# HG changeset patch
# User haftmann
# Date 1383216260 -3600
# Node ID 0e6645622f22ad199ce50b72fa3aff0637d88818
# Parent  63fe59f64578b098dbec416fe82f9931ee4d7697
more convenient place for a theory in solitariness

diff -r 63fe59f64578 -r 0e6645622f22 src/HOL/Decision_Procs/Rat_Pair.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Decision_Procs/Rat_Pair.thy	Thu Oct 31 11:44:20 2013 +0100
@@ -0,0 +1,521 @@
+(*  Title:      HOL/Decision_Procs/Rat_Pair.thy
+    Author:     Amine Chaieb
+*)
+
+header {* Rational numbers as pairs *}
+
+theory Rat_Pair
+imports Complex_Main
+begin
+
+type_synonym 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] gcd_dvd2_int[presburger]
+
+lemma normNum_isnormNum [simp]: "isnormNum (normNum x)"
+proof -
+  obtain a b where x: "x = (a, b)" by (cases x)
+  { assume "a=0 \<or> b = 0" hence ?thesis by (simp add: x 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 dvd_mult_div_cancel[OF gdvd(1)] dvd_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: x isnormNum_def normNum_def Let_def split_def) }
+    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'] dvd_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: x 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 = (\<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 = (\<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 -
+  obtain a b where x: "x = (a, b)" by (cases x)
+  obtain a' b' where y: "y = (a', b')" by (cases y)
+  { assume "a = 0"
+    hence ?thesis using xn x y
+      by (simp add: isnormNum_def Let_def Nmul_def split_def) }
+  moreover
+  { assume "a' = 0"
+    hence ?thesis using yn x y
+      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 x y by (simp_all add: isnormNum_def)
+    from mult_pos_pos[OF bp] have "x *\<^sub>N y = normNum (a * a', b * b')"
+      using x y 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
+  obtain a b where x: "x = (a, b)" by (cases x)
+  obtain a' b' where y: "y = (a', b')" by (cases y)
+  assume H: ?lhs
+  { assume "a = 0 \<or> b = 0 \<or> a' = 0 \<or> b' = 0"
+    hence ?rhs using na nb H
+      by (simp add: x y 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: x y isnormNum_def)
+    from H bz b'z have eq: "a * b' = a'*b"
+      by (simp add: x y INum_def eq_divide_eq divide_eq_eq of_int_mult[symmetric] del: of_int_mult)
+    from az a'z na nb have gcd1: "gcd a b = 1" "gcd b a = 1" "gcd a' b' = 1" "gcd b' a' = 1"
+      by (simp_all add: x y 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 add: x y) }
+  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"
+  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 `d ~= 0`)
+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 -
+  obtain a b where x: "x = (a, b)" by (cases x)
+  { assume "a = 0 \<or> b = 0"
+    hence ?thesis by (simp add: x 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: x 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"
+  obtain a b where x: "x = (a, b)" by (cases x)
+  obtain a' b' where y: "y = (a', b')" by (cases y)
+  { assume "a=0 \<or> a'= 0 \<or> b =0 \<or> b' = 0"
+    hence ?thesis
+      apply (cases "a=0", simp_all add: x y 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: x y 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: x y Nadd_def INum_def normNum_def 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"
+  obtain a b where x: "x = (a, b)" by (cases x)
+  obtain a' b' where y: "y = (a', b')" by (cases y)
+  { 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]: "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 -
+  obtain a b where x: "x = (a, b)" by (cases x)
+  { assume "a = 0" hence ?thesis by (simp add: x Nlt0_def INum_def) }
+  moreover
+  { assume a: "a \<noteq> 0" hence b: "(of_int b::'a) > 0"
+      using nx by (simp add: x isnormNum_def)
+    from pos_divide_less_eq[OF b, where b="of_int a" and a="0::'a"]
+    have ?thesis by (simp add: x 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 -
+  obtain a b where x: "x = (a, b)" by (cases x)
+  { assume "a = 0" hence ?thesis by (simp add: x Nle0_def INum_def) }
+  moreover
+  { assume a: "a \<noteq> 0" hence b: "(of_int b :: 'a) > 0"
+      using nx by (simp add: x isnormNum_def)
+    from pos_divide_le_eq[OF b, where b="of_int a" and a="0::'a"]
+    have ?thesis by (simp add: x 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 -
+  obtain a b where x: "x = (a, b)" by (cases x)
+  { assume "a = 0" hence ?thesis by (simp add: x Ngt0_def INum_def) }
+  moreover
+  { assume a: "a \<noteq> 0" hence b: "(of_int b::'a) > 0" using nx
+      by (simp add: x isnormNum_def)
+    from pos_less_divide_eq[OF b, where b="of_int a" and a="0::'a"]
+    have ?thesis by (simp add: x 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 -
+  obtain a b where x: "x = (a, b)" by (cases x)
+  { assume "a = 0" hence ?thesis by (simp add: x Nge0_def INum_def) }
+  moreover
+  { assume "a \<noteq> 0" hence b: "(of_int b::'a) > 0" using nx
+      by (simp add: x isnormNum_def)
+    from pos_le_divide_eq[OF b, where b="of_int a" and a="0::'a"]
+    have ?thesis by (simp add: x 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 \<longleftrightarrow> 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 \<longleftrightarrow> x = 0\<^sub>N \<or> y = 0\<^sub>N"
+proof -
+  fix h :: 'a
+  obtain a b where x: "x = (a, b)" by (cases x)
+  obtain a' b' where y: "y = (a', b')" by (cases y)
+  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])
+    apply (simp add: x y INum_def split_def isnormNum_def split: split_if_asm)
+    done
+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
diff -r 63fe59f64578 -r 0e6645622f22 src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy
--- a/src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy	Thu Oct 31 11:44:20 2013 +0100
+++ b/src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy	Thu Oct 31 11:44:20 2013 +0100
@@ -5,7 +5,7 @@
 header {* Implementation and verification of multivariate polynomials *}
 
 theory Reflected_Multivariate_Polynomial
-imports Complex_Main "~~/src/HOL/Library/Abstract_Rat" Polynomial_List
+imports Complex_Main Rat_Pair Polynomial_List
 begin
 
 subsection{* Datatype of polynomial expressions *}
diff -r 63fe59f64578 -r 0e6645622f22 src/HOL/Library/Abstract_Rat.thy
--- a/src/HOL/Library/Abstract_Rat.thy	Thu Oct 31 11:44:20 2013 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,521 +0,0 @@
-(*  Title:      HOL/Library/Abstract_Rat.thy
-    Author:     Amine Chaieb
-*)
-
-header {* Abstract rational numbers *}
-
-theory Abstract_Rat
-imports Complex_Main
-begin
-
-type_synonym 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] gcd_dvd2_int[presburger]
-
-lemma normNum_isnormNum [simp]: "isnormNum (normNum x)"
-proof -
-  obtain a b where x: "x = (a, b)" by (cases x)
-  { assume "a=0 \<or> b = 0" hence ?thesis by (simp add: x 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 dvd_mult_div_cancel[OF gdvd(1)] dvd_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: x isnormNum_def normNum_def Let_def split_def) }
-    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'] dvd_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: x 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 = (\<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 = (\<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 -
-  obtain a b where x: "x = (a, b)" by (cases x)
-  obtain a' b' where y: "y = (a', b')" by (cases y)
-  { assume "a = 0"
-    hence ?thesis using xn x y
-      by (simp add: isnormNum_def Let_def Nmul_def split_def) }
-  moreover
-  { assume "a' = 0"
-    hence ?thesis using yn x y
-      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 x y by (simp_all add: isnormNum_def)
-    from mult_pos_pos[OF bp] have "x *\<^sub>N y = normNum (a * a', b * b')"
-      using x y 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
-  obtain a b where x: "x = (a, b)" by (cases x)
-  obtain a' b' where y: "y = (a', b')" by (cases y)
-  assume H: ?lhs
-  { assume "a = 0 \<or> b = 0 \<or> a' = 0 \<or> b' = 0"
-    hence ?rhs using na nb H
-      by (simp add: x y 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: x y isnormNum_def)
-    from H bz b'z have eq: "a * b' = a'*b"
-      by (simp add: x y INum_def eq_divide_eq divide_eq_eq of_int_mult[symmetric] del: of_int_mult)
-    from az a'z na nb have gcd1: "gcd a b = 1" "gcd b a = 1" "gcd a' b' = 1" "gcd b' a' = 1"
-      by (simp_all add: x y 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 add: x y) }
-  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"
-  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 `d ~= 0`)
-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 -
-  obtain a b where x: "x = (a, b)" by (cases x)
-  { assume "a = 0 \<or> b = 0"
-    hence ?thesis by (simp add: x 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: x 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"
-  obtain a b where x: "x = (a, b)" by (cases x)
-  obtain a' b' where y: "y = (a', b')" by (cases y)
-  { assume "a=0 \<or> a'= 0 \<or> b =0 \<or> b' = 0"
-    hence ?thesis
-      apply (cases "a=0", simp_all add: x y 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: x y 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: x y Nadd_def INum_def normNum_def 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"
-  obtain a b where x: "x = (a, b)" by (cases x)
-  obtain a' b' where y: "y = (a', b')" by (cases y)
-  { 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]: "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 -
-  obtain a b where x: "x = (a, b)" by (cases x)
-  { assume "a = 0" hence ?thesis by (simp add: x Nlt0_def INum_def) }
-  moreover
-  { assume a: "a \<noteq> 0" hence b: "(of_int b::'a) > 0"
-      using nx by (simp add: x isnormNum_def)
-    from pos_divide_less_eq[OF b, where b="of_int a" and a="0::'a"]
-    have ?thesis by (simp add: x 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 -
-  obtain a b where x: "x = (a, b)" by (cases x)
-  { assume "a = 0" hence ?thesis by (simp add: x Nle0_def INum_def) }
-  moreover
-  { assume a: "a \<noteq> 0" hence b: "(of_int b :: 'a) > 0"
-      using nx by (simp add: x isnormNum_def)
-    from pos_divide_le_eq[OF b, where b="of_int a" and a="0::'a"]
-    have ?thesis by (simp add: x 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 -
-  obtain a b where x: "x = (a, b)" by (cases x)
-  { assume "a = 0" hence ?thesis by (simp add: x Ngt0_def INum_def) }
-  moreover
-  { assume a: "a \<noteq> 0" hence b: "(of_int b::'a) > 0" using nx
-      by (simp add: x isnormNum_def)
-    from pos_less_divide_eq[OF b, where b="of_int a" and a="0::'a"]
-    have ?thesis by (simp add: x 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 -
-  obtain a b where x: "x = (a, b)" by (cases x)
-  { assume "a = 0" hence ?thesis by (simp add: x Nge0_def INum_def) }
-  moreover
-  { assume "a \<noteq> 0" hence b: "(of_int b::'a) > 0" using nx
-      by (simp add: x isnormNum_def)
-    from pos_le_divide_eq[OF b, where b="of_int a" and a="0::'a"]
-    have ?thesis by (simp add: x 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 \<longleftrightarrow> 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 \<longleftrightarrow> x = 0\<^sub>N \<or> y = 0\<^sub>N"
-proof -
-  fix h :: 'a
-  obtain a b where x: "x = (a, b)" by (cases x)
-  obtain a' b' where y: "y = (a', b')" by (cases y)
-  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])
-    apply (simp add: x y INum_def split_def isnormNum_def split: split_if_asm)
-    done
-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
diff -r 63fe59f64578 -r 0e6645622f22 src/HOL/Library/Library.thy
--- a/src/HOL/Library/Library.thy	Thu Oct 31 11:44:20 2013 +0100
+++ b/src/HOL/Library/Library.thy	Thu Oct 31 11:44:20 2013 +0100
@@ -1,7 +1,6 @@
 (*<*)
 theory Library
 imports
-  Abstract_Rat
   AList
   BigO
   Binomial