src/HOL/Library/Primes.thy
 author haftmann Fri Jul 18 18:25:53 2008 +0200 (2008-07-18) changeset 27651 16a26996c30e parent 27567 e3fe9a327c63 child 27670 3b5425dead98 permissions -rw-r--r--
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
```     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 Plain "~~/src/HOL/ATP_Linkup" 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   [code func del]: "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"
```
```   392   "gcd(b + a) b = gcd a b" "gcd a (a + b) = gcd a b" "gcd a (b + a) = gcd a b"
```
```   393 apply (simp_all add: gcd_add1)
```
```   394 by (simp add: gcd_commute gcd_add1)
```
```   395
```
```   396 lemma gcd_sub: "b <= a ==> gcd(a - b) b = gcd a b" "a <= b ==> gcd a (b - a) = gcd a b"
```
```   397 proof-
```
```   398   {fix a b assume H: "b \<le> (a::nat)"
```
```   399     hence th: "a - b + b = a" by arith
```
```   400     from gcd_add(1)[of "a - b" b] th  have "gcd(a - b) b = gcd a b" by simp}
```
```   401   note th = this
```
```   402 {
```
```   403   assume ab: "b \<le> a"
```
```   404   from th[OF ab] show "gcd (a - b)  b = gcd a b" by blast
```
```   405 next
```
```   406   assume ab: "a \<le> b"
```
```   407   from th[OF ab] show "gcd a (b - a) = gcd a b"
```
```   408     by (simp add: gcd_commute)}
```
```   409 qed
```
```   410
```
```   411 text {* Coprimality *}
```
```   412
```
```   413 lemma coprime: "coprime a b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
```
```   414 using gcd_unique[of 1 a b, simplified] by (auto simp add: coprime_def)
```
```   415 lemma coprime_commute: "coprime a b \<longleftrightarrow> coprime b a" by (simp add: coprime_def gcd_commute)
```
```   416
```
```   417 lemma coprime_bezout: "coprime a b \<longleftrightarrow> (\<exists>x y. a * x - b * y = 1 \<or> b * x - a * y = 1)"
```
```   418 using coprime_def gcd_bezout by auto
```
```   419
```
```   420 lemma coprime_divprod: "d dvd a * b  \<Longrightarrow> coprime d a \<Longrightarrow> d dvd b"
```
```   421   using relprime_dvd_mult_iff[of d a b] by (auto simp add: coprime_def mult_commute)
```
```   422
```
```   423 lemma coprime_1[simp]: "coprime a 1" by (simp add: coprime_def)
```
```   424 lemma coprime_1'[simp]: "coprime 1 a" by (simp add: coprime_def)
```
```   425 lemma coprime_Suc0[simp]: "coprime a (Suc 0)" by (simp add: coprime_def)
```
```   426 lemma coprime_Suc0'[simp]: "coprime (Suc 0) a" by (simp add: coprime_def)
```
```   427
```
```   428 lemma gcd_coprime:
```
```   429   assumes z: "gcd a b \<noteq> 0" and a: "a = a' * gcd a b" and b: "b = b' * gcd a b"
```
```   430   shows    "coprime a' b'"
```
```   431 proof-
```
```   432   let ?g = "gcd a b"
```
```   433   {assume bz: "a = 0" from b bz z a have ?thesis by (simp add: gcd_zero coprime_def)}
```
```   434   moreover
```
```   435   {assume az: "a\<noteq> 0"
```
```   436     from z have z': "?g > 0" by simp
```
```   437     from bezout_gcd_strong[OF az, of b]
```
```   438     obtain x y where xy: "a*x = b*y + ?g" by blast
```
```   439     from xy a b have "?g * a'*x = ?g * (b'*y + 1)" by (simp add: ring_simps)
```
```   440     hence "?g * (a'*x) = ?g * (b'*y + 1)" by (simp add: mult_assoc)
```
```   441     hence "a'*x = (b'*y + 1)"
```
```   442       by (simp only: nat_mult_eq_cancel1[OF z'])
```
```   443     hence "a'*x - b'*y = 1" by simp
```
```   444     with coprime_bezout[of a' b'] have ?thesis by auto}
```
```   445   ultimately show ?thesis by blast
```
```   446 qed
```
```   447 lemma coprime_0: "coprime d 0 \<longleftrightarrow> d = 1" by (simp add: coprime_def)
```
```   448 lemma coprime_mul: assumes da: "coprime d a" and db: "coprime d b"
```
```   449   shows "coprime d (a * b)"
```
```   450 proof-
```
```   451   from da have th: "gcd a d = 1" by (simp add: coprime_def gcd_commute)
```
```   452   from gcd_mult_cancel[of a d b, OF th] db[unfolded coprime_def] have "gcd d (a*b) = 1"
```
```   453     by (simp add: gcd_commute)
```
```   454   thus ?thesis unfolding coprime_def .
```
```   455 qed
```
```   456 lemma coprime_lmul2: assumes dab: "coprime d (a * b)" shows "coprime d b"
```
```   457 using prems unfolding coprime_bezout
```
```   458 apply clarsimp
```
```   459 apply (case_tac "d * x - a * b * y = Suc 0 ", simp_all)
```
```   460 apply (rule_tac x="x" in exI)
```
```   461 apply (rule_tac x="a*y" in exI)
```
```   462 apply (simp add: mult_ac)
```
```   463 apply (rule_tac x="a*x" in exI)
```
```   464 apply (rule_tac x="y" in exI)
```
```   465 apply (simp add: mult_ac)
```
```   466 done
```
```   467
```
```   468 lemma coprime_rmul2: "coprime d (a * b) \<Longrightarrow> coprime d a"
```
```   469 unfolding coprime_bezout
```
```   470 apply clarsimp
```
```   471 apply (case_tac "d * x - a * b * y = Suc 0 ", simp_all)
```
```   472 apply (rule_tac x="x" in exI)
```
```   473 apply (rule_tac x="b*y" in exI)
```
```   474 apply (simp add: mult_ac)
```
```   475 apply (rule_tac x="b*x" in exI)
```
```   476 apply (rule_tac x="y" in exI)
```
```   477 apply (simp add: mult_ac)
```
```   478 done
```
```   479 lemma coprime_mul_eq: "coprime d (a * b) \<longleftrightarrow> coprime d a \<and>  coprime d b"
```
```   480   using coprime_rmul2[of d a b] coprime_lmul2[of d a b] coprime_mul[of d a b]
```
```   481   by blast
```
```   482
```
```   483 lemma gcd_coprime_exists:
```
```   484   assumes nz: "gcd a b \<noteq> 0"
```
```   485   shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
```
```   486 proof-
```
```   487   let ?g = "gcd a b"
```
```   488   from gcd_dvd1[of a b] gcd_dvd2[of a b]
```
```   489   obtain a' b' where "a = ?g*a'"  "b = ?g*b'" unfolding dvd_def by blast
```
```   490   hence ab': "a = a'*?g" "b = b'*?g" by algebra+
```
```   491   from ab' gcd_coprime[OF nz ab'] show ?thesis by blast
```
```   492 qed
```
```   493
```
```   494 lemma coprime_exp: "coprime d a ==> coprime d (a^n)"
```
```   495   by(induct n, simp_all add: coprime_mul)
```
```   496
```
```   497 lemma coprime_exp_imp: "coprime a b ==> coprime (a ^n) (b ^n)"
```
```   498   by (induct n, simp_all add: coprime_mul_eq coprime_commute coprime_exp)
```
```   499 lemma coprime_refl[simp]: "coprime n n \<longleftrightarrow> n = 1" by (simp add: coprime_def)
```
```   500 lemma coprime_plus1[simp]: "coprime (n + 1) n"
```
```   501   apply (simp add: coprime_bezout)
```
```   502   apply (rule exI[where x=1])
```
```   503   apply (rule exI[where x=1])
```
```   504   apply simp
```
```   505   done
```
```   506 lemma coprime_minus1: "n \<noteq> 0 ==> coprime (n - 1) n"
```
```   507   using coprime_plus1[of "n - 1"] coprime_commute[of "n - 1" n] by auto
```
```   508
```
```   509 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"
```
```   510 proof-
```
```   511   let ?g = "gcd a b"
```
```   512   {assume z: "?g = 0" hence ?thesis
```
```   513       apply (cases n, simp)
```
```   514       apply arith
```
```   515       apply (simp only: z power_0_Suc)
```
```   516       apply (rule exI[where x=0])
```
```   517       apply (rule exI[where x=0])
```
```   518       by simp}
```
```   519   moreover
```
```   520   {assume z: "?g \<noteq> 0"
```
```   521     from gcd_dvd1[of a b] gcd_dvd2[of a b] obtain a' b' where
```
```   522       ab': "a = a'*?g" "b = b'*?g" unfolding dvd_def by (auto simp add: mult_ac)
```
```   523     hence ab'': "?g*a' = a" "?g * b' = b" by algebra+
```
```   524     from coprime_exp_imp[OF gcd_coprime[OF z ab'], unfolded coprime_bezout, of n]
```
```   525     obtain x y where "a'^n * x - b'^n * y = 1 \<or> b'^n * x - a'^n * y = 1"  by blast
```
```   526     hence "?g^n * (a'^n * x - b'^n * y) = ?g^n \<or> ?g^n*(b'^n * x - a'^n * y) = ?g^n"
```
```   527       using z by auto
```
```   528     then have "a^n * x - b^n * y = ?g^n \<or> b^n * x - a^n * y = ?g^n"
```
```   529       using z ab'' by (simp only: power_mult_distrib[symmetric]
```
```   530 	diff_mult_distrib2 mult_assoc[symmetric])
```
```   531     hence  ?thesis by blast }
```
```   532   ultimately show ?thesis by blast
```
```   533 qed
```
```   534
```
```   535 lemma gcd_exp: "gcd (a^n) (b^n) = gcd a b^n"
```
```   536 proof-
```
```   537   let ?g = "gcd (a^n) (b^n)"
```
```   538   let ?gn = "gcd a b^n"
```
```   539   {fix e assume H: "e dvd a^n" "e dvd b^n"
```
```   540     from bezout_gcd_pow[of a n b] obtain x y
```
```   541       where xy: "a ^ n * x - b ^ n * y = ?gn \<or> b ^ n * x - a ^ n * y = ?gn" by blast
```
```   542     from dvd_diff [OF dvd_mult2[OF H(1), of x] dvd_mult2[OF H(2), of y]]
```
```   543       dvd_diff [OF dvd_mult2[OF H(2), of x] dvd_mult2[OF H(1), of y]] xy
```
```   544     have "e dvd ?gn" by (cases "a ^ n * x - b ^ n * y = gcd a b ^ n", simp_all)}
```
```   545   hence th:  "\<forall>e. e dvd a^n \<and> e dvd b^n \<longrightarrow> e dvd ?gn" by blast
```
```   546   from divides_exp[OF gcd_dvd1[of a b], of n] divides_exp[OF gcd_dvd2[of a b], of n] th
```
```   547     gcd_unique have "?gn = ?g" by blast thus ?thesis by simp
```
```   548 qed
```
```   549
```
```   550 lemma coprime_exp2:  "coprime (a ^ Suc n) (b^ Suc n) \<longleftrightarrow> coprime a b"
```
```   551 by (simp only: coprime_def gcd_exp exp_eq_1) simp
```
```   552
```
```   553 lemma division_decomp: assumes dc: "(a::nat) dvd b * c"
```
```   554   shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
```
```   555 proof-
```
```   556   let ?g = "gcd a b"
```
```   557   {assume "?g = 0" with dc have ?thesis apply (simp add: gcd_zero)
```
```   558       apply (rule exI[where x="0"])
```
```   559       by (rule exI[where x="c"], simp)}
```
```   560   moreover
```
```   561   {assume z: "?g \<noteq> 0"
```
```   562     from gcd_coprime_exists[OF z]
```
```   563     obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'" by blast
```
```   564     from gcd_dvd2[of a b] have thb: "?g dvd b" .
```
```   565     from ab'(1) have "a' dvd a"  unfolding dvd_def by blast
```
```   566     with dc have th0: "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
```
```   567     from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
```
```   568     hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult_assoc)
```
```   569     with z have th_1: "a' dvd b'*c" by simp
```
```   570     from coprime_divprod[OF th_1 ab'(3)] have thc: "a' dvd c" .
```
```   571     from ab' have "a = ?g*a'" by algebra
```
```   572     with thb thc have ?thesis by blast }
```
```   573   ultimately show ?thesis by blast
```
```   574 qed
```
```   575
```
```   576 lemma nat_power_eq_0_iff: "(m::nat) ^ n = 0 \<longleftrightarrow> n \<noteq> 0 \<and> m = 0" by (induct n, auto)
```
```   577
```
```   578 lemma divides_rev: assumes ab: "(a::nat) ^ n dvd b ^n" and n:"n \<noteq> 0" shows "a dvd b"
```
```   579 proof-
```
```   580   let ?g = "gcd a b"
```
```   581   from n obtain m where m: "n = Suc m" by (cases n, simp_all)
```
```   582   {assume "?g = 0" with ab n have ?thesis by (simp add: gcd_zero)}
```
```   583   moreover
```
```   584   {assume z: "?g \<noteq> 0"
```
```   585     hence zn: "?g ^ n \<noteq> 0" using n by (simp add: neq0_conv)
```
```   586     from gcd_coprime_exists[OF z]
```
```   587     obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'" by blast
```
```   588     from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n" by (simp add: ab'(1,2)[symmetric])
```
```   589     hence "?g^n*a'^n dvd ?g^n *b'^n" by (simp only: power_mult_distrib mult_commute)
```
```   590     with zn z n have th0:"a'^n dvd b'^n" by (auto simp add: nat_power_eq_0_iff)
```
```   591     have "a' dvd a'^n" by (simp add: m)
```
```   592     with th0 have "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by simp
```
```   593     hence th1: "a' dvd b'^m * b'" by (simp add: m mult_commute)
```
```   594     from coprime_divprod[OF th1 coprime_exp[OF ab'(3), of m]]
```
```   595     have "a' dvd b'" .
```
```   596     hence "a'*?g dvd b'*?g" by simp
```
```   597     with ab'(1,2)  have ?thesis by simp }
```
```   598   ultimately show ?thesis by blast
```
```   599 qed
```
```   600
```
```   601 lemma divides_mul: assumes mr: "m dvd r" and nr: "n dvd r" and mn:"coprime m n"
```
```   602   shows "m * n dvd r"
```
```   603 proof-
```
```   604   from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
```
```   605     unfolding dvd_def by blast
```
```   606   from mr n' have "m dvd n'*n" by (simp add: mult_commute)
```
```   607   hence "m dvd n'" using relprime_dvd_mult_iff[OF mn[unfolded coprime_def]] by simp
```
```   608   then obtain k where k: "n' = m*k" unfolding dvd_def by blast
```
```   609   from n' k show ?thesis unfolding dvd_def by auto
```
```   610 qed
```
```   611
```
```   612
```
```   613 text {* A binary form of the Chinese Remainder Theorem. *}
```
```   614
```
```   615 lemma chinese_remainder: assumes ab: "coprime a b" and a:"a \<noteq> 0" and b:"b \<noteq> 0"
```
```   616   shows "\<exists>x q1 q2. x = u + q1 * a \<and> x = v + q2 * b"
```
```   617 proof-
```
```   618   from bezout_add_strong[OF a, of b] bezout_add_strong[OF b, of a]
```
```   619   obtain d1 x1 y1 d2 x2 y2 where dxy1: "d1 dvd a" "d1 dvd b" "a * x1 = b * y1 + d1"
```
```   620     and dxy2: "d2 dvd b" "d2 dvd a" "b * x2 = a * y2 + d2" by blast
```
```   621   from gcd_unique[of 1 a b, simplified ab[unfolded coprime_def], simplified]
```
```   622     dxy1(1,2) dxy2(1,2) have d12: "d1 = 1" "d2 =1" by auto
```
```   623   let ?x = "v * a * x1 + u * b * x2"
```
```   624   let ?q1 = "v * x1 + u * y2"
```
```   625   let ?q2 = "v * y1 + u * x2"
```
```   626   from dxy2(3)[simplified d12] dxy1(3)[simplified d12]
```
```   627   have "?x = u + ?q1 * a" "?x = v + ?q2 * b" by algebra+
```
```   628   thus ?thesis by blast
```
```   629 qed
```
```   630
```
```   631 text {* Primality *}
```
```   632
```
```   633 text {* A few useful theorems about primes *}
```
```   634
```
```   635 lemma prime_0[simp]: "~prime 0" by (simp add: prime_def)
```
```   636 lemma prime_1[simp]: "~ prime 1"  by (simp add: prime_def)
```
```   637 lemma prime_Suc0[simp]: "~ prime (Suc 0)"  by (simp add: prime_def)
```
```   638
```
```   639 lemma prime_ge_2: "prime p ==> p \<ge> 2" by (simp add: prime_def)
```
```   640 lemma prime_factor: assumes n: "n \<noteq> 1" shows "\<exists> p. prime p \<and> p dvd n"
```
```   641 using n
```
```   642 proof(induct n rule: nat_less_induct)
```
```   643   fix n
```
```   644   assume H: "\<forall>m<n. m \<noteq> 1 \<longrightarrow> (\<exists>p. prime p \<and> p dvd m)" "n \<noteq> 1"
```
```   645   let ?ths = "\<exists>p. prime p \<and> p dvd n"
```
```   646   {assume "n=0" hence ?ths using two_is_prime by auto}
```
```   647   moreover
```
```   648   {assume nz: "n\<noteq>0"
```
```   649     {assume "prime n" hence ?ths by - (rule exI[where x="n"], simp)}
```
```   650     moreover
```
```   651     {assume n: "\<not> prime n"
```
```   652       with nz H(2)
```
```   653       obtain k where k:"k dvd n" "k \<noteq> 1" "k \<noteq> n" by (auto simp add: prime_def)
```
```   654       from dvd_imp_le[OF k(1)] nz k(3) have kn: "k < n" by simp
```
```   655       from H(1)[rule_format, OF kn k(2)] obtain p where p: "prime p" "p dvd k" by blast
```
```   656       from dvd_trans[OF p(2) k(1)] p(1) have ?ths by blast}
```
```   657     ultimately have ?ths by blast}
```
```   658   ultimately show ?ths by blast
```
```   659 qed
```
```   660
```
```   661 lemma prime_factor_lt: assumes p: "prime p" and n: "n \<noteq> 0" and npm:"n = p * m"
```
```   662   shows "m < n"
```
```   663 proof-
```
```   664   {assume "m=0" with n have ?thesis by simp}
```
```   665   moreover
```
```   666   {assume m: "m \<noteq> 0"
```
```   667     from npm have mn: "m dvd n" unfolding dvd_def by auto
```
```   668     from npm m have "n \<noteq> m" using p by auto
```
```   669     with dvd_imp_le[OF mn] n have ?thesis by simp}
```
```   670   ultimately show ?thesis by blast
```
```   671 qed
```
```   672
```
```   673 lemma euclid_bound: "\<exists>p. prime p \<and> n < p \<and>  p <= Suc (fact n)"
```
```   674 proof-
```
```   675   have f1: "fact n + 1 \<noteq> 1" using fact_le[of n] by arith
```
```   676   from prime_factor[OF f1] obtain p where p: "prime p" "p dvd fact n + 1" by blast
```
```   677   from dvd_imp_le[OF p(2)] have pfn: "p \<le> fact n + 1" by simp
```
```   678   {assume np: "p \<le> n"
```
```   679     from p(1) have p1: "p \<ge> 1" by (cases p, simp_all)
```
```   680     from divides_fact[OF p1 np] have pfn': "p dvd fact n" .
```
```   681     from divides_add_revr[OF pfn' p(2)] p(1) have False by simp}
```
```   682   hence "n < p" by arith
```
```   683   with p(1) pfn show ?thesis by auto
```
```   684 qed
```
```   685
```
```   686 lemma euclid: "\<exists>p. prime p \<and> p > n" using euclid_bound by auto
```
```   687 lemma primes_infinite: "\<not> (finite {p. prime p})"
```
```   688 proof (auto simp add: finite_conv_nat_seg_image)
```
```   689   fix n f
```
```   690   assume H: "Collect prime = f ` {i. i < (n::nat)}"
```
```   691   let ?P = "Collect prime"
```
```   692   let ?m = "Max ?P"
```
```   693   have P0: "?P \<noteq> {}" using two_is_prime by auto
```
```   694   from H have fP: "finite ?P" using finite_conv_nat_seg_image by blast
```
```   695   from Max_in [OF fP P0] have "?m \<in> ?P" .
```
```   696   from Max_ge [OF fP] have contr: "\<forall> p. prime p \<longrightarrow> p \<le> ?m" by blast
```
```   697   from euclid [of ?m] obtain q where q: "prime q" "q > ?m" by blast
```
```   698   with contr show False by auto
```
```   699 qed
```
```   700
```
```   701 lemma coprime_prime: assumes ab: "coprime a b"
```
```   702   shows "~(prime p \<and> p dvd a \<and> p dvd b)"
```
```   703 proof
```
```   704   assume "prime p \<and> p dvd a \<and> p dvd b"
```
```   705   thus False using ab gcd_greatest[of p a b] by (simp add: coprime_def)
```
```   706 qed
```
```   707 lemma coprime_prime_eq: "coprime a b \<longleftrightarrow> (\<forall>p. ~(prime p \<and> p dvd a \<and> p dvd b))"
```
```   708   (is "?lhs = ?rhs")
```
```   709 proof-
```
```   710   {assume "?lhs" with coprime_prime  have ?rhs by blast}
```
```   711   moreover
```
```   712   {assume r: "?rhs" and c: "\<not> ?lhs"
```
```   713     then obtain g where g: "g\<noteq>1" "g dvd a" "g dvd b" unfolding coprime_def by blast
```
```   714     from prime_factor[OF g(1)] obtain p where p: "prime p" "p dvd g" by blast
```
```   715     from dvd_trans [OF p(2) g(2)] dvd_trans [OF p(2) g(3)]
```
```   716     have "p dvd a" "p dvd b" . with p(1) r have False by blast}
```
```   717   ultimately show ?thesis by blast
```
```   718 qed
```
```   719
```
```   720 lemma prime_coprime: assumes p: "prime p"
```
```   721   shows "n = 1 \<or> p dvd n \<or> coprime p n"
```
```   722 using p prime_imp_relprime[of p n] by (auto simp add: coprime_def)
```
```   723
```
```   724 lemma prime_coprime_strong: "prime p \<Longrightarrow> p dvd n \<or> coprime p n"
```
```   725   using prime_coprime[of p n] by auto
```
```   726
```
```   727 declare  coprime_0[simp]
```
```   728
```
```   729 lemma coprime_0'[simp]: "coprime 0 d \<longleftrightarrow> d = 1" by (simp add: coprime_commute[of 0 d])
```
```   730 lemma coprime_bezout_strong: assumes ab: "coprime a b" and b: "b \<noteq> 1"
```
```   731   shows "\<exists>x y. a * x = b * y + 1"
```
```   732 proof-
```
```   733   from ab b have az: "a \<noteq> 0" by - (rule ccontr, auto)
```
```   734   from bezout_gcd_strong[OF az, of b] ab[unfolded coprime_def]
```
```   735   show ?thesis by auto
```
```   736 qed
```
```   737
```
```   738 lemma bezout_prime: assumes p: "prime p"  and pa: "\<not> p dvd a"
```
```   739   shows "\<exists>x y. a*x = p*y + 1"
```
```   740 proof-
```
```   741   from p have p1: "p \<noteq> 1" using prime_1 by blast
```
```   742   from prime_coprime[OF p, of a] p1 pa have ap: "coprime a p"
```
```   743     by (auto simp add: coprime_commute)
```
```   744   from coprime_bezout_strong[OF ap p1] show ?thesis .
```
```   745 qed
```
```   746 lemma prime_divprod: assumes p: "prime p" and pab: "p dvd a*b"
```
```   747   shows "p dvd a \<or> p dvd b"
```
```   748 proof-
```
```   749   {assume "a=1" hence ?thesis using pab by simp }
```
```   750   moreover
```
```   751   {assume "p dvd a" hence ?thesis by blast}
```
```   752   moreover
```
```   753   {assume pa: "coprime p a" from coprime_divprod[OF pab pa]  have ?thesis .. }
```
```   754   ultimately show ?thesis using prime_coprime[OF p, of a] by blast
```
```   755 qed
```
```   756
```
```   757 lemma prime_divprod_eq: assumes p: "prime p"
```
```   758   shows "p dvd a*b \<longleftrightarrow> p dvd a \<or> p dvd b"
```
```   759 using p prime_divprod dvd_mult dvd_mult2 by auto
```
```   760
```
```   761 lemma prime_divexp: assumes p:"prime p" and px: "p dvd x^n"
```
```   762   shows "p dvd x"
```
```   763 using px
```
```   764 proof(induct n)
```
```   765   case 0 thus ?case by simp
```
```   766 next
```
```   767   case (Suc n)
```
```   768   hence th: "p dvd x*x^n" by simp
```
```   769   {assume H: "p dvd x^n"
```
```   770     from Suc.hyps[OF H] have ?case .}
```
```   771   with prime_divprod[OF p th] show ?case by blast
```
```   772 qed
```
```   773
```
```   774 lemma prime_divexp_n: "prime p \<Longrightarrow> p dvd x^n \<Longrightarrow> p^n dvd x^n"
```
```   775   using prime_divexp[of p x n] divides_exp[of p x n] by blast
```
```   776
```
```   777 lemma coprime_prime_dvd_ex: assumes xy: "\<not>coprime x y"
```
```   778   shows "\<exists>p. prime p \<and> p dvd x \<and> p dvd y"
```
```   779 proof-
```
```   780   from xy[unfolded coprime_def] obtain g where g: "g \<noteq> 1" "g dvd x" "g dvd y"
```
```   781     by blast
```
```   782   from prime_factor[OF g(1)] obtain p where p: "prime p" "p dvd g" by blast
```
```   783   from g(2,3) dvd_trans[OF p(2)] p(1) show ?thesis by auto
```
```   784 qed
```
```   785 lemma coprime_sos: assumes xy: "coprime x y"
```
```   786   shows "coprime (x * y) (x^2 + y^2)"
```
```   787 proof-
```
```   788   {assume c: "\<not> coprime (x * y) (x^2 + y^2)"
```
```   789     from coprime_prime_dvd_ex[OF c] obtain p
```
```   790       where p: "prime p" "p dvd x*y" "p dvd x^2 + y^2" by blast
```
```   791     {assume px: "p dvd x"
```
```   792       from dvd_mult[OF px, of x] p(3)
```
```   793         obtain r s where "x * x = p * r" and "x^2 + y^2 = p * s"
```
```   794           by (auto elim!: dvdE)
```
```   795         then have "y^2 = p * (s - r)"
```
```   796           by (auto simp add: power2_eq_square diff_mult_distrib2)
```
```   797         then have "p dvd y^2" ..
```
```   798       with prime_divexp[OF p(1), of y 2] have py: "p dvd y" .
```
```   799       from p(1) px py xy[unfolded coprime, rule_format, of p] prime_1
```
```   800       have False by simp }
```
```   801     moreover
```
```   802     {assume py: "p dvd y"
```
```   803       from dvd_mult[OF py, of y] p(3)
```
```   804         obtain r s where "y * y = p * r" and "x^2 + y^2 = p * s"
```
```   805           by (auto elim!: dvdE)
```
```   806         then have "x^2 = p * (s - r)"
```
```   807           by (auto simp add: power2_eq_square diff_mult_distrib2)
```
```   808         then have "p dvd x^2" ..
```
```   809       with prime_divexp[OF p(1), of x 2] have px: "p dvd x" .
```
```   810       from p(1) px py xy[unfolded coprime, rule_format, of p] prime_1
```
```   811       have False by simp }
```
```   812     ultimately have False using prime_divprod[OF p(1,2)] by blast}
```
```   813   thus ?thesis by blast
```
```   814 qed
```
```   815
```
```   816 lemma distinct_prime_coprime: "prime p \<Longrightarrow> prime q \<Longrightarrow> p \<noteq> q \<Longrightarrow> coprime p q"
```
```   817   unfolding prime_def coprime_prime_eq by blast
```
```   818
```
```   819 lemma prime_coprime_lt: assumes p: "prime p" and x: "0 < x" and xp: "x < p"
```
```   820   shows "coprime x p"
```
```   821 proof-
```
```   822   {assume c: "\<not> coprime x p"
```
```   823     then obtain g where g: "g \<noteq> 1" "g dvd x" "g dvd p" unfolding coprime_def by blast
```
```   824   from dvd_imp_le[OF g(2)] x xp have gp: "g < p" by arith
```
```   825   from g(2) x have "g \<noteq> 0" by - (rule ccontr, simp)
```
```   826   with g gp p[unfolded prime_def] have False by blast}
```
```   827 thus ?thesis by blast
```
```   828 qed
```
```   829
```
```   830 lemma even_dvd[simp]: "even (n::nat) \<longleftrightarrow> 2 dvd n" by presburger
```
```   831 lemma prime_odd: "prime p \<Longrightarrow> p = 2 \<or> odd p" unfolding prime_def by auto
```
```   832
```
```   833
```
```   834 text {* One property of coprimality is easier to prove via prime factors. *}
```
```   835
```
```   836 lemma prime_divprod_pow:
```
```   837   assumes p: "prime p" and ab: "coprime a b" and pab: "p^n dvd a * b"
```
```   838   shows "p^n dvd a \<or> p^n dvd b"
```
```   839 proof-
```
```   840   {assume "n = 0 \<or> a = 1 \<or> b = 1" with pab have ?thesis
```
```   841       apply (cases "n=0", simp_all)
```
```   842       apply (cases "a=1", simp_all) done}
```
```   843   moreover
```
```   844   {assume n: "n \<noteq> 0" and a: "a\<noteq>1" and b: "b\<noteq>1"
```
```   845     then obtain m where m: "n = Suc m" by (cases n, auto)
```
```   846     from divides_exp2[OF n pab] have pab': "p dvd a*b" .
```
```   847     from prime_divprod[OF p pab']
```
```   848     have "p dvd a \<or> p dvd b" .
```
```   849     moreover
```
```   850     {assume pa: "p dvd a"
```
```   851       have pnba: "p^n dvd b*a" using pab by (simp add: mult_commute)
```
```   852       from coprime_prime[OF ab, of p] p pa have "\<not> p dvd b" by blast
```
```   853       with prime_coprime[OF p, of b] b
```
```   854       have cpb: "coprime b p" using coprime_commute by blast
```
```   855       from coprime_exp[OF cpb] have pnb: "coprime (p^n) b"
```
```   856 	by (simp add: coprime_commute)
```
```   857       from coprime_divprod[OF pnba pnb] have ?thesis by blast }
```
```   858     moreover
```
```   859     {assume pb: "p dvd b"
```
```   860       have pnba: "p^n dvd b*a" using pab by (simp add: mult_commute)
```
```   861       from coprime_prime[OF ab, of p] p pb have "\<not> p dvd a" by blast
```
```   862       with prime_coprime[OF p, of a] a
```
```   863       have cpb: "coprime a p" using coprime_commute by blast
```
```   864       from coprime_exp[OF cpb] have pnb: "coprime (p^n) a"
```
```   865 	by (simp add: coprime_commute)
```
```   866       from coprime_divprod[OF pab pnb] have ?thesis by blast }
```
```   867     ultimately have ?thesis by blast}
```
```   868   ultimately show ?thesis by blast
```
```   869 qed
```
```   870
```
```   871 lemma nat_mult_eq_one: "(n::nat) * m = 1 \<longleftrightarrow> n = 1 \<and> m = 1" (is "?lhs \<longleftrightarrow> ?rhs")
```
```   872 proof
```
```   873   assume H: "?lhs"
```
```   874   hence "n dvd 1" "m dvd 1" unfolding dvd_def by (auto simp add: mult_commute)
```
```   875   thus ?rhs by auto
```
```   876 next
```
```   877   assume ?rhs then show ?lhs by auto
```
```   878 qed
```
```   879
```
```   880 lemma power_Suc0[simp]: "Suc 0 ^ n = Suc 0"
```
```   881   unfolding One_nat_def[symmetric] power_one ..
```
```   882 lemma coprime_pow: assumes ab: "coprime a b" and abcn: "a * b = c ^n"
```
```   883   shows "\<exists>r s. a = r^n  \<and> b = s ^n"
```
```   884   using ab abcn
```
```   885 proof(induct c arbitrary: a b rule: nat_less_induct)
```
```   886   fix c a b
```
```   887   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"
```
```   888   let ?ths = "\<exists>r s. a = r^n  \<and> b = s ^n"
```
```   889   {assume n: "n = 0"
```
```   890     with H(3) power_one have "a*b = 1" by simp
```
```   891     hence "a = 1 \<and> b = 1" by simp
```
```   892     hence ?ths
```
```   893       apply -
```
```   894       apply (rule exI[where x=1])
```
```   895       apply (rule exI[where x=1])
```
```   896       using power_one[of  n]
```
```   897       by simp}
```
```   898   moreover
```
```   899   {assume n: "n \<noteq> 0" then obtain m where m: "n = Suc m" by (cases n, auto)
```
```   900     {assume c: "c = 0"
```
```   901       with H(3) m H(2) have ?ths apply simp
```
```   902 	apply (cases "a=0", simp_all)
```
```   903 	apply (rule exI[where x="0"], simp)
```
```   904 	apply (rule exI[where x="0"], simp)
```
```   905 	done}
```
```   906     moreover
```
```   907     {assume "c=1" with H(3) power_one have "a*b = 1" by simp
```
```   908 	hence "a = 1 \<and> b = 1" by simp
```
```   909 	hence ?ths
```
```   910       apply -
```
```   911       apply (rule exI[where x=1])
```
```   912       apply (rule exI[where x=1])
```
```   913       using power_one[of  n]
```
```   914       by simp}
```
```   915   moreover
```
```   916   {assume c: "c\<noteq>1" "c \<noteq> 0"
```
```   917     from prime_factor[OF c(1)] obtain p where p: "prime p" "p dvd c" by blast
```
```   918     from prime_divprod_pow[OF p(1) H(2), unfolded H(3), OF divides_exp[OF p(2), of n]]
```
```   919     have pnab: "p ^ n dvd a \<or> p^n dvd b" .
```
```   920     from p(2) obtain l where l: "c = p*l" unfolding dvd_def by blast
```
```   921     have pn0: "p^n \<noteq> 0" using n prime_ge_2 [OF p(1)] by (simp add: neq0_conv)
```
```   922     {assume pa: "p^n dvd a"
```
```   923       then obtain k where k: "a = p^n * k" unfolding dvd_def by blast
```
```   924       from l have "l dvd c" by auto
```
```   925       with dvd_imp_le[of l c] c have "l \<le> c" by auto
```
```   926       moreover {assume "l = c" with l c  have "p = 1" by simp with p have False by simp}
```
```   927       ultimately have lc: "l < c" by arith
```
```   928       from coprime_lmul2 [OF H(2)[unfolded k coprime_commute[of "p^n*k" b]]]
```
```   929       have kb: "coprime k b" by (simp add: coprime_commute)
```
```   930       from H(3) l k pn0 have kbln: "k * b = l ^ n"
```
```   931 	by (auto simp add: power_mult_distrib)
```
```   932       from H(1)[rule_format, OF lc kb kbln]
```
```   933       obtain r s where rs: "k = r ^n" "b = s^n" by blast
```
```   934       from k rs(1) have "a = (p*r)^n" by (simp add: power_mult_distrib)
```
```   935       with rs(2) have ?ths by blast }
```
```   936     moreover
```
```   937     {assume pb: "p^n dvd b"
```
```   938       then obtain k where k: "b = p^n * k" unfolding dvd_def by blast
```
```   939       from l have "l dvd c" by auto
```
```   940       with dvd_imp_le[of l c] c have "l \<le> c" by auto
```
```   941       moreover {assume "l = c" with l c  have "p = 1" by simp with p have False by simp}
```
```   942       ultimately have lc: "l < c" by arith
```
```   943       from coprime_lmul2 [OF H(2)[unfolded k coprime_commute[of "p^n*k" a]]]
```
```   944       have kb: "coprime k a" by (simp add: coprime_commute)
```
```   945       from H(3) l k pn0 n have kbln: "k * a = l ^ n"
```
```   946 	by (simp add: power_mult_distrib mult_commute)
```
```   947       from H(1)[rule_format, OF lc kb kbln]
```
```   948       obtain r s where rs: "k = r ^n" "a = s^n" by blast
```
```   949       from k rs(1) have "b = (p*r)^n" by (simp add: power_mult_distrib)
```
```   950       with rs(2) have ?ths by blast }
```
```   951     ultimately have ?ths using pnab by blast}
```
```   952   ultimately have ?ths by blast}
```
```   953 ultimately show ?ths by blast
```
```   954 qed
```
```   955
```
```   956 text {* More useful lemmas. *}
```
```   957 lemma prime_product:
```
```   958   assumes "prime (p * q)"
```
```   959   shows "p = 1 \<or> q = 1"
```
```   960 proof -
```
```   961   from assms have
```
```   962     "1 < p * q" and P: "\<And>m. m dvd p * q \<Longrightarrow> m = 1 \<or> m = p * q"
```
```   963     unfolding prime_def by auto
```
```   964   from `1 < p * q` have "p \<noteq> 0" by (cases p) auto
```
```   965   then have Q: "p = p * q \<longleftrightarrow> q = 1" by auto
```
```   966   have "p dvd p * q" by simp
```
```   967   then have "p = 1 \<or> p = p * q" by (rule P)
```
```   968   then show ?thesis by (simp add: Q)
```
```   969 qed
```
```   970
```
```   971 lemma prime_exp: "prime (p^n) \<longleftrightarrow> prime p \<and> n = 1"
```
```   972 proof(induct n)
```
```   973   case 0 thus ?case by simp
```
```   974 next
```
```   975   case (Suc n)
```
```   976   {assume "p = 0" hence ?case by simp}
```
```   977   moreover
```
```   978   {assume "p=1" hence ?case by simp}
```
```   979   moreover
```
```   980   {assume p: "p \<noteq> 0" "p\<noteq>1"
```
```   981     {assume pp: "prime (p^Suc n)"
```
```   982       hence "p = 1 \<or> p^n = 1" using prime_product[of p "p^n"] by simp
```
```   983       with p have n: "n = 0"
```
```   984 	by (simp only: exp_eq_1 ) simp
```
```   985       with pp have "prime p \<and> Suc n = 1" by simp}
```
```   986     moreover
```
```   987     {assume n: "prime p \<and> Suc n = 1" hence "prime (p^Suc n)" by simp}
```
```   988     ultimately have ?case by blast}
```
```   989   ultimately show ?case by blast
```
```   990 qed
```
```   991
```
```   992 lemma prime_power_mult:
```
```   993   assumes p: "prime p" and xy: "x * y = p ^ k"
```
```   994   shows "\<exists>i j. x = p ^i \<and> y = p^ j"
```
```   995   using xy
```
```   996 proof(induct k arbitrary: x y)
```
```   997   case 0 thus ?case apply simp by (rule exI[where x="0"], simp)
```
```   998 next
```
```   999   case (Suc k x y)
```
```  1000   from Suc.prems have pxy: "p dvd x*y" by auto
```
```  1001   from prime_divprod[OF p pxy] have pxyc: "p dvd x \<or> p dvd y" .
```
```  1002   from p have p0: "p \<noteq> 0" by - (rule ccontr, simp)
```
```  1003   {assume px: "p dvd x"
```
```  1004     then obtain d where d: "x = p*d" unfolding dvd_def by blast
```
```  1005     from Suc.prems d  have "p*d*y = p^Suc k" by simp
```
```  1006     hence th: "d*y = p^k" using p0 by simp
```
```  1007     from Suc.hyps[OF th] obtain i j where ij: "d = p^i" "y = p^j" by blast
```
```  1008     with d have "x = p^Suc i" by simp
```
```  1009     with ij(2) have ?case by blast}
```
```  1010   moreover
```
```  1011   {assume px: "p dvd y"
```
```  1012     then obtain d where d: "y = p*d" unfolding dvd_def by blast
```
```  1013     from Suc.prems d  have "p*d*x = p^Suc k" by (simp add: mult_commute)
```
```  1014     hence th: "d*x = p^k" using p0 by simp
```
```  1015     from Suc.hyps[OF th] obtain i j where ij: "d = p^i" "x = p^j" by blast
```
```  1016     with d have "y = p^Suc i" by simp
```
```  1017     with ij(2) have ?case by blast}
```
```  1018   ultimately show ?case  using pxyc by blast
```
```  1019 qed
```
```  1020
```
```  1021 lemma prime_power_exp: assumes p: "prime p" and n:"n \<noteq> 0"
```
```  1022   and xn: "x^n = p^k" shows "\<exists>i. x = p^i"
```
```  1023   using n xn
```
```  1024 proof(induct n arbitrary: k)
```
```  1025   case 0 thus ?case by simp
```
```  1026 next
```
```  1027   case (Suc n k) hence th: "x*x^n = p^k" by simp
```
```  1028   {assume "n = 0" with prems have ?case apply simp
```
```  1029       by (rule exI[where x="k"],simp)}
```
```  1030   moreover
```
```  1031   {assume n: "n \<noteq> 0"
```
```  1032     from prime_power_mult[OF p th]
```
```  1033     obtain i j where ij: "x = p^i" "x^n = p^j"by blast
```
```  1034     from Suc.hyps[OF n ij(2)] have ?case .}
```
```  1035   ultimately show ?case by blast
```
```  1036 qed
```
```  1037
```
```  1038 lemma divides_primepow: assumes p: "prime p"
```
```  1039   shows "d dvd p^k \<longleftrightarrow> (\<exists> i. i \<le> k \<and> d = p ^i)"
```
```  1040 proof
```
```  1041   assume H: "d dvd p^k" then obtain e where e: "d*e = p^k"
```
```  1042     unfolding dvd_def  apply (auto simp add: mult_commute) by blast
```
```  1043   from prime_power_mult[OF p e] obtain i j where ij: "d = p^i" "e=p^j" by blast
```
```  1044   from prime_ge_2[OF p] have p1: "p > 1" by arith
```
```  1045   from e ij have "p^(i + j) = p^k" by (simp add: power_add)
```
```  1046   hence "i + j = k" using power_inject_exp[of p "i+j" k, OF p1] by simp
```
```  1047   hence "i \<le> k" by arith
```
```  1048   with ij(1) show "\<exists>i\<le>k. d = p ^ i" by blast
```
```  1049 next
```
```  1050   {fix i assume H: "i \<le> k" "d = p^i"
```
```  1051     hence "\<exists>j. k = i + j" by arith
```
```  1052     then obtain j where j: "k = i + j" by blast
```
```  1053     hence "p^k = p^j*d" using H(2) by (simp add: power_add)
```
```  1054     hence "d dvd p^k" unfolding dvd_def by auto}
```
```  1055   thus "\<exists>i\<le>k. d = p ^ i \<Longrightarrow> d dvd p ^ k" by blast
```
```  1056 qed
```
```  1057
```
```  1058 lemma coprime_divisors: "d dvd a \<Longrightarrow> e dvd b \<Longrightarrow> coprime a b \<Longrightarrow> coprime d e"
```
```  1059   by (auto simp add: dvd_def coprime)
```
```  1060
```
```  1061 declare power_Suc0[simp del]
```
```  1062 declare even_dvd[simp del]
```
```  1063
```
```  1064 end
```