paulson@1804: (*  Title:      HOL/ex/Primes.ML
paulson@1804:     ID:         $Id$
paulson@1804:     Author:     Christophe Tabacznyj and Lawrence C Paulson
paulson@1804:     Copyright   1996  University of Cambridge
paulson@1804: 
paulson@1804: The "divides" relation, the greatest common divisor and Euclid's algorithm
paulson@3377: 
paulson@3377: See H. Davenport, "The Higher Arithmetic".  6th edition.  (CUP, 1992)
paulson@1804: *)
paulson@1804: 
paulson@1804: eta_contract:=false;
paulson@1804: 
paulson@1804: open Primes;
paulson@1804: 
paulson@1804: (************************************************)
paulson@1804: (** Greatest Common Divisor                    **)
paulson@1804: (************************************************)
paulson@1804: 
paulson@3495: (*** Euclid's Algorithm ***)
paulson@3242: 
paulson@1804: 
paulson@3495: (** Prove the termination condition and remove it from the recursion equations
paulson@3495:     and induction rule **)
paulson@3495: 
paulson@3242: Tfl.tgoalw thy [] gcd.rules;
wenzelm@4089: by (simp_tac (simpset() addsimps [mod_less_divisor, zero_less_eq]) 1);
paulson@3242: val tc = result();
paulson@1804: 
paulson@3242: val gcd_eq = tc RS hd gcd.rules;
paulson@3270: val gcd_induct = tc RS gcd.induct;
paulson@1804: 
paulson@3242: goal thy "gcd(m,0) = m";
paulson@3242: by (rtac (gcd_eq RS trans) 1);
paulson@3242: by (Simp_tac 1);
paulson@3242: qed "gcd_0";
paulson@1804: 
paulson@3242: goal thy "!!m. 0<n ==> gcd(m,n) = gcd (n, m mod n)";
paulson@3242: by (rtac (gcd_eq RS trans) 1);
wenzelm@4089: by (asm_simp_tac (simpset() addsplits [expand_if]) 1);
paulson@3242: qed "gcd_less_0";
paulson@3242: Addsimps [gcd_0, gcd_less_0];
paulson@1804: 
paulson@3242: goal thy "gcd(m,0) dvd m";
paulson@1804: by (Simp_tac 1);
paulson@3242: qed "gcd_0_dvd_m";
paulson@1804: 
paulson@3242: goal thy "gcd(m,0) dvd 0";
paulson@1804: by (Simp_tac 1);
paulson@3242: qed "gcd_0_dvd_0";
paulson@1804: 
paulson@3242: (*gcd(m,n) divides m and n.  The conjunctions don't seem provable separately*)
paulson@3242: goal thy "(gcd(m,n) dvd m) & (gcd(m,n) dvd n)";
paulson@3301: by (res_inst_tac [("u","m"),("v","n")] gcd_induct 1);
paulson@2102: by (case_tac "n=0" 1);
paulson@3242: by (ALLGOALS 
wenzelm@4089:     (asm_simp_tac (simpset() addsimps [mod_less_divisor,zero_less_eq])));
wenzelm@4089: by (blast_tac (claset() addDs [dvd_mod_imp_dvd]) 1);
paulson@3242: qed "gcd_divides_both";
paulson@1804: 
paulson@3242: (*Maximality: for all m,n,f naturals, 
paulson@3242:                 if f divides m and f divides n then f divides gcd(m,n)*)
paulson@3377: goal thy "!!k. (f dvd m) --> (f dvd n) --> f dvd gcd(m,n)";
paulson@3301: by (res_inst_tac [("u","m"),("v","n")] gcd_induct 1);
paulson@2102: by (case_tac "n=0" 1);
paulson@3242: by (ALLGOALS 
wenzelm@4089:     (asm_simp_tac (simpset() addsimps [dvd_mod, mod_less_divisor,
paulson@3242: 				      zero_less_eq])));
paulson@3242: qed_spec_mp "gcd_greatest";
paulson@1804: 
paulson@3377: (*Function gcd yields the Greatest Common Divisor*)
paulson@3242: goalw thy [is_gcd_def] "is_gcd (gcd(m,n)) m n";
wenzelm@4089: by (asm_simp_tac (simpset() addsimps [gcd_greatest, gcd_divides_both]) 1);
paulson@3242: qed "is_gcd";
paulson@1804: 
paulson@3377: (*uniqueness of GCDs*)
paulson@3242: goalw thy [is_gcd_def] "is_gcd m a b & is_gcd n a b --> m=n";
wenzelm@4089: by (blast_tac (claset() addIs [dvd_anti_sym]) 1);
paulson@3242: qed "is_gcd_unique";
paulson@1804: 
paulson@3377: (*Davenport, page 27*)
paulson@3377: goal thy "k * gcd(m,n) = gcd(k*m, k*n)";
paulson@3377: by (res_inst_tac [("u","m"),("v","n")] gcd_induct 1);
paulson@3377: by (case_tac "k=0" 1);
paulson@3377: by (case_tac "n=0" 2);
paulson@3377: by (ALLGOALS 
wenzelm@4089:     (asm_simp_tac (simpset() addsimps [mod_less_divisor, zero_less_eq,
paulson@3377: 				      mod_geq, mod_mult_distrib2])));
paulson@3377: qed "gcd_mult_distrib2";
paulson@3377: 
paulson@3377: (*This theorem leads immediately to a proof of the uniqueness of factorization.
paulson@3377:   If p divides a product of primes then it is one of those primes.*)
paulson@3377: goalw thy [prime_def] "!!p. [| p: prime; p dvd (m*n) |] ==> p dvd m | p dvd n";
paulson@3718: by (Clarify_tac 1);
paulson@3377: by (subgoal_tac "m = gcd(m*p, m*n)" 1);
paulson@3457: by (etac ssubst 1);
paulson@3457: by (rtac gcd_greatest 1);
wenzelm@4089: by (ALLGOALS (asm_simp_tac (simpset() addsimps [gcd_mult_distrib2 RS sym])));
paulson@3377: (*Now deduce  gcd(p,n)=1  to finish the proof*)
paulson@3377: by (cut_inst_tac [("m","p"),("n","n")] gcd_divides_both 1);
wenzelm@4089: by (fast_tac (claset() addSss (simpset())) 1);
paulson@3377: qed "prime_dvd_mult";