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Mon, 15 Jun 2009 17:59:36 0700  
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permissions  rwrr 
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(* Title: HOL/Divides.thy 
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory 

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Copyright 1999 University of Cambridge 
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*) 
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header {* The division operators div and mod *} 
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theory Divides 
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imports Nat 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  13 
subsection {* Syntactic division operations *} 
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class div = dvd + 
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and mod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "mod" 70) 
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subsection {* Abstract division in commutative semirings. *} 
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30930  22 
class semiring_div = comm_semiring_1_cancel + no_zero_divisors + div + 
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assumes mod_div_equality: "a div b * b + a mod b = a" 
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and div_by_0 [simp]: "a div 0 = 0" 
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and div_0 [simp]: "0 div a = 0" 
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and div_mult_self1 [simp]: "b \<noteq> 0 \<Longrightarrow> (a + c * b) div b = c + a div b" 
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and div_mult_mult1 [simp]: "c \<noteq> 0 \<Longrightarrow> (c * a) div (c * b) = a div b" 
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begin 
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text {* @{const div} and @{const mod} *} 
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26062  32 
lemma mod_div_equality2: "b * (a div b) + a mod b = a" 
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unfolding mult_commute [of b] 

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by (rule mod_div_equality) 

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lemma mod_div_equality': "a mod b + a div b * b = a" 
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using mod_div_equality [of a b] 
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by (simp only: add_ac) 
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lemma div_mod_equality: "((a div b) * b + a mod b) + c = a + c" 
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by (simp add: mod_div_equality) 
26062  42 

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

30934  44 
by (simp add: mod_div_equality2) 
26062  45 

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lemma mod_by_0 [simp]: "a mod 0 = a" 
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lemma mod_0 [simp]: "0 mod a = 0" 
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lemma div_mult_self2 [simp]: 
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assumes "b \<noteq> 0" 
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shows "(a + b * c) div b = c + a div b" 
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using assms div_mult_self1 [of b a c] by (simp add: mult_commute) 
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lemma mod_mult_self1 [simp]: "(a + c * b) mod b = a mod b" 
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proof (cases "b = 0") 
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case True then show ?thesis by simp 
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next 
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case False 
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have "a + c * b = (a + c * b) div b * b + (a + c * b) mod b" 
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by (simp add: mod_div_equality) 
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also from False div_mult_self1 [of b a c] have 
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"\<dots> = (c + a div b) * b + (a + c * b) mod b" 
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by (simp add: algebra_simps) 
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finally have "a = a div b * b + (a + c * b) mod b" 
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by (simp add: add_commute [of a] add_assoc left_distrib) 
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then have "a div b * b + (a + c * b) mod b = a div b * b + a mod b" 
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by (simp add: mod_div_equality) 
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then show ?thesis by simp 
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qed 
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lemma mod_mult_self2 [simp]: "(a + b * c) mod b = a mod b" 
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by (simp add: mult_commute [of b]) 
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lemma div_mult_self1_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> b * a div b = a" 
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using div_mult_self2 [of b 0 a] by simp 
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lemma div_mult_self2_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> a * b div b = a" 
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lemma mod_mult_self1_is_0 [simp]: "b * a mod b = 0" 
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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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using div_mult_self2_is_id [of _ 1] by simp 
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27676  105 
lemma div_add_self1 [simp]: 
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assumes "b \<noteq> 0" 
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shows "(b + a) div b = a div b + 1" 
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using assms div_mult_self1 [of b a 1] by (simp add: add_commute) 
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27676  110 
lemma div_add_self2 [simp]: 
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shows "(a + b) div b = a div b + 1" 
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using assms div_add_self1 [of b a] by (simp add: add_commute) 
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27676  115 
lemma mod_add_self1 [simp]: 
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"(b + a) mod b = a mod b" 
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using mod_mult_self1 [of a 1 b] by (simp add: add_commute) 
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27676  119 
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 
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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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by simp 
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162 
finally show ?thesis 
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163 
by (rule add_left_imp_eq) 
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164 
qed 
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165 

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

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

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

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

180 

30052  181 
lemma dvd_div_mult: "a dvd b \<Longrightarrow> (b div a) * c = b * c div a" 
182 
apply (cases "a = 0") 

183 
apply simp 

184 
apply (auto simp: dvd_def mult_assoc) 

185 
done 

186 

29925  187 
lemma div_dvd_div[simp]: 
188 
"a dvd b \<Longrightarrow> a dvd c \<Longrightarrow> (b div a dvd c div a) = (b dvd c)" 

189 
apply (cases "a = 0") 

190 
apply simp 

191 
apply (unfold dvd_def) 

192 
apply auto 

193 
apply(blast intro:mult_assoc[symmetric]) 

194 
apply(fastsimp simp add: mult_assoc) 

195 
done 

196 

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lemma dvd_mod_imp_dvd: "[ k dvd m mod n; k dvd n ] ==> k dvd m" 
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198 
apply (subgoal_tac "k dvd (m div n) *n + m mod n") 
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199 
apply (simp add: mod_div_equality) 
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200 
apply (simp only: dvd_add dvd_mult) 
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201 
done 
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202 

29403
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203 
text {* Addition respects modular equivalence. *} 
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204 

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205 
lemma mod_add_left_eq: "(a + b) mod c = (a mod c + b) mod c" 
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206 
proof  
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207 
have "(a + b) mod c = (a div c * c + a mod c + b) mod c" 
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208 
by (simp only: mod_div_equality) 
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209 
also have "\<dots> = (a mod c + b + a div c * c) mod c" 
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210 
by (simp only: add_ac) 
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211 
also have "\<dots> = (a mod c + b) mod c" 
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212 
by (rule mod_mult_self1) 
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213 
finally show ?thesis . 
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214 
qed 
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215 

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216 
lemma mod_add_right_eq: "(a + b) mod c = (a + b mod c) mod c" 
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217 
proof  
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218 
have "(a + b) mod c = (a + (b div c * c + b mod c)) mod c" 
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219 
by (simp only: mod_div_equality) 
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220 
also have "\<dots> = (a + b mod c + b div c * c) mod c" 
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221 
by (simp only: add_ac) 
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222 
also have "\<dots> = (a + b mod c) mod c" 
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223 
by (rule mod_mult_self1) 
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224 
finally show ?thesis . 
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225 
qed 
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226 

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227 
lemma mod_add_eq: "(a + b) mod c = (a mod c + b mod c) mod c" 
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228 
by (rule trans [OF mod_add_left_eq mod_add_right_eq]) 
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229 

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230 
lemma mod_add_cong: 
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231 
assumes "a mod c = a' mod c" 
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232 
assumes "b mod c = b' mod c" 
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233 
shows "(a + b) mod c = (a' + b') mod c" 
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234 
proof  
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235 
have "(a mod c + b mod c) mod c = (a' mod c + b' mod c) mod c" 
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236 
unfolding assms .. 
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237 
thus ?thesis 
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238 
by (simp only: mod_add_eq [symmetric]) 
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239 
qed 
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240 

30923
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241 
lemma div_add [simp]: "z dvd x \<Longrightarrow> z dvd y 
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\<Longrightarrow> (x + y) div z = x div z + y div z" 
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243 
by (cases "z = 0", simp, unfold dvd_def, auto simp add: algebra_simps) 
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244 

29403
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245 
text {* Multiplication respects modular equivalence. *} 
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246 

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247 
lemma mod_mult_left_eq: "(a * b) mod c = ((a mod c) * b) mod c" 
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248 
proof  
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249 
have "(a * b) mod c = ((a div c * c + a mod c) * b) mod c" 
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250 
by (simp only: mod_div_equality) 
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251 
also have "\<dots> = (a mod c * b + a div c * b * c) mod c" 
29667  252 
by (simp only: algebra_simps) 
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253 
also have "\<dots> = (a mod c * b) mod c" 
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254 
by (rule mod_mult_self1) 
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255 
finally show ?thesis . 
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256 
qed 
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257 

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258 
lemma mod_mult_right_eq: "(a * b) mod c = (a * (b mod c)) mod c" 
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259 
proof  
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260 
have "(a * b) mod c = (a * (b div c * c + b mod c)) mod c" 
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261 
by (simp only: mod_div_equality) 
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262 
also have "\<dots> = (a * (b mod c) + a * (b div c) * c) mod c" 
29667  263 
by (simp only: algebra_simps) 
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264 
also have "\<dots> = (a * (b mod c)) mod c" 
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265 
by (rule mod_mult_self1) 
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266 
finally show ?thesis . 
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267 
qed 
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268 

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269 
lemma mod_mult_eq: "(a * b) mod c = ((a mod c) * (b mod c)) mod c" 
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270 
by (rule trans [OF mod_mult_left_eq mod_mult_right_eq]) 
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271 

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272 
lemma mod_mult_cong: 
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273 
assumes "a mod c = a' mod c" 
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274 
assumes "b mod c = b' mod c" 
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275 
shows "(a * b) mod c = (a' * b') mod c" 
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276 
proof  
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277 
have "(a mod c * (b mod c)) mod c = (a' mod c * (b' mod c)) mod c" 
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278 
unfolding assms .. 
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279 
thus ?thesis 
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280 
by (simp only: mod_mult_eq [symmetric]) 
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281 
qed 
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282 

29404  283 
lemma mod_mod_cancel: 
284 
assumes "c dvd b" 

285 
shows "a mod b mod c = a mod c" 

286 
proof  

287 
from `c dvd b` obtain k where "b = c * k" 

288 
by (rule dvdE) 

289 
have "a mod b mod c = a mod (c * k) mod c" 

290 
by (simp only: `b = c * k`) 

291 
also have "\<dots> = (a mod (c * k) + a div (c * k) * k * c) mod c" 

292 
by (simp only: mod_mult_self1) 

293 
also have "\<dots> = (a div (c * k) * (c * k) + a mod (c * k)) mod c" 

294 
by (simp only: add_ac mult_ac) 

295 
also have "\<dots> = a mod c" 

296 
by (simp only: mod_div_equality) 

297 
finally show ?thesis . 

298 
qed 

299 

30930  300 
lemma div_mult_div_if_dvd: 
301 
"y dvd x \<Longrightarrow> z dvd w \<Longrightarrow> (x div y) * (w div z) = (x * w) div (y * z)" 

302 
apply (cases "y = 0", simp) 

303 
apply (cases "z = 0", simp) 

304 
apply (auto elim!: dvdE simp add: algebra_simps) 

30476  305 
apply (subst mult_assoc [symmetric]) 
306 
apply (simp add: no_zero_divisors) 

30930  307 
done 
308 

309 
lemma div_mult_mult2 [simp]: 

310 
"c \<noteq> 0 \<Longrightarrow> (a * c) div (b * c) = a div b" 

311 
by (drule div_mult_mult1) (simp add: mult_commute) 

312 

313 
lemma div_mult_mult1_if [simp]: 

314 
"(c * a) div (c * b) = (if c = 0 then 0 else a div b)" 

315 
by simp_all 

30476  316 

30930  317 
lemma mod_mult_mult1: 
318 
"(c * a) mod (c * b) = c * (a mod b)" 

319 
proof (cases "c = 0") 

320 
case True then show ?thesis by simp 

321 
next 

322 
case False 

323 
from mod_div_equality 

324 
have "((c * a) div (c * b)) * (c * b) + (c * a) mod (c * b) = c * a" . 

325 
with False have "c * ((a div b) * b + a mod b) + (c * a) mod (c * b) 

326 
= c * a + c * (a mod b)" by (simp add: algebra_simps) 

327 
with mod_div_equality show ?thesis by simp 

328 
qed 

329 

330 
lemma mod_mult_mult2: 

331 
"(a * c) mod (b * c) = (a mod b) * c" 

332 
using mod_mult_mult1 [of c a b] by (simp add: mult_commute) 

333 

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334 
lemma div_power: 
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335 
"y dvd x \<Longrightarrow> (x div y) ^ n = x ^ n div y ^ n" 
30476  336 
apply (induct n) 
337 
apply simp 

338 
apply(simp add: div_mult_div_if_dvd dvd_power_same) 

339 
done 

340 

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341 
end 
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342 

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343 
class ring_div = semiring_div + idom 
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344 
begin 
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345 

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346 
text {* Negation respects modular equivalence. *} 
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347 

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348 
lemma mod_minus_eq: "( a) mod b = ( (a mod b)) mod b" 
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349 
proof  
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350 
have "( a) mod b = ( (a div b * b + a mod b)) mod b" 
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351 
by (simp only: mod_div_equality) 
98ab21b14f09
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huffman
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diff
changeset

352 
also have "\<dots> = ( (a mod b) +  (a div b) * b) mod b" 
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add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
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diff
changeset

353 
by (simp only: minus_add_distrib minus_mult_left add_ac) 
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add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
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diff
changeset

354 
also have "\<dots> = ( (a mod b)) mod b" 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
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diff
changeset

355 
by (rule mod_mult_self1) 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
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diff
changeset

356 
finally show ?thesis . 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
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diff
changeset

357 
qed 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
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diff
changeset

358 

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

359 
lemma mod_minus_cong: 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
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diff
changeset

360 
assumes "a mod b = a' mod b" 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
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diff
changeset

361 
shows "( a) mod b = ( a') mod b" 
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add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
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diff
changeset

362 
proof  
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

363 
have "( (a mod b)) mod b = ( (a' mod b)) mod b" 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

364 
unfolding assms .. 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

365 
thus ?thesis 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

366 
by (simp only: mod_minus_eq [symmetric]) 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
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diff
changeset

367 
qed 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

368 

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

369 
text {* Subtraction respects modular equivalence. *} 
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add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
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diff
changeset

370 

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

371 
lemma mod_diff_left_eq: "(a  b) mod c = (a mod c  b) mod c" 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
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diff
changeset

372 
unfolding diff_minus 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

373 
by (intro mod_add_cong mod_minus_cong) simp_all 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

374 

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

375 
lemma mod_diff_right_eq: "(a  b) mod c = (a  b mod c) mod c" 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
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diff
changeset

376 
unfolding diff_minus 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

377 
by (intro mod_add_cong mod_minus_cong) simp_all 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

378 

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

379 
lemma mod_diff_eq: "(a  b) mod c = (a mod c  b mod c) mod c" 
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add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
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diff
changeset

380 
unfolding diff_minus 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

381 
by (intro mod_add_cong mod_minus_cong) simp_all 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

382 

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

383 
lemma mod_diff_cong: 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
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diff
changeset

384 
assumes "a mod c = a' mod c" 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

385 
assumes "b mod c = b' mod c" 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

386 
shows "(a  b) mod c = (a'  b') mod c" 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

387 
unfolding diff_minus using assms 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

388 
by (intro mod_add_cong mod_minus_cong) 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

389 

30180  390 
lemma dvd_neg_div: "y dvd x \<Longrightarrow> x div y =  (x div y)" 
391 
apply (case_tac "y = 0") apply simp 

392 
apply (auto simp add: dvd_def) 

393 
apply (subgoal_tac "(y * k) = y *  k") 

394 
apply (erule ssubst) 

395 
apply (erule div_mult_self1_is_id) 

396 
apply simp 

397 
done 

398 

399 
lemma dvd_div_neg: "y dvd x \<Longrightarrow> x div y =  (x div y)" 

400 
apply (case_tac "y = 0") apply simp 

401 
apply (auto simp add: dvd_def) 

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

403 
apply (erule ssubst) 

404 
apply (rule div_mult_self1_is_id) 

405 
apply simp 

406 
apply simp 

407 
done 

408 

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

409 
end 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

410 

25942  411 

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

412 
subsection {* Division on @{typ nat} *} 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

413 

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

414 
text {* 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

415 
We define @{const div} and @{const mod} on @{typ nat} by means 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

416 
of a characteristic relation with two input arguments 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

417 
@{term "m\<Colon>nat"}, @{term "n\<Colon>nat"} and two output arguments 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

418 
@{term "q\<Colon>nat"}(uotient) and @{term "r\<Colon>nat"}(emainder). 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

419 
*} 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

420 

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

421 
definition divmod_rel :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat \<Rightarrow> bool" where 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
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diff
changeset

422 
"divmod_rel m n qr \<longleftrightarrow> 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

423 
m = fst qr * n + snd qr \<and> 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

424 
(if n = 0 then fst qr = 0 else if n > 0 then 0 \<le> snd qr \<and> snd qr < n else n < snd qr \<and> snd qr \<le> 0)" 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

425 

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

426 
text {* @{const divmod_rel} is total: *} 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

427 

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

428 
lemma divmod_rel_ex: 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

429 
obtains q r where "divmod_rel m n (q, r)" 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

430 
proof (cases "n = 0") 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

431 
case True with that show thesis 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

432 
by (auto simp add: divmod_rel_def) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

433 
next 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

434 
case False 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

435 
have "\<exists>q r. m = q * n + r \<and> r < n" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

436 
proof (induct m) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

437 
case 0 with `n \<noteq> 0` 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

438 
have "(0\<Colon>nat) = 0 * n + 0 \<and> 0 < n" by simp 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

439 
then show ?case by blast 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

440 
next 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

441 
case (Suc m) then obtain q' r' 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

442 
where m: "m = q' * n + r'" and n: "r' < n" by auto 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

443 
then show ?case proof (cases "Suc r' < n") 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

444 
case True 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

445 
from m n have "Suc m = q' * n + Suc r'" by simp 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

446 
with True show ?thesis by blast 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

447 
next 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

448 
case False then have "n \<le> Suc r'" by auto 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

449 
moreover from n have "Suc r' \<le> n" by auto 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

450 
ultimately have "n = Suc r'" by auto 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

451 
with m have "Suc m = Suc q' * n + 0" by simp 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

452 
with `n \<noteq> 0` show ?thesis by blast 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

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

454 
qed 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

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

456 
using `n \<noteq> 0` by (auto simp add: divmod_rel_def) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

457 
qed 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

458 

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

459 
text {* @{const divmod_rel} is injective: *} 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

460 

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

461 
lemma divmod_rel_unique: 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

462 
assumes "divmod_rel m n qr" 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

463 
and "divmod_rel m n qr'" 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

464 
shows "qr = qr'" 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

465 
proof (cases "n = 0") 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

466 
case True with assms show ?thesis 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

467 
by (cases qr, cases qr') 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

468 
(simp add: divmod_rel_def) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

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

470 
case False 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

471 
have aux: "\<And>q r q' r'. q' * n + r' = q * n + r \<Longrightarrow> r < n \<Longrightarrow> q' \<le> (q\<Colon>nat)" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

472 
apply (rule leI) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

473 
apply (subst less_iff_Suc_add) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

474 
apply (auto simp add: add_mult_distrib) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

475 
done 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

476 
from `n \<noteq> 0` assms have "fst qr = fst qr'" 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

477 
by (auto simp add: divmod_rel_def intro: order_antisym dest: aux sym) 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

478 
moreover from this assms have "snd qr = snd qr'" 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

479 
by (simp add: divmod_rel_def) 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

480 
ultimately show ?thesis by (cases qr, cases qr') simp 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

481 
qed 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

482 

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

483 
text {* 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

484 
We instantiate divisibility on the natural numbers by 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

485 
means of @{const divmod_rel}: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

486 
*} 
25942  487 

488 
instantiation nat :: semiring_div 

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

489 
begin 
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25162
diff
changeset

490 

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

491 
definition divmod :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

492 
[code del]: "divmod m n = (THE qr. divmod_rel m n qr)" 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

493 

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

494 
lemma divmod_rel_divmod: 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

495 
"divmod_rel m n (divmod m n)" 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

496 
proof  
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

497 
from divmod_rel_ex 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

498 
obtain qr where rel: "divmod_rel m n qr" . 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

499 
then show ?thesis 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

500 
by (auto simp add: divmod_def intro: theI elim: divmod_rel_unique) 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

501 
qed 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

502 

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

503 
lemma divmod_eq: 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

504 
assumes "divmod_rel m n qr" 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

505 
shows "divmod m n = qr" 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

506 
using assms by (auto intro: divmod_rel_unique divmod_rel_divmod) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

507 

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

508 
definition div_nat where 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

509 
"m div n = fst (divmod m n)" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

510 

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

511 
definition mod_nat where 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

512 
"m mod n = snd (divmod m n)" 
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25162
diff
changeset

513 

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

514 
lemma divmod_div_mod: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

515 
"divmod m n = (m div n, m mod n)" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

516 
unfolding div_nat_def mod_nat_def by simp 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

517 

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

518 
lemma div_eq: 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

519 
assumes "divmod_rel m n (q, r)" 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

520 
shows "m div n = q" 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

521 
using assms by (auto dest: divmod_eq simp add: divmod_div_mod) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

522 

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

523 
lemma mod_eq: 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

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

525 
shows "m mod n = r" 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

526 
using assms by (auto dest: divmod_eq simp add: divmod_div_mod) 
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25162
diff
changeset

527 

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

528 
lemma divmod_rel: "divmod_rel m n (m div n, m mod n)" 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

529 
by (simp add: div_nat_def mod_nat_def divmod_rel_divmod) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

530 

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

531 
lemma divmod_zero: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

532 
"divmod m 0 = (0, m)" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

533 
proof  
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

534 
from divmod_rel [of m 0] show ?thesis 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

535 
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

536 
qed 
25942  537 

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

538 
lemma divmod_base: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

539 
assumes "m < n" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

540 
shows "divmod m n = (0, m)" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

541 
proof  
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

542 
from divmod_rel [of m n] show ?thesis 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

543 
unfolding divmod_div_mod divmod_rel_def 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

544 
using assms by (cases "m div n = 0") 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

545 
(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

546 
qed 
25942  547 

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

548 
lemma divmod_step: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

549 
assumes "0 < n" and "n \<le> m" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

550 
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

551 
proof  
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

552 
from divmod_rel have divmod_m_n: "divmod_rel m n (m div n, m mod n)" . 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

553 
with assms have m_div_n: "m div n \<ge> 1" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

554 
by (cases "m div n") (auto simp add: divmod_rel_def) 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

555 
from assms divmod_m_n have "divmod_rel (m  n) n (m div n  Suc 0, m mod n)" 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

556 
by (cases "m div n") (auto simp add: divmod_rel_def) 
30079
293b896b9c25
make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents:
30078
diff
changeset

557 
with divmod_eq have "divmod (m  n) n = (m div n  Suc 0, m mod n)" by simp 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

558 
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

559 
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

560 
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

561 
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

562 
qed 
25942  563 

26300  564 
text {* The ''recursion'' equations for @{const div} and @{const mod} *} 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

565 

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

566 
lemma div_less [simp]: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

567 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

568 
assumes "m < n" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

569 
shows "m div n = 0" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

570 
using assms divmod_base divmod_div_mod by simp 
25942  571 

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

572 
lemma le_div_geq: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

573 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

574 
assumes "0 < n" and "n \<le> m" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

575 
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

576 
using assms divmod_step divmod_div_mod by simp 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

577 

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

578 
lemma mod_less [simp]: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

579 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

580 
assumes "m < n" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

581 
shows "m mod n = m" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

582 
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

583 

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

584 
lemma le_mod_geq: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

585 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

586 
assumes "n \<le> m" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

587 
shows "m mod n = (m  n) mod n" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

588 
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

589 

30930  590 
instance proof  
591 
have [simp]: "\<And>n::nat. n div 0 = 0" 

592 
by (simp add: div_nat_def divmod_zero) 

593 
have [simp]: "\<And>n::nat. 0 div n = 0" 

594 
proof  

595 
fix n :: nat 

596 
show "0 div n = 0" 

597 
by (cases "n = 0") simp_all 

598 
qed 

599 
show "OFCLASS(nat, semiring_div_class)" proof 

600 
fix m n :: nat 

601 
show "m div n * n + m mod n = m" 

602 
using divmod_rel [of m n] by (simp add: divmod_rel_def) 

603 
next 

604 
fix m n q :: nat 

605 
assume "n \<noteq> 0" 

606 
then show "(q + m * n) div n = m + q div n" 

607 
by (induct m) (simp_all add: le_div_geq) 

608 
next 

609 
fix m n q :: nat 

610 
assume "m \<noteq> 0" 

611 
then show "(m * n) div (m * q) = n div q" 

612 
proof (cases "n \<noteq> 0 \<and> q \<noteq> 0") 

613 
case False then show ?thesis by auto 

614 
next 

615 
case True with `m \<noteq> 0` 

616 
have "m > 0" and "n > 0" and "q > 0" by auto 

617 
then have "\<And>a b. divmod_rel n q (a, b) \<Longrightarrow> divmod_rel (m * n) (m * q) (a, m * b)" 

618 
by (auto simp add: divmod_rel_def) (simp_all add: algebra_simps) 

619 
moreover from divmod_rel have "divmod_rel n q (n div q, n mod q)" . 

620 
ultimately have "divmod_rel (m * n) (m * q) (n div q, m * (n mod q))" . 

621 
then show ?thesis by (simp add: div_eq) 

622 
qed 

623 
qed simp_all 

25942  624 
qed 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

625 

25942  626 
end 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

627 

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

628 
text {* Simproc for cancelling @{const div} and @{const mod} *} 
25942  629 

30934  630 
ML {* 
631 
local 

632 

633 
structure CancelDivMod = CancelDivModFun(struct 

25942  634 

30934  635 
val div_name = @{const_name div}; 
636 
val mod_name = @{const_name mod}; 

637 
val mk_binop = HOLogic.mk_binop; 

638 
val mk_sum = Nat_Arith.mk_sum; 

639 
val dest_sum = Nat_Arith.dest_sum; 

25942  640 

30934  641 
val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}]; 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

642 

30934  643 
val trans = trans; 
25942  644 

30934  645 
val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac 
646 
(@{thm monoid_add_class.add_0_left} :: @{thm monoid_add_class.add_0_right} :: @{thms add_ac})) 

25942  647 

30934  648 
end) 
25942  649 

30934  650 
in 
25942  651 

30934  652 
val cancel_div_mod_nat_proc = Simplifier.simproc (the_context ()) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

653 
"cancel_div_mod" ["(m::nat) + n"] (K CancelDivMod.proc); 
25942  654 

30934  655 
val _ = Addsimprocs [cancel_div_mod_nat_proc]; 
656 

657 
end 

25942  658 
*} 
659 

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

660 
text {* code generator setup *} 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

661 

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

662 
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

663 
let (q, r) = divmod (m  n) n in (Suc q, r))" 
29667  664 
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

665 
(simp add: divmod_div_mod) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

666 

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

667 
code_modulename SML 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

668 
Divides Nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

669 

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

670 
code_modulename OCaml 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

671 
Divides Nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

672 

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

673 
code_modulename Haskell 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

674 
Divides Nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

675 

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

676 

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

677 
subsubsection {* Quotient *} 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

678 

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

679 
lemma div_geq: "0 < n \<Longrightarrow> \<not> m < n \<Longrightarrow> m div n = Suc ((m  n) div n)" 
29667  680 
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

681 

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

682 
lemma div_if: "0 < n \<Longrightarrow> m div n = (if m < n then 0 else Suc ((m  n) div n))" 
29667  683 
by (simp add: div_geq) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

684 

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

685 
lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)" 
29667  686 
by simp 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

687 

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

688 
lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)" 
29667  689 
by simp 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

690 

25942  691 

692 
subsubsection {* Remainder *} 

693 

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

694 
lemma mod_less_divisor [simp]: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

695 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

696 
assumes "n > 0" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

697 
shows "m mod n < (n::nat)" 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

698 
using assms divmod_rel [of m n] 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

699 

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

700 
lemma mod_less_eq_dividend [simp]: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

701 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

702 
shows "m mod n \<le> m" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

703 
proof (rule add_leD2) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

704 
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

705 
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

706 
qed 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

707 

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

708 
lemma mod_geq: "\<not> m < (n\<Colon>nat) \<Longrightarrow> m mod n = (m  n) mod n" 
29667  709 
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

710 

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

711 
lemma mod_if: "m mod (n\<Colon>nat) = (if m < n then m else (m  n) mod n)" 
29667  712 
by (simp add: le_mod_geq) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

713 

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

714 
lemma mod_1 [simp]: "m mod Suc 0 = 0" 
29667  715 
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

716 

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

717 
lemma mod_mult_distrib: "(m mod n) * (k\<Colon>nat) = (m * k) mod (n * k)" 
22718  718 
apply (cases "n = 0", simp) 
719 
apply (cases "k = 0", simp) 

720 
apply (induct m rule: nat_less_induct) 

721 
apply (subst mod_if, simp) 

722 
apply (simp add: mod_geq diff_mult_distrib) 

723 
done 

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

724 

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

725 
lemma mod_mult_distrib2: "(k::nat) * (m mod n) = (k*m) mod (k*n)" 
29667  726 
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

727 

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

728 
(* 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

729 
lemma mult_div_cancel: "(n::nat) * (m div n) = m  (m mod n)" 
29667  730 
by (cut_tac a = m and b = n in mod_div_equality2, arith) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

731 

15439  732 
lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)" 
22718  733 
apply (drule mod_less_divisor [where m = m]) 
734 
apply simp 

735 
done 

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

736 

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

737 
subsubsection {* Quotient and Remainder *} 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

738 

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

739 
lemma divmod_rel_mult1_eq: 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

740 
"divmod_rel b c (q, r) \<Longrightarrow> c > 0 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

741 
\<Longrightarrow> divmod_rel (a * b) c (a * q + a * r div c, a * r mod c)" 
29667  742 
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

743 

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

744 
lemma div_mult1_eq: 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

745 
"(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

746 
apply (cases "c = 0", simp) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

747 
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

748 
done 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

749 

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

750 
lemma divmod_rel_add1_eq: 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

751 
"divmod_rel a c (aq, ar) \<Longrightarrow> divmod_rel b c (bq, br) \<Longrightarrow> c > 0 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

752 
\<Longrightarrow> divmod_rel (a + b) c (aq + bq + (ar + br) div c, (ar + br) mod c)" 
29667  753 
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

754 

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

755 
(*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

756 
lemma div_add1_eq: 
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

757 
"(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

758 
apply (cases "c = 0", simp) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

759 
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

760 
done 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

761 

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

762 
lemma mod_lemma: "[ (0::nat) < c; r < b ] ==> b * (q mod c) + r < b * c" 
22718  763 
apply (cut_tac m = q and n = c in mod_less_divisor) 
764 
apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto) 

765 
apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst) 

766 
apply (simp add: add_mult_distrib2) 

767 
done 

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

768 

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

769 
lemma divmod_rel_mult2_eq: 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

770 
"divmod_rel a b (q, r) \<Longrightarrow> 0 < b \<Longrightarrow> 0 < c 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

771 
\<Longrightarrow> divmod_rel a (b * c) (q div c, b *(q mod c) + r)" 
29667  772 
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

773 

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

774 
lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)" 
22718  775 
apply (cases "b = 0", simp) 
776 
apply (cases "c = 0", simp) 

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

777 
apply (force simp add: divmod_rel [THEN divmod_rel_mult2_eq, THEN div_eq]) 
22718  778 
done 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

779 

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

780 
lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)" 
22718  781 
apply (cases "b = 0", simp) 
782 
apply (cases "c = 0", simp) 

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

783 
apply (auto simp add: mult_commute divmod_rel [THEN divmod_rel_mult2_eq, THEN mod_eq]) 
22718  784 
done 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

785 

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

786 

25942  787 
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

788 

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

789 
lemma div_1 [simp]: "m div Suc 0 = m" 
29667  790 
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

791 

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

792 

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

793 
(* Monotonicity of div in first argument *) 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

794 
lemma div_le_mono [rule_format (no_asm)]: 
22718  795 
"\<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

796 
apply (case_tac "k=0", simp) 
15251  797 
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

798 
apply (case_tac "n<k") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

799 
(* 1 case n<k *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

800 
apply simp 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

801 
(* 2 case n >= k *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

802 
apply (case_tac "m<k") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

803 
(* 2.1 case m<k *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

804 
apply simp 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

805 
(* 2.2 case m>=k *) 
15439  806 
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

807 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

808 

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

809 
(* Antimonotonicity of div in second argument *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

810 
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

811 
apply (subgoal_tac "0<n") 
22718  812 
prefer 2 apply simp 
15251  813 
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

814 
apply (rename_tac "k") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

815 
apply (case_tac "k<n", simp) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

816 
apply (subgoal_tac "~ (k<m) ") 
22718  817 
prefer 2 apply simp 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

818 
apply (simp add: div_geq) 
15251  819 
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

820 
prefer 2 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

821 
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

822 
apply (rule le_trans, simp) 
15439  823 
apply (simp) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

824 
done 
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 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

827 
apply (case_tac "n=0", simp) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

828 
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

829 
apply (rule div_le_mono2) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

830 
apply (simp_all (no_asm_simp)) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

831 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

832 

22718  833 
(* Similar for "less than" *) 
17085  834 
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

835 
"!!n::nat. 1<n ==> 0 < m > m div n < m" 
15251  836 
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

837 
apply (rename_tac "m") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

838 
apply (case_tac "m<n", simp) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

839 
apply (subgoal_tac "0<n") 
22718  840 
prefer 2 apply simp 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

841 
apply (simp add: div_geq) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

842 
apply (case_tac "n<m") 
15251  843 
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

844 
apply (rule impI less_trans_Suc)+ 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

845 
apply assumption 
15439  846 
apply (simp_all) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

847 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

848 

17085  849 
declare div_less_dividend [simp] 
850 

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

851 
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

852 
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

853 
apply (case_tac "n=0", simp) 
15251  854 
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

855 
apply (case_tac "Suc (na) <n") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

856 
(* case Suc(na) < n *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

857 
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

858 
(* case n \<le> Suc(na) *) 
16796  859 
apply (simp add: linorder_not_less le_Suc_eq mod_geq) 
15439  860 
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

861 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

862 

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

863 

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

864 
subsubsection {* The Divides Relation *} 
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24268
diff
changeset

865 

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

866 
lemma dvd_1_left [iff]: "Suc 0 dvd k" 
22718  867 
unfolding dvd_def by simp 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

868 

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

869 
lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)" 
29667  870 
by (simp add: dvd_def) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

871 

30079
293b896b9c25
make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents:
30078
diff
changeset

872 
lemma nat_dvd_1_iff_1 [simp]: "m dvd (1::nat) \<longleftrightarrow> m = 1" 
293b896b9c25
make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents:
30078
diff
changeset

873 
by (simp add: dvd_def) 
293b896b9c25
make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents:
30078
diff
changeset

874 

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

875 
lemma dvd_anti_sym: "[ m dvd n; n dvd m ] ==> m = (n::nat)" 
22718  876 
unfolding dvd_def 
877 
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

878 

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

879 
text {* @{term "op dvd"} is a partial order *} 
8c508c4dc53b
introduced (auxiliary) class dvd_mod for more convenient code generation
haftmann
parents:
23162
diff
changeset

880 

30729
461ee3e49ad3
interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents:
30653
diff
changeset

881 
interpretation dvd: order "op dvd" "\<lambda>n m \<Colon> nat. n dvd m \<and> \<not> m dvd n" 
28823  882 
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

883 

30042  884 
lemma nat_dvd_diff[simp]: "[ k dvd m; k dvd n ] ==> k dvd (mn :: nat)" 
885 
unfolding dvd_def 

886 
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

887 

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

888 
lemma dvd_diffD: "[ k dvd mn; k dvd n; n\<le>m ] ==> k dvd (m::nat)" 
22718  889 
apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst]) 
890 
apply (blast intro: dvd_add) 

891 
done 

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

892 

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

893 
lemma dvd_diffD1: "[ k dvd mn; k dvd m; n\<le>m ] ==> k dvd (n::nat)" 
30042  894 
by (drule_tac m = m in nat_dvd_diff, auto) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

895 

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

896 
lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))" 
22718  897 
apply (rule iffI) 
898 
apply (erule_tac [2] dvd_add) 

899 
apply (rule_tac [2] dvd_refl) 

900 
apply (subgoal_tac "n = (n+k) k") 

901 
prefer 2 apply simp 

902 
apply (erule ssubst) 

30042  903 
apply (erule nat_dvd_diff) 
22718  904 
apply (rule dvd_refl) 
905 
done 

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

906 

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

907 
lemma dvd_mod: "!!n::nat. [ f dvd m; f dvd n ] ==> f dvd m mod n" 
22718  908 
unfolding dvd_def 
909 
apply (case_tac "n = 0", auto) 

910 
apply (blast intro: mod_mult_distrib2 [symmetric]) 

911 
done 

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

912 

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

913 
lemma dvd_mod_iff: "k dvd n ==> ((k::nat) dvd m mod n) = (k dvd m)" 
29667  914 
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

915 

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

916 
lemma dvd_mult_cancel: "!!k::nat. [ k*m dvd k*n; 0<k ] ==> m dvd n" 
22718  917 
unfolding dvd_def 
918 
apply (erule exE) 

919 
apply (simp add: mult_ac) 

920 
done 

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

921 

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

922 
lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))" 
22718  923 
apply auto 
924 
apply (subgoal_tac "m*n dvd m*1") 

925 
apply (drule dvd_mult_cancel, auto) 

926 
done 

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

927 

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

928 
lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))" 
22718  929 
apply (subst mult_commute) 
930 
apply (erule dvd_mult_cancel1) 

931 
done 

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

932 

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

933 
lemma dvd_imp_le: "[ k dvd n; 0 < n ] ==> k \<le> (n::nat)" 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

934 
by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

935 

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

936 
lemma dvd_mult_div_cancel: "n dvd m ==> n * (m div n) = (m::nat)" 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

937 
by (simp add: dvd_eq_mod_eq_0 mult_div_cancel) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

938 

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

939 
lemma power_dvd_imp_le: 
41fd307cab30
dropped reference to class recpower and lemma duplicate
haftmann
parents:
30934
diff
changeset

940 
"i ^ m dvd i ^ n \<Longrightarrow> (1::nat) < i \<Longrightarrow> m \<le> n" 
22718  941 
apply (rule power_le_imp_le_exp, assumption) 
942 
apply (erule dvd_imp_le, simp) 

943 
done 

21408  944 

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

945 
lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)" 
29667  946 
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

947 

22718  948 
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

949 

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

950 
(*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

951 
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

952 
apply (cut_tac a = m in mod_div_equality) 
22718  953 
apply (simp only: add_ac) 
954 
apply (blast intro: sym) 

955 
done 

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

956 

13152  957 
lemma split_div: 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

958 
"P(n div k :: nat) = 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

959 
((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

960 
(is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))") 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

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

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

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

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

965 
assume "k = 0" 
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

966 
with P show ?Q by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

967 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

968 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

969 
thus ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

970 
proof (simp, intro allI impI) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

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

972 
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

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

974 
proof (cases) 
22718  975 
assume "i = 0" 
976 
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

977 
next 
22718  978 
assume "i \<noteq> 0" 
979 
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

980 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

981 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

982 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

983 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

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

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

986 
proof (cases) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

987 
assume "k = 0" 
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

988 
with Q show ?P by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

989 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

990 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

991 
with Q have R: ?R by simp 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

992 
from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"] 
13517  993 
show ?P by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

994 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

995 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

996 

13882  997 
lemma split_div_lemma: 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

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

999 
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

1000 
proof 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1001 
assume ?rhs 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1002 
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

1003 
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

1004 
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

1005 
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

1006 
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

1007 
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

1008 
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

1009 
from A B show ?lhs .. 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1010 
next 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1011 
assume P: ?lhs 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

1012 
then have "divmod_rel m n (q, m  n * q)" 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1013 
unfolding divmod_rel_def by (auto simp add: mult_ac) 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

1014 
with divmod_rel_unique divmod_rel [of m n] 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

1015 
have "(q, m  n * q) = (m div n, m mod n)" by auto 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

1016 
then show ?rhs by simp 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

1017 
qed 
13882  1018 

1019 
theorem split_div': 

1020 
"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

1021 
(\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))" 
13882  1022 
apply (case_tac "0 < n") 
1023 
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

1024 
apply simp_all 
13882  1025 
done 
1026 

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

1027 
lemma split_mod: 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1028 
"P(n mod k :: nat) = 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1029 
((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

1030 
(is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))") 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1031 
proof 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1032 
assume P: ?P 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1033 
show ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1034 
proof (cases) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1035 
assume "k = 0" 
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

1036 
with P show ?Q by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1037 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1038 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1039 
thus ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1040 
proof (simp, intro allI impI) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1041 
fix i j 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1042 
assume "n = k*i + j" "j < k" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1043 
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

1044 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1045 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1046 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1047 
assume Q: ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1048 
show ?P 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1049 
proof (cases) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1050 
assume "k = 0" 
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

1051 
with Q show ?P by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1052 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1053 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1054 
with Q have R: ?R by simp 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1055 
from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"] 
13517  1056 
show ?P by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1057 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1058 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1059 

13882  1060 
theorem mod_div_equality': "(m::nat) mod n = m  (m div n) * n" 
1061 
apply (rule_tac P="%x. m mod n = x  (m div n) * n" in 

1062 
subst [OF mod_div_equality [of _ n]]) 

1063 
apply arith 

1064 
done 

1065 

22800  1066 
lemma div_mod_equality': 
1067 
fixes m n :: nat 

1068 
shows "m div n * n = m  m mod n" 

1069 
proof  

1070 
have "m mod n \<le> m mod n" .. 

1071 
from div_mod_equality have 

1072 
"m div n * n + m mod n  m mod n = m  m mod n" by simp 

1073 
with diff_add_assoc [OF `m mod n \<le> m mod n`, of "m div n * n"] have 

1074 
"m div n * n + (m mod n  m mod n) = m  m mod n" 

1075 
by simp 

1076 
then show ?thesis by simp 

1077 
qed 

1078 

1079 

25942  1080 
subsubsection {*An ``induction'' law for modulus arithmetic.*} 
14640  1081 

1082 
lemma mod_induct_0: 

1083 
assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)" 

1084 
and base: "P i" and i: "i<p" 

1085 
shows "P 0" 

1086 
proof (rule ccontr) 

1087 
assume contra: "\<not>(P 0)" 

1088 
from i have p: "0<p" by simp 

1089 
have "\<forall>k. 0<k \<longrightarrow> \<not> P (pk)" (is "\<forall>k. ?A k") 

1090 
proof 

1091 
fix k 

1092 
show "?A k" 

1093 
proof (induct k) 

1094 
show "?A 0" by simp  "by contradiction" 

1095 
next 

1096 
fix n 

1097 
assume ih: "?A n" 

1098 
show "?A (Suc n)" 

1099 
proof (clarsimp) 

22718  1100 
assume y: "P (p  Suc n)" 
1101 
have n: "Suc n < p" 

1102 
proof (rule ccontr) 

1103 
assume "\<not>(Suc n < p)" 

1104 
hence "p  Suc n = 0" 

1105 
by simp 

1106 
with y contra show "False" 

1107 
by simp 

1108 
qed 

1109 
hence n2: "Suc (p  Suc n) = pn" by arith 

1110 
from p have "p  Suc n < p" by arith 

1111 
with y step have z: "P ((Suc (p  Suc n)) mod p)" 

1112 
by blast 

1113 
show "False" 

1114 
proof (cases "n=0") 

1115 
case True 

1116 
with z n2 contra show ?thesis by simp 

1117 
next 

1118 
case False 

1119 
with p have "pn < p" by arith 

1120 
with z n2 False ih show ?thesis by simp 

1121 
qed 

14640  1122 
qed 
1123 
qed 

1124 
qed 

1125 
moreover 

1126 
from i obtain k where "0<k \<and> i+k=p" 

1127 
by (blast dest: less_imp_add_positive) 

1128 
hence "0<k \<and> i=pk" by auto 

1129 
moreover 

1130 
note base 

1131 
ultimately 

1132 
show "False" by blast 

1133 
qed 

1134 

1135 
lemma mod_induct: 

1136 
assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)" 

1137 
and base: "P i" and i: "i<p" and j: "j<p" 

1138 
shows "P j" 

1139 
proof  

1140 
have "\<forall>j<p. P j" 

1141 
proof 

1142 
fix j 

1143 
show "j<p \<longrightarrow> P j" (is "?A j") 

1144 
proof (induct j) 

1145 
from step base i show "?A 0" 

22718  1146 
by (auto elim: mod_induct_0) 
14640  1147 
next 
1148 
fix k 

1149 
assume ih: "?A k" 

1150 
show "?A (Suc k)" 

1151 
proof 

22718  1152 
assume suc: "Suc k < p" 
1153 
hence k: "k<p" by simp 

1154 
with ih have "P k" .. 

1155 
with step k have "P (Suc k mod p)" 

1156 
by blast 

1157 
moreover 

1158 
from suc have "Suc k mod p = Suc k" 

1159 
by simp 

1160 
ultimately 

1161 
show "P (Suc k)" by simp 

14640  1162 
qed 
1163 
qed 

1164 
qed 

1165 
with j show ?thesis by blast 

1166 
qed 

1167 

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

1168 
lemma nat_dvd_not_less: 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

1169 
fixes m n :: nat 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

1170 
shows "0 < m \<Longrightarrow> m < n \<Longrightarrow> \<not> n dvd m" 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

1171 
by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc) 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

1172 

3366  1173 
end 