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> \<up> n * u [n]" and
13 rule5: "|| z || < || \<alpha> || ==> z / (z - \<alpha>) = -(\<alpha> \<up> n) * u [-n - 1]" and
14 rule6: "|| z || > 1 ==> z/(z - 1) \<up> 2 = n * u [n]" (*and
15 rule42: "(a * (z/(z-b)) + c * (z/(z-d))) = (a * b \<up> n * u [n] + c * d \<up> 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_Def.Repeat {id = "inverse_z", preconds = [], rew_ord = ("dummy_ord",Rewrite_Ord.dummy_ord),
31 erls = Rule_Set.Empty, srls = Rule_Set.Empty, calc = [], errpatts = [],
34 \<^rule_thm>\<open>rule4\<close>
36 scr = Rule.Empty_Prog});
40 text \<open>store the rule set for math engine\<close>
42 rule_set_knowledge inverse_z = inverse_z
44 subsection\<open>Define the Specification\<close>
46 problem pbl_SP : "SignalProcessing" = \<open>Rule_Set.empty\<close>
48 problem pbl_SP_Ztrans : "Z_Transform/SignalProcessing" = \<open>Rule_Set.empty\<close>
50 problem pbl_SP_Ztrans_inv : "Inverse/Z_Transform/SignalProcessing" =
51 \<open>Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)]\<close>
52 Method_Ref: "SignalProcessing/Z_Transform/Inverse"
53 Given: "filterExpression X_eq"
54 Find: "stepResponse n_eq" \<comment> \<open>TODO: unused, "u [n]" is introduced by rule1..6\<close>
57 subsection \<open>Setup Parent Nodes in Hierarchy of MethodC\<close>
59 method met_SP : "SignalProcessing" =
60 \<open>{rew_ord'="tless_true", rls'= Rule_Set.empty, calc = [], srls = Rule_Set.empty, prls = Rule_Set.empty, crls = Rule_Set.empty,
61 errpats = [], nrls = Rule_Set.empty}\<close>
63 method met_SP_Ztrans : "SignalProcessing/Z_Transform" =
64 \<open>{rew_ord'="tless_true", rls'= Rule_Set.empty, calc = [], srls = Rule_Set.empty, prls = Rule_Set.empty, crls = Rule_Set.empty,
65 errpats = [], nrls = Rule_Set.empty}\<close>
67 partial_function (tailrec) inverse_ztransform :: "bool \<Rightarrow> real \<Rightarrow> bool"
69 "inverse_ztransform X_eq X_z = (
72 X' = Rewrite ''ruleZY'' X; \<comment> \<open>z * denominator\<close>
73 X' = (Rewrite_Set ''norm_Rational'' ) X'; \<comment> \<open>simplify\<close>
74 funterm = Take (rhs X'); \<comment> \<open>drop X' z = for equation solving\<close>
75 denom = (Rewrite_Set ''partial_fraction'' ) funterm; \<comment> \<open>get_denominator\<close>
76 equ = (denom = (0::real));
77 fun_arg = Take (lhs X');
78 arg = (Rewrite_Set ''partial_fraction'' ) X'; \<comment> \<open>get_argument TODO\<close>
79 (L_L::bool list) = \<comment> \<open>'bool list' inhibits:
80 WARNING: Additional type variable(s) in specification of inverse_ztransform: 'a\<close>
81 SubProblem (''Test'', [''LINEAR'',''univariate'',''equation'',''test''], [''Test'',''solve_linear''])
85 method met_SP_Ztrans_inv : "SignalProcessing/Z_Transform/Inverse" =
86 \<open>{rew_ord'="tless_true", rls'= Rule_Set.empty, calc = [], srls = Rule_Set.empty, prls = Rule_Set.empty, crls = Rule_Set.empty,
87 errpats = [], nrls = Rule_Set.empty}\<close>
88 Program: inverse_ztransform.simps
89 Given: "filterExpression X_eq" "functionName X_z"
90 Find: "stepResponse n_eq" \<comment> \<open>TODO: unused, "u [n]" is introduced by rule1..6\<close>
92 partial_function (tailrec) inverse_ztransform2 :: "bool \<Rightarrow> real \<Rightarrow> bool"
94 "inverse_ztransform2 X_eq X_z = (
97 X' = Rewrite ''ruleZY'' X;
99 zzz = argument_in X_z;
101 pbz = SubProblem (''Isac_Knowledge'',
102 [''partial_fraction'',''rational'',''simplification''],
103 [''simplification'',''of_rationals'',''to_partial_fraction''])
104 [REAL funterm, REAL zzz];
105 pbz_eq = Take (X_z = pbz);
106 pbz_eq = Rewrite ''ruleYZ'' pbz_eq;
107 X_zeq = Take (X_z = rhs pbz_eq);
108 n_eq = (Rewrite_Set ''inverse_z'' ) X_zeq
111 method met_SP_Ztrans_inv_sub : "SignalProcessing/Z_Transform/Inverse_sub" =
112 \<open>{rew_ord'="tless_true", rls'= Rule_Set.empty, calc = [],
113 srls = Rule_Def.Repeat {
114 id = "srls_partial_fraction", preconds = [], rew_ord = ("termlessI",termlessI),
115 erls = Rule_Set.append_rules "erls_in_srls_partial_fraction" Rule_Set.empty [
116 \<^rule_eval>\<open>less\<close> (Prog_Expr.eval_equ "#less_"), (* ...for asm in NTH_CONS*)
117 (*2nd NTH_CONS pushes n+-1 into asms*)
118 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_")],
119 srls = Rule_Set.Empty, calc = [], errpatts = [],
120 rules = [\<^rule_thm>\<open>NTH_CONS\<close>,
121 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_"),
122 \<^rule_thm>\<open>NTH_NIL\<close>,
123 \<^rule_eval>\<open>Prog_Expr.lhs\<close> (Prog_Expr.eval_lhs "eval_lhs_"),
124 \<^rule_eval>\<open>Prog_Expr.rhs\<close> (Prog_Expr.eval_rhs"eval_rhs_"),
125 \<^rule_eval>\<open>Prog_Expr.argument_in\<close> (Prog_Expr.eval_argument_in "Prog_Expr.argument_in"),
126 \<^rule_eval>\<open>get_denominator\<close> (eval_get_denominator "#get_denominator"),
127 \<^rule_eval>\<open>get_numerator\<close> (eval_get_numerator "#get_numerator"),
128 \<^rule_eval>\<open>factors_from_solution\<close> (eval_factors_from_solution "#factors_from_solution")],
129 scr = Rule.Empty_Prog},
130 prls = Rule_Set.empty, crls = Rule_Set.empty, errpats = [], nrls = norm_Rational}\<close>
131 Program: inverse_ztransform2.simps
132 Given: "filterExpression X_eq" "functionName X_z"
133 Find: "stepResponse n_eq" \<comment> \<open>TODO: unused, "u [n]" is introduced by rule1..6\<close>