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

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Author: Lawrence C Paulson, Cambridge University Computer Laboratory 

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Copyright 1999 University of Cambridge 
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*) 
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header {* The division operators div and mod *} 
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theory Divides 
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imports Nat Power Product_Type 
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uses "~~/src/Provers/Arith/cancel_div_mod.ML" 
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begin 
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subsection {* Syntactic division operations *} 
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class div = dvd + 
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fixes div :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "div" 70) 
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and mod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "mod" 70) 
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subsection {* Abstract division in commutative semirings. *} 
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class semiring_div = comm_semiring_1_cancel + div + 
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assumes mod_div_equality: "a div b * b + a mod b = a" 
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and div_by_0 [simp]: "a div 0 = 0" 
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and div_0 [simp]: "0 div a = 0" 
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and div_mult_self1 [simp]: "b \<noteq> 0 \<Longrightarrow> (a + c * b) div b = c + a div b" 
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begin 
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text {* @{const div} and @{const mod} *} 
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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) 

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

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by (simp add: mod_div_equality2) 

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lemma mod_by_0 [simp]: "a mod 0 = a" 
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using mod_div_equality [of a zero] by simp 
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lemma mod_0 [simp]: "0 mod a = 0" 
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using mod_div_equality [of zero a] div_0 by simp 
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lemma div_mult_self2 [simp]: 
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assumes "b \<noteq> 0" 
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shows "(a + b * c) div b = c + a div b" 
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using assms div_mult_self1 [of b a c] by (simp add: mult_commute) 
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lemma mod_mult_self1 [simp]: "(a + c * b) mod b = a mod b" 
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proof (cases "b = 0") 
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case True then show ?thesis by simp 
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next 
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case False 
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have "a + c * b = (a + c * b) div b * b + (a + c * b) mod b" 
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by (simp add: mod_div_equality) 
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also from False div_mult_self1 [of b a c] have 
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"\<dots> = (c + a div b) * b + (a + c * b) mod b" 
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by (simp add: left_distrib add_ac) 
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finally have "a = a div b * b + (a + c * b) mod b" 
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by (simp add: add_commute [of a] add_assoc left_distrib) 
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then have "a div b * b + (a + c * b) mod b = a div b * b + a mod b" 
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by (simp add: mod_div_equality) 
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then show ?thesis by simp 
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qed 
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lemma mod_mult_self2 [simp]: "(a + b * c) mod b = a mod b" 
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by (simp add: mult_commute [of b]) 
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lemma div_mult_self1_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> b * a div b = a" 
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using div_mult_self2 [of b 0 a] by simp 
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lemma div_mult_self2_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> a * b div b = a" 
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using div_mult_self1 [of b 0 a] by simp 
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lemma mod_mult_self1_is_0 [simp]: "b * a mod b = 0" 
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using mod_mult_self2 [of 0 b a] by simp 
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lemma mod_mult_self2_is_0 [simp]: "a * b mod b = 0" 
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using mod_mult_self1 [of 0 a b] by simp 
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lemma div_by_1 [simp]: "a div 1 = a" 
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using div_mult_self2_is_id [of 1 a] zero_neq_one by simp 
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lemma mod_by_1 [simp]: "a mod 1 = 0" 
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proof  
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from mod_div_equality [of a one] div_by_1 have "a + a mod 1 = a" by simp 
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then have "a + a mod 1 = a + 0" by simp 
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then show ?thesis by (rule add_left_imp_eq) 
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qed 
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lemma mod_self [simp]: "a mod a = 0" 
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using mod_mult_self2_is_0 [of 1] by simp 
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lemma div_self [simp]: "a \<noteq> 0 \<Longrightarrow> a div a = 1" 
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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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lemma div_add_self2 [simp]: 
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assumes "b \<noteq> 0" 
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shows "(a + b) div b = a div b + 1" 
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using assms div_add_self1 [of b a] by (simp add: add_commute) 
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lemma mod_add_self1 [simp]: 
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"(b + a) mod b = a mod b" 
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using mod_mult_self1 [of a 1 b] by (simp add: add_commute) 
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lemma mod_add_self2 [simp]: 
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"(a + b) mod b = a mod b" 
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using mod_mult_self1 [of a 1 b] by simp 
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lemma mod_div_decomp: 
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fixes a b 
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obtains q r where "q = a div b" and "r = a mod b" 
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and "a = q * b + r" 
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proof  
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from mod_div_equality have "a = a div b * b + a mod b" by simp 
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moreover have "a div b = a div b" .. 
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moreover have "a mod b = a mod b" .. 
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note that ultimately show thesis by blast 
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qed 
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29108  134 
lemma dvd_eq_mod_eq_0 [code unfold]: "a dvd b \<longleftrightarrow> b mod a = 0" 
25942  135 
proof 
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assume "b mod a = 0" 

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

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

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

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then show "a dvd b" unfolding dvd_def . 

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next 

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assume "a dvd b" 

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

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

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

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

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

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175 
text {* Addition respects modular equivalence. *} 
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176 

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177 
lemma mod_add_left_eq: "(a + b) mod c = (a mod c + b) mod c" 
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178 
proof  
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179 
have "(a + b) mod c = (a div c * c + a mod c + b) mod c" 
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180 
by (simp only: mod_div_equality) 
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181 
also have "\<dots> = (a mod c + b + a div c * c) mod c" 
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182 
by (simp only: add_ac) 
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183 
also have "\<dots> = (a mod c + b) mod c" 
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184 
by (rule mod_mult_self1) 
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185 
finally show ?thesis . 
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186 
qed 
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187 

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188 
lemma mod_add_right_eq: "(a + b) mod c = (a + b mod c) mod c" 
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189 
proof  
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190 
have "(a + b) mod c = (a + (b div c * c + b mod c)) mod c" 
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191 
by (simp only: mod_div_equality) 
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192 
also have "\<dots> = (a + b mod c + b div c * c) mod c" 
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193 
by (simp only: add_ac) 
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194 
also have "\<dots> = (a + b mod c) mod c" 
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195 
by (rule mod_mult_self1) 
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196 
finally show ?thesis . 
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197 
qed 
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198 

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199 
lemma mod_add_eq: "(a + b) mod c = (a mod c + b mod c) mod c" 
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200 
by (rule trans [OF mod_add_left_eq mod_add_right_eq]) 
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201 

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202 
lemma mod_add_cong: 
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203 
assumes "a mod c = a' mod c" 
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204 
assumes "b mod c = b' mod c" 
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205 
shows "(a + b) mod c = (a' + b') mod c" 
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206 
proof  
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207 
have "(a mod c + b mod c) mod c = (a' mod c + b' mod c) mod c" 
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208 
unfolding assms .. 
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209 
thus ?thesis 
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210 
by (simp only: mod_add_eq [symmetric]) 
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211 
qed 
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212 

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213 
text {* Multiplication respects modular equivalence. *} 
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214 

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215 
lemma mod_mult_left_eq: "(a * b) mod c = ((a mod c) * b) mod c" 
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216 
proof  
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217 
have "(a * b) mod c = ((a div c * c + a mod c) * b) mod c" 
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218 
by (simp only: mod_div_equality) 
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219 
also have "\<dots> = (a mod c * b + a div c * b * c) mod c" 
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220 
by (simp only: left_distrib right_distrib add_ac mult_ac) 
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221 
also have "\<dots> = (a mod c * b) mod c" 
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222 
by (rule mod_mult_self1) 
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223 
finally show ?thesis . 
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224 
qed 
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225 

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226 
lemma mod_mult_right_eq: "(a * b) mod c = (a * (b mod c)) mod c" 
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227 
proof  
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228 
have "(a * b) mod c = (a * (b div c * c + b mod c)) mod c" 
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229 
by (simp only: mod_div_equality) 
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230 
also have "\<dots> = (a * (b mod c) + a * (b div c) * c) mod c" 
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231 
by (simp only: left_distrib right_distrib add_ac mult_ac) 
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232 
also have "\<dots> = (a * (b mod c)) mod c" 
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233 
by (rule mod_mult_self1) 
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234 
finally show ?thesis . 
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235 
qed 
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236 

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237 
lemma mod_mult_eq: "(a * b) mod c = ((a mod c) * (b mod c)) mod c" 
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238 
by (rule trans [OF mod_mult_left_eq mod_mult_right_eq]) 
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239 

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240 
lemma mod_mult_cong: 
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241 
assumes "a mod c = a' mod c" 
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242 
assumes "b mod c = b' mod c" 
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243 
shows "(a * b) mod c = (a' * b') mod c" 
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244 
proof  
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245 
have "(a mod c * (b mod c)) mod c = (a' mod c * (b' mod c)) mod c" 
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246 
unfolding assms .. 
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247 
thus ?thesis 
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248 
by (simp only: mod_mult_eq [symmetric]) 
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249 
qed 
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250 

29404  251 
lemma mod_mod_cancel: 
252 
assumes "c dvd b" 

253 
shows "a mod b mod c = a mod c" 

254 
proof  

255 
from `c dvd b` obtain k where "b = c * k" 

256 
by (rule dvdE) 

257 
have "a mod b mod c = a mod (c * k) mod c" 

258 
by (simp only: `b = c * k`) 

259 
also have "\<dots> = (a mod (c * k) + a div (c * k) * k * c) mod c" 

260 
by (simp only: mod_mult_self1) 

261 
also have "\<dots> = (a div (c * k) * (c * k) + a mod (c * k)) mod c" 

262 
by (simp only: add_ac mult_ac) 

263 
also have "\<dots> = a mod c" 

264 
by (simp only: mod_div_equality) 

265 
finally show ?thesis . 

266 
qed 

267 

25942  268 
end 
269 

29405
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270 
class ring_div = semiring_div + comm_ring_1 
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271 
begin 
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272 

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273 
text {* Negation respects modular equivalence. *} 
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274 

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275 
lemma mod_minus_eq: "( a) mod b = ( (a mod b)) mod b" 
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276 
proof  
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277 
have "( a) mod b = ( (a div b * b + a mod b)) mod b" 
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278 
by (simp only: mod_div_equality) 
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279 
also have "\<dots> = ( (a mod b) +  (a div b) * b) mod b" 
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280 
by (simp only: minus_add_distrib minus_mult_left add_ac) 
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281 
also have "\<dots> = ( (a mod b)) mod b" 
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282 
by (rule mod_mult_self1) 
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283 
finally show ?thesis . 
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284 
qed 
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285 

98ab21b14f09
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286 
lemma mod_minus_cong: 
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287 
assumes "a mod b = a' mod b" 
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288 
shows "( a) mod b = ( a') mod b" 
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289 
proof  
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290 
have "( (a mod b)) mod b = ( (a' mod b)) mod b" 
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291 
unfolding assms .. 
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292 
thus ?thesis 
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293 
by (simp only: mod_minus_eq [symmetric]) 
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294 
qed 
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295 

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296 
text {* Subtraction respects modular equivalence. *} 
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297 

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298 
lemma mod_diff_left_eq: "(a  b) mod c = (a mod c  b) mod c" 
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299 
unfolding diff_minus 
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300 
by (intro mod_add_cong mod_minus_cong) simp_all 
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301 

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302 
lemma mod_diff_right_eq: "(a  b) mod c = (a  b mod c) mod c" 
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303 
unfolding diff_minus 
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304 
by (intro mod_add_cong mod_minus_cong) simp_all 
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305 

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lemma mod_diff_eq: "(a  b) mod c = (a mod c  b mod c) mod c" 
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307 
unfolding diff_minus 
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308 
by (intro mod_add_cong mod_minus_cong) simp_all 
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309 

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lemma mod_diff_cong: 
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311 
assumes "a mod c = a' mod c" 
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312 
assumes "b mod c = b' mod c" 
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313 
shows "(a  b) mod c = (a'  b') mod c" 
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314 
unfolding diff_minus using assms 
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315 
by (intro mod_add_cong mod_minus_cong) 
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316 

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317 
end 
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318 

25942  319 

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320 
subsection {* Division on @{typ nat} *} 
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321 

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322 
text {* 
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We define @{const div} and @{const mod} on @{typ nat} by means 
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324 
of a characteristic relation with two input arguments 
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325 
@{term "m\<Colon>nat"}, @{term "n\<Colon>nat"} and two output arguments 
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@{term "q\<Colon>nat"}(uotient) and @{term "r\<Colon>nat"}(emainder). 
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327 
*} 
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328 

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definition divmod_rel :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" where 
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"divmod_rel m n q r \<longleftrightarrow> m = q * n + r \<and> (if n > 0 then 0 \<le> r \<and> r < n else q = 0)" 
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331 

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332 
text {* @{const divmod_rel} is total: *} 
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333 

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334 
lemma divmod_rel_ex: 
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335 
obtains q r where "divmod_rel m n q r" 
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336 
proof (cases "n = 0") 
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337 
case True with that show thesis 
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338 
by (auto simp add: divmod_rel_def) 
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339 
next 
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340 
case False 
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341 
have "\<exists>q r. m = q * n + r \<and> r < n" 
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342 
proof (induct m) 
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343 
case 0 with `n \<noteq> 0` 
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344 
have "(0\<Colon>nat) = 0 * n + 0 \<and> 0 < n" by simp 
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345 
then show ?case by blast 
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346 
next 
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347 
case (Suc m) then obtain q' r' 
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348 
where m: "m = q' * n + r'" and n: "r' < n" by auto 
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349 
then show ?case proof (cases "Suc r' < n") 
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350 
case True 
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351 
from m n have "Suc m = q' * n + Suc r'" by simp 
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352 
with True show ?thesis by blast 
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353 
next 
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354 
case False then have "n \<le> Suc r'" by auto 
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355 
moreover from n have "Suc r' \<le> n" by auto 
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356 
ultimately have "n = Suc r'" by auto 
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357 
with m have "Suc m = Suc q' * n + 0" by simp 
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358 
with `n \<noteq> 0` show ?thesis by blast 
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359 
qed 
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360 
qed 
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361 
with that show thesis 
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362 
using `n \<noteq> 0` by (auto simp add: divmod_rel_def) 
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363 
qed 
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364 

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365 
text {* @{const divmod_rel} is injective: *} 
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366 

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367 
lemma divmod_rel_unique_div: 
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368 
assumes "divmod_rel m n q r" 
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369 
and "divmod_rel m n q' r'" 
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370 
shows "q = q'" 
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371 
proof (cases "n = 0") 
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372 
case True with assms show ?thesis 
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373 
by (simp add: divmod_rel_def) 
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374 
next 
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375 
case False 
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376 
have aux: "\<And>q r q' r'. q' * n + r' = q * n + r \<Longrightarrow> r < n \<Longrightarrow> q' \<le> (q\<Colon>nat)" 
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377 
apply (rule leI) 
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378 
apply (subst less_iff_Suc_add) 
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379 
apply (auto simp add: add_mult_distrib) 
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380 
done 
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381 
from `n \<noteq> 0` assms show ?thesis 
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382 
by (auto simp add: divmod_rel_def 
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383 
intro: order_antisym dest: aux sym) 
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384 
qed 
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385 

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386 
lemma divmod_rel_unique_mod: 
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387 
assumes "divmod_rel m n q r" 
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388 
and "divmod_rel m n q' r'" 
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389 
shows "r = r'" 
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390 
proof  
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391 
from assms have "q = q'" by (rule divmod_rel_unique_div) 
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392 
with assms show ?thesis by (simp add: divmod_rel_def) 
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393 
qed 
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394 

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395 
text {* 
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396 
We instantiate divisibility on the natural numbers by 
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397 
means of @{const divmod_rel}: 
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398 
*} 
25942  399 

400 
instantiation nat :: semiring_div 

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401 
begin 
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402 

26100
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403 
definition divmod :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where 
28562  404 
[code del]: "divmod m n = (THE (q, r). divmod_rel m n q r)" 
26100
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405 

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406 
definition div_nat where 
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407 
"m div n = fst (divmod m n)" 
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408 

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409 
definition mod_nat where 
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410 
"m mod n = snd (divmod m n)" 
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411 

26100
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412 
lemma divmod_div_mod: 
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413 
"divmod m n = (m div n, m mod n)" 
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414 
unfolding div_nat_def mod_nat_def by simp 
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415 

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416 
lemma divmod_eq: 
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417 
assumes "divmod_rel m n q r" 
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418 
shows "divmod m n = (q, r)" 
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419 
using assms by (auto simp add: divmod_def 
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420 
dest: divmod_rel_unique_div divmod_rel_unique_mod) 
25942  421 

26100
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422 
lemma div_eq: 
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423 
assumes "divmod_rel m n q r" 
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424 
shows "m div n = q" 
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425 
using assms by (auto dest: divmod_eq simp add: div_nat_def) 
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426 

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427 
lemma mod_eq: 
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428 
assumes "divmod_rel m n q r" 
fbc60cd02ae2
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haftmann
parents:
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diff
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429 
shows "m mod n = r" 
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430 
using assms by (auto dest: divmod_eq simp add: mod_nat_def) 
25571
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haftmann
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diff
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431 

26100
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432 
lemma divmod_rel: "divmod_rel m n (m div n) (m mod n)" 
fbc60cd02ae2
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433 
proof  
fbc60cd02ae2
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haftmann
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434 
from divmod_rel_ex 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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435 
obtain q r where rel: "divmod_rel m n q r" . 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
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436 
moreover with div_eq mod_eq have "m div n = q" and "m mod n = r" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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437 
by simp_all 
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haftmann
parents:
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diff
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438 
ultimately show ?thesis by simp 
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using only an relation predicate to construct div and mod
haftmann
parents:
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diff
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439 
qed 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

440 

26100
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haftmann
parents:
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diff
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441 
lemma divmod_zero: 
fbc60cd02ae2
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haftmann
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442 
"divmod m 0 = (0, m)" 
fbc60cd02ae2
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haftmann
parents:
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diff
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443 
proof  
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
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444 
from divmod_rel [of m 0] show ?thesis 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

445 
unfolding divmod_div_mod divmod_rel_def by simp 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

446 
qed 
25942  447 

26100
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diff
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448 
lemma divmod_base: 
fbc60cd02ae2
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haftmann
parents:
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diff
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449 
assumes "m < n" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
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450 
shows "divmod m n = (0, m)" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
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451 
proof  
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
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452 
from divmod_rel [of m n] show ?thesis 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

453 
unfolding divmod_div_mod divmod_rel_def 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

454 
using assms by (cases "m div n = 0") 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

455 
(auto simp add: gr0_conv_Suc [of "m div n"]) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

456 
qed 
25942  457 

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

458 
lemma divmod_step: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

459 
assumes "0 < n" and "n \<le> m" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

460 
shows "divmod m n = (Suc ((m  n) div n), (m  n) mod n)" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

461 
proof  
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

462 
from divmod_rel have divmod_m_n: "divmod_rel m n (m div n) (m mod n)" . 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
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463 
with assms have m_div_n: "m div n \<ge> 1" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
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464 
by (cases "m div n") (auto simp add: divmod_rel_def) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

465 
from assms divmod_m_n have "divmod_rel (m  n) n (m div n  1) (m mod n)" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
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466 
by (cases "m div n") (auto simp add: divmod_rel_def) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

467 
with divmod_eq have "divmod (m  n) n = (m div n  1, m mod n)" by simp 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

468 
moreover from divmod_div_mod have "divmod (m  n) n = ((m  n) div n, (m  n) mod n)" . 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

469 
ultimately have "m div n = Suc ((m  n) div n)" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

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

471 
then show ?thesis using divmod_div_mod by simp 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

472 
qed 
25942  473 

26300  474 
text {* The ''recursion'' equations for @{const div} and @{const mod} *} 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
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diff
changeset

475 

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

476 
lemma div_less [simp]: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

477 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

478 
assumes "m < n" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

479 
shows "m div n = 0" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

480 
using assms divmod_base divmod_div_mod by simp 
25942  481 

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

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

483 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

484 
assumes "0 < n" and "n \<le> m" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

485 
shows "m div n = Suc ((m  n) div n)" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

486 
using assms divmod_step divmod_div_mod by simp 
14267
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

487 

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

488 
lemma mod_less [simp]: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

489 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

490 
assumes "m < n" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

491 
shows "m mod n = m" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

492 
using assms divmod_base divmod_div_mod by simp 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

493 

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

494 
lemma le_mod_geq: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

495 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

496 
assumes "n \<le> m" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

497 
shows "m mod n = (m  n) mod n" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

498 
using assms divmod_step divmod_div_mod by (cases "n = 0") simp_all 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

499 

25942  500 
instance proof 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

501 
fix m n :: nat show "m div n * n + m mod n = m" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

502 
using divmod_rel [of m n] by (simp add: divmod_rel_def) 
25942  503 
next 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

504 
fix n :: nat show "n div 0 = 0" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

505 
using divmod_zero divmod_div_mod [of n 0] by simp 
25942  506 
next 
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

507 
fix n :: nat show "0 div n = 0" 
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

508 
using divmod_rel [of 0 n] by (cases n) (simp_all add: divmod_rel_def) 
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

509 
next 
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

510 
fix m n q :: nat assume "n \<noteq> 0" then show "(q + m * n) div n = m + q div n" 
25942  511 
by (induct m) (simp_all add: le_div_geq) 
512 
qed 

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

513 

25942  514 
end 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

515 

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

516 
text {* Simproc for cancelling @{const div} and @{const mod} *} 
25942  517 

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

518 
(*lemmas mod_div_equality_nat = semiring_div_class.times_div_mod_plus_zero_one.mod_div_equality [of "m\<Colon>nat" n, standard] 
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

519 
lemmas mod_div_equality2_nat = mod_div_equality2 [of "n\<Colon>nat" m, standard*) 
25942  520 

521 
ML {* 

522 
structure CancelDivModData = 

523 
struct 

524 

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

525 
val div_name = @{const_name div}; 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

526 
val mod_name = @{const_name mod}; 
25942  527 
val mk_binop = HOLogic.mk_binop; 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

528 
val mk_sum = ArithData.mk_sum; 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

529 
val dest_sum = ArithData.dest_sum; 
25942  530 

531 
(*logic*) 

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

532 

25942  533 
val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}] 
534 

535 
val trans = trans 

536 

537 
val prove_eq_sums = 

538 
let val simps = @{thm add_0} :: @{thm add_0_right} :: @{thms add_ac} 

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

539 
in ArithData.prove_conv all_tac (ArithData.simp_all_tac simps) end; 
25942  540 

541 
end; 

542 

543 
structure CancelDivMod = CancelDivModFun(CancelDivModData); 

544 

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

545 
val cancel_div_mod_proc = Simplifier.simproc (the_context ()) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

546 
"cancel_div_mod" ["(m::nat) + n"] (K CancelDivMod.proc); 
25942  547 

548 
Addsimprocs[cancel_div_mod_proc]; 

549 
*} 

550 

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

551 
text {* code generator setup *} 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

552 

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

553 
lemma divmod_if [code]: "divmod m n = (if n = 0 \<or> m < n then (0, m) else 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

554 
let (q, r) = divmod (m  n) n in (Suc q, r))" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

555 
by (simp add: divmod_zero divmod_base divmod_step) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

556 
(simp add: divmod_div_mod) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

557 

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

558 
code_modulename SML 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

559 
Divides Nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

560 

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

561 
code_modulename OCaml 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

562 
Divides Nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

563 

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

564 
code_modulename Haskell 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

565 
Divides Nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

566 

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

567 

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

568 
subsubsection {* Quotient *} 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

569 

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

570 
lemma div_geq: "0 < n \<Longrightarrow> \<not> m < n \<Longrightarrow> m div n = Suc ((m  n) div n)" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

571 
by (simp add: le_div_geq linorder_not_less) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

572 

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

573 
lemma div_if: "0 < n \<Longrightarrow> m div n = (if m < n then 0 else Suc ((m  n) div n))" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

574 
by (simp add: div_geq) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

575 

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

576 
lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)" 
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

577 
by simp 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

578 

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

579 
lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)" 
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

580 
by simp 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

581 

25942  582 

583 
subsubsection {* Remainder *} 

584 

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

585 
lemma mod_less_divisor [simp]: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

586 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

587 
assumes "n > 0" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

588 
shows "m mod n < (n::nat)" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

589 
using assms divmod_rel unfolding divmod_rel_def by auto 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

590 

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

591 
lemma mod_less_eq_dividend [simp]: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

592 
fixes m n :: nat 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

593 
shows "m mod n \<le> m" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

594 
proof (rule add_leD2) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

595 
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

596 
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

597 
qed 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

598 

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

599 
lemma mod_geq: "\<not> m < (n\<Colon>nat) \<Longrightarrow> m mod n = (m  n) mod n" 
25942  600 
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

601 

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

602 
lemma mod_if: "m mod (n\<Colon>nat) = (if m < n then m else (m  n) mod n)" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

603 
by (simp add: le_mod_geq) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

604 

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

605 
lemma mod_1 [simp]: "m mod Suc 0 = 0" 
22718  606 
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

607 

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

608 
lemma mod_mult_distrib: "(m mod n) * (k\<Colon>nat) = (m * k) mod (n * k)" 
22718  609 
apply (cases "n = 0", simp) 
610 
apply (cases "k = 0", simp) 

611 
apply (induct m rule: nat_less_induct) 

612 
apply (subst mod_if, simp) 

613 
apply (simp add: mod_geq diff_mult_distrib) 

614 
done 

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

615 

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

616 
lemma mod_mult_distrib2: "(k::nat) * (m mod n) = (k*m) mod (k*n)" 
22718  617 
by (simp add: mult_commute [of k] mod_mult_distrib) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

618 

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

619 
(* 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

620 
lemma mult_div_cancel: "(n::nat) * (m div n) = m  (m mod n)" 
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

621 
by (cut_tac a = m and b = n in mod_div_equality2, arith) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

622 

15439  623 
lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)" 
22718  624 
apply (drule mod_less_divisor [where m = m]) 
625 
apply simp 

626 
done 

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

627 

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

628 
subsubsection {* Quotient and Remainder *} 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

629 

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

630 
lemma divmod_rel_mult1_eq: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

631 
"[ divmod_rel b c q r; c > 0 ] 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

632 
==> divmod_rel (a*b) c (a*q + a*r div c) (a*r mod c)" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

633 
by (auto simp add: split_ifs mult_ac divmod_rel_def add_mult_distrib2) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

634 

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

635 
lemma div_mult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::nat)" 
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

636 
apply (cases "c = 0", simp) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

637 
apply (blast intro: divmod_rel [THEN divmod_rel_mult1_eq, THEN div_eq]) 
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

638 
done 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

639 

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

640 
lemma mod_mult1_eq: "(a*b) mod c = a*(b mod c) mod (c::nat)" 
29403
fe17df4e4ab3
generalize some div/mod lemmas; remove typespecific proofs
huffman
parents:
29252
diff
changeset

641 
by (rule mod_mult_right_eq) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

642 

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

643 
lemma mod_mult1_eq': "(a*b) mod (c::nat) = ((a mod c) * b) mod c" 
29403
fe17df4e4ab3
generalize some div/mod lemmas; remove typespecific proofs
huffman
parents:
29252
diff
changeset

644 
by (rule mod_mult_left_eq) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

645 

25162  646 
lemma mod_mult_distrib_mod: 
647 
"(a*b) mod (c::nat) = ((a mod c) * (b mod c)) mod c" 

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

648 
by (rule mod_mult_eq) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

649 

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

650 
lemma divmod_rel_add1_eq: 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

651 
"[ divmod_rel a c aq ar; divmod_rel b c bq br; c > 0 ] 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

652 
==> divmod_rel (a + b) c (aq + bq + (ar+br) div c) ((ar + br) mod c)" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

653 
by (auto simp add: split_ifs mult_ac divmod_rel_def add_mult_distrib2) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

654 

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

655 
(*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

656 
lemma div_add1_eq: 
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

657 
"(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)" 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

658 
apply (cases "c = 0", simp) 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

659 
apply (blast intro: divmod_rel_add1_eq [THEN div_eq] divmod_rel) 
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

660 
done 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

661 

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

662 
lemma mod_add1_eq: "(a+b) mod (c::nat) = (a mod c + b mod c) mod c" 
29403
fe17df4e4ab3
generalize some div/mod lemmas; remove typespecific proofs
huffman
parents:
29252
diff
changeset

663 
by (rule mod_add_eq) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

664 

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

665 
lemma mod_lemma: "[ (0::nat) < c; r < b ] ==> b * (q mod c) + r < b * c" 
22718  666 
apply (cut_tac m = q and n = c in mod_less_divisor) 
667 
apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto) 

668 
apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst) 

669 
apply (simp add: add_mult_distrib2) 

670 
done 

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

671 

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

672 
lemma divmod_rel_mult2_eq: "[ divmod_rel a b q r; 0 < b; 0 < c ] 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

673 
==> divmod_rel a (b*c) (q div c) (b*(q mod c) + r)" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

674 
by (auto simp add: mult_ac divmod_rel_def add_mult_distrib2 [symmetric] mod_lemma) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

675 

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

676 
lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)" 
22718  677 
apply (cases "b = 0", simp) 
678 
apply (cases "c = 0", simp) 

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

679 
apply (force simp add: divmod_rel [THEN divmod_rel_mult2_eq, THEN div_eq]) 
22718  680 
done 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

681 

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

682 
lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)" 
22718  683 
apply (cases "b = 0", simp) 
684 
apply (cases "c = 0", simp) 

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

685 
apply (auto simp add: mult_commute divmod_rel [THEN divmod_rel_mult2_eq, THEN mod_eq]) 
22718  686 
done 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

687 

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

688 

25942  689 
subsubsection{*Cancellation of Common Factors in Division*} 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

690 

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

691 
lemma div_mult_mult_lemma: 
22718  692 
"[ (0::nat) < b; 0 < c ] ==> (c*a) div (c*b) = a div b" 
693 
by (auto simp add: div_mult2_eq) 

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

694 

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

695 
lemma div_mult_mult1 [simp]: "(0::nat) < c ==> (c*a) div (c*b) = a div b" 
22718  696 
apply (cases "b = 0") 
697 
apply (auto simp add: linorder_neq_iff [of b] div_mult_mult_lemma) 

698 
done 

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

699 

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

700 
lemma div_mult_mult2 [simp]: "(0::nat) < c ==> (a*c) div (b*c) = a div b" 
22718  701 
apply (drule div_mult_mult1) 
702 
apply (auto simp add: mult_commute) 

703 
done 

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

704 

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

705 

25942  706 
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

707 

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

708 
lemma div_1 [simp]: "m div Suc 0 = m" 
22718  709 
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

710 

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

711 

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

712 
(* Monotonicity of div in first argument *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

713 
lemma div_le_mono [rule_format (no_asm)]: 
22718  714 
"\<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

715 
apply (case_tac "k=0", simp) 
15251  716 
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

717 
apply (case_tac "n<k") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

718 
(* 1 case n<k *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

719 
apply simp 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

720 
(* 2 case n >= k *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

721 
apply (case_tac "m<k") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

722 
(* 2.1 case m<k *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

723 
apply simp 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

724 
(* 2.2 case m>=k *) 
15439  725 
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

726 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

727 

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

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

729 
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

730 
apply (subgoal_tac "0<n") 
22718  731 
prefer 2 apply simp 
15251  732 
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

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

734 
apply (case_tac "k<n", simp) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

735 
apply (subgoal_tac "~ (k<m) ") 
22718  736 
prefer 2 apply simp 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

737 
apply (simp add: div_geq) 
15251  738 
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

739 
prefer 2 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

740 
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

741 
apply (rule le_trans, simp) 
15439  742 
apply (simp) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

743 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

744 

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

745 
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

746 
apply (case_tac "n=0", simp) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

747 
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

748 
apply (rule div_le_mono2) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

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

750 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

751 

22718  752 
(* Similar for "less than" *) 
17085  753 
lemma div_less_dividend [rule_format]: 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

754 
"!!n::nat. 1<n ==> 0 < m > m div n < m" 
15251  755 
apply (induct_tac m rule: nat_less_induct) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

756 
apply (rename_tac "m") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

757 
apply (case_tac "m<n", simp) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

758 
apply (subgoal_tac "0<n") 
22718  759 
prefer 2 apply simp 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

760 
apply (simp add: div_geq) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

761 
apply (case_tac "n<m") 
15251  762 
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

763 
apply (rule impI less_trans_Suc)+ 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

764 
apply assumption 
15439  765 
apply (simp_all) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

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

767 

17085  768 
declare div_less_dividend [simp] 
769 

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

770 
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

771 
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

772 
apply (case_tac "n=0", simp) 
15251  773 
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

774 
apply (case_tac "Suc (na) <n") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

775 
(* case Suc(na) < n *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

776 
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

777 
(* case n \<le> Suc(na) *) 
16796  778 
apply (simp add: linorder_not_less le_Suc_eq mod_geq) 
15439  779 
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

780 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

781 

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

782 
lemma nat_mod_div_trivial: "m mod n div n = (0 :: nat)" 
fe17df4e4ab3
generalize some div/mod lemmas; remove typespecific proofs
huffman
parents:
29252
diff
changeset

783 
by simp 
14437  784 

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

785 
lemma nat_mod_mod_trivial: "m mod n mod n = (m mod n :: nat)" 
fe17df4e4ab3
generalize some div/mod lemmas; remove typespecific proofs
huffman
parents:
29252
diff
changeset

786 
by simp 
14437  787 

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

788 

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

789 
subsubsection {* The Divides Relation *} 
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24268
diff
changeset

790 

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

791 
lemma dvd_1_left [iff]: "Suc 0 dvd k" 
22718  792 
unfolding dvd_def by simp 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

793 

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

794 
lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)" 
22718  795 
by (simp add: dvd_def) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

796 

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

797 
lemma dvd_anti_sym: "[ m dvd n; n dvd m ] ==> m = (n::nat)" 
22718  798 
unfolding dvd_def 
799 
by (force dest: mult_eq_self_implies_10 simp add: mult_assoc mult_eq_1_iff) 

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

800 

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

801 
text {* @{term "op dvd"} is a partial order *} 
8c508c4dc53b
introduced (auxiliary) class dvd_mod for more convenient code generation
haftmann
parents:
23162
diff
changeset

802 

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

803 
interpretation dvd!: order "op dvd" "\<lambda>n m \<Colon> nat. n dvd m \<and> \<not> m dvd n" 
28823  804 
proof qed (auto intro: dvd_refl dvd_trans dvd_anti_sym) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

805 

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

806 
lemma dvd_diff: "[ k dvd m; k dvd n ] ==> k dvd (mn :: nat)" 
22718  807 
unfolding dvd_def 
808 
by (blast intro: diff_mult_distrib2 [symmetric]) 

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

809 

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

810 
lemma dvd_diffD: "[ k dvd mn; k dvd n; n\<le>m ] ==> k dvd (m::nat)" 
22718  811 
apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst]) 
812 
apply (blast intro: dvd_add) 

813 
done 

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

814 

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

815 
lemma dvd_diffD1: "[ k dvd mn; k dvd m; n\<le>m ] ==> k dvd (n::nat)" 
22718  816 
by (drule_tac m = m in dvd_diff, auto) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

817 

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

818 
lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))" 
22718  819 
apply (rule iffI) 
820 
apply (erule_tac [2] dvd_add) 

821 
apply (rule_tac [2] dvd_refl) 

822 
apply (subgoal_tac "n = (n+k) k") 

823 
prefer 2 apply simp 

824 
apply (erule ssubst) 

825 
apply (erule dvd_diff) 

826 
apply (rule dvd_refl) 

827 
done 

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

828 

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

829 
lemma dvd_mod: "!!n::nat. [ f dvd m; f dvd n ] ==> f dvd m mod n" 
22718  830 
unfolding dvd_def 
831 
apply (case_tac "n = 0", auto) 

832 
apply (blast intro: mod_mult_distrib2 [symmetric]) 

833 
done 

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

834 

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

835 
lemma dvd_mod_imp_dvd: "[ (k::nat) dvd m mod n; k dvd n ] ==> k dvd m" 
22718  836 
apply (subgoal_tac "k dvd (m div n) *n + m mod n") 
837 
apply (simp add: mod_div_equality) 

838 
apply (simp only: dvd_add dvd_mult) 

839 
done 

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

840 

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

841 
lemma dvd_mod_iff: "k dvd n ==> ((k::nat) dvd m mod n) = (k dvd m)" 
22718  842 
by (blast intro: dvd_mod_imp_dvd dvd_mod) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

843 

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

844 
lemma dvd_mult_cancel: "!!k::nat. [ k*m dvd k*n; 0<k ] ==> m dvd n" 
22718  845 
unfolding dvd_def 
846 
apply (erule exE) 

847 
apply (simp add: mult_ac) 

848 
done 

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

849 

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

850 
lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))" 
22718  851 
apply auto 
852 
apply (subgoal_tac "m*n dvd m*1") 

853 
apply (drule dvd_mult_cancel, auto) 

854 
done 

14267
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 dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))" 
22718  857 
apply (subst mult_commute) 
858 
apply (erule dvd_mult_cancel1) 

859 
done 

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 
lemma dvd_imp_le: "[ k dvd n; 0 < n ] ==> k \<le> (n::nat)" 
22718  862 
apply (unfold dvd_def, clarify) 
863 
apply (simp_all (no_asm_use) add: zero_less_mult_iff) 

864 
apply (erule conjE) 

865 
apply (rule le_trans) 

866 
apply (rule_tac [2] le_refl [THEN mult_le_mono]) 

867 
apply (erule_tac [2] Suc_leI, simp) 

868 
done 

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

869 

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

870 
lemma dvd_mult_div_cancel: "n dvd m ==> n * (m div n) = (m::nat)" 
22718  871 
apply (subgoal_tac "m mod n = 0") 
872 
apply (simp add: mult_div_cancel) 

873 
apply (simp only: dvd_eq_mod_eq_0) 

874 
done 

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

875 

21408  876 
lemma le_imp_power_dvd: "!!i::nat. m \<le> n ==> i^m dvd i^n" 
22718  877 
apply (unfold dvd_def) 
878 
apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst]) 

879 
apply (simp add: power_add) 

880 
done 

21408  881 

25162  882 
lemma nat_zero_less_power_iff [simp]: "(x^n > 0) = (x > (0::nat)  n=0)" 
22718  883 
by (induct n) auto 
21408  884 

885 
lemma power_le_dvd [rule_format]: "k^j dvd n > i\<le>j > k^i dvd (n::nat)" 

22718  886 
apply (induct j) 
887 
apply (simp_all add: le_Suc_eq) 

888 
apply (blast dest!: dvd_mult_right) 

889 
done 

21408  890 

891 
lemma power_dvd_imp_le: "[i^m dvd i^n; (1::nat) < i] ==> m \<le> n" 

22718  892 
apply (rule power_le_imp_le_exp, assumption) 
893 
apply (erule dvd_imp_le, simp) 

894 
done 

21408  895 

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

896 
lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)" 
22718  897 
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

898 

22718  899 
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

900 

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

901 
(*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

902 
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

903 
apply (cut_tac a = m in mod_div_equality) 
22718  904 
apply (simp only: add_ac) 
905 
apply (blast intro: sym) 

906 
done 

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

907 

13152  908 
lemma split_div: 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

909 
"P(n div k :: nat) = 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

910 
((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

911 
(is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))") 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

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

913 
assume P: ?P 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

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

915 
proof (cases) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

916 
assume "k = 0" 
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

917 
with P show ?Q by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

918 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

919 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

920 
thus ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

921 
proof (simp, intro allI impI) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

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

923 
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

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

925 
proof (cases) 
22718  926 
assume "i = 0" 
927 
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

928 
next 
22718  929 
assume "i \<noteq> 0" 
930 
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

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

932 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

933 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

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

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

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

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

938 
assume "k = 0" 
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

939 
with Q show ?P by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

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

941 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

942 
with Q have R: ?R by simp 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

943 
from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"] 
13517  944 
show ?P by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

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

946 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

947 

13882  948 
lemma split_div_lemma: 
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

949 
assumes "0 < n" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

950 
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

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

952 
assume ?rhs 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

953 
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

954 
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

955 
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

956 
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

957 
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

958 
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

959 
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

960 
from A B show ?lhs .. 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

961 
next 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

962 
assume P: ?lhs 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

963 
then have "divmod_rel m n q (m  n * q)" 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

964 
unfolding divmod_rel_def by (auto simp add: mult_ac) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

965 
then show ?rhs using divmod_rel by (rule divmod_rel_unique_div) 
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset

966 
qed 
13882  967 

968 
theorem split_div': 

969 
"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

970 
(\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))" 
13882  971 
apply (case_tac "0 < n") 
972 
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

973 
apply simp_all 
13882  974 
done 
975 

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

976 
lemma split_mod: 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

977 
"P(n mod k :: nat) = 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

978 
((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

979 
(is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))") 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

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

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

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

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

984 
assume "k = 0" 
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

985 
with P show ?Q by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

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

987 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

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

989 
proof (simp, intro allI impI) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

990 
fix i j 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

991 
assume "n = k*i + j" "j < k" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

992 
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

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

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

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

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

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

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

999 
assume "k = 0" 
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

1000 
with Q show ?P by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1001 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1002 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1003 
with Q have R: ?R by simp 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1004 
from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"] 
13517  1005 
show ?P by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

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

1007 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

1008 

13882  1009 
theorem mod_div_equality': "(m::nat) mod n = m  (m div n) * n" 
1010 
apply (rule_tac P="%x. m mod n = x  (m div n) * n" in 

1011 
subst [OF mod_div_equality [of _ n]]) 

1012 
apply arith 

1013 
done 

1014 

22800  1015 
lemma div_mod_equality': 
1016 
fixes m n :: nat 

1017 
shows "m div n * n = m  m mod n" 

1018 
proof  

1019 
have "m mod n \<le> m mod n" .. 

1020 
from div_mod_equality have 

1021 
"m div n * n + m mod n  m mod n = m  m mod n" by simp 

1022 
with diff_add_assoc [OF `m mod n \<le> m mod n`, of "m div n * n"] have 

1023 
"m div n * n + (m mod n  m mod n) = m  m mod n" 

1024 
by simp 

1025 
then show ?thesis by simp 

1026 
qed 

1027 

1028 

25942  1029 
subsubsection {*An ``induction'' law for modulus arithmetic.*} 
14640  1030 

1031 
lemma mod_induct_0: 

1032 
assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)" 

1033 
and base: "P i" and i: "i<p" 

1034 
shows "P 0" 

1035 
proof (rule ccontr) 

1036 
assume contra: "\<not>(P 0)" 

1037 
from i have p: "0<p" by simp 

1038 
have "\<forall>k. 0<k \<longrightarrow> \<not> P (pk)" (is "\<forall>k. ?A k") 

1039 
proof 

1040 
fix k 

1041 
show "?A k" 

1042 
proof (induct k) 

1043 
show "?A 0" by simp  "by contradiction" 

1044 
next 

1045 
fix n 

1046 
assume ih: "?A n" 

1047 
show "?A (Suc n)" 

1048 
proof (clarsimp) 

22718  1049 
assume y: "P (p  Suc n)" 
1050 
have n: "Suc n < p" 

1051 
proof (rule ccontr) 

1052 
assume "\<not>(Suc n < p)" 

1053 
hence "p  Suc n = 0" 

1054 
by simp 

1055 
with y contra show "False" 

1056 
by simp 

1057 
qed 

1058 
hence n2: "Suc (p  Suc n) = pn" by arith 

1059 
from p have "p  Suc n < p" by arith 

1060 
with y step have z: "P ((Suc (p  Suc n)) mod p)" 

1061 
by blast 

1062 
show "False" 

1063 
proof (cases "n=0") 

1064 
case True 

1065 
with z n2 contra show ?thesis by simp 

1066 
next 

1067 
case False 

1068 
with p have "pn < p" by arith 

1069 
with z n2 False ih show ?thesis by simp 

1070 
qed 

14640  1071 
qed 
1072 
qed 

1073 
qed 

1074 
moreover 

1075 
from i obtain k where "0<k \<and> i+k=p" 

1076 
by (blast dest: less_imp_add_positive) 

1077 
hence "0<k \<and> i=pk" by auto 

1078 
moreover 

1079 
note base 

1080 
ultimately 

1081 
show "False" by blast 

1082 
qed 

1083 

1084 
lemma mod_induct: 

1085 
assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)" 

1086 
and base: "P i" and i: "i<p" and j: "j<p" 

1087 
shows "P j" 

1088 
proof  

1089 
have "\<forall>j<p. P j" 

1090 
proof 

1091 
fix j 

1092 
show "j<p \<longrightarrow> P j" (is "?A j") 

1093 
proof (induct j) 

1094 
from step base i show "?A 0" 

22718  1095 
by (auto elim: mod_induct_0) 
14640  1096 
next 
1097 
fix k 

1098 
assume ih: "?A k" 

1099 
show "?A (Suc k)" 

1100 
proof 

22718  1101 
assume suc: "Suc k < p" 
1102 
hence k: "k<p" by simp 

1103 
with ih have "P k" .. 

1104 
with step k have "P (Suc k mod p)" 

1105 
by blast 

1106 
moreover 

1107 
from suc have "Suc k mod p = Suc k" 

1108 
by simp 

1109 
ultimately 

1110 
show "P (Suc k)" by simp 

14640  1111 
qed 
1112 
qed 

1113 
qed 

1114 
with j show ?thesis by blast 

1115 
qed 

1116 

3366  1117 
end 