author  haftmann 
Thu, 29 Dec 2011 10:47:54 +0100  
changeset 46026  83caa4f4bd56 
parent 45607  16b4f5774621 
child 46551  866bce5442a3 
permissions  rwrr 
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(* Title: HOL/Divides.thy 
2 
Author: Lawrence C Paulson, Cambridge University Computer Laboratory 

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Copyright 1999 University of Cambridge 
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*) 
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header {* The division operators div and mod *} 
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theory Divides 
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imports Nat_Numeral Nat_Transfer 
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uses "~~/src/Provers/Arith/cancel_div_mod.ML" 
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begin 
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25942  13 
subsection {* Syntactic division operations *} 
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class div = dvd + 
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and mod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "mod" 70) 
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subsection {* Abstract division in commutative semirings. *} 
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class semiring_div = comm_semiring_1_cancel + no_zero_divisors + div + 
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and div_by_0 [simp]: "a div 0 = 0" 
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and div_0 [simp]: "0 div a = 0" 
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and div_mult_self1 [simp]: "b \<noteq> 0 \<Longrightarrow> (a + c * b) div b = c + a div b" 
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and div_mult_mult1 [simp]: "c \<noteq> 0 \<Longrightarrow> (c * a) div (c * b) = a div b" 
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begin 
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text {* @{const div} and @{const mod} *} 
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26062  32 
lemma mod_div_equality2: "b * (a div b) + a mod b = a" 
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unfolding mult_commute [of b] 

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by (rule mod_div_equality) 

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

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

30934  44 
by (simp add: mod_div_equality2) 
26062  45 

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lemma mod_by_0 [simp]: "a mod 0 = a" 
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lemma mod_0 [simp]: "0 mod a = 0" 
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lemma div_mult_self2 [simp]: 
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assumes "b \<noteq> 0" 
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shows "(a + b * c) div b = c + a div b" 
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using assms div_mult_self1 [of b a c] by (simp add: mult_commute) 
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lemma mod_mult_self1 [simp]: "(a + c * b) mod b = a mod b" 
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proof (cases "b = 0") 
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case True then show ?thesis by simp 
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next 
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case False 
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have "a + c * b = (a + c * b) div b * b + (a + c * b) mod b" 
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by (simp add: mod_div_equality) 
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also from False div_mult_self1 [of b a c] have 
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"\<dots> = (c + a div b) * b + (a + c * b) mod b" 
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by (simp add: algebra_simps) 
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finally have "a = a div b * b + (a + c * b) mod b" 
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by (simp add: add_commute [of a] add_assoc left_distrib) 
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then have "a div b * b + (a + c * b) mod b = a div b * b + a mod b" 
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by (simp add: mod_div_equality) 
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then show ?thesis by simp 
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qed 
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lemma mod_mult_self2 [simp]: "(a + b * c) mod b = a mod b" 
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by (simp add: mult_commute [of b]) 
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lemma div_mult_self1_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> b * a div b = a" 
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using div_mult_self2 [of b 0 a] by simp 
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lemma div_mult_self2_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> a * b div b = a" 
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lemma mod_mult_self1_is_0 [simp]: "b * a mod b = 0" 
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lemma mod_mult_self2_is_0 [simp]: "a * b mod b = 0" 
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using mod_mult_self1 [of 0 a b] by simp 
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lemma div_by_1 [simp]: "a div 1 = a" 
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lemma mod_by_1 [simp]: "a mod 1 = 0" 
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proof  
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from mod_div_equality [of a one] div_by_1 have "a + a mod 1 = a" by simp 
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then have "a + a mod 1 = a + 0" by simp 
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then show ?thesis by (rule add_left_imp_eq) 
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qed 
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lemma mod_self [simp]: "a mod a = 0" 
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lemma div_self [simp]: "a \<noteq> 0 \<Longrightarrow> a div a = 1" 
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27676  105 
lemma div_add_self1 [simp]: 
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assumes "b \<noteq> 0" 
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shows "(b + a) div b = a div b + 1" 
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using assms div_mult_self1 [of b a 1] by (simp add: add_commute) 
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27676  110 
lemma div_add_self2 [simp]: 
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shows "(a + b) div b = a div b + 1" 
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using assms div_add_self1 [of b a] by (simp add: add_commute) 
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27676  115 
lemma mod_add_self1 [simp]: 
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"(b + a) mod b = a mod b" 
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lemma mod_add_self2 [simp]: 
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"(a + b) mod b = a mod b" 
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lemma mod_div_decomp: 
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fixes a b 
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obtains q r where "q = a div b" and "r = a mod b" 
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and "a = q * b + r" 
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proof  
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moreover have "a div b = a div b" .. 
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moreover have "a mod b = a mod b" .. 
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note that ultimately show thesis by blast 
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qed 
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lemma dvd_eq_mod_eq_0 [code]: "a dvd b \<longleftrightarrow> b mod a = 0" 
25942  135 
proof 
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assume "b mod a = 0" 

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

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

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

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then show "a dvd b" unfolding dvd_def . 

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next 

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assume "a dvd b" 

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

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

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

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

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then show "b mod a = 0" by simp 
25942  148 
qed 
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lemma mod_div_trivial [simp]: "a mod b div b = 0" 
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proof (cases "b = 0") 
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assume "b = 0" 
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thus ?thesis by simp 
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next 
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assume "b \<noteq> 0" 
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hence "a div b + a mod b div b = (a mod b + a div b * b) div b" 
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by (rule div_mult_self1 [symmetric]) 
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also have "\<dots> = a div b" 
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by (simp only: mod_div_equality') 
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160 
also have "\<dots> = a div b + 0" 
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161 
by simp 
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162 
finally show ?thesis 
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163 
by (rule add_left_imp_eq) 
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164 
qed 
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165 

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

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

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

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

180 

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

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

186 
apply simp 

187 
apply (auto simp: dvd_def mult_assoc) 

188 
done 

189 

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

192 
apply (cases "a = 0") 

193 
apply simp 

194 
apply (unfold dvd_def) 

195 
apply auto 

196 
apply(blast intro:mult_assoc[symmetric]) 

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197 
apply(fastforce simp add: mult_assoc) 
29925  198 
done 
199 

30078
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200 
lemma dvd_mod_imp_dvd: "[ k dvd m mod n; k dvd n ] ==> k dvd m" 
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201 
apply (subgoal_tac "k dvd (m div n) *n + m mod n") 
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202 
apply (simp add: mod_div_equality) 
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203 
apply (simp only: dvd_add dvd_mult) 
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204 
done 
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205 

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

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

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

fe17df4e4ab3
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230 
lemma mod_add_eq: "(a + b) mod c = (a mod c + b mod c) mod c" 
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231 
by (rule trans [OF mod_add_left_eq mod_add_right_eq]) 
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232 

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

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

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

fe17df4e4ab3
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250 
lemma mod_mult_left_eq: "(a * b) mod c = ((a mod c) * b) mod c" 
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251 
proof  
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252 
have "(a * b) mod c = ((a div c * c + a mod c) * b) mod c" 
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253 
by (simp only: mod_div_equality) 
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254 
also have "\<dots> = (a mod c * b + a div c * b * c) mod c" 
29667  255 
by (simp only: algebra_simps) 
29403
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256 
also have "\<dots> = (a mod c * b) mod c" 
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257 
by (rule mod_mult_self1) 
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258 
finally show ?thesis . 
fe17df4e4ab3
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259 
qed 
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260 

fe17df4e4ab3
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261 
lemma mod_mult_right_eq: "(a * b) mod c = (a * (b mod c)) mod c" 
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262 
proof  
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263 
have "(a * b) mod c = (a * (b div c * c + b mod c)) mod c" 
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264 
by (simp only: mod_div_equality) 
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265 
also have "\<dots> = (a * (b mod c) + a * (b div c) * c) mod c" 
29667  266 
by (simp only: algebra_simps) 
29403
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267 
also have "\<dots> = (a * (b mod c)) mod c" 
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268 
by (rule mod_mult_self1) 
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269 
finally show ?thesis . 
fe17df4e4ab3
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270 
qed 
fe17df4e4ab3
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271 

fe17df4e4ab3
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272 
lemma mod_mult_eq: "(a * b) mod c = ((a mod c) * (b mod c)) mod c" 
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273 
by (rule trans [OF mod_mult_left_eq mod_mult_right_eq]) 
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274 

fe17df4e4ab3
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275 
lemma mod_mult_cong: 
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276 
assumes "a mod c = a' mod c" 
fe17df4e4ab3
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277 
assumes "b mod c = b' mod c" 
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278 
shows "(a * b) mod c = (a' * b') mod c" 
fe17df4e4ab3
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279 
proof  
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280 
have "(a mod c * (b mod c)) mod c = (a' mod c * (b' mod c)) mod c" 
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281 
unfolding assms .. 
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282 
thus ?thesis 
fe17df4e4ab3
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283 
by (simp only: mod_mult_eq [symmetric]) 
fe17df4e4ab3
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284 
qed 
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285 

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

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

289 
proof  

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

291 
by (rule dvdE) 

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

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

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

295 
by (simp only: mod_mult_self1) 

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

297 
by (simp only: add_ac mult_ac) 

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

299 
by (simp only: mod_div_equality) 

300 
finally show ?thesis . 

301 
qed 

302 

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

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

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

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

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

30930  310 
done 
311 

35367
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lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
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312 
lemma div_mult_swap: 
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lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
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313 
assumes "c dvd b" 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
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314 
shows "a * (b div c) = (a * b) div c" 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
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315 
proof  
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
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316 
from assms have "b div c * (a div 1) = b * a div (c * 1)" 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
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317 
by (simp only: div_mult_div_if_dvd one_dvd) 
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lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
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318 
then show ?thesis by (simp add: mult_commute) 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
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319 
qed 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
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320 

30930  321 
lemma div_mult_mult2 [simp]: 
322 
"c \<noteq> 0 \<Longrightarrow> (a * c) div (b * c) = a div b" 

323 
by (drule div_mult_mult1) (simp add: mult_commute) 

324 

325 
lemma div_mult_mult1_if [simp]: 

326 
"(c * a) div (c * b) = (if c = 0 then 0 else a div b)" 

327 
by simp_all 

30476  328 

30930  329 
lemma mod_mult_mult1: 
330 
"(c * a) mod (c * b) = c * (a mod b)" 

331 
proof (cases "c = 0") 

332 
case True then show ?thesis by simp 

333 
next 

334 
case False 

335 
from mod_div_equality 

336 
have "((c * a) div (c * b)) * (c * b) + (c * a) mod (c * b) = c * a" . 

337 
with False have "c * ((a div b) * b + a mod b) + (c * a) mod (c * b) 

338 
= c * a + c * (a mod b)" by (simp add: algebra_simps) 

339 
with mod_div_equality show ?thesis by simp 

340 
qed 

341 

342 
lemma mod_mult_mult2: 

343 
"(a * c) mod (b * c) = (a mod b) * c" 

344 
using mod_mult_mult1 [of c a b] by (simp add: mult_commute) 

345 

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

346 
lemma dvd_mod: "k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd (m mod n)" 
57f7ef0dba8e
generalize lemmas dvd_mod and dvd_mod_iff to class semiring_div
huffman
parents:
31661
diff
changeset

347 
unfolding dvd_def by (auto simp add: mod_mult_mult1) 
57f7ef0dba8e
generalize lemmas dvd_mod and dvd_mod_iff to class semiring_div
huffman
parents:
31661
diff
changeset

348 

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

349 
lemma dvd_mod_iff: "k dvd n \<Longrightarrow> k dvd (m mod n) \<longleftrightarrow> k dvd m" 
57f7ef0dba8e
generalize lemmas dvd_mod and dvd_mod_iff to class semiring_div
huffman
parents:
31661
diff
changeset

350 
by (blast intro: dvd_mod_imp_dvd dvd_mod) 
57f7ef0dba8e
generalize lemmas dvd_mod and dvd_mod_iff to class semiring_div
huffman
parents:
31661
diff
changeset

351 

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

352 
lemma div_power: 
31661
1e252b8b2334
move lemma div_power into semiring_div context; class ring_div inherits from idom
huffman
parents:
31009
diff
changeset

353 
"y dvd x \<Longrightarrow> (x div y) ^ n = x ^ n div y ^ n" 
30476  354 
apply (induct n) 
355 
apply simp 

356 
apply(simp add: div_mult_div_if_dvd dvd_power_same) 

357 
done 

358 

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

359 
lemma dvd_div_eq_mult: 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset

360 
assumes "a \<noteq> 0" and "a dvd b" 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset

361 
shows "b div a = c \<longleftrightarrow> b = c * a" 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset

362 
proof 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset

363 
assume "b = c * a" 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset

364 
then show "b div a = c" by (simp add: assms) 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset

365 
next 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset

366 
assume "b div a = c" 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset

367 
then have "b div a * a = c * a" by simp 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset

368 
moreover from `a dvd b` have "b div a * a = b" by (simp add: dvd_div_mult_self) 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset

369 
ultimately show "b = c * a" by simp 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset

370 
qed 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset

371 

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

372 
lemma dvd_div_div_eq_mult: 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset

373 
assumes "a \<noteq> 0" "c \<noteq> 0" and "a dvd b" "c dvd d" 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset

374 
shows "b div a = d div c \<longleftrightarrow> b * c = a * d" 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset

375 
using assms by (auto simp add: mult_commute [of _ a] dvd_div_mult_self dvd_div_eq_mult div_mult_swap intro: sym) 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset

376 

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

377 
end 
1e252b8b2334
move lemma div_power into semiring_div context; class ring_div inherits from idom
huffman
parents:
31009
diff
changeset

378 

35673  379 
class ring_div = semiring_div + comm_ring_1 
29405
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

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

381 

36634  382 
subclass ring_1_no_zero_divisors .. 
383 

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

384 
text {* Negation respects modular equivalence. *} 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

385 

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

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

387 
proof  
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

388 
have "( a) mod b = ( (a div b * b + a mod b)) mod b" 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

389 
by (simp only: mod_div_equality) 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

390 
also have "\<dots> = ( (a mod b) +  (a div b) * b) mod b" 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

391 
by (simp only: minus_add_distrib minus_mult_left add_ac) 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

392 
also have "\<dots> = ( (a mod b)) mod b" 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

393 
by (rule mod_mult_self1) 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

394 
finally show ?thesis . 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

395 
qed 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

396 

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

397 
lemma mod_minus_cong: 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

398 
assumes "a mod b = a' mod b" 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

399 
shows "( a) mod b = ( a') mod b" 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

400 
proof  
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

401 
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

402 
unfolding assms .. 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

403 
thus ?thesis 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

404 
by (simp only: mod_minus_eq [symmetric]) 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

405 
qed 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

406 

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

407 
text {* Subtraction respects modular equivalence. *} 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

408 

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

409 
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:
29404
diff
changeset

410 
unfolding diff_minus 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

411 
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

412 

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

413 
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:
29404
diff
changeset

414 
unfolding diff_minus 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

415 
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

416 

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

417 
lemma mod_diff_eq: "(a  b) mod c = (a mod c  b mod c) mod c" 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

418 
unfolding diff_minus 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

419 
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

420 

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

421 
lemma mod_diff_cong: 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

422 
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

423 
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

424 
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

425 
unfolding diff_minus using assms 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

426 
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

427 

30180  428 
lemma dvd_neg_div: "y dvd x \<Longrightarrow> x div y =  (x div y)" 
429 
apply (case_tac "y = 0") apply simp 

430 
apply (auto simp add: dvd_def) 

431 
apply (subgoal_tac "(y * k) = y *  k") 

432 
apply (erule ssubst) 

433 
apply (erule div_mult_self1_is_id) 

434 
apply simp 

435 
done 

436 

437 
lemma dvd_div_neg: "y dvd x \<Longrightarrow> x div y =  (x div y)" 

438 
apply (case_tac "y = 0") apply simp 

439 
apply (auto simp add: dvd_def) 

440 
apply (subgoal_tac "y * k = y * k") 

441 
apply (erule ssubst) 

442 
apply (rule div_mult_self1_is_id) 

443 
apply simp 

444 
apply simp 

445 
done 

446 

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

447 
end 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

448 

25942  449 

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

450 
subsection {* Division on @{typ nat} *} 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

451 

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

452 
text {* 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

453 
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

454 
of a characteristic relation with two input arguments 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

455 
@{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

456 
@{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

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

458 

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

459 
definition divmod_nat_rel :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat \<Rightarrow> bool" where 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

460 
"divmod_nat_rel m n qr \<longleftrightarrow> 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

461 
m = fst qr * n + snd qr \<and> 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

462 
(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

463 

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

464 
text {* @{const divmod_nat_rel} is total: *} 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

465 

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

466 
lemma divmod_nat_rel_ex: 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

467 
obtains q r where "divmod_nat_rel m n (q, r)" 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

468 
proof (cases "n = 0") 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

469 
case True with that show thesis 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

470 
by (auto simp add: divmod_nat_rel_def) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

471 
next 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

472 
case False 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

473 
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

474 
proof (induct m) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

475 
case 0 with `n \<noteq> 0` 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

476 
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

477 
then show ?case by blast 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

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

479 
case (Suc m) then obtain q' r' 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

480 
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

481 
then show ?case proof (cases "Suc r' < n") 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

482 
case True 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

483 
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

484 
with True show ?thesis by blast 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

485 
next 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

486 
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

487 
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

488 
ultimately have "n = Suc r'" by auto 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

489 
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

490 
with `n \<noteq> 0` show ?thesis by blast 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

491 
qed 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

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

493 
with that show thesis 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

494 
using `n \<noteq> 0` by (auto simp add: divmod_nat_rel_def) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

495 
qed 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

496 

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

497 
text {* @{const divmod_nat_rel} is injective: *} 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

498 

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

499 
lemma divmod_nat_rel_unique: 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

500 
assumes "divmod_nat_rel m n qr" 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

501 
and "divmod_nat_rel m n qr'" 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

502 
shows "qr = qr'" 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

503 
proof (cases "n = 0") 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

504 
case True with assms show ?thesis 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

505 
by (cases qr, cases qr') 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

506 
(simp add: divmod_nat_rel_def) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

507 
next 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

508 
case False 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

509 
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

510 
apply (rule leI) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

511 
apply (subst less_iff_Suc_add) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

512 
apply (auto simp add: add_mult_distrib) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

513 
done 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

514 
from `n \<noteq> 0` assms have "fst qr = fst qr'" 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

515 
by (auto simp add: divmod_nat_rel_def intro: order_antisym dest: aux sym) 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

516 
moreover from this assms have "snd qr = snd qr'" 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

517 
by (simp add: divmod_nat_rel_def) 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

518 
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

519 
qed 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

520 

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

521 
text {* 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

522 
We instantiate divisibility on the natural numbers by 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

523 
means of @{const divmod_nat_rel}: 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

524 
*} 
25942  525 

526 
instantiation nat :: semiring_div 

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

527 
begin 
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25162
diff
changeset

528 

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

529 
definition divmod_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where 
37767  530 
"divmod_nat m n = (THE qr. divmod_nat_rel m n qr)" 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

531 

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

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

533 
"divmod_nat_rel m n (divmod_nat m n)" 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

534 
proof  
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

535 
from divmod_nat_rel_ex 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

536 
obtain qr where rel: "divmod_nat_rel m n qr" . 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

537 
then show ?thesis 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

538 
by (auto simp add: divmod_nat_def intro: theI elim: divmod_nat_rel_unique) 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

539 
qed 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

540 

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

541 
lemma divmod_nat_eq: 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

542 
assumes "divmod_nat_rel m n qr" 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

543 
shows "divmod_nat m n = qr" 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

544 
using assms by (auto intro: divmod_nat_rel_unique divmod_nat_rel_divmod_nat) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

545 

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

546 
definition div_nat where 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

547 
"m div n = fst (divmod_nat m n)" 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

548 

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

549 
definition mod_nat where 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

550 
"m mod n = snd (divmod_nat m n)" 
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25162
diff
changeset

551 

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

552 
lemma divmod_nat_div_mod: 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

553 
"divmod_nat m n = (m div n, m mod n)" 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

554 
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

555 

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

556 
lemma div_eq: 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

557 
assumes "divmod_nat_rel m n (q, r)" 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

558 
shows "m div n = q" 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

559 
using assms by (auto dest: divmod_nat_eq simp add: divmod_nat_div_mod) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

560 

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

561 
lemma mod_eq: 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

562 
assumes "divmod_nat_rel m n (q, r)" 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

563 
shows "m mod n = r" 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

564 
using assms by (auto dest: divmod_nat_eq simp add: divmod_nat_div_mod) 
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25162
diff
changeset

565 

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

566 
lemma divmod_nat_rel: "divmod_nat_rel m n (m div n, m mod n)" 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

567 
by (simp add: div_nat_def mod_nat_def divmod_nat_rel_divmod_nat) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

568 

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

569 
lemma divmod_nat_zero: 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

570 
"divmod_nat m 0 = (0, m)" 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

571 
proof  
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

572 
from divmod_nat_rel [of m 0] show ?thesis 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

573 
unfolding divmod_nat_div_mod divmod_nat_rel_def by simp 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

574 
qed 
25942  575 

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

576 
lemma divmod_nat_base: 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

577 
assumes "m < n" 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

578 
shows "divmod_nat m n = (0, m)" 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

579 
proof  
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

580 
from divmod_nat_rel [of m n] show ?thesis 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

581 
unfolding divmod_nat_div_mod divmod_nat_rel_def 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

582 
using assms by (cases "m div n = 0") 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

583 
(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

584 
qed 
25942  585 

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

586 
lemma divmod_nat_step: 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

587 
assumes "0 < n" and "n \<le> m" 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

588 
shows "divmod_nat m n = (Suc ((m  n) div n), (m  n) mod n)" 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

589 
proof  
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

590 
from divmod_nat_rel have divmod_nat_m_n: "divmod_nat_rel m n (m div n, m mod n)" . 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

591 
with assms have m_div_n: "m div n \<ge> 1" 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

592 
by (cases "m div n") (auto simp add: divmod_nat_rel_def) 
35815
10e723e54076
tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents:
35673
diff
changeset

593 
have "divmod_nat_rel (m  n) n (m div n  Suc 0, m mod n)" 
10e723e54076
tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents:
35673
diff
changeset

594 
proof  
10e723e54076
tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents:
35673
diff
changeset

595 
from assms have 
10e723e54076
tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents:
35673
diff
changeset

596 
"n \<noteq> 0" 
10e723e54076
tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents:
35673
diff
changeset

597 
"\<And>k. m = Suc k * n + m mod n ==> m  n = (Suc k  Suc 0) * n + m mod n" 
10e723e54076
tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents:
35673
diff
changeset

598 
by simp_all 
10e723e54076
tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents:
35673
diff
changeset

599 
then show ?thesis using assms divmod_nat_m_n 
10e723e54076
tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents:
35673
diff
changeset

600 
by (cases "m div n") 
10e723e54076
tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents:
35673
diff
changeset

601 
(simp_all only: divmod_nat_rel_def fst_conv snd_conv, simp_all) 
10e723e54076
tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents:
35673
diff
changeset

602 
qed 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

603 
with divmod_nat_eq have "divmod_nat (m  n) n = (m div n  Suc 0, m mod n)" by simp 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

604 
moreover from divmod_nat_div_mod have "divmod_nat (m  n) n = ((m  n) div n, (m  n) mod n)" . 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

605 
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

606 
and "m mod n = (m  n) mod n" using m_div_n by simp_all 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

607 
then show ?thesis using divmod_nat_div_mod by simp 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

608 
qed 
25942  609 

26300  610 
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

611 

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

612 
lemma div_less [simp]: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

613 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

614 
assumes "m < n" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

615 
shows "m div n = 0" 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

616 
using assms divmod_nat_base divmod_nat_div_mod by simp 
25942  617 

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

618 
lemma le_div_geq: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

619 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

620 
assumes "0 < n" and "n \<le> m" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

621 
shows "m div n = Suc ((m  n) div n)" 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

622 
using assms divmod_nat_step divmod_nat_div_mod by simp 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

623 

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

624 
lemma mod_less [simp]: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

625 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

626 
assumes "m < n" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

627 
shows "m mod n = m" 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

628 
using assms divmod_nat_base divmod_nat_div_mod by simp 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

629 

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

630 
lemma le_mod_geq: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

631 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

632 
assumes "n \<le> m" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

633 
shows "m mod n = (m  n) mod n" 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

634 
using assms divmod_nat_step divmod_nat_div_mod by (cases "n = 0") simp_all 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

635 

30930  636 
instance proof  
637 
have [simp]: "\<And>n::nat. n div 0 = 0" 

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

638 
by (simp add: div_nat_def divmod_nat_zero) 
30930  639 
have [simp]: "\<And>n::nat. 0 div n = 0" 
640 
proof  

641 
fix n :: nat 

642 
show "0 div n = 0" 

643 
by (cases "n = 0") simp_all 

644 
qed 

645 
show "OFCLASS(nat, semiring_div_class)" proof 

646 
fix m n :: nat 

647 
show "m div n * n + m mod n = m" 

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

648 
using divmod_nat_rel [of m n] by (simp add: divmod_nat_rel_def) 
30930  649 
next 
650 
fix m n q :: nat 

651 
assume "n \<noteq> 0" 

652 
then show "(q + m * n) div n = m + q div n" 

653 
by (induct m) (simp_all add: le_div_geq) 

654 
next 

655 
fix m n q :: nat 

656 
assume "m \<noteq> 0" 

657 
then show "(m * n) div (m * q) = n div q" 

658 
proof (cases "n \<noteq> 0 \<and> q \<noteq> 0") 

659 
case False then show ?thesis by auto 

660 
next 

661 
case True with `m \<noteq> 0` 

662 
have "m > 0" and "n > 0" and "q > 0" by auto 

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

663 
then have "\<And>a b. divmod_nat_rel n q (a, b) \<Longrightarrow> divmod_nat_rel (m * n) (m * q) (a, m * b)" 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

664 
by (auto simp add: divmod_nat_rel_def) (simp_all add: algebra_simps) 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

665 
moreover from divmod_nat_rel have "divmod_nat_rel n q (n div q, n mod q)" . 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

666 
ultimately have "divmod_nat_rel (m * n) (m * q) (n div q, m * (n mod q))" . 
30930  667 
then show ?thesis by (simp add: div_eq) 
668 
qed 

669 
qed simp_all 

25942  670 
qed 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

671 

25942  672 
end 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

673 

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

674 
lemma divmod_nat_if [code]: "divmod_nat m n = (if n = 0 \<or> m < n then (0, m) else 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

675 
let (q, r) = divmod_nat (m  n) n in (Suc q, r))" 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

676 
by (simp add: divmod_nat_zero divmod_nat_base divmod_nat_step) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

677 
(simp add: divmod_nat_div_mod) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

678 

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

679 
text {* Simproc for cancelling @{const div} and @{const mod} *} 
25942  680 

30934  681 
ML {* 
43594  682 
structure Cancel_Div_Mod_Nat = Cancel_Div_Mod 
41550  683 
( 
30934  684 
val div_name = @{const_name div}; 
685 
val mod_name = @{const_name mod}; 

686 
val mk_binop = HOLogic.mk_binop; 

687 
val mk_sum = Nat_Arith.mk_sum; 

688 
val dest_sum = Nat_Arith.dest_sum; 

25942  689 

30934  690 
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

691 

30934  692 
val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac 
35050
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
34982
diff
changeset

693 
(@{thm add_0_left} :: @{thm add_0_right} :: @{thms add_ac})) 
41550  694 
) 
25942  695 
*} 
696 

43594  697 
simproc_setup cancel_div_mod_nat ("(m::nat) + n") = {* K Cancel_Div_Mod_Nat.proc *} 
698 

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

699 

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

700 
subsubsection {* Quotient *} 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

701 

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

702 
lemma div_geq: "0 < n \<Longrightarrow> \<not> m < n \<Longrightarrow> m div n = Suc ((m  n) div n)" 
29667  703 
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

704 

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

705 
lemma div_if: "0 < n \<Longrightarrow> m div n = (if m < n then 0 else Suc ((m  n) div n))" 
29667  706 
by (simp add: div_geq) 
26100
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 div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)" 
29667  709 
by simp 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

710 

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

711 
lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)" 
29667  712 
by simp 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

713 

25942  714 

715 
subsubsection {* Remainder *} 

716 

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

717 
lemma mod_less_divisor [simp]: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

718 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

719 
assumes "n > 0" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

720 
shows "m mod n < (n::nat)" 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

721 
using assms divmod_nat_rel [of m n] unfolding divmod_nat_rel_def by auto 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

722 

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

723 
lemma mod_less_eq_dividend [simp]: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

724 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

725 
shows "m mod n \<le> m" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

726 
proof (rule add_leD2) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

727 
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

728 
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

729 
qed 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

730 

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

731 
lemma mod_geq: "\<not> m < (n\<Colon>nat) \<Longrightarrow> m mod n = (m  n) mod n" 
29667  732 
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

733 

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

734 
lemma mod_if: "m mod (n\<Colon>nat) = (if m < n then m else (m  n) mod n)" 
29667  735 
by (simp add: le_mod_geq) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

736 

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

737 
lemma mod_1 [simp]: "m mod Suc 0 = 0" 
29667  738 
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

739 

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

740 
lemma mod_mult_distrib: "(m mod n) * (k\<Colon>nat) = (m * k) mod (n * k)" 
22718  741 
apply (cases "n = 0", simp) 
742 
apply (cases "k = 0", simp) 

743 
apply (induct m rule: nat_less_induct) 

744 
apply (subst mod_if, simp) 

745 
apply (simp add: mod_geq diff_mult_distrib) 

746 
done 

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

747 

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

748 
lemma mod_mult_distrib2: "(k::nat) * (m mod n) = (k*m) mod (k*n)" 
29667  749 
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

750 

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

751 
(* 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

752 
lemma mult_div_cancel: "(n::nat) * (m div n) = m  (m mod n)" 
29667  753 
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

754 

15439  755 
lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)" 
22718  756 
apply (drule mod_less_divisor [where m = m]) 
757 
apply simp 

758 
done 

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

759 

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

760 
subsubsection {* Quotient and Remainder *} 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

761 

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

762 
lemma divmod_nat_rel_mult1_eq: 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

763 
"divmod_nat_rel b c (q, r) \<Longrightarrow> c > 0 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

764 
\<Longrightarrow> divmod_nat_rel (a * b) c (a * q + a * r div c, a * r mod c)" 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

765 
by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

766 

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

767 
lemma div_mult1_eq: 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

768 
"(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

769 
apply (cases "c = 0", simp) 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

770 
apply (blast intro: divmod_nat_rel [THEN divmod_nat_rel_mult1_eq, THEN div_eq]) 
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

771 
done 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

772 

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

773 
lemma divmod_nat_rel_add1_eq: 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

774 
"divmod_nat_rel a c (aq, ar) \<Longrightarrow> divmod_nat_rel b c (bq, br) \<Longrightarrow> c > 0 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

775 
\<Longrightarrow> divmod_nat_rel (a + b) c (aq + bq + (ar + br) div c, (ar + br) mod c)" 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

776 
by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

777 

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

778 
(*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

779 
lemma div_add1_eq: 
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

780 
"(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

781 
apply (cases "c = 0", simp) 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

782 
apply (blast intro: divmod_nat_rel_add1_eq [THEN div_eq] divmod_nat_rel) 
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

783 
done 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

784 

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

785 
lemma mod_lemma: "[ (0::nat) < c; r < b ] ==> b * (q mod c) + r < b * c" 
22718  786 
apply (cut_tac m = q and n = c in mod_less_divisor) 
787 
apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto) 

788 
apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst) 

789 
apply (simp add: add_mult_distrib2) 

790 
done 

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

791 

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

792 
lemma divmod_nat_rel_mult2_eq: 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

793 
"divmod_nat_rel a b (q, r) \<Longrightarrow> 0 < b \<Longrightarrow> 0 < c 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

794 
\<Longrightarrow> divmod_nat_rel a (b * c) (q div c, b *(q mod c) + r)" 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

795 
by (auto simp add: mult_ac divmod_nat_rel_def add_mult_distrib2 [symmetric] mod_lemma) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

796 

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

797 
lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)" 
22718  798 
apply (cases "b = 0", simp) 
799 
apply (cases "c = 0", simp) 

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

800 
apply (force simp add: divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN div_eq]) 
22718  801 
done 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

802 

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

803 
lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)" 
22718  804 
apply (cases "b = 0", simp) 
805 
apply (cases "c = 0", simp) 

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

806 
apply (auto simp add: mult_commute divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN mod_eq]) 
22718  807 
done 
14267
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 

25942  810 
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

811 

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

812 
lemma div_1 [simp]: "m div Suc 0 = m" 
29667  813 
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

814 

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

815 

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

816 
(* Monotonicity of div in first argument *) 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

817 
lemma div_le_mono [rule_format (no_asm)]: 
22718  818 
"\<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

819 
apply (case_tac "k=0", simp) 
15251  820 
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

821 
apply (case_tac "n<k") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

822 
(* 1 case n<k *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

823 
apply simp 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

824 
(* 2 case n >= k *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

825 
apply (case_tac "m<k") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

826 
(* 2.1 case m<k *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

827 
apply simp 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

828 
(* 2.2 case m>=k *) 
15439  829 
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

830 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

831 

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

832 
(* Antimonotonicity of div in second argument *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

833 
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

834 
apply (subgoal_tac "0<n") 
22718  835 
prefer 2 apply simp 
15251  836 
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

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

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

839 
apply (subgoal_tac "~ (k<m) ") 
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) 
15251  842 
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

843 
prefer 2 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

844 
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

845 
apply (rule le_trans, simp) 
15439  846 
apply (simp) 
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 

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

849 
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

850 
apply (case_tac "n=0", simp) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

851 
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

852 
apply (rule div_le_mono2) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

853 
apply (simp_all (no_asm_simp)) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

854 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

855 

22718  856 
(* Similar for "less than" *) 
17085  857 
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

858 
"!!n::nat. 1<n ==> 0 < m > m div n < m" 
15251  859 
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

860 
apply (rename_tac "m") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

861 
apply (case_tac "m<n", simp) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

862 
apply (subgoal_tac "0<n") 
22718  863 
prefer 2 apply simp 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

864 
apply (simp add: div_geq) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

865 
apply (case_tac "n<m") 
15251  866 
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

867 
apply (rule impI less_trans_Suc)+ 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

868 
apply assumption 
15439  869 
apply (simp_all) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

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

871 

17085  872 
declare div_less_dividend [simp] 
873 

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

874 
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

875 
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

876 
apply (case_tac "n=0", simp) 
15251  877 
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

878 
apply (case_tac "Suc (na) <n") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

879 
(* case Suc(na) < n *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

880 
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

881 
(* case n \<le> Suc(na) *) 
16796  882 
apply (simp add: linorder_not_less le_Suc_eq mod_geq) 
15439  883 
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

884 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

885 

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

886 
lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)" 
29667  887 
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

888 

22718  889 
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

890 

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

891 
(*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

892 
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

893 
apply (cut_tac a = m in mod_div_equality) 
22718  894 
apply (simp only: add_ac) 
895 
apply (blast intro: sym) 

896 
done 

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

897 

13152  898 
lemma split_div: 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

899 
"P(n div k :: nat) = 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

900 
((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

901 
(is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))") 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

902 
proof 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

903 
assume P: ?P 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

904 
show ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

905 
proof (cases) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

906 
assume "k = 0" 
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

907 
with P show ?Q by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

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

909 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

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

911 
proof (simp, intro allI impI) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

912 
fix i j 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

913 
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

914 
show "P i" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

915 
proof (cases) 
22718  916 
assume "i = 0" 
917 
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

918 
next 
22718  919 
assume "i \<noteq> 0" 
920 
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

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

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

923 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

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

925 
assume Q: ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

926 
show ?P 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

927 
proof (cases) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

928 
assume "k = 0" 
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

929 
with Q show ?P by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

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

931 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

932 
with Q have R: ?R by simp 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

933 
from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"] 
13517  934 
show ?P by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

935 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

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

937 

13882  938 
lemma split_div_lemma: 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

939 
assumes "0 < n" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

940 
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

941 
proof 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

942 
assume ?rhs 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

943 
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

944 
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

945 
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

946 
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

947 
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

948 
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

949 
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

950 
from A B show ?lhs .. 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

951 
next 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

952 
assume P: ?lhs 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

953 
then have "divmod_nat_rel m n (q, m  n * q)" 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

954 
unfolding divmod_nat_rel_def by (auto simp add: mult_ac) 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

955 
with divmod_nat_rel_unique divmod_nat_rel [of m n] 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

956 
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

957 
then show ?rhs by simp 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

958 
qed 
13882  959 

960 
theorem split_div': 

961 
"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

962 
(\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))" 
13882  963 
apply (case_tac "0 < n") 
964 
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

965 
apply simp_all 
13882  966 
done 
967 

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

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

969 
"P(n mod k :: nat) = 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

970 
((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

971 
(is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))") 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

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

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

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

975 
proof (cases) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

976 
assume "k = 0" 
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

977 
with P show ?Q by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

978 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

979 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

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

981 
proof (simp, intro allI impI) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

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

983 
assume "n = k*i + j" "j < k" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

984 
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

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

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

987 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

988 
assume Q: ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

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

990 
proof (cases) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

991 
assume "k = 0" 
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

992 
with Q show ?P by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

993 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

994 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

995 
with Q have R: ?R by simp 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

996 
from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"] 
13517  997 
show ?P by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

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

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

1000 

13882  1001 
theorem mod_div_equality': "(m::nat) mod n = m  (m div n) * n" 
1002 
apply (rule_tac P="%x. m mod n = x  (m div n) * n" in 

1003 
subst [OF mod_div_equality [of _ n]]) 

1004 
apply arith 

1005 
done 

1006 

22800  1007 
lemma div_mod_equality': 
1008 
fixes m n :: nat 

1009 
shows "m div n * n = m  m mod n" 

1010 
proof  

1011 
have "m mod n \<le> m mod n" .. 

1012 
from div_mod_equality have 

1013 
"m div n * n + m mod n  m mod n = m  m mod n" by simp 

1014 
with diff_add_assoc [OF `m mod n \<le> m mod n`, of "m div n * n"] have 

1015 
"m div n * n + (m mod n  m mod n) = m  m mod n" 

1016 
by simp 

1017 
then show ?thesis by simp 

1018 
qed 

1019 

1020 

25942  1021 
subsubsection {*An ``induction'' law for modulus arithmetic.*} 
14640  1022 

1023 
lemma mod_induct_0: 

1024 
assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)" 

1025 
and base: "P i" and i: "i<p" 

1026 
shows "P 0" 

1027 
proof (rule ccontr) 

1028 
assume contra: "\<not>(P 0)" 

1029 
from i have p: "0<p" by simp 

1030 
have "\<forall>k. 0<k \<longrightarrow> \<not> P (pk)" (is "\<forall>k. ?A k") 

1031 
proof 

1032 
fix k 

1033 
show "?A k" 

1034 
proof (induct k) 

1035 
show "?A 0" by simp  "by contradiction" 

1036 
next 

1037 
fix n 

1038 
assume ih: "?A n" 

1039 
show "?A (Suc n)" 

1040 
proof (clarsimp) 

22718  1041 
assume y: "P (p  Suc n)" 
1042 
have n: "Suc n < p" 

1043 
proof (rule ccontr) 

1044 
assume "\<not>(Suc n < p)" 

1045 
hence "p  Suc n = 0" 

1046 
by simp 

1047 
with y contra show "False" 

1048 
by simp 

1049 
qed 

1050 
hence n2: "Suc (p  Suc n) = pn" by arith 

1051 
from p have "p  Suc n < p" by arith 

1052 
with y step have z: "P ((Suc (p  Suc n)) mod p)" 

1053 
by blast 

1054 
show "False" 

1055 
proof (cases "n=0") 

1056 
case True 

1057 
with z n2 contra show ?thesis by simp 

1058 
next 

1059 
case False 

1060 
with p have "pn < p" by arith 

1061 
with z n2 False ih show ?thesis by simp 

1062 
qed 

14640  1063 
qed 
1064 
qed 

1065 
qed 

1066 
moreover 

1067 
from i obtain k where "0<k \<and> i+k=p" 

1068 
by (blast dest: less_imp_add_positive) 

1069 
hence "0<k \<and> i=pk" by auto 

1070 
moreover 

1071 
note base 

1072 
ultimately 

1073 
show "False" by blast 

1074 
qed 

1075 

1076 
lemma mod_induct: 

1077 
assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)" 

1078 
and base: "P i" and i: "i<p" and j: "j<p" 

1079 
shows "P j" 

1080 
proof  

1081 
have "\<forall>j<p. P j" 

1082 
proof 

1083 
fix j 

1084 
show "j<p \<longrightarrow> P j" (is "?A j") 

1085 
proof (induct j) 

1086 
from step base i show "?A 0" 

22718  1087 
by (auto elim: mod_induct_0) 
14640  1088 
next 
1089 
fix k 

1090 
assume ih: "?A k" 

1091 
show "?A (Suc k)" 

1092 
proof 

22718  1093 
assume suc: "Suc k < p" 
1094 
hence k: "k<p" by simp 

1095 
with ih have "P k" .. 

1096 
with step k have "P (Suc k mod p)" 

1097 
by blast 

1098 
moreover 

1099 
from suc have "Suc k mod p = Suc k" 

1100 
by simp 

1101 
ultimately 

1102 
show "P (Suc k)" by simp 

14640  1103 
qed 
1104 
qed 

1105 
qed 

1106 
with j show ?thesis by blast 

1107 
qed 

1108 

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

1109 
lemma div2_Suc_Suc [simp]: "Suc (Suc m) div 2 = Suc (m div 2)" 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1110 
by (auto simp add: numeral_2_eq_2 le_div_geq) 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1111 

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

1112 
lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)" 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1113 
by (simp add: nat_mult_2 [symmetric]) 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1114 

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

1115 
lemma mod2_Suc_Suc [simp]: "Suc(Suc(m)) mod 2 = m mod 2" 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1116 
apply (subgoal_tac "m mod 2 < 2") 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1117 
apply (erule less_2_cases [THEN disjE]) 
35216  1118 
apply (simp_all (no_asm_simp) add: Let_def mod_Suc) 
33296
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1119 
done 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1120 

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

1121 
lemma mod2_gr_0 [simp]: "0 < (m\<Colon>nat) mod 2 \<longleftrightarrow> m mod 2 = 1" 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1122 
proof  
35815
10e723e54076
tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents:
35673
diff
changeset

1123 
{ fix n :: nat have "(n::nat) < 2 \<Longrightarrow> n = 0 \<or> n = 1" by (cases n) simp_all } 
33296
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1124 
moreover have "m mod 2 < 2" by simp 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1125 
ultimately have "m mod 2 = 0 \<or> m mod 2 = 1" . 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1126 
then show ?thesis by auto 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1127 
qed 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1128 

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

1129 
text{*These lemmas collapse some needless occurrences of Suc: 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1130 
at least three Sucs, since two and fewer are rewritten back to Suc again! 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1131 
We already have some rules to simplify operands smaller than 3.*} 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1132 

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

1133 
lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)" 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1134 
by (simp add: Suc3_eq_add_3) 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1135 

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

1136 
lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)" 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1137 
by (simp add: Suc3_eq_add_3) 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1138 

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

1139 
lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n" 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1140 
by (simp add: Suc3_eq_add_3) 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1141 

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

1142 
lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n" 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1143 
by (simp add: Suc3_eq_add_3) 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1144 

45607  1145 
lemmas Suc_div_eq_add3_div_number_of [simp] = Suc_div_eq_add3_div [of _ "number_of v"] for v 
1146 
lemmas Suc_mod_eq_add3_mod_number_of [simp] = Suc_mod_eq_add3_mod [of _ "number_of v"] for v 

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

1147 

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

1148 

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

1149 
lemma Suc_times_mod_eq: "1<k ==> Suc (k * m) mod k = 1" 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1150 
apply (induct "m") 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1151 
apply (simp_all add: mod_Suc) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1152 
done 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1153 

45607  1154 
declare Suc_times_mod_eq [of "number_of w", simp] for w 
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1155 

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

1156 
lemma [simp]: "n div k \<le> (Suc n) div k" 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1157 
by (simp add: div_le_mono) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1158 

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

1159 
lemma Suc_n_div_2_gt_zero [simp]: "(0::nat) < n ==> 0 < (n + 1) div 2" 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1160 
by (cases n) simp_all 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1161 

35815
10e723e54076
tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents:
35673
diff
changeset

1162 
lemma div_2_gt_zero [simp]: assumes A: "(1::nat) < n" shows "0 < n div 2" 
10e723e54076
tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents:
35673
diff
changeset

1163 
proof  
10e723e54076
tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents:
35673
diff
changeset

1164 
from A have B: "0 < n  1" and C: "n  1 + 1 = n" by simp_all 
10e723e54076
tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents:
35673
diff
changeset

1165 
from Suc_n_div_2_gt_zero [OF B] C show ?thesis by simp 
10e723e54076
tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents:
35673
diff
changeset

1166 
qed 
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1167 

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

1168 
(* Potential use of algebra : Equality modulo n*) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1169 
lemma mod_mult_self3 [simp]: "(k*n + m) mod n = m mod (n::nat)" 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1170 
by (simp add: mult_ac add_ac) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1171 

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

1172 
lemma mod_mult_self4 [simp]: "Suc (k*n + m) mod n = Suc m mod n" 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1173 
proof  
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1174 
have "Suc (k * n + m) mod n = (k * n + Suc m) mod n" by simp 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1175 
also have "... = Suc m mod n" by (rule mod_mult_self3) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1176 
finally show ?thesis . 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1177 
qed 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1178 

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

1179 
lemma mod_Suc_eq_Suc_mod: "Suc m mod n = Suc (m mod n) mod n" 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1180 
apply (subst mod_Suc [of m]) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1181 
apply (subst mod_Suc [of "m mod n"], simp) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1182 
done 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1183 

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

1184 

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

1185 
subsection {* Division on @{typ int} *} 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1186 

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

1187 
definition divmod_int_rel :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool" where 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1188 
{*definition of quotient and remainder*} 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1189 
[code]: "divmod_int_rel a b = (\<lambda>(q, r). a = b * q + r \<and> 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1190 
(if 0 < b then 0 \<le> r \<and> r < b else b < r \<and> r \<le> 0))" 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1191 

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

1192 
definition adjust :: "int \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int" where 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1193 
{*for the division algorithm*} 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1194 
[code]: "adjust b = (\<lambda>(q, r). if 0 \<le> r  b then (2 * q + 1, r  b) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1195 
else (2 * q, r))" 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1196 

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

1197 
text{*algorithm for the case @{text "a\<ge>0, b>0"}*} 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1198 
function posDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1199 
"posDivAlg a b = (if a < b \<or> b \<le> 0 then (0, a) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1200 
else adjust b (posDivAlg a (2 * b)))" 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1201 
by auto 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1202 
termination by (relation "measure (\<lambda>(a, b). nat (a  b + 1))") 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1203 
(auto simp add: mult_2) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1204 

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

1205 
text{*algorithm for the case @{text "a<0, b>0"}*} 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1206 
function negDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1207 
"negDivAlg a b = (if 0 \<le>a + b \<or> b \<le> 0 then (1, a + b) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1208 
else adjust b (negDivAlg a (2 * b)))" 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1209 
by auto 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1210 
termination by (relation "measure (\<lambda>(a, b). nat ( a  b))") 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1211 
(auto simp add: mult_2) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1212 

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

1213 
text{*algorithm for the general case @{term "b\<noteq>0"}*} 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1214 
definition negateSnd :: "int \<times> int \<Rightarrow> int \<times> int" where 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1215 
[code_unfold]: "negateSnd = apsnd uminus" 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1216 

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

1217 
definition divmod_int :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1218 
{*The full division algorithm considers all possible signs for a, b 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1219 
including the special case @{text "a=0, b<0"} because 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1220 
@{term negDivAlg} requires @{term "a<0"}.*} 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1221 
"divmod_int a b = (if 0 \<le> a then if 0 \<le> b then posDivAlg a b 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1222 
else if a = 0 then (0, 0) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1223 
else negateSnd (negDivAlg (a) (b)) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1224 
else 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1225 
if 0 < b then negDivAlg a b 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1226 
else negateSnd (posDivAlg (a) (b)))" 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1227 

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

1228 
instantiation int :: Divides.div 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1229 
begin 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1230 

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

1231 
definition 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1232 
"a div b = fst (divmod_int a b)" 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1233 

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

1234 
definition 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1235 
"a mod b = snd (divmod_int a b)" 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1236 

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

1237 
instance .. 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1238 

3366  1239 
end 
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1240 

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

1241 
lemma divmod_int_mod_div: 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1242 
"divmod_int p q = (p div q, p mod q)" 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1243 
by (auto simp add: div_int_def mod_int_def) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1244 

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

1245 
text{* 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1246 
Here is the division algorithm in ML: 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1247 

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

1248 
\begin{verbatim} 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1249 
fun posDivAlg (a,b) = 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1250 
if a<b then (0,a) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1251 
else let val (q,r) = posDivAlg(a, 2*b) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1252 
in if 0\<le>rb then (2*q+1, rb) else (2*q, r) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1253 
end 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1254 

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

1255 
fun negDivAlg (a,b) = 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1256 
if 0\<le>a+b then (~1,a+b) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1257 
else let val (q,r) = negDivAlg(a, 2*b) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1258 
in if 0\<le>rb then (2*q+1, rb) else (2*q, r) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1259 
end; 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1260 

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

1261 
fun negateSnd (q,r:int) = (q,~r); 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1262 

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

1263 
fun divmod (a,b) = if 0\<le>a then 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1264 
if b>0 then posDivAlg (a,b) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1265 
else if a=0 then (0,0) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1266 
else negateSnd (negDivAlg (~a,~b)) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1267 
else 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1268 
if 0<b then negDivAlg (a,b) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1269 
else negateSnd (posDivAlg (~a,~b)); 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1270 
\end{verbatim} 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1271 
*} 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1272 

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

1273 

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

1274 

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

1275 
subsubsection{*Uniqueness and Monotonicity of Quotients and Remainders*} 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1276 

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

1277 
lemma unique_quotient_lemma: 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1278 
"[ b*q' + r' \<le> b*q + r; 0 \<le> r'; r' < b; r < b ] 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1279 
==> q' \<le> (q::int)" 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1280 
apply (subgoal_tac "r' + b * (q'q) \<le> r") 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1281 
prefer 2 apply (simp add: right_diff_distrib) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1282 
apply (subgoal_tac "0 < b * (1 + q  q') ") 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1283 
apply (erule_tac [2] order_le_less_trans) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1284 
prefer 2 apply (simp add: right_diff_distrib right_distrib) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1285 
apply (subgoal_tac "b * q' < b * (1 + q) ") 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1286 
prefer 2 apply (simp add: right_diff_distrib right_distrib) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1287 
apply (simp add: mult_less_cancel_left) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1288 
done 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1289 

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

1290 
lemma unique_quotient_lemma_neg: 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1291 
"[ b*q' + r' \<le> b*q + r; r \<le> 0; b < r; b < r' ] 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1292 
==> q \<le> (q'::int)" 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1293 
by (rule_tac b = "b" and r = "r'" and r' = "r" in unique_quotient_lemma, 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1294 
auto) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1295 

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

1296 
lemma unique_quotient: 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1297 
"[ divmod_int_rel a b (q, r); divmod_int_rel a b (q', r'); b \<noteq> 0 ] 
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1298 
==> q = q'" 
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combined former theories Divides and IntDiv to one theory Divides
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1299 
apply (simp add: divmod_int_rel_def linorder_neq_iff split: split_if_asm) 
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combined former theories Divides and IntDiv to one theory Divides
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1300 
apply (blast intro: order_antisym 
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1301 
dest: order_eq_refl [THEN unique_quotient_lemma] 
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1302 
order_eq_refl [THEN unique_quotient_lemma_neg] sym)+ 
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changeset

1303 
done 
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1304 

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1305 

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combined former theories Divides and IntDiv to one theory Divides
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1306 
lemma unique_remainder: 