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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 V6.1                                 *)
(*                               Oct 1st 1996                               *)
(*                                                                          *)
(****************************************************************************)
(*                               Coherence.v                                *)
(****************************************************************************)
Hint Resolve refl_equal. (* TO BE REMOVED *)
Set Implicit Arguments.
Unset Strict Implicit.

(* Specifications: *)
Theorem Rstar_reflexive :
forall (U : Type) (R : Relation U), Reflexive (Rstar R).
Theorem Rplus_contains_R :
forall (U : Type) (R : Relation U), Contains (Rplus R) R.
Theorem Rstar_contains_R :
forall (U : Type) (R : Relation U), Contains (Rstar R) R.
Theorem Rstar_contains_Rplus :
forall (U : Type) (R : Relation U), Contains (Rstar R) (Rplus R).
Theorem Rstar_transitive :
forall (U : Type) (R : Relation U), Transitive (Rstar R).
Theorem Rstar_cases :
forall (U : Type) (R : Relation U) (x y : U),
Rstar R x y -> x = y \/ (exists u : U, R x u /\ Rstar R u y).
Theorem Rstar_equiv_Rstar1 :
forall (U : Type) (R : Relation U), Same_relation (Rstar R) (Rstar1 R).
Theorem Rsym_imp_Rstarsym :
forall (U : Type) (R : Relation U), Symmetric R -> Symmetric (Rstar R).
Theorem Sstar_contains_Rstar :
forall (U : Type) (R S : Relation U),
Contains (Rstar S) R -> Contains (Rstar S) (Rstar R).
Theorem star_monotone :
forall (U : Type) (R S : Relation U),
Contains S R -> Contains (Rstar S) (Rstar R).
Theorem RstarRplus_RRstar :
forall (U : Type) (R : Relation U) (x y z : U),
Rstar R x y -> Rplus R y z -> exists u : U, R x u /\ Rstar R u z.
Theorem Rstar_imp_coherent :
forall (U : Type) (R : Relation U) (x y : U), Rstar R x y -> coherent R x y.