Library Functions
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).
Composition
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)
end.
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))
end.
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.