src/ZF/Integ/IntDiv.ML

better rules for cancellation of common factors across comparisons

(*  Title:      HOL/IntDiv.ML
    ID:         $Id$
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1999  University of Cambridge

The division operators div, mod and the divides relation "dvd"

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));
*)

Goal "[| #0 $< k; k: int |] ==> 0 < zmagnitude(k)";
by (dtac zero_zless_imp_znegative_zminus 1);
by (dtac zneg_int_of 2);
by (auto_tac (claset(), simpset() addsimps [inst "x" "k" zminus_equation]));  
by (subgoal_tac "0 < zmagnitude($# succ(x))" 1);
by (Asm_full_simp_tac 1);
by (asm_full_simp_tac (simpset_of Arith.thy addsimps [zmagnitude_int_of]) 1);
qed "zero_lt_zmagnitude";


(*** Inequality lemmas involving $#succ(m) ***)

Goal "(w $< z $+ $# succ(m)) <-> (w $< z $+ $#m | intify(w) = z $+ $#m)"; 
by (auto_tac (claset(),
	      simpset() addsimps [zless_iff_succ_zadd, zadd_assoc, 
                                  int_of_add RS sym]));
by (res_inst_tac [("x","0")] bexI 3);
by (TRYALL (dtac sym));
by (cut_inst_tac [("m","m")] int_succ_int_1 1);
by (cut_inst_tac [("m","n")] int_succ_int_1 1);
by (Asm_full_simp_tac 1);
by (eres_inst_tac [("n","x")] natE 1);
by Auto_tac;
by (res_inst_tac [("x","succ(x)")] bexI 1);
by Auto_tac;  
qed "zless_add_succ_iff";

Goal "z : int ==> (w $+ $# succ(m) $<= z) <-> (w $+ $#m $< z)";
by (asm_simp_tac (simpset_of Int.thy addsimps
                  [not_zless_iff_zle RS iff_sym, zless_add_succ_iff]) 1);
by (auto_tac (claset() addIs [zle_anti_sym] addEs [zless_asym],
	      simpset() addsimps [zless_imp_zle, not_zless_iff_zle]));
qed "lemma";

Goal "(w $+ $# succ(m) $<= z) <-> (w $+ $#m $< z)";
by (cut_inst_tac [("z","intify(z)")] lemma 1);
by Auto_tac;  
qed "zadd_succ_zle_iff";

(** Inequality reasoning **)

Goal "(w $< z $+ #1) <-> (w$<=z)";
by (subgoal_tac "#1 = $# 1" 1);
by (asm_simp_tac (simpset_of Int.thy 
                  addsimps [zless_add_succ_iff, zle_def]) 1);
by Auto_tac;  
qed "zless_add1_iff_zle";

Goal "(w $+ #1 $<= z) <-> (w $< z)";
by (subgoal_tac "#1 = $# 1" 1);
by (asm_simp_tac (simpset_of Int.thy addsimps [zadd_succ_zle_iff]) 1);
by Auto_tac;  
qed "add1_zle_iff";

Goal "(#1 $+ w $<= z) <-> (w $< z)";
by (stac zadd_commute 1);
by (rtac add1_zle_iff 1);
qed "add1_left_zle_iff";


(*** Monotonicity results ***)

Goal "(v$+z $< w$+z) <-> (v $< w)";
by (Simp_tac 1);
qed "zadd_right_cancel_zless";

Goal "(z$+v $< z$+w) <-> (v $< w)";
by (Simp_tac 1);
qed "zadd_left_cancel_zless";

Addsimps [zadd_right_cancel_zless, zadd_left_cancel_zless];

Goal "(v$+z $<= w$+z) <-> (v $<= w)";
by (Simp_tac 1);
qed "zadd_right_cancel_zle";

Goal "(z$+v $<= z$+w) <-> (v $<= w)";
by (Simp_tac 1);
qed "zadd_left_cancel_zle";

Addsimps [zadd_right_cancel_zle, zadd_left_cancel_zle];

(*"v $<= w ==> v$+z $<= w$+z"*)
bind_thm ("zadd_zless_mono1", zadd_right_cancel_zless RS iffD2);

(*"v $<= w ==> z$+v $<= z$+w"*)
bind_thm ("zadd_zless_mono2", zadd_left_cancel_zless RS iffD2);

(*"v $<= w ==> v$+z $<= w$+z"*)
bind_thm ("zadd_zle_mono1", zadd_right_cancel_zle RS iffD2);

(*"v $<= w ==> z$+v $<= z$+w"*)
bind_thm ("zadd_zle_mono2", zadd_left_cancel_zle RS iffD2);

Goal "[| w' $<= w; z' $<= z |] ==> w' $+ z' $<= w $+ z";
by (etac (zadd_zle_mono1 RS zle_trans) 1);
by (Simp_tac 1);
qed "zadd_zle_mono";

Goal "[| w' $< w; z' $<= z |] ==> w' $+ z' $< w $+ z";
by (etac (zadd_zless_mono1 RS zless_zle_trans) 1);
by (Simp_tac 1);
qed "zadd_zless_mono";


(*** Monotonicity of Multiplication ***)

Goal "k : nat ==> i $<= j ==> i $* $#k $<= j $* $#k";
by (induct_tac "k" 1);
by (stac int_succ_int_1 2);
by (ALLGOALS 
    (asm_simp_tac (simpset() addsimps [zadd_zmult_distrib2, zadd_zle_mono])));
val lemma = result();

Goal "[| i $<= j;  #0 $<= k |] ==> i$*k $<= j$*k";
by (subgoal_tac "i $* intify(k) $<= j $* intify(k)" 1);
by (Full_simp_tac 1);
by (res_inst_tac [("b", "intify(k)")] (not_zneg_mag RS subst) 1);
by (rtac lemma 3);
by Auto_tac;
by (dtac znegative_imp_zless_0 1);
by (dtac zless_zle_trans 2);
by Auto_tac;  
qed "zmult_zle_mono1";

Goal "[| i $<= j;  k $<= #0 |] ==> j$*k $<= i$*k";
by (rtac (zminus_zle_zminus RS iffD1) 1);
by (asm_simp_tac (simpset() delsimps [zmult_zminus_right]
                            addsimps [zmult_zminus_right RS sym,
				      zmult_zle_mono1, zle_zminus]) 1);
qed "zmult_zle_mono1_neg";

Goal "[| i $<= j;  #0 $<= k |] ==> k$*i $<= k$*j";
by (dtac zmult_zle_mono1 1);
by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [zmult_commute])));
qed "zmult_zle_mono2";

Goal "[| i $<= j;  k $<= #0 |] ==> k$*j $<= k$*i";
by (dtac zmult_zle_mono1_neg 1);
by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [zmult_commute])));
qed "zmult_zle_mono2_neg";

(* $<= monotonicity, BOTH arguments*)
Goal "[| i $<= j;  k $<= l;  #0 $<= j;  #0 $<= k |] ==> i$*k $<= j$*l";
by (etac (zmult_zle_mono1 RS zle_trans) 1);
by (assume_tac 1);
by (etac zmult_zle_mono2 1);
by (assume_tac 1);
qed "zmult_zle_mono";


(** strict, in 1st argument; proof is by induction on k>0 **)

Goal "[| i$<j; k : nat |] ==> 0<k --> $#k $* i $< $#k $* j";
by (induct_tac "k" 1);
by (stac int_succ_int_1 2);
by (etac natE 2);
by (ALLGOALS (asm_full_simp_tac
	      (simpset() addsimps [zadd_zmult_distrib, zadd_zless_mono, 
				   zle_def])));
by (ftac nat_0_le 1);
by (mp_tac 1);
by (subgoal_tac "i $+ (i $+ $# xa $* i) $< j $+ (j $+ $# xa $* j)" 1);
by (Full_simp_tac 1);
by (rtac zadd_zless_mono 1);
by (ALLGOALS (asm_simp_tac (simpset() addsimps [zle_def])));
val lemma = result() RS mp;

Goal "[| i$<j;  #0 $< k |] ==> k$*i $< k$*j";
by (subgoal_tac "intify(k) $* i $< intify(k) $* j" 1);
by (Full_simp_tac 1);
by (res_inst_tac [("b", "intify(k)")] (not_zneg_mag RS subst) 1);
by (rtac lemma 3);
by Auto_tac;
by (dtac znegative_imp_zless_0 1);
by (dtac zless_trans 2 THEN assume_tac 2);
by (auto_tac (claset(), simpset() addsimps [zero_lt_zmagnitude]));  
qed "zmult_zless_mono2";

Goal "[| i$<j;  #0 $< k |] ==> i$*k $< j$*k";
by (dtac zmult_zless_mono2 1);
by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [zmult_commute])));
qed "zmult_zless_mono1";

(* < monotonicity, BOTH arguments*)
Goal "[| i $< j;  k $< l;  #0 $< j;  #0 $< k |] ==> i$*k $< j$*l";
by (etac (zmult_zless_mono1 RS zless_trans) 1);
by (assume_tac 1);
by (etac zmult_zless_mono2 1);
by (assume_tac 1);
qed "zmult_zless_mono";

Goal "[| i $< j;  k $< #0 |] ==> j$*k $< i$*k";
by (rtac (zminus_zless_zminus RS iffD1) 1);
by (asm_simp_tac (simpset() delsimps [zmult_zminus_right]
                            addsimps [zmult_zminus_right RS sym,
				      zmult_zless_mono1, zless_zminus]) 1);
qed "zmult_zless_mono1_neg";

Goal "[| i $< j;  k $< #0 |] ==> k$*j $< k$*i";
by (rtac (zminus_zless_zminus RS iffD1) 1);
by (asm_simp_tac (simpset() delsimps [zmult_zminus]
                            addsimps [zmult_zminus RS sym,
				      zmult_zless_mono2, zless_zminus]) 1);
qed "zmult_zless_mono2_neg";

Goal "[| m: int; n: int |] ==> (m$*n = #0) <-> (m = #0 | n = #0)";
by (case_tac "m $< #0" 1);
by (auto_tac (claset(), 
     simpset() addsimps [not_zless_iff_zle, zle_def, neq_iff_zless])); 
by (REPEAT 
    (force_tac (claset() addDs [zmult_zless_mono1_neg, zmult_zless_mono1], 
		simpset()) 1));
val lemma = result();

Goal "(m$*n = #0) <-> (intify(m) = #0 | intify(n) = #0)";
by (asm_full_simp_tac (simpset() addsimps [lemma RS iff_sym]) 1);
qed "zmult_eq_0_iff";


(** Cancellation laws for k*m < k*n and m*k < n*k, also for <= and =,
    but not (yet?) for k*m < n*k. **)

Goal "[| k: int; m: int; n: int |] \
\     ==> (m$*k $< n$*k) <-> ((#0 $< k & m$<n) | (k $< #0 & n$<m))";
by (case_tac "k = #0" 1);
by (auto_tac (claset(), simpset() addsimps [neq_iff_zless, 
                              zmult_zless_mono1, zmult_zless_mono1_neg]));  
by (auto_tac (claset(), 
      simpset() addsimps [not_zless_iff_zle,
			  inst "w1" "m$*k" (not_zle_iff_zless RS iff_sym),
			  inst "w1" "m" (not_zle_iff_zless RS iff_sym)]));
by (ALLGOALS (etac notE));
by (auto_tac (claset(), simpset() addsimps [zless_imp_zle, zmult_zle_mono1,
                                            zmult_zle_mono1_neg]));  
val lemma = result();

Goal "(m$*k $< n$*k) <-> ((#0 $< k & m$<n) | (k $< #0 & n$<m))";
by (cut_inst_tac [("k","intify(k)"),("m","intify(m)"),("n","intify(n)")]
                 lemma 1);
by Auto_tac;  
qed "zmult_zless_cancel2";

Goal "(k$*m $< k$*n) <-> ((#0 $< k & m$<n) | (k $< #0 & n$<m))";
by (simp_tac (simpset() addsimps [inst "z" "k" zmult_commute, 
                                  zmult_zless_cancel2]) 1);
qed "zmult_zless_cancel1";

Goal "(m$*k $<= n$*k) <-> ((#0 $< k --> m$<=n) & (k $< #0 --> n$<=m))";
by (simp_tac (simpset() addsimps [not_zless_iff_zle RS iff_sym, 
                                  zmult_zless_cancel2]) 1);
by Auto_tac;  
qed "zmult_zle_cancel2";

Goal "(k$*m $<= k$*n) <-> ((#0 $< k --> m$<=n) & (k $< #0 --> n$<=m))";
by (simp_tac (simpset() addsimps [not_zless_iff_zle RS iff_sym, 
                                  zmult_zless_cancel1]) 1);
by Auto_tac;  
qed "zmult_zle_cancel1";

Goal "[| m: int; n: int |] ==> m=n <-> (m $<= n & n $<= m)";
by (blast_tac (claset() addIs [zle_refl,zle_anti_sym]) 1); 
qed "int_eq_iff_zle";

Goal "[| k: int; m: int; n: int |] ==> (m$*k = n$*k) <-> (k=#0 | m=n)";
by (asm_simp_tac (simpset() addsimps [inst "m" "m$*k" int_eq_iff_zle,
                                      inst "m" "m" int_eq_iff_zle]) 1); 
by (auto_tac (claset(), 
              simpset() addsimps [zmult_zle_cancel2, neq_iff_zless]));  
val lemma = result();

Goal "(m$*k = n$*k) <-> (intify(k) = #0 | intify(m) = intify(n))";
by (rtac iff_trans 1);
by (rtac lemma 2);
by Auto_tac;  
qed "zmult_cancel2";

Goal "(k$*m = k$*n) <-> (intify(k) = #0 | intify(m) = intify(n))";
by (simp_tac (simpset() addsimps [inst "z" "k" zmult_commute, 
                                  zmult_cancel2]) 1);
qed "zmult_cancel1";
Addsimps [zmult_cancel1, zmult_cancel2];


(*** Uniqueness and monotonicity of quotients and remainders ***)


Goal "[| b$*q' $+ r' $<= b$*q $+ r;  #0 $<= r';  #0 $< b;  r $< b |] \
\     ==> q' $<= q";
by (subgoal_tac "r' $+ b $* (q'$-q) $<= r" 1);
by (full_simp_tac 
    (simpset() addsimps [zdiff_zmult_distrib2]@zadd_ac@zcompare_rls) 2);
by (subgoal_tac "#0 $< b $* (#1 $+ q $- q')" 1);
by (etac zle_zless_trans 2);
by (full_simp_tac 
    (simpset() addsimps [zdiff_zmult_distrib2,
			 zadd_zmult_distrib2]@zadd_ac@zcompare_rls) 2);
by (etac zle_zless_trans 2);
by (Asm_simp_tac 2);
by (subgoal_tac "b $* q' $< b $* (#1 $+ q)" 1);
by (full_simp_tac
    (simpset() addsimps [zdiff_zmult_distrib2,
			 zadd_zmult_distrib2]@zadd_ac@zcompare_rls) 2);
by (auto_tac (claset() addEs [zless_asym], 
              simpset() addsimps [zmult_zless_cancel1, zless_add1_iff_zle]@
                                 zadd_ac@zcompare_rls));
qed "unique_quotient_lemma";

Goal "[| b$*q' $+ r' $<= b$*q $+ r;  r $<= #0;  b $< #0;  b $< r' |] \
\     ==> q $<= q'";
by (res_inst_tac [("b", "$-b"), ("r", "$-r'"), ("r'", "$-r")] 
    unique_quotient_lemma 1);
by (auto_tac (claset(), 
	      simpset() delsimps [zminus_zadd_distrib]
			addsimps [zminus_zadd_distrib RS sym,
	                          zle_zminus, zless_zminus])); 
qed "unique_quotient_lemma_neg";


Goal "[| quorem (<a,b>, <q,r>);  quorem (<a,b>, <q',r'>);  b: int; b ~= #0; \
\        q: int; q' : int |] ==> q = q'";
by (asm_full_simp_tac 
    (simpset() addsimps split_ifs@
                        [quorem_def, neq_iff_zless]) 1);
by Safe_tac; 
by (ALLGOALS Asm_full_simp_tac);
by (REPEAT 
    (blast_tac (claset() addIs [zle_anti_sym]
			 addDs [zle_eq_refl RS unique_quotient_lemma, 
				zle_eq_refl RS unique_quotient_lemma_neg,
				sym]) 1));
qed "unique_quotient";

Goal "[| quorem (<a,b>, <q,r>);  quorem (<a,b>, <q',r'>);  b: int; b ~= #0; \
\        q: int; q' : int; \
\        r: int; r' : int |] ==> r = r'";
by (subgoal_tac "q = q'" 1);
by (blast_tac (claset() addIs [unique_quotient]) 2);
by (asm_full_simp_tac (simpset() addsimps [quorem_def]) 1);
by Auto_tac;  
qed "unique_remainder";


(*** THE REST TO PORT LATER.  The division algorithm can wait; most properties
     of division flow from the uniqueness properties proved above...


 (*** Correctness of posDivAlg, the division algorithm for a>=0 and b>0 ***)


 Goal "adjust a b <q,r> = (let diff = r$-b in \
 \                         if #0 $<= diff then <#2$*q $+ #1,diff>  \
 \                                       else <#2$*q,r>)";
 by (simp_tac (simpset() addsimps [Let_def,adjust_def]) 1);
 qed "adjust_eq";
 Addsimps [adjust_eq];

 (*Proving posDivAlg's termination condition*)
 val [tc] = posDivAlg.tcs;
 goalw_cterm [] (cterm_of (sign_of (the_context ())) (HOLogic.mk_Trueprop tc));
 by Auto_tac;
 val lemma = result();

 (* removing the termination condition from the generated theorems *)

 bind_thm ("posDivAlg_raw_eqn", lemma RS hd posDivAlg.simps);

 (**use with simproc to avoid re-proving the premise*)
 Goal "#0 $< b ==> \
 \     posDivAlg <a,b> =      \
 \      (if a$<b then <#0,a> else adjust a b (posDivAlg<a,#2$*b>))";
 by (rtac (posDivAlg_raw_eqn RS trans) 1);
 by (Asm_simp_tac 1);
 qed "posDivAlg_eqn";

 bind_thm ("posDivAlg_induct", lemma RS posDivAlg.induct);


 (*Correctness of posDivAlg: it computes quotients correctly*)
 Goal "#0 $<= a --> #0 $< b --> quorem (<a,b>, posDivAlg <a,b>)";
 by (res_inst_tac [("u", "a"), ("v", "b")] posDivAlg_induct 1);
 by Auto_tac;
 by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [quorem_def])));
 (*base case: a<b*)
 by (asm_full_simp_tac (simpset() addsimps [posDivAlg_eqn]) 1);
 (*main argument*)
 by (stac posDivAlg_eqn 1);
 by (ALLGOALS Asm_simp_tac);
 by (etac splitE 1);
 by (auto_tac (claset(), simpset() addsimps [zadd_zmult_distrib2, Let_def]));
 qed_spec_mp "posDivAlg_correct";


 (*** Correctness of negDivAlg, the division algorithm for a<0 and b>0 ***)

 (*Proving negDivAlg's termination condition*)
 val [tc] = negDivAlg.tcs;
 goalw_cterm [] (cterm_of (sign_of (the_context ())) (HOLogic.mk_Trueprop tc));
 by Auto_tac;
 val lemma = result();

 (* removing the termination condition from the generated theorems *)

 bind_thm ("negDivAlg_raw_eqn", lemma RS hd negDivAlg.simps);

 (**use with simproc to avoid re-proving the premise*)
 Goal "#0 $< b ==> \
 \     negDivAlg <a,b> =      \
 \      (if #0$<=a$+b then <#-1,a$+b> else adjust a b (negDivAlg<a,#2$*b>))";
 by (rtac (negDivAlg_raw_eqn RS trans) 1);
 by (Asm_simp_tac 1);
 qed "negDivAlg_eqn";

 bind_thm ("negDivAlg_induct", lemma RS negDivAlg.induct);


 (*Correctness of negDivAlg: it computes quotients correctly
   It doesn't work if a=0 because the 0/b=0 rather than -1*)
 Goal "a $< #0 --> #0 $< b --> quorem (<a,b>, negDivAlg <a,b>)";
 by (res_inst_tac [("u", "a"), ("v", "b")] negDivAlg_induct 1);
 by Auto_tac;
 by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [quorem_def])));
 (*base case: 0$<=a$+b*)
 by (asm_full_simp_tac (simpset() addsimps [negDivAlg_eqn]) 1);
 (*main argument*)
 by (stac negDivAlg_eqn 1);
 by (ALLGOALS Asm_simp_tac);
 by (etac splitE 1);
 by (auto_tac (claset(), simpset() addsimps [zadd_zmult_distrib2, Let_def]));
 qed_spec_mp "negDivAlg_correct";


 (*** Existence shown by proving the division algorithm to be correct ***)

 (*the case a=0*)
 Goal "b ~= #0 ==> quorem (<#0,b>, <#0,#0>)";
 by (auto_tac (claset(), 
	       simpset() addsimps [quorem_def, neq_iff_zless]));
 qed "quorem_0";

 Goal "posDivAlg <#0,b> = <#0,#0>";
 by (stac posDivAlg_raw_eqn 1);
 by Auto_tac;
 qed "posDivAlg_0";
 Addsimps [posDivAlg_0];

 Goal "negDivAlg <#-1,b> = <#-1,b-#1>";
 by (stac negDivAlg_raw_eqn 1);
 by Auto_tac;
 qed "negDivAlg_minus1";
 Addsimps [negDivAlg_minus1];

 Goalw [negateSnd_def] "negateSnd<q,r> = <q,-r>";
 by Auto_tac;
 qed "negateSnd_eq";
 Addsimps [negateSnd_eq];

 Goal "quorem (<-a,-b>, qr) ==> quorem (<a,b>, negateSnd qr)";
 by (auto_tac (claset(), simpset() addsimps split_ifs@[quorem_def]));
 qed "quorem_neg";

 Goal "b ~= #0 ==> quorem (<a,b>, divAlg<a,b>)";
 by (auto_tac (claset(), 
	       simpset() addsimps [quorem_0, divAlg_def]));
 by (REPEAT_FIRST (resolve_tac [quorem_neg, posDivAlg_correct,
				negDivAlg_correct]));
 by (auto_tac (claset(), 
	       simpset() addsimps [quorem_def, neq_iff_zless]));
 qed "divAlg_correct";

 (** Aribtrary definitions for division by zero.  Useful to simplify 
     certain equations **)

 Goal "a div (#0::int) = #0";
 by (simp_tac (simpset() addsimps [div_def, divAlg_def, posDivAlg_raw_eqn]) 1);
 qed "DIVISION_BY_ZERO_ZDIV";  (*NOT for adding to default simpset*)

 Goal "a mod (#0::int) = a";
 by (simp_tac (simpset() addsimps [mod_def, divAlg_def, posDivAlg_raw_eqn]) 1);
 qed "DIVISION_BY_ZERO_ZMOD";  (*NOT for adding to default simpset*)

 fun zdiv_undefined_case_tac s i =
   case_tac s i THEN 
   asm_simp_tac (simpset() addsimps [DIVISION_BY_ZERO_ZDIV, 
				     DIVISION_BY_ZERO_ZMOD]) i;

 (** Basic laws about division and remainder **)

 Goal "a = b $* (a div b) $+ (a mod b)";
 by (zdiv_undefined_case_tac "b = #0" 1);
 by (cut_inst_tac [("a","a"),("b","b")] divAlg_correct 1);
 by (auto_tac (claset(), 
	       simpset() addsimps [quorem_def, div_def, mod_def]));
 qed "zmod_zdiv_equality";  

 Goal "(#0::int) $< b ==> #0 $<= a mod b & a mod b $< b";
 by (cut_inst_tac [("a","a"),("b","b")] divAlg_correct 1);
 by (auto_tac (claset(), 
	       simpset() addsimps [quorem_def, mod_def]));
 bind_thm ("pos_mod_sign", result() RS conjunct1);
 bind_thm ("pos_mod_bound", result() RS conjunct2);

 Goal "b $< (#0::int) ==> a mod b $<= #0 & b $< a mod b";
 by (cut_inst_tac [("a","a"),("b","b")] divAlg_correct 1);
 by (auto_tac (claset(), 
	       simpset() addsimps [quorem_def, div_def, mod_def]));
 bind_thm ("neg_mod_sign", result() RS conjunct1);
 bind_thm ("neg_mod_bound", result() RS conjunct2);


 (** proving general properties of div and mod **)

 Goal "b ~= #0 ==> quorem (<a,b>, <a div b,a mod b>)";
 by (cut_inst_tac [("a","a"),("b","b")] zmod_zdiv_equality 1);
 by (auto_tac
     (claset(),
      simpset() addsimps [quorem_def, neq_iff_zless, 
			  pos_mod_sign,pos_mod_bound,
			  neg_mod_sign, neg_mod_bound]));
 qed "quorem_div_mod";

 Goal "[| quorem(<a,b>,<q,r>);  b ~= #0 |] ==> a div b = q";
 by (asm_simp_tac (simpset() addsimps [quorem_div_mod RS unique_quotient]) 1);
 qed "quorem_div";

 Goal "[| quorem(<a,b>,<q,r>);  b ~= #0 |] ==> a mod b = r";
 by (asm_simp_tac (simpset() addsimps [quorem_div_mod RS unique_remainder]) 1);
 qed "quorem_mod";

 Goal "[| (#0::int) $<= a;  a $< b |] ==> a div b = #0";
 by (rtac quorem_div 1);
 by (auto_tac (claset(), simpset() addsimps [quorem_def]));
 qed "div_pos_pos_trivial";

 Goal "[| a $<= (#0::int);  b $< a |] ==> a div b = #0";
 by (rtac quorem_div 1);
 by (auto_tac (claset(), simpset() addsimps [quorem_def]));
 qed "div_neg_neg_trivial";

 Goal "[| (#0::int) $< a;  a$+b $<= #0 |] ==> a div b = #-1";
 by (rtac quorem_div 1);
 by (auto_tac (claset(), simpset() addsimps [quorem_def]));
 qed "div_pos_neg_trivial";

 (*There is no div_neg_pos_trivial because  #0 div b = #0 would supersede it*)

 Goal "[| (#0::int) $<= a;  a $< b |] ==> a mod b = a";
 by (res_inst_tac [("q","#0")] quorem_mod 1);
 by (auto_tac (claset(), simpset() addsimps [quorem_def]));
 qed "mod_pos_pos_trivial";

 Goal "[| a $<= (#0::int);  b $< a |] ==> a mod b = a";
 by (res_inst_tac [("q","#0")] quorem_mod 1);
 by (auto_tac (claset(), simpset() addsimps [quorem_def]));
 qed "mod_neg_neg_trivial";

 Goal "[| (#0::int) $< a;  a$+b $<= #0 |] ==> a mod b = a$+b";
 by (res_inst_tac [("q","#-1")] quorem_mod 1);
 by (auto_tac (claset(), simpset() addsimps [quorem_def]));
 qed "mod_pos_neg_trivial";

 (*There is no mod_neg_pos_trivial...*)


 (*Simpler laws such as -a div b = -(a div b) FAIL*)
 Goal "(-a) div (-b) = a div b";
 by (zdiv_undefined_case_tac "b = #0" 1);
 by (stac ((simplify(simpset()) (quorem_div_mod RS quorem_neg)) 
	   RS quorem_div) 1);
 by Auto_tac;
 qed "zdiv_zminus_zminus";
 Addsimps [zdiv_zminus_zminus];

 (*Simpler laws such as -a mod b = -(a mod b) FAIL*)
 Goal "(-a) mod (-b) = - (a mod b)";
 by (zdiv_undefined_case_tac "b = #0" 1);
 by (stac ((simplify(simpset()) (quorem_div_mod RS quorem_neg)) 
	   RS quorem_mod) 1);
 by Auto_tac;
 qed "zmod_zminus_zminus";
 Addsimps [zmod_zminus_zminus];


 (*** division of a number by itself ***)

 Goal "[| (#0::int) $< a; a = r $+ a$*q; r $< a |] ==> #1 $<= q";
 by (subgoal_tac "#0 $< a$*q" 1);
 by (arith_tac 2);
 by (asm_full_simp_tac (simpset() addsimps [int_0_less_mult_iff]) 1);
 val lemma1 = result();

 Goal "[| (#0::int) $< a; a = r $+ a$*q; #0 $<= r |] ==> q $<= #1";
 by (subgoal_tac "#0 $<= a$*(#1$-q)" 1);
 by (asm_simp_tac (simpset() addsimps [zdiff_zmult_distrib2]) 2);
 by (asm_full_simp_tac (simpset() addsimps [int_0_le_mult_iff]) 1);
 val lemma2 = result();

 Goal "[| quorem(<a,a>,<q,r>);  a ~= (#0::int) |] ==> q = #1";
 by (asm_full_simp_tac 
     (simpset() addsimps split_ifs@[quorem_def, neq_iff_zless]) 1);
 by (rtac order_antisym 1);
 by Safe_tac;
 by Auto_tac;
 by (res_inst_tac [("a", "-a"),("r", "-r")] lemma1 3);
 by (res_inst_tac [("a", "-a"),("r", "-r")] lemma2 1);
 by (REPEAT (force_tac  (claset() addIs [lemma1,lemma2], 
	       simpset() addsimps [zadd_commute, zmult_zminus]) 1));
 qed "self_quotient";

 Goal "[| quorem(<a,a>,<q,r>);  a ~= (#0::int) |] ==> r = #0";
 by (ftac self_quotient 1);
 by (assume_tac 1);
 by (asm_full_simp_tac (simpset() addsimps [quorem_def]) 1);
 qed "self_remainder";

 Goal "a ~= #0 ==> a div a = (#1::int)";
 by (asm_simp_tac (simpset() addsimps [quorem_div_mod RS self_quotient]) 1);
 qed "zdiv_self";
 Addsimps [zdiv_self];

 (*Here we have 0 mod 0 = 0, also assumed by Knuth (who puts m mod 0 = 0) *)
 Goal "a mod a = (#0::int)";
 by (zdiv_undefined_case_tac "a = #0" 1);
 by (asm_simp_tac (simpset() addsimps [quorem_div_mod RS self_remainder]) 1);
 qed "zmod_self";
 Addsimps [zmod_self];


 (*** Computation of division and remainder ***)

 Goal "(#0::int) div b = #0";
 by (simp_tac (simpset() addsimps [div_def, divAlg_def]) 1);
 qed "zdiv_zero";

 Goal "(#0::int) $< b ==> #-1 div b = #-1";
 by (asm_simp_tac (simpset() addsimps [div_def, divAlg_def]) 1);
 qed "div_eq_minus1";

 Goal "(#0::int) mod b = #0";
 by (simp_tac (simpset() addsimps [mod_def, divAlg_def]) 1);
 qed "zmod_zero";

 Addsimps [zdiv_zero, zmod_zero];

 Goal "(#0::int) $< b ==> #-1 div b = #-1";
 by (asm_simp_tac (simpset() addsimps [div_def, divAlg_def]) 1);
 qed "zdiv_minus1";

 Goal "(#0::int) $< b ==> #-1 mod b = b-#1";
 by (asm_simp_tac (simpset() addsimps [mod_def, divAlg_def]) 1);
 qed "zmod_minus1";

 (** a positive, b positive **)

 Goal "[| #0 $< a;  #0 $<= b |] ==> a div b = fst (posDivAlg<a,b>)";
 by (asm_simp_tac (simpset() addsimps [div_def, divAlg_def]) 1);
 qed "div_pos_pos";

 Goal "[| #0 $< a;  #0 $<= b |] ==> a mod b = snd (posDivAlg<a,b>)";
 by (asm_simp_tac (simpset() addsimps [mod_def, divAlg_def]) 1);
 qed "mod_pos_pos";

 (** a negative, b positive **)

 Goal "[| a $< #0;  #0 $< b |] ==> a div b = fst (negDivAlg<a,b>)";
 by (asm_simp_tac (simpset() addsimps [div_def, divAlg_def]) 1);
 qed "div_neg_pos";

 Goal "[| a $< #0;  #0 $< b |] ==> a mod b = snd (negDivAlg<a,b>)";
 by (asm_simp_tac (simpset() addsimps [mod_def, divAlg_def]) 1);
 qed "mod_neg_pos";

 (** a positive, b negative **)

 Goal "[| #0 $< a;  b $< #0 |] ==> a div b = fst (negateSnd(negDivAlg<-a,-b>))";
 by (asm_simp_tac (simpset() addsimps [div_def, divAlg_def]) 1);
 qed "div_pos_neg";

 Goal "[| #0 $< a;  b $< #0 |] ==> a mod b = snd (negateSnd(negDivAlg<-a,-b>))";
 by (asm_simp_tac (simpset() addsimps [mod_def, divAlg_def]) 1);
 qed "mod_pos_neg";

 (** a negative, b negative **)

 Goal "[| a $< #0;  b $<= #0 |] ==> a div b = fst (negateSnd(posDivAlg<-a,-b>))";
 by (asm_simp_tac (simpset() addsimps [div_def, divAlg_def]) 1);
 qed "div_neg_neg";

 Goal "[| a $< #0;  b $<= #0 |] ==> a mod b = snd (negateSnd(posDivAlg<-a,-b>))";
 by (asm_simp_tac (simpset() addsimps [mod_def, divAlg_def]) 1);
 qed "mod_neg_neg";

 Addsimps (map (read_instantiate_sg (sign_of (the_context ()))
		[("a", "number_of ?v"), ("b", "number_of ?w")])
	   [div_pos_pos, div_neg_pos, div_pos_neg, div_neg_neg,
	    mod_pos_pos, mod_neg_pos, mod_pos_neg, mod_neg_neg,
	    posDivAlg_eqn, negDivAlg_eqn]);


 (** Special-case simplification **)

 Goal "a mod (#1::int) = #0";
 by (cut_inst_tac [("a","a"),("b","#1")] pos_mod_sign 1);
 by (cut_inst_tac [("a","a"),("b","#1")] pos_mod_bound 2);
 by Auto_tac;
 qed "zmod_1";
 Addsimps [zmod_1];

 Goal "a div (#1::int) = a";
 by (cut_inst_tac [("a","a"),("b","#1")] zmod_zdiv_equality 1);
 by Auto_tac;
 qed "zdiv_1";
 Addsimps [zdiv_1];

 Goal "a mod (#-1::int) = #0";
 by (cut_inst_tac [("a","a"),("b","#-1")] neg_mod_sign 1);
 by (cut_inst_tac [("a","a"),("b","#-1")] neg_mod_bound 2);
 by Auto_tac;
 qed "zmod_minus1_right";
 Addsimps [zmod_minus1_right];

 Goal "a div (#-1::int) = -a";
 by (cut_inst_tac [("a","a"),("b","#-1")] zmod_zdiv_equality 1);
 by Auto_tac;
 qed "zdiv_minus1_right";
 Addsimps [zdiv_minus1_right];


 (*** Monotonicity in the first argument (divisor) ***)

 Goal "[| a $<= a';  #0 $< b |] ==> a div b $<= a' div b";
 by (cut_inst_tac [("a","a"),("b","b")] zmod_zdiv_equality 1);
 by (cut_inst_tac [("a","a'"),("b","b")] zmod_zdiv_equality 1);
 by (rtac unique_quotient_lemma 1);
 by (etac subst 1);
 by (etac subst 1);
 by (ALLGOALS (asm_simp_tac (simpset() addsimps [pos_mod_sign,pos_mod_bound])));
 qed "zdiv_mono1";

 Goal "[| a $<= a';  b $< #0 |] ==> a' div b $<= a div b";
 by (cut_inst_tac [("a","a"),("b","b")] zmod_zdiv_equality 1);
 by (cut_inst_tac [("a","a'"),("b","b")] zmod_zdiv_equality 1);
 by (rtac unique_quotient_lemma_neg 1);
 by (etac subst 1);
 by (etac subst 1);
 by (ALLGOALS (asm_simp_tac (simpset() addsimps [neg_mod_sign,neg_mod_bound])));
 qed "zdiv_mono1_neg";


 (*** Monotonicity in the second argument (dividend) ***)

 Goal "[| b$*q $+ r = b'$*q' $+ r';  #0 $<= b'$*q' $+ r';  \
 \        r' $< b';  #0 $<= r;  #0 $< b';  b' $<= b |]  \
 \     ==> q $<= q'";
 by (subgoal_tac "#0 $<= q'" 1);
  by (subgoal_tac "#0 $< b'$*(q' $+ #1)" 2);
   by (asm_simp_tac (simpset() addsimps [zadd_zmult_distrib2]) 3);
  by (asm_full_simp_tac (simpset() addsimps [int_0_less_mult_iff]) 2);
 by (subgoal_tac "b$*q $< b$*(q' $+ #1)" 1);
  by (Asm_full_simp_tac 1);
 by (subgoal_tac "b$*q = r' $- r $+ b'$*q'" 1);
  by (Simp_tac 2);
 by (asm_simp_tac (simpset() addsimps [zadd_zmult_distrib2]) 1);
 by (stac zadd_commute 1 THEN rtac zadd_zless_mono 1 THEN arith_tac 1);
 by (rtac zmult_zle_mono1 1);
 by Auto_tac;
 qed "zdiv_mono2_lemma";

 Goal "[| (#0::int) $<= a;  #0 $< b';  b' $<= b |]  \
 \     ==> a div b $<= a div b'";
 by (subgoal_tac "b ~= #0" 1);
 by (arith_tac 2);
 by (cut_inst_tac [("a","a"),("b","b")] zmod_zdiv_equality 1);
 by (cut_inst_tac [("a","a"),("b","b'")] zmod_zdiv_equality 1);
 by (rtac zdiv_mono2_lemma 1);
 by (etac subst 1);
 by (etac subst 1);
 by (ALLGOALS (asm_simp_tac (simpset() addsimps [pos_mod_sign,pos_mod_bound])));
 qed "zdiv_mono2";

 Goal "[| b$*q $+ r = b'$*q' $+ r';  b'$*q' $+ r' $< #0;  \
 \        r $< b;  #0 $<= r';  #0 $< b';  b' $<= b |]  \
 \     ==> q' $<= q";
 by (subgoal_tac "q' $< #0" 1);
  by (subgoal_tac "b'$*q' $< #0" 2);
   by (arith_tac 3);
  by (asm_full_simp_tac (simpset() addsimps [zmult_less_0_iff]) 2);
 by (subgoal_tac "b$*q' $< b$*(q $+ #1)" 1);
  by (Asm_full_simp_tac 1);
 by (asm_simp_tac (simpset() addsimps [zadd_zmult_distrib2]) 1);
 by (subgoal_tac "b$*q' $<= b'$*q'" 1);
  by (asm_simp_tac (simpset() addsimps [zmult_zle_mono1_neg]) 2);
 by (subgoal_tac "b'$*q' $< b $+ b$*q" 1);
  by (Asm_simp_tac 2);
 by (arith_tac 1);
 qed "zdiv_mono2_neg_lemma";

 Goal "[| a $< (#0::int);  #0 $< b';  b' $<= b |]  \
 \     ==> a div b' $<= a div b";
 by (cut_inst_tac [("a","a"),("b","b")] zmod_zdiv_equality 1);
 by (cut_inst_tac [("a","a"),("b","b'")] zmod_zdiv_equality 1);
 by (rtac zdiv_mono2_neg_lemma 1);
 by (etac subst 1);
 by (etac subst 1);
 by (ALLGOALS (asm_simp_tac (simpset() addsimps [pos_mod_sign,pos_mod_bound])));
 qed "zdiv_mono2_neg";


 (*** More algebraic laws for div and mod ***)

 (** proving (a*b) div c = a $* (b div c) $+ a * (b mod c) **)

 Goal "[| quorem(<b,c>,<q,r>);  c ~= #0 |] \
 \     ==> quorem (<a$*b,c>, <a$*q $+ a$*r div c,a$*r mod c>)";
 by (auto_tac
     (claset(),
      simpset() addsimps split_ifs@
			 [quorem_def, neq_iff_zless, 
			  zadd_zmult_distrib2,
			  pos_mod_sign,pos_mod_bound,
			  neg_mod_sign, neg_mod_bound]));
 by (ALLGOALS(rtac zmod_zdiv_equality));
 val lemma = result();

 Goal "(a$*b) div c = a$*(b div c) $+ a$*(b mod c) div c";
 by (zdiv_undefined_case_tac "c = #0" 1);
 by (blast_tac (claset() addIs [quorem_div_mod RS lemma RS quorem_div]) 1);
 qed "zdiv_zmult1_eq";

 Goal "(a$*b) mod c = a$*(b mod c) mod c";
 by (zdiv_undefined_case_tac "c = #0" 1);
 by (blast_tac (claset() addIs [quorem_div_mod RS lemma RS quorem_mod]) 1);
 qed "zmod_zmult1_eq";

 Goal "(a$*b) mod c = ((a mod c) $* b) mod c";
 by (rtac trans 1);
 by (res_inst_tac [("s","b$*a mod c")] trans 1);
 by (rtac zmod_zmult1_eq 2);
 by (ALLGOALS (simp_tac (simpset() addsimps [zmult_commute])));
 qed "zmod_zmult1_eq'";

 Goal "(a$*b) mod c = ((a mod c) $* (b mod c)) mod c";
 by (rtac (zmod_zmult1_eq' RS trans) 1);
 by (rtac zmod_zmult1_eq 1);
 qed "zmod_zmult_distrib";

 Goal "b ~= (#0::int) ==> (a$*b) div b = a";
 by (asm_simp_tac (simpset() addsimps [zdiv_zmult1_eq]) 1);
 qed "zdiv_zmult_self1";

 Goal "b ~= (#0::int) ==> (b$*a) div b = a";
 by (stac zmult_commute 1 THEN etac zdiv_zmult_self1 1);
 qed "zdiv_zmult_self2";

 Addsimps [zdiv_zmult_self1, zdiv_zmult_self2];

 Goal "(a$*b) mod b = (#0::int)";
 by (simp_tac (simpset() addsimps [zmod_zmult1_eq]) 1);
 qed "zmod_zmult_self1";

 Goal "(b$*a) mod b = (#0::int)";
 by (simp_tac (simpset() addsimps [zmult_commute, zmod_zmult1_eq]) 1);
 qed "zmod_zmult_self2";

 Addsimps [zmod_zmult_self1, zmod_zmult_self2];


 (** proving (a$+b) div c = a div c $+ b div c $+ ((a mod c $+ b mod c) div c) **)

 Goal "[| 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 (auto_tac
     (claset(),
      simpset() addsimps split_ifs@
			 [quorem_def, neq_iff_zless, 
			  zadd_zmult_distrib2,
			  pos_mod_sign,pos_mod_bound,
			  neg_mod_sign, neg_mod_bound]));
 by (ALLGOALS(rtac zmod_zdiv_equality));
 val lemma = result();

 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)
 Goal "(a$+b) div c = a div c $+ b div c $+ ((a mod c $+ b mod c) div c)";
 by (zdiv_undefined_case_tac "c = #0" 1);
 by (blast_tac (claset() addIs [[quorem_div_mod,quorem_div_mod]
				MRS lemma RS quorem_div]) 1);
 qed "zdiv_zadd1_eq";

 Goal "(a$+b) mod c = (a mod c $+ b mod c) mod c";
 by (zdiv_undefined_case_tac "c = #0" 1);
 by (blast_tac (claset() addIs [[quorem_div_mod,quorem_div_mod]
				MRS lemma RS quorem_mod]) 1);
 qed "zmod_zadd1_eq";

 Goal "(a mod b) div b = (#0::int)";
 by (zdiv_undefined_case_tac "b = #0" 1);
 by (auto_tac (claset(), 
	simpset() addsimps [neq_iff_zless, 
			    pos_mod_sign, pos_mod_bound, div_pos_pos_trivial, 
			    neg_mod_sign, neg_mod_bound, div_neg_neg_trivial]));
 qed "mod_div_trivial";
 Addsimps [mod_div_trivial];

 Goal "(a mod b) mod b = a mod b";
 by (zdiv_undefined_case_tac "b = #0" 1);
 by (auto_tac (claset(), 
	simpset() addsimps [neq_iff_zless, 
			    pos_mod_sign, pos_mod_bound, mod_pos_pos_trivial, 
			    neg_mod_sign, neg_mod_bound, mod_neg_neg_trivial]));
 qed "mod_mod_trivial";
 Addsimps [mod_mod_trivial];

 Goal "(a$+b) mod c = ((a mod c) $+ b) mod c";
 by (rtac (trans RS sym) 1);
 by (rtac zmod_zadd1_eq 1);
 by (Simp_tac 1);
 by (rtac (zmod_zadd1_eq RS sym) 1);
 qed "zmod_zadd_left_eq";

 Goal "(a$+b) mod c = (a $+ (b mod c)) mod c";
 by (rtac (trans RS sym) 1);
 by (rtac zmod_zadd1_eq 1);
 by (Simp_tac 1);
 by (rtac (zmod_zadd1_eq RS sym) 1);
 qed "zmod_zadd_right_eq";


 Goal "a ~= (#0::int) ==> (a$+b) div a = b div a $+ #1";
 by (asm_simp_tac (simpset() addsimps [zdiv_zadd1_eq]) 1);
 qed "zdiv_zadd_self1";

 Goal "a ~= (#0::int) ==> (b$+a) div a = b div a $+ #1";
 by (asm_simp_tac (simpset() addsimps [zdiv_zadd1_eq]) 1);
 qed "zdiv_zadd_self2";
 Addsimps [zdiv_zadd_self1, zdiv_zadd_self2];

 Goal "(a$+b) mod a = b mod a";
 by (zdiv_undefined_case_tac "a = #0" 1);
 by (asm_simp_tac (simpset() addsimps [zmod_zadd1_eq]) 1);
 qed "zmod_zadd_self1";

 Goal "(b$+a) mod a = b mod a";
 by (zdiv_undefined_case_tac "a = #0" 1);
 by (asm_simp_tac (simpset() addsimps [zmod_zadd1_eq]) 1);
 qed "zmod_zadd_self2";
 Addsimps [zmod_zadd_self1, zmod_zadd_self2];


 (*** proving  a div (b*c) = (a div b) div c ***)

 (*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but
   7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems
   to cause particular problems.*)

 (** first, four lemmas to bound the remainder for the cases b<0 and b>0 **)

 Goal "[| (#0::int) $< c;  b $< r;  r $<= #0 |] ==> b$*c $< b$*(q mod c) $+ r";
 by (subgoal_tac "b $* (c $- q mod c) $< r $* #1" 1);
 by (asm_full_simp_tac (simpset() addsimps [zdiff_zmult_distrib2]) 1);
 by (rtac order_le_less_trans 1);
 by (etac zmult_zless_mono1 2);
 by (rtac zmult_zle_mono2_neg 1);
 by (auto_tac
     (claset(),
      simpset() addsimps zcompare_rls@
			 [zadd_commute, add1_zle_eq, pos_mod_bound]));
 val lemma1 = result();

 Goal "[| (#0::int) $< c;   b $< r;  r $<= #0 |] ==> b $* (q mod c) $+ r $<= #0";
 by (subgoal_tac "b $* (q mod c) $<= #0" 1);
 by (arith_tac 1);
 by (asm_simp_tac (simpset() addsimps [zmult_le_0_iff, pos_mod_sign]) 1);
 val lemma2 = result();

 Goal "[| (#0::int) $< c;  #0 $<= r;  r $< b |] ==> #0 $<= b $* (q mod c) $+ r";
 by (subgoal_tac "#0 $<= b $* (q mod c)" 1);
 by (arith_tac 1);
 by (asm_simp_tac (simpset() addsimps [int_0_le_mult_iff, pos_mod_sign]) 1);
 val lemma3 = result();

 Goal "[| (#0::int) $< c; #0 $<= r; r $< b |] ==> b $* (q mod c) $+ r $< b $* c";
 by (subgoal_tac "r $* #1 $< b $* (c $- q mod c)" 1);
 by (asm_full_simp_tac (simpset() addsimps [zdiff_zmult_distrib2]) 1);
 by (rtac order_less_le_trans 1);
 by (etac zmult_zless_mono1 1);
 by (rtac zmult_zle_mono2 2);
 by (auto_tac
     (claset(),
      simpset() addsimps zcompare_rls@
			 [zadd_commute, add1_zle_eq, pos_mod_bound]));
 val lemma4 = result();

 Goal "[| quorem (<a,b>, <q,r>);  b ~= #0;  #0 $< c |] \
 \     ==> quorem (<a,b$*c>, (q div c, b$*(q mod c) $+ r))";
 by (auto_tac  
     (claset(),
      simpset() addsimps zmult_ac@
			 [zmod_zdiv_equality, quorem_def, neq_iff_zless,
			  int_0_less_mult_iff,
			  zadd_zmult_distrib2 RS sym,
			  lemma1, lemma2, lemma3, lemma4]));
 val lemma = result();

 Goal "(#0::int) $< c ==> a div (b$*c) = (a div b) div c";
 by (zdiv_undefined_case_tac "b = #0" 1);
 by (force_tac (claset(),
		simpset() addsimps [quorem_div_mod RS lemma RS quorem_div, 
				    zmult_eq_0_iff]) 1);
 qed "zdiv_zmult2_eq";

 Goal "(#0::int) $< c ==> a mod (b$*c) = b$*(a div b mod c) $+ a mod b";
 by (zdiv_undefined_case_tac "b = #0" 1);
 by (force_tac (claset(),
		simpset() addsimps [quorem_div_mod RS lemma RS quorem_mod, 
				    zmult_eq_0_iff]) 1);
 qed "zmod_zmult2_eq";


 (*** Cancellation of common factors in "div" ***)

 Goal "[| (#0::int) $< b;  c ~= #0 |] ==> (c$*a) div (c$*b) = a div b";
 by (stac zdiv_zmult2_eq 1);
 by Auto_tac;
 val lemma1 = result();

 Goal "[| b $< (#0::int);  c ~= #0 |] ==> (c$*a) div (c$*b) = a div b";
 by (subgoal_tac "(c $* (-a)) div (c $* (-b)) = (-a) div (-b)" 1);
 by (rtac lemma1 2);
 by Auto_tac;
 val lemma2 = result();

 Goal "c ~= (#0::int) ==> (c$*a) div (c$*b) = a div b";
 by (zdiv_undefined_case_tac "b = #0" 1);
 by (auto_tac
     (claset(), 
      simpset() addsimps [read_instantiate [("x", "b")] neq_iff_zless, 
			  lemma1, lemma2]));
 qed "zdiv_zmult_zmult1";

 Goal "c ~= (#0::int) ==> (a$*c) div (b$*c) = a div b";
 by (dtac zdiv_zmult_zmult1 1);
 by (auto_tac (claset(), simpset() addsimps [zmult_commute]));
 qed "zdiv_zmult_zmult2";



 (*** Distribution of factors over "mod" ***)

 Goal "[| (#0::int) $< b;  c ~= #0 |] ==> (c$*a) mod (c$*b) = c $* (a mod b)";
 by (stac zmod_zmult2_eq 1);
 by Auto_tac;
 val lemma1 = result();

 Goal "[| b $< (#0::int);  c ~= #0 |] ==> (c$*a) mod (c$*b) = c $* (a mod b)";
 by (subgoal_tac "(c $* (-a)) mod (c $* (-b)) = c $* ((-a) mod (-b))" 1);
 by (rtac lemma1 2);
 by Auto_tac;
 val lemma2 = result();

 Goal "(c$*a) mod (c$*b) = c $* (a mod b)";
 by (zdiv_undefined_case_tac "b = #0" 1);
 by (zdiv_undefined_case_tac "c = #0" 1);
 by (auto_tac
     (claset(), 
      simpset() addsimps [read_instantiate [("x", "b")] neq_iff_zless, 
			  lemma1, lemma2]));
 qed "zmod_zmult_zmult1";

 Goal "(a$*c) mod (b$*c) = (a mod b) $* c";
 by (cut_inst_tac [("c","c")] zmod_zmult_zmult1 1);
 by (auto_tac (claset(), simpset() addsimps [zmult_commute]));
 qed "zmod_zmult_zmult2";


 (*** Speeding up the division algorithm with shifting ***)

 (** computing "div" by shifting **)

 Goal "(#0::int) $<= a ==> (#1 $+ #2$*b) div (#2$*a) = b div a";
 by (zdiv_undefined_case_tac "a = #0" 1);
 by (subgoal_tac "#1 $<= a" 1);
  by (arith_tac 2);
 by (subgoal_tac "#1 $< a $* #2" 1);
  by (arith_tac 2);
 by (subgoal_tac "#2$*(#1 $+ b mod a) $<= #2$*a" 1);
  by (rtac zmult_zle_mono2 2);
 by (auto_tac (claset(),
	       simpset() addsimps [zadd_commute, zmult_commute, 
				   add1_zle_eq, pos_mod_bound]));
 by (stac zdiv_zadd1_eq 1);
 by (asm_simp_tac (simpset() addsimps [zdiv_zmult_zmult2, zmod_zmult_zmult2, 
				       div_pos_pos_trivial]) 1);
 by (stac div_pos_pos_trivial 1);
 by (asm_simp_tac (simpset() 
	    addsimps [mod_pos_pos_trivial,
		     pos_mod_sign RS zadd_zle_mono1 RSN (2,order_trans)]) 1);
 by (auto_tac (claset(),
	       simpset() addsimps [mod_pos_pos_trivial]));
 by (subgoal_tac "#0 $<= b mod a" 1);
  by (asm_simp_tac (simpset() addsimps [pos_mod_sign]) 2);
 by (arith_tac 1);
 qed "pos_zdiv_mult_2";


 Goal "a $<= (#0::int) ==> (#1 $+ #2$*b) div (#2$*a) = (b$+#1) div a";
 by (subgoal_tac "(#1 $+ #2$*(-b-#1)) div (#2 $* (-a)) = (-b-#1) div (-a)" 1);
 by (rtac pos_zdiv_mult_2 2);
 by (auto_tac (claset(),
	       simpset() addsimps [zmult_zminus_right]));
 by (subgoal_tac "(#-1 - (#2 $* b)) = - (#1 $+ (#2 $* b))" 1);
 by (Simp_tac 2);
 by (asm_full_simp_tac (HOL_ss
			addsimps [zdiv_zminus_zminus, zdiff_def,
				  zminus_zadd_distrib RS sym]) 1);
 qed "neg_zdiv_mult_2";


 (*Not clear why this must be proved separately; probably number_of causes
   simplification problems*)
 Goal "~ #0 $<= x ==> x $<= (#0::int)";
 by Auto_tac;
 val lemma = result();

 Goal "number_of (v BIT b) div number_of (w BIT False) = \
 \         (if ~b | (#0::int) $<= number_of w                   \
 \          then number_of v div (number_of w)    \
 \          else (number_of v $+ (#1::int)) div (number_of w))";
 by (simp_tac (simpset_of Int.thy addsimps [zadd_assoc, number_of_BIT]) 1);
 by (asm_simp_tac (simpset()
		   delsimps bin_arith_extra_simps@bin_rel_simps
		   addsimps [zdiv_zmult_zmult1,
			     pos_zdiv_mult_2, lemma, neg_zdiv_mult_2]) 1);
 qed "zdiv_number_of_BIT";

 Addsimps [zdiv_number_of_BIT];


 (** computing "mod" by shifting (proofs resemble those for "div") **)

 Goal "(#0::int) $<= a ==> (#1 $+ #2$*b) mod (#2$*a) = #1 $+ #2 $* (b mod a)";
 by (zdiv_undefined_case_tac "a = #0" 1);
 by (subgoal_tac "#1 $<= a" 1);
  by (arith_tac 2);
 by (subgoal_tac "#1 $< a $* #2" 1);
  by (arith_tac 2);
 by (subgoal_tac "#2$*(#1 $+ b mod a) $<= #2$*a" 1);
  by (rtac zmult_zle_mono2 2);
 by (auto_tac (claset(),
	       simpset() addsimps [zadd_commute, zmult_commute, 
				   add1_zle_eq, pos_mod_bound]));
 by (stac zmod_zadd1_eq 1);
 by (asm_simp_tac (simpset() addsimps [zmod_zmult_zmult2, 
				       mod_pos_pos_trivial]) 1);
 by (rtac mod_pos_pos_trivial 1);
 by (asm_simp_tac (simpset() 
		   addsimps [mod_pos_pos_trivial,
		     pos_mod_sign RS zadd_zle_mono1 RSN (2,order_trans)]) 1);
 by (auto_tac (claset(),
	       simpset() addsimps [mod_pos_pos_trivial]));
 by (subgoal_tac "#0 $<= b mod a" 1);
  by (asm_simp_tac (simpset() addsimps [pos_mod_sign]) 2);
 by (arith_tac 1);
 qed "pos_zmod_mult_2";


 Goal "a $<= (#0::int) ==> (#1 $+ #2$*b) mod (#2$*a) = #2 $* ((b$+#1) mod a) - #1";
 by (subgoal_tac 
     "(#1 $+ #2$*(-b-#1)) mod (#2$*(-a)) = #1 $+ #2$*((-b-#1) mod (-a))" 1);
 by (rtac pos_zmod_mult_2 2);
 by (auto_tac (claset(),
	       simpset() addsimps [zmult_zminus_right]));
 by (subgoal_tac "(#-1 - (#2 $* b)) = - (#1 $+ (#2 $* b))" 1);
 by (Simp_tac 2);
 by (asm_full_simp_tac (HOL_ss
			addsimps [zmod_zminus_zminus, zdiff_def,
				  zminus_zadd_distrib RS sym]) 1);
 by (dtac (zminus_equation RS iffD1 RS sym) 1);
 by Auto_tac;
 qed "neg_zmod_mult_2";

 Goal "number_of (v BIT b) mod number_of (w BIT False) = \
 \         (if b then \
 \               if (#0::int) $<= number_of w \
 \               then #2 $* (number_of v mod number_of w) $+ #1    \
 \               else #2 $* ((number_of v $+ (#1::int)) mod number_of w) - #1  \
 \          else #2 $* (number_of v mod number_of w))";
 by (simp_tac (simpset_of Int.thy addsimps [zadd_assoc, number_of_BIT]) 1);
 by (asm_simp_tac (simpset()
		   delsimps bin_arith_extra_simps@bin_rel_simps
		   addsimps [zmod_zmult_zmult1,
			     pos_zmod_mult_2, lemma, neg_zmod_mult_2]) 1);
 qed "zmod_number_of_BIT";

 Addsimps [zmod_number_of_BIT];


 (** Quotients of signs **)

 Goal "[| a $< (#0::int);  #0 $< b |] ==> a div b $< #0";
 by (subgoal_tac "a div b $<= #-1" 1);
 by (Force_tac 1);
 by (rtac order_trans 1);
 by (res_inst_tac [("a'","#-1")]  zdiv_mono1 1);
 by (auto_tac (claset(), simpset() addsimps [zdiv_minus1]));
 qed "div_neg_pos_less0";

 Goal "[| (#0::int) $<= a;  b $< #0 |] ==> a div b $<= #0";
 by (dtac zdiv_mono1_neg 1);
 by Auto_tac;
 qed "div_nonneg_neg_le0";

 Goal "(#0::int) $< b ==> (#0 $<= a div b) = (#0 $<= a)";
 by Auto_tac;
 by (dtac zdiv_mono1 2);
 by (auto_tac (claset(), simpset() addsimps [neq_iff_zless]));
 by (full_simp_tac (simpset() addsimps [not_zless_iff_zle RS sym]) 1);
 by (blast_tac (claset() addIs [div_neg_pos_less0]) 1);
 qed "pos_imp_zdiv_nonneg_iff";

 Goal "b $< (#0::int) ==> (#0 $<= a div b) = (a $<= (#0::int))";
 by (stac (zdiv_zminus_zminus RS sym) 1);
 by (stac pos_imp_zdiv_nonneg_iff 1);
 by Auto_tac;
 qed "neg_imp_zdiv_nonneg_iff";

 (*But not (a div b $<= 0 iff a$<=0); consider a=1, b=2 when a div b = 0.*)
 Goal "(#0::int) $< b ==> (a div b $< #0) = (a $< #0)";
 by (asm_simp_tac (simpset() addsimps [linorder_not_le RS sym,
				       pos_imp_zdiv_nonneg_iff]) 1);
 qed "pos_imp_zdiv_neg_iff";

 (*Again the law fails for $<=: consider a = -1, b = -2 when a div b = 0*)
 Goal "b $< (#0::int) ==> (a div b $< #0) = (#0 $< a)";
 by (asm_simp_tac (simpset() addsimps [linorder_not_le RS sym,
				       neg_imp_zdiv_nonneg_iff]) 1);
 qed "neg_imp_zdiv_neg_iff";
*)