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+++ b/src/HOL/NumberTheory/Primes.thy Wed Sep 13 18:46:45 2000 +0200
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+(* Title: HOL/ex/Primes.thy
+ ID: $Id$
+ Author: Christophe Tabacznyj and Lawrence C Paulson
+ Copyright 1996 University of Cambridge
+
+The Greatest Common Divisor and Euclid's algorithm
+
+See H. Davenport, "The Higher Arithmetic". 6th edition. (CUP, 1992)
+*)
+
+theory Primes = Main:
+consts
+ gcd :: "nat*nat=>nat" (*Euclid's algorithm *)
+
+recdef gcd "measure ((%(m,n).n) ::nat*nat=>nat)"
+ "gcd (m, n) = (if n=0 then m else gcd(n, m mod n))"
+
+constdefs
+ is_gcd :: "[nat,nat,nat]=>bool" (*gcd as a relation*)
+ "is_gcd p m n == p dvd m & p dvd n &
+ (ALL d. d dvd m & d dvd n --> d dvd p)"
+
+ coprime :: "[nat,nat]=>bool"
+ "coprime m n == gcd(m,n) = 1"
+
+ prime :: "nat set"
+ "prime == {p. 1<p & (ALL m. m dvd p --> m=1 | m=p)}"
+
+
+(************************************************)
+(** Greatest Common Divisor **)
+(************************************************)
+
+(*** Euclid's Algorithm ***)
+
+
+lemma gcd_induct:
+ "[| !!m. P m 0;
+ !!m n. [| 0<n; P n (m mod n) |] ==> P m n
+ |] ==> P (m::nat) (n::nat)"
+ apply (induct_tac m n rule: gcd.induct)
+ apply (case_tac "n=0")
+ apply (simp_all)
+ done
+
+
+lemma gcd_0 [simp]: "gcd(m,0) = m"
+ apply (simp);
+ done
+
+lemma gcd_non_0: "0<n ==> gcd(m,n) = gcd (n, m mod n)"
+ apply (simp)
+ done;
+
+declare gcd.simps [simp del];
+
+lemma gcd_1 [simp]: "gcd(m,1) = 1"
+ apply (simp add: gcd_non_0)
+ done
+
+(*gcd(m,n) divides m and n. The conjunctions don't seem provable separately*)
+lemma gcd_dvd_both: "(gcd(m,n) dvd m) & (gcd(m,n) dvd n)"
+ apply (induct_tac m n rule: gcd_induct)
+ apply (simp_all add: gcd_non_0)
+ apply (blast dest: dvd_mod_imp_dvd)
+ done
+
+lemmas gcd_dvd1 [iff] = gcd_dvd_both [THEN conjunct1]
+lemmas gcd_dvd2 [iff] = gcd_dvd_both [THEN conjunct2];
+
+
+(*Maximality: for all m,n,k naturals,
+ if k divides m and k divides n then k divides gcd(m,n)*)
+lemma gcd_greatest [rule_format]: "(k dvd m) --> (k dvd n) --> k dvd gcd(m,n)"
+ apply (induct_tac m n rule: gcd_induct)
+ apply (simp_all add: gcd_non_0 dvd_mod);
+ done;
+
+lemma gcd_greatest_iff [iff]: "k dvd gcd(m,n) = (k dvd m & k dvd n)"
+ apply (blast intro!: gcd_greatest intro: dvd_trans);
+ done;
+
+(*Function gcd yields the Greatest Common Divisor*)
+lemma is_gcd: "is_gcd (gcd(m,n)) m n"
+ apply (simp add: is_gcd_def gcd_greatest)
+ done
+
+(*uniqueness of GCDs*)
+lemma is_gcd_unique: "[| is_gcd m a b; is_gcd n a b |] ==> m=n"
+ apply (simp add: is_gcd_def);
+ apply (blast intro: dvd_anti_sym)
+ done
+
+lemma is_gcd_dvd: "[| is_gcd m a b; k dvd a; k dvd b |] ==> k dvd m"
+ apply (auto simp add: is_gcd_def);
+ done
+
+(** Commutativity **)
+
+lemma is_gcd_commute: "is_gcd k m n = is_gcd k n m"
+ apply (auto simp add: is_gcd_def);
+ done
+
+lemma gcd_commute: "gcd(m,n) = gcd(n,m)"
+ apply (rule is_gcd_unique)
+ apply (rule is_gcd)
+ apply (subst is_gcd_commute)
+ apply (simp add: is_gcd)
+ done
+
+lemma gcd_assoc: "gcd(gcd(k,m),n) = gcd(k,gcd(m,n))"
+ apply (rule is_gcd_unique)
+ apply (rule is_gcd)
+ apply (simp add: is_gcd_def);
+ apply (blast intro: dvd_trans);
+ done
+
+lemma gcd_0_left [simp]: "gcd(0,m) = m"
+ apply (simp add: gcd_commute [of 0])
+ done
+
+lemma gcd_1_left [simp]: "gcd(1,m) = 1"
+ apply (simp add: gcd_commute [of 1])
+ done
+
+
+(** Multiplication laws **)
+
+(*Davenport, page 27*)
+lemma gcd_mult_distrib2: "k * gcd(m,n) = gcd(k*m, k*n)"
+ apply (induct_tac m n rule: gcd_induct)
+ apply (simp)
+ apply (case_tac "k=0")
+ apply (simp_all add: mod_geq gcd_non_0 mod_mult_distrib2)
+ done
+
+lemma gcd_mult [simp]: "gcd(k, k*n) = k"
+ apply (rule gcd_mult_distrib2 [of k 1 n, simplified, THEN sym])
+ done
+
+lemma gcd_self [simp]: "gcd(k,k) = k"
+ apply (rule gcd_mult [of k 1, simplified])
+ done
+
+lemma relprime_dvd_mult: "[| gcd(k,n)=1; k dvd (m*n) |] ==> k dvd m";
+ apply (insert gcd_mult_distrib2 [of m k n])
+ apply (simp)
+ apply (erule_tac t="m" in ssubst);
+ apply (simp)
+ done
+
+lemma relprime_dvd_mult_iff: "gcd(k,n)=1 \<Longrightarrow> k dvd (m*n) = k dvd m";
+ apply (blast intro: relprime_dvd_mult dvd_trans)
+ done
+
+lemma prime_imp_relprime: "[| p: prime; ~ p dvd n |] ==> gcd (p, n) = 1"
+ apply (auto simp add: prime_def)
+ apply (drule_tac x="gcd(p,n)" in spec)
+ apply auto
+ apply (insert gcd_dvd2 [of p n])
+ apply (simp)
+ done
+
+(*This theorem leads immediately to a proof of the uniqueness of factorization.
+ If p divides a product of primes then it is one of those primes.*)
+lemma prime_dvd_mult: "[| p: prime; p dvd (m*n) |] ==> p dvd m | p dvd n"
+ apply (blast intro: relprime_dvd_mult prime_imp_relprime)
+ done
+
+
+(** Addition laws **)
+
+lemma gcd_add1 [simp]: "gcd(m+n, n) = gcd(m,n)"
+ apply (case_tac "n=0")
+ apply (simp_all add: gcd_non_0)
+ done
+
+lemma gcd_add2 [simp]: "gcd(m, m+n) = gcd(m,n)"
+ apply (rule gcd_commute [THEN trans])
+ apply (subst add_commute)
+ apply (simp add: gcd_add1)
+ apply (rule gcd_commute)
+ done
+
+lemma gcd_add2' [simp]: "gcd(m, n+m) = gcd(m,n)"
+ apply (subst add_commute)
+ apply (rule gcd_add2)
+ done
+
+lemma gcd_add_mult: "gcd(m, k*m+n) = gcd(m,n)"
+ apply (induct_tac "k")
+ apply (simp_all add: gcd_add2 add_assoc)
+ done
+
+
+(** More multiplication laws **)
+
+lemma gcd_mult_cancel: "gcd(k,n) = 1 ==> gcd(k*m, n) = gcd(m,n)"
+ apply (rule dvd_anti_sym)
+ apply (rule gcd_greatest)
+ apply (rule_tac n="k" in relprime_dvd_mult)
+ apply (simp add: gcd_assoc)
+ apply (simp add: gcd_commute)
+ apply (simp_all add: mult_commute gcd_dvd1 gcd_dvd2)
+ apply (blast intro: gcd_dvd1 dvd_trans);
+ done
+
+end