1 (* Title: Test_Z_Transform
3 (c) copyright due to lincense terms.
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8 theory Inverse_Z_Transform imports PolyEq DiffApp Partial_Fractions begin
11 rule1: "1 = \<delta>[n]" and
12 rule2: "|| z || > 1 ==> z / (z - 1) = u [n]" and
13 rule3: "|| z || < 1 ==> z / (z - 1) = -u [-n - 1]" and
14 rule4: "c * (z / (z - \<alpha>)) = c * \<alpha>^^^n * u [n]" and
15 rule5: "|| z || < || \<alpha> || ==> z / (z - \<alpha>) = -(\<alpha>^^^n) * u [-n - 1]" and
16 rule6: "|| z || > 1 ==> z/(z - 1)^^^2 = n * u [n]" (*and
17 rule42: "(a * (z/(z-b)) + c * (z/(z-d))) = (a * b^^^n * u [n] + c * d^^^n * u [n])"*)
20 ruleZY: "(X z = a / b) = (X' z = a / (z * b))" and
21 ruleYZ: "(a/b + c/d) = (a*(z/b) + c*(z/d))"
23 subsection{*Define the Field Descriptions for the specification*}
25 filterExpression :: "bool => una"
26 stepResponse :: "bool => una"
30 val inverse_z = prep_rls(
31 Rls {id = "inverse_z", preconds = [], rew_ord = ("dummy_ord",dummy_ord),
32 erls = Erls, srls = Erls, calc = [], errpatts = [],
35 Thm ("rule4",num_str @{thm rule4})
41 text {*store the rule set for math engine*}
44 ruleset' := overwritelthy @{theory} (!ruleset',
45 [("inverse_z", inverse_z)
49 subsection{*Define the Specification*}
54 (prep_pbt thy "pbl_SP" [] e_pblID
55 (["SignalProcessing"], [], e_rls, NONE, []));
57 (prep_pbt thy "pbl_SP_Ztrans" [] e_pblID
58 (["Z_Transform","SignalProcessing"], [], e_rls, NONE, []));
60 (prep_pbt thy "pbl_SP_Ztrans_inv" [] e_pblID
61 (["Inverse", "Z_Transform", "SignalProcessing"],
62 (*^ capital letter breaks coding standard
63 because "inverse" = Const ("Rings.inverse_class.inverse", ..*)
64 [("#Given" ,["filterExpression (X_eq::bool)"]),
65 ("#Find" ,["stepResponse (n_eq::bool)"])
67 append_rls "e_rls" e_rls [(*for preds in where_*)], NONE,
68 [["SignalProcessing","Z_Transform","Inverse"]]));
72 (prep_pbt thy "pbl_SP_Ztrans_inv" [] e_pblID
73 (["Inverse", "Z_Transform", "SignalProcessing"],
74 [("#Given" ,["filterExpression X_eq"]),
75 ("#Find" ,["stepResponse n_eq"])
77 append_rls "e_rls" e_rls [(*for preds in where_*)], NONE,
78 [["SignalProcessing","Z_Transform","Inverse"]]));
81 subsection {*Define Name and Signature for the Method*}
83 InverseZTransform :: "[bool, bool] => bool"
84 ("((Script InverseZTransform (_ =))// (_))" 9)
86 subsection {*Setup Parent Nodes in Hierarchy of Method*}
89 (prep_met thy "met_SP" [] e_metID
90 (["SignalProcessing"], [],
91 {rew_ord'="tless_true", rls'= e_rls, calc = [], srls = e_rls, prls = e_rls,
92 crls = e_rls, errpats = [], nrls = e_rls}, "empty_script"));
94 (prep_met thy "met_SP_Ztrans" [] e_metID
95 (["SignalProcessing", "Z_Transform"], [],
96 {rew_ord'="tless_true", rls'= e_rls, calc = [], srls = e_rls, prls = e_rls,
97 crls = e_rls, errpats = [], nrls = e_rls}, "empty_script"));
98 val thy = @{theory}; (*latest version of thy required*)
100 (prep_met thy "met_SP_Ztrans_inv" [] e_metID
101 (["SignalProcessing", "Z_Transform", "Inverse"],
102 [("#Given" ,["filterExpression (X_eq::bool)"]),
103 ("#Find" ,["stepResponse (n_eq::bool)"])
105 {rew_ord'="tless_true", rls'= e_rls, calc = [], srls = e_rls, prls = e_rls,
106 crls = e_rls, errpats = [], nrls = e_rls},
107 "Script InverseZTransform (X_eq::bool) =" ^ (*(1/z) instead of z ^^^ -1*)
108 " (let X = Take X_eq;" ^
109 " X' = Rewrite ruleZY False X;" ^ (*z * denominator*)
110 " X' = (Rewrite_Set norm_Rational False) X';" ^ (*simplify*)
111 " funterm = Take (rhs X');" ^ (*drop X' z = for equation solving*)
112 " denom = (Rewrite_Set partial_fraction False) funterm;" ^ (*get_denominator*)
113 " equ = (denom = (0::real));" ^
114 " fun_arg = Take (lhs X');" ^
115 " arg = (Rewrite_Set partial_fraction False) X';" ^ (*get_argument TODO*)
116 " (L_L::bool list) = " ^
117 " (SubProblem (Test', " ^
118 " [linear,univariate,equation,test]," ^
119 " [Test,solve_linear]) " ^
120 " [BOOL equ, REAL z]) " ^
126 prep_met thy "met_SP_Ztrans_inv" [] e_metID
127 (["SignalProcessing",
131 ("#Given" ,["filterExpression X_eq"]),
132 ("#Find" ,["stepResponse n_eq"])
135 rew_ord'="tless_true",
136 rls'= e_rls, calc = [],
137 srls = srls_partial_fraction,
139 crls = e_rls, errpats = [], nrls = e_rls
141 "Script InverseZTransform (X_eq::bool) = "^
142 (*(1/z) instead of z ^^^ -1*)
143 "(let X = Take X_eq; "^
144 " X' = Rewrite ruleZY False X; "^
146 " (num_orig::real) = get_numerator (rhs X'); "^
147 " X' = (Rewrite_Set norm_Rational False) X'; "^
149 " (X'_z::real) = lhs X'; "^
150 " (zzz::real) = argument_in X'_z; "^
151 " (funterm::real) = rhs X'; "^
152 (*drop X' z = for equation solving*)
153 " (denom::real) = get_denominator funterm; "^
155 " (num::real) = get_numerator funterm; "^
157 " (equ::bool) = (denom = (0::real)); "^
158 " (L_L::bool list) = (SubProblem (PolyEq', "^
159 " [abcFormula,degree_2,polynomial,univariate,equation], "^
161 " [BOOL equ, REAL zzz]); "^
162 " (facs::real) = factors_from_solution L_L; "^
163 " (eql::real) = Take (num_orig / facs); "^
165 " (eqr::real) = (Try (Rewrite_Set ansatz_rls False)) eql; "^
167 " (eq::bool) = Take (eql = eqr); "^
168 (*Maybe possible to use HOLogic.mk_eq ??*)
169 " eq = (Try (Rewrite_Set equival_trans False)) eq; "^
171 " eq = drop_questionmarks eq; "^
172 " (z1::real) = (rhs (NTH 1 L_L)); "^
174 * prepare equation for a - eq_a
175 * therefor substitute z with solution 1 - z1
177 " (z2::real) = (rhs (NTH 2 L_L)); "^
179 " (eq_a::bool) = Take eq; "^
180 " eq_a = (Substitute [zzz=z1]) eq; "^
181 " eq_a = (Rewrite_Set norm_Rational False) eq_a; "^
182 " (sol_a::bool list) = "^
183 " (SubProblem (Isac', "^
184 " [univariate,equation],[no_met]) "^
185 " [BOOL eq_a, REAL (A::real)]); "^
186 " (a::real) = (rhs(NTH 1 sol_a)); "^
188 " (eq_b::bool) = Take eq; "^
189 " eq_b = (Substitute [zzz=z2]) eq_b; "^
190 " eq_b = (Rewrite_Set norm_Rational False) eq_b; "^
191 " (sol_b::bool list) = "^
192 " (SubProblem (Isac', "^
193 " [univariate,equation],[no_met]) "^
194 " [BOOL eq_b, REAL (B::real)]); "^
195 " (b::real) = (rhs(NTH 1 sol_b)); "^
197 " eqr = drop_questionmarks eqr; "^
198 " (pbz::real) = Take eqr; "^
199 " pbz = ((Substitute [A=a, B=b]) pbz); "^
201 " pbz = Rewrite ruleYZ False pbz; "^
202 " pbz = drop_questionmarks pbz; "^
204 " (X_z::bool) = Take (X_z = pbz); "^
205 " (n_eq::bool) = (Rewrite_Set inverse_z False) X_z; "^
206 " n_eq = drop_questionmarks n_eq "^
211 store_met (prep_met thy "met_SP_Ztrans_inv_sub" [] e_metID
212 (["SignalProcessing", "Z_Transform", "Inverse_sub"],
213 [("#Given" ,["filterExpression X_eq"]),
214 ("#Find" ,["stepResponse n_eq"])],
215 {rew_ord'="tless_true",
216 rls'= e_rls, calc = [],
217 srls = Rls {id="srls_partial_fraction",
219 rew_ord = ("termlessI",termlessI),
220 erls = append_rls "erls_in_srls_partial_fraction" e_rls
221 [(*for asm in NTH_CONS ...*)
222 Calc ("Orderings.ord_class.less",eval_equ "#less_"),
223 (*2nd NTH_CONS pushes n+-1 into asms*)
224 Calc("Groups.plus_class.plus", eval_binop "#add_")],
225 srls = Erls, calc = [], errpatts = [],
227 Thm ("NTH_CONS",num_str @{thm NTH_CONS}),
228 Calc("Groups.plus_class.plus", eval_binop "#add_"),
229 Thm ("NTH_NIL",num_str @{thm NTH_NIL}),
230 Calc("Tools.lhs", eval_lhs "eval_lhs_"),
231 Calc("Tools.rhs", eval_rhs"eval_rhs_"),
232 Calc("Atools.argument'_in", eval_argument_in "Atools.argument'_in"),
233 Calc("Rational.get_denominator", eval_get_denominator "#get_denominator"),
234 Calc("Rational.get_numerator", eval_get_numerator "#get_numerator"),
235 Calc("Partial_Fractions.factors_from_solution",
236 eval_factors_from_solution "#factors_from_solution"),
237 Calc("Partial_Fractions.drop_questionmarks", eval_drop_questionmarks "#drop_?")],
239 prls = e_rls, crls = e_rls, errpats = [], nrls = norm_Rational},
240 "Script InverseZTransform (X_eq::bool) = "^(*([], Frm), Problem (Isac, [Inverse, Z_Transform, SignalProcessing])*)
241 "(let X = Take X_eq; "^(*([1], Frm), X z = 3 / (z - 1 / 4 + -1 / 8 * (1 / z))*)
242 " X' = Rewrite ruleZY False X; "^(*([1], Res), ?X' z = 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))*)
243 " (X'_z::real) = lhs X'; "^(* ?X' z*)
244 " (zzz::real) = argument_in X'_z; "^(* z *)
245 " (funterm::real) = rhs X'; "^(* 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))*)
247 " (pbz::real) = (SubProblem (Isac', "^(**)
248 " [partial_fraction,rational,simplification], "^
249 " [simplification,of_rationals,to_partial_fraction]) "^
250 " [REAL funterm, REAL zzz]); "^(*([2], Res), 4 / (z - 1 / 2) + -4 / (z - -1 / 4)*)
252 " (pbz_eq::bool) = Take (X'_z = pbz); "^(*([3], Frm), ?X' z = 4 / (z - 1 / 2) + -4 / (z - -1 / 4)*)
253 " pbz_eq = Rewrite ruleYZ False pbz_eq; "^(*([3], Res), ?X' z = 4 * (?z / (z - 1 / 2)) + -4 * (?z / (z - -1 / 4))*)
254 " pbz_eq = drop_questionmarks pbz_eq; "^(* 4 * (z / (z - 1 / 2)) + -4 * (z / (z - -1 / 4))*)
255 " (X_zeq::bool) = Take (X_z = rhs pbz_eq); "^(*([4], Frm), X_z = 4 * (z / (z - 1 / 2)) + -4 * (z / (z - -1 / 4))*)
256 " 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]*)
257 " n_eq = drop_questionmarks n_eq "^(* X_z = 4 * (1 / 2) ^^^ n * u [n] + -4 * (-1 / 4) ^^^ n * u [n]*)
258 "in n_eq)" (*([], Res), X_z = 4 * (1 / 2) ^^^ n * u [n] + -4 * (-1 / 4) ^^^ n * u [n]*)