(* Title: HOL/Computational_Algebra/Group_Closure.thy
Author: Johannes Hoelzl, TU Muenchen
Author: Florian Haftmann, TU Muenchen
*)
theory Group_Closure
imports
Main
begin
context ab_group_add
begin
inductive_set group_closure :: "'a set \ 'a set" for S
where base: "s \ insert 0 S \ s \ group_closure S"
| diff: "s \ group_closure S \ t \ group_closure S \ s - t \ group_closure S"
lemma zero_in_group_closure [simp]:
"0 \ group_closure S"
using group_closure.base [of 0 S] by simp
lemma group_closure_minus_iff [simp]:
"- s \ group_closure S \ s \ group_closure S"
using group_closure.diff [of 0 S s] group_closure.diff [of 0 S "- s"] by auto
lemma group_closure_add:
"s + t \ group_closure S" if "s \ group_closure S" and "t \ group_closure S"
using that group_closure.diff [of s S "- t"] by auto
lemma group_closure_empty [simp]:
"group_closure {} = {0}"
by (rule ccontr) (auto elim: group_closure.induct)
lemma group_closure_insert_zero [simp]:
"group_closure (insert 0 S) = group_closure S"
by (auto elim: group_closure.induct intro: group_closure.intros)
end
context comm_ring_1
begin
lemma group_closure_scalar_mult_left:
"of_nat n * s \ group_closure S" if "s \ group_closure S"
using that by (induction n) (auto simp add: algebra_simps intro: group_closure_add)
lemma group_closure_scalar_mult_right:
"s * of_nat n \ group_closure S" if "s \ group_closure S"
using that group_closure_scalar_mult_left [of s S n] by (simp add: ac_simps)
end
lemma group_closure_abs_iff [simp]:
"\s\ \ group_closure S \ s \ group_closure S" for s :: int
by (simp add: abs_if)
lemma group_closure_mult_left:
"s * t \ group_closure S" if "s \ group_closure S" for s t :: int
proof -
from that group_closure_scalar_mult_right [of s S "nat \t\"]
have "s * int (nat \t\) \ group_closure S"
by (simp only:)
then show ?thesis
by (cases "t \ 0") simp_all
qed
lemma group_closure_mult_right:
"s * t \ group_closure S" if "t \ group_closure S" for s t :: int
using that group_closure_mult_left [of t S s] by (simp add: ac_simps)
context idom
begin
lemma group_closure_mult_all_eq:
"group_closure (times k ` S) = times k ` group_closure S"
proof (rule; rule)
fix s
have *: "k * a + k * b = k * (a + b)"
"k * a - k * b = k * (a - b)" for a b
by (simp_all add: algebra_simps)
assume "s \ group_closure (times k ` S)"
then show "s \ times k ` group_closure S"
by induction (auto simp add: * image_iff intro: group_closure.base group_closure.diff bexI [of _ 0])
next
fix s
assume "s \ times k ` group_closure S"
then obtain r where r: "r \ group_closure S" and s: "s = k * r"
by auto
from r have "k * r \ group_closure (times k ` S)"
by (induction arbitrary: s) (auto simp add: algebra_simps intro: group_closure.intros)
with s show "s \ group_closure (times k ` S)"
by simp
qed
end
lemma Gcd_group_closure_eq_Gcd:
"Gcd (group_closure S) = Gcd S" for S :: "int set"
proof (rule associated_eqI)
have "Gcd S dvd s" if "s \ group_closure S" for s
using that by induction auto
then show "Gcd S dvd Gcd (group_closure S)"
by auto
have "Gcd (group_closure S) dvd s" if "s \ S" for s
proof -
from that have "s \ group_closure S"
by (simp add: group_closure.base)
then show ?thesis
by (rule Gcd_dvd)
qed
then show "Gcd (group_closure S) dvd Gcd S"
by auto
qed simp_all
lemma group_closure_sum:
fixes S :: "int set"
assumes X: "finite X" "X \ {}" "X \ S"
shows "(\x\X. a x * x) \ group_closure S"
using X by (induction X rule: finite_ne_induct)
(auto intro: group_closure_mult_right group_closure.base group_closure_add)
lemma Gcd_group_closure_in_group_closure:
"Gcd (group_closure S) \ group_closure S" for S :: "int set"
proof (cases "S \ {0}")
case True
then have "S = {} \ S = {0}"
by auto
then show ?thesis
by auto
next
case False
then obtain s where s: "s \ 0" "s \ S"
by auto
then have s': "\s\ \ 0" "\s\ \ group_closure S"
by (auto intro: group_closure.base)
define m where "m = (LEAST n. n > 0 \ int n \ group_closure S)"
have "m > 0 \ int m \ group_closure S"
unfolding m_def
apply (rule LeastI [of _ "nat \s\"])
using s'
by simp
then have m: "int m \ group_closure S" and "0 < m"
by auto
have "Gcd (group_closure S) = int m"
proof (rule associated_eqI)
from m show "Gcd (group_closure S) dvd int m"
by (rule Gcd_dvd)
show "int m dvd Gcd (group_closure S)"
proof (rule Gcd_greatest)
fix s
assume s: "s \ group_closure S"
show "int m dvd s"
proof (rule ccontr)
assume "\ int m dvd s"
then have *: "0 < s mod int m"
using \<open>0 < m\<close> le_less by fastforce
have "m \ nat (s mod int m)"
proof (subst m_def, rule Least_le, rule)
from * show "0 < nat (s mod int m)"
by simp
from minus_div_mult_eq_mod [symmetric, of s "int m"]
have "s mod int m = s - s div int m * int m"
by auto
also have "s - s div int m * int m \ group_closure S"
by (auto intro: group_closure.diff s group_closure_mult_right m)
finally show "int (nat (s mod int m)) \ group_closure S"
by simp
qed
with * have "int m \ s mod int m"
by simp
moreover have "s mod int m < int m"
using \<open>0 < m\<close> by simp
ultimately show False
by auto
qed
qed
qed simp_all
with m show ?thesis
by simp
qed
lemma Gcd_in_group_closure:
"Gcd S \ group_closure S" for S :: "int set"
using Gcd_group_closure_in_group_closure [of S]
by (simp add: Gcd_group_closure_eq_Gcd)
lemma group_closure_eq:
"group_closure S = range (times (Gcd S))" for S :: "int set"
proof (auto intro: Gcd_in_group_closure group_closure_mult_left)
fix s
assume "s \ group_closure S"
then show "s \ range (times (Gcd S))"
proof induction
case (base s)
then have "Gcd S dvd s"
by (auto intro: Gcd_dvd)
then obtain t where "s = Gcd S * t" ..
then show ?case
by auto
next
case (diff s t)
moreover have "Gcd S * a - Gcd S * b = Gcd S * (a - b)" for a b
by (simp add: algebra_simps)
ultimately show ?case
by auto
qed
qed
end
¤ Dauer der Verarbeitung: 0.1 Sekunden
(vorverarbeitet)
¤
|
Haftungshinweis
Die Informationen auf dieser Webseite wurden
nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit,
noch Qualität der bereit gestellten Informationen zugesichert.
Bemerkung:
Die farbliche Syntaxdarstellung ist noch experimentell.
|
