--- a/src/HOL/Algebra/abstract/Ring2.ML Sun Nov 19 13:02:55 2006 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,350 +0,0 @@
-(*
- Abstract class ring (commutative, with 1)
- $Id$
- Author: Clemens Ballarin, started 9 December 1996
-*)
-
-(*
-val a_assoc = thm "semigroup_add.add_assoc";
-val l_zero = thm "comm_monoid_add.add_0";
-val a_comm = thm "ab_semigroup_add.add_commute";
-
-section "Rings";
-
-fun make_left_commute assoc commute s =
- [rtac (commute RS trans) 1, rtac (assoc RS trans) 1,
- rtac (commute RS arg_cong) 1];
-
-(* addition *)
-
-qed_goal "a_lcomm" Ring2.thy "!!a::'a::ring. a+(b+c) = b+(a+c)"
- (make_left_commute a_assoc a_comm);
-
-val a_ac = [a_assoc, a_comm, a_lcomm];
-
-Goal "!!a::'a::ring. a + 0 = a";
-by (rtac (a_comm RS trans) 1);
-by (rtac l_zero 1);
-qed "r_zero";
-
-Goal "!!a::'a::ring. a + (-a) = 0";
-by (rtac (a_comm RS trans) 1);
-by (rtac l_neg 1);
-qed "r_neg";
-
-Goal "!! a::'a::ring. a + b = a + c ==> b = c";
-by (rtac box_equals 1);
-by (rtac l_zero 2);
-by (rtac l_zero 2);
-by (res_inst_tac [("a1", "a")] (l_neg RS subst) 1);
-by (asm_simp_tac (simpset() addsimps [a_assoc]) 1);
-qed "a_lcancel";
-
-Goal "!! a::'a::ring. b + a = c + a ==> b = c";
-by (rtac a_lcancel 1);
-by (asm_simp_tac (simpset() addsimps a_ac) 1);
-qed "a_rcancel";
-
-Goal "!! a::'a::ring. (a + b = a + c) = (b = c)";
-by (auto_tac (claset() addSDs [a_lcancel], simpset()));
-qed "a_lcancel_eq";
-
-Goal "!! a::'a::ring. (b + a = c + a) = (b = c)";
-by (simp_tac (simpset() addsimps [a_lcancel_eq, a_comm]) 1);
-qed "a_rcancel_eq";
-
-Addsimps [a_lcancel_eq, a_rcancel_eq];
-
-Goal "!!a::'a::ring. -(-a) = a";
-by (rtac a_lcancel 1);
-by (rtac (r_neg RS trans) 1);
-by (rtac (l_neg RS sym) 1);
-qed "minus_minus";
-
-Goal "- 0 = (0::'a::ring)";
-by (rtac a_lcancel 1);
-by (rtac (r_neg RS trans) 1);
-by (rtac (l_zero RS sym) 1);
-qed "minus0";
-
-Goal "!!a::'a::ring. -(a + b) = (-a) + (-b)";
-by (res_inst_tac [("a", "a+b")] a_lcancel 1);
-by (simp_tac (simpset() addsimps ([r_neg, l_neg, l_zero]@a_ac)) 1);
-qed "minus_add";
-
-(* multiplication *)
-
-qed_goal "m_lcomm" Ring2.thy "!!a::'a::ring. a*(b*c) = b*(a*c)"
- (make_left_commute m_assoc m_comm);
-
-val m_ac = [m_assoc, m_comm, m_lcomm];
-
-Goal "!!a::'a::ring. a * 1 = a";
-by (rtac (m_comm RS trans) 1);
-by (rtac l_one 1);
-qed "r_one";
-
-(* distributive and derived *)
-
-Goal "!!a::'a::ring. a * (b + c) = a * b + a * c";
-by (rtac (m_comm RS trans) 1);
-by (rtac (l_distr RS trans) 1);
-by (simp_tac (simpset() addsimps [m_comm]) 1);
-qed "r_distr";
-
-val m_distr = m_ac @ [l_distr, r_distr];
-
-(* the following two proofs can be found in
- Jacobson, Basic Algebra I, pp. 88-89 *)
-
-Goal "!!a::'a::ring. 0 * a = 0";
-by (rtac a_lcancel 1);
-by (rtac (l_distr RS sym RS trans) 1);
-by (simp_tac (simpset() addsimps [r_zero]) 1);
-qed "l_null";
-
-Goal "!!a::'a::ring. a * 0 = 0";
-by (rtac (m_comm RS trans) 1);
-by (rtac l_null 1);
-qed "r_null";
-
-Goal "!!a::'a::ring. (-a) * b = - (a * b)";
-by (rtac a_lcancel 1);
-by (rtac (r_neg RS sym RSN (2, trans)) 1);
-by (rtac (l_distr RS sym RS trans) 1);
-by (simp_tac (simpset() addsimps [l_null, r_neg]) 1);
-qed "l_minus";
-
-Goal "!!a::'a::ring. a * (-b) = - (a * b)";
-by (rtac a_lcancel 1);
-by (rtac (r_neg RS sym RSN (2, trans)) 1);
-by (rtac (r_distr RS sym RS trans) 1);
-by (simp_tac (simpset() addsimps [r_null, r_neg]) 1);
-qed "r_minus";
-
-val m_minus = [l_minus, r_minus];
-
-Addsimps [l_zero, r_zero, l_neg, r_neg, minus_minus, minus0,
- l_one, r_one, l_null, r_null];
-
-(* further rules *)
-
-Goal "!!a::'a::ring. -a = 0 ==> a = 0";
-by (res_inst_tac [("t", "a")] (minus_minus RS subst) 1);
-by (Asm_simp_tac 1);
-qed "uminus_monom";
-
-Goal "!!a::'a::ring. a ~= 0 ==> -a ~= 0";
-by (blast_tac (claset() addIs [uminus_monom]) 1);
-qed "uminus_monom_neq";
-
-Goal "!!a::'a::ring. a * b ~= 0 ==> a ~= 0";
-by Auto_tac;
-qed "l_nullD";
-
-Goal "!!a::'a::ring. a * b ~= 0 ==> b ~= 0";
-by Auto_tac;
-qed "r_nullD";
-
-(* reflection between a = b and a -- b = 0 *)
-
-Goal "!!a::'a::ring. a = b ==> a + (-b) = 0";
-by (Asm_simp_tac 1);
-qed "eq_imp_diff_zero";
-
-Goal "!!a::'a::ring. a + (-b) = 0 ==> a = b";
-by (res_inst_tac [("a", "-b")] a_rcancel 1);
-by (Asm_simp_tac 1);
-qed "diff_zero_imp_eq";
-
-(* this could be a rewrite rule, but won't terminate
- ==> make it a simproc?
-Goal "!!a::'a::ring. (a = b) = (a -- b = 0)";
-*)
-
-*)
-
-val dvd_def = thm "dvd_def'";
-
-Goalw [dvd_def]
- "!!a::'a::ring. [| a dvd 1; b dvd 1 |] ==> a * b dvd 1";
-by (Clarify_tac 1);
-by (res_inst_tac [("x", "k * ka")] exI 1);
-by (Asm_full_simp_tac 1);
-qed "unit_mult";
-
-Goal "!!a::'a::ring. a dvd 1 ==> a^n dvd 1";
-by (induct_tac "n" 1);
-by (Simp_tac 1);
-by (asm_simp_tac (simpset() addsimps [unit_mult]) 1);
-qed "unit_power";
-
-Goalw [dvd_def]
- "!! a::'a::ring. [| a dvd b; a dvd c |] ==> a dvd b + c";
-by (Clarify_tac 1);
-by (res_inst_tac [("x", "k + ka")] exI 1);
-by (simp_tac (simpset() addsimps [r_distr]) 1);
-qed "dvd_add_right";
-
-Goalw [dvd_def]
- "!! a::'a::ring. a dvd b ==> a dvd -b";
-by (Clarify_tac 1);
-by (res_inst_tac [("x", "-k")] exI 1);
-by (simp_tac (simpset() addsimps [r_minus]) 1);
-qed "dvd_uminus_right";
-
-Goalw [dvd_def]
- "!! a::'a::ring. a dvd b ==> a dvd c*b";
-by (Clarify_tac 1);
-by (res_inst_tac [("x", "c * k")] exI 1);
-by (Simp_tac 1);
-qed "dvd_l_mult_right";
-
-Goalw [dvd_def]
- "!! a::'a::ring. a dvd b ==> a dvd b*c";
-by (Clarify_tac 1);
-by (res_inst_tac [("x", "k * c")] exI 1);
-by (Simp_tac 1);
-qed "dvd_r_mult_right";
-
-Addsimps [dvd_add_right, dvd_uminus_right, dvd_l_mult_right, dvd_r_mult_right];
-
-(* Inverse of multiplication *)
-
-section "inverse";
-
-Goal "!! a::'a::ring. [| a * x = 1; a * y = 1 |] ==> x = y";
-by (res_inst_tac [("a", "(a*y)*x"), ("b", "y*(a*x)")] box_equals 1);
-by (Simp_tac 1);
-by Auto_tac;
-qed "inverse_unique";
-
-Goal "!! a::'a::ring. a dvd 1 ==> a * inverse a = 1";
-by (asm_full_simp_tac (simpset () addsimps [inverse_def, dvd_def]
- delsimprocs [ring_simproc]) 1);
-by (Clarify_tac 1);
-by (rtac theI 1);
-by (atac 1);
-by (rtac inverse_unique 1);
-by (atac 1);
-by (atac 1);
-qed "r_inverse_ring";
-
-Goal "!! a::'a::ring. a dvd 1 ==> inverse a * a= 1";
-by (asm_simp_tac (simpset() addsimps [r_inverse_ring]) 1);
-qed "l_inverse_ring";
-
-(* Integral domain *)
-
-(*
-section "Integral domains";
-
-Goal "0 ~= (1::'a::domain)";
-by (rtac not_sym 1);
-by (rtac one_not_zero 1);
-qed "zero_not_one";
-
-Goal "!! a::'a::domain. a dvd 1 ==> a ~= 0";
-by (auto_tac (claset() addDs [dvd_zero_left],
- simpset() addsimps [one_not_zero] ));
-qed "unit_imp_nonzero";
-
-Goal "[| a * b = 0; a ~= 0 |] ==> (b::'a::domain) = 0";
-by (dtac integral 1);
-by (Fast_tac 1);
-qed "r_integral";
-
-Goal "[| a * b = 0; b ~= 0 |] ==> (a::'a::domain) = 0";
-by (dtac integral 1);
-by (Fast_tac 1);
-qed "l_integral";
-
-Goal "!! a::'a::domain. [| a ~= 0; b ~= 0 |] ==> a * b ~= 0";
-by (blast_tac (claset() addIs [l_integral]) 1);
-qed "not_integral";
-
-Addsimps [not_integral, one_not_zero, zero_not_one];
-
-Goal "!! a::'a::domain. [| a * x = x; x ~= 0 |] ==> a = 1";
-by (res_inst_tac [("a", "- 1")] a_lcancel 1);
-by (Simp_tac 1);
-by (rtac l_integral 1);
-by (assume_tac 2);
-by (asm_simp_tac (simpset() addsimps [l_distr, l_minus]) 1);
-qed "l_one_integral";
-
-Goal "!! a::'a::domain. [| x * a = x; x ~= 0 |] ==> a = 1";
-by (res_inst_tac [("a", "- 1")] a_rcancel 1);
-by (Simp_tac 1);
-by (rtac r_integral 1);
-by (assume_tac 2);
-by (asm_simp_tac (simpset() addsimps [r_distr, r_minus]) 1);
-qed "r_one_integral";
-
-(* cancellation laws for multiplication *)
-
-Goal "!! a::'a::domain. [| a ~= 0; a * b = a * c |] ==> b = c";
-by (rtac diff_zero_imp_eq 1);
-by (dtac eq_imp_diff_zero 1);
-by (full_simp_tac (simpset() addsimps [r_minus RS sym, r_distr RS sym]) 1);
-by (fast_tac (claset() addIs [l_integral]) 1);
-qed "m_lcancel";
-
-Goal "!! a::'a::domain. [| a ~= 0; b * a = c * a |] ==> b = c";
-by (rtac m_lcancel 1);
-by (assume_tac 1);
-by (Asm_full_simp_tac 1);
-qed "m_rcancel";
-
-Goal "!! a::'a::domain. a ~= 0 ==> (a * b = a * c) = (b = c)";
-by (auto_tac (claset() addDs [m_lcancel], simpset()));
-qed "m_lcancel_eq";
-
-Goal "!! a::'a::domain. a ~= 0 ==> (b * a = c * a) = (b = c)";
-by (asm_simp_tac (simpset() addsimps [m_lcancel_eq, m_comm]) 1);
-qed "m_rcancel_eq";
-
-Addsimps [m_lcancel_eq, m_rcancel_eq];
-*)
-
-(* Fields *)
-
-section "Fields";
-
-Goal "!! a::'a::field. (a dvd 1) = (a ~= 0)";
-by (auto_tac (claset() addDs [thm "field_ax", thm "dvd_zero_left"],
- simpset() addsimps [thm "field_one_not_zero"]));
-qed "field_unit";
-
-(* required for instantiation of field < domain in file Field.thy *)
-
-Addsimps [field_unit];
-
-val field_one_not_zero = thm "field_one_not_zero";
-
-Goal "!! a::'a::field. a ~= 0 ==> a * inverse a = 1";
-by (asm_full_simp_tac (simpset() addsimps [r_inverse_ring]) 1);
-qed "r_inverse";
-
-Goal "!! a::'a::field. a ~= 0 ==> inverse a * a= 1";
-by (asm_full_simp_tac (simpset() addsimps [l_inverse_ring]) 1);
-qed "l_inverse";
-
-Addsimps [l_inverse, r_inverse];
-
-(* fields are integral domains *)
-
-Goal "!! a::'a::field. a * b = 0 ==> a = 0 | b = 0";
-by (Step_tac 1);
-by (res_inst_tac [("a", "(a*b)*inverse b")] box_equals 1);
-by (rtac refl 3);
-by (Simp_tac 2);
-by Auto_tac;
-qed "field_integral";
-
-(* fields are factorial domains *)
-
-Goalw [thm "prime_def", thm "irred_def"]
- "!! a::'a::field. irred a ==> prime a";
-by (blast_tac (claset() addIs [thm "field_ax"]) 1);
-qed "field_fact_prime";