src/HOL/Semiring_Normalization.thy
 changeset 36751 7f1da69cacb3 parent 36720 41da7025e59c child 36753 5cf4e9128f22
1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/src/HOL/Semiring_Normalization.thy	Fri May 07 15:05:52 2010 +0200
1.3 @@ -0,0 +1,336 @@
1.4 +(*  Title:      HOL/Semiring_Normalization.thy
1.5 +    Author:     Amine Chaieb, TU Muenchen
1.6 +*)
1.7 +
1.8 +header {* Semiring normalization *}
1.9 +
1.10 +theory Semiring_Normalization
1.11 +imports Numeral_Simprocs Nat_Transfer
1.12 +uses
1.13 +  "Tools/Groebner_Basis/normalizer.ML"
1.14 +begin
1.15 +
1.16 +setup Normalizer.setup
1.17 +
1.18 +locale normalizing_semiring =
1.19 +  fixes add mul pwr r0 r1
1.21 +    and add_c: "add x y = add y x" and add_0:"add r0 x = x"
1.22 +    and mul_a:"mul x (mul y z) = mul (mul x y) z" and mul_c:"mul x y = mul y x"
1.23 +    and mul_1:"mul r1 x = x" and  mul_0:"mul r0 x = r0"
1.24 +    and mul_d:"mul x (add y z) = add (mul x y) (mul x z)"
1.25 +    and pwr_0:"pwr x 0 = r1" and pwr_Suc:"pwr x (Suc n) = mul x (pwr x n)"
1.26 +begin
1.27 +
1.28 +lemma mul_pwr:"mul (pwr x p) (pwr x q) = pwr x (p + q)"
1.29 +proof (induct p)
1.30 +  case 0
1.31 +  then show ?case by (auto simp add: pwr_0 mul_1)
1.32 +next
1.33 +  case Suc
1.34 +  from this [symmetric] show ?case
1.35 +    by (auto simp add: pwr_Suc mul_1 mul_a)
1.36 +qed
1.37 +
1.38 +lemma pwr_mul: "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
1.39 +proof (induct q arbitrary: x y, auto simp add:pwr_0 pwr_Suc mul_1)
1.40 +  fix q x y
1.41 +  assume "\<And>x y. pwr (mul x y) q = mul (pwr x q) (pwr y q)"
1.42 +  have "mul (mul x y) (mul (pwr x q) (pwr y q)) = mul x (mul y (mul (pwr x q) (pwr y q)))"
1.43 +    by (simp add: mul_a)
1.44 +  also have "\<dots> = (mul (mul y (mul (pwr y q) (pwr x q))) x)" by (simp add: mul_c)
1.45 +  also have "\<dots> = (mul (mul y (pwr y q)) (mul (pwr x q) x))" by (simp add: mul_a)
1.46 +  finally show "mul (mul x y) (mul (pwr x q) (pwr y q)) =
1.47 +    mul (mul x (pwr x q)) (mul y (pwr y q))" by (simp add: mul_c)
1.48 +qed
1.49 +
1.50 +lemma pwr_pwr: "pwr (pwr x p) q = pwr x (p * q)"
1.51 +proof (induct p arbitrary: q)
1.52 +  case 0
1.53 +  show ?case using pwr_Suc mul_1 pwr_0 by (induct q) auto
1.54 +next
1.55 +  case Suc
1.56 +  thus ?case by (auto simp add: mul_pwr [symmetric] pwr_mul pwr_Suc)
1.57 +qed
1.58 +
1.59 +lemma semiring_ops:
1.60 +  shows "TERM (add x y)" and "TERM (mul x y)" and "TERM (pwr x n)"
1.61 +    and "TERM r0" and "TERM r1" .
1.62 +
1.63 +lemma semiring_rules:
1.64 +  "add (mul a m) (mul b m) = mul (add a b) m"
1.65 +  "add (mul a m) m = mul (add a r1) m"
1.66 +  "add m (mul a m) = mul (add a r1) m"
1.67 +  "add m m = mul (add r1 r1) m"
1.68 +  "add r0 a = a"
1.69 +  "add a r0 = a"
1.70 +  "mul a b = mul b a"
1.71 +  "mul (add a b) c = add (mul a c) (mul b c)"
1.72 +  "mul r0 a = r0"
1.73 +  "mul a r0 = r0"
1.74 +  "mul r1 a = a"
1.75 +  "mul a r1 = a"
1.76 +  "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
1.77 +  "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
1.78 +  "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
1.79 +  "mul (mul lx ly) rx = mul (mul lx rx) ly"
1.80 +  "mul (mul lx ly) rx = mul lx (mul ly rx)"
1.81 +  "mul lx (mul rx ry) = mul (mul lx rx) ry"
1.82 +  "mul lx (mul rx ry) = mul rx (mul lx ry)"
1.87 +  "add a c = add c a"
1.89 +  "mul (pwr x p) (pwr x q) = pwr x (p + q)"
1.90 +  "mul x (pwr x q) = pwr x (Suc q)"
1.91 +  "mul (pwr x q) x = pwr x (Suc q)"
1.92 +  "mul x x = pwr x 2"
1.93 +  "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
1.94 +  "pwr (pwr x p) q = pwr x (p * q)"
1.95 +  "pwr x 0 = r1"
1.96 +  "pwr x 1 = x"
1.97 +  "mul x (add y z) = add (mul x y) (mul x z)"
1.98 +  "pwr x (Suc q) = mul x (pwr x q)"
1.99 +  "pwr x (2*n) = mul (pwr x n) (pwr x n)"
1.100 +  "pwr x (Suc (2*n)) = mul x (mul (pwr x n) (pwr x n))"
1.101 +proof -
1.102 +  show "add (mul a m) (mul b m) = mul (add a b) m" using mul_d mul_c by simp
1.103 +next show"add (mul a m) m = mul (add a r1) m" using mul_d mul_c mul_1 by simp
1.104 +next show "add m (mul a m) = mul (add a r1) m" using mul_c mul_d mul_1 add_c by simp
1.105 +next show "add m m = mul (add r1 r1) m" using mul_c mul_d mul_1 by simp
1.106 +next show "add r0 a = a" using add_0 by simp
1.107 +next show "add a r0 = a" using add_0 add_c by simp
1.108 +next show "mul a b = mul b a" using mul_c by simp
1.109 +next show "mul (add a b) c = add (mul a c) (mul b c)" using mul_c mul_d by simp
1.110 +next show "mul r0 a = r0" using mul_0 by simp
1.111 +next show "mul a r0 = r0" using mul_0 mul_c by simp
1.112 +next show "mul r1 a = a" using mul_1 by simp
1.113 +next show "mul a r1 = a" using mul_1 mul_c by simp
1.114 +next show "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
1.115 +    using mul_c mul_a by simp
1.116 +next show "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
1.117 +    using mul_a by simp
1.118 +next
1.119 +  have "mul (mul lx ly) (mul rx ry) = mul (mul rx ry) (mul lx ly)" by (rule mul_c)
1.120 +  also have "\<dots> = mul rx (mul ry (mul lx ly))" using mul_a by simp
1.121 +  finally
1.122 +  show "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
1.123 +    using mul_c by simp
1.124 +next show "mul (mul lx ly) rx = mul (mul lx rx) ly" using mul_c mul_a by simp
1.125 +next
1.126 +  show "mul (mul lx ly) rx = mul lx (mul ly rx)" by (simp add: mul_a)
1.127 +next show "mul lx (mul rx ry) = mul (mul lx rx) ry" by (simp add: mul_a )
1.128 +next show "mul lx (mul rx ry) = mul rx (mul lx ry)" by (simp add: mul_a,simp add: mul_c)
1.130 +    using add_c add_a by simp
1.131 +next show "add (add a b) c = add a (add b c)" using add_a by simp
1.132 +next show "add a (add c d) = add c (add a d)"
1.133 +    apply (simp add: add_a) by (simp only: add_c)
1.135 +next show "add a c = add c a" by (rule add_c)
1.136 +next show "add a (add c d) = add (add a c) d" using add_a by simp
1.137 +next show "mul (pwr x p) (pwr x q) = pwr x (p + q)" by (rule mul_pwr)
1.138 +next show "mul x (pwr x q) = pwr x (Suc q)" using pwr_Suc by simp
1.139 +next show "mul (pwr x q) x = pwr x (Suc q)" using pwr_Suc mul_c by simp
1.140 +next show "mul x x = pwr x 2" by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c)
1.141 +next show "pwr (mul x y) q = mul (pwr x q) (pwr y q)" by (rule pwr_mul)
1.142 +next show "pwr (pwr x p) q = pwr x (p * q)" by (rule pwr_pwr)
1.143 +next show "pwr x 0 = r1" using pwr_0 .
1.144 +next show "pwr x 1 = x" unfolding One_nat_def by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c)
1.145 +next show "mul x (add y z) = add (mul x y) (mul x z)" using mul_d by simp
1.146 +next show "pwr x (Suc q) = mul x (pwr x q)" using pwr_Suc by simp
1.147 +next show "pwr x (2 * n) = mul (pwr x n) (pwr x n)" by (simp add: nat_number' mul_pwr)
1.148 +next show "pwr x (Suc (2 * n)) = mul x (mul (pwr x n) (pwr x n))"
1.149 +    by (simp add: nat_number' pwr_Suc mul_pwr)
1.150 +qed
1.153 +lemmas normalizing_semiring_axioms' =
1.154 +  normalizing_semiring_axioms [normalizer
1.155 +    semiring ops: semiring_ops
1.156 +    semiring rules: semiring_rules]
1.158 +end
1.160 +sublocale comm_semiring_1
1.161 +  < normalizing!: normalizing_semiring plus times power zero one
1.162 +proof
1.163 +qed (simp_all add: algebra_simps)
1.165 +declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_semiring_axioms'} *}
1.167 +locale normalizing_ring = normalizing_semiring +
1.168 +  fixes sub :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
1.169 +    and neg :: "'a \<Rightarrow> 'a"
1.170 +  assumes neg_mul: "neg x = mul (neg r1) x"
1.171 +    and sub_add: "sub x y = add x (neg y)"
1.172 +begin
1.174 +lemma ring_ops: shows "TERM (sub x y)" and "TERM (neg x)" .
1.176 +lemmas ring_rules = neg_mul sub_add
1.178 +lemmas normalizing_ring_axioms' =
1.179 +  normalizing_ring_axioms [normalizer
1.180 +    semiring ops: semiring_ops
1.181 +    semiring rules: semiring_rules
1.182 +    ring ops: ring_ops
1.183 +    ring rules: ring_rules]
1.185 +end
1.187 +sublocale comm_ring_1
1.188 +  < normalizing!: normalizing_ring plus times power zero one minus uminus
1.189 +proof
1.190 +qed (simp_all add: diff_minus)
1.192 +declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_ring_axioms'} *}
1.194 +locale normalizing_field = normalizing_ring +
1.195 +  fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
1.196 +    and inverse:: "'a \<Rightarrow> 'a"
1.197 +  assumes divide_inverse: "divide x y = mul x (inverse y)"
1.198 +     and inverse_divide: "inverse x = divide r1 x"
1.199 +begin
1.201 +lemma field_ops: shows "TERM (divide x y)" and "TERM (inverse x)" .
1.203 +lemmas field_rules = divide_inverse inverse_divide
1.205 +lemmas normalizing_field_axioms' =
1.206 +  normalizing_field_axioms [normalizer
1.207 +    semiring ops: semiring_ops
1.208 +    semiring rules: semiring_rules
1.209 +    ring ops: ring_ops
1.210 +    ring rules: ring_rules
1.211 +    field ops: field_ops
1.212 +    field rules: field_rules]
1.214 +end
1.216 +locale normalizing_semiring_cancel = normalizing_semiring +
1.217 +  assumes add_cancel: "add (x::'a) y = add x z \<longleftrightarrow> y = z"
1.218 +  and add_mul_solve: "add (mul w y) (mul x z) =
1.219 +    add (mul w z) (mul x y) \<longleftrightarrow> w = x \<or> y = z"
1.220 +begin
1.222 +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)"
1.223 +proof-
1.224 +  have "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> \<not> (a = b \<or> c = d)" by simp
1.225 +  also have "\<dots> \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
1.226 +    using add_mul_solve by blast
1.227 +  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)"
1.228 +    by simp
1.229 +qed
1.231 +lemma add_scale_eq_noteq: "\<lbrakk>r \<noteq> r0 ; (a = b) \<and> ~(c = d)\<rbrakk>
1.232 +  \<Longrightarrow> add a (mul r c) \<noteq> add b (mul r d)"
1.233 +proof(clarify)
1.234 +  assume nz: "r\<noteq> r0" and cnd: "c\<noteq>d"
1.235 +    and eq: "add b (mul r c) = add b (mul r d)"
1.236 +  hence "mul r c = mul r d" using cnd add_cancel by simp
1.237 +  hence "add (mul r0 d) (mul r c) = add (mul r0 c) (mul r d)"
1.238 +    using mul_0 add_cancel by simp
1.239 +  thus "False" using add_mul_solve nz cnd by simp
1.240 +qed
1.242 +lemma add_r0_iff: " x = add x a \<longleftrightarrow> a = r0"
1.243 +proof-
1.244 +  have "a = r0 \<longleftrightarrow> add x a = add x r0" by (simp add: add_cancel)
1.245 +  thus "x = add x a \<longleftrightarrow> a = r0" by (auto simp add: add_c add_0)
1.246 +qed
1.248 +declare normalizing_semiring_axioms' [normalizer del]
1.250 +lemmas normalizing_semiring_cancel_axioms' =
1.251 +  normalizing_semiring_cancel_axioms [normalizer
1.252 +    semiring ops: semiring_ops
1.253 +    semiring rules: semiring_rules
1.254 +    idom rules: noteq_reduce add_scale_eq_noteq]
1.256 +end
1.258 +locale normalizing_ring_cancel = normalizing_semiring_cancel + normalizing_ring +
1.259 +  assumes subr0_iff: "sub x y = r0 \<longleftrightarrow> x = y"
1.260 +begin
1.262 +declare normalizing_ring_axioms' [normalizer del]
1.264 +lemmas normalizing_ring_cancel_axioms' = normalizing_ring_cancel_axioms [normalizer
1.265 +  semiring ops: semiring_ops
1.266 +  semiring rules: semiring_rules
1.267 +  ring ops: ring_ops
1.268 +  ring rules: ring_rules
1.269 +  idom rules: noteq_reduce add_scale_eq_noteq
1.270 +  ideal rules: subr0_iff add_r0_iff]
1.272 +end
1.274 +sublocale idom
1.275 +  < normalizing!: normalizing_ring_cancel plus times power zero one minus uminus
1.276 +proof
1.277 +  fix w x y z
1.278 +  show "w * y + x * z = w * z + x * y \<longleftrightarrow> w = x \<or> y = z"
1.279 +  proof
1.280 +    assume "w * y + x * z = w * z + x * y"
1.281 +    then have "w * y + x * z - w * z - x * y = 0" by (simp add: algebra_simps)
1.282 +    then have "w * (y - z) - x * (y - z) = 0" by (simp add: algebra_simps)
1.283 +    then have "(y - z) * (w - x) = 0" by (simp add: algebra_simps)
1.284 +    then have "y - z = 0 \<or> w - x = 0" by (rule divisors_zero)
1.285 +    then show "w = x \<or> y = z" by auto
1.286 +  qed (auto simp add: add_ac)
1.287 +qed (simp_all add: algebra_simps)
1.289 +declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_ring_cancel_axioms'} *}
1.291 +interpretation normalizing_nat!: normalizing_semiring_cancel
1.292 +  "op +" "op *" "op ^" "0::nat" "1"
1.293 +proof (unfold_locales, simp add: algebra_simps)
1.294 +  fix w x y z ::"nat"
1.295 +  { assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
1.296 +    hence "y < z \<or> y > z" by arith
1.297 +    moreover {
1.298 +      assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z - y" in exI, auto)
1.299 +      then obtain k where kp: "k>0" and yz:"z = y + k" by blast
1.300 +      from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz algebra_simps)
1.301 +      hence "x*k = w*k" by simp
1.302 +      hence "w = x" using kp by simp }
1.303 +    moreover {
1.304 +      assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y - z" in exI, auto)
1.305 +      then obtain k where kp: "k>0" and yz:"y = z + k" by blast
1.306 +      from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz algebra_simps)
1.307 +      hence "w*k = x*k" by simp
1.308 +      hence "w = x" using kp by simp }
1.309 +    ultimately have "w=x" by blast }
1.310 +  thus "(w * y + x * z = w * z + x * y) = (w = x \<or> y = z)" by auto
1.311 +qed
1.313 +declaration {* Normalizer.semiring_funs @{thm normalizing_nat.normalizing_semiring_cancel_axioms'} *}
1.315 +locale normalizing_field_cancel = normalizing_ring_cancel + normalizing_field
1.316 +begin
1.318 +declare normalizing_field_axioms' [normalizer del]
1.320 +lemmas normalizing_field_cancel_axioms' = normalizing_field_cancel_axioms [normalizer
1.321 +  semiring ops: semiring_ops
1.322 +  semiring rules: semiring_rules
1.323 +  ring ops: ring_ops
1.324 +  ring rules: ring_rules
1.325 +  field ops: field_ops
1.326 +  field rules: field_rules
1.327 +  idom rules: noteq_reduce add_scale_eq_noteq
1.328 +  ideal rules: subr0_iff add_r0_iff]
1.330 +end
1.332 +sublocale field
1.333 +  < normalizing!: normalizing_field_cancel plus times power zero one minus uminus divide inverse
1.334 +proof
1.335 +qed (simp_all add: divide_inverse)
1.337 +declaration {* Normalizer.field_funs @{thm normalizing.normalizing_field_cancel_axioms'} *}
1.339 +end