--- a/src/HOL/Library/Lattice_Algebras.thy Thu Mar 20 12:43:48 2014 +0000
+++ b/src/HOL/Library/Lattice_Algebras.thy Thu Mar 20 15:38:49 2014 +0100
@@ -18,9 +18,10 @@
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)"
+ have "c + inf a b = inf (c + a) (c + b)"
by (simp add: add_inf_distrib_left)
- thus ?thesis by (simp add: add_commute)
+ then show ?thesis
+ by (simp add: add_commute)
qed
end
@@ -37,10 +38,12 @@
apply (rule add_le_imp_le_left [of "a"], simp only: add_assoc[symmetric], simp)+
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)
+ have "c + sup a b = sup (c+a) (c+b)"
+ by (simp add: add_sup_distrib_left)
+ then show ?thesis
+ by (simp add: add_commute)
qed
end
@@ -54,10 +57,10 @@
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)"
+lemma inf_eq_neg_sup: "inf a b = - sup (- a) (- b)"
proof (rule inf_unique)
fix a b c :: 'a
- show "- sup (-a) (-b) \<le> 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)
show "- sup (-a) (-b) \<le> b"
@@ -68,26 +71,27 @@
by (subst neg_le_iff_le [symmetric]) (simp add: le_supI)
qed
-lemma sup_eq_neg_inf: "sup a b = - inf (-a) (-b)"
+lemma sup_eq_neg_inf: "sup a b = - inf (- a) (- b)"
proof (rule sup_unique)
fix a b c :: 'a
- show "a \<le> - inf (-a) (-b)"
+ 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)
- show "b \<le> - inf (-a) (-b)"
+ 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)
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)"
+lemma neg_inf_eq_sup: "- inf a b = sup (- a) (- b)"
by (simp add: inf_eq_neg_sup)
lemma diff_inf_eq_sup: "a - inf b c = a + sup (- b) (- c)"
using neg_inf_eq_sup [of b c, symmetric] by simp
-lemma neg_sup_eq_inf: "- sup a b = inf (-a) (-b)"
+lemma neg_sup_eq_inf: "- sup a b = inf (- a) (- b)"
by (simp add: sup_eq_neg_inf)
lemma diff_sup_eq_inf: "a - sup b c = a + inf (- b) (- c)"
@@ -95,13 +99,14 @@
lemma add_eq_inf_sup: "a + b = sup a b + inf a b"
proof -
- have "0 = - inf 0 (a-b) + inf (a-b) 0"
+ 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"
+ then have "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))"
+ then have "0 = (- a + sup a b) + (inf a b + (- b))"
by (simp only: add_sup_distrib_left add_inf_distrib_right) simp
- then show ?thesis by (simp add: algebra_simps)
+ then show ?thesis
+ by (simp add: algebra_simps)
qed
@@ -115,10 +120,13 @@
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 ..
+ 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 .
+ then show ?thesis
+ unfolding pprt_def nprt_def .
qed
lemma nprt_neg: "nprt (- x) = - pprt x"
@@ -172,20 +180,26 @@
lemma sup_0_imp_0: "sup a (- a) = 0 \<Longrightarrow> a = 0"
proof -
{
- 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)
- hence "0 <= a" by (blast intro: order_trans inf_sup_ord)
+ fix a :: 'a
+ assume hyp: "sup a (- a) = 0"
+ then have "sup a (- a) + a = a"
+ by simp
+ then have "sup (a + a) 0 = a"
+ by (simp add: add_sup_distrib_right)
+ then have "sup (a + a) 0 \<le> a"
+ by simp
+ then have "0 \<le> a"
+ by (blast intro: order_trans inf_sup_ord)
}
note p = this
assume hyp:"sup a (-a) = 0"
- hence hyp2:"sup (-a) (-(-a)) = 0" by (simp add: sup_commute)
- from p[OF hyp] p[OF hyp2] show "a = 0" by simp
+ then have hyp2:"sup (-a) (-(-a)) = 0"
+ by (simp add: sup_commute)
+ from p[OF hyp] p[OF hyp2] show "a = 0"
+ by simp
qed
-lemma inf_0_imp_0: "inf a (-a) = 0 \<Longrightarrow> a = 0"
+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)
@@ -206,24 +220,32 @@
lemma zero_le_double_add_iff_zero_le_single_add [simp]:
"0 \<le> a + a \<longleftrightarrow> 0 \<le> a"
proof
- assume "0 <= a + a"
- hence a:"inf (a+a) 0 = 0" by (simp add: inf_commute inf_absorb1)
- have "(inf a 0)+(inf a 0) = inf (inf (a+a) 0) a" (is "?l=_")
+ assume "0 \<le> a + a"
+ then have a: "inf (a + a) 0 = 0"
+ by (simp add: inf_commute inf_absorb1)
+ have "inf a 0 + inf a 0 = inf (inf (a + a) 0) a" (is "?l=_")
by (simp add: add_sup_inf_distribs inf_aci)
- hence "?l = 0 + inf a 0" by (simp add: a, simp add: inf_commute)
- hence "inf a 0 = 0" by (simp only: add_right_cancel)
- then show "0 <= a" unfolding le_iff_inf by (simp add: inf_commute)
+ then have "?l = 0 + inf a 0"
+ by (simp add: a, simp add: inf_commute)
+ then have "inf a 0 = 0"
+ by (simp only: add_right_cancel)
+ then show "0 \<le> a"
+ unfolding le_iff_inf by (simp add: inf_commute)
next
- assume a: "0 <= a"
- show "0 <= a + a" by (simp add: add_mono[OF a a, simplified])
+ assume a: "0 \<le> a"
+ show "0 \<le> a + a"
+ by (simp add: add_mono[OF a a, simplified])
qed
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
+ 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])
@@ -236,7 +258,8 @@
done
next
assume "a = 0"
- then show "a + a = 0" by simp
+ 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"
@@ -261,19 +284,23 @@
lemma double_add_le_zero_iff_single_add_le_zero [simp]:
"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)
+ 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 only: minus_add_distrib zero_le_double_add_iff_zero_le_single_add) simp
- ultimately show ?thesis by blast
+ ultimately show ?thesis
+ by blast
qed
lemma double_add_less_zero_iff_single_less_zero [simp]:
"a + a < 0 \<longleftrightarrow> a < 0"
proof -
- have "a + a < 0 \<longleftrightarrow> 0 < - (a + a)" by (subst less_minus_iff, simp)
+ have "a + a < 0 \<longleftrightarrow> 0 < - (a + a)"
+ by (subst less_minus_iff) simp
moreover have "\<dots> \<longleftrightarrow> a < 0"
by (simp only: minus_add_distrib zero_less_double_add_iff_zero_less_single_add) simp
- ultimately show ?thesis by blast
+ ultimately show ?thesis
+ by blast
qed
declare neg_inf_eq_sup [simp] neg_sup_eq_inf [simp] diff_inf_eq_sup [simp] diff_sup_eq_inf [simp]
@@ -281,17 +308,19 @@
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 \<le> - a \<longleftrightarrow> a + a \<le> 0"
by (simp add: add_assoc[symmetric])
- thus ?thesis by simp
+ then show ?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)"
+ have "- a \<le> a \<longleftrightarrow> 0 \<le> a + a"
by (simp add: add_assoc[symmetric])
- thus ?thesis by simp
+ then show ?thesis
+ by simp
qed
lemma zero_le_iff_zero_nprt: "0 \<le> a \<longleftrightarrow> nprt a = 0"
@@ -314,7 +343,8 @@
end
-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
class lattice_ab_group_add_abs = lattice_ab_group_add + abs +
@@ -325,11 +355,15 @@
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])
+ 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 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 ac_simps pprt_def nprt_def abs_lattice)
qed
@@ -347,7 +381,8 @@
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)
fix a b
- show "0 \<le> \<bar>a\<bar>" by simp
+ show "0 \<le> \<bar>a\<bar>"
+ by simp
show "a \<le> \<bar>a\<bar>"
by (auto simp add: abs_lattice)
show "\<bar>-a\<bar> = \<bar>a\<bar>"
@@ -359,14 +394,20 @@
}
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")
+ have g: "\<bar>a\<bar> + \<bar>b\<bar> = 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 ac_simps)
- 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
- from a d e have "abs(a+b) <= sup ?m ?n"
+ have a: "a + b \<le> sup ?m ?n"
+ by simp
+ have b: "- a - b \<le> ?n"
+ by simp
+ have c: "?n \<le> sup ?m ?n"
+ by simp
+ from b c have d: "- a - b \<le> sup ?m ?n"
+ by (rule order_trans)
+ have e: "- a - b = - (a + b)"
+ by simp
+ from a d e have "\<bar>a + b\<bar> \<le> sup ?m ?n"
apply -
apply (drule abs_leI)
apply (simp_all only: algebra_simps ac_simps minus_add)
@@ -379,7 +420,7 @@
end
lemma sup_eq_if:
- fixes a :: "'a\<Colon>{lattice_ab_group_add, linorder}"
+ fixes a :: "'a::{lattice_ab_group_add, linorder}"
shows "sup a (- a) = (if a < 0 then - a else a)"
proof -
note add_le_cancel_right [of a a "- a", symmetric, simplified]
@@ -388,18 +429,23 @@
qed
lemma abs_if_lattice:
- fixes a :: "'a\<Colon>{lattice_ab_group_add_abs, linorder}"
+ fixes a :: "'a::{lattice_ab_group_add_abs, linorder}"
shows "\<bar>a\<bar> = (if a < 0 then - a else a)"
by auto
lemma estimate_by_abs:
- "a + b <= (c::'a::lattice_ab_group_add_abs) \<Longrightarrow> a <= c + abs b"
+ fixes a b c :: "'a::lattice_ab_group_add_abs"
+ shows "a + b \<le> c \<Longrightarrow> a \<le> c + \<bar>b\<bar>"
proof -
- assume "a+b <= c"
- then have "a <= c+(-b)" by (simp add: algebra_simps)
- have "(-b) <= abs b" by (rule abs_ge_minus_self)
- then have "c + (- b) \<le> c + \<bar>b\<bar>" by (rule add_left_mono)
- with `a \<le> c + (- b)` show ?thesis by (rule order_trans)
+ assume "a + b \<le> c"
+ then have "a \<le> c + (- b)"
+ by (simp add: algebra_simps)
+ have "- b \<le> \<bar>b\<bar>"
+ by (rule abs_ge_minus_self)
+ then have "c + (- b) \<le> c + \<bar>b\<bar>"
+ by (rule add_left_mono)
+ with `a \<le> c + (- b)` show ?thesis
+ by (rule order_trans)
qed
class lattice_ring = ordered_ring + lattice_ab_group_add_abs
@@ -410,15 +456,17 @@
end
-lemma abs_le_mult: "abs (a * b) \<le> (abs a) * (abs (b::'a::lattice_ring))"
+lemma abs_le_mult:
+ fixes a b :: "'a::lattice_ring"
+ shows "\<bar>a * b\<bar> \<le> \<bar>a\<bar> * \<bar>b\<bar>"
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"
- have a: "(abs a) * (abs b) = ?x"
+ have a: "\<bar>a\<bar> * \<bar>b\<bar> = ?x"
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>
+ have bh: "u = a \<Longrightarrow> v = b \<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])
@@ -426,16 +474,22 @@
done
}
note b = this[OF refl[of a] refl[of b]]
- have xy: "- ?x <= ?y"
+ have xy: "- ?x \<le> ?y"
apply simp
- apply (metis (full_types) add_increasing add_uminus_conv_diff lattice_ab_group_add_class.minus_le_self_iff minus_add_distrib mult_nonneg_nonneg mult_nonpos_nonpos nprt_le_zero zero_le_pprt)
+ apply (metis (full_types) add_increasing add_uminus_conv_diff
+ lattice_ab_group_add_class.minus_le_self_iff minus_add_distrib mult_nonneg_nonneg
+ mult_nonpos_nonpos nprt_le_zero zero_le_pprt)
done
- have yx: "?y <= ?x"
+ have yx: "?y \<le> ?x"
apply simp
- apply (metis (full_types) add_nonpos_nonpos add_uminus_conv_diff lattice_ab_group_add_class.le_minus_self_iff minus_add_distrib mult_nonneg_nonpos mult_nonpos_nonneg nprt_le_zero zero_le_pprt)
+ apply (metis (full_types) add_nonpos_nonpos add_uminus_conv_diff
+ lattice_ab_group_add_class.le_minus_self_iff minus_add_distrib mult_nonneg_nonpos
+ mult_nonpos_nonneg nprt_le_zero zero_le_pprt)
done
- have i1: "a*b <= abs a * abs b" by (simp only: a b yx)
- have i2: "- (abs a * abs b) <= a*b" by (simp only: a b xy)
+ have i1: "a * b \<le> \<bar>a\<bar> * \<bar>b\<bar>"
+ by (simp only: a b yx)
+ have i2: "- (\<bar>a\<bar> * \<bar>b\<bar>) \<le> a * b"
+ by (simp only: a b xy)
show ?thesis
apply (rule abs_leI)
apply (simp add: i1)
@@ -445,37 +499,38 @@
instance lattice_ring \<subseteq> ordered_ring_abs
proof
- fix a b :: "'a\<Colon> lattice_ring"
+ fix a b :: "'a::lattice_ring"
assume a: "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0)"
- show "abs (a*b) = abs a * abs b"
+ show "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"
proof -
- have s: "(0 <= a*b) | (a*b <= 0)"
- apply (auto)
+ have s: "(0 \<le> a * b) \<or> (a * b \<le> 0)"
+ apply auto
apply (rule_tac split_mult_pos_le)
- apply (rule_tac contrapos_np[of "a*b <= 0"])
- apply (simp)
+ apply (rule_tac contrapos_np[of "a * b \<le> 0"])
+ apply simp
apply (rule_tac split_mult_neg_le)
- apply (insert a)
- apply (blast)
+ using a
+ apply blast
done
have mulprts: "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
by (simp add: prts[symmetric])
show ?thesis
- proof cases
- assume "0 <= a * b"
+ proof (cases "0 \<le> a * b")
+ case True
then show ?thesis
apply (simp_all add: mulprts abs_prts)
- apply (insert a)
+ using a
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)
- apply(drule (1) mult_nonneg_nonpos2[of b a], simp)
+ apply(drule (1) mult_nonneg_nonpos[of a b], simp)
+ apply(drule (1) mult_nonneg_nonpos2[of b a], simp)
done
next
- assume "~(0 <= a*b)"
- with s have "a*b <= 0" by simp
+ case False
+ with s have "a * b \<le> 0"
+ by simp
then show ?thesis
apply (simp_all add: mulprts abs_prts)
apply (insert a)
@@ -488,11 +543,12 @@
qed
lemma mult_le_prts:
- assumes "a1 <= (a::'a::lattice_ring)"
- and "a <= a2"
- and "b1 <= b"
- and "b <= b2"
- shows "a * b <=
+ fixes a b :: "'a::lattice_ring"
+ assumes "a1 \<le> a"
+ and "a \<le> a2"
+ and "b1 \<le> b"
+ and "b \<le> b2"
+ shows "a * b \<le>
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)"
@@ -501,31 +557,31 @@
done
then have "a * b = pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
by (simp add: algebra_simps)
- moreover have "pprt a * pprt b <= pprt a2 * pprt b2"
+ moreover have "pprt a * pprt b \<le> pprt a2 * pprt b2"
by (simp_all add: assms mult_mono)
- moreover have "pprt a * nprt b <= pprt a1 * nprt b2"
+ moreover have "pprt a * nprt b \<le> pprt a1 * nprt b2"
proof -
- have "pprt a * nprt b <= pprt a * nprt b2"
+ have "pprt a * nprt b \<le> pprt a * nprt b2"
by (simp add: mult_left_mono assms)
- moreover have "pprt a * nprt b2 <= pprt a1 * nprt b2"
+ moreover have "pprt a * nprt b2 \<le> pprt a1 * nprt b2"
by (simp add: mult_right_mono_neg assms)
ultimately show ?thesis
by simp
qed
- moreover have "nprt a * pprt b <= nprt a2 * pprt b1"
+ moreover have "nprt a * pprt b \<le> nprt a2 * pprt b1"
proof -
- have "nprt a * pprt b <= nprt a2 * pprt b"
+ have "nprt a * pprt b \<le> nprt a2 * pprt b"
by (simp add: mult_right_mono assms)
- moreover have "nprt a2 * pprt b <= nprt a2 * pprt b1"
+ moreover have "nprt a2 * pprt b \<le> nprt a2 * pprt b1"
by (simp add: mult_left_mono_neg assms)
ultimately show ?thesis
by simp
qed
- moreover have "nprt a * nprt b <= nprt a1 * nprt b1"
+ moreover have "nprt a * nprt b \<le> nprt a1 * nprt b1"
proof -
- have "nprt a * nprt b <= nprt a * nprt b1"
+ have "nprt a * nprt b \<le> nprt a * nprt b1"
by (simp add: mult_left_mono_neg assms)
- moreover have "nprt a * nprt b1 <= nprt a1 * nprt b1"
+ moreover have "nprt a * nprt b1 \<le> nprt a1 * nprt b1"
by (simp add: mult_right_mono_neg assms)
ultimately show ?thesis
by simp
@@ -537,36 +593,41 @@
qed
lemma mult_ge_prts:
- assumes "a1 <= (a::'a::lattice_ring)"
- and "a <= a2"
- and "b1 <= b"
- and "b <= b2"
- shows "a * b >=
+ fixes a b :: "'a::lattice_ring"
+ assumes "a1 \<le> a"
+ and "a \<le> a2"
+ and "b1 \<le> b"
+ and "b \<le> b2"
+ shows "a * b \<ge>
nprt a1 * pprt b2 + nprt a2 * nprt b2 + pprt a1 * pprt b1 + pprt a2 * nprt b1"
proof -
- from assms have a1:"- a2 <= -a"
+ from assms have a1: "- a2 \<le> -a"
by auto
- from assms have a2: "-a <= -a1"
+ from assms have a2: "- a \<le> -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"
+ 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) \<le> - 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"
+ then have "- (- nprt a1 * pprt b2 + - nprt a2 * nprt b2 +
+ - pprt a1 * pprt b1 + - pprt a2 * nprt b1) \<le> a * b"
by (simp only: minus_le_iff)
- then show ?thesis by (simp add: algebra_simps)
+ then show ?thesis
+ by (simp add: algebra_simps)
qed
instance int :: lattice_ring
proof
fix k :: int
- show "abs k = sup k (- k)"
+ show "\<bar>k\<bar> = sup k (- k)"
by (auto simp add: sup_int_def)
qed
instance real :: lattice_ring
proof
fix a :: real
- show "abs a = sup a (- a)"
+ show "\<bar>a\<bar> = sup a (- a)"
by (auto simp add: sup_real_def)
qed