--- a/src/HOL/Lattices.thy Sun Jun 19 22:51:42 2016 +0200
+++ b/src/HOL/Lattices.thy Mon Jun 20 17:03:50 2016 +0200
@@ -21,24 +21,23 @@
begin
lemma left_idem [simp]: "a \<^bold>* (a \<^bold>* b) = a \<^bold>* b"
-by (simp add: assoc [symmetric])
+ by (simp add: assoc [symmetric])
lemma right_idem [simp]: "(a \<^bold>* b) \<^bold>* b = a \<^bold>* b"
-by (simp add: assoc)
+ by (simp add: assoc)
end
locale semilattice_neutr = semilattice + comm_monoid
locale semilattice_order = semilattice +
- fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<^bold>\<le>" 50)
- and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<^bold><" 50)
+ fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<^bold>\<le>" 50)
+ and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<^bold><" 50)
assumes order_iff: "a \<^bold>\<le> b \<longleftrightarrow> a = a \<^bold>* b"
and strict_order_iff: "a \<^bold>< b \<longleftrightarrow> a = a \<^bold>* b \<and> a \<noteq> b"
begin
-lemma orderI:
- "a = a \<^bold>* b \<Longrightarrow> a \<^bold>\<le> b"
+lemma orderI: "a = a \<^bold>* b \<Longrightarrow> a \<^bold>\<le> b"
by (simp add: order_iff)
lemma orderE:
@@ -49,7 +48,7 @@
sublocale ordering less_eq less
proof
fix a b
- show "a \<^bold>< b \<longleftrightarrow> a \<^bold>\<le> b \<and> a \<noteq> b"
+ show "a \<^bold>< b \<longleftrightarrow> a \<^bold>\<le> b \<and> a \<noteq> b" for a b
by (simp add: order_iff strict_order_iff)
next
fix a
@@ -74,12 +73,10 @@
then show "a \<^bold>\<le> c" by (rule orderI)
qed
-lemma cobounded1 [simp]:
- "a \<^bold>* b \<^bold>\<le> a"
- by (simp add: order_iff commute)
+lemma cobounded1 [simp]: "a \<^bold>* b \<^bold>\<le> a"
+ by (simp add: order_iff commute)
-lemma cobounded2 [simp]:
- "a \<^bold>* b \<^bold>\<le> b"
+lemma cobounded2 [simp]: "a \<^bold>* b \<^bold>\<le> b"
by (simp add: order_iff)
lemma boundedI:
@@ -95,8 +92,7 @@
obtains "a \<^bold>\<le> b" and "a \<^bold>\<le> c"
using assms by (blast intro: trans cobounded1 cobounded2)
-lemma bounded_iff [simp]:
- "a \<^bold>\<le> b \<^bold>* c \<longleftrightarrow> a \<^bold>\<le> b \<and> a \<^bold>\<le> c"
+lemma bounded_iff [simp]: "a \<^bold>\<le> b \<^bold>* c \<longleftrightarrow> a \<^bold>\<le> b \<and> a \<^bold>\<le> c"
by (blast intro: boundedI elim: boundedE)
lemma strict_boundedE:
@@ -104,21 +100,17 @@
obtains "a \<^bold>< b" and "a \<^bold>< c"
using assms by (auto simp add: commute strict_iff_order elim: orderE intro!: that)+
-lemma coboundedI1:
- "a \<^bold>\<le> c \<Longrightarrow> a \<^bold>* b \<^bold>\<le> c"
+lemma coboundedI1: "a \<^bold>\<le> c \<Longrightarrow> a \<^bold>* b \<^bold>\<le> c"
by (rule trans) auto
-lemma coboundedI2:
- "b \<^bold>\<le> c \<Longrightarrow> a \<^bold>* b \<^bold>\<le> c"
+lemma coboundedI2: "b \<^bold>\<le> c \<Longrightarrow> a \<^bold>* b \<^bold>\<le> c"
by (rule trans) auto
-lemma strict_coboundedI1:
- "a \<^bold>< c \<Longrightarrow> a \<^bold>* b \<^bold>< c"
+lemma strict_coboundedI1: "a \<^bold>< c \<Longrightarrow> a \<^bold>* b \<^bold>< c"
using irrefl
by (auto intro: not_eq_order_implies_strict coboundedI1 strict_implies_order elim: strict_boundedE)
-lemma strict_coboundedI2:
- "b \<^bold>< c \<Longrightarrow> a \<^bold>* b \<^bold>< c"
+lemma strict_coboundedI2: "b \<^bold>< c \<Longrightarrow> a \<^bold>* b \<^bold>< c"
using strict_coboundedI1 [of b c a] by (simp add: commute)
lemma mono: "a \<^bold>\<le> c \<Longrightarrow> b \<^bold>\<le> d \<Longrightarrow> a \<^bold>* b \<^bold>\<le> c \<^bold>* d"
@@ -152,7 +144,7 @@
class inf =
fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<sqinter>" 70)
-class sup =
+class sup =
fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<squnion>" 65)
@@ -175,10 +167,9 @@
text \<open>Dual lattice\<close>
-lemma dual_semilattice:
- "class.semilattice_inf sup greater_eq greater"
-by (rule class.semilattice_inf.intro, rule dual_order)
- (unfold_locales, simp_all add: sup_least)
+lemma dual_semilattice: "class.semilattice_inf sup greater_eq greater"
+ by (rule class.semilattice_inf.intro, rule dual_order)
+ (unfold_locales, simp_all add: sup_least)
end
@@ -190,12 +181,10 @@
context semilattice_inf
begin
-lemma le_infI1:
- "a \<sqsubseteq> x \<Longrightarrow> a \<sqinter> b \<sqsubseteq> x"
+lemma le_infI1: "a \<sqsubseteq> x \<Longrightarrow> a \<sqinter> b \<sqsubseteq> x"
by (rule order_trans) auto
-lemma le_infI2:
- "b \<sqsubseteq> x \<Longrightarrow> a \<sqinter> b \<sqsubseteq> x"
+lemma le_infI2: "b \<sqsubseteq> x \<Longrightarrow> a \<sqinter> b \<sqsubseteq> x"
by (rule order_trans) auto
lemma le_infI: "x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> b \<Longrightarrow> x \<sqsubseteq> a \<sqinter> b"
@@ -204,20 +193,16 @@
lemma le_infE: "x \<sqsubseteq> a \<sqinter> b \<Longrightarrow> (x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> b \<Longrightarrow> P) \<Longrightarrow> P"
by (blast intro: order_trans inf_le1 inf_le2)
-lemma le_inf_iff:
- "x \<sqsubseteq> y \<sqinter> z \<longleftrightarrow> x \<sqsubseteq> y \<and> x \<sqsubseteq> z"
+lemma le_inf_iff: "x \<sqsubseteq> y \<sqinter> z \<longleftrightarrow> x \<sqsubseteq> y \<and> x \<sqsubseteq> z"
by (blast intro: le_infI elim: le_infE)
-lemma le_iff_inf:
- "x \<sqsubseteq> y \<longleftrightarrow> x \<sqinter> y = x"
+lemma le_iff_inf: "x \<sqsubseteq> y \<longleftrightarrow> x \<sqinter> y = x"
by (auto intro: le_infI1 antisym dest: eq_iff [THEN iffD1] simp add: le_inf_iff)
lemma inf_mono: "a \<sqsubseteq> c \<Longrightarrow> b \<sqsubseteq> d \<Longrightarrow> a \<sqinter> b \<sqsubseteq> c \<sqinter> d"
by (fast intro: inf_greatest le_infI1 le_infI2)
-lemma mono_inf:
- fixes f :: "'a \<Rightarrow> 'b::semilattice_inf"
- shows "mono f \<Longrightarrow> f (A \<sqinter> B) \<sqsubseteq> f A \<sqinter> f B"
+lemma mono_inf: "mono f \<Longrightarrow> f (A \<sqinter> B) \<sqsubseteq> f A \<sqinter> f B" for f :: "'a \<Rightarrow> 'b::semilattice_inf"
by (auto simp add: mono_def intro: Lattices.inf_greatest)
end
@@ -225,36 +210,28 @@
context semilattice_sup
begin
-lemma le_supI1:
- "x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> a \<squnion> b"
+lemma le_supI1: "x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> a \<squnion> b"
+ by (rule order_trans) auto
+
+lemma le_supI2: "x \<sqsubseteq> b \<Longrightarrow> x \<sqsubseteq> a \<squnion> b"
by (rule order_trans) auto
-lemma le_supI2:
- "x \<sqsubseteq> b \<Longrightarrow> x \<sqsubseteq> a \<squnion> b"
- by (rule order_trans) auto
-
-lemma le_supI:
- "a \<sqsubseteq> x \<Longrightarrow> b \<sqsubseteq> x \<Longrightarrow> a \<squnion> b \<sqsubseteq> x"
+lemma le_supI: "a \<sqsubseteq> x \<Longrightarrow> b \<sqsubseteq> x \<Longrightarrow> a \<squnion> b \<sqsubseteq> x"
by (fact sup_least) (* FIXME: duplicate lemma *)
-lemma le_supE:
- "a \<squnion> b \<sqsubseteq> x \<Longrightarrow> (a \<sqsubseteq> x \<Longrightarrow> b \<sqsubseteq> x \<Longrightarrow> P) \<Longrightarrow> P"
+lemma le_supE: "a \<squnion> b \<sqsubseteq> x \<Longrightarrow> (a \<sqsubseteq> x \<Longrightarrow> b \<sqsubseteq> x \<Longrightarrow> P) \<Longrightarrow> P"
by (blast intro: order_trans sup_ge1 sup_ge2)
-lemma le_sup_iff:
- "x \<squnion> y \<sqsubseteq> z \<longleftrightarrow> x \<sqsubseteq> z \<and> y \<sqsubseteq> z"
+lemma le_sup_iff: "x \<squnion> y \<sqsubseteq> z \<longleftrightarrow> x \<sqsubseteq> z \<and> y \<sqsubseteq> z"
by (blast intro: le_supI elim: le_supE)
-lemma le_iff_sup:
- "x \<sqsubseteq> y \<longleftrightarrow> x \<squnion> y = y"
+lemma le_iff_sup: "x \<sqsubseteq> y \<longleftrightarrow> x \<squnion> y = y"
by (auto intro: le_supI2 antisym dest: eq_iff [THEN iffD1] simp add: le_sup_iff)
lemma sup_mono: "a \<sqsubseteq> c \<Longrightarrow> b \<sqsubseteq> d \<Longrightarrow> a \<squnion> b \<sqsubseteq> c \<squnion> d"
by (fast intro: sup_least le_supI1 le_supI2)
-lemma mono_sup:
- fixes f :: "'a \<Rightarrow> 'b::semilattice_sup"
- shows "mono f \<Longrightarrow> f A \<squnion> f B \<sqsubseteq> f (A \<squnion> B)"
+lemma mono_sup: "mono f \<Longrightarrow> f A \<squnion> f B \<sqsubseteq> f (A \<squnion> B)" for f :: "'a \<Rightarrow> 'b::semilattice_sup"
by (auto simp add: mono_def intro: Lattices.sup_least)
end
@@ -302,7 +279,7 @@
lemma inf_absorb2: "y \<sqsubseteq> x \<Longrightarrow> x \<sqinter> y = y"
by (rule antisym) auto
-
+
lemmas inf_aci = inf_commute inf_assoc inf_left_commute inf_left_idem
end
@@ -352,8 +329,7 @@
context lattice
begin
-lemma dual_lattice:
- "class.lattice sup (op \<ge>) (op >) inf"
+lemma dual_lattice: "class.lattice sup (op \<ge>) (op >) inf"
by (rule class.lattice.intro, rule dual_semilattice, rule class.semilattice_sup.intro, rule dual_order)
(unfold_locales, auto)
@@ -375,47 +351,48 @@
lemma distrib_inf_le: "(x \<sqinter> y) \<squnion> (x \<sqinter> z) \<sqsubseteq> x \<sqinter> (y \<squnion> z)"
by (auto intro: le_infI1 le_infI2 le_supI1 le_supI2)
-text\<open>If you have one of them, you have them all.\<close>
+text \<open>If you have one of them, you have them all.\<close>
lemma distrib_imp1:
-assumes D: "!!x y z. x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
-shows "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
+ assumes distrib: "\<And>x y z. x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
+ shows "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
proof-
- have "x \<squnion> (y \<sqinter> z) = (x \<squnion> (x \<sqinter> z)) \<squnion> (y \<sqinter> z)" by simp
+ have "x \<squnion> (y \<sqinter> z) = (x \<squnion> (x \<sqinter> z)) \<squnion> (y \<sqinter> z)"
+ by simp
also have "\<dots> = x \<squnion> (z \<sqinter> (x \<squnion> y))"
- by (simp add: D inf_commute sup_assoc del: sup_inf_absorb)
+ by (simp add: distrib inf_commute sup_assoc del: sup_inf_absorb)
also have "\<dots> = ((x \<squnion> y) \<sqinter> x) \<squnion> ((x \<squnion> y) \<sqinter> z)"
- by(simp add: inf_commute)
- also have "\<dots> = (x \<squnion> y) \<sqinter> (x \<squnion> z)" by(simp add:D)
+ by (simp add: inf_commute)
+ also have "\<dots> = (x \<squnion> y) \<sqinter> (x \<squnion> z)" by(simp add:distrib)
finally show ?thesis .
qed
lemma distrib_imp2:
-assumes D: "!!x y z. x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
-shows "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
+ assumes distrib: "\<And>x y z. x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
+ shows "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
proof-
- have "x \<sqinter> (y \<squnion> z) = (x \<sqinter> (x \<squnion> z)) \<sqinter> (y \<squnion> z)" by simp
+ have "x \<sqinter> (y \<squnion> z) = (x \<sqinter> (x \<squnion> z)) \<sqinter> (y \<squnion> z)"
+ by simp
also have "\<dots> = x \<sqinter> (z \<squnion> (x \<sqinter> y))"
- by (simp add: D sup_commute inf_assoc del: inf_sup_absorb)
+ by (simp add: distrib sup_commute inf_assoc del: inf_sup_absorb)
also have "\<dots> = ((x \<sqinter> y) \<squnion> x) \<sqinter> ((x \<sqinter> y) \<squnion> z)"
- by(simp add: sup_commute)
- also have "\<dots> = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" by(simp add:D)
+ by (simp add: sup_commute)
+ also have "\<dots> = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" by (simp add:distrib)
finally show ?thesis .
qed
end
+
subsubsection \<open>Strict order\<close>
context semilattice_inf
begin
-lemma less_infI1:
- "a \<sqsubset> x \<Longrightarrow> a \<sqinter> b \<sqsubset> x"
+lemma less_infI1: "a \<sqsubset> x \<Longrightarrow> a \<sqinter> b \<sqsubset> x"
by (auto simp add: less_le inf_absorb1 intro: le_infI1)
-lemma less_infI2:
- "b \<sqsubset> x \<Longrightarrow> a \<sqinter> b \<sqsubset> x"
+lemma less_infI2: "b \<sqsubset> x \<Longrightarrow> a \<sqinter> b \<sqsubset> x"
by (auto simp add: less_le inf_absorb2 intro: le_infI2)
end
@@ -423,13 +400,11 @@
context semilattice_sup
begin
-lemma less_supI1:
- "x \<sqsubset> a \<Longrightarrow> x \<sqsubset> a \<squnion> b"
+lemma less_supI1: "x \<sqsubset> a \<Longrightarrow> x \<sqsubset> a \<squnion> b"
using dual_semilattice
by (rule semilattice_inf.less_infI1)
-lemma less_supI2:
- "x \<sqsubset> b \<Longrightarrow> x \<sqsubset> a \<squnion> b"
+lemma less_supI2: "x \<sqsubset> b \<Longrightarrow> x \<sqsubset> a \<squnion> b"
using dual_semilattice
by (rule semilattice_inf.less_infI2)
@@ -444,31 +419,24 @@
context distrib_lattice
begin
-lemma sup_inf_distrib2:
- "(y \<sqinter> z) \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)"
+lemma sup_inf_distrib2: "(y \<sqinter> z) \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)"
by (simp add: sup_commute sup_inf_distrib1)
-lemma inf_sup_distrib1:
- "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
+lemma inf_sup_distrib1: "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
by (rule distrib_imp2 [OF sup_inf_distrib1])
-lemma inf_sup_distrib2:
- "(y \<squnion> z) \<sqinter> x = (y \<sqinter> x) \<squnion> (z \<sqinter> x)"
+lemma inf_sup_distrib2: "(y \<squnion> z) \<sqinter> x = (y \<sqinter> x) \<squnion> (z \<sqinter> x)"
by (simp add: inf_commute inf_sup_distrib1)
-lemma dual_distrib_lattice:
- "class.distrib_lattice sup (op \<ge>) (op >) inf"
+lemma dual_distrib_lattice: "class.distrib_lattice sup (op \<ge>) (op >) inf"
by (rule class.distrib_lattice.intro, rule dual_lattice)
(unfold_locales, fact inf_sup_distrib1)
-lemmas sup_inf_distrib =
- sup_inf_distrib1 sup_inf_distrib2
+lemmas sup_inf_distrib = sup_inf_distrib1 sup_inf_distrib2
-lemmas inf_sup_distrib =
- inf_sup_distrib1 inf_sup_distrib2
+lemmas inf_sup_distrib = inf_sup_distrib1 inf_sup_distrib2
-lemmas distrib =
- sup_inf_distrib1 sup_inf_distrib2 inf_sup_distrib1 inf_sup_distrib2
+lemmas distrib = sup_inf_distrib1 sup_inf_distrib2 inf_sup_distrib1 inf_sup_distrib2
end
@@ -481,8 +449,7 @@
sublocale inf_top: semilattice_neutr inf top
+ inf_top: semilattice_neutr_order inf top less_eq less
proof
- fix x
- show "x \<sqinter> \<top> = x"
+ show "x \<sqinter> \<top> = x" for x
by (rule inf_absorb1) simp
qed
@@ -494,8 +461,7 @@
sublocale sup_bot: semilattice_neutr sup bot
+ sup_bot: semilattice_neutr_order sup bot greater_eq greater
proof
- fix x
- show "x \<squnion> \<bottom> = x"
+ show "x \<squnion> \<bottom> = x" for x
by (rule sup_absorb1) simp
qed
@@ -506,28 +472,22 @@
subclass bounded_semilattice_sup_bot ..
-lemma inf_bot_left [simp]:
- "\<bottom> \<sqinter> x = \<bottom>"
+lemma inf_bot_left [simp]: "\<bottom> \<sqinter> x = \<bottom>"
by (rule inf_absorb1) simp
-lemma inf_bot_right [simp]:
- "x \<sqinter> \<bottom> = \<bottom>"
+lemma inf_bot_right [simp]: "x \<sqinter> \<bottom> = \<bottom>"
by (rule inf_absorb2) simp
-lemma sup_bot_left:
- "\<bottom> \<squnion> x = x"
+lemma sup_bot_left: "\<bottom> \<squnion> x = x"
by (fact sup_bot.left_neutral)
-lemma sup_bot_right:
- "x \<squnion> \<bottom> = x"
+lemma sup_bot_right: "x \<squnion> \<bottom> = x"
by (fact sup_bot.right_neutral)
-lemma sup_eq_bot_iff [simp]:
- "x \<squnion> y = \<bottom> \<longleftrightarrow> x = \<bottom> \<and> y = \<bottom>"
+lemma sup_eq_bot_iff [simp]: "x \<squnion> y = \<bottom> \<longleftrightarrow> x = \<bottom> \<and> y = \<bottom>"
by (simp add: eq_iff)
-lemma bot_eq_sup_iff [simp]:
- "\<bottom> = x \<squnion> y \<longleftrightarrow> x = \<bottom> \<and> y = \<bottom>"
+lemma bot_eq_sup_iff [simp]: "\<bottom> = x \<squnion> y \<longleftrightarrow> x = \<bottom> \<and> y = \<bottom>"
by (simp add: eq_iff)
end
@@ -537,24 +497,19 @@
subclass bounded_semilattice_inf_top ..
-lemma sup_top_left [simp]:
- "\<top> \<squnion> x = \<top>"
+lemma sup_top_left [simp]: "\<top> \<squnion> x = \<top>"
by (rule sup_absorb1) simp
-lemma sup_top_right [simp]:
- "x \<squnion> \<top> = \<top>"
+lemma sup_top_right [simp]: "x \<squnion> \<top> = \<top>"
by (rule sup_absorb2) simp
-lemma inf_top_left:
- "\<top> \<sqinter> x = x"
+lemma inf_top_left: "\<top> \<sqinter> x = x"
by (fact inf_top.left_neutral)
-lemma inf_top_right:
- "x \<sqinter> \<top> = x"
+lemma inf_top_right: "x \<sqinter> \<top> = x"
by (fact inf_top.right_neutral)
-lemma inf_eq_top_iff [simp]:
- "x \<sqinter> y = \<top> \<longleftrightarrow> x = \<top> \<and> y = \<top>"
+lemma inf_eq_top_iff [simp]: "x \<sqinter> y = \<top> \<longleftrightarrow> x = \<top> \<and> y = \<top>"
by (simp add: eq_iff)
end
@@ -565,8 +520,7 @@
subclass bounded_lattice_bot ..
subclass bounded_lattice_top ..
-lemma dual_bounded_lattice:
- "class.bounded_lattice sup greater_eq greater inf \<top> \<bottom>"
+lemma dual_bounded_lattice: "class.bounded_lattice sup greater_eq greater inf \<top> \<bottom>"
by unfold_locales (auto simp add: less_le_not_le)
end
@@ -582,12 +536,10 @@
by (rule class.boolean_algebra.intro, rule dual_bounded_lattice, rule dual_distrib_lattice)
(unfold_locales, auto simp add: inf_compl_bot sup_compl_top diff_eq)
-lemma compl_inf_bot [simp]:
- "- x \<sqinter> x = \<bottom>"
+lemma compl_inf_bot [simp]: "- x \<sqinter> x = \<bottom>"
by (simp add: inf_commute inf_compl_bot)
-lemma compl_sup_top [simp]:
- "- x \<squnion> x = \<top>"
+lemma compl_sup_top [simp]: "- x \<squnion> x = \<top>"
by (simp add: sup_commute sup_compl_top)
lemma compl_unique:
@@ -606,12 +558,10 @@
then show "- x = y" by simp
qed
-lemma double_compl [simp]:
- "- (- x) = x"
+lemma double_compl [simp]: "- (- x) = x"
using compl_inf_bot compl_sup_top by (rule compl_unique)
-lemma compl_eq_compl_iff [simp]:
- "- x = - y \<longleftrightarrow> x = y"
+lemma compl_eq_compl_iff [simp]: "- x = - y \<longleftrightarrow> x = y"
proof
assume "- x = - y"
then have "- (- x) = - (- y)" by (rule arg_cong)
@@ -621,22 +571,19 @@
then show "- x = - y" by simp
qed
-lemma compl_bot_eq [simp]:
- "- \<bottom> = \<top>"
+lemma compl_bot_eq [simp]: "- \<bottom> = \<top>"
proof -
from sup_compl_top have "\<bottom> \<squnion> - \<bottom> = \<top>" .
then show ?thesis by simp
qed
-lemma compl_top_eq [simp]:
- "- \<top> = \<bottom>"
+lemma compl_top_eq [simp]: "- \<top> = \<bottom>"
proof -
from inf_compl_bot have "\<top> \<sqinter> - \<top> = \<bottom>" .
then show ?thesis by simp
qed
-lemma compl_inf [simp]:
- "- (x \<sqinter> y) = - x \<squnion> - y"
+lemma compl_inf [simp]: "- (x \<sqinter> y) = - x \<squnion> - y"
proof (rule compl_unique)
have "(x \<sqinter> y) \<sqinter> (- x \<squnion> - y) = (y \<sqinter> (x \<sqinter> - x)) \<squnion> (x \<sqinter> (y \<sqinter> - y))"
by (simp only: inf_sup_distrib inf_aci)
@@ -649,86 +596,87 @@
by (simp add: sup_compl_top)
qed
-lemma compl_sup [simp]:
- "- (x \<squnion> y) = - x \<sqinter> - y"
+lemma compl_sup [simp]: "- (x \<squnion> y) = - x \<sqinter> - y"
using dual_boolean_algebra
by (rule boolean_algebra.compl_inf)
lemma compl_mono:
- "x \<sqsubseteq> y \<Longrightarrow> - y \<sqsubseteq> - x"
+ assumes "x \<sqsubseteq> y"
+ shows "- y \<sqsubseteq> - x"
proof -
- assume "x \<sqsubseteq> y"
- then have "x \<squnion> y = y" by (simp only: le_iff_sup)
+ from assms have "x \<squnion> y = y" by (simp only: le_iff_sup)
then have "- (x \<squnion> y) = - y" by simp
then have "- x \<sqinter> - y = - y" by simp
then have "- y \<sqinter> - x = - y" by (simp only: inf_commute)
- then show "- y \<sqsubseteq> - x" by (simp only: le_iff_inf)
+ then show ?thesis by (simp only: le_iff_inf)
qed
-lemma compl_le_compl_iff [simp]:
- "- x \<sqsubseteq> - y \<longleftrightarrow> y \<sqsubseteq> x"
+lemma compl_le_compl_iff [simp]: "- x \<sqsubseteq> - y \<longleftrightarrow> y \<sqsubseteq> x"
by (auto dest: compl_mono)
lemma compl_le_swap1:
- assumes "y \<sqsubseteq> - x" shows "x \<sqsubseteq> -y"
+ assumes "y \<sqsubseteq> - x"
+ shows "x \<sqsubseteq> -y"
proof -
from assms have "- (- x) \<sqsubseteq> - y" by (simp only: compl_le_compl_iff)
then show ?thesis by simp
qed
lemma compl_le_swap2:
- assumes "- y \<sqsubseteq> x" shows "- x \<sqsubseteq> y"
+ assumes "- y \<sqsubseteq> x"
+ shows "- x \<sqsubseteq> y"
proof -
from assms have "- x \<sqsubseteq> - (- y)" by (simp only: compl_le_compl_iff)
then show ?thesis by simp
qed
-lemma compl_less_compl_iff: (* TODO: declare [simp] ? *)
- "- x \<sqsubset> - y \<longleftrightarrow> y \<sqsubset> x"
+lemma compl_less_compl_iff: "- x \<sqsubset> - y \<longleftrightarrow> y \<sqsubset> x" (* TODO: declare [simp] ? *)
by (auto simp add: less_le)
lemma compl_less_swap1:
- assumes "y \<sqsubset> - x" shows "x \<sqsubset> - y"
+ assumes "y \<sqsubset> - x"
+ shows "x \<sqsubset> - y"
proof -
from assms have "- (- x) \<sqsubset> - y" by (simp only: compl_less_compl_iff)
then show ?thesis by simp
qed
lemma compl_less_swap2:
- assumes "- y \<sqsubset> x" shows "- x \<sqsubset> y"
+ assumes "- y \<sqsubset> x"
+ shows "- x \<sqsubset> y"
proof -
from assms have "- x \<sqsubset> - (- y)" by (simp only: compl_less_compl_iff)
then show ?thesis by simp
qed
lemma sup_cancel_left1: "sup (sup x a) (sup (- x) b) = top"
-by(simp add: inf_sup_aci sup_compl_top)
+ by (simp add: inf_sup_aci sup_compl_top)
lemma sup_cancel_left2: "sup (sup (- x) a) (sup x b) = top"
-by(simp add: inf_sup_aci sup_compl_top)
+ by (simp add: inf_sup_aci sup_compl_top)
lemma inf_cancel_left1: "inf (inf x a) (inf (- x) b) = bot"
-by(simp add: inf_sup_aci inf_compl_bot)
+ by (simp add: inf_sup_aci inf_compl_bot)
lemma inf_cancel_left2: "inf (inf (- x) a) (inf x b) = bot"
-by(simp add: inf_sup_aci inf_compl_bot)
+ by (simp add: inf_sup_aci inf_compl_bot)
-declare inf_compl_bot [simp] sup_compl_top [simp]
+declare inf_compl_bot [simp] and sup_compl_top [simp]
lemma sup_compl_top_left1 [simp]: "sup (- x) (sup x y) = top"
-by(simp add: sup_assoc[symmetric])
+ by (simp add: sup_assoc[symmetric])
lemma sup_compl_top_left2 [simp]: "sup x (sup (- x) y) = top"
-using sup_compl_top_left1[of "- x" y] by simp
+ using sup_compl_top_left1[of "- x" y] by simp
lemma inf_compl_bot_left1 [simp]: "inf (- x) (inf x y) = bot"
-by(simp add: inf_assoc[symmetric])
+ by (simp add: inf_assoc[symmetric])
lemma inf_compl_bot_left2 [simp]: "inf x (inf (- x) y) = bot"
-using inf_compl_bot_left1[of "- x" y] by simp
+ using inf_compl_bot_left1[of "- x" y] by simp
lemma inf_compl_bot_right [simp]: "inf x (inf y (- x)) = bot"
-by(subst inf_left_commute) simp
+ by (subst inf_left_commute) simp
end
@@ -740,6 +688,7 @@
simproc_setup boolean_algebra_cancel_inf ("inf a b::'a::boolean_algebra") =
\<open>fn phi => fn ss => try Boolean_Algebra_Cancel.cancel_inf_conv\<close>
+
subsection \<open>\<open>min/max\<close> as special case of lattice\<close>
context linorder
@@ -749,64 +698,48 @@
+ max: semilattice_order max greater_eq greater
by standard (auto simp add: min_def max_def)
-lemma min_le_iff_disj:
- "min x y \<le> z \<longleftrightarrow> x \<le> z \<or> y \<le> z"
+lemma min_le_iff_disj: "min x y \<le> z \<longleftrightarrow> x \<le> z \<or> y \<le> z"
unfolding min_def using linear by (auto intro: order_trans)
-lemma le_max_iff_disj:
- "z \<le> max x y \<longleftrightarrow> z \<le> x \<or> z \<le> y"
+lemma le_max_iff_disj: "z \<le> max x y \<longleftrightarrow> z \<le> x \<or> z \<le> y"
unfolding max_def using linear by (auto intro: order_trans)
-lemma min_less_iff_disj:
- "min x y < z \<longleftrightarrow> x < z \<or> y < z"
+lemma min_less_iff_disj: "min x y < z \<longleftrightarrow> x < z \<or> y < z"
unfolding min_def le_less using less_linear by (auto intro: less_trans)
-lemma less_max_iff_disj:
- "z < max x y \<longleftrightarrow> z < x \<or> z < y"
+lemma less_max_iff_disj: "z < max x y \<longleftrightarrow> z < x \<or> z < y"
unfolding max_def le_less using less_linear by (auto intro: less_trans)
-lemma min_less_iff_conj [simp]:
- "z < min x y \<longleftrightarrow> z < x \<and> z < y"
+lemma min_less_iff_conj [simp]: "z < min x y \<longleftrightarrow> z < x \<and> z < y"
unfolding min_def le_less using less_linear by (auto intro: less_trans)
-lemma max_less_iff_conj [simp]:
- "max x y < z \<longleftrightarrow> x < z \<and> y < z"
+lemma max_less_iff_conj [simp]: "max x y < z \<longleftrightarrow> x < z \<and> y < z"
unfolding max_def le_less using less_linear by (auto intro: less_trans)
-lemma min_max_distrib1:
- "min (max b c) a = max (min b a) (min c a)"
+lemma min_max_distrib1: "min (max b c) a = max (min b a) (min c a)"
by (auto simp add: min_def max_def not_le dest: le_less_trans less_trans intro: antisym)
-lemma min_max_distrib2:
- "min a (max b c) = max (min a b) (min a c)"
+lemma min_max_distrib2: "min a (max b c) = max (min a b) (min a c)"
by (auto simp add: min_def max_def not_le dest: le_less_trans less_trans intro: antisym)
-lemma max_min_distrib1:
- "max (min b c) a = min (max b a) (max c a)"
+lemma max_min_distrib1: "max (min b c) a = min (max b a) (max c a)"
by (auto simp add: min_def max_def not_le dest: le_less_trans less_trans intro: antisym)
-lemma max_min_distrib2:
- "max a (min b c) = min (max a b) (max a c)"
+lemma max_min_distrib2: "max a (min b c) = min (max a b) (max a c)"
by (auto simp add: min_def max_def not_le dest: le_less_trans less_trans intro: antisym)
lemmas min_max_distribs = min_max_distrib1 min_max_distrib2 max_min_distrib1 max_min_distrib2
-lemma split_min [no_atp]:
- "P (min i j) \<longleftrightarrow> (i \<le> j \<longrightarrow> P i) \<and> (\<not> i \<le> j \<longrightarrow> P j)"
+lemma split_min [no_atp]: "P (min i j) \<longleftrightarrow> (i \<le> j \<longrightarrow> P i) \<and> (\<not> i \<le> j \<longrightarrow> P j)"
by (simp add: min_def)
-lemma split_max [no_atp]:
- "P (max i j) \<longleftrightarrow> (i \<le> j \<longrightarrow> P j) \<and> (\<not> i \<le> j \<longrightarrow> P i)"
+lemma split_max [no_atp]: "P (max i j) \<longleftrightarrow> (i \<le> j \<longrightarrow> P j) \<and> (\<not> i \<le> j \<longrightarrow> P i)"
by (simp add: max_def)
-lemma min_of_mono:
- fixes f :: "'a \<Rightarrow> 'b::linorder"
- shows "mono f \<Longrightarrow> min (f m) (f n) = f (min m n)"
+lemma min_of_mono: "mono f \<Longrightarrow> min (f m) (f n) = f (min m n)" for f :: "'a \<Rightarrow> 'b::linorder"
by (auto simp: mono_def Orderings.min_def min_def intro: Orderings.antisym)
-lemma max_of_mono:
- fixes f :: "'a \<Rightarrow> 'b::linorder"
- shows "mono f \<Longrightarrow> max (f m) (f n) = f (max m n)"
+lemma max_of_mono: "mono f \<Longrightarrow> max (f m) (f n) = f (max m n)" for f :: "'a \<Rightarrow> 'b::linorder"
by (auto simp: mono_def Orderings.max_def max_def intro: Orderings.antisym)
end
@@ -821,27 +754,33 @@
subsection \<open>Uniqueness of inf and sup\<close>
lemma (in semilattice_inf) inf_unique:
- fixes f (infixl "\<triangle>" 70)
- assumes le1: "\<And>x y. x \<triangle> y \<sqsubseteq> x" and le2: "\<And>x y. x \<triangle> y \<sqsubseteq> y"
- and greatest: "\<And>x y z. x \<sqsubseteq> y \<Longrightarrow> x \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> y \<triangle> z"
+ fixes f (infixl "\<triangle>" 70)
+ assumes le1: "\<And>x y. x \<triangle> y \<sqsubseteq> x"
+ and le2: "\<And>x y. x \<triangle> y \<sqsubseteq> y"
+ and greatest: "\<And>x y z. x \<sqsubseteq> y \<Longrightarrow> x \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> y \<triangle> z"
shows "x \<sqinter> y = x \<triangle> y"
proof (rule antisym)
- show "x \<triangle> y \<sqsubseteq> x \<sqinter> y" by (rule le_infI) (rule le1, rule le2)
-next
- have leI: "\<And>x y z. x \<sqsubseteq> y \<Longrightarrow> x \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> y \<triangle> z" by (blast intro: greatest)
- show "x \<sqinter> y \<sqsubseteq> x \<triangle> y" by (rule leI) simp_all
+ show "x \<triangle> y \<sqsubseteq> x \<sqinter> y"
+ by (rule le_infI) (rule le1, rule le2)
+ have leI: "\<And>x y z. x \<sqsubseteq> y \<Longrightarrow> x \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> y \<triangle> z"
+ by (blast intro: greatest)
+ show "x \<sqinter> y \<sqsubseteq> x \<triangle> y"
+ by (rule leI) simp_all
qed
lemma (in semilattice_sup) sup_unique:
- fixes f (infixl "\<nabla>" 70)
- assumes ge1 [simp]: "\<And>x y. x \<sqsubseteq> x \<nabla> y" and ge2: "\<And>x y. y \<sqsubseteq> x \<nabla> y"
- and least: "\<And>x y z. y \<sqsubseteq> x \<Longrightarrow> z \<sqsubseteq> x \<Longrightarrow> y \<nabla> z \<sqsubseteq> x"
+ fixes f (infixl "\<nabla>" 70)
+ assumes ge1 [simp]: "\<And>x y. x \<sqsubseteq> x \<nabla> y"
+ and ge2: "\<And>x y. y \<sqsubseteq> x \<nabla> y"
+ and least: "\<And>x y z. y \<sqsubseteq> x \<Longrightarrow> z \<sqsubseteq> x \<Longrightarrow> y \<nabla> z \<sqsubseteq> x"
shows "x \<squnion> y = x \<nabla> y"
proof (rule antisym)
- show "x \<squnion> y \<sqsubseteq> x \<nabla> y" by (rule le_supI) (rule ge1, rule ge2)
-next
- have leI: "\<And>x y z. x \<sqsubseteq> z \<Longrightarrow> y \<sqsubseteq> z \<Longrightarrow> x \<nabla> y \<sqsubseteq> z" by (blast intro: least)
- show "x \<nabla> y \<sqsubseteq> x \<squnion> y" by (rule leI) simp_all
+ show "x \<squnion> y \<sqsubseteq> x \<nabla> y"
+ by (rule le_supI) (rule ge1, rule ge2)
+ have leI: "\<And>x y z. x \<sqsubseteq> z \<Longrightarrow> y \<sqsubseteq> z \<Longrightarrow> x \<nabla> y \<sqsubseteq> z"
+ by (blast intro: least)
+ show "x \<nabla> y \<sqsubseteq> x \<squnion> y"
+ by (rule leI) simp_all
qed
@@ -850,33 +789,25 @@
instantiation bool :: boolean_algebra
begin
-definition
- bool_Compl_def [simp]: "uminus = Not"
+definition bool_Compl_def [simp]: "uminus = Not"
-definition
- bool_diff_def [simp]: "A - B \<longleftrightarrow> A \<and> \<not> B"
+definition bool_diff_def [simp]: "A - B \<longleftrightarrow> A \<and> \<not> B"
-definition
- [simp]: "P \<sqinter> Q \<longleftrightarrow> P \<and> Q"
+definition [simp]: "P \<sqinter> Q \<longleftrightarrow> P \<and> Q"
-definition
- [simp]: "P \<squnion> Q \<longleftrightarrow> P \<or> Q"
+definition [simp]: "P \<squnion> Q \<longleftrightarrow> P \<or> Q"
-instance proof
-qed auto
+instance by standard auto
end
-lemma sup_boolI1:
- "P \<Longrightarrow> P \<squnion> Q"
+lemma sup_boolI1: "P \<Longrightarrow> P \<squnion> Q"
by simp
-lemma sup_boolI2:
- "Q \<Longrightarrow> P \<squnion> Q"
+lemma sup_boolI2: "Q \<Longrightarrow> P \<squnion> Q"
by simp
-lemma sup_boolE:
- "P \<squnion> Q \<Longrightarrow> (P \<Longrightarrow> R) \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
+lemma sup_boolE: "P \<squnion> Q \<Longrightarrow> (P \<Longrightarrow> R) \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
by auto
@@ -885,48 +816,40 @@
instantiation "fun" :: (type, semilattice_sup) semilattice_sup
begin
-definition
- "f \<squnion> g = (\<lambda>x. f x \<squnion> g x)"
+definition "f \<squnion> g = (\<lambda>x. f x \<squnion> g x)"
-lemma sup_apply [simp, code]:
- "(f \<squnion> g) x = f x \<squnion> g x"
+lemma sup_apply [simp, code]: "(f \<squnion> g) x = f x \<squnion> g x"
by (simp add: sup_fun_def)
-instance proof
-qed (simp_all add: le_fun_def)
+instance by standard (simp_all add: le_fun_def)
end
instantiation "fun" :: (type, semilattice_inf) semilattice_inf
begin
-definition
- "f \<sqinter> g = (\<lambda>x. f x \<sqinter> g x)"
+definition "f \<sqinter> g = (\<lambda>x. f x \<sqinter> g x)"
-lemma inf_apply [simp, code]:
- "(f \<sqinter> g) x = f x \<sqinter> g x"
+lemma inf_apply [simp, code]: "(f \<sqinter> g) x = f x \<sqinter> g x"
by (simp add: inf_fun_def)
-instance proof
-qed (simp_all add: le_fun_def)
+instance by standard (simp_all add: le_fun_def)
end
instance "fun" :: (type, lattice) lattice ..
-instance "fun" :: (type, distrib_lattice) distrib_lattice proof
-qed (rule ext, simp add: sup_inf_distrib1)
+instance "fun" :: (type, distrib_lattice) distrib_lattice
+ by standard (rule ext, simp add: sup_inf_distrib1)
instance "fun" :: (type, bounded_lattice) bounded_lattice ..
instantiation "fun" :: (type, uminus) uminus
begin
-definition
- fun_Compl_def: "- A = (\<lambda>x. - A x)"
+definition fun_Compl_def: "- A = (\<lambda>x. - A x)"
-lemma uminus_apply [simp, code]:
- "(- A) x = - (A x)"
+lemma uminus_apply [simp, code]: "(- A) x = - (A x)"
by (simp add: fun_Compl_def)
instance ..
@@ -936,19 +859,17 @@
instantiation "fun" :: (type, minus) minus
begin
-definition
- fun_diff_def: "A - B = (\<lambda>x. A x - B x)"
+definition fun_diff_def: "A - B = (\<lambda>x. A x - B x)"
-lemma minus_apply [simp, code]:
- "(A - B) x = A x - B x"
+lemma minus_apply [simp, code]: "(A - B) x = A x - B x"
by (simp add: fun_diff_def)
instance ..
end
-instance "fun" :: (type, boolean_algebra) boolean_algebra proof
-qed (rule ext, simp_all add: inf_compl_bot sup_compl_top diff_eq)+
+instance "fun" :: (type, boolean_algebra) boolean_algebra
+ by standard (rule ext, simp_all add: inf_compl_bot sup_compl_top diff_eq)+
subsection \<open>Lattice on unary and binary predicates\<close>
@@ -995,10 +916,7 @@
lemma sup2E: "(A \<squnion> B) x y \<Longrightarrow> (A x y \<Longrightarrow> P) \<Longrightarrow> (B x y \<Longrightarrow> P) \<Longrightarrow> P"
by (simp add: sup_fun_def) iprover
-text \<open>
- \medskip Classical introduction rule: no commitment to \<open>A\<close> vs
- \<open>B\<close>.
-\<close>
+text \<open> \<^medskip> Classical introduction rule: no commitment to \<open>A\<close> vs \<open>B\<close>.\<close>
lemma sup1CI: "(\<not> B x \<Longrightarrow> A x) \<Longrightarrow> (A \<squnion> B) x"
by (auto simp add: sup_fun_def)
@@ -1012,4 +930,3 @@
less (infix "\<sqsubset>" 50)
end
-