Coq.Num.NeqProps

# Module Coq.Num.NeqProps

``Require` `Export` NeqParams.
`Require` `Export` NeqAxioms.
`Require` `Export` EqParams.
`Require` `Export` EqAxioms.

`

This file contains basic properties of disequality

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`Lemma` neq_antirefl : (x:N)~(x<>x).
Auto with num.
`Save`.
`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.
`Save`.
`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.
`Save`.
`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.
`Save`.
`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.
`Save`.
`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.
`Save`.

`

Index