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(* Title: HOL/Divides.thy 
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ID: $Id$ 

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Author: Lawrence C Paulson, Cambridge University Computer Laboratory 

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Copyright 1999 University of Cambridge 
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*) 
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header {* The division operators div and mod *} 
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theory Divides 
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imports Nat Power Product_Type 
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uses "~~/src/Provers/Arith/cancel_div_mod.ML" 
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begin 
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25942  14 
subsection {* Syntactic division operations *} 
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class div = dvd + 
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fixes div :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "div" 70) 
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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 + 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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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" 

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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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using mod_div_equality [of a zero] by simp 
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lemma mod_0 [simp]: "0 mod a = 0" 
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using mod_div_equality [of zero a] div_0 by simp 
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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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using div_mult_self1 [of b 0 a] by simp 
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lemma mod_mult_self1_is_0 [simp]: "b * a mod b = 0" 
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using mod_mult_self2 [of 0 b a] by simp 
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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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using div_mult_self2_is_id [of 1 a] zero_neq_one by simp 
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lemma mod_by_1 [simp]: "a mod 1 = 0" 
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proof  
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from mod_div_equality [of a one] div_by_1 have "a + a mod 1 = a" by simp 
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then have "a + a mod 1 = a + 0" by simp 
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then show ?thesis by (rule add_left_imp_eq) 
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qed 
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lemma mod_self [simp]: "a mod a = 0" 
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using mod_mult_self2_is_0 [of 1] by simp 
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lemma div_self [simp]: "a \<noteq> 0 \<Longrightarrow> a div a = 1" 
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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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assumes "b \<noteq> 0" 
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shows "(a + b) div b = a div b + 1" 
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using assms div_add_self1 [of b a] by (simp add: add_commute) 
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lemma mod_add_self1 [simp]: 
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"(b + a) mod b = a mod b" 
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using mod_mult_self1 [of a 1 b] by (simp add: add_commute) 
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lemma mod_add_self2 [simp]: 
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"(a + b) mod b = a mod b" 
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using mod_mult_self1 [of a 1 b] by simp 
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lemma mod_div_decomp: 
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fixes a b 
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obtains q r where "q = a div b" and "r = a mod b" 
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and "a = q * b + r" 
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proof  
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from mod_div_equality have "a = a div b * b + a mod b" by simp 
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moreover have "a div b = a div b" .. 
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moreover have "a mod b = a mod b" .. 
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note that ultimately show thesis by blast 
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qed 
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29108  134 
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" 
176 
by (unfold dvd_def, auto) 

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 

181 
lemma div_dvd_div[simp]: 

182 
"a dvd b \<Longrightarrow> a dvd c \<Longrightarrow> (b div a dvd c div a) = (b dvd c)" 

183 
apply (cases "a = 0") 

184 
apply simp 

185 
apply (unfold dvd_def) 

186 
apply auto 

187 
apply(blast intro:mult_assoc[symmetric]) 

188 
apply(fastsimp simp add: mult_assoc) 

189 
done 

190 

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191 
text {* Addition respects modular equivalence. *} 
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192 

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193 
lemma mod_add_left_eq: "(a + b) mod c = (a mod c + b) mod c" 
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194 
proof  
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195 
have "(a + b) mod c = (a div c * c + a mod c + b) mod c" 
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196 
by (simp only: mod_div_equality) 
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197 
also have "\<dots> = (a mod c + b + a div c * c) mod c" 
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198 
by (simp only: add_ac) 
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199 
also have "\<dots> = (a mod c + b) mod c" 
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200 
by (rule mod_mult_self1) 
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201 
finally show ?thesis . 
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202 
qed 
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203 

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204 
lemma mod_add_right_eq: "(a + b) mod c = (a + b mod c) mod c" 
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205 
proof  
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206 
have "(a + b) mod c = (a + (b div c * c + b mod c)) mod c" 
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207 
by (simp only: mod_div_equality) 
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208 
also have "\<dots> = (a + b mod c + b div c * c) mod c" 
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209 
by (simp only: add_ac) 
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210 
also have "\<dots> = (a + b mod c) mod c" 
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211 
by (rule mod_mult_self1) 
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212 
finally show ?thesis . 
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213 
qed 
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214 

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215 
lemma mod_add_eq: "(a + b) mod c = (a mod c + b mod c) mod c" 
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216 
by (rule trans [OF mod_add_left_eq mod_add_right_eq]) 
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217 

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218 
lemma mod_add_cong: 
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219 
assumes "a mod c = a' mod c" 
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220 
assumes "b mod c = b' mod c" 
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221 
shows "(a + b) mod c = (a' + b') mod c" 
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222 
proof  
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223 
have "(a mod c + b mod c) mod c = (a' mod c + b' mod c) mod c" 
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224 
unfolding assms .. 
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225 
thus ?thesis 
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226 
by (simp only: mod_add_eq [symmetric]) 
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227 
qed 
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228 

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229 
text {* Multiplication respects modular equivalence. *} 
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230 

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231 
lemma mod_mult_left_eq: "(a * b) mod c = ((a mod c) * b) mod c" 
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232 
proof  
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233 
have "(a * b) mod c = ((a div c * c + a mod c) * b) mod c" 
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234 
by (simp only: mod_div_equality) 
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235 
also have "\<dots> = (a mod c * b + a div c * b * c) mod c" 
29667  236 
by (simp only: algebra_simps) 
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237 
also have "\<dots> = (a mod c * b) mod c" 
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238 
by (rule mod_mult_self1) 
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239 
finally show ?thesis . 
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240 
qed 
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241 

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242 
lemma mod_mult_right_eq: "(a * b) mod c = (a * (b mod c)) mod c" 
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243 
proof  
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244 
have "(a * b) mod c = (a * (b div c * c + b mod c)) mod c" 
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245 
by (simp only: mod_div_equality) 
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246 
also have "\<dots> = (a * (b mod c) + a * (b div c) * c) mod c" 
29667  247 
by (simp only: algebra_simps) 
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248 
also have "\<dots> = (a * (b mod c)) mod c" 
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249 
by (rule mod_mult_self1) 
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250 
finally show ?thesis . 
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251 
qed 
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252 

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253 
lemma mod_mult_eq: "(a * b) mod c = ((a mod c) * (b mod c)) mod c" 
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254 
by (rule trans [OF mod_mult_left_eq mod_mult_right_eq]) 
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255 

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256 
lemma mod_mult_cong: 
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257 
assumes "a mod c = a' mod c" 
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258 
assumes "b mod c = b' mod c" 
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259 
shows "(a * b) mod c = (a' * b') mod c" 
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260 
proof  
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261 
have "(a mod c * (b mod c)) mod c = (a' mod c * (b' mod c)) mod c" 
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262 
unfolding assms .. 
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263 
thus ?thesis 
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264 
by (simp only: mod_mult_eq [symmetric]) 
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265 
qed 
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266 

29404  267 
lemma mod_mod_cancel: 
268 
assumes "c dvd b" 

269 
shows "a mod b mod c = a mod c" 

270 
proof  

271 
from `c dvd b` obtain k where "b = c * k" 

272 
by (rule dvdE) 

273 
have "a mod b mod c = a mod (c * k) mod c" 

274 
by (simp only: `b = c * k`) 

275 
also have "\<dots> = (a mod (c * k) + a div (c * k) * k * c) mod c" 

276 
by (simp only: mod_mult_self1) 

277 
also have "\<dots> = (a div (c * k) * (c * k) + a mod (c * k)) mod c" 

278 
by (simp only: add_ac mult_ac) 

279 
also have "\<dots> = a mod c" 

280 
by (simp only: mod_div_equality) 

281 
finally show ?thesis . 

282 
qed 

283 

25942  284 
end 
285 

29405
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286 
class ring_div = semiring_div + comm_ring_1 
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287 
begin 
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288 

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289 
text {* Negation respects modular equivalence. *} 
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290 

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291 
lemma mod_minus_eq: "( a) mod b = ( (a mod b)) mod b" 
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292 
proof  
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293 
have "( a) mod b = ( (a div b * b + a mod b)) mod b" 
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294 
by (simp only: mod_div_equality) 
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295 
also have "\<dots> = ( (a mod b) +  (a div b) * b) mod b" 
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296 
by (simp only: minus_add_distrib minus_mult_left add_ac) 
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297 
also have "\<dots> = ( (a mod b)) mod b" 
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298 
by (rule mod_mult_self1) 
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299 
finally show ?thesis . 
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300 
qed 
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301 

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302 
lemma mod_minus_cong: 
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303 
assumes "a mod b = a' mod b" 
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304 
shows "( a) mod b = ( a') mod b" 
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305 
proof  
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306 
have "( (a mod b)) mod b = ( (a' mod b)) mod b" 
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307 
unfolding assms .. 
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308 
thus ?thesis 
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309 
by (simp only: mod_minus_eq [symmetric]) 
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310 
qed 
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311 

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312 
text {* Subtraction respects modular equivalence. *} 
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313 

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314 
lemma mod_diff_left_eq: "(a  b) mod c = (a mod c  b) mod c" 
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315 
unfolding diff_minus 
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316 
by (intro mod_add_cong mod_minus_cong) simp_all 
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317 

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318 
lemma mod_diff_right_eq: "(a  b) mod c = (a  b mod c) mod c" 
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319 
unfolding diff_minus 
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by (intro mod_add_cong mod_minus_cong) simp_all 
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321 

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322 
lemma mod_diff_eq: "(a  b) mod c = (a mod c  b mod c) mod c" 
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323 
unfolding diff_minus 
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324 
by (intro mod_add_cong mod_minus_cong) simp_all 
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325 

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326 
lemma mod_diff_cong: 
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327 
assumes "a mod c = a' mod c" 
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328 
assumes "b mod c = b' mod c" 
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329 
shows "(a  b) mod c = (a'  b') mod c" 
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330 
unfolding diff_minus using assms 
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add class ring_div; generalize mod/diff/minus proofs for class ring_div
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331 
by (intro mod_add_cong mod_minus_cong) 
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332 

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333 
end 
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334 

25942  335 

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336 
subsection {* Division on @{typ nat} *} 
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337 

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338 
text {* 
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339 
We define @{const div} and @{const mod} on @{typ nat} by means 
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340 
of a characteristic relation with two input arguments 
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341 
@{term "m\<Colon>nat"}, @{term "n\<Colon>nat"} and two output arguments 
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342 
@{term "q\<Colon>nat"}(uotient) and @{term "r\<Colon>nat"}(emainder). 
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343 
*} 
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344 

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345 
definition divmod_rel :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" where 
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"divmod_rel m n q r \<longleftrightarrow> m = q * n + r \<and> (if n > 0 then 0 \<le> r \<and> r < n else q = 0)" 
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347 

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348 
text {* @{const divmod_rel} is total: *} 
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349 

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350 
lemma divmod_rel_ex: 
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351 
obtains q r where "divmod_rel m n q r" 
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352 
proof (cases "n = 0") 
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353 
case True with that show thesis 
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354 
by (auto simp add: divmod_rel_def) 
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355 
next 
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356 
case False 
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357 
have "\<exists>q r. m = q * n + r \<and> r < n" 
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358 
proof (induct m) 
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359 
case 0 with `n \<noteq> 0` 
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360 
have "(0\<Colon>nat) = 0 * n + 0 \<and> 0 < n" by simp 
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361 
then show ?case by blast 
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362 
next 
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363 
case (Suc m) then obtain q' r' 
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364 
where m: "m = q' * n + r'" and n: "r' < n" by auto 
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365 
then show ?case proof (cases "Suc r' < n") 
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366 
case True 
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367 
from m n have "Suc m = q' * n + Suc r'" by simp 
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368 
with True show ?thesis by blast 
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369 
next 
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370 
case False then have "n \<le> Suc r'" by auto 
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371 
moreover from n have "Suc r' \<le> n" by auto 
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372 
ultimately have "n = Suc r'" by auto 
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373 
with m have "Suc m = Suc q' * n + 0" by simp 
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374 
with `n \<noteq> 0` show ?thesis by blast 
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375 
qed 
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parents:
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376 
qed 
fbc60cd02ae2
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parents:
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377 
with that show thesis 
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378 
using `n \<noteq> 0` by (auto simp add: divmod_rel_def) 
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379 
qed 
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parents:
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380 

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381 
text {* @{const divmod_rel} is injective: *} 
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382 

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383 
lemma divmod_rel_unique_div: 
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384 
assumes "divmod_rel m n q r" 
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385 
and "divmod_rel m n q' r'" 
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parents:
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386 
shows "q = q'" 
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387 
proof (cases "n = 0") 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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388 
case True with assms show ?thesis 
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389 
by (simp add: divmod_rel_def) 
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haftmann
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390 
next 
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parents:
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diff
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391 
case False 
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using only an relation predicate to construct div and mod
haftmann
parents:
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diff
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392 
have aux: "\<And>q r q' r'. q' * n + r' = q * n + r \<Longrightarrow> r < n \<Longrightarrow> q' \<le> (q\<Colon>nat)" 
fbc60cd02ae2
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haftmann
parents:
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393 
apply (rule leI) 
fbc60cd02ae2
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haftmann
parents:
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diff
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394 
apply (subst less_iff_Suc_add) 
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haftmann
parents:
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395 
apply (auto simp add: add_mult_distrib) 
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using only an relation predicate to construct div and mod
haftmann
parents:
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396 
done 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
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397 
from `n \<noteq> 0` assms show ?thesis 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

398 
by (auto simp add: divmod_rel_def 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
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399 
intro: order_antisym dest: aux sym) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
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400 
qed 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

401 

fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
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402 
lemma divmod_rel_unique_mod: 
fbc60cd02ae2
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haftmann
parents:
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diff
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403 
assumes "divmod_rel m n q r" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
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404 
and "divmod_rel m n q' r'" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

405 
shows "r = r'" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
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406 
proof  
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
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407 
from assms have "q = q'" by (rule divmod_rel_unique_div) 
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using only an relation predicate to construct div and mod
haftmann
parents:
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diff
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408 
with assms show ?thesis by (simp add: divmod_rel_def) 
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using only an relation predicate to construct div and mod
haftmann
parents:
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diff
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409 
qed 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

410 

fbc60cd02ae2
using only an relation predicate to construct div and mod
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parents:
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411 
text {* 
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using only an relation predicate to construct div and mod
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412 
We instantiate divisibility on the natural numbers by 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
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413 
means of @{const divmod_rel}: 
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haftmann
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414 
*} 
25942  415 

416 
instantiation nat :: semiring_div 

25571
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diff
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417 
begin 
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
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diff
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418 

26100
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419 
definition divmod :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where 
28562  420 
[code del]: "divmod m n = (THE (q, r). divmod_rel m n q r)" 
26100
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diff
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421 

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422 
definition div_nat where 
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423 
"m div n = fst (divmod m n)" 
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424 

fbc60cd02ae2
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425 
definition mod_nat where 
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426 
"m mod n = snd (divmod m n)" 
25571
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instantiation target rather than legacy instance
haftmann
parents:
25162
diff
changeset

427 

26100
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428 
lemma divmod_div_mod: 
fbc60cd02ae2
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haftmann
parents:
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diff
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429 
"divmod m n = (m div n, m mod n)" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

430 
unfolding div_nat_def mod_nat_def by simp 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

431 

fbc60cd02ae2
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haftmann
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432 
lemma divmod_eq: 
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haftmann
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diff
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433 
assumes "divmod_rel m n q r" 
fbc60cd02ae2
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haftmann
parents:
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diff
changeset

434 
shows "divmod m n = (q, r)" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

435 
using assms by (auto simp add: divmod_def 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

436 
dest: divmod_rel_unique_div divmod_rel_unique_mod) 
25942  437 

26100
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haftmann
parents:
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diff
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438 
lemma div_eq: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

439 
assumes "divmod_rel m n q r" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

440 
shows "m div n = q" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

441 
using assms by (auto dest: divmod_eq simp add: div_nat_def) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

442 

fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
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443 
lemma mod_eq: 
fbc60cd02ae2
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haftmann
parents:
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diff
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444 
assumes "divmod_rel m n q r" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

445 
shows "m mod n = r" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
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446 
using assms by (auto dest: divmod_eq simp add: mod_nat_def) 
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25162
diff
changeset

447 

26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
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448 
lemma divmod_rel: "divmod_rel m n (m div n) (m mod n)" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
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449 
proof  
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

450 
from divmod_rel_ex 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
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451 
obtain q r where rel: "divmod_rel m n q r" . 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

452 
moreover with div_eq mod_eq have "m div n = q" and "m mod n = r" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

453 
by simp_all 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

454 
ultimately show ?thesis by simp 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

455 
qed 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

456 

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

457 
lemma divmod_zero: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

458 
"divmod m 0 = (0, m)" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

459 
proof  
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

460 
from divmod_rel [of m 0] show ?thesis 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

461 
unfolding divmod_div_mod divmod_rel_def by simp 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

462 
qed 
25942  463 

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

464 
lemma divmod_base: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

465 
assumes "m < n" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

466 
shows "divmod m n = (0, m)" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

467 
proof  
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

468 
from divmod_rel [of m n] show ?thesis 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

469 
unfolding divmod_div_mod divmod_rel_def 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

470 
using assms by (cases "m div n = 0") 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

471 
(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

472 
qed 
25942  473 

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

474 
lemma divmod_step: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

475 
assumes "0 < n" and "n \<le> m" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

476 
shows "divmod m n = (Suc ((m  n) div n), (m  n) mod n)" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

477 
proof  
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

478 
from divmod_rel have divmod_m_n: "divmod_rel m n (m div n) (m mod n)" . 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

479 
with assms have m_div_n: "m div n \<ge> 1" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
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480 
by (cases "m div n") (auto simp add: divmod_rel_def) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

481 
from assms divmod_m_n have "divmod_rel (m  n) n (m div n  1) (m mod n)" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

482 
by (cases "m div n") (auto simp add: divmod_rel_def) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

483 
with divmod_eq have "divmod (m  n) n = (m div n  1, m mod n)" by simp 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

484 
moreover from divmod_div_mod have "divmod (m  n) n = ((m  n) div n, (m  n) mod n)" . 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

485 
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

486 
and "m mod n = (m  n) mod n" using m_div_n by simp_all 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

487 
then show ?thesis using divmod_div_mod by simp 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

488 
qed 
25942  489 

26300  490 
text {* The ''recursion'' equations for @{const div} and @{const mod} *} 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

491 

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

492 
lemma div_less [simp]: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

493 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

494 
assumes "m < n" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

495 
shows "m div n = 0" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

496 
using assms divmod_base divmod_div_mod by simp 
25942  497 

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

498 
lemma le_div_geq: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

499 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

500 
assumes "0 < n" and "n \<le> m" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

501 
shows "m div n = Suc ((m  n) div n)" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

502 
using assms divmod_step divmod_div_mod by simp 
14267
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

503 

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

504 
lemma mod_less [simp]: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

505 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

506 
assumes "m < n" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

507 
shows "m mod n = m" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

508 
using assms divmod_base divmod_div_mod by simp 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

509 

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

510 
lemma le_mod_geq: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

511 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

512 
assumes "n \<le> m" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

513 
shows "m mod n = (m  n) mod n" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

514 
using assms divmod_step divmod_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

515 

25942  516 
instance proof 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

517 
fix m n :: nat show "m div n * n + m mod n = m" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

518 
using divmod_rel [of m n] by (simp add: divmod_rel_def) 
25942  519 
next 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

520 
fix n :: nat show "n div 0 = 0" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

521 
using divmod_zero divmod_div_mod [of n 0] by simp 
25942  522 
next 
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

523 
fix n :: nat show "0 div n = 0" 
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

524 
using divmod_rel [of 0 n] by (cases n) (simp_all add: divmod_rel_def) 
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

525 
next 
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

526 
fix m n q :: nat assume "n \<noteq> 0" then show "(q + m * n) div n = m + q div n" 
25942  527 
by (induct m) (simp_all add: le_div_geq) 
528 
qed 

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

529 

25942  530 
end 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

531 

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

532 
text {* Simproc for cancelling @{const div} and @{const mod} *} 
25942  533 

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

534 
(*lemmas mod_div_equality_nat = semiring_div_class.times_div_mod_plus_zero_one.mod_div_equality [of "m\<Colon>nat" n, standard] 
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

535 
lemmas mod_div_equality2_nat = mod_div_equality2 [of "n\<Colon>nat" m, standard*) 
25942  536 

537 
ML {* 

538 
structure CancelDivModData = 

539 
struct 

540 

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

541 
val div_name = @{const_name div}; 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

542 
val mod_name = @{const_name mod}; 
25942  543 
val mk_binop = HOLogic.mk_binop; 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

544 
val mk_sum = ArithData.mk_sum; 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

545 
val dest_sum = ArithData.dest_sum; 
25942  546 

547 
(*logic*) 

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

548 

25942  549 
val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}] 
550 

551 
val trans = trans 

552 

553 
val prove_eq_sums = 

554 
let val simps = @{thm add_0} :: @{thm add_0_right} :: @{thms add_ac} 

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

555 
in ArithData.prove_conv all_tac (ArithData.simp_all_tac simps) end; 
25942  556 

557 
end; 

558 

559 
structure CancelDivMod = CancelDivModFun(CancelDivModData); 

560 

28262
aa7ca36d67fd
back to dynamic the_context(), because static @{theory} is invalidated if ML environment changes within the same code block;
wenzelm
parents:
27676
diff
changeset

561 
val cancel_div_mod_proc = Simplifier.simproc (the_context ()) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

562 
"cancel_div_mod" ["(m::nat) + n"] (K CancelDivMod.proc); 
25942  563 

564 
Addsimprocs[cancel_div_mod_proc]; 

565 
*} 

566 

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

567 
text {* code generator setup *} 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

568 

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

569 
lemma divmod_if [code]: "divmod m n = (if n = 0 \<or> m < n then (0, m) else 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

570 
let (q, r) = divmod (m  n) n in (Suc q, r))" 
29667  571 
by (simp add: divmod_zero divmod_base divmod_step) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

572 
(simp add: divmod_div_mod) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

573 

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

574 
code_modulename SML 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

575 
Divides Nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

576 

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

577 
code_modulename OCaml 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

578 
Divides Nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

579 

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

580 
code_modulename Haskell 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

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

582 

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

583 

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

584 
subsubsection {* Quotient *} 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

585 

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

586 
lemma div_geq: "0 < n \<Longrightarrow> \<not> m < n \<Longrightarrow> m div n = Suc ((m  n) div n)" 
29667  587 
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

588 

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

589 
lemma div_if: "0 < n \<Longrightarrow> m div n = (if m < n then 0 else Suc ((m  n) div n))" 
29667  590 
by (simp add: div_geq) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

591 

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

592 
lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)" 
29667  593 
by simp 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

594 

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

595 
lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)" 
29667  596 
by simp 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

597 

25942  598 

599 
subsubsection {* Remainder *} 

600 

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

601 
lemma mod_less_divisor [simp]: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

602 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

603 
assumes "n > 0" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

604 
shows "m mod n < (n::nat)" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

605 
using assms divmod_rel unfolding divmod_rel_def by auto 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

606 

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

607 
lemma mod_less_eq_dividend [simp]: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

608 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

609 
shows "m mod n \<le> m" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

610 
proof (rule add_leD2) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

611 
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

612 
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

613 
qed 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

614 

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

615 
lemma mod_geq: "\<not> m < (n\<Colon>nat) \<Longrightarrow> m mod n = (m  n) mod n" 
29667  616 
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

617 

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

618 
lemma mod_if: "m mod (n\<Colon>nat) = (if m < n then m else (m  n) mod n)" 
29667  619 
by (simp add: le_mod_geq) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

620 

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

621 
lemma mod_1 [simp]: "m mod Suc 0 = 0" 
29667  622 
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

623 

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

624 
lemma mod_mult_distrib: "(m mod n) * (k\<Colon>nat) = (m * k) mod (n * k)" 
22718  625 
apply (cases "n = 0", simp) 
626 
apply (cases "k = 0", simp) 

627 
apply (induct m rule: nat_less_induct) 

628 
apply (subst mod_if, simp) 

629 
apply (simp add: mod_geq diff_mult_distrib) 

630 
done 

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

631 

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

632 
lemma mod_mult_distrib2: "(k::nat) * (m mod n) = (k*m) mod (k*n)" 
29667  633 
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

634 

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

635 
(* 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

636 
lemma mult_div_cancel: "(n::nat) * (m div n) = m  (m mod n)" 
29667  637 
by (cut_tac a = m and b = n in mod_div_equality2, arith) 
14267
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

638 

15439  639 
lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)" 
22718  640 
apply (drule mod_less_divisor [where m = m]) 
641 
apply simp 

642 
done 

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

643 

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

644 
subsubsection {* Quotient and Remainder *} 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

645 

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

646 
lemma divmod_rel_mult1_eq: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

647 
"[ divmod_rel b c q r; c > 0 ] 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

648 
==> divmod_rel (a*b) c (a*q + a*r div c) (a*r mod c)" 
29667  649 
by (auto simp add: split_ifs divmod_rel_def algebra_simps) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

650 

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

651 
lemma div_mult1_eq: "(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

652 
apply (cases "c = 0", simp) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

653 
apply (blast intro: divmod_rel [THEN divmod_rel_mult1_eq, THEN div_eq]) 
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

654 
done 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

655 

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

656 
lemma mod_mult1_eq: "(a*b) mod c = a*(b mod c) mod (c::nat)" 
29667  657 
by (rule mod_mult_right_eq) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

658 

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

659 
lemma mod_mult1_eq': "(a*b) mod (c::nat) = ((a mod c) * b) mod c" 
29667  660 
by (rule mod_mult_left_eq) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

661 

25162  662 
lemma mod_mult_distrib_mod: 
663 
"(a*b) mod (c::nat) = ((a mod c) * (b mod c)) mod c" 

29667  664 
by (rule mod_mult_eq) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

665 

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

666 
lemma divmod_rel_add1_eq: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

667 
"[ divmod_rel a c aq ar; divmod_rel b c bq br; c > 0 ] 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

668 
==> divmod_rel (a + b) c (aq + bq + (ar+br) div c) ((ar + br) mod c)" 
29667  669 
by (auto simp add: split_ifs divmod_rel_def algebra_simps) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

670 

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

671 
(*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

672 
lemma div_add1_eq: 
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

673 
"(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

674 
apply (cases "c = 0", simp) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

675 
apply (blast intro: divmod_rel_add1_eq [THEN div_eq] divmod_rel) 
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

676 
done 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

677 

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

678 
lemma mod_add1_eq: "(a+b) mod (c::nat) = (a mod c + b mod c) mod c" 
29667  679 
by (rule mod_add_eq) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

680 

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

681 
lemma mod_lemma: "[ (0::nat) < c; r < b ] ==> b * (q mod c) + r < b * c" 
22718  682 
apply (cut_tac m = q and n = c in mod_less_divisor) 
683 
apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto) 

684 
apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst) 

685 
apply (simp add: add_mult_distrib2) 

686 
done 

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

687 

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

688 
lemma divmod_rel_mult2_eq: "[ divmod_rel a b q r; 0 < b; 0 < c ] 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

689 
==> divmod_rel a (b*c) (q div c) (b*(q mod c) + r)" 
29667  690 
by (auto simp add: mult_ac divmod_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

691 

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

692 
lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)" 
22718  693 
apply (cases "b = 0", simp) 
694 
apply (cases "c = 0", simp) 

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

695 
apply (force simp add: divmod_rel [THEN divmod_rel_mult2_eq, THEN div_eq]) 
22718  696 
done 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

697 

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

698 
lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)" 
22718  699 
apply (cases "b = 0", simp) 
700 
apply (cases "c = 0", simp) 

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

701 
apply (auto simp add: mult_commute divmod_rel [THEN divmod_rel_mult2_eq, THEN mod_eq]) 
22718  702 
done 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

703 

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

704 

25942  705 
subsubsection{*Cancellation of Common Factors in Division*} 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

706 

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

707 
lemma div_mult_mult_lemma: 
22718  708 
"[ (0::nat) < b; 0 < c ] ==> (c*a) div (c*b) = a div b" 
29667  709 
by (auto simp add: div_mult2_eq) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

710 

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

711 
lemma div_mult_mult1 [simp]: "(0::nat) < c ==> (c*a) div (c*b) = a div b" 
22718  712 
apply (cases "b = 0") 
713 
apply (auto simp add: linorder_neq_iff [of b] div_mult_mult_lemma) 

714 
done 

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

715 

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

716 
lemma div_mult_mult2 [simp]: "(0::nat) < c ==> (a*c) div (b*c) = a div b" 
22718  717 
apply (drule div_mult_mult1) 
718 
apply (auto simp add: mult_commute) 

719 
done 

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

720 

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

721 

25942  722 
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

723 

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

724 
lemma div_1 [simp]: "m div Suc 0 = m" 
29667  725 
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

726 

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

727 

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

728 
(* Monotonicity of div in first argument *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

729 
lemma div_le_mono [rule_format (no_asm)]: 
22718  730 
"\<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

731 
apply (case_tac "k=0", simp) 
15251  732 
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

733 
apply (case_tac "n<k") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

734 
(* 1 case n<k *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

735 
apply simp 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

736 
(* 2 case n >= k *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

737 
apply (case_tac "m<k") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

738 
(* 2.1 case m<k *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

739 
apply simp 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

740 
(* 2.2 case m>=k *) 
15439  741 
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

742 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

743 

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

744 
(* Antimonotonicity of div in second argument *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

745 
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

746 
apply (subgoal_tac "0<n") 
22718  747 
prefer 2 apply simp 
15251  748 
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

749 
apply (rename_tac "k") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

750 
apply (case_tac "k<n", simp) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

751 
apply (subgoal_tac "~ (k<m) ") 
22718  752 
prefer 2 apply simp 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

753 
apply (simp add: div_geq) 
15251  754 
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

755 
prefer 2 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

756 
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

757 
apply (rule le_trans, simp) 
15439  758 
apply (simp) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

759 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

760 

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

761 
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

762 
apply (case_tac "n=0", simp) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

763 
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

764 
apply (rule div_le_mono2) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

765 
apply (simp_all (no_asm_simp)) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

766 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

767 

22718  768 
(* Similar for "less than" *) 
17085  769 
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

770 
"!!n::nat. 1<n ==> 0 < m > m div n < m" 
15251  771 
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

772 
apply (rename_tac "m") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

773 
apply (case_tac "m<n", simp) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

774 
apply (subgoal_tac "0<n") 
22718  775 
prefer 2 apply simp 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

776 
apply (simp add: div_geq) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

777 
apply (case_tac "n<m") 
15251  778 
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

779 
apply (rule impI less_trans_Suc)+ 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

780 
apply assumption 
15439  781 
apply (simp_all) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

782 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

783 

17085  784 
declare div_less_dividend [simp] 
785 

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

786 
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

787 
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

788 
apply (case_tac "n=0", simp) 
15251  789 
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

790 
apply (case_tac "Suc (na) <n") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

791 
(* case Suc(na) < n *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

792 
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

793 
(* case n \<le> Suc(na) *) 
16796  794 
apply (simp add: linorder_not_less le_Suc_eq mod_geq) 
15439  795 
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

796 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

797 

29403
fe17df4e4ab3
generalize some div/mod lemmas; remove typespecific proofs
huffman
parents:
29252
diff
changeset

798 
lemma nat_mod_div_trivial: "m mod n div n = (0 :: nat)" 
29667  799 
by simp 
14437  800 

29403
fe17df4e4ab3
generalize some div/mod lemmas; remove typespecific proofs
huffman
parents:
29252
diff
changeset

801 
lemma nat_mod_mod_trivial: "m mod n mod n = (m mod n :: nat)" 
29667  802 
by simp 
14437  803 

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

804 

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

805 
subsubsection {* The Divides Relation *} 
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24268
diff
changeset

806 

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

807 
lemma dvd_1_left [iff]: "Suc 0 dvd k" 
22718  808 
unfolding dvd_def by simp 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

809 

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

810 
lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)" 
29667  811 
by (simp add: dvd_def) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

812 

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

813 
lemma dvd_anti_sym: "[ m dvd n; n dvd m ] ==> m = (n::nat)" 
22718  814 
unfolding dvd_def 
815 
by (force dest: mult_eq_self_implies_10 simp add: mult_assoc mult_eq_1_iff) 

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

816 

23684
8c508c4dc53b
introduced (auxiliary) class dvd_mod for more convenient code generation
haftmann
parents:
23162
diff
changeset

817 
text {* @{term "op dvd"} is a partial order *} 
8c508c4dc53b
introduced (auxiliary) class dvd_mod for more convenient code generation
haftmann
parents:
23162
diff
changeset

818 

29509
1ff0f3f08a7b
migrated class package to new locale implementation
haftmann
parents:
29405
diff
changeset

819 
interpretation dvd!: order "op dvd" "\<lambda>n m \<Colon> nat. n dvd m \<and> \<not> m dvd n" 
28823  820 
proof qed (auto intro: dvd_refl dvd_trans dvd_anti_sym) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

821 

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

822 
lemma dvd_diff: "[ k dvd m; k dvd n ] ==> k dvd (mn :: nat)" 
22718  823 
unfolding dvd_def 
824 
by (blast intro: diff_mult_distrib2 [symmetric]) 

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

825 

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

826 
lemma dvd_diffD: "[ k dvd mn; k dvd n; n\<le>m ] ==> k dvd (m::nat)" 
22718  827 
apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst]) 
828 
apply (blast intro: dvd_add) 

829 
done 

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

830 

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

831 
lemma dvd_diffD1: "[ k dvd mn; k dvd m; n\<le>m ] ==> k dvd (n::nat)" 
29667  832 
by (drule_tac m = m in dvd_diff, auto) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

833 

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

834 
lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))" 
22718  835 
apply (rule iffI) 
836 
apply (erule_tac [2] dvd_add) 

837 
apply (rule_tac [2] dvd_refl) 

838 
apply (subgoal_tac "n = (n+k) k") 

839 
prefer 2 apply simp 

840 
apply (erule ssubst) 

841 
apply (erule dvd_diff) 

842 
apply (rule dvd_refl) 

843 
done 

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

844 

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

845 
lemma dvd_mod: "!!n::nat. [ f dvd m; f dvd n ] ==> f dvd m mod n" 
22718  846 
unfolding dvd_def 
847 
apply (case_tac "n = 0", auto) 

848 
apply (blast intro: mod_mult_distrib2 [symmetric]) 

849 
done 

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

850 

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

851 
lemma dvd_mod_imp_dvd: "[ (k::nat) dvd m mod n; k dvd n ] ==> k dvd m" 
22718  852 
apply (subgoal_tac "k dvd (m div n) *n + m mod n") 
853 
apply (simp add: mod_div_equality) 

854 
apply (simp only: dvd_add dvd_mult) 

855 
done 

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

856 

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

857 
lemma dvd_mod_iff: "k dvd n ==> ((k::nat) dvd m mod n) = (k dvd m)" 
29667  858 
by (blast intro: dvd_mod_imp_dvd dvd_mod) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

859 

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

860 
lemma dvd_mult_cancel: "!!k::nat. [ k*m dvd k*n; 0<k ] ==> m dvd n" 
22718  861 
unfolding dvd_def 
862 
apply (erule exE) 

863 
apply (simp add: mult_ac) 

864 
done 

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

865 

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

866 
lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))" 
22718  867 
apply auto 
868 
apply (subgoal_tac "m*n dvd m*1") 

869 
apply (drule dvd_mult_cancel, auto) 

870 
done 

14267
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 dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))" 
22718  873 
apply (subst mult_commute) 
874 
apply (erule dvd_mult_cancel1) 

875 
done 

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 
lemma dvd_imp_le: "[ k dvd n; 0 < n ] ==> k \<le> (n::nat)" 
22718  878 
apply (unfold dvd_def, clarify) 
879 
apply (simp_all (no_asm_use) add: zero_less_mult_iff) 

880 
apply (erule conjE) 

881 
apply (rule le_trans) 

882 
apply (rule_tac [2] le_refl [THEN mult_le_mono]) 

883 
apply (erule_tac [2] Suc_leI, simp) 

884 
done 

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

885 

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

886 
lemma dvd_mult_div_cancel: "n dvd m ==> n * (m div n) = (m::nat)" 
22718  887 
apply (subgoal_tac "m mod n = 0") 
888 
apply (simp add: mult_div_cancel) 

889 
apply (simp only: dvd_eq_mod_eq_0) 

890 
done 

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

891 

21408  892 
lemma le_imp_power_dvd: "!!i::nat. m \<le> n ==> i^m dvd i^n" 
22718  893 
apply (unfold dvd_def) 
894 
apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst]) 

895 
apply (simp add: power_add) 

896 
done 

21408  897 

25162  898 
lemma nat_zero_less_power_iff [simp]: "(x^n > 0) = (x > (0::nat)  n=0)" 
22718  899 
by (induct n) auto 
21408  900 

901 
lemma power_le_dvd [rule_format]: "k^j dvd n > i\<le>j > k^i dvd (n::nat)" 

22718  902 
apply (induct j) 
903 
apply (simp_all add: le_Suc_eq) 

904 
apply (blast dest!: dvd_mult_right) 

905 
done 

21408  906 

907 
lemma power_dvd_imp_le: "[i^m dvd i^n; (1::nat) < i] ==> m \<le> n" 

22718  908 
apply (rule power_le_imp_le_exp, assumption) 
909 
apply (erule dvd_imp_le, simp) 

910 
done 

21408  911 

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

912 
lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)" 
29667  913 
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

914 

22718  915 
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

916 

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

917 
(*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

918 
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

919 
apply (cut_tac a = m in mod_div_equality) 
22718  920 
apply (simp only: add_ac) 
921 
apply (blast intro: sym) 

922 
done 

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

923 

13152  924 
lemma split_div: 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

925 
"P(n div k :: nat) = 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

926 
((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

927 
(is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))") 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

928 
proof 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

929 
assume P: ?P 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

930 
show ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

931 
proof (cases) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

932 
assume "k = 0" 
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

933 
with P show ?Q by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

934 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

935 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

936 
thus ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

937 
proof (simp, intro allI impI) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

938 
fix i j 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

939 
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

940 
show "P i" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

941 
proof (cases) 
22718  942 
assume "i = 0" 
943 
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

944 
next 
22718  945 
assume "i \<noteq> 0" 
946 
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

947 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

948 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

949 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

950 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

951 
assume Q: ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

952 
show ?P 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

953 
proof (cases) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

954 
assume "k = 0" 
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

955 
with Q show ?P by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

956 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

957 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

958 
with Q have R: ?R by simp 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

959 
from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"] 
13517  960 
show ?P by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

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

962 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

963 

13882  964 
lemma split_div_lemma: 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

965 
assumes "0 < n" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

966 
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

967 
proof 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

968 
assume ?rhs 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

969 
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

970 
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

971 
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

972 
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

973 
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

974 
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

975 
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

976 
from A B show ?lhs .. 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

977 
next 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

978 
assume P: ?lhs 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

979 
then have "divmod_rel m n q (m  n * q)" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

980 
unfolding divmod_rel_def by (auto simp add: mult_ac) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

981 
then show ?rhs using divmod_rel by (rule divmod_rel_unique_div) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

982 
qed 
13882  983 

984 
theorem split_div': 

985 
"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

986 
(\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))" 
13882  987 
apply (case_tac "0 < n") 
988 
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

989 
apply simp_all 
13882  990 
done 
991 

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

992 
lemma split_mod: 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

993 
"P(n mod k :: nat) = 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

994 
((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

995 
(is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))") 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

996 
proof 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

997 
assume P: ?P 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

998 
show ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

999 
proof (cases) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1000 
assume "k = 0" 
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

1001 
with P show ?Q by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1002 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1003 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1004 
thus ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1005 
proof (simp, intro allI impI) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1006 
fix i j 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1007 
assume "n = k*i + j" "j < k" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1008 
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

1009 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1010 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1011 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1012 
assume Q: ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1013 
show ?P 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1014 
proof (cases) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1015 
assume "k = 0" 
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

1016 
with Q show ?P by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1017 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1018 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1019 
with Q have R: ?R by simp 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1020 
from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"] 
13517  1021 
show ?P by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1022 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1023 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1024 

13882  1025 
theorem mod_div_equality': "(m::nat) mod n = m  (m div n) * n" 
1026 
apply (rule_tac P="%x. m mod n = x  (m div n) * n" in 

1027 
subst [OF mod_div_equality [of _ n]]) 

1028 
apply arith 

1029 
done 

1030 

22800  1031 
lemma div_mod_equality': 
1032 
fixes m n :: nat 

1033 
shows "m div n * n = m  m mod n" 

1034 
proof  

1035 
have "m mod n \<le> m mod n" .. 

1036 
from div_mod_equality have 

1037 
"m div n * n + m mod n  m mod n = m  m mod n" by simp 

1038 
with diff_add_assoc [OF `m mod n \<le> m mod n`, of "m div n * n"] have 

1039 
"m div n * n + (m mod n  m mod n) = m  m mod n" 

1040 
by simp 

1041 
then show ?thesis by simp 

1042 
qed 

1043 

1044 

25942  1045 
subsubsection {*An ``induction'' law for modulus arithmetic.*} 
14640  1046 

1047 
lemma mod_induct_0: 

1048 
assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)" 

1049 
and base: "P i" and i: "i<p" 

1050 
shows "P 0" 

1051 
proof (rule ccontr) 

1052 
assume contra: "\<not>(P 0)" 

1053 
from i have p: "0<p" by simp 

1054 
have "\<forall>k. 0<k \<longrightarrow> \<not> P (pk)" (is "\<forall>k. ?A k") 

1055 
proof 

1056 
fix k 

1057 
show "?A k" 

1058 
proof (induct k) 

1059 
show "?A 0" by simp  "by contradiction" 

1060 
next 

1061 
fix n 

1062 
assume ih: "?A n" 

1063 
show "?A (Suc n)" 

1064 
proof (clarsimp) 

22718  1065 
assume y: "P (p  Suc n)" 
1066 
have n: "Suc n < p" 

1067 
proof (rule ccontr) 

1068 
assume "\<not>(Suc n < p)" 

1069 
hence "p  Suc n = 0" 

1070 
by simp 

1071 
with y contra show "False" 

1072 
by simp 

1073 
qed 

1074 
hence n2: "Suc (p  Suc n) = pn" by arith 

1075 
from p have "p  Suc n < p" by arith 

1076 
with y step have z: "P ((Suc (p  Suc n)) mod p)" 

1077 
by blast 

1078 
show "False" 

1079 
proof (cases "n=0") 

1080 
case True 

1081 
with z n2 contra show ?thesis by simp 

1082 
next 

1083 
case False 

1084 
with p have "pn < p" by arith 

1085 
with z n2 False ih show ?thesis by simp 

1086 
qed 

14640  1087 
qed 
1088 
qed 

1089 
qed 

1090 
moreover 

1091 
from i obtain k where "0<k \<and> i+k=p" 

1092 
by (blast dest: less_imp_add_positive) 

1093 
hence "0<k \<and> i=pk" by auto 

1094 
moreover 

1095 
note base 

1096 
ultimately 

1097 
show "False" by blast 

1098 
qed 

1099 

1100 
lemma mod_induct: 

1101 
assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)" 

1102 
and base: "P i" and i: "i<p" and j: "j<p" 

1103 
shows "P j" 

1104 
proof  

1105 
have "\<forall>j<p. P j" 

1106 
proof 

1107 
fix j 

1108 
show "j<p \<longrightarrow> P j" (is "?A j") 

1109 
proof (induct j) 

1110 
from step base i show "?A 0" 

22718  1111 
by (auto elim: mod_induct_0) 
14640  1112 
next 
1113 
fix k 

1114 
assume ih: "?A k" 

1115 
show "?A (Suc k)" 

1116 
proof 

22718  1117 
assume suc: "Suc k < p" 
1118 
hence k: "k<p" by simp 

1119 
with ih have "P k" .. 

1120 
with step k have "P (Suc k mod p)" 

1121 
by blast 

1122 
moreover 

1123 
from suc have "Suc k mod p = Suc k" 

1124 
by simp 

1125 
ultimately 

1126 
show "P (Suc k)" by simp 

14640  1127 
qed 
1128 
qed 

1129 
qed 

1130 
with j show ?thesis by blast 

1131 
qed 

1132 

3366  1133 
end 