(*  Title:      HOL/Library/Abstract_Rat.thy

    Author:     Amine Chaieb

*)

header {* Abstract rational numbers *}

theory Abstract_Rat

imports GCD 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::{ring_char_0,field, division_by_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::{ring_char_0, field,division_by_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, ring_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, ring_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::{ring_char_0,field, division_by_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, division_by_zero, ring_char_0}) = 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 :: {ring_char_0,division_by_zero,field})"

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 :: {ring_char_0,division_by_zero,field}) "

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 :: {ring_char_0,division_by_zero,field})"

by (simp add: Nsub_def split_def)

lemma Ninv[simp]: "INum (Ninv x) = (1::'a :: {division_by_zero,field}) / (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 :: {ring_char_0, division_by_zero,field})" by (simp add: Ndiv_def)

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

  shows "((INum x :: 'a :: {ring_char_0,division_by_zero,ordered_field})< 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 :: {ring_char_0,division_by_zero,ordered_field}) \<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 :: {ring_char_0,division_by_zero,ordered_field})> 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 :: {ring_char_0,division_by_zero,ordered_field}) \<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 :: {ring_char_0,division_by_zero,ordered_field}) < 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 :: {ring_char_0,division_by_zero,ordered_field})\<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::{ring_char_0,division_by_zero,field})"

  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::{ring_char_0,division_by_zero,field})"

  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::{ring_char_0,division_by_zero,field})"

  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::{ring_char_0,division_by_zero,field})"

  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::{ring_char_0,division_by_zero,field})"

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

lemma Nadd_normNum1[simp]:

  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"

  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::{ring_char_0,division_by_zero,field})"

  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::{ring_char_0,division_by_zero,field})"

  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::{ring_char_0,division_by_zero,field})"

  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::{ring_char_0,division_by_zero,field})"

  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::{ring_char_0,division_by_zero,field})"

  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