1 (* Title: partial fraction decomposition
3 (c) due to copyright terms
6 "--------------------------------------------------------";
7 "table of contents --------------------------------------";
8 "--------------------------------------------------------";
9 "----------- why helpless here ? ------------------------";
10 "----------- why not nxt = Model_Problem here ? ---------";
11 "----------- fun factors_from_solution ------------------";
12 "----------- Logic.unvarify_global ----------------------";
13 "----------- eval_drop_questionmarks --------------------";
14 "----------- = me for met_partial_fraction --------------";
15 "----------- progr.vers.2: check erls for multiply_ansatz";
16 "----------- progr.vers.2: improve program --------------";
17 "--------------------------------------------------------";
18 "--------------------------------------------------------";
19 "--------------------------------------------------------";
22 "----------- why helpless here ? ------------------------";
23 "----------- why helpless here ? ------------------------";
24 "----------- why helpless here ? ------------------------";
25 val fmz = ["filterExpression (X z = 3 / (z - 1/4 + -1/8 * (1/(z::real))))",
26 "stepResponse (x[n::real]::bool)"];
27 val (dI,pI,mI) = ("Isac", ["Inverse", "Z_Transform", "SignalProcessing"],
28 ["SignalProcessing","Z_Transform","Inverse"]);
29 val (p,_,f,nxt,_,pt) = CalcTreeTEST [(fmz, (dI,pI,mI))];
30 val (p,_,f,nxt,_,pt) = me nxt p [] pt; "Add_Given";
31 val (p,_,f,nxt,_,pt) = me nxt p [] pt; "Add_Find";
32 val (p,_,f,nxt,_,pt) = me nxt p [] pt; "Specify_Theory";
33 val (p,_,f,nxt,_,pt) = me nxt p [] pt; "Specify_Problem";
34 val (p,_,f,nxt,_,pt) = me nxt p [] pt; "Specify_Method";
35 val (p,_,f,nxt,_,pt) = me nxt p [] pt; "nxt = Apply_Method";
36 val (p,_,f,nxt,_,pt) = me nxt p [] pt; "nxt = Rewrite (ruleZY, Inverse_Z_Transform.ruleZY) --> X z = 3 / (z - 1 / 4 + -1 / 8 * (1 / z))";
37 val (p''',_,f,nxt''',_,pt''') = me nxt p [] pt; "nxt = Rewrite_Set norm_Rational --> X' z = 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))";
38 "~~~~~ fun me, args:"; val ((_,tac), (p:pos'), _, (pt:ptree)) = (nxt, p, [], pt);
39 val ("ok", (_, _, ptp)) = locatetac tac (pt,p)
41 "~~~~~ fun step, args:"; val (((ip as (_,p_)):pos'), ((ptp as (pt,p), tacis):calcstate)) =
42 (p, ((pt, e_pos'),[]));
43 val pIopt = get_pblID (pt,ip);
44 ip = ([],Res); "false";
46 pIopt; (* = SOME ["Inverse", "Z_Transform", "SignalProcessing"]*)
47 member op = [Pbl,Met] p_ andalso is_none (get_obj g_env pt (fst p)); "false";
48 (*nxt_solve_ (pt,ip); "WAS isalist2list applied to NON-list 'no_meth'"
49 THIS MEANS: replace no_meth by [no_meth] in Script.*)
50 (*WAS val ("helpless",_) = step p ((pt, e_pos'),[]) *)
51 (*WAS val (p,_,f,nxt,_,pt) = me nxt p [] pt; "Empty_Tac instead SubProblem";*)
53 "----------- why not nxt = Model_Problem here ? ---------";
54 "----------- why not nxt = Model_Problem here ? ---------";
55 "----------- why not nxt = Model_Problem here ? ---------";
56 val (p,_,f,nxt,_,pt) = me nxt''' p''' [] pt''';
57 "~~~~~ fun me, args:"; val (((_,tac):tac'_), (p:pos'), _, (pt:ptree)) = (nxt, p, c, pt);
58 val ("ok", (_, _, ptp)) = locatetac tac (pt,p);
60 "~~~~~ fun step, args:"; val (((ip as (_,p_)):pos'), ((ptp as (pt,p), tacis):calcstate)) =
61 (p, ((pt, e_pos'),[]));
62 val pIopt = get_pblID (pt,ip);
63 ip = ([],Res); " = false";
65 pIopt (* = SOME ["Inverse", "Z_Transform", "SignalProcessing"]*);
66 member op = [Pbl,Met] p_ andalso is_none (get_obj g_env pt (fst p)); " = false";
67 (* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ leads into
68 nxt_solve_, which is definitely WRONG (should be nxt_specify_ for FIND_ADD).
69 This ERROR seems to be introduced together with ctxt, concerns Apply_Method without init_form.
73 "----------- fun factors_from_solution ------------------";
74 "----------- fun factors_from_solution ------------------";
75 "----------- fun factors_from_solution ------------------";
76 val ctxt = ProofContext.init_global @{theory Isac};
77 val SOME t = parseNEW ctxt "factors_from_solution [(z::real) = 1 / 2, z = -1 / 4]";
78 val SOME (_, t') = eval_factors_from_solution "" 0 t thy;
80 "factors_from_solution [z = 1 / 2, z = -1 / 4] = (z - 1 / 2) * (z - -1 / 4)"
81 then () else error "factors_from_solution broken";
83 "----------- Logic.unvarify_global ----------------------";
84 "----------- Logic.unvarify_global ----------------------";
85 "----------- Logic.unvarify_global ----------------------";
86 val A_var = parseNEW ctxt "A::real" |> the |> Logic.varify_global
87 val B_var = parseNEW ctxt "B::real" |> the |> Logic.varify_global
89 val denom1 = parseNEW ctxt "(z + -1 * (1 / 2))::real" |> the;
90 val denom2 = parseNEW ctxt "(z + -1 * (-1 / 4))::real" |> the;
92 val test = HOLogic.mk_binop "Groups.plus_class.plus"
93 (HOLogic.mk_binop "Rings.inverse_class.divide" (A_var, denom1),
94 HOLogic.mk_binop "Rings.inverse_class.divide" (B_var, denom2));
96 if term2str test = "?A / (z + -1 * (1 / 2)) + ?B / (z + -1 * (-1 / 4))" then ()
97 else error "HOLogic.mk_binop broken ?";
99 (* Logic.unvarify_global test
100 ---> exception TERM raised (line 539 of "logic.ML"): Illegal fixed variable: "z"
101 thus we need another fun var2free in termC.sml*)
103 "----------- eval_drop_questionmarks --------------------";
104 "----------- eval_drop_questionmarks --------------------";
105 "----------- eval_drop_questionmarks --------------------";
106 if term2str test = "?A / (z + -1 * (1 / 2)) + ?B / (z + -1 * (-1 / 4))"
107 then () else error "test setup (ctxt!) probably changed";
109 val t = Const ("Partial_Fractions.drop_questionmarks", HOLogic.realT --> HOLogic.realT) $ test;
111 val SOME (_, t') = eval_drop_questionmarks "drop_questionmarks" 0 t @{theory Isac};
112 if term2str t' = "drop_questionmarks (?A / (z + -1 * (1 / 2)) + ?B / (z + -1 * (-1 / 4))) =" ^
113 "\nA / (z + -1 * (1 / 2)) + B / (z + -1 * (-1 / 4))"
114 then () else error "eval_drop_questionmarks broken";
116 "----------- = me for met_partial_fraction --------------";
117 "----------- = me for met_partial_fraction --------------";
118 "----------- = me for met_partial_fraction --------------";
120 ["functionTerm (3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z))))",
121 "solveFor z", "decomposedFunction p_p"];
123 ("Partial_Fractions",
124 ["partial_fraction", "rational", "simplification"],
125 ["simplification","of_rationals","to_partial_fraction"]);
126 val (p,_,f,nxt,_,pt) = CalcTreeTEST [(fmz, (dI',pI',mI'))];
127 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
128 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
129 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
130 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
131 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
132 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
133 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
134 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
135 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
136 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
137 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
138 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
139 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
140 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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;
153 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
154 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
155 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
156 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
157 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
158 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
159 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
160 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
161 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
162 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
163 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
164 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
165 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
166 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
167 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
168 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
169 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
170 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
171 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
172 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
173 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
174 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
175 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
176 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
177 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
178 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
179 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
180 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
181 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
182 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
183 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
184 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
185 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
186 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
187 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
188 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
189 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
190 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
191 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
192 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
193 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
194 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
195 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
196 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
197 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
198 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
199 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
200 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
201 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
202 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
203 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
204 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
205 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
206 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
207 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
208 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
209 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
210 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
211 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
212 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
213 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
214 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
215 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
216 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
217 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
218 val (p,_,f,nxt,_,pt) = me nxt p [] pt;
221 if f2str f = "4 / (z - 1 / 2) + -4 / (z - -1 / 4)" then()
222 else error "= me .. met_partial_fraction broken";
224 "----------- progr.vers.2: check erls for multiply_ansatz";
225 "----------- progr.vers.2: check erls for multiply_ansatz";
226 "----------- progr.vers.2: check erls for multiply_ansatz";
227 (*test for outcommented 3 lines in script: is norm_Rational strong enough?*)
228 val t = str2term "(3 / ((-1 + -2 * z + 8 * z ^^^ 2) *3/24)) = (3 / ((z - 1 / 2) * (z - -1 / 4)))";
229 val SOME (t', _) = rewrite_set_ @{theory Isac} true ansatz_rls t;
230 term2str t' = "3 / ((-1 + -2 * z + 8 * z ^^^ 2) * 3 / 24) =\n?A / (z - 1 / 2) + ?B / (z - -1 / 4)";
232 val SOME (t'', _) = rewrite_set_ @{theory Isac} true multiply_ansatz t'; (*true*)
233 term2str t'' = "(z - 1 / 2) * (z - -1 / 4) * 3 / ((-1 + -2 * z + 8 * z ^^^ 2) * 3 / 24) =\n" ^
234 "?A * (z - -1 / 4) + ?B * (z - 1 / 2)"; (*true*)
236 val SOME (t''', _) = rewrite_set_ @{theory Isac} true norm_Rational t'';
237 if term2str t''' = "3 = ?A * (1 + 4 * z) / 4 + ?B * (-1 + 2 * z) / 2" then ()
238 else error "ansatz_rls - multiply_ansatz - norm_Rational broken";
240 (*test for outcommented 3 lines in script: empower erls for x = a*b ==> ...*)
241 val xxx = append_rls "multiply_ansatz_erls" norm_Rational
242 [Calc ("HOL.eq",eval_equal "#equal_")];
244 val multiply_ansatz = prep_rls(
245 Rls {id = "multiply_ansatz", preconds = [], rew_ord = ("dummy_ord",dummy_ord),
247 srls = Erls, calc = [],
249 [Thm ("multiply_2nd_order",num_str @{thm multiply_2nd_order})
251 scr = EmptyScr}:rls);
253 rewrite_set_ thy true xxx @{term "a+b = a+(b::real)"}; (*SOME ok*)
254 rewrite_set_ thy true multiply_ansatz @{term "a+b = a+(b::real)"}; (*why NONE?: GOON*)
256 "----------- progr.vers.2: improve program --------------";
257 "----------- progr.vers.2: improve program --------------";
258 "----------- progr.vers.2: improve program --------------";
259 (*WN120318 stopped due to much effort with the test above*)
260 "Script PartFracScript (f_f::real) (zzz::real) = " ^(*f_f: 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)), zzz: z*)
261 " (let f_f = Take f_f; " ^
262 " (num_orig::real) = get_numerator f_f; " ^(*num_orig: 3*)
263 " f_f = (Rewrite_Set norm_Rational False) f_f; " ^(*f_f: 24 / (-1 + -2 * z + 8 * z ^^^ 2)*)
264 " (numer::real) = get_numerator f_f; " ^(*numer: 24*)
265 " (denom::real) = get_denominator f_f; " ^(*denom: -1 + -2 * z + 8 * z ^^^ 2*)
266 " (equ::bool) = (denom = (0::real)); " ^(*equ: -1 + -2 * z + 8 * z ^^^ 2 = 0*)
267 " (L_L::bool list) = (SubProblem (PolyEq', " ^
268 " [abcFormula, degree_2, polynomial, univariate, equation], " ^
269 " [no_met]) [BOOL equ, REAL zzz]); " ^(*L_L: [z = 1 / 2, z = -1 / 4]*)
270 " (facs::real) = factors_from_solution L_L; " ^(*facs: (z - 1 / 2) * (z - -1 / 4)*)
271 " (eql::real) = Take (num_orig / facs); " ^(*eql: 3 / ((z - 1 / 2) * (z - -1 / 4)) *)
272 " (eqr::real) = (Try (Rewrite_Set ansatz_rls False)) eql; " ^(*eqr: ?A / (z - 1 / 2) + ?B / (z - -1 / 4)*)
273 " (eq::bool) = Take (eql = eqr); " ^(*eq: 3 / ((z - 1 / 2) * (z - -1 / 4)) = ?A / (z - 1 / 2) + ?B / (z - -1 / 4)*)
274 (*this has been tested below by rewrite_set_
275 " (eeeee::bool) = Take ((num_orig / denom * denom / numer) = (num_orig / facs));" ^(**)
276 " (eeeee::bool) = (Rewrite_Set ansatz_rls False) eeeee;" ^(**)
277 " eq = (Try (Rewrite_Set multiply_ansatz False)) eeeee; " ^(*eq: 3 = ?A * (z - -1 / 4) + ?B * (z - 1 / 2)*)
278 NEXT try to outcomment the very next line..*)
279 " eq = (Try (Rewrite_Set equival_trans False)) eeeee; " ^(*eq: 3 = ?A * (z - -1 / 4) + ?B * (z - 1 / 2)*)
280 " eq = drop_questionmarks eq; " ^(*eq: 3 = A * (z - -1 / 4) + B * (z - 1 / 2)*)
281 " (z1::real) = (rhs (NTH 1 L_L)); " ^(*z1: 1 / 2*)
282 " (z2::real) = (rhs (NTH 2 L_L)); " ^(*z2: -1 / 4*)
283 " (eq_a::bool) = Take eq; " ^(*eq_a: 3 = A * (z - -1 / 4) + B * (z - 1 / 2)*)
284 " eq_a = (Substitute [zzz = z1]) eq; " ^(*eq_a: 3 = A * (1 / 2 - -1 / 4) + B * (1 / 2 - 1 / 2)*)
285 " eq_a = (Rewrite_Set norm_Rational False) eq_a; " ^(*eq_a: 3 = 3 * A / 4*)
286 " (sol_a::bool list) = " ^
287 " (SubProblem (Isac', [univariate,equation], [no_met]) " ^
288 " [BOOL eq_a, REAL (A::real)]); " ^(*sol_a: [A = 4]*)
289 " (a::real) = (rhs (NTH 1 sol_a)); " ^(*a: 4*)
290 " (eq_b::bool) = Take eq; " ^(*eq_b: 3 = A * (z - -1 / 4) + B * (z - 1 / 2)*)
291 " eq_b = (Substitute [zzz = z2]) eq_b; " ^(*eq_b: *)
292 " eq_b = (Rewrite_Set norm_Rational False) eq_b; " ^(*eq_b: *)
293 " (sol_b::bool list) = " ^
294 " (SubProblem (Isac', [univariate,equation], [no_met]) " ^
295 " [BOOL eq_b, REAL (B::real)]); " ^(*sol_b: [B = -4]*)
296 " (b::real) = (rhs (NTH 1 sol_b)); " ^(*b: -4*)
297 " eqr = drop_questionmarks eqr; " ^(*eqr: A / (z - 1 / 2) + B / (z - -1 / 4)*)
298 " (pbz::real) = Take eqr; " ^(*pbz: A / (z - 1 / 2) + B / (z - -1 / 4)*)
299 " pbz = ((Substitute [A = a, B = b]) pbz) " ^(*pbz: 4 / (z - 1 / 2) + -4 / (z - -1 / 4)*)