author  haftmann 
Thu, 29 Oct 2009 22:13:09 +0100  
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child 33361  1f18de40b43f 
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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theory Divides 
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imports Nat_Numeral Nat_Transfer 
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uses "~~/src/Provers/Arith/cancel_div_mod.ML" 
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begin 
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25942  13 
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. *} 
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class semiring_div = comm_semiring_1_cancel + no_zero_divisors + div + 
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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" 
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begin 
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text {* @{const div} and @{const mod} *} 
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26062  32 
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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lemma div_mod_equality: "((a div b) * b + a mod b) + c = a + c" 
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by (simp add: mod_div_equality) 
26062  42 

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

30934  44 
by (simp add: mod_div_equality2) 
26062  45 

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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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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 left_distrib) 
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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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lemma mod_mult_self2 [simp]: "(a + b * c) mod b = a mod b" 
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by (simp add: mult_commute [of b]) 
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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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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 
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lemma div_by_1 [simp]: "a div 1 = a" 
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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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lemma div_self [simp]: "a \<noteq> 0 \<Longrightarrow> a div a = 1" 
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27676  105 
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) 
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27676  110 
lemma div_add_self2 [simp]: 
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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  115 
lemma mod_add_self1 [simp]: 
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"(b + a) mod b = a mod b" 
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lemma mod_add_self2 [simp]: 
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"(a + b) mod b = a mod b" 
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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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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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31998
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lemma dvd_eq_mod_eq_0 [code_unfold]: "a dvd b \<longleftrightarrow> b mod a = 0" 
25942  135 
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 

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

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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 

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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  148 
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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also have "\<dots> = a div b" 
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by (simp only: mod_div_equality') 
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also have "\<dots> = a div b + 0" 
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161 
by simp 
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162 
finally show ?thesis 
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163 
by (rule add_left_imp_eq) 
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164 
qed 
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165 

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166 
lemma mod_mod_trivial [simp]: "a mod b mod b = a mod b" 
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167 
proof  
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168 
have "a mod b mod b = (a mod b + a div b * b) mod b" 
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169 
by (simp only: mod_mult_self1) 
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170 
also have "\<dots> = a mod b" 
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171 
by (simp only: mod_div_equality') 
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172 
finally show ?thesis . 
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173 
qed 
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174 

29925  175 
lemma dvd_imp_mod_0: "a dvd b \<Longrightarrow> b mod a = 0" 
29948  176 
by (rule dvd_eq_mod_eq_0[THEN iffD1]) 
29925  177 

178 
lemma dvd_div_mult_self: "a dvd b \<Longrightarrow> (b div a) * a = b" 

179 
by (subst (2) mod_div_equality [of b a, symmetric]) (simp add:dvd_imp_mod_0) 

180 

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181 
lemma dvd_mult_div_cancel: "a dvd b \<Longrightarrow> a * (b div a) = b" 
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182 
by (drule dvd_div_mult_self) (simp add: mult_commute) 
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183 

30052  184 
lemma dvd_div_mult: "a dvd b \<Longrightarrow> (b div a) * c = b * c div a" 
185 
apply (cases "a = 0") 

186 
apply simp 

187 
apply (auto simp: dvd_def mult_assoc) 

188 
done 

189 

29925  190 
lemma div_dvd_div[simp]: 
191 
"a dvd b \<Longrightarrow> a dvd c \<Longrightarrow> (b div a dvd c div a) = (b dvd c)" 

192 
apply (cases "a = 0") 

193 
apply simp 

194 
apply (unfold dvd_def) 

195 
apply auto 

196 
apply(blast intro:mult_assoc[symmetric]) 

197 
apply(fastsimp simp add: mult_assoc) 

198 
done 

199 

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200 
lemma dvd_mod_imp_dvd: "[ k dvd m mod n; k dvd n ] ==> k dvd m" 
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201 
apply (subgoal_tac "k dvd (m div n) *n + m mod n") 
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202 
apply (simp add: mod_div_equality) 
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203 
apply (simp only: dvd_add dvd_mult) 
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204 
done 
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205 

29403
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206 
text {* Addition respects modular equivalence. *} 
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207 

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208 
lemma mod_add_left_eq: "(a + b) mod c = (a mod c + b) mod c" 
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209 
proof  
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210 
have "(a + b) mod c = (a div c * c + a mod c + b) mod c" 
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211 
by (simp only: mod_div_equality) 
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212 
also have "\<dots> = (a mod c + b + a div c * c) mod c" 
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213 
by (simp only: add_ac) 
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214 
also have "\<dots> = (a mod c + b) mod c" 
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215 
by (rule mod_mult_self1) 
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216 
finally show ?thesis . 
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217 
qed 
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218 

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219 
lemma mod_add_right_eq: "(a + b) mod c = (a + b mod c) mod c" 
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220 
proof  
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221 
have "(a + b) mod c = (a + (b div c * c + b mod c)) mod c" 
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222 
by (simp only: mod_div_equality) 
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223 
also have "\<dots> = (a + b mod c + b div c * c) mod c" 
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224 
by (simp only: add_ac) 
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225 
also have "\<dots> = (a + b mod c) mod c" 
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226 
by (rule mod_mult_self1) 
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227 
finally show ?thesis . 
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228 
qed 
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229 

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230 
lemma mod_add_eq: "(a + b) mod c = (a mod c + b mod c) mod c" 
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231 
by (rule trans [OF mod_add_left_eq mod_add_right_eq]) 
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232 

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233 
lemma mod_add_cong: 
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234 
assumes "a mod c = a' mod c" 
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235 
assumes "b mod c = b' mod c" 
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236 
shows "(a + b) mod c = (a' + b') mod c" 
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237 
proof  
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238 
have "(a mod c + b mod c) mod c = (a' mod c + b' mod c) mod c" 
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239 
unfolding assms .. 
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240 
thus ?thesis 
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241 
by (simp only: mod_add_eq [symmetric]) 
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242 
qed 
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243 

30923
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244 
lemma div_add [simp]: "z dvd x \<Longrightarrow> z dvd y 
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245 
\<Longrightarrow> (x + y) div z = x div z + y div z" 
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246 
by (cases "z = 0", simp, unfold dvd_def, auto simp add: algebra_simps) 
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247 

29403
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248 
text {* Multiplication respects modular equivalence. *} 
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249 

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250 
lemma mod_mult_left_eq: "(a * b) mod c = ((a mod c) * b) mod c" 
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251 
proof  
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252 
have "(a * b) mod c = ((a div c * c + a mod c) * b) mod c" 
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253 
by (simp only: mod_div_equality) 
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254 
also have "\<dots> = (a mod c * b + a div c * b * c) mod c" 
29667  255 
by (simp only: algebra_simps) 
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256 
also have "\<dots> = (a mod c * b) mod c" 
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257 
by (rule mod_mult_self1) 
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258 
finally show ?thesis . 
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259 
qed 
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260 

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261 
lemma mod_mult_right_eq: "(a * b) mod c = (a * (b mod c)) mod c" 
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262 
proof  
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263 
have "(a * b) mod c = (a * (b div c * c + b mod c)) mod c" 
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264 
by (simp only: mod_div_equality) 
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265 
also have "\<dots> = (a * (b mod c) + a * (b div c) * c) mod c" 
29667  266 
by (simp only: algebra_simps) 
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267 
also have "\<dots> = (a * (b mod c)) mod c" 
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268 
by (rule mod_mult_self1) 
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269 
finally show ?thesis . 
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270 
qed 
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271 

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272 
lemma mod_mult_eq: "(a * b) mod c = ((a mod c) * (b mod c)) mod c" 
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273 
by (rule trans [OF mod_mult_left_eq mod_mult_right_eq]) 
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274 

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275 
lemma mod_mult_cong: 
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276 
assumes "a mod c = a' mod c" 
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277 
assumes "b mod c = b' mod c" 
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278 
shows "(a * b) mod c = (a' * b') mod c" 
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279 
proof  
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280 
have "(a mod c * (b mod c)) mod c = (a' mod c * (b' mod c)) mod c" 
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281 
unfolding assms .. 
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282 
thus ?thesis 
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283 
by (simp only: mod_mult_eq [symmetric]) 
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284 
qed 
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285 

29404  286 
lemma mod_mod_cancel: 
287 
assumes "c dvd b" 

288 
shows "a mod b mod c = a mod c" 

289 
proof  

290 
from `c dvd b` obtain k where "b = c * k" 

291 
by (rule dvdE) 

292 
have "a mod b mod c = a mod (c * k) mod c" 

293 
by (simp only: `b = c * k`) 

294 
also have "\<dots> = (a mod (c * k) + a div (c * k) * k * c) mod c" 

295 
by (simp only: mod_mult_self1) 

296 
also have "\<dots> = (a div (c * k) * (c * k) + a mod (c * k)) mod c" 

297 
by (simp only: add_ac mult_ac) 

298 
also have "\<dots> = a mod c" 

299 
by (simp only: mod_div_equality) 

300 
finally show ?thesis . 

301 
qed 

302 

30930  303 
lemma div_mult_div_if_dvd: 
304 
"y dvd x \<Longrightarrow> z dvd w \<Longrightarrow> (x div y) * (w div z) = (x * w) div (y * z)" 

305 
apply (cases "y = 0", simp) 

306 
apply (cases "z = 0", simp) 

307 
apply (auto elim!: dvdE simp add: algebra_simps) 

30476  308 
apply (subst mult_assoc [symmetric]) 
309 
apply (simp add: no_zero_divisors) 

30930  310 
done 
311 

312 
lemma div_mult_mult2 [simp]: 

313 
"c \<noteq> 0 \<Longrightarrow> (a * c) div (b * c) = a div b" 

314 
by (drule div_mult_mult1) (simp add: mult_commute) 

315 

316 
lemma div_mult_mult1_if [simp]: 

317 
"(c * a) div (c * b) = (if c = 0 then 0 else a div b)" 

318 
by simp_all 

30476  319 

30930  320 
lemma mod_mult_mult1: 
321 
"(c * a) mod (c * b) = c * (a mod b)" 

322 
proof (cases "c = 0") 

323 
case True then show ?thesis by simp 

324 
next 

325 
case False 

326 
from mod_div_equality 

327 
have "((c * a) div (c * b)) * (c * b) + (c * a) mod (c * b) = c * a" . 

328 
with False have "c * ((a div b) * b + a mod b) + (c * a) mod (c * b) 

329 
= c * a + c * (a mod b)" by (simp add: algebra_simps) 

330 
with mod_div_equality show ?thesis by simp 

331 
qed 

332 

333 
lemma mod_mult_mult2: 

334 
"(a * c) mod (b * c) = (a mod b) * c" 

335 
using mod_mult_mult1 [of c a b] by (simp add: mult_commute) 

336 

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337 
lemma dvd_mod: "k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd (m mod n)" 
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338 
unfolding dvd_def by (auto simp add: mod_mult_mult1) 
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339 

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340 
lemma dvd_mod_iff: "k dvd n \<Longrightarrow> k dvd (m mod n) \<longleftrightarrow> k dvd m" 
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341 
by (blast intro: dvd_mod_imp_dvd dvd_mod) 
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342 

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343 
lemma div_power: 
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344 
"y dvd x \<Longrightarrow> (x div y) ^ n = x ^ n div y ^ n" 
30476  345 
apply (induct n) 
346 
apply simp 

347 
apply(simp add: div_mult_div_if_dvd dvd_power_same) 

348 
done 

349 

31661
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350 
end 
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351 

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352 
class ring_div = semiring_div + idom 
29405
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353 
begin 
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354 

98ab21b14f09
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355 
text {* Negation respects modular equivalence. *} 
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356 

98ab21b14f09
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357 
lemma mod_minus_eq: "( a) mod b = ( (a mod b)) mod b" 
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358 
proof  
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359 
have "( a) mod b = ( (a div b * b + a mod b)) mod b" 
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360 
by (simp only: mod_div_equality) 
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361 
also have "\<dots> = ( (a mod b) +  (a div b) * b) mod b" 
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362 
by (simp only: minus_add_distrib minus_mult_left add_ac) 
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changeset

363 
also have "\<dots> = ( (a mod b)) mod b" 
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diff
changeset

364 
by (rule mod_mult_self1) 
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huffman
parents:
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changeset

365 
finally show ?thesis . 
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366 
qed 
98ab21b14f09
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huffman
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diff
changeset

367 

98ab21b14f09
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changeset

368 
lemma mod_minus_cong: 
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369 
assumes "a mod b = a' mod b" 
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370 
shows "( a) mod b = ( a') mod b" 
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371 
proof  
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372 
have "( (a mod b)) mod b = ( (a' mod b)) mod b" 
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changeset

373 
unfolding assms .. 
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changeset

374 
thus ?thesis 
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375 
by (simp only: mod_minus_eq [symmetric]) 
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376 
qed 
98ab21b14f09
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huffman
parents:
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diff
changeset

377 

98ab21b14f09
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parents:
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378 
text {* Subtraction respects modular equivalence. *} 
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changeset

379 

98ab21b14f09
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380 
lemma mod_diff_left_eq: "(a  b) mod c = (a mod c  b) mod c" 
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changeset

381 
unfolding diff_minus 
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huffman
parents:
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diff
changeset

382 
by (intro mod_add_cong mod_minus_cong) simp_all 
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huffman
parents:
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diff
changeset

383 

98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
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parents:
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384 
lemma mod_diff_right_eq: "(a  b) mod c = (a  b mod c) mod c" 
98ab21b14f09
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huffman
parents:
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diff
changeset

385 
unfolding diff_minus 
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add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
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diff
changeset

386 
by (intro mod_add_cong mod_minus_cong) simp_all 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
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diff
changeset

387 

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

388 
lemma mod_diff_eq: "(a  b) mod c = (a mod c  b mod c) mod c" 
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huffman
parents:
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diff
changeset

389 
unfolding diff_minus 
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huffman
parents:
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diff
changeset

390 
by (intro mod_add_cong mod_minus_cong) simp_all 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

391 

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

392 
lemma mod_diff_cong: 
98ab21b14f09
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parents:
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diff
changeset

393 
assumes "a mod c = a' mod c" 
98ab21b14f09
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huffman
parents:
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diff
changeset

394 
assumes "b mod c = b' mod c" 
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huffman
parents:
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diff
changeset

395 
shows "(a  b) mod c = (a'  b') mod c" 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

396 
unfolding diff_minus using assms 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

397 
by (intro mod_add_cong mod_minus_cong) 
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huffman
parents:
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diff
changeset

398 

30180  399 
lemma dvd_neg_div: "y dvd x \<Longrightarrow> x div y =  (x div y)" 
400 
apply (case_tac "y = 0") apply simp 

401 
apply (auto simp add: dvd_def) 

402 
apply (subgoal_tac "(y * k) = y *  k") 

403 
apply (erule ssubst) 

404 
apply (erule div_mult_self1_is_id) 

405 
apply simp 

406 
done 

407 

408 
lemma dvd_div_neg: "y dvd x \<Longrightarrow> x div y =  (x div y)" 

409 
apply (case_tac "y = 0") apply simp 

410 
apply (auto simp add: dvd_def) 

411 
apply (subgoal_tac "y * k = y * k") 

412 
apply (erule ssubst) 

413 
apply (rule div_mult_self1_is_id) 

414 
apply simp 

415 
apply simp 

416 
done 

417 

29405
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huffman
parents:
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diff
changeset

418 
end 
98ab21b14f09
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huffman
parents:
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diff
changeset

419 

25942  420 

26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
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421 
subsection {* Division on @{typ nat} *} 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

422 

fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
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423 
text {* 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
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diff
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424 
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:
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diff
changeset

425 
of a characteristic relation with two input arguments 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
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426 
@{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:
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diff
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427 
@{term "q\<Colon>nat"}(uotient) and @{term "r\<Colon>nat"}(emainder). 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

428 
*} 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

429 

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

430 
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

431 
"divmod_nat_rel m n qr \<longleftrightarrow> 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

432 
m = fst qr * n + snd qr \<and> 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

433 
(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:
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diff
changeset

434 

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

435 
text {* @{const divmod_nat_rel} is total: *} 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

436 

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

437 
lemma divmod_nat_rel_ex: 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

438 
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

439 
proof (cases "n = 0") 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

440 
case True with that show thesis 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

441 
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

442 
next 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

443 
case False 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

444 
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

445 
proof (induct m) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

446 
case 0 with `n \<noteq> 0` 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

447 
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:
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diff
changeset

448 
then show ?case by blast 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

449 
next 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

450 
case (Suc m) then obtain q' r' 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

451 
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

452 
then show ?case proof (cases "Suc r' < n") 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

453 
case True 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

454 
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

455 
with True show ?thesis by blast 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

456 
next 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

457 
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

458 
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

459 
ultimately have "n = Suc r'" by auto 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

460 
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

461 
with `n \<noteq> 0` show ?thesis by blast 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

462 
qed 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

463 
qed 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

464 
with that show thesis 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

465 
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

466 
qed 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

467 

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

468 
text {* @{const divmod_nat_rel} is injective: *} 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

469 

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

470 
lemma divmod_nat_rel_unique: 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

471 
assumes "divmod_nat_rel m n qr" 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

472 
and "divmod_nat_rel m n qr'" 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

473 
shows "qr = qr'" 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

474 
proof (cases "n = 0") 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

475 
case True with assms show ?thesis 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

476 
by (cases qr, cases qr') 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

477 
(simp add: divmod_nat_rel_def) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

478 
next 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

479 
case False 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

480 
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

481 
apply (rule leI) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

482 
apply (subst less_iff_Suc_add) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

483 
apply (auto simp add: add_mult_distrib) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

484 
done 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

485 
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

486 
by (auto simp add: divmod_nat_rel_def intro: order_antisym dest: aux sym) 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

487 
moreover from this assms have "snd qr = snd qr'" 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

488 
by (simp add: divmod_nat_rel_def) 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

489 
ultimately 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

490 
qed 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

491 

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

492 
text {* 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

493 
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

494 
means of @{const divmod_nat_rel}: 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

495 
*} 
25942  496 

497 
instantiation nat :: semiring_div 

25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25162
diff
changeset

498 
begin 
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25162
diff
changeset

499 

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

500 
definition divmod_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

501 
[code del]: "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

502 

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

503 
lemma divmod_nat_rel_divmod_nat: 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

504 
"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

505 
proof  
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

506 
from divmod_nat_rel_ex 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

507 
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

508 
then show ?thesis 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

509 
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

510 
qed 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

511 

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

512 
lemma divmod_nat_eq: 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

513 
assumes "divmod_nat_rel m n qr" 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

514 
shows "divmod_nat m n = qr" 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

515 
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

516 

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

517 
definition div_nat where 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

518 
"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

519 

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

520 
definition mod_nat where 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

521 
"m mod n = snd (divmod_nat m n)" 
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25162
diff
changeset

522 

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

523 
lemma divmod_nat_div_mod: 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

524 
"divmod_nat m n = (m div n, m mod n)" 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

525 
unfolding div_nat_def mod_nat_def by simp 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

526 

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

527 
lemma div_eq: 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

528 
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

529 
shows "m div n = q" 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

530 
using assms by (auto dest: divmod_nat_eq simp add: divmod_nat_div_mod) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

531 

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

532 
lemma mod_eq: 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

533 
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

534 
shows "m mod n = r" 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

535 
using assms by (auto dest: divmod_nat_eq simp add: divmod_nat_div_mod) 
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25162
diff
changeset

536 

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

537 
lemma divmod_nat_rel: "divmod_nat_rel m n (m div n, m mod n)" 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

538 
by (simp add: div_nat_def mod_nat_def divmod_nat_rel_divmod_nat) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

539 

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

540 
lemma divmod_nat_zero: 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

541 
"divmod_nat m 0 = (0, m)" 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

542 
proof  
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

543 
from divmod_nat_rel [of m 0] show ?thesis 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

544 
unfolding divmod_nat_div_mod divmod_nat_rel_def by simp 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

545 
qed 
25942  546 

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

547 
lemma divmod_nat_base: 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

548 
assumes "m < n" 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

549 
shows "divmod_nat m n = (0, m)" 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

550 
proof  
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

551 
from divmod_nat_rel [of m n] show ?thesis 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

552 
unfolding divmod_nat_div_mod divmod_nat_rel_def 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

553 
using assms by (cases "m div n = 0") 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

554 
(auto simp add: gr0_conv_Suc [of "m div n"]) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

555 
qed 
25942  556 

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

557 
lemma divmod_nat_step: 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

558 
assumes "0 < n" and "n \<le> m" 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

559 
shows "divmod_nat m n = (Suc ((m  n) div n), (m  n) mod n)" 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

560 
proof  
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

561 
from divmod_nat_rel have divmod_nat_m_n: "divmod_nat_rel m n (m div n, m mod n)" . 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

562 
with assms have m_div_n: "m div n \<ge> 1" 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

563 
by (cases "m div n") (auto simp add: divmod_nat_rel_def) 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

564 
from assms divmod_nat_m_n have "divmod_nat_rel (m  n) n (m div n  Suc 0, m mod n)" 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

565 
by (cases "m div n") (auto simp add: divmod_nat_rel_def) 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

566 
with divmod_nat_eq have "divmod_nat (m  n) n = (m div n  Suc 0, m mod n)" by simp 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

567 
moreover from divmod_nat_div_mod have "divmod_nat (m  n) n = ((m  n) div n, (m  n) mod n)" . 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

568 
ultimately have "m div n = Suc ((m  n) div n)" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

569 
and "m mod n = (m  n) mod n" using m_div_n by simp_all 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

570 
then show ?thesis using divmod_nat_div_mod by simp 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

571 
qed 
25942  572 

26300  573 
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

574 

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

575 
lemma div_less [simp]: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

576 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

577 
assumes "m < n" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

578 
shows "m div n = 0" 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

579 
using assms divmod_nat_base divmod_nat_div_mod by simp 
25942  580 

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

581 
lemma le_div_geq: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

582 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

583 
assumes "0 < n" and "n \<le> m" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

584 
shows "m div n = Suc ((m  n) div n)" 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

585 
using assms divmod_nat_step divmod_nat_div_mod by simp 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

586 

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

587 
lemma mod_less [simp]: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

588 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

589 
assumes "m < n" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

590 
shows "m mod n = m" 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

591 
using assms divmod_nat_base divmod_nat_div_mod by simp 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

592 

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

593 
lemma le_mod_geq: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

594 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

595 
assumes "n \<le> m" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

596 
shows "m mod n = (m  n) mod n" 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

597 
using assms divmod_nat_step divmod_nat_div_mod by (cases "n = 0") simp_all 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

598 

30930  599 
instance proof  
600 
have [simp]: "\<And>n::nat. n div 0 = 0" 

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

601 
by (simp add: div_nat_def divmod_nat_zero) 
30930  602 
have [simp]: "\<And>n::nat. 0 div n = 0" 
603 
proof  

604 
fix n :: nat 

605 
show "0 div n = 0" 

606 
by (cases "n = 0") simp_all 

607 
qed 

608 
show "OFCLASS(nat, semiring_div_class)" proof 

609 
fix m n :: nat 

610 
show "m div n * n + m mod n = m" 

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

611 
using divmod_nat_rel [of m n] by (simp add: divmod_nat_rel_def) 
30930  612 
next 
613 
fix m n q :: nat 

614 
assume "n \<noteq> 0" 

615 
then show "(q + m * n) div n = m + q div n" 

616 
by (induct m) (simp_all add: le_div_geq) 

617 
next 

618 
fix m n q :: nat 

619 
assume "m \<noteq> 0" 

620 
then show "(m * n) div (m * q) = n div q" 

621 
proof (cases "n \<noteq> 0 \<and> q \<noteq> 0") 

622 
case False then show ?thesis by auto 

623 
next 

624 
case True with `m \<noteq> 0` 

625 
have "m > 0" and "n > 0" and "q > 0" by auto 

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

626 
then have "\<And>a b. divmod_nat_rel n q (a, b) \<Longrightarrow> divmod_nat_rel (m * n) (m * q) (a, m * b)" 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

627 
by (auto simp add: divmod_nat_rel_def) (simp_all add: algebra_simps) 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

628 
moreover from divmod_nat_rel have "divmod_nat_rel n q (n div q, n mod q)" . 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

629 
ultimately have "divmod_nat_rel (m * n) (m * q) (n div q, m * (n mod q))" . 
30930  630 
then show ?thesis by (simp add: div_eq) 
631 
qed 

632 
qed simp_all 

25942  633 
qed 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

634 

25942  635 
end 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

636 

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

637 
text {* Simproc for cancelling @{const div} and @{const mod} *} 
25942  638 

30934  639 
ML {* 
640 
local 

641 

642 
structure CancelDivMod = CancelDivModFun(struct 

25942  643 

30934  644 
val div_name = @{const_name div}; 
645 
val mod_name = @{const_name mod}; 

646 
val mk_binop = HOLogic.mk_binop; 

647 
val mk_sum = Nat_Arith.mk_sum; 

648 
val dest_sum = Nat_Arith.dest_sum; 

25942  649 

30934  650 
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

651 

30934  652 
val trans = trans; 
25942  653 

30934  654 
val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac 
655 
(@{thm monoid_add_class.add_0_left} :: @{thm monoid_add_class.add_0_right} :: @{thms add_ac})) 

25942  656 

30934  657 
end) 
25942  658 

30934  659 
in 
25942  660 

32010  661 
val cancel_div_mod_nat_proc = Simplifier.simproc @{theory} 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

662 
"cancel_div_mod" ["(m::nat) + n"] (K CancelDivMod.proc); 
25942  663 

30934  664 
val _ = Addsimprocs [cancel_div_mod_nat_proc]; 
665 

666 
end 

25942  667 
*} 
668 

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

669 
text {* code generator setup *} 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

670 

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

671 
lemma divmod_nat_if [code]: "divmod_nat m n = (if n = 0 \<or> m < n then (0, m) else 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

672 
let (q, r) = divmod_nat (m  n) n in (Suc q, r))" 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

673 
by (simp add: divmod_nat_zero divmod_nat_base divmod_nat_step) 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

674 
(simp add: divmod_nat_div_mod) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

675 

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

676 
code_modulename SML 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

677 
Divides Nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

678 

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

679 
code_modulename OCaml 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

680 
Divides Nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

681 

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

682 
code_modulename Haskell 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

683 
Divides Nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

684 

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

685 

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

686 
subsubsection {* Quotient *} 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

687 

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

688 
lemma div_geq: "0 < n \<Longrightarrow> \<not> m < n \<Longrightarrow> m div n = Suc ((m  n) div n)" 
29667  689 
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

690 

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

691 
lemma div_if: "0 < n \<Longrightarrow> m div n = (if m < n then 0 else Suc ((m  n) div n))" 
29667  692 
by (simp add: div_geq) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

693 

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

694 
lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)" 
29667  695 
by simp 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

696 

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

697 
lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)" 
29667  698 
by simp 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

699 

25942  700 

701 
subsubsection {* Remainder *} 

702 

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

703 
lemma mod_less_divisor [simp]: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

704 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

705 
assumes "n > 0" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

706 
shows "m mod n < (n::nat)" 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

707 
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

708 

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

709 
lemma mod_less_eq_dividend [simp]: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

710 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

711 
shows "m mod n \<le> m" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

712 
proof (rule add_leD2) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

713 
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

714 
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

715 
qed 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

716 

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

717 
lemma mod_geq: "\<not> m < (n\<Colon>nat) \<Longrightarrow> m mod n = (m  n) mod n" 
29667  718 
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

719 

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

720 
lemma mod_if: "m mod (n\<Colon>nat) = (if m < n then m else (m  n) mod n)" 
29667  721 
by (simp add: le_mod_geq) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

722 

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

723 
lemma mod_1 [simp]: "m mod Suc 0 = 0" 
29667  724 
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

725 

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

726 
lemma mod_mult_distrib: "(m mod n) * (k\<Colon>nat) = (m * k) mod (n * k)" 
22718  727 
apply (cases "n = 0", simp) 
728 
apply (cases "k = 0", simp) 

729 
apply (induct m rule: nat_less_induct) 

730 
apply (subst mod_if, simp) 

731 
apply (simp add: mod_geq diff_mult_distrib) 

732 
done 

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

733 

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

734 
lemma mod_mult_distrib2: "(k::nat) * (m mod n) = (k*m) mod (k*n)" 
29667  735 
by (simp add: mult_commute [of k] mod_mult_distrib) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

736 

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

737 
(* 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

738 
lemma mult_div_cancel: "(n::nat) * (m div n) = m  (m mod n)" 
29667  739 
by (cut_tac a = m and b = n in mod_div_equality2, arith) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

740 

15439  741 
lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)" 
22718  742 
apply (drule mod_less_divisor [where m = m]) 
743 
apply simp 

744 
done 

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

745 

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

746 
subsubsection {* Quotient and Remainder *} 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

747 

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

748 
lemma divmod_nat_rel_mult1_eq: 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

749 
"divmod_nat_rel b c (q, r) \<Longrightarrow> c > 0 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

750 
\<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

751 
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

752 

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

753 
lemma div_mult1_eq: 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

754 
"(a * b) div c = a * (b div c) + a * (b mod c) div (c::nat)" 
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

755 
apply (cases "c = 0", simp) 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

756 
apply (blast intro: divmod_nat_rel [THEN divmod_nat_rel_mult1_eq, THEN div_eq]) 
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

757 
done 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

758 

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

759 
lemma divmod_nat_rel_add1_eq: 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

760 
"divmod_nat_rel a c (aq, ar) \<Longrightarrow> divmod_nat_rel b c (bq, br) \<Longrightarrow> c > 0 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

761 
\<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

762 
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

763 

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

764 
(*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

765 
lemma div_add1_eq: 
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

766 
"(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)" 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

767 
apply (cases "c = 0", simp) 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

768 
apply (blast intro: divmod_nat_rel_add1_eq [THEN div_eq] divmod_nat_rel) 
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

769 
done 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

770 

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

771 
lemma mod_lemma: "[ (0::nat) < c; r < b ] ==> b * (q mod c) + r < b * c" 
22718  772 
apply (cut_tac m = q and n = c in mod_less_divisor) 
773 
apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto) 

774 
apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst) 

775 
apply (simp add: add_mult_distrib2) 

776 
done 

10559
d3fd54fc659b
many new div and mod properties (borrowed from Integ/IntDiv)
paulson
parents:
10214
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_mult2_eq: 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

779 
"divmod_nat_rel a b (q, r) \<Longrightarrow> 0 < b \<Longrightarrow> 0 < c 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

780 
\<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

781 
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

782 

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

783 
lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)" 
22718  784 
apply (cases "b = 0", simp) 
785 
apply (cases "c = 0", simp) 

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

786 
apply (force simp add: divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN div_eq]) 
22718  787 
done 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

788 

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

789 
lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)" 
22718  790 
apply (cases "b = 0", simp) 
791 
apply (cases "c = 0", simp) 

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

792 
apply (auto simp add: mult_commute divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN mod_eq]) 
22718  793 
done 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

794 

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

795 

25942  796 
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

797 

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

798 
lemma div_1 [simp]: "m div Suc 0 = m" 
29667  799 
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

800 

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

801 

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

802 
(* Monotonicity of div in first argument *) 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

803 
lemma div_le_mono [rule_format (no_asm)]: 
22718  804 
"\<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

805 
apply (case_tac "k=0", simp) 
15251  806 
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

807 
apply (case_tac "n<k") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

808 
(* 1 case n<k *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

809 
apply simp 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

810 
(* 2 case n >= k *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

811 
apply (case_tac "m<k") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

812 
(* 2.1 case m<k *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

813 
apply simp 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

814 
(* 2.2 case m>=k *) 
15439  815 
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

816 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

817 

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

818 
(* Antimonotonicity of div in second argument *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

819 
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

820 
apply (subgoal_tac "0<n") 
22718  821 
prefer 2 apply simp 
15251  822 
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

823 
apply (rename_tac "k") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

824 
apply (case_tac "k<n", simp) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

825 
apply (subgoal_tac "~ (k<m) ") 
22718  826 
prefer 2 apply simp 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

827 
apply (simp add: div_geq) 
15251  828 
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

829 
prefer 2 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

830 
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

831 
apply (rule le_trans, simp) 
15439  832 
apply (simp) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

833 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

834 

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

835 
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

836 
apply (case_tac "n=0", simp) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

837 
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

838 
apply (rule div_le_mono2) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

839 
apply (simp_all (no_asm_simp)) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

840 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

841 

22718  842 
(* Similar for "less than" *) 
17085  843 
lemma div_less_dividend [rule_format]: 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

844 
"!!n::nat. 1<n ==> 0 < m > m div n < m" 
15251  845 
apply (induct_tac m rule: nat_less_induct) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

846 
apply (rename_tac "m") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

847 
apply (case_tac "m<n", simp) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

848 
apply (subgoal_tac "0<n") 
22718  849 
prefer 2 apply simp 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

850 
apply (simp add: div_geq) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

851 
apply (case_tac "n<m") 
15251  852 
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

853 
apply (rule impI less_trans_Suc)+ 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

854 
apply assumption 
15439  855 
apply (simp_all) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

856 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

857 

17085  858 
declare div_less_dividend [simp] 
859 

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

860 
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

861 
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

862 
apply (case_tac "n=0", simp) 
15251  863 
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

864 
apply (case_tac "Suc (na) <n") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

865 
(* case Suc(na) < n *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

866 
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

867 
(* case n \<le> Suc(na) *) 
16796  868 
apply (simp add: linorder_not_less le_Suc_eq mod_geq) 
15439  869 
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

870 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

871 

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

872 
lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)" 
29667  873 
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

874 

22718  875 
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

876 

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

877 
(*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

878 
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

879 
apply (cut_tac a = m in mod_div_equality) 
22718  880 
apply (simp only: add_ac) 
881 
apply (blast intro: sym) 

882 
done 

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

883 

13152  884 
lemma split_div: 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

885 
"P(n div k :: nat) = 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

886 
((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

887 
(is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))") 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

888 
proof 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

889 
assume P: ?P 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

890 
show ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

891 
proof (cases) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

892 
assume "k = 0" 
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

893 
with P show ?Q by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

894 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

895 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

896 
thus ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

897 
proof (simp, intro allI impI) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

898 
fix i j 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

899 
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

900 
show "P i" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

901 
proof (cases) 
22718  902 
assume "i = 0" 
903 
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

904 
next 
22718  905 
assume "i \<noteq> 0" 
906 
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

907 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

908 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

909 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

910 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

911 
assume Q: ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

912 
show ?P 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

913 
proof (cases) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

914 
assume "k = 0" 
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

915 
with Q show ?P by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

916 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

917 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

918 
with Q have R: ?R by simp 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

919 
from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"] 
13517  920 
show ?P by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

921 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

922 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

923 

13882  924 
lemma split_div_lemma: 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

925 
assumes "0 < n" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

926 
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

927 
proof 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

928 
assume ?rhs 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

929 
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

930 
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

931 
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

932 
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

933 
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

934 
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

935 
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

936 
from A B show ?lhs .. 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

937 
next 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

938 
assume P: ?lhs 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

939 
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

940 
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

941 
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

942 
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

943 
then show ?rhs by simp 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

944 
qed 
13882  945 

946 
theorem split_div': 

947 
"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

948 
(\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))" 
13882  949 
apply (case_tac "0 < n") 
950 
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

951 
apply simp_all 
13882  952 
done 
953 

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

954 
lemma split_mod: 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

955 
"P(n mod k :: nat) = 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

956 
((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

957 
(is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))") 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

958 
proof 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

959 
assume P: ?P 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

960 
show ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

961 
proof (cases) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

962 
assume "k = 0" 
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

963 
with P show ?Q by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

964 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

965 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

966 
thus ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

967 
proof (simp, intro allI impI) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

968 
fix i j 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

969 
assume "n = k*i + j" "j < k" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

970 
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

971 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

972 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

973 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

974 
assume Q: ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

975 
show ?P 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

976 
proof (cases) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

977 
assume "k = 0" 
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

978 
with Q show ?P by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

979 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

980 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

981 
with Q have R: ?R by simp 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

982 
from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"] 
13517  983 
show ?P by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

984 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

985 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

986 

13882  987 
theorem mod_div_equality': "(m::nat) mod n = m  (m div n) * n" 
988 
apply (rule_tac P="%x. m mod n = x  (m div n) * n" in 

989 
subst [OF mod_div_equality [of _ n]]) 

990 
apply arith 

991 
done 

992 

22800  993 
lemma div_mod_equality': 
994 
fixes m n :: nat 

995 
shows "m div n * n = m  m mod n" 

996 
proof  

997 
have "m mod n \<le> m mod n" .. 

998 
from div_mod_equality have 

999 
"m div n * n + m mod n  m mod n = m  m mod n" by simp 

1000 
with diff_add_assoc [OF `m mod n \<le> m mod n`, of "m div n * n"] have 

1001 
"m div n * n + (m mod n  m mod n) = m  m mod n" 

1002 
by simp 

1003 
then show ?thesis by simp 

1004 
qed 

1005 

1006 

25942  1007 
subsubsection {*An ``induction'' law for modulus arithmetic.*} 
14640  1008 

1009 
lemma mod_induct_0: 

1010 
assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)" 

1011 
and base: "P i" and i: "i<p" 

1012 
shows "P 0" 

1013 
proof (rule ccontr) 

1014 
assume contra: "\<not>(P 0)" 

1015 
from i have p: "0<p" by simp 

1016 
have "\<forall>k. 0<k \<longrightarrow> \<not> P (pk)" (is "\<forall>k. ?A k") 

1017 
proof 

1018 
fix k 

1019 
show "?A k" 

1020 
proof (induct k) 

1021 
show "?A 0" by simp  "by contradiction" 

1022 
next 

1023 
fix n 

1024 
assume ih: "?A n" 

1025 
show "?A (Suc n)" 

1026 
proof (clarsimp) 

22718  1027 
assume y: "P (p  Suc n)" 
1028 
have n: "Suc n < p" 

1029 
proof (rule ccontr) 

1030 
assume "\<not>(Suc n < p)" 

1031 
hence "p  Suc n = 0" 

1032 
by simp 

1033 
with y contra show "False" 

1034 
by simp 

1035 
qed 

1036 
hence n2: "Suc (p  Suc n) = pn" by arith 

1037 
from p have "p  Suc n < p" by arith 

1038 
with y step have z: "P ((Suc (p  Suc n)) mod p)" 

1039 
by blast 

1040 
show "False" 

1041 
proof (cases "n=0") 

1042 
case True 

1043 
with z n2 contra show ?thesis by simp 

1044 
next 

1045 
case False 

1046 
with p have "pn < p" by arith 

1047 
with z n2 False ih show ?thesis by simp 

1048 
qed 

14640  1049 
qed 
1050 
qed 

1051 
qed 

1052 
moreover 

1053 
from i obtain k where "0<k \<and> i+k=p" 

1054 
by (blast dest: less_imp_add_positive) 

1055 
hence "0<k \<and> i=pk" by auto 

1056 
moreover 

1057 
note base 

1058 
ultimately 

1059 
show "False" by blast 

1060 
qed 

1061 

1062 
lemma mod_induct: 

1063 
assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)" 

1064 
and base: "P i" and i: "i<p" and j: "j<p" 

1065 
shows "P j" 

1066 
proof  

1067 
have "\<forall>j<p. P j" 

1068 
proof 

1069 
fix j 

1070 
show "j<p \<longrightarrow> P j" (is "?A j") 

1071 
proof (induct j) 

1072 
from step base i show "?A 0" 

22718  1073 
by (auto elim: mod_induct_0) 
14640  1074 
next 
1075 
fix k 

1076 
assume ih: "?A k" 

1077 
show "?A (Suc k)" 

1078 
proof 

22718  1079 
assume suc: "Suc k < p" 
1080 
hence k: "k<p" by simp 

1081 
with ih have "P k" .. 

1082 
with step k have "P (Suc k mod p)" 

1083 
by blast 

1084 
moreover 

1085 
from suc have "Suc k mod p = Suc k" 

1086 
by simp 

1087 
ultimately 

1088 
show "P (Suc k)" by simp 

14640  1089 
qed 
1090 
qed 

1091 
qed 

1092 
with j show ?thesis by blast 

1093 
qed 

1094 

33296
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1095 
lemma div2_Suc_Suc [simp]: "Suc (Suc m) div 2 = Suc (m div 2)" 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1096 
by (auto simp add: numeral_2_eq_2 le_div_geq) 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1097 

a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1098 
lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)" 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1099 
by (simp add: nat_mult_2 [symmetric]) 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1100 

a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1101 
lemma mod2_Suc_Suc [simp]: "Suc(Suc(m)) mod 2 = m mod 2" 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1102 
apply (subgoal_tac "m mod 2 < 2") 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1103 
apply (erule less_2_cases [THEN disjE]) 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1104 
apply (simp_all (no_asm_simp) add: Let_def mod_Suc nat_1) 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1105 
done 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1106 

a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1107 
lemma mod2_gr_0 [simp]: "0 < (m\<Colon>nat) mod 2 \<longleftrightarrow> m mod 2 = 1" 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1108 
proof  
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1109 
{ fix n :: nat have "(n::nat) < 2 \<Longrightarrow> n = 0 \<or> n = 1" by (induct n) simp_all } 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1110 
moreover have "m mod 2 < 2" by simp 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1111 
ultimately have "m mod 2 = 0 \<or> m mod 2 = 1" . 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1112 
then show ?thesis by auto 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1113 
qed 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1114 

a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1115 
text{*These lemmas collapse some needless occurrences of Suc: 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1116 
at least three Sucs, since two and fewer are rewritten back to Suc again! 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1117 
We already have some rules to simplify operands smaller than 3.*} 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1118 

a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1119 
lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)" 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1120 
by (simp add: Suc3_eq_add_3) 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1121 

a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1122 
lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)" 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1123 
by (simp add: Suc3_eq_add_3) 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1124 

a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1125 
lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n" 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1126 
by (simp add: Suc3_eq_add_3) 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1127 

a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1128 
lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n" 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1129 
by (simp add: Suc3_eq_add_3) 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1130 

a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1131 
lemmas Suc_div_eq_add3_div_number_of = 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1132 
Suc_div_eq_add3_div [of _ "number_of v", standard] 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1133 
declare Suc_div_eq_add3_div_number_of [simp] 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1134 

a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1135 
lemmas Suc_mod_eq_add3_mod_number_of = 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1136 
Suc_mod_eq_add3_mod [of _ "number_of v", standard] 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1137 
declare Suc_mod_eq_add3_mod_number_of [simp] 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1138 

3366  1139 
end 