# HG changeset patch
# User paulson
# Date 1080121829 -3600
# Node ID 0eca4aabf371a3987d46308987c7af1498845772
# Parent  bdf6b7adc3ec3b81e8b546ef01756627a896848b
streamlined treatment of quotients for the integers

diff -r bdf6b7adc3ec -r 0eca4aabf371 src/HOL/Integ/Bin.thy
--- a/src/HOL/Integ/Bin.thy	Fri Mar 19 11:06:53 2004 +0100
+++ b/src/HOL/Integ/Bin.thy	Wed Mar 24 10:50:29 2004 +0100
@@ -137,7 +137,7 @@
 lemma odd_nonzero: "1 + z + z \<noteq> (0::int)";
 proof (cases z rule: int_cases)
   case (nonneg n)
-  have le: "0 \<le> z+z" by (simp add: prems add_increasing) 
+  have le: "0 \<le> z+z" by (simp add: nonneg add_increasing) 
   thus ?thesis using  le_imp_0_less [OF le]
     by (auto simp add: add_assoc) 
 next
@@ -147,7 +147,8 @@
     assume eq: "1 + z + z = 0"
     have "0 < 1 + (int n + int n)"
       by (simp add: le_imp_0_less add_increasing) 
-    also have "... = - (1 + z + z)" by (simp add: prems int_Suc add_ac) 
+    also have "... = - (1 + z + z)"
+      by (simp add: neg add_assoc [symmetric], simp add: add_commute) 
     also have "... = 0" by (simp add: eq) 
     finally have "0<0" ..
     thus False by blast
@@ -235,9 +236,9 @@
   assume "a \<in> range of_int"
   then obtain z where a: "a = of_int z" ..
   hence "((1::'a) + a + a < 0) = (of_int (1 + z + z) < (of_int 0 :: 'a))"
-    by (simp add: prems)
+    by (simp add: a)
   also have "... = (z < 0)" by (simp only: of_int_less_iff odd_less_0)
-  also have "... = (a < 0)" by (simp add: prems)
+  also have "... = (a < 0)" by (simp add: a)
   finally show ?thesis .
 qed
 
diff -r bdf6b7adc3ec -r 0eca4aabf371 src/HOL/Integ/IntArith.thy
--- a/src/HOL/Integ/IntArith.thy	Fri Mar 19 11:06:53 2004 +0100
+++ b/src/HOL/Integ/IntArith.thy	Wed Mar 24 10:50:29 2004 +0100
@@ -40,17 +40,6 @@
 
 subsection{*Inequality Reasoning for the Arithmetic Simproc*}
 
-lemma zero_less_nat_eq [simp]: "(0 < nat z) = (0 < z)"
-by (cut_tac w = 0 in zless_nat_conj, auto)
-
-lemma zless_imp_add1_zle: "w<z ==> w + (1::int) \<le> z"
-apply (rule eq_Abs_Integ [of z])
-apply (rule eq_Abs_Integ [of w])
-apply (simp add: linorder_not_le [symmetric] zle int_def zadd One_int_def)
-done
-
-
-
 lemma add_numeral_0: "Numeral0 + a = (a::'a::number_ring)"
 by simp 
 
@@ -170,10 +159,10 @@
 lemma add1_zle_eq: "(w + (1::int) \<le> z) = (w<z)"
 by arith
 
-lemma zle_diff1_eq [simp]: "(w \<le> z - (1::int)) = (w<(z::int))"
+lemma zle_diff1_eq [simp]: "(w \<le> z - (1::int)) = (w<z)"
 by arith
 
-lemma zle_add1_eq_le [simp]: "(w < z + 1) = (w\<le>(z::int))"
+lemma zle_add1_eq_le [simp]: "(w < z + (1::int)) = (w\<le>z)"
 by arith
 
 lemma int_one_le_iff_zero_less: "((1::int) \<le> z) = (0 < z)"
@@ -182,32 +171,13 @@
 
 subsection{*The Functions @{term nat} and @{term int}*}
 
-lemma nonneg_eq_int: "[| 0 \<le> z;  !!m. z = int m ==> P |] ==> P"
-by (blast dest: nat_0_le sym)
-
-lemma nat_eq_iff: "(nat w = m) = (if 0 \<le> w then w = int m else m=0)"
-by auto
-
-lemma nat_eq_iff2: "(m = nat w) = (if 0 \<le> w then w = int m else m=0)"
-by auto
-
-lemma nat_less_iff: "0 \<le> w ==> (nat w < m) = (w < int m)"
-apply (rule iffI)
-apply (erule nat_0_le [THEN subst])
-apply (simp_all del: zless_int add: zless_int [symmetric]) 
-done
-
-lemma int_eq_iff: "(int m = z) = (m = nat z & 0 \<le> z)"
-by (auto simp add: nat_eq_iff2)
-
-
 text{*Simplify the terms @{term "int 0"}, @{term "int(Suc 0)"} and
   @{term "w + - z"}*}
 declare Zero_int_def [symmetric, simp]
 declare One_int_def [symmetric, simp]
 
 text{*cooper.ML refers to this theorem*}
-lemmas zdiff_def_symmetric = zdiff_def [symmetric, simp]
+lemmas diff_int_def_symmetric = diff_int_def [symmetric, simp]
 
 lemma nat_0: "nat 0 = 0"
 by (simp add: nat_eq_iff)
@@ -218,22 +188,10 @@
 lemma nat_2: "nat 2 = Suc (Suc 0)"
 by (subst nat_eq_iff, simp)
 
-lemma nat_less_eq_zless: "0 \<le> w ==> (nat w < nat z) = (w<z)"
-apply (case_tac "z < 0")
-apply (auto simp add: nat_less_iff)
-done
-
-lemma nat_le_eq_zle: "0 < w | 0 \<le> z ==> (nat w \<le> nat z) = (w\<le>z)"
-by (auto simp add: linorder_not_less [symmetric] zless_nat_conj)
-
-
 text{*This simplifies expressions of the form @{term "int n = z"} where
       z is an integer literal.*}
 declare int_eq_iff [of _ "number_of v", standard, simp]
 
-lemma int_nat_eq [simp]: "int (nat z) = (if 0 \<le> z then z else 0)"
-  by simp
-
 lemma split_nat [arith_split]:
   "P(nat(i::int)) = ((\<forall>n. i = int n \<longrightarrow> P n) & (i < 0 \<longrightarrow> P 0))"
   (is "?P = (?L & ?R)")
@@ -251,6 +209,29 @@
 qed
 
 
+(*Analogous to zadd_int*)
+lemma zdiff_int: "n \<le> m ==> int m - int n = int (m-n)"
+by (induct m n rule: diff_induct, simp_all)
+
+lemma nat_mult_distrib: "(0::int) \<le> z ==> nat (z*z') = nat z * nat z'"
+apply (case_tac "0 \<le> z'")
+apply (rule inj_int [THEN injD])
+apply (simp add: zmult_int [symmetric] zero_le_mult_iff)
+apply (simp add: mult_le_0_iff)
+done
+
+lemma nat_mult_distrib_neg: "z \<le> (0::int) ==> nat(z*z') = nat(-z) * nat(-z')"
+apply (rule trans)
+apply (rule_tac [2] nat_mult_distrib, auto)
+done
+
+lemma nat_abs_mult_distrib: "nat (abs (w * z)) = nat (abs w) * nat (abs z)"
+apply (case_tac "z=0 | w=0")
+apply (auto simp add: zabs_def nat_mult_distrib [symmetric] 
+                      nat_mult_distrib_neg [symmetric] mult_less_0_iff)
+done
+
+
 subsubsection "Induction principles for int"
 
                      (* `set:int': dummy construction *)
@@ -390,42 +371,6 @@
 done
 
 
-subsection{*More about nat*}
-
-(*Analogous to zadd_int*)
-lemma zdiff_int: "n \<le> m ==> int m - int n = int (m-n)"
-by (induct m n rule: diff_induct, simp_all)
-
-lemma nat_add_distrib:
-     "[| (0::int) \<le> z;  0 \<le> z' |] ==> nat (z+z') = nat z + nat z'"
-apply (rule inj_int [THEN injD])
-apply (simp add: zadd_int [symmetric])
-done
-
-lemma nat_diff_distrib:
-     "[| (0::int) \<le> z';  z' \<le> z |] ==> nat (z-z') = nat z - nat z'"
-apply (rule inj_int [THEN injD])
-apply (simp add: zdiff_int [symmetric] nat_le_eq_zle)
-done
-
-lemma nat_mult_distrib: "(0::int) \<le> z ==> nat (z*z') = nat z * nat z'"
-apply (case_tac "0 \<le> z'")
-apply (rule inj_int [THEN injD])
-apply (simp add: zmult_int [symmetric] zero_le_mult_iff)
-apply (simp add: mult_le_0_iff)
-done
-
-lemma nat_mult_distrib_neg: "z \<le> (0::int) ==> nat(z*z') = nat(-z) * nat(-z')"
-apply (rule trans)
-apply (rule_tac [2] nat_mult_distrib, auto)
-done
-
-lemma nat_abs_mult_distrib: "nat (abs (w * z)) = nat (abs w) * nat (abs z)"
-apply (case_tac "z=0 | w=0")
-apply (auto simp add: zabs_def nat_mult_distrib [symmetric] 
-                      nat_mult_distrib_neg [symmetric] mult_less_0_iff)
-done
-
 
 ML
 {*
diff -r bdf6b7adc3ec -r 0eca4aabf371 src/HOL/Integ/IntDef.thy
--- a/src/HOL/Integ/IntDef.thy	Fri Mar 19 11:06:53 2004 +0100
+++ b/src/HOL/Integ/IntDef.thy	Wed Mar 24 10:50:29 2004 +0100
@@ -8,13 +8,15 @@
 header{*The Integers as Equivalence Classes over Pairs of Natural Numbers*}
 
 theory IntDef = Equiv + NatArith:
+
 constdefs
   intrel :: "((nat * nat) * (nat * nat)) set"
-    "intrel == {p. \<exists>x1 y1 x2 y2. p = ((x1,y1),(x2,y2)) & x1+y2 = x2+y1}"
+    --{*the equivalence relation underlying the integers*}
+    "intrel == {p. \<exists>x y u v. p = ((x,y),(u,v)) & x+v = u+y}"
 
 typedef (Integ)
   int = "UNIV//intrel"
-    by (auto simp add: quotient_def) 
+    by (auto simp add: quotient_def)
 
 instance int :: ord ..
 instance int :: zero ..
@@ -27,49 +29,57 @@
 
   int :: "nat => int"
   "int m == Abs_Integ(intrel `` {(m,0)})"
-  
+
+
 defs (overloaded)
-  
-  zminus_def:    "- Z == Abs_Integ(\<Union>(x,y) \<in> Rep_Integ(Z). intrel``{(y,x)})"
 
   Zero_int_def:  "0 == int 0"
   One_int_def:   "1 == int 1"
 
-  zadd_def:
-   "z + w == 
-       Abs_Integ(\<Union>(x1,y1) \<in> Rep_Integ(z). \<Union>(x2,y2) \<in> Rep_Integ(w).   
-		 intrel``{(x1+x2, y1+y2)})"
+  minus_int_def:
+    "- z ==  contents (\<Union>(x,y) \<in> Rep_Integ(z). { Abs_Integ(intrel``{(y,x)}) })"
+
+  add_int_def:
+   "z + w ==
+       contents (\<Union>(x,y) \<in> Rep_Integ(z). \<Union>(u,v) \<in> Rep_Integ(w).
+		 { Abs_Integ(intrel``{(x+u, y+v)}) })"
+
+  diff_int_def:  "z - (w::int) == z + (-w)"
 
-  zdiff_def:  "z - (w::int) == z + (-w)"
-  zmult_def:
-   "z * w == 
-       Abs_Integ(\<Union>(x1,y1) \<in> Rep_Integ(z). \<Union>(x2,y2) \<in> Rep_Integ(w).   
-		 intrel``{(x1*x2 + y1*y2, x1*y2 + y1*x2)})"
+  mult_int_def:
+   "z * w ==
+       contents (\<Union>(x,y) \<in> Rep_Integ(z). \<Union>(u,v) \<in> Rep_Integ(w).
+		 { Abs_Integ(intrel``{(x*u + y*v, x*v + y*u)}) })"
+
+  le_int_def:
+   "z \<le> (w::int) == 
+    \<exists>x y u v. x+v \<le> u+y & (x,y) \<in> Rep_Integ z & (u,v) \<in> Rep_Integ w"
 
-  zless_def: "(z < (w::int)) == (z \<le> w & z \<noteq> w)"
+  less_int_def: "(z < (w::int)) == (z \<le> w & z \<noteq> w)"
+
+
+subsection{*Construction of the Integers*}
 
-  zle_def:
-  "z \<le> (w::int) == \<exists>x1 y1 x2 y2. x1 + y2 \<le> x2 + y1 &
-                            (x1,y1) \<in> Rep_Integ z & (x2,y2) \<in> Rep_Integ w"
+subsubsection{*Preliminary Lemmas about the Equivalence Relation*}
 
-lemma intrel_iff [simp]: "(((x1,y1),(x2,y2)) \<in>  intrel) = (x1+y2 = x2+y1)"
-by (unfold intrel_def, blast)
+lemma intrel_iff [simp]: "(((x,y),(u,v)) \<in>  intrel) = (x+v = u+y)"
+by (simp add: intrel_def)
 
 lemma equiv_intrel: "equiv UNIV intrel"
-by (unfold intrel_def equiv_def refl_def sym_def trans_def, auto)
+by (simp add: intrel_def equiv_def refl_def sym_def trans_def)
 
 lemmas equiv_intrel_iff =
        eq_equiv_class_iff [OF equiv_intrel UNIV_I UNIV_I, simp]
 
-lemma intrel_in_integ [simp]: "intrel``{(x,y)}:Integ"
-by (unfold Integ_def intrel_def quotient_def, fast)
+lemma intrel_in_integ [simp]: "intrel``{(x,y)} \<in> Integ"
+by (simp add: Integ_def intrel_def quotient_def, fast)
 
 lemma inj_on_Abs_Integ: "inj_on Abs_Integ Integ"
 apply (rule inj_on_inverseI)
 apply (erule Abs_Integ_inverse)
 done
 
-declare inj_on_Abs_Integ [THEN inj_on_iff, simp] 
+declare inj_on_Abs_Integ [THEN inj_on_iff, simp]
         Abs_Integ_inverse [simp]
 
 lemma inj_Rep_Integ: "inj(Rep_Integ)"
@@ -77,85 +87,64 @@
 apply (rule Rep_Integ_inverse)
 done
 
+text{*Case analysis on the representation of an integer as an equivalence
+      class.*}
+lemma eq_Abs_Integ [case_names Abs_Integ, cases type: int]:
+     "(!!x y. z = Abs_Integ(intrel``{(x,y)}) ==> P) ==> P"
+apply (rule Rep_Integ [of z, unfolded Integ_def, THEN quotientE])
+apply (drule arg_cong [where f=Abs_Integ])
+apply (auto simp add: Rep_Integ_inverse)
+done
 
-(** int: the injection from "nat" to "int" **)
+
+subsubsection{*@{term int}: Embedding the Naturals into the Integers*}
 
 lemma inj_int: "inj int"
-apply (rule inj_onI)
-apply (unfold int_def)
-apply (drule inj_on_Abs_Integ [THEN inj_onD])
-apply (rule intrel_in_integ)+
-apply (drule eq_equiv_class)
-apply (rule equiv_intrel, fast)
-apply (simp add: intrel_def)
-done
+by (simp add: inj_on_def int_def)
 
 lemma int_int_eq [iff]: "(int m = int n) = (m = n)"
 by (fast elim!: inj_int [THEN injD])
 
 
-
-subsection{*zminus: unary negation on Integ*}
-
-lemma zminus_congruent: "congruent intrel (%(x,y). intrel``{(y,x)})"
-apply (unfold congruent_def intrel_def)
-apply (auto simp add: add_ac)
-done
-
-lemma zminus: "- Abs_Integ(intrel``{(x,y)}) = Abs_Integ(intrel `` {(y,x)})"
-by (simp add: zminus_def equiv_intrel [THEN UN_equiv_class] zminus_congruent)
+subsubsection{*Integer Unary Negation*}
 
-(*Every integer can be written in the form Abs_Integ(...) *)
-lemma eq_Abs_Integ: 
-     "(!!x y. z = Abs_Integ(intrel``{(x,y)}) ==> P) ==> P"
-apply (rule_tac x1=z in Rep_Integ [unfolded Integ_def, THEN quotientE]) 
-apply (drule_tac f = Abs_Integ in arg_cong)
-apply (rule_tac p = x in PairE)
-apply (simp add: Rep_Integ_inverse)
-done
+lemma minus: "- Abs_Integ(intrel``{(x,y)}) = Abs_Integ(intrel `` {(y,x)})"
+proof -
+  have "congruent intrel (\<lambda>(x,y). {Abs_Integ (intrel``{(y,x)})})"
+    by (simp add: congruent_def) 
+  thus ?thesis
+    by (simp add: minus_int_def UN_equiv_class [OF equiv_intrel])
+qed
 
-lemma zminus_zminus [simp]: "- (- z) = (z::int)"
-apply (rule eq_Abs_Integ [of z])
-apply (simp add: zminus)
-done
+lemma zminus_zminus: "- (- z) = (z::int)"
+by (cases z, simp add: minus)
 
-lemma inj_zminus: "inj(%z::int. -z)"
-apply (rule inj_onI)
-apply (drule_tac f = uminus in arg_cong, simp)
-done
-
-lemma zminus_0 [simp]: "- 0 = (0::int)"
-by (simp add: int_def Zero_int_def zminus)
+lemma zminus_0: "- 0 = (0::int)"
+by (simp add: int_def Zero_int_def minus)
 
 
-subsection{*zadd: addition on Integ*}
+subsection{*Integer Addition*}
 
-lemma zadd: 
-  "Abs_Integ(intrel``{(x1,y1)}) + Abs_Integ(intrel``{(x2,y2)}) =  
-   Abs_Integ(intrel``{(x1+x2, y1+y2)})"
-apply (simp add: zadd_def UN_UN_split_split_eq)
-apply (subst equiv_intrel [THEN UN_equiv_class2])
-apply (auto simp add: congruent2_def)
-done
+lemma add:
+     "Abs_Integ (intrel``{(x,y)}) + Abs_Integ (intrel``{(u,v)}) =
+      Abs_Integ (intrel``{(x+u, y+v)})"
+proof -
+  have "congruent2 intrel
+        (\<lambda>z w. (\<lambda>(x,y). (\<lambda>(u,v). {Abs_Integ (intrel `` {(x+u, y+v)})}) w) z)"
+    by (simp add: congruent2_def)
+  thus ?thesis
+    by (simp add: add_int_def UN_UN_split_split_eq
+                  UN_equiv_class2 [OF equiv_intrel])
+qed
 
-lemma zminus_zadd_distrib [simp]: "- (z + w) = (- z) + (- w::int)"
-apply (rule eq_Abs_Integ [of z])
-apply (rule eq_Abs_Integ [of w])
-apply (simp add: zminus zadd)
-done
+lemma zminus_zadd_distrib: "- (z + w) = (- z) + (- w::int)"
+by (cases z, cases w, simp add: minus add)
 
 lemma zadd_commute: "(z::int) + w = w + z"
-apply (rule eq_Abs_Integ [of z])
-apply (rule eq_Abs_Integ [of w])
-apply (simp add: add_ac zadd)
-done
+by (cases z, cases w, simp add: add_ac add)
 
 lemma zadd_assoc: "((z1::int) + z2) + z3 = z1 + (z2 + z3)"
-apply (rule eq_Abs_Integ [of z1])
-apply (rule eq_Abs_Integ [of z2])
-apply (rule eq_Abs_Integ [of z3])
-apply (simp add: zadd add_assoc)
-done
+by (cases z1, cases z2, cases z3, simp add: add add_assoc)
 
 (*For AC rewriting*)
 lemma zadd_left_commute: "x + (y + z) = y + ((x + z)  ::int)"
@@ -164,13 +153,12 @@
   apply (rule zadd_commute)
   done
 
-(*Integer addition is an AC operator*)
 lemmas zadd_ac = zadd_assoc zadd_commute zadd_left_commute
 
 lemmas zmult_ac = Ring_and_Field.mult_ac
 
 lemma zadd_int: "(int m) + (int n) = int (m + n)"
-by (simp add: int_def zadd)
+by (simp add: int_def add)
 
 lemma zadd_int_left: "(int m) + (int n + z) = int (m + n) + z"
 by (simp add: zadd_int zadd_assoc [symmetric])
@@ -179,104 +167,56 @@
 by (simp add: One_int_def zadd_int)
 
 (*also for the instance declaration int :: plus_ac0*)
-lemma zadd_0 [simp]: "(0::int) + z = z"
-apply (unfold Zero_int_def int_def)
-apply (rule eq_Abs_Integ [of z])
-apply (simp add: zadd)
-done
-
-lemma zadd_0_right [simp]: "z + (0::int) = z"
-by (rule trans [OF zadd_commute zadd_0])
-
-lemma zadd_zminus_inverse [simp]: "z + (- z) = (0::int)"
-apply (unfold int_def Zero_int_def)
-apply (rule eq_Abs_Integ [of z])
-apply (simp add: zminus zadd add_commute)
-done
-
-lemma zadd_zminus_inverse2 [simp]: "(- z) + z = (0::int)"
-apply (rule zadd_commute [THEN trans])
-apply (rule zadd_zminus_inverse)
+lemma zadd_0: "(0::int) + z = z"
+apply (simp add: Zero_int_def int_def)
+apply (cases z, simp add: add)
 done
 
-lemma zadd_zminus_cancel [simp]: "z + (- z + w) = (w::int)"
-by (simp add: zadd_assoc [symmetric] zadd_0)
-
-lemma zminus_zadd_cancel [simp]: "(-z) + (z + w) = (w::int)"
-by (simp add: zadd_assoc [symmetric] zadd_0)
+lemma zadd_0_right: "z + (0::int) = z"
+by (rule trans [OF zadd_commute zadd_0])
 
-lemma zdiff0 [simp]: "(0::int) - x = -x"
-by (simp add: zdiff_def)
-
-lemma zdiff0_right [simp]: "x - (0::int) = x"
-by (simp add: zdiff_def)
-
-lemma zdiff_self [simp]: "x - x = (0::int)"
-by (simp add: zdiff_def Zero_int_def)
+lemma zadd_zminus_inverse2: "(- z) + z = (0::int)"
+by (cases z, simp add: int_def Zero_int_def minus add)
 
 
-(** Lemmas **)
-
-lemma zadd_assoc_cong: "(z::int) + v = z' + v' ==> z + (v + w) = z' + (v' + w)"
-by (simp add: zadd_assoc [symmetric])
-
-
-subsection{*zmult: multiplication on Integ*}
+subsection{*Integer Multiplication*}
 
 text{*Congruence property for multiplication*}
-lemma zmult_congruent2: "congruent2 intrel  
-        (%p1 p2. (%(x1,y1). (%(x2,y2).    
-                    intrel``{(x1*x2 + y1*y2, x1*y2 + y1*x2)}) p2) p1)"
+lemma mult_congruent2:
+     "congruent2 intrel
+        (%p1 p2. (%(x,y). (%(u,v).
+                    { Abs_Integ(intrel``{(x*u + y*v, x*v + y*u)}) }) p2) p1)"
 apply (rule equiv_intrel [THEN congruent2_commuteI])
- apply (force simp add: add_ac mult_ac) 
-apply (clarify, simp del: equiv_intrel_iff add: add_ac mult_ac)
-apply (rename_tac x1 x2 y1 y2 z1 z2)
-apply (rule equiv_class_eq [OF equiv_intrel intrel_iff [THEN iffD2]])
-apply (subgoal_tac "x1*z1 + y2*z1 = y1*z1 + x2*z1 & x1*z2 + y2*z2 = y1*z2 + x2*z2")
-apply (simp add: mult_ac, arith) 
+apply (auto simp add: congruent_def mult_ac)  
+apply (rename_tac u v w x y z)
+apply (subgoal_tac "u*y + x*y = w*y + v*y  &  u*z + x*z = w*z + v*z")
+apply (simp add: mult_ac, arith)
 apply (simp add: add_mult_distrib [symmetric])
 done
 
-lemma zmult: 
-   "Abs_Integ((intrel``{(x1,y1)})) * Abs_Integ((intrel``{(x2,y2)})) =    
-    Abs_Integ(intrel `` {(x1*x2 + y1*y2, x1*y2 + y1*x2)})"
-by (simp add: zmult_def UN_UN_split_split_eq zmult_congruent2 
+lemma mult:
+     "Abs_Integ((intrel``{(x,y)})) * Abs_Integ((intrel``{(u,v)})) =
+      Abs_Integ(intrel `` {(x*u + y*v, x*v + y*u)})"
+by (simp add: mult_int_def UN_UN_split_split_eq mult_congruent2
               equiv_intrel [THEN UN_equiv_class2])
 
 lemma zmult_zminus: "(- z) * w = - (z * (w::int))"
-apply (rule eq_Abs_Integ [of z])
-apply (rule eq_Abs_Integ [of w])
-apply (simp add: zminus zmult add_ac)
-done
+by (cases z, cases w, simp add: minus mult add_ac)
 
 lemma zmult_commute: "(z::int) * w = w * z"
-apply (rule eq_Abs_Integ [of z])
-apply (rule eq_Abs_Integ [of w])
-apply (simp add: zmult add_ac mult_ac)
-done
+by (cases z, cases w, simp add: mult add_ac mult_ac)
 
 lemma zmult_assoc: "((z1::int) * z2) * z3 = z1 * (z2 * z3)"
-apply (rule eq_Abs_Integ [of z1])
-apply (rule eq_Abs_Integ [of z2])
-apply (rule eq_Abs_Integ [of z3])
-apply (simp add: add_mult_distrib2 zmult add_ac mult_ac)
-done
+by (cases z1, cases z2, cases z3, simp add: mult add_mult_distrib2 mult_ac)
 
 lemma zadd_zmult_distrib: "((z1::int) + z2) * w = (z1 * w) + (z2 * w)"
-apply (rule eq_Abs_Integ [of z1])
-apply (rule eq_Abs_Integ [of z2])
-apply (rule eq_Abs_Integ [of w])
-apply (simp add: add_mult_distrib2 zadd zmult add_ac mult_ac)
-done
-
-lemma zmult_zminus_right: "w * (- z) = - (w * (z::int))"
-by (simp add: zmult_commute [of w] zmult_zminus)
+by (cases z1, cases z2, cases w, simp add: add mult add_mult_distrib2 mult_ac)
 
 lemma zadd_zmult_distrib2: "(w::int) * (z1 + z2) = (w * z1) + (w * z2)"
 by (simp add: zmult_commute [of w] zadd_zmult_distrib)
 
 lemma zdiff_zmult_distrib: "((z1::int) - z2) * w = (z1 * w) - (z2 * w)"
-apply (unfold zdiff_def)
+apply (simp add: diff_int_def)
 apply (subst zadd_zmult_distrib)
 apply (simp add: zmult_zminus)
 done
@@ -285,28 +225,16 @@
 by (simp add: zmult_commute [of w] zdiff_zmult_distrib)
 
 lemmas int_distrib =
-  zadd_zmult_distrib zadd_zmult_distrib2 
+  zadd_zmult_distrib zadd_zmult_distrib2
   zdiff_zmult_distrib zdiff_zmult_distrib2
 
 lemma zmult_int: "(int m) * (int n) = int (m * n)"
-by (simp add: int_def zmult)
-
-lemma zmult_0 [simp]: "0 * z = (0::int)"
-apply (unfold Zero_int_def int_def)
-apply (rule eq_Abs_Integ [of z])
-apply (simp add: zmult)
-done
+by (simp add: int_def mult)
 
-lemma zmult_1 [simp]: "(1::int) * z = z"
-apply (unfold One_int_def int_def)
-apply (rule eq_Abs_Integ [of z])
-apply (simp add: zmult)
-done
+lemma zmult_1: "(1::int) * z = z"
+by (cases z, simp add: One_int_def int_def mult)
 
-lemma zmult_0_right [simp]: "z * 0 = (0::int)"
-by (rule trans [OF zmult_commute zmult_0])
-
-lemma zmult_1_right [simp]: "z * (1::int) = z"
+lemma zmult_1_right: "z * (1::int) = z"
 by (rule trans [OF zmult_commute zmult_1])
 
 
@@ -316,46 +244,36 @@
   fix i j k :: int
   show "(i + j) + k = i + (j + k)" by (simp add: zadd_assoc)
   show "i + j = j + i" by (simp add: zadd_commute)
-  show "0 + i = i" by simp
-  show "- i + i = 0" by simp
-  show "i - j = i + (-j)" by (simp add: zdiff_def)
+  show "0 + i = i" by (rule zadd_0)
+  show "- i + i = 0" by (rule zadd_zminus_inverse2)
+  show "i - j = i + (-j)" by (simp add: diff_int_def)
   show "(i * j) * k = i * (j * k)" by (rule zmult_assoc)
   show "i * j = j * i" by (rule zmult_commute)
-  show "1 * i = i" by simp
+  show "1 * i = i" by (rule zmult_1) 
   show "(i + j) * k = i * k + j * k" by (simp add: int_distrib)
-  show "0 \<noteq> (1::int)" 
+  show "0 \<noteq> (1::int)"
     by (simp only: Zero_int_def One_int_def One_nat_def int_int_eq)
 qed
 
 
 subsection{*The @{text "\<le>"} Ordering*}
 
-lemma zle: 
-  "(Abs_Integ(intrel``{(x1,y1)}) \<le> Abs_Integ(intrel``{(x2,y2)})) =  
-   (x1 + y2 \<le> x2 + y1)"
-by (force simp add: zle_def)
+lemma le:
+  "(Abs_Integ(intrel``{(x,y)}) \<le> Abs_Integ(intrel``{(u,v)})) = (x+v \<le> u+y)"
+by (force simp add: le_int_def)
 
 lemma zle_refl: "w \<le> (w::int)"
-apply (rule eq_Abs_Integ [of w])
-apply (force simp add: zle)
-done
+by (cases w, simp add: le)
 
 lemma zle_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::int)"
-apply (rule eq_Abs_Integ [of i]) 
-apply (rule eq_Abs_Integ [of j]) 
-apply (rule eq_Abs_Integ [of k]) 
-apply (simp add: zle) 
-done
+by (cases i, cases j, cases k, simp add: le)
 
 lemma zle_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::int)"
-apply (rule eq_Abs_Integ [of w]) 
-apply (rule eq_Abs_Integ [of z]) 
-apply (simp add: zle) 
-done
+by (cases w, cases z, simp add: le)
 
 (* Axiom 'order_less_le' of class 'order': *)
 lemma zless_le: "((w::int) < z) = (w \<le> z & w \<noteq> z)"
-by (simp add: zless_def)
+by (simp add: less_int_def)
 
 instance int :: order
 proof qed
@@ -364,10 +282,7 @@
 
 (* Axiom 'linorder_linear' of class 'linorder': *)
 lemma zle_linear: "(z::int) \<le> w | w \<le> z"
-apply (rule eq_Abs_Integ [of z])
-apply (rule eq_Abs_Integ [of w])
-apply (simp add: zle linorder_linear) 
-done
+by (cases z, cases w, simp add: le linorder_linear)
 
 instance int :: linorder
 proof qed (rule zle_linear)
@@ -379,21 +294,8 @@
 lemma int_eq_0_conv [simp]: "(int n = 0) = (n = 0)"
 by (simp add: Zero_int_def)
 
-(*This lemma allows direct proofs of other <-properties*)
-lemma zless_iff_Suc_zadd: 
-    "(w < z) = (\<exists>n. z = w + int(Suc n))"
-apply (rule eq_Abs_Integ [of z])
-apply (rule eq_Abs_Integ [of w])
-apply (simp add: linorder_not_le [where 'a = int, symmetric] 
-                 linorder_not_le [where 'a = nat] 
-                 zle int_def zdiff_def zadd zminus) 
-apply (safe dest!: less_imp_Suc_add)
-apply (rule_tac x = k in exI)
-apply (simp_all add: add_ac)
-done
-
 lemma zless_int [simp]: "(int m < int n) = (m<n)"
-by (simp add: less_iff_Suc_add zless_iff_Suc_zadd zadd_int)
+by (simp add: le add int_def linorder_not_le [symmetric]) 
 
 lemma int_less_0_conv [simp]: "~ (int k < 0)"
 by (simp add: Zero_int_def)
@@ -425,25 +327,20 @@
 lemma int_Suc0_eq_1: "int (Suc 0) = 1"
 by (simp add: One_int_def One_nat_def)
 
+
 subsection{*Monotonicity results*}
 
-lemma zadd_left_mono: "i \<le> j ==> k + i \<le> k + (j::int)" 
-apply (rule eq_Abs_Integ [of i]) 
-apply (rule eq_Abs_Integ [of j]) 
-apply (rule eq_Abs_Integ [of k]) 
-apply (simp add: zle zadd) 
-done
+lemma zadd_left_mono: "i \<le> j ==> k + i \<le> k + (j::int)"
+by (cases i, cases j, cases k, simp add: le add)
 
-lemma zadd_strict_right_mono: "i < j ==> i + k < j + (k::int)" 
-apply (rule eq_Abs_Integ [of i]) 
-apply (rule eq_Abs_Integ [of j]) 
-apply (rule eq_Abs_Integ [of k]) 
-apply (simp add: linorder_not_le [where 'a = int, symmetric] 
-                 linorder_not_le [where 'a = nat]  zle zadd)
+lemma zadd_strict_right_mono: "i < j ==> i + k < j + (k::int)"
+apply (cases i, cases j, cases k)
+apply (simp add: linorder_not_le [where 'a = int, symmetric]
+                 linorder_not_le [where 'a = nat]  le add)
 done
 
 lemma zadd_zless_mono: "[| w'<w; z'\<le>z |] ==> w' + z' < w + (z::int)"
-by (rule order_less_le_trans [OF zadd_strict_right_mono zadd_left_mono]) 
+by (rule order_less_le_trans [OF zadd_strict_right_mono zadd_left_mono])
 
 
 subsection{*Strict Monotonicity of Multiplication*}
@@ -451,7 +348,7 @@
 text{*strict, in 1st argument; proof is by induction on k>0*}
 lemma zmult_zless_mono2_lemma [rule_format]:
      "i<j ==> 0<k --> int k * i < int k * j"
-apply (induct_tac "k", simp) 
+apply (induct_tac "k", simp)
 apply (simp add: int_Suc)
 apply (case_tac "n=0")
 apply (simp_all add: zadd_zmult_distrib int_Suc0_eq_1 order_le_less)
@@ -459,14 +356,14 @@
 done
 
 lemma zero_le_imp_eq_int: "0 \<le> k ==> \<exists>n. k = int n"
-apply (rule eq_Abs_Integ [of k]) 
-apply (auto simp add: zle zadd int_def Zero_int_def)
-apply (rule_tac x="x-y" in exI, simp) 
+apply (cases k)
+apply (auto simp add: le add int_def Zero_int_def)
+apply (rule_tac x="x-y" in exI, simp)
 done
 
 lemma zmult_zless_mono2: "[| i<j;  (0::int) < k |] ==> k*i < k*j"
-apply (frule order_less_imp_le [THEN zero_le_imp_eq_int]) 
-apply (auto simp add: zmult_zless_mono2_lemma) 
+apply (frule order_less_imp_le [THEN zero_le_imp_eq_int])
+apply (auto simp add: zmult_zless_mono2_lemma)
 done
 
 
@@ -484,49 +381,100 @@
 qed
 
 
+lemma zless_imp_add1_zle: "w<z ==> w + (1::int) \<le> z"
+apply (cases w, cases z) 
+apply (simp add: linorder_not_le [symmetric] le int_def add One_int_def)
+done
+
 subsection{*Magnitide of an Integer, as a Natural Number: @{term nat}*}
 
 constdefs
    nat  :: "int => nat"
-    "nat(Z) == if Z<0 then 0 else (THE m. Z = int m)"
+    "nat z == contents (\<Union>(x,y) \<in> Rep_Integ(z). { x-y })"
+
+lemma nat: "nat (Abs_Integ (intrel``{(x,y)})) = x-y"
+proof -
+  have "congruent intrel (\<lambda>(x,y). {x-y})"
+    by (simp add: congruent_def, arith) 
+  thus ?thesis
+    by (simp add: nat_def UN_equiv_class [OF equiv_intrel])
+qed
 
 lemma nat_int [simp]: "nat(int n) = n"
-by (unfold nat_def, auto)
+by (simp add: nat int_def) 
 
 lemma nat_zero [simp]: "nat 0 = 0"
-apply (unfold Zero_int_def)
-apply (rule nat_int)
-done
+by (simp only: Zero_int_def nat_int)
 
-lemma nat_0_le [simp]: "0 \<le> z ==> int (nat z) = z"
-apply (rule eq_Abs_Integ [of z]) 
-apply (simp add: nat_def linorder_not_le [symmetric] zle int_def Zero_int_def)
-apply (subgoal_tac "(THE m. x = m + y) = x-y")
-apply (auto simp add: the_equality) 
+lemma int_nat_eq [simp]: "int (nat z) = (if 0 \<le> z then z else 0)"
+by (cases z, simp add: nat le int_def Zero_int_def)
+
+corollary nat_0_le: "0 \<le> z ==> int (nat z) = z"
+apply simp 
 done
 
 lemma nat_le_0 [simp]: "z \<le> 0 ==> nat z = 0"
-by (simp add: nat_def  order_less_le eq_commute [of 0])
+by (cases z, simp add: nat le int_def Zero_int_def)
+
+lemma nat_le_eq_zle: "0 < w | 0 \<le> z ==> (nat w \<le> nat z) = (w\<le>z)"
+apply (cases w, cases z) 
+apply (simp add: nat le linorder_not_le [symmetric] int_def Zero_int_def, arith)
+done
 
 text{*An alternative condition is @{term "0 \<le> w"} *}
-lemma nat_mono_iff: "0 < z ==> (nat w < nat z) = (w < z)"
-apply (subst zless_int [symmetric])
-apply (simp add: order_le_less)
-apply (case_tac "w < 0")
- apply (simp add: order_less_imp_le)
-apply (simp add: linorder_not_less)
+corollary nat_mono_iff: "0 < z ==> (nat w < nat z) = (w < z)"
+by (simp add: nat_le_eq_zle linorder_not_le [symmetric]) 
+
+corollary nat_less_eq_zless: "0 \<le> w ==> (nat w < nat z) = (w<z)"
+by (simp add: nat_le_eq_zle linorder_not_le [symmetric]) 
+
+lemma zless_nat_conj: "(nat w < nat z) = (0 < z & w < z)"
+apply (cases w, cases z) 
+apply (simp add: nat le int_def Zero_int_def linorder_not_le [symmetric], arith)
 done
 
-lemma zless_nat_conj: "(nat w < nat z) = (0 < z & w < z)"
-apply (case_tac "0 < z")
-apply (auto simp add: nat_mono_iff linorder_not_less)
+lemma nonneg_eq_int: "[| 0 \<le> z;  !!m. z = int m ==> P |] ==> P"
+by (blast dest: nat_0_le sym)
+
+lemma nat_eq_iff: "(nat w = m) = (if 0 \<le> w then w = int m else m=0)"
+by (cases w, simp add: nat le int_def Zero_int_def, arith)
+
+corollary nat_eq_iff2: "(m = nat w) = (if 0 \<le> w then w = int m else m=0)"
+by (simp only: eq_commute [of m] nat_eq_iff) 
+
+lemma nat_less_iff: "0 \<le> w ==> (nat w < m) = (w < int m)"
+apply (cases w)
+apply (simp add: nat le int_def Zero_int_def linorder_not_le [symmetric], arith)
 done
 
+lemma int_eq_iff: "(int m = z) = (m = nat z & 0 \<le> z)"
+by (auto simp add: nat_eq_iff2)
+
+lemma zero_less_nat_eq [simp]: "(0 < nat z) = (0 < z)"
+by (insert zless_nat_conj [of 0], auto)
+
+
+lemma nat_add_distrib:
+     "[| (0::int) \<le> z;  0 \<le> z' |] ==> nat (z+z') = nat z + nat z'"
+by (cases z, cases z', simp add: nat add le int_def Zero_int_def)
+
+lemma nat_diff_distrib:
+     "[| (0::int) \<le> z';  z' \<le> z |] ==> nat (z-z') = nat z - nat z'"
+by (cases z, cases z', 
+    simp add: nat add minus diff_minus le int_def Zero_int_def)
+
+
+lemma nat_zminus_int [simp]: "nat (- (int n)) = 0"
+by (simp add: int_def minus nat Zero_int_def) 
+
+lemma zless_nat_eq_int_zless: "(m < nat z) = (int m < z)"
+by (cases z, simp add: nat le int_def  linorder_not_le [symmetric], arith)
+
 
 subsection{*Lemmas about the Function @{term int} and Orderings*}
 
 lemma negative_zless_0: "- (int (Suc n)) < 0"
-by (simp add: zless_def)
+by (simp add: order_less_le)
 
 lemma negative_zless [iff]: "- (int (Suc n)) < int m"
 by (rule negative_zless_0 [THEN order_less_le_trans], simp)
@@ -541,10 +489,7 @@
 by (subst le_minus_iff, simp)
 
 lemma int_zle_neg: "(int n \<le> - int m) = (n = 0 & m = 0)"
-apply safe 
-apply (drule_tac [2] le_minus_iff [THEN iffD1])
-apply (auto dest: zle_trans [OF _ negative_zle_0]) 
-done
+by (simp add: int_def le minus Zero_int_def) 
 
 lemma not_int_zless_negative [simp]: "~ (int n < - int m)"
 by (simp add: linorder_not_less)
@@ -553,10 +498,15 @@
 by (force simp add: order_eq_iff [of "- int n"] int_zle_neg)
 
 lemma zle_iff_zadd: "(w \<le> z) = (\<exists>n. z = w + int n)"
-by (force intro: exI [of _ "0::nat"] 
-            intro!: not_sym [THEN not0_implies_Suc]
-            simp add: zless_iff_Suc_zadd order_le_less)
+apply (cases w, cases z)
+apply (auto simp add: le add int_def) 
+apply (rename_tac a b c d) 
+apply (rule_tac x="c+b - (a+d)" in exI) 
+apply arith
+done
 
+lemma abs_int_eq [simp]: "abs (int m) = int m"
+by (simp add: zabs_def)
 
 text{*This version is proved for all ordered rings, not just integers!
       It is proved here because attribute @{text arith_split} is not available
@@ -566,38 +516,6 @@
      "P(abs(a::'a::ordered_ring)) = ((0 \<le> a --> P a) & (a < 0 --> P(-a)))"
 by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
 
-lemma abs_int_eq [simp]: "abs (int m) = int m"
-by (simp add: zabs_def)
-
-
-subsection{*Misc Results*}
-
-lemma nat_zminus_int [simp]: "nat(- (int n)) = 0"
-by (auto simp add: nat_def zero_reorient minus_less_iff)
-
-lemma zless_nat_eq_int_zless: "(m < nat z) = (int m < z)"
-apply (case_tac "0 \<le> z")
-apply (erule nat_0_le [THEN subst], simp) 
-apply (simp add: linorder_not_le)
-apply (auto dest: order_less_trans simp add: order_less_imp_le)
-done
-
-text{*A case theorem distinguishing non-negative and negative int*}
-
-lemma negD: "x<0 ==> \<exists>n. x = - (int (Suc n))"
-by (auto simp add: zless_iff_Suc_zadd 
-                   diff_eq_eq [symmetric] zdiff_def)
-
-lemma int_cases [cases type: int, case_names nonneg neg]: 
-     "[|!! n. z = int n ==> P;  !! n. z =  - (int (Suc n)) ==> P |] ==> P"
-apply (case_tac "z < 0", blast dest!: negD)
-apply (simp add: linorder_not_less)
-apply (blast dest: nat_0_le [THEN sym])
-done
-
-lemma int_induct [induct type: int, case_names nonneg neg]: 
-     "[|!! n. P (int n);  !!n. P (- (int (Suc n))) |] ==> P z"
-  by (cases z) auto
 
 
 subsection{*The Constants @{term neg} and @{term iszero}*}
@@ -610,7 +528,7 @@
   (*For simplifying equalities*)
   iszero :: "'a::semiring => bool"
   "iszero z == z = (0)"
-  
+
 
 lemma not_neg_int [simp]: "~ neg(int n)"
 by (simp add: neg_def)
@@ -623,22 +541,23 @@
 lemma not_neg_eq_ge_0: "(~neg x) = (0 \<le> x)"
 by (simp add: neg_def linorder_not_less)
 
+
 subsection{*To simplify inequalities when Numeral1 can get simplified to 1*}
 
 lemma not_neg_0: "~ neg 0"
 by (simp add: One_int_def neg_def)
 
 lemma not_neg_1: "~ neg 1"
-by (simp add: neg_def linorder_not_less zero_le_one) 
+by (simp add: neg_def linorder_not_less zero_le_one)
 
 lemma iszero_0: "iszero 0"
 by (simp add: iszero_def)
 
 lemma not_iszero_1: "~ iszero 1"
-by (simp add: iszero_def eq_commute) 
+by (simp add: iszero_def eq_commute)
 
 lemma neg_nat: "neg z ==> nat z = 0"
-by (simp add: nat_def neg_def) 
+by (simp add: neg_def order_less_imp_le) 
 
 lemma not_neg_nat: "~ neg z ==> int (nat z) = z"
 by (simp add: linorder_not_less neg_def)
@@ -657,36 +576,36 @@
 
 lemma of_nat_add [simp]: "of_nat (m+n) = of_nat m + of_nat n"
 apply (induct m)
-apply (simp_all add: add_ac) 
+apply (simp_all add: add_ac)
 done
 
 lemma of_nat_mult [simp]: "of_nat (m*n) = of_nat m * of_nat n"
-apply (induct m) 
-apply (simp_all add: mult_ac add_ac right_distrib) 
+apply (induct m)
+apply (simp_all add: mult_ac add_ac right_distrib)
 done
 
 lemma zero_le_imp_of_nat: "0 \<le> (of_nat m::'a::ordered_semiring)"
-apply (induct m, simp_all) 
-apply (erule order_trans) 
-apply (rule less_add_one [THEN order_less_imp_le]) 
+apply (induct m, simp_all)
+apply (erule order_trans)
+apply (rule less_add_one [THEN order_less_imp_le])
 done
 
 lemma less_imp_of_nat_less:
      "m < n ==> of_nat m < (of_nat n::'a::ordered_semiring)"
-apply (induct m n rule: diff_induct, simp_all) 
-apply (insert add_le_less_mono [OF zero_le_imp_of_nat zero_less_one], force) 
+apply (induct m n rule: diff_induct, simp_all)
+apply (insert add_le_less_mono [OF zero_le_imp_of_nat zero_less_one], force)
 done
 
 lemma of_nat_less_imp_less:
      "of_nat m < (of_nat n::'a::ordered_semiring) ==> m < n"
-apply (induct m n rule: diff_induct, simp_all) 
-apply (insert zero_le_imp_of_nat) 
-apply (force simp add: linorder_not_less [symmetric]) 
+apply (induct m n rule: diff_induct, simp_all)
+apply (insert zero_le_imp_of_nat)
+apply (force simp add: linorder_not_less [symmetric])
 done
 
 lemma of_nat_less_iff [simp]:
      "(of_nat m < (of_nat n::'a::ordered_semiring)) = (m<n)"
-by (blast intro: of_nat_less_imp_less less_imp_of_nat_less ) 
+by (blast intro: of_nat_less_imp_less less_imp_of_nat_less)
 
 text{*Special cases where either operand is zero*}
 declare of_nat_less_iff [of 0, simplified, simp]
@@ -694,7 +613,7 @@
 
 lemma of_nat_le_iff [simp]:
      "(of_nat m \<le> (of_nat n::'a::ordered_semiring)) = (m \<le> n)"
-by (simp add: linorder_not_less [symmetric]) 
+by (simp add: linorder_not_less [symmetric])
 
 text{*Special cases where either operand is zero*}
 declare of_nat_le_iff [of 0, simplified, simp]
@@ -704,7 +623,7 @@
 a finite field, which indeed wraps back to zero.*}
 lemma of_nat_eq_iff [simp]:
      "(of_nat m = (of_nat n::'a::ordered_semiring)) = (m = n)"
-by (simp add: order_eq_iff) 
+by (simp add: order_eq_iff)
 
 text{*Special cases where either operand is zero*}
 declare of_nat_eq_iff [of 0, simplified, simp]
@@ -713,7 +632,7 @@
 lemma of_nat_diff [simp]:
      "n \<le> m ==> of_nat (m - n) = of_nat m - (of_nat n :: 'a::ring)"
 by (simp del: of_nat_add
-	 add: compare_rls of_nat_add [symmetric] split add: nat_diff_split) 
+	 add: compare_rls of_nat_add [symmetric] split add: nat_diff_split)
 
 
 subsection{*The Set of Natural Numbers*}
@@ -725,29 +644,29 @@
 syntax (xsymbols)    Nats :: "'a set"   ("\<nat>")
 
 lemma of_nat_in_Nats [simp]: "of_nat n \<in> Nats"
-by (simp add: Nats_def) 
+by (simp add: Nats_def)
 
 lemma Nats_0 [simp]: "0 \<in> Nats"
-apply (simp add: Nats_def) 
-apply (rule range_eqI) 
+apply (simp add: Nats_def)
+apply (rule range_eqI)
 apply (rule of_nat_0 [symmetric])
 done
 
 lemma Nats_1 [simp]: "1 \<in> Nats"
-apply (simp add: Nats_def) 
-apply (rule range_eqI) 
+apply (simp add: Nats_def)
+apply (rule range_eqI)
 apply (rule of_nat_1 [symmetric])
 done
 
 lemma Nats_add [simp]: "[|a \<in> Nats; b \<in> Nats|] ==> a+b \<in> Nats"
-apply (auto simp add: Nats_def) 
-apply (rule range_eqI) 
+apply (auto simp add: Nats_def)
+apply (rule range_eqI)
 apply (rule of_nat_add [symmetric])
 done
 
 lemma Nats_mult [simp]: "[|a \<in> Nats; b \<in> Nats|] ==> a*b \<in> Nats"
-apply (auto simp add: Nats_def) 
-apply (rule range_eqI) 
+apply (auto simp add: Nats_def)
+apply (rule range_eqI)
 apply (rule of_nat_mult [symmetric])
 done
 
@@ -755,7 +674,7 @@
 lemma int_eq_of_nat: "int = (of_nat :: nat => int)"
 proof
   fix n
-  show "int n = of_nat n"  by (induct n, simp_all add: int_Suc add_ac) 
+  show "int n = of_nat n"  by (induct n, simp_all add: int_Suc add_ac)
 qed
 
 
@@ -769,9 +688,9 @@
 
 lemma of_int: "of_int (Abs_Integ (intrel `` {(i,j)})) = of_nat i - of_nat j"
 apply (simp add: of_int_def)
-apply (rule the_equality, auto) 
+apply (rule the_equality, auto)
 apply (simp add: compare_rls add_ac of_nat_add [symmetric]
-            del: of_nat_add) 
+            del: of_nat_add)
 done
 
 lemma of_int_0 [simp]: "of_int 0 = 0"
@@ -781,32 +700,26 @@
 by (simp add: of_int One_int_def int_def)
 
 lemma of_int_add [simp]: "of_int (w+z) = of_int w + of_int z"
-apply (rule eq_Abs_Integ [of w])
-apply (rule eq_Abs_Integ [of z])
-apply (simp add: compare_rls of_int zadd) 
-done
+by (cases w, cases z, simp add: compare_rls of_int add)
 
 lemma of_int_minus [simp]: "of_int (-z) = - (of_int z)"
-apply (rule eq_Abs_Integ [of z])
-apply (simp add: compare_rls of_int zminus) 
-done
+by (cases z, simp add: compare_rls of_int minus)
 
 lemma of_int_diff [simp]: "of_int (w-z) = of_int w - of_int z"
 by (simp add: diff_minus)
 
 lemma of_int_mult [simp]: "of_int (w*z) = of_int w * of_int z"
-apply (rule eq_Abs_Integ [of w])
-apply (rule eq_Abs_Integ [of z])
-apply (simp add: compare_rls of_int left_diff_distrib right_diff_distrib 
-                 zmult add_ac) 
+apply (cases w, cases z)
+apply (simp add: compare_rls of_int left_diff_distrib right_diff_distrib
+                 mult add_ac)
 done
 
 lemma of_int_le_iff [simp]:
      "(of_int w \<le> (of_int z::'a::ordered_ring)) = (w \<le> z)"
-apply (rule eq_Abs_Integ [of w])
-apply (rule eq_Abs_Integ [of z])
-apply (simp add: compare_rls of_int zle zdiff_def zadd zminus 
-                 of_nat_add [symmetric]   del: of_nat_add) 
+apply (cases w)
+apply (cases z)
+apply (simp add: compare_rls of_int le diff_int_def add minus
+                 of_nat_add [symmetric]   del: of_nat_add)
 done
 
 text{*Special cases where either operand is zero*}
@@ -824,7 +737,7 @@
 text{*The ordering on the ring is necessary. See @{text of_nat_eq_iff} above.*}
 lemma of_int_eq_iff [simp]:
      "(of_int w = (of_int z::'a::ordered_ring)) = (w = z)"
-by (simp add: order_eq_iff) 
+by (simp add: order_eq_iff)
 
 text{*Special cases where either operand is zero*}
 declare of_int_eq_iff [of 0, simplified, simp]
@@ -842,52 +755,52 @@
   Ints      :: "'a set"                   ("\<int>")
 
 lemma Ints_0 [simp]: "0 \<in> Ints"
-apply (simp add: Ints_def) 
-apply (rule range_eqI) 
+apply (simp add: Ints_def)
+apply (rule range_eqI)
 apply (rule of_int_0 [symmetric])
 done
 
 lemma Ints_1 [simp]: "1 \<in> Ints"
-apply (simp add: Ints_def) 
-apply (rule range_eqI) 
+apply (simp add: Ints_def)
+apply (rule range_eqI)
 apply (rule of_int_1 [symmetric])
 done
 
 lemma Ints_add [simp]: "[|a \<in> Ints; b \<in> Ints|] ==> a+b \<in> Ints"
-apply (auto simp add: Ints_def) 
-apply (rule range_eqI) 
+apply (auto simp add: Ints_def)
+apply (rule range_eqI)
 apply (rule of_int_add [symmetric])
 done
 
 lemma Ints_minus [simp]: "a \<in> Ints ==> -a \<in> Ints"
-apply (auto simp add: Ints_def) 
-apply (rule range_eqI) 
+apply (auto simp add: Ints_def)
+apply (rule range_eqI)
 apply (rule of_int_minus [symmetric])
 done
 
 lemma Ints_diff [simp]: "[|a \<in> Ints; b \<in> Ints|] ==> a-b \<in> Ints"
-apply (auto simp add: Ints_def) 
-apply (rule range_eqI) 
+apply (auto simp add: Ints_def)
+apply (rule range_eqI)
 apply (rule of_int_diff [symmetric])
 done
 
 lemma Ints_mult [simp]: "[|a \<in> Ints; b \<in> Ints|] ==> a*b \<in> Ints"
-apply (auto simp add: Ints_def) 
-apply (rule range_eqI) 
+apply (auto simp add: Ints_def)
+apply (rule range_eqI)
 apply (rule of_int_mult [symmetric])
 done
 
 text{*Collapse nested embeddings*}
 lemma of_int_of_nat_eq [simp]: "of_int (of_nat n) = of_nat n"
-by (induct n, auto) 
+by (induct n, auto)
 
 lemma of_int_int_eq [simp]: "of_int (int n) = int n"
-by (simp add: int_eq_of_nat) 
+by (simp add: int_eq_of_nat)
 
 
 lemma Ints_cases [case_names of_int, cases set: Ints]:
   "q \<in> \<int> ==> (!!z. q = of_int z ==> C) ==> C"
-proof (unfold Ints_def)
+proof (simp add: Ints_def)
   assume "!!z. q = of_int z ==> C"
   assume "q \<in> range of_int" thus C ..
 qed
@@ -963,11 +876,41 @@
   by (rule setprod_zero_eq, auto)
 
 
+text{*Now we replace the case analysis rule by a more conventional one:
+whether an integer is negative or not.*}
+
+lemma zless_iff_Suc_zadd:
+    "(w < z) = (\<exists>n. z = w + int(Suc n))"
+apply (cases z, cases w)
+apply (auto simp add: le add int_def linorder_not_le [symmetric]) 
+apply (rename_tac a b c d) 
+apply (rule_tac x="a+d - Suc(c+b)" in exI) 
+apply arith
+done
+
+lemma negD: "x<0 ==> \<exists>n. x = - (int (Suc n))"
+apply (cases x)
+apply (auto simp add: le minus Zero_int_def int_def order_less_le) 
+apply (rule_tac x="y - Suc x" in exI) 
+apply arith
+done
+
+theorem int_cases [cases type: int, case_names nonneg neg]:
+     "[|!! n. z = int n ==> P;  !! n. z =  - (int (Suc n)) ==> P |] ==> P"
+apply (case_tac "z < 0", blast dest!: negD)
+apply (simp add: linorder_not_less)
+apply (blast dest: nat_0_le [THEN sym])
+done
+
+theorem int_induct [induct type: int, case_names nonneg neg]:
+     "[|!! n. P (int n);  !!n. P (- (int (Suc n))) |] ==> P z"
+  by (cases z) auto
+
+
 (*Legacy ML bindings, but no longer the structure Int.*)
 ML
 {*
 val zabs_def = thm "zabs_def"
-val nat_def  = thm "nat_def"
 
 val int_0 = thm "int_0";
 val int_1 = thm "int_1";
@@ -1002,11 +945,7 @@
 val int_def = thm "int_def";
 val Zero_int_def = thm "Zero_int_def";
 val One_int_def = thm "One_int_def";
-val zadd_def = thm "zadd_def";
-val zdiff_def = thm "zdiff_def";
-val zless_def = thm "zless_def";
-val zle_def = thm "zle_def";
-val zmult_def = thm "zmult_def";
+val diff_int_def = thm "diff_int_def";
 
 val intrel_iff = thm "intrel_iff";
 val equiv_intrel = thm "equiv_intrel";
@@ -1015,13 +954,8 @@
 val inj_on_Abs_Integ = thm "inj_on_Abs_Integ";
 val inj_Rep_Integ = thm "inj_Rep_Integ";
 val inj_int = thm "inj_int";
-val zminus_congruent = thm "zminus_congruent";
-val zminus = thm "zminus";
-val eq_Abs_Integ = thm "eq_Abs_Integ";
 val zminus_zminus = thm "zminus_zminus";
-val inj_zminus = thm "inj_zminus";
 val zminus_0 = thm "zminus_0";
-val zadd = thm "zadd";
 val zminus_zadd_distrib = thm "zminus_zadd_distrib";
 val zadd_commute = thm "zadd_commute";
 val zadd_assoc = thm "zadd_assoc";
@@ -1033,30 +967,17 @@
 val int_Suc = thm "int_Suc";
 val zadd_0 = thm "zadd_0";
 val zadd_0_right = thm "zadd_0_right";
-val zadd_zminus_inverse = thm "zadd_zminus_inverse";
-val zadd_zminus_inverse2 = thm "zadd_zminus_inverse2";
-val zadd_zminus_cancel = thm "zadd_zminus_cancel";
-val zminus_zadd_cancel = thm "zminus_zadd_cancel";
-val zdiff0 = thm "zdiff0";
-val zdiff0_right = thm "zdiff0_right";
-val zdiff_self = thm "zdiff_self";
-val zmult_congruent2 = thm "zmult_congruent2";
-val zmult = thm "zmult";
 val zmult_zminus = thm "zmult_zminus";
 val zmult_commute = thm "zmult_commute";
 val zmult_assoc = thm "zmult_assoc";
 val zadd_zmult_distrib = thm "zadd_zmult_distrib";
-val zmult_zminus_right = thm "zmult_zminus_right";
 val zadd_zmult_distrib2 = thm "zadd_zmult_distrib2";
 val zdiff_zmult_distrib = thm "zdiff_zmult_distrib";
 val zdiff_zmult_distrib2 = thm "zdiff_zmult_distrib2";
 val int_distrib = thms "int_distrib";
 val zmult_int = thm "zmult_int";
-val zmult_0 = thm "zmult_0";
 val zmult_1 = thm "zmult_1";
-val zmult_0_right = thm "zmult_0_right";
 val zmult_1_right = thm "zmult_1_right";
-val zless_iff_Suc_zadd = thm "zless_iff_Suc_zadd";
 val int_int_eq = thm "int_int_eq";
 val int_eq_0_conv = thm "int_eq_0_conv";
 val zless_int = thm "zless_int";
diff -r bdf6b7adc3ec -r 0eca4aabf371 src/HOL/Integ/IntDiv.thy
--- a/src/HOL/Integ/IntDiv.thy	Fri Mar 19 11:06:53 2004 +0100
+++ b/src/HOL/Integ/IntDiv.thy	Wed Mar 24 10:50:29 2004 +0100
@@ -95,12 +95,12 @@
      "[| b*q' + r'  \<le> b*q + r;  0 \<le> r';  0 < b;  r < b |]  
       ==> q' \<le> (q::int)"
 apply (subgoal_tac "r' + b * (q'-q) \<le> r")
- prefer 2 apply (simp add: zdiff_zmult_distrib2)
+ prefer 2 apply (simp add: right_diff_distrib)
 apply (subgoal_tac "0 < b * (1 + q - q') ")
 apply (erule_tac [2] order_le_less_trans)
- prefer 2 apply (simp add: zdiff_zmult_distrib2 zadd_zmult_distrib2)
+ prefer 2 apply (simp add: right_diff_distrib right_distrib)
 apply (subgoal_tac "b * q' < b * (1 + q) ")
- prefer 2 apply (simp add: zdiff_zmult_distrib2 zadd_zmult_distrib2)
+ prefer 2 apply (simp add: right_diff_distrib right_distrib)
 apply (simp add: mult_less_cancel_left)
 done
 
@@ -158,7 +158,7 @@
 (*main argument*)
 apply (subst posDivAlg_eqn, simp_all)
 apply (erule splitE)
-apply (auto simp add: zadd_zmult_distrib2 Let_def)
+apply (auto simp add: right_distrib Let_def)
 done
 
 
@@ -186,7 +186,7 @@
 (*main argument*)
 apply (subst negDivAlg_eqn, assumption)
 apply (erule splitE)
-apply (auto simp add: zadd_zmult_distrib2 Let_def)
+apply (auto simp add: right_distrib Let_def)
 done
 
 
@@ -321,7 +321,7 @@
      "quorem((a,b),(q,r))  
       ==> quorem ((-a,b), (if r=0 then -q else -q - 1),  
                           (if r=0 then 0 else b-r))"
-by (force simp add: split_ifs quorem_def linorder_neq_iff zdiff_zmult_distrib2)
+by (force simp add: split_ifs quorem_def linorder_neq_iff right_diff_distrib)
 
 
 lemma zdiv_zminus1_eq_if:
@@ -363,7 +363,7 @@
 lemma self_quotient_aux2: "[| (0::int) < a; a = r + a*q; 0 \<le> r |] ==> q \<le> 1"
 apply (subgoal_tac "0 \<le> a* (1-q) ")
  apply (simp add: zero_le_mult_iff)
-apply (simp add: zdiff_zmult_distrib2)
+apply (simp add: right_diff_distrib)
 done
 
 lemma self_quotient: "[| quorem((a,a),(q,r));  a ~= (0::int) |] ==> q = 1"
@@ -517,7 +517,7 @@
      "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"
 apply (subgoal_tac "0 < b'* (q' + 1) ")
  apply (simp add: zero_less_mult_iff)
-apply (simp add: zadd_zmult_distrib2)
+apply (simp add: right_distrib)
 done
 
 lemma zdiv_mono2_lemma:
@@ -529,7 +529,7 @@
  apply (simp add: mult_less_cancel_left)
 apply (subgoal_tac "b*q = r' - r + b'*q'")
  prefer 2 apply simp
-apply (simp (no_asm_simp) add: zadd_zmult_distrib2)
+apply (simp (no_asm_simp) add: right_distrib)
 apply (subst zadd_commute, rule zadd_zless_mono, arith)
 apply (rule mult_right_mono, auto)
 done
@@ -559,7 +559,7 @@
 apply (frule q_neg_lemma, assumption+) 
 apply (subgoal_tac "b*q' < b* (q + 1) ")
  apply (simp add: mult_less_cancel_left)
-apply (simp add: zadd_zmult_distrib2)
+apply (simp add: right_distrib)
 apply (subgoal_tac "b*q' \<le> b'*q'")
  prefer 2 apply (simp add: mult_right_mono_neg)
 apply (subgoal_tac "b'*q' < b + b*q")
@@ -585,7 +585,7 @@
 lemma zmult1_lemma:
      "[| quorem((b,c),(q,r));  c ~= 0 |]  
       ==> quorem ((a*b, c), (a*q + a*r div c, a*r mod c))"
-by (force simp add: split_ifs quorem_def linorder_neq_iff zadd_zmult_distrib2)
+by (force simp add: split_ifs quorem_def linorder_neq_iff right_distrib)
 
 lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"
 apply (case_tac "c = 0", simp add: DIVISION_BY_ZERO)
@@ -637,7 +637,7 @@
 lemma zadd1_lemma:
      "[| quorem((a,c),(aq,ar));  quorem((b,c),(bq,br));  c ~= 0 |]  
       ==> quorem ((a+b, c), (aq + bq + (ar+br) div c, (ar+br) mod c))"
-by (force simp add: split_ifs quorem_def linorder_neq_iff zadd_zmult_distrib2)
+by (force simp add: split_ifs quorem_def linorder_neq_iff right_distrib)
 
 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)
 lemma zdiv_zadd1_eq:
@@ -700,7 +700,7 @@
 
 lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b*c < b*(q mod c) + r"
 apply (subgoal_tac "b * (c - q mod c) < r * 1")
-apply (simp add: zdiff_zmult_distrib2)
+apply (simp add: right_diff_distrib)
 apply (rule order_le_less_trans)
 apply (erule_tac [2] mult_strict_right_mono)
 apply (rule mult_left_mono_neg)
@@ -722,7 +722,7 @@
 
 lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"
 apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
-apply (simp add: zdiff_zmult_distrib2)
+apply (simp add: right_diff_distrib)
 apply (rule order_less_le_trans)
 apply (erule mult_strict_right_mono)
 apply (rule_tac [2] mult_left_mono)
@@ -733,7 +733,7 @@
 lemma zmult2_lemma: "[| quorem ((a,b), (q,r));  b ~= 0;  0 < c |]  
       ==> quorem ((a, b*c), (q div c, b*(q mod c) + r))"
 by (auto simp add: mult_ac quorem_def linorder_neq_iff
-                   zero_less_mult_iff zadd_zmult_distrib2 [symmetric] 
+                   zero_less_mult_iff right_distrib [symmetric] 
                    zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4)
 
 lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c"
@@ -887,9 +887,9 @@
 lemma neg_zdiv_mult_2: "a \<le> (0::int) ==> (1 + 2*b) div (2*a) = (b+1) div a"
 apply (subgoal_tac " (1 + 2* (-b - 1)) div (2 * (-a)) = (-b - 1) div (-a) ")
 apply (rule_tac [2] pos_zdiv_mult_2)
-apply (auto simp add: zmult_zminus_right right_diff_distrib)
+apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib)
 apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")
-apply (simp only: zdiv_zminus_zminus zdiff_def zminus_zadd_distrib [symmetric],
+apply (simp only: zdiv_zminus_zminus diff_minus minus_add_distrib [symmetric],
        simp) 
 done
 
@@ -934,16 +934,15 @@
 apply (simp)
 done
 
-
 lemma neg_zmod_mult_2:
      "a \<le> (0::int) ==> (1 + 2*b) mod (2*a) = 2 * ((b+1) mod a) - 1"
 apply (subgoal_tac "(1 + 2* (-b - 1)) mod (2* (-a)) = 
                     1 + 2* ((-b - 1) mod (-a))")
 apply (rule_tac [2] pos_zmod_mult_2)
-apply (auto simp add: zmult_zminus_right right_diff_distrib)
+apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib)
 apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")
  prefer 2 apply simp 
-apply (simp only: zmod_zminus_zminus zdiff_def zminus_zadd_distrib [symmetric])
+apply (simp only: zmod_zminus_zminus diff_minus minus_add_distrib [symmetric])
 done
 
 lemma zmod_number_of_BIT [simp]:
@@ -1003,7 +1002,7 @@
 
 lemma zdvd_0_right [iff]: "(m::int) dvd 0"
   apply (unfold dvd_def)
-  apply (blast intro: zmult_0_right [symmetric])
+  apply (blast intro: mult_zero_right [symmetric])
   done
 
 lemma zdvd_0_left [iff]: "(0 dvd (m::int)) = (m = 0)"
@@ -1040,12 +1039,12 @@
 
 lemma zdvd_zadd: "k dvd m ==> k dvd n ==> k dvd (m + n :: int)"
   apply (unfold dvd_def)
-  apply (blast intro: zadd_zmult_distrib2 [symmetric])
+  apply (blast intro: right_distrib [symmetric])
   done
 
 lemma zdvd_zdiff: "k dvd m ==> k dvd n ==> k dvd (m - n :: int)"
   apply (unfold dvd_def)
-  apply (blast intro: zdiff_zmult_distrib2 [symmetric])
+  apply (blast intro: right_diff_distrib [symmetric])
   done
 
 lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"
diff -r bdf6b7adc3ec -r 0eca4aabf371 src/HOL/Integ/IntDiv_setup.ML
--- a/src/HOL/Integ/IntDiv_setup.ML	Fri Mar 19 11:06:53 2004 +0100
+++ b/src/HOL/Integ/IntDiv_setup.ML	Wed Mar 24 10:50:29 2004 +0100
@@ -24,7 +24,7 @@
 val trans = trans
 
 val prove_eq_sums =
-  let val simps = zdiff_def :: Int_Numeral_Simprocs.add_0s @ zadd_ac
+  let val simps = diff_int_def :: Int_Numeral_Simprocs.add_0s @ zadd_ac
   in NatArithUtils.prove_conv all_tac (NatArithUtils.simp_all simps) end
 
 end;
diff -r bdf6b7adc3ec -r 0eca4aabf371 src/HOL/Integ/cooper_proof.ML
--- a/src/HOL/Integ/cooper_proof.ML	Fri Mar 19 11:06:53 2004 +0100
+++ b/src/HOL/Integ/cooper_proof.ML	Wed Mar 24 10:50:29 2004 +0100
@@ -44,7 +44,7 @@
 (*-----------------------------------------------------------------*)
 
 val presburger_ss = simpset_of (theory "Presburger")
-  addsimps [zdiff_def] delsimps [thm"zdiff_def_symmetric"];
+  addsimps [diff_int_def] delsimps [thm"diff_int_def_symmetric"];
 val cboolT = ctyp_of (sign_of HOL.thy) HOLogic.boolT;
 
 (*Theorems that will be used later for the proofgeneration*)
diff -r bdf6b7adc3ec -r 0eca4aabf371 src/HOL/Set.thy
--- a/src/HOL/Set.thy	Fri Mar 19 11:06:53 2004 +0100
+++ b/src/HOL/Set.thy	Wed Mar 24 10:50:29 2004 +0100
@@ -1949,6 +1949,16 @@
   by blast
 
 
+subsection {* Getting the Contents of a Singleton Set *}
+
+constdefs
+  contents :: "'a set => 'a"
+   "contents X == THE x. X = {x}"
+
+lemma contents_eq [simp]: "contents {x} = x"
+by (simp add: contents_def)
+
+
 subsection {* Transitivity rules for calculational reasoning *}
 
 lemma forw_subst: "a = b ==> P b ==> P a"
diff -r bdf6b7adc3ec -r 0eca4aabf371 src/HOL/Tools/Presburger/cooper_proof.ML
--- a/src/HOL/Tools/Presburger/cooper_proof.ML	Fri Mar 19 11:06:53 2004 +0100
+++ b/src/HOL/Tools/Presburger/cooper_proof.ML	Wed Mar 24 10:50:29 2004 +0100
@@ -44,7 +44,7 @@
 (*-----------------------------------------------------------------*)
 
 val presburger_ss = simpset_of (theory "Presburger")
-  addsimps [zdiff_def] delsimps [thm"zdiff_def_symmetric"];
+  addsimps [diff_int_def] delsimps [thm"diff_int_def_symmetric"];
 val cboolT = ctyp_of (sign_of HOL.thy) HOLogic.boolT;
 
 (*Theorems that will be used later for the proofgeneration*)