src/HOL/IntDiv.thy
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tightended specification of class semiring_div
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(*  Title:      HOL/IntDiv.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1999  University of Cambridge
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*)
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header{* The Division Operators div and mod *}
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theory IntDiv
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imports Int Divides FunDef
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uses
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  "~~/src/Provers/Arith/cancel_numeral_factor.ML"
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  "~~/src/Provers/Arith/extract_common_term.ML"
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  ("Tools/int_factor_simprocs.ML")
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begin
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definition divmod_rel :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool" where
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    --{*definition of quotient and remainder*}
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    [code]: "divmod_rel a b = (\<lambda>(q, r). a = b * q + r \<and>
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               (if 0 < b then 0 \<le> r \<and> r < b else b < r \<and> r \<le> 0))"
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definition adjust :: "int \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int" where
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    --{*for the division algorithm*}
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    [code]: "adjust b = (\<lambda>(q, r). if 0 \<le> r - b then (2 * q + 1, r - b)
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                         else (2 * q, r))"
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text{*algorithm for the case @{text "a\<ge>0, b>0"}*}
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function posDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
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  "posDivAlg a b = (if a < b \<or>  b \<le> 0 then (0, a)
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     else adjust b (posDivAlg a (2 * b)))"
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by auto
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termination by (relation "measure (\<lambda>(a, b). nat (a - b + 1))") auto
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text{*algorithm for the case @{text "a<0, b>0"}*}
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function negDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
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  "negDivAlg a b = (if 0 \<le>a + b \<or> b \<le> 0  then (-1, a + b)
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     else adjust b (negDivAlg a (2 * b)))"
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by auto
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termination by (relation "measure (\<lambda>(a, b). nat (- a - b))") auto
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text{*algorithm for the general case @{term "b\<noteq>0"}*}
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definition negateSnd :: "int \<times> int \<Rightarrow> int \<times> int" where
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  [code inline]: "negateSnd = apsnd uminus"
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definition divmod :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
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    --{*The full division algorithm considers all possible signs for a, b
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       including the special case @{text "a=0, b<0"} because 
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       @{term negDivAlg} requires @{term "a<0"}.*}
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  "divmod a b = (if 0 \<le> a then if 0 \<le> b then posDivAlg a b
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                  else if a = 0 then (0, 0)
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                       else negateSnd (negDivAlg (-a) (-b))
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               else 
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                  if 0 < b then negDivAlg a b
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                  else negateSnd (posDivAlg (-a) (-b)))"
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instantiation int :: Divides.div
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begin
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definition
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  div_def: "a div b = fst (divmod a b)"
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definition
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  mod_def: "a mod b = snd (divmod a b)"
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instance ..
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end
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lemma divmod_mod_div:
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  "divmod p q = (p div q, p mod q)"
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  by (auto simp add: div_def mod_def)
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text{*
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Here is the division algorithm in ML:
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\begin{verbatim}
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    fun posDivAlg (a,b) =
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      if a<b then (0,a)
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      else let val (q,r) = posDivAlg(a, 2*b)
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	       in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
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	   end
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    fun negDivAlg (a,b) =
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      if 0\<le>a+b then (~1,a+b)
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      else let val (q,r) = negDivAlg(a, 2*b)
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	       in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
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	   end;
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    fun negateSnd (q,r:int) = (q,~r);
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    fun divmod (a,b) = if 0\<le>a then 
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			  if b>0 then posDivAlg (a,b) 
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			   else if a=0 then (0,0)
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				else negateSnd (negDivAlg (~a,~b))
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		       else 
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			  if 0<b then negDivAlg (a,b)
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			  else        negateSnd (posDivAlg (~a,~b));
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\end{verbatim}
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*}
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subsection{*Uniqueness and Monotonicity of Quotients and Remainders*}
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lemma unique_quotient_lemma:
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     "[| b*q' + r'  \<le> b*q + r;  0 \<le> r';  r' < b;  r < b |]  
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      ==> q' \<le> (q::int)"
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apply (subgoal_tac "r' + b * (q'-q) \<le> r")
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 prefer 2 apply (simp add: right_diff_distrib)
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apply (subgoal_tac "0 < b * (1 + q - q') ")
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apply (erule_tac [2] order_le_less_trans)
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 prefer 2 apply (simp add: right_diff_distrib right_distrib)
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apply (subgoal_tac "b * q' < b * (1 + q) ")
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 prefer 2 apply (simp add: right_diff_distrib right_distrib)
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apply (simp add: mult_less_cancel_left)
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done
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lemma unique_quotient_lemma_neg:
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     "[| b*q' + r' \<le> b*q + r;  r \<le> 0;  b < r;  b < r' |]  
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      ==> q \<le> (q'::int)"
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by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma, 
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    auto)
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lemma unique_quotient:
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     "[| divmod_rel a b (q, r); divmod_rel a b (q', r');  b \<noteq> 0 |]  
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      ==> q = q'"
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apply (simp add: divmod_rel_def linorder_neq_iff split: split_if_asm)
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apply (blast intro: order_antisym
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             dest: order_eq_refl [THEN unique_quotient_lemma] 
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             order_eq_refl [THEN unique_quotient_lemma_neg] sym)+
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done
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lemma unique_remainder:
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     "[| divmod_rel a b (q, r); divmod_rel a b (q', r');  b \<noteq> 0 |]  
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      ==> r = r'"
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apply (subgoal_tac "q = q'")
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 apply (simp add: divmod_rel_def)
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apply (blast intro: unique_quotient)
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done
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subsection{*Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends*}
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text{*And positive divisors*}
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lemma adjust_eq [simp]:
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     "adjust b (q,r) = 
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      (let diff = r-b in  
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	if 0 \<le> diff then (2*q + 1, diff)   
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                     else (2*q, r))"
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by (simp add: Let_def adjust_def)
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declare posDivAlg.simps [simp del]
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text{*use with a simproc to avoid repeatedly proving the premise*}
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lemma posDivAlg_eqn:
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     "0 < b ==>  
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      posDivAlg a b = (if a<b then (0,a) else adjust b (posDivAlg a (2*b)))"
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by (rule posDivAlg.simps [THEN trans], simp)
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text{*Correctness of @{term posDivAlg}: it computes quotients correctly*}
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theorem posDivAlg_correct:
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  assumes "0 \<le> a" and "0 < b"
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  shows "divmod_rel a b (posDivAlg a b)"
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using prems apply (induct a b rule: posDivAlg.induct)
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apply auto
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apply (simp add: divmod_rel_def)
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apply (subst posDivAlg_eqn, simp add: right_distrib)
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apply (case_tac "a < b")
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apply simp_all
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apply (erule splitE)
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apply (auto simp add: right_distrib Let_def)
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done
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subsection{*Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends*}
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text{*And positive divisors*}
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declare negDivAlg.simps [simp del]
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text{*use with a simproc to avoid repeatedly proving the premise*}
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lemma negDivAlg_eqn:
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     "0 < b ==>  
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      negDivAlg a b =       
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       (if 0\<le>a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))"
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by (rule negDivAlg.simps [THEN trans], simp)
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(*Correctness of negDivAlg: it computes quotients correctly
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  It doesn't work if a=0 because the 0/b equals 0, not -1*)
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lemma negDivAlg_correct:
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  assumes "a < 0" and "b > 0"
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  shows "divmod_rel a b (negDivAlg a b)"
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using prems apply (induct a b rule: negDivAlg.induct)
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apply (auto simp add: linorder_not_le)
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apply (simp add: divmod_rel_def)
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apply (subst negDivAlg_eqn, assumption)
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apply (case_tac "a + b < (0\<Colon>int)")
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apply simp_all
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apply (erule splitE)
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apply (auto simp add: right_distrib Let_def)
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done
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subsection{*Existence Shown by Proving the Division Algorithm to be Correct*}
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(*the case a=0*)
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lemma divmod_rel_0: "b \<noteq> 0 ==> divmod_rel 0 b (0, 0)"
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by (auto simp add: divmod_rel_def linorder_neq_iff)
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lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)"
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by (subst posDivAlg.simps, auto)
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lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)"
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by (subst negDivAlg.simps, auto)
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lemma negateSnd_eq [simp]: "negateSnd(q,r) = (q,-r)"
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by (simp add: negateSnd_def)
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lemma divmod_rel_neg: "divmod_rel (-a) (-b) qr ==> divmod_rel a b (negateSnd qr)"
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by (auto simp add: split_ifs divmod_rel_def)
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lemma divmod_correct: "b \<noteq> 0 ==> divmod_rel a b (divmod a b)"
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by (force simp add: linorder_neq_iff divmod_rel_0 divmod_def divmod_rel_neg
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                    posDivAlg_correct negDivAlg_correct)
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text{*Arbitrary definitions for division by zero.  Useful to simplify 
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    certain equations.*}
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lemma DIVISION_BY_ZERO [simp]: "a div (0::int) = 0 & a mod (0::int) = a"
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by (simp add: div_def mod_def divmod_def posDivAlg.simps)  
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text{*Basic laws about division and remainder*}
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lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"
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apply (case_tac "b = 0", simp)
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apply (cut_tac a = a and b = b in divmod_correct)
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apply (auto simp add: divmod_rel_def div_def mod_def)
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done
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lemma zdiv_zmod_equality: "(b * (a div b) + (a mod b)) + k = (a::int)+k"
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by(simp add: zmod_zdiv_equality[symmetric])
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lemma zdiv_zmod_equality2: "((a div b) * b + (a mod b)) + k = (a::int)+k"
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by(simp add: mult_commute zmod_zdiv_equality[symmetric])
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text {* Tool setup *}
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ML {*
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local 
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structure CancelDivMod = CancelDivModFun(
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struct
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  val div_name = @{const_name Divides.div};
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  val mod_name = @{const_name Divides.mod};
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  val mk_binop = HOLogic.mk_binop;
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  val mk_sum = Int_Numeral_Simprocs.mk_sum HOLogic.intT;
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  val dest_sum = Int_Numeral_Simprocs.dest_sum;
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  val div_mod_eqs =
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    map mk_meta_eq [@{thm zdiv_zmod_equality},
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      @{thm zdiv_zmod_equality2}];
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  val trans = trans;
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  val prove_eq_sums =
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    let
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      val simps = @{thm diff_int_def} :: Int_Numeral_Simprocs.add_0s @ @{thms zadd_ac}
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    in Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac simps) end;
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end)
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in
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val cancel_zdiv_zmod_proc = Simplifier.simproc (the_context ())
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  "cancel_zdiv_zmod" ["(m::int) + n"] (K CancelDivMod.proc)
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end;
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Addsimprocs [cancel_zdiv_zmod_proc]
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*}
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lemma pos_mod_conj : "(0::int) < b ==> 0 \<le> a mod b & a mod b < b"
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apply (cut_tac a = a and b = b in divmod_correct)
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apply (auto simp add: divmod_rel_def mod_def)
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done
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lemmas pos_mod_sign  [simp] = pos_mod_conj [THEN conjunct1, standard]
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   and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2, standard]
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lemma neg_mod_conj : "b < (0::int) ==> a mod b \<le> 0 & b < a mod b"
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apply (cut_tac a = a and b = b in divmod_correct)
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apply (auto simp add: divmod_rel_def div_def mod_def)
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done
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lemmas neg_mod_sign  [simp] = neg_mod_conj [THEN conjunct1, standard]
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   and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2, standard]
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subsection{*General Properties of div and mod*}
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lemma divmod_rel_div_mod: "b \<noteq> 0 ==> divmod_rel a b (a div b, a mod b)"
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apply (cut_tac a = a and b = b in zmod_zdiv_equality)
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apply (force simp add: divmod_rel_def linorder_neq_iff)
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done
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lemma divmod_rel_div: "[| divmod_rel a b (q, r);  b \<noteq> 0 |] ==> a div b = q"
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by (simp add: divmod_rel_div_mod [THEN unique_quotient])
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lemma divmod_rel_mod: "[| divmod_rel a b (q, r);  b \<noteq> 0 |] ==> a mod b = r"
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by (simp add: divmod_rel_div_mod [THEN unique_remainder])
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lemma div_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a div b = 0"
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apply (rule divmod_rel_div)
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apply (auto simp add: divmod_rel_def)
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done
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lemma div_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a div b = 0"
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apply (rule divmod_rel_div)
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apply (auto simp add: divmod_rel_def)
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done
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lemma div_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a div b = -1"
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apply (rule divmod_rel_div)
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apply (auto simp add: divmod_rel_def)
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done
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(*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 divmod_rel_mod)
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apply (auto simp add: divmod_rel_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 divmod_rel_mod)
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apply (auto simp add: divmod_rel_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 divmod_rel_mod)
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apply (auto simp add: divmod_rel_def)
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done
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text{*There is no @{text mod_neg_pos_trivial}.*}
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(*Simpler laws such as -a div b = -(a div b) FAIL, but see just below*)
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lemma zdiv_zminus_zminus [simp]: "(-a) div (-b) = a div (b::int)"
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apply (case_tac "b = 0", simp)
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apply (simp add: divmod_rel_div_mod [THEN divmod_rel_neg, simplified, 
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                                 THEN divmod_rel_div, THEN sym])
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done
69e55066dbca moved Integ files to canonical place;
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   354
69e55066dbca moved Integ files to canonical place;
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   355
(*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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   358
apply (subst divmod_rel_div_mod [THEN divmod_rel_neg, simplified, THEN divmod_rel_mod],
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       auto)
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   360
done
69e55066dbca moved Integ files to canonical place;
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   361
69e55066dbca moved Integ files to canonical place;
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   362
69e55066dbca moved Integ files to canonical place;
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   363
subsection{*Laws for div and mod with Unary Minus*}
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diff changeset
   364
69e55066dbca moved Integ files to canonical place;
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lemma zminus1_lemma:
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     "divmod_rel a b (q, r)
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parents: 29410
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   367
      ==> divmod_rel (-a) b (if r=0 then -q else -q - 1,  
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diff changeset
   368
                          if r=0 then 0 else b-r)"
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by (force simp add: split_ifs divmod_rel_def linorder_neq_iff right_diff_distrib)
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69e55066dbca moved Integ files to canonical place;
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   371
69e55066dbca moved Integ files to canonical place;
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   372
lemma zdiv_zminus1_eq_if:
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parents:
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   373
     "b \<noteq> (0::int)  
69e55066dbca moved Integ files to canonical place;
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parents:
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   374
      ==> (-a) div b =  
69e55066dbca moved Integ files to canonical place;
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          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
29651
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parents: 29410
diff changeset
   376
by (blast intro: divmod_rel_div_mod [THEN zminus1_lemma, THEN divmod_rel_div])
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parents:
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   377
69e55066dbca moved Integ files to canonical place;
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parents:
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   378
lemma zmod_zminus1_eq_if:
69e55066dbca moved Integ files to canonical place;
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parents:
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   379
     "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))"
69e55066dbca moved Integ files to canonical place;
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parents:
diff changeset
   380
apply (case_tac "b = 0", simp)
29651
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parents: 29410
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   381
apply (blast intro: divmod_rel_div_mod [THEN zminus1_lemma, THEN divmod_rel_mod])
23164
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parents:
diff changeset
   382
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   383
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diff changeset
   384
lemma zmod_zminus1_not_zero:
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
   385
  fixes k l :: int
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
   386
  shows "- k mod l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
   387
  unfolding zmod_zminus1_eq_if by auto
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
   388
23164
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   389
lemma zdiv_zminus2: "a div (-b) = (-a::int) div b"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   390
by (cut_tac a = "-a" in zdiv_zminus_zminus, auto)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   391
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   392
lemma zmod_zminus2: "a mod (-b) = - ((-a::int) mod b)"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   393
by (cut_tac a = "-a" and b = b in zmod_zminus_zminus, auto)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   394
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   395
lemma zdiv_zminus2_eq_if:
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   396
     "b \<noteq> (0::int)  
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   397
      ==> a div (-b) =  
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   398
          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   399
by (simp add: zdiv_zminus1_eq_if zdiv_zminus2)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   400
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   401
lemma zmod_zminus2_eq_if:
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   402
     "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   403
by (simp add: zmod_zminus1_eq_if zmod_zminus2)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   404
29936
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
   405
lemma zmod_zminus2_not_zero:
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
   406
  fixes k l :: int
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
   407
  shows "k mod - l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
   408
  unfolding zmod_zminus2_eq_if by auto 
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
   409
23164
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wenzelm
parents:
diff changeset
   410
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   411
subsection{*Division of a Number by Itself*}
69e55066dbca moved Integ files to canonical place;
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parents:
diff changeset
   412
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   413
lemma self_quotient_aux1: "[| (0::int) < a; a = r + a*q; r < a |] ==> 1 \<le> q"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   414
apply (subgoal_tac "0 < a*q")
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   415
 apply (simp add: zero_less_mult_iff, arith)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   416
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   417
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   418
lemma self_quotient_aux2: "[| (0::int) < a; a = r + a*q; 0 \<le> r |] ==> q \<le> 1"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   419
apply (subgoal_tac "0 \<le> a* (1-q) ")
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   420
 apply (simp add: zero_le_mult_iff)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   421
apply (simp add: right_diff_distrib)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   422
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   423
29651
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haftmann
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diff changeset
   424
lemma self_quotient: "[| divmod_rel a a (q, r);  a \<noteq> (0::int) |] ==> q = 1"
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   425
apply (simp add: split_ifs divmod_rel_def linorder_neq_iff)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   426
apply (rule order_antisym, safe, simp_all)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   427
apply (rule_tac [3] a = "-a" and r = "-r" in self_quotient_aux1)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   428
apply (rule_tac a = "-a" and r = "-r" in self_quotient_aux2)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   429
apply (force intro: self_quotient_aux1 self_quotient_aux2 simp add: add_commute)+
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   430
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   431
29651
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   432
lemma self_remainder: "[| divmod_rel a a (q, r);  a \<noteq> (0::int) |] ==> r = 0"
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   433
apply (frule self_quotient, assumption)
29651
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   434
apply (simp add: divmod_rel_def)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   435
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   436
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   437
lemma zdiv_self [simp]: "a \<noteq> 0 ==> a div a = (1::int)"
29651
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   438
by (simp add: divmod_rel_div_mod [THEN self_quotient])
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   439
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   440
(*Here we have 0 mod 0 = 0, also assumed by Knuth (who puts m mod 0 = 0) *)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   441
lemma zmod_self [simp]: "a mod a = (0::int)"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   442
apply (case_tac "a = 0", simp)
29651
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   443
apply (simp add: divmod_rel_div_mod [THEN self_remainder])
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   444
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   445
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   446
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   447
subsection{*Computation of Division and Remainder*}
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   448
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   449
lemma zdiv_zero [simp]: "(0::int) div b = 0"
29651
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   450
by (simp add: div_def divmod_def)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   451
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   452
lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"
29651
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   453
by (simp add: div_def divmod_def)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   454
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   455
lemma zmod_zero [simp]: "(0::int) mod b = 0"
29651
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   456
by (simp add: mod_def divmod_def)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   457
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   458
lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"
29651
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   459
by (simp add: mod_def divmod_def)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   460
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   461
text{*a positive, b positive *}
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   462
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   463
lemma div_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a div b = fst (posDivAlg a b)"
29651
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   464
by (simp add: div_def divmod_def)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   465
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   466
lemma mod_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a mod b = snd (posDivAlg a b)"
29651
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   467
by (simp add: mod_def divmod_def)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   468
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   469
text{*a negative, b positive *}
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   470
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   471
lemma div_neg_pos: "[| a < 0;  0 < b |] ==> a div b = fst (negDivAlg a b)"
29651
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   472
by (simp add: div_def divmod_def)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   473
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   474
lemma mod_neg_pos: "[| a < 0;  0 < b |] ==> a mod b = snd (negDivAlg a b)"
29651
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   475
by (simp add: mod_def divmod_def)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   476
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   477
text{*a positive, b negative *}
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   478
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   479
lemma div_pos_neg:
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   480
     "[| 0 < a;  b < 0 |] ==> a div b = fst (negateSnd (negDivAlg (-a) (-b)))"
29651
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   481
by (simp add: div_def divmod_def)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   482
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   483
lemma mod_pos_neg:
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   484
     "[| 0 < a;  b < 0 |] ==> a mod b = snd (negateSnd (negDivAlg (-a) (-b)))"
29651
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   485
by (simp add: mod_def divmod_def)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   486
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   487
text{*a negative, b negative *}
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   488
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   489
lemma div_neg_neg:
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   490
     "[| a < 0;  b \<le> 0 |] ==> a div b = fst (negateSnd (posDivAlg (-a) (-b)))"
29651
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   491
by (simp add: div_def divmod_def)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   492
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   493
lemma mod_neg_neg:
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   494
     "[| a < 0;  b \<le> 0 |] ==> a mod b = snd (negateSnd (posDivAlg (-a) (-b)))"
29651
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   495
by (simp add: mod_def divmod_def)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   496
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   497
text {*Simplify expresions in which div and mod combine numerical constants*}
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   498
29651
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   499
lemma divmod_relI:
24481
c3a4a289decc ported div/mod simprocs from HOL/ex/Binary.thy
huffman
parents: 24391
diff changeset
   500
  "\<lbrakk>a == b * q + r; if 0 < b then 0 \<le> r \<and> r < b else b < r \<and> r \<le> 0\<rbrakk>
29651
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   501
    \<Longrightarrow> divmod_rel a b (q, r)"
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   502
  unfolding divmod_rel_def by simp
24481
c3a4a289decc ported div/mod simprocs from HOL/ex/Binary.thy
huffman
parents: 24391
diff changeset
   503
29651
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   504
lemmas divmod_rel_div_eq = divmod_relI [THEN divmod_rel_div, THEN eq_reflection]
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   505
lemmas divmod_rel_mod_eq = divmod_relI [THEN divmod_rel_mod, THEN eq_reflection]
24481
c3a4a289decc ported div/mod simprocs from HOL/ex/Binary.thy
huffman
parents: 24391
diff changeset
   506
lemmas arithmetic_simps =
c3a4a289decc ported div/mod simprocs from HOL/ex/Binary.thy
huffman
parents: 24391
diff changeset
   507
  arith_simps
c3a4a289decc ported div/mod simprocs from HOL/ex/Binary.thy
huffman
parents: 24391
diff changeset
   508
  add_special
c3a4a289decc ported div/mod simprocs from HOL/ex/Binary.thy
huffman
parents: 24391
diff changeset
   509
  OrderedGroup.add_0_left
c3a4a289decc ported div/mod simprocs from HOL/ex/Binary.thy
huffman
parents: 24391
diff changeset
   510
  OrderedGroup.add_0_right
c3a4a289decc ported div/mod simprocs from HOL/ex/Binary.thy
huffman
parents: 24391
diff changeset
   511
  mult_zero_left
c3a4a289decc ported div/mod simprocs from HOL/ex/Binary.thy
huffman
parents: 24391
diff changeset
   512
  mult_zero_right
c3a4a289decc ported div/mod simprocs from HOL/ex/Binary.thy
huffman
parents: 24391
diff changeset
   513
  mult_1_left
c3a4a289decc ported div/mod simprocs from HOL/ex/Binary.thy
huffman
parents: 24391
diff changeset
   514
  mult_1_right
c3a4a289decc ported div/mod simprocs from HOL/ex/Binary.thy
huffman
parents: 24391
diff changeset
   515
c3a4a289decc ported div/mod simprocs from HOL/ex/Binary.thy
huffman
parents: 24391
diff changeset
   516
(* simprocs adapted from HOL/ex/Binary.thy *)
c3a4a289decc ported div/mod simprocs from HOL/ex/Binary.thy
huffman
parents: 24391
diff changeset
   517
ML {*
c3a4a289decc ported div/mod simprocs from HOL/ex/Binary.thy
huffman
parents: 24391
diff changeset
   518
local
30517
51a39ed24c0f tuned ML code
haftmann
parents: 30496
diff changeset
   519
  val mk_number = HOLogic.mk_number HOLogic.intT;
51a39ed24c0f tuned ML code
haftmann
parents: 30496
diff changeset
   520
  fun mk_cert u k l = @{term "plus :: int \<Rightarrow> int \<Rightarrow> int"} $
51a39ed24c0f tuned ML code
haftmann
parents: 30496
diff changeset
   521
    (@{term "times :: int \<Rightarrow> int \<Rightarrow> int"} $ u $ mk_number k) $
51a39ed24c0f tuned ML code
haftmann
parents: 30496
diff changeset
   522
      mk_number l;
51a39ed24c0f tuned ML code
haftmann
parents: 30496
diff changeset
   523
  fun prove ctxt prop = Goal.prove ctxt [] [] prop
51a39ed24c0f tuned ML code
haftmann
parents: 30496
diff changeset
   524
    (K (ALLGOALS (full_simp_tac (HOL_basic_ss addsimps @{thms arithmetic_simps}))));
24481
c3a4a289decc ported div/mod simprocs from HOL/ex/Binary.thy
huffman
parents: 24391
diff changeset
   525
  fun binary_proc proc ss ct =
c3a4a289decc ported div/mod simprocs from HOL/ex/Binary.thy
huffman
parents: 24391
diff changeset
   526
    (case Thm.term_of ct of
c3a4a289decc ported div/mod simprocs from HOL/ex/Binary.thy
huffman
parents: 24391
diff changeset
   527
      _ $ t $ u =>
c3a4a289decc ported div/mod simprocs from HOL/ex/Binary.thy
huffman
parents: 24391
diff changeset
   528
      (case try (pairself (`(snd o HOLogic.dest_number))) (t, u) of
c3a4a289decc ported div/mod simprocs from HOL/ex/Binary.thy
huffman
parents: 24391
diff changeset
   529
        SOME args => proc (Simplifier.the_context ss) args
c3a4a289decc ported div/mod simprocs from HOL/ex/Binary.thy
huffman
parents: 24391
diff changeset
   530
      | NONE => NONE)
c3a4a289decc ported div/mod simprocs from HOL/ex/Binary.thy
huffman
parents: 24391
diff changeset
   531
    | _ => NONE);
c3a4a289decc ported div/mod simprocs from HOL/ex/Binary.thy
huffman
parents: 24391
diff changeset
   532
in
30517
51a39ed24c0f tuned ML code
haftmann
parents: 30496
diff changeset
   533
  fun divmod_proc rule = binary_proc (fn ctxt => fn ((m, t), (n, u)) =>
51a39ed24c0f tuned ML code
haftmann
parents: 30496
diff changeset
   534
    if n = 0 then NONE
51a39ed24c0f tuned ML code
haftmann
parents: 30496
diff changeset
   535
    else let val (k, l) = Integer.div_mod m n;
51a39ed24c0f tuned ML code
haftmann
parents: 30496
diff changeset
   536
    in SOME (rule OF [prove ctxt (Logic.mk_equals (t, mk_cert u k l))]) end);
51a39ed24c0f tuned ML code
haftmann
parents: 30496
diff changeset
   537
end
24481
c3a4a289decc ported div/mod simprocs from HOL/ex/Binary.thy
huffman
parents: 24391
diff changeset
   538
*}
c3a4a289decc ported div/mod simprocs from HOL/ex/Binary.thy
huffman
parents: 24391
diff changeset
   539
c3a4a289decc ported div/mod simprocs from HOL/ex/Binary.thy
huffman
parents: 24391
diff changeset
   540
simproc_setup binary_int_div ("number_of m div number_of n :: int") =
29651
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   541
  {* K (divmod_proc (@{thm divmod_rel_div_eq})) *}
24481
c3a4a289decc ported div/mod simprocs from HOL/ex/Binary.thy
huffman
parents: 24391
diff changeset
   542
c3a4a289decc ported div/mod simprocs from HOL/ex/Binary.thy
huffman
parents: 24391
diff changeset
   543
simproc_setup binary_int_mod ("number_of m mod number_of n :: int") =
29651
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   544
  {* K (divmod_proc (@{thm divmod_rel_mod_eq})) *}
24481
c3a4a289decc ported div/mod simprocs from HOL/ex/Binary.thy
huffman
parents: 24391
diff changeset
   545
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   546
lemmas posDivAlg_eqn_number_of [simp] =
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   547
    posDivAlg_eqn [of "number_of v" "number_of w", standard]
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   548
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   549
lemmas negDivAlg_eqn_number_of [simp] =
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   550
    negDivAlg_eqn [of "number_of v" "number_of w", standard]
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   551
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   552
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   553
text{*Special-case simplification *}
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   554
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   555
lemma zmod_minus1_right [simp]: "a mod (-1::int) = 0"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   556
apply (cut_tac a = a and b = "-1" in neg_mod_sign)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   557
apply (cut_tac [2] a = a and b = "-1" in neg_mod_bound)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   558
apply (auto simp del: neg_mod_sign neg_mod_bound)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   559
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   560
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   561
lemma zdiv_minus1_right [simp]: "a div (-1::int) = -a"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   562
by (cut_tac a = a and b = "-1" in zmod_zdiv_equality, auto)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   563
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   564
(** The last remaining special cases for constant arithmetic:
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   565
    1 div z and 1 mod z **)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   566
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   567
lemmas div_pos_pos_1_number_of [simp] =
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   568
    div_pos_pos [OF int_0_less_1, of "number_of w", standard]
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   569
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   570
lemmas div_pos_neg_1_number_of [simp] =
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   571
    div_pos_neg [OF int_0_less_1, of "number_of w", standard]
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   572
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   573
lemmas mod_pos_pos_1_number_of [simp] =
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   574
    mod_pos_pos [OF int_0_less_1, of "number_of w", standard]
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   575
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   576
lemmas mod_pos_neg_1_number_of [simp] =
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   577
    mod_pos_neg [OF int_0_less_1, of "number_of w", standard]
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   578
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   579
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   580
lemmas posDivAlg_eqn_1_number_of [simp] =
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   581
    posDivAlg_eqn [of concl: 1 "number_of w", standard]
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   582
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   583
lemmas negDivAlg_eqn_1_number_of [simp] =
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   584
    negDivAlg_eqn [of concl: 1 "number_of w", standard]
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   585
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   586
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   587
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   588
subsection{*Monotonicity in the First Argument (Dividend)*}
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   589
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   590
lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div b"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   591
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   592
apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   593
apply (rule unique_quotient_lemma)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   594
apply (erule subst)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   595
apply (erule subst, simp_all)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   596
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   597
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   598
lemma zdiv_mono1_neg: "[| a \<le> a';  (b::int) < 0 |] ==> a' div b \<le> a div b"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   599
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   600
apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   601
apply (rule unique_quotient_lemma_neg)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   602
apply (erule subst)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   603
apply (erule subst, simp_all)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   604
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   605
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   606
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   607
subsection{*Monotonicity in the Second Argument (Divisor)*}
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   608
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   609
lemma q_pos_lemma:
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   610
     "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   611
apply (subgoal_tac "0 < b'* (q' + 1) ")
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   612
 apply (simp add: zero_less_mult_iff)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   613
apply (simp add: right_distrib)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   614
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   615
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   616
lemma zdiv_mono2_lemma:
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   617
     "[| b*q + r = b'*q' + r';  0 \<le> b'*q' + r';   
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   618
         r' < b';  0 \<le> r;  0 < b';  b' \<le> b |]   
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   619
      ==> q \<le> (q'::int)"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   620
apply (frule q_pos_lemma, assumption+) 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   621
apply (subgoal_tac "b*q < b* (q' + 1) ")
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   622
 apply (simp add: mult_less_cancel_left)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   623
apply (subgoal_tac "b*q = r' - r + b'*q'")
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   624
 prefer 2 apply simp
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   625
apply (simp (no_asm_simp) add: right_distrib)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   626
apply (subst add_commute, rule zadd_zless_mono, arith)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   627
apply (rule mult_right_mono, auto)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   628
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   629
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   630
lemma zdiv_mono2:
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   631
     "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   632
apply (subgoal_tac "b \<noteq> 0")
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   633
 prefer 2 apply arith
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   634
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   635
apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   636
apply (rule zdiv_mono2_lemma)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   637
apply (erule subst)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   638
apply (erule subst, simp_all)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   639
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   640
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   641
lemma q_neg_lemma:
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   642
     "[| b'*q' + r' < 0;  0 \<le> r';  0 < b' |] ==> q' \<le> (0::int)"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   643
apply (subgoal_tac "b'*q' < 0")
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   644
 apply (simp add: mult_less_0_iff, arith)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   645
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   646
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   647
lemma zdiv_mono2_neg_lemma:
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   648
     "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;   
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   649
         r < b;  0 \<le> r';  0 < b';  b' \<le> b |]   
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   650
      ==> q' \<le> (q::int)"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   651
apply (frule q_neg_lemma, assumption+) 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   652
apply (subgoal_tac "b*q' < b* (q + 1) ")
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   653
 apply (simp add: mult_less_cancel_left)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   654
apply (simp add: right_distrib)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   655
apply (subgoal_tac "b*q' \<le> b'*q'")
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   656
 prefer 2 apply (simp add: mult_right_mono_neg, arith)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   657
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   658
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   659
lemma zdiv_mono2_neg:
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   660
     "[| a < (0::int);  0 < b';  b' \<le> b |] ==> a div b' \<le> a div b"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   661
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   662
apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   663
apply (rule zdiv_mono2_neg_lemma)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   664
apply (erule subst)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   665
apply (erule subst, simp_all)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   666
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   667
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25919
diff changeset
   668
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   669
subsection{*More Algebraic Laws for div and mod*}
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   670
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   671
text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   672
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   673
lemma zmult1_lemma:
29651
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   674
     "[| divmod_rel b c (q, r);  c \<noteq> 0 |]  
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   675
      ==> divmod_rel (a * b) c (a*q + a*r div c, a*r mod c)"
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   676
by (force simp add: split_ifs divmod_rel_def linorder_neq_iff right_distrib)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   677
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   678
lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   679
apply (case_tac "c = 0", simp)
29651
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   680
apply (blast intro: divmod_rel_div_mod [THEN zmult1_lemma, THEN divmod_rel_div])
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   681
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   682
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   683
lemma zmod_zmult1_eq: "(a*b) mod c = a*(b mod c) mod (c::int)"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   684
apply (case_tac "c = 0", simp)
29651
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   685
apply (blast intro: divmod_rel_div_mod [THEN zmult1_lemma, THEN divmod_rel_mod])
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   686
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   687
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29045
diff changeset
   688
lemma zmod_zdiv_trivial: "(a mod b) div b = (0::int)"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26507
diff changeset
   689
apply (case_tac "b = 0", simp)
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26507
diff changeset
   690
apply (auto simp add: linorder_neq_iff div_pos_pos_trivial div_neg_neg_trivial)
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26507
diff changeset
   691
done
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26507
diff changeset
   692
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26507
diff changeset
   693
text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26507
diff changeset
   694
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26507
diff changeset
   695
lemma zadd1_lemma:
29651
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   696
     "[| divmod_rel a c (aq, ar);  divmod_rel b c (bq, br);  c \<noteq> 0 |]  
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   697
      ==> divmod_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   698
by (force simp add: split_ifs divmod_rel_def linorder_neq_iff right_distrib)
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26507
diff changeset
   699
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26507
diff changeset
   700
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26507
diff changeset
   701
lemma zdiv_zadd1_eq:
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26507
diff changeset
   702
     "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26507
diff changeset
   703
apply (case_tac "c = 0", simp)
29651
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   704
apply (blast intro: zadd1_lemma [OF divmod_rel_div_mod divmod_rel_div_mod] divmod_rel_div)
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26507
diff changeset
   705
done
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26507
diff changeset
   706
29405
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
   707
instance int :: ring_div
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26507
diff changeset
   708
proof
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26507
diff changeset
   709
  fix a b c :: int
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26507
diff changeset
   710
  assume not0: "b \<noteq> 0"
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26507
diff changeset
   711
  show "(a + c * b) div b = c + a div b"
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26507
diff changeset
   712
    unfolding zdiv_zadd1_eq [of a "c * b"] using not0 
30181
05629f28f0f7 removed redundant lemmas
nipkow
parents: 30180
diff changeset
   713
      by (simp add: zmod_zmult1_eq zmod_zdiv_trivial zdiv_zmult1_eq)
30930
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30652
diff changeset
   714
next
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30652
diff changeset
   715
  fix a b c :: int
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30652
diff changeset
   716
  assume "a \<noteq> 0"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30652
diff changeset
   717
  then show "(a * b) div (a * c) = b div c"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30652
diff changeset
   718
  proof (cases "b \<noteq> 0 \<and> c \<noteq> 0")
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30652
diff changeset
   719
    case False then show ?thesis by auto
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30652
diff changeset
   720
  next
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30652
diff changeset
   721
    case True then have "b \<noteq> 0" and "c \<noteq> 0" by auto
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30652
diff changeset
   722
    with `a \<noteq> 0`
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30652
diff changeset
   723
    have "\<And>q r. divmod_rel b c (q, r) \<Longrightarrow> divmod_rel (a * b) (a * c) (q, a * r)"
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30652
diff changeset
   724
      apply (auto simp add: divmod_rel_def) 
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30652
diff changeset
   725
      apply (auto simp add: algebra_simps)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30652
diff changeset
   726
      apply (auto simp add: zero_less_mult_iff zero_le_mult_iff mult_le_0_iff)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30652
diff changeset
   727
      apply (simp_all add: mult_less_cancel_left_disj mult_commute [of _ a])
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30652
diff changeset
   728
      done
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30652
diff changeset
   729
    moreover with `c \<noteq> 0` divmod_rel_div_mod have "divmod_rel b c (b div c, b mod c)" by auto
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30652
diff changeset
   730
    ultimately have "divmod_rel (a * b) (a * c) (b div c, a * (b mod c))" .
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30652
diff changeset
   731
    moreover from  `a \<noteq> 0` `c \<noteq> 0` have "a * c \<noteq> 0" by simp
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30652
diff changeset
   732
    ultimately show ?thesis by (rule divmod_rel_div)
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30652
diff changeset
   733
  qed
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26507
diff changeset
   734
qed auto
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25919
diff changeset
   735
29651
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   736
lemma posDivAlg_div_mod:
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   737
  assumes "k \<ge> 0"
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   738
  and "l \<ge> 0"
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   739
  shows "posDivAlg k l = (k div l, k mod l)"
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   740
proof (cases "l = 0")
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   741
  case True then show ?thesis by (simp add: posDivAlg.simps)
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   742
next
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   743
  case False with assms posDivAlg_correct
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   744
    have "divmod_rel k l (fst (posDivAlg k l), snd (posDivAlg k l))"
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   745
    by simp
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   746
  from divmod_rel_div [OF this `l \<noteq> 0`] divmod_rel_mod [OF this `l \<noteq> 0`]
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   747
  show ?thesis by simp
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   748
qed
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   749
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   750
lemma negDivAlg_div_mod:
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   751
  assumes "k < 0"
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   752
  and "l > 0"
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   753
  shows "negDivAlg k l = (k div l, k mod l)"
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   754
proof -
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   755
  from assms have "l \<noteq> 0" by simp
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   756
  from assms negDivAlg_correct
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   757
    have "divmod_rel k l (fst (negDivAlg k l), snd (negDivAlg k l))"
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   758
    by simp
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   759
  from divmod_rel_div [OF this `l \<noteq> 0`] divmod_rel_mod [OF this `l \<noteq> 0`]
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   760
  show ?thesis by simp
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   761
qed
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   762
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   763
lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29045
diff changeset
   764
by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   765
29403
fe17df4e4ab3 generalize some div/mod lemmas; remove type-specific proofs
huffman
parents: 29045
diff changeset
   766
(* REVISIT: should this be generalized to all semiring_div types? *)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   767
lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   768
23983
79dc793bec43 Added lemmas
nipkow
parents: 23969
diff changeset
   769
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   770
subsection{*Proving  @{term "a div (b*c) = (a div b) div c"} *}
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   771
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   772
(*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   773
  7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   774
  to cause particular problems.*)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   775
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   776
text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   777
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   778
lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b*c < b*(q mod c) + r"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   779
apply (subgoal_tac "b * (c - q mod c) < r * 1")
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29410
diff changeset
   780
 apply (simp add: algebra_simps)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   781
apply (rule order_le_less_trans)
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29410
diff changeset
   782
 apply (erule_tac [2] mult_strict_right_mono)
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29410
diff changeset
   783
 apply (rule mult_left_mono_neg)
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29410
diff changeset
   784
  using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps pos_mod_bound)
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29410
diff changeset
   785
 apply (simp)
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29410
diff changeset
   786
apply (simp)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   787
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   788
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   789
lemma zmult2_lemma_aux2:
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   790
     "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   791
apply (subgoal_tac "b * (q mod c) \<le> 0")
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   792
 apply arith
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   793
apply (simp add: mult_le_0_iff)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   794
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   795
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   796
lemma zmult2_lemma_aux3: "[| (0::int) < c;  0 \<le> r;  r < b |] ==> 0 \<le> b * (q mod c) + r"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   797
apply (subgoal_tac "0 \<le> b * (q mod c) ")
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   798
apply arith
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   799
apply (simp add: zero_le_mult_iff)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   800
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   801
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   802
lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   803
apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29410
diff changeset
   804
 apply (simp add: right_diff_distrib)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   805
apply (rule order_less_le_trans)
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29410
diff changeset
   806
 apply (erule mult_strict_right_mono)
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29410
diff changeset
   807
 apply (rule_tac [2] mult_left_mono)
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29410
diff changeset
   808
  apply simp
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29410
diff changeset
   809
 using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps pos_mod_bound)
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29410
diff changeset
   810
apply simp
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   811
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   812
29651
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   813
lemma zmult2_lemma: "[| divmod_rel a b (q, r);  b \<noteq> 0;  0 < c |]  
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   814
      ==> divmod_rel a (b * c) (q div c, b*(q mod c) + r)"
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   815
by (auto simp add: mult_ac divmod_rel_def linorder_neq_iff
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   816
                   zero_less_mult_iff right_distrib [symmetric] 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   817
                   zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   818
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   819
lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   820
apply (case_tac "b = 0", simp)
29651
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   821
apply (force simp add: divmod_rel_div_mod [THEN zmult2_lemma, THEN divmod_rel_div])
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   822
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   823
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   824
lemma zmod_zmult2_eq:
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   825
     "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   826
apply (case_tac "b = 0", simp)
29651
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
   827
apply (force simp add: divmod_rel_div_mod [THEN zmult2_lemma, THEN divmod_rel_mod])
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   828
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   829
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   830
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   831
subsection {*Splitting Rules for div and mod*}
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   832
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   833
text{*The proofs of the two lemmas below are essentially identical*}
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   834
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   835
lemma split_pos_lemma:
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   836
 "0<k ==> 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   837
    P(n div k :: int)(n mod k) = (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i j)"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   838
apply (rule iffI, clarify)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   839
 apply (erule_tac P="P ?x ?y" in rev_mp)  
29948
cdf12a1cb963 Cleaned up IntDiv and removed subsumed lemmas.
nipkow
parents: 29936
diff changeset
   840
 apply (subst mod_add_eq) 
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   841
 apply (subst zdiv_zadd1_eq) 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   842
 apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)  
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   843
txt{*converse direction*}
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   844
apply (drule_tac x = "n div k" in spec) 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   845
apply (drule_tac x = "n mod k" in spec, simp)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   846
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   847
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   848
lemma split_neg_lemma:
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   849
 "k<0 ==>
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   850
    P(n div k :: int)(n mod k) = (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i j)"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   851
apply (rule iffI, clarify)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   852
 apply (erule_tac P="P ?x ?y" in rev_mp)  
29948
cdf12a1cb963 Cleaned up IntDiv and removed subsumed lemmas.
nipkow
parents: 29936
diff changeset
   853
 apply (subst mod_add_eq) 
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   854
 apply (subst zdiv_zadd1_eq) 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   855
 apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)  
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   856
txt{*converse direction*}
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   857
apply (drule_tac x = "n div k" in spec) 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   858
apply (drule_tac x = "n mod k" in spec, simp)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   859
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   860
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   861
lemma split_zdiv:
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   862
 "P(n div k :: int) =
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   863
  ((k = 0 --> P 0) & 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   864
   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) & 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   865
   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   866
apply (case_tac "k=0", simp)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   867
apply (simp only: linorder_neq_iff)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   868
apply (erule disjE) 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   869
 apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"] 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   870
                      split_neg_lemma [of concl: "%x y. P x"])
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   871
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   872
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   873
lemma split_zmod:
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   874
 "P(n mod k :: int) =
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   875
  ((k = 0 --> P n) & 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   876
   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) & 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   877
   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   878
apply (case_tac "k=0", simp)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   879
apply (simp only: linorder_neq_iff)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   880
apply (erule disjE) 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   881
 apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"] 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   882
                      split_neg_lemma [of concl: "%x y. P y"])
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   883
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   884
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   885
(* Enable arith to deal with div 2 and mod 2: *)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   886
declare split_zdiv [of _ _ "number_of k", simplified, standard, arith_split]
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   887
declare split_zmod [of _ _ "number_of k", simplified, standard, arith_split]
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   888
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   889
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   890
subsection{*Speeding up the Division Algorithm with Shifting*}
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   891
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   892
text{*computing div by shifting *}
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   893
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   894
lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   895
proof cases
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   896
  assume "a=0"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   897
    thus ?thesis by simp
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   898
next
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   899
  assume "a\<noteq>0" and le_a: "0\<le>a"   
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   900
  hence a_pos: "1 \<le> a" by arith
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30517
diff changeset
   901
  hence one_less_a2: "1 < 2 * a" by arith
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   902
  hence le_2a: "2 * (1 + b mod a) \<le> 2 * a"
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30517
diff changeset
   903
    unfolding mult_le_cancel_left
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30517
diff changeset
   904
    by (simp add: add1_zle_eq add_commute [of 1])
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   905
  with a_pos have "0 \<le> b mod a" by simp
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   906
  hence le_addm: "0 \<le> 1 mod (2*a) + 2*(b mod a)"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   907
    by (simp add: mod_pos_pos_trivial one_less_a2)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   908
  with  le_2a
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   909
  have "(1 mod (2*a) + 2*(b mod a)) div (2*a) = 0"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   910
    by (simp add: div_pos_pos_trivial le_addm mod_pos_pos_trivial one_less_a2
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   911
                  right_distrib) 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   912
  thus ?thesis
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   913
    by (subst zdiv_zadd1_eq,
30930
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30652
diff changeset
   914
        simp add: mod_mult_mult1 one_less_a2
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   915
                  div_pos_pos_trivial)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   916
qed
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   917
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   918
lemma neg_zdiv_mult_2: "a \<le> (0::int) ==> (1 + 2*b) div (2*a) = (b+1) div a"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   919
apply (subgoal_tac " (1 + 2* (-b - 1)) div (2 * (-a)) = (-b - 1) div (-a) ")
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   920
apply (rule_tac [2] pos_zdiv_mult_2)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   921
apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   922
apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   923
apply (simp only: zdiv_zminus_zminus diff_minus minus_add_distrib [symmetric],
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   924
       simp) 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   925
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   926
26086
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25961
diff changeset
   927
lemma zdiv_number_of_Bit0 [simp]:
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25961
diff changeset
   928
     "number_of (Int.Bit0 v) div number_of (Int.Bit0 w) =  
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25961
diff changeset
   929
          number_of v div (number_of w :: int)"
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25961
diff changeset
   930
by (simp only: number_of_eq numeral_simps) simp
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25961
diff changeset
   931
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25961
diff changeset
   932
lemma zdiv_number_of_Bit1 [simp]:
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25961
diff changeset
   933
     "number_of (Int.Bit1 v) div number_of (Int.Bit0 w) =  
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25961
diff changeset
   934
          (if (0::int) \<le> number_of w                    
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   935
           then number_of v div (number_of w)     
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   936
           else (number_of v + (1::int)) div (number_of w))"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   937
apply (simp only: number_of_eq numeral_simps UNIV_I split: split_if) 
30930
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30652
diff changeset
   938
apply (simp add: pos_zdiv_mult_2 neg_zdiv_mult_2 add_ac)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   939
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   940
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   941
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   942
subsection{*Computing mod by Shifting (proofs resemble those for div)*}
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   943
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   944
lemma pos_zmod_mult_2:
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   945
     "(0::int) \<le> a ==> (1 + 2*b) mod (2*a) = 1 + 2 * (b mod a)"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   946
apply (case_tac "a = 0", simp)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   947
apply (subgoal_tac "1 < a * 2")
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   948
 prefer 2 apply arith
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   949
apply (subgoal_tac "2* (1 + b mod a) \<le> 2*a")
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   950
 apply (rule_tac [2] mult_left_mono)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   951
apply (auto simp add: add_commute [of 1] mult_commute add1_zle_eq 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   952
                      pos_mod_bound)
29948
cdf12a1cb963 Cleaned up IntDiv and removed subsumed lemmas.
nipkow
parents: 29936
diff changeset
   953
apply (subst mod_add_eq)
30930
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30652
diff changeset
   954
apply (simp add: mod_mult_mult2 mod_pos_pos_trivial)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   955
apply (rule mod_pos_pos_trivial)
26086
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25961
diff changeset
   956
apply (auto simp add: mod_pos_pos_trivial ring_distribs)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   957
apply (subgoal_tac "0 \<le> b mod a", arith, simp)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   958
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   959
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   960
lemma neg_zmod_mult_2:
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   961
     "a \<le> (0::int) ==> (1 + 2*b) mod (2*a) = 2 * ((b+1) mod a) - 1"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   962
apply (subgoal_tac "(1 + 2* (-b - 1)) mod (2* (-a)) = 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   963
                    1 + 2* ((-b - 1) mod (-a))")
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   964
apply (rule_tac [2] pos_zmod_mult_2)
30042
31039ee583fa Removed subsumed lemmas
nipkow
parents: 30034
diff changeset
   965
apply (auto simp add: right_diff_distrib)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   966
apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   967
 prefer 2 apply simp 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   968
apply (simp only: zmod_zminus_zminus diff_minus minus_add_distrib [symmetric])
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   969
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   970
26086
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25961
diff changeset
   971
lemma zmod_number_of_Bit0 [simp]:
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25961
diff changeset
   972
     "number_of (Int.Bit0 v) mod number_of (Int.Bit0 w) =  
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25961
diff changeset
   973
      (2::int) * (number_of v mod number_of w)"
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25961
diff changeset
   974
apply (simp only: number_of_eq numeral_simps) 
30930
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30652
diff changeset
   975
apply (simp add: mod_mult_mult1 pos_zmod_mult_2 
29948
cdf12a1cb963 Cleaned up IntDiv and removed subsumed lemmas.
nipkow
parents: 29936
diff changeset
   976
                 neg_zmod_mult_2 add_ac)
26086
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25961
diff changeset
   977
done
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25961
diff changeset
   978
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25961
diff changeset
   979
lemma zmod_number_of_Bit1 [simp]:
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25961
diff changeset
   980
     "number_of (Int.Bit1 v) mod number_of (Int.Bit0 w) =  
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25961
diff changeset
   981
      (if (0::int) \<le> number_of w  
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   982
                then 2 * (number_of v mod number_of w) + 1     
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   983
                else 2 * ((number_of v + (1::int)) mod number_of w) - 1)"
26086
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25961
diff changeset
   984
apply (simp only: number_of_eq numeral_simps) 
30930
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30652
diff changeset
   985
apply (simp add: mod_mult_mult1 pos_zmod_mult_2 
29948
cdf12a1cb963 Cleaned up IntDiv and removed subsumed lemmas.
nipkow
parents: 29936
diff changeset
   986
                 neg_zmod_mult_2 add_ac)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   987
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   988
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   989
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   990
subsection{*Quotients of Signs*}
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   991
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   992
lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   993
apply (subgoal_tac "a div b \<le> -1", force)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   994
apply (rule order_trans)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   995
apply (rule_tac a' = "-1" in zdiv_mono1)
29948
cdf12a1cb963 Cleaned up IntDiv and removed subsumed lemmas.
nipkow
parents: 29936
diff changeset
   996
apply (auto simp add: div_eq_minus1)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   997
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
   998
30323
6a02238da8e9 added lemma
nipkow
parents: 30242
diff changeset
   999
lemma div_nonneg_neg_le0: "[| (0::int) \<le> a; b < 0 |] ==> a div b \<le> 0"
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1000
by (drule zdiv_mono1_neg, auto)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1001
30323
6a02238da8e9 added lemma
nipkow
parents: 30242
diff changeset
  1002
lemma div_nonpos_pos_le0: "[| (a::int) \<le> 0; b > 0 |] ==> a div b \<le> 0"
6a02238da8e9 added lemma
nipkow
parents: 30242
diff changeset
  1003
by (drule zdiv_mono1, auto)
6a02238da8e9 added lemma
nipkow
parents: 30242
diff changeset
  1004
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1005
lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1006
apply auto
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1007
apply (drule_tac [2] zdiv_mono1)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1008
apply (auto simp add: linorder_neq_iff)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1009
apply (simp (no_asm_use) add: linorder_not_less [symmetric])
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1010
apply (blast intro: div_neg_pos_less0)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1011
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1012
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1013
lemma neg_imp_zdiv_nonneg_iff:
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1014
     "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1015
apply (subst zdiv_zminus_zminus [symmetric])
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1016
apply (subst pos_imp_zdiv_nonneg_iff, auto)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1017
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1018
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1019
(*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1020
lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1021
by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1022
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1023
(*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1024
lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1025
by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1026
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1027
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1028
subsection {* The Divides Relation *}
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1029
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1030
lemmas zdvd_iff_zmod_eq_0_number_of [simp] =
30042
31039ee583fa Removed subsumed lemmas
nipkow
parents: 30034
diff changeset
  1031
  dvd_eq_mod_eq_0 [of "number_of x::int" "number_of y::int", standard]
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1032
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1033
lemma zdvd_anti_sym:
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1034
    "0 < m ==> 0 < n ==> m dvd n ==> n dvd m ==> m = (n::int)"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1035
  apply (simp add: dvd_def, auto)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1036
  apply (simp add: mult_assoc zero_less_mult_iff zmult_eq_1_iff)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1037
  done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1038
30042
31039ee583fa Removed subsumed lemmas
nipkow
parents: 30034
diff changeset
  1039
lemma zdvd_dvd_eq: assumes "a \<noteq> 0" and "(a::int) dvd b" and "b dvd a" 
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1040
  shows "\<bar>a\<bar> = \<bar>b\<bar>"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1041
proof-
30042
31039ee583fa Removed subsumed lemmas
nipkow
parents: 30034
diff changeset
  1042
  from `a dvd b` obtain k where k:"b = a*k" unfolding dvd_def by blast 
31039ee583fa Removed subsumed lemmas
nipkow
parents: 30034
diff changeset
  1043
  from `b dvd a` obtain k' where k':"a = b*k'" unfolding dvd_def by blast 
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1044
  from k k' have "a = a*k*k'" by simp
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1045
  with mult_cancel_left1[where c="a" and b="k*k'"]
30042
31039ee583fa Removed subsumed lemmas
nipkow
parents: 30034
diff changeset
  1046
  have kk':"k*k' = 1" using `a\<noteq>0` by (simp add: mult_assoc)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1047
  hence "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1" by (simp add: zmult_eq_1_iff)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1048
  thus ?thesis using k k' by auto
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1049
qed
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1050
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1051
lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1052
  apply (subgoal_tac "m = n + (m - n)")
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1053
   apply (erule ssubst)
30042
31039ee583fa Removed subsumed lemmas
nipkow
parents: 30034
diff changeset
  1054
   apply (blast intro: dvd_add, simp)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1055
  done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1056
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1057
lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"
30042
31039ee583fa Removed subsumed lemmas
nipkow
parents: 30034
diff changeset
  1058
apply (rule iffI)
31039ee583fa Removed subsumed lemmas
nipkow
parents: 30034
diff changeset
  1059
 apply (erule_tac [2] dvd_add)
31039ee583fa Removed subsumed lemmas
nipkow
parents: 30034
diff changeset
  1060
 apply (subgoal_tac "n = (n + k * m) - k * m")
31039ee583fa Removed subsumed lemmas
nipkow
parents: 30034
diff changeset
  1061
  apply (erule ssubst)
31039ee583fa Removed subsumed lemmas
nipkow
parents: 30034
diff changeset
  1062
  apply (erule dvd_diff)
31039ee583fa Removed subsumed lemmas
nipkow
parents: 30034
diff changeset
  1063
  apply(simp_all)
31039ee583fa Removed subsumed lemmas
nipkow
parents: 30034
diff changeset
  1064
done
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1065
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1066
lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"
30930
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30652
diff changeset
  1067
  by (auto elim!: dvdE simp add: mod_mult_mult1)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1068
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1069
lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1070
  apply (subgoal_tac "k dvd n * (m div n) + m mod n")
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1071
   apply (simp add: zmod_zdiv_equality [symmetric])
30042
31039ee583fa Removed subsumed lemmas
nipkow
parents: 30034
diff changeset
  1072
  apply (simp only: dvd_add dvd_mult2)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1073
  done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1074
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1075
lemma zdvd_not_zless: "0 < m ==> m < n ==> \<not> n dvd (m::int)"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26507
diff changeset
  1076
  apply (auto elim!: dvdE)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1077
  apply (subgoal_tac "0 < n")
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1078
   prefer 2
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1079
   apply (blast intro: order_less_trans)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1080
  apply (simp add: zero_less_mult_iff)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1081
  apply (subgoal_tac "n * k < n * 1")
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1082
   apply (drule mult_less_cancel_left [THEN iffD1], auto)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1083
  done
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26507
diff changeset
  1084
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1085
lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1086
  using zmod_zdiv_equality[where a="m" and b="n"]
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29410
diff changeset
  1087
  by (simp add: algebra_simps)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1088
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1089
lemma zdvd_mult_div_cancel:"(n::int) dvd m \<Longrightarrow> n * (m div n) = m"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1090
apply (subgoal_tac "m mod n = 0")
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1091
 apply (simp add: zmult_div_cancel)
30042
31039ee583fa Removed subsumed lemmas
nipkow
parents: 30034
diff changeset
  1092
apply (simp only: dvd_eq_mod_eq_0)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1093
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1094
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1095
lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k \<noteq> (0::int)"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1096
  shows "m dvd n"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1097
proof-
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1098
  from d obtain h where h: "k*n = k*m * h" unfolding dvd_def by blast
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1099
  {assume "n \<noteq> m*h" hence "k* n \<noteq> k* (m*h)" using kz by simp
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1100
    with h have False by (simp add: mult_assoc)}
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1101
  hence "n = m * h" by blast
29410
97916a925a69 remove type-specific proofs
huffman
parents: 29405
diff changeset
  1102
  thus ?thesis by simp
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1103
qed
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1104
23969
ef782bbf2d09 Added cancel simprocs for dvd on nat and int
nipkow
parents: 23853
diff changeset
  1105
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1106
theorem ex_nat: "(\<exists>x::nat. P x) = (\<exists>x::int. 0 <= x \<and> P (nat x))"
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
  1107
apply (simp split add: split_nat)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
  1108
apply (rule iffI)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
  1109
apply (erule exE)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
  1110
apply (rule_tac x = "int x" in exI)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
  1111
apply simp
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
  1112
apply (erule exE)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
  1113
apply (rule_tac x = "nat x" in exI)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
  1114
apply (erule conjE)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
  1115
apply (erule_tac x = "nat x" in allE)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
  1116
apply simp
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
  1117
done
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1118
23365
f31794033ae1 removed constant int :: nat => int;
huffman
parents: 23307
diff changeset
  1119
theorem zdvd_int: "(x dvd y) = (int x dvd int y)"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26507
diff changeset
  1120
proof -
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26507
diff changeset
  1121
  have "\<And>k. int y = int x * k \<Longrightarrow> x dvd y"
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26507
diff changeset
  1122
  proof -
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26507
diff changeset
  1123
    fix k
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26507
diff changeset
  1124
    assume A: "int y = int x * k"
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26507
diff changeset
  1125
    then show "x dvd y" proof (cases k)
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26507
diff changeset
  1126
      case (1 n) with A have "y = x * n" by (simp add: zmult_int)
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26507
diff changeset
  1127
      then show ?thesis ..
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26507
diff changeset
  1128
    next
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26507
diff changeset
  1129
      case (2 n) with A have "int y = int x * (- int (Suc n))" by simp
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26507
diff changeset
  1130
      also have "\<dots> = - (int x * int (Suc n))" by (simp only: mult_minus_right)
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26507
diff changeset
  1131
      also have "\<dots> = - int (x * Suc n)" by (simp only: zmult_int)
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26507
diff changeset
  1132
      finally have "- int (x * Suc n) = int y" ..
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26507
diff changeset
  1133
      then show ?thesis by (simp only: negative_eq_positive) auto
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26507
diff changeset
  1134
    qed
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26507
diff changeset
  1135
  qed
30042
31039ee583fa Removed subsumed lemmas
nipkow
parents: 30034
diff changeset
  1136
  then show ?thesis by (auto elim!: dvdE simp only: dvd_triv_left int_mult)
29410
97916a925a69 remove type-specific proofs
huffman
parents: 29405
diff changeset
  1137
qed
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1138
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1139
lemma zdvd1_eq[simp]: "(x::int) dvd 1 = ( \<bar>x\<bar> = 1)"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1140
proof
30042
31039ee583fa Removed subsumed lemmas
nipkow
parents: 30034
diff changeset
  1141
  assume d: "x dvd 1" hence "int (nat \<bar>x\<bar>) dvd int (nat 1)" by simp
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1142
  hence "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1143
  hence "nat \<bar>x\<bar> = 1"  by simp
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1144
  thus "\<bar>x\<bar> = 1" by (cases "x < 0", auto)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1145
next
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1146
  assume "\<bar>x\<bar>=1" thus "x dvd 1" 
30042
31039ee583fa Removed subsumed lemmas
nipkow
parents: 30034
diff changeset
  1147
    by(cases "x < 0",simp_all add: minus_equation_iff dvd_eq_mod_eq_0)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1148
qed
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1149
lemma zdvd_mult_cancel1: 
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1150
  assumes mp:"m \<noteq>(0::int)" shows "(m * n dvd m) = (\<bar>n\<bar> = 1)"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1151
proof
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1152
  assume n1: "\<bar>n\<bar> = 1" thus "m * n dvd m" 
30042
31039ee583fa Removed subsumed lemmas
nipkow
parents: 30034
diff changeset
  1153
    by (cases "n >0", auto simp add: minus_dvd_iff minus_equation_iff)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1154
next
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1155
  assume H: "m * n dvd m" hence H2: "m * n dvd m * 1" by simp
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1156
  from zdvd_mult_cancel[OF H2 mp] show "\<bar>n\<bar> = 1" by (simp only: zdvd1_eq)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1157
qed
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1158
23365
f31794033ae1 removed constant int :: nat => int;
huffman
parents: 23307
diff changeset
  1159
lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))"
30042
31039ee583fa Removed subsumed lemmas
nipkow
parents: 30034
diff changeset
  1160
  unfolding zdvd_int by (cases "z \<ge> 0") simp_all
23306
cdb027d0637e add int_of_nat versions of lemmas about int::nat=>int
huffman
parents: 23164
diff changeset
  1161
23365
f31794033ae1 removed constant int :: nat => int;
huffman
parents: 23307
diff changeset
  1162
lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)"
30042
31039ee583fa Removed subsumed lemmas
nipkow
parents: 30034
diff changeset
  1163
  unfolding zdvd_int by (cases "z \<ge> 0") simp_all
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1165
lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \<le> z then (z dvd int m) else m = 0)"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 26507
diff changeset
  1166
  by (auto simp add: dvd_int_iff)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1167
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1168
lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \<le> (n::int)"
23365
f31794033ae1 removed constant int :: nat => int;
huffman
parents: 23307
diff changeset
  1169
  apply (rule_tac z=n in int_cases)
f31794033ae1 removed constant int :: nat => int;
huffman
parents: 23307
diff changeset
  1170
  apply (auto simp add: dvd_int_iff)
f31794033ae1 removed constant int :: nat => int;
huffman
parents: 23307
diff changeset
  1171
  apply (rule_tac z=z in int_cases)
23307
2fe3345035c7 modify proofs to avoid referring to int::nat=>int
huffman
parents: 23306
diff changeset
  1172
  apply (auto simp add: dvd_imp_le)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1173
  done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1174
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1175
lemma zpower_zmod: "((x::int) mod m)^y mod m = x^y mod m"
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1176
apply (induct "y", auto)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1177
apply (rule zmod_zmult1_eq [THEN trans])
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1178
apply (simp (no_asm_simp))
29948
cdf12a1cb963 Cleaned up IntDiv and removed subsumed lemmas.
nipkow
parents: 29936
diff changeset
  1179
apply (rule mod_mult_eq [symmetric])
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1180
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1181
23365
f31794033ae1 removed constant int :: nat => int;
huffman
parents: 23307
diff changeset
  1182
lemma zdiv_int: "int (a div b) = (int a) div (int b)"
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1183
apply (subst split_div, auto)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1184
apply (subst split_zdiv, auto)
23365
f31794033ae1 removed constant int :: nat => int;
huffman
parents: 23307
diff changeset
  1185
apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in IntDiv.unique_quotient)
29651
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
  1186
apply (auto simp add: IntDiv.divmod_rel_def of_nat_mult)
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1187
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1188
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1189
lemma zmod_int: "int (a mod b) = (int a) mod (int b)"
23365
f31794033ae1 removed constant int :: nat => int;
huffman
parents: 23307
diff changeset
  1190
apply (subst split_mod, auto)
f31794033ae1 removed constant int :: nat => int;
huffman
parents: 23307
diff changeset
  1191
apply (subst split_zmod, auto)
f31794033ae1 removed constant int :: nat => int;
huffman
parents: 23307
diff changeset
  1192
apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia 
f31794033ae1 removed constant int :: nat => int;
huffman
parents: 23307
diff changeset
  1193
       in unique_remainder)
29651
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
  1194
apply (auto simp add: IntDiv.divmod_rel_def of_nat_mult)
23365
f31794033ae1 removed constant int :: nat => int;
huffman
parents: 23307
diff changeset
  1195
done
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1196
30180
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
  1197
lemma abs_div: "(y::int) dvd x \<Longrightarrow> abs (x div y) = abs x div abs y"
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
  1198
by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)
6d29a873141f added lemmas by Jeremy Avigad
nipkow
parents: 30079
diff changeset
  1199
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1200
text{*Suggested by Matthias Daum*}
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1201
lemma int_power_div_base:
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1202
     "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30042
diff changeset
  1203
apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1204
 apply (erule ssubst)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1205
 apply (simp only: power_add)
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1206
 apply simp_all
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1207
done
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1208
23853
2c69bb1374b8 added lemmas by Brian Huffman
haftmann
parents: 23684
diff changeset
  1209
text {* by Brian Huffman *}
2c69bb1374b8 added lemmas by Brian Huffman
haftmann
parents: 23684
diff changeset
  1210
lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"
29405
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
  1211
by (rule mod_minus_eq [symmetric])
23853
2c69bb1374b8 added lemmas by Brian Huffman
haftmann
parents: 23684
diff changeset
  1212
2c69bb1374b8 added lemmas by Brian Huffman
haftmann
parents: 23684
diff changeset
  1213
lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"
29405
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
  1214
by (rule mod_diff_left_eq [symmetric])
23853
2c69bb1374b8 added lemmas by Brian Huffman
haftmann
parents: 23684
diff changeset
  1215
2c69bb1374b8 added lemmas by Brian Huffman
haftmann
parents: 23684
diff changeset
  1216
lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"
29405
98ab21b14f09 add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents: 29404
diff changeset
  1217
by (rule mod_diff_right_eq [symmetric])
23853
2c69bb1374b8 added lemmas by Brian Huffman
haftmann
parents: 23684
diff changeset
  1218
2c69bb1374b8 added lemmas by Brian Huffman
haftmann
parents: 23684
diff changeset
  1219
lemmas zmod_simps =
30034
60f64f112174 removed redundant thms
nipkow
parents: 30031
diff changeset
  1220
  mod_add_left_eq  [symmetric]
60f64f112174 removed redundant thms
nipkow
parents: 30031
diff changeset
  1221
  mod_add_right_eq [symmetric]
30930
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30652
diff changeset
  1222
  zmod_zmult1_eq   [symmetric]
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30652
diff changeset
  1223
  mod_mult_left_eq [symmetric]
11010e5f18f0 tightended specification of class semiring_div
haftmann
parents: 30652
diff changeset
  1224
  zpower_zmod
23853
2c69bb1374b8 added lemmas by Brian Huffman
haftmann
parents: 23684
diff changeset
  1225
  zminus_zmod zdiff_zmod_left zdiff_zmod_right
2c69bb1374b8 added lemmas by Brian Huffman
haftmann
parents: 23684
diff changeset
  1226
29045
3c8f48333731 move nat_{div,mod}_distrib from NatBin to IntDiv, simplified proofs
huffman
parents: 28562
diff changeset
  1227
text {* Distributive laws for function @{text nat}. *}
3c8f48333731 move nat_{div,mod}_distrib from NatBin to IntDiv, simplified proofs
huffman
parents: 28562
diff changeset
  1228
3c8f48333731 move nat_{div,mod}_distrib from NatBin to IntDiv, simplified proofs
huffman
parents: 28562
diff changeset
  1229
lemma nat_div_distrib: "0 \<le> x \<Longrightarrow> nat (x div y) = nat x div nat y"
3c8f48333731 move nat_{div,mod}_distrib from NatBin to IntDiv, simplified proofs
huffman
parents: 28562
diff changeset
  1230
apply (rule linorder_cases [of y 0])
3c8f48333731 move nat_{div,mod}_distrib from NatBin to IntDiv, simplified proofs
huffman
parents: 28562
diff changeset
  1231
apply (simp add: div_nonneg_neg_le0)
3c8f48333731 move nat_{div,mod}_distrib from NatBin to IntDiv, simplified proofs
huffman
parents: 28562
diff changeset
  1232
apply simp
3c8f48333731 move nat_{div,mod}_distrib from NatBin to IntDiv, simplified proofs
huffman
parents: 28562
diff changeset
  1233
apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)
3c8f48333731 move nat_{div,mod}_distrib from NatBin to IntDiv, simplified proofs
huffman
parents: 28562
diff changeset
  1234
done
3c8f48333731 move nat_{div,mod}_distrib from NatBin to IntDiv, simplified proofs
huffman
parents: 28562
diff changeset
  1235
3c8f48333731 move nat_{div,mod}_distrib from NatBin to IntDiv, simplified proofs
huffman
parents: 28562
diff changeset
  1236
(*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)
3c8f48333731 move nat_{div,mod}_distrib from NatBin to IntDiv, simplified proofs
huffman
parents: 28562
diff changeset
  1237
lemma nat_mod_distrib:
3c8f48333731 move nat_{div,mod}_distrib from NatBin to IntDiv, simplified proofs
huffman
parents: 28562
diff changeset
  1238
  "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x mod y) = nat x mod nat y"
3c8f48333731 move nat_{div,mod}_distrib from NatBin to IntDiv, simplified proofs
huffman
parents: 28562
diff changeset
  1239
apply (case_tac "y = 0", simp add: DIVISION_BY_ZERO)
3c8f48333731 move nat_{div,mod}_distrib from NatBin to IntDiv, simplified proofs
huffman
parents: 28562
diff changeset
  1240
apply (simp add: nat_eq_iff zmod_int)
3c8f48333731 move nat_{div,mod}_distrib from NatBin to IntDiv, simplified proofs
huffman
parents: 28562
diff changeset
  1241
done
3c8f48333731 move nat_{div,mod}_distrib from NatBin to IntDiv, simplified proofs
huffman
parents: 28562
diff changeset
  1242
3c8f48333731 move nat_{div,mod}_distrib from NatBin to IntDiv, simplified proofs
huffman
parents: 28562
diff changeset
  1243
text{*Suggested by Matthias Daum*}
3c8f48333731 move nat_{div,mod}_distrib from NatBin to IntDiv, simplified proofs
huffman
parents: 28562
diff changeset
  1244
lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)"
3c8f48333731 move nat_{div,mod}_distrib from NatBin to IntDiv, simplified proofs
huffman
parents: 28562
diff changeset
  1245
apply (subgoal_tac "nat x div nat k < nat x")
3c8f48333731 move nat_{div,mod}_distrib from NatBin to IntDiv, simplified proofs
huffman
parents: 28562
diff changeset
  1246
 apply (simp (asm_lr) add: nat_div_distrib [symmetric])
3c8f48333731 move nat_{div,mod}_distrib from NatBin to IntDiv, simplified proofs
huffman
parents: 28562
diff changeset
  1247
apply (rule Divides.div_less_dividend, simp_all)
3c8f48333731 move nat_{div,mod}_distrib from NatBin to IntDiv, simplified proofs
huffman
parents: 28562
diff changeset
  1248
done
3c8f48333731 move nat_{div,mod}_distrib from NatBin to IntDiv, simplified proofs
huffman
parents: 28562
diff changeset
  1249
23853
2c69bb1374b8 added lemmas by Brian Huffman
haftmann
parents: 23684
diff changeset
  1250
text {* code generator setup *}
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1251
26507
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26480
diff changeset
  1252
context ring_1
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26480
diff changeset
  1253
begin
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26480
diff changeset
  1254
28562
4e74209f113e `code func` now just `code`
haftmann
parents: 28262
diff changeset
  1255
lemma of_int_num [code]:
26507
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26480
diff changeset
  1256
  "of_int k = (if k = 0 then 0 else if k < 0 then
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26480
diff changeset
  1257
     - of_int (- k) else let
29651
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
  1258
       (l, m) = divmod k 2;
26507
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26480
diff changeset
  1259
       l' = of_int l
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26480
diff changeset
  1260
     in if m = 0 then l' + l' else l' + l' + 1)"
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26480
diff changeset
  1261
proof -
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26480
diff changeset
  1262
  have aux1: "k mod (2\<Colon>int) \<noteq> (0\<Colon>int) \<Longrightarrow> 
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26480
diff changeset
  1263
    of_int k = of_int (k div 2 * 2 + 1)"
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26480
diff changeset
  1264
  proof -
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26480
diff changeset
  1265
    have "k mod 2 < 2" by (auto intro: pos_mod_bound)
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26480
diff changeset
  1266
    moreover have "0 \<le> k mod 2" by (auto intro: pos_mod_sign)
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26480
diff changeset
  1267
    moreover assume "k mod 2 \<noteq> 0"
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26480
diff changeset
  1268
    ultimately have "k mod 2 = 1" by arith
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26480
diff changeset
  1269
    moreover have "of_int k = of_int (k div 2 * 2 + k mod 2)" by simp
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26480
diff changeset
  1270
    ultimately show ?thesis by auto
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26480
diff changeset
  1271
  qed
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26480
diff changeset
  1272
  have aux2: "\<And>x. of_int 2 * x = x + x"
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26480
diff changeset
  1273
  proof -
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26480
diff changeset
  1274
    fix x
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26480
diff changeset
  1275
    have int2: "(2::int) = 1 + 1" by arith
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26480
diff changeset
  1276
    show "of_int 2 * x = x + x"
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26480
diff changeset
  1277
    unfolding int2 of_int_add left_distrib by simp
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26480
diff changeset
  1278
  qed
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26480
diff changeset
  1279
  have aux3: "\<And>x. x * of_int 2 = x + x"
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26480
diff changeset
  1280
  proof -
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26480
diff changeset
  1281
    fix x
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26480
diff changeset
  1282
    have int2: "(2::int) = 1 + 1" by arith
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26480
diff changeset
  1283
    show "x * of_int 2 = x + x" 
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26480
diff changeset
  1284
    unfolding int2 of_int_add right_distrib by simp
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26480
diff changeset
  1285
  qed
29651
16a19466bf81 slightly adapted towards more uniformity with div/mod on nat
haftmann
parents: 29410
diff changeset
  1286
  from aux1 show ?thesis by (auto simp add: divmod_mod_div Let_def aux2 aux3)
26507
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26480
diff changeset
  1287
qed
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26480
diff changeset
  1288
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26480
diff changeset
  1289
end
6da615cef733 moved some code lemmas for Numerals here
haftmann
parents: 26480
diff changeset
  1290
27667
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1291
lemma zmod_eq_dvd_iff: "(x::int) mod n = y mod n \<longleftrightarrow> n dvd x - y"
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1292
proof
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1293
  assume H: "x mod n = y mod n"
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1294
  hence "x mod n - y mod n = 0" by simp
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1295
  hence "(x mod n - y mod n) mod n = 0" by simp 
30034
60f64f112174 removed redundant thms
nipkow
parents: 30031
diff changeset
  1296
  hence "(x - y) mod n = 0" by (simp add: mod_diff_eq[symmetric])
30042
31039ee583fa Removed subsumed lemmas
nipkow
parents: 30034
diff changeset
  1297
  thus "n dvd x - y" by (simp add: dvd_eq_mod_eq_0)
27667
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1298
next
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1299
  assume H: "n dvd x - y"
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1300
  then obtain k where k: "x-y = n*k" unfolding dvd_def by blast
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1301
  hence "x = n*k + y" by simp
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1302
  hence "x mod n = (n*k + y) mod n" by simp
30034
60f64f112174 removed redundant thms
nipkow
parents: 30031
diff changeset
  1303
  thus "x mod n = y mod n" by (simp add: mod_add_left_eq)
27667
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1304
qed
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1305
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1306
lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y  mod n" and xy:"y \<le> x"
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1307
  shows "\<exists>q. x = y + n * q"
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1308
proof-
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1309
  from xy have th: "int x - int y = int (x - y)" by simp 
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1310
  from xyn have "int x mod int n = int y mod int n" 
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1311
    by (simp add: zmod_int[symmetric])
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1312
  hence "int n dvd int x - int y" by (simp only: zmod_eq_dvd_iff[symmetric]) 
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1313
  hence "n dvd x - y" by (simp add: th zdvd_int)
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1314
  then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1315
qed
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1316
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1317
lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)" 
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1318
  (is "?lhs = ?rhs")
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1319
proof
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1320
  assume H: "x mod n = y mod n"
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1321
  {assume xy: "x \<le> y"
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1322
    from H have th: "y mod n = x mod n" by simp
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1323
    from nat_mod_eq_lemma[OF th xy] have ?rhs 
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1324
      apply clarify  apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1325
  moreover
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1326
  {assume xy: "y \<le> x"
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1327
    from nat_mod_eq_lemma[OF H xy] have ?rhs 
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1328
      apply clarify  apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1329
  ultimately  show ?rhs using linear[of x y] by blast  
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1330
next
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1331
  assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1332
  hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1333
  thus  ?lhs by simp
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1334
qed
62500b980749 Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents: 27651
diff changeset
  1335
29936
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
  1336
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30517
diff changeset
  1337
subsection {* Simproc setup *}
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30517
diff changeset
  1338
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30517
diff changeset
  1339
use "Tools/int_factor_simprocs.ML"
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30517
diff changeset
  1340
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30517
diff changeset
  1341
29936
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
  1342
subsection {* Code generation *}
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
  1343
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
  1344
definition pdivmod :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
  1345
  "pdivmod k l = (\<bar>k\<bar> div \<bar>l\<bar>, \<bar>k\<bar> mod \<bar>l\<bar>)"
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
  1346
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
  1347
lemma pdivmod_posDivAlg [code]:
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
  1348
  "pdivmod k l = (if l = 0 then (0, \<bar>k\<bar>) else posDivAlg \<bar>k\<bar> \<bar>l\<bar>)"
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
  1349
by (subst posDivAlg_div_mod) (simp_all add: pdivmod_def)
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
  1350
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
  1351
lemma divmod_pdivmod: "divmod k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
  1352
  apsnd ((op *) (sgn l)) (if 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
  1353
    then pdivmod k l
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
  1354
    else (let (r, s) = pdivmod k l in
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
  1355
      if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
  1356
proof -
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
  1357
  have aux: "\<And>q::int. - k = l * q \<longleftrightarrow> k = l * - q" by auto
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
  1358
  show ?thesis
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
  1359
    by (simp add: divmod_mod_div pdivmod_def)
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
  1360
      (auto simp add: aux not_less not_le zdiv_zminus1_eq_if
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
  1361
      zmod_zminus1_eq_if zdiv_zminus2_eq_if zmod_zminus2_eq_if)
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
  1362
qed
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
  1363
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
  1364
lemma divmod_code [code]: "divmod k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
  1365
  apsnd ((op *) (sgn l)) (if sgn k = sgn l
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
  1366
    then pdivmod k l
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
  1367
    else (let (r, s) = pdivmod k l in
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
  1368
      if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
  1369
proof -
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
  1370
  have "k \<noteq> 0 \<Longrightarrow> l \<noteq> 0 \<Longrightarrow> 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0 \<longleftrightarrow> sgn k = sgn l"
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
  1371
    by (auto simp add: not_less sgn_if)
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
  1372
  then show ?thesis by (simp add: divmod_pdivmod)
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
  1373
qed
d3dfb67f0f59 added pdivmod on int (for code generation)
haftmann
parents: 29700
diff changeset
  1374
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1375
code_modulename SML
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1376
  IntDiv Integer
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1377
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1378
code_modulename OCaml
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1379
  IntDiv Integer
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1380
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1381
code_modulename Haskell
24195
haftmann
parents: 23983
diff changeset
  1382
  IntDiv Integer
23164
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1383
69e55066dbca moved Integ files to canonical place;
wenzelm
parents:
diff changeset
  1384
end