wneuper/isa: doc-src/isac/jrocnik/Test_Z_Transform.thy@6201b34bd323

     1 (* Title:  Test_Z_Transform

     2    Author: Jan Rocnik

     3    (c) copyright due to lincense terms.

     4 12345678901234567890123456789012345678901234567890123456789012345678901234567890

     5         10        20        30        40        50        60        70        80

     6 *)

     8 theory Test_Z_Transform imports Isac begin

    10 section {*trials towards Z transform *}

    11 text{*===============================*}

    12 subsection {*terms*}

    13 ML {*

    14 @{term "1 < || z ||"};

    15 @{term "z / (z - 1)"};

    16 @{term "-u -n - 1"};

    17 @{term "-u [-n - 1]"}; (*[ ] denotes lists !!!*)

    18 @{term "z /(z - 1) = -u [-n - 1]"};Isac

    19 @{term "1 < || z || ==> z / (z - 1) = -u [-n - 1]"};

    20 term2str @{term "1 < || z || ==> z / (z - 1) = -u [-n - 1]"};

    21 *}

    22 ML {*

    23 (*alpha -->  "</alpha>" *)

    25 @{term "\<alpha> "};

    26 @{term "\<delta> "};

    27 @{term "\<phi> "};

    28 @{term "\<rho> "};

    29 term2str @{term "\<rho> "};

    30 *}

    32 subsection {*rules*}

    33 (*axiomatization "z / (z - 1) = -u [-n - 1]" Illegal variable name: "z / (z - 1) = -u [-n - 1]" *)

    34 (*definition     "z / (z - 1) = -u [-n - 1]" Bad head of lhs: existing constant "op /"*)

    35 axiomatization where

    36   rule1: "1 = \<delta>[n]" and

    37   rule2: "|| z || > 1 ==> z / (z - 1) = u [n]" and

    38   rule3: "|| z || < 1 ==> z / (z - 1) = -u [-n - 1]" and

    39   rule4: "|| z || > || \<alpha> || ==> z / (z - \<alpha>) = \<alpha>^^^n * u [n]" and

    40   rule5: "|| z || < || \<alpha> || ==> z / (z - \<alpha>) = -(\<alpha>^^^n) * u [-n - 1]" and

    41   rule6: "|| z || > 1 ==> z/(z - 1)^^^2 = n * u [n]"

    42 ML {*

    43 @{thm rule1};

    44 @{thm rule2};

    45 @{thm rule3};

    46 @{thm rule4};

    47 *}

    49 subsection {*apply rules*}

    50 ML {*

    51 val inverse_Z = append_rls "inverse_Z" e_rls

    52   [ Thm  ("rule3",num_str @{thm rule3}),

    53     Thm  ("rule4",num_str @{thm rule4}),

    54     Thm  ("rule1",num_str @{thm rule1})

    55   ];

    57 val t = str2term "z / (z - 1) + z / (z - \<alpha>) + 1";

    58 val SOME (t', asm) = rewrite_set_ thy true inverse_Z t;

    59 term2str t' = "z / (z - ?\<delta> [?n]) + z / (z - \<alpha>) + ?\<delta> [?n]"; (*attention rule1 !!!*)

    60 *}

    61 ML {*

    62 val (thy, ro, er) = (@{theory}, tless_true, eval_rls);

    63 *}

    64 ML {*

    65 val SOME (t, asm1) = rewrite_ thy ro er true (num_str @{thm rule3}) t;

    66 term2str t = "- ?u [- ?n - 1] + z / (z - \<alpha>) + 1"; (*- real *)

    67 term2str t;

    68 *}

    69 ML {*

    70 val SOME (t, asm2) = rewrite_ thy ro er true (num_str @{thm rule4}) t;

    71 term2str t = "- ?u [- ?n - 1] + \<alpha> ^^^ ?n * ?u [?n] + 1"; (*- real *)

    72 term2str t;

    73 *}

    74 ML {*

    75 val SOME (t, asm3) = rewrite_ thy ro er true (num_str @{thm rule1}) t;

    76 term2str t = "- ?u [- ?n - 1] + \<alpha> ^^^ ?n * ?u [?n] + ?\<delta> [?n]"; (*- real *)

    77 term2str t;

    78 *}

    79 ML {*

    80 terms2str (asm1 @ asm2 @ asm3);

    81 *}

    83 section {*Prepare steps in CTP-based programming language*}

    84 text{*===================================================*}

    85 subsection {*prepare expression*}

    86 ML {*

    87 val ctxt = ProofContext.init_global @{theory};

    88 val ctxt = declare_constraints' [@{term "z::real"}] ctxt;

    90 val SOME fun1 = parseNEW ctxt "X z = 3 / (z - 1/4 + -1/8 * z ^^^ -1)"; term2str fun1;

    91 val SOME fun1' = parseNEW ctxt "X z = 3 / (z - 1/4 + -1/8 * (1/z))"; term2str fun1';

    92 *}

    94 axiomatization where

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

    97 ML {*

    98 val (thy, ro, er) = (@{theory}, tless_true, eval_rls);

    99 val SOME (fun2, asm1) = rewrite_ thy ro er true  @{thm ruleZY} fun1; term2str fun2;

   100 val SOME (fun2', asm1) = rewrite_ thy ro er true  @{thm ruleZY} fun1'; term2str fun2';

   102 val SOME (fun3,_) = rewrite_set_ @{theory Isac} false norm_Rational fun2;

   103 term2str fun3; (*fails on x^^^(-1) TODO*)

   104 val SOME (fun3',_) = rewrite_set_ @{theory Isac} false norm_Rational fun2';

   105 term2str fun3'; (*OK*)

   107 val (_, expr) = HOLogic.dest_eq fun3'; term2str expr;

   108 *}

   110 subsection {*solve equation*}

   111 text {*this type of equation if too general for the present program*}

   112 ML {*

   113 "----------- Minisubplb/100-init-rootp (*OK*)bl.sml ---------------------";

   114 val denominator = parseNEW ctxt "z^^^2 - 1/4*z - 1/8 = 0";

   115 val fmz = ["equality (z^^^2 - 1/4*z - 1/8 = (0::real))", "solveFor z","solutions L"];

   116 val (dI',pI',mI') =("Isac", ["univariate","equation"], ["no_met"]);

   117 (*                           ^^^^^^^^^^^^^^^^^^^^^^ TODO: ISAC determines type of eq*)

   118 *}

   119 text {*Does the Equation Match the Specification ?*}

   120 ML {*

   121 match_pbl fmz (get_pbt ["univariate","equation"]);

   122 *}

   124 ML {*

   125 val denominator = parseNEW ctxt "-1/8 + -1/4*z + z^^^2 = 0";

   126 val fmz =                                            (*specification*)

   127   ["equality (-1/8 + (-1/4)*z + z^^^2 = (0::real))", (*equality*)

   128    "solveFor z",                                     (*bound variable*)

   129    "solutions L"];                                   (*identifier for solution*)

   130 (*liste der theoreme die zum lösen benötigt werden, aus isac, keine spezielle methode (no met)*)

   131 val (dI',pI',mI') =

   132   ("Isac", ["pqFormula","degree_2","polynomial","univariate","equation"], ["no_met"]);

   133 *}

   134 text {*Does the Other Equation Match the Specification ?*}

   135 ML {*

   136 match_pbl fmz (get_pbt ["pqFormula","degree_2","polynomial","univariate","equation"]);

   137 *}

   138 text {*Solve Equation Stepwise*}

   139 ML {*

   140 val (p,_,f,nxt,_,pt) = CalcTreeTEST [(fmz, (dI',pI',mI'))];

   141 val (p,_,f,nxt,_,pt) = me nxt p [] pt;

   142 val (p,_,f,nxt,_,pt) = me nxt p [] pt;

   143 val (p,_,f,nxt,_,pt) = me nxt p [] pt;

   144 val (p,_,f,nxt,_,pt) = me nxt p [] pt;

   145 val (p,_,f,nxt,_,pt) = me nxt p [] pt;

   146 val (p,_,f,nxt,_,pt) = me nxt p [] pt;

   147 val (p,_,f,nxt,_,pt) = me nxt p [] pt;

   148 val (p,_,f,nxt,_,pt) = me nxt p [] pt;

   149 val (p,_,f,nxt,_,pt) = me nxt p [] pt;

   150 val (p,_,f,nxt,_,pt) = me nxt p [] pt;

   151 val (p,_,f,nxt,_,pt) = me nxt p [] pt;

   152 val (p,_,f,nxt,_,pt) = me nxt p [] pt; (*nxt =..,Check_elementwise "Assumptions")*)

   153 val (p,_,f,nxt,_,pt) = me nxt p [] pt;

   154 val (p,_,f,nxt,_,pt) = me nxt p [] pt; f2str f;

   155 (*[z = 1 / 8 + sqrt (9 / 16) / 2, z = 1 / 8 + -1 * sqrt (9 / 16) / 2] TODO sqrt*)

   156 show_pt pt;

   157 val SOME f = parseNEW ctxt "[z = 1 / 8 + 3 / 8, z = 1 / 8 + -3 / 8]";

   158 *}

   160 subsection {*partial fraction decomposition*}

   161 subsubsection {*solution of the equation*}

   162 ML {*

   163 val SOME solutions = parseNEW ctxt "[z=1/2, z=-1/4]";

   164 term2str solutions;

   165 atomty solutions;

   166 *}

   168 subsubsection {*get solutions out of list*}

   169 text {*in isac's CTP-based programming language: let s_1 = NTH 1 solutions; s_2 = NTH 2...*}

   170 ML {*

   171 val Const ("List.list.Cons", _) $ s_1 $ (Const ("List.list.Cons", _) $

   172       s_2 $ Const ("List.list.Nil", _)) = solutions;

   173 term2str s_1;

   174 term2str s_2;

   175 *}

   177 ML {* (*Solutions as Denominator --> Denominator1 = z - Zeropoint1, Denominator2 = z-Zeropoint2,...*)

   178 val xx = HOLogic.dest_eq s_1;

   179 val s_1' = HOLogic.mk_binop "Groups.minus_class.minus" xx;

   180 val xx = HOLogic.dest_eq s_2;

   181 val s_2' = HOLogic.mk_binop "Groups.minus_class.minus" xx;

   182 term2str s_1';

   183 term2str s_2';

   184 *}

   186 subsubsection {*build expression*}

   187 text {*in isac's CTP-based programming language: let s_1 = Take numerator / (s_1 * s_2)*}

   188 ML {*

   189 (*The Main Denominator is the multiplikation of the partial fraction denominators*)

   190 val denominator' = HOLogic.mk_binop "Groups.times_class.times" (s_1', s_2') ;

   191 val SOME numerator = parseNEW ctxt "3::real";

   193 val expr' = HOLogic.mk_binop "Rings.inverse_class.divide" (numerator, denominator');

   194 term2str expr';

   195 *}

   197 subsubsection {*Ansatz - create partial fractions out of our expression*}

   199 axiomatization where

   200   ansatz2: "n / (a*b) = A/a + B/(b::real)" and

   201   multiply_eq2: "(n / (a*b) = A/a + B/b) = (a*b*(n  / (a*b)) = a*b*(A/a + B/b))"

   203 ML {*

   204 (*we use our ansatz2 to rewrite our expression and get an equilation with our expression on the left and the partial fractions of it on the right side*)

   205 val SOME (t1,_) = rewrite_ @{theory Isac} e_rew_ord e_rls false @{thm ansatz2} expr';

   206 term2str t1; atomty t1;

   207 val eq1 = HOLogic.mk_eq (expr', t1);

   208 term2str eq1;

   209 *}

   210 ML {*

   211 (*eliminate the demoninators by multiplying the left and the right side with the main denominator*)

   212 val SOME (eq2,_) = rewrite_ @{theory Isac} e_rew_ord e_rls false @{thm multiply_eq2} eq1;

   213 term2str eq2;

   214 *}

   215 ML {*

   216 (*simplificatoin*)

   217 val SOME (eq3,_) = rewrite_set_ @{theory Isac} false norm_Rational eq2;

   218 term2str eq3; (*?A ?B not simplified*)

   219 *}

   220 ML {*

   221 val SOME fract1 =

   222   parseNEW ctxt "(z - 1 / 2) * (z - -1 / 4) * (A / (z - 1 / 2) + B / (z - -1 / 4))"; (*A B !*)

   223 val SOME (fract2,_) = rewrite_set_ @{theory Isac} false norm_Rational fract1;

   224 term2str fract2 = "(A + -2 * B + 4 * A * z + 4 * B * z) / 4";

   225 (*term2str fract2 = "A * (1 / 4 + z) + B * (-1 / 2 + z)" would be more traditional*)

   226 *}

   227 ML {*

   228 val (numerator, denominator) = HOLogic.dest_eq eq3;

   229 val eq3' = HOLogic.mk_eq (numerator, fract1); (*A B !*)

   230 term2str eq3';

   231 (*MANDATORY: simplify (and remove denominator) otherwise 3 = 0*)

   232 val SOME (eq3'' ,_) = rewrite_set_ @{theory Isac} false norm_Rational eq3';

   233 term2str eq3'';

   234 *}

   236 subsubsection {*get first koeffizient*}

   238 ML {*

   239 (*substitude z with the first zeropoint to get A*)

   240 val SOME (eq4_1,_) = rewrite_terms_ @{theory Isac} e_rew_ord e_rls [s_1] eq3'';

   241 term2str eq4_1;

   243 val SOME (eq4_2,_) = rewrite_set_ @{theory Isac} false norm_Rational eq4_1;

   244 term2str eq4_2;

   246 val fmz = ["equality (3 = 3 * A / (4::real))", "solveFor A","solutions L"];

   247 val (dI',pI',mI') =("Isac", ["univariate","equation"], ["no_met"]);

   248 (*solve the simple linear equilation for A TODO: return eq, not list of eq*)

   249 val (p,_,fa,nxt,_,pt) = CalcTreeTEST [(fmz, (dI',pI',mI'))];

   250 val (p,_,fa,nxt,_,pt) = me nxt p [] pt;

   251 val (p,_,fa,nxt,_,pt) = me nxt p [] pt;

   252 val (p,_,fa,nxt,_,pt) = me nxt p [] pt;

   253 val (p,_,fa,nxt,_,pt) = me nxt p [] pt;

   254 val (p,_,fa,nxt,_,pt) = me nxt p [] pt;

   255 val (p,_,fa,nxt,_,pt) = me nxt p [] pt;

   256 val (p,_,fa,nxt,_,pt) = me nxt p [] pt;

   257 val (p,_,fa,nxt,_,pt) = me nxt p [] pt;

   258 val (p,_,fa,nxt,_,pt) = me nxt p [] pt;

   259 val (p,_,fa,nxt,_,pt) = me nxt p [] pt;

   260 val (p,_,fa,nxt,_,pt) = me nxt p [] pt;

   261 val (p,_,fa,nxt,_,pt) = me nxt p [] pt;

   262 val (p,_,fa,nxt,_,pt) = me nxt p [] pt;

   263 val (p,_,fa,nxt,_,pt) = me nxt p [] pt;

   264 val (p,_,fa,nxt,_,pt) = me nxt p [] pt;

   265 val (p,_,fa,nxt,_,pt) = me nxt p [] pt;

   266 val (p,_,fa,nxt,_,pt) = me nxt p [] pt;

   267 val (p,_,fa,nxt,_,pt) = me nxt p [] pt;

   268 val (p,_,fa,nxt,_,pt) = me nxt p [] pt;

   269 val (p,_,fa,nxt,_,pt) = me nxt p [] pt;

   270 val (p,_,fa,nxt,_,pt) = me nxt p [] pt;

   271 val (p,_,fa,nxt,_,pt) = me nxt p [] pt;

   272 val (p,_,fa,nxt,_,pt) = me nxt p [] pt;

   273 val (p,_,fa,nxt,_,pt) = me nxt p [] pt;

   274 val (p,_,fa,nxt,_,pt) = me nxt p [] pt;

   275 val (p,_,fa,nxt,_,pt) = me nxt p [] pt;

   276 val (p,_,fa,nxt,_,pt) = me nxt p [] pt;

   277 f2str fa;

   278 *}

   280 subsubsection {*get second koeffizient*}

   282 ML {*

   283 (*substitude z with the second zeropoint to get B*)

   284 val SOME (eq4b_1,_) = rewrite_terms_ @{theory Isac} e_rew_ord e_rls [s_2] eq3'';

   285 term2str eq4b_1;

   287 val SOME (eq4b_2,_) = rewrite_set_ @{theory Isac} false norm_Rational eq4b_1;

   288 term2str eq4b_2;

   289 *}

   290 ML {*

   291 (*solve the simple linear equilation for B TODO: return eq, not list of eq*)

   292 val fmz = ["equality (3 = -3 * B / (4::real))", "solveFor B","solutions L"];

   293 val (dI',pI',mI') =("Isac", ["univariate","equation"], ["no_met"]);

   294 val (p,_,fb,nxt,_,pt) = CalcTreeTEST [(fmz, (dI',pI',mI'))];

   295 val (p,_,fb,nxt,_,pt) = me nxt p [] pt;

   296 val (p,_,fb,nxt,_,pt) = me nxt p [] pt;

   297 val (p,_,fb,nxt,_,pt) = me nxt p [] pt;

   298 val (p,_,fb,nxt,_,pt) = me nxt p [] pt;

   299 val (p,_,fb,nxt,_,pt) = me nxt p [] pt;

   300 val (p,_,fb,nxt,_,pt) = me nxt p [] pt;

   301 val (p,_,fb,nxt,_,pt) = me nxt p [] pt;

   302 val (p,_,fb,nxt,_,pt) = me nxt p [] pt;

   303 val (p,_,fb,nxt,_,pt) = me nxt p [] pt;

   304 val (p,_,fb,nxt,_,pt) = me nxt p [] pt;

   305 val (p,_,fb,nxt,_,pt) = me nxt p [] pt;

   306 val (p,_,fb,nxt,_,pt) = me nxt p [] pt;

   307 val (p,_,fb,nxt,_,pt) = me nxt p [] pt;

   308 val (p,_,fb,nxt,_,pt) = me nxt p [] pt;

   309 val (p,_,fb,nxt,_,pt) = me nxt p [] pt;

   310 val (p,_,fb,nxt,_,pt) = me nxt p [] pt;

   311 val (p,_,fb,nxt,_,pt) = me nxt p [] pt;

   312 val (p,_,fb,nxt,_,pt) = me nxt p [] pt;

   313 val (p,_,fb,nxt,_,pt) = me nxt p [] pt;

   314 val (p,_,fb,nxt,_,pt) = me nxt p [] pt;

   315 val (p,_,fb,nxt,_,pt) = me nxt p [] pt;

   316 val (p,_,fb,nxt,_,pt) = me nxt p [] pt;

   317 val (p,_,fb,nxt,_,pt) = me nxt p [] pt;

   318 val (p,_,fb,nxt,_,pt) = me nxt p [] pt;

   319 val (p,_,fb,nxt,_,pt) = me nxt p [] pt;

   320 val (p,_,fb,nxt,_,pt) = me nxt p [] pt;

   321 val (p,_,fb,nxt,_,pt) = me nxt p [] pt;

   322 f2str fb;

   323 *}

   325 ML {* (*check koeffizients*)

   326 if f2str fa = "[A = 4]" then () else error "part.fract. eq4_1";

   327 if f2str fb = "[B = -4]" then () else error "part.fract. eq4_1";

   328 *}

   330 subsubsection {*substitute expression with solutions*}

   331 ML {*

   332 *}

   334 section {*Implement the Specification and the Method*}

   335 text{*==============================================*}

   336 subsection{*Define the Specification*}

   337 ML {*

   338 val thy = @{theory};

   339 *}

   340 ML {*

   341 store_pbt

   342  (prep_pbt thy "pbl_SP" [] e_pblID

   343  (["SignalProcessing"], [], e_rls, NONE, []));

   344 store_pbt

   345  (prep_pbt thy "pbl_SP_Ztrans" [] e_pblID

   346  (["Z_Transform","SignalProcessing"], [], e_rls, NONE, []));

   347 store_pbt

   348  (prep_pbt thy "pbl_SP_Ztrans_inv" [] e_pblID

   349  (["inverse", "Z_Transform", "SignalProcessing"],

   350   [("#Given" ,["equality X_eq"]),

   351    ("#Find"  ,["equality n_eq"])

   352   ],

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

   354   [["TODO: path to method"]]));

   356 show_ptyps();

   357 get_pbt ["inverse","Z_Transform","SignalProcessing"];

   358 *}

   360 subsection{*Define the (Dummy-)Method*}

   361 subsection {*Define Name and Signature for the Method*}

   362 consts

   363   InverseZTransform :: "[bool, bool] => bool"

   364     ("((Script InverseZTransform (_ =))// (_))" 9)

   366 ML {*

   367 store_met

   368  (prep_met thy "met_SP" [] e_metID

   369  (["SignalProcessing"], [],

   370    {rew_ord'="tless_true", rls'= e_rls, calc = [], srls = e_rls, prls = e_rls,

   371     crls = e_rls, nrls = e_rls}, "empty_script"));

   372 store_met

   373  (prep_met thy "met_SP_Ztrans" [] e_metID

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

   375    {rew_ord'="tless_true", rls'= e_rls, calc = [], srls = e_rls, prls = e_rls,

   376     crls = e_rls, nrls = e_rls}, "empty_script"));

   377 *}

   378 ML {*

   379 store_met

   380  (prep_met thy "met_SP_Ztrans_inv" [] e_metID

   381  (["SignalProcessing", "Z_Transform", "inverse"], [],

   382    {rew_ord'="tless_true", rls'= e_rls, calc = [], srls = e_rls, prls = e_rls,

   383     crls = e_rls, nrls = e_rls},

   384   "empty_script"

   385  ));

   386 val thy = @{theory}; (*latest version of thy required*)

   387 store_met

   388  (prep_met thy "met_SP_Ztrans_inv" [] e_metID

   389  (["SignalProcessing", "Z_Transform", "inverse"],

   390   [("#Given" ,["equality X_eq"]),

   391    ("#Find"  ,["equality n_eq"])

   392   ],

   393    {rew_ord'="tless_true", rls'= e_rls, calc = [], srls = e_rls, prls = e_rls,

   394     crls = e_rls, nrls = e_rls},

   395   "Script InverseZTransform (Xeq::bool) =" ^

   396   " (let X = Take Xeq;" ^

   397   "      X = Rewrite ruleZY False X" ^

   398   "  in X)"

   399  ));

   401 show_mets();

   402 get_met ["SignalProcessing","Z_Transform","inverse"];

   403 *}

   406 section {*Program in CTP-based language*}

   407 text{*=================================*}

   408 subsection {*Stepwise extend Program*}

   409 ML {*

   410 val str =

   411 "Script InverseZTransform (Xeq::bool) =" ^

   412 " Xeq";

   413 *}

   414 ML {*

   415 val str =

   416 "Script InverseZTransform (Xeq::bool) =" ^

   417 " (let X = Take Xeq;" ^

   418 "      X = Rewrite ruleZY False X" ^

   419 "  in X)";

   420 *}

   421 ML {*

   422 val thy = @{theory};

   423 val sc = ((inst_abs thy) o term_of o the o (parse thy)) str;

   424 *}

   425 ML {*

   426 term2str sc;

   427 atomty sc

   428 *}

   431 subsection {*Store Final Version of Program for Execution*}

   432 ML {*

   433 store_met

   434  (prep_met thy "met_SP_Ztrans_inv" [] e_metID

   435  (["SignalProcessing", "Z_Transform", "inverse"],

   436   [("#Given" ,["equality X_eq"]),

   437    ("#Find"  ,["equality n_eq"])

   438   ],

   439    {rew_ord'="tless_true", rls'= e_rls, calc = [], srls = e_rls, prls = e_rls,

   440     crls = e_rls, nrls = e_rls},

   441   "Script InverseZTransform (Xeq::bool) =" ^

   442   " (let X = Take Xeq;" ^

   443   "      X = Rewrite ruleZY False X" ^

   444   "  in X)"

   445  ));

   446 *}

   449 subsection {*Stepwise Execute the Program*}

   458 section {*Write Tests for Crucial Details*}

   459 text{*===================================*}

   460 ML {*

   462 *}

   464 section {*Integrate Program into Knowledge*}

   465 ML {*

   467 *}

   469 end

author	Walther Neuper <neuper@ist.tugraz.at>
	Sat, 10 Sep 2011 10:37:24 +0200
branch	decompose-isar
changeset 42255	6201b34bd323
parent 42250	fdc85b0443e6
permissions	-rwxr-xr-x