src/HOL/Groebner_Basis.thy
 author chaieb Mon Jul 21 13:36:39 2008 +0200 (2008-07-21) changeset 27666 1436d81d1294 parent 26462 dac4e2bce00d child 28402 09e4aa3ddc25 permissions -rw-r--r--
Relevant rules added to algebra's context
1 (*  Title:      HOL/Groebner_Basis.thy
2     ID:         \$Id\$
3     Author:     Amine Chaieb, TU Muenchen
4 *)
6 header {* Semiring normalization and Groebner Bases *}
7 theory Groebner_Basis
8 imports NatBin
9 uses
10   "Tools/Groebner_Basis/misc.ML"
11   "Tools/Groebner_Basis/normalizer_data.ML"
12   ("Tools/Groebner_Basis/normalizer.ML")
13   ("Tools/Groebner_Basis/groebner.ML")
14 begin
16 subsection {* Semiring normalization *}
18 setup NormalizerData.setup
21 locale gb_semiring =
22   fixes add mul pwr r0 r1
25     and mul_a:"mul x (mul y z) = mul (mul x y) z" and mul_c:"mul x y = mul y x"
26     and mul_1:"mul r1 x = x" and  mul_0:"mul r0 x = r0"
27     and mul_d:"mul x (add y z) = add (mul x y) (mul x z)"
28     and pwr_0:"pwr x 0 = r1" and pwr_Suc:"pwr x (Suc n) = mul x (pwr x n)"
29 begin
31 lemma mul_pwr:"mul (pwr x p) (pwr x q) = pwr x (p + q)"
32 proof (induct p)
33   case 0
34   then show ?case by (auto simp add: pwr_0 mul_1)
35 next
36   case Suc
37   from this [symmetric] show ?case
38     by (auto simp add: pwr_Suc mul_1 mul_a)
39 qed
41 lemma pwr_mul: "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
42 proof (induct q arbitrary: x y, auto simp add:pwr_0 pwr_Suc mul_1)
43   fix q x y
44   assume "\<And>x y. pwr (mul x y) q = mul (pwr x q) (pwr y q)"
45   have "mul (mul x y) (mul (pwr x q) (pwr y q)) = mul x (mul y (mul (pwr x q) (pwr y q)))"
47   also have "\<dots> = (mul (mul y (mul (pwr y q) (pwr x q))) x)" by (simp add: mul_c)
48   also have "\<dots> = (mul (mul y (pwr y q)) (mul (pwr x q) x))" by (simp add: mul_a)
49   finally show "mul (mul x y) (mul (pwr x q) (pwr y q)) =
50     mul (mul x (pwr x q)) (mul y (pwr y q))" by (simp add: mul_c)
51 qed
53 lemma pwr_pwr: "pwr (pwr x p) q = pwr x (p * q)"
54 proof (induct p arbitrary: q)
55   case 0
56   show ?case using pwr_Suc mul_1 pwr_0 by (induct q) auto
57 next
58   case Suc
59   thus ?case by (auto simp add: mul_pwr [symmetric] pwr_mul pwr_Suc)
60 qed
63 subsubsection {* Declaring the abstract theory *}
65 lemma semiring_ops:
66   includes meta_term_syntax
67   shows "TERM (add x y)" and "TERM (mul x y)" and "TERM (pwr x n)"
68     and "TERM r0" and "TERM r1"
69   by rule+
71 lemma semiring_rules:
72   "add (mul a m) (mul b m) = mul (add a b) m"
73   "add (mul a m) m = mul (add a r1) m"
74   "add m (mul a m) = mul (add a r1) m"
76   "add r0 a = a"
77   "add a r0 = a"
78   "mul a b = mul b a"
79   "mul (add a b) c = add (mul a c) (mul b c)"
80   "mul r0 a = r0"
81   "mul a r0 = r0"
82   "mul r1 a = a"
83   "mul a r1 = a"
84   "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
85   "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
86   "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
87   "mul (mul lx ly) rx = mul (mul lx rx) ly"
88   "mul (mul lx ly) rx = mul lx (mul ly rx)"
89   "mul lx (mul rx ry) = mul (mul lx rx) ry"
90   "mul lx (mul rx ry) = mul rx (mul lx ry)"
97   "mul (pwr x p) (pwr x q) = pwr x (p + q)"
98   "mul x (pwr x q) = pwr x (Suc q)"
99   "mul (pwr x q) x = pwr x (Suc q)"
100   "mul x x = pwr x 2"
101   "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
102   "pwr (pwr x p) q = pwr x (p * q)"
103   "pwr x 0 = r1"
104   "pwr x 1 = x"
105   "mul x (add y z) = add (mul x y) (mul x z)"
106   "pwr x (Suc q) = mul x (pwr x q)"
107   "pwr x (2*n) = mul (pwr x n) (pwr x n)"
108   "pwr x (Suc (2*n)) = mul x (mul (pwr x n) (pwr x n))"
109 proof -
110   show "add (mul a m) (mul b m) = mul (add a b) m" using mul_d mul_c by simp
111 next show"add (mul a m) m = mul (add a r1) m" using mul_d mul_c mul_1 by simp
112 next show "add m (mul a m) = mul (add a r1) m" using mul_c mul_d mul_1 add_c by simp
113 next show "add m m = mul (add r1 r1) m" using mul_c mul_d mul_1 by simp
114 next show "add r0 a = a" using add_0 by simp
116 next show "mul a b = mul b a" using mul_c by simp
117 next show "mul (add a b) c = add (mul a c) (mul b c)" using mul_c mul_d by simp
118 next show "mul r0 a = r0" using mul_0 by simp
119 next show "mul a r0 = r0" using mul_0 mul_c by simp
120 next show "mul r1 a = a" using mul_1 by simp
121 next show "mul a r1 = a" using mul_1 mul_c by simp
122 next show "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
123     using mul_c mul_a by simp
124 next show "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
125     using mul_a by simp
126 next
127   have "mul (mul lx ly) (mul rx ry) = mul (mul rx ry) (mul lx ly)" by (rule mul_c)
128   also have "\<dots> = mul rx (mul ry (mul lx ly))" using mul_a by simp
129   finally
130   show "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
131     using mul_c by simp
132 next show "mul (mul lx ly) rx = mul (mul lx rx) ly" using mul_c mul_a by simp
133 next
134   show "mul (mul lx ly) rx = mul lx (mul ly rx)" by (simp add: mul_a)
135 next show "mul lx (mul rx ry) = mul (mul lx rx) ry" by (simp add: mul_a )
136 next show "mul lx (mul rx ry) = mul rx (mul lx ry)" by (simp add: mul_a,simp add: mul_c)
145 next show "mul (pwr x p) (pwr x q) = pwr x (p + q)" by (rule mul_pwr)
146 next show "mul x (pwr x q) = pwr x (Suc q)" using pwr_Suc by simp
147 next show "mul (pwr x q) x = pwr x (Suc q)" using pwr_Suc mul_c by simp
148 next show "mul x x = pwr x 2" by (simp add: nat_number pwr_Suc pwr_0 mul_1 mul_c)
149 next show "pwr (mul x y) q = mul (pwr x q) (pwr y q)" by (rule pwr_mul)
150 next show "pwr (pwr x p) q = pwr x (p * q)" by (rule pwr_pwr)
151 next show "pwr x 0 = r1" using pwr_0 .
152 next show "pwr x 1 = x" by (simp add: nat_number pwr_Suc pwr_0 mul_1 mul_c)
153 next show "mul x (add y z) = add (mul x y) (mul x z)" using mul_d by simp
154 next show "pwr x (Suc q) = mul x (pwr x q)" using pwr_Suc by simp
155 next show "pwr x (2 * n) = mul (pwr x n) (pwr x n)" by (simp add: nat_number mul_pwr)
156 next show "pwr x (Suc (2 * n)) = mul x (mul (pwr x n) (pwr x n))"
157     by (simp add: nat_number pwr_Suc mul_pwr)
158 qed
161 lemmas gb_semiring_axioms' =
162   gb_semiring_axioms [normalizer
163     semiring ops: semiring_ops
164     semiring rules: semiring_rules]
166 end
168 interpretation class_semiring: gb_semiring
169     ["op +" "op *" "op ^" "0::'a::{comm_semiring_1, recpower}" "1"]
170   by unfold_locales (auto simp add: ring_simps power_Suc)
172 lemmas nat_arith =
173   add_nat_number_of diff_nat_number_of mult_nat_number_of eq_nat_number_of less_nat_number_of
175 lemma not_iszero_Numeral1: "\<not> iszero (Numeral1::'a::number_ring)"
177 lemmas comp_arith = Let_def arith_simps nat_arith rel_simps if_False
180   numeral_0_eq_0[symmetric] numerals[symmetric] not_iszero_1
181   iszero_number_of_Bit1 iszero_number_of_Bit0 nonzero_number_of_Min
182   iszero_number_of_Pls iszero_0 not_iszero_Numeral1
184 lemmas semiring_norm = comp_arith
186 ML {*
187 local
189 open Conv;
191 fun numeral_is_const ct =
192   can HOLogic.dest_number (Thm.term_of ct);
194 fun int_of_rat x =
195   (case Rat.quotient_of_rat x of (i, 1) => i
196   | _ => error "int_of_rat: bad int");
198 val numeral_conv =
199   Simplifier.rewrite (HOL_basic_ss addsimps @{thms semiring_norm}) then_conv
201     (@{thms numeral_1_eq_1} @ @{thms numeral_0_eq_0} @ @{thms numerals(1-2)}));
203 in
205 fun normalizer_funs key =
206   NormalizerData.funs key
207    {is_const = fn phi => numeral_is_const,
208     dest_const = fn phi => fn ct =>
209       Rat.rat_of_int (snd
210         (HOLogic.dest_number (Thm.term_of ct)
211           handle TERM _ => error "ring_dest_const")),
212     mk_const = fn phi => fn cT => fn x => Numeral.mk_cnumber cT (int_of_rat x),
213     conv = fn phi => K numeral_conv}
215 end
216 *}
218 declaration {* normalizer_funs @{thm class_semiring.gb_semiring_axioms'} *}
221 locale gb_ring = gb_semiring +
222   fixes sub :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
223     and neg :: "'a \<Rightarrow> 'a"
224   assumes neg_mul: "neg x = mul (neg r1) x"
226 begin
228 lemma ring_ops:
229   includes meta_term_syntax
230   shows "TERM (sub x y)" and "TERM (neg x)" .
232 lemmas ring_rules = neg_mul sub_add
234 lemmas gb_ring_axioms' =
235   gb_ring_axioms [normalizer
236     semiring ops: semiring_ops
237     semiring rules: semiring_rules
238     ring ops: ring_ops
239     ring rules: ring_rules]
241 end
244 interpretation class_ring: gb_ring ["op +" "op *" "op ^"
245     "0::'a::{comm_semiring_1,recpower,number_ring}" 1 "op -" "uminus"]
246   by unfold_locales simp_all
249 declaration {* normalizer_funs @{thm class_ring.gb_ring_axioms'} *}
251 use "Tools/Groebner_Basis/normalizer.ML"
254 method_setup sring_norm = {*
255   Method.ctxt_args (fn ctxt => Method.SIMPLE_METHOD' (Normalizer.semiring_normalize_tac ctxt))
256 *} "semiring normalizer"
259 locale gb_field = gb_ring +
260   fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
261     and inverse:: "'a \<Rightarrow> 'a"
262   assumes divide: "divide x y = mul x (inverse y)"
263      and inverse: "inverse x = divide r1 x"
264 begin
266 lemmas gb_field_axioms' =
267   gb_field_axioms [normalizer
268     semiring ops: semiring_ops
269     semiring rules: semiring_rules
270     ring ops: ring_ops
271     ring rules: ring_rules]
273 end
276 subsection {* Groebner Bases *}
278 locale semiringb = gb_semiring +
281     add (mul w z) (mul x y) \<longleftrightarrow> w = x \<or> y = z"
282 begin
284 lemma noteq_reduce: "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
285 proof-
286   have "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> \<not> (a = b \<or> c = d)" by simp
287   also have "\<dots> \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
289   finally show "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
290     by simp
291 qed
293 lemma add_scale_eq_noteq: "\<lbrakk>r \<noteq> r0 ; (a = b) \<and> ~(c = d)\<rbrakk>
294   \<Longrightarrow> add a (mul r c) \<noteq> add b (mul r d)"
295 proof(clarify)
296   assume nz: "r\<noteq> r0" and cnd: "c\<noteq>d"
297     and eq: "add b (mul r c) = add b (mul r d)"
298   hence "mul r c = mul r d" using cnd add_cancel by simp
299   hence "add (mul r0 d) (mul r c) = add (mul r0 c) (mul r d)"
300     using mul_0 add_cancel by simp
301   thus "False" using add_mul_solve nz cnd by simp
302 qed
304 lemma add_r0_iff: " x = add x a \<longleftrightarrow> a = r0"
305 proof-
308 qed
310 declare gb_semiring_axioms' [normalizer del]
312 lemmas semiringb_axioms' = semiringb_axioms [normalizer
313   semiring ops: semiring_ops
314   semiring rules: semiring_rules
317 end
319 locale ringb = semiringb + gb_ring +
320   assumes subr0_iff: "sub x y = r0 \<longleftrightarrow> x = y"
321 begin
323 declare gb_ring_axioms' [normalizer del]
325 lemmas ringb_axioms' = ringb_axioms [normalizer
326   semiring ops: semiring_ops
327   semiring rules: semiring_rules
328   ring ops: ring_ops
329   ring rules: ring_rules
333 end
336 lemma no_zero_divirors_neq0:
337   assumes az: "(a::'a::no_zero_divisors) \<noteq> 0"
338     and ab: "a*b = 0" shows "b = 0"
339 proof -
340   { assume bz: "b \<noteq> 0"
341     from no_zero_divisors [OF az bz] ab have False by blast }
342   thus "b = 0" by blast
343 qed
345 interpretation class_ringb: ringb
346   ["op +" "op *" "op ^" "0::'a::{idom,recpower,number_ring}" "1" "op -" "uminus"]
347 proof(unfold_locales, simp add: ring_simps power_Suc, auto)
348   fix w x y z ::"'a::{idom,recpower,number_ring}"
349   assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
350   hence ynz': "y - z \<noteq> 0" by simp
351   from p have "w * y + x* z - w*z - x*y = 0" by simp
352   hence "w* (y - z) - x * (y - z) = 0" by (simp add: ring_simps)
353   hence "(y - z) * (w - x) = 0" by (simp add: ring_simps)
354   with  no_zero_divirors_neq0 [OF ynz']
355   have "w - x = 0" by blast
356   thus "w = x"  by simp
357 qed
359 declaration {* normalizer_funs @{thm class_ringb.ringb_axioms'} *}
361 interpretation natgb: semiringb
362   ["op +" "op *" "op ^" "0::nat" "1"]
363 proof (unfold_locales, simp add: ring_simps power_Suc)
364   fix w x y z ::"nat"
365   { assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
366     hence "y < z \<or> y > z" by arith
367     moreover {
368       assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z - y" in exI, auto)
369       then obtain k where kp: "k>0" and yz:"z = y + k" by blast
370       from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz ring_simps)
371       hence "x*k = w*k" by simp
372       hence "w = x" using kp by (simp add: mult_cancel2) }
373     moreover {
374       assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y - z" in exI, auto)
375       then obtain k where kp: "k>0" and yz:"y = z + k" by blast
376       from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz ring_simps)
377       hence "w*k = x*k" by simp
378       hence "w = x" using kp by (simp add: mult_cancel2)}
379     ultimately have "w=x" by blast }
380   thus "(w * y + x * z = w * z + x * y) = (w = x \<or> y = z)" by auto
381 qed
383 declaration {* normalizer_funs @{thm natgb.semiringb_axioms'} *}
385 locale fieldgb = ringb + gb_field
386 begin
388 declare gb_field_axioms' [normalizer del]
390 lemmas fieldgb_axioms' = fieldgb_axioms [normalizer
391   semiring ops: semiring_ops
392   semiring rules: semiring_rules
393   ring ops: ring_ops
394   ring rules: ring_rules
398 end
401 lemmas bool_simps = simp_thms(1-34)
402 lemma dnf:
403     "(P & (Q | R)) = ((P&Q) | (P&R))" "((Q | R) & P) = ((Q&P) | (R&P))"
404     "(P \<and> Q) = (Q \<and> P)" "(P \<or> Q) = (Q \<or> P)"
405   by blast+
407 lemmas weak_dnf_simps = dnf bool_simps
409 lemma nnf_simps:
410     "(\<not>(P \<and> Q)) = (\<not>P \<or> \<not>Q)" "(\<not>(P \<or> Q)) = (\<not>P \<and> \<not>Q)" "(P \<longrightarrow> Q) = (\<not>P \<or> Q)"
411     "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not> \<not>(P)) = P"
412   by blast+
414 lemma PFalse:
415     "P \<equiv> False \<Longrightarrow> \<not> P"
416     "\<not> P \<Longrightarrow> (P \<equiv> False)"
417   by auto
418 use "Tools/Groebner_Basis/groebner.ML"
420 method_setup algebra =
421 {*
422 let
423  fun keyword k = Scan.lift (Args.\$\$\$ k -- Args.colon) >> K ()
425  val delN = "del"
426  val any_keyword = keyword addN || keyword delN
427  val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
428 in
429 fn src => Method.syntax
430     ((Scan.optional (keyword addN |-- thms) []) --
431     (Scan.optional (keyword delN |-- thms) [])) src
432  #> (fn ((add_ths, del_ths), ctxt) =>
433        Method.SIMPLE_METHOD' (Groebner.algebra_tac add_ths del_ths ctxt))
434 end
435 *} "solve polynomial equations over (semi)rings and ideal membership problems using Groebner bases"
436 declare dvd_def[algebra]
437 declare dvd_eq_mod_eq_0[symmetric, algebra]
438 declare nat_mod_div_trivial[algebra]
439 declare nat_mod_mod_trivial[algebra]
440 declare conjunct1[OF DIVISION_BY_ZERO, algebra]
441 declare conjunct2[OF DIVISION_BY_ZERO, algebra]
442 declare zmod_zdiv_equality[symmetric,algebra]
443 declare zdiv_zmod_equality[symmetric, algebra]
444 declare zdiv_zminus_zminus[algebra]
445 declare zmod_zminus_zminus[algebra]
446 declare zdiv_zminus2[algebra]
447 declare zmod_zminus2[algebra]
448 declare zdiv_zero[algebra]
449 declare zmod_zero[algebra]
450 declare zmod_1[algebra]
451 declare zdiv_1[algebra]
452 declare zmod_minus1_right[algebra]
453 declare zdiv_minus1_right[algebra]
454 declare mod_div_trivial[algebra]
455 declare mod_mod_trivial[algebra]
456 declare zmod_zmult_self1[algebra]
457 declare zmod_zmult_self2[algebra]
458 declare zmod_eq_0_iff[algebra]
459 declare zdvd_0_left[algebra]
460 declare zdvd1_eq[algebra]
461 declare zmod_eq_dvd_iff[algebra]
462 declare nat_mod_eq_iff[algebra]
464 end