src/HOL/Integ/IntDiv.thy
author paulson
Tue Jun 28 15:27:45 2005 +0200 (2005-06-28)
changeset 16587 b34c8aa657a5
parent 16417 9bc16273c2d4
child 16733 236dfafbeb63
permissions -rw-r--r--
Constant "If" is now local
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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 Recdef
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uses ("IntDiv_setup.ML")
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begin
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declare zless_nat_conj [simp]
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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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    "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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    "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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consts posDivAlg :: "int*int => int*int"
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recdef posDivAlg "measure (%(a,b). nat(a - b + 1))"
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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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text{*algorithm for the case @{text "a<0, b>0"}*}
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consts negDivAlg :: "int*int => int*int"
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recdef negDivAlg "measure (%(a,b). nat(- a - b))"
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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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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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    "negateSnd == %(q,r). (q,-r)"
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  divAlg :: "int*int => int*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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    "divAlg ==
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       %(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
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  int :: "Divides.div" ..       --{*avoid clash with 'div' token*}
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text{*The operators are defined with reference to the algorithm, which is
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proved to satisfy the specification.*}
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defs
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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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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';  0 < 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 < 0;  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 [rule_format]:
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     "0 \<le> a --> 0 < b --> quorem ((a, b), posDivAlg (a, b))"
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apply (induct_tac a b rule: posDivAlg.induct, auto)
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 apply (simp_all add: quorem_def)
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 (*base case: a<b*)
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 apply (simp add: posDivAlg_eqn)
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(*main argument*)
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apply (subst posDivAlg_eqn, 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 [rule_format]:
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     "a < 0 --> 0 < b --> quorem ((a, b), negDivAlg (a, b))"
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apply (induct_tac a b rule: negDivAlg.induct, auto)
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 apply (simp_all add: quorem_def)
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 (*base case: 0\<le>a+b*)
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 apply (simp add: negDivAlg_eqn)
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(*main argument*)
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apply (subst negDivAlg_eqn, assumption)
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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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use "IntDiv_setup.ML"
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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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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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apply (rule quorem_div)
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apply (auto simp add: quorem_def)
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done
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(*There is no div_neg_pos_trivial because  0 div b = 0 would supersede it*)
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lemma mod_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a mod b = a"
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apply (rule_tac q = 0 in quorem_mod)
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apply (auto simp add: quorem_def)
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done
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lemma mod_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a mod b = a"
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apply (rule_tac q = 0 in quorem_mod)
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apply (auto simp add: quorem_def)
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done
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lemma mod_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a mod b = a+b"
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apply (rule_tac q = "-1" in quorem_mod)
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apply (auto simp add: quorem_def)
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done
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text{*There is no @{text mod_neg_pos_trivial}.*}
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(*Simpler laws such as -a div b = -(a div b) FAIL, but see just below*)
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lemma zdiv_zminus_zminus [simp]: "(-a) div (-b) = a div (b::int)"
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apply (case_tac "b = 0", simp)
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   317
apply (simp add: quorem_div_mod [THEN quorem_neg, simplified, 
paulson@13183
   318
                                 THEN quorem_div, THEN sym])
paulson@13183
   319
paulson@13183
   320
done
paulson@13183
   321
paulson@13183
   322
(*Simpler laws such as -a mod b = -(a mod b) FAIL, but see just below*)
paulson@13183
   323
lemma zmod_zminus_zminus [simp]: "(-a) mod (-b) = - (a mod (b::int))"
paulson@15013
   324
apply (case_tac "b = 0", simp)
paulson@13183
   325
apply (subst quorem_div_mod [THEN quorem_neg, simplified, THEN quorem_mod],
paulson@13183
   326
       auto)
paulson@13183
   327
done
paulson@13183
   328
paulson@15221
   329
paulson@15221
   330
subsection{*Laws for div and mod with Unary Minus*}
paulson@13183
   331
paulson@13183
   332
lemma zminus1_lemma:
paulson@13183
   333
     "quorem((a,b),(q,r))  
paulson@13183
   334
      ==> quorem ((-a,b), (if r=0 then -q else -q - 1),  
paulson@13183
   335
                          (if r=0 then 0 else b-r))"
paulson@14479
   336
by (force simp add: split_ifs quorem_def linorder_neq_iff right_diff_distrib)
paulson@13183
   337
paulson@13183
   338
paulson@13183
   339
lemma zdiv_zminus1_eq_if:
paulson@15221
   340
     "b \<noteq> (0::int)  
paulson@13183
   341
      ==> (-a) div b =  
paulson@13183
   342
          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
paulson@13183
   343
by (blast intro: quorem_div_mod [THEN zminus1_lemma, THEN quorem_div])
paulson@13183
   344
paulson@13183
   345
lemma zmod_zminus1_eq_if:
paulson@13183
   346
     "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))"
paulson@15013
   347
apply (case_tac "b = 0", simp)
paulson@13183
   348
apply (blast intro: quorem_div_mod [THEN zminus1_lemma, THEN quorem_mod])
paulson@13183
   349
done
paulson@13183
   350
paulson@13183
   351
lemma zdiv_zminus2: "a div (-b) = (-a::int) div b"
paulson@13183
   352
by (cut_tac a = "-a" in zdiv_zminus_zminus, auto)
paulson@13183
   353
paulson@13183
   354
lemma zmod_zminus2: "a mod (-b) = - ((-a::int) mod b)"
paulson@13183
   355
by (cut_tac a = "-a" and b = b in zmod_zminus_zminus, auto)
paulson@13183
   356
paulson@13183
   357
lemma zdiv_zminus2_eq_if:
paulson@15221
   358
     "b \<noteq> (0::int)  
paulson@13183
   359
      ==> a div (-b) =  
paulson@13183
   360
          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
paulson@13183
   361
by (simp add: zdiv_zminus1_eq_if zdiv_zminus2)
paulson@13183
   362
paulson@13183
   363
lemma zmod_zminus2_eq_if:
paulson@13183
   364
     "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"
paulson@13183
   365
by (simp add: zmod_zminus1_eq_if zmod_zminus2)
paulson@13183
   366
paulson@13183
   367
paulson@14271
   368
subsection{*Division of a Number by Itself*}
paulson@13183
   369
paulson@14288
   370
lemma self_quotient_aux1: "[| (0::int) < a; a = r + a*q; r < a |] ==> 1 \<le> q"
paulson@13183
   371
apply (subgoal_tac "0 < a*q")
paulson@14353
   372
 apply (simp add: zero_less_mult_iff, arith)
paulson@13183
   373
done
paulson@13183
   374
paulson@14288
   375
lemma self_quotient_aux2: "[| (0::int) < a; a = r + a*q; 0 \<le> r |] ==> q \<le> 1"
paulson@14288
   376
apply (subgoal_tac "0 \<le> a* (1-q) ")
paulson@14353
   377
 apply (simp add: zero_le_mult_iff)
paulson@14479
   378
apply (simp add: right_diff_distrib)
paulson@13183
   379
done
paulson@13183
   380
paulson@15221
   381
lemma self_quotient: "[| quorem((a,a),(q,r));  a \<noteq> (0::int) |] ==> q = 1"
paulson@13183
   382
apply (simp add: split_ifs quorem_def linorder_neq_iff)
paulson@15221
   383
apply (rule order_antisym, safe, simp_all)
wenzelm@13524
   384
apply (rule_tac [3] a = "-a" and r = "-r" in self_quotient_aux1)
wenzelm@13524
   385
apply (rule_tac a = "-a" and r = "-r" in self_quotient_aux2)
paulson@15221
   386
apply (force intro: self_quotient_aux1 self_quotient_aux2 simp add: add_commute)+
paulson@13183
   387
done
paulson@13183
   388
paulson@15221
   389
lemma self_remainder: "[| quorem((a,a),(q,r));  a \<noteq> (0::int) |] ==> r = 0"
paulson@13183
   390
apply (frule self_quotient, assumption)
paulson@13183
   391
apply (simp add: quorem_def)
paulson@13183
   392
done
paulson@13183
   393
paulson@15221
   394
lemma zdiv_self [simp]: "a \<noteq> 0 ==> a div a = (1::int)"
paulson@13183
   395
by (simp add: quorem_div_mod [THEN self_quotient])
paulson@13183
   396
paulson@13183
   397
(*Here we have 0 mod 0 = 0, also assumed by Knuth (who puts m mod 0 = 0) *)
paulson@13183
   398
lemma zmod_self [simp]: "a mod a = (0::int)"
paulson@15013
   399
apply (case_tac "a = 0", simp)
paulson@13183
   400
apply (simp add: quorem_div_mod [THEN self_remainder])
paulson@13183
   401
done
paulson@13183
   402
paulson@13183
   403
paulson@14271
   404
subsection{*Computation of Division and Remainder*}
paulson@13183
   405
paulson@13183
   406
lemma zdiv_zero [simp]: "(0::int) div b = 0"
paulson@13183
   407
by (simp add: div_def divAlg_def)
paulson@13183
   408
paulson@13183
   409
lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"
paulson@13183
   410
by (simp add: div_def divAlg_def)
paulson@13183
   411
paulson@13183
   412
lemma zmod_zero [simp]: "(0::int) mod b = 0"
paulson@13183
   413
by (simp add: mod_def divAlg_def)
paulson@13183
   414
paulson@13183
   415
lemma zdiv_minus1: "(0::int) < b ==> -1 div b = -1"
paulson@13183
   416
by (simp add: div_def divAlg_def)
paulson@13183
   417
paulson@13183
   418
lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"
paulson@13183
   419
by (simp add: mod_def divAlg_def)
paulson@13183
   420
paulson@15221
   421
text{*a positive, b positive *}
paulson@13183
   422
paulson@14288
   423
lemma div_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a div b = fst (posDivAlg(a,b))"
paulson@13183
   424
by (simp add: div_def divAlg_def)
paulson@13183
   425
paulson@14288
   426
lemma mod_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a mod b = snd (posDivAlg(a,b))"
paulson@13183
   427
by (simp add: mod_def divAlg_def)
paulson@13183
   428
paulson@15221
   429
text{*a negative, b positive *}
paulson@13183
   430
paulson@13183
   431
lemma div_neg_pos: "[| a < 0;  0 < b |] ==> a div b = fst (negDivAlg(a,b))"
paulson@13183
   432
by (simp add: div_def divAlg_def)
paulson@13183
   433
paulson@13183
   434
lemma mod_neg_pos: "[| a < 0;  0 < b |] ==> a mod b = snd (negDivAlg(a,b))"
paulson@13183
   435
by (simp add: mod_def divAlg_def)
paulson@13183
   436
paulson@15221
   437
text{*a positive, b negative *}
paulson@13183
   438
paulson@13183
   439
lemma div_pos_neg:
paulson@13183
   440
     "[| 0 < a;  b < 0 |] ==> a div b = fst (negateSnd(negDivAlg(-a,-b)))"
paulson@13183
   441
by (simp add: div_def divAlg_def)
paulson@13183
   442
paulson@13183
   443
lemma mod_pos_neg:
paulson@13183
   444
     "[| 0 < a;  b < 0 |] ==> a mod b = snd (negateSnd(negDivAlg(-a,-b)))"
paulson@13183
   445
by (simp add: mod_def divAlg_def)
paulson@13183
   446
paulson@15221
   447
text{*a negative, b negative *}
paulson@13183
   448
paulson@13183
   449
lemma div_neg_neg:
paulson@14288
   450
     "[| a < 0;  b \<le> 0 |] ==> a div b = fst (negateSnd(posDivAlg(-a,-b)))"
paulson@13183
   451
by (simp add: div_def divAlg_def)
paulson@13183
   452
paulson@13183
   453
lemma mod_neg_neg:
paulson@14288
   454
     "[| a < 0;  b \<le> 0 |] ==> a mod b = snd (negateSnd(posDivAlg(-a,-b)))"
paulson@13183
   455
by (simp add: mod_def divAlg_def)
paulson@13183
   456
paulson@13183
   457
text {*Simplify expresions in which div and mod combine numerical constants*}
paulson@13183
   458
paulson@13183
   459
declare div_pos_pos [of "number_of v" "number_of w", standard, simp]
paulson@13183
   460
declare div_neg_pos [of "number_of v" "number_of w", standard, simp]
paulson@13183
   461
declare div_pos_neg [of "number_of v" "number_of w", standard, simp]
paulson@13183
   462
declare div_neg_neg [of "number_of v" "number_of w", standard, simp]
paulson@13183
   463
paulson@13183
   464
declare mod_pos_pos [of "number_of v" "number_of w", standard, simp]
paulson@13183
   465
declare mod_neg_pos [of "number_of v" "number_of w", standard, simp]
paulson@13183
   466
declare mod_pos_neg [of "number_of v" "number_of w", standard, simp]
paulson@13183
   467
declare mod_neg_neg [of "number_of v" "number_of w", standard, simp]
paulson@13183
   468
paulson@13183
   469
declare posDivAlg_eqn [of "number_of v" "number_of w", standard, simp]
paulson@13183
   470
declare negDivAlg_eqn [of "number_of v" "number_of w", standard, simp]
paulson@13183
   471
paulson@13183
   472
paulson@15221
   473
text{*Special-case simplification *}
paulson@13183
   474
paulson@13183
   475
lemma zmod_1 [simp]: "a mod (1::int) = 0"
paulson@13183
   476
apply (cut_tac a = a and b = 1 in pos_mod_sign)
nipkow@13788
   477
apply (cut_tac [2] a = a and b = 1 in pos_mod_bound)
nipkow@13788
   478
apply (auto simp del:pos_mod_bound pos_mod_sign)
nipkow@13788
   479
done
paulson@13183
   480
paulson@13183
   481
lemma zdiv_1 [simp]: "a div (1::int) = a"
paulson@13183
   482
by (cut_tac a = a and b = 1 in zmod_zdiv_equality, auto)
paulson@13183
   483
paulson@13183
   484
lemma zmod_minus1_right [simp]: "a mod (-1::int) = 0"
paulson@13183
   485
apply (cut_tac a = a and b = "-1" in neg_mod_sign)
nipkow@13788
   486
apply (cut_tac [2] a = a and b = "-1" in neg_mod_bound)
nipkow@13788
   487
apply (auto simp del: neg_mod_sign neg_mod_bound)
paulson@13183
   488
done
paulson@13183
   489
paulson@13183
   490
lemma zdiv_minus1_right [simp]: "a div (-1::int) = -a"
paulson@13183
   491
by (cut_tac a = a and b = "-1" in zmod_zdiv_equality, auto)
paulson@13183
   492
paulson@13183
   493
(** The last remaining special cases for constant arithmetic:
paulson@13183
   494
    1 div z and 1 mod z **)
paulson@13183
   495
paulson@13183
   496
declare div_pos_pos [OF int_0_less_1, of "number_of w", standard, simp]
paulson@13183
   497
declare div_pos_neg [OF int_0_less_1, of "number_of w", standard, simp]
paulson@13183
   498
declare mod_pos_pos [OF int_0_less_1, of "number_of w", standard, simp]
paulson@13183
   499
declare mod_pos_neg [OF int_0_less_1, of "number_of w", standard, simp]
paulson@13183
   500
paulson@13183
   501
declare posDivAlg_eqn [of concl: 1 "number_of w", standard, simp]
paulson@13183
   502
declare negDivAlg_eqn [of concl: 1 "number_of w", standard, simp]
paulson@13183
   503
paulson@13183
   504
paulson@14271
   505
subsection{*Monotonicity in the First Argument (Dividend)*}
paulson@13183
   506
paulson@14288
   507
lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div b"
paulson@13183
   508
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
paulson@13183
   509
apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
paulson@13183
   510
apply (rule unique_quotient_lemma)
paulson@13183
   511
apply (erule subst)
paulson@15221
   512
apply (erule subst, simp_all)
paulson@13183
   513
done
paulson@13183
   514
paulson@14288
   515
lemma zdiv_mono1_neg: "[| a \<le> a';  (b::int) < 0 |] ==> a' div b \<le> a div b"
paulson@13183
   516
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
paulson@13183
   517
apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
paulson@13183
   518
apply (rule unique_quotient_lemma_neg)
paulson@13183
   519
apply (erule subst)
paulson@15221
   520
apply (erule subst, simp_all)
paulson@13183
   521
done
paulson@6917
   522
paulson@6917
   523
paulson@14271
   524
subsection{*Monotonicity in the Second Argument (Divisor)*}
paulson@13183
   525
paulson@13183
   526
lemma q_pos_lemma:
paulson@14288
   527
     "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"
paulson@13183
   528
apply (subgoal_tac "0 < b'* (q' + 1) ")
paulson@14353
   529
 apply (simp add: zero_less_mult_iff)
paulson@14479
   530
apply (simp add: right_distrib)
paulson@13183
   531
done
paulson@13183
   532
paulson@13183
   533
lemma zdiv_mono2_lemma:
paulson@14288
   534
     "[| b*q + r = b'*q' + r';  0 \<le> b'*q' + r';   
paulson@14288
   535
         r' < b';  0 \<le> r;  0 < b';  b' \<le> b |]   
paulson@14288
   536
      ==> q \<le> (q'::int)"
paulson@13183
   537
apply (frule q_pos_lemma, assumption+) 
paulson@13183
   538
apply (subgoal_tac "b*q < b* (q' + 1) ")
paulson@14387
   539
 apply (simp add: mult_less_cancel_left)
paulson@13183
   540
apply (subgoal_tac "b*q = r' - r + b'*q'")
paulson@13183
   541
 prefer 2 apply simp
paulson@14479
   542
apply (simp (no_asm_simp) add: right_distrib)
paulson@15221
   543
apply (subst add_commute, rule zadd_zless_mono, arith)
paulson@14378
   544
apply (rule mult_right_mono, auto)
paulson@13183
   545
done
paulson@13183
   546
paulson@13183
   547
lemma zdiv_mono2:
paulson@14288
   548
     "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"
paulson@15221
   549
apply (subgoal_tac "b \<noteq> 0")
paulson@13183
   550
 prefer 2 apply arith
paulson@13183
   551
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
paulson@13183
   552
apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
paulson@13183
   553
apply (rule zdiv_mono2_lemma)
paulson@13183
   554
apply (erule subst)
paulson@15221
   555
apply (erule subst, simp_all)
paulson@13183
   556
done
paulson@13183
   557
paulson@13183
   558
lemma q_neg_lemma:
paulson@14288
   559
     "[| b'*q' + r' < 0;  0 \<le> r';  0 < b' |] ==> q' \<le> (0::int)"
paulson@13183
   560
apply (subgoal_tac "b'*q' < 0")
paulson@14353
   561
 apply (simp add: mult_less_0_iff, arith)
paulson@13183
   562
done
paulson@13183
   563
paulson@13183
   564
lemma zdiv_mono2_neg_lemma:
paulson@13183
   565
     "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;   
paulson@14288
   566
         r < b;  0 \<le> r';  0 < b';  b' \<le> b |]   
paulson@14288
   567
      ==> q' \<le> (q::int)"
paulson@13183
   568
apply (frule q_neg_lemma, assumption+) 
paulson@13183
   569
apply (subgoal_tac "b*q' < b* (q + 1) ")
paulson@14387
   570
 apply (simp add: mult_less_cancel_left)
paulson@14479
   571
apply (simp add: right_distrib)
paulson@14288
   572
apply (subgoal_tac "b*q' \<le> b'*q'")
paulson@15221
   573
 prefer 2 apply (simp add: mult_right_mono_neg, arith)
paulson@13183
   574
done
paulson@13183
   575
paulson@13183
   576
lemma zdiv_mono2_neg:
paulson@14288
   577
     "[| a < (0::int);  0 < b';  b' \<le> b |] ==> a div b' \<le> a div b"
paulson@13183
   578
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
paulson@13183
   579
apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
paulson@13183
   580
apply (rule zdiv_mono2_neg_lemma)
paulson@13183
   581
apply (erule subst)
paulson@15221
   582
apply (erule subst, simp_all)
paulson@13183
   583
done
paulson@13183
   584
paulson@14271
   585
subsection{*More Algebraic Laws for div and mod*}
paulson@13183
   586
paulson@15221
   587
text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}
paulson@13183
   588
paulson@13183
   589
lemma zmult1_lemma:
paulson@15221
   590
     "[| quorem((b,c),(q,r));  c \<noteq> 0 |]  
paulson@13183
   591
      ==> quorem ((a*b, c), (a*q + a*r div c, a*r mod c))"
paulson@14479
   592
by (force simp add: split_ifs quorem_def linorder_neq_iff right_distrib)
paulson@13183
   593
paulson@13183
   594
lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"
paulson@15013
   595
apply (case_tac "c = 0", simp)
paulson@13183
   596
apply (blast intro: quorem_div_mod [THEN zmult1_lemma, THEN quorem_div])
paulson@13183
   597
done
paulson@13183
   598
paulson@13183
   599
lemma zmod_zmult1_eq: "(a*b) mod c = a*(b mod c) mod (c::int)"
paulson@15013
   600
apply (case_tac "c = 0", simp)
paulson@13183
   601
apply (blast intro: quorem_div_mod [THEN zmult1_lemma, THEN quorem_mod])
paulson@13183
   602
done
paulson@13183
   603
paulson@13183
   604
lemma zmod_zmult1_eq': "(a*b) mod (c::int) = ((a mod c) * b) mod c"
paulson@13183
   605
apply (rule trans)
paulson@13183
   606
apply (rule_tac s = "b*a mod c" in trans)
paulson@13183
   607
apply (rule_tac [2] zmod_zmult1_eq)
paulson@15234
   608
apply (simp_all add: mult_commute)
paulson@13183
   609
done
paulson@13183
   610
paulson@13183
   611
lemma zmod_zmult_distrib: "(a*b) mod (c::int) = ((a mod c) * (b mod c)) mod c"
paulson@13183
   612
apply (rule zmod_zmult1_eq' [THEN trans])
paulson@13183
   613
apply (rule zmod_zmult1_eq)
paulson@13183
   614
done
paulson@13183
   615
paulson@15221
   616
lemma zdiv_zmult_self1 [simp]: "b \<noteq> (0::int) ==> (a*b) div b = a"
paulson@13183
   617
by (simp add: zdiv_zmult1_eq)
paulson@13183
   618
paulson@15221
   619
lemma zdiv_zmult_self2 [simp]: "b \<noteq> (0::int) ==> (b*a) div b = a"
paulson@15234
   620
by (subst mult_commute, erule zdiv_zmult_self1)
paulson@13183
   621
paulson@13183
   622
lemma zmod_zmult_self1 [simp]: "(a*b) mod b = (0::int)"
paulson@13183
   623
by (simp add: zmod_zmult1_eq)
paulson@13183
   624
paulson@13183
   625
lemma zmod_zmult_self2 [simp]: "(b*a) mod b = (0::int)"
paulson@15234
   626
by (simp add: mult_commute zmod_zmult1_eq)
paulson@13183
   627
paulson@13183
   628
lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"
nipkow@13517
   629
proof
nipkow@13517
   630
  assume "m mod d = 0"
paulson@14473
   631
  with zmod_zdiv_equality[of m d] show "EX q::int. m = d*q" by auto
nipkow@13517
   632
next
nipkow@13517
   633
  assume "EX q::int. m = d*q"
nipkow@13517
   634
  thus "m mod d = 0" by auto
nipkow@13517
   635
qed
paulson@13183
   636
paulson@13183
   637
declare zmod_eq_0_iff [THEN iffD1, dest!]
paulson@13183
   638
paulson@15221
   639
text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}
paulson@13183
   640
paulson@13183
   641
lemma zadd1_lemma:
paulson@15221
   642
     "[| quorem((a,c),(aq,ar));  quorem((b,c),(bq,br));  c \<noteq> 0 |]  
paulson@13183
   643
      ==> quorem ((a+b, c), (aq + bq + (ar+br) div c, (ar+br) mod c))"
paulson@14479
   644
by (force simp add: split_ifs quorem_def linorder_neq_iff right_distrib)
paulson@13183
   645
paulson@13183
   646
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
paulson@13183
   647
lemma zdiv_zadd1_eq:
paulson@13183
   648
     "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"
paulson@15013
   649
apply (case_tac "c = 0", simp)
paulson@13183
   650
apply (blast intro: zadd1_lemma [OF quorem_div_mod quorem_div_mod] quorem_div)
paulson@13183
   651
done
paulson@13183
   652
paulson@13183
   653
lemma zmod_zadd1_eq: "(a+b) mod (c::int) = (a mod c + b mod c) mod c"
paulson@15013
   654
apply (case_tac "c = 0", simp)
paulson@13183
   655
apply (blast intro: zadd1_lemma [OF quorem_div_mod quorem_div_mod] quorem_mod)
paulson@13183
   656
done
paulson@13183
   657
paulson@13183
   658
lemma mod_div_trivial [simp]: "(a mod b) div b = (0::int)"
paulson@15013
   659
apply (case_tac "b = 0", simp)
nipkow@13788
   660
apply (auto simp add: linorder_neq_iff div_pos_pos_trivial div_neg_neg_trivial)
paulson@13183
   661
done
paulson@13183
   662
paulson@13183
   663
lemma mod_mod_trivial [simp]: "(a mod b) mod b = a mod (b::int)"
paulson@15013
   664
apply (case_tac "b = 0", simp)
nipkow@13788
   665
apply (force simp add: linorder_neq_iff mod_pos_pos_trivial mod_neg_neg_trivial)
paulson@13183
   666
done
paulson@13183
   667
paulson@13183
   668
lemma zmod_zadd_left_eq: "(a+b) mod (c::int) = ((a mod c) + b) mod c"
paulson@13183
   669
apply (rule trans [symmetric])
paulson@13183
   670
apply (rule zmod_zadd1_eq, simp)
paulson@13183
   671
apply (rule zmod_zadd1_eq [symmetric])
paulson@13183
   672
done
paulson@13183
   673
paulson@13183
   674
lemma zmod_zadd_right_eq: "(a+b) mod (c::int) = (a + (b mod c)) mod c"
paulson@13183
   675
apply (rule trans [symmetric])
paulson@13183
   676
apply (rule zmod_zadd1_eq, simp)
paulson@13183
   677
apply (rule zmod_zadd1_eq [symmetric])
paulson@13183
   678
done
paulson@13183
   679
paulson@15221
   680
lemma zdiv_zadd_self1[simp]: "a \<noteq> (0::int) ==> (a+b) div a = b div a + 1"
paulson@13183
   681
by (simp add: zdiv_zadd1_eq)
paulson@13183
   682
paulson@15221
   683
lemma zdiv_zadd_self2[simp]: "a \<noteq> (0::int) ==> (b+a) div a = b div a + 1"
paulson@13183
   684
by (simp add: zdiv_zadd1_eq)
paulson@13183
   685
paulson@13183
   686
lemma zmod_zadd_self1[simp]: "(a+b) mod a = b mod (a::int)"
paulson@15013
   687
apply (case_tac "a = 0", simp)
paulson@13183
   688
apply (simp add: zmod_zadd1_eq)
paulson@13183
   689
done
paulson@13183
   690
paulson@13183
   691
lemma zmod_zadd_self2[simp]: "(b+a) mod a = b mod (a::int)"
paulson@15013
   692
apply (case_tac "a = 0", simp)
paulson@13183
   693
apply (simp add: zmod_zadd1_eq)
paulson@13183
   694
done
paulson@13183
   695
paulson@13183
   696
paulson@14271
   697
subsection{*Proving  @{term "a div (b*c) = (a div b) div c"} *}
paulson@13183
   698
paulson@13183
   699
(*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but
paulson@13183
   700
  7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems
paulson@13183
   701
  to cause particular problems.*)
paulson@13183
   702
paulson@15221
   703
text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}
paulson@13183
   704
paulson@14288
   705
lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b*c < b*(q mod c) + r"
paulson@13183
   706
apply (subgoal_tac "b * (c - q mod c) < r * 1")
paulson@14479
   707
apply (simp add: right_diff_distrib)
paulson@13183
   708
apply (rule order_le_less_trans)
paulson@14378
   709
apply (erule_tac [2] mult_strict_right_mono)
paulson@14378
   710
apply (rule mult_left_mono_neg)
paulson@15221
   711
apply (auto simp add: compare_rls add_commute [of 1]
paulson@13183
   712
                      add1_zle_eq pos_mod_bound)
paulson@13183
   713
done
paulson@13183
   714
paulson@15221
   715
lemma zmult2_lemma_aux2:
paulson@15221
   716
     "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"
paulson@14288
   717
apply (subgoal_tac "b * (q mod c) \<le> 0")
paulson@13183
   718
 apply arith
paulson@14353
   719
apply (simp add: mult_le_0_iff)
paulson@13183
   720
done
paulson@13183
   721
paulson@14288
   722
lemma zmult2_lemma_aux3: "[| (0::int) < c;  0 \<le> r;  r < b |] ==> 0 \<le> b * (q mod c) + r"
paulson@14288
   723
apply (subgoal_tac "0 \<le> b * (q mod c) ")
paulson@13183
   724
apply arith
paulson@14353
   725
apply (simp add: zero_le_mult_iff)
paulson@13183
   726
done
paulson@13183
   727
paulson@14288
   728
lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"
paulson@13183
   729
apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
paulson@14479
   730
apply (simp add: right_diff_distrib)
paulson@13183
   731
apply (rule order_less_le_trans)
paulson@14378
   732
apply (erule mult_strict_right_mono)
paulson@14387
   733
apply (rule_tac [2] mult_left_mono)
paulson@15221
   734
apply (auto simp add: compare_rls add_commute [of 1]
paulson@13183
   735
                      add1_zle_eq pos_mod_bound)
paulson@13183
   736
done
paulson@13183
   737
paulson@15221
   738
lemma zmult2_lemma: "[| quorem ((a,b), (q,r));  b \<noteq> 0;  0 < c |]  
paulson@13183
   739
      ==> quorem ((a, b*c), (q div c, b*(q mod c) + r))"
paulson@14271
   740
by (auto simp add: mult_ac quorem_def linorder_neq_iff
paulson@14479
   741
                   zero_less_mult_iff right_distrib [symmetric] 
wenzelm@13524
   742
                   zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4)
paulson@13183
   743
paulson@13183
   744
lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c"
paulson@15013
   745
apply (case_tac "b = 0", simp)
paulson@13183
   746
apply (force simp add: quorem_div_mod [THEN zmult2_lemma, THEN quorem_div])
paulson@13183
   747
done
paulson@13183
   748
paulson@13183
   749
lemma zmod_zmult2_eq:
paulson@13183
   750
     "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"
paulson@15013
   751
apply (case_tac "b = 0", simp)
paulson@13183
   752
apply (force simp add: quorem_div_mod [THEN zmult2_lemma, THEN quorem_mod])
paulson@13183
   753
done
paulson@13183
   754
paulson@13183
   755
paulson@14271
   756
subsection{*Cancellation of Common Factors in div*}
paulson@13183
   757
paulson@15221
   758
lemma zdiv_zmult_zmult1_aux1:
paulson@15221
   759
     "[| (0::int) < b;  c \<noteq> 0 |] ==> (c*a) div (c*b) = a div b"
paulson@13183
   760
by (subst zdiv_zmult2_eq, auto)
paulson@13183
   761
paulson@15221
   762
lemma zdiv_zmult_zmult1_aux2:
paulson@15221
   763
     "[| b < (0::int);  c \<noteq> 0 |] ==> (c*a) div (c*b) = a div b"
paulson@13183
   764
apply (subgoal_tac " (c * (-a)) div (c * (-b)) = (-a) div (-b) ")
wenzelm@13524
   765
apply (rule_tac [2] zdiv_zmult_zmult1_aux1, auto)
paulson@13183
   766
done
paulson@13183
   767
paulson@15221
   768
lemma zdiv_zmult_zmult1: "c \<noteq> (0::int) ==> (c*a) div (c*b) = a div b"
paulson@15013
   769
apply (case_tac "b = 0", simp)
wenzelm@13524
   770
apply (auto simp add: linorder_neq_iff zdiv_zmult_zmult1_aux1 zdiv_zmult_zmult1_aux2)
paulson@13183
   771
done
paulson@13183
   772
paulson@15221
   773
lemma zdiv_zmult_zmult2: "c \<noteq> (0::int) ==> (a*c) div (b*c) = a div b"
paulson@13183
   774
apply (drule zdiv_zmult_zmult1)
paulson@15234
   775
apply (auto simp add: mult_commute)
paulson@13183
   776
done
paulson@13183
   777
paulson@13183
   778
paulson@13183
   779
paulson@14271
   780
subsection{*Distribution of Factors over mod*}
paulson@13183
   781
paulson@15221
   782
lemma zmod_zmult_zmult1_aux1:
paulson@15221
   783
     "[| (0::int) < b;  c \<noteq> 0 |] ==> (c*a) mod (c*b) = c * (a mod b)"
paulson@13183
   784
by (subst zmod_zmult2_eq, auto)
paulson@13183
   785
paulson@15221
   786
lemma zmod_zmult_zmult1_aux2:
paulson@15221
   787
     "[| b < (0::int);  c \<noteq> 0 |] ==> (c*a) mod (c*b) = c * (a mod b)"
paulson@13183
   788
apply (subgoal_tac " (c * (-a)) mod (c * (-b)) = c * ((-a) mod (-b))")
wenzelm@13524
   789
apply (rule_tac [2] zmod_zmult_zmult1_aux1, auto)
paulson@13183
   790
done
paulson@13183
   791
paulson@13183
   792
lemma zmod_zmult_zmult1: "(c*a) mod (c*b) = (c::int) * (a mod b)"
paulson@15013
   793
apply (case_tac "b = 0", simp)
paulson@15013
   794
apply (case_tac "c = 0", simp)
wenzelm@13524
   795
apply (auto simp add: linorder_neq_iff zmod_zmult_zmult1_aux1 zmod_zmult_zmult1_aux2)
paulson@13183
   796
done
paulson@13183
   797
paulson@13183
   798
lemma zmod_zmult_zmult2: "(a*c) mod (b*c) = (a mod b) * (c::int)"
paulson@13183
   799
apply (cut_tac c = c in zmod_zmult_zmult1)
paulson@15234
   800
apply (auto simp add: mult_commute)
paulson@13183
   801
done
paulson@13183
   802
paulson@13183
   803
paulson@14271
   804
subsection {*Splitting Rules for div and mod*}
paulson@13260
   805
paulson@13260
   806
text{*The proofs of the two lemmas below are essentially identical*}
paulson@13260
   807
paulson@13260
   808
lemma split_pos_lemma:
paulson@13260
   809
 "0<k ==> 
paulson@14288
   810
    P(n div k :: int)(n mod k) = (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i j)"
paulson@15221
   811
apply (rule iffI, clarify)
paulson@13260
   812
 apply (erule_tac P="P ?x ?y" in rev_mp)  
paulson@13260
   813
 apply (subst zmod_zadd1_eq) 
paulson@13260
   814
 apply (subst zdiv_zadd1_eq) 
paulson@13260
   815
 apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)  
paulson@13260
   816
txt{*converse direction*}
paulson@13260
   817
apply (drule_tac x = "n div k" in spec) 
paulson@15221
   818
apply (drule_tac x = "n mod k" in spec, simp)
paulson@13260
   819
done
paulson@13260
   820
paulson@13260
   821
lemma split_neg_lemma:
paulson@13260
   822
 "k<0 ==>
paulson@14288
   823
    P(n div k :: int)(n mod k) = (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i j)"
paulson@15221
   824
apply (rule iffI, clarify)
paulson@13260
   825
 apply (erule_tac P="P ?x ?y" in rev_mp)  
paulson@13260
   826
 apply (subst zmod_zadd1_eq) 
paulson@13260
   827
 apply (subst zdiv_zadd1_eq) 
paulson@13260
   828
 apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)  
paulson@13260
   829
txt{*converse direction*}
paulson@13260
   830
apply (drule_tac x = "n div k" in spec) 
paulson@15221
   831
apply (drule_tac x = "n mod k" in spec, simp)
paulson@13260
   832
done
paulson@13260
   833
paulson@13260
   834
lemma split_zdiv:
paulson@13260
   835
 "P(n div k :: int) =
paulson@13260
   836
  ((k = 0 --> P 0) & 
paulson@14288
   837
   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) & 
paulson@14288
   838
   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"
paulson@15221
   839
apply (case_tac "k=0", simp)
paulson@13260
   840
apply (simp only: linorder_neq_iff)
paulson@13260
   841
apply (erule disjE) 
paulson@13260
   842
 apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"] 
paulson@13260
   843
                      split_neg_lemma [of concl: "%x y. P x"])
paulson@13260
   844
done
paulson@13260
   845
paulson@13260
   846
lemma split_zmod:
paulson@13260
   847
 "P(n mod k :: int) =
paulson@13260
   848
  ((k = 0 --> P n) & 
paulson@14288
   849
   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) & 
paulson@14288
   850
   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"
paulson@15221
   851
apply (case_tac "k=0", simp)
paulson@13260
   852
apply (simp only: linorder_neq_iff)
paulson@13260
   853
apply (erule disjE) 
paulson@13260
   854
 apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"] 
paulson@13260
   855
                      split_neg_lemma [of concl: "%x y. P y"])
paulson@13260
   856
done
paulson@13260
   857
paulson@13260
   858
(* Enable arith to deal with div 2 and mod 2: *)
nipkow@13266
   859
declare split_zdiv [of _ _ "number_of k", simplified, standard, arith_split]
nipkow@13266
   860
declare split_zmod [of _ _ "number_of k", simplified, standard, arith_split]
paulson@13260
   861
paulson@13260
   862
paulson@14271
   863
subsection{*Speeding up the Division Algorithm with Shifting*}
paulson@13183
   864
paulson@15221
   865
text{*computing div by shifting *}
paulson@13183
   866
paulson@14288
   867
lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
paulson@14288
   868
proof cases
paulson@14288
   869
  assume "a=0"
paulson@14288
   870
    thus ?thesis by simp
paulson@14288
   871
next
paulson@14288
   872
  assume "a\<noteq>0" and le_a: "0\<le>a"   
paulson@14288
   873
  hence a_pos: "1 \<le> a" by arith
paulson@14288
   874
  hence one_less_a2: "1 < 2*a" by arith
paulson@14288
   875
  hence le_2a: "2 * (1 + b mod a) \<le> 2 * a"
paulson@15221
   876
    by (simp add: mult_le_cancel_left add_commute [of 1] add1_zle_eq)
paulson@14288
   877
  with a_pos have "0 \<le> b mod a" by simp
paulson@14288
   878
  hence le_addm: "0 \<le> 1 mod (2*a) + 2*(b mod a)"
paulson@14288
   879
    by (simp add: mod_pos_pos_trivial one_less_a2)
paulson@14288
   880
  with  le_2a
paulson@14288
   881
  have "(1 mod (2*a) + 2*(b mod a)) div (2*a) = 0"
paulson@14288
   882
    by (simp add: div_pos_pos_trivial le_addm mod_pos_pos_trivial one_less_a2
paulson@14288
   883
                  right_distrib) 
paulson@14288
   884
  thus ?thesis
paulson@14288
   885
    by (subst zdiv_zadd1_eq,
paulson@14288
   886
        simp add: zdiv_zmult_zmult1 zmod_zmult_zmult1 one_less_a2
paulson@14288
   887
                  div_pos_pos_trivial)
paulson@14288
   888
qed
paulson@13183
   889
paulson@14288
   890
lemma neg_zdiv_mult_2: "a \<le> (0::int) ==> (1 + 2*b) div (2*a) = (b+1) div a"
paulson@13183
   891
apply (subgoal_tac " (1 + 2* (-b - 1)) div (2 * (-a)) = (-b - 1) div (-a) ")
paulson@13183
   892
apply (rule_tac [2] pos_zdiv_mult_2)
paulson@14479
   893
apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib)
paulson@13183
   894
apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")
paulson@14479
   895
apply (simp only: zdiv_zminus_zminus diff_minus minus_add_distrib [symmetric],
paulson@13183
   896
       simp) 
paulson@13183
   897
done
paulson@13183
   898
paulson@13183
   899
paulson@13183
   900
(*Not clear why this must be proved separately; probably number_of causes
paulson@13183
   901
  simplification problems*)
paulson@14288
   902
lemma not_0_le_lemma: "~ 0 \<le> x ==> x \<le> (0::int)"
paulson@13183
   903
by auto
paulson@13183
   904
paulson@13183
   905
lemma zdiv_number_of_BIT[simp]:
paulson@15620
   906
     "number_of (v BIT b) div number_of (w BIT bit.B0) =  
paulson@15620
   907
          (if b=bit.B0 | (0::int) \<le> number_of w                    
paulson@13183
   908
           then number_of v div (number_of w)     
paulson@13183
   909
           else (number_of v + (1::int)) div (number_of w))"
paulson@15013
   910
apply (simp only: number_of_eq Bin_simps UNIV_I split: split_if) 
paulson@15620
   911
apply (simp add: zdiv_zmult_zmult1 pos_zdiv_mult_2 neg_zdiv_mult_2 add_ac 
paulson@15620
   912
            split: bit.split)
paulson@13183
   913
done
paulson@13183
   914
paulson@13183
   915
paulson@15013
   916
subsection{*Computing mod by Shifting (proofs resemble those for div)*}
paulson@13183
   917
paulson@13183
   918
lemma pos_zmod_mult_2:
paulson@14288
   919
     "(0::int) \<le> a ==> (1 + 2*b) mod (2*a) = 1 + 2 * (b mod a)"
paulson@15013
   920
apply (case_tac "a = 0", simp)
paulson@13183
   921
apply (subgoal_tac "1 < a * 2")
paulson@13183
   922
 prefer 2 apply arith
paulson@14288
   923
apply (subgoal_tac "2* (1 + b mod a) \<le> 2*a")
paulson@14387
   924
 apply (rule_tac [2] mult_left_mono)
paulson@15234
   925
apply (auto simp add: add_commute [of 1] mult_commute add1_zle_eq 
paulson@13183
   926
                      pos_mod_bound)
paulson@13183
   927
apply (subst zmod_zadd1_eq)
paulson@13183
   928
apply (simp add: zmod_zmult_zmult2 mod_pos_pos_trivial)
paulson@13183
   929
apply (rule mod_pos_pos_trivial)
paulson@14288
   930
apply (auto simp add: mod_pos_pos_trivial left_distrib)
paulson@15221
   931
apply (subgoal_tac "0 \<le> b mod a", arith, simp)
paulson@13183
   932
done
paulson@13183
   933
paulson@13183
   934
lemma neg_zmod_mult_2:
paulson@14288
   935
     "a \<le> (0::int) ==> (1 + 2*b) mod (2*a) = 2 * ((b+1) mod a) - 1"
paulson@13183
   936
apply (subgoal_tac "(1 + 2* (-b - 1)) mod (2* (-a)) = 
paulson@13183
   937
                    1 + 2* ((-b - 1) mod (-a))")
paulson@13183
   938
apply (rule_tac [2] pos_zmod_mult_2)
paulson@14479
   939
apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib)
paulson@13183
   940
apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")
paulson@13183
   941
 prefer 2 apply simp 
paulson@14479
   942
apply (simp only: zmod_zminus_zminus diff_minus minus_add_distrib [symmetric])
paulson@13183
   943
done
paulson@13183
   944
paulson@13183
   945
lemma zmod_number_of_BIT [simp]:
paulson@15620
   946
     "number_of (v BIT b) mod number_of (w BIT bit.B0) =  
paulson@15620
   947
      (case b of
paulson@15620
   948
          bit.B0 => 2 * (number_of v mod number_of w)
paulson@15620
   949
        | bit.B1 => if (0::int) \<le> number_of w  
paulson@13183
   950
                then 2 * (number_of v mod number_of w) + 1     
paulson@15620
   951
                else 2 * ((number_of v + (1::int)) mod number_of w) - 1)"
paulson@15620
   952
apply (simp only: number_of_eq Bin_simps UNIV_I split: bit.split) 
paulson@15013
   953
apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2 
paulson@15013
   954
                 not_0_le_lemma neg_zmod_mult_2 add_ac)
paulson@13183
   955
done
paulson@13183
   956
paulson@13183
   957
paulson@15013
   958
subsection{*Quotients of Signs*}
paulson@13183
   959
paulson@13183
   960
lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"
paulson@14288
   961
apply (subgoal_tac "a div b \<le> -1", force)
paulson@13183
   962
apply (rule order_trans)
paulson@13183
   963
apply (rule_tac a' = "-1" in zdiv_mono1)
paulson@13183
   964
apply (auto simp add: zdiv_minus1)
paulson@13183
   965
done
paulson@13183
   966
paulson@14288
   967
lemma div_nonneg_neg_le0: "[| (0::int) \<le> a;  b < 0 |] ==> a div b \<le> 0"
paulson@13183
   968
by (drule zdiv_mono1_neg, auto)
paulson@13183
   969
paulson@14288
   970
lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"
paulson@13183
   971
apply auto
paulson@13183
   972
apply (drule_tac [2] zdiv_mono1)
paulson@13183
   973
apply (auto simp add: linorder_neq_iff)
paulson@13183
   974
apply (simp (no_asm_use) add: linorder_not_less [symmetric])
paulson@13183
   975
apply (blast intro: div_neg_pos_less0)
paulson@13183
   976
done
paulson@13183
   977
paulson@13183
   978
lemma neg_imp_zdiv_nonneg_iff:
paulson@14288
   979
     "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"
paulson@13183
   980
apply (subst zdiv_zminus_zminus [symmetric])
paulson@13183
   981
apply (subst pos_imp_zdiv_nonneg_iff, auto)
paulson@13183
   982
done
paulson@13183
   983
paulson@14288
   984
(*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
paulson@13183
   985
lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
paulson@13183
   986
by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
paulson@13183
   987
paulson@14288
   988
(*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
paulson@13183
   989
lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
paulson@13183
   990
by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
paulson@13183
   991
paulson@13837
   992
paulson@14271
   993
subsection {* The Divides Relation *}
paulson@13837
   994
paulson@13837
   995
lemma zdvd_iff_zmod_eq_0: "(m dvd n) = (n mod m = (0::int))"
paulson@13837
   996
by(simp add:dvd_def zmod_eq_0_iff)
paulson@13837
   997
paulson@13837
   998
lemma zdvd_0_right [iff]: "(m::int) dvd 0"
paulson@15221
   999
by (simp add: dvd_def)
paulson@13837
  1000
paulson@13837
  1001
lemma zdvd_0_left [iff]: "(0 dvd (m::int)) = (m = 0)"
paulson@15221
  1002
  by (simp add: dvd_def)
paulson@13837
  1003
paulson@13837
  1004
lemma zdvd_1_left [iff]: "1 dvd (m::int)"
paulson@15221
  1005
  by (simp add: dvd_def)
paulson@13837
  1006
paulson@13837
  1007
lemma zdvd_refl [simp]: "m dvd (m::int)"
paulson@15221
  1008
by (auto simp add: dvd_def intro: zmult_1_right [symmetric])
paulson@13837
  1009
paulson@13837
  1010
lemma zdvd_trans: "m dvd n ==> n dvd k ==> m dvd (k::int)"
paulson@15234
  1011
by (auto simp add: dvd_def intro: mult_assoc)
paulson@13837
  1012
paulson@13837
  1013
lemma zdvd_zminus_iff: "(m dvd -n) = (m dvd (n::int))"
paulson@15221
  1014
  apply (simp add: dvd_def, auto)
paulson@13837
  1015
   apply (rule_tac [!] x = "-k" in exI, auto)
paulson@13837
  1016
  done
paulson@13837
  1017
paulson@13837
  1018
lemma zdvd_zminus2_iff: "(-m dvd n) = (m dvd (n::int))"
paulson@15221
  1019
  apply (simp add: dvd_def, auto)
paulson@13837
  1020
   apply (rule_tac [!] x = "-k" in exI, auto)
paulson@13837
  1021
  done
paulson@13837
  1022
paulson@13837
  1023
lemma zdvd_anti_sym:
paulson@13837
  1024
    "0 < m ==> 0 < n ==> m dvd n ==> n dvd m ==> m = (n::int)"
paulson@15221
  1025
  apply (simp add: dvd_def, auto)
paulson@15234
  1026
  apply (simp add: mult_assoc zero_less_mult_iff zmult_eq_1_iff)
paulson@13837
  1027
  done
paulson@13837
  1028
paulson@13837
  1029
lemma zdvd_zadd: "k dvd m ==> k dvd n ==> k dvd (m + n :: int)"
paulson@15221
  1030
  apply (simp add: dvd_def)
paulson@14479
  1031
  apply (blast intro: right_distrib [symmetric])
paulson@13837
  1032
  done
paulson@13837
  1033
paulson@13837
  1034
lemma zdvd_zdiff: "k dvd m ==> k dvd n ==> k dvd (m - n :: int)"
paulson@15221
  1035
  apply (simp add: dvd_def)
paulson@14479
  1036
  apply (blast intro: right_diff_distrib [symmetric])
paulson@13837
  1037
  done
paulson@13837
  1038
paulson@13837
  1039
lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"
paulson@13837
  1040
  apply (subgoal_tac "m = n + (m - n)")
paulson@13837
  1041
   apply (erule ssubst)
paulson@13837
  1042
   apply (blast intro: zdvd_zadd, simp)
paulson@13837
  1043
  done
paulson@13837
  1044
paulson@13837
  1045
lemma zdvd_zmult: "k dvd (n::int) ==> k dvd m * n"
paulson@15221
  1046
  apply (simp add: dvd_def)
paulson@14271
  1047
  apply (blast intro: mult_left_commute)
paulson@13837
  1048
  done
paulson@13837
  1049
paulson@13837
  1050
lemma zdvd_zmult2: "k dvd (m::int) ==> k dvd m * n"
paulson@15234
  1051
  apply (subst mult_commute)
paulson@13837
  1052
  apply (erule zdvd_zmult)
paulson@13837
  1053
  done
paulson@13837
  1054
paulson@13837
  1055
lemma [iff]: "(k::int) dvd m * k"
paulson@13837
  1056
  apply (rule zdvd_zmult)
paulson@13837
  1057
  apply (rule zdvd_refl)
paulson@13837
  1058
  done
paulson@13837
  1059
paulson@13837
  1060
lemma [iff]: "(k::int) dvd k * m"
paulson@13837
  1061
  apply (rule zdvd_zmult2)
paulson@13837
  1062
  apply (rule zdvd_refl)
paulson@13837
  1063
  done
paulson@13837
  1064
paulson@13837
  1065
lemma zdvd_zmultD2: "j * k dvd n ==> j dvd (n::int)"
paulson@15221
  1066
  apply (simp add: dvd_def)
paulson@15234
  1067
  apply (simp add: mult_assoc, blast)
paulson@13837
  1068
  done
paulson@13837
  1069
paulson@13837
  1070
lemma zdvd_zmultD: "j * k dvd n ==> k dvd (n::int)"
paulson@13837
  1071
  apply (rule zdvd_zmultD2)
paulson@15234
  1072
  apply (subst mult_commute, assumption)
paulson@13837
  1073
  done
paulson@13837
  1074
paulson@13837
  1075
lemma zdvd_zmult_mono: "i dvd m ==> j dvd (n::int) ==> i * j dvd m * n"
paulson@15221
  1076
  apply (simp add: dvd_def, clarify)
paulson@13837
  1077
  apply (rule_tac x = "k * ka" in exI)
paulson@14271
  1078
  apply (simp add: mult_ac)
paulson@13837
  1079
  done
paulson@13837
  1080
paulson@13837
  1081
lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"
paulson@13837
  1082
  apply (rule iffI)
paulson@13837
  1083
   apply (erule_tac [2] zdvd_zadd)
paulson@13837
  1084
   apply (subgoal_tac "n = (n + k * m) - k * m")
paulson@13837
  1085
    apply (erule ssubst)
paulson@13837
  1086
    apply (erule zdvd_zdiff, simp_all)
paulson@13837
  1087
  done
paulson@13837
  1088
paulson@13837
  1089
lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"
paulson@15221
  1090
  apply (simp add: dvd_def)
paulson@13837
  1091
  apply (auto simp add: zmod_zmult_zmult1)
paulson@13837
  1092
  done
paulson@13837
  1093
paulson@13837
  1094
lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)"
paulson@13837
  1095
  apply (subgoal_tac "k dvd n * (m div n) + m mod n")
paulson@13837
  1096
   apply (simp add: zmod_zdiv_equality [symmetric])
paulson@13837
  1097
  apply (simp only: zdvd_zadd zdvd_zmult2)
paulson@13837
  1098
  done
paulson@13837
  1099
paulson@13837
  1100
lemma zdvd_not_zless: "0 < m ==> m < n ==> \<not> n dvd (m::int)"
paulson@15221
  1101
  apply (simp add: dvd_def, auto)
paulson@13837
  1102
  apply (subgoal_tac "0 < n")
paulson@13837
  1103
   prefer 2
paulson@14378
  1104
   apply (blast intro: order_less_trans)
paulson@14353
  1105
  apply (simp add: zero_less_mult_iff)
paulson@13837
  1106
  apply (subgoal_tac "n * k < n * 1")
paulson@14387
  1107
   apply (drule mult_less_cancel_left [THEN iffD1], auto)
paulson@13837
  1108
  done
paulson@13837
  1109
paulson@13837
  1110
lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))"
paulson@13837
  1111
  apply (auto simp add: dvd_def nat_abs_mult_distrib)
paulson@14353
  1112
  apply (auto simp add: nat_eq_iff abs_if split add: split_if_asm)
paulson@14353
  1113
   apply (rule_tac x = "-(int k)" in exI)
paulson@16413
  1114
  apply (auto simp add: int_mult)
paulson@13837
  1115
  done
paulson@13837
  1116
paulson@13837
  1117
lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)"
paulson@16413
  1118
  apply (auto simp add: dvd_def abs_if int_mult)
paulson@13837
  1119
    apply (rule_tac [3] x = "nat k" in exI)
paulson@13837
  1120
    apply (rule_tac [2] x = "-(int k)" in exI)
paulson@13837
  1121
    apply (rule_tac x = "nat (-k)" in exI)
paulson@13837
  1122
    apply (cut_tac [3] k = m in int_less_0_conv)
paulson@13837
  1123
    apply (cut_tac k = m in int_less_0_conv)
paulson@14353
  1124
    apply (auto simp add: zero_le_mult_iff mult_less_0_iff
paulson@13837
  1125
      nat_mult_distrib [symmetric] nat_eq_iff2)
paulson@13837
  1126
  done
paulson@13837
  1127
paulson@13837
  1128
lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \<le> z then (z dvd int m) else m = 0)"
paulson@16413
  1129
  apply (auto simp add: dvd_def int_mult)
paulson@13837
  1130
  apply (rule_tac x = "nat k" in exI)
paulson@13837
  1131
  apply (cut_tac k = m in int_less_0_conv)
paulson@14353
  1132
  apply (auto simp add: zero_le_mult_iff mult_less_0_iff
paulson@13837
  1133
    nat_mult_distrib [symmetric] nat_eq_iff2)
paulson@13837
  1134
  done
paulson@13837
  1135
paulson@13837
  1136
lemma zminus_dvd_iff [iff]: "(-z dvd w) = (z dvd (w::int))"
paulson@13837
  1137
  apply (auto simp add: dvd_def)
paulson@13837
  1138
   apply (rule_tac [!] x = "-k" in exI, auto)
paulson@13837
  1139
  done
paulson@13837
  1140
paulson@13837
  1141
lemma dvd_zminus_iff [iff]: "(z dvd -w) = (z dvd (w::int))"
paulson@13837
  1142
  apply (auto simp add: dvd_def)
paulson@14378
  1143
   apply (drule minus_equation_iff [THEN iffD1])
paulson@13837
  1144
   apply (rule_tac [!] x = "-k" in exI, auto)
paulson@13837
  1145
  done
paulson@13837
  1146
paulson@14288
  1147
lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \<le> (n::int)"
paulson@13837
  1148
  apply (rule_tac z=n in int_cases)
paulson@13837
  1149
  apply (auto simp add: dvd_int_iff) 
paulson@13837
  1150
  apply (rule_tac z=z in int_cases) 
paulson@13837
  1151
  apply (auto simp add: dvd_imp_le) 
paulson@13837
  1152
  done
paulson@13837
  1153
paulson@13837
  1154
paulson@14353
  1155
subsection{*Integer Powers*} 
paulson@14353
  1156
paulson@14353
  1157
instance int :: power ..
paulson@14353
  1158
paulson@14353
  1159
primrec
paulson@14353
  1160
  "p ^ 0 = 1"
paulson@14353
  1161
  "p ^ (Suc n) = (p::int) * (p ^ n)"
paulson@14353
  1162
paulson@14353
  1163
paulson@15003
  1164
instance int :: recpower
paulson@14353
  1165
proof
paulson@14353
  1166
  fix z :: int
paulson@14353
  1167
  fix n :: nat
paulson@14353
  1168
  show "z^0 = 1" by simp
paulson@14353
  1169
  show "z^(Suc n) = z * (z^n)" by simp
paulson@14353
  1170
qed
paulson@14353
  1171
paulson@14353
  1172
paulson@14353
  1173
lemma zpower_zmod: "((x::int) mod m)^y mod m = x^y mod m"
paulson@15251
  1174
apply (induct "y", auto)
paulson@14353
  1175
apply (rule zmod_zmult1_eq [THEN trans])
paulson@14353
  1176
apply (simp (no_asm_simp))
paulson@14353
  1177
apply (rule zmod_zmult_distrib [symmetric])
paulson@14353
  1178
done
paulson@14353
  1179
paulson@14353
  1180
lemma zpower_zadd_distrib: "x^(y+z) = ((x^y)*(x^z)::int)"
paulson@14353
  1181
  by (rule Power.power_add)
paulson@14353
  1182
paulson@14353
  1183
lemma zpower_zpower: "(x^y)^z = (x^(y*z)::int)"
paulson@14353
  1184
  by (rule Power.power_mult [symmetric])
paulson@14353
  1185
paulson@14353
  1186
lemma zero_less_zpower_abs_iff [simp]:
paulson@14353
  1187
     "(0 < (abs x)^n) = (x \<noteq> (0::int) | n=0)"
paulson@15251
  1188
apply (induct "n")
paulson@14353
  1189
apply (auto simp add: zero_less_mult_iff)
paulson@14353
  1190
done
paulson@14353
  1191
paulson@14353
  1192
lemma zero_le_zpower_abs [simp]: "(0::int) <= (abs x)^n"
paulson@15251
  1193
apply (induct "n")
paulson@14353
  1194
apply (auto simp add: zero_le_mult_iff)
paulson@14353
  1195
done
paulson@14353
  1196
paulson@16413
  1197
lemma int_power: "int (m^n) = (int m) ^ n"
paulson@16413
  1198
  by (induct n, simp_all add: int_mult)
paulson@16413
  1199
paulson@16413
  1200
text{*Compatibility binding*}
paulson@16413
  1201
lemmas zpower_int = int_power [symmetric]
berghofe@15320
  1202
obua@15101
  1203
lemma zdiv_int: "int (a div b) = (int a) div (int b)"
obua@15101
  1204
apply (subst split_div, auto)
obua@15101
  1205
apply (subst split_zdiv, auto)
obua@15101
  1206
apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in IntDiv.unique_quotient)
obua@15101
  1207
apply (auto simp add: IntDiv.quorem_def int_eq_of_nat)
obua@15101
  1208
done
obua@15101
  1209
obua@15101
  1210
lemma zmod_int: "int (a mod b) = (int a) mod (int b)"
obua@15101
  1211
apply (subst split_mod, auto)
obua@15101
  1212
apply (subst split_zmod, auto)
paulson@16413
  1213
apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia 
paulson@16413
  1214
       in unique_remainder)
obua@15101
  1215
apply (auto simp add: IntDiv.quorem_def int_eq_of_nat)
obua@15101
  1216
done
paulson@14353
  1217
paulson@16413
  1218
text{*Suggested by Matthias Daum*}
paulson@16413
  1219
lemma int_power_div_base:
paulson@16413
  1220
     "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"
paulson@16413
  1221
apply (subgoal_tac "k ^ m = k ^ ((m - 1) + 1)")
paulson@16413
  1222
 apply (erule ssubst)
paulson@16413
  1223
 apply (simp only: power_add)
paulson@16413
  1224
 apply simp_all
paulson@16413
  1225
done
paulson@16413
  1226
paulson@13183
  1227
ML
paulson@13183
  1228
{*
paulson@13183
  1229
val quorem_def = thm "quorem_def";
paulson@13183
  1230
paulson@13183
  1231
val unique_quotient = thm "unique_quotient";
paulson@13183
  1232
val unique_remainder = thm "unique_remainder";
paulson@13183
  1233
val adjust_eq = thm "adjust_eq";
paulson@13183
  1234
val posDivAlg_eqn = thm "posDivAlg_eqn";
paulson@13183
  1235
val posDivAlg_correct = thm "posDivAlg_correct";
paulson@13183
  1236
val negDivAlg_eqn = thm "negDivAlg_eqn";
paulson@13183
  1237
val negDivAlg_correct = thm "negDivAlg_correct";
paulson@13183
  1238
val quorem_0 = thm "quorem_0";
paulson@13183
  1239
val posDivAlg_0 = thm "posDivAlg_0";
paulson@13183
  1240
val negDivAlg_minus1 = thm "negDivAlg_minus1";
paulson@13183
  1241
val negateSnd_eq = thm "negateSnd_eq";
paulson@13183
  1242
val quorem_neg = thm "quorem_neg";
paulson@13183
  1243
val divAlg_correct = thm "divAlg_correct";
paulson@13183
  1244
val DIVISION_BY_ZERO = thm "DIVISION_BY_ZERO";
paulson@13183
  1245
val zmod_zdiv_equality = thm "zmod_zdiv_equality";
paulson@13183
  1246
val pos_mod_conj = thm "pos_mod_conj";
paulson@13183
  1247
val pos_mod_sign = thm "pos_mod_sign";
paulson@13183
  1248
val neg_mod_conj = thm "neg_mod_conj";
paulson@13183
  1249
val neg_mod_sign = thm "neg_mod_sign";
paulson@13183
  1250
val quorem_div_mod = thm "quorem_div_mod";
paulson@13183
  1251
val quorem_div = thm "quorem_div";
paulson@13183
  1252
val quorem_mod = thm "quorem_mod";
paulson@13183
  1253
val div_pos_pos_trivial = thm "div_pos_pos_trivial";
paulson@13183
  1254
val div_neg_neg_trivial = thm "div_neg_neg_trivial";
paulson@13183
  1255
val div_pos_neg_trivial = thm "div_pos_neg_trivial";
paulson@13183
  1256
val mod_pos_pos_trivial = thm "mod_pos_pos_trivial";
paulson@13183
  1257
val mod_neg_neg_trivial = thm "mod_neg_neg_trivial";
paulson@13183
  1258
val mod_pos_neg_trivial = thm "mod_pos_neg_trivial";
paulson@13183
  1259
val zdiv_zminus_zminus = thm "zdiv_zminus_zminus";
paulson@13183
  1260
val zmod_zminus_zminus = thm "zmod_zminus_zminus";
paulson@13183
  1261
val zdiv_zminus1_eq_if = thm "zdiv_zminus1_eq_if";
paulson@13183
  1262
val zmod_zminus1_eq_if = thm "zmod_zminus1_eq_if";
paulson@13183
  1263
val zdiv_zminus2 = thm "zdiv_zminus2";
paulson@13183
  1264
val zmod_zminus2 = thm "zmod_zminus2";
paulson@13183
  1265
val zdiv_zminus2_eq_if = thm "zdiv_zminus2_eq_if";
paulson@13183
  1266
val zmod_zminus2_eq_if = thm "zmod_zminus2_eq_if";
paulson@13183
  1267
val self_quotient = thm "self_quotient";
paulson@13183
  1268
val self_remainder = thm "self_remainder";
paulson@13183
  1269
val zdiv_self = thm "zdiv_self";
paulson@13183
  1270
val zmod_self = thm "zmod_self";
paulson@13183
  1271
val zdiv_zero = thm "zdiv_zero";
paulson@13183
  1272
val div_eq_minus1 = thm "div_eq_minus1";
paulson@13183
  1273
val zmod_zero = thm "zmod_zero";
paulson@13183
  1274
val zdiv_minus1 = thm "zdiv_minus1";
paulson@13183
  1275
val zmod_minus1 = thm "zmod_minus1";
paulson@13183
  1276
val div_pos_pos = thm "div_pos_pos";
paulson@13183
  1277
val mod_pos_pos = thm "mod_pos_pos";
paulson@13183
  1278
val div_neg_pos = thm "div_neg_pos";
paulson@13183
  1279
val mod_neg_pos = thm "mod_neg_pos";
paulson@13183
  1280
val div_pos_neg = thm "div_pos_neg";
paulson@13183
  1281
val mod_pos_neg = thm "mod_pos_neg";
paulson@13183
  1282
val div_neg_neg = thm "div_neg_neg";
paulson@13183
  1283
val mod_neg_neg = thm "mod_neg_neg";
paulson@13183
  1284
val zmod_1 = thm "zmod_1";
paulson@13183
  1285
val zdiv_1 = thm "zdiv_1";
paulson@13183
  1286
val zmod_minus1_right = thm "zmod_minus1_right";
paulson@13183
  1287
val zdiv_minus1_right = thm "zdiv_minus1_right";
paulson@13183
  1288
val zdiv_mono1 = thm "zdiv_mono1";
paulson@13183
  1289
val zdiv_mono1_neg = thm "zdiv_mono1_neg";
paulson@13183
  1290
val zdiv_mono2 = thm "zdiv_mono2";
paulson@13183
  1291
val zdiv_mono2_neg = thm "zdiv_mono2_neg";
paulson@13183
  1292
val zdiv_zmult1_eq = thm "zdiv_zmult1_eq";
paulson@13183
  1293
val zmod_zmult1_eq = thm "zmod_zmult1_eq";
paulson@13183
  1294
val zmod_zmult1_eq' = thm "zmod_zmult1_eq'";
paulson@13183
  1295
val zmod_zmult_distrib = thm "zmod_zmult_distrib";
paulson@13183
  1296
val zdiv_zmult_self1 = thm "zdiv_zmult_self1";
paulson@13183
  1297
val zdiv_zmult_self2 = thm "zdiv_zmult_self2";
paulson@13183
  1298
val zmod_zmult_self1 = thm "zmod_zmult_self1";
paulson@13183
  1299
val zmod_zmult_self2 = thm "zmod_zmult_self2";
paulson@13183
  1300
val zmod_eq_0_iff = thm "zmod_eq_0_iff";
paulson@13183
  1301
val zdiv_zadd1_eq = thm "zdiv_zadd1_eq";
paulson@13183
  1302
val zmod_zadd1_eq = thm "zmod_zadd1_eq";
paulson@13183
  1303
val mod_div_trivial = thm "mod_div_trivial";
paulson@13183
  1304
val mod_mod_trivial = thm "mod_mod_trivial";
paulson@13183
  1305
val zmod_zadd_left_eq = thm "zmod_zadd_left_eq";
paulson@13183
  1306
val zmod_zadd_right_eq = thm "zmod_zadd_right_eq";
paulson@13183
  1307
val zdiv_zadd_self1 = thm "zdiv_zadd_self1";
paulson@13183
  1308
val zdiv_zadd_self2 = thm "zdiv_zadd_self2";
paulson@13183
  1309
val zmod_zadd_self1 = thm "zmod_zadd_self1";
paulson@13183
  1310
val zmod_zadd_self2 = thm "zmod_zadd_self2";
paulson@13183
  1311
val zdiv_zmult2_eq = thm "zdiv_zmult2_eq";
paulson@13183
  1312
val zmod_zmult2_eq = thm "zmod_zmult2_eq";
paulson@13183
  1313
val zdiv_zmult_zmult1 = thm "zdiv_zmult_zmult1";
paulson@13183
  1314
val zdiv_zmult_zmult2 = thm "zdiv_zmult_zmult2";
paulson@13183
  1315
val zmod_zmult_zmult1 = thm "zmod_zmult_zmult1";
paulson@13183
  1316
val zmod_zmult_zmult2 = thm "zmod_zmult_zmult2";
paulson@13183
  1317
val pos_zdiv_mult_2 = thm "pos_zdiv_mult_2";
paulson@13183
  1318
val neg_zdiv_mult_2 = thm "neg_zdiv_mult_2";
paulson@13183
  1319
val zdiv_number_of_BIT = thm "zdiv_number_of_BIT";
paulson@13183
  1320
val pos_zmod_mult_2 = thm "pos_zmod_mult_2";
paulson@13183
  1321
val neg_zmod_mult_2 = thm "neg_zmod_mult_2";
paulson@13183
  1322
val zmod_number_of_BIT = thm "zmod_number_of_BIT";
paulson@13183
  1323
val div_neg_pos_less0 = thm "div_neg_pos_less0";
paulson@13183
  1324
val div_nonneg_neg_le0 = thm "div_nonneg_neg_le0";
paulson@13183
  1325
val pos_imp_zdiv_nonneg_iff = thm "pos_imp_zdiv_nonneg_iff";
paulson@13183
  1326
val neg_imp_zdiv_nonneg_iff = thm "neg_imp_zdiv_nonneg_iff";
paulson@13183
  1327
val pos_imp_zdiv_neg_iff = thm "pos_imp_zdiv_neg_iff";
paulson@13183
  1328
val neg_imp_zdiv_neg_iff = thm "neg_imp_zdiv_neg_iff";
paulson@14353
  1329
paulson@14353
  1330
val zpower_zmod = thm "zpower_zmod";
paulson@14353
  1331
val zpower_zadd_distrib = thm "zpower_zadd_distrib";
paulson@14353
  1332
val zpower_zpower = thm "zpower_zpower";
paulson@14353
  1333
val zero_less_zpower_abs_iff = thm "zero_less_zpower_abs_iff";
paulson@14353
  1334
val zero_le_zpower_abs = thm "zero_le_zpower_abs";
berghofe@15320
  1335
val zpower_int = thm "zpower_int";
paulson@13183
  1336
*}
paulson@13183
  1337
paulson@6917
  1338
end