src/HOL/Library/Primes.thy

 
 lemma prime_dvd_power_two: "prime p ==> p dvd m\<twosuperior> ==> p dvd m"
   by (rule prime_dvd_square) (simp_all add: power2_eq_square)
 
 
-lemma exp_eq_1:"(x::nat)^n = 1 \<longleftrightarrow> x = 1 \<or> n = 0" by (induct n, auto)
+lemma exp_eq_1:"(x::nat)^n = 1 \<longleftrightarrow> x = 1 \<or> n = 0"
+by (induct n, auto)
+
 lemma exp_mono_lt: "(x::nat) ^ (Suc n) < y ^ (Suc n) \<longleftrightarrow> x < y"
-  using power_less_imp_less_base[of x "Suc n" y] power_strict_mono[of x y "Suc n"]
-    by auto
+by(metis linorder_not_less not_less0 power_le_imp_le_base power_less_imp_less_base)
+
 lemma exp_mono_le: "(x::nat) ^ (Suc n) \<le> y ^ (Suc n) \<longleftrightarrow> x \<le> y"
-  by (simp only: linorder_not_less[symmetric] exp_mono_lt)
+by (simp only: linorder_not_less[symmetric] exp_mono_lt)
 
 lemma exp_mono_eq: "(x::nat) ^ Suc n = y ^ Suc n \<longleftrightarrow> x = y"
 using power_inject_base[of x n y] by auto
 
 

   let ?g = "gcd (a^n) (b^n)"
   let ?gn = "gcd a b^n"
   {fix e assume H: "e dvd a^n" "e dvd b^n"
     from bezout_gcd_pow[of a n b] obtain x y 
       where xy: "a ^ n * x - b ^ n * y = ?gn \<or> b ^ n * x - a ^ n * y = ?gn" by blast
-    from dvd_diff [OF dvd_mult2[OF H(1), of x] dvd_mult2[OF H(2), of y]]
-      dvd_diff [OF dvd_mult2[OF H(2), of x] dvd_mult2[OF H(1), of y]] xy
+    from nat_dvd_diff [OF dvd_mult2[OF H(1), of x] dvd_mult2[OF H(2), of y]]
+      nat_dvd_diff [OF dvd_mult2[OF H(2), of x] dvd_mult2[OF H(1), of y]] xy
     have "e dvd ?gn" by (cases "a ^ n * x - b ^ n * y = gcd a b ^ n", simp_all)}
   hence th:  "\<forall>e. e dvd a^n \<and> e dvd b^n \<longrightarrow> e dvd ?gn" by blast
   from divides_exp[OF gcd_dvd1[of a b], of n] divides_exp[OF gcd_dvd2[of a b], of n] th
     gcd_unique have "?gn = ?g" by blast thus ?thesis by simp 
 qed