--- a/src/HOL/OrderedGroup.thy Mon Oct 29 17:08:01 2007 +0100
+++ b/src/HOL/OrderedGroup.thy Tue Oct 30 08:45:54 2007 +0100
@@ -535,8 +535,21 @@
lemma le_iff_diff_le_0: "a \<le> b \<longleftrightarrow> a - b \<le> 0"
by (simp add: compare_rls)
+lemmas group_simps =
+ add_ac
+ add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2
+ diff_eq_eq eq_diff_eq diff_minus [symmetric] uminus_add_conv_diff
+ diff_less_eq less_diff_eq diff_le_eq le_diff_eq
+
end
+lemmas group_simps =
+ mult_ac
+ add_ac
+ add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2
+ diff_eq_eq eq_diff_eq diff_minus [symmetric] uminus_add_conv_diff
+ diff_less_eq less_diff_eq diff_le_eq le_diff_eq
+
class ordered_ab_semigroup_add =
linorder + pordered_ab_semigroup_add
@@ -565,6 +578,12 @@
qed
qed
+class ordered_ab_group_add =
+ linorder + pordered_ab_group_add
+
+subclass (in ordered_ab_group_add) ordered_cancel_ab_semigroup_add
+ by unfold_locales
+
-- {* FIXME localize the following *}
lemma add_increasing:
@@ -752,6 +771,12 @@
(simp add: le_infI)
qed
+lemma neg_inf_eq_sup: "- inf a b = sup (-a) (-b)"
+ by (simp add: inf_eq_neg_sup)
+
+lemma neg_sup_eq_inf: "- sup a b = inf (-a) (-b)"
+ by (simp add: sup_eq_neg_inf)
+
lemma add_eq_inf_sup: "a + b = sup a b + inf a b"
proof -
have "0 = - inf 0 (a-b) + inf (a-b) 0" by (simp add: inf_commute)
@@ -776,6 +801,21 @@
pprt :: "'a \<Rightarrow> 'a" where
"pprt x = sup x 0"
+lemma pprt_neg: "pprt (- x) = - nprt x"
+proof -
+ have "sup (- x) 0 = sup (- x) (- 0)" unfolding minus_zero ..
+ also have "\<dots> = - inf x 0" unfolding neg_inf_eq_sup ..
+ finally have "sup (- x) 0 = - inf x 0" .
+ then show ?thesis unfolding pprt_def nprt_def .
+qed
+
+lemma nprt_neg: "nprt (- x) = - pprt x"
+proof -
+ from pprt_neg have "pprt (- (- x)) = - nprt (- x)" .
+ then have "pprt x = - nprt (- x)" by simp
+ then show ?thesis by simp
+qed
+
lemma prts: "a = pprt a + nprt a"
by (simp add: pprt_def nprt_def add_eq_inf_sup[symmetric])
@@ -904,38 +944,67 @@
ultimately show ?thesis by blast
qed
+declare neg_inf_eq_sup [simp] neg_sup_eq_inf [simp]
+
+lemma le_minus_self_iff: "a \<le> - a \<longleftrightarrow> a \<le> 0"
+proof -
+ from add_le_cancel_left [of "uminus a" "plus a a" zero]
+ have "(a <= -a) = (a+a <= 0)"
+ by (simp add: add_assoc[symmetric])
+ thus ?thesis by simp
+qed
+
+lemma minus_le_self_iff: "- a \<le> a \<longleftrightarrow> 0 \<le> a"
+proof -
+ from add_le_cancel_left [of "uminus a" zero "plus a a"]
+ have "(-a <= a) = (0 <= a+a)"
+ by (simp add: add_assoc[symmetric])
+ thus ?thesis by simp
+qed
+
+lemma zero_le_iff_zero_nprt: "0 \<le> a \<longleftrightarrow> nprt a = 0"
+ by (simp add: le_iff_inf nprt_def inf_commute)
+
+lemma le_zero_iff_zero_pprt: "a \<le> 0 \<longleftrightarrow> pprt a = 0"
+ by (simp add: le_iff_sup pprt_def sup_commute)
+
+lemma le_zero_iff_pprt_id: "0 \<le> a \<longleftrightarrow> pprt a = a"
+ by (simp add: le_iff_sup pprt_def sup_commute)
+
+lemma zero_le_iff_nprt_id: "a \<le> 0 \<longleftrightarrow> nprt a = a"
+ by (simp add: le_iff_inf nprt_def inf_commute)
+
+lemma pprt_mono [simp, noatp]: "a \<le> b \<Longrightarrow> pprt a \<le> pprt b"
+ by (simp add: le_iff_sup pprt_def sup_ACI sup_assoc [symmetric, of a])
+
+lemma nprt_mono [simp, noatp]: "a \<le> b \<Longrightarrow> nprt a \<le> nprt b"
+ by (simp add: le_iff_inf nprt_def inf_ACI inf_assoc [symmetric, of a])
+
end
lemmas add_sup_inf_distribs = add_inf_distrib_right add_inf_distrib_left add_sup_distrib_right add_sup_distrib_left
-class lordered_ab_group_abs = lordered_ab_group + abs +
- assumes abs_lattice: "\<bar>x\<bar> = sup x (- x)"
+
+class pordered_ab_group_add_abs = pordered_ab_group_add + abs +
+ assumes abs_ge_zero [simp]: "\<bar>a\<bar> \<ge> 0"
+ and abs_ge_self: "a \<le> \<bar>a\<bar>"
+ and abs_of_nonneg [simp]: "0 \<le> a \<Longrightarrow> \<bar>a\<bar> = a"
+ and abs_leI: "a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b"
+ and abs_eq_0 [simp]: "\<bar>a\<bar> = 0 \<longleftrightarrow> a = 0"
+ and abs_minus_cancel [simp]: "\<bar>-a\<bar> = \<bar>a\<bar>"
+ and abs_idempotent [simp]: "\<bar>\<bar>a\<bar>\<bar> = \<bar>a\<bar>"
+ and abs_triangle_ineq: "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
begin
lemma abs_zero [simp]: "\<bar>0\<bar> = 0"
- by (simp add: abs_lattice)
-
-lemma abs_eq_0 [simp]: "\<bar>a\<bar> = 0 \<longleftrightarrow> a = 0"
- by (simp add: abs_lattice)
+ by simp
lemma abs_0_eq [simp, noatp]: "0 = \<bar>a\<bar> \<longleftrightarrow> a = 0"
proof -
- have "(0 = abs a) = (abs a = 0)" by (simp only: eq_ac)
+ have "0 = \<bar>a\<bar> \<longleftrightarrow> \<bar>a\<bar> = 0" by (simp only: eq_ac)
thus ?thesis by simp
qed
-lemma neg_inf_eq_sup [simp]: "- inf a b = sup (-a) (-b)"
- by (simp add: inf_eq_neg_sup)
-
-lemma neg_sup_eq_inf [simp]: "- sup a b = inf (-a) (-b)"
- by (simp del: neg_inf_eq_sup add: sup_eq_neg_inf)
-
-lemma abs_ge_zero [simp]: "0 \<le> \<bar>a\<bar>"
-proof -
- have a: "a \<le> \<bar>a\<bar>" and b: "- a \<le> \<bar>a\<bar>" by (auto simp add: abs_lattice)
- show ?thesis by (rule add_mono [OF a b, simplified])
-qed
-
lemma abs_le_zero_iff [simp]: "\<bar>a\<bar> \<le> 0 \<longleftrightarrow> a = 0"
proof
assume "\<bar>a\<bar> \<le> 0"
@@ -955,20 +1024,11 @@
show ?thesis by (simp add: a)
qed
-lemma abs_ge_self: "a \<le> \<bar>a\<bar>"
- by (auto simp add: abs_lattice)
-
lemma abs_ge_minus_self: "- a \<le> \<bar>a\<bar>"
- by (auto simp add: abs_lattice)
-
-lemma abs_minus_cancel [simp]: "\<bar>-a\<bar> = \<bar>a\<bar>"
- by (simp add: abs_lattice sup_commute)
-
-lemma abs_idempotent [simp]: "\<bar>\<bar>a\<bar>\<bar> = \<bar>a\<bar>"
-apply (simp add: abs_lattice [of "abs a"])
-apply (subst sup_absorb1)
-apply (rule order_trans [of _ zero])
-by auto
+proof -
+ have "- a \<le> \<bar>-a\<bar>" by (rule abs_ge_self)
+ then show ?thesis by simp
+qed
lemma abs_minus_commute:
"\<bar>a - b\<bar> = \<bar>b - a\<bar>"
@@ -978,73 +1038,24 @@
finally show ?thesis .
qed
-lemma abs_prts: "\<bar>a\<bar> = pprt a - nprt a"
+lemma abs_of_pos: "0 < a \<Longrightarrow> \<bar>a\<bar> = a"
+ by (rule abs_of_nonneg, rule less_imp_le)
+
+lemma abs_of_nonpos [simp]:
+ assumes "a \<le> 0"
+ shows "\<bar>a\<bar> = - a"
proof -
- have "0 \<le> \<bar>a\<bar>" by simp
- then have "0 \<le> sup a (- a)" unfolding abs_lattice .
- then have "sup (sup a (- a)) 0 = sup a (- a)" by (rule sup_absorb1)
- then show ?thesis
- by (simp add: add_sup_inf_distribs sup_ACI
- pprt_def nprt_def diff_minus abs_lattice)
+ let ?b = "- a"
+ have "- ?b \<le> 0 \<Longrightarrow> \<bar>- ?b\<bar> = - (- ?b)"
+ unfolding abs_minus_cancel [of "?b"]
+ unfolding neg_le_0_iff_le [of "?b"]
+ unfolding minus_minus by (erule abs_of_nonneg)
+ then show ?thesis using assms by auto
qed
-lemma zero_le_iff_zero_nprt: "0 \<le> a \<longleftrightarrow> nprt a = 0"
- by (simp add: le_iff_inf nprt_def inf_commute)
-
-lemma le_zero_iff_zero_pprt: "a \<le> 0 \<longleftrightarrow> pprt a = 0"
- by (simp add: le_iff_sup pprt_def sup_commute)
-
-lemma le_zero_iff_pprt_id: "0 \<le> a \<longleftrightarrow> pprt a = a"
- by (simp add: le_iff_sup pprt_def sup_commute)
-
-lemma zero_le_iff_nprt_id: "a \<le> 0 \<longleftrightarrow> nprt a = a"
-by (simp add: le_iff_inf nprt_def inf_commute)
-
-lemma pprt_mono [simp, noatp]: "a \<le> b \<Longrightarrow> pprt a \<le> pprt b"
- by (simp add: le_iff_sup pprt_def sup_ACI sup_assoc [symmetric, of a])
-
-lemma nprt_mono [simp, noatp]: "a \<le> b \<Longrightarrow> nprt a \<le> nprt b"
- by (simp add: le_iff_inf nprt_def inf_ACI inf_assoc [symmetric, of a])
-
-lemma pprt_neg: "pprt (- x) = - nprt x"
- by (simp add: pprt_def nprt_def)
-
-lemma nprt_neg: "nprt (- x) = - pprt x"
- by (simp add: pprt_def nprt_def)
-
-lemma abs_of_nonneg [simp]: "0 \<le> a \<Longrightarrow> \<bar>a\<bar> = a"
- by (simp add: iffD1 [OF zero_le_iff_zero_nprt]
- iffD1[OF le_zero_iff_pprt_id] abs_prts)
-
-lemma abs_of_pos: "0 < x \<Longrightarrow> \<bar>x\<bar> = x"
- by (rule abs_of_nonneg, rule less_imp_le)
-
-lemma abs_of_nonpos [simp]: "a \<le> 0 \<Longrightarrow> \<bar>a\<bar> = - a"
- by (simp add: iffD1 [OF le_zero_iff_zero_pprt]
- iffD1 [OF zero_le_iff_nprt_id] abs_prts)
-
-lemma abs_of_neg: "x < 0 \<Longrightarrow> \<bar>x\<bar> = - x"
+lemma abs_of_neg: "a < 0 \<Longrightarrow> \<bar>a\<bar> = - a"
by (rule abs_of_nonpos, rule less_imp_le)
-lemma abs_leI: "a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b"
- by (simp add: abs_lattice le_supI)
-
-lemma le_minus_self_iff: "a \<le> - a \<longleftrightarrow> a \<le> 0"
-proof -
- from add_le_cancel_left [of "uminus a" "plus a a" zero]
- have "(a <= -a) = (a+a <= 0)"
- by (simp add: add_assoc[symmetric])
- thus ?thesis by simp
-qed
-
-lemma minus_le_self_iff: "- a \<le> a \<longleftrightarrow> 0 \<le> a"
-proof -
- from add_le_cancel_left [of "uminus a" zero "plus a a"]
- have "(-a <= a) = (0 <= a+a)"
- by (simp add: add_assoc[symmetric])
- thus ?thesis by simp
-qed
-
lemma abs_le_D1: "\<bar>a\<bar> \<le> b \<Longrightarrow> a \<le> b"
by (insert abs_ge_self, blast intro: order_trans)
@@ -1054,20 +1065,6 @@
lemma abs_le_iff: "\<bar>a\<bar> \<le> b \<longleftrightarrow> a \<le> b \<and> - a \<le> b"
by (blast intro: abs_leI dest: abs_le_D1 abs_le_D2)
-lemma abs_triangle_ineq: "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
-proof -
- have g:"abs a + abs b = sup (a+b) (sup (-a-b) (sup (-a+b) (a + (-b))))" (is "_=sup ?m ?n")
- by (simp add: abs_lattice add_sup_inf_distribs sup_ACI diff_minus)
- have a:"a+b <= sup ?m ?n" by (simp)
- have b:"-a-b <= ?n" by (simp)
- have c:"?n <= sup ?m ?n" by (simp)
- from b c have d: "-a-b <= sup ?m ?n" by(rule order_trans)
- have e:"-a-b = -(a+b)" by (simp add: diff_minus)
- from a d e have "abs(a+b) <= sup ?m ?n"
- by (drule_tac abs_leI, auto)
- with g[symmetric] show ?thesis by simp
-qed
-
lemma abs_triangle_ineq2: "\<bar>a\<bar> - \<bar>b\<bar> \<le> \<bar>a - b\<bar>"
apply (simp add: compare_rls)
apply (subgoal_tac "abs a = abs (plus (minus a b) b)")
@@ -1102,7 +1099,7 @@
finally show ?thesis .
qed
-lemma abs_add_abs[simp]:
+lemma abs_add_abs [simp]:
"\<bar>\<bar>a\<bar> + \<bar>b\<bar>\<bar> = \<bar>a\<bar> + \<bar>b\<bar>" (is "?L = ?R")
proof (rule antisym)
show "?L \<ge> ?R" by(rule abs_ge_self)
@@ -1114,6 +1111,87 @@
end
+
+class lordered_ab_group_abs = lordered_ab_group + abs +
+ assumes abs_lattice: "\<bar>a\<bar> = sup a (- a)"
+begin
+
+lemma abs_prts: "\<bar>a\<bar> = pprt a - nprt a"
+proof -
+ have "0 \<le> \<bar>a\<bar>"
+ proof -
+ have a: "a \<le> \<bar>a\<bar>" and b: "- a \<le> \<bar>a\<bar>" by (auto simp add: abs_lattice)
+ show ?thesis by (rule add_mono [OF a b, simplified])
+ qed
+ then have "0 \<le> sup a (- a)" unfolding abs_lattice .
+ then have "sup (sup a (- a)) 0 = sup a (- a)" by (rule sup_absorb1)
+ then show ?thesis
+ by (simp add: add_sup_inf_distribs sup_ACI
+ pprt_def nprt_def diff_minus abs_lattice)
+qed
+
+subclass pordered_ab_group_add_abs
+proof -
+ have abs_ge_zero [simp]: "\<And>a. 0 \<le> \<bar>a\<bar>"
+ proof -
+ fix a b
+ have a: "a \<le> \<bar>a\<bar>" and b: "- a \<le> \<bar>a\<bar>" by (auto simp add: abs_lattice)
+ show "0 \<le> \<bar>a\<bar>" by (rule add_mono [OF a b, simplified])
+ qed
+ have abs_leI: "\<And>a b. a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b"
+ by (simp add: abs_lattice le_supI)
+ show ?thesis
+ proof unfold_locales
+ fix a
+ show "0 \<le> \<bar>a\<bar>" by simp
+ next
+ fix a
+ show "a \<le> \<bar>a\<bar>"
+ by (auto simp add: abs_lattice)
+ next
+ fix a
+ show "\<bar>a\<bar> = 0 \<longleftrightarrow> a = 0"
+ by (simp add: abs_lattice)
+ next
+ fix a
+ show "\<bar>-a\<bar> = \<bar>a\<bar>"
+ by (simp add: abs_lattice sup_commute)
+ next
+ fix a
+ show "\<bar>\<bar>a\<bar>\<bar> = \<bar>a\<bar>"
+ apply (simp add: abs_lattice [of "abs a"])
+ apply (subst sup_absorb1)
+ apply (rule order_trans [of _ zero])
+ apply auto
+ done
+ next
+ fix a
+ show "0 \<le> a \<Longrightarrow> \<bar>a\<bar> = a"
+ by (simp add: iffD1 [OF zero_le_iff_zero_nprt]
+ iffD1[OF le_zero_iff_pprt_id] abs_prts)
+ next
+ fix a b
+ show "a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b" by (erule abs_leI)
+ next
+ fix a b
+ show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
+ proof -
+ have g:"abs a + abs b = sup (a+b) (sup (-a-b) (sup (-a+b) (a + (-b))))" (is "_=sup ?m ?n")
+ by (simp add: abs_lattice add_sup_inf_distribs sup_ACI diff_minus)
+ have a:"a+b <= sup ?m ?n" by (simp)
+ have b:"-a-b <= ?n" by (simp)
+ have c:"?n <= sup ?m ?n" by (simp)
+ from b c have d: "-a-b <= sup ?m ?n" by(rule order_trans)
+ have e:"-a-b = -(a+b)" by (simp add: diff_minus)
+ from a d e have "abs(a+b) <= sup ?m ?n"
+ by (drule_tac abs_leI, auto)
+ with g[symmetric] show ?thesis by simp
+ qed
+ qed auto
+qed
+
+end
+
lemma sup_eq_if:
fixes a :: "'a\<Colon>{lordered_ab_group, linorder}"
shows "sup a (- a) = (if a < 0 then - a else a)"
@@ -1168,13 +1246,6 @@
apply (simp_all add: prems)
done
-lemmas group_simps =
- mult_ac
- add_ac
- add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2
- diff_eq_eq eq_diff_eq diff_minus [symmetric] uminus_add_conv_diff
- diff_less_eq less_diff_eq diff_le_eq le_diff_eq
-
lemma estimate_by_abs:
"a + b <= (c::'a::lordered_ab_group_abs) \<Longrightarrow> a <= c + abs b"
proof -