author  huffman 
Tue, 27 Mar 2012 15:27:49 +0200  
changeset 47159  978c00c20a59 
parent 47142  d64fa2ca54b8 
child 47160  8ada79014cb2 
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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27676  119 
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 

47159  346 
lemma mult_mod_left: "(a mod b) * c = (a * c) mod (b * c)" 
347 
by (fact mod_mult_mult2 [symmetric]) 

348 

349 
lemma mult_mod_right: "c * (a mod b) = (c * a) mod (c * b)" 

350 
by (fact mod_mult_mult1 [symmetric]) 

351 

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

352 
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:
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diff
changeset

353 
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

354 

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

355 
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

356 
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

357 

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

358 
lemma div_power: 
31661
1e252b8b2334
move lemma div_power into semiring_div context; class ring_div inherits from idom
huffman
parents:
31009
diff
changeset

359 
"y dvd x \<Longrightarrow> (x div y) ^ n = x ^ n div y ^ n" 
30476  360 
apply (induct n) 
361 
apply simp 

362 
apply(simp add: div_mult_div_if_dvd dvd_power_same) 

363 
done 

364 

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

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

366 
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

367 
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

368 
proof 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset

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

370 
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

371 
next 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset

372 
assume "b div a = c" 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset

373 
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

374 
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

375 
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

376 
qed 
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset

377 

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

378 
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

379 
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

380 
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

381 
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

382 

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

383 
end 
1e252b8b2334
move lemma div_power into semiring_div context; class ring_div inherits from idom
huffman
parents:
31009
diff
changeset

384 

35673  385 
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

386 
begin 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

387 

36634  388 
subclass ring_1_no_zero_divisors .. 
389 

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

390 
text {* Negation respects modular equivalence. *} 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

391 

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

392 
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

393 
proof  
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

394 
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

395 
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

396 
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

397 
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

398 
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

399 
by (rule mod_mult_self1) 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

400 
finally show ?thesis . 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

401 
qed 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

402 

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

403 
lemma mod_minus_cong: 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

404 
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

405 
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

406 
proof  
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

407 
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

408 
unfolding assms .. 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

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

410 
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

411 
qed 
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 
text {* Subtraction respects modular equivalence. *} 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

414 

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

415 
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

416 
unfolding diff_minus 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

417 
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

418 

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

419 
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

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

421 
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

422 

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

423 
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

424 
unfolding diff_minus 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

425 
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

426 

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

427 
lemma mod_diff_cong: 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

428 
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

429 
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

430 
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

431 
unfolding diff_minus using assms 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

432 
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

433 

30180  434 
lemma dvd_neg_div: "y dvd x \<Longrightarrow> x div y =  (x div y)" 
435 
apply (case_tac "y = 0") apply simp 

436 
apply (auto simp add: dvd_def) 

437 
apply (subgoal_tac "(y * k) = y *  k") 

438 
apply (erule ssubst) 

439 
apply (erule div_mult_self1_is_id) 

440 
apply simp 

441 
done 

442 

443 
lemma dvd_div_neg: "y dvd x \<Longrightarrow> x div y =  (x div y)" 

444 
apply (case_tac "y = 0") apply simp 

445 
apply (auto simp add: dvd_def) 

446 
apply (subgoal_tac "y * k = y * k") 

447 
apply (erule ssubst) 

448 
apply (rule div_mult_self1_is_id) 

449 
apply simp 

450 
apply simp 

451 
done 

452 

47159  453 
lemma div_minus_minus [simp]: "(a) div (b) = a div b" 
454 
using div_mult_mult1 [of " 1" a b] 

455 
unfolding neg_equal_0_iff_equal by simp 

456 

457 
lemma mod_minus_minus [simp]: "(a) mod (b) =  (a mod b)" 

458 
using mod_mult_mult1 [of " 1" a b] by simp 

459 

460 
lemma div_minus_right: "a div (b) = (a) div b" 

461 
using div_minus_minus [of "a" b] by simp 

462 

463 
lemma mod_minus_right: "a mod (b) =  ((a) mod b)" 

464 
using mod_minus_minus [of "a" b] by simp 

465 

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

466 
end 
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset

467 

25942  468 

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

469 
subsection {* Division on @{typ nat} *} 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

470 

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

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

472 
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

473 
of a characteristic relation with two input arguments 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

474 
@{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

475 
@{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

476 
*} 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

477 

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

478 
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

479 
"divmod_nat_rel m n qr \<longleftrightarrow> 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

480 
m = fst qr * n + snd qr \<and> 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

481 
(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

482 

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

483 
text {* @{const divmod_nat_rel} is total: *} 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

484 

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

485 
lemma divmod_nat_rel_ex: 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

486 
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

487 
proof (cases "n = 0") 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

488 
case True with that show thesis 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

489 
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

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

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

492 
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

493 
proof (induct m) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

494 
case 0 with `n \<noteq> 0` 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

495 
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

496 
then show ?case by blast 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

497 
next 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

498 
case (Suc m) then obtain q' r' 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

499 
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

500 
then show ?case proof (cases "Suc r' < n") 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

501 
case True 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

502 
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

503 
with True show ?thesis by blast 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

504 
next 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

505 
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

506 
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

507 
ultimately have "n = Suc r'" by auto 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

508 
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

509 
with `n \<noteq> 0` show ?thesis by blast 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

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

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

512 
with that show thesis 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

513 
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

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

515 

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

516 
text {* @{const divmod_nat_rel} is injective: *} 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

517 

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

518 
lemma divmod_nat_rel_unique: 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

519 
assumes "divmod_nat_rel m n qr" 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

520 
and "divmod_nat_rel m n qr'" 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

521 
shows "qr = qr'" 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

522 
proof (cases "n = 0") 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

523 
case True with assms show ?thesis 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

524 
by (cases qr, cases qr') 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

525 
(simp add: divmod_nat_rel_def) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

526 
next 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

527 
case False 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

528 
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

529 
apply (rule leI) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

530 
apply (subst less_iff_Suc_add) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

531 
apply (auto simp add: add_mult_distrib) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

532 
done 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

533 
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

534 
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

535 
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

536 
by (simp add: divmod_nat_rel_def) 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

537 
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

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

539 

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

540 
text {* 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

541 
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

542 
means of @{const divmod_nat_rel}: 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

543 
*} 
25942  544 

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

545 
definition divmod_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where 
37767  546 
"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

547 

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

548 
lemma divmod_nat_rel_divmod_nat: 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

549 
"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

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

551 
from divmod_nat_rel_ex 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

552 
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

553 
then show ?thesis 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

554 
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

555 
qed 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

556 

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

557 
lemma divmod_nat_unique: 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

558 
assumes "divmod_nat_rel m n qr" 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

559 
shows "divmod_nat m n = qr" 
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

560 
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

561 

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

562 
instantiation nat :: semiring_div 
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

563 
begin 
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

564 

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

565 
definition div_nat where 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

566 
"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

567 

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

568 
lemma fst_divmod_nat [simp]: 
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

569 
"fst (divmod_nat m n) = m div n" 
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

570 
by (simp add: div_nat_def) 
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

571 

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

572 
definition mod_nat where 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

573 
"m mod n = snd (divmod_nat m n)" 
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25162
diff
changeset

574 

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

575 
lemma snd_divmod_nat [simp]: 
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

576 
"snd (divmod_nat m n) = m mod n" 
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

577 
by (simp add: mod_nat_def) 
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

578 

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

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

580 
"divmod_nat m n = (m div n, m mod n)" 
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

581 
by (simp add: prod_eq_iff) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

582 

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

583 
lemma div_nat_unique: 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

584 
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

585 
shows "m div n = q" 
47135
fb67b596067f
rename lemmas {div,mod}_eq > {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents:
47134
diff
changeset

586 
using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff) 
fb67b596067f
rename lemmas {div,mod}_eq > {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents:
47134
diff
changeset

587 

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

588 
lemma mod_nat_unique: 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

589 
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

590 
shows "m mod n = r" 
47135
fb67b596067f
rename lemmas {div,mod}_eq > {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents:
47134
diff
changeset

591 
using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff) 
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25162
diff
changeset

592 

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

593 
lemma divmod_nat_rel: "divmod_nat_rel m n (m div n, m mod n)" 
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

594 
using divmod_nat_rel_divmod_nat by (simp add: divmod_nat_div_mod) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

595 

47136  596 
lemma divmod_nat_zero: "divmod_nat m 0 = (0, m)" 
597 
by (simp add: divmod_nat_unique divmod_nat_rel_def) 

598 

599 
lemma divmod_nat_zero_left: "divmod_nat 0 n = (0, 0)" 

600 
by (simp add: divmod_nat_unique divmod_nat_rel_def) 

25942  601 

47137  602 
lemma divmod_nat_base: "m < n \<Longrightarrow> divmod_nat m n = (0, m)" 
603 
by (simp add: divmod_nat_unique divmod_nat_rel_def) 

25942  604 

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

605 
lemma divmod_nat_step: 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

606 
assumes "0 < n" and "n \<le> m" 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

607 
shows "divmod_nat m n = (Suc ((m  n) div n), (m  n) mod n)" 
47135
fb67b596067f
rename lemmas {div,mod}_eq > {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents:
47134
diff
changeset

608 
proof (rule divmod_nat_unique) 
47134  609 
have "divmod_nat_rel (m  n) n ((m  n) div n, (m  n) mod n)" 
610 
by (rule divmod_nat_rel) 

611 
thus "divmod_nat_rel m n (Suc ((m  n) div n), (m  n) mod n)" 

612 
unfolding divmod_nat_rel_def using assms by auto 

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

613 
qed 
25942  614 

26300  615 
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

616 

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

617 
lemma div_less [simp]: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

618 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

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

620 
shows "m div n = 0" 
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

621 
using assms divmod_nat_base by (simp add: prod_eq_iff) 
25942  622 

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

623 
lemma le_div_geq: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

624 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

625 
assumes "0 < n" and "n \<le> m" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

626 
shows "m div n = Suc ((m  n) div n)" 
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

627 
using assms divmod_nat_step by (simp add: prod_eq_iff) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

628 

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

629 
lemma mod_less [simp]: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

630 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

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

632 
shows "m mod n = m" 
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

633 
using assms divmod_nat_base by (simp add: prod_eq_iff) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

634 

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

635 
lemma le_mod_geq: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

636 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

637 
assumes "n \<le> m" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

638 
shows "m mod n = (m  n) mod n" 
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

639 
using assms divmod_nat_step by (cases "n = 0") (simp_all add: prod_eq_iff) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

640 

47136  641 
instance proof 
642 
fix m n :: nat 

643 
show "m div n * n + m mod n = m" 

644 
using divmod_nat_rel [of m n] by (simp add: divmod_nat_rel_def) 

645 
next 

646 
fix m n q :: nat 

647 
assume "n \<noteq> 0" 

648 
then show "(q + m * n) div n = m + q div n" 

649 
by (induct m) (simp_all add: le_div_geq) 

650 
next 

651 
fix m n q :: nat 

652 
assume "m \<noteq> 0" 

653 
hence "\<And>a b. divmod_nat_rel n q (a, b) \<Longrightarrow> divmod_nat_rel (m * n) (m * q) (a, m * b)" 

654 
unfolding divmod_nat_rel_def 

655 
by (auto split: split_if_asm, simp_all add: algebra_simps) 

656 
moreover from divmod_nat_rel have "divmod_nat_rel n q (n div q, n mod q)" . 

657 
ultimately have "divmod_nat_rel (m * n) (m * q) (n div q, m * (n mod q))" . 

658 
thus "(m * n) div (m * q) = n div q" by (rule div_nat_unique) 

659 
next 

660 
fix n :: nat show "n div 0 = 0" 

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

661 
by (simp add: div_nat_def divmod_nat_zero) 
47136  662 
next 
663 
fix n :: nat show "0 div n = 0" 

664 
by (simp add: div_nat_def divmod_nat_zero_left) 

25942  665 
qed 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

666 

25942  667 
end 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

668 

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

669 
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

670 
let (q, r) = divmod_nat (m  n) n in (Suc q, r))" 
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

671 
by (simp add: prod_eq_iff prod_case_beta not_less le_div_geq le_mod_geq) 
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

672 

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

673 
text {* Simproc for cancelling @{const div} and @{const mod} *} 
25942  674 

30934  675 
ML {* 
43594  676 
structure Cancel_Div_Mod_Nat = Cancel_Div_Mod 
41550  677 
( 
30934  678 
val div_name = @{const_name div}; 
679 
val mod_name = @{const_name mod}; 

680 
val mk_binop = HOLogic.mk_binop; 

681 
val mk_sum = Nat_Arith.mk_sum; 

682 
val dest_sum = Nat_Arith.dest_sum; 

25942  683 

30934  684 
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

685 

30934  686 
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

687 
(@{thm add_0_left} :: @{thm add_0_right} :: @{thms add_ac})) 
41550  688 
) 
25942  689 
*} 
690 

43594  691 
simproc_setup cancel_div_mod_nat ("(m::nat) + n") = {* K Cancel_Div_Mod_Nat.proc *} 
692 

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

693 

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

694 
subsubsection {* Quotient *} 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

695 

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

696 
lemma div_geq: "0 < n \<Longrightarrow> \<not> m < n \<Longrightarrow> m div n = Suc ((m  n) div n)" 
29667  697 
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

698 

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

699 
lemma div_if: "0 < n \<Longrightarrow> m div n = (if m < n then 0 else Suc ((m  n) div n))" 
29667  700 
by (simp add: div_geq) 
26100
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_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)" 
29667  703 
by simp 
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_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)" 
29667  706 
by simp 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

707 

25942  708 

709 
subsubsection {* Remainder *} 

710 

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

711 
lemma mod_less_divisor [simp]: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

712 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

713 
assumes "n > 0" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

714 
shows "m mod n < (n::nat)" 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

715 
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

716 

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

717 
lemma mod_less_eq_dividend [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 
shows "m mod n \<le> m" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

720 
proof (rule add_leD2) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

721 
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

722 
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

723 
qed 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

724 

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

725 
lemma mod_geq: "\<not> m < (n\<Colon>nat) \<Longrightarrow> m mod n = (m  n) mod n" 
29667  726 
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

727 

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

728 
lemma mod_if: "m mod (n\<Colon>nat) = (if m < n then m else (m  n) mod n)" 
29667  729 
by (simp add: le_mod_geq) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

730 

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

731 
lemma mod_1 [simp]: "m mod Suc 0 = 0" 
29667  732 
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

733 

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

734 
(* 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

735 
lemma mult_div_cancel: "(n::nat) * (m div n) = m  (m mod n)" 
47138  736 
using mod_div_equality2 [of n m] by arith 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

737 

15439  738 
lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)" 
22718  739 
apply (drule mod_less_divisor [where m = m]) 
740 
apply simp 

741 
done 

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

742 

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

743 
subsubsection {* Quotient and Remainder *} 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

744 

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

745 
lemma divmod_nat_rel_mult1_eq: 
46552  746 
"divmod_nat_rel b c (q, r) 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

747 
\<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

748 
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

749 

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

750 
lemma div_mult1_eq: 
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

751 
"(a * b) div c = a * (b div c) + a * (b mod c) div (c::nat)" 
47135
fb67b596067f
rename lemmas {div,mod}_eq > {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents:
47134
diff
changeset

752 
by (blast intro: divmod_nat_rel_mult1_eq [THEN div_nat_unique] divmod_nat_rel) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

753 

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

754 
lemma divmod_nat_rel_add1_eq: 
46552  755 
"divmod_nat_rel a c (aq, ar) \<Longrightarrow> divmod_nat_rel b c (bq, br) 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

756 
\<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

757 
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

758 

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

759 
(*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

760 
lemma div_add1_eq: 
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

761 
"(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)" 
47135
fb67b596067f
rename lemmas {div,mod}_eq > {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents:
47134
diff
changeset

762 
by (blast intro: divmod_nat_rel_add1_eq [THEN div_nat_unique] divmod_nat_rel) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

763 

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

764 
lemma mod_lemma: "[ (0::nat) < c; r < b ] ==> b * (q mod c) + r < b * c" 
22718  765 
apply (cut_tac m = q and n = c in mod_less_divisor) 
766 
apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto) 

767 
apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst) 

768 
apply (simp add: add_mult_distrib2) 

769 
done 

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

770 

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

771 
lemma divmod_nat_rel_mult2_eq: 
46552  772 
"divmod_nat_rel a b (q, r) 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

773 
\<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

774 
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

775 

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

776 
lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)" 
47135
fb67b596067f
rename lemmas {div,mod}_eq > {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents:
47134
diff
changeset

777 
by (force simp add: divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN div_nat_unique]) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

778 

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

779 
lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)" 
47135
fb67b596067f
rename lemmas {div,mod}_eq > {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents:
47134
diff
changeset

780 
by (auto simp add: mult_commute divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN mod_nat_unique]) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

781 

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

782 

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

783 
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

784 

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

785 
lemma div_1 [simp]: "m div Suc 0 = m" 
29667  786 
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

787 

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

788 
(* Monotonicity of div in first argument *) 
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset

789 
lemma div_le_mono [rule_format (no_asm)]: 
22718  790 
"\<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

791 
apply (case_tac "k=0", simp) 
15251  792 
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

793 
apply (case_tac "n<k") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

794 
(* 1 case n<k *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

795 
apply simp 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

796 
(* 2 case n >= k *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

797 
apply (case_tac "m<k") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

798 
(* 2.1 case m<k *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

799 
apply simp 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

800 
(* 2.2 case m>=k *) 
15439  801 
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

802 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

803 

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

804 
(* Antimonotonicity of div in second argument *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

805 
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

806 
apply (subgoal_tac "0<n") 
22718  807 
prefer 2 apply simp 
15251  808 
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

809 
apply (rename_tac "k") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

810 
apply (case_tac "k<n", simp) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

811 
apply (subgoal_tac "~ (k<m) ") 
22718  812 
prefer 2 apply simp 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

813 
apply (simp add: div_geq) 
15251  814 
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

815 
prefer 2 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

816 
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

817 
apply (rule le_trans, simp) 
15439  818 
apply (simp) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

819 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

820 

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

821 
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

822 
apply (case_tac "n=0", simp) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

823 
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

824 
apply (rule div_le_mono2) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

825 
apply (simp_all (no_asm_simp)) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

826 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

827 

22718  828 
(* Similar for "less than" *) 
47138  829 
lemma div_less_dividend [simp]: 
830 
"\<lbrakk>(1::nat) < n; 0 < m\<rbrakk> \<Longrightarrow> m div n < m" 

831 
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

832 
apply (rename_tac "m") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

833 
apply (case_tac "m<n", simp) 
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 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

836 
apply (simp add: div_geq) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

837 
apply (case_tac "n<m") 
15251  838 
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

839 
apply (rule impI less_trans_Suc)+ 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

840 
apply assumption 
15439  841 
apply (simp_all) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

842 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

843 

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

844 
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

845 
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

846 
apply (case_tac "n=0", simp) 
15251  847 
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

848 
apply (case_tac "Suc (na) <n") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

849 
(* case Suc(na) < n *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

850 
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

851 
(* case n \<le> Suc(na) *) 
16796  852 
apply (simp add: linorder_not_less le_Suc_eq mod_geq) 
15439  853 
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

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

855 

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

856 
lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)" 
29667  857 
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

858 

22718  859 
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

860 

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

861 
(*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

862 
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

863 
apply (cut_tac a = m in mod_div_equality) 
22718  864 
apply (simp only: add_ac) 
865 
apply (blast intro: sym) 

866 
done 

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

867 

13152  868 
lemma split_div: 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

869 
"P(n div k :: nat) = 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

870 
((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

871 
(is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))") 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

872 
proof 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

873 
assume P: ?P 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

874 
show ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

875 
proof (cases) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

876 
assume "k = 0" 
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

877 
with P show ?Q by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

878 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

879 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

880 
thus ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

881 
proof (simp, intro allI impI) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

882 
fix i j 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

883 
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

884 
show "P i" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

885 
proof (cases) 
22718  886 
assume "i = 0" 
887 
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

888 
next 
22718  889 
assume "i \<noteq> 0" 
890 
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

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

892 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

893 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

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

895 
assume Q: ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

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

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

898 
assume "k = 0" 
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

899 
with Q show ?P by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

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

901 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

902 
with Q have R: ?R by simp 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

903 
from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"] 
13517  904 
show ?P by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

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

906 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

907 

13882  908 
lemma split_div_lemma: 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

909 
assumes "0 < n" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

910 
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

911 
proof 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

912 
assume ?rhs 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

913 
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

914 
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

915 
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

916 
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

917 
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

918 
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

919 
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

920 
from A B show ?lhs .. 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

921 
next 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

922 
assume P: ?lhs 
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset

923 
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

924 
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

925 
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

926 
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

927 
then show ?rhs by simp 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

928 
qed 
13882  929 

930 
theorem split_div': 

931 
"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

932 
(\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))" 
13882  933 
apply (case_tac "0 < n") 
934 
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

935 
apply simp_all 
13882  936 
done 
937 

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

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

939 
"P(n mod k :: nat) = 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

940 
((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

941 
(is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))") 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

942 
proof 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

943 
assume P: ?P 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

944 
show ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

945 
proof (cases) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

946 
assume "k = 0" 
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

947 
with P show ?Q by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

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

949 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

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

951 
proof (simp, intro allI impI) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

952 
fix i j 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

953 
assume "n = k*i + j" "j < k" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

954 
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

955 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

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

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

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

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

960 
proof (cases) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

961 
assume "k = 0" 
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

962 
with Q show ?P by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

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

964 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

965 
with Q have R: ?R by simp 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

966 
from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"] 
13517  967 
show ?P by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

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

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

970 

13882  971 
theorem mod_div_equality': "(m::nat) mod n = m  (m div n) * n" 
47138  972 
using mod_div_equality [of m n] by arith 
973 

974 
lemma div_mod_equality': "(m::nat) div n * n = m  m mod n" 

975 
using mod_div_equality [of m n] by arith 

976 
(* FIXME: very similar to mult_div_cancel *) 

22800  977 

978 

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

979 
subsubsection {* An ``induction'' law for modulus arithmetic. *} 
14640  980 

981 
lemma mod_induct_0: 

982 
assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)" 

983 
and base: "P i" and i: "i<p" 

984 
shows "P 0" 

985 
proof (rule ccontr) 

986 
assume contra: "\<not>(P 0)" 

987 
from i have p: "0<p" by simp 

988 
have "\<forall>k. 0<k \<longrightarrow> \<not> P (pk)" (is "\<forall>k. ?A k") 

989 
proof 

990 
fix k 

991 
show "?A k" 

992 
proof (induct k) 

993 
show "?A 0" by simp  "by contradiction" 

994 
next 

995 
fix n 

996 
assume ih: "?A n" 

997 
show "?A (Suc n)" 

998 
proof (clarsimp) 

22718  999 
assume y: "P (p  Suc n)" 
1000 
have n: "Suc n < p" 

1001 
proof (rule ccontr) 

1002 
assume "\<not>(Suc n < p)" 

1003 
hence "p  Suc n = 0" 

1004 
by simp 

1005 
with y contra show "False" 

1006 
by simp 

1007 
qed 

1008 
hence n2: "Suc (p  Suc n) = pn" by arith 

1009 
from p have "p  Suc n < p" by arith 

1010 
with y step have z: "P ((Suc (p  Suc n)) mod p)" 

1011 
by blast 

1012 
show "False" 

1013 
proof (cases "n=0") 

1014 
case True 

1015 
with z n2 contra show ?thesis by simp 

1016 
next 

1017 
case False 

1018 
with p have "pn < p" by arith 

1019 
with z n2 False ih show ?thesis by simp 

1020 
qed 

14640  1021 
qed 
1022 
qed 

1023 
qed 

1024 
moreover 

1025 
from i obtain k where "0<k \<and> i+k=p" 

1026 
by (blast dest: less_imp_add_positive) 

1027 
hence "0<k \<and> i=pk" by auto 

1028 
moreover 

1029 
note base 

1030 
ultimately 

1031 
show "False" by blast 

1032 
qed 

1033 

1034 
lemma mod_induct: 

1035 
assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)" 

1036 
and base: "P i" and i: "i<p" and j: "j<p" 

1037 
shows "P j" 

1038 
proof  

1039 
have "\<forall>j<p. P j" 

1040 
proof 

1041 
fix j 

1042 
show "j<p \<longrightarrow> P j" (is "?A j") 

1043 
proof (induct j) 

1044 
from step base i show "?A 0" 

22718  1045 
by (auto elim: mod_induct_0) 
14640  1046 
next 
1047 
fix k 

1048 
assume ih: "?A k" 

1049 
show "?A (Suc k)" 

1050 
proof 

22718  1051 
assume suc: "Suc k < p" 
1052 
hence k: "k<p" by simp 

1053 
with ih have "P k" .. 

1054 
with step k have "P (Suc k mod p)" 

1055 
by blast 

1056 
moreover 

1057 
from suc have "Suc k mod p = Suc k" 

1058 
by simp 

1059 
ultimately 

1060 
show "P (Suc k)" by simp 

14640  1061 
qed 
1062 
qed 

1063 
qed 

1064 
with j show ?thesis by blast 

1065 
qed 

1066 

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

1067 
lemma div2_Suc_Suc [simp]: "Suc (Suc m) div 2 = Suc (m div 2)" 
47138  1068 
by (simp add: numeral_2_eq_2 le_div_geq) 
1069 

1070 
lemma mod2_Suc_Suc [simp]: "Suc (Suc m) mod 2 = m mod 2" 

1071 
by (simp add: numeral_2_eq_2 le_mod_geq) 

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

1072 

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

1073 
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

1074 
by (simp add: nat_mult_2 [symmetric]) 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1075 

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

1076 
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

1077 
proof  
35815
10e723e54076
tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents:
35673
diff
changeset

1078 
{ 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

1079 
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

1080 
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

1081 
then show ?thesis by auto 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1082 
qed 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1083 

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

1084 
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

1085 
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

1086 
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

1087 

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

1088 
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

1089 
by (simp add: Suc3_eq_add_3) 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1090 

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

1091 
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

1092 
by (simp add: Suc3_eq_add_3) 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1093 

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

1094 
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

1095 
by (simp add: Suc3_eq_add_3) 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1096 

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

1097 
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

1098 
by (simp add: Suc3_eq_add_3) 
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1099 

47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset

1100 
lemmas Suc_div_eq_add3_div_numeral [simp] = Suc_div_eq_add3_div [of _ "numeral v"] for v 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset

1101 
lemmas Suc_mod_eq_add3_mod_numeral [simp] = Suc_mod_eq_add3_mod [of _ "numeral v"] for v 
33296
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset

1102 

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

1103 

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

1104 
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

1105 
apply (induct "m") 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1106 
apply (simp_all add: mod_Suc) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1107 
done 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1108 

47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset

1109 
declare Suc_times_mod_eq [of "numeral w", simp] for w 
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1110 

47138  1111 
lemma Suc_div_le_mono [simp]: "n div k \<le> (Suc n) div k" 
1112 
by (simp add: div_le_mono) 

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

1113 

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

1114 
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

1115 
by (cases n) simp_all 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1116 

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

1117 
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

1118 
proof  
10e723e54076
tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents:
35673
diff
changeset

1119 
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

1120 
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

1121 
qed 
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1122 

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

1123 
(* Potential use of algebra : Equality modulo n*) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1124 
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

1125 
by (simp add: mult_ac add_ac) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1126 

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

1127 
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

1128 
proof  
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1129 
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

1130 
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

1131 
finally show ?thesis . 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1132 
qed 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1133 

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

1134 
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

1135 
apply (subst mod_Suc [of m]) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1136 
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

1137 
done 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1138 

47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset

1139 
lemma mod_2_not_eq_zero_eq_one_nat: 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset

1140 
fixes n :: nat 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset

1141 
shows "n mod 2 \<noteq> 0 \<longleftrightarrow> n mod 2 = 1" 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset

1142 
by simp 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset

1143 

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

1144 

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

1145 
subsection {* Division on @{typ int} *} 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1146 

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

1147 
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

1148 
{*definition of quotient and remainder*} 
47139
98bddfa0f717
extend definition of divmod_int_rel to handle denominator=0 case
huffman
parents:
47138
diff
changeset

1149 
"divmod_int_rel a b = (\<lambda>(q, r). a = b * q + r \<and> 
98bddfa0f717
extend definition of divmod_int_rel to handle denominator=0 case
huffman
parents:
47138
diff
changeset

1150 
(if 0 < b then 0 \<le> r \<and> r < b else if b < 0 then b < r \<and> r \<le> 0 else q = 0))" 
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1151 

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

1152 
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

1153 
{*for the division algorithm*} 
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset

1154 
"adjust b = (\<lambda>(q, r). if 0 \<le> r  b then (2 * q + 1, r  b) 
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1155 
else (2 * q, r))" 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1156 

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

1157 
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

1158 
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

1159 
"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

1160 
else adjust b (posDivAlg a (2 * b)))" 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1161 
by auto 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1162 
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

1163 
(auto simp add: mult_2) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1164 

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

1165 
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

1166 
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

1167 
"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

1168 
else adjust b (negDivAlg a (2 * b)))" 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1169 
by auto 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1170 
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

1171 
(auto simp add: mult_2) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1172 

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

1173 
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

1174 

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

1175 
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

1176 
{*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

1177 
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

1178 
@{term negDivAlg} requires @{term "a<0"}.*} 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1179 
"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

1180 
else if a = 0 then (0, 0) 
46560
8e252a608765
remove constant negateSnd in favor of 'apsnd uminus' (from Florian Haftmann)
huffman
parents:
46552
diff
changeset

1181 
else apsnd uminus (negDivAlg (a) (b)) 
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

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

1183 
if 0 < b then negDivAlg a b 
46560
8e252a608765
remove constant negateSnd in favor of 'apsnd uminus' (from Florian Haftmann)
huffman
parents:
46552
diff
changeset

1184 
else apsnd uminus (posDivAlg (a) (b)))" 
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1185 

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

1186 
instantiation int :: Divides.div 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1187 
begin 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1188 

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

1189 
definition div_int where 
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1190 
"a div b = fst (divmod_int a b)" 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1191 

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

1192 
lemma fst_divmod_int [simp]: 
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

1193 
"fst (divmod_int a b) = a div b" 
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

1194 
by (simp add: div_int_def) 
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

1195 

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

1196 
definition mod_int where 
46560
8e252a608765
remove constant negateSnd in favor of 'apsnd uminus' (from Florian Haftmann)
huffman
parents:
46552
diff
changeset

1197 
"a mod b = snd (divmod_int a b)" 
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1198 

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

1199 
lemma snd_divmod_int [simp]: 
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

1200 
"snd (divmod_int a b) = a mod b" 
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

1201 
by (simp add: mod_int_def) 
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

1202 

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

1203 
instance .. 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1204 

3366  1205 
end 
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1206 

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

1207 
lemma divmod_int_mod_div: 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1208 
"divmod_int p q = (p div q, p mod q)" 
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset

1209 
by (simp add: prod_eq_iff) 
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1210 

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

1211 
text{* 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1212 
Here is the division algorithm in ML: 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1213 

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

1214 
\begin{verbatim} 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1215 
fun posDivAlg (a,b) = 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1216 
if a<b then (0,a) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1217 
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

1218 
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

1219 
end 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1220 

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

1221 
fun negDivAlg (a,b) = 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1222 
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

1223 
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

1224 
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

1225 
end; 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1226 

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

1227 
fun negateSnd (q,r:int) = (q,~r); 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1228 

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

1229 
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

1230 
if b>0 then posDivAlg (a,b) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1231 
else if a=0 then (0,0) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1232 
else negateSnd (negDivAlg (~a,~b)) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1233 
else 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1234 
if 0<b then negDivAlg (a,b) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1235 
else negateSnd (posDivAlg (~a,~b)); 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1236 
\end{verbatim} 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

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

1238 

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

1239 

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

1240 
subsubsection {* Uniqueness and Monotonicity of Quotients and Remainders *} 
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1241 

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

1242 
lemma unique_quotient_lemma: 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1243 
"[ 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

1244 
==> q' \<le> (q::int)" 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1245 
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

1246 
prefer 2 apply (simp add: right_diff_distrib) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1247 
apply (subgoal_tac "0 < b * (1 + q  q') ") 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1248 
apply (erule_tac [2] order_le_less_trans) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1249 
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

1250 
apply (subgoal_tac "b * q' < b * (1 + q) ") 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1251 
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

1252 
apply (simp add: mult_less_cancel_left) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

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

1256 
"[ 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

1257 
==> q \<le> (q'::int)" 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1258 
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

1259 
auto) 
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 
lemma unique_quotient: 
46552  1262 
"[ divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') ] 
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1263 
==> q = q'" 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1264 
apply (simp add: divmod_int_rel_def linorder_neq_iff split: split_if_asm) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1265 
apply (blast intro: order_antisym 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1266 
dest: order_eq_refl [THEN unique_quotient_lemma] 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1267 
order_eq_refl [THEN unique_quotient_lemma_neg] sym)+ 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1268 
done 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1269 

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

1270 

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

1271 
lemma unique_remainder: 
46552  1272 
"[ divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') ] 
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1273 
==> r = r'" 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1274 
apply (subgoal_tac "q = q'") 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1275 
apply (simp add: divmod_int_rel_def) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1276 
apply (blast intro: unique_quotient) 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

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

1278 

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

1279 

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

1280 
subsubsection {* Correctness of @{term posDivAlg}, the Algorithm for NonNegative Dividends *} 
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1281 

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

1282 
text{*And positive divisors*} 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1283 

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

1284 
lemma adjust_eq [simp]: 
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset

1285 
"adjust b (q, r) = 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset

1286 
(let diff = r  b in 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset

1287 
if 0 \<le> diff then (2 * q + 1, diff) 
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1288 
else (2*q, r))" 
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset

1289 
by (simp add: Let_def adjust_def) 
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1290 

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

1291 
declare posDivAlg.simps [simp del] 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset

1292 

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

1293 
text{*use with a simproc to avoid repeatedly proving the premise*} 
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
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1294 
lemma posDivAlg_eqn: 
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combined former theories Divides and IntDiv to one theory Divides
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1295 
"0 < b ==> 
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combined former theories Divides and IntDiv to one theory Divides
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1296 
posDivAlg a b = (if a<b then (0,a) else adjust b (posDivAlg a (2*b)))" 
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combined former theories Divides and IntDiv to one theory Divides
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1297 
by (rule posDivAlg.simps [THEN trans], simp) 
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combined former theories Divides and IntDiv to one theory Divides
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1298 

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combined former theories Divides and IntDiv to one theory Divides
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1299 
text{*Correctness of @{term posDivAlg}: it computes quotients correctly*} 
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combined former theories Divides and IntDiv to one theory Divides
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1300 
theorem posDivAlg_correct: 
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1301 
assumes "0 \<le> a" and "0 < b" 
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1302 
shows "divmod_int_rel a b (posDivAlg a b)" 
41550  1303 
using assms 
1304 
apply (induct a b rule: posDivAlg.induct) 
