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(* 02110-1301 USA                                                     *)
(* Contribution to the Coq Library   V6.3 (July 1999)                    *)
(****************************************************************************)
(*                 The Calculus of Inductive Constructions                  *)
(*                                                                          *)
(*                                Projet Coq                                *)
(*                                                                          *)
(*                     INRIA                        ENS-CNRS                *)
(*              Rocquencourt                        Lyon                    *)
(*                                                                          *)
(*                                Coq V5.10                                 *)
(*                              Nov 25th 1994                               *)
(*                                                                          *)
(****************************************************************************)
(****************************************************************************)
(*                                  Cube.v                                  *)
(*                                                                          *)
(*                                Gerard Huet                               *)
(*                                                                          *)
(* Developed in V5.8  January 1993                                          *)
(* Ported    to V5.10 January 1995                                          *)
(****************************************************************************)
(*****************)
(* Prism Theorem *)
(*****************)
(* Auxiliary notion : compat Used to generate the right simultaneous induction on U, V and W *)
(* (compat U V W) iff U,V,W are compatible markings, and (sub V U) *)
Inductive compat : redexes -> redexes -> redexes -> Prop :=
| Compat_Var : forall n : nat, compat (Var n) (Var n) (Var n)
| Compat_Fun :
forall U V W : redexes, compat U V W -> compat (Fun U) (Fun V) (Fun W)
| Compat_Ap1 :
forall U1 V1 W1 : redexes,
compat U1 V1 W1 ->
forall U2 V2 W2 : redexes,
compat U2 V2 W2 ->
forall b : bool, compat (Ap false U1 U2) (Ap false V1 V2) (Ap b W1 W2)
| Compat_Ap2 :
forall U1 V1 W1 : redexes,
compat U1 V1 W1 ->
forall U2 V2 W2 : redexes,
compat U2 V2 W2 ->
forall b b' : bool,
compat (Ap true (Fun U1) U2) (Ap b (Fun V1) V2) (Ap b' (Fun W1) W2).
(* Remark - It should be possible to get rid of auxiliary compat and to combine prism0 with compat_intro in one lemma *)
(*****************************************************************)
(* Theorem prism : (U,V,W:redexes)(sub V U) ->                   *)
(*     (UV:redexes)(residuals U V UV) ->                         *)
(*     (WV:redexes)(residuals W V WV) ->                         *)
(*     (WU:redexes)(residuals W U WU) <-> (residuals WV UV WU).  *)
(*****************************************************************)
(**************************************************************************)
(*  Levy's cube lemma :                                                   *)
(*  (U,V,UV,VU:redexes)  (residuals U V UV) -> (residuals V U VU) ->      *)
(*  (W,WU,WV,WUV:redexes)(residuals W U WU) -> (residuals WU VU WUV) ->   *)
(*                       (residuals W V WV) -> (residuals WV UV WUV).     *)
(**************************************************************************)

(* 3-dimensional paving diagram *)
Lemma paving :
forall U V W WU WV : redexes,
residuals W U WU ->
residuals W V WV ->
exists UV : redexes,
(exists VU : redexes,
(exists WUV : redexes, residuals WU VU WUV /\ residuals WV UV WUV)).