src/HOL/Groebner_Basis.thy
 author haftmann Wed Feb 10 08:49:26 2010 +0100 (2010-02-10) changeset 35084 e25eedfc15ce parent 35050 9f841f20dca6 child 35092 cfe605c54e50 permissions -rw-r--r--
moved constants inverse and divide to Ring.thy
1 (*  Title:      HOL/Groebner_Basis.thy
2     Author:     Amine Chaieb, TU Muenchen
3 *)
5 header {* Semiring normalization and Groebner Bases *}
7 theory Groebner_Basis
8 imports Numeral_Simprocs
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   shows "TERM (add x y)" and "TERM (mul x y)" and "TERM (pwr x n)"
67     and "TERM r0" and "TERM r1" .
69 lemma semiring_rules:
70   "add (mul a m) (mul b m) = mul (add a b) m"
71   "add (mul a m) m = mul (add a r1) m"
72   "add m (mul a m) = mul (add a r1) m"
74   "add r0 a = a"
75   "add a r0 = a"
76   "mul a b = mul b a"
77   "mul (add a b) c = add (mul a c) (mul b c)"
78   "mul r0 a = r0"
79   "mul a r0 = r0"
80   "mul r1 a = a"
81   "mul a r1 = a"
82   "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
83   "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
84   "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
85   "mul (mul lx ly) rx = mul (mul lx rx) ly"
86   "mul (mul lx ly) rx = mul lx (mul ly rx)"
87   "mul lx (mul rx ry) = mul (mul lx rx) ry"
88   "mul lx (mul rx ry) = mul rx (mul lx ry)"
95   "mul (pwr x p) (pwr x q) = pwr x (p + q)"
96   "mul x (pwr x q) = pwr x (Suc q)"
97   "mul (pwr x q) x = pwr x (Suc q)"
98   "mul x x = pwr x 2"
99   "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
100   "pwr (pwr x p) q = pwr x (p * q)"
101   "pwr x 0 = r1"
102   "pwr x 1 = x"
103   "mul x (add y z) = add (mul x y) (mul x z)"
104   "pwr x (Suc q) = mul x (pwr x q)"
105   "pwr x (2*n) = mul (pwr x n) (pwr x n)"
106   "pwr x (Suc (2*n)) = mul x (mul (pwr x n) (pwr x n))"
107 proof -
108   show "add (mul a m) (mul b m) = mul (add a b) m" using mul_d mul_c by simp
109 next show"add (mul a m) m = mul (add a r1) m" using mul_d mul_c mul_1 by simp
110 next show "add m (mul a m) = mul (add a r1) m" using mul_c mul_d mul_1 add_c by simp
111 next show "add m m = mul (add r1 r1) m" using mul_c mul_d mul_1 by simp
112 next show "add r0 a = a" using add_0 by simp
114 next show "mul a b = mul b a" using mul_c by simp
115 next show "mul (add a b) c = add (mul a c) (mul b c)" using mul_c mul_d by simp
116 next show "mul r0 a = r0" using mul_0 by simp
117 next show "mul a r0 = r0" using mul_0 mul_c by simp
118 next show "mul r1 a = a" using mul_1 by simp
119 next show "mul a r1 = a" using mul_1 mul_c by simp
120 next show "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
121     using mul_c mul_a by simp
122 next show "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
123     using mul_a by simp
124 next
125   have "mul (mul lx ly) (mul rx ry) = mul (mul rx ry) (mul lx ly)" by (rule mul_c)
126   also have "\<dots> = mul rx (mul ry (mul lx ly))" using mul_a by simp
127   finally
128   show "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
129     using mul_c by simp
130 next show "mul (mul lx ly) rx = mul (mul lx rx) ly" using mul_c mul_a by simp
131 next
132   show "mul (mul lx ly) rx = mul lx (mul ly rx)" by (simp add: mul_a)
133 next show "mul lx (mul rx ry) = mul (mul lx rx) ry" by (simp add: mul_a )
134 next show "mul lx (mul rx ry) = mul rx (mul lx ry)" by (simp add: mul_a,simp add: mul_c)
143 next show "mul (pwr x p) (pwr x q) = pwr x (p + q)" by (rule mul_pwr)
144 next show "mul x (pwr x q) = pwr x (Suc q)" using pwr_Suc by simp
145 next show "mul (pwr x q) x = pwr x (Suc q)" using pwr_Suc mul_c by simp
146 next show "mul x x = pwr x 2" by (simp add: nat_number pwr_Suc pwr_0 mul_1 mul_c)
147 next show "pwr (mul x y) q = mul (pwr x q) (pwr y q)" by (rule pwr_mul)
148 next show "pwr (pwr x p) q = pwr x (p * q)" by (rule pwr_pwr)
149 next show "pwr x 0 = r1" using pwr_0 .
150 next show "pwr x 1 = x" unfolding One_nat_def by (simp add: nat_number pwr_Suc pwr_0 mul_1 mul_c)
151 next show "mul x (add y z) = add (mul x y) (mul x z)" using mul_d by simp
152 next show "pwr x (Suc q) = mul x (pwr x q)" using pwr_Suc by simp
153 next show "pwr x (2 * n) = mul (pwr x n) (pwr x n)" by (simp add: nat_number mul_pwr)
154 next show "pwr x (Suc (2 * n)) = mul x (mul (pwr x n) (pwr x n))"
155     by (simp add: nat_number pwr_Suc mul_pwr)
156 qed
159 lemmas gb_semiring_axioms' =
160   gb_semiring_axioms [normalizer
161     semiring ops: semiring_ops
162     semiring rules: semiring_rules]
164 end
166 interpretation class_semiring: gb_semiring
167     "op +" "op *" "op ^" "0::'a::{comm_semiring_1}" "1"
168   proof qed (auto simp add: algebra_simps power_Suc)
170 lemmas nat_arith =
172   diff_nat_number_of
173   mult_nat_number_of
174   eq_nat_number_of
175   less_nat_number_of
177 lemma not_iszero_Numeral1: "\<not> iszero (Numeral1::'a::number_ring)"
180 lemmas comp_arith =
181   Let_def arith_simps nat_arith rel_simps neg_simps if_False
183   numeral_1_eq_1[symmetric] Suc_eq_plus1
184   numeral_0_eq_0[symmetric] numerals[symmetric]
185   iszero_simps not_iszero_Numeral1
187 lemmas semiring_norm = comp_arith
189 ML {*
190 local
192 open Conv;
194 fun numeral_is_const ct = can HOLogic.dest_number (Thm.term_of ct);
196 fun int_of_rat x =
197   (case Rat.quotient_of_rat x of (i, 1) => i
198   | _ => error "int_of_rat: bad int");
200 val numeral_conv =
201   Simplifier.rewrite (HOL_basic_ss addsimps @{thms semiring_norm}) then_conv
203     (@{thms numeral_1_eq_1} @ @{thms numeral_0_eq_0} @ @{thms numerals(1-2)}));
205 in
207 fun normalizer_funs key =
208   NormalizerData.funs key
209    {is_const = fn phi => numeral_is_const,
210     dest_const = fn phi => fn ct =>
211       Rat.rat_of_int (snd
212         (HOLogic.dest_number (Thm.term_of ct)
213           handle TERM _ => error "ring_dest_const")),
214     mk_const = fn phi => fn cT => fn x => Numeral.mk_cnumber cT (int_of_rat x),
215     conv = fn phi => K numeral_conv}
217 end
218 *}
220 declaration {* normalizer_funs @{thm class_semiring.gb_semiring_axioms'} *}
223 locale gb_ring = gb_semiring +
224   fixes sub :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
225     and neg :: "'a \<Rightarrow> 'a"
226   assumes neg_mul: "neg x = mul (neg r1) x"
228 begin
230 lemma ring_ops: 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,number_ring}" 1 "op -" "uminus"
246   proof qed 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   Scan.succeed (SIMPLE_METHOD' o Normalizer.semiring_normalize_tac)
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_inverse: "divide x y = mul x (inverse y)"
263      and inverse_divide: "inverse x = divide r1 x"
264 begin
266 lemma field_ops: shows "TERM (divide x y)" and "TERM (inverse x)" .
268 lemmas field_rules = divide_inverse inverse_divide
270 lemmas gb_field_axioms' =
271   gb_field_axioms [normalizer
272     semiring ops: semiring_ops
273     semiring rules: semiring_rules
274     ring ops: ring_ops
275     ring rules: ring_rules
276     field ops: field_ops
277     field rules: field_rules]
279 end
282 subsection {* Groebner Bases *}
284 locale semiringb = gb_semiring +
287     add (mul w z) (mul x y) \<longleftrightarrow> w = x \<or> y = z"
288 begin
290 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)"
291 proof-
292   have "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> \<not> (a = b \<or> c = d)" by simp
293   also have "\<dots> \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
295   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)"
296     by simp
297 qed
299 lemma add_scale_eq_noteq: "\<lbrakk>r \<noteq> r0 ; (a = b) \<and> ~(c = d)\<rbrakk>
300   \<Longrightarrow> add a (mul r c) \<noteq> add b (mul r d)"
301 proof(clarify)
302   assume nz: "r\<noteq> r0" and cnd: "c\<noteq>d"
303     and eq: "add b (mul r c) = add b (mul r d)"
304   hence "mul r c = mul r d" using cnd add_cancel by simp
305   hence "add (mul r0 d) (mul r c) = add (mul r0 c) (mul r d)"
306     using mul_0 add_cancel by simp
307   thus "False" using add_mul_solve nz cnd by simp
308 qed
310 lemma add_r0_iff: " x = add x a \<longleftrightarrow> a = r0"
311 proof-
314 qed
316 declare gb_semiring_axioms' [normalizer del]
318 lemmas semiringb_axioms' = semiringb_axioms [normalizer
319   semiring ops: semiring_ops
320   semiring rules: semiring_rules
323 end
325 locale ringb = semiringb + gb_ring +
326   assumes subr0_iff: "sub x y = r0 \<longleftrightarrow> x = y"
327 begin
329 declare gb_ring_axioms' [normalizer del]
331 lemmas ringb_axioms' = ringb_axioms [normalizer
332   semiring ops: semiring_ops
333   semiring rules: semiring_rules
334   ring ops: ring_ops
335   ring rules: ring_rules
339 end
342 lemma no_zero_divirors_neq0:
343   assumes az: "(a::'a::no_zero_divisors) \<noteq> 0"
344     and ab: "a*b = 0" shows "b = 0"
345 proof -
346   { assume bz: "b \<noteq> 0"
347     from no_zero_divisors [OF az bz] ab have False by blast }
348   thus "b = 0" by blast
349 qed
351 interpretation class_ringb: ringb
352   "op +" "op *" "op ^" "0::'a::{idom,number_ring}" "1" "op -" "uminus"
353 proof(unfold_locales, simp add: algebra_simps power_Suc, auto)
354   fix w x y z ::"'a::{idom,number_ring}"
355   assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
356   hence ynz': "y - z \<noteq> 0" by simp
357   from p have "w * y + x* z - w*z - x*y = 0" by simp
358   hence "w* (y - z) - x * (y - z) = 0" by (simp add: algebra_simps)
359   hence "(y - z) * (w - x) = 0" by (simp add: algebra_simps)
360   with  no_zero_divirors_neq0 [OF ynz']
361   have "w - x = 0" by blast
362   thus "w = x"  by simp
363 qed
365 declaration {* normalizer_funs @{thm class_ringb.ringb_axioms'} *}
367 interpretation natgb: semiringb
368   "op +" "op *" "op ^" "0::nat" "1"
369 proof (unfold_locales, simp add: algebra_simps power_Suc)
370   fix w x y z ::"nat"
371   { assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
372     hence "y < z \<or> y > z" by arith
373     moreover {
374       assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z - y" in exI, auto)
375       then obtain k where kp: "k>0" and yz:"z = y + k" by blast
376       from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz algebra_simps)
377       hence "x*k = w*k" by simp
378       hence "w = x" using kp by (simp add: mult_cancel2) }
379     moreover {
380       assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y - z" in exI, auto)
381       then obtain k where kp: "k>0" and yz:"y = z + k" by blast
382       from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz algebra_simps)
383       hence "w*k = x*k" by simp
384       hence "w = x" using kp by (simp add: mult_cancel2)}
385     ultimately have "w=x" by blast }
386   thus "(w * y + x * z = w * z + x * y) = (w = x \<or> y = z)" by auto
387 qed
389 declaration {* normalizer_funs @{thm natgb.semiringb_axioms'} *}
391 locale fieldgb = ringb + gb_field
392 begin
394 declare gb_field_axioms' [normalizer del]
396 lemmas fieldgb_axioms' = fieldgb_axioms [normalizer
397   semiring ops: semiring_ops
398   semiring rules: semiring_rules
399   ring ops: ring_ops
400   ring rules: ring_rules
401   field ops: field_ops
402   field rules: field_rules
406 end
409 lemmas bool_simps = simp_thms(1-34)
410 lemma dnf:
411     "(P & (Q | R)) = ((P&Q) | (P&R))" "((Q | R) & P) = ((Q&P) | (R&P))"
412     "(P \<and> Q) = (Q \<and> P)" "(P \<or> Q) = (Q \<or> P)"
413   by blast+
415 lemmas weak_dnf_simps = dnf bool_simps
417 lemma nnf_simps:
418     "(\<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)"
419     "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not> \<not>(P)) = P"
420   by blast+
422 lemma PFalse:
423     "P \<equiv> False \<Longrightarrow> \<not> P"
424     "\<not> P \<Longrightarrow> (P \<equiv> False)"
425   by auto
426 use "Tools/Groebner_Basis/groebner.ML"
428 method_setup algebra =
429 {*
430 let
431  fun keyword k = Scan.lift (Args.\$\$\$ k -- Args.colon) >> K ()
433  val delN = "del"
434  val any_keyword = keyword addN || keyword delN
435  val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
436 in
437   ((Scan.optional (keyword addN |-- thms) []) --
438    (Scan.optional (keyword delN |-- thms) [])) >>
439   (fn (add_ths, del_ths) => fn ctxt =>
440        SIMPLE_METHOD' (Groebner.algebra_tac add_ths del_ths ctxt))
441 end
442 *} "solve polynomial equations over (semi)rings and ideal membership problems using Groebner bases"
443 declare dvd_def[algebra]
444 declare dvd_eq_mod_eq_0[symmetric, algebra]
445 declare mod_div_trivial[algebra]
446 declare mod_mod_trivial[algebra]
447 declare conjunct1[OF DIVISION_BY_ZERO, algebra]
448 declare conjunct2[OF DIVISION_BY_ZERO, algebra]
449 declare zmod_zdiv_equality[symmetric,algebra]
450 declare zdiv_zmod_equality[symmetric, algebra]
451 declare zdiv_zminus_zminus[algebra]
452 declare zmod_zminus_zminus[algebra]
453 declare zdiv_zminus2[algebra]
454 declare zmod_zminus2[algebra]
455 declare zdiv_zero[algebra]
456 declare zmod_zero[algebra]
457 declare mod_by_1[algebra]
458 declare div_by_1[algebra]
459 declare zmod_minus1_right[algebra]
460 declare zdiv_minus1_right[algebra]
461 declare mod_div_trivial[algebra]
462 declare mod_mod_trivial[algebra]
463 declare mod_mult_self2_is_0[algebra]
464 declare mod_mult_self1_is_0[algebra]
465 declare zmod_eq_0_iff[algebra]
466 declare dvd_0_left_iff[algebra]
467 declare zdvd1_eq[algebra]
468 declare zmod_eq_dvd_iff[algebra]
469 declare nat_mod_eq_iff[algebra]
471 subsection{* Groebner Bases for fields *}
473 interpretation class_fieldgb:
474   fieldgb "op +" "op *" "op ^" "0::'a::{field,number_ring}" "1" "op -" "uminus" "op /" "inverse" apply (unfold_locales) by (simp_all add: divide_inverse)
476 lemma divide_Numeral1: "(x::'a::{field,number_ring}) / Numeral1 = x" by simp
477 lemma divide_Numeral0: "(x::'a::{field,number_ring, division_by_zero}) / Numeral0 = 0"
478   by simp
479 lemma mult_frac_frac: "((x::'a::{field,division_by_zero}) / y) * (z / w) = (x*z) / (y*w)"
480   by simp
481 lemma mult_frac_num: "((x::'a::{field, division_by_zero}) / y) * z  = (x*z) / y"
482   by simp
483 lemma mult_num_frac: "((x::'a::{field, division_by_zero}) / y) * z  = (x*z) / y"
484   by simp
486 lemma Numeral1_eq1_nat: "(1::nat) = Numeral1" by simp
488 lemma add_frac_num: "y\<noteq> 0 \<Longrightarrow> (x::'a::{field, division_by_zero}) / y + z = (x + z*y) / y"
490 lemma add_num_frac: "y\<noteq> 0 \<Longrightarrow> z + (x::'a::{field, division_by_zero}) / y = (x + z*y) / y"
493 ML {*
494 let open Conv
495 in fconv_rule (arg_conv (arg1_conv (rewr_conv (mk_meta_eq @{thm mult_commute})))) (@{thm field_divide_inverse} RS sym)
496 end
497 *}
499 ML{*
500 local
501  val zr = @{cpat "0"}
502  val zT = ctyp_of_term zr
503  val geq = @{cpat "op ="}
504  val eqT = Thm.dest_ctyp (ctyp_of_term geq) |> hd
509  fun prove_nz ss T t =
510     let
511       val z = instantiate_cterm ([(zT,T)],[]) zr
512       val eq = instantiate_cterm ([(eqT,T)],[]) geq
513       val th = Simplifier.rewrite (ss addsimps simp_thms)
514            (Thm.capply @{cterm "Trueprop"} (Thm.capply @{cterm "Not"}
515                   (Thm.capply (Thm.capply eq t) z)))
516     in equal_elim (symmetric th) TrueI
517     end
519  fun proc phi ss ct =
520   let
521     val ((x,y),(w,z)) =
522          (Thm.dest_binop #> (fn (a,b) => (Thm.dest_binop a, Thm.dest_binop b))) ct
523     val _ = map (HOLogic.dest_number o term_of) [x,y,z,w]
524     val T = ctyp_of_term x
525     val [y_nz, z_nz] = map (prove_nz ss T) [y, z]
526     val th = instantiate' [SOME T] (map SOME [y,z,x,w]) add_frac_eq
527   in SOME (implies_elim (implies_elim th y_nz) z_nz)
528   end
529   handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE
531  fun proc2 phi ss ct =
532   let
533     val (l,r) = Thm.dest_binop ct
534     val T = ctyp_of_term l
535   in (case (term_of l, term_of r) of
536       (Const(@{const_name Rings.divide},_)\$_\$_, _) =>
537         let val (x,y) = Thm.dest_binop l val z = r
538             val _ = map (HOLogic.dest_number o term_of) [x,y,z]
539             val ynz = prove_nz ss T y
540         in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,x,z]) add_frac_num) ynz)
541         end
542      | (_, Const (@{const_name Rings.divide},_)\$_\$_) =>
543         let val (x,y) = Thm.dest_binop r val z = l
544             val _ = map (HOLogic.dest_number o term_of) [x,y,z]
545             val ynz = prove_nz ss T y
546         in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,z,x]) add_num_frac) ynz)
547         end
548      | _ => NONE)
549   end
550   handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE
552  fun is_number (Const(@{const_name Rings.divide},_)\$a\$b) = is_number a andalso is_number b
553    | is_number t = can HOLogic.dest_number t
555  val is_number = is_number o term_of
557  fun proc3 phi ss ct =
558   (case term_of ct of
559     Const(@{const_name Algebras.less},_)\$(Const(@{const_name Rings.divide},_)\$_\$_)\$_ =>
560       let
561         val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
562         val _ = map is_number [a,b,c]
563         val T = ctyp_of_term c
564         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_less_eq"}
565       in SOME (mk_meta_eq th) end
566   | Const(@{const_name Algebras.less_eq},_)\$(Const(@{const_name Rings.divide},_)\$_\$_)\$_ =>
567       let
568         val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
569         val _ = map is_number [a,b,c]
570         val T = ctyp_of_term c
571         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_le_eq"}
572       in SOME (mk_meta_eq th) end
573   | Const("op =",_)\$(Const(@{const_name Rings.divide},_)\$_\$_)\$_ =>
574       let
575         val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
576         val _ = map is_number [a,b,c]
577         val T = ctyp_of_term c
578         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_eq_eq"}
579       in SOME (mk_meta_eq th) end
580   | Const(@{const_name Algebras.less},_)\$_\$(Const(@{const_name Rings.divide},_)\$_\$_) =>
581     let
582       val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
583         val _ = map is_number [a,b,c]
584         val T = ctyp_of_term c
585         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "less_divide_eq"}
586       in SOME (mk_meta_eq th) end
587   | Const(@{const_name Algebras.less_eq},_)\$_\$(Const(@{const_name Rings.divide},_)\$_\$_) =>
588     let
589       val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
590         val _ = map is_number [a,b,c]
591         val T = ctyp_of_term c
592         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "le_divide_eq"}
593       in SOME (mk_meta_eq th) end
594   | Const("op =",_)\$_\$(Const(@{const_name Rings.divide},_)\$_\$_) =>
595     let
596       val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
597         val _ = map is_number [a,b,c]
598         val T = ctyp_of_term c
599         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "eq_divide_eq"}
600       in SOME (mk_meta_eq th) end
601   | _ => NONE)
602   handle TERM _ => NONE | CTERM _ => NONE | THM _ => NONE
605        make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + (?w::?'a::field)/?z"}],
607                      proc = proc, identifier = []}
610        make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + ?z"}, @{cpat "?z + (?x::?'a::field)/?y"}],
612                      proc = proc2, identifier = []}
614 val ord_frac_simproc =
615   make_simproc
616     {lhss = [@{cpat "(?a::(?'a::{field, ord}))/?b < ?c"},
617              @{cpat "(?a::(?'a::{field, ord}))/?b \<le> ?c"},
618              @{cpat "?c < (?a::(?'a::{field, ord}))/?b"},
619              @{cpat "?c \<le> (?a::(?'a::{field, ord}))/?b"},
620              @{cpat "?c = ((?a::(?'a::{field, ord}))/?b)"},
621              @{cpat "((?a::(?'a::{field, ord}))/ ?b) = ?c"}],
622              name = "ord_frac_simproc", proc = proc3, identifier = []}
624 local
625 open Conv
626 in
628 val ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"},
629            @{thm "divide_Numeral1"},
630            @{thm "Fields.divide_zero"}, @{thm "divide_Numeral0"},
631            @{thm "divide_divide_eq_left"}, @{thm "mult_frac_frac"},
632            @{thm "mult_num_frac"}, @{thm "mult_frac_num"},
633            @{thm "mult_frac_frac"}, @{thm "times_divide_eq_right"},
634            @{thm "times_divide_eq_left"}, @{thm "divide_divide_eq_right"},
635            @{thm "diff_def"}, @{thm "minus_divide_left"},
636            @{thm "Numeral1_eq1_nat"}, @{thm "add_divide_distrib"} RS sym,
637            @{thm field_divide_inverse} RS sym, @{thm inverse_divide},
638            fconv_rule (arg_conv (arg1_conv (rewr_conv (mk_meta_eq @{thm mult_commute}))))
639            (@{thm field_divide_inverse} RS sym)]
641 val comp_conv = (Simplifier.rewrite
646                             ord_frac_simproc]
649   [@{thm numeral_1_eq_1},@{thm numeral_0_eq_0}] @ @{thms numerals(1-2)}))
650 end
652 fun numeral_is_const ct =
653   case term_of ct of
654    Const (@{const_name Rings.divide},_) \$ a \$ b =>
655      can HOLogic.dest_number a andalso can HOLogic.dest_number b
656  | Const (@{const_name Rings.inverse},_)\$t => can HOLogic.dest_number t
657  | t => can HOLogic.dest_number t
659 fun dest_const ct = ((case term_of ct of
660    Const (@{const_name Rings.divide},_) \$ a \$ b=>
661     Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b))
662  | Const (@{const_name Rings.inverse},_)\$t =>
663                Rat.inv (Rat.rat_of_int (snd (HOLogic.dest_number t)))
664  | t => Rat.rat_of_int (snd (HOLogic.dest_number t)))
665    handle TERM _ => error "ring_dest_const")
667 fun mk_const phi cT x =
668  let val (a, b) = Rat.quotient_of_rat x
669  in if b = 1 then Numeral.mk_cnumber cT a
670     else Thm.capply
671          (Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"})
672                      (Numeral.mk_cnumber cT a))
673          (Numeral.mk_cnumber cT b)
674   end
676 in
677  val field_comp_conv = comp_conv;
678  val fieldgb_declaration =
679   NormalizerData.funs @{thm class_fieldgb.fieldgb_axioms'}
680    {is_const = K numeral_is_const,
681     dest_const = K dest_const,
682     mk_const = mk_const,
683     conv = K (K comp_conv)}
684 end;
685 *}
687 declaration fieldgb_declaration
689 end