src/HOL/Groebner_Basis.thy
author haftmann
Thu May 06 23:11:57 2010 +0200 (2010-05-06)
changeset 36720 41da7025e59c
parent 36716 b09f3ad3208f
child 36751 7f1da69cacb3
permissions -rw-r--r--
proper sublocales; no free-floating ML sections
     1 (*  Title:      HOL/Groebner_Basis.thy
     2     Author:     Amine Chaieb, TU Muenchen
     3 *)
     4 
     5 header {* Semiring normalization and Groebner Bases *}
     6 
     7 theory Groebner_Basis
     8 imports Numeral_Simprocs Nat_Transfer
     9 uses
    10   "Tools/Groebner_Basis/normalizer.ML"
    11   ("Tools/Groebner_Basis/groebner.ML")
    12 begin
    13 
    14 subsection {* Semiring normalization *}
    15 
    16 setup Normalizer.setup
    17 
    18 locale normalizing_semiring =
    19   fixes add mul pwr r0 r1
    20   assumes add_a:"(add x (add y z) = add (add x y) z)"
    21     and add_c: "add x y = add y x" and add_0:"add r0 x = x"
    22     and mul_a:"mul x (mul y z) = mul (mul x y) z" and mul_c:"mul x y = mul y x"
    23     and mul_1:"mul r1 x = x" and  mul_0:"mul r0 x = r0"
    24     and mul_d:"mul x (add y z) = add (mul x y) (mul x z)"
    25     and pwr_0:"pwr x 0 = r1" and pwr_Suc:"pwr x (Suc n) = mul x (pwr x n)"
    26 begin
    27 
    28 lemma mul_pwr:"mul (pwr x p) (pwr x q) = pwr x (p + q)"
    29 proof (induct p)
    30   case 0
    31   then show ?case by (auto simp add: pwr_0 mul_1)
    32 next
    33   case Suc
    34   from this [symmetric] show ?case
    35     by (auto simp add: pwr_Suc mul_1 mul_a)
    36 qed
    37 
    38 lemma pwr_mul: "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
    39 proof (induct q arbitrary: x y, auto simp add:pwr_0 pwr_Suc mul_1)
    40   fix q x y
    41   assume "\<And>x y. pwr (mul x y) q = mul (pwr x q) (pwr y q)"
    42   have "mul (mul x y) (mul (pwr x q) (pwr y q)) = mul x (mul y (mul (pwr x q) (pwr y q)))"
    43     by (simp add: mul_a)
    44   also have "\<dots> = (mul (mul y (mul (pwr y q) (pwr x q))) x)" by (simp add: mul_c)
    45   also have "\<dots> = (mul (mul y (pwr y q)) (mul (pwr x q) x))" by (simp add: mul_a)
    46   finally show "mul (mul x y) (mul (pwr x q) (pwr y q)) =
    47     mul (mul x (pwr x q)) (mul y (pwr y q))" by (simp add: mul_c)
    48 qed
    49 
    50 lemma pwr_pwr: "pwr (pwr x p) q = pwr x (p * q)"
    51 proof (induct p arbitrary: q)
    52   case 0
    53   show ?case using pwr_Suc mul_1 pwr_0 by (induct q) auto
    54 next
    55   case Suc
    56   thus ?case by (auto simp add: mul_pwr [symmetric] pwr_mul pwr_Suc)
    57 qed
    58 
    59 lemma semiring_ops:
    60   shows "TERM (add x y)" and "TERM (mul x y)" and "TERM (pwr x n)"
    61     and "TERM r0" and "TERM r1" .
    62 
    63 lemma semiring_rules:
    64   "add (mul a m) (mul b m) = mul (add a b) m"
    65   "add (mul a m) m = mul (add a r1) m"
    66   "add m (mul a m) = mul (add a r1) m"
    67   "add m m = mul (add r1 r1) m"
    68   "add r0 a = a"
    69   "add a r0 = a"
    70   "mul a b = mul b a"
    71   "mul (add a b) c = add (mul a c) (mul b c)"
    72   "mul r0 a = r0"
    73   "mul a r0 = r0"
    74   "mul r1 a = a"
    75   "mul a r1 = a"
    76   "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
    77   "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
    78   "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
    79   "mul (mul lx ly) rx = mul (mul lx rx) ly"
    80   "mul (mul lx ly) rx = mul lx (mul ly rx)"
    81   "mul lx (mul rx ry) = mul (mul lx rx) ry"
    82   "mul lx (mul rx ry) = mul rx (mul lx ry)"
    83   "add (add a b) (add c d) = add (add a c) (add b d)"
    84   "add (add a b) c = add a (add b c)"
    85   "add a (add c d) = add c (add a d)"
    86   "add (add a b) c = add (add a c) b"
    87   "add a c = add c a"
    88   "add a (add c d) = add (add a c) d"
    89   "mul (pwr x p) (pwr x q) = pwr x (p + q)"
    90   "mul x (pwr x q) = pwr x (Suc q)"
    91   "mul (pwr x q) x = pwr x (Suc q)"
    92   "mul x x = pwr x 2"
    93   "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
    94   "pwr (pwr x p) q = pwr x (p * q)"
    95   "pwr x 0 = r1"
    96   "pwr x 1 = x"
    97   "mul x (add y z) = add (mul x y) (mul x z)"
    98   "pwr x (Suc q) = mul x (pwr x q)"
    99   "pwr x (2*n) = mul (pwr x n) (pwr x n)"
   100   "pwr x (Suc (2*n)) = mul x (mul (pwr x n) (pwr x n))"
   101 proof -
   102   show "add (mul a m) (mul b m) = mul (add a b) m" using mul_d mul_c by simp
   103 next show"add (mul a m) m = mul (add a r1) m" using mul_d mul_c mul_1 by simp
   104 next show "add m (mul a m) = mul (add a r1) m" using mul_c mul_d mul_1 add_c by simp
   105 next show "add m m = mul (add r1 r1) m" using mul_c mul_d mul_1 by simp
   106 next show "add r0 a = a" using add_0 by simp
   107 next show "add a r0 = a" using add_0 add_c by simp
   108 next show "mul a b = mul b a" using mul_c by simp
   109 next show "mul (add a b) c = add (mul a c) (mul b c)" using mul_c mul_d by simp
   110 next show "mul r0 a = r0" using mul_0 by simp
   111 next show "mul a r0 = r0" using mul_0 mul_c by simp
   112 next show "mul r1 a = a" using mul_1 by simp
   113 next show "mul a r1 = a" using mul_1 mul_c by simp
   114 next show "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
   115     using mul_c mul_a by simp
   116 next show "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
   117     using mul_a by simp
   118 next
   119   have "mul (mul lx ly) (mul rx ry) = mul (mul rx ry) (mul lx ly)" by (rule mul_c)
   120   also have "\<dots> = mul rx (mul ry (mul lx ly))" using mul_a by simp
   121   finally
   122   show "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
   123     using mul_c by simp
   124 next show "mul (mul lx ly) rx = mul (mul lx rx) ly" using mul_c mul_a by simp
   125 next
   126   show "mul (mul lx ly) rx = mul lx (mul ly rx)" by (simp add: mul_a)
   127 next show "mul lx (mul rx ry) = mul (mul lx rx) ry" by (simp add: mul_a )
   128 next show "mul lx (mul rx ry) = mul rx (mul lx ry)" by (simp add: mul_a,simp add: mul_c)
   129 next show "add (add a b) (add c d) = add (add a c) (add b d)"
   130     using add_c add_a by simp
   131 next show "add (add a b) c = add a (add b c)" using add_a by simp
   132 next show "add a (add c d) = add c (add a d)"
   133     apply (simp add: add_a) by (simp only: add_c)
   134 next show "add (add a b) c = add (add a c) b" using add_a add_c by simp
   135 next show "add a c = add c a" by (rule add_c)
   136 next show "add a (add c d) = add (add a c) d" using add_a by simp
   137 next show "mul (pwr x p) (pwr x q) = pwr x (p + q)" by (rule mul_pwr)
   138 next show "mul x (pwr x q) = pwr x (Suc q)" using pwr_Suc by simp
   139 next show "mul (pwr x q) x = pwr x (Suc q)" using pwr_Suc mul_c by simp
   140 next show "mul x x = pwr x 2" by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c)
   141 next show "pwr (mul x y) q = mul (pwr x q) (pwr y q)" by (rule pwr_mul)
   142 next show "pwr (pwr x p) q = pwr x (p * q)" by (rule pwr_pwr)
   143 next show "pwr x 0 = r1" using pwr_0 .
   144 next show "pwr x 1 = x" unfolding One_nat_def by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c)
   145 next show "mul x (add y z) = add (mul x y) (mul x z)" using mul_d by simp
   146 next show "pwr x (Suc q) = mul x (pwr x q)" using pwr_Suc by simp
   147 next show "pwr x (2 * n) = mul (pwr x n) (pwr x n)" by (simp add: nat_number' mul_pwr)
   148 next show "pwr x (Suc (2 * n)) = mul x (mul (pwr x n) (pwr x n))"
   149     by (simp add: nat_number' pwr_Suc mul_pwr)
   150 qed
   151 
   152 
   153 lemmas normalizing_semiring_axioms' =
   154   normalizing_semiring_axioms [normalizer
   155     semiring ops: semiring_ops
   156     semiring rules: semiring_rules]
   157 
   158 end
   159 
   160 sublocale comm_semiring_1
   161   < normalizing!: normalizing_semiring plus times power zero one
   162 proof
   163 qed (simp_all add: algebra_simps)
   164 
   165 declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_semiring_axioms'} *}
   166 
   167 locale normalizing_ring = normalizing_semiring +
   168   fixes sub :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
   169     and neg :: "'a \<Rightarrow> 'a"
   170   assumes neg_mul: "neg x = mul (neg r1) x"
   171     and sub_add: "sub x y = add x (neg y)"
   172 begin
   173 
   174 lemma ring_ops: shows "TERM (sub x y)" and "TERM (neg x)" .
   175 
   176 lemmas ring_rules = neg_mul sub_add
   177 
   178 lemmas normalizing_ring_axioms' =
   179   normalizing_ring_axioms [normalizer
   180     semiring ops: semiring_ops
   181     semiring rules: semiring_rules
   182     ring ops: ring_ops
   183     ring rules: ring_rules]
   184 
   185 end
   186 
   187 sublocale comm_ring_1
   188   < normalizing!: normalizing_ring plus times power zero one minus uminus
   189 proof
   190 qed (simp_all add: diff_minus)
   191 
   192 declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_ring_axioms'} *}
   193 
   194 locale normalizing_field = normalizing_ring +
   195   fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
   196     and inverse:: "'a \<Rightarrow> 'a"
   197   assumes divide_inverse: "divide x y = mul x (inverse y)"
   198      and inverse_divide: "inverse x = divide r1 x"
   199 begin
   200 
   201 lemma field_ops: shows "TERM (divide x y)" and "TERM (inverse x)" .
   202 
   203 lemmas field_rules = divide_inverse inverse_divide
   204 
   205 lemmas normalizing_field_axioms' =
   206   normalizing_field_axioms [normalizer
   207     semiring ops: semiring_ops
   208     semiring rules: semiring_rules
   209     ring ops: ring_ops
   210     ring rules: ring_rules
   211     field ops: field_ops
   212     field rules: field_rules]
   213 
   214 end
   215 
   216 locale normalizing_semiring_cancel = normalizing_semiring +
   217   assumes add_cancel: "add (x::'a) y = add x z \<longleftrightarrow> y = z"
   218   and add_mul_solve: "add (mul w y) (mul x z) =
   219     add (mul w z) (mul x y) \<longleftrightarrow> w = x \<or> y = z"
   220 begin
   221 
   222 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)"
   223 proof-
   224   have "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> \<not> (a = b \<or> c = d)" by simp
   225   also have "\<dots> \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
   226     using add_mul_solve by blast
   227   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)"
   228     by simp
   229 qed
   230 
   231 lemma add_scale_eq_noteq: "\<lbrakk>r \<noteq> r0 ; (a = b) \<and> ~(c = d)\<rbrakk>
   232   \<Longrightarrow> add a (mul r c) \<noteq> add b (mul r d)"
   233 proof(clarify)
   234   assume nz: "r\<noteq> r0" and cnd: "c\<noteq>d"
   235     and eq: "add b (mul r c) = add b (mul r d)"
   236   hence "mul r c = mul r d" using cnd add_cancel by simp
   237   hence "add (mul r0 d) (mul r c) = add (mul r0 c) (mul r d)"
   238     using mul_0 add_cancel by simp
   239   thus "False" using add_mul_solve nz cnd by simp
   240 qed
   241 
   242 lemma add_r0_iff: " x = add x a \<longleftrightarrow> a = r0"
   243 proof-
   244   have "a = r0 \<longleftrightarrow> add x a = add x r0" by (simp add: add_cancel)
   245   thus "x = add x a \<longleftrightarrow> a = r0" by (auto simp add: add_c add_0)
   246 qed
   247 
   248 declare normalizing_semiring_axioms' [normalizer del]
   249 
   250 lemmas normalizing_semiring_cancel_axioms' =
   251   normalizing_semiring_cancel_axioms [normalizer
   252     semiring ops: semiring_ops
   253     semiring rules: semiring_rules
   254     idom rules: noteq_reduce add_scale_eq_noteq]
   255 
   256 end
   257 
   258 locale normalizing_ring_cancel = normalizing_semiring_cancel + normalizing_ring + 
   259   assumes subr0_iff: "sub x y = r0 \<longleftrightarrow> x = y"
   260 begin
   261 
   262 declare normalizing_ring_axioms' [normalizer del]
   263 
   264 lemmas normalizing_ring_cancel_axioms' = normalizing_ring_cancel_axioms [normalizer
   265   semiring ops: semiring_ops
   266   semiring rules: semiring_rules
   267   ring ops: ring_ops
   268   ring rules: ring_rules
   269   idom rules: noteq_reduce add_scale_eq_noteq
   270   ideal rules: subr0_iff add_r0_iff]
   271 
   272 end
   273 
   274 sublocale idom
   275   < normalizing!: normalizing_ring_cancel plus times power zero one minus uminus
   276 proof
   277   fix w x y z
   278   show "w * y + x * z = w * z + x * y \<longleftrightarrow> w = x \<or> y = z"
   279   proof
   280     assume "w * y + x * z = w * z + x * y"
   281     then have "w * y + x * z - w * z - x * y = 0" by (simp add: algebra_simps)
   282     then have "w * (y - z) - x * (y - z) = 0" by (simp add: algebra_simps)
   283     then have "(y - z) * (w - x) = 0" by (simp add: algebra_simps)
   284     then have "y - z = 0 \<or> w - x = 0" by (rule divisors_zero)
   285     then show "w = x \<or> y = z" by auto
   286   qed (auto simp add: add_ac)
   287 qed (simp_all add: algebra_simps)
   288 
   289 declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_ring_cancel_axioms'} *}
   290 
   291 interpretation normalizing_nat!: normalizing_semiring_cancel
   292   "op +" "op *" "op ^" "0::nat" "1"
   293 proof (unfold_locales, simp add: algebra_simps)
   294   fix w x y z ::"nat"
   295   { assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
   296     hence "y < z \<or> y > z" by arith
   297     moreover {
   298       assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z - y" in exI, auto)
   299       then obtain k where kp: "k>0" and yz:"z = y + k" by blast
   300       from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz algebra_simps)
   301       hence "x*k = w*k" by simp
   302       hence "w = x" using kp by simp }
   303     moreover {
   304       assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y - z" in exI, auto)
   305       then obtain k where kp: "k>0" and yz:"y = z + k" by blast
   306       from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz algebra_simps)
   307       hence "w*k = x*k" by simp
   308       hence "w = x" using kp by simp }
   309     ultimately have "w=x" by blast }
   310   thus "(w * y + x * z = w * z + x * y) = (w = x \<or> y = z)" by auto
   311 qed
   312 
   313 declaration {* Normalizer.semiring_funs @{thm normalizing_nat.normalizing_semiring_cancel_axioms'} *}
   314 
   315 locale normalizing_field_cancel = normalizing_ring_cancel + normalizing_field
   316 begin
   317 
   318 declare normalizing_field_axioms' [normalizer del]
   319 
   320 lemmas normalizing_field_cancel_axioms' = normalizing_field_cancel_axioms [normalizer
   321   semiring ops: semiring_ops
   322   semiring rules: semiring_rules
   323   ring ops: ring_ops
   324   ring rules: ring_rules
   325   field ops: field_ops
   326   field rules: field_rules
   327   idom rules: noteq_reduce add_scale_eq_noteq
   328   ideal rules: subr0_iff add_r0_iff]
   329 
   330 end
   331 
   332 sublocale field 
   333   < normalizing!: normalizing_field_cancel plus times power zero one minus uminus divide inverse
   334 proof
   335 qed (simp_all add: divide_inverse)
   336 
   337 declaration {* Normalizer.field_funs @{thm normalizing.normalizing_field_cancel_axioms'} *}
   338  
   339 
   340 subsection {* Groebner Bases *}
   341 
   342 lemmas bool_simps = simp_thms(1-34)
   343 
   344 lemma dnf:
   345     "(P & (Q | R)) = ((P&Q) | (P&R))" "((Q | R) & P) = ((Q&P) | (R&P))"
   346     "(P \<and> Q) = (Q \<and> P)" "(P \<or> Q) = (Q \<or> P)"
   347   by blast+
   348 
   349 lemmas weak_dnf_simps = dnf bool_simps
   350 
   351 lemma nnf_simps:
   352     "(\<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)"
   353     "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not> \<not>(P)) = P"
   354   by blast+
   355 
   356 lemma PFalse:
   357     "P \<equiv> False \<Longrightarrow> \<not> P"
   358     "\<not> P \<Longrightarrow> (P \<equiv> False)"
   359   by auto
   360 
   361 ML {*
   362 structure Algebra_Simplification = Named_Thms(
   363   val name = "algebra"
   364   val description = "pre-simplification rules for algebraic methods"
   365 )
   366 *}
   367 
   368 setup Algebra_Simplification.setup
   369 
   370 declare dvd_def[algebra]
   371 declare dvd_eq_mod_eq_0[symmetric, algebra]
   372 declare mod_div_trivial[algebra]
   373 declare mod_mod_trivial[algebra]
   374 declare conjunct1[OF DIVISION_BY_ZERO, algebra]
   375 declare conjunct2[OF DIVISION_BY_ZERO, algebra]
   376 declare zmod_zdiv_equality[symmetric,algebra]
   377 declare zdiv_zmod_equality[symmetric, algebra]
   378 declare zdiv_zminus_zminus[algebra]
   379 declare zmod_zminus_zminus[algebra]
   380 declare zdiv_zminus2[algebra]
   381 declare zmod_zminus2[algebra]
   382 declare zdiv_zero[algebra]
   383 declare zmod_zero[algebra]
   384 declare mod_by_1[algebra]
   385 declare div_by_1[algebra]
   386 declare zmod_minus1_right[algebra]
   387 declare zdiv_minus1_right[algebra]
   388 declare mod_div_trivial[algebra]
   389 declare mod_mod_trivial[algebra]
   390 declare mod_mult_self2_is_0[algebra]
   391 declare mod_mult_self1_is_0[algebra]
   392 declare zmod_eq_0_iff[algebra]
   393 declare dvd_0_left_iff[algebra]
   394 declare zdvd1_eq[algebra]
   395 declare zmod_eq_dvd_iff[algebra]
   396 declare nat_mod_eq_iff[algebra]
   397 
   398 use "Tools/Groebner_Basis/groebner.ML"
   399 
   400 method_setup algebra = Groebner.algebra_method
   401   "solve polynomial equations over (semi)rings and ideal membership problems using Groebner bases"
   402 
   403 end