(** ** Less than or equal *)
Definition le0 (p q : rel0) :=
(forall (PR: p : Prop), q : Prop).
Arguments le0 : clear implicits.
Definition le1 T0 (p q : rel1 T0) :=
(forall x0 (PR: p x0 : Prop), q x0 : Prop).
Arguments le1 [ T0] p q.
Definition le2 T0 T1 (p q : rel2 T0 T1) :=
(forall x0 x1 (PR: p x0 x1 : Prop), q x0 x1 : Prop).
Arguments le2 [ T0 T1] p q.
Definition le3 T0 T1 T2 (p q : rel3 T0 T1 T2) :=
(forall x0 x1 x2 (PR: p x0 x1 x2 : Prop), q x0 x1 x2 : Prop).
Arguments le3 [ T0 T1 T2] p q.
Definition le4 T0 T1 T2 T3 (p q : rel4 T0 T1 T2 T3) :=
(forall x0 x1 x2 x3 (PR: p x0 x1 x2 x3 : Prop), q x0 x1 x2 x3 : Prop).
Arguments le4 [ T0 T1 T2 T3] p q.
Definition le5 T0 T1 T2 T3 T4 (p q : rel5 T0 T1 T2 T3 T4) :=
(forall x0 x1 x2 x3 x4 (PR: p x0 x1 x2 x3 x4 : Prop), q x0 x1 x2 x3 x4 : Prop).
Arguments le5 [ T0 T1 T2 T3 T4] p q.
Definition le6 T0 T1 T2 T3 T4 T5 (p q : rel6 T0 T1 T2 T3 T4 T5) :=
(forall x0 x1 x2 x3 x4 x5 (PR: p x0 x1 x2 x3 x4 x5 : Prop), q x0 x1 x2 x3 x4 x5 : Prop).
Arguments le6 [ T0 T1 T2 T3 T4 T5] p q.
Definition le7 T0 T1 T2 T3 T4 T5 T6 (p q : rel7 T0 T1 T2 T3 T4 T5 T6) :=
(forall x0 x1 x2 x3 x4 x5 x6 (PR: p x0 x1 x2 x3 x4 x5 x6 : Prop), q x0 x1 x2 x3 x4 x5 x6 : Prop).
Arguments le7 [ T0 T1 T2 T3 T4 T5 T6] p q.
Definition le8 T0 T1 T2 T3 T4 T5 T6 T7 (p q : rel8 T0 T1 T2 T3 T4 T5 T6 T7) :=
(forall x0 x1 x2 x3 x4 x5 x6 x7 (PR: p x0 x1 x2 x3 x4 x5 x6 x7 : Prop), q x0 x1 x2 x3 x4 x5 x6 x7 : Prop).
Arguments le8 [ T0 T1 T2 T3 T4 T5 T6 T7] p q.
Definition le9 T0 T1 T2 T3 T4 T5 T6 T7 T8 (p q : rel9 T0 T1 T2 T3 T4 T5 T6 T7 T8) :=
(forall x0 x1 x2 x3 x4 x5 x6 x7 x8 (PR: p x0 x1 x2 x3 x4 x5 x6 x7 x8 : Prop), q x0 x1 x2 x3 x4 x5 x6 x7 x8 : Prop).
Arguments le9 [ T0 T1 T2 T3 T4 T5 T6 T7 T8] p q.
Definition le10 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 (p q : rel10 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9) :=
(forall x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 (PR: p x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 : Prop), q x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 : Prop).
Arguments le10 [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9] p q.
Definition le11 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 (p q : rel11 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10) :=
(forall x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 (PR: p x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 : Prop), q x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 : Prop).
Arguments le11 [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10] p q.
Definition le12 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 (p q : rel12 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11) :=
(forall x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 (PR: p x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 : Prop), q x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 : Prop).
Arguments le12 [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11] p q.
Definition le13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 (p q : rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12) :=
(forall x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 (PR: p x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 : Prop), q x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 : Prop).
Arguments le13 [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12] p q.
Definition le14 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 T13 (p q : rel14 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 T13) :=
(forall x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 (PR: p x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 : Prop), q x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 : Prop).
Arguments le14 [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 T13] p q.
Notation "p <0== q" :=
(le0 p q)
(at level 50, no associativity, only parsing).
Notation "p <1== q" :=
(le1 p q)
(at level 50, no associativity).
Notation "p <2== q" :=
(le2 p q)
(at level 50, no associativity).
Notation "p <3== q" :=
(le3 p q)
(at level 50, no associativity).
Notation "p <4== q" :=
(le4 p q)
(at level 50, no associativity).
Notation "p <5== q" :=
(le5 p q)
(at level 50, no associativity).
Notation "p <6== q" :=
(le6 p q)
(at level 50, no associativity).
Notation "p <7== q" :=
(le7 p q)
(at level 50, no associativity).
Notation "p <8== q" :=
(le8 p q)
(at level 50, no associativity).
Notation "p <9== q" :=
(le9 p q)
(at level 50, no associativity).
Notation "p <10== q" :=
(le10 p q)
(at level 50, no associativity).
Notation "p <11== q" :=
(le11 p q)
(at level 50, no associativity).
Notation "p <12== q" :=
(le12 p q)
(at level 50, no associativity).
Notation "p <13== q" :=
(le13 p q)
(at level 50, no associativity).
Notation "p <14== q" :=
(le14 p q)
(at level 50, no associativity).
(** ** Tranisitivity and Reflexivity *)

Lemma le0_trans(r0 r1 r2 : rel0)
(LE0 : r0 <0== r1) (LE1 : r1 <0== r2) :
r0 <0== r2.