1.1 --- a/src/Tools/isac/Knowledge/Partial_Fractions.thy Thu Mar 07 16:50:20 2019 +0100
1.2 +++ b/src/Tools/isac/Knowledge/Partial_Fractions.thy Thu Mar 07 17:22:20 2019 +0100
1.3 @@ -21,11 +21,8 @@
1.4 subsection \<open>eval_ functions\<close>
1.5 consts
1.6 factors_from_solution :: "bool list => real"
1.7 - drop_questionmarks :: "'a => 'a"
1.8 -(* version for later switch to partial_function
1.9 - A :: real \<comment> \<open>PROG redesign (Substitute [A = a, B = b]) pbz ?\<close>
1.10 - B :: real \<comment> \<open>PROG redesign (Substitute [A = a, B = b]) pbz ?\<close>
1.11 -*)
1.12 + AA :: real
1.13 + BB :: real
1.14
1.15 text \<open>these might be used for variants of fac_from_sol\<close>
1.16 ML \<open>
1.17 @@ -70,56 +67,36 @@
1.18 | eval_factors_from_solution _ _ _ _ = NONE;
1.19 \<close>
1.20
1.21 -text \<open>'ansatz' introduces '?Vars' (questionable design); drop these again\<close>
1.22 -ML \<open>
1.23 -(*("drop_questionmarks", ("Partial_Fractions.drop_questionmarks", eval_drop_questionmarks ""))*)
1.24 -fun eval_drop_questionmarks (thmid:string) _
1.25 - (t as Const ("Partial_Fractions.drop_questionmarks", _) $ tm) thy =
1.26 - if TermC.contains_Var tm
1.27 - then
1.28 - let
1.29 - val tm' = TermC.var2free tm
1.30 - in SOME (TermC.mk_thmid thmid (Rule.term_to_string''' thy tm') "",
1.31 - HOLogic.Trueprop $ (TermC.mk_equality (t, tm')))
1.32 - end
1.33 - else NONE
1.34 - | eval_drop_questionmarks _ _ _ _ = NONE;
1.35 -\<close>
1.36 -
1.37 -text \<open>store eval_ functions for calls from Scripts\<close>
1.38 -setup \<open>KEStore_Elems.add_calcs
1.39 - [("drop_questionmarks", ("Partial_Fractions.drop'_questionmarks", eval_drop_questionmarks ""))]\<close>
1.40 -
1.41 subsection \<open>'ansatz' for partial fractions\<close>
1.42 axiomatization where
1.43 - ansatz_2nd_order: "n / (a*b) = A/a + B/b" and
1.44 - ansatz_3rd_order: "n / (a*b*c) = A/a + B/b + C/c" and
1.45 - ansatz_4th_order: "n / (a*b*c*d) = A/a + B/b + C/c + D/d" and
1.46 + ansatz_2nd_order: "n / (a*b) = AA/a + BB/b" and
1.47 + ansatz_3rd_order: "n / (a*b*c) = AA/a + BB/b + C/c" and
1.48 + ansatz_4th_order: "n / (a*b*c*d) = AA/a + BB/b + C/c + D/d" and
1.49 (*version 1*)
1.50 - equival_trans_2nd_order: "(n/(a*b) = A/a + B/b) = (n = A*b + B*a)" and
1.51 - equival_trans_3rd_order: "(n/(a*b*c) = A/a + B/b + C/c) = (n = A*b*c + B*a*c + C*a*b)" and
1.52 - equival_trans_4th_order: "(n/(a*b*c*d) = A/a + B/b + C/c + D/d) =
1.53 - (n = A*b*c*d + B*a*c*d + C*a*b*d + D*a*b*c)" and
1.54 + equival_trans_2nd_order: "(n/(a*b) = AA/a + BB/b) = (n = AA*b + BB*a)" and
1.55 + 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
1.56 + equival_trans_4th_order: "(n/(a*b*c*d) = AA/a + BB/b + C/c + D/d) =
1.57 + (n = AA*b*c*d + BB*a*c*d + C*a*b*d + D*a*b*c)" and
1.58 (*version 2: not yet used, see partial_fractions.sml*)
1.59 - multiply_2nd_order: "(n/x = A/a + B/b) = (a*b*n/x = A*b + B*a)" and
1.60 - multiply_3rd_order: "(n/x = A/a + B/b + C/c) = (a*b*c*n/x = A*b*c + B*a*c + C*a*b)" and
1.61 + multiply_2nd_order: "(n/x = AA/a + BB/b) = (a*b*n/x = AA*b + BB*a)" and
1.62 + 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
1.63 multiply_4th_order:
1.64 - "(n/x = A/a + B/b + C/c + D/d) = (a*b*c*d*n/x = A*b*c*d + B*a*c*d + C*a*b*d + D*a*b*c)"
1.65 + "(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)"
1.66
1.67 text \<open>Probably the optimal formalization woudl be ...
1.68
1.69 - multiply_2nd_order: "x = a*b ==> (n/x = A/a + B/b) = (a*b*n/x = A*b + B*a)" and
1.70 + multiply_2nd_order: "x = a*b ==> (n/x = AA/a + BB/b) = (a*b*n/x = AA*b + BB*a)" and
1.71 multiply_3rd_order: "x = a*b*c ==>
1.72 - (n/x = A/a + B/b + C/c) = (a*b*c*n/x = A*b*c + B*a*c + C*a*b)" and
1.73 + (n/x = AA/a + BB/b + C/c) = (a*b*c*n/x = AA*b*c + BB*a*c + C*a*b)" and
1.74 multiply_4th_order: "x = a*b*c*d ==>
1.75 - (n/x = A/a + B/b + C/c + D/d) = (a*b*c*d*n/x = A*b*c*d + B*a*c*d + C*a*b*d + D*a*b*c)"
1.76 + (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)"
1.77
1.78 ... because it would allow to start the ansatz as follows
1.79 (1) 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z))) = 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))
1.80 (2) 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z))) = AA / (z - 1 / 2) + BB / (z - -1 / 4)
1.81 (3) (z - 1 / 2) * (z - -1 / 4) * 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z))) =
1.82 (z - 1 / 2) * (z - -1 / 4) * AA / (z - 1 / 2) + BB / (z - -1 / 4)
1.83 -(4) 3 = A * (z - -1 / 4) + B * (z - 1 / 2)
1.84 +(4) 3 = AA * (z - -1 / 4) + BB * (z - 1 / 2)
1.85
1.86 ... (1==>2) ansatz
1.87 (2==>3) multiply_*
1.88 @@ -209,21 +186,10 @@
1.89 Rule.Calc("Rational.get_denominator", eval_get_denominator "#get_denominator"),
1.90 Rule.Calc("Rational.get_numerator", eval_get_numerator "#get_numerator"),
1.91 Rule.Calc("Partial_Fractions.factors_from_solution",
1.92 - eval_factors_from_solution "#factors_from_solution"),
1.93 - Rule.Calc("Partial_Fractions.drop_questionmarks", eval_drop_questionmarks "#drop_?")],
1.94 + eval_factors_from_solution "#factors_from_solution")
1.95 + ],
1.96 scr = Rule.EmptyScr};
1.97 \<close>
1.98 -ML \<open>
1.99 -eval_drop_questionmarks;
1.100 -\<close>
1.101 -ML \<open>
1.102 -val ctxt = Proof_Context.init_global @{theory};
1.103 -val SOME t = TermC.parseNEW ctxt "eqr = drop_questionmarks eqr";
1.104 -\<close>
1.105 -ML \<open>
1.106 -TermC.parseNEW ctxt "decomposedFunction p_p'''";
1.107 -TermC.parseNEW ctxt "decomposedFunction";
1.108 -\<close>
1.109
1.110 (* current version, error outcommented *)
1.111 (*ok
1.112 @@ -245,28 +211,26 @@
1.113 \<comment> \<open>([4], Frm), 3 / ((z - 1 / 2) * (z - -1 / 4)) = ?A / (z - 1 / 2) + ?B / (z - -1 / 4)\<close>
1.114 eq = Take (eql = eqr); \<comment> \<open>([4], Res), 3 = ?A * (z - -1 / 4) + ?B * (z - 1 / 2)\<close>
1.115 eq = (Try (Rewrite_Set ''equival_trans'' False)) eq;
1.116 - \<comment> \<open>eq = 3 = A * (z - -1 / 4) + B * (z - 1 / 2)\<close>
1.117 - eq = drop_questionmarks eq; \<comment> \<open>z1 = 1 / 2)\<close>
1.118 + \<comment> \<open>eq = 3 = AA * (z - -1 / 4) + BB * (z - 1 / 2)\<close>
1.119 z1 = rhs (NTH 1 L_L); \<comment> \<open>z2 = -1 / 4\<close>
1.120 - z2 = rhs (NTH 2 L_L); \<comment> \<open>([5], Frm), 3 = A * (z - -1 / 4) + B * (z - 1 / 2)\<close>
1.121 - eq_a = Take eq; \<comment> \<open>([5], Res), 3 = A * (1 / 2 - -1 / 4) + B * (1 / 2 - 1 / 2)\<close>
1.122 - eq_a = Substitute [zzz = z1] eq; \<comment> \<open>([6], Res), 3 = 3 * A / 4\<close>
1.123 + z2 = rhs (NTH 2 L_L); \<comment> \<open>([5], Frm), 3 = AA * (z - -1 / 4) + BB * (z - 1 / 2)\<close>
1.124 + eq_a = Take eq; \<comment> \<open>([5], Res), 3 = AA * (1 / 2 - -1 / 4) + BB * (1 / 2 - 1 / 2)\<close>
1.125 + eq_a = Substitute [zzz = z1] eq; \<comment> \<open>([6], Res), 3 = 3 * AA / 4\<close>
1.126 eq_a = (Rewrite_Set ''norm_Rational'' False) eq_a;
1.127 -\<comment> \<open>----- ([7], Pbl), solve (3 = 3 * A / 4, A)\<close>
1.128 - \<comment> \<open>([7], Res), [A = 4]\<close>
1.129 +\<comment> \<open>----- ([7], Pbl), solve (3 = 3 * AA / 4, AA)\<close>
1.130 + \<comment> \<open>([7], Res), [AA = 4]\<close>
1.131 sol_a = SubProblem (''Isac'', [''univariate'',''equation''], [''no_met''])
1.132 - [BOOL eq_a, REAL (A::real)] ; \<comment> \<open>a = 4\<close>
1.133 - a = rhs (NTH 1 sol_a); \<comment> \<open>([8], Frm), 3 = A * (z - -1 / 4) + B * (z - 1 / 2)\<close>
1.134 - eq_b = Take eq; \<comment> \<open>([8], Res), 3 = A * (-1 / 4 - -1 / 4) + B * (-1 / 4 - 1 / 2)\<close>
1.135 - eq_b = Substitute [zzz = z2] eq_b; \<comment> \<open>([9], Res), 3 = -3 * B / 4\<close>
1.136 - eq_b = (Rewrite_Set ''norm_Rational'' False) eq_b; \<comment> \<open>([10], Pbl), solve (3 = -3 * B / 4, B)\<close>
1.137 - sol_b = SubProblem (''Isac'', \<comment> \<open>([10], Res), [B = -4]\<close>
1.138 + [BOOL eq_a, REAL (AA::real)] ; \<comment> \<open>a = 4\<close>
1.139 + a = rhs (NTH 1 sol_a); \<comment> \<open>([8], Frm), 3 = AA * (z - -1 / 4) + BB * (z - 1 / 2)\<close>
1.140 + eq_b = Take eq; \<comment> \<open>([8], Res), 3 = AA * (-1 / 4 - -1 / 4) + BB * (-1 / 4 - 1 / 2)\<close>
1.141 + eq_b = Substitute [zzz = z2] eq_b; \<comment> \<open>([9], Res), 3 = -3 * BB / 4\<close>
1.142 + eq_b = (Rewrite_Set ''norm_Rational'' False) eq_b; \<comment> \<open>([10], Pbl), solve (3 = -3 * BB / 4, BB)\<close>
1.143 + sol_b = SubProblem (''Isac'', \<comment> \<open>([10], Res), [BB = -4]\<close>
1.144 [''univariate'',''equation''], [''no_met''])
1.145 - [BOOL eq_b, REAL (B::real)]; \<comment> \<open>b = -4\<close>
1.146 - b = rhs (NTH 1 sol_b); \<comment> \<open>eqr = A / (z - 1 / 2) + B / (z - -1 / 4)\<close>
1.147 - eqr = drop_questionmarks eqr; \<comment> \<open>([11], Frm), A / (z - 1 / 2) + B / (z - -1 / 4)\<close>
1.148 + [BOOL eq_b, REAL (BB::real)]; \<comment> \<open>b = -4\<close>
1.149 + b = rhs (NTH 1 sol_b); \<comment> \<open>eqr = AA / (z - 1 / 2) + BB / (z - -1 / 4)\<close>
1.150 pbz = Take eqr; \<comment> \<open>([11], Res), 4 / (z - 1 / 2) + -4 / (z - -1 / 4)\<close>
1.151 - pbz = Substitute [A = a, B = b] pbz \<comment> \<open>([], Res), 4 / (z - 1 / 2) + -4 / (z - -1 / 4)\<close>
1.152 + pbz = Substitute [AA = a, BB = b] pbz \<comment> \<open>([], Res), 4 / (z - 1 / 2) + -4 / (z - -1 / 4)\<close>
1.153 in pbz) "
1.154 *)
1.155 setup \<open>KEStore_Elems.add_mets
1.156 @@ -307,45 +271,43 @@
1.157 " (eq::bool) = Take (eql = eqr); " ^
1.158 (*([4], Res), 3 = ?A * (z - -1 / 4) + ?B * (z - 1 / 2)*)
1.159 " eq = (Try (Rewrite_Set ''equival_trans'' False)) eq;" ^
1.160 - (* eq = 3 = A * (z - -1 / 4) + B * (z - 1 / 2)*)
1.161 - " eq = drop_questionmarks eq; " ^
1.162 + (* eq = 3 = AA * (z - -1 / 4) + BB * (z - 1 / 2)*)
1.163 (* z1 = 1 / 2*)
1.164 " (z1::real) = (rhs (NTH 1 L_L)); " ^
1.165 (* z2 = -1 / 4*)
1.166 " (z2::real) = (rhs (NTH 2 L_L)); " ^
1.167 - (*([5], Frm), 3 = A * (z - -1 / 4) + B * (z - 1 / 2)*)
1.168 + (*([5], Frm), 3 = AA * (z - -1 / 4) + BB * (z - 1 / 2)*)
1.169 " (eq_a::bool) = Take eq; " ^
1.170 - (*([5], Res), 3 = A * (1 / 2 - -1 / 4) + B * (1 / 2 - 1 / 2)*)
1.171 + (*([5], Res), 3 = AA * (1 / 2 - -1 / 4) + BB * (1 / 2 - 1 / 2)*)
1.172 " eq_a = (Substitute [zzz = z1]) eq; " ^
1.173 - (*([6], Res), 3 = 3 * A / 4*)
1.174 + (*([6], Res), 3 = 3 * AA / 4*)
1.175 " eq_a = (Rewrite_Set ''norm_Rational'' False) eq_a; " ^
1.176
1.177 - (*([7], Pbl), solve (3 = 3 * A / 4, A)*)
1.178 + (*([7], Pbl), solve (3 = 3 * AA / 4, AA)*)
1.179 " (sol_a::bool list) = " ^
1.180 " (SubProblem (''Isac'', [''univariate'',''equation''], [''no_met'']) " ^
1.181 - (*([7], Res), [A = 4]*)
1.182 - " [BOOL eq_a, REAL (A::real)]); " ^
1.183 + (*([7], Res), [AA = 4]*)
1.184 + " [BOOL eq_a, REAL (AA::real)]); " ^
1.185 (* a = 4*)
1.186 " (a::real) = (rhs (NTH 1 sol_a)); " ^
1.187 - (*([8], Frm), 3 = A * (z - -1 / 4) + B * (z - 1 / 2)*)
1.188 + (*([8], Frm), 3 = AA * (z - -1 / 4) + BB * (z - 1 / 2)*)
1.189 " (eq_b::bool) = Take eq; " ^
1.190 - (*([8], Res), 3 = A * (-1 / 4 - -1 / 4) + B * (-1 / 4 - 1 / 2)*)
1.191 + (*([8], Res), 3 = AA * (-1 / 4 - -1 / 4) + BB * (-1 / 4 - 1 / 2)*)
1.192 " eq_b = (Substitute [zzz = z2]) eq_b; " ^
1.193 - (*([9], Res), 3 = -3 * B / 4*)
1.194 + (*([9], Res), 3 = -3 * BB / 4*)
1.195 " eq_b = (Rewrite_Set ''norm_Rational'' False) eq_b; " ^
1.196 - (*([10], Pbl), solve (3 = -3 * B / 4, B)*)
1.197 + (*([10], Pbl), solve (3 = -3 * BB / 4, BB)*)
1.198 " (sol_b::bool list) = " ^
1.199 " (SubProblem (''Isac'', [''univariate'',''equation''], [''no_met'']) " ^
1.200 - (*([10], Res), [B = -4]*)
1.201 - " [BOOL eq_b, REAL (B::real)]); " ^
1.202 + (*([10], Res), [BB = -4]*)
1.203 + " [BOOL eq_b, REAL (BB::real)]); " ^
1.204 (* b = -4*)
1.205 " (b::real) = (rhs (NTH 1 sol_b)); " ^
1.206 - (* eqr = A / (z - 1 / 2) + B / (z - -1 / 4)*)
1.207 - " eqr = drop_questionmarks eqr; " ^
1.208 - (*([11], Frm), A / (z - 1 / 2) + B / (z - -1 / 4)*)
1.209 + (* eqr = AA / (z - 1 / 2) + BB / (z - -1 / 4)*)
1.210 + (*([11], Frm), AA / (z - 1 / 2) + BB / (z - -1 / 4)*)
1.211 " (pbz::real) = Take eqr; " ^
1.212 (*([11], Res), 4 / (z - 1 / 2) + -4 / (z - -1 / 4)*)
1.213 - " pbz = ((Substitute [A = a, B = b]) pbz) " ^
1.214 + " pbz = ((Substitute [AA = a, BB = b]) pbz) " ^
1.215 (*([], Res), 4 / (z - 1 / 2) + -4 / (z - -1 / 4)*)
1.216 "in pbz)"
1.217 )]