src/HOL/Groebner_Basis.thy
 author ballarin Fri Dec 12 20:10:22 2008 +0100 (2008-12-12) changeset 29230 155f6c110dfc parent 29223 e09c53289830 parent 29039 8b9207f82a78 child 29233 ce6d35a0bed6 permissions -rw-r--r--
Merged.
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 Arith_Tools
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" 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, recpower}" "1"
168   proof qed (auto simp add: ring_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
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 =
195   can HOLogic.dest_number (Thm.term_of ct);
197 fun int_of_rat x =
198   (case Rat.quotient_of_rat x of (i, 1) => i
199   | _ => error "int_of_rat: bad int");
201 val numeral_conv =
202   Simplifier.rewrite (HOL_basic_ss addsimps @{thms semiring_norm}) then_conv
204     (@{thms numeral_1_eq_1} @ @{thms numeral_0_eq_0} @ @{thms numerals(1-2)}));
206 in
208 fun normalizer_funs key =
209   NormalizerData.funs key
210    {is_const = fn phi => numeral_is_const,
211     dest_const = fn phi => fn ct =>
212       Rat.rat_of_int (snd
213         (HOLogic.dest_number (Thm.term_of ct)
214           handle TERM _ => error "ring_dest_const")),
215     mk_const = fn phi => fn cT => fn x => Numeral.mk_cnumber cT (int_of_rat x),
216     conv = fn phi => K numeral_conv}
218 end
219 *}
221 declaration {* normalizer_funs @{thm class_semiring.gb_semiring_axioms'} *}
224 locale gb_ring = gb_semiring +
225   fixes sub :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
226     and neg :: "'a \<Rightarrow> 'a"
227   assumes neg_mul: "neg x = mul (neg r1) x"
229 begin
231 lemma ring_ops: shows "TERM (sub x y)" and "TERM (neg x)" .
233 lemmas ring_rules = neg_mul sub_add
235 lemmas gb_ring_axioms' =
236   gb_ring_axioms [normalizer
237     semiring ops: semiring_ops
238     semiring rules: semiring_rules
239     ring ops: ring_ops
240     ring rules: ring_rules]
242 end
245 interpretation class_ring: gb_ring "op +" "op *" "op ^"
246     "0::'a::{comm_semiring_1,recpower,number_ring}" 1 "op -" "uminus"
247   proof qed simp_all
250 declaration {* normalizer_funs @{thm class_ring.gb_ring_axioms'} *}
252 use "Tools/Groebner_Basis/normalizer.ML"
255 method_setup sring_norm = {*
256   Method.ctxt_args (fn ctxt => Method.SIMPLE_METHOD' (Normalizer.semiring_normalize_tac ctxt))
257 *} "semiring normalizer"
260 locale gb_field = gb_ring +
261   fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
262     and inverse:: "'a \<Rightarrow> 'a"
263   assumes divide: "divide x y = mul x (inverse y)"
264      and inverse: "inverse x = divide r1 x"
265 begin
267 lemmas gb_field_axioms' =
268   gb_field_axioms [normalizer
269     semiring ops: semiring_ops
270     semiring rules: semiring_rules
271     ring ops: ring_ops
272     ring rules: ring_rules]
274 end
277 subsection {* Groebner Bases *}
279 locale semiringb = gb_semiring +
282     add (mul w z) (mul x y) \<longleftrightarrow> w = x \<or> y = z"
283 begin
285 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)"
286 proof-
287   have "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> \<not> (a = b \<or> c = d)" by simp
288   also have "\<dots> \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
290   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)"
291     by simp
292 qed
294 lemma add_scale_eq_noteq: "\<lbrakk>r \<noteq> r0 ; (a = b) \<and> ~(c = d)\<rbrakk>
295   \<Longrightarrow> add a (mul r c) \<noteq> add b (mul r d)"
296 proof(clarify)
297   assume nz: "r\<noteq> r0" and cnd: "c\<noteq>d"
298     and eq: "add b (mul r c) = add b (mul r d)"
299   hence "mul r c = mul r d" using cnd add_cancel by simp
300   hence "add (mul r0 d) (mul r c) = add (mul r0 c) (mul r d)"
301     using mul_0 add_cancel by simp
302   thus "False" using add_mul_solve nz cnd by simp
303 qed
305 lemma add_r0_iff: " x = add x a \<longleftrightarrow> a = r0"
306 proof-
309 qed
311 declare gb_semiring_axioms' [normalizer del]
313 lemmas semiringb_axioms' = semiringb_axioms [normalizer
314   semiring ops: semiring_ops
315   semiring rules: semiring_rules
318 end
320 locale ringb = semiringb + gb_ring +
321   assumes subr0_iff: "sub x y = r0 \<longleftrightarrow> x = y"
322 begin
324 declare gb_ring_axioms' [normalizer del]
326 lemmas ringb_axioms' = ringb_axioms [normalizer
327   semiring ops: semiring_ops
328   semiring rules: semiring_rules
329   ring ops: ring_ops
330   ring rules: ring_rules
334 end
337 lemma no_zero_divirors_neq0:
338   assumes az: "(a::'a::no_zero_divisors) \<noteq> 0"
339     and ab: "a*b = 0" shows "b = 0"
340 proof -
341   { assume bz: "b \<noteq> 0"
342     from no_zero_divisors [OF az bz] ab have False by blast }
343   thus "b = 0" by blast
344 qed
346 interpretation class_ringb: ringb
347   "op +" "op *" "op ^" "0::'a::{idom,recpower,number_ring}" "1" "op -" "uminus"
348 proof(unfold_locales, simp add: ring_simps power_Suc, auto)
349   fix w x y z ::"'a::{idom,recpower,number_ring}"
350   assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
351   hence ynz': "y - z \<noteq> 0" by simp
352   from p have "w * y + x* z - w*z - x*y = 0" by simp
353   hence "w* (y - z) - x * (y - z) = 0" by (simp add: ring_simps)
354   hence "(y - z) * (w - x) = 0" by (simp add: ring_simps)
355   with  no_zero_divirors_neq0 [OF ynz']
356   have "w - x = 0" by blast
357   thus "w = x"  by simp
358 qed
360 declaration {* normalizer_funs @{thm class_ringb.ringb_axioms'} *}
362 interpretation natgb: semiringb
363   "op +" "op *" "op ^" "0::nat" "1"
364 proof (unfold_locales, simp add: ring_simps power_Suc)
365   fix w x y z ::"nat"
366   { assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
367     hence "y < z \<or> y > z" by arith
368     moreover {
369       assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z - y" in exI, auto)
370       then obtain k where kp: "k>0" and yz:"z = y + k" by blast
371       from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz ring_simps)
372       hence "x*k = w*k" by simp
373       hence "w = x" using kp by (simp add: mult_cancel2) }
374     moreover {
375       assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y - z" in exI, auto)
376       then obtain k where kp: "k>0" and yz:"y = z + k" by blast
377       from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz ring_simps)
378       hence "w*k = x*k" by simp
379       hence "w = x" using kp by (simp add: mult_cancel2)}
380     ultimately have "w=x" by blast }
381   thus "(w * y + x * z = w * z + x * y) = (w = x \<or> y = z)" by auto
382 qed
384 declaration {* normalizer_funs @{thm natgb.semiringb_axioms'} *}
386 locale fieldgb = ringb + gb_field
387 begin
389 declare gb_field_axioms' [normalizer del]
391 lemmas fieldgb_axioms' = fieldgb_axioms [normalizer
392   semiring ops: semiring_ops
393   semiring rules: semiring_rules
394   ring ops: ring_ops
395   ring rules: ring_rules
399 end
402 lemmas bool_simps = simp_thms(1-34)
403 lemma dnf:
404     "(P & (Q | R)) = ((P&Q) | (P&R))" "((Q | R) & P) = ((Q&P) | (R&P))"
405     "(P \<and> Q) = (Q \<and> P)" "(P \<or> Q) = (Q \<or> P)"
406   by blast+
408 lemmas weak_dnf_simps = dnf bool_simps
410 lemma nnf_simps:
411     "(\<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)"
412     "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not> \<not>(P)) = P"
413   by blast+
415 lemma PFalse:
416     "P \<equiv> False \<Longrightarrow> \<not> P"
417     "\<not> P \<Longrightarrow> (P \<equiv> False)"
418   by auto
419 use "Tools/Groebner_Basis/groebner.ML"
421 method_setup algebra =
422 {*
423 let
424  fun keyword k = Scan.lift (Args.\$\$\$ k -- Args.colon) >> K ()
426  val delN = "del"
427  val any_keyword = keyword addN || keyword delN
428  val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
429 in
430 fn src => Method.syntax
431     ((Scan.optional (keyword addN |-- thms) []) --
432     (Scan.optional (keyword delN |-- thms) [])) src
433  #> (fn ((add_ths, del_ths), ctxt) =>
434        Method.SIMPLE_METHOD' (Groebner.algebra_tac add_ths del_ths ctxt))
435 end
436 *} "solve polynomial equations over (semi)rings and ideal membership problems using Groebner bases"
437 declare dvd_def[algebra]
438 declare dvd_eq_mod_eq_0[symmetric, algebra]
439 declare nat_mod_div_trivial[algebra]
440 declare nat_mod_mod_trivial[algebra]
441 declare conjunct1[OF DIVISION_BY_ZERO, algebra]
442 declare conjunct2[OF DIVISION_BY_ZERO, algebra]
443 declare zmod_zdiv_equality[symmetric,algebra]
444 declare zdiv_zmod_equality[symmetric, algebra]
445 declare zdiv_zminus_zminus[algebra]
446 declare zmod_zminus_zminus[algebra]
447 declare zdiv_zminus2[algebra]
448 declare zmod_zminus2[algebra]
449 declare zdiv_zero[algebra]
450 declare zmod_zero[algebra]
451 declare zmod_1[algebra]
452 declare zdiv_1[algebra]
453 declare zmod_minus1_right[algebra]
454 declare zdiv_minus1_right[algebra]
455 declare mod_div_trivial[algebra]
456 declare mod_mod_trivial[algebra]
457 declare zmod_zmult_self1[algebra]
458 declare zmod_zmult_self2[algebra]
459 declare zmod_eq_0_iff[algebra]
460 declare zdvd_0_left[algebra]
461 declare zdvd1_eq[algebra]
462 declare zmod_eq_dvd_iff[algebra]
463 declare nat_mod_eq_iff[algebra]
465 subsection{* Groebner Bases for fields *}
467 interpretation class_fieldgb:
468   fieldgb "op +" "op *" "op ^" "0::'a::{field,recpower,number_ring}" "1" "op -" "uminus" "op /" "inverse" apply (unfold_locales) by (simp_all add: divide_inverse)
470 lemma divide_Numeral1: "(x::'a::{field,number_ring}) / Numeral1 = x" by simp
471 lemma divide_Numeral0: "(x::'a::{field,number_ring, division_by_zero}) / Numeral0 = 0"
472   by simp
473 lemma mult_frac_frac: "((x::'a::{field,division_by_zero}) / y) * (z / w) = (x*z) / (y*w)"
474   by simp
475 lemma mult_frac_num: "((x::'a::{field, division_by_zero}) / y) * z  = (x*z) / y"
476   by simp
477 lemma mult_num_frac: "((x::'a::{field, division_by_zero}) / y) * z  = (x*z) / y"
478   by simp
480 lemma Numeral1_eq1_nat: "(1::nat) = Numeral1" by simp
482 lemma add_frac_num: "y\<noteq> 0 \<Longrightarrow> (x::'a::{field, division_by_zero}) / y + z = (x + z*y) / y"
484 lemma add_num_frac: "y\<noteq> 0 \<Longrightarrow> z + (x::'a::{field, division_by_zero}) / y = (x + z*y) / y"
488 ML{*
489 local
490  val zr = @{cpat "0"}
491  val zT = ctyp_of_term zr
492  val geq = @{cpat "op ="}
493  val eqT = Thm.dest_ctyp (ctyp_of_term geq) |> hd
498  fun prove_nz ss T t =
499     let
500       val z = instantiate_cterm ([(zT,T)],[]) zr
501       val eq = instantiate_cterm ([(eqT,T)],[]) geq
502       val th = Simplifier.rewrite (ss addsimps simp_thms)
503            (Thm.capply @{cterm "Trueprop"} (Thm.capply @{cterm "Not"}
504                   (Thm.capply (Thm.capply eq t) z)))
505     in equal_elim (symmetric th) TrueI
506     end
508  fun proc phi ss ct =
509   let
510     val ((x,y),(w,z)) =
511          (Thm.dest_binop #> (fn (a,b) => (Thm.dest_binop a, Thm.dest_binop b))) ct
512     val _ = map (HOLogic.dest_number o term_of) [x,y,z,w]
513     val T = ctyp_of_term x
514     val [y_nz, z_nz] = map (prove_nz ss T) [y, z]
515     val th = instantiate' [SOME T] (map SOME [y,z,x,w]) add_frac_eq
516   in SOME (implies_elim (implies_elim th y_nz) z_nz)
517   end
518   handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE
520  fun proc2 phi ss ct =
521   let
522     val (l,r) = Thm.dest_binop ct
523     val T = ctyp_of_term l
524   in (case (term_of l, term_of r) of
525       (Const(@{const_name "HOL.divide"},_)\$_\$_, _) =>
526         let val (x,y) = Thm.dest_binop l val z = r
527             val _ = map (HOLogic.dest_number o term_of) [x,y,z]
528             val ynz = prove_nz ss T y
529         in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,x,z]) add_frac_num) ynz)
530         end
531      | (_, Const (@{const_name "HOL.divide"},_)\$_\$_) =>
532         let val (x,y) = Thm.dest_binop r val z = l
533             val _ = map (HOLogic.dest_number o term_of) [x,y,z]
534             val ynz = prove_nz ss T y
535         in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,z,x]) add_num_frac) ynz)
536         end
537      | _ => NONE)
538   end
539   handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE
541  fun is_number (Const(@{const_name "HOL.divide"},_)\$a\$b) = is_number a andalso is_number b
542    | is_number t = can HOLogic.dest_number t
544  val is_number = is_number o term_of
546  fun proc3 phi ss ct =
547   (case term_of ct of
548     Const(@{const_name HOL.less},_)\$(Const(@{const_name "HOL.divide"},_)\$_\$_)\$_ =>
549       let
550         val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
551         val _ = map is_number [a,b,c]
552         val T = ctyp_of_term c
553         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_less_eq"}
554       in SOME (mk_meta_eq th) end
555   | Const(@{const_name HOL.less_eq},_)\$(Const(@{const_name "HOL.divide"},_)\$_\$_)\$_ =>
556       let
557         val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
558         val _ = map is_number [a,b,c]
559         val T = ctyp_of_term c
560         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_le_eq"}
561       in SOME (mk_meta_eq th) end
562   | Const("op =",_)\$(Const(@{const_name "HOL.divide"},_)\$_\$_)\$_ =>
563       let
564         val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
565         val _ = map is_number [a,b,c]
566         val T = ctyp_of_term c
567         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_eq_eq"}
568       in SOME (mk_meta_eq th) end
569   | Const(@{const_name HOL.less},_)\$_\$(Const(@{const_name "HOL.divide"},_)\$_\$_) =>
570     let
571       val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
572         val _ = map is_number [a,b,c]
573         val T = ctyp_of_term c
574         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "less_divide_eq"}
575       in SOME (mk_meta_eq th) end
576   | Const(@{const_name HOL.less_eq},_)\$_\$(Const(@{const_name "HOL.divide"},_)\$_\$_) =>
577     let
578       val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
579         val _ = map is_number [a,b,c]
580         val T = ctyp_of_term c
581         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "le_divide_eq"}
582       in SOME (mk_meta_eq th) end
583   | Const("op =",_)\$_\$(Const(@{const_name "HOL.divide"},_)\$_\$_) =>
584     let
585       val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
586         val _ = map is_number [a,b,c]
587         val T = ctyp_of_term c
588         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "eq_divide_eq"}
589       in SOME (mk_meta_eq th) end
590   | _ => NONE)
591   handle TERM _ => NONE | CTERM _ => NONE | THM _ => NONE
594        make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + (?w::?'a::field)/?z"}],
596                      proc = proc, identifier = []}
599        make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + ?z"}, @{cpat "?z + (?x::?'a::field)/?y"}],
601                      proc = proc2, identifier = []}
603 val ord_frac_simproc =
604   make_simproc
605     {lhss = [@{cpat "(?a::(?'a::{field, ord}))/?b < ?c"},
606              @{cpat "(?a::(?'a::{field, ord}))/?b \<le> ?c"},
607              @{cpat "?c < (?a::(?'a::{field, ord}))/?b"},
608              @{cpat "?c \<le> (?a::(?'a::{field, ord}))/?b"},
609              @{cpat "?c = ((?a::(?'a::{field, ord}))/?b)"},
610              @{cpat "((?a::(?'a::{field, ord}))/ ?b) = ?c"}],
611              name = "ord_frac_simproc", proc = proc3, identifier = []}
613 val ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"},
614            @{thm "divide_Numeral1"},
615            @{thm "Ring_and_Field.divide_zero"}, @{thm "divide_Numeral0"},
616            @{thm "divide_divide_eq_left"}, @{thm "mult_frac_frac"},
617            @{thm "mult_num_frac"}, @{thm "mult_frac_num"},
618            @{thm "mult_frac_frac"}, @{thm "times_divide_eq_right"},
619            @{thm "times_divide_eq_left"}, @{thm "divide_divide_eq_right"},
620            @{thm "diff_def"}, @{thm "minus_divide_left"},
621            @{thm "Numeral1_eq1_nat"}, @{thm "add_divide_distrib"} RS sym]
623 local
624 open Conv
625 in
626 val comp_conv = (Simplifier.rewrite
631                             ord_frac_simproc]
634   [@{thm numeral_1_eq_1},@{thm numeral_0_eq_0}] @ @{thms numerals(1-2)}))
635 end
637 fun numeral_is_const ct =
638   case term_of ct of
639    Const (@{const_name "HOL.divide"},_) \$ a \$ b =>
640      numeral_is_const (Thm.dest_arg1 ct) andalso numeral_is_const (Thm.dest_arg ct)
641  | Const (@{const_name "HOL.uminus"},_)\$t => numeral_is_const (Thm.dest_arg ct)
642  | t => can HOLogic.dest_number t
644 fun dest_const ct = ((case term_of ct of
645    Const (@{const_name "HOL.divide"},_) \$ a \$ b=>
646     Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b))
647  | t => Rat.rat_of_int (snd (HOLogic.dest_number t)))
648    handle TERM _ => error "ring_dest_const")
650 fun mk_const phi cT x =
651  let val (a, b) = Rat.quotient_of_rat x
652  in if b = 1 then Numeral.mk_cnumber cT a
653     else Thm.capply
654          (Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"})
655                      (Numeral.mk_cnumber cT a))
656          (Numeral.mk_cnumber cT b)
657   end
659 in
660  val field_comp_conv = comp_conv;
661  val fieldgb_declaration =
662   NormalizerData.funs @{thm class_fieldgb.fieldgb_axioms'}
663    {is_const = K numeral_is_const,
664     dest_const = K dest_const,
665     mk_const = mk_const,
666     conv = K (K comp_conv)}
667 end;
668 *}
670 declaration fieldgb_declaration
672 end