# HG changeset patch
# User wenzelm
# Date 1315407229 -7200
# Node ID a13cdb1e9e08811dfc4a7861f7a1e27a76e2ebff
# Parent  98d597c4193d4d6656a2fd9926eb9db9630ad35f
tuned/simplified proofs;

diff -r 98d597c4193d -r a13cdb1e9e08 src/HOL/Library/Abstract_Rat.thy
--- a/src/HOL/Library/Abstract_Rat.thy	Wed Sep 07 16:37:50 2011 +0200
+++ b/src/HOL/Library/Abstract_Rat.thy	Wed Sep 07 16:53:49 2011 +0200
@@ -10,10 +10,10 @@
 
 type_synonym Num = "int \<times> int"
 
-abbreviation Num0_syn :: Num ("0\<^sub>N")
+abbreviation Num0_syn :: Num  ("0\<^sub>N")
   where "0\<^sub>N \<equiv> (0, 0)"
 
-abbreviation Numi_syn :: "int \<Rightarrow> Num" ("_\<^sub>N")
+abbreviation Numi_syn :: "int \<Rightarrow> Num"  ("_\<^sub>N")
   where "i\<^sub>N \<equiv> (i, 1)"
 
 definition isnormNum :: "Num \<Rightarrow> bool" where
@@ -22,16 +22,15 @@
 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 
+      (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 -
-  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) }
+  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"
@@ -42,7 +41,7 @@
     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)+
+    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
@@ -50,18 +49,18 @@
     { 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)}
+      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" 
+      { 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) }
+      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
@@ -69,25 +68,25 @@
 
 text {* Arithmetic over Num *}
 
-definition Nadd :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "+\<^sub>N" 60) where
+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 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') 
+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)
+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)
+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)"
@@ -100,24 +99,24 @@
   by (simp add: Nsub_def split_def)
 
 lemma Nmul_normN[simp]:
-  assumes xn:"isnormNum x" and yn: "isnormNum y"
+  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)}
+  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 ab ab' 
-      by (simp add: isnormNum_def Let_def Nmul_def split_def)}
+  { 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 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}
+  { 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
 
@@ -125,26 +124,26 @@
   by (simp add: Ninv_def isnormNum_def split_def)
     (cases "fst x = 0", auto simp add: gcd_commute_int)
 
-lemma isnormNum_int[simp]: 
+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")
+definition Nlt0:: "Num \<Rightarrow> bool"  ("0>\<^sub>N")
   where "Nlt0 = (\<lambda>(a,b). a < 0)"
 
-definition Nle0:: "Num \<Rightarrow> bool" ("0\<ge>\<^sub>N")
+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")
+definition Ngt0:: "Num \<Rightarrow> bool"  ("0<\<^sub>N")
   where "Ngt0 = (\<lambda>(a,b). a > 0)"
 
-definition Nge0:: "Num \<Rightarrow> bool" ("0\<le>\<^sub>N")
+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)
+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)
@@ -155,35 +154,35 @@
 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" 
+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 
+  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: INum_def split_def isnormNum_def split: split_if_asm) }
+      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: isnormNum_def)
-    from H bz b'z 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 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: 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 - 
+    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)]]
+        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}
+      with eq1 have ?rhs by (simp add: x y) }
   ultimately show ?rhs by blast
 next
   assume ?rhs thus ?lhs by simp
@@ -195,7 +194,7 @@
   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) = 
+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"
@@ -205,7 +204,7 @@
     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" 
+  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
@@ -220,12 +219,11 @@
 
 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: "x = (a,b)" by blast
-  { 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"
+  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]
@@ -246,26 +244,26 @@
 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: "x = (a,b)" 
-    and y[simp]: "y = (a',b')" by blast
+  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 Nadd_def)
+    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" 
+  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 "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 th Nadd_def normNum_def INum_def split_def) }
+        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')"
@@ -273,29 +271,29 @@
       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 Nadd_def INum_def normNum_def Let_def add_divide_distrib)}
+        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 Nadd_def INum_def normNum_def Let_def) }
+      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"
-  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
+  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 
+    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) 
+      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'"]] 
+      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
@@ -313,16 +311,16 @@
   by (simp add: Ndiv_def)
 
 lemma Nlt0_iff[simp]:
-  assumes nx: "isnormNum x" 
+  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) }
+  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: isnormNum_def)
+  { 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: Nlt0_def INum_def) }
+    have ?thesis by (simp add: x Nlt0_def INum_def) }
   ultimately show ?thesis by blast
 qed
 
@@ -330,13 +328,13 @@
   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) }
+  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: isnormNum_def)
+  { 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: Nle0_def INum_def)}
+    have ?thesis by (simp add: x Nle0_def INum_def) }
   ultimately show ?thesis by blast
 qed
 
@@ -344,14 +342,13 @@
   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) }
+  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: isnormNum_def)
+  { 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: Ngt0_def INum_def) }
+    have ?thesis by (simp add: x Ngt0_def INum_def) }
   ultimately show ?thesis by blast
 qed
 
@@ -359,14 +356,13 @@
   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) }
+  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: isnormNum_def)
+      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: Nge0_def INum_def) }
+    have ?thesis by (simp add: x Nge0_def INum_def) }
   ultimately show ?thesis by blast
 qed
 
@@ -405,7 +401,7 @@
 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 "(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)
@@ -416,7 +412,7 @@
 
 lemma normNum_nilpotent_aux[simp]:
   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
-  assumes nx: "isnormNum x" 
+  assumes nx: "isnormNum x"
   shows "normNum x = x"
 proof -
   let ?a = "normNum x"
@@ -471,10 +467,10 @@
 
 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"
+  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))" 
+  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
@@ -482,10 +478,11 @@
 
 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)"
+  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"] 
+  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
@@ -497,26 +494,26 @@
 
 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)"
+  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
-  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
+  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 
+  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: INum_def split_def isnormNum_def split: split_if_asm)
+    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" 
+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