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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
Require Export Axioms.
Require Export AddProps.
Require Export NeqProps.
(** This file contains basic properties of the less than relation *)
Lemma lt_anti_sym : (x,y:N)x<y->~(y<x).
Red; Intros x y lt1 lt2; Apply (lt_anti_refl x); EAuto with num.
Qed.
Hints Resolve lt_anti_refl : num.
Lemma eq_not_lt : (x,y:N)(x=y)->~(x<y).
Red; Intros x y eq1 lt1; Apply (lt_anti_refl x); EAuto with num.
Qed.
Hints Resolve eq_not_lt : num.
Lemma lt_0_1 : (zero<one).
EAuto with num.
Qed.
Hints Resolve lt_0_1 : num.
Lemma eq_lt_x_Sy : (x,y:N)(x=y)->(x<(S y)).
EAuto with num.
Qed.
Hints Resolve eq_lt_x_Sy : num.
Lemma lt_lt_x_Sy : (x,y:N)(x<y)->(x<(S y)).
EAuto with num.
Qed.
Hints Immediate lt_lt_x_Sy : num.
Lemma lt_Sx_y_lt : (x,y:N)((S x)<y)->(x<y).
EAuto with num.
Qed.
Hints Immediate lt_Sx_y_lt : num.
(** Relating [<] and [=] *)
Lemma lt_neq : (x,y:N)(x<y)->(x<>y).
Red; Intros x y lt1 eq1; Apply (lt_anti_refl x); EAuto with num.
Qed.
Hints Immediate lt_neq : num.
Lemma lt_neq_sym : (x,y:N)(y<x)->(x<>y).
Intros x y lt1 ; Apply neq_sym; Auto with num.
Qed.
Hints Immediate lt_neq_sym : num.
(** Application to inequalities properties *)
Lemma neq_x_Sx : (x:N)x<>(S x).
Auto with num.
Qed.
Hints Resolve neq_x_Sx : num.
Lemma neq_0_1 : zero<>one.
Auto with num.
Qed.
Hints Resolve neq_0_1 : num.
(** Relating [<] and [+] *)
Lemma lt_add_compat_r : (x,y,z:N)(x<y)->((z+x)<(z+y)).
Intros x y z H; Apply lt_eq_compat with (x+z) (y+z); Auto with num.
Qed.
Hints Resolve lt_add_compat_r : num.
Lemma lt_add_compat : (x1,x2,y1,y2:N)(x1<x2)->(y1<y2)->((x1+y1)<(x2+y2)).
Intros; Apply lt_trans with (x1+y2); Auto with num.
Qed.
Hints Immediate lt_add_compat : num.
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