(** * Comparisons *)

Inductive comparison : Type :=
  | Ceq : comparison               (**r same *)
  | Cne : comparison               (**r different *)
  | Clt : comparison               (**r less than *)
  | Cle : comparison               (**r less than or equal *)
  | Cgt : comparison               (**r greater than *)
  | Cge : comparison.(** * Comparisons *)

Inductive comparison : Type :=
  | Ceq : comparison               (**r same *)
  | Cne : comparison               (**r different *)
  | Clt : comparison               (**r less than *)
  | Cle : comparison               (**r less than or equal *)
  | Cgt : comparison               (**r greater than *)
  | Cge : comparison.(** * Comparisons *)

Inductive comparison : Type :=
  | Ceq : comparison               (**r same *)
  | Cne : comparison               (**r different *)
  | Clt : comparison               (**r less than *)
  | Cle : comparison               (**r less than or equal *)
  | Cgt : comparison               (**r greater than *)
  | Cge : comparison.(** * Comparisons *)

Inductive comparison : Type :=
  | Ceq : comparison               (**r same *)
  | Cne : comparison               (**r different *)
  | Clt : comparison               (**r less than *)
  | Cle : comparison               (**r less than or equal *)
  | Cgt : comparison               (**r greater than *)
  | Cge : comparison.(** * Comparisons *)

Inductive comparison : Type :=
  | Ceq : comparison               (**r same *)
  | Cne : comparison               (**r different *)
  | Clt : comparison               (**r less than *)
  | Cle : comparison               (**r less than or equal *)
  | Cgt : comparison               (**r greater than *)
  | Cge : comparison.(** * Comparisons *)

Inductive comparison : Type :=
  | Ceq : comparison               (**r same *)
  | Cne : comparison               (**r different *)
  | Clt : comparison               (**r less than *)
  | Cle : comparison               (**r less than or equal *)
  | Cgt : comparison               (**r greater than *)
  | Cge : comparison.(** * Comparisons *)

Inductive comparison : Type :=
  | Ceq : comparison               (**r same *)
  | Cne : comparison               (**r different *)
  | Clt : comparison               (**r less than *)
  | Cle : comparison               (**r less than or equal *)
  | Cgt : comparison               (**r greater than *)
  | Cge : comparison.(**r greater than or equal *)

Definition negate_comparison (c: comparison): comparison :=
  match c with
  | Ceq => Cne
  | Cne => Ceq
  | Clt => Cge
  | Cle => Cgt
  | Cgt => Cle
  | Cge => Clt
  end.Definition swap_comparison (c: comparison): comparison :=
  match c with
  | Ceq => Ceq
  | Cne => Cne
  | Clt => Cgt
  | Cle => Cge
  | Cgt => Clt
  | Cge => Cle
  end.Definition wordsize: nat := WS.wordsize.Definition zwordsize: Z := Z.of_nat wordsize.Definition modulus : Z := two_power_nat wordsize.Definition half_modulus : Z := modulus / 2.Definition max_unsigned : Z := modulus - 1.Definition max_signed : Z := half_modulus - 1.Definition min_signed : Z := - half_modulus.Remark wordsize_pos: zwordsize > 0.Remark modulus_power: modulus = two_p zwordsize.Remark modulus_pos: modulus > 0.(** * Representation of machine integers *)

(** A machine integer (type [int]) is represented as a Coq arbitrary-precision
  integer (type [Z]) plus a proof that it is in the range 0 (included) to
  [modulus] (excluded). *)

Record int: Type := mkint { intval: Z; intrange: -1 < intval < modulus }.(** * Representation of machine integers *)

(** A machine integer (type [int]) is represented as a Coq arbitrary-precision
  integer (type [Z]) plus a proof that it is in the range 0 (included) to
  [modulus] (excluded). *)

Record int: Type := mkint { intval: Z; intrange: -1 < intval < modulus }.(** * Representation of machine integers *)

(** A machine integer (type [int]) is represented as a Coq arbitrary-precision
  integer (type [Z]) plus a proof that it is in the range 0 (included) to
  [modulus] (excluded). *)

Record int: Type := mkint { intval: Z; intrange: -1 < intval < modulus }.(** * Representation of machine integers *)

(** A machine integer (type [int]) is represented as a Coq arbitrary-precision
  integer (type [Z]) plus a proof that it is in the range 0 (included) to
  [modulus] (excluded). *)

Record int: Type := mkint { intval: Z; intrange: -1 < intval < modulus }.(** Fast normalization modulo [2^wordsize] *)

Fixpoint P_mod_two_p (p: positive) (n: nat) {struct n} : Z :=
  match n with
  | O => 0
  | S m =>
      match p with
      | xH => 1
      | xO q => Z.double (P_mod_two_p q m)
      | xI q => Z.succ_double (P_mod_two_p q m)
      end
  end.Definition Z_mod_modulus (x: Z) : Z :=
  match x with
  | Z0 => 0
  | Zpos p => P_mod_two_p p wordsize
  | Zneg p => let r := P_mod_two_p p wordsize in if zeq r 0 then 0 else modulus - r
  end.

(* Specifications: *)
Lemma P_mod_two_p_range: forall n p, 0 <= P_mod_two_p p n < two_power_nat n.
Lemma P_mod_two_p_eq: forall n p, P_mod_two_p p n = (Zpos p) mod (two_power_nat n).
Lemma Z_mod_modulus_range: forall x, 0 <= Z_mod_modulus x < modulus.
Lemma Z_mod_modulus_range': forall x, -1 < Z_mod_modulus x < modulus.
Lemma Z_mod_modulus_eq: forall x, Z_mod_modulus x = x mod modulus.
Lemma P_mod_two_p_range: forall n p, 0 <= P_mod_two_p p n < two_power_nat n.
Lemma P_mod_two_p_eq: forall n p, P_mod_two_p p n = (Zpos p) mod (two_power_nat n).
Lemma Z_mod_modulus_range: forall x, 0 <= Z_mod_modulus x < modulus.
Lemma Z_mod_modulus_range': forall x, -1 < Z_mod_modulus x < modulus.
Lemma Z_mod_modulus_eq: forall x, Z_mod_modulus x = x mod modulus.