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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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joined theories IntDef, Numeral, IntArith to theory Int
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imports Int Divides FunDef |
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begin |
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definition divmod_rel :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool" where |
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--{*definition of quotient and remainder*} |
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[code]: "divmod_rel a b = (\<lambda>(q, r). a = b * q + r \<and> |
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(if 0 < b then 0 \<le> r \<and> r < b else b < r \<and> r \<le> 0))" |
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definition adjust :: "int \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int" where |
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--{*for the division algorithm*} |
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[code]: "adjust b = (\<lambda>(q, r). if 0 \<le> r - b then (2 * q + 1, r - b) |
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else (2 * q, r))" |
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text{*algorithm for the case @{text "a\<ge>0, b>0"}*} |
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function posDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where |
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"posDivAlg a b = (if a < b \<or> b \<le> 0 then (0, a) |
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else adjust b (posDivAlg a (2 * b)))" |
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by auto |
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termination by (relation "measure (\<lambda>(a, b). nat (a - b + 1))") auto |
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text{*algorithm for the case @{text "a<0, b>0"}*} |
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function negDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where |
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"negDivAlg a b = (if 0 \<le>a + b \<or> b \<le> 0 then (-1, a + b) |
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else adjust b (negDivAlg a (2 * b)))" |
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by auto |
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termination by (relation "measure (\<lambda>(a, b). nat (- a - b))") auto |
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text{*algorithm for the general case @{term "b\<noteq>0"}*} |
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definition negateSnd :: "int \<times> int \<Rightarrow> int \<times> int" where |
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[code inline]: "negateSnd = apsnd uminus" |
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definition divmod :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where |
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--{*The full division algorithm considers all possible signs for a, b |
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including the special case @{text "a=0, b<0"} because |
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@{term negDivAlg} requires @{term "a<0"}.*} |
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"divmod a b = (if 0 \<le> a then if 0 \<le> b then posDivAlg a b |
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else if a = 0 then (0, 0) |
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else negateSnd (negDivAlg (-a) (-b)) |
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else |
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if 0 < b then negDivAlg a b |
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else negateSnd (posDivAlg (-a) (-b)))" |
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instantiation int :: Divides.div |
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begin |
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definition |
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div_def: "a div b = fst (divmod a b)" |
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definition |
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mod_def: "a mod b = snd (divmod a b)" |
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instance .. |
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end |
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lemma divmod_mod_div: |
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"divmod p q = (p div q, p mod q)" |
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by (auto simp add: div_def mod_def) |
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text{* |
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Here is the division algorithm in ML: |
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\begin{verbatim} |
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fun posDivAlg (a,b) = |
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if a<b then (0,a) |
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else let val (q,r) = posDivAlg(a, 2*b) |
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in if 0\<le>r-b then (2*q+1, r-b) else (2*q, r) |
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end |
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fun negDivAlg (a,b) = |
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if 0\<le>a+b then (~1,a+b) |
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else let val (q,r) = negDivAlg(a, 2*b) |
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in if 0\<le>r-b then (2*q+1, r-b) else (2*q, r) |
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end; |
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fun negateSnd (q,r:int) = (q,~r); |
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fun divmod (a,b) = if 0\<le>a then |
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if b>0 then posDivAlg (a,b) |
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else if a=0 then (0,0) |
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else negateSnd (negDivAlg (~a,~b)) |
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else |
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if 0<b then negDivAlg (a,b) |
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else negateSnd (posDivAlg (~a,~b)); |
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\end{verbatim} |
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*} |
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subsection{*Uniqueness and Monotonicity of Quotients and Remainders*} |
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lemma unique_quotient_lemma: |
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"[| b*q' + r' \<le> b*q + r; 0 \<le> r'; r' < b; r < b |] |
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==> q' \<le> (q::int)" |
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apply (subgoal_tac "r' + b * (q'-q) \<le> r") |
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prefer 2 apply (simp add: right_diff_distrib) |
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apply (subgoal_tac "0 < b * (1 + q - q') ") |
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apply (erule_tac [2] order_le_less_trans) |
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prefer 2 apply (simp add: right_diff_distrib right_distrib) |
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apply (subgoal_tac "b * q' < b * (1 + q) ") |
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prefer 2 apply (simp add: right_diff_distrib right_distrib) |
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apply (simp add: mult_less_cancel_left) |
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done |
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lemma unique_quotient_lemma_neg: |
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"[| b*q' + r' \<le> b*q + r; r \<le> 0; b < r; b < r' |] |
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==> q \<le> (q'::int)" |
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by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma, |
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auto) |
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lemma unique_quotient: |
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"[| divmod_rel a b (q, r); divmod_rel a b (q', r'); b \<noteq> 0 |] |
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==> q = q'" |
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apply (simp add: divmod_rel_def linorder_neq_iff split: split_if_asm) |
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apply (blast intro: order_antisym |
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dest: order_eq_refl [THEN unique_quotient_lemma] |
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order_eq_refl [THEN unique_quotient_lemma_neg] sym)+ |
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done |
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lemma unique_remainder: |
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"[| divmod_rel a b (q, r); divmod_rel a b (q', r'); b \<noteq> 0 |] |
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==> r = r'" |
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apply (subgoal_tac "q = q'") |
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apply (simp add: divmod_rel_def) |
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apply (blast intro: unique_quotient) |
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done |
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subsection{*Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends*} |
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text{*And positive divisors*} |
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lemma adjust_eq [simp]: |
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"adjust b (q,r) = |
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(let diff = r-b in |
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if 0 \<le> diff then (2*q + 1, diff) |
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else (2*q, r))" |
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by (simp add: Let_def adjust_def) |
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declare posDivAlg.simps [simp del] |
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text{*use with a simproc to avoid repeatedly proving the premise*} |
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lemma posDivAlg_eqn: |
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"0 < b ==> |
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posDivAlg a b = (if a<b then (0,a) else adjust b (posDivAlg a (2*b)))" |
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by (rule posDivAlg.simps [THEN trans], simp) |
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text{*Correctness of @{term posDivAlg}: it computes quotients correctly*} |
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theorem posDivAlg_correct: |
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assumes "0 \<le> a" and "0 < b" |
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shows "divmod_rel a b (posDivAlg a b)" |
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using prems apply (induct a b rule: posDivAlg.induct) |
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apply auto |
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apply (simp add: divmod_rel_def) |
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apply (subst posDivAlg_eqn, simp add: right_distrib) |
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apply (case_tac "a < b") |
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apply simp_all |
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apply (erule splitE) |
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apply (auto simp add: right_distrib Let_def) |
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done |
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subsection{*Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends*} |
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text{*And positive divisors*} |
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declare negDivAlg.simps [simp del] |
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text{*use with a simproc to avoid repeatedly proving the premise*} |
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lemma negDivAlg_eqn: |
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"0 < b ==> |
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negDivAlg a b = |
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(if 0\<le>a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))" |
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by (rule negDivAlg.simps [THEN trans], simp) |
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(*Correctness of negDivAlg: it computes quotients correctly |
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It doesn't work if a=0 because the 0/b equals 0, not -1*) |
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lemma negDivAlg_correct: |
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assumes "a < 0" and "b > 0" |
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shows "divmod_rel a b (negDivAlg a b)" |
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using prems apply (induct a b rule: negDivAlg.induct) |
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apply (auto simp add: linorder_not_le) |
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apply (simp add: divmod_rel_def) |
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apply (subst negDivAlg_eqn, assumption) |
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apply (case_tac "a + b < (0\<Colon>int)") |
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apply simp_all |
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apply (erule splitE) |
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apply (auto simp add: right_distrib Let_def) |
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done |
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subsection{*Existence Shown by Proving the Division Algorithm to be Correct*} |
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(*the case a=0*) |
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lemma divmod_rel_0: "b \<noteq> 0 ==> divmod_rel 0 b (0, 0)" |
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by (auto simp add: divmod_rel_def linorder_neq_iff) |
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lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)" |
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by (subst posDivAlg.simps, auto) |
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lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)" |
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by (subst negDivAlg.simps, auto) |
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lemma negateSnd_eq [simp]: "negateSnd(q,r) = (q,-r)" |
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by (simp add: negateSnd_def) |
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lemma divmod_rel_neg: "divmod_rel (-a) (-b) qr ==> divmod_rel a b (negateSnd qr)" |
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by (auto simp add: split_ifs divmod_rel_def) |
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lemma divmod_correct: "b \<noteq> 0 ==> divmod_rel a b (divmod a b)" |
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by (force simp add: linorder_neq_iff divmod_rel_0 divmod_def divmod_rel_neg |
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posDivAlg_correct negDivAlg_correct) |
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text{*Arbitrary definitions for division by zero. Useful to simplify |
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certain equations.*} |
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lemma DIVISION_BY_ZERO [simp]: "a div (0::int) = 0 & a mod (0::int) = a" |
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by (simp add: div_def mod_def divmod_def posDivAlg.simps) |
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text{*Basic laws about division and remainder*} |
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lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)" |
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apply (case_tac "b = 0", simp) |
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apply (cut_tac a = a and b = b in divmod_correct) |
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apply (auto simp add: divmod_rel_def div_def mod_def) |
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done |
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lemma zdiv_zmod_equality: "(b * (a div b) + (a mod b)) + k = (a::int)+k" |
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by(simp add: zmod_zdiv_equality[symmetric]) |
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lemma zdiv_zmod_equality2: "((a div b) * b + (a mod b)) + k = (a::int)+k" |
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by(simp add: mult_commute zmod_zdiv_equality[symmetric]) |
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text {* Tool setup *} |
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ML {* |
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local |
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structure CancelDivMod = CancelDivModFun( |
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struct |
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val div_name = @{const_name Divides.div}; |
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val mod_name = @{const_name Divides.mod}; |
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val mk_binop = HOLogic.mk_binop; |
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val mk_sum = Int_Numeral_Simprocs.mk_sum HOLogic.intT; |
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val dest_sum = Int_Numeral_Simprocs.dest_sum; |
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val div_mod_eqs = |
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map mk_meta_eq [@{thm zdiv_zmod_equality}, |
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@{thm zdiv_zmod_equality2}]; |
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val trans = trans; |
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val prove_eq_sums = |
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let |
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val simps = @{thm diff_int_def} :: Int_Numeral_Simprocs.add_0s @ @{thms zadd_ac} |
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in ArithData.prove_conv all_tac (ArithData.simp_all_tac simps) end; |
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end) |
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in |
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val cancel_zdiv_zmod_proc = Simplifier.simproc (the_context ()) |
26101 | 270 |
"cancel_zdiv_zmod" ["(m::int) + n"] (K CancelDivMod.proc) |
23164 | 271 |
|
272 |
end; |
|
273 |
||
274 |
Addsimprocs [cancel_zdiv_zmod_proc] |
|
275 |
*} |
|
276 |
||
277 |
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) |
23164 | 280 |
done |
281 |
||
282 |
lemmas pos_mod_sign [simp] = pos_mod_conj [THEN conjunct1, standard] |
|
283 |
and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2, standard] |
|
284 |
||
285 |
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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287 |
apply (auto simp add: divmod_rel_def div_def mod_def) |
23164 | 288 |
done |
289 |
||
290 |
lemmas neg_mod_sign [simp] = neg_mod_conj [THEN conjunct1, standard] |
|
291 |
and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2, standard] |
|
292 |
||
293 |
||
294 |
||
295 |
subsection{*General Properties of div and mod*} |
|
296 |
||
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lemma divmod_rel_div_mod: "b \<noteq> 0 ==> divmod_rel a b (a div b, a mod b)" |
23164 | 298 |
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) |
23164 | 300 |
done |
301 |
||
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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]) |
23164 | 304 |
|
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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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306 |
by (simp add: divmod_rel_div_mod [THEN unique_remainder]) |
23164 | 307 |
|
308 |
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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310 |
apply (auto simp add: divmod_rel_def) |
23164 | 311 |
done |
312 |
||
313 |
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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315 |
apply (auto simp add: divmod_rel_def) |
23164 | 316 |
done |
317 |
||
318 |
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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320 |
apply (auto simp add: divmod_rel_def) |
23164 | 321 |
done |
322 |
||
323 |
(*There is no div_neg_pos_trivial because 0 div b = 0 would supersede it*) |
|
324 |
||
325 |
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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|
327 |
apply (auto simp add: divmod_rel_def) |
23164 | 328 |
done |
329 |
||
330 |
lemma mod_neg_neg_trivial: "[| a \<le> (0::int); b < a |] ==> a mod b = a" |
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331 |
apply (rule_tac q = 0 in divmod_rel_mod) |
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|
332 |
apply (auto simp add: divmod_rel_def) |
23164 | 333 |
done |
334 |
||
335 |
lemma mod_pos_neg_trivial: "[| (0::int) < a; a+b \<le> 0 |] ==> a mod b = a+b" |
|
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336 |
apply (rule_tac q = "-1" in divmod_rel_mod) |
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|
337 |
apply (auto simp add: divmod_rel_def) |
23164 | 338 |
done |
339 |
||
340 |
text{*There is no @{text mod_neg_pos_trivial}.*} |
|
341 |
||
342 |
||
343 |
(*Simpler laws such as -a div b = -(a div b) FAIL, but see just below*) |
|
344 |
lemma zdiv_zminus_zminus [simp]: "(-a) div (-b) = a div (b::int)" |
|
345 |
apply (case_tac "b = 0", simp) |
|
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346 |
apply (simp add: divmod_rel_div_mod [THEN divmod_rel_neg, simplified, |
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347 |
THEN divmod_rel_div, THEN sym]) |
23164 | 348 |
|
349 |
done |
|
350 |
||
351 |
(*Simpler laws such as -a mod b = -(a mod b) FAIL, but see just below*) |
|
352 |
lemma zmod_zminus_zminus [simp]: "(-a) mod (-b) = - (a mod (b::int))" |
|
353 |
apply (case_tac "b = 0", simp) |
|
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354 |
apply (subst divmod_rel_div_mod [THEN divmod_rel_neg, simplified, THEN divmod_rel_mod], |
23164 | 355 |
auto) |
356 |
done |
|
357 |
||
358 |
||
359 |
subsection{*Laws for div and mod with Unary Minus*} |
|
360 |
||
361 |
lemma zminus1_lemma: |
|
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|
362 |
"divmod_rel a b (q, r) |
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|
363 |
==> divmod_rel (-a) b (if r=0 then -q else -q - 1, |
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|
364 |
if r=0 then 0 else b-r)" |
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|
365 |
by (force simp add: split_ifs divmod_rel_def linorder_neq_iff right_diff_distrib) |
23164 | 366 |
|
367 |
||
368 |
lemma zdiv_zminus1_eq_if: |
|
369 |
"b \<noteq> (0::int) |
|
370 |
==> (-a) div b = |
|
371 |
(if a mod b = 0 then - (a div b) else - (a div b) - 1)" |
|
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|
372 |
by (blast intro: divmod_rel_div_mod [THEN zminus1_lemma, THEN divmod_rel_div]) |
23164 | 373 |
|
374 |
lemma zmod_zminus1_eq_if: |
|
375 |
"(-a::int) mod b = (if a mod b = 0 then 0 else b - (a mod b))" |
|
376 |
apply (case_tac "b = 0", simp) |
|
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|
377 |
apply (blast intro: divmod_rel_div_mod [THEN zminus1_lemma, THEN divmod_rel_mod]) |
23164 | 378 |
done |
379 |
||
380 |
lemma zdiv_zminus2: "a div (-b) = (-a::int) div b" |
|
381 |
by (cut_tac a = "-a" in zdiv_zminus_zminus, auto) |
|
382 |
||
383 |
lemma zmod_zminus2: "a mod (-b) = - ((-a::int) mod b)" |
|
384 |
by (cut_tac a = "-a" and b = b in zmod_zminus_zminus, auto) |
|
385 |
||
386 |
lemma zdiv_zminus2_eq_if: |
|
387 |
"b \<noteq> (0::int) |
|
388 |
==> a div (-b) = |
|
389 |
(if a mod b = 0 then - (a div b) else - (a div b) - 1)" |
|
390 |
by (simp add: zdiv_zminus1_eq_if zdiv_zminus2) |
|
391 |
||
392 |
lemma zmod_zminus2_eq_if: |
|
393 |
"a mod (-b::int) = (if a mod b = 0 then 0 else (a mod b) - b)" |
|
394 |
by (simp add: zmod_zminus1_eq_if zmod_zminus2) |
|
395 |
||
396 |
||
397 |
subsection{*Division of a Number by Itself*} |
|
398 |
||
399 |
lemma self_quotient_aux1: "[| (0::int) < a; a = r + a*q; r < a |] ==> 1 \<le> q" |
|
400 |
apply (subgoal_tac "0 < a*q") |
|
401 |
apply (simp add: zero_less_mult_iff, arith) |
|
402 |
done |
|
403 |
||
404 |
lemma self_quotient_aux2: "[| (0::int) < a; a = r + a*q; 0 \<le> r |] ==> q \<le> 1" |
|
405 |
apply (subgoal_tac "0 \<le> a* (1-q) ") |
|
406 |
apply (simp add: zero_le_mult_iff) |
|
407 |
apply (simp add: right_diff_distrib) |
|
408 |
done |
|
409 |
||
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|
410 |
lemma self_quotient: "[| divmod_rel a a (q, r); a \<noteq> (0::int) |] ==> q = 1" |
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|
411 |
apply (simp add: split_ifs divmod_rel_def linorder_neq_iff) |
23164 | 412 |
apply (rule order_antisym, safe, simp_all) |
413 |
apply (rule_tac [3] a = "-a" and r = "-r" in self_quotient_aux1) |
|
414 |
apply (rule_tac a = "-a" and r = "-r" in self_quotient_aux2) |
|
415 |
apply (force intro: self_quotient_aux1 self_quotient_aux2 simp add: add_commute)+ |
|
416 |
done |
|
417 |
||
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|
418 |
lemma self_remainder: "[| divmod_rel a a (q, r); a \<noteq> (0::int) |] ==> r = 0" |
23164 | 419 |
apply (frule self_quotient, assumption) |
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|
420 |
apply (simp add: divmod_rel_def) |
23164 | 421 |
done |
422 |
||
423 |
lemma zdiv_self [simp]: "a \<noteq> 0 ==> a div a = (1::int)" |
|
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|
424 |
by (simp add: divmod_rel_div_mod [THEN self_quotient]) |
23164 | 425 |
|
426 |
(*Here we have 0 mod 0 = 0, also assumed by Knuth (who puts m mod 0 = 0) *) |
|
427 |
lemma zmod_self [simp]: "a mod a = (0::int)" |
|
428 |
apply (case_tac "a = 0", simp) |
|
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|
429 |
apply (simp add: divmod_rel_div_mod [THEN self_remainder]) |
23164 | 430 |
done |
431 |
||
432 |
||
433 |
subsection{*Computation of Division and Remainder*} |
|
434 |
||
435 |
lemma zdiv_zero [simp]: "(0::int) div b = 0" |
|
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|
436 |
by (simp add: div_def divmod_def) |
23164 | 437 |
|
438 |
lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1" |
|
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|
439 |
by (simp add: div_def divmod_def) |
23164 | 440 |
|
441 |
lemma zmod_zero [simp]: "(0::int) mod b = 0" |
|
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|
442 |
by (simp add: mod_def divmod_def) |
23164 | 443 |
|
444 |
lemma zdiv_minus1: "(0::int) < b ==> -1 div b = -1" |
|
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|
445 |
by (simp add: div_def divmod_def) |
23164 | 446 |
|
447 |
lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1" |
|
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|
448 |
by (simp add: mod_def divmod_def) |
23164 | 449 |
|
450 |
text{*a positive, b positive *} |
|
451 |
||
452 |
lemma div_pos_pos: "[| 0 < a; 0 \<le> b |] ==> a div b = fst (posDivAlg a b)" |
|
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|
453 |
by (simp add: div_def divmod_def) |
23164 | 454 |
|
455 |
lemma mod_pos_pos: "[| 0 < a; 0 \<le> b |] ==> a mod b = snd (posDivAlg a b)" |
|
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|
456 |
by (simp add: mod_def divmod_def) |
23164 | 457 |
|
458 |
text{*a negative, b positive *} |
|
459 |
||
460 |
lemma div_neg_pos: "[| a < 0; 0 < b |] ==> a div b = fst (negDivAlg a b)" |
|
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|
461 |
by (simp add: div_def divmod_def) |
23164 | 462 |
|
463 |
lemma mod_neg_pos: "[| a < 0; 0 < b |] ==> a mod b = snd (negDivAlg a b)" |
|
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|
464 |
by (simp add: mod_def divmod_def) |
23164 | 465 |
|
466 |
text{*a positive, b negative *} |
|
467 |
||
468 |
lemma div_pos_neg: |
|
469 |
"[| 0 < a; b < 0 |] ==> a div b = fst (negateSnd (negDivAlg (-a) (-b)))" |
|
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|
470 |
by (simp add: div_def divmod_def) |
23164 | 471 |
|
472 |
lemma mod_pos_neg: |
|
473 |
"[| 0 < a; b < 0 |] ==> a mod b = snd (negateSnd (negDivAlg (-a) (-b)))" |
|
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|
474 |
by (simp add: mod_def divmod_def) |
23164 | 475 |
|
476 |
text{*a negative, b negative *} |
|
477 |
||
478 |
lemma div_neg_neg: |
|
479 |
"[| a < 0; b \<le> 0 |] ==> a div b = fst (negateSnd (posDivAlg (-a) (-b)))" |
|
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|
480 |
by (simp add: div_def divmod_def) |
23164 | 481 |
|
482 |
lemma mod_neg_neg: |
|
483 |
"[| a < 0; b \<le> 0 |] ==> a mod b = snd (negateSnd (posDivAlg (-a) (-b)))" |
|
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|
484 |
by (simp add: mod_def divmod_def) |
23164 | 485 |
|
486 |
text {*Simplify expresions in which div and mod combine numerical constants*} |
|
487 |
||
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|
488 |
lemma divmod_relI: |
24481 | 489 |
"\<lbrakk>a == b * q + r; if 0 < b then 0 \<le> r \<and> r < b else b < r \<and> r \<le> 0\<rbrakk> |
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|
490 |
\<Longrightarrow> divmod_rel a b (q, r)" |
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|
491 |
unfolding divmod_rel_def by simp |
24481 | 492 |
|
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|
493 |
lemmas divmod_rel_div_eq = divmod_relI [THEN divmod_rel_div, THEN eq_reflection] |
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|
494 |
lemmas divmod_rel_mod_eq = divmod_relI [THEN divmod_rel_mod, THEN eq_reflection] |
24481 | 495 |
lemmas arithmetic_simps = |
496 |
arith_simps |
|
497 |
add_special |
|
498 |
OrderedGroup.add_0_left |
|
499 |
OrderedGroup.add_0_right |
|
500 |
mult_zero_left |
|
501 |
mult_zero_right |
|
502 |
mult_1_left |
|
503 |
mult_1_right |
|
504 |
||
505 |
(* simprocs adapted from HOL/ex/Binary.thy *) |
|
506 |
ML {* |
|
507 |
local |
|
508 |
infix ==; |
|
509 |
val op == = Logic.mk_equals; |
|
510 |
fun plus m n = @{term "plus :: int \<Rightarrow> int \<Rightarrow> int"} $ m $ n; |
|
511 |
fun mult m n = @{term "times :: int \<Rightarrow> int \<Rightarrow> int"} $ m $ n; |
|
512 |
||
513 |
val binary_ss = HOL_basic_ss addsimps @{thms arithmetic_simps}; |
|
514 |
fun prove ctxt prop = |
|
515 |
Goal.prove ctxt [] [] prop (fn _ => ALLGOALS (full_simp_tac binary_ss)); |
|
516 |
||
517 |
fun binary_proc proc ss ct = |
|
518 |
(case Thm.term_of ct of |
|
519 |
_ $ t $ u => |
|
520 |
(case try (pairself (`(snd o HOLogic.dest_number))) (t, u) of |
|
521 |
SOME args => proc (Simplifier.the_context ss) args |
|
522 |
| NONE => NONE) |
|
523 |
| _ => NONE); |
|
524 |
in |
|
525 |
||
526 |
fun divmod_proc rule = binary_proc (fn ctxt => fn ((m, t), (n, u)) => |
|
527 |
if n = 0 then NONE |
|
528 |
else |
|
24630
351a308ab58d
simplified type int (eliminated IntInf.int, integer);
wenzelm
parents:
24490
diff
changeset
|
529 |
let val (k, l) = Integer.div_mod m n; |
24481 | 530 |
fun mk_num x = HOLogic.mk_number HOLogic.intT x; |
531 |
in SOME (rule OF [prove ctxt (t == plus (mult u (mk_num k)) (mk_num l))]) |
|
532 |
end); |
|
533 |
||
534 |
end; |
|
535 |
*} |
|
536 |
||
537 |
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
|
538 |
{* K (divmod_proc (@{thm divmod_rel_div_eq})) *} |
24481 | 539 |
|
540 |
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
|
541 |
{* K (divmod_proc (@{thm divmod_rel_mod_eq})) *} |
24481 | 542 |
|
543 |
(* The following 8 lemmas are made unnecessary by the above simprocs: *) |
|
544 |
||
545 |
lemmas div_pos_pos_number_of = |
|
23164 | 546 |
div_pos_pos [of "number_of v" "number_of w", standard] |
547 |
||
24481 | 548 |
lemmas div_neg_pos_number_of = |
23164 | 549 |
div_neg_pos [of "number_of v" "number_of w", standard] |
550 |
||
24481 | 551 |
lemmas div_pos_neg_number_of = |
23164 | 552 |
div_pos_neg [of "number_of v" "number_of w", standard] |
553 |
||
24481 | 554 |
lemmas div_neg_neg_number_of = |
23164 | 555 |
div_neg_neg [of "number_of v" "number_of w", standard] |
556 |
||
557 |
||
24481 | 558 |
lemmas mod_pos_pos_number_of = |
23164 | 559 |
mod_pos_pos [of "number_of v" "number_of w", standard] |
560 |
||
24481 | 561 |
lemmas mod_neg_pos_number_of = |
23164 | 562 |
mod_neg_pos [of "number_of v" "number_of w", standard] |
563 |
||
24481 | 564 |
lemmas mod_pos_neg_number_of = |
23164 | 565 |
mod_pos_neg [of "number_of v" "number_of w", standard] |
566 |
||
24481 | 567 |
lemmas mod_neg_neg_number_of = |
23164 | 568 |
mod_neg_neg [of "number_of v" "number_of w", standard] |
569 |
||
570 |
||
571 |
lemmas posDivAlg_eqn_number_of [simp] = |
|
572 |
posDivAlg_eqn [of "number_of v" "number_of w", standard] |
|
573 |
||
574 |
lemmas negDivAlg_eqn_number_of [simp] = |
|
575 |
negDivAlg_eqn [of "number_of v" "number_of w", standard] |
|
576 |
||
577 |
||
578 |
text{*Special-case simplification *} |
|
579 |
||
580 |
lemma zmod_1 [simp]: "a mod (1::int) = 0" |
|
581 |
apply (cut_tac a = a and b = 1 in pos_mod_sign) |
|
582 |
apply (cut_tac [2] a = a and b = 1 in pos_mod_bound) |
|
583 |
apply (auto simp del:pos_mod_bound pos_mod_sign) |
|
584 |
done |
|
585 |
||
586 |
lemma zdiv_1 [simp]: "a div (1::int) = a" |
|
587 |
by (cut_tac a = a and b = 1 in zmod_zdiv_equality, auto) |
|
588 |
||
589 |
lemma zmod_minus1_right [simp]: "a mod (-1::int) = 0" |
|
590 |
apply (cut_tac a = a and b = "-1" in neg_mod_sign) |
|
591 |
apply (cut_tac [2] a = a and b = "-1" in neg_mod_bound) |
|
592 |
apply (auto simp del: neg_mod_sign neg_mod_bound) |
|
593 |
done |
|
594 |
||
595 |
lemma zdiv_minus1_right [simp]: "a div (-1::int) = -a" |
|
596 |
by (cut_tac a = a and b = "-1" in zmod_zdiv_equality, auto) |
|
597 |
||
598 |
(** The last remaining special cases for constant arithmetic: |
|
599 |
1 div z and 1 mod z **) |
|
600 |
||
601 |
lemmas div_pos_pos_1_number_of [simp] = |
|
602 |
div_pos_pos [OF int_0_less_1, of "number_of w", standard] |
|
603 |
||
604 |
lemmas div_pos_neg_1_number_of [simp] = |
|
605 |
div_pos_neg [OF int_0_less_1, of "number_of w", standard] |
|
606 |
||
607 |
lemmas mod_pos_pos_1_number_of [simp] = |
|
608 |
mod_pos_pos [OF int_0_less_1, of "number_of w", standard] |
|
609 |
||
610 |
lemmas mod_pos_neg_1_number_of [simp] = |
|
611 |
mod_pos_neg [OF int_0_less_1, of "number_of w", standard] |
|
612 |
||
613 |
||
614 |
lemmas posDivAlg_eqn_1_number_of [simp] = |
|
615 |
posDivAlg_eqn [of concl: 1 "number_of w", standard] |
|
616 |
||
617 |
lemmas negDivAlg_eqn_1_number_of [simp] = |
|
618 |
negDivAlg_eqn [of concl: 1 "number_of w", standard] |
|
619 |
||
620 |
||
621 |
||
622 |
subsection{*Monotonicity in the First Argument (Dividend)*} |
|
623 |
||
624 |
lemma zdiv_mono1: "[| a \<le> a'; 0 < (b::int) |] ==> a div b \<le> a' div b" |
|
625 |
apply (cut_tac a = a and b = b in zmod_zdiv_equality) |
|
626 |
apply (cut_tac a = a' and b = b in zmod_zdiv_equality) |
|
627 |
apply (rule unique_quotient_lemma) |
|
628 |
apply (erule subst) |
|
629 |
apply (erule subst, simp_all) |
|
630 |
done |
|
631 |
||
632 |
lemma zdiv_mono1_neg: "[| a \<le> a'; (b::int) < 0 |] ==> a' div b \<le> a div b" |
|
633 |
apply (cut_tac a = a and b = b in zmod_zdiv_equality) |
|
634 |
apply (cut_tac a = a' and b = b in zmod_zdiv_equality) |
|
635 |
apply (rule unique_quotient_lemma_neg) |
|
636 |
apply (erule subst) |
|
637 |
apply (erule subst, simp_all) |
|
638 |
done |
|
639 |
||
640 |
||
641 |
subsection{*Monotonicity in the Second Argument (Divisor)*} |
|
642 |
||
643 |
lemma q_pos_lemma: |
|
644 |
"[| 0 \<le> b'*q' + r'; r' < b'; 0 < b' |] ==> 0 \<le> (q'::int)" |
|
645 |
apply (subgoal_tac "0 < b'* (q' + 1) ") |
|
646 |
apply (simp add: zero_less_mult_iff) |
|
647 |
apply (simp add: right_distrib) |
|
648 |
done |
|
649 |
||
650 |
lemma zdiv_mono2_lemma: |
|
651 |
"[| b*q + r = b'*q' + r'; 0 \<le> b'*q' + r'; |
|
652 |
r' < b'; 0 \<le> r; 0 < b'; b' \<le> b |] |
|
653 |
==> q \<le> (q'::int)" |
|
654 |
apply (frule q_pos_lemma, assumption+) |
|
655 |
apply (subgoal_tac "b*q < b* (q' + 1) ") |
|
656 |
apply (simp add: mult_less_cancel_left) |
|
657 |
apply (subgoal_tac "b*q = r' - r + b'*q'") |
|
658 |
prefer 2 apply simp |
|
659 |
apply (simp (no_asm_simp) add: right_distrib) |
|
660 |
apply (subst add_commute, rule zadd_zless_mono, arith) |
|
661 |
apply (rule mult_right_mono, auto) |
|
662 |
done |
|
663 |
||
664 |
lemma zdiv_mono2: |
|
665 |
"[| (0::int) \<le> a; 0 < b'; b' \<le> b |] ==> a div b \<le> a div b'" |
|
666 |
apply (subgoal_tac "b \<noteq> 0") |
|
667 |
prefer 2 apply arith |
|
668 |
apply (cut_tac a = a and b = b in zmod_zdiv_equality) |
|
669 |
apply (cut_tac a = a and b = b' in zmod_zdiv_equality) |
|
670 |
apply (rule zdiv_mono2_lemma) |
|
671 |
apply (erule subst) |
|
672 |
apply (erule subst, simp_all) |
|
673 |
done |
|
674 |
||
675 |
lemma q_neg_lemma: |
|
676 |
"[| b'*q' + r' < 0; 0 \<le> r'; 0 < b' |] ==> q' \<le> (0::int)" |
|
677 |
apply (subgoal_tac "b'*q' < 0") |
|
678 |
apply (simp add: mult_less_0_iff, arith) |
|
679 |
done |
|
680 |
||
681 |
lemma zdiv_mono2_neg_lemma: |
|
682 |
"[| b*q + r = b'*q' + r'; b'*q' + r' < 0; |
|
683 |
r < b; 0 \<le> r'; 0 < b'; b' \<le> b |] |
|
684 |
==> q' \<le> (q::int)" |
|
685 |
apply (frule q_neg_lemma, assumption+) |
|
686 |
apply (subgoal_tac "b*q' < b* (q + 1) ") |
|
687 |
apply (simp add: mult_less_cancel_left) |
|
688 |
apply (simp add: right_distrib) |
|
689 |
apply (subgoal_tac "b*q' \<le> b'*q'") |
|
690 |
prefer 2 apply (simp add: mult_right_mono_neg, arith) |
|
691 |
done |
|
692 |
||
693 |
lemma zdiv_mono2_neg: |
|
694 |
"[| a < (0::int); 0 < b'; b' \<le> b |] ==> a div b' \<le> a div b" |
|
695 |
apply (cut_tac a = a and b = b in zmod_zdiv_equality) |
|
696 |
apply (cut_tac a = a and b = b' in zmod_zdiv_equality) |
|
697 |
apply (rule zdiv_mono2_neg_lemma) |
|
698 |
apply (erule subst) |
|
699 |
apply (erule subst, simp_all) |
|
700 |
done |
|
701 |
||
25942 | 702 |
|
23164 | 703 |
subsection{*More Algebraic Laws for div and mod*} |
704 |
||
705 |
text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *} |
|
706 |
||
707 |
lemma zmult1_lemma: |
|
29651
16a19466bf81
slightly adapted towards more uniformity with div/mod on nat
haftmann
parents:
29410
diff
changeset
|
708 |
"[| 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
|
709 |
==> 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
|
710 |
by (force simp add: split_ifs divmod_rel_def linorder_neq_iff right_distrib) |
23164 | 711 |
|
712 |
lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)" |
|
713 |
apply (case_tac "c = 0", simp) |
|
29651
16a19466bf81
slightly adapted towards more uniformity with div/mod on nat
haftmann
parents:
29410
diff
changeset
|
714 |
apply (blast intro: divmod_rel_div_mod [THEN zmult1_lemma, THEN divmod_rel_div]) |
23164 | 715 |
done |
716 |
||
717 |
lemma zmod_zmult1_eq: "(a*b) mod c = a*(b mod c) mod (c::int)" |
|
718 |
apply (case_tac "c = 0", simp) |
|
29651
16a19466bf81
slightly adapted towards more uniformity with div/mod on nat
haftmann
parents:
29410
diff
changeset
|
719 |
apply (blast intro: divmod_rel_div_mod [THEN zmult1_lemma, THEN divmod_rel_mod]) |
23164 | 720 |
done |
721 |
||
722 |
lemma zmod_zmult1_eq': "(a*b) mod (c::int) = ((a mod c) * b) mod c" |
|
723 |
apply (rule trans) |
|
724 |
apply (rule_tac s = "b*a mod c" in trans) |
|
725 |
apply (rule_tac [2] zmod_zmult1_eq) |
|
726 |
apply (simp_all add: mult_commute) |
|
727 |
done |
|
728 |
||
729 |
lemma zmod_zmult_distrib: "(a*b) mod (c::int) = ((a mod c) * (b mod c)) mod c" |
|
730 |
apply (rule zmod_zmult1_eq' [THEN trans]) |
|
731 |
apply (rule zmod_zmult1_eq) |
|
732 |
done |
|
733 |
||
734 |
lemma zdiv_zmult_self1 [simp]: "b \<noteq> (0::int) ==> (a*b) div b = a" |
|
735 |
by (simp add: zdiv_zmult1_eq) |
|
736 |
||
29403
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29045
diff
changeset
|
737 |
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
|
738 |
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
|
739 |
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
|
740 |
done |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
26507
diff
changeset
|
741 |
|
29403
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29045
diff
changeset
|
742 |
lemma zmod_zmod_trivial: "(a mod b) mod b = a mod (b::int)" |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
26507
diff
changeset
|
743 |
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
|
744 |
apply (force simp add: linorder_neq_iff mod_pos_pos_trivial mod_neg_neg_trivial) |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
26507
diff
changeset
|
745 |
done |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
26507
diff
changeset
|
746 |
|
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
26507
diff
changeset
|
747 |
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
|
748 |
|
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
26507
diff
changeset
|
749 |
lemma zadd1_lemma: |
29651
16a19466bf81
slightly adapted towards more uniformity with div/mod on nat
haftmann
parents:
29410
diff
changeset
|
750 |
"[| 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
|
751 |
==> 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
|
752 |
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
|
753 |
|
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
26507
diff
changeset
|
754 |
(*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
|
755 |
lemma zdiv_zadd1_eq: |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
26507
diff
changeset
|
756 |
"(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
|
757 |
apply (case_tac "c = 0", simp) |
29651
16a19466bf81
slightly adapted towards more uniformity with div/mod on nat
haftmann
parents:
29410
diff
changeset
|
758 |
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
|
759 |
done |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
26507
diff
changeset
|
760 |
|
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
26507
diff
changeset
|
761 |
lemma zmod_zadd1_eq: "(a+b) mod (c::int) = (a mod c + b mod c) mod c" |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
26507
diff
changeset
|
762 |
apply (case_tac "c = 0", simp) |
29651
16a19466bf81
slightly adapted towards more uniformity with div/mod on nat
haftmann
parents:
29410
diff
changeset
|
763 |
apply (blast intro: zadd1_lemma [OF divmod_rel_div_mod divmod_rel_div_mod] divmod_rel_mod) |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
26507
diff
changeset
|
764 |
done |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
26507
diff
changeset
|
765 |
|
29405
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
766 |
instance int :: ring_div |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
26507
diff
changeset
|
767 |
proof |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
26507
diff
changeset
|
768 |
fix a b c :: int |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
26507
diff
changeset
|
769 |
assume not0: "b \<noteq> 0" |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
26507
diff
changeset
|
770 |
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
|
771 |
unfolding zdiv_zadd1_eq [of a "c * b"] using not0 |
29403
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29045
diff
changeset
|
772 |
by (simp add: zmod_zmult1_eq zmod_zdiv_trivial) |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
26507
diff
changeset
|
773 |
qed auto |
25942 | 774 |
|
29651
16a19466bf81
slightly adapted towards more uniformity with div/mod on nat
haftmann
parents:
29410
diff
changeset
|
775 |
lemma posDivAlg_div_mod: |
16a19466bf81
slightly adapted towards more uniformity with div/mod on nat
haftmann
parents:
29410
diff
changeset
|
776 |
assumes "k \<ge> 0" |
16a19466bf81
slightly adapted towards more uniformity with div/mod on nat
haftmann
parents:
29410
diff
changeset
|
777 |
and "l \<ge> 0" |
16a19466bf81
slightly adapted towards more uniformity with div/mod on nat
haftmann
parents:
29410
diff
changeset
|
778 |
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
|
779 |
proof (cases "l = 0") |
16a19466bf81
slightly adapted towards more uniformity with div/mod on nat
haftmann
parents:
29410
diff
changeset
|
780 |
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
|
781 |
next |
16a19466bf81
slightly adapted towards more uniformity with div/mod on nat
haftmann
parents:
29410
diff
changeset
|
782 |
case False with assms posDivAlg_correct |
16a19466bf81
slightly adapted towards more uniformity with div/mod on nat
haftmann
parents:
29410
diff
changeset
|
783 |
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
|
784 |
by simp |
16a19466bf81
slightly adapted towards more uniformity with div/mod on nat
haftmann
parents:
29410
diff
changeset
|
785 |
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
|
786 |
show ?thesis by simp |
16a19466bf81
slightly adapted towards more uniformity with div/mod on nat
haftmann
parents:
29410
diff
changeset
|
787 |
qed |
16a19466bf81
slightly adapted towards more uniformity with div/mod on nat
haftmann
parents:
29410
diff
changeset
|
788 |
|
16a19466bf81
slightly adapted towards more uniformity with div/mod on nat
haftmann
parents:
29410
diff
changeset
|
789 |
lemma negDivAlg_div_mod: |
16a19466bf81
slightly adapted towards more uniformity with div/mod on nat
haftmann
parents:
29410
diff
changeset
|
790 |
assumes "k < 0" |
16a19466bf81
slightly adapted towards more uniformity with div/mod on nat
haftmann
parents:
29410
diff
changeset
|
791 |
and "l > 0" |
16a19466bf81
slightly adapted towards more uniformity with div/mod on nat
haftmann
parents:
29410
diff
changeset
|
792 |
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
|
793 |
proof - |
16a19466bf81
slightly adapted towards more uniformity with div/mod on nat
haftmann
parents:
29410
diff
changeset
|
794 |
from assms have "l \<noteq> 0" by simp |
16a19466bf81
slightly adapted towards more uniformity with div/mod on nat
haftmann
parents:
29410
diff
changeset
|
795 |
from assms negDivAlg_correct |
16a19466bf81
slightly adapted towards more uniformity with div/mod on nat
haftmann
parents:
29410
diff
changeset
|
796 |
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
|
797 |
by simp |
16a19466bf81
slightly adapted towards more uniformity with div/mod on nat
haftmann
parents:
29410
diff
changeset
|
798 |
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
|
799 |
show ?thesis by simp |
16a19466bf81
slightly adapted towards more uniformity with div/mod on nat
haftmann
parents:
29410
diff
changeset
|
800 |
qed |
16a19466bf81
slightly adapted towards more uniformity with div/mod on nat
haftmann
parents:
29410
diff
changeset
|
801 |
|
29403
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29045
diff
changeset
|
802 |
lemma zdiv_zadd_self1: "a \<noteq> (0::int) ==> (a+b) div a = b div a + 1" |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29045
diff
changeset
|
803 |
by (rule div_add_self1) (* already declared [simp] *) |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29045
diff
changeset
|
804 |
|
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29045
diff
changeset
|
805 |
lemma zdiv_zadd_self2: "a \<noteq> (0::int) ==> (b+a) div a = b div a + 1" |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29045
diff
changeset
|
806 |
by (rule div_add_self2) (* already declared [simp] *) |
23164 | 807 |
|
29403
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29045
diff
changeset
|
808 |
lemma zdiv_zmult_self2: "b \<noteq> (0::int) ==> (b*a) div b = a" |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29045
diff
changeset
|
809 |
by (rule div_mult_self1_is_id) (* already declared [simp] *) |
23164 | 810 |
|
29403
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29045
diff
changeset
|
811 |
lemma zmod_zmult_self1: "(a*b) mod b = (0::int)" |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29045
diff
changeset
|
812 |
by (rule mod_mult_self2_is_0) (* already declared [simp] *) |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29045
diff
changeset
|
813 |
|
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29045
diff
changeset
|
814 |
lemma zmod_zmult_self2: "(b*a) mod b = (0::int)" |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29045
diff
changeset
|
815 |
by (rule mod_mult_self1_is_0) (* already declared [simp] *) |
23164 | 816 |
|
817 |
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
|
818 |
by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def) |
23164 | 819 |
|
29403
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29045
diff
changeset
|
820 |
(* REVISIT: should this be generalized to all semiring_div types? *) |
23164 | 821 |
lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1] |
822 |
||
823 |
lemma zmod_zadd_left_eq: "(a+b) mod (c::int) = ((a mod c) + b) mod c" |
|
29403
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29045
diff
changeset
|
824 |
by (rule mod_add_left_eq) |
23164 | 825 |
|
826 |
lemma zmod_zadd_right_eq: "(a+b) mod (c::int) = (a + (b mod c)) mod c" |
|
29403
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29045
diff
changeset
|
827 |
by (rule mod_add_right_eq) |
23164 | 828 |
|
29403
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29045
diff
changeset
|
829 |
lemma zmod_zadd_self1: "(a+b) mod a = b mod (a::int)" |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29045
diff
changeset
|
830 |
by (rule mod_add_self1) (* already declared [simp] *) |
23164 | 831 |
|
29403
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29045
diff
changeset
|
832 |
lemma zmod_zadd_self2: "(b+a) mod a = b mod (a::int)" |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29045
diff
changeset
|
833 |
by (rule mod_add_self2) (* already declared [simp] *) |
23164 | 834 |
|
29405
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
835 |
lemma zmod_zdiff1_eq: "(a - b) mod c = (a mod c - b mod c) mod (c::int)" |
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
836 |
by (rule mod_diff_eq) |
23983 | 837 |
|
23164 | 838 |
subsection{*Proving @{term "a div (b*c) = (a div b) div c"} *} |
839 |
||
840 |
(*The condition c>0 seems necessary. Consider that 7 div ~6 = ~2 but |
|
841 |
7 div 2 div ~3 = 3 div ~3 = ~1. The subcase (a div b) mod c = 0 seems |
|
842 |
to cause particular problems.*) |
|
843 |
||
844 |
text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *} |
|
845 |
||
846 |
lemma zmult2_lemma_aux1: "[| (0::int) < c; b < r; r \<le> 0 |] ==> b*c < b*(q mod c) + r" |
|
847 |
apply (subgoal_tac "b * (c - q mod c) < r * 1") |
|
29667 | 848 |
apply (simp add: algebra_simps) |
23164 | 849 |
apply (rule order_le_less_trans) |
29667 | 850 |
apply (erule_tac [2] mult_strict_right_mono) |
851 |
apply (rule mult_left_mono_neg) |
|
852 |
using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps pos_mod_bound) |
|
853 |
apply (simp) |
|
854 |
apply (simp) |
|
23164 | 855 |
done |
856 |
||
857 |
lemma zmult2_lemma_aux2: |
|
858 |
"[| (0::int) < c; b < r; r \<le> 0 |] ==> b * (q mod c) + r \<le> 0" |
|
859 |
apply (subgoal_tac "b * (q mod c) \<le> 0") |
|
860 |
apply arith |
|
861 |
apply (simp add: mult_le_0_iff) |
|
862 |
done |
|
863 |
||
864 |
lemma zmult2_lemma_aux3: "[| (0::int) < c; 0 \<le> r; r < b |] ==> 0 \<le> b * (q mod c) + r" |
|
865 |
apply (subgoal_tac "0 \<le> b * (q mod c) ") |
|
866 |
apply arith |
|
867 |
apply (simp add: zero_le_mult_iff) |
|
868 |
done |
|
869 |
||
870 |
lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c" |
|
871 |
apply (subgoal_tac "r * 1 < b * (c - q mod c) ") |
|
29667 | 872 |
apply (simp add: right_diff_distrib) |
23164 | 873 |
apply (rule order_less_le_trans) |
29667 | 874 |
apply (erule mult_strict_right_mono) |
875 |
apply (rule_tac [2] mult_left_mono) |
|
876 |
apply simp |
|
877 |
using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps pos_mod_bound) |
|
878 |
apply simp |
|
23164 | 879 |
done |
880 |
||
29651
16a19466bf81
slightly adapted towards more uniformity with div/mod on nat
haftmann
parents:
29410
diff
changeset
|
881 |
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
|
882 |
==> 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
|
883 |
by (auto simp add: mult_ac divmod_rel_def linorder_neq_iff |
23164 | 884 |
zero_less_mult_iff right_distrib [symmetric] |
885 |
zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4) |
|
886 |
||
887 |
lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c" |
|
888 |
apply (case_tac "b = 0", simp) |
|
29651
16a19466bf81
slightly adapted towards more uniformity with div/mod on nat
haftmann
parents:
29410
diff
changeset
|
889 |
apply (force simp add: divmod_rel_div_mod [THEN zmult2_lemma, THEN divmod_rel_div]) |
23164 | 890 |
done |
891 |
||
892 |
lemma zmod_zmult2_eq: |
|
893 |
"(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b" |
|
894 |
apply (case_tac "b = 0", simp) |
|
29651
16a19466bf81
slightly adapted towards more uniformity with div/mod on nat
haftmann
parents:
29410
diff
changeset
|
895 |
apply (force simp add: divmod_rel_div_mod [THEN zmult2_lemma, THEN divmod_rel_mod]) |
23164 | 896 |
done |
897 |
||
898 |
||
899 |
subsection{*Cancellation of Common Factors in div*} |
|
900 |
||
901 |
lemma zdiv_zmult_zmult1_aux1: |
|
902 |
"[| (0::int) < b; c \<noteq> 0 |] ==> (c*a) div (c*b) = a div b" |
|
903 |
by (subst zdiv_zmult2_eq, auto) |
|
904 |
||
905 |
lemma zdiv_zmult_zmult1_aux2: |
|
906 |
"[| b < (0::int); c \<noteq> 0 |] ==> (c*a) div (c*b) = a div b" |
|
907 |
apply (subgoal_tac " (c * (-a)) div (c * (-b)) = (-a) div (-b) ") |
|
908 |
apply (rule_tac [2] zdiv_zmult_zmult1_aux1, auto) |
|
909 |
done |
|
910 |
||
911 |
lemma zdiv_zmult_zmult1: "c \<noteq> (0::int) ==> (c*a) div (c*b) = a div b" |
|
912 |
apply (case_tac "b = 0", simp) |
|
913 |
apply (auto simp add: linorder_neq_iff zdiv_zmult_zmult1_aux1 zdiv_zmult_zmult1_aux2) |
|
914 |
done |
|
915 |
||
23401 | 916 |
lemma zdiv_zmult_zmult1_if[simp]: |
917 |
"(k*m) div (k*n) = (if k = (0::int) then 0 else m div n)" |
|
918 |
by (simp add:zdiv_zmult_zmult1) |
|
919 |
||
920 |
(* |
|
23164 | 921 |
lemma zdiv_zmult_zmult2: "c \<noteq> (0::int) ==> (a*c) div (b*c) = a div b" |
922 |
apply (drule zdiv_zmult_zmult1) |
|
923 |
apply (auto simp add: mult_commute) |
|
924 |
done |
|
23401 | 925 |
*) |
23164 | 926 |
|
927 |
||
928 |
subsection{*Distribution of Factors over mod*} |
|
929 |
||
930 |
lemma zmod_zmult_zmult1_aux1: |
|
931 |
"[| (0::int) < b; c \<noteq> 0 |] ==> (c*a) mod (c*b) = c * (a mod b)" |
|
932 |
by (subst zmod_zmult2_eq, auto) |
|
933 |
||
934 |
lemma zmod_zmult_zmult1_aux2: |
|
935 |
"[| b < (0::int); c \<noteq> 0 |] ==> (c*a) mod (c*b) = c * (a mod b)" |
|
936 |
apply (subgoal_tac " (c * (-a)) mod (c * (-b)) = c * ((-a) mod (-b))") |
|
937 |
apply (rule_tac [2] zmod_zmult_zmult1_aux1, auto) |
|
938 |
done |
|
939 |
||
940 |
lemma zmod_zmult_zmult1: "(c*a) mod (c*b) = (c::int) * (a mod b)" |
|
941 |
apply (case_tac "b = 0", simp) |
|
942 |
apply (case_tac "c = 0", simp) |
|
943 |
apply (auto simp add: linorder_neq_iff zmod_zmult_zmult1_aux1 zmod_zmult_zmult1_aux2) |
|
944 |
done |
|
945 |
||
946 |
lemma zmod_zmult_zmult2: "(a*c) mod (b*c) = (a mod b) * (c::int)" |
|
947 |
apply (cut_tac c = c in zmod_zmult_zmult1) |
|
948 |
apply (auto simp add: mult_commute) |
|
949 |
done |
|
950 |
||
29404 | 951 |
lemma zmod_zmod_cancel: "n dvd m \<Longrightarrow> (k::int) mod m mod n = k mod n" |
952 |
by (rule mod_mod_cancel) |
|
24490 | 953 |
|
23164 | 954 |
|
955 |
subsection {*Splitting Rules for div and mod*} |
|
956 |
||
957 |
text{*The proofs of the two lemmas below are essentially identical*} |
|
958 |
||
959 |
lemma split_pos_lemma: |
|
960 |
"0<k ==> |
|
961 |
P(n div k :: int)(n mod k) = (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i j)" |
|
962 |
apply (rule iffI, clarify) |
|
963 |
apply (erule_tac P="P ?x ?y" in rev_mp) |
|
964 |
apply (subst zmod_zadd1_eq) |
|
965 |
apply (subst zdiv_zadd1_eq) |
|
966 |
apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial) |
|
967 |
txt{*converse direction*} |
|
968 |
apply (drule_tac x = "n div k" in spec) |
|
969 |
apply (drule_tac x = "n mod k" in spec, simp) |
|
970 |
done |
|
971 |
||
972 |
lemma split_neg_lemma: |
|
973 |
"k<0 ==> |
|
974 |
P(n div k :: int)(n mod k) = (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i j)" |
|
975 |
apply (rule iffI, clarify) |
|
976 |
apply (erule_tac P="P ?x ?y" in rev_mp) |
|
977 |
apply (subst zmod_zadd1_eq) |
|
978 |
apply (subst zdiv_zadd1_eq) |
|
979 |
apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial) |
|
980 |
txt{*converse direction*} |
|
981 |
apply (drule_tac x = "n div k" in spec) |
|
982 |
apply (drule_tac x = "n mod k" in spec, simp) |
|
983 |
done |
|
984 |
||
985 |
lemma split_zdiv: |
|
986 |
"P(n div k :: int) = |
|
987 |
((k = 0 --> P 0) & |
|
988 |
(0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) & |
|
989 |
(k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))" |
|
990 |
apply (case_tac "k=0", simp) |
|
991 |
apply (simp only: linorder_neq_iff) |
|
992 |
apply (erule disjE) |
|
993 |
apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"] |
|
994 |
split_neg_lemma [of concl: "%x y. P x"]) |
|
995 |
done |
|
996 |
||
997 |
lemma split_zmod: |
|
998 |
"P(n mod k :: int) = |
|
999 |
((k = 0 --> P n) & |
|
1000 |
(0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) & |
|
1001 |
(k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))" |
|
1002 |
apply (case_tac "k=0", simp) |
|
1003 |
apply (simp only: linorder_neq_iff) |
|
1004 |
apply (erule disjE) |
|
1005 |
apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"] |
|
1006 |
split_neg_lemma [of concl: "%x y. P y"]) |
|
1007 |
done |
|
1008 |
||
1009 |
(* Enable arith to deal with div 2 and mod 2: *) |
|
1010 |
declare split_zdiv [of _ _ "number_of k", simplified, standard, arith_split] |
|
1011 |
declare split_zmod [of _ _ "number_of k", simplified, standard, arith_split] |
|
1012 |
||
1013 |
||
1014 |
subsection{*Speeding up the Division Algorithm with Shifting*} |
|
1015 |
||
1016 |
text{*computing div by shifting *} |
|
1017 |
||
1018 |
lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a" |
|
1019 |
proof cases |
|
1020 |
assume "a=0" |
|
1021 |
thus ?thesis by simp |
|
1022 |
next |
|
1023 |
assume "a\<noteq>0" and le_a: "0\<le>a" |
|
1024 |
hence a_pos: "1 \<le> a" by arith |
|
1025 |
hence one_less_a2: "1 < 2*a" by arith |
|
1026 |
hence le_2a: "2 * (1 + b mod a) \<le> 2 * a" |
|
1027 |
by (simp add: mult_le_cancel_left add_commute [of 1] add1_zle_eq) |
|
1028 |
with a_pos have "0 \<le> b mod a" by simp |
|
1029 |
hence le_addm: "0 \<le> 1 mod (2*a) + 2*(b mod a)" |
|
1030 |
by (simp add: mod_pos_pos_trivial one_less_a2) |
|
1031 |
with le_2a |
|
1032 |
have "(1 mod (2*a) + 2*(b mod a)) div (2*a) = 0" |
|
1033 |
by (simp add: div_pos_pos_trivial le_addm mod_pos_pos_trivial one_less_a2 |
|
1034 |
right_distrib) |
|
1035 |
thus ?thesis |
|
1036 |
by (subst zdiv_zadd1_eq, |
|
1037 |
simp add: zdiv_zmult_zmult1 zmod_zmult_zmult1 one_less_a2 |
|
1038 |
div_pos_pos_trivial) |
|
1039 |
qed |
|
1040 |
||
1041 |
lemma neg_zdiv_mult_2: "a \<le> (0::int) ==> (1 + 2*b) div (2*a) = (b+1) div a" |
|
1042 |
apply (subgoal_tac " (1 + 2* (-b - 1)) div (2 * (-a)) = (-b - 1) div (-a) ") |
|
1043 |
apply (rule_tac [2] pos_zdiv_mult_2) |
|
1044 |
apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib) |
|
1045 |
apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))") |
|
1046 |
apply (simp only: zdiv_zminus_zminus diff_minus minus_add_distrib [symmetric], |
|
1047 |
simp) |
|
1048 |
done |
|
1049 |
||
1050 |
(*Not clear why this must be proved separately; probably number_of causes |
|
1051 |
simplification problems*) |
|
1052 |
lemma not_0_le_lemma: "~ 0 \<le> x ==> x \<le> (0::int)" |
|
1053 |
by auto |
|
1054 |
||
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
25961
diff
changeset
|
1055 |
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
|
1056 |
"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
|
1057 |
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
|
1058 |
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
|
1059 |
|
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
25961
diff
changeset
|
1060 |
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
|
1061 |
"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
|
1062 |
(if (0::int) \<le> number_of w |
23164 | 1063 |
then number_of v div (number_of w) |
1064 |
else (number_of v + (1::int)) div (number_of w))" |
|
1065 |
apply (simp only: number_of_eq numeral_simps UNIV_I split: split_if) |
|
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
25961
diff
changeset
|
1066 |
apply (simp add: zdiv_zmult_zmult1 pos_zdiv_mult_2 neg_zdiv_mult_2 add_ac) |
23164 | 1067 |
done |
1068 |
||
1069 |
||
1070 |
subsection{*Computing mod by Shifting (proofs resemble those for div)*} |
|
1071 |
||
1072 |
lemma pos_zmod_mult_2: |
|
1073 |
"(0::int) \<le> a ==> (1 + 2*b) mod (2*a) = 1 + 2 * (b mod a)" |
|
1074 |
apply (case_tac "a = 0", simp) |
|
1075 |
apply (subgoal_tac "1 < a * 2") |
|
1076 |
prefer 2 apply arith |
|
1077 |
apply (subgoal_tac "2* (1 + b mod a) \<le> 2*a") |
|
1078 |
apply (rule_tac [2] mult_left_mono) |
|
1079 |
apply (auto simp add: add_commute [of 1] mult_commute add1_zle_eq |
|
1080 |
pos_mod_bound) |
|
1081 |
apply (subst zmod_zadd1_eq) |
|
1082 |
apply (simp add: zmod_zmult_zmult2 mod_pos_pos_trivial) |
|
1083 |
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
|
1084 |
apply (auto simp add: mod_pos_pos_trivial ring_distribs) |
23164 | 1085 |
apply (subgoal_tac "0 \<le> b mod a", arith, simp) |
1086 |
done |
|
1087 |
||
1088 |
lemma neg_zmod_mult_2: |
|
1089 |
"a \<le> (0::int) ==> (1 + 2*b) mod (2*a) = 2 * ((b+1) mod a) - 1" |
|
1090 |
apply (subgoal_tac "(1 + 2* (-b - 1)) mod (2* (-a)) = |
|
1091 |
1 + 2* ((-b - 1) mod (-a))") |
|
1092 |
apply (rule_tac [2] pos_zmod_mult_2) |
|
1093 |
apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib) |
|
1094 |
apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))") |
|
1095 |
prefer 2 apply simp |
|
1096 |
apply (simp only: zmod_zminus_zminus diff_minus minus_add_distrib [symmetric]) |
|
1097 |
done |
|
1098 |
||
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
25961
diff
changeset
|
1099 |
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
|
1100 |
"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
|
1101 |
(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
|
1102 |
apply (simp only: number_of_eq numeral_simps) |
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
25961
diff
changeset
|
1103 |
apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2 |
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
25961
diff
changeset
|
1104 |
not_0_le_lemma neg_zmod_mult_2 add_ac) |
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
25961
diff
changeset
|
1105 |
done |
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
25961
diff
changeset
|
1106 |
|
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
25961
diff
changeset
|
1107 |
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
|
1108 |
"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
|
1109 |
(if (0::int) \<le> number_of w |
23164 | 1110 |
then 2 * (number_of v mod number_of w) + 1 |
1111 |
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
|
1112 |
apply (simp only: number_of_eq numeral_simps) |
23164 | 1113 |
apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2 |
1114 |
not_0_le_lemma neg_zmod_mult_2 add_ac) |
|
1115 |
done |
|
1116 |
||
1117 |
||
1118 |
subsection{*Quotients of Signs*} |
|
1119 |
||
1120 |
lemma div_neg_pos_less0: "[| a < (0::int); 0 < b |] ==> a div b < 0" |
|
1121 |
apply (subgoal_tac "a div b \<le> -1", force) |
|
1122 |
apply (rule order_trans) |
|
1123 |
apply (rule_tac a' = "-1" in zdiv_mono1) |
|
1124 |
apply (auto simp add: zdiv_minus1) |
|
1125 |
done |
|
1126 |
||
1127 |
lemma div_nonneg_neg_le0: "[| (0::int) \<le> a; b < 0 |] ==> a div b \<le> 0" |
|
1128 |
by (drule zdiv_mono1_neg, auto) |
|
1129 |
||
1130 |
lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)" |
|
1131 |
apply auto |
|
1132 |
apply (drule_tac [2] zdiv_mono1) |
|
1133 |
apply (auto simp add: linorder_neq_iff) |
|
1134 |
apply (simp (no_asm_use) add: linorder_not_less [symmetric]) |
|
1135 |
apply (blast intro: div_neg_pos_less0) |
|
1136 |
done |
|
1137 |
||
1138 |
lemma neg_imp_zdiv_nonneg_iff: |
|
1139 |
"b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))" |
|
1140 |
apply (subst zdiv_zminus_zminus [symmetric]) |
|
1141 |
apply (subst pos_imp_zdiv_nonneg_iff, auto) |
|
1142 |
done |
|
1143 |
||
1144 |
(*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*) |
|
1145 |
lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)" |
|
1146 |
by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff) |
|
1147 |
||
1148 |
(*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*) |
|
1149 |
lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)" |
|
1150 |
by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff) |
|
1151 |
||
1152 |
||
1153 |
subsection {* The Divides Relation *} |
|
1154 |
||
1155 |
lemma zdvd_iff_zmod_eq_0: "(m dvd n) = (n mod m = (0::int))" |
|
29410 | 1156 |
by (rule dvd_eq_mod_eq_0) |
23512 | 1157 |
|
23164 | 1158 |
lemmas zdvd_iff_zmod_eq_0_number_of [simp] = |
1159 |
zdvd_iff_zmod_eq_0 [of "number_of x" "number_of y", standard] |
|
1160 |
||
29410 | 1161 |
lemma zdvd_0_right: "(m::int) dvd 0" |
1162 |
by (rule dvd_0_right) (* already declared [iff] *) |
|
23164 | 1163 |
|
29410 | 1164 |
lemma zdvd_0_left: "(0 dvd (m::int)) = (m = 0)" |
1165 |
by (rule dvd_0_left_iff) (* already declared [noatp,simp] *) |
|
23164 | 1166 |
|
29410 | 1167 |
lemma zdvd_1_left: "1 dvd (m::int)" |
1168 |
by (rule one_dvd) (* already declared [simp] *) |
|
23164 | 1169 |
|
1170 |
lemma zdvd_refl [simp]: "m dvd (m::int)" |
|
29410 | 1171 |
by (rule dvd_refl) (* TODO: declare generic dvd_refl [simp] *) |
23164 | 1172 |
|
1173 |
lemma zdvd_trans: "m dvd n ==> n dvd k ==> m dvd (k::int)" |
|
29410 | 1174 |
by (rule dvd_trans) |
23164 | 1175 |
|
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
26507
diff
changeset
|
1176 |
lemma zdvd_zminus_iff: "m dvd -n \<longleftrightarrow> m dvd (n::int)" |
29410 | 1177 |
by (rule dvd_minus_iff) |
23164 | 1178 |
|
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
26507
diff
changeset
|
1179 |
lemma zdvd_zminus2_iff: "-m dvd n \<longleftrightarrow> m dvd (n::int)" |
29410 | 1180 |
by (rule minus_dvd_iff) |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
26507
diff
changeset
|
1181 |
|
23164 | 1182 |
lemma zdvd_abs1: "( \<bar>i::int\<bar> dvd j) = (i dvd j)" |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
26507
diff
changeset
|
1183 |
by (cases "i > 0") (simp_all add: zdvd_zminus2_iff) |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
26507
diff
changeset
|
1184 |
|
23164 | 1185 |
lemma zdvd_abs2: "( (i::int) dvd \<bar>j\<bar>) = (i dvd j)" |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
26507
diff
changeset
|
1186 |
by (cases "j > 0") (simp_all add: zdvd_zminus_iff) |
23164 | 1187 |
|
1188 |
lemma zdvd_anti_sym: |
|
1189 |
"0 < m ==> 0 < n ==> m dvd n ==> n dvd m ==> m = (n::int)" |
|
1190 |
apply (simp add: dvd_def, auto) |
|
1191 |
apply (simp add: mult_assoc zero_less_mult_iff zmult_eq_1_iff) |
|
1192 |
done |
|
1193 |
||
1194 |
lemma zdvd_zadd: "k dvd m ==> k dvd n ==> k dvd (m + n :: int)" |
|
29410 | 1195 |
by (rule dvd_add) |
23164 | 1196 |
|
1197 |
lemma zdvd_dvd_eq: assumes anz:"a \<noteq> 0" and ab: "(a::int) dvd b" and ba:"b dvd a" |
|
1198 |
shows "\<bar>a\<bar> = \<bar>b\<bar>" |
|
1199 |
proof- |
|
1200 |
from ab obtain k where k:"b = a*k" unfolding dvd_def by blast |
|
1201 |
from ba obtain k' where k':"a = b*k'" unfolding dvd_def by blast |
|
1202 |
from k k' have "a = a*k*k'" by simp |
|
1203 |
with mult_cancel_left1[where c="a" and b="k*k'"] |
|
1204 |
have kk':"k*k' = 1" using anz by (simp add: mult_assoc) |
|
1205 |
hence "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1" by (simp add: zmult_eq_1_iff) |
|
1206 |
thus ?thesis using k k' by auto |
|
1207 |
qed |
|
1208 |
||
1209 |
lemma zdvd_zdiff: "k dvd m ==> k dvd n ==> k dvd (m - n :: int)" |
|
29410 | 1210 |
by (rule Ring_and_Field.dvd_diff) |
23164 | 1211 |
|
1212 |
lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)" |
|
1213 |
apply (subgoal_tac "m = n + (m - n)") |
|
1214 |
apply (erule ssubst) |
|
1215 |
apply (blast intro: zdvd_zadd, simp) |
|
1216 |
done |
|
1217 |
||
1218 |
lemma zdvd_zmult: "k dvd (n::int) ==> k dvd m * n" |
|
29410 | 1219 |
by (rule dvd_mult) |
23164 | 1220 |
|
1221 |
lemma zdvd_zmult2: "k dvd (m::int) ==> k dvd m * n" |
|
29410 | 1222 |
by (rule dvd_mult2) |
23164 | 1223 |
|
29410 | 1224 |
lemma zdvd_triv_right: "(k::int) dvd m * k" |
1225 |
by (rule dvd_triv_right) (* already declared [simp] *) |
|
23164 | 1226 |
|
29410 | 1227 |
lemma zdvd_triv_left: "(k::int) dvd k * m" |
1228 |
by (rule dvd_triv_left) (* already declared [simp] *) |
|
23164 | 1229 |
|
1230 |
lemma zdvd_zmultD2: "j * k dvd n ==> j dvd (n::int)" |
|
29410 | 1231 |
by (rule dvd_mult_left) |
23164 | 1232 |
|
1233 |
lemma zdvd_zmultD: "j * k dvd n ==> k dvd (n::int)" |
|
29410 | 1234 |
by (rule dvd_mult_right) |
23164 | 1235 |
|
1236 |
lemma zdvd_zmult_mono: "i dvd m ==> j dvd (n::int) ==> i * j dvd m * n" |
|
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
26507
diff
changeset
|
1237 |
by (rule mult_dvd_mono) |
23164 | 1238 |
|
1239 |
lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))" |
|
1240 |
apply (rule iffI) |
|
1241 |
apply (erule_tac [2] zdvd_zadd) |
|
1242 |
apply (subgoal_tac "n = (n + k * m) - k * m") |
|
1243 |
apply (erule ssubst) |
|
1244 |
apply (erule zdvd_zdiff, simp_all) |
|
1245 |
done |
|
1246 |
||
1247 |
lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n" |
|
1248 |
apply (simp add: dvd_def) |
|
1249 |
apply (auto simp add: zmod_zmult_zmult1) |
|
1250 |
done |
|
1251 |
||
1252 |
lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)" |
|
1253 |
apply (subgoal_tac "k dvd n * (m div n) + m mod n") |
|
1254 |
apply (simp add: zmod_zdiv_equality [symmetric]) |
|
1255 |
apply (simp only: zdvd_zadd zdvd_zmult2) |
|
1256 |
done |
|
1257 |
||
1258 |
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
|
1259 |
apply (auto elim!: dvdE) |
23164 | 1260 |
apply (subgoal_tac "0 < n") |
1261 |
prefer 2 |
|
1262 |
apply (blast intro: order_less_trans) |
|
1263 |
apply (simp add: zero_less_mult_iff) |
|
1264 |
apply (subgoal_tac "n * k < n * 1") |
|
1265 |
apply (drule mult_less_cancel_left [THEN iffD1], auto) |
|
1266 |
done |
|
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
26507
diff
changeset
|
1267 |
|
23164 | 1268 |
lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)" |
1269 |
using zmod_zdiv_equality[where a="m" and b="n"] |
|
29667 | 1270 |
by (simp add: algebra_simps) |
23164 | 1271 |
|
1272 |
lemma zdvd_mult_div_cancel:"(n::int) dvd m \<Longrightarrow> n * (m div n) = m" |
|
1273 |
apply (subgoal_tac "m mod n = 0") |
|
1274 |
apply (simp add: zmult_div_cancel) |
|
1275 |
apply (simp only: zdvd_iff_zmod_eq_0) |
|
1276 |
done |
|
1277 |
||
1278 |
lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k \<noteq> (0::int)" |
|
1279 |
shows "m dvd n" |
|
1280 |
proof- |
|
1281 |
from d obtain h where h: "k*n = k*m * h" unfolding dvd_def by blast |
|
1282 |
{assume "n \<noteq> m*h" hence "k* n \<noteq> k* (m*h)" using kz by simp |
|
1283 |
with h have False by (simp add: mult_assoc)} |
|
1284 |
hence "n = m * h" by blast |
|
29410 | 1285 |
thus ?thesis by simp |
23164 | 1286 |
qed |
1287 |
||
23969 | 1288 |
lemma zdvd_zmult_cancel_disj[simp]: |
1289 |
"(k*m) dvd (k*n) = (k=0 | m dvd (n::int))" |
|
1290 |
by (auto simp: zdvd_zmult_mono dest: zdvd_mult_cancel) |
|
1291 |
||
1292 |
||
23164 | 1293 |
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
|
1294 |
apply (simp split add: split_nat) |
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
1295 |
apply (rule iffI) |
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
1296 |
apply (erule exE) |
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
1297 |
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
|
1298 |
apply simp |
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
1299 |
apply (erule exE) |
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
1300 |
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
|
1301 |
apply (erule conjE) |
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
1302 |
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
|
1303 |
apply simp |
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
1304 |
done |
23164 | 1305 |
|
23365 | 1306 |
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
|
1307 |
proof - |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
26507
diff
changeset
|
1308 |
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
|
1309 |
proof - |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
26507
diff
changeset
|
1310 |
fix k |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
26507
diff
changeset
|
1311 |
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
|
1312 |
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
|
1313 |
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
|
1314 |
then show ?thesis .. |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
26507
diff
changeset
|
1315 |
next |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
26507
diff
changeset
|
1316 |
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
|
1317 |
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
|
1318 |
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
|
1319 |
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
|
1320 |
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
|
1321 |
qed |
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
26507
diff
changeset
|
1322 |
qed |
29410 | 1323 |
then show ?thesis by (auto elim!: dvdE simp only: zdvd_triv_left int_mult) |
1324 |
qed |
|
23164 | 1325 |
|
1326 |
lemma zdvd1_eq[simp]: "(x::int) dvd 1 = ( \<bar>x\<bar> = 1)" |
|
1327 |
proof |
|
1328 |
assume d: "x dvd 1" hence "int (nat \<bar>x\<bar>) dvd int (nat 1)" by (simp add: zdvd_abs1) |
|
1329 |
hence "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int) |
|
1330 |
hence "nat \<bar>x\<bar> = 1" by simp |
|
1331 |
thus "\<bar>x\<bar> = 1" by (cases "x < 0", auto) |
|
1332 |
next |
|
1333 |
assume "\<bar>x\<bar>=1" thus "x dvd 1" |
|
1334 |
by(cases "x < 0",simp_all add: minus_equation_iff zdvd_iff_zmod_eq_0) |
|
1335 |
qed |
|
1336 |
lemma zdvd_mult_cancel1: |
|
1337 |
assumes mp:"m \<noteq>(0::int)" shows "(m * n dvd m) = (\<bar>n\<bar> = 1)" |
|
1338 |
proof |
|
1339 |
assume n1: "\<bar>n\<bar> = 1" thus "m * n dvd m" |
|
1340 |
by (cases "n >0", auto simp add: zdvd_zminus2_iff minus_equation_iff) |
|
1341 |
next |
|
1342 |
assume H: "m * n dvd m" hence H2: "m * n dvd m * 1" by simp |
|
1343 |
from zdvd_mult_cancel[OF H2 mp] show "\<bar>n\<bar> = 1" by (simp only: zdvd1_eq) |
|
1344 |
qed |
|
1345 |
||
23365 | 1346 |
lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))" |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
26507
diff
changeset
|
1347 |
unfolding zdvd_int by (cases "z \<ge> 0") (simp_all add: zdvd_zminus_iff) |
23306
cdb027d0637e
add int_of_nat versions of lemmas about int::nat=>int
huffman
parents:
23164
diff
changeset
|
1348 |
|
23365 | 1349 |
lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)" |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
26507
diff
changeset
|
1350 |
unfolding zdvd_int by (cases "z \<ge> 0") (simp_all add: zdvd_zminus2_iff) |
23164 | 1351 |
|
1352 |
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
|
1353 |
by (auto simp add: dvd_int_iff) |
23164 | 1354 |
|
1355 |
lemma zminus_dvd_iff [iff]: "(-z dvd w) = (z dvd (w::int))" |
|
29410 | 1356 |
by (rule minus_dvd_iff) |
23164 | 1357 |
|
1358 |
lemma dvd_zminus_iff [iff]: "(z dvd -w) = (z dvd (w::int))" |
|
29410 | 1359 |
by (rule dvd_minus_iff) |
23164 | 1360 |
|
1361 |
lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \<le> (n::int)" |
|
23365 | 1362 |
apply (rule_tac z=n in int_cases) |
1363 |
apply (auto simp add: dvd_int_iff) |
|
1364 |
apply (rule_tac z=z in int_cases) |
|
23307
2fe3345035c7
modify proofs to avoid referring to int::nat=>int
huffman
parents:
23306
diff
changeset
|
1365 |
apply (auto simp add: dvd_imp_le) |
23164 | 1366 |
done |
1367 |
||
1368 |
lemma zpower_zmod: "((x::int) mod m)^y mod m = x^y mod m" |
|
1369 |
apply (induct "y", auto) |
|
1370 |
apply (rule zmod_zmult1_eq [THEN trans]) |
|
1371 |
apply (simp (no_asm_simp)) |
|
1372 |
apply (rule zmod_zmult_distrib [symmetric]) |
|
1373 |
done |
|
1374 |
||
23365 | 1375 |
lemma zdiv_int: "int (a div b) = (int a) div (int b)" |
23164 | 1376 |
apply (subst split_div, auto) |
1377 |
apply (subst split_zdiv, auto) |
|
23365 | 1378 |
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
|
1379 |
apply (auto simp add: IntDiv.divmod_rel_def of_nat_mult) |
23164 | 1380 |
done |
1381 |
||
1382 |
lemma zmod_int: "int (a mod b) = (int a) mod (int b)" |
|
23365 | 1383 |
apply (subst split_mod, auto) |
1384 |
apply (subst split_zmod, auto) |
|
1385 |
apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia |
|
1386 |
in unique_remainder) |
|
29651
16a19466bf81
slightly adapted towards more uniformity with div/mod on nat
haftmann
parents:
29410
diff
changeset
|
1387 |
apply (auto simp add: IntDiv.divmod_rel_def of_nat_mult) |
23365 | 1388 |
done |
23164 | 1389 |
|
1390 |
text{*Suggested by Matthias Daum*} |
|
1391 |
lemma int_power_div_base: |
|
1392 |
"\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)" |
|
1393 |
apply (subgoal_tac "k ^ m = k ^ ((m - 1) + 1)") |
|
1394 |
apply (erule ssubst) |
|
1395 |
apply (simp only: power_add) |
|
1396 |
apply simp_all |
|
1397 |
done |
|
1398 |
||
23853 | 1399 |
text {* by Brian Huffman *} |
1400 |
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
|
1401 |
by (rule mod_minus_eq [symmetric]) |
23853 | 1402 |
|
1403 |
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
|
1404 |
by (rule mod_diff_left_eq [symmetric]) |
23853 | 1405 |
|
1406 |
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
|
1407 |
by (rule mod_diff_right_eq [symmetric]) |
23853 | 1408 |
|
1409 |
lemmas zmod_simps = |
|
1410 |
IntDiv.zmod_zadd_left_eq [symmetric] |
|
1411 |
IntDiv.zmod_zadd_right_eq [symmetric] |
|
1412 |
IntDiv.zmod_zmult1_eq [symmetric] |
|
1413 |
IntDiv.zmod_zmult1_eq' [symmetric] |
|
1414 |
IntDiv.zpower_zmod |
|
1415 |
zminus_zmod zdiff_zmod_left zdiff_zmod_right |
|
1416 |
||
29045
3c8f48333731
move nat_{div,mod}_distrib from NatBin to IntDiv, simplified proofs
huffman
parents:
28562
diff
changeset
|
1417 |
text {* Distributive laws for function @{text nat}. *} |
3c8f48333731
move nat_{div,mod}_distrib from NatBin to IntDiv, simplified proofs
huffman
parents:
28562
diff
changeset
|
1418 |
|
3c8f48333731
move nat_{div,mod}_distrib from NatBin to IntDiv, simplified proofs
huffman
parents:
28562
diff
changeset
|
1419 |
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
|
1420 |
apply (rule linorder_cases [of y 0]) |
3c8f48333731
move nat_{div,mod}_distrib from NatBin to IntDiv, simplified proofs
huffman
parents:
28562
diff
changeset
|
1421 |
apply (simp add: div_nonneg_neg_le0) |
3c8f48333731
move nat_{div,mod}_distrib from NatBin to IntDiv, simplified proofs
huffman
parents:
28562
diff
changeset
|
1422 |
apply simp |
3c8f48333731
move nat_{div,mod}_distrib from NatBin to IntDiv, simplified proofs
huffman
parents:
28562
diff
changeset
|
1423 |
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
|
1424 |
done |
3c8f48333731
move nat_{div,mod}_distrib from NatBin to IntDiv, simplified proofs
huffman
parents:
28562
diff
changeset
|
1425 |
|
3c8f48333731
move nat_{div,mod}_distrib from NatBin to IntDiv, simplified proofs
huffman
parents:
28562
diff
changeset
|
1426 |
(*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
|
1427 |
lemma nat_mod_distrib: |
3c8f48333731
move nat_{div,mod}_distrib from NatBin to IntDiv, simplified proofs
huffman
parents:
28562
diff
changeset
|
1428 |
"\<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
|
1429 |
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
|
1430 |
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
|
1431 |
done |
3c8f48333731
move nat_{div,mod}_distrib from NatBin to IntDiv, simplified proofs
huffman
parents:
28562
diff
changeset
|
1432 |
|
3c8f48333731
move nat_{div,mod}_distrib from NatBin to IntDiv, simplified proofs
huffman
parents:
28562
diff
changeset
|
1433 |
text{*Suggested by Matthias Daum*} |
3c8f48333731
move nat_{div,mod}_distrib from NatBin to IntDiv, simplified proofs
huffman
parents:
28562
diff
changeset
|
1434 |
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
|
1435 |
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
|
1436 |
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
|
1437 |
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
|
1438 |
done |
3c8f48333731
move nat_{div,mod}_distrib from NatBin to IntDiv, simplified proofs
huffman
parents:
28562
diff
changeset
|
1439 |
|
23853 | 1440 |
text {* code generator setup *} |
23164 | 1441 |
|
26507 | 1442 |
context ring_1 |
1443 |
begin |
|
1444 |
||
28562 | 1445 |
lemma of_int_num [code]: |
26507 | 1446 |
"of_int k = (if k = 0 then 0 else if k < 0 then |
1447 |
- of_int (- k) else let |
|
29651
16a19466bf81
slightly adapted towards more uniformity with div/mod on nat
haftmann
parents:
29410
diff
changeset
|
1448 |
(l, m) = divmod k 2; |
26507 | 1449 |
l' = of_int l |
1450 |
in if m = 0 then l' + l' else l' + l' + 1)" |
|
1451 |
proof - |
|
1452 |
have aux1: "k mod (2\<Colon>int) \<noteq> (0\<Colon>int) \<Longrightarrow> |
|
1453 |
of_int k = of_int (k div 2 * 2 + 1)" |
|
1454 |
proof - |
|
1455 |
have "k mod 2 < 2" by (auto intro: pos_mod_bound) |
|
1456 |
moreover have "0 \<le> k mod 2" by (auto intro: pos_mod_sign) |
|
1457 |
moreover assume "k mod 2 \<noteq> 0" |
|
1458 |
ultimately have "k mod 2 = 1" by arith |
|
1459 |
moreover have "of_int k = of_int (k div 2 * 2 + k mod 2)" by simp |
|
1460 |
ultimately show ?thesis by auto |
|
1461 |
qed |
|
1462 |
have aux2: "\<And>x. of_int 2 * x = x + x" |
|
1463 |
proof - |
|
1464 |
fix x |
|
1465 |
have int2: "(2::int) = 1 + 1" by arith |
|
1466 |
show "of_int 2 * x = x + x" |
|
1467 |
unfolding int2 of_int_add left_distrib by simp |
|
1468 |
qed |
|
1469 |
have aux3: "\<And>x. x * of_int 2 = x + x" |
|
1470 |
proof - |
|
1471 |
fix x |
|
1472 |
have int2: "(2::int) = 1 + 1" by arith |
|
1473 |
show "x * of_int 2 = x + x" |
|
1474 |
unfolding int2 of_int_add right_distrib by simp |
|
1475 |
qed |
|
29651
16a19466bf81
slightly adapted towards more uniformity with div/mod on nat
haftmann
parents:
29410
diff
changeset
|
1476 |
from aux1 show ?thesis by (auto simp add: divmod_mod_div Let_def aux2 aux3) |
26507 | 1477 |
qed |
1478 |
||
1479 |
end |
|
1480 |
||
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
|
1481 |
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
|
1482 |
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
|
1483 |
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
|
1484 |
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
|
1485 |
hence "(x mod n - y mod n) 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
|
1486 |
hence "(x - y) mod n = 0" by (simp add: zmod_zdiff1_eq[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
|
1487 |
thus "n dvd x - y" by (simp add: zdvd_iff_zmod_eq_0) |
62500b980749
Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents:
27651
diff
changeset
|
1488 |
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
|
1489 |
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
|
1490 |
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
|
1491 |
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
|
1492 |
hence "x mod n = (n*k + y) 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
|
1493 |
thus "x mod n = y mod n" by (simp add: zmod_zadd_left_eq) |
62500b980749
Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents:
27651
diff
changeset
|
1494 |
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
|
1495 |
|
62500b980749
Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents:
27651
diff
changeset
|
1496 |
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
|
1497 |
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
|
1498 |
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
|
1499 |
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
|
1500 |
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
|
1501 |
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
|
1502 |
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
|
1503 |
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
|
1504 |
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
|
1505 |
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
|
1506 |
|
62500b980749
Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
chaieb
parents:
27651
diff
changeset
|
1507 |
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
|
1508 |
(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
|
1509 |
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
|
1510 |
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
|
1511 |
{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
|
1512 |
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
|
1513 |
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
|
1514 |
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
|
1515 |
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
|
1516 |
{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
|
1517 |
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
|
1518 |
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
|
1519 |
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
|
1520 |
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
|
1521 |
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
|
1522 |
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
|
1523 |
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
|
1524 |
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
|
1525 |
|
23164 | 1526 |
code_modulename SML |
1527 |
IntDiv Integer |
|
1528 |
||
1529 |
code_modulename OCaml |
|
1530 |
IntDiv Integer |
|
1531 |
||
1532 |
code_modulename Haskell |
|
24195 | 1533 |
IntDiv Integer |
23164 | 1534 |
|
1535 |
end |