(* Title:  partial fraction decomposition

   Author: Jan Rocnik

   (c) due to copyright terms

*)

"--------------------------------------------------------";

"table of contents --------------------------------------";

"--------------------------------------------------------";

"----------- why helpless here ? ------------------------";

"----------- why not nxt = Model_Problem here ? ---------";

"----------- fun factors_from_solution ------------------";

"----------- Logic.unvarify_global ----------------------";

"----------- eval_drop_questionmarks --------------------";

"----------- = me for met_partial_fraction --------------";

"----------- autoCalculate for met_partial_fraction -----";

"----------- progr.vers.2: check erls for multiply_ansatz";

"----------- progr.vers.2: improve program --------------";

"--------------------------------------------------------";

"--------------------------------------------------------";

"--------------------------------------------------------";

"----------- why helpless here ? ------------------------";

"----------- why helpless here ? ------------------------";

"----------- why helpless here ? ------------------------";

val fmz = ["filterExpression (X z = 3 / (z - 1/4 + -1/8 * (1/(z::real))))", 

  "stepResponse (x[n::real]::bool)"];

val (dI,pI,mI) = ("Isac", ["Inverse", "Z_Transform", "SignalProcessing"], 

  ["SignalProcessing","Z_Transform","Inverse"]);

val (p,_,f,nxt,_,pt)  = CalcTreeTEST [(fmz, (dI,pI,mI))]; 

val (p,_,f,nxt,_,pt) = me nxt p [] pt; "Add_Given";

val (p,_,f,nxt,_,pt) = me nxt p [] pt; "Add_Find";

val (p,_,f,nxt,_,pt) = me nxt p [] pt; "Specify_Theory";

val (p,_,f,nxt,_,pt) = me nxt p [] pt; "Specify_Problem";

val (p,_,f,nxt,_,pt) = me nxt p [] pt; "Specify_Method";

val (p,_,f,nxt,_,pt) = me nxt p [] pt; "nxt = Apply_Method";

val (p,_,f,nxt,_,pt) = me nxt p [] pt; "nxt = Rewrite (ruleZY, Inverse_Z_Transform.ruleZY) --> X z = 3 / (z - 1 / 4 + -1 / 8 * (1 / z))";

val (p''',_,f,nxt''',_,pt''') = me nxt p [] pt; "nxt = Rewrite_Set norm_Rational --> X' z = 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))";

"~~~~~ fun me, args:"; val ((_,tac), (p:pos'), _, (pt:ptree)) = (nxt, p, [], pt);

val ("ok", (_, _, ptp)) = locatetac tac (pt,p)

val (pt, p) = ptp;

"~~~~~ fun step, args:"; val (((ip as (_,p_)):pos'), ((ptp as (pt,p), tacis):calcstate)) = 

                           (p, ((pt, e_pos'),[]));

val pIopt = get_pblID (pt,ip);

ip = ([],Res); "false";

tacis; " = []";

pIopt; (* = SOME ["Inverse", "Z_Transform", "SignalProcessing"]*)

member op = [Pbl,Met] p_ andalso is_none (get_obj g_env pt (fst p)); "false";

(*nxt_solve_ (pt,ip); "WAS isalist2list applied to NON-list 'no_meth'"

   THIS MEANS: replace no_meth by [no_meth] in Script.*)

(*WAS val ("helpless",_) = step p ((pt, e_pos'),[]) *)

(*WAS val (p,_,f,nxt,_,pt) = me nxt p [] pt; "Empty_Tac instead SubProblem";*)

"----------- why not nxt = Model_Problem here ? ---------";

"----------- why not nxt = Model_Problem here ? ---------";

"----------- why not nxt = Model_Problem here ? ---------";

val (p,_,f,nxt,_,pt) = me nxt''' p''' [] pt'''; 

"~~~~~ fun me, args:"; val (((_,tac):tac'_), (p:pos'), _, (pt:ptree)) = (nxt, p, c, pt);

val ("ok", (_, _, ptp)) = locatetac tac (pt,p);

val (pt, p) = ptp;

"~~~~~ fun step, args:"; val (((ip as (_,p_)):pos'), ((ptp as (pt,p), tacis):calcstate)) =

                           (p, ((pt, e_pos'),[]));

val pIopt = get_pblID (pt,ip);

ip = ([],Res); " = false";

tacis; " = []";                                         

pIopt (* = SOME ["Inverse", "Z_Transform", "SignalProcessing"]*);

member op = [Pbl,Met] p_ andalso is_none (get_obj g_env pt (fst p)); " = false";

(*                               ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ leads into

nxt_solve_, which is definitely WRONG (should be nxt_specify_ for FIND_ADD).

This ERROR seems to be introduced together with ctxt, concerns Apply_Method without init_form.

See TODO.txt

*)

"----------- fun factors_from_solution ------------------";

"----------- fun factors_from_solution ------------------";

"----------- fun factors_from_solution ------------------";

val ctxt = Proof_Context.init_global @{theory Isac};

val SOME t = parseNEW ctxt "factors_from_solution [(z::real) = 1 / 2, z = -1 / 4]";

val SOME (_, t') = eval_factors_from_solution "" 0 t thy;

if term2str t' =

 "factors_from_solution [z = 1 / 2, z = -1 / 4] = (z - 1 / 2) * (z - -1 / 4)"

then () else error "factors_from_solution broken";

"----------- Logic.unvarify_global ----------------------";

"----------- Logic.unvarify_global ----------------------";

"----------- Logic.unvarify_global ----------------------";

val A_var = parseNEW ctxt "A::real" |> the |> Logic.varify_global

val B_var = parseNEW ctxt "B::real" |> the |> Logic.varify_global

val denom1 = parseNEW ctxt "(z + -1 * (1 / 2))::real" |> the;

val denom2 = parseNEW ctxt "(z + -1 * (-1 / 4))::real" |> the;

val test = HOLogic.mk_binop "Groups.plus_class.plus"

  (HOLogic.mk_binop "Fields.inverse_class.divide" (A_var, denom1),

   HOLogic.mk_binop "Fields.inverse_class.divide" (B_var, denom2));

if term2str test = "?A / (z + -1 * (1 / 2)) + ?B / (z + -1 * (-1 / 4))" then ()

  else error "HOLogic.mk_binop broken ?";

(* Logic.unvarify_global test

---> exception TERM raised (line 539 of "logic.ML"): Illegal fixed variable: "z"

thus we need another fun var2free in termC.sml*)

"----------- eval_drop_questionmarks --------------------";

"----------- eval_drop_questionmarks --------------------";

"----------- eval_drop_questionmarks --------------------";

if term2str test = "?A / (z + -1 * (1 / 2)) + ?B / (z + -1 * (-1 / 4))"

then () else error "test setup (ctxt!) probably changed";

val t = Const ("Partial_Fractions.drop_questionmarks", HOLogic.realT --> HOLogic.realT) $ test;

val SOME (_, t') = eval_drop_questionmarks "drop_questionmarks" 0 t @{theory Isac};

if term2str t' = "drop_questionmarks (?A / (z + -1 * (1 / 2)) + ?B / (z + -1 * (-1 / 4))) =" ^

  "\nA / (z + -1 * (1 / 2)) + B / (z + -1 * (-1 / 4))"

then () else error "eval_drop_questionmarks broken";

"----------- = me for met_partial_fraction --------------";

"----------- = me for met_partial_fraction --------------";

"----------- = me for met_partial_fraction --------------";

  val fmz =                                             

    ["functionTerm (3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z))))", 

      "solveFor z", "decomposedFunction p_p"];

  val (dI',pI',mI') =

    ("Partial_Fractions", 

      ["partial_fraction", "rational", "simplification"],

      ["simplification","of_rationals","to_partial_fraction"]);

  val (p,_,f,nxt,_,pt) = CalcTreeTEST [(fmz, (dI',pI',mI'))];

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt; 

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt;

  val (p,_,f,nxt,_,pt) = me nxt p [] pt; 

show_pt pt;

if f2str f = "4 / (z - 1 / 2) + -4 / (z - -1 / 4)" then() 

else error "= me .. met_partial_fraction broken";

"----------- autoCalculate for met_partial_fraction -----";

"----------- autoCalculate for met_partial_fraction -----";

"----------- autoCalculate for met_partial_fraction -----";

states:= [];

  val fmz =                                             

    ["functionTerm (3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z))))", 

      "solveFor z", "decomposedFunction p_p"];

  val (dI', pI', mI') =

    ("Partial_Fractions", 

      ["partial_fraction", "rational", "simplification"],

      ["simplification","of_rationals","to_partial_fraction"]);

CalcTree [(fmz, (dI' ,pI' ,mI'))];

Iterator 1;

moveActiveRoot 1;

autoCalculate 1 CompleteCalc; 

val ((pt,p),_) = get_calc 1; val ip = get_pos 1 1;

if p = ip andalso ip = ([], Res) then ()

  else error "autoCalculate for met_partial_fraction changed: final pos'";

val (Form f, tac, asms) = pt_extract (pt, p);

if term2str f = "4 / (z - 1 / 2) + -4 / (z - -1 / 4)" andalso

  tac = NONE andalso

  terms2str' asms = "[~ matches (?a = 0) (3 = -3 * B / 4) | ~ lhs (3 = -3 * B / 4) is_poly_in B," ^

  "B = -4,lhs (3 + 3 / 4 * B = 0) is_poly_in B,lhs (3 + 3 / 4 * B = 0) has_degree_in B = 1," ^

  "B is_polyexp,A is_polyexp," ^

  "~ matches (?a = 0) (3 = 3 * A / 4) | ~ lhs (3 = 3 * A / 4) is_poly_in A," ^

  "A = 4,lhs (3 + -3 / 4 * A = 0) is_poly_in A,lhs (3 + -3 / 4 * A = 0) has_degree_in A = 1," ^

  "lhs (-1 + -2 * z + 8 * z ^^^ 2 = 0) has_degree_in z = 2," ^

  "lhs (-1 + -2 * z + 8 * z ^^^ 2 = 0) is_poly_in z,z = 1 / 2,z = -1 / 4,z ~= 0,z is_polyexp]"

then ()

  else error "autoCalculate for met_partial_fraction changed: final result"

show_pt pt;

(*[

(([], Frm), Problem (Partial_Fractions, [partial_fraction, rational, simplification])),

(([1], Frm), 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))),

(([1], Res), 24 / (-1 + -2 * z + 8 * z ^^^ 2)),

(([2], Pbl), solve (-1 + -2 * z + 8 * z ^^^ 2 = 0, z)),

(([2,1], Frm), -1 + -2 * z + 8 * z ^^^ 2 = 0),

(([2,1], Res), z = (- -2 + sqrt (-2 ^^^ 2 - 4 * 8 * -1)) / (2 * 8) |

z = (- -2 - sqrt (-2 ^^^ 2 - 4 * 8 * -1)) / (2 * 8)),

(([2,2], Res), z = 1 / 2 | z = -1 / 4),

(([2,3], Res), [z = 1 / 2, z = -1 / 4]),

(([2,4], Res), [z = 1 / 2, z = -1 / 4]),

(([2], Res), [z = 1 / 2, z = -1 / 4]),

(([3], Frm), 3 / ((z - 1 / 2) * (z - -1 / 4))),

(([3], Res), ?A / (z - 1 / 2) + ?B / (z - -1 / 4)),

(([4], Frm), 3 / ((z - 1 / 2) * (z - -1 / 4)) = ?A / (z - 1 / 2) + ?B / (z - -1 / 4)),

(([4], Res), 3 = ?A * (z - -1 / 4) + ?B * (z - 1 / 2)),

(([5], Frm), 3 = A * (z - -1 / 4) + B * (z - 1 / 2)),

(([5], Res), 3 = A * (1 / 2 - -1 / 4) + B * (1 / 2 - 1 / 2)),

(([6], Res), 3 = 3 * A / 4),

(([7], Pbl), solve (3 = 3 * A / 4, A)),

(([7,1], Frm), 3 = 3 * A / 4),

(([7,1], Res), 3 - 3 * A / 4 = 0),

(([7,2], Res), 3 / 1 + -3 / 4 * A = 0),

(([7,3], Res), 3 + -3 / 4 * A = 0),

(([7,4], Pbl), solve (3 + -3 / 4 * A = 0, A)),

(([7,4,1], Frm), 3 + -3 / 4 * A = 0),

(([7,4,1], Res), A = -1 * 3 / (-3 / 4)),

(([7,4,2], Res), A = -3 / (-3 / 4)),

(([7,4,3], Res), A = 4),

(([7,4,4], Res), [A = 4]),

(([7,4,5], Res), [A = 4]),

(([7,4], Res), [A = 4]),

(([7], Res), [A = 4]),

(([8], Frm), 3 = A * (z - -1 / 4) + B * (z - 1 / 2)),

(([8], Res), 3 = A * (-1 / 4 - -1 / 4) + B * (-1 / 4 - 1 / 2)),

(([9], Res), 3 = -3 * B / 4),

(([10], Pbl), solve (3 = -3 * B / 4, B)),

(([10,1], Frm), 3 = -3 * B / 4),

(([10,1], Res), 3 - -3 * B / 4 = 0),

(([10,2], Res), 3 / 1 + 3 / 4 * B = 0),

(([10,3], Res), 3 + 3 / 4 * B = 0),

(([10,4], Pbl), solve (3 + 3 / 4 * B = 0, B)),

(([10,4,1], Frm), 3 + 3 / 4 * B = 0),

(([10,4,1], Res), B = -1 * 3 / (3 / 4)),

(([10,4,2], Res), B = -3 / (3 / 4)),

(([10,4,3], Res), B = -4),

(([10,4,4], Res), [B = -4]),

(([10,4,5], Res), [B = -4]),

(([10,4], Res), [B = -4]),

(([10], Res), [B = -4]),

(([11], Frm), A / (z - 1 / 2) + B / (z - -1 / 4)),

(([11], Res), 4 / (z - 1 / 2) + -4 / (z - -1 / 4)),

(([], Res), 4 / (z - 1 / 2) + -4 / (z - -1 / 4))] *)

"----------- progr.vers.2: check erls for multiply_ansatz";

"----------- progr.vers.2: check erls for multiply_ansatz";

"----------- progr.vers.2: check erls for multiply_ansatz";

(*test for outcommented 3 lines in script: is norm_Rational strong enough?*)

val t = str2term "(3 / ((-1 + -2 * z + 8 * z ^^^ 2) *3/24)) = (3 / ((z - 1 / 2) * (z - -1 / 4)))";

val SOME (t', _) = rewrite_set_ @{theory Isac} true ansatz_rls t;

term2str t' = "3 / ((-1 + -2 * z + 8 * z ^^^ 2) * 3 / 24) =\n?A / (z - 1 / 2) + ?B / (z - -1 / 4)";

val SOME (t'', _) = rewrite_set_ @{theory Isac} true multiply_ansatz t'; (*true*)

term2str t'' = "(z - 1 / 2) * (z - -1 / 4) * 3 / ((-1 + -2 * z + 8 * z ^^^ 2) * 3 / 24) =\n" ^

  "?A * (z - -1 / 4) + ?B * (z - 1 / 2)"; (*true*)

val SOME (t''', _) = rewrite_set_ @{theory Isac} true norm_Rational t'';

if term2str t''' = "3 = ?A * (1 + 4 * z) / 4 + ?B * (-1 + 2 * z) / 2" then ()

else error "ansatz_rls - multiply_ansatz - norm_Rational broken";

(*test for outcommented 3 lines in script: empower erls for x = a*b ==> ...*)

val xxx = append_rls "multiply_ansatz_erls" norm_Rational 

  [Calc ("HOL.eq",eval_equal "#equal_")];

val multiply_ansatz = prep_rls(

  Rls {id = "multiply_ansatz", preconds = [], rew_ord = ("dummy_ord",dummy_ord), 

	  erls = xxx,

	  srls = Erls, calc = [], errpatts = [],

	  rules = 

	   [Thm ("multiply_2nd_order",num_str @{thm multiply_2nd_order})

	   ], 

	 scr = EmptyScr}:rls);

rewrite_set_ thy true xxx @{term "a+b = a+(b::real)"}; (*SOME ok*)

rewrite_set_ thy true multiply_ansatz @{term "a+b = a+(b::real)"}; (*why NONE?: GOON*)

"----------- progr.vers.2: improve program --------------";

"----------- progr.vers.2: improve program --------------";

"----------- progr.vers.2: improve program --------------";

(*WN120318 stopped due to much effort with the test above*)

     "Script PartFracScript (f_f::real) (zzz::real) =                    " ^(*f_f: 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)), zzz: z*)

     " (let f_f = Take f_f;                                       " ^

     "   (num_orig::real) = get_numerator f_f;                    " ^(*num_orig: 3*)

     "   f_f = (Rewrite_Set norm_Rational False) f_f;             " ^(*f_f: 24 / (-1 + -2 * z + 8 * z ^^^ 2)*)

     "   (numer::real) = get_numerator f_f;                       " ^(*numer: 24*)

     "   (denom::real) = get_denominator f_f;                     " ^(*denom: -1 + -2 * z + 8 * z ^^^ 2*)

     "   (equ::bool) = (denom = (0::real));                       " ^(*equ: -1 + -2 * z + 8 * z ^^^ 2 = 0*)

     "   (L_L::bool list) = (SubProblem (PolyEq',                 " ^

     "     [abcFormula, degree_2, polynomial, univariate, equation], " ^

     "     [no_met]) [BOOL equ, REAL zzz]);                       " ^(*L_L: [z = 1 / 2, z = -1 / 4]*)

     "   (facs::real) = factors_from_solution L_L;                " ^(*facs: (z - 1 / 2) * (z - -1 / 4)*)

     "   (eql::real) = Take (num_orig / facs);                    " ^(*eql: 3 / ((z - 1 / 2) * (z - -1 / 4)) *) 

     "   (eqr::real) = (Try (Rewrite_Set ansatz_rls False)) eql;  " ^(*eqr: ?A / (z - 1 / 2) + ?B / (z - -1 / 4)*)

     "   (eq::bool) = Take (eql = eqr);                           " ^(*eq:  3 / ((z - 1 / 2) * (z - -1 / 4)) = ?A / (z - 1 / 2) + ?B / (z - -1 / 4)*)

(*this has been tested below by rewrite_set_

     "   (eeeee::bool) = Take ((num_orig / denom * denom / numer) = (num_orig / facs));" ^(**)

     "   (eeeee::bool) = (Rewrite_Set ansatz_rls False) eeeee;" ^(**)

     "   eq = (Try (Rewrite_Set multiply_ansatz False)) eeeee;         " ^(*eq: 3 = ?A * (z - -1 / 4) + ?B * (z - 1 / 2)*) 

NEXT try to outcomment the very next line..*)

     "   eq = (Try (Rewrite_Set equival_trans False)) eeeee;         " ^(*eq: 3 = ?A * (z - -1 / 4) + ?B * (z - 1 / 2)*) 

     "   eq = drop_questionmarks eq;                              " ^(*eq: 3 = A * (z - -1 / 4) + B * (z - 1 / 2)*)

     "   (z1::real) = (rhs (NTH 1 L_L));                          " ^(*z1: 1 / 2*)

     "   (z2::real) = (rhs (NTH 2 L_L));                          " ^(*z2: -1 / 4*)

     "   (eq_a::bool) = Take eq;                                  " ^(*eq_a: 3 = A * (z - -1 / 4) + B * (z - 1 / 2)*)

     "   eq_a = (Substitute [zzz = z1]) eq;                       " ^(*eq_a: 3 = A * (1 / 2 - -1 / 4) + B * (1 / 2 - 1 / 2)*)

     "   eq_a = (Rewrite_Set norm_Rational False) eq_a;           " ^(*eq_a: 3 = 3 * A / 4*)

     "   (sol_a::bool list) =                                     " ^

     "     (SubProblem (Isac', [univariate,equation], [no_met])   " ^

     "     [BOOL eq_a, REAL (A::real)]);                          " ^(*sol_a: [A = 4]*)

     "   (a::real) = (rhs (NTH 1 sol_a));                         " ^(*a: 4*)

     "   (eq_b::bool) = Take eq;                                  " ^(*eq_b: 3 = A * (z - -1 / 4) + B * (z - 1 / 2)*)

     "   eq_b = (Substitute [zzz = z2]) eq_b;                     " ^(*eq_b: *)

     "   eq_b = (Rewrite_Set norm_Rational False) eq_b;           " ^(*eq_b: *)

     "   (sol_b::bool list) =                                     " ^

     "     (SubProblem (Isac', [univariate,equation], [no_met])   " ^

     "     [BOOL eq_b, REAL (B::real)]);                          " ^(*sol_b: [B = -4]*)

     "   (b::real) = (rhs (NTH 1 sol_b));                         " ^(*b: -4*)

     "   eqr = drop_questionmarks eqr;                            " ^(*eqr: A / (z - 1 / 2) + B / (z - -1 / 4)*)

     "   (pbz::real) = Take eqr;                                  " ^(*pbz: A / (z - 1 / 2) + B / (z - -1 / 4)*)

     "   pbz = ((Substitute [A = a, B = b]) pbz)                  " ^(*pbz: 4 / (z - 1 / 2) + -4 / (z - -1 / 4)*)

     " in pbz)"