src/HOL/Semiring_Normalization.thy
 author haftmann Fri May 07 16:12:26 2010 +0200 (2010-05-07) changeset 36753 5cf4e9128f22 parent 36751 7f1da69cacb3 child 36756 c1ae8a0b4265 permissions -rw-r--r--
renamed Normalizer to the more specific Semiring_Normalizer
```     1 (*  Title:      HOL/Semiring_Normalization.thy
```
```     2     Author:     Amine Chaieb, TU Muenchen
```
```     3 *)
```
```     4
```
```     5 header {* Semiring normalization *}
```
```     6
```
```     7 theory Semiring_Normalization
```
```     8 imports Numeral_Simprocs Nat_Transfer
```
```     9 uses
```
```    10   "Tools/semiring_normalizer.ML"
```
```    11 begin
```
```    12
```
```    13 setup Semiring_Normalizer.setup
```
```    14
```
```    15 locale normalizing_semiring =
```
```    16   fixes add mul pwr r0 r1
```
```    17   assumes add_a:"(add x (add y z) = add (add x y) z)"
```
```    18     and add_c: "add x y = add y x" and add_0:"add r0 x = x"
```
```    19     and mul_a:"mul x (mul y z) = mul (mul x y) z" and mul_c:"mul x y = mul y x"
```
```    20     and mul_1:"mul r1 x = x" and  mul_0:"mul r0 x = r0"
```
```    21     and mul_d:"mul x (add y z) = add (mul x y) (mul x z)"
```
```    22     and pwr_0:"pwr x 0 = r1" and pwr_Suc:"pwr x (Suc n) = mul x (pwr x n)"
```
```    23 begin
```
```    24
```
```    25 lemma mul_pwr:"mul (pwr x p) (pwr x q) = pwr x (p + q)"
```
```    26 proof (induct p)
```
```    27   case 0
```
```    28   then show ?case by (auto simp add: pwr_0 mul_1)
```
```    29 next
```
```    30   case Suc
```
```    31   from this [symmetric] show ?case
```
```    32     by (auto simp add: pwr_Suc mul_1 mul_a)
```
```    33 qed
```
```    34
```
```    35 lemma pwr_mul: "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
```
```    36 proof (induct q arbitrary: x y, auto simp add:pwr_0 pwr_Suc mul_1)
```
```    37   fix q x y
```
```    38   assume "\<And>x y. pwr (mul x y) q = mul (pwr x q) (pwr y q)"
```
```    39   have "mul (mul x y) (mul (pwr x q) (pwr y q)) = mul x (mul y (mul (pwr x q) (pwr y q)))"
```
```    40     by (simp add: mul_a)
```
```    41   also have "\<dots> = (mul (mul y (mul (pwr y q) (pwr x q))) x)" by (simp add: mul_c)
```
```    42   also have "\<dots> = (mul (mul y (pwr y q)) (mul (pwr x q) x))" by (simp add: mul_a)
```
```    43   finally show "mul (mul x y) (mul (pwr x q) (pwr y q)) =
```
```    44     mul (mul x (pwr x q)) (mul y (pwr y q))" by (simp add: mul_c)
```
```    45 qed
```
```    46
```
```    47 lemma pwr_pwr: "pwr (pwr x p) q = pwr x (p * q)"
```
```    48 proof (induct p arbitrary: q)
```
```    49   case 0
```
```    50   show ?case using pwr_Suc mul_1 pwr_0 by (induct q) auto
```
```    51 next
```
```    52   case Suc
```
```    53   thus ?case by (auto simp add: mul_pwr [symmetric] pwr_mul pwr_Suc)
```
```    54 qed
```
```    55
```
```    56 lemma semiring_ops:
```
```    57   shows "TERM (add x y)" and "TERM (mul x y)" and "TERM (pwr x n)"
```
```    58     and "TERM r0" and "TERM r1" .
```
```    59
```
```    60 lemma semiring_rules:
```
```    61   "add (mul a m) (mul b m) = mul (add a b) m"
```
```    62   "add (mul a m) m = mul (add a r1) m"
```
```    63   "add m (mul a m) = mul (add a r1) m"
```
```    64   "add m m = mul (add r1 r1) m"
```
```    65   "add r0 a = a"
```
```    66   "add a r0 = a"
```
```    67   "mul a b = mul b a"
```
```    68   "mul (add a b) c = add (mul a c) (mul b c)"
```
```    69   "mul r0 a = r0"
```
```    70   "mul a r0 = r0"
```
```    71   "mul r1 a = a"
```
```    72   "mul a r1 = a"
```
```    73   "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
```
```    74   "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
```
```    75   "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
```
```    76   "mul (mul lx ly) rx = mul (mul lx rx) ly"
```
```    77   "mul (mul lx ly) rx = mul lx (mul ly rx)"
```
```    78   "mul lx (mul rx ry) = mul (mul lx rx) ry"
```
```    79   "mul lx (mul rx ry) = mul rx (mul lx ry)"
```
```    80   "add (add a b) (add c d) = add (add a c) (add b d)"
```
```    81   "add (add a b) c = add a (add b c)"
```
```    82   "add a (add c d) = add c (add a d)"
```
```    83   "add (add a b) c = add (add a c) b"
```
```    84   "add a c = add c a"
```
```    85   "add a (add c d) = add (add a c) d"
```
```    86   "mul (pwr x p) (pwr x q) = pwr x (p + q)"
```
```    87   "mul x (pwr x q) = pwr x (Suc q)"
```
```    88   "mul (pwr x q) x = pwr x (Suc q)"
```
```    89   "mul x x = pwr x 2"
```
```    90   "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
```
```    91   "pwr (pwr x p) q = pwr x (p * q)"
```
```    92   "pwr x 0 = r1"
```
```    93   "pwr x 1 = x"
```
```    94   "mul x (add y z) = add (mul x y) (mul x z)"
```
```    95   "pwr x (Suc q) = mul x (pwr x q)"
```
```    96   "pwr x (2*n) = mul (pwr x n) (pwr x n)"
```
```    97   "pwr x (Suc (2*n)) = mul x (mul (pwr x n) (pwr x n))"
```
```    98 proof -
```
```    99   show "add (mul a m) (mul b m) = mul (add a b) m" using mul_d mul_c by simp
```
```   100 next show"add (mul a m) m = mul (add a r1) m" using mul_d mul_c mul_1 by simp
```
```   101 next show "add m (mul a m) = mul (add a r1) m" using mul_c mul_d mul_1 add_c by simp
```
```   102 next show "add m m = mul (add r1 r1) m" using mul_c mul_d mul_1 by simp
```
```   103 next show "add r0 a = a" using add_0 by simp
```
```   104 next show "add a r0 = a" using add_0 add_c by simp
```
```   105 next show "mul a b = mul b a" using mul_c by simp
```
```   106 next show "mul (add a b) c = add (mul a c) (mul b c)" using mul_c mul_d by simp
```
```   107 next show "mul r0 a = r0" using mul_0 by simp
```
```   108 next show "mul a r0 = r0" using mul_0 mul_c by simp
```
```   109 next show "mul r1 a = a" using mul_1 by simp
```
```   110 next show "mul a r1 = a" using mul_1 mul_c by simp
```
```   111 next show "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
```
```   112     using mul_c mul_a by simp
```
```   113 next show "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
```
```   114     using mul_a by simp
```
```   115 next
```
```   116   have "mul (mul lx ly) (mul rx ry) = mul (mul rx ry) (mul lx ly)" by (rule mul_c)
```
```   117   also have "\<dots> = mul rx (mul ry (mul lx ly))" using mul_a by simp
```
```   118   finally
```
```   119   show "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
```
```   120     using mul_c by simp
```
```   121 next show "mul (mul lx ly) rx = mul (mul lx rx) ly" using mul_c mul_a by simp
```
```   122 next
```
```   123   show "mul (mul lx ly) rx = mul lx (mul ly rx)" by (simp add: mul_a)
```
```   124 next show "mul lx (mul rx ry) = mul (mul lx rx) ry" by (simp add: mul_a )
```
```   125 next show "mul lx (mul rx ry) = mul rx (mul lx ry)" by (simp add: mul_a,simp add: mul_c)
```
```   126 next show "add (add a b) (add c d) = add (add a c) (add b d)"
```
```   127     using add_c add_a by simp
```
```   128 next show "add (add a b) c = add a (add b c)" using add_a by simp
```
```   129 next show "add a (add c d) = add c (add a d)"
```
```   130     apply (simp add: add_a) by (simp only: add_c)
```
```   131 next show "add (add a b) c = add (add a c) b" using add_a add_c by simp
```
```   132 next show "add a c = add c a" by (rule add_c)
```
```   133 next show "add a (add c d) = add (add a c) d" using add_a by simp
```
```   134 next show "mul (pwr x p) (pwr x q) = pwr x (p + q)" by (rule mul_pwr)
```
```   135 next show "mul x (pwr x q) = pwr x (Suc q)" using pwr_Suc by simp
```
```   136 next show "mul (pwr x q) x = pwr x (Suc q)" using pwr_Suc mul_c by simp
```
```   137 next show "mul x x = pwr x 2" by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c)
```
```   138 next show "pwr (mul x y) q = mul (pwr x q) (pwr y q)" by (rule pwr_mul)
```
```   139 next show "pwr (pwr x p) q = pwr x (p * q)" by (rule pwr_pwr)
```
```   140 next show "pwr x 0 = r1" using pwr_0 .
```
```   141 next show "pwr x 1 = x" unfolding One_nat_def by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c)
```
```   142 next show "mul x (add y z) = add (mul x y) (mul x z)" using mul_d by simp
```
```   143 next show "pwr x (Suc q) = mul x (pwr x q)" using pwr_Suc by simp
```
```   144 next show "pwr x (2 * n) = mul (pwr x n) (pwr x n)" by (simp add: nat_number' mul_pwr)
```
```   145 next show "pwr x (Suc (2 * n)) = mul x (mul (pwr x n) (pwr x n))"
```
```   146     by (simp add: nat_number' pwr_Suc mul_pwr)
```
```   147 qed
```
```   148
```
```   149
```
```   150 lemmas normalizing_semiring_axioms' =
```
```   151   normalizing_semiring_axioms [normalizer
```
```   152     semiring ops: semiring_ops
```
```   153     semiring rules: semiring_rules]
```
```   154
```
```   155 end
```
```   156
```
```   157 sublocale comm_semiring_1
```
```   158   < normalizing!: normalizing_semiring plus times power zero one
```
```   159 proof
```
```   160 qed (simp_all add: algebra_simps)
```
```   161
```
```   162 declaration {* Semiring_Normalizer.semiring_funs @{thm normalizing.normalizing_semiring_axioms'} *}
```
```   163
```
```   164 locale normalizing_ring = normalizing_semiring +
```
```   165   fixes sub :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
```
```   166     and neg :: "'a \<Rightarrow> 'a"
```
```   167   assumes neg_mul: "neg x = mul (neg r1) x"
```
```   168     and sub_add: "sub x y = add x (neg y)"
```
```   169 begin
```
```   170
```
```   171 lemma ring_ops: shows "TERM (sub x y)" and "TERM (neg x)" .
```
```   172
```
```   173 lemmas ring_rules = neg_mul sub_add
```
```   174
```
```   175 lemmas normalizing_ring_axioms' =
```
```   176   normalizing_ring_axioms [normalizer
```
```   177     semiring ops: semiring_ops
```
```   178     semiring rules: semiring_rules
```
```   179     ring ops: ring_ops
```
```   180     ring rules: ring_rules]
```
```   181
```
```   182 end
```
```   183
```
```   184 sublocale comm_ring_1
```
```   185   < normalizing!: normalizing_ring plus times power zero one minus uminus
```
```   186 proof
```
```   187 qed (simp_all add: diff_minus)
```
```   188
```
```   189 declaration {* Semiring_Normalizer.semiring_funs @{thm normalizing.normalizing_ring_axioms'} *}
```
```   190
```
```   191 locale normalizing_field = normalizing_ring +
```
```   192   fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
```
```   193     and inverse:: "'a \<Rightarrow> 'a"
```
```   194   assumes divide_inverse: "divide x y = mul x (inverse y)"
```
```   195      and inverse_divide: "inverse x = divide r1 x"
```
```   196 begin
```
```   197
```
```   198 lemma field_ops: shows "TERM (divide x y)" and "TERM (inverse x)" .
```
```   199
```
```   200 lemmas field_rules = divide_inverse inverse_divide
```
```   201
```
```   202 lemmas normalizing_field_axioms' =
```
```   203   normalizing_field_axioms [normalizer
```
```   204     semiring ops: semiring_ops
```
```   205     semiring rules: semiring_rules
```
```   206     ring ops: ring_ops
```
```   207     ring rules: ring_rules
```
```   208     field ops: field_ops
```
```   209     field rules: field_rules]
```
```   210
```
```   211 end
```
```   212
```
```   213 locale normalizing_semiring_cancel = normalizing_semiring +
```
```   214   assumes add_cancel: "add (x::'a) y = add x z \<longleftrightarrow> y = z"
```
```   215   and add_mul_solve: "add (mul w y) (mul x z) =
```
```   216     add (mul w z) (mul x y) \<longleftrightarrow> w = x \<or> y = z"
```
```   217 begin
```
```   218
```
```   219 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)"
```
```   220 proof-
```
```   221   have "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> \<not> (a = b \<or> c = d)" by simp
```
```   222   also have "\<dots> \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
```
```   223     using add_mul_solve by blast
```
```   224   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)"
```
```   225     by simp
```
```   226 qed
```
```   227
```
```   228 lemma add_scale_eq_noteq: "\<lbrakk>r \<noteq> r0 ; (a = b) \<and> ~(c = d)\<rbrakk>
```
```   229   \<Longrightarrow> add a (mul r c) \<noteq> add b (mul r d)"
```
```   230 proof(clarify)
```
```   231   assume nz: "r\<noteq> r0" and cnd: "c\<noteq>d"
```
```   232     and eq: "add b (mul r c) = add b (mul r d)"
```
```   233   hence "mul r c = mul r d" using cnd add_cancel by simp
```
```   234   hence "add (mul r0 d) (mul r c) = add (mul r0 c) (mul r d)"
```
```   235     using mul_0 add_cancel by simp
```
```   236   thus "False" using add_mul_solve nz cnd by simp
```
```   237 qed
```
```   238
```
```   239 lemma add_r0_iff: " x = add x a \<longleftrightarrow> a = r0"
```
```   240 proof-
```
```   241   have "a = r0 \<longleftrightarrow> add x a = add x r0" by (simp add: add_cancel)
```
```   242   thus "x = add x a \<longleftrightarrow> a = r0" by (auto simp add: add_c add_0)
```
```   243 qed
```
```   244
```
```   245 declare normalizing_semiring_axioms' [normalizer del]
```
```   246
```
```   247 lemmas normalizing_semiring_cancel_axioms' =
```
```   248   normalizing_semiring_cancel_axioms [normalizer
```
```   249     semiring ops: semiring_ops
```
```   250     semiring rules: semiring_rules
```
```   251     idom rules: noteq_reduce add_scale_eq_noteq]
```
```   252
```
```   253 end
```
```   254
```
```   255 locale normalizing_ring_cancel = normalizing_semiring_cancel + normalizing_ring +
```
```   256   assumes subr0_iff: "sub x y = r0 \<longleftrightarrow> x = y"
```
```   257 begin
```
```   258
```
```   259 declare normalizing_ring_axioms' [normalizer del]
```
```   260
```
```   261 lemmas normalizing_ring_cancel_axioms' = normalizing_ring_cancel_axioms [normalizer
```
```   262   semiring ops: semiring_ops
```
```   263   semiring rules: semiring_rules
```
```   264   ring ops: ring_ops
```
```   265   ring rules: ring_rules
```
```   266   idom rules: noteq_reduce add_scale_eq_noteq
```
```   267   ideal rules: subr0_iff add_r0_iff]
```
```   268
```
```   269 end
```
```   270
```
```   271 sublocale idom
```
```   272   < normalizing!: normalizing_ring_cancel plus times power zero one minus uminus
```
```   273 proof
```
```   274   fix w x y z
```
```   275   show "w * y + x * z = w * z + x * y \<longleftrightarrow> w = x \<or> y = z"
```
```   276   proof
```
```   277     assume "w * y + x * z = w * z + x * y"
```
```   278     then have "w * y + x * z - w * z - x * y = 0" by (simp add: algebra_simps)
```
```   279     then have "w * (y - z) - x * (y - z) = 0" by (simp add: algebra_simps)
```
```   280     then have "(y - z) * (w - x) = 0" by (simp add: algebra_simps)
```
```   281     then have "y - z = 0 \<or> w - x = 0" by (rule divisors_zero)
```
```   282     then show "w = x \<or> y = z" by auto
```
```   283   qed (auto simp add: add_ac)
```
```   284 qed (simp_all add: algebra_simps)
```
```   285
```
```   286 declaration {* Semiring_Normalizer.semiring_funs @{thm normalizing.normalizing_ring_cancel_axioms'} *}
```
```   287
```
```   288 interpretation normalizing_nat!: normalizing_semiring_cancel
```
```   289   "op +" "op *" "op ^" "0::nat" "1"
```
```   290 proof (unfold_locales, simp add: algebra_simps)
```
```   291   fix w x y z ::"nat"
```
```   292   { assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
```
```   293     hence "y < z \<or> y > z" by arith
```
```   294     moreover {
```
```   295       assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z - y" in exI, auto)
```
```   296       then obtain k where kp: "k>0" and yz:"z = y + k" by blast
```
```   297       from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz algebra_simps)
```
```   298       hence "x*k = w*k" by simp
```
```   299       hence "w = x" using kp by simp }
```
```   300     moreover {
```
```   301       assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y - z" in exI, auto)
```
```   302       then obtain k where kp: "k>0" and yz:"y = z + k" by blast
```
```   303       from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz algebra_simps)
```
```   304       hence "w*k = x*k" by simp
```
```   305       hence "w = x" using kp by simp }
```
```   306     ultimately have "w=x" by blast }
```
```   307   thus "(w * y + x * z = w * z + x * y) = (w = x \<or> y = z)" by auto
```
```   308 qed
```
```   309
```
```   310 declaration {* Semiring_Normalizer.semiring_funs @{thm normalizing_nat.normalizing_semiring_cancel_axioms'} *}
```
```   311
```
```   312 locale normalizing_field_cancel = normalizing_ring_cancel + normalizing_field
```
```   313 begin
```
```   314
```
```   315 declare normalizing_field_axioms' [normalizer del]
```
```   316
```
```   317 lemmas normalizing_field_cancel_axioms' = normalizing_field_cancel_axioms [normalizer
```
```   318   semiring ops: semiring_ops
```
```   319   semiring rules: semiring_rules
```
```   320   ring ops: ring_ops
```
```   321   ring rules: ring_rules
```
```   322   field ops: field_ops
```
```   323   field rules: field_rules
```
```   324   idom rules: noteq_reduce add_scale_eq_noteq
```
```   325   ideal rules: subr0_iff add_r0_iff]
```
```   326
```
```   327 end
```
```   328
```
```   329 sublocale field
```
```   330   < normalizing!: normalizing_field_cancel plus times power zero one minus uminus divide inverse
```
```   331 proof
```
```   332 qed (simp_all add: divide_inverse)
```
```   333
```
```   334 declaration {* Semiring_Normalizer.field_funs @{thm normalizing.normalizing_field_cancel_axioms'} *}
```
```   335
```
```   336 end
```