src/HOL/Integ/IntDiv.thy
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New theorem zpower_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 Recdef
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files ("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 "inv_image less_than (%(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 "inv_image less_than (%(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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c7290200b3f4 conversion of IntDiv.thy to Isar format
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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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apply (simp add: quorem_div_mod [THEN quorem_neg, simplified, 
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                                 THEN quorem_div, THEN sym])
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c7290200b3f4 conversion of IntDiv.thy to Isar format
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done
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   321
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(*Simpler laws such as -a mod b = -(a mod b) FAIL, but see just below*)
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lemma zmod_zminus_zminus [simp]: "(-a) mod (-b) = - (a mod (b::int))"
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apply (case_tac "b = 0", simp)
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apply (subst quorem_div_mod [THEN quorem_neg, simplified, THEN quorem_mod],
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       auto)
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done
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   328
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subsection{*Laws for div and mod with Unary Minus*}
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lemma zminus1_lemma:
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     "quorem((a,b),(q,r))  
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      ==> quorem ((-a,b), (if r=0 then -q else -q - 1),  
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                          (if r=0 then 0 else b-r))"
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by (force simp add: split_ifs quorem_def linorder_neq_iff right_diff_distrib)
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   337
c7290200b3f4 conversion of IntDiv.thy to Isar format
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   338
c7290200b3f4 conversion of IntDiv.thy to Isar format
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lemma zdiv_zminus1_eq_if:
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     "b \<noteq> (0::int)  
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      ==> (-a) div b =  
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          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
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by (blast intro: quorem_div_mod [THEN zminus1_lemma, THEN quorem_div])
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   344
c7290200b3f4 conversion of IntDiv.thy to Isar format
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lemma zmod_zminus1_eq_if:
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     "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))"
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apply (case_tac "b = 0", simp)
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apply (blast intro: quorem_div_mod [THEN zminus1_lemma, THEN quorem_mod])
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done
c7290200b3f4 conversion of IntDiv.thy to Isar format
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   350
c7290200b3f4 conversion of IntDiv.thy to Isar format
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lemma zdiv_zminus2: "a div (-b) = (-a::int) div b"
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   352
by (cut_tac a = "-a" in zdiv_zminus_zminus, auto)
c7290200b3f4 conversion of IntDiv.thy to Isar format
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diff changeset
   353
c7290200b3f4 conversion of IntDiv.thy to Isar format
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   354
lemma zmod_zminus2: "a mod (-b) = - ((-a::int) mod b)"
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   355
by (cut_tac a = "-a" and b = b in zmod_zminus_zminus, auto)
c7290200b3f4 conversion of IntDiv.thy to Isar format
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diff changeset
   356
c7290200b3f4 conversion of IntDiv.thy to Isar format
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   357
lemma zdiv_zminus2_eq_if:
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     "b \<noteq> (0::int)  
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      ==> a div (-b) =  
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          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
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parents: 11868
diff changeset
   361
by (simp add: zdiv_zminus1_eq_if zdiv_zminus2)
c7290200b3f4 conversion of IntDiv.thy to Isar format
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diff changeset
   362
c7290200b3f4 conversion of IntDiv.thy to Isar format
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lemma zmod_zminus2_eq_if:
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     "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"
c7290200b3f4 conversion of IntDiv.thy to Isar format
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parents: 11868
diff changeset
   365
by (simp add: zmod_zminus1_eq_if zmod_zminus2)
c7290200b3f4 conversion of IntDiv.thy to Isar format
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diff changeset
   366
c7290200b3f4 conversion of IntDiv.thy to Isar format
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   367
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   368
subsection{*Division of a Number by Itself*}
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   369
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lemma self_quotient_aux1: "[| (0::int) < a; a = r + a*q; r < a |] ==> 1 \<le> q"
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   371
apply (subgoal_tac "0 < a*q")
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   372
 apply (simp add: zero_less_mult_iff, arith)
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   373
done
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   374
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lemma self_quotient_aux2: "[| (0::int) < a; a = r + a*q; 0 \<le> r |] ==> q \<le> 1"
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   376
apply (subgoal_tac "0 \<le> a* (1-q) ")
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   377
 apply (simp add: zero_le_mult_iff)
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   378
apply (simp add: right_diff_distrib)
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   379
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
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   380
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lemma self_quotient: "[| quorem((a,a),(q,r));  a \<noteq> (0::int) |] ==> q = 1"
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   382
apply (simp add: split_ifs quorem_def linorder_neq_iff)
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   383
apply (rule order_antisym, safe, simp_all)
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604d0f3622d6 *** empty log message ***
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parents: 13517
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   384
apply (rule_tac [3] a = "-a" and r = "-r" in self_quotient_aux1)
604d0f3622d6 *** empty log message ***
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parents: 13517
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   385
apply (rule_tac a = "-a" and r = "-r" in self_quotient_aux2)
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parents: 15140
diff changeset
   386
apply (force intro: self_quotient_aux1 self_quotient_aux2 simp add: add_commute)+
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   387
done
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   388
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   389
lemma self_remainder: "[| quorem((a,a),(q,r));  a \<noteq> (0::int) |] ==> r = 0"
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   390
apply (frule self_quotient, assumption)
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   391
apply (simp add: quorem_def)
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diff changeset
   392
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
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diff changeset
   393
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   394
lemma zdiv_self [simp]: "a \<noteq> 0 ==> a div a = (1::int)"
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c7290200b3f4 conversion of IntDiv.thy to Isar format
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parents: 11868
diff changeset
   395
by (simp add: quorem_div_mod [THEN self_quotient])
c7290200b3f4 conversion of IntDiv.thy to Isar format
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diff changeset
   396
c7290200b3f4 conversion of IntDiv.thy to Isar format
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   397
(*Here we have 0 mod 0 = 0, also assumed by Knuth (who puts m mod 0 = 0) *)
c7290200b3f4 conversion of IntDiv.thy to Isar format
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diff changeset
   398
lemma zmod_self [simp]: "a mod a = (0::int)"
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parents: 15003
diff changeset
   399
apply (case_tac "a = 0", simp)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   400
apply (simp add: quorem_div_mod [THEN self_remainder])
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   401
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   402
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   403
14271
8ed6989228bb Simplification of the development of Integers
paulson
parents: 13837
diff changeset
   404
subsection{*Computation of Division and Remainder*}
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   405
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   406
lemma zdiv_zero [simp]: "(0::int) div b = 0"
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   407
by (simp add: div_def divAlg_def)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   408
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   409
lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   410
by (simp add: div_def divAlg_def)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   411
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   412
lemma zmod_zero [simp]: "(0::int) mod b = 0"
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   413
by (simp add: mod_def divAlg_def)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   414
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   415
lemma zdiv_minus1: "(0::int) < b ==> -1 div b = -1"
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   416
by (simp add: div_def divAlg_def)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   417
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   418
lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   419
by (simp add: mod_def divAlg_def)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   420
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   421
text{*a positive, b positive *}
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   422
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   423
lemma div_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a div b = fst (posDivAlg(a,b))"
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   424
by (simp add: div_def divAlg_def)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   425
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   426
lemma mod_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a mod b = snd (posDivAlg(a,b))"
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   427
by (simp add: mod_def divAlg_def)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   428
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   429
text{*a negative, b positive *}
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   430
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   431
lemma div_neg_pos: "[| a < 0;  0 < b |] ==> a div b = fst (negDivAlg(a,b))"
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   432
by (simp add: div_def divAlg_def)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   433
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   434
lemma mod_neg_pos: "[| a < 0;  0 < b |] ==> a mod b = snd (negDivAlg(a,b))"
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   435
by (simp add: mod_def divAlg_def)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   436
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   437
text{*a positive, b negative *}
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   438
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   439
lemma div_pos_neg:
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   440
     "[| 0 < a;  b < 0 |] ==> a div b = fst (negateSnd(negDivAlg(-a,-b)))"
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   441
by (simp add: div_def divAlg_def)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   442
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   443
lemma mod_pos_neg:
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   444
     "[| 0 < a;  b < 0 |] ==> a mod b = snd (negateSnd(negDivAlg(-a,-b)))"
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   445
by (simp add: mod_def divAlg_def)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   446
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   447
text{*a negative, b negative *}
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   448
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   449
lemma div_neg_neg:
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   450
     "[| a < 0;  b \<le> 0 |] ==> a div b = fst (negateSnd(posDivAlg(-a,-b)))"
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   451
by (simp add: div_def divAlg_def)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   452
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   453
lemma mod_neg_neg:
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   454
     "[| a < 0;  b \<le> 0 |] ==> a mod b = snd (negateSnd(posDivAlg(-a,-b)))"
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   455
by (simp add: mod_def divAlg_def)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   456
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   457
text {*Simplify expresions in which div and mod combine numerical constants*}
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   458
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   459
declare div_pos_pos [of "number_of v" "number_of w", standard, simp]
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   460
declare div_neg_pos [of "number_of v" "number_of w", standard, simp]
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   461
declare div_pos_neg [of "number_of v" "number_of w", standard, simp]
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   462
declare div_neg_neg [of "number_of v" "number_of w", standard, simp]
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   463
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   464
declare mod_pos_pos [of "number_of v" "number_of w", standard, simp]
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   465
declare mod_neg_pos [of "number_of v" "number_of w", standard, simp]
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   466
declare mod_pos_neg [of "number_of v" "number_of w", standard, simp]
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   467
declare mod_neg_neg [of "number_of v" "number_of w", standard, simp]
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   468
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   469
declare posDivAlg_eqn [of "number_of v" "number_of w", standard, simp]
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   470
declare negDivAlg_eqn [of "number_of v" "number_of w", standard, simp]
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   471
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   472
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   473
text{*Special-case simplification *}
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   474
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   475
lemma zmod_1 [simp]: "a mod (1::int) = 0"
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   476
apply (cut_tac a = a and b = 1 in pos_mod_sign)
13788
fd03c4ab89d4 pos/neg_mod_sign/bound are now simp rules.
nipkow
parents: 13716
diff changeset
   477
apply (cut_tac [2] a = a and b = 1 in pos_mod_bound)
fd03c4ab89d4 pos/neg_mod_sign/bound are now simp rules.
nipkow
parents: 13716
diff changeset
   478
apply (auto simp del:pos_mod_bound pos_mod_sign)
fd03c4ab89d4 pos/neg_mod_sign/bound are now simp rules.
nipkow
parents: 13716
diff changeset
   479
done
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   480
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   481
lemma zdiv_1 [simp]: "a div (1::int) = a"
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   482
by (cut_tac a = a and b = 1 in zmod_zdiv_equality, auto)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   483
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   484
lemma zmod_minus1_right [simp]: "a mod (-1::int) = 0"
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   485
apply (cut_tac a = a and b = "-1" in neg_mod_sign)
13788
fd03c4ab89d4 pos/neg_mod_sign/bound are now simp rules.
nipkow
parents: 13716
diff changeset
   486
apply (cut_tac [2] a = a and b = "-1" in neg_mod_bound)
fd03c4ab89d4 pos/neg_mod_sign/bound are now simp rules.
nipkow
parents: 13716
diff changeset
   487
apply (auto simp del: neg_mod_sign neg_mod_bound)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   488
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   489
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   490
lemma zdiv_minus1_right [simp]: "a div (-1::int) = -a"
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   491
by (cut_tac a = a and b = "-1" in zmod_zdiv_equality, auto)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   492
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   493
(** The last remaining special cases for constant arithmetic:
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   494
    1 div z and 1 mod z **)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   495
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   496
declare div_pos_pos [OF int_0_less_1, of "number_of w", standard, simp]
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   497
declare div_pos_neg [OF int_0_less_1, of "number_of w", standard, simp]
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   498
declare mod_pos_pos [OF int_0_less_1, of "number_of w", standard, simp]
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   499
declare mod_pos_neg [OF int_0_less_1, of "number_of w", standard, simp]
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   500
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   501
declare posDivAlg_eqn [of concl: 1 "number_of w", standard, simp]
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   502
declare negDivAlg_eqn [of concl: 1 "number_of w", standard, simp]
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   503
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   504
14271
8ed6989228bb Simplification of the development of Integers
paulson
parents: 13837
diff changeset
   505
subsection{*Monotonicity in the First Argument (Dividend)*}
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   506
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   507
lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div b"
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   508
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   509
apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   510
apply (rule unique_quotient_lemma)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   511
apply (erule subst)
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   512
apply (erule subst, simp_all)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   513
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   514
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   515
lemma zdiv_mono1_neg: "[| a \<le> a';  (b::int) < 0 |] ==> a' div b \<le> a div b"
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   516
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   517
apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   518
apply (rule unique_quotient_lemma_neg)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   519
apply (erule subst)
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   520
apply (erule subst, simp_all)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   521
done
6917
eba301caceea Introduction of integer division algorithm
paulson
parents:
diff changeset
   522
eba301caceea Introduction of integer division algorithm
paulson
parents:
diff changeset
   523
14271
8ed6989228bb Simplification of the development of Integers
paulson
parents: 13837
diff changeset
   524
subsection{*Monotonicity in the Second Argument (Divisor)*}
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   525
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   526
lemma q_pos_lemma:
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   527
     "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   528
apply (subgoal_tac "0 < b'* (q' + 1) ")
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   529
 apply (simp add: zero_less_mult_iff)
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14473
diff changeset
   530
apply (simp add: right_distrib)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   531
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   532
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   533
lemma zdiv_mono2_lemma:
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   534
     "[| b*q + r = b'*q' + r';  0 \<le> b'*q' + r';   
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   535
         r' < b';  0 \<le> r;  0 < b';  b' \<le> b |]   
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   536
      ==> q \<le> (q'::int)"
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   537
apply (frule q_pos_lemma, assumption+) 
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   538
apply (subgoal_tac "b*q < b* (q' + 1) ")
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   539
 apply (simp add: mult_less_cancel_left)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   540
apply (subgoal_tac "b*q = r' - r + b'*q'")
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   541
 prefer 2 apply simp
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14473
diff changeset
   542
apply (simp (no_asm_simp) add: right_distrib)
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   543
apply (subst add_commute, rule zadd_zless_mono, arith)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14353
diff changeset
   544
apply (rule mult_right_mono, auto)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   545
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   546
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   547
lemma zdiv_mono2:
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   548
     "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   549
apply (subgoal_tac "b \<noteq> 0")
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   550
 prefer 2 apply arith
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   551
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   552
apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   553
apply (rule zdiv_mono2_lemma)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   554
apply (erule subst)
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   555
apply (erule subst, simp_all)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   556
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   557
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   558
lemma q_neg_lemma:
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   559
     "[| b'*q' + r' < 0;  0 \<le> r';  0 < b' |] ==> q' \<le> (0::int)"
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   560
apply (subgoal_tac "b'*q' < 0")
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   561
 apply (simp add: mult_less_0_iff, arith)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   562
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   563
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   564
lemma zdiv_mono2_neg_lemma:
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   565
     "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;   
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   566
         r < b;  0 \<le> r';  0 < b';  b' \<le> b |]   
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   567
      ==> q' \<le> (q::int)"
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   568
apply (frule q_neg_lemma, assumption+) 
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   569
apply (subgoal_tac "b*q' < b* (q + 1) ")
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   570
 apply (simp add: mult_less_cancel_left)
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14473
diff changeset
   571
apply (simp add: right_distrib)
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   572
apply (subgoal_tac "b*q' \<le> b'*q'")
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   573
 prefer 2 apply (simp add: mult_right_mono_neg, arith)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   574
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   575
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   576
lemma zdiv_mono2_neg:
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   577
     "[| a < (0::int);  0 < b';  b' \<le> b |] ==> a div b' \<le> a div b"
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   578
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   579
apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   580
apply (rule zdiv_mono2_neg_lemma)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   581
apply (erule subst)
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   582
apply (erule subst, simp_all)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   583
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   584
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   585
14271
8ed6989228bb Simplification of the development of Integers
paulson
parents: 13837
diff changeset
   586
subsection{*More Algebraic Laws for div and mod*}
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   587
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   588
text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   589
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   590
lemma zmult1_lemma:
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   591
     "[| quorem((b,c),(q,r));  c \<noteq> 0 |]  
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   592
      ==> quorem ((a*b, c), (a*q + a*r div c, a*r mod c))"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14473
diff changeset
   593
by (force simp add: split_ifs quorem_def linorder_neq_iff right_distrib)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   594
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   595
lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   596
apply (case_tac "c = 0", simp)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   597
apply (blast intro: quorem_div_mod [THEN zmult1_lemma, THEN quorem_div])
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   598
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   599
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   600
lemma zmod_zmult1_eq: "(a*b) mod c = a*(b mod c) mod (c::int)"
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   601
apply (case_tac "c = 0", simp)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   602
apply (blast intro: quorem_div_mod [THEN zmult1_lemma, THEN quorem_mod])
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   603
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   604
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   605
lemma zmod_zmult1_eq': "(a*b) mod (c::int) = ((a mod c) * b) mod c"
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   606
apply (rule trans)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   607
apply (rule_tac s = "b*a mod c" in trans)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   608
apply (rule_tac [2] zmod_zmult1_eq)
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15221
diff changeset
   609
apply (simp_all add: mult_commute)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   610
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   611
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   612
lemma zmod_zmult_distrib: "(a*b) mod (c::int) = ((a mod c) * (b mod c)) mod c"
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   613
apply (rule zmod_zmult1_eq' [THEN trans])
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   614
apply (rule zmod_zmult1_eq)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   615
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   616
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   617
lemma zdiv_zmult_self1 [simp]: "b \<noteq> (0::int) ==> (a*b) div b = a"
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   618
by (simp add: zdiv_zmult1_eq)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   619
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   620
lemma zdiv_zmult_self2 [simp]: "b \<noteq> (0::int) ==> (b*a) div b = a"
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15221
diff changeset
   621
by (subst mult_commute, erule zdiv_zmult_self1)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   622
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   623
lemma zmod_zmult_self1 [simp]: "(a*b) mod b = (0::int)"
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   624
by (simp add: zmod_zmult1_eq)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   625
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   626
lemma zmod_zmult_self2 [simp]: "(b*a) mod b = (0::int)"
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15221
diff changeset
   627
by (simp add: mult_commute zmod_zmult1_eq)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   628
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   629
lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"
13517
42efec18f5b2 Added div+mod cancelling simproc
nipkow
parents: 13266
diff changeset
   630
proof
42efec18f5b2 Added div+mod cancelling simproc
nipkow
parents: 13266
diff changeset
   631
  assume "m mod d = 0"
14473
846c237bd9b3 stylistic tweaks
paulson
parents: 14387
diff changeset
   632
  with zmod_zdiv_equality[of m d] show "EX q::int. m = d*q" by auto
13517
42efec18f5b2 Added div+mod cancelling simproc
nipkow
parents: 13266
diff changeset
   633
next
42efec18f5b2 Added div+mod cancelling simproc
nipkow
parents: 13266
diff changeset
   634
  assume "EX q::int. m = d*q"
42efec18f5b2 Added div+mod cancelling simproc
nipkow
parents: 13266
diff changeset
   635
  thus "m mod d = 0" by auto
42efec18f5b2 Added div+mod cancelling simproc
nipkow
parents: 13266
diff changeset
   636
qed
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   637
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   638
declare zmod_eq_0_iff [THEN iffD1, dest!]
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   639
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   640
text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   641
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   642
lemma zadd1_lemma:
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   643
     "[| quorem((a,c),(aq,ar));  quorem((b,c),(bq,br));  c \<noteq> 0 |]  
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   644
      ==> quorem ((a+b, c), (aq + bq + (ar+br) div c, (ar+br) mod c))"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14473
diff changeset
   645
by (force simp add: split_ifs quorem_def linorder_neq_iff right_distrib)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   646
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   647
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   648
lemma zdiv_zadd1_eq:
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   649
     "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   650
apply (case_tac "c = 0", simp)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   651
apply (blast intro: zadd1_lemma [OF quorem_div_mod quorem_div_mod] quorem_div)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   652
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   653
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   654
lemma zmod_zadd1_eq: "(a+b) mod (c::int) = (a mod c + b mod c) mod c"
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   655
apply (case_tac "c = 0", simp)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   656
apply (blast intro: zadd1_lemma [OF quorem_div_mod quorem_div_mod] quorem_mod)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   657
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   658
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   659
lemma mod_div_trivial [simp]: "(a mod b) div b = (0::int)"
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   660
apply (case_tac "b = 0", simp)
13788
fd03c4ab89d4 pos/neg_mod_sign/bound are now simp rules.
nipkow
parents: 13716
diff changeset
   661
apply (auto simp add: linorder_neq_iff div_pos_pos_trivial div_neg_neg_trivial)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   662
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   663
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   664
lemma mod_mod_trivial [simp]: "(a mod b) mod b = a mod (b::int)"
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   665
apply (case_tac "b = 0", simp)
13788
fd03c4ab89d4 pos/neg_mod_sign/bound are now simp rules.
nipkow
parents: 13716
diff changeset
   666
apply (force simp add: linorder_neq_iff mod_pos_pos_trivial mod_neg_neg_trivial)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   667
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   668
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   669
lemma zmod_zadd_left_eq: "(a+b) mod (c::int) = ((a mod c) + b) mod c"
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   670
apply (rule trans [symmetric])
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   671
apply (rule zmod_zadd1_eq, simp)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   672
apply (rule zmod_zadd1_eq [symmetric])
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   673
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   674
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   675
lemma zmod_zadd_right_eq: "(a+b) mod (c::int) = (a + (b mod c)) mod c"
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   676
apply (rule trans [symmetric])
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   677
apply (rule zmod_zadd1_eq, simp)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   678
apply (rule zmod_zadd1_eq [symmetric])
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   679
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   680
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   681
lemma zdiv_zadd_self1[simp]: "a \<noteq> (0::int) ==> (a+b) div a = b div a + 1"
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   682
by (simp add: zdiv_zadd1_eq)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   683
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   684
lemma zdiv_zadd_self2[simp]: "a \<noteq> (0::int) ==> (b+a) div a = b div a + 1"
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   685
by (simp add: zdiv_zadd1_eq)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   686
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   687
lemma zmod_zadd_self1[simp]: "(a+b) mod a = b mod (a::int)"
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   688
apply (case_tac "a = 0", simp)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   689
apply (simp add: zmod_zadd1_eq)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   690
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   691
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   692
lemma zmod_zadd_self2[simp]: "(b+a) mod a = b mod (a::int)"
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   693
apply (case_tac "a = 0", simp)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   694
apply (simp add: zmod_zadd1_eq)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   695
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   696
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   697
14271
8ed6989228bb Simplification of the development of Integers
paulson
parents: 13837
diff changeset
   698
subsection{*Proving  @{term "a div (b*c) = (a div b) div c"} *}
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   699
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   700
(*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   701
  7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   702
  to cause particular problems.*)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   703
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   704
text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   705
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   706
lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b*c < b*(q mod c) + r"
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   707
apply (subgoal_tac "b * (c - q mod c) < r * 1")
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14473
diff changeset
   708
apply (simp add: right_diff_distrib)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   709
apply (rule order_le_less_trans)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14353
diff changeset
   710
apply (erule_tac [2] mult_strict_right_mono)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14353
diff changeset
   711
apply (rule mult_left_mono_neg)
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   712
apply (auto simp add: compare_rls add_commute [of 1]
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   713
                      add1_zle_eq pos_mod_bound)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   714
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   715
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   716
lemma zmult2_lemma_aux2:
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   717
     "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   718
apply (subgoal_tac "b * (q mod c) \<le> 0")
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   719
 apply arith
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   720
apply (simp add: mult_le_0_iff)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   721
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   722
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   723
lemma zmult2_lemma_aux3: "[| (0::int) < c;  0 \<le> r;  r < b |] ==> 0 \<le> b * (q mod c) + r"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   724
apply (subgoal_tac "0 \<le> b * (q mod c) ")
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   725
apply arith
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
   726
apply (simp add: zero_le_mult_iff)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   727
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   728
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   729
lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   730
apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14473
diff changeset
   731
apply (simp add: right_diff_distrib)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   732
apply (rule order_less_le_trans)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14353
diff changeset
   733
apply (erule mult_strict_right_mono)
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   734
apply (rule_tac [2] mult_left_mono)
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   735
apply (auto simp add: compare_rls add_commute [of 1]
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   736
                      add1_zle_eq pos_mod_bound)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   737
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   738
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   739
lemma zmult2_lemma: "[| quorem ((a,b), (q,r));  b \<noteq> 0;  0 < c |]  
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   740
      ==> quorem ((a, b*c), (q div c, b*(q mod c) + r))"
14271
8ed6989228bb Simplification of the development of Integers
paulson
parents: 13837
diff changeset
   741
by (auto simp add: mult_ac quorem_def linorder_neq_iff
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14473
diff changeset
   742
                   zero_less_mult_iff right_distrib [symmetric] 
13524
604d0f3622d6 *** empty log message ***
wenzelm
parents: 13517
diff changeset
   743
                   zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   744
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   745
lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c"
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   746
apply (case_tac "b = 0", simp)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   747
apply (force simp add: quorem_div_mod [THEN zmult2_lemma, THEN quorem_div])
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   748
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   749
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   750
lemma zmod_zmult2_eq:
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   751
     "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   752
apply (case_tac "b = 0", simp)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   753
apply (force simp add: quorem_div_mod [THEN zmult2_lemma, THEN quorem_mod])
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   754
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   755
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   756
14271
8ed6989228bb Simplification of the development of Integers
paulson
parents: 13837
diff changeset
   757
subsection{*Cancellation of Common Factors in div*}
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   758
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   759
lemma zdiv_zmult_zmult1_aux1:
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   760
     "[| (0::int) < b;  c \<noteq> 0 |] ==> (c*a) div (c*b) = a div b"
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   761
by (subst zdiv_zmult2_eq, auto)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   762
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   763
lemma zdiv_zmult_zmult1_aux2:
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   764
     "[| b < (0::int);  c \<noteq> 0 |] ==> (c*a) div (c*b) = a div b"
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   765
apply (subgoal_tac " (c * (-a)) div (c * (-b)) = (-a) div (-b) ")
13524
604d0f3622d6 *** empty log message ***
wenzelm
parents: 13517
diff changeset
   766
apply (rule_tac [2] zdiv_zmult_zmult1_aux1, auto)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   767
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   768
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   769
lemma zdiv_zmult_zmult1: "c \<noteq> (0::int) ==> (c*a) div (c*b) = a div b"
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   770
apply (case_tac "b = 0", simp)
13524
604d0f3622d6 *** empty log message ***
wenzelm
parents: 13517
diff changeset
   771
apply (auto simp add: linorder_neq_iff zdiv_zmult_zmult1_aux1 zdiv_zmult_zmult1_aux2)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   772
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   773
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   774
lemma zdiv_zmult_zmult2: "c \<noteq> (0::int) ==> (a*c) div (b*c) = a div b"
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   775
apply (drule zdiv_zmult_zmult1)
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15221
diff changeset
   776
apply (auto simp add: mult_commute)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   777
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   778
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   779
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   780
14271
8ed6989228bb Simplification of the development of Integers
paulson
parents: 13837
diff changeset
   781
subsection{*Distribution of Factors over mod*}
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   782
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   783
lemma zmod_zmult_zmult1_aux1:
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   784
     "[| (0::int) < b;  c \<noteq> 0 |] ==> (c*a) mod (c*b) = c * (a mod b)"
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   785
by (subst zmod_zmult2_eq, auto)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   786
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   787
lemma zmod_zmult_zmult1_aux2:
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   788
     "[| b < (0::int);  c \<noteq> 0 |] ==> (c*a) mod (c*b) = c * (a mod b)"
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   789
apply (subgoal_tac " (c * (-a)) mod (c * (-b)) = c * ((-a) mod (-b))")
13524
604d0f3622d6 *** empty log message ***
wenzelm
parents: 13517
diff changeset
   790
apply (rule_tac [2] zmod_zmult_zmult1_aux1, auto)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   791
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   792
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   793
lemma zmod_zmult_zmult1: "(c*a) mod (c*b) = (c::int) * (a mod b)"
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   794
apply (case_tac "b = 0", simp)
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   795
apply (case_tac "c = 0", simp)
13524
604d0f3622d6 *** empty log message ***
wenzelm
parents: 13517
diff changeset
   796
apply (auto simp add: linorder_neq_iff zmod_zmult_zmult1_aux1 zmod_zmult_zmult1_aux2)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   797
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   798
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   799
lemma zmod_zmult_zmult2: "(a*c) mod (b*c) = (a mod b) * (c::int)"
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   800
apply (cut_tac c = c in zmod_zmult_zmult1)
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15221
diff changeset
   801
apply (auto simp add: mult_commute)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   802
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   803
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   804
14271
8ed6989228bb Simplification of the development of Integers
paulson
parents: 13837
diff changeset
   805
subsection {*Splitting Rules for div and mod*}
13260
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   806
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   807
text{*The proofs of the two lemmas below are essentially identical*}
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   808
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   809
lemma split_pos_lemma:
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   810
 "0<k ==> 
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   811
    P(n div k :: int)(n mod k) = (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i j)"
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   812
apply (rule iffI, clarify)
13260
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   813
 apply (erule_tac P="P ?x ?y" in rev_mp)  
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   814
 apply (subst zmod_zadd1_eq) 
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   815
 apply (subst zdiv_zadd1_eq) 
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   816
 apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)  
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   817
txt{*converse direction*}
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   818
apply (drule_tac x = "n div k" in spec) 
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   819
apply (drule_tac x = "n mod k" in spec, simp)
13260
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   820
done
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   821
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   822
lemma split_neg_lemma:
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   823
 "k<0 ==>
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   824
    P(n div k :: int)(n mod k) = (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i j)"
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   825
apply (rule iffI, clarify)
13260
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   826
 apply (erule_tac P="P ?x ?y" in rev_mp)  
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   827
 apply (subst zmod_zadd1_eq) 
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   828
 apply (subst zdiv_zadd1_eq) 
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   829
 apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)  
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   830
txt{*converse direction*}
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   831
apply (drule_tac x = "n div k" in spec) 
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   832
apply (drule_tac x = "n mod k" in spec, simp)
13260
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   833
done
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   834
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   835
lemma split_zdiv:
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   836
 "P(n div k :: int) =
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   837
  ((k = 0 --> P 0) & 
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   838
   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) & 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   839
   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   840
apply (case_tac "k=0", simp)
13260
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   841
apply (simp only: linorder_neq_iff)
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   842
apply (erule disjE) 
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   843
 apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"] 
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   844
                      split_neg_lemma [of concl: "%x y. P x"])
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   845
done
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   846
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   847
lemma split_zmod:
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   848
 "P(n mod k :: int) =
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   849
  ((k = 0 --> P n) & 
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   850
   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) & 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   851
   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   852
apply (case_tac "k=0", simp)
13260
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   853
apply (simp only: linorder_neq_iff)
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   854
apply (erule disjE) 
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   855
 apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"] 
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   856
                      split_neg_lemma [of concl: "%x y. P y"])
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   857
done
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   858
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   859
(* Enable arith to deal with div 2 and mod 2: *)
13266
2a6ad4357d72 modified Larry's changes to make div/mod a numeral work in arith.
nipkow
parents: 13260
diff changeset
   860
declare split_zdiv [of _ _ "number_of k", simplified, standard, arith_split]
2a6ad4357d72 modified Larry's changes to make div/mod a numeral work in arith.
nipkow
parents: 13260
diff changeset
   861
declare split_zmod [of _ _ "number_of k", simplified, standard, arith_split]
13260
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   862
ea36a40c004f new splitting rules for zdiv, zmod
paulson
parents: 13183
diff changeset
   863
14271
8ed6989228bb Simplification of the development of Integers
paulson
parents: 13837
diff changeset
   864
subsection{*Speeding up the Division Algorithm with Shifting*}
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   865
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   866
text{*computing div by shifting *}
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   867
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   868
lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   869
proof cases
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   870
  assume "a=0"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   871
    thus ?thesis by simp
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   872
next
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   873
  assume "a\<noteq>0" and le_a: "0\<le>a"   
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   874
  hence a_pos: "1 \<le> a" by arith
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   875
  hence one_less_a2: "1 < 2*a" by arith
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   876
  hence le_2a: "2 * (1 + b mod a) \<le> 2 * a"
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   877
    by (simp add: mult_le_cancel_left add_commute [of 1] add1_zle_eq)
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   878
  with a_pos have "0 \<le> b mod a" by simp
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   879
  hence le_addm: "0 \<le> 1 mod (2*a) + 2*(b mod a)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   880
    by (simp add: mod_pos_pos_trivial one_less_a2)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   881
  with  le_2a
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   882
  have "(1 mod (2*a) + 2*(b mod a)) div (2*a) = 0"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   883
    by (simp add: div_pos_pos_trivial le_addm mod_pos_pos_trivial one_less_a2
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   884
                  right_distrib) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   885
  thus ?thesis
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   886
    by (subst zdiv_zadd1_eq,
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   887
        simp add: zdiv_zmult_zmult1 zmod_zmult_zmult1 one_less_a2
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   888
                  div_pos_pos_trivial)
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   889
qed
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   890
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   891
lemma neg_zdiv_mult_2: "a \<le> (0::int) ==> (1 + 2*b) div (2*a) = (b+1) div a"
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   892
apply (subgoal_tac " (1 + 2* (-b - 1)) div (2 * (-a)) = (-b - 1) div (-a) ")
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   893
apply (rule_tac [2] pos_zdiv_mult_2)
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14473
diff changeset
   894
apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   895
apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14473
diff changeset
   896
apply (simp only: zdiv_zminus_zminus diff_minus minus_add_distrib [symmetric],
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   897
       simp) 
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   898
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   899
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   900
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   901
(*Not clear why this must be proved separately; probably number_of causes
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   902
  simplification problems*)
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   903
lemma not_0_le_lemma: "~ 0 \<le> x ==> x \<le> (0::int)"
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   904
by auto
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   905
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   906
lemma zdiv_number_of_BIT[simp]:
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   907
     "number_of (v BIT b) div number_of (w BIT False) =  
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   908
          (if ~b | (0::int) \<le> number_of w                    
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   909
           then number_of v div (number_of w)     
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   910
           else (number_of v + (1::int)) div (number_of w))"
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   911
apply (simp only: number_of_eq Bin_simps UNIV_I split: split_if) 
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   912
apply (simp add: zdiv_zmult_zmult1 pos_zdiv_mult_2 neg_zdiv_mult_2 add_ac)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   913
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   914
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   915
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   916
subsection{*Computing mod by Shifting (proofs resemble those for div)*}
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   917
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   918
lemma pos_zmod_mult_2:
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   919
     "(0::int) \<le> a ==> (1 + 2*b) mod (2*a) = 1 + 2 * (b mod a)"
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   920
apply (case_tac "a = 0", simp)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   921
apply (subgoal_tac "1 < a * 2")
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   922
 prefer 2 apply arith
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   923
apply (subgoal_tac "2* (1 + b mod a) \<le> 2*a")
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   924
 apply (rule_tac [2] mult_left_mono)
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15221
diff changeset
   925
apply (auto simp add: add_commute [of 1] mult_commute add1_zle_eq 
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   926
                      pos_mod_bound)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   927
apply (subst zmod_zadd1_eq)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   928
apply (simp add: zmod_zmult_zmult2 mod_pos_pos_trivial)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   929
apply (rule mod_pos_pos_trivial)
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   930
apply (auto simp add: mod_pos_pos_trivial left_distrib)
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
   931
apply (subgoal_tac "0 \<le> b mod a", arith, simp)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   932
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   933
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   934
lemma neg_zmod_mult_2:
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   935
     "a \<le> (0::int) ==> (1 + 2*b) mod (2*a) = 2 * ((b+1) mod a) - 1"
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   936
apply (subgoal_tac "(1 + 2* (-b - 1)) mod (2* (-a)) = 
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   937
                    1 + 2* ((-b - 1) mod (-a))")
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   938
apply (rule_tac [2] pos_zmod_mult_2)
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14473
diff changeset
   939
apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   940
apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   941
 prefer 2 apply simp 
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14473
diff changeset
   942
apply (simp only: zmod_zminus_zminus diff_minus minus_add_distrib [symmetric])
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   943
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   944
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   945
lemma zmod_number_of_BIT [simp]:
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   946
     "number_of (v BIT b) mod number_of (w BIT False) =  
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   947
          (if b then  
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   948
                if (0::int) \<le> number_of w  
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   949
                then 2 * (number_of v mod number_of w) + 1     
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   950
                else 2 * ((number_of v + (1::int)) mod number_of w) - 1   
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   951
           else 2 * (number_of v mod number_of w))"
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   952
apply (simp only: number_of_eq Bin_simps UNIV_I split: split_if) 
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   953
apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2 
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   954
                 not_0_le_lemma neg_zmod_mult_2 add_ac)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   955
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   956
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   957
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   958
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   959
subsection{*Quotients of Signs*}
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   960
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   961
lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   962
apply (subgoal_tac "a div b \<le> -1", force)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   963
apply (rule order_trans)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   964
apply (rule_tac a' = "-1" in zdiv_mono1)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   965
apply (auto simp add: zdiv_minus1)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   966
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   967
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   968
lemma div_nonneg_neg_le0: "[| (0::int) \<le> a;  b < 0 |] ==> a div b \<le> 0"
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   969
by (drule zdiv_mono1_neg, auto)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   970
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   971
lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   972
apply auto
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   973
apply (drule_tac [2] zdiv_mono1)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   974
apply (auto simp add: linorder_neq_iff)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   975
apply (simp (no_asm_use) add: linorder_not_less [symmetric])
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   976
apply (blast intro: div_neg_pos_less0)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   977
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   978
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   979
lemma neg_imp_zdiv_nonneg_iff:
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   980
     "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   981
apply (subst zdiv_zminus_zminus [symmetric])
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   982
apply (subst pos_imp_zdiv_nonneg_iff, auto)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   983
done
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   984
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   985
(*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   986
lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   987
by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   988
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
   989
(*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   990
lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   991
by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   992
13837
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
   993
14271
8ed6989228bb Simplification of the development of Integers
paulson
parents: 13837
diff changeset
   994
subsection {* The Divides Relation *}
13837
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
   995
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
   996
lemma zdvd_iff_zmod_eq_0: "(m dvd n) = (n mod m = (0::int))"
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
   997
by(simp add:dvd_def zmod_eq_0_iff)
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
   998
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
   999
lemma zdvd_0_right [iff]: "(m::int) dvd 0"
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
  1000
by (simp add: dvd_def)
13837
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1001
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1002
lemma zdvd_0_left [iff]: "(0 dvd (m::int)) = (m = 0)"
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
  1003
  by (simp add: dvd_def)
13837
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1004
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1005
lemma zdvd_1_left [iff]: "1 dvd (m::int)"
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
  1006
  by (simp add: dvd_def)
13837
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1007
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1008
lemma zdvd_refl [simp]: "m dvd (m::int)"
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
  1009
by (auto simp add: dvd_def intro: zmult_1_right [symmetric])
13837
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1010
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1011
lemma zdvd_trans: "m dvd n ==> n dvd k ==> m dvd (k::int)"
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15221
diff changeset
  1012
by (auto simp add: dvd_def intro: mult_assoc)
13837
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1013
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1014
lemma zdvd_zminus_iff: "(m dvd -n) = (m dvd (n::int))"
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
  1015
  apply (simp add: dvd_def, auto)
13837
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1016
   apply (rule_tac [!] x = "-k" in exI, auto)
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1017
  done
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1018
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1019
lemma zdvd_zminus2_iff: "(-m dvd n) = (m dvd (n::int))"
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
  1020
  apply (simp add: dvd_def, auto)
13837
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1021
   apply (rule_tac [!] x = "-k" in exI, auto)
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1022
  done
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1023
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1024
lemma zdvd_anti_sym:
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1025
    "0 < m ==> 0 < n ==> m dvd n ==> n dvd m ==> m = (n::int)"
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
  1026
  apply (simp add: dvd_def, auto)
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15221
diff changeset
  1027
  apply (simp add: mult_assoc zero_less_mult_iff zmult_eq_1_iff)
13837
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1028
  done
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1029
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1030
lemma zdvd_zadd: "k dvd m ==> k dvd n ==> k dvd (m + n :: int)"
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
  1031
  apply (simp add: dvd_def)
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14473
diff changeset
  1032
  apply (blast intro: right_distrib [symmetric])
13837
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1033
  done
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1034
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1035
lemma zdvd_zdiff: "k dvd m ==> k dvd n ==> k dvd (m - n :: int)"
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
  1036
  apply (simp add: dvd_def)
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14473
diff changeset
  1037
  apply (blast intro: right_diff_distrib [symmetric])
13837
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1038
  done
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1039
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1040
lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1041
  apply (subgoal_tac "m = n + (m - n)")
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1042
   apply (erule ssubst)
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1043
   apply (blast intro: zdvd_zadd, simp)
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1044
  done
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1045
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1046
lemma zdvd_zmult: "k dvd (n::int) ==> k dvd m * n"
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
  1047
  apply (simp add: dvd_def)
14271
8ed6989228bb Simplification of the development of Integers
paulson
parents: 13837
diff changeset
  1048
  apply (blast intro: mult_left_commute)
13837
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1049
  done
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1050
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1051
lemma zdvd_zmult2: "k dvd (m::int) ==> k dvd m * n"
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15221
diff changeset
  1052
  apply (subst mult_commute)
13837
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1053
  apply (erule zdvd_zmult)
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1054
  done
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1055
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1056
lemma [iff]: "(k::int) dvd m * k"
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1057
  apply (rule zdvd_zmult)
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1058
  apply (rule zdvd_refl)
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1059
  done
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1060
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1061
lemma [iff]: "(k::int) dvd k * m"
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1062
  apply (rule zdvd_zmult2)
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1063
  apply (rule zdvd_refl)
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1064
  done
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1065
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1066
lemma zdvd_zmultD2: "j * k dvd n ==> j dvd (n::int)"
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
  1067
  apply (simp add: dvd_def)
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15221
diff changeset
  1068
  apply (simp add: mult_assoc, blast)
13837
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1069
  done
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1070
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1071
lemma zdvd_zmultD: "j * k dvd n ==> k dvd (n::int)"
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1072
  apply (rule zdvd_zmultD2)
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15221
diff changeset
  1073
  apply (subst mult_commute, assumption)
13837
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1074
  done
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1075
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1076
lemma zdvd_zmult_mono: "i dvd m ==> j dvd (n::int) ==> i * j dvd m * n"
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
  1077
  apply (simp add: dvd_def, clarify)
13837
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1078
  apply (rule_tac x = "k * ka" in exI)
14271
8ed6989228bb Simplification of the development of Integers
paulson
parents: 13837
diff changeset
  1079
  apply (simp add: mult_ac)
13837
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1080
  done
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1081
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1082
lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1083
  apply (rule iffI)
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1084
   apply (erule_tac [2] zdvd_zadd)
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1085
   apply (subgoal_tac "n = (n + k * m) - k * m")
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1086
    apply (erule ssubst)
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1087
    apply (erule zdvd_zdiff, simp_all)
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1088
  done
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1089
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1090
lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
  1091
  apply (simp add: dvd_def)
13837
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1092
  apply (auto simp add: zmod_zmult_zmult1)
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1093
  done
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1094
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1095
lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)"
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1096
  apply (subgoal_tac "k dvd n * (m div n) + m mod n")
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1097
   apply (simp add: zmod_zdiv_equality [symmetric])
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1098
  apply (simp only: zdvd_zadd zdvd_zmult2)
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1099
  done
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1100
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1101
lemma zdvd_not_zless: "0 < m ==> m < n ==> \<not> n dvd (m::int)"
15221
8412cfdf3287 tweaking of arithmetic proofs
paulson
parents: 15140
diff changeset
  1102
  apply (simp add: dvd_def, auto)
13837
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1103
  apply (subgoal_tac "0 < n")
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1104
   prefer 2
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14353
diff changeset
  1105
   apply (blast intro: order_less_trans)
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1106
  apply (simp add: zero_less_mult_iff)
13837
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1107
  apply (subgoal_tac "n * k < n * 1")
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1108
   apply (drule mult_less_cancel_left [THEN iffD1], auto)
13837
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1109
  done
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1110
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1111
lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))"
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1112
  apply (auto simp add: dvd_def nat_abs_mult_distrib)
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1113
  apply (auto simp add: nat_eq_iff abs_if split add: split_if_asm)
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1114
   apply (rule_tac x = "-(int k)" in exI)
13837
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1115
  apply (auto simp add: zmult_int [symmetric])
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1116
  done
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1117
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1118
lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)"
15003
6145dd7538d7 replaced monomorphic abs definitions by abs_if
paulson
parents: 14479
diff changeset
  1119
  apply (auto simp add: dvd_def abs_if zmult_int [symmetric])
13837
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1120
    apply (rule_tac [3] x = "nat k" in exI)
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1121
    apply (rule_tac [2] x = "-(int k)" in exI)
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1122
    apply (rule_tac x = "nat (-k)" in exI)
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1123
    apply (cut_tac [3] k = m in int_less_0_conv)
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1124
    apply (cut_tac k = m in int_less_0_conv)
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1125
    apply (auto simp add: zero_le_mult_iff mult_less_0_iff
13837
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1126
      nat_mult_distrib [symmetric] nat_eq_iff2)
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1127
  done
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1128
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1129
lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \<le> z then (z dvd int m) else m = 0)"
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1130
  apply (auto simp add: dvd_def zmult_int [symmetric])
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1131
  apply (rule_tac x = "nat k" in exI)
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1132
  apply (cut_tac k = m in int_less_0_conv)
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1133
  apply (auto simp add: zero_le_mult_iff mult_less_0_iff
13837
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1134
    nat_mult_distrib [symmetric] nat_eq_iff2)
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1135
  done
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1136
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1137
lemma zminus_dvd_iff [iff]: "(-z dvd w) = (z dvd (w::int))"
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1138
  apply (auto simp add: dvd_def)
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1139
   apply (rule_tac [!] x = "-k" in exI, auto)
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1140
  done
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1141
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1142
lemma dvd_zminus_iff [iff]: "(z dvd -w) = (z dvd (w::int))"
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1143
  apply (auto simp add: dvd_def)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14353
diff changeset
  1144
   apply (drule minus_equation_iff [THEN iffD1])
13837
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1145
   apply (rule_tac [!] x = "-k" in exI, auto)
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1146
  done
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1147
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14271
diff changeset
  1148
lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \<le> (n::int)"
13837
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1149
  apply (rule_tac z=n in int_cases)
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1150
  apply (auto simp add: dvd_int_iff) 
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1151
  apply (rule_tac z=z in int_cases) 
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1152
  apply (auto simp add: dvd_imp_le) 
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1153
  done
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1154
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13788
diff changeset
  1155
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1156
subsection{*Integer Powers*} 
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1157
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1158
instance int :: power ..
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1159
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1160
primrec
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1161
  "p ^ 0 = 1"
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1162
  "p ^ (Suc n) = (p::int) * (p ^ n)"
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1163
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1164
15003
6145dd7538d7 replaced monomorphic abs definitions by abs_if
paulson
parents: 14479
diff changeset
  1165
instance int :: recpower
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1166
proof
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1167
  fix z :: int
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1168
  fix n :: nat
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1169
  show "z^0 = 1" by simp
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1170
  show "z^(Suc n) = z * (z^n)" by simp
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1171
qed
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1172
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1173
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1174
lemma zpower_zmod: "((x::int) mod m)^y mod m = x^y mod m"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15234
diff changeset
  1175
apply (induct "y", auto)
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1176
apply (rule zmod_zmult1_eq [THEN trans])
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1177
apply (simp (no_asm_simp))
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1178
apply (rule zmod_zmult_distrib [symmetric])
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1179
done
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1180
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1181
lemma zpower_zadd_distrib: "x^(y+z) = ((x^y)*(x^z)::int)"
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1182
  by (rule Power.power_add)
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1183
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1184
lemma zpower_zpower: "(x^y)^z = (x^(y*z)::int)"
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1185
  by (rule Power.power_mult [symmetric])
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1186
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1187
lemma zero_less_zpower_abs_iff [simp]:
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1188
     "(0 < (abs x)^n) = (x \<noteq> (0::int) | n=0)"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15234
diff changeset
  1189
apply (induct "n")
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1190
apply (auto simp add: zero_less_mult_iff)
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1191
done
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1192
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1193
lemma zero_le_zpower_abs [simp]: "(0::int) <= (abs x)^n"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15234
diff changeset
  1194
apply (induct "n")
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1195
apply (auto simp add: zero_le_mult_iff)
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1196
done
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1197
15320
dfc2654eea9f New theorem zpower_int
berghofe
parents: 15251
diff changeset
  1198
theorem zpower_int: "(int m)^n = int (m^n)"
dfc2654eea9f New theorem zpower_int
berghofe
parents: 15251
diff changeset
  1199
  by (induct n) (simp_all add: zmult_int)
dfc2654eea9f New theorem zpower_int
berghofe
parents: 15251
diff changeset
  1200
15101
d027515e2aa6 zdiv_int, zmod_int
obua
parents: 15013
diff changeset
  1201
lemma zdiv_int: "int (a div b) = (int a) div (int b)"
d027515e2aa6 zdiv_int, zmod_int
obua
parents: 15013
diff changeset
  1202
apply (subst split_div, auto)
d027515e2aa6 zdiv_int, zmod_int
obua
parents: 15013
diff changeset
  1203
apply (subst split_zdiv, auto)
d027515e2aa6 zdiv_int, zmod_int
obua
parents: 15013
diff changeset
  1204
apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in IntDiv.unique_quotient)
d027515e2aa6 zdiv_int, zmod_int
obua
parents: 15013
diff changeset
  1205
apply (auto simp add: IntDiv.quorem_def int_eq_of_nat)
d027515e2aa6 zdiv_int, zmod_int
obua
parents: 15013
diff changeset
  1206
done
d027515e2aa6 zdiv_int, zmod_int
obua
parents: 15013
diff changeset
  1207
d027515e2aa6 zdiv_int, zmod_int
obua
parents: 15013
diff changeset
  1208
lemma zmod_int: "int (a mod b) = (int a) mod (int b)"
d027515e2aa6 zdiv_int, zmod_int
obua
parents: 15013
diff changeset
  1209
apply (subst split_mod, auto)
d027515e2aa6 zdiv_int, zmod_int
obua
parents: 15013
diff changeset
  1210
apply (subst split_zmod, auto)
d027515e2aa6 zdiv_int, zmod_int
obua
parents: 15013
diff changeset
  1211
apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia in IntDiv.unique_remainder)
d027515e2aa6 zdiv_int, zmod_int
obua
parents: 15013
diff changeset
  1212
apply (auto simp add: IntDiv.quorem_def int_eq_of_nat)
d027515e2aa6 zdiv_int, zmod_int
obua
parents: 15013
diff changeset
  1213
done
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14288
diff changeset
  1214
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1215
ML
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1216
{*
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1217
val quorem_def = thm "quorem_def";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1218
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1219
val unique_quotient = thm "unique_quotient";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1220
val unique_remainder = thm "unique_remainder";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1221
val adjust_eq = thm "adjust_eq";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1222
val posDivAlg_eqn = thm "posDivAlg_eqn";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1223
val posDivAlg_correct = thm "posDivAlg_correct";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1224
val negDivAlg_eqn = thm "negDivAlg_eqn";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1225
val negDivAlg_correct = thm "negDivAlg_correct";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1226
val quorem_0 = thm "quorem_0";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1227
val posDivAlg_0 = thm "posDivAlg_0";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1228
val negDivAlg_minus1 = thm "negDivAlg_minus1";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1229
val negateSnd_eq = thm "negateSnd_eq";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1230
val quorem_neg = thm "quorem_neg";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1231
val divAlg_correct = thm "divAlg_correct";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1232
val DIVISION_BY_ZERO = thm "DIVISION_BY_ZERO";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1233
val zmod_zdiv_equality = thm "zmod_zdiv_equality";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1234
val pos_mod_conj = thm "pos_mod_conj";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1235
val pos_mod_sign = thm "pos_mod_sign";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1236
val neg_mod_conj = thm "neg_mod_conj";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1237
val neg_mod_sign = thm "neg_mod_sign";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1238
val quorem_div_mod = thm "quorem_div_mod";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1239
val quorem_div = thm "quorem_div";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1240
val quorem_mod = thm "quorem_mod";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1241
val div_pos_pos_trivial = thm "div_pos_pos_trivial";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1242
val div_neg_neg_trivial = thm "div_neg_neg_trivial";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1243
val div_pos_neg_trivial = thm "div_pos_neg_trivial";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1244
val mod_pos_pos_trivial = thm "mod_pos_pos_trivial";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1245
val mod_neg_neg_trivial = thm "mod_neg_neg_trivial";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1246
val mod_pos_neg_trivial = thm "mod_pos_neg_trivial";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1247
val zdiv_zminus_zminus = thm "zdiv_zminus_zminus";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1248
val zmod_zminus_zminus = thm "zmod_zminus_zminus";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1249
val zdiv_zminus1_eq_if = thm "zdiv_zminus1_eq_if";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1250
val zmod_zminus1_eq_if = thm "zmod_zminus1_eq_if";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1251
val zdiv_zminus2 = thm "zdiv_zminus2";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1252
val zmod_zminus2 = thm "zmod_zminus2";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1253
val zdiv_zminus2_eq_if = thm "zdiv_zminus2_eq_if";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1254
val zmod_zminus2_eq_if = thm "zmod_zminus2_eq_if";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1255
val self_quotient = thm "self_quotient";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1256
val self_remainder = thm "self_remainder";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1257
val zdiv_self = thm "zdiv_self";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1258
val zmod_self = thm "zmod_self";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1259
val zdiv_zero = thm "zdiv_zero";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1260
val div_eq_minus1 = thm "div_eq_minus1";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1261
val zmod_zero = thm "zmod_zero";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1262
val zdiv_minus1 = thm "zdiv_minus1";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1263
val zmod_minus1 = thm "zmod_minus1";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1264
val div_pos_pos = thm "div_pos_pos";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1265
val mod_pos_pos = thm "mod_pos_pos";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1266
val div_neg_pos = thm "div_neg_pos";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1267
val mod_neg_pos = thm "mod_neg_pos";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1268
val div_pos_neg = thm "div_pos_neg";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1269
val mod_pos_neg = thm "mod_pos_neg";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1270
val div_neg_neg = thm "div_neg_neg";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1271
val mod_neg_neg = thm "mod_neg_neg";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1272
val zmod_1 = thm "zmod_1";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1273
val zdiv_1 = thm "zdiv_1";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1274
val zmod_minus1_right = thm "zmod_minus1_right";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1275
val zdiv_minus1_right = thm "zdiv_minus1_right";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1276
val zdiv_mono1 = thm "zdiv_mono1";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1277
val zdiv_mono1_neg = thm "zdiv_mono1_neg";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1278
val zdiv_mono2 = thm "zdiv_mono2";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1279
val zdiv_mono2_neg = thm "zdiv_mono2_neg";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1280
val zdiv_zmult1_eq = thm "zdiv_zmult1_eq";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1281
val zmod_zmult1_eq = thm "zmod_zmult1_eq";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1282
val zmod_zmult1_eq' = thm "zmod_zmult1_eq'";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1283
val zmod_zmult_distrib = thm "zmod_zmult_distrib";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1284
val zdiv_zmult_self1 = thm "zdiv_zmult_self1";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1285
val zdiv_zmult_self2 = thm "zdiv_zmult_self2";
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
  1286
val zmod_zmult_self1 = thm "zmod_zmult_self1";