# File Arith2.ML

    (*  Title:      HOL/Hoare/Arith2.ML
        ID:         $Id: Arith2.ML,v 1.23 2005/09/06 14:59:53 wenzelm Exp $
        Author:     Norbert Galm
        Copyright   1995 TUM

    More arithmetic lemmas.
    *)

    (*** cd ***)

    Goalw [cd_def] "0<n ==> cd n n n";
    by (Asm_simp_tac 1);
    qed "cd_nnn";

    Goalw [cd_def] "[| cd x m n; 0<m; 0<n |] ==> x<=m & x<=n";
    by (blast_tac (claset() addIs [dvd_imp_le]) 1);
    qed "cd_le";

    Goalw [cd_def] "cd x m n = cd x n m";
    by (Fast_tac 1);
    qed "cd_swap";

    Goalw [cd_def] "n<=m ==> cd x m n = cd x (m-n) n";
    by (blast_tac (claset() addIs [dvd_diff] addDs [dvd_diffD]) 1);
    qed "cd_diff_l";

    Goalw [cd_def] "m<=n ==> cd x m n = cd x m (n-m)";
    by (blast_tac (claset() addIs [dvd_diff] addDs [dvd_diffD]) 1);
    qed "cd_diff_r";

    (*** gcd ***)

    Goalw [gcd_def] "0<n ==> n = gcd n n";
    by (ftac cd_nnn 1);
    by (rtac (some_equality RS sym) 1);
    by (blast_tac (claset() addDs [cd_le]) 1);
    by (blast_tac (claset() addIs [le_anti_sym] addDs [cd_le]) 1);
    qed "gcd_nnn";

    val prems = goalw (the_context ()) [gcd_def] "gcd m n = gcd n m";
    by (simp_tac (simpset() addsimps [cd_swap]) 1);
    qed "gcd_swap";

    Goalw [gcd_def] "n<=m ==> gcd m n = gcd (m-n) n";
    by (subgoal_tac "n<=m ==> !x. cd x m n = cd x (m-n) n" 1);
    by (Asm_simp_tac 1);
    by (rtac allI 1);
    by (etac cd_diff_l 1);
    qed "gcd_diff_l";

    Goalw [gcd_def] "m<=n ==> gcd m n = gcd m (n-m)";
    by (subgoal_tac "m<=n ==> !x. cd x m n = cd x m (n-m)" 1);
    by (Asm_simp_tac 1);
    by (rtac allI 1);
    by (etac cd_diff_r 1);
    qed "gcd_diff_r";

    (*** pow ***)

    Goal "m mod 2 = 0 ==> ((n::nat)*n)^(m div 2) = n^m";
    by (asm_simp_tac (simpset() addsimps [power2_eq_square RS sym,
                       power_mult RS sym, mult_div_cancel]) 1);
    qed "sq_pow_div2";
    Addsimps [sq_pow_div2];