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(* Title: Build_Inverse_Z_Transform
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Author: Jan Rocnik
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(c) copyright due to license terms.
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*)
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theory Build_Inverse_Z_Transform imports Inverse_Z_Transform
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begin
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text{* We stepwise build \ttfamily Inverse\_Z\_Transform.thy \normalfont as an
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exercise. Because Subsection~\ref{sec:stepcheck} requires
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\ttfamily Inverse\_Z\_Transform.thy \normalfont as a subtheory of \ttfamily
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Isac.thy\normalfont, the setup has been changed from \ttfamily theory
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Inverse\_Z\_Transform imports Isac \normalfont to the above one.
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\par \noindent
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\begin{center}
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\textbf{Attention with the names of identifiers when going into internals!}
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\end{center}
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Here in this theory there are the internal names twice, for instance we have
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\ttfamily (Thm.derivation\_name @{thm rule1} =
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"Build\_Inverse\_Z\_Transform.rule1") = true; \normalfont
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but actually in us will be \ttfamily Inverse\_Z\_Transform.rule1 \normalfont
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*}
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section {*Trials towards the Z-Transform\label{sec:trials}*}
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ML {*val thy = @{theory};*}
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subsection {*Notations and Terms*}
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text{*\noindent Try which notations we are able to use.*}
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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]"};
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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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text{*\noindent Try which symbols we are able to use and how we generate them.*}
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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]"
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Illegal variable name: "z / (z - 1) = -u [-n - 1]" *)
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(*definition "z / (z - 1) = -u [-n - 1]"
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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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text{*\noindent Check the rules for their correct notation.
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(See the machine output.)*}
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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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text{*\noindent We try to apply the rules to a given expression.*}
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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]";
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(*
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* Attention rule1 is applied before the expression is
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* checked for rule4 which would be correct!!!
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*)
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*}
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ML {* val (thy, ro, er) = (@{theory}, tless_true, eval_rls); *}
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ML {*
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val SOME (t, asm1) =
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rewrite_ thy ro er true (num_str @{thm rule3}) t;
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term2str t = "- ?u [- ?n - 1] + z / (z - \<alpha>) + 1";
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(*- real *)
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term2str t;
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val SOME (t, asm2) =
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rewrite_ thy ro er true (num_str @{thm rule4}) t;
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term2str t = "- ?u [- ?n - 1] + \<alpha> ^^^ ?n * ?u [?n] + 1";
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(*- real *)
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term2str t;
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val SOME (t, asm3) =
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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]";
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(*- real *)
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term2str t;
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*}
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ML {* terms2str (asm1 @ asm2 @ asm3); *}
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section{*Prepare Steps for TP-based programming Language\label{sec:prepstep}*}
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text{*
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\par \noindent The following sections are challenging with the CTP-based
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possibilities of building the program. The goal is realized in
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Section~\ref{spec-meth} and Section~\ref{prog-steps}.
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\par The reader is advised to jump between the subsequent subsections and
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the respective steps in Section~\ref{prog-steps}. By comparing
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Section~\ref{sec:calc:ztrans} the calculation can be comprehended step
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by step.
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*}
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subsection {*Prepare Expression\label{prep-expr}*}
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text{*\noindent We try two different notations and check which of them
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Isabelle can handle best.*}
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ML {*
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val ctxt = Proof_Context.init_global @{theory};
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val ctxt = declare_constraints' [@{term "z::real"}] ctxt;
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val SOME fun1 =
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parseNEW ctxt "X z = 3 / (z - 1/4 + -1/8 * z ^^^ -1)"; term2str fun1;
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val SOME fun1' =
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parseNEW ctxt "X z = 3 / (z - 1/4 + -1/8 * (1/z))"; term2str fun1';
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*}
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subsubsection {*Prepare Numerator and Denominator*}
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text{*\noindent The partial fraction decomposition is only possible if we
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get the bound variable out of the numerator. Therefor we divide
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the expression by $z$. Follow up the Calculation at
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Section~\ref{sec:calc:ztrans} line number 02.*}
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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) =
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rewrite_ thy ro er true @{thm ruleZY} fun1; term2str fun2;
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val SOME (fun2', asm1) =
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rewrite_ thy ro er true @{thm ruleZY} fun1'; term2str fun2';
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val SOME (fun3,_) =
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rewrite_set_ @{theory} false norm_Rational fun2;
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term2str fun3;
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(*
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* Fails on x^^^(-1)
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* We solve this problem by using 1/x as a workaround.
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*)
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val SOME (fun3',_) =
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rewrite_set_ @{theory} false norm_Rational fun2';
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term2str fun3';
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(*
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* OK - workaround!
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*)
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*}
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subsubsection {*Get the Argument of the Expression X'*}
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text{*\noindent We use \texttt{grep} for finding possibilities how we can
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extract the bound variable in the expression. \ttfamily Atools.thy,
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Tools.thy \normalfont contain general utilities: \ttfamily
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eval\_argument\_in, eval\_rhs, eval\_lhs,\ldots \normalfont
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\ttfamily grep -r "fun eva\_" * \normalfont shows all functions
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witch can be used in a script. Lookup this files how to build
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and handle such functions.
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\par The next section shows how to introduce such a function.
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*}
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subsubsection{*Decompose the Given Term Into lhs and rhs*}
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ML {*
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val (_, expr) = HOLogic.dest_eq fun3'; term2str expr;
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val (_, denom) =
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HOLogic.dest_bin "Rings.divide_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{*\noindent We have rhs\footnote{Note: lhs means \em Left Hand Side
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\normalfont and rhs means \em Right Hand Side \normalfont and indicates
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the left or the right part of an equation.} in the Script language, but
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we need a function which gets the denominator of a fraction.*}
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subsubsection{*Get the Denominator and Numerator out of a Fraction*}
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text{*\noindent The self written functions in e.g. \texttt{get\_denominator}
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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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get_numerator :: "real => real"
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text {*\noindent With the above definition we run into problems when we parse
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the Script \texttt{InverseZTransform}. This leads to \em ambiguous
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parse trees. \normalfont We avoid this by moving the definition
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to \ttfamily Rational.thy \normalfont and re-building {\sisac}.
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\par \noindent ATTENTION: From now on \ttfamily
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Build\_Inverse\_Z\_Transform \normalfont mimics a build from scratch;
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it only works due to re-building {\sisac} several times (indicated
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explicitly).
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*}
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ML {*
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(*
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*("get_denominator",
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* ("Rational.get_denominator", eval_get_denominator ""))
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*)
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fun eval_get_denominator (thmid:string) _
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(t as Const ("Rational.get_denominator", _) $
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(Const ("Rings.divide_class.divide", _) $num
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$denom)) thy =
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SOME (mk_thmid thmid "" (term_to_string''' 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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text {*\noindent For the tests of \ttfamily eval\_get\_denominator \normalfont
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see \ttfamily test/Knowledge/rational.sml\normalfont*}
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text {*\noindent \ttfamily get\_numerator \normalfont should also become a
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constant for the Isabelle parser, follow up the \texttt{const}
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declaration above.*}
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ML {*
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(*
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*("get_numerator",
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* ("Rational.get_numerator", eval_get_numerator ""))
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*)
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fun eval_get_numerator (thmid:string) _
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(t as Const ("Rational.get_numerator", _) $
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(Const ("Rings.divide_class.divide", _) $num
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$denom )) thy =
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SOME (mk_thmid thmid "" (term_to_string''' thy num) "",
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Trueprop $ (mk_equality (t, num)))
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| eval_get_numerator _ _ _ _ = NONE;
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*}
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text {*\noindent We discovered several problems by implementing the
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\ttfamily get\_numerator \normalfont function. Remember when
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putting new functions to {\sisac}, put them in a thy file and rebuild
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{\sisac}, also put them in the ruleset for the script!*}
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subsection {*Solve Equation\label{sec:solveq}*}
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text {*\noindent We have to find the zeros of the term, therefor we use our
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\ttfamily get\_denominator \normalfont function from the step before
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and try to solve the second order equation. (Follow up the Calculation
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in Section~\ref{sec:calc:ztrans} Subproblem 2) Note: This type of
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equation is too general for the present program.
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\par We know that this equation can be categorized as \em univariate
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equation \normalfont and solved with the functions {\sisac} provides
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for this equation type. Later on {\sisac} should determine the type
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of the given equation self.*}
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ML {*
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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))",
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"solveFor z",
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"solutions L"];
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val (dI',pI',mI') =("Isac", ["univariate","equation"], ["no_met"]);
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*}
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text {*\noindent Check if the given equation matches the specification of this
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equation type.*}
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ML {*
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match_pbl fmz (get_pbt ["univariate","equation"]);
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*}
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text{*\noindent We switch up to the {\sisac} Context and try to solve the
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equation with a more specific type definition.*}
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ML {*
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Context.theory_name thy = "Isac";
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val denominator = parseNEW ctxt "-1 + -2 * z + 8 * z ^^^ 2 = 0";
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val fmz = (*specification*)
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["equality (-1 + -2 * z + 8 * z ^^^ 2 = (0::real))",(*equality*)
|
jan@42369
|
282 |
"solveFor z", (*bound variable*)
|
jan@42369
|
283 |
"solutions L"]; (*identifier for
|
jan@42369
|
284 |
solution*)
|
jan@42369
|
285 |
val (dI',pI',mI') =
|
jan@42369
|
286 |
("Isac",
|
jan@42369
|
287 |
["abcFormula","degree_2","polynomial","univariate","equation"],
|
jan@42369
|
288 |
["no_met"]);
|
neuper@42279
|
289 |
*}
|
neuper@42279
|
290 |
|
jan@42369
|
291 |
text {*\noindent Check if the (other) given equation matches the
|
neuper@42376
|
292 |
specification of this equation type.*}
|
jan@42369
|
293 |
|
neuper@42279
|
294 |
ML {*
|
s1210629013@55355
|
295 |
match_pbl fmz (get_pbt
|
jan@42369
|
296 |
["abcFormula","degree_2","polynomial","univariate","equation"]);
|
neuper@42279
|
297 |
*}
|
neuper@42279
|
298 |
|
jan@42369
|
299 |
text {*\noindent We stepwise solve the equation. This is done by the
|
jan@42369
|
300 |
use of a so called calc tree seen downwards.*}
|
jan@42369
|
301 |
|
neuper@42279
|
302 |
ML {*
|
jan@42369
|
303 |
val (p,_,f,nxt,_,pt) = CalcTreeTEST [(fmz, (dI',pI',mI'))];
|
jan@42369
|
304 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
305 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
306 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
307 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
308 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
309 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
310 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
311 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
312 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
313 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
314 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
315 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
316 |
(*
|
jan@42369
|
317 |
* nxt =..,Check_elementwise "Assumptions")
|
jan@42369
|
318 |
*)
|
jan@42369
|
319 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
320 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt; f2str f;
|
jan@42369
|
321 |
(*
|
jan@42369
|
322 |
* [z = 1 / 2, z = -1 / 4]
|
jan@42369
|
323 |
*)
|
wneuper@59265
|
324 |
Chead.show_pt pt;
|
jan@42369
|
325 |
val SOME f = parseNEW ctxt "[z=1/2, z=-1/4]";
|
neuper@42279
|
326 |
*}
|
neuper@42279
|
327 |
|
jan@42369
|
328 |
subsection {*Partial Fraction Decomposition\label{sec:pbz}*}
|
neuper@42376
|
329 |
text{*\noindent We go on with the decomposition of our expression. Follow up the
|
jan@42369
|
330 |
Calculation in Section~\ref{sec:calc:ztrans} Step~3 and later on
|
jan@42369
|
331 |
Subproblem~1.*}
|
jan@42369
|
332 |
subsubsection {*Solutions of the Equation*}
|
jan@42369
|
333 |
text{*\noindent We get the solutions of the before solved equation in a list.*}
|
jan@42369
|
334 |
|
jan@42369
|
335 |
ML {*
|
jan@42369
|
336 |
val SOME solutions = parseNEW ctxt "[z=1/2, z=-1/4]";
|
jan@42369
|
337 |
term2str solutions;
|
jan@42369
|
338 |
atomty solutions;
|
neuper@42279
|
339 |
*}
|
jan@42369
|
340 |
|
jan@42369
|
341 |
subsubsection {*Get Solutions out of a List*}
|
jan@42374
|
342 |
text {*\noindent In {\sisac}'s TP-based programming language:
|
jan@42381
|
343 |
\begin{verbatim}
|
jan@42381
|
344 |
let $ $ s_1 = NTH 1 $ solutions; $ s_2 = NTH 2... $
|
jan@42381
|
345 |
\end{verbatim}
|
jan@42381
|
346 |
can be useful.
|
jan@42381
|
347 |
*}
|
jan@42369
|
348 |
|
neuper@42335
|
349 |
ML {*
|
jan@42369
|
350 |
val Const ("List.list.Cons", _) $ s_1 $ (Const ("List.list.Cons", _)
|
jan@42369
|
351 |
$ s_2 $ Const ("List.list.Nil", _)) = solutions;
|
jan@42369
|
352 |
term2str s_1;
|
jan@42369
|
353 |
term2str s_2;
|
neuper@42335
|
354 |
*}
|
jan@42369
|
355 |
|
neuper@42376
|
356 |
text{*\noindent The ansatz for the \em Partial Fraction Decomposition \normalfont
|
jan@42369
|
357 |
requires to get the denominators of the partial fractions out of the
|
jan@42369
|
358 |
Solutions as:
|
jan@42369
|
359 |
\begin{itemize}
|
jan@42381
|
360 |
\item $Denominator_{1}=z-Zeropoint_{1}$
|
jan@42381
|
361 |
\item $Denominator_{2}=z-Zeropoint_{2}$
|
jan@42381
|
362 |
\item \ldots
|
jan@42381
|
363 |
\end{itemize}
|
jan@42381
|
364 |
*}
|
jan@42369
|
365 |
|
neuper@42335
|
366 |
ML {*
|
jan@42369
|
367 |
val xx = HOLogic.dest_eq s_1;
|
jan@42369
|
368 |
val s_1' = HOLogic.mk_binop "Groups.minus_class.minus" xx;
|
jan@42369
|
369 |
val xx = HOLogic.dest_eq s_2;
|
jan@42369
|
370 |
val s_2' = HOLogic.mk_binop "Groups.minus_class.minus" xx;
|
jan@42369
|
371 |
term2str s_1';
|
jan@42369
|
372 |
term2str s_2';
|
neuper@42335
|
373 |
*}
|
jan@42369
|
374 |
|
jan@42369
|
375 |
text {*\noindent For the programming language a function collecting all the
|
jan@42369
|
376 |
above manipulations is helpful.*}
|
jan@42369
|
377 |
|
neuper@42335
|
378 |
ML {*
|
jan@42369
|
379 |
fun fac_from_sol s =
|
jan@42369
|
380 |
let val (lhs, rhs) = HOLogic.dest_eq s
|
jan@42369
|
381 |
in HOLogic.mk_binop "Groups.minus_class.minus" (lhs, rhs) end;
|
neuper@42335
|
382 |
*}
|
jan@42369
|
383 |
|
neuper@42335
|
384 |
ML {*
|
jan@42369
|
385 |
fun mk_prod prod [] =
|
jan@42369
|
386 |
if prod = e_term
|
jan@42369
|
387 |
then error "mk_prod called with []"
|
jan@42369
|
388 |
else prod
|
jan@42369
|
389 |
| mk_prod prod (t :: []) =
|
jan@42369
|
390 |
if prod = e_term
|
jan@42369
|
391 |
then t
|
jan@42369
|
392 |
else HOLogic.mk_binop "Groups.times_class.times" (prod, t)
|
jan@42369
|
393 |
| mk_prod prod (t1 :: t2 :: ts) =
|
jan@42369
|
394 |
if prod = e_term
|
jan@42369
|
395 |
then
|
jan@42369
|
396 |
let val p =
|
jan@42369
|
397 |
HOLogic.mk_binop "Groups.times_class.times" (t1, t2)
|
jan@42369
|
398 |
in mk_prod p ts end
|
jan@42369
|
399 |
else
|
jan@42369
|
400 |
let val p =
|
jan@42369
|
401 |
HOLogic.mk_binop "Groups.times_class.times" (prod, t1)
|
jan@42369
|
402 |
in mk_prod p (t2 :: ts) end
|
neuper@42335
|
403 |
*}
|
jan@42369
|
404 |
(* ML {*
|
neuper@42376
|
405 |
probably keep these test in test/Tools/isac/...
|
neuper@42335
|
406 |
(*mk_prod e_term [];*)
|
neuper@42335
|
407 |
|
neuper@42335
|
408 |
val prod = mk_prod e_term [str2term "x + 123"];
|
neuper@42335
|
409 |
term2str prod = "x + 123";
|
neuper@42335
|
410 |
|
neuper@42335
|
411 |
val sol = str2term "[z = 1 / 2, z = -1 / 4]";
|
neuper@42335
|
412 |
val sols = HOLogic.dest_list sol;
|
neuper@42335
|
413 |
val facs = map fac_from_sol sols;
|
neuper@42335
|
414 |
val prod = mk_prod e_term facs;
|
neuper@42335
|
415 |
term2str prod = "(z + -1 * (1 / 2)) * (z + -1 * (-1 / 4))";
|
neuper@42335
|
416 |
|
jan@42369
|
417 |
val prod =
|
jan@42369
|
418 |
mk_prod e_term [str2term "x + 1", str2term "x + 2", str2term "x + 3"];
|
neuper@42335
|
419 |
term2str prod = "(x + 1) * (x + 2) * (x + 3)";
|
jan@42369
|
420 |
*} *)
|
jan@42369
|
421 |
ML {*
|
jan@42369
|
422 |
fun factors_from_solution sol =
|
jan@42369
|
423 |
let val ts = HOLogic.dest_list sol
|
jan@42369
|
424 |
in mk_prod e_term (map fac_from_sol ts) end;
|
jan@42369
|
425 |
*}
|
jan@42369
|
426 |
(* ML {*
|
neuper@42335
|
427 |
val sol = str2term "[z = 1 / 2, z = -1 / 4]";
|
neuper@42335
|
428 |
val fs = factors_from_solution sol;
|
neuper@42335
|
429 |
term2str fs = "(z + -1 * (1 / 2)) * (z + -1 * (-1 / 4))"
|
jan@42369
|
430 |
*} *)
|
jan@42369
|
431 |
text {*\noindent This function needs to be packed such that it can be evaluated
|
jan@42369
|
432 |
by the Lucas-Interpreter. Therefor we moved the function to the
|
jan@42369
|
433 |
corresponding \ttfamily Equation.thy \normalfont in our case
|
neuper@42376
|
434 |
\ttfamily PartialFractions.thy \normalfont. The necessary steps
|
jan@42381
|
435 |
are quit the same as we have done with \ttfamily get\_denominator
|
jan@42369
|
436 |
\normalfont before.*}
|
neuper@42335
|
437 |
ML {*
|
jan@42369
|
438 |
(*("factors_from_solution",
|
jan@42369
|
439 |
("Partial_Fractions.factors_from_solution",
|
jan@42369
|
440 |
eval_factors_from_solution ""))*)
|
jan@42369
|
441 |
|
jan@42369
|
442 |
fun eval_factors_from_solution (thmid:string) _
|
jan@42369
|
443 |
(t as Const ("Partial_Fractions.factors_from_solution", _) $ sol)
|
jan@42369
|
444 |
thy = ((let val prod = factors_from_solution sol
|
neuper@52070
|
445 |
in SOME (mk_thmid thmid "" (term_to_string''' thy prod) "",
|
jan@42369
|
446 |
Trueprop $ (mk_equality (t, prod)))
|
jan@42369
|
447 |
end)
|
jan@42369
|
448 |
handle _ => NONE)
|
jan@42369
|
449 |
| eval_factors_from_solution _ _ _ _ = NONE;
|
jan@42352
|
450 |
*}
|
jan@42352
|
451 |
|
neuper@42376
|
452 |
text {*\noindent The tracing output of the calc tree after applying this
|
jan@42381
|
453 |
function was:
|
jan@42381
|
454 |
\begin{verbatim}
|
jan@42381
|
455 |
24 / factors_from_solution [z = 1/ 2, z = -1 / 4])]
|
jan@42381
|
456 |
\end{verbatim}
|
jan@42381
|
457 |
and the next step:
|
jan@42381
|
458 |
\begin{verbatim}
|
jan@42381
|
459 |
val nxt = ("Empty_Tac", ...): tac'_)
|
jan@42381
|
460 |
\end{verbatim}
|
jan@42381
|
461 |
These observations indicate, that the Lucas-Interpreter (LIP)
|
jan@42381
|
462 |
does not know how to evaluate \ttfamily factors\_from\_solution
|
jan@42381
|
463 |
\normalfont, so we knew that there is something wrong or missing.
|
jan@42381
|
464 |
*}
|
jan@42369
|
465 |
|
jan@42381
|
466 |
text{*\noindent First we isolate the difficulty in the program as follows:
|
jan@42381
|
467 |
\begin{verbatim}
|
jan@42381
|
468 |
" (L_L::bool list) = (SubProblem (PolyEq', " ^
|
jan@42381
|
469 |
" [abcFormula, degree_2, polynomial, " ^
|
jan@42381
|
470 |
" univariate,equation], " ^
|
jan@42381
|
471 |
" [no_met]) " ^
|
jan@42381
|
472 |
" [BOOL equ, REAL zzz]); " ^
|
jan@42381
|
473 |
" (facs::real) = factors_from_solution L_L; " ^
|
jan@42381
|
474 |
" (foo::real) = Take facs " ^
|
jan@42381
|
475 |
\end{verbatim}
|
jan@42381
|
476 |
|
jan@42381
|
477 |
\par \noindent And see the tracing output:
|
jan@42381
|
478 |
|
jan@42381
|
479 |
\begin{verbatim}
|
jan@42381
|
480 |
[(([], Frm), Problem (Isac, [inverse,
|
jan@42381
|
481 |
Z_Transform,
|
jan@42381
|
482 |
SignalProcessing])),
|
jan@42381
|
483 |
(([1], Frm), X z = 3 / (z - 1 / 4 + -1 / 8 * (1 / z))),
|
jan@42381
|
484 |
(([1], Res), ?X' z = 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))),
|
jan@42381
|
485 |
(([2], Res), ?X' z = 24 / (-1 + -2 * z + 8 * z ^^^ 2)),
|
jan@42381
|
486 |
(([3], Pbl), solve (-1 + -2 * z + 8 * z ^^^ 2 = 0, z)),
|
jan@42381
|
487 |
(([3,1], Frm), -1 + -2 * z + 8 * z ^^^ 2 = 0),
|
jan@42381
|
488 |
(([3,1], Res), z = (- -2 + sqrt (-2 ^^^ 2 - 4 * 8 * -1)) / (2 * 8)|
|
jan@42381
|
489 |
z = (- -2 - sqrt (-2 ^^^ 2 - 4 * 8 * -1)) / (2 * 8)),
|
jan@42381
|
490 |
(([3,2], Res), z = 1 / 2 | z = -1 / 4),
|
jan@42381
|
491 |
(([3,3], Res), [ z = 1 / 2, z = -1 / 4]),
|
jan@42381
|
492 |
(([3,4], Res), [ z = 1 / 2, z = -1 / 4]),
|
jan@42381
|
493 |
(([3], Res), [ z = 1 / 2, z = -1 / 4]),
|
jan@42381
|
494 |
(([4], Frm), factors_from_solution [z = 1 / 2, z = -1 / 4])]
|
jan@42381
|
495 |
\end{verbatim}
|
jan@42381
|
496 |
|
jan@42381
|
497 |
\par \noindent In particular that:
|
jan@42381
|
498 |
|
jan@42381
|
499 |
\begin{verbatim}
|
jan@42381
|
500 |
(([3], Pbl), solve (-1 + -2 * z + 8 * z ^^^ 2 = 0, z)),
|
jan@42381
|
501 |
\end{verbatim}
|
jan@42381
|
502 |
\par \noindent Shows the equation which has been created in
|
jan@42381
|
503 |
the program by:
|
jan@42381
|
504 |
\begin{verbatim}
|
jan@42381
|
505 |
"(denom::real) = get_denominator funterm; " ^
|
jan@42381
|
506 |
(* get_denominator *)
|
jan@42381
|
507 |
"(equ::bool) = (denom = (0::real)); " ^
|
jan@42381
|
508 |
\end{verbatim}
|
jan@42369
|
509 |
|
jan@42381
|
510 |
\noindent \ttfamily get\_denominator \normalfont has been evaluated successfully,
|
jan@42369
|
511 |
but not\\ \ttfamily factors\_from\_solution.\normalfont
|
jan@42369
|
512 |
So we stepwise compare with an analogous case, \ttfamily get\_denominator
|
jan@42369
|
513 |
\normalfont successfully done above: We know that LIP evaluates
|
jan@42369
|
514 |
expressions in the program by use of the \emph{srls}, so we try to get
|
jan@42369
|
515 |
the original \emph{srls}.\\
|
jan@42352
|
516 |
|
jan@42381
|
517 |
\begin{verbatim}
|
jan@42381
|
518 |
val {srls,...} = get_met ["SignalProcessing",
|
jan@42381
|
519 |
"Z_Transform",
|
neuper@42405
|
520 |
"Inverse"];
|
jan@42381
|
521 |
\end{verbatim}
|
jan@42369
|
522 |
|
jan@42381
|
523 |
\par \noindent Create 2 good example terms:
|
jan@42352
|
524 |
|
jan@42381
|
525 |
\begin{verbatim}
|
jan@42381
|
526 |
val SOME t1 =
|
jan@42381
|
527 |
parseNEW ctxt "get_denominator ((111::real) / 222)";
|
jan@42381
|
528 |
val SOME t2 =
|
jan@42381
|
529 |
parseNEW ctxt "factors_from_solution [(z::real)=1/2, z=-1/4]";
|
jan@42381
|
530 |
\end{verbatim}
|
jan@42381
|
531 |
|
jan@42381
|
532 |
\par \noindent Rewrite the terms using srls:\\
|
jan@42369
|
533 |
\ttfamily \par \noindent rewrite\_set\_ thy true srls t1;\\
|
jan@42369
|
534 |
rewrite\_set\_ thy true srls t2;\\
|
jan@42369
|
535 |
\par \noindent \normalfont Now we see a difference: \texttt{t1} gives
|
jan@42369
|
536 |
\texttt{SOME} but \texttt{t2} gives \texttt{NONE}. We look at the
|
jan@42381
|
537 |
\emph{srls}:
|
jan@42381
|
538 |
\begin{verbatim}
|
jan@42381
|
539 |
val srls =
|
jan@42381
|
540 |
Rls{id = "srls_InverseZTransform",
|
jan@42381
|
541 |
rules = [Calc("Rational.get_numerator",
|
jan@42381
|
542 |
eval_get_numerator "Rational.get_numerator"),
|
jan@42381
|
543 |
Calc("Partial_Fractions.factors_from_solution",
|
jan@42381
|
544 |
eval_factors_from_solution
|
jan@42381
|
545 |
"Partial_Fractions.factors_from_solution")]}
|
jan@42381
|
546 |
\end{verbatim}
|
jan@42381
|
547 |
\par \noindent Here everthing is perfect. So the error can
|
jan@42369
|
548 |
only be in the SML code of \ttfamily eval\_factors\_from\_solution.
|
jan@42369
|
549 |
\normalfont We try to check the code with an existing test; since the
|
jan@42369
|
550 |
\emph{code} is in
|
jan@42369
|
551 |
\begin{center}\ttfamily src/Tools/isac/Knowledge/Partial\_Fractions.thy
|
jan@42369
|
552 |
\normalfont\end{center}
|
jan@42369
|
553 |
the \emph{test} should be in
|
jan@42369
|
554 |
\begin{center}\ttfamily test/Tools/isac/Knowledge/partial\_fractions.sml
|
jan@42369
|
555 |
\normalfont\end{center}
|
jan@42369
|
556 |
\par \noindent After updating the function \ttfamily
|
jan@42369
|
557 |
factors\_from\_solution \normalfont to a new version and putting a
|
neuper@42376
|
558 |
test-case to \ttfamily Partial\_Fractions.sml \normalfont we tried again
|
jan@42369
|
559 |
to evaluate the term with the same result.
|
jan@42369
|
560 |
\par We opened the test \ttfamily Test\_Isac.thy \normalfont and saw that
|
jan@42369
|
561 |
everything is working fine. Also we checked that the test \ttfamily
|
jan@42369
|
562 |
partial\_fractions.sml \normalfont is used in \ttfamily Test\_Isac.thy
|
jan@42369
|
563 |
\normalfont
|
jan@42369
|
564 |
\begin{center}use \ttfamily "Knowledge/partial\_fractions.sml"
|
jan@42369
|
565 |
\normalfont \end{center}
|
jan@42369
|
566 |
and \ttfamily Partial\_Fractions.thy \normalfont is part is part of
|
jan@42381
|
567 |
{\sisac} by evaluating
|
jan@42352
|
568 |
|
jan@42381
|
569 |
\begin{verbatim}
|
neuper@42389
|
570 |
val thy = @{theory "Inverse_Z_Transform"};
|
jan@42381
|
571 |
\end{verbatim}
|
jan@42352
|
572 |
|
jan@42381
|
573 |
After rebuilding {\sisac} again it worked.
|
neuper@42335
|
574 |
*}
|
neuper@42279
|
575 |
|
jan@42369
|
576 |
subsubsection {*Build Expression*}
|
jan@42374
|
577 |
text {*\noindent In {\sisac}'s TP-based programming language we can build
|
jan@42369
|
578 |
expressions by:\\
|
jan@42369
|
579 |
\ttfamily let s\_1 = Take numerator / (s\_1 * s\_2) \normalfont*}
|
jan@42369
|
580 |
|
neuper@42279
|
581 |
ML {*
|
jan@42369
|
582 |
(*
|
neuper@42376
|
583 |
* The main denominator is the multiplication of the denominators of
|
jan@42369
|
584 |
* all partial fractions.
|
jan@42369
|
585 |
*)
|
jan@42369
|
586 |
|
jan@42369
|
587 |
val denominator' = HOLogic.mk_binop
|
jan@42369
|
588 |
"Groups.times_class.times" (s_1', s_2') ;
|
jan@42369
|
589 |
val SOME numerator = parseNEW ctxt "3::real";
|
neuper@42279
|
590 |
|
jan@42369
|
591 |
val expr' = HOLogic.mk_binop
|
wneuper@59360
|
592 |
"Rings.divide_class.divide" (numerator, denominator');
|
jan@42369
|
593 |
term2str expr';
|
neuper@42279
|
594 |
*}
|
neuper@42279
|
595 |
|
jan@42369
|
596 |
subsubsection {*Apply the Partial Fraction Decomposion Ansatz*}
|
jan@42369
|
597 |
|
neuper@42376
|
598 |
text{*\noindent We use the Ansatz of the Partial Fraction Decomposition for our
|
jan@42369
|
599 |
expression 2nd order. Follow up the calculation in
|
jan@42369
|
600 |
Section~\ref{sec:calc:ztrans} Step~03.*}
|
jan@42369
|
601 |
|
neuper@42302
|
602 |
ML {*Context.theory_name thy = "Isac"*}
|
neuper@42279
|
603 |
|
neuper@42376
|
604 |
text{*\noindent We define two axiomatization, the first one is the main ansatz,
|
neuper@42376
|
605 |
the next one is just an equivalent transformation of the resulting
|
jan@42369
|
606 |
equation. Both axiomatizations were moved to \ttfamily
|
jan@42369
|
607 |
Partial\_Fractions.thy \normalfont and got their own rulesets. In later
|
neuper@42376
|
608 |
programs it is possible to use the rulesets and the machine will find
|
jan@42369
|
609 |
the correct ansatz and equivalent transformation itself.*}
|
jan@42369
|
610 |
|
neuper@42279
|
611 |
axiomatization where
|
jan@42369
|
612 |
ansatz_2nd_order: "n / (a*b) = A/a + B/b" and
|
jan@42369
|
613 |
equival_trans_2nd_order: "(n/(a*b) = A/a + B/b) = (n = A*b + B*a)"
|
jan@42369
|
614 |
|
jan@42369
|
615 |
text{*\noindent We use our \ttfamily ansatz\_2nd\_order \normalfont to rewrite
|
neuper@42376
|
616 |
our expression and get an equation with our expression on the left
|
jan@42369
|
617 |
and the partial fractions of it on the right hand side.*}
|
jan@42369
|
618 |
|
jan@42369
|
619 |
ML {*
|
jan@42369
|
620 |
val SOME (t1,_) =
|
neuper@42384
|
621 |
rewrite_ @{theory} e_rew_ord e_rls false
|
jan@42369
|
622 |
@{thm ansatz_2nd_order} expr';
|
jan@42369
|
623 |
term2str t1; atomty t1;
|
jan@42369
|
624 |
val eq1 = HOLogic.mk_eq (expr', t1);
|
jan@42369
|
625 |
term2str eq1;
|
jan@42369
|
626 |
*}
|
jan@42369
|
627 |
|
neuper@42376
|
628 |
text{*\noindent Eliminate the denominators by multiplying the left and the
|
jan@42369
|
629 |
right hand side of the equation with the main denominator. This is an
|
jan@42369
|
630 |
simple equivalent transformation. Later on we use an own ruleset
|
jan@42369
|
631 |
defined in \ttfamily Partial\_Fractions.thy \normalfont for doing this.
|
jan@42369
|
632 |
Follow up the calculation in Section~\ref{sec:calc:ztrans} Step~04.*}
|
neuper@42279
|
633 |
|
neuper@42279
|
634 |
ML {*
|
jan@42369
|
635 |
val SOME (eq2,_) =
|
neuper@42384
|
636 |
rewrite_ @{theory} e_rew_ord e_rls false
|
jan@42369
|
637 |
@{thm equival_trans_2nd_order} eq1;
|
jan@42369
|
638 |
term2str eq2;
|
neuper@42342
|
639 |
*}
|
neuper@42342
|
640 |
|
jan@42369
|
641 |
text{*\noindent We use the existing ruleset \ttfamily norm\_Rational \normalfont
|
jan@42369
|
642 |
for simplifications on expressions.*}
|
neuper@42279
|
643 |
|
neuper@42279
|
644 |
ML {*
|
neuper@42384
|
645 |
val SOME (eq3,_) = rewrite_set_ @{theory} false norm_Rational eq2;
|
jan@42369
|
646 |
term2str eq3;
|
jan@42369
|
647 |
(*
|
jan@42369
|
648 |
* ?A ?B not simplified
|
jan@42369
|
649 |
*)
|
neuper@42279
|
650 |
*}
|
neuper@42279
|
651 |
|
neuper@42376
|
652 |
text{*\noindent In Example~\ref{eg:gap} of my thesis I'm describing a problem about
|
jan@42369
|
653 |
simplifications. The problem that we would like to have only a specific degree
|
neuper@42376
|
654 |
of simplification occurs right here, in the next step.*}
|
jan@42369
|
655 |
|
jan@42369
|
656 |
ML {*
|
neuper@52065
|
657 |
trace_rewrite := false;
|
jan@42369
|
658 |
val SOME fract1 =
|
jan@42369
|
659 |
parseNEW ctxt "(z - 1/2)*(z - -1/4) * (A/(z - 1/2) + B/(z - -1/4))";
|
jan@42369
|
660 |
(*
|
jan@42369
|
661 |
* A B !
|
jan@42369
|
662 |
*)
|
jan@42369
|
663 |
val SOME (fract2,_) =
|
neuper@42384
|
664 |
rewrite_set_ @{theory} false norm_Rational fract1;
|
jan@42369
|
665 |
term2str fract2 = "(A + -2 * B + 4 * A * z + 4 * B * z) / 4";
|
jan@42369
|
666 |
(*
|
jan@42369
|
667 |
* term2str fract2 = "A * (1 / 4 + z) + B * (-1 / 2 + z)"
|
jan@42369
|
668 |
* would be more traditional...
|
jan@42369
|
669 |
*)
|
jan@42369
|
670 |
*}
|
jan@42369
|
671 |
|
jan@42369
|
672 |
text{*\noindent We walk around this problem by generating our new equation first.*}
|
jan@42369
|
673 |
|
jan@42369
|
674 |
ML {*
|
jan@42369
|
675 |
val (numerator, denominator) = HOLogic.dest_eq eq3;
|
jan@42369
|
676 |
val eq3' = HOLogic.mk_eq (numerator, fract1);
|
jan@42369
|
677 |
(*
|
jan@42369
|
678 |
* A B !
|
jan@42369
|
679 |
*)
|
jan@42369
|
680 |
term2str eq3';
|
jan@42369
|
681 |
(*
|
jan@42369
|
682 |
* MANDATORY: simplify (and remove denominator) otherwise 3 = 0
|
jan@42369
|
683 |
*)
|
jan@42369
|
684 |
val SOME (eq3'' ,_) =
|
neuper@42384
|
685 |
rewrite_set_ @{theory} false norm_Rational eq3';
|
jan@42369
|
686 |
term2str eq3'';
|
jan@42369
|
687 |
*}
|
jan@42369
|
688 |
|
jan@42369
|
689 |
text{*\noindent Still working at {\sisac}\ldots*}
|
jan@42369
|
690 |
|
jan@42369
|
691 |
ML {* Context.theory_name thy = "Isac" *}
|
jan@42369
|
692 |
|
jan@42369
|
693 |
subsubsection {*Build a Rule-Set for the Ansatz*}
|
jan@42369
|
694 |
text {*\noindent The \emph{ansatz} rules violate the principle that each
|
jan@42369
|
695 |
variable on the right-hand-side must also occur on the
|
jan@42369
|
696 |
left-hand-side of the rule: A, B, etc. don't do that. Thus the
|
jan@42369
|
697 |
rewriter marks these variables with question marks: ?A, ?B, etc.
|
jan@42369
|
698 |
These question marks can be dropped by \ttfamily fun
|
jan@42369
|
699 |
drop\_questionmarks\normalfont.*}
|
jan@42369
|
700 |
|
jan@42369
|
701 |
ML {*
|
s1210629013@55444
|
702 |
val ansatz_rls = prep_rls @{theory} (
|
jan@42369
|
703 |
Rls {id = "ansatz_rls", preconds = [], rew_ord = ("dummy_ord",dummy_ord),
|
neuper@42451
|
704 |
erls = e_rls, srls = Erls, calc = [], errpatts = [],
|
jan@42369
|
705 |
rules = [
|
jan@42369
|
706 |
Thm ("ansatz_2nd_order",num_str @{thm ansatz_2nd_order}),
|
jan@42369
|
707 |
Thm ("equival_trans_2nd_order",num_str @{thm equival_trans_2nd_order})
|
jan@42369
|
708 |
],
|
jan@42369
|
709 |
scr = EmptyScr});
|
jan@42369
|
710 |
*}
|
jan@42369
|
711 |
|
jan@42381
|
712 |
text{*\noindent We apply the ruleset\ldots*}
|
jan@42369
|
713 |
|
jan@42369
|
714 |
ML {*
|
jan@42369
|
715 |
val SOME (ttttt,_) =
|
neuper@42384
|
716 |
rewrite_set_ @{theory} false ansatz_rls expr';
|
jan@42369
|
717 |
*}
|
jan@42369
|
718 |
|
jan@42369
|
719 |
text{*\noindent And check the output\ldots*}
|
jan@42369
|
720 |
|
jan@42369
|
721 |
ML {*
|
jan@42369
|
722 |
term2str expr' = "3 / ((z - 1 / 2) * (z - -1 / 4))";
|
jan@42369
|
723 |
term2str ttttt = "?A / (z - 1 / 2) + ?B / (z - -1 / 4)";
|
jan@42369
|
724 |
*}
|
jan@42369
|
725 |
|
neuper@42376
|
726 |
subsubsection {*Get the First Coefficient*}
|
jan@42369
|
727 |
|
neuper@42376
|
728 |
text{*\noindent Now it's up to get the two coefficients A and B, which will be
|
neuper@42376
|
729 |
the numerators of our partial fractions. Continue following up the
|
jan@42369
|
730 |
Calculation in Section~\ref{sec:calc:ztrans} Subproblem~1.*}
|
jan@42369
|
731 |
|
neuper@42376
|
732 |
text{*\noindent To get the first coefficient we substitute $z$ with the first
|
jan@42381
|
733 |
zero-point we determined in Section~\ref{sec:solveq}.*}
|
jan@42369
|
734 |
|
jan@42369
|
735 |
ML {*
|
jan@42369
|
736 |
val SOME (eq4_1,_) =
|
neuper@42384
|
737 |
rewrite_terms_ @{theory} e_rew_ord e_rls [s_1] eq3'';
|
jan@42369
|
738 |
term2str eq4_1;
|
jan@42369
|
739 |
val SOME (eq4_2,_) =
|
neuper@42384
|
740 |
rewrite_set_ @{theory} false norm_Rational eq4_1;
|
jan@42369
|
741 |
term2str eq4_2;
|
jan@42369
|
742 |
|
jan@42369
|
743 |
val fmz = ["equality (3=3*A/(4::real))", "solveFor A","solutions L"];
|
jan@42369
|
744 |
val (dI',pI',mI') =("Isac", ["univariate","equation"], ["no_met"]);
|
jan@42369
|
745 |
(*
|
neuper@42377
|
746 |
* Solve the simple linear equation for A:
|
jan@42369
|
747 |
* Return eq, not list of eq's
|
jan@42369
|
748 |
*)
|
jan@42369
|
749 |
val (p,_,fa,nxt,_,pt) = CalcTreeTEST [(fmz, (dI',pI',mI'))];
|
jan@42369
|
750 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
751 |
(*Add_Given "equality (3=3*A/4)"*)
|
jan@42369
|
752 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
753 |
(*Add_Given "solveFor A"*)
|
jan@42369
|
754 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
755 |
(*Add_Find "solutions L"*)
|
jan@42369
|
756 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
757 |
(*Specify_Theory "Isac"*)
|
jan@42369
|
758 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
wneuper@59367
|
759 |
(*Specify_Problem ["normalise","polynomial",
|
jan@42369
|
760 |
"univariate","equation"])*)
|
jan@42369
|
761 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
wneuper@59370
|
762 |
(* Specify_Method["PolyEq","normalise_poly"]*)
|
jan@42369
|
763 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
wneuper@59370
|
764 |
(*Apply_Method["PolyEq","normalise_poly"]*)
|
jan@42369
|
765 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
766 |
(*Rewrite ("all_left","PolyEq.all_left")*)
|
jan@42369
|
767 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
768 |
(*Rewrite_Set_Inst(["(bdv,A)"],"make_ratpoly_in")*)
|
jan@42369
|
769 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
770 |
(*Rewrite_Set "polyeq_simplify"*)
|
jan@42369
|
771 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
772 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
773 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
774 |
(*Add_Given "equality (3 + -3 / 4 * A =0)"*)
|
jan@42369
|
775 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
776 |
(*Add_Given "solveFor A"*)
|
jan@42369
|
777 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
778 |
(*Add_Find "solutions A_i"*)
|
jan@42369
|
779 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
780 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
781 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
782 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
783 |
(*Apply_Method ["PolyEq","solve_d1_polyeq_equation"]*)
|
jan@42369
|
784 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
785 |
(*Rewrite_Set_Inst(["(bdv,A)"],"d1_polyeq_simplify")*)
|
jan@42369
|
786 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
787 |
(*Rewrite_Set "polyeq_simplify"*)
|
jan@42369
|
788 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
789 |
(*Rewrite_Set "norm_Rational_parenthesized"*)
|
jan@42369
|
790 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
791 |
(*Or_to_List*)
|
jan@42369
|
792 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
793 |
(*Check_elementwise "Assumptions"*)
|
jan@42369
|
794 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
795 |
(*Check_Postcond ["degree_1","polynomial",
|
jan@42369
|
796 |
"univariate","equation"]*)
|
jan@42369
|
797 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
wneuper@59367
|
798 |
(*Check_Postcond ["normalise","polynomial",
|
jan@42369
|
799 |
"univariate","equation"]*)
|
jan@42369
|
800 |
val (p,_,fa,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
801 |
(*End_Proof'*)
|
jan@42369
|
802 |
f2str fa;
|
jan@42369
|
803 |
*}
|
jan@42369
|
804 |
|
neuper@42376
|
805 |
subsubsection {*Get Second Coefficient*}
|
jan@42369
|
806 |
|
jan@42369
|
807 |
text{*\noindent With the use of \texttt{thy} we check which theories the
|
jan@42369
|
808 |
interpreter knows.*}
|
jan@42369
|
809 |
|
neuper@42279
|
810 |
ML {*thy*}
|
neuper@42279
|
811 |
|
neuper@42376
|
812 |
text{*\noindent To get the second coefficient we substitute $z$ with the second
|
jan@42381
|
813 |
zero-point we determined in Section~\ref{sec:solveq}.*}
|
jan@42369
|
814 |
|
neuper@42279
|
815 |
ML {*
|
jan@42369
|
816 |
val SOME (eq4b_1,_) =
|
neuper@42384
|
817 |
rewrite_terms_ @{theory} e_rew_ord e_rls [s_2] eq3'';
|
jan@42369
|
818 |
term2str eq4b_1;
|
jan@42369
|
819 |
val SOME (eq4b_2,_) =
|
neuper@42384
|
820 |
rewrite_set_ @{theory} false norm_Rational eq4b_1;
|
jan@42369
|
821 |
term2str eq4b_2;
|
neuper@42279
|
822 |
|
jan@42369
|
823 |
val fmz = ["equality (3= -3*B/(4::real))", "solveFor B","solutions L"];
|
jan@42369
|
824 |
val (dI',pI',mI') =("Isac", ["univariate","equation"], ["no_met"]);
|
jan@42369
|
825 |
val (p,_,fb,nxt,_,pt) = CalcTreeTEST [(fmz, (dI',pI',mI'))];
|
jan@42369
|
826 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
827 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
828 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
829 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
830 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
831 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
832 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
833 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
834 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
835 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
836 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
837 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
838 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
839 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
840 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
841 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
842 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
843 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
844 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
845 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
846 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
847 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
848 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
849 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
850 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
851 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
852 |
val (p,_,fb,nxt,_,pt) = me nxt p [] pt;
|
jan@42369
|
853 |
f2str fb;
|
neuper@42279
|
854 |
*}
|
neuper@42279
|
855 |
|
jan@42369
|
856 |
text{*\noindent We compare our results with the pre calculated upshot.*}
|
jan@42369
|
857 |
|
jan@42369
|
858 |
ML {*
|
jan@42369
|
859 |
if f2str fa = "[A = 4]" then () else error "part.fract. eq4_1";
|
jan@42369
|
860 |
if f2str fb = "[B = -4]" then () else error "part.fract. eq4_1";
|
neuper@42279
|
861 |
*}
|
neuper@42279
|
862 |
|
jan@42369
|
863 |
section {*Implement the Specification and the Method \label{spec-meth}*}
|
neuper@42279
|
864 |
|
jan@42369
|
865 |
text{*\noindent Now everything we need for solving the problem has been
|
jan@42369
|
866 |
tested out. We now start by creating new nodes for our methods and
|
neuper@42376
|
867 |
further on our new program in the interpreter.*}
|
jan@42369
|
868 |
|
jan@42369
|
869 |
subsection{*Define the Field Descriptions for the
|
jan@42369
|
870 |
Specification\label{sec:deffdes}*}
|
jan@42369
|
871 |
|
jan@42369
|
872 |
text{*\noindent We define the fields \em filterExpression \normalfont and
|
neuper@42376
|
873 |
\em stepResponse \normalfont both as equations, they represent the in- and
|
jan@42369
|
874 |
output of the program.*}
|
jan@42369
|
875 |
|
neuper@42279
|
876 |
consts
|
neuper@42279
|
877 |
filterExpression :: "bool => una"
|
neuper@42279
|
878 |
stepResponse :: "bool => una"
|
neuper@42279
|
879 |
|
neuper@42279
|
880 |
subsection{*Define the Specification*}
|
jan@42369
|
881 |
|
jan@42369
|
882 |
text{*\noindent The next step is defining the specifications as nodes in the
|
neuper@42376
|
883 |
designated part. We have to create the hierarchy node by node and start
|
jan@42369
|
884 |
with \em SignalProcessing \normalfont and go on by creating the node
|
jan@42369
|
885 |
\em Z\_Transform\normalfont.*}
|
jan@42369
|
886 |
|
s1210629013@55359
|
887 |
setup {* KEStore_Elems.add_pbts
|
s1210629013@55355
|
888 |
[prep_pbt thy "pbl_SP" [] e_pblID (["SignalProcessing"], [], e_rls, NONE, []),
|
s1210629013@55355
|
889 |
prep_pbt thy "pbl_SP_Ztrans" [] e_pblID
|
s1210629013@55355
|
890 |
(["Z_Transform","SignalProcessing"], [], e_rls, NONE, [])] *}
|
jan@42369
|
891 |
|
jan@42369
|
892 |
text{*\noindent For the suddenly created node we have to define the input
|
neuper@42376
|
893 |
and output parameters. We already prepared their definition in
|
jan@42381
|
894 |
Section~\ref{sec:deffdes}.*}
|
jan@42369
|
895 |
|
s1210629013@55359
|
896 |
setup {* KEStore_Elems.add_pbts
|
s1210629013@55355
|
897 |
[prep_pbt thy "pbl_SP_Ztrans_inv" [] e_pblID
|
s1210629013@55355
|
898 |
(["Inverse", "Z_Transform", "SignalProcessing"],
|
s1210629013@55355
|
899 |
[("#Given", ["filterExpression X_eq"]),
|
s1210629013@55355
|
900 |
("#Find", ["stepResponse n_eq"])],
|
s1210629013@55355
|
901 |
append_rls "e_rls" e_rls [(*for preds in where_*)],
|
s1210629013@55355
|
902 |
NONE,
|
s1210629013@55355
|
903 |
[["SignalProcessing","Z_Transform","Inverse"]])] *}
|
s1210629013@55355
|
904 |
ML {*
|
s1210629013@55355
|
905 |
show_ptyps ();
|
neuper@42405
|
906 |
get_pbt ["Inverse","Z_Transform","SignalProcessing"];
|
neuper@42279
|
907 |
*}
|
neuper@42279
|
908 |
|
neuper@42279
|
909 |
subsection {*Define Name and Signature for the Method*}
|
jan@42369
|
910 |
|
jan@42369
|
911 |
text{*\noindent As a next step we store the definition of our new method as a
|
jan@42369
|
912 |
constant for the interpreter.*}
|
jan@42369
|
913 |
|
neuper@42279
|
914 |
consts
|
neuper@42279
|
915 |
InverseZTransform :: "[bool, bool] => bool"
|
neuper@42279
|
916 |
("((Script InverseZTransform (_ =))// (_))" 9)
|
neuper@42279
|
917 |
|
jan@42370
|
918 |
subsection {*Setup Parent Nodes in Hierarchy of Method\label{sec:cparentnode}*}
|
jan@42369
|
919 |
|
jan@42369
|
920 |
text{*\noindent Again we have to generate the nodes step by step, first the
|
jan@42369
|
921 |
parent node and then the originally \em Z\_Transformation
|
jan@42369
|
922 |
\normalfont node. We have to define both nodes first with an empty script
|
jan@42369
|
923 |
as content.*}
|
jan@42369
|
924 |
|
s1210629013@55377
|
925 |
setup {* KEStore_Elems.add_mets
|
s1210629013@55377
|
926 |
[prep_met thy "met_SP" [] e_metID
|
s1210629013@55377
|
927 |
(["SignalProcessing"], [],
|
s1210629013@55377
|
928 |
{rew_ord'="tless_true", rls'= e_rls, calc = [], srls = e_rls, prls = e_rls, crls = e_rls,
|
s1210629013@55377
|
929 |
errpats = [], nrls = e_rls},
|
s1210629013@55377
|
930 |
"empty_script"),
|
s1210629013@55377
|
931 |
prep_met thy "met_SP_Ztrans" [] e_metID
|
s1210629013@55377
|
932 |
(["SignalProcessing", "Z_Transform"], [],
|
s1210629013@55377
|
933 |
{rew_ord'="tless_true", rls'= e_rls, calc = [], srls = e_rls, prls = e_rls, crls = e_rls,
|
s1210629013@55377
|
934 |
errpats = [], nrls = e_rls},
|
s1210629013@55377
|
935 |
"empty_script")]
|
s1210629013@55377
|
936 |
*}
|
jan@42369
|
937 |
|
jan@42369
|
938 |
text{*\noindent After we generated both nodes, we can fill the containing
|
jan@42369
|
939 |
script we want to implement later. First we define the specifications
|
jan@42369
|
940 |
of the script in e.g. the in- and output.*}
|
jan@42369
|
941 |
|
s1210629013@55377
|
942 |
setup {* KEStore_Elems.add_mets
|
s1210629013@55377
|
943 |
[prep_met thy "met_SP_Ztrans_inv" [] e_metID
|
s1210629013@55377
|
944 |
(["SignalProcessing", "Z_Transform", "Inverse"],
|
s1210629013@55377
|
945 |
[("#Given" ,["filterExpression X_eq"]), ("#Find" ,["stepResponse n_eq"])],
|
s1210629013@55377
|
946 |
{rew_ord'="tless_true", rls'= e_rls, calc = [], srls = e_rls, prls = e_rls, crls = e_rls,
|
s1210629013@55377
|
947 |
errpats = [], nrls = e_rls},
|
s1210629013@55377
|
948 |
"empty_script")]
|
s1210629013@55377
|
949 |
*}
|
jan@42369
|
950 |
|
jan@42369
|
951 |
text{*\noindent After we stored the definition we can start implementing the
|
jan@42369
|
952 |
script itself. As a first try we define only three rows containing one
|
jan@42369
|
953 |
simple operation.*}
|
jan@42369
|
954 |
|
s1210629013@55377
|
955 |
setup {* KEStore_Elems.add_mets
|
s1210629013@55377
|
956 |
[prep_met thy "met_SP_Ztrans_inv" [] e_metID
|
s1210629013@55377
|
957 |
(["SignalProcessing", "Z_Transform", "Inverse"],
|
s1210629013@55377
|
958 |
[("#Given" ,["filterExpression X_eq"]), ("#Find" ,["stepResponse n_eq"])],
|
s1210629013@55377
|
959 |
{rew_ord'="tless_true", rls'= e_rls, calc = [], srls = e_rls, prls = e_rls, crls = e_rls,
|
s1210629013@55377
|
960 |
errpats = [], nrls = e_rls},
|
s1210629013@55377
|
961 |
"Script InverseZTransform (Xeq::bool) =" ^
|
s1210629013@55377
|
962 |
" (let X = Take Xeq;" ^
|
s1210629013@55377
|
963 |
" X = Rewrite ruleZY False X" ^
|
s1210629013@55377
|
964 |
" in X)")]
|
s1210629013@55377
|
965 |
*}
|
jan@42369
|
966 |
|
jan@42369
|
967 |
text{*\noindent Check if the method has been stored correctly\ldots*}
|
jan@42369
|
968 |
|
jan@42299
|
969 |
ML {*
|
jan@42369
|
970 |
show_mets();
|
jan@42299
|
971 |
*}
|
jan@42369
|
972 |
|
neuper@42376
|
973 |
text{*\noindent If yes we can get the method by stepping backwards through
|
neuper@42376
|
974 |
the hierarchy.*}
|
jan@42369
|
975 |
|
jan@42299
|
976 |
ML {*
|
neuper@42405
|
977 |
get_met ["SignalProcessing","Z_Transform","Inverse"];
|
neuper@42279
|
978 |
*}
|
neuper@42279
|
979 |
|
jan@42374
|
980 |
section {*Program in TP-based language \label{prog-steps}*}
|
jan@42369
|
981 |
|
neuper@42376
|
982 |
text{*\noindent We start stepwise expanding our program. The script is a
|
neuper@42376
|
983 |
simple string containing several manipulation instructions.
|
jan@42370
|
984 |
\par The first script we try contains no instruction, we only test if
|
jan@42370
|
985 |
building scripts that way work.*}
|
jan@42369
|
986 |
|
jan@42370
|
987 |
subsection {*Stepwise Extend the Program*}
|
neuper@42279
|
988 |
ML {*
|
jan@42370
|
989 |
val str =
|
jan@42381
|
990 |
"Script InverseZTransform (Xeq::bool) = "^
|
jan@42370
|
991 |
" Xeq";
|
neuper@42279
|
992 |
*}
|
jan@42300
|
993 |
|
jan@42370
|
994 |
text{*\noindent Next we put some instructions in the script and try if we are
|
jan@42370
|
995 |
able to solve our first equation.*}
|
jan@42370
|
996 |
|
jan@42370
|
997 |
ML {*
|
jan@42370
|
998 |
val str =
|
jan@42381
|
999 |
"Script InverseZTransform (Xeq::bool) = "^
|
jan@42370
|
1000 |
(*
|
jan@42370
|
1001 |
* 1/z) instead of z ^^^ -1
|
jan@42370
|
1002 |
*)
|
jan@42381
|
1003 |
" (let X = Take Xeq; "^
|
jan@42381
|
1004 |
" X' = Rewrite ruleZY False X; "^
|
jan@42370
|
1005 |
(*
|
jan@42370
|
1006 |
* z * denominator
|
jan@42370
|
1007 |
*)
|
jan@42381
|
1008 |
" X' = (Rewrite_Set norm_Rational False) X' "^
|
jan@42370
|
1009 |
(*
|
jan@42370
|
1010 |
* simplify
|
jan@42370
|
1011 |
*)
|
jan@42370
|
1012 |
" in X)";
|
jan@42370
|
1013 |
(*
|
jan@42370
|
1014 |
* NONE
|
jan@42370
|
1015 |
*)
|
jan@42381
|
1016 |
"Script InverseZTransform (Xeq::bool) = "^
|
jan@42370
|
1017 |
(*
|
jan@42370
|
1018 |
* (1/z) instead of z ^^^ -1
|
jan@42370
|
1019 |
*)
|
jan@42381
|
1020 |
" (let X = Take Xeq; "^
|
jan@42381
|
1021 |
" X' = Rewrite ruleZY False X; "^
|
jan@42370
|
1022 |
(*
|
jan@42370
|
1023 |
* z * denominator
|
jan@42370
|
1024 |
*)
|
jan@42381
|
1025 |
" X' = (Rewrite_Set norm_Rational False) X'; "^
|
jan@42370
|
1026 |
(*
|
jan@42370
|
1027 |
* simplify
|
jan@42370
|
1028 |
*)
|
jan@42381
|
1029 |
" X' = (SubProblem (Isac',[pqFormula,degree_2, "^
|
jan@42381
|
1030 |
" polynomial,univariate,equation], "^
|
jan@42381
|
1031 |
" [no_met]) "^
|
jan@42381
|
1032 |
" [BOOL e_e, REAL v_v]) "^
|
jan@42370
|
1033 |
" in X)";
|
jan@42370
|
1034 |
*}
|
jan@42370
|
1035 |
|
neuper@42376
|
1036 |
text{*\noindent To solve the equation it is necessary to drop the left hand
|
jan@42370
|
1037 |
side, now we only need the denominator of the right hand side. The first
|
jan@42370
|
1038 |
equation solves the zeros of our expression.*}
|
jan@42370
|
1039 |
|
jan@42370
|
1040 |
ML {*
|
jan@42370
|
1041 |
val str =
|
jan@42381
|
1042 |
"Script InverseZTransform (Xeq::bool) = "^
|
jan@42381
|
1043 |
" (let X = Take Xeq; "^
|
jan@42381
|
1044 |
" X' = Rewrite ruleZY False X; "^
|
jan@42381
|
1045 |
" X' = (Rewrite_Set norm_Rational False) X'; "^
|
jan@42381
|
1046 |
" funterm = rhs X' "^
|
jan@42370
|
1047 |
(*
|
jan@42370
|
1048 |
* drop X'= for equation solving
|
jan@42370
|
1049 |
*)
|
jan@42370
|
1050 |
" in X)";
|
jan@42370
|
1051 |
*}
|
jan@42370
|
1052 |
|
jan@42370
|
1053 |
text{*\noindent As mentioned above, we need the denominator of the right hand
|
jan@42370
|
1054 |
side. The equation itself consists of this denominator and tries to find
|
jan@42370
|
1055 |
a $z$ for this the denominator is equal to zero.*}
|
jan@42370
|
1056 |
|
jan@42370
|
1057 |
ML {*
|
jan@42370
|
1058 |
val str =
|
jan@42381
|
1059 |
"Script InverseZTransform (X_eq::bool) = "^
|
jan@42381
|
1060 |
" (let X = Take X_eq; "^
|
jan@42381
|
1061 |
" X' = Rewrite ruleZY False X; "^
|
jan@42381
|
1062 |
" X' = (Rewrite_Set norm_Rational False) X'; "^
|
jan@42381
|
1063 |
" (X'_z::real) = lhs X'; "^
|
jan@42381
|
1064 |
" (z::real) = argument_in X'_z; "^
|
jan@42381
|
1065 |
" (funterm::real) = rhs X'; "^
|
jan@42381
|
1066 |
" (denom::real) = get_denominator funterm; "^
|
jan@42370
|
1067 |
(*
|
jan@42370
|
1068 |
* get_denominator
|
jan@42370
|
1069 |
*)
|
jan@42381
|
1070 |
" (equ::bool) = (denom = (0::real)); "^
|
jan@42381
|
1071 |
" (L_L::bool list) = "^
|
jan@42381
|
1072 |
" (SubProblem (Test', "^
|
neuper@55279
|
1073 |
" [LINEAR,univariate,equation,test], "^
|
jan@42381
|
1074 |
" [Test,solve_linear]) "^
|
jan@42381
|
1075 |
" [BOOL equ, REAL z]) "^
|
jan@42370
|
1076 |
" in X)";
|
jan@42370
|
1077 |
|
jan@42370
|
1078 |
parse thy str;
|
wneuper@59188
|
1079 |
val sc = ((inst_abs thy) o Thm.term_of o the o (parse thy)) str;
|
jan@42370
|
1080 |
atomty sc;
|
jan@42370
|
1081 |
*}
|
jan@42370
|
1082 |
|
jan@42370
|
1083 |
text {*\noindent This ruleset contains all functions that are in the script;
|
jan@42370
|
1084 |
The evaluation of the functions is done by rewriting using this ruleset.*}
|
jan@42370
|
1085 |
|
jan@42370
|
1086 |
ML {*
|
jan@42381
|
1087 |
val srls =
|
jan@42381
|
1088 |
Rls {id="srls_InverseZTransform",
|
jan@42381
|
1089 |
preconds = [],
|
jan@42381
|
1090 |
rew_ord = ("termlessI",termlessI),
|
jan@42381
|
1091 |
erls = append_rls "erls_in_srls_InverseZTransform" e_rls
|
jan@42381
|
1092 |
[(*for asm in NTH_CONS ...*)
|
jan@42381
|
1093 |
Calc ("Orderings.ord_class.less",eval_equ "#less_"),
|
jan@42381
|
1094 |
(*2nd NTH_CONS pushes n+-1 into asms*)
|
jan@42381
|
1095 |
Calc("Groups.plus_class.plus", eval_binop "#add_")
|
jan@42381
|
1096 |
],
|
neuper@42451
|
1097 |
srls = Erls, calc = [], errpatts = [],
|
jan@42381
|
1098 |
rules = [
|
jan@42381
|
1099 |
Thm ("NTH_CONS",num_str @{thm NTH_CONS}),
|
jan@42381
|
1100 |
Calc("Groups.plus_class.plus",
|
jan@42381
|
1101 |
eval_binop "#add_"),
|
jan@42381
|
1102 |
Thm ("NTH_NIL",num_str @{thm NTH_NIL}),
|
jan@42381
|
1103 |
Calc("Tools.lhs", eval_lhs"eval_lhs_"),
|
jan@42381
|
1104 |
Calc("Tools.rhs", eval_rhs"eval_rhs_"),
|
jan@42381
|
1105 |
Calc("Atools.argument'_in",
|
jan@42381
|
1106 |
eval_argument_in "Atools.argument'_in"),
|
jan@42381
|
1107 |
Calc("Rational.get_denominator",
|
jan@42381
|
1108 |
eval_get_denominator "#get_denominator"),
|
jan@42381
|
1109 |
Calc("Rational.get_numerator",
|
jan@42381
|
1110 |
eval_get_numerator "#get_numerator"),
|
jan@42381
|
1111 |
Calc("Partial_Fractions.factors_from_solution",
|
jan@42381
|
1112 |
eval_factors_from_solution
|
jan@42381
|
1113 |
"#factors_from_solution"),
|
jan@42381
|
1114 |
Calc("Partial_Fractions.drop_questionmarks",
|
jan@42381
|
1115 |
eval_drop_questionmarks "#drop_?")
|
jan@42381
|
1116 |
],
|
jan@42381
|
1117 |
scr = EmptyScr};
|
jan@42370
|
1118 |
*}
|
jan@42370
|
1119 |
|
jan@42370
|
1120 |
|
jan@42370
|
1121 |
subsection {*Store Final Version of Program for Execution*}
|
jan@42370
|
1122 |
|
jan@42370
|
1123 |
text{*\noindent After we also tested how to write scripts and run them,
|
jan@42370
|
1124 |
we start creating the final version of our script and store it into
|
jan@42381
|
1125 |
the method for which we created a node in Section~\ref{sec:cparentnode}
|
jan@42370
|
1126 |
Note that we also did this stepwise, but we didn't kept every
|
jan@42370
|
1127 |
subversion.*}
|
jan@42370
|
1128 |
|
s1210629013@55377
|
1129 |
setup {* KEStore_Elems.add_mets
|
s1210629013@55377
|
1130 |
[prep_met thy "met_SP_Ztrans_inv" [] e_metID
|
s1210629013@55377
|
1131 |
(["SignalProcessing", "Z_Transform", "Inverse"],
|
s1210629013@55377
|
1132 |
[("#Given" ,["filterExpression X_eq"]), ("#Find" ,["stepResponse n_eq"])],
|
s1210629013@55377
|
1133 |
{rew_ord'="tless_true", rls'= e_rls, calc = [], srls = srls, prls = e_rls, crls = e_rls,
|
s1210629013@55377
|
1134 |
errpats = [], nrls = e_rls},
|
s1210629013@55377
|
1135 |
"Script InverseZTransform (X_eq::bool) = "^
|
s1210629013@55377
|
1136 |
(*(1/z) instead of z ^^^ -1*)
|
s1210629013@55377
|
1137 |
"(let X = Take X_eq; "^
|
s1210629013@55377
|
1138 |
" X' = Rewrite ruleZY False X; "^
|
s1210629013@55377
|
1139 |
(*z * denominator*)
|
s1210629013@55377
|
1140 |
" (num_orig::real) = get_numerator (rhs X'); "^
|
s1210629013@55377
|
1141 |
" X' = (Rewrite_Set norm_Rational False) X'; "^
|
s1210629013@55377
|
1142 |
(*simplify*)
|
s1210629013@55377
|
1143 |
" (X'_z::real) = lhs X'; "^
|
s1210629013@55377
|
1144 |
" (zzz::real) = argument_in X'_z; "^
|
s1210629013@55377
|
1145 |
" (funterm::real) = rhs X'; "^
|
s1210629013@55377
|
1146 |
(*drop X' z = for equation solving*)
|
s1210629013@55377
|
1147 |
" (denom::real) = get_denominator funterm; "^
|
s1210629013@55377
|
1148 |
(*get_denominator*)
|
s1210629013@55377
|
1149 |
" (num::real) = get_numerator funterm; "^
|
s1210629013@55377
|
1150 |
(*get_numerator*)
|
s1210629013@55377
|
1151 |
" (equ::bool) = (denom = (0::real)); "^
|
s1210629013@55377
|
1152 |
" (L_L::bool list) = (SubProblem (PolyEq', "^
|
s1210629013@55377
|
1153 |
" [abcFormula,degree_2,polynomial,univariate,equation], "^
|
s1210629013@55377
|
1154 |
" [no_met]) "^
|
s1210629013@55377
|
1155 |
" [BOOL equ, REAL zzz]); "^
|
s1210629013@55377
|
1156 |
" (facs::real) = factors_from_solution L_L; "^
|
s1210629013@55377
|
1157 |
" (eql::real) = Take (num_orig / facs); "^
|
s1210629013@55377
|
1158 |
|
s1210629013@55377
|
1159 |
" (eqr::real) = (Try (Rewrite_Set ansatz_rls False)) eql; "^
|
s1210629013@55377
|
1160 |
|
s1210629013@55377
|
1161 |
" (eq::bool) = Take (eql = eqr); "^
|
s1210629013@55377
|
1162 |
(*Maybe possible to use HOLogic.mk_eq ??*)
|
s1210629013@55377
|
1163 |
" eq = (Try (Rewrite_Set equival_trans False)) eq; "^
|
s1210629013@55377
|
1164 |
|
s1210629013@55377
|
1165 |
" eq = drop_questionmarks eq; "^
|
s1210629013@55377
|
1166 |
" (z1::real) = (rhs (NTH 1 L_L)); "^
|
s1210629013@55377
|
1167 |
(*
|
s1210629013@55377
|
1168 |
* prepare equation for a - eq_a
|
s1210629013@55377
|
1169 |
* therefor substitute z with solution 1 - z1
|
s1210629013@55377
|
1170 |
*)
|
s1210629013@55377
|
1171 |
" (z2::real) = (rhs (NTH 2 L_L)); "^
|
s1210629013@55377
|
1172 |
|
s1210629013@55377
|
1173 |
" (eq_a::bool) = Take eq; "^
|
s1210629013@55377
|
1174 |
" eq_a = (Substitute [zzz=z1]) eq; "^
|
s1210629013@55377
|
1175 |
" eq_a = (Rewrite_Set norm_Rational False) eq_a; "^
|
s1210629013@55377
|
1176 |
" (sol_a::bool list) = "^
|
s1210629013@55377
|
1177 |
" (SubProblem (Isac', "^
|
s1210629013@55377
|
1178 |
" [univariate,equation],[no_met]) "^
|
s1210629013@55377
|
1179 |
" [BOOL eq_a, REAL (A::real)]); "^
|
s1210629013@55377
|
1180 |
" (a::real) = (rhs(NTH 1 sol_a)); "^
|
s1210629013@55377
|
1181 |
|
s1210629013@55377
|
1182 |
" (eq_b::bool) = Take eq; "^
|
s1210629013@55377
|
1183 |
" eq_b = (Substitute [zzz=z2]) eq_b; "^
|
s1210629013@55377
|
1184 |
" eq_b = (Rewrite_Set norm_Rational False) eq_b; "^
|
s1210629013@55377
|
1185 |
" (sol_b::bool list) = "^
|
s1210629013@55377
|
1186 |
" (SubProblem (Isac', "^
|
s1210629013@55377
|
1187 |
" [univariate,equation],[no_met]) "^
|
s1210629013@55377
|
1188 |
" [BOOL eq_b, REAL (B::real)]); "^
|
s1210629013@55377
|
1189 |
" (b::real) = (rhs(NTH 1 sol_b)); "^
|
s1210629013@55377
|
1190 |
|
s1210629013@55377
|
1191 |
" eqr = drop_questionmarks eqr; "^
|
s1210629013@55377
|
1192 |
" (pbz::real) = Take eqr; "^
|
s1210629013@55377
|
1193 |
" pbz = ((Substitute [A=a, B=b]) pbz); "^
|
s1210629013@55377
|
1194 |
|
s1210629013@55377
|
1195 |
" pbz = Rewrite ruleYZ False pbz; "^
|
s1210629013@55377
|
1196 |
" pbz = drop_questionmarks pbz; "^
|
s1210629013@55377
|
1197 |
|
s1210629013@55377
|
1198 |
" (X_z::bool) = Take (X_z = pbz); "^
|
s1210629013@55377
|
1199 |
" (n_eq::bool) = (Rewrite_Set inverse_z False) X_z; "^
|
s1210629013@55377
|
1200 |
" n_eq = drop_questionmarks n_eq "^
|
s1210629013@55377
|
1201 |
"in n_eq)")]
|
s1210629013@55377
|
1202 |
*}
|
jan@42370
|
1203 |
|
jan@42370
|
1204 |
|
jan@42370
|
1205 |
subsection {*Check the Program*}
|
jan@42370
|
1206 |
text{*\noindent When the script is ready we can start checking our work.*}
|
jan@42370
|
1207 |
subsubsection {*Check the Formalization*}
|
jan@42370
|
1208 |
text{*\noindent First we want to check the formalization of the in and
|
neuper@42376
|
1209 |
output of our program.*}
|
jan@42370
|
1210 |
|
jan@42370
|
1211 |
ML {*
|
jan@42370
|
1212 |
val fmz =
|
jan@42370
|
1213 |
["filterExpression (X = 3 / (z - 1/4 + -1/8 * (1/(z::real))))",
|
jan@42370
|
1214 |
"stepResponse (x[n::real]::bool)"];
|
jan@42370
|
1215 |
val (dI,pI,mI) =
|
neuper@42405
|
1216 |
("Isac", ["Inverse", "Z_Transform", "SignalProcessing"],
|
neuper@42405
|
1217 |
["SignalProcessing","Z_Transform","Inverse"]);
|
jan@42370
|
1218 |
|
jan@42370
|
1219 |
val ([
|
jan@42370
|
1220 |
(
|
jan@42370
|
1221 |
1,
|
jan@42370
|
1222 |
[1],
|
jan@42370
|
1223 |
"#Given",
|
jan@42370
|
1224 |
Const ("Inverse_Z_Transform.filterExpression", _),
|
jan@42370
|
1225 |
[Const ("HOL.eq", _) $ _ $ _]
|
jan@42370
|
1226 |
),
|
jan@42370
|
1227 |
(
|
jan@42370
|
1228 |
2,
|
jan@42370
|
1229 |
[1],
|
jan@42370
|
1230 |
"#Find",
|
jan@42370
|
1231 |
Const ("Inverse_Z_Transform.stepResponse", _),
|
jan@42370
|
1232 |
[Free ("x", _) $ _]
|
jan@42370
|
1233 |
)
|
jan@42370
|
1234 |
],_
|
jan@42370
|
1235 |
) = prep_ori fmz thy ((#ppc o get_pbt) pI);
|
jan@42370
|
1236 |
|
neuper@48790
|
1237 |
val Prog sc
|
jan@42370
|
1238 |
= (#scr o get_met) ["SignalProcessing",
|
jan@42370
|
1239 |
"Z_Transform",
|
neuper@42405
|
1240 |
"Inverse"];
|
jan@42370
|
1241 |
atomty sc;
|
jan@42370
|
1242 |
*}
|
jan@42370
|
1243 |
|
jan@42370
|
1244 |
subsubsection {*Stepwise Check the Program\label{sec:stepcheck}*}
|
neuper@42376
|
1245 |
text{*\noindent We start to stepwise execute our new program in a calculation
|
jan@42370
|
1246 |
tree and check if every node implements that what we have wanted.*}
|
jan@42370
|
1247 |
|
jan@42370
|
1248 |
ML {*
|
neuper@52101
|
1249 |
trace_rewrite := false; (*true*)
|
neuper@52101
|
1250 |
trace_script := false; (*true*)
|
jan@42370
|
1251 |
print_depth 9;
|
jan@42370
|
1252 |
|
jan@42370
|
1253 |
val fmz =
|
jan@42418
|
1254 |
["filterExpression (X z = 3 / ((z::real) + 1/10 - 1/50*(1/z)))",
|
jan@42370
|
1255 |
"stepResponse (x[n::real]::bool)"];
|
jan@42370
|
1256 |
|
jan@42370
|
1257 |
val (dI,pI,mI) =
|
neuper@42405
|
1258 |
("Isac", ["Inverse", "Z_Transform", "SignalProcessing"],
|
neuper@42405
|
1259 |
["SignalProcessing","Z_Transform","Inverse"]);
|
jan@42370
|
1260 |
|
jan@42370
|
1261 |
val (p,_,f,nxt,_,pt) = CalcTreeTEST [(fmz, (dI,pI,mI))];
|
jan@42370
|
1262 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1263 |
"Add_Given";
|
jan@42370
|
1264 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1265 |
"Add_Find";
|
jan@42370
|
1266 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1267 |
"Specify_Theory";
|
jan@42370
|
1268 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1269 |
"Specify_Problem";
|
jan@42370
|
1270 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1271 |
"Specify_Method";
|
jan@42370
|
1272 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1273 |
"Apply_Method";
|
jan@42370
|
1274 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1275 |
"Rewrite (ruleZY, Inverse_Z_Transform.ruleZY)";
|
jan@42370
|
1276 |
"--> X z = 3 / (z - 1 / 4 + -1 / 8 * (1 / z))";
|
jan@42370
|
1277 |
(*
|
jan@42370
|
1278 |
* TODO naming!
|
jan@42370
|
1279 |
*)
|
jan@42370
|
1280 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1281 |
"Rewrite_Set norm_Rational";
|
jan@42370
|
1282 |
" --> X' z = 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))";
|
jan@42371
|
1283 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1284 |
"SubProblem";
|
neuper@52101
|
1285 |
print_depth 3;
|
jan@42370
|
1286 |
*}
|
jan@42370
|
1287 |
|
jan@42370
|
1288 |
text {*\noindent Instead of \ttfamily nxt = Subproblem \normalfont above we had
|
jan@42370
|
1289 |
\ttfamily Empty\_Tac; \normalfont the search for the reason considered
|
jan@42370
|
1290 |
the following points:\begin{itemize}
|
jan@42381
|
1291 |
\item What shows \ttfamily show\_pt pt;\normalfont\ldots?
|
jan@42381
|
1292 |
\begin{verbatim}(([2], Res), ?X' z = 24 / (-1 + -2 * z + 8 * z ^^^ 2))]\end{verbatim}
|
jan@42370
|
1293 |
The calculation is ok but no \ttfamily next \normalfont step found:
|
jan@42370
|
1294 |
Should be\\ \ttfamily nxt = Subproblem\normalfont!
|
jan@42370
|
1295 |
\item What shows \ttfamily trace\_script := true; \normalfont we read
|
jan@42381
|
1296 |
bottom up\ldots
|
jan@42381
|
1297 |
\begin{verbatim}
|
jan@42381
|
1298 |
@@@next leaf 'SubProblem
|
jan@42381
|
1299 |
(PolyEq',[abcFormula, degree_2, polynomial,
|
jan@42381
|
1300 |
univariate, equation], no_meth)
|
jan@42381
|
1301 |
[BOOL equ, REAL z]'
|
jan@42381
|
1302 |
---> STac 'SubProblem (PolyEq',
|
jan@42381
|
1303 |
[abcFormula, degree_2, polynomial,
|
jan@42381
|
1304 |
univariate, equation], no_meth)
|
jan@42381
|
1305 |
[BOOL (-1 + -2 * z + 8 * z \^\^\^ ~2 = 0), REAL z]'
|
jan@42381
|
1306 |
\end{verbatim}
|
jan@42370
|
1307 |
We see the SubProblem with correct arguments from searching next
|
jan@42370
|
1308 |
step (program text !!!--->!!! STac (script tactic) with arguments
|
jan@42370
|
1309 |
evaluated.)
|
jan@42370
|
1310 |
\item Do we have the right Script \ldots difference in the
|
jan@42381
|
1311 |
arguments in the arguments\ldots
|
jan@42381
|
1312 |
\begin{verbatim}
|
neuper@48790
|
1313 |
val Prog s =
|
jan@42381
|
1314 |
(#scr o get_met) ["SignalProcessing",
|
jan@42381
|
1315 |
"Z_Transform",
|
neuper@42405
|
1316 |
"Inverse"];
|
jan@42381
|
1317 |
writeln (term2str s);
|
jan@42381
|
1318 |
\end{verbatim}
|
jan@42370
|
1319 |
\ldots shows the right script. Difference in the arguments.
|
jan@42370
|
1320 |
\item Test --- Why helpless here ? --- shows: \ttfamily replace
|
jan@42370
|
1321 |
no\_meth by [no\_meth] \normalfont in Script
|
jan@42370
|
1322 |
\end{itemize}
|
jan@42300
|
1323 |
*}
|
jan@42300
|
1324 |
|
neuper@42279
|
1325 |
ML {*
|
jan@42370
|
1326 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1327 |
(*Model_Problem";*)
|
neuper@42279
|
1328 |
*}
|
neuper@42279
|
1329 |
|
jan@42370
|
1330 |
text {*\noindent Instead of \ttfamily nxt = Model\_Problem \normalfont above
|
jan@42370
|
1331 |
we had \ttfamily Empty\_Tac; \normalfont the search for the reason
|
jan@42370
|
1332 |
considered the following points:\begin{itemize}
|
jan@42370
|
1333 |
\item Difference in the arguments
|
jan@42381
|
1334 |
\item Comparison with Subsection~\ref{sec:solveq}: There solving this
|
jan@42370
|
1335 |
equation works, so there must be some difference in the arguments
|
jan@42370
|
1336 |
of the Subproblem: RIGHT: we had \ttfamily [no\_meth] \normalfont
|
jan@42370
|
1337 |
instead of \ttfamily [no\_met] \normalfont ;-)
|
jan@42370
|
1338 |
\end{itemize}*}
|
jan@42338
|
1339 |
|
neuper@42279
|
1340 |
ML {*
|
jan@42370
|
1341 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1342 |
(*Add_Given equality (-1 + -2 * z + 8 * z ^^^ 2 = 0)";*)
|
jan@42370
|
1343 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1344 |
(*Add_Given solveFor z";*)
|
jan@42370
|
1345 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1346 |
(*Add_Find solutions z_i";*)
|
jan@42370
|
1347 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1348 |
(*Specify_Theory Isac";*)
|
neuper@42279
|
1349 |
*}
|
neuper@42279
|
1350 |
|
jan@42370
|
1351 |
text {*\noindent We had \ttfamily nxt = Empty\_Tac instead Specify\_Theory;
|
jan@42370
|
1352 |
\normalfont The search for the reason considered the following points:
|
jan@42370
|
1353 |
\begin{itemize}
|
jan@42370
|
1354 |
\item Was there an error message? NO -- ok
|
jan@42370
|
1355 |
\item Has \ttfamily nxt = Add\_Find \normalfont been inserted in pt:\\
|
jan@42370
|
1356 |
\ttfamily get\_obj g\_pbl pt (fst p);\normalfont? YES -- ok
|
jan@42381
|
1357 |
\item What is the returned formula:
|
jan@42381
|
1358 |
\begin{verbatim}
|
neuper@52101
|
1359 |
print_depth 999; f; print_depth 3;
|
jan@42381
|
1360 |
{ Find = [ Correct "solutions z_i"],
|
jan@42381
|
1361 |
With = [],
|
jan@42381
|
1362 |
Given = [Correct "equality (-1 + -2*z + 8*z ^^^ 2 = 0)",
|
jan@42381
|
1363 |
Correct "solveFor z"],
|
jan@42381
|
1364 |
Where = [...],
|
jan@42381
|
1365 |
Relate = [] }
|
jan@42381
|
1366 |
\end{verbatim}
|
jan@42370
|
1367 |
\normalfont The only False is the reason: the Where (the precondition) is
|
jan@42370
|
1368 |
False for good reasons: The precondition seems to check for linear
|
jan@42370
|
1369 |
equations, not for the one we want to solve! Removed this error by
|
jan@42370
|
1370 |
correcting the Script from \ttfamily SubProblem (PolyEq',
|
jan@42370
|
1371 |
\lbrack linear,univariate,equation,
|
jan@42370
|
1372 |
test\rbrack, \lbrack Test,solve\_linear\rbrack \normalfont to
|
jan@42370
|
1373 |
\ttfamily SubProblem (PolyEq',\\ \lbrack abcFormula,degree\_2,
|
jan@42370
|
1374 |
polynomial,univariate,equation\rbrack,\\
|
jan@42370
|
1375 |
\lbrack PolyEq,solve\_d2\_polyeq\_abc\_equation
|
jan@42370
|
1376 |
\rbrack\normalfont
|
jan@42370
|
1377 |
You find the appropriate type of equation at the
|
jan@42370
|
1378 |
{\sisac}-WEB-Page\footnote{
|
jan@42370
|
1379 |
\href{http://www.ist.tugraz.at/projects/isac/www/kbase/pbl/index\_pbl.html}
|
jan@42370
|
1380 |
{http://www.ist.tugraz.at/projects/isac/www/kbase/pbl/index\_pbl.html}
|
jan@42370
|
1381 |
}
|
jan@42370
|
1382 |
And the respective method in \ttfamily Knowledge/PolyEq.thy \normalfont
|
jan@42370
|
1383 |
at the respective \ttfamily store\_pbt. \normalfont Or you leave the
|
jan@42370
|
1384 |
selection of the appropriate type to isac as done in the final Script ;-))
|
jan@42370
|
1385 |
\end{itemize}*}
|
jan@42370
|
1386 |
|
neuper@42279
|
1387 |
ML {*
|
jan@42370
|
1388 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1389 |
(*Specify_Problem [abcFormula,...";*)
|
jan@42370
|
1390 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1391 |
(*Specify_Method [PolyEq,solve_d2_polyeq_abc_equation";*)
|
jan@42370
|
1392 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1393 |
(*Apply_Method [PolyEq,solve_d2_polyeq_abc_equation";*)
|
jan@42370
|
1394 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42371
|
1395 |
(*Rewrite_Set_Inst ([(bdv, z)], d2_polyeq_abcFormula_simplify";*)
|
jan@42370
|
1396 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1397 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1398 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1399 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1400 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1401 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1402 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1403 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1404 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1405 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1406 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1407 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1408 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1409 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1410 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1411 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1412 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
wneuper@59367
|
1413 |
(*Specify_Problem ["normalise","polynomial","univariate","equation"]*)
|
jan@42370
|
1414 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
wneuper@59370
|
1415 |
(*Specify_Method ["PolyEq", "normalise_poly"]*)
|
jan@42370
|
1416 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
wneuper@59370
|
1417 |
(*Apply_Method ["PolyEq", "normalise_poly"]*)
|
jan@42370
|
1418 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1419 |
(*Rewrite ("all_left", "PolyEq.all_left")*)
|
jan@42370
|
1420 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1421 |
(*Rewrite_Set_Inst (["(bdv, A)"], "make_ratpoly_in")*)
|
jan@42370
|
1422 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1423 |
(*Rewrite_Set "polyeq_simplify"*)
|
jan@42370
|
1424 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1425 |
(*Subproblem("Isac",["degree_1","polynomial","univariate","equation"])*)
|
jan@42370
|
1426 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1427 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1428 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1429 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1430 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1431 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1432 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1433 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1434 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1435 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1436 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1437 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1438 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1439 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1440 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1441 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1442 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1443 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1444 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1445 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1446 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1447 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1448 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1449 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1450 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1451 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1452 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1453 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1454 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1455 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1456 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1457 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1458 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1459 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1460 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1461 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1462 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1463 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1464 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1465 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1466 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1467 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1468 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1469 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
jan@42370
|
1470 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
neuper@42451
|
1471 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt; (*([11, 4, 5], Res) Check_Postcond*)
|
neuper@42451
|
1472 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt; (*([11, 4], Res) Check_Postcond*)
|
neuper@42451
|
1473 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt; (*([11], Res) Take*)
|
neuper@42451
|
1474 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt; (*([12], Frm) Substitute*)
|
neuper@42451
|
1475 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt; (*([12], Res) Rewrite*)
|
neuper@42451
|
1476 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt; (*([13], Res) Take*)
|
neuper@42451
|
1477 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt; (*([14], Frm) Empty_Tac*)
|
neuper@42451
|
1478 |
*}
|
neuper@42451
|
1479 |
ML {*
|
jan@42370
|
1480 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
neuper@42451
|
1481 |
*}
|
neuper@42451
|
1482 |
ML {*
|
jan@42370
|
1483 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
neuper@42451
|
1484 |
*}
|
neuper@42451
|
1485 |
ML {*
|
jan@42370
|
1486 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
neuper@42451
|
1487 |
*}
|
neuper@42451
|
1488 |
ML {*
|
jan@42370
|
1489 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
neuper@42451
|
1490 |
*}
|
neuper@42451
|
1491 |
ML {*
|
jan@42370
|
1492 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
neuper@42451
|
1493 |
*}
|
neuper@42451
|
1494 |
ML {*
|
jan@42370
|
1495 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
neuper@42290
|
1496 |
*}
|
neuper@42281
|
1497 |
|
jan@42418
|
1498 |
ML {*
|
jan@42418
|
1499 |
trace_script := true;
|
neuper@42451
|
1500 |
*}
|
neuper@42451
|
1501 |
ML {*
|
jan@42418
|
1502 |
val (p,_,f,nxt,_,pt) = me nxt p [] pt;
|
neuper@42451
|
1503 |
*}
|
neuper@42451
|
1504 |
ML {*
|
wneuper@59265
|
1505 |
Chead.show_pt pt;
|
neuper@42451
|
1506 |
*}
|
neuper@42451
|
1507 |
ML {*
|
neuper@42451
|
1508 |
*}
|
neuper@42451
|
1509 |
ML {*
|
neuper@42451
|
1510 |
*}
|
neuper@42451
|
1511 |
ML {*
|
neuper@42451
|
1512 |
*}
|
neuper@42451
|
1513 |
ML {*
|
neuper@42451
|
1514 |
*}
|
neuper@42451
|
1515 |
ML {*
|
neuper@42451
|
1516 |
*}
|
neuper@42451
|
1517 |
ML {*
|
jan@42418
|
1518 |
*}
|
jan@42418
|
1519 |
|
jan@42370
|
1520 |
text{*\noindent As a last step we check the tracing output of the last calc
|
jan@42370
|
1521 |
tree instruction and compare it with the pre-calculated result.*}
|
neuper@42315
|
1522 |
|
neuper@42376
|
1523 |
section {* Improve and Transfer into Knowledge *}
|
neuper@42376
|
1524 |
text {*
|
neuper@42376
|
1525 |
We want to improve the very long program \ttfamily InverseZTransform
|
neuper@42376
|
1526 |
\normalfont by modularisation: partial fraction decomposition shall
|
neuper@42376
|
1527 |
become a sub-problem.
|
neuper@42376
|
1528 |
|
neuper@42376
|
1529 |
We could transfer all knowledge in \ttfamily Build\_Inverse\_Z\_Transform.thy
|
neuper@42376
|
1530 |
\normalfont first to the \ttfamily Knowledge/Inverse\_Z\_Transform.thy
|
neuper@42376
|
1531 |
\normalfont and then modularise. In this case TODO problems?!?
|
neuper@42376
|
1532 |
|
neuper@42376
|
1533 |
We chose another way and go bottom up: first we build the sub-problem in
|
jan@42381
|
1534 |
\ttfamily Partial\_Fractions.thy \normalfont with the term:
|
neuper@42376
|
1535 |
|
jan@42381
|
1536 |
$$\frac{3}{x\cdot(z - \frac{1}{4} + \frac{-1}{8}\cdot\frac{1}{z})}$$
|
neuper@42376
|
1537 |
|
jan@42381
|
1538 |
\noindent (how this still can be improved see \ttfamily Partial\_Fractions.thy\normalfont),
|
neuper@42376
|
1539 |
and re-use all stuff prepared in \ttfamily Build\_Inverse\_Z\_Transform.thy:
|
jan@42381
|
1540 |
\normalfont The knowledge will be transferred to \ttfamily src/../Partial\_Fractions.thy
|
jan@42381
|
1541 |
\normalfont and the respective tests to:
|
jan@42381
|
1542 |
\begin{center}\ttfamily test/../sartial\_fractions.sml\normalfont\end{center}
|
neuper@42279
|
1543 |
*}
|
neuper@42279
|
1544 |
|
neuper@42376
|
1545 |
subsection {* Transfer to Partial\_Fractions.thy *}
|
neuper@42376
|
1546 |
text {*
|
jan@42381
|
1547 |
First we transfer both, knowledge and tests into:
|
jan@42381
|
1548 |
\begin{center}\ttfamily src/../Partial\_Fractions.thy\normalfont\end{center}
|
jan@42381
|
1549 |
in order to immediately have the test results.
|
neuper@42376
|
1550 |
|
jan@42381
|
1551 |
We copy \ttfamily factors\_from\_solution, drop\_questionmarks,\\
|
jan@42381
|
1552 |
ansatz\_2nd\_order \normalfont and rule-sets --- no problem.
|
jan@42381
|
1553 |
|
jan@42381
|
1554 |
Also \ttfamily store\_pbt ..\\ "pbl\_simp\_rat\_partfrac"
|
neuper@42376
|
1555 |
\normalfont is easy.
|
neuper@42376
|
1556 |
|
jan@42381
|
1557 |
But then we copy from:\\
|
jan@42381
|
1558 |
(1) \ttfamily Build\_Inverse\_Z\_Transform.thy store\_met\ldots "met\_SP\_Ztrans\_inv"
|
jan@42381
|
1559 |
\normalfont\\ to\\
|
jan@42381
|
1560 |
(2) \ttfamily Partial\_Fractions.thy store\_met\ldots "met\_SP\_Ztrans\_inv"
|
jan@42381
|
1561 |
\normalfont\\ and cut out the respective part from the program. First we ensure that
|
neuper@42376
|
1562 |
the string is correct. When we insert the string into (2)
|
jan@42381
|
1563 |
\ttfamily store\_met .. "met\_partial\_fraction" \normalfont --- and get an error.
|
neuper@42376
|
1564 |
*}
|
neuper@42376
|
1565 |
|
jan@42381
|
1566 |
subsubsection {* 'Programming' in ISAC's TP-based Language *}
|
neuper@42376
|
1567 |
text {*
|
neuper@42376
|
1568 |
At the present state writing programs in {\sisac} is particularly cumbersome.
|
neuper@42376
|
1569 |
So we give hints how to cope with the many obstacles. Below we describe the
|
neuper@42376
|
1570 |
steps we did in making (2) run.
|
neuper@42376
|
1571 |
|
neuper@42376
|
1572 |
\begin{enumerate}
|
neuper@42376
|
1573 |
\item We check if the \textbf{string} containing the program is correct.
|
neuper@42376
|
1574 |
\item We check if the \textbf{types in the program} are correct.
|
neuper@42376
|
1575 |
For this purpose we start start with the first and last lines
|
jan@42381
|
1576 |
\begin{verbatim}
|
jan@42381
|
1577 |
"PartFracScript (f_f::real) (v_v::real) = " ^
|
jan@42381
|
1578 |
" (let X = Take f_f; " ^
|
jan@42381
|
1579 |
" pbz = ((Substitute []) X) " ^
|
jan@42381
|
1580 |
" in pbz)"
|
jan@42381
|
1581 |
\end{verbatim}
|
neuper@42376
|
1582 |
The last but one line helps not to bother with ';'.
|
neuper@42376
|
1583 |
\item Then we add line by line. Already the first line causes the error.
|
neuper@42376
|
1584 |
So we investigate it by
|
jan@42381
|
1585 |
\begin{verbatim}
|
neuper@48761
|
1586 |
val ctxt = Proof_Context.init_global @{theory "Inverse_Z_Transform"} ;
|
jan@42381
|
1587 |
val SOME t =
|
jan@42381
|
1588 |
parseNEW ctxt "(num_orig::real) =
|
jan@42381
|
1589 |
get_numerator(rhs f_f)";
|
jan@42381
|
1590 |
\end{verbatim}
|
neuper@42376
|
1591 |
and see a type clash: \ttfamily rhs \normalfont from (1) requires type
|
jan@42381
|
1592 |
\ttfamily bool \normalfont while (2) wants to have \ttfamily (f\_f::real).
|
neuper@42376
|
1593 |
\normalfont Of course, we don't need \ttfamily rhs \normalfont anymore.
|
neuper@42376
|
1594 |
\item Type-checking can be very tedious. One might even inspect the
|
jan@42381
|
1595 |
parse-tree of the program with {\sisac}'s specific debug tools:
|
jan@42381
|
1596 |
\begin{verbatim}
|
neuper@48790
|
1597 |
val {scr = Prog t,...} =
|
jan@42381
|
1598 |
get_met ["simplification",
|
jan@42381
|
1599 |
"of_rationals",
|
jan@42381
|
1600 |
"to_partial_fraction"];
|
neuper@42389
|
1601 |
atomty_thy @{theory "Inverse_Z_Transform"} t ;
|
jan@42381
|
1602 |
\end{verbatim}
|
neuper@42376
|
1603 |
\item We check if the \textbf{semantics of the program} by stepwise evaluation
|
neuper@42376
|
1604 |
of the program. Evaluation is done by the Lucas-Interpreter, which works
|
neuper@42376
|
1605 |
using the knowledge in theory Isac; so we have to re-build Isac. And the
|
neuper@42376
|
1606 |
test are performed simplest in a file which is loaded with Isac.
|
jan@42381
|
1607 |
See \ttfamily tests/../partial\_fractions.sml \normalfont.
|
neuper@42376
|
1608 |
\end{enumerate}
|
neuper@42376
|
1609 |
*}
|
neuper@42376
|
1610 |
|
neuper@42376
|
1611 |
subsection {* Transfer to Inverse\_Z\_Transform.thy *}
|
neuper@42376
|
1612 |
text {*
|
neuper@42388
|
1613 |
It was not possible to complete this task, because we ran out of time.
|
neuper@42376
|
1614 |
*}
|
neuper@42376
|
1615 |
|
neuper@42376
|
1616 |
|
neuper@42279
|
1617 |
end
|
neuper@42279
|
1618 |
|