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+++ b/src/HOL/Algebra/CRing.thy Thu Feb 27 15:12:29 2003 +0100
@@ -0,0 +1,278 @@
+(*
+ Title: The algebraic hierarchy of rings
+ Id: $Id$
+ Author: Clemens Ballarin, started 9 December 1996
+ Copyright: Clemens Ballarin
+*)
+
+header {* The algebraic hierarchy of rings as axiomatic classes *}
+
+theory Ring = Group
+
+section {* The Algebraic Hierarchy of Rings *}
+
+subsection {* Basic Definitions *}
+
+record 'a ring = "'a group" +
+ zero :: 'a ("\<zero>\<index>")
+ add :: "['a, 'a] => 'a" (infixl "\<oplus>\<index>" 65)
+ a_inv :: "'a => 'a" ("\<ominus>\<index> _" [81] 80)
+
+locale cring = abelian_monoid R +
+ assumes a_abelian_group: "abelian_group (| carrier = carrier R,
+ mult = add R, one = zero R, m_inv = a_inv R |)"
+ and m_inv_def: "[| EX y. y \<in> carrier R & x \<otimes> y = \<one>; x \<in> carrier R |]
+ ==> inv x = (THE y. y \<in> carrier R & x \<otimes> y = \<one>)"
+ and l_distr: "[| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
+ ==> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
+
+ML {*
+ thm "cring.l_distr"
+*}
+
+(*
+locale cring = struct R +
+ assumes a_group: "abelian_group (| carrier = carrier R,
+ mult = add R, one = zero R, m_inv = a_inv R |)"
+ and m_monoid: "abelian_monoid (| carrier = carrier R - {zero R},
+ mult = mult R, one = one R |)"
+ and l_distr: "[| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
+ ==> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
+
+locale field = struct R +
+ assumes a_group: "abelian_group (| carrier = carrier R,
+ mult = add R, one = zero R, m_inv = a_inv R |)"
+ and m_group: "monoid (| carrier = carrier R - {zero R},
+ mult = mult R, one = one R |)"
+ and l_distr: "[| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
+ ==> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
+*)
+(*
+ a_assoc: "(a + b) + c = a + (b + c)"
+ l_zero: "0 + a = a"
+ l_neg: "(-a) + a = 0"
+ a_comm: "a + b = b + a"
+
+ m_assoc: "(a * b) * c = a * (b * c)"
+ l_one: "1 * a = a"
+
+ l_distr: "(a + b) * c = a * c + b * c"
+
+ m_comm: "a * b = b * a"
+
+ -- {* Definition of derived operations *}
+
+ minus_def: "a - b = a + (-b)"
+ inverse_def: "inverse a = (if a dvd 1 then THE x. a*x = 1 else 0)"
+ divide_def: "a / b = a * inverse b"
+ power_def: "a ^ n = nat_rec 1 (%u b. b * a) n"
+*)
+subsection {* Basic Facts *}
+
+lemma (in cring) a_magma [simp, intro]:
+ "magma (| carrier = carrier R,
+ mult = add R, one = zero R, m_inv = a_inv R |)"
+ using a_abelian_group by (simp only: abelian_group_def)
+
+lemma (in cring) a_l_one [simp, intro]:
+ "l_one (| carrier = carrier R,
+ mult = add R, one = zero R, m_inv = a_inv R |)"
+ using a_abelian_group by (simp only: abelian_group_def)
+
+lemma (in cring) a_abelian_group_parts [simp, intro]:
+ "semigroup_axioms (| carrier = carrier R,
+ mult = add R, one = zero R, m_inv = a_inv R |)"
+ "group_axioms (| carrier = carrier R,
+ mult = add R, one = zero R, m_inv = a_inv R |)"
+ "abelian_semigroup_axioms (| carrier = carrier R,
+ mult = add R, one = zero R, m_inv = a_inv R |)"
+ using a_abelian_group by (simp_all only: abelian_group_def)
+
+lemma (in cring) a_semigroup:
+ "semigroup (| carrier = carrier R,
+ mult = add R, one = zero R, m_inv = a_inv R |)"
+ by (simp add: semigroup_def)
+
+lemma (in cring) a_group:
+ "group (| carrier = carrier R,
+ mult = add R, one = zero R, m_inv = a_inv R |)"
+ by (simp add: group_def)
+
+lemma (in cring) a_abelian_semigroup:
+ "abelian_semigroup (| carrier = carrier R,
+ mult = add R, one = zero R, m_inv = a_inv R |)"
+ by (simp add: abelian_semigroup_def)
+
+lemma (in cring) a_abelian_group:
+ "abelian_group (| carrier = carrier R,
+ mult = add R, one = zero R, m_inv = a_inv R |)"
+ by (simp add: abelian_group_def)
+
+lemma (in cring) a_assoc:
+ "[| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
+ ==> (x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)"
+ using semigroup.m_assoc [OF a_semigroup]
+ by simp
+
+lemma (in cring) l_zero:
+ "x \<in> carrier R ==> \<zero> \<oplus> x = x"
+ using l_one.l_one [OF a_l_one]
+ by simp
+
+lemma (in cring) l_neg:
+ "x \<in> carrier R ==> (\<ominus> x) \<oplus> x = \<zero>"
+ using group.l_inv [OF a_group]
+ by simp
+
+lemma (in cring) a_comm:
+ "[| x \<in> carrier R; y \<in> carrier R |]
+ ==> x \<oplus> y = y \<oplus> x"
+ using abelian_semigroup.m_comm [OF a_abelian_semigroup]
+ by simp
+
+
+
+
+qed
+
+ l_zero: "0 + a = a"
+ l_neg: "(-a) + a = 0"
+ a_comm: "a + b = b + a"
+
+ m_assoc: "(a * b) * c = a * (b * c)"
+ l_one: "1 * a = a"
+
+ l_distr: "(a + b) * c = a * c + b * c"
+
+ m_comm: "a * b = b * a"
+text {* Normaliser for Commutative Rings *}
+
+use "order.ML"
+
+method_setup ring =
+ {* Method.no_args (Method.SIMPLE_METHOD' HEADGOAL (full_simp_tac ring_ss)) *}
+ {* computes distributive normal form in rings *}
+
+subsection {* Rings and the summation operator *}
+
+(* Basic facts --- move to HOL!!! *)
+
+lemma natsum_0 [simp]: "setsum f {..(0::nat)} = (f 0::'a::plus_ac0)"
+by simp
+
+lemma natsum_Suc [simp]:
+ "setsum f {..Suc n} = (f (Suc n) + setsum f {..n}::'a::plus_ac0)"
+by (simp add: atMost_Suc)
+
+lemma natsum_Suc2:
+ "setsum f {..Suc n} = (setsum (%i. f (Suc i)) {..n} + f 0::'a::plus_ac0)"
+proof (induct n)
+ case 0 show ?case by simp
+next
+ case Suc thus ?case by (simp add: assoc)
+qed
+
+lemma natsum_cong [cong]:
+ "!!k. [| j = k; !!i::nat. i <= k ==> f i = (g i::'a::plus_ac0) |] ==>
+ setsum f {..j} = setsum g {..k}"
+by (induct j) auto
+
+lemma natsum_zero [simp]: "setsum (%i. 0) {..n::nat} = (0::'a::plus_ac0)"
+by (induct n) simp_all
+
+lemma natsum_add [simp]:
+ "!!f::nat=>'a::plus_ac0.
+ setsum (%i. f i + g i) {..n::nat} = setsum f {..n} + setsum g {..n}"
+by (induct n) (simp_all add: plus_ac0)
+
+(* Facts specific to rings *)
+
+instance ring < plus_ac0
+proof
+ fix x y z
+ show "(x::'a::ring) + y = y + x" by (rule a_comm)
+ show "((x::'a::ring) + y) + z = x + (y + z)" by (rule a_assoc)
+ show "0 + (x::'a::ring) = x" by (rule l_zero)
+qed
+
+ML {*
+ local
+ val lhss =
+ [read_cterm (sign_of (the_context ()))
+ ("?t + ?u::'a::ring", TVar (("'z", 0), [])),
+ read_cterm (sign_of (the_context ()))
+ ("?t - ?u::'a::ring", TVar (("'z", 0), [])),
+ read_cterm (sign_of (the_context ()))
+ ("?t * ?u::'a::ring", TVar (("'z", 0), [])),
+ read_cterm (sign_of (the_context ()))
+ ("- ?t::'a::ring", TVar (("'z", 0), []))
+ ];
+ fun proc sg _ t =
+ let val rew = Tactic.prove sg [] []
+ (HOLogic.mk_Trueprop
+ (HOLogic.mk_eq (t, Var (("x", Term.maxidx_of_term t + 1), fastype_of t))))
+ (fn _ => simp_tac ring_ss 1)
+ |> mk_meta_eq;
+ val (t', u) = Logic.dest_equals (Thm.prop_of rew);
+ in if t' aconv u
+ then None
+ else Some rew
+ end;
+ in
+ val ring_simproc = mk_simproc "ring" lhss proc;
+ end;
+*}
+
+ML_setup {* Addsimprocs [ring_simproc] *}
+
+lemma natsum_ldistr:
+ "!!a::'a::ring. setsum f {..n::nat} * a = setsum (%i. f i * a) {..n}"
+by (induct n) simp_all
+
+lemma natsum_rdistr:
+ "!!a::'a::ring. a * setsum f {..n::nat} = setsum (%i. a * f i) {..n}"
+by (induct n) simp_all
+
+subsection {* Integral Domains *}
+
+declare one_not_zero [simp]
+
+lemma zero_not_one [simp]:
+ "0 ~= (1::'a::domain)"
+by (rule not_sym) simp
+
+lemma integral_iff: (* not by default a simp rule! *)
+ "(a * b = (0::'a::domain)) = (a = 0 | b = 0)"
+proof
+ assume "a * b = 0" then show "a = 0 | b = 0" by (simp add: integral)
+next
+ assume "a = 0 | b = 0" then show "a * b = 0" by auto
+qed
+
+(*
+lemma "(a::'a::ring) - (a - b) = b" apply simp
+ simproc seems to fail on this example (fixed with new term order)
+*)
+(*
+lemma bug: "(b::'a::ring) - (b - a) = a" by simp
+ simproc for rings cannot prove "(a::'a::ring) - (a - b) = b"
+*)
+lemma m_lcancel:
+ assumes prem: "(a::'a::domain) ~= 0" shows conc: "(a * b = a * c) = (b = c)"
+proof
+ assume eq: "a * b = a * c"
+ then have "a * (b - c) = 0" by simp
+ then have "a = 0 | (b - c) = 0" by (simp only: integral_iff)
+ with prem have "b - c = 0" by auto
+ then have "b = b - (b - c)" by simp
+ also have "b - (b - c) = c" by simp
+ finally show "b = c" .
+next
+ assume "b = c" then show "a * b = a * c" by simp
+qed
+
+lemma m_rcancel:
+ "(a::'a::domain) ~= 0 ==> (b * a = c * a) = (b = c)"
+by (simp add: m_lcancel)
+
+end