src/HOL/Semiring_Normalization.thy
author haftmann
Sat, 08 May 2010 18:52:38 +0200
changeset 36756 c1ae8a0b4265
parent 36753 5cf4e9128f22
child 36845 d778c64fc35d
permissions -rw-r--r--
moved normalization proof tool infrastructure to canonical algebraic classes
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
36751
7f1da69cacb3 split of semiring normalization from Groebner theory; moved field_comp_conv to Numeral_Simproces
haftmann
parents: 36720
diff changeset
     1
(*  Title:      HOL/Semiring_Normalization.thy
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
     2
    Author:     Amine Chaieb, TU Muenchen
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
     3
*)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
     4
36751
7f1da69cacb3 split of semiring normalization from Groebner theory; moved field_comp_conv to Numeral_Simproces
haftmann
parents: 36720
diff changeset
     5
header {* Semiring normalization *}
28402
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
     6
36751
7f1da69cacb3 split of semiring normalization from Groebner theory; moved field_comp_conv to Numeral_Simproces
haftmann
parents: 36720
diff changeset
     7
theory Semiring_Normalization
36699
816da1023508 moved nat_arith ot Nat_Numeral: clarified normalizer setup
haftmann
parents: 36698
diff changeset
     8
imports Numeral_Simprocs Nat_Transfer
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
     9
uses
36753
5cf4e9128f22 renamed Normalizer to the more specific Semiring_Normalizer
haftmann
parents: 36751
diff changeset
    10
  "Tools/semiring_normalizer.ML"
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    11
begin
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    12
36756
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    13
text {* FIXME prelude *}
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    14
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    15
class comm_semiring_1_cancel_norm (*FIXME name*) = comm_semiring_1_cancel +
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    16
  assumes add_mult_solve: "w * y + x * z = w * z + x * y \<longleftrightarrow> w = x \<or> y = z"
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    17
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    18
sublocale idom < comm_semiring_1_cancel_norm
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    19
proof
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    20
  fix w x y z
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    21
  show "w * y + x * z = w * z + x * y \<longleftrightarrow> w = x \<or> y = z"
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    22
  proof
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    23
    assume "w * y + x * z = w * z + x * y"
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    24
    then have "w * y + x * z - w * z - x * y = 0" by (simp add: algebra_simps)
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    25
    then have "w * (y - z) - x * (y - z) = 0" by (simp add: algebra_simps)
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    26
    then have "(y - z) * (w - x) = 0" by (simp add: algebra_simps)
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    27
    then have "y - z = 0 \<or> w - x = 0" by (rule divisors_zero)
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    28
    then show "w = x \<or> y = z" by auto
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    29
  qed (auto simp add: add_ac)
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    30
qed
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    31
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    32
instance nat :: comm_semiring_1_cancel_norm
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    33
proof
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    34
  fix w x y z :: nat
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    35
  { assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    36
    hence "y < z \<or> y > z" by arith
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    37
    moreover {
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    38
      assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z - y" in exI, auto)
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    39
      then obtain k where kp: "k>0" and yz:"z = y + k" by blast
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    40
      from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz algebra_simps)
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    41
      hence "x*k = w*k" by simp
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    42
      hence "w = x" using kp by simp }
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    43
    moreover {
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    44
      assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y - z" in exI, auto)
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    45
      then obtain k where kp: "k>0" and yz:"y = z + k" by blast
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    46
      from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz algebra_simps)
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    47
      hence "w*k = x*k" by simp
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    48
      hence "w = x" using kp by simp }
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    49
    ultimately have "w=x" by blast }
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    50
  then show "w * y + x * z = w * z + x * y \<longleftrightarrow> w = x \<or> y = z" by auto
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    51
qed
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
    52
36753
5cf4e9128f22 renamed Normalizer to the more specific Semiring_Normalizer
haftmann
parents: 36751
diff changeset
    53
setup Semiring_Normalizer.setup
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    54
36712
2f4c318861b3 avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents: 36702
diff changeset
    55
locale normalizing_semiring =
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    56
  fixes add mul pwr r0 r1
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    57
  assumes add_a:"(add x (add y z) = add (add x y) z)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    58
    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
    59
    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
    60
    and mul_1:"mul r1 x = x" and  mul_0:"mul r0 x = r0"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    61
    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
    62
    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
    63
begin
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    64
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    65
lemma mul_pwr:"mul (pwr x p) (pwr x q) = pwr x (p + q)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    66
proof (induct p)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    67
  case 0
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    68
  then show ?case by (auto simp add: pwr_0 mul_1)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    69
next
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    70
  case Suc
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    71
  from this [symmetric] show ?case
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    72
    by (auto simp add: pwr_Suc mul_1 mul_a)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    73
qed
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    74
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    75
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
    76
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
    77
  fix q x y
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    78
  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
    79
  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
    80
    by (simp add: mul_a)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    81
  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
    82
  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
    83
  finally show "mul (mul x y) (mul (pwr x q) (pwr y q)) =
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    84
    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
    85
qed
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    86
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    87
lemma pwr_pwr: "pwr (pwr x p) q = pwr x (p * q)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    88
proof (induct p arbitrary: q)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    89
  case 0
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    90
  show ?case using pwr_Suc mul_1 pwr_0 by (induct q) auto
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    91
next
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    92
  case Suc
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    93
  thus ?case by (auto simp add: mul_pwr [symmetric] pwr_mul pwr_Suc)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    94
qed
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    95
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    96
lemma semiring_ops:
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    97
  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
    98
    and "TERM r0" and "TERM r1" .
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
    99
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   100
lemma semiring_rules:
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   101
  "add (mul a m) (mul b m) = mul (add a b) m"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   102
  "add (mul a m) m = mul (add a r1) m"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   103
  "add m (mul a m) = mul (add a r1) m"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   104
  "add m m = mul (add r1 r1) m"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   105
  "add r0 a = a"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   106
  "add a r0 = a"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   107
  "mul a b = mul b a"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   108
  "mul (add a b) c = add (mul a c) (mul b c)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   109
  "mul r0 a = r0"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   110
  "mul a r0 = r0"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   111
  "mul r1 a = a"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   112
  "mul a r1 = a"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   113
  "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   114
  "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   115
  "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   116
  "mul (mul lx ly) rx = mul (mul lx rx) ly"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   117
  "mul (mul lx ly) rx = mul lx (mul ly rx)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   118
  "mul lx (mul rx ry) = mul (mul lx rx) ry"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   119
  "mul lx (mul rx ry) = mul rx (mul lx ry)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   120
  "add (add a b) (add c d) = add (add a c) (add b d)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   121
  "add (add a b) c = add a (add b c)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   122
  "add a (add c d) = add c (add a d)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   123
  "add (add a b) c = add (add a c) b"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   124
  "add a c = add c a"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   125
  "add a (add c d) = add (add a c) d"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   126
  "mul (pwr x p) (pwr x q) = pwr x (p + q)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   127
  "mul x (pwr x q) = pwr x (Suc q)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   128
  "mul (pwr x q) x = pwr x (Suc q)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   129
  "mul x x = pwr x 2"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   130
  "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   131
  "pwr (pwr x p) q = pwr x (p * q)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   132
  "pwr x 0 = r1"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   133
  "pwr x 1 = x"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   134
  "mul x (add y z) = add (mul x y) (mul x z)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   135
  "pwr x (Suc q) = mul x (pwr x q)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   136
  "pwr x (2*n) = mul (pwr x n) (pwr x n)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   137
  "pwr x (Suc (2*n)) = mul x (mul (pwr x n) (pwr x n))"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   138
proof -
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   139
  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
   140
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
   141
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
   142
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
   143
next show "add r0 a = a" using add_0 by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   144
next show "add a r0 = a" using add_0 add_c by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   145
next show "mul a b = mul b a" using mul_c by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   146
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
   147
next show "mul r0 a = r0" using mul_0 by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   148
next show "mul a r0 = r0" using mul_0 mul_c by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   149
next show "mul r1 a = a" using mul_1 by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   150
next show "mul a r1 = a" using mul_1 mul_c by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   151
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
   152
    using mul_c mul_a by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   153
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
   154
    using mul_a by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   155
next
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   156
  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
   157
  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
   158
  finally
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   159
  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
   160
    using mul_c by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   161
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
   162
next
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   163
  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
   164
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
   165
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
   166
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
   167
    using add_c add_a by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   168
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
   169
next show "add a (add c d) = add c (add a d)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   170
    apply (simp add: add_a) by (simp only: add_c)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   171
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
   172
next show "add a c = add c a" by (rule add_c)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   173
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
   174
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
   175
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
   176
next show "mul (pwr x q) x = pwr x (Suc q)" using pwr_Suc mul_c by simp
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35092
diff changeset
   177
next show "mul x x = pwr x 2" by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c)
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   178
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
   179
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
   180
next show "pwr x 0 = r1" using pwr_0 .
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35092
diff changeset
   181
next show "pwr x 1 = x" unfolding One_nat_def by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c)
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   182
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
   183
next show "pwr x (Suc q) = mul x (pwr x q)" using pwr_Suc by simp
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35092
diff changeset
   184
next show "pwr x (2 * n) = mul (pwr x n) (pwr x n)" by (simp add: nat_number' mul_pwr)
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   185
next show "pwr x (Suc (2 * n)) = mul x (mul (pwr x n) (pwr x n))"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35092
diff changeset
   186
    by (simp add: nat_number' pwr_Suc mul_pwr)
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   187
qed
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   188
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   189
end
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   190
36712
2f4c318861b3 avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents: 36702
diff changeset
   191
sublocale comm_semiring_1
2f4c318861b3 avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents: 36702
diff changeset
   192
  < normalizing!: normalizing_semiring plus times power zero one
2f4c318861b3 avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents: 36702
diff changeset
   193
proof
2f4c318861b3 avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents: 36702
diff changeset
   194
qed (simp_all add: algebra_simps)
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   195
36756
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   196
lemmas (in comm_semiring_1) normalizing_comm_semiring_1_axioms =
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   197
  comm_semiring_1_axioms [normalizer
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   198
    semiring ops: normalizing.semiring_ops
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   199
    semiring rules: normalizing.semiring_rules]
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   200
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   201
declaration (in comm_semiring_1)
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   202
  {* Semiring_Normalizer.semiring_funs @{thm normalizing_comm_semiring_1_axioms} *}
23573
d85a277f90fd common normalizer_funs, avoid cterm_of;
wenzelm
parents: 23477
diff changeset
   203
36712
2f4c318861b3 avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents: 36702
diff changeset
   204
locale normalizing_ring = normalizing_semiring +
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   205
  fixes sub :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   206
    and neg :: "'a \<Rightarrow> 'a"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   207
  assumes neg_mul: "neg x = mul (neg r1) x"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   208
    and sub_add: "sub x y = add x (neg y)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   209
begin
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   210
28856
5e009a80fe6d Pure syntax: more coherent treatment of aprop, permanent TERM and &&&;
wenzelm
parents: 28823
diff changeset
   211
lemma ring_ops: shows "TERM (sub x y)" and "TERM (neg x)" .
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   212
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   213
lemmas ring_rules = neg_mul sub_add
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   214
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   215
end
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   216
36720
41da7025e59c proper sublocales; no free-floating ML sections
haftmann
parents: 36716
diff changeset
   217
sublocale comm_ring_1
41da7025e59c proper sublocales; no free-floating ML sections
haftmann
parents: 36716
diff changeset
   218
  < normalizing!: normalizing_ring plus times power zero one minus uminus
41da7025e59c proper sublocales; no free-floating ML sections
haftmann
parents: 36716
diff changeset
   219
proof
41da7025e59c proper sublocales; no free-floating ML sections
haftmann
parents: 36716
diff changeset
   220
qed (simp_all add: diff_minus)
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   221
36756
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   222
lemmas (in comm_ring_1) normalizing_comm_ring_1_axioms =
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   223
  comm_ring_1_axioms [normalizer
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   224
    semiring ops: normalizing.semiring_ops
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   225
    semiring rules: normalizing.semiring_rules
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   226
    ring ops: normalizing.ring_ops
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   227
    ring rules: normalizing.ring_rules]
30866
dd5117e2d73e now deals with devision in fields
chaieb
parents: 30729
diff changeset
   228
36756
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   229
declaration (in comm_ring_1)
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   230
  {* Semiring_Normalizer.semiring_funs @{thm normalizing_comm_ring_1_axioms} *}
23327
1654013ec97c Added instantiation of algebra method to fields
chaieb
parents: 23312
diff changeset
   231
36712
2f4c318861b3 avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents: 36702
diff changeset
   232
locale normalizing_semiring_cancel = normalizing_semiring +
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   233
  assumes add_cancel: "add (x::'a) y = add x z \<longleftrightarrow> y = z"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   234
  and add_mul_solve: "add (mul w y) (mul x z) =
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   235
    add (mul w z) (mul x y) \<longleftrightarrow> w = x \<or> y = z"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   236
begin
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   237
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   238
lemma noteq_reduce: "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   239
proof-
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   240
  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
   241
  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
   242
    using add_mul_solve by blast
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   243
  finally show "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   244
    by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   245
qed
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   246
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   247
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
   248
  \<Longrightarrow> add a (mul r c) \<noteq> add b (mul r d)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   249
proof(clarify)
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   250
  assume nz: "r\<noteq> r0" and cnd: "c\<noteq>d"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   251
    and eq: "add b (mul r c) = add b (mul r d)"
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   252
  hence "mul r c = mul r d" using cnd add_cancel by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   253
  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
   254
    using mul_0 add_cancel by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   255
  thus "False" using add_mul_solve nz cnd by simp
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   256
qed
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   257
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
   258
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
   259
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
   260
  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
   261
  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
   262
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
   263
36756
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   264
end
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   265
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   266
sublocale comm_semiring_1_cancel_norm
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   267
  < normalizing!: normalizing_semiring_cancel plus times power zero one
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   268
proof
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   269
qed (simp_all add: add_mult_solve)
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   270
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   271
declare (in comm_semiring_1_cancel_norm)
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   272
  normalizing_comm_semiring_1_axioms [normalizer del]
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   273
36756
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   274
lemmas (in comm_semiring_1_cancel_norm)
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   275
  normalizing_comm_semiring_1_cancel_norm_axioms =
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   276
  comm_semiring_1_cancel_norm_axioms [normalizer
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   277
    semiring ops: normalizing.semiring_ops
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   278
    semiring rules: normalizing.semiring_rules
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   279
    idom rules: normalizing.noteq_reduce normalizing.add_scale_eq_noteq]
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   280
36756
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   281
declaration (in comm_semiring_1_cancel_norm)
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   282
  {* Semiring_Normalizer.semiring_funs @{thm normalizing_comm_semiring_1_cancel_norm_axioms} *}
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   283
36712
2f4c318861b3 avoid references to groebner bases in names which have no references to groebner bases
haftmann
parents: 36702
diff changeset
   284
locale normalizing_ring_cancel = normalizing_semiring_cancel + normalizing_ring + 
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
   285
  assumes subr0_iff: "sub x y = r0 \<longleftrightarrow> x = y"
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   286
36720
41da7025e59c proper sublocales; no free-floating ML sections
haftmann
parents: 36716
diff changeset
   287
sublocale idom
41da7025e59c proper sublocales; no free-floating ML sections
haftmann
parents: 36716
diff changeset
   288
  < normalizing!: normalizing_ring_cancel plus times power zero one minus uminus
41da7025e59c proper sublocales; no free-floating ML sections
haftmann
parents: 36716
diff changeset
   289
proof
36756
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   290
qed simp
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   291
36756
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   292
declare (in idom) normalizing_comm_ring_1_axioms [normalizer del]
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   293
36756
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   294
lemmas (in idom) normalizing_idom_axioms = idom_axioms [normalizer
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   295
  semiring ops: normalizing.semiring_ops
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   296
  semiring rules: normalizing.semiring_rules
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   297
  ring ops: normalizing.ring_ops
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   298
  ring rules: normalizing.ring_rules
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   299
  idom rules: normalizing.noteq_reduce normalizing.add_scale_eq_noteq
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   300
  ideal rules: normalizing.subr0_iff normalizing.add_r0_iff]
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   301
36756
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   302
declaration (in idom)
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   303
  {* Semiring_Normalizer.semiring_funs @{thm normalizing_idom_axioms} *}
23252
67268bb40b21 Semiring normalization and Groebner Bases.
wenzelm
parents:
diff changeset
   304
36756
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   305
locale normalizing_field = normalizing_ring_cancel +
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   306
  fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   307
    and inverse:: "'a \<Rightarrow> 'a"
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   308
  assumes divide_inverse: "divide x y = mul x (inverse y)"
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   309
     and inverse_divide: "inverse x = divide r1 x"
23327
1654013ec97c Added instantiation of algebra method to fields
chaieb
parents: 23312
diff changeset
   310
begin
1654013ec97c Added instantiation of algebra method to fields
chaieb
parents: 23312
diff changeset
   311
36756
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   312
lemma field_ops: shows "TERM (divide x y)" and "TERM (inverse x)" .
23327
1654013ec97c Added instantiation of algebra method to fields
chaieb
parents: 23312
diff changeset
   313
36756
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   314
lemmas field_rules = divide_inverse inverse_divide
26314
9c39fc898fff avoid rebinding of existing facts;
wenzelm
parents: 26199
diff changeset
   315
23327
1654013ec97c Added instantiation of algebra method to fields
chaieb
parents: 23312
diff changeset
   316
end
1654013ec97c Added instantiation of algebra method to fields
chaieb
parents: 23312
diff changeset
   317
36720
41da7025e59c proper sublocales; no free-floating ML sections
haftmann
parents: 36716
diff changeset
   318
sublocale field 
36756
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   319
  < normalizing!: normalizing_field plus times power zero one minus uminus divide inverse
36720
41da7025e59c proper sublocales; no free-floating ML sections
haftmann
parents: 36716
diff changeset
   320
proof
41da7025e59c proper sublocales; no free-floating ML sections
haftmann
parents: 36716
diff changeset
   321
qed (simp_all add: divide_inverse)
28402
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   322
36756
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   323
lemmas (in field) normalizing_field_axioms =
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   324
  field_axioms [normalizer
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   325
    semiring ops: normalizing.semiring_ops
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   326
    semiring rules: normalizing.semiring_rules
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   327
    ring ops: normalizing.ring_ops
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   328
    ring rules: normalizing.ring_rules
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   329
    field ops: normalizing.field_ops
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   330
    field rules: normalizing.field_rules
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   331
    idom rules: normalizing.noteq_reduce normalizing.add_scale_eq_noteq
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   332
    ideal rules: normalizing.subr0_iff normalizing.add_r0_iff]
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   333
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   334
declaration (in field)
c1ae8a0b4265 moved normalization proof tool infrastructure to canonical algebraic classes
haftmann
parents: 36753
diff changeset
   335
  {* Semiring_Normalizer.field_funs @{thm normalizing_field_axioms} *}
28402
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   336
09e4aa3ddc25 clarified dependencies between arith tools
haftmann
parents: 27666
diff changeset
   337
end