--- a/src/HOL/Library/Primes.thy Fri Jul 08 11:38:53 2005 +0200
+++ b/src/HOL/Library/Primes.thy Fri Jul 08 11:39:08 2005 +0200
@@ -4,28 +4,13 @@
Copyright 1996 University of Cambridge
*)
-header {* The Greatest Common Divisor and Euclid's algorithm *}
+header {* Primality on nat *}
theory Primes
imports Main
begin
-text {*
- See \cite{davenport92}.
- \bigskip
-*}
-
-consts
- gcd :: "nat \<times> nat => nat" -- {* Euclid's algorithm *}
-
-recdef gcd "measure ((\<lambda>(m, n). n) :: nat \<times> nat => nat)"
- "gcd (m, n) = (if n = 0 then m else gcd (n, m mod n))"
-
constdefs
- is_gcd :: "nat => nat => nat => bool" -- {* @{term gcd} as a relation *}
- "is_gcd p m n == p dvd m \<and> p dvd n \<and>
- (\<forall>d. d dvd m \<and> d dvd n --> d dvd p)"
-
coprime :: "nat => nat => bool"
"coprime m n == gcd (m, n) = 1"
@@ -33,143 +18,10 @@
"prime p == 1 < p \<and> (\<forall>m. m dvd p --> m = 1 \<or> m = p)"
-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 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, Suc 0) = 1"
- apply (simp add: gcd_non_0)
- done
-
-text {*
- \medskip @{term "gcd (m, n)"} divides @{text m} and @{text n}. The
- conjunctions don't seem provable separately.
-*}
-
-lemma gcd_dvd1 [iff]: "gcd (m, n) dvd m"
- and gcd_dvd2 [iff]: "gcd (m, n) dvd n"
- apply (induct m n rule: gcd_induct)
- apply (simp_all add: gcd_non_0)
- apply (blast dest: dvd_mod_imp_dvd)
- done
-
-text {*
- \medskip Maximality: for all @{term m}, @{term n}, @{term k}
- naturals, if @{term k} divides @{term m} and @{term k} divides
- @{term n} then @{term k} divides @{term "gcd (m, n)"}.
-*}
-
-lemma gcd_greatest: "k dvd m ==> k dvd n ==> k dvd gcd (m, n)"
- apply (induct 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 \<and> k dvd n)"
- apply (blast intro!: gcd_greatest intro: dvd_trans)
- done
-
-lemma gcd_zero: "(gcd (m, n) = 0) = (m = 0 \<and> n = 0)"
- by (simp only: dvd_0_left_iff [THEN sym] gcd_greatest_iff)
-
-
-text {*
- \medskip 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
-
-text {*
- \medskip 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
-
-
-text {*
- \medskip 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 (Suc 0, m) = 1"
- apply (simp add: gcd_commute [of "Suc 0"])
- done
-
-
-text {*
- \medskip Multiplication laws
-*}
-
-lemma gcd_mult_distrib2: "k * gcd (m, n) = gcd (k * m, k * n)"
- -- {* \cite[page 27]{davenport92} *}
- apply (induct 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, symmetric])
- 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)
+lemma two_is_prime: "prime 2"
+ apply (auto simp add: prime_def)
+ apply (case_tac m)
+ apply (auto dest!: dvd_imp_le)
done
lemma prime_imp_relprime: "prime p ==> \<not> p dvd n ==> gcd (p, n) = 1"
@@ -180,12 +32,6 @@
apply simp
done
-lemma two_is_prime: "prime 2"
- apply (auto simp add: prime_def)
- apply (case_tac m)
- apply (auto dest!: dvd_imp_le)
- done
-
text {*
This theorem leads immediately to a proof of the uniqueness of
factorization. If @{term p} divides a product of primes then it is
@@ -202,43 +48,4 @@
by (rule prime_dvd_square) (simp_all add: power2_eq_square)
-text {* \medskip 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)"
-proof -
- have "gcd (m, m + n) = gcd (m + n, m)" by (rule gcd_commute)
- also have "... = gcd (n + m, m)" by (simp add: add_commute)
- also have "... = gcd (n, m)" by simp
- also have "... = gcd (m, n)" by (rule gcd_commute)
- finally show ?thesis .
-qed
-
-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 k)
- apply (simp_all add: add_assoc)
- done
-
-
-text {* \medskip 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)
- apply (blast intro: dvd_trans)
- done
-
end