(* Title:  Inverse_Z_Transform

   Author: Jan Rocnik

   (c) copyright due to lincense terms.

*)

theory Inverse_Z_Transform imports PolyEq DiffApp Partial_Fractions begin

axiomatization where       \<comment> \<open>TODO: new variables on the rhs enforce replacement by substitution\<close>

  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: "c * (z / (z - \<alpha>)) = c * \<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]" (*and

  rule42: "(a * (z/(z-b)) + c * (z/(z-d))) = (a * b^^^n * u [n] + c * d^^^n * u [n])"*)

axiomatization where

(*ruleZY: "(X z = a / b) = (d_d z X = a / (z * b))"         ..looks better, but types are flawed*)

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

  ruleYZ: "a / (z - b) + c / (z - d) = a * (z / (z - b)) + c * (z / (z - d))" and

  ruleYZa: "(a / b + c / d) = (a * (z / b) + c * (z / d))"        \<comment> \<open>that is what students learn\<close>

subsection\<open>Define the Field Descriptions for the specification\<close>

consts

  filterExpression  :: "bool => una"

  stepResponse      :: "bool => una"    \<comment> \<open>TODO: unused, "u [n]" is introduced by rule1..6 above\<close>

ML \<open>

val inverse_z = prep_rls'(

  Rule.Rls {id = "inverse_z", preconds = [], rew_ord = ("dummy_ord",Rule.dummy_ord), 

	  erls = Rule.Erls, srls = Rule.Erls, calc = [], errpatts = [],

	  rules = 

	   [

    Rule.Thm ("rule4", @{thm rule4})

	   ], 

	 scr = Rule.EmptyScr});

\<close>

text \<open>store the rule set for math engine\<close>

setup \<open>KEStore_Elems.add_rlss [("inverse_z", (Context.theory_name @{theory}, inverse_z))]\<close>

subsection\<open>Define the Specification\<close>

ML \<open>

val thy = @{theory};

\<close>

setup \<open>KEStore_Elems.add_pbts

  [(Specify.prep_pbt thy "pbl_SP" [] Celem.e_pblID (["SignalProcessing"], [], Rule.e_rls, NONE, [])),

    (Specify.prep_pbt thy "pbl_SP_Ztrans" [] Celem.e_pblID

      (["Z_Transform","SignalProcessing"], [], Rule.e_rls, NONE, [])),

    (Specify.prep_pbt thy "pbl_SP_Ztrans_inv" [] Celem.e_pblID

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

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

          ("#Find"  ,["stepResponse n_eq"])], \<comment> \<open>TODO: unused, "u [n]" is introduced by rule1..6\<close>

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

        [["SignalProcessing","Z_Transform","Inverse"]]))]\<close>

subsection \<open>Define Name and Signature for the Method\<close>

consts

  InverseZTransform1 :: "[bool, bool] => bool"

    ("((Script InverseZTransform1 (_ =))// (_))" 9)

  InverseZTransform2 :: "[bool, real, bool] => bool"

    ("((Script InverseZTransform2 (_ _ =))// (_))" 9)

subsection \<open>Setup Parent Nodes in Hierarchy of Method\<close>

ML \<open>val thy = @{theory}; (*latest version of thy required*)\<close>

setup \<open>KEStore_Elems.add_mets

    [Specify.prep_met thy "met_SP" [] Celem.e_metID

      (["SignalProcessing"], [],

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

          errpats = [], nrls = Rule.e_rls}, @{thm refl}),

    Specify.prep_met thy "met_SP_Ztrans" [] Celem.e_metID

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

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

          errpats = [], nrls = Rule.e_rls}, @{thm refl})]

\<close>

partial_function (tailrec) inverse_ztransform :: "bool \<Rightarrow> real \<Rightarrow> bool"

  where

"inverse_ztransform X_eq X_z =                                       \<comment> \<open>(1/z) instead of z ^^^ -1\<close>

 (let X = Take X_eq;

      X' = Rewrite ''ruleZY'' False X;                                         \<comment> \<open>z * denominator\<close>

      X' = (Rewrite_Set ''norm_Rational'' False) X';                                  \<comment> \<open>simplify\<close>

      funterm = Take (rhs X');                                \<comment> \<open>drop X' z = for equation solving\<close>

      denom = (Rewrite_Set ''partial_fraction'' False) funterm;                \<comment> \<open>get_denominator\<close>

      equ = (denom = (0::real));

      fun_arg = Take (lhs X');

      arg = (Rewrite_Set ''partial_fraction'' False) X';                     \<comment> \<open>get_argument TODO\<close>

      (L_L::bool list) = \<comment> \<open>'bool list' inhibits (?!?):

                  WARNING: Additional type variable(s) in specification of inverse_ztransform: 'a\<close>

        SubProblem (''Test'', [''LINEAR'',''univariate'',''equation'',''test''],

          [''Test'',''solve_linear'']) [BOOL equ, REAL X_z]

  in X) "

setup \<open>KEStore_Elems.add_mets

    [Specify.prep_met thy "met_SP_Ztrans_inv" [] Celem.e_metID

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

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

          ("#Find"  ,["stepResponse n_eq"])], \<comment> \<open>TODO: unused, "u [n]" is introduced by rule1..6\<close>

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

          errpats = [], nrls = Rule.e_rls},

        @{thm inverse_ztransform.simps}

	    (*"Script InverseZTransform1 (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*)

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

          "      denom = (Rewrite_Set ''partial_fraction'' False) funterm;" ^ (*get_denominator*)

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

          "      fun_arg = Take (lhs X');" ^

          "      arg = (Rewrite_Set ''partial_fraction'' False) X';" ^ (*get_argument TODO*)

          "      (L_L::bool list) =                                    " ^

          "            (SubProblem (''Test'',                            " ^

          "                         [''LINEAR'',''univariate'',''equation'',''test'']," ^

          "                         [''Test'',''solve_linear''])              " ^

          "                        [BOOL equ, REAL z])              " ^

          "  in X)"*))]

\<close>

partial_function (tailrec) inverse_ztransform2 :: "bool \<Rightarrow> real \<Rightarrow> bool"

  where

"inverse_ztransform2 X_eq X_z =

  (let X = Take X_eq;

    X' = Rewrite ''ruleZY'' False X;

    X'_z = lhs X';

    zzz = argument_in X'_z;

    funterm = rhs X';

    pbz = SubProblem (''Isac'',

      [''partial_fraction'',''rational'',''simplification''],

      [''simplification'',''of_rationals'',''to_partial_fraction''])

      [REAL funterm, REAL zzz];

    pbz_eq = Take (X'_z = pbz);

    pbz_eq = Rewrite ''ruleYZ'' False pbz_eq;

    X_zeq = Take (X_z = rhs pbz_eq);

    n_eq = (Rewrite_Set ''inverse_z'' False) X_zeq

  in n_eq)"

setup \<open>KEStore_Elems.add_mets

    [Specify.prep_met thy "met_SP_Ztrans_inv_sub" [] Celem.e_metID

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

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

          ("#Find"  ,["stepResponse n_eq"])], \<comment> \<open>TODO: unused, "u [n]" is introduced by rule1..6\<close>

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

          srls = Rule.Rls {id="srls_partial_fraction", 

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

              erls = Rule.append_rls "erls_in_srls_partial_fraction" Rule.e_rls

                  [(*for asm in NTH_CONS ...*)

                    Rule.Calc ("Orderings.ord_class.less",eval_equ "#less_"),

                    (*2nd NTH_CONS pushes n+-1 into asms*)

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

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

              rules = [Rule.Thm ("NTH_CONS", @{thm NTH_CONS}),

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

                  Rule.Thm ("NTH_NIL", @{thm NTH_NIL}),

                  Rule.Calc ("Tools.lhs", Tools.eval_lhs "eval_lhs_"),

                  Rule.Calc ("Tools.rhs", Tools.eval_rhs"eval_rhs_"),

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

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

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

                  Rule.Calc ("Partial_Fractions.factors_from_solution",

                    eval_factors_from_solution "#factors_from_solution")

                  ], scr = Rule.EmptyScr},

          prls = Rule.e_rls, crls = Rule.e_rls, errpats = [], nrls = norm_Rational},

        @{thm inverse_ztransform2.simps}

	    (*" Script InverseZTransform2 (X_eq::bool) (X_z::real) =               "^ (*([], Frm), Problem (Isac, [Inverse, Z_Transform, SignalProcessing])*)

        " (let X = Take X_eq;                                                "^ (*([1], Frm), X z = 3 / (z - 1 / 4 + -1 / 8 * (1 / z))*)

        "   X' = Rewrite ''ruleZY'' False X;                                 "^ (*([1], Res), ?X' z = 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))*)

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

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

        "   (funterm::real) = rhs X';                                        "^ (*            3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))*)

        "   (pbz::real) = (SubProblem (''Isac'',                             "^ (*([2], Res), 4 / (z - 1 / 2) + -4 / (z - -1 / 4)*)

        "     [''partial_fraction'',''rational'',''simplification''],        "^

        "     [''simplification'',''of_rationals'',''to_partial_fraction'']) "^

        "     [REAL funterm, REAL zzz]);                                     "^

        "   (pbz_eq::bool) = Take (X'_z = pbz);                              "^ (*([3], Frm), ?X' z = 4 / (z - 1 / 2) + -4 / (z - -1 / 4)*)

        "   pbz_eq = Rewrite ''ruleYZ'' False pbz_eq;                        "^ (*([3], Res), ?X' z = 4 * (?z / (z - 1 / 2)) + -4 * (?z / (z - -1 / 4))*)

        "   (X_zeq::bool) = Take (X_z = rhs pbz_eq);                         "^ (*([4], Frm), X_z = 4 * (z / (z - 1 / 2)) + -4 * (z / (z - -1 / 4))*)

        "   n_eq = (Rewrite_Set ''inverse_z'' False) X_zeq                   "^ (*([4], Res), X_z = 4 * (1 / 2) ^^^ ?n * ?u [?n] + -4 * (-1 / 4) ^^^ ?n * ?u [?n]*)

        " in n_eq)                                                           "*))](*            X_z = 4 * (1 / 2) ^^^ n * u [n] + -4 * (-1 / 4) ^^^ n * u [n]*)

\<close>

ML \<open>

\<close> ML \<open>

\<close>

end