Library Coq.Num.Axioms

Axioms for the basic numerical operations


Require Export Params.

Require Export EqParams.

Require Export NSyntax.

Axioms for eq





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).

Axioms for add





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.

Axioms for S


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

Axioms for one


Axiom S_0_1 : (S zero)=one.

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





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.