Library Functions

Definition of functions and constructors

Require Export Relations.

Section Global.

  Section Def.

    Variables X Y X' Y': Type.

    Definition function2 := relation2 X Y -> relation2 X' Y'.
    Definition function := relation2 X Y -> relation2 X Y.

    Definition increasing (F: function) :=
      forall R S, incl R S -> incl (F R) (F S).

    Definition contains (F G: function2) := forall R, incl (F R) (G R).

Constant and identity functions
    Definition constant S: function2 := fun _ => S.
    Definition identity: function := fun R => R.

Binary and general union functions
    Definition Union2 F G: function2 := fun R => union2 (F R) (G R).
    Definition Union I H: function2 := fun R => union (fun i: I => H i R).

  End Def.

  Section Def'.

    Variables X Y Z X' Y' Z' X'' Y'': Type.

    Definition transparent (B: relation X) (F: function2 X Y X Y') :=
      forall R, incl (F (comp (star B) R)) (comp (star B) (F R)).

Chaining functions
    Definition chaining_l (S: relation2 X Y): function2 Y Z X Z := comp S.
    Definition chaining_r (S: relation2 Y Z): function2 X Y X Z := fun R => comp R S.
    Definition Chain (F: function2 X Y X' Y') (G: function2 X Y Y' Z') := fun R => comp (F R) (G R).

    Definition Comp (G: function2 X' Y' X'' Y'') (F: function2 X Y X' Y') := fun R => G (F R).

    Variable F: function X Y.
    Variable R: relation2 X Y.

Simple iteration function
    Fixpoint Exp(n: nat): relation2 X Y :=
      match n with
        | O => R
        | S n => F (Exp n)
    Definition Iter := union Exp.

Increasing iteration function

    Fixpoint UExp(n: nat): relation2 X Y :=
      match n with
        | O => R
        | S n => union2 (UExp n) (F (UExp n))
    Definition UIter := union UExp.

    Lemma UExp_incl: forall n, incl (UExp n) (UExp (S n)).

    Lemma UIter_incl: incl R UIter.

  End Def'.

  Section UIter.
    Variables X Y: Type.
    Variable F: function X Y.
    Hypothesis HF: increasing F.

    Lemma UExp_inc: forall n R S, incl R S -> incl (UExp F R n) (UExp F S n).

  End UIter.

  Section UIter'.
    Variable X: Type.
    Variable F: function X X.
    Hypothesis HF: forall R, eeq (trans (F R)) (F (trans R)) .
    Hypothesis HF': increasing F.

    Lemma UExp_trans: forall n R, eeq (trans (UExp F R n)) (UExp F (trans R) n).

    Lemma UIter_trans: forall R, eeq (trans (UIter F R)) (UIter F (trans R)).
  End UIter'.
End Global.

Hint Immediate UExp_incl.
Hint Immediate UIter_incl.