src/HOL/Library/Abstract_Rat.thy
 author wenzelm Thu Feb 16 22:53:24 2012 +0100 (2012-02-16) changeset 46507 1b24c24017dd parent 44780 a13cdb1e9e08 child 47162 9d7d919b9fd8 permissions -rw-r--r--
tuned proofs;
```     1 (*  Title:      HOL/Library/Abstract_Rat.thy
```
```     2     Author:     Amine Chaieb
```
```     3 *)
```
```     4
```
```     5 header {* Abstract rational numbers *}
```
```     6
```
```     7 theory Abstract_Rat
```
```     8 imports Complex_Main
```
```     9 begin
```
```    10
```
```    11 type_synonym Num = "int \<times> int"
```
```    12
```
```    13 abbreviation Num0_syn :: Num  ("0\<^sub>N")
```
```    14   where "0\<^sub>N \<equiv> (0, 0)"
```
```    15
```
```    16 abbreviation Numi_syn :: "int \<Rightarrow> Num"  ("_\<^sub>N")
```
```    17   where "i\<^sub>N \<equiv> (i, 1)"
```
```    18
```
```    19 definition isnormNum :: "Num \<Rightarrow> bool" where
```
```    20   "isnormNum = (\<lambda>(a,b). (if a = 0 then b = 0 else b > 0 \<and> gcd a b = 1))"
```
```    21
```
```    22 definition normNum :: "Num \<Rightarrow> Num" where
```
```    23   "normNum = (\<lambda>(a,b).
```
```    24     (if a=0 \<or> b = 0 then (0,0) else
```
```    25       (let g = gcd a b
```
```    26        in if b > 0 then (a div g, b div g) else (- (a div g), - (b div g)))))"
```
```    27
```
```    28 declare gcd_dvd1_int[presburger] gcd_dvd2_int[presburger]
```
```    29
```
```    30 lemma normNum_isnormNum [simp]: "isnormNum (normNum x)"
```
```    31 proof -
```
```    32   obtain a b where x: "x = (a, b)" by (cases x)
```
```    33   { assume "a=0 \<or> b = 0" hence ?thesis by (simp add: x normNum_def isnormNum_def) }
```
```    34   moreover
```
```    35   { assume anz: "a \<noteq> 0" and bnz: "b \<noteq> 0"
```
```    36     let ?g = "gcd a b"
```
```    37     let ?a' = "a div ?g"
```
```    38     let ?b' = "b div ?g"
```
```    39     let ?g' = "gcd ?a' ?b'"
```
```    40     from anz bnz have "?g \<noteq> 0" by simp  with gcd_ge_0_int[of a b]
```
```    41     have gpos: "?g > 0" by arith
```
```    42     have gdvd: "?g dvd a" "?g dvd b" by arith+
```
```    43     from zdvd_mult_div_cancel[OF gdvd(1)] zdvd_mult_div_cancel[OF gdvd(2)] anz bnz
```
```    44     have nz': "?a' \<noteq> 0" "?b' \<noteq> 0" by - (rule notI, simp)+
```
```    45     from anz bnz have stupid: "a \<noteq> 0 \<or> b \<noteq> 0" by arith
```
```    46     from div_gcd_coprime_int[OF stupid] have gp1: "?g' = 1" .
```
```    47     from bnz have "b < 0 \<or> b > 0" by arith
```
```    48     moreover
```
```    49     { assume b: "b > 0"
```
```    50       from b have "?b' \<ge> 0"
```
```    51         by (presburger add: pos_imp_zdiv_nonneg_iff[OF gpos])
```
```    52       with nz' have b': "?b' > 0" by arith
```
```    53       from b b' anz bnz nz' gp1 have ?thesis
```
```    54         by (simp add: x isnormNum_def normNum_def Let_def split_def) }
```
```    55     moreover {
```
```    56       assume b: "b < 0"
```
```    57       { assume b': "?b' \<ge> 0"
```
```    58         from gpos have th: "?g \<ge> 0" by arith
```
```    59         from mult_nonneg_nonneg[OF th b'] zdvd_mult_div_cancel[OF gdvd(2)]
```
```    60         have False using b by arith }
```
```    61       hence b': "?b' < 0" by (presburger add: linorder_not_le[symmetric])
```
```    62       from anz bnz nz' b b' gp1 have ?thesis
```
```    63         by (simp add: x isnormNum_def normNum_def Let_def split_def) }
```
```    64     ultimately have ?thesis by blast
```
```    65   }
```
```    66   ultimately show ?thesis by blast
```
```    67 qed
```
```    68
```
```    69 text {* Arithmetic over Num *}
```
```    70
```
```    71 definition Nadd :: "Num \<Rightarrow> Num \<Rightarrow> Num"  (infixl "+\<^sub>N" 60) where
```
```    72   "Nadd = (\<lambda>(a,b) (a',b'). if a = 0 \<or> b = 0 then normNum(a',b')
```
```    73     else if a'=0 \<or> b' = 0 then normNum(a,b)
```
```    74     else normNum(a*b' + b*a', b*b'))"
```
```    75
```
```    76 definition Nmul :: "Num \<Rightarrow> Num \<Rightarrow> Num"  (infixl "*\<^sub>N" 60) where
```
```    77   "Nmul = (\<lambda>(a,b) (a',b'). let g = gcd (a*a') (b*b')
```
```    78     in (a*a' div g, b*b' div g))"
```
```    79
```
```    80 definition Nneg :: "Num \<Rightarrow> Num" ("~\<^sub>N")
```
```    81   where "Nneg \<equiv> (\<lambda>(a,b). (-a,b))"
```
```    82
```
```    83 definition Nsub :: "Num \<Rightarrow> Num \<Rightarrow> Num"  (infixl "-\<^sub>N" 60)
```
```    84   where "Nsub = (\<lambda>a b. a +\<^sub>N ~\<^sub>N b)"
```
```    85
```
```    86 definition Ninv :: "Num \<Rightarrow> Num"
```
```    87   where "Ninv = (\<lambda>(a,b). if a < 0 then (-b, \<bar>a\<bar>) else (b,a))"
```
```    88
```
```    89 definition Ndiv :: "Num \<Rightarrow> Num \<Rightarrow> Num"  (infixl "\<div>\<^sub>N" 60)
```
```    90   where "Ndiv = (\<lambda>a b. a *\<^sub>N Ninv b)"
```
```    91
```
```    92 lemma Nneg_normN[simp]: "isnormNum x \<Longrightarrow> isnormNum (~\<^sub>N x)"
```
```    93   by (simp add: isnormNum_def Nneg_def split_def)
```
```    94
```
```    95 lemma Nadd_normN[simp]: "isnormNum (x +\<^sub>N y)"
```
```    96   by (simp add: Nadd_def split_def)
```
```    97
```
```    98 lemma Nsub_normN[simp]: "\<lbrakk> isnormNum y\<rbrakk> \<Longrightarrow> isnormNum (x -\<^sub>N y)"
```
```    99   by (simp add: Nsub_def split_def)
```
```   100
```
```   101 lemma Nmul_normN[simp]:
```
```   102   assumes xn: "isnormNum x" and yn: "isnormNum y"
```
```   103   shows "isnormNum (x *\<^sub>N y)"
```
```   104 proof -
```
```   105   obtain a b where x: "x = (a, b)" by (cases x)
```
```   106   obtain a' b' where y: "y = (a', b')" by (cases y)
```
```   107   { assume "a = 0"
```
```   108     hence ?thesis using xn x y
```
```   109       by (simp add: isnormNum_def Let_def Nmul_def split_def) }
```
```   110   moreover
```
```   111   { assume "a' = 0"
```
```   112     hence ?thesis using yn x y
```
```   113       by (simp add: isnormNum_def Let_def Nmul_def split_def) }
```
```   114   moreover
```
```   115   { assume a: "a \<noteq>0" and a': "a'\<noteq>0"
```
```   116     hence bp: "b > 0" "b' > 0" using xn yn x y by (simp_all add: isnormNum_def)
```
```   117     from mult_pos_pos[OF bp] have "x *\<^sub>N y = normNum (a * a', b * b')"
```
```   118       using x y a a' bp by (simp add: Nmul_def Let_def split_def normNum_def)
```
```   119     hence ?thesis by simp }
```
```   120   ultimately show ?thesis by blast
```
```   121 qed
```
```   122
```
```   123 lemma Ninv_normN[simp]: "isnormNum x \<Longrightarrow> isnormNum (Ninv x)"
```
```   124   by (simp add: Ninv_def isnormNum_def split_def)
```
```   125     (cases "fst x = 0", auto simp add: gcd_commute_int)
```
```   126
```
```   127 lemma isnormNum_int[simp]:
```
```   128   "isnormNum 0\<^sub>N" "isnormNum ((1::int)\<^sub>N)" "i \<noteq> 0 \<Longrightarrow> isnormNum (i\<^sub>N)"
```
```   129   by (simp_all add: isnormNum_def)
```
```   130
```
```   131
```
```   132 text {* Relations over Num *}
```
```   133
```
```   134 definition Nlt0:: "Num \<Rightarrow> bool"  ("0>\<^sub>N")
```
```   135   where "Nlt0 = (\<lambda>(a,b). a < 0)"
```
```   136
```
```   137 definition Nle0:: "Num \<Rightarrow> bool"  ("0\<ge>\<^sub>N")
```
```   138   where "Nle0 = (\<lambda>(a,b). a \<le> 0)"
```
```   139
```
```   140 definition Ngt0:: "Num \<Rightarrow> bool"  ("0<\<^sub>N")
```
```   141   where "Ngt0 = (\<lambda>(a,b). a > 0)"
```
```   142
```
```   143 definition Nge0:: "Num \<Rightarrow> bool"  ("0\<le>\<^sub>N")
```
```   144   where "Nge0 = (\<lambda>(a,b). a \<ge> 0)"
```
```   145
```
```   146 definition Nlt :: "Num \<Rightarrow> Num \<Rightarrow> bool"  (infix "<\<^sub>N" 55)
```
```   147   where "Nlt = (\<lambda>a b. 0>\<^sub>N (a -\<^sub>N b))"
```
```   148
```
```   149 definition Nle :: "Num \<Rightarrow> Num \<Rightarrow> bool"  (infix "\<le>\<^sub>N" 55)
```
```   150   where "Nle = (\<lambda>a b. 0\<ge>\<^sub>N (a -\<^sub>N b))"
```
```   151
```
```   152 definition "INum = (\<lambda>(a,b). of_int a / of_int b)"
```
```   153
```
```   154 lemma INum_int [simp]: "INum (i\<^sub>N) = ((of_int i) ::'a::field)" "INum 0\<^sub>N = (0::'a::field)"
```
```   155   by (simp_all add: INum_def)
```
```   156
```
```   157 lemma isnormNum_unique[simp]:
```
```   158   assumes na: "isnormNum x" and nb: "isnormNum y"
```
```   159   shows "((INum x ::'a::{field_char_0, field_inverse_zero}) = INum y) = (x = y)" (is "?lhs = ?rhs")
```
```   160 proof
```
```   161   obtain a b where x: "x = (a, b)" by (cases x)
```
```   162   obtain a' b' where y: "y = (a', b')" by (cases y)
```
```   163   assume H: ?lhs
```
```   164   { assume "a = 0 \<or> b = 0 \<or> a' = 0 \<or> b' = 0"
```
```   165     hence ?rhs using na nb H
```
```   166       by (simp add: x y INum_def split_def isnormNum_def split: split_if_asm) }
```
```   167   moreover
```
```   168   { assume az: "a \<noteq> 0" and bz: "b \<noteq> 0" and a'z: "a'\<noteq>0" and b'z: "b'\<noteq>0"
```
```   169     from az bz a'z b'z na nb have pos: "b > 0" "b' > 0" by (simp_all add: x y isnormNum_def)
```
```   170     from H bz b'z have eq: "a * b' = a'*b"
```
```   171       by (simp add: x y INum_def eq_divide_eq divide_eq_eq of_int_mult[symmetric] del: of_int_mult)
```
```   172     from az a'z na nb have gcd1: "gcd a b = 1" "gcd b a = 1" "gcd a' b' = 1" "gcd b' a' = 1"
```
```   173       by (simp_all add: x y isnormNum_def add: gcd_commute_int)
```
```   174     from eq have raw_dvd: "a dvd a' * b" "b dvd b' * a" "a' dvd a * b'" "b' dvd b * a'"
```
```   175       apply -
```
```   176       apply algebra
```
```   177       apply algebra
```
```   178       apply simp
```
```   179       apply algebra
```
```   180       done
```
```   181     from zdvd_antisym_abs[OF coprime_dvd_mult_int[OF gcd1(2) raw_dvd(2)]
```
```   182         coprime_dvd_mult_int[OF gcd1(4) raw_dvd(4)]]
```
```   183       have eq1: "b = b'" using pos by arith
```
```   184       with eq have "a = a'" using pos by simp
```
```   185       with eq1 have ?rhs by (simp add: x y) }
```
```   186   ultimately show ?rhs by blast
```
```   187 next
```
```   188   assume ?rhs thus ?lhs by simp
```
```   189 qed
```
```   190
```
```   191
```
```   192 lemma isnormNum0[simp]:
```
```   193     "isnormNum x \<Longrightarrow> (INum x = (0::'a::{field_char_0, field_inverse_zero})) = (x = 0\<^sub>N)"
```
```   194   unfolding INum_int(2)[symmetric]
```
```   195   by (rule isnormNum_unique) simp_all
```
```   196
```
```   197 lemma of_int_div_aux: "d ~= 0 ==> ((of_int x)::'a::field_char_0) / (of_int d) =
```
```   198     of_int (x div d) + (of_int (x mod d)) / ((of_int d)::'a)"
```
```   199 proof -
```
```   200   assume "d ~= 0"
```
```   201   let ?t = "of_int (x div d) * ((of_int d)::'a) + of_int(x mod d)"
```
```   202   let ?f = "\<lambda>x. x / of_int d"
```
```   203   have "x = (x div d) * d + x mod d"
```
```   204     by auto
```
```   205   then have eq: "of_int x = ?t"
```
```   206     by (simp only: of_int_mult[symmetric] of_int_add [symmetric])
```
```   207   then have "of_int x / of_int d = ?t / of_int d"
```
```   208     using cong[OF refl[of ?f] eq] by simp
```
```   209   then show ?thesis by (simp add: add_divide_distrib algebra_simps `d ~= 0`)
```
```   210 qed
```
```   211
```
```   212 lemma of_int_div: "(d::int) ~= 0 ==> d dvd n ==>
```
```   213     (of_int(n div d)::'a::field_char_0) = of_int n / of_int d"
```
```   214   apply (frule of_int_div_aux [of d n, where ?'a = 'a])
```
```   215   apply simp
```
```   216   apply (simp add: dvd_eq_mod_eq_0)
```
```   217   done
```
```   218
```
```   219
```
```   220 lemma normNum[simp]: "INum (normNum x) = (INum x :: 'a::{field_char_0, field_inverse_zero})"
```
```   221 proof -
```
```   222   obtain a b where x: "x = (a, b)" by (cases x)
```
```   223   { assume "a = 0 \<or> b = 0"
```
```   224     hence ?thesis by (simp add: x INum_def normNum_def split_def Let_def) }
```
```   225   moreover
```
```   226   { assume a: "a \<noteq> 0" and b: "b \<noteq> 0"
```
```   227     let ?g = "gcd a b"
```
```   228     from a b have g: "?g \<noteq> 0"by simp
```
```   229     from of_int_div[OF g, where ?'a = 'a]
```
```   230     have ?thesis by (auto simp add: x INum_def normNum_def split_def Let_def) }
```
```   231   ultimately show ?thesis by blast
```
```   232 qed
```
```   233
```
```   234 lemma INum_normNum_iff:
```
```   235   "(INum x ::'a::{field_char_0, field_inverse_zero}) = INum y \<longleftrightarrow> normNum x = normNum y"
```
```   236   (is "?lhs = ?rhs")
```
```   237 proof -
```
```   238   have "normNum x = normNum y \<longleftrightarrow> (INum (normNum x) :: 'a) = INum (normNum y)"
```
```   239     by (simp del: normNum)
```
```   240   also have "\<dots> = ?lhs" by simp
```
```   241   finally show ?thesis by simp
```
```   242 qed
```
```   243
```
```   244 lemma Nadd[simp]: "INum (x +\<^sub>N y) = INum x + (INum y :: 'a :: {field_char_0, field_inverse_zero})"
```
```   245 proof -
```
```   246   let ?z = "0:: 'a"
```
```   247   obtain a b where x: "x = (a, b)" by (cases x)
```
```   248   obtain a' b' where y: "y = (a', b')" by (cases y)
```
```   249   { assume "a=0 \<or> a'= 0 \<or> b =0 \<or> b' = 0"
```
```   250     hence ?thesis
```
```   251       apply (cases "a=0", simp_all add: x y Nadd_def)
```
```   252       apply (cases "b= 0", simp_all add: INum_def)
```
```   253        apply (cases "a'= 0", simp_all)
```
```   254        apply (cases "b'= 0", simp_all)
```
```   255        done }
```
```   256   moreover
```
```   257   { assume aa': "a \<noteq> 0" "a'\<noteq> 0" and bb': "b \<noteq> 0" "b' \<noteq> 0"
```
```   258     { assume z: "a * b' + b * a' = 0"
```
```   259       hence "of_int (a*b' + b*a') / (of_int b* of_int b') = ?z" by simp
```
```   260       hence "of_int b' * of_int a / (of_int b * of_int b') +
```
```   261           of_int b * of_int a' / (of_int b * of_int b') = ?z"
```
```   262         by (simp add:add_divide_distrib)
```
```   263       hence th: "of_int a / of_int b + of_int a' / of_int b' = ?z" using bb' aa'
```
```   264         by simp
```
```   265       from z aa' bb' have ?thesis
```
```   266         by (simp add: x y th Nadd_def normNum_def INum_def split_def) }
```
```   267     moreover {
```
```   268       assume z: "a * b' + b * a' \<noteq> 0"
```
```   269       let ?g = "gcd (a * b' + b * a') (b*b')"
```
```   270       have gz: "?g \<noteq> 0" using z by simp
```
```   271       have ?thesis using aa' bb' z gz
```
```   272         of_int_div[where ?'a = 'a, OF gz gcd_dvd1_int[where x="a * b' + b * a'" and y="b*b'"]]
```
```   273         of_int_div[where ?'a = 'a, OF gz gcd_dvd2_int[where x="a * b' + b * a'" and y="b*b'"]]
```
```   274         by (simp add: x y Nadd_def INum_def normNum_def Let_def add_divide_distrib) }
```
```   275     ultimately have ?thesis using aa' bb'
```
```   276       by (simp add: x y Nadd_def INum_def normNum_def Let_def) }
```
```   277   ultimately show ?thesis by blast
```
```   278 qed
```
```   279
```
```   280 lemma Nmul[simp]: "INum (x *\<^sub>N y) = INum x * (INum y:: 'a :: {field_char_0, field_inverse_zero})"
```
```   281 proof -
```
```   282   let ?z = "0::'a"
```
```   283   obtain a b where x: "x = (a, b)" by (cases x)
```
```   284   obtain a' b' where y: "y = (a', b')" by (cases y)
```
```   285   { assume "a=0 \<or> a'= 0 \<or> b = 0 \<or> b' = 0"
```
```   286     hence ?thesis
```
```   287       apply (cases "a=0", simp_all add: x y Nmul_def INum_def Let_def)
```
```   288       apply (cases "b=0", simp_all)
```
```   289       apply (cases "a'=0", simp_all)
```
```   290       done }
```
```   291   moreover
```
```   292   { assume z: "a \<noteq> 0" "a' \<noteq> 0" "b \<noteq> 0" "b' \<noteq> 0"
```
```   293     let ?g="gcd (a*a') (b*b')"
```
```   294     have gz: "?g \<noteq> 0" using z by simp
```
```   295     from z of_int_div[where ?'a = 'a, OF gz gcd_dvd1_int[where x="a*a'" and y="b*b'"]]
```
```   296       of_int_div[where ?'a = 'a , OF gz gcd_dvd2_int[where x="a*a'" and y="b*b'"]]
```
```   297     have ?thesis by (simp add: Nmul_def x y Let_def INum_def) }
```
```   298   ultimately show ?thesis by blast
```
```   299 qed
```
```   300
```
```   301 lemma Nneg[simp]: "INum (~\<^sub>N x) = - (INum x ::'a:: field)"
```
```   302   by (simp add: Nneg_def split_def INum_def)
```
```   303
```
```   304 lemma Nsub[simp]: "INum (x -\<^sub>N y) = INum x - (INum y:: 'a :: {field_char_0, field_inverse_zero})"
```
```   305   by (simp add: Nsub_def split_def)
```
```   306
```
```   307 lemma Ninv[simp]: "INum (Ninv x) = (1::'a :: field_inverse_zero) / (INum x)"
```
```   308   by (simp add: Ninv_def INum_def split_def)
```
```   309
```
```   310 lemma Ndiv[simp]: "INum (x \<div>\<^sub>N y) = INum x / (INum y ::'a :: {field_char_0, field_inverse_zero})"
```
```   311   by (simp add: Ndiv_def)
```
```   312
```
```   313 lemma Nlt0_iff[simp]:
```
```   314   assumes nx: "isnormNum x"
```
```   315   shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero})< 0) = 0>\<^sub>N x"
```
```   316 proof -
```
```   317   obtain a b where x: "x = (a, b)" by (cases x)
```
```   318   { assume "a = 0" hence ?thesis by (simp add: x Nlt0_def INum_def) }
```
```   319   moreover
```
```   320   { assume a: "a \<noteq> 0" hence b: "(of_int b::'a) > 0"
```
```   321       using nx by (simp add: x isnormNum_def)
```
```   322     from pos_divide_less_eq[OF b, where b="of_int a" and a="0::'a"]
```
```   323     have ?thesis by (simp add: x Nlt0_def INum_def) }
```
```   324   ultimately show ?thesis by blast
```
```   325 qed
```
```   326
```
```   327 lemma Nle0_iff[simp]:
```
```   328   assumes nx: "isnormNum x"
```
```   329   shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero}) \<le> 0) = 0\<ge>\<^sub>N x"
```
```   330 proof -
```
```   331   obtain a b where x: "x = (a, b)" by (cases x)
```
```   332   { assume "a = 0" hence ?thesis by (simp add: x Nle0_def INum_def) }
```
```   333   moreover
```
```   334   { assume a: "a \<noteq> 0" hence b: "(of_int b :: 'a) > 0"
```
```   335       using nx by (simp add: x isnormNum_def)
```
```   336     from pos_divide_le_eq[OF b, where b="of_int a" and a="0::'a"]
```
```   337     have ?thesis by (simp add: x Nle0_def INum_def) }
```
```   338   ultimately show ?thesis by blast
```
```   339 qed
```
```   340
```
```   341 lemma Ngt0_iff[simp]:
```
```   342   assumes nx: "isnormNum x"
```
```   343   shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero})> 0) = 0<\<^sub>N x"
```
```   344 proof -
```
```   345   obtain a b where x: "x = (a, b)" by (cases x)
```
```   346   { assume "a = 0" hence ?thesis by (simp add: x Ngt0_def INum_def) }
```
```   347   moreover
```
```   348   { assume a: "a \<noteq> 0" hence b: "(of_int b::'a) > 0" using nx
```
```   349       by (simp add: x isnormNum_def)
```
```   350     from pos_less_divide_eq[OF b, where b="of_int a" and a="0::'a"]
```
```   351     have ?thesis by (simp add: x Ngt0_def INum_def) }
```
```   352   ultimately show ?thesis by blast
```
```   353 qed
```
```   354
```
```   355 lemma Nge0_iff[simp]:
```
```   356   assumes nx: "isnormNum x"
```
```   357   shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero}) \<ge> 0) = 0\<le>\<^sub>N x"
```
```   358 proof -
```
```   359   obtain a b where x: "x = (a, b)" by (cases x)
```
```   360   { assume "a = 0" hence ?thesis by (simp add: x Nge0_def INum_def) }
```
```   361   moreover
```
```   362   { assume "a \<noteq> 0" hence b: "(of_int b::'a) > 0" using nx
```
```   363       by (simp add: x isnormNum_def)
```
```   364     from pos_le_divide_eq[OF b, where b="of_int a" and a="0::'a"]
```
```   365     have ?thesis by (simp add: x Nge0_def INum_def) }
```
```   366   ultimately show ?thesis by blast
```
```   367 qed
```
```   368
```
```   369 lemma Nlt_iff[simp]:
```
```   370   assumes nx: "isnormNum x" and ny: "isnormNum y"
```
```   371   shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero}) < INum y) = (x <\<^sub>N y)"
```
```   372 proof -
```
```   373   let ?z = "0::'a"
```
```   374   have "((INum x ::'a) < INum y) = (INum (x -\<^sub>N y) < ?z)"
```
```   375     using nx ny by simp
```
```   376   also have "\<dots> = (0>\<^sub>N (x -\<^sub>N y))"
```
```   377     using Nlt0_iff[OF Nsub_normN[OF ny]] by simp
```
```   378   finally show ?thesis by (simp add: Nlt_def)
```
```   379 qed
```
```   380
```
```   381 lemma Nle_iff[simp]:
```
```   382   assumes nx: "isnormNum x" and ny: "isnormNum y"
```
```   383   shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero})\<le> INum y) = (x \<le>\<^sub>N y)"
```
```   384 proof -
```
```   385   have "((INum x ::'a) \<le> INum y) = (INum (x -\<^sub>N y) \<le> (0::'a))"
```
```   386     using nx ny by simp
```
```   387   also have "\<dots> = (0\<ge>\<^sub>N (x -\<^sub>N y))"
```
```   388     using Nle0_iff[OF Nsub_normN[OF ny]] by simp
```
```   389   finally show ?thesis by (simp add: Nle_def)
```
```   390 qed
```
```   391
```
```   392 lemma Nadd_commute:
```
```   393   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   394   shows "x +\<^sub>N y = y +\<^sub>N x"
```
```   395 proof -
```
```   396   have n: "isnormNum (x +\<^sub>N y)" "isnormNum (y +\<^sub>N x)" by simp_all
```
```   397   have "(INum (x +\<^sub>N y)::'a) = INum (y +\<^sub>N x)" by simp
```
```   398   with isnormNum_unique[OF n] show ?thesis by simp
```
```   399 qed
```
```   400
```
```   401 lemma [simp]:
```
```   402   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   403   shows "(0, b) +\<^sub>N y = normNum y"
```
```   404     and "(a, 0) +\<^sub>N y = normNum y"
```
```   405     and "x +\<^sub>N (0, b) = normNum x"
```
```   406     and "x +\<^sub>N (a, 0) = normNum x"
```
```   407   apply (simp add: Nadd_def split_def)
```
```   408   apply (simp add: Nadd_def split_def)
```
```   409   apply (subst Nadd_commute, simp add: Nadd_def split_def)
```
```   410   apply (subst Nadd_commute, simp add: Nadd_def split_def)
```
```   411   done
```
```   412
```
```   413 lemma normNum_nilpotent_aux[simp]:
```
```   414   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   415   assumes nx: "isnormNum x"
```
```   416   shows "normNum x = x"
```
```   417 proof -
```
```   418   let ?a = "normNum x"
```
```   419   have n: "isnormNum ?a" by simp
```
```   420   have th: "INum ?a = (INum x ::'a)" by simp
```
```   421   with isnormNum_unique[OF n nx] show ?thesis by simp
```
```   422 qed
```
```   423
```
```   424 lemma normNum_nilpotent[simp]:
```
```   425   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   426   shows "normNum (normNum x) = normNum x"
```
```   427   by simp
```
```   428
```
```   429 lemma normNum0[simp]: "normNum (0,b) = 0\<^sub>N" "normNum (a,0) = 0\<^sub>N"
```
```   430   by (simp_all add: normNum_def)
```
```   431
```
```   432 lemma normNum_Nadd:
```
```   433   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   434   shows "normNum (x +\<^sub>N y) = x +\<^sub>N y" by simp
```
```   435
```
```   436 lemma Nadd_normNum1[simp]:
```
```   437   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   438   shows "normNum x +\<^sub>N y = x +\<^sub>N y"
```
```   439 proof -
```
```   440   have n: "isnormNum (normNum x +\<^sub>N y)" "isnormNum (x +\<^sub>N y)" by simp_all
```
```   441   have "INum (normNum x +\<^sub>N y) = INum x + (INum y :: 'a)" by simp
```
```   442   also have "\<dots> = INum (x +\<^sub>N y)" by simp
```
```   443   finally show ?thesis using isnormNum_unique[OF n] by simp
```
```   444 qed
```
```   445
```
```   446 lemma Nadd_normNum2[simp]:
```
```   447   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   448   shows "x +\<^sub>N normNum y = x +\<^sub>N y"
```
```   449 proof -
```
```   450   have n: "isnormNum (x +\<^sub>N normNum y)" "isnormNum (x +\<^sub>N y)" by simp_all
```
```   451   have "INum (x +\<^sub>N normNum y) = INum x + (INum y :: 'a)" by simp
```
```   452   also have "\<dots> = INum (x +\<^sub>N y)" by simp
```
```   453   finally show ?thesis using isnormNum_unique[OF n] by simp
```
```   454 qed
```
```   455
```
```   456 lemma Nadd_assoc:
```
```   457   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   458   shows "x +\<^sub>N y +\<^sub>N z = x +\<^sub>N (y +\<^sub>N z)"
```
```   459 proof -
```
```   460   have n: "isnormNum (x +\<^sub>N y +\<^sub>N z)" "isnormNum (x +\<^sub>N (y +\<^sub>N z))" by simp_all
```
```   461   have "INum (x +\<^sub>N y +\<^sub>N z) = (INum (x +\<^sub>N (y +\<^sub>N z)) :: 'a)" by simp
```
```   462   with isnormNum_unique[OF n] show ?thesis by simp
```
```   463 qed
```
```   464
```
```   465 lemma Nmul_commute: "isnormNum x \<Longrightarrow> isnormNum y \<Longrightarrow> x *\<^sub>N y = y *\<^sub>N x"
```
```   466   by (simp add: Nmul_def split_def Let_def gcd_commute_int mult_commute)
```
```   467
```
```   468 lemma Nmul_assoc:
```
```   469   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   470   assumes nx: "isnormNum x" and ny: "isnormNum y" and nz: "isnormNum z"
```
```   471   shows "x *\<^sub>N y *\<^sub>N z = x *\<^sub>N (y *\<^sub>N z)"
```
```   472 proof -
```
```   473   from nx ny nz have n: "isnormNum (x *\<^sub>N y *\<^sub>N z)" "isnormNum (x *\<^sub>N (y *\<^sub>N z))"
```
```   474     by simp_all
```
```   475   have "INum (x +\<^sub>N y +\<^sub>N z) = (INum (x +\<^sub>N (y +\<^sub>N z)) :: 'a)" by simp
```
```   476   with isnormNum_unique[OF n] show ?thesis by simp
```
```   477 qed
```
```   478
```
```   479 lemma Nsub0:
```
```   480   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   481   assumes x: "isnormNum x" and y: "isnormNum y"
```
```   482   shows "x -\<^sub>N y = 0\<^sub>N \<longleftrightarrow> x = y"
```
```   483 proof -
```
```   484   fix h :: 'a
```
```   485   from isnormNum_unique[where 'a = 'a, OF Nsub_normN[OF y], where y="0\<^sub>N"]
```
```   486   have "(x -\<^sub>N y = 0\<^sub>N) = (INum (x -\<^sub>N y) = (INum 0\<^sub>N :: 'a)) " by simp
```
```   487   also have "\<dots> = (INum x = (INum y :: 'a))" by simp
```
```   488   also have "\<dots> = (x = y)" using x y by simp
```
```   489   finally show ?thesis .
```
```   490 qed
```
```   491
```
```   492 lemma Nmul0[simp]: "c *\<^sub>N 0\<^sub>N = 0\<^sub>N" " 0\<^sub>N *\<^sub>N c = 0\<^sub>N"
```
```   493   by (simp_all add: Nmul_def Let_def split_def)
```
```   494
```
```   495 lemma Nmul_eq0[simp]:
```
```   496   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   497   assumes nx: "isnormNum x" and ny: "isnormNum y"
```
```   498   shows "x*\<^sub>N y = 0\<^sub>N \<longleftrightarrow> x = 0\<^sub>N \<or> y = 0\<^sub>N"
```
```   499 proof -
```
```   500   fix h :: 'a
```
```   501   obtain a b where x: "x = (a, b)" by (cases x)
```
```   502   obtain a' b' where y: "y = (a', b')" by (cases y)
```
```   503   have n0: "isnormNum 0\<^sub>N" by simp
```
```   504   show ?thesis using nx ny
```
```   505     apply (simp only: isnormNum_unique[where ?'a = 'a, OF  Nmul_normN[OF nx ny] n0, symmetric]
```
```   506       Nmul[where ?'a = 'a])
```
```   507     apply (simp add: x y INum_def split_def isnormNum_def split: split_if_asm)
```
```   508     done
```
```   509 qed
```
```   510
```
```   511 lemma Nneg_Nneg[simp]: "~\<^sub>N (~\<^sub>N c) = c"
```
```   512   by (simp add: Nneg_def split_def)
```
```   513
```
```   514 lemma Nmul1[simp]:
```
```   515     "isnormNum c \<Longrightarrow> 1\<^sub>N *\<^sub>N c = c"
```
```   516     "isnormNum c \<Longrightarrow> c *\<^sub>N (1\<^sub>N) = c"
```
```   517   apply (simp_all add: Nmul_def Let_def split_def isnormNum_def)
```
```   518   apply (cases "fst c = 0", simp_all, cases c, simp_all)+
```
```   519   done
```
```   520
```
```   521 end
```