2 author: Jan Rocnik, isac team
3 Copyright (c) isac team 2011
4 Use is subject to license terms.
6 (* Partial Fraction Decomposition *)
9 theory Partial_Fractions imports RootRatEq begin
16 val ansatz_rls_ : theory -> term -> (term * term list) option
21 subsection \<open>eval_ functions\<close>
23 factors_from_solution :: "bool list => real"
27 text \<open>these might be used for variants of fac_from_sol\<close>
29 fun mk_minus_1 T = Free("-1", T); (*TODO DELETE WITH numbers_to_string*)
30 fun flip_sign t = (*TODO improve for use in factors_from_solution: -(-1) etc*)
31 let val minus_1 = t |> type_of |> mk_minus_1
32 in HOLogic.mk_binop "Groups.times_class.times" (minus_1, t) end;
35 text \<open>from solutions (e.g. [z = 1, z = -2]) make linear factors (e.g. (z - 1)*(z - -2))\<close>
38 let val (lhs, rhs) = HOLogic.dest_eq s
39 in HOLogic.mk_binop "Groups.minus_class.minus" (lhs, rhs) end;
42 if prod = Rule.e_term then error "mk_prod called with []" else prod
43 | mk_prod prod (t :: []) =
44 if prod = Rule.e_term then t else HOLogic.mk_binop "Groups.times_class.times" (prod, t)
45 | mk_prod prod (t1 :: t2 :: ts) =
48 let val p = HOLogic.mk_binop "Groups.times_class.times" (t1, t2)
51 let val p = HOLogic.mk_binop "Groups.times_class.times" (prod, t1)
52 in mk_prod p (t2 :: ts) end
54 fun factors_from_solution sol =
55 let val ts = HOLogic.dest_list sol
56 in mk_prod Rule.e_term (map fac_from_sol ts) end;
58 (*("factors_from_solution", ("Partial_Fractions.factors_from_solution",
59 eval_factors_from_solution ""))*)
60 fun eval_factors_from_solution (thmid:string) _
61 (t as Const ("Partial_Fractions.factors_from_solution", _) $ sol) thy =
62 ((let val prod = factors_from_solution sol
63 in SOME (TermC.mk_thmid thmid (Rule.term_to_string''' thy prod) "",
64 HOLogic.Trueprop $ (TermC.mk_equality (t, prod)))
67 | eval_factors_from_solution _ _ _ _ = NONE;
70 subsection \<open>'ansatz' for partial fractions\<close>
72 ansatz_2nd_order: "n / (a*b) = AA/a + BB/b" and
73 ansatz_3rd_order: "n / (a*b*c) = AA/a + BB/b + C/c" and
74 ansatz_4th_order: "n / (a*b*c*d) = AA/a + BB/b + C/c + D/d" and
76 equival_trans_2nd_order: "(n/(a*b) = AA/a + BB/b) = (n = AA*b + BB*a)" and
77 equival_trans_3rd_order: "(n/(a*b*c) = AA/a + BB/b + C/c) = (n = AA*b*c + BB*a*c + C*a*b)" and
78 equival_trans_4th_order: "(n/(a*b*c*d) = AA/a + BB/b + C/c + D/d) =
79 (n = AA*b*c*d + BB*a*c*d + C*a*b*d + D*a*b*c)" and
80 (*version 2: not yet used, see partial_fractions.sml*)
81 multiply_2nd_order: "(n/x = AA/a + BB/b) = (a*b*n/x = AA*b + BB*a)" and
82 multiply_3rd_order: "(n/x = AA/a + BB/b + C/c) = (a*b*c*n/x = AA*b*c + BB*a*c + C*a*b)" and
84 "(n/x = AA/a + BB/b + C/c + D/d) = (a*b*c*d*n/x = AA*b*c*d + BB*a*c*d + C*a*b*d + D*a*b*c)"
86 text \<open>Probably the optimal formalization woudl be ...
88 multiply_2nd_order: "x = a*b ==> (n/x = AA/a + BB/b) = (a*b*n/x = AA*b + BB*a)" and
89 multiply_3rd_order: "x = a*b*c ==>
90 (n/x = AA/a + BB/b + C/c) = (a*b*c*n/x = AA*b*c + BB*a*c + C*a*b)" and
91 multiply_4th_order: "x = a*b*c*d ==>
92 (n/x = AA/a + BB/b + C/c + D/d) = (a*b*c*d*n/x = AA*b*c*d + BB*a*c*d + C*a*b*d + D*a*b*c)"
94 ... because it would allow to start the ansatz as follows
95 (1) 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z))) = 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))
96 (2) 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z))) = AA / (z - 1 / 2) + BB / (z - -1 / 4)
97 (3) (z - 1 / 2) * (z - -1 / 4) * 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z))) =
98 (z - 1 / 2) * (z - -1 / 4) * AA / (z - 1 / 2) + BB / (z - -1 / 4)
99 (4) 3 = AA * (z - -1 / 4) + BB * (z - 1 / 2)
103 (3==>4) norm_Rational
104 TODOs for this version are in partial_fractions.sml "--- progr.vers.2: "
108 val ansatz_rls = prep_rls'(
109 Rule.Rls {id = "ansatz_rls", preconds = [], rew_ord = ("dummy_ord",Rule.dummy_ord),
110 erls = Rule.Erls, srls = Rule.Erls, calc = [], errpatts = [],
112 [Rule.Thm ("ansatz_2nd_order",TermC.num_str @{thm ansatz_2nd_order}),
113 Rule.Thm ("ansatz_3rd_order",TermC.num_str @{thm ansatz_3rd_order})
115 scr = Rule.EmptyScr});
117 val equival_trans = prep_rls'(
118 Rule.Rls {id = "equival_trans", preconds = [], rew_ord = ("dummy_ord",Rule.dummy_ord),
119 erls = Rule.Erls, srls = Rule.Erls, calc = [], errpatts = [],
121 [Rule.Thm ("equival_trans_2nd_order",TermC.num_str @{thm equival_trans_2nd_order}),
122 Rule.Thm ("equival_trans_3rd_order",TermC.num_str @{thm equival_trans_3rd_order})
124 scr = Rule.EmptyScr});
126 val multiply_ansatz = prep_rls'(
127 Rule.Rls {id = "multiply_ansatz", preconds = [], rew_ord = ("dummy_ord",Rule.dummy_ord),
129 srls = Rule.Erls, calc = [], errpatts = [],
131 [Rule.Thm ("multiply_2nd_order",TermC.num_str @{thm multiply_2nd_order})
133 scr = Rule.EmptyScr});
136 text \<open>store the rule set for math engine\<close>
137 setup \<open>KEStore_Elems.add_rlss
138 [("ansatz_rls", (Context.theory_name @{theory}, ansatz_rls)),
139 ("multiply_ansatz", (Context.theory_name @{theory}, multiply_ansatz)),
140 ("equival_trans", (Context.theory_name @{theory}, equival_trans))]\<close>
142 subsection \<open>Specification\<close>
145 decomposedFunction :: "real => una"
148 Celem.check_guhs_unique := false; (*WN120307 REMOVE after editing*)
150 setup \<open>KEStore_Elems.add_pbts
151 [(Specify.prep_pbt @{theory} "pbl_simp_rat_partfrac" [] Celem.e_pblID
152 (["partial_fraction", "rational", "simplification"],
153 [("#Given" ,["functionTerm t_t", "solveFor v_v"]),
154 (* TODO: call this sub-problem with appropriate functionTerm:
155 leading coefficient of the denominator is 1: to be checked here! and..
156 ("#Where" ,["((get_numerator t_t) has_degree_in v_v) <
157 ((get_denominator t_t) has_degree_in v_v)"]), TODO*)
158 ("#Find" ,["decomposedFunction p_p'''"])],
159 Rule.append_rls "e_rls" Rule.e_rls [(*for preds in where_ TODO*)],
161 [["simplification","of_rationals","to_partial_fraction"]]))]\<close>
163 subsection \<open>Method\<close>
164 text \<open>rule set for functions called in the Program\<close>
166 val srls_partial_fraction = Rule.Rls {id="srls_partial_fraction",
168 rew_ord = ("termlessI",termlessI),
169 erls = Rule.append_rls "erls_in_srls_partial_fraction" Rule.e_rls
170 [(*for asm in NTH_CONS ...*)
171 Rule.Calc ("Orderings.ord_class.less",eval_equ "#less_"),
172 (*2nd NTH_CONS pushes n+-1 into asms*)
173 Rule.Calc("Groups.plus_class.plus", eval_binop "#add_")],
174 srls = Rule.Erls, calc = [], errpatts = [],
176 Rule.Thm ("NTH_CONS",TermC.num_str @{thm NTH_CONS}),
177 Rule.Calc("Groups.plus_class.plus", eval_binop "#add_"),
178 Rule.Thm ("NTH_NIL",TermC.num_str @{thm NTH_NIL}),
179 Rule.Calc("Tools.lhs", Tools.eval_lhs "eval_lhs_"),
180 Rule.Calc("Tools.rhs", Tools.eval_rhs"eval_rhs_"),
181 Rule.Calc("Atools.argument'_in", eval_argument_in "Atools.argument'_in"),
182 Rule.Calc("Rational.get_denominator", eval_get_denominator "#get_denominator"),
183 Rule.Calc("Rational.get_numerator", eval_get_numerator "#get_numerator"),
184 Rule.Calc("Partial_Fractions.factors_from_solution",
185 eval_factors_from_solution "#factors_from_solution")
187 scr = Rule.EmptyScr};
190 (* current version, error outcommented *)
191 partial_function (tailrec) partial_fraction :: "real \<Rightarrow> real \<Rightarrow> real"
193 "partial_fraction f_f zzz = \<comment> \<open>([1], Frm), 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))\<close>
194 (let f_f = Take f_f; \<comment> \<open>num_orig = 3\<close>
195 num_orig = get_numerator f_f; \<comment> \<open>([1], Res), 24 / (-1 + -2 * z + 8 * z ^^^ 2)\<close>
196 f_f = (Rewrite_Set ''norm_Rational'' False) f_f; \<comment> \<open>denom = -1 + -2 * z + 8 * z ^^^ 2\<close>
197 denom = get_denominator f_f; \<comment> \<open>equ = -1 + -2 * z + 8 * z ^^^ 2 = 0\<close>
198 equ = denom = (0::real);
199 \<comment> \<open>----- ([2], Pbl), solve (-1 + -2 * z + 8 * z ^^^ 2 = 0, z)\<close>
200 L_L = SubProblem (''Partial_Fractions'', \<comment> \<open>([2], Res), [z = 1 / 2, z = -1 / 4\<close>
201 [''abcFormula'', ''degree_2'', ''polynomial'', ''univariate'', ''equation''],
202 [''no_met'']) [BOOL equ, REAL zzz]; \<comment> \<open>facs: (z - 1 / 2) * (z - -1 / 4)\<close>
203 facs = factors_from_solution L_L; \<comment> \<open>([3], Frm), 33 / ((z - 1 / 2) * (z - -1 / 4))\<close>
204 eql = Take (num_orig / facs); \<comment> \<open>([3], Res), ?A / (z - 1 / 2) + ?B / (z - -1 / 4)\<close>
205 eqr = (Try (Rewrite_Set ''ansatz_rls'' False)) eql;
206 \<comment> \<open>([4], Frm), 3 / ((z - 1 / 2) * (z - -1 / 4)) = ?A / (z - 1 / 2) + ?B / (z - -1 / 4)\<close>
207 eq = Take (eql = eqr); \<comment> \<open>([4], Res), 3 = ?A * (z - -1 / 4) + ?B * (z - 1 / 2)\<close>
208 eq = (Try (Rewrite_Set ''equival_trans'' False)) eq;
209 \<comment> \<open>eq = 3 = AA * (z - -1 / 4) + BB * (z - 1 / 2)\<close>
210 z1 = rhs (NTH 1 L_L); \<comment> \<open>z2 = -1 / 4\<close>
211 z2 = rhs (NTH 2 L_L); \<comment> \<open>([5], Frm), 3 = AA * (z - -1 / 4) + BB * (z - 1 / 2)\<close>
212 eq_a = Take eq; \<comment> \<open>([5], Res), 3 = AA * (1 / 2 - -1 / 4) + BB * (1 / 2 - 1 / 2)\<close>
213 eq_a = Substitute [zzz = z1] eq; \<comment> \<open>([6], Res), 3 = 3 * AA / 4\<close>
214 eq_a = (Rewrite_Set ''norm_Rational'' False) eq_a;
215 \<comment> \<open>----- ([7], Pbl), solve (3 = 3 * AA / 4, AA)\<close>
216 \<comment> \<open>([7], Res), [AA = 4]\<close>
217 sol_a = SubProblem (''Isac'', [''univariate'',''equation''], [''no_met''])
218 [BOOL eq_a, REAL (AA::real)] ; \<comment> \<open>a = 4\<close>
219 a = rhs (NTH 1 sol_a); \<comment> \<open>([8], Frm), 3 = AA * (z - -1 / 4) + BB * (z - 1 / 2)\<close>
220 eq_b = Take eq; \<comment> \<open>([8], Res), 3 = AA * (-1 / 4 - -1 / 4) + BB * (-1 / 4 - 1 / 2)\<close>
221 eq_b = Substitute [zzz = z2] eq_b; \<comment> \<open>([9], Res), 3 = -3 * BB / 4\<close>
222 eq_b = (Rewrite_Set ''norm_Rational'' False) eq_b; \<comment> \<open>([10], Pbl), solve (3 = -3 * BB / 4, BB)\<close>
223 sol_b = SubProblem (''Isac'', \<comment> \<open>([10], Res), [BB = -4]\<close>
224 [''univariate'',''equation''], [''no_met''])
225 [BOOL eq_b, REAL (BB::real)]; \<comment> \<open>b = -4\<close>
226 b = rhs (NTH 1 sol_b); \<comment> \<open>eqr = AA / (z - 1 / 2) + BB / (z - -1 / 4)\<close>
227 pbz = Take eqr; \<comment> \<open>([11], Res), 4 / (z - 1 / 2) + -4 / (z - -1 / 4)\<close>
228 pbz = Substitute [AA = a, BB = b] pbz \<comment> \<open>([], Res), 4 / (z - 1 / 2) + -4 / (z - -1 / 4)\<close>
230 setup \<open>KEStore_Elems.add_mets
231 [Specify.prep_met @{theory} "met_partial_fraction" [] Celem.e_metID
232 (["simplification","of_rationals","to_partial_fraction"],
233 [("#Given" ,["functionTerm t_t", "solveFor v_v"]),
234 (*("#Where" ,["((get_numerator t_t) has_degree_in v_v) <
235 ((get_denominator t_t) has_degree_in v_v)"]), TODO*)
236 ("#Find" ,["decomposedFunction p_p'''"])],
237 (*f_f = 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)), zzz: z*)
238 {rew_ord'="tless_true", rls'= Rule.e_rls, calc = [], srls = srls_partial_fraction, prls = Rule.e_rls,
239 crls = Rule.e_rls, errpats = [], nrls = Rule.e_rls},
240 (*([], Frm), Problem (Partial_Fractions, [partial_fraction, rational, simplification])*)
241 @{thm partial_fraction.simps})]
246 ["functionTerm (3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z))))",
247 "solveFor z", "functionTerm p_p"];
249 ("Partial_Fractions",
250 ["partial_fraction", "rational", "simplification"],
251 ["simplification","of_rationals","to_partial_fraction"]);
252 val (p,_,f,nxt,_,pt) = CalcTreeTEST [(fmz, (dI',pI',mI'))];