Library Coq.micromega.ZifyNat

Require Import BinInt Znat Zdiv.
Require Import ZifyClasses ZifyInst Zify.

Ltac zify_convert_to_euclidean_division_equations_flag ::= constr:(true).

#[local]
Existing Instance Inj_nat_Z.

#[global]
Instance Op_mod : BinOp Nat.modulo :=
  {| TBOp := Z.modulo ; TBOpInj := Nat2Z.inj_mod |}.
Add Zify BinOp Op_mod.

#[global]
Instance Op_div : BinOp Nat.div :=
  {| TBOp := Z.div ; TBOpInj := Nat2Z.inj_div |}.
Add Zify BinOp Op_div.

#[global]
Instance Op_pow : BinOp Nat.pow :=
  {| TBOp := Z.pow ; TBOpInj := Nat2Z.inj_pow |}.
Add Zify BinOp Op_pow.

#[local] Open Scope Z_scope.

#[global]
Instance SatDiv : Saturate Z.div :=
  {|
    PArg1 := fun x => 0 <= x;
    PArg2 := fun y => 0 <= y;
    PRes := fun r => 0 <= r;
    SatOk := Z_div_nonneg_nonneg
  |}.
Add Zify Saturate SatDiv.

#[global]
Instance SatMod : Saturate Z.modulo :=
  {|
    PArg1 := fun x => 0 <= x;
    PArg2 := fun y => 0 <= y;
    PRes := fun r => 0 <= r;
    SatOk := Z_mod_nonneg_nonneg
  |}.
Add Zify Saturate SatMod.