src/HOL/Semiring_Normalization.thy
 author paulson Mon Feb 22 14:37:56 2016 +0000 (2016-02-22) changeset 62379 340738057c8c parent 61153 3d5e01b427cb child 66836 4eb431c3f974 permissions -rw-r--r--
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
```     1 (*  Title:      HOL/Semiring_Normalization.thy
```
```     2     Author:     Amine Chaieb, TU Muenchen
```
```     3 *)
```
```     4
```
```     5 section \<open>Semiring normalization\<close>
```
```     6
```
```     7 theory Semiring_Normalization
```
```     8 imports Numeral_Simprocs Nat_Transfer
```
```     9 begin
```
```    10
```
```    11 text \<open>Prelude\<close>
```
```    12
```
```    13 class comm_semiring_1_cancel_crossproduct = comm_semiring_1_cancel +
```
```    14   assumes crossproduct_eq: "w * y + x * z = w * z + x * y \<longleftrightarrow> w = x \<or> y = z"
```
```    15 begin
```
```    16
```
```    17 lemma crossproduct_noteq:
```
```    18   "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> a * c + b * d \<noteq> a * d + b * c"
```
```    19   by (simp add: crossproduct_eq)
```
```    20
```
```    21 lemma add_scale_eq_noteq:
```
```    22   "r \<noteq> 0 \<Longrightarrow> a = b \<and> c \<noteq> d \<Longrightarrow> a + r * c \<noteq> b + r * d"
```
```    23 proof (rule notI)
```
```    24   assume nz: "r\<noteq> 0" and cnd: "a = b \<and> c\<noteq>d"
```
```    25     and eq: "a + (r * c) = b + (r * d)"
```
```    26   have "(0 * d) + (r * c) = (0 * c) + (r * d)"
```
```    27     using add_left_imp_eq eq mult_zero_left by (simp add: cnd)
```
```    28   then show False using crossproduct_eq [of 0 d] nz cnd by simp
```
```    29 qed
```
```    30
```
```    31 lemma add_0_iff:
```
```    32   "b = b + a \<longleftrightarrow> a = 0"
```
```    33   using add_left_imp_eq [of b a 0] by auto
```
```    34
```
```    35 end
```
```    36
```
```    37 subclass (in idom) comm_semiring_1_cancel_crossproduct
```
```    38 proof
```
```    39   fix w x y z
```
```    40   show "w * y + x * z = w * z + x * y \<longleftrightarrow> w = x \<or> y = z"
```
```    41   proof
```
```    42     assume "w * y + x * z = w * z + x * y"
```
```    43     then have "w * y + x * z - w * z - x * y = 0" by (simp add: algebra_simps)
```
```    44     then have "w * (y - z) - x * (y - z) = 0" by (simp add: algebra_simps)
```
```    45     then have "(y - z) * (w - x) = 0" by (simp add: algebra_simps)
```
```    46     then have "y - z = 0 \<or> w - x = 0" by (rule divisors_zero)
```
```    47     then show "w = x \<or> y = z" by auto
```
```    48   qed (auto simp add: ac_simps)
```
```    49 qed
```
```    50
```
```    51 instance nat :: comm_semiring_1_cancel_crossproduct
```
```    52 proof
```
```    53   fix w x y z :: nat
```
```    54   have aux: "\<And>y z. y < z \<Longrightarrow> w * y + x * z = w * z + x * y \<Longrightarrow> w = x"
```
```    55   proof -
```
```    56     fix y z :: nat
```
```    57     assume "y < z" then have "\<exists>k. z = y + k \<and> k \<noteq> 0" by (intro exI [of _ "z - y"]) auto
```
```    58     then obtain k where "z = y + k" and "k \<noteq> 0" by blast
```
```    59     assume "w * y + x * z = w * z + x * y"
```
```    60     then have "(w * y + x * y) + x * k = (w * y + x * y) + w * k" by (simp add: \<open>z = y + k\<close> algebra_simps)
```
```    61     then have "x * k = w * k" by simp
```
```    62     then show "w = x" using \<open>k \<noteq> 0\<close> by simp
```
```    63   qed
```
```    64   show "w * y + x * z = w * z + x * y \<longleftrightarrow> w = x \<or> y = z"
```
```    65     by (auto simp add: neq_iff dest!: aux)
```
```    66 qed
```
```    67
```
```    68 text \<open>Semiring normalization proper\<close>
```
```    69
```
```    70 ML_file "Tools/semiring_normalizer.ML"
```
```    71
```
```    72 context comm_semiring_1
```
```    73 begin
```
```    74
```
```    75 lemma semiring_normalization_rules:
```
```    76   "(a * m) + (b * m) = (a + b) * m"
```
```    77   "(a * m) + m = (a + 1) * m"
```
```    78   "m + (a * m) = (a + 1) * m"
```
```    79   "m + m = (1 + 1) * m"
```
```    80   "0 + a = a"
```
```    81   "a + 0 = a"
```
```    82   "a * b = b * a"
```
```    83   "(a + b) * c = (a * c) + (b * c)"
```
```    84   "0 * a = 0"
```
```    85   "a * 0 = 0"
```
```    86   "1 * a = a"
```
```    87   "a * 1 = a"
```
```    88   "(lx * ly) * (rx * ry) = (lx * rx) * (ly * ry)"
```
```    89   "(lx * ly) * (rx * ry) = lx * (ly * (rx * ry))"
```
```    90   "(lx * ly) * (rx * ry) = rx * ((lx * ly) * ry)"
```
```    91   "(lx * ly) * rx = (lx * rx) * ly"
```
```    92   "(lx * ly) * rx = lx * (ly * rx)"
```
```    93   "lx * (rx * ry) = (lx * rx) * ry"
```
```    94   "lx * (rx * ry) = rx * (lx * ry)"
```
```    95   "(a + b) + (c + d) = (a + c) + (b + d)"
```
```    96   "(a + b) + c = a + (b + c)"
```
```    97   "a + (c + d) = c + (a + d)"
```
```    98   "(a + b) + c = (a + c) + b"
```
```    99   "a + c = c + a"
```
```   100   "a + (c + d) = (a + c) + d"
```
```   101   "(x ^ p) * (x ^ q) = x ^ (p + q)"
```
```   102   "x * (x ^ q) = x ^ (Suc q)"
```
```   103   "(x ^ q) * x = x ^ (Suc q)"
```
```   104   "x * x = x\<^sup>2"
```
```   105   "(x * y) ^ q = (x ^ q) * (y ^ q)"
```
```   106   "(x ^ p) ^ q = x ^ (p * q)"
```
```   107   "x ^ 0 = 1"
```
```   108   "x ^ 1 = x"
```
```   109   "x * (y + z) = (x * y) + (x * z)"
```
```   110   "x ^ (Suc q) = x * (x ^ q)"
```
```   111   "x ^ (2*n) = (x ^ n) * (x ^ n)"
```
```   112   by (simp_all add: algebra_simps power_add power2_eq_square
```
```   113     power_mult_distrib power_mult del: one_add_one)
```
```   114
```
```   115 local_setup \<open>
```
```   116   Semiring_Normalizer.declare @{thm comm_semiring_1_axioms}
```
```   117     {semiring = ([@{term "x + y"}, @{term "x * y"}, @{term "x ^ n"}, @{term 0}, @{term 1}],
```
```   118       @{thms semiring_normalization_rules}),
```
```   119      ring = ([], []),
```
```   120      field = ([], []),
```
```   121      idom = [],
```
```   122      ideal = []}
```
```   123 \<close>
```
```   124
```
```   125 end
```
```   126
```
```   127 context comm_ring_1
```
```   128 begin
```
```   129
```
```   130 lemma ring_normalization_rules:
```
```   131   "- x = (- 1) * x"
```
```   132   "x - y = x + (- y)"
```
```   133   by simp_all
```
```   134
```
```   135 local_setup \<open>
```
```   136   Semiring_Normalizer.declare @{thm comm_ring_1_axioms}
```
```   137     {semiring = ([@{term "x + y"}, @{term "x * y"}, @{term "x ^ n"}, @{term 0}, @{term 1}],
```
```   138       @{thms semiring_normalization_rules}),
```
```   139       ring = ([@{term "x - y"}, @{term "- x"}], @{thms ring_normalization_rules}),
```
```   140       field = ([], []),
```
```   141       idom = [],
```
```   142       ideal = []}
```
```   143 \<close>
```
```   144
```
```   145 end
```
```   146
```
```   147 context comm_semiring_1_cancel_crossproduct
```
```   148 begin
```
```   149
```
```   150 local_setup \<open>
```
```   151   Semiring_Normalizer.declare @{thm comm_semiring_1_cancel_crossproduct_axioms}
```
```   152     {semiring = ([@{term "x + y"}, @{term "x * y"}, @{term "x ^ n"}, @{term 0}, @{term 1}],
```
```   153       @{thms semiring_normalization_rules}),
```
```   154      ring = ([], []),
```
```   155      field = ([], []),
```
```   156      idom = @{thms crossproduct_noteq add_scale_eq_noteq},
```
```   157      ideal = []}
```
```   158 \<close>
```
```   159
```
```   160 end
```
```   161
```
```   162 context idom
```
```   163 begin
```
```   164
```
```   165 local_setup \<open>
```
```   166   Semiring_Normalizer.declare @{thm idom_axioms}
```
```   167    {semiring = ([@{term "x + y"}, @{term "x * y"}, @{term "x ^ n"}, @{term 0}, @{term 1}],
```
```   168       @{thms semiring_normalization_rules}),
```
```   169     ring = ([@{term "x - y"}, @{term "- x"}], @{thms ring_normalization_rules}),
```
```   170     field = ([], []),
```
```   171     idom = @{thms crossproduct_noteq add_scale_eq_noteq},
```
```   172     ideal = @{thms right_minus_eq add_0_iff}}
```
```   173 \<close>
```
```   174
```
```   175 end
```
```   176
```
```   177 context field
```
```   178 begin
```
```   179
```
```   180 local_setup \<open>
```
```   181   Semiring_Normalizer.declare @{thm field_axioms}
```
```   182    {semiring = ([@{term "x + y"}, @{term "x * y"}, @{term "x ^ n"}, @{term 0}, @{term 1}],
```
```   183       @{thms semiring_normalization_rules}),
```
```   184     ring = ([@{term "x - y"}, @{term "- x"}], @{thms ring_normalization_rules}),
```
```   185     field = ([@{term "x / y"}, @{term "inverse x"}], @{thms divide_inverse inverse_eq_divide}),
```
```   186     idom = @{thms crossproduct_noteq add_scale_eq_noteq},
```
```   187     ideal = @{thms right_minus_eq add_0_iff}}
```
```   188 \<close>
```
```   189
```
```   190 end
```
```   191
```
```   192 code_identifier
```
```   193   code_module Semiring_Normalization \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
```
```   194
```
```   195 end
```