src/HOL/Decision_Procs/Rat_Pair.thy
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```     1 (*  Title:      HOL/Decision_Procs/Rat_Pair.thy
```
```     2     Author:     Amine Chaieb
```
```     3 *)
```
```     4
```
```     5 section \<open>Rational numbers as pairs\<close>
```
```     6
```
```     7 theory Rat_Pair
```
```     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"
```
```    20   where "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"
```
```    23   where "normNum = (\<lambda>(a,b).
```
```    24     (if a = 0 \<or> b = 0 then (0, 0)
```
```    25      else
```
```    26       (let g = gcd a b
```
```    27        in if b > 0 then (a div g, b div g) else (- (a div g), - (b div g)))))"
```
```    28
```
```    29 declare gcd_dvd1[presburger] gcd_dvd2[presburger]
```
```    30
```
```    31 lemma normNum_isnormNum [simp]: "isnormNum (normNum x)"
```
```    32 proof -
```
```    33   obtain a b where x: "x = (a, b)" by (cases x)
```
```    34   consider "a = 0 \<or> b = 0" | "a \<noteq> 0" "b \<noteq> 0"
```
```    35     by blast
```
```    36   then show ?thesis
```
```    37   proof cases
```
```    38     case 1
```
```    39     then show ?thesis
```
```    40       by (simp add: x normNum_def isnormNum_def)
```
```    41   next
```
```    42     case ab: 2
```
```    43     let ?g = "gcd a b"
```
```    44     let ?a' = "a div ?g"
```
```    45     let ?b' = "b div ?g"
```
```    46     let ?g' = "gcd ?a' ?b'"
```
```    47     from ab have "?g \<noteq> 0" by simp
```
```    48     with gcd_ge_0_int[of a b] have gpos: "?g > 0" by arith
```
```    49     have gdvd: "?g dvd a" "?g dvd b" by arith+
```
```    50     from dvd_mult_div_cancel[OF gdvd(1)] dvd_mult_div_cancel[OF gdvd(2)] ab
```
```    51     have nz': "?a' \<noteq> 0" "?b' \<noteq> 0" by - (rule notI, simp)+
```
```    52     from ab have stupid: "a \<noteq> 0 \<or> b \<noteq> 0" by arith
```
```    53     from div_gcd_coprime[OF stupid] have gp1: "?g' = 1"
```
```    54       by simp
```
```    55     from ab consider "b < 0" | "b > 0" by arith
```
```    56     then show ?thesis
```
```    57     proof cases
```
```    58       case b: 1
```
```    59       have False if b': "?b' \<ge> 0"
```
```    60       proof -
```
```    61         from gpos have th: "?g \<ge> 0" by arith
```
```    62         from mult_nonneg_nonneg[OF th b'] dvd_mult_div_cancel[OF gdvd(2)]
```
```    63         show ?thesis using b by arith
```
```    64       qed
```
```    65       then have b': "?b' < 0" by (presburger add: linorder_not_le[symmetric])
```
```    66       from ab(1) nz' b b' gp1 show ?thesis
```
```    67         by (simp add: x isnormNum_def normNum_def Let_def split_def)
```
```    68     next
```
```    69       case b: 2
```
```    70       then have "?b' \<ge> 0"
```
```    71         by (presburger add: pos_imp_zdiv_nonneg_iff[OF gpos])
```
```    72       with nz' have b': "?b' > 0" by arith
```
```    73       from b b' ab(1) nz' gp1 show ?thesis
```
```    74         by (simp add: x isnormNum_def normNum_def Let_def split_def)
```
```    75     qed
```
```    76   qed
```
```    77 qed
```
```    78
```
```    79 text \<open>Arithmetic over Num\<close>
```
```    80
```
```    81 definition Nadd :: "Num \<Rightarrow> Num \<Rightarrow> Num"  (infixl "+\<^sub>N" 60)
```
```    82 where
```
```    83   "Nadd = (\<lambda>(a, b) (a', b').
```
```    84     if a = 0 \<or> b = 0 then normNum (a', b')
```
```    85     else if a' = 0 \<or> b' = 0 then normNum (a, b)
```
```    86     else normNum (a * b' + b * a', b * b'))"
```
```    87
```
```    88 definition Nmul :: "Num \<Rightarrow> Num \<Rightarrow> Num"  (infixl "*\<^sub>N" 60)
```
```    89 where
```
```    90   "Nmul = (\<lambda>(a, b) (a', b').
```
```    91     let g = gcd (a * a') (b * b')
```
```    92     in (a * a' div g, b * b' div g))"
```
```    93
```
```    94 definition Nneg :: "Num \<Rightarrow> Num" ("~\<^sub>N")
```
```    95   where "Nneg = (\<lambda>(a, b). (- a, b))"
```
```    96
```
```    97 definition Nsub :: "Num \<Rightarrow> Num \<Rightarrow> Num"  (infixl "-\<^sub>N" 60)
```
```    98   where "Nsub = (\<lambda>a b. a +\<^sub>N ~\<^sub>N b)"
```
```    99
```
```   100 definition Ninv :: "Num \<Rightarrow> Num"
```
```   101   where "Ninv = (\<lambda>(a, b). if a < 0 then (- b, \<bar>a\<bar>) else (b, a))"
```
```   102
```
```   103 definition Ndiv :: "Num \<Rightarrow> Num \<Rightarrow> Num"  (infixl "\<div>\<^sub>N" 60)
```
```   104   where "Ndiv = (\<lambda>a b. a *\<^sub>N Ninv b)"
```
```   105
```
```   106 lemma Nneg_normN[simp]: "isnormNum x \<Longrightarrow> isnormNum (~\<^sub>N x)"
```
```   107   by (simp add: isnormNum_def Nneg_def split_def)
```
```   108
```
```   109 lemma Nadd_normN[simp]: "isnormNum (x +\<^sub>N y)"
```
```   110   by (simp add: Nadd_def split_def)
```
```   111
```
```   112 lemma Nsub_normN[simp]: "isnormNum y \<Longrightarrow> isnormNum (x -\<^sub>N y)"
```
```   113   by (simp add: Nsub_def split_def)
```
```   114
```
```   115 lemma Nmul_normN[simp]:
```
```   116   assumes xn: "isnormNum x"
```
```   117     and yn: "isnormNum y"
```
```   118   shows "isnormNum (x *\<^sub>N y)"
```
```   119 proof -
```
```   120   obtain a b where x: "x = (a, b)" by (cases x)
```
```   121   obtain a' b' where y: "y = (a', b')" by (cases y)
```
```   122   consider "a = 0" | "a' = 0" | "a \<noteq> 0" "a' \<noteq> 0" by blast
```
```   123   then show ?thesis
```
```   124   proof cases
```
```   125     case 1
```
```   126     then show ?thesis
```
```   127       using xn x y by (simp add: isnormNum_def Let_def Nmul_def split_def)
```
```   128   next
```
```   129     case 2
```
```   130     then show ?thesis
```
```   131       using yn x y by (simp add: isnormNum_def Let_def Nmul_def split_def)
```
```   132   next
```
```   133     case aa': 3
```
```   134     then have bp: "b > 0" "b' > 0"
```
```   135       using xn yn x y by (simp_all add: isnormNum_def)
```
```   136     from bp have "x *\<^sub>N y = normNum (a * a', b * b')"
```
```   137       using x y aa' bp by (simp add: Nmul_def Let_def split_def normNum_def)
```
```   138     then show ?thesis by simp
```
```   139   qed
```
```   140 qed
```
```   141
```
```   142 lemma Ninv_normN[simp]: "isnormNum x \<Longrightarrow> isnormNum (Ninv x)"
```
```   143   apply (simp add: Ninv_def isnormNum_def split_def)
```
```   144   apply (cases "fst x = 0")
```
```   145   apply (auto simp add: gcd.commute)
```
```   146   done
```
```   147
```
```   148 lemma isnormNum_int[simp]: "isnormNum 0\<^sub>N" "isnormNum ((1::int)\<^sub>N)" "i \<noteq> 0 \<Longrightarrow> isnormNum (i)\<^sub>N"
```
```   149   by (simp_all add: isnormNum_def)
```
```   150
```
```   151
```
```   152 text \<open>Relations over Num\<close>
```
```   153
```
```   154 definition Nlt0:: "Num \<Rightarrow> bool"  ("0>\<^sub>N")
```
```   155   where "Nlt0 = (\<lambda>(a, b). a < 0)"
```
```   156
```
```   157 definition Nle0:: "Num \<Rightarrow> bool"  ("0\<ge>\<^sub>N")
```
```   158   where "Nle0 = (\<lambda>(a, b). a \<le> 0)"
```
```   159
```
```   160 definition Ngt0:: "Num \<Rightarrow> bool"  ("0<\<^sub>N")
```
```   161   where "Ngt0 = (\<lambda>(a, b). a > 0)"
```
```   162
```
```   163 definition Nge0:: "Num \<Rightarrow> bool"  ("0\<le>\<^sub>N")
```
```   164   where "Nge0 = (\<lambda>(a, b). a \<ge> 0)"
```
```   165
```
```   166 definition Nlt :: "Num \<Rightarrow> Num \<Rightarrow> bool"  (infix "<\<^sub>N" 55)
```
```   167   where "Nlt = (\<lambda>a b. 0>\<^sub>N (a -\<^sub>N b))"
```
```   168
```
```   169 definition Nle :: "Num \<Rightarrow> Num \<Rightarrow> bool"  (infix "\<le>\<^sub>N" 55)
```
```   170   where "Nle = (\<lambda>a b. 0\<ge>\<^sub>N (a -\<^sub>N b))"
```
```   171
```
```   172 definition "INum = (\<lambda>(a, b). of_int a / of_int b)"
```
```   173
```
```   174 lemma INum_int [simp]: "INum (i)\<^sub>N = (of_int i ::'a::field)" "INum 0\<^sub>N = (0::'a::field)"
```
```   175   by (simp_all add: INum_def)
```
```   176
```
```   177 lemma isnormNum_unique[simp]:
```
```   178   assumes na: "isnormNum x"
```
```   179     and nb: "isnormNum y"
```
```   180   shows "(INum x ::'a::field_char_0) = INum y \<longleftrightarrow> x = y"
```
```   181   (is "?lhs = ?rhs")
```
```   182 proof
```
```   183   obtain a b where x: "x = (a, b)" by (cases x)
```
```   184   obtain a' b' where y: "y = (a', b')" by (cases y)
```
```   185   consider "a = 0 \<or> b = 0 \<or> a' = 0 \<or> b' = 0" | "a \<noteq> 0" "b \<noteq> 0" "a' \<noteq> 0" "b' \<noteq> 0"
```
```   186     by blast
```
```   187   then show ?rhs if H: ?lhs
```
```   188   proof cases
```
```   189     case 1
```
```   190     then show ?thesis
```
```   191       using na nb H by (simp add: x y INum_def split_def isnormNum_def split: if_split_asm)
```
```   192   next
```
```   193     case 2
```
```   194     with na nb have pos: "b > 0" "b' > 0"
```
```   195       by (simp_all add: x y isnormNum_def)
```
```   196     from H \<open>b \<noteq> 0\<close> \<open>b' \<noteq> 0\<close> have eq: "a * b' = a' * b"
```
```   197       by (simp add: x y INum_def eq_divide_eq divide_eq_eq of_int_mult[symmetric] del: of_int_mult)
```
```   198     from \<open>a \<noteq> 0\<close> \<open>a' \<noteq> 0\<close> na nb
```
```   199     have gcd1: "gcd a b = 1" "gcd b a = 1" "gcd a' b' = 1" "gcd b' a' = 1"
```
```   200       by (simp_all add: x y isnormNum_def add: gcd.commute)
```
```   201     then have "coprime a b" "coprime b a" "coprime a' b'" "coprime b' a'"
```
```   202       by (simp_all add: coprime_iff_gcd_eq_1)
```
```   203     from eq have raw_dvd: "a dvd a' * b" "b dvd b' * a" "a' dvd a * b'" "b' dvd b * a'"
```
```   204       apply -
```
```   205       apply algebra
```
```   206       apply algebra
```
```   207       apply simp
```
```   208       apply algebra
```
```   209       done
```
```   210     then have eq1: "b = b'"
```
```   211       using pos \<open>coprime b a\<close> \<open>coprime b' a'\<close>
```
```   212       by (auto simp add: coprime_dvd_mult_left_iff intro: associated_eqI)
```
```   213     with eq have "a = a'" using pos by simp
```
```   214     with \<open>b = b'\<close> show ?thesis by (simp add: x y)
```
```   215   qed
```
```   216   show ?lhs if ?rhs
```
```   217     using that by simp
```
```   218 qed
```
```   219
```
```   220 lemma isnormNum0[simp]: "isnormNum x \<Longrightarrow> INum x = (0::'a::field_char_0) \<longleftrightarrow> x = 0\<^sub>N"
```
```   221   unfolding INum_int(2)[symmetric]
```
```   222   by (rule isnormNum_unique) simp_all
```
```   223
```
```   224 lemma of_int_div_aux:
```
```   225   assumes "d \<noteq> 0"
```
```   226   shows "(of_int x ::'a::field_char_0) / of_int d =
```
```   227     of_int (x div d) + (of_int (x mod d)) / of_int d"
```
```   228 proof -
```
```   229   let ?t = "of_int (x div d) * (of_int d ::'a) + of_int (x mod d)"
```
```   230   let ?f = "\<lambda>x. x / of_int d"
```
```   231   have "x = (x div d) * d + x mod d"
```
```   232     by auto
```
```   233   then have eq: "of_int x = ?t"
```
```   234     by (simp only: of_int_mult[symmetric] of_int_add [symmetric])
```
```   235   then have "of_int x / of_int d = ?t / of_int d"
```
```   236     using cong[OF refl[of ?f] eq] by simp
```
```   237   then show ?thesis
```
```   238     by (simp add: add_divide_distrib algebra_simps \<open>d \<noteq> 0\<close>)
```
```   239 qed
```
```   240
```
```   241 lemma of_int_div:
```
```   242   fixes d :: int
```
```   243   assumes "d \<noteq> 0" "d dvd n"
```
```   244   shows "(of_int (n div d) ::'a::field_char_0) = of_int n / of_int d"
```
```   245   using assms of_int_div_aux [of d n, where ?'a = 'a] by simp
```
```   246
```
```   247 lemma normNum[simp]: "INum (normNum x) = (INum x :: 'a::field_char_0)"
```
```   248 proof -
```
```   249   obtain a b where x: "x = (a, b)" by (cases x)
```
```   250   consider "a = 0 \<or> b = 0" | "a \<noteq> 0" "b \<noteq> 0" by blast
```
```   251   then show ?thesis
```
```   252   proof cases
```
```   253     case 1
```
```   254     then show ?thesis
```
```   255       by (simp add: x INum_def normNum_def split_def Let_def)
```
```   256   next
```
```   257     case ab: 2
```
```   258     let ?g = "gcd a b"
```
```   259     from ab have g: "?g \<noteq> 0"by simp
```
```   260     from of_int_div[OF g, where ?'a = 'a] show ?thesis
```
```   261       by (auto simp add: x INum_def normNum_def split_def Let_def)
```
```   262   qed
```
```   263 qed
```
```   264
```
```   265 lemma INum_normNum_iff: "(INum x ::'a::field_char_0) = INum y \<longleftrightarrow> normNum x = normNum y"
```
```   266   (is "?lhs \<longleftrightarrow> _")
```
```   267 proof -
```
```   268   have "normNum x = normNum y \<longleftrightarrow> (INum (normNum x) :: 'a) = INum (normNum y)"
```
```   269     by (simp del: normNum)
```
```   270   also have "\<dots> = ?lhs" by simp
```
```   271   finally show ?thesis by simp
```
```   272 qed
```
```   273
```
```   274 lemma Nadd[simp]: "INum (x +\<^sub>N y) = INum x + (INum y :: 'a :: field_char_0)"
```
```   275 proof -
```
```   276   let ?z = "0::'a"
```
```   277   obtain a b where x: "x = (a, b)" by (cases x)
```
```   278   obtain a' b' where y: "y = (a', b')" by (cases y)
```
```   279   consider "a = 0 \<or> a'= 0 \<or> b =0 \<or> b' = 0" | "a \<noteq> 0" "a'\<noteq> 0" "b \<noteq> 0" "b' \<noteq> 0"
```
```   280     by blast
```
```   281   then show ?thesis
```
```   282   proof cases
```
```   283     case 1
```
```   284     then show ?thesis
```
```   285       apply (cases "a = 0")
```
```   286       apply (simp_all add: x y Nadd_def)
```
```   287       apply (cases "b = 0")
```
```   288       apply (simp_all add: INum_def)
```
```   289       apply (cases "a'= 0")
```
```   290       apply simp_all
```
```   291       apply (cases "b'= 0")
```
```   292       apply simp_all
```
```   293       done
```
```   294   next
```
```   295     case neq: 2
```
```   296     show ?thesis
```
```   297     proof (cases "a * b' + b * a' = 0")
```
```   298       case True
```
```   299       then have "of_int (a * b' + b * a') / (of_int b * of_int b') = ?z"
```
```   300         by simp
```
```   301       then have "of_int b' * of_int a / (of_int b * of_int b') +
```
```   302           of_int b * of_int a' / (of_int b * of_int b') = ?z"
```
```   303         by (simp add: add_divide_distrib)
```
```   304       then have th: "of_int a / of_int b + of_int a' / of_int b' = ?z"
```
```   305         using neq by simp
```
```   306       from True neq show ?thesis
```
```   307         by (simp add: x y th Nadd_def normNum_def INum_def split_def)
```
```   308     next
```
```   309       case False
```
```   310       let ?g = "gcd (a * b' + b * a') (b * b')"
```
```   311       have gz: "?g \<noteq> 0"
```
```   312         using False by simp
```
```   313       show ?thesis
```
```   314         using neq False gz
```
```   315           of_int_div [where ?'a = 'a, OF gz gcd_dvd1 [of "a * b' + b * a'" "b * b'"]]
```
```   316           of_int_div [where ?'a = 'a, OF gz gcd_dvd2 [of "a * b' + b * a'" "b * b'"]]
```
```   317         by (simp add: x y Nadd_def INum_def normNum_def Let_def) (simp add: field_simps)
```
```   318     qed
```
```   319   qed
```
```   320 qed
```
```   321
```
```   322 lemma Nmul[simp]: "INum (x *\<^sub>N y) = INum x * (INum y:: 'a::field_char_0)"
```
```   323 proof -
```
```   324   let ?z = "0::'a"
```
```   325   obtain a b where x: "x = (a, b)" by (cases x)
```
```   326   obtain a' b' where y: "y = (a', b')" by (cases y)
```
```   327   consider "a = 0 \<or> a' = 0 \<or> b = 0 \<or> b' = 0" | "a \<noteq> 0" "a' \<noteq> 0" "b \<noteq> 0" "b' \<noteq> 0"
```
```   328     by blast
```
```   329   then show ?thesis
```
```   330   proof cases
```
```   331     case 1
```
```   332     then show ?thesis
```
```   333       by (auto simp add: x y Nmul_def INum_def)
```
```   334   next
```
```   335     case neq: 2
```
```   336     let ?g = "gcd (a * a') (b * b')"
```
```   337     have gz: "?g \<noteq> 0"
```
```   338       using neq by simp
```
```   339     from neq of_int_div [where ?'a = 'a, OF gz gcd_dvd1 [of "a * a'" "b * b'"]]
```
```   340       of_int_div [where ?'a = 'a , OF gz gcd_dvd2 [of "a * a'" "b * b'"]]
```
```   341     show ?thesis
```
```   342       by (simp add: Nmul_def x y Let_def INum_def)
```
```   343   qed
```
```   344 qed
```
```   345
```
```   346 lemma Nneg[simp]: "INum (~\<^sub>N x) = - (INum x :: 'a::field)"
```
```   347   by (simp add: Nneg_def split_def INum_def)
```
```   348
```
```   349 lemma Nsub[simp]: "INum (x -\<^sub>N y) = INum x - (INum y:: 'a::field_char_0)"
```
```   350   by (simp add: Nsub_def split_def)
```
```   351
```
```   352 lemma Ninv[simp]: "INum (Ninv x) = (1 :: 'a::field) / INum x"
```
```   353   by (simp add: Ninv_def INum_def split_def)
```
```   354
```
```   355 lemma Ndiv[simp]: "INum (x \<div>\<^sub>N y) = INum x / (INum y :: 'a::field_char_0)"
```
```   356   by (simp add: Ndiv_def)
```
```   357
```
```   358 lemma Nlt0_iff[simp]:
```
```   359   assumes nx: "isnormNum x"
```
```   360   shows "((INum x :: 'a::{field_char_0,linordered_field}) < 0) = 0>\<^sub>N x"
```
```   361 proof -
```
```   362   obtain a b where x: "x = (a, b)" by (cases x)
```
```   363   show ?thesis
```
```   364   proof (cases "a = 0")
```
```   365     case True
```
```   366     then show ?thesis
```
```   367       by (simp add: x Nlt0_def INum_def)
```
```   368   next
```
```   369     case False
```
```   370     then have b: "(of_int b::'a) > 0"
```
```   371       using nx by (simp add: x isnormNum_def)
```
```   372     from pos_divide_less_eq[OF b, where b="of_int a" and a="0::'a"]
```
```   373     show ?thesis
```
```   374       by (simp add: x Nlt0_def INum_def)
```
```   375   qed
```
```   376 qed
```
```   377
```
```   378 lemma Nle0_iff[simp]:
```
```   379   assumes nx: "isnormNum x"
```
```   380   shows "((INum x :: 'a::{field_char_0,linordered_field}) \<le> 0) = 0\<ge>\<^sub>N x"
```
```   381 proof -
```
```   382   obtain a b where x: "x = (a, b)" by (cases x)
```
```   383   show ?thesis
```
```   384   proof (cases "a = 0")
```
```   385     case True
```
```   386     then show ?thesis
```
```   387       by (simp add: x Nle0_def INum_def)
```
```   388   next
```
```   389     case False
```
```   390     then have b: "(of_int b :: 'a) > 0"
```
```   391       using nx by (simp add: x isnormNum_def)
```
```   392     from pos_divide_le_eq[OF b, where b="of_int a" and a="0::'a"]
```
```   393     show ?thesis
```
```   394       by (simp add: x Nle0_def INum_def)
```
```   395   qed
```
```   396 qed
```
```   397
```
```   398 lemma Ngt0_iff[simp]:
```
```   399   assumes nx: "isnormNum x"
```
```   400   shows "((INum x :: 'a::{field_char_0,linordered_field}) > 0) = 0<\<^sub>N x"
```
```   401 proof -
```
```   402   obtain a b where x: "x = (a, b)" by (cases x)
```
```   403   show ?thesis
```
```   404   proof (cases "a = 0")
```
```   405     case True
```
```   406     then show ?thesis
```
```   407       by (simp add: x Ngt0_def INum_def)
```
```   408   next
```
```   409     case False
```
```   410     then have b: "(of_int b::'a) > 0"
```
```   411       using nx by (simp add: x isnormNum_def)
```
```   412     from pos_less_divide_eq[OF b, where b="of_int a" and a="0::'a"]
```
```   413     show ?thesis
```
```   414       by (simp add: x Ngt0_def INum_def)
```
```   415   qed
```
```   416 qed
```
```   417
```
```   418 lemma Nge0_iff[simp]:
```
```   419   assumes nx: "isnormNum x"
```
```   420   shows "(INum x :: 'a::{field_char_0,linordered_field}) \<ge> 0 \<longleftrightarrow> 0\<le>\<^sub>N x"
```
```   421 proof -
```
```   422   obtain a b where x: "x = (a, b)" by (cases x)
```
```   423   show ?thesis
```
```   424   proof (cases "a = 0")
```
```   425     case True
```
```   426     then show ?thesis
```
```   427       by (simp add: x Nge0_def INum_def)
```
```   428   next
```
```   429     case False
```
```   430     then have b: "(of_int b::'a) > 0"
```
```   431       using nx by (simp add: x isnormNum_def)
```
```   432     from pos_le_divide_eq[OF b, where b="of_int a" and a="0::'a"]
```
```   433     show ?thesis
```
```   434       by (simp add: x Nge0_def INum_def)
```
```   435   qed
```
```   436 qed
```
```   437
```
```   438 lemma Nlt_iff[simp]:
```
```   439   assumes nx: "isnormNum x"
```
```   440     and ny: "isnormNum y"
```
```   441   shows "((INum x :: 'a::{field_char_0,linordered_field}) < INum y) \<longleftrightarrow> x <\<^sub>N y"
```
```   442 proof -
```
```   443   let ?z = "0::'a"
```
```   444   have "((INum x ::'a) < INum y) \<longleftrightarrow> INum (x -\<^sub>N y) < ?z"
```
```   445     using nx ny by simp
```
```   446   also have "\<dots> \<longleftrightarrow> 0>\<^sub>N (x -\<^sub>N y)"
```
```   447     using Nlt0_iff[OF Nsub_normN[OF ny]] by simp
```
```   448   finally show ?thesis
```
```   449     by (simp add: Nlt_def)
```
```   450 qed
```
```   451
```
```   452 lemma Nle_iff[simp]:
```
```   453   assumes nx: "isnormNum x"
```
```   454     and ny: "isnormNum y"
```
```   455   shows "((INum x :: 'a::{field_char_0,linordered_field}) \<le> INum y) \<longleftrightarrow> x \<le>\<^sub>N y"
```
```   456 proof -
```
```   457   have "((INum x ::'a) \<le> INum y) \<longleftrightarrow> INum (x -\<^sub>N y) \<le> (0::'a)"
```
```   458     using nx ny by simp
```
```   459   also have "\<dots> \<longleftrightarrow> 0\<ge>\<^sub>N (x -\<^sub>N y)"
```
```   460     using Nle0_iff[OF Nsub_normN[OF ny]] by simp
```
```   461   finally show ?thesis
```
```   462     by (simp add: Nle_def)
```
```   463 qed
```
```   464
```
```   465 lemma Nadd_commute:
```
```   466   assumes "SORT_CONSTRAINT('a::field_char_0)"
```
```   467   shows "x +\<^sub>N y = y +\<^sub>N x"
```
```   468 proof -
```
```   469   have n: "isnormNum (x +\<^sub>N y)" "isnormNum (y +\<^sub>N x)"
```
```   470     by simp_all
```
```   471   have "(INum (x +\<^sub>N y)::'a) = INum (y +\<^sub>N x)"
```
```   472     by simp
```
```   473   with isnormNum_unique[OF n] show ?thesis
```
```   474     by simp
```
```   475 qed
```
```   476
```
```   477 lemma [simp]:
```
```   478   assumes "SORT_CONSTRAINT('a::field_char_0)"
```
```   479   shows "(0, b) +\<^sub>N y = normNum y"
```
```   480     and "(a, 0) +\<^sub>N y = normNum y"
```
```   481     and "x +\<^sub>N (0, b) = normNum x"
```
```   482     and "x +\<^sub>N (a, 0) = normNum x"
```
```   483   apply (simp add: Nadd_def split_def)
```
```   484   apply (simp add: Nadd_def split_def)
```
```   485   apply (subst Nadd_commute)
```
```   486   apply (simp add: Nadd_def split_def)
```
```   487   apply (subst Nadd_commute)
```
```   488   apply (simp add: Nadd_def split_def)
```
```   489   done
```
```   490
```
```   491 lemma normNum_nilpotent_aux[simp]:
```
```   492   assumes "SORT_CONSTRAINT('a::field_char_0)"
```
```   493   assumes nx: "isnormNum x"
```
```   494   shows "normNum x = x"
```
```   495 proof -
```
```   496   let ?a = "normNum x"
```
```   497   have n: "isnormNum ?a" by simp
```
```   498   have th: "INum ?a = (INum x ::'a)" by simp
```
```   499   with isnormNum_unique[OF n nx] show ?thesis by simp
```
```   500 qed
```
```   501
```
```   502 lemma normNum_nilpotent[simp]:
```
```   503   assumes "SORT_CONSTRAINT('a::field_char_0)"
```
```   504   shows "normNum (normNum x) = normNum x"
```
```   505   by simp
```
```   506
```
```   507 lemma normNum0[simp]: "normNum (0, b) = 0\<^sub>N" "normNum (a, 0) = 0\<^sub>N"
```
```   508   by (simp_all add: normNum_def)
```
```   509
```
```   510 lemma normNum_Nadd:
```
```   511   assumes "SORT_CONSTRAINT('a::field_char_0)"
```
```   512   shows "normNum (x +\<^sub>N y) = x +\<^sub>N y"
```
```   513   by simp
```
```   514
```
```   515 lemma Nadd_normNum1[simp]:
```
```   516   assumes "SORT_CONSTRAINT('a::field_char_0)"
```
```   517   shows "normNum x +\<^sub>N y = x +\<^sub>N y"
```
```   518 proof -
```
```   519   have n: "isnormNum (normNum x +\<^sub>N y)" "isnormNum (x +\<^sub>N y)"
```
```   520     by simp_all
```
```   521   have "INum (normNum x +\<^sub>N y) = INum x + (INum y :: 'a)"
```
```   522     by simp
```
```   523   also have "\<dots> = INum (x +\<^sub>N y)"
```
```   524     by simp
```
```   525   finally show ?thesis
```
```   526     using isnormNum_unique[OF n] by simp
```
```   527 qed
```
```   528
```
```   529 lemma Nadd_normNum2[simp]:
```
```   530   assumes "SORT_CONSTRAINT('a::field_char_0)"
```
```   531   shows "x +\<^sub>N normNum y = x +\<^sub>N y"
```
```   532 proof -
```
```   533   have n: "isnormNum (x +\<^sub>N normNum y)" "isnormNum (x +\<^sub>N y)"
```
```   534     by simp_all
```
```   535   have "INum (x +\<^sub>N normNum y) = INum x + (INum y :: 'a)"
```
```   536     by simp
```
```   537   also have "\<dots> = INum (x +\<^sub>N y)"
```
```   538     by simp
```
```   539   finally show ?thesis
```
```   540     using isnormNum_unique[OF n] by simp
```
```   541 qed
```
```   542
```
```   543 lemma Nadd_assoc:
```
```   544   assumes "SORT_CONSTRAINT('a::field_char_0)"
```
```   545   shows "x +\<^sub>N y +\<^sub>N z = x +\<^sub>N (y +\<^sub>N z)"
```
```   546 proof -
```
```   547   have n: "isnormNum (x +\<^sub>N y +\<^sub>N z)" "isnormNum (x +\<^sub>N (y +\<^sub>N z))"
```
```   548     by simp_all
```
```   549   have "INum (x +\<^sub>N y +\<^sub>N z) = (INum (x +\<^sub>N (y +\<^sub>N z)) :: 'a)"
```
```   550     by simp
```
```   551   with isnormNum_unique[OF n] show ?thesis
```
```   552     by simp
```
```   553 qed
```
```   554
```
```   555 lemma Nmul_commute: "isnormNum x \<Longrightarrow> isnormNum y \<Longrightarrow> x *\<^sub>N y = y *\<^sub>N x"
```
```   556   by (simp add: Nmul_def split_def Let_def gcd.commute mult.commute)
```
```   557
```
```   558 lemma Nmul_assoc:
```
```   559   assumes "SORT_CONSTRAINT('a::field_char_0)"
```
```   560   assumes nx: "isnormNum x"
```
```   561     and ny: "isnormNum y"
```
```   562     and nz: "isnormNum z"
```
```   563   shows "x *\<^sub>N y *\<^sub>N z = x *\<^sub>N (y *\<^sub>N z)"
```
```   564 proof -
```
```   565   from nx ny nz have n: "isnormNum (x *\<^sub>N y *\<^sub>N z)" "isnormNum (x *\<^sub>N (y *\<^sub>N z))"
```
```   566     by simp_all
```
```   567   have "INum (x +\<^sub>N y +\<^sub>N z) = (INum (x +\<^sub>N (y +\<^sub>N z)) :: 'a)"
```
```   568     by simp
```
```   569   with isnormNum_unique[OF n] show ?thesis
```
```   570     by simp
```
```   571 qed
```
```   572
```
```   573 lemma Nsub0:
```
```   574   assumes "SORT_CONSTRAINT('a::field_char_0)"
```
```   575   assumes x: "isnormNum x"
```
```   576     and y: "isnormNum y"
```
```   577   shows "x -\<^sub>N y = 0\<^sub>N \<longleftrightarrow> x = y"
```
```   578 proof -
```
```   579   from isnormNum_unique[where 'a = 'a, OF Nsub_normN[OF y], where y="0\<^sub>N"]
```
```   580   have "x -\<^sub>N y = 0\<^sub>N \<longleftrightarrow> INum (x -\<^sub>N y) = (INum 0\<^sub>N :: 'a)"
```
```   581     by simp
```
```   582   also have "\<dots> \<longleftrightarrow> INum x = (INum y :: 'a)"
```
```   583     by simp
```
```   584   also have "\<dots> \<longleftrightarrow> x = y"
```
```   585     using x y by simp
```
```   586   finally show ?thesis .
```
```   587 qed
```
```   588
```
```   589 lemma Nmul0[simp]: "c *\<^sub>N 0\<^sub>N = 0\<^sub>N" " 0\<^sub>N *\<^sub>N c = 0\<^sub>N"
```
```   590   by (simp_all add: Nmul_def Let_def split_def)
```
```   591
```
```   592 lemma Nmul_eq0[simp]:
```
```   593   assumes "SORT_CONSTRAINT('a::field_char_0)"
```
```   594   assumes nx: "isnormNum x"
```
```   595     and ny: "isnormNum y"
```
```   596   shows "x*\<^sub>N y = 0\<^sub>N \<longleftrightarrow> x = 0\<^sub>N \<or> y = 0\<^sub>N"
```
```   597 proof -
```
```   598   obtain a b where x: "x = (a, b)" by (cases x)
```
```   599   obtain a' b' where y: "y = (a', b')" by (cases y)
```
```   600   have n0: "isnormNum 0\<^sub>N" by simp
```
```   601   show ?thesis using nx ny
```
```   602     apply (simp only: isnormNum_unique[where ?'a = 'a, OF  Nmul_normN[OF nx ny] n0, symmetric]
```
```   603       Nmul[where ?'a = 'a])
```
```   604     apply (simp add: x y INum_def split_def isnormNum_def split: if_split_asm)
```
```   605     done
```
```   606 qed
```
```   607
```
```   608 lemma Nneg_Nneg[simp]: "~\<^sub>N (~\<^sub>N c) = c"
```
```   609   by (simp add: Nneg_def split_def)
```
```   610
```
```   611 lemma Nmul1[simp]: "isnormNum c \<Longrightarrow> (1)\<^sub>N *\<^sub>N c = c" "isnormNum c \<Longrightarrow> c *\<^sub>N (1)\<^sub>N = c"
```
```   612   apply (simp_all add: Nmul_def Let_def split_def isnormNum_def)
```
```   613   apply (cases "fst c = 0", simp_all, cases c, simp_all)+
```
```   614   done
```
```   615
```
```   616 end
```