--- a/src/HOL/Library/Lattice_Algebras.thy Tue Aug 27 23:21:12 2013 +0200
+++ b/src/HOL/Library/Lattice_Algebras.thy Tue Aug 27 23:54:23 2013 +0200
@@ -9,20 +9,19 @@
class semilattice_inf_ab_group_add = ordered_ab_group_add + semilattice_inf
begin
-lemma add_inf_distrib_left:
- "a + inf b c = inf (a + b) (a + c)"
-apply (rule antisym)
-apply (simp_all add: le_infI)
-apply (rule add_le_imp_le_left [of "uminus a"])
-apply (simp only: add_assoc [symmetric], simp)
-apply rule
-apply (rule add_le_imp_le_left[of "a"], simp only: add_assoc[symmetric], simp)+
-done
+lemma add_inf_distrib_left: "a + inf b c = inf (a + b) (a + c)"
+ apply (rule antisym)
+ apply (simp_all add: le_infI)
+ apply (rule add_le_imp_le_left [of "uminus a"])
+ apply (simp only: add_assoc [symmetric], simp)
+ apply rule
+ apply (rule add_le_imp_le_left[of "a"], simp only: add_assoc[symmetric], simp)+
+ done
-lemma add_inf_distrib_right:
- "inf a b + c = inf (a + c) (b + c)"
+lemma add_inf_distrib_right: "inf a b + c = inf (a + c) (b + c)"
proof -
- have "c + inf a b = inf (c+a) (c+b)" by (simp add: add_inf_distrib_left)
+ have "c + inf a b = inf (c+a) (c+b)"
+ by (simp add: add_inf_distrib_left)
thus ?thesis by (simp add: add_commute)
qed
@@ -31,19 +30,17 @@
class semilattice_sup_ab_group_add = ordered_ab_group_add + semilattice_sup
begin
-lemma add_sup_distrib_left:
- "a + sup b c = sup (a + b) (a + c)"
-apply (rule antisym)
-apply (rule add_le_imp_le_left [of "uminus a"])
-apply (simp only: add_assoc[symmetric], simp)
-apply rule
-apply (rule add_le_imp_le_left [of "a"], simp only: add_assoc[symmetric], simp)+
-apply (rule le_supI)
-apply (simp_all)
-done
+lemma add_sup_distrib_left: "a + sup b c = sup (a + b) (a + c)"
+ apply (rule antisym)
+ apply (rule add_le_imp_le_left [of "uminus a"])
+ apply (simp only: add_assoc[symmetric], simp)
+ apply rule
+ apply (rule add_le_imp_le_left [of "a"], simp only: add_assoc[symmetric], simp)+
+ apply (rule le_supI)
+ apply (simp_all)
+ done
-lemma add_sup_distrib_right:
- "sup a b + c = sup (a+c) (b+c)"
+lemma add_sup_distrib_right: "sup a b + c = sup (a+c) (b+c)"
proof -
have "c + sup a b = sup (c+a) (c+b)" by (simp add: add_sup_distrib_left)
thus ?thesis by (simp add: add_commute)
@@ -57,69 +54,61 @@
subclass semilattice_inf_ab_group_add ..
subclass semilattice_sup_ab_group_add ..
-lemmas add_sup_inf_distribs = add_inf_distrib_right add_inf_distrib_left add_sup_distrib_right add_sup_distrib_left
+lemmas add_sup_inf_distribs =
+ add_inf_distrib_right add_inf_distrib_left add_sup_distrib_right add_sup_distrib_left
lemma inf_eq_neg_sup: "inf a b = - sup (-a) (-b)"
proof (rule inf_unique)
- fix a b :: 'a
+ fix a b c :: 'a
show "- sup (-a) (-b) \<le> a"
by (rule add_le_imp_le_right [of _ "sup (uminus a) (uminus b)"])
(simp, simp add: add_sup_distrib_left)
-next
- fix a b :: 'a
show "- sup (-a) (-b) \<le> b"
by (rule add_le_imp_le_right [of _ "sup (uminus a) (uminus b)"])
(simp, simp add: add_sup_distrib_left)
-next
- fix a b c :: 'a
assume "a \<le> b" "a \<le> c"
- then show "a \<le> - sup (-b) (-c)" by (subst neg_le_iff_le [symmetric])
- (simp add: le_supI)
+ then show "a \<le> - sup (-b) (-c)"
+ by (subst neg_le_iff_le [symmetric]) (simp add: le_supI)
qed
-
+
lemma sup_eq_neg_inf: "sup a b = - inf (-a) (-b)"
proof (rule sup_unique)
- fix a b :: 'a
+ fix a b c :: 'a
show "a \<le> - inf (-a) (-b)"
by (rule add_le_imp_le_right [of _ "inf (uminus a) (uminus b)"])
(simp, simp add: add_inf_distrib_left)
-next
- fix a b :: 'a
show "b \<le> - inf (-a) (-b)"
by (rule add_le_imp_le_right [of _ "inf (uminus a) (uminus b)"])
(simp, simp add: add_inf_distrib_left)
-next
- fix a b c :: 'a
assume "a \<le> c" "b \<le> c"
- then show "- inf (-a) (-b) \<le> c" by (subst neg_le_iff_le [symmetric])
- (simp add: le_infI)
+ then show "- inf (-a) (-b) \<le> c" by (subst neg_le_iff_le [symmetric]) (simp add: le_infI)
qed
lemma neg_inf_eq_sup: "- inf a b = sup (-a) (-b)"
-by (simp add: inf_eq_neg_sup)
+ 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)
+ 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)
- hence "0 = sup 0 (b-a) + inf (a-b) 0" by (simp add: inf_eq_neg_sup)
+ have "0 = - inf 0 (a-b) + inf (a-b) 0"
+ by (simp add: inf_commute)
+ hence "0 = sup 0 (b-a) + inf (a-b) 0"
+ by (simp add: inf_eq_neg_sup)
hence "0 = (-a + sup a b) + (inf a b + (-b))"
- by (simp add: add_sup_distrib_left add_inf_distrib_right)
- (simp add: algebra_simps)
+ by (simp add: add_sup_distrib_left add_inf_distrib_right) (simp add: algebra_simps)
thus ?thesis by (simp add: algebra_simps)
qed
+
subsection {* Positive Part, Negative Part, Absolute Value *}
-definition
- nprt :: "'a \<Rightarrow> 'a" where
- "nprt x = inf x 0"
+definition nprt :: "'a \<Rightarrow> 'a"
+ where "nprt x = inf x 0"
-definition
- pprt :: "'a \<Rightarrow> 'a" where
- "pprt x = sup x 0"
+definition pprt :: "'a \<Rightarrow> 'a"
+ where "pprt x = sup x 0"
lemma pprt_neg: "pprt (- x) = - nprt x"
proof -
@@ -137,27 +126,29 @@
qed
lemma prts: "a = pprt a + nprt a"
-by (simp add: pprt_def nprt_def add_eq_inf_sup[symmetric])
+ by (simp add: pprt_def nprt_def add_eq_inf_sup[symmetric])
lemma zero_le_pprt[simp]: "0 \<le> pprt a"
-by (simp add: pprt_def)
+ by (simp add: pprt_def)
lemma nprt_le_zero[simp]: "nprt a \<le> 0"
-by (simp add: nprt_def)
+ by (simp add: nprt_def)
lemma le_eq_neg: "a \<le> - b \<longleftrightarrow> a + b \<le> 0" (is "?l = ?r")
-proof -
- have a: "?l \<longrightarrow> ?r"
- apply (auto)
+proof
+ assume ?l
+ then show ?r
+ apply -
apply (rule add_le_imp_le_right[of _ "uminus b" _])
apply (simp add: add_assoc)
done
- have b: "?r \<longrightarrow> ?l"
- apply (auto)
+next
+ assume ?r
+ then show ?l
+ apply -
apply (rule add_le_imp_le_right[of _ "b" _])
- apply (simp)
+ apply simp
done
- from a b show ?thesis by blast
qed
lemma pprt_0[simp]: "pprt 0 = 0" by (simp add: pprt_def)
@@ -181,7 +172,7 @@
fix a::'a
assume hyp: "sup a (-a) = 0"
hence "sup a (-a) + a = a" by (simp)
- hence "sup (a+a) 0 = a" by (simp add: add_sup_distrib_right)
+ hence "sup (a+a) 0 = a" by (simp add: add_sup_distrib_right)
hence "sup (a+a) 0 <= a" by (simp)
hence "0 <= a" by (blast intro: order_trans inf_sup_ord)
}
@@ -192,16 +183,22 @@
qed
lemma inf_0_imp_0: "inf a (-a) = 0 \<Longrightarrow> a = 0"
-apply (simp add: inf_eq_neg_sup)
-apply (simp add: sup_commute)
-apply (erule sup_0_imp_0)
-done
+ apply (simp add: inf_eq_neg_sup)
+ apply (simp add: sup_commute)
+ apply (erule sup_0_imp_0)
+ done
lemma inf_0_eq_0 [simp, no_atp]: "inf a (- a) = 0 \<longleftrightarrow> a = 0"
-by (rule, erule inf_0_imp_0) simp
+ apply rule
+ apply (erule inf_0_imp_0)
+ apply simp
+ done
lemma sup_0_eq_0 [simp, no_atp]: "sup a (- a) = 0 \<longleftrightarrow> a = 0"
-by (rule, erule sup_0_imp_0) simp
+ apply rule
+ apply (erule sup_0_imp_0)
+ apply simp
+ done
lemma zero_le_double_add_iff_zero_le_single_add [simp]:
"0 \<le> a + a \<longleftrightarrow> 0 \<le> a"
@@ -218,39 +215,48 @@
show "0 <= a + a" by (simp add: add_mono[OF a a, simplified])
qed
-lemma double_zero [simp]:
- "a + a = 0 \<longleftrightarrow> a = 0"
+lemma double_zero [simp]: "a + a = 0 \<longleftrightarrow> a = 0"
proof
assume assm: "a + a = 0"
then have "a + a + - a = - a" by simp
then have "a + (a + - a) = - a" by (simp only: add_assoc)
then have a: "- a = a" by simp
- show "a = 0" apply (rule antisym)
- apply (unfold neg_le_iff_le [symmetric, of a])
- unfolding a apply simp
- unfolding zero_le_double_add_iff_zero_le_single_add [symmetric, of a]
- unfolding assm unfolding le_less apply simp_all done
+ show "a = 0"
+ apply (rule antisym)
+ apply (unfold neg_le_iff_le [symmetric, of a])
+ unfolding a
+ apply simp
+ unfolding zero_le_double_add_iff_zero_le_single_add [symmetric, of a]
+ unfolding assm
+ unfolding le_less
+ apply simp_all
+ done
next
- assume "a = 0" then show "a + a = 0" by simp
+ assume "a = 0"
+ then show "a + a = 0" by simp
qed
-lemma zero_less_double_add_iff_zero_less_single_add [simp]:
- "0 < a + a \<longleftrightarrow> 0 < a"
+lemma zero_less_double_add_iff_zero_less_single_add [simp]: "0 < a + a \<longleftrightarrow> 0 < a"
proof (cases "a = 0")
- case True then show ?thesis by auto
+ case True
+ then show ?thesis by auto
next
- case False then show ?thesis (*FIXME tune proof*)
- unfolding less_le apply simp apply rule
- apply clarify
- apply rule
- apply assumption
- apply (rule notI)
- unfolding double_zero [symmetric, of a] apply simp
- done
+ case False
+ then show ?thesis
+ unfolding less_le
+ apply simp
+ apply rule
+ apply clarify
+ apply rule
+ apply assumption
+ apply (rule notI)
+ unfolding double_zero [symmetric, of a]
+ apply simp
+ done
qed
lemma double_add_le_zero_iff_single_add_le_zero [simp]:
- "a + a \<le> 0 \<longleftrightarrow> a \<le> 0"
+ "a + a \<le> 0 \<longleftrightarrow> a \<le> 0"
proof -
have "a + a \<le> 0 \<longleftrightarrow> 0 \<le> - (a + a)" by (subst le_minus_iff, simp)
moreover have "\<dots> \<longleftrightarrow> a \<le> 0" by simp
@@ -270,7 +276,7 @@
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)"
+ have "(a <= -a) = (a+a <= 0)"
by (simp add: add_assoc[symmetric])
thus ?thesis by simp
qed
@@ -278,28 +284,28 @@
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)"
+ 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"
-unfolding le_iff_inf by (simp add: nprt_def inf_commute)
+ unfolding le_iff_inf by (simp add: nprt_def inf_commute)
lemma le_zero_iff_zero_pprt: "a \<le> 0 \<longleftrightarrow> pprt a = 0"
-unfolding le_iff_sup by (simp add: pprt_def sup_commute)
+ unfolding le_iff_sup by (simp add: pprt_def sup_commute)
lemma le_zero_iff_pprt_id: "0 \<le> a \<longleftrightarrow> pprt a = a"
-unfolding le_iff_sup by (simp add: pprt_def sup_commute)
+ unfolding le_iff_sup by (simp add: pprt_def sup_commute)
lemma zero_le_iff_nprt_id: "a \<le> 0 \<longleftrightarrow> nprt a = a"
-unfolding le_iff_inf by (simp add: nprt_def inf_commute)
+ unfolding le_iff_inf by (simp add: nprt_def inf_commute)
lemma pprt_mono [simp, no_atp]: "a \<le> b \<Longrightarrow> pprt a \<le> pprt b"
-unfolding le_iff_sup by (simp add: pprt_def sup_aci sup_assoc [symmetric, of a])
+ unfolding le_iff_sup by (simp add: pprt_def sup_aci sup_assoc [symmetric, of a])
lemma nprt_mono [simp, no_atp]: "a \<le> b \<Longrightarrow> nprt a \<le> nprt b"
-unfolding le_iff_inf by (simp add: nprt_def inf_aci inf_assoc [symmetric, of a])
+ unfolding le_iff_inf by (simp add: nprt_def inf_aci inf_assoc [symmetric, of a])
end
@@ -320,8 +326,7 @@
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)
+ by (simp add: add_sup_inf_distribs sup_aci pprt_def nprt_def diff_minus abs_lattice)
qed
subclass ordered_ab_group_add_abs
@@ -329,8 +334,10 @@
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])
+ 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)
@@ -340,18 +347,25 @@
by (auto simp add: abs_lattice)
show "\<bar>-a\<bar> = \<bar>a\<bar>"
by (simp add: abs_lattice sup_commute)
- show "a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b" by (fact abs_leI)
+ {
+ assume "a \<le> b"
+ then show "- a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b"
+ by (rule abs_leI)
+ }
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 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)
+ from a d e have "abs(a+b) <= sup ?m ?n"
+ apply -
+ apply (drule abs_leI)
+ apply auto
+ done
with g[symmetric] show ?thesis by simp
qed
qed
@@ -370,10 +384,10 @@
lemma abs_if_lattice:
fixes a :: "'a\<Colon>{lattice_ab_group_add_abs, linorder}"
shows "\<bar>a\<bar> = (if a < 0 then - a else a)"
-by auto
+ by auto
lemma estimate_by_abs:
- "a + b <= (c::'a::lattice_ab_group_add_abs) \<Longrightarrow> a <= c + abs b"
+ "a + b <= (c::'a::lattice_ab_group_add_abs) \<Longrightarrow> a <= c + abs b"
proof -
assume "a+b <= c"
then have "a <= c+(-b)" by (simp add: algebra_simps)
@@ -390,7 +404,7 @@
end
-lemma abs_le_mult: "abs (a * b) \<le> (abs a) * (abs (b::'a::lattice_ring))"
+lemma abs_le_mult: "abs (a * b) \<le> (abs a) * (abs (b::'a::lattice_ring))"
proof -
let ?x = "pprt a * pprt b - pprt a * nprt b - nprt a * pprt b + nprt a * nprt b"
let ?y = "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
@@ -398,11 +412,11 @@
by (simp only: abs_prts[of a] abs_prts[of b] algebra_simps)
{
fix u v :: 'a
- have bh: "\<lbrakk>u = a; v = b\<rbrakk> \<Longrightarrow>
- u * v = pprt a * pprt b + pprt a * nprt b +
+ have bh: "\<lbrakk>u = a; v = b\<rbrakk> \<Longrightarrow>
+ u * v = pprt a * pprt b + pprt a * nprt b +
nprt a * pprt b + nprt a * nprt b"
apply (subst prts[of u], subst prts[of v])
- apply (simp add: algebra_simps)
+ apply (simp add: algebra_simps)
done
}
note b = this[OF refl[of a] refl[of b]]
@@ -432,7 +446,7 @@
show "abs (a*b) = abs a * abs b"
proof -
have s: "(0 <= a*b) | (a*b <= 0)"
- apply (auto)
+ apply (auto)
apply (rule_tac split_mult_pos_le)
apply (rule_tac contrapos_np[of "a*b <= 0"])
apply (simp)
@@ -448,8 +462,8 @@
then show ?thesis
apply (simp_all add: mulprts abs_prts)
apply (insert a)
- apply (auto simp add:
- algebra_simps
+ apply (auto simp add:
+ algebra_simps
iffD1[OF zero_le_iff_zero_nprt] iffD1[OF le_zero_iff_zero_pprt]
iffD1[OF le_zero_iff_pprt_id] iffD1[OF zero_le_iff_nprt_id])
apply(drule (1) mult_nonneg_nonpos[of a b], simp)
@@ -470,15 +484,14 @@
qed
lemma mult_le_prts:
- assumes
- "a1 <= (a::'a::lattice_ring)"
- "a <= a2"
- "b1 <= b"
- "b <= b2"
- shows
- "a * b <= pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1"
-proof -
- have "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
+ assumes "a1 <= (a::'a::lattice_ring)"
+ and "a <= a2"
+ and "b1 <= b"
+ and "b <= b2"
+ shows "a * b <=
+ pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1"
+proof -
+ have "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
apply (subst prts[symmetric])+
apply simp
done
@@ -496,7 +509,7 @@
by simp
qed
moreover have "nprt a * pprt b <= nprt a2 * pprt b1"
- proof -
+ proof -
have "nprt a * pprt b <= nprt a2 * pprt b"
by (simp add: mult_right_mono assms)
moreover have "nprt a2 * pprt b <= nprt a2 * pprt b1"
@@ -514,29 +527,33 @@
by simp
qed
ultimately show ?thesis
- by - (rule add_mono | simp)+
+ apply -
+ apply (rule add_mono | simp)+
+ done
qed
lemma mult_ge_prts:
- assumes
- "a1 <= (a::'a::lattice_ring)"
- "a <= a2"
- "b1 <= b"
- "b <= b2"
- shows
- "a * b >= nprt a1 * pprt b2 + nprt a2 * nprt b2 + pprt a1 * pprt b1 + pprt a2 * nprt b1"
-proof -
- from assms have a1:"- a2 <= -a" by auto
- from assms have a2: "-a <= -a1" by auto
- from mult_le_prts[of "-a2" "-a" "-a1" "b1" b "b2", OF a1 a2 assms(3) assms(4), simplified nprt_neg pprt_neg]
- have le: "- (a * b) <= - nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1" by simp
+ assumes "a1 <= (a::'a::lattice_ring)"
+ and "a <= a2"
+ and "b1 <= b"
+ and "b <= b2"
+ shows "a * b >=
+ nprt a1 * pprt b2 + nprt a2 * nprt b2 + pprt a1 * pprt b1 + pprt a2 * nprt b1"
+proof -
+ from assms have a1:"- a2 <= -a"
+ by auto
+ from assms have a2: "-a <= -a1"
+ by auto
+ from mult_le_prts[of "-a2" "-a" "-a1" "b1" b "b2", OF a1 a2 assms(3) assms(4), simplified nprt_neg pprt_neg]
+ have le: "- (a * b) <= - nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1"
+ by simp
then have "-(- nprt a1 * pprt b2 + - nprt a2 * nprt b2 + - pprt a1 * pprt b1 + - pprt a2 * nprt b1) <= a * b"
by (simp only: minus_le_iff)
then show ?thesis by simp
qed
instance int :: lattice_ring
-proof
+proof
fix k :: int
show "abs k = sup k (- k)"
by (auto simp add: sup_int_def)