(*  Title:      ZF/ex/Primes.thy

     ID:         $Id$

     Author:     Christophe Tabacznyj and Lawrence C Paulson

     Copyright   1996  University of Cambridge

 *)

 header{*The Divides Relation and Euclid's algorithm for the GCD*}

 theory Primes = Main:

 constdefs

   divides :: "[i,i]=>o"              (infixl "dvd" 50) 

     "m dvd n == m \<in> nat & n \<in> nat & (\<exists>k \<in> nat. n = m#*k)"

   is_gcd  :: "[i,i,i]=>o"     --{*definition of great common divisor*}

     "is_gcd(p,m,n) == ((p dvd m) & (p dvd n))   &

                        (\<forall>d\<in>nat. (d dvd m) & (d dvd n) --> d dvd p)"

   gcd     :: "[i,i]=>i"       --{*Euclid's algorithm for the gcd*}

     "gcd(m,n) == transrec(natify(n),

 			%n f. \<lambda>m \<in> nat.

 			        if n=0 then m else f`(m mod n)`n) ` natify(m)"

   coprime :: "[i,i]=>o"       --{*the coprime relation*}

     "coprime(m,n) == gcd(m,n) = 1"

   prime   :: i                --{*the set of prime numbers*}

    "prime == {p \<in> nat. 1<p & (\<forall>m \<in> nat. m dvd p --> m=1 | m=p)}"

 subsection{*The Divides Relation*}

 lemma dvdD: "m dvd n ==> m \<in> nat & n \<in> nat & (\<exists>k \<in> nat. n = m#*k)"

 by (unfold divides_def, assumption)

 lemma dvdE:

      "[|m dvd n;  !!k. [|m \<in> nat; n \<in> nat; k \<in> nat; n = m#*k|] ==> P|] ==> P"

 by (blast dest!: dvdD)

 lemmas dvd_imp_nat1 = dvdD [THEN conjunct1, standard]

 lemmas dvd_imp_nat2 = dvdD [THEN conjunct2, THEN conjunct1, standard]

 lemma dvd_0_right [simp]: "m \<in> nat ==> m dvd 0"

 apply (simp add: divides_def)

 apply (fast intro: nat_0I mult_0_right [symmetric])

 done

 lemma dvd_0_left: "0 dvd m ==> m = 0"

 by (simp add: divides_def)

 lemma dvd_refl [simp]: "m \<in> nat ==> m dvd m"

 apply (simp add: divides_def)

 apply (fast intro: nat_1I mult_1_right [symmetric])

 done

 lemma dvd_trans: "[| m dvd n; n dvd p |] ==> m dvd p"

 by (auto simp add: divides_def intro: mult_assoc mult_type)

 lemma dvd_anti_sym: "[| m dvd n; n dvd m |] ==> m=n"

 apply (simp add: divides_def)

 apply (force dest: mult_eq_self_implies_10

              simp add: mult_assoc mult_eq_1_iff)

 done

 lemma dvd_mult_left: "[|(i#*j) dvd k; i \<in> nat|] ==> i dvd k"

 by (auto simp add: divides_def mult_assoc)

 lemma dvd_mult_right: "[|(i#*j) dvd k; j \<in> nat|] ==> j dvd k"

 apply (simp add: divides_def, clarify)

 apply (rule_tac x = "i#*k" in bexI)

 apply (simp add: mult_ac)

 apply (rule mult_type)

 done

 subsection{*Euclid's Algorithm for the GCD*}

 lemma gcd_0 [simp]: "gcd(m,0) = natify(m)"

 apply (simp add: gcd_def)

 apply (subst transrec, simp)

 done

 lemma gcd_natify1 [simp]: "gcd(natify(m),n) = gcd(m,n)"

 by (simp add: gcd_def)

 lemma gcd_natify2 [simp]: "gcd(m, natify(n)) = gcd(m,n)"

 by (simp add: gcd_def)

 lemma gcd_non_0_raw: 

     "[| 0<n;  n \<in> nat |] ==> gcd(m,n) = gcd(n, m mod n)"

 apply (simp add: gcd_def)

 apply (rule_tac P = "%z. ?left (z) = ?right" in transrec [THEN ssubst])

 apply (simp add: ltD [THEN mem_imp_not_eq, THEN not_sym] 

                  mod_less_divisor [THEN ltD])

 done

 lemma gcd_non_0: "0 < natify(n) ==> gcd(m,n) = gcd(n, m mod n)"

 apply (cut_tac m = m and n = "natify (n) " in gcd_non_0_raw)

 apply auto

 done

 lemma gcd_1 [simp]: "gcd(m,1) = 1"

 by (simp (no_asm_simp) add: gcd_non_0)

 lemma dvd_add: "[| k dvd a; k dvd b |] ==> k dvd (a #+ b)"

 apply (simp add: divides_def)

 apply (fast intro: add_mult_distrib_left [symmetric] add_type)

 done

 lemma dvd_mult: "k dvd n ==> k dvd (m #* n)"

 apply (simp add: divides_def)

 apply (fast intro: mult_left_commute mult_type)

 done

 lemma dvd_mult2: "k dvd m ==> k dvd (m #* n)"

 apply (subst mult_commute)

 apply (blast intro: dvd_mult)

 done

 (* k dvd (m*k) *)

 lemmas dvdI1 [simp] = dvd_refl [THEN dvd_mult, standard]

 lemmas dvdI2 [simp] = dvd_refl [THEN dvd_mult2, standard]

 lemma dvd_mod_imp_dvd_raw:

      "[| a \<in> nat; b \<in> nat; k dvd b; k dvd (a mod b) |] ==> k dvd a"

 apply (case_tac "b=0") 

  apply (simp add: DIVISION_BY_ZERO_MOD)

 apply (blast intro: mod_div_equality [THEN subst]

              elim: dvdE 

              intro!: dvd_add dvd_mult mult_type mod_type div_type)

 done

 lemma dvd_mod_imp_dvd: "[| k dvd (a mod b); k dvd b; a \<in> nat |] ==> k dvd a"

 apply (cut_tac b = "natify (b)" in dvd_mod_imp_dvd_raw)

 apply auto

 apply (simp add: divides_def)

 done

 (*Imitating TFL*)

 lemma gcd_induct_lemma [rule_format (no_asm)]: "[| n \<in> nat;  

          \<forall>m \<in> nat. P(m,0);  

          \<forall>m \<in> nat. \<forall>n \<in> nat. 0<n --> P(n, m mod n) --> P(m,n) |]  

       ==> \<forall>m \<in> nat. P (m,n)"

 apply (erule_tac i = n in complete_induct)

 apply (case_tac "x=0")

 apply (simp (no_asm_simp))

 apply clarify

 apply (drule_tac x1 = m and x = x in bspec [THEN bspec])

 apply (simp_all add: Ord_0_lt_iff)

 apply (blast intro: mod_less_divisor [THEN ltD])

 done

 lemma gcd_induct: "!!P. [| m \<in> nat; n \<in> nat;  

          !!m. m \<in> nat ==> P(m,0);  

          !!m n. [|m \<in> nat; n \<in> nat; 0<n; P(n, m mod n)|] ==> P(m,n) |]  

       ==> P (m,n)"

 by (blast intro: gcd_induct_lemma)

 subsection{*Basic Properties of @{term gcd}*}

 text{*type of gcd*}

 lemma gcd_type [simp,TC]: "gcd(m, n) \<in> nat"

 apply (subgoal_tac "gcd (natify (m), natify (n)) \<in> nat")

 apply simp

 apply (rule_tac m = "natify (m)" and n = "natify (n)" in gcd_induct)

 apply auto

 apply (simp add: gcd_non_0)

 done

 text{* Property 1: gcd(a,b) divides a and b *}

 lemma gcd_dvd_both:

      "[| m \<in> nat; n \<in> nat |] ==> gcd (m, n) dvd m & gcd (m, n) dvd n"

 apply (rule_tac m = m and n = n in gcd_induct)

 apply (simp_all add: gcd_non_0)

 apply (blast intro: dvd_mod_imp_dvd_raw nat_into_Ord [THEN Ord_0_lt])

 done

 lemma gcd_dvd1 [simp]: "m \<in> nat ==> gcd(m,n) dvd m"

 apply (cut_tac m = "natify (m)" and n = "natify (n)" in gcd_dvd_both)

 apply auto

 done

 lemma gcd_dvd2 [simp]: "n \<in> nat ==> gcd(m,n) dvd n"

 apply (cut_tac m = "natify (m)" and n = "natify (n)" in gcd_dvd_both)

 apply auto

 done

 text{* if f divides a and b then f divides gcd(a,b) *}

 lemma dvd_mod: "[| f dvd a; f dvd b |] ==> f dvd (a mod b)"

 apply (simp add: divides_def)

 apply (case_tac "b=0")

  apply (simp add: DIVISION_BY_ZERO_MOD, auto)

 apply (blast intro: mod_mult_distrib2 [symmetric])

 done

 text{* Property 2: for all a,b,f naturals, 

                if f divides a and f divides b then f divides gcd(a,b)*}

 lemma gcd_greatest_raw [rule_format]:

      "[| m \<in> nat; n \<in> nat; f \<in> nat |]    

       ==> (f dvd m) --> (f dvd n) --> f dvd gcd(m,n)"

 apply (rule_tac m = m and n = n in gcd_induct)

 apply (simp_all add: gcd_non_0 dvd_mod)

 done

 lemma gcd_greatest: "[| f dvd m;  f dvd n;  f \<in> nat |] ==> f dvd gcd(m,n)"

 apply (rule gcd_greatest_raw)

 apply (auto simp add: divides_def)

 done

 lemma gcd_greatest_iff [simp]: "[| k \<in> nat; m \<in> nat; n \<in> nat |]  

       ==> (k dvd gcd (m, n)) <-> (k dvd m & k dvd n)"

 by (blast intro!: gcd_greatest gcd_dvd1 gcd_dvd2 intro: dvd_trans)

 subsection{*The Greatest Common Divisor*}

 text{*The GCD exists and function gcd computes it.*}

 lemma is_gcd: "[| m \<in> nat; n \<in> nat |] ==> is_gcd(gcd(m,n), m, n)"

 by (simp add: is_gcd_def)

 text{*The GCD is unique*}

 lemma is_gcd_unique: "[|is_gcd(m,a,b); is_gcd(n,a,b); m\<in>nat; n\<in>nat|] ==> m=n"

 apply (simp add: is_gcd_def)

 apply (blast intro: dvd_anti_sym)

 done

 lemma is_gcd_commute: "is_gcd(k,m,n) <-> is_gcd(k,n,m)"

 by (simp add: is_gcd_def, blast)

 lemma gcd_commute_raw: "[| m \<in> nat; n \<in> nat |] ==> gcd(m,n) = gcd(n,m)"

 apply (rule is_gcd_unique)

 apply (rule is_gcd)

 apply (rule_tac [3] is_gcd_commute [THEN iffD1])

 apply (rule_tac [3] is_gcd, auto)

 done

 lemma gcd_commute: "gcd(m,n) = gcd(n,m)"

 apply (cut_tac m = "natify (m)" and n = "natify (n)" in gcd_commute_raw)

 apply auto

 done

 lemma gcd_assoc_raw: "[| k \<in> nat; m \<in> nat; n \<in> nat |]  

       ==> gcd (gcd (k, m), n) = gcd (k, gcd (m, n))"

 apply (rule is_gcd_unique)

 apply (rule is_gcd)

 apply (simp_all add: is_gcd_def)

 apply (blast intro: gcd_dvd1 gcd_dvd2 gcd_type intro: dvd_trans)

 done

 lemma gcd_assoc: "gcd (gcd (k, m), n) = gcd (k, gcd (m, n))"

 apply (cut_tac k = "natify (k)" and m = "natify (m)" and n = "natify (n) " 

        in gcd_assoc_raw)

 apply auto

 done

 lemma gcd_0_left [simp]: "gcd (0, m) = natify(m)"

 by (simp add: gcd_commute [of 0])

 lemma gcd_1_left [simp]: "gcd (1, m) = 1"

 by (simp add: gcd_commute [of 1])

 subsection{*Addition laws*}

 lemma gcd_add1 [simp]: "gcd (m #+ n, n) = gcd (m, n)"

 apply (subgoal_tac "gcd (m #+ natify (n), natify (n)) = gcd (m, natify (n))")

 apply simp

 apply (case_tac "natify (n) = 0")

 apply (auto simp add: Ord_0_lt_iff 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, simp)

 apply (rule gcd_commute)

 done

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

 by (subst add_commute, rule gcd_add2)

 lemma gcd_add_mult_raw: "k \<in> nat ==> gcd (m, k #* m #+ n) = gcd (m, n)"

 apply (erule nat_induct)

 apply (auto simp add: gcd_add2 add_assoc)

 done

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

 apply (cut_tac k = "natify (k)" in gcd_add_mult_raw)

 apply auto

 done

 subsection{* Multiplication Laws*}

 lemma gcd_mult_distrib2_raw:

      "[| k \<in> nat; m \<in> nat; n \<in> nat |]  

       ==> k #* gcd (m, n) = gcd (k #* m, k #* n)"

 apply (erule_tac m = m and n = n in gcd_induct, assumption)

 apply simp

 apply (case_tac "k = 0", simp)

 apply (simp add: mod_geq gcd_non_0 mod_mult_distrib2 Ord_0_lt_iff)

 done

 lemma gcd_mult_distrib2: "k #* gcd (m, n) = gcd (k #* m, k #* n)"

 apply (cut_tac k = "natify (k)" and m = "natify (m)" and n = "natify (n) " 

        in gcd_mult_distrib2_raw)

 apply auto

 done

 lemma gcd_mult [simp]: "gcd (k, k #* n) = natify(k)"

 by (cut_tac k = k and m = 1 and n = n in gcd_mult_distrib2, auto)

 lemma gcd_self [simp]: "gcd (k, k) = natify(k)"

 by (cut_tac k = k and n = 1 in gcd_mult, auto)

 lemma relprime_dvd_mult:

      "[| gcd (k,n) = 1;  k dvd (m #* n);  m \<in> nat |] ==> k dvd m"

 apply (cut_tac k = m and m = k and n = n in gcd_mult_distrib2, auto)

 apply (erule_tac b = m in ssubst)

 apply (simp add: dvd_imp_nat1)

 done

 lemma relprime_dvd_mult_iff:

      "[| gcd (k,n) = 1;  m \<in> nat |] ==> k dvd (m #* n) <-> k dvd m"

 by (blast intro: dvdI2 relprime_dvd_mult dvd_trans)

 lemma prime_imp_relprime: 

      "[| p \<in> prime;  ~ (p dvd n);  n \<in> nat |] ==> gcd (p, n) = 1"

 apply (simp add: prime_def, clarify)

 apply (drule_tac x = "gcd (p,n)" in bspec)

 apply auto

 apply (cut_tac m = p and n = n in gcd_dvd2, auto)

 done

 lemma prime_into_nat: "p \<in> prime ==> p \<in> nat"

 by (simp add: prime_def)

 lemma prime_nonzero: "p \<in> prime \<Longrightarrow> p\<noteq>0"

 by (auto simp add: prime_def)

 text{*This theorem leads immediately to a proof of the uniqueness of

   factorization.  If @{term p} divides a product of primes then it is

   one of those primes.*}

 lemma prime_dvd_mult:

      "[|p dvd m #* n; p \<in> prime; m \<in> nat; n \<in> nat |] ==> p dvd m \<or> p dvd n"

 by (blast intro: relprime_dvd_mult prime_imp_relprime prime_into_nat)

 lemma gcd_mult_cancel_raw:

      "[|gcd (k,n) = 1; m \<in> nat; n \<in> nat|] ==> gcd (k #* m, n) = gcd (m, n)"

 apply (rule dvd_anti_sym)

  apply (rule gcd_greatest)

   apply (rule relprime_dvd_mult [of _ k])

 apply (simp add: gcd_assoc)

 apply (simp add: gcd_commute)

 apply (simp_all add: mult_commute)

 apply (blast intro: dvdI1 gcd_dvd1 dvd_trans)

 done

 lemma gcd_mult_cancel: "gcd (k,n) = 1 ==> gcd (k #* m, n) = gcd (m, n)"

 apply (cut_tac m = "natify (m)" and n = "natify (n)" in gcd_mult_cancel_raw)

 apply auto

 done

 subsection{*The Square Root of a Prime is Irrational: Key Lemma*}

 lemma prime_dvd_other_side:

      "\<lbrakk>n#*n = p#*(k#*k); p \<in> prime; n \<in> nat\<rbrakk> \<Longrightarrow> p dvd n"

 apply (subgoal_tac "p dvd n#*n")

  apply (blast dest: prime_dvd_mult)

 apply (rule_tac j = "k#*k" in dvd_mult_left)

  apply (auto simp add: prime_def)

 done

 lemma reduction:

      "\<lbrakk>k#*k = p#*(j#*j); p \<in> prime; 0 < k; j \<in> nat; k \<in> nat\<rbrakk>  

       \<Longrightarrow> k < p#*j & 0 < j"

 apply (rule ccontr)

 apply (simp add: not_lt_iff_le prime_into_nat)

 apply (erule disjE)

  apply (frule mult_le_mono, assumption+)

 apply (simp add: mult_ac)

 apply (auto dest!: natify_eqE 

             simp add: not_lt_iff_le prime_into_nat mult_le_cancel_le1)

 apply (simp add: prime_def)

 apply (blast dest: lt_trans1)

 done

 lemma rearrange: "j #* (p#*j) = k#*k \<Longrightarrow> k#*k = p#*(j#*j)"

 by (simp add: mult_ac)

 lemma prime_not_square:

      "\<lbrakk>m \<in> nat; p \<in> prime\<rbrakk> \<Longrightarrow> \<forall>k \<in> nat. 0<k \<longrightarrow> m#*m \<noteq> p#*(k#*k)"

 apply (erule complete_induct, clarify)

 apply (frule prime_dvd_other_side, assumption)

 apply assumption

 apply (erule dvdE)

 apply (simp add: mult_assoc mult_cancel1 prime_nonzero prime_into_nat)

 apply (blast dest: rearrange reduction ltD)

 done

 end