src/HOL/Semiring_Normalization.thy
author hoelzl
Tue May 11 19:21:39 2010 +0200 (2010-05-11)
changeset 36845 d778c64fc35d
parent 36756 c1ae8a0b4265
child 36871 3763c349c8c1
permissions -rw-r--r--
Add rules directly to the corresponding class locales instead.
     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 lemma (in comm_semiring_1) semiring_ops:
    56   shows "TERM (x + y)" and "TERM (x * y)" and "TERM (x ^ n)"
    57     and "TERM 0" and "TERM 1" .
    58 
    59 lemma (in comm_semiring_1) semiring_rules:
    60   "(a * m) + (b * m) = (a + b) * m"
    61   "(a * m) + m = (a + 1) * m"
    62   "m + (a * m) = (a + 1) * m"
    63   "m + m = (1 + 1) * m"
    64   "0 + a = a"
    65   "a + 0 = a"
    66   "a * b = b * a"
    67   "(a + b) * c = (a * c) + (b * c)"
    68   "0 * a = 0"
    69   "a * 0 = 0"
    70   "1 * a = a"
    71   "a * 1 = a"
    72   "(lx * ly) * (rx * ry) = (lx * rx) * (ly * ry)"
    73   "(lx * ly) * (rx * ry) = lx * (ly * (rx * ry))"
    74   "(lx * ly) * (rx * ry) = rx * ((lx * ly) * ry)"
    75   "(lx * ly) * rx = (lx * rx) * ly"
    76   "(lx * ly) * rx = lx * (ly * rx)"
    77   "lx * (rx * ry) = (lx * rx) * ry"
    78   "lx * (rx * ry) = rx * (lx * ry)"
    79   "(a + b) + (c + d) = (a + c) + (b + d)"
    80   "(a + b) + c = a + (b + c)"
    81   "a + (c + d) = c + (a + d)"
    82   "(a + b) + c = (a + c) + b"
    83   "a + c = c + a"
    84   "a + (c + d) = (a + c) + d"
    85   "(x ^ p) * (x ^ q) = x ^ (p + q)"
    86   "x * (x ^ q) = x ^ (Suc q)"
    87   "(x ^ q) * x = x ^ (Suc q)"
    88   "x * x = x ^ 2"
    89   "(x * y) ^ q = (x ^ q) * (y ^ q)"
    90   "(x ^ p) ^ q = x ^ (p * q)"
    91   "x ^ 0 = 1"
    92   "x ^ 1 = x"
    93   "x * (y + z) = (x * y) + (x * z)"
    94   "x ^ (Suc q) = x * (x ^ q)"
    95   "x ^ (2*n) = (x ^ n) * (x ^ n)"
    96   "x ^ (Suc (2*n)) = x * ((x ^ n) * (x ^ n))"
    97   by (simp_all add: algebra_simps power_add power2_eq_square power_mult_distrib power_mult)
    98 
    99 lemmas (in comm_semiring_1) normalizing_comm_semiring_1_axioms =
   100   comm_semiring_1_axioms [normalizer
   101     semiring ops: semiring_ops
   102     semiring rules: semiring_rules]
   103 
   104 declaration (in comm_semiring_1)
   105   {* Semiring_Normalizer.semiring_funs @{thm normalizing_comm_semiring_1_axioms} *}
   106 
   107 lemma (in comm_ring_1) ring_ops: shows "TERM (x- y)" and "TERM (- x)" .
   108 
   109 lemma (in comm_ring_1) ring_rules:
   110   "- x = (- 1) * x"
   111   "x - y = x + (- y)"
   112   by (simp_all add: diff_minus)
   113 
   114 lemmas (in comm_ring_1) normalizing_comm_ring_1_axioms =
   115   comm_ring_1_axioms [normalizer
   116     semiring ops: semiring_ops
   117     semiring rules: semiring_rules
   118     ring ops: ring_ops
   119     ring rules: ring_rules]
   120 
   121 declaration (in comm_ring_1)
   122   {* Semiring_Normalizer.semiring_funs @{thm normalizing_comm_ring_1_axioms} *}
   123 
   124 lemma (in comm_semiring_1_cancel_norm) noteq_reduce:
   125   "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> (a * c) + (b * d) \<noteq> (a * d) + (b * c)"
   126 proof-
   127   have "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> \<not> (a = b \<or> c = d)" by simp
   128   also have "\<dots> \<longleftrightarrow> (a * c) + (b * d) \<noteq> (a * d) + (b * c)"
   129     using add_mult_solve by blast
   130   finally show "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> (a * c) + (b * d) \<noteq> (a * d) + (b * c)"
   131     by simp
   132 qed
   133 
   134 lemma (in comm_semiring_1_cancel_norm) add_scale_eq_noteq:
   135   "\<lbrakk>r \<noteq> 0 ; a = b \<and> c \<noteq> d\<rbrakk> \<Longrightarrow> a + (r * c) \<noteq> b + (r * d)"
   136 proof(clarify)
   137   assume nz: "r\<noteq> 0" and cnd: "c\<noteq>d"
   138     and eq: "b + (r * c) = b + (r * d)"
   139   have "(0 * d) + (r * c) = (0 * c) + (r * d)"
   140     using add_imp_eq eq mult_zero_left by simp
   141   thus "False" using add_mult_solve[of 0 d] nz cnd by simp
   142 qed
   143 
   144 lemma (in comm_semiring_1_cancel_norm) add_0_iff:
   145   "x = x + a \<longleftrightarrow> a = 0"
   146 proof-
   147   have "a = 0 \<longleftrightarrow> x + a = x + 0" using add_imp_eq[of x a 0] by auto
   148   thus "x = x + a \<longleftrightarrow> a = 0" by (auto simp add: add_commute)
   149 qed
   150 
   151 declare (in comm_semiring_1_cancel_norm)
   152   normalizing_comm_semiring_1_axioms [normalizer del]
   153 
   154 lemmas (in comm_semiring_1_cancel_norm)
   155   normalizing_comm_semiring_1_cancel_norm_axioms =
   156   comm_semiring_1_cancel_norm_axioms [normalizer
   157     semiring ops: semiring_ops
   158     semiring rules: semiring_rules
   159     idom rules: noteq_reduce add_scale_eq_noteq]
   160 
   161 declaration (in comm_semiring_1_cancel_norm)
   162   {* Semiring_Normalizer.semiring_funs @{thm normalizing_comm_semiring_1_cancel_norm_axioms} *}
   163 
   164 declare (in idom) normalizing_comm_ring_1_axioms [normalizer del]
   165 
   166 lemmas (in idom) normalizing_idom_axioms = idom_axioms [normalizer
   167   semiring ops: semiring_ops
   168   semiring rules: semiring_rules
   169   ring ops: ring_ops
   170   ring rules: ring_rules
   171   idom rules: noteq_reduce add_scale_eq_noteq
   172   ideal rules: right_minus_eq add_0_iff]
   173 
   174 declaration (in idom)
   175   {* Semiring_Normalizer.semiring_funs @{thm normalizing_idom_axioms} *}
   176 
   177 lemma (in field) field_ops:
   178   shows "TERM (x / y)" and "TERM (inverse x)" .
   179 
   180 lemmas (in field) field_rules = divide_inverse inverse_eq_divide
   181 
   182 lemmas (in field) normalizing_field_axioms =
   183   field_axioms [normalizer
   184     semiring ops: semiring_ops
   185     semiring rules: semiring_rules
   186     ring ops: ring_ops
   187     ring rules: ring_rules
   188     field ops: field_ops
   189     field rules: field_rules
   190     idom rules: noteq_reduce add_scale_eq_noteq
   191     ideal rules: right_minus_eq add_0_iff]
   192 
   193 declaration (in field)
   194   {* Semiring_Normalizer.field_funs @{thm normalizing_field_axioms} *}
   195 
   196 hide_fact (open) normalizing_comm_semiring_1_axioms
   197   normalizing_comm_semiring_1_cancel_norm_axioms semiring_ops semiring_rules
   198 
   199 hide_fact (open) normalizing_comm_ring_1_axioms
   200   normalizing_idom_axioms ring_ops ring_rules
   201 
   202 hide_fact (open)  normalizing_field_axioms field_ops field_rules
   203 
   204 hide_fact (open) add_scale_eq_noteq noteq_reduce
   205 
   206 end