(* Title:  Build_Inverse_Z_Transform

    Author: Jan Rocnik

    (c) copyright due to lincense terms.

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         10        20        30        40        50        60        70        80

 *)

 theory Build_Inverse_Z_Transform imports Isac

 begin

 text{* We stepwise build \ttfamily Inverse\_Z\_Transform.thy \normalfont as an 

   exercise. Because subsection~\ref{sec:stepcheck} requires 

   \ttfamily Inverse\_Z\_Transform.thy \normalfont as a subtheory of \ttfamily 

   Isac.thy\normalfont, the setup has been changed from \ttfamily theory 

   Inverse\_Z\_Transform imports Isac \normalfont to the above one.

   \par \noindent

   \begin{center} 

   \textbf{ATTENTION WITH NAMES OF IDENTIFIERS WHEN GOING INTO INTERNALS}

   \end{center}

   Here in this theory there are the internal names twice, for instance we have

   \ttfamily (Thm.derivation\_name @{thm rule1} = 

   "Build\_Inverse\_Z\_Transform.rule1") = true; \normalfont

   but actually in us will be \ttfamily Inverse\_Z\_Transform.rule1 \normalfont

 *}

 section {*Trials towards the Z-Transform\label{sec:trials}*}

 ML {*val thy = @{theory Isac};*}

 subsection {*Notations and Terms*}

 text{*\noindent Try which notations we are able to use.*}

 ML {*

   @{term "1 < || z ||"};

   @{term "z / (z - 1)"};

   @{term "-u -n - 1"};

   @{term "-u [-n - 1]"}; (*[ ] denotes lists !!!*)

   @{term "z /(z - 1) = -u [-n - 1]"};Isac

   @{term "1 < || z || ==> z / (z - 1) = -u [-n - 1]"};

   term2str @{term "1 < || z || ==> z / (z - 1) = -u [-n - 1]"};

 *}

 text{*\noindent Try which symbols we are able to use and how we generate them.*}

 ML {*

   (*alpha -->  "</alpha>" *)

   @{term "\<alpha> "};

   @{term "\<delta> "};

   @{term "\<phi> "};

   @{term "\<rho> "};

   term2str @{term "\<rho> "};

 *}

 subsection {*Rules*}

 (*axiomatization "z / (z - 1) = -u [-n - 1]"

   Illegal variable name: "z / (z - 1) = -u [-n - 1]" *)

 (*definition     "z / (z - 1) = -u [-n - 1]"

   Bad head of lhs: existing constant "op /"*)

 axiomatization where 

   rule1: "1 = \<delta>[n]" and

   rule2: "|| z || > 1 ==> z / (z - 1) = u [n]" and

   rule3: "|| z || < 1 ==> z / (z - 1) = -u [-n - 1]" and 

   rule4: "|| z || > || \<alpha> || ==> z / (z - \<alpha>) = \<alpha>^^^n * u [n]" and

   rule5: "|| z || < || \<alpha> || ==> z / (z - \<alpha>) = -(\<alpha>^^^n) * u [-n - 1]" and

   rule6: "|| z || > 1 ==> z/(z - 1)^^^2 = n * u [n]"

 text{*\noindent Check the rules for their correct notation. 

       (See the machine output.)*}

 ML {*

   @{thm rule1};

   @{thm rule2};

   @{thm rule3};

   @{thm rule4};

 *}

 subsection {*Apply Rules*}

 text{*\noindent We try to apply the rules to a given expression.*}

 ML {*

   val inverse_Z = append_rls "inverse_Z" e_rls

     [ Thm  ("rule3",num_str @{thm rule3}),

       Thm  ("rule4",num_str @{thm rule4}),

       Thm  ("rule1",num_str @{thm rule1})   

     ];

   val t = str2term "z / (z - 1) + z / (z - \<alpha>) + 1";

   val SOME (t', asm) = rewrite_set_ thy true inverse_Z t;

   term2str t' = "z / (z - ?\<delta> [?n]) + z / (z - \<alpha>) + ?\<delta> [?n]";

   (*

    * Attention rule1 is applied before the expression is 

    * checked for rule4 which would be correct!!!

    *)

 *}

 ML {* val (thy, ro, er) = (@{theory Isac}, tless_true, eval_rls); *}

 ML {*

   val SOME (t, asm1) = 

     rewrite_ thy ro er true (num_str @{thm rule3}) t;

   term2str t = "- ?u [- ?n - 1] + z / (z - \<alpha>) + 1";

   (*- real *)

   term2str t;

   val SOME (t, asm2) = 

     rewrite_ thy ro er true (num_str @{thm rule4}) t;

   term2str t = "- ?u [- ?n - 1] + \<alpha> ^^^ ?n * ?u [?n] + 1";

   (*- real *)

   term2str t;

   val SOME (t, asm3) = 

     rewrite_ thy ro er true (num_str @{thm rule1}) t;

   term2str t = "- ?u [- ?n - 1] + \<alpha> ^^^ ?n * ?u [?n] + ?\<delta> [?n]";

   (*- real *)

   term2str t;

 *}

 ML {* terms2str (asm1 @ asm2 @ asm3); *}

 section{*Prepare Steps for CTP-based programming Language\label{sec:prepstep}*}

 text{*

       \par \noindent The following sections are challanging with the CTP-based 

       possibilities of building the programm. The goal is realized in 

       Section~\ref{spec-meth} and Section~\ref{prog-steps}.

       \par The reader is advised to jump between the subsequent subsections and 

       the respective steps in Section~\ref{prog-steps}. By comparing 

       Section~\ref{sec:calc:ztrans} the calculation can be comprehended step 

       by step.

 *}

 subsection {*Prepare Expression\label{prep-expr}*}

 ML {*

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

   val ctxt = declare_constraints' [@{term "z::real"}] ctxt;

   val SOME fun1 = 

     parseNEW ctxt "X z = 3 / (z - 1/4 + -1/8 * z ^^^ -1)"; term2str fun1;

   val SOME fun1' = 

     parseNEW ctxt "X z = 3 / (z - 1/4 + -1/8 * (1/z))"; term2str fun1';

 *}

 subsubsection {*Prepare Numerator and Denominator*}

 text{*\noindent The partial fraction decomposion is only possible ig we

        get the bound variable out of the numerator. Therefor we divide

        the expression by $z$. Follow up the Calculation at 

        Section~\ref{sec:calc:ztrans} line number 02.*}

 axiomatization where

   ruleZY: "(X z = a / b) = (X' z = a / (z * b))"

 ML {*

   val (thy, ro, er) = (@{theory Isac}, tless_true, eval_rls);

   val SOME (fun2, asm1) = 

     rewrite_ thy ro er true  @{thm ruleZY} fun1; term2str fun2;

   val SOME (fun2', asm1) = 

     rewrite_ thy ro er true  @{thm ruleZY} fun1'; term2str fun2';

   val SOME (fun3,_) = 

     rewrite_set_ @{theory Isac} false norm_Rational fun2;

   term2str fun3;

   (*

    * Fails on x^^^(-1)

    * We solve this problem by using 1/x as a workaround.

    *)

   val SOME (fun3',_) = 

     rewrite_set_ @{theory Isac} false norm_Rational fun2';

   term2str fun3';

   (*

    * OK - workaround!

    *)

 *}

 subsubsection {*Get the Argument of the Expression X'*}

 text{*\noindent We use \texttt{grep} for finding possibilities how we can

        extract the bound variable in the expression. \ttfamily Atools.thy, 

        Tools.thy \normalfont contain general utilities: \ttfamily 

        eval\_argument\_in, eval\_rhs, eval\_lhs,\ldots \normalfont

        \ttfamily grep -r "fun eva\_" * \normalfont shows all functions 

        witch can be used in a script. Lookup this files how to build 

        and handle such functions.

        \par The next section shows how to introduce such a function.

 *}

 subsubsection{*Decompose the Given Term Into lhs and rhs\footnote{Note:

                lhs means \em Left Hand Side \normalfont and rhs means 

                \em Right Hand Side \normalfont and indicates the left or 

                the right part of an equation.}*}

 ML {*

   val (_, expr) = HOLogic.dest_eq fun3'; term2str expr;

   val (_, denom) = 

     HOLogic.dest_bin "Rings.inverse_class.divide" (type_of expr) expr;

   term2str denom = "-1 + -2 * z + 8 * z ^^^ 2";

 *}

 text{*\noindent We have rhs in the Script language, but we need a function 

        which gets the denominator of a fraction.*}

 subsubsection{*Get the Denominator and Numerator out of a Fraction*}

 text{*\noindent The selv written functions in e.g. \texttt{get\_denominator}

        should become a constant for the isabelle parser:*}

 consts

   get_denominator :: "real => real"

   get_numerator :: "real => real"

 text {*\noindent With the above definition we run into problems when we parse

         the Script \texttt{InverseZTransform}. This leads to \em ambiguous

         parse trees. \normalfont We avoid this by moving the definition

         to \ttfamily Rational.thy \normalfont and re-building Isac.

         \par \noindent ATTENTION: From now on \ttfamily 

         Build\_Inverse\_Z\_Transform \normalfont mimics a build from scratch;

         it only works due to re-building Isac several times (indicated 

         explicityl).

 *}

 ML {*

 (*

  *("get_denominator",

  *  ("Rational.get_denominator", eval_get_denominator ""))

  *)

 fun eval_get_denominator (thmid:string) _ 

 		      (t as Const ("Rational.get_denominator", _) $

               (Const ("Rings.inverse_class.divide", _) $ num $

                 denom)) thy = 

         SOME (mk_thmid thmid "" 

             (Print_Mode.setmp [] 

               (Syntax.string_of_term (thy2ctxt thy)) denom) "", 

 	          Trueprop $ (mk_equality (t, denom)))

   | eval_get_denominator _ _ _ _ = NONE; 

 *}

 text {* tests of eval_get_denominator see test/Knowledge/rational.sml*}

 text {*get numerator should also become a constant for the isabelle parser: *}

 ML {*

 fun eval_get_numerator (thmid:string) _ 

 		      (t as Const ("Rational.get_numerator", _) $

               (Const ("Rings.inverse_class.divide", _) $num

                 $denom )) thy = 

         SOME (mk_thmid thmid "" 

             (Print_Mode.setmp [] (Syntax.string_of_term (thy2ctxt thy)) num) "", 

 	          Trueprop $ (mk_equality (t, num)))

   | eval_get_numerator _ _ _ _ = NONE; 

 *}

 text {*

 We discovered severell problems by implementing the get_numerator function.

 Remember when putting new functions to Isac, put them in a thy file and rebuild 

 isac, also put them in the ruleset for the script!

 *}

 subsection {*solve equation*}

 text {*this type of equation if too general for the present program*}

 ML {*

 "----------- Minisubplb/100-init-rootp (*OK*)bl.sml ---------------------";

 val denominator = parseNEW ctxt "z^^^2 - 1/4*z - 1/8 = 0";

 val fmz = ["equality (z^^^2 - 1/4*z - 1/8 = (0::real))", "solveFor z","solutions L"];

 val (dI',pI',mI') =("Isac", ["univariate","equation"], ["no_met"]);

 (*                           ^^^^^^^^^^^^^^^^^^^^^^ TODO: ISAC determines type of eq*)

 *}

 text {*Does the Equation Match the Specification ?*}

 ML {*

 match_pbl fmz (get_pbt ["univariate","equation"]);

 *}

 ML {*Context.theory_name thy = "Isac"(*==================================================*)*}

 ML {*

 val denominator = parseNEW ctxt "-1 + -2 * z + 8 * z ^^^ 2 = 0";

 val fmz =                                            (*specification*)

   ["equality (-1 + -2 * z + 8 * z ^^^ 2 = (0::real))", (*equality*)

    "solveFor z",                                     (*bound variable*)

    "solutions L"];                                   (*identifier for solution*)

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

   ("Isac", ["abcFormula","degree_2","polynomial","univariate","equation"], ["no_met"]);

 *}

 text {*Does the Other Equation Match the Specification ?*}

 ML {*

 match_pbl fmz (get_pbt ["abcFormula","degree_2","polynomial","univariate","equation"]);

 *}

 text {*Solve Equation Stepwise*}

 ML {*

 *}

 ML {*

 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; (*nxt =..,Check_elementwise "Assumptions")*)

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

 val (p,_,f,nxt,_,pt) = me nxt p [] pt; f2str f;

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

 show_pt pt; 

 val SOME f = parseNEW ctxt "[z=1/2, z=-1/4]";

 *}

 subsection {*partial fraction decomposition*}

 subsubsection {*solution of the equation*}

 ML {*

 val SOME solutions = parseNEW ctxt "[z=1/2, z=-1/4]";

 term2str solutions;

 atomty solutions;

 *}

 subsubsection {*get solutions out of list*}

 text {*in isac's CTP-based programming language: let$ $s_1 = NTH 1$ solutions; $s_2 = NTH 2...$*}

 ML {*

 val Const ("List.list.Cons", _) $ s_1 $ (Const ("List.list.Cons", _) $

       s_2 $ Const ("List.list.Nil", _)) = solutions;

 term2str s_1;

 term2str s_2;

 *}

 ML {* (*Solutions as Denominator --> Denominator1 = z - Zeropoint1, Denominator2 = z-Zeropoint2,...*)

 val xx = HOLogic.dest_eq s_1;

 val s_1' = HOLogic.mk_binop "Groups.minus_class.minus" xx;

 val xx = HOLogic.dest_eq s_2;

 val s_2' = HOLogic.mk_binop "Groups.minus_class.minus" xx;

 term2str s_1';

 term2str s_2';

 *}

 text {* for the programming language a function 

   collecting all the above manipulations is helpful*}

 ML {*

 fun mk_minus_1 T = Free("-1", T); (*TODO DELETE WITH numbers_to_string*)

 fun flip_sign t = (*TODO improve for use in factors_from_solution: -(-1) etc*)

   let val minus_1 = t |> type_of |> mk_minus_1

   in HOLogic.mk_binop "Groups.times_class.times" (minus_1, t) end;

 fun fac_from_sol s =

   let val (lhs, rhs) = HOLogic.dest_eq s

   in HOLogic.mk_binop "Groups.minus_class.minus" (lhs, rhs) end;

 *}

 ML {*

 e_term

 *}

 ML {*

 fun mk_prod prod [] =

       if prod = e_term then error "mk_prod called with []" else prod

   | mk_prod prod (t :: []) =

       if prod = e_term then t else HOLogic.mk_binop "Groups.times_class.times" (prod, t)

   | mk_prod prod (t1 :: t2 :: ts) =

         if prod = e_term 

         then 

            let val p = HOLogic.mk_binop "Groups.times_class.times" (t1, t2)

            in mk_prod p ts end 

         else 

            let val p = HOLogic.mk_binop "Groups.times_class.times" (prod, t1)

            in mk_prod p (t2 :: ts) end 

 *}

 ML {*

 *}

 ML {*

 (*probably keept these test in test/Tools/isac/...

 (*mk_prod e_term [];*)

 val prod = mk_prod e_term [str2term "x + 123"]; 

 term2str prod = "x + 123";

 val sol = str2term "[z = 1 / 2, z = -1 / 4]";

 val sols = HOLogic.dest_list sol;

 val facs = map fac_from_sol sols;

 val prod = mk_prod e_term facs; 

 term2str prod = "(z + -1 * (1 / 2)) * (z + -1 * (-1 / 4))";

 val prod = mk_prod e_term [str2term "x + 1", str2term "x + 2", str2term "x + 3"]; 

 term2str prod = "(x + 1) * (x + 2) * (x + 3)";

 *)

 fun factors_from_solution sol = 

   let val ts = HOLogic.dest_list sol

   in mk_prod e_term (map fac_from_sol ts) end;

 (*

 val sol = str2term "[z = 1 / 2, z = -1 / 4]";

 val fs = factors_from_solution sol;

 term2str fs = "(z + -1 * (1 / 2)) * (z + -1 * (-1 / 4))"

 *)

 *}

 text {* This function needs to be packed such that it can be evaluated by the Lucas-Interpreter:

   # shift these functions into the related Equation.thy

   #  -- compare steps done with get_denominator above

   # done 02.12.2011 moved to PartialFractions.thy

   *}

 ML {*

 (*("factors_from_solution", ("Partial_Fractions.factors_from_solution", eval_factors_from_solution ""))*)

 fun eval_factors_from_solution (thmid:string) _

      (t as Const ("Partial_Fractions.factors_from_solution", _) $ sol) thy =

        ((let val prod = factors_from_solution sol

          in SOME (mk_thmid thmid ""

            (Print_Mode.setmp [] (Syntax.string_of_term (thy2ctxt thy)) prod) "",

                Trueprop $ (mk_equality (t, prod)))

          end)

        handle _ => NONE)

  | eval_factors_from_solution _ _ _ _ = NONE;

 *}

 text {*

 The tracing output of the calc tree after apllying this function was

 24 / factors_from_solution [z = 1/ 2, z = -1 / 4])]  and the next step

  val nxt = ("Empty_Tac", ...): tac'_).

 These observations indicate, that the Lucas-Interpreter (LIP) does 

 not know how to evaluate factors_from_solution, so there is something 

 wrong or missing.

 # First we isolate the difficulty in the program as follows:

   :

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

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

   "        [BOOL equ, REAL zzz]);              " ^

   "      (facs::real) = factors_from_solution L_L;" ^

   "      (foo::real) = Take facs" ^

   :

 and see

   [

   (([], Frm), Problem (Isac, [inverse, Z_Transform, SignalProcessing])),

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

   (([1], Res), ?X' z = 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))),

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

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

   (([3,1], Frm), -1 + -2 * z + 8 * z ^^^ 2 = 0),

   (([3,1], Res), z = (- -2 + sqrt (-2 ^^^ 2 - 4 * 8 * -1)) / (2 * 8) |

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

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

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

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

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

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

 in particular that

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

 shows the equation which has been created in the program by

   "      (denom::real) = get_denominator funterm;" ^ (*get_denominator*)

   "      (equ::bool) = (denom = (0::real));" ^

 # 'get_denominator' has been evaluated successfully, but not factors_from_solution.

 So we stepwise compare with an analogous case, get_denominator 

 successfully done above: We know that LIP evaluates expressions in the 

 program by use of the "srls", so we

 # try to get the original srls

   val {srls, ...} = get_met ["SignalProcessing","Z_Transform","inverse"];

 # create 2 good example terms

   val SOME t1 = parseNEW ctxt "get_denominator ((111::real) / 222)";

   val SOME t2 = parseNEW ctxt "factors_from_solution [(z::real) = 1 / 2, z = -1 / 4]";

 # rewrite the terms using srls

   rewrite_set_ thy true srls t1;

   rewrite_set_ thy true srls t2;

 and we see a difference: t1 gives SOME, t2 gives NONE.

 Now we look at the srls:

   val srls = Rls {id="srls_InverseZTransform",

     :

     rules =

       [

         :

         Calc("Rational.get_numerator",

           eval_get_numerator "Rational.get_numerator"),

         Calc("Partial_Fractions.factors_from_solution",

           eval_factors_from_solution "Partial_Fractions.factors_from_solution")

       ],

     :

 Here everthing is perfect. So the error can only be in the SML code of eval_factors_from_solution.

 We try to check the code with an existing test; since the code is in

   src/Tools/isac/Knowledge/Partial_Fractions.thy

 the test should be in

   test/Tools/isac/Knowledge/partial_fractions.sml

 -------------------------------------------------------------------------------

 After updating the function get_factors_from solution to a new version and 

 putting a testcase to Partial_Fractions.sml we tried again to evaluate the

 term with the same result.

 We opened the test Test_Isac.thy and saw that everything is working fine.

 Also we checked that the test partial_fractions.sml is used in Test_Isac.thy 

 -->  use "Knowledge/partial_fractions.sml"

 and Partial_Fractions.thy is part is part of isac by evaluating

 val thy = @{theory Isac};

 after rebuilding isac again it worked

 *}

 subsubsection {*build expression*}

 text {*in isac's CTP-based programming language: let s_1 = Take numerator / (s_1 * s_2)*}

 ML {*

 (*The Main Denominator is the multiplikation of the partial fraction denominators*)

 val denominator' = HOLogic.mk_binop "Groups.times_class.times" (s_1', s_2') ;

 val SOME numerator = parseNEW ctxt "3::real";

 val expr' = HOLogic.mk_binop "Rings.inverse_class.divide" (numerator, denominator');

 term2str expr';

 *}

 subsubsection {*Ansatz - create partial fractions out of our expression*}

 ML {*Context.theory_name thy = "Isac"*}

 axiomatization where

   ansatz2: "n / (a*b) = A/a + B/(b::real)" and

   multiply_eq2: "((n::real) / (a*b) = A/a + B/b) = (a*b*(n  / (a*b)) = a*b*(A/a + B/b::real))"  

 ML {*

 (*we use our ansatz2 to rewrite our expression and get an equilation with our expression on the left and the partial fractions of it on the right side*)

 val SOME (t1,_) = rewrite_ @{theory Isac} e_rew_ord e_rls false @{thm ansatz2} expr';

 term2str t1; atomty t1;

 val eq1 = HOLogic.mk_eq (expr', t1);

 term2str eq1;

 *}

 ML {*

 (*eliminate the demoninators by multiplying the left and the right side with the main denominator*)

 val SOME (eq2,_) = rewrite_ @{theory Isac} e_rew_ord e_rls false @{thm multiply_eq2} eq1;

 term2str eq2;

 *}

 ML {*

 (*simplificatoin*)

 val SOME (eq3,_) = rewrite_set_ @{theory Isac} false norm_Rational eq2;

 term2str eq3; (*?A ?B not simplified*)

 *}

 ML {*

 val SOME fract1 =

   parseNEW ctxt "(z - 1 / 2) * (z - -1 / 4) * (A / (z - 1 / 2) + B / (z - -1 / 4))"; (*A B !*)

 val SOME (fract2,_) = rewrite_set_ @{theory Isac} false norm_Rational fract1;

 term2str fract2 = "(A + -2 * B + 4 * A * z + 4 * B * z) / 4";

 (*term2str fract2 = "A * (1 / 4 + z) + B * (-1 / 2 + z)" would be more traditional*)

 *}

 ML {*

 val (numerator, denominator) = HOLogic.dest_eq eq3;

 val eq3' = HOLogic.mk_eq (numerator, fract1); (*A B !*)

 term2str eq3';

 (*MANDATORY: simplify (and remove denominator) otherwise 3 = 0*)

 val SOME (eq3'' ,_) = rewrite_set_ @{theory Isac} false norm_Rational eq3';

 term2str eq3'';

 *}

 ML {*Context.theory_name thy = "Isac"(*==================================================*)*}

 subsubsection {*Build a rule-set for ansatz*}

 text {* the "ansatz" rules violate the principle that each variable on 

   the right-hand-side must also occur on the left-hand-side of the rule:

   A, B, etc don't.

   Thus the rewriter marks these variables with question marks: ?A, ?B, etc.

   These question marks can be dropped by "fun drop_questionmarks".

   *}

 ML {*

 val ansatz_rls = prep_rls(

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

 	  erls = e_rls, srls = Erls, calc = [],

 	  rules = 

 	   [Thm ("ansatz2",num_str @{thm ansatz2}),

 	    Thm ("multiply_eq2",num_str @{thm multiply_eq2})

 	   ], 

 	 scr = EmptyScr});

 *}

 ML {*

 val SOME (ttttt,_) = rewrite_set_ @{theory Isac} false ansatz_rls expr';

 *}

 ML {*

 term2str expr' = "3 / ((z - 1 / 2) * (z - -1 / 4))";

 term2str ttttt = "?A / (z - 1 / 2) + ?B / (z - -1 / 4)";

 *}

 subsubsection {*get first koeffizient*}

 ML {*

 (*substitude z with the first zeropoint to get A*)

 val SOME (eq4_1,_) = rewrite_terms_ @{theory Isac} e_rew_ord e_rls [s_1] eq3'';

 term2str eq4_1;

 val SOME (eq4_2,_) = rewrite_set_ @{theory Isac} false norm_Rational eq4_1;

 term2str eq4_2;

 val fmz = ["equality (3 = 3 * A / (4::real))", "solveFor A","solutions L"];

 val (dI',pI',mI') =("Isac", ["univariate","equation"], ["no_met"]);

 (*solve the simple linear equilation for A TODO: return eq, not list of eq*)

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

 val (p,_,fa,nxt,_,pt) = me nxt p [] pt; (*Add_Given "equality (3 = 3 * A / 4)"*)

 val (p,_,fa,nxt,_,pt) = me nxt p [] pt; (* Add_Given "solveFor A"*)

 val (p,_,fa,nxt,_,pt) = me nxt p [] pt; (*Add_Find "solutions L"*)

 val (p,_,fa,nxt,_,pt) = me nxt p [] pt; (*Specify_Theory "Isac"*)

 val (p,_,fa,nxt,_,pt) = me nxt p [] pt; (*Specify_Problem ["normalize", "polynomial", 

                                           "univariate", "equation"])*)

 val (p,_,fa,nxt,_,pt) = me nxt p [] pt; (* Specify_Method ["PolyEq", "normalize_poly"]*)

 val (p,_,fa,nxt,_,pt) = me nxt p [] pt; (*Apply_Method ["PolyEq", "normalize_poly"]*)

 val (p,_,fa,nxt,_,pt) = me nxt p [] pt; (*Rewrite ("all_left", "PolyEq.all_left")*)

 val (p,_,fa,nxt,_,pt) = me nxt p [] pt; (*Rewrite_Set_Inst (["(bdv, A)"], "make_ratpoly_in")*)

 val (p,_,fa,nxt,_,pt) = me nxt p [] pt; (*Rewrite_Set "polyeq_simplify"*)

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

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

 val (p,_,fa,nxt,_,pt) = me nxt p [] pt; (*Add_Given "equality (3 + -3 / 4 * A = 0)"*)

 val (p,_,fa,nxt,_,pt) = me nxt p [] pt; (*Add_Given "solveFor A"*)

 val (p,_,fa,nxt,_,pt) = me nxt p [] pt; (*Add_Find "solutions A_i"*)

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

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

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

 val (p,_,fa,nxt,_,pt) = me nxt p [] pt; (*Apply_Method ["PolyEq", "solve_d1_polyeq_equation"]*)

 val (p,_,fa,nxt,_,pt) = me nxt p [] pt; (*Rewrite_Set_Inst (["(bdv, A)"], "d1_polyeq_simplify")*)

 val (p,_,fa,nxt,_,pt) = me nxt p [] pt; (*Rewrite_Set "polyeq_simplify"*)

 val (p,_,fa,nxt,_,pt) = me nxt p [] pt; (*Rewrite_Set "norm_Rational_parenthesized"*)

 val (p,_,fa,nxt,_,pt) = me nxt p [] pt; (*Or_to_List*)

 val (p,_,fa,nxt,_,pt) = me nxt p [] pt; (*Check_elementwise "Assumptions"*)

 val (p,_,fa,nxt,_,pt) = me nxt p [] pt; (*Check_Postcond ["degree_1", "polynomial", 

                                           "univariate", "equation"]*)

 val (p,_,fa,nxt,_,pt) = me nxt p [] pt; (*Check_Postcond ["normalize", "polynomial", 

                                           "univariate", "equation"]*)

 val (p,_,fa,nxt,_,pt) = me nxt p [] pt; (*End_Proof'*)

 f2str fa;

 *}

 subsubsection {*get second koeffizient*}

 ML {*thy*}

 ML {*

 (*substitude z with the second zeropoint to get B*)

 val SOME (eq4b_1,_) = rewrite_terms_ @{theory Isac} e_rew_ord e_rls [s_2] eq3'';

 term2str eq4b_1;

 val SOME (eq4b_2,_) = rewrite_set_ @{theory Isac} false norm_Rational eq4b_1;

 term2str eq4b_2;

 *}

 ML {*

 (*solve the simple linear equilation for B TODO: return eq, not list of eq*)

 val fmz = ["equality (3 = -3 * B / (4::real))", "solveFor B","solutions L"];

 val (dI',pI',mI') =("Isac", ["univariate","equation"], ["no_met"]);

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 f2str fb;

 *}

 ML {* (*check koeffizients*)

 if f2str fa = "[A = 4]" then () else error "part.fract. eq4_1";

 if f2str fb = "[B = -4]" then () else error "part.fract. eq4_1";

 *}

 subsubsection {*substitute expression with solutions*}

 ML {*

 *}

 ML {*thy*}

 section {*Implement the Specification and the Method \label{spec-meth}*}

 text{*==============================================*}

 subsection{*Define the Field Descriptions for the specification*}

 consts

   filterExpression  :: "bool => una"

   stepResponse      :: "bool => una"

 subsection{*Define the Specification*}

 ML {*

 store_pbt

  (prep_pbt thy "pbl_SP" [] e_pblID

  (["SignalProcessing"], [], e_rls, NONE, []));

 store_pbt

  (prep_pbt thy "pbl_SP_Ztrans" [] e_pblID

  (["Z_Transform","SignalProcessing"], [], e_rls, NONE, []));

 *}

 ML {*thy*}

 ML {*

 store_pbt

  (prep_pbt thy "pbl_SP_Ztrans_inv" [] e_pblID

  (["inverse", "Z_Transform", "SignalProcessing"],

   [("#Given" ,["filterExpression X_eq"]),

    ("#Find"  ,["stepResponse n_eq"])

   ],

   append_rls "e_rls" e_rls [(*for preds in where_*)], NONE, 

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

 show_ptyps();

 get_pbt ["inverse","Z_Transform","SignalProcessing"];

 *}

 subsection {*Define Name and Signature for the Method*}

 consts

   InverseZTransform :: "[bool, bool] => bool"

     ("((Script InverseZTransform (_ =))// (_))" 9)

 subsection {*Setup Parent Nodes in Hierarchy of Method*}

 ML {*

 store_met

  (prep_met thy "met_SP" [] e_metID

  (["SignalProcessing"], [],

    {rew_ord'="tless_true", rls'= e_rls, calc = [], srls = e_rls, prls = e_rls,

     crls = e_rls, nrls = e_rls}, "empty_script"));

 store_met

  (prep_met thy "met_SP_Ztrans" [] e_metID

  (["SignalProcessing", "Z_Transform"], [],

    {rew_ord'="tless_true", rls'= e_rls, calc = [], srls = e_rls, prls = e_rls,

     crls = e_rls, nrls = e_rls}, "empty_script"));

 *}

 ML {*

 store_met

  (prep_met thy "met_SP_Ztrans_inv" [] e_metID

  (["SignalProcessing", "Z_Transform", "inverse"], 

   [("#Given" ,["filterExpression X_eq"]),

    ("#Find"  ,["stepResponse n_eq"])

   ],

    {rew_ord'="tless_true", rls'= e_rls, calc = [], srls = e_rls, prls = e_rls,

     crls = e_rls, nrls = e_rls},

   "empty_script"

  ));

 *}

 ML {*

 store_met

  (prep_met thy "met_SP_Ztrans_inv" [] e_metID

  (["SignalProcessing", "Z_Transform", "inverse"], 

   [("#Given" ,["filterExpression X_eq"]),

    ("#Find"  ,["stepResponse n_eq"])

   ],

    {rew_ord'="tless_true", rls'= e_rls, calc = [], srls = e_rls, prls = e_rls,

     crls = e_rls, nrls = e_rls},

   "Script InverseZTransform (Xeq::bool) =" ^

   " (let X = Take Xeq;" ^

   "      X = Rewrite ruleZY False X" ^

   "  in X)"

  ));

 *}

 ML {*

 show_mets();

 *}

 ML {*

 get_met ["SignalProcessing","Z_Transform","inverse"];

 *}

 section {*Program in CTP-based language \label{prog-steps}*}

 text{*=================================*}

 subsection {*Stepwise extend Program*}

 ML {*

 val str = 

 "Script InverseZTransform (Xeq::bool) =" ^

 " Xeq";

 *}

 ML {*

 val str = 

 "Script InverseZTransform (Xeq::bool) =" ^ (*(1/z) instead of z ^^^ -1*)

 " (let X = Take Xeq;" ^

 "      X' = Rewrite ruleZY False X;" ^ (*z * denominator*)

 "      X' = (Rewrite_Set norm_Rational False) X'" ^ (*simplify*)

 "  in X)";

 (*NONE*)

 "Script InverseZTransform (Xeq::bool) =" ^ (*(1/z) instead of z ^^^ -1*)

 " (let X = Take Xeq;" ^

 "      X' = Rewrite ruleZY False X;" ^ (*z * denominator*)

 "      X' = (Rewrite_Set norm_Rational False) X';" ^ (*simplify*)

 "      X' = (SubProblem (Isac',[pqFormula,degree_2,polynomial,univariate,equation], [no_met])   " ^

     "                 [BOOL e_e, REAL v_v])" ^

 "  in X)";

 *}

 ML {*

 val str = 

 "Script InverseZTransform (Xeq::bool) =" ^ (*(1/z) instead of z ^^^ -1*)

 " (let X = Take Xeq;" ^

 "      X' = Rewrite ruleZY False X;" ^ (*z * denominator*)

 "      X' = (Rewrite_Set norm_Rational False) X';" ^ (*simplify*)

 "      funterm = rhs X'" ^ (*drop X'= for equation solving*)

 "  in X)";

 *}

 ML {*

 val str = 

 "Script InverseZTransform (X_eq::bool) =" ^ (*(1/z) instead of z ^^^ -1*)

 " (let X = Take X_eq;" ^

 "      X' = Rewrite ruleZY False X;" ^ (*z * denominator*)

 "      X' = (Rewrite_Set norm_Rational False) X';" ^ (*simplify*)

 "      (X'_z::real) = lhs X';" ^

 "      (z::real) = argument_in X'_z;" ^

 "      (funterm::real) = rhs X';" ^ (*drop X' z = for equation solving*)

 "      (denom::real) = get_denominator funterm;" ^ (*get_denominator*)

 "      (equ::bool) = (denom = (0::real));" ^

 "      (L_L::bool list) =                                    " ^

 "            (SubProblem (Test',                            " ^

 "                         [linear,univariate,equation,test]," ^

 "                         [Test,solve_linear])              " ^

 "                        [BOOL equ, REAL z])              " ^

 "  in X)"

 ;

 parse thy str;

 val sc = ((inst_abs thy) o term_of o the o (parse thy)) str;

 atomty sc;

 *}

 text {*

 This ruleset contains all functions that are in the script; 

 The evaluation of the functions is done by rewriting using this ruleset.

 *}

 ML {*

 val srls = Rls {id="srls_InverseZTransform", 

 		  preconds = [], rew_ord = ("termlessI",termlessI), 

 		  erls = append_rls "erls_in_srls_InverseZTransform" e_rls

 				    [(*for asm in NTH_CONS ...*) Calc ("Orderings.ord_class.less",eval_equ "#less_"),

 				     (*2nd NTH_CONS pushes n+-1 into asms*) Calc("Groups.plus_class.plus", eval_binop "#add_")

 				    ], 

   srls = Erls, calc = [],

 		  rules =

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

 			     Calc("Groups.plus_class.plus", eval_binop "#add_"),

 			     Thm ("NTH_NIL",num_str @{thm NTH_NIL}),

 			     Calc("Tools.lhs", eval_lhs"eval_lhs_"), (*<=== ONLY USED*)

 			     Calc("Tools.rhs", eval_rhs"eval_rhs_"), (*<=== ONLY USED*)

 			     Calc("Atools.argument'_in", eval_argument_in "Atools.argument'_in"),

      Calc("Rational.get_denominator", eval_get_denominator "#get_denominator"),

      Calc("Rational.get_numerator", eval_get_numerator "#get_numerator"),

      Calc("Partial_Fractions.factors_from_solution",

        eval_factors_from_solution "#factors_from_solution"),

      Calc("Partial_Fractions.drop_questionmarks", eval_drop_questionmarks "#drop_?")

 			    ],

 		  scr = EmptyScr};

 *}

 subsection {*Store Final Version of Program for Execution*}

 ML {*

 store_met

  (prep_met thy "met_SP_Ztrans_inv" [] e_metID

  (["SignalProcessing", "Z_Transform", "inverse"], 

   [("#Given" ,["filterExpression X_eq"]),

    ("#Find"  ,["stepResponse n_eq"])

   ],

    {rew_ord'="tless_true", rls'= e_rls, calc = [], srls = srls, 

     prls = e_rls,

     crls = e_rls, nrls = e_rls},

       "Script InverseZTransform (X_eq::bool) =" ^ (*(1/z) instead of z ^^^ -1*)

       " (let X = Take X_eq;" ^

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

       "      X' = Rewrite ruleZY False X;" ^ (*z * denominator*)

 (*([1], Res), ?X' z = 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))*)

       "      (num_orig::real) = get_numerator (rhs X');"^

       "      X' = (Rewrite_Set norm_Rational False) X';" ^ (*simplify*)

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

       "      (X'_z::real) = lhs X';" ^ (**)

       "      (zzz::real) = argument_in X'_z;" ^ (**)

       "      (funterm::real) = rhs X';" ^ (*drop X' z = for equation solving*)

       "      (denom::real) = get_denominator funterm;" ^ (*get_denominator*)

       "      (num::real) = get_numerator funterm; " ^ (*get_numerator*)

       "      (equ::bool) = (denom = (0::real));" ^

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

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

       "         [BOOL equ, REAL zzz]);              " ^

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

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

       "      (facs::real) = factors_from_solution L_L;" ^

       "      (eql::real) = Take (num_orig / facs);" ^                                 (*([4], Frm), 24 / ((z + -1 * (1 / 2)) * (z + -1 * (-1 / 4)))*)

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

       "      (eq::bool) = Take (eql = eqr);"^ (*Maybe possible to use HOLogic.mk_eq ??*) (*([5], Frm), 24 / ((z + -1 * (1 / 2)) * (z + -1 * (-1 / 4))) = ?A / (z + -1 * (1 / 2)) + ?B / (z + -1 * (-1 / 4))*)

       "      eq = (Try (Rewrite_Set equival_trans False)) eq;"^                  (*([5], Res), 24 = ?A * (z + -1 * (-1 / 4)) + ?B * (z + -1 * (1 / 2))*)

       "      eq = drop_questionmarks eq;"^

       "      (z1::real) = (rhs (NTH 1 L_L));"^                                   (*prepare equliation for a - eq_a therfor subsitude z with solution 1 - z1*)

       "      (z2::real) = (rhs (NTH 2 L_L));"^

       "      (eq_a::bool) = Take eq;"^

       "      eq_a = (Substitute [zzz=z1]) eq;"^                          (*([6], Res), 24 = ?A * (1 / 2 + -1 * (-1 / 4)) + ?B * (1 / 2 + -1 * (1 / 2))*)

       "      eq_a = (Rewrite_Set norm_Rational False) eq_a;"^                    (*([7], Res), 24 = ?A * 3 / 4*)

       "      (sol_a::bool list) = (SubProblem (Isac'," ^

       "          [univariate,equation],[no_met])" ^

       "        [BOOL eq_a, REAL (A::real)]);"^

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

       "      (eq_b::bool) = Take eq;"^

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

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

       "      (sol_b::bool list) = (SubProblem (Isac'," ^

       "          [univariate,equation],[no_met])" ^

       "        [BOOL eq_b, REAL (B::real)]);"^

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

       "      eqr = drop_questionmarks eqr;"^

       "      (pbz::real) = Take eqr;"^

       "      pbz = ((Substitute [A=a]) pbz);"^

       "      pbz = ((Substitute [B=b]) pbz);"^

       "      pbz = Rewrite ruleYZ False pbz;"^

       "      pbz = drop_questionmarks pbz;"^

       "      (iztrans::real) = Take pbz;"^

       "      iztrans = (Rewrite_Set inverse_z False) iztrans;"^

       "      iztrans = drop_questionmarks iztrans;"^

       "      (n_eq::bool) = Take (X_n = iztrans)"^ 

       "  in n_eq)" 

       ));

 *}

 subsection {*Check the Program*}

 subsubsection {*Check the formalization*}

 ML {*

 val fmz = ["filterExpression (X  = 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 ([(1, [1], "#Given", Const ("Inverse_Z_Transform.filterExpression", _),

             [Const ("HOL.eq", _) $ _ $ _]),

            (2, [1], "#Find", Const ("Inverse_Z_Transform.stepResponse", _),

             [Free ("x", _) $ _])],

           _) = prep_ori fmz thy ((#ppc o get_pbt) pI);

 *}

 ML {*

 val Script sc = (#scr o get_met) ["SignalProcessing","Z_Transform","inverse"];

 atomty sc;

 *}

 subsubsection {*Stepwise check the program\label{sec:stepcheck}*}

 ML {*

 trace_rewrite := false;

 trace_script := false; print_depth 9;

 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))];

 (*([], Frm), Problem (Isac, [inverse, Z_Transform, SignalProcessing])*)

 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))"; (*TODO naming!*)

 (*([1], Frm), 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)))";

 (*([1], Res), ?X' z = 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))*)

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

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

 *}

 text {* Instead of nxt = Subproblem above we had Empty_Tac; the search for the reason 

   considered the following points:

   # what shows show_pt pt; ...

     (([2], Res), ?X' z = 24 / (-1 + -2 * z + 8 * z ^^^ 2))] ..calculation ok,

     but no "next" step found: should be "nxt = Subproblem" ?!?

   # what shows trace_script := true; we read bottom up ...

     @@@ next   leaf 'SubProbfrom

      (PolyEq', [abcFormula, degree_2, polynomial, univariate, equation],

       no_meth)

      [BOOL equ, REAL z]' ---> STac 'SubProblem

      (PolyEq', [abcFormula, degree_2, polynomial, univariate, equation],

       no_meth)

      [BOOL (-1 + -2 * z + 8 * z ^^^ 2 = 0), REAL z]'

     ... and see the SubProblem with correct arguments from searching next step

     (program text !!!--->!!! STac (script tactic) with arguments evaluated.)

   # do we have the right Script ...difference in the argumentsdifference in the arguments

     val Script s = (#scr o get_met) ["SignalProcessing","Z_Transform","inverse"];

     writeln (term2str s);

     ... shows the right script.difference in the arguments

   # test --- why helpless here ? --- shows: replace no_meth by [no_meth] in Script

 *}

 ML {*

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

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

 *}

 text {* Instead of nxt = Model_Problem above we had Empty_Tac; the search for the reason 

   considered the following points:difference in the arguments

   # comparison with subsection { *solve equation* }: there solving this equation works,

     so there must be some difference in the arguments of the Subproblem:

     RIGHT: we had [no_meth] instead of [no_met] ;-))

 *}

 ML {*

 val (p,_,f,nxt,_,pt) = me nxt p [] pt; "nxt = Add_Given equality (-1 + -2 * z + 8 * z ^^^ 2 = 0)";

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

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

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

 *}

 text {* We had "nxt = Empty_Tac instead Specify_Theory; 

   the search for the reason considered the following points:

   # was there an error message ? NO --ok

   # has "nxt = Add_Find" been inserted in pt: get_obj g_pbl pt (fst p); YES --ok

   # what is the returned "formula": print_depth 999; f; print_depth 999; --

     {Find = [Correct "solutions z_i"], With = [], 

      Given = [Correct "equality (-1 + -2 * z + 8 * z ^^^ 2 = 0)", Correct "solveFor z"],

      Where = [False "matches (z = 0) (-1 + -2 * z + 8 * z ^^^ 2 = 0) |\n

                      matches (?b * z = 0) (-1 + -2 * z + 8 * z ^^^ 2 = 0) |\n

                      matches (?a + z = 0) (-1 + -2 * z + 8 * z ^^^ 2 = 0) |\n

                      matches (?a + ?b * z = 0) (-1 + -2 * z + 8 * z ^^^ 2 = 0)"],

      Relate = []}

      -- the only False is the reason: the Where (the precondition) is False for good reasons:

      the precondition seems to check for linear equations, not for the one we want to solve!

   Removed this error by correcting the Script

   from SubProblem (PolyEq', [linear,univariate,equation,test], [Test,solve_linear]

   to   SubProblem (PolyEq', [abcFormula,degree_2,polynomial,univariate,equation],

                    [PolyEq,solve_d2_polyeq_abc_equation]

   You find the appropriate type of equation at

     http://www.ist.tugraz.at/projects/isac/www/kbase/pbl/index_pbl.html

   and the respective method in Knowledge/PolyEq.thy at the respective store_pbt.

   Or you leave the selection of the appropriate type to isac as done in the final Script ;-))

 *}

 ML {*

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

 val (p,_,f,nxt,_,pt) = me nxt p [] pt; "nxt = Specify_Method [PolyEq,solve_d2_polyeq_abc_equation";

 val (p,_,f,nxt,_,pt) = me nxt p [] pt; "nxt = Apply_Method [PolyEq,solve_d2_polyeq_abc_equation";

 val (p,_,f,nxt,_,pt) = me nxt p [] pt; "nxt = Rewrite_Set_Inst ([(bdv, z)], d2_polyeq_abcFormula_simplify";

 (*([3,1], Frm), -1 + -2 * z + 8 * z ^^^ 2 = 0)*)

 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;

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

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

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

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

 (*([4], Frm), 24 / ((z + -1 * (1 / 2)) * (z + -1 * (-1 / 4)))*)

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

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

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

 (*([5], Frm), 24 / ((z + -1 * (1 / 2)) * (z + -1 * (-1 / 4))) = ?A / (z + -1 * (1 / 2)) + ?B / (z + -1 * (-1 / 4))*)

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

 (*([5], Res), 24 = ?A * (z + -1 * (-1 / 4)) + ?B * (z + -1 * (1 / 2))*)

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

 (*([6], Res), 24 = A * (1 / 2 + -1 * (-1 / 4)) + B * (1 / 2 + -1 * (1 / 2))*)

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

 (*([7], Res), 24 = A * 3 / 4*)

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

 (*([8], Pbl), solve (24 = 3 * A / 4, A)*)

 val (p,_,f,nxt,_,pt) = me nxt p [] pt; (*Add_Given "equality (24 = 3 * A / 4)"*)

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

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

 val (p,_,f,nxt,_,pt) = me nxt p [] pt; (*Specify_Theory "Isac"*)val (p,_,f,nxt,_,pt) = me nxt p [] pt; (*Specify_Problem ["normalize", "polynomial", 

                                          "univariate", "equation"]*)

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

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

 val (p,_,f,nxt,_,pt) = me nxt p [] pt; (*Rewrite ("all_left", "PolyEq.all_left")*)

 val (p,_,f,nxt,_,pt) = me nxt p [] pt; (*Rewrite_Set_Inst (["(bdv, A)"], "make_ratpoly_in")*)

 val (p,_,f,nxt,_,pt) = me nxt p [] pt; (*Rewrite_Set "polyeq_simplify"*)

 val (p,_,f,nxt,_,pt) = me nxt p [] pt; (*Subproblem ("Isac", ["degree_1", "polynomial", 

                                          "univariate", "equation"])*)

 *}

 ML {*

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

 show_pt pt;

 *}

 ML {*

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

 show_pt pt;

 *}

 ML {*

 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;

 *}

 ML {*

 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;

 *}

 ML {*

 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;

 *}

 ML {*

 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;

 *}

 ML {*

 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;

 *}

 ML {*

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

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

 *}

 ML {*

 trace_script := true;

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

 show_pt pt;

 *}

 section {*Write Tests for Crucial Details*}

 text{*===================================*}

 ML {*

 *}

 section {*Integrate Program into Knowledge*}

 ML {*

 print_depth 999;

 *}

 end