# Library Coq.Numbers.Cyclic.Int31.Int31

Require Import NaryFunctions.
Require Import Wf_nat.
Require Export ZArith.
Require Export DoubleType.

# 31-bit integers

This file contains basic definitions of a 31-bit integer arithmetic. In fact it is more general than that. The only reason for this use of 31 is the underlying mecanism for hardware-efficient computations by A. Spiwack. Apart from this, a switch to, say, 63-bit integers is now just a matter of replacing every occurences of 31 by 63. This is actually made possible by the use of dependently-typed n-ary constructions for the inductive type int31, its constructor I31 and any pattern matching on it. If you modify this file, please preserve this genericity.

Definition size := 31%nat.

Digits

Inductive digits : Type := D0 | D1.

The type of 31-bit integers

The type int31 has a unique constructor I31 that expects 31 arguments of type digits.

Inductive int31 : Type := I31 : nfun digits size int31.

Delimit Scope int31_scope with int31.
Open Scope int31_scope.

# Constants

Zero is I31 D0 ... D0
Definition On : int31 := Eval compute in napply_cst _ _ D0 size I31.

One is I31 D0 ... D0 D1
Definition In : int31 := Eval compute in (napply_cst _ _ D0 (size-1) I31) D1.

The biggest integer is I31 D1 ... D1, corresponding to (2^size)-1
Definition Tn : int31 := Eval compute in napply_cst _ _ D1 size I31.

Two is I31 D0 ... D0 D1 D0
Definition Twon : int31 := Eval compute in (napply_cst _ _ D0 (size-2) I31) D1 D0.

# Bits manipulation

sneakr b x shifts x to the right by one bit. Rightmost digit is lost while leftmost digit becomes b. Pseudo-code is match x with (I31 d0 ... dN) => I31 b d0 ... d(N-1) end

Definition sneakr : digits -> int31 -> int31 := Eval compute in
fun b => int31_rect _ (napply_except_last _ _ (size-1) (I31 b)).

sneakl b x shifts x to the left by one bit. Leftmost digit is lost while rightmost digit becomes b. Pseudo-code is match x with (I31 d0 ... dN) => I31 d1 ... dN b end

Definition sneakl : digits -> int31 -> int31 := Eval compute in
fun b => int31_rect _ (fun _ => napply_then_last _ _ b (size-1) I31).

shiftl, shiftr, twice and twice_plus_one are direct consequences of sneakl and sneakr.

Definition shiftl := sneakl D0.
Definition shiftr := sneakr D0.
Definition twice := sneakl D0.
Definition twice_plus_one := sneakl D1.

firstl x returns the leftmost digit of number x. Pseudo-code is match x with (I31 d0 ... dN) => d0 end

Definition firstl : int31 -> digits := Eval compute in
int31_rect _ (fun d => napply_discard _ _ d (size-1)).

firstr x returns the rightmost digit of number x. Pseudo-code is match x with (I31 d0 ... dN) => dN end

Definition firstr : int31 -> digits := Eval compute in
int31_rect _ (napply_discard _ _ (fun d=>d) (size-1)).

iszero x is true iff x = I31 D0 ... D0. Pseudo-code is match x with (I31 D0 ... D0) => true | _ => false end

Definition iszero : int31 -> bool := Eval compute in
let f d b := match d with D0 => b | D1 => false end
in int31_rect _ (nfold_bis _ _ f true size).

base is 2^31, obtained via iterations of Zdouble. It can also be seen as the smallest b > 0 s.t. phi_inv b = 0 (see below)

Definition base := Eval compute in
iter_nat size Z Zdouble 1%Z.

# Recursors

Fixpoint recl_aux (n:nat)(A:Type)(case0:A)(caserec:digits->int31->A->A)
(i:int31) : A :=
match n with
| O => case0
| S next =>
if iszero i then
case0
else
let si := shiftl i in
caserec (firstl i) si (recl_aux next A case0 caserec si)
end.

Fixpoint recr_aux (n:nat)(A:Type)(case0:A)(caserec:digits->int31->A->A)
(i:int31) : A :=
match n with
| O => case0
| S next =>
if iszero i then
case0
else
let si := shiftr i in
caserec (firstr i) si (recr_aux next A case0 caserec si)
end.

Definition recl := recl_aux size.
Definition recr := recr_aux size.

# Conversions

From int31 to Z, we simply iterates Zdouble or Zdouble_plus_one.

Definition phi : int31 -> Z :=
recr Z (0%Z)
(fun b _ => match b with D0 => Zdouble | D1 => Zdouble_plus_one end).

From positive to int31. An abstract definition could be : phi_inv (2n) = 2*(phi_inv n) /\ phi_inv 2n+1 = 2*(phi_inv n) + 1

Fixpoint phi_inv_positive p :=
match p with
| xI q => twice_plus_one (phi_inv_positive q)
| xO q => twice (phi_inv_positive q)
| xH => In
end.

The negative part : 2-complement

Fixpoint complement_negative p :=
match p with
| xI q => twice (complement_negative q)
| xO q => twice_plus_one (complement_negative q)
| xH => twice Tn
end.

A simple incrementation function

Definition incr : int31 -> int31 :=
recr int31 In
(fun b si rec => match b with
| D0 => sneakl D1 si
| D1 => sneakl D0 rec end).

We can now define the conversion from Z to int31.

Definition phi_inv : Z -> int31 := fun n =>
match n with
| Z0 => On
| Zpos p => phi_inv_positive p
| Zneg p => incr (complement_negative p)
end.

phi_inv2 is similar to phi_inv but returns a double word zn2z int31

Definition phi_inv2 n :=
match n with
| Z0 => W0
| _ => WW (phi_inv (n/base)%Z) (phi_inv n)
end.

phi2 is similar to phi but takes a double word (two args)

Definition phi2 nh nl :=
((phi nh)*base+(phi nl))%Z.

Definition add31 (n m : int31) := phi_inv ((phi n)+(phi m)).
Notation "n + m" := (add31 n m) : int31_scope.

Addition with carry (the result is thus exact)

Definition add31c (n m : int31) :=
let npm := n+m in
match (phi npm ?= (phi n)+(phi m))%Z with
| Eq => C0 npm
| _ => C1 npm
end.
Notation "n '+c' m" := (add31c n m) (at level 50, no associativity) : int31_scope.

Addition plus one with carry (the result is thus exact)

Definition add31carryc (n m : int31) :=
let npmpone_exact := ((phi n)+(phi m)+1)%Z in
let npmpone := phi_inv npmpone_exact in
match (phi npmpone ?= npmpone_exact)%Z with
| Eq => C0 npmpone
| _ => C1 npmpone
end.

# Substraction

Subtraction modulo 2^31

Definition sub31 (n m : int31) := phi_inv ((phi n)-(phi m)).
Notation "n - m" := (sub31 n m) : int31_scope.

Subtraction with carry (thus exact)

Definition sub31c (n m : int31) :=
let nmm := n-m in
match (phi nmm ?= (phi n)-(phi m))%Z with
| Eq => C0 nmm
| _ => C1 nmm
end.
Notation "n '-c' m" := (sub31c n m) (at level 50, no associativity) : int31_scope.

subtraction minus one with carry (thus exact)

Definition sub31carryc (n m : int31) :=
let nmmmone_exact := ((phi n)-(phi m)-1)%Z in
let nmmmone := phi_inv nmmmone_exact in
match (phi nmmmone ?= nmmmone_exact)%Z with
| Eq => C0 nmmmone
| _ => C1 nmmmone
end.

Multiplication

multiplication modulo 2^31

Definition mul31 (n m : int31) := phi_inv ((phi n)*(phi m)).
Notation "n * m" := (mul31 n m) : int31_scope.

multiplication with double word result (thus exact)

Definition mul31c (n m : int31) := phi_inv2 ((phi n)*(phi m)).
Notation "n '*c' m" := (mul31c n m) (at level 40, no associativity) : int31_scope.

# Division

Division of a double size word modulo 2^31

Definition div3121 (nh nl m : int31) :=
let (q,r) := Zdiv_eucl (phi2 nh nl) (phi m) in
(phi_inv q, phi_inv r).

Division modulo 2^31

Definition div31 (n m : int31) :=
let (q,r) := Zdiv_eucl (phi n) (phi m) in
(phi_inv q, phi_inv r).
Notation "n / m" := (div31 n m) : int31_scope.

# Unsigned comparison

Definition compare31 (n m : int31) := ((phi n)?=(phi m))%Z.
Notation "n ?= m" := (compare31 n m) (at level 70, no associativity) : int31_scope.

Computing the i-th iterate of a function: iter_int31 i A f = f^i

Definition iter_int31 i A f :=
recr (A->A) (fun x => x)
(fun b si rec => match b with
| D0 => fun x => rec (rec x)
| D1 => fun x => f (rec (rec x))
end)
i.

Combining the (31-p) low bits of i above the p high bits of j: addmuldiv31 p i j = i*2^p+j/2^(31-p) (modulo 2^31)

Definition addmuldiv31 p i j :=
let (res, _ ) :=
iter_int31 p (int31*int31)
(fun ij => let (i,j) := ij in (sneakl (firstl j) i, shiftl j))
(i,j)
in
res.

Definition gcd31 (i j:int31) :=
(fix euler (guard:nat) (i j:int31) {struct guard} :=
match guard with
| O => In
| S p => match j ?= On with
| Eq => i
| _ => euler p j (let (_, r ) := i/j in r)
end
end)
(2*size)%nat i j.

Square root functions using newton iteration we use a very naive upper-bound on the iteration 2^31 instead of the usual 31.

Definition sqrt31_step (rec: int31 -> int31 -> int31) (i j: int31) :=
Eval lazy delta [Twon] in
let (quo,_) := i/j in
match quo ?= j with
Lt => rec i (fst ((j + quo)/Twon))
| _ => j
end.

Fixpoint iter31_sqrt (n: nat) (rec: int31 -> int31 -> int31)
(i j: int31) {struct n} : int31 :=
sqrt31_step
(match n with
O => rec
| S n => (iter31_sqrt n (iter31_sqrt n rec))
end) i j.

Definition sqrt31 i :=
Eval lazy delta [On In Twon] in
match compare31 In i with
Gt => On
| Eq => In
| Lt => iter31_sqrt 31 (fun i j => j) i (fst (i/Twon))
end.

Definition v30 := Eval compute in (addmuldiv31 (phi_inv (Z_of_nat size - 1)) In On).

Definition sqrt312_step (rec: int31 -> int31 -> int31 -> int31)
(ih il j: int31) :=
Eval lazy delta [Twon v30] in
match ih ?= j with Eq => j | Gt => j | _ =>
let (quo,_) := div3121 ih il j in
match quo ?= j with
Lt => let m := match j +c quo with
C0 m1 => fst (m1/Twon)
| C1 m1 => fst (m1/Twon) + v30
end in rec ih il m
| _ => j
end end.

Fixpoint iter312_sqrt (n: nat)
(rec: int31 -> int31 -> int31 -> int31)
(ih il j: int31) {struct n} : int31 :=
sqrt312_step
(match n with
O => rec
| S n => (iter312_sqrt n (iter312_sqrt n rec))
end) ih il j.

Definition sqrt312 ih il :=
Eval lazy delta [On In] in
let s := iter312_sqrt 31 (fun ih il j => j) ih il Tn in
match s *c s with
W0 => (On, C0 On)
| WW ih1 il1 =>
match il -c il1 with
C0 il2 =>
match ih ?= ih1 with
Gt => (s, C1 il2)
| _ => (s, C0 il2)
end
| C1 il2 =>
match (ih - In) ?= ih1 with
Gt => (s, C1 il2)
| _ => (s, C0 il2)
end
end
end.

Fixpoint p2i n p : (N*int31)%type :=
match n with
| O => (Npos p, On)
| S n => match p with
| xO p => let (r,i) := p2i n p in (r, Twon*i)
| xI p => let (r,i) := p2i n p in (r, Twon*i+In)
| xH => (N0, In)
end
end.

Definition positive_to_int31 (p:positive) := p2i size p.

Constant 31 converted into type int31. It is used as default answer for numbers of zeros in head0 and tail0

Definition T31 : int31 := Eval compute in phi_inv (Z_of_nat size).

recl _ (fun _ => T31)
(fun b si rec n => match b with
| D0 => rec (add31 n In)
| D1 => n
end)
i On.

Definition tail031 (i:int31) :=
recr _ (fun _ => T31)
(fun b si rec n => match b with
| D0 => rec (add31 n In)
| D1 => n
end)
i On.