2 cd /usr/local/Isabelle/test/Tools/isac/ADDTESTS/course/
3 /usr/local/Isabelle/bin/isabelle jedit -l Isac T2_Rewriting.thy &
6 theory T2_Rewriting imports Isac.Isac_Knowledge
9 chapter \<open>Rewriting\<close>
11 text \<open>\emph{Rewriting} is a technique of Symbolic Computation, which is
12 appropriate to make a 'transparent system', because it is intuitively
13 comprehensible. For a thourogh introduction see the book of Tobias Nipkow,
14 http://www4.informatik.tu-muenchen.de/~nipkow/TRaAT/
16 section {* Introduction to rewriting\<close>
18 text \<open>Rewriting creates calculations which look like written by hand; see the
19 ISAC tutoring system ! ISAC finds the rules automatically. Here we start by
20 telling the rules ourselves.
21 Let's differentiate after we have identified the rules for differentiation, as
22 found in ~~/src/Tools/isac/Knowledge/Diff.thy:
25 val diff_sum = ThmC.numerals_to_Free @{thm diff_sum};
26 val diff_pow = ThmC.numerals_to_Free @{thm diff_pow};
27 val diff_var = ThmC.numerals_to_Free @{thm diff_var};
28 val diff_const = ThmC.numerals_to_Free @{thm diff_const};
30 text \<open>Looking at the rules (abbreviated by 'thm' above), we see the
31 differential operator abbreviated by 'd_d ?bdv', where '?bdv' is the bound
33 Can you read diff_const in the Ouput window ?
35 Please, skip this introductory ML-section in the first go ...\<close>
37 (*default_print_depth 1;*)
38 val (thy, ro, er, inst) =
39 (@{theory "Isac_Knowledge"}, tless_true, eval_rls, [(@{term "bdv::real"}, @{term "x::real"})]);
41 text \<open>... and let us differentiate the term t:\<close>
43 val t = (Thm.term_of o the o (TermC.parse thy)) "d_d x (x^^^2 + x + y)";
45 val SOME (t, _) = Rewrite.rewrite_inst_ thy ro er true inst diff_sum t; UnparseC.term t;
46 val SOME (t, _) = Rewrite.rewrite_inst_ thy ro er true inst diff_sum t; UnparseC.term t;
47 val SOME (t, _) = Rewrite.rewrite_inst_ thy ro er true inst diff_pow t; UnparseC.term t;
48 val SOME (t, _) = Rewrite.rewrite_inst_ thy ro er true inst diff_var t; UnparseC.term t;
49 val SOME (t, _) = Rewrite.rewrite_inst_ thy ro er true inst diff_const t; UnparseC.term t;
51 text \<open>Please, scoll up the Output-window to check the 5 steps of rewriting !
52 You might not be satisfied by the result "2 * x ^^^ (2 - 1) + 1 + 0".
54 ISAC has a set of rules called 'make_polynomial', which simplifies the result:
57 val SOME (t, _) = Rewrite.rewrite_set_ thy true make_polynomial t; UnparseC.term t;
58 Rewrite.trace_on := false;
61 section \<open>Note on bound variables\<close>
62 text \<open>You may have noticed that rewrite_ above could distinguish between x
63 (d_d x x = 1) and y (d_d x y = 0). ISAC does this by instantiating theorems
64 before application: given [(@{term "bdv::real"}, @{term "x::real"})] the
65 theorem diff_sum becomes "d_d x (?u + ?v) = d_d x ?u + d_d x ?v".
67 Isabelle does this differently by variables bound by a 'lambda' %, see
68 http://isabelle.in.tum.de/dist/library/HOL/HOL-Multivariate_Analysis/Derivative.html
71 val t = @{term "%x. x^2 + x + y"}; TermC.atomwy t; UnparseC.term t;
73 text \<open>Since this notation does not conform to present high-school matheatics
74 ISAC introduced the 'bdv' mechanism. This mechanism is also used for equation
78 section \<open>Conditional and ordered rewriting\<close>
79 text \<open>We have already seen conditional rewriting above when we used the rule
80 diff_const; let us try:\<close>
82 val t1 = (Thm.term_of o the o (TermC.parse thy)) "d_d x (a*BC*x*z)";
83 Rewrite.rewrite_inst_ thy ro er true inst diff_const t1;
85 val t2 = (Thm.term_of o the o (TermC.parse thy)) "d_d x (a*BC*y*z)";
86 Rewrite.rewrite_inst_ thy ro er true inst diff_const t2;
88 text \<open>For term t1 the assumption 'not (x occurs_in "a*BC*x*z")' is false,
89 since x occurs in t1 actually; thus the rule following implication '==>' is
90 not applied and rewrite_inst_ returns NONE.
91 For term t2 is 'not (x occurs_in "a*BC*y*z")' true, thus the rule is applied.
94 subsection \<open>ordered rewriting\<close>
95 text \<open>Let us start with an example; in order to see what is going on we tell
96 Isabelle to show all brackets:
99 (*show_brackets := true; TODO*)
100 val t0 = (Thm.term_of o the o (TermC.parse thy)) "2*a + 3*b + 4*a"; UnparseC.term t0;
101 (*show_brackets := false;*)
103 text \<open>Now we want to bring 4*a close to 2*a in order to get 6*a:
106 val SOME (t, _) = Rewrite.rewrite_ thy ro er true @{thm add.assoc} t0; UnparseC.term t;
107 val SOME (t, _) = Rewrite.rewrite_ thy ro er true @{thm add.left_commute} t; UnparseC.term t;
108 val SOME (t, _) = Rewrite.rewrite_ thy ro er true @{thm add.commute} t; UnparseC.term t;
109 val SOME (t, _) = Rewrite.rewrite_ thy ro er true @{thm real_num_collect} t; UnparseC.term t;
111 text \<open>That is fine, we just need to add 2+4 !!!!! See the next section below.
113 But we cannot automate such ordering with what we know so far: If we put
114 add.assoc, add.left_commute and add.commute into one ruleset (your have used
115 ruleset 'make_polynomial' already), then all the rules are applied as long
116 as one rule is applicable (that is the way such rulesets work).
117 Try to step through the ML-sections without skipping one of them ...
119 ML \<open>val SOME (t, _) = Rewrite.rewrite_ thy ro er true @{thm add.commute} t; UnparseC.term t\<close>
120 ML \<open>val SOME (t, _) = Rewrite.rewrite_ thy ro er true @{thm add.commute} t; UnparseC.term t\<close>
121 ML \<open>val SOME (t, _) = Rewrite.rewrite_ thy ro er true @{thm add.commute} t; UnparseC.term t\<close>
122 ML \<open>val SOME (t, _) = Rewrite.rewrite_ thy ro er true @{thm add.commute} t; UnparseC.term t\<close>
123 text \<open>... you can go forever, the ruleset is 'not terminating'.
124 The theory of rewriting makes this kind of rulesets terminate by the use of
126 Given two terms t1 and t2 we describe rewriting by: t1 ~~> t2. Then
127 'ordered rewriting' is: t2 < t1 ==> t1 ~~> t2. That means, a rule is only
128 applied, when the result t2 is 'smaller', '<', than the one to be rewritten.
129 Defining such a '<' is not trivial, see ~~/src/Tools/isacKnowledge/Poly.thy
130 for 'fun has_degree_in' etc.
133 subsection \<open>Simplification in ISAC\<close>
135 With the introduction into rewriting, ordered rewriting and conditional
136 rewriting we have seen all what is necessary for the practice of rewriting.
138 One basic technique of 'symbolic computation' which uses rewriting is
139 simplification, that means: transform terms into an equivalent form which is
140 as simple as possible.
141 Isabelle has powerful and efficient simplifiers. Nevertheless, ISAC has another
142 kind of simplifiers, which groups rulesets such that the trace of rewrites is
143 more readable in ISAC's worksheets.
145 Here are examples of of how ISAC's simplifier work:
148 (*show_brackets := false; TODO*)
149 val t1 = (Thm.term_of o the o (TermC.parse thy)) "(a - b) * (a^^^2 + a*b + b^^^2)";
150 val SOME (t, _) = Rewrite.rewrite_set_ thy true make_polynomial t1; UnparseC.term t;
153 val t2 = (Thm.term_of o the o (TermC.parse thy))
154 "(2 / (x + 3) + 2 / (x - 3)) / (8 * x / (x ^^^ 2 - 9))";
155 val SOME (t, _) = Rewrite.rewrite_set_ thy true norm_Rational t2; UnparseC.term t;
157 text \<open>The simplifiers are quite busy when finding the above results. you can
158 watch them at work by setting the switch 'Rewrite.trace_on:\<close>
160 Rewrite.trace_on := false;
161 tracing "+++begin++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++";
162 val SOME (t, _) = Rewrite.rewrite_set_ thy true norm_Rational t2; UnparseC.term t;
163 tracing "+++end++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++";
164 Rewrite.trace_on := false;
166 text \<open>You see what happend when you click the checkbox <Tracing> on the bar
167 separating this window from the Output-window.
169 So, it might be better to take simpler examples for watching the simplifiers.
173 section \<open>Experiments with a simplifier conserving minus\<close>
175 text \<open>We conclude the section on rewriting with starting into an experimental
176 development. This development has been urged by teachers using ISAC for
177 introduction to algebra with students at the age of 12-14.
179 The teachers requested ISAC to keep the minus, for instance in the above
180 result "a^3 + -1 * b^3" (here ISAC should write "a^3 - * b^3") and also
181 in all intermediate steps.
183 So we started to develop (in German !) such a simplifier in
184 ~~/src/Tools/isac/Knowledge/PolyMinus.thy
187 subsection \<open>What already works\<close>
189 val t2 = (Thm.term_of o the o (TermC.parse thy))
190 "5*e + 6*f - 8*g - 9 - 7*e - 4*f + 10*g + 12";
191 val SOME (t, _) = Rewrite.rewrite_set_ thy true rls_p_33 t2; UnparseC.term t;
193 text \<open>Try your own examples !\<close>
195 subsection \<open>This raises questions about didactics\<close>
196 text \<open>Oberserve the '-' ! That works out. But see the efforts in PolyMinus.thy
200 @{thm subtrahiere_von_1};
201 @{thm subtrahiere_1};
203 text \<open>That would not be so bad. But it is only a little part of what else is
207 @{thm subtrahiere_x_plus_minus};
208 @{thm subtrahiere_x_plus1_minus};
209 @{thm subtrahiere_x_plus_minus1};
210 @{thm subtrahiere_x_minus_plus};
211 @{thm subtrahiere_x_minus1_plus};
212 @{thm subtrahiere_x_minus_plus1};
213 @{thm subtrahiere_x_minus_minus};
214 @{thm subtrahiere_x_minus1_minus};
215 @{thm subtrahiere_x_minus_minus1};
217 text \<open>So, learning so many rules, and learning to apply these rules, is futile.
218 Actually, most of our students are unable to apply theorems.
220 But if you look at 'make_polynomial' or even 'norm_Rational' you see,
221 that these general simplifiers require about 10% than rulesets for '-'.
223 So, we might have better chances to teach our student to apply theorems
227 subsection \<open>This does not yet work\<close>
229 val t = (Thm.term_of o the o (TermC.parse thy))
230 "(2*a - 5*b) * (2*a + 5*b)";
231 Rewrite.rewrite_set_ thy true rls_p_33 t; UnparseC.term t;