Theory Arith2

theory Arith2
imports Main
(*  Title:      HOL/Hoare/Arith2.thy
Author: Norbert Galm
Copyright 1995 TUM

More arithmetic. Much of this duplicates ex/Primes.
*)


theory Arith2
imports Main
begin

definition "cd" :: "[nat, nat, nat] => bool"
where "cd x m n <-> x dvd m & x dvd n"

definition gcd :: "[nat, nat] => nat"
where "gcd m n = (SOME x. cd x m n & (!y.(cd y m n) --> y<=x))"

primrec fac :: "nat => nat"
where
"fac 0 = Suc 0"
| "fac (Suc n) = Suc n * fac n"


subsubsection {* cd *}

lemma cd_nnn: "0<n ==> cd n n n"
apply (simp add: cd_def)
done

lemma cd_le: "[| cd x m n; 0<m; 0<n |] ==> x<=m & x<=n"
apply (unfold cd_def)
apply (blast intro: dvd_imp_le)
done

lemma cd_swap: "cd x m n = cd x n m"
apply (unfold cd_def)
apply blast
done

lemma cd_diff_l: "n<=m ==> cd x m n = cd x (m-n) n"
apply (unfold cd_def)
apply (fastforce dest: dvd_diffD)
done

lemma cd_diff_r: "m<=n ==> cd x m n = cd x m (n-m)"
apply (unfold cd_def)
apply (fastforce dest: dvd_diffD)
done


subsubsection {* gcd *}

lemma gcd_nnn: "0<n ==> n = gcd n n"
apply (unfold gcd_def)
apply (frule cd_nnn)
apply (rule some_equality [symmetric])
apply (blast dest: cd_le)
apply (blast intro: le_antisym dest: cd_le)
done

lemma gcd_swap: "gcd m n = gcd n m"
apply (simp add: gcd_def cd_swap)
done

lemma gcd_diff_l: "n<=m ==> gcd m n = gcd (m-n) n"
apply (unfold gcd_def)
apply (subgoal_tac "n<=m ==> !x. cd x m n = cd x (m-n) n")
apply simp
apply (rule allI)
apply (erule cd_diff_l)
done

lemma gcd_diff_r: "m<=n ==> gcd m n = gcd m (n-m)"
apply (unfold gcd_def)
apply (subgoal_tac "m<=n ==> !x. cd x m n = cd x m (n-m) ")
apply simp
apply (rule allI)
apply (erule cd_diff_r)
done


subsubsection {* pow *}

lemma sq_pow_div2 [simp]:
"m mod 2 = 0 ==> ((n::nat)*n)^(m div 2) = n^m"
apply (simp add: power2_eq_square [symmetric] power_mult [symmetric] mult_div_cancel)
done

end