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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 *)
(************************************************************************)
(* $Id$ *)
Require Omega.
Require Export ZArith_base.
Require Export ZArithRing.
V7only [Import Z_scope.].
Open Local Scope Z_scope.
(**********************************************************************)
(** Definition and properties of square root on Z *)
(** The following tactic replaces all instances of (POS (xI ...)) by
`2*(POS ...)+1` , but only when ... is not made only with xO, XI, or xH. *)
Tactic Definition compute_POS :=
Match Context With
| [|- [(POS (xI ?1))]] ->
(Match ?1 With
| [[xH]] -> Fail 1
| _ -> Rewrite (POS_xI ?1))
| [|- [(POS (xO ?1))]] ->
(Match ?1 With
| [[xH]] -> Fail 1
| _ -> Rewrite (POS_xO ?1)).
Inductive sqrt_data [n : Z] : Set :=
c_sqrt: (s, r :Z)`n=s*s+r`->`0<=r<=2*s`->(sqrt_data n) .
Definition sqrtrempos: (p : positive) (sqrt_data (POS p)).
Refine (Fix sqrtrempos {
sqrtrempos [p : positive] : (sqrt_data (POS p)) :=
<[p : ?] (sqrt_data (POS p))> Cases p of
xH => (c_sqrt `1` `1` `0` ? ?)
| (xO xH) => (c_sqrt `2` `1` `1` ? ?)
| (xI xH) => (c_sqrt `3` `1` `2` ? ?)
| (xO (xO p')) =>
Cases (sqrtrempos p') of
(c_sqrt s' r' Heq Hint) =>
Cases (Z_le_gt_dec `4*s'+1` `4*r'`) of
(left Hle) =>
(c_sqrt (POS (xO (xO p'))) `2*s'+1` `4*r'-(4*s'+1)` ? ?)
| (right Hgt) =>
(c_sqrt (POS (xO (xO p'))) `2*s'` `4*r'` ? ?)
end
end
| (xO (xI p')) =>
Cases (sqrtrempos p') of
(c_sqrt s' r' Heq Hint) =>
Cases
(Z_le_gt_dec `4*s'+1` `4*r'+2`) of
(left Hle) =>
(c_sqrt
(POS (xO (xI p'))) `2*s'+1` `4*r'+2-(4*s'+1)` ? ?)
| (right Hgt) =>
(c_sqrt (POS (xO (xI p'))) `2*s'` `4*r'+2` ? ?)
end
end
| (xI (xO p')) =>
Cases (sqrtrempos p') of
(c_sqrt s' r' Heq Hint) =>
Cases
(Z_le_gt_dec `4*s'+1` `4*r'+1`) of
(left Hle) =>
(c_sqrt
(POS (xI (xO p'))) `2*s'+1` `4*r'+1-(4*s'+1)` ? ?)
| (right Hgt) =>
(c_sqrt (POS (xI (xO p'))) `2*s'` `4*r'+1` ? ?)
end
end
| (xI (xI p')) =>
Cases (sqrtrempos p') of
(c_sqrt s' r' Heq Hint) =>
Cases
(Z_le_gt_dec `4*s'+1` `4*r'+3`) of
(left Hle) =>
(c_sqrt
(POS (xI (xI p'))) `2*s'+1` `4*r'+3-(4*s'+1)` ? ?)
| (right Hgt) =>
(c_sqrt (POS (xI (xI p'))) `2*s'` `4*r'+3` ? ?)
end
end
end
}); Clear sqrtrempos; Repeat compute_POS;
Try (Try Rewrite Heq; Ring; Fail); Try Omega.
Defined.
(** Define with integer input, but with a strong (readable) specification. *)
Definition Zsqrt : (x:Z)`0<=x`->{s:Z & {r:Z | x=`s*s+r` /\ `s*s<=x<(s+1)*(s+1)`}}.
Refine [x]
<[x:Z]`0<=x`->{s:Z & {r:Z | x=`s*s+r` /\ `s*s<=x<(s+1)*(s+1)`}}>Cases x of
(POS p) => [h]Cases (sqrtrempos p) of
(c_sqrt s r Heq Hint) =>
(existS ? [s:Z]{r:Z | `(POS p)=s*s+r` /\
`s*s<=(POS p)<(s+1)*(s+1)`}
s
(exist Z [r:Z]((POS p)=`s*s+r` /\ `s*s<=(POS p)<(s+1)*(s+1)`)
r ?))
end
| (NEG p) => [h](False_rec
{s:Z & {r:Z |
(NEG p)=`s*s+r` /\ `s*s<=(NEG p)<(s+1)*(s+1)`}}
(h (refl_equal ? SUPERIEUR)))
| ZERO => [h](existS ? [s:Z]{r:Z | `0=s*s+r` /\ `s*s<=0<(s+1)*(s+1)`}
`0` (exist Z [r:Z](`0=0*0+r`/\`0*0<=0<(0+1)*(0+1)`)
`0` ?))
end;Try Omega.
Split;[Omega|Rewrite Heq;Ring `(s+1)*(s+1)`;Omega].
Defined.
(** Define a function of type Z->Z that computes the integer square root,
but only for positive numbers, and 0 for others. *)
Definition Zsqrt_plain : Z->Z :=
[x]Cases x of
(POS p)=>Cases (Zsqrt (POS p) (ZERO_le_POS p)) of (existS s _) => s end
|(NEG p)=>`0`
|ZERO=>`0`
end.
(** A basic theorem about Zsqrt_plain *)
Theorem Zsqrt_interval :(x:Z)`0<=x`->
`(Zsqrt_plain x)*(Zsqrt_plain x)<= x < ((Zsqrt_plain x)+1)*((Zsqrt_plain x)+1)`.
Intros x;Case x.
Unfold Zsqrt_plain;Omega.
Intros p;Unfold Zsqrt_plain;Case (Zsqrt (POS p) (ZERO_le_POS p)).
Intros s (r,(Heq,Hint)) Hle;Assumption.
Intros p Hle;Elim Hle;Auto.
Qed.
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