src/HOL/IntDiv.thy
author nipkow
Fri Mar 06 17:38:47 2009 +0100 (2009-03-06)
changeset 30313 b2441b0c8d38
parent 30242 aea5d7fa7ef5
child 30323 6a02238da8e9
permissions -rw-r--r--
added lemmas
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(*  Title:      HOL/IntDiv.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1999  University of Cambridge
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*)
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header{* The Division Operators div and mod *}
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theory IntDiv
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imports Int Divides FunDef
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begin
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definition divmod_rel :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool" where
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    --{*definition of quotient and remainder*}
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    [code]: "divmod_rel a b = (\<lambda>(q, r). a = b * q + r \<and>
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               (if 0 < b then 0 \<le> r \<and> r < b else b < r \<and> r \<le> 0))"
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definition adjust :: "int \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int" where
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    --{*for the division algorithm*}
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    [code]: "adjust b = (\<lambda>(q, r). if 0 \<le> r - b then (2 * q + 1, r - b)
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                         else (2 * q, r))"
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text{*algorithm for the case @{text "a\<ge>0, b>0"}*}
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function posDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
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  "posDivAlg a b = (if a < b \<or>  b \<le> 0 then (0, a)
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     else adjust b (posDivAlg a (2 * b)))"
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by auto
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termination by (relation "measure (\<lambda>(a, b). nat (a - b + 1))") auto
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text{*algorithm for the case @{text "a<0, b>0"}*}
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function negDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
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  "negDivAlg a b = (if 0 \<le>a + b \<or> b \<le> 0  then (-1, a + b)
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     else adjust b (negDivAlg a (2 * b)))"
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by auto
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termination by (relation "measure (\<lambda>(a, b). nat (- a - b))") auto
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text{*algorithm for the general case @{term "b\<noteq>0"}*}
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definition negateSnd :: "int \<times> int \<Rightarrow> int \<times> int" where
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  [code inline]: "negateSnd = apsnd uminus"
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definition divmod :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
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    --{*The full division algorithm considers all possible signs for a, b
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       including the special case @{text "a=0, b<0"} because 
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       @{term negDivAlg} requires @{term "a<0"}.*}
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  "divmod a b = (if 0 \<le> a then if 0 \<le> b then posDivAlg a b
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                  else if a = 0 then (0, 0)
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                       else negateSnd (negDivAlg (-a) (-b))
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               else 
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                  if 0 < b then negDivAlg a b
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                  else negateSnd (posDivAlg (-a) (-b)))"
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instantiation int :: Divides.div
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begin
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definition
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  div_def: "a div b = fst (divmod a b)"
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definition
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  mod_def: "a mod b = snd (divmod a b)"
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instance ..
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end
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lemma divmod_mod_div:
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  "divmod p q = (p div q, p mod q)"
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  by (auto simp add: div_def mod_def)
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text{*
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Here is the division algorithm in ML:
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\begin{verbatim}
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    fun posDivAlg (a,b) =
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      if a<b then (0,a)
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      else let val (q,r) = posDivAlg(a, 2*b)
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	       in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
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	   end
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    fun negDivAlg (a,b) =
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      if 0\<le>a+b then (~1,a+b)
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      else let val (q,r) = negDivAlg(a, 2*b)
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	       in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
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	   end;
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    fun negateSnd (q,r:int) = (q,~r);
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    fun divmod (a,b) = if 0\<le>a then 
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			  if b>0 then posDivAlg (a,b) 
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			   else if a=0 then (0,0)
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				else negateSnd (negDivAlg (~a,~b))
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		       else 
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			  if 0<b then negDivAlg (a,b)
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			  else        negateSnd (posDivAlg (~a,~b));
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\end{verbatim}
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*}
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subsection{*Uniqueness and Monotonicity of Quotients and Remainders*}
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lemma unique_quotient_lemma:
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     "[| b*q' + r'  \<le> b*q + r;  0 \<le> r';  r' < b;  r < b |]  
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      ==> q' \<le> (q::int)"
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apply (subgoal_tac "r' + b * (q'-q) \<le> r")
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 prefer 2 apply (simp add: right_diff_distrib)
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apply (subgoal_tac "0 < b * (1 + q - q') ")
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apply (erule_tac [2] order_le_less_trans)
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 prefer 2 apply (simp add: right_diff_distrib right_distrib)
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apply (subgoal_tac "b * q' < b * (1 + q) ")
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 prefer 2 apply (simp add: right_diff_distrib right_distrib)
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apply (simp add: mult_less_cancel_left)
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done
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lemma unique_quotient_lemma_neg:
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     "[| b*q' + r' \<le> b*q + r;  r \<le> 0;  b < r;  b < r' |]  
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      ==> q \<le> (q'::int)"
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by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma, 
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    auto)
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lemma unique_quotient:
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     "[| divmod_rel a b (q, r); divmod_rel a b (q', r');  b \<noteq> 0 |]  
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      ==> q = q'"
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apply (simp add: divmod_rel_def linorder_neq_iff split: split_if_asm)
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apply (blast intro: order_antisym
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             dest: order_eq_refl [THEN unique_quotient_lemma] 
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             order_eq_refl [THEN unique_quotient_lemma_neg] sym)+
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done
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lemma unique_remainder:
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     "[| divmod_rel a b (q, r); divmod_rel a b (q', r');  b \<noteq> 0 |]  
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      ==> r = r'"
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apply (subgoal_tac "q = q'")
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 apply (simp add: divmod_rel_def)
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apply (blast intro: unique_quotient)
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done
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subsection{*Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends*}
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text{*And positive divisors*}
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lemma adjust_eq [simp]:
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     "adjust b (q,r) = 
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      (let diff = r-b in  
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	if 0 \<le> diff then (2*q + 1, diff)   
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                     else (2*q, r))"
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by (simp add: Let_def adjust_def)
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declare posDivAlg.simps [simp del]
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text{*use with a simproc to avoid repeatedly proving the premise*}
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lemma posDivAlg_eqn:
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     "0 < b ==>  
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      posDivAlg a b = (if a<b then (0,a) else adjust b (posDivAlg a (2*b)))"
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by (rule posDivAlg.simps [THEN trans], simp)
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text{*Correctness of @{term posDivAlg}: it computes quotients correctly*}
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theorem posDivAlg_correct:
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  assumes "0 \<le> a" and "0 < b"
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  shows "divmod_rel a b (posDivAlg a b)"
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using prems apply (induct a b rule: posDivAlg.induct)
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apply auto
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apply (simp add: divmod_rel_def)
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apply (subst posDivAlg_eqn, simp add: right_distrib)
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apply (case_tac "a < b")
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apply simp_all
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apply (erule splitE)
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apply (auto simp add: right_distrib Let_def)
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done
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subsection{*Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends*}
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text{*And positive divisors*}
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declare negDivAlg.simps [simp del]
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text{*use with a simproc to avoid repeatedly proving the premise*}
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lemma negDivAlg_eqn:
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     "0 < b ==>  
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      negDivAlg a b =       
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       (if 0\<le>a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))"
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by (rule negDivAlg.simps [THEN trans], simp)
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(*Correctness of negDivAlg: it computes quotients correctly
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  It doesn't work if a=0 because the 0/b equals 0, not -1*)
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lemma negDivAlg_correct:
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  assumes "a < 0" and "b > 0"
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  shows "divmod_rel a b (negDivAlg a b)"
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using prems apply (induct a b rule: negDivAlg.induct)
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apply (auto simp add: linorder_not_le)
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apply (simp add: divmod_rel_def)
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apply (subst negDivAlg_eqn, assumption)
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apply (case_tac "a + b < (0\<Colon>int)")
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apply simp_all
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apply (erule splitE)
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apply (auto simp add: right_distrib Let_def)
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done
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subsection{*Existence Shown by Proving the Division Algorithm to be Correct*}
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(*the case a=0*)
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lemma divmod_rel_0: "b \<noteq> 0 ==> divmod_rel 0 b (0, 0)"
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by (auto simp add: divmod_rel_def linorder_neq_iff)
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lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)"
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by (subst posDivAlg.simps, auto)
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lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)"
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by (subst negDivAlg.simps, auto)
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lemma negateSnd_eq [simp]: "negateSnd(q,r) = (q,-r)"
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by (simp add: negateSnd_def)
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lemma divmod_rel_neg: "divmod_rel (-a) (-b) qr ==> divmod_rel a b (negateSnd qr)"
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by (auto simp add: split_ifs divmod_rel_def)
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lemma divmod_correct: "b \<noteq> 0 ==> divmod_rel a b (divmod a b)"
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by (force simp add: linorder_neq_iff divmod_rel_0 divmod_def divmod_rel_neg
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                    posDivAlg_correct negDivAlg_correct)
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text{*Arbitrary definitions for division by zero.  Useful to simplify 
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    certain equations.*}
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lemma DIVISION_BY_ZERO [simp]: "a div (0::int) = 0 & a mod (0::int) = a"
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by (simp add: div_def mod_def divmod_def posDivAlg.simps)  
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text{*Basic laws about division and remainder*}
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lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"
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apply (case_tac "b = 0", simp)
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apply (cut_tac a = a and b = b in divmod_correct)
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apply (auto simp add: divmod_rel_def div_def mod_def)
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done
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lemma zdiv_zmod_equality: "(b * (a div b) + (a mod b)) + k = (a::int)+k"
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by(simp add: zmod_zdiv_equality[symmetric])
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lemma zdiv_zmod_equality2: "((a div b) * b + (a mod b)) + k = (a::int)+k"
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by(simp add: mult_commute zmod_zdiv_equality[symmetric])
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text {* Tool setup *}
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ML {*
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local 
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structure CancelDivMod = CancelDivModFun(
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struct
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  val div_name = @{const_name Divides.div};
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  val mod_name = @{const_name Divides.mod};
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  val mk_binop = HOLogic.mk_binop;
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  val mk_sum = Int_Numeral_Simprocs.mk_sum HOLogic.intT;
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  val dest_sum = Int_Numeral_Simprocs.dest_sum;
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  val div_mod_eqs =
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    map mk_meta_eq [@{thm zdiv_zmod_equality},
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      @{thm zdiv_zmod_equality2}];
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  val trans = trans;
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  val prove_eq_sums =
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    let
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      val simps = @{thm diff_int_def} :: Int_Numeral_Simprocs.add_0s @ @{thms zadd_ac}
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    in ArithData.prove_conv all_tac (ArithData.simp_all_tac simps) end;
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end)
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in
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val cancel_zdiv_zmod_proc = Simplifier.simproc (the_context ())
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  "cancel_zdiv_zmod" ["(m::int) + n"] (K CancelDivMod.proc)
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end;
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Addsimprocs [cancel_zdiv_zmod_proc]
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*}
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lemma pos_mod_conj : "(0::int) < b ==> 0 \<le> a mod b & a mod b < b"
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apply (cut_tac a = a and b = b in divmod_correct)
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apply (auto simp add: divmod_rel_def mod_def)
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done
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lemmas pos_mod_sign  [simp] = pos_mod_conj [THEN conjunct1, standard]
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   and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2, standard]
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lemma neg_mod_conj : "b < (0::int) ==> a mod b \<le> 0 & b < a mod b"
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apply (cut_tac a = a and b = b in divmod_correct)
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apply (auto simp add: divmod_rel_def div_def mod_def)
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done
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lemmas neg_mod_sign  [simp] = neg_mod_conj [THEN conjunct1, standard]
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   and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2, standard]
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subsection{*General Properties of div and mod*}
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lemma divmod_rel_div_mod: "b \<noteq> 0 ==> divmod_rel a b (a div b, a mod b)"
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apply (cut_tac a = a and b = b in zmod_zdiv_equality)
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apply (force simp add: divmod_rel_def linorder_neq_iff)
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done
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lemma divmod_rel_div: "[| divmod_rel a b (q, r);  b \<noteq> 0 |] ==> a div b = q"
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by (simp add: divmod_rel_div_mod [THEN unique_quotient])
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lemma divmod_rel_mod: "[| divmod_rel a b (q, r);  b \<noteq> 0 |] ==> a mod b = r"
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by (simp add: divmod_rel_div_mod [THEN unique_remainder])
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lemma div_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a div b = 0"
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apply (rule divmod_rel_div)
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apply (auto simp add: divmod_rel_def)
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done
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lemma div_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a div b = 0"
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apply (rule divmod_rel_div)
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apply (auto simp add: divmod_rel_def)
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done
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lemma div_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a div b = -1"
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   319
apply (rule divmod_rel_div)
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   320
apply (auto simp add: divmod_rel_def)
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   321
done
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   322
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   323
(*There is no div_neg_pos_trivial because  0 div b = 0 would supersede it*)
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   324
wenzelm@23164
   325
lemma mod_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a mod b = a"
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   326
apply (rule_tac q = 0 in divmod_rel_mod)
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   327
apply (auto simp add: divmod_rel_def)
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   328
done
wenzelm@23164
   329
wenzelm@23164
   330
lemma mod_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a mod b = a"
haftmann@29651
   331
apply (rule_tac q = 0 in divmod_rel_mod)
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   332
apply (auto simp add: divmod_rel_def)
wenzelm@23164
   333
done
wenzelm@23164
   334
wenzelm@23164
   335
lemma mod_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a mod b = a+b"
haftmann@29651
   336
apply (rule_tac q = "-1" in divmod_rel_mod)
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   337
apply (auto simp add: divmod_rel_def)
wenzelm@23164
   338
done
wenzelm@23164
   339
wenzelm@23164
   340
text{*There is no @{text mod_neg_pos_trivial}.*}
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   341
wenzelm@23164
   342
wenzelm@23164
   343
(*Simpler laws such as -a div b = -(a div b) FAIL, but see just below*)
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   344
lemma zdiv_zminus_zminus [simp]: "(-a) div (-b) = a div (b::int)"
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   345
apply (case_tac "b = 0", simp)
haftmann@29651
   346
apply (simp add: divmod_rel_div_mod [THEN divmod_rel_neg, simplified, 
haftmann@29651
   347
                                 THEN divmod_rel_div, THEN sym])
wenzelm@23164
   348
wenzelm@23164
   349
done
wenzelm@23164
   350
wenzelm@23164
   351
(*Simpler laws such as -a mod b = -(a mod b) FAIL, but see just below*)
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   352
lemma zmod_zminus_zminus [simp]: "(-a) mod (-b) = - (a mod (b::int))"
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   353
apply (case_tac "b = 0", simp)
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   354
apply (subst divmod_rel_div_mod [THEN divmod_rel_neg, simplified, THEN divmod_rel_mod],
wenzelm@23164
   355
       auto)
wenzelm@23164
   356
done
wenzelm@23164
   357
wenzelm@23164
   358
wenzelm@23164
   359
subsection{*Laws for div and mod with Unary Minus*}
wenzelm@23164
   360
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   361
lemma zminus1_lemma:
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   362
     "divmod_rel a b (q, r)
haftmann@29651
   363
      ==> divmod_rel (-a) b (if r=0 then -q else -q - 1,  
haftmann@29651
   364
                          if r=0 then 0 else b-r)"
haftmann@29651
   365
by (force simp add: split_ifs divmod_rel_def linorder_neq_iff right_diff_distrib)
wenzelm@23164
   366
wenzelm@23164
   367
wenzelm@23164
   368
lemma zdiv_zminus1_eq_if:
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   369
     "b \<noteq> (0::int)  
wenzelm@23164
   370
      ==> (-a) div b =  
wenzelm@23164
   371
          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
haftmann@29651
   372
by (blast intro: divmod_rel_div_mod [THEN zminus1_lemma, THEN divmod_rel_div])
wenzelm@23164
   373
wenzelm@23164
   374
lemma zmod_zminus1_eq_if:
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   375
     "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))"
wenzelm@23164
   376
apply (case_tac "b = 0", simp)
haftmann@29651
   377
apply (blast intro: divmod_rel_div_mod [THEN zminus1_lemma, THEN divmod_rel_mod])
wenzelm@23164
   378
done
wenzelm@23164
   379
haftmann@29936
   380
lemma zmod_zminus1_not_zero:
haftmann@29936
   381
  fixes k l :: int
haftmann@29936
   382
  shows "- k mod l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
haftmann@29936
   383
  unfolding zmod_zminus1_eq_if by auto
haftmann@29936
   384
wenzelm@23164
   385
lemma zdiv_zminus2: "a div (-b) = (-a::int) div b"
wenzelm@23164
   386
by (cut_tac a = "-a" in zdiv_zminus_zminus, auto)
wenzelm@23164
   387
wenzelm@23164
   388
lemma zmod_zminus2: "a mod (-b) = - ((-a::int) mod b)"
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   389
by (cut_tac a = "-a" and b = b in zmod_zminus_zminus, auto)
wenzelm@23164
   390
wenzelm@23164
   391
lemma zdiv_zminus2_eq_if:
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   392
     "b \<noteq> (0::int)  
wenzelm@23164
   393
      ==> a div (-b) =  
wenzelm@23164
   394
          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
wenzelm@23164
   395
by (simp add: zdiv_zminus1_eq_if zdiv_zminus2)
wenzelm@23164
   396
wenzelm@23164
   397
lemma zmod_zminus2_eq_if:
wenzelm@23164
   398
     "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"
wenzelm@23164
   399
by (simp add: zmod_zminus1_eq_if zmod_zminus2)
wenzelm@23164
   400
haftmann@29936
   401
lemma zmod_zminus2_not_zero:
haftmann@29936
   402
  fixes k l :: int
haftmann@29936
   403
  shows "k mod - l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
haftmann@29936
   404
  unfolding zmod_zminus2_eq_if by auto 
haftmann@29936
   405
wenzelm@23164
   406
wenzelm@23164
   407
subsection{*Division of a Number by Itself*}
wenzelm@23164
   408
wenzelm@23164
   409
lemma self_quotient_aux1: "[| (0::int) < a; a = r + a*q; r < a |] ==> 1 \<le> q"
wenzelm@23164
   410
apply (subgoal_tac "0 < a*q")
wenzelm@23164
   411
 apply (simp add: zero_less_mult_iff, arith)
wenzelm@23164
   412
done
wenzelm@23164
   413
wenzelm@23164
   414
lemma self_quotient_aux2: "[| (0::int) < a; a = r + a*q; 0 \<le> r |] ==> q \<le> 1"
wenzelm@23164
   415
apply (subgoal_tac "0 \<le> a* (1-q) ")
wenzelm@23164
   416
 apply (simp add: zero_le_mult_iff)
wenzelm@23164
   417
apply (simp add: right_diff_distrib)
wenzelm@23164
   418
done
wenzelm@23164
   419
haftmann@29651
   420
lemma self_quotient: "[| divmod_rel a a (q, r);  a \<noteq> (0::int) |] ==> q = 1"
haftmann@29651
   421
apply (simp add: split_ifs divmod_rel_def linorder_neq_iff)
wenzelm@23164
   422
apply (rule order_antisym, safe, simp_all)
wenzelm@23164
   423
apply (rule_tac [3] a = "-a" and r = "-r" in self_quotient_aux1)
wenzelm@23164
   424
apply (rule_tac a = "-a" and r = "-r" in self_quotient_aux2)
wenzelm@23164
   425
apply (force intro: self_quotient_aux1 self_quotient_aux2 simp add: add_commute)+
wenzelm@23164
   426
done
wenzelm@23164
   427
haftmann@29651
   428
lemma self_remainder: "[| divmod_rel a a (q, r);  a \<noteq> (0::int) |] ==> r = 0"
wenzelm@23164
   429
apply (frule self_quotient, assumption)
haftmann@29651
   430
apply (simp add: divmod_rel_def)
wenzelm@23164
   431
done
wenzelm@23164
   432
wenzelm@23164
   433
lemma zdiv_self [simp]: "a \<noteq> 0 ==> a div a = (1::int)"
haftmann@29651
   434
by (simp add: divmod_rel_div_mod [THEN self_quotient])
wenzelm@23164
   435
wenzelm@23164
   436
(*Here we have 0 mod 0 = 0, also assumed by Knuth (who puts m mod 0 = 0) *)
wenzelm@23164
   437
lemma zmod_self [simp]: "a mod a = (0::int)"
wenzelm@23164
   438
apply (case_tac "a = 0", simp)
haftmann@29651
   439
apply (simp add: divmod_rel_div_mod [THEN self_remainder])
wenzelm@23164
   440
done
wenzelm@23164
   441
wenzelm@23164
   442
wenzelm@23164
   443
subsection{*Computation of Division and Remainder*}
wenzelm@23164
   444
wenzelm@23164
   445
lemma zdiv_zero [simp]: "(0::int) div b = 0"
haftmann@29651
   446
by (simp add: div_def divmod_def)
wenzelm@23164
   447
wenzelm@23164
   448
lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"
haftmann@29651
   449
by (simp add: div_def divmod_def)
wenzelm@23164
   450
wenzelm@23164
   451
lemma zmod_zero [simp]: "(0::int) mod b = 0"
haftmann@29651
   452
by (simp add: mod_def divmod_def)
wenzelm@23164
   453
wenzelm@23164
   454
lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"
haftmann@29651
   455
by (simp add: mod_def divmod_def)
wenzelm@23164
   456
wenzelm@23164
   457
text{*a positive, b positive *}
wenzelm@23164
   458
wenzelm@23164
   459
lemma div_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a div b = fst (posDivAlg a b)"
haftmann@29651
   460
by (simp add: div_def divmod_def)
wenzelm@23164
   461
wenzelm@23164
   462
lemma mod_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a mod b = snd (posDivAlg a b)"
haftmann@29651
   463
by (simp add: mod_def divmod_def)
wenzelm@23164
   464
wenzelm@23164
   465
text{*a negative, b positive *}
wenzelm@23164
   466
wenzelm@23164
   467
lemma div_neg_pos: "[| a < 0;  0 < b |] ==> a div b = fst (negDivAlg a b)"
haftmann@29651
   468
by (simp add: div_def divmod_def)
wenzelm@23164
   469
wenzelm@23164
   470
lemma mod_neg_pos: "[| a < 0;  0 < b |] ==> a mod b = snd (negDivAlg a b)"
haftmann@29651
   471
by (simp add: mod_def divmod_def)
wenzelm@23164
   472
wenzelm@23164
   473
text{*a positive, b negative *}
wenzelm@23164
   474
wenzelm@23164
   475
lemma div_pos_neg:
wenzelm@23164
   476
     "[| 0 < a;  b < 0 |] ==> a div b = fst (negateSnd (negDivAlg (-a) (-b)))"
haftmann@29651
   477
by (simp add: div_def divmod_def)
wenzelm@23164
   478
wenzelm@23164
   479
lemma mod_pos_neg:
wenzelm@23164
   480
     "[| 0 < a;  b < 0 |] ==> a mod b = snd (negateSnd (negDivAlg (-a) (-b)))"
haftmann@29651
   481
by (simp add: mod_def divmod_def)
wenzelm@23164
   482
wenzelm@23164
   483
text{*a negative, b negative *}
wenzelm@23164
   484
wenzelm@23164
   485
lemma div_neg_neg:
wenzelm@23164
   486
     "[| a < 0;  b \<le> 0 |] ==> a div b = fst (negateSnd (posDivAlg (-a) (-b)))"
haftmann@29651
   487
by (simp add: div_def divmod_def)
wenzelm@23164
   488
wenzelm@23164
   489
lemma mod_neg_neg:
wenzelm@23164
   490
     "[| a < 0;  b \<le> 0 |] ==> a mod b = snd (negateSnd (posDivAlg (-a) (-b)))"
haftmann@29651
   491
by (simp add: mod_def divmod_def)
wenzelm@23164
   492
wenzelm@23164
   493
text {*Simplify expresions in which div and mod combine numerical constants*}
wenzelm@23164
   494
haftmann@29651
   495
lemma divmod_relI:
huffman@24481
   496
  "\<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
   497
    \<Longrightarrow> divmod_rel a b (q, r)"
haftmann@29651
   498
  unfolding divmod_rel_def by simp
huffman@24481
   499
haftmann@29651
   500
lemmas divmod_rel_div_eq = divmod_relI [THEN divmod_rel_div, THEN eq_reflection]
haftmann@29651
   501
lemmas divmod_rel_mod_eq = divmod_relI [THEN divmod_rel_mod, THEN eq_reflection]
huffman@24481
   502
lemmas arithmetic_simps =
huffman@24481
   503
  arith_simps
huffman@24481
   504
  add_special
huffman@24481
   505
  OrderedGroup.add_0_left
huffman@24481
   506
  OrderedGroup.add_0_right
huffman@24481
   507
  mult_zero_left
huffman@24481
   508
  mult_zero_right
huffman@24481
   509
  mult_1_left
huffman@24481
   510
  mult_1_right
huffman@24481
   511
huffman@24481
   512
(* simprocs adapted from HOL/ex/Binary.thy *)
huffman@24481
   513
ML {*
huffman@24481
   514
local
huffman@24481
   515
  infix ==;
huffman@24481
   516
  val op == = Logic.mk_equals;
huffman@24481
   517
  fun plus m n = @{term "plus :: int \<Rightarrow> int \<Rightarrow> int"} $ m $ n;
huffman@24481
   518
  fun mult m n = @{term "times :: int \<Rightarrow> int \<Rightarrow> int"} $ m $ n;
huffman@24481
   519
huffman@24481
   520
  val binary_ss = HOL_basic_ss addsimps @{thms arithmetic_simps};
huffman@24481
   521
  fun prove ctxt prop =
huffman@24481
   522
    Goal.prove ctxt [] [] prop (fn _ => ALLGOALS (full_simp_tac binary_ss));
huffman@24481
   523
huffman@24481
   524
  fun binary_proc proc ss ct =
huffman@24481
   525
    (case Thm.term_of ct of
huffman@24481
   526
      _ $ t $ u =>
huffman@24481
   527
      (case try (pairself (`(snd o HOLogic.dest_number))) (t, u) of
huffman@24481
   528
        SOME args => proc (Simplifier.the_context ss) args
huffman@24481
   529
      | NONE => NONE)
huffman@24481
   530
    | _ => NONE);
huffman@24481
   531
in
huffman@24481
   532
huffman@24481
   533
fun divmod_proc rule = binary_proc (fn ctxt => fn ((m, t), (n, u)) =>
huffman@24481
   534
  if n = 0 then NONE
huffman@24481
   535
  else
wenzelm@24630
   536
    let val (k, l) = Integer.div_mod m n;
huffman@24481
   537
        fun mk_num x = HOLogic.mk_number HOLogic.intT x;
huffman@24481
   538
    in SOME (rule OF [prove ctxt (t == plus (mult u (mk_num k)) (mk_num l))])
huffman@24481
   539
    end);
huffman@24481
   540
huffman@24481
   541
end;
huffman@24481
   542
*}
huffman@24481
   543
huffman@24481
   544
simproc_setup binary_int_div ("number_of m div number_of n :: int") =
haftmann@29651
   545
  {* K (divmod_proc (@{thm divmod_rel_div_eq})) *}
huffman@24481
   546
huffman@24481
   547
simproc_setup binary_int_mod ("number_of m mod number_of n :: int") =
haftmann@29651
   548
  {* K (divmod_proc (@{thm divmod_rel_mod_eq})) *}
huffman@24481
   549
wenzelm@23164
   550
lemmas posDivAlg_eqn_number_of [simp] =
wenzelm@23164
   551
    posDivAlg_eqn [of "number_of v" "number_of w", standard]
wenzelm@23164
   552
wenzelm@23164
   553
lemmas negDivAlg_eqn_number_of [simp] =
wenzelm@23164
   554
    negDivAlg_eqn [of "number_of v" "number_of w", standard]
wenzelm@23164
   555
wenzelm@23164
   556
wenzelm@23164
   557
text{*Special-case simplification *}
wenzelm@23164
   558
wenzelm@23164
   559
lemma zmod_minus1_right [simp]: "a mod (-1::int) = 0"
wenzelm@23164
   560
apply (cut_tac a = a and b = "-1" in neg_mod_sign)
wenzelm@23164
   561
apply (cut_tac [2] a = a and b = "-1" in neg_mod_bound)
wenzelm@23164
   562
apply (auto simp del: neg_mod_sign neg_mod_bound)
wenzelm@23164
   563
done
wenzelm@23164
   564
wenzelm@23164
   565
lemma zdiv_minus1_right [simp]: "a div (-1::int) = -a"
wenzelm@23164
   566
by (cut_tac a = a and b = "-1" in zmod_zdiv_equality, auto)
wenzelm@23164
   567
wenzelm@23164
   568
(** The last remaining special cases for constant arithmetic:
wenzelm@23164
   569
    1 div z and 1 mod z **)
wenzelm@23164
   570
wenzelm@23164
   571
lemmas div_pos_pos_1_number_of [simp] =
wenzelm@23164
   572
    div_pos_pos [OF int_0_less_1, of "number_of w", standard]
wenzelm@23164
   573
wenzelm@23164
   574
lemmas div_pos_neg_1_number_of [simp] =
wenzelm@23164
   575
    div_pos_neg [OF int_0_less_1, of "number_of w", standard]
wenzelm@23164
   576
wenzelm@23164
   577
lemmas mod_pos_pos_1_number_of [simp] =
wenzelm@23164
   578
    mod_pos_pos [OF int_0_less_1, of "number_of w", standard]
wenzelm@23164
   579
wenzelm@23164
   580
lemmas mod_pos_neg_1_number_of [simp] =
wenzelm@23164
   581
    mod_pos_neg [OF int_0_less_1, of "number_of w", standard]
wenzelm@23164
   582
wenzelm@23164
   583
wenzelm@23164
   584
lemmas posDivAlg_eqn_1_number_of [simp] =
wenzelm@23164
   585
    posDivAlg_eqn [of concl: 1 "number_of w", standard]
wenzelm@23164
   586
wenzelm@23164
   587
lemmas negDivAlg_eqn_1_number_of [simp] =
wenzelm@23164
   588
    negDivAlg_eqn [of concl: 1 "number_of w", standard]
wenzelm@23164
   589
wenzelm@23164
   590
wenzelm@23164
   591
wenzelm@23164
   592
subsection{*Monotonicity in the First Argument (Dividend)*}
wenzelm@23164
   593
wenzelm@23164
   594
lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div b"
wenzelm@23164
   595
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
wenzelm@23164
   596
apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
wenzelm@23164
   597
apply (rule unique_quotient_lemma)
wenzelm@23164
   598
apply (erule subst)
wenzelm@23164
   599
apply (erule subst, simp_all)
wenzelm@23164
   600
done
wenzelm@23164
   601
wenzelm@23164
   602
lemma zdiv_mono1_neg: "[| a \<le> a';  (b::int) < 0 |] ==> a' div b \<le> a div b"
wenzelm@23164
   603
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
wenzelm@23164
   604
apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
wenzelm@23164
   605
apply (rule unique_quotient_lemma_neg)
wenzelm@23164
   606
apply (erule subst)
wenzelm@23164
   607
apply (erule subst, simp_all)
wenzelm@23164
   608
done
wenzelm@23164
   609
wenzelm@23164
   610
wenzelm@23164
   611
subsection{*Monotonicity in the Second Argument (Divisor)*}
wenzelm@23164
   612
wenzelm@23164
   613
lemma q_pos_lemma:
wenzelm@23164
   614
     "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"
wenzelm@23164
   615
apply (subgoal_tac "0 < b'* (q' + 1) ")
wenzelm@23164
   616
 apply (simp add: zero_less_mult_iff)
wenzelm@23164
   617
apply (simp add: right_distrib)
wenzelm@23164
   618
done
wenzelm@23164
   619
wenzelm@23164
   620
lemma zdiv_mono2_lemma:
wenzelm@23164
   621
     "[| b*q + r = b'*q' + r';  0 \<le> b'*q' + r';   
wenzelm@23164
   622
         r' < b';  0 \<le> r;  0 < b';  b' \<le> b |]   
wenzelm@23164
   623
      ==> q \<le> (q'::int)"
wenzelm@23164
   624
apply (frule q_pos_lemma, assumption+) 
wenzelm@23164
   625
apply (subgoal_tac "b*q < b* (q' + 1) ")
wenzelm@23164
   626
 apply (simp add: mult_less_cancel_left)
wenzelm@23164
   627
apply (subgoal_tac "b*q = r' - r + b'*q'")
wenzelm@23164
   628
 prefer 2 apply simp
wenzelm@23164
   629
apply (simp (no_asm_simp) add: right_distrib)
wenzelm@23164
   630
apply (subst add_commute, rule zadd_zless_mono, arith)
wenzelm@23164
   631
apply (rule mult_right_mono, auto)
wenzelm@23164
   632
done
wenzelm@23164
   633
wenzelm@23164
   634
lemma zdiv_mono2:
wenzelm@23164
   635
     "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"
wenzelm@23164
   636
apply (subgoal_tac "b \<noteq> 0")
wenzelm@23164
   637
 prefer 2 apply arith
wenzelm@23164
   638
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
wenzelm@23164
   639
apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
wenzelm@23164
   640
apply (rule zdiv_mono2_lemma)
wenzelm@23164
   641
apply (erule subst)
wenzelm@23164
   642
apply (erule subst, simp_all)
wenzelm@23164
   643
done
wenzelm@23164
   644
wenzelm@23164
   645
lemma q_neg_lemma:
wenzelm@23164
   646
     "[| b'*q' + r' < 0;  0 \<le> r';  0 < b' |] ==> q' \<le> (0::int)"
wenzelm@23164
   647
apply (subgoal_tac "b'*q' < 0")
wenzelm@23164
   648
 apply (simp add: mult_less_0_iff, arith)
wenzelm@23164
   649
done
wenzelm@23164
   650
wenzelm@23164
   651
lemma zdiv_mono2_neg_lemma:
wenzelm@23164
   652
     "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;   
wenzelm@23164
   653
         r < b;  0 \<le> r';  0 < b';  b' \<le> b |]   
wenzelm@23164
   654
      ==> q' \<le> (q::int)"
wenzelm@23164
   655
apply (frule q_neg_lemma, assumption+) 
wenzelm@23164
   656
apply (subgoal_tac "b*q' < b* (q + 1) ")
wenzelm@23164
   657
 apply (simp add: mult_less_cancel_left)
wenzelm@23164
   658
apply (simp add: right_distrib)
wenzelm@23164
   659
apply (subgoal_tac "b*q' \<le> b'*q'")
wenzelm@23164
   660
 prefer 2 apply (simp add: mult_right_mono_neg, arith)
wenzelm@23164
   661
done
wenzelm@23164
   662
wenzelm@23164
   663
lemma zdiv_mono2_neg:
wenzelm@23164
   664
     "[| a < (0::int);  0 < b';  b' \<le> b |] ==> a div b' \<le> a div b"
wenzelm@23164
   665
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
wenzelm@23164
   666
apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
wenzelm@23164
   667
apply (rule zdiv_mono2_neg_lemma)
wenzelm@23164
   668
apply (erule subst)
wenzelm@23164
   669
apply (erule subst, simp_all)
wenzelm@23164
   670
done
wenzelm@23164
   671
haftmann@25942
   672
wenzelm@23164
   673
subsection{*More Algebraic Laws for div and mod*}
wenzelm@23164
   674
wenzelm@23164
   675
text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}
wenzelm@23164
   676
wenzelm@23164
   677
lemma zmult1_lemma:
haftmann@29651
   678
     "[| divmod_rel b c (q, r);  c \<noteq> 0 |]  
haftmann@29651
   679
      ==> divmod_rel (a * b) c (a*q + a*r div c, a*r mod c)"
haftmann@29651
   680
by (force simp add: split_ifs divmod_rel_def linorder_neq_iff right_distrib)
wenzelm@23164
   681
wenzelm@23164
   682
lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"
wenzelm@23164
   683
apply (case_tac "c = 0", simp)
haftmann@29651
   684
apply (blast intro: divmod_rel_div_mod [THEN zmult1_lemma, THEN divmod_rel_div])
wenzelm@23164
   685
done
wenzelm@23164
   686
wenzelm@23164
   687
lemma zmod_zmult1_eq: "(a*b) mod c = a*(b mod c) mod (c::int)"
wenzelm@23164
   688
apply (case_tac "c = 0", simp)
haftmann@29651
   689
apply (blast intro: divmod_rel_div_mod [THEN zmult1_lemma, THEN divmod_rel_mod])
wenzelm@23164
   690
done
wenzelm@23164
   691
huffman@29403
   692
lemma zmod_zdiv_trivial: "(a mod b) div b = (0::int)"
haftmann@27651
   693
apply (case_tac "b = 0", simp)
haftmann@27651
   694
apply (auto simp add: linorder_neq_iff div_pos_pos_trivial div_neg_neg_trivial)
haftmann@27651
   695
done
haftmann@27651
   696
haftmann@27651
   697
text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}
haftmann@27651
   698
haftmann@27651
   699
lemma zadd1_lemma:
haftmann@29651
   700
     "[| divmod_rel a c (aq, ar);  divmod_rel b c (bq, br);  c \<noteq> 0 |]  
haftmann@29651
   701
      ==> divmod_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"
haftmann@29651
   702
by (force simp add: split_ifs divmod_rel_def linorder_neq_iff right_distrib)
haftmann@27651
   703
haftmann@27651
   704
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
haftmann@27651
   705
lemma zdiv_zadd1_eq:
haftmann@27651
   706
     "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"
haftmann@27651
   707
apply (case_tac "c = 0", simp)
haftmann@29651
   708
apply (blast intro: zadd1_lemma [OF divmod_rel_div_mod divmod_rel_div_mod] divmod_rel_div)
haftmann@27651
   709
done
haftmann@27651
   710
huffman@29405
   711
instance int :: ring_div
haftmann@27651
   712
proof
haftmann@27651
   713
  fix a b c :: int
haftmann@27651
   714
  assume not0: "b \<noteq> 0"
haftmann@27651
   715
  show "(a + c * b) div b = c + a div b"
haftmann@27651
   716
    unfolding zdiv_zadd1_eq [of a "c * b"] using not0 
nipkow@30181
   717
      by (simp add: zmod_zmult1_eq zmod_zdiv_trivial zdiv_zmult1_eq)
haftmann@27651
   718
qed auto
haftmann@25942
   719
haftmann@29651
   720
lemma posDivAlg_div_mod:
haftmann@29651
   721
  assumes "k \<ge> 0"
haftmann@29651
   722
  and "l \<ge> 0"
haftmann@29651
   723
  shows "posDivAlg k l = (k div l, k mod l)"
haftmann@29651
   724
proof (cases "l = 0")
haftmann@29651
   725
  case True then show ?thesis by (simp add: posDivAlg.simps)
haftmann@29651
   726
next
haftmann@29651
   727
  case False with assms posDivAlg_correct
haftmann@29651
   728
    have "divmod_rel k l (fst (posDivAlg k l), snd (posDivAlg k l))"
haftmann@29651
   729
    by simp
haftmann@29651
   730
  from divmod_rel_div [OF this `l \<noteq> 0`] divmod_rel_mod [OF this `l \<noteq> 0`]
haftmann@29651
   731
  show ?thesis by simp
haftmann@29651
   732
qed
haftmann@29651
   733
haftmann@29651
   734
lemma negDivAlg_div_mod:
haftmann@29651
   735
  assumes "k < 0"
haftmann@29651
   736
  and "l > 0"
haftmann@29651
   737
  shows "negDivAlg k l = (k div l, k mod l)"
haftmann@29651
   738
proof -
haftmann@29651
   739
  from assms have "l \<noteq> 0" by simp
haftmann@29651
   740
  from assms negDivAlg_correct
haftmann@29651
   741
    have "divmod_rel k l (fst (negDivAlg k l), snd (negDivAlg k l))"
haftmann@29651
   742
    by simp
haftmann@29651
   743
  from divmod_rel_div [OF this `l \<noteq> 0`] divmod_rel_mod [OF this `l \<noteq> 0`]
haftmann@29651
   744
  show ?thesis by simp
haftmann@29651
   745
qed
haftmann@29651
   746
wenzelm@23164
   747
lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"
huffman@29403
   748
by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
wenzelm@23164
   749
huffman@29403
   750
(* REVISIT: should this be generalized to all semiring_div types? *)
wenzelm@23164
   751
lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]
wenzelm@23164
   752
nipkow@23983
   753
wenzelm@23164
   754
subsection{*Proving  @{term "a div (b*c) = (a div b) div c"} *}
wenzelm@23164
   755
wenzelm@23164
   756
(*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but
wenzelm@23164
   757
  7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems
wenzelm@23164
   758
  to cause particular problems.*)
wenzelm@23164
   759
wenzelm@23164
   760
text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}
wenzelm@23164
   761
wenzelm@23164
   762
lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b*c < b*(q mod c) + r"
wenzelm@23164
   763
apply (subgoal_tac "b * (c - q mod c) < r * 1")
nipkow@29667
   764
 apply (simp add: algebra_simps)
wenzelm@23164
   765
apply (rule order_le_less_trans)
nipkow@29667
   766
 apply (erule_tac [2] mult_strict_right_mono)
nipkow@29667
   767
 apply (rule mult_left_mono_neg)
nipkow@29667
   768
  using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps pos_mod_bound)
nipkow@29667
   769
 apply (simp)
nipkow@29667
   770
apply (simp)
wenzelm@23164
   771
done
wenzelm@23164
   772
wenzelm@23164
   773
lemma zmult2_lemma_aux2:
wenzelm@23164
   774
     "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"
wenzelm@23164
   775
apply (subgoal_tac "b * (q mod c) \<le> 0")
wenzelm@23164
   776
 apply arith
wenzelm@23164
   777
apply (simp add: mult_le_0_iff)
wenzelm@23164
   778
done
wenzelm@23164
   779
wenzelm@23164
   780
lemma zmult2_lemma_aux3: "[| (0::int) < c;  0 \<le> r;  r < b |] ==> 0 \<le> b * (q mod c) + r"
wenzelm@23164
   781
apply (subgoal_tac "0 \<le> b * (q mod c) ")
wenzelm@23164
   782
apply arith
wenzelm@23164
   783
apply (simp add: zero_le_mult_iff)
wenzelm@23164
   784
done
wenzelm@23164
   785
wenzelm@23164
   786
lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"
wenzelm@23164
   787
apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
nipkow@29667
   788
 apply (simp add: right_diff_distrib)
wenzelm@23164
   789
apply (rule order_less_le_trans)
nipkow@29667
   790
 apply (erule mult_strict_right_mono)
nipkow@29667
   791
 apply (rule_tac [2] mult_left_mono)
nipkow@29667
   792
  apply simp
nipkow@29667
   793
 using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps pos_mod_bound)
nipkow@29667
   794
apply simp
wenzelm@23164
   795
done
wenzelm@23164
   796
haftmann@29651
   797
lemma zmult2_lemma: "[| divmod_rel a b (q, r);  b \<noteq> 0;  0 < c |]  
haftmann@29651
   798
      ==> divmod_rel a (b * c) (q div c, b*(q mod c) + r)"
haftmann@29651
   799
by (auto simp add: mult_ac divmod_rel_def linorder_neq_iff
wenzelm@23164
   800
                   zero_less_mult_iff right_distrib [symmetric] 
wenzelm@23164
   801
                   zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4)
wenzelm@23164
   802
wenzelm@23164
   803
lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c"
wenzelm@23164
   804
apply (case_tac "b = 0", simp)
haftmann@29651
   805
apply (force simp add: divmod_rel_div_mod [THEN zmult2_lemma, THEN divmod_rel_div])
wenzelm@23164
   806
done
wenzelm@23164
   807
wenzelm@23164
   808
lemma zmod_zmult2_eq:
wenzelm@23164
   809
     "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"
wenzelm@23164
   810
apply (case_tac "b = 0", simp)
haftmann@29651
   811
apply (force simp add: divmod_rel_div_mod [THEN zmult2_lemma, THEN divmod_rel_mod])
wenzelm@23164
   812
done
wenzelm@23164
   813
wenzelm@23164
   814
wenzelm@23164
   815
subsection{*Cancellation of Common Factors in div*}
wenzelm@23164
   816
wenzelm@23164
   817
lemma zdiv_zmult_zmult1_aux1:
wenzelm@23164
   818
     "[| (0::int) < b;  c \<noteq> 0 |] ==> (c*a) div (c*b) = a div b"
wenzelm@23164
   819
by (subst zdiv_zmult2_eq, auto)
wenzelm@23164
   820
wenzelm@23164
   821
lemma zdiv_zmult_zmult1_aux2:
wenzelm@23164
   822
     "[| b < (0::int);  c \<noteq> 0 |] ==> (c*a) div (c*b) = a div b"
wenzelm@23164
   823
apply (subgoal_tac " (c * (-a)) div (c * (-b)) = (-a) div (-b) ")
wenzelm@23164
   824
apply (rule_tac [2] zdiv_zmult_zmult1_aux1, auto)
wenzelm@23164
   825
done
wenzelm@23164
   826
wenzelm@23164
   827
lemma zdiv_zmult_zmult1: "c \<noteq> (0::int) ==> (c*a) div (c*b) = a div b"
wenzelm@23164
   828
apply (case_tac "b = 0", simp)
wenzelm@23164
   829
apply (auto simp add: linorder_neq_iff zdiv_zmult_zmult1_aux1 zdiv_zmult_zmult1_aux2)
wenzelm@23164
   830
done
wenzelm@23164
   831
nipkow@23401
   832
lemma zdiv_zmult_zmult1_if[simp]:
nipkow@23401
   833
  "(k*m) div (k*n) = (if k = (0::int) then 0 else m div n)"
nipkow@23401
   834
by (simp add:zdiv_zmult_zmult1)
nipkow@23401
   835
wenzelm@23164
   836
wenzelm@23164
   837
subsection{*Distribution of Factors over mod*}
wenzelm@23164
   838
wenzelm@23164
   839
lemma zmod_zmult_zmult1_aux1:
wenzelm@23164
   840
     "[| (0::int) < b;  c \<noteq> 0 |] ==> (c*a) mod (c*b) = c * (a mod b)"
wenzelm@23164
   841
by (subst zmod_zmult2_eq, auto)
wenzelm@23164
   842
wenzelm@23164
   843
lemma zmod_zmult_zmult1_aux2:
wenzelm@23164
   844
     "[| b < (0::int);  c \<noteq> 0 |] ==> (c*a) mod (c*b) = c * (a mod b)"
wenzelm@23164
   845
apply (subgoal_tac " (c * (-a)) mod (c * (-b)) = c * ((-a) mod (-b))")
wenzelm@23164
   846
apply (rule_tac [2] zmod_zmult_zmult1_aux1, auto)
wenzelm@23164
   847
done
wenzelm@23164
   848
wenzelm@23164
   849
lemma zmod_zmult_zmult1: "(c*a) mod (c*b) = (c::int) * (a mod b)"
wenzelm@23164
   850
apply (case_tac "b = 0", simp)
wenzelm@23164
   851
apply (case_tac "c = 0", simp)
wenzelm@23164
   852
apply (auto simp add: linorder_neq_iff zmod_zmult_zmult1_aux1 zmod_zmult_zmult1_aux2)
wenzelm@23164
   853
done
wenzelm@23164
   854
wenzelm@23164
   855
lemma zmod_zmult_zmult2: "(a*c) mod (b*c) = (a mod b) * (c::int)"
wenzelm@23164
   856
apply (cut_tac c = c in zmod_zmult_zmult1)
wenzelm@23164
   857
apply (auto simp add: mult_commute)
wenzelm@23164
   858
done
wenzelm@23164
   859
wenzelm@23164
   860
wenzelm@23164
   861
subsection {*Splitting Rules for div and mod*}
wenzelm@23164
   862
wenzelm@23164
   863
text{*The proofs of the two lemmas below are essentially identical*}
wenzelm@23164
   864
wenzelm@23164
   865
lemma split_pos_lemma:
wenzelm@23164
   866
 "0<k ==> 
wenzelm@23164
   867
    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
   868
apply (rule iffI, clarify)
wenzelm@23164
   869
 apply (erule_tac P="P ?x ?y" in rev_mp)  
nipkow@29948
   870
 apply (subst mod_add_eq) 
wenzelm@23164
   871
 apply (subst zdiv_zadd1_eq) 
wenzelm@23164
   872
 apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)  
wenzelm@23164
   873
txt{*converse direction*}
wenzelm@23164
   874
apply (drule_tac x = "n div k" in spec) 
wenzelm@23164
   875
apply (drule_tac x = "n mod k" in spec, simp)
wenzelm@23164
   876
done
wenzelm@23164
   877
wenzelm@23164
   878
lemma split_neg_lemma:
wenzelm@23164
   879
 "k<0 ==>
wenzelm@23164
   880
    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
   881
apply (rule iffI, clarify)
wenzelm@23164
   882
 apply (erule_tac P="P ?x ?y" in rev_mp)  
nipkow@29948
   883
 apply (subst mod_add_eq) 
wenzelm@23164
   884
 apply (subst zdiv_zadd1_eq) 
wenzelm@23164
   885
 apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)  
wenzelm@23164
   886
txt{*converse direction*}
wenzelm@23164
   887
apply (drule_tac x = "n div k" in spec) 
wenzelm@23164
   888
apply (drule_tac x = "n mod k" in spec, simp)
wenzelm@23164
   889
done
wenzelm@23164
   890
wenzelm@23164
   891
lemma split_zdiv:
wenzelm@23164
   892
 "P(n div k :: int) =
wenzelm@23164
   893
  ((k = 0 --> P 0) & 
wenzelm@23164
   894
   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) & 
wenzelm@23164
   895
   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"
wenzelm@23164
   896
apply (case_tac "k=0", simp)
wenzelm@23164
   897
apply (simp only: linorder_neq_iff)
wenzelm@23164
   898
apply (erule disjE) 
wenzelm@23164
   899
 apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"] 
wenzelm@23164
   900
                      split_neg_lemma [of concl: "%x y. P x"])
wenzelm@23164
   901
done
wenzelm@23164
   902
wenzelm@23164
   903
lemma split_zmod:
wenzelm@23164
   904
 "P(n mod k :: int) =
wenzelm@23164
   905
  ((k = 0 --> P n) & 
wenzelm@23164
   906
   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) & 
wenzelm@23164
   907
   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"
wenzelm@23164
   908
apply (case_tac "k=0", simp)
wenzelm@23164
   909
apply (simp only: linorder_neq_iff)
wenzelm@23164
   910
apply (erule disjE) 
wenzelm@23164
   911
 apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"] 
wenzelm@23164
   912
                      split_neg_lemma [of concl: "%x y. P y"])
wenzelm@23164
   913
done
wenzelm@23164
   914
wenzelm@23164
   915
(* Enable arith to deal with div 2 and mod 2: *)
wenzelm@23164
   916
declare split_zdiv [of _ _ "number_of k", simplified, standard, arith_split]
wenzelm@23164
   917
declare split_zmod [of _ _ "number_of k", simplified, standard, arith_split]
wenzelm@23164
   918
wenzelm@23164
   919
wenzelm@23164
   920
subsection{*Speeding up the Division Algorithm with Shifting*}
wenzelm@23164
   921
wenzelm@23164
   922
text{*computing div by shifting *}
wenzelm@23164
   923
wenzelm@23164
   924
lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
wenzelm@23164
   925
proof cases
wenzelm@23164
   926
  assume "a=0"
wenzelm@23164
   927
    thus ?thesis by simp
wenzelm@23164
   928
next
wenzelm@23164
   929
  assume "a\<noteq>0" and le_a: "0\<le>a"   
wenzelm@23164
   930
  hence a_pos: "1 \<le> a" by arith
wenzelm@23164
   931
  hence one_less_a2: "1 < 2*a" by arith
wenzelm@23164
   932
  hence le_2a: "2 * (1 + b mod a) \<le> 2 * a"
wenzelm@23164
   933
    by (simp add: mult_le_cancel_left add_commute [of 1] add1_zle_eq)
wenzelm@23164
   934
  with a_pos have "0 \<le> b mod a" by simp
wenzelm@23164
   935
  hence le_addm: "0 \<le> 1 mod (2*a) + 2*(b mod a)"
wenzelm@23164
   936
    by (simp add: mod_pos_pos_trivial one_less_a2)
wenzelm@23164
   937
  with  le_2a
wenzelm@23164
   938
  have "(1 mod (2*a) + 2*(b mod a)) div (2*a) = 0"
wenzelm@23164
   939
    by (simp add: div_pos_pos_trivial le_addm mod_pos_pos_trivial one_less_a2
wenzelm@23164
   940
                  right_distrib) 
wenzelm@23164
   941
  thus ?thesis
wenzelm@23164
   942
    by (subst zdiv_zadd1_eq,
wenzelm@23164
   943
        simp add: zdiv_zmult_zmult1 zmod_zmult_zmult1 one_less_a2
wenzelm@23164
   944
                  div_pos_pos_trivial)
wenzelm@23164
   945
qed
wenzelm@23164
   946
wenzelm@23164
   947
lemma neg_zdiv_mult_2: "a \<le> (0::int) ==> (1 + 2*b) div (2*a) = (b+1) div a"
wenzelm@23164
   948
apply (subgoal_tac " (1 + 2* (-b - 1)) div (2 * (-a)) = (-b - 1) div (-a) ")
wenzelm@23164
   949
apply (rule_tac [2] pos_zdiv_mult_2)
wenzelm@23164
   950
apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib)
wenzelm@23164
   951
apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")
wenzelm@23164
   952
apply (simp only: zdiv_zminus_zminus diff_minus minus_add_distrib [symmetric],
wenzelm@23164
   953
       simp) 
wenzelm@23164
   954
done
wenzelm@23164
   955
huffman@26086
   956
lemma zdiv_number_of_Bit0 [simp]:
huffman@26086
   957
     "number_of (Int.Bit0 v) div number_of (Int.Bit0 w) =  
huffman@26086
   958
          number_of v div (number_of w :: int)"
huffman@26086
   959
by (simp only: number_of_eq numeral_simps) simp
huffman@26086
   960
huffman@26086
   961
lemma zdiv_number_of_Bit1 [simp]:
huffman@26086
   962
     "number_of (Int.Bit1 v) div number_of (Int.Bit0 w) =  
huffman@26086
   963
          (if (0::int) \<le> number_of w                    
wenzelm@23164
   964
           then number_of v div (number_of w)     
wenzelm@23164
   965
           else (number_of v + (1::int)) div (number_of w))"
wenzelm@23164
   966
apply (simp only: number_of_eq numeral_simps UNIV_I split: split_if) 
huffman@26086
   967
apply (simp add: zdiv_zmult_zmult1 pos_zdiv_mult_2 neg_zdiv_mult_2 add_ac)
wenzelm@23164
   968
done
wenzelm@23164
   969
wenzelm@23164
   970
wenzelm@23164
   971
subsection{*Computing mod by Shifting (proofs resemble those for div)*}
wenzelm@23164
   972
wenzelm@23164
   973
lemma pos_zmod_mult_2:
wenzelm@23164
   974
     "(0::int) \<le> a ==> (1 + 2*b) mod (2*a) = 1 + 2 * (b mod a)"
wenzelm@23164
   975
apply (case_tac "a = 0", simp)
wenzelm@23164
   976
apply (subgoal_tac "1 < a * 2")
wenzelm@23164
   977
 prefer 2 apply arith
wenzelm@23164
   978
apply (subgoal_tac "2* (1 + b mod a) \<le> 2*a")
wenzelm@23164
   979
 apply (rule_tac [2] mult_left_mono)
wenzelm@23164
   980
apply (auto simp add: add_commute [of 1] mult_commute add1_zle_eq 
wenzelm@23164
   981
                      pos_mod_bound)
nipkow@29948
   982
apply (subst mod_add_eq)
wenzelm@23164
   983
apply (simp add: zmod_zmult_zmult2 mod_pos_pos_trivial)
wenzelm@23164
   984
apply (rule mod_pos_pos_trivial)
huffman@26086
   985
apply (auto simp add: mod_pos_pos_trivial ring_distribs)
wenzelm@23164
   986
apply (subgoal_tac "0 \<le> b mod a", arith, simp)
wenzelm@23164
   987
done
wenzelm@23164
   988
wenzelm@23164
   989
lemma neg_zmod_mult_2:
wenzelm@23164
   990
     "a \<le> (0::int) ==> (1 + 2*b) mod (2*a) = 2 * ((b+1) mod a) - 1"
wenzelm@23164
   991
apply (subgoal_tac "(1 + 2* (-b - 1)) mod (2* (-a)) = 
wenzelm@23164
   992
                    1 + 2* ((-b - 1) mod (-a))")
wenzelm@23164
   993
apply (rule_tac [2] pos_zmod_mult_2)
nipkow@30042
   994
apply (auto simp add: right_diff_distrib)
wenzelm@23164
   995
apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")
wenzelm@23164
   996
 prefer 2 apply simp 
wenzelm@23164
   997
apply (simp only: zmod_zminus_zminus diff_minus minus_add_distrib [symmetric])
wenzelm@23164
   998
done
wenzelm@23164
   999
huffman@26086
  1000
lemma zmod_number_of_Bit0 [simp]:
huffman@26086
  1001
     "number_of (Int.Bit0 v) mod number_of (Int.Bit0 w) =  
huffman@26086
  1002
      (2::int) * (number_of v mod number_of w)"
huffman@26086
  1003
apply (simp only: number_of_eq numeral_simps) 
huffman@26086
  1004
apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2 
nipkow@29948
  1005
                 neg_zmod_mult_2 add_ac)
huffman@26086
  1006
done
huffman@26086
  1007
huffman@26086
  1008
lemma zmod_number_of_Bit1 [simp]:
huffman@26086
  1009
     "number_of (Int.Bit1 v) mod number_of (Int.Bit0 w) =  
huffman@26086
  1010
      (if (0::int) \<le> number_of w  
wenzelm@23164
  1011
                then 2 * (number_of v mod number_of w) + 1     
wenzelm@23164
  1012
                else 2 * ((number_of v + (1::int)) mod number_of w) - 1)"
huffman@26086
  1013
apply (simp only: number_of_eq numeral_simps) 
wenzelm@23164
  1014
apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2 
nipkow@29948
  1015
                 neg_zmod_mult_2 add_ac)
wenzelm@23164
  1016
done
wenzelm@23164
  1017
wenzelm@23164
  1018
wenzelm@23164
  1019
subsection{*Quotients of Signs*}
wenzelm@23164
  1020
wenzelm@23164
  1021
lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"
wenzelm@23164
  1022
apply (subgoal_tac "a div b \<le> -1", force)
wenzelm@23164
  1023
apply (rule order_trans)
wenzelm@23164
  1024
apply (rule_tac a' = "-1" in zdiv_mono1)
nipkow@29948
  1025
apply (auto simp add: div_eq_minus1)
wenzelm@23164
  1026
done
wenzelm@23164
  1027
wenzelm@23164
  1028
lemma div_nonneg_neg_le0: "[| (0::int) \<le> a;  b < 0 |] ==> a div b \<le> 0"
wenzelm@23164
  1029
by (drule zdiv_mono1_neg, auto)
wenzelm@23164
  1030
wenzelm@23164
  1031
lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"
wenzelm@23164
  1032
apply auto
wenzelm@23164
  1033
apply (drule_tac [2] zdiv_mono1)
wenzelm@23164
  1034
apply (auto simp add: linorder_neq_iff)
wenzelm@23164
  1035
apply (simp (no_asm_use) add: linorder_not_less [symmetric])
wenzelm@23164
  1036
apply (blast intro: div_neg_pos_less0)
wenzelm@23164
  1037
done
wenzelm@23164
  1038
wenzelm@23164
  1039
lemma neg_imp_zdiv_nonneg_iff:
wenzelm@23164
  1040
     "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"
wenzelm@23164
  1041
apply (subst zdiv_zminus_zminus [symmetric])
wenzelm@23164
  1042
apply (subst pos_imp_zdiv_nonneg_iff, auto)
wenzelm@23164
  1043
done
wenzelm@23164
  1044
wenzelm@23164
  1045
(*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
wenzelm@23164
  1046
lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
wenzelm@23164
  1047
by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
wenzelm@23164
  1048
wenzelm@23164
  1049
(*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
wenzelm@23164
  1050
lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
wenzelm@23164
  1051
by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
wenzelm@23164
  1052
wenzelm@23164
  1053
wenzelm@23164
  1054
subsection {* The Divides Relation *}
wenzelm@23164
  1055
wenzelm@23164
  1056
lemmas zdvd_iff_zmod_eq_0_number_of [simp] =
nipkow@30042
  1057
  dvd_eq_mod_eq_0 [of "number_of x::int" "number_of y::int", standard]
wenzelm@23164
  1058
wenzelm@23164
  1059
lemma zdvd_anti_sym:
wenzelm@23164
  1060
    "0 < m ==> 0 < n ==> m dvd n ==> n dvd m ==> m = (n::int)"
wenzelm@23164
  1061
  apply (simp add: dvd_def, auto)
wenzelm@23164
  1062
  apply (simp add: mult_assoc zero_less_mult_iff zmult_eq_1_iff)
wenzelm@23164
  1063
  done
wenzelm@23164
  1064
nipkow@30042
  1065
lemma zdvd_dvd_eq: assumes "a \<noteq> 0" and "(a::int) dvd b" and "b dvd a" 
wenzelm@23164
  1066
  shows "\<bar>a\<bar> = \<bar>b\<bar>"
wenzelm@23164
  1067
proof-
nipkow@30042
  1068
  from `a dvd b` obtain k where k:"b = a*k" unfolding dvd_def by blast 
nipkow@30042
  1069
  from `b dvd a` obtain k' where k':"a = b*k'" unfolding dvd_def by blast 
wenzelm@23164
  1070
  from k k' have "a = a*k*k'" by simp
wenzelm@23164
  1071
  with mult_cancel_left1[where c="a" and b="k*k'"]
nipkow@30042
  1072
  have kk':"k*k' = 1" using `a\<noteq>0` by (simp add: mult_assoc)
wenzelm@23164
  1073
  hence "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1" by (simp add: zmult_eq_1_iff)
wenzelm@23164
  1074
  thus ?thesis using k k' by auto
wenzelm@23164
  1075
qed
wenzelm@23164
  1076
wenzelm@23164
  1077
lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"
wenzelm@23164
  1078
  apply (subgoal_tac "m = n + (m - n)")
wenzelm@23164
  1079
   apply (erule ssubst)
nipkow@30042
  1080
   apply (blast intro: dvd_add, simp)
wenzelm@23164
  1081
  done
wenzelm@23164
  1082
wenzelm@23164
  1083
lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"
nipkow@30042
  1084
apply (rule iffI)
nipkow@30042
  1085
 apply (erule_tac [2] dvd_add)
nipkow@30042
  1086
 apply (subgoal_tac "n = (n + k * m) - k * m")
nipkow@30042
  1087
  apply (erule ssubst)
nipkow@30042
  1088
  apply (erule dvd_diff)
nipkow@30042
  1089
  apply(simp_all)
nipkow@30042
  1090
done
wenzelm@23164
  1091
wenzelm@23164
  1092
lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"
wenzelm@23164
  1093
  apply (simp add: dvd_def)
wenzelm@23164
  1094
  apply (auto simp add: zmod_zmult_zmult1)
wenzelm@23164
  1095
  done
wenzelm@23164
  1096
wenzelm@23164
  1097
lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)"
wenzelm@23164
  1098
  apply (subgoal_tac "k dvd n * (m div n) + m mod n")
wenzelm@23164
  1099
   apply (simp add: zmod_zdiv_equality [symmetric])
nipkow@30042
  1100
  apply (simp only: dvd_add dvd_mult2)
wenzelm@23164
  1101
  done
wenzelm@23164
  1102
wenzelm@23164
  1103
lemma zdvd_not_zless: "0 < m ==> m < n ==> \<not> n dvd (m::int)"
haftmann@27651
  1104
  apply (auto elim!: dvdE)
wenzelm@23164
  1105
  apply (subgoal_tac "0 < n")
wenzelm@23164
  1106
   prefer 2
wenzelm@23164
  1107
   apply (blast intro: order_less_trans)
wenzelm@23164
  1108
  apply (simp add: zero_less_mult_iff)
wenzelm@23164
  1109
  apply (subgoal_tac "n * k < n * 1")
wenzelm@23164
  1110
   apply (drule mult_less_cancel_left [THEN iffD1], auto)
wenzelm@23164
  1111
  done
haftmann@27651
  1112
wenzelm@23164
  1113
lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"
wenzelm@23164
  1114
  using zmod_zdiv_equality[where a="m" and b="n"]
nipkow@29667
  1115
  by (simp add: algebra_simps)
wenzelm@23164
  1116
wenzelm@23164
  1117
lemma zdvd_mult_div_cancel:"(n::int) dvd m \<Longrightarrow> n * (m div n) = m"
wenzelm@23164
  1118
apply (subgoal_tac "m mod n = 0")
wenzelm@23164
  1119
 apply (simp add: zmult_div_cancel)
nipkow@30042
  1120
apply (simp only: dvd_eq_mod_eq_0)
wenzelm@23164
  1121
done
wenzelm@23164
  1122
wenzelm@23164
  1123
lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k \<noteq> (0::int)"
wenzelm@23164
  1124
  shows "m dvd n"
wenzelm@23164
  1125
proof-
wenzelm@23164
  1126
  from d obtain h where h: "k*n = k*m * h" unfolding dvd_def by blast
wenzelm@23164
  1127
  {assume "n \<noteq> m*h" hence "k* n \<noteq> k* (m*h)" using kz by simp
wenzelm@23164
  1128
    with h have False by (simp add: mult_assoc)}
wenzelm@23164
  1129
  hence "n = m * h" by blast
huffman@29410
  1130
  thus ?thesis by simp
wenzelm@23164
  1131
qed
wenzelm@23164
  1132
nipkow@23969
  1133
wenzelm@23164
  1134
theorem ex_nat: "(\<exists>x::nat. P x) = (\<exists>x::int. 0 <= x \<and> P (nat x))"
nipkow@25134
  1135
apply (simp split add: split_nat)
nipkow@25134
  1136
apply (rule iffI)
nipkow@25134
  1137
apply (erule exE)
nipkow@25134
  1138
apply (rule_tac x = "int x" in exI)
nipkow@25134
  1139
apply simp
nipkow@25134
  1140
apply (erule exE)
nipkow@25134
  1141
apply (rule_tac x = "nat x" in exI)
nipkow@25134
  1142
apply (erule conjE)
nipkow@25134
  1143
apply (erule_tac x = "nat x" in allE)
nipkow@25134
  1144
apply simp
nipkow@25134
  1145
done
wenzelm@23164
  1146
huffman@23365
  1147
theorem zdvd_int: "(x dvd y) = (int x dvd int y)"
haftmann@27651
  1148
proof -
haftmann@27651
  1149
  have "\<And>k. int y = int x * k \<Longrightarrow> x dvd y"
haftmann@27651
  1150
  proof -
haftmann@27651
  1151
    fix k
haftmann@27651
  1152
    assume A: "int y = int x * k"
haftmann@27651
  1153
    then show "x dvd y" proof (cases k)
haftmann@27651
  1154
      case (1 n) with A have "y = x * n" by (simp add: zmult_int)
haftmann@27651
  1155
      then show ?thesis ..
haftmann@27651
  1156
    next
haftmann@27651
  1157
      case (2 n) with A have "int y = int x * (- int (Suc n))" by simp
haftmann@27651
  1158
      also have "\<dots> = - (int x * int (Suc n))" by (simp only: mult_minus_right)
haftmann@27651
  1159
      also have "\<dots> = - int (x * Suc n)" by (simp only: zmult_int)
haftmann@27651
  1160
      finally have "- int (x * Suc n) = int y" ..
haftmann@27651
  1161
      then show ?thesis by (simp only: negative_eq_positive) auto
haftmann@27651
  1162
    qed
haftmann@27651
  1163
  qed
nipkow@30042
  1164
  then show ?thesis by (auto elim!: dvdE simp only: dvd_triv_left int_mult)
huffman@29410
  1165
qed
wenzelm@23164
  1166
wenzelm@23164
  1167
lemma zdvd1_eq[simp]: "(x::int) dvd 1 = ( \<bar>x\<bar> = 1)"
wenzelm@23164
  1168
proof
nipkow@30042
  1169
  assume d: "x dvd 1" hence "int (nat \<bar>x\<bar>) dvd int (nat 1)" by simp
wenzelm@23164
  1170
  hence "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int)
wenzelm@23164
  1171
  hence "nat \<bar>x\<bar> = 1"  by simp
wenzelm@23164
  1172
  thus "\<bar>x\<bar> = 1" by (cases "x < 0", auto)
wenzelm@23164
  1173
next
wenzelm@23164
  1174
  assume "\<bar>x\<bar>=1" thus "x dvd 1" 
nipkow@30042
  1175
    by(cases "x < 0",simp_all add: minus_equation_iff dvd_eq_mod_eq_0)
wenzelm@23164
  1176
qed
wenzelm@23164
  1177
lemma zdvd_mult_cancel1: 
wenzelm@23164
  1178
  assumes mp:"m \<noteq>(0::int)" shows "(m * n dvd m) = (\<bar>n\<bar> = 1)"
wenzelm@23164
  1179
proof
wenzelm@23164
  1180
  assume n1: "\<bar>n\<bar> = 1" thus "m * n dvd m" 
nipkow@30042
  1181
    by (cases "n >0", auto simp add: minus_dvd_iff minus_equation_iff)
wenzelm@23164
  1182
next
wenzelm@23164
  1183
  assume H: "m * n dvd m" hence H2: "m * n dvd m * 1" by simp
wenzelm@23164
  1184
  from zdvd_mult_cancel[OF H2 mp] show "\<bar>n\<bar> = 1" by (simp only: zdvd1_eq)
wenzelm@23164
  1185
qed
wenzelm@23164
  1186
huffman@23365
  1187
lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))"
nipkow@30042
  1188
  unfolding zdvd_int by (cases "z \<ge> 0") simp_all
huffman@23306
  1189
huffman@23365
  1190
lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)"
nipkow@30042
  1191
  unfolding zdvd_int by (cases "z \<ge> 0") simp_all
wenzelm@23164
  1192
wenzelm@23164
  1193
lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \<le> z then (z dvd int m) else m = 0)"
haftmann@27651
  1194
  by (auto simp add: dvd_int_iff)
wenzelm@23164
  1195
wenzelm@23164
  1196
lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \<le> (n::int)"
huffman@23365
  1197
  apply (rule_tac z=n in int_cases)
huffman@23365
  1198
  apply (auto simp add: dvd_int_iff)
huffman@23365
  1199
  apply (rule_tac z=z in int_cases)
huffman@23307
  1200
  apply (auto simp add: dvd_imp_le)
wenzelm@23164
  1201
  done
wenzelm@23164
  1202
wenzelm@23164
  1203
lemma zpower_zmod: "((x::int) mod m)^y mod m = x^y mod m"
wenzelm@23164
  1204
apply (induct "y", auto)
wenzelm@23164
  1205
apply (rule zmod_zmult1_eq [THEN trans])
wenzelm@23164
  1206
apply (simp (no_asm_simp))
nipkow@29948
  1207
apply (rule mod_mult_eq [symmetric])
wenzelm@23164
  1208
done
wenzelm@23164
  1209
huffman@23365
  1210
lemma zdiv_int: "int (a div b) = (int a) div (int b)"
wenzelm@23164
  1211
apply (subst split_div, auto)
wenzelm@23164
  1212
apply (subst split_zdiv, auto)
huffman@23365
  1213
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
  1214
apply (auto simp add: IntDiv.divmod_rel_def of_nat_mult)
wenzelm@23164
  1215
done
wenzelm@23164
  1216
wenzelm@23164
  1217
lemma zmod_int: "int (a mod b) = (int a) mod (int b)"
huffman@23365
  1218
apply (subst split_mod, auto)
huffman@23365
  1219
apply (subst split_zmod, auto)
huffman@23365
  1220
apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia 
huffman@23365
  1221
       in unique_remainder)
haftmann@29651
  1222
apply (auto simp add: IntDiv.divmod_rel_def of_nat_mult)
huffman@23365
  1223
done
wenzelm@23164
  1224
nipkow@30180
  1225
lemma abs_div: "(y::int) dvd x \<Longrightarrow> abs (x div y) = abs x div abs y"
nipkow@30180
  1226
by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)
nipkow@30180
  1227
wenzelm@23164
  1228
text{*Suggested by Matthias Daum*}
wenzelm@23164
  1229
lemma int_power_div_base:
wenzelm@23164
  1230
     "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"
huffman@30079
  1231
apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")
wenzelm@23164
  1232
 apply (erule ssubst)
wenzelm@23164
  1233
 apply (simp only: power_add)
wenzelm@23164
  1234
 apply simp_all
wenzelm@23164
  1235
done
wenzelm@23164
  1236
haftmann@23853
  1237
text {* by Brian Huffman *}
haftmann@23853
  1238
lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"
huffman@29405
  1239
by (rule mod_minus_eq [symmetric])
haftmann@23853
  1240
haftmann@23853
  1241
lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"
huffman@29405
  1242
by (rule mod_diff_left_eq [symmetric])
haftmann@23853
  1243
haftmann@23853
  1244
lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"
huffman@29405
  1245
by (rule mod_diff_right_eq [symmetric])
haftmann@23853
  1246
haftmann@23853
  1247
lemmas zmod_simps =
nipkow@30034
  1248
  mod_add_left_eq  [symmetric]
nipkow@30034
  1249
  mod_add_right_eq [symmetric]
haftmann@23853
  1250
  IntDiv.zmod_zmult1_eq     [symmetric]
nipkow@29948
  1251
  mod_mult_left_eq          [symmetric]
haftmann@23853
  1252
  IntDiv.zpower_zmod
haftmann@23853
  1253
  zminus_zmod zdiff_zmod_left zdiff_zmod_right
haftmann@23853
  1254
huffman@29045
  1255
text {* Distributive laws for function @{text nat}. *}
huffman@29045
  1256
huffman@29045
  1257
lemma nat_div_distrib: "0 \<le> x \<Longrightarrow> nat (x div y) = nat x div nat y"
huffman@29045
  1258
apply (rule linorder_cases [of y 0])
huffman@29045
  1259
apply (simp add: div_nonneg_neg_le0)
huffman@29045
  1260
apply simp
huffman@29045
  1261
apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)
huffman@29045
  1262
done
huffman@29045
  1263
huffman@29045
  1264
(*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)
huffman@29045
  1265
lemma nat_mod_distrib:
huffman@29045
  1266
  "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x mod y) = nat x mod nat y"
huffman@29045
  1267
apply (case_tac "y = 0", simp add: DIVISION_BY_ZERO)
huffman@29045
  1268
apply (simp add: nat_eq_iff zmod_int)
huffman@29045
  1269
done
huffman@29045
  1270
huffman@29045
  1271
text{*Suggested by Matthias Daum*}
huffman@29045
  1272
lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)"
huffman@29045
  1273
apply (subgoal_tac "nat x div nat k < nat x")
huffman@29045
  1274
 apply (simp (asm_lr) add: nat_div_distrib [symmetric])
huffman@29045
  1275
apply (rule Divides.div_less_dividend, simp_all)
huffman@29045
  1276
done
huffman@29045
  1277
haftmann@23853
  1278
text {* code generator setup *}
wenzelm@23164
  1279
haftmann@26507
  1280
context ring_1
haftmann@26507
  1281
begin
haftmann@26507
  1282
haftmann@28562
  1283
lemma of_int_num [code]:
haftmann@26507
  1284
  "of_int k = (if k = 0 then 0 else if k < 0 then
haftmann@26507
  1285
     - of_int (- k) else let
haftmann@29651
  1286
       (l, m) = divmod k 2;
haftmann@26507
  1287
       l' = of_int l
haftmann@26507
  1288
     in if m = 0 then l' + l' else l' + l' + 1)"
haftmann@26507
  1289
proof -
haftmann@26507
  1290
  have aux1: "k mod (2\<Colon>int) \<noteq> (0\<Colon>int) \<Longrightarrow> 
haftmann@26507
  1291
    of_int k = of_int (k div 2 * 2 + 1)"
haftmann@26507
  1292
  proof -
haftmann@26507
  1293
    have "k mod 2 < 2" by (auto intro: pos_mod_bound)
haftmann@26507
  1294
    moreover have "0 \<le> k mod 2" by (auto intro: pos_mod_sign)
haftmann@26507
  1295
    moreover assume "k mod 2 \<noteq> 0"
haftmann@26507
  1296
    ultimately have "k mod 2 = 1" by arith
haftmann@26507
  1297
    moreover have "of_int k = of_int (k div 2 * 2 + k mod 2)" by simp
haftmann@26507
  1298
    ultimately show ?thesis by auto
haftmann@26507
  1299
  qed
haftmann@26507
  1300
  have aux2: "\<And>x. of_int 2 * x = x + x"
haftmann@26507
  1301
  proof -
haftmann@26507
  1302
    fix x
haftmann@26507
  1303
    have int2: "(2::int) = 1 + 1" by arith
haftmann@26507
  1304
    show "of_int 2 * x = x + x"
haftmann@26507
  1305
    unfolding int2 of_int_add left_distrib by simp
haftmann@26507
  1306
  qed
haftmann@26507
  1307
  have aux3: "\<And>x. x * of_int 2 = x + x"
haftmann@26507
  1308
  proof -
haftmann@26507
  1309
    fix x
haftmann@26507
  1310
    have int2: "(2::int) = 1 + 1" by arith
haftmann@26507
  1311
    show "x * of_int 2 = x + x" 
haftmann@26507
  1312
    unfolding int2 of_int_add right_distrib by simp
haftmann@26507
  1313
  qed
haftmann@29651
  1314
  from aux1 show ?thesis by (auto simp add: divmod_mod_div Let_def aux2 aux3)
haftmann@26507
  1315
qed
haftmann@26507
  1316
haftmann@26507
  1317
end
haftmann@26507
  1318
chaieb@27667
  1319
lemma zmod_eq_dvd_iff: "(x::int) mod n = y mod n \<longleftrightarrow> n dvd x - y"
chaieb@27667
  1320
proof
chaieb@27667
  1321
  assume H: "x mod n = y mod n"
chaieb@27667
  1322
  hence "x mod n - y mod n = 0" by simp
chaieb@27667
  1323
  hence "(x mod n - y mod n) mod n = 0" by simp 
nipkow@30034
  1324
  hence "(x - y) mod n = 0" by (simp add: mod_diff_eq[symmetric])
nipkow@30042
  1325
  thus "n dvd x - y" by (simp add: dvd_eq_mod_eq_0)
chaieb@27667
  1326
next
chaieb@27667
  1327
  assume H: "n dvd x - y"
chaieb@27667
  1328
  then obtain k where k: "x-y = n*k" unfolding dvd_def by blast
chaieb@27667
  1329
  hence "x = n*k + y" by simp
chaieb@27667
  1330
  hence "x mod n = (n*k + y) mod n" by simp
nipkow@30034
  1331
  thus "x mod n = y mod n" by (simp add: mod_add_left_eq)
chaieb@27667
  1332
qed
chaieb@27667
  1333
chaieb@27667
  1334
lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y  mod n" and xy:"y \<le> x"
chaieb@27667
  1335
  shows "\<exists>q. x = y + n * q"
chaieb@27667
  1336
proof-
chaieb@27667
  1337
  from xy have th: "int x - int y = int (x - y)" by simp 
chaieb@27667
  1338
  from xyn have "int x mod int n = int y mod int n" 
chaieb@27667
  1339
    by (simp add: zmod_int[symmetric])
chaieb@27667
  1340
  hence "int n dvd int x - int y" by (simp only: zmod_eq_dvd_iff[symmetric]) 
chaieb@27667
  1341
  hence "n dvd x - y" by (simp add: th zdvd_int)
chaieb@27667
  1342
  then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith
chaieb@27667
  1343
qed
chaieb@27667
  1344
chaieb@27667
  1345
lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)" 
chaieb@27667
  1346
  (is "?lhs = ?rhs")
chaieb@27667
  1347
proof
chaieb@27667
  1348
  assume H: "x mod n = y mod n"
chaieb@27667
  1349
  {assume xy: "x \<le> y"
chaieb@27667
  1350
    from H have th: "y mod n = x mod n" by simp
chaieb@27667
  1351
    from nat_mod_eq_lemma[OF th xy] have ?rhs 
chaieb@27667
  1352
      apply clarify  apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}
chaieb@27667
  1353
  moreover
chaieb@27667
  1354
  {assume xy: "y \<le> x"
chaieb@27667
  1355
    from nat_mod_eq_lemma[OF H xy] have ?rhs 
chaieb@27667
  1356
      apply clarify  apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}
chaieb@27667
  1357
  ultimately  show ?rhs using linear[of x y] by blast  
chaieb@27667
  1358
next
chaieb@27667
  1359
  assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast
chaieb@27667
  1360
  hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp
chaieb@27667
  1361
  thus  ?lhs by simp
chaieb@27667
  1362
qed
chaieb@27667
  1363
haftmann@29936
  1364
haftmann@29936
  1365
subsection {* Code generation *}
haftmann@29936
  1366
haftmann@29936
  1367
definition pdivmod :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
haftmann@29936
  1368
  "pdivmod k l = (\<bar>k\<bar> div \<bar>l\<bar>, \<bar>k\<bar> mod \<bar>l\<bar>)"
haftmann@29936
  1369
haftmann@29936
  1370
lemma pdivmod_posDivAlg [code]:
haftmann@29936
  1371
  "pdivmod k l = (if l = 0 then (0, \<bar>k\<bar>) else posDivAlg \<bar>k\<bar> \<bar>l\<bar>)"
haftmann@29936
  1372
by (subst posDivAlg_div_mod) (simp_all add: pdivmod_def)
haftmann@29936
  1373
haftmann@29936
  1374
lemma divmod_pdivmod: "divmod k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else
haftmann@29936
  1375
  apsnd ((op *) (sgn l)) (if 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0
haftmann@29936
  1376
    then pdivmod k l
haftmann@29936
  1377
    else (let (r, s) = pdivmod k l in
haftmann@29936
  1378
      if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
haftmann@29936
  1379
proof -
haftmann@29936
  1380
  have aux: "\<And>q::int. - k = l * q \<longleftrightarrow> k = l * - q" by auto
haftmann@29936
  1381
  show ?thesis
haftmann@29936
  1382
    by (simp add: divmod_mod_div pdivmod_def)
haftmann@29936
  1383
      (auto simp add: aux not_less not_le zdiv_zminus1_eq_if
haftmann@29936
  1384
      zmod_zminus1_eq_if zdiv_zminus2_eq_if zmod_zminus2_eq_if)
haftmann@29936
  1385
qed
haftmann@29936
  1386
haftmann@29936
  1387
lemma divmod_code [code]: "divmod k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else
haftmann@29936
  1388
  apsnd ((op *) (sgn l)) (if sgn k = sgn l
haftmann@29936
  1389
    then pdivmod k l
haftmann@29936
  1390
    else (let (r, s) = pdivmod k l in
haftmann@29936
  1391
      if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
haftmann@29936
  1392
proof -
haftmann@29936
  1393
  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
  1394
    by (auto simp add: not_less sgn_if)
haftmann@29936
  1395
  then show ?thesis by (simp add: divmod_pdivmod)
haftmann@29936
  1396
qed
haftmann@29936
  1397
wenzelm@23164
  1398
code_modulename SML
wenzelm@23164
  1399
  IntDiv Integer
wenzelm@23164
  1400
wenzelm@23164
  1401
code_modulename OCaml
wenzelm@23164
  1402
  IntDiv Integer
wenzelm@23164
  1403
wenzelm@23164
  1404
code_modulename Haskell
haftmann@24195
  1405
  IntDiv Integer
wenzelm@23164
  1406
wenzelm@23164
  1407
end