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(* Title: Test_Z_Transform
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Author: Jan Rocnik
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(c) copyright due to lincense terms.
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*)
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theory Inverse_Z_Transform imports Isac begin
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section {*trials towards Z transform *}
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text{*===============================*}
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subsection {*terms*}
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ML {*
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@{term "1 < || z ||"};
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@{term "z / (z - 1)"};
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@{term "-u -n - 1"};
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@{term "-u [-n - 1]"}; (*[ ] denotes lists !!!*)
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@{term "z /(z - 1) = -u [-n - 1]"};Isac
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@{term "1 < || z || ==> z / (z - 1) = -u [-n - 1]"};
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term2str @{term "1 < || z || ==> z / (z - 1) = -u [-n - 1]"};
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*}
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ML {*
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(*alpha --> "</alpha>" *)
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@{term "\<alpha> "};
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@{term "\<delta> "};
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@{term "\<phi> "};
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@{term "\<rho> "};
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term2str @{term "\<rho> "};
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*}
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subsection {*rules*}
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(*axiomatization "z / (z - 1) = -u [-n - 1]" Illegal variable name: "z / (z - 1) = -u [-n - 1]" *)
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(*definition "z / (z - 1) = -u [-n - 1]" Bad head of lhs: existing constant "op /"*)
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axiomatization where
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rule1: "1 = \<delta>[n]" and
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rule2: "|| z || > 1 ==> z / (z - 1) = u [n]" and
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rule3: "|| z || < 1 ==> z / (z - 1) = -u [-n - 1]" and
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rule4: "|| z || > || \<alpha> || ==> z / (z - \<alpha>) = \<alpha>^n * u [n]" and
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rule5: "|| z || < || \<alpha> || ==> z / (z - \<alpha>) = -(\<alpha>^n) * u [-n - 1]" and
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rule6: "|| z || > 1 ==> z/(z - 1)^2 = n * u [n]"
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ML {*
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@{thm rule1};
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@{thm rule2};
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@{thm rule3};
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@{thm rule4};
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*}
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subsection {*apply rules*}
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ML {*
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val inverse_Z = append_rls "inverse_Z" e_rls
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[ Thm ("rule3",num_str @{thm rule3}),
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Thm ("rule4",num_str @{thm rule4}),
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Thm ("rule1",num_str @{thm rule1})
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];
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val t = str2term "z / (z - 1) + z / (z - \<alpha>) + 1";
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val SOME (t', asm) = rewrite_set_ thy true inverse_Z t;
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term2str t' = "z / (z - ?\<delta> [?n]) + z / (z - \<alpha>) + ?\<delta> [?n]"; (*attention rule1 !!!*)
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*}
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ML {*
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val (thy, ro, er) = (@{theory}, tless_true, eval_rls);
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*}
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ML {*
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val SOME (t, asm1) = rewrite_ thy ro er true (num_str @{thm rule3}) t;
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term2str t = "- ?u [- ?n - 1] + z / (z - \<alpha>) + 1"; (*- real *)
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term2str t;
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*}
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ML {*
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val SOME (t, asm2) = rewrite_ thy ro er true (num_str @{thm rule4}) t;
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term2str t = "- ?u [- ?n - 1] + \<alpha> ^ ?n * ?u [?n] + 1"; (*- real *)
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term2str t;
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*}
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ML {*
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val SOME (t, asm3) = rewrite_ thy ro er true (num_str @{thm rule1}) t;
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term2str t = "- ?u [- ?n - 1] + \<alpha> ^ ?n * ?u [?n] + ?\<delta> [?n]"; (*- real *)
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term2str t;
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*}
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ML {*
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terms2str (asm1 @ asm2 @ asm3);
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*}
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section {*Prepare steps in CTP-based programming language*}
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text{*===================================================*}
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subsection {*prepare expression*}
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ML {*
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val ctxt = ProofContext.init_global @{theory};
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val ctxt = declare_constraints' [@{term "z::real"}] ctxt;
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val SOME fun1 = parseNEW ctxt "X z = 3 / (z - 1/4 + -1/8 * z ^ -1)"; term2str fun1;
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val SOME fun1' = parseNEW ctxt "X z = 3 / (z - 1/4 + -1/8 * (1/z))"; term2str fun1';
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*}
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axiomatization where
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ruleZY: "(X z = a / b) = (X' z = a / (z * b))"
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ML {*
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val (thy, ro, er) = (@{theory}, tless_true, eval_rls);
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val SOME (fun2, asm1) = rewrite_ thy ro er true @{thm ruleZY} fun1; term2str fun2;
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val SOME (fun2', asm1) = rewrite_ thy ro er true @{thm ruleZY} fun1'; term2str fun2';
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val SOME (fun3,_) = rewrite_set_ @{theory Isac} false norm_Rational fun2;
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term2str fun3; (*fails on x^(-1) TODO*)
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val SOME (fun3',_) = rewrite_set_ @{theory Isac} false norm_Rational fun2';
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term2str fun3'; (*OK*)
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val (_, expr) = HOLogic.dest_eq fun3'; term2str expr;
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*}
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subsection {*solve equation*}
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ML {*(*from test/Tools/isac/Minisubpbl/100-init-rootpbl.sml*)
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"----------- Minisubplb/100-init-rootp (*OK*)bl.sml ---------------------";
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val denominator = parseNEW ctxt "z^2 - 1/4*z - 1/8 = 0";
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val fmz = ["equality (z^2 - 1/4*z - 1/8 = (0::real))", "solveFor z","solutions L"];
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val (dI',pI',mI') =("Isac", ["univariate","equation"], ["no_met"]);
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(* ^^^^^^^^^^^^^^^^^^^^^^ TODO: ISAC determines type of eq*)
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*}
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ML {*
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val (p,_,f,nxt,_,pt) = CalcTreeTEST [(fmz, (dI',pI',mI'))];
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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(*[
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(([], Frm), solve (z ^ 2 - 1 / 4 * z - 1 / 8 = 0, z)),
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(([1], Frm), z ^ 2 - 1 / 4 * z - 1 / 8 = 0), bad rewrite order
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(([1], Res), -1 / 8 + z ^ 2 + -1 / 4 * z = 0), bad pattern
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(([2], Pbl), solve (-1 / 8 + z ^ 2 + -1 / 4 * z = 0, z)),
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(([2,1], Pbl), solve (-1 / 8 + z ^ 2 + -1 / 4 * z = 0, z)),
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(([2,1,1], Pbl), solve (-1 / 8 + z ^ 2 + -1 / 4 * z = 0, z)),
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(([2,1,1,1], Frm), -1 / 8 + z ^ 2 + -1 / 4 * z = 0)]
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*)
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*}
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ML {*
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val denominator = parseNEW ctxt "-1/8 + -1/4*z + z^2 = 0";
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(*ergebnis: [gleichung, was tun?, lösung]*)
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val fmz = ["equality (-1/8 + -1/4*z + z^2 = (0::real))", "solveFor z","solutions L"];
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(*liste der theoreme die zum lösen benötigt werden, aus isac, keine spezielle methode (no met)*)
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val (dI',pI',mI') =
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("Isac", ["pqFormula","degree_2","polynomial","univariate","equation"], ["no_met"]);
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(*schritte abarbeiten*)
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val (p,_,f,nxt,_,pt) = CalcTreeTEST [(fmz, (dI',pI',mI'))];
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt; (*val nxt = ("Empty_Tac", ...): tac'_*)
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show_pt pt;
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*}
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subsection {*partial fraction decomposition*}
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subsubsection {*solution of the equation*}
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ML {*
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val SOME solutions = parseNEW ctxt "[z=1/2, z=-1/4]";
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term2str solutions;
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atomty solutions;
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*}
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subsubsection {*get solutions out of list*}
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text {*in isac's CTP-based programming language: $let s_1 = NTH 1 solutions; s_2 = NTH 2...$*}
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ML {*
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val Const ("List.list.Cons", _) $ s_1 $ (Const ("List.list.Cons", _) $
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s_2 $ Const ("List.list.Nil", _)) = solutions;
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term2str s_1;
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term2str s_2;
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*}
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ML {* (*Solutions as Denominator --> Denominator1 = z - Zeropoint1, Denominator2 = z-Zeropoint2,...*)
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val xx = HOLogic.dest_eq s_1;
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val s_1' = HOLogic.mk_binop "Groups.minus_class.minus" xx;
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val xx = HOLogic.dest_eq s_2;
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val s_2' = HOLogic.mk_binop "Groups.minus_class.minus" xx;
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term2str s_1';
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term2str s_2';
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*}
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subsubsection {*build expression*}
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text {*in isac's CTP-based programming language: $let s_1 = Take numerator / (s_1 * s_2)$*}
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ML {*
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(*The Main Denominator is the multiplikation of the partial fraction denominators*)
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val denominator' = HOLogic.mk_binop "Groups.times_class.times" (s_1', s_2') ;
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val SOME numerator = parseNEW ctxt "3::real";
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val expr' = HOLogic.mk_binop "Rings.inverse_class.divide" (numerator, denominator');
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term2str expr';
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*}
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subsubsection {*Ansatz - create partial fractions out of our expression*}
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axiomatization where
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ansatz2: "n / (a*b) = A/a + B/(b::real)" and
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multiply_eq2: "(n / (a*b) = A/a + B/b) = (a*b*(n / (a*b)) = a*b*(A/a + B/b))"
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ML {*
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(*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*)
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val SOME (t1,_) = rewrite_ @{theory Isac} e_rew_ord e_rls false @{thm ansatz2} expr';
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term2str t1;
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atomty t1;
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val eq1 = HOLogic.mk_eq (expr', t1);
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term2str eq1;
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*}
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ML {*
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(*eliminate the demoninators by multiplying the left and the right side with the main denominator*)
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val SOME (eq2,_) = rewrite_ @{theory Isac} e_rew_ord e_rls false @{thm multiply_eq2} eq1;
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term2str eq2;
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*}
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ML {*
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(*simplificatoin*)
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val SOME (eq3,_) = rewrite_set_ @{theory Isac} false norm_Rational eq2;
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term2str eq3; (*?A ?B not simplified*)
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*}
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ML {*
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val SOME fract1 =
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parseNEW ctxt "(z - 1 / 2) * (z - -1 / 4) * (A / (z - 1 / 2) + B / (z - -1 / 4))"; (*A B !*)
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val SOME (fract2,_) = rewrite_set_ @{theory Isac} false norm_Rational fract1;
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term2str fract2 = "(A + -2 * B + 4 * A * z + 4 * B * z) / 4";
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(*term2str fract2 = "A * (1 / 4 + z) + B * (-1 / 2 + z)" would be more traditional*)
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*}
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ML {*
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val (numerator, denominator) = HOLogic.dest_eq eq3;
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val eq3' = HOLogic.mk_eq (numerator, fract1); (*A B !*)
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term2str eq3';
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*}
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ML {* (*MANDATORY: otherwise 3 = 0*)
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val SOME (eq3'' ,_) = rewrite_set_ @{theory Isac} false norm_Rational eq3';
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term2str eq3'';
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*}
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subsubsection {*get first koeffizient*}
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ML {*
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(*substitude z with the first zeropoint to get A*)
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val SOME (eq4_1,_) = rewrite_terms_ @{theory Isac} e_rew_ord e_rls [s_1] eq3'';
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term2str eq4_1;
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*}
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ML {*
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val SOME (eq4_2,_) = rewrite_set_ @{theory Isac} false norm_Rational eq4_1;
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term2str eq4_2;
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*}
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ML {*
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val fmz = ["equality (3 = 3 * A / (4::real))", "solveFor A","solutions L"];
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val (dI',pI',mI') =("Isac", ["univariate","equation"], ["no_met"]);
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*}
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ML {*
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(*solve the simple linear equilation for A TODO: return eq, not list of eq*)
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val (p,_,fa,nxt,_,pt) = CalcTreeTEST [(fmz, (dI',pI',mI'))];
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val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
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val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
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val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
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val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
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val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
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val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
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val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
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val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
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val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
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val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
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val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
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val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
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val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
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val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
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val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
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val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
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val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
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val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
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val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
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val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
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val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
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val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
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val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
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val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
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val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
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val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
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270 |
*}
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ML {*
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val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
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f2str fa;
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*}
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275 |
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276 |
subsubsection {*get second koeffizient*}
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277 |
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278 |
ML {*
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|
279 |
(*substitude z with the second zeropoint to get B*)
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280 |
val SOME (eq4b_1,_) = rewrite_terms_ @{theory Isac} e_rew_ord e_rls [s_2] eq3'';
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term2str eq4b_1;
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282 |
*}
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283 |
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284 |
ML {*
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val SOME (eq4b_2,_) = rewrite_set_ @{theory Isac} false norm_Rational eq4b_1;
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term2str eq4b_2;
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287 |
*}
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|
288 |
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289 |
ML {*
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290 |
(*solve the simple linear equilation for B TODO: return eq, not list of eq*)
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val fmz = ["equality (3 = -3 * B / (4::real))", "solveFor B","solutions L"];
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val (dI',pI',mI') =("Isac", ["univariate","equation"], ["no_met"]);
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val (p,_,fb,nxt,_,pt) = CalcTreeTEST [(fmz, (dI',pI',mI'))];
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val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
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|
295 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
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jan@42252
|
296 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
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jan@42252
|
297 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
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jan@42252
|
298 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
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jan@42252
|
299 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
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jan@42252
|
300 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
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jan@42252
|
301 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
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jan@42252
|
302 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
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jan@42252
|
303 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
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jan@42252
|
304 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
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jan@42252
|
305 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
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jan@42252
|
306 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
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jan@42252
|
307 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42252
|
308 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42252
|
309 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42252
|
310 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42252
|
311 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42252
|
312 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42252
|
313 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42252
|
314 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42252
|
315 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42252
|
316 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42252
|
317 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42252
|
318 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42252
|
319 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42252
|
320 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42252
|
321 |
f2str fb;
|
jan@42252
|
322 |
*}
|
jan@42252
|
323 |
|
jan@42252
|
324 |
ML {* (*check koeffizients*)
|
jan@42252
|
325 |
if f2str fa = "[A = 4]" then () else error "part.fract. eq4_1";
|
jan@42252
|
326 |
if f2str fb = "[B = -4]" then () else error "part.fract. eq4_1";
|
jan@42252
|
327 |
*}
|
jan@42252
|
328 |
|
jan@42252
|
329 |
subsubsection {*substitute expression with solutions*}
|
jan@42252
|
330 |
ML {*
|
jan@42252
|
331 |
*}
|
jan@42252
|
332 |
|
jan@42252
|
333 |
section {*Implement the Specification and the Method*}
|
jan@42252
|
334 |
text{*==============================================*}
|
jan@42252
|
335 |
subsection{*Define the Specification*}
|
jan@42252
|
336 |
ML {*
|
jan@42252
|
337 |
val thy = @{theory};
|
jan@42252
|
338 |
*}
|
jan@42252
|
339 |
ML {*
|
jan@42252
|
340 |
store_pbt
|
jan@42252
|
341 |
(prep_pbt thy "pbl_SP" [] e_pblID
|
jan@42252
|
342 |
(["SignalProcessing"], [], e_rls, NONE, []));
|
jan@42252
|
343 |
store_pbt
|
jan@42252
|
344 |
(prep_pbt thy "pbl_SP_Ztrans" [] e_pblID
|
jan@42252
|
345 |
(["Z_Transform","SignalProcessing"], [], e_rls, NONE, []));
|
jan@42252
|
346 |
store_pbt
|
jan@42252
|
347 |
(prep_pbt thy "pbl_SP_Ztrans_inv" [] e_pblID
|
jan@42252
|
348 |
(["inverse", "Z_Transform", "SignalProcessing"],
|
jan@42252
|
349 |
[("#Given" ,["equality X_eq"]),
|
jan@42252
|
350 |
("#Find" ,["equality n_eq"])
|
jan@42252
|
351 |
],
|
jan@42252
|
352 |
append_rls "e_rls" e_rls [(*for preds in where_*)], NONE,
|
jan@42252
|
353 |
[["TODO: path to method"]]));
|
jan@42252
|
354 |
|
jan@42252
|
355 |
show_ptyps();
|
jan@42252
|
356 |
get_pbt ["inverse","Z_Transform","SignalProcessing"];
|
jan@42252
|
357 |
*}
|
jan@42252
|
358 |
|
jan@42252
|
359 |
subsection{*Define the (Dummy-)Method*}
|
jan@42252
|
360 |
subsection {*Define Name and Signature for the Method*}
|
jan@42252
|
361 |
consts
|
jan@42252
|
362 |
InverseZTransform :: "[bool, bool] => bool"
|
jan@42252
|
363 |
("((Script InverseZTransform (_ =))// (_))" 9)
|
jan@42252
|
364 |
|
jan@42252
|
365 |
ML {*
|
jan@42252
|
366 |
store_met
|
jan@42252
|
367 |
(prep_met thy "met_SP" [] e_metID
|
jan@42252
|
368 |
(["SignalProcessing"], [],
|
jan@42252
|
369 |
{rew_ord'="tless_true", rls'= e_rls, calc = [], srls = e_rls, prls = e_rls,
|
jan@42252
|
370 |
crls = e_rls, nrls = e_rls}, "empty_script"));
|
jan@42252
|
371 |
store_met
|
jan@42252
|
372 |
(prep_met thy "met_SP_Ztrans" [] e_metID
|
jan@42252
|
373 |
(["SignalProcessing", "Z_Transform"], [],
|
jan@42252
|
374 |
{rew_ord'="tless_true", rls'= e_rls, calc = [], srls = e_rls, prls = e_rls,
|
jan@42252
|
375 |
crls = e_rls, nrls = e_rls}, "empty_script"));
|
jan@42252
|
376 |
*}
|
jan@42252
|
377 |
ML {*
|
jan@42252
|
378 |
store_met
|
jan@42252
|
379 |
(prep_met thy "met_SP_Ztrans_inv" [] e_metID
|
jan@42252
|
380 |
(["SignalProcessing", "Z_Transform", "inverse"], [],
|
jan@42252
|
381 |
{rew_ord'="tless_true", rls'= e_rls, calc = [], srls = e_rls, prls = e_rls,
|
jan@42252
|
382 |
crls = e_rls, nrls = e_rls},
|
jan@42252
|
383 |
"empty_script"
|
jan@42252
|
384 |
));
|
jan@42252
|
385 |
*}
|
jan@42252
|
386 |
ML {*(*
|
jan@42252
|
387 |
store_met
|
jan@42252
|
388 |
(prep_met thy "met_SP_Ztrans_inv" [] e_metID
|
jan@42252
|
389 |
(["SignalProcessing", "Z_Transform", "inverse"], [],
|
jan@42252
|
390 |
{rew_ord'="tless_true", rls'= e_rls, calc = [], srls = e_rls, prls = e_rls,
|
jan@42252
|
391 |
crls = e_rls, nrls = e_rls},
|
jan@42252
|
392 |
"Script InverseZTransform (Xeq::bool) =" ^
|
jan@42252
|
393 |
" (let X = Take Xeq;" ^
|
jan@42252
|
394 |
" X = Rewrite ruleZY False X" ^
|
jan@42252
|
395 |
" in X)"
|
jan@42252
|
396 |
));
|
jan@42252
|
397 |
*)*}
|
jan@42252
|
398 |
ML {*
|
jan@42252
|
399 |
show_mets();
|
jan@42252
|
400 |
get_met ["SignalProcessing","Z_Transform","inverse"];
|
jan@42252
|
401 |
*}
|
jan@42252
|
402 |
|
jan@42252
|
403 |
|
jan@42252
|
404 |
section {*Program in CTP-based language*}
|
jan@42252
|
405 |
text{*=================================*}
|
jan@42252
|
406 |
subsection {*Stepwise extend Program*}
|
jan@42252
|
407 |
ML {*
|
jan@42252
|
408 |
val str =
|
jan@42252
|
409 |
"Script InverseZTransform (Xeq::bool) =" ^
|
jan@42252
|
410 |
" Xeq";
|
jan@42252
|
411 |
*}
|
jan@42252
|
412 |
ML {*
|
jan@42252
|
413 |
val str =
|
jan@42252
|
414 |
"Script InverseZTransform (Xeq::bool) =" ^
|
jan@42252
|
415 |
" (let X = Take Xeq;" ^
|
jan@42252
|
416 |
" X = Rewrite ruleZY False X" ^
|
jan@42252
|
417 |
" in X)";
|
jan@42252
|
418 |
*}
|
jan@42252
|
419 |
ML {*
|
jan@42252
|
420 |
val thy = @{theory};
|
jan@42252
|
421 |
val sc = ((inst_abs thy) o term_of o the o (parse thy)) str;
|
jan@42252
|
422 |
*}
|
jan@42252
|
423 |
ML {*
|
jan@42252
|
424 |
term2str sc;
|
jan@42252
|
425 |
atomty sc
|
jan@42252
|
426 |
*}
|
jan@42252
|
427 |
|
jan@42252
|
428 |
subsection {*Stepwise Execute the Program*}
|
jan@42252
|
429 |
|
jan@42252
|
430 |
|
jan@42252
|
431 |
|
jan@42252
|
432 |
|
jan@42252
|
433 |
|
jan@42252
|
434 |
|
jan@42252
|
435 |
|
jan@42252
|
436 |
|
jan@42252
|
437 |
section {*Write Tests for Crucial Details*}
|
jan@42252
|
438 |
text{*===================================*}
|
jan@42252
|
439 |
ML {*
|
jan@42252
|
440 |
|
jan@42252
|
441 |
*}
|
jan@42252
|
442 |
|
jan@42252
|
443 |
section {*Integrate Program into Knowledge*}
|
jan@42252
|
444 |
ML {*
|
jan@42252
|
445 |
|
jan@42252
|
446 |
*}
|
jan@42252
|
447 |
|
jan@42252
|
448 |
end
|
jan@42252
|
449 |
|