Change to meta simplifier: congruence rules may now have frees as head of term.
(*
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