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1<!--Converted with LaTeX2HTML 97.1 (release) (July 13th, 1997)
2 by Nikos Drakos (nikos@cbl.leeds.ac.uk), CBLU, University of Leeds
3* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
4* with significant contributions from:
5 Jens Lippman, Marek Rouchal, Martin Wilck and others -->
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8<TITLE>All examples produced by the all-examples utility</TITLE>
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44<H1><A NAME="SECTION000130000000000000000">
45All examples produced by the <TT>all-examples</TT> utility</A>
46</H1>
47<SMALL>
48<PRE>
49;;----------------------------------------------------------------
50;;
51;; (STRING-GROBNER &quot;[x^2+y,x-y]&quot; '(X Y))
52;;
53;;----------------------------------------------------------------
54Args:[ X^2 + Y, X - Y ]
55[ X - Y, Y^2 + Y ]
56;;----------------------------------------------------------------
57;;
58;; (STRING-GROBNER &quot;[y-x^2,z-x^3]&quot; '(X Y Z) :ORDER #'GREVLEX&gt;)
59;;
60;;----------------------------------------------------------------
61Args:[ - X^2 + Y, - X^3 + Z ]
62[ X^2 - Y, X * Y - Z, Y^2 - X * Z ]
63;;----------------------------------------------------------------
64;;
65;; (STRING-GROBNER-SYSTEM &quot;[u*x+y,x+y]&quot; '(X Y) '(U))
66;;
67;;----------------------------------------------------------------
68------------------- CASE 1 -------------------
69Condition:
70 Green list: [ U ]
71 Red list: [ ]
72 Basis: [ (1) * Y, (1) * X ]
73------------------- CASE 2 -------------------
74Condition:
75 Green list: [ ]
76 Red list: [ U, U - 1 ]
77 Basis: [ (U - 1) * X, ( - U + 1) * Y ]
78------------------- CASE 3 -------------------
79Condition:
80 Green list: [ U - 1 ]
81 Red list: [ U ]
82 Basis: [ (1) * X + (1) * Y ]
83;;----------------------------------------------------------------
84;;
85;; (STRING-GROBNER-SYSTEM &quot;[u*x+y,x+y]&quot; '(X Y) '(U) :COVER '((&quot;[u-1]&quot; &quot;[]&quot;)))
86;;
87;;----------------------------------------------------------------
88------------------- CASE 1 -------------------
89Condition:
90 Green list: [ U - 1 ]
91 Red list: [ U ]
92 Basis: [ (1) * X + (1) * Y ]
93;;----------------------------------------------------------------
94;;
95;; (STRING-READ-POLY &quot;[x^3+3*x^2+3*x+1]&quot; '(X))
96;;
97;;----------------------------------------------------------------
98Args:[ X^3 + 3 * X^2 + 3 * X + 1 ]
99[ RETURN VALUE 1]--&gt;&gt; ([ (((3) . 1) ((2) . 3) ((1) . 3) ((0) . 1)))
100
101;;----------------------------------------------------------------
102;;
103;; (STRING-ELIMINATION-IDEAL &quot;[x^2+y^2-2,x*y-1]&quot; '(X Y) 1)
104;;
105;;----------------------------------------------------------------
106Args:[ X^2 + Y^2 - 2, X * Y - 1 ]
107[ Y^4 - 2 * Y^2 + 1 ]
108;;----------------------------------------------------------------
109;;
110;; (STRING-IDEAL-SATURATION-1 &quot;[x^2*y,y^3]&quot; &quot;x&quot; '(X Y))
111;;
112;;----------------------------------------------------------------
113[ Y ]
114;;----------------------------------------------------------------
115;;
116;; (STRING-IDEAL-POLYSATURATION-1 &quot;[x^2*y,y^3]&quot; &quot;[x,y]&quot; '(X Y))
117;;
118;;----------------------------------------------------------------
119Args1:[ X^2 * Y, Y^3 ]
120Args2:[ X, Y ]
121[ 1 ]
122;;----------------------------------------------------------------
123;;
124;; (STRING-COND '(&quot;[u^2-v]&quot; &quot;[v-1]&quot;) '(U V) #'GREVLEX&gt;)
125;;
126;;----------------------------------------------------------------
127[ RETURN VALUE 1]--&gt;&gt; (((((2 0) . 1) ((0 1) . -1))) ((((0 1) . 1) ((0 0) . -1))))
128
129;;----------------------------------------------------------------
130;;
131;; (STRING-COVER '((&quot;[u^2-v]&quot; &quot;[u]&quot;) (&quot;[u+v]&quot; &quot;[]&quot;)) '(U V) #'GREVLEX&gt;)
132;;
133;;----------------------------------------------------------------
134[ RETURN VALUE 1]--&gt;&gt; ((((((2 0) . 1) ((0 1) . -1))) ((((1 0) . 1)))) (((((1 0) . 1) ((0 1) . 1))) NIL))
135
136;;----------------------------------------------------------------
137;;
138;; (STRING-DETERMINE &quot;[u*x+y,v*x^2+y^2]&quot; '(X Y) '(U V) :COND '(&quot;[u,v]&quot; &quot;[v-1]&quot;) :MAIN-ORDER #'LEX&gt;)
139;;
140;;----------------------------------------------------------------
141------------------- CASE 1 -------------------
142Condition:
143 Green list: [ U, V ]
144 Red list: [ V - 1, 1 ]
145 Basis: [ (1) * Y, (1) * Y^2 ]
146;;----------------------------------------------------------------
147;;
148;; (PARSE-STRING-TO-SORTED-ALIST &quot;x^2+y^3&quot; '(X Y) #'GREVLEX&gt;)
149;;
150;;----------------------------------------------------------------
151[ RETURN VALUE 1]--&gt;&gt; (((0 3) . 1) ((2 0) . 1))
152
153;;----------------------------------------------------------------
154;;
155;; (PARSE-STRING-TO-SORTED-ALIST &quot;[x^2+y^3,x-y]&quot; '(X Y) #'GREVLEX&gt;)
156;;
157;;----------------------------------------------------------------
158[ RETURN VALUE 1]--&gt;&gt; ([ (((0 3) . 1) ((2 0) . 1)) (((1 0) . 1) ((0 1) . -1)))
159
160;;----------------------------------------------------------------
161;;
162;; (TRANSLATE-STATEMENTS (COLLINEAR A B C) (PERPENDICULAR A B A C))
163;;
164;;----------------------------------------------------------------
165[ RETURN VALUE 1]--&gt;&gt; ((((+ (- (* B1 C2) (* B2 C1)) (- (* A2 C1) (* A1 C2)) (- (* A1 B2) (* A2 B1))))
166 ((+ (* (- A1 B1) (- A1 C1)) (* (- A2 B2) (- A2 C2)))))
167 (B1 B2 A1 A2 C1 C2))
168
169;;----------------------------------------------------------------
170;;
171;; (TRANSLATE-THEOREM ((PERPENDICULAR A B C D) (PERPENDICULAR C D E F))
172 ((PARALLEL A B E F) (IDENTICAL-POINTS C D)))
173;;
174;;----------------------------------------------------------------
175[ RETURN VALUE 1]--&gt;&gt; (((+ (* (- A1 B1) (- C1 D1)) (* (- A2 B2) (- C2 D2)))
176 (+ (* (- C1 D1) (- E1 F1)) (* (- C2 D2) (- E2 F2))))
177 (A1 A2 B1 B2 C1 C2 D1 D2 E1 E2 F1 F2))
178[ RETURN VALUE 2]--&gt;&gt; ((((- (* (- A1 B1) (- E2 F2)) (* (- A2 B2) (- E1 F1)))) ((- C1 D1) (- C2 D2)))
179 (A1 A2 B1 B2 E1 E2 F1 F2 C1 C2 D1 D2))
180
181;;----------------------------------------------------------------
182;;
183;; (TRANSLATE-THEOREM ((PERPENDICULAR A B A C) (MIDPOINT B C M) (MIDPOINT A M O) (COLLINEAR B H C)
184 (PERPENDICULAR A H B C))
185 ((EQUIDISTANT M O H O) (IDENTICAL-POINTS B C)))
186;;
187;;----------------------------------------------------------------
188[ RETURN VALUE 1]--&gt;&gt; (((+ (* (- A1 B1) (- A1 C1)) (* (- A2 B2) (- A2 C2))) (- (* 2 M1) B1 C1)
189 (- (* 2 M2) B2 C2) (- (* 2 O1) A1 M1) (- (* 2 O2) A2 M2)
190 (+ (- (* H1 C2) (* H2 C1)) (- (* B2 C1) (* B1 C2)) (- (* B1 H2) (* B2 H1)))
191 (+ (* (- A1 H1) (- B1 C1)) (* (- A2 H2) (- B2 C2))))
192 (M1 M2 O1 O2 A1 A2 H1 H2 B1 B2 C1 C2))
193[ RETURN VALUE 2]--&gt;&gt; ((((- (+ (EXPT (- M1 O1) 2) (EXPT (- M2 O2) 2))
194 (+ (EXPT (- H1 O1) 2) (EXPT (- H2 O2) 2))))
195 ((- B1 C1) (- B2 C2)))
196 (M1 M2 H1 H2 O1 O2 B1 B2 C1 C2))
197
198;;----------------------------------------------------------------
199;;
200;; (PROVE-THEOREM ((PERPENDICULAR A B C D) (PERPENDICULAR C D E F)) ((PARALLEL A B E F) (IDENTICAL-POINTS C D)))
201;;
202;;----------------------------------------------------------------
203[ 1 ]
204;;----------------------------------------------------------------
205;;
206;; (PROVE-THEOREM ((PERPENDICULAR A B A C) (MIDPOINT B C M) (MIDPOINT A M O) (COLLINEAR B H C)
207 (PERPENDICULAR A H B C))
208 ((EQUIDISTANT M O H O) (IDENTICAL-POINTS B C)))
209;;
210;;----------------------------------------------------------------
211[ 1 ]
212;;----------------------------------------------------------------
213;;
214;; (PROVE-THEOREM ((PERPENDICULAR A B A C) (IDENTICAL-POINTS B C))
215 ((IDENTICAL-POINTS A B) (IDENTICAL-POINTS A C)))
216;;
217;;----------------------------------------------------------------
218[ B1 - C1, B2 - C2, A1^2 + A2^2 - 2 * A1 * C1 + C1^2 - 2 * A2 * C2 + C2^2 ]
219;;----------------------------------------------------------------
220;;
221;; (PROVE-THEOREM ((PERPENDICULAR A B A C) (IDENTICAL-POINTS B C))
222 ((IDENTICAL-POINTS A B) (REAL-IDENTICAL-POINTS A C)))
223;;
224;;----------------------------------------------------------------
225[ 1 ]
226</PRE>
227</SMALL>
228<BR><HR>
229<ADDRESS>
230<I>Marek Rychlik</I>
231<BR><I>3/21/1998</I>
232</ADDRESS>
233</BODY>
234</HTML>
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