(* PURE TYPE SYSTEMS OF THE λ-CUBE ********************************************)
-inductive Cube_Ax (i,j:nat): Prop ≝
- | star_box: i = 0 → j = 1 → Cube_Ax i j
+inductive Cube_Ax: nat → nat → Prop ≝
+ | star_box: Cube_Ax 0 1
.
(* The λPω pure type system (a.k.a. λC or CC) *********************************)
-inductive CC_Re (i,j,k:nat): Prop ≝
- | star_star: i = 0 → j = 0 → k = 0 → CC_Re i j k
- | box_star : i = 1 → j = 0 → k = 0 → CC_Re i j k
- | box_box : i = 1 → j = 1 → k = 1 → CC_Re i j k
- | star_box : i = 0 → j = 1 → k = 1 → CC_Re i j k
+inductive CC_Re: nat → nat → nat → Prop ≝
+ | star_star: CC_Re 0 0 0
+ | box_star : CC_Re 1 0 0
+ | box_box : CC_Re 1 1 1
+ | star_box : CC_Re 0 1 1
.
definition CC: pts ≝ mk_pts Cube_Ax CC_Re conv.
(* The λω pure type system (a.k.a. Fω) ****************************************)
-inductive FO_Re (i,j,k:nat): Prop ≝
- | star_star: i = 0 → j = 0 → k = 0 → FO_Re i j k
- | box_star : i = 1 → j = 0 → k = 0 → FO_Re i j k
- | box_box : i = 1 → j = 1 → k = 1 → FO_Re i j k
+inductive FO_Re: nat → nat → nat → Prop ≝
+ | star_star: FO_Re 0 0 0
+ | box_star : FO_Re 1 0 0
+ | box_box : FO_Re 1 1 1
.
definition FO: pts ≝ mk_pts Cube_Ax FO_Re conv.