theory Pr

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

begin

fun 

  add :: "nat \<Rightarrow> nat \<Rightarrow> nat" 

where

  "add 0 n = n" |

  "add (Suc m) n = Suc (add m n)"

fun 

  doub :: "nat \<Rightarrow> nat" 

where

  "doub 0 = 0" |

  "doub n = n + n"

lemma add_lem:

  "add m n = m + n"

apply(induct m arbitrary: )

apply(auto)

done

fun 

  count :: "'a \<Rightarrow> 'a list \<Rightarrow> nat"

where

  "count n Nil = 0" |

  "count n (x # xs) = (if n = x then (add 1 (count n xs)) else count n xs)"

value "count 3 [1,2,3,3,4,3,5]"

fun 

  count2 :: "nat \<Rightarrow> nat list \<Rightarrow> nat" 

where

"count2 n Nil = 0" |

"count2 n (Cons x xs) = (if n = x then (add 1 (count2 n xs)) else count2 n xs)"

value "count2 (2::nat) [2,2,3::nat]"

lemma

  "count2 x xs \<le> length xs"

apply(induct xs)

apply(simp)

apply(simp)

apply(auto)

done

fun 

  sum :: "nat \<Rightarrow> nat"

where

  "sum 0 = 0"

| "sum (Suc n) = (Suc n) + sum n"

lemma

  "sum n = (n * (Suc n)) div 2"

apply(induct n)

apply(auto)

done

lemma

  "doub n = add n n"

apply(induct n)

apply(simp)

apply(simp add: add_lem)

done

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