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-rw-r--r--theories/Numbers/Cyclic/Int31/Int31.v141
1 files changed, 75 insertions, 66 deletions
diff --git a/theories/Numbers/Cyclic/Int31/Int31.v b/theories/Numbers/Cyclic/Int31/Int31.v
index 154b436b..cc224254 100644
--- a/theories/Numbers/Cyclic/Int31/Int31.v
+++ b/theories/Numbers/Cyclic/Int31/Int31.v
@@ -8,7 +8,7 @@
(* Benjamin Gregoire, Laurent Thery, INRIA, 2007 *)
(************************************************************************)
-(*i $Id: Int31.v 11072 2008-06-08 16:13:37Z herbelin $ i*)
+(*i $Id$ i*)
Require Import NaryFunctions.
Require Import Wf_nat.
@@ -17,7 +17,7 @@ Require Export DoubleType.
Unset Boxed Definitions.
-(** * 31-bit integers *)
+(** * 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
@@ -36,11 +36,13 @@ Definition size := 31%nat.
Inductive digits : Type := D0 | D1.
(** The type of 31-bit integers *)
-
-(** The type [int31] has a unique constructor [I31] that expects
+
+(** The type [int31] has a unique constructor [I31] that expects
31 arguments of type [digits]. *)
-Inductive int31 : Type := I31 : nfun digits size int31.
+Definition digits31 t := Eval compute in nfun digits size t.
+
+Inductive int31 : Type := I31 : digits31 int31.
(* spiwack: Registration of the type of integers, so that the matchs in
the functions below perform dynamic decompilation (otherwise some segfault
@@ -50,7 +52,7 @@ Register int31 as int31 type in "coq_int31" by True.
Delimit Scope int31_scope with int31.
Bind Scope int31_scope with int31.
-Open Scope int31_scope.
+Local Open Scope int31_scope.
(** * Constants *)
@@ -69,26 +71,26 @@ Definition Twon : int31 := Eval compute in (napply_cst _ _ D0 (size-2) I31) D1 D
(** * Bits manipulation *)
-(** [sneakr b x] shifts [x] to the right by one bit.
+(** [sneakr b x] shifts [x] to the right by one bit.
Rightmost digit is lost while leftmost digit becomes [b].
- Pseudo-code is
+ 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.
+(** [sneakl b x] shifts [x] to the left by one bit.
Leftmost digit is lost while rightmost digit becomes [b].
- Pseudo-code is
+ Pseudo-code is
[ match x with (I31 d0 ... dN) => I31 d1 ... dN b end ]
*)
-Definition sneakl : digits -> int31 -> int31 := Eval compute in
+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
+(** [shiftl], [shiftr], [twice] and [twice_plus_one] are direct
consequences of [sneakl] and [sneakr]. *)
Definition shiftl := sneakl D0.
@@ -96,31 +98,31 @@ Definition shiftr := sneakr D0.
Definition twice := sneakl D0.
Definition twice_plus_one := sneakl D1.
-(** [firstl x] returns the leftmost digit of number [x].
+(** [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
+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].
+(** [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
+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
+(** [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
+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).
-(* NB: DO NOT transform the above match in a nicer (if then else).
+(* NB: DO NOT transform the above match in a nicer (if then else).
It seems to work, but later "unfold iszero" takes forever. *)
-(** [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
+(** [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
@@ -140,7 +142,7 @@ Fixpoint recl_aux (n:nat)(A:Type)(case0:A)(caserec:digits->int31->A->A)
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)
+Fixpoint recr_aux (n:nat)(A:Type)(case0:A)(caserec:digits->int31->A->A)
(i:int31) : A :=
match n with
| O => case0
@@ -159,22 +161,22 @@ Definition recr := recr_aux size.
(** From int31 to Z, we simply iterates [Zdouble] or [Zdouble_plus_one]. *)
-Definition phi : int31 -> Z :=
+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) /\
+(** 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 :=
+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 *)
+(** The negative part : 2-complement *)
Fixpoint complement_negative p :=
match p with
@@ -186,9 +188,9 @@ Fixpoint complement_negative p :=
(** A simple incrementation function *)
Definition incr : int31 -> int31 :=
- recr int31 In
- (fun b si rec => match b with
- | D0 => sneakl D1 si
+ 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. *)
@@ -196,11 +198,11 @@ Definition incr : int31 -> int31 :=
Definition phi_inv : Z -> int31 := fun n =>
match n with
| Z0 => On
- | Zpos p => phi_inv_positive p
+ | 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
+(** [phi_inv2] is similar to [phi_inv] but returns a double word
[zn2z int31] *)
Definition phi_inv2 n :=
@@ -211,7 +213,7 @@ Definition phi_inv2 n :=
(** [phi2] is similar to [phi] but takes a double word (two args) *)
-Definition phi2 nh nl :=
+Definition phi2 nh nl :=
((phi nh)*base+(phi nl))%Z.
(** * Addition *)
@@ -227,11 +229,11 @@ Notation "n + m" := (add31 n m) : int31_scope.
(* mode, (phi n)+(phi m) is computed twice*)
(* it may be considered to optimize it *)
-Definition add31c (n m : int31) :=
+Definition add31c (n m : int31) :=
let npm := n+m in
- match (phi npm ?= (phi n)+(phi m))%Z with
- | Eq => C0 npm
- | _ => C1 npm
+ 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.
@@ -254,7 +256,7 @@ Notation "n - m" := (sub31 n m) : int31_scope.
(** Subtraction with carry (thus exact) *)
-Definition sub31c (n m : int31) :=
+Definition sub31c (n m : int31) :=
let nmm := n-m in
match (phi nmm ?= (phi n)-(phi m))%Z with
| Eq => C0 nmm
@@ -272,6 +274,10 @@ Definition sub31carryc (n m : int31) :=
| _ => C1 nmmmone
end.
+(** Opposite *)
+
+Definition opp31 x := On - x.
+Notation "- x" := (opp31 x) : int31_scope.
(** Multiplication *)
@@ -290,13 +296,13 @@ Notation "n '*c' m" := (mul31c n m) (at level 40, no associativity) : int31_scop
(** Division of a double size word modulo [2^31] *)
-Definition div3121 (nh nl m : int31) :=
+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) :=
+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.
@@ -307,13 +313,16 @@ Notation "n / m" := (div31 n m) : int31_scope.
Definition compare31 (n m : int31) := ((phi n)?=(phi m))%Z.
Notation "n ?= m" := (compare31 n m) (at level 70, no associativity) : int31_scope.
+Definition eqb31 (n m : int31) :=
+ match n ?= m with Eq => true | _ => false end.
+
-(** Computing the [i]-th iterate of a function:
+(** 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
+ 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)
@@ -322,9 +331,9 @@ Definition iter_int31 i A f :=
(** 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)
+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
@@ -346,7 +355,7 @@ Register addmuldiv31 as int31 addmuldiv in "coq_int31" by True.
Definition gcd31 (i j:int31) :=
(fix euler (guard:nat) (i j:int31) {struct guard} :=
- match guard with
+ match guard with
| O => In
| S p => match j ?= On with
| Eq => i
@@ -370,17 +379,17 @@ Eval lazy delta [Twon] in
| _ => j
end.
-Fixpoint iter31_sqrt (n: nat) (rec: int31 -> int31 -> int31)
+Fixpoint iter31_sqrt (n: nat) (rec: int31 -> int31 -> int31)
(i j: int31) {struct n} : int31 :=
- sqrt31_step
+ sqrt31_step
(match n with
O => rec
| S n => (iter31_sqrt n (iter31_sqrt n rec))
end) i j.
-Definition sqrt31 i :=
+Definition sqrt31 i :=
Eval lazy delta [On In Twon] in
- match compare31 In i with
+ match compare31 In i with
Gt => On
| Eq => In
| Lt => iter31_sqrt 31 (fun i j => j) i (fst (i/Twon))
@@ -388,7 +397,7 @@ Eval lazy delta [On In Twon] in
Definition v30 := Eval compute in (addmuldiv31 (phi_inv (Z_of_nat size - 1)) In On).
-Definition sqrt312_step (rec: int31 -> int31 -> int31 -> int31)
+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 | _ =>
@@ -401,28 +410,28 @@ Eval lazy delta [Twon v30] in
| _ => j
end end.
-Fixpoint iter312_sqrt (n: nat)
- (rec: int31 -> int31 -> int31 -> int31)
+Fixpoint iter312_sqrt (n: nat)
+ (rec: int31 -> int31 -> int31 -> int31)
(ih il j: int31) {struct n} : int31 :=
- sqrt312_step
+ sqrt312_step
(match n with
O => rec
| S n => (iter312_sqrt n (iter312_sqrt n rec))
end) ih il j.
-Definition sqrt312 ih il :=
+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) (* impossible *)
| WW ih1 il1 =>
match il -c il1 with
- C0 il2 =>
+ C0 il2 =>
match ih ?= ih1 with
Gt => (s, C1 il2)
| _ => (s, C0 il2)
end
- | C1 il2 =>
+ | C1 il2 =>
match (ih - In) ?= ih1 with (* we could parametrize ih - 1 *)
Gt => (s, C1 il2)
| _ => (s, C0 il2)
@@ -431,7 +440,7 @@ Eval lazy delta [On In] in
end.
-Fixpoint p2i n p : (N*int31)%type :=
+Fixpoint p2i n p : (N*int31)%type :=
match n with
| O => (Npos p, On)
| S n => match p with
@@ -444,26 +453,26 @@ Fixpoint p2i n p : (N*int31)%type :=
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
+ 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).
Definition head031 (i:int31) :=
- recl _ (fun _ => T31)
- (fun b si rec n => match b with
+ 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
+ recr _ (fun _ => T31)
+ (fun b si rec n => match b with
| D0 => rec (add31 n In)
| D1 => n
end)
i On.
Register head031 as int31 head0 in "coq_int31" by True.
-Register tail031 as int31 tail0 in "coq_int31" by True.
+Register tail031 as int31 tail0 in "coq_int31" by True.
generated by cgit v1.2.3 (git 2.39.1) at 2024-05-18 05:15:55 +0000