
diff -r 5d8faadf3ecf -r 0bc590051d95 src/ZF/IntDiv.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/ZF/IntDiv.thy	Thu May 31 12:06:31 2007 +0200
@@ -0,0 +1,1925 @@
+(*  Title:      ZF/IntDiv.thy
+    ID:         $Id$
+    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
+    Copyright   1999  University of Cambridge
+
+Here is the division algorithm in ML:
+
+    fun posDivAlg (a,b) =
+      if a<b then (0,a)
+      else let val (q,r) = posDivAlg(a, 2*b)
+	       in  if 0<=r-b then (2*q+1, r-b) else (2*q, r)
+	   end
+
+    fun negDivAlg (a,b) =
+      if 0<=a+b then (~1,a+b)
+      else let val (q,r) = negDivAlg(a, 2*b)
+	       in  if 0<=r-b then (2*q+1, r-b) else (2*q, r)
+	   end;
+
+    fun negateSnd (q,r:int) = (q,~r);
+
+    fun divAlg (a,b) = if 0<=a then 
+			  if b>0 then posDivAlg (a,b) 
+			   else if a=0 then (0,0)
+				else negateSnd (negDivAlg (~a,~b))
+		       else 
+			  if 0<b then negDivAlg (a,b)
+			  else        negateSnd (posDivAlg (~a,~b));
+
+*)
+
+header{*The Division Operators Div and Mod*}
+
+theory IntDiv imports IntArith OrderArith begin
+
+constdefs
+  quorem :: "[i,i] => o"
+    "quorem == %<a,b> <q,r>.
+                      a = b$*q $+ r &
+                      (#0$<b & #0$<=r & r$<b | ~(#0$<b) & b$<r & r $<= #0)"
+
+  adjust :: "[i,i] => i"
+    "adjust(b) == %<q,r>. if #0 $<= r$-b then <#2$*q $+ #1,r$-b>
+                          else <#2$*q,r>"
+
+
+(** the division algorithm **)
+
+constdefs posDivAlg :: "i => i"
+(*for the case a>=0, b>0*)
+(*recdef posDivAlg "inv_image less_than (%(a,b). nat_of(a $- b $+ #1))"*)
+    "posDivAlg(ab) ==
+       wfrec(measure(int*int, %<a,b>. nat_of (a $- b $+ #1)),
+	     ab,
+	     %<a,b> f. if (a$<b | b$<=#0) then <#0,a>
+                       else adjust(b, f ` <a,#2$*b>))"
+
+
+(*for the case a<0, b>0*)
+constdefs negDivAlg :: "i => i"
+(*recdef negDivAlg "inv_image less_than (%(a,b). nat_of(- a $- b))"*)
+    "negDivAlg(ab) ==
+       wfrec(measure(int*int, %<a,b>. nat_of ($- a $- b)),
+	     ab,
+	     %<a,b> f. if (#0 $<= a$+b | b$<=#0) then <#-1,a$+b>
+                       else adjust(b, f ` <a,#2$*b>))"
+
+(*for the general case b\<noteq>0*)
+
+constdefs
+  negateSnd :: "i => i"
+    "negateSnd == %<q,r>. <q, $-r>"
+
+  (*The full division algorithm considers all possible signs for a, b
+    including the special case a=0, b<0, because negDivAlg requires a<0*)
+  divAlg :: "i => i"
+    "divAlg ==
+       %<a,b>. if #0 $<= a then
+                  if #0 $<= b then posDivAlg (<a,b>)
+                  else if a=#0 then <#0,#0>
+                       else negateSnd (negDivAlg (<$-a,$-b>))
+               else 
+                  if #0$<b then negDivAlg (<a,b>)
+                  else         negateSnd (posDivAlg (<$-a,$-b>))"
+
+  zdiv  :: "[i,i]=>i"                    (infixl "zdiv" 70) 
+    "a zdiv b == fst (divAlg (<intify(a), intify(b)>))"
+
+  zmod  :: "[i,i]=>i"                    (infixl "zmod" 70)
+    "a zmod b == snd (divAlg (<intify(a), intify(b)>))"
+
+
+(** Some basic laws by Sidi Ehmety (need linear arithmetic!) **)
+
+lemma zspos_add_zspos_imp_zspos: "[| #0 $< x;  #0 $< y |] ==> #0 $< x $+ y"
+apply (rule_tac y = "y" in zless_trans)
+apply (rule_tac [2] zdiff_zless_iff [THEN iffD1])
+apply auto
+done
+
+lemma zpos_add_zpos_imp_zpos: "[| #0 $<= x;  #0 $<= y |] ==> #0 $<= x $+ y"
+apply (rule_tac y = "y" in zle_trans)
+apply (rule_tac [2] zdiff_zle_iff [THEN iffD1])
+apply auto
+done
+
+lemma zneg_add_zneg_imp_zneg: "[| x $< #0;  y $< #0 |] ==> x $+ y $< #0"
+apply (rule_tac y = "y" in zless_trans)
+apply (rule zless_zdiff_iff [THEN iffD1])
+apply auto
+done
+
+(* this theorem is used below *)
+lemma zneg_or_0_add_zneg_or_0_imp_zneg_or_0:
+     "[| x $<= #0;  y $<= #0 |] ==> x $+ y $<= #0"
+apply (rule_tac y = "y" in zle_trans)
+apply (rule zle_zdiff_iff [THEN iffD1])
+apply auto
+done
+
+lemma zero_lt_zmagnitude: "[| #0 $< k; k \<in> int |] ==> 0 < zmagnitude(k)"
+apply (drule zero_zless_imp_znegative_zminus)
+apply (drule_tac [2] zneg_int_of)
+apply (auto simp add: zminus_equation [of k])
+apply (subgoal_tac "0 < zmagnitude ($# succ (n))")
+ apply simp
+apply (simp only: zmagnitude_int_of)
+apply simp
+done
+
+
+(*** Inequality lemmas involving $#succ(m) ***)
+
+lemma zless_add_succ_iff:
+     "(w $< z $+ $# succ(m)) <-> (w $< z $+ $#m | intify(w) = z $+ $#m)"
+apply (auto simp add: zless_iff_succ_zadd zadd_assoc int_of_add [symmetric])
+apply (rule_tac [3] x = "0" in bexI)
+apply (cut_tac m = "m" in int_succ_int_1)
+apply (cut_tac m = "n" in int_succ_int_1)
+apply simp
+apply (erule natE)
+apply auto
+apply (rule_tac x = "succ (n) " in bexI)
+apply auto
+done
+
+lemma zadd_succ_lemma:
+     "z \<in> int ==> (w $+ $# succ(m) $<= z) <-> (w $+ $#m $< z)"
+apply (simp only: not_zless_iff_zle [THEN iff_sym] zless_add_succ_iff)
+apply (auto intro: zle_anti_sym elim: zless_asym
+            simp add: zless_imp_zle not_zless_iff_zle)
+done
+
+lemma zadd_succ_zle_iff: "(w $+ $# succ(m) $<= z) <-> (w $+ $#m $< z)"
+apply (cut_tac z = "intify (z)" in zadd_succ_lemma)
+apply auto
+done
+
+(** Inequality reasoning **)
+
+lemma zless_add1_iff_zle: "(w $< z $+ #1) <-> (w$<=z)"
+apply (subgoal_tac "#1 = $# 1")
+apply (simp only: zless_add_succ_iff zle_def)
+apply auto
+done
+
+lemma add1_zle_iff: "(w $+ #1 $<= z) <-> (w $< z)"
+apply (subgoal_tac "#1 = $# 1")
+apply (simp only: zadd_succ_zle_iff)
+apply auto
+done
+
+lemma add1_left_zle_iff: "(#1 $+ w $<= z) <-> (w $< z)"
+apply (subst zadd_commute)
+apply (rule add1_zle_iff)
+done
+
+
+(*** Monotonicity of Multiplication ***)
+
+lemma zmult_mono_lemma: "k \<in> nat ==> i $<= j ==> i $* $#k $<= j $* $#k"
+apply (induct_tac "k")
+ prefer 2 apply (subst int_succ_int_1)
+apply (simp_all (no_asm_simp) add: zadd_zmult_distrib2 zadd_zle_mono)
+done
+
+lemma zmult_zle_mono1: "[| i $<= j;  #0 $<= k |] ==> i$*k $<= j$*k"
+apply (subgoal_tac "i $* intify (k) $<= j $* intify (k) ")
+apply (simp (no_asm_use))
+apply (rule_tac b = "intify (k)" in not_zneg_mag [THEN subst])
+apply (rule_tac [3] zmult_mono_lemma)
+apply auto
+apply (simp add: znegative_iff_zless_0 not_zless_iff_zle [THEN iff_sym])
+done
+
+lemma zmult_zle_mono1_neg: "[| i $<= j;  k $<= #0 |] ==> j$*k $<= i$*k"
+apply (rule zminus_zle_zminus [THEN iffD1])
+apply (simp del: zmult_zminus_right
+            add: zmult_zminus_right [symmetric] zmult_zle_mono1 zle_zminus)
+done
+
+lemma zmult_zle_mono2: "[| i $<= j;  #0 $<= k |] ==> k$*i $<= k$*j"
+apply (drule zmult_zle_mono1)
+apply (simp_all add: zmult_commute)
+done
+
+lemma zmult_zle_mono2_neg: "[| i $<= j;  k $<= #0 |] ==> k$*j $<= k$*i"
+apply (drule zmult_zle_mono1_neg)
+apply (simp_all add: zmult_commute)
+done
+
+(* $<= monotonicity, BOTH arguments*)
+lemma zmult_zle_mono:
+     "[| i $<= j;  k $<= l;  #0 $<= j;  #0 $<= k |] ==> i$*k $<= j$*l"
+apply (erule zmult_zle_mono1 [THEN zle_trans])
+apply assumption
+apply (erule zmult_zle_mono2)
+apply assumption
+done
+
+
+(** strict, in 1st argument; proof is by induction on k>0 **)
+
+lemma zmult_zless_mono2_lemma [rule_format]:
+     "[| i$<j; k \<in> nat |] ==> 0<k --> $#k $* i $< $#k $* j"
+apply (induct_tac "k")
+ prefer 2
+ apply (subst int_succ_int_1)
+ apply (erule natE)
+apply (simp_all add: zadd_zmult_distrib zadd_zless_mono zle_def)
+apply (frule nat_0_le)
+apply (subgoal_tac "i $+ (i $+ $# xa $* i) $< j $+ (j $+ $# xa $* j) ")
+apply (simp (no_asm_use))
+apply (rule zadd_zless_mono)
+apply (simp_all (no_asm_simp) add: zle_def)
+done
+
+lemma zmult_zless_mono2: "[| i$<j;  #0 $< k |] ==> k$*i $< k$*j"
+apply (subgoal_tac "intify (k) $* i $< intify (k) $* j")
+apply (simp (no_asm_use))
+apply (rule_tac b = "intify (k)" in not_zneg_mag [THEN subst])
+apply (rule_tac [3] zmult_zless_mono2_lemma)
+apply auto
+apply (simp add: znegative_iff_zless_0)
+apply (drule zless_trans, assumption)
+apply (auto simp add: zero_lt_zmagnitude)
+done
+
+lemma zmult_zless_mono1: "[| i$<j;  #0 $< k |] ==> i$*k $< j$*k"
+apply (drule zmult_zless_mono2)
+apply (simp_all add: zmult_commute)
+done
+
+(* < monotonicity, BOTH arguments*)
+lemma zmult_zless_mono:
+     "[| i $< j;  k $< l;  #0 $< j;  #0 $< k |] ==> i$*k $< j$*l"
+apply (erule zmult_zless_mono1 [THEN zless_trans])
+apply assumption
+apply (erule zmult_zless_mono2)
+apply assumption
+done
+
+lemma zmult_zless_mono1_neg: "[| i $< j;  k $< #0 |] ==> j$*k $< i$*k"
+apply (rule zminus_zless_zminus [THEN iffD1])
+apply (simp del: zmult_zminus_right 
+            add: zmult_zminus_right [symmetric] zmult_zless_mono1 zless_zminus)
+done
+
+lemma zmult_zless_mono2_neg: "[| i $< j;  k $< #0 |] ==> k$*j $< k$*i"
+apply (rule zminus_zless_zminus [THEN iffD1])
+apply (simp del: zmult_zminus 
+            add: zmult_zminus [symmetric] zmult_zless_mono2 zless_zminus)
+done
+
+
+(** Products of zeroes **)
+
+lemma zmult_eq_lemma:
+     "[| m \<in> int; n \<in> int |] ==> (m = #0 | n = #0) <-> (m$*n = #0)"
+apply (case_tac "m $< #0")
+apply (auto simp add: not_zless_iff_zle zle_def neq_iff_zless)
+apply (force dest: zmult_zless_mono1_neg zmult_zless_mono1)+
+done
+
+lemma zmult_eq_0_iff [iff]: "(m$*n = #0) <-> (intify(m) = #0 | intify(n) = #0)"
+apply (simp add: zmult_eq_lemma)
+done
+
+
+(** Cancellation laws for k*m < k*n and m*k < n*k, also for <= and =,
+    but not (yet?) for k*m < n*k. **)
+
+lemma zmult_zless_lemma:
+     "[| k \<in> int; m \<in> int; n \<in> int |]  
+      ==> (m$*k $< n$*k) <-> ((#0 $< k & m$<n) | (k $< #0 & n$<m))"
+apply (case_tac "k = #0")
+apply (auto simp add: neq_iff_zless zmult_zless_mono1 zmult_zless_mono1_neg)
+apply (auto simp add: not_zless_iff_zle 
+                      not_zle_iff_zless [THEN iff_sym, of "m$*k"] 
+                      not_zle_iff_zless [THEN iff_sym, of m])
+apply (auto elim: notE
+            simp add: zless_imp_zle zmult_zle_mono1 zmult_zle_mono1_neg)
+done
+
+lemma zmult_zless_cancel2:
+     "(m$*k $< n$*k) <-> ((#0 $< k & m$<n) | (k $< #0 & n$<m))"
+apply (cut_tac k = "intify (k)" and m = "intify (m)" and n = "intify (n)" 
+       in zmult_zless_lemma)
+apply auto
+done
+
+lemma zmult_zless_cancel1:
+     "(k$*m $< k$*n) <-> ((#0 $< k & m$<n) | (k $< #0 & n$<m))"
+by (simp add: zmult_commute [of k] zmult_zless_cancel2)
+
+lemma zmult_zle_cancel2:
+     "(m$*k $<= n$*k) <-> ((#0 $< k --> m$<=n) & (k $< #0 --> n$<=m))"
+by (auto simp add: not_zless_iff_zle [THEN iff_sym] zmult_zless_cancel2)
+
+lemma zmult_zle_cancel1:
+     "(k$*m $<= k$*n) <-> ((#0 $< k --> m$<=n) & (k $< #0 --> n$<=m))"
+by (auto simp add: not_zless_iff_zle [THEN iff_sym] zmult_zless_cancel1)
+
+lemma int_eq_iff_zle: "[| m \<in> int; n \<in> int |] ==> m=n <-> (m $<= n & n $<= m)"
+apply (blast intro: zle_refl zle_anti_sym)
+done
+
+lemma zmult_cancel2_lemma:
+     "[| k \<in> int; m \<in> int; n \<in> int |] ==> (m$*k = n$*k) <-> (k=#0 | m=n)"
+apply (simp add: int_eq_iff_zle [of "m$*k"] int_eq_iff_zle [of m])
+apply (auto simp add: zmult_zle_cancel2 neq_iff_zless)
+done
+
+lemma zmult_cancel2 [simp]:
+     "(m$*k = n$*k) <-> (intify(k) = #0 | intify(m) = intify(n))"
+apply (rule iff_trans)
+apply (rule_tac [2] zmult_cancel2_lemma)
+apply auto
+done
+
+lemma zmult_cancel1 [simp]:
+     "(k$*m = k$*n) <-> (intify(k) = #0 | intify(m) = intify(n))"
+by (simp add: zmult_commute [of k] zmult_cancel2)
+
+
+subsection{* Uniqueness and monotonicity of quotients and remainders *}
+
+lemma unique_quotient_lemma:
+     "[| b$*q' $+ r' $<= b$*q $+ r;  #0 $<= r';  #0 $< b;  r $< b |]  
+      ==> q' $<= q"
+apply (subgoal_tac "r' $+ b $* (q'$-q) $<= r")
+ prefer 2 apply (simp add: zdiff_zmult_distrib2 zadd_ac zcompare_rls)
+apply (subgoal_tac "#0 $< b $* (#1 $+ q $- q') ")
+ prefer 2
+ apply (erule zle_zless_trans)
+ apply (simp add: zdiff_zmult_distrib2 zadd_zmult_distrib2 zadd_ac zcompare_rls)
+ apply (erule zle_zless_trans)
+ apply (simp add: ); 
+apply (subgoal_tac "b $* q' $< b $* (#1 $+ q)")
+ prefer 2 
+ apply (simp add: zdiff_zmult_distrib2 zadd_zmult_distrib2 zadd_ac zcompare_rls)
+apply (auto elim: zless_asym
+        simp add: zmult_zless_cancel1 zless_add1_iff_zle zadd_ac zcompare_rls)
+done
+
+lemma unique_quotient_lemma_neg:
+     "[| b$*q' $+ r' $<= b$*q $+ r;  r $<= #0;  b $< #0;  b $< r' |]  
+      ==> q $<= q'"
+apply (rule_tac b = "$-b" and r = "$-r'" and r' = "$-r" 
+       in unique_quotient_lemma)
+apply (auto simp del: zminus_zadd_distrib
+            simp add: zminus_zadd_distrib [symmetric] zle_zminus zless_zminus)
+done
+
+
+lemma unique_quotient:
+     "[| quorem (<a,b>, <q,r>);  quorem (<a,b>, <q',r'>);  b \<in> int; b ~= #0;  
+         q \<in> int; q' \<in> int |] ==> q = q'"
+apply (simp add: split_ifs quorem_def neq_iff_zless)
+apply safe
+apply simp_all
+apply (blast intro: zle_anti_sym
+             dest: zle_eq_refl [THEN unique_quotient_lemma] 
+                   zle_eq_refl [THEN unique_quotient_lemma_neg] sym)+
+done
+
+lemma unique_remainder:
+     "[| quorem (<a,b>, <q,r>);  quorem (<a,b>, <q',r'>);  b \<in> int; b ~= #0;  
+         q \<in> int; q' \<in> int;  
+         r \<in> int; r' \<in> int |] ==> r = r'"
+apply (subgoal_tac "q = q'")
+ prefer 2 apply (blast intro: unique_quotient)
+apply (simp add: quorem_def)
+done
+
+
+subsection{*Correctness of posDivAlg, 
+           the Division Algorithm for @{text "a\<ge>0"} and @{text "b>0"} *}
+
+lemma adjust_eq [simp]:
+     "adjust(b, <q,r>) = (let diff = r$-b in  
+                          if #0 $<= diff then <#2$*q $+ #1,diff>   
+                                         else <#2$*q,r>)"
+by (simp add: Let_def adjust_def)
+
+
+lemma posDivAlg_termination:
+     "[| #0 $< b; ~ a $< b |]    
+      ==> nat_of(a $- #2 $\<times> b $+ #1) < nat_of(a $- b $+ #1)"
+apply (simp (no_asm) add: zless_nat_conj)
+apply (simp add: not_zless_iff_zle zless_add1_iff_zle zcompare_rls)
+done
+
+lemmas posDivAlg_unfold = def_wfrec [OF posDivAlg_def wf_measure]
+
+lemma posDivAlg_eqn:
+     "[| #0 $< b; a \<in> int; b \<in> int |] ==>  
+      posDivAlg(<a,b>) =       
+       (if a$<b then <#0,a> else adjust(b, posDivAlg (<a, #2$*b>)))"
+apply (rule posDivAlg_unfold [THEN trans])
+apply (simp add: vimage_iff not_zless_iff_zle [THEN iff_sym])
+apply (blast intro: posDivAlg_termination)
+done
+
+lemma posDivAlg_induct_lemma [rule_format]:
+  assumes prem:
+        "!!a b. [| a \<in> int; b \<in> int;  
+                   ~ (a $< b | b $<= #0) --> P(<a, #2 $* b>) |] ==> P(<a,b>)"
+  shows "<u,v> \<in> int*int --> P(<u,v>)"
+apply (rule_tac a = "<u,v>" in wf_induct)
+apply (rule_tac A = "int*int" and f = "%<a,b>.nat_of (a $- b $+ #1)" 
+       in wf_measure)
+apply clarify
+apply (rule prem)
+apply (drule_tac [3] x = "<xa, #2 $\<times> y>" in spec)
+apply auto
+apply (simp add: not_zle_iff_zless posDivAlg_termination)
+done
+
+
+lemma posDivAlg_induct [consumes 2]:
+  assumes u_int: "u \<in> int"
+      and v_int: "v \<in> int"
+      and ih: "!!a b. [| a \<in> int; b \<in> int;
+                     ~ (a $< b | b $<= #0) --> P(a, #2 $* b) |] ==> P(a,b)"
+  shows "P(u,v)"
+apply (subgoal_tac "(%<x,y>. P (x,y)) (<u,v>)")
+apply simp
+apply (rule posDivAlg_induct_lemma)
+apply (simp (no_asm_use))
+apply (rule ih)
+apply (auto simp add: u_int v_int)
+done
+
+(*FIXME: use intify in integ_of so that we always have integ_of w \<in> int.
+    then this rewrite can work for ALL constants!!*)
+lemma intify_eq_0_iff_zle: "intify(m) = #0 <-> (m $<= #0 & #0 $<= m)"
+apply (simp (no_asm) add: int_eq_iff_zle)
+done
+
+
+subsection{* Some convenient biconditionals for products of signs *}
+
+lemma zmult_pos: "[| #0 $< i; #0 $< j |] ==> #0 $< i $* j"
+apply (drule zmult_zless_mono1)
+apply auto
+done
+
+lemma zmult_neg: "[| i $< #0; j $< #0 |] ==> #0 $< i $* j"
+apply (drule zmult_zless_mono1_neg)
+apply auto
+done
+
+lemma zmult_pos_neg: "[| #0 $< i; j $< #0 |] ==> i $* j $< #0"
+apply (drule zmult_zless_mono1_neg)
+apply auto
+done
+
+(** Inequality reasoning **)
+
+lemma int_0_less_lemma:
+     "[| x \<in> int; y \<in> int |]  
+      ==> (#0 $< x $* y) <-> (#0 $< x & #0 $< y | x $< #0 & y $< #0)"
+apply (auto simp add: zle_def not_zless_iff_zle zmult_pos zmult_neg)
+apply (rule ccontr) 
+apply (rule_tac [2] ccontr) 
+apply (auto simp add: zle_def not_zless_iff_zle)
+apply (erule_tac P = "#0$< x$* y" in rev_mp)
+apply (erule_tac [2] P = "#0$< x$* y" in rev_mp)
+apply (drule zmult_pos_neg, assumption) 
+ prefer 2
+ apply (drule zmult_pos_neg, assumption) 
+apply (auto dest: zless_not_sym simp add: zmult_commute)
+done
+
+lemma int_0_less_mult_iff:
+     "(#0 $< x $* y) <-> (#0 $< x & #0 $< y | x $< #0 & y $< #0)"
+apply (cut_tac x = "intify (x)" and y = "intify (y)" in int_0_less_lemma)
+apply auto
+done
+
+lemma int_0_le_lemma:
+     "[| x \<in> int; y \<in> int |]  
+      ==> (#0 $<= x $* y) <-> (#0 $<= x & #0 $<= y | x $<= #0 & y $<= #0)"
+by (auto simp add: zle_def not_zless_iff_zle int_0_less_mult_iff)
+
+lemma int_0_le_mult_iff:
+     "(#0 $<= x $* y) <-> ((#0 $<= x & #0 $<= y) | (x $<= #0 & y $<= #0))"
+apply (cut_tac x = "intify (x)" and y = "intify (y)" in int_0_le_lemma)
+apply auto
+done
+
+lemma zmult_less_0_iff:
+     "(x $* y $< #0) <-> (#0 $< x & y $< #0 | x $< #0 & #0 $< y)"
+apply (auto simp add: int_0_le_mult_iff not_zle_iff_zless [THEN iff_sym])
+apply (auto dest: zless_not_sym simp add: not_zle_iff_zless)
+done
+
+lemma zmult_le_0_iff:
+     "(x $* y $<= #0) <-> (#0 $<= x & y $<= #0 | x $<= #0 & #0 $<= y)"
+by (auto dest: zless_not_sym
+         simp add: int_0_less_mult_iff not_zless_iff_zle [THEN iff_sym])
+
+
+(*Typechecking for posDivAlg*)
+lemma posDivAlg_type [rule_format]:
+     "[| a \<in> int; b \<in> int |] ==> posDivAlg(<a,b>) \<in> int * int"
+apply (rule_tac u = "a" and v = "b" in posDivAlg_induct)
+apply assumption+
+apply (case_tac "#0 $< ba")
+ apply (simp add: posDivAlg_eqn adjust_def integ_of_type 
+             split add: split_if_asm)
+ apply clarify
+ apply (simp add: int_0_less_mult_iff not_zle_iff_zless)
+apply (simp add: not_zless_iff_zle)
+apply (subst posDivAlg_unfold)
+apply simp
+done
+
+(*Correctness of posDivAlg: it computes quotients correctly*)
+lemma posDivAlg_correct [rule_format]:
+     "[| a \<in> int; b \<in> int |]  
+      ==> #0 $<= a --> #0 $< b --> quorem (<a,b>, posDivAlg(<a,b>))"
+apply (rule_tac u = "a" and v = "b" in posDivAlg_induct)
+apply auto
+   apply (simp_all add: quorem_def)
+   txt{*base case: a<b*}
+   apply (simp add: posDivAlg_eqn)
+  apply (simp add: not_zless_iff_zle [THEN iff_sym])
+ apply (simp add: int_0_less_mult_iff)
+txt{*main argument*}
+apply (subst posDivAlg_eqn)
+apply (simp_all (no_asm_simp))
+apply (erule splitE)
+apply (rule posDivAlg_type)
+apply (simp_all add: int_0_less_mult_iff)
+apply (auto simp add: zadd_zmult_distrib2 Let_def)
+txt{*now just linear arithmetic*}
+apply (simp add: not_zle_iff_zless zdiff_zless_iff)
+done
+
+
+subsection{*Correctness of negDivAlg, the division algorithm for a<0 and b>0*}
+
+lemma negDivAlg_termination:
+     "[| #0 $< b; a $+ b $< #0 |] 
+      ==> nat_of($- a $- #2 $* b) < nat_of($- a $- b)"
+apply (simp (no_asm) add: zless_nat_conj)
+apply (simp add: zcompare_rls not_zle_iff_zless zless_zdiff_iff [THEN iff_sym]
+                 zless_zminus)
+done
+
+lemmas negDivAlg_unfold = def_wfrec [OF negDivAlg_def wf_measure]
+
+lemma negDivAlg_eqn:
+     "[| #0 $< b; a : int; b : int |] ==>  
+      negDivAlg(<a,b>) =       
+       (if #0 $<= a$+b then <#-1,a$+b>  
+                       else adjust(b, negDivAlg (<a, #2$*b>)))"
+apply (rule negDivAlg_unfold [THEN trans])
+apply (simp (no_asm_simp) add: vimage_iff not_zless_iff_zle [THEN iff_sym])
+apply (blast intro: negDivAlg_termination)
+done
+
+lemma negDivAlg_induct_lemma [rule_format]:
+  assumes prem:
+        "!!a b. [| a \<in> int; b \<in> int;  
+                   ~ (#0 $<= a $+ b | b $<= #0) --> P(<a, #2 $* b>) |]  
+                ==> P(<a,b>)"
+  shows "<u,v> \<in> int*int --> P(<u,v>)"
+apply (rule_tac a = "<u,v>" in wf_induct)
+apply (rule_tac A = "int*int" and f = "%<a,b>.nat_of ($- a $- b)" 
+       in wf_measure)
+apply clarify
+apply (rule prem)
+apply (drule_tac [3] x = "<xa, #2 $\<times> y>" in spec)
+apply auto
+apply (simp add: not_zle_iff_zless negDivAlg_termination)
+done
+
+lemma negDivAlg_induct [consumes 2]:
+  assumes u_int: "u \<in> int"
+      and v_int: "v \<in> int"
+      and ih: "!!a b. [| a \<in> int; b \<in> int;  
+                         ~ (#0 $<= a $+ b | b $<= #0) --> P(a, #2 $* b) |]  
+                      ==> P(a,b)"
+  shows "P(u,v)"
+apply (subgoal_tac " (%<x,y>. P (x,y)) (<u,v>)")
+apply simp
+apply (rule negDivAlg_induct_lemma)
+apply (simp (no_asm_use))
+apply (rule ih)
+apply (auto simp add: u_int v_int)
+done
+
+
+(*Typechecking for negDivAlg*)
+lemma negDivAlg_type:
+     "[| a \<in> int; b \<in> int |] ==> negDivAlg(<a,b>) \<in> int * int"
+apply (rule_tac u = "a" and v = "b" in negDivAlg_induct)
+apply assumption+
+apply (case_tac "#0 $< ba")
+ apply (simp add: negDivAlg_eqn adjust_def integ_of_type 
+             split add: split_if_asm)
+ apply clarify
+ apply (simp add: int_0_less_mult_iff not_zle_iff_zless)
+apply (simp add: not_zless_iff_zle)
+apply (subst negDivAlg_unfold)
+apply simp
+done
+
+
+(*Correctness of negDivAlg: it computes quotients correctly
+  It doesn't work if a=0 because the 0/b=0 rather than -1*)
+lemma negDivAlg_correct [rule_format]:
+     "[| a \<in> int; b \<in> int |]  
+      ==> a $< #0 --> #0 $< b --> quorem (<a,b>, negDivAlg(<a,b>))"
+apply (rule_tac u = "a" and v = "b" in negDivAlg_induct)
+  apply auto
+   apply (simp_all add: quorem_def)
+   txt{*base case: @{term "0$<=a$+b"}*}
+   apply (simp add: negDivAlg_eqn)
+  apply (simp add: not_zless_iff_zle [THEN iff_sym])
+ apply (simp add: int_0_less_mult_iff)
+txt{*main argument*}
+apply (subst negDivAlg_eqn)
+apply (simp_all (no_asm_simp))
+apply (erule splitE)
+apply (rule negDivAlg_type)
+apply (simp_all add: int_0_less_mult_iff)
+apply (auto simp add: zadd_zmult_distrib2 Let_def)
+txt{*now just linear arithmetic*}
+apply (simp add: not_zle_iff_zless zdiff_zless_iff)
+done
+
+
+subsection{* Existence shown by proving the division algorithm to be correct *}
+
+(*the case a=0*)
+lemma quorem_0: "[|b \<noteq> #0;  b \<in> int|] ==> quorem (<#0,b>, <#0,#0>)"
+by (force simp add: quorem_def neq_iff_zless)
+
+lemma posDivAlg_zero_divisor: "posDivAlg(<a,#0>) = <#0,a>"
+apply (subst posDivAlg_unfold)
+apply simp
+done
+
+lemma posDivAlg_0 [simp]: "posDivAlg (<#0,b>) = <#0,#0>"
+apply (subst posDivAlg_unfold)
+apply (simp add: not_zle_iff_zless)
+done
+
+
+(*Needed below.  Actually it's an equivalence.*)
+lemma linear_arith_lemma: "~ (#0 $<= #-1 $+ b) ==> (b $<= #0)"
+apply (simp add: not_zle_iff_zless)
+apply (drule zminus_zless_zminus [THEN iffD2])
+apply (simp add: zadd_commute zless_add1_iff_zle zle_zminus)
+done
+
+lemma negDivAlg_minus1 [simp]: "negDivAlg (<#-1,b>) = <#-1, b$-#1>"
+apply (subst negDivAlg_unfold)
+apply (simp add: linear_arith_lemma integ_of_type vimage_iff)
+done
+
+lemma negateSnd_eq [simp]: "negateSnd (<q,r>) = <q, $-r>"
+apply (unfold negateSnd_def)
+apply auto
+done
+
+lemma negateSnd_type: "qr \<in> int * int ==> negateSnd (qr) \<in> int * int"
+apply (unfold negateSnd_def)
+apply auto
+done
+
+lemma quorem_neg:
+     "[|quorem (<$-a,$-b>, qr);  a \<in> int;  b \<in> int;  qr \<in> int * int|]   
+      ==> quorem (<a,b>, negateSnd(qr))"
+apply clarify
+apply (auto elim: zless_asym simp add: quorem_def zless_zminus)
+txt{*linear arithmetic from here on*}
+apply (simp_all add: zminus_equation [of a] zminus_zless)
+apply (cut_tac [2] z = "b" and w = "#0" in zless_linear)
+apply (cut_tac [1] z = "b" and w = "#0" in zless_linear)
+apply auto
+apply (blast dest: zle_zless_trans)+
+done
+
+lemma divAlg_correct:
+     "[|b \<noteq> #0;  a \<in> int;  b \<in> int|] ==> quorem (<a,b>, divAlg(<a,b>))"
+apply (auto simp add: quorem_0 divAlg_def)
+apply (safe intro!: quorem_neg posDivAlg_correct negDivAlg_correct
+                    posDivAlg_type negDivAlg_type) 
+apply (auto simp add: quorem_def neq_iff_zless)
+txt{*linear arithmetic from here on*}
+apply (auto simp add: zle_def)
+done
+
+lemma divAlg_type: "[|a \<in> int;  b \<in> int|] ==> divAlg(<a,b>) \<in> int * int"
+apply (auto simp add: divAlg_def)
+apply (auto simp add: posDivAlg_type negDivAlg_type negateSnd_type)
+done
+
+
+(** intify cancellation **)
+
+lemma zdiv_intify1 [simp]: "intify(x) zdiv y = x zdiv y"
+apply (simp (no_asm) add: zdiv_def)
+done
+
+lemma zdiv_intify2 [simp]: "x zdiv intify(y) = x zdiv y"
+apply (simp (no_asm) add: zdiv_def)
+done
+
+lemma zdiv_type [iff,TC]: "z zdiv w \<in> int"
+apply (unfold zdiv_def)
+apply (blast intro: fst_type divAlg_type)
+done
+
+lemma zmod_intify1 [simp]: "intify(x) zmod y = x zmod y"
+apply (simp (no_asm) add: zmod_def)
+done
+
+lemma zmod_intify2 [simp]: "x zmod intify(y) = x zmod y"
+apply (simp (no_asm) add: zmod_def)
+done
+
+lemma zmod_type [iff,TC]: "z zmod w \<in> int"
+apply (unfold zmod_def)
+apply (rule snd_type)
+apply (blast intro: divAlg_type)
+done
+
+
+(** Arbitrary definitions for division by zero.  Useful to simplify 
+    certain equations **)
+
+lemma DIVISION_BY_ZERO_ZDIV: "a zdiv #0 = #0"
+apply (simp (no_asm) add: zdiv_def divAlg_def posDivAlg_zero_divisor)
+done  (*NOT for adding to default simpset*)
+
+lemma DIVISION_BY_ZERO_ZMOD: "a zmod #0 = intify(a)"
+apply (simp (no_asm) add: zmod_def divAlg_def posDivAlg_zero_divisor)
+done  (*NOT for adding to default simpset*)
+
+
+
+(** Basic laws about division and remainder **)
+
+lemma raw_zmod_zdiv_equality:
+     "[| a \<in> int; b \<in> int |] ==> a = b $* (a zdiv b) $+ (a zmod b)"
+apply (case_tac "b = #0")
+ apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD) 
+apply (cut_tac a = "a" and b = "b" in divAlg_correct)
+apply (auto simp add: quorem_def zdiv_def zmod_def split_def)
+done
+
+lemma zmod_zdiv_equality: "intify(a) = b $* (a zdiv b) $+ (a zmod b)"
+apply (rule trans)
+apply (rule_tac b = "intify (b)" in raw_zmod_zdiv_equality)
+apply auto
+done
+
+lemma pos_mod: "#0 $< b ==> #0 $<= a zmod b & a zmod b $< b"
+apply (cut_tac a = "intify (a)" and b = "intify (b)" in divAlg_correct)
+apply (auto simp add: intify_eq_0_iff_zle quorem_def zmod_def split_def)
+apply (blast dest: zle_zless_trans)+
+done
+
+lemmas pos_mod_sign = pos_mod [THEN conjunct1, standard]
+and    pos_mod_bound = pos_mod [THEN conjunct2, standard]
+
+lemma neg_mod: "b $< #0 ==> a zmod b $<= #0 & b $< a zmod b"
+apply (cut_tac a = "intify (a)" and b = "intify (b)" in divAlg_correct)
+apply (auto simp add: intify_eq_0_iff_zle quorem_def zmod_def split_def)
+apply (blast dest: zle_zless_trans)
+apply (blast dest: zless_trans)+
+done
+
+lemmas neg_mod_sign = neg_mod [THEN conjunct1, standard]
+and    neg_mod_bound = neg_mod [THEN conjunct2, standard]
+
+
+(** proving general properties of zdiv and zmod **)
+
+lemma quorem_div_mod:
+     "[|b \<noteq> #0;  a \<in> int;  b \<in> int |]  
+      ==> quorem (<a,b>, <a zdiv b, a zmod b>)"
+apply (cut_tac a = "a" and b = "b" in zmod_zdiv_equality)
+apply (auto simp add: quorem_def neq_iff_zless pos_mod_sign pos_mod_bound 
+                      neg_mod_sign neg_mod_bound)
+done
+
+(*Surely quorem(<a,b>,<q,r>) implies a \<in> int, but it doesn't matter*)
+lemma quorem_div:
+     "[| quorem(<a,b>,<q,r>);  b \<noteq> #0;  a \<in> int;  b \<in> int;  q \<in> int |]  
+      ==> a zdiv b = q"
+by (blast intro: quorem_div_mod [THEN unique_quotient])
+
+lemma quorem_mod:
+     "[| quorem(<a,b>,<q,r>); b \<noteq> #0; a \<in> int; b \<in> int; q \<in> int; r \<in> int |] 
+      ==> a zmod b = r"
+by (blast intro: quorem_div_mod [THEN unique_remainder])
+
+lemma zdiv_pos_pos_trivial_raw:
+     "[| a \<in> int;  b \<in> int;  #0 $<= a;  a $< b |] ==> a zdiv b = #0"
+apply (rule quorem_div)
+apply (auto simp add: quorem_def)
+(*linear arithmetic*)
+apply (blast dest: zle_zless_trans)+
+done
+
+lemma zdiv_pos_pos_trivial: "[| #0 $<= a;  a $< b |] ==> a zdiv b = #0"
+apply (cut_tac a = "intify (a)" and b = "intify (b)" 
+       in zdiv_pos_pos_trivial_raw)
+apply auto
+done
+
+lemma zdiv_neg_neg_trivial_raw:
+     "[| a \<in> int;  b \<in> int;  a $<= #0;  b $< a |] ==> a zdiv b = #0"
+apply (rule_tac r = "a" in quorem_div)
+apply (auto simp add: quorem_def)
+(*linear arithmetic*)
+apply (blast dest: zle_zless_trans zless_trans)+
+done
+
+lemma zdiv_neg_neg_trivial: "[| a $<= #0;  b $< a |] ==> a zdiv b = #0"
+apply (cut_tac a = "intify (a)" and b = "intify (b)" 
+       in zdiv_neg_neg_trivial_raw)
+apply auto
+done
+
+lemma zadd_le_0_lemma: "[| a$+b $<= #0;  #0 $< a;  #0 $< b |] ==> False"
+apply (drule_tac z' = "#0" and z = "b" in zadd_zless_mono)
+apply (auto simp add: zle_def)
+apply (blast dest: zless_trans)
+done
+
+lemma zdiv_pos_neg_trivial_raw:
+     "[| a \<in> int;  b \<in> int;  #0 $< a;  a$+b $<= #0 |] ==> a zdiv b = #-1"
+apply (rule_tac r = "a $+ b" in quorem_div)
+apply (auto simp add: quorem_def)
+(*linear arithmetic*)
+apply (blast dest: zadd_le_0_lemma zle_zless_trans)+
+done
+
+lemma zdiv_pos_neg_trivial: "[| #0 $< a;  a$+b $<= #0 |] ==> a zdiv b = #-1"
+apply (cut_tac a = "intify (a)" and b = "intify (b)" 
+       in zdiv_pos_neg_trivial_raw)
+apply auto
+done
+
+(*There is no zdiv_neg_pos_trivial because  #0 zdiv b = #0 would supersede it*)
+
+
+lemma zmod_pos_pos_trivial_raw:
+     "[| a \<in> int;  b \<in> int;  #0 $<= a;  a $< b |] ==> a zmod b = a"
+apply (rule_tac q = "#0" in quorem_mod)
+apply (auto simp add: quorem_def)
+(*linear arithmetic*)
+apply (blast dest: zle_zless_trans)+
+done
+
+lemma zmod_pos_pos_trivial: "[| #0 $<= a;  a $< b |] ==> a zmod b = intify(a)"
+apply (cut_tac a = "intify (a)" and b = "intify (b)" 
+       in zmod_pos_pos_trivial_raw)
+apply auto
+done
+
+lemma zmod_neg_neg_trivial_raw:
+     "[| a \<in> int;  b \<in> int;  a $<= #0;  b $< a |] ==> a zmod b = a"
+apply (rule_tac q = "#0" in quorem_mod)
+apply (auto simp add: quorem_def)
+(*linear arithmetic*)
+apply (blast dest: zle_zless_trans zless_trans)+
+done
+
+lemma zmod_neg_neg_trivial: "[| a $<= #0;  b $< a |] ==> a zmod b = intify(a)"
+apply (cut_tac a = "intify (a)" and b = "intify (b)" 
+       in zmod_neg_neg_trivial_raw)
+apply auto
+done
+
+lemma zmod_pos_neg_trivial_raw:
+     "[| a \<in> int;  b \<in> int;  #0 $< a;  a$+b $<= #0 |] ==> a zmod b = a$+b"
+apply (rule_tac q = "#-1" in quorem_mod)
+apply (auto simp add: quorem_def)
+(*linear arithmetic*)
+apply (blast dest: zadd_le_0_lemma zle_zless_trans)+
+done
+
+lemma zmod_pos_neg_trivial: "[| #0 $< a;  a$+b $<= #0 |] ==> a zmod b = a$+b"
+apply (cut_tac a = "intify (a)" and b = "intify (b)" 
+       in zmod_pos_neg_trivial_raw)
+apply auto
+done
+
+(*There is no zmod_neg_pos_trivial...*)
+
+
+(*Simpler laws such as -a zdiv b = -(a zdiv b) FAIL*)
+
+lemma zdiv_zminus_zminus_raw:
+     "[|a \<in> int;  b \<in> int|] ==> ($-a) zdiv ($-b) = a zdiv b"
+apply (case_tac "b = #0")
+ apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD) 
+apply (subst quorem_div_mod [THEN quorem_neg, simplified, THEN quorem_div])
+apply auto
+done
+
+lemma zdiv_zminus_zminus [simp]: "($-a) zdiv ($-b) = a zdiv b"
+apply (cut_tac a = "intify (a)" and b = "intify (b)" in zdiv_zminus_zminus_raw)
+apply auto
+done
+
+(*Simpler laws such as -a zmod b = -(a zmod b) FAIL*)
+lemma zmod_zminus_zminus_raw:
+     "[|a \<in> int;  b \<in> int|] ==> ($-a) zmod ($-b) = $- (a zmod b)"
+apply (case_tac "b = #0")
+ apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD) 
+apply (subst quorem_div_mod [THEN quorem_neg, simplified, THEN quorem_mod])
+apply auto
+done
+
+lemma zmod_zminus_zminus [simp]: "($-a) zmod ($-b) = $- (a zmod b)"
+apply (cut_tac a = "intify (a)" and b = "intify (b)" in zmod_zminus_zminus_raw)
+apply auto
+done
+
+
+subsection{* division of a number by itself *}
+
+lemma self_quotient_aux1: "[| #0 $< a; a = r $+ a$*q; r $< a |] ==> #1 $<= q"
+apply (subgoal_tac "#0 $< a$*q")
+apply (cut_tac w = "#0" and z = "q" in add1_zle_iff)
+apply (simp add: int_0_less_mult_iff)
+apply (blast dest: zless_trans)
+(*linear arithmetic...*)
+apply (drule_tac t = "%x. x $- r" in subst_context)
+apply (drule sym)
+apply (simp add: zcompare_rls)
+done
+
+lemma self_quotient_aux2: "[| #0 $< a; a = r $+ a$*q; #0 $<= r |] ==> q $<= #1"
+apply (subgoal_tac "#0 $<= a$* (#1$-q)")
+ apply (simp add: int_0_le_mult_iff zcompare_rls)
+ apply (blast dest: zle_zless_trans)
+apply (simp add: zdiff_zmult_distrib2)
+apply (drule_tac t = "%x. x $- a $* q" in subst_context)
+apply (simp add: zcompare_rls)
+done
+
+lemma self_quotient:
+     "[| quorem(<a,a>,<q,r>);  a \<in> int;  q \<in> int;  a \<noteq> #0|] ==> q = #1"
+apply (simp add: split_ifs quorem_def neq_iff_zless)
+apply (rule zle_anti_sym)
+apply safe
+apply auto
+prefer 4 apply (blast dest: zless_trans)
+apply (blast dest: zless_trans)
+apply (rule_tac [3] a = "$-a" and r = "$-r" in self_quotient_aux1)
+apply (rule_tac a = "$-a" and r = "$-r" in self_quotient_aux2)
+apply (rule_tac [6] zminus_equation [THEN iffD1])
+apply (rule_tac [2] zminus_equation [THEN iffD1])
+apply (force intro: self_quotient_aux1 self_quotient_aux2
+  simp add: zadd_commute zmult_zminus)+
+done
+
+lemma self_remainder:
+     "[|quorem(<a,a>,<q,r>); a \<in> int; q \<in> int; r \<in> int; a \<noteq> #0|] ==> r = #0"
+apply (frule self_quotient)
+apply (auto simp add: quorem_def)
+done
+
+lemma zdiv_self_raw: "[|a \<noteq> #0; a \<in> int|] ==> a zdiv a = #1"
+apply (blast intro: quorem_div_mod [THEN self_quotient])
+done
+
+lemma zdiv_self [simp]: "intify(a) \<noteq> #0 ==> a zdiv a = #1"
+apply (drule zdiv_self_raw)
+apply auto
+done
+
+(*Here we have 0 zmod 0 = 0, also assumed by Knuth (who puts m zmod 0 = 0) *)
+lemma zmod_self_raw: "a \<in> int ==> a zmod a = #0"
+apply (case_tac "a = #0")
+ apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD) 
+apply (blast intro: quorem_div_mod [THEN self_remainder])
+done
+
+lemma zmod_self [simp]: "a zmod a = #0"
+apply (cut_tac a = "intify (a)" in zmod_self_raw)
+apply auto
+done
+
+
+subsection{* Computation of division and remainder *}
+
+lemma zdiv_zero [simp]: "#0 zdiv b = #0"
+apply (simp (no_asm) add: zdiv_def divAlg_def)
+done
+
+lemma zdiv_eq_minus1: "#0 $< b ==> #-1 zdiv b = #-1"
+apply (simp (no_asm_simp) add: zdiv_def divAlg_def)
+done
+
+lemma zmod_zero [simp]: "#0 zmod b = #0"
+apply (simp (no_asm) add: zmod_def divAlg_def)
+done
+
+lemma zdiv_minus1: "#0 $< b ==> #-1 zdiv b = #-1"
+apply (simp (no_asm_simp) add: zdiv_def divAlg_def)
+done
+
+lemma zmod_minus1: "#0 $< b ==> #-1 zmod b = b $- #1"
+apply (simp (no_asm_simp) add: zmod_def divAlg_def)
+done
+
+(** a positive, b positive **)
+
+lemma zdiv_pos_pos: "[| #0 $< a;  #0 $<= b |]  
+      ==> a zdiv b = fst (posDivAlg(<intify(a), intify(b)>))"
+apply (simp (no_asm_simp) add: zdiv_def divAlg_def)
+apply (auto simp add: zle_def)
+done
+
+lemma zmod_pos_pos:
+     "[| #0 $< a;  #0 $<= b |]  
+      ==> a zmod b = snd (posDivAlg(<intify(a), intify(b)>))"
+apply (simp (no_asm_simp) add: zmod_def divAlg_def)
+apply (auto simp add: zle_def)
+done
+
+(** a negative, b positive **)
+
+lemma zdiv_neg_pos:
+     "[| a $< #0;  #0 $< b |]  
+      ==> a zdiv b = fst (negDivAlg(<intify(a), intify(b)>))"
+apply (simp (no_asm_simp) add: zdiv_def divAlg_def)
+apply (blast dest: zle_zless_trans)
+done
+
+lemma zmod_neg_pos:
+     "[| a $< #0;  #0 $< b |]  
+      ==> a zmod b = snd (negDivAlg(<intify(a), intify(b)>))"
+apply (simp (no_asm_simp) add: zmod_def divAlg_def)
+apply (blast dest: zle_zless_trans)
+done
+
+(** a positive, b negative **)
+
+lemma zdiv_pos_neg:
+     "[| #0 $< a;  b $< #0 |]  
+      ==> a zdiv b = fst (negateSnd(negDivAlg (<$-a, $-b>)))"
+apply (simp (no_asm_simp) add: zdiv_def divAlg_def intify_eq_0_iff_zle)
+apply auto
+apply (blast dest: zle_zless_trans)+
+apply (blast dest: zless_trans)
+apply (blast intro: zless_imp_zle)
+done
+
+lemma zmod_pos_neg:
+     "[| #0 $< a;  b $< #0 |]  
+      ==> a zmod b = snd (negateSnd(negDivAlg (<$-a, $-b>)))"
+apply (simp (no_asm_simp) add: zmod_def divAlg_def intify_eq_0_iff_zle)
+apply auto
+apply (blast dest: zle_zless_trans)+
+apply (blast dest: zless_trans)
+apply (blast intro: zless_imp_zle)
+done
+
+(** a negative, b negative **)
+
+lemma zdiv_neg_neg:
+     "[| a $< #0;  b $<= #0 |]  
+      ==> a zdiv b = fst (negateSnd(posDivAlg(<$-a, $-b>)))"
+apply (simp (no_asm_simp) add: zdiv_def divAlg_def)
+apply auto
+apply (blast dest!: zle_zless_trans)+
+done
+
+lemma zmod_neg_neg:
+     "[| a $< #0;  b $<= #0 |]  
+      ==> a zmod b = snd (negateSnd(posDivAlg(<$-a, $-b>)))"
+apply (simp (no_asm_simp) add: zmod_def divAlg_def)
+apply auto
+apply (blast dest!: zle_zless_trans)+
+done
+
+declare zdiv_pos_pos [of "integ_of (v)" "integ_of (w)", standard, simp]
+declare zdiv_neg_pos [of "integ_of (v)" "integ_of (w)", standard, simp]
+declare zdiv_pos_neg [of "integ_of (v)" "integ_of (w)", standard, simp]
+declare zdiv_neg_neg [of "integ_of (v)" "integ_of (w)", standard, simp]
+declare zmod_pos_pos [of "integ_of (v)" "integ_of (w)", standard, simp]
+declare zmod_neg_pos [of "integ_of (v)" "integ_of (w)", standard, simp]
+declare zmod_pos_neg [of "integ_of (v)" "integ_of (w)", standard, simp]
+declare zmod_neg_neg [of "integ_of (v)" "integ_of (w)", standard, simp]
+declare posDivAlg_eqn [of concl: "integ_of (v)" "integ_of (w)", standard, simp]
+declare negDivAlg_eqn [of concl: "integ_of (v)" "integ_of (w)", standard, simp]
+
+
+(** Special-case simplification **)
+
+lemma zmod_1 [simp]: "a zmod #1 = #0"
+apply (cut_tac a = "a" and b = "#1" in pos_mod_sign)
+apply (cut_tac [2] a = "a" and b = "#1" in pos_mod_bound)
+apply auto
+(*arithmetic*)
+apply (drule add1_zle_iff [THEN iffD2])
+apply (rule zle_anti_sym)
+apply auto
+done
+
+lemma zdiv_1 [simp]: "a zdiv #1 = intify(a)"
+apply (cut_tac a = "a" and b = "#1" in zmod_zdiv_equality)
+apply auto
+done
+
+lemma zmod_minus1_right [simp]: "a zmod #-1 = #0"
+apply (cut_tac a = "a" and b = "#-1" in neg_mod_sign)
+apply (cut_tac [2] a = "a" and b = "#-1" in neg_mod_bound)
+apply auto
+(*arithmetic*)
+apply (drule add1_zle_iff [THEN iffD2])
+apply (rule zle_anti_sym)
+apply auto
+done
+
+lemma zdiv_minus1_right_raw: "a \<in> int ==> a zdiv #-1 = $-a"
+apply (cut_tac a = "a" and b = "#-1" in zmod_zdiv_equality)
+apply auto
+apply (rule equation_zminus [THEN iffD2])
+apply auto
+done
+
+lemma zdiv_minus1_right: "a zdiv #-1 = $-a"
+apply (cut_tac a = "intify (a)" in zdiv_minus1_right_raw)
+apply auto
+done
+declare zdiv_minus1_right [simp]
+
+
+subsection{* Monotonicity in the first argument (divisor) *}
+
+lemma zdiv_mono1: "[| a $<= a';  #0 $< b |] ==> a zdiv b $<= a' zdiv b"
+apply (cut_tac a = "a" and b = "b" in zmod_zdiv_equality)
+apply (cut_tac a = "a'" and b = "b" in zmod_zdiv_equality)
+apply (rule unique_quotient_lemma)
+apply (erule subst)
+apply (erule subst)
+apply (simp_all (no_asm_simp) add: pos_mod_sign pos_mod_bound)
+done
+
+lemma zdiv_mono1_neg: "[| a $<= a';  b $< #0 |] ==> a' zdiv b $<= a zdiv b"
+apply (cut_tac a = "a" and b = "b" in zmod_zdiv_equality)
+apply (cut_tac a = "a'" and b = "b" in zmod_zdiv_equality)
+apply (rule unique_quotient_lemma_neg)
+apply (erule subst)
+apply (erule subst)
+apply (simp_all (no_asm_simp) add: neg_mod_sign neg_mod_bound)
+done
+
+
+subsection{* Monotonicity in the second argument (dividend) *}
+
+lemma q_pos_lemma:
+     "[| #0 $<= b'$*q' $+ r'; r' $< b';  #0 $< b' |] ==> #0 $<= q'"
+apply (subgoal_tac "#0 $< b'$* (q' $+ #1)")
+ apply (simp add: int_0_less_mult_iff)
+ apply (blast dest: zless_trans intro: zless_add1_iff_zle [THEN iffD1])
+apply (simp add: zadd_zmult_distrib2)
+apply (erule zle_zless_trans)
+apply (erule zadd_zless_mono2)
+done
+
+lemma zdiv_mono2_lemma:
+     "[| b$*q $+ r = b'$*q' $+ r';  #0 $<= b'$*q' $+ r';   
+         r' $< b';  #0 $<= r;  #0 $< b';  b' $<= b |]   
+      ==> q $<= q'"
+apply (frule q_pos_lemma, assumption+) 
+apply (subgoal_tac "b$*q $< b$* (q' $+ #1)")
+ apply (simp add: zmult_zless_cancel1)
+ apply (force dest: zless_add1_iff_zle [THEN iffD1] zless_trans zless_zle_trans)
+apply (subgoal_tac "b$*q = r' $- r $+ b'$*q'")
+ prefer 2 apply (simp add: zcompare_rls)
+apply (simp (no_asm_simp) add: zadd_zmult_distrib2)
+apply (subst zadd_commute [of "b $\<times> q'"], rule zadd_zless_mono)
+ prefer 2 apply (blast intro: zmult_zle_mono1)
+apply (subgoal_tac "r' $+ #0 $< b $+ r")
+ apply (simp add: zcompare_rls)
+apply (rule zadd_zless_mono)
+ apply auto
+apply (blast dest: zless_zle_trans)
+done
+
+
+lemma zdiv_mono2_raw:
+     "[| #0 $<= a;  #0 $< b';  b' $<= b;  a \<in> int |]   
+      ==> a zdiv b $<= a zdiv b'"
+apply (subgoal_tac "#0 $< b")
+ prefer 2 apply (blast dest: zless_zle_trans)
+apply (cut_tac a = "a" and b = "b" in zmod_zdiv_equality)
+apply (cut_tac a = "a" and b = "b'" in zmod_zdiv_equality)
+apply (rule zdiv_mono2_lemma)
+apply (erule subst)
+apply (erule subst)
+apply (simp_all add: pos_mod_sign pos_mod_bound)
+done
+
+lemma zdiv_mono2:
+     "[| #0 $<= a;  #0 $< b';  b' $<= b |]   
+      ==> a zdiv b $<= a zdiv b'"
+apply (cut_tac a = "intify (a)" in zdiv_mono2_raw)
+apply auto
+done
+
+lemma q_neg_lemma:
+     "[| b'$*q' $+ r' $< #0;  #0 $<= r';  #0 $< b' |] ==> q' $< #0"
+apply (subgoal_tac "b'$*q' $< #0")
+ prefer 2 apply (force intro: zle_zless_trans)
+apply (simp add: zmult_less_0_iff)
+apply (blast dest: zless_trans)
+done
+
+
+
+lemma zdiv_mono2_neg_lemma:
+     "[| b$*q $+ r = b'$*q' $+ r';  b'$*q' $+ r' $< #0;   
+         r $< b;  #0 $<= r';  #0 $< b';  b' $<= b |]   
+      ==> q' $<= q"
+apply (subgoal_tac "#0 $< b")
+ prefer 2 apply (blast dest: zless_zle_trans)
+apply (frule q_neg_lemma, assumption+) 
+apply (subgoal_tac "b$*q' $< b$* (q $+ #1)")
+ apply (simp add: zmult_zless_cancel1)
+ apply (blast dest: zless_trans zless_add1_iff_zle [THEN iffD1])
+apply (simp (no_asm_simp) add: zadd_zmult_distrib2)
+apply (subgoal_tac "b$*q' $<= b'$*q'")
+ prefer 2
+ apply (simp add: zmult_zle_cancel2)
+ apply (blast dest: zless_trans)
+apply (subgoal_tac "b'$*q' $+ r $< b $+ (b$*q $+ r)")
+ prefer 2
+ apply (erule ssubst)
+ apply simp
+ apply (drule_tac w' = "r" and z' = "#0" in zadd_zless_mono)
+  apply (assumption)
+ apply simp
+apply (simp (no_asm_use) add: zadd_commute)
+apply (rule zle_zless_trans)
+ prefer 2 apply (assumption)
+apply (simp (no_asm_simp) add: zmult_zle_cancel2)
+apply (blast dest: zless_trans)
+done
+
+lemma zdiv_mono2_neg_raw:
+     "[| a $< #0;  #0 $< b';  b' $<= b;  a \<in> int |]   
+      ==> a zdiv b' $<= a zdiv b"
+apply (subgoal_tac "#0 $< b")
+ prefer 2 apply (blast dest: zless_zle_trans)
+apply (cut_tac a = "a" and b = "b" in zmod_zdiv_equality)
+apply (cut_tac a = "a" and b = "b'" in zmod_zdiv_equality)
+apply (rule zdiv_mono2_neg_lemma)
+apply (erule subst)
+apply (erule subst)
+apply (simp_all add: pos_mod_sign pos_mod_bound)
+done
+
+lemma zdiv_mono2_neg: "[| a $< #0;  #0 $< b';  b' $<= b |]   
+      ==> a zdiv b' $<= a zdiv b"
+apply (cut_tac a = "intify (a)" in zdiv_mono2_neg_raw)
+apply auto
+done
+
+
+
+subsection{* More algebraic laws for zdiv and zmod *}
+
+(** proving (a*b) zdiv c = a $* (b zdiv c) $+ a * (b zmod c) **)
+
+lemma zmult1_lemma:
+     "[| quorem(<b,c>, <q,r>);  c \<in> int;  c \<noteq> #0 |]  
+      ==> quorem (<a$*b, c>, <a$*q $+ (a$*r) zdiv c, (a$*r) zmod c>)"
+apply (auto simp add: split_ifs quorem_def neq_iff_zless zadd_zmult_distrib2
+                      pos_mod_sign pos_mod_bound neg_mod_sign neg_mod_bound)
+apply (auto intro: raw_zmod_zdiv_equality) 
+done
+
+lemma zdiv_zmult1_eq_raw:
+     "[|b \<in> int;  c \<in> int|]  
+      ==> (a$*b) zdiv c = a$*(b zdiv c) $+ a$*(b zmod c) zdiv c"
+apply (case_tac "c = #0")
+ apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD) 
+apply (rule quorem_div_mod [THEN zmult1_lemma, THEN quorem_div])
+apply auto
+done
+
+lemma zdiv_zmult1_eq: "(a$*b) zdiv c = a$*(b zdiv c) $+ a$*(b zmod c) zdiv c"
+apply (cut_tac b = "intify (b)" and c = "intify (c)" in zdiv_zmult1_eq_raw)
+apply auto
+done
+
+lemma zmod_zmult1_eq_raw:
+     "[|b \<in> int;  c \<in> int|] ==> (a$*b) zmod c = a$*(b zmod c) zmod c"
+apply (case_tac "c = #0")
+ apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD) 
+apply (rule quorem_div_mod [THEN zmult1_lemma, THEN quorem_mod])
+apply auto
+done
+
+lemma zmod_zmult1_eq: "(a$*b) zmod c = a$*(b zmod c) zmod c"
+apply (cut_tac b = "intify (b)" and c = "intify (c)" in zmod_zmult1_eq_raw)
+apply auto
+done
+
+lemma zmod_zmult1_eq': "(a$*b) zmod c = ((a zmod c) $* b) zmod c"
+apply (rule trans)
+apply (rule_tac b = " (b $* a) zmod c" in trans)
+apply (rule_tac [2] zmod_zmult1_eq)
+apply (simp_all (no_asm) add: zmult_commute)
+done
+
+lemma zmod_zmult_distrib: "(a$*b) zmod c = ((a zmod c) $* (b zmod c)) zmod c"
+apply (rule zmod_zmult1_eq' [THEN trans])
+apply (rule zmod_zmult1_eq)
+done
+
+lemma zdiv_zmult_self1 [simp]: "intify(b) \<noteq> #0 ==> (a$*b) zdiv b = intify(a)"
+apply (simp (no_asm_simp) add: zdiv_zmult1_eq)
+done
+
+lemma zdiv_zmult_self2 [simp]: "intify(b) \<noteq> #0 ==> (b$*a) zdiv b = intify(a)"
+apply (subst zmult_commute , erule zdiv_zmult_self1)
+done
+
+lemma zmod_zmult_self1 [simp]: "(a$*b) zmod b = #0"
+apply (simp (no_asm) add: zmod_zmult1_eq)
+done
+
+lemma zmod_zmult_self2 [simp]: "(b$*a) zmod b = #0"
+apply (simp (no_asm) add: zmult_commute zmod_zmult1_eq)
+done
+
+
+(** proving (a$+b) zdiv c = 
+            a zdiv c $+ b zdiv c $+ ((a zmod c $+ b zmod c) zdiv c) **)
+
+lemma zadd1_lemma:
+     "[| quorem(<a,c>, <aq,ar>);  quorem(<b,c>, <bq,br>);   
+         c \<in> int;  c \<noteq> #0 |]  
+      ==> quorem (<a$+b, c>, <aq $+ bq $+ (ar$+br) zdiv c, (ar$+br) zmod c>)"
+apply (auto simp add: split_ifs quorem_def neq_iff_zless zadd_zmult_distrib2
+                      pos_mod_sign pos_mod_bound neg_mod_sign neg_mod_bound)
+apply (auto intro: raw_zmod_zdiv_equality)
+done
+
+(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
+lemma zdiv_zadd1_eq_raw:
+     "[|a \<in> int; b \<in> int; c \<in> int|] ==>  
+      (a$+b) zdiv c = a zdiv c $+ b zdiv c $+ ((a zmod c $+ b zmod c) zdiv c)"
+apply (case_tac "c = #0")
+ apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD) 
+apply (blast intro: zadd1_lemma [OF quorem_div_mod quorem_div_mod,
+                                 THEN quorem_div])
+done
+
+lemma zdiv_zadd1_eq:
+     "(a$+b) zdiv c = a zdiv c $+ b zdiv c $+ ((a zmod c $+ b zmod c) zdiv c)"
+apply (cut_tac a = "intify (a)" and b = "intify (b)" and c = "intify (c)" 
+       in zdiv_zadd1_eq_raw)
+apply auto
+done
+
+lemma zmod_zadd1_eq_raw:
+     "[|a \<in> int; b \<in> int; c \<in> int|]   
+      ==> (a$+b) zmod c = (a zmod c $+ b zmod c) zmod c"
+apply (case_tac "c = #0")
+ apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD) 
+apply (blast intro: zadd1_lemma [OF quorem_div_mod quorem_div_mod, 
+                                 THEN quorem_mod])
+done
+
+lemma zmod_zadd1_eq: "(a$+b) zmod c = (a zmod c $+ b zmod c) zmod c"
+apply (cut_tac a = "intify (a)" and b = "intify (b)" and c = "intify (c)" 
+       in zmod_zadd1_eq_raw)
+apply auto
+done
+
+lemma zmod_div_trivial_raw:
+     "[|a \<in> int; b \<in> int|] ==> (a zmod b) zdiv b = #0"
+apply (case_tac "b = #0")
+ apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD) 
+apply (auto simp add: neq_iff_zless pos_mod_sign pos_mod_bound
+         zdiv_pos_pos_trivial neg_mod_sign neg_mod_bound zdiv_neg_neg_trivial)
+done
+
+lemma zmod_div_trivial [simp]: "(a zmod b) zdiv b = #0"
+apply (cut_tac a = "intify (a)" and b = "intify (b)" in zmod_div_trivial_raw)
+apply auto
+done
+
+lemma zmod_mod_trivial_raw:
+     "[|a \<in> int; b \<in> int|] ==> (a zmod b) zmod b = a zmod b"
+apply (case_tac "b = #0")
+ apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD) 
+apply (auto simp add: neq_iff_zless pos_mod_sign pos_mod_bound 
+       zmod_pos_pos_trivial neg_mod_sign neg_mod_bound zmod_neg_neg_trivial)
+done
+
+lemma zmod_mod_trivial [simp]: "(a zmod b) zmod b = a zmod b"
+apply (cut_tac a = "intify (a)" and b = "intify (b)" in zmod_mod_trivial_raw)
+apply auto
+done
+
+lemma zmod_zadd_left_eq: "(a$+b) zmod c = ((a zmod c) $+ b) zmod c"
+apply (rule trans [symmetric])
+apply (rule zmod_zadd1_eq)
+apply (simp (no_asm))
+apply (rule zmod_zadd1_eq [symmetric])
+done
+
+lemma zmod_zadd_right_eq: "(a$+b) zmod c = (a $+ (b zmod c)) zmod c"
+apply (rule trans [symmetric])
+apply (rule zmod_zadd1_eq)
+apply (simp (no_asm))
+apply (rule zmod_zadd1_eq [symmetric])
+done
+
+
+lemma zdiv_zadd_self1 [simp]:
+     "intify(a) \<noteq> #0 ==> (a$+b) zdiv a = b zdiv a $+ #1"
+by (simp (no_asm_simp) add: zdiv_zadd1_eq)
+
+lemma zdiv_zadd_self2 [simp]:
+     "intify(a) \<noteq> #0 ==> (b$+a) zdiv a = b zdiv a $+ #1"
+by (simp (no_asm_simp) add: zdiv_zadd1_eq)
+
+lemma zmod_zadd_self1 [simp]: "(a$+b) zmod a = b zmod a"
+apply (case_tac "a = #0")
+ apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD) 
+apply (simp (no_asm_simp) add: zmod_zadd1_eq)
+done
+
+lemma zmod_zadd_self2 [simp]: "(b$+a) zmod a = b zmod a"
+apply (case_tac "a = #0")
+ apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD) 
+apply (simp (no_asm_simp) add: zmod_zadd1_eq)
+done
+
+
+subsection{* proving  a zdiv (b*c) = (a zdiv b) zdiv c *}
+
+(*The condition c>0 seems necessary.  Consider that 7 zdiv ~6 = ~2 but
+  7 zdiv 2 zdiv ~3 = 3 zdiv ~3 = ~1.  The subcase (a zdiv b) zmod c = 0 seems
+  to cause particular problems.*)
+
+(** first, four lemmas to bound the remainder for the cases b<0 and b>0 **)
+
+lemma zdiv_zmult2_aux1:
+     "[| #0 $< c;  b $< r;  r $<= #0 |] ==> b$*c $< b$*(q zmod c) $+ r"
+apply (subgoal_tac "b $* (c $- q zmod c) $< r $* #1")
+apply (simp add: zdiff_zmult_distrib2 zadd_commute zcompare_rls)
+apply (rule zle_zless_trans)
+apply (erule_tac [2] zmult_zless_mono1)
+apply (rule zmult_zle_mono2_neg)
+apply (auto simp add: zcompare_rls zadd_commute add1_zle_iff pos_mod_bound)
+apply (blast intro: zless_imp_zle dest: zless_zle_trans)
+done
+
+lemma zdiv_zmult2_aux2:
+     "[| #0 $< c;   b $< r;  r $<= #0 |] ==> b $* (q zmod c) $+ r $<= #0"
+apply (subgoal_tac "b $* (q zmod c) $<= #0")
+ prefer 2
+ apply (simp add: zmult_le_0_iff pos_mod_sign) 
+ apply (blast intro: zless_imp_zle dest: zless_zle_trans)
+(*arithmetic*)
+apply (drule zadd_zle_mono)
+apply assumption
+apply (simp add: zadd_commute)
+done
+
+lemma zdiv_zmult2_aux3:
+     "[| #0 $< c;  #0 $<= r;  r $< b |] ==> #0 $<= b $* (q zmod c) $+ r"
+apply (subgoal_tac "#0 $<= b $* (q zmod c)")
+ prefer 2
+ apply (simp add: int_0_le_mult_iff pos_mod_sign) 
+ apply (blast intro: zless_imp_zle dest: zle_zless_trans)
+(*arithmetic*)
+apply (drule zadd_zle_mono)
+apply assumption
+apply (simp add: zadd_commute)
+done
+
+lemma zdiv_zmult2_aux4:
+     "[| #0 $< c; #0 $<= r; r $< b |] ==> b $* (q zmod c) $+ r $< b $* c"
+apply (subgoal_tac "r $* #1 $< b $* (c $- q zmod c)")
+apply (simp add: zdiff_zmult_distrib2 zadd_commute zcompare_rls)
+apply (rule zless_zle_trans)
+apply (erule zmult_zless_mono1)
+apply (rule_tac [2] zmult_zle_mono2)
+apply (auto simp add: zcompare_rls zadd_commute add1_zle_iff pos_mod_bound)
+apply (blast intro: zless_imp_zle dest: zle_zless_trans)
+done
+
+lemma zdiv_zmult2_lemma:
+     "[| quorem (<a,b>, <q,r>);  a \<in> int;  b \<in> int;  b \<noteq> #0;  #0 $< c |]  
+      ==> quorem (<a,b$*c>, <q zdiv c, b$*(q zmod c) $+ r>)"
+apply (auto simp add: zmult_ac zmod_zdiv_equality [symmetric] quorem_def
+               neq_iff_zless int_0_less_mult_iff 
+               zadd_zmult_distrib2 [symmetric] zdiv_zmult2_aux1 zdiv_zmult2_aux2
+               zdiv_zmult2_aux3 zdiv_zmult2_aux4)
+apply (blast dest: zless_trans)+
+done
+
+lemma zdiv_zmult2_eq_raw:
+     "[|#0 $< c;  a \<in> int;  b \<in> int|] ==> a zdiv (b$*c) = (a zdiv b) zdiv c"
+apply (case_tac "b = #0")
+ apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD) 
+apply (rule quorem_div_mod [THEN zdiv_zmult2_lemma, THEN quorem_div])
+apply (auto simp add: intify_eq_0_iff_zle)
+apply (blast dest: zle_zless_trans)
+done
+
+lemma zdiv_zmult2_eq: "#0 $< c ==> a zdiv (b$*c) = (a zdiv b) zdiv c"
+apply (cut_tac a = "intify (a)" and b = "intify (b)" in zdiv_zmult2_eq_raw)
+apply auto
+done
+
+lemma zmod_zmult2_eq_raw:
+     "[|#0 $< c;  a \<in> int;  b \<in> int|]  
+      ==> a zmod (b$*c) = b$*(a zdiv b zmod c) $+ a zmod b"
+apply (case_tac "b = #0")
+ apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD) 
+apply (rule quorem_div_mod [THEN zdiv_zmult2_lemma, THEN quorem_mod])
+apply (auto simp add: intify_eq_0_iff_zle)
+apply (blast dest: zle_zless_trans)
+done
+
+lemma zmod_zmult2_eq:
+     "#0 $< c ==> a zmod (b$*c) = b$*(a zdiv b zmod c) $+ a zmod b"
+apply (cut_tac a = "intify (a)" and b = "intify (b)" in zmod_zmult2_eq_raw)
+apply auto
+done
+
+subsection{* Cancellation of common factors in "zdiv" *}
+
+lemma zdiv_zmult_zmult1_aux1:
+     "[| #0 $< b;  intify(c) \<noteq> #0 |] ==> (c$*a) zdiv (c$*b) = a zdiv b"
+apply (subst zdiv_zmult2_eq)
+apply auto
+done
+
+lemma zdiv_zmult_zmult1_aux2:
+     "[| b $< #0;  intify(c) \<noteq> #0 |] ==> (c$*a) zdiv (c$*b) = a zdiv b"
+apply (subgoal_tac " (c $* ($-a)) zdiv (c $* ($-b)) = ($-a) zdiv ($-b)")
+apply (rule_tac [2] zdiv_zmult_zmult1_aux1)
+apply auto
+done
+
+lemma zdiv_zmult_zmult1_raw:
+     "[|intify(c) \<noteq> #0; b \<in> int|] ==> (c$*a) zdiv (c$*b) = a zdiv b"
+apply (case_tac "b = #0")
+ apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD) 
+apply (auto simp add: neq_iff_zless [of b]
+  zdiv_zmult_zmult1_aux1 zdiv_zmult_zmult1_aux2)
+done
+
+lemma zdiv_zmult_zmult1: "intify(c) \<noteq> #0 ==> (c$*a) zdiv (c$*b) = a zdiv b"
+apply (cut_tac b = "intify (b)" in zdiv_zmult_zmult1_raw)
+apply auto
+done
+
+lemma zdiv_zmult_zmult2: "intify(c) \<noteq> #0 ==> (a$*c) zdiv (b$*c) = a zdiv b"
+apply (drule zdiv_zmult_zmult1)
+apply (auto simp add: zmult_commute)
+done
+
+
+subsection{* Distribution of factors over "zmod" *}
+
+lemma zmod_zmult_zmult1_aux1:
+     "[| #0 $< b;  intify(c) \<noteq> #0 |]  
+      ==> (c$*a) zmod (c$*b) = c $* (a zmod b)"
+apply (subst zmod_zmult2_eq)
+apply auto
+done
+
+lemma zmod_zmult_zmult1_aux2:
+     "[| b $< #0;  intify(c) \<noteq> #0 |]  
+      ==> (c$*a) zmod (c$*b) = c $* (a zmod b)"
+apply (subgoal_tac " (c $* ($-a)) zmod (c $* ($-b)) = c $* (($-a) zmod ($-b))")
+apply (rule_tac [2] zmod_zmult_zmult1_aux1)
+apply auto
+done
+
+lemma zmod_zmult_zmult1_raw:
+     "[|b \<in> int; c \<in> int|] ==> (c$*a) zmod (c$*b) = c $* (a zmod b)"
+apply (case_tac "b = #0")
+ apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD) 
+apply (case_tac "c = #0")
+ apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD) 
+apply (auto simp add: neq_iff_zless [of b]
+  zmod_zmult_zmult1_aux1 zmod_zmult_zmult1_aux2)
+done
+
+lemma zmod_zmult_zmult1: "(c$*a) zmod (c$*b) = c $* (a zmod b)"
+apply (cut_tac b = "intify (b)" and c = "intify (c)" in zmod_zmult_zmult1_raw)
+apply auto
+done
+
+lemma zmod_zmult_zmult2: "(a$*c) zmod (b$*c) = (a zmod b) $* c"
+apply (cut_tac c = "c" in zmod_zmult_zmult1)
+apply (auto simp add: zmult_commute)
+done
+
+
+(** Quotients of signs **)
+
+lemma zdiv_neg_pos_less0: "[| a $< #0;  #0 $< b |] ==> a zdiv b $< #0"
+apply (subgoal_tac "a zdiv b $<= #-1")
+apply (erule zle_zless_trans)
+apply (simp (no_asm))
+apply (rule zle_trans)
+apply (rule_tac a' = "#-1" in zdiv_mono1)
+apply (rule zless_add1_iff_zle [THEN iffD1])
+apply (simp (no_asm))
+apply (auto simp add: zdiv_minus1)
+done
+
+lemma zdiv_nonneg_neg_le0: "[| #0 $<= a;  b $< #0 |] ==> a zdiv b $<= #0"
+apply (drule zdiv_mono1_neg)
+apply auto
+done
+
+lemma pos_imp_zdiv_nonneg_iff: "#0 $< b ==> (#0 $<= a zdiv b) <-> (#0 $<= a)"
+apply auto
+apply (drule_tac [2] zdiv_mono1)
+apply (auto simp add: neq_iff_zless)
+apply (simp (no_asm_use) add: not_zless_iff_zle [THEN iff_sym])
+apply (blast intro: zdiv_neg_pos_less0)
+done
+
+lemma neg_imp_zdiv_nonneg_iff: "b $< #0 ==> (#0 $<= a zdiv b) <-> (a $<= #0)"
+apply (subst zdiv_zminus_zminus [symmetric])
+apply (rule iff_trans)
+apply (rule pos_imp_zdiv_nonneg_iff)
+apply auto
+done
+
+(*But not (a zdiv b $<= 0 iff a$<=0); consider a=1, b=2 when a zdiv b = 0.*)
+lemma pos_imp_zdiv_neg_iff: "#0 $< b ==> (a zdiv b $< #0) <-> (a $< #0)"
+apply (simp (no_asm_simp) add: not_zle_iff_zless [THEN iff_sym])
+apply (erule pos_imp_zdiv_nonneg_iff)
+done
+
+(*Again the law fails for $<=: consider a = -1, b = -2 when a zdiv b = 0*)
+lemma neg_imp_zdiv_neg_iff: "b $< #0 ==> (a zdiv b $< #0) <-> (#0 $< a)"
+apply (simp (no_asm_simp) add: not_zle_iff_zless [THEN iff_sym])
+apply (erule neg_imp_zdiv_nonneg_iff)
+done
+
+(*
+ THESE REMAIN TO BE CONVERTED -- but aren't that useful!
+
+ subsection{* Speeding up the division algorithm with shifting *}
+
+ (** computing "zdiv" by shifting **)
+
+ lemma pos_zdiv_mult_2: "#0 $<= a ==> (#1 $+ #2$*b) zdiv (#2$*a) = b zdiv a"
+ apply (case_tac "a = #0")
+ apply (subgoal_tac "#1 $<= a")
+  apply (arith_tac 2)
+ apply (subgoal_tac "#1 $< a $* #2")
+  apply (arith_tac 2)
+ apply (subgoal_tac "#2$* (#1 $+ b zmod a) $<= #2$*a")
+  apply (rule_tac [2] zmult_zle_mono2)
+ apply (auto simp add: zadd_commute zmult_commute add1_zle_iff pos_mod_bound)
+ apply (subst zdiv_zadd1_eq)
+ apply (simp (no_asm_simp) add: zdiv_zmult_zmult2 zmod_zmult_zmult2 zdiv_pos_pos_trivial)
+ apply (subst zdiv_pos_pos_trivial)
+ apply (simp (no_asm_simp) add: [zmod_pos_pos_trivial pos_mod_sign [THEN zadd_zle_mono1] RSN (2,zle_trans) ])
+ apply (auto simp add: zmod_pos_pos_trivial)
+ apply (subgoal_tac "#0 $<= b zmod a")
+  apply (asm_simp_tac (simpset () add: pos_mod_sign) 2)
+ apply arith
+ done
+
+
+ lemma neg_zdiv_mult_2: "a $<= #0 ==> (#1 $+ #2$*b) zdiv (#2$*a) <-> (b$+#1) zdiv a"
+ apply (subgoal_tac " (#1 $+ #2$* ($-b-#1)) zdiv (#2 $* ($-a)) <-> ($-b-#1) zdiv ($-a)")
+ apply (rule_tac [2] pos_zdiv_mult_2)
+ apply (auto simp add: zmult_zminus_right)
+ apply (subgoal_tac " (#-1 - (#2 $* b)) = - (#1 $+ (#2 $* b))")
+ apply (Simp_tac 2)
+ apply (asm_full_simp_tac (HOL_ss add: zdiv_zminus_zminus zdiff_def zminus_zadd_distrib [symmetric])
+ done
+
+
+ (*Not clear why this must be proved separately; probably integ_of causes
+   simplification problems*)
+ lemma lemma: "~ #0 $<= x ==> x $<= #0"
+ apply auto
+ done
+
+ lemma zdiv_integ_of_BIT: "integ_of (v BIT b) zdiv integ_of (w BIT False) =  
+           (if ~b | #0 $<= integ_of w                    
+            then integ_of v zdiv (integ_of w)     
+            else (integ_of v $+ #1) zdiv (integ_of w))"
+ apply (simp_tac (simpset_of Int.thy add: zadd_assoc integ_of_BIT)
+ apply (simp (no_asm_simp) del: bin_arith_extra_simps@bin_rel_simps add: zdiv_zmult_zmult1 pos_zdiv_mult_2 lemma neg_zdiv_mult_2)
+ done
+
+ declare zdiv_integ_of_BIT [simp]
+
+
+ (** computing "zmod" by shifting (proofs resemble those for "zdiv") **)
+
+ lemma pos_zmod_mult_2: "#0 $<= a ==> (#1 $+ #2$*b) zmod (#2$*a) = #1 $+ #2 $* (b zmod a)"
+ apply (case_tac "a = #0")
+ apply (subgoal_tac "#1 $<= a")
+  apply (arith_tac 2)
+ apply (subgoal_tac "#1 $< a $* #2")
+  apply (arith_tac 2)
+ apply (subgoal_tac "#2$* (#1 $+ b zmod a) $<= #2$*a")
+  apply (rule_tac [2] zmult_zle_mono2)
+ apply (auto simp add: zadd_commute zmult_commute add1_zle_iff pos_mod_bound)
+ apply (subst zmod_zadd1_eq)
+ apply (simp (no_asm_simp) add: zmod_zmult_zmult2 zmod_pos_pos_trivial)
+ apply (rule zmod_pos_pos_trivial)
+ apply (simp (no_asm_simp) # add: [zmod_pos_pos_trivial pos_mod_sign [THEN zadd_zle_mono1] RSN (2,zle_trans) ])
+ apply (auto simp add: zmod_pos_pos_trivial)
+ apply (subgoal_tac "#0 $<= b zmod a")
+  apply (asm_simp_tac (simpset () add: pos_mod_sign) 2)
+ apply arith
+ done
+
+
+ lemma neg_zmod_mult_2: "a $<= #0 ==> (#1 $+ #2$*b) zmod (#2$*a) = #2 $* ((b$+#1) zmod a) - #1"
+ apply (subgoal_tac " (#1 $+ #2$* ($-b-#1)) zmod (#2$* ($-a)) = #1 $+ #2$* (($-b-#1) zmod ($-a))")
+ apply (rule_tac [2] pos_zmod_mult_2)
+ apply (auto simp add: zmult_zminus_right)
+ apply (subgoal_tac " (#-1 - (#2 $* b)) = - (#1 $+ (#2 $* b))")
+ apply (Simp_tac 2)
+ apply (asm_full_simp_tac (HOL_ss add: zmod_zminus_zminus zdiff_def zminus_zadd_distrib [symmetric])
+ apply (dtac (zminus_equation [THEN iffD1, symmetric])
+ apply auto
+ done
+
+ lemma zmod_integ_of_BIT: "integ_of (v BIT b) zmod integ_of (w BIT False) =  
+           (if b then  
+                 if #0 $<= integ_of w  
+                 then #2 $* (integ_of v zmod integ_of w) $+ #1     
+                 else #2 $* ((integ_of v $+ #1) zmod integ_of w) - #1   
+            else #2 $* (integ_of v zmod integ_of w))"
+ apply (simp_tac (simpset_of Int.thy add: zadd_assoc integ_of_BIT)
+ apply (simp (no_asm_simp) del: bin_arith_extra_simps@bin_rel_simps add: zmod_zmult_zmult1 pos_zmod_mult_2 lemma neg_zmod_mult_2)
+ done
+
+ declare zmod_integ_of_BIT [simp]
+*)
+
+ML{*
+val zspos_add_zspos_imp_zspos = thm "zspos_add_zspos_imp_zspos";
+val zpos_add_zpos_imp_zpos = thm "zpos_add_zpos_imp_zpos";
+val zneg_add_zneg_imp_zneg = thm "zneg_add_zneg_imp_zneg";
+val zneg_or_0_add_zneg_or_0_imp_zneg_or_0 = thm "zneg_or_0_add_zneg_or_0_imp_zneg_or_0";
+val zero_lt_zmagnitude = thm "zero_lt_zmagnitude";
+val zless_add_succ_iff = thm "zless_add_succ_iff";
+val zadd_succ_zle_iff = thm "zadd_succ_zle_iff";
+val zless_add1_iff_zle = thm "zless_add1_iff_zle";
+val add1_zle_iff = thm "add1_zle_iff";
+val add1_left_zle_iff = thm "add1_left_zle_iff";
+val zmult_zle_mono1 = thm "zmult_zle_mono1";
+val zmult_zle_mono1_neg = thm "zmult_zle_mono1_neg";
+val zmult_zle_mono2 = thm "zmult_zle_mono2";
+val zmult_zle_mono2_neg = thm "zmult_zle_mono2_neg";
+val zmult_zle_mono = thm "zmult_zle_mono";
+val zmult_zless_mono2 = thm "zmult_zless_mono2";
+val zmult_zless_mono1 = thm "zmult_zless_mono1";
+val zmult_zless_mono = thm "zmult_zless_mono";
+val zmult_zless_mono1_neg = thm "zmult_zless_mono1_neg";
+val zmult_zless_mono2_neg = thm "zmult_zless_mono2_neg";
+val zmult_eq_0_iff = thm "zmult_eq_0_iff";
+val zmult_zless_cancel2 = thm "zmult_zless_cancel2";
+val zmult_zless_cancel1 = thm "zmult_zless_cancel1";
+val zmult_zle_cancel2 = thm "zmult_zle_cancel2";
+val zmult_zle_cancel1 = thm "zmult_zle_cancel1";
+val int_eq_iff_zle = thm "int_eq_iff_zle";
+val zmult_cancel2 = thm "zmult_cancel2";
+val zmult_cancel1 = thm "zmult_cancel1";
+val unique_quotient = thm "unique_quotient";
+val unique_remainder = thm "unique_remainder";
+val adjust_eq = thm "adjust_eq";
+val posDivAlg_termination = thm "posDivAlg_termination";
+val posDivAlg_unfold = thm "posDivAlg_unfold";
+val posDivAlg_eqn = thm "posDivAlg_eqn";
+val posDivAlg_induct = thm "posDivAlg_induct";
+val intify_eq_0_iff_zle = thm "intify_eq_0_iff_zle";
+val zmult_pos = thm "zmult_pos";
+val zmult_neg = thm "zmult_neg";
+val zmult_pos_neg = thm "zmult_pos_neg";
+val int_0_less_mult_iff = thm "int_0_less_mult_iff";
+val int_0_le_mult_iff = thm "int_0_le_mult_iff";
+val zmult_less_0_iff = thm "zmult_less_0_iff";
+val zmult_le_0_iff = thm "zmult_le_0_iff";
+val posDivAlg_type = thm "posDivAlg_type";
+val posDivAlg_correct = thm "posDivAlg_correct";
+val negDivAlg_termination = thm "negDivAlg_termination";
+val negDivAlg_unfold = thm "negDivAlg_unfold";
+val negDivAlg_eqn = thm "negDivAlg_eqn";
+val negDivAlg_induct = thm "negDivAlg_induct";
+val negDivAlg_type = thm "negDivAlg_type";
+val negDivAlg_correct = thm "negDivAlg_correct";
+val quorem_0 = thm "quorem_0";
+val posDivAlg_zero_divisor = thm "posDivAlg_zero_divisor";
+val posDivAlg_0 = thm "posDivAlg_0";
+val negDivAlg_minus1 = thm "negDivAlg_minus1";
+val negateSnd_eq = thm "negateSnd_eq";
+val negateSnd_type = thm "negateSnd_type";
+val quorem_neg = thm "quorem_neg";
+val divAlg_correct = thm "divAlg_correct";
+val divAlg_type = thm "divAlg_type";
+val zdiv_intify1 = thm "zdiv_intify1";
+val zdiv_intify2 = thm "zdiv_intify2";
+val zdiv_type = thm "zdiv_type";
+val zmod_intify1 = thm "zmod_intify1";
+val zmod_intify2 = thm "zmod_intify2";
+val zmod_type = thm "zmod_type";
+val DIVISION_BY_ZERO_ZDIV = thm "DIVISION_BY_ZERO_ZDIV";
+val DIVISION_BY_ZERO_ZMOD = thm "DIVISION_BY_ZERO_ZMOD";
+val zmod_zdiv_equality = thm "zmod_zdiv_equality";
+val pos_mod = thm "pos_mod";
+val pos_mod_sign = thm "pos_mod_sign";
+val neg_mod = thm "neg_mod";
+val neg_mod_sign = thm "neg_mod_sign";
+val quorem_div_mod = thm "quorem_div_mod";
+val quorem_div = thm "quorem_div";
+val quorem_mod = thm "quorem_mod";
+val zdiv_pos_pos_trivial = thm "zdiv_pos_pos_trivial";
+val zdiv_neg_neg_trivial = thm "zdiv_neg_neg_trivial";
+val zdiv_pos_neg_trivial = thm "zdiv_pos_neg_trivial";
+val zmod_pos_pos_trivial = thm "zmod_pos_pos_trivial";
+val zmod_neg_neg_trivial = thm "zmod_neg_neg_trivial";
+val zmod_pos_neg_trivial = thm "zmod_pos_neg_trivial";
+val zdiv_zminus_zminus = thm "zdiv_zminus_zminus";
+val zmod_zminus_zminus = thm "zmod_zminus_zminus";
+val self_quotient = thm "self_quotient";
+val self_remainder = thm "self_remainder";
+val zdiv_self = thm "zdiv_self";
+val zmod_self = thm "zmod_self";
+val zdiv_zero = thm "zdiv_zero";
+val zdiv_eq_minus1 = thm "zdiv_eq_minus1";
+val zmod_zero = thm "zmod_zero";
+val zdiv_minus1 = thm "zdiv_minus1";
+val zmod_minus1 = thm "zmod_minus1";
+val zdiv_pos_pos = thm "zdiv_pos_pos";
+val zmod_pos_pos = thm "zmod_pos_pos";
+val zdiv_neg_pos = thm "zdiv_neg_pos";
+val zmod_neg_pos = thm "zmod_neg_pos";
+val zdiv_pos_neg = thm "zdiv_pos_neg";
+val zmod_pos_neg = thm "zmod_pos_neg";
+val zdiv_neg_neg = thm "zdiv_neg_neg";
+val zmod_neg_neg = thm "zmod_neg_neg";
+val zmod_1 = thm "zmod_1";
+val zdiv_1 = thm "zdiv_1";
+val zmod_minus1_right = thm "zmod_minus1_right";
+val zdiv_minus1_right = thm "zdiv_minus1_right";
+val zdiv_mono1 = thm "zdiv_mono1";
+val zdiv_mono1_neg = thm "zdiv_mono1_neg";
+val zdiv_mono2 = thm "zdiv_mono2";
+val zdiv_mono2_neg = thm "zdiv_mono2_neg";
+val zdiv_zmult1_eq = thm "zdiv_zmult1_eq";
+val zmod_zmult1_eq = thm "zmod_zmult1_eq";
+val zmod_zmult1_eq' = thm "zmod_zmult1_eq'";
+val zmod_zmult_distrib = thm "zmod_zmult_distrib";
+val zdiv_zmult_self1 = thm "zdiv_zmult_self1";
+val zdiv_zmult_self2 = thm "zdiv_zmult_self2";
+val zmod_zmult_self1 = thm "zmod_zmult_self1";
+val zmod_zmult_self2 = thm "zmod_zmult_self2";
+val zdiv_zadd1_eq = thm "zdiv_zadd1_eq";
+val zmod_zadd1_eq = thm "zmod_zadd1_eq";
+val zmod_div_trivial = thm "zmod_div_trivial";
+val zmod_mod_trivial = thm "zmod_mod_trivial";
+val zmod_zadd_left_eq = thm "zmod_zadd_left_eq";
+val zmod_zadd_right_eq = thm "zmod_zadd_right_eq";
+val zdiv_zadd_self1 = thm "zdiv_zadd_self1";
+val zdiv_zadd_self2 = thm "zdiv_zadd_self2";
+val zmod_zadd_self1 = thm "zmod_zadd_self1";
+val zmod_zadd_self2 = thm "zmod_zadd_self2";
+val zdiv_zmult2_eq = thm "zdiv_zmult2_eq";
+val zmod_zmult2_eq = thm "zmod_zmult2_eq";
+val zdiv_zmult_zmult1 = thm "zdiv_zmult_zmult1";
+val zdiv_zmult_zmult2 = thm "zdiv_zmult_zmult2";
+val zmod_zmult_zmult1 = thm "zmod_zmult_zmult1";
+val zmod_zmult_zmult2 = thm "zmod_zmult_zmult2";
+val zdiv_neg_pos_less0 = thm "zdiv_neg_pos_less0";
+val zdiv_nonneg_neg_le0 = thm "zdiv_nonneg_neg_le0";
+val pos_imp_zdiv_nonneg_iff = thm "pos_imp_zdiv_nonneg_iff";
+val neg_imp_zdiv_nonneg_iff = thm "neg_imp_zdiv_nonneg_iff";
+val pos_imp_zdiv_neg_iff = thm "pos_imp_zdiv_neg_iff";
+val neg_imp_zdiv_neg_iff = thm "neg_imp_zdiv_neg_iff";
+*}
+
+end
+




