src/HOL/IntDiv.thy
author huffman
Wed Jun 13 03:31:11 2007 +0200 (2007-06-13)
changeset 23365 f31794033ae1
parent 23307 2fe3345035c7
child 23401 8c5532263ba9
permissions -rw-r--r--
removed constant int :: nat => int;
int is now an abbreviation for of_nat :: nat => int
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(*  Title:      HOL/IntDiv.thy
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    ID:         $Id$
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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; the Divides Relation dvd*}
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theory IntDiv
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imports IntArith Divides FunDef
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begin
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constdefs
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  quorem :: "(int*int) * (int*int) => bool"
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    --{*definition of quotient and remainder*}
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    [code func]: "quorem == %((a,b), (q,r)).
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                      a = b*q + r &
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                      (if 0 < b then 0\<le>r & r<b else b<r & r \<le> 0)"
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  adjust :: "[int, int*int] => int*int"
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    --{*for the division algorithm*}
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    [code func]: "adjust b == %(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
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  posDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int"
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where
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  "posDivAlg a b =
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     (if (a<b | 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 (%(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
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  negDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int"
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where
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  "negDivAlg a b  =
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     (if (0\<le>a+b | 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 (%(a,b). nat(- a - b))") auto
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text{*algorithm for the general case @{term "b\<noteq>0"}*}
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constdefs
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  negateSnd :: "int*int => int*int"
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    [code func]: "negateSnd == %(q,r). (q,-r)"
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definition
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  divAlg :: "int \<times> int \<Rightarrow> int \<times> int"
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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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where
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  "divAlg = (\<lambda>(a, b). (if 0\<le>a then
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                  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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instance int :: Divides.div
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  div_def: "a div b == fst (divAlg (a, b))"
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  mod_def: "a mod b == snd (divAlg (a, b))" ..
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lemma divAlg_mod_div:
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  "divAlg (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 divAlg (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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     "[| quorem ((a,b), (q,r));  quorem ((a,b), (q',r'));  b \<noteq> 0 |]  
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      ==> q = q'"
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apply (simp add: quorem_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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     "[| quorem ((a,b), (q,r));  quorem ((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: quorem_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 "quorem ((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: quorem_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 "quorem ((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: quorem_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 quorem_0: "b \<noteq> 0 ==> quorem ((0,b), (0,0))"
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by (auto simp add: quorem_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 quorem_neg: "quorem ((-a,-b), qr) ==> quorem ((a,b), negateSnd qr)"
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by (auto simp add: split_ifs quorem_def)
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lemma divAlg_correct: "b \<noteq> 0 ==> quorem ((a,b), divAlg (a, b))"
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by (force simp add: linorder_neq_iff quorem_0 divAlg_def quorem_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 divAlg_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 divAlg_correct)
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apply (auto simp add: quorem_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_setup {*
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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 NatArithUtils.prove_conv all_tac (NatArithUtils.simp_all_tac simps) end;
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end)
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in
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val cancel_zdiv_zmod_proc = NatArithUtils.prep_simproc
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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 divAlg_correct)
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apply (auto simp add: quorem_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 divAlg_correct)
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apply (auto simp add: quorem_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 quorem_div_mod: "b \<noteq> 0 ==> quorem ((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: quorem_def linorder_neq_iff)
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done
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lemma quorem_div: "[| quorem((a,b),(q,r));  b \<noteq> 0 |] ==> a div b = q"
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by (simp add: quorem_div_mod [THEN unique_quotient])
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lemma quorem_mod: "[| quorem((a,b),(q,r));  b \<noteq> 0 |] ==> a mod b = r"
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by (simp add: quorem_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 quorem_div)
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apply (auto simp add: quorem_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 quorem_div)
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apply (auto simp add: quorem_def)
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   320
done
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   321
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   322
lemma div_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a div b = -1"
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   323
apply (rule quorem_div)
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   324
apply (auto simp add: quorem_def)
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   325
done
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   326
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   327
(*There is no div_neg_pos_trivial because  0 div b = 0 would supersede it*)
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   328
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   329
lemma mod_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a mod b = a"
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   330
apply (rule_tac q = 0 in quorem_mod)
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   331
apply (auto simp add: quorem_def)
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   332
done
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   333
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   334
lemma mod_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a mod b = a"
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   335
apply (rule_tac q = 0 in quorem_mod)
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   336
apply (auto simp add: quorem_def)
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   337
done
wenzelm@23164
   338
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   339
lemma mod_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a mod b = a+b"
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   340
apply (rule_tac q = "-1" in quorem_mod)
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   341
apply (auto simp add: quorem_def)
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   342
done
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   343
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   344
text{*There is no @{text mod_neg_pos_trivial}.*}
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   345
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   346
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   347
(*Simpler laws such as -a div b = -(a div b) FAIL, but see just below*)
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   348
lemma zdiv_zminus_zminus [simp]: "(-a) div (-b) = a div (b::int)"
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   349
apply (case_tac "b = 0", simp)
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   350
apply (simp add: quorem_div_mod [THEN quorem_neg, simplified, 
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   351
                                 THEN quorem_div, THEN sym])
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   352
wenzelm@23164
   353
done
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   354
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   355
(*Simpler laws such as -a mod b = -(a mod b) FAIL, but see just below*)
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   356
lemma zmod_zminus_zminus [simp]: "(-a) mod (-b) = - (a mod (b::int))"
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   357
apply (case_tac "b = 0", simp)
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   358
apply (subst quorem_div_mod [THEN quorem_neg, simplified, THEN quorem_mod],
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   359
       auto)
wenzelm@23164
   360
done
wenzelm@23164
   361
wenzelm@23164
   362
wenzelm@23164
   363
subsection{*Laws for div and mod with Unary Minus*}
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   364
wenzelm@23164
   365
lemma zminus1_lemma:
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   366
     "quorem((a,b),(q,r))  
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   367
      ==> quorem ((-a,b), (if r=0 then -q else -q - 1),  
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   368
                          (if r=0 then 0 else b-r))"
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   369
by (force simp add: split_ifs quorem_def linorder_neq_iff right_diff_distrib)
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   370
wenzelm@23164
   371
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   372
lemma zdiv_zminus1_eq_if:
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   373
     "b \<noteq> (0::int)  
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   374
      ==> (-a) div b =  
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   375
          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
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   376
by (blast intro: quorem_div_mod [THEN zminus1_lemma, THEN quorem_div])
wenzelm@23164
   377
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   378
lemma zmod_zminus1_eq_if:
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   379
     "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))"
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   380
apply (case_tac "b = 0", simp)
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   381
apply (blast intro: quorem_div_mod [THEN zminus1_lemma, THEN quorem_mod])
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   382
done
wenzelm@23164
   383
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   384
lemma zdiv_zminus2: "a div (-b) = (-a::int) div b"
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   385
by (cut_tac a = "-a" in zdiv_zminus_zminus, auto)
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   386
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   387
lemma zmod_zminus2: "a mod (-b) = - ((-a::int) mod b)"
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   388
by (cut_tac a = "-a" and b = b in zmod_zminus_zminus, auto)
wenzelm@23164
   389
wenzelm@23164
   390
lemma zdiv_zminus2_eq_if:
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   391
     "b \<noteq> (0::int)  
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   392
      ==> a div (-b) =  
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   393
          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
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   394
by (simp add: zdiv_zminus1_eq_if zdiv_zminus2)
wenzelm@23164
   395
wenzelm@23164
   396
lemma zmod_zminus2_eq_if:
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   397
     "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"
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   398
by (simp add: zmod_zminus1_eq_if zmod_zminus2)
wenzelm@23164
   399
wenzelm@23164
   400
wenzelm@23164
   401
subsection{*Division of a Number by Itself*}
wenzelm@23164
   402
wenzelm@23164
   403
lemma self_quotient_aux1: "[| (0::int) < a; a = r + a*q; r < a |] ==> 1 \<le> q"
wenzelm@23164
   404
apply (subgoal_tac "0 < a*q")
wenzelm@23164
   405
 apply (simp add: zero_less_mult_iff, arith)
wenzelm@23164
   406
done
wenzelm@23164
   407
wenzelm@23164
   408
lemma self_quotient_aux2: "[| (0::int) < a; a = r + a*q; 0 \<le> r |] ==> q \<le> 1"
wenzelm@23164
   409
apply (subgoal_tac "0 \<le> a* (1-q) ")
wenzelm@23164
   410
 apply (simp add: zero_le_mult_iff)
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   411
apply (simp add: right_diff_distrib)
wenzelm@23164
   412
done
wenzelm@23164
   413
wenzelm@23164
   414
lemma self_quotient: "[| quorem((a,a),(q,r));  a \<noteq> (0::int) |] ==> q = 1"
wenzelm@23164
   415
apply (simp add: split_ifs quorem_def linorder_neq_iff)
wenzelm@23164
   416
apply (rule order_antisym, safe, simp_all)
wenzelm@23164
   417
apply (rule_tac [3] a = "-a" and r = "-r" in self_quotient_aux1)
wenzelm@23164
   418
apply (rule_tac a = "-a" and r = "-r" in self_quotient_aux2)
wenzelm@23164
   419
apply (force intro: self_quotient_aux1 self_quotient_aux2 simp add: add_commute)+
wenzelm@23164
   420
done
wenzelm@23164
   421
wenzelm@23164
   422
lemma self_remainder: "[| quorem((a,a),(q,r));  a \<noteq> (0::int) |] ==> r = 0"
wenzelm@23164
   423
apply (frule self_quotient, assumption)
wenzelm@23164
   424
apply (simp add: quorem_def)
wenzelm@23164
   425
done
wenzelm@23164
   426
wenzelm@23164
   427
lemma zdiv_self [simp]: "a \<noteq> 0 ==> a div a = (1::int)"
wenzelm@23164
   428
by (simp add: quorem_div_mod [THEN self_quotient])
wenzelm@23164
   429
wenzelm@23164
   430
(*Here we have 0 mod 0 = 0, also assumed by Knuth (who puts m mod 0 = 0) *)
wenzelm@23164
   431
lemma zmod_self [simp]: "a mod a = (0::int)"
wenzelm@23164
   432
apply (case_tac "a = 0", simp)
wenzelm@23164
   433
apply (simp add: quorem_div_mod [THEN self_remainder])
wenzelm@23164
   434
done
wenzelm@23164
   435
wenzelm@23164
   436
wenzelm@23164
   437
subsection{*Computation of Division and Remainder*}
wenzelm@23164
   438
wenzelm@23164
   439
lemma zdiv_zero [simp]: "(0::int) div b = 0"
wenzelm@23164
   440
by (simp add: div_def divAlg_def)
wenzelm@23164
   441
wenzelm@23164
   442
lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"
wenzelm@23164
   443
by (simp add: div_def divAlg_def)
wenzelm@23164
   444
wenzelm@23164
   445
lemma zmod_zero [simp]: "(0::int) mod b = 0"
wenzelm@23164
   446
by (simp add: mod_def divAlg_def)
wenzelm@23164
   447
wenzelm@23164
   448
lemma zdiv_minus1: "(0::int) < b ==> -1 div b = -1"
wenzelm@23164
   449
by (simp add: div_def divAlg_def)
wenzelm@23164
   450
wenzelm@23164
   451
lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"
wenzelm@23164
   452
by (simp add: mod_def divAlg_def)
wenzelm@23164
   453
wenzelm@23164
   454
text{*a positive, b positive *}
wenzelm@23164
   455
wenzelm@23164
   456
lemma div_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a div b = fst (posDivAlg a b)"
wenzelm@23164
   457
by (simp add: div_def divAlg_def)
wenzelm@23164
   458
wenzelm@23164
   459
lemma mod_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a mod b = snd (posDivAlg a b)"
wenzelm@23164
   460
by (simp add: mod_def divAlg_def)
wenzelm@23164
   461
wenzelm@23164
   462
text{*a negative, b positive *}
wenzelm@23164
   463
wenzelm@23164
   464
lemma div_neg_pos: "[| a < 0;  0 < b |] ==> a div b = fst (negDivAlg a b)"
wenzelm@23164
   465
by (simp add: div_def divAlg_def)
wenzelm@23164
   466
wenzelm@23164
   467
lemma mod_neg_pos: "[| a < 0;  0 < b |] ==> a mod b = snd (negDivAlg a b)"
wenzelm@23164
   468
by (simp add: mod_def divAlg_def)
wenzelm@23164
   469
wenzelm@23164
   470
text{*a positive, b negative *}
wenzelm@23164
   471
wenzelm@23164
   472
lemma div_pos_neg:
wenzelm@23164
   473
     "[| 0 < a;  b < 0 |] ==> a div b = fst (negateSnd (negDivAlg (-a) (-b)))"
wenzelm@23164
   474
by (simp add: div_def divAlg_def)
wenzelm@23164
   475
wenzelm@23164
   476
lemma mod_pos_neg:
wenzelm@23164
   477
     "[| 0 < a;  b < 0 |] ==> a mod b = snd (negateSnd (negDivAlg (-a) (-b)))"
wenzelm@23164
   478
by (simp add: mod_def divAlg_def)
wenzelm@23164
   479
wenzelm@23164
   480
text{*a negative, b negative *}
wenzelm@23164
   481
wenzelm@23164
   482
lemma div_neg_neg:
wenzelm@23164
   483
     "[| a < 0;  b \<le> 0 |] ==> a div b = fst (negateSnd (posDivAlg (-a) (-b)))"
wenzelm@23164
   484
by (simp add: div_def divAlg_def)
wenzelm@23164
   485
wenzelm@23164
   486
lemma mod_neg_neg:
wenzelm@23164
   487
     "[| a < 0;  b \<le> 0 |] ==> a mod b = snd (negateSnd (posDivAlg (-a) (-b)))"
wenzelm@23164
   488
by (simp add: mod_def divAlg_def)
wenzelm@23164
   489
wenzelm@23164
   490
text {*Simplify expresions in which div and mod combine numerical constants*}
wenzelm@23164
   491
wenzelm@23164
   492
lemmas div_pos_pos_number_of [simp] =
wenzelm@23164
   493
    div_pos_pos [of "number_of v" "number_of w", standard]
wenzelm@23164
   494
wenzelm@23164
   495
lemmas div_neg_pos_number_of [simp] =
wenzelm@23164
   496
    div_neg_pos [of "number_of v" "number_of w", standard]
wenzelm@23164
   497
wenzelm@23164
   498
lemmas div_pos_neg_number_of [simp] =
wenzelm@23164
   499
    div_pos_neg [of "number_of v" "number_of w", standard]
wenzelm@23164
   500
wenzelm@23164
   501
lemmas div_neg_neg_number_of [simp] =
wenzelm@23164
   502
    div_neg_neg [of "number_of v" "number_of w", standard]
wenzelm@23164
   503
wenzelm@23164
   504
wenzelm@23164
   505
lemmas mod_pos_pos_number_of [simp] =
wenzelm@23164
   506
    mod_pos_pos [of "number_of v" "number_of w", standard]
wenzelm@23164
   507
wenzelm@23164
   508
lemmas mod_neg_pos_number_of [simp] =
wenzelm@23164
   509
    mod_neg_pos [of "number_of v" "number_of w", standard]
wenzelm@23164
   510
wenzelm@23164
   511
lemmas mod_pos_neg_number_of [simp] =
wenzelm@23164
   512
    mod_pos_neg [of "number_of v" "number_of w", standard]
wenzelm@23164
   513
wenzelm@23164
   514
lemmas mod_neg_neg_number_of [simp] =
wenzelm@23164
   515
    mod_neg_neg [of "number_of v" "number_of w", standard]
wenzelm@23164
   516
wenzelm@23164
   517
wenzelm@23164
   518
lemmas posDivAlg_eqn_number_of [simp] =
wenzelm@23164
   519
    posDivAlg_eqn [of "number_of v" "number_of w", standard]
wenzelm@23164
   520
wenzelm@23164
   521
lemmas negDivAlg_eqn_number_of [simp] =
wenzelm@23164
   522
    negDivAlg_eqn [of "number_of v" "number_of w", standard]
wenzelm@23164
   523
wenzelm@23164
   524
wenzelm@23164
   525
text{*Special-case simplification *}
wenzelm@23164
   526
wenzelm@23164
   527
lemma zmod_1 [simp]: "a mod (1::int) = 0"
wenzelm@23164
   528
apply (cut_tac a = a and b = 1 in pos_mod_sign)
wenzelm@23164
   529
apply (cut_tac [2] a = a and b = 1 in pos_mod_bound)
wenzelm@23164
   530
apply (auto simp del:pos_mod_bound pos_mod_sign)
wenzelm@23164
   531
done
wenzelm@23164
   532
wenzelm@23164
   533
lemma zdiv_1 [simp]: "a div (1::int) = a"
wenzelm@23164
   534
by (cut_tac a = a and b = 1 in zmod_zdiv_equality, auto)
wenzelm@23164
   535
wenzelm@23164
   536
lemma zmod_minus1_right [simp]: "a mod (-1::int) = 0"
wenzelm@23164
   537
apply (cut_tac a = a and b = "-1" in neg_mod_sign)
wenzelm@23164
   538
apply (cut_tac [2] a = a and b = "-1" in neg_mod_bound)
wenzelm@23164
   539
apply (auto simp del: neg_mod_sign neg_mod_bound)
wenzelm@23164
   540
done
wenzelm@23164
   541
wenzelm@23164
   542
lemma zdiv_minus1_right [simp]: "a div (-1::int) = -a"
wenzelm@23164
   543
by (cut_tac a = a and b = "-1" in zmod_zdiv_equality, auto)
wenzelm@23164
   544
wenzelm@23164
   545
(** The last remaining special cases for constant arithmetic:
wenzelm@23164
   546
    1 div z and 1 mod z **)
wenzelm@23164
   547
wenzelm@23164
   548
lemmas div_pos_pos_1_number_of [simp] =
wenzelm@23164
   549
    div_pos_pos [OF int_0_less_1, of "number_of w", standard]
wenzelm@23164
   550
wenzelm@23164
   551
lemmas div_pos_neg_1_number_of [simp] =
wenzelm@23164
   552
    div_pos_neg [OF int_0_less_1, of "number_of w", standard]
wenzelm@23164
   553
wenzelm@23164
   554
lemmas mod_pos_pos_1_number_of [simp] =
wenzelm@23164
   555
    mod_pos_pos [OF int_0_less_1, of "number_of w", standard]
wenzelm@23164
   556
wenzelm@23164
   557
lemmas mod_pos_neg_1_number_of [simp] =
wenzelm@23164
   558
    mod_pos_neg [OF int_0_less_1, of "number_of w", standard]
wenzelm@23164
   559
wenzelm@23164
   560
wenzelm@23164
   561
lemmas posDivAlg_eqn_1_number_of [simp] =
wenzelm@23164
   562
    posDivAlg_eqn [of concl: 1 "number_of w", standard]
wenzelm@23164
   563
wenzelm@23164
   564
lemmas negDivAlg_eqn_1_number_of [simp] =
wenzelm@23164
   565
    negDivAlg_eqn [of concl: 1 "number_of w", standard]
wenzelm@23164
   566
wenzelm@23164
   567
wenzelm@23164
   568
wenzelm@23164
   569
subsection{*Monotonicity in the First Argument (Dividend)*}
wenzelm@23164
   570
wenzelm@23164
   571
lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div b"
wenzelm@23164
   572
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
wenzelm@23164
   573
apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
wenzelm@23164
   574
apply (rule unique_quotient_lemma)
wenzelm@23164
   575
apply (erule subst)
wenzelm@23164
   576
apply (erule subst, simp_all)
wenzelm@23164
   577
done
wenzelm@23164
   578
wenzelm@23164
   579
lemma zdiv_mono1_neg: "[| a \<le> a';  (b::int) < 0 |] ==> a' div b \<le> a div b"
wenzelm@23164
   580
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
wenzelm@23164
   581
apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
wenzelm@23164
   582
apply (rule unique_quotient_lemma_neg)
wenzelm@23164
   583
apply (erule subst)
wenzelm@23164
   584
apply (erule subst, simp_all)
wenzelm@23164
   585
done
wenzelm@23164
   586
wenzelm@23164
   587
wenzelm@23164
   588
subsection{*Monotonicity in the Second Argument (Divisor)*}
wenzelm@23164
   589
wenzelm@23164
   590
lemma q_pos_lemma:
wenzelm@23164
   591
     "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"
wenzelm@23164
   592
apply (subgoal_tac "0 < b'* (q' + 1) ")
wenzelm@23164
   593
 apply (simp add: zero_less_mult_iff)
wenzelm@23164
   594
apply (simp add: right_distrib)
wenzelm@23164
   595
done
wenzelm@23164
   596
wenzelm@23164
   597
lemma zdiv_mono2_lemma:
wenzelm@23164
   598
     "[| b*q + r = b'*q' + r';  0 \<le> b'*q' + r';   
wenzelm@23164
   599
         r' < b';  0 \<le> r;  0 < b';  b' \<le> b |]   
wenzelm@23164
   600
      ==> q \<le> (q'::int)"
wenzelm@23164
   601
apply (frule q_pos_lemma, assumption+) 
wenzelm@23164
   602
apply (subgoal_tac "b*q < b* (q' + 1) ")
wenzelm@23164
   603
 apply (simp add: mult_less_cancel_left)
wenzelm@23164
   604
apply (subgoal_tac "b*q = r' - r + b'*q'")
wenzelm@23164
   605
 prefer 2 apply simp
wenzelm@23164
   606
apply (simp (no_asm_simp) add: right_distrib)
wenzelm@23164
   607
apply (subst add_commute, rule zadd_zless_mono, arith)
wenzelm@23164
   608
apply (rule mult_right_mono, auto)
wenzelm@23164
   609
done
wenzelm@23164
   610
wenzelm@23164
   611
lemma zdiv_mono2:
wenzelm@23164
   612
     "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"
wenzelm@23164
   613
apply (subgoal_tac "b \<noteq> 0")
wenzelm@23164
   614
 prefer 2 apply arith
wenzelm@23164
   615
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
wenzelm@23164
   616
apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
wenzelm@23164
   617
apply (rule zdiv_mono2_lemma)
wenzelm@23164
   618
apply (erule subst)
wenzelm@23164
   619
apply (erule subst, simp_all)
wenzelm@23164
   620
done
wenzelm@23164
   621
wenzelm@23164
   622
lemma q_neg_lemma:
wenzelm@23164
   623
     "[| b'*q' + r' < 0;  0 \<le> r';  0 < b' |] ==> q' \<le> (0::int)"
wenzelm@23164
   624
apply (subgoal_tac "b'*q' < 0")
wenzelm@23164
   625
 apply (simp add: mult_less_0_iff, arith)
wenzelm@23164
   626
done
wenzelm@23164
   627
wenzelm@23164
   628
lemma zdiv_mono2_neg_lemma:
wenzelm@23164
   629
     "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;   
wenzelm@23164
   630
         r < b;  0 \<le> r';  0 < b';  b' \<le> b |]   
wenzelm@23164
   631
      ==> q' \<le> (q::int)"
wenzelm@23164
   632
apply (frule q_neg_lemma, assumption+) 
wenzelm@23164
   633
apply (subgoal_tac "b*q' < b* (q + 1) ")
wenzelm@23164
   634
 apply (simp add: mult_less_cancel_left)
wenzelm@23164
   635
apply (simp add: right_distrib)
wenzelm@23164
   636
apply (subgoal_tac "b*q' \<le> b'*q'")
wenzelm@23164
   637
 prefer 2 apply (simp add: mult_right_mono_neg, arith)
wenzelm@23164
   638
done
wenzelm@23164
   639
wenzelm@23164
   640
lemma zdiv_mono2_neg:
wenzelm@23164
   641
     "[| a < (0::int);  0 < b';  b' \<le> b |] ==> a div b' \<le> a div b"
wenzelm@23164
   642
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
wenzelm@23164
   643
apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
wenzelm@23164
   644
apply (rule zdiv_mono2_neg_lemma)
wenzelm@23164
   645
apply (erule subst)
wenzelm@23164
   646
apply (erule subst, simp_all)
wenzelm@23164
   647
done
wenzelm@23164
   648
wenzelm@23164
   649
subsection{*More Algebraic Laws for div and mod*}
wenzelm@23164
   650
wenzelm@23164
   651
text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}
wenzelm@23164
   652
wenzelm@23164
   653
lemma zmult1_lemma:
wenzelm@23164
   654
     "[| quorem((b,c),(q,r));  c \<noteq> 0 |]  
wenzelm@23164
   655
      ==> quorem ((a*b, c), (a*q + a*r div c, a*r mod c))"
wenzelm@23164
   656
by (force simp add: split_ifs quorem_def linorder_neq_iff right_distrib)
wenzelm@23164
   657
wenzelm@23164
   658
lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"
wenzelm@23164
   659
apply (case_tac "c = 0", simp)
wenzelm@23164
   660
apply (blast intro: quorem_div_mod [THEN zmult1_lemma, THEN quorem_div])
wenzelm@23164
   661
done
wenzelm@23164
   662
wenzelm@23164
   663
lemma zmod_zmult1_eq: "(a*b) mod c = a*(b mod c) mod (c::int)"
wenzelm@23164
   664
apply (case_tac "c = 0", simp)
wenzelm@23164
   665
apply (blast intro: quorem_div_mod [THEN zmult1_lemma, THEN quorem_mod])
wenzelm@23164
   666
done
wenzelm@23164
   667
wenzelm@23164
   668
lemma zmod_zmult1_eq': "(a*b) mod (c::int) = ((a mod c) * b) mod c"
wenzelm@23164
   669
apply (rule trans)
wenzelm@23164
   670
apply (rule_tac s = "b*a mod c" in trans)
wenzelm@23164
   671
apply (rule_tac [2] zmod_zmult1_eq)
wenzelm@23164
   672
apply (simp_all add: mult_commute)
wenzelm@23164
   673
done
wenzelm@23164
   674
wenzelm@23164
   675
lemma zmod_zmult_distrib: "(a*b) mod (c::int) = ((a mod c) * (b mod c)) mod c"
wenzelm@23164
   676
apply (rule zmod_zmult1_eq' [THEN trans])
wenzelm@23164
   677
apply (rule zmod_zmult1_eq)
wenzelm@23164
   678
done
wenzelm@23164
   679
wenzelm@23164
   680
lemma zdiv_zmult_self1 [simp]: "b \<noteq> (0::int) ==> (a*b) div b = a"
wenzelm@23164
   681
by (simp add: zdiv_zmult1_eq)
wenzelm@23164
   682
wenzelm@23164
   683
lemma zdiv_zmult_self2 [simp]: "b \<noteq> (0::int) ==> (b*a) div b = a"
wenzelm@23164
   684
by (subst mult_commute, erule zdiv_zmult_self1)
wenzelm@23164
   685
wenzelm@23164
   686
lemma zmod_zmult_self1 [simp]: "(a*b) mod b = (0::int)"
wenzelm@23164
   687
by (simp add: zmod_zmult1_eq)
wenzelm@23164
   688
wenzelm@23164
   689
lemma zmod_zmult_self2 [simp]: "(b*a) mod b = (0::int)"
wenzelm@23164
   690
by (simp add: mult_commute zmod_zmult1_eq)
wenzelm@23164
   691
wenzelm@23164
   692
lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"
wenzelm@23164
   693
proof
wenzelm@23164
   694
  assume "m mod d = 0"
wenzelm@23164
   695
  with zmod_zdiv_equality[of m d] show "EX q::int. m = d*q" by auto
wenzelm@23164
   696
next
wenzelm@23164
   697
  assume "EX q::int. m = d*q"
wenzelm@23164
   698
  thus "m mod d = 0" by auto
wenzelm@23164
   699
qed
wenzelm@23164
   700
wenzelm@23164
   701
lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]
wenzelm@23164
   702
wenzelm@23164
   703
text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}
wenzelm@23164
   704
wenzelm@23164
   705
lemma zadd1_lemma:
wenzelm@23164
   706
     "[| quorem((a,c),(aq,ar));  quorem((b,c),(bq,br));  c \<noteq> 0 |]  
wenzelm@23164
   707
      ==> quorem ((a+b, c), (aq + bq + (ar+br) div c, (ar+br) mod c))"
wenzelm@23164
   708
by (force simp add: split_ifs quorem_def linorder_neq_iff right_distrib)
wenzelm@23164
   709
wenzelm@23164
   710
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
wenzelm@23164
   711
lemma zdiv_zadd1_eq:
wenzelm@23164
   712
     "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"
wenzelm@23164
   713
apply (case_tac "c = 0", simp)
wenzelm@23164
   714
apply (blast intro: zadd1_lemma [OF quorem_div_mod quorem_div_mod] quorem_div)
wenzelm@23164
   715
done
wenzelm@23164
   716
wenzelm@23164
   717
lemma zmod_zadd1_eq: "(a+b) mod (c::int) = (a mod c + b mod c) mod c"
wenzelm@23164
   718
apply (case_tac "c = 0", simp)
wenzelm@23164
   719
apply (blast intro: zadd1_lemma [OF quorem_div_mod quorem_div_mod] quorem_mod)
wenzelm@23164
   720
done
wenzelm@23164
   721
wenzelm@23164
   722
lemma mod_div_trivial [simp]: "(a mod b) div b = (0::int)"
wenzelm@23164
   723
apply (case_tac "b = 0", simp)
wenzelm@23164
   724
apply (auto simp add: linorder_neq_iff div_pos_pos_trivial div_neg_neg_trivial)
wenzelm@23164
   725
done
wenzelm@23164
   726
wenzelm@23164
   727
lemma mod_mod_trivial [simp]: "(a mod b) mod b = a mod (b::int)"
wenzelm@23164
   728
apply (case_tac "b = 0", simp)
wenzelm@23164
   729
apply (force simp add: linorder_neq_iff mod_pos_pos_trivial mod_neg_neg_trivial)
wenzelm@23164
   730
done
wenzelm@23164
   731
wenzelm@23164
   732
lemma zmod_zadd_left_eq: "(a+b) mod (c::int) = ((a mod c) + b) mod c"
wenzelm@23164
   733
apply (rule trans [symmetric])
wenzelm@23164
   734
apply (rule zmod_zadd1_eq, simp)
wenzelm@23164
   735
apply (rule zmod_zadd1_eq [symmetric])
wenzelm@23164
   736
done
wenzelm@23164
   737
wenzelm@23164
   738
lemma zmod_zadd_right_eq: "(a+b) mod (c::int) = (a + (b mod c)) mod c"
wenzelm@23164
   739
apply (rule trans [symmetric])
wenzelm@23164
   740
apply (rule zmod_zadd1_eq, simp)
wenzelm@23164
   741
apply (rule zmod_zadd1_eq [symmetric])
wenzelm@23164
   742
done
wenzelm@23164
   743
wenzelm@23164
   744
lemma zdiv_zadd_self1[simp]: "a \<noteq> (0::int) ==> (a+b) div a = b div a + 1"
wenzelm@23164
   745
by (simp add: zdiv_zadd1_eq)
wenzelm@23164
   746
wenzelm@23164
   747
lemma zdiv_zadd_self2[simp]: "a \<noteq> (0::int) ==> (b+a) div a = b div a + 1"
wenzelm@23164
   748
by (simp add: zdiv_zadd1_eq)
wenzelm@23164
   749
wenzelm@23164
   750
lemma zmod_zadd_self1[simp]: "(a+b) mod a = b mod (a::int)"
wenzelm@23164
   751
apply (case_tac "a = 0", simp)
wenzelm@23164
   752
apply (simp add: zmod_zadd1_eq)
wenzelm@23164
   753
done
wenzelm@23164
   754
wenzelm@23164
   755
lemma zmod_zadd_self2[simp]: "(b+a) mod a = b mod (a::int)"
wenzelm@23164
   756
apply (case_tac "a = 0", simp)
wenzelm@23164
   757
apply (simp add: zmod_zadd1_eq)
wenzelm@23164
   758
done
wenzelm@23164
   759
wenzelm@23164
   760
wenzelm@23164
   761
subsection{*Proving  @{term "a div (b*c) = (a div b) div c"} *}
wenzelm@23164
   762
wenzelm@23164
   763
(*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but
wenzelm@23164
   764
  7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems
wenzelm@23164
   765
  to cause particular problems.*)
wenzelm@23164
   766
wenzelm@23164
   767
text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}
wenzelm@23164
   768
wenzelm@23164
   769
lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b*c < b*(q mod c) + r"
wenzelm@23164
   770
apply (subgoal_tac "b * (c - q mod c) < r * 1")
wenzelm@23164
   771
apply (simp add: right_diff_distrib)
wenzelm@23164
   772
apply (rule order_le_less_trans)
wenzelm@23164
   773
apply (erule_tac [2] mult_strict_right_mono)
wenzelm@23164
   774
apply (rule mult_left_mono_neg)
wenzelm@23164
   775
apply (auto simp add: compare_rls add_commute [of 1]
wenzelm@23164
   776
                      add1_zle_eq pos_mod_bound)
wenzelm@23164
   777
done
wenzelm@23164
   778
wenzelm@23164
   779
lemma zmult2_lemma_aux2:
wenzelm@23164
   780
     "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"
wenzelm@23164
   781
apply (subgoal_tac "b * (q mod c) \<le> 0")
wenzelm@23164
   782
 apply arith
wenzelm@23164
   783
apply (simp add: mult_le_0_iff)
wenzelm@23164
   784
done
wenzelm@23164
   785
wenzelm@23164
   786
lemma zmult2_lemma_aux3: "[| (0::int) < c;  0 \<le> r;  r < b |] ==> 0 \<le> b * (q mod c) + r"
wenzelm@23164
   787
apply (subgoal_tac "0 \<le> b * (q mod c) ")
wenzelm@23164
   788
apply arith
wenzelm@23164
   789
apply (simp add: zero_le_mult_iff)
wenzelm@23164
   790
done
wenzelm@23164
   791
wenzelm@23164
   792
lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"
wenzelm@23164
   793
apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
wenzelm@23164
   794
apply (simp add: right_diff_distrib)
wenzelm@23164
   795
apply (rule order_less_le_trans)
wenzelm@23164
   796
apply (erule mult_strict_right_mono)
wenzelm@23164
   797
apply (rule_tac [2] mult_left_mono)
wenzelm@23164
   798
apply (auto simp add: compare_rls add_commute [of 1]
wenzelm@23164
   799
                      add1_zle_eq pos_mod_bound)
wenzelm@23164
   800
done
wenzelm@23164
   801
wenzelm@23164
   802
lemma zmult2_lemma: "[| quorem ((a,b), (q,r));  b \<noteq> 0;  0 < c |]  
wenzelm@23164
   803
      ==> quorem ((a, b*c), (q div c, b*(q mod c) + r))"
wenzelm@23164
   804
by (auto simp add: mult_ac quorem_def linorder_neq_iff
wenzelm@23164
   805
                   zero_less_mult_iff right_distrib [symmetric] 
wenzelm@23164
   806
                   zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4)
wenzelm@23164
   807
wenzelm@23164
   808
lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c"
wenzelm@23164
   809
apply (case_tac "b = 0", simp)
wenzelm@23164
   810
apply (force simp add: quorem_div_mod [THEN zmult2_lemma, THEN quorem_div])
wenzelm@23164
   811
done
wenzelm@23164
   812
wenzelm@23164
   813
lemma zmod_zmult2_eq:
wenzelm@23164
   814
     "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"
wenzelm@23164
   815
apply (case_tac "b = 0", simp)
wenzelm@23164
   816
apply (force simp add: quorem_div_mod [THEN zmult2_lemma, THEN quorem_mod])
wenzelm@23164
   817
done
wenzelm@23164
   818
wenzelm@23164
   819
wenzelm@23164
   820
subsection{*Cancellation of Common Factors in div*}
wenzelm@23164
   821
wenzelm@23164
   822
lemma zdiv_zmult_zmult1_aux1:
wenzelm@23164
   823
     "[| (0::int) < b;  c \<noteq> 0 |] ==> (c*a) div (c*b) = a div b"
wenzelm@23164
   824
by (subst zdiv_zmult2_eq, auto)
wenzelm@23164
   825
wenzelm@23164
   826
lemma zdiv_zmult_zmult1_aux2:
wenzelm@23164
   827
     "[| b < (0::int);  c \<noteq> 0 |] ==> (c*a) div (c*b) = a div b"
wenzelm@23164
   828
apply (subgoal_tac " (c * (-a)) div (c * (-b)) = (-a) div (-b) ")
wenzelm@23164
   829
apply (rule_tac [2] zdiv_zmult_zmult1_aux1, auto)
wenzelm@23164
   830
done
wenzelm@23164
   831
wenzelm@23164
   832
lemma zdiv_zmult_zmult1: "c \<noteq> (0::int) ==> (c*a) div (c*b) = a div b"
wenzelm@23164
   833
apply (case_tac "b = 0", simp)
wenzelm@23164
   834
apply (auto simp add: linorder_neq_iff zdiv_zmult_zmult1_aux1 zdiv_zmult_zmult1_aux2)
wenzelm@23164
   835
done
wenzelm@23164
   836
wenzelm@23164
   837
lemma zdiv_zmult_zmult2: "c \<noteq> (0::int) ==> (a*c) div (b*c) = a div b"
wenzelm@23164
   838
apply (drule zdiv_zmult_zmult1)
wenzelm@23164
   839
apply (auto simp add: mult_commute)
wenzelm@23164
   840
done
wenzelm@23164
   841
wenzelm@23164
   842
wenzelm@23164
   843
wenzelm@23164
   844
subsection{*Distribution of Factors over mod*}
wenzelm@23164
   845
wenzelm@23164
   846
lemma zmod_zmult_zmult1_aux1:
wenzelm@23164
   847
     "[| (0::int) < b;  c \<noteq> 0 |] ==> (c*a) mod (c*b) = c * (a mod b)"
wenzelm@23164
   848
by (subst zmod_zmult2_eq, auto)
wenzelm@23164
   849
wenzelm@23164
   850
lemma zmod_zmult_zmult1_aux2:
wenzelm@23164
   851
     "[| b < (0::int);  c \<noteq> 0 |] ==> (c*a) mod (c*b) = c * (a mod b)"
wenzelm@23164
   852
apply (subgoal_tac " (c * (-a)) mod (c * (-b)) = c * ((-a) mod (-b))")
wenzelm@23164
   853
apply (rule_tac [2] zmod_zmult_zmult1_aux1, auto)
wenzelm@23164
   854
done
wenzelm@23164
   855
wenzelm@23164
   856
lemma zmod_zmult_zmult1: "(c*a) mod (c*b) = (c::int) * (a mod b)"
wenzelm@23164
   857
apply (case_tac "b = 0", simp)
wenzelm@23164
   858
apply (case_tac "c = 0", simp)
wenzelm@23164
   859
apply (auto simp add: linorder_neq_iff zmod_zmult_zmult1_aux1 zmod_zmult_zmult1_aux2)
wenzelm@23164
   860
done
wenzelm@23164
   861
wenzelm@23164
   862
lemma zmod_zmult_zmult2: "(a*c) mod (b*c) = (a mod b) * (c::int)"
wenzelm@23164
   863
apply (cut_tac c = c in zmod_zmult_zmult1)
wenzelm@23164
   864
apply (auto simp add: mult_commute)
wenzelm@23164
   865
done
wenzelm@23164
   866
wenzelm@23164
   867
wenzelm@23164
   868
subsection {*Splitting Rules for div and mod*}
wenzelm@23164
   869
wenzelm@23164
   870
text{*The proofs of the two lemmas below are essentially identical*}
wenzelm@23164
   871
wenzelm@23164
   872
lemma split_pos_lemma:
wenzelm@23164
   873
 "0<k ==> 
wenzelm@23164
   874
    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
   875
apply (rule iffI, clarify)
wenzelm@23164
   876
 apply (erule_tac P="P ?x ?y" in rev_mp)  
wenzelm@23164
   877
 apply (subst zmod_zadd1_eq) 
wenzelm@23164
   878
 apply (subst zdiv_zadd1_eq) 
wenzelm@23164
   879
 apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)  
wenzelm@23164
   880
txt{*converse direction*}
wenzelm@23164
   881
apply (drule_tac x = "n div k" in spec) 
wenzelm@23164
   882
apply (drule_tac x = "n mod k" in spec, simp)
wenzelm@23164
   883
done
wenzelm@23164
   884
wenzelm@23164
   885
lemma split_neg_lemma:
wenzelm@23164
   886
 "k<0 ==>
wenzelm@23164
   887
    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
   888
apply (rule iffI, clarify)
wenzelm@23164
   889
 apply (erule_tac P="P ?x ?y" in rev_mp)  
wenzelm@23164
   890
 apply (subst zmod_zadd1_eq) 
wenzelm@23164
   891
 apply (subst zdiv_zadd1_eq) 
wenzelm@23164
   892
 apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)  
wenzelm@23164
   893
txt{*converse direction*}
wenzelm@23164
   894
apply (drule_tac x = "n div k" in spec) 
wenzelm@23164
   895
apply (drule_tac x = "n mod k" in spec, simp)
wenzelm@23164
   896
done
wenzelm@23164
   897
wenzelm@23164
   898
lemma split_zdiv:
wenzelm@23164
   899
 "P(n div k :: int) =
wenzelm@23164
   900
  ((k = 0 --> P 0) & 
wenzelm@23164
   901
   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) & 
wenzelm@23164
   902
   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"
wenzelm@23164
   903
apply (case_tac "k=0", simp)
wenzelm@23164
   904
apply (simp only: linorder_neq_iff)
wenzelm@23164
   905
apply (erule disjE) 
wenzelm@23164
   906
 apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"] 
wenzelm@23164
   907
                      split_neg_lemma [of concl: "%x y. P x"])
wenzelm@23164
   908
done
wenzelm@23164
   909
wenzelm@23164
   910
lemma split_zmod:
wenzelm@23164
   911
 "P(n mod k :: int) =
wenzelm@23164
   912
  ((k = 0 --> P n) & 
wenzelm@23164
   913
   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) & 
wenzelm@23164
   914
   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"
wenzelm@23164
   915
apply (case_tac "k=0", simp)
wenzelm@23164
   916
apply (simp only: linorder_neq_iff)
wenzelm@23164
   917
apply (erule disjE) 
wenzelm@23164
   918
 apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"] 
wenzelm@23164
   919
                      split_neg_lemma [of concl: "%x y. P y"])
wenzelm@23164
   920
done
wenzelm@23164
   921
wenzelm@23164
   922
(* Enable arith to deal with div 2 and mod 2: *)
wenzelm@23164
   923
declare split_zdiv [of _ _ "number_of k", simplified, standard, arith_split]
wenzelm@23164
   924
declare split_zmod [of _ _ "number_of k", simplified, standard, arith_split]
wenzelm@23164
   925
wenzelm@23164
   926
wenzelm@23164
   927
subsection{*Speeding up the Division Algorithm with Shifting*}
wenzelm@23164
   928
wenzelm@23164
   929
text{*computing div by shifting *}
wenzelm@23164
   930
wenzelm@23164
   931
lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
wenzelm@23164
   932
proof cases
wenzelm@23164
   933
  assume "a=0"
wenzelm@23164
   934
    thus ?thesis by simp
wenzelm@23164
   935
next
wenzelm@23164
   936
  assume "a\<noteq>0" and le_a: "0\<le>a"   
wenzelm@23164
   937
  hence a_pos: "1 \<le> a" by arith
wenzelm@23164
   938
  hence one_less_a2: "1 < 2*a" by arith
wenzelm@23164
   939
  hence le_2a: "2 * (1 + b mod a) \<le> 2 * a"
wenzelm@23164
   940
    by (simp add: mult_le_cancel_left add_commute [of 1] add1_zle_eq)
wenzelm@23164
   941
  with a_pos have "0 \<le> b mod a" by simp
wenzelm@23164
   942
  hence le_addm: "0 \<le> 1 mod (2*a) + 2*(b mod a)"
wenzelm@23164
   943
    by (simp add: mod_pos_pos_trivial one_less_a2)
wenzelm@23164
   944
  with  le_2a
wenzelm@23164
   945
  have "(1 mod (2*a) + 2*(b mod a)) div (2*a) = 0"
wenzelm@23164
   946
    by (simp add: div_pos_pos_trivial le_addm mod_pos_pos_trivial one_less_a2
wenzelm@23164
   947
                  right_distrib) 
wenzelm@23164
   948
  thus ?thesis
wenzelm@23164
   949
    by (subst zdiv_zadd1_eq,
wenzelm@23164
   950
        simp add: zdiv_zmult_zmult1 zmod_zmult_zmult1 one_less_a2
wenzelm@23164
   951
                  div_pos_pos_trivial)
wenzelm@23164
   952
qed
wenzelm@23164
   953
wenzelm@23164
   954
lemma neg_zdiv_mult_2: "a \<le> (0::int) ==> (1 + 2*b) div (2*a) = (b+1) div a"
wenzelm@23164
   955
apply (subgoal_tac " (1 + 2* (-b - 1)) div (2 * (-a)) = (-b - 1) div (-a) ")
wenzelm@23164
   956
apply (rule_tac [2] pos_zdiv_mult_2)
wenzelm@23164
   957
apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib)
wenzelm@23164
   958
apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")
wenzelm@23164
   959
apply (simp only: zdiv_zminus_zminus diff_minus minus_add_distrib [symmetric],
wenzelm@23164
   960
       simp) 
wenzelm@23164
   961
done
wenzelm@23164
   962
wenzelm@23164
   963
wenzelm@23164
   964
(*Not clear why this must be proved separately; probably number_of causes
wenzelm@23164
   965
  simplification problems*)
wenzelm@23164
   966
lemma not_0_le_lemma: "~ 0 \<le> x ==> x \<le> (0::int)"
wenzelm@23164
   967
by auto
wenzelm@23164
   968
wenzelm@23164
   969
lemma zdiv_number_of_BIT[simp]:
wenzelm@23164
   970
     "number_of (v BIT b) div number_of (w BIT bit.B0) =  
wenzelm@23164
   971
          (if b=bit.B0 | (0::int) \<le> number_of w                    
wenzelm@23164
   972
           then number_of v div (number_of w)     
wenzelm@23164
   973
           else (number_of v + (1::int)) div (number_of w))"
wenzelm@23164
   974
apply (simp only: number_of_eq numeral_simps UNIV_I split: split_if) 
wenzelm@23164
   975
apply (simp add: zdiv_zmult_zmult1 pos_zdiv_mult_2 neg_zdiv_mult_2 add_ac 
wenzelm@23164
   976
            split: bit.split)
wenzelm@23164
   977
done
wenzelm@23164
   978
wenzelm@23164
   979
wenzelm@23164
   980
subsection{*Computing mod by Shifting (proofs resemble those for div)*}
wenzelm@23164
   981
wenzelm@23164
   982
lemma pos_zmod_mult_2:
wenzelm@23164
   983
     "(0::int) \<le> a ==> (1 + 2*b) mod (2*a) = 1 + 2 * (b mod a)"
wenzelm@23164
   984
apply (case_tac "a = 0", simp)
wenzelm@23164
   985
apply (subgoal_tac "1 < a * 2")
wenzelm@23164
   986
 prefer 2 apply arith
wenzelm@23164
   987
apply (subgoal_tac "2* (1 + b mod a) \<le> 2*a")
wenzelm@23164
   988
 apply (rule_tac [2] mult_left_mono)
wenzelm@23164
   989
apply (auto simp add: add_commute [of 1] mult_commute add1_zle_eq 
wenzelm@23164
   990
                      pos_mod_bound)
wenzelm@23164
   991
apply (subst zmod_zadd1_eq)
wenzelm@23164
   992
apply (simp add: zmod_zmult_zmult2 mod_pos_pos_trivial)
wenzelm@23164
   993
apply (rule mod_pos_pos_trivial)
wenzelm@23164
   994
apply (auto simp add: mod_pos_pos_trivial left_distrib)
wenzelm@23164
   995
apply (subgoal_tac "0 \<le> b mod a", arith, simp)
wenzelm@23164
   996
done
wenzelm@23164
   997
wenzelm@23164
   998
lemma neg_zmod_mult_2:
wenzelm@23164
   999
     "a \<le> (0::int) ==> (1 + 2*b) mod (2*a) = 2 * ((b+1) mod a) - 1"
wenzelm@23164
  1000
apply (subgoal_tac "(1 + 2* (-b - 1)) mod (2* (-a)) = 
wenzelm@23164
  1001
                    1 + 2* ((-b - 1) mod (-a))")
wenzelm@23164
  1002
apply (rule_tac [2] pos_zmod_mult_2)
wenzelm@23164
  1003
apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib)
wenzelm@23164
  1004
apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")
wenzelm@23164
  1005
 prefer 2 apply simp 
wenzelm@23164
  1006
apply (simp only: zmod_zminus_zminus diff_minus minus_add_distrib [symmetric])
wenzelm@23164
  1007
done
wenzelm@23164
  1008
wenzelm@23164
  1009
lemma zmod_number_of_BIT [simp]:
wenzelm@23164
  1010
     "number_of (v BIT b) mod number_of (w BIT bit.B0) =  
wenzelm@23164
  1011
      (case b of
wenzelm@23164
  1012
          bit.B0 => 2 * (number_of v mod number_of w)
wenzelm@23164
  1013
        | bit.B1 => if (0::int) \<le> number_of w  
wenzelm@23164
  1014
                then 2 * (number_of v mod number_of w) + 1     
wenzelm@23164
  1015
                else 2 * ((number_of v + (1::int)) mod number_of w) - 1)"
wenzelm@23164
  1016
apply (simp only: number_of_eq numeral_simps UNIV_I split: bit.split) 
wenzelm@23164
  1017
apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2 
wenzelm@23164
  1018
                 not_0_le_lemma neg_zmod_mult_2 add_ac)
wenzelm@23164
  1019
done
wenzelm@23164
  1020
wenzelm@23164
  1021
wenzelm@23164
  1022
subsection{*Quotients of Signs*}
wenzelm@23164
  1023
wenzelm@23164
  1024
lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"
wenzelm@23164
  1025
apply (subgoal_tac "a div b \<le> -1", force)
wenzelm@23164
  1026
apply (rule order_trans)
wenzelm@23164
  1027
apply (rule_tac a' = "-1" in zdiv_mono1)
wenzelm@23164
  1028
apply (auto simp add: zdiv_minus1)
wenzelm@23164
  1029
done
wenzelm@23164
  1030
wenzelm@23164
  1031
lemma div_nonneg_neg_le0: "[| (0::int) \<le> a;  b < 0 |] ==> a div b \<le> 0"
wenzelm@23164
  1032
by (drule zdiv_mono1_neg, auto)
wenzelm@23164
  1033
wenzelm@23164
  1034
lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"
wenzelm@23164
  1035
apply auto
wenzelm@23164
  1036
apply (drule_tac [2] zdiv_mono1)
wenzelm@23164
  1037
apply (auto simp add: linorder_neq_iff)
wenzelm@23164
  1038
apply (simp (no_asm_use) add: linorder_not_less [symmetric])
wenzelm@23164
  1039
apply (blast intro: div_neg_pos_less0)
wenzelm@23164
  1040
done
wenzelm@23164
  1041
wenzelm@23164
  1042
lemma neg_imp_zdiv_nonneg_iff:
wenzelm@23164
  1043
     "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"
wenzelm@23164
  1044
apply (subst zdiv_zminus_zminus [symmetric])
wenzelm@23164
  1045
apply (subst pos_imp_zdiv_nonneg_iff, auto)
wenzelm@23164
  1046
done
wenzelm@23164
  1047
wenzelm@23164
  1048
(*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
wenzelm@23164
  1049
lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
wenzelm@23164
  1050
by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
wenzelm@23164
  1051
wenzelm@23164
  1052
(*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
wenzelm@23164
  1053
lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
wenzelm@23164
  1054
by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
wenzelm@23164
  1055
wenzelm@23164
  1056
wenzelm@23164
  1057
subsection {* The Divides Relation *}
wenzelm@23164
  1058
wenzelm@23164
  1059
lemma zdvd_iff_zmod_eq_0: "(m dvd n) = (n mod m = (0::int))"
wenzelm@23164
  1060
by(simp add:dvd_def zmod_eq_0_iff)
wenzelm@23164
  1061
wenzelm@23164
  1062
lemmas zdvd_iff_zmod_eq_0_number_of [simp] =
wenzelm@23164
  1063
  zdvd_iff_zmod_eq_0 [of "number_of x" "number_of y", standard]
wenzelm@23164
  1064
wenzelm@23164
  1065
lemma zdvd_0_right [iff]: "(m::int) dvd 0"
wenzelm@23164
  1066
by (simp add: dvd_def)
wenzelm@23164
  1067
wenzelm@23164
  1068
lemma zdvd_0_left [iff]: "(0 dvd (m::int)) = (m = 0)"
wenzelm@23164
  1069
  by (simp add: dvd_def)
wenzelm@23164
  1070
wenzelm@23164
  1071
lemma zdvd_1_left [iff]: "1 dvd (m::int)"
wenzelm@23164
  1072
  by (simp add: dvd_def)
wenzelm@23164
  1073
wenzelm@23164
  1074
lemma zdvd_refl [simp]: "m dvd (m::int)"
wenzelm@23164
  1075
by (auto simp add: dvd_def intro: zmult_1_right [symmetric])
wenzelm@23164
  1076
wenzelm@23164
  1077
lemma zdvd_trans: "m dvd n ==> n dvd k ==> m dvd (k::int)"
wenzelm@23164
  1078
by (auto simp add: dvd_def intro: mult_assoc)
wenzelm@23164
  1079
wenzelm@23164
  1080
lemma zdvd_zminus_iff: "(m dvd -n) = (m dvd (n::int))"
wenzelm@23164
  1081
  apply (simp add: dvd_def, auto)
wenzelm@23164
  1082
   apply (rule_tac [!] x = "-k" in exI, auto)
wenzelm@23164
  1083
  done
wenzelm@23164
  1084
wenzelm@23164
  1085
lemma zdvd_zminus2_iff: "(-m dvd n) = (m dvd (n::int))"
wenzelm@23164
  1086
  apply (simp add: dvd_def, auto)
wenzelm@23164
  1087
   apply (rule_tac [!] x = "-k" in exI, auto)
wenzelm@23164
  1088
  done
wenzelm@23164
  1089
lemma zdvd_abs1: "( \<bar>i::int\<bar> dvd j) = (i dvd j)" 
wenzelm@23164
  1090
  apply (cases "i > 0", simp)
wenzelm@23164
  1091
  apply (simp add: dvd_def)
wenzelm@23164
  1092
  apply (rule iffI)
wenzelm@23164
  1093
  apply (erule exE)
wenzelm@23164
  1094
  apply (rule_tac x="- k" in exI, simp)
wenzelm@23164
  1095
  apply (erule exE)
wenzelm@23164
  1096
  apply (rule_tac x="- k" in exI, simp)
wenzelm@23164
  1097
  done
wenzelm@23164
  1098
lemma zdvd_abs2: "( (i::int) dvd \<bar>j\<bar>) = (i dvd j)" 
wenzelm@23164
  1099
  apply (cases "j > 0", simp)
wenzelm@23164
  1100
  apply (simp add: dvd_def)
wenzelm@23164
  1101
  apply (rule iffI)
wenzelm@23164
  1102
  apply (erule exE)
wenzelm@23164
  1103
  apply (rule_tac x="- k" in exI, simp)
wenzelm@23164
  1104
  apply (erule exE)
wenzelm@23164
  1105
  apply (rule_tac x="- k" in exI, simp)
wenzelm@23164
  1106
  done
wenzelm@23164
  1107
wenzelm@23164
  1108
lemma zdvd_anti_sym:
wenzelm@23164
  1109
    "0 < m ==> 0 < n ==> m dvd n ==> n dvd m ==> m = (n::int)"
wenzelm@23164
  1110
  apply (simp add: dvd_def, auto)
wenzelm@23164
  1111
  apply (simp add: mult_assoc zero_less_mult_iff zmult_eq_1_iff)
wenzelm@23164
  1112
  done
wenzelm@23164
  1113
wenzelm@23164
  1114
lemma zdvd_zadd: "k dvd m ==> k dvd n ==> k dvd (m + n :: int)"
wenzelm@23164
  1115
  apply (simp add: dvd_def)
wenzelm@23164
  1116
  apply (blast intro: right_distrib [symmetric])
wenzelm@23164
  1117
  done
wenzelm@23164
  1118
wenzelm@23164
  1119
lemma zdvd_dvd_eq: assumes anz:"a \<noteq> 0" and ab: "(a::int) dvd b" and ba:"b dvd a" 
wenzelm@23164
  1120
  shows "\<bar>a\<bar> = \<bar>b\<bar>"
wenzelm@23164
  1121
proof-
wenzelm@23164
  1122
  from ab obtain k where k:"b = a*k" unfolding dvd_def by blast 
wenzelm@23164
  1123
  from ba obtain k' where k':"a = b*k'" unfolding dvd_def by blast 
wenzelm@23164
  1124
  from k k' have "a = a*k*k'" by simp
wenzelm@23164
  1125
  with mult_cancel_left1[where c="a" and b="k*k'"]
wenzelm@23164
  1126
  have kk':"k*k' = 1" using anz by (simp add: mult_assoc)
wenzelm@23164
  1127
  hence "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1" by (simp add: zmult_eq_1_iff)
wenzelm@23164
  1128
  thus ?thesis using k k' by auto
wenzelm@23164
  1129
qed
wenzelm@23164
  1130
wenzelm@23164
  1131
lemma zdvd_zdiff: "k dvd m ==> k dvd n ==> k dvd (m - n :: int)"
wenzelm@23164
  1132
  apply (simp add: dvd_def)
wenzelm@23164
  1133
  apply (blast intro: right_diff_distrib [symmetric])
wenzelm@23164
  1134
  done
wenzelm@23164
  1135
wenzelm@23164
  1136
lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"
wenzelm@23164
  1137
  apply (subgoal_tac "m = n + (m - n)")
wenzelm@23164
  1138
   apply (erule ssubst)
wenzelm@23164
  1139
   apply (blast intro: zdvd_zadd, simp)
wenzelm@23164
  1140
  done
wenzelm@23164
  1141
wenzelm@23164
  1142
lemma zdvd_zmult: "k dvd (n::int) ==> k dvd m * n"
wenzelm@23164
  1143
  apply (simp add: dvd_def)
wenzelm@23164
  1144
  apply (blast intro: mult_left_commute)
wenzelm@23164
  1145
  done
wenzelm@23164
  1146
wenzelm@23164
  1147
lemma zdvd_zmult2: "k dvd (m::int) ==> k dvd m * n"
wenzelm@23164
  1148
  apply (subst mult_commute)
wenzelm@23164
  1149
  apply (erule zdvd_zmult)
wenzelm@23164
  1150
  done
wenzelm@23164
  1151
wenzelm@23164
  1152
lemma zdvd_triv_right [iff]: "(k::int) dvd m * k"
wenzelm@23164
  1153
  apply (rule zdvd_zmult)
wenzelm@23164
  1154
  apply (rule zdvd_refl)
wenzelm@23164
  1155
  done
wenzelm@23164
  1156
wenzelm@23164
  1157
lemma zdvd_triv_left [iff]: "(k::int) dvd k * m"
wenzelm@23164
  1158
  apply (rule zdvd_zmult2)
wenzelm@23164
  1159
  apply (rule zdvd_refl)
wenzelm@23164
  1160
  done
wenzelm@23164
  1161
wenzelm@23164
  1162
lemma zdvd_zmultD2: "j * k dvd n ==> j dvd (n::int)"
wenzelm@23164
  1163
  apply (simp add: dvd_def)
wenzelm@23164
  1164
  apply (simp add: mult_assoc, blast)
wenzelm@23164
  1165
  done
wenzelm@23164
  1166
wenzelm@23164
  1167
lemma zdvd_zmultD: "j * k dvd n ==> k dvd (n::int)"
wenzelm@23164
  1168
  apply (rule zdvd_zmultD2)
wenzelm@23164
  1169
  apply (subst mult_commute, assumption)
wenzelm@23164
  1170
  done
wenzelm@23164
  1171
wenzelm@23164
  1172
lemma zdvd_zmult_mono: "i dvd m ==> j dvd (n::int) ==> i * j dvd m * n"
wenzelm@23164
  1173
  apply (simp add: dvd_def, clarify)
wenzelm@23164
  1174
  apply (rule_tac x = "k * ka" in exI)
wenzelm@23164
  1175
  apply (simp add: mult_ac)
wenzelm@23164
  1176
  done
wenzelm@23164
  1177
wenzelm@23164
  1178
lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"
wenzelm@23164
  1179
  apply (rule iffI)
wenzelm@23164
  1180
   apply (erule_tac [2] zdvd_zadd)
wenzelm@23164
  1181
   apply (subgoal_tac "n = (n + k * m) - k * m")
wenzelm@23164
  1182
    apply (erule ssubst)
wenzelm@23164
  1183
    apply (erule zdvd_zdiff, simp_all)
wenzelm@23164
  1184
  done
wenzelm@23164
  1185
wenzelm@23164
  1186
lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"
wenzelm@23164
  1187
  apply (simp add: dvd_def)
wenzelm@23164
  1188
  apply (auto simp add: zmod_zmult_zmult1)
wenzelm@23164
  1189
  done
wenzelm@23164
  1190
wenzelm@23164
  1191
lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)"
wenzelm@23164
  1192
  apply (subgoal_tac "k dvd n * (m div n) + m mod n")
wenzelm@23164
  1193
   apply (simp add: zmod_zdiv_equality [symmetric])
wenzelm@23164
  1194
  apply (simp only: zdvd_zadd zdvd_zmult2)
wenzelm@23164
  1195
  done
wenzelm@23164
  1196
wenzelm@23164
  1197
lemma zdvd_not_zless: "0 < m ==> m < n ==> \<not> n dvd (m::int)"
wenzelm@23164
  1198
  apply (simp add: dvd_def, auto)
wenzelm@23164
  1199
  apply (subgoal_tac "0 < n")
wenzelm@23164
  1200
   prefer 2
wenzelm@23164
  1201
   apply (blast intro: order_less_trans)
wenzelm@23164
  1202
  apply (simp add: zero_less_mult_iff)
wenzelm@23164
  1203
  apply (subgoal_tac "n * k < n * 1")
wenzelm@23164
  1204
   apply (drule mult_less_cancel_left [THEN iffD1], auto)
wenzelm@23164
  1205
  done
wenzelm@23164
  1206
lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"
wenzelm@23164
  1207
  using zmod_zdiv_equality[where a="m" and b="n"]
wenzelm@23164
  1208
  by (simp add: ring_eq_simps)
wenzelm@23164
  1209
wenzelm@23164
  1210
lemma zdvd_mult_div_cancel:"(n::int) dvd m \<Longrightarrow> n * (m div n) = m"
wenzelm@23164
  1211
apply (subgoal_tac "m mod n = 0")
wenzelm@23164
  1212
 apply (simp add: zmult_div_cancel)
wenzelm@23164
  1213
apply (simp only: zdvd_iff_zmod_eq_0)
wenzelm@23164
  1214
done
wenzelm@23164
  1215
wenzelm@23164
  1216
lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k \<noteq> (0::int)"
wenzelm@23164
  1217
  shows "m dvd n"
wenzelm@23164
  1218
proof-
wenzelm@23164
  1219
  from d obtain h where h: "k*n = k*m * h" unfolding dvd_def by blast
wenzelm@23164
  1220
  {assume "n \<noteq> m*h" hence "k* n \<noteq> k* (m*h)" using kz by simp
wenzelm@23164
  1221
    with h have False by (simp add: mult_assoc)}
wenzelm@23164
  1222
  hence "n = m * h" by blast
wenzelm@23164
  1223
  thus ?thesis by blast
wenzelm@23164
  1224
qed
wenzelm@23164
  1225
wenzelm@23164
  1226
theorem ex_nat: "(\<exists>x::nat. P x) = (\<exists>x::int. 0 <= x \<and> P (nat x))"
wenzelm@23164
  1227
  apply (simp split add: split_nat)
wenzelm@23164
  1228
  apply (rule iffI)
wenzelm@23164
  1229
  apply (erule exE)
wenzelm@23164
  1230
  apply (rule_tac x = "int x" in exI)
wenzelm@23164
  1231
  apply simp
wenzelm@23164
  1232
  apply (erule exE)
wenzelm@23164
  1233
  apply (rule_tac x = "nat x" in exI)
wenzelm@23164
  1234
  apply (erule conjE)
wenzelm@23164
  1235
  apply (erule_tac x = "nat x" in allE)
wenzelm@23164
  1236
  apply simp
wenzelm@23164
  1237
  done
wenzelm@23164
  1238
huffman@23365
  1239
theorem zdvd_int: "(x dvd y) = (int x dvd int y)"
huffman@23306
  1240
  unfolding dvd_def
huffman@23365
  1241
  apply (rule_tac s="\<exists>k. int y = int x * int k" in trans)
huffman@23306
  1242
  apply (simp only: of_nat_mult [symmetric] of_nat_eq_iff)
huffman@23306
  1243
  apply (simp add: ex_nat cong add: conj_cong)
wenzelm@23164
  1244
  apply (rule iffI)
wenzelm@23164
  1245
  apply iprover
wenzelm@23164
  1246
  apply (erule exE)
wenzelm@23164
  1247
  apply (case_tac "x=0")
wenzelm@23164
  1248
  apply (rule_tac x=0 in exI)
wenzelm@23164
  1249
  apply simp
wenzelm@23164
  1250
  apply (case_tac "0 \<le> k")
wenzelm@23164
  1251
  apply iprover
wenzelm@23164
  1252
  apply (simp add: linorder_not_le)
huffman@23306
  1253
  apply (drule mult_strict_left_mono_neg [OF iffD2 [OF of_nat_0_less_iff]])
wenzelm@23164
  1254
  apply assumption
wenzelm@23164
  1255
  apply (simp add: mult_ac)
wenzelm@23164
  1256
  done
wenzelm@23164
  1257
wenzelm@23164
  1258
lemma zdvd1_eq[simp]: "(x::int) dvd 1 = ( \<bar>x\<bar> = 1)"
wenzelm@23164
  1259
proof
wenzelm@23164
  1260
  assume d: "x dvd 1" hence "int (nat \<bar>x\<bar>) dvd int (nat 1)" by (simp add: zdvd_abs1)
wenzelm@23164
  1261
  hence "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int)
wenzelm@23164
  1262
  hence "nat \<bar>x\<bar> = 1"  by simp
wenzelm@23164
  1263
  thus "\<bar>x\<bar> = 1" by (cases "x < 0", auto)
wenzelm@23164
  1264
next
wenzelm@23164
  1265
  assume "\<bar>x\<bar>=1" thus "x dvd 1" 
wenzelm@23164
  1266
    by(cases "x < 0",simp_all add: minus_equation_iff zdvd_iff_zmod_eq_0)
wenzelm@23164
  1267
qed
wenzelm@23164
  1268
lemma zdvd_mult_cancel1: 
wenzelm@23164
  1269
  assumes mp:"m \<noteq>(0::int)" shows "(m * n dvd m) = (\<bar>n\<bar> = 1)"
wenzelm@23164
  1270
proof
wenzelm@23164
  1271
  assume n1: "\<bar>n\<bar> = 1" thus "m * n dvd m" 
wenzelm@23164
  1272
    by (cases "n >0", auto simp add: zdvd_zminus2_iff minus_equation_iff)
wenzelm@23164
  1273
next
wenzelm@23164
  1274
  assume H: "m * n dvd m" hence H2: "m * n dvd m * 1" by simp
wenzelm@23164
  1275
  from zdvd_mult_cancel[OF H2 mp] show "\<bar>n\<bar> = 1" by (simp only: zdvd1_eq)
wenzelm@23164
  1276
qed
wenzelm@23164
  1277
huffman@23365
  1278
lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))"
wenzelm@23164
  1279
  apply (auto simp add: dvd_def nat_abs_mult_distrib)
huffman@23365
  1280
  apply (auto simp add: nat_eq_iff abs_if split add: split_if_asm)
huffman@23365
  1281
   apply (rule_tac x = "-(int k)" in exI)
huffman@23306
  1282
  apply auto
huffman@23306
  1283
  done
huffman@23306
  1284
huffman@23365
  1285
lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)"
huffman@23306
  1286
  apply (auto simp add: dvd_def abs_if)
huffman@23306
  1287
    apply (rule_tac [3] x = "nat k" in exI)
huffman@23365
  1288
    apply (rule_tac [2] x = "-(int k)" in exI)
huffman@23306
  1289
    apply (rule_tac x = "nat (-k)" in exI)
huffman@23365
  1290
    apply (cut_tac [3] m = m in int_less_0_conv)
huffman@23365
  1291
    apply (cut_tac m = m in int_less_0_conv)
huffman@23306
  1292
    apply (auto simp add: zero_le_mult_iff mult_less_0_iff
huffman@23365
  1293
      nat_mult_distrib [symmetric] nat_eq_iff2)
wenzelm@23164
  1294
  done
wenzelm@23164
  1295
wenzelm@23164
  1296
lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \<le> z then (z dvd int m) else m = 0)"
huffman@23365
  1297
  apply (auto simp add: dvd_def)
huffman@23365
  1298
  apply (rule_tac x = "nat k" in exI)
huffman@23365
  1299
  apply (cut_tac m = m in int_less_0_conv)
huffman@23365
  1300
  apply (auto simp add: zero_le_mult_iff mult_less_0_iff
huffman@23365
  1301
    nat_mult_distrib [symmetric] nat_eq_iff2)
huffman@23365
  1302
  done
wenzelm@23164
  1303
wenzelm@23164
  1304
lemma zminus_dvd_iff [iff]: "(-z dvd w) = (z dvd (w::int))"
wenzelm@23164
  1305
  apply (auto simp add: dvd_def)
wenzelm@23164
  1306
   apply (rule_tac [!] x = "-k" in exI, auto)
wenzelm@23164
  1307
  done
wenzelm@23164
  1308
wenzelm@23164
  1309
lemma dvd_zminus_iff [iff]: "(z dvd -w) = (z dvd (w::int))"
wenzelm@23164
  1310
  apply (auto simp add: dvd_def)
wenzelm@23164
  1311
   apply (drule minus_equation_iff [THEN iffD1])
wenzelm@23164
  1312
   apply (rule_tac [!] x = "-k" in exI, auto)
wenzelm@23164
  1313
  done
wenzelm@23164
  1314
wenzelm@23164
  1315
lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \<le> (n::int)"
huffman@23365
  1316
  apply (rule_tac z=n in int_cases)
huffman@23365
  1317
  apply (auto simp add: dvd_int_iff)
huffman@23365
  1318
  apply (rule_tac z=z in int_cases)
huffman@23307
  1319
  apply (auto simp add: dvd_imp_le)
wenzelm@23164
  1320
  done
wenzelm@23164
  1321
wenzelm@23164
  1322
wenzelm@23164
  1323
subsection{*Integer Powers*} 
wenzelm@23164
  1324
wenzelm@23164
  1325
instance int :: power ..
wenzelm@23164
  1326
wenzelm@23164
  1327
primrec
wenzelm@23164
  1328
  "p ^ 0 = 1"
wenzelm@23164
  1329
  "p ^ (Suc n) = (p::int) * (p ^ n)"
wenzelm@23164
  1330
wenzelm@23164
  1331
wenzelm@23164
  1332
instance int :: recpower
wenzelm@23164
  1333
proof
wenzelm@23164
  1334
  fix z :: int
wenzelm@23164
  1335
  fix n :: nat
wenzelm@23164
  1336
  show "z^0 = 1" by simp
wenzelm@23164
  1337
  show "z^(Suc n) = z * (z^n)" by simp
wenzelm@23164
  1338
qed
wenzelm@23164
  1339
wenzelm@23164
  1340
wenzelm@23164
  1341
lemma zpower_zmod: "((x::int) mod m)^y mod m = x^y mod m"
wenzelm@23164
  1342
apply (induct "y", auto)
wenzelm@23164
  1343
apply (rule zmod_zmult1_eq [THEN trans])
wenzelm@23164
  1344
apply (simp (no_asm_simp))
wenzelm@23164
  1345
apply (rule zmod_zmult_distrib [symmetric])
wenzelm@23164
  1346
done
wenzelm@23164
  1347
wenzelm@23164
  1348
lemma zpower_zadd_distrib: "x^(y+z) = ((x^y)*(x^z)::int)"
wenzelm@23164
  1349
  by (rule Power.power_add)
wenzelm@23164
  1350
wenzelm@23164
  1351
lemma zpower_zpower: "(x^y)^z = (x^(y*z)::int)"
wenzelm@23164
  1352
  by (rule Power.power_mult [symmetric])
wenzelm@23164
  1353
wenzelm@23164
  1354
lemma zero_less_zpower_abs_iff [simp]:
wenzelm@23164
  1355
     "(0 < (abs x)^n) = (x \<noteq> (0::int) | n=0)"
wenzelm@23164
  1356
apply (induct "n")
wenzelm@23164
  1357
apply (auto simp add: zero_less_mult_iff)
wenzelm@23164
  1358
done
wenzelm@23164
  1359
wenzelm@23164
  1360
lemma zero_le_zpower_abs [simp]: "(0::int) <= (abs x)^n"
wenzelm@23164
  1361
apply (induct "n")
wenzelm@23164
  1362
apply (auto simp add: zero_le_mult_iff)
wenzelm@23164
  1363
done
wenzelm@23164
  1364
wenzelm@23164
  1365
lemma int_power: "int (m^n) = (int m) ^ n"
huffman@23365
  1366
  by (rule of_nat_power)
wenzelm@23164
  1367
wenzelm@23164
  1368
text{*Compatibility binding*}
wenzelm@23164
  1369
lemmas zpower_int = int_power [symmetric]
wenzelm@23164
  1370
huffman@23365
  1371
lemma zdiv_int: "int (a div b) = (int a) div (int b)"
wenzelm@23164
  1372
apply (subst split_div, auto)
wenzelm@23164
  1373
apply (subst split_zdiv, auto)
huffman@23365
  1374
apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in IntDiv.unique_quotient)
huffman@23306
  1375
apply (auto simp add: IntDiv.quorem_def)
wenzelm@23164
  1376
done
wenzelm@23164
  1377
wenzelm@23164
  1378
lemma zmod_int: "int (a mod b) = (int a) mod (int b)"
huffman@23365
  1379
apply (subst split_mod, auto)
huffman@23365
  1380
apply (subst split_zmod, auto)
huffman@23365
  1381
apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia 
huffman@23365
  1382
       in unique_remainder)
huffman@23365
  1383
apply (auto simp add: IntDiv.quorem_def)
huffman@23365
  1384
done
wenzelm@23164
  1385
wenzelm@23164
  1386
text{*Suggested by Matthias Daum*}
wenzelm@23164
  1387
lemma int_power_div_base:
wenzelm@23164
  1388
     "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"
wenzelm@23164
  1389
apply (subgoal_tac "k ^ m = k ^ ((m - 1) + 1)")
wenzelm@23164
  1390
 apply (erule ssubst)
wenzelm@23164
  1391
 apply (simp only: power_add)
wenzelm@23164
  1392
 apply simp_all
wenzelm@23164
  1393
done
wenzelm@23164
  1394
wenzelm@23164
  1395
text {* code serializer setup *}
wenzelm@23164
  1396
wenzelm@23164
  1397
code_modulename SML
wenzelm@23164
  1398
  IntDiv Integer
wenzelm@23164
  1399
wenzelm@23164
  1400
code_modulename OCaml
wenzelm@23164
  1401
  IntDiv Integer
wenzelm@23164
  1402
wenzelm@23164
  1403
code_modulename Haskell
wenzelm@23164
  1404
  IntDiv Divides
wenzelm@23164
  1405
wenzelm@23164
  1406
end