Re: Bug#224417: Problem with the old Num implementation of ocaml - more details

[Thomas Fischbacher <tf@cip.physik.uni-muenchen.de>]

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[Maybe I should file this as a separate bugreport]

On Thu, 18 Dec 2003, Sven Luther wrote:

> tags 224417 + upstream
> thanks
> On Thu, Dec 18, 2003 at 09:15:40PM +0100, Thomas Fischbacher wrote:
> > 
> > Package:  ocaml 
> > Version: 3.06-21
> > Severity: grave
> 
> Huh ...
> 
> BTW, 3.06-21 is not part of debian anymore, but i guess 3.07-7 and the
> woody (and older) version are vulnerable.
> 
> I will transmit this to upstream too, altough i somewhat doubt they will
> fix it, since old-bignum is going to be replaced with the new
> implementation. But i think it will interest them still.

Yes, I suppose so. But strangely enough, the problem _did_ re-occur - 
albeit with very different values - with

Package: ocaml
Version: 3.07.2a-1

and

Package: libnums-ocaml
Version: 3.07-1

Here, I get:

0 ==> -63739694011402166976051874041676063865784980095311212432062478926751299450139305457948134038790907007782943115234375/2294287488095403763073974596800604256904498055825734116841519453845801891422853587965749144293625676308261780152348780234845939207145908383623706636688843241160704
1 ==> 0
2 ==> -12747938802280433395210374808335212773156996019062242486412495785350259890027861091589626807758181401556588623046875/11301908808351742675241254171431548063568955940028246880992706669191142322280066935791867705879929439942176256908122070122393789197763095485831067175807109562368
3 ==> 0
4 ==> -12747938802280433395210374808335212773156996019062242486412495785350259890027861091589626807758181401556588623046875/1280294357196095849929673324107480054076170790081324841987455052369309091195788832570172513556710756868449654102873203256052421432559100660504300578509399130112
5 ==> 0
6 ==> -12747938802280433395210374808335212773156996019062242486412495785350259890027861091589626807758181401556588623046875/11301908808351742675241254171431548063568955940028246880992706669191142322280066935791867705879929439942176256908122070122393789197763095485831067175807109562368
7 ==> -39148920061803210956691061036397438426365134774540146675772774556810648122275561412271743926625375084180283661376953125/2294287488095403763073974596800604256904498055825734116841519453845801891422853587965749144293625676308261780152348780234845939207145908383623706636688843241160704
8 ==> -570392894161458354485388740908288985192455282044195662517309148957502839351474204759670942202082333316731293944994384765625/95595312003975156794748941533358510704354085659405588201729977243575078809285566165239547678901069846177574173014532509785247466964412849317654443195368468381696
9 ==> -39148920061803210956691061036397438426365134774540146675772774556810648122275561412271743926625375084180283661376953125/2294287488095403763073974596800604256904498055825734116841519453845801891422853587965749144293625676308261780152348780234845939207145908383623706636688843241160704
- : unit = ()

Again, of course, zero is the correct value. At least, it does occur more 
frequently.


For sure, this made me wonder whether the bug may nevertheless be on my 
side, considering that I am terribly overworked due to this research paper 
at the moment, but I stared at the source over and over again, and I 
simply do not use global state / non-functional features outside of the 
main functions that could cause such behaviour... And even if I did, how 
could the results depend on the particular implementation of ratio 
arithmetics I am using? Yet to me it seems plausible that some serious 
issue may have been overlooked here, since I suppose very few people use 
ratio arith (in contrast to bignum-only arith, which has applications for, 
e.g., RSA).

This is all very puzzling at the moment...

-- 
regards,                  tf@cip.physik.uni-muenchen.de              (o_
Dr. Thomas Fischbacher -  http://www.cip.physik.uni-muenchen.de/~tf  //\
(lambda (n) ((lambda (p q r) (p p q r)) (lambda (g x y)              V_/_
(if (= x 0) y (g g (- x 1) (* x y)))) n 1))                      (Debian GNU)

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open Num;;

let int1=Int 1;; (* To avoid number consing later on... *)
let int0=Int 0;; 
let int2=Int 2;; 

type lpower =
    {mutable power: int;
     mutable vars: int; (* bit vector *)
     mutable coeffs: int array (* always make denominators rational *)
   };;

let power_of_2 n =
  let rec trav p ppow =
    if ppow>n
    then -1
    else if ppow<n
    then trav (1+p) (ppow+ppow)
    else p
  in trav 0 1;;

(*
   We need an order on lpowers...
 *)

let v_cmp cmp a b =
  let len_a = Array.length a in
  let len_b = Array.length b in
  let rec trav pos =
    if pos == len_a
    then
      if len_a == len_b then 0 else -1
    else
      if pos == len_b
      then 1
      else
	let xa = Array.get a pos in
	let xb = Array.get b pos in
	let cmp_xa_xb = cmp xa xb in
	if cmp_xa_xb != 0 then cmp_xa_xb else trav (1+pos)
  in trav 0;;


(* 
   Example:

   v_cmp (fun x y -> if x < y then -1 else if x > y then 1 else 0) [|1;2;3;4;8|] [|1;2;3;4|];;
 *)


let lpower_cmp also_cmp_powers lp_a lp_b =
  if lp_a.vars < lp_b.vars then -1
  else if lp_a.vars > lp_b.vars then 1
  else let cmp =
    v_cmp (fun x y -> if x < y
                       then -1
                       else if x > y then 1
                       else 0)
      lp_a.coeffs lp_b.coeffs in
  if cmp != 0 then cmp
      else
       if also_cmp_powers
       then if lp_a.power < lp_b.power then -1
            else if lp_a.power > lp_b.power then 1
            else 0
       else 0;;

(*
   When combining lpower vectors,
   we first have to count the number of different entries...
 *)


let lpower_nr_different_noncancellating v_lp_a v_lp_b =
  let len_a = Array.length v_lp_a in
  let len_b = Array.length v_lp_b in
  let rec trav n_so_far pos_a pos_b =
    if pos_a == len_a then n_so_far + len_b-pos_b
	else if pos_b == len_b then n_so_far + len_a-pos_a
	else
	  let entry_a = Array.get v_lp_a pos_a in
	  let entry_b = Array.get v_lp_b pos_b in
	  let cmp_here = lpower_cmp false entry_a entry_b in
	  if cmp_here == 0 then
	    if entry_a.power + entry_b.power == 0
	    then trav n_so_far (1+pos_a) (1+pos_b)
	    else trav (1+n_so_far) (1+pos_a) (1+pos_b)
	  else if cmp_here == -1 then trav (1+n_so_far) (1+pos_a) pos_b
	  else trav (1+n_so_far) pos_a (1+pos_b)
  in trav 0 0 0;;


(*
   We will make good use of lpower substitution not only
   for doing calculations, but also for constructing
   the initial terms.
 *)

(*

   First of all, we need integer bitmask functions...

 *)

let integer_length n =
  let rec trav rest_n bit_now =
  if rest_n == 0
  then bit_now
  else
    let contrib = 1 lsl bit_now in
    if 0 == contrib land rest_n
    then trav rest_n (1+bit_now)
    else trav (rest_n-contrib) (1+bit_now)
  in trav n 0;;

let nr_bits_in n =
  let rec trav rest_n bit_now nr =
  if rest_n == 0
  then nr
  else
    let contrib = 1 lsl bit_now in
    if 0 == contrib land rest_n
    then trav rest_n (1+bit_now) nr
    else trav (rest_n-contrib) (1+bit_now) (1+nr)
  in trav n 0 0;;


(* Given a number and a value 2^bit, tell me the how manyth set bit
   in the number it is, starting to count at 0... *)

let nr_bits_set_before num bitpow =
  let rec walk nr val_now =
    if val_now == bitpow
    then nr
    else walk (if (0==val_now land num) then nr else nr+1)
	(val_now+val_now)
  in walk 0 1;;

(* ... and some other helpers... *)

let gcd a b =
  let rec work p q =
    let mpq = p mod q in
    if mpq==0 then q else work q mpq
  in
  let aa = abs a in
  let ab = abs b in
  if aa > ab then work aa ab else work ab aa;;


let v_gcd v_int =
  let len = Array.length v_int in
  if len == 0
   then 1
   else if len == 1 then abs (Array.get v_int 0)
   else let rec trav pos gcd_now =
     if gcd_now==1 then 1
     else if pos==len then gcd_now
     else trav (1+pos) (gcd gcd_now (Array.get v_int pos))
   in trav 1 (Array.get v_int 0);;


let lcm a b = abs(a*b/(gcd a b));;

(*

   Here, quite some interesting things do happen...

   Note that the given var MUST occur in the substitution rule...

   We desructively change the extra factors in ref_numer and ref_denom
   in order to be able to keep our coeffs integral and stay fast...

   Note that one has to convince oneself that, by looking at the given
   structures of the terms showing up, we do not leave integer range
   by applying #r substitutions of linear functions with coeffs max. 2
   (and what one can get from that in consecutive subs) as well as
   powers up to 2^r...
   
   Now, how big can we make r before violating these constraints...?

   XXX Note: need another variant, where we do not cons the result,
   but instead just eval it!

 *)


let get_coeff lp coeff_bitpow =
  let pos = nr_bits_set_before lp.vars coeff_bitpow in
  Array.get lp.coeffs pos;;


let lpower_subs ref_xfactor var_bitpow to_subs lp =
  if (var_bitpow land lp.vars) == 0 then lp (* nothing to substitute *)
  else
    let subs_other_vars = var_bitpow lxor to_subs.vars in
    let lp_other_vars = var_bitpow lxor lp.vars in
    let total_vars = subs_other_vars lor lp_other_vars in
    let pos_var_in_lp = nr_bits_set_before lp.vars var_bitpow in
    let pos_var_in_subs = nr_bits_set_before to_subs.vars var_bitpow in
    let var_coeff_in_lp = Array.get lp.coeffs pos_var_in_lp in
    let var_coeff_in_subs = Array.get to_subs.coeffs pos_var_in_subs in
    let var_coeff = lcm var_coeff_in_lp var_coeff_in_subs in
    let factor_subs = -var_coeff/var_coeff_in_subs in
    let factor_lp = var_coeff/var_coeff_in_lp in
    let nr_total_vars = nr_bits_in total_vars in (* May still be reduced by additional cancellations! *)
    let result_coeffs = Array.make nr_total_vars 0 in
    let rec fill_result bitpow result_pos true_result_vars =
      if bitpow > total_vars
      then (* finished -- but may have to take sub-range! *)
       let final_coeffs =
	 if result_pos==nr_total_vars
 	  then result_coeffs
	  else Array.sub result_coeffs 0 result_pos in
       {power=lp.power;vars=true_result_vars;coeffs=final_coeffs}
      else if 0 != (lp_other_vars land bitpow)
	   then if 0 != (subs_other_vars land bitpow)
	        then let val_to_set =  (get_coeff lp bitpow)*factor_lp
		                     + (get_coeff to_subs bitpow)*factor_subs in
		  if val_to_set==0
		   then fill_result (bitpow+bitpow) result_pos true_result_vars
		   else (Array.set result_coeffs result_pos val_to_set;
			 fill_result (bitpow+bitpow) (1+result_pos) (true_result_vars+bitpow))
	        else (* contrib from lp, but not from subs... *)
    	          (Array.set result_coeffs result_pos
		     ((get_coeff lp bitpow)*factor_lp);
		   fill_result (bitpow+bitpow) (1+result_pos) (true_result_vars+bitpow))
           else if 0 != (subs_other_vars land bitpow)
		(* Contrib from subs, but not from lp *)
	         then (Array.set result_coeffs result_pos
			 ((get_coeff to_subs bitpow)*factor_subs);
		       fill_result (bitpow+bitpow) (1+result_pos) (true_result_vars+bitpow))
	         else (* both contribs zero... *)
	          fill_result (bitpow+bitpow) result_pos true_result_vars in
    let result_lp = fill_result 1 0 0 in
    if Array.length result_lp.coeffs == 0
	(* if we substituted X by X, we get out an empty factor...
	   For term normalization purposes, we make 0^0=1.
	 *)
    then
      (result_lp.power <- 0;
       result_lp.vars <- 0;
       result_lp)
    else
    let coeffs_gcd0 = v_gcd result_lp.coeffs in
    let coeffs_gcd = if (Array.get result_lp.coeffs 0) < 0 then (-coeffs_gcd0) else coeffs_gcd0 in
    (* Printf.printf "coeffs_gcd0=%d coeffs_gcd=%d\n" coeffs_gcd0 coeffs_gcd; DDD *)
    (
     if coeffs_gcd != 1
     then Array.iteri (fun pos coeff -> Array.set result_lp.coeffs pos (coeff/coeffs_gcd)) result_lp.coeffs
     else ();
     ref_xfactor:= !ref_xfactor */ (power_num  ((Int factor_lp)//(Int coeffs_gcd)) (Int (-lp.power)));
     result_lp
    );;


let v_lpower_subs_uniq numcoeff_plus_v_lp var_bitpow to_subs =
  let (nc,v_lp) = numcoeff_plus_v_lp in
  let rnc = ref nc in
  let len_v_lp = Array.length v_lp in
  let v_lp_subs = Array.map (lpower_subs rnc var_bitpow to_subs) v_lp in
  (Array.sort (lpower_cmp true) v_lp_subs;
   let rec trav_uniq pos_src pos_dst =
     if pos_src == len_v_lp
      then if pos_dst == pos_src
            then v_lp_subs
	    else Array.sub v_lp_subs 0 pos_dst
      else (* still work to be done *)
       if pos_dst==0
        then (* no terms on stack *)
	 (Array.set v_lp_subs pos_dst (Array.get v_lp_subs pos_src);
	  trav_uniq (1+pos_src) (1+pos_dst))
	else
	 let lp1 = Array.get v_lp_subs (pos_dst-1) in
	 let lp2 = Array.get v_lp_subs pos_src in
	 let cmp = lpower_cmp false lp1 lp2 in
	 if cmp==0 (* combinable terms *)
	  then let total_power =  lp1.power + lp2.power in
	   if total_power == 0
	    then trav_uniq (1+pos_src) (pos_dst-1) (* Term on stack was eaten *)
	    else (Array.set v_lp_subs (pos_dst-1)
		    {power=total_power;
		     vars=lp1.vars;
		     coeffs=lp1.coeffs}
		    (* This conses more often than strictly necessary...
		       However, I do not want to worry about when I already
		       have copied a value and may do destructive
		       modification...
		     *);
		  trav_uniq (1+pos_src) pos_dst)
	  else (* cmp!=0, which means that we have to transfer next term *)
	   (Array.set v_lp_subs pos_dst lp2;
	    trav_uniq (1+pos_src) (1+pos_dst)) in
   let v_lp_subs_uniq = trav_uniq 0 0 in
   (!rnc,v_lp_subs_uniq));;


let eval_nc_v_lp nc_v_lp v_Ex =
  let (nc,v_lp)=nc_v_lp in
  let nr_factors=Array.length v_lp in
  let bitmask = (1 lsl (Array.length v_Ex))-1 in
  (* quick hack: use a v_lp to retrieve coeff data... *)
  let eval_lp={power=0;vars=bitmask;coeffs=v_Ex} in
  let rec trav so_far n =
    if n == nr_factors then so_far else
    let factor_n = Array.get v_lp n in
    let coeffs_n = factor_n.coeffs in
    let nr_coeffs=Array.length coeffs_n in
    let rec trav_vars rest_vars bitpow sum_now =
      if rest_vars==0
      then sum_now
      else if (bitpow land rest_vars) == 0
            then trav_vars rest_vars (bitpow+bitpow) sum_now
	    else
	     let coeff_j = get_coeff factor_n bitpow in
	     let var_j = get_coeff eval_lp bitpow in
	     trav_vars (rest_vars-bitpow) (bitpow+bitpow) (sum_now+/ (Int coeff_j) */ (Int var_j))
    in let val_factor_n = trav_vars factor_n.vars 1 (Int 0) in
    trav (so_far*/(power_num val_factor_n (Int factor_n.power))) (1+ n) in
  trav nc 0;;

let check_substitution nc_v_lp =
  let debug_lp_res = {power = -1; vars = 80; coeffs = [|3; -2|]} in
  let v_eval =  [|0;1000;1100;1110;-1000-1100-1110;101;3*(-1000-1100-1110)/2|] in
  let var_bitpow = 1 lsl ((integer_length debug_lp_res.vars)-1) in
  let substituted = v_lpower_subs_uniq nc_v_lp var_bitpow debug_lp_res in
  let val_substituted = eval_nc_v_lp substituted v_eval in
  let val_direct = eval_nc_v_lp nc_v_lp v_eval in
  let difference = val_substituted-/val_direct in
  (* List.map string_of_num [val_substituted;val_direct;difference] *)
  string_of_num difference;;

let debug_v_lp=
((Int (-16))//(Int 3),
 [|{power = 0; vars = 0; coeffs = [||]};
   {power = 3; vars = 16; coeffs = [|1|]};
   {power = -1; vars = 18; coeffs = [|1; -2|]};
   {power = -3; vars = 18; coeffs = [|1; -1|]};
   {power = -3; vars = 18; coeffs = [|1; 1|]};
   {power = -1; vars = 18; coeffs = [|1; 2|]};
   {power = -1; vars = 20; coeffs = [|1; -2|]};
   {power = -3; vars = 20; coeffs = [|1; -1|]};
   {power = -3; vars = 20; coeffs = [|1; 1|]};
   {power = -1; vars = 20; coeffs = [|1; 2|]};
   {power = 1; vars = 22; coeffs = [|1; 1; -2|]};
   {power = 3; vars = 22; coeffs = [|1; 1; -1|]};
   {power = 3; vars = 22; coeffs = [|1; 1; 1|]};
   {power = 1; vars = 22; coeffs = [|1; 1; 2|]};
   {power = -1; vars = 24; coeffs = [|1; -2|]};
   {power = -3; vars = 24; coeffs = [|1; -1|]};
   {power = -3; vars = 24; coeffs = [|1; 1|]};
   {power = -1; vars = 24; coeffs = [|1; 2|]};
   {power = 1; vars = 26; coeffs = [|1; 1; -2|]};
   {power = 3; vars = 26; coeffs = [|1; 1; -1|]};
   {power = 3; vars = 26; coeffs = [|1; 1; 1|]};
   {power = 1; vars = 26; coeffs = [|1; 1; 2|]};
   {power = 1; vars = 28; coeffs = [|1; 1; -2|]};
   {power = 3; vars = 28; coeffs = [|1; 1; -1|]};
   {power = 3; vars = 28; coeffs = [|1; 1; 1|]};
   {power = 1; vars = 28; coeffs = [|1; 1; 2|]};
   {power = 2; vars = 32; coeffs = [|1|]};
   {power = -2; vars = 34; coeffs = [|1; -2|]};
   {power = -2; vars = 34; coeffs = [|1; 2|]};
   {power = -2; vars = 36; coeffs = [|1; -2|]};
   {power = -2; vars = 36; coeffs = [|1; 2|]};
   {power = 2; vars = 38; coeffs = [|1; 1; -2|]};
   {power = 2; vars = 38; coeffs = [|1; 1; 2|]};
   {power = -2; vars = 40; coeffs = [|1; -2|]};
   {power = -2; vars = 40; coeffs = [|1; 2|]};
   {power = 2; vars = 42; coeffs = [|1; 1; -2|]};
   {power = 2; vars = 42; coeffs = [|1; 1; 2|]};
   {power = 2; vars = 44; coeffs = [|1; 1; -2|]};
   {power = 2; vars = 44; coeffs = [|1; 1; 2|]};
   {power = -1; vars = 48; coeffs = [|1; -2|]};	
   {power = 1; vars = 48; coeffs = [|1; -1|]};
   {power = 1; vars = 48; coeffs = [|1; 1|]};
   {power = -1; vars = 48; coeffs = [|1; 2|]};
   {power = -1; vars = 48; coeffs = [|3; -2|]};
   {power = -1; vars = 48; coeffs = [|3; 2|]};
   {power = -1; vars = 50; coeffs = [|1; -2; -2|]};
   {power = -1; vars = 50; coeffs = [|1; -2; 2|]};
   {power = -1; vars = 50; coeffs = [|1; -1; -2|]};
   {power = -1; vars = 50; coeffs = [|1; -1; 2|]};
   {power = -1; vars = 50; coeffs = [|1; 1; -2|]};
   {power = -1; vars = 50; coeffs = [|1; 1; 2|]};
   {power = -1; vars = 50; coeffs = [|1; 2; -2|]};
   {power = -1; vars = 50; coeffs = [|1; 2; 2|]};
   {power = -1; vars = 52; coeffs = [|1; -2; -2|]};
   {power = -1; vars = 52; coeffs = [|1; -2; 2|]};
   {power = -1; vars = 52; coeffs = [|1; -1; -2|]};
   {power = -1; vars = 52; coeffs = [|1; -1; 2|]};
   {power = -1; vars = 52; coeffs = [|1; 1; -2|]};
   {power = -1; vars = 52; coeffs = [|1; 1; 2|]};
   {power = -1; vars = 52; coeffs = [|1; 2; -2|]};
   {power = -1; vars = 52; coeffs = [|1; 2; 2|]};
   {power = 1; vars = 54; coeffs = [|1; 1; -2; -2|]};
   {power = 1; vars = 54; coeffs = [|1; 1; -2; 2|]};
   {power = 1; vars = 54; coeffs = [|1; 1; -1; -2|]};
   {power = 1; vars = 54; coeffs = [|1; 1; -1; 2|]};
   {power = 1; vars = 54; coeffs = [|1; 1; 1; -2|]};
   {power = 1; vars = 54; coeffs = [|1; 1; 1; 2|]};
   {power = 1; vars = 54; coeffs = [|1; 1; 2; -2|]};
   {power = 1; vars = 54; coeffs = [|1; 1; 2; 2|]};
   {power = -1; vars = 56; coeffs = [|1; -2; -2|]};
   {power = -1; vars = 56; coeffs = [|1; -2; 2|]};
   {power = -1; vars = 56; coeffs = [|1; -1; -2|]};
   {power = -1; vars = 56; coeffs = [|1; -1; 2|]};
   {power = -1; vars = 56; coeffs = [|1; 1; -2|]};
   {power = -1; vars = 56; coeffs = [|1; 1; 2|]};
   {power = -1; vars = 56; coeffs = [|1; 2; -2|]};
   {power = -1; vars = 56; coeffs = [|1; 2; 2|]};
   {power = 1; vars = 58; coeffs = [|1; 1; -2; -2|]};
   {power = 1; vars = 58; coeffs = [|1; 1; -2; 2|]};
   {power = 1; vars = 58; coeffs = [|1; 1; -1; -2|]};
   {power = 1; vars = 58; coeffs = [|1; 1; -1; 2|]};
   {power = 1; vars = 58; coeffs = [|1; 1; 1; -2|]};
   {power = 1; vars = 58; coeffs = [|1; 1; 1; 2|]};
   {power = 1; vars = 58; coeffs = [|1; 1; 2; -2|]};
   {power = 1; vars = 58; coeffs = [|1; 1; 2; 2|]};
   {power = 1; vars = 60; coeffs = [|1; 1; -2; -2|]};
   {power = 1; vars = 60; coeffs = [|1; 1; -2; 2|]};
   {power = 1; vars = 60; coeffs = [|1; 1; -1; -2|]};
   {power = 1; vars = 60; coeffs = [|1; 1; -1; 2|]};
   {power = 1; vars = 60; coeffs = [|1; 1; 1; -2|]};
   {power = 1; vars = 60; coeffs = [|1; 1; 1; 2|]};
   {power = 1; vars = 60; coeffs = [|1; 1; 2; -2|]};
   {power = 1; vars = 60; coeffs = [|1; 1; 2; 2|]};
   {power = 1; vars = 64; coeffs = [|1|]};
   {power = -1; vars = 66; coeffs = [|1; -2|]};
   {power = -1; vars = 66; coeffs = [|1; 2|]};
   {power = -1; vars = 68; coeffs = [|1; -2|]};
   {power = -1; vars = 68; coeffs = [|1; 2|]};
   {power = 1; vars = 70; coeffs = [|1; 1; -2|]};
   {power = 1; vars = 70; coeffs = [|1; 1; 2|]};
   {power = -1; vars = 72; coeffs = [|1; -2|]};
   {power = -1; vars = 72; coeffs = [|1; 2|]};
   {power = 1; vars = 74; coeffs = [|1; 1; -2|]};
   {power = 1; vars = 74; coeffs = [|1; 1; 2|]};
   {power = 1; vars = 76; coeffs = [|1; 1; -2|]};
   {power = 1; vars = 76; coeffs = [|1; 1; 2|]};
   {power = 1; vars = 80; coeffs = [|1; -1|]};
   {power = -1; vars = 80; coeffs = [|1; 2|]};
   {power = 0; vars = 80; coeffs = [|3; -2|]};
   {power = -1; vars = 82; coeffs = [|1; -2; 2|]};
   {power = -1; vars = 82; coeffs = [|1; -1; 2|]};
   {power = -1; vars = 82; coeffs = [|1; 1; -2|]};
   {power = -1; vars = 82; coeffs = [|1; 2; -2|]};
   {power = -1; vars = 84; coeffs = [|1; -2; 2|]};
   {power = -1; vars = 84; coeffs = [|1; -1; 2|]};
   {power = -1; vars = 84; coeffs = [|1; 1; -2|]};
   {power = -1; vars = 84; coeffs = [|1; 2; -2|]};
   {power = 1; vars = 86; coeffs = [|1; 1; -2; 2|]};
   {power = 1; vars = 86; coeffs = [|1; 1; -1; 2|]};
   {power = 1; vars = 86; coeffs = [|1; 1; 1; -2|]};
   {power = 1; vars = 86; coeffs = [|1; 1; 2; -2|]};
   {power = -1; vars = 88; coeffs = [|1; -2; 2|]};
   {power = -1; vars = 88; coeffs = [|1; -1; 2|]};
   {power = -1; vars = 88; coeffs = [|1; 1; -2|]};
   {power = -1; vars = 88; coeffs = [|1; 2; -2|]};
   {power = 1; vars = 90; coeffs = [|1; 1; -2; 2|]};
   {power = 1; vars = 90; coeffs = [|1; 1; -1; 2|]};
   {power = 1; vars = 90; coeffs = [|1; 1; 1; -2|]};
   {power = 1; vars = 90; coeffs = [|1; 1; 2; -2|]};
   {power = 1; vars = 92; coeffs = [|1; 1; -2; 2|]};
   {power = 1; vars = 92; coeffs = [|1; 1; -1; 2|]};
   {power = 1; vars = 92; coeffs = [|1; 1; 1; -2|]};
   {power = 1; vars = 92; coeffs = [|1; 1; 2; -2|]};
   {power = 1; vars = 96; coeffs = [|1; -1|]};
   {power = 1; vars = 96; coeffs = [|1; 1|]};
   {power = -2; vars = 98; coeffs = [|1; -1; -1|]};
   {power = -2; vars = 98; coeffs = [|1; -1; 1|]};
   {power = -2; vars = 98; coeffs = [|1; 1; -1|]};
   {power = -2; vars = 98; coeffs = [|1; 1; 1|]};
   {power = -2; vars = 100; coeffs = [|1; -1; -1|]};
   {power = -2; vars = 100; coeffs = [|1; -1; 1|]};
   {power = -2; vars = 100; coeffs = [|1; 1; -1|]};
   {power = -2; vars = 100; coeffs = [|1; 1; 1|]};
   {power = 2; vars = 102; coeffs = [|1; 1; -1; -1|]};
   {power = 2; vars = 102; coeffs = [|1; 1; -1; 1|]};
   {power = 2; vars = 102; coeffs = [|1; 1; 1; -1|]};
   {power = 2; vars = 102; coeffs = [|1; 1; 1; 1|]};
   {power = -2; vars = 104; coeffs = [|1; -1; -1|]};
   {power = -2; vars = 104; coeffs = [|1; -1; 1|]};
   {power = -2; vars = 104; coeffs = [|1; 1; -1|]};
   {power = -2; vars = 104; coeffs = [|1; 1; 1|]};
   {power = 2; vars = 106; coeffs = [|1; 1; -1; -1|]};
   {power = 2; vars = 106; coeffs = [|1; 1; -1; 1|]};
   {power = 2; vars = 106; coeffs = [|1; 1; 1; -1|]};
   {power = 2; vars = 106; coeffs = [|1; 1; 1; 1|]};
   {power = 2; vars = 108; coeffs = [|1; 1; -1; -1|]};
   {power = 2; vars = 108; coeffs = [|1; 1; -1; 1|]};
   {power = 2; vars = 108; coeffs = [|1; 1; 1; -1|]};
   {power = 2; vars = 108; coeffs = [|1; 1; 1; 1|]};
   {power = 1; vars = 112; coeffs = [|1; -1; -1|]};
   {power = 1; vars = 112; coeffs = [|1; 1; -1|]};
   {power = -1; vars = 112; coeffs = [|2; -1; 1|]};
   {power = -1; vars = 112; coeffs = [|2; 1; 1|]};
   {power = -1; vars = 112; coeffs = [|3; -1; -1|]};
   {power = -1; vars = 112; coeffs = [|3; 1; -1|]};
   {power = -1; vars = 114; coeffs = [|1; -2; -1; 1|]};
   {power = -1; vars = 114; coeffs = [|1; -2; 1; 1|]};
   {power = -1; vars = 114; coeffs = [|1; -1; -1; -1|]};
   {power = -2; vars = 114; coeffs = [|1; -1; -1; 1|]};
   {power = -1; vars = 114; coeffs = [|1; -1; 1; -1|]};
   {power = -2; vars = 114; coeffs = [|1; -1; 1; 1|]};
   {power = -2; vars = 114; coeffs = [|1; 1; -1; -1|]};
   {power = -1; vars = 114; coeffs = [|1; 1; -1; 1|]};
   {power = -2; vars = 114; coeffs = [|1; 1; 1; -1|]};
   {power = -1; vars = 114; coeffs = [|1; 1; 1; 1|]};
   {power = -1; vars = 114; coeffs = [|1; 2; -1; -1|]};
   {power = -1; vars = 114; coeffs = [|1; 2; 1; -1|]};
   {power = -1; vars = 116; coeffs = [|1; -2; -1; 1|]};
   {power = -1; vars = 116; coeffs = [|1; -2; 1; 1|]};
   {power = -1; vars = 116; coeffs = [|1; -1; -1; -1|]};
   {power = -2; vars = 116; coeffs = [|1; -1; -1; 1|]};
   {power = -1; vars = 116; coeffs = [|1; -1; 1; -1|]};
   {power = -2; vars = 116; coeffs = [|1; -1; 1; 1|]};
   {power = -2; vars = 116; coeffs = [|1; 1; -1; -1|]};
   {power = -1; vars = 116; coeffs = [|1; 1; -1; 1|]};
   {power = -2; vars = 116; coeffs = [|1; 1; 1; -1|]};
   {power = -1; vars = 116; coeffs = [|1; 1; 1; 1|]};
   {power = -1; vars = 116; coeffs = [|1; 2; -1; -1|]};
   {power = -1; vars = 116; coeffs = [|1; 2; 1; -1|]};
   {power = 1; vars = 118; coeffs = [|1; 1; -2; -1; 1|]};
   {power = 1; vars = 118; coeffs = [|1; 1; -2; 1; 1|]};
   {power = 1; vars = 118; coeffs = [|1; 1; -1; -1; -1|]};
   {power = 2; vars = 118; coeffs = [|1; 1; -1; -1; 1|]};
   {power = 1; vars = 118; coeffs = [|1; 1; -1; 1; -1|]};
   {power = 2; vars = 118; coeffs = [|1; 1; -1; 1; 1|]};
   {power = 2; vars = 118; coeffs = [|1; 1; 1; -1; -1|]};
   {power = 1; vars = 118; coeffs = [|1; 1; 1; -1; 1|]};
   {power = 2; vars = 118; coeffs = [|1; 1; 1; 1; -1|]};
   {power = 1; vars = 118; coeffs = [|1; 1; 1; 1; 1|]};
   {power = 1; vars = 118; coeffs = [|1; 1; 2; -1; -1|]}|]);;

let rec trav n =
  if n == 10 then () else
  (Printf.printf "%d ==> %s\n" n (check_substitution debug_v_lp);
   trav (1+ n))
    in trav 0;;