# Library Coq.Numbers.Integer.Abstract.ZAxioms

Require Export NZAxioms.
Require Import Bool NZParity NZPow NZSqrt NZLog NZGcd NZDiv NZBits.
We obtain integers by postulating that successor of predecessor is identity.
Module Type ZAxiom (Import Z : NZAxiomsSig').
Axiom succ_pred : forall n, S (P n) == n.
End ZAxiom.
For historical reasons, ZAxiomsMiniSig isn't just NZ + ZAxiom, we also add an opp function, that can be seen as a shortcut for sub 0.
Module Type Opp (Import T:Typ).
Parameter Inline opp : t -> t.
End Opp.
Module Type OppNotation (T:Typ)(Import O : Opp T).
Notation "- x" := (opp x) (at level 35, right associativity).
End OppNotation.
Module Type Opp' (T:Typ) := Opp T <+ OppNotation T.
Module Type IsOpp (Import Z : NZAxiomsSig')(Import O : Opp' Z).
Declare Instance opp_wd : Proper (eq ==> eq) opp.
Axiom opp_0 : - 0 == 0.
Axiom opp_succ : forall n, - ( S n ) == P (- n).
End IsOpp.
Module Type OppCstNotation (Import A : NZAxiomsSig)(Import B : Opp A).
Notation "- 1" := (opp one).
Notation "- 2" := (opp two).
End OppCstNotation.
Module Type ZAxiomsMiniSig := NZOrdAxiomsSig <+ ZAxiom <+ Opp <+ IsOpp.
Module Type ZAxiomsMiniSig' := NZOrdAxiomsSig' <+ ZAxiom <+ Opp' <+ IsOpp
<+ OppCstNotation.
Other functions and their specifications
Absolute value
Module Type HasAbs(Import Z : ZAxiomsMiniSig').
Parameter Inline abs : t -> t.
Axiom abs_eq : forall n, 0<= n -> abs n == n.
Axiom abs_neq : forall n, n <=0 -> abs n == - n.
End HasAbs.
A sign function
Module Type HasSgn (Import Z : ZAxiomsMiniSig').
Parameter Inline sgn : t -> t.
Axiom sgn_null : forall n, n ==0 -> sgn n == 0.
Axiom sgn_pos : forall n, 0< n -> sgn n == 1.
Axiom sgn_neg : forall n, n <0 -> sgn n == -1.
End HasSgn.
Divisions
First, the usual Coq convention of Truncated-Toward-Bottom (a.k.a Floor). We simply extend the NZ signature.
Module Type ZDivSpecific (Import A:ZAxiomsMiniSig')(Import B : DivMod' A).
Axiom mod_pos_bound : forall a b, 0 < b -> 0 <= a mod b < b.
Axiom mod_neg_bound : forall a b, b < 0 -> b < a mod b <= 0.
End ZDivSpecific.
Module Type ZDiv (Z:ZAxiomsMiniSig) := NZDiv.NZDiv Z <+ ZDivSpecific Z.
Module Type ZDiv' (Z:ZAxiomsMiniSig) := NZDiv.NZDiv' Z <+ ZDivSpecific Z.
Then, the Truncated-Toward-Zero convention. For not colliding with Floor operations, we use different names
Module Type QuotRem (Import A : Typ).
Parameters Inline quot rem : t -> t -> t.
End QuotRem.
Module Type QuotRemNotation (A : Typ)(Import B : QuotRem A).
Infix "÷" := quot (at level 40, left associativity).
Infix "rem" := rem (at level 40, no associativity).
End QuotRemNotation.
Module Type QuotRem' (A : Typ) := QuotRem A <+ QuotRemNotation A.
Module Type QuotRemSpec (Import A : ZAxiomsMiniSig')(Import B : QuotRem' A).
Declare Instance quot_wd : Proper (eq ==> eq ==> eq) quot.
Declare Instance rem_wd : Proper (eq ==> eq ==> eq) B.rem.
Axiom quot_rem : forall a b, b ~= 0 -> a == b *( a ÷ b ) + ( a rem b ).
Axiom rem_bound_pos : forall a b, 0<= a -> 0< b -> 0 <= a rem b < b.
Axiom rem_opp_l : forall a b, b ~= 0 -> ( - a ) rem b == - ( a rem b ).
Axiom rem_opp_r : forall a b, b ~= 0 -> a rem ( - b ) == a rem b.
End QuotRemSpec.
Module Type ZQuot (Z:ZAxiomsMiniSig) := QuotRem Z <+ QuotRemSpec Z.
Module Type ZQuot' (Z:ZAxiomsMiniSig) := QuotRem' Z <+ QuotRemSpec Z.
For all other functions, the NZ axiomatizations are enough.
Let's group everything
Module Type ZAxiomsSig := ZAxiomsMiniSig <+ OrderFunctions
<+ HasAbs <+ HasSgn <+ NZParity.NZParity
<+ NZPow.NZPow <+ NZSqrt.NZSqrt <+ NZLog.NZLog2 <+ NZGcd.NZGcd
<+ ZDiv <+ ZQuot <+ NZBits.NZBits <+ NZSquare.
Module Type ZAxiomsSig' := ZAxiomsMiniSig' <+ OrderFunctions'
<+ HasAbs <+ HasSgn <+ NZParity.NZParity
<+ NZPow.NZPow' <+ NZSqrt.NZSqrt' <+ NZLog.NZLog2 <+ NZGcd.NZGcd'
<+ ZDiv' <+ ZQuot' <+ NZBits.NZBits' <+ NZSquare.