Axioms for the basic numerical operations |
Require
Export
Params.
Require
Export
EqParams.
Require
Export
NSyntax.
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).
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.
Axioms for S
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Axiom
add_Sx_y : (x,y:N)((S x)+y)=(S (x+y)).
Axioms for one
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Axiom
S_0_1 : (S zero)=one.
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.