations *)
Section Symmetric_Product.
Variable A : Type.
Variable B : Type.
Variable leA : A -> A -> Prop.
Variable leB : B -> B -> Prop.
Inductive symprod : A * B -> A * B -> Prop :=
| left_sym :
forall x x':A, leA x x' -> forall y:B, symprod (x, y) (x', y)
| right_sym :
forall y y':B, leB y y' -> forall x:A, symprod (x, y) (x, y').
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 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.
Local Open Scope list_scope.
Section Lexicographic_Exponentiation.
Variable A : Set.
Variable leA : A -> A -> Prop.
Let Nil := nil (A:=A).
Let List := list A.
Inductive Ltl : List -> List -> Prop :=
| 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 (x:A) : Desc (x :: Nil)
| d_conc (x y:A) (l:List) :
clos_refl A leA x y -> Desc (l ++ y :: Nil) -> Desc ((l ++ y :: Nil) ++ x :: Nil).
Definition Pow : Set := sig Desc.
Definition lex_exp (a b:Pow) : Prop := Ltl (proj1_sig a) (proj1_sig b).
End Lexicographic_Exponentiation.
Hint Unfold transp union: sets.
Hint Resolve t_step rt_step rt_refl rst_step rst_refl: sets.
Hint Immediate rst_sym: sets.
(* begin hide *)
(* Compatibility *)
Notation rts1n_refl := rst1n_refl (only parsing).
Notation rts1n_trans := rst1n_trans (only parsing).
Notation rtsn1_refl := rstn1_refl (only parsing).
Notation rtsn1_trans := rstn1_trans (only parsing).
(* end hide *)
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