theory Pr

imports Main "~~/src/HOL/Number_Theory/Primes" "~~/src/HOL/Real"

begin

lemma 

  fixes a b::nat

  shows "(a + b) ^ 2 = a ^ 2 + 2 * a * b + b ^ 2"

apply(simp add: power2_sum)

done

lemma

  fixes a b c::"real"

  assumes eq: "a * c \<le> b * c" and ineq: "b < a"

  shows "c \<le> 0"

proof -

  {

    assume  "0 < c" 

    then have "b * c < a * c" using ineq by(auto)

    then have "False" using eq by auto

  } then show "c \<le> 0" by (auto simp add: not_le[symmetric]) 

qed

lemma "n > 1 \<Longrightarrow> \<not>(prime (2 * n))"

by (metis One_nat_def Suc_leI less_Suc0 not_le numeral_eq_one_iff prime_product semiring_norm(85))

lemma 

 fixes n::"nat"

 assumes a: "n > 1"

    and  b: "\<not>(prime n)"

 shows "\<not>(prime ((2 ^ n) - 1))"   

using a b

apply(induct n)

apply(simp)

apply(simp)

end