
neuper@42256: (* Title:  Test_Z_Transform
neuper@42256:    Author: Jan Rocnik
neuper@42256:    (c) copyright due to lincense terms.
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neuper@42256: *)
neuper@42256: 
neuper@42290: theory Inverse_Z_Transform imports PolyEq DiffApp Partial_Fractions begin
neuper@42256: 
neuper@42256: axiomatization where 
neuper@42256:   rule1: "1 = \<delta>[n]" and
neuper@42256:   rule2: "|| z || > 1 ==> z / (z - 1) = u [n]" and
neuper@42256:   rule3: "|| z || < 1 ==> z / (z - 1) = -u [-n - 1]" and 
jan@42367:   rule4: "c * (z / (z - \<alpha>)) = c * \<alpha>^^^n * u [n]" and
neuper@42256:   rule5: "|| z || < || \<alpha> || ==> z / (z - \<alpha>) = -(\<alpha>^^^n) * u [-n - 1]" and
jan@42367:   rule6: "|| z || > 1 ==> z/(z - 1)^^^2 = n * u [n]" (*and
jan@42367:   rule42: "(a * (z/(z-b)) + c * (z/(z-d))) = (a * b^^^n * u [n] + c * d^^^n * u [n])"*)
neuper@42256: 
neuper@42256: axiomatization where
jan@42367:   ruleZY: "(X z = a / b) = (X' z = a / (z * b))" and
jan@42367:   ruleYZ: "(a/b + c/d) = (a*(z/b) + c*(z/d))" 
neuper@42256: 
neuper@42279: subsection{*Define the Field Descriptions for the specification*}
neuper@42279: consts
neuper@42279:   filterExpression  :: "bool => una"
neuper@42279:   stepResponse      :: "bool => una"
neuper@42279: 
jan@42366: 
jan@42366: ML {*
jan@42366: val inverse_z = prep_rls(
jan@42366:   Rls {id = "inverse_z", preconds = [], rew_ord = ("dummy_ord",dummy_ord), 
neuper@42451: 	  erls = Erls, srls = Erls, calc = [], errpatts = [],
jan@42366: 	  rules = 
jan@42367: 	   [
jan@42367:     Thm ("rule4",num_str @{thm rule4})
jan@42366: 	   ], 
jan@42366: 	 scr = EmptyScr}:rls);
jan@42366: *}
jan@42366: 
jan@42366: 
jan@42366: text {*store the rule set for math engine*}
jan@42366: 
neuper@52125: setup {* KEStore_Elems.add_rlss [("inverse_z", (Context.theory_name @{theory}, inverse_z))] *}
jan@42366: 
neuper@42256: subsection{*Define the Specification*}
neuper@42256: ML {*
neuper@42256: val thy = @{theory};
neuper@42412: *}
s1210629013@55359: setup {* KEStore_Elems.add_pbts
s1210629013@55339:   [(prep_pbt thy "pbl_SP" [] e_pblID (["SignalProcessing"], [], e_rls, NONE, [])),
s1210629013@55339:     (prep_pbt thy "pbl_SP_Ztrans" [] e_pblID
s1210629013@55339:       (["Z_Transform","SignalProcessing"], [], e_rls, NONE, [])),
s1210629013@55339:     (prep_pbt thy "pbl_SP_Ztrans_inv" [] e_pblID
s1210629013@55339:       (["Inverse", "Z_Transform", "SignalProcessing"],
s1210629013@55339:         (*^ capital letter breaks coding standard
s1210629013@55339:           because "inverse" = Const ("Rings.inverse_class.inverse", ..*)
s1210629013@55339:         [("#Given" ,["filterExpression (X_eq::bool)"]),
s1210629013@55339:           ("#Find"  ,["stepResponse (n_eq::bool)"])],
s1210629013@55339:         append_rls "e_rls" e_rls [(*for preds in where_*)], NONE, 
s1210629013@55339:         [["SignalProcessing","Z_Transform","Inverse"]])),
s1210629013@55339:     (prep_pbt thy "pbl_SP_Ztrans_inv" [] e_pblID
s1210629013@55339:       (["Inverse", "Z_Transform", "SignalProcessing"],
s1210629013@55339:         [("#Given" ,["filterExpression X_eq"]),
s1210629013@55339:           ("#Find"  ,["stepResponse n_eq"])],
s1210629013@55339:         append_rls "e_rls" e_rls [(*for preds in where_*)], NONE, 
s1210629013@55339:         [["SignalProcessing","Z_Transform","Inverse"]]))] *}
neuper@42256: 
neuper@42256: subsection {*Define Name and Signature for the Method*}
neuper@42256: consts
neuper@42256:   InverseZTransform :: "[bool, bool] => bool"
neuper@42256:     ("((Script InverseZTransform (_ =))// (_))" 9)
neuper@42256: 
neuper@42277: subsection {*Setup Parent Nodes in Hierarchy of Method*}
neuper@42256: ML {*
neuper@42256: store_met
neuper@42256:  (prep_met thy "met_SP" [] e_metID
neuper@42256:  (["SignalProcessing"], [],
neuper@42256:    {rew_ord'="tless_true", rls'= e_rls, calc = [], srls = e_rls, prls = e_rls,
neuper@42425:     crls = e_rls, errpats = [], nrls = e_rls}, "empty_script"));
neuper@42256: store_met
neuper@42256:  (prep_met thy "met_SP_Ztrans" [] e_metID
neuper@42256:  (["SignalProcessing", "Z_Transform"], [],
neuper@42256:    {rew_ord'="tless_true", rls'= e_rls, calc = [], srls = e_rls, prls = e_rls,
neuper@42425:     crls = e_rls, errpats = [], nrls = e_rls}, "empty_script"));
neuper@42256: val thy = @{theory}; (*latest version of thy required*)
neuper@42256: store_met
neuper@42256:  (prep_met thy "met_SP_Ztrans_inv" [] e_metID
neuper@42405:  (["SignalProcessing", "Z_Transform", "Inverse"], 
neuper@42278:   [("#Given" ,["filterExpression (X_eq::bool)"]),
neuper@42278:    ("#Find"  ,["stepResponse (n_eq::bool)"])
neuper@42256:   ],
neuper@42256:    {rew_ord'="tless_true", rls'= e_rls, calc = [], srls = e_rls, prls = e_rls,
neuper@42425:     crls = e_rls, errpats = [], nrls = e_rls},
neuper@42281: "Script InverseZTransform (X_eq::bool) =" ^ (*(1/z) instead of z ^^^ -1*)
neuper@42281: " (let X = Take X_eq;" ^
neuper@42281: "      X' = Rewrite ruleZY False X;" ^ (*z * denominator*)
neuper@42281: "      X' = (Rewrite_Set norm_Rational False) X';" ^ (*simplify*)
neuper@42290: "      funterm = Take (rhs X');" ^ (*drop X' z = for equation solving*)
neuper@42290: "      denom = (Rewrite_Set partial_fraction False) funterm;" ^ (*get_denominator*)
neuper@42290: "      equ = (denom = (0::real));" ^
neuper@42290: "      fun_arg = Take (lhs X');" ^
neuper@42290: "      arg = (Rewrite_Set partial_fraction False) X';" ^ (*get_argument TODO*)
neuper@42290: "      (L_L::bool list) =                                    " ^
neuper@42290: "            (SubProblem (Test',                            " ^
neuper@55276: "                         [LINEAR,univariate,equation,test]," ^
neuper@42290: "                         [Test,solve_linear])              " ^
neuper@42290: "                        [BOOL equ, REAL z])              " ^
neuper@42281: "  in X)"
neuper@42256:  ));
neuper@42256: *}
neuper@42412: ML {*
neuper@42412:   store_met(
neuper@42412:     prep_met thy "met_SP_Ztrans_inv" [] e_metID
neuper@42412:     (["SignalProcessing",
neuper@42412:       "Z_Transform",
neuper@42412:       "Inverse"], 
neuper@42412:      [
neuper@42412:        ("#Given" ,["filterExpression X_eq"]),
neuper@42412:        ("#Find"  ,["stepResponse n_eq"])
neuper@42412:      ],
neuper@42412:      {
neuper@42412:        rew_ord'="tless_true",
neuper@42412:        rls'= e_rls, calc = [],
neuper@42413:        srls = srls_partial_fraction, 
neuper@42412:        prls = e_rls,
neuper@42425:        crls = e_rls, errpats = [], nrls = e_rls
neuper@42412:      },
neuper@42412:      "Script InverseZTransform (X_eq::bool) =                        "^
neuper@42412:      (*(1/z) instead of z ^^^ -1*)
neuper@42412:      "(let X = Take X_eq;                                            "^
neuper@42412:      "      X' = Rewrite ruleZY False X;                             "^
neuper@42412:      (*z * denominator*)
neuper@42412:      "      (num_orig::real) = get_numerator (rhs X');               "^
neuper@42412:      "      X' = (Rewrite_Set norm_Rational False) X';               "^
neuper@42412:      (*simplify*)
neuper@42412:      "      (X'_z::real) = lhs X';                                   "^
neuper@42412:      "      (zzz::real) = argument_in X'_z;                          "^
neuper@42412:      "      (funterm::real) = rhs X';                                "^
neuper@42412:      (*drop X' z = for equation solving*)
neuper@42412:      "      (denom::real) = get_denominator funterm;                 "^
neuper@42412:      (*get_denominator*)
neuper@42412:      "      (num::real) = get_numerator funterm;                     "^
neuper@42412:      (*get_numerator*)
neuper@42412:      "      (equ::bool) = (denom = (0::real));                       "^
neuper@42412:      "      (L_L::bool list) = (SubProblem (PolyEq',                 "^
neuper@42412:      "         [abcFormula,degree_2,polynomial,univariate,equation], "^
neuper@42412:      "         [no_met])                                             "^
neuper@42412:      "         [BOOL equ, REAL zzz]);                                "^
neuper@42412:      "      (facs::real) = factors_from_solution L_L;                "^
neuper@42412:      "      (eql::real) = Take (num_orig / facs);                    "^ 
neuper@42412: 
neuper@42412:      "      (eqr::real) = (Try (Rewrite_Set ansatz_rls False)) eql;  "^
neuper@42412: 
neuper@42412:      "      (eq::bool) = Take (eql = eqr);                           "^
neuper@42412:      (*Maybe possible to use HOLogic.mk_eq ??*)
neuper@42412:      "      eq = (Try (Rewrite_Set equival_trans False)) eq;         "^ 
neuper@42412: 
neuper@42412:      "      eq = drop_questionmarks eq;                              "^
neuper@42412:      "      (z1::real) = (rhs (NTH 1 L_L));                          "^
neuper@42412:      (* 
neuper@42412:       * prepare equation for a - eq_a
neuper@42412:       * therefor substitute z with solution 1 - z1
neuper@42412:       *)
neuper@42412:      "      (z2::real) = (rhs (NTH 2 L_L));                          "^
neuper@42412:  
neuper@42412:      "      (eq_a::bool) = Take eq;                                  "^
neuper@42412:      "      eq_a = (Substitute [zzz=z1]) eq;                         "^
neuper@42412:      "      eq_a = (Rewrite_Set norm_Rational False) eq_a;           "^
neuper@42412:      "      (sol_a::bool list) =                                     "^
neuper@42412:      "                 (SubProblem (Isac',                           "^
neuper@42412:      "                              [univariate,equation],[no_met])  "^
neuper@42412:      "                              [BOOL eq_a, REAL (A::real)]);    "^
neuper@42412:      "      (a::real) = (rhs(NTH 1 sol_a));                          "^
neuper@42412: 
neuper@42412:      "      (eq_b::bool) = Take eq;                                  "^
neuper@42412:      "      eq_b =  (Substitute [zzz=z2]) eq_b;                      "^
neuper@42412:      "      eq_b = (Rewrite_Set norm_Rational False) eq_b;           "^
neuper@42412:      "      (sol_b::bool list) =                                     "^
neuper@42412:      "                 (SubProblem (Isac',                           "^
neuper@42412:      "                              [univariate,equation],[no_met])  "^
neuper@42412:      "                              [BOOL eq_b, REAL (B::real)]);    "^
neuper@42412:      "      (b::real) = (rhs(NTH 1 sol_b));                          "^
neuper@42412: 
neuper@42412:      "      eqr = drop_questionmarks eqr;                            "^
neuper@42412:      "      (pbz::real) = Take eqr;                                  "^
neuper@42412:      "      pbz = ((Substitute [A=a, B=b]) pbz);                     "^
neuper@42412: 
neuper@42412:      "      pbz = Rewrite ruleYZ False pbz;                          "^
neuper@42412:      "      pbz = drop_questionmarks pbz;                            "^
neuper@42412: 
neuper@42412:      "      (X_z::bool) = Take (X_z = pbz);                          "^
neuper@42412:      "      (n_eq::bool) = (Rewrite_Set inverse_z False) X_z;        "^
neuper@42412:      "      n_eq = drop_questionmarks n_eq                           "^
neuper@42412:      "in n_eq)" 
neuper@42412:     )
neuper@42412:            );
neuper@42417: 
neuper@42417: store_met (prep_met thy "met_SP_Ztrans_inv_sub" [] e_metID
neuper@42417:   (["SignalProcessing", "Z_Transform", "Inverse_sub"], 
neuper@42417:    [("#Given" ,["filterExpression X_eq"]),
neuper@42417:     ("#Find"  ,["stepResponse n_eq"])],
neuper@42417:    {rew_ord'="tless_true",
neuper@42417:     rls'= e_rls, calc = [],
neuper@42417:     srls = Rls {id="srls_partial_fraction", 
neuper@42417:       preconds = [],
neuper@42417:       rew_ord = ("termlessI",termlessI),
neuper@42417:       erls = append_rls "erls_in_srls_partial_fraction" e_rls
neuper@42417:        [(*for asm in NTH_CONS ...*)
neuper@42417:         Calc ("Orderings.ord_class.less",eval_equ "#less_"),
neuper@42417:         (*2nd NTH_CONS pushes n+-1 into asms*)
neuper@42417:         Calc("Groups.plus_class.plus", eval_binop "#add_")], 
neuper@42451:         srls = Erls, calc = [], errpatts = [],
neuper@42417:         rules = [
neuper@42417:           Thm ("NTH_CONS",num_str @{thm NTH_CONS}),
neuper@42417:           Calc("Groups.plus_class.plus", eval_binop "#add_"),
neuper@42417:           Thm ("NTH_NIL",num_str @{thm NTH_NIL}),
neuper@42417:           Calc("Tools.lhs", eval_lhs "eval_lhs_"),
neuper@42417:           Calc("Tools.rhs", eval_rhs"eval_rhs_"),
neuper@42417:           Calc("Atools.argument'_in", eval_argument_in "Atools.argument'_in"),
neuper@42417:           Calc("Rational.get_denominator", eval_get_denominator "#get_denominator"),
neuper@42417:           Calc("Rational.get_numerator", eval_get_numerator "#get_numerator"),
neuper@42417:           Calc("Partial_Fractions.factors_from_solution",
neuper@42417:             eval_factors_from_solution "#factors_from_solution"),
neuper@42417:           Calc("Partial_Fractions.drop_questionmarks", eval_drop_questionmarks "#drop_?")],
neuper@42417:           scr = EmptyScr},
neuper@42425:     prls = e_rls, crls = e_rls, errpats = [], nrls = norm_Rational},
neuper@42417:    "Script InverseZTransform (X_eq::bool) =            "^(*([], Frm), Problem (Isac, [Inverse, Z_Transform, SignalProcessing])*)
neuper@42417:    "(let X = Take X_eq;                                "^(*([1], Frm), X z = 3 / (z - 1 / 4 + -1 / 8 * (1 / z))*)
neuper@42417:    "  X' = Rewrite ruleZY False X;                     "^(*([1], Res), ?X' z = 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))*)
neuper@42417:    "  (X'_z::real) = lhs X';                           "^(*            ?X' z*)
neuper@42417:    "  (zzz::real) = argument_in X'_z;                  "^(*            z *)
neuper@42417:    "  (funterm::real) = rhs X';                        "^(*            3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))*)
neuper@42417: 
neuper@42417:    "  (pbz::real) = (SubProblem (Isac',                "^(**)
neuper@42417:    "    [partial_fraction,rational,simplification],    "^
neuper@42417:    "    [simplification,of_rationals,to_partial_fraction]) "^
neuper@42421:    "    [REAL funterm, REAL zzz]);                     "^(*([2], Res), 4 / (z - 1 / 2) + -4 / (z - -1 / 4)*)
neuper@42417: 
neuper@42421:    "  (pbz_eq::bool) = Take (X'_z = pbz);              "^(*([3], Frm), ?X' z = 4 / (z - 1 / 2) + -4 / (z - -1 / 4)*)
neuper@42421:    "  pbz_eq = Rewrite ruleYZ False pbz_eq;            "^(*([3], Res), ?X' z = 4 * (?z / (z - 1 / 2)) + -4 * (?z / (z - -1 / 4))*)
neuper@42421:    "  pbz_eq = drop_questionmarks pbz_eq;              "^(*               4 * (z / (z - 1 / 2)) + -4 * (z / (z - -1 / 4))*)
neuper@42421:    "  (X_zeq::bool) = Take (X_z = rhs pbz_eq);         "^(*([4], Frm), X_z = 4 * (z / (z - 1 / 2)) + -4 * (z / (z - -1 / 4))*)
neuper@42421:    "  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]*)
neuper@42421:    "  n_eq = drop_questionmarks n_eq                   "^(*            X_z = 4 * (1 / 2) ^^^ n * u [n] + -4 * (-1 / 4) ^^^ n * u [n]*)
neuper@42417:    "in n_eq)"                                            (*([], Res), X_z = 4 * (1 / 2) ^^^ n * u [n] + -4 * (-1 / 4) ^^^ n * u [n]*)
neuper@42417:   ));
neuper@42417: 
neuper@42412: *}
neuper@42256: 
neuper@42256: end
neuper@42256: 




