split of semiring normalization from Groebner theory; moved field_comp_conv to Numeral_Simproces
authorhaftmann
Fri May 07 15:05:52 2010 +0200 (2010-05-07)
changeset 367517f1da69cacb3
parent 36750 912080b2c449
child 36752 cf558aeb35b0
split of semiring normalization from Groebner theory; moved field_comp_conv to Numeral_Simproces
src/HOL/Groebner_Basis.thy
src/HOL/IsaMakefile
src/HOL/Library/Sum_Of_Squares/sum_of_squares.ML
src/HOL/Library/normarith.ML
src/HOL/Library/positivstellensatz.ML
src/HOL/Semiring_Normalization.thy
src/HOL/Tools/Groebner_Basis/normalizer.ML
src/HOL/Tools/numeral_simprocs.ML
     1.1 --- a/src/HOL/Groebner_Basis.thy	Fri May 07 10:00:24 2010 +0200
     1.2 +++ b/src/HOL/Groebner_Basis.thy	Fri May 07 15:05:52 2010 +0200
     1.3 @@ -2,341 +2,14 @@
     1.4      Author:     Amine Chaieb, TU Muenchen
     1.5  *)
     1.6  
     1.7 -header {* Semiring normalization and Groebner Bases *}
     1.8 +header {* Groebner bases *}
     1.9  
    1.10  theory Groebner_Basis
    1.11 -imports Numeral_Simprocs Nat_Transfer
    1.12 +imports Semiring_Normalization
    1.13  uses
    1.14 -  "Tools/Groebner_Basis/normalizer.ML"
    1.15    ("Tools/Groebner_Basis/groebner.ML")
    1.16  begin
    1.17  
    1.18 -subsection {* Semiring normalization *}
    1.19 -
    1.20 -setup Normalizer.setup
    1.21 -
    1.22 -locale normalizing_semiring =
    1.23 -  fixes add mul pwr r0 r1
    1.24 -  assumes add_a:"(add x (add y z) = add (add x y) z)"
    1.25 -    and add_c: "add x y = add y x" and add_0:"add r0 x = x"
    1.26 -    and mul_a:"mul x (mul y z) = mul (mul x y) z" and mul_c:"mul x y = mul y x"
    1.27 -    and mul_1:"mul r1 x = x" and  mul_0:"mul r0 x = r0"
    1.28 -    and mul_d:"mul x (add y z) = add (mul x y) (mul x z)"
    1.29 -    and pwr_0:"pwr x 0 = r1" and pwr_Suc:"pwr x (Suc n) = mul x (pwr x n)"
    1.30 -begin
    1.31 -
    1.32 -lemma mul_pwr:"mul (pwr x p) (pwr x q) = pwr x (p + q)"
    1.33 -proof (induct p)
    1.34 -  case 0
    1.35 -  then show ?case by (auto simp add: pwr_0 mul_1)
    1.36 -next
    1.37 -  case Suc
    1.38 -  from this [symmetric] show ?case
    1.39 -    by (auto simp add: pwr_Suc mul_1 mul_a)
    1.40 -qed
    1.41 -
    1.42 -lemma pwr_mul: "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
    1.43 -proof (induct q arbitrary: x y, auto simp add:pwr_0 pwr_Suc mul_1)
    1.44 -  fix q x y
    1.45 -  assume "\<And>x y. pwr (mul x y) q = mul (pwr x q) (pwr y q)"
    1.46 -  have "mul (mul x y) (mul (pwr x q) (pwr y q)) = mul x (mul y (mul (pwr x q) (pwr y q)))"
    1.47 -    by (simp add: mul_a)
    1.48 -  also have "\<dots> = (mul (mul y (mul (pwr y q) (pwr x q))) x)" by (simp add: mul_c)
    1.49 -  also have "\<dots> = (mul (mul y (pwr y q)) (mul (pwr x q) x))" by (simp add: mul_a)
    1.50 -  finally show "mul (mul x y) (mul (pwr x q) (pwr y q)) =
    1.51 -    mul (mul x (pwr x q)) (mul y (pwr y q))" by (simp add: mul_c)
    1.52 -qed
    1.53 -
    1.54 -lemma pwr_pwr: "pwr (pwr x p) q = pwr x (p * q)"
    1.55 -proof (induct p arbitrary: q)
    1.56 -  case 0
    1.57 -  show ?case using pwr_Suc mul_1 pwr_0 by (induct q) auto
    1.58 -next
    1.59 -  case Suc
    1.60 -  thus ?case by (auto simp add: mul_pwr [symmetric] pwr_mul pwr_Suc)
    1.61 -qed
    1.62 -
    1.63 -lemma semiring_ops:
    1.64 -  shows "TERM (add x y)" and "TERM (mul x y)" and "TERM (pwr x n)"
    1.65 -    and "TERM r0" and "TERM r1" .
    1.66 -
    1.67 -lemma semiring_rules:
    1.68 -  "add (mul a m) (mul b m) = mul (add a b) m"
    1.69 -  "add (mul a m) m = mul (add a r1) m"
    1.70 -  "add m (mul a m) = mul (add a r1) m"
    1.71 -  "add m m = mul (add r1 r1) m"
    1.72 -  "add r0 a = a"
    1.73 -  "add a r0 = a"
    1.74 -  "mul a b = mul b a"
    1.75 -  "mul (add a b) c = add (mul a c) (mul b c)"
    1.76 -  "mul r0 a = r0"
    1.77 -  "mul a r0 = r0"
    1.78 -  "mul r1 a = a"
    1.79 -  "mul a r1 = a"
    1.80 -  "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
    1.81 -  "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
    1.82 -  "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
    1.83 -  "mul (mul lx ly) rx = mul (mul lx rx) ly"
    1.84 -  "mul (mul lx ly) rx = mul lx (mul ly rx)"
    1.85 -  "mul lx (mul rx ry) = mul (mul lx rx) ry"
    1.86 -  "mul lx (mul rx ry) = mul rx (mul lx ry)"
    1.87 -  "add (add a b) (add c d) = add (add a c) (add b d)"
    1.88 -  "add (add a b) c = add a (add b c)"
    1.89 -  "add a (add c d) = add c (add a d)"
    1.90 -  "add (add a b) c = add (add a c) b"
    1.91 -  "add a c = add c a"
    1.92 -  "add a (add c d) = add (add a c) d"
    1.93 -  "mul (pwr x p) (pwr x q) = pwr x (p + q)"
    1.94 -  "mul x (pwr x q) = pwr x (Suc q)"
    1.95 -  "mul (pwr x q) x = pwr x (Suc q)"
    1.96 -  "mul x x = pwr x 2"
    1.97 -  "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
    1.98 -  "pwr (pwr x p) q = pwr x (p * q)"
    1.99 -  "pwr x 0 = r1"
   1.100 -  "pwr x 1 = x"
   1.101 -  "mul x (add y z) = add (mul x y) (mul x z)"
   1.102 -  "pwr x (Suc q) = mul x (pwr x q)"
   1.103 -  "pwr x (2*n) = mul (pwr x n) (pwr x n)"
   1.104 -  "pwr x (Suc (2*n)) = mul x (mul (pwr x n) (pwr x n))"
   1.105 -proof -
   1.106 -  show "add (mul a m) (mul b m) = mul (add a b) m" using mul_d mul_c by simp
   1.107 -next show"add (mul a m) m = mul (add a r1) m" using mul_d mul_c mul_1 by simp
   1.108 -next show "add m (mul a m) = mul (add a r1) m" using mul_c mul_d mul_1 add_c by simp
   1.109 -next show "add m m = mul (add r1 r1) m" using mul_c mul_d mul_1 by simp
   1.110 -next show "add r0 a = a" using add_0 by simp
   1.111 -next show "add a r0 = a" using add_0 add_c by simp
   1.112 -next show "mul a b = mul b a" using mul_c by simp
   1.113 -next show "mul (add a b) c = add (mul a c) (mul b c)" using mul_c mul_d by simp
   1.114 -next show "mul r0 a = r0" using mul_0 by simp
   1.115 -next show "mul a r0 = r0" using mul_0 mul_c by simp
   1.116 -next show "mul r1 a = a" using mul_1 by simp
   1.117 -next show "mul a r1 = a" using mul_1 mul_c by simp
   1.118 -next show "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
   1.119 -    using mul_c mul_a by simp
   1.120 -next show "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
   1.121 -    using mul_a by simp
   1.122 -next
   1.123 -  have "mul (mul lx ly) (mul rx ry) = mul (mul rx ry) (mul lx ly)" by (rule mul_c)
   1.124 -  also have "\<dots> = mul rx (mul ry (mul lx ly))" using mul_a by simp
   1.125 -  finally
   1.126 -  show "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
   1.127 -    using mul_c by simp
   1.128 -next show "mul (mul lx ly) rx = mul (mul lx rx) ly" using mul_c mul_a by simp
   1.129 -next
   1.130 -  show "mul (mul lx ly) rx = mul lx (mul ly rx)" by (simp add: mul_a)
   1.131 -next show "mul lx (mul rx ry) = mul (mul lx rx) ry" by (simp add: mul_a )
   1.132 -next show "mul lx (mul rx ry) = mul rx (mul lx ry)" by (simp add: mul_a,simp add: mul_c)
   1.133 -next show "add (add a b) (add c d) = add (add a c) (add b d)"
   1.134 -    using add_c add_a by simp
   1.135 -next show "add (add a b) c = add a (add b c)" using add_a by simp
   1.136 -next show "add a (add c d) = add c (add a d)"
   1.137 -    apply (simp add: add_a) by (simp only: add_c)
   1.138 -next show "add (add a b) c = add (add a c) b" using add_a add_c by simp
   1.139 -next show "add a c = add c a" by (rule add_c)
   1.140 -next show "add a (add c d) = add (add a c) d" using add_a by simp
   1.141 -next show "mul (pwr x p) (pwr x q) = pwr x (p + q)" by (rule mul_pwr)
   1.142 -next show "mul x (pwr x q) = pwr x (Suc q)" using pwr_Suc by simp
   1.143 -next show "mul (pwr x q) x = pwr x (Suc q)" using pwr_Suc mul_c by simp
   1.144 -next show "mul x x = pwr x 2" by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c)
   1.145 -next show "pwr (mul x y) q = mul (pwr x q) (pwr y q)" by (rule pwr_mul)
   1.146 -next show "pwr (pwr x p) q = pwr x (p * q)" by (rule pwr_pwr)
   1.147 -next show "pwr x 0 = r1" using pwr_0 .
   1.148 -next show "pwr x 1 = x" unfolding One_nat_def by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c)
   1.149 -next show "mul x (add y z) = add (mul x y) (mul x z)" using mul_d by simp
   1.150 -next show "pwr x (Suc q) = mul x (pwr x q)" using pwr_Suc by simp
   1.151 -next show "pwr x (2 * n) = mul (pwr x n) (pwr x n)" by (simp add: nat_number' mul_pwr)
   1.152 -next show "pwr x (Suc (2 * n)) = mul x (mul (pwr x n) (pwr x n))"
   1.153 -    by (simp add: nat_number' pwr_Suc mul_pwr)
   1.154 -qed
   1.155 -
   1.156 -
   1.157 -lemmas normalizing_semiring_axioms' =
   1.158 -  normalizing_semiring_axioms [normalizer
   1.159 -    semiring ops: semiring_ops
   1.160 -    semiring rules: semiring_rules]
   1.161 -
   1.162 -end
   1.163 -
   1.164 -sublocale comm_semiring_1
   1.165 -  < normalizing!: normalizing_semiring plus times power zero one
   1.166 -proof
   1.167 -qed (simp_all add: algebra_simps)
   1.168 -
   1.169 -declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_semiring_axioms'} *}
   1.170 -
   1.171 -locale normalizing_ring = normalizing_semiring +
   1.172 -  fixes sub :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
   1.173 -    and neg :: "'a \<Rightarrow> 'a"
   1.174 -  assumes neg_mul: "neg x = mul (neg r1) x"
   1.175 -    and sub_add: "sub x y = add x (neg y)"
   1.176 -begin
   1.177 -
   1.178 -lemma ring_ops: shows "TERM (sub x y)" and "TERM (neg x)" .
   1.179 -
   1.180 -lemmas ring_rules = neg_mul sub_add
   1.181 -
   1.182 -lemmas normalizing_ring_axioms' =
   1.183 -  normalizing_ring_axioms [normalizer
   1.184 -    semiring ops: semiring_ops
   1.185 -    semiring rules: semiring_rules
   1.186 -    ring ops: ring_ops
   1.187 -    ring rules: ring_rules]
   1.188 -
   1.189 -end
   1.190 -
   1.191 -sublocale comm_ring_1
   1.192 -  < normalizing!: normalizing_ring plus times power zero one minus uminus
   1.193 -proof
   1.194 -qed (simp_all add: diff_minus)
   1.195 -
   1.196 -declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_ring_axioms'} *}
   1.197 -
   1.198 -locale normalizing_field = normalizing_ring +
   1.199 -  fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
   1.200 -    and inverse:: "'a \<Rightarrow> 'a"
   1.201 -  assumes divide_inverse: "divide x y = mul x (inverse y)"
   1.202 -     and inverse_divide: "inverse x = divide r1 x"
   1.203 -begin
   1.204 -
   1.205 -lemma field_ops: shows "TERM (divide x y)" and "TERM (inverse x)" .
   1.206 -
   1.207 -lemmas field_rules = divide_inverse inverse_divide
   1.208 -
   1.209 -lemmas normalizing_field_axioms' =
   1.210 -  normalizing_field_axioms [normalizer
   1.211 -    semiring ops: semiring_ops
   1.212 -    semiring rules: semiring_rules
   1.213 -    ring ops: ring_ops
   1.214 -    ring rules: ring_rules
   1.215 -    field ops: field_ops
   1.216 -    field rules: field_rules]
   1.217 -
   1.218 -end
   1.219 -
   1.220 -locale normalizing_semiring_cancel = normalizing_semiring +
   1.221 -  assumes add_cancel: "add (x::'a) y = add x z \<longleftrightarrow> y = z"
   1.222 -  and add_mul_solve: "add (mul w y) (mul x z) =
   1.223 -    add (mul w z) (mul x y) \<longleftrightarrow> w = x \<or> y = z"
   1.224 -begin
   1.225 -
   1.226 -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)"
   1.227 -proof-
   1.228 -  have "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> \<not> (a = b \<or> c = d)" by simp
   1.229 -  also have "\<dots> \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
   1.230 -    using add_mul_solve by blast
   1.231 -  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)"
   1.232 -    by simp
   1.233 -qed
   1.234 -
   1.235 -lemma add_scale_eq_noteq: "\<lbrakk>r \<noteq> r0 ; (a = b) \<and> ~(c = d)\<rbrakk>
   1.236 -  \<Longrightarrow> add a (mul r c) \<noteq> add b (mul r d)"
   1.237 -proof(clarify)
   1.238 -  assume nz: "r\<noteq> r0" and cnd: "c\<noteq>d"
   1.239 -    and eq: "add b (mul r c) = add b (mul r d)"
   1.240 -  hence "mul r c = mul r d" using cnd add_cancel by simp
   1.241 -  hence "add (mul r0 d) (mul r c) = add (mul r0 c) (mul r d)"
   1.242 -    using mul_0 add_cancel by simp
   1.243 -  thus "False" using add_mul_solve nz cnd by simp
   1.244 -qed
   1.245 -
   1.246 -lemma add_r0_iff: " x = add x a \<longleftrightarrow> a = r0"
   1.247 -proof-
   1.248 -  have "a = r0 \<longleftrightarrow> add x a = add x r0" by (simp add: add_cancel)
   1.249 -  thus "x = add x a \<longleftrightarrow> a = r0" by (auto simp add: add_c add_0)
   1.250 -qed
   1.251 -
   1.252 -declare normalizing_semiring_axioms' [normalizer del]
   1.253 -
   1.254 -lemmas normalizing_semiring_cancel_axioms' =
   1.255 -  normalizing_semiring_cancel_axioms [normalizer
   1.256 -    semiring ops: semiring_ops
   1.257 -    semiring rules: semiring_rules
   1.258 -    idom rules: noteq_reduce add_scale_eq_noteq]
   1.259 -
   1.260 -end
   1.261 -
   1.262 -locale normalizing_ring_cancel = normalizing_semiring_cancel + normalizing_ring + 
   1.263 -  assumes subr0_iff: "sub x y = r0 \<longleftrightarrow> x = y"
   1.264 -begin
   1.265 -
   1.266 -declare normalizing_ring_axioms' [normalizer del]
   1.267 -
   1.268 -lemmas normalizing_ring_cancel_axioms' = normalizing_ring_cancel_axioms [normalizer
   1.269 -  semiring ops: semiring_ops
   1.270 -  semiring rules: semiring_rules
   1.271 -  ring ops: ring_ops
   1.272 -  ring rules: ring_rules
   1.273 -  idom rules: noteq_reduce add_scale_eq_noteq
   1.274 -  ideal rules: subr0_iff add_r0_iff]
   1.275 -
   1.276 -end
   1.277 -
   1.278 -sublocale idom
   1.279 -  < normalizing!: normalizing_ring_cancel plus times power zero one minus uminus
   1.280 -proof
   1.281 -  fix w x y z
   1.282 -  show "w * y + x * z = w * z + x * y \<longleftrightarrow> w = x \<or> y = z"
   1.283 -  proof
   1.284 -    assume "w * y + x * z = w * z + x * y"
   1.285 -    then have "w * y + x * z - w * z - x * y = 0" by (simp add: algebra_simps)
   1.286 -    then have "w * (y - z) - x * (y - z) = 0" by (simp add: algebra_simps)
   1.287 -    then have "(y - z) * (w - x) = 0" by (simp add: algebra_simps)
   1.288 -    then have "y - z = 0 \<or> w - x = 0" by (rule divisors_zero)
   1.289 -    then show "w = x \<or> y = z" by auto
   1.290 -  qed (auto simp add: add_ac)
   1.291 -qed (simp_all add: algebra_simps)
   1.292 -
   1.293 -declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_ring_cancel_axioms'} *}
   1.294 -
   1.295 -interpretation normalizing_nat!: normalizing_semiring_cancel
   1.296 -  "op +" "op *" "op ^" "0::nat" "1"
   1.297 -proof (unfold_locales, simp add: algebra_simps)
   1.298 -  fix w x y z ::"nat"
   1.299 -  { assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
   1.300 -    hence "y < z \<or> y > z" by arith
   1.301 -    moreover {
   1.302 -      assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z - y" in exI, auto)
   1.303 -      then obtain k where kp: "k>0" and yz:"z = y + k" by blast
   1.304 -      from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz algebra_simps)
   1.305 -      hence "x*k = w*k" by simp
   1.306 -      hence "w = x" using kp by simp }
   1.307 -    moreover {
   1.308 -      assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y - z" in exI, auto)
   1.309 -      then obtain k where kp: "k>0" and yz:"y = z + k" by blast
   1.310 -      from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz algebra_simps)
   1.311 -      hence "w*k = x*k" by simp
   1.312 -      hence "w = x" using kp by simp }
   1.313 -    ultimately have "w=x" by blast }
   1.314 -  thus "(w * y + x * z = w * z + x * y) = (w = x \<or> y = z)" by auto
   1.315 -qed
   1.316 -
   1.317 -declaration {* Normalizer.semiring_funs @{thm normalizing_nat.normalizing_semiring_cancel_axioms'} *}
   1.318 -
   1.319 -locale normalizing_field_cancel = normalizing_ring_cancel + normalizing_field
   1.320 -begin
   1.321 -
   1.322 -declare normalizing_field_axioms' [normalizer del]
   1.323 -
   1.324 -lemmas normalizing_field_cancel_axioms' = normalizing_field_cancel_axioms [normalizer
   1.325 -  semiring ops: semiring_ops
   1.326 -  semiring rules: semiring_rules
   1.327 -  ring ops: ring_ops
   1.328 -  ring rules: ring_rules
   1.329 -  field ops: field_ops
   1.330 -  field rules: field_rules
   1.331 -  idom rules: noteq_reduce add_scale_eq_noteq
   1.332 -  ideal rules: subr0_iff add_r0_iff]
   1.333 -
   1.334 -end
   1.335 -
   1.336 -sublocale field 
   1.337 -  < normalizing!: normalizing_field_cancel plus times power zero one minus uminus divide inverse
   1.338 -proof
   1.339 -qed (simp_all add: divide_inverse)
   1.340 -
   1.341 -declaration {* Normalizer.field_funs @{thm normalizing.normalizing_field_cancel_axioms'} *}
   1.342 - 
   1.343 -
   1.344  subsection {* Groebner Bases *}
   1.345  
   1.346  lemmas bool_simps = simp_thms(1-34)
   1.347 @@ -367,6 +40,11 @@
   1.348  
   1.349  setup Algebra_Simplification.setup
   1.350  
   1.351 +use "Tools/Groebner_Basis/groebner.ML"
   1.352 +
   1.353 +method_setup algebra = Groebner.algebra_method
   1.354 +  "solve polynomial equations over (semi)rings and ideal membership problems using Groebner bases"
   1.355 +
   1.356  declare dvd_def[algebra]
   1.357  declare dvd_eq_mod_eq_0[symmetric, algebra]
   1.358  declare mod_div_trivial[algebra]
   1.359 @@ -395,9 +73,4 @@
   1.360  declare zmod_eq_dvd_iff[algebra]
   1.361  declare nat_mod_eq_iff[algebra]
   1.362  
   1.363 -use "Tools/Groebner_Basis/groebner.ML"
   1.364 -
   1.365 -method_setup algebra = Groebner.algebra_method
   1.366 -  "solve polynomial equations over (semi)rings and ideal membership problems using Groebner bases"
   1.367 -
   1.368  end
     2.1 --- a/src/HOL/IsaMakefile	Fri May 07 10:00:24 2010 +0200
     2.2 +++ b/src/HOL/IsaMakefile	Fri May 07 15:05:52 2010 +0200
     2.3 @@ -271,6 +271,7 @@
     2.4    Random.thy \
     2.5    Random_Sequence.thy \
     2.6    Recdef.thy \
     2.7 +  Semiring_Normalization.thy \
     2.8    SetInterval.thy \
     2.9    Sledgehammer.thy \
    2.10    String.thy \
     3.1 --- a/src/HOL/Library/Sum_Of_Squares/sum_of_squares.ML	Fri May 07 10:00:24 2010 +0200
     3.2 +++ b/src/HOL/Library/Sum_Of_Squares/sum_of_squares.ML	Fri May 07 15:05:52 2010 +0200
     3.3 @@ -1222,7 +1222,7 @@
     3.4     in
     3.5    (let val th = tryfind trivial_axiom (keq @ klep @ kltp)
     3.6     in
     3.7 -    (fconv_rule (arg_conv (arg1_conv real_poly_conv) then_conv Normalizer.field_comp_conv) th, RealArith.Trivial)
     3.8 +    (fconv_rule (arg_conv (arg1_conv real_poly_conv) then_conv Numeral_Simprocs.field_comp_conv) th, RealArith.Trivial)
     3.9     end)
    3.10     handle Failure _ =>
    3.11       (let val proof =
     4.1 --- a/src/HOL/Library/normarith.ML	Fri May 07 10:00:24 2010 +0200
     4.2 +++ b/src/HOL/Library/normarith.ML	Fri May 07 15:05:52 2010 +0200
     4.3 @@ -168,7 +168,7 @@
     4.4    val real_poly_conv = 
     4.5      Normalizer.semiring_normalize_wrapper ctxt
     4.6       (the (Normalizer.match ctxt @{cterm "(0::real) + 1"}))
     4.7 - in fconv_rule (arg_conv ((rewr_conv @{thm ge_iff_diff_ge_0}) then_conv arg_conv (Normalizer.field_comp_conv then_conv real_poly_conv)))
     4.8 + in fconv_rule (arg_conv ((rewr_conv @{thm ge_iff_diff_ge_0}) then_conv arg_conv (Numeral_Simprocs.field_comp_conv then_conv real_poly_conv)))
     4.9  end;
    4.10  
    4.11   fun absc cv ct = case term_of ct of 
    4.12 @@ -190,8 +190,8 @@
    4.13   val apply_pth5 = rewr_conv @{thm pth_5};
    4.14   val apply_pth6 = rewr_conv @{thm pth_6};
    4.15   val apply_pth7 = rewrs_conv @{thms pth_7};
    4.16 - val apply_pth8 = rewr_conv @{thm pth_8} then_conv arg1_conv Normalizer.field_comp_conv then_conv (try_conv (rewr_conv (mk_meta_eq @{thm scaleR_zero_left})));
    4.17 - val apply_pth9 = rewrs_conv @{thms pth_9} then_conv arg1_conv (arg1_conv Normalizer.field_comp_conv);
    4.18 + val apply_pth8 = rewr_conv @{thm pth_8} then_conv arg1_conv Numeral_Simprocs.field_comp_conv then_conv (try_conv (rewr_conv (mk_meta_eq @{thm scaleR_zero_left})));
    4.19 + val apply_pth9 = rewrs_conv @{thms pth_9} then_conv arg1_conv (arg1_conv Numeral_Simprocs.field_comp_conv);
    4.20   val apply_ptha = rewr_conv @{thm pth_a};
    4.21   val apply_pthb = rewrs_conv @{thms pth_b};
    4.22   val apply_pthc = rewrs_conv @{thms pth_c};
    4.23 @@ -204,7 +204,7 @@
    4.24   | _ => error "headvector: non-canonical term"
    4.25  
    4.26  fun vector_cmul_conv ct =
    4.27 -   ((apply_pth5 then_conv arg1_conv Normalizer.field_comp_conv) else_conv
    4.28 +   ((apply_pth5 then_conv arg1_conv Numeral_Simprocs.field_comp_conv) else_conv
    4.29      (apply_pth6 then_conv binop_conv vector_cmul_conv)) ct
    4.30  
    4.31  fun vector_add_conv ct = apply_pth7 ct 
    4.32 @@ -396,7 +396,7 @@
    4.33    fun init_conv ctxt = 
    4.34     Simplifier.rewrite (Simplifier.context ctxt 
    4.35       (HOL_basic_ss addsimps ([(*@{thm vec_0}, @{thm vec_1},*) @{thm dist_norm}, @{thm diff_0_right}, @{thm right_minus}, @{thm diff_self}, @{thm norm_zero}] @ @{thms arithmetic_simps} @ @{thms norm_pths})))
    4.36 -   then_conv Normalizer.field_comp_conv 
    4.37 +   then_conv Numeral_Simprocs.field_comp_conv 
    4.38     then_conv nnf_conv
    4.39  
    4.40   fun pure ctxt = fst o RealArith.gen_prover_real_arith ctxt (real_vector_prover ctxt);
     5.1 --- a/src/HOL/Library/positivstellensatz.ML	Fri May 07 10:00:24 2010 +0200
     5.2 +++ b/src/HOL/Library/positivstellensatz.ML	Fri May 07 15:05:52 2010 +0200
     5.3 @@ -751,7 +751,7 @@
     5.4        (the (Normalizer.match ctxt @{cterm "(0::real) + 1"})) 
     5.5       simple_cterm_ord
     5.6  in gen_real_arith ctxt
     5.7 -   (cterm_of_rat, Normalizer.field_comp_conv, Normalizer.field_comp_conv, Normalizer.field_comp_conv,
     5.8 +   (cterm_of_rat, Numeral_Simprocs.field_comp_conv, Numeral_Simprocs.field_comp_conv, Numeral_Simprocs.field_comp_conv,
     5.9      main,neg,add,mul, prover)
    5.10  end;
    5.11  
     6.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     6.2 +++ b/src/HOL/Semiring_Normalization.thy	Fri May 07 15:05:52 2010 +0200
     6.3 @@ -0,0 +1,336 @@
     6.4 +(*  Title:      HOL/Semiring_Normalization.thy
     6.5 +    Author:     Amine Chaieb, TU Muenchen
     6.6 +*)
     6.7 +
     6.8 +header {* Semiring normalization *}
     6.9 +
    6.10 +theory Semiring_Normalization
    6.11 +imports Numeral_Simprocs Nat_Transfer
    6.12 +uses
    6.13 +  "Tools/Groebner_Basis/normalizer.ML"
    6.14 +begin
    6.15 +
    6.16 +setup Normalizer.setup
    6.17 +
    6.18 +locale normalizing_semiring =
    6.19 +  fixes add mul pwr r0 r1
    6.20 +  assumes add_a:"(add x (add y z) = add (add x y) z)"
    6.21 +    and add_c: "add x y = add y x" and add_0:"add r0 x = x"
    6.22 +    and mul_a:"mul x (mul y z) = mul (mul x y) z" and mul_c:"mul x y = mul y x"
    6.23 +    and mul_1:"mul r1 x = x" and  mul_0:"mul r0 x = r0"
    6.24 +    and mul_d:"mul x (add y z) = add (mul x y) (mul x z)"
    6.25 +    and pwr_0:"pwr x 0 = r1" and pwr_Suc:"pwr x (Suc n) = mul x (pwr x n)"
    6.26 +begin
    6.27 +
    6.28 +lemma mul_pwr:"mul (pwr x p) (pwr x q) = pwr x (p + q)"
    6.29 +proof (induct p)
    6.30 +  case 0
    6.31 +  then show ?case by (auto simp add: pwr_0 mul_1)
    6.32 +next
    6.33 +  case Suc
    6.34 +  from this [symmetric] show ?case
    6.35 +    by (auto simp add: pwr_Suc mul_1 mul_a)
    6.36 +qed
    6.37 +
    6.38 +lemma pwr_mul: "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
    6.39 +proof (induct q arbitrary: x y, auto simp add:pwr_0 pwr_Suc mul_1)
    6.40 +  fix q x y
    6.41 +  assume "\<And>x y. pwr (mul x y) q = mul (pwr x q) (pwr y q)"
    6.42 +  have "mul (mul x y) (mul (pwr x q) (pwr y q)) = mul x (mul y (mul (pwr x q) (pwr y q)))"
    6.43 +    by (simp add: mul_a)
    6.44 +  also have "\<dots> = (mul (mul y (mul (pwr y q) (pwr x q))) x)" by (simp add: mul_c)
    6.45 +  also have "\<dots> = (mul (mul y (pwr y q)) (mul (pwr x q) x))" by (simp add: mul_a)
    6.46 +  finally show "mul (mul x y) (mul (pwr x q) (pwr y q)) =
    6.47 +    mul (mul x (pwr x q)) (mul y (pwr y q))" by (simp add: mul_c)
    6.48 +qed
    6.49 +
    6.50 +lemma pwr_pwr: "pwr (pwr x p) q = pwr x (p * q)"
    6.51 +proof (induct p arbitrary: q)
    6.52 +  case 0
    6.53 +  show ?case using pwr_Suc mul_1 pwr_0 by (induct q) auto
    6.54 +next
    6.55 +  case Suc
    6.56 +  thus ?case by (auto simp add: mul_pwr [symmetric] pwr_mul pwr_Suc)
    6.57 +qed
    6.58 +
    6.59 +lemma semiring_ops:
    6.60 +  shows "TERM (add x y)" and "TERM (mul x y)" and "TERM (pwr x n)"
    6.61 +    and "TERM r0" and "TERM r1" .
    6.62 +
    6.63 +lemma semiring_rules:
    6.64 +  "add (mul a m) (mul b m) = mul (add a b) m"
    6.65 +  "add (mul a m) m = mul (add a r1) m"
    6.66 +  "add m (mul a m) = mul (add a r1) m"
    6.67 +  "add m m = mul (add r1 r1) m"
    6.68 +  "add r0 a = a"
    6.69 +  "add a r0 = a"
    6.70 +  "mul a b = mul b a"
    6.71 +  "mul (add a b) c = add (mul a c) (mul b c)"
    6.72 +  "mul r0 a = r0"
    6.73 +  "mul a r0 = r0"
    6.74 +  "mul r1 a = a"
    6.75 +  "mul a r1 = a"
    6.76 +  "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
    6.77 +  "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
    6.78 +  "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
    6.79 +  "mul (mul lx ly) rx = mul (mul lx rx) ly"
    6.80 +  "mul (mul lx ly) rx = mul lx (mul ly rx)"
    6.81 +  "mul lx (mul rx ry) = mul (mul lx rx) ry"
    6.82 +  "mul lx (mul rx ry) = mul rx (mul lx ry)"
    6.83 +  "add (add a b) (add c d) = add (add a c) (add b d)"
    6.84 +  "add (add a b) c = add a (add b c)"
    6.85 +  "add a (add c d) = add c (add a d)"
    6.86 +  "add (add a b) c = add (add a c) b"
    6.87 +  "add a c = add c a"
    6.88 +  "add a (add c d) = add (add a c) d"
    6.89 +  "mul (pwr x p) (pwr x q) = pwr x (p + q)"
    6.90 +  "mul x (pwr x q) = pwr x (Suc q)"
    6.91 +  "mul (pwr x q) x = pwr x (Suc q)"
    6.92 +  "mul x x = pwr x 2"
    6.93 +  "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
    6.94 +  "pwr (pwr x p) q = pwr x (p * q)"
    6.95 +  "pwr x 0 = r1"
    6.96 +  "pwr x 1 = x"
    6.97 +  "mul x (add y z) = add (mul x y) (mul x z)"
    6.98 +  "pwr x (Suc q) = mul x (pwr x q)"
    6.99 +  "pwr x (2*n) = mul (pwr x n) (pwr x n)"
   6.100 +  "pwr x (Suc (2*n)) = mul x (mul (pwr x n) (pwr x n))"
   6.101 +proof -
   6.102 +  show "add (mul a m) (mul b m) = mul (add a b) m" using mul_d mul_c by simp
   6.103 +next show"add (mul a m) m = mul (add a r1) m" using mul_d mul_c mul_1 by simp
   6.104 +next show "add m (mul a m) = mul (add a r1) m" using mul_c mul_d mul_1 add_c by simp
   6.105 +next show "add m m = mul (add r1 r1) m" using mul_c mul_d mul_1 by simp
   6.106 +next show "add r0 a = a" using add_0 by simp
   6.107 +next show "add a r0 = a" using add_0 add_c by simp
   6.108 +next show "mul a b = mul b a" using mul_c by simp
   6.109 +next show "mul (add a b) c = add (mul a c) (mul b c)" using mul_c mul_d by simp
   6.110 +next show "mul r0 a = r0" using mul_0 by simp
   6.111 +next show "mul a r0 = r0" using mul_0 mul_c by simp
   6.112 +next show "mul r1 a = a" using mul_1 by simp
   6.113 +next show "mul a r1 = a" using mul_1 mul_c by simp
   6.114 +next show "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
   6.115 +    using mul_c mul_a by simp
   6.116 +next show "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
   6.117 +    using mul_a by simp
   6.118 +next
   6.119 +  have "mul (mul lx ly) (mul rx ry) = mul (mul rx ry) (mul lx ly)" by (rule mul_c)
   6.120 +  also have "\<dots> = mul rx (mul ry (mul lx ly))" using mul_a by simp
   6.121 +  finally
   6.122 +  show "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
   6.123 +    using mul_c by simp
   6.124 +next show "mul (mul lx ly) rx = mul (mul lx rx) ly" using mul_c mul_a by simp
   6.125 +next
   6.126 +  show "mul (mul lx ly) rx = mul lx (mul ly rx)" by (simp add: mul_a)
   6.127 +next show "mul lx (mul rx ry) = mul (mul lx rx) ry" by (simp add: mul_a )
   6.128 +next show "mul lx (mul rx ry) = mul rx (mul lx ry)" by (simp add: mul_a,simp add: mul_c)
   6.129 +next show "add (add a b) (add c d) = add (add a c) (add b d)"
   6.130 +    using add_c add_a by simp
   6.131 +next show "add (add a b) c = add a (add b c)" using add_a by simp
   6.132 +next show "add a (add c d) = add c (add a d)"
   6.133 +    apply (simp add: add_a) by (simp only: add_c)
   6.134 +next show "add (add a b) c = add (add a c) b" using add_a add_c by simp
   6.135 +next show "add a c = add c a" by (rule add_c)
   6.136 +next show "add a (add c d) = add (add a c) d" using add_a by simp
   6.137 +next show "mul (pwr x p) (pwr x q) = pwr x (p + q)" by (rule mul_pwr)
   6.138 +next show "mul x (pwr x q) = pwr x (Suc q)" using pwr_Suc by simp
   6.139 +next show "mul (pwr x q) x = pwr x (Suc q)" using pwr_Suc mul_c by simp
   6.140 +next show "mul x x = pwr x 2" by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c)
   6.141 +next show "pwr (mul x y) q = mul (pwr x q) (pwr y q)" by (rule pwr_mul)
   6.142 +next show "pwr (pwr x p) q = pwr x (p * q)" by (rule pwr_pwr)
   6.143 +next show "pwr x 0 = r1" using pwr_0 .
   6.144 +next show "pwr x 1 = x" unfolding One_nat_def by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c)
   6.145 +next show "mul x (add y z) = add (mul x y) (mul x z)" using mul_d by simp
   6.146 +next show "pwr x (Suc q) = mul x (pwr x q)" using pwr_Suc by simp
   6.147 +next show "pwr x (2 * n) = mul (pwr x n) (pwr x n)" by (simp add: nat_number' mul_pwr)
   6.148 +next show "pwr x (Suc (2 * n)) = mul x (mul (pwr x n) (pwr x n))"
   6.149 +    by (simp add: nat_number' pwr_Suc mul_pwr)
   6.150 +qed
   6.151 +
   6.152 +
   6.153 +lemmas normalizing_semiring_axioms' =
   6.154 +  normalizing_semiring_axioms [normalizer
   6.155 +    semiring ops: semiring_ops
   6.156 +    semiring rules: semiring_rules]
   6.157 +
   6.158 +end
   6.159 +
   6.160 +sublocale comm_semiring_1
   6.161 +  < normalizing!: normalizing_semiring plus times power zero one
   6.162 +proof
   6.163 +qed (simp_all add: algebra_simps)
   6.164 +
   6.165 +declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_semiring_axioms'} *}
   6.166 +
   6.167 +locale normalizing_ring = normalizing_semiring +
   6.168 +  fixes sub :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
   6.169 +    and neg :: "'a \<Rightarrow> 'a"
   6.170 +  assumes neg_mul: "neg x = mul (neg r1) x"
   6.171 +    and sub_add: "sub x y = add x (neg y)"
   6.172 +begin
   6.173 +
   6.174 +lemma ring_ops: shows "TERM (sub x y)" and "TERM (neg x)" .
   6.175 +
   6.176 +lemmas ring_rules = neg_mul sub_add
   6.177 +
   6.178 +lemmas normalizing_ring_axioms' =
   6.179 +  normalizing_ring_axioms [normalizer
   6.180 +    semiring ops: semiring_ops
   6.181 +    semiring rules: semiring_rules
   6.182 +    ring ops: ring_ops
   6.183 +    ring rules: ring_rules]
   6.184 +
   6.185 +end
   6.186 +
   6.187 +sublocale comm_ring_1
   6.188 +  < normalizing!: normalizing_ring plus times power zero one minus uminus
   6.189 +proof
   6.190 +qed (simp_all add: diff_minus)
   6.191 +
   6.192 +declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_ring_axioms'} *}
   6.193 +
   6.194 +locale normalizing_field = normalizing_ring +
   6.195 +  fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
   6.196 +    and inverse:: "'a \<Rightarrow> 'a"
   6.197 +  assumes divide_inverse: "divide x y = mul x (inverse y)"
   6.198 +     and inverse_divide: "inverse x = divide r1 x"
   6.199 +begin
   6.200 +
   6.201 +lemma field_ops: shows "TERM (divide x y)" and "TERM (inverse x)" .
   6.202 +
   6.203 +lemmas field_rules = divide_inverse inverse_divide
   6.204 +
   6.205 +lemmas normalizing_field_axioms' =
   6.206 +  normalizing_field_axioms [normalizer
   6.207 +    semiring ops: semiring_ops
   6.208 +    semiring rules: semiring_rules
   6.209 +    ring ops: ring_ops
   6.210 +    ring rules: ring_rules
   6.211 +    field ops: field_ops
   6.212 +    field rules: field_rules]
   6.213 +
   6.214 +end
   6.215 +
   6.216 +locale normalizing_semiring_cancel = normalizing_semiring +
   6.217 +  assumes add_cancel: "add (x::'a) y = add x z \<longleftrightarrow> y = z"
   6.218 +  and add_mul_solve: "add (mul w y) (mul x z) =
   6.219 +    add (mul w z) (mul x y) \<longleftrightarrow> w = x \<or> y = z"
   6.220 +begin
   6.221 +
   6.222 +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)"
   6.223 +proof-
   6.224 +  have "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> \<not> (a = b \<or> c = d)" by simp
   6.225 +  also have "\<dots> \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
   6.226 +    using add_mul_solve by blast
   6.227 +  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)"
   6.228 +    by simp
   6.229 +qed
   6.230 +
   6.231 +lemma add_scale_eq_noteq: "\<lbrakk>r \<noteq> r0 ; (a = b) \<and> ~(c = d)\<rbrakk>
   6.232 +  \<Longrightarrow> add a (mul r c) \<noteq> add b (mul r d)"
   6.233 +proof(clarify)
   6.234 +  assume nz: "r\<noteq> r0" and cnd: "c\<noteq>d"
   6.235 +    and eq: "add b (mul r c) = add b (mul r d)"
   6.236 +  hence "mul r c = mul r d" using cnd add_cancel by simp
   6.237 +  hence "add (mul r0 d) (mul r c) = add (mul r0 c) (mul r d)"
   6.238 +    using mul_0 add_cancel by simp
   6.239 +  thus "False" using add_mul_solve nz cnd by simp
   6.240 +qed
   6.241 +
   6.242 +lemma add_r0_iff: " x = add x a \<longleftrightarrow> a = r0"
   6.243 +proof-
   6.244 +  have "a = r0 \<longleftrightarrow> add x a = add x r0" by (simp add: add_cancel)
   6.245 +  thus "x = add x a \<longleftrightarrow> a = r0" by (auto simp add: add_c add_0)
   6.246 +qed
   6.247 +
   6.248 +declare normalizing_semiring_axioms' [normalizer del]
   6.249 +
   6.250 +lemmas normalizing_semiring_cancel_axioms' =
   6.251 +  normalizing_semiring_cancel_axioms [normalizer
   6.252 +    semiring ops: semiring_ops
   6.253 +    semiring rules: semiring_rules
   6.254 +    idom rules: noteq_reduce add_scale_eq_noteq]
   6.255 +
   6.256 +end
   6.257 +
   6.258 +locale normalizing_ring_cancel = normalizing_semiring_cancel + normalizing_ring + 
   6.259 +  assumes subr0_iff: "sub x y = r0 \<longleftrightarrow> x = y"
   6.260 +begin
   6.261 +
   6.262 +declare normalizing_ring_axioms' [normalizer del]
   6.263 +
   6.264 +lemmas normalizing_ring_cancel_axioms' = normalizing_ring_cancel_axioms [normalizer
   6.265 +  semiring ops: semiring_ops
   6.266 +  semiring rules: semiring_rules
   6.267 +  ring ops: ring_ops
   6.268 +  ring rules: ring_rules
   6.269 +  idom rules: noteq_reduce add_scale_eq_noteq
   6.270 +  ideal rules: subr0_iff add_r0_iff]
   6.271 +
   6.272 +end
   6.273 +
   6.274 +sublocale idom
   6.275 +  < normalizing!: normalizing_ring_cancel plus times power zero one minus uminus
   6.276 +proof
   6.277 +  fix w x y z
   6.278 +  show "w * y + x * z = w * z + x * y \<longleftrightarrow> w = x \<or> y = z"
   6.279 +  proof
   6.280 +    assume "w * y + x * z = w * z + x * y"
   6.281 +    then have "w * y + x * z - w * z - x * y = 0" by (simp add: algebra_simps)
   6.282 +    then have "w * (y - z) - x * (y - z) = 0" by (simp add: algebra_simps)
   6.283 +    then have "(y - z) * (w - x) = 0" by (simp add: algebra_simps)
   6.284 +    then have "y - z = 0 \<or> w - x = 0" by (rule divisors_zero)
   6.285 +    then show "w = x \<or> y = z" by auto
   6.286 +  qed (auto simp add: add_ac)
   6.287 +qed (simp_all add: algebra_simps)
   6.288 +
   6.289 +declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_ring_cancel_axioms'} *}
   6.290 +
   6.291 +interpretation normalizing_nat!: normalizing_semiring_cancel
   6.292 +  "op +" "op *" "op ^" "0::nat" "1"
   6.293 +proof (unfold_locales, simp add: algebra_simps)
   6.294 +  fix w x y z ::"nat"
   6.295 +  { assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
   6.296 +    hence "y < z \<or> y > z" by arith
   6.297 +    moreover {
   6.298 +      assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z - y" in exI, auto)
   6.299 +      then obtain k where kp: "k>0" and yz:"z = y + k" by blast
   6.300 +      from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz algebra_simps)
   6.301 +      hence "x*k = w*k" by simp
   6.302 +      hence "w = x" using kp by simp }
   6.303 +    moreover {
   6.304 +      assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y - z" in exI, auto)
   6.305 +      then obtain k where kp: "k>0" and yz:"y = z + k" by blast
   6.306 +      from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz algebra_simps)
   6.307 +      hence "w*k = x*k" by simp
   6.308 +      hence "w = x" using kp by simp }
   6.309 +    ultimately have "w=x" by blast }
   6.310 +  thus "(w * y + x * z = w * z + x * y) = (w = x \<or> y = z)" by auto
   6.311 +qed
   6.312 +
   6.313 +declaration {* Normalizer.semiring_funs @{thm normalizing_nat.normalizing_semiring_cancel_axioms'} *}
   6.314 +
   6.315 +locale normalizing_field_cancel = normalizing_ring_cancel + normalizing_field
   6.316 +begin
   6.317 +
   6.318 +declare normalizing_field_axioms' [normalizer del]
   6.319 +
   6.320 +lemmas normalizing_field_cancel_axioms' = normalizing_field_cancel_axioms [normalizer
   6.321 +  semiring ops: semiring_ops
   6.322 +  semiring rules: semiring_rules
   6.323 +  ring ops: ring_ops
   6.324 +  ring rules: ring_rules
   6.325 +  field ops: field_ops
   6.326 +  field rules: field_rules
   6.327 +  idom rules: noteq_reduce add_scale_eq_noteq
   6.328 +  ideal rules: subr0_iff add_r0_iff]
   6.329 +
   6.330 +end
   6.331 +
   6.332 +sublocale field 
   6.333 +  < normalizing!: normalizing_field_cancel plus times power zero one minus uminus divide inverse
   6.334 +proof
   6.335 +qed (simp_all add: divide_inverse)
   6.336 +
   6.337 +declaration {* Normalizer.field_funs @{thm normalizing.normalizing_field_cancel_axioms'} *}
   6.338 +
   6.339 +end
     7.1 --- a/src/HOL/Tools/Groebner_Basis/normalizer.ML	Fri May 07 10:00:24 2010 +0200
     7.2 +++ b/src/HOL/Tools/Groebner_Basis/normalizer.ML	Fri May 07 15:05:52 2010 +0200
     7.3 @@ -31,7 +31,6 @@
     7.4    val semiring_normalizers_ord_wrapper:  Proof.context -> entry ->
     7.5      (cterm -> cterm -> bool) ->
     7.6        {add: conv, mul: conv, neg: conv, main: conv, pow: conv, sub: conv}
     7.7 -  val field_comp_conv: conv
     7.8  
     7.9    val setup: theory -> theory
    7.10  end
    7.11 @@ -41,156 +40,6 @@
    7.12  
    7.13  (** some conversion **)
    7.14  
    7.15 -local
    7.16 - val zr = @{cpat "0"}
    7.17 - val zT = ctyp_of_term zr
    7.18 - val geq = @{cpat "op ="}
    7.19 - val eqT = Thm.dest_ctyp (ctyp_of_term geq) |> hd
    7.20 - val add_frac_eq = mk_meta_eq @{thm "add_frac_eq"}
    7.21 - val add_frac_num = mk_meta_eq @{thm "add_frac_num"}
    7.22 - val add_num_frac = mk_meta_eq @{thm "add_num_frac"}
    7.23 -
    7.24 - fun prove_nz ss T t =
    7.25 -    let
    7.26 -      val z = instantiate_cterm ([(zT,T)],[]) zr
    7.27 -      val eq = instantiate_cterm ([(eqT,T)],[]) geq
    7.28 -      val th = Simplifier.rewrite (ss addsimps @{thms simp_thms})
    7.29 -           (Thm.capply @{cterm "Trueprop"} (Thm.capply @{cterm "Not"}
    7.30 -                  (Thm.capply (Thm.capply eq t) z)))
    7.31 -    in equal_elim (symmetric th) TrueI
    7.32 -    end
    7.33 -
    7.34 - fun proc phi ss ct =
    7.35 -  let
    7.36 -    val ((x,y),(w,z)) =
    7.37 -         (Thm.dest_binop #> (fn (a,b) => (Thm.dest_binop a, Thm.dest_binop b))) ct
    7.38 -    val _ = map (HOLogic.dest_number o term_of) [x,y,z,w]
    7.39 -    val T = ctyp_of_term x
    7.40 -    val [y_nz, z_nz] = map (prove_nz ss T) [y, z]
    7.41 -    val th = instantiate' [SOME T] (map SOME [y,z,x,w]) add_frac_eq
    7.42 -  in SOME (implies_elim (implies_elim th y_nz) z_nz)
    7.43 -  end
    7.44 -  handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE
    7.45 -
    7.46 - fun proc2 phi ss ct =
    7.47 -  let
    7.48 -    val (l,r) = Thm.dest_binop ct
    7.49 -    val T = ctyp_of_term l
    7.50 -  in (case (term_of l, term_of r) of
    7.51 -      (Const(@{const_name Rings.divide},_)$_$_, _) =>
    7.52 -        let val (x,y) = Thm.dest_binop l val z = r
    7.53 -            val _ = map (HOLogic.dest_number o term_of) [x,y,z]
    7.54 -            val ynz = prove_nz ss T y
    7.55 -        in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,x,z]) add_frac_num) ynz)
    7.56 -        end
    7.57 -     | (_, Const (@{const_name Rings.divide},_)$_$_) =>
    7.58 -        let val (x,y) = Thm.dest_binop r val z = l
    7.59 -            val _ = map (HOLogic.dest_number o term_of) [x,y,z]
    7.60 -            val ynz = prove_nz ss T y
    7.61 -        in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,z,x]) add_num_frac) ynz)
    7.62 -        end
    7.63 -     | _ => NONE)
    7.64 -  end
    7.65 -  handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE
    7.66 -
    7.67 - fun is_number (Const(@{const_name Rings.divide},_)$a$b) = is_number a andalso is_number b
    7.68 -   | is_number t = can HOLogic.dest_number t
    7.69 -
    7.70 - val is_number = is_number o term_of
    7.71 -
    7.72 - fun proc3 phi ss ct =
    7.73 -  (case term_of ct of
    7.74 -    Const(@{const_name Orderings.less},_)$(Const(@{const_name Rings.divide},_)$_$_)$_ =>
    7.75 -      let
    7.76 -        val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
    7.77 -        val _ = map is_number [a,b,c]
    7.78 -        val T = ctyp_of_term c
    7.79 -        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_less_eq"}
    7.80 -      in SOME (mk_meta_eq th) end
    7.81 -  | Const(@{const_name Orderings.less_eq},_)$(Const(@{const_name Rings.divide},_)$_$_)$_ =>
    7.82 -      let
    7.83 -        val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
    7.84 -        val _ = map is_number [a,b,c]
    7.85 -        val T = ctyp_of_term c
    7.86 -        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_le_eq"}
    7.87 -      in SOME (mk_meta_eq th) end
    7.88 -  | Const("op =",_)$(Const(@{const_name Rings.divide},_)$_$_)$_ =>
    7.89 -      let
    7.90 -        val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
    7.91 -        val _ = map is_number [a,b,c]
    7.92 -        val T = ctyp_of_term c
    7.93 -        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_eq_eq"}
    7.94 -      in SOME (mk_meta_eq th) end
    7.95 -  | Const(@{const_name Orderings.less},_)$_$(Const(@{const_name Rings.divide},_)$_$_) =>
    7.96 -    let
    7.97 -      val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
    7.98 -        val _ = map is_number [a,b,c]
    7.99 -        val T = ctyp_of_term c
   7.100 -        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "less_divide_eq"}
   7.101 -      in SOME (mk_meta_eq th) end
   7.102 -  | Const(@{const_name Orderings.less_eq},_)$_$(Const(@{const_name Rings.divide},_)$_$_) =>
   7.103 -    let
   7.104 -      val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
   7.105 -        val _ = map is_number [a,b,c]
   7.106 -        val T = ctyp_of_term c
   7.107 -        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "le_divide_eq"}
   7.108 -      in SOME (mk_meta_eq th) end
   7.109 -  | Const("op =",_)$_$(Const(@{const_name Rings.divide},_)$_$_) =>
   7.110 -    let
   7.111 -      val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
   7.112 -        val _ = map is_number [a,b,c]
   7.113 -        val T = ctyp_of_term c
   7.114 -        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "eq_divide_eq"}
   7.115 -      in SOME (mk_meta_eq th) end
   7.116 -  | _ => NONE)
   7.117 -  handle TERM _ => NONE | CTERM _ => NONE | THM _ => NONE
   7.118 -
   7.119 -val add_frac_frac_simproc =
   7.120 -       make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + (?w::?'a::field)/?z"}],
   7.121 -                     name = "add_frac_frac_simproc",
   7.122 -                     proc = proc, identifier = []}
   7.123 -
   7.124 -val add_frac_num_simproc =
   7.125 -       make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + ?z"}, @{cpat "?z + (?x::?'a::field)/?y"}],
   7.126 -                     name = "add_frac_num_simproc",
   7.127 -                     proc = proc2, identifier = []}
   7.128 -
   7.129 -val ord_frac_simproc =
   7.130 -  make_simproc
   7.131 -    {lhss = [@{cpat "(?a::(?'a::{field, ord}))/?b < ?c"},
   7.132 -             @{cpat "(?a::(?'a::{field, ord}))/?b <= ?c"},
   7.133 -             @{cpat "?c < (?a::(?'a::{field, ord}))/?b"},
   7.134 -             @{cpat "?c <= (?a::(?'a::{field, ord}))/?b"},
   7.135 -             @{cpat "?c = ((?a::(?'a::{field, ord}))/?b)"},
   7.136 -             @{cpat "((?a::(?'a::{field, ord}))/ ?b) = ?c"}],
   7.137 -             name = "ord_frac_simproc", proc = proc3, identifier = []}
   7.138 -
   7.139 -val ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"},
   7.140 -           @{thm "divide_Numeral1"},
   7.141 -           @{thm "divide_zero"}, @{thm "divide_Numeral0"},
   7.142 -           @{thm "divide_divide_eq_left"}, 
   7.143 -           @{thm "times_divide_eq_left"}, @{thm "times_divide_eq_right"},
   7.144 -           @{thm "times_divide_times_eq"},
   7.145 -           @{thm "divide_divide_eq_right"},
   7.146 -           @{thm "diff_def"}, @{thm "minus_divide_left"},
   7.147 -           @{thm "Numeral1_eq1_nat"}, @{thm "add_divide_distrib"} RS sym,
   7.148 -           @{thm field_divide_inverse} RS sym, @{thm inverse_divide}, 
   7.149 -           Conv.fconv_rule (Conv.arg_conv (Conv.arg1_conv (Conv.rewr_conv (mk_meta_eq @{thm mult_commute}))))   
   7.150 -           (@{thm field_divide_inverse} RS sym)]
   7.151 -
   7.152 -in
   7.153 -
   7.154 -val field_comp_conv = (Simplifier.rewrite
   7.155 -(HOL_basic_ss addsimps @{thms "semiring_norm"}
   7.156 -              addsimps ths addsimps @{thms simp_thms}
   7.157 -              addsimprocs Numeral_Simprocs.field_cancel_numeral_factors
   7.158 -               addsimprocs [add_frac_frac_simproc, add_frac_num_simproc,
   7.159 -                            ord_frac_simproc]
   7.160 -                addcongs [@{thm "if_weak_cong"}]))
   7.161 -then_conv (Simplifier.rewrite (HOL_basic_ss addsimps
   7.162 -  [@{thm numeral_1_eq_1},@{thm numeral_0_eq_0}] @ @{thms numerals(1-2)}))
   7.163 -
   7.164 -end
   7.165  
   7.166  
   7.167  (** data **)
   7.168 @@ -365,7 +214,7 @@
   7.169       {is_const = K numeral_is_const,
   7.170        dest_const = K dest_const,
   7.171        mk_const = mk_const,
   7.172 -      conv = K (K field_comp_conv)}
   7.173 +      conv = K (K Numeral_Simprocs.field_comp_conv)}
   7.174    end;
   7.175  
   7.176  
     8.1 --- a/src/HOL/Tools/numeral_simprocs.ML	Fri May 07 10:00:24 2010 +0200
     8.2 +++ b/src/HOL/Tools/numeral_simprocs.ML	Fri May 07 15:05:52 2010 +0200
     8.3 @@ -1,7 +1,7 @@
     8.4  (* Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     8.5     Copyright   2000  University of Cambridge
     8.6  
     8.7 -Simprocs for the integer numerals.
     8.8 +Simprocs for the (integer) numerals.
     8.9  *)
    8.10  
    8.11  (*To quote from Provers/Arith/cancel_numeral_factor.ML:
    8.12 @@ -24,6 +24,7 @@
    8.13    val field_combine_numerals: simproc
    8.14    val field_cancel_numeral_factors: simproc list
    8.15    val num_ss: simpset
    8.16 +  val field_comp_conv: conv
    8.17  end;
    8.18  
    8.19  structure Numeral_Simprocs : NUMERAL_SIMPROCS =
    8.20 @@ -602,6 +603,157 @@
    8.21        "(l::'a::field_inverse_zero) / (m * n)"],
    8.22       K DivideCancelFactor.proc)];
    8.23  
    8.24 +local
    8.25 + val zr = @{cpat "0"}
    8.26 + val zT = ctyp_of_term zr
    8.27 + val geq = @{cpat "op ="}
    8.28 + val eqT = Thm.dest_ctyp (ctyp_of_term geq) |> hd
    8.29 + val add_frac_eq = mk_meta_eq @{thm "add_frac_eq"}
    8.30 + val add_frac_num = mk_meta_eq @{thm "add_frac_num"}
    8.31 + val add_num_frac = mk_meta_eq @{thm "add_num_frac"}
    8.32 +
    8.33 + fun prove_nz ss T t =
    8.34 +    let
    8.35 +      val z = instantiate_cterm ([(zT,T)],[]) zr
    8.36 +      val eq = instantiate_cterm ([(eqT,T)],[]) geq
    8.37 +      val th = Simplifier.rewrite (ss addsimps @{thms simp_thms})
    8.38 +           (Thm.capply @{cterm "Trueprop"} (Thm.capply @{cterm "Not"}
    8.39 +                  (Thm.capply (Thm.capply eq t) z)))
    8.40 +    in equal_elim (symmetric th) TrueI
    8.41 +    end
    8.42 +
    8.43 + fun proc phi ss ct =
    8.44 +  let
    8.45 +    val ((x,y),(w,z)) =
    8.46 +         (Thm.dest_binop #> (fn (a,b) => (Thm.dest_binop a, Thm.dest_binop b))) ct
    8.47 +    val _ = map (HOLogic.dest_number o term_of) [x,y,z,w]
    8.48 +    val T = ctyp_of_term x
    8.49 +    val [y_nz, z_nz] = map (prove_nz ss T) [y, z]
    8.50 +    val th = instantiate' [SOME T] (map SOME [y,z,x,w]) add_frac_eq
    8.51 +  in SOME (implies_elim (implies_elim th y_nz) z_nz)
    8.52 +  end
    8.53 +  handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE
    8.54 +
    8.55 + fun proc2 phi ss ct =
    8.56 +  let
    8.57 +    val (l,r) = Thm.dest_binop ct
    8.58 +    val T = ctyp_of_term l
    8.59 +  in (case (term_of l, term_of r) of
    8.60 +      (Const(@{const_name Rings.divide},_)$_$_, _) =>
    8.61 +        let val (x,y) = Thm.dest_binop l val z = r
    8.62 +            val _ = map (HOLogic.dest_number o term_of) [x,y,z]
    8.63 +            val ynz = prove_nz ss T y
    8.64 +        in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,x,z]) add_frac_num) ynz)
    8.65 +        end
    8.66 +     | (_, Const (@{const_name Rings.divide},_)$_$_) =>
    8.67 +        let val (x,y) = Thm.dest_binop r val z = l
    8.68 +            val _ = map (HOLogic.dest_number o term_of) [x,y,z]
    8.69 +            val ynz = prove_nz ss T y
    8.70 +        in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,z,x]) add_num_frac) ynz)
    8.71 +        end
    8.72 +     | _ => NONE)
    8.73 +  end
    8.74 +  handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE
    8.75 +
    8.76 + fun is_number (Const(@{const_name Rings.divide},_)$a$b) = is_number a andalso is_number b
    8.77 +   | is_number t = can HOLogic.dest_number t
    8.78 +
    8.79 + val is_number = is_number o term_of
    8.80 +
    8.81 + fun proc3 phi ss ct =
    8.82 +  (case term_of ct of
    8.83 +    Const(@{const_name Orderings.less},_)$(Const(@{const_name Rings.divide},_)$_$_)$_ =>
    8.84 +      let
    8.85 +        val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
    8.86 +        val _ = map is_number [a,b,c]
    8.87 +        val T = ctyp_of_term c
    8.88 +        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_less_eq"}
    8.89 +      in SOME (mk_meta_eq th) end
    8.90 +  | Const(@{const_name Orderings.less_eq},_)$(Const(@{const_name Rings.divide},_)$_$_)$_ =>
    8.91 +      let
    8.92 +        val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
    8.93 +        val _ = map is_number [a,b,c]
    8.94 +        val T = ctyp_of_term c
    8.95 +        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_le_eq"}
    8.96 +      in SOME (mk_meta_eq th) end
    8.97 +  | Const("op =",_)$(Const(@{const_name Rings.divide},_)$_$_)$_ =>
    8.98 +      let
    8.99 +        val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
   8.100 +        val _ = map is_number [a,b,c]
   8.101 +        val T = ctyp_of_term c
   8.102 +        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_eq_eq"}
   8.103 +      in SOME (mk_meta_eq th) end
   8.104 +  | Const(@{const_name Orderings.less},_)$_$(Const(@{const_name Rings.divide},_)$_$_) =>
   8.105 +    let
   8.106 +      val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
   8.107 +        val _ = map is_number [a,b,c]
   8.108 +        val T = ctyp_of_term c
   8.109 +        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "less_divide_eq"}
   8.110 +      in SOME (mk_meta_eq th) end
   8.111 +  | Const(@{const_name Orderings.less_eq},_)$_$(Const(@{const_name Rings.divide},_)$_$_) =>
   8.112 +    let
   8.113 +      val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
   8.114 +        val _ = map is_number [a,b,c]
   8.115 +        val T = ctyp_of_term c
   8.116 +        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "le_divide_eq"}
   8.117 +      in SOME (mk_meta_eq th) end
   8.118 +  | Const("op =",_)$_$(Const(@{const_name Rings.divide},_)$_$_) =>
   8.119 +    let
   8.120 +      val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
   8.121 +        val _ = map is_number [a,b,c]
   8.122 +        val T = ctyp_of_term c
   8.123 +        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "eq_divide_eq"}
   8.124 +      in SOME (mk_meta_eq th) end
   8.125 +  | _ => NONE)
   8.126 +  handle TERM _ => NONE | CTERM _ => NONE | THM _ => NONE
   8.127 +
   8.128 +val add_frac_frac_simproc =
   8.129 +       make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + (?w::?'a::field)/?z"}],
   8.130 +                     name = "add_frac_frac_simproc",
   8.131 +                     proc = proc, identifier = []}
   8.132 +
   8.133 +val add_frac_num_simproc =
   8.134 +       make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + ?z"}, @{cpat "?z + (?x::?'a::field)/?y"}],
   8.135 +                     name = "add_frac_num_simproc",
   8.136 +                     proc = proc2, identifier = []}
   8.137 +
   8.138 +val ord_frac_simproc =
   8.139 +  make_simproc
   8.140 +    {lhss = [@{cpat "(?a::(?'a::{field, ord}))/?b < ?c"},
   8.141 +             @{cpat "(?a::(?'a::{field, ord}))/?b <= ?c"},
   8.142 +             @{cpat "?c < (?a::(?'a::{field, ord}))/?b"},
   8.143 +             @{cpat "?c <= (?a::(?'a::{field, ord}))/?b"},
   8.144 +             @{cpat "?c = ((?a::(?'a::{field, ord}))/?b)"},
   8.145 +             @{cpat "((?a::(?'a::{field, ord}))/ ?b) = ?c"}],
   8.146 +             name = "ord_frac_simproc", proc = proc3, identifier = []}
   8.147 +
   8.148 +val ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"},
   8.149 +           @{thm "divide_Numeral1"},
   8.150 +           @{thm "divide_zero"}, @{thm "divide_Numeral0"},
   8.151 +           @{thm "divide_divide_eq_left"}, 
   8.152 +           @{thm "times_divide_eq_left"}, @{thm "times_divide_eq_right"},
   8.153 +           @{thm "times_divide_times_eq"},
   8.154 +           @{thm "divide_divide_eq_right"},
   8.155 +           @{thm "diff_def"}, @{thm "minus_divide_left"},
   8.156 +           @{thm "Numeral1_eq1_nat"}, @{thm "add_divide_distrib"} RS sym,
   8.157 +           @{thm field_divide_inverse} RS sym, @{thm inverse_divide}, 
   8.158 +           Conv.fconv_rule (Conv.arg_conv (Conv.arg1_conv (Conv.rewr_conv (mk_meta_eq @{thm mult_commute}))))   
   8.159 +           (@{thm field_divide_inverse} RS sym)]
   8.160 +
   8.161 +in
   8.162 +
   8.163 +val field_comp_conv = (Simplifier.rewrite
   8.164 +(HOL_basic_ss addsimps @{thms "semiring_norm"}
   8.165 +              addsimps ths addsimps @{thms simp_thms}
   8.166 +              addsimprocs field_cancel_numeral_factors
   8.167 +               addsimprocs [add_frac_frac_simproc, add_frac_num_simproc,
   8.168 +                            ord_frac_simproc]
   8.169 +                addcongs [@{thm "if_weak_cong"}]))
   8.170 +then_conv (Simplifier.rewrite (HOL_basic_ss addsimps
   8.171 +  [@{thm numeral_1_eq_1},@{thm numeral_0_eq_0}] @ @{thms numerals(1-2)}))
   8.172 +
   8.173 +end
   8.174 +
   8.175  end;
   8.176  
   8.177  Addsimprocs Numeral_Simprocs.cancel_numerals;