src/HOL/ex/Primes.ML

-(*  Title:      HOL/ex/Primes.ML
-    ID:         $Id$
-    Author:     Christophe Tabacznyj and Lawrence C Paulson
-    Copyright   1996  University of Cambridge
-
-The "divides" relation, the greatest common divisor and Euclid's algorithm
-
-See H. Davenport, "The Higher Arithmetic".  6th edition.  (CUP, 1992)
-*)
-
-eta_contract:=false;
-
-(************************************************)
-(** Greatest Common Divisor                    **)
-(************************************************)
-
-(*** Euclid's Algorithm ***)
-
-
-val [gcd_eq] = gcd.simps;
-
-
-val prems = goal thy
-     "[| !!m. P m 0;     \
-\        !!m n. [| 0<n;  P n (m mod n) |] ==> P m n  \
-\     |] ==> P (m::nat) (n::nat)";
-by (induct_thm_tac gcd.induct "m n" 1);
-by (case_tac "n=0" 1);
-by (asm_simp_tac (simpset() addsimps prems) 1);
-by Safe_tac;
-by (asm_simp_tac (simpset() addsimps prems) 1);
-qed "gcd_induct";
-
-
-Goal "gcd(m,0) = m";
-by (Simp_tac 1);
-qed "gcd_0";
-Addsimps [gcd_0];
-
-Goal "0<n ==> gcd(m,n) = gcd (n, m mod n)";
-by (Asm_simp_tac 1);
-qed "gcd_non_0";
-
-Delsimps gcd.simps;
-
-Goal "gcd(m,1) = 1";
-by (simp_tac (simpset() addsimps [gcd_non_0]) 1);
-qed "gcd_1";
-Addsimps [gcd_1];
-
-(*gcd(m,n) divides m and n.  The conjunctions don't seem provable separately*)
-Goal "(gcd(m,n) dvd m) & (gcd(m,n) dvd n)";
-by (induct_thm_tac gcd_induct "m n" 1);
-by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [gcd_non_0])));
-by (blast_tac (claset() addDs [dvd_mod_imp_dvd]) 1);
-qed "gcd_dvd_both";
-
-bind_thm ("gcd_dvd1", gcd_dvd_both RS conjunct1);
-bind_thm ("gcd_dvd2", gcd_dvd_both RS conjunct2);
-
-
-(*Maximality: for all m,n,f naturals, 
-                if f divides m and f divides n then f divides gcd(m,n)*)
-Goal "(f dvd m) --> (f dvd n) --> f dvd gcd(m,n)";
-by (induct_thm_tac gcd_induct "m n" 1);
-by (ALLGOALS (asm_full_simp_tac (simpset() addsimps[gcd_non_0, dvd_mod])));
-qed_spec_mp "gcd_greatest";
-
-(*Function gcd yields the Greatest Common Divisor*)
-Goalw [is_gcd_def] "is_gcd (gcd(m,n)) m n";
-by (asm_simp_tac (simpset() addsimps [gcd_greatest, gcd_dvd_both]) 1);
-qed "is_gcd";
-
-(*uniqueness of GCDs*)
-Goalw [is_gcd_def] "[| is_gcd m a b; is_gcd n a b |] ==> m=n";
-by (blast_tac (claset() addIs [dvd_anti_sym]) 1);
-qed "is_gcd_unique";
-
-(*USED??*)
-Goalw [is_gcd_def]
-    "[| is_gcd m a b; k dvd a; k dvd b |] ==> k dvd m";
-by (Blast_tac 1);
-qed "is_gcd_dvd";
-
-(** Commutativity **)
-
-Goalw [is_gcd_def] "is_gcd k m n = is_gcd k n m";
-by (Blast_tac 1);
-qed "is_gcd_commute";
-
-Goal "gcd(m,n) = gcd(n,m)";
-by (rtac is_gcd_unique 1);
-by (rtac is_gcd 2);
-by (asm_simp_tac (simpset() addsimps [is_gcd, is_gcd_commute]) 1);
-qed "gcd_commute";
-
-Goal "gcd(gcd(k,m),n) = gcd(k,gcd(m,n))";
-by (rtac is_gcd_unique 1);
-by (rtac is_gcd 2);
-by (rewtac is_gcd_def);
-by (blast_tac (claset() addSIs [gcd_dvd1, gcd_dvd2]
-   	                addIs  [gcd_greatest, dvd_trans]) 1);
-qed "gcd_assoc";
-
-Goal "gcd(0,m) = m";
-by (stac gcd_commute 1);
-by (rtac gcd_0 1);
-qed "gcd_0_left";
-
-Goal "gcd(1,m) = 1";
-by (stac gcd_commute 1);
-by (rtac gcd_1 1);
-qed "gcd_1_left";
-Addsimps [gcd_0_left, gcd_1_left];
-
-
-(** Multiplication laws **)
-
-(*Davenport, page 27*)
-Goal "k * gcd(m,n) = gcd(k*m, k*n)";
-by (induct_thm_tac gcd_induct "m n" 1);
-by (Asm_full_simp_tac 1);
-by (case_tac "k=0" 1);
- by (Asm_full_simp_tac 1);
-by (asm_full_simp_tac
-    (simpset() addsimps [mod_geq, gcd_non_0, mod_mult_distrib2]) 1);
-qed "gcd_mult_distrib2";
-
-Goal "gcd(m,m) = m";
-by (cut_inst_tac [("k","m"),("m","1"),("n","1")] gcd_mult_distrib2 1);
-by (Asm_full_simp_tac 1);
-qed "gcd_self";
-Addsimps [gcd_self];
-
-Goal "gcd(k, k*n) = k";
-by (cut_inst_tac [("k","k"),("m","1"),("n","n")] gcd_mult_distrib2 1);
-by (Asm_full_simp_tac 1);
-qed "gcd_mult";
-Addsimps [gcd_mult];
-
-Goal "[| gcd(k,n)=1; k dvd (m*n) |] ==> k dvd m";
-by (subgoal_tac "m = gcd(m*k, m*n)" 1);
-by (etac ssubst 1 THEN rtac gcd_greatest 1);
-by (ALLGOALS (asm_simp_tac (simpset() addsimps [gcd_mult_distrib2 RS sym])));
-qed "relprime_dvd_mult";
-
-Goalw [prime_def] "[| p: prime;  ~ p dvd n |] ==> gcd (p, n) = 1";
-by (cut_inst_tac [("m","p"),("n","n")] gcd_dvd_both 1);
-by Auto_tac;
-qed "prime_imp_relprime";
-
-(*This theorem leads immediately to a proof of the uniqueness of factorization.
-  If p divides a product of primes then it is one of those primes.*)
-Goal "[| p: prime; p dvd (m*n) |] ==> p dvd m | p dvd n";
-by (blast_tac (claset() addIs [relprime_dvd_mult, prime_imp_relprime]) 1);
-qed "prime_dvd_mult";
-
-
-(** Addition laws **)
-
-Goal "gcd(m, m+n) = gcd(m,n)";
-by (res_inst_tac [("n1", "m+n")] (gcd_commute RS ssubst) 1);
-by (rtac (gcd_eq RS trans) 1);
-by Auto_tac;
-by (asm_simp_tac (simpset() addsimps [mod_add_self1]) 1);
-by (stac gcd_commute 1);
-by (stac gcd_non_0 1);
-by Safe_tac;
-qed "gcd_add";
-
-Goal "gcd(m, n+m) = gcd(m,n)";
-by (asm_simp_tac (simpset() addsimps [add_commute, gcd_add]) 1);
-qed "gcd_add2";
-
-Goal "gcd(m, k*m+n) = gcd(m,n)";
-by (induct_tac "k" 1);
-by (asm_simp_tac (simpset() addsimps [gcd_add, add_assoc]) 2); 
-by (Simp_tac 1);
-qed "gcd_add_mult";
-
-
-(** More multiplication laws **)
-
-Goal "gcd(m,n) dvd gcd(k*m, n)";
-by (blast_tac (claset() addIs [gcd_greatest, dvd_trans, 
-                               gcd_dvd1, gcd_dvd2]) 1);
-qed "gcd_dvd_gcd_mult";
-
-Goal "gcd(k,n) = 1 ==> gcd(k*m, n) = gcd(m,n)";
-by (rtac dvd_anti_sym 1);
-by (rtac gcd_dvd_gcd_mult 2);
-by (rtac ([relprime_dvd_mult, gcd_dvd2] MRS gcd_greatest) 1);
-by (stac mult_commute 2);
-by (rtac gcd_dvd1 2);
-by (stac gcd_commute 1);
-by (asm_simp_tac (simpset() addsimps [gcd_assoc RS sym]) 1);
-qed "gcd_mult_cancel";