src/HOL/Groebner_Basis.thy
author wenzelm
Tue Jun 05 16:26:04 2007 +0200 (2007-06-05)
changeset 23252 67268bb40b21
child 23258 9062e98fdab1
permissions -rw-r--r--
Semiring normalization and Groebner Bases.
     1 (*  Title:      HOL/Groebner_Basis.thy
     2     ID:         $Id$
     3     Author:     Amine Chaieb, TU Muenchen
     4 *)
     5 
     6 header {* Semiring normalization and Groebner Bases *}
     7 
     8 theory Groebner_Basis
     9 imports NatBin
    10 uses
    11   "Tools/Groebner_Basis/misc.ML"
    12   "Tools/Groebner_Basis/normalizer_data.ML"
    13   ("Tools/Groebner_Basis/normalizer.ML")
    14 begin
    15 
    16 subsection {* Semiring normalization *}
    17 
    18 setup NormalizerData.setup
    19 
    20 
    21 locale semiring =
    22   fixes add mul pwr r0 r1
    23   assumes add_a:"(add x (add y z) = add (add x y) z)"
    24     and add_c: "add x y = add y x" and add_0:"add r0 x = x"
    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
    30 
    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
    40 
    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)))"
    46     by (simp add: mul_a)
    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
    52 
    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
    61 
    62 
    63 subsubsection {* Declaring the abstract theory *}
    64 
    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+
    70 
    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"
    75   "add m m = mul (add r1 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)"
    91   "add (add a b) (add c d) = add (add a c) (add b d)"
    92   "add (add a b) c = add a (add b c)"
    93   "add a (add c d) = add c (add a d)"
    94   "add (add a b) c = add (add a c) b"
    95   "add a c = add c a"
    96   "add a (add c d) = add (add a c) d"
    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
   115 next show "add a r0 = a" using add_0 add_c 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)
   137 next show "add (add a b) (add c d) = add (add a c) (add b d)"
   138     using add_c add_a by simp
   139 next show "add (add a b) c = add a (add b c)" using add_a by simp
   140 next show "add a (add c d) = add c (add a d)"
   141     apply (simp add: add_a) by (simp only: add_c)
   142 next show "add (add a b) c = add (add a c) b" using add_a add_c by simp
   143 next show "add a c = add c a" by (rule add_c)
   144 next show "add a (add c d) = add (add a c) d" using add_a by simp
   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
   159 
   160 
   161 lemma "axioms" [normalizer
   162     semiring ops: semiring_ops
   163     semiring rules: semiring_rules]:
   164   "semiring add mul pwr r0 r1" .
   165 
   166 end
   167 
   168 interpretation class_semiring: semiring
   169     ["op +" "op *" "op ^" "0::'a::{comm_semiring_1, recpower}" "1"]
   170   by unfold_locales (auto simp add: ring_eq_simps power_Suc)
   171 
   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
   174 
   175 lemma not_iszero_Numeral1: "\<not> iszero (Numeral1::'a::number_ring)"
   176   by (simp add: numeral_1_eq_1)
   177 lemmas comp_arith = Let_def arith_simps nat_arith rel_simps if_False
   178   if_True add_0 add_Suc add_number_of_left mult_number_of_left
   179   numeral_1_eq_1[symmetric] Suc_eq_add_numeral_1
   180   numeral_0_eq_0[symmetric] numerals[symmetric] not_iszero_1
   181   iszero_number_of_1 iszero_number_of_0 nonzero_number_of_Min
   182   iszero_number_of_Pls iszero_0 not_iszero_Numeral1
   183 
   184 lemmas semiring_norm = comp_arith
   185 
   186 ML {*
   187   fun numeral_is_const ct =
   188     can HOLogic.dest_number (Thm.term_of ct);
   189 
   190   val numeral_conv =
   191     Conv.then_conv (Simplifier.rewrite (HOL_basic_ss addsimps @{thms semiring_norm}),
   192    Simplifier.rewrite (HOL_basic_ss addsimps
   193   [@{thm numeral_1_eq_1},@{thm numeral_0_eq_0}] @ @{thms numerals(1-2)}));
   194 *}
   195 
   196 ML {*
   197   fun int_of_rat x =
   198     (case Rat.quotient_of_rat x of (i, 1) => i
   199     | _ => error "int_of_rat: bad int")
   200 *}
   201 
   202 declaration {*
   203   NormalizerData.funs @{thm class_semiring.axioms}
   204    {is_const = fn phi => numeral_is_const,
   205     dest_const = fn phi => fn ct =>
   206       Rat.rat_of_int (snd
   207         (HOLogic.dest_number (Thm.term_of ct)
   208           handle TERM _ => error "ring_dest_const")),
   209     mk_const = fn phi => fn cT => fn x =>
   210       Thm.cterm_of (Thm.theory_of_ctyp cT) (HOLogic.mk_number (typ_of cT) (int_of_rat x)),
   211     conv = fn phi => numeral_conv}
   212 *}
   213 
   214 
   215 locale ring = semiring +
   216   fixes sub :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
   217     and neg :: "'a \<Rightarrow> 'a"
   218   assumes neg_mul: "neg x = mul (neg r1) x"
   219     and sub_add: "sub x y = add x (neg y)"
   220 begin
   221 
   222 lemma ring_ops:
   223   includes meta_term_syntax
   224   shows "TERM (sub x y)" and "TERM (neg x)" .
   225 
   226 lemmas ring_rules = neg_mul sub_add
   227 
   228 lemma "axioms" [normalizer
   229   semiring ops: semiring_ops
   230   semiring rules: semiring_rules
   231   ring ops: ring_ops
   232   ring rules: ring_rules]:
   233   "ring add mul pwr r0 r1 sub neg" .
   234 
   235 end
   236 
   237 
   238 interpretation class_ring: ring ["op +" "op *" "op ^"
   239     "0::'a::{comm_semiring_1,recpower,number_ring}" 1 "op -" "uminus"]
   240   by unfold_locales simp_all
   241 
   242 
   243 declaration {*
   244   NormalizerData.funs @{thm class_ring.axioms}
   245    {is_const = fn phi => numeral_is_const,
   246     dest_const = fn phi => fn ct =>
   247       Rat.rat_of_int (snd
   248         (HOLogic.dest_number (Thm.term_of ct)
   249           handle TERM _ => error "ring_dest_const")),
   250     mk_const = fn phi => fn cT => fn x =>
   251       Thm.cterm_of (Thm.theory_of_ctyp cT) (HOLogic.mk_number (typ_of cT) (int_of_rat x)),
   252     conv = fn phi => numeral_conv}
   253 *}
   254 
   255 use "Tools/Groebner_Basis/normalizer.ML"
   256 
   257 method_setup sring_norm = {*
   258   Method.ctxt_args (fn ctxt => Method.SIMPLE_METHOD' (Normalizer.semiring_normalize_tac ctxt))
   259 *} "Semiring_normalizer"
   260 
   261 
   262 subsection {* Gröbner Bases *}
   263 
   264 locale semiringb = semiring +
   265   assumes add_cancel: "add (x::'a) y = add x z \<longleftrightarrow> y = z"
   266   and add_mul_solve: "add (mul w y) (mul x z) =
   267     add (mul w z) (mul x y) \<longleftrightarrow> w = x \<or> y = z"
   268 begin
   269 
   270 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)"
   271 proof-
   272   have "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> \<not> (a = b \<or> c = d)" by simp
   273   also have "\<dots> \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
   274     using add_mul_solve by blast
   275   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)"
   276     by simp
   277 qed
   278 
   279 lemma add_scale_eq_noteq: "\<lbrakk>r \<noteq> r0 ; (a = b) \<and> ~(c = d)\<rbrakk>
   280   \<Longrightarrow> add a (mul r c) \<noteq> add b (mul r d)"
   281 proof(clarify)
   282   assume nz: "r\<noteq> r0" and cnd: "c\<noteq>d"
   283     and eq: "add b (mul r c) = add b (mul r d)"
   284   hence "mul r c = mul r d" using cnd add_cancel by simp
   285   hence "add (mul r0 d) (mul r c) = add (mul r0 c) (mul r d)"
   286     using mul_0 add_cancel by simp
   287   thus "False" using add_mul_solve nz cnd by simp
   288 qed
   289 
   290 declare "axioms" [normalizer del]
   291 
   292 lemma "axioms" [normalizer
   293   semiring ops: semiring_ops
   294   semiring rules: semiring_rules
   295   idom rules: noteq_reduce add_scale_eq_noteq]:
   296   "semiringb add mul pwr r0 r1" .
   297 
   298 end
   299 
   300 locale ringb = semiringb + ring
   301 begin
   302 
   303 declare "axioms" [normalizer del]
   304 
   305 lemma "axioms" [normalizer
   306   semiring ops: semiring_ops
   307   semiring rules: semiring_rules
   308   ring ops: ring_ops
   309   ring rules: ring_rules
   310   idom rules: noteq_reduce add_scale_eq_noteq]:
   311   "ringb add mul pwr r0 r1 sub neg" .
   312 
   313 end
   314 
   315 lemma no_zero_divirors_neq0:
   316   assumes az: "(a::'a::no_zero_divisors) \<noteq> 0"
   317     and ab: "a*b = 0" shows "b = 0"
   318 proof -
   319   { assume bz: "b \<noteq> 0"
   320     from no_zero_divisors [OF az bz] ab have False by blast }
   321   thus "b = 0" by blast
   322 qed
   323 
   324 interpretation class_ringb: ringb
   325   ["op +" "op *" "op ^" "0::'a::{idom,recpower,number_ring}" "1" "op -" "uminus"]
   326 proof(unfold_locales, simp add: ring_eq_simps power_Suc, auto)
   327   fix w x y z ::"'a::{idom,recpower,number_ring}"
   328   assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
   329   hence ynz': "y - z \<noteq> 0" by simp
   330   from p have "w * y + x* z - w*z - x*y = 0" by simp
   331   hence "w* (y - z) - x * (y - z) = 0" by (simp add: ring_eq_simps)
   332   hence "(y - z) * (w - x) = 0" by (simp add: ring_eq_simps)
   333   with  no_zero_divirors_neq0 [OF ynz']
   334   have "w - x = 0" by blast
   335   thus "w = x"  by simp
   336 qed
   337 
   338 
   339 declaration {*
   340   NormalizerData.funs @{thm class_ringb.axioms}
   341    {is_const = fn phi => numeral_is_const,
   342     dest_const = fn phi => fn ct =>
   343       Rat.rat_of_int (snd
   344         (HOLogic.dest_number (Thm.term_of ct)
   345           handle TERM _ => error "ring_dest_const")),
   346     mk_const = fn phi => fn cT => fn x =>
   347       Thm.cterm_of (Thm.theory_of_ctyp cT) (HOLogic.mk_number (typ_of cT) (int_of_rat x)),
   348     conv = fn phi => numeral_conv}
   349 *}
   350 
   351 
   352 interpretation natgb: semiringb
   353   ["op +" "op *" "op ^" "0::nat" "1"]
   354 proof (unfold_locales, simp add: ring_eq_simps power_Suc)
   355   fix w x y z ::"nat"
   356   { assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
   357     hence "y < z \<or> y > z" by arith
   358     moreover {
   359       assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z - y" in exI, auto)
   360       then obtain k where kp: "k>0" and yz:"z = y + k" by blast
   361       from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz ring_eq_simps)
   362       hence "x*k = w*k" by simp
   363       hence "w = x" using kp by (simp add: mult_cancel2) }
   364     moreover {
   365       assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y - z" in exI, auto)
   366       then obtain k where kp: "k>0" and yz:"y = z + k" by blast
   367       from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz ring_eq_simps)
   368       hence "w*k = x*k" by simp
   369       hence "w = x" using kp by (simp add: mult_cancel2)}
   370     ultimately have "w=x" by blast }
   371   thus "(w * y + x * z = w * z + x * y) = (w = x \<or> y = z)" by auto
   372 qed
   373 
   374 declaration {*
   375   NormalizerData.funs @{thm natgb.axioms}
   376    {is_const = fn phi => numeral_is_const,
   377     dest_const = fn phi => fn ct =>
   378       Rat.rat_of_int (snd
   379         (HOLogic.dest_number (Thm.term_of ct)
   380           handle TERM _ => error "ring_dest_const")),
   381     mk_const = fn phi => fn cT => fn x =>
   382       Thm.cterm_of (Thm.theory_of_ctyp cT) (HOLogic.mk_number (typ_of cT) (int_of_rat x)),
   383     conv = fn phi => numeral_conv}
   384 *}
   385 
   386 
   387 lemmas bool_simps =  simp_thms(1-34)
   388 lemma dnf:
   389     "(P & (Q | R)) = ((P&Q) | (P&R))" "((Q | R) & P) = ((Q&P) | (R&P))"
   390     "(P \<and> Q) = (Q \<and> P)" "(P \<or> Q) = (Q \<or> P)"
   391   by blast+
   392 
   393 lemmas weak_dnf_simps = dnf bool_simps
   394 
   395 lemma nnf_simps:
   396     "(\<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)"
   397     "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not> \<not>(P)) = P"
   398   by blast+
   399 
   400 lemma PFalse:
   401     "P \<equiv> False \<Longrightarrow> \<not> P"
   402     "\<not> P \<Longrightarrow> (P \<equiv> False)"
   403   by auto
   404 
   405 use "Tools/Groebner_Basis/groebner.ML"
   406 
   407 ML {*
   408   fun algebra_tac ctxt i = ObjectLogic.full_atomize_tac i THEN (fn st =>
   409   rtac (Groebner.ring_conv ctxt (Thm.dest_arg (nth (cprems_of st) (i - 1)))) i st);
   410 *}
   411 
   412 method_setup algebra = {*
   413   Method.ctxt_args (Method.SIMPLE_METHOD' o algebra_tac)
   414 *} ""
   415 
   416 end