Library Coq.Num.NeqProps


Require Export NeqParams.

Require Export NeqAxioms.

Require Export EqParams.

Require Export EqAxioms.

This file contains basic properties of disequality





Lemma neq_antirefl : (x:N)~(x<>x).

Auto with num.

Qed.

Hints Resolve neq_antirefl : num.




Lemma eq_not_neq_y_x : (x,y:N)(x=y)->~(y<>x).

Intros; Apply eq_not_neq; Auto with num.

Qed.

Hints Immediate eq_not_neq_y_x : num.




Lemma neq_not_eq : (x,y:N)(x<>y)->~(x=y).

Red; Intros; Apply (eq_not_neq x y); Trivial.

Qed.

Hints Immediate neq_not_eq : num.




Lemma neq_not_eq_y_x : (x,y:N)(x<>y)->~(y=x).

Intros; Apply neq_not_eq; Auto with num.

Qed.

Hints Immediate neq_not_eq_y_x : num.




Lemma not_neq_neq_trans : (x,y,z:N)~(x<>y)->(y<>z)->(x<>z).

Intros; Apply neq_sym; Apply neq_not_neq_trans with y; Auto with num.

Qed.

Hints Resolve not_neq_neq_trans : num.




Lemma neq_eq_compat : (x1,x2,y1,y2:N)(x1=y1)->(x2=y2)->(x1<>x2)->(y1<>y2).

Intros.

EAuto with num.

Qed.