1 (* Title: Inverse_Z_Transform
3 (c) copyright due to lincense terms.
6 theory Inverse_Z_Transform imports PolyEq DiffApp Partial_Fractions begin
8 axiomatization where \<comment> \<open>TODO: new variables on the rhs enforce replacement by substitution\<close>
9 rule1: "1 = \<delta>[n]" and
10 rule2: "|| z || > 1 ==> z / (z - 1) = u [n]" and
11 rule3: "|| z || < 1 ==> z / (z - 1) = -u [-n - 1]" and
12 rule4: "c * (z / (z - \<alpha>)) = c * \<alpha>^^^n * u [n]" and
13 rule5: "|| z || < || \<alpha> || ==> z / (z - \<alpha>) = -(\<alpha>^^^n) * u [-n - 1]" and
14 rule6: "|| z || > 1 ==> z/(z - 1)^^^2 = n * u [n]" (*and
15 rule42: "(a * (z/(z-b)) + c * (z/(z-d))) = (a * b^^^n * u [n] + c * d^^^n * u [n])"*)
18 (*ruleZY: "(X z = a / b) = (d_d z X = a / (z * b))" ..looks better, but types are flawed*)
19 ruleZY: "(X z = a / b) = (X' z = a / (z * b))" and
20 ruleYZ: "a / (z - b) + c / (z - d) = a * (z / (z - b)) + c * (z / (z - d))" and
21 ruleYZa: "(a / b + c / d) = (a * (z / b) + c * (z / d))" \<comment> \<open>that is what students learn\<close>
23 subsection\<open>Define the Field Descriptions for the specification\<close>
25 filterExpression :: "bool => una"
26 stepResponse :: "bool => una" \<comment> \<open>TODO: unused, "u [n]" is introduced by rule1..6 above\<close>
29 val inverse_z = prep_rls'(
30 Rule.Rls {id = "inverse_z", preconds = [], rew_ord = ("dummy_ord",Rule.dummy_ord),
31 erls = Rule.Erls, srls = Rule.Erls, calc = [], errpatts = [],
34 Rule.Thm ("rule4", @{thm rule4})
36 scr = Rule.EmptyScr});
40 text \<open>store the rule set for math engine\<close>
42 setup \<open>KEStore_Elems.add_rlss [("inverse_z", (Context.theory_name @{theory}, inverse_z))]\<close>
44 subsection\<open>Define the Specification\<close>
48 setup \<open>KEStore_Elems.add_pbts
49 [(Specify.prep_pbt thy "pbl_SP" [] Celem.e_pblID (["SignalProcessing"], [], Rule.e_rls, NONE, [])),
50 (Specify.prep_pbt thy "pbl_SP_Ztrans" [] Celem.e_pblID
51 (["Z_Transform","SignalProcessing"], [], Rule.e_rls, NONE, [])),
52 (Specify.prep_pbt thy "pbl_SP_Ztrans_inv" [] Celem.e_pblID
53 (["Inverse", "Z_Transform", "SignalProcessing"],
54 [("#Given" , ["filterExpression X_eq"]),
55 ("#Find" ,["stepResponse n_eq"])], \<comment> \<open>TODO: unused, "u [n]" is introduced by rule1..6\<close>
56 Rule.append_rls "e_rls" Rule.e_rls [(*for preds in where_*)], NONE,
57 [["SignalProcessing","Z_Transform","Inverse"]]))]\<close>
59 subsection \<open>Setup Parent Nodes in Hierarchy of Method\<close>
60 ML \<open>val thy = @{theory}; (*latest version of thy required*)\<close>
61 setup \<open>KEStore_Elems.add_mets
62 [Specify.prep_met thy "met_SP" [] Celem.e_metID
63 (["SignalProcessing"], [],
64 {rew_ord'="tless_true", rls'= Rule.e_rls, calc = [], srls = Rule.e_rls, prls = Rule.e_rls, crls = Rule.e_rls,
65 errpats = [], nrls = Rule.e_rls}, @{thm refl}),
66 Specify.prep_met thy "met_SP_Ztrans" [] Celem.e_metID
67 (["SignalProcessing", "Z_Transform"], [],
68 {rew_ord'="tless_true", rls'= Rule.e_rls, calc = [], srls = Rule.e_rls, prls = Rule.e_rls, crls = Rule.e_rls,
69 errpats = [], nrls = Rule.e_rls}, @{thm refl})]
72 partial_function (tailrec) inverse_ztransform :: "bool \<Rightarrow> real \<Rightarrow> bool"
74 "inverse_ztransform X_eq X_z = \<comment> \<open>(1/z) instead of z ^^^ -1\<close>
76 X' = Rewrite ''ruleZY'' False X; \<comment> \<open>z * denominator\<close>
77 X' = (Rewrite_Set ''norm_Rational'' False) X'; \<comment> \<open>simplify\<close>
78 funterm = Take (rhs X'); \<comment> \<open>drop X' z = for equation solving\<close>
79 denom = (Rewrite_Set ''partial_fraction'' False) funterm; \<comment> \<open>get_denominator\<close>
80 equ = (denom = (0::real));
81 fun_arg = Take (lhs X');
82 arg = (Rewrite_Set ''partial_fraction'' False) X'; \<comment> \<open>get_argument TODO\<close>
83 (L_L::bool list) = \<comment> \<open>'bool list' inhibits (?!?):
84 WARNING: Additional type variable(s) in specification of inverse_ztransform: 'a\<close>
85 SubProblem (''Test'', [''LINEAR'',''univariate'',''equation'',''test''],
86 [''Test'',''solve_linear'']) [BOOL equ, REAL X_z]
88 setup \<open>KEStore_Elems.add_mets
89 [Specify.prep_met thy "met_SP_Ztrans_inv" [] Celem.e_metID
90 (["SignalProcessing", "Z_Transform", "Inverse"],
91 [("#Given" ,["filterExpression X_eq", "functionName X_z"]),
92 ("#Find" ,["stepResponse n_eq"])], \<comment> \<open>TODO: unused, "u [n]" is introduced by rule1..6\<close>
93 {rew_ord'="tless_true", rls'= Rule.e_rls, calc = [], srls = Rule.e_rls, prls = Rule.e_rls, crls = Rule.e_rls,
94 errpats = [], nrls = Rule.e_rls},
95 @{thm inverse_ztransform.simps})]
98 partial_function (tailrec) inverse_ztransform2 :: "bool \<Rightarrow> real \<Rightarrow> bool"
100 "inverse_ztransform2 X_eq X_z =
102 X' = Rewrite ''ruleZY'' False X;
104 zzz = argument_in X'_z;
106 pbz = SubProblem (''Isac'',
107 [''partial_fraction'',''rational'',''simplification''],
108 [''simplification'',''of_rationals'',''to_partial_fraction''])
109 [REAL funterm, REAL zzz];
110 pbz_eq = Take (X'_z = pbz);
111 pbz_eq = Rewrite ''ruleYZ'' False pbz_eq;
112 X_zeq = Take (X_z = rhs pbz_eq);
113 n_eq = (Rewrite_Set ''inverse_z'' False) X_zeq
115 setup \<open>KEStore_Elems.add_mets
116 [Specify.prep_met thy "met_SP_Ztrans_inv_sub" [] Celem.e_metID
117 (["SignalProcessing", "Z_Transform", "Inverse_sub"],
118 [("#Given" ,["filterExpression X_eq", "functionName X_z"]),
119 ("#Find" ,["stepResponse n_eq"])], \<comment> \<open>TODO: unused, "u [n]" is introduced by rule1..6\<close>
120 {rew_ord'="tless_true", rls'= Rule.e_rls, calc = [],
121 srls = Rule.Rls {id="srls_partial_fraction",
122 preconds = [], rew_ord = ("termlessI",termlessI),
123 erls = Rule.append_rls "erls_in_srls_partial_fraction" Rule.e_rls
124 [(*for asm in NTH_CONS ...*)
125 Rule.Calc ("Orderings.ord_class.less",eval_equ "#less_"),
126 (*2nd NTH_CONS pushes n+-1 into asms*)
127 Rule.Calc("Groups.plus_class.plus", eval_binop "#add_")],
128 srls = Rule.Erls, calc = [], errpatts = [],
129 rules = [Rule.Thm ("NTH_CONS", @{thm NTH_CONS}),
130 Rule.Calc ("Groups.plus_class.plus", eval_binop "#add_"),
131 Rule.Thm ("NTH_NIL", @{thm NTH_NIL}),
132 Rule.Calc ("Tools.lhs", Tools.eval_lhs "eval_lhs_"),
133 Rule.Calc ("Tools.rhs", Tools.eval_rhs"eval_rhs_"),
134 Rule.Calc ("Atools.argument'_in", eval_argument_in "Atools.argument'_in"),
135 Rule.Calc ("Rational.get_denominator", eval_get_denominator "#get_denominator"),
136 Rule.Calc ("Rational.get_numerator", eval_get_numerator "#get_numerator"),
137 Rule.Calc ("Partial_Fractions.factors_from_solution",
138 eval_factors_from_solution "#factors_from_solution")
139 ], scr = Rule.EmptyScr},
140 prls = Rule.e_rls, crls = Rule.e_rls, errpats = [], nrls = norm_Rational},
141 @{thm inverse_ztransform2.simps})]