(*
 - N: https://coq.inria.fr/library/Coq.Numbers.BinNums.html
 - Z: https://coq.inria.fr/library/Coq.Numbers.BinNums.html
 - positive: https://coq.inria.fr/library/Coq.Numbers.BinNums.html
 *)

Require Import ZArith.
Require Import QArith.
Require Import Reals.

Check INR.
(*
INR
     : nat -> R
*)
Compute 12%R.
(*
= ((R1 + R1) * ((R1 + R1) * (R1 + (R1 + R1))))%R
     : R
*)
Compute INR 5.
(*
= (R1 + R1 + R1 + R1 + R1)%R
     : R
*)















Check Q.

Compute Qmake 2 3.
(*
= 2 # 3
     : Q
*)

Definition a:Q := Qmake 10 5.
Definition b:Q := Qmake 5 10.
Compute a*b.
(*
= 50 # 50
     : Q
*)

Search (Q -> _).

Compute Qred (50 # 25).
(*
= 2
     : Q
*)
Compute Qopp (50 # 25).
Compute Qopp (-1 # 25).
Compute Qinv (-1 # 25).
Compute Qmult (20 # 1) (1 # 2).
Compute Qmult' (20 # 1) (1 # 2).
Compute Qminus (10 # 2) (6 # 3).
Compute Qminus' (10 # 2) (6 # 3).















Check Z.

Compute Z0.
(*
= 0%Z
     : Z
*)
Compute Zpos 8%positive.
(*
= 8%Z
     : Z
*)
Compute Zneg 53%positive.
(*
= (-53)%Z
     : Z
*)










Check N.

Compute N0.
(*
= 0%N
     : N
*)
Compute Npos 1%positive.
(*
= 1%N
     : N
*)
Compute Npos 45%positive.
(*
= 45%N
     : N
*)














Check positive.
Check N.
Check Z.


Compute xH.
(*
= 1%positive
     : positive
*)

Compute xO xH.
(*
= 2%positive
     : positive
*)

Compute xI xH.
(*
= 3%positive
     : positive
*)

Compute xO (xI (xO xH)).
(*
The term "xO" has type "positive -> positive"
while it is expected to have type "positive".
*)


Compute xO (xI (xO xO)).

Unset Printing Notations.

Compute xI xH.
Set Printing Notations.
Compute xI xH.
Compute (Pos.to_nat 6%positive).

Search (positive -> nat).