(* Title:  Build_Inverse_Z_Transform

   Author: Jan Rocnik

   (c) copyright due to lincense terms.

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*)

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