(** * Study of the class of monotonic functions (Lemma 1.7 and 1.8) *)
Require Export Settings.
Set Implicit Arguments.
Section Global.
Variable A: Type.
Section A.
Variables X Y: Type.
Variable TX: reduction_t A X.
Variable TY: reduction_t A Y.
Section A'.
Variable R: relation2 X Y.
Variable E: relation2 X X.
Variable T: relation2 Y Y.

(* Specifications: *)
(** identity is monotonic *)
Lemma identity_mon: monotonic TX TY (identity (X:=X) (Y:=Y)).
(** simulation-right-chaining is monotonic *)
Lemma chaining_r_mon: simulation TY TY T -> monotonic TX TY (chaining_r T).
(** expansion-left-chaining is monotonic *)
Lemma chaining_l_mon: expansion1 TX TX E -> monotonic TX TY (chaining_l E).
End A'.