Library Coq.Numbers.Integer.BigZ.BigZ


Require Export BigN.
Require Import ZProperties ZDivFloor ZSig ZSigZAxioms ZMake.

BigZ : arbitrary large efficient integers.

The following BigZ module regroups both the operations and all the abstract properties:
  • ZMake.Make BigN provides the operations and basic specs w.r.t. ZArith
  • ZTypeIsZAxioms shows (mainly) that these operations implement the interface ZAxioms
  • ZProp adds all generic properties derived from ZAxioms
  • MinMax*Properties provides properties of min and max

Delimit Scope bigZ_scope with bigZ.

Module BigZ <: ZType <: OrderedTypeFull <: TotalOrder :=
 ZMake.Make BigN
 <+ ZTypeIsZAxioms
 <+ ZBasicProp [no inline] <+ ZExtraProp [no inline]
 <+ HasEqBool2Dec [no inline]
 <+ MinMaxLogicalProperties [no inline]
 <+ MinMaxDecProperties [no inline].

For precision concerning the above scope handling, see comment in BigN
Notations about BigZ

Local Open Scope bigZ_scope.

Notation bigZ := BigZ.t.
Infix "+" := BigZ.add : bigZ_scope.
Infix "-" := BigZ.sub : bigZ_scope.
Notation "- x" := (BigZ.opp x) : bigZ_scope.
Infix "*" := BigZ.mul : bigZ_scope.
Infix "/" := BigZ.div : bigZ_scope.
Infix "^" := BigZ.pow : bigZ_scope.
Infix "?=" := BigZ.compare : bigZ_scope.
Infix "=?" := BigZ.eqb (at level 70, no associativity) : bigZ_scope.
Infix "<=?" := BigZ.leb (at level 70, no associativity) : bigZ_scope.
Infix "<?" := BigZ.ltb (at level 70, no associativity) : bigZ_scope.
Infix "==" := BigZ.eq (at level 70, no associativity) : bigZ_scope.
Notation "x != y" := (~x==y) (at level 70, no associativity) : bigZ_scope.
Infix "<" := BigZ.lt : bigZ_scope.
Infix "<=" := BigZ.le : bigZ_scope.
Notation "x > y" := (y < x) (only parsing) : bigZ_scope.
Notation "x >= y" := (y <= x) (only parsing) : bigZ_scope.
Notation "x < y < z" := (x<y /\ y<z) : bigZ_scope.
Notation "x < y <= z" := (x<y /\ y<=z) : bigZ_scope.
Notation "x <= y < z" := (x<=y /\ y<z) : bigZ_scope.
Notation "x <= y <= z" := (x<=y /\ y<=z) : bigZ_scope.
Notation "[ i ]" := (BigZ.to_Z i) : bigZ_scope.
Infix "mod" := BigZ.modulo (at level 40, no associativity) : bigZ_scope.
Infix "รท" := BigZ.quot (at level 40, left associativity) : bigZ_scope.

Some additional results about BigZ

Theorem spec_to_Z: forall n : bigZ,
  BigN.to_Z (BigZ.to_N n) = ((Z.sgn [n]) * [n])%Z.

Theorem spec_to_N n:
 ([n] = Z.sgn [n] * (BigN.to_Z (BigZ.to_N n)))%Z.

Theorem spec_to_Z_pos: forall n, (0 <= [n])%Z ->
  BigN.to_Z (BigZ.to_N n) = [n].

BigZ is a ring
Detection of constants

Ltac isBigZcst t :=
 match t with
 | BigZ.Pos ?t => isBigNcst t
 | BigZ.Neg ?t => isBigNcst t
 | BigZ.zero => constr:(true)
 | BigZ.one => constr:(true)
 | BigZ.two => constr:(true)
 | BigZ.minus_one => constr:(true)
 | _ => constr:(false)
 end.

Ltac BigZcst t :=
 match isBigZcst t with
 | true => constr:(t)
 | false => constr:(NotConstant)
 end.

Ltac BigZ_to_N t :=
 match t with
 | BigZ.Pos ?t => BigN_to_N t
 | BigZ.zero => constr:(0%N)
 | BigZ.one => constr:(1%N)
 | BigZ.two => constr:(2%N)
 | _ => constr:(NotConstant)
 end.

Registration for the "ring" tactic

Add Ring BigZr : BigZring
 (decidable BigZeqb_correct,
  constants [BigZcst],
  power_tac BigZpower [BigZ_to_N],
  div BigZdiv).

Section TestRing.
Let test : forall x y, 1 + x*y + x^2 + 1 == 1*1 + 1 + (y + 1*x)*x.
Let test' : forall x y, 1 + x*y + x^2 - 1*1 - y*x + 1*(-x)*x == 0.
End TestRing.

BigZ also benefits from an "order" tactic

Ltac bigZ_order := BigZ.order.

Section TestOrder.
Let test : forall x y : bigZ, x<=y -> y<=x -> x==y.
End TestOrder.

We can use at least a bit of (r)omega by translating to Z.

Section TestOmega.
Let test : forall x y : bigZ, x<=y -> y<=x -> x==y.
End TestOmega.

Todo: micromega