
wenzelm@23164: (*  Title:      HOL/IntDiv.thy
wenzelm@23164:     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
wenzelm@23164:     Copyright   1999  University of Cambridge
wenzelm@23164: 
wenzelm@23164: *)
wenzelm@23164: 
haftmann@29651: header{* The Division Operators div and mod *}
wenzelm@23164: 
wenzelm@23164: theory IntDiv
haftmann@25919: imports Int Divides FunDef
wenzelm@23164: begin
wenzelm@23164: 
haftmann@29651: definition divmod_rel :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool" where
wenzelm@23164:     --{*definition of quotient and remainder*}
haftmann@29651:     [code]: "divmod_rel a b = (\<lambda>(q, r). a = b * q + r \<and>
haftmann@29651:                (if 0 < b then 0 \<le> r \<and> r < b else b < r \<and> r \<le> 0))"
wenzelm@23164: 
haftmann@29651: definition adjust :: "int \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int" where
wenzelm@23164:     --{*for the division algorithm*}
haftmann@29651:     [code]: "adjust b = (\<lambda>(q, r). if 0 \<le> r - b then (2 * q + 1, r - b)
haftmann@29651:                          else (2 * q, r))"
wenzelm@23164: 
wenzelm@23164: text{*algorithm for the case @{text "a\<ge>0, b>0"}*}
haftmann@29651: function posDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
haftmann@29651:   "posDivAlg a b = (if a < b \<or>  b \<le> 0 then (0, a)
haftmann@29651:      else adjust b (posDivAlg a (2 * b)))"
wenzelm@23164: by auto
haftmann@29651: termination by (relation "measure (\<lambda>(a, b). nat (a - b + 1))") auto
wenzelm@23164: 
wenzelm@23164: text{*algorithm for the case @{text "a<0, b>0"}*}
haftmann@29651: function negDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
haftmann@29651:   "negDivAlg a b = (if 0 \<le>a + b \<or> b \<le> 0  then (-1, a + b)
haftmann@29651:      else adjust b (negDivAlg a (2 * b)))"
wenzelm@23164: by auto
haftmann@29651: termination by (relation "measure (\<lambda>(a, b). nat (- a - b))") auto
wenzelm@23164: 
wenzelm@23164: text{*algorithm for the general case @{term "b\<noteq>0"}*}
haftmann@29651: definition negateSnd :: "int \<times> int \<Rightarrow> int \<times> int" where
haftmann@29651:   [code inline]: "negateSnd = apsnd uminus"
wenzelm@23164: 
haftmann@29651: definition divmod :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
wenzelm@23164:     --{*The full division algorithm considers all possible signs for a, b
wenzelm@23164:        including the special case @{text "a=0, b<0"} because 
wenzelm@23164:        @{term negDivAlg} requires @{term "a<0"}.*}
haftmann@29651:   "divmod a b = (if 0 \<le> a then if 0 \<le> b then posDivAlg a b
haftmann@29651:                   else if a = 0 then (0, 0)
wenzelm@23164:                        else negateSnd (negDivAlg (-a) (-b))
wenzelm@23164:                else 
haftmann@29651:                   if 0 < b then negDivAlg a b
haftmann@29651:                   else negateSnd (posDivAlg (-a) (-b)))"
wenzelm@23164: 
haftmann@25571: instantiation int :: Divides.div
haftmann@25571: begin
haftmann@25571: 
haftmann@25571: definition
haftmann@29651:   div_def: "a div b = fst (divmod a b)"
haftmann@25571: 
haftmann@25571: definition
haftmann@29651:   mod_def: "a mod b = snd (divmod a b)"
haftmann@25571: 
haftmann@25571: instance ..
haftmann@25571: 
haftmann@25571: end
wenzelm@23164: 
haftmann@29651: lemma divmod_mod_div:
haftmann@29651:   "divmod p q = (p div q, p mod q)"
wenzelm@23164:   by (auto simp add: div_def mod_def)
wenzelm@23164: 
wenzelm@23164: text{*
wenzelm@23164: Here is the division algorithm in ML:
wenzelm@23164: 
wenzelm@23164: \begin{verbatim}
wenzelm@23164:     fun posDivAlg (a,b) =
wenzelm@23164:       if a<b then (0,a)
wenzelm@23164:       else let val (q,r) = posDivAlg(a, 2*b)
wenzelm@23164: 	       in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
wenzelm@23164: 	   end
wenzelm@23164: 
wenzelm@23164:     fun negDivAlg (a,b) =
wenzelm@23164:       if 0\<le>a+b then (~1,a+b)
wenzelm@23164:       else let val (q,r) = negDivAlg(a, 2*b)
wenzelm@23164: 	       in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
wenzelm@23164: 	   end;
wenzelm@23164: 
wenzelm@23164:     fun negateSnd (q,r:int) = (q,~r);
wenzelm@23164: 
haftmann@29651:     fun divmod (a,b) = if 0\<le>a then 
wenzelm@23164: 			  if b>0 then posDivAlg (a,b) 
wenzelm@23164: 			   else if a=0 then (0,0)
wenzelm@23164: 				else negateSnd (negDivAlg (~a,~b))
wenzelm@23164: 		       else 
wenzelm@23164: 			  if 0<b then negDivAlg (a,b)
wenzelm@23164: 			  else        negateSnd (posDivAlg (~a,~b));
wenzelm@23164: \end{verbatim}
wenzelm@23164: *}
wenzelm@23164: 
wenzelm@23164: 
wenzelm@23164: 
wenzelm@23164: subsection{*Uniqueness and Monotonicity of Quotients and Remainders*}
wenzelm@23164: 
wenzelm@23164: lemma unique_quotient_lemma:
wenzelm@23164:      "[| b*q' + r'  \<le> b*q + r;  0 \<le> r';  r' < b;  r < b |]  
wenzelm@23164:       ==> q' \<le> (q::int)"
wenzelm@23164: apply (subgoal_tac "r' + b * (q'-q) \<le> r")
wenzelm@23164:  prefer 2 apply (simp add: right_diff_distrib)
wenzelm@23164: apply (subgoal_tac "0 < b * (1 + q - q') ")
wenzelm@23164: apply (erule_tac [2] order_le_less_trans)
wenzelm@23164:  prefer 2 apply (simp add: right_diff_distrib right_distrib)
wenzelm@23164: apply (subgoal_tac "b * q' < b * (1 + q) ")
wenzelm@23164:  prefer 2 apply (simp add: right_diff_distrib right_distrib)
wenzelm@23164: apply (simp add: mult_less_cancel_left)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: lemma unique_quotient_lemma_neg:
wenzelm@23164:      "[| b*q' + r' \<le> b*q + r;  r \<le> 0;  b < r;  b < r' |]  
wenzelm@23164:       ==> q \<le> (q'::int)"
wenzelm@23164: by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma, 
wenzelm@23164:     auto)
wenzelm@23164: 
wenzelm@23164: lemma unique_quotient:
haftmann@29651:      "[| divmod_rel a b (q, r); divmod_rel a b (q', r');  b \<noteq> 0 |]  
wenzelm@23164:       ==> q = q'"
haftmann@29651: apply (simp add: divmod_rel_def linorder_neq_iff split: split_if_asm)
wenzelm@23164: apply (blast intro: order_antisym
wenzelm@23164:              dest: order_eq_refl [THEN unique_quotient_lemma] 
wenzelm@23164:              order_eq_refl [THEN unique_quotient_lemma_neg] sym)+
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: 
wenzelm@23164: lemma unique_remainder:
haftmann@29651:      "[| divmod_rel a b (q, r); divmod_rel a b (q', r');  b \<noteq> 0 |]  
wenzelm@23164:       ==> r = r'"
wenzelm@23164: apply (subgoal_tac "q = q'")
haftmann@29651:  apply (simp add: divmod_rel_def)
wenzelm@23164: apply (blast intro: unique_quotient)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: 
wenzelm@23164: subsection{*Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends*}
wenzelm@23164: 
wenzelm@23164: text{*And positive divisors*}
wenzelm@23164: 
wenzelm@23164: lemma adjust_eq [simp]:
wenzelm@23164:      "adjust b (q,r) = 
wenzelm@23164:       (let diff = r-b in  
wenzelm@23164: 	if 0 \<le> diff then (2*q + 1, diff)   
wenzelm@23164:                      else (2*q, r))"
wenzelm@23164: by (simp add: Let_def adjust_def)
wenzelm@23164: 
wenzelm@23164: declare posDivAlg.simps [simp del]
wenzelm@23164: 
wenzelm@23164: text{*use with a simproc to avoid repeatedly proving the premise*}
wenzelm@23164: lemma posDivAlg_eqn:
wenzelm@23164:      "0 < b ==>  
wenzelm@23164:       posDivAlg a b = (if a<b then (0,a) else adjust b (posDivAlg a (2*b)))"
wenzelm@23164: by (rule posDivAlg.simps [THEN trans], simp)
wenzelm@23164: 
wenzelm@23164: text{*Correctness of @{term posDivAlg}: it computes quotients correctly*}
wenzelm@23164: theorem posDivAlg_correct:
wenzelm@23164:   assumes "0 \<le> a" and "0 < b"
haftmann@29651:   shows "divmod_rel a b (posDivAlg a b)"
wenzelm@23164: using prems apply (induct a b rule: posDivAlg.induct)
wenzelm@23164: apply auto
haftmann@29651: apply (simp add: divmod_rel_def)
wenzelm@23164: apply (subst posDivAlg_eqn, simp add: right_distrib)
wenzelm@23164: apply (case_tac "a < b")
wenzelm@23164: apply simp_all
wenzelm@23164: apply (erule splitE)
wenzelm@23164: apply (auto simp add: right_distrib Let_def)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: 
wenzelm@23164: subsection{*Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends*}
wenzelm@23164: 
wenzelm@23164: text{*And positive divisors*}
wenzelm@23164: 
wenzelm@23164: declare negDivAlg.simps [simp del]
wenzelm@23164: 
wenzelm@23164: text{*use with a simproc to avoid repeatedly proving the premise*}
wenzelm@23164: lemma negDivAlg_eqn:
wenzelm@23164:      "0 < b ==>  
wenzelm@23164:       negDivAlg a b =       
wenzelm@23164:        (if 0\<le>a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))"
wenzelm@23164: by (rule negDivAlg.simps [THEN trans], simp)
wenzelm@23164: 
wenzelm@23164: (*Correctness of negDivAlg: it computes quotients correctly
wenzelm@23164:   It doesn't work if a=0 because the 0/b equals 0, not -1*)
wenzelm@23164: lemma negDivAlg_correct:
wenzelm@23164:   assumes "a < 0" and "b > 0"
haftmann@29651:   shows "divmod_rel a b (negDivAlg a b)"
wenzelm@23164: using prems apply (induct a b rule: negDivAlg.induct)
wenzelm@23164: apply (auto simp add: linorder_not_le)
haftmann@29651: apply (simp add: divmod_rel_def)
wenzelm@23164: apply (subst negDivAlg_eqn, assumption)
wenzelm@23164: apply (case_tac "a + b < (0\<Colon>int)")
wenzelm@23164: apply simp_all
wenzelm@23164: apply (erule splitE)
wenzelm@23164: apply (auto simp add: right_distrib Let_def)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: 
wenzelm@23164: subsection{*Existence Shown by Proving the Division Algorithm to be Correct*}
wenzelm@23164: 
wenzelm@23164: (*the case a=0*)
haftmann@29651: lemma divmod_rel_0: "b \<noteq> 0 ==> divmod_rel 0 b (0, 0)"
haftmann@29651: by (auto simp add: divmod_rel_def linorder_neq_iff)
wenzelm@23164: 
wenzelm@23164: lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)"
wenzelm@23164: by (subst posDivAlg.simps, auto)
wenzelm@23164: 
wenzelm@23164: lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)"
wenzelm@23164: by (subst negDivAlg.simps, auto)
wenzelm@23164: 
wenzelm@23164: lemma negateSnd_eq [simp]: "negateSnd(q,r) = (q,-r)"
wenzelm@23164: by (simp add: negateSnd_def)
wenzelm@23164: 
haftmann@29651: lemma divmod_rel_neg: "divmod_rel (-a) (-b) qr ==> divmod_rel a b (negateSnd qr)"
haftmann@29651: by (auto simp add: split_ifs divmod_rel_def)
wenzelm@23164: 
haftmann@29651: lemma divmod_correct: "b \<noteq> 0 ==> divmod_rel a b (divmod a b)"
haftmann@29651: by (force simp add: linorder_neq_iff divmod_rel_0 divmod_def divmod_rel_neg
wenzelm@23164:                     posDivAlg_correct negDivAlg_correct)
wenzelm@23164: 
wenzelm@23164: text{*Arbitrary definitions for division by zero.  Useful to simplify 
wenzelm@23164:     certain equations.*}
wenzelm@23164: 
wenzelm@23164: lemma DIVISION_BY_ZERO [simp]: "a div (0::int) = 0 & a mod (0::int) = a"
haftmann@29651: by (simp add: div_def mod_def divmod_def posDivAlg.simps)  
wenzelm@23164: 
wenzelm@23164: 
wenzelm@23164: text{*Basic laws about division and remainder*}
wenzelm@23164: 
wenzelm@23164: lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"
wenzelm@23164: apply (case_tac "b = 0", simp)
haftmann@29651: apply (cut_tac a = a and b = b in divmod_correct)
haftmann@29651: apply (auto simp add: divmod_rel_def div_def mod_def)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: lemma zdiv_zmod_equality: "(b * (a div b) + (a mod b)) + k = (a::int)+k"
wenzelm@23164: by(simp add: zmod_zdiv_equality[symmetric])
wenzelm@23164: 
wenzelm@23164: lemma zdiv_zmod_equality2: "((a div b) * b + (a mod b)) + k = (a::int)+k"
wenzelm@23164: by(simp add: mult_commute zmod_zdiv_equality[symmetric])
wenzelm@23164: 
wenzelm@23164: text {* Tool setup *}
wenzelm@23164: 
wenzelm@26480: ML {*
wenzelm@23164: local 
wenzelm@23164: 
wenzelm@23164: structure CancelDivMod = CancelDivModFun(
wenzelm@23164: struct
wenzelm@23164:   val div_name = @{const_name Divides.div};
wenzelm@23164:   val mod_name = @{const_name Divides.mod};
wenzelm@23164:   val mk_binop = HOLogic.mk_binop;
wenzelm@23164:   val mk_sum = Int_Numeral_Simprocs.mk_sum HOLogic.intT;
wenzelm@23164:   val dest_sum = Int_Numeral_Simprocs.dest_sum;
wenzelm@23164:   val div_mod_eqs =
wenzelm@23164:     map mk_meta_eq [@{thm zdiv_zmod_equality},
wenzelm@23164:       @{thm zdiv_zmod_equality2}];
wenzelm@23164:   val trans = trans;
wenzelm@23164:   val prove_eq_sums =
wenzelm@23164:     let
huffman@23365:       val simps = @{thm diff_int_def} :: Int_Numeral_Simprocs.add_0s @ @{thms zadd_ac}
haftmann@26101:     in ArithData.prove_conv all_tac (ArithData.simp_all_tac simps) end;
wenzelm@23164: end)
wenzelm@23164: 
wenzelm@23164: in
wenzelm@23164: 
wenzelm@28262: val cancel_zdiv_zmod_proc = Simplifier.simproc (the_context ())
haftmann@26101:   "cancel_zdiv_zmod" ["(m::int) + n"] (K CancelDivMod.proc)
wenzelm@23164: 
wenzelm@23164: end;
wenzelm@23164: 
wenzelm@23164: Addsimprocs [cancel_zdiv_zmod_proc]
wenzelm@23164: *}
wenzelm@23164: 
wenzelm@23164: lemma pos_mod_conj : "(0::int) < b ==> 0 \<le> a mod b & a mod b < b"
haftmann@29651: apply (cut_tac a = a and b = b in divmod_correct)
haftmann@29651: apply (auto simp add: divmod_rel_def mod_def)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: lemmas pos_mod_sign  [simp] = pos_mod_conj [THEN conjunct1, standard]
wenzelm@23164:    and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2, standard]
wenzelm@23164: 
wenzelm@23164: lemma neg_mod_conj : "b < (0::int) ==> a mod b \<le> 0 & b < a mod b"
haftmann@29651: apply (cut_tac a = a and b = b in divmod_correct)
haftmann@29651: apply (auto simp add: divmod_rel_def div_def mod_def)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: lemmas neg_mod_sign  [simp] = neg_mod_conj [THEN conjunct1, standard]
wenzelm@23164:    and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2, standard]
wenzelm@23164: 
wenzelm@23164: 
wenzelm@23164: 
wenzelm@23164: subsection{*General Properties of div and mod*}
wenzelm@23164: 
haftmann@29651: lemma divmod_rel_div_mod: "b \<noteq> 0 ==> divmod_rel a b (a div b, a mod b)"
wenzelm@23164: apply (cut_tac a = a and b = b in zmod_zdiv_equality)
haftmann@29651: apply (force simp add: divmod_rel_def linorder_neq_iff)
wenzelm@23164: done
wenzelm@23164: 
haftmann@29651: lemma divmod_rel_div: "[| divmod_rel a b (q, r);  b \<noteq> 0 |] ==> a div b = q"
haftmann@29651: by (simp add: divmod_rel_div_mod [THEN unique_quotient])
wenzelm@23164: 
haftmann@29651: lemma divmod_rel_mod: "[| divmod_rel a b (q, r);  b \<noteq> 0 |] ==> a mod b = r"
haftmann@29651: by (simp add: divmod_rel_div_mod [THEN unique_remainder])
wenzelm@23164: 
wenzelm@23164: lemma div_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a div b = 0"
haftmann@29651: apply (rule divmod_rel_div)
haftmann@29651: apply (auto simp add: divmod_rel_def)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: lemma div_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a div b = 0"
haftmann@29651: apply (rule divmod_rel_div)
haftmann@29651: apply (auto simp add: divmod_rel_def)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: lemma div_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a div b = -1"
haftmann@29651: apply (rule divmod_rel_div)
haftmann@29651: apply (auto simp add: divmod_rel_def)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: (*There is no div_neg_pos_trivial because  0 div b = 0 would supersede it*)
wenzelm@23164: 
wenzelm@23164: lemma mod_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a mod b = a"
haftmann@29651: apply (rule_tac q = 0 in divmod_rel_mod)
haftmann@29651: apply (auto simp add: divmod_rel_def)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: lemma mod_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a mod b = a"
haftmann@29651: apply (rule_tac q = 0 in divmod_rel_mod)
haftmann@29651: apply (auto simp add: divmod_rel_def)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: lemma mod_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a mod b = a+b"
haftmann@29651: apply (rule_tac q = "-1" in divmod_rel_mod)
haftmann@29651: apply (auto simp add: divmod_rel_def)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: text{*There is no @{text mod_neg_pos_trivial}.*}
wenzelm@23164: 
wenzelm@23164: 
wenzelm@23164: (*Simpler laws such as -a div b = -(a div b) FAIL, but see just below*)
wenzelm@23164: lemma zdiv_zminus_zminus [simp]: "(-a) div (-b) = a div (b::int)"
wenzelm@23164: apply (case_tac "b = 0", simp)
haftmann@29651: apply (simp add: divmod_rel_div_mod [THEN divmod_rel_neg, simplified, 
haftmann@29651:                                  THEN divmod_rel_div, THEN sym])
wenzelm@23164: 
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: (*Simpler laws such as -a mod b = -(a mod b) FAIL, but see just below*)
wenzelm@23164: lemma zmod_zminus_zminus [simp]: "(-a) mod (-b) = - (a mod (b::int))"
wenzelm@23164: apply (case_tac "b = 0", simp)
haftmann@29651: apply (subst divmod_rel_div_mod [THEN divmod_rel_neg, simplified, THEN divmod_rel_mod],
wenzelm@23164:        auto)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: 
wenzelm@23164: subsection{*Laws for div and mod with Unary Minus*}
wenzelm@23164: 
wenzelm@23164: lemma zminus1_lemma:
haftmann@29651:      "divmod_rel a b (q, r)
haftmann@29651:       ==> divmod_rel (-a) b (if r=0 then -q else -q - 1,  
haftmann@29651:                           if r=0 then 0 else b-r)"
haftmann@29651: by (force simp add: split_ifs divmod_rel_def linorder_neq_iff right_diff_distrib)
wenzelm@23164: 
wenzelm@23164: 
wenzelm@23164: lemma zdiv_zminus1_eq_if:
wenzelm@23164:      "b \<noteq> (0::int)  
wenzelm@23164:       ==> (-a) div b =  
wenzelm@23164:           (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
haftmann@29651: by (blast intro: divmod_rel_div_mod [THEN zminus1_lemma, THEN divmod_rel_div])
wenzelm@23164: 
wenzelm@23164: lemma zmod_zminus1_eq_if:
wenzelm@23164:      "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))"
wenzelm@23164: apply (case_tac "b = 0", simp)
haftmann@29651: apply (blast intro: divmod_rel_div_mod [THEN zminus1_lemma, THEN divmod_rel_mod])
wenzelm@23164: done
wenzelm@23164: 
haftmann@29936: lemma zmod_zminus1_not_zero:
haftmann@29936:   fixes k l :: int
haftmann@29936:   shows "- k mod l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
haftmann@29936:   unfolding zmod_zminus1_eq_if by auto
haftmann@29936: 
wenzelm@23164: lemma zdiv_zminus2: "a div (-b) = (-a::int) div b"
wenzelm@23164: by (cut_tac a = "-a" in zdiv_zminus_zminus, auto)
wenzelm@23164: 
wenzelm@23164: lemma zmod_zminus2: "a mod (-b) = - ((-a::int) mod b)"
wenzelm@23164: by (cut_tac a = "-a" and b = b in zmod_zminus_zminus, auto)
wenzelm@23164: 
wenzelm@23164: lemma zdiv_zminus2_eq_if:
wenzelm@23164:      "b \<noteq> (0::int)  
wenzelm@23164:       ==> a div (-b) =  
wenzelm@23164:           (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
wenzelm@23164: by (simp add: zdiv_zminus1_eq_if zdiv_zminus2)
wenzelm@23164: 
wenzelm@23164: lemma zmod_zminus2_eq_if:
wenzelm@23164:      "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"
wenzelm@23164: by (simp add: zmod_zminus1_eq_if zmod_zminus2)
wenzelm@23164: 
haftmann@29936: lemma zmod_zminus2_not_zero:
haftmann@29936:   fixes k l :: int
haftmann@29936:   shows "k mod - l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
haftmann@29936:   unfolding zmod_zminus2_eq_if by auto 
haftmann@29936: 
wenzelm@23164: 
wenzelm@23164: subsection{*Division of a Number by Itself*}
wenzelm@23164: 
wenzelm@23164: lemma self_quotient_aux1: "[| (0::int) < a; a = r + a*q; r < a |] ==> 1 \<le> q"
wenzelm@23164: apply (subgoal_tac "0 < a*q")
wenzelm@23164:  apply (simp add: zero_less_mult_iff, arith)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: lemma self_quotient_aux2: "[| (0::int) < a; a = r + a*q; 0 \<le> r |] ==> q \<le> 1"
wenzelm@23164: apply (subgoal_tac "0 \<le> a* (1-q) ")
wenzelm@23164:  apply (simp add: zero_le_mult_iff)
wenzelm@23164: apply (simp add: right_diff_distrib)
wenzelm@23164: done
wenzelm@23164: 
haftmann@29651: lemma self_quotient: "[| divmod_rel a a (q, r);  a \<noteq> (0::int) |] ==> q = 1"
haftmann@29651: apply (simp add: split_ifs divmod_rel_def linorder_neq_iff)
wenzelm@23164: apply (rule order_antisym, safe, simp_all)
wenzelm@23164: apply (rule_tac [3] a = "-a" and r = "-r" in self_quotient_aux1)
wenzelm@23164: apply (rule_tac a = "-a" and r = "-r" in self_quotient_aux2)
wenzelm@23164: apply (force intro: self_quotient_aux1 self_quotient_aux2 simp add: add_commute)+
wenzelm@23164: done
wenzelm@23164: 
haftmann@29651: lemma self_remainder: "[| divmod_rel a a (q, r);  a \<noteq> (0::int) |] ==> r = 0"
wenzelm@23164: apply (frule self_quotient, assumption)
haftmann@29651: apply (simp add: divmod_rel_def)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: lemma zdiv_self [simp]: "a \<noteq> 0 ==> a div a = (1::int)"
haftmann@29651: by (simp add: divmod_rel_div_mod [THEN self_quotient])
wenzelm@23164: 
wenzelm@23164: (*Here we have 0 mod 0 = 0, also assumed by Knuth (who puts m mod 0 = 0) *)
wenzelm@23164: lemma zmod_self [simp]: "a mod a = (0::int)"
wenzelm@23164: apply (case_tac "a = 0", simp)
haftmann@29651: apply (simp add: divmod_rel_div_mod [THEN self_remainder])
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: 
wenzelm@23164: subsection{*Computation of Division and Remainder*}
wenzelm@23164: 
wenzelm@23164: lemma zdiv_zero [simp]: "(0::int) div b = 0"
haftmann@29651: by (simp add: div_def divmod_def)
wenzelm@23164: 
wenzelm@23164: lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"
haftmann@29651: by (simp add: div_def divmod_def)
wenzelm@23164: 
wenzelm@23164: lemma zmod_zero [simp]: "(0::int) mod b = 0"
haftmann@29651: by (simp add: mod_def divmod_def)
wenzelm@23164: 
wenzelm@23164: lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"
haftmann@29651: by (simp add: mod_def divmod_def)
wenzelm@23164: 
wenzelm@23164: text{*a positive, b positive *}
wenzelm@23164: 
wenzelm@23164: lemma div_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a div b = fst (posDivAlg a b)"
haftmann@29651: by (simp add: div_def divmod_def)
wenzelm@23164: 
wenzelm@23164: lemma mod_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a mod b = snd (posDivAlg a b)"
haftmann@29651: by (simp add: mod_def divmod_def)
wenzelm@23164: 
wenzelm@23164: text{*a negative, b positive *}
wenzelm@23164: 
wenzelm@23164: lemma div_neg_pos: "[| a < 0;  0 < b |] ==> a div b = fst (negDivAlg a b)"
haftmann@29651: by (simp add: div_def divmod_def)
wenzelm@23164: 
wenzelm@23164: lemma mod_neg_pos: "[| a < 0;  0 < b |] ==> a mod b = snd (negDivAlg a b)"
haftmann@29651: by (simp add: mod_def divmod_def)
wenzelm@23164: 
wenzelm@23164: text{*a positive, b negative *}
wenzelm@23164: 
wenzelm@23164: lemma div_pos_neg:
wenzelm@23164:      "[| 0 < a;  b < 0 |] ==> a div b = fst (negateSnd (negDivAlg (-a) (-b)))"
haftmann@29651: by (simp add: div_def divmod_def)
wenzelm@23164: 
wenzelm@23164: lemma mod_pos_neg:
wenzelm@23164:      "[| 0 < a;  b < 0 |] ==> a mod b = snd (negateSnd (negDivAlg (-a) (-b)))"
haftmann@29651: by (simp add: mod_def divmod_def)
wenzelm@23164: 
wenzelm@23164: text{*a negative, b negative *}
wenzelm@23164: 
wenzelm@23164: lemma div_neg_neg:
wenzelm@23164:      "[| a < 0;  b \<le> 0 |] ==> a div b = fst (negateSnd (posDivAlg (-a) (-b)))"
haftmann@29651: by (simp add: div_def divmod_def)
wenzelm@23164: 
wenzelm@23164: lemma mod_neg_neg:
wenzelm@23164:      "[| a < 0;  b \<le> 0 |] ==> a mod b = snd (negateSnd (posDivAlg (-a) (-b)))"
haftmann@29651: by (simp add: mod_def divmod_def)
wenzelm@23164: 
wenzelm@23164: text {*Simplify expresions in which div and mod combine numerical constants*}
wenzelm@23164: 
haftmann@29651: lemma divmod_relI:
huffman@24481:   "\<lbrakk>a == b * q + r; if 0 < b then 0 \<le> r \<and> r < b else b < r \<and> r \<le> 0\<rbrakk>
haftmann@29651:     \<Longrightarrow> divmod_rel a b (q, r)"
haftmann@29651:   unfolding divmod_rel_def by simp
huffman@24481: 
haftmann@29651: lemmas divmod_rel_div_eq = divmod_relI [THEN divmod_rel_div, THEN eq_reflection]
haftmann@29651: lemmas divmod_rel_mod_eq = divmod_relI [THEN divmod_rel_mod, THEN eq_reflection]
huffman@24481: lemmas arithmetic_simps =
huffman@24481:   arith_simps
huffman@24481:   add_special
huffman@24481:   OrderedGroup.add_0_left
huffman@24481:   OrderedGroup.add_0_right
huffman@24481:   mult_zero_left
huffman@24481:   mult_zero_right
huffman@24481:   mult_1_left
huffman@24481:   mult_1_right
huffman@24481: 
huffman@24481: (* simprocs adapted from HOL/ex/Binary.thy *)
huffman@24481: ML {*
huffman@24481: local
huffman@24481:   infix ==;
huffman@24481:   val op == = Logic.mk_equals;
huffman@24481:   fun plus m n = @{term "plus :: int \<Rightarrow> int \<Rightarrow> int"} $ m $ n;
huffman@24481:   fun mult m n = @{term "times :: int \<Rightarrow> int \<Rightarrow> int"} $ m $ n;
huffman@24481: 
huffman@24481:   val binary_ss = HOL_basic_ss addsimps @{thms arithmetic_simps};
huffman@24481:   fun prove ctxt prop =
huffman@24481:     Goal.prove ctxt [] [] prop (fn _ => ALLGOALS (full_simp_tac binary_ss));
huffman@24481: 
huffman@24481:   fun binary_proc proc ss ct =
huffman@24481:     (case Thm.term_of ct of
huffman@24481:       _ $ t $ u =>
huffman@24481:       (case try (pairself (`(snd o HOLogic.dest_number))) (t, u) of
huffman@24481:         SOME args => proc (Simplifier.the_context ss) args
huffman@24481:       | NONE => NONE)
huffman@24481:     | _ => NONE);
huffman@24481: in
huffman@24481: 
huffman@24481: fun divmod_proc rule = binary_proc (fn ctxt => fn ((m, t), (n, u)) =>
huffman@24481:   if n = 0 then NONE
huffman@24481:   else
wenzelm@24630:     let val (k, l) = Integer.div_mod m n;
huffman@24481:         fun mk_num x = HOLogic.mk_number HOLogic.intT x;
huffman@24481:     in SOME (rule OF [prove ctxt (t == plus (mult u (mk_num k)) (mk_num l))])
huffman@24481:     end);
huffman@24481: 
huffman@24481: end;
huffman@24481: *}
huffman@24481: 
huffman@24481: simproc_setup binary_int_div ("number_of m div number_of n :: int") =
haftmann@29651:   {* K (divmod_proc (@{thm divmod_rel_div_eq})) *}
huffman@24481: 
huffman@24481: simproc_setup binary_int_mod ("number_of m mod number_of n :: int") =
haftmann@29651:   {* K (divmod_proc (@{thm divmod_rel_mod_eq})) *}
huffman@24481: 
wenzelm@23164: lemmas posDivAlg_eqn_number_of [simp] =
wenzelm@23164:     posDivAlg_eqn [of "number_of v" "number_of w", standard]
wenzelm@23164: 
wenzelm@23164: lemmas negDivAlg_eqn_number_of [simp] =
wenzelm@23164:     negDivAlg_eqn [of "number_of v" "number_of w", standard]
wenzelm@23164: 
wenzelm@23164: 
wenzelm@23164: text{*Special-case simplification *}
wenzelm@23164: 
wenzelm@23164: lemma zmod_minus1_right [simp]: "a mod (-1::int) = 0"
wenzelm@23164: apply (cut_tac a = a and b = "-1" in neg_mod_sign)
wenzelm@23164: apply (cut_tac [2] a = a and b = "-1" in neg_mod_bound)
wenzelm@23164: apply (auto simp del: neg_mod_sign neg_mod_bound)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: lemma zdiv_minus1_right [simp]: "a div (-1::int) = -a"
wenzelm@23164: by (cut_tac a = a and b = "-1" in zmod_zdiv_equality, auto)
wenzelm@23164: 
wenzelm@23164: (** The last remaining special cases for constant arithmetic:
wenzelm@23164:     1 div z and 1 mod z **)
wenzelm@23164: 
wenzelm@23164: lemmas div_pos_pos_1_number_of [simp] =
wenzelm@23164:     div_pos_pos [OF int_0_less_1, of "number_of w", standard]
wenzelm@23164: 
wenzelm@23164: lemmas div_pos_neg_1_number_of [simp] =
wenzelm@23164:     div_pos_neg [OF int_0_less_1, of "number_of w", standard]
wenzelm@23164: 
wenzelm@23164: lemmas mod_pos_pos_1_number_of [simp] =
wenzelm@23164:     mod_pos_pos [OF int_0_less_1, of "number_of w", standard]
wenzelm@23164: 
wenzelm@23164: lemmas mod_pos_neg_1_number_of [simp] =
wenzelm@23164:     mod_pos_neg [OF int_0_less_1, of "number_of w", standard]
wenzelm@23164: 
wenzelm@23164: 
wenzelm@23164: lemmas posDivAlg_eqn_1_number_of [simp] =
wenzelm@23164:     posDivAlg_eqn [of concl: 1 "number_of w", standard]
wenzelm@23164: 
wenzelm@23164: lemmas negDivAlg_eqn_1_number_of [simp] =
wenzelm@23164:     negDivAlg_eqn [of concl: 1 "number_of w", standard]
wenzelm@23164: 
wenzelm@23164: 
wenzelm@23164: 
wenzelm@23164: subsection{*Monotonicity in the First Argument (Dividend)*}
wenzelm@23164: 
wenzelm@23164: lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div b"
wenzelm@23164: apply (cut_tac a = a and b = b in zmod_zdiv_equality)
wenzelm@23164: apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
wenzelm@23164: apply (rule unique_quotient_lemma)
wenzelm@23164: apply (erule subst)
wenzelm@23164: apply (erule subst, simp_all)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: lemma zdiv_mono1_neg: "[| a \<le> a';  (b::int) < 0 |] ==> a' div b \<le> a div b"
wenzelm@23164: apply (cut_tac a = a and b = b in zmod_zdiv_equality)
wenzelm@23164: apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
wenzelm@23164: apply (rule unique_quotient_lemma_neg)
wenzelm@23164: apply (erule subst)
wenzelm@23164: apply (erule subst, simp_all)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: 
wenzelm@23164: subsection{*Monotonicity in the Second Argument (Divisor)*}
wenzelm@23164: 
wenzelm@23164: lemma q_pos_lemma:
wenzelm@23164:      "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"
wenzelm@23164: apply (subgoal_tac "0 < b'* (q' + 1) ")
wenzelm@23164:  apply (simp add: zero_less_mult_iff)
wenzelm@23164: apply (simp add: right_distrib)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: lemma zdiv_mono2_lemma:
wenzelm@23164:      "[| b*q + r = b'*q' + r';  0 \<le> b'*q' + r';   
wenzelm@23164:          r' < b';  0 \<le> r;  0 < b';  b' \<le> b |]   
wenzelm@23164:       ==> q \<le> (q'::int)"
wenzelm@23164: apply (frule q_pos_lemma, assumption+) 
wenzelm@23164: apply (subgoal_tac "b*q < b* (q' + 1) ")
wenzelm@23164:  apply (simp add: mult_less_cancel_left)
wenzelm@23164: apply (subgoal_tac "b*q = r' - r + b'*q'")
wenzelm@23164:  prefer 2 apply simp
wenzelm@23164: apply (simp (no_asm_simp) add: right_distrib)
wenzelm@23164: apply (subst add_commute, rule zadd_zless_mono, arith)
wenzelm@23164: apply (rule mult_right_mono, auto)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: lemma zdiv_mono2:
wenzelm@23164:      "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"
wenzelm@23164: apply (subgoal_tac "b \<noteq> 0")
wenzelm@23164:  prefer 2 apply arith
wenzelm@23164: apply (cut_tac a = a and b = b in zmod_zdiv_equality)
wenzelm@23164: apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
wenzelm@23164: apply (rule zdiv_mono2_lemma)
wenzelm@23164: apply (erule subst)
wenzelm@23164: apply (erule subst, simp_all)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: lemma q_neg_lemma:
wenzelm@23164:      "[| b'*q' + r' < 0;  0 \<le> r';  0 < b' |] ==> q' \<le> (0::int)"
wenzelm@23164: apply (subgoal_tac "b'*q' < 0")
wenzelm@23164:  apply (simp add: mult_less_0_iff, arith)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: lemma zdiv_mono2_neg_lemma:
wenzelm@23164:      "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;   
wenzelm@23164:          r < b;  0 \<le> r';  0 < b';  b' \<le> b |]   
wenzelm@23164:       ==> q' \<le> (q::int)"
wenzelm@23164: apply (frule q_neg_lemma, assumption+) 
wenzelm@23164: apply (subgoal_tac "b*q' < b* (q + 1) ")
wenzelm@23164:  apply (simp add: mult_less_cancel_left)
wenzelm@23164: apply (simp add: right_distrib)
wenzelm@23164: apply (subgoal_tac "b*q' \<le> b'*q'")
wenzelm@23164:  prefer 2 apply (simp add: mult_right_mono_neg, arith)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: lemma zdiv_mono2_neg:
wenzelm@23164:      "[| a < (0::int);  0 < b';  b' \<le> b |] ==> a div b' \<le> a div b"
wenzelm@23164: apply (cut_tac a = a and b = b in zmod_zdiv_equality)
wenzelm@23164: apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
wenzelm@23164: apply (rule zdiv_mono2_neg_lemma)
wenzelm@23164: apply (erule subst)
wenzelm@23164: apply (erule subst, simp_all)
wenzelm@23164: done
wenzelm@23164: 
haftmann@25942: 
wenzelm@23164: subsection{*More Algebraic Laws for div and mod*}
wenzelm@23164: 
wenzelm@23164: text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}
wenzelm@23164: 
wenzelm@23164: lemma zmult1_lemma:
haftmann@29651:      "[| divmod_rel b c (q, r);  c \<noteq> 0 |]  
haftmann@29651:       ==> divmod_rel (a * b) c (a*q + a*r div c, a*r mod c)"
haftmann@29651: by (force simp add: split_ifs divmod_rel_def linorder_neq_iff right_distrib)
wenzelm@23164: 
wenzelm@23164: lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"
wenzelm@23164: apply (case_tac "c = 0", simp)
haftmann@29651: apply (blast intro: divmod_rel_div_mod [THEN zmult1_lemma, THEN divmod_rel_div])
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: lemma zmod_zmult1_eq: "(a*b) mod c = a*(b mod c) mod (c::int)"
wenzelm@23164: apply (case_tac "c = 0", simp)
haftmann@29651: apply (blast intro: divmod_rel_div_mod [THEN zmult1_lemma, THEN divmod_rel_mod])
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: lemma zdiv_zmult_self1 [simp]: "b \<noteq> (0::int) ==> (a*b) div b = a"
wenzelm@23164: by (simp add: zdiv_zmult1_eq)
wenzelm@23164: 
huffman@29403: lemma zmod_zdiv_trivial: "(a mod b) div b = (0::int)"
haftmann@27651: apply (case_tac "b = 0", simp)
haftmann@27651: apply (auto simp add: linorder_neq_iff div_pos_pos_trivial div_neg_neg_trivial)
haftmann@27651: done
haftmann@27651: 
haftmann@27651: text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}
haftmann@27651: 
haftmann@27651: lemma zadd1_lemma:
haftmann@29651:      "[| divmod_rel a c (aq, ar);  divmod_rel b c (bq, br);  c \<noteq> 0 |]  
haftmann@29651:       ==> divmod_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"
haftmann@29651: by (force simp add: split_ifs divmod_rel_def linorder_neq_iff right_distrib)
haftmann@27651: 
haftmann@27651: (*NOT suitable for rewriting: the RHS has an instance of the LHS*)
haftmann@27651: lemma zdiv_zadd1_eq:
haftmann@27651:      "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"
haftmann@27651: apply (case_tac "c = 0", simp)
haftmann@29651: apply (blast intro: zadd1_lemma [OF divmod_rel_div_mod divmod_rel_div_mod] divmod_rel_div)
haftmann@27651: done
haftmann@27651: 
huffman@29405: instance int :: ring_div
haftmann@27651: proof
haftmann@27651:   fix a b c :: int
haftmann@27651:   assume not0: "b \<noteq> 0"
haftmann@27651:   show "(a + c * b) div b = c + a div b"
haftmann@27651:     unfolding zdiv_zadd1_eq [of a "c * b"] using not0 
huffman@29403:       by (simp add: zmod_zmult1_eq zmod_zdiv_trivial)
haftmann@27651: qed auto
haftmann@25942: 
haftmann@29651: lemma posDivAlg_div_mod:
haftmann@29651:   assumes "k \<ge> 0"
haftmann@29651:   and "l \<ge> 0"
haftmann@29651:   shows "posDivAlg k l = (k div l, k mod l)"
haftmann@29651: proof (cases "l = 0")
haftmann@29651:   case True then show ?thesis by (simp add: posDivAlg.simps)
haftmann@29651: next
haftmann@29651:   case False with assms posDivAlg_correct
haftmann@29651:     have "divmod_rel k l (fst (posDivAlg k l), snd (posDivAlg k l))"
haftmann@29651:     by simp
haftmann@29651:   from divmod_rel_div [OF this `l \<noteq> 0`] divmod_rel_mod [OF this `l \<noteq> 0`]
haftmann@29651:   show ?thesis by simp
haftmann@29651: qed
haftmann@29651: 
haftmann@29651: lemma negDivAlg_div_mod:
haftmann@29651:   assumes "k < 0"
haftmann@29651:   and "l > 0"
haftmann@29651:   shows "negDivAlg k l = (k div l, k mod l)"
haftmann@29651: proof -
haftmann@29651:   from assms have "l \<noteq> 0" by simp
haftmann@29651:   from assms negDivAlg_correct
haftmann@29651:     have "divmod_rel k l (fst (negDivAlg k l), snd (negDivAlg k l))"
haftmann@29651:     by simp
haftmann@29651:   from divmod_rel_div [OF this `l \<noteq> 0`] divmod_rel_mod [OF this `l \<noteq> 0`]
haftmann@29651:   show ?thesis by simp
haftmann@29651: qed
haftmann@29651: 
wenzelm@23164: lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"
huffman@29403: by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
wenzelm@23164: 
huffman@29403: (* REVISIT: should this be generalized to all semiring_div types? *)
wenzelm@23164: lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]
wenzelm@23164: 
nipkow@23983: 
wenzelm@23164: subsection{*Proving  @{term "a div (b*c) = (a div b) div c"} *}
wenzelm@23164: 
wenzelm@23164: (*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but
wenzelm@23164:   7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems
wenzelm@23164:   to cause particular problems.*)
wenzelm@23164: 
wenzelm@23164: text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}
wenzelm@23164: 
wenzelm@23164: lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b*c < b*(q mod c) + r"
wenzelm@23164: apply (subgoal_tac "b * (c - q mod c) < r * 1")
nipkow@29667:  apply (simp add: algebra_simps)
wenzelm@23164: apply (rule order_le_less_trans)
nipkow@29667:  apply (erule_tac [2] mult_strict_right_mono)
nipkow@29667:  apply (rule mult_left_mono_neg)
nipkow@29667:   using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps pos_mod_bound)
nipkow@29667:  apply (simp)
nipkow@29667: apply (simp)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: lemma zmult2_lemma_aux2:
wenzelm@23164:      "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"
wenzelm@23164: apply (subgoal_tac "b * (q mod c) \<le> 0")
wenzelm@23164:  apply arith
wenzelm@23164: apply (simp add: mult_le_0_iff)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: lemma zmult2_lemma_aux3: "[| (0::int) < c;  0 \<le> r;  r < b |] ==> 0 \<le> b * (q mod c) + r"
wenzelm@23164: apply (subgoal_tac "0 \<le> b * (q mod c) ")
wenzelm@23164: apply arith
wenzelm@23164: apply (simp add: zero_le_mult_iff)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"
wenzelm@23164: apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
nipkow@29667:  apply (simp add: right_diff_distrib)
wenzelm@23164: apply (rule order_less_le_trans)
nipkow@29667:  apply (erule mult_strict_right_mono)
nipkow@29667:  apply (rule_tac [2] mult_left_mono)
nipkow@29667:   apply simp
nipkow@29667:  using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps pos_mod_bound)
nipkow@29667: apply simp
wenzelm@23164: done
wenzelm@23164: 
haftmann@29651: lemma zmult2_lemma: "[| divmod_rel a b (q, r);  b \<noteq> 0;  0 < c |]  
haftmann@29651:       ==> divmod_rel a (b * c) (q div c, b*(q mod c) + r)"
haftmann@29651: by (auto simp add: mult_ac divmod_rel_def linorder_neq_iff
wenzelm@23164:                    zero_less_mult_iff right_distrib [symmetric] 
wenzelm@23164:                    zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4)
wenzelm@23164: 
wenzelm@23164: lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c"
wenzelm@23164: apply (case_tac "b = 0", simp)
haftmann@29651: apply (force simp add: divmod_rel_div_mod [THEN zmult2_lemma, THEN divmod_rel_div])
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: lemma zmod_zmult2_eq:
wenzelm@23164:      "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"
wenzelm@23164: apply (case_tac "b = 0", simp)
haftmann@29651: apply (force simp add: divmod_rel_div_mod [THEN zmult2_lemma, THEN divmod_rel_mod])
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: 
wenzelm@23164: subsection{*Cancellation of Common Factors in div*}
wenzelm@23164: 
wenzelm@23164: lemma zdiv_zmult_zmult1_aux1:
wenzelm@23164:      "[| (0::int) < b;  c \<noteq> 0 |] ==> (c*a) div (c*b) = a div b"
wenzelm@23164: by (subst zdiv_zmult2_eq, auto)
wenzelm@23164: 
wenzelm@23164: lemma zdiv_zmult_zmult1_aux2:
wenzelm@23164:      "[| b < (0::int);  c \<noteq> 0 |] ==> (c*a) div (c*b) = a div b"
wenzelm@23164: apply (subgoal_tac " (c * (-a)) div (c * (-b)) = (-a) div (-b) ")
wenzelm@23164: apply (rule_tac [2] zdiv_zmult_zmult1_aux1, auto)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: lemma zdiv_zmult_zmult1: "c \<noteq> (0::int) ==> (c*a) div (c*b) = a div b"
wenzelm@23164: apply (case_tac "b = 0", simp)
wenzelm@23164: apply (auto simp add: linorder_neq_iff zdiv_zmult_zmult1_aux1 zdiv_zmult_zmult1_aux2)
wenzelm@23164: done
wenzelm@23164: 
nipkow@23401: lemma zdiv_zmult_zmult1_if[simp]:
nipkow@23401:   "(k*m) div (k*n) = (if k = (0::int) then 0 else m div n)"
nipkow@23401: by (simp add:zdiv_zmult_zmult1)
nipkow@23401: 
nipkow@23401: (*
wenzelm@23164: lemma zdiv_zmult_zmult2: "c \<noteq> (0::int) ==> (a*c) div (b*c) = a div b"
wenzelm@23164: apply (drule zdiv_zmult_zmult1)
wenzelm@23164: apply (auto simp add: mult_commute)
wenzelm@23164: done
nipkow@23401: *)
wenzelm@23164: 
wenzelm@23164: 
wenzelm@23164: subsection{*Distribution of Factors over mod*}
wenzelm@23164: 
wenzelm@23164: lemma zmod_zmult_zmult1_aux1:
wenzelm@23164:      "[| (0::int) < b;  c \<noteq> 0 |] ==> (c*a) mod (c*b) = c * (a mod b)"
wenzelm@23164: by (subst zmod_zmult2_eq, auto)
wenzelm@23164: 
wenzelm@23164: lemma zmod_zmult_zmult1_aux2:
wenzelm@23164:      "[| b < (0::int);  c \<noteq> 0 |] ==> (c*a) mod (c*b) = c * (a mod b)"
wenzelm@23164: apply (subgoal_tac " (c * (-a)) mod (c * (-b)) = c * ((-a) mod (-b))")
wenzelm@23164: apply (rule_tac [2] zmod_zmult_zmult1_aux1, auto)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: lemma zmod_zmult_zmult1: "(c*a) mod (c*b) = (c::int) * (a mod b)"
wenzelm@23164: apply (case_tac "b = 0", simp)
wenzelm@23164: apply (case_tac "c = 0", simp)
wenzelm@23164: apply (auto simp add: linorder_neq_iff zmod_zmult_zmult1_aux1 zmod_zmult_zmult1_aux2)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: lemma zmod_zmult_zmult2: "(a*c) mod (b*c) = (a mod b) * (c::int)"
wenzelm@23164: apply (cut_tac c = c in zmod_zmult_zmult1)
wenzelm@23164: apply (auto simp add: mult_commute)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: 
wenzelm@23164: subsection {*Splitting Rules for div and mod*}
wenzelm@23164: 
wenzelm@23164: text{*The proofs of the two lemmas below are essentially identical*}
wenzelm@23164: 
wenzelm@23164: lemma split_pos_lemma:
wenzelm@23164:  "0<k ==> 
wenzelm@23164:     P(n div k :: int)(n mod k) = (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i j)"
wenzelm@23164: apply (rule iffI, clarify)
wenzelm@23164:  apply (erule_tac P="P ?x ?y" in rev_mp)  
nipkow@29948:  apply (subst mod_add_eq) 
wenzelm@23164:  apply (subst zdiv_zadd1_eq) 
wenzelm@23164:  apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)  
wenzelm@23164: txt{*converse direction*}
wenzelm@23164: apply (drule_tac x = "n div k" in spec) 
wenzelm@23164: apply (drule_tac x = "n mod k" in spec, simp)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: lemma split_neg_lemma:
wenzelm@23164:  "k<0 ==>
wenzelm@23164:     P(n div k :: int)(n mod k) = (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i j)"
wenzelm@23164: apply (rule iffI, clarify)
wenzelm@23164:  apply (erule_tac P="P ?x ?y" in rev_mp)  
nipkow@29948:  apply (subst mod_add_eq) 
wenzelm@23164:  apply (subst zdiv_zadd1_eq) 
wenzelm@23164:  apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)  
wenzelm@23164: txt{*converse direction*}
wenzelm@23164: apply (drule_tac x = "n div k" in spec) 
wenzelm@23164: apply (drule_tac x = "n mod k" in spec, simp)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: lemma split_zdiv:
wenzelm@23164:  "P(n div k :: int) =
wenzelm@23164:   ((k = 0 --> P 0) & 
wenzelm@23164:    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) & 
wenzelm@23164:    (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"
wenzelm@23164: apply (case_tac "k=0", simp)
wenzelm@23164: apply (simp only: linorder_neq_iff)
wenzelm@23164: apply (erule disjE) 
wenzelm@23164:  apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"] 
wenzelm@23164:                       split_neg_lemma [of concl: "%x y. P x"])
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: lemma split_zmod:
wenzelm@23164:  "P(n mod k :: int) =
wenzelm@23164:   ((k = 0 --> P n) & 
wenzelm@23164:    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) & 
wenzelm@23164:    (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"
wenzelm@23164: apply (case_tac "k=0", simp)
wenzelm@23164: apply (simp only: linorder_neq_iff)
wenzelm@23164: apply (erule disjE) 
wenzelm@23164:  apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"] 
wenzelm@23164:                       split_neg_lemma [of concl: "%x y. P y"])
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: (* Enable arith to deal with div 2 and mod 2: *)
wenzelm@23164: declare split_zdiv [of _ _ "number_of k", simplified, standard, arith_split]
wenzelm@23164: declare split_zmod [of _ _ "number_of k", simplified, standard, arith_split]
wenzelm@23164: 
wenzelm@23164: 
wenzelm@23164: subsection{*Speeding up the Division Algorithm with Shifting*}
wenzelm@23164: 
wenzelm@23164: text{*computing div by shifting *}
wenzelm@23164: 
wenzelm@23164: lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
wenzelm@23164: proof cases
wenzelm@23164:   assume "a=0"
wenzelm@23164:     thus ?thesis by simp
wenzelm@23164: next
wenzelm@23164:   assume "a\<noteq>0" and le_a: "0\<le>a"   
wenzelm@23164:   hence a_pos: "1 \<le> a" by arith
wenzelm@23164:   hence one_less_a2: "1 < 2*a" by arith
wenzelm@23164:   hence le_2a: "2 * (1 + b mod a) \<le> 2 * a"
wenzelm@23164:     by (simp add: mult_le_cancel_left add_commute [of 1] add1_zle_eq)
wenzelm@23164:   with a_pos have "0 \<le> b mod a" by simp
wenzelm@23164:   hence le_addm: "0 \<le> 1 mod (2*a) + 2*(b mod a)"
wenzelm@23164:     by (simp add: mod_pos_pos_trivial one_less_a2)
wenzelm@23164:   with  le_2a
wenzelm@23164:   have "(1 mod (2*a) + 2*(b mod a)) div (2*a) = 0"
wenzelm@23164:     by (simp add: div_pos_pos_trivial le_addm mod_pos_pos_trivial one_less_a2
wenzelm@23164:                   right_distrib) 
wenzelm@23164:   thus ?thesis
wenzelm@23164:     by (subst zdiv_zadd1_eq,
wenzelm@23164:         simp add: zdiv_zmult_zmult1 zmod_zmult_zmult1 one_less_a2
wenzelm@23164:                   div_pos_pos_trivial)
wenzelm@23164: qed
wenzelm@23164: 
wenzelm@23164: lemma neg_zdiv_mult_2: "a \<le> (0::int) ==> (1 + 2*b) div (2*a) = (b+1) div a"
wenzelm@23164: apply (subgoal_tac " (1 + 2* (-b - 1)) div (2 * (-a)) = (-b - 1) div (-a) ")
wenzelm@23164: apply (rule_tac [2] pos_zdiv_mult_2)
wenzelm@23164: apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib)
wenzelm@23164: apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")
wenzelm@23164: apply (simp only: zdiv_zminus_zminus diff_minus minus_add_distrib [symmetric],
wenzelm@23164:        simp) 
wenzelm@23164: done
wenzelm@23164: 
huffman@26086: lemma zdiv_number_of_Bit0 [simp]:
huffman@26086:      "number_of (Int.Bit0 v) div number_of (Int.Bit0 w) =  
huffman@26086:           number_of v div (number_of w :: int)"
huffman@26086: by (simp only: number_of_eq numeral_simps) simp
huffman@26086: 
huffman@26086: lemma zdiv_number_of_Bit1 [simp]:
huffman@26086:      "number_of (Int.Bit1 v) div number_of (Int.Bit0 w) =  
huffman@26086:           (if (0::int) \<le> number_of w                    
wenzelm@23164:            then number_of v div (number_of w)     
wenzelm@23164:            else (number_of v + (1::int)) div (number_of w))"
wenzelm@23164: apply (simp only: number_of_eq numeral_simps UNIV_I split: split_if) 
huffman@26086: apply (simp add: zdiv_zmult_zmult1 pos_zdiv_mult_2 neg_zdiv_mult_2 add_ac)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: 
wenzelm@23164: subsection{*Computing mod by Shifting (proofs resemble those for div)*}
wenzelm@23164: 
wenzelm@23164: lemma pos_zmod_mult_2:
wenzelm@23164:      "(0::int) \<le> a ==> (1 + 2*b) mod (2*a) = 1 + 2 * (b mod a)"
wenzelm@23164: apply (case_tac "a = 0", simp)
wenzelm@23164: apply (subgoal_tac "1 < a * 2")
wenzelm@23164:  prefer 2 apply arith
wenzelm@23164: apply (subgoal_tac "2* (1 + b mod a) \<le> 2*a")
wenzelm@23164:  apply (rule_tac [2] mult_left_mono)
wenzelm@23164: apply (auto simp add: add_commute [of 1] mult_commute add1_zle_eq 
wenzelm@23164:                       pos_mod_bound)
nipkow@29948: apply (subst mod_add_eq)
wenzelm@23164: apply (simp add: zmod_zmult_zmult2 mod_pos_pos_trivial)
wenzelm@23164: apply (rule mod_pos_pos_trivial)
huffman@26086: apply (auto simp add: mod_pos_pos_trivial ring_distribs)
wenzelm@23164: apply (subgoal_tac "0 \<le> b mod a", arith, simp)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: lemma neg_zmod_mult_2:
wenzelm@23164:      "a \<le> (0::int) ==> (1 + 2*b) mod (2*a) = 2 * ((b+1) mod a) - 1"
wenzelm@23164: apply (subgoal_tac "(1 + 2* (-b - 1)) mod (2* (-a)) = 
wenzelm@23164:                     1 + 2* ((-b - 1) mod (-a))")
wenzelm@23164: apply (rule_tac [2] pos_zmod_mult_2)
wenzelm@23164: apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib)
wenzelm@23164: apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")
wenzelm@23164:  prefer 2 apply simp 
wenzelm@23164: apply (simp only: zmod_zminus_zminus diff_minus minus_add_distrib [symmetric])
wenzelm@23164: done
wenzelm@23164: 
huffman@26086: lemma zmod_number_of_Bit0 [simp]:
huffman@26086:      "number_of (Int.Bit0 v) mod number_of (Int.Bit0 w) =  
huffman@26086:       (2::int) * (number_of v mod number_of w)"
huffman@26086: apply (simp only: number_of_eq numeral_simps) 
huffman@26086: apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2 
nipkow@29948:                  neg_zmod_mult_2 add_ac)
huffman@26086: done
huffman@26086: 
huffman@26086: lemma zmod_number_of_Bit1 [simp]:
huffman@26086:      "number_of (Int.Bit1 v) mod number_of (Int.Bit0 w) =  
huffman@26086:       (if (0::int) \<le> number_of w  
wenzelm@23164:                 then 2 * (number_of v mod number_of w) + 1     
wenzelm@23164:                 else 2 * ((number_of v + (1::int)) mod number_of w) - 1)"
huffman@26086: apply (simp only: number_of_eq numeral_simps) 
wenzelm@23164: apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2 
nipkow@29948:                  neg_zmod_mult_2 add_ac)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: 
wenzelm@23164: subsection{*Quotients of Signs*}
wenzelm@23164: 
wenzelm@23164: lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"
wenzelm@23164: apply (subgoal_tac "a div b \<le> -1", force)
wenzelm@23164: apply (rule order_trans)
wenzelm@23164: apply (rule_tac a' = "-1" in zdiv_mono1)
nipkow@29948: apply (auto simp add: div_eq_minus1)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: lemma div_nonneg_neg_le0: "[| (0::int) \<le> a;  b < 0 |] ==> a div b \<le> 0"
wenzelm@23164: by (drule zdiv_mono1_neg, auto)
wenzelm@23164: 
wenzelm@23164: lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"
wenzelm@23164: apply auto
wenzelm@23164: apply (drule_tac [2] zdiv_mono1)
wenzelm@23164: apply (auto simp add: linorder_neq_iff)
wenzelm@23164: apply (simp (no_asm_use) add: linorder_not_less [symmetric])
wenzelm@23164: apply (blast intro: div_neg_pos_less0)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: lemma neg_imp_zdiv_nonneg_iff:
wenzelm@23164:      "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"
wenzelm@23164: apply (subst zdiv_zminus_zminus [symmetric])
wenzelm@23164: apply (subst pos_imp_zdiv_nonneg_iff, auto)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: (*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
wenzelm@23164: lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
wenzelm@23164: by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
wenzelm@23164: 
wenzelm@23164: (*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
wenzelm@23164: lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
wenzelm@23164: by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
wenzelm@23164: 
wenzelm@23164: 
wenzelm@23164: subsection {* The Divides Relation *}
wenzelm@23164: 
wenzelm@23164: lemma zdvd_iff_zmod_eq_0: "(m dvd n) = (n mod m = (0::int))"
huffman@29410:   by (rule dvd_eq_mod_eq_0)
haftmann@23512: 
wenzelm@23164: lemmas zdvd_iff_zmod_eq_0_number_of [simp] =
wenzelm@23164:   zdvd_iff_zmod_eq_0 [of "number_of x" "number_of y", standard]
wenzelm@23164: 
huffman@29410: lemma zdvd_0_right: "(m::int) dvd 0"
huffman@29410:   by (rule dvd_0_right) (* already declared [iff] *)
wenzelm@23164: 
huffman@29410: lemma zdvd_0_left: "(0 dvd (m::int)) = (m = 0)"
huffman@29410:   by (rule dvd_0_left_iff) (* already declared [noatp,simp] *)
wenzelm@23164: 
huffman@29410: lemma zdvd_1_left: "1 dvd (m::int)"
huffman@29410:   by (rule one_dvd) (* already declared [simp] *)
wenzelm@23164: 
huffman@29950: lemma zdvd_refl: "m dvd (m::int)"
huffman@29950:   by (rule dvd_refl) (* already declared [simp] *)
wenzelm@23164: 
wenzelm@23164: lemma zdvd_trans: "m dvd n ==> n dvd k ==> m dvd (k::int)"
huffman@29410:   by (rule dvd_trans)
wenzelm@23164: 
huffman@29950: lemma zdvd_zminus_iff: "m dvd -n \<longleftrightarrow> m dvd (n::int)"
huffman@29950:   by (rule dvd_minus_iff) (* already declared [simp] *)
wenzelm@23164: 
huffman@29950: lemma zdvd_zminus2_iff: "-m dvd n \<longleftrightarrow> m dvd (n::int)"
huffman@29950:   by (rule minus_dvd_iff) (* already declared [simp] *)
haftmann@27651: 
huffman@29950: lemma zdvd_abs1: "( \<bar>i::int\<bar> dvd j) = (i dvd j)"
huffman@29950:   by (rule abs_dvd_iff) (* already declared [simp] *)
haftmann@27651: 
huffman@29950: lemma zdvd_abs2: "( (i::int) dvd \<bar>j\<bar>) = (i dvd j)" 
huffman@29950:   by (rule dvd_abs_iff) (* already declared [simp] *)
wenzelm@23164: 
wenzelm@23164: lemma zdvd_anti_sym:
wenzelm@23164:     "0 < m ==> 0 < n ==> m dvd n ==> n dvd m ==> m = (n::int)"
wenzelm@23164:   apply (simp add: dvd_def, auto)
wenzelm@23164:   apply (simp add: mult_assoc zero_less_mult_iff zmult_eq_1_iff)
wenzelm@23164:   done
wenzelm@23164: 
wenzelm@23164: lemma zdvd_zadd: "k dvd m ==> k dvd n ==> k dvd (m + n :: int)"
huffman@29410:   by (rule dvd_add)
wenzelm@23164: 
wenzelm@23164: lemma zdvd_dvd_eq: assumes anz:"a \<noteq> 0" and ab: "(a::int) dvd b" and ba:"b dvd a" 
wenzelm@23164:   shows "\<bar>a\<bar> = \<bar>b\<bar>"
wenzelm@23164: proof-
wenzelm@23164:   from ab obtain k where k:"b = a*k" unfolding dvd_def by blast 
wenzelm@23164:   from ba obtain k' where k':"a = b*k'" unfolding dvd_def by blast 
wenzelm@23164:   from k k' have "a = a*k*k'" by simp
wenzelm@23164:   with mult_cancel_left1[where c="a" and b="k*k'"]
wenzelm@23164:   have kk':"k*k' = 1" using anz by (simp add: mult_assoc)
wenzelm@23164:   hence "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1" by (simp add: zmult_eq_1_iff)
wenzelm@23164:   thus ?thesis using k k' by auto
wenzelm@23164: qed
wenzelm@23164: 
wenzelm@23164: lemma zdvd_zdiff: "k dvd m ==> k dvd n ==> k dvd (m - n :: int)"
huffman@29410:   by (rule Ring_and_Field.dvd_diff)
wenzelm@23164: 
wenzelm@23164: lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"
wenzelm@23164:   apply (subgoal_tac "m = n + (m - n)")
wenzelm@23164:    apply (erule ssubst)
wenzelm@23164:    apply (blast intro: zdvd_zadd, simp)
wenzelm@23164:   done
wenzelm@23164: 
wenzelm@23164: lemma zdvd_zmult: "k dvd (n::int) ==> k dvd m * n"
huffman@29410:   by (rule dvd_mult)
wenzelm@23164: 
wenzelm@23164: lemma zdvd_zmult2: "k dvd (m::int) ==> k dvd m * n"
huffman@29410:   by (rule dvd_mult2)
wenzelm@23164: 
huffman@29410: lemma zdvd_triv_right: "(k::int) dvd m * k"
huffman@29410:   by (rule dvd_triv_right) (* already declared [simp] *)
wenzelm@23164: 
huffman@29410: lemma zdvd_triv_left: "(k::int) dvd k * m"
huffman@29410:   by (rule dvd_triv_left) (* already declared [simp] *)
wenzelm@23164: 
wenzelm@23164: lemma zdvd_zmultD2: "j * k dvd n ==> j dvd (n::int)"
huffman@29410:   by (rule dvd_mult_left)
wenzelm@23164: 
wenzelm@23164: lemma zdvd_zmultD: "j * k dvd n ==> k dvd (n::int)"
huffman@29410:   by (rule dvd_mult_right)
wenzelm@23164: 
wenzelm@23164: lemma zdvd_zmult_mono: "i dvd m ==> j dvd (n::int) ==> i * j dvd m * n"
haftmann@27651:   by (rule mult_dvd_mono)
wenzelm@23164: 
wenzelm@23164: lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"
wenzelm@23164:   apply (rule iffI)
wenzelm@23164:    apply (erule_tac [2] zdvd_zadd)
wenzelm@23164:    apply (subgoal_tac "n = (n + k * m) - k * m")
wenzelm@23164:     apply (erule ssubst)
wenzelm@23164:     apply (erule zdvd_zdiff, simp_all)
wenzelm@23164:   done
wenzelm@23164: 
wenzelm@23164: lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"
wenzelm@23164:   apply (simp add: dvd_def)
wenzelm@23164:   apply (auto simp add: zmod_zmult_zmult1)
wenzelm@23164:   done
wenzelm@23164: 
wenzelm@23164: lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)"
wenzelm@23164:   apply (subgoal_tac "k dvd n * (m div n) + m mod n")
wenzelm@23164:    apply (simp add: zmod_zdiv_equality [symmetric])
wenzelm@23164:   apply (simp only: zdvd_zadd zdvd_zmult2)
wenzelm@23164:   done
wenzelm@23164: 
wenzelm@23164: lemma zdvd_not_zless: "0 < m ==> m < n ==> \<not> n dvd (m::int)"
haftmann@27651:   apply (auto elim!: dvdE)
wenzelm@23164:   apply (subgoal_tac "0 < n")
wenzelm@23164:    prefer 2
wenzelm@23164:    apply (blast intro: order_less_trans)
wenzelm@23164:   apply (simp add: zero_less_mult_iff)
wenzelm@23164:   apply (subgoal_tac "n * k < n * 1")
wenzelm@23164:    apply (drule mult_less_cancel_left [THEN iffD1], auto)
wenzelm@23164:   done
haftmann@27651: 
wenzelm@23164: lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"
wenzelm@23164:   using zmod_zdiv_equality[where a="m" and b="n"]
nipkow@29667:   by (simp add: algebra_simps)
wenzelm@23164: 
wenzelm@23164: lemma zdvd_mult_div_cancel:"(n::int) dvd m \<Longrightarrow> n * (m div n) = m"
wenzelm@23164: apply (subgoal_tac "m mod n = 0")
wenzelm@23164:  apply (simp add: zmult_div_cancel)
wenzelm@23164: apply (simp only: zdvd_iff_zmod_eq_0)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k \<noteq> (0::int)"
wenzelm@23164:   shows "m dvd n"
wenzelm@23164: proof-
wenzelm@23164:   from d obtain h where h: "k*n = k*m * h" unfolding dvd_def by blast
wenzelm@23164:   {assume "n \<noteq> m*h" hence "k* n \<noteq> k* (m*h)" using kz by simp
wenzelm@23164:     with h have False by (simp add: mult_assoc)}
wenzelm@23164:   hence "n = m * h" by blast
huffman@29410:   thus ?thesis by simp
wenzelm@23164: qed
wenzelm@23164: 
huffman@29981: lemma zdvd_zmult_cancel_disj:
nipkow@23969:   "(k*m) dvd (k*n) = (k=0 | m dvd (n::int))"
huffman@29981: by (rule dvd_mult_cancel_left) (* already declared [simp] *)
nipkow@23969: 
nipkow@23969: 
wenzelm@23164: theorem ex_nat: "(\<exists>x::nat. P x) = (\<exists>x::int. 0 <= x \<and> P (nat x))"
nipkow@25134: apply (simp split add: split_nat)
nipkow@25134: apply (rule iffI)
nipkow@25134: apply (erule exE)
nipkow@25134: apply (rule_tac x = "int x" in exI)
nipkow@25134: apply simp
nipkow@25134: apply (erule exE)
nipkow@25134: apply (rule_tac x = "nat x" in exI)
nipkow@25134: apply (erule conjE)
nipkow@25134: apply (erule_tac x = "nat x" in allE)
nipkow@25134: apply simp
nipkow@25134: done
wenzelm@23164: 
huffman@23365: theorem zdvd_int: "(x dvd y) = (int x dvd int y)"
haftmann@27651: proof -
haftmann@27651:   have "\<And>k. int y = int x * k \<Longrightarrow> x dvd y"
haftmann@27651:   proof -
haftmann@27651:     fix k
haftmann@27651:     assume A: "int y = int x * k"
haftmann@27651:     then show "x dvd y" proof (cases k)
haftmann@27651:       case (1 n) with A have "y = x * n" by (simp add: zmult_int)
haftmann@27651:       then show ?thesis ..
haftmann@27651:     next
haftmann@27651:       case (2 n) with A have "int y = int x * (- int (Suc n))" by simp
haftmann@27651:       also have "\<dots> = - (int x * int (Suc n))" by (simp only: mult_minus_right)
haftmann@27651:       also have "\<dots> = - int (x * Suc n)" by (simp only: zmult_int)
haftmann@27651:       finally have "- int (x * Suc n) = int y" ..
haftmann@27651:       then show ?thesis by (simp only: negative_eq_positive) auto
haftmann@27651:     qed
haftmann@27651:   qed
huffman@29410:   then show ?thesis by (auto elim!: dvdE simp only: zdvd_triv_left int_mult)
huffman@29410: qed
wenzelm@23164: 
wenzelm@23164: lemma zdvd1_eq[simp]: "(x::int) dvd 1 = ( \<bar>x\<bar> = 1)"
wenzelm@23164: proof
wenzelm@23164:   assume d: "x dvd 1" hence "int (nat \<bar>x\<bar>) dvd int (nat 1)" by (simp add: zdvd_abs1)
wenzelm@23164:   hence "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int)
wenzelm@23164:   hence "nat \<bar>x\<bar> = 1"  by simp
wenzelm@23164:   thus "\<bar>x\<bar> = 1" by (cases "x < 0", auto)
wenzelm@23164: next
wenzelm@23164:   assume "\<bar>x\<bar>=1" thus "x dvd 1" 
wenzelm@23164:     by(cases "x < 0",simp_all add: minus_equation_iff zdvd_iff_zmod_eq_0)
wenzelm@23164: qed
wenzelm@23164: lemma zdvd_mult_cancel1: 
wenzelm@23164:   assumes mp:"m \<noteq>(0::int)" shows "(m * n dvd m) = (\<bar>n\<bar> = 1)"
wenzelm@23164: proof
wenzelm@23164:   assume n1: "\<bar>n\<bar> = 1" thus "m * n dvd m" 
wenzelm@23164:     by (cases "n >0", auto simp add: zdvd_zminus2_iff minus_equation_iff)
wenzelm@23164: next
wenzelm@23164:   assume H: "m * n dvd m" hence H2: "m * n dvd m * 1" by simp
wenzelm@23164:   from zdvd_mult_cancel[OF H2 mp] show "\<bar>n\<bar> = 1" by (simp only: zdvd1_eq)
wenzelm@23164: qed
wenzelm@23164: 
huffman@23365: lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))"
haftmann@27651:   unfolding zdvd_int by (cases "z \<ge> 0") (simp_all add: zdvd_zminus_iff)
huffman@23306: 
huffman@23365: lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)"
haftmann@27651:   unfolding zdvd_int by (cases "z \<ge> 0") (simp_all add: zdvd_zminus2_iff)
wenzelm@23164: 
wenzelm@23164: lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \<le> z then (z dvd int m) else m = 0)"
haftmann@27651:   by (auto simp add: dvd_int_iff)
wenzelm@23164: 
wenzelm@23164: lemma zminus_dvd_iff [iff]: "(-z dvd w) = (z dvd (w::int))"
huffman@29410:   by (rule minus_dvd_iff)
wenzelm@23164: 
wenzelm@23164: lemma dvd_zminus_iff [iff]: "(z dvd -w) = (z dvd (w::int))"
huffman@29410:   by (rule dvd_minus_iff)
wenzelm@23164: 
wenzelm@23164: lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \<le> (n::int)"
huffman@23365:   apply (rule_tac z=n in int_cases)
huffman@23365:   apply (auto simp add: dvd_int_iff)
huffman@23365:   apply (rule_tac z=z in int_cases)
huffman@23307:   apply (auto simp add: dvd_imp_le)
wenzelm@23164:   done
wenzelm@23164: 
wenzelm@23164: lemma zpower_zmod: "((x::int) mod m)^y mod m = x^y mod m"
wenzelm@23164: apply (induct "y", auto)
wenzelm@23164: apply (rule zmod_zmult1_eq [THEN trans])
wenzelm@23164: apply (simp (no_asm_simp))
nipkow@29948: apply (rule mod_mult_eq [symmetric])
wenzelm@23164: done
wenzelm@23164: 
huffman@23365: lemma zdiv_int: "int (a div b) = (int a) div (int b)"
wenzelm@23164: apply (subst split_div, auto)
wenzelm@23164: apply (subst split_zdiv, auto)
huffman@23365: apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in IntDiv.unique_quotient)
haftmann@29651: apply (auto simp add: IntDiv.divmod_rel_def of_nat_mult)
wenzelm@23164: done
wenzelm@23164: 
wenzelm@23164: lemma zmod_int: "int (a mod b) = (int a) mod (int b)"
huffman@23365: apply (subst split_mod, auto)
huffman@23365: apply (subst split_zmod, auto)
huffman@23365: apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia 
huffman@23365:        in unique_remainder)
haftmann@29651: apply (auto simp add: IntDiv.divmod_rel_def of_nat_mult)
huffman@23365: done
wenzelm@23164: 
wenzelm@23164: text{*Suggested by Matthias Daum*}
wenzelm@23164: lemma int_power_div_base:
wenzelm@23164:      "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"
wenzelm@23164: apply (subgoal_tac "k ^ m = k ^ ((m - 1) + 1)")
wenzelm@23164:  apply (erule ssubst)
wenzelm@23164:  apply (simp only: power_add)
wenzelm@23164:  apply simp_all
wenzelm@23164: done
wenzelm@23164: 
haftmann@23853: text {* by Brian Huffman *}
haftmann@23853: lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"
huffman@29405: by (rule mod_minus_eq [symmetric])
haftmann@23853: 
haftmann@23853: lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"
huffman@29405: by (rule mod_diff_left_eq [symmetric])
haftmann@23853: 
haftmann@23853: lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"
huffman@29405: by (rule mod_diff_right_eq [symmetric])
haftmann@23853: 
haftmann@23853: lemmas zmod_simps =
nipkow@30034:   mod_add_left_eq  [symmetric]
nipkow@30034:   mod_add_right_eq [symmetric]
haftmann@23853:   IntDiv.zmod_zmult1_eq     [symmetric]
nipkow@29948:   mod_mult_left_eq          [symmetric]
haftmann@23853:   IntDiv.zpower_zmod
haftmann@23853:   zminus_zmod zdiff_zmod_left zdiff_zmod_right
haftmann@23853: 
huffman@29045: text {* Distributive laws for function @{text nat}. *}
huffman@29045: 
huffman@29045: lemma nat_div_distrib: "0 \<le> x \<Longrightarrow> nat (x div y) = nat x div nat y"
huffman@29045: apply (rule linorder_cases [of y 0])
huffman@29045: apply (simp add: div_nonneg_neg_le0)
huffman@29045: apply simp
huffman@29045: apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)
huffman@29045: done
huffman@29045: 
huffman@29045: (*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)
huffman@29045: lemma nat_mod_distrib:
huffman@29045:   "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x mod y) = nat x mod nat y"
huffman@29045: apply (case_tac "y = 0", simp add: DIVISION_BY_ZERO)
huffman@29045: apply (simp add: nat_eq_iff zmod_int)
huffman@29045: done
huffman@29045: 
huffman@29045: text{*Suggested by Matthias Daum*}
huffman@29045: lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)"
huffman@29045: apply (subgoal_tac "nat x div nat k < nat x")
huffman@29045:  apply (simp (asm_lr) add: nat_div_distrib [symmetric])
huffman@29045: apply (rule Divides.div_less_dividend, simp_all)
huffman@29045: done
huffman@29045: 
haftmann@23853: text {* code generator setup *}
wenzelm@23164: 
haftmann@26507: context ring_1
haftmann@26507: begin
haftmann@26507: 
haftmann@28562: lemma of_int_num [code]:
haftmann@26507:   "of_int k = (if k = 0 then 0 else if k < 0 then
haftmann@26507:      - of_int (- k) else let
haftmann@29651:        (l, m) = divmod k 2;
haftmann@26507:        l' = of_int l
haftmann@26507:      in if m = 0 then l' + l' else l' + l' + 1)"
haftmann@26507: proof -
haftmann@26507:   have aux1: "k mod (2\<Colon>int) \<noteq> (0\<Colon>int) \<Longrightarrow> 
haftmann@26507:     of_int k = of_int (k div 2 * 2 + 1)"
haftmann@26507:   proof -
haftmann@26507:     have "k mod 2 < 2" by (auto intro: pos_mod_bound)
haftmann@26507:     moreover have "0 \<le> k mod 2" by (auto intro: pos_mod_sign)
haftmann@26507:     moreover assume "k mod 2 \<noteq> 0"
haftmann@26507:     ultimately have "k mod 2 = 1" by arith
haftmann@26507:     moreover have "of_int k = of_int (k div 2 * 2 + k mod 2)" by simp
haftmann@26507:     ultimately show ?thesis by auto
haftmann@26507:   qed
haftmann@26507:   have aux2: "\<And>x. of_int 2 * x = x + x"
haftmann@26507:   proof -
haftmann@26507:     fix x
haftmann@26507:     have int2: "(2::int) = 1 + 1" by arith
haftmann@26507:     show "of_int 2 * x = x + x"
haftmann@26507:     unfolding int2 of_int_add left_distrib by simp
haftmann@26507:   qed
haftmann@26507:   have aux3: "\<And>x. x * of_int 2 = x + x"
haftmann@26507:   proof -
haftmann@26507:     fix x
haftmann@26507:     have int2: "(2::int) = 1 + 1" by arith
haftmann@26507:     show "x * of_int 2 = x + x" 
haftmann@26507:     unfolding int2 of_int_add right_distrib by simp
haftmann@26507:   qed
haftmann@29651:   from aux1 show ?thesis by (auto simp add: divmod_mod_div Let_def aux2 aux3)
haftmann@26507: qed
haftmann@26507: 
haftmann@26507: end
haftmann@26507: 
chaieb@27667: lemma zmod_eq_dvd_iff: "(x::int) mod n = y mod n \<longleftrightarrow> n dvd x - y"
chaieb@27667: proof
chaieb@27667:   assume H: "x mod n = y mod n"
chaieb@27667:   hence "x mod n - y mod n = 0" by simp
chaieb@27667:   hence "(x mod n - y mod n) mod n = 0" by simp 
nipkow@30034:   hence "(x - y) mod n = 0" by (simp add: mod_diff_eq[symmetric])
chaieb@27667:   thus "n dvd x - y" by (simp add: zdvd_iff_zmod_eq_0)
chaieb@27667: next
chaieb@27667:   assume H: "n dvd x - y"
chaieb@27667:   then obtain k where k: "x-y = n*k" unfolding dvd_def by blast
chaieb@27667:   hence "x = n*k + y" by simp
chaieb@27667:   hence "x mod n = (n*k + y) mod n" by simp
nipkow@30034:   thus "x mod n = y mod n" by (simp add: mod_add_left_eq)
chaieb@27667: qed
chaieb@27667: 
chaieb@27667: lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y  mod n" and xy:"y \<le> x"
chaieb@27667:   shows "\<exists>q. x = y + n * q"
chaieb@27667: proof-
chaieb@27667:   from xy have th: "int x - int y = int (x - y)" by simp 
chaieb@27667:   from xyn have "int x mod int n = int y mod int n" 
chaieb@27667:     by (simp add: zmod_int[symmetric])
chaieb@27667:   hence "int n dvd int x - int y" by (simp only: zmod_eq_dvd_iff[symmetric]) 
chaieb@27667:   hence "n dvd x - y" by (simp add: th zdvd_int)
chaieb@27667:   then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith
chaieb@27667: qed
chaieb@27667: 
chaieb@27667: lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)" 
chaieb@27667:   (is "?lhs = ?rhs")
chaieb@27667: proof
chaieb@27667:   assume H: "x mod n = y mod n"
chaieb@27667:   {assume xy: "x \<le> y"
chaieb@27667:     from H have th: "y mod n = x mod n" by simp
chaieb@27667:     from nat_mod_eq_lemma[OF th xy] have ?rhs 
chaieb@27667:       apply clarify  apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}
chaieb@27667:   moreover
chaieb@27667:   {assume xy: "y \<le> x"
chaieb@27667:     from nat_mod_eq_lemma[OF H xy] have ?rhs 
chaieb@27667:       apply clarify  apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}
chaieb@27667:   ultimately  show ?rhs using linear[of x y] by blast  
chaieb@27667: next
chaieb@27667:   assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast
chaieb@27667:   hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp
chaieb@27667:   thus  ?lhs by simp
chaieb@27667: qed
chaieb@27667: 
haftmann@29936: 
haftmann@29936: subsection {* Code generation *}
haftmann@29936: 
haftmann@29936: definition pdivmod :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
haftmann@29936:   "pdivmod k l = (\<bar>k\<bar> div \<bar>l\<bar>, \<bar>k\<bar> mod \<bar>l\<bar>)"
haftmann@29936: 
haftmann@29936: lemma pdivmod_posDivAlg [code]:
haftmann@29936:   "pdivmod k l = (if l = 0 then (0, \<bar>k\<bar>) else posDivAlg \<bar>k\<bar> \<bar>l\<bar>)"
haftmann@29936: by (subst posDivAlg_div_mod) (simp_all add: pdivmod_def)
haftmann@29936: 
haftmann@29936: lemma divmod_pdivmod: "divmod k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else
haftmann@29936:   apsnd ((op *) (sgn l)) (if 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0
haftmann@29936:     then pdivmod k l
haftmann@29936:     else (let (r, s) = pdivmod k l in
haftmann@29936:       if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
haftmann@29936: proof -
haftmann@29936:   have aux: "\<And>q::int. - k = l * q \<longleftrightarrow> k = l * - q" by auto
haftmann@29936:   show ?thesis
haftmann@29936:     by (simp add: divmod_mod_div pdivmod_def)
haftmann@29936:       (auto simp add: aux not_less not_le zdiv_zminus1_eq_if
haftmann@29936:       zmod_zminus1_eq_if zdiv_zminus2_eq_if zmod_zminus2_eq_if)
haftmann@29936: qed
haftmann@29936: 
haftmann@29936: lemma divmod_code [code]: "divmod k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else
haftmann@29936:   apsnd ((op *) (sgn l)) (if sgn k = sgn l
haftmann@29936:     then pdivmod k l
haftmann@29936:     else (let (r, s) = pdivmod k l in
haftmann@29936:       if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
haftmann@29936: proof -
haftmann@29936:   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"
haftmann@29936:     by (auto simp add: not_less sgn_if)
haftmann@29936:   then show ?thesis by (simp add: divmod_pdivmod)
haftmann@29936: qed
haftmann@29936: 
wenzelm@23164: code_modulename SML
wenzelm@23164:   IntDiv Integer
wenzelm@23164: 
wenzelm@23164: code_modulename OCaml
wenzelm@23164:   IntDiv Integer
wenzelm@23164: 
wenzelm@23164: code_modulename Haskell
haftmann@24195:   IntDiv Integer
wenzelm@23164: 
wenzelm@23164: end




