(*
Title: The algebraic hierarchy of rings as axiomatic classes
Id: $Id$
Author: Clemens Ballarin, started 9 December 1996
Copyright: Clemens Ballarin
*)
header {* The algebraic hierarchy of rings as axiomatic classes *}
theory Ring2 imports Main
begin
section {* Constants *}
text {* Most constants are already declared by HOL. *}
consts
assoc :: "['a::times, 'a] => bool" (infixl 50)
irred :: "'a::{zero, one, times} => bool"
prime :: "'a::{zero, one, times} => bool"
section {* Axioms *}
subsection {* Ring axioms *}
axclass ring < zero, one, plus, minus, times, inverse, power
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"
defs
assoc_def: "a assoc b == a dvd b & b dvd a"
irred_def: "irred a == a ~= 0 & ~ a dvd 1
& (ALL d. d dvd a --> d dvd 1 | a dvd d)"
prime_def: "prime p == p ~= 0 & ~ p dvd 1
& (ALL a b. p dvd (a*b) --> p dvd a | p dvd b)"
subsection {* Integral domains *}
axclass
"domain" < ring
one_not_zero: "1 ~= 0"
integral: "a * b = 0 ==> a = 0 | b = 0"
subsection {* Factorial domains *}
axclass
factorial < "domain"
(*
Proper definition using divisor chain condition currently not supported.
factorial_divisor: "wf {(a, b). a dvd b & ~ (b dvd a)}"
*)
factorial_divisor: "True"
factorial_prime: "irred a ==> prime a"
subsection {* Euclidean domains *}
(*
axclass
euclidean < "domain"
euclidean_ax: "b ~= 0 ==> Ex (% (q, r, e_size::('a::ringS)=>nat).
a = b * q + r & e_size r < e_size b)"
Nothing has been proved about Euclidean domains, yet.
Design question:
Fix quo, rem and e_size as constants that are axiomatised with
euclidean_ax?
- advantage: more pragmatic and easier to use
- disadvantage: for every type, one definition of quo and rem will
be fixed, users may want to use differing ones;
also, it seems not possible to prove that fields are euclidean
domains, because that would require generic (type-independent)
definitions of quo and rem.
*)
subsection {* Fields *}
axclass
field < ring
field_one_not_zero: "1 ~= 0"
(* Avoid a common superclass as the first thing we will
prove about fields is that they are domains. *)
field_ax: "a ~= 0 ==> a dvd 1"
section {* Basic facts *}
subsection {* Normaliser for rings *}
(* derived rewrite rules *)
lemma a_lcomm: "(a::'a::ring)+(b+c) = b+(a+c)"
apply (rule a_comm [THEN trans])
apply (rule a_assoc [THEN trans])
apply (rule a_comm [THEN arg_cong])
done
lemma r_zero: "(a::'a::ring) + 0 = a"
apply (rule a_comm [THEN trans])
apply (rule l_zero)
done
lemma r_neg: "(a::'a::ring) + (-a) = 0"
apply (rule a_comm [THEN trans])
apply (rule l_neg)
done
lemma r_neg2: "(a::'a::ring) + (-a + b) = b"
apply (rule a_assoc [symmetric, THEN trans])
apply (simp add: r_neg l_zero)
done
lemma r_neg1: "-(a::'a::ring) + (a + b) = b"
apply (rule a_assoc [symmetric, THEN trans])
apply (simp add: l_neg l_zero)
done
(* auxiliary *)
lemma a_lcancel: "!! a::'a::ring. a + b = a + c ==> b = c"
apply (rule box_equals)
prefer 2
apply (rule l_zero)
prefer 2
apply (rule l_zero)
apply (rule_tac a1 = a in l_neg [THEN subst])
apply (simp add: a_assoc)
done
lemma minus_add: "-((a::'a::ring) + b) = (-a) + (-b)"
apply (rule_tac a = "a + b" in a_lcancel)
apply (simp add: r_neg l_neg l_zero a_assoc a_comm a_lcomm)
done
lemma minus_minus: "-(-(a::'a::ring)) = a"
apply (rule a_lcancel)
apply (rule r_neg [THEN trans])
apply (rule l_neg [symmetric])
done
lemma minus0: "- 0 = (0::'a::ring)"
apply (rule a_lcancel)
apply (rule r_neg [THEN trans])
apply (rule l_zero [symmetric])
done
(* derived rules for multiplication *)
lemma m_lcomm: "(a::'a::ring)*(b*c) = b*(a*c)"
apply (rule m_comm [THEN trans])
apply (rule m_assoc [THEN trans])
apply (rule m_comm [THEN arg_cong])
done
lemma r_one: "(a::'a::ring) * 1 = a"
apply (rule m_comm [THEN trans])
apply (rule l_one)
done
lemma r_distr: "(a::'a::ring) * (b + c) = a * b + a * c"
apply (rule m_comm [THEN trans])
apply (rule l_distr [THEN trans])
apply (simp add: m_comm)
done
(* the following proof is from Jacobson, Basic Algebra I, pp. 88-89 *)
lemma l_null: "0 * (a::'a::ring) = 0"
apply (rule a_lcancel)
apply (rule l_distr [symmetric, THEN trans])
apply (simp add: r_zero)
done
lemma r_null: "(a::'a::ring) * 0 = 0"
apply (rule m_comm [THEN trans])
apply (rule l_null)
done
lemma l_minus: "(-(a::'a::ring)) * b = - (a * b)"
apply (rule a_lcancel)
apply (rule r_neg [symmetric, THEN [2] trans])
apply (rule l_distr [symmetric, THEN trans])
apply (simp add: l_null r_neg)
done
lemma r_minus: "(a::'a::ring) * (-b) = - (a * b)"
apply (rule a_lcancel)
apply (rule r_neg [symmetric, THEN [2] trans])
apply (rule r_distr [symmetric, THEN trans])
apply (simp add: r_null r_neg)
done
(*** Term order for commutative rings ***)
ML {*
fun ring_ord (Const (a, _)) =
find_index (fn a' => a = a')
["HOL.zero", "HOL.plus", "HOL.uminus", "HOL.minus", "HOL.one", "HOL.times"]
| ring_ord _ = ~1;
fun termless_ring (a, b) = (Term.term_lpo ring_ord (a, b) = LESS);
val ring_ss = HOL_basic_ss settermless termless_ring addsimps
[thm "a_assoc", thm "l_zero", thm "l_neg", thm "a_comm", thm "m_assoc",
thm "l_one", thm "l_distr", thm "m_comm", thm "minus_def",
thm "r_zero", thm "r_neg", thm "r_neg2", thm "r_neg1", thm "minus_add",
thm "minus_minus", thm "minus0", thm "a_lcomm", thm "m_lcomm", (*thm "r_one",*)
thm "r_distr", thm "l_null", thm "r_null", thm "l_minus", thm "r_minus"];
*} (* Note: r_one is not necessary in ring_ss *)
method_setup ring =
{* Method.no_args (Method.SIMPLE_METHOD' (full_simp_tac ring_ss)) *}
{* computes distributive normal form in rings *}
lemmas ring_simps =
l_zero r_zero l_neg r_neg minus_minus minus0
l_one r_one l_null r_null l_minus r_minus
subsection {* Rings and the summation operator *}
(* Basic facts --- move to HOL!!! *)
(* needed because natsum_cong (below) disables atMost_0 *)
lemma natsum_0 [simp]: "setsum f {..(0::nat)} = (f 0::'a::comm_monoid_add)"
by simp
(*
lemma natsum_Suc [simp]:
"setsum f {..Suc n} = (f (Suc n) + setsum f {..n}::'a::comm_monoid_add)"
by (simp add: atMost_Suc)
*)
lemma natsum_Suc2:
"setsum f {..Suc n} = (f 0::'a::comm_monoid_add) + (setsum (%i. f (Suc i)) {..n})"
proof (induct n)
case 0 show ?case by simp
next
case Suc thus ?case by (simp add: semigroup_add_class.add_assoc)
qed
lemma natsum_cong [cong]:
"!!k. [| j = k; !!i::nat. i <= k ==> f i = (g i::'a::comm_monoid_add) |] ==>
setsum f {..j} = setsum g {..k}"
by (induct j) auto
lemma natsum_zero [simp]: "setsum (%i. 0) {..n::nat} = (0::'a::comm_monoid_add)"
by (induct n) simp_all
lemma natsum_add [simp]:
"!!f::nat=>'a::comm_monoid_add.
setsum (%i. f i + g i) {..n::nat} = setsum f {..n} + setsum g {..n}"
by (induct n) (simp_all add: add_ac)
(* Facts specific to rings *)
instance ring < comm_monoid_add
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 =
["t + u::'a::ring",
"t - u::'a::ring",
"t * u::'a::ring",
"- t::'a::ring"];
fun proc ss t =
let val rew = Goal.prove (Simplifier.the_context ss) [] []
(HOLogic.mk_Trueprop
(HOLogic.mk_eq (t, Var (("x", Term.maxidx_of_term t + 1), fastype_of t))))
(fn _ => simp_tac (Simplifier.inherit_context ss 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 = Simplifier.simproc (the_context ()) "ring" lhss (K 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)
lemma power_0 [simp]:
"(a::'a::ring) ^ 0 = 1" unfolding power_def by simp
lemma power_Suc [simp]:
"(a::'a::ring) ^ Suc n = a ^ n * a" unfolding power_def by simp
lemma power_one [simp]:
"1 ^ n = (1::'a::ring)" by (induct n) simp_all
lemma power_zero [simp]:
"n \<noteq> 0 \<Longrightarrow> 0 ^ n = (0::'a::ring)" by (induct n) simp_all
lemma power_mult [simp]:
"(a::'a::ring) ^ m * a ^ n = a ^ (m + n)"
by (induct m) simp_all
section "Divisibility"
lemma dvd_zero_right [simp]:
"(a::'a::ring) dvd 0"
proof
show "0 = a * 0" by simp
qed
lemma dvd_zero_left:
"0 dvd (a::'a::ring) \<Longrightarrow> a = 0" unfolding dvd_def by simp
lemma dvd_refl_ring [simp]:
"(a::'a::ring) dvd a"
proof
show "a = a * 1" by simp
qed
lemma dvd_trans_ring:
fixes a b c :: "'a::ring"
assumes a_dvd_b: "a dvd b"
and b_dvd_c: "b dvd c"
shows "a dvd c"
proof -
from a_dvd_b obtain l where "b = a * l" using dvd_def by blast
moreover from b_dvd_c obtain j where "c = b * j" using dvd_def by blast
ultimately have "c = a * (l * j)" by simp
then have "\<exists>k. c = a * k" ..
then show ?thesis using dvd_def by blast
qed
lemma unit_mult:
"!!a::'a::ring. [| a dvd 1; b dvd 1 |] ==> a * b dvd 1"
apply (unfold dvd_def)
apply clarify
apply (rule_tac x = "k * ka" in exI)
apply simp
done
lemma unit_power: "!!a::'a::ring. a dvd 1 ==> a^n dvd 1"
apply (induct_tac n)
apply simp
apply (simp add: unit_mult)
done
lemma dvd_add_right [simp]:
"!! a::'a::ring. [| a dvd b; a dvd c |] ==> a dvd b + c"
apply (unfold dvd_def)
apply clarify
apply (rule_tac x = "k + ka" in exI)
apply (simp add: r_distr)
done
lemma dvd_uminus_right [simp]:
"!! a::'a::ring. a dvd b ==> a dvd -b"
apply (unfold dvd_def)
apply clarify
apply (rule_tac x = "-k" in exI)
apply (simp add: r_minus)
done
lemma dvd_l_mult_right [simp]:
"!! a::'a::ring. a dvd b ==> a dvd c*b"
apply (unfold dvd_def)
apply clarify
apply (rule_tac x = "c * k" in exI)
apply simp
done
lemma dvd_r_mult_right [simp]:
"!! a::'a::ring. a dvd b ==> a dvd b*c"
apply (unfold dvd_def)
apply clarify
apply (rule_tac x = "k * c" in exI)
apply simp
done
(* Inverse of multiplication *)
section "inverse"
lemma inverse_unique: "!! a::'a::ring. [| a * x = 1; a * y = 1 |] ==> x = y"
apply (rule_tac a = "(a*y) * x" and b = "y * (a*x)" in box_equals)
apply (simp (no_asm))
apply auto
done
lemma r_inverse_ring: "!! a::'a::ring. a dvd 1 ==> a * inverse a = 1"
apply (unfold inverse_def dvd_def)
apply (tactic {* asm_full_simp_tac (simpset () delsimprocs [ring_simproc]) 1 *})
apply clarify
apply (rule theI)
apply assumption
apply (rule inverse_unique)
apply assumption
apply assumption
done
lemma l_inverse_ring: "!! a::'a::ring. a dvd 1 ==> inverse a * a = 1"
by (simp add: r_inverse_ring)
(* Fields *)
section "Fields"
lemma field_unit [simp]: "!! a::'a::field. (a dvd 1) = (a ~= 0)"
by (auto dest: field_ax dvd_zero_left simp add: field_one_not_zero)
lemma r_inverse [simp]: "!! a::'a::field. a ~= 0 ==> a * inverse a = 1"
by (simp add: r_inverse_ring)
lemma l_inverse [simp]: "!! a::'a::field. a ~= 0 ==> inverse a * a= 1"
by (simp add: l_inverse_ring)
(* fields are integral domains *)
lemma field_integral: "!! a::'a::field. a * b = 0 ==> a = 0 | b = 0"
apply (tactic "Step_tac 1")
apply (rule_tac a = " (a*b) * inverse b" in box_equals)
apply (rule_tac [3] refl)
prefer 2
apply (simp (no_asm))
apply auto
done
(* fields are factorial domains *)
lemma field_fact_prime: "!! a::'a::field. irred a ==> prime a"
unfolding prime_def irred_def by (blast intro: field_ax)
end