src/HOL/NumberTheory/Primes.thy
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(*  Title:      HOL/NumberTheory/Primes.thy
ID:         \$Id\$
Author:     Christophe Tabacznyj and Lawrence C Paulson

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"
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 (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)
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"
done

(*uniqueness of GCDs*)
lemma is_gcd_unique: "[| is_gcd m a b; is_gcd n a b |] ==> m=n"
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"
done

(** Commutativity **)

lemma is_gcd_commute: "is_gcd k m n = is_gcd k n m"
done

lemma gcd_commute: "gcd(m,n) = gcd(n,m)"
apply (rule is_gcd_unique)
apply (rule is_gcd)
apply (subst is_gcd_commute)
done

lemma gcd_assoc: "gcd(gcd(k,m),n) = gcd(k,gcd(m,n))"
apply (rule is_gcd_unique)
apply (rule is_gcd)
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 ==> 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 (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

lemma gcd_add1 [simp]: "gcd(m+n, n) = gcd(m,n)"
apply (case_tac "n=0")
done

lemma gcd_add2 [simp]: "gcd(m, m+n) = gcd(m,n)"
apply (rule gcd_commute [THEN trans])
apply (rule gcd_commute)
done

lemma gcd_add2' [simp]: "gcd(m, n+m) = gcd(m,n)"
done

lemma gcd_add_mult: "gcd(m, k*m+n) = gcd(m,n)"
apply (induct_tac "k")
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)