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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 Build_Inverse_Z_Transform imports Isac
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begin
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text{* We stepwise build Inverse_Z_Transform.thy as an exercise.
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Because subsection "Stepwise Check the Program" requires
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Inverse_Z_Transform.thy as a subtheory of Isac.thy, the setup has been changed
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from "theory Inverse_Z_Transform imports Isac begin.." to the above.
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ATTENTION WITH NAMES OF IDENTIFIERS WHEN GOING INTO INTERNALS:
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Here in this theory there are the internal names twice, for instance we have
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(Thm.derivation_name @{thm rule1} = "Build_Inverse_Z_Transform.rule1") = true;
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but actually in us will be "Inverse_Z_Transform.rule1"
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*}
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ML {*val thy = @{theory Isac};*}
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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 Isac}, 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 for CTP-based programming language*}
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text{*TODO insert Calculation (Referenz?!)
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The goal... realized in sections below, in Sect.\ref{spec-meth} and Sect.\ref{prog-steps}
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the reader is advised to jump between the subsequent subsections and the respective steps in Sect.\ref{prog-steps}
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*}
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subsection {*prepare expression \label{prep-expr}*}
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ML {*
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val ctxt = ProofContext.init_global @{theory Isac};
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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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subsubsection {*multply with z*}
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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 Isac}, 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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*}
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subsubsection {*get argument of X': z is the variable the equation is solved for*}
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text{*grep... Atools.thy, Tools.thy contain general utilities: eval_argument_in, eval_rhs, eval_lhs,...
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grep -r "fun eva_" ... shows all functions witch can be used in a script.
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lookup this files how to build and handle such functions.
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the next section shows how to introduce such a function.
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*}
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text{*---------------------------begin partial fractions snip--------------------------*}
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subsubsection {*get the denominator out of a fraction*}
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text {*get denominator should become a constant for the isabelle parser: *}
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consts
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get_denominator :: "real => real"
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text {*Attention Build_Inverse_Z_Transform
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works only because definition was copied into Rationals.thy
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*}
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ML {*
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(*("get_denominator", ("Rational.get'_denominator", eval_get_denominator ""))*)
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fun eval_get_denominator (thmid:string) _
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(t as Const ("Build_Inverse_Z_Transform.get_denominator", _) $
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(Const ("Rings.inverse_class.divide", _) $ num $
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denom)) thy =
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SOME (mk_thmid thmid ""
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(Print_Mode.setmp [] (Syntax.string_of_term (thy2ctxt thy)) denom) "",
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Trueprop $ (mk_equality (t, denom)))
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| eval_get_denominator _ _ _ _ = NONE;
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*}
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ML {*
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val t = @{term "get_denominator ((a +x)/b)"};
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val SOME (_, t') = eval_get_denominator "" 0 t @{theory};
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term2str t';
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*}
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ML {*
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val t as Const ("Build_Inverse_Z_Transform.get_denominator", _) $
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(Const ("Rings.inverse_class.divide", _) $ num $
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denom) = t;
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*}
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ML {*
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(*
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if term2s t' = "(argument_in M_b x) = x" then ()
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else error "atools.sml:(argument_in M_b x) = x ???";
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*)
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*}
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subsubsection {*build equation from given term*}
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ML {*
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val (_, expr) = HOLogic.dest_eq fun3'; term2str expr;
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val (_, denom) = HOLogic.dest_bin "Rings.inverse_class.divide" (type_of expr) expr;
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term2str denom = "-1 + -2 * z + 8 * z ^^^ 2";
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*}
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text {*we have rhs in the language, but we need a function
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which gets the denominator of a fraction*}
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text{*---------------------------end partial fractions snip--------------------------*}
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subsection {*solve equation*}
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text {*this type of equation if too general for the present program*}
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ML {*
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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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text {*Does the Equation Match the Specification ?*}
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ML {*
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match_pbl fmz (get_pbt ["univariate","equation"]);
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*}
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ML {*Context.theory_name thy = "Isac"(*==================================================*)*}
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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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val fmz = (*specification*)
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["equality (-1/8 + (-1/4)*z + z^^^2 = (0::real))", (*equality*)
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"solveFor z", (*bound variable*)
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"solutions L"]; (*identifier for solution*)
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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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*}
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text {*Does the Other Equation Match the Specification ?*}
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ML {*
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match_pbl fmz (get_pbt ["pqFormula","degree_2","polynomial","univariate","equation"]);
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*}
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text {*Solve Equation Stepwise*}
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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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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;
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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;
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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; (*nxt =..,Check_elementwise "Assumptions")*)
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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; f2str f;
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(*[z = 1 / 8 + sqrt (9 / 16) / 2, z = 1 / 8 + -1 * sqrt (9 / 16) / 2] TODO sqrt*)
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show_pt pt;
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val SOME f = parseNEW ctxt "[z=1/2, z=-1/4]";
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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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|
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term2str s_1;
|
neuper@42279
|
276 |
term2str s_2;
|
neuper@42279
|
277 |
*}
|
neuper@42279
|
278 |
|
neuper@42279
|
279 |
ML {* (*Solutions as Denominator --> Denominator1 = z - Zeropoint1, Denominator2 = z-Zeropoint2,...*)
|
neuper@42279
|
280 |
val xx = HOLogic.dest_eq s_1;
|
neuper@42279
|
281 |
val s_1' = HOLogic.mk_binop "Groups.minus_class.minus" xx;
|
neuper@42279
|
282 |
val xx = HOLogic.dest_eq s_2;
|
neuper@42279
|
283 |
val s_2' = HOLogic.mk_binop "Groups.minus_class.minus" xx;
|
neuper@42279
|
284 |
term2str s_1';
|
neuper@42279
|
285 |
term2str s_2';
|
neuper@42279
|
286 |
*}
|
neuper@42279
|
287 |
|
neuper@42279
|
288 |
subsubsection {*build expression*}
|
neuper@42279
|
289 |
text {*in isac's CTP-based programming language: let s_1 = Take numerator / (s_1 * s_2)*}
|
neuper@42279
|
290 |
ML {*
|
neuper@42279
|
291 |
(*The Main Denominator is the multiplikation of the partial fraction denominators*)
|
neuper@42279
|
292 |
val denominator' = HOLogic.mk_binop "Groups.times_class.times" (s_1', s_2') ;
|
neuper@42279
|
293 |
val SOME numerator = parseNEW ctxt "3::real";
|
neuper@42279
|
294 |
|
neuper@42279
|
295 |
val expr' = HOLogic.mk_binop "Rings.inverse_class.divide" (numerator, denominator');
|
neuper@42279
|
296 |
term2str expr';
|
neuper@42279
|
297 |
*}
|
neuper@42279
|
298 |
|
neuper@42279
|
299 |
subsubsection {*Ansatz - create partial fractions out of our expression*}
|
neuper@42279
|
300 |
ML {*Context.theory_name thy = "Isac"(*==================================================*)*}
|
neuper@42279
|
301 |
|
neuper@42279
|
302 |
axiomatization where
|
neuper@42279
|
303 |
ansatz2: "n / (a*b) = A/a + B/(b::real)" and
|
neuper@42279
|
304 |
multiply_eq2: "(n / (a*b) = A/a + B/b) = (a*b*(n / (a*b)) = a*b*(A/a + B/b))"
|
neuper@42279
|
305 |
|
neuper@42279
|
306 |
ML {*
|
neuper@42279
|
307 |
(*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*)
|
neuper@42279
|
308 |
val SOME (t1,_) = rewrite_ @{theory Isac} e_rew_ord e_rls false @{thm ansatz2} expr';
|
neuper@42279
|
309 |
term2str t1; atomty t1;
|
neuper@42279
|
310 |
val eq1 = HOLogic.mk_eq (expr', t1);
|
neuper@42279
|
311 |
term2str eq1;
|
neuper@42279
|
312 |
*}
|
neuper@42279
|
313 |
ML {*
|
neuper@42279
|
314 |
(*eliminate the demoninators by multiplying the left and the right side with the main denominator*)
|
neuper@42279
|
315 |
val SOME (eq2,_) = rewrite_ @{theory Isac} e_rew_ord e_rls false @{thm multiply_eq2} eq1;
|
neuper@42279
|
316 |
term2str eq2;
|
neuper@42279
|
317 |
*}
|
neuper@42279
|
318 |
ML {*
|
neuper@42279
|
319 |
(*simplificatoin*)
|
neuper@42279
|
320 |
val SOME (eq3,_) = rewrite_set_ @{theory Isac} false norm_Rational eq2;
|
neuper@42279
|
321 |
term2str eq3; (*?A ?B not simplified*)
|
neuper@42279
|
322 |
*}
|
neuper@42279
|
323 |
ML {*
|
neuper@42279
|
324 |
val SOME fract1 =
|
neuper@42279
|
325 |
parseNEW ctxt "(z - 1 / 2) * (z - -1 / 4) * (A / (z - 1 / 2) + B / (z - -1 / 4))"; (*A B !*)
|
neuper@42279
|
326 |
val SOME (fract2,_) = rewrite_set_ @{theory Isac} false norm_Rational fract1;
|
neuper@42279
|
327 |
term2str fract2 = "(A + -2 * B + 4 * A * z + 4 * B * z) / 4";
|
neuper@42279
|
328 |
(*term2str fract2 = "A * (1 / 4 + z) + B * (-1 / 2 + z)" would be more traditional*)
|
neuper@42279
|
329 |
*}
|
neuper@42279
|
330 |
ML {*
|
neuper@42279
|
331 |
val (numerator, denominator) = HOLogic.dest_eq eq3;
|
neuper@42279
|
332 |
val eq3' = HOLogic.mk_eq (numerator, fract1); (*A B !*)
|
neuper@42279
|
333 |
term2str eq3';
|
neuper@42279
|
334 |
(*MANDATORY: simplify (and remove denominator) otherwise 3 = 0*)
|
neuper@42279
|
335 |
val SOME (eq3'' ,_) = rewrite_set_ @{theory Isac} false norm_Rational eq3';
|
neuper@42279
|
336 |
term2str eq3'';
|
neuper@42279
|
337 |
*}
|
neuper@42279
|
338 |
ML {*Context.theory_name thy = "Isac"(*==================================================*)*}
|
neuper@42279
|
339 |
|
neuper@42279
|
340 |
subsubsection {*get first koeffizient*}
|
neuper@42279
|
341 |
|
neuper@42279
|
342 |
ML {*
|
neuper@42279
|
343 |
(*substitude z with the first zeropoint to get A*)
|
neuper@42279
|
344 |
val SOME (eq4_1,_) = rewrite_terms_ @{theory Isac} e_rew_ord e_rls [s_1] eq3'';
|
neuper@42279
|
345 |
term2str eq4_1;
|
neuper@42279
|
346 |
|
neuper@42279
|
347 |
val SOME (eq4_2,_) = rewrite_set_ @{theory Isac} false norm_Rational eq4_1;
|
neuper@42279
|
348 |
term2str eq4_2;
|
neuper@42279
|
349 |
|
neuper@42279
|
350 |
val fmz = ["equality (3 = 3 * A / (4::real))", "solveFor A","solutions L"];
|
neuper@42279
|
351 |
val (dI',pI',mI') =("Isac", ["univariate","equation"], ["no_met"]);
|
neuper@42279
|
352 |
(*solve the simple linear equilation for A TODO: return eq, not list of eq*)
|
neuper@42279
|
353 |
val (p,_,fa,nxt,_,pt) = CalcTreeTEST [(fmz, (dI',pI',mI'))];
|
neuper@42279
|
354 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
355 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
356 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
357 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
358 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
359 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
360 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
361 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
362 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
363 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
364 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
365 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
366 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
367 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
368 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
369 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
370 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
371 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
372 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
373 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
374 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
375 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
376 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
377 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
378 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
379 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
380 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
381 |
f2str fa;
|
neuper@42279
|
382 |
*}
|
neuper@42279
|
383 |
|
neuper@42279
|
384 |
subsubsection {*get second koeffizient*}
|
neuper@42279
|
385 |
ML {*thy*}
|
neuper@42279
|
386 |
|
neuper@42279
|
387 |
ML {*
|
neuper@42279
|
388 |
(*substitude z with the second zeropoint to get B*)
|
neuper@42279
|
389 |
val SOME (eq4b_1,_) = rewrite_terms_ @{theory Isac} e_rew_ord e_rls [s_2] eq3'';
|
neuper@42279
|
390 |
term2str eq4b_1;
|
neuper@42279
|
391 |
|
neuper@42279
|
392 |
val SOME (eq4b_2,_) = rewrite_set_ @{theory Isac} false norm_Rational eq4b_1;
|
neuper@42279
|
393 |
term2str eq4b_2;
|
neuper@42279
|
394 |
*}
|
neuper@42279
|
395 |
ML {*
|
neuper@42279
|
396 |
(*solve the simple linear equilation for B TODO: return eq, not list of eq*)
|
neuper@42279
|
397 |
val fmz = ["equality (3 = -3 * B / (4::real))", "solveFor B","solutions L"];
|
neuper@42279
|
398 |
val (dI',pI',mI') =("Isac", ["univariate","equation"], ["no_met"]);
|
neuper@42279
|
399 |
val (p,_,fb,nxt,_,pt) = CalcTreeTEST [(fmz, (dI',pI',mI'))];
|
neuper@42279
|
400 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
401 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
402 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
403 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
404 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
405 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
406 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
407 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
408 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
409 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
410 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
411 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
412 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
413 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
414 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
415 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
416 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
417 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
418 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
419 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
420 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
421 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
422 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
423 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
424 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
425 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
426 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
427 |
f2str fb;
|
neuper@42279
|
428 |
*}
|
neuper@42279
|
429 |
|
neuper@42279
|
430 |
ML {* (*check koeffizients*)
|
neuper@42279
|
431 |
if f2str fa = "[A = 4]" then () else error "part.fract. eq4_1";
|
neuper@42279
|
432 |
if f2str fb = "[B = -4]" then () else error "part.fract. eq4_1";
|
neuper@42279
|
433 |
*}
|
neuper@42279
|
434 |
|
neuper@42279
|
435 |
subsubsection {*substitute expression with solutions*}
|
neuper@42279
|
436 |
ML {*
|
neuper@42279
|
437 |
*}
|
neuper@42279
|
438 |
ML {*thy*}
|
neuper@42279
|
439 |
|
jan@42296
|
440 |
section {*Implement the Specification and the Method \label{spec-meth}*}
|
neuper@42279
|
441 |
text{*==============================================*}
|
neuper@42279
|
442 |
subsection{*Define the Field Descriptions for the specification*}
|
neuper@42279
|
443 |
consts
|
neuper@42279
|
444 |
filterExpression :: "bool => una"
|
neuper@42279
|
445 |
stepResponse :: "bool => una"
|
neuper@42279
|
446 |
|
neuper@42279
|
447 |
subsection{*Define the Specification*}
|
neuper@42279
|
448 |
ML {*
|
neuper@42279
|
449 |
store_pbt
|
neuper@42279
|
450 |
(prep_pbt thy "pbl_SP" [] e_pblID
|
neuper@42279
|
451 |
(["SignalProcessing"], [], e_rls, NONE, []));
|
neuper@42279
|
452 |
store_pbt
|
neuper@42279
|
453 |
(prep_pbt thy "pbl_SP_Ztrans" [] e_pblID
|
neuper@42279
|
454 |
(["Z_Transform","SignalProcessing"], [], e_rls, NONE, []));
|
neuper@42279
|
455 |
*}
|
neuper@42279
|
456 |
ML {*thy*}
|
neuper@42279
|
457 |
ML {*
|
neuper@42279
|
458 |
store_pbt
|
neuper@42279
|
459 |
(prep_pbt thy "pbl_SP_Ztrans_inv" [] e_pblID
|
neuper@42279
|
460 |
(["inverse", "Z_Transform", "SignalProcessing"],
|
neuper@42279
|
461 |
[("#Given" ,["filterExpression X_eq"]),
|
neuper@42279
|
462 |
("#Find" ,["stepResponse n_eq"])
|
neuper@42279
|
463 |
],
|
neuper@42279
|
464 |
append_rls "e_rls" e_rls [(*for preds in where_*)], NONE,
|
neuper@42279
|
465 |
[["SignalProcessing","Z_Transform","inverse"]]));
|
neuper@42279
|
466 |
|
neuper@42279
|
467 |
show_ptyps();
|
neuper@42279
|
468 |
get_pbt ["inverse","Z_Transform","SignalProcessing"];
|
neuper@42279
|
469 |
*}
|
neuper@42279
|
470 |
|
neuper@42279
|
471 |
subsection {*Define Name and Signature for the Method*}
|
neuper@42279
|
472 |
consts
|
neuper@42279
|
473 |
InverseZTransform :: "[bool, bool] => bool"
|
neuper@42279
|
474 |
("((Script InverseZTransform (_ =))// (_))" 9)
|
neuper@42279
|
475 |
|
neuper@42279
|
476 |
subsection {*Setup Parent Nodes in Hierarchy of Method*}
|
neuper@42279
|
477 |
ML {*
|
neuper@42279
|
478 |
store_met
|
neuper@42279
|
479 |
(prep_met thy "met_SP" [] e_metID
|
neuper@42279
|
480 |
(["SignalProcessing"], [],
|
neuper@42279
|
481 |
{rew_ord'="tless_true", rls'= e_rls, calc = [], srls = e_rls, prls = e_rls,
|
neuper@42279
|
482 |
crls = e_rls, nrls = e_rls}, "empty_script"));
|
neuper@42279
|
483 |
store_met
|
neuper@42279
|
484 |
(prep_met thy "met_SP_Ztrans" [] e_metID
|
neuper@42279
|
485 |
(["SignalProcessing", "Z_Transform"], [],
|
neuper@42279
|
486 |
{rew_ord'="tless_true", rls'= e_rls, calc = [], srls = e_rls, prls = e_rls,
|
neuper@42279
|
487 |
crls = e_rls, nrls = e_rls}, "empty_script"));
|
neuper@42279
|
488 |
*}
|
neuper@42279
|
489 |
ML {*
|
neuper@42279
|
490 |
store_met
|
neuper@42279
|
491 |
(prep_met thy "met_SP_Ztrans_inv" [] e_metID
|
neuper@42279
|
492 |
(["SignalProcessing", "Z_Transform", "inverse"],
|
neuper@42279
|
493 |
[("#Given" ,["filterExpression X_eq"]),
|
neuper@42279
|
494 |
("#Find" ,["stepResponse n_eq"])
|
neuper@42279
|
495 |
],
|
neuper@42279
|
496 |
{rew_ord'="tless_true", rls'= e_rls, calc = [], srls = e_rls, prls = e_rls,
|
neuper@42279
|
497 |
crls = e_rls, nrls = e_rls},
|
neuper@42279
|
498 |
"empty_script"
|
neuper@42279
|
499 |
));
|
neuper@42279
|
500 |
*}
|
neuper@42279
|
501 |
ML {*
|
neuper@42279
|
502 |
store_met
|
neuper@42279
|
503 |
(prep_met thy "met_SP_Ztrans_inv" [] e_metID
|
neuper@42279
|
504 |
(["SignalProcessing", "Z_Transform", "inverse"],
|
neuper@42279
|
505 |
[("#Given" ,["filterExpression X_eq"]),
|
neuper@42279
|
506 |
("#Find" ,["stepResponse n_eq"])
|
neuper@42279
|
507 |
],
|
neuper@42279
|
508 |
{rew_ord'="tless_true", rls'= e_rls, calc = [], srls = e_rls, prls = e_rls,
|
neuper@42279
|
509 |
crls = e_rls, nrls = e_rls},
|
neuper@42279
|
510 |
"Script InverseZTransform (Xeq::bool) =" ^
|
neuper@42279
|
511 |
" (let X = Take Xeq;" ^
|
neuper@42279
|
512 |
" X = Rewrite ruleZY False X" ^
|
neuper@42279
|
513 |
" in X)"
|
neuper@42279
|
514 |
));
|
jan@42299
|
515 |
*}
|
jan@42299
|
516 |
ML {*
|
neuper@42279
|
517 |
show_mets();
|
jan@42299
|
518 |
*}
|
jan@42299
|
519 |
ML {*
|
neuper@42279
|
520 |
get_met ["SignalProcessing","Z_Transform","inverse"];
|
neuper@42279
|
521 |
*}
|
neuper@42279
|
522 |
|
jan@42296
|
523 |
section {*Program in CTP-based language \label{prog-steps}*}
|
neuper@42279
|
524 |
text{*=================================*}
|
neuper@42279
|
525 |
subsection {*Stepwise extend Program*}
|
neuper@42279
|
526 |
ML {*
|
neuper@42279
|
527 |
val str =
|
neuper@42279
|
528 |
"Script InverseZTransform (Xeq::bool) =" ^
|
neuper@42279
|
529 |
" Xeq";
|
neuper@42279
|
530 |
*}
|
neuper@42279
|
531 |
ML {*
|
neuper@42279
|
532 |
val str =
|
neuper@42279
|
533 |
"Script InverseZTransform (Xeq::bool) =" ^ (*(1/z) instead of z ^^^ -1*)
|
neuper@42279
|
534 |
" (let X = Take Xeq;" ^
|
neuper@42279
|
535 |
" X' = Rewrite ruleZY False X;" ^ (*z * denominator*)
|
neuper@42279
|
536 |
" X' = (Rewrite_Set norm_Rational False) X'" ^ (*simplify*)
|
neuper@42279
|
537 |
" in X)";
|
neuper@42279
|
538 |
(*NONE*)
|
neuper@42279
|
539 |
"Script InverseZTransform (Xeq::bool) =" ^ (*(1/z) instead of z ^^^ -1*)
|
neuper@42279
|
540 |
" (let X = Take Xeq;" ^
|
neuper@42279
|
541 |
" X' = Rewrite ruleZY False X;" ^ (*z * denominator*)
|
neuper@42279
|
542 |
" X' = (Rewrite_Set norm_Rational False) X';" ^ (*simplify*)
|
neuper@42279
|
543 |
" X' = (SubProblem (Isac',[pqFormula,degree_2,polynomial,univariate,equation], [no_met]) " ^
|
neuper@42279
|
544 |
" [BOOL e_e, REAL v_v])" ^
|
neuper@42279
|
545 |
" in X)";
|
neuper@42279
|
546 |
*}
|
neuper@42279
|
547 |
ML {*
|
neuper@42279
|
548 |
val str =
|
neuper@42279
|
549 |
"Script InverseZTransform (Xeq::bool) =" ^ (*(1/z) instead of z ^^^ -1*)
|
neuper@42279
|
550 |
" (let X = Take Xeq;" ^
|
neuper@42279
|
551 |
" X' = Rewrite ruleZY False X;" ^ (*z * denominator*)
|
neuper@42279
|
552 |
" X' = (Rewrite_Set norm_Rational False) X';" ^ (*simplify*)
|
neuper@42279
|
553 |
" funterm = rhs X'" ^ (*drop X'= for equation solving*)
|
neuper@42279
|
554 |
" in X)";
|
neuper@42279
|
555 |
*}
|
neuper@42279
|
556 |
ML {*
|
neuper@42290
|
557 |
val str =
|
neuper@42290
|
558 |
"Script InverseZTransform (X_eq::bool) =" ^ (*(1/z) instead of z ^^^ -1*)
|
neuper@42290
|
559 |
" (let X = Take X_eq;" ^
|
neuper@42290
|
560 |
" X' = Rewrite ruleZY False X;" ^ (*z * denominator*)
|
neuper@42290
|
561 |
" X' = (Rewrite_Set norm_Rational False) X';" ^ (*simplify*)
|
jan@42298
|
562 |
" (X'_z::real) = lhs X';" ^
|
jan@42298
|
563 |
" (z::real) = argument_in X'_z;" ^
|
jan@42298
|
564 |
" (funterm::real) = rhs X';" ^ (*drop X' z = for equation solving*)
|
jan@42298
|
565 |
" (denom::real) = get_denominator funterm;" ^ (*get_denominator*)
|
jan@42298
|
566 |
" (equ::bool) = (denom = (0::real));" ^
|
neuper@42290
|
567 |
" (L_L::bool list) = " ^
|
neuper@42290
|
568 |
" (SubProblem (Test', " ^
|
neuper@42290
|
569 |
" [linear,univariate,equation,test]," ^
|
neuper@42290
|
570 |
" [Test,solve_linear]) " ^
|
neuper@42290
|
571 |
" [BOOL equ, REAL z]) " ^
|
neuper@42290
|
572 |
" in X)"
|
neuper@42290
|
573 |
;
|
neuper@42290
|
574 |
|
neuper@42279
|
575 |
parse thy str;
|
neuper@42279
|
576 |
val sc = ((inst_abs thy) o term_of o the o (parse thy)) str;
|
neuper@42279
|
577 |
atomty sc;
|
neuper@42279
|
578 |
|
neuper@42279
|
579 |
*}
|
jan@42300
|
580 |
|
jan@42300
|
581 |
text {*
|
jan@42300
|
582 |
This ruleset contains all functions that are in the script;
|
jan@42300
|
583 |
The evaluation of the functions is done by rewriting using this ruleset.
|
jan@42300
|
584 |
*}
|
jan@42300
|
585 |
|
neuper@42279
|
586 |
ML {*
|
neuper@42290
|
587 |
val srls = Rls {id="srls_InverseZTransform",
|
neuper@42290
|
588 |
preconds = [], rew_ord = ("termlessI",termlessI),
|
neuper@42290
|
589 |
erls = append_rls "erls_in_srls_InverseZTransform" e_rls
|
neuper@42290
|
590 |
[(*for asm in NTH_CONS ...*) Calc ("Orderings.ord_class.less",eval_equ "#less_"),
|
neuper@42290
|
591 |
(*2nd NTH_CONS pushes n+-1 into asms*) Calc("Groups.plus_class.plus", eval_binop "#add_")
|
neuper@42290
|
592 |
],
|
neuper@42290
|
593 |
srls = Erls, calc = [],
|
neuper@42290
|
594 |
rules =
|
neuper@42290
|
595 |
[Thm ("NTH_CONS",num_str @{thm NTH_CONS}),
|
neuper@42290
|
596 |
Calc("Groups.plus_class.plus", eval_binop "#add_"),
|
neuper@42290
|
597 |
Thm ("NTH_NIL",num_str @{thm NTH_NIL}),
|
neuper@42290
|
598 |
Calc("Tools.lhs", eval_lhs"eval_lhs_"), (*<=== ONLY USED*)
|
neuper@42290
|
599 |
Calc("Tools.rhs", eval_rhs"eval_rhs_"), (*<=== ONLY USED*)
|
jan@42300
|
600 |
Calc("Atools.argument'_in", eval_argument_in "Atools.argument'_in"),
|
jan@42300
|
601 |
Calc("Rationals.get'_denominator", eval_get_denominator "Rationals.get'_denominator")
|
neuper@42290
|
602 |
],
|
neuper@42290
|
603 |
scr = EmptyScr};
|
neuper@42279
|
604 |
*}
|
neuper@42279
|
605 |
|
neuper@42279
|
606 |
|
neuper@42279
|
607 |
subsection {*Store Final Version of Program for Execution*}
|
neuper@42279
|
608 |
ML {*
|
neuper@42279
|
609 |
store_met
|
neuper@42279
|
610 |
(prep_met thy "met_SP_Ztrans_inv" [] e_metID
|
neuper@42279
|
611 |
(["SignalProcessing", "Z_Transform", "inverse"],
|
neuper@42279
|
612 |
[("#Given" ,["filterExpression X_eq"]),
|
neuper@42279
|
613 |
("#Find" ,["stepResponse n_eq"])
|
neuper@42279
|
614 |
],
|
neuper@42290
|
615 |
{rew_ord'="tless_true", rls'= e_rls, calc = [], srls = srls,
|
neuper@42290
|
616 |
prls = e_rls,
|
neuper@42279
|
617 |
crls = e_rls, nrls = e_rls},
|
neuper@42289
|
618 |
"Script InverseZTransform (X_eq::bool) =" ^ (*(1/z) instead of z ^^^ -1*)
|
neuper@42289
|
619 |
" (let X = Take X_eq;" ^
|
neuper@42279
|
620 |
" X' = Rewrite ruleZY False X;" ^ (*z * denominator*)
|
neuper@42279
|
621 |
" X' = (Rewrite_Set norm_Rational False) X';" ^ (*simplify*)
|
jan@42298
|
622 |
" (X'_z::real) = lhs X';" ^ (**)
|
jan@42298
|
623 |
" (z::real) = argument_in X'_z;" ^ (**)
|
jan@42298
|
624 |
" (funterm::real) = rhs X';" ^ (*drop X' z = for equation solving*)
|
jan@42298
|
625 |
" (denom::real) = get_denominator funterm;" ^ (*get_denominator*)
|
jan@42298
|
626 |
" (equ::bool) = (denom = (0::real));" ^
|
neuper@42290
|
627 |
" (L_L::bool list) = " ^
|
neuper@42290
|
628 |
" (SubProblem (Test', " ^
|
neuper@42290
|
629 |
" [linear,univariate,equation,test]," ^
|
neuper@42290
|
630 |
" [Test,solve_linear]) " ^
|
neuper@42290
|
631 |
" [BOOL equ, REAL z]) " ^
|
neuper@42279
|
632 |
" in X)"
|
neuper@42279
|
633 |
));
|
neuper@42279
|
634 |
*}
|
neuper@42290
|
635 |
ML {*
|
jan@42300
|
636 |
|
jan@42300
|
637 |
val Script scr = (#scr o get_met) ["SignalProcessing", "Z_Transform", "inverse"];
|
jan@42300
|
638 |
atomty scr
|
neuper@42279
|
639 |
|
neuper@42290
|
640 |
*}
|
neuper@42279
|
641 |
|
neuper@42281
|
642 |
subsection {*Check the Program*}
|
neuper@42279
|
643 |
|
neuper@42281
|
644 |
subsubsection {*Check the formalization*}
|
neuper@42279
|
645 |
ML {*
|
neuper@42279
|
646 |
val fmz = ["filterExpression (X = 3 / (z - 1/4 + -1/8 * (1/(z::real))))",
|
neuper@42279
|
647 |
"stepResponse (x[n::real]::bool)"];
|
neuper@42279
|
648 |
val (dI,pI,mI) = ("Isac", ["inverse", "Z_Transform", "SignalProcessing"],
|
neuper@42279
|
649 |
["SignalProcessing","Z_Transform","inverse"]);
|
neuper@42281
|
650 |
|
neuper@42281
|
651 |
val ([(1, [1], "#Given", Const ("Inverse_Z_Transform.filterExpression", _),
|
neuper@42281
|
652 |
[Const ("HOL.eq", _) $ _ $ _]),
|
neuper@42281
|
653 |
(2, [1], "#Find", Const ("Inverse_Z_Transform.stepResponse", _),
|
neuper@42281
|
654 |
[Free ("x", _) $ _])],
|
neuper@42281
|
655 |
_) = prep_ori fmz thy ((#ppc o get_pbt) pI);
|
neuper@42281
|
656 |
*}
|
neuper@42290
|
657 |
ML {*
|
neuper@42290
|
658 |
val Script sc = (#scr o get_met) ["SignalProcessing","Z_Transform","inverse"];
|
neuper@42290
|
659 |
atomty sc;
|
neuper@42290
|
660 |
*}
|
neuper@42281
|
661 |
|
neuper@42281
|
662 |
subsubsection {*Stepwise check the program*}
|
neuper@42281
|
663 |
ML {*
|
jan@42296
|
664 |
trace_script := false; print_depth 999;
|
neuper@42281
|
665 |
val fmz = ["filterExpression (X z = 3 / (z - 1/4 + -1/8 * (1/(z::real))))",
|
neuper@42281
|
666 |
"stepResponse (x[n::real]::bool)"];
|
neuper@42281
|
667 |
val (dI,pI,mI) = ("Isac", ["inverse", "Z_Transform", "SignalProcessing"],
|
neuper@42281
|
668 |
["SignalProcessing","Z_Transform","inverse"]);
|
jan@42296
|
669 |
val (p,_,f,nxt,_,pt) = CalcTreeTEST [(fmz, (dI,pI,mI))];
|
jan@42296
|
670 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42296
|
671 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42296
|
672 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42296
|
673 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42296
|
674 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42296
|
675 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt; "nxt = Apply_Method";
|
jan@42297
|
676 |
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))"; (*TODO naming!*)
|
jan@42296
|
677 |
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)))";
|
jan@42300
|
678 |
*}
|
jan@42300
|
679 |
ML {*
|
jan@42296
|
680 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt; "nxt = Take 24 / (-1 + -2 * z + 8 * z ^^^ 2)";
|
jan@42300
|
681 |
*}
|
jan@42300
|
682 |
ML {*
|
jan@42300
|
683 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
neuper@42279
|
684 |
*}
|
neuper@42279
|
685 |
ML {*
|
neuper@42289
|
686 |
show_pt pt;
|
neuper@42279
|
687 |
*}
|
neuper@42279
|
688 |
ML {*
|
neuper@42279
|
689 |
*}
|
neuper@42279
|
690 |
ML {*
|
neuper@42279
|
691 |
|
neuper@42279
|
692 |
*}
|
neuper@42279
|
693 |
ML {*
|
neuper@42279
|
694 |
|
neuper@42279
|
695 |
*}
|
neuper@42279
|
696 |
ML {*
|
neuper@42279
|
697 |
|
neuper@42279
|
698 |
*}
|
neuper@42279
|
699 |
ML {*
|
neuper@42279
|
700 |
|
neuper@42279
|
701 |
*}
|
neuper@42279
|
702 |
|
neuper@42279
|
703 |
ML {*
|
neuper@42279
|
704 |
|
neuper@42279
|
705 |
*}
|
neuper@42279
|
706 |
ML {*
|
neuper@42290
|
707 |
@{theory Isac}
|
neuper@42279
|
708 |
*}
|
neuper@42279
|
709 |
|
neuper@42279
|
710 |
ML {*
|
neuper@42279
|
711 |
*}
|
neuper@42279
|
712 |
ML {*
|
neuper@42279
|
713 |
*}
|
neuper@42279
|
714 |
ML {*
|
neuper@42279
|
715 |
*}
|
neuper@42279
|
716 |
|
neuper@42279
|
717 |
|
neuper@42279
|
718 |
|
neuper@42279
|
719 |
|
neuper@42279
|
720 |
|
neuper@42279
|
721 |
|
neuper@42279
|
722 |
|
neuper@42279
|
723 |
|
neuper@42279
|
724 |
section {*Write Tests for Crucial Details*}
|
neuper@42279
|
725 |
text{*===================================*}
|
neuper@42279
|
726 |
ML {*
|
neuper@42279
|
727 |
|
neuper@42279
|
728 |
*}
|
neuper@42279
|
729 |
|
neuper@42279
|
730 |
section {*Integrate Program into Knowledge*}
|
neuper@42279
|
731 |
ML {*
|
neuper@42290
|
732 |
@{theory Isac}
|
neuper@42279
|
733 |
*}
|
neuper@42279
|
734 |
|
neuper@42279
|
735 |
end
|
neuper@42279
|
736 |
|