author  haftmann 
Fri, 01 Nov 2013 18:51:14 +0100  
changeset 54230  b1d955791529 
parent 54227  63b441f49645 
child 54489  03ff4d1e6784 
permissions  rwrr 
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(* Title: HOL/Divides.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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15131  8 
theory Divides 
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imports Nat_Transfer 
15131  10 
begin 
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25942  12 
subsection {* Syntactic division operations *} 
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class div = dvd + 
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and mod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "mod" 70) 
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subsection {* Abstract division in commutative semirings. *} 
25942  20 

30930  21 
class semiring_div = comm_semiring_1_cancel + no_zero_divisors + div + 
25942  22 
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and div_by_0 [simp]: "a div 0 = 0" 
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and div_0 [simp]: "0 div a = 0" 
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and div_mult_self1 [simp]: "b \<noteq> 0 \<Longrightarrow> (a + c * b) div b = c + a div b" 
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and div_mult_mult1 [simp]: "c \<noteq> 0 \<Longrightarrow> (c * a) div (c * b) = a div b" 
25942  27 
begin 
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text {* @{const div} and @{const mod} *} 
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26062  31 
lemma mod_div_equality2: "b * (a div b) + a mod b = a" 
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unfolding mult_commute [of b] 

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by (rule mod_div_equality) 

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lemma mod_div_equality': "a mod b + a div b * b = a" 
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using mod_div_equality [of a b] 
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by (simp only: add_ac) 
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26062  39 
lemma div_mod_equality: "((a div b) * b + a mod b) + c = a + c" 
30934  40 
by (simp add: mod_div_equality) 
26062  41 

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lemma div_mod_equality2: "(b * (a div b) + a mod b) + c = a + c" 

30934  43 
by (simp add: mod_div_equality2) 
26062  44 

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lemma mod_by_0 [simp]: "a mod 0 = a" 
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lemma mod_0 [simp]: "0 mod a = 0" 
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lemma div_mult_self2 [simp]: 
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assumes "b \<noteq> 0" 
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shows "(a + b * c) div b = c + a div b" 
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using assms div_mult_self1 [of b a c] by (simp add: mult_commute) 
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54221  56 
lemma div_mult_self3 [simp]: 
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assumes "b \<noteq> 0" 

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shows "(c * b + a) div b = c + a div b" 

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using assms by (simp add: add.commute) 

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61 
lemma div_mult_self4 [simp]: 

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assumes "b \<noteq> 0" 

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shows "(b * c + a) div b = c + a div b" 

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using assms by (simp add: add.commute) 

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lemma mod_mult_self1 [simp]: "(a + c * b) mod b = a mod b" 
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proof (cases "b = 0") 
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case True then show ?thesis by simp 
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next 
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case False 
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have "a + c * b = (a + c * b) div b * b + (a + c * b) mod b" 
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by (simp add: mod_div_equality) 
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also from False div_mult_self1 [of b a c] have 
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"\<dots> = (c + a div b) * b + (a + c * b) mod b" 
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by (simp add: algebra_simps) 
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finally have "a = a div b * b + (a + c * b) mod b" 
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by (simp add: add_commute [of a] add_assoc distrib_right) 
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then have "a div b * b + (a + c * b) mod b = a div b * b + a mod b" 
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by (simp add: mod_div_equality) 
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then show ?thesis by simp 
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qed 
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54221  83 
lemma mod_mult_self2 [simp]: 
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"(a + b * c) mod b = a mod b" 

30934  85 
by (simp add: mult_commute [of b]) 
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54221  87 
lemma mod_mult_self3 [simp]: 
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"(c * b + a) mod b = a mod b" 

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by (simp add: add.commute) 

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lemma mod_mult_self4 [simp]: 

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"(b * c + a) mod b = a mod b" 

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by (simp add: add.commute) 

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lemma div_mult_self1_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> b * a div b = a" 
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using div_mult_self2 [of b 0 a] by simp 
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lemma div_mult_self2_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> a * b div b = a" 
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using div_mult_self1 [of b 0 a] by simp 
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lemma mod_mult_self1_is_0 [simp]: "b * a mod b = 0" 
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lemma mod_mult_self2_is_0 [simp]: "a * b mod b = 0" 
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using mod_mult_self1 [of 0 a b] by simp 
26062  106 

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lemma div_by_1 [simp]: "a div 1 = a" 
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using div_mult_self2_is_id [of 1 a] zero_neq_one by simp 
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lemma mod_by_1 [simp]: "a mod 1 = 0" 
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proof  
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from mod_div_equality [of a one] div_by_1 have "a + a mod 1 = a" by simp 
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then have "a + a mod 1 = a + 0" by simp 
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then show ?thesis by (rule add_left_imp_eq) 
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qed 
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lemma mod_self [simp]: "a mod a = 0" 
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using mod_mult_self2_is_0 [of 1] by simp 
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lemma div_self [simp]: "a \<noteq> 0 \<Longrightarrow> a div a = 1" 
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using div_mult_self2_is_id [of _ 1] by simp 
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27676  123 
lemma div_add_self1 [simp]: 
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assumes "b \<noteq> 0" 
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shows "(b + a) div b = a div b + 1" 
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using assms div_mult_self1 [of b a 1] by (simp add: add_commute) 
26062  127 

27676  128 
lemma div_add_self2 [simp]: 
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assumes "b \<noteq> 0" 
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shows "(a + b) div b = a div b + 1" 
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using assms div_add_self1 [of b a] by (simp add: add_commute) 
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27676  133 
lemma mod_add_self1 [simp]: 
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"(b + a) mod b = a mod b" 
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using mod_mult_self1 [of a 1 b] by (simp add: add_commute) 
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27676  137 
lemma mod_add_self2 [simp]: 
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"(a + b) mod b = a mod b" 
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using mod_mult_self1 [of a 1 b] by simp 
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lemma mod_div_decomp: 
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fixes a b 
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obtains q r where "q = a div b" and "r = a mod b" 
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and "a = q * b + r" 
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proof  
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from mod_div_equality have "a = a div b * b + a mod b" by simp 
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moreover have "a div b = a div b" .. 
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moreover have "a mod b = a mod b" .. 
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note that ultimately show thesis by blast 
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qed 
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lemma dvd_eq_mod_eq_0 [code]: "a dvd b \<longleftrightarrow> b mod a = 0" 
25942  153 
proof 
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assume "b mod a = 0" 

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with mod_div_equality [of b a] have "b div a * a = b" by simp 

156 
then have "b = a * (b div a)" unfolding mult_commute .. 

157 
then have "\<exists>c. b = a * c" .. 

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then show "a dvd b" unfolding dvd_def . 

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next 

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assume "a dvd b" 

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then have "\<exists>c. b = a * c" unfolding dvd_def . 

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then obtain c where "b = a * c" .. 

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then have "b mod a = a * c mod a" by simp 

164 
then have "b mod a = c * a mod a" by (simp add: mult_commute) 

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then show "b mod a = 0" by simp 
25942  166 
qed 
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lemma mod_div_trivial [simp]: "a mod b div b = 0" 
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proof (cases "b = 0") 
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assume "b = 0" 
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thus ?thesis by simp 
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next 
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assume "b \<noteq> 0" 
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hence "a div b + a mod b div b = (a mod b + a div b * b) div b" 
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by (rule div_mult_self1 [symmetric]) 
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176 
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177 
by (simp only: mod_div_equality') 
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178 
also have "\<dots> = a div b + 0" 
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179 
by simp 
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180 
finally show ?thesis 
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181 
by (rule add_left_imp_eq) 
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182 
qed 
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183 

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184 
lemma mod_mod_trivial [simp]: "a mod b mod b = a mod b" 
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185 
proof  
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186 
have "a mod b mod b = (a mod b + a div b * b) mod b" 
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187 
by (simp only: mod_mult_self1) 
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188 
also have "\<dots> = a mod b" 
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189 
by (simp only: mod_div_equality') 
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190 
finally show ?thesis . 
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191 
qed 
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192 

29925  193 
lemma dvd_imp_mod_0: "a dvd b \<Longrightarrow> b mod a = 0" 
29948  194 
by (rule dvd_eq_mod_eq_0[THEN iffD1]) 
29925  195 

196 
lemma dvd_div_mult_self: "a dvd b \<Longrightarrow> (b div a) * a = b" 

197 
by (subst (2) mod_div_equality [of b a, symmetric]) (simp add:dvd_imp_mod_0) 

198 

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199 
lemma dvd_mult_div_cancel: "a dvd b \<Longrightarrow> a * (b div a) = b" 
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200 
by (drule dvd_div_mult_self) (simp add: mult_commute) 
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201 

30052  202 
lemma dvd_div_mult: "a dvd b \<Longrightarrow> (b div a) * c = b * c div a" 
203 
apply (cases "a = 0") 

204 
apply simp 

205 
apply (auto simp: dvd_def mult_assoc) 

206 
done 

207 

29925  208 
lemma div_dvd_div[simp]: 
209 
"a dvd b \<Longrightarrow> a dvd c \<Longrightarrow> (b div a dvd c div a) = (b dvd c)" 

210 
apply (cases "a = 0") 

211 
apply simp 

212 
apply (unfold dvd_def) 

213 
apply auto 

214 
apply(blast intro:mult_assoc[symmetric]) 

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215 
apply(fastforce simp add: mult_assoc) 
29925  216 
done 
217 

30078
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218 
lemma dvd_mod_imp_dvd: "[ k dvd m mod n; k dvd n ] ==> k dvd m" 
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219 
apply (subgoal_tac "k dvd (m div n) *n + m mod n") 
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220 
apply (simp add: mod_div_equality) 
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221 
apply (simp only: dvd_add dvd_mult) 
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222 
done 
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223 

29403
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224 
text {* Addition respects modular equivalence. *} 
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225 

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226 
lemma mod_add_left_eq: "(a + b) mod c = (a mod c + b) mod c" 
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227 
proof  
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228 
have "(a + b) mod c = (a div c * c + a mod c + b) mod c" 
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229 
by (simp only: mod_div_equality) 
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230 
also have "\<dots> = (a mod c + b + a div c * c) mod c" 
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231 
by (simp only: add_ac) 
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232 
also have "\<dots> = (a mod c + b) mod c" 
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233 
by (rule mod_mult_self1) 
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234 
finally show ?thesis . 
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235 
qed 
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236 

fe17df4e4ab3
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237 
lemma mod_add_right_eq: "(a + b) mod c = (a + b mod c) mod c" 
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238 
proof  
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239 
have "(a + b) mod c = (a + (b div c * c + b mod c)) mod c" 
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240 
by (simp only: mod_div_equality) 
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241 
also have "\<dots> = (a + b mod c + b div c * c) mod c" 
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242 
by (simp only: add_ac) 
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243 
also have "\<dots> = (a + b mod c) mod c" 
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244 
by (rule mod_mult_self1) 
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245 
finally show ?thesis . 
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246 
qed 
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247 

fe17df4e4ab3
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248 
lemma mod_add_eq: "(a + b) mod c = (a mod c + b mod c) mod c" 
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249 
by (rule trans [OF mod_add_left_eq mod_add_right_eq]) 
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250 

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251 
lemma mod_add_cong: 
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252 
assumes "a mod c = a' mod c" 
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253 
assumes "b mod c = b' mod c" 
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254 
shows "(a + b) mod c = (a' + b') mod c" 
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255 
proof  
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256 
have "(a mod c + b mod c) mod c = (a' mod c + b' mod c) mod c" 
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257 
unfolding assms .. 
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258 
thus ?thesis 
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259 
by (simp only: mod_add_eq [symmetric]) 
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260 
qed 
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261 

30923
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262 
lemma div_add [simp]: "z dvd x \<Longrightarrow> z dvd y 
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263 
\<Longrightarrow> (x + y) div z = x div z + y div z" 
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264 
by (cases "z = 0", simp, unfold dvd_def, auto simp add: algebra_simps) 
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265 

29403
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266 
text {* Multiplication respects modular equivalence. *} 
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267 

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268 
lemma mod_mult_left_eq: "(a * b) mod c = ((a mod c) * b) mod c" 
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269 
proof  
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270 
have "(a * b) mod c = ((a div c * c + a mod c) * b) mod c" 
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271 
by (simp only: mod_div_equality) 
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272 
also have "\<dots> = (a mod c * b + a div c * b * c) mod c" 
29667  273 
by (simp only: algebra_simps) 
29403
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274 
also have "\<dots> = (a mod c * b) mod c" 
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275 
by (rule mod_mult_self1) 
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276 
finally show ?thesis . 
fe17df4e4ab3
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277 
qed 
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278 

fe17df4e4ab3
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279 
lemma mod_mult_right_eq: "(a * b) mod c = (a * (b mod c)) mod c" 
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280 
proof  
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281 
have "(a * b) mod c = (a * (b div c * c + b mod c)) mod c" 
fe17df4e4ab3
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282 
by (simp only: mod_div_equality) 
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283 
also have "\<dots> = (a * (b mod c) + a * (b div c) * c) mod c" 
29667  284 
by (simp only: algebra_simps) 
29403
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285 
also have "\<dots> = (a * (b mod c)) mod c" 
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286 
by (rule mod_mult_self1) 
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287 
finally show ?thesis . 
fe17df4e4ab3
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288 
qed 
fe17df4e4ab3
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289 

fe17df4e4ab3
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290 
lemma mod_mult_eq: "(a * b) mod c = ((a mod c) * (b mod c)) mod c" 
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291 
by (rule trans [OF mod_mult_left_eq mod_mult_right_eq]) 
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292 

fe17df4e4ab3
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293 
lemma mod_mult_cong: 
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294 
assumes "a mod c = a' mod c" 
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295 
assumes "b mod c = b' mod c" 
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296 
shows "(a * b) mod c = (a' * b') mod c" 
fe17df4e4ab3
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297 
proof  
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298 
have "(a mod c * (b mod c)) mod c = (a' mod c * (b' mod c)) mod c" 
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299 
unfolding assms .. 
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300 
thus ?thesis 
fe17df4e4ab3
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301 
by (simp only: mod_mult_eq [symmetric]) 
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302 
qed 
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303 

47164  304 
text {* Exponentiation respects modular equivalence. *} 
305 

306 
lemma power_mod: "(a mod b)^n mod b = a^n mod b" 

307 
apply (induct n, simp_all) 

308 
apply (rule mod_mult_right_eq [THEN trans]) 

309 
apply (simp (no_asm_simp)) 

310 
apply (rule mod_mult_eq [symmetric]) 

311 
done 

312 

29404  313 
lemma mod_mod_cancel: 
314 
assumes "c dvd b" 

315 
shows "a mod b mod c = a mod c" 

316 
proof  

317 
from `c dvd b` obtain k where "b = c * k" 

318 
by (rule dvdE) 

319 
have "a mod b mod c = a mod (c * k) mod c" 

320 
by (simp only: `b = c * k`) 

321 
also have "\<dots> = (a mod (c * k) + a div (c * k) * k * c) mod c" 

322 
by (simp only: mod_mult_self1) 

323 
also have "\<dots> = (a div (c * k) * (c * k) + a mod (c * k)) mod c" 

324 
by (simp only: add_ac mult_ac) 

325 
also have "\<dots> = a mod c" 

326 
by (simp only: mod_div_equality) 

327 
finally show ?thesis . 

328 
qed 

329 

30930  330 
lemma div_mult_div_if_dvd: 
331 
"y dvd x \<Longrightarrow> z dvd w \<Longrightarrow> (x div y) * (w div z) = (x * w) div (y * z)" 

332 
apply (cases "y = 0", simp) 

333 
apply (cases "z = 0", simp) 

334 
apply (auto elim!: dvdE simp add: algebra_simps) 

30476  335 
apply (subst mult_assoc [symmetric]) 
336 
apply (simp add: no_zero_divisors) 

30930  337 
done 
338 

35367
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
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339 
lemma div_mult_swap: 
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lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
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340 
assumes "c dvd b" 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
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341 
shows "a * (b div c) = (a * b) div c" 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
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342 
proof  
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
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343 
from assms have "b div c * (a div 1) = b * a div (c * 1)" 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
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344 
by (simp only: div_mult_div_if_dvd one_dvd) 
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lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
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345 
then show ?thesis by (simp add: mult_commute) 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
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346 
qed 
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lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
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347 

30930  348 
lemma div_mult_mult2 [simp]: 
349 
"c \<noteq> 0 \<Longrightarrow> (a * c) div (b * c) = a div b" 

350 
by (drule div_mult_mult1) (simp add: mult_commute) 

351 

352 
lemma div_mult_mult1_if [simp]: 

353 
"(c * a) div (c * b) = (if c = 0 then 0 else a div b)" 

354 
by simp_all 

30476  355 

30930  356 
lemma mod_mult_mult1: 
357 
"(c * a) mod (c * b) = c * (a mod b)" 

358 
proof (cases "c = 0") 

359 
case True then show ?thesis by simp 

360 
next 

361 
case False 

362 
from mod_div_equality 

363 
have "((c * a) div (c * b)) * (c * b) + (c * a) mod (c * b) = c * a" . 

364 
with False have "c * ((a div b) * b + a mod b) + (c * a) mod (c * b) 

365 
= c * a + c * (a mod b)" by (simp add: algebra_simps) 

366 
with mod_div_equality show ?thesis by simp 

367 
qed 

368 

369 
lemma mod_mult_mult2: 

370 
"(a * c) mod (b * c) = (a mod b) * c" 

371 
using mod_mult_mult1 [of c a b] by (simp add: mult_commute) 

372 

47159  373 
lemma mult_mod_left: "(a mod b) * c = (a * c) mod (b * c)" 
374 
by (fact mod_mult_mult2 [symmetric]) 

375 

376 
lemma mult_mod_right: "c * (a mod b) = (c * a) mod (c * b)" 

377 
by (fact mod_mult_mult1 [symmetric]) 

378 

31662
57f7ef0dba8e
generalize lemmas dvd_mod and dvd_mod_iff to class semiring_div
huffman
parents:
31661
diff
changeset

379 
lemma dvd_mod: "k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd (m mod n)" 
57f7ef0dba8e
generalize lemmas dvd_mod and dvd_mod_iff to class semiring_div
huffman
parents:
31661
diff
changeset

380 
unfolding dvd_def by (auto simp add: mod_mult_mult1) 
57f7ef0dba8e
generalize lemmas dvd_mod and dvd_mod_iff to class semiring_div
huffman
parents:
31661
diff
changeset

381 

57f7ef0dba8e
generalize lemmas dvd_mod and dvd_mod_iff to class semiring_div
huffman
parents:
31661
diff
changeset

382 
lemma dvd_mod_iff: "k dvd n \<Longrightarrow> k dvd (m mod n) \<longleftrightarrow> k dvd m" 
57f7ef0dba8e
generalize lemmas dvd_mod and dvd_mod_iff to class semiring_div
huffman
parents:
31661
diff
changeset

383 
by (blast intro: dvd_mod_imp_dvd dvd_mod) 
57f7ef0dba8e
generalize lemmas dvd_mod and dvd_mod_iff to class semiring_div
huffman
parents:
31661
diff
changeset

384 

31009
41fd307cab30
dropped reference to class recpower and lemma duplicate
haftmann
parents:
30934
diff
changeset

385 
lemma div_power: 
31661
1e252b8b2334
move lemma div_power into semiring_div context; class ring_div inherits from idom
huffman
parents:
31009
diff
changeset

386 
"y dvd x \<Longrightarrow> (x div y) ^ n = x ^ n div y ^ n" 
30476  387 
apply (induct n) 
388 
apply simp 

389 
apply(simp add: div_mult_div_if_dvd dvd_power_same) 

390 
done 

391 

35367
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset

392 
lemma dvd_div_eq_mult: 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset

393 
assumes "a \<noteq> 0" and "a dvd b" 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset

394 
shows "b div a = c \<longleftrightarrow> b = c * a" 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset

395 
proof 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset

396 
assume "b = c * a" 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset

397 
then show "b div a = c" by (simp add: assms) 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset

398 
next 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset

399 
assume "b div a = c" 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset

400 
then have "b div a * a = c * a" by simp 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset

401 
moreover from `a dvd b` have "b div a * a = b" by (simp add: dvd_div_mult_self) 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset

402 
ultimately show "b = c * a" by simp 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset

403 
qed 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset

404 

45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset

405 
lemma dvd_div_div_eq_mult: 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset

406 
assumes "a \<noteq> 0" "c \<noteq> 0" and "a dvd b" "c dvd d" 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset

407 
shows "b div a = d div c \<longleftrightarrow> b * c = a * d" 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset

408 
using assms by (auto simp add: mult_commute [of _ a] dvd_div_mult_self dvd_div_eq_mult div_mult_swap intro: sym) 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset

409 

31661
1e252b8b2334
move lemma div_power into semiring_div context; class ring_div inherits from idom
huffman
parents:
31009
diff
changeset

410 
end 
1e252b8b2334
move lemma div_power into semiring_div context; class ring_div inherits from idom
huffman
parents:
31009
diff
changeset

411 

35673  412 
class ring_div = semiring_div + comm_ring_1 
29405
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

413 
begin 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

414 

36634  415 
subclass ring_1_no_zero_divisors .. 
416 

29405
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

417 
text {* Negation respects modular equivalence. *} 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

418 

98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

419 
lemma mod_minus_eq: "( a) mod b = ( (a mod b)) mod b" 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

420 
proof  
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

421 
have "( a) mod b = ( (a div b * b + a mod b)) mod b" 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

422 
by (simp only: mod_div_equality) 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

423 
also have "\<dots> = ( (a mod b) +  (a div b) * b) mod b" 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

424 
by (simp only: minus_add_distrib minus_mult_left add_ac) 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

425 
also have "\<dots> = ( (a mod b)) mod b" 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

426 
by (rule mod_mult_self1) 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

427 
finally show ?thesis . 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

428 
qed 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

429 

98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

430 
lemma mod_minus_cong: 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

431 
assumes "a mod b = a' mod b" 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

432 
shows "( a) mod b = ( a') mod b" 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

433 
proof  
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

434 
have "( (a mod b)) mod b = ( (a' mod b)) mod b" 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

435 
unfolding assms .. 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

436 
thus ?thesis 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

437 
by (simp only: mod_minus_eq [symmetric]) 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

438 
qed 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

439 

98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

440 
text {* Subtraction respects modular equivalence. *} 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

441 

54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54227
diff
changeset

442 
lemma mod_diff_left_eq: 
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54227
diff
changeset

443 
"(a  b) mod c = (a mod c  b) mod c" 
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54227
diff
changeset

444 
using mod_add_cong [of a c "a mod c" " b" " b"] by simp 
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54227
diff
changeset

445 

b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54227
diff
changeset

446 
lemma mod_diff_right_eq: 
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54227
diff
changeset

447 
"(a  b) mod c = (a  b mod c) mod c" 
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54227
diff
changeset

448 
using mod_add_cong [of a c a " b" " (b mod c)"] mod_minus_cong [of "b mod c" c b] by simp 
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54227
diff
changeset

449 

b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54227
diff
changeset

450 
lemma mod_diff_eq: 
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54227
diff
changeset

451 
"(a  b) mod c = (a mod c  b mod c) mod c" 
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54227
diff
changeset

452 
using mod_add_cong [of a c "a mod c" " b" " (b mod c)"] mod_minus_cong [of "b mod c" c b] by simp 
29405
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

453 

98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

454 
lemma mod_diff_cong: 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

455 
assumes "a mod c = a' mod c" 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

456 
assumes "b mod c = b' mod c" 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

457 
shows "(a  b) mod c = (a'  b') mod c" 
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54227
diff
changeset

458 
using assms mod_add_cong [of a c a' " b" " b'"] mod_minus_cong [of b c "b'"] by simp 
29405
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

459 

30180  460 
lemma dvd_neg_div: "y dvd x \<Longrightarrow> x div y =  (x div y)" 
461 
apply (case_tac "y = 0") apply simp 

462 
apply (auto simp add: dvd_def) 

463 
apply (subgoal_tac "(y * k) = y *  k") 

464 
apply (erule ssubst) 

465 
apply (erule div_mult_self1_is_id) 

466 
apply simp 

467 
done 

468 

469 
lemma dvd_div_neg: "y dvd x \<Longrightarrow> x div y =  (x div y)" 

470 
apply (case_tac "y = 0") apply simp 

471 
apply (auto simp add: dvd_def) 

472 
apply (subgoal_tac "y * k = y * k") 

473 
apply (erule ssubst) 

474 
apply (rule div_mult_self1_is_id) 

475 
apply simp 

476 
apply simp 

477 
done 

478 

47159  479 
lemma div_minus_minus [simp]: "(a) div (b) = a div b" 
480 
using div_mult_mult1 [of " 1" a b] 

481 
unfolding neg_equal_0_iff_equal by simp 

482 

483 
lemma mod_minus_minus [simp]: "(a) mod (b) =  (a mod b)" 

484 
using mod_mult_mult1 [of " 1" a b] by simp 

485 

486 
lemma div_minus_right: "a div (b) = (a) div b" 

487 
using div_minus_minus [of "a" b] by simp 

488 

489 
lemma mod_minus_right: "a mod (b) =  ((a) mod b)" 

490 
using mod_minus_minus [of "a" b] by simp 

491 

47160  492 
lemma div_minus1_right [simp]: "a div (1) = a" 
493 
using div_minus_right [of a 1] by simp 

494 

495 
lemma mod_minus1_right [simp]: "a mod (1) = 0" 

496 
using mod_minus_right [of a 1] by simp 

497 

54221  498 
lemma minus_mod_self2 [simp]: 
499 
"(a  b) mod b = a mod b" 

500 
by (simp add: mod_diff_right_eq) 

501 

502 
lemma minus_mod_self1 [simp]: 

503 
"(b  a) mod b =  a mod b" 

54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
54227
diff
changeset

504 
using mod_add_self2 [of " a" b] by simp 
54221  505 

29405
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

506 
end 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

507 

54226
e3df2a4e02fc
explicit type class for modelling even/odd parity
haftmann
parents:
54221
diff
changeset

508 
class semiring_div_parity = semiring_div + semiring_numeral + 
e3df2a4e02fc
explicit type class for modelling even/odd parity
haftmann
parents:
54221
diff
changeset

509 
assumes parity: "a mod 2 = 0 \<or> a mod 2 = 1" 
e3df2a4e02fc
explicit type class for modelling even/odd parity
haftmann
parents:
54221
diff
changeset

510 
begin 
e3df2a4e02fc
explicit type class for modelling even/odd parity
haftmann
parents:
54221
diff
changeset

511 

e3df2a4e02fc
explicit type class for modelling even/odd parity
haftmann
parents:
54221
diff
changeset

512 
lemma parity_cases [case_names even odd]: 
e3df2a4e02fc
explicit type class for modelling even/odd parity
haftmann
parents:
54221
diff
changeset

513 
assumes "a mod 2 = 0 \<Longrightarrow> P" 
e3df2a4e02fc
explicit type class for modelling even/odd parity
haftmann
parents:
54221
diff
changeset

514 
assumes "a mod 2 = 1 \<Longrightarrow> P" 
e3df2a4e02fc
explicit type class for modelling even/odd parity
haftmann
parents:
54221
diff
changeset

515 
shows P 
e3df2a4e02fc
explicit type class for modelling even/odd parity
haftmann
parents:
54221
diff
changeset

516 
using assms parity by blast 
e3df2a4e02fc
explicit type class for modelling even/odd parity
haftmann
parents:
54221
diff
changeset

517 

e3df2a4e02fc
explicit type class for modelling even/odd parity
haftmann
parents:
54221
diff
changeset

518 
lemma not_mod_2_eq_0_eq_1 [simp]: 
e3df2a4e02fc
explicit type class for modelling even/odd parity
haftmann
parents:
54221
diff
changeset

519 
"a mod 2 \<noteq> 0 \<longleftrightarrow> a mod 2 = 1" 
e3df2a4e02fc
explicit type class for modelling even/odd parity
haftmann
parents:
54221
diff
changeset

520 
by (cases a rule: parity_cases) simp_all 
e3df2a4e02fc
explicit type class for modelling even/odd parity
haftmann
parents:
54221
diff
changeset

521 

e3df2a4e02fc
explicit type class for modelling even/odd parity
haftmann
parents:
54221
diff
changeset

522 
lemma not_mod_2_eq_1_eq_0 [simp]: 
e3df2a4e02fc
explicit type class for modelling even/odd parity
haftmann
parents:
54221
diff
changeset

523 
"a mod 2 \<noteq> 1 \<longleftrightarrow> a mod 2 = 0" 
e3df2a4e02fc
explicit type class for modelling even/odd parity
haftmann
parents:
54221
diff
changeset

524 
by (cases a rule: parity_cases) simp_all 
e3df2a4e02fc
explicit type class for modelling even/odd parity
haftmann
parents:
54221
diff
changeset

525 

e3df2a4e02fc
explicit type class for modelling even/odd parity
haftmann
parents:
54221
diff
changeset

526 
end 
e3df2a4e02fc
explicit type class for modelling even/odd parity
haftmann
parents:
54221
diff
changeset

527 

25942  528 

53067
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

529 
subsection {* Generic numeral division with a pragmatic type class *} 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

530 

ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

531 
text {* 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

532 
The following type class contains everything necessary to formulate 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

533 
a division algorithm in ring structures with numerals, restricted 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

534 
to its positive segments. This is its primary motiviation, and it 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

535 
could surely be formulated using a more finegrained, more algebraic 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

536 
and less technical class hierarchy. 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

537 
*} 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

538 

ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

539 
class semiring_numeral_div = linordered_semidom + minus + semiring_div + 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

540 
assumes diff_invert_add1: "a + b = c \<Longrightarrow> a = c  b" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

541 
and le_add_diff_inverse2: "b \<le> a \<Longrightarrow> a  b + b = a" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

542 
assumes mult_div_cancel: "b * (a div b) = a  a mod b" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

543 
and div_less: "0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a div b = 0" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

544 
and mod_less: " 0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a mod b = a" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

545 
and div_positive: "0 < b \<Longrightarrow> b \<le> a \<Longrightarrow> a div b > 0" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

546 
and mod_less_eq_dividend: "0 \<le> a \<Longrightarrow> a mod b \<le> a" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

547 
and pos_mod_bound: "0 < b \<Longrightarrow> a mod b < b" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

548 
and pos_mod_sign: "0 < b \<Longrightarrow> 0 \<le> a mod b" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

549 
and mod_mult2_eq: "0 \<le> c \<Longrightarrow> a mod (b * c) = b * (a div b mod c) + a mod b" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

550 
and div_mult2_eq: "0 \<le> c \<Longrightarrow> a div (b * c) = a div b div c" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

551 
assumes discrete: "a < b \<longleftrightarrow> a + 1 \<le> b" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

552 
begin 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

553 

ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

554 
lemma diff_zero [simp]: 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

555 
"a  0 = a" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

556 
by (rule diff_invert_add1 [symmetric]) simp 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

557 

54226
e3df2a4e02fc
explicit type class for modelling even/odd parity
haftmann
parents:
54221
diff
changeset

558 
subclass semiring_div_parity 
e3df2a4e02fc
explicit type class for modelling even/odd parity
haftmann
parents:
54221
diff
changeset

559 
proof 
e3df2a4e02fc
explicit type class for modelling even/odd parity
haftmann
parents:
54221
diff
changeset

560 
fix a 
e3df2a4e02fc
explicit type class for modelling even/odd parity
haftmann
parents:
54221
diff
changeset

561 
show "a mod 2 = 0 \<or> a mod 2 = 1" 
e3df2a4e02fc
explicit type class for modelling even/odd parity
haftmann
parents:
54221
diff
changeset

562 
proof (rule ccontr) 
e3df2a4e02fc
explicit type class for modelling even/odd parity
haftmann
parents:
54221
diff
changeset

563 
assume "\<not> (a mod 2 = 0 \<or> a mod 2 = 1)" 
e3df2a4e02fc
explicit type class for modelling even/odd parity
haftmann
parents:
54221
diff
changeset

564 
then have "a mod 2 \<noteq> 0" and "a mod 2 \<noteq> 1" by simp_all 
e3df2a4e02fc
explicit type class for modelling even/odd parity
haftmann
parents:
54221
diff
changeset

565 
have "0 < 2" by simp 
e3df2a4e02fc
explicit type class for modelling even/odd parity
haftmann
parents:
54221
diff
changeset

566 
with pos_mod_bound pos_mod_sign have "0 \<le> a mod 2" "a mod 2 < 2" by simp_all 
e3df2a4e02fc
explicit type class for modelling even/odd parity
haftmann
parents:
54221
diff
changeset

567 
with `a mod 2 \<noteq> 0` have "0 < a mod 2" by simp 
e3df2a4e02fc
explicit type class for modelling even/odd parity
haftmann
parents:
54221
diff
changeset

568 
with discrete have "1 \<le> a mod 2" by simp 
e3df2a4e02fc
explicit type class for modelling even/odd parity
haftmann
parents:
54221
diff
changeset

569 
with `a mod 2 \<noteq> 1` have "1 < a mod 2" by simp 
e3df2a4e02fc
explicit type class for modelling even/odd parity
haftmann
parents:
54221
diff
changeset

570 
with discrete have "2 \<le> a mod 2" by simp 
e3df2a4e02fc
explicit type class for modelling even/odd parity
haftmann
parents:
54221
diff
changeset

571 
with `a mod 2 < 2` show False by simp 
e3df2a4e02fc
explicit type class for modelling even/odd parity
haftmann
parents:
54221
diff
changeset

572 
qed 
53067
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

573 
qed 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

574 

ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

575 
lemma divmod_digit_1: 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

576 
assumes "0 \<le> a" "0 < b" and "b \<le> a mod (2 * b)" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

577 
shows "2 * (a div (2 * b)) + 1 = a div b" (is "?P") 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

578 
and "a mod (2 * b)  b = a mod b" (is "?Q") 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

579 
proof  
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

580 
from assms mod_less_eq_dividend [of a "2 * b"] have "b \<le> a" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

581 
by (auto intro: trans) 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

582 
with `0 < b` have "0 < a div b" by (auto intro: div_positive) 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

583 
then have [simp]: "1 \<le> a div b" by (simp add: discrete) 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

584 
with `0 < b` have mod_less: "a mod b < b" by (simp add: pos_mod_bound) 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

585 
def w \<equiv> "a div b mod 2" with parity have w_exhaust: "w = 0 \<or> w = 1" by auto 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

586 
have mod_w: "a mod (2 * b) = a mod b + b * w" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

587 
by (simp add: w_def mod_mult2_eq ac_simps) 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

588 
from assms w_exhaust have "w = 1" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

589 
by (auto simp add: mod_w) (insert mod_less, auto) 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

590 
with mod_w have mod: "a mod (2 * b) = a mod b + b" by simp 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

591 
have "2 * (a div (2 * b)) = a div b  w" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

592 
by (simp add: w_def div_mult2_eq mult_div_cancel ac_simps) 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

593 
with `w = 1` have div: "2 * (a div (2 * b)) = a div b  1" by simp 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

594 
then show ?P and ?Q 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

595 
by (simp_all add: div mod diff_invert_add1 [symmetric] le_add_diff_inverse2) 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

596 
qed 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

597 

ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

598 
lemma divmod_digit_0: 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

599 
assumes "0 < b" and "a mod (2 * b) < b" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

600 
shows "2 * (a div (2 * b)) = a div b" (is "?P") 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

601 
and "a mod (2 * b) = a mod b" (is "?Q") 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

602 
proof  
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

603 
def w \<equiv> "a div b mod 2" with parity have w_exhaust: "w = 0 \<or> w = 1" by auto 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

604 
have mod_w: "a mod (2 * b) = a mod b + b * w" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

605 
by (simp add: w_def mod_mult2_eq ac_simps) 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

606 
moreover have "b \<le> a mod b + b" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

607 
proof  
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

608 
from `0 < b` pos_mod_sign have "0 \<le> a mod b" by blast 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

609 
then have "0 + b \<le> a mod b + b" by (rule add_right_mono) 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

610 
then show ?thesis by simp 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

611 
qed 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

612 
moreover note assms w_exhaust 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

613 
ultimately have "w = 0" by auto 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

614 
with mod_w have mod: "a mod (2 * b) = a mod b" by simp 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

615 
have "2 * (a div (2 * b)) = a div b  w" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

616 
by (simp add: w_def div_mult2_eq mult_div_cancel ac_simps) 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

617 
with `w = 0` have div: "2 * (a div (2 * b)) = a div b" by simp 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

618 
then show ?P and ?Q 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

619 
by (simp_all add: div mod) 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

620 
qed 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

621 

ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

622 
definition divmod :: "num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

623 
where 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

624 
"divmod m n = (numeral m div numeral n, numeral m mod numeral n)" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

625 

ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

626 
lemma fst_divmod [simp]: 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

627 
"fst (divmod m n) = numeral m div numeral n" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

628 
by (simp add: divmod_def) 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

629 

ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

630 
lemma snd_divmod [simp]: 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

631 
"snd (divmod m n) = numeral m mod numeral n" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

632 
by (simp add: divmod_def) 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

633 

ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

634 
definition divmod_step :: "num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

635 
where 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

636 
"divmod_step l qr = (let (q, r) = qr 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

637 
in if r \<ge> numeral l then (2 * q + 1, r  numeral l) 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

638 
else (2 * q, r))" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

639 

ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

640 
text {* 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

641 
This is a formulation of one step (referring to one digit position) 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

642 
in schoolmethod division: compare the dividend at the current 
53070  643 
digit position with the remainder from previous division steps 
53067
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

644 
and evaluate accordingly. 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

645 
*} 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

646 

ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

647 
lemma divmod_step_eq [code]: 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

648 
"divmod_step l (q, r) = (if numeral l \<le> r 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

649 
then (2 * q + 1, r  numeral l) else (2 * q, r))" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

650 
by (simp add: divmod_step_def) 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

651 

ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

652 
lemma divmod_step_simps [simp]: 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

653 
"r < numeral l \<Longrightarrow> divmod_step l (q, r) = (2 * q, r)" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

654 
"numeral l \<le> r \<Longrightarrow> divmod_step l (q, r) = (2 * q + 1, r  numeral l)" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

655 
by (auto simp add: divmod_step_eq not_le) 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

656 

ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

657 
text {* 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

658 
This is a formulation of schoolmethod division. 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

659 
If the divisor is smaller than the dividend, terminate. 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

660 
If not, shift the dividend to the right until termination 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

661 
occurs and then reiterate single division steps in the 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

662 
opposite direction. 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

663 
*} 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

664 

ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

665 
lemma divmod_divmod_step [code]: 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

666 
"divmod m n = (if m < n then (0, numeral m) 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

667 
else divmod_step n (divmod m (Num.Bit0 n)))" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

668 
proof (cases "m < n") 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

669 
case True then have "numeral m < numeral n" by simp 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

670 
then show ?thesis 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

671 
by (simp add: prod_eq_iff div_less mod_less) 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

672 
next 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

673 
case False 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

674 
have "divmod m n = 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

675 
divmod_step n (numeral m div (2 * numeral n), 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

676 
numeral m mod (2 * numeral n))" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

677 
proof (cases "numeral n \<le> numeral m mod (2 * numeral n)") 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

678 
case True 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

679 
with divmod_step_simps 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

680 
have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) = 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

681 
(2 * (numeral m div (2 * numeral n)) + 1, numeral m mod (2 * numeral n)  numeral n)" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

682 
by blast 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

683 
moreover from True divmod_digit_1 [of "numeral m" "numeral n"] 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

684 
have "2 * (numeral m div (2 * numeral n)) + 1 = numeral m div numeral n" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

685 
and "numeral m mod (2 * numeral n)  numeral n = numeral m mod numeral n" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

686 
by simp_all 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

687 
ultimately show ?thesis by (simp only: divmod_def) 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

688 
next 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

689 
case False then have *: "numeral m mod (2 * numeral n) < numeral n" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

690 
by (simp add: not_le) 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

691 
with divmod_step_simps 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

692 
have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) = 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

693 
(2 * (numeral m div (2 * numeral n)), numeral m mod (2 * numeral n))" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

694 
by blast 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

695 
moreover from * divmod_digit_0 [of "numeral n" "numeral m"] 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

696 
have "2 * (numeral m div (2 * numeral n)) = numeral m div numeral n" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

697 
and "numeral m mod (2 * numeral n) = numeral m mod numeral n" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

698 
by (simp_all only: zero_less_numeral) 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

699 
ultimately show ?thesis by (simp only: divmod_def) 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

700 
qed 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

701 
then have "divmod m n = 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

702 
divmod_step n (numeral m div numeral (Num.Bit0 n), 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

703 
numeral m mod numeral (Num.Bit0 n))" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

704 
by (simp only: numeral.simps distrib mult_1) 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

705 
then have "divmod m n = divmod_step n (divmod m (Num.Bit0 n))" 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

706 
by (simp add: divmod_def) 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

707 
with False show ?thesis by simp 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

708 
qed 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

709 

53069
d165213e3924
execution of int division by class semiring_numeral_div, replacing pdivmod by divmod_abs
haftmann
parents:
53068
diff
changeset

710 
lemma divmod_cancel [code]: 
d165213e3924
execution of int division by class semiring_numeral_div, replacing pdivmod by divmod_abs
haftmann
parents:
53068
diff
changeset

711 
"divmod (Num.Bit0 m) (Num.Bit0 n) = (case divmod m n of (q, r) \<Rightarrow> (q, 2 * r))" (is ?P) 
d165213e3924
execution of int division by class semiring_numeral_div, replacing pdivmod by divmod_abs
haftmann
parents:
53068
diff
changeset

712 
"divmod (Num.Bit1 m) (Num.Bit0 n) = (case divmod m n of (q, r) \<Rightarrow> (q, 2 * r + 1))" (is ?Q) 
d165213e3924
execution of int division by class semiring_numeral_div, replacing pdivmod by divmod_abs
haftmann
parents:
53068
diff
changeset

713 
proof  
d165213e3924
execution of int division by class semiring_numeral_div, replacing pdivmod by divmod_abs
haftmann
parents:
53068
diff
changeset

714 
have *: "\<And>q. numeral (Num.Bit0 q) = 2 * numeral q" 
d165213e3924
execution of int division by class semiring_numeral_div, replacing pdivmod by divmod_abs
haftmann
parents:
53068
diff
changeset

715 
"\<And>q. numeral (Num.Bit1 q) = 2 * numeral q + 1" 
d165213e3924
execution of int division by class semiring_numeral_div, replacing pdivmod by divmod_abs
haftmann
parents:
53068
diff
changeset

716 
by (simp_all only: numeral_mult numeral.simps distrib) simp_all 
d165213e3924
execution of int division by class semiring_numeral_div, replacing pdivmod by divmod_abs
haftmann
parents:
53068
diff
changeset

717 
have "1 div 2 = 0" "1 mod 2 = 1" by (auto intro: div_less mod_less) 
d165213e3924
execution of int division by class semiring_numeral_div, replacing pdivmod by divmod_abs
haftmann
parents:
53068
diff
changeset

718 
then show ?P and ?Q 
d165213e3924
execution of int division by class semiring_numeral_div, replacing pdivmod by divmod_abs
haftmann
parents:
53068
diff
changeset

719 
by (simp_all add: prod_eq_iff split_def * [of m] * [of n] mod_mult_mult1 
d165213e3924
execution of int division by class semiring_numeral_div, replacing pdivmod by divmod_abs
haftmann
parents:
53068
diff
changeset

720 
div_mult2_eq [of _ _ 2] mod_mult2_eq [of _ _ 2] add.commute del: numeral_times_numeral) 
d165213e3924
execution of int division by class semiring_numeral_div, replacing pdivmod by divmod_abs
haftmann
parents:
53068
diff
changeset

721 
qed 
d165213e3924
execution of int division by class semiring_numeral_div, replacing pdivmod by divmod_abs
haftmann
parents:
53068
diff
changeset

722 

53067
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

723 
end 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

724 

ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

725 
hide_fact (open) diff_invert_add1 le_add_diff_inverse2 diff_zero 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

726 
 {* restore simple accesses for more general variants of theorems *} 
ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

727 

ee0b7c2315d2
type class for generic division algorithm on numerals
haftmann
parents:
53066
diff
changeset

728 

26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

729 
subsection {* Division on @{typ nat} *} 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

730 

fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

731 
text {* 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

732 
We define @{const div} and @{const mod} on @{typ nat} by means 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

733 
of a characteristic relation with two input arguments 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

734 
@{term "m\<Colon>nat"}, @{term "n\<Colon>nat"} and two output arguments 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

735 
@{term "q\<Colon>nat"}(uotient) and @{term "r\<Colon>nat"}(emainder). 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

736 
*} 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

737 

33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

738 
definition divmod_nat_rel :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat \<Rightarrow> bool" where 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

739 
"divmod_nat_rel m n qr \<longleftrightarrow> 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

740 
m = fst qr * n + snd qr \<and> 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

741 
(if n = 0 then fst qr = 0 else if n > 0 then 0 \<le> snd qr \<and> snd qr < n else n < snd qr \<and> snd qr \<le> 0)" 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

742 

33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

743 
text {* @{const divmod_nat_rel} is total: *} 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

744 

33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

745 
lemma divmod_nat_rel_ex: 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

746 
obtains q r where "divmod_nat_rel m n (q, r)" 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

747 
proof (cases "n = 0") 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

748 
case True with that show thesis 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

749 
by (auto simp add: divmod_nat_rel_def) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

750 
next 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

751 
case False 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

752 
have "\<exists>q r. m = q * n + r \<and> r < n" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

753 
proof (induct m) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

754 
case 0 with `n \<noteq> 0` 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

755 
have "(0\<Colon>nat) = 0 * n + 0 \<and> 0 < n" by simp 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

756 
then show ?case by blast 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

757 
next 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

758 
case (Suc m) then obtain q' r' 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

759 
where m: "m = q' * n + r'" and n: "r' < n" by auto 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

760 
then show ?case proof (cases "Suc r' < n") 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

761 
case True 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

762 
from m n have "Suc m = q' * n + Suc r'" by simp 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

763 
with True show ?thesis by blast 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

764 
next 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

765 
case False then have "n \<le> Suc r'" by auto 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

766 
moreover from n have "Suc r' \<le> n" by auto 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

767 
ultimately have "n = Suc r'" by auto 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

768 
with m have "Suc m = Suc q' * n + 0" by simp 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

769 
with `n \<noteq> 0` show ?thesis by blast 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

770 
qed 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

771 
qed 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

772 
with that show thesis 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

773 
using `n \<noteq> 0` by (auto simp add: divmod_nat_rel_def) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

774 
qed 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

775 

33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

776 
text {* @{const divmod_nat_rel} is injective: *} 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

777 

33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

778 
lemma divmod_nat_rel_unique: 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

779 
assumes "divmod_nat_rel m n qr" 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

780 
and "divmod_nat_rel m n qr'" 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

781 
shows "qr = qr'" 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

782 
proof (cases "n = 0") 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

783 
case True with assms show ?thesis 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

784 
by (cases qr, cases qr') 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

785 
(simp add: divmod_nat_rel_def) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

786 
next 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

787 
case False 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

788 
have aux: "\<And>q r q' r'. q' * n + r' = q * n + r \<Longrightarrow> r < n \<Longrightarrow> q' \<le> (q\<Colon>nat)" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

789 
apply (rule leI) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

790 
apply (subst less_iff_Suc_add) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

791 
apply (auto simp add: add_mult_distrib) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

792 
done 
53374
a14d2a854c02
tuned proofs  clarified flow of facts wrt. calculation;
wenzelm
parents:
53199
diff
changeset

793 
from `n \<noteq> 0` assms have *: "fst qr = fst qr'" 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

794 
by (auto simp add: divmod_nat_rel_def intro: order_antisym dest: aux sym) 
53374
a14d2a854c02
tuned proofs  clarified flow of facts wrt. calculation;
wenzelm
parents:
53199
diff
changeset

795 
with assms have "snd qr = snd qr'" 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

796 
by (simp add: divmod_nat_rel_def) 
53374
a14d2a854c02
tuned proofs  clarified flow of facts wrt. calculation;
wenzelm
parents:
53199
diff
changeset

797 
with * show ?thesis by (cases qr, cases qr') simp 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

798 
qed 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

799 

fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

800 
text {* 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

801 
We instantiate divisibility on the natural numbers by 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

802 
means of @{const divmod_nat_rel}: 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

803 
*} 
25942  804 

33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

805 
definition divmod_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where 
37767  806 
"divmod_nat m n = (THE qr. divmod_nat_rel m n qr)" 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

807 

33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

808 
lemma divmod_nat_rel_divmod_nat: 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

809 
"divmod_nat_rel m n (divmod_nat m n)" 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

810 
proof  
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

811 
from divmod_nat_rel_ex 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

812 
obtain qr where rel: "divmod_nat_rel m n qr" . 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

813 
then show ?thesis 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

814 
by (auto simp add: divmod_nat_def intro: theI elim: divmod_nat_rel_unique) 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

815 
qed 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

816 

47135
fb67b596067f
rename lemmas {div,mod}_eq > {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents:
47134
diff
changeset

817 
lemma divmod_nat_unique: 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

818 
assumes "divmod_nat_rel m n qr" 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

819 
shows "divmod_nat m n = qr" 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

820 
using assms by (auto intro: divmod_nat_rel_unique divmod_nat_rel_divmod_nat) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

821 

46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

822 
instantiation nat :: semiring_div 
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

823 
begin 
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

824 

26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

825 
definition div_nat where 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

826 
"m div n = fst (divmod_nat m n)" 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

827 

46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

828 
lemma fst_divmod_nat [simp]: 
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

829 
"fst (divmod_nat m n) = m div n" 
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

830 
by (simp add: div_nat_def) 
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

831 

26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

832 
definition mod_nat where 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

833 
"m mod n = snd (divmod_nat m n)" 
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25162
diff
changeset

834 

46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

835 
lemma snd_divmod_nat [simp]: 
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

836 
"snd (divmod_nat m n) = m mod n" 
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

837 
by (simp add: mod_nat_def) 
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

838 

33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

839 
lemma divmod_nat_div_mod: 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

840 
"divmod_nat m n = (m div n, m mod n)" 
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

841 
by (simp add: prod_eq_iff) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

842 

47135
fb67b596067f
rename lemmas {div,mod}_eq > {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents:
47134
diff
changeset

843 
lemma div_nat_unique: 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

844 
assumes "divmod_nat_rel m n (q, r)" 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

845 
shows "m div n = q" 
47135
fb67b596067f
rename lemmas {div,mod}_eq > {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents:
47134
diff
changeset

846 
using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff) 
fb67b596067f
rename lemmas {div,mod}_eq > {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents:
47134
diff
changeset

847 

fb67b596067f
rename lemmas {div,mod}_eq > {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents:
47134
diff
changeset

848 
lemma mod_nat_unique: 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

849 
assumes "divmod_nat_rel m n (q, r)" 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

850 
shows "m mod n = r" 
47135
fb67b596067f
rename lemmas {div,mod}_eq > {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents:
47134
diff
changeset

851 
using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff) 
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25162
diff
changeset

852 

33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

853 
lemma divmod_nat_rel: "divmod_nat_rel m n (m div n, m mod n)" 
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

854 
using divmod_nat_rel_divmod_nat by (simp add: divmod_nat_div_mod) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

855 

47136  856 
lemma divmod_nat_zero: "divmod_nat m 0 = (0, m)" 
857 
by (simp add: divmod_nat_unique divmod_nat_rel_def) 

858 

859 
lemma divmod_nat_zero_left: "divmod_nat 0 n = (0, 0)" 

860 
by (simp add: divmod_nat_unique divmod_nat_rel_def) 

25942  861 

47137  862 
lemma divmod_nat_base: "m < n \<Longrightarrow> divmod_nat m n = (0, m)" 
863 
by (simp add: divmod_nat_unique divmod_nat_rel_def) 

25942  864 

33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

865 
lemma divmod_nat_step: 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

866 
assumes "0 < n" and "n \<le> m" 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

867 
shows "divmod_nat m n = (Suc ((m  n) div n), (m  n) mod n)" 
47135
fb67b596067f
rename lemmas {div,mod}_eq > {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents:
47134
diff
changeset

868 
proof (rule divmod_nat_unique) 
47134  869 
have "divmod_nat_rel (m  n) n ((m  n) div n, (m  n) mod n)" 
870 
by (rule divmod_nat_rel) 

871 
thus "divmod_nat_rel m n (Suc ((m  n) div n), (m  n) mod n)" 

872 
unfolding divmod_nat_rel_def using assms by auto 

26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

873 
qed 
25942  874 

26300  875 
text {* The ''recursion'' equations for @{const div} and @{const mod} *} 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

876 

fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

877 
lemma div_less [simp]: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

878 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

879 
assumes "m < n" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

880 
shows "m div n = 0" 
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

881 
using assms divmod_nat_base by (simp add: prod_eq_iff) 
25942  882 

26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

883 
lemma le_div_geq: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

884 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

885 
assumes "0 < n" and "n \<le> m" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

886 
shows "m div n = Suc ((m  n) div n)" 
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

887 
using assms divmod_nat_step by (simp add: prod_eq_iff) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

888 

26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

889 
lemma mod_less [simp]: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

890 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

891 
assumes "m < n" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

892 
shows "m mod n = m" 
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

893 
using assms divmod_nat_base by (simp add: prod_eq_iff) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

894 

fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

895 
lemma le_mod_geq: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

896 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

897 
assumes "n \<le> m" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

898 
shows "m mod n = (m  n) mod n" 
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

899 
using assms divmod_nat_step by (cases "n = 0") (simp_all add: prod_eq_iff) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

900 

47136  901 
instance proof 
902 
fix m n :: nat 

903 
show "m div n * n + m mod n = m" 

904 
using divmod_nat_rel [of m n] by (simp add: divmod_nat_rel_def) 

905 
next 

906 
fix m n q :: nat 

907 
assume "n \<noteq> 0" 

908 
then show "(q + m * n) div n = m + q div n" 

909 
by (induct m) (simp_all add: le_div_geq) 

910 
next 

911 
fix m n q :: nat 

912 
assume "m \<noteq> 0" 

913 
hence "\<And>a b. divmod_nat_rel n q (a, b) \<Longrightarrow> divmod_nat_rel (m * n) (m * q) (a, m * b)" 

914 
unfolding divmod_nat_rel_def 

915 
by (auto split: split_if_asm, simp_all add: algebra_simps) 

916 
moreover from divmod_nat_rel have "divmod_nat_rel n q (n div q, n mod q)" . 

917 
ultimately have "divmod_nat_rel (m * n) (m * q) (n div q, m * (n mod q))" . 

918 
thus "(m * n) div (m * q) = n div q" by (rule div_nat_unique) 

919 
next 

920 
fix n :: nat show "n div 0 = 0" 

33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

921 
by (simp add: div_nat_def divmod_nat_zero) 
47136  922 
next 
923 
fix n :: nat show "0 div n = 0" 

924 
by (simp add: div_nat_def divmod_nat_zero_left) 

25942  925 
qed 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

926 

25942  927 
end 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

928 

33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

929 
lemma divmod_nat_if [code]: "divmod_nat m n = (if n = 0 \<or> m < n then (0, m) else 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

930 
let (q, r) = divmod_nat (m  n) n in (Suc q, r))" 
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

931 
by (simp add: prod_eq_iff prod_case_beta not_less le_div_geq le_mod_geq) 
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

932 

26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

933 
text {* Simproc for cancelling @{const div} and @{const mod} *} 
25942  934 

51299
30b014246e21
proper place for cancel_div_mod.ML (see also ee729dbd1b7f and ec7f10155389);
wenzelm
parents:
51173
diff
changeset

935 
ML_file "~~/src/Provers/Arith/cancel_div_mod.ML" 
30b014246e21
proper place for cancel_div_mod.ML (see also ee729dbd1b7f and ec7f10155389);
wenzelm
parents:
51173
diff
changeset

936 

30934  937 
ML {* 
43594  938 
structure Cancel_Div_Mod_Nat = Cancel_Div_Mod 
41550  939 
( 
30934  940 
val div_name = @{const_name div}; 
941 
val mod_name = @{const_name mod}; 

942 
val mk_binop = HOLogic.mk_binop; 

48561
12aa0cb2b447
move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents:
47268
diff
changeset

943 
val mk_plus = HOLogic.mk_binop @{const_name Groups.plus}; 
12aa0cb2b447
move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents:
47268
diff
changeset

944 
val dest_plus = HOLogic.dest_bin @{const_name Groups.plus} HOLogic.natT; 
12aa0cb2b447
move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents:
47268
diff
changeset

945 
fun mk_sum [] = HOLogic.zero 
12aa0cb2b447
move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents:
47268
diff
changeset

946 
 mk_sum [t] = t 
12aa0cb2b447
move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents:
47268
diff
changeset

947 
 mk_sum (t :: ts) = mk_plus (t, mk_sum ts); 
12aa0cb2b447
move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents:
47268
diff
changeset

948 
fun dest_sum tm = 
12aa0cb2b447
move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents:
47268
diff
changeset

949 
if HOLogic.is_zero tm then [] 
12aa0cb2b447
move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents:
47268
diff
changeset

950 
else 
12aa0cb2b447
move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents:
47268
diff
changeset

951 
(case try HOLogic.dest_Suc tm of 
12aa0cb2b447
move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents:
47268
diff
changeset

952 
SOME t => HOLogic.Suc_zero :: dest_sum t 
12aa0cb2b447
move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents:
47268
diff
changeset

953 
 NONE => 
12aa0cb2b447
move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents:
47268
diff
changeset

954 
(case try dest_plus tm of 
12aa0cb2b447
move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents:
47268
diff
changeset

955 
SOME (t, u) => dest_sum t @ dest_sum u 
12aa0cb2b447
move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents:
47268
diff
changeset

956 
 NONE => [tm])); 
25942  957 

30934  958 
val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}]; 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

959 

30934  960 
val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac 
35050
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
34982
diff
changeset

961 
(@{thm add_0_left} :: @{thm add_0_right} :: @{thms add_ac})) 
41550  962 
) 
25942  963 
*} 
964 

43594  965 
simproc_setup cancel_div_mod_nat ("(m::nat) + n") = {* K Cancel_Div_Mod_Nat.proc *} 
966 

26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

967 

fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

968 
subsubsection {* Quotient *} 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

969 

fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

970 
lemma div_geq: "0 < n \<Longrightarrow> \<not> m < n \<Longrightarrow> m div n = Suc ((m  n) div n)" 
29667  971 
by (simp add: le_div_geq linorder_not_less) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

972 

fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

973 
lemma div_if: "0 < n \<Longrightarrow> m div n = (if m < n then 0 else Suc ((m  n) div n))" 
29667  974 
by (simp add: div_geq) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

975 

fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

976 
lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)" 
29667  977 
by simp 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

978 

fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

979 
lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)" 
29667  980 
by simp 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

981 

53066  982 
lemma div_positive: 
983 
fixes m n :: nat 

984 
assumes "n > 0" 

985 
assumes "m \<ge> n" 

986 
shows "m div n > 0" 

987 
proof  

988 
from `m \<ge> n` obtain q where "m = n + q" 

989 
by (auto simp add: le_iff_add) 

990 
with `n > 0` show ?thesis by simp 

991 
qed 

992 

25942  993 

994 
subsubsection {* Remainder *} 

995 

26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

996 
lemma mod_less_divisor [simp]: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

997 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

998 
assumes "n > 0" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

999 
shows "m mod n < (n::nat)" 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

1000 
using assms divmod_nat_rel [of m n] unfolding divmod_nat_rel_def by auto 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1001 

51173  1002 
lemma mod_Suc_le_divisor [simp]: 
1003 
"m mod Suc n \<le> n" 

1004 
using mod_less_divisor [of "Suc n" m] by arith 

1005 

26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1006 
lemma mod_less_eq_dividend [simp]: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1007 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1008 
shows "m mod n \<le> m" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1009 
proof (rule add_leD2) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1010 
from mod_div_equality have "m div n * n + m mod n = m" . 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1011 
then show "m div n * n + m mod n \<le> m" by auto 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1012 
qed 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1013 

fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1014 
lemma mod_geq: "\<not> m < (n\<Colon>nat) \<Longrightarrow> m mod n = (m  n) mod n" 
29667  1015 
by (simp add: le_mod_geq linorder_not_less) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1016 

26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1017 
lemma mod_if: "m mod (n\<Colon>nat) = (if m < n then m else (m  n) mod n)" 
29667  1018 
by (simp add: le_mod_geq) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1019 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1020 
lemma mod_1 [simp]: "m mod Suc 0 = 0" 
29667  1021 
by (induct m) (simp_all add: mod_geq) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1022 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1023 
(* a simple rearrangement of mod_div_equality: *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1024 
lemma mult_div_cancel: "(n::nat) * (m div n) = m  (m mod n)" 
47138  1025 
using mod_div_equality2 [of n m] by arith 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1026 

15439  1027 
lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)" 
22718  1028 
apply (drule mod_less_divisor [where m = m]) 
1029 
apply simp 

1030 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1031 

26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1032 
subsubsection {* Quotient and Remainder *} 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1033 

33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

1034 
lemma divmod_nat_rel_mult1_eq: 
46552  1035 
"divmod_nat_rel b c (q, r) 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

1036 
\<Longrightarrow> divmod_nat_rel (a * b) c (a * q + a * r div c, a * r mod c)" 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

1037 
by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1038 

30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

1039 
lemma div_mult1_eq: 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

1040 
"(a * b) div c = a * (b div c) + a * (b mod c) div (c::nat)" 
47135
fb67b596067f
rename lemmas {div,mod}_eq > {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents:
47134
diff
changeset

1041 
by (blast intro: divmod_nat_rel_mult1_eq [THEN div_nat_unique] divmod_nat_rel) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1042 

33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

1043 
lemma divmod_nat_rel_add1_eq: 
46552  1044 
"divmod_nat_rel a c (aq, ar) \<Longrightarrow> divmod_nat_rel b c (bq, br) 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

1045 
\<Longrightarrow> divmod_nat_rel (a + b) c (aq + bq + (ar + br) div c, (ar + br) mod c)" 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

1046 
by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1047 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1048 
(*NOT suitable for rewriting: the RHS has an instance of the LHS*) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1049 
lemma div_add1_eq: 
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

1050 
"(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)" 
47135
fb67b596067f
rename lemmas {div,mod}_eq > {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents:
47134
diff
changeset

1051 
by (blast intro: divmod_nat_rel_add1_eq [THEN div_nat_unique] divmod_nat_rel) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1052 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1053 
lemma mod_lemma: "[ (0::nat) < c; r < b ] ==> b * (q mod c) + r < b * c" 
22718  1054 
apply (cut_tac m = q and n = c in mod_less_divisor) 
1055 
apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto) 

1056 
apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst) 

1057 
apply (simp add: add_mult_distrib2) 

1058 
done 

10559
d3fd54fc659b
many new div and mod properties (borrowed from Integ/IntDiv)
paulson
parents:
10214
diff
changeset

1059 

33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

1060 
lemma divmod_nat_rel_mult2_eq: 
46552  1061 
"divmod_nat_rel a b (q, r) 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

1062 
\<Longrightarrow> divmod_nat_rel a (b * c) (q div c, b *(q mod c) + r)" 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

1063 
by (auto simp add: mult_ac divmod_nat_rel_def add_mult_distrib2 [symmetric] mod_lemma) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1064 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1065 
lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)" 
47135
fb67b596067f
rename lemmas {div,mod}_eq > {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents:
47134
diff
changeset

1066 
by (force simp add: divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN div_nat_unique]) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1067 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1068 
lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)" 
47135
fb67b596067f
rename lemmas {div,mod}_eq > {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents:
47134
diff
changeset

1069 
by (auto simp add: mult_commute divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN mod_nat_unique]) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1070 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1071 

46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

1072 
subsubsection {* Further Facts about Quotient and Remainder *} 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1073 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1074 
lemma div_1 [simp]: "m div Suc 0 = m" 
29667  1075 
by (induct m) (simp_all add: div_geq) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1076 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1077 
(* Monotonicity of div in first argument *) 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

1078 
lemma div_le_mono [rule_format (no_asm)]: 
22718  1079 
"\<forall>m::nat. m \<le> n > (m div k) \<le> (n div k)" 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1080 
apply (case_tac "k=0", simp) 
15251  1081 
apply (induct "n" rule: nat_less_induct, clarify) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1082 
apply (case_tac "n<k") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1083 
(* 1 case n<k *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1084 
apply simp 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1085 
(* 2 case n >= k *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1086 
apply (case_tac "m<k") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1087 
(* 2.1 case m<k *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1088 
apply simp 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1089 
(* 2.2 case m>=k *) 
15439  1090 
apply (simp add: div_geq diff_le_mono) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1091 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1092 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1093 
(* Antimonotonicity of div in second argument *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1094 
lemma div_le_mono2: "!!m::nat. [ 0<m; m\<le>n ] ==> (k div n) \<le> (k div m)" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1095 
apply (subgoal_tac "0<n") 
22718  1096 
prefer 2 apply simp 
15251  1097 
apply (induct_tac k rule: nat_less_induct) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1098 
apply (rename_tac "k") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1099 
apply (case_tac "k<n", simp) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1100 
apply (subgoal_tac "~ (k<m) ") 
22718  1101 
prefer 2 apply simp 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1102 
apply (simp add: div_geq) 
15251  1103 
apply (subgoal_tac "(kn) div n \<le> (km) div n") 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1104 
prefer 2 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1105 
apply (blast intro: div_le_mono diff_le_mono2) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1106 
apply (rule le_trans, simp) 
15439  1107 
apply (simp) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1108 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1109 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1110 
lemma div_le_dividend [simp]: "m div n \<le> (m::nat)" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1111 
apply (case_tac "n=0", simp) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1112 
apply (subgoal_tac "m div n \<le> m div 1", simp) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1113 
apply (rule div_le_mono2) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1114 
apply (simp_all (no_asm_simp)) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1115 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1116 

22718  1117 
(* Similar for "less than" *) 
47138  1118 
lemma div_less_dividend [simp]: 
1119 
"\<lbrakk>(1::nat) < n; 0 < m\<rbrakk> \<Longrightarrow> m div n < m" 

1120 
apply (induct m rule: nat_less_induct) 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1121 
apply (rename_tac "m") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1122 
apply (case_tac "m<n", simp) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1123 
apply (subgoal_tac "0<n") 
22718  1124 
prefer 2 apply simp 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1125 
apply (simp add: div_geq) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1126 
apply (case_tac "n<m") 
15251  1127 
apply (subgoal_tac "(mn) div n < (mn) ") 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1128 
apply (rule impI less_trans_Suc)+ 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1129 
apply assumption 
15439  1130 
apply (simp_all) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1131 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1132 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1133 
text{*A fact for the mutilated chess board*} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1134 
lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1135 
apply (case_tac "n=0", simp) 
15251  1136 
apply (induct "m" rule: nat_less_induct) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1137 
apply (case_tac "Suc (na) <n") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1138 
(* case Suc(na) < n *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1139 
apply (frule lessI [THEN less_trans], simp add: less_not_refl3) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1140 
(* case n \<le> Suc(na) *) 
16796  1141 
apply (simp add: linorder_not_less le_Suc_eq mod_geq) 
15439  1142 
apply (auto simp add: Suc_diff_le le_mod_geq) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1143 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1144 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1145 
lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)" 
29667  1146 
by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def) 
17084
fb0a80aef0be
classical rules must have names for ATP integration
paulson
parents:
16796
diff
changeset

1147 

22718  1148 
lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1] 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1149 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1150 
(*Loses information, namely we also have r<d provided d is nonzero*) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1151 
lemma mod_eqD: "(m mod d = r) ==> \<exists>q::nat. m = r + q*d" 
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

1152 
apply (cut_tac a = m in mod_div_equality) 
22718  1153 
apply (simp only: add_ac) 
1154 
apply (blast intro: sym) 

1155 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1156 

13152  1157 
lemma split_div: 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1158 
"P(n div k :: nat) = 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1159 
((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1160 
(is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))") 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1161 
proof 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1162 
assume P: ?P 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1163 
show ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1164 
proof (cases) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1165 
assume "k = 0" 
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

1166 
with P show ?Q by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1167 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1168 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1169 
thus ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1170 
proof (simp, intro allI impI) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1171 
fix i j 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1172 
assume n: "n = k*i + j" and j: "j < k" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1173 
show "P i" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1174 
proof (cases) 
22718  1175 
assume "i = 0" 
1176 
with n j P show "P i" by simp 

13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1177 
next 
22718  1178 
assume "i \<noteq> 0" 
1179 
with not0 n j P show "P i" by(simp add:add_ac) 

13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1180 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1181 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1182 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1183 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1184 
assume Q: ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1185 
show ?P 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1186 
proof (cases) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1187 
assume "k = 0" 
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

1188 
with Q show ?P by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1189 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1190 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1191 
with Q have R: ?R by simp 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1192 
from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"] 
13517  1193 
show ?P by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1194 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1195 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1196 

13882  1197 
lemma split_div_lemma: 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1198 
assumes "0 < n" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1199 
shows "n * q \<le> m \<and> m < n * Suc q \<longleftrightarrow> q = ((m\<Colon>nat) div n)" (is "?lhs \<longleftrightarrow> ?rhs") 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1200 
proof 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1201 
assume ?rhs 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1202 
with mult_div_cancel have nq: "n * q = m  (m mod n)" by simp 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1203 
then have A: "n * q \<le> m" by simp 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1204 
have "n  (m mod n) > 0" using mod_less_divisor assms by auto 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1205 
then have "m < m + (n  (m mod n))" by simp 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1206 
then have "m < n + (m  (m mod n))" by simp 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1207 
with nq have "m < n + n * q" by simp 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1208 
then have B: "m < n * Suc q" by simp 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1209 
from A B show ?lhs .. 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1210 
next 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1211 
assume P: ?lhs 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

1212 
then have "divmod_nat_rel m n (q, m  n * q)" 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

1213 
unfolding divmod_nat_rel_def by (auto simp add: mult_ac) 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

1214 
with divmod_nat_rel_unique divmod_nat_rel [of m n] 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

1215 
have "(q, m  n * q) = (m div n, m mod n)" by auto 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

1216 
then show ?rhs by simp 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1217 
qed 
13882  1218 

1219 
theorem split_div': 

1220 
"P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or> 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1221 
(\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))" 
13882  1222 
apply (case_tac "0 < n") 
1223 
apply (simp only: add: split_div_lemma) 

27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

1224 
apply simp_all 
13882  1225 
done 
1226 

13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1227 
lemma split_mod: 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1228 
"P(n mod k :: nat) = 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1229 
((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1230 
(is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))") 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1231 
proof 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1232 
assume P: ?P 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1233 
show ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1234 
proof (cases) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1235 
assume "k = 0" 
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

1236 
with P show ?Q by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1237 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1238 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1239 
thus ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1240 
proof (simp, intro allI impI) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1241 
fix i j 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1242 
assume "n = k*i + j" "j < k" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1243 
thus "P j" using not0 P by(simp add:add_ac mult_ac) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1244 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1245 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1246 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1247 
assume Q: ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1248 
show ?P 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1249 
proof (cases) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1250 
assume "k = 0" 
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

1251 
with Q show ?P by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1252 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1253 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1254 
with Q have R: ?R by simp 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1255 
from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"] 
13517  1256 
show ?P by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1257 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1258 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1259 

13882  1260 
theorem mod_div_equality': "(m::nat) mod n = m  (m div n) * n" 
47138  1261 
using mod_div_equality [of m n] by arith 
1262 

1263 
lemma div_mod_equality': "(m::nat) div n * n = m  m mod n" 

1264 
using mod_div_equality [of m n] by arith 

1265 
(* FIXME: very similar to mult_div_cancel *) 

22800  1266 

52398  1267 
lemma div_eq_dividend_iff: "a \<noteq> 0 \<Longrightarrow> (a :: nat) div b = a \<longleftrightarrow> b = 1" 
1268 
apply rule 

1269 
apply (cases "b = 0") 

1270 
apply simp_all 

1271 
apply (metis (full_types) One_nat_def Suc_lessI div_less_dividend less_not_refl3) 

1272 
done 

1273 

22800  1274 

46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

1275 
subsubsection {* An ``induction'' law for modulus arithmetic. *} 
14640  1276 

1277 
lemma mod_induct_0: 

1278 
assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)" 

1279 
and base: "P i" and i: "i<p" 

1280 
shows "P 0" 

1281 
proof (rule ccontr) 

1282 
assume contra: "\<not>(P 0)" 

1283 
from i have p: "0<p" by simp 

1284 
have "\<forall>k. 0<k \<longrightarrow> \<not> P (pk)" (is "\<forall>k. ?A k") 

1285 
proof 

1286 
fix k 

1287 
show "?A k" 

1288 
proof (induct k) 

1289 
show "?A 0" by simp  "by contradiction" 

1290 
next 

1291 
fix n 

1292 
assume ih: "?A n" 

1293 
show "?A (Suc n)" 

1294 
proof (clarsimp) 

22718  1295 
assume y: "P (p  Suc n)" 
1296 
have n: "Suc n < p" 

1297 
proof (rule ccontr) 

1298 
assume "\<not>(Suc n < p)" 

1299 
hence "p  Suc n = 0" 

1300 
by simp 

1301 
with y contra show "False" 

1302 
by simp 

1303 
qed 

1304 
hence n2: "Suc (p  Suc n) = pn" by arith 

1305 
from p have "p  Suc n < p" by arith 

1306 
with y step have z: "P ((Suc (p  Suc n)) mod p)" 

1307 
by blast 
