
diff -r 5cb663aa48ee -r 6a21ced199e3 src/HOL/IntDiv.thy
--- a/src/HOL/IntDiv.thy	Sat Oct 31 10:02:37 2009 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,1474 +0,0 @@
-(*  Title:      HOL/IntDiv.thy
-    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
-    Copyright   1999  University of Cambridge
-*)
-
-header{* The Division Operators div and mod *}
-
-theory IntDiv
-imports Int Divides FunDef
-uses
-  "~~/src/Provers/Arith/assoc_fold.ML"
-  "~~/src/Provers/Arith/cancel_numerals.ML"
-  "~~/src/Provers/Arith/combine_numerals.ML"
-  "~~/src/Provers/Arith/cancel_numeral_factor.ML"
-  "~~/src/Provers/Arith/extract_common_term.ML"
-  ("Tools/numeral_simprocs.ML")
-  ("Tools/nat_numeral_simprocs.ML")
-begin
-
-definition divmod_int_rel :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool" where
-    --{*definition of quotient and remainder*}
-    [code]: "divmod_int_rel a b = (\<lambda>(q, r). a = b * q + r \<and>
-               (if 0 < b then 0 \<le> r \<and> r < b else b < r \<and> r \<le> 0))"
-
-definition adjust :: "int \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int" where
-    --{*for the division algorithm*}
-    [code]: "adjust b = (\<lambda>(q, r). if 0 \<le> r - b then (2 * q + 1, r - b)
-                         else (2 * q, r))"
-
-text{*algorithm for the case @{text "a\<ge>0, b>0"}*}
-function posDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
-  "posDivAlg a b = (if a < b \<or>  b \<le> 0 then (0, a)
-     else adjust b (posDivAlg a (2 * b)))"
-by auto
-termination by (relation "measure (\<lambda>(a, b). nat (a - b + 1))")
-  (auto simp add: mult_2)
-
-text{*algorithm for the case @{text "a<0, b>0"}*}
-function negDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
-  "negDivAlg a b = (if 0 \<le>a + b \<or> b \<le> 0  then (-1, a + b)
-     else adjust b (negDivAlg a (2 * b)))"
-by auto
-termination by (relation "measure (\<lambda>(a, b). nat (- a - b))")
-  (auto simp add: mult_2)
-
-text{*algorithm for the general case @{term "b\<noteq>0"}*}
-definition negateSnd :: "int \<times> int \<Rightarrow> int \<times> int" where
-  [code_unfold]: "negateSnd = apsnd uminus"
-
-definition divmod_int :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
-    --{*The full division algorithm considers all possible signs for a, b
-       including the special case @{text "a=0, b<0"} because 
-       @{term negDivAlg} requires @{term "a<0"}.*}
-  "divmod_int a b = (if 0 \<le> a then if 0 \<le> 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)))"
-
-instantiation int :: Divides.div
-begin
-
-definition
-  "a div b = fst (divmod_int a b)"
-
-definition
- "a mod b = snd (divmod_int a b)"
-
-instance ..
-
-end
-
-lemma divmod_int_mod_div:
-  "divmod_int p q = (p div q, p mod q)"
-  by (auto simp add: div_int_def mod_int_def)
-
-text{*
-Here is the division algorithm in ML:
-
-\begin{verbatim}
-    fun posDivAlg (a,b) =
-      if a<b then (0,a)
-      else let val (q,r) = posDivAlg(a, 2*b)
-               in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
-           end
-
-    fun negDivAlg (a,b) =
-      if 0\<le>a+b then (~1,a+b)
-      else let val (q,r) = negDivAlg(a, 2*b)
-               in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
-           end;
-
-    fun negateSnd (q,r:int) = (q,~r);
-
-    fun divmod (a,b) = if 0\<le>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));
-\end{verbatim}
-*}
-
-
-
-subsection{*Uniqueness and Monotonicity of Quotients and Remainders*}
-
-lemma unique_quotient_lemma:
-     "[| b*q' + r'  \<le> b*q + r;  0 \<le> r';  r' < b;  r < b |]  
-      ==> q' \<le> (q::int)"
-apply (subgoal_tac "r' + b * (q'-q) \<le> r")
- prefer 2 apply (simp add: right_diff_distrib)
-apply (subgoal_tac "0 < b * (1 + q - q') ")
-apply (erule_tac [2] order_le_less_trans)
- prefer 2 apply (simp add: right_diff_distrib right_distrib)
-apply (subgoal_tac "b * q' < b * (1 + q) ")
- prefer 2 apply (simp add: right_diff_distrib right_distrib)
-apply (simp add: mult_less_cancel_left)
-done
-
-lemma unique_quotient_lemma_neg:
-     "[| b*q' + r' \<le> b*q + r;  r \<le> 0;  b < r;  b < r' |]  
-      ==> q \<le> (q'::int)"
-by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma, 
-    auto)
-
-lemma unique_quotient:
-     "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r');  b \<noteq> 0 |]  
-      ==> q = q'"
-apply (simp add: divmod_int_rel_def linorder_neq_iff split: split_if_asm)
-apply (blast intro: order_antisym
-             dest: order_eq_refl [THEN unique_quotient_lemma] 
-             order_eq_refl [THEN unique_quotient_lemma_neg] sym)+
-done
-
-
-lemma unique_remainder:
-     "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r');  b \<noteq> 0 |]  
-      ==> r = r'"
-apply (subgoal_tac "q = q'")
- apply (simp add: divmod_int_rel_def)
-apply (blast intro: unique_quotient)
-done
-
-
-subsection{*Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends*}
-
-text{*And positive divisors*}
-
-lemma adjust_eq [simp]:
-     "adjust b (q,r) = 
-      (let diff = r-b in  
-        if 0 \<le> diff then (2*q + 1, diff)   
-                     else (2*q, r))"
-by (simp add: Let_def adjust_def)
-
-declare posDivAlg.simps [simp del]
-
-text{*use with a simproc to avoid repeatedly proving the premise*}
-lemma posDivAlg_eqn:
-     "0 < b ==>  
-      posDivAlg a b = (if a<b then (0,a) else adjust b (posDivAlg a (2*b)))"
-by (rule posDivAlg.simps [THEN trans], simp)
-
-text{*Correctness of @{term posDivAlg}: it computes quotients correctly*}
-theorem posDivAlg_correct:
-  assumes "0 \<le> a" and "0 < b"
-  shows "divmod_int_rel a b (posDivAlg a b)"
-using prems apply (induct a b rule: posDivAlg.induct)
-apply auto
-apply (simp add: divmod_int_rel_def)
-apply (subst posDivAlg_eqn, simp add: right_distrib)
-apply (case_tac "a < b")
-apply simp_all
-apply (erule splitE)
-apply (auto simp add: right_distrib Let_def mult_ac mult_2_right)
-done
-
-
-subsection{*Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends*}
-
-text{*And positive divisors*}
-
-declare negDivAlg.simps [simp del]
-
-text{*use with a simproc to avoid repeatedly proving the premise*}
-lemma negDivAlg_eqn:
-     "0 < b ==>  
-      negDivAlg a b =       
-       (if 0\<le>a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))"
-by (rule negDivAlg.simps [THEN trans], simp)
-
-(*Correctness of negDivAlg: it computes quotients correctly
-  It doesn't work if a=0 because the 0/b equals 0, not -1*)
-lemma negDivAlg_correct:
-  assumes "a < 0" and "b > 0"
-  shows "divmod_int_rel a b (negDivAlg a b)"
-using prems apply (induct a b rule: negDivAlg.induct)
-apply (auto simp add: linorder_not_le)
-apply (simp add: divmod_int_rel_def)
-apply (subst negDivAlg_eqn, assumption)
-apply (case_tac "a + b < (0\<Colon>int)")
-apply simp_all
-apply (erule splitE)
-apply (auto simp add: right_distrib Let_def mult_ac mult_2_right)
-done
-
-
-subsection{*Existence Shown by Proving the Division Algorithm to be Correct*}
-
-(*the case a=0*)
-lemma divmod_int_rel_0: "b \<noteq> 0 ==> divmod_int_rel 0 b (0, 0)"
-by (auto simp add: divmod_int_rel_def linorder_neq_iff)
-
-lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)"
-by (subst posDivAlg.simps, auto)
-
-lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)"
-by (subst negDivAlg.simps, auto)
-
-lemma negateSnd_eq [simp]: "negateSnd(q,r) = (q,-r)"
-by (simp add: negateSnd_def)
-
-lemma divmod_int_rel_neg: "divmod_int_rel (-a) (-b) qr ==> divmod_int_rel a b (negateSnd qr)"
-by (auto simp add: split_ifs divmod_int_rel_def)
-
-lemma divmod_int_correct: "b \<noteq> 0 ==> divmod_int_rel a b (divmod_int a b)"
-by (force simp add: linorder_neq_iff divmod_int_rel_0 divmod_int_def divmod_int_rel_neg
-                    posDivAlg_correct negDivAlg_correct)
-
-text{*Arbitrary definitions for division by zero.  Useful to simplify 
-    certain equations.*}
-
-lemma DIVISION_BY_ZERO [simp]: "a div (0::int) = 0 & a mod (0::int) = a"
-by (simp add: div_int_def mod_int_def divmod_int_def posDivAlg.simps)  
-
-
-text{*Basic laws about division and remainder*}
-
-lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"
-apply (case_tac "b = 0", simp)
-apply (cut_tac a = a and b = b in divmod_int_correct)
-apply (auto simp add: divmod_int_rel_def div_int_def mod_int_def)
-done
-
-lemma zdiv_zmod_equality: "(b * (a div b) + (a mod b)) + k = (a::int)+k"
-by(simp add: zmod_zdiv_equality[symmetric])
-
-lemma zdiv_zmod_equality2: "((a div b) * b + (a mod b)) + k = (a::int)+k"
-by(simp add: mult_commute zmod_zdiv_equality[symmetric])
-
-text {* Tool setup *}
-
-ML {*
-local
-
-fun mk_number T n = HOLogic.number_of_const T $ HOLogic.mk_numeral n;
-
-fun find_first_numeral past (t::terms) =
-        ((snd (HOLogic.dest_number t), rev past @ terms)
-         handle TERM _ => find_first_numeral (t::past) terms)
-  | find_first_numeral past [] = raise TERM("find_first_numeral", []);
-
-val mk_plus = HOLogic.mk_binop @{const_name HOL.plus};
-
-fun mk_minus t = 
-  let val T = Term.fastype_of t
-  in Const (@{const_name HOL.uminus}, T --> T) $ t end;
-
-(*Thus mk_sum[t] yields t+0; longer sums don't have a trailing zero*)
-fun mk_sum T []        = mk_number T 0
-  | mk_sum T [t,u]     = mk_plus (t, u)
-  | mk_sum T (t :: ts) = mk_plus (t, mk_sum T ts);
-
-(*this version ALWAYS includes a trailing zero*)
-fun long_mk_sum T []        = mk_number T 0
-  | long_mk_sum T (t :: ts) = mk_plus (t, mk_sum T ts);
-
-val dest_plus = HOLogic.dest_bin @{const_name HOL.plus} Term.dummyT;
-
-(*decompose additions AND subtractions as a sum*)
-fun dest_summing (pos, Const (@{const_name HOL.plus}, _) $ t $ u, ts) =
-        dest_summing (pos, t, dest_summing (pos, u, ts))
-  | dest_summing (pos, Const (@{const_name HOL.minus}, _) $ t $ u, ts) =
-        dest_summing (pos, t, dest_summing (not pos, u, ts))
-  | dest_summing (pos, t, ts) =
-        if pos then t::ts else mk_minus t :: ts;
-
-fun dest_sum t = dest_summing (true, t, []);
-
-structure CancelDivMod = CancelDivModFun(struct
-
-  val div_name = @{const_name div};
-  val mod_name = @{const_name mod};
-  val mk_binop = HOLogic.mk_binop;
-  val mk_sum = mk_sum HOLogic.intT;
-  val dest_sum = dest_sum;
-
-  val div_mod_eqs = map mk_meta_eq [@{thm zdiv_zmod_equality}, @{thm zdiv_zmod_equality2}];
-
-  val trans = trans;
-
-  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac 
-    (@{thm diff_minus} :: @{thms add_0s} @ @{thms add_ac}))
-
-end)
-
-in
-
-val cancel_div_mod_int_proc = Simplifier.simproc @{theory}
-  "cancel_zdiv_zmod" ["(k::int) + l"] (K CancelDivMod.proc);
-
-val _ = Addsimprocs [cancel_div_mod_int_proc];
-
-end
-*}
-
-lemma pos_mod_conj : "(0::int) < b ==> 0 \<le> a mod b & a mod b < b"
-apply (cut_tac a = a and b = b in divmod_int_correct)
-apply (auto simp add: divmod_int_rel_def mod_int_def)
-done
-
-lemmas pos_mod_sign  [simp] = pos_mod_conj [THEN conjunct1, standard]
-   and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2, standard]
-
-lemma neg_mod_conj : "b < (0::int) ==> a mod b \<le> 0 & b < a mod b"
-apply (cut_tac a = a and b = b in divmod_int_correct)
-apply (auto simp add: divmod_int_rel_def div_int_def mod_int_def)
-done
-
-lemmas neg_mod_sign  [simp] = neg_mod_conj [THEN conjunct1, standard]
-   and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2, standard]
-
-
-
-subsection{*General Properties of div and mod*}
-
-lemma divmod_int_rel_div_mod: "b \<noteq> 0 ==> divmod_int_rel a b (a div b, a mod b)"
-apply (cut_tac a = a and b = b in zmod_zdiv_equality)
-apply (force simp add: divmod_int_rel_def linorder_neq_iff)
-done
-
-lemma divmod_int_rel_div: "[| divmod_int_rel a b (q, r);  b \<noteq> 0 |] ==> a div b = q"
-by (simp add: divmod_int_rel_div_mod [THEN unique_quotient])
-
-lemma divmod_int_rel_mod: "[| divmod_int_rel a b (q, r);  b \<noteq> 0 |] ==> a mod b = r"
-by (simp add: divmod_int_rel_div_mod [THEN unique_remainder])
-
-lemma div_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a div b = 0"
-apply (rule divmod_int_rel_div)
-apply (auto simp add: divmod_int_rel_def)
-done
-
-lemma div_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a div b = 0"
-apply (rule divmod_int_rel_div)
-apply (auto simp add: divmod_int_rel_def)
-done
-
-lemma div_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a div b = -1"
-apply (rule divmod_int_rel_div)
-apply (auto simp add: divmod_int_rel_def)
-done
-
-(*There is no div_neg_pos_trivial because  0 div b = 0 would supersede it*)
-
-lemma mod_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a mod b = a"
-apply (rule_tac q = 0 in divmod_int_rel_mod)
-apply (auto simp add: divmod_int_rel_def)
-done
-
-lemma mod_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a mod b = a"
-apply (rule_tac q = 0 in divmod_int_rel_mod)
-apply (auto simp add: divmod_int_rel_def)
-done
-
-lemma mod_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a mod b = a+b"
-apply (rule_tac q = "-1" in divmod_int_rel_mod)
-apply (auto simp add: divmod_int_rel_def)
-done
-
-text{*There is no @{text mod_neg_pos_trivial}.*}
-
-
-(*Simpler laws such as -a div b = -(a div b) FAIL, but see just below*)
-lemma zdiv_zminus_zminus [simp]: "(-a) div (-b) = a div (b::int)"
-apply (case_tac "b = 0", simp)
-apply (simp add: divmod_int_rel_div_mod [THEN divmod_int_rel_neg, simplified, 
-                                 THEN divmod_int_rel_div, THEN sym])
-
-done
-
-(*Simpler laws such as -a mod b = -(a mod b) FAIL, but see just below*)
-lemma zmod_zminus_zminus [simp]: "(-a) mod (-b) = - (a mod (b::int))"
-apply (case_tac "b = 0", simp)
-apply (subst divmod_int_rel_div_mod [THEN divmod_int_rel_neg, simplified, THEN divmod_int_rel_mod],
-       auto)
-done
-
-
-subsection{*Laws for div and mod with Unary Minus*}
-
-lemma zminus1_lemma:
-     "divmod_int_rel a b (q, r)
-      ==> divmod_int_rel (-a) b (if r=0 then -q else -q - 1,  
-                          if r=0 then 0 else b-r)"
-by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_diff_distrib)
-
-
-lemma zdiv_zminus1_eq_if:
-     "b \<noteq> (0::int)  
-      ==> (-a) div b =  
-          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
-by (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN divmod_int_rel_div])
-
-lemma zmod_zminus1_eq_if:
-     "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))"
-apply (case_tac "b = 0", simp)
-apply (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN divmod_int_rel_mod])
-done
-
-lemma zmod_zminus1_not_zero:
-  fixes k l :: int
-  shows "- k mod l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
-  unfolding zmod_zminus1_eq_if by auto
-
-lemma zdiv_zminus2: "a div (-b) = (-a::int) div b"
-by (cut_tac a = "-a" in zdiv_zminus_zminus, auto)
-
-lemma zmod_zminus2: "a mod (-b) = - ((-a::int) mod b)"
-by (cut_tac a = "-a" and b = b in zmod_zminus_zminus, auto)
-
-lemma zdiv_zminus2_eq_if:
-     "b \<noteq> (0::int)  
-      ==> a div (-b) =  
-          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
-by (simp add: zdiv_zminus1_eq_if zdiv_zminus2)
-
-lemma zmod_zminus2_eq_if:
-     "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"
-by (simp add: zmod_zminus1_eq_if zmod_zminus2)
-
-lemma zmod_zminus2_not_zero:
-  fixes k l :: int
-  shows "k mod - l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
-  unfolding zmod_zminus2_eq_if by auto 
-
-
-subsection{*Division of a Number by Itself*}
-
-lemma self_quotient_aux1: "[| (0::int) < a; a = r + a*q; r < a |] ==> 1 \<le> q"
-apply (subgoal_tac "0 < a*q")
- apply (simp add: zero_less_mult_iff, arith)
-done
-
-lemma self_quotient_aux2: "[| (0::int) < a; a = r + a*q; 0 \<le> r |] ==> q \<le> 1"
-apply (subgoal_tac "0 \<le> a* (1-q) ")
- apply (simp add: zero_le_mult_iff)
-apply (simp add: right_diff_distrib)
-done
-
-lemma self_quotient: "[| divmod_int_rel a a (q, r);  a \<noteq> (0::int) |] ==> q = 1"
-apply (simp add: split_ifs divmod_int_rel_def linorder_neq_iff)
-apply (rule order_antisym, safe, simp_all)
-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 (force intro: self_quotient_aux1 self_quotient_aux2 simp add: add_commute)+
-done
-
-lemma self_remainder: "[| divmod_int_rel a a (q, r);  a \<noteq> (0::int) |] ==> r = 0"
-apply (frule self_quotient, assumption)
-apply (simp add: divmod_int_rel_def)
-done
-
-lemma zdiv_self [simp]: "a \<noteq> 0 ==> a div a = (1::int)"
-by (simp add: divmod_int_rel_div_mod [THEN self_quotient])
-
-(*Here we have 0 mod 0 = 0, also assumed by Knuth (who puts m mod 0 = 0) *)
-lemma zmod_self [simp]: "a mod a = (0::int)"
-apply (case_tac "a = 0", simp)
-apply (simp add: divmod_int_rel_div_mod [THEN self_remainder])
-done
-
-
-subsection{*Computation of Division and Remainder*}
-
-lemma zdiv_zero [simp]: "(0::int) div b = 0"
-by (simp add: div_int_def divmod_int_def)
-
-lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"
-by (simp add: div_int_def divmod_int_def)
-
-lemma zmod_zero [simp]: "(0::int) mod b = 0"
-by (simp add: mod_int_def divmod_int_def)
-
-lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"
-by (simp add: mod_int_def divmod_int_def)
-
-text{*a positive, b positive *}
-
-lemma div_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a div b = fst (posDivAlg a b)"
-by (simp add: div_int_def divmod_int_def)
-
-lemma mod_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a mod b = snd (posDivAlg a b)"
-by (simp add: mod_int_def divmod_int_def)
-
-text{*a negative, b positive *}
-
-lemma div_neg_pos: "[| a < 0;  0 < b |] ==> a div b = fst (negDivAlg a b)"
-by (simp add: div_int_def divmod_int_def)
-
-lemma mod_neg_pos: "[| a < 0;  0 < b |] ==> a mod b = snd (negDivAlg a b)"
-by (simp add: mod_int_def divmod_int_def)
-
-text{*a positive, b negative *}
-
-lemma div_pos_neg:
-     "[| 0 < a;  b < 0 |] ==> a div b = fst (negateSnd (negDivAlg (-a) (-b)))"
-by (simp add: div_int_def divmod_int_def)
-
-lemma mod_pos_neg:
-     "[| 0 < a;  b < 0 |] ==> a mod b = snd (negateSnd (negDivAlg (-a) (-b)))"
-by (simp add: mod_int_def divmod_int_def)
-
-text{*a negative, b negative *}
-
-lemma div_neg_neg:
-     "[| a < 0;  b \<le> 0 |] ==> a div b = fst (negateSnd (posDivAlg (-a) (-b)))"
-by (simp add: div_int_def divmod_int_def)
-
-lemma mod_neg_neg:
-     "[| a < 0;  b \<le> 0 |] ==> a mod b = snd (negateSnd (posDivAlg (-a) (-b)))"
-by (simp add: mod_int_def divmod_int_def)
-
-text {*Simplify expresions in which div and mod combine numerical constants*}
-
-lemma divmod_int_relI:
-  "\<lbrakk>a == b * q + r; if 0 < b then 0 \<le> r \<and> r < b else b < r \<and> r \<le> 0\<rbrakk>
-    \<Longrightarrow> divmod_int_rel a b (q, r)"
-  unfolding divmod_int_rel_def by simp
-
-lemmas divmod_int_rel_div_eq = divmod_int_relI [THEN divmod_int_rel_div, THEN eq_reflection]
-lemmas divmod_int_rel_mod_eq = divmod_int_relI [THEN divmod_int_rel_mod, THEN eq_reflection]
-lemmas arithmetic_simps =
-  arith_simps
-  add_special
-  OrderedGroup.add_0_left
-  OrderedGroup.add_0_right
-  mult_zero_left
-  mult_zero_right
-  mult_1_left
-  mult_1_right
-
-(* simprocs adapted from HOL/ex/Binary.thy *)
-ML {*
-local
-  val mk_number = HOLogic.mk_number HOLogic.intT;
-  fun mk_cert u k l = @{term "plus :: int \<Rightarrow> int \<Rightarrow> int"} $
-    (@{term "times :: int \<Rightarrow> int \<Rightarrow> int"} $ u $ mk_number k) $
-      mk_number l;
-  fun prove ctxt prop = Goal.prove ctxt [] [] prop
-    (K (ALLGOALS (full_simp_tac (HOL_basic_ss addsimps @{thms arithmetic_simps}))));
-  fun binary_proc proc ss ct =
-    (case Thm.term_of ct of
-      _ $ t $ u =>
-      (case try (pairself (`(snd o HOLogic.dest_number))) (t, u) of
-        SOME args => proc (Simplifier.the_context ss) args
-      | NONE => NONE)
-    | _ => NONE);
-in
-  fun divmod_proc rule = binary_proc (fn ctxt => fn ((m, t), (n, u)) =>
-    if n = 0 then NONE
-    else let val (k, l) = Integer.div_mod m n;
-    in SOME (rule OF [prove ctxt (Logic.mk_equals (t, mk_cert u k l))]) end);
-end
-*}
-
-simproc_setup binary_int_div ("number_of m div number_of n :: int") =
-  {* K (divmod_proc (@{thm divmod_int_rel_div_eq})) *}
-
-simproc_setup binary_int_mod ("number_of m mod number_of n :: int") =
-  {* K (divmod_proc (@{thm divmod_int_rel_mod_eq})) *}
-
-lemmas posDivAlg_eqn_number_of [simp] =
-    posDivAlg_eqn [of "number_of v" "number_of w", standard]
-
-lemmas negDivAlg_eqn_number_of [simp] =
-    negDivAlg_eqn [of "number_of v" "number_of w", standard]
-
-
-text{*Special-case simplification *}
-
-lemma zmod_minus1_right [simp]: "a mod (-1::int) = 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 simp del: neg_mod_sign neg_mod_bound)
-done
-
-lemma zdiv_minus1_right [simp]: "a div (-1::int) = -a"
-by (cut_tac a = a and b = "-1" in zmod_zdiv_equality, auto)
-
-(** The last remaining special cases for constant arithmetic:
-    1 div z and 1 mod z **)
-
-lemmas div_pos_pos_1_number_of [simp] =
-    div_pos_pos [OF int_0_less_1, of "number_of w", standard]
-
-lemmas div_pos_neg_1_number_of [simp] =
-    div_pos_neg [OF int_0_less_1, of "number_of w", standard]
-
-lemmas mod_pos_pos_1_number_of [simp] =
-    mod_pos_pos [OF int_0_less_1, of "number_of w", standard]
-
-lemmas mod_pos_neg_1_number_of [simp] =
-    mod_pos_neg [OF int_0_less_1, of "number_of w", standard]
-
-
-lemmas posDivAlg_eqn_1_number_of [simp] =
-    posDivAlg_eqn [of concl: 1 "number_of w", standard]
-
-lemmas negDivAlg_eqn_1_number_of [simp] =
-    negDivAlg_eqn [of concl: 1 "number_of w", standard]
-
-
-
-subsection{*Monotonicity in the First Argument (Dividend)*}
-
-lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div 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, simp_all)
-done
-
-lemma zdiv_mono1_neg: "[| a \<le> a';  (b::int) < 0 |] ==> a' div b \<le> a div 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, simp_all)
-done
-
-
-subsection{*Monotonicity in the Second Argument (Divisor)*}
-
-lemma q_pos_lemma:
-     "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"
-apply (subgoal_tac "0 < b'* (q' + 1) ")
- apply (simp add: zero_less_mult_iff)
-apply (simp add: right_distrib)
-done
-
-lemma zdiv_mono2_lemma:
-     "[| b*q + r = b'*q' + r';  0 \<le> b'*q' + r';   
-         r' < b';  0 \<le> r;  0 < b';  b' \<le> b |]   
-      ==> q \<le> (q'::int)"
-apply (frule q_pos_lemma, assumption+) 
-apply (subgoal_tac "b*q < b* (q' + 1) ")
- apply (simp add: mult_less_cancel_left)
-apply (subgoal_tac "b*q = r' - r + b'*q'")
- prefer 2 apply simp
-apply (simp (no_asm_simp) add: right_distrib)
-apply (subst add_commute, rule zadd_zless_mono, arith)
-apply (rule mult_right_mono, auto)
-done
-
-lemma zdiv_mono2:
-     "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"
-apply (subgoal_tac "b \<noteq> 0")
- prefer 2 apply arith
-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, simp_all)
-done
-
-lemma q_neg_lemma:
-     "[| b'*q' + r' < 0;  0 \<le> r';  0 < b' |] ==> q' \<le> (0::int)"
-apply (subgoal_tac "b'*q' < 0")
- apply (simp add: mult_less_0_iff, arith)
-done
-
-lemma zdiv_mono2_neg_lemma:
-     "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;   
-         r < b;  0 \<le> r';  0 < b';  b' \<le> b |]   
-      ==> q' \<le> (q::int)"
-apply (frule q_neg_lemma, assumption+) 
-apply (subgoal_tac "b*q' < b* (q + 1) ")
- apply (simp add: mult_less_cancel_left)
-apply (simp add: right_distrib)
-apply (subgoal_tac "b*q' \<le> b'*q'")
- prefer 2 apply (simp add: mult_right_mono_neg, arith)
-done
-
-lemma zdiv_mono2_neg:
-     "[| a < (0::int);  0 < b';  b' \<le> b |] ==> a div b' \<le> a div 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 zdiv_mono2_neg_lemma)
-apply (erule subst)
-apply (erule subst, simp_all)
-done
-
-
-subsection{*More Algebraic Laws for div and mod*}
-
-text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}
-
-lemma zmult1_lemma:
-     "[| divmod_int_rel b c (q, r);  c \<noteq> 0 |]  
-      ==> divmod_int_rel (a * b) c (a*q + a*r div c, a*r mod c)"
-by (auto simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_distrib mult_ac)
-
-lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"
-apply (case_tac "c = 0", simp)
-apply (blast intro: divmod_int_rel_div_mod [THEN zmult1_lemma, THEN divmod_int_rel_div])
-done
-
-lemma zmod_zmult1_eq: "(a*b) mod c = a*(b mod c) mod (c::int)"
-apply (case_tac "c = 0", simp)
-apply (blast intro: divmod_int_rel_div_mod [THEN zmult1_lemma, THEN divmod_int_rel_mod])
-done
-
-lemma zmod_zdiv_trivial: "(a mod b) div b = (0::int)"
-apply (case_tac "b = 0", simp)
-apply (auto simp add: linorder_neq_iff div_pos_pos_trivial div_neg_neg_trivial)
-done
-
-text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}
-
-lemma zadd1_lemma:
-     "[| divmod_int_rel a c (aq, ar);  divmod_int_rel b c (bq, br);  c \<noteq> 0 |]  
-      ==> divmod_int_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"
-by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_distrib)
-
-(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
-lemma zdiv_zadd1_eq:
-     "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"
-apply (case_tac "c = 0", simp)
-apply (blast intro: zadd1_lemma [OF divmod_int_rel_div_mod divmod_int_rel_div_mod] divmod_int_rel_div)
-done
-
-instance int :: ring_div
-proof
-  fix a b c :: int
-  assume not0: "b \<noteq> 0"
-  show "(a + c * b) div b = c + a div b"
-    unfolding zdiv_zadd1_eq [of a "c * b"] using not0 
-      by (simp add: zmod_zmult1_eq zmod_zdiv_trivial zdiv_zmult1_eq)
-next
-  fix a b c :: int
-  assume "a \<noteq> 0"
-  then show "(a * b) div (a * c) = b div c"
-  proof (cases "b \<noteq> 0 \<and> c \<noteq> 0")
-    case False then show ?thesis by auto
-  next
-    case True then have "b \<noteq> 0" and "c \<noteq> 0" by auto
-    with `a \<noteq> 0`
-    have "\<And>q r. divmod_int_rel b c (q, r) \<Longrightarrow> divmod_int_rel (a * b) (a * c) (q, a * r)"
-      apply (auto simp add: divmod_int_rel_def) 
-      apply (auto simp add: algebra_simps)
-      apply (auto simp add: zero_less_mult_iff zero_le_mult_iff mult_le_0_iff mult_commute [of a] mult_less_cancel_right)
-      done
-    moreover with `c \<noteq> 0` divmod_int_rel_div_mod have "divmod_int_rel b c (b div c, b mod c)" by auto
-    ultimately have "divmod_int_rel (a * b) (a * c) (b div c, a * (b mod c))" .
-    moreover from  `a \<noteq> 0` `c \<noteq> 0` have "a * c \<noteq> 0" by simp
-    ultimately show ?thesis by (rule divmod_int_rel_div)
-  qed
-qed auto
-
-lemma posDivAlg_div_mod:
-  assumes "k \<ge> 0"
-  and "l \<ge> 0"
-  shows "posDivAlg k l = (k div l, k mod l)"
-proof (cases "l = 0")
-  case True then show ?thesis by (simp add: posDivAlg.simps)
-next
-  case False with assms posDivAlg_correct
-    have "divmod_int_rel k l (fst (posDivAlg k l), snd (posDivAlg k l))"
-    by simp
-  from divmod_int_rel_div [OF this `l \<noteq> 0`] divmod_int_rel_mod [OF this `l \<noteq> 0`]
-  show ?thesis by simp
-qed
-
-lemma negDivAlg_div_mod:
-  assumes "k < 0"
-  and "l > 0"
-  shows "negDivAlg k l = (k div l, k mod l)"
-proof -
-  from assms have "l \<noteq> 0" by simp
-  from assms negDivAlg_correct
-    have "divmod_int_rel k l (fst (negDivAlg k l), snd (negDivAlg k l))"
-    by simp
-  from divmod_int_rel_div [OF this `l \<noteq> 0`] divmod_int_rel_mod [OF this `l \<noteq> 0`]
-  show ?thesis by simp
-qed
-
-lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"
-by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
-
-(* REVISIT: should this be generalized to all semiring_div types? *)
-lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]
-
-
-subsection{*Proving  @{term "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.*)
-
-text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}
-
-lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b*c < b*(q mod c) + r"
-apply (subgoal_tac "b * (c - q mod c) < r * 1")
- apply (simp add: algebra_simps)
-apply (rule order_le_less_trans)
- apply (erule_tac [2] mult_strict_right_mono)
- apply (rule mult_left_mono_neg)
-  using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps pos_mod_bound)
- apply (simp)
-apply (simp)
-done
-
-lemma zmult2_lemma_aux2:
-     "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"
-apply (subgoal_tac "b * (q mod c) \<le> 0")
- apply arith
-apply (simp add: mult_le_0_iff)
-done
-
-lemma zmult2_lemma_aux3: "[| (0::int) < c;  0 \<le> r;  r < b |] ==> 0 \<le> b * (q mod c) + r"
-apply (subgoal_tac "0 \<le> b * (q mod c) ")
-apply arith
-apply (simp add: zero_le_mult_iff)
-done
-
-lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"
-apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
- apply (simp add: right_diff_distrib)
-apply (rule order_less_le_trans)
- apply (erule mult_strict_right_mono)
- apply (rule_tac [2] mult_left_mono)
-  apply simp
- using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps pos_mod_bound)
-apply simp
-done
-
-lemma zmult2_lemma: "[| divmod_int_rel a b (q, r);  b \<noteq> 0;  0 < c |]  
-      ==> divmod_int_rel a (b * c) (q div c, b*(q mod c) + r)"
-by (auto simp add: mult_ac divmod_int_rel_def linorder_neq_iff
-                   zero_less_mult_iff right_distrib [symmetric] 
-                   zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4)
-
-lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c"
-apply (case_tac "b = 0", simp)
-apply (force simp add: divmod_int_rel_div_mod [THEN zmult2_lemma, THEN divmod_int_rel_div])
-done
-
-lemma zmod_zmult2_eq:
-     "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"
-apply (case_tac "b = 0", simp)
-apply (force simp add: divmod_int_rel_div_mod [THEN zmult2_lemma, THEN divmod_int_rel_mod])
-done
-
-
-subsection {*Splitting Rules for div and mod*}
-
-text{*The proofs of the two lemmas below are essentially identical*}
-
-lemma split_pos_lemma:
- "0<k ==> 
-    P(n div k :: int)(n mod k) = (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i j)"
-apply (rule iffI, clarify)
- apply (erule_tac P="P ?x ?y" in rev_mp)  
- apply (subst mod_add_eq) 
- apply (subst zdiv_zadd1_eq) 
- apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)  
-txt{*converse direction*}
-apply (drule_tac x = "n div k" in spec) 
-apply (drule_tac x = "n mod k" in spec, simp)
-done
-
-lemma split_neg_lemma:
- "k<0 ==>
-    P(n div k :: int)(n mod k) = (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i j)"
-apply (rule iffI, clarify)
- apply (erule_tac P="P ?x ?y" in rev_mp)  
- apply (subst mod_add_eq) 
- apply (subst zdiv_zadd1_eq) 
- apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)  
-txt{*converse direction*}
-apply (drule_tac x = "n div k" in spec) 
-apply (drule_tac x = "n mod k" in spec, simp)
-done
-
-lemma split_zdiv:
- "P(n div k :: int) =
-  ((k = 0 --> P 0) & 
-   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) & 
-   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"
-apply (case_tac "k=0", simp)
-apply (simp only: linorder_neq_iff)
-apply (erule disjE) 
- apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"] 
-                      split_neg_lemma [of concl: "%x y. P x"])
-done
-
-lemma split_zmod:
- "P(n mod k :: int) =
-  ((k = 0 --> P n) & 
-   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) & 
-   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"
-apply (case_tac "k=0", simp)
-apply (simp only: linorder_neq_iff)
-apply (erule disjE) 
- apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"] 
-                      split_neg_lemma [of concl: "%x y. P y"])
-done
-
-(* Enable arith to deal with div 2 and mod 2: *)
-declare split_zdiv [of _ _ "number_of k", simplified, standard, arith_split]
-declare split_zmod [of _ _ "number_of k", simplified, standard, arith_split]
-
-
-subsection{*Speeding up the Division Algorithm with Shifting*}
-
-text{*computing div by shifting *}
-
-lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
-proof cases
-  assume "a=0"
-    thus ?thesis by simp
-next
-  assume "a\<noteq>0" and le_a: "0\<le>a"   
-  hence a_pos: "1 \<le> a" by arith
-  hence one_less_a2: "1 < 2 * a" by arith
-  hence le_2a: "2 * (1 + b mod a) \<le> 2 * a"
-    unfolding mult_le_cancel_left
-    by (simp add: add1_zle_eq add_commute [of 1])
-  with a_pos have "0 \<le> b mod a" by simp
-  hence le_addm: "0 \<le> 1 mod (2*a) + 2*(b mod a)"
-    by (simp add: mod_pos_pos_trivial one_less_a2)
-  with  le_2a
-  have "(1 mod (2*a) + 2*(b mod a)) div (2*a) = 0"
-    by (simp add: div_pos_pos_trivial le_addm mod_pos_pos_trivial one_less_a2
-                  right_distrib) 
-  thus ?thesis
-    by (subst zdiv_zadd1_eq,
-        simp add: mod_mult_mult1 one_less_a2
-                  div_pos_pos_trivial)
-qed
-
-lemma neg_zdiv_mult_2: "a \<le> (0::int) ==> (1 + 2*b) div (2*a) = (b+1) div a"
-apply (subgoal_tac " (1 + 2* (-b - 1)) div (2 * (-a)) = (-b - 1) div (-a) ")
-apply (rule_tac [2] pos_zdiv_mult_2)
-apply (auto simp add: right_diff_distrib)
-apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")
-apply (simp only: zdiv_zminus_zminus diff_minus minus_add_distrib [symmetric])
-apply (simp_all add: algebra_simps)
-apply (simp only: ab_diff_minus minus_add_distrib [symmetric] number_of_Min zdiv_zminus_zminus)
-done
-
-lemma zdiv_number_of_Bit0 [simp]:
-     "number_of (Int.Bit0 v) div number_of (Int.Bit0 w) =  
-          number_of v div (number_of w :: int)"
-by (simp only: number_of_eq numeral_simps) (simp add: mult_2 [symmetric])
-
-lemma zdiv_number_of_Bit1 [simp]:
-     "number_of (Int.Bit1 v) div number_of (Int.Bit0 w) =  
-          (if (0::int) \<le> number_of w                    
-           then number_of v div (number_of w)     
-           else (number_of v + (1::int)) div (number_of w))"
-apply (simp only: number_of_eq numeral_simps UNIV_I split: split_if) 
-apply (simp add: pos_zdiv_mult_2 neg_zdiv_mult_2 add_ac mult_2 [symmetric])
-done
-
-
-subsection{*Computing mod by Shifting (proofs resemble those for div)*}
-
-lemma pos_zmod_mult_2:
-  fixes a b :: int
-  assumes "0 \<le> a"
-  shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)"
-proof (cases "0 < a")
-  case False with assms show ?thesis by simp
-next
-  case True
-  then have "b mod a < a" by (rule pos_mod_bound)
-  then have "1 + b mod a \<le> a" by simp
-  then have A: "2 * (1 + b mod a) \<le> 2 * a" by simp
-  from `0 < a` have "0 \<le> b mod a" by (rule pos_mod_sign)
-  then have B: "0 \<le> 1 + 2 * (b mod a)" by simp
-  have "((1\<Colon>int) mod ((2\<Colon>int) * a) + (2\<Colon>int) * b mod ((2\<Colon>int) * a)) mod ((2\<Colon>int) * a) = (1\<Colon>int) + (2\<Colon>int) * (b mod a)"
-    using `0 < a` and A
-    by (auto simp add: mod_mult_mult1 mod_pos_pos_trivial ring_distribs intro!: mod_pos_pos_trivial B)
-  then show ?thesis by (subst mod_add_eq)
-qed
-
-lemma neg_zmod_mult_2:
-  fixes a b :: int
-  assumes "a \<le> 0"
-  shows "(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1"
-proof -
-  from assms have "0 \<le> - a" by auto
-  then have "(1 + 2 * (- b - 1)) mod (2 * (- a)) = 1 + 2 * ((- b - 1) mod (- a))"
-    by (rule pos_zmod_mult_2)
-  then show ?thesis by (simp add: zmod_zminus2 algebra_simps)
-     (simp add: diff_minus add_ac)
-qed
-
-lemma zmod_number_of_Bit0 [simp]:
-     "number_of (Int.Bit0 v) mod number_of (Int.Bit0 w) =  
-      (2::int) * (number_of v mod number_of w)"
-apply (simp only: number_of_eq numeral_simps) 
-apply (simp add: mod_mult_mult1 pos_zmod_mult_2 
-                 neg_zmod_mult_2 add_ac mult_2 [symmetric])
-done
-
-lemma zmod_number_of_Bit1 [simp]:
-     "number_of (Int.Bit1 v) mod number_of (Int.Bit0 w) =  
-      (if (0::int) \<le> 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)"
-apply (simp only: number_of_eq numeral_simps) 
-apply (simp add: mod_mult_mult1 pos_zmod_mult_2 
-                 neg_zmod_mult_2 add_ac mult_2 [symmetric])
-done
-
-
-subsection{*Quotients of Signs*}
-
-lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"
-apply (subgoal_tac "a div b \<le> -1", force)
-apply (rule order_trans)
-apply (rule_tac a' = "-1" in zdiv_mono1)
-apply (auto simp add: div_eq_minus1)
-done
-
-lemma div_nonneg_neg_le0: "[| (0::int) \<le> a; b < 0 |] ==> a div b \<le> 0"
-by (drule zdiv_mono1_neg, auto)
-
-lemma div_nonpos_pos_le0: "[| (a::int) \<le> 0; b > 0 |] ==> a div b \<le> 0"
-by (drule zdiv_mono1, auto)
-
-lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"
-apply auto
-apply (drule_tac [2] zdiv_mono1)
-apply (auto simp add: linorder_neq_iff)
-apply (simp (no_asm_use) add: linorder_not_less [symmetric])
-apply (blast intro: div_neg_pos_less0)
-done
-
-lemma neg_imp_zdiv_nonneg_iff:
-     "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"
-apply (subst zdiv_zminus_zminus [symmetric])
-apply (subst pos_imp_zdiv_nonneg_iff, auto)
-done
-
-(*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
-lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
-by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
-
-(*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
-lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
-by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
-
-
-subsection {* The Divides Relation *}
-
-lemmas zdvd_iff_zmod_eq_0_number_of [simp] =
-  dvd_eq_mod_eq_0 [of "number_of x::int" "number_of y::int", standard]
-
-lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"
-  by (rule dvd_mod) (* TODO: remove *)
-
-lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)"
-  by (rule dvd_mod_imp_dvd) (* TODO: remove *)
-
-lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"
-  using zmod_zdiv_equality[where a="m" and b="n"]
-  by (simp add: algebra_simps)
-
-lemma zpower_zmod: "((x::int) mod m)^y mod m = x^y mod m"
-apply (induct "y", auto)
-apply (rule zmod_zmult1_eq [THEN trans])
-apply (simp (no_asm_simp))
-apply (rule mod_mult_eq [symmetric])
-done
-
-lemma zdiv_int: "int (a div b) = (int a) div (int b)"
-apply (subst split_div, auto)
-apply (subst split_zdiv, auto)
-apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in IntDiv.unique_quotient)
-apply (auto simp add: IntDiv.divmod_int_rel_def of_nat_mult)
-done
-
-lemma zmod_int: "int (a mod b) = (int a) mod (int b)"
-apply (subst split_mod, auto)
-apply (subst split_zmod, auto)
-apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia 
-       in unique_remainder)
-apply (auto simp add: IntDiv.divmod_int_rel_def of_nat_mult)
-done
-
-lemma abs_div: "(y::int) dvd x \<Longrightarrow> abs (x div y) = abs x div abs y"
-by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)
-
-lemma zdvd_mult_div_cancel:"(n::int) dvd m \<Longrightarrow> n * (m div n) = m"
-apply (subgoal_tac "m mod n = 0")
- apply (simp add: zmult_div_cancel)
-apply (simp only: dvd_eq_mod_eq_0)
-done
-
-text{*Suggested by Matthias Daum*}
-lemma int_power_div_base:
-     "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"
-apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")
- apply (erule ssubst)
- apply (simp only: power_add)
- apply simp_all
-done
-
-text {* by Brian Huffman *}
-lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"
-by (rule mod_minus_eq [symmetric])
-
-lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"
-by (rule mod_diff_left_eq [symmetric])
-
-lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"
-by (rule mod_diff_right_eq [symmetric])
-
-lemmas zmod_simps =
-  mod_add_left_eq  [symmetric]
-  mod_add_right_eq [symmetric]
-  zmod_zmult1_eq   [symmetric]
-  mod_mult_left_eq [symmetric]
-  zpower_zmod
-  zminus_zmod zdiff_zmod_left zdiff_zmod_right
-
-text {* Distributive laws for function @{text nat}. *}
-
-lemma nat_div_distrib: "0 \<le> x \<Longrightarrow> nat (x div y) = nat x div nat y"
-apply (rule linorder_cases [of y 0])
-apply (simp add: div_nonneg_neg_le0)
-apply simp
-apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)
-done
-
-(*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)
-lemma nat_mod_distrib:
-  "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x mod y) = nat x mod nat y"
-apply (case_tac "y = 0", simp add: DIVISION_BY_ZERO)
-apply (simp add: nat_eq_iff zmod_int)
-done
-
-text{*Suggested by Matthias Daum*}
-lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)"
-apply (subgoal_tac "nat x div nat k < nat x")
- apply (simp (asm_lr) add: nat_div_distrib [symmetric])
-apply (rule Divides.div_less_dividend, simp_all)
-done
-
-text {* code generator setup *}
-
-context ring_1
-begin
-
-lemma of_int_num [code]:
-  "of_int k = (if k = 0 then 0 else if k < 0 then
-     - of_int (- k) else let
-       (l, m) = divmod_int k 2;
-       l' = of_int l
-     in if m = 0 then l' + l' else l' + l' + 1)"
-proof -
-  have aux1: "k mod (2\<Colon>int) \<noteq> (0\<Colon>int) \<Longrightarrow> 
-    of_int k = of_int (k div 2 * 2 + 1)"
-  proof -
-    have "k mod 2 < 2" by (auto intro: pos_mod_bound)
-    moreover have "0 \<le> k mod 2" by (auto intro: pos_mod_sign)
-    moreover assume "k mod 2 \<noteq> 0"
-    ultimately have "k mod 2 = 1" by arith
-    moreover have "of_int k = of_int (k div 2 * 2 + k mod 2)" by simp
-    ultimately show ?thesis by auto
-  qed
-  have aux2: "\<And>x. of_int 2 * x = x + x"
-  proof -
-    fix x
-    have int2: "(2::int) = 1 + 1" by arith
-    show "of_int 2 * x = x + x"
-    unfolding int2 of_int_add left_distrib by simp
-  qed
-  have aux3: "\<And>x. x * of_int 2 = x + x"
-  proof -
-    fix x
-    have int2: "(2::int) = 1 + 1" by arith
-    show "x * of_int 2 = x + x" 
-    unfolding int2 of_int_add right_distrib by simp
-  qed
-  from aux1 show ?thesis by (auto simp add: divmod_int_mod_div Let_def aux2 aux3)
-qed
-
-end
-
-lemma zmod_eq_dvd_iff: "(x::int) mod n = y mod n \<longleftrightarrow> n dvd x - y"
-proof
-  assume H: "x mod n = y mod n"
-  hence "x mod n - y mod n = 0" by simp
-  hence "(x mod n - y mod n) mod n = 0" by simp 
-  hence "(x - y) mod n = 0" by (simp add: mod_diff_eq[symmetric])
-  thus "n dvd x - y" by (simp add: dvd_eq_mod_eq_0)
-next
-  assume H: "n dvd x - y"
-  then obtain k where k: "x-y = n*k" unfolding dvd_def by blast
-  hence "x = n*k + y" by simp
-  hence "x mod n = (n*k + y) mod n" by simp
-  thus "x mod n = y mod n" by (simp add: mod_add_left_eq)
-qed
-
-lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y  mod n" and xy:"y \<le> x"
-  shows "\<exists>q. x = y + n * q"
-proof-
-  from xy have th: "int x - int y = int (x - y)" by simp 
-  from xyn have "int x mod int n = int y mod int n" 
-    by (simp add: zmod_int[symmetric])
-  hence "int n dvd int x - int y" by (simp only: zmod_eq_dvd_iff[symmetric]) 
-  hence "n dvd x - y" by (simp add: th zdvd_int)
-  then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith
-qed
-
-lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)" 
-  (is "?lhs = ?rhs")
-proof
-  assume H: "x mod n = y mod n"
-  {assume xy: "x \<le> y"
-    from H have th: "y mod n = x mod n" by simp
-    from nat_mod_eq_lemma[OF th xy] have ?rhs 
-      apply clarify  apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}
-  moreover
-  {assume xy: "y \<le> x"
-    from nat_mod_eq_lemma[OF H xy] have ?rhs 
-      apply clarify  apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}
-  ultimately  show ?rhs using linear[of x y] by blast  
-next
-  assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast
-  hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp
-  thus  ?lhs by simp
-qed
-
-lemma div_nat_number_of [simp]:
-     "(number_of v :: nat)  div  number_of v' =  
-          (if neg (number_of v :: int) then 0  
-           else nat (number_of v div number_of v'))"
-  unfolding nat_number_of_def number_of_is_id neg_def
-  by (simp add: nat_div_distrib)
-
-lemma one_div_nat_number_of [simp]:
-     "Suc 0 div number_of v' = nat (1 div number_of v')" 
-by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric]) 
-
-lemma mod_nat_number_of [simp]:
-     "(number_of v :: nat)  mod  number_of v' =  
-        (if neg (number_of v :: int) then 0  
-         else if neg (number_of v' :: int) then number_of v  
-         else nat (number_of v mod number_of v'))"
-  unfolding nat_number_of_def number_of_is_id neg_def
-  by (simp add: nat_mod_distrib)
-
-lemma one_mod_nat_number_of [simp]:
-     "Suc 0 mod number_of v' =  
-        (if neg (number_of v' :: int) then Suc 0
-         else nat (1 mod number_of v'))"
-by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric]) 
-
-lemmas dvd_eq_mod_eq_0_number_of =
-  dvd_eq_mod_eq_0 [of "number_of x" "number_of y", standard]
-
-declare dvd_eq_mod_eq_0_number_of [simp]
-
-
-subsection {* Transfer setup *}
-
-lemma transfer_nat_int_functions:
-    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) div (nat y) = nat (x div y)"
-    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) mod (nat y) = nat (x mod y)"
-  by (auto simp add: nat_div_distrib nat_mod_distrib)
-
-lemma transfer_nat_int_function_closures:
-    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x div y >= 0"
-    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x mod y >= 0"
-  apply (cases "y = 0")
-  apply (auto simp add: pos_imp_zdiv_nonneg_iff)
-  apply (cases "y = 0")
-  apply auto
-done
-
-declare TransferMorphism_nat_int[transfer add return:
-  transfer_nat_int_functions
-  transfer_nat_int_function_closures
-]
-
-lemma transfer_int_nat_functions:
-    "(int x) div (int y) = int (x div y)"
-    "(int x) mod (int y) = int (x mod y)"
-  by (auto simp add: zdiv_int zmod_int)
-
-lemma transfer_int_nat_function_closures:
-    "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x div y)"
-    "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x mod y)"
-  by (simp_all only: is_nat_def transfer_nat_int_function_closures)
-
-declare TransferMorphism_int_nat[transfer add return:
-  transfer_int_nat_functions
-  transfer_int_nat_function_closures
-]
-
-
-subsection {* Code generation *}
-
-definition pdivmod :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
-  "pdivmod k l = (\<bar>k\<bar> div \<bar>l\<bar>, \<bar>k\<bar> mod \<bar>l\<bar>)"
-
-lemma pdivmod_posDivAlg [code]:
-  "pdivmod k l = (if l = 0 then (0, \<bar>k\<bar>) else posDivAlg \<bar>k\<bar> \<bar>l\<bar>)"
-by (subst posDivAlg_div_mod) (simp_all add: pdivmod_def)
-
-lemma divmod_int_pdivmod: "divmod_int k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else
-  apsnd ((op *) (sgn l)) (if 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0
-    then pdivmod k l
-    else (let (r, s) = pdivmod k l in
-      if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
-proof -
-  have aux: "\<And>q::int. - k = l * q \<longleftrightarrow> k = l * - q" by auto
-  show ?thesis
-    by (simp add: divmod_int_mod_div pdivmod_def)
-      (auto simp add: aux not_less not_le zdiv_zminus1_eq_if
-      zmod_zminus1_eq_if zdiv_zminus2_eq_if zmod_zminus2_eq_if)
-qed
-
-lemma divmod_int_code [code]: "divmod_int k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else
-  apsnd ((op *) (sgn l)) (if sgn k = sgn l
-    then pdivmod k l
-    else (let (r, s) = pdivmod k l in
-      if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
-proof -
-  have "k \<noteq> 0 \<Longrightarrow> l \<noteq> 0 \<Longrightarrow> 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0 \<longleftrightarrow> sgn k = sgn l"
-    by (auto simp add: not_less sgn_if)
-  then show ?thesis by (simp add: divmod_int_pdivmod)
-qed
-
-code_modulename SML
-  IntDiv Integer
-
-code_modulename OCaml
-  IntDiv Integer
-
-code_modulename Haskell
-  IntDiv Integer
-
-
-
-subsection {* Proof Tools setup; Combination and Cancellation Simprocs *}
-
-declare split_div[of _ _ "number_of k", standard, arith_split]
-declare split_mod[of _ _ "number_of k", standard, arith_split]
-
-
-subsubsection{*For @{text combine_numerals}*}
-
-lemma left_add_mult_distrib: "i*u + (j*u + k) = (i+j)*u + (k::nat)"
-by (simp add: add_mult_distrib)
-
-
-subsubsection{*For @{text cancel_numerals}*}
-
-lemma nat_diff_add_eq1:
-     "j <= (i::nat) ==> ((i*u + m) - (j*u + n)) = (((i-j)*u + m) - n)"
-by (simp split add: nat_diff_split add: add_mult_distrib)
-
-lemma nat_diff_add_eq2:
-     "i <= (j::nat) ==> ((i*u + m) - (j*u + n)) = (m - ((j-i)*u + n))"
-by (simp split add: nat_diff_split add: add_mult_distrib)
-
-lemma nat_eq_add_iff1:
-     "j <= (i::nat) ==> (i*u + m = j*u + n) = ((i-j)*u + m = n)"
-by (auto split add: nat_diff_split simp add: add_mult_distrib)
-
-lemma nat_eq_add_iff2:
-     "i <= (j::nat) ==> (i*u + m = j*u + n) = (m = (j-i)*u + n)"
-by (auto split add: nat_diff_split simp add: add_mult_distrib)
-
-lemma nat_less_add_iff1:
-     "j <= (i::nat) ==> (i*u + m < j*u + n) = ((i-j)*u + m < n)"
-by (auto split add: nat_diff_split simp add: add_mult_distrib)
-
-lemma nat_less_add_iff2:
-     "i <= (j::nat) ==> (i*u + m < j*u + n) = (m < (j-i)*u + n)"
-by (auto split add: nat_diff_split simp add: add_mult_distrib)
-
-lemma nat_le_add_iff1:
-     "j <= (i::nat) ==> (i*u + m <= j*u + n) = ((i-j)*u + m <= n)"
-by (auto split add: nat_diff_split simp add: add_mult_distrib)
-
-lemma nat_le_add_iff2:
-     "i <= (j::nat) ==> (i*u + m <= j*u + n) = (m <= (j-i)*u + n)"
-by (auto split add: nat_diff_split simp add: add_mult_distrib)
-
-
-subsubsection{*For @{text cancel_numeral_factors} *}
-
-lemma nat_mult_le_cancel1: "(0::nat) < k ==> (k*m <= k*n) = (m<=n)"
-by auto
-
-lemma nat_mult_less_cancel1: "(0::nat) < k ==> (k*m < k*n) = (m<n)"
-by auto
-
-lemma nat_mult_eq_cancel1: "(0::nat) < k ==> (k*m = k*n) = (m=n)"
-by auto
-
-lemma nat_mult_div_cancel1: "(0::nat) < k ==> (k*m) div (k*n) = (m div n)"
-by auto
-
-lemma nat_mult_dvd_cancel_disj[simp]:
-  "(k*m) dvd (k*n) = (k=0 | m dvd (n::nat))"
-by(auto simp: dvd_eq_mod_eq_0 mod_mult_distrib2[symmetric])
-
-lemma nat_mult_dvd_cancel1: "0 < k \<Longrightarrow> (k*m) dvd (k*n::nat) = (m dvd n)"
-by(auto)
-
-
-subsubsection{*For @{text cancel_factor} *}
-
-lemma nat_mult_le_cancel_disj: "(k*m <= k*n) = ((0::nat) < k --> m<=n)"
-by auto
-
-lemma nat_mult_less_cancel_disj: "(k*m < k*n) = ((0::nat) < k & m<n)"
-by auto
-
-lemma nat_mult_eq_cancel_disj: "(k*m = k*n) = (k = (0::nat) | m=n)"
-by auto
-
-lemma nat_mult_div_cancel_disj[simp]:
-     "(k*m) div (k*n) = (if k = (0::nat) then 0 else m div n)"
-by (simp add: nat_mult_div_cancel1)
-
-
-use "Tools/numeral_simprocs.ML"
-
-use "Tools/nat_numeral_simprocs.ML"
-
-declaration {* 
-  K (Lin_Arith.add_simps (@{thms neg_simps} @ [@{thm Suc_nat_number_of}, @{thm int_nat_number_of}])
-  #> Lin_Arith.add_simps (@{thms ring_distribs} @ [@{thm Let_number_of}, @{thm Let_0}, @{thm Let_1},
-     @{thm nat_0}, @{thm nat_1},
-     @{thm add_nat_number_of}, @{thm diff_nat_number_of}, @{thm mult_nat_number_of},
-     @{thm eq_nat_number_of}, @{thm less_nat_number_of}, @{thm le_number_of_eq_not_less},
-     @{thm le_Suc_number_of}, @{thm le_number_of_Suc},
-     @{thm less_Suc_number_of}, @{thm less_number_of_Suc},
-     @{thm Suc_eq_number_of}, @{thm eq_number_of_Suc},
-     @{thm mult_Suc}, @{thm mult_Suc_right},
-     @{thm add_Suc}, @{thm add_Suc_right},
-     @{thm eq_number_of_0}, @{thm eq_0_number_of}, @{thm less_0_number_of},
-     @{thm of_int_number_of_eq}, @{thm of_nat_number_of_eq}, @{thm nat_number_of},
-     @{thm if_True}, @{thm if_False}])
-  #> Lin_Arith.add_simprocs (Numeral_Simprocs.assoc_fold_simproc
-      :: Numeral_Simprocs.combine_numerals
-      :: Numeral_Simprocs.cancel_numerals)
-  #> Lin_Arith.add_simprocs (Nat_Numeral_Simprocs.combine_numerals :: Nat_Numeral_Simprocs.cancel_numerals))
-*}
-
-end




