src/HOL/Library/Abstract_Rat.thy
author chaieb
Mon Feb 09 17:21:46 2009 +0000 (2009-02-09)
changeset 29847 af32126ee729
parent 29667 53103fc8ffa3
child 30042 31039ee583fa
permissions -rw-r--r--
added Determinants to Library
haftmann@24197
     1
(*  Title:      HOL/Library/Abstract_Rat.thy
haftmann@24197
     2
    ID:         $Id$
haftmann@24197
     3
    Author:     Amine Chaieb
haftmann@24197
     4
*)
haftmann@24197
     5
haftmann@24197
     6
header {* Abstract rational numbers *}
haftmann@24197
     7
haftmann@24197
     8
theory Abstract_Rat
haftmann@27368
     9
imports Plain GCD
haftmann@24197
    10
begin
haftmann@24197
    11
haftmann@24197
    12
types Num = "int \<times> int"
wenzelm@25005
    13
wenzelm@25005
    14
abbreviation
wenzelm@25005
    15
  Num0_syn :: Num ("0\<^sub>N")
wenzelm@25005
    16
where "0\<^sub>N \<equiv> (0, 0)"
wenzelm@25005
    17
wenzelm@25005
    18
abbreviation
wenzelm@25005
    19
  Numi_syn :: "int \<Rightarrow> Num" ("_\<^sub>N")
wenzelm@25005
    20
where "i\<^sub>N \<equiv> (i, 1)"
haftmann@24197
    21
haftmann@24197
    22
definition
haftmann@24197
    23
  isnormNum :: "Num \<Rightarrow> bool"
haftmann@24197
    24
where
haftmann@27556
    25
  "isnormNum = (\<lambda>(a,b). (if a = 0 then b = 0 else b > 0 \<and> zgcd a b = 1))"
haftmann@24197
    26
haftmann@24197
    27
definition
haftmann@24197
    28
  normNum :: "Num \<Rightarrow> Num"
haftmann@24197
    29
where
haftmann@24197
    30
  "normNum = (\<lambda>(a,b). (if a=0 \<or> b = 0 then (0,0) else 
haftmann@27556
    31
  (let g = zgcd a b 
haftmann@24197
    32
   in if b > 0 then (a div g, b div g) else (- (a div g), - (b div g)))))"
haftmann@24197
    33
chaieb@27668
    34
declare zgcd_zdvd1[presburger] 
chaieb@27668
    35
declare zgcd_zdvd2[presburger]
haftmann@24197
    36
lemma normNum_isnormNum [simp]: "isnormNum (normNum x)"
haftmann@24197
    37
proof -
haftmann@24197
    38
  have " \<exists> a b. x = (a,b)" by auto
haftmann@24197
    39
  then obtain a b where x[simp]: "x = (a,b)" by blast
haftmann@24197
    40
  {assume "a=0 \<or> b = 0" hence ?thesis by (simp add: normNum_def isnormNum_def)}  
haftmann@24197
    41
  moreover
haftmann@24197
    42
  {assume anz: "a \<noteq> 0" and bnz: "b \<noteq> 0" 
haftmann@27556
    43
    let ?g = "zgcd a b"
haftmann@24197
    44
    let ?a' = "a div ?g"
haftmann@24197
    45
    let ?b' = "b div ?g"
haftmann@27556
    46
    let ?g' = "zgcd ?a' ?b'"
haftmann@27556
    47
    from anz bnz have "?g \<noteq> 0" by simp  with zgcd_pos[of a b] 
haftmann@24197
    48
    have gpos: "?g > 0"  by arith
chaieb@27668
    49
    have gdvd: "?g dvd a" "?g dvd b" by arith+ 
haftmann@24197
    50
    from zdvd_mult_div_cancel[OF gdvd(1)] zdvd_mult_div_cancel[OF gdvd(2)]
haftmann@24197
    51
    anz bnz
haftmann@24197
    52
    have nz':"?a' \<noteq> 0" "?b' \<noteq> 0" 
haftmann@27556
    53
      by - (rule notI,simp add:zgcd_def)+
chaieb@27668
    54
    from anz bnz have stupid: "a \<noteq> 0 \<or> b \<noteq> 0" by arith 
haftmann@27556
    55
    from div_zgcd_relprime[OF stupid] have gp1: "?g' = 1" .
haftmann@24197
    56
    from bnz have "b < 0 \<or> b > 0" by arith
haftmann@24197
    57
    moreover
haftmann@24197
    58
    {assume b: "b > 0"
chaieb@27668
    59
      from b have "?b' \<ge> 0" 
chaieb@27668
    60
	by (presburger add: pos_imp_zdiv_nonneg_iff[OF gpos])  
chaieb@27668
    61
      with nz' have b': "?b' > 0" by arith 
haftmann@24197
    62
      from b b' anz bnz nz' gp1 have ?thesis 
haftmann@24197
    63
	by (simp add: isnormNum_def normNum_def Let_def split_def fst_conv snd_conv)}
haftmann@24197
    64
    moreover {assume b: "b < 0"
haftmann@24197
    65
      {assume b': "?b' \<ge> 0" 
haftmann@24197
    66
	from gpos have th: "?g \<ge> 0" by arith
haftmann@24197
    67
	from mult_nonneg_nonneg[OF th b'] zdvd_mult_div_cancel[OF gdvd(2)]
chaieb@27668
    68
	have False using b by arith }
haftmann@24197
    69
      hence b': "?b' < 0" by (presburger add: linorder_not_le[symmetric]) 
haftmann@24197
    70
      from anz bnz nz' b b' gp1 have ?thesis 
haftmann@24197
    71
	by (simp add: isnormNum_def normNum_def Let_def split_def fst_conv snd_conv)}
haftmann@24197
    72
    ultimately have ?thesis by blast
haftmann@24197
    73
  }
haftmann@24197
    74
  ultimately show ?thesis by blast
haftmann@24197
    75
qed
haftmann@24197
    76
haftmann@24197
    77
text {* Arithmetic over Num *}
haftmann@24197
    78
haftmann@24197
    79
definition
haftmann@24197
    80
  Nadd :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "+\<^sub>N" 60)
haftmann@24197
    81
where
haftmann@24197
    82
  "Nadd = (\<lambda>(a,b) (a',b'). if a = 0 \<or> b = 0 then normNum(a',b') 
haftmann@24197
    83
    else if a'=0 \<or> b' = 0 then normNum(a,b) 
haftmann@24197
    84
    else normNum(a*b' + b*a', b*b'))"
haftmann@24197
    85
haftmann@24197
    86
definition
haftmann@24197
    87
  Nmul :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "*\<^sub>N" 60)
haftmann@24197
    88
where
haftmann@27556
    89
  "Nmul = (\<lambda>(a,b) (a',b'). let g = zgcd (a*a') (b*b') 
haftmann@24197
    90
    in (a*a' div g, b*b' div g))"
haftmann@24197
    91
haftmann@24197
    92
definition
haftmann@24197
    93
  Nneg :: "Num \<Rightarrow> Num" ("~\<^sub>N")
haftmann@24197
    94
where
haftmann@24197
    95
  "Nneg \<equiv> (\<lambda>(a,b). (-a,b))"
haftmann@24197
    96
haftmann@24197
    97
definition
haftmann@24197
    98
  Nsub :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "-\<^sub>N" 60)
haftmann@24197
    99
where
haftmann@24197
   100
  "Nsub = (\<lambda>a b. a +\<^sub>N ~\<^sub>N b)"
haftmann@24197
   101
haftmann@24197
   102
definition
haftmann@24197
   103
  Ninv :: "Num \<Rightarrow> Num" 
haftmann@24197
   104
where
haftmann@24197
   105
  "Ninv \<equiv> \<lambda>(a,b). if a < 0 then (-b, \<bar>a\<bar>) else (b,a)"
haftmann@24197
   106
haftmann@24197
   107
definition
haftmann@24197
   108
  Ndiv :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "\<div>\<^sub>N" 60)
haftmann@24197
   109
where
haftmann@24197
   110
  "Ndiv \<equiv> \<lambda>a b. a *\<^sub>N Ninv b"
haftmann@24197
   111
haftmann@24197
   112
lemma Nneg_normN[simp]: "isnormNum x \<Longrightarrow> isnormNum (~\<^sub>N x)"
haftmann@24197
   113
  by(simp add: isnormNum_def Nneg_def split_def)
haftmann@24197
   114
lemma Nadd_normN[simp]: "isnormNum (x +\<^sub>N y)"
haftmann@24197
   115
  by (simp add: Nadd_def split_def)
haftmann@24197
   116
lemma Nsub_normN[simp]: "\<lbrakk> isnormNum y\<rbrakk> \<Longrightarrow> isnormNum (x -\<^sub>N y)"
haftmann@24197
   117
  by (simp add: Nsub_def split_def)
haftmann@24197
   118
lemma Nmul_normN[simp]: assumes xn:"isnormNum x" and yn: "isnormNum y"
haftmann@24197
   119
  shows "isnormNum (x *\<^sub>N y)"
haftmann@24197
   120
proof-
haftmann@24197
   121
  have "\<exists>a b. x = (a,b)" and "\<exists> a' b'. y = (a',b')" by auto
haftmann@24197
   122
  then obtain a b a' b' where ab: "x = (a,b)"  and ab': "y = (a',b')" by blast 
haftmann@24197
   123
  {assume "a = 0"
haftmann@24197
   124
    hence ?thesis using xn ab ab'
haftmann@27556
   125
      by (simp add: zgcd_def isnormNum_def Let_def Nmul_def split_def)}
haftmann@24197
   126
  moreover
haftmann@24197
   127
  {assume "a' = 0"
haftmann@24197
   128
    hence ?thesis using yn ab ab' 
haftmann@27556
   129
      by (simp add: zgcd_def isnormNum_def Let_def Nmul_def split_def)}
haftmann@24197
   130
  moreover
haftmann@24197
   131
  {assume a: "a \<noteq>0" and a': "a'\<noteq>0"
haftmann@24197
   132
    hence bp: "b > 0" "b' > 0" using xn yn ab ab' by (simp_all add: isnormNum_def)
haftmann@24197
   133
    from mult_pos_pos[OF bp] have "x *\<^sub>N y = normNum (a*a', b*b')" 
haftmann@24197
   134
      using ab ab' a a' bp by (simp add: Nmul_def Let_def split_def normNum_def)
haftmann@24197
   135
    hence ?thesis by simp}
haftmann@24197
   136
  ultimately show ?thesis by blast
haftmann@24197
   137
qed
haftmann@24197
   138
haftmann@24197
   139
lemma Ninv_normN[simp]: "isnormNum x \<Longrightarrow> isnormNum (Ninv x)"
wenzelm@25005
   140
  by (simp add: Ninv_def isnormNum_def split_def)
haftmann@27556
   141
    (cases "fst x = 0", auto simp add: zgcd_commute)
haftmann@24197
   142
haftmann@24197
   143
lemma isnormNum_int[simp]: 
haftmann@24197
   144
  "isnormNum 0\<^sub>N" "isnormNum (1::int)\<^sub>N" "i \<noteq> 0 \<Longrightarrow> isnormNum i\<^sub>N"
haftmann@27556
   145
  by (simp_all add: isnormNum_def zgcd_def)
haftmann@24197
   146
haftmann@24197
   147
haftmann@24197
   148
text {* Relations over Num *}
haftmann@24197
   149
haftmann@24197
   150
definition
haftmann@24197
   151
  Nlt0:: "Num \<Rightarrow> bool" ("0>\<^sub>N")
haftmann@24197
   152
where
haftmann@24197
   153
  "Nlt0 = (\<lambda>(a,b). a < 0)"
haftmann@24197
   154
haftmann@24197
   155
definition
haftmann@24197
   156
  Nle0:: "Num \<Rightarrow> bool" ("0\<ge>\<^sub>N")
haftmann@24197
   157
where
haftmann@24197
   158
  "Nle0 = (\<lambda>(a,b). a \<le> 0)"
haftmann@24197
   159
haftmann@24197
   160
definition
haftmann@24197
   161
  Ngt0:: "Num \<Rightarrow> bool" ("0<\<^sub>N")
haftmann@24197
   162
where
haftmann@24197
   163
  "Ngt0 = (\<lambda>(a,b). a > 0)"
haftmann@24197
   164
haftmann@24197
   165
definition
haftmann@24197
   166
  Nge0:: "Num \<Rightarrow> bool" ("0\<le>\<^sub>N")
haftmann@24197
   167
where
haftmann@24197
   168
  "Nge0 = (\<lambda>(a,b). a \<ge> 0)"
haftmann@24197
   169
haftmann@24197
   170
definition
haftmann@24197
   171
  Nlt :: "Num \<Rightarrow> Num \<Rightarrow> bool" (infix "<\<^sub>N" 55)
haftmann@24197
   172
where
haftmann@24197
   173
  "Nlt = (\<lambda>a b. 0>\<^sub>N (a -\<^sub>N b))"
haftmann@24197
   174
haftmann@24197
   175
definition
haftmann@24197
   176
  Nle :: "Num \<Rightarrow> Num \<Rightarrow> bool" (infix "\<le>\<^sub>N" 55)
haftmann@24197
   177
where
haftmann@24197
   178
  "Nle = (\<lambda>a b. 0\<ge>\<^sub>N (a -\<^sub>N b))"
haftmann@24197
   179
haftmann@24197
   180
definition
haftmann@24197
   181
  "INum = (\<lambda>(a,b). of_int a / of_int b)"
haftmann@24197
   182
haftmann@24197
   183
lemma INum_int [simp]: "INum i\<^sub>N = ((of_int i) ::'a::field)" "INum 0\<^sub>N = (0::'a::field)"
haftmann@24197
   184
  by (simp_all add: INum_def)
haftmann@24197
   185
haftmann@24197
   186
lemma isnormNum_unique[simp]: 
haftmann@24197
   187
  assumes na: "isnormNum x" and nb: "isnormNum y" 
haftmann@24197
   188
  shows "((INum x ::'a::{ring_char_0,field, division_by_zero}) = INum y) = (x = y)" (is "?lhs = ?rhs")
haftmann@24197
   189
proof
haftmann@24197
   190
  have "\<exists> a b a' b'. x = (a,b) \<and> y = (a',b')" by auto
haftmann@24197
   191
  then obtain a b a' b' where xy[simp]: "x = (a,b)" "y=(a',b')" by blast
haftmann@24197
   192
  assume H: ?lhs 
haftmann@24197
   193
  {assume "a = 0 \<or> b = 0 \<or> a' = 0 \<or> b' = 0" hence ?rhs
haftmann@24197
   194
      using na nb H
haftmann@24197
   195
      apply (simp add: INum_def split_def isnormNum_def)
haftmann@24197
   196
      apply (cases "a = 0", simp_all)
haftmann@24197
   197
      apply (cases "b = 0", simp_all)
haftmann@24197
   198
      apply (cases "a' = 0", simp_all)
haftmann@24197
   199
      apply (cases "a' = 0", simp_all add: of_int_eq_0_iff)
haftmann@24197
   200
      done}
haftmann@24197
   201
  moreover
haftmann@24197
   202
  { assume az: "a \<noteq> 0" and bz: "b \<noteq> 0" and a'z: "a'\<noteq>0" and b'z: "b'\<noteq>0"
haftmann@24197
   203
    from az bz a'z b'z na nb have pos: "b > 0" "b' > 0" by (simp_all add: isnormNum_def)
haftmann@24197
   204
    from prems have eq:"a * b' = a'*b" 
haftmann@24197
   205
      by (simp add: INum_def  eq_divide_eq divide_eq_eq of_int_mult[symmetric] del: of_int_mult)
haftmann@27556
   206
    from prems have gcd1: "zgcd a b = 1" "zgcd b a = 1" "zgcd a' b' = 1" "zgcd b' a' = 1"       
haftmann@27556
   207
      by (simp_all add: isnormNum_def add: zgcd_commute)
chaieb@27668
   208
    from eq have raw_dvd: "a dvd a'*b" "b dvd b'*a" "a' dvd a*b'" "b' dvd b*a'"
chaieb@27668
   209
      apply - 
chaieb@27668
   210
      apply algebra
chaieb@27668
   211
      apply algebra
chaieb@27668
   212
      apply simp
chaieb@27668
   213
      apply algebra
haftmann@24197
   214
      done
haftmann@24197
   215
    from zdvd_dvd_eq[OF bz zrelprime_dvd_mult[OF gcd1(2) raw_dvd(2)]
haftmann@24197
   216
      zrelprime_dvd_mult[OF gcd1(4) raw_dvd(4)]]
chaieb@27668
   217
      have eq1: "b = b'" using pos by arith  
haftmann@24197
   218
      with eq have "a = a'" using pos by simp
haftmann@24197
   219
      with eq1 have ?rhs by simp}
haftmann@24197
   220
  ultimately show ?rhs by blast
haftmann@24197
   221
next
haftmann@24197
   222
  assume ?rhs thus ?lhs by simp
haftmann@24197
   223
qed
haftmann@24197
   224
haftmann@24197
   225
haftmann@24197
   226
lemma isnormNum0[simp]: "isnormNum x \<Longrightarrow> (INum x = (0::'a::{ring_char_0, field,division_by_zero})) = (x = 0\<^sub>N)"
haftmann@24197
   227
  unfolding INum_int(2)[symmetric]
haftmann@24197
   228
  by (rule isnormNum_unique, simp_all)
haftmann@24197
   229
haftmann@24197
   230
lemma of_int_div_aux: "d ~= 0 ==> ((of_int x)::'a::{field, ring_char_0}) / (of_int d) = 
haftmann@24197
   231
    of_int (x div d) + (of_int (x mod d)) / ((of_int d)::'a)"
haftmann@24197
   232
proof -
haftmann@24197
   233
  assume "d ~= 0"
haftmann@24197
   234
  hence dz: "of_int d \<noteq> (0::'a)" by (simp add: of_int_eq_0_iff)
haftmann@24197
   235
  let ?t = "of_int (x div d) * ((of_int d)::'a) + of_int(x mod d)"
haftmann@24197
   236
  let ?f = "\<lambda>x. x / of_int d"
haftmann@24197
   237
  have "x = (x div d) * d + x mod d"
haftmann@24197
   238
    by auto
haftmann@24197
   239
  then have eq: "of_int x = ?t"
haftmann@24197
   240
    by (simp only: of_int_mult[symmetric] of_int_add [symmetric])
haftmann@24197
   241
  then have "of_int x / of_int d = ?t / of_int d" 
haftmann@24197
   242
    using cong[OF refl[of ?f] eq] by simp
nipkow@29667
   243
  then show ?thesis by (simp add: add_divide_distrib algebra_simps prems)
haftmann@24197
   244
qed
haftmann@24197
   245
haftmann@24197
   246
lemma of_int_div: "(d::int) ~= 0 ==> d dvd n ==>
haftmann@24197
   247
    (of_int(n div d)::'a::{field, ring_char_0}) = of_int n / of_int d"
haftmann@24197
   248
  apply (frule of_int_div_aux [of d n, where ?'a = 'a])
haftmann@24197
   249
  apply simp
haftmann@24197
   250
  apply (simp add: zdvd_iff_zmod_eq_0)
haftmann@24197
   251
done
haftmann@24197
   252
haftmann@24197
   253
haftmann@24197
   254
lemma normNum[simp]: "INum (normNum x) = (INum x :: 'a::{ring_char_0,field, division_by_zero})"
haftmann@24197
   255
proof-
haftmann@24197
   256
  have "\<exists> a b. x = (a,b)" by auto
haftmann@24197
   257
  then obtain a b where x[simp]: "x = (a,b)" by blast
haftmann@24197
   258
  {assume "a=0 \<or> b = 0" hence ?thesis
haftmann@24197
   259
      by (simp add: INum_def normNum_def split_def Let_def)}
haftmann@24197
   260
  moreover 
haftmann@24197
   261
  {assume a: "a\<noteq>0" and b: "b\<noteq>0"
haftmann@27556
   262
    let ?g = "zgcd a b"
haftmann@24197
   263
    from a b have g: "?g \<noteq> 0"by simp
haftmann@24197
   264
    from of_int_div[OF g, where ?'a = 'a]
haftmann@24197
   265
    have ?thesis by (auto simp add: INum_def normNum_def split_def Let_def)}
haftmann@24197
   266
  ultimately show ?thesis by blast
haftmann@24197
   267
qed
haftmann@24197
   268
haftmann@26509
   269
lemma INum_normNum_iff: "(INum x ::'a::{field, division_by_zero, ring_char_0}) = INum y \<longleftrightarrow> normNum x = normNum y" (is "?lhs = ?rhs")
haftmann@24197
   270
proof -
haftmann@24197
   271
  have "normNum x = normNum y \<longleftrightarrow> (INum (normNum x) :: 'a) = INum (normNum y)"
haftmann@24197
   272
    by (simp del: normNum)
haftmann@24197
   273
  also have "\<dots> = ?lhs" by simp
haftmann@24197
   274
  finally show ?thesis by simp
haftmann@24197
   275
qed
haftmann@24197
   276
haftmann@24197
   277
lemma Nadd[simp]: "INum (x +\<^sub>N y) = INum x + (INum y :: 'a :: {ring_char_0,division_by_zero,field})"
haftmann@24197
   278
proof-
haftmann@24197
   279
let ?z = "0:: 'a"
haftmann@24197
   280
  have " \<exists> a b. x = (a,b)" " \<exists> a' b'. y = (a',b')" by auto
haftmann@24197
   281
  then obtain a b a' b' where x[simp]: "x = (a,b)" 
haftmann@24197
   282
    and y[simp]: "y = (a',b')" by blast
haftmann@24197
   283
  {assume "a=0 \<or> a'= 0 \<or> b =0 \<or> b' = 0" hence ?thesis 
haftmann@24197
   284
      apply (cases "a=0",simp_all add: Nadd_def)
haftmann@24197
   285
      apply (cases "b= 0",simp_all add: INum_def)
haftmann@24197
   286
       apply (cases "a'= 0",simp_all)
haftmann@24197
   287
       apply (cases "b'= 0",simp_all)
haftmann@24197
   288
       done }
haftmann@24197
   289
  moreover 
haftmann@24197
   290
  {assume aa':"a \<noteq> 0" "a'\<noteq> 0" and bb': "b \<noteq> 0" "b' \<noteq> 0" 
haftmann@24197
   291
    {assume z: "a * b' + b * a' = 0"
haftmann@24197
   292
      hence "of_int (a*b' + b*a') / (of_int b* of_int b') = ?z" by simp
haftmann@24197
   293
      hence "of_int b' * of_int a / (of_int b * of_int b') + of_int b * of_int a' / (of_int b * of_int b') = ?z"  by (simp add:add_divide_distrib) 
haftmann@24197
   294
      hence th: "of_int a / of_int b + of_int a' / of_int b' = ?z" using bb' aa' by simp 
haftmann@24197
   295
      from z aa' bb' have ?thesis 
haftmann@24197
   296
	by (simp add: th Nadd_def normNum_def INum_def split_def)}
haftmann@24197
   297
    moreover {assume z: "a * b' + b * a' \<noteq> 0"
haftmann@27556
   298
      let ?g = "zgcd (a * b' + b * a') (b*b')"
haftmann@24197
   299
      have gz: "?g \<noteq> 0" using z by simp
haftmann@24197
   300
      have ?thesis using aa' bb' z gz
chaieb@27567
   301
	of_int_div[where ?'a = 'a, OF gz zgcd_zdvd1[where i="a * b' + b * a'" and j="b*b'"]]	of_int_div[where ?'a = 'a,
chaieb@27567
   302
	OF gz zgcd_zdvd2[where i="a * b' + b * a'" and j="b*b'"]]
haftmann@24197
   303
	by (simp add: x y Nadd_def INum_def normNum_def Let_def add_divide_distrib)}
haftmann@24197
   304
    ultimately have ?thesis using aa' bb' 
haftmann@24197
   305
      by (simp add: Nadd_def INum_def normNum_def x y Let_def) }
haftmann@24197
   306
  ultimately show ?thesis by blast
haftmann@24197
   307
qed
haftmann@24197
   308
haftmann@24197
   309
lemma Nmul[simp]: "INum (x *\<^sub>N y) = INum x * (INum y:: 'a :: {ring_char_0,division_by_zero,field}) "
haftmann@24197
   310
proof-
haftmann@24197
   311
  let ?z = "0::'a"
haftmann@24197
   312
  have " \<exists> a b. x = (a,b)" " \<exists> a' b'. y = (a',b')" by auto
haftmann@24197
   313
  then obtain a b a' b' where x: "x = (a,b)" and y: "y = (a',b')" by blast
haftmann@24197
   314
  {assume "a=0 \<or> a'= 0 \<or> b = 0 \<or> b' = 0" hence ?thesis 
haftmann@24197
   315
      apply (cases "a=0",simp_all add: x y Nmul_def INum_def Let_def)
haftmann@24197
   316
      apply (cases "b=0",simp_all)
haftmann@24197
   317
      apply (cases "a'=0",simp_all) 
haftmann@24197
   318
      done }
haftmann@24197
   319
  moreover
haftmann@24197
   320
  {assume z: "a \<noteq> 0" "a' \<noteq> 0" "b \<noteq> 0" "b' \<noteq> 0"
haftmann@27556
   321
    let ?g="zgcd (a*a') (b*b')"
haftmann@24197
   322
    have gz: "?g \<noteq> 0" using z by simp
chaieb@27567
   323
    from z of_int_div[where ?'a = 'a, OF gz zgcd_zdvd1[where i="a*a'" and j="b*b'"]] 
chaieb@27567
   324
      of_int_div[where ?'a = 'a , OF gz zgcd_zdvd2[where i="a*a'" and j="b*b'"]] 
haftmann@24197
   325
    have ?thesis by (simp add: Nmul_def x y Let_def INum_def)}
haftmann@24197
   326
  ultimately show ?thesis by blast
haftmann@24197
   327
qed
haftmann@24197
   328
haftmann@24197
   329
lemma Nneg[simp]: "INum (~\<^sub>N x) = - (INum x ::'a:: field)"
haftmann@24197
   330
  by (simp add: Nneg_def split_def INum_def)
haftmann@24197
   331
haftmann@24197
   332
lemma Nsub[simp]: shows "INum (x -\<^sub>N y) = INum x - (INum y:: 'a :: {ring_char_0,division_by_zero,field})"
haftmann@24197
   333
by (simp add: Nsub_def split_def)
haftmann@24197
   334
haftmann@24197
   335
lemma Ninv[simp]: "INum (Ninv x) = (1::'a :: {division_by_zero,field}) / (INum x)"
haftmann@24197
   336
  by (simp add: Ninv_def INum_def split_def)
haftmann@24197
   337
haftmann@24197
   338
lemma Ndiv[simp]: "INum (x \<div>\<^sub>N y) = INum x / (INum y ::'a :: {ring_char_0, division_by_zero,field})" by (simp add: Ndiv_def)
haftmann@24197
   339
haftmann@24197
   340
lemma Nlt0_iff[simp]: assumes nx: "isnormNum x" 
haftmann@24197
   341
  shows "((INum x :: 'a :: {ring_char_0,division_by_zero,ordered_field})< 0) = 0>\<^sub>N x "
haftmann@24197
   342
proof-
haftmann@24197
   343
  have " \<exists> a b. x = (a,b)" by simp
haftmann@24197
   344
  then obtain a b where x[simp]:"x = (a,b)" by blast
haftmann@24197
   345
  {assume "a = 0" hence ?thesis by (simp add: Nlt0_def INum_def) }
haftmann@24197
   346
  moreover
haftmann@24197
   347
  {assume a: "a\<noteq>0" hence b: "(of_int b::'a) > 0" using nx by (simp add: isnormNum_def)
haftmann@24197
   348
    from pos_divide_less_eq[OF b, where b="of_int a" and a="0::'a"]
haftmann@24197
   349
    have ?thesis by (simp add: Nlt0_def INum_def)}
haftmann@24197
   350
  ultimately show ?thesis by blast
haftmann@24197
   351
qed
haftmann@24197
   352
haftmann@24197
   353
lemma Nle0_iff[simp]:assumes nx: "isnormNum x" 
haftmann@24197
   354
  shows "((INum x :: 'a :: {ring_char_0,division_by_zero,ordered_field}) \<le> 0) = 0\<ge>\<^sub>N x"
haftmann@24197
   355
proof-
haftmann@24197
   356
  have " \<exists> a b. x = (a,b)" by simp
haftmann@24197
   357
  then obtain a b where x[simp]:"x = (a,b)" by blast
haftmann@24197
   358
  {assume "a = 0" hence ?thesis by (simp add: Nle0_def INum_def) }
haftmann@24197
   359
  moreover
haftmann@24197
   360
  {assume a: "a\<noteq>0" hence b: "(of_int b :: 'a) > 0" using nx by (simp add: isnormNum_def)
haftmann@24197
   361
    from pos_divide_le_eq[OF b, where b="of_int a" and a="0::'a"]
haftmann@24197
   362
    have ?thesis by (simp add: Nle0_def INum_def)}
haftmann@24197
   363
  ultimately show ?thesis by blast
haftmann@24197
   364
qed
haftmann@24197
   365
haftmann@24197
   366
lemma Ngt0_iff[simp]:assumes nx: "isnormNum x" shows "((INum x :: 'a :: {ring_char_0,division_by_zero,ordered_field})> 0) = 0<\<^sub>N x"
haftmann@24197
   367
proof-
haftmann@24197
   368
  have " \<exists> a b. x = (a,b)" by simp
haftmann@24197
   369
  then obtain a b where x[simp]:"x = (a,b)" by blast
haftmann@24197
   370
  {assume "a = 0" hence ?thesis by (simp add: Ngt0_def INum_def) }
haftmann@24197
   371
  moreover
haftmann@24197
   372
  {assume a: "a\<noteq>0" hence b: "(of_int b::'a) > 0" using nx by (simp add: isnormNum_def)
haftmann@24197
   373
    from pos_less_divide_eq[OF b, where b="of_int a" and a="0::'a"]
haftmann@24197
   374
    have ?thesis by (simp add: Ngt0_def INum_def)}
haftmann@24197
   375
  ultimately show ?thesis by blast
haftmann@24197
   376
qed
haftmann@24197
   377
lemma Nge0_iff[simp]:assumes nx: "isnormNum x" 
haftmann@24197
   378
  shows "((INum x :: 'a :: {ring_char_0,division_by_zero,ordered_field}) \<ge> 0) = 0\<le>\<^sub>N x"
haftmann@24197
   379
proof-
haftmann@24197
   380
  have " \<exists> a b. x = (a,b)" by simp
haftmann@24197
   381
  then obtain a b where x[simp]:"x = (a,b)" by blast
haftmann@24197
   382
  {assume "a = 0" hence ?thesis by (simp add: Nge0_def INum_def) }
haftmann@24197
   383
  moreover
haftmann@24197
   384
  {assume a: "a\<noteq>0" hence b: "(of_int b::'a) > 0" using nx by (simp add: isnormNum_def)
haftmann@24197
   385
    from pos_le_divide_eq[OF b, where b="of_int a" and a="0::'a"]
haftmann@24197
   386
    have ?thesis by (simp add: Nge0_def INum_def)}
haftmann@24197
   387
  ultimately show ?thesis by blast
haftmann@24197
   388
qed
haftmann@24197
   389
haftmann@24197
   390
lemma Nlt_iff[simp]: assumes nx: "isnormNum x" and ny: "isnormNum y"
haftmann@24197
   391
  shows "((INum x :: 'a :: {ring_char_0,division_by_zero,ordered_field}) < INum y) = (x <\<^sub>N y)"
haftmann@24197
   392
proof-
haftmann@24197
   393
  let ?z = "0::'a"
haftmann@24197
   394
  have "((INum x ::'a) < INum y) = (INum (x -\<^sub>N y) < ?z)" using nx ny by simp
haftmann@24197
   395
  also have "\<dots> = (0>\<^sub>N (x -\<^sub>N y))" using Nlt0_iff[OF Nsub_normN[OF ny]] by simp
haftmann@24197
   396
  finally show ?thesis by (simp add: Nlt_def)
haftmann@24197
   397
qed
haftmann@24197
   398
haftmann@24197
   399
lemma Nle_iff[simp]: assumes nx: "isnormNum x" and ny: "isnormNum y"
haftmann@24197
   400
  shows "((INum x :: 'a :: {ring_char_0,division_by_zero,ordered_field})\<le> INum y) = (x \<le>\<^sub>N y)"
haftmann@24197
   401
proof-
haftmann@24197
   402
  have "((INum x ::'a) \<le> INum y) = (INum (x -\<^sub>N y) \<le> (0::'a))" using nx ny by simp
haftmann@24197
   403
  also have "\<dots> = (0\<ge>\<^sub>N (x -\<^sub>N y))" using Nle0_iff[OF Nsub_normN[OF ny]] by simp
haftmann@24197
   404
  finally show ?thesis by (simp add: Nle_def)
haftmann@24197
   405
qed
haftmann@24197
   406
wenzelm@28615
   407
lemma Nadd_commute:
wenzelm@28615
   408
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
wenzelm@28615
   409
  shows "x +\<^sub>N y = y +\<^sub>N x"
haftmann@24197
   410
proof-
haftmann@24197
   411
  have n: "isnormNum (x +\<^sub>N y)" "isnormNum (y +\<^sub>N x)" by simp_all
wenzelm@28615
   412
  have "(INum (x +\<^sub>N y)::'a) = INum (y +\<^sub>N x)" by simp
haftmann@24197
   413
  with isnormNum_unique[OF n] show ?thesis by simp
haftmann@24197
   414
qed
haftmann@24197
   415
wenzelm@28615
   416
lemma [simp]:
wenzelm@28615
   417
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
wenzelm@28615
   418
  shows "(0, b) +\<^sub>N y = normNum y"
wenzelm@28615
   419
    and "(a, 0) +\<^sub>N y = normNum y" 
wenzelm@28615
   420
    and "x +\<^sub>N (0, b) = normNum x"
wenzelm@28615
   421
    and "x +\<^sub>N (a, 0) = normNum x"
wenzelm@28615
   422
  apply (simp add: Nadd_def split_def)
wenzelm@28615
   423
  apply (simp add: Nadd_def split_def)
wenzelm@28615
   424
  apply (subst Nadd_commute, simp add: Nadd_def split_def)
wenzelm@28615
   425
  apply (subst Nadd_commute, simp add: Nadd_def split_def)
haftmann@24197
   426
  done
haftmann@24197
   427
wenzelm@28615
   428
lemma normNum_nilpotent_aux[simp]:
wenzelm@28615
   429
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
wenzelm@28615
   430
  assumes nx: "isnormNum x" 
haftmann@24197
   431
  shows "normNum x = x"
haftmann@24197
   432
proof-
haftmann@24197
   433
  let ?a = "normNum x"
haftmann@24197
   434
  have n: "isnormNum ?a" by simp
wenzelm@28615
   435
  have th:"INum ?a = (INum x ::'a)" by simp
haftmann@24197
   436
  with isnormNum_unique[OF n nx]  
haftmann@24197
   437
  show ?thesis by simp
haftmann@24197
   438
qed
haftmann@24197
   439
wenzelm@28615
   440
lemma normNum_nilpotent[simp]:
wenzelm@28615
   441
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
wenzelm@28615
   442
  shows "normNum (normNum x) = normNum x"
haftmann@24197
   443
  by simp
wenzelm@28615
   444
haftmann@24197
   445
lemma normNum0[simp]: "normNum (0,b) = 0\<^sub>N" "normNum (a,0) = 0\<^sub>N"
haftmann@24197
   446
  by (simp_all add: normNum_def)
wenzelm@28615
   447
wenzelm@28615
   448
lemma normNum_Nadd:
wenzelm@28615
   449
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
wenzelm@28615
   450
  shows "normNum (x +\<^sub>N y) = x +\<^sub>N y" by simp
wenzelm@28615
   451
wenzelm@28615
   452
lemma Nadd_normNum1[simp]:
wenzelm@28615
   453
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
wenzelm@28615
   454
  shows "normNum x +\<^sub>N y = x +\<^sub>N y"
haftmann@24197
   455
proof-
haftmann@24197
   456
  have n: "isnormNum (normNum x +\<^sub>N y)" "isnormNum (x +\<^sub>N y)" by simp_all
wenzelm@28615
   457
  have "INum (normNum x +\<^sub>N y) = INum x + (INum y :: 'a)" by simp
haftmann@24197
   458
  also have "\<dots> = INum (x +\<^sub>N y)" by simp
haftmann@24197
   459
  finally show ?thesis using isnormNum_unique[OF n] by simp
haftmann@24197
   460
qed
haftmann@24197
   461
wenzelm@28615
   462
lemma Nadd_normNum2[simp]:
wenzelm@28615
   463
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
wenzelm@28615
   464
  shows "x +\<^sub>N normNum y = x +\<^sub>N y"
wenzelm@28615
   465
proof-
wenzelm@28615
   466
  have n: "isnormNum (x +\<^sub>N normNum y)" "isnormNum (x +\<^sub>N y)" by simp_all
wenzelm@28615
   467
  have "INum (x +\<^sub>N normNum y) = INum x + (INum y :: 'a)" by simp
wenzelm@28615
   468
  also have "\<dots> = INum (x +\<^sub>N y)" by simp
wenzelm@28615
   469
  finally show ?thesis using isnormNum_unique[OF n] by simp
wenzelm@28615
   470
qed
wenzelm@28615
   471
wenzelm@28615
   472
lemma Nadd_assoc:
wenzelm@28615
   473
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
wenzelm@28615
   474
  shows "x +\<^sub>N y +\<^sub>N z = x +\<^sub>N (y +\<^sub>N z)"
haftmann@24197
   475
proof-
haftmann@24197
   476
  have n: "isnormNum (x +\<^sub>N y +\<^sub>N z)" "isnormNum (x +\<^sub>N (y +\<^sub>N z))" by simp_all
wenzelm@28615
   477
  have "INum (x +\<^sub>N y +\<^sub>N z) = (INum (x +\<^sub>N (y +\<^sub>N z)) :: 'a)" by simp
haftmann@24197
   478
  with isnormNum_unique[OF n] show ?thesis by simp
haftmann@24197
   479
qed
haftmann@24197
   480
haftmann@24197
   481
lemma Nmul_commute: "isnormNum x \<Longrightarrow> isnormNum y \<Longrightarrow> x *\<^sub>N y = y *\<^sub>N x"
haftmann@27556
   482
  by (simp add: Nmul_def split_def Let_def zgcd_commute mult_commute)
haftmann@24197
   483
wenzelm@28615
   484
lemma Nmul_assoc:
wenzelm@28615
   485
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
wenzelm@28615
   486
  assumes nx: "isnormNum x" and ny:"isnormNum y" and nz:"isnormNum z"
haftmann@24197
   487
  shows "x *\<^sub>N y *\<^sub>N z = x *\<^sub>N (y *\<^sub>N z)"
haftmann@24197
   488
proof-
haftmann@24197
   489
  from nx ny nz have n: "isnormNum (x *\<^sub>N y *\<^sub>N z)" "isnormNum (x *\<^sub>N (y *\<^sub>N z))" 
haftmann@24197
   490
    by simp_all
wenzelm@28615
   491
  have "INum (x +\<^sub>N y +\<^sub>N z) = (INum (x +\<^sub>N (y +\<^sub>N z)) :: 'a)" by simp
haftmann@24197
   492
  with isnormNum_unique[OF n] show ?thesis by simp
haftmann@24197
   493
qed
haftmann@24197
   494
wenzelm@28615
   495
lemma Nsub0:
wenzelm@28615
   496
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
wenzelm@28615
   497
  assumes x: "isnormNum x" and y:"isnormNum y" shows "(x -\<^sub>N y = 0\<^sub>N) = (x = y)"
haftmann@24197
   498
proof-
wenzelm@28615
   499
  { fix h :: 'a
wenzelm@28615
   500
    from isnormNum_unique[where 'a = 'a, OF Nsub_normN[OF y], where y="0\<^sub>N"] 
haftmann@24197
   501
    have "(x -\<^sub>N y = 0\<^sub>N) = (INum (x -\<^sub>N y) = (INum 0\<^sub>N :: 'a)) " by simp
wenzelm@28615
   502
    also have "\<dots> = (INum x = (INum y :: 'a))" by simp
haftmann@24197
   503
    also have "\<dots> = (x = y)" using x y by simp
wenzelm@28615
   504
    finally show ?thesis . }
haftmann@24197
   505
qed
haftmann@24197
   506
haftmann@24197
   507
lemma Nmul0[simp]: "c *\<^sub>N 0\<^sub>N = 0\<^sub>N" " 0\<^sub>N *\<^sub>N c = 0\<^sub>N"
haftmann@24197
   508
  by (simp_all add: Nmul_def Let_def split_def)
haftmann@24197
   509
wenzelm@28615
   510
lemma Nmul_eq0[simp]:
wenzelm@28615
   511
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
wenzelm@28615
   512
  assumes nx:"isnormNum x" and ny: "isnormNum y"
haftmann@24197
   513
  shows "(x*\<^sub>N y = 0\<^sub>N) = (x = 0\<^sub>N \<or> y = 0\<^sub>N)"
haftmann@24197
   514
proof-
wenzelm@28615
   515
  { fix h :: 'a
wenzelm@28615
   516
    have " \<exists> a b a' b'. x = (a,b) \<and> y= (a',b')" by auto
wenzelm@28615
   517
    then obtain a b a' b' where xy[simp]: "x = (a,b)" "y = (a',b')" by blast
wenzelm@28615
   518
    have n0: "isnormNum 0\<^sub>N" by simp
wenzelm@28615
   519
    show ?thesis using nx ny 
wenzelm@28615
   520
      apply (simp only: isnormNum_unique[where ?'a = 'a, OF  Nmul_normN[OF nx ny] n0, symmetric] Nmul[where ?'a = 'a])
wenzelm@28615
   521
      apply (simp add: INum_def split_def isnormNum_def fst_conv snd_conv)
wenzelm@28615
   522
      apply (cases "a=0",simp_all)
wenzelm@28615
   523
      apply (cases "a'=0",simp_all)
wenzelm@28615
   524
      done
wenzelm@28615
   525
  }
haftmann@24197
   526
qed
haftmann@24197
   527
lemma Nneg_Nneg[simp]: "~\<^sub>N (~\<^sub>N c) = c"
haftmann@24197
   528
  by (simp add: Nneg_def split_def)
haftmann@24197
   529
haftmann@24197
   530
lemma Nmul1[simp]: 
haftmann@24197
   531
  "isnormNum c \<Longrightarrow> 1\<^sub>N *\<^sub>N c = c" 
haftmann@24197
   532
  "isnormNum c \<Longrightarrow> c *\<^sub>N 1\<^sub>N  = c" 
haftmann@24197
   533
  apply (simp_all add: Nmul_def Let_def split_def isnormNum_def)
wenzelm@28615
   534
  apply (cases "fst c = 0", simp_all, cases c, simp_all)+
wenzelm@28615
   535
  done
haftmann@24197
   536
wenzelm@28615
   537
end