src/HOL/Lattices.thy
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distributed theory Algebras to theories Groups and Lattices
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(*  Title:      HOL/Lattices.thy
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    Author:     Tobias Nipkow
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*)
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header {* Abstract lattices *}
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theory Lattices
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imports Orderings Groups
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begin
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subsection {* Abstract semilattice *}
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text {*
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  This locales provide a basic structure for interpretation into
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  bigger structures;  extensions require careful thinking, otherwise
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  undesired effects may occur due to interpretation.
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*}
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locale semilattice = abel_semigroup +
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  assumes idem [simp]: "f a a = a"
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begin
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lemma left_idem [simp]:
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  "f a (f a b) = f a b"
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  by (simp add: assoc [symmetric])
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end
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subsection {* Idempotent semigroup *}
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class ab_semigroup_idem_mult = ab_semigroup_mult +
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  assumes mult_idem: "x * x = x"
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sublocale ab_semigroup_idem_mult < times!: semilattice times proof
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qed (fact mult_idem)
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context ab_semigroup_idem_mult
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begin
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lemmas mult_left_idem = times.left_idem
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end
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subsection {* Conrete lattices *}
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notation
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  less_eq  (infix "\<sqsubseteq>" 50) and
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  less  (infix "\<sqsubset>" 50) and
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  top ("\<top>") and
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  bot ("\<bottom>")
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class semilattice_inf = order +
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  fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<sqinter>" 70)
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  assumes inf_le1 [simp]: "x \<sqinter> y \<sqsubseteq> x"
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  and inf_le2 [simp]: "x \<sqinter> y \<sqsubseteq> y"
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  and inf_greatest: "x \<sqsubseteq> y \<Longrightarrow> x \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> y \<sqinter> z"
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class semilattice_sup = order +
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  assumes sup_ge1 [simp]: "x \<sqsubseteq> x \<squnion> y"
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  and sup_ge2 [simp]: "y \<sqsubseteq> x \<squnion> y"
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  and sup_least: "y \<sqsubseteq> x \<Longrightarrow> z \<sqsubseteq> x \<Longrightarrow> y \<squnion> z \<sqsubseteq> x"
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begin
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text {* Dual lattice *}
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lemma dual_semilattice:
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  "semilattice_inf (op \<ge>) (op >) sup"
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by (rule semilattice_inf.intro, rule dual_order)
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  (unfold_locales, simp_all add: sup_least)
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end
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class lattice = semilattice_inf + semilattice_sup
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subsubsection {* Intro and elim rules*}
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context semilattice_inf
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begin
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lemma le_infI1:
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  "a \<sqsubseteq> x \<Longrightarrow> a \<sqinter> b \<sqsubseteq> x"
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  by (rule order_trans) auto
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lemma le_infI2:
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  "b \<sqsubseteq> x \<Longrightarrow> a \<sqinter> b \<sqsubseteq> x"
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  by (rule order_trans) auto
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lemma le_infI: "x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> b \<Longrightarrow> x \<sqsubseteq> a \<sqinter> b"
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  by (blast intro: inf_greatest)
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lemma le_infE: "x \<sqsubseteq> a \<sqinter> b \<Longrightarrow> (x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> b \<Longrightarrow> P) \<Longrightarrow> P"
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  by (blast intro: order_trans le_infI1 le_infI2)
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lemma le_inf_iff [simp]:
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  "x \<sqsubseteq> y \<sqinter> z \<longleftrightarrow> x \<sqsubseteq> y \<and> x \<sqsubseteq> z"
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  by (blast intro: le_infI elim: le_infE)
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lemma le_iff_inf:
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  "x \<sqsubseteq> y \<longleftrightarrow> x \<sqinter> y = x"
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  by (auto intro: le_infI1 antisym dest: eq_iff [THEN iffD1])
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lemma mono_inf:
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  fixes f :: "'a \<Rightarrow> 'b\<Colon>semilattice_inf"
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  shows "mono f \<Longrightarrow> f (A \<sqinter> B) \<sqsubseteq> f A \<sqinter> f B"
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  by (auto simp add: mono_def intro: Lattices.inf_greatest)
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end
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context semilattice_sup
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begin
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lemma le_supI1:
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  "x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> a \<squnion> b"
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  by (rule order_trans) auto
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lemma le_supI2:
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  "x \<sqsubseteq> b \<Longrightarrow> x \<sqsubseteq> a \<squnion> b"
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  by (rule order_trans) auto 
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lemma le_supI:
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  "a \<sqsubseteq> x \<Longrightarrow> b \<sqsubseteq> x \<Longrightarrow> a \<squnion> b \<sqsubseteq> x"
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  by (blast intro: sup_least)
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lemma le_supE:
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  "a \<squnion> b \<sqsubseteq> x \<Longrightarrow> (a \<sqsubseteq> x \<Longrightarrow> b \<sqsubseteq> x \<Longrightarrow> P) \<Longrightarrow> P"
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  by (blast intro: le_supI1 le_supI2 order_trans)
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lemma le_sup_iff [simp]:
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  "x \<squnion> y \<sqsubseteq> z \<longleftrightarrow> x \<sqsubseteq> z \<and> y \<sqsubseteq> z"
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  by (blast intro: le_supI elim: le_supE)
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lemma le_iff_sup:
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  "x \<sqsubseteq> y \<longleftrightarrow> x \<squnion> y = y"
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  by (auto intro: le_supI2 antisym dest: eq_iff [THEN iffD1])
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lemma mono_sup:
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  fixes f :: "'a \<Rightarrow> 'b\<Colon>semilattice_sup"
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  shows "mono f \<Longrightarrow> f A \<squnion> f B \<sqsubseteq> f (A \<squnion> B)"
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  by (auto simp add: mono_def intro: Lattices.sup_least)
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end
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subsubsection {* Equational laws *}
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sublocale semilattice_inf < inf!: semilattice inf
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proof
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  fix a b c
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  show "(a \<sqinter> b) \<sqinter> c = a \<sqinter> (b \<sqinter> c)"
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    by (rule antisym) (auto intro: le_infI1 le_infI2)
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  show "a \<sqinter> b = b \<sqinter> a"
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    by (rule antisym) auto
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  show "a \<sqinter> a = a"
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    by (rule antisym) auto
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qed
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context semilattice_inf
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begin
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lemma inf_assoc: "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
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  by (fact inf.assoc)
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lemma inf_commute: "(x \<sqinter> y) = (y \<sqinter> x)"
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  by (fact inf.commute)
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lemma inf_left_commute: "x \<sqinter> (y \<sqinter> z) = y \<sqinter> (x \<sqinter> z)"
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  by (fact inf.left_commute)
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   172
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lemma inf_idem: "x \<sqinter> x = x"
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  by (fact inf.idem)
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   175
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lemma inf_left_idem: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"
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  by (fact inf.left_idem)
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lemma inf_absorb1: "x \<sqsubseteq> y \<Longrightarrow> x \<sqinter> y = x"
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  by (rule antisym) auto
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lemma inf_absorb2: "y \<sqsubseteq> x \<Longrightarrow> x \<sqinter> y = y"
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  by (rule antisym) auto
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   184
 
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lemmas inf_aci = inf_commute inf_assoc inf_left_commute inf_left_idem
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end
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sublocale semilattice_sup < sup!: semilattice sup
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proof
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  fix a b c
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  show "(a \<squnion> b) \<squnion> c = a \<squnion> (b \<squnion> c)"
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    by (rule antisym) (auto intro: le_supI1 le_supI2)
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  show "a \<squnion> b = b \<squnion> a"
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    by (rule antisym) auto
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  show "a \<squnion> a = a"
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    by (rule antisym) auto
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qed
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context semilattice_sup
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begin
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lemma sup_assoc: "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
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  by (fact sup.assoc)
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lemma sup_commute: "(x \<squnion> y) = (y \<squnion> x)"
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  by (fact sup.commute)
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   208
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lemma sup_left_commute: "x \<squnion> (y \<squnion> z) = y \<squnion> (x \<squnion> z)"
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  by (fact sup.left_commute)
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   211
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lemma sup_idem: "x \<squnion> x = x"
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  by (fact sup.idem)
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   214
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lemma sup_left_idem: "x \<squnion> (x \<squnion> y) = x \<squnion> y"
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  by (fact sup.left_idem)
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   217
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lemma sup_absorb1: "y \<sqsubseteq> x \<Longrightarrow> x \<squnion> y = x"
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  by (rule antisym) auto
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lemma sup_absorb2: "x \<sqsubseteq> y \<Longrightarrow> x \<squnion> y = y"
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  by (rule antisym) auto
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lemmas sup_aci = sup_commute sup_assoc sup_left_commute sup_left_idem
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131dd2a27137 Modified lattice locale
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   226
end
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context lattice
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begin
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31991
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lemma dual_lattice:
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  "lattice (op \<ge>) (op >) sup inf"
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  by (rule lattice.intro, rule dual_semilattice, rule semilattice_sup.intro, rule dual_order)
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    (unfold_locales, auto)
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   235
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lemma inf_sup_absorb: "x \<sqinter> (x \<squnion> y) = x"
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  by (blast intro: antisym inf_le1 inf_greatest sup_ge1)
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   238
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lemma sup_inf_absorb: "x \<squnion> (x \<sqinter> y) = x"
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  by (blast intro: antisym sup_ge1 sup_least inf_le1)
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   241
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lemmas inf_sup_aci = inf_aci sup_aci
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lemmas inf_sup_ord = inf_le1 inf_le2 sup_ge1 sup_ge2
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text{* Towards distributivity *}
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lemma distrib_sup_le: "x \<squnion> (y \<sqinter> z) \<sqsubseteq> (x \<squnion> y) \<sqinter> (x \<squnion> z)"
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  by (auto intro: le_infI1 le_infI2 le_supI1 le_supI2)
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   250
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lemma distrib_inf_le: "(x \<sqinter> y) \<squnion> (x \<sqinter> z) \<sqsubseteq> x \<sqinter> (y \<squnion> z)"
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  by (auto intro: le_infI1 le_infI2 le_supI1 le_supI2)
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text{* If you have one of them, you have them all. *}
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lemma distrib_imp1:
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assumes D: "!!x y z. x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
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shows "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
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   259
proof-
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  have "x \<squnion> (y \<sqinter> z) = (x \<squnion> (x \<sqinter> z)) \<squnion> (y \<sqinter> z)" by(simp add:sup_inf_absorb)
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  also have "\<dots> = x \<squnion> (z \<sqinter> (x \<squnion> y))" by(simp add:D inf_commute sup_assoc)
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  also have "\<dots> = ((x \<squnion> y) \<sqinter> x) \<squnion> ((x \<squnion> y) \<sqinter> z)"
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    by(simp add:inf_sup_absorb inf_commute)
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   264
  also have "\<dots> = (x \<squnion> y) \<sqinter> (x \<squnion> z)" by(simp add:D)
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  finally show ?thesis .
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qed
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   267
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lemma distrib_imp2:
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assumes D: "!!x y z. x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
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shows "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
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   271
proof-
d594c58e24ed renamed Lattice_Locales to Lattices
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   272
  have "x \<sqinter> (y \<squnion> z) = (x \<sqinter> (x \<squnion> z)) \<sqinter> (y \<squnion> z)" by(simp add:inf_sup_absorb)
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   273
  also have "\<dots> = x \<sqinter> (z \<squnion> (x \<sqinter> y))" by(simp add:D sup_commute inf_assoc)
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  also have "\<dots> = ((x \<sqinter> y) \<squnion> x) \<sqinter> ((x \<sqinter> y) \<squnion> z)"
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   275
    by(simp add:sup_inf_absorb sup_commute)
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   276
  also have "\<dots> = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" by(simp add:D)
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  finally show ?thesis .
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qed
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   279
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   280
end
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   282
subsubsection {* Strict order *}
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   283
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context semilattice_inf
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   285
begin
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   286
89518a3074e1 some lemmas about strict order in lattices
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   287
lemma less_infI1:
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   288
  "a \<sqsubset> x \<Longrightarrow> a \<sqinter> b \<sqsubset> x"
32642
026e7c6a6d08 be more cautious wrt. simp rules: inf_absorb1, inf_absorb2, sup_absorb1, sup_absorb2 are no simp rules by default any longer
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   289
  by (auto simp add: less_le inf_absorb1 intro: le_infI1)
32568
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   290
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   291
lemma less_infI2:
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   292
  "b \<sqsubset> x \<Longrightarrow> a \<sqinter> b \<sqsubset> x"
32642
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   293
  by (auto simp add: less_le inf_absorb2 intro: le_infI2)
32568
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   294
89518a3074e1 some lemmas about strict order in lattices
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   295
end
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   296
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   297
context semilattice_sup
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   298
begin
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   299
89518a3074e1 some lemmas about strict order in lattices
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   300
lemma less_supI1:
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   301
  "x \<sqsubset> a \<Longrightarrow> x \<sqsubset> a \<squnion> b"
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   302
proof -
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   303
  interpret dual: semilattice_inf "op \<ge>" "op >" sup
32568
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   304
    by (fact dual_semilattice)
34007
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haftmann
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diff changeset
   305
  assume "x \<sqsubset> a"
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
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diff changeset
   306
  then show "x \<sqsubset> a \<squnion> b"
32568
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haftmann
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diff changeset
   307
    by (fact dual.less_infI1)
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   308
qed
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   309
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   310
lemma less_supI2:
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   311
  "x \<sqsubset> b \<Longrightarrow> x \<sqsubset> a \<squnion> b"
32568
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   312
proof -
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34973
diff changeset
   313
  interpret dual: semilattice_inf "op \<ge>" "op >" sup
32568
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   314
    by (fact dual_semilattice)
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   315
  assume "x \<sqsubset> b"
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   316
  then show "x \<sqsubset> a \<squnion> b"
32568
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   317
    by (fact dual.less_infI2)
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   318
qed
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   319
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   320
end
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   321
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   322
24164
haftmann
parents: 23948
diff changeset
   323
subsection {* Distributive lattices *}
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   324
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   325
class distrib_lattice = lattice +
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   326
  assumes sup_inf_distrib1: "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   327
21733
131dd2a27137 Modified lattice locale
nipkow
parents: 21619
diff changeset
   328
context distrib_lattice
131dd2a27137 Modified lattice locale
nipkow
parents: 21619
diff changeset
   329
begin
131dd2a27137 Modified lattice locale
nipkow
parents: 21619
diff changeset
   330
131dd2a27137 Modified lattice locale
nipkow
parents: 21619
diff changeset
   331
lemma sup_inf_distrib2:
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   332
 "(y \<sqinter> z) \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)"
32064
53ca12ff305d refinement of lattice classes
haftmann
parents: 32063
diff changeset
   333
by(simp add: inf_sup_aci sup_inf_distrib1)
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   334
21733
131dd2a27137 Modified lattice locale
nipkow
parents: 21619
diff changeset
   335
lemma inf_sup_distrib1:
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   336
 "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   337
by(rule distrib_imp2[OF sup_inf_distrib1])
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   338
21733
131dd2a27137 Modified lattice locale
nipkow
parents: 21619
diff changeset
   339
lemma inf_sup_distrib2:
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   340
 "(y \<squnion> z) \<sqinter> x = (y \<sqinter> x) \<squnion> (z \<sqinter> x)"
32064
53ca12ff305d refinement of lattice classes
haftmann
parents: 32063
diff changeset
   341
by(simp add: inf_sup_aci inf_sup_distrib1)
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   342
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   343
lemma dual_distrib_lattice:
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   344
  "distrib_lattice (op \<ge>) (op >) sup inf"
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   345
  by (rule distrib_lattice.intro, rule dual_lattice)
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   346
    (unfold_locales, fact inf_sup_distrib1)
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   347
21733
131dd2a27137 Modified lattice locale
nipkow
parents: 21619
diff changeset
   348
lemmas distrib =
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   349
  sup_inf_distrib1 sup_inf_distrib2 inf_sup_distrib1 inf_sup_distrib2
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   350
21733
131dd2a27137 Modified lattice locale
nipkow
parents: 21619
diff changeset
   351
end
131dd2a27137 Modified lattice locale
nipkow
parents: 21619
diff changeset
   352
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   353
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   354
subsection {* Bounded lattices and boolean algebras *}
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   355
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   356
class bounded_lattice = lattice + top + bot
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   357
begin
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   358
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   359
lemma dual_bounded_lattice:
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   360
  "bounded_lattice (op \<ge>) (op >) (op \<squnion>) (op \<sqinter>) \<top> \<bottom>"
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   361
  by (rule bounded_lattice.intro, rule dual_lattice)
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   362
    (unfold_locales, auto simp add: less_le_not_le)
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   363
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   364
lemma inf_bot_left [simp]:
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   365
  "\<bottom> \<sqinter> x = \<bottom>"
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   366
  by (rule inf_absorb1) simp
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   367
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   368
lemma inf_bot_right [simp]:
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   369
  "x \<sqinter> \<bottom> = \<bottom>"
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   370
  by (rule inf_absorb2) simp
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   371
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   372
lemma sup_top_left [simp]:
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   373
  "\<top> \<squnion> x = \<top>"
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   374
  by (rule sup_absorb1) simp
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   375
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   376
lemma sup_top_right [simp]:
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   377
  "x \<squnion> \<top> = \<top>"
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   378
  by (rule sup_absorb2) simp
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   379
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   380
lemma inf_top_left [simp]:
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   381
  "\<top> \<sqinter> x = x"
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   382
  by (rule inf_absorb2) simp
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   383
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   384
lemma inf_top_right [simp]:
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   385
  "x \<sqinter> \<top> = x"
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   386
  by (rule inf_absorb1) simp
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   387
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   388
lemma sup_bot_left [simp]:
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   389
  "\<bottom> \<squnion> x = x"
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   390
  by (rule sup_absorb2) simp
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   391
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   392
lemma sup_bot_right [simp]:
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   393
  "x \<squnion> \<bottom> = x"
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   394
  by (rule sup_absorb1) simp
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   395
32568
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   396
lemma inf_eq_top_eq1:
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   397
  assumes "A \<sqinter> B = \<top>"
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   398
  shows "A = \<top>"
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   399
proof (cases "B = \<top>")
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   400
  case True with assms show ?thesis by simp
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   401
next
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   402
  case False with top_greatest have "B \<sqsubset> \<top>" by (auto intro: neq_le_trans)
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   403
  then have "A \<sqinter> B \<sqsubset> \<top>" by (rule less_infI2)
32568
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   404
  with assms show ?thesis by simp
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   405
qed
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   406
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   407
lemma inf_eq_top_eq2:
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   408
  assumes "A \<sqinter> B = \<top>"
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   409
  shows "B = \<top>"
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   410
  by (rule inf_eq_top_eq1, unfold inf_commute [of B]) (fact assms)
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   411
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   412
lemma sup_eq_bot_eq1:
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   413
  assumes "A \<squnion> B = \<bottom>"
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   414
  shows "A = \<bottom>"
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   415
proof -
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   416
  interpret dual: bounded_lattice "op \<ge>" "op >" "op \<squnion>" "op \<sqinter>" \<top> \<bottom>
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   417
    by (rule dual_bounded_lattice)
32568
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   418
  from dual.inf_eq_top_eq1 assms show ?thesis .
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   419
qed
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   420
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   421
lemma sup_eq_bot_eq2:
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   422
  assumes "A \<squnion> B = \<bottom>"
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   423
  shows "B = \<bottom>"
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   424
proof -
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   425
  interpret dual: bounded_lattice "op \<ge>" "op >" "op \<squnion>" "op \<sqinter>" \<top> \<bottom>
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   426
    by (rule dual_bounded_lattice)
32568
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   427
  from dual.inf_eq_top_eq2 assms show ?thesis .
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   428
qed
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   429
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   430
end
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   431
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   432
class boolean_algebra = distrib_lattice + bounded_lattice + minus + uminus +
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   433
  assumes inf_compl_bot: "x \<sqinter> - x = \<bottom>"
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   434
    and sup_compl_top: "x \<squnion> - x = \<top>"
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   435
  assumes diff_eq: "x - y = x \<sqinter> - y"
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   436
begin
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   437
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   438
lemma dual_boolean_algebra:
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   439
  "boolean_algebra (\<lambda>x y. x \<squnion> - y) uminus (op \<ge>) (op >) (op \<squnion>) (op \<sqinter>) \<top> \<bottom>"
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   440
  by (rule boolean_algebra.intro, rule dual_bounded_lattice, rule dual_distrib_lattice)
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   441
    (unfold_locales, auto simp add: inf_compl_bot sup_compl_top diff_eq)
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   442
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   443
lemma compl_inf_bot:
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   444
  "- x \<sqinter> x = \<bottom>"
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   445
  by (simp add: inf_commute inf_compl_bot)
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   446
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   447
lemma compl_sup_top:
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   448
  "- x \<squnion> x = \<top>"
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   449
  by (simp add: sup_commute sup_compl_top)
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   450
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   451
lemma compl_unique:
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   452
  assumes "x \<sqinter> y = \<bottom>"
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   453
    and "x \<squnion> y = \<top>"
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   454
  shows "- x = y"
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   455
proof -
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   456
  have "(x \<sqinter> - x) \<squnion> (- x \<sqinter> y) = (x \<sqinter> y) \<squnion> (- x \<sqinter> y)"
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   457
    using inf_compl_bot assms(1) by simp
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   458
  then have "(- x \<sqinter> x) \<squnion> (- x \<sqinter> y) = (y \<sqinter> x) \<squnion> (y \<sqinter> - x)"
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   459
    by (simp add: inf_commute)
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   460
  then have "- x \<sqinter> (x \<squnion> y) = y \<sqinter> (x \<squnion> - x)"
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   461
    by (simp add: inf_sup_distrib1)
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   462
  then have "- x \<sqinter> \<top> = y \<sqinter> \<top>"
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   463
    using sup_compl_top assms(2) by simp
34209
c7f621786035 killed a few warnings
krauss
parents: 34007
diff changeset
   464
  then show "- x = y" by simp
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   465
qed
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   466
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   467
lemma double_compl [simp]:
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   468
  "- (- x) = x"
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   469
  using compl_inf_bot compl_sup_top by (rule compl_unique)
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   470
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   471
lemma compl_eq_compl_iff [simp]:
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   472
  "- x = - y \<longleftrightarrow> x = y"
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   473
proof
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   474
  assume "- x = - y"
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   475
  then have "- x \<sqinter> y = \<bottom>"
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   476
    and "- x \<squnion> y = \<top>"
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   477
    by (simp_all add: compl_inf_bot compl_sup_top)
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   478
  then have "- (- x) = y" by (rule compl_unique)
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   479
  then show "x = y" by simp
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   480
next
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   481
  assume "x = y"
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   482
  then show "- x = - y" by simp
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   483
qed
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   484
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   485
lemma compl_bot_eq [simp]:
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   486
  "- \<bottom> = \<top>"
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   487
proof -
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   488
  from sup_compl_top have "\<bottom> \<squnion> - \<bottom> = \<top>" .
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   489
  then show ?thesis by simp
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   490
qed
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   491
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   492
lemma compl_top_eq [simp]:
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   493
  "- \<top> = \<bottom>"
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   494
proof -
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   495
  from inf_compl_bot have "\<top> \<sqinter> - \<top> = \<bottom>" .
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   496
  then show ?thesis by simp
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   497
qed
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   498
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   499
lemma compl_inf [simp]:
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   500
  "- (x \<sqinter> y) = - x \<squnion> - y"
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   501
proof (rule compl_unique)
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   502
  have "(x \<sqinter> y) \<sqinter> (- x \<squnion> - y) = ((x \<sqinter> y) \<sqinter> - x) \<squnion> ((x \<sqinter> y) \<sqinter> - y)"
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   503
    by (rule inf_sup_distrib1)
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   504
  also have "... = (y \<sqinter> (x \<sqinter> - x)) \<squnion> (x \<sqinter> (y \<sqinter> - y))"
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   505
    by (simp only: inf_commute inf_assoc inf_left_commute)
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   506
  finally show "(x \<sqinter> y) \<sqinter> (- x \<squnion> - y) = \<bottom>"
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   507
    by (simp add: inf_compl_bot)
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   508
next
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   509
  have "(x \<sqinter> y) \<squnion> (- x \<squnion> - y) = (x \<squnion> (- x \<squnion> - y)) \<sqinter> (y \<squnion> (- x \<squnion> - y))"
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   510
    by (rule sup_inf_distrib2)
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   511
  also have "... = (- y \<squnion> (x \<squnion> - x)) \<sqinter> (- x \<squnion> (y \<squnion> - y))"
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   512
    by (simp only: sup_commute sup_assoc sup_left_commute)
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   513
  finally show "(x \<sqinter> y) \<squnion> (- x \<squnion> - y) = \<top>"
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   514
    by (simp add: sup_compl_top)
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   515
qed
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   516
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   517
lemma compl_sup [simp]:
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   518
  "- (x \<squnion> y) = - x \<sqinter> - y"
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   519
proof -
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   520
  interpret boolean_algebra "\<lambda>x y. x \<squnion> - y" uminus "op \<ge>" "op >" "op \<squnion>" "op \<sqinter>" \<top> \<bottom>
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   521
    by (rule dual_boolean_algebra)
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   522
  then show ?thesis by simp
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   523
qed
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   524
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   525
end
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   526
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   527
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   528
subsection {* Uniqueness of inf and sup *}
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   529
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34973
diff changeset
   530
lemma (in semilattice_inf) inf_unique:
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   531
  fixes f (infixl "\<triangle>" 70)
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   532
  assumes le1: "\<And>x y. x \<triangle> y \<sqsubseteq> x" and le2: "\<And>x y. x \<triangle> y \<sqsubseteq> y"
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   533
  and greatest: "\<And>x y z. x \<sqsubseteq> y \<Longrightarrow> x \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> y \<triangle> z"
22737
haftmann
parents: 22548
diff changeset
   534
  shows "x \<sqinter> y = x \<triangle> y"
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   535
proof (rule antisym)
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   536
  show "x \<triangle> y \<sqsubseteq> x \<sqinter> y" by (rule le_infI) (rule le1, rule le2)
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   537
next
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   538
  have leI: "\<And>x y z. x \<sqsubseteq> y \<Longrightarrow> x \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> y \<triangle> z" by (blast intro: greatest)
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   539
  show "x \<sqinter> y \<sqsubseteq> x \<triangle> y" by (rule leI) simp_all
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   540
qed
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   541
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34973
diff changeset
   542
lemma (in semilattice_sup) sup_unique:
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   543
  fixes f (infixl "\<nabla>" 70)
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   544
  assumes ge1 [simp]: "\<And>x y. x \<sqsubseteq> x \<nabla> y" and ge2: "\<And>x y. y \<sqsubseteq> x \<nabla> y"
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   545
  and least: "\<And>x y z. y \<sqsubseteq> x \<Longrightarrow> z \<sqsubseteq> x \<Longrightarrow> y \<nabla> z \<sqsubseteq> x"
22737
haftmann
parents: 22548
diff changeset
   546
  shows "x \<squnion> y = x \<nabla> y"
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   547
proof (rule antisym)
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   548
  show "x \<squnion> y \<sqsubseteq> x \<nabla> y" by (rule le_supI) (rule ge1, rule ge2)
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   549
next
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   550
  have leI: "\<And>x y z. x \<sqsubseteq> z \<Longrightarrow> y \<sqsubseteq> z \<Longrightarrow> x \<nabla> y \<sqsubseteq> z" by (blast intro: least)
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   551
  show "x \<nabla> y \<sqsubseteq> x \<squnion> y" by (rule leI) simp_all
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   552
qed
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   553
  
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   554
22916
haftmann
parents: 22737
diff changeset
   555
subsection {* @{const min}/@{const max} on linear orders as
haftmann
parents: 22737
diff changeset
   556
  special case of @{const inf}/@{const sup} *}
haftmann
parents: 22737
diff changeset
   557
32512
d14762642cdd proper class syntax for sublocale class < expr
haftmann
parents: 32436
diff changeset
   558
sublocale linorder < min_max!: distrib_lattice less_eq less min max
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28692
diff changeset
   559
proof
22916
haftmann
parents: 22737
diff changeset
   560
  fix x y z
32512
d14762642cdd proper class syntax for sublocale class < expr
haftmann
parents: 32436
diff changeset
   561
  show "max x (min y z) = min (max x y) (max x z)"
d14762642cdd proper class syntax for sublocale class < expr
haftmann
parents: 32436
diff changeset
   562
    by (auto simp add: min_def max_def)
22916
haftmann
parents: 22737
diff changeset
   563
qed (auto simp add: min_def max_def not_le less_imp_le)
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   564
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34973
diff changeset
   565
lemma inf_min: "inf = (min \<Colon> 'a\<Colon>{semilattice_inf, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)"
25102
db3e412c4cb1 antisymmetry not a default intro rule any longer
haftmann
parents: 25062
diff changeset
   566
  by (rule ext)+ (auto intro: antisym)
21733
131dd2a27137 Modified lattice locale
nipkow
parents: 21619
diff changeset
   567
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34973
diff changeset
   568
lemma sup_max: "sup = (max \<Colon> 'a\<Colon>{semilattice_sup, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)"
25102
db3e412c4cb1 antisymmetry not a default intro rule any longer
haftmann
parents: 25062
diff changeset
   569
  by (rule ext)+ (auto intro: antisym)
21733
131dd2a27137 Modified lattice locale
nipkow
parents: 21619
diff changeset
   570
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   571
lemmas le_maxI1 = min_max.sup_ge1
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   572
lemmas le_maxI2 = min_max.sup_ge2
21381
79e065f2be95 reworking of min/max lemmas
haftmann
parents: 21312
diff changeset
   573
 
34973
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents: 34209
diff changeset
   574
lemmas min_ac = min_max.inf_assoc min_max.inf_commute
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents: 34209
diff changeset
   575
  min_max.inf.left_commute
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   576
34973
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents: 34209
diff changeset
   577
lemmas max_ac = min_max.sup_assoc min_max.sup_commute
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents: 34209
diff changeset
   578
  min_max.sup.left_commute
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents: 34209
diff changeset
   579
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   580
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   581
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   582
subsection {* Bool as lattice *}
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   583
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   584
instantiation bool :: boolean_algebra
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   585
begin
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   586
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   587
definition
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   588
  bool_Compl_def: "uminus = Not"
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   589
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   590
definition
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   591
  bool_diff_def: "A - B \<longleftrightarrow> A \<and> \<not> B"
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   592
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   593
definition
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   594
  inf_bool_eq: "P \<sqinter> Q \<longleftrightarrow> P \<and> Q"
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   595
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   596
definition
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   597
  sup_bool_eq: "P \<squnion> Q \<longleftrightarrow> P \<or> Q"
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   598
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   599
instance proof
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   600
qed (simp_all add: inf_bool_eq sup_bool_eq le_bool_def
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   601
  bot_bool_eq top_bool_eq bool_Compl_def bool_diff_def, auto)
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   602
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   603
end
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   604
32781
19c01bd7f6ae moved lemmas about sup on bool to Lattices.thy
haftmann
parents: 32780
diff changeset
   605
lemma sup_boolI1:
19c01bd7f6ae moved lemmas about sup on bool to Lattices.thy
haftmann
parents: 32780
diff changeset
   606
  "P \<Longrightarrow> P \<squnion> Q"
19c01bd7f6ae moved lemmas about sup on bool to Lattices.thy
haftmann
parents: 32780
diff changeset
   607
  by (simp add: sup_bool_eq)
19c01bd7f6ae moved lemmas about sup on bool to Lattices.thy
haftmann
parents: 32780
diff changeset
   608
19c01bd7f6ae moved lemmas about sup on bool to Lattices.thy
haftmann
parents: 32780
diff changeset
   609
lemma sup_boolI2:
19c01bd7f6ae moved lemmas about sup on bool to Lattices.thy
haftmann
parents: 32780
diff changeset
   610
  "Q \<Longrightarrow> P \<squnion> Q"
19c01bd7f6ae moved lemmas about sup on bool to Lattices.thy
haftmann
parents: 32780
diff changeset
   611
  by (simp add: sup_bool_eq)
19c01bd7f6ae moved lemmas about sup on bool to Lattices.thy
haftmann
parents: 32780
diff changeset
   612
19c01bd7f6ae moved lemmas about sup on bool to Lattices.thy
haftmann
parents: 32780
diff changeset
   613
lemma sup_boolE:
19c01bd7f6ae moved lemmas about sup on bool to Lattices.thy
haftmann
parents: 32780
diff changeset
   614
  "P \<squnion> Q \<Longrightarrow> (P \<Longrightarrow> R) \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
19c01bd7f6ae moved lemmas about sup on bool to Lattices.thy
haftmann
parents: 32780
diff changeset
   615
  by (auto simp add: sup_bool_eq)
19c01bd7f6ae moved lemmas about sup on bool to Lattices.thy
haftmann
parents: 32780
diff changeset
   616
23878
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   617
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   618
subsection {* Fun as lattice *}
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   619
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   620
instantiation "fun" :: (type, lattice) lattice
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   621
begin
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   622
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   623
definition
28562
4e74209f113e `code func` now just `code`
haftmann
parents: 27682
diff changeset
   624
  inf_fun_eq [code del]: "f \<sqinter> g = (\<lambda>x. f x \<sqinter> g x)"
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   625
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   626
definition
28562
4e74209f113e `code func` now just `code`
haftmann
parents: 27682
diff changeset
   627
  sup_fun_eq [code del]: "f \<squnion> g = (\<lambda>x. f x \<squnion> g x)"
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   628
32780
be337ec31268 tuned proofs
haftmann
parents: 32642
diff changeset
   629
instance proof
be337ec31268 tuned proofs
haftmann
parents: 32642
diff changeset
   630
qed (simp_all add: le_fun_def inf_fun_eq sup_fun_eq)
23878
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   631
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   632
end
23878
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   633
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   634
instance "fun" :: (type, distrib_lattice) distrib_lattice
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   635
proof
32780
be337ec31268 tuned proofs
haftmann
parents: 32642
diff changeset
   636
qed (simp_all add: inf_fun_eq sup_fun_eq sup_inf_distrib1)
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   637
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   638
instance "fun" :: (type, bounded_lattice) bounded_lattice ..
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   639
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   640
instantiation "fun" :: (type, uminus) uminus
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   641
begin
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   642
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   643
definition
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   644
  fun_Compl_def: "- A = (\<lambda>x. - A x)"
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   645
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   646
instance ..
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   647
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   648
end
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   649
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   650
instantiation "fun" :: (type, minus) minus
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   651
begin
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   652
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   653
definition
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   654
  fun_diff_def: "A - B = (\<lambda>x. A x - B x)"
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   655
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   656
instance ..
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   657
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   658
end
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   659
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   660
instance "fun" :: (type, boolean_algebra) boolean_algebra
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   661
proof
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   662
qed (simp_all add: inf_fun_eq sup_fun_eq bot_fun_eq top_fun_eq fun_Compl_def fun_diff_def
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   663
  inf_compl_bot sup_compl_top diff_eq)
23878
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   664
26794
354c3844dfde - Now imports Fun rather than Orderings
berghofe
parents: 26233
diff changeset
   665
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24749
diff changeset
   666
no_notation
25382
72cfe89f7b21 tuned specifications of 'notation';
wenzelm
parents: 25206
diff changeset
   667
  less_eq  (infix "\<sqsubseteq>" 50) and
72cfe89f7b21 tuned specifications of 'notation';
wenzelm
parents: 25206
diff changeset
   668
  less (infix "\<sqsubset>" 50) and
72cfe89f7b21 tuned specifications of 'notation';
wenzelm
parents: 25206
diff changeset
   669
  inf  (infixl "\<sqinter>" 70) and
32568
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   670
  sup  (infixl "\<squnion>" 65) and
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   671
  top ("\<top>") and
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   672
  bot ("\<bottom>")
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24749
diff changeset
   673
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   674
end