(* Ideals for commutative rings $Id: Ideal.ML,v 1.11 2005/09/17 18:49:14 wenzelm Exp $ Author: Clemens Ballarin, started 24 September 1999 *) (* is_ideal *) Goalw [thm "is_ideal_def"] "!! I. [| !! a b. [| a:I; b:I |] ==> a + b : I; \ \ !! a. a:I ==> - a : I; 0 : I; \ \ !! a r. a:I ==> a * r : I |] ==> is_ideal I"; by Auto_tac; qed "is_idealI"; Goalw [thm "is_ideal_def"] "[| is_ideal I; a:I; b:I |] ==> a + b : I"; by (Fast_tac 1); qed "is_ideal_add"; Goalw [thm "is_ideal_def"] "[| is_ideal I; a:I |] ==> - a : I"; by (Fast_tac 1); qed "is_ideal_uminus"; Goalw [thm "is_ideal_def"] "[| is_ideal I |] ==> 0 : I"; by (Fast_tac 1); qed "is_ideal_zero"; Goalw [thm "is_ideal_def"] "[| is_ideal I; a:I |] ==> a * r : I"; by (Fast_tac 1); qed "is_ideal_mult"; Goalw [dvd_def, thm "is_ideal_def"] "[| a dvd b; is_ideal I; a:I |] ==> b:I"; by (Fast_tac 1); qed "is_ideal_dvd"; Goalw [thm "is_ideal_def"] "is_ideal (UNIV::('a::ring) set)"; by Auto_tac; qed "UNIV_is_ideal"; Goalw [thm "is_ideal_def"] "is_ideal {0::'a::ring}"; by Auto_tac; qed "zero_is_ideal"; Addsimps [is_ideal_add, is_ideal_uminus, is_ideal_zero, is_ideal_mult, UNIV_is_ideal, zero_is_ideal]; Goal "is_ideal {x::'a::ring. a dvd x}"; by (rtac is_idealI 1); by Auto_tac; qed "is_ideal_1"; Goal "is_ideal {x::'a::ring. EX u v. x = a * u + b * v}"; by (rtac is_idealI 1); (* by Auto_tac; FIXME: makes Zumkeller's order fail (raises exn Domain) *) by (Clarify_tac 1); by (Clarify_tac 2); by (Clarify_tac 3); by (Clarify_tac 4); by (res_inst_tac [("x", "u+ua")] exI 1); by (res_inst_tac [("x", "v+va")] exI 1); by (res_inst_tac [("x", "-u")] exI 2); by (res_inst_tac [("x", "-v")] exI 2); by (res_inst_tac [("x", "0")] exI 3); by (res_inst_tac [("x", "0")] exI 3); by (res_inst_tac [("x", "u * r")] exI 4); by (res_inst_tac [("x", "v * r")] exI 4); by (REPEAT (Simp_tac 1)); qed "is_ideal_2"; (* ideal_of *) Goalw [thm "is_ideal_def", thm "ideal_of_def"] "is_ideal (ideal_of S)"; by (Blast_tac 1); (* Here, blast_tac is much superior to fast_tac! *) qed "ideal_of_is_ideal"; Goalw [thm "ideal_of_def"] "a:S ==> a : (ideal_of S)"; by Auto_tac; qed "generator_in_ideal"; Goalw [thm "ideal_of_def"] "ideal_of {1::'a::ring} = UNIV"; by (force_tac (claset() addDs [is_ideal_mult], simpset() addsimps [l_one] delsimprocs [ring_simproc]) 1); (* FIXME: Zumkeller's order raises Domain exn *) qed "ideal_of_one_eq"; Goalw [thm "ideal_of_def"] "ideal_of {} = {0::'a::ring}"; by (rtac subset_antisym 1); by (rtac subsetI 1); by (dtac InterD 1); by (assume_tac 2); by (auto_tac (claset(), simpset() addsimps [is_ideal_zero])); qed "ideal_of_empty_eq"; Goalw [thm "ideal_of_def"] "ideal_of {a} = {x::'a::ring. a dvd x}"; by (rtac subset_antisym 1); by (rtac subsetI 1); by (dtac InterD 1); by (assume_tac 2); by (auto_tac (claset() addIs [is_ideal_1], simpset())); by (asm_simp_tac (simpset() addsimps [is_ideal_dvd]) 1); qed "pideal_structure"; Goalw [thm "ideal_of_def"] "ideal_of {a, b} = {x::'a::ring. EX u v. x = a * u + b * v}"; by (rtac subset_antisym 1); by (rtac subsetI 1); by (dtac InterD 1); by (assume_tac 2); by (auto_tac (claset() addIs [is_ideal_2], simpset() delsimprocs [ring_simproc])); (* FIXME: Zumkeller's order *) by (res_inst_tac [("x", "1")] exI 1); by (res_inst_tac [("x", "0")] exI 1); by (res_inst_tac [("x", "0")] exI 2); by (res_inst_tac [("x", "1")] exI 2); by (Simp_tac 1); by (Simp_tac 1); qed "ideal_of_2_structure"; Goalw [thm "ideal_of_def"] "A <= B ==> ideal_of A <= ideal_of B"; by Auto_tac; qed "ideal_of_mono"; Goal "ideal_of {0::'a::ring} = {0}"; by (simp_tac (simpset() addsimps [pideal_structure]) 1); by (rtac subset_antisym 1); by (auto_tac (claset() addIs [dvd_zero_left], simpset())); qed "ideal_of_zero_eq"; Goal "[| is_ideal I; a : I |] ==> ideal_of {a::'a::ring} <= I"; by (auto_tac (claset(), simpset() addsimps [pideal_structure, is_ideal_dvd])); qed "element_generates_subideal"; (* is_pideal *) Goalw [thm "is_pideal_def"] "is_pideal (I::('a::ring) set) ==> is_ideal I"; by (fast_tac (claset() addIs [ideal_of_is_ideal]) 1); qed "is_pideal_imp_is_ideal"; Goalw [thm "is_pideal_def"] "is_pideal (ideal_of {a::'a::ring})"; by (Fast_tac 1); qed "pideal_is_pideal"; Goalw [thm "is_pideal_def"] "is_pideal I ==> EX a. I = ideal_of {a}"; by (assume_tac 1); qed "is_pidealD"; (* Ideals and divisibility *) Goal "b dvd a ==> ideal_of {a::'a::ring} <= ideal_of {b}"; by (auto_tac (claset() addIs [dvd_trans_ring], simpset() addsimps [pideal_structure])); qed "dvd_imp_subideal"; Goal "ideal_of {a::'a::ring} <= ideal_of {b} ==> b dvd a"; by (auto_tac (claset() addSDs [subsetD], simpset() addsimps [pideal_structure])); qed "subideal_imp_dvd"; Goal "(ideal_of {a::'a::ring} <= ideal_of {b}) = (b dvd a)"; by (rtac iffI 1); by (REPEAT (ares_tac [subideal_imp_dvd, dvd_imp_subideal] 1)); qed "subideal_is_dvd"; Goal "(ideal_of {a::'a::ring} < ideal_of {b}) ==> ~ a dvd b"; by (full_simp_tac (simpset() addsimps [psubset_eq, pideal_structure]) 1); by (etac conjE 1); by (dres_inst_tac [("c", "a")] subsetD 1); by (auto_tac (claset() addIs [dvd_trans_ring], simpset())); qed "psubideal_not_dvd"; Goal "[| b dvd a; ~ a dvd b |] ==> ideal_of {a::'a::ring} < ideal_of {b}"; by (rtac psubsetI 1); by (etac dvd_imp_subideal 1); by (blast_tac (claset() addIs [dvd_imp_subideal, subideal_imp_dvd]) 1); qed "not_dvd_psubideal_singleton"; Goal "(ideal_of {a::'a::ring} < ideal_of {b}) = (b dvd a & ~ a dvd b)"; by (rtac iffI 1); by (REPEAT (ares_tac [conjI, psubideal_not_dvd, psubset_imp_subset RS subideal_imp_dvd] 1)); by (etac conjE 1); by (REPEAT (ares_tac [not_dvd_psubideal_singleton] 1)); qed "psubideal_is_dvd"; Goal "[| a dvd b; b dvd a |] ==> ideal_of {a::'a::ring} = ideal_of {b}"; by (rtac subset_antisym 1); by (REPEAT (ares_tac [dvd_imp_subideal] 1)); qed "assoc_pideal_eq"; AddIffs [subideal_is_dvd, psubideal_is_dvd]; Goal "!!a::'a::ring. a dvd b ==> b : (ideal_of {a})"; by (rtac is_ideal_dvd 1); by (assume_tac 1); by (rtac ideal_of_is_ideal 1); by (rtac generator_in_ideal 1); by (Simp_tac 1); qed "dvd_imp_in_pideal"; Goal "!!a::'a::ring. b : (ideal_of {a}) ==> a dvd b"; by (full_simp_tac (simpset() addsimps [pideal_structure]) 1); qed "in_pideal_imp_dvd"; Goal "~ (a dvd b) ==> ideal_of {a::'a::ring} < ideal_of {a, b}"; by (asm_simp_tac (simpset() addsimps [psubset_eq, ideal_of_mono]) 1); by (etac contrapos_pp 1); by (full_simp_tac (simpset() addsimps [ideal_of_2_structure]) 1); by (rtac in_pideal_imp_dvd 1); by (Asm_simp_tac 1); by (res_inst_tac [("x", "0")] exI 1); by (res_inst_tac [("x", "1")] exI 1); by (Simp_tac 1); qed "not_dvd_psubideal"; Goalw [thm "irred_def"] "[| irred (a::'a::ring); is_pideal I; ideal_of {a} < I |] ==> x : I"; by (dtac is_pidealD 1); by (etac exE 1); by (Clarify_tac 1); by (eres_inst_tac [("x", "aa")] allE 1); by (Clarify_tac 1); by (dres_inst_tac [("a", "1")] dvd_imp_subideal 1); by (auto_tac (claset(), simpset() addsimps [ideal_of_one_eq])); qed "irred_imp_max_ideal"; (* Pid are factorial *) (* Divisor chain condition *) (* proofs not finished *) Goal "(ALL i. I i <= I (Suc i)) ==> (n <= m & a : I n --> a : I m)"; by (induct_tac "m" 1); by (Blast_tac 1); (* induction step *) by (rename_tac "m" 1); by (case_tac "n <= m" 1); by Auto_tac; by (subgoal_tac "n = Suc m" 1); by (arith_tac 2); by (Force_tac 1); qed_spec_mp "subset_chain_lemma"; Goal "[| ALL i. is_ideal (I i); ALL i. I i <= I (Suc i) |] \ \ ==> is_ideal (Union (I`UNIV))"; by (rtac is_idealI 1); by Auto_tac; by (res_inst_tac [("x", "max x xa")] exI 1); by (rtac is_ideal_add 1); by (Asm_simp_tac 1); by (rtac subset_chain_lemma 1); by (assume_tac 1); by (rtac conjI 1); by (assume_tac 2); by (arith_tac 1); by (rtac subset_chain_lemma 1); by (assume_tac 1); by (rtac conjI 1); by (assume_tac 2); by (arith_tac 1); by (res_inst_tac [("x", "x")] exI 1); by (Asm_simp_tac 1); by (res_inst_tac [("x", "x")] exI 1); by (Asm_simp_tac 1); qed "chain_is_ideal"; (* Goal "ALL i. ideal_of {a i} < ideal_of {a (Suc i)} ==> \ \ EX n. Union ((ideal_of o (%a. {a}))`UNIV) = ideal_of {a n}"; Goal "wf {(a::'a::pid, b). a dvd b & ~ b dvd a}"; by (simp_tac (simpset() addsimps [psubideal_is_dvd RS sym, wf_iff_no_infinite_down_chain] delsimps [psubideal_is_dvd]) 1); *) (* Primeness condition *) Goalw [thm "prime_def"] "irred a ==> prime (a::'a::pid)"; by (rtac conjI 1); by (rtac conjI 2); by (Clarify_tac 3); by (dres_inst_tac [("I", "ideal_of {a, b}"), ("x", "1")] irred_imp_max_ideal 3); by (auto_tac (claset() addIs [ideal_of_is_ideal, thm "pid_ax"], simpset() addsimps [thm "irred_def", not_dvd_psubideal, thm "pid_ax"])); by (full_simp_tac (simpset() addsimps [ideal_of_2_structure]) 1); by (Clarify_tac 1); by (dres_inst_tac [("f", "op* aa")] arg_cong 1); by (full_simp_tac (simpset() addsimps [r_distr]) 1); by (etac subst 1); by (asm_simp_tac (simpset() addsimps [m_assoc RS sym] delsimprocs [ring_simproc]) 1); qed "pid_irred_imp_prime"; (* Fields are Pid *) Goal "a ~= 0 ==> ideal_of {a::'a::field} = UNIV"; by (rtac subset_antisym 1); by (Simp_tac 1); by (rtac subset_trans 1); by (rtac dvd_imp_subideal 2); by (rtac (thm "field_ax") 2); by (assume_tac 2); by (simp_tac (simpset() addsimps [ideal_of_one_eq]) 1); qed "field_pideal_univ"; Goal "[| is_ideal I; I ~= {0} |] ==> {0} < I"; by (asm_simp_tac (simpset() addsimps [psubset_eq, not_sym, is_ideal_zero]) 1); qed "proper_ideal"; Goalw [thm "is_pideal_def"] "is_ideal (I::('a::field) set) ==> is_pideal I"; by (case_tac "I = {0}" 1); by (res_inst_tac [("x", "0")] exI 1); by (asm_simp_tac (simpset() addsimps [ideal_of_zero_eq]) 1); (* case "I ~= {0}" *) by (ftac proper_ideal 1); by (assume_tac 1); by (dtac psubset_imp_ex_mem 1); by (etac exE 1); by (res_inst_tac [("x", "b")] exI 1); by (cut_inst_tac [("a", "b")] element_generates_subideal 1); by (assume_tac 1); by (Blast_tac 1); by (auto_tac (claset(), simpset() addsimps [field_pideal_univ])); qed "field_pid";