src/HOL/Semiring_Normalization.thy
 author haftmann Sat May 08 18:52:38 2010 +0200 (2010-05-08) changeset 36756 c1ae8a0b4265 parent 36753 5cf4e9128f22 child 36845 d778c64fc35d permissions -rw-r--r--
moved normalization proof tool infrastructure to canonical algebraic classes
```     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 text {* FIXME prelude *}
```
```    14
```
```    15 class comm_semiring_1_cancel_norm (*FIXME name*) = comm_semiring_1_cancel +
```
```    16   assumes add_mult_solve: "w * y + x * z = w * z + x * y \<longleftrightarrow> w = x \<or> y = z"
```
```    17
```
```    18 sublocale idom < comm_semiring_1_cancel_norm
```
```    19 proof
```
```    20   fix w x y z
```
```    21   show "w * y + x * z = w * z + x * y \<longleftrightarrow> w = x \<or> y = z"
```
```    22   proof
```
```    23     assume "w * y + x * z = w * z + x * y"
```
```    24     then have "w * y + x * z - w * z - x * y = 0" by (simp add: algebra_simps)
```
```    25     then have "w * (y - z) - x * (y - z) = 0" by (simp add: algebra_simps)
```
```    26     then have "(y - z) * (w - x) = 0" by (simp add: algebra_simps)
```
```    27     then have "y - z = 0 \<or> w - x = 0" by (rule divisors_zero)
```
```    28     then show "w = x \<or> y = z" by auto
```
```    29   qed (auto simp add: add_ac)
```
```    30 qed
```
```    31
```
```    32 instance nat :: comm_semiring_1_cancel_norm
```
```    33 proof
```
```    34   fix w x y z :: nat
```
```    35   { assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
```
```    36     hence "y < z \<or> y > z" by arith
```
```    37     moreover {
```
```    38       assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z - y" in exI, auto)
```
```    39       then obtain k where kp: "k>0" and yz:"z = y + k" by blast
```
```    40       from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz algebra_simps)
```
```    41       hence "x*k = w*k" by simp
```
```    42       hence "w = x" using kp by simp }
```
```    43     moreover {
```
```    44       assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y - z" in exI, auto)
```
```    45       then obtain k where kp: "k>0" and yz:"y = z + k" by blast
```
```    46       from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz algebra_simps)
```
```    47       hence "w*k = x*k" by simp
```
```    48       hence "w = x" using kp by simp }
```
```    49     ultimately have "w=x" by blast }
```
```    50   then show "w * y + x * z = w * z + x * y \<longleftrightarrow> w = x \<or> y = z" by auto
```
```    51 qed
```
```    52
```
```    53 setup Semiring_Normalizer.setup
```
```    54
```
```    55 locale normalizing_semiring =
```
```    56   fixes add mul pwr r0 r1
```
```    57   assumes add_a:"(add x (add y z) = add (add x y) z)"
```
```    58     and add_c: "add x y = add y x" and add_0:"add r0 x = x"
```
```    59     and mul_a:"mul x (mul y z) = mul (mul x y) z" and mul_c:"mul x y = mul y x"
```
```    60     and mul_1:"mul r1 x = x" and  mul_0:"mul r0 x = r0"
```
```    61     and mul_d:"mul x (add y z) = add (mul x y) (mul x z)"
```
```    62     and pwr_0:"pwr x 0 = r1" and pwr_Suc:"pwr x (Suc n) = mul x (pwr x n)"
```
```    63 begin
```
```    64
```
```    65 lemma mul_pwr:"mul (pwr x p) (pwr x q) = pwr x (p + q)"
```
```    66 proof (induct p)
```
```    67   case 0
```
```    68   then show ?case by (auto simp add: pwr_0 mul_1)
```
```    69 next
```
```    70   case Suc
```
```    71   from this [symmetric] show ?case
```
```    72     by (auto simp add: pwr_Suc mul_1 mul_a)
```
```    73 qed
```
```    74
```
```    75 lemma pwr_mul: "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
```
```    76 proof (induct q arbitrary: x y, auto simp add:pwr_0 pwr_Suc mul_1)
```
```    77   fix q x y
```
```    78   assume "\<And>x y. pwr (mul x y) q = mul (pwr x q) (pwr y q)"
```
```    79   have "mul (mul x y) (mul (pwr x q) (pwr y q)) = mul x (mul y (mul (pwr x q) (pwr y q)))"
```
```    80     by (simp add: mul_a)
```
```    81   also have "\<dots> = (mul (mul y (mul (pwr y q) (pwr x q))) x)" by (simp add: mul_c)
```
```    82   also have "\<dots> = (mul (mul y (pwr y q)) (mul (pwr x q) x))" by (simp add: mul_a)
```
```    83   finally show "mul (mul x y) (mul (pwr x q) (pwr y q)) =
```
```    84     mul (mul x (pwr x q)) (mul y (pwr y q))" by (simp add: mul_c)
```
```    85 qed
```
```    86
```
```    87 lemma pwr_pwr: "pwr (pwr x p) q = pwr x (p * q)"
```
```    88 proof (induct p arbitrary: q)
```
```    89   case 0
```
```    90   show ?case using pwr_Suc mul_1 pwr_0 by (induct q) auto
```
```    91 next
```
```    92   case Suc
```
```    93   thus ?case by (auto simp add: mul_pwr [symmetric] pwr_mul pwr_Suc)
```
```    94 qed
```
```    95
```
```    96 lemma semiring_ops:
```
```    97   shows "TERM (add x y)" and "TERM (mul x y)" and "TERM (pwr x n)"
```
```    98     and "TERM r0" and "TERM r1" .
```
```    99
```
```   100 lemma semiring_rules:
```
```   101   "add (mul a m) (mul b m) = mul (add a b) m"
```
```   102   "add (mul a m) m = mul (add a r1) m"
```
```   103   "add m (mul a m) = mul (add a r1) m"
```
```   104   "add m m = mul (add r1 r1) m"
```
```   105   "add r0 a = a"
```
```   106   "add a r0 = a"
```
```   107   "mul a b = mul b a"
```
```   108   "mul (add a b) c = add (mul a c) (mul b c)"
```
```   109   "mul r0 a = r0"
```
```   110   "mul a r0 = r0"
```
```   111   "mul r1 a = a"
```
```   112   "mul a r1 = a"
```
```   113   "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
```
```   114   "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
```
```   115   "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
```
```   116   "mul (mul lx ly) rx = mul (mul lx rx) ly"
```
```   117   "mul (mul lx ly) rx = mul lx (mul ly rx)"
```
```   118   "mul lx (mul rx ry) = mul (mul lx rx) ry"
```
```   119   "mul lx (mul rx ry) = mul rx (mul lx ry)"
```
```   120   "add (add a b) (add c d) = add (add a c) (add b d)"
```
```   121   "add (add a b) c = add a (add b c)"
```
```   122   "add a (add c d) = add c (add a d)"
```
```   123   "add (add a b) c = add (add a c) b"
```
```   124   "add a c = add c a"
```
```   125   "add a (add c d) = add (add a c) d"
```
```   126   "mul (pwr x p) (pwr x q) = pwr x (p + q)"
```
```   127   "mul x (pwr x q) = pwr x (Suc q)"
```
```   128   "mul (pwr x q) x = pwr x (Suc q)"
```
```   129   "mul x x = pwr x 2"
```
```   130   "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
```
```   131   "pwr (pwr x p) q = pwr x (p * q)"
```
```   132   "pwr x 0 = r1"
```
```   133   "pwr x 1 = x"
```
```   134   "mul x (add y z) = add (mul x y) (mul x z)"
```
```   135   "pwr x (Suc q) = mul x (pwr x q)"
```
```   136   "pwr x (2*n) = mul (pwr x n) (pwr x n)"
```
```   137   "pwr x (Suc (2*n)) = mul x (mul (pwr x n) (pwr x n))"
```
```   138 proof -
```
```   139   show "add (mul a m) (mul b m) = mul (add a b) m" using mul_d mul_c by simp
```
```   140 next show"add (mul a m) m = mul (add a r1) m" using mul_d mul_c mul_1 by simp
```
```   141 next show "add m (mul a m) = mul (add a r1) m" using mul_c mul_d mul_1 add_c by simp
```
```   142 next show "add m m = mul (add r1 r1) m" using mul_c mul_d mul_1 by simp
```
```   143 next show "add r0 a = a" using add_0 by simp
```
```   144 next show "add a r0 = a" using add_0 add_c by simp
```
```   145 next show "mul a b = mul b a" using mul_c by simp
```
```   146 next show "mul (add a b) c = add (mul a c) (mul b c)" using mul_c mul_d by simp
```
```   147 next show "mul r0 a = r0" using mul_0 by simp
```
```   148 next show "mul a r0 = r0" using mul_0 mul_c by simp
```
```   149 next show "mul r1 a = a" using mul_1 by simp
```
```   150 next show "mul a r1 = a" using mul_1 mul_c by simp
```
```   151 next show "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
```
```   152     using mul_c mul_a by simp
```
```   153 next show "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
```
```   154     using mul_a by simp
```
```   155 next
```
```   156   have "mul (mul lx ly) (mul rx ry) = mul (mul rx ry) (mul lx ly)" by (rule mul_c)
```
```   157   also have "\<dots> = mul rx (mul ry (mul lx ly))" using mul_a by simp
```
```   158   finally
```
```   159   show "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
```
```   160     using mul_c by simp
```
```   161 next show "mul (mul lx ly) rx = mul (mul lx rx) ly" using mul_c mul_a by simp
```
```   162 next
```
```   163   show "mul (mul lx ly) rx = mul lx (mul ly rx)" by (simp add: mul_a)
```
```   164 next show "mul lx (mul rx ry) = mul (mul lx rx) ry" by (simp add: mul_a )
```
```   165 next show "mul lx (mul rx ry) = mul rx (mul lx ry)" by (simp add: mul_a,simp add: mul_c)
```
```   166 next show "add (add a b) (add c d) = add (add a c) (add b d)"
```
```   167     using add_c add_a by simp
```
```   168 next show "add (add a b) c = add a (add b c)" using add_a by simp
```
```   169 next show "add a (add c d) = add c (add a d)"
```
```   170     apply (simp add: add_a) by (simp only: add_c)
```
```   171 next show "add (add a b) c = add (add a c) b" using add_a add_c by simp
```
```   172 next show "add a c = add c a" by (rule add_c)
```
```   173 next show "add a (add c d) = add (add a c) d" using add_a by simp
```
```   174 next show "mul (pwr x p) (pwr x q) = pwr x (p + q)" by (rule mul_pwr)
```
```   175 next show "mul x (pwr x q) = pwr x (Suc q)" using pwr_Suc by simp
```
```   176 next show "mul (pwr x q) x = pwr x (Suc q)" using pwr_Suc mul_c by simp
```
```   177 next show "mul x x = pwr x 2" by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c)
```
```   178 next show "pwr (mul x y) q = mul (pwr x q) (pwr y q)" by (rule pwr_mul)
```
```   179 next show "pwr (pwr x p) q = pwr x (p * q)" by (rule pwr_pwr)
```
```   180 next show "pwr x 0 = r1" using pwr_0 .
```
```   181 next show "pwr x 1 = x" unfolding One_nat_def by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c)
```
```   182 next show "mul x (add y z) = add (mul x y) (mul x z)" using mul_d by simp
```
```   183 next show "pwr x (Suc q) = mul x (pwr x q)" using pwr_Suc by simp
```
```   184 next show "pwr x (2 * n) = mul (pwr x n) (pwr x n)" by (simp add: nat_number' mul_pwr)
```
```   185 next show "pwr x (Suc (2 * n)) = mul x (mul (pwr x n) (pwr x n))"
```
```   186     by (simp add: nat_number' pwr_Suc mul_pwr)
```
```   187 qed
```
```   188
```
```   189 end
```
```   190
```
```   191 sublocale comm_semiring_1
```
```   192   < normalizing!: normalizing_semiring plus times power zero one
```
```   193 proof
```
```   194 qed (simp_all add: algebra_simps)
```
```   195
```
```   196 lemmas (in comm_semiring_1) normalizing_comm_semiring_1_axioms =
```
```   197   comm_semiring_1_axioms [normalizer
```
```   198     semiring ops: normalizing.semiring_ops
```
```   199     semiring rules: normalizing.semiring_rules]
```
```   200
```
```   201 declaration (in comm_semiring_1)
```
```   202   {* Semiring_Normalizer.semiring_funs @{thm normalizing_comm_semiring_1_axioms} *}
```
```   203
```
```   204 locale normalizing_ring = normalizing_semiring +
```
```   205   fixes sub :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
```
```   206     and neg :: "'a \<Rightarrow> 'a"
```
```   207   assumes neg_mul: "neg x = mul (neg r1) x"
```
```   208     and sub_add: "sub x y = add x (neg y)"
```
```   209 begin
```
```   210
```
```   211 lemma ring_ops: shows "TERM (sub x y)" and "TERM (neg x)" .
```
```   212
```
```   213 lemmas ring_rules = neg_mul sub_add
```
```   214
```
```   215 end
```
```   216
```
```   217 sublocale comm_ring_1
```
```   218   < normalizing!: normalizing_ring plus times power zero one minus uminus
```
```   219 proof
```
```   220 qed (simp_all add: diff_minus)
```
```   221
```
```   222 lemmas (in comm_ring_1) normalizing_comm_ring_1_axioms =
```
```   223   comm_ring_1_axioms [normalizer
```
```   224     semiring ops: normalizing.semiring_ops
```
```   225     semiring rules: normalizing.semiring_rules
```
```   226     ring ops: normalizing.ring_ops
```
```   227     ring rules: normalizing.ring_rules]
```
```   228
```
```   229 declaration (in comm_ring_1)
```
```   230   {* Semiring_Normalizer.semiring_funs @{thm normalizing_comm_ring_1_axioms} *}
```
```   231
```
```   232 locale normalizing_semiring_cancel = normalizing_semiring +
```
```   233   assumes add_cancel: "add (x::'a) y = add x z \<longleftrightarrow> y = z"
```
```   234   and add_mul_solve: "add (mul w y) (mul x z) =
```
```   235     add (mul w z) (mul x y) \<longleftrightarrow> w = x \<or> y = z"
```
```   236 begin
```
```   237
```
```   238 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)"
```
```   239 proof-
```
```   240   have "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> \<not> (a = b \<or> c = d)" by simp
```
```   241   also have "\<dots> \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
```
```   242     using add_mul_solve by blast
```
```   243   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)"
```
```   244     by simp
```
```   245 qed
```
```   246
```
```   247 lemma add_scale_eq_noteq: "\<lbrakk>r \<noteq> r0 ; (a = b) \<and> ~(c = d)\<rbrakk>
```
```   248   \<Longrightarrow> add a (mul r c) \<noteq> add b (mul r d)"
```
```   249 proof(clarify)
```
```   250   assume nz: "r\<noteq> r0" and cnd: "c\<noteq>d"
```
```   251     and eq: "add b (mul r c) = add b (mul r d)"
```
```   252   hence "mul r c = mul r d" using cnd add_cancel by simp
```
```   253   hence "add (mul r0 d) (mul r c) = add (mul r0 c) (mul r d)"
```
```   254     using mul_0 add_cancel by simp
```
```   255   thus "False" using add_mul_solve nz cnd by simp
```
```   256 qed
```
```   257
```
```   258 lemma add_r0_iff: " x = add x a \<longleftrightarrow> a = r0"
```
```   259 proof-
```
```   260   have "a = r0 \<longleftrightarrow> add x a = add x r0" by (simp add: add_cancel)
```
```   261   thus "x = add x a \<longleftrightarrow> a = r0" by (auto simp add: add_c add_0)
```
```   262 qed
```
```   263
```
```   264 end
```
```   265
```
```   266 sublocale comm_semiring_1_cancel_norm
```
```   267   < normalizing!: normalizing_semiring_cancel plus times power zero one
```
```   268 proof
```
```   269 qed (simp_all add: add_mult_solve)
```
```   270
```
```   271 declare (in comm_semiring_1_cancel_norm)
```
```   272   normalizing_comm_semiring_1_axioms [normalizer del]
```
```   273
```
```   274 lemmas (in comm_semiring_1_cancel_norm)
```
```   275   normalizing_comm_semiring_1_cancel_norm_axioms =
```
```   276   comm_semiring_1_cancel_norm_axioms [normalizer
```
```   277     semiring ops: normalizing.semiring_ops
```
```   278     semiring rules: normalizing.semiring_rules
```
```   279     idom rules: normalizing.noteq_reduce normalizing.add_scale_eq_noteq]
```
```   280
```
```   281 declaration (in comm_semiring_1_cancel_norm)
```
```   282   {* Semiring_Normalizer.semiring_funs @{thm normalizing_comm_semiring_1_cancel_norm_axioms} *}
```
```   283
```
```   284 locale normalizing_ring_cancel = normalizing_semiring_cancel + normalizing_ring +
```
```   285   assumes subr0_iff: "sub x y = r0 \<longleftrightarrow> x = y"
```
```   286
```
```   287 sublocale idom
```
```   288   < normalizing!: normalizing_ring_cancel plus times power zero one minus uminus
```
```   289 proof
```
```   290 qed simp
```
```   291
```
```   292 declare (in idom) normalizing_comm_ring_1_axioms [normalizer del]
```
```   293
```
```   294 lemmas (in idom) normalizing_idom_axioms = idom_axioms [normalizer
```
```   295   semiring ops: normalizing.semiring_ops
```
```   296   semiring rules: normalizing.semiring_rules
```
```   297   ring ops: normalizing.ring_ops
```
```   298   ring rules: normalizing.ring_rules
```
```   299   idom rules: normalizing.noteq_reduce normalizing.add_scale_eq_noteq
```
```   300   ideal rules: normalizing.subr0_iff normalizing.add_r0_iff]
```
```   301
```
```   302 declaration (in idom)
```
```   303   {* Semiring_Normalizer.semiring_funs @{thm normalizing_idom_axioms} *}
```
```   304
```
```   305 locale normalizing_field = normalizing_ring_cancel +
```
```   306   fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
```
```   307     and inverse:: "'a \<Rightarrow> 'a"
```
```   308   assumes divide_inverse: "divide x y = mul x (inverse y)"
```
```   309      and inverse_divide: "inverse x = divide r1 x"
```
```   310 begin
```
```   311
```
```   312 lemma field_ops: shows "TERM (divide x y)" and "TERM (inverse x)" .
```
```   313
```
```   314 lemmas field_rules = divide_inverse inverse_divide
```
```   315
```
```   316 end
```
```   317
```
```   318 sublocale field
```
```   319   < normalizing!: normalizing_field plus times power zero one minus uminus divide inverse
```
```   320 proof
```
```   321 qed (simp_all add: divide_inverse)
```
```   322
```
```   323 lemmas (in field) normalizing_field_axioms =
```
```   324   field_axioms [normalizer
```
```   325     semiring ops: normalizing.semiring_ops
```
```   326     semiring rules: normalizing.semiring_rules
```
```   327     ring ops: normalizing.ring_ops
```
```   328     ring rules: normalizing.ring_rules
```
```   329     field ops: normalizing.field_ops
```
```   330     field rules: normalizing.field_rules
```
```   331     idom rules: normalizing.noteq_reduce normalizing.add_scale_eq_noteq
```
```   332     ideal rules: normalizing.subr0_iff normalizing.add_r0_iff]
```
```   333
```
```   334 declaration (in field)
```
```   335   {* Semiring_Normalizer.field_funs @{thm normalizing_field_axioms} *}
```
```   336
```
```   337 end
```