src/HOL/Groebner_Basis.thy
author wenzelm
Thu, 20 Nov 2008 00:03:47 +0100
changeset 28856 5e009a80fe6d
parent 28823 dcbef866c9e2
child 28986 1ff53ff7041d
permissions -rw-r--r--
Pure syntax: more coherent treatment of aprop, permanent TERM and &&&;
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
     1
(*  Title:      HOL/Groebner_Basis.thy
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
     2
    ID:         $Id$
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
     3
    Author:     Amine Chaieb, TU Muenchen
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
     4
*)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
     5
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
     6
header {* Semiring normalization and Groebner Bases *}
28402
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
     7
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
     8
theory Groebner_Basis
28402
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
     9
imports Arith_Tools
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    10
uses
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    11
  "Tools/Groebner_Basis/misc.ML"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    12
  "Tools/Groebner_Basis/normalizer_data.ML"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    13
  ("Tools/Groebner_Basis/normalizer.ML")
23312
6e32a5bfc30f explicitely depends on file groebner.ML
chaieb
parents: 23266
diff changeset
    14
  ("Tools/Groebner_Basis/groebner.ML")
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    15
begin
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    16
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    17
subsection {* Semiring normalization *}
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    18
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    19
setup NormalizerData.setup
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    20
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    21
23258
9062e98fdab1 renamed locale ring/semiring to gb_ring/gb_semiring to avoid clash with Ring_and_Field versions;
wenzelm
parents: 23252
diff changeset
    22
locale gb_semiring =
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    23
  fixes add mul pwr r0 r1
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    24
  assumes add_a:"(add x (add y z) = add (add x y) z)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    25
    and add_c: "add x y = add y x" and add_0:"add r0 x = x"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    26
    and mul_a:"mul x (mul y z) = mul (mul x y) z" and mul_c:"mul x y = mul y x"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    27
    and mul_1:"mul r1 x = x" and  mul_0:"mul r0 x = r0"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    28
    and mul_d:"mul x (add y z) = add (mul x y) (mul x z)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    29
    and pwr_0:"pwr x 0 = r1" and pwr_Suc:"pwr x (Suc n) = mul x (pwr x n)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    30
begin
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    31
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    32
lemma mul_pwr:"mul (pwr x p) (pwr x q) = pwr x (p + q)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    33
proof (induct p)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    34
  case 0
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    35
  then show ?case by (auto simp add: pwr_0 mul_1)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    36
next
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    37
  case Suc
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    38
  from this [symmetric] show ?case
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    39
    by (auto simp add: pwr_Suc mul_1 mul_a)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    40
qed
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    41
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    42
lemma pwr_mul: "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    43
proof (induct q arbitrary: x y, auto simp add:pwr_0 pwr_Suc mul_1)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    44
  fix q x y
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    45
  assume "\<And>x y. pwr (mul x y) q = mul (pwr x q) (pwr y q)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    46
  have "mul (mul x y) (mul (pwr x q) (pwr y q)) = mul x (mul y (mul (pwr x q) (pwr y q)))"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    47
    by (simp add: mul_a)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    48
  also have "\<dots> = (mul (mul y (mul (pwr y q) (pwr x q))) x)" by (simp add: mul_c)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    49
  also have "\<dots> = (mul (mul y (pwr y q)) (mul (pwr x q) x))" by (simp add: mul_a)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    50
  finally show "mul (mul x y) (mul (pwr x q) (pwr y q)) =
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    51
    mul (mul x (pwr x q)) (mul y (pwr y q))" by (simp add: mul_c)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    52
qed
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    53
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    54
lemma pwr_pwr: "pwr (pwr x p) q = pwr x (p * q)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    55
proof (induct p arbitrary: q)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    56
  case 0
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    57
  show ?case using pwr_Suc mul_1 pwr_0 by (induct q) auto
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    58
next
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    59
  case Suc
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    60
  thus ?case by (auto simp add: mul_pwr [symmetric] pwr_mul pwr_Suc)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    61
qed
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    62
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    63
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    64
subsubsection {* Declaring the abstract theory *}
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    65
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    66
lemma semiring_ops:
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    67
  shows "TERM (add x y)" and "TERM (mul x y)" and "TERM (pwr x n)"
28856
5e009a80fe6d Pure syntax: more coherent treatment of aprop, permanent TERM and &&&;
wenzelm
parents: 28823
diff changeset
    68
    and "TERM r0" and "TERM r1" .
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    69
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    70
lemma semiring_rules:
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    71
  "add (mul a m) (mul b m) = mul (add a b) m"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    72
  "add (mul a m) m = mul (add a r1) m"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    73
  "add m (mul a m) = mul (add a r1) m"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    74
  "add m m = mul (add r1 r1) m"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    75
  "add r0 a = a"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    76
  "add a r0 = a"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    77
  "mul a b = mul b a"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    78
  "mul (add a b) c = add (mul a c) (mul b c)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    79
  "mul r0 a = r0"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    80
  "mul a r0 = r0"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    81
  "mul r1 a = a"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    82
  "mul a r1 = a"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    83
  "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    84
  "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    85
  "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    86
  "mul (mul lx ly) rx = mul (mul lx rx) ly"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    87
  "mul (mul lx ly) rx = mul lx (mul ly rx)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    88
  "mul lx (mul rx ry) = mul (mul lx rx) ry"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    89
  "mul lx (mul rx ry) = mul rx (mul lx ry)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    90
  "add (add a b) (add c d) = add (add a c) (add b d)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    91
  "add (add a b) c = add a (add b c)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    92
  "add a (add c d) = add c (add a d)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    93
  "add (add a b) c = add (add a c) b"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    94
  "add a c = add c a"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    95
  "add a (add c d) = add (add a c) d"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    96
  "mul (pwr x p) (pwr x q) = pwr x (p + q)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    97
  "mul x (pwr x q) = pwr x (Suc q)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    98
  "mul (pwr x q) x = pwr x (Suc q)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    99
  "mul x x = pwr x 2"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   100
  "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   101
  "pwr (pwr x p) q = pwr x (p * q)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   102
  "pwr x 0 = r1"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   103
  "pwr x 1 = x"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   104
  "mul x (add y z) = add (mul x y) (mul x z)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   105
  "pwr x (Suc q) = mul x (pwr x q)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   106
  "pwr x (2*n) = mul (pwr x n) (pwr x n)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   107
  "pwr x (Suc (2*n)) = mul x (mul (pwr x n) (pwr x n))"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   108
proof -
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   109
  show "add (mul a m) (mul b m) = mul (add a b) m" using mul_d mul_c by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   110
next show"add (mul a m) m = mul (add a r1) m" using mul_d mul_c mul_1 by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   111
next show "add m (mul a m) = mul (add a r1) m" using mul_c mul_d mul_1 add_c by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   112
next show "add m m = mul (add r1 r1) m" using mul_c mul_d mul_1 by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   113
next show "add r0 a = a" using add_0 by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   114
next show "add a r0 = a" using add_0 add_c by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   115
next show "mul a b = mul b a" using mul_c by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   116
next show "mul (add a b) c = add (mul a c) (mul b c)" using mul_c mul_d by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   117
next show "mul r0 a = r0" using mul_0 by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   118
next show "mul a r0 = r0" using mul_0 mul_c by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   119
next show "mul r1 a = a" using mul_1 by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   120
next show "mul a r1 = a" using mul_1 mul_c by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   121
next show "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   122
    using mul_c mul_a by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   123
next show "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   124
    using mul_a by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   125
next
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   126
  have "mul (mul lx ly) (mul rx ry) = mul (mul rx ry) (mul lx ly)" by (rule mul_c)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   127
  also have "\<dots> = mul rx (mul ry (mul lx ly))" using mul_a by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   128
  finally
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   129
  show "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   130
    using mul_c by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   131
next show "mul (mul lx ly) rx = mul (mul lx rx) ly" using mul_c mul_a by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   132
next
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   133
  show "mul (mul lx ly) rx = mul lx (mul ly rx)" by (simp add: mul_a)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   134
next show "mul lx (mul rx ry) = mul (mul lx rx) ry" by (simp add: mul_a )
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   135
next show "mul lx (mul rx ry) = mul rx (mul lx ry)" by (simp add: mul_a,simp add: mul_c)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   136
next show "add (add a b) (add c d) = add (add a c) (add b d)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   137
    using add_c add_a by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   138
next show "add (add a b) c = add a (add b c)" using add_a by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   139
next show "add a (add c d) = add c (add a d)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   140
    apply (simp add: add_a) by (simp only: add_c)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   141
next show "add (add a b) c = add (add a c) b" using add_a add_c by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   142
next show "add a c = add c a" by (rule add_c)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   143
next show "add a (add c d) = add (add a c) d" using add_a by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   144
next show "mul (pwr x p) (pwr x q) = pwr x (p + q)" by (rule mul_pwr)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   145
next show "mul x (pwr x q) = pwr x (Suc q)" using pwr_Suc by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   146
next show "mul (pwr x q) x = pwr x (Suc q)" using pwr_Suc mul_c by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   147
next show "mul x x = pwr x 2" by (simp add: nat_number pwr_Suc pwr_0 mul_1 mul_c)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   148
next show "pwr (mul x y) q = mul (pwr x q) (pwr y q)" by (rule pwr_mul)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   149
next show "pwr (pwr x p) q = pwr x (p * q)" by (rule pwr_pwr)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   150
next show "pwr x 0 = r1" using pwr_0 .
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   151
next show "pwr x 1 = x" by (simp add: nat_number pwr_Suc pwr_0 mul_1 mul_c)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   152
next show "mul x (add y z) = add (mul x y) (mul x z)" using mul_d by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   153
next show "pwr x (Suc q) = mul x (pwr x q)" using pwr_Suc by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   154
next show "pwr x (2 * n) = mul (pwr x n) (pwr x n)" by (simp add: nat_number mul_pwr)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   155
next show "pwr x (Suc (2 * n)) = mul x (mul (pwr x n) (pwr x n))"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   156
    by (simp add: nat_number pwr_Suc mul_pwr)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   157
qed
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   158
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   159
26462
dac4e2bce00d avoid rebinding of existing facts;
wenzelm
parents: 26314
diff changeset
   160
lemmas gb_semiring_axioms' =
26314
9c39fc898fff avoid rebinding of existing facts;
wenzelm
parents: 26199
diff changeset
   161
  gb_semiring_axioms [normalizer
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   162
    semiring ops: semiring_ops
26314
9c39fc898fff avoid rebinding of existing facts;
wenzelm
parents: 26199
diff changeset
   163
    semiring rules: semiring_rules]
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   164
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   165
end
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   166
23258
9062e98fdab1 renamed locale ring/semiring to gb_ring/gb_semiring to avoid clash with Ring_and_Field versions;
wenzelm
parents: 23252
diff changeset
   167
interpretation class_semiring: gb_semiring
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   168
    ["op +" "op *" "op ^" "0::'a::{comm_semiring_1, recpower}" "1"]
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28699
diff changeset
   169
  proof qed (auto simp add: ring_simps power_Suc)
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   170
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   171
lemmas nat_arith =
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   172
  add_nat_number_of diff_nat_number_of mult_nat_number_of eq_nat_number_of less_nat_number_of
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   173
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   174
lemma not_iszero_Numeral1: "\<not> iszero (Numeral1::'a::number_ring)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   175
  by (simp add: numeral_1_eq_1)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   176
lemmas comp_arith = Let_def arith_simps nat_arith rel_simps if_False
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   177
  if_True add_0 add_Suc add_number_of_left mult_number_of_left
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   178
  numeral_1_eq_1[symmetric] Suc_eq_add_numeral_1
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   179
  numeral_0_eq_0[symmetric] numerals[symmetric] not_iszero_1
26086
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25250
diff changeset
   180
  iszero_number_of_Bit1 iszero_number_of_Bit0 nonzero_number_of_Min
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   181
  iszero_number_of_Pls iszero_0 not_iszero_Numeral1
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   182
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   183
lemmas semiring_norm = comp_arith
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   184
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   185
ML {*
23573
d85a277f90fd common normalizer_funs, avoid cterm_of;
wenzelm
parents: 23477
diff changeset
   186
local
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   187
23573
d85a277f90fd common normalizer_funs, avoid cterm_of;
wenzelm
parents: 23477
diff changeset
   188
open Conv;
d85a277f90fd common normalizer_funs, avoid cterm_of;
wenzelm
parents: 23477
diff changeset
   189
d85a277f90fd common normalizer_funs, avoid cterm_of;
wenzelm
parents: 23477
diff changeset
   190
fun numeral_is_const ct =
d85a277f90fd common normalizer_funs, avoid cterm_of;
wenzelm
parents: 23477
diff changeset
   191
  can HOLogic.dest_number (Thm.term_of ct);
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   192
23573
d85a277f90fd common normalizer_funs, avoid cterm_of;
wenzelm
parents: 23477
diff changeset
   193
fun int_of_rat x =
d85a277f90fd common normalizer_funs, avoid cterm_of;
wenzelm
parents: 23477
diff changeset
   194
  (case Rat.quotient_of_rat x of (i, 1) => i
d85a277f90fd common normalizer_funs, avoid cterm_of;
wenzelm
parents: 23477
diff changeset
   195
  | _ => error "int_of_rat: bad int");
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   196
23573
d85a277f90fd common normalizer_funs, avoid cterm_of;
wenzelm
parents: 23477
diff changeset
   197
val numeral_conv =
d85a277f90fd common normalizer_funs, avoid cterm_of;
wenzelm
parents: 23477
diff changeset
   198
  Simplifier.rewrite (HOL_basic_ss addsimps @{thms semiring_norm}) then_conv
d85a277f90fd common normalizer_funs, avoid cterm_of;
wenzelm
parents: 23477
diff changeset
   199
  Simplifier.rewrite (HOL_basic_ss addsimps
d85a277f90fd common normalizer_funs, avoid cterm_of;
wenzelm
parents: 23477
diff changeset
   200
    (@{thms numeral_1_eq_1} @ @{thms numeral_0_eq_0} @ @{thms numerals(1-2)}));
d85a277f90fd common normalizer_funs, avoid cterm_of;
wenzelm
parents: 23477
diff changeset
   201
d85a277f90fd common normalizer_funs, avoid cterm_of;
wenzelm
parents: 23477
diff changeset
   202
in
d85a277f90fd common normalizer_funs, avoid cterm_of;
wenzelm
parents: 23477
diff changeset
   203
d85a277f90fd common normalizer_funs, avoid cterm_of;
wenzelm
parents: 23477
diff changeset
   204
fun normalizer_funs key =
d85a277f90fd common normalizer_funs, avoid cterm_of;
wenzelm
parents: 23477
diff changeset
   205
  NormalizerData.funs key
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   206
   {is_const = fn phi => numeral_is_const,
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   207
    dest_const = fn phi => fn ct =>
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   208
      Rat.rat_of_int (snd
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   209
        (HOLogic.dest_number (Thm.term_of ct)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   210
          handle TERM _ => error "ring_dest_const")),
23573
d85a277f90fd common normalizer_funs, avoid cterm_of;
wenzelm
parents: 23477
diff changeset
   211
    mk_const = fn phi => fn cT => fn x => Numeral.mk_cnumber cT (int_of_rat x),
23330
01c09922ce59 Conversion for computation on constants now depends on the context
chaieb
parents: 23327
diff changeset
   212
    conv = fn phi => K numeral_conv}
23573
d85a277f90fd common normalizer_funs, avoid cterm_of;
wenzelm
parents: 23477
diff changeset
   213
d85a277f90fd common normalizer_funs, avoid cterm_of;
wenzelm
parents: 23477
diff changeset
   214
end
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   215
*}
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   216
26462
dac4e2bce00d avoid rebinding of existing facts;
wenzelm
parents: 26314
diff changeset
   217
declaration {* normalizer_funs @{thm class_semiring.gb_semiring_axioms'} *}
23573
d85a277f90fd common normalizer_funs, avoid cterm_of;
wenzelm
parents: 23477
diff changeset
   218
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   219
23258
9062e98fdab1 renamed locale ring/semiring to gb_ring/gb_semiring to avoid clash with Ring_and_Field versions;
wenzelm
parents: 23252
diff changeset
   220
locale gb_ring = gb_semiring +
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   221
  fixes sub :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   222
    and neg :: "'a \<Rightarrow> 'a"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   223
  assumes neg_mul: "neg x = mul (neg r1) x"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   224
    and sub_add: "sub x y = add x (neg y)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   225
begin
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   226
28856
5e009a80fe6d Pure syntax: more coherent treatment of aprop, permanent TERM and &&&;
wenzelm
parents: 28823
diff changeset
   227
lemma ring_ops: shows "TERM (sub x y)" and "TERM (neg x)" .
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   228
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   229
lemmas ring_rules = neg_mul sub_add
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   230
26462
dac4e2bce00d avoid rebinding of existing facts;
wenzelm
parents: 26314
diff changeset
   231
lemmas gb_ring_axioms' =
26314
9c39fc898fff avoid rebinding of existing facts;
wenzelm
parents: 26199
diff changeset
   232
  gb_ring_axioms [normalizer
9c39fc898fff avoid rebinding of existing facts;
wenzelm
parents: 26199
diff changeset
   233
    semiring ops: semiring_ops
9c39fc898fff avoid rebinding of existing facts;
wenzelm
parents: 26199
diff changeset
   234
    semiring rules: semiring_rules
9c39fc898fff avoid rebinding of existing facts;
wenzelm
parents: 26199
diff changeset
   235
    ring ops: ring_ops
9c39fc898fff avoid rebinding of existing facts;
wenzelm
parents: 26199
diff changeset
   236
    ring rules: ring_rules]
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   237
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   238
end
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   239
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   240
23258
9062e98fdab1 renamed locale ring/semiring to gb_ring/gb_semiring to avoid clash with Ring_and_Field versions;
wenzelm
parents: 23252
diff changeset
   241
interpretation class_ring: gb_ring ["op +" "op *" "op ^"
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   242
    "0::'a::{comm_semiring_1,recpower,number_ring}" 1 "op -" "uminus"]
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28699
diff changeset
   243
  proof qed simp_all
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   244
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   245
26462
dac4e2bce00d avoid rebinding of existing facts;
wenzelm
parents: 26314
diff changeset
   246
declaration {* normalizer_funs @{thm class_ring.gb_ring_axioms'} *}
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   247
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   248
use "Tools/Groebner_Basis/normalizer.ML"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   249
27666
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   250
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   251
method_setup sring_norm = {*
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   252
  Method.ctxt_args (fn ctxt => Method.SIMPLE_METHOD' (Normalizer.semiring_normalize_tac ctxt))
23458
b2267a9e9e28 tuned comments;
wenzelm
parents: 23389
diff changeset
   253
*} "semiring normalizer"
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   254
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   255
23327
1654013ec97c Added instantiation of algebra method to fields
chaieb
parents: 23312
diff changeset
   256
locale gb_field = gb_ring +
1654013ec97c Added instantiation of algebra method to fields
chaieb
parents: 23312
diff changeset
   257
  fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
1654013ec97c Added instantiation of algebra method to fields
chaieb
parents: 23312
diff changeset
   258
    and inverse:: "'a \<Rightarrow> 'a"
1654013ec97c Added instantiation of algebra method to fields
chaieb
parents: 23312
diff changeset
   259
  assumes divide: "divide x y = mul x (inverse y)"
1654013ec97c Added instantiation of algebra method to fields
chaieb
parents: 23312
diff changeset
   260
     and inverse: "inverse x = divide r1 x"
1654013ec97c Added instantiation of algebra method to fields
chaieb
parents: 23312
diff changeset
   261
begin
1654013ec97c Added instantiation of algebra method to fields
chaieb
parents: 23312
diff changeset
   262
26462
dac4e2bce00d avoid rebinding of existing facts;
wenzelm
parents: 26314
diff changeset
   263
lemmas gb_field_axioms' =
26314
9c39fc898fff avoid rebinding of existing facts;
wenzelm
parents: 26199
diff changeset
   264
  gb_field_axioms [normalizer
9c39fc898fff avoid rebinding of existing facts;
wenzelm
parents: 26199
diff changeset
   265
    semiring ops: semiring_ops
9c39fc898fff avoid rebinding of existing facts;
wenzelm
parents: 26199
diff changeset
   266
    semiring rules: semiring_rules
9c39fc898fff avoid rebinding of existing facts;
wenzelm
parents: 26199
diff changeset
   267
    ring ops: ring_ops
9c39fc898fff avoid rebinding of existing facts;
wenzelm
parents: 26199
diff changeset
   268
    ring rules: ring_rules]
23327
1654013ec97c Added instantiation of algebra method to fields
chaieb
parents: 23312
diff changeset
   269
1654013ec97c Added instantiation of algebra method to fields
chaieb
parents: 23312
diff changeset
   270
end
1654013ec97c Added instantiation of algebra method to fields
chaieb
parents: 23312
diff changeset
   271
23458
b2267a9e9e28 tuned comments;
wenzelm
parents: 23389
diff changeset
   272
23266
50f0a4f12ed3 tuned document;
wenzelm
parents: 23258
diff changeset
   273
subsection {* Groebner Bases *}
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   274
23258
9062e98fdab1 renamed locale ring/semiring to gb_ring/gb_semiring to avoid clash with Ring_and_Field versions;
wenzelm
parents: 23252
diff changeset
   275
locale semiringb = gb_semiring +
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   276
  assumes add_cancel: "add (x::'a) y = add x z \<longleftrightarrow> y = z"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   277
  and add_mul_solve: "add (mul w y) (mul x z) =
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   278
    add (mul w z) (mul x y) \<longleftrightarrow> w = x \<or> y = z"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   279
begin
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   280
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   281
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)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   282
proof-
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   283
  have "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> \<not> (a = b \<or> c = d)" by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   284
  also have "\<dots> \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   285
    using add_mul_solve by blast
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   286
  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)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   287
    by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   288
qed
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   289
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   290
lemma add_scale_eq_noteq: "\<lbrakk>r \<noteq> r0 ; (a = b) \<and> ~(c = d)\<rbrakk>
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   291
  \<Longrightarrow> add a (mul r c) \<noteq> add b (mul r d)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   292
proof(clarify)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   293
  assume nz: "r\<noteq> r0" and cnd: "c\<noteq>d"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   294
    and eq: "add b (mul r c) = add b (mul r d)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   295
  hence "mul r c = mul r d" using cnd add_cancel by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   296
  hence "add (mul r0 d) (mul r c) = add (mul r0 c) (mul r d)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   297
    using mul_0 add_cancel by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   298
  thus "False" using add_mul_solve nz cnd by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   299
qed
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   300
25250
b3a485b98963 (1) added axiom to ringb and theorems to enable algebra to prove the ideal membership problem; (2) Method algebra now calls algebra_tac which first tries to solve a universal formula, then in case of failure trie to solve the ideal membership problem (see HOL/Tools/Groebner_Basis/groebner.ML)
chaieb
parents: 23573
diff changeset
   301
lemma add_r0_iff: " x = add x a \<longleftrightarrow> a = r0"
b3a485b98963 (1) added axiom to ringb and theorems to enable algebra to prove the ideal membership problem; (2) Method algebra now calls algebra_tac which first tries to solve a universal formula, then in case of failure trie to solve the ideal membership problem (see HOL/Tools/Groebner_Basis/groebner.ML)
chaieb
parents: 23573
diff changeset
   302
proof-
b3a485b98963 (1) added axiom to ringb and theorems to enable algebra to prove the ideal membership problem; (2) Method algebra now calls algebra_tac which first tries to solve a universal formula, then in case of failure trie to solve the ideal membership problem (see HOL/Tools/Groebner_Basis/groebner.ML)
chaieb
parents: 23573
diff changeset
   303
  have "a = r0 \<longleftrightarrow> add x a = add x r0" by (simp add: add_cancel)
b3a485b98963 (1) added axiom to ringb and theorems to enable algebra to prove the ideal membership problem; (2) Method algebra now calls algebra_tac which first tries to solve a universal formula, then in case of failure trie to solve the ideal membership problem (see HOL/Tools/Groebner_Basis/groebner.ML)
chaieb
parents: 23573
diff changeset
   304
  thus "x = add x a \<longleftrightarrow> a = r0" by (auto simp add: add_c add_0)
b3a485b98963 (1) added axiom to ringb and theorems to enable algebra to prove the ideal membership problem; (2) Method algebra now calls algebra_tac which first tries to solve a universal formula, then in case of failure trie to solve the ideal membership problem (see HOL/Tools/Groebner_Basis/groebner.ML)
chaieb
parents: 23573
diff changeset
   305
qed
b3a485b98963 (1) added axiom to ringb and theorems to enable algebra to prove the ideal membership problem; (2) Method algebra now calls algebra_tac which first tries to solve a universal formula, then in case of failure trie to solve the ideal membership problem (see HOL/Tools/Groebner_Basis/groebner.ML)
chaieb
parents: 23573
diff changeset
   306
26462
dac4e2bce00d avoid rebinding of existing facts;
wenzelm
parents: 26314
diff changeset
   307
declare gb_semiring_axioms' [normalizer del]
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   308
26462
dac4e2bce00d avoid rebinding of existing facts;
wenzelm
parents: 26314
diff changeset
   309
lemmas semiringb_axioms' = semiringb_axioms [normalizer
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   310
  semiring ops: semiring_ops
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   311
  semiring rules: semiring_rules
26314
9c39fc898fff avoid rebinding of existing facts;
wenzelm
parents: 26199
diff changeset
   312
  idom rules: noteq_reduce add_scale_eq_noteq]
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   313
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   314
end
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   315
25250
b3a485b98963 (1) added axiom to ringb and theorems to enable algebra to prove the ideal membership problem; (2) Method algebra now calls algebra_tac which first tries to solve a universal formula, then in case of failure trie to solve the ideal membership problem (see HOL/Tools/Groebner_Basis/groebner.ML)
chaieb
parents: 23573
diff changeset
   316
locale ringb = semiringb + gb_ring + 
b3a485b98963 (1) added axiom to ringb and theorems to enable algebra to prove the ideal membership problem; (2) Method algebra now calls algebra_tac which first tries to solve a universal formula, then in case of failure trie to solve the ideal membership problem (see HOL/Tools/Groebner_Basis/groebner.ML)
chaieb
parents: 23573
diff changeset
   317
  assumes subr0_iff: "sub x y = r0 \<longleftrightarrow> x = y"
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   318
begin
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   319
26462
dac4e2bce00d avoid rebinding of existing facts;
wenzelm
parents: 26314
diff changeset
   320
declare gb_ring_axioms' [normalizer del]
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   321
26462
dac4e2bce00d avoid rebinding of existing facts;
wenzelm
parents: 26314
diff changeset
   322
lemmas ringb_axioms' = ringb_axioms [normalizer
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   323
  semiring ops: semiring_ops
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   324
  semiring rules: semiring_rules
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   325
  ring ops: ring_ops
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   326
  ring rules: ring_rules
25250
b3a485b98963 (1) added axiom to ringb and theorems to enable algebra to prove the ideal membership problem; (2) Method algebra now calls algebra_tac which first tries to solve a universal formula, then in case of failure trie to solve the ideal membership problem (see HOL/Tools/Groebner_Basis/groebner.ML)
chaieb
parents: 23573
diff changeset
   327
  idom rules: noteq_reduce add_scale_eq_noteq
26314
9c39fc898fff avoid rebinding of existing facts;
wenzelm
parents: 26199
diff changeset
   328
  ideal rules: subr0_iff add_r0_iff]
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   329
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   330
end
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   331
25250
b3a485b98963 (1) added axiom to ringb and theorems to enable algebra to prove the ideal membership problem; (2) Method algebra now calls algebra_tac which first tries to solve a universal formula, then in case of failure trie to solve the ideal membership problem (see HOL/Tools/Groebner_Basis/groebner.ML)
chaieb
parents: 23573
diff changeset
   332
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   333
lemma no_zero_divirors_neq0:
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   334
  assumes az: "(a::'a::no_zero_divisors) \<noteq> 0"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   335
    and ab: "a*b = 0" shows "b = 0"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   336
proof -
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   337
  { assume bz: "b \<noteq> 0"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   338
    from no_zero_divisors [OF az bz] ab have False by blast }
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   339
  thus "b = 0" by blast
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   340
qed
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   341
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   342
interpretation class_ringb: ringb
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   343
  ["op +" "op *" "op ^" "0::'a::{idom,recpower,number_ring}" "1" "op -" "uminus"]
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23458
diff changeset
   344
proof(unfold_locales, simp add: ring_simps power_Suc, auto)
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   345
  fix w x y z ::"'a::{idom,recpower,number_ring}"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   346
  assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   347
  hence ynz': "y - z \<noteq> 0" by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   348
  from p have "w * y + x* z - w*z - x*y = 0" by simp
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23458
diff changeset
   349
  hence "w* (y - z) - x * (y - z) = 0" by (simp add: ring_simps)
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23458
diff changeset
   350
  hence "(y - z) * (w - x) = 0" by (simp add: ring_simps)
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   351
  with  no_zero_divirors_neq0 [OF ynz']
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   352
  have "w - x = 0" by blast
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   353
  thus "w = x"  by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   354
qed
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   355
26462
dac4e2bce00d avoid rebinding of existing facts;
wenzelm
parents: 26314
diff changeset
   356
declaration {* normalizer_funs @{thm class_ringb.ringb_axioms'} *}
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   357
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   358
interpretation natgb: semiringb
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   359
  ["op +" "op *" "op ^" "0::nat" "1"]
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23458
diff changeset
   360
proof (unfold_locales, simp add: ring_simps power_Suc)
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   361
  fix w x y z ::"nat"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   362
  { assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   363
    hence "y < z \<or> y > z" by arith
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   364
    moreover {
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   365
      assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z - y" in exI, auto)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   366
      then obtain k where kp: "k>0" and yz:"z = y + k" by blast
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23458
diff changeset
   367
      from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz ring_simps)
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   368
      hence "x*k = w*k" by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   369
      hence "w = x" using kp by (simp add: mult_cancel2) }
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   370
    moreover {
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   371
      assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y - z" in exI, auto)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   372
      then obtain k where kp: "k>0" and yz:"y = z + k" by blast
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23458
diff changeset
   373
      from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz ring_simps)
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   374
      hence "w*k = x*k" by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   375
      hence "w = x" using kp by (simp add: mult_cancel2)}
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   376
    ultimately have "w=x" by blast }
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   377
  thus "(w * y + x * z = w * z + x * y) = (w = x \<or> y = z)" by auto
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   378
qed
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   379
26462
dac4e2bce00d avoid rebinding of existing facts;
wenzelm
parents: 26314
diff changeset
   380
declaration {* normalizer_funs @{thm natgb.semiringb_axioms'} *}
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   381
23327
1654013ec97c Added instantiation of algebra method to fields
chaieb
parents: 23312
diff changeset
   382
locale fieldgb = ringb + gb_field
1654013ec97c Added instantiation of algebra method to fields
chaieb
parents: 23312
diff changeset
   383
begin
1654013ec97c Added instantiation of algebra method to fields
chaieb
parents: 23312
diff changeset
   384
26462
dac4e2bce00d avoid rebinding of existing facts;
wenzelm
parents: 26314
diff changeset
   385
declare gb_field_axioms' [normalizer del]
23327
1654013ec97c Added instantiation of algebra method to fields
chaieb
parents: 23312
diff changeset
   386
26462
dac4e2bce00d avoid rebinding of existing facts;
wenzelm
parents: 26314
diff changeset
   387
lemmas fieldgb_axioms' = fieldgb_axioms [normalizer
23327
1654013ec97c Added instantiation of algebra method to fields
chaieb
parents: 23312
diff changeset
   388
  semiring ops: semiring_ops
1654013ec97c Added instantiation of algebra method to fields
chaieb
parents: 23312
diff changeset
   389
  semiring rules: semiring_rules
1654013ec97c Added instantiation of algebra method to fields
chaieb
parents: 23312
diff changeset
   390
  ring ops: ring_ops
1654013ec97c Added instantiation of algebra method to fields
chaieb
parents: 23312
diff changeset
   391
  ring rules: ring_rules
25250
b3a485b98963 (1) added axiom to ringb and theorems to enable algebra to prove the ideal membership problem; (2) Method algebra now calls algebra_tac which first tries to solve a universal formula, then in case of failure trie to solve the ideal membership problem (see HOL/Tools/Groebner_Basis/groebner.ML)
chaieb
parents: 23573
diff changeset
   392
  idom rules: noteq_reduce add_scale_eq_noteq
26314
9c39fc898fff avoid rebinding of existing facts;
wenzelm
parents: 26199
diff changeset
   393
  ideal rules: subr0_iff add_r0_iff]
9c39fc898fff avoid rebinding of existing facts;
wenzelm
parents: 26199
diff changeset
   394
23327
1654013ec97c Added instantiation of algebra method to fields
chaieb
parents: 23312
diff changeset
   395
end
1654013ec97c Added instantiation of algebra method to fields
chaieb
parents: 23312
diff changeset
   396
1654013ec97c Added instantiation of algebra method to fields
chaieb
parents: 23312
diff changeset
   397
23258
9062e98fdab1 renamed locale ring/semiring to gb_ring/gb_semiring to avoid clash with Ring_and_Field versions;
wenzelm
parents: 23252
diff changeset
   398
lemmas bool_simps = simp_thms(1-34)
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   399
lemma dnf:
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   400
    "(P & (Q | R)) = ((P&Q) | (P&R))" "((Q | R) & P) = ((Q&P) | (R&P))"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   401
    "(P \<and> Q) = (Q \<and> P)" "(P \<or> Q) = (Q \<or> P)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   402
  by blast+
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   403
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   404
lemmas weak_dnf_simps = dnf bool_simps
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   405
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   406
lemma nnf_simps:
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   407
    "(\<not>(P \<and> Q)) = (\<not>P \<or> \<not>Q)" "(\<not>(P \<or> Q)) = (\<not>P \<and> \<not>Q)" "(P \<longrightarrow> Q) = (\<not>P \<or> Q)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   408
    "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not> \<not>(P)) = P"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   409
  by blast+
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   410
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   411
lemma PFalse:
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   412
    "P \<equiv> False \<Longrightarrow> \<not> P"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   413
    "\<not> P \<Longrightarrow> (P \<equiv> False)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   414
  by auto
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   415
use "Tools/Groebner_Basis/groebner.ML"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   416
23332
b91295432e6d algebra_tac moved to file Tools/Groebner_Basis/groebner.ML; Method now takes theorems to be added or deleted from a simpset for simplificatio *before* the core method starts
chaieb
parents: 23330
diff changeset
   417
method_setup algebra =
23458
b2267a9e9e28 tuned comments;
wenzelm
parents: 23389
diff changeset
   418
{*
23332
b91295432e6d algebra_tac moved to file Tools/Groebner_Basis/groebner.ML; Method now takes theorems to be added or deleted from a simpset for simplificatio *before* the core method starts
chaieb
parents: 23330
diff changeset
   419
let
b91295432e6d algebra_tac moved to file Tools/Groebner_Basis/groebner.ML; Method now takes theorems to be added or deleted from a simpset for simplificatio *before* the core method starts
chaieb
parents: 23330
diff changeset
   420
 fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ()
b91295432e6d algebra_tac moved to file Tools/Groebner_Basis/groebner.ML; Method now takes theorems to be added or deleted from a simpset for simplificatio *before* the core method starts
chaieb
parents: 23330
diff changeset
   421
 val addN = "add"
b91295432e6d algebra_tac moved to file Tools/Groebner_Basis/groebner.ML; Method now takes theorems to be added or deleted from a simpset for simplificatio *before* the core method starts
chaieb
parents: 23330
diff changeset
   422
 val delN = "del"
b91295432e6d algebra_tac moved to file Tools/Groebner_Basis/groebner.ML; Method now takes theorems to be added or deleted from a simpset for simplificatio *before* the core method starts
chaieb
parents: 23330
diff changeset
   423
 val any_keyword = keyword addN || keyword delN
b91295432e6d algebra_tac moved to file Tools/Groebner_Basis/groebner.ML; Method now takes theorems to be added or deleted from a simpset for simplificatio *before* the core method starts
chaieb
parents: 23330
diff changeset
   424
 val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
b91295432e6d algebra_tac moved to file Tools/Groebner_Basis/groebner.ML; Method now takes theorems to be added or deleted from a simpset for simplificatio *before* the core method starts
chaieb
parents: 23330
diff changeset
   425
in
b91295432e6d algebra_tac moved to file Tools/Groebner_Basis/groebner.ML; Method now takes theorems to be added or deleted from a simpset for simplificatio *before* the core method starts
chaieb
parents: 23330
diff changeset
   426
fn src => Method.syntax 
b91295432e6d algebra_tac moved to file Tools/Groebner_Basis/groebner.ML; Method now takes theorems to be added or deleted from a simpset for simplificatio *before* the core method starts
chaieb
parents: 23330
diff changeset
   427
    ((Scan.optional (keyword addN |-- thms) []) -- 
b91295432e6d algebra_tac moved to file Tools/Groebner_Basis/groebner.ML; Method now takes theorems to be added or deleted from a simpset for simplificatio *before* the core method starts
chaieb
parents: 23330
diff changeset
   428
    (Scan.optional (keyword delN |-- thms) [])) src 
b91295432e6d algebra_tac moved to file Tools/Groebner_Basis/groebner.ML; Method now takes theorems to be added or deleted from a simpset for simplificatio *before* the core method starts
chaieb
parents: 23330
diff changeset
   429
 #> (fn ((add_ths, del_ths), ctxt) => 
25250
b3a485b98963 (1) added axiom to ringb and theorems to enable algebra to prove the ideal membership problem; (2) Method algebra now calls algebra_tac which first tries to solve a universal formula, then in case of failure trie to solve the ideal membership problem (see HOL/Tools/Groebner_Basis/groebner.ML)
chaieb
parents: 23573
diff changeset
   430
       Method.SIMPLE_METHOD' (Groebner.algebra_tac add_ths del_ths ctxt))
23332
b91295432e6d algebra_tac moved to file Tools/Groebner_Basis/groebner.ML; Method now takes theorems to be added or deleted from a simpset for simplificatio *before* the core method starts
chaieb
parents: 23330
diff changeset
   431
end
25250
b3a485b98963 (1) added axiom to ringb and theorems to enable algebra to prove the ideal membership problem; (2) Method algebra now calls algebra_tac which first tries to solve a universal formula, then in case of failure trie to solve the ideal membership problem (see HOL/Tools/Groebner_Basis/groebner.ML)
chaieb
parents: 23573
diff changeset
   432
*} "solve polynomial equations over (semi)rings and ideal membership problems using Groebner bases"
27666
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   433
declare dvd_def[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   434
declare dvd_eq_mod_eq_0[symmetric, algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   435
declare nat_mod_div_trivial[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   436
declare nat_mod_mod_trivial[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   437
declare conjunct1[OF DIVISION_BY_ZERO, algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   438
declare conjunct2[OF DIVISION_BY_ZERO, algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   439
declare zmod_zdiv_equality[symmetric,algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   440
declare zdiv_zmod_equality[symmetric, algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   441
declare zdiv_zminus_zminus[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   442
declare zmod_zminus_zminus[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   443
declare zdiv_zminus2[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   444
declare zmod_zminus2[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   445
declare zdiv_zero[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   446
declare zmod_zero[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   447
declare zmod_1[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   448
declare zdiv_1[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   449
declare zmod_minus1_right[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   450
declare zdiv_minus1_right[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   451
declare mod_div_trivial[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   452
declare mod_mod_trivial[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   453
declare zmod_zmult_self1[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   454
declare zmod_zmult_self2[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   455
declare zmod_eq_0_iff[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   456
declare zdvd_0_left[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   457
declare zdvd1_eq[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   458
declare zmod_eq_dvd_iff[algebra]
1436d81d1294 Relevant rules added to algebra's context
chaieb
parents: 26462
diff changeset
   459
declare nat_mod_eq_iff[algebra]
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   460
28402
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   461
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   462
subsection{* Groebner Bases for fields *}
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   463
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   464
interpretation class_fieldgb:
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   465
  fieldgb["op +" "op *" "op ^" "0::'a::{field,recpower,number_ring}" "1" "op -" "uminus" "op /" "inverse"] apply (unfold_locales) by (simp_all add: divide_inverse)
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   466
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   467
lemma divide_Numeral1: "(x::'a::{field,number_ring}) / Numeral1 = x" by simp
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   468
lemma divide_Numeral0: "(x::'a::{field,number_ring, division_by_zero}) / Numeral0 = 0"
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   469
  by simp
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   470
lemma mult_frac_frac: "((x::'a::{field,division_by_zero}) / y) * (z / w) = (x*z) / (y*w)"
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   471
  by simp
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   472
lemma mult_frac_num: "((x::'a::{field, division_by_zero}) / y) * z  = (x*z) / y"
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   473
  by simp
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   474
lemma mult_num_frac: "((x::'a::{field, division_by_zero}) / y) * z  = (x*z) / y"
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   475
  by simp
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   476
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   477
lemma Numeral1_eq1_nat: "(1::nat) = Numeral1" by simp
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   478
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   479
lemma add_frac_num: "y\<noteq> 0 \<Longrightarrow> (x::'a::{field, division_by_zero}) / y + z = (x + z*y) / y"
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   480
  by (simp add: add_divide_distrib)
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   481
lemma add_num_frac: "y\<noteq> 0 \<Longrightarrow> z + (x::'a::{field, division_by_zero}) / y = (x + z*y) / y"
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   482
  by (simp add: add_divide_distrib)
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   483
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   484
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   485
ML{* 
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   486
local
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   487
 val zr = @{cpat "0"}
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   488
 val zT = ctyp_of_term zr
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   489
 val geq = @{cpat "op ="}
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   490
 val eqT = Thm.dest_ctyp (ctyp_of_term geq) |> hd
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   491
 val add_frac_eq = mk_meta_eq @{thm "add_frac_eq"}
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   492
 val add_frac_num = mk_meta_eq @{thm "add_frac_num"}
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   493
 val add_num_frac = mk_meta_eq @{thm "add_num_frac"}
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   494
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   495
 fun prove_nz ss T t =
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   496
    let
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   497
      val z = instantiate_cterm ([(zT,T)],[]) zr
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   498
      val eq = instantiate_cterm ([(eqT,T)],[]) geq
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   499
      val th = Simplifier.rewrite (ss addsimps simp_thms)
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   500
           (Thm.capply @{cterm "Trueprop"} (Thm.capply @{cterm "Not"}
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   501
                  (Thm.capply (Thm.capply eq t) z)))
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   502
    in equal_elim (symmetric th) TrueI
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   503
    end
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   504
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   505
 fun proc phi ss ct =
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   506
  let
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   507
    val ((x,y),(w,z)) =
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   508
         (Thm.dest_binop #> (fn (a,b) => (Thm.dest_binop a, Thm.dest_binop b))) ct
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   509
    val _ = map (HOLogic.dest_number o term_of) [x,y,z,w]
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   510
    val T = ctyp_of_term x
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   511
    val [y_nz, z_nz] = map (prove_nz ss T) [y, z]
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   512
    val th = instantiate' [SOME T] (map SOME [y,z,x,w]) add_frac_eq
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   513
  in SOME (implies_elim (implies_elim th y_nz) z_nz)
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   514
  end
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   515
  handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   516
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   517
 fun proc2 phi ss ct =
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   518
  let
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   519
    val (l,r) = Thm.dest_binop ct
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   520
    val T = ctyp_of_term l
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   521
  in (case (term_of l, term_of r) of
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   522
      (Const(@{const_name "HOL.divide"},_)$_$_, _) =>
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   523
        let val (x,y) = Thm.dest_binop l val z = r
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   524
            val _ = map (HOLogic.dest_number o term_of) [x,y,z]
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   525
            val ynz = prove_nz ss T y
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   526
        in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,x,z]) add_frac_num) ynz)
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   527
        end
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   528
     | (_, Const (@{const_name "HOL.divide"},_)$_$_) =>
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   529
        let val (x,y) = Thm.dest_binop r val z = l
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   530
            val _ = map (HOLogic.dest_number o term_of) [x,y,z]
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   531
            val ynz = prove_nz ss T y
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   532
        in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,z,x]) add_num_frac) ynz)
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   533
        end
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   534
     | _ => NONE)
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   535
  end
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   536
  handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   537
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   538
 fun is_number (Const(@{const_name "HOL.divide"},_)$a$b) = is_number a andalso is_number b
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   539
   | is_number t = can HOLogic.dest_number t
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   540
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   541
 val is_number = is_number o term_of
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   542
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   543
 fun proc3 phi ss ct =
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   544
  (case term_of ct of
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   545
    Const(@{const_name HOL.less},_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ =>
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   546
      let
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   547
        val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   548
        val _ = map is_number [a,b,c]
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   549
        val T = ctyp_of_term c
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   550
        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_less_eq"}
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   551
      in SOME (mk_meta_eq th) end
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   552
  | Const(@{const_name HOL.less_eq},_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ =>
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   553
      let
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   554
        val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   555
        val _ = map is_number [a,b,c]
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   556
        val T = ctyp_of_term c
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   557
        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_le_eq"}
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   558
      in SOME (mk_meta_eq th) end
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   559
  | Const("op =",_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ =>
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   560
      let
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   561
        val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   562
        val _ = map is_number [a,b,c]
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   563
        val T = ctyp_of_term c
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   564
        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_eq_eq"}
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   565
      in SOME (mk_meta_eq th) end
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   566
  | Const(@{const_name HOL.less},_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) =>
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   567
    let
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   568
      val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   569
        val _ = map is_number [a,b,c]
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   570
        val T = ctyp_of_term c
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   571
        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "less_divide_eq"}
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   572
      in SOME (mk_meta_eq th) end
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   573
  | Const(@{const_name HOL.less_eq},_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) =>
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   574
    let
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   575
      val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   576
        val _ = map is_number [a,b,c]
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   577
        val T = ctyp_of_term c
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   578
        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "le_divide_eq"}
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   579
      in SOME (mk_meta_eq th) end
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   580
  | Const("op =",_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) =>
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   581
    let
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   582
      val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   583
        val _ = map is_number [a,b,c]
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   584
        val T = ctyp_of_term c
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   585
        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "eq_divide_eq"}
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   586
      in SOME (mk_meta_eq th) end
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   587
  | _ => NONE)
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   588
  handle TERM _ => NONE | CTERM _ => NONE | THM _ => NONE
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   589
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   590
val add_frac_frac_simproc =
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   591
       make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + (?w::?'a::field)/?z"}],
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   592
                     name = "add_frac_frac_simproc",
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   593
                     proc = proc, identifier = []}
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   594
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   595
val add_frac_num_simproc =
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   596
       make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + ?z"}, @{cpat "?z + (?x::?'a::field)/?y"}],
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   597
                     name = "add_frac_num_simproc",
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   598
                     proc = proc2, identifier = []}
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   599
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   600
val ord_frac_simproc =
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   601
  make_simproc
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   602
    {lhss = [@{cpat "(?a::(?'a::{field, ord}))/?b < ?c"},
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   603
             @{cpat "(?a::(?'a::{field, ord}))/?b \<le> ?c"},
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   604
             @{cpat "?c < (?a::(?'a::{field, ord}))/?b"},
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   605
             @{cpat "?c \<le> (?a::(?'a::{field, ord}))/?b"},
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   606
             @{cpat "?c = ((?a::(?'a::{field, ord}))/?b)"},
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   607
             @{cpat "((?a::(?'a::{field, ord}))/ ?b) = ?c"}],
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   608
             name = "ord_frac_simproc", proc = proc3, identifier = []}
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   609
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   610
val nat_arith = map thm ["add_nat_number_of", "diff_nat_number_of",
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   611
               "mult_nat_number_of", "eq_nat_number_of", "less_nat_number_of"]
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   612
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   613
val comp_arith = (map thm ["Let_def", "if_False", "if_True", "add_0",
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   614
                 "add_Suc", "add_number_of_left", "mult_number_of_left",
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   615
                 "Suc_eq_add_numeral_1"])@
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   616
                 (map (fn s => thm s RS sym) ["numeral_1_eq_1", "numeral_0_eq_0"])
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   617
                 @ @{thms arith_simps} @ nat_arith @ @{thms rel_simps}
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   618
val ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"},
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   619
           @{thm "divide_Numeral1"},
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   620
           @{thm "Ring_and_Field.divide_zero"}, @{thm "divide_Numeral0"},
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   621
           @{thm "divide_divide_eq_left"}, @{thm "mult_frac_frac"},
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   622
           @{thm "mult_num_frac"}, @{thm "mult_frac_num"},
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   623
           @{thm "mult_frac_frac"}, @{thm "times_divide_eq_right"},
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   624
           @{thm "times_divide_eq_left"}, @{thm "divide_divide_eq_right"},
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   625
           @{thm "diff_def"}, @{thm "minus_divide_left"},
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   626
           @{thm "Numeral1_eq1_nat"}, @{thm "add_divide_distrib"} RS sym]
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   627
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   628
local
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   629
open Conv
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   630
in
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   631
val comp_conv = (Simplifier.rewrite
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   632
(HOL_basic_ss addsimps @{thms "Groebner_Basis.comp_arith"}
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   633
              addsimps ths addsimps comp_arith addsimps simp_thms
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   634
              addsimprocs field_cancel_numeral_factors
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   635
               addsimprocs [add_frac_frac_simproc, add_frac_num_simproc,
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   636
                            ord_frac_simproc]
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   637
                addcongs [@{thm "if_weak_cong"}]))
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   638
then_conv (Simplifier.rewrite (HOL_basic_ss addsimps
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   639
  [@{thm numeral_1_eq_1},@{thm numeral_0_eq_0}] @ @{thms numerals(1-2)}))
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   640
end
28402
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   641
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   642
fun numeral_is_const ct =
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   643
  case term_of ct of
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   644
   Const (@{const_name "HOL.divide"},_) $ a $ b =>
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   645
     numeral_is_const (Thm.dest_arg1 ct) andalso numeral_is_const (Thm.dest_arg ct)
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   646
 | Const (@{const_name "HOL.uminus"},_)$t => numeral_is_const (Thm.dest_arg ct)
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   647
 | t => can HOLogic.dest_number t
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   648
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   649
fun dest_const ct = ((case term_of ct of
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   650
   Const (@{const_name "HOL.divide"},_) $ a $ b=>
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   651
    Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b))
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   652
 | t => Rat.rat_of_int (snd (HOLogic.dest_number t))) 
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   653
   handle TERM _ => error "ring_dest_const")
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   654
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   655
fun mk_const phi cT x =
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   656
 let val (a, b) = Rat.quotient_of_rat x
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   657
 in if b = 1 then Numeral.mk_cnumber cT a
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   658
    else Thm.capply
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   659
         (Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"})
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   660
                     (Numeral.mk_cnumber cT a))
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   661
         (Numeral.mk_cnumber cT b)
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   662
  end
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   663
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   664
in
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   665
 val field_comp_conv = comp_conv;
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   666
 val fieldgb_declaration = 
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   667
  NormalizerData.funs @{thm class_fieldgb.fieldgb_axioms'}
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   668
   {is_const = K numeral_is_const,
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   669
    dest_const = K dest_const,
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   670
    mk_const = mk_const,
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   671
    conv = K (K comp_conv)}
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   672
end;
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   673
*}
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   674
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   675
declaration fieldgb_declaration
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   676
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   677
end