Library Coq.Numbers.Natural.Abstract.NProperties


Require Export NAxioms.
Require Import NMaxMin NParity NPow NSqrt NLog NDiv NDiv0 NGcd NLcm NLcm0 NBits.

The two following functors summarize all known facts about N.
  • NBasicProp provides properties of basic functions: + - * min max <= <
  • NExtraProp provides properties of advanced functions: pow, sqrt, log2, div, gcd, and bitwise functions. NExtraProp0 additionally assumes a / 0 == 0 and a mod 0 == a.
If necessary, the earlier all-in-one functor NProp could be re-obtained via NBasicProp <+ NExtraProp

Module Type NBasicProp (N:NAxiomsMiniSig) := NMaxMinProp N.

Module Type NExtraPreProp (N:NAxiomsSig)(P:NBasicProp N) :=
 NParityProp N P <+ NPowProp N P <+ NSqrtProp N P <+ NLog2Prop N P <+ NGcdProp N P.

Module Type NExtraProp (N:NAxiomsSig)(P:NBasicProp N) :=
 NExtraPreProp N P <+ NDivProp N P <+ NLcmProp N P <+ NBitsProp N P.

Module Type NExtraProp0 (N:NAxiomsSig)(P:NBasicProp N)(D0:NZDivSpec0 N N N)(E:NExtraPreProp N P).
 Module Private_NDivProp := Nop <+ NDivProp N P.
 Include NDivProp0 N P D0.
 Module Private_NLcmProp := Nop <+ NLcmProp N P Private_NDivProp E.
 Include NLcmProp0 N P D0 E.
 Include NBitsProp N P E E Private_NDivProp E.
End NExtraProp0.