src/HOL/Library/Primes.thy
author wenzelm
Tue Feb 26 16:10:54 2008 +0100 (2008-02-26)
changeset 26144 98d23fc02585
parent 26125 345465cc9e79
child 26159 ff372ff5cc34
permissions -rw-r--r--
tuned document;
     1 (*  Title:      HOL/Library/Primes.thy
     2     ID:         $Id$
     3     Author:     Amine Chaieb Christophe Tabacznyj and Lawrence C Paulson
     4     Copyright   1996  University of Cambridge
     5 *)
     6 
     7 header {* Primality on nat *}
     8 
     9 theory Primes
    10 imports GCD Parity
    11 begin
    12 
    13 definition
    14   coprime :: "nat => nat => bool" where
    15   "coprime m n \<longleftrightarrow> (gcd (m, n) = 1)"
    16 
    17 definition
    18   prime :: "nat \<Rightarrow> bool" where
    19   "prime p \<longleftrightarrow> (1 < p \<and> (\<forall>m. m dvd p --> m = 1 \<or> m = p))"
    20 
    21 
    22 lemma two_is_prime: "prime 2"
    23   apply (auto simp add: prime_def)
    24   apply (case_tac m)
    25    apply (auto dest!: dvd_imp_le)
    26   done
    27 
    28 lemma prime_imp_relprime: "prime p ==> \<not> p dvd n ==> gcd (p, n) = 1"
    29   apply (auto simp add: prime_def)
    30   apply (metis One_nat_def gcd_dvd1 gcd_dvd2)
    31   done
    32 
    33 text {*
    34   This theorem leads immediately to a proof of the uniqueness of
    35   factorization.  If @{term p} divides a product of primes then it is
    36   one of those primes.
    37 *}
    38 
    39 lemma prime_dvd_mult: "prime p ==> p dvd m * n ==> p dvd m \<or> p dvd n"
    40   by (blast intro: relprime_dvd_mult prime_imp_relprime)
    41 
    42 lemma prime_dvd_square: "prime p ==> p dvd m^Suc (Suc 0) ==> p dvd m"
    43   by (auto dest: prime_dvd_mult)
    44 
    45 lemma prime_dvd_power_two: "prime p ==> p dvd m\<twosuperior> ==> p dvd m"
    46   by (rule prime_dvd_square) (simp_all add: power2_eq_square)
    47 
    48 
    49 lemma exp_eq_1:"(x::nat)^n = 1 \<longleftrightarrow> x = 1 \<or> n = 0" by (induct n, auto)
    50 lemma exp_mono_lt: "(x::nat) ^ (Suc n) < y ^ (Suc n) \<longleftrightarrow> x < y"
    51   using power_less_imp_less_base[of x "Suc n" y] power_strict_mono[of x y "Suc n"]
    52     by auto
    53 lemma exp_mono_le: "(x::nat) ^ (Suc n) \<le> y ^ (Suc n) \<longleftrightarrow> x \<le> y"
    54   by (simp only: linorder_not_less[symmetric] exp_mono_lt)
    55 
    56 lemma exp_mono_eq: "(x::nat) ^ Suc n = y ^ Suc n \<longleftrightarrow> x = y"
    57 using power_inject_base[of x n y] by auto
    58 
    59 
    60 lemma even_square: assumes e: "even (n::nat)" shows "\<exists>x. n ^ 2 = 4*x"
    61 proof-
    62   from e have "2 dvd n" by presburger
    63   then obtain k where k: "n = 2*k" using dvd_def by auto
    64   hence "n^2 = 4* (k^2)" by (simp add: power2_eq_square)
    65   thus ?thesis by blast
    66 qed
    67 
    68 lemma odd_square: assumes e: "odd (n::nat)" shows "\<exists>x. n ^ 2 = 4*x + 1"
    69 proof-
    70   from e have np: "n > 0" by presburger
    71   from e have "2 dvd (n - 1)" by presburger
    72   then obtain k where "n - 1 = 2*k" using dvd_def by auto
    73   hence k: "n = 2*k + 1"  using e by presburger 
    74   hence "n^2 = 4* (k^2 + k) + 1" by algebra   
    75   thus ?thesis by blast
    76 qed
    77 
    78 lemma diff_square: "(x::nat)^2 - y^2 = (x+y)*(x - y)" 
    79 proof-
    80   have "x \<le> y \<or> y \<le> x" by (rule nat_le_linear)
    81   moreover
    82   {assume le: "x \<le> y"
    83     hence "x ^2 \<le> y^2" by (simp only: numeral_2_eq_2 exp_mono_le Let_def)
    84     with le have ?thesis by simp }
    85   moreover
    86   {assume le: "y \<le> x"
    87     hence le2: "y ^2 \<le> x^2" by (simp only: numeral_2_eq_2 exp_mono_le Let_def)
    88     from le have "\<exists>z. y + z = x" by presburger
    89     then obtain z where z: "x = y + z" by blast 
    90     from le2 have "\<exists>z. x^2 = y^2 + z" by presburger
    91     then obtain z2 where z2: "x^2 = y^2 + z2"  by blast
    92     from z z2 have ?thesis apply simp by algebra }
    93   ultimately show ?thesis by blast  
    94 qed
    95 
    96 text {* Elementary theory of divisibility *}
    97 lemma divides_ge: "(a::nat) dvd b \<Longrightarrow> b = 0 \<or> a \<le> b" unfolding dvd_def by auto
    98 lemma divides_antisym: "(x::nat) dvd y \<and> y dvd x \<longleftrightarrow> x = y"
    99   using dvd_anti_sym[of x y] by auto
   100 
   101 lemma divides_add_revr: assumes da: "(d::nat) dvd a" and dab:"d dvd (a + b)"
   102   shows "d dvd b"
   103 proof-
   104   from da obtain k where k:"a = d*k" by (auto simp add: dvd_def)
   105   from dab obtain k' where k': "a + b = d*k'" by (auto simp add: dvd_def)
   106   from k k' have "b = d *(k' - k)" by (simp add : diff_mult_distrib2)
   107   thus ?thesis unfolding dvd_def by blast
   108 qed
   109 
   110 declare nat_mult_dvd_cancel_disj[presburger]
   111 lemma nat_mult_dvd_cancel_disj'[presburger]: 
   112   "(m\<Colon>nat)*k dvd n*k \<longleftrightarrow> k = 0 \<or> m dvd n" unfolding mult_commute[of m k] mult_commute[of n k] by presburger 
   113 
   114 lemma divides_mul_l: "(a::nat) dvd b ==> (c * a) dvd (c * b)"
   115   by presburger
   116 
   117 lemma divides_mul_r: "(a::nat) dvd b ==> (a * c) dvd (b * c)" by presburger
   118 lemma divides_cases: "(n::nat) dvd m ==> m = 0 \<or> m = n \<or> 2 * n <= m" 
   119   by (auto simp add: dvd_def)
   120 lemma divides_le: "m dvd n ==> m <= n \<or> n = (0::nat)" by (auto simp add: dvd_def)
   121 
   122 lemma divides_div_not: "(x::nat) = (q * n) + r \<Longrightarrow> 0 < r \<Longrightarrow> r < n ==> ~(n dvd x)"
   123 proof(auto simp add: dvd_def)
   124   fix k assume H: "0 < r" "r < n" "q * n + r = n * k"
   125   from H(3) have r: "r = n* (k -q)" by(simp add: diff_mult_distrib2 mult_commute)
   126   {assume "k - q = 0" with r H(1) have False by simp}
   127   moreover
   128   {assume "k - q \<noteq> 0" with r have "r \<ge> n" by auto
   129     with H(2) have False by simp}
   130   ultimately show False by blast
   131 qed
   132 lemma divides_exp: "(x::nat) dvd y ==> x ^ n dvd y ^ n"
   133   by (auto simp add: power_mult_distrib dvd_def)
   134 
   135 lemma divides_exp2: "n \<noteq> 0 \<Longrightarrow> (x::nat) ^ n dvd y \<Longrightarrow> x dvd y" 
   136   by (induct n ,auto simp add: dvd_def)
   137 
   138 fun fact :: "nat \<Rightarrow> nat" where
   139   "fact 0 = 1"
   140 | "fact (Suc n) = Suc n * fact n"	
   141 
   142 lemma fact_lt: "0 < fact n" by(induct n, simp_all)
   143 lemma fact_le: "fact n \<ge> 1" using fact_lt[of n] by simp 
   144 lemma fact_mono: assumes le: "m \<le> n" shows "fact m \<le> fact n"
   145 proof-
   146   from le have "\<exists>i. n = m+i" by presburger
   147   then obtain i where i: "n = m+i" by blast 
   148   have "fact m \<le> fact (m + i)"
   149   proof(induct m)
   150     case 0 thus ?case using fact_le[of i] by simp
   151   next
   152     case (Suc m)
   153     have "fact (Suc m) = Suc m * fact m" by simp
   154     have th1: "Suc m \<le> Suc (m + i)" by simp
   155     from mult_le_mono[of "Suc m" "Suc (m+i)" "fact m" "fact (m+i)", OF th1 Suc.hyps]
   156     show ?case by simp
   157   qed
   158   thus ?thesis using i by simp
   159 qed
   160 
   161 lemma divides_fact: "1 <= p \<Longrightarrow> p <= n ==> p dvd fact n"
   162 proof(induct n arbitrary: p)
   163   case 0 thus ?case by simp
   164 next
   165   case (Suc n p)
   166   from Suc.prems have "p = Suc n \<or> p \<le> n" by presburger 
   167   moreover
   168   {assume "p = Suc n" hence ?case  by (simp only: fact.simps dvd_triv_left)}
   169   moreover
   170   {assume "p \<le> n"
   171     with Suc.prems(1) Suc.hyps have th: "p dvd fact n" by simp
   172     from dvd_mult[OF th] have ?case by (simp only: fact.simps) }
   173   ultimately show ?case by blast
   174 qed
   175 
   176 declare dvd_triv_left[presburger]
   177 declare dvd_triv_right[presburger]
   178 lemma divides_rexp: 
   179   "x dvd y \<Longrightarrow> (x::nat) dvd (y^(Suc n))" by (simp add: dvd_mult2[of x y])
   180 
   181 text {* The Bezout theorem is a bit ugly for N; it'd be easier for Z *}
   182 lemma ind_euclid: 
   183   assumes c: " \<forall>a b. P (a::nat) b \<longleftrightarrow> P b a" and z: "\<forall>a. P a 0" 
   184   and add: "\<forall>a b. P a b \<longrightarrow> P a (a + b)" 
   185   shows "P a b"
   186 proof(induct n\<equiv>"a+b" arbitrary: a b rule: nat_less_induct)
   187   fix n a b
   188   assume H: "\<forall>m < n. \<forall>a b. m = a + b \<longrightarrow> P a b" "n = a + b"
   189   have "a = b \<or> a < b \<or> b < a" by arith
   190   moreover {assume eq: "a= b"
   191     from add[rule_format, OF z[rule_format, of a]] have "P a b" using eq by simp}
   192   moreover
   193   {assume lt: "a < b"
   194     hence "a + b - a < n \<or> a = 0"  using H(2) by arith
   195     moreover
   196     {assume "a =0" with z c have "P a b" by blast }
   197     moreover
   198     {assume ab: "a + b - a < n"
   199       have th0: "a + b - a = a + (b - a)" using lt by arith
   200       from add[rule_format, OF H(1)[rule_format, OF ab th0]]
   201       have "P a b" by (simp add: th0[symmetric])}
   202     ultimately have "P a b" by blast}
   203   moreover
   204   {assume lt: "a > b"
   205     hence "b + a - b < n \<or> b = 0"  using H(2) by arith
   206     moreover
   207     {assume "b =0" with z c have "P a b" by blast }
   208     moreover
   209     {assume ab: "b + a - b < n"
   210       have th0: "b + a - b = b + (a - b)" using lt by arith
   211       from add[rule_format, OF H(1)[rule_format, OF ab th0]]
   212       have "P b a" by (simp add: th0[symmetric])
   213       hence "P a b" using c by blast }
   214     ultimately have "P a b" by blast}
   215 ultimately  show "P a b" by blast
   216 qed
   217 
   218 lemma bezout_lemma: 
   219   assumes ex: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and> (a * x = b * y + d \<or> b * x = a * y + d)"
   220   shows "\<exists>d x y. d dvd a \<and> d dvd a + b \<and> (a * x = (a + b) * y + d \<or> (a + b) * x = a * y + d)"
   221 using ex
   222 apply clarsimp
   223 apply (rule_tac x="d" in exI, simp add: dvd_add)
   224 apply (case_tac "a * x = b * y + d" , simp_all)
   225 apply (rule_tac x="x + y" in exI)
   226 apply (rule_tac x="y" in exI)
   227 apply algebra
   228 apply (rule_tac x="x" in exI)
   229 apply (rule_tac x="x + y" in exI)
   230 apply algebra
   231 done
   232 
   233 lemma bezout_add: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and> (a * x = b * y + d \<or> b * x = a * y + d)"
   234 apply(induct a b rule: ind_euclid)
   235 apply blast
   236 apply clarify
   237 apply (rule_tac x="a" in exI, simp add: dvd_add)
   238 apply clarsimp
   239 apply (rule_tac x="d" in exI)
   240 apply (case_tac "a * x = b * y + d", simp_all add: dvd_add)
   241 apply (rule_tac x="x+y" in exI)
   242 apply (rule_tac x="y" in exI)
   243 apply algebra
   244 apply (rule_tac x="x" in exI)
   245 apply (rule_tac x="x+y" in exI)
   246 apply algebra
   247 done
   248 
   249 lemma bezout: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and> (a * x - b * y = d \<or> b * x - a * y = d)"
   250 using bezout_add[of a b]
   251 apply clarsimp
   252 apply (rule_tac x="d" in exI, simp)
   253 apply (rule_tac x="x" in exI)
   254 apply (rule_tac x="y" in exI)
   255 apply auto
   256 done
   257 
   258 text {* We can get a stronger version with a nonzeroness assumption. *}
   259 
   260 lemma bezout_add_strong: assumes nz: "a \<noteq> (0::nat)"
   261   shows "\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d"
   262 proof-
   263   from nz have ap: "a > 0" by simp
   264  from bezout_add[of a b] 
   265  have "(\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d) \<or> (\<exists>d x y. d dvd a \<and> d dvd b \<and> b * x = a * y + d)" by blast
   266  moreover
   267  {fix d x y assume H: "d dvd a" "d dvd b" "a * x = b * y + d"
   268    from H have ?thesis by blast }
   269  moreover
   270  {fix d x y assume H: "d dvd a" "d dvd b" "b * x = a * y + d"
   271    {assume b0: "b = 0" with H  have ?thesis by simp}
   272    moreover 
   273    {assume b: "b \<noteq> 0" hence bp: "b > 0" by simp
   274      from divides_le[OF H(2)] b have "d < b \<or> d = b" using le_less by blast
   275      moreover
   276      {assume db: "d=b"
   277        from prems have ?thesis apply simp
   278 	 apply (rule exI[where x = b], simp)
   279 	 apply (rule exI[where x = b])
   280 	by (rule exI[where x = "a - 1"], simp add: diff_mult_distrib2)}
   281     moreover
   282     {assume db: "d < b" 
   283 	{assume "x=0" hence ?thesis  using prems by simp }
   284 	moreover
   285 	{assume x0: "x \<noteq> 0" hence xp: "x > 0" by simp
   286 	  
   287 	  from db have "d \<le> b - 1" by simp
   288 	  hence "d*b \<le> b*(b - 1)" by simp
   289 	  with xp mult_mono[of "1" "x" "d*b" "b*(b - 1)"]
   290 	  have dble: "d*b \<le> x*b*(b - 1)" using bp by simp
   291 	  from H (3) have "d + (b - 1) * (b*x) = d + (b - 1) * (a*y + d)" by simp
   292 	  hence "d + (b - 1) * a * y + (b - 1) * d = d + (b - 1) * b * x"
   293 	    by (simp only: mult_assoc right_distrib)
   294 	  hence "a * ((b - 1) * y) + d * (b - 1 + 1) = d + x*b*(b - 1)" by algebra
   295 	  hence "a * ((b - 1) * y) = d + x*b*(b - 1) - d*b" using bp by simp
   296 	  hence "a * ((b - 1) * y) = d + (x*b*(b - 1) - d*b)"
   297 	    by (simp only: diff_add_assoc[OF dble, of d, symmetric])
   298 	  hence "a * ((b - 1) * y) = b*(x*(b - 1) - d) + d"
   299 	    by (simp only: diff_mult_distrib2 add_commute mult_ac)
   300 	  hence ?thesis using H(1,2)
   301 	    apply -
   302 	    apply (rule exI[where x=d], simp)
   303 	    apply (rule exI[where x="(b - 1) * y"])
   304 	    by (rule exI[where x="x*(b - 1) - d"], simp)}
   305 	ultimately have ?thesis by blast}
   306     ultimately have ?thesis by blast}
   307   ultimately have ?thesis by blast}
   308  ultimately show ?thesis by blast
   309 qed
   310 
   311 text {* Greatest common divisor. *}
   312 lemma gcd_unique: "d dvd a\<and>d dvd b \<and> (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd(a,b)"
   313 proof(auto)
   314   assume H: "d dvd a" "d dvd b" "\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d"
   315   from H(3)[rule_format] gcd_dvd1[of a b] gcd_dvd2[of a b] 
   316   have th: "gcd (a,b) dvd d" by blast
   317   from dvd_anti_sym[OF th gcd_greatest[OF H(1,2)]]  show "d = gcd (a,b)" by blast 
   318 qed
   319 
   320 lemma gcd_eq: assumes H: "\<forall>d. d dvd x \<and> d dvd y \<longleftrightarrow> d dvd u \<and> d dvd v"
   321   shows "gcd (x,y) = gcd(u,v)"
   322 proof-
   323   from H have "\<forall>d. d dvd x \<and> d dvd y \<longleftrightarrow> d dvd gcd (u,v)" by simp
   324   with gcd_unique[of "gcd(u,v)" x y]  show ?thesis by auto
   325 qed
   326 
   327 lemma bezout_gcd: "\<exists>x y. a * x - b * y = gcd(a,b) \<or> b * x - a * y = gcd(a,b)"
   328 proof-
   329   let ?g = "gcd (a,b)"
   330   from bezout[of a b] obtain d x y where d: "d dvd a" "d dvd b" "a * x - b * y = d \<or> b * x - a * y = d" by blast
   331   from d(1,2) have "d dvd ?g" by simp
   332   then obtain k where k: "?g = d*k" unfolding dvd_def by blast
   333   from d(3) have "(a * x - b * y)*k = d*k \<or> (b * x - a * y)*k = d*k" by blast 
   334   hence "a * x * k - b * y*k = d*k \<or> b * x * k - a * y*k = d*k"
   335     by (simp only: diff_mult_distrib)
   336   hence "a * (x * k) - b * (y*k) = ?g \<or> b * (x * k) - a * (y*k) = ?g"
   337     by (simp add: k mult_assoc)
   338   thus ?thesis by blast
   339 qed
   340 
   341 lemma bezout_gcd_strong: assumes a: "a \<noteq> 0" 
   342   shows "\<exists>x y. a * x = b * y + gcd(a,b)"
   343 proof-
   344   let ?g = "gcd (a,b)"
   345   from bezout_add_strong[OF a, of b]
   346   obtain d x y where d: "d dvd a" "d dvd b" "a * x = b * y + d" by blast
   347   from d(1,2) have "d dvd ?g" by simp
   348   then obtain k where k: "?g = d*k" unfolding dvd_def by blast
   349   from d(3) have "a * x * k = (b * y + d) *k " by auto 
   350   hence "a * (x * k) = b * (y*k) + ?g" by (algebra add: k)
   351   thus ?thesis by blast
   352 qed
   353 
   354 lemma gcd_mult_distrib: "gcd(a * c, b * c) = c * gcd(a,b)"
   355 by(simp add: gcd_mult_distrib2 mult_commute)
   356 
   357 lemma gcd_bezout: "(\<exists>x y. a * x - b * y = d \<or> b * x - a * y = d) \<longleftrightarrow> gcd(a,b) dvd d"
   358   (is "?lhs \<longleftrightarrow> ?rhs")
   359 proof-
   360   let ?g = "gcd (a,b)"
   361   {assume H: ?rhs then obtain k where k: "d = ?g*k" unfolding dvd_def by blast
   362     from bezout_gcd[of a b] obtain x y where xy: "a * x - b * y = ?g \<or> b * x - a * y = ?g"
   363       by blast
   364     hence "(a * x - b * y)*k = ?g*k \<or> (b * x - a * y)*k = ?g*k" by auto
   365     hence "a * x*k - b * y*k = ?g*k \<or> b * x * k - a * y*k = ?g*k" 
   366       by (simp only: diff_mult_distrib)
   367     hence "a * (x*k) - b * (y*k) = d \<or> b * (x * k) - a * (y*k) = d" 
   368       by (simp add: k[symmetric] mult_assoc)
   369     hence ?lhs by blast}
   370   moreover
   371   {fix x y assume H: "a * x - b * y = d \<or> b * x - a * y = d"
   372     have dv: "?g dvd a*x" "?g dvd b * y" "?g dvd b*x" "?g dvd a * y" 
   373       using dvd_mult2[OF gcd_dvd1[of a b]] dvd_mult2[OF gcd_dvd2[of a b]] by simp_all
   374     from dvd_diff[OF dv(1,2)] dvd_diff[OF dv(3,4)] H
   375     have ?rhs by auto}
   376   ultimately show ?thesis by blast
   377 qed
   378 
   379 lemma gcd_bezout_sum: assumes H:"a * x + b * y = d" shows "gcd(a,b) dvd d"
   380 proof-
   381   let ?g = "gcd (a,b)"
   382     have dv: "?g dvd a*x" "?g dvd b * y" 
   383       using dvd_mult2[OF gcd_dvd1[of a b]] dvd_mult2[OF gcd_dvd2[of a b]] by simp_all
   384     from dvd_add[OF dv] H
   385     show ?thesis by auto
   386 qed
   387 
   388 lemma gcd_mult': "gcd(b,a * b) = b"
   389 by (simp add: gcd_mult mult_commute[of a b]) 
   390 
   391 lemma gcd_add: "gcd(a + b,b) = gcd(a,b)" "gcd(b + a,b) = gcd(a,b)" "gcd(a,a + b) = gcd(a,b)" "gcd(a,b + a) = gcd(a,b)"
   392 apply (simp_all add: gcd_add1)
   393 by (simp add: gcd_commute gcd_add1)
   394 
   395 lemma gcd_sub: "b <= a ==> gcd(a - b,b) = gcd(a,b)" "a <= b ==> gcd(a,b - a) = gcd(a,b)"
   396 proof-
   397   {fix a b assume H: "b \<le> (a::nat)"
   398     hence th: "a - b + b = a" by arith
   399     from gcd_add(1)[of "a - b" b] th  have "gcd(a - b,b) = gcd(a,b)" by simp}
   400   note th = this
   401 {
   402   assume ab: "b \<le> a"
   403   from th[OF ab] show "gcd (a - b, b) = gcd (a, b)" by blast
   404 next
   405   assume ab: "a \<le> b"
   406   from th[OF ab] show "gcd (a,b - a) = gcd (a, b)" 
   407     by (simp add: gcd_commute)}
   408 qed
   409 
   410 text {* Coprimality *}
   411 
   412 lemma coprime: "coprime a b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
   413 using gcd_unique[of 1 a b, simplified] by (auto simp add: coprime_def)
   414 lemma coprime_commute: "coprime a b \<longleftrightarrow> coprime b a" by (simp add: coprime_def gcd_commute)
   415 
   416 lemma coprime_bezout: "coprime a b \<longleftrightarrow> (\<exists>x y. a * x - b * y = 1 \<or> b * x - a * y = 1)"
   417 using coprime_def gcd_bezout by auto
   418 
   419 lemma coprime_divprod: "d dvd a * b  \<Longrightarrow> coprime d a \<Longrightarrow> d dvd b"
   420   using relprime_dvd_mult_iff[of d a b] by (auto simp add: coprime_def mult_commute)
   421 
   422 lemma coprime_1[simp]: "coprime a 1" by (simp add: coprime_def)
   423 lemma coprime_1'[simp]: "coprime 1 a" by (simp add: coprime_def)
   424 lemma coprime_Suc0[simp]: "coprime a (Suc 0)" by (simp add: coprime_def)
   425 lemma coprime_Suc0'[simp]: "coprime (Suc 0) a" by (simp add: coprime_def)
   426 
   427 lemma gcd_coprime: 
   428   assumes z: "gcd(a,b) \<noteq> 0" and a: "a = a' * gcd(a,b)" and b: "b = b' * gcd(a,b)" 
   429   shows    "coprime a' b'"
   430 proof-
   431   let ?g = "gcd(a,b)"
   432   {assume bz: "a = 0" from b bz z a have ?thesis by (simp add: gcd_zero coprime_def)}
   433   moreover 
   434   {assume az: "a\<noteq> 0" 
   435     from z have z': "?g > 0" by simp
   436     from bezout_gcd_strong[OF az, of b] 
   437     obtain x y where xy: "a*x = b*y + ?g" by blast
   438     from xy a b have "?g * a'*x = ?g * (b'*y + 1)" by (simp add: ring_simps)
   439     hence "?g * (a'*x) = ?g * (b'*y + 1)" by (simp add: mult_assoc)
   440     hence "a'*x = (b'*y + 1)"
   441       by (simp only: nat_mult_eq_cancel1[OF z']) 
   442     hence "a'*x - b'*y = 1" by simp
   443     with coprime_bezout[of a' b'] have ?thesis by auto}
   444   ultimately show ?thesis by blast
   445 qed
   446 lemma coprime_0: "coprime d 0 \<longleftrightarrow> d = 1" by (simp add: coprime_def)
   447 lemma coprime_mul: assumes da: "coprime d a" and db: "coprime d b"
   448   shows "coprime d (a * b)"
   449 proof-
   450   from da have th: "gcd(a, d) = 1" by (simp add: coprime_def gcd_commute)
   451   from gcd_mult_cancel[of a d b, OF th] db[unfolded coprime_def] have "gcd (d, a*b) = 1"
   452     by (simp add: gcd_commute)
   453   thus ?thesis unfolding coprime_def .
   454 qed
   455 lemma coprime_lmul2: assumes dab: "coprime d (a * b)" shows "coprime d b"
   456 using prems unfolding coprime_bezout
   457 apply clarsimp
   458 apply (case_tac "d * x - a * b * y = Suc 0 ", simp_all)
   459 apply (rule_tac x="x" in exI)
   460 apply (rule_tac x="a*y" in exI)
   461 apply (simp add: mult_ac)
   462 apply (rule_tac x="a*x" in exI)
   463 apply (rule_tac x="y" in exI)
   464 apply (simp add: mult_ac)
   465 done
   466 
   467 lemma coprime_rmul2: "coprime d (a * b) \<Longrightarrow> coprime d a"
   468 unfolding coprime_bezout
   469 apply clarsimp
   470 apply (case_tac "d * x - a * b * y = Suc 0 ", simp_all)
   471 apply (rule_tac x="x" in exI)
   472 apply (rule_tac x="b*y" in exI)
   473 apply (simp add: mult_ac)
   474 apply (rule_tac x="b*x" in exI)
   475 apply (rule_tac x="y" in exI)
   476 apply (simp add: mult_ac)
   477 done
   478 lemma coprime_mul_eq: "coprime d (a * b) \<longleftrightarrow> coprime d a \<and>  coprime d b"
   479   using coprime_rmul2[of d a b] coprime_lmul2[of d a b] coprime_mul[of d a b] 
   480   by blast
   481 
   482 lemma gcd_coprime_exists:
   483   assumes nz: "gcd(a,b) \<noteq> 0" 
   484   shows "\<exists>a' b'. a = a' * gcd(a,b) \<and> b = b' * gcd(a,b) \<and> coprime a' b'"
   485 proof-
   486   let ?g = "gcd (a,b)"
   487   from gcd_dvd1[of a b] gcd_dvd2[of a b] 
   488   obtain a' b' where "a = ?g*a'"  "b = ?g*b'" unfolding dvd_def by blast
   489   hence ab': "a = a'*?g" "b = b'*?g" by algebra+
   490   from ab' gcd_coprime[OF nz ab'] show ?thesis by blast
   491 qed
   492 
   493 lemma coprime_exp: "coprime d a ==> coprime d (a^n)" 
   494   by(induct n, simp_all add: coprime_mul)
   495 
   496 lemma coprime_exp_imp: "coprime a b ==> coprime (a ^n) (b ^n)"
   497   by (induct n, simp_all add: coprime_mul_eq coprime_commute coprime_exp)
   498 lemma coprime_refl[simp]: "coprime n n \<longleftrightarrow> n = 1" by (simp add: coprime_def)
   499 lemma coprime_plus1[simp]: "coprime (n + 1) n"
   500   apply (simp add: coprime_bezout)
   501   apply (rule exI[where x=1])
   502   apply (rule exI[where x=1])
   503   apply simp
   504   done
   505 lemma coprime_minus1: "n \<noteq> 0 ==> coprime (n - 1) n"
   506   using coprime_plus1[of "n - 1"] coprime_commute[of "n - 1" n] by auto
   507 
   508 lemma bezout_gcd_pow: "\<exists>x y. a ^n * x - b ^ n * y = gcd(a,b) ^ n \<or> b ^ n * x - a ^ n * y = gcd(a,b) ^ n"
   509 proof-
   510   let ?g = "gcd (a,b)"
   511   {assume z: "?g = 0" hence ?thesis 
   512       apply (cases n, simp)
   513       apply arith
   514       apply (simp only: z power_0_Suc)
   515       apply (rule exI[where x=0])
   516       apply (rule exI[where x=0])
   517       by simp}
   518   moreover
   519   {assume z: "?g \<noteq> 0"
   520     from gcd_dvd1[of a b] gcd_dvd2[of a b] obtain a' b' where
   521       ab': "a = a'*?g" "b = b'*?g" unfolding dvd_def by (auto simp add: mult_ac)
   522     hence ab'': "?g*a' = a" "?g * b' = b" by algebra+
   523     from coprime_exp_imp[OF gcd_coprime[OF z ab'], unfolded coprime_bezout, of n]
   524     obtain x y where "a'^n * x - b'^n * y = 1 \<or> b'^n * x - a'^n * y = 1"  by blast
   525     hence "?g^n * (a'^n * x - b'^n * y) = ?g^n \<or> ?g^n*(b'^n * x - a'^n * y) = ?g^n"
   526       using z by auto 
   527     then have "a^n * x - b^n * y = ?g^n \<or> b^n * x - a^n * y = ?g^n"
   528       using z ab'' by (simp only: power_mult_distrib[symmetric] 
   529 	diff_mult_distrib2 mult_assoc[symmetric])
   530     hence  ?thesis by blast }
   531   ultimately show ?thesis by blast
   532 qed
   533 lemma gcd_exp: "gcd (a^n, b^n) = gcd(a,b)^n"
   534 proof-
   535   let ?g = "gcd(a^n,b^n)"
   536   let ?gn = "gcd(a,b)^n"
   537   {fix e assume H: "e dvd a^n" "e dvd b^n"
   538     from bezout_gcd_pow[of a n b] obtain x y 
   539       where xy: "a ^ n * x - b ^ n * y = ?gn \<or> b ^ n * x - a ^ n * y = ?gn" by blast
   540     from dvd_diff [OF dvd_mult2[OF H(1), of x] dvd_mult2[OF H(2), of y]]
   541       dvd_diff [OF dvd_mult2[OF H(2), of x] dvd_mult2[OF H(1), of y]] xy
   542     have "e dvd ?gn" by (cases "a ^ n * x - b ^ n * y = gcd (a, b) ^ n", simp_all)}
   543   hence th:  "\<forall>e. e dvd a^n \<and> e dvd b^n \<longrightarrow> e dvd ?gn" by blast
   544   from divides_exp[OF gcd_dvd1[of a b], of n] divides_exp[OF gcd_dvd2[of a b], of n] th
   545     gcd_unique have "?gn = ?g" by blast thus ?thesis by simp 
   546 qed
   547 
   548 lemma coprime_exp2:  "coprime (a ^ Suc n) (b^ Suc n) \<longleftrightarrow> coprime a b"
   549 by (simp only: coprime_def gcd_exp exp_eq_1) simp
   550 
   551 lemma division_decomp: assumes dc: "(a::nat) dvd b * c"
   552   shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
   553 proof-
   554   let ?g = "gcd (a,b)"
   555   {assume "?g = 0" with dc have ?thesis apply (simp add: gcd_zero)
   556       apply (rule exI[where x="0"])
   557       by (rule exI[where x="c"], simp)}
   558   moreover
   559   {assume z: "?g \<noteq> 0"
   560     from gcd_coprime_exists[OF z]
   561     obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'" by blast
   562     from gcd_dvd2[of a b] have thb: "?g dvd b" .
   563     from ab'(1) have "a' dvd a"  unfolding dvd_def by blast  
   564     with dc have th0: "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
   565     from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
   566     hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult_assoc)
   567     with z have th_1: "a' dvd b'*c" by simp
   568     from coprime_divprod[OF th_1 ab'(3)] have thc: "a' dvd c" . 
   569     from ab' have "a = ?g*a'" by algebra
   570     with thb thc have ?thesis by blast }
   571   ultimately show ?thesis by blast
   572 qed
   573 
   574 lemma nat_power_eq_0_iff: "(m::nat) ^ n = 0 \<longleftrightarrow> n \<noteq> 0 \<and> m = 0" by (induct n, auto)
   575 
   576 lemma divides_rev: assumes ab: "(a::nat) ^ n dvd b ^n" and n:"n \<noteq> 0" shows "a dvd b"
   577 proof-
   578   let ?g = "gcd (a,b)"
   579   from n obtain m where m: "n = Suc m" by (cases n, simp_all)
   580   {assume "?g = 0" with ab n have ?thesis by (simp add: gcd_zero)}
   581   moreover
   582   {assume z: "?g \<noteq> 0"
   583     hence zn: "?g ^ n \<noteq> 0" using n by (simp add: neq0_conv)
   584     from gcd_coprime_exists[OF z] 
   585     obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'" by blast
   586     from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n" by (simp add: ab'(1,2)[symmetric])
   587     hence "?g^n*a'^n dvd ?g^n *b'^n" by (simp only: power_mult_distrib mult_commute)
   588     with zn z n have th0:"a'^n dvd b'^n" by (auto simp add: nat_power_eq_0_iff)
   589     have "a' dvd a'^n" by (simp add: m)
   590     with th0 have "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by simp
   591     hence th1: "a' dvd b'^m * b'" by (simp add: m mult_commute)
   592     from coprime_divprod[OF th1 coprime_exp[OF ab'(3), of m]]
   593     have "a' dvd b'" .
   594     hence "a'*?g dvd b'*?g" by simp
   595     with ab'(1,2)  have ?thesis by simp }
   596   ultimately show ?thesis by blast
   597 qed
   598 
   599 lemma divides_mul: assumes mr: "m dvd r" and nr: "n dvd r" and mn:"coprime m n" 
   600   shows "m * n dvd r"
   601 proof-
   602   from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
   603     unfolding dvd_def by blast
   604   from mr n' have "m dvd n'*n" by (simp add: mult_commute)
   605   hence "m dvd n'" using relprime_dvd_mult_iff[OF mn[unfolded coprime_def]] by simp
   606   then obtain k where k: "n' = m*k" unfolding dvd_def by blast
   607   from n' k show ?thesis unfolding dvd_def by auto
   608 qed
   609 
   610 
   611 text {* A binary form of the Chinese Remainder Theorem. *}
   612 
   613 lemma chinese_remainder: assumes ab: "coprime a b" and a:"a \<noteq> 0" and b:"b \<noteq> 0"
   614   shows "\<exists>x q1 q2. x = u + q1 * a \<and> x = v + q2 * b"
   615 proof-
   616   from bezout_add_strong[OF a, of b] bezout_add_strong[OF b, of a]
   617   obtain d1 x1 y1 d2 x2 y2 where dxy1: "d1 dvd a" "d1 dvd b" "a * x1 = b * y1 + d1" 
   618     and dxy2: "d2 dvd b" "d2 dvd a" "b * x2 = a * y2 + d2" by blast
   619   from gcd_unique[of 1 a b, simplified ab[unfolded coprime_def], simplified] 
   620     dxy1(1,2) dxy2(1,2) have d12: "d1 = 1" "d2 =1" by auto
   621   let ?x = "v * a * x1 + u * b * x2"
   622   let ?q1 = "v * x1 + u * y2"
   623   let ?q2 = "v * y1 + u * x2"
   624   from dxy2(3)[simplified d12] dxy1(3)[simplified d12] 
   625   have "?x = u + ?q1 * a" "?x = v + ?q2 * b" by algebra+ 
   626   thus ?thesis by blast
   627 qed
   628 
   629 text {* Primality *}
   630 
   631 text {* A few useful theorems about primes *}
   632 
   633 lemma prime_0[simp]: "~prime 0" by (simp add: prime_def)
   634 lemma prime_1[simp]: "~ prime 1"  by (simp add: prime_def)
   635 lemma prime_Suc0[simp]: "~ prime (Suc 0)"  by (simp add: prime_def)
   636 
   637 lemma prime_ge_2: "prime p ==> p \<ge> 2" by (simp add: prime_def)
   638 lemma prime_factor: assumes n: "n \<noteq> 1" shows "\<exists> p. prime p \<and> p dvd n"
   639 using n
   640 proof(induct n rule: nat_less_induct)
   641   fix n
   642   assume H: "\<forall>m<n. m \<noteq> 1 \<longrightarrow> (\<exists>p. prime p \<and> p dvd m)" "n \<noteq> 1"
   643   let ?ths = "\<exists>p. prime p \<and> p dvd n"
   644   {assume "n=0" hence ?ths using two_is_prime by auto}
   645   moreover
   646   {assume nz: "n\<noteq>0" 
   647     {assume "prime n" hence ?ths by - (rule exI[where x="n"], simp)}
   648     moreover
   649     {assume n: "\<not> prime n"
   650       with nz H(2) 
   651       obtain k where k:"k dvd n" "k \<noteq> 1" "k \<noteq> n" by (auto simp add: prime_def) 
   652       from dvd_imp_le[OF k(1)] nz k(3) have kn: "k < n" by simp
   653       from H(1)[rule_format, OF kn k(2)] obtain p where p: "prime p" "p dvd k" by blast
   654       from dvd_trans[OF p(2) k(1)] p(1) have ?ths by blast}
   655     ultimately have ?ths by blast}
   656   ultimately show ?ths by blast
   657 qed
   658 
   659 lemma prime_factor_lt: assumes p: "prime p" and n: "n \<noteq> 0" and npm:"n = p * m"
   660   shows "m < n"
   661 proof-
   662   {assume "m=0" with n have ?thesis by simp}
   663   moreover
   664   {assume m: "m \<noteq> 0"
   665     from npm have mn: "m dvd n" unfolding dvd_def by auto
   666     from npm m have "n \<noteq> m" using p by auto
   667     with dvd_imp_le[OF mn] n have ?thesis by simp}
   668   ultimately show ?thesis by blast
   669 qed
   670 
   671 lemma euclid_bound: "\<exists>p. prime p \<and> n < p \<and>  p <= Suc (fact n)"
   672 proof-
   673   have f1: "fact n + 1 \<noteq> 1" using fact_le[of n] by arith 
   674   from prime_factor[OF f1] obtain p where p: "prime p" "p dvd fact n + 1" by blast
   675   from dvd_imp_le[OF p(2)] have pfn: "p \<le> fact n + 1" by simp
   676   {assume np: "p \<le> n"
   677     from p(1) have p1: "p \<ge> 1" by (cases p, simp_all)
   678     from divides_fact[OF p1 np] have pfn': "p dvd fact n" .
   679     from divides_add_revr[OF pfn' p(2)] p(1) have False by simp}
   680   hence "n < p" by arith
   681   with p(1) pfn show ?thesis by auto
   682 qed
   683 
   684 lemma euclid: "\<exists>p. prime p \<and> p > n" using euclid_bound by auto
   685 lemma primes_infinite: "\<not> (finite {p. prime p})"
   686 proof (auto simp add: finite_conv_nat_seg_image)
   687   fix n f 
   688   assume H: "Collect prime = f ` {i. i < (n::nat)}"
   689   let ?P = "Collect prime"
   690   let ?m = "Max ?P"
   691   have P0: "?P \<noteq> {}" using two_is_prime by auto
   692   from H have fP: "finite ?P" using finite_conv_nat_seg_image by blast 
   693   from Max_in[OF fP P0]  have "?m \<in> ?P" . 
   694   from Max_ge[OF fP P0] have contr: "\<forall> p. prime p \<longrightarrow> p \<le> ?m" by blast
   695   from euclid[of ?m] obtain q where q: "prime q" "q > ?m" by blast
   696   with contr show False by auto
   697 qed
   698 
   699 lemma coprime_prime: assumes ab: "coprime a b"
   700   shows "~(prime p \<and> p dvd a \<and> p dvd b)"
   701 proof
   702   assume "prime p \<and> p dvd a \<and> p dvd b"
   703   thus False using ab gcd_greatest[of p a b] by (simp add: coprime_def)
   704 qed
   705 lemma coprime_prime_eq: "coprime a b \<longleftrightarrow> (\<forall>p. ~(prime p \<and> p dvd a \<and> p dvd b))" 
   706   (is "?lhs = ?rhs")
   707 proof-
   708   {assume "?lhs" with coprime_prime  have ?rhs by blast}
   709   moreover
   710   {assume r: "?rhs" and c: "\<not> ?lhs"
   711     then obtain g where g: "g\<noteq>1" "g dvd a" "g dvd b" unfolding coprime_def by blast
   712     from prime_factor[OF g(1)] obtain p where p: "prime p" "p dvd g" by blast
   713     from dvd_trans [OF p(2) g(2)] dvd_trans [OF p(2) g(3)] 
   714     have "p dvd a" "p dvd b" . with p(1) r have False by blast}
   715   ultimately show ?thesis by blast
   716 qed
   717 
   718 lemma prime_coprime: assumes p: "prime p" 
   719   shows "n = 1 \<or> p dvd n \<or> coprime p n"
   720 using p prime_imp_relprime[of p n] by (auto simp add: coprime_def)
   721 
   722 lemma prime_coprime_strong: "prime p \<Longrightarrow> p dvd n \<or> coprime p n"
   723   using prime_coprime[of p n] by auto
   724 
   725 declare  coprime_0[simp]
   726 
   727 lemma coprime_0'[simp]: "coprime 0 d \<longleftrightarrow> d = 1" by (simp add: coprime_commute[of 0 d])
   728 lemma coprime_bezout_strong: assumes ab: "coprime a b" and b: "b \<noteq> 1"
   729   shows "\<exists>x y. a * x = b * y + 1"
   730 proof-
   731   from ab b have az: "a \<noteq> 0" by - (rule ccontr, auto)
   732   from bezout_gcd_strong[OF az, of b] ab[unfolded coprime_def]
   733   show ?thesis by auto
   734 qed
   735 
   736 lemma bezout_prime: assumes p: "prime p"  and pa: "\<not> p dvd a"
   737   shows "\<exists>x y. a*x = p*y + 1"
   738 proof-
   739   from p have p1: "p \<noteq> 1" using prime_1 by blast 
   740   from prime_coprime[OF p, of a] p1 pa have ap: "coprime a p" 
   741     by (auto simp add: coprime_commute)
   742   from coprime_bezout_strong[OF ap p1] show ?thesis . 
   743 qed
   744 lemma prime_divprod: assumes p: "prime p" and pab: "p dvd a*b"
   745   shows "p dvd a \<or> p dvd b"
   746 proof-
   747   {assume "a=1" hence ?thesis using pab by simp }
   748   moreover
   749   {assume "p dvd a" hence ?thesis by blast}
   750   moreover
   751   {assume pa: "coprime p a" from coprime_divprod[OF pab pa]  have ?thesis .. }
   752   ultimately show ?thesis using prime_coprime[OF p, of a] by blast
   753 qed
   754 
   755 lemma prime_divprod_eq: assumes p: "prime p"
   756   shows "p dvd a*b \<longleftrightarrow> p dvd a \<or> p dvd b"
   757 using p prime_divprod dvd_mult dvd_mult2 by auto
   758 
   759 lemma prime_divexp: assumes p:"prime p" and px: "p dvd x^n"
   760   shows "p dvd x"
   761 using px
   762 proof(induct n)
   763   case 0 thus ?case by simp
   764 next
   765   case (Suc n) 
   766   hence th: "p dvd x*x^n" by simp
   767   {assume H: "p dvd x^n"
   768     from Suc.hyps[OF H] have ?case .}
   769   with prime_divprod[OF p th] show ?case by blast
   770 qed
   771 
   772 lemma prime_divexp_n: "prime p \<Longrightarrow> p dvd x^n \<Longrightarrow> p^n dvd x^n"
   773   using prime_divexp[of p x n] divides_exp[of p x n] by blast
   774 
   775 lemma coprime_prime_dvd_ex: assumes xy: "\<not>coprime x y"
   776   shows "\<exists>p. prime p \<and> p dvd x \<and> p dvd y"
   777 proof-
   778   from xy[unfolded coprime_def] obtain g where g: "g \<noteq> 1" "g dvd x" "g dvd y" 
   779     by blast
   780   from prime_factor[OF g(1)] obtain p where p: "prime p" "p dvd g" by blast
   781   from g(2,3) dvd_trans[OF p(2)] p(1) show ?thesis by auto
   782 qed
   783 lemma coprime_sos: assumes xy: "coprime x y" 
   784   shows "coprime (x * y) (x^2 + y^2)"
   785 proof-
   786   {assume c: "\<not> coprime (x * y) (x^2 + y^2)"
   787     from coprime_prime_dvd_ex[OF c] obtain p 
   788       where p: "prime p" "p dvd x*y" "p dvd x^2 + y^2" by blast
   789     {assume px: "p dvd x"
   790       from dvd_mult[OF px, of x] p(3) have "p dvd y^2" 
   791 	unfolding dvd_def 
   792 	apply (auto simp add: power2_eq_square)
   793 	apply (rule_tac x= "ka - k" in exI)
   794 	by (simp add: diff_mult_distrib2)
   795       with prime_divexp[OF p(1), of y 2] have py: "p dvd y" .
   796       from p(1) px py xy[unfolded coprime, rule_format, of p] prime_1  
   797       have False by simp }
   798     moreover
   799     {assume py: "p dvd y"
   800       from dvd_mult[OF py, of y] p(3) have "p dvd x^2" 
   801 	unfolding dvd_def 
   802 	apply (auto simp add: power2_eq_square)
   803 	apply (rule_tac x= "ka - k" in exI)
   804 	by (simp add: diff_mult_distrib2)
   805       with prime_divexp[OF p(1), of x 2] have px: "p dvd x" .
   806       from p(1) px py xy[unfolded coprime, rule_format, of p] prime_1  
   807       have False by simp }
   808     ultimately have False using prime_divprod[OF p(1,2)] by blast}
   809   thus ?thesis by blast
   810 qed
   811 
   812 lemma distinct_prime_coprime: "prime p \<Longrightarrow> prime q \<Longrightarrow> p \<noteq> q \<Longrightarrow> coprime p q"
   813   unfolding prime_def coprime_prime_eq by blast
   814 
   815 lemma prime_coprime_lt: assumes p: "prime p" and x: "0 < x" and xp: "x < p"
   816   shows "coprime x p"
   817 proof-
   818   {assume c: "\<not> coprime x p"
   819     then obtain g where g: "g \<noteq> 1" "g dvd x" "g dvd p" unfolding coprime_def by blast
   820   from dvd_imp_le[OF g(2)] x xp have gp: "g < p" by arith
   821   from g(2) x have "g \<noteq> 0" by - (rule ccontr, simp)
   822   with g gp p[unfolded prime_def] have False by blast}
   823 thus ?thesis by blast
   824 qed
   825 
   826 lemma even_dvd[simp]: "even (n::nat) \<longleftrightarrow> 2 dvd n" by presburger
   827 lemma prime_odd: "prime p \<Longrightarrow> p = 2 \<or> odd p" unfolding prime_def by auto
   828 
   829 
   830 text {* One property of coprimality is easier to prove via prime factors. *}
   831 
   832 lemma prime_divprod_pow: 
   833   assumes p: "prime p" and ab: "coprime a b" and pab: "p^n dvd a * b"
   834   shows "p^n dvd a \<or> p^n dvd b"
   835 proof-
   836   {assume "n = 0 \<or> a = 1 \<or> b = 1" with pab have ?thesis 
   837       apply (cases "n=0", simp_all)
   838       apply (cases "a=1", simp_all) done}
   839   moreover
   840   {assume n: "n \<noteq> 0" and a: "a\<noteq>1" and b: "b\<noteq>1" 
   841     then obtain m where m: "n = Suc m" by (cases n, auto)
   842     from divides_exp2[OF n pab] have pab': "p dvd a*b" .
   843     from prime_divprod[OF p pab'] 
   844     have "p dvd a \<or> p dvd b" .
   845     moreover
   846     {assume pa: "p dvd a"
   847       have pnba: "p^n dvd b*a" using pab by (simp add: mult_commute)
   848       from coprime_prime[OF ab, of p] p pa have "\<not> p dvd b" by blast
   849       with prime_coprime[OF p, of b] b 
   850       have cpb: "coprime b p" using coprime_commute by blast 
   851       from coprime_exp[OF cpb] have pnb: "coprime (p^n) b" 
   852 	by (simp add: coprime_commute)
   853       from coprime_divprod[OF pnba pnb] have ?thesis by blast }
   854     moreover
   855     {assume pb: "p dvd b"
   856       have pnba: "p^n dvd b*a" using pab by (simp add: mult_commute)
   857       from coprime_prime[OF ab, of p] p pb have "\<not> p dvd a" by blast
   858       with prime_coprime[OF p, of a] a
   859       have cpb: "coprime a p" using coprime_commute by blast 
   860       from coprime_exp[OF cpb] have pnb: "coprime (p^n) a" 
   861 	by (simp add: coprime_commute)
   862       from coprime_divprod[OF pab pnb] have ?thesis by blast }
   863     ultimately have ?thesis by blast}
   864   ultimately show ?thesis by blast
   865 qed
   866 
   867 lemma nat_mult_eq_one: "(n::nat) * m = 1 \<longleftrightarrow> n = 1 \<and> m = 1" (is "?lhs \<longleftrightarrow> ?rhs")
   868 proof
   869   assume H: "?lhs"
   870   hence "n dvd 1" "m dvd 1" unfolding dvd_def by (auto simp add: mult_commute)
   871   thus ?rhs by auto
   872 next
   873   assume ?rhs then show ?lhs by auto
   874 qed
   875   
   876 lemma power_Suc0[simp]: "Suc 0 ^ n = Suc 0" 
   877   unfolding One_nat_def[symmetric] power_one ..
   878 lemma coprime_pow: assumes ab: "coprime a b" and abcn: "a * b = c ^n"
   879   shows "\<exists>r s. a = r^n  \<and> b = s ^n"
   880   using ab abcn
   881 proof(induct c arbitrary: a b rule: nat_less_induct)
   882   fix c a b
   883   assume H: "\<forall>m<c. \<forall>a b. coprime a b \<longrightarrow> a * b = m ^ n \<longrightarrow> (\<exists>r s. a = r ^ n \<and> b = s ^ n)" "coprime a b" "a * b = c ^ n" 
   884   let ?ths = "\<exists>r s. a = r^n  \<and> b = s ^n"
   885   {assume n: "n = 0"
   886     with H(3) power_one have "a*b = 1" by simp
   887     hence "a = 1 \<and> b = 1" by simp
   888     hence ?ths 
   889       apply -
   890       apply (rule exI[where x=1])
   891       apply (rule exI[where x=1])
   892       using power_one[of  n]
   893       by simp}
   894   moreover
   895   {assume n: "n \<noteq> 0" then obtain m where m: "n = Suc m" by (cases n, auto)
   896     {assume c: "c = 0"
   897       with H(3) m H(2) have ?ths apply simp 
   898 	apply (cases "a=0", simp_all) 
   899 	apply (rule exI[where x="0"], simp)
   900 	apply (rule exI[where x="0"], simp)
   901 	done}
   902     moreover
   903     {assume "c=1" with H(3) power_one have "a*b = 1" by simp 
   904 	hence "a = 1 \<and> b = 1" by simp
   905 	hence ?ths 
   906       apply -
   907       apply (rule exI[where x=1])
   908       apply (rule exI[where x=1])
   909       using power_one[of  n]
   910       by simp}
   911   moreover
   912   {assume c: "c\<noteq>1" "c \<noteq> 0"
   913     from prime_factor[OF c(1)] obtain p where p: "prime p" "p dvd c" by blast
   914     from prime_divprod_pow[OF p(1) H(2), unfolded H(3), OF divides_exp[OF p(2), of n]] 
   915     have pnab: "p ^ n dvd a \<or> p^n dvd b" . 
   916     from p(2) obtain l where l: "c = p*l" unfolding dvd_def by blast
   917     have pn0: "p^n \<noteq> 0" using n prime_ge_2 [OF p(1)] by (simp add: neq0_conv)
   918     {assume pa: "p^n dvd a"
   919       then obtain k where k: "a = p^n * k" unfolding dvd_def by blast
   920       from l have "l dvd c" by auto
   921       with dvd_imp_le[of l c] c have "l \<le> c" by auto
   922       moreover {assume "l = c" with l c  have "p = 1" by simp with p have False by simp}
   923       ultimately have lc: "l < c" by arith
   924       from coprime_lmul2 [OF H(2)[unfolded k coprime_commute[of "p^n*k" b]]]
   925       have kb: "coprime k b" by (simp add: coprime_commute) 
   926       from H(3) l k pn0 have kbln: "k * b = l ^ n" 
   927 	by (auto simp add: power_mult_distrib)
   928       from H(1)[rule_format, OF lc kb kbln]
   929       obtain r s where rs: "k = r ^n" "b = s^n" by blast
   930       from k rs(1) have "a = (p*r)^n" by (simp add: power_mult_distrib)
   931       with rs(2) have ?ths by blast }
   932     moreover
   933     {assume pb: "p^n dvd b"
   934       then obtain k where k: "b = p^n * k" unfolding dvd_def by blast
   935       from l have "l dvd c" by auto
   936       with dvd_imp_le[of l c] c have "l \<le> c" by auto
   937       moreover {assume "l = c" with l c  have "p = 1" by simp with p have False by simp}
   938       ultimately have lc: "l < c" by arith
   939       from coprime_lmul2 [OF H(2)[unfolded k coprime_commute[of "p^n*k" a]]]
   940       have kb: "coprime k a" by (simp add: coprime_commute) 
   941       from H(3) l k pn0 n have kbln: "k * a = l ^ n" 
   942 	by (simp add: power_mult_distrib mult_commute)
   943       from H(1)[rule_format, OF lc kb kbln]
   944       obtain r s where rs: "k = r ^n" "a = s^n" by blast
   945       from k rs(1) have "b = (p*r)^n" by (simp add: power_mult_distrib)
   946       with rs(2) have ?ths by blast }
   947     ultimately have ?ths using pnab by blast}
   948   ultimately have ?ths by blast}
   949 ultimately show ?ths by blast
   950 qed
   951 
   952 text {* More useful lemmas. *}
   953 lemma prime_product: 
   954   "prime (p*q) \<Longrightarrow> p = 1 \<or> q  = 1" unfolding prime_def by auto
   955 
   956 lemma prime_exp: "prime (p^n) \<longleftrightarrow> prime p \<and> n = 1"
   957 proof(induct n)
   958   case 0 thus ?case by simp
   959 next
   960   case (Suc n)
   961   {assume "p = 0" hence ?case by simp}
   962   moreover
   963   {assume "p=1" hence ?case by simp}
   964   moreover
   965   {assume p: "p \<noteq> 0" "p\<noteq>1"
   966     {assume pp: "prime (p^Suc n)"
   967       hence "p = 1 \<or> p^n = 1" using prime_product[of p "p^n"] by simp
   968       with p have n: "n = 0" 
   969 	by (simp only: exp_eq_1 ) simp
   970       with pp have "prime p \<and> Suc n = 1" by simp}
   971     moreover
   972     {assume n: "prime p \<and> Suc n = 1" hence "prime (p^Suc n)" by simp}
   973     ultimately have ?case by blast}
   974   ultimately show ?case by blast
   975 qed
   976 
   977 lemma prime_power_mult: 
   978   assumes p: "prime p" and xy: "x * y = p ^ k"
   979   shows "\<exists>i j. x = p ^i \<and> y = p^ j"
   980   using xy
   981 proof(induct k arbitrary: x y)
   982   case 0 thus ?case apply simp by (rule exI[where x="0"], simp)
   983 next
   984   case (Suc k x y)
   985   from Suc.prems have pxy: "p dvd x*y" by auto
   986   from prime_divprod[OF p pxy] have pxyc: "p dvd x \<or> p dvd y" .
   987   from p have p0: "p \<noteq> 0" by - (rule ccontr, simp) 
   988   {assume px: "p dvd x"
   989     then obtain d where d: "x = p*d" unfolding dvd_def by blast
   990     from Suc.prems d  have "p*d*y = p^Suc k" by simp
   991     hence th: "d*y = p^k" using p0 by simp
   992     from Suc.hyps[OF th] obtain i j where ij: "d = p^i" "y = p^j" by blast
   993     with d have "x = p^Suc i" by simp 
   994     with ij(2) have ?case by blast}
   995   moreover 
   996   {assume px: "p dvd y"
   997     then obtain d where d: "y = p*d" unfolding dvd_def by blast
   998     from Suc.prems d  have "p*d*x = p^Suc k" by (simp add: mult_commute)
   999     hence th: "d*x = p^k" using p0 by simp
  1000     from Suc.hyps[OF th] obtain i j where ij: "d = p^i" "x = p^j" by blast
  1001     with d have "y = p^Suc i" by simp 
  1002     with ij(2) have ?case by blast}
  1003   ultimately show ?case  using pxyc by blast
  1004 qed
  1005 
  1006 lemma prime_power_exp: assumes p: "prime p" and n:"n \<noteq> 0" 
  1007   and xn: "x^n = p^k" shows "\<exists>i. x = p^i"
  1008   using n xn
  1009 proof(induct n arbitrary: k)
  1010   case 0 thus ?case by simp
  1011 next
  1012   case (Suc n k) hence th: "x*x^n = p^k" by simp
  1013   {assume "n = 0" with prems have ?case apply simp 
  1014       by (rule exI[where x="k"],simp)}
  1015   moreover
  1016   {assume n: "n \<noteq> 0"
  1017     from prime_power_mult[OF p th] 
  1018     obtain i j where ij: "x = p^i" "x^n = p^j"by blast
  1019     from Suc.hyps[OF n ij(2)] have ?case .}
  1020   ultimately show ?case by blast
  1021 qed
  1022 
  1023 lemma divides_primepow: assumes p: "prime p" 
  1024   shows "d dvd p^k \<longleftrightarrow> (\<exists> i. i \<le> k \<and> d = p ^i)"
  1025 proof
  1026   assume H: "d dvd p^k" then obtain e where e: "d*e = p^k" 
  1027     unfolding dvd_def  apply (auto simp add: mult_commute) by blast
  1028   from prime_power_mult[OF p e] obtain i j where ij: "d = p^i" "e=p^j" by blast
  1029   from prime_ge_2[OF p] have p1: "p > 1" by arith
  1030   from e ij have "p^(i + j) = p^k" by (simp add: power_add)
  1031   hence "i + j = k" using power_inject_exp[of p "i+j" k, OF p1] by simp 
  1032   hence "i \<le> k" by arith
  1033   with ij(1) show "\<exists>i\<le>k. d = p ^ i" by blast
  1034 next
  1035   {fix i assume H: "i \<le> k" "d = p^i"
  1036     hence "\<exists>j. k = i + j" by arith
  1037     then obtain j where j: "k = i + j" by blast
  1038     hence "p^k = p^j*d" using H(2) by (simp add: power_add)
  1039     hence "d dvd p^k" unfolding dvd_def by auto}
  1040   thus "\<exists>i\<le>k. d = p ^ i \<Longrightarrow> d dvd p ^ k" by blast
  1041 qed
  1042 
  1043 lemma coprime_divisors: "d dvd a \<Longrightarrow> e dvd b \<Longrightarrow> coprime a b \<Longrightarrow> coprime d e"
  1044   by (auto simp add: dvd_def coprime)
  1045 
  1046 end