Coq.Num.Axioms

# Module Coq.Num.Axioms

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Axioms for the basic numerical operations

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`Require` `Export` Params.
`Require` `Export` EqParams.
`Require` `Export` NSyntax.

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Axioms for `eq`

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`Axiom` eq_refl : (x:N)(x=x).
`Axiom` eq_sym : (x,y:N)(x=y)->(y=x).
`Axiom` eq_trans : (x,y,z:N)(x=y)->(y=z)->(x=z).

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Axioms for `add`

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`Axiom` add_sym : (x,y:N)(x+y)=(y+x).
`Axiom` add_assoc_l : (x,y,z:N)((x+y)+z)=(x+(y+z)).
`Axiom` add_0_x : (x:N)(zero+x)=x.

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Axioms for `S`

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`Axiom` add_Sx_y : (x,y:N)((S x)+y)=(S (x+y)).

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Axioms for `one`

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`Axiom` S_0_1 : (S zero)=one.

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Axioms for `<`, properties of `>`, `<=` and `>=` will be derived from `<`

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`Axiom` lt_trans : (x,y,z:N)x<y->y<z->x<z.
`Axiom` lt_anti_refl : (x:N)~(x<x).

`Axiom` lt_x_Sx : (x:N)x<(S x).
`Axiom` lt_S_compat : (x,y:N)(x<y)->(S x)<(S y).

`Axiom` lt_add_compat_l : (x,y,z:N)(x<y)->((x+z)<(y+z)).

`Hints` Resolve add_sym add_assoc_l add_0_x add_Sx_y S_0_1 lt_x_Sx lt_S_compat
lt_trans lt_anti_refl lt_add_compat_l : num.

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Index