--- a/src/HOL/Lattices.thy Fri Apr 02 13:33:48 2010 +0200
+++ b/src/HOL/Lattices.thy Wed Apr 07 19:17:10 2010 +0200
@@ -5,10 +5,45 @@
header {* Abstract lattices *}
theory Lattices
-imports Orderings
+imports Orderings Groups
+begin
+
+subsection {* Abstract semilattice *}
+
+text {*
+ This locales provide a basic structure for interpretation into
+ bigger structures; extensions require careful thinking, otherwise
+ undesired effects may occur due to interpretation.
+*}
+
+locale semilattice = abel_semigroup +
+ assumes idem [simp]: "f a a = a"
begin
-subsection {* Lattices *}
+lemma left_idem [simp]:
+ "f a (f a b) = f a b"
+ by (simp add: assoc [symmetric])
+
+end
+
+
+subsection {* Idempotent semigroup *}
+
+class ab_semigroup_idem_mult = ab_semigroup_mult +
+ assumes mult_idem: "x * x = x"
+
+sublocale ab_semigroup_idem_mult < times!: semilattice times proof
+qed (fact mult_idem)
+
+context ab_semigroup_idem_mult
+begin
+
+lemmas mult_left_idem = times.left_idem
+
+end
+
+
+subsection {* Concrete lattices *}
notation
less_eq (infix "\<sqsubseteq>" 50) and
@@ -16,13 +51,13 @@
top ("\<top>") and
bot ("\<bottom>")
-class lower_semilattice = order +
+class semilattice_inf = order +
fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<sqinter>" 70)
assumes inf_le1 [simp]: "x \<sqinter> y \<sqsubseteq> x"
and inf_le2 [simp]: "x \<sqinter> y \<sqsubseteq> y"
and inf_greatest: "x \<sqsubseteq> y \<Longrightarrow> x \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> y \<sqinter> z"
-class upper_semilattice = order +
+class semilattice_sup = order +
fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<squnion>" 65)
assumes sup_ge1 [simp]: "x \<sqsubseteq> x \<squnion> y"
and sup_ge2 [simp]: "y \<sqsubseteq> x \<squnion> y"
@@ -32,18 +67,18 @@
text {* Dual lattice *}
lemma dual_semilattice:
- "lower_semilattice (op \<ge>) (op >) sup"
-by (rule lower_semilattice.intro, rule dual_order)
+ "semilattice_inf (op \<ge>) (op >) sup"
+by (rule semilattice_inf.intro, rule dual_order)
(unfold_locales, simp_all add: sup_least)
end
-class lattice = lower_semilattice + upper_semilattice
+class lattice = semilattice_inf + semilattice_sup
subsubsection {* Intro and elim rules*}
-context lower_semilattice
+context semilattice_inf
begin
lemma le_infI1:
@@ -55,10 +90,10 @@
by (rule order_trans) auto
lemma le_infI: "x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> b \<Longrightarrow> x \<sqsubseteq> a \<sqinter> b"
- by (blast intro: inf_greatest)
+ by (rule inf_greatest) (* FIXME: duplicate lemma *)
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 le_infI1 le_infI2)
+ by (blast intro: order_trans inf_le1 inf_le2)
lemma le_inf_iff [simp]:
"x \<sqsubseteq> y \<sqinter> z \<longleftrightarrow> x \<sqsubseteq> y \<and> x \<sqsubseteq> z"
@@ -68,14 +103,17 @@
"x \<sqsubseteq> y \<longleftrightarrow> x \<sqinter> y = x"
by (auto intro: le_infI1 antisym dest: eq_iff [THEN iffD1])
+lemma inf_mono: "a \<sqsubseteq> c \<Longrightarrow> b \<le> 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\<Colon>lower_semilattice"
+ fixes f :: "'a \<Rightarrow> 'b\<Colon>semilattice_inf"
shows "mono f \<Longrightarrow> f (A \<sqinter> B) \<sqsubseteq> f A \<sqinter> f B"
by (auto simp add: mono_def intro: Lattices.inf_greatest)
end
-context upper_semilattice
+context semilattice_sup
begin
lemma le_supI1:
@@ -88,11 +126,11 @@
lemma le_supI:
"a \<sqsubseteq> x \<Longrightarrow> b \<sqsubseteq> x \<Longrightarrow> a \<squnion> b \<sqsubseteq> x"
- by (blast intro: sup_least)
+ by (rule 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"
- by (blast intro: le_supI1 le_supI2 order_trans)
+ by (blast intro: order_trans sup_ge1 sup_ge2)
lemma le_sup_iff [simp]:
"x \<squnion> y \<sqsubseteq> z \<longleftrightarrow> x \<sqsubseteq> z \<and> y \<sqsubseteq> z"
@@ -102,8 +140,11 @@
"x \<sqsubseteq> y \<longleftrightarrow> x \<squnion> y = y"
by (auto intro: le_supI2 antisym dest: eq_iff [THEN iffD1])
+lemma sup_mono: "a \<sqsubseteq> c \<Longrightarrow> b \<le> 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\<Colon>upper_semilattice"
+ fixes f :: "'a \<Rightarrow> 'b\<Colon>semilattice_sup"
shows "mono f \<Longrightarrow> f A \<squnion> f B \<sqsubseteq> f (A \<squnion> B)"
by (auto simp add: mono_def intro: Lattices.sup_least)
@@ -112,48 +153,73 @@
subsubsection {* Equational laws *}
-context lower_semilattice
-begin
+sublocale semilattice_inf < inf!: semilattice inf
+proof
+ fix a b c
+ show "(a \<sqinter> b) \<sqinter> c = a \<sqinter> (b \<sqinter> c)"
+ by (rule antisym) (auto intro: le_infI1 le_infI2)
+ show "a \<sqinter> b = b \<sqinter> a"
+ by (rule antisym) auto
+ show "a \<sqinter> a = a"
+ by (rule antisym) auto
+qed
-lemma inf_commute: "(x \<sqinter> y) = (y \<sqinter> x)"
- by (rule antisym) auto
+context semilattice_inf
+begin
lemma inf_assoc: "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
- by (rule antisym) (auto intro: le_infI1 le_infI2)
+ by (fact inf.assoc)
+
+lemma inf_commute: "(x \<sqinter> y) = (y \<sqinter> x)"
+ by (fact inf.commute)
-lemma inf_idem[simp]: "x \<sqinter> x = x"
- by (rule antisym) auto
+lemma inf_left_commute: "x \<sqinter> (y \<sqinter> z) = y \<sqinter> (x \<sqinter> z)"
+ by (fact inf.left_commute)
-lemma inf_left_idem[simp]: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"
- by (rule antisym) (auto intro: le_infI2)
+lemma inf_idem: "x \<sqinter> x = x"
+ by (fact inf.idem)
+
+lemma inf_left_idem: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"
+ by (fact inf.left_idem)
lemma inf_absorb1: "x \<sqsubseteq> y \<Longrightarrow> x \<sqinter> y = x"
by (rule antisym) auto
lemma inf_absorb2: "y \<sqsubseteq> x \<Longrightarrow> x \<sqinter> y = y"
by (rule antisym) auto
-
-lemma inf_left_commute: "x \<sqinter> (y \<sqinter> z) = y \<sqinter> (x \<sqinter> z)"
- by (rule mk_left_commute [of inf]) (fact inf_assoc inf_commute)+
-
+
lemmas inf_aci = inf_commute inf_assoc inf_left_commute inf_left_idem
end
-context upper_semilattice
-begin
+sublocale semilattice_sup < sup!: semilattice sup
+proof
+ fix a b c
+ show "(a \<squnion> b) \<squnion> c = a \<squnion> (b \<squnion> c)"
+ by (rule antisym) (auto intro: le_supI1 le_supI2)
+ show "a \<squnion> b = b \<squnion> a"
+ by (rule antisym) auto
+ show "a \<squnion> a = a"
+ by (rule antisym) auto
+qed
-lemma sup_commute: "(x \<squnion> y) = (y \<squnion> x)"
- by (rule antisym) auto
+context semilattice_sup
+begin
lemma sup_assoc: "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
- by (rule antisym) (auto intro: le_supI1 le_supI2)
+ by (fact sup.assoc)
+
+lemma sup_commute: "(x \<squnion> y) = (y \<squnion> x)"
+ by (fact sup.commute)
-lemma sup_idem[simp]: "x \<squnion> x = x"
- by (rule antisym) auto
+lemma sup_left_commute: "x \<squnion> (y \<squnion> z) = y \<squnion> (x \<squnion> z)"
+ by (fact sup.left_commute)
-lemma sup_left_idem[simp]: "x \<squnion> (x \<squnion> y) = x \<squnion> y"
- by (rule antisym) (auto intro: le_supI2)
+lemma sup_idem: "x \<squnion> x = x"
+ by (fact sup.idem)
+
+lemma sup_left_idem: "x \<squnion> (x \<squnion> y) = x \<squnion> y"
+ by (fact sup.left_idem)
lemma sup_absorb1: "y \<sqsubseteq> x \<Longrightarrow> x \<squnion> y = x"
by (rule antisym) auto
@@ -161,9 +227,6 @@
lemma sup_absorb2: "x \<sqsubseteq> y \<Longrightarrow> x \<squnion> y = y"
by (rule antisym) auto
-lemma sup_left_commute: "x \<squnion> (y \<squnion> z) = y \<squnion> (x \<squnion> z)"
- by (rule mk_left_commute [of sup]) (fact sup_assoc sup_commute)+
-
lemmas sup_aci = sup_commute sup_assoc sup_left_commute sup_left_idem
end
@@ -173,7 +236,7 @@
lemma dual_lattice:
"lattice (op \<ge>) (op >) sup inf"
- by (rule lattice.intro, rule dual_semilattice, rule upper_semilattice.intro, rule dual_order)
+ by (rule lattice.intro, rule dual_semilattice, rule semilattice_sup.intro, rule dual_order)
(unfold_locales, auto)
lemma inf_sup_absorb: "x \<sqinter> (x \<squnion> y) = x"
@@ -224,7 +287,7 @@
subsubsection {* Strict order *}
-context lower_semilattice
+context semilattice_inf
begin
lemma less_infI1:
@@ -237,13 +300,13 @@
end
-context upper_semilattice
+context semilattice_sup
begin
lemma less_supI1:
"x \<sqsubset> a \<Longrightarrow> x \<sqsubset> a \<squnion> b"
proof -
- interpret dual: lower_semilattice "op \<ge>" "op >" sup
+ interpret dual: semilattice_inf "op \<ge>" "op >" sup
by (fact dual_semilattice)
assume "x \<sqsubset> a"
then show "x \<sqsubset> a \<squnion> b"
@@ -253,7 +316,7 @@
lemma less_supI2:
"x \<sqsubset> b \<Longrightarrow> x \<sqsubset> a \<squnion> b"
proof -
- interpret dual: lower_semilattice "op \<ge>" "op >" sup
+ interpret dual: semilattice_inf "op \<ge>" "op >" sup
by (fact dual_semilattice)
assume "x \<sqsubset> b"
then show "x \<sqsubset> a \<squnion> b"
@@ -288,6 +351,12 @@
by (rule distrib_lattice.intro, rule dual_lattice)
(unfold_locales, fact inf_sup_distrib1)
+lemmas sup_inf_distrib =
+ sup_inf_distrib1 sup_inf_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
@@ -335,39 +404,13 @@
"x \<squnion> \<bottom> = x"
by (rule sup_absorb1) simp
-lemma inf_eq_top_eq1:
- assumes "A \<sqinter> B = \<top>"
- shows "A = \<top>"
-proof (cases "B = \<top>")
- case True with assms show ?thesis by simp
-next
- case False with top_greatest have "B \<sqsubset> \<top>" by (auto intro: neq_le_trans)
- then have "A \<sqinter> B \<sqsubset> \<top>" by (rule less_infI2)
- with assms show ?thesis by simp
-qed
-
-lemma inf_eq_top_eq2:
- assumes "A \<sqinter> B = \<top>"
- shows "B = \<top>"
- by (rule inf_eq_top_eq1, unfold inf_commute [of B]) (fact assms)
+lemma inf_eq_top_iff [simp]:
+ "x \<sqinter> y = \<top> \<longleftrightarrow> x = \<top> \<and> y = \<top>"
+ by (simp add: eq_iff)
-lemma sup_eq_bot_eq1:
- assumes "A \<squnion> B = \<bottom>"
- shows "A = \<bottom>"
-proof -
- interpret dual: bounded_lattice "op \<ge>" "op >" "op \<squnion>" "op \<sqinter>" \<top> \<bottom>
- by (rule dual_bounded_lattice)
- from dual.inf_eq_top_eq1 assms show ?thesis .
-qed
-
-lemma sup_eq_bot_eq2:
- assumes "A \<squnion> B = \<bottom>"
- shows "B = \<bottom>"
-proof -
- interpret dual: bounded_lattice "op \<ge>" "op >" "op \<squnion>" "op \<sqinter>" \<top> \<bottom>
- by (rule dual_bounded_lattice)
- from dual.inf_eq_top_eq2 assms show ?thesis .
-qed
+lemma sup_eq_bot_iff [simp]:
+ "x \<squnion> y = \<bottom> \<longleftrightarrow> x = \<bottom> \<and> y = \<bottom>"
+ by (simp add: eq_iff)
end
@@ -414,10 +457,7 @@
"- x = - y \<longleftrightarrow> x = y"
proof
assume "- x = - y"
- then have "- x \<sqinter> y = \<bottom>"
- and "- x \<squnion> y = \<top>"
- by (simp_all add: compl_inf_bot compl_sup_top)
- then have "- (- x) = y" by (rule compl_unique)
+ then have "- (- x) = - (- y)" by (rule arg_cong)
then show "x = y" by simp
next
assume "x = y"
@@ -441,18 +481,14 @@
lemma compl_inf [simp]:
"- (x \<sqinter> y) = - x \<squnion> - y"
proof (rule compl_unique)
- have "(x \<sqinter> y) \<sqinter> (- x \<squnion> - y) = ((x \<sqinter> y) \<sqinter> - x) \<squnion> ((x \<sqinter> y) \<sqinter> - y)"
- by (rule inf_sup_distrib1)
- also have "... = (y \<sqinter> (x \<sqinter> - x)) \<squnion> (x \<sqinter> (y \<sqinter> - y))"
- by (simp only: inf_commute inf_assoc inf_left_commute)
- finally show "(x \<sqinter> y) \<sqinter> (- x \<squnion> - y) = \<bottom>"
+ 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)
+ then show "(x \<sqinter> y) \<sqinter> (- x \<squnion> - y) = \<bottom>"
by (simp add: inf_compl_bot)
next
- have "(x \<sqinter> y) \<squnion> (- x \<squnion> - y) = (x \<squnion> (- x \<squnion> - y)) \<sqinter> (y \<squnion> (- x \<squnion> - y))"
- by (rule sup_inf_distrib2)
- also have "... = (- y \<squnion> (x \<squnion> - x)) \<sqinter> (- x \<squnion> (y \<squnion> - y))"
- by (simp only: sup_commute sup_assoc sup_left_commute)
- finally show "(x \<sqinter> y) \<squnion> (- x \<squnion> - y) = \<top>"
+ have "(x \<sqinter> y) \<squnion> (- x \<squnion> - y) = (- y \<squnion> (x \<squnion> - x)) \<sqinter> (- x \<squnion> (y \<squnion> - y))"
+ by (simp only: sup_inf_distrib sup_aci)
+ then show "(x \<sqinter> y) \<squnion> (- x \<squnion> - y) = \<top>"
by (simp add: sup_compl_top)
qed
@@ -464,12 +500,27 @@
then show ?thesis by simp
qed
+lemma compl_mono:
+ "x \<sqsubseteq> y \<Longrightarrow> - y \<sqsubseteq> - x"
+proof -
+ assume "x \<sqsubseteq> y"
+ then 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)
+qed
+
+lemma compl_le_compl_iff: (* TODO: declare [simp] ? *)
+ "- x \<le> - y \<longleftrightarrow> y \<le> x"
+by (auto dest: compl_mono)
+
end
subsection {* Uniqueness of inf and sup *}
-lemma (in lower_semilattice) inf_unique:
+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"
@@ -481,7 +532,7 @@
show "x \<sqinter> y \<sqsubseteq> x \<triangle> y" by (rule leI) simp_all
qed
-lemma (in upper_semilattice) sup_unique:
+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"
@@ -492,7 +543,7 @@
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
-
+
subsection {* @{const min}/@{const max} on linear orders as
special case of @{const inf}/@{const sup} *}
@@ -504,20 +555,21 @@
by (auto simp add: min_def max_def)
qed (auto simp add: min_def max_def not_le less_imp_le)
-lemma inf_min: "inf = (min \<Colon> 'a\<Colon>{lower_semilattice, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)"
+lemma inf_min: "inf = (min \<Colon> 'a\<Colon>{semilattice_inf, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)"
by (rule ext)+ (auto intro: antisym)
-lemma sup_max: "sup = (max \<Colon> 'a\<Colon>{upper_semilattice, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)"
+lemma sup_max: "sup = (max \<Colon> 'a\<Colon>{semilattice_sup, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)"
by (rule ext)+ (auto intro: antisym)
lemmas le_maxI1 = min_max.sup_ge1
lemmas le_maxI2 = min_max.sup_ge2
-lemmas max_ac = min_max.sup_assoc min_max.sup_commute
- mk_left_commute [of max, OF min_max.sup_assoc min_max.sup_commute]
+lemmas min_ac = min_max.inf_assoc min_max.inf_commute
+ min_max.inf.left_commute
-lemmas min_ac = min_max.inf_assoc min_max.inf_commute
- mk_left_commute [of min, OF min_max.inf_assoc min_max.inf_commute]
+lemmas max_ac = min_max.sup_assoc min_max.sup_commute
+ min_max.sup.left_commute
+
subsection {* Bool as lattice *}