Coq.Num.NeqDef

# Module Coq.Num.NeqDef

``

DisEquality is defined as the negation of equality

`

`Require` Params.
`Require` EqParams.
`Require` EqAxioms.

`Definition` neq : N -> N -> Prop := [x,y] ~(x=y).
`Infix` 6 "<>" neq.

`Lemma` eq_not_neq : (x,y:N)x=y->~(x<>y).
Unfold neq; Auto with num.
`Save`.
`Hints` `Immediate` eq_not_neq : num.

`Lemma` neq_sym : (x,y:N)(x<>y)->(y<>x).
Unfold neq; Auto with num.
`Save`.
`Hints` Resolve neq_sym : num.

`Lemma` neq_not_neq_trans : (x,y,z:N)(x<>y)->~(y<>z)->(x<>z).
Unfold neq; EAuto with num.
`Save`.
`Hints` Resolve neq_not_neq_trans : num.

`

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