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|
(**************************************************************************)
(* *)
(* SMTCoq *)
(* Copyright (C) 2011 - 2015 *)
(* *)
(* Chantal Keller *)
(* *)
(* from the Int63 library of native-coq *)
(* by Benjamin Gregoire and Laurent Thery *)
(* *)
(* Inria - École Polytechnique - MSR-Inria Joint Lab *)
(* *)
(* This file is distributed under the terms of the CeCILL-C licence *)
(* *)
(**************************************************************************)
Require Import Int63Lib Cyclic63.
Require Export Int63Native.
Require Import BigNumPrelude.
Require Import Bvector.
Local Open Scope int63_scope.
(** The number of digits as a int *)
Definition digits := zdigits.
(** The bigger int *)
Definition max_int := Eval vm_compute in 0 - 1.
(** Access to the nth digits *)
Definition get_digit x p := (0 < (x land (1 << p))).
Definition set_digit x p (b:bool) :=
if (0 <= p) && (p < digits) then
if b then x lor (1 << p)
else x land (max_int lxor (1 << p))
else x.
(** Equality to 0 *)
Definition is_zero (i:int) := i == 0.
(** Parity *)
Definition is_even (i:int) := is_zero (i land 1).
(** Bit *)
Definition bit i n := negb (is_zero ((i >> n) << (digits - 1))).
(* Register bit as PrimInline. *)
(** Extra modulo operations *)
Definition opp (i:int) := 0 - i.
Notation "- x" := (opp x) : int63_scope.
Definition oppcarry i := max_int - i.
Definition succ i := i + 1.
Definition pred i := i - 1.
Definition addcarry i j := i + j + 1.
Definition subcarry i j := i - j - 1.
(** Exact arithmetic operations *)
Definition addc_def x y :=
let r := x + y in
if r < x then C1 r else C0 r.
(* the same but direct implementation for efficiancy *)
Definition addc : int -> int -> carry int := add63c.
Notation "n '+c' m" := (addc n m) (at level 50, no associativity) : int63_scope.
Definition addcarryc_def x y :=
let r := addcarry x y in
if r <= x then C1 r else C0 r.
(* the same but direct implementation for efficiancy *)
Definition addcarryc : int -> int -> carry int := add63carryc.
Definition subc_def x y :=
if y <= x then C0 (x - y) else C1 (x - y).
(* the same but direct implementation for efficiancy *)
Definition subc : int -> int -> carry int := sub63c.
Notation "n '-c' m" := (subc n m) (at level 50, no associativity) : int63_scope.
Definition subcarryc_def x y :=
if y < x then C0 (x - y - 1) else C1 (x - y - 1).
(* the same but direct implementation for efficiancy *)
Definition subcarryc : int -> int -> carry int := sub63carryc.
Definition diveucl_def x y := (x/y, x\%y).
(* the same but direct implementation for efficiancy *)
Definition diveucl : int -> int -> int * int := div63.
Definition diveucl_21 : int -> int -> int -> int * int := div6321.
Definition addmuldiv_def p x y :=
(x << p) lor (y >> (digits - p)).
(* the same but direct implementation for efficiancy *)
Definition addmuldiv : int -> int -> int -> int := addmuldiv63.
Definition oppc (i:int) := 0 -c i.
Definition succc i := i +c 1.
Definition predc i := i -c 1.
(** Comparison *)
Definition compare_def x y :=
if x < y then Lt
else if (x == y) then Eq else Gt.
Definition compare : int -> int -> comparison := compare63.
Notation "n ?= m" := (compare n m) (at level 70, no associativity) : int63_scope.
(** Exotic operations *)
(** I should add the definition (like for compare) *)
Definition head0 : int -> int := head063.
Definition tail0 : int -> int := tail063.
(** Iterators *)
Definition foldi {A} (f:int -> A -> A) from to :=
foldi_cont (fun i cont a => cont (f i a)) from to (fun a => a).
Definition fold {A} (f: A -> A) from to :=
foldi_cont (fun i cont a => cont (f a)) from to (fun a => a).
Definition foldi_down {A} (f:int -> A -> A) from downto :=
foldi_down_cont (fun i cont a => cont (f i a)) from downto (fun a => a).
Definition fold_down {A} (f:A -> A) from downto :=
foldi_down_cont (fun i cont a => cont (f a)) from downto (fun a => a).
Definition forallb (f:int -> bool) from to :=
foldi_cont (fun i cont _ => if f i then cont tt else false) from to (fun _ => true) tt.
Definition existsb (f:int -> bool) from to :=
foldi_cont (fun i cont _ => if f i then true else cont tt) from to (fun _ => false) tt.
(** Translation to Z *)
(* Fixpoint to_Z_rec (n:nat) (i:int) := *)
(* match n with *)
(* | O => 0%Z *)
(* | S n => *)
(* (if is_even i then Zdouble else Zdouble_plus_one) (to_Z_rec n (i >> 1)) *)
(* end. *)
(* Definition to_Z := to_Z_rec size. *)
Definition to_Z := phi.
Fixpoint of_pos_rec (n:nat) (p:positive) :=
match n, p with
| O, _ => 0
| S n, xH => 1
| S n, xO p => (of_pos_rec n p) << 1
| S n, xI p => (of_pos_rec n p) << 1 lor 1
end.
Definition of_pos := of_pos_rec size.
Definition of_Z z :=
match z with
| Zpos p => of_pos p
| Z0 => 0
| Zneg p => - (of_pos p)
end.
(** Gcd **)
Fixpoint gcd_rec (guard:nat) (i j:int) {struct guard} :=
match guard with
| O => 1
| S p => if j == 0 then i else gcd_rec p j (i \% j)
end.
Definition gcd := gcd_rec (2*size).
(** Square root functions using newton iteration **)
Definition sqrt_step (rec: int -> int -> int) (i j: int) :=
let quo := i/j in
if quo < j then rec i ((j + i/j) >> 1)
else j.
Definition iter_sqrt :=
Eval lazy beta delta [sqrt_step] in
fix iter_sqrt (n: nat) (rec: int -> int -> int)
(i j: int) {struct n} : int :=
sqrt_step
(fun i j => match n with
O => rec i j
| S n => (iter_sqrt n (iter_sqrt n rec)) i j
end) i j.
Definition sqrt i :=
match compare 1 i with
Gt => 0
| Eq => 1
| Lt => iter_sqrt size (fun i j => j) i (i >> 1)
end.
Definition high_bit := 1 << (digits - 1).
Definition sqrt2_step (rec: int -> int -> int -> int)
(ih il j: int) :=
if ih < j then
let (quo,_) := diveucl_21 ih il j in
if quo < j then
match j +c quo with
| C0 m1 => rec ih il (m1 >> 1)
| C1 m1 => rec ih il ((m1 >> 1) + high_bit)
end
else j
else j.
Definition iter2_sqrt :=
Eval lazy beta delta [sqrt2_step] in
fix iter2_sqrt (n: nat)
(rec: int -> int -> int -> int)
(ih il j: int) {struct n} : int :=
sqrt2_step
(fun ih il j =>
match n with
| O => rec ih il j
| S n => (iter2_sqrt n (iter2_sqrt n rec)) ih il j
end) ih il j.
Definition sqrt2 ih il :=
let s := iter2_sqrt size (fun ih il j => j) ih il max_int in
let (ih1, il1) := mulc s s in
match il -c il1 with
| C0 il2 =>
if ih1 < ih then (s, C1 il2) else (s, C0 il2)
| C1 il2 =>
if ih1 < (ih - 1) then (s, C1 il2) else (s, C0 il2)
end.
(* Extra function on equality *)
Definition cast i j :=
(if i == j as b return ((b = true -> i = j) -> option (forall P : int -> Type, P i -> P j))
then fun Heq : true = true -> i = j =>
Some
(fun (P : int -> Type) (Hi : P i) =>
match Heq (eq_refl true) in (_ = y) return (P y) with
| eq_refl => Hi
end)
else fun _ : false = true -> i = j => None) (eqb_correct i j).
Definition eqo i j :=
(if i == j as b return ((b = true -> i = j) -> option (i=j))
then fun Heq : true = true -> i = j =>
Some (Heq (eq_refl true))
else fun _ : false = true -> i = j => None) (eqb_correct i j).
|
