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author | wenzelm |

Sun, 11 Sep 2016 00:14:37 +0200 | |

changeset 63833 | 4aaeb9427c96 |

parent 63832 | a400b127853c |

child 63834 | 6a757f36997e |

misc tuning and modernization;

src/HOL/Old_Number_Theory/Pocklington.thy | file | annotate | diff | comparison | revisions | |

src/HOL/Old_Number_Theory/Primes.thy | file | annotate | diff | comparison | revisions |

--- a/src/HOL/Old_Number_Theory/Pocklington.thy Sun Sep 11 00:13:25 2016 +0200 +++ b/src/HOL/Old_Number_Theory/Pocklington.thy Sun Sep 11 00:14:37 2016 +0200 @@ -492,8 +492,8 @@ {fix m assume m: "0 < m" "m < n" "\<not> coprime m n" hence mS': "m \<notin> ?S'" by auto have "insert m ?S' \<le> ?S" using m by auto - from m have "card (insert m ?S') \<le> card ?S" - by - (rule card_mono[of ?S "insert m ?S'"], auto) + have "card (insert m ?S') \<le> card ?S" + by (rule card_mono[of ?S "insert m ?S'"]) (use m in auto) hence False unfolding card_insert_disjoint[of "?S'" m, OF fS' mS'] ceq by simp } @@ -767,7 +767,7 @@ hence "(n - 1) mod m = 0" by auto then have mn: "m dvd n - 1" by presburger then obtain r where r: "n - 1 = m*r" unfolding dvd_def by blast - from n01 r m(2) have r01: "r\<noteq>0" "r\<noteq>1" by - (rule ccontr, simp)+ + from n01 r m(2) have r01: "r\<noteq>0" "r\<noteq>1" by auto from prime_factor[OF r01(2)] obtain p where p: "prime p" "p dvd r" by blast hence th: "prime p \<and> p dvd n - 1" unfolding r by (auto intro: dvd_mult) have "(a ^ ((n - 1) div p)) mod n = (a^(m*r div p)) mod n" using r @@ -800,8 +800,8 @@ moreover {assume "n\<noteq>0 \<and> n\<noteq>1" hence n2:"n \<ge> 2" by arith from na have na': "coprime a n" by (simp add: coprime_commute) - from phi_lowerbound_1[OF n2] fermat_little[OF na'] - have ex: "\<exists>m>0. ?P m" by - (rule exI[where x="\<phi> n"], auto) } + have ex: "\<exists>m>0. ?P m" + by (rule exI[where x="\<phi> n"]) (use phi_lowerbound_1[OF n2] fermat_little[OF na'] in auto) } ultimately have ex: "\<exists>m>0. ?P m" by blast from nat_exists_least_iff'[of ?P] ex na show ?thesis unfolding o[symmetric] by auto @@ -992,7 +992,7 @@ from prime_factor[OF d(3)] obtain p where p: "prime p" "p dvd d" by blast from n have np: "n > 0" by arith - from d(1) n have "d \<noteq> 0" by - (rule ccontr, auto) + have "d \<noteq> 0" by (rule ccontr) (use d(1) n in auto) hence dp: "d > 0" by arith from mult_mono[OF dvd_imp_le[OF p(2) dp] dvd_imp_le[OF p(2) dp]] d(2) have "p\<^sup>2 \<le> n" unfolding power2_eq_square by arith @@ -1029,7 +1029,7 @@ from d dvd_mult[OF P(2), of "ord p (a^r)"] have Pq: "P dvd q" by simp from aq P(1) Pq have caP:"coprime (a^ ((n - 1) div P) - 1) n" by blast from Pq obtain s where s: "q = P*s" unfolding dvd_def by blast - have P0: "P \<noteq> 0" using P(1) prime_0 by - (rule ccontr, simp) + have P0: "P \<noteq> 0" by (rule ccontr) (use P(1) prime_0 in simp) from P(2) obtain t where t: "d = P*t" unfolding dvd_def by blast from d s t P0 have s': "ord p (a^r) * t = s" by algebra have "ord p (a^r) * t*r = r * ord p (a^r) * t" by algebra @@ -1052,8 +1052,8 @@ with divides_rexp[OF d(2)[unfolded dp], of "n - 2"] have th0: "p dvd a ^ (n - 1)" by simp from n have n0: "n \<noteq> 0" by simp - from d(2) an n12[symmetric] have a0: "a \<noteq> 0" - by - (rule ccontr, simp add: modeq_def) + have a0: "a \<noteq> 0" + by (rule ccontr) (use d(2) an n12[symmetric] in \<open>simp add: modeq_def\<close>) have th1: "a^ (n - 1) \<noteq> 0" using n d(2) dp a0 by auto from coprime_minus1[OF th1, unfolded coprime] dvd_trans[OF pn cong_1_divides[OF an]] th0 d(3) dp @@ -1064,7 +1064,7 @@ from fermat_little[OF arp, simplified ord_divides] o phip have "q dvd (p - 1)" by simp then obtain d where d:"p - 1 = q * d" unfolding dvd_def by blast - from prime_0 pp have p0:"p \<noteq> 0" by - (rule ccontr, auto) + have p0: "p \<noteq> 0" by (rule ccontr) (use prime_0 pp in auto) from p0 d have "p + q * 0 = 1 + q * d" by simp with nat_mod[of p 1 q, symmetric] show ?thesis by blast

--- a/src/HOL/Old_Number_Theory/Primes.thy Sun Sep 11 00:13:25 2016 +0200 +++ b/src/HOL/Old_Number_Theory/Primes.thy Sun Sep 11 00:14:37 2016 +0200 @@ -406,37 +406,44 @@ lemma prime_Suc0[simp]: "~ prime (Suc 0)" by (simp add: prime_def) lemma prime_ge_2: "prime p ==> p \<ge> 2" by (simp add: prime_def) -lemma prime_factor: assumes n: "n \<noteq> 1" shows "\<exists> p. prime p \<and> p dvd n" -using n -proof(induct n rule: nat_less_induct) + +lemma prime_factor: "n \<noteq> 1 \<Longrightarrow> \<exists>p. prime p \<and> p dvd n" +proof (induct n rule: nat_less_induct) fix n assume H: "\<forall>m<n. m \<noteq> 1 \<longrightarrow> (\<exists>p. prime p \<and> p dvd m)" "n \<noteq> 1" - let ?ths = "\<exists>p. prime p \<and> p dvd n" - {assume "n=0" hence ?ths using two_is_prime by auto} - moreover - {assume nz: "n\<noteq>0" - {assume "prime n" hence ?ths by - (rule exI[where x="n"], simp)} - moreover - {assume n: "\<not> prime n" - with nz H(2) - obtain k where k:"k dvd n" "k \<noteq> 1" "k \<noteq> n" by (auto simp add: prime_def) + show "\<exists>p. prime p \<and> p dvd n" + proof (cases "n = 0") + case True + with two_is_prime show ?thesis by auto + next + case nz: False + show ?thesis + proof (cases "prime n") + case True + then have "prime n \<and> n dvd n" by simp + then show ?thesis .. + next + case n: False + with nz H(2) obtain k where k: "k dvd n" "k \<noteq> 1" "k \<noteq> n" + by (auto simp: prime_def) from dvd_imp_le[OF k(1)] nz k(3) have kn: "k < n" by simp from H(1)[rule_format, OF kn k(2)] obtain p where p: "prime p" "p dvd k" by blast - from dvd_trans[OF p(2) k(1)] p(1) have ?ths by blast} - ultimately have ?ths by blast} - ultimately show ?ths by blast + from dvd_trans[OF p(2) k(1)] p(1) show ?thesis by blast + qed + qed qed -lemma prime_factor_lt: assumes p: "prime p" and n: "n \<noteq> 0" and npm:"n = p * m" +lemma prime_factor_lt: + assumes p: "prime p" and n: "n \<noteq> 0" and npm:"n = p * m" shows "m < n" -proof- - {assume "m=0" with n have ?thesis by simp} - moreover - {assume m: "m \<noteq> 0" - from npm have mn: "m dvd n" unfolding dvd_def by auto - from npm m have "n \<noteq> m" using p by auto - with dvd_imp_le[OF mn] n have ?thesis by simp} - ultimately show ?thesis by blast +proof (cases "m = 0") + case True + with n show ?thesis by simp +next + case m: False + from npm have mn: "m dvd n" unfolding dvd_def by auto + from npm m have "n \<noteq> m" using p by auto + with dvd_imp_le[OF mn] n show ?thesis by simp qed lemma euclid_bound: "\<exists>p. prime p \<and> n < p \<and> p <= Suc (fact n)" @@ -491,7 +498,7 @@ lemma coprime_bezout_strong: assumes ab: "coprime a b" and b: "b \<noteq> 1" shows "\<exists>x y. a * x = b * y + 1" proof- - from ab b have az: "a \<noteq> 0" by - (rule ccontr, auto) + have az: "a \<noteq> 0" by (rule ccontr) (use ab b in auto) from bezout_gcd_strong[OF az, of b] ab[unfolded coprime_def] show ?thesis by auto qed @@ -577,15 +584,15 @@ lemma distinct_prime_coprime: "prime p \<Longrightarrow> prime q \<Longrightarrow> p \<noteq> q \<Longrightarrow> coprime p q" unfolding prime_def coprime_prime_eq by blast -lemma prime_coprime_lt: assumes p: "prime p" and x: "0 < x" and xp: "x < p" +lemma prime_coprime_lt: + assumes p: "prime p" and x: "0 < x" and xp: "x < p" shows "coprime x p" -proof- - {assume c: "\<not> coprime x p" - then obtain g where g: "g \<noteq> 1" "g dvd x" "g dvd p" unfolding coprime_def by blast +proof (rule ccontr) + assume c: "\<not> ?thesis" + then obtain g where g: "g \<noteq> 1" "g dvd x" "g dvd p" unfolding coprime_def by blast from dvd_imp_le[OF g(2)] x xp have gp: "g < p" by arith - from g(2) x have "g \<noteq> 0" by - (rule ccontr, simp) - with g gp p[unfolded prime_def] have False by blast} -thus ?thesis by blast + have "g \<noteq> 0" by (rule ccontr) (use g(2) x in simp) + with g gp p[unfolded prime_def] show False by blast qed lemma prime_odd: "prime p \<Longrightarrow> p = 2 \<or> odd p" unfolding prime_def by auto @@ -755,28 +762,30 @@ shows "\<exists>i j. x = p ^i \<and> y = p^ j" using xy proof(induct k arbitrary: x y) - case 0 thus ?case apply simp by (rule exI[where x="0"], simp) + case 0 + thus ?case apply simp by (rule exI[where x="0"], simp) next case (Suc k x y) + have p0: "p \<noteq> 0" by (rule ccontr) (use p in simp) from Suc.prems have pxy: "p dvd x*y" by auto - from prime_divprod[OF p pxy] have pxyc: "p dvd x \<or> p dvd y" . - from p have p0: "p \<noteq> 0" by - (rule ccontr, simp) - {assume px: "p dvd x" + from prime_divprod[OF p pxy] show ?case + proof + assume px: "p dvd x" then obtain d where d: "x = p*d" unfolding dvd_def by blast from Suc.prems d have "p*d*y = p^Suc k" by simp hence th: "d*y = p^k" using p0 by simp from Suc.hyps[OF th] obtain i j where ij: "d = p^i" "y = p^j" by blast with d have "x = p^Suc i" by simp - with ij(2) have ?case by blast} - moreover - {assume px: "p dvd y" + with ij(2) show ?thesis by blast + next + assume px: "p dvd y" then obtain d where d: "y = p*d" unfolding dvd_def by blast from Suc.prems d have "p*d*x = p^Suc k" by (simp add: mult.commute) hence th: "d*x = p^k" using p0 by simp from Suc.hyps[OF th] obtain i j where ij: "d = p^i" "x = p^j" by blast with d have "y = p^Suc i" by simp - with ij(2) have ?case by blast} - ultimately show ?case using pxyc by blast + with ij(2) show ?thesis by blast + qed qed lemma prime_power_exp: assumes p: "prime p" and n:"n \<noteq> 0"