(* 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 Inverse_Z_Transform.thy as an exercise.

   Because subsection "Stepwise Check the Program" requires 

   Inverse_Z_Transform.thy as a subtheory of Isac.thy, the setup has been changed 

   from "theory Inverse_Z_Transform imports Isac begin.." to the above.

   ATTENTION WITH NAMES OF IDENTIFIERS WHEN GOING INTO INTERNALS:

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

   (Thm.derivation_name @{thm rule1} = "Build_Inverse_Z_Transform.rule1") = true;

   but actually in us will be "Inverse_Z_Transform.rule1"

 *}

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

 section {*trials towards Z transform *}

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

 subsection {*terms*}

 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]"};

 *}

 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]"

 ML {*

 @{thm rule1};

 @{thm rule2};

 @{thm rule3};

 @{thm rule4};

 *}

 subsection {*apply rules*}

 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 !!!*)

 *}

 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;*}

 ML {*

 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;

 *}

 ML {*

 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*}

 text{*TODO insert Calculation (Referenz?!)

 The goal... realized in sections below, in Sect.\ref{spec-meth} and Sect.\ref{prog-steps} 

 the reader is advised to jump between the subsequent subsections and the respective steps in Sect.\ref{prog-steps} 

 *}

 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 {*multply with z*}

 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) TODO*)

 val SOME (fun3',_) = rewrite_set_ @{theory Isac} false norm_Rational fun2';

 term2str fun3'; (*OK*)

 *}

 subsubsection {*get argument of X': z is the variable the equation is solved for*}

 text{*grep... Atools.thy, Tools.thy contain general utilities: eval_argument_in, eval_rhs, eval_lhs,...

 grep -r "fun eva_" ... shows all functions witch can be used in a script.

 lookup this files how to build and handle such functions.

 the next section shows how to introduce such a function.

 *}

 subsubsection {*Decompose given term into lhs = rhs*}

 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 {*we have rhs in the Script language, but we need a function 

   which gets the denominator of a fraction*}

 text{*---------------------------begin partial fractions snip--------------------------*}

 subsubsection {*get the denominator out of a fraction*}

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

 consts

   get_denominator :: "real => real"

 text {* With the above definition we run into problems with parsing the Script InverseZTransform:

   This leads to "ambiguous parse trees" and we avoid this by shifting the definition

   to Rational.thy and re-building Isac.

   ATTENTION: from now on Build_Inverse_Z_Transform 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{*---------------------------end partial fractions snip--------------------------*}

 subsubsection {*get the numerator out of a fraction*}

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

 consts

   get_numerator :: "real => real"

 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; 

 *}

 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.plus_class.plus" (lhs, flip_sign 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

   *}

 ML {*

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

 fun eval_factors_from_solution (thmid:string) _ t thy =

     (let val prod = factors_from_solution t

      in SOME (mk_thmid thmid "" 

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

 	          Trueprop $ (mk_equality (t, prod)))

      end)

      handle _ => NONE; 

 *}

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

 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 {*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;

 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;

 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;

 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;

 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;

 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;

 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;

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

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

 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 "Rational.get_denominator"),

      Calc("Rational.get_numerator",

           eval_get_numerator "Rational.get_numerator")

 			    ],

 		  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;" ^

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

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

 "      (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; " ^

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

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

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

 "        [BOOL equ, REAL zzz])              " ^

 "  in X)" 

  ));

 *}

 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*}

 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))];

 *}

 ML {*

 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!*)

 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)))";

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

 *}

 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";

 *}

 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";

 show_pt pt;

 *}

 ML {*

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

 *}

 ML {*

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

 *}

 ML {*

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

 *}

 ML {*

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

 *}

 ML {*

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

 *}

 ML {*

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

 show_pt pt;

 *}

 ML {*

 *}

 section {*Write Tests for Crucial Details*}

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

 ML {*

 *}

 section {*Integrate Program into Knowledge*}

 ML {*

 @{theory Isac}

 *}

 end