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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(** * Natural numbers in base 2^31 *)
(**
Author: Arnaud Spiwack
*)
Require Export Int31.
Require Import NMake.
Require Import ZnZ.
Open Scope int31_scope.
Definition int31_op : znz_op int31.
split.
(* Conversion functions with Z *)
exact (31%positive). (* number of digits *)
exact (31). (* number of digits *)
exact (phi). (* conversion to Z *)
exact (positive_to_int31). (* positive -> N*int31 : p => N,i where p = N*2^31+phi i *)
exact head031. (* number of head 0 *)
exact tail031. (* number of tail 0 *)
(* Basic constructors *)
exact 0. (* 0 *)
exact 1. (* 1 *)
exact Tn. (* 2^31 - 1 *)
(* A function which given two int31 i and j, returns a double word
which is worth i*2^31+j *)
exact (fun i j => match (match i ?= 0 with | Eq => j ?= 0 | not0 => not0 end) with | Eq => W0 | _ => WW i j end).
(* two special cases where i and j are respectively taken equal to 0 *)
exact (fun i => match i ?= 0 with | Eq => W0 | _ => WW i 0 end).
exact (fun j => match j ?= 0 with | Eq => W0 | _ => WW 0 j end).
(* Comparison *)
exact compare31.
exact (fun i => match i ?= 0 with | Eq => true | _ => false end).
(* Basic arithmetic operations *)
(* opposite functions *)
exact (fun i => 0 -c i).
exact (fun i => 0 - i).
exact (fun i => 0-i-1). (* the carry is always -1*)
(* successor and addition functions *)
exact (fun i => i +c 1).
exact add31c.
exact add31carryc.
exact (fun i => i + 1).
exact add31.
exact (fun i j => i + j + 1).
(* predecessor and subtraction functions *)
exact (fun i => i -c 1).
exact sub31c.
exact sub31carryc.
exact (fun i => i - 1).
exact sub31.
exact (fun i j => i - j - 1).
(* multiplication functions *)
exact mul31c.
exact mul31.
exact (fun x => x *c x).
(* special (euclidian) division operations *)
exact div3121.
exact div31. (* this is supposed to be the special case of division a/b where a > b *)
exact div31.
(* euclidian division remainder *)
(* again special case for a > b *)
exact (fun i j => let (_,r) := i/j in r).
exact (fun i j => let (_,r) := i/j in r).
(* gcd functions *)
exact gcd31. (*gcd_gt*)
exact gcd31. (*gcd*)
(* shift operations *)
exact addmuldiv31. (*add_mul_div *)
(*modulo 2^p *)
exact (fun p i =>
match compare31 p 32 with
| Lt => addmuldiv31 p 0 (addmuldiv31 (31-p) i 0)
| _ => i
end).
(* is i even ? *)
exact (fun i => let (_,r) := i/2 in
match r ?= 0 with
| Eq => true
| _ => false
end).
(* square root operations *)
exact sqrt312. (* sqrt2 *)
exact sqrt31. (* sqr *)
Defined.
Definition int31_spec : znz_spec int31_op.
Admitted.
Module Int31_words <: W0Type.
Definition w := int31.
Definition w_op := int31_op.
Definition w_spec := int31_spec.
End Int31_words.
Module BigN := NMake.Make Int31_words.
Definition bigN := BigN.t.
Delimit Scope bigN_scope with bigN.
Bind Scope bigN_scope with bigN.
Bind Scope bigN_scope with BigN.t.
Bind Scope bigN_scope with BigN.t_.
Notation " i + j " := (BigN.add i j) : bigN_scope.
Notation " i - j " := (BigN.sub i j) : bigN_scope.
Notation " i * j " := (BigN.mul i j) : bigN_scope.
Notation " i / j " := (BigN.div i j) : bigN_scope.
Notation " i ?= j " := (BigN.compare i j) : bigN_scope.
Theorem succ_pred: forall q,
(0 < BigN.to_Z q ->
BigN.to_Z (BigN.succ (BigN.pred q)) = BigN.to_Z q)%Z.
intros q Hq.
rewrite BigN.spec_succ.
rewrite BigN.spec_pred; auto.
generalize Hq; set (a := BigN.to_Z q).
ring_simplify (a - 1 + 1)%Z; auto.
Qed.
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