(*  Title:      HOL/Library/Abstract_Rat.thy

     Author:     Amine Chaieb

 *)

 header {* Abstract rational numbers *}

 theory Abstract_Rat

 imports Complex_Main

 begin

 types Num = "int \<times> int"

 abbreviation

   Num0_syn :: Num ("0\<^sub>N")

 where "0\<^sub>N \<equiv> (0, 0)"

 abbreviation

   Numi_syn :: "int \<Rightarrow> Num" ("_\<^sub>N")

 where "i\<^sub>N \<equiv> (i, 1)"

 definition

   isnormNum :: "Num \<Rightarrow> bool"

 where

   "isnormNum = (\<lambda>(a,b). (if a = 0 then b = 0 else b > 0 \<and> gcd a b = 1))"

 definition

   normNum :: "Num \<Rightarrow> Num"

 where

   "normNum = (\<lambda>(a,b). (if a=0 \<or> b = 0 then (0,0) else 

   (let g = gcd a b 

    in if b > 0 then (a div g, b div g) else (- (a div g), - (b div g)))))"

 declare gcd_dvd1_int[presburger]

 declare gcd_dvd2_int[presburger]

 lemma normNum_isnormNum [simp]: "isnormNum (normNum x)"

 proof -

   have " \<exists> a b. x = (a,b)" by auto

   then obtain a b where x[simp]: "x = (a,b)" by blast

   {assume "a=0 \<or> b = 0" hence ?thesis by (simp add: normNum_def isnormNum_def)}  

   moreover

   {assume anz: "a \<noteq> 0" and bnz: "b \<noteq> 0" 

     let ?g = "gcd a b"

     let ?a' = "a div ?g"

     let ?b' = "b div ?g"

     let ?g' = "gcd ?a' ?b'"

     from anz bnz have "?g \<noteq> 0" by simp  with gcd_ge_0_int[of a b] 

     have gpos: "?g > 0"  by arith

     have gdvd: "?g dvd a" "?g dvd b" by arith+ 

     from zdvd_mult_div_cancel[OF gdvd(1)] zdvd_mult_div_cancel[OF gdvd(2)]

     anz bnz

     have nz':"?a' \<noteq> 0" "?b' \<noteq> 0"

       by - (rule notI, simp)+

     from anz bnz have stupid: "a \<noteq> 0 \<or> b \<noteq> 0" by arith 

     from div_gcd_coprime_int[OF stupid] have gp1: "?g' = 1" .

     from bnz have "b < 0 \<or> b > 0" by arith

     moreover

     {assume b: "b > 0"

       from b have "?b' \<ge> 0" 

         by (presburger add: pos_imp_zdiv_nonneg_iff[OF gpos])  

       with nz' have b': "?b' > 0" by arith 

       from b b' anz bnz nz' gp1 have ?thesis 

         by (simp add: isnormNum_def normNum_def Let_def split_def fst_conv snd_conv)}

     moreover {assume b: "b < 0"

       {assume b': "?b' \<ge> 0" 

         from gpos have th: "?g \<ge> 0" by arith

         from mult_nonneg_nonneg[OF th b'] zdvd_mult_div_cancel[OF gdvd(2)]

         have False using b by arith }

       hence b': "?b' < 0" by (presburger add: linorder_not_le[symmetric]) 

       from anz bnz nz' b b' gp1 have ?thesis 

         by (simp add: isnormNum_def normNum_def Let_def split_def)}

     ultimately have ?thesis by blast

   }

   ultimately show ?thesis by blast

 qed

 text {* Arithmetic over Num *}

 definition

   Nadd :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "+\<^sub>N" 60)

 where

   "Nadd = (\<lambda>(a,b) (a',b'). if a = 0 \<or> b = 0 then normNum(a',b') 

     else if a'=0 \<or> b' = 0 then normNum(a,b) 

     else normNum(a*b' + b*a', b*b'))"

 definition

   Nmul :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "*\<^sub>N" 60)

 where

   "Nmul = (\<lambda>(a,b) (a',b'). let g = gcd (a*a') (b*b') 

     in (a*a' div g, b*b' div g))"

 definition

   Nneg :: "Num \<Rightarrow> Num" ("~\<^sub>N")

 where

   "Nneg \<equiv> (\<lambda>(a,b). (-a,b))"

 definition

   Nsub :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "-\<^sub>N" 60)

 where

   "Nsub = (\<lambda>a b. a +\<^sub>N ~\<^sub>N b)"

 definition

   Ninv :: "Num \<Rightarrow> Num" 

 where

   "Ninv \<equiv> \<lambda>(a,b). if a < 0 then (-b, \<bar>a\<bar>) else (b,a)"

 definition

   Ndiv :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "\<div>\<^sub>N" 60)

 where

   "Ndiv \<equiv> \<lambda>a b. a *\<^sub>N Ninv b"

 lemma Nneg_normN[simp]: "isnormNum x \<Longrightarrow> isnormNum (~\<^sub>N x)"

   by(simp add: isnormNum_def Nneg_def split_def)

 lemma Nadd_normN[simp]: "isnormNum (x +\<^sub>N y)"

   by (simp add: Nadd_def split_def)

 lemma Nsub_normN[simp]: "\<lbrakk> isnormNum y\<rbrakk> \<Longrightarrow> isnormNum (x -\<^sub>N y)"

   by (simp add: Nsub_def split_def)

 lemma Nmul_normN[simp]: assumes xn:"isnormNum x" and yn: "isnormNum y"

   shows "isnormNum (x *\<^sub>N y)"

 proof-

   have "\<exists>a b. x = (a,b)" and "\<exists> a' b'. y = (a',b')" by auto

   then obtain a b a' b' where ab: "x = (a,b)"  and ab': "y = (a',b')" by blast 

   {assume "a = 0"

     hence ?thesis using xn ab ab'

       by (simp add: isnormNum_def Let_def Nmul_def split_def)}

   moreover

   {assume "a' = 0"

     hence ?thesis using yn ab ab' 

       by (simp add: isnormNum_def Let_def Nmul_def split_def)}

   moreover

   {assume a: "a \<noteq>0" and a': "a'\<noteq>0"

     hence bp: "b > 0" "b' > 0" using xn yn ab ab' by (simp_all add: isnormNum_def)

     from mult_pos_pos[OF bp] have "x *\<^sub>N y = normNum (a*a', b*b')" 

       using ab ab' a a' bp by (simp add: Nmul_def Let_def split_def normNum_def)

     hence ?thesis by simp}

   ultimately show ?thesis by blast

 qed

 lemma Ninv_normN[simp]: "isnormNum x \<Longrightarrow> isnormNum (Ninv x)"

   by (simp add: Ninv_def isnormNum_def split_def)

     (cases "fst x = 0", auto simp add: gcd_commute_int)

 lemma isnormNum_int[simp]: 

   "isnormNum 0\<^sub>N" "isnormNum (1::int)\<^sub>N" "i \<noteq> 0 \<Longrightarrow> isnormNum i\<^sub>N"

   by (simp_all add: isnormNum_def)

 text {* Relations over Num *}

 definition

   Nlt0:: "Num \<Rightarrow> bool" ("0>\<^sub>N")

 where

   "Nlt0 = (\<lambda>(a,b). a < 0)"

 definition

   Nle0:: "Num \<Rightarrow> bool" ("0\<ge>\<^sub>N")

 where

   "Nle0 = (\<lambda>(a,b). a \<le> 0)"

 definition

   Ngt0:: "Num \<Rightarrow> bool" ("0<\<^sub>N")

 where

   "Ngt0 = (\<lambda>(a,b). a > 0)"

 definition

   Nge0:: "Num \<Rightarrow> bool" ("0\<le>\<^sub>N")

 where

   "Nge0 = (\<lambda>(a,b). a \<ge> 0)"

 definition

   Nlt :: "Num \<Rightarrow> Num \<Rightarrow> bool" (infix "<\<^sub>N" 55)

 where

   "Nlt = (\<lambda>a b. 0>\<^sub>N (a -\<^sub>N b))"

 definition

   Nle :: "Num \<Rightarrow> Num \<Rightarrow> bool" (infix "\<le>\<^sub>N" 55)

 where

   "Nle = (\<lambda>a b. 0\<ge>\<^sub>N (a -\<^sub>N b))"

 definition

   "INum = (\<lambda>(a,b). of_int a / of_int b)"

 lemma INum_int [simp]: "INum i\<^sub>N = ((of_int i) ::'a::field)" "INum 0\<^sub>N = (0::'a::field)"

   by (simp_all add: INum_def)

 lemma isnormNum_unique[simp]: 

   assumes na: "isnormNum x" and nb: "isnormNum y" 

   shows "((INum x ::'a::{field_char_0, field_inverse_zero}) = INum y) = (x = y)" (is "?lhs = ?rhs")

 proof

   have "\<exists> a b a' b'. x = (a,b) \<and> y = (a',b')" by auto

   then obtain a b a' b' where xy[simp]: "x = (a,b)" "y=(a',b')" by blast

   assume H: ?lhs 

   {assume "a = 0 \<or> b = 0 \<or> a' = 0 \<or> b' = 0"

     hence ?rhs using na nb H

       by (simp add: INum_def split_def isnormNum_def split: split_if_asm)}

   moreover

   { assume az: "a \<noteq> 0" and bz: "b \<noteq> 0" and a'z: "a'\<noteq>0" and b'z: "b'\<noteq>0"

     from az bz a'z b'z na nb have pos: "b > 0" "b' > 0" by (simp_all add: isnormNum_def)

     from prems have eq:"a * b' = a'*b" 

       by (simp add: INum_def  eq_divide_eq divide_eq_eq of_int_mult[symmetric] del: of_int_mult)

     from prems have gcd1: "gcd a b = 1" "gcd b a = 1" "gcd a' b' = 1" "gcd b' a' = 1"       

       by (simp_all add: isnormNum_def add: gcd_commute_int)

     from eq have raw_dvd: "a dvd a'*b" "b dvd b'*a" "a' dvd a*b'" "b' dvd b*a'"

       apply - 

       apply algebra

       apply algebra

       apply simp

       apply algebra

       done

     from zdvd_antisym_abs[OF coprime_dvd_mult_int[OF gcd1(2) raw_dvd(2)]

       coprime_dvd_mult_int[OF gcd1(4) raw_dvd(4)]]

       have eq1: "b = b'" using pos by arith  

       with eq have "a = a'" using pos by simp

       with eq1 have ?rhs by simp}

   ultimately show ?rhs by blast

 next

   assume ?rhs thus ?lhs by simp

 qed

 lemma isnormNum0[simp]: "isnormNum x \<Longrightarrow> (INum x = (0::'a::{field_char_0, field_inverse_zero})) = (x = 0\<^sub>N)"

   unfolding INum_int(2)[symmetric]

   by (rule isnormNum_unique, simp_all)

 lemma of_int_div_aux: "d ~= 0 ==> ((of_int x)::'a::field_char_0) / (of_int d) = 

     of_int (x div d) + (of_int (x mod d)) / ((of_int d)::'a)"

 proof -

   assume "d ~= 0"

   hence dz: "of_int d \<noteq> (0::'a)" by (simp add: of_int_eq_0_iff)

   let ?t = "of_int (x div d) * ((of_int d)::'a) + of_int(x mod d)"

   let ?f = "\<lambda>x. x / of_int d"

   have "x = (x div d) * d + x mod d"

     by auto

   then have eq: "of_int x = ?t"

     by (simp only: of_int_mult[symmetric] of_int_add [symmetric])

   then have "of_int x / of_int d = ?t / of_int d" 

     using cong[OF refl[of ?f] eq] by simp

   then show ?thesis by (simp add: add_divide_distrib algebra_simps prems)

 qed

 lemma of_int_div: "(d::int) ~= 0 ==> d dvd n ==>

     (of_int(n div d)::'a::field_char_0) = of_int n / of_int d"

   apply (frule of_int_div_aux [of d n, where ?'a = 'a])

   apply simp

   apply (simp add: dvd_eq_mod_eq_0)

 done

 lemma normNum[simp]: "INum (normNum x) = (INum x :: 'a::{field_char_0, field_inverse_zero})"

 proof-

   have "\<exists> a b. x = (a,b)" by auto

   then obtain a b where x[simp]: "x = (a,b)" by blast

   {assume "a=0 \<or> b = 0" hence ?thesis

       by (simp add: INum_def normNum_def split_def Let_def)}

   moreover 

   {assume a: "a\<noteq>0" and b: "b\<noteq>0"

     let ?g = "gcd a b"

     from a b have g: "?g \<noteq> 0"by simp

     from of_int_div[OF g, where ?'a = 'a]

     have ?thesis by (auto simp add: INum_def normNum_def split_def Let_def)}

   ultimately show ?thesis by blast

 qed

 lemma INum_normNum_iff: "(INum x ::'a::{field_char_0, field_inverse_zero}) = INum y \<longleftrightarrow> normNum x = normNum y" (is "?lhs = ?rhs")

 proof -

   have "normNum x = normNum y \<longleftrightarrow> (INum (normNum x) :: 'a) = INum (normNum y)"

     by (simp del: normNum)

   also have "\<dots> = ?lhs" by simp

   finally show ?thesis by simp

 qed

 lemma Nadd[simp]: "INum (x +\<^sub>N y) = INum x + (INum y :: 'a :: {field_char_0, field_inverse_zero})"

 proof-

 let ?z = "0:: 'a"

   have " \<exists> a b. x = (a,b)" " \<exists> a' b'. y = (a',b')" by auto

   then obtain a b a' b' where x[simp]: "x = (a,b)" 

     and y[simp]: "y = (a',b')" by blast

   {assume "a=0 \<or> a'= 0 \<or> b =0 \<or> b' = 0" hence ?thesis 

       apply (cases "a=0",simp_all add: Nadd_def)

       apply (cases "b= 0",simp_all add: INum_def)

        apply (cases "a'= 0",simp_all)

        apply (cases "b'= 0",simp_all)

        done }

   moreover 

   {assume aa':"a \<noteq> 0" "a'\<noteq> 0" and bb': "b \<noteq> 0" "b' \<noteq> 0" 

     {assume z: "a * b' + b * a' = 0"

       hence "of_int (a*b' + b*a') / (of_int b* of_int b') = ?z" by simp

       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) 

       hence th: "of_int a / of_int b + of_int a' / of_int b' = ?z" using bb' aa' by simp 

       from z aa' bb' have ?thesis 

         by (simp add: th Nadd_def normNum_def INum_def split_def)}

     moreover {assume z: "a * b' + b * a' \<noteq> 0"

       let ?g = "gcd (a * b' + b * a') (b*b')"

       have gz: "?g \<noteq> 0" using z by simp

       have ?thesis using aa' bb' z gz

         of_int_div[where ?'a = 'a, OF gz gcd_dvd1_int[where x="a * b' + b * a'" and y="b*b'"]]  of_int_div[where ?'a = 'a,

         OF gz gcd_dvd2_int[where x="a * b' + b * a'" and y="b*b'"]]

         by (simp add: x y Nadd_def INum_def normNum_def Let_def add_divide_distrib)}

     ultimately have ?thesis using aa' bb' 

       by (simp add: Nadd_def INum_def normNum_def x y Let_def) }

   ultimately show ?thesis by blast

 qed

 lemma Nmul[simp]: "INum (x *\<^sub>N y) = INum x * (INum y:: 'a :: {field_char_0, field_inverse_zero}) "

 proof-

   let ?z = "0::'a"

   have " \<exists> a b. x = (a,b)" " \<exists> a' b'. y = (a',b')" by auto

   then obtain a b a' b' where x: "x = (a,b)" and y: "y = (a',b')" by blast

   {assume "a=0 \<or> a'= 0 \<or> b = 0 \<or> b' = 0" hence ?thesis 

       apply (cases "a=0",simp_all add: x y Nmul_def INum_def Let_def)

       apply (cases "b=0",simp_all)

       apply (cases "a'=0",simp_all) 

       done }

   moreover

   {assume z: "a \<noteq> 0" "a' \<noteq> 0" "b \<noteq> 0" "b' \<noteq> 0"

     let ?g="gcd (a*a') (b*b')"

     have gz: "?g \<noteq> 0" using z by simp

     from z of_int_div[where ?'a = 'a, OF gz gcd_dvd1_int[where x="a*a'" and y="b*b'"]] 

       of_int_div[where ?'a = 'a , OF gz gcd_dvd2_int[where x="a*a'" and y="b*b'"]] 

     have ?thesis by (simp add: Nmul_def x y Let_def INum_def)}

   ultimately show ?thesis by blast

 qed

 lemma Nneg[simp]: "INum (~\<^sub>N x) = - (INum x ::'a:: field)"

   by (simp add: Nneg_def split_def INum_def)

 lemma Nsub[simp]: shows "INum (x -\<^sub>N y) = INum x - (INum y:: 'a :: {field_char_0, field_inverse_zero})"

 by (simp add: Nsub_def split_def)

 lemma Ninv[simp]: "INum (Ninv x) = (1::'a :: field_inverse_zero) / (INum x)"

   by (simp add: Ninv_def INum_def split_def)

 lemma Ndiv[simp]: "INum (x \<div>\<^sub>N y) = INum x / (INum y ::'a :: {field_char_0, field_inverse_zero})" by (simp add: Ndiv_def)

 lemma Nlt0_iff[simp]: assumes nx: "isnormNum x" 

   shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero})< 0) = 0>\<^sub>N x "

 proof-

   have " \<exists> a b. x = (a,b)" by simp

   then obtain a b where x[simp]:"x = (a,b)" by blast

   {assume "a = 0" hence ?thesis by (simp add: Nlt0_def INum_def) }

   moreover

   {assume a: "a\<noteq>0" hence b: "(of_int b::'a) > 0" using nx by (simp add: isnormNum_def)

     from pos_divide_less_eq[OF b, where b="of_int a" and a="0::'a"]

     have ?thesis by (simp add: Nlt0_def INum_def)}

   ultimately show ?thesis by blast

 qed

 lemma Nle0_iff[simp]:assumes nx: "isnormNum x" 

   shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero}) \<le> 0) = 0\<ge>\<^sub>N x"

 proof-

   have " \<exists> a b. x = (a,b)" by simp

   then obtain a b where x[simp]:"x = (a,b)" by blast

   {assume "a = 0" hence ?thesis by (simp add: Nle0_def INum_def) }

   moreover

   {assume a: "a\<noteq>0" hence b: "(of_int b :: 'a) > 0" using nx by (simp add: isnormNum_def)

     from pos_divide_le_eq[OF b, where b="of_int a" and a="0::'a"]

     have ?thesis by (simp add: Nle0_def INum_def)}

   ultimately show ?thesis by blast

 qed

 lemma Ngt0_iff[simp]:assumes nx: "isnormNum x" shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero})> 0) = 0<\<^sub>N x"

 proof-

   have " \<exists> a b. x = (a,b)" by simp

   then obtain a b where x[simp]:"x = (a,b)" by blast

   {assume "a = 0" hence ?thesis by (simp add: Ngt0_def INum_def) }

   moreover

   {assume a: "a\<noteq>0" hence b: "(of_int b::'a) > 0" using nx by (simp add: isnormNum_def)

     from pos_less_divide_eq[OF b, where b="of_int a" and a="0::'a"]

     have ?thesis by (simp add: Ngt0_def INum_def)}

   ultimately show ?thesis by blast

 qed

 lemma Nge0_iff[simp]:assumes nx: "isnormNum x" 

   shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero}) \<ge> 0) = 0\<le>\<^sub>N x"

 proof-

   have " \<exists> a b. x = (a,b)" by simp

   then obtain a b where x[simp]:"x = (a,b)" by blast

   {assume "a = 0" hence ?thesis by (simp add: Nge0_def INum_def) }

   moreover

   {assume a: "a\<noteq>0" hence b: "(of_int b::'a) > 0" using nx by (simp add: isnormNum_def)

     from pos_le_divide_eq[OF b, where b="of_int a" and a="0::'a"]

     have ?thesis by (simp add: Nge0_def INum_def)}

   ultimately show ?thesis by blast

 qed

 lemma Nlt_iff[simp]: assumes nx: "isnormNum x" and ny: "isnormNum y"

   shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero}) < INum y) = (x <\<^sub>N y)"

 proof-

   let ?z = "0::'a"

   have "((INum x ::'a) < INum y) = (INum (x -\<^sub>N y) < ?z)" using nx ny by simp

   also have "\<dots> = (0>\<^sub>N (x -\<^sub>N y))" using Nlt0_iff[OF Nsub_normN[OF ny]] by simp

   finally show ?thesis by (simp add: Nlt_def)

 qed

 lemma Nle_iff[simp]: assumes nx: "isnormNum x" and ny: "isnormNum y"

   shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero})\<le> INum y) = (x \<le>\<^sub>N y)"

 proof-

   have "((INum x ::'a) \<le> INum y) = (INum (x -\<^sub>N y) \<le> (0::'a))" using nx ny by simp

   also have "\<dots> = (0\<ge>\<^sub>N (x -\<^sub>N y))" using Nle0_iff[OF Nsub_normN[OF ny]] by simp

   finally show ?thesis by (simp add: Nle_def)

 qed

 lemma Nadd_commute:

   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"

   shows "x +\<^sub>N y = y +\<^sub>N x"

 proof-

   have n: "isnormNum (x +\<^sub>N y)" "isnormNum (y +\<^sub>N x)" by simp_all

   have "(INum (x +\<^sub>N y)::'a) = INum (y +\<^sub>N x)" by simp

   with isnormNum_unique[OF n] show ?thesis by simp

 qed

 lemma [simp]:

   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"

   shows "(0, b) +\<^sub>N y = normNum y"

     and "(a, 0) +\<^sub>N y = normNum y" 

     and "x +\<^sub>N (0, b) = normNum x"

     and "x +\<^sub>N (a, 0) = normNum x"

   apply (simp add: Nadd_def split_def)

   apply (simp add: Nadd_def split_def)

   apply (subst Nadd_commute, simp add: Nadd_def split_def)

   apply (subst Nadd_commute, simp add: Nadd_def split_def)

   done

 lemma normNum_nilpotent_aux[simp]:

   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"

   assumes nx: "isnormNum x" 

   shows "normNum x = x"

 proof-

   let ?a = "normNum x"

   have n: "isnormNum ?a" by simp

   have th:"INum ?a = (INum x ::'a)" by simp

   with isnormNum_unique[OF n nx]  

   show ?thesis by simp

 qed

 lemma normNum_nilpotent[simp]:

   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"

   shows "normNum (normNum x) = normNum x"

   by simp

 lemma normNum0[simp]: "normNum (0,b) = 0\<^sub>N" "normNum (a,0) = 0\<^sub>N"

   by (simp_all add: normNum_def)

 lemma normNum_Nadd:

   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"

   shows "normNum (x +\<^sub>N y) = x +\<^sub>N y" by simp

 lemma Nadd_normNum1[simp]:

   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"

   shows "normNum x +\<^sub>N y = x +\<^sub>N y"

 proof-

   have n: "isnormNum (normNum x +\<^sub>N y)" "isnormNum (x +\<^sub>N y)" by simp_all

   have "INum (normNum x +\<^sub>N y) = INum x + (INum y :: 'a)" by simp

   also have "\<dots> = INum (x +\<^sub>N y)" by simp

   finally show ?thesis using isnormNum_unique[OF n] by simp

 qed

 lemma Nadd_normNum2[simp]:

   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"

   shows "x +\<^sub>N normNum y = x +\<^sub>N y"

 proof-

   have n: "isnormNum (x +\<^sub>N normNum y)" "isnormNum (x +\<^sub>N y)" by simp_all

   have "INum (x +\<^sub>N normNum y) = INum x + (INum y :: 'a)" by simp

   also have "\<dots> = INum (x +\<^sub>N y)" by simp

   finally show ?thesis using isnormNum_unique[OF n] by simp

 qed

 lemma Nadd_assoc:

   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"

   shows "x +\<^sub>N y +\<^sub>N z = x +\<^sub>N (y +\<^sub>N z)"

 proof-

   have n: "isnormNum (x +\<^sub>N y +\<^sub>N z)" "isnormNum (x +\<^sub>N (y +\<^sub>N z))" by simp_all

   have "INum (x +\<^sub>N y +\<^sub>N z) = (INum (x +\<^sub>N (y +\<^sub>N z)) :: 'a)" by simp

   with isnormNum_unique[OF n] show ?thesis by simp

 qed

 lemma Nmul_commute: "isnormNum x \<Longrightarrow> isnormNum y \<Longrightarrow> x *\<^sub>N y = y *\<^sub>N x"

   by (simp add: Nmul_def split_def Let_def gcd_commute_int mult_commute)

 lemma Nmul_assoc:

   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"

   assumes nx: "isnormNum x" and ny:"isnormNum y" and nz:"isnormNum z"

   shows "x *\<^sub>N y *\<^sub>N z = x *\<^sub>N (y *\<^sub>N z)"

 proof-

   from nx ny nz have n: "isnormNum (x *\<^sub>N y *\<^sub>N z)" "isnormNum (x *\<^sub>N (y *\<^sub>N z))" 

     by simp_all

   have "INum (x +\<^sub>N y +\<^sub>N z) = (INum (x +\<^sub>N (y +\<^sub>N z)) :: 'a)" by simp

   with isnormNum_unique[OF n] show ?thesis by simp

 qed

 lemma Nsub0:

   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"

   assumes x: "isnormNum x" and y:"isnormNum y" shows "(x -\<^sub>N y = 0\<^sub>N) = (x = y)"

 proof-

   { fix h :: 'a

     from isnormNum_unique[where 'a = 'a, OF Nsub_normN[OF y], where y="0\<^sub>N"] 

     have "(x -\<^sub>N y = 0\<^sub>N) = (INum (x -\<^sub>N y) = (INum 0\<^sub>N :: 'a)) " by simp

     also have "\<dots> = (INum x = (INum y :: 'a))" by simp

     also have "\<dots> = (x = y)" using x y by simp

     finally show ?thesis . }

 qed

 lemma Nmul0[simp]: "c *\<^sub>N 0\<^sub>N = 0\<^sub>N" " 0\<^sub>N *\<^sub>N c = 0\<^sub>N"

   by (simp_all add: Nmul_def Let_def split_def)

 lemma Nmul_eq0[simp]:

   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"

   assumes nx:"isnormNum x" and ny: "isnormNum y"

   shows "(x*\<^sub>N y = 0\<^sub>N) = (x = 0\<^sub>N \<or> y = 0\<^sub>N)"

 proof-

   { fix h :: 'a

     have " \<exists> a b a' b'. x = (a,b) \<and> y= (a',b')" by auto

     then obtain a b a' b' where xy[simp]: "x = (a,b)" "y = (a',b')" by blast

     have n0: "isnormNum 0\<^sub>N" by simp

     show ?thesis using nx ny 

       apply (simp only: isnormNum_unique[where ?'a = 'a, OF  Nmul_normN[OF nx ny] n0, symmetric] Nmul[where ?'a = 'a])

       by (simp add: INum_def split_def isnormNum_def fst_conv snd_conv split: split_if_asm)

   }

 qed

 lemma Nneg_Nneg[simp]: "~\<^sub>N (~\<^sub>N c) = c"

   by (simp add: Nneg_def split_def)

 lemma Nmul1[simp]: 

   "isnormNum c \<Longrightarrow> 1\<^sub>N *\<^sub>N c = c" 

   "isnormNum c \<Longrightarrow> c *\<^sub>N 1\<^sub>N  = c" 

   apply (simp_all add: Nmul_def Let_def split_def isnormNum_def)

   apply (cases "fst c = 0", simp_all, cases c, simp_all)+

   done

 end