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 begin
8 chapter {* Rewriting *}
10 text {* \emph{Rewriting} is a technique of Symbolic Computation, which is
11 appropriate to make a 'transparent system', because it is intuitively
12 comprehensible. For a thourogh introduction see the book of Tobias Nipkow,
13 http://www4.informatik.tu-muenchen.de/~nipkow/TRaAT/
15 section {* Introduction to rewriting *}
17 text {* Rewriting creates calculations which look like written by hand; see the
18 ISAC tutoring system ! ISAC finds the rules automatically. Here we start by
19 telling the rules ourselves.
20 Let's differentiate after we have identified the rules for differentiation, as
21 found in ~~/src/Tools/isac/Knowledge/Diff.thy:
24 val diff_sum = num_str @{thm diff_sum};
25 val diff_pow = num_str @{thm diff_pow};
26 val diff_var = num_str @{thm diff_var};
27 val diff_const = num_str @{thm diff_const};
29 text {* Looking at the rules (abbreviated by 'thm' above), we see the
30 differential operator abbreviated by 'd_d ?bdv', where '?bdv' is the bound
32 Can you read diff_const in the Ouput window ?
34 Please, skip this introductory ML-section in the first go ...*}
36 (*default_print_depth 1;*)
37 val (thy, ro, er, inst) =
38 (@{theory "Isac"}, tless_true, eval_rls, [(@{term "bdv::real"}, @{term "x::real"})]);
40 text {* ... and let us differentiate the term t: *}
42 val t = (Thm.term_of o the o (parse thy)) "d_d x (x^^^2 + x + y)";
44 val SOME (t, _) = Rewrite.rewrite_inst_ thy ro er true inst diff_sum t; term2str t;
45 val SOME (t, _) = Rewrite.rewrite_inst_ thy ro er true inst diff_sum t; term2str t;
46 val SOME (t, _) = Rewrite.rewrite_inst_ thy ro er true inst diff_pow t; term2str t;
47 val SOME (t, _) = Rewrite.rewrite_inst_ thy ro er true inst diff_var t; term2str t;
48 val SOME (t, _) = Rewrite.rewrite_inst_ thy ro er true inst diff_const t; term2str t;
50 text {* Please, scoll up the Output-window to check the 5 steps of rewriting !
51 You might not be satisfied by the result "2 * x ^^^ (2 - 1) + 1 + 0".
53 ISAC has a set of rules called 'make_polynomial', which simplifies the result:
56 val SOME (t, _) = Rewrite.rewrite_set_ thy true make_polynomial t; term2str t;
57 trace_rewrite := false;
60 section {* Note on bound variables *}
61 text {* You may have noticed that rewrite_ above could distinguish between x
62 (d_d x x = 1) and y (d_d x y = 0). ISAC does this by instantiating theorems
63 before application: given [(@{term "bdv::real"}, @{term "x::real"})] the
64 theorem diff_sum becomes "d_d x (?u + ?v) = d_d x ?u + d_d x ?v".
66 Isabelle does this differently by variables bound by a 'lambda' %, see
67 http://isabelle.in.tum.de/dist/library/HOL/HOL-Multivariate_Analysis/Derivative.html
70 val t = @{term "%x. x^2 + x + y"}; atomwy t; term2str t;
72 text {* Since this notation does not conform to present high-school matheatics
73 ISAC introduced the 'bdv' mechanism. This mechanism is also used for equation
77 section {* Conditional and ordered rewriting *}
78 text {* We have already seen conditional rewriting above when we used the rule
79 diff_const; let us try: *}
81 val t1 = (Thm.term_of o the o (parse thy)) "d_d x (a*BC*x*z)";
82 Rewrite.rewrite_inst_ thy ro er true inst diff_const t1;
84 val t2 = (Thm.term_of o the o (parse thy)) "d_d x (a*BC*y*z)";
85 Rewrite.rewrite_inst_ thy ro er true inst diff_const t2;
87 text {* For term t1 the assumption 'not (x occurs_in "a*BC*x*z")' is false,
88 since x occurs in t1 actually; thus the rule following implication '==>' is
89 not applied and rewrite_inst_ returns NONE.
90 For term t2 is 'not (x occurs_in "a*BC*y*z")' true, thus the rule is applied.
93 subsection {* ordered rewriting *}
94 text {* Let us start with an example; in order to see what is going on we tell
95 Isabelle to show all brackets:
98 (*show_brackets := true; TODO*)
99 val t0 = (Thm.term_of o the o (parse thy)) "2*a + 3*b + 4*a"; term2str t0;
100 (*show_brackets := false;*)
102 text {* Now we want to bring 4*a close to 2*a in order to get 6*a:
105 val SOME (t, _) = Rewrite.rewrite_ thy ro er true @{thm add.assoc} t0; term2str t;
106 val SOME (t, _) = Rewrite.rewrite_ thy ro er true @{thm add.left_commute} t; term2str t;
107 val SOME (t, _) = Rewrite.rewrite_ thy ro er true @{thm add.commute} t; term2str t;
108 val SOME (t, _) = Rewrite.rewrite_ thy ro er true @{thm real_num_collect} t; term2str t;
110 text {* That is fine, we just need to add 2+4 !!!!! See the next section below.
112 But we cannot automate such ordering with what we know so far: If we put
113 add.assoc, add.left_commute and add.commute into one ruleset (your have used
114 ruleset 'make_polynomial' already), then all the rules are applied as long
115 as one rule is applicable (that is the way such rulesets work).
116 Try to step through the ML-sections without skipping one of them ...
118 ML {*val SOME (t, _) = Rewrite.rewrite_ thy ro er true @{thm add.commute} t; term2str t*}
119 ML {*val SOME (t, _) = Rewrite.rewrite_ thy ro er true @{thm add.commute} t; term2str t*}
120 ML {*val SOME (t, _) = Rewrite.rewrite_ thy ro er true @{thm add.commute} t; term2str t*}
121 ML {*val SOME (t, _) = Rewrite.rewrite_ thy ro er true @{thm add.commute} t; term2str t*}
122 text {* ... you can go forever, the ruleset is 'not terminating'.
123 The theory of rewriting makes this kind of rulesets terminate by the use of
125 Given two terms t1 and t2 we describe rewriting by: t1 ~~> t2. Then
126 'ordered rewriting' is: t2 < t1 ==> t1 ~~> t2. That means, a rule is only
127 applied, when the result t2 is 'smaller', '<', than the one to be rewritten.
128 Defining such a '<' is not trivial, see ~~/src/Tools/isacKnowledge/Poly.thy
129 for 'fun has_degree_in' etc.
132 subsection {* Simplification in ISAC *}
134 With the introduction into rewriting, ordered rewriting and conditional
135 rewriting we have seen all what is necessary for the practice of rewriting.
137 One basic technique of 'symbolic computation' which uses rewriting is
138 simplification, that means: transform terms into an equivalent form which is
139 as simple as possible.
140 Isabelle has powerful and efficient simplifiers. Nevertheless, ISAC has another
141 kind of simplifiers, which groups rulesets such that the trace of rewrites is
142 more readable in ISAC's worksheets.
144 Here are examples of of how ISAC's simplifier work:
147 (*show_brackets := false; TODO*)
148 val t1 = (Thm.term_of o the o (parse thy)) "(a - b) * (a^^^2 + a*b + b^^^2)";
149 val SOME (t, _) = Rewrite.rewrite_set_ thy true make_polynomial t1; term2str t;
152 val t2 = (Thm.term_of o the o (parse thy))
153 "(2 / (x + 3) + 2 / (x - 3)) / (8 * x / (x ^^^ 2 - 9))";
154 val SOME (t, _) = Rewrite.rewrite_set_ thy true norm_Rational t2; term2str t;
156 text {* The simplifiers are quite busy when finding the above results. you can
157 watch them at work by setting the switch 'trace_rewrite:*}
159 trace_rewrite := false;
160 tracing "+++begin++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++";
161 val SOME (t, _) = Rewrite.rewrite_set_ thy true norm_Rational t2; term2str t;
162 tracing "+++end++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++";
163 trace_rewrite := false;
165 text {* You see what happend when you click the checkbox <Tracing> on the bar
166 separating this window from the Output-window.
168 So, it might be better to take simpler examples for watching the simplifiers.
172 section {* Experiments with a simplifier conserving minus *}
174 text {* We conclude the section on rewriting with starting into an experimental
175 development. This development has been urged by teachers using ISAC for
176 introduction to algebra with students at the age of 12-14.
178 The teachers requested ISAC to keep the minus, for instance in the above
179 result "a^3 + -1 * b^3" (here ISAC should write "a^3 - * b^3") and also
180 in all intermediate steps.
182 So we started to develop (in German !) such a simplifier in
183 ~~/src/Tools/isac/Knowledge/PolyMinus.thy
186 subsection {* What already works *}
188 val t2 = (Thm.term_of o the o (parse thy))
189 "5*e + 6*f - 8*g - 9 - 7*e - 4*f + 10*g + 12";
190 val SOME (t, _) = Rewrite.rewrite_set_ thy true rls_p_33 t2; term2str t;
192 text {* Try your own examples ! *}
194 subsection {* This raises questions about didactics *}
195 text {* Oberserve the '-' ! That works out. But see the efforts in PolyMinus.thy
199 @{thm subtrahiere_von_1};
200 @{thm subtrahiere_1};
202 text {* That would not be so bad. But it is only a little part of what else is
206 @{thm subtrahiere_x_plus_minus};
207 @{thm subtrahiere_x_plus1_minus};
208 @{thm subtrahiere_x_plus_minus1};
209 @{thm subtrahiere_x_minus_plus};
210 @{thm subtrahiere_x_minus1_plus};
211 @{thm subtrahiere_x_minus_plus1};
212 @{thm subtrahiere_x_minus_minus};
213 @{thm subtrahiere_x_minus1_minus};
214 @{thm subtrahiere_x_minus_minus1};
216 text {* So, learning so many rules, and learning to apply these rules, is futile.
217 Actually, most of our students are unable to apply theorems.
219 But if you look at 'make_polynomial' or even 'norm_Rational' you see,
220 that these general simplifiers require about 10% than rulesets for '-'.
222 So, we might have better chances to teach our student to apply theorems
226 subsection {* This does not yet work *}
228 val t = (Thm.term_of o the o (parse thy))
229 "(2*a - 5*b) * (2*a + 5*b)";
230 Rewrite.rewrite_set_ thy true rls_p_33 t; term2str t;