Require Import Relation_Definitions.
Require Import List.
| Some operators to build relations |
Section Transitive_Closure.
Variable A : Set.
Variable R : relation A.
Inductive clos_trans : A -> A -> Prop :=
| t_step : forall x y:A, R x y -> clos_trans x y
| t_trans :
forall x y z:A, clos_trans x y -> clos_trans y z -> clos_trans x z.
End Transitive_Closure.
Section Reflexive_Transitive_Closure.
Variable A : Set.
Variable R : relation A.
Inductive clos_refl_trans : relation A :=
| rt_step : forall x y:A, R x y -> clos_refl_trans x y
| rt_refl : forall x:A, clos_refl_trans x x
| rt_trans :
forall x y z:A,
clos_refl_trans x y -> clos_refl_trans y z -> clos_refl_trans x z.
End Reflexive_Transitive_Closure.
Section Reflexive_Symetric_Transitive_Closure.
Variable A : Set.
Variable R : relation A.
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,
clos_refl_sym_trans x y ->
clos_refl_sym_trans y z -> clos_refl_sym_trans x z.
End Reflexive_Symetric_Transitive_Closure.
Section Transposee.
Variable A : Set.
Variable R : relation A.
Definition transp (x y:A) := R y x.
End Transposee.
Section Union.
Variable A : Set.
Variables R1 R2 : relation A.
Definition union (x y:A) := R1 x y \/ R2 x y.
End Union.
Section Disjoint_Union.
Variables A B : Set.
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 B x) (inl B y)
| le_ab : forall (x:A) (y:B), le_AsB (inl B x) (inr A y)
| le_bb : forall x y:B, leB x y -> le_AsB (inr A x) (inr A y).
End Disjoint_Union.
Section Lexicographic_Product.
Variable A : Set.
Variable B : A -> Set.
Variable leA : A -> A -> Prop.
Variable leB : forall x:A, B x -> B x -> Prop.
Inductive lexprod : sigS B -> sigS B -> Prop :=
| left_lex :
forall (x x':A) (y:B x) (y':B x'),
leA x x' -> lexprod (existS B x y) (existS B x' y')
| 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.
Section Symmetric_Product.
Variable A : Set.
Variable B : Set.
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.
Section Swap.
Variable A : Set.
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.
End Swap.
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 : 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).
Inductive Desc : List -> Prop :=
| d_nil : Desc Nil
| d_one : forall x:A, Desc (x :: Nil)
| d_conc :
forall (x y:A) (l:List),
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 v62.
Hint Resolve t_step rt_step rt_refl rst_step rst_refl: sets v62.
Hint Immediate rst_sym: sets v62.