src/HOL/Lattices.thy
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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 {* Concrete 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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  bot ("\<bottom>") and
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  top ("\<top>")
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class inf =
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class sup = 
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class semilattice_inf =  order + inf +
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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 + sup +
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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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  "class.semilattice_inf sup greater_eq greater"
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by (rule class.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 (rule inf_greatest) (* FIXME: duplicate lemma *)
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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 inf_le1 inf_le2)
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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 inf_mono: "a \<sqsubseteq> c \<Longrightarrow> b \<sqsubseteq> d \<Longrightarrow> a \<sqinter> b \<sqsubseteq> c \<sqinter> d"
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  by (fast intro: inf_greatest le_infI1 le_infI2)
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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 (rule sup_least) (* FIXME: duplicate lemma *)
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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: order_trans sup_ge1 sup_ge2)
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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 sup_mono: "a \<sqsubseteq> c \<Longrightarrow> b \<sqsubseteq> d \<Longrightarrow> a \<squnion> b \<sqsubseteq> c \<squnion> d"
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  by (fast intro: sup_least le_supI1 le_supI2)
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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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   162
  fix a b c
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   163
  show "(a \<sqinter> b) \<sqinter> c = a \<sqinter> (b \<sqinter> c)"
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
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   164
    by (rule antisym) (auto intro: le_infI1 le_infI2)
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   165
  show "a \<sqinter> b = b \<sqinter> a"
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   166
    by (rule antisym) auto
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haftmann
parents: 34209
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   167
  show "a \<sqinter> a = a"
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haftmann
parents: 34209
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   168
    by (rule antisym) auto
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   169
qed
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   170
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   171
context semilattice_inf
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   172
begin
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   173
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   174
lemma inf_assoc: "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
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   175
  by (fact inf.assoc)
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   176
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   177
lemma inf_commute: "(x \<sqinter> y) = (y \<sqinter> x)"
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   178
  by (fact inf.commute)
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   179
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   180
lemma inf_left_commute: "x \<sqinter> (y \<sqinter> z) = y \<sqinter> (x \<sqinter> z)"
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   181
  by (fact inf.left_commute)
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   182
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   183
lemma inf_idem (*[simp]*): "x \<sqinter> x = x"
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   184
  by (fact inf.idem)
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   185
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   186
lemma inf_left_idem (*[simp]*): "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"
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   187
  by (fact inf.left_idem)
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   188
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   189
lemma inf_absorb1: "x \<sqsubseteq> y \<Longrightarrow> x \<sqinter> y = x"
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   190
  by (rule antisym) auto
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   191
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   192
lemma inf_absorb2: "y \<sqsubseteq> x \<Longrightarrow> x \<sqinter> y = y"
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   193
  by (rule antisym) auto
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   194
 
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   195
lemmas inf_aci = inf_commute inf_assoc inf_left_commute inf_left_idem
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   196
131dd2a27137 Modified lattice locale
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   197
end
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   198
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   199
sublocale semilattice_sup < sup!: semilattice sup
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   200
proof
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   201
  fix a b c
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   202
  show "(a \<squnion> b) \<squnion> c = a \<squnion> (b \<squnion> c)"
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
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diff changeset
   203
    by (rule antisym) (auto intro: le_supI1 le_supI2)
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
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   204
  show "a \<squnion> b = b \<squnion> a"
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
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   205
    by (rule antisym) auto
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   206
  show "a \<squnion> a = a"
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
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   207
    by (rule antisym) auto
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   208
qed
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   209
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   210
context semilattice_sup
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   211
begin
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   212
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   213
lemma sup_assoc: "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
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   214
  by (fact sup.assoc)
21733
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diff changeset
   215
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   216
lemma sup_commute: "(x \<squnion> y) = (y \<squnion> x)"
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   217
  by (fact sup.commute)
21733
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   218
34973
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   219
lemma sup_left_commute: "x \<squnion> (y \<squnion> z) = y \<squnion> (x \<squnion> z)"
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   220
  by (fact sup.left_commute)
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   221
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   222
lemma sup_idem (*[simp]*): "x \<squnion> x = x"
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   223
  by (fact sup.idem)
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   224
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   225
lemma sup_left_idem (*[simp]*): "x \<squnion> (x \<squnion> y) = x \<squnion> y"
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   226
  by (fact sup.left_idem)
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   227
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   228
lemma sup_absorb1: "y \<sqsubseteq> x \<Longrightarrow> x \<squnion> y = x"
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   229
  by (rule antisym) auto
21733
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diff changeset
   230
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   231
lemma sup_absorb2: "x \<sqsubseteq> y \<Longrightarrow> x \<squnion> y = y"
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   232
  by (rule antisym) auto
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   233
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   234
lemmas sup_aci = sup_commute sup_assoc sup_left_commute sup_left_idem
21733
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   235
131dd2a27137 Modified lattice locale
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   236
end
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   237
21733
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   238
context lattice
131dd2a27137 Modified lattice locale
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   239
begin
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   240
31991
37390299214a added boolean_algebra type class; tuned lattice duals
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diff changeset
   241
lemma dual_lattice:
44845
5e51075cbd97 added syntactic classes for "inf" and "sup"
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   242
  "class.lattice sup (op \<ge>) (op >) inf"
36635
080b755377c0 locale predicates of classes carry a mandatory "class" prefix
haftmann
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   243
  by (rule class.lattice.intro, rule dual_semilattice, rule class.semilattice_sup.intro, rule dual_order)
31991
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   244
    (unfold_locales, auto)
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
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diff changeset
   245
44085
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diff changeset
   246
lemma inf_sup_absorb (*[simp]*): "x \<sqinter> (x \<squnion> y) = x"
25102
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   247
  by (blast intro: antisym inf_le1 inf_greatest sup_ge1)
21733
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diff changeset
   248
44085
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   249
lemma sup_inf_absorb (*[simp]*): "x \<squnion> (x \<sqinter> y) = x"
25102
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diff changeset
   250
  by (blast intro: antisym sup_ge1 sup_least inf_le1)
21733
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diff changeset
   251
32064
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   252
lemmas inf_sup_aci = inf_aci sup_aci
21734
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diff changeset
   253
22454
c3654ba76a09 integrated with LOrder.thy
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   254
lemmas inf_sup_ord = inf_le1 inf_le2 sup_ge1 sup_ge2
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   255
21734
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   256
text{* Towards distributivity *}
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   257
21734
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   258
lemma distrib_sup_le: "x \<squnion> (y \<sqinter> z) \<sqsubseteq> (x \<squnion> y) \<sqinter> (x \<squnion> z)"
32064
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diff changeset
   259
  by (auto intro: le_infI1 le_infI2 le_supI1 le_supI2)
21734
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diff changeset
   260
283461c15fa7 renaming
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diff changeset
   261
lemma distrib_inf_le: "(x \<sqinter> y) \<squnion> (x \<sqinter> z) \<sqsubseteq> x \<sqinter> (y \<squnion> z)"
32064
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diff changeset
   262
  by (auto intro: le_infI1 le_infI2 le_supI1 le_supI2)
21734
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diff changeset
   263
283461c15fa7 renaming
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   264
text{* If you have one of them, you have them all. *}
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parents:
diff changeset
   265
21733
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   266
lemma distrib_imp1:
21249
d594c58e24ed renamed Lattice_Locales to Lattices
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parents:
diff changeset
   267
assumes D: "!!x y z. x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
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parents:
diff changeset
   268
shows "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   269
proof-
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   270
  have "x \<squnion> (y \<sqinter> z) = (x \<squnion> (x \<sqinter> z)) \<squnion> (y \<sqinter> z)" by(simp add:sup_inf_absorb)
34209
c7f621786035 killed a few warnings
krauss
parents: 34007
diff changeset
   271
  also have "\<dots> = x \<squnion> (z \<sqinter> (x \<squnion> y))" by(simp add:D inf_commute sup_assoc)
21249
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haftmann
parents:
diff changeset
   272
  also have "\<dots> = ((x \<squnion> y) \<sqinter> x) \<squnion> ((x \<squnion> y) \<sqinter> z)"
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   273
    by(simp add:inf_sup_absorb inf_commute)
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   274
  also have "\<dots> = (x \<squnion> y) \<sqinter> (x \<squnion> z)" by(simp add:D)
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   275
  finally show ?thesis .
d594c58e24ed renamed Lattice_Locales to Lattices
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parents:
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   276
qed
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   277
21733
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diff changeset
   278
lemma distrib_imp2:
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   279
assumes D: "!!x y z. x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   280
shows "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   281
proof-
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   282
  have "x \<sqinter> (y \<squnion> z) = (x \<sqinter> (x \<squnion> z)) \<sqinter> (y \<squnion> z)" by(simp add:inf_sup_absorb)
34209
c7f621786035 killed a few warnings
krauss
parents: 34007
diff changeset
   283
  also have "\<dots> = x \<sqinter> (z \<squnion> (x \<sqinter> y))" by(simp add:D sup_commute inf_assoc)
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   284
  also have "\<dots> = ((x \<sqinter> y) \<squnion> x) \<sqinter> ((x \<sqinter> y) \<squnion> z)"
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   285
    by(simp add:sup_inf_absorb sup_commute)
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   286
  also have "\<dots> = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" by(simp add:D)
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   287
  finally show ?thesis .
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   288
qed
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   289
21733
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diff changeset
   290
end
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   291
32568
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   292
subsubsection {* Strict order *}
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   293
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
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parents: 34973
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   294
context semilattice_inf
32568
89518a3074e1 some lemmas about strict order in lattices
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diff changeset
   295
begin
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   296
89518a3074e1 some lemmas about strict order in lattices
haftmann
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diff changeset
   297
lemma less_infI1:
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   298
  "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
haftmann
parents: 32568
diff changeset
   299
  by (auto simp add: less_le inf_absorb1 intro: le_infI1)
32568
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   300
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   301
lemma less_infI2:
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   302
  "b \<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
haftmann
parents: 32568
diff changeset
   303
  by (auto simp add: less_le inf_absorb2 intro: le_infI2)
32568
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   304
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   305
end
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   306
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
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   307
context semilattice_sup
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   308
begin
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   309
89518a3074e1 some lemmas about strict order in lattices
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   310
lemma less_supI1:
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  "x \<sqsubset> a \<Longrightarrow> x \<sqsubset> a \<squnion> b"
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   312
proof -
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  interpret dual: semilattice_inf sup "op \<ge>" "op >"
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    by (fact dual_semilattice)
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  assume "x \<sqsubset> a"
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
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   316
  then show "x \<sqsubset> a \<squnion> b"
32568
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diff changeset
   317
    by (fact dual.less_infI1)
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   318
qed
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diff changeset
   319
89518a3074e1 some lemmas about strict order in lattices
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diff changeset
   320
lemma less_supI2:
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diff changeset
   321
  "x \<sqsubset> b \<Longrightarrow> x \<sqsubset> a \<squnion> b"
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diff changeset
   322
proof -
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5e51075cbd97 added syntactic classes for "inf" and "sup"
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diff changeset
   323
  interpret dual: semilattice_inf sup "op \<ge>" "op >"
32568
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haftmann
parents: 32512
diff changeset
   324
    by (fact dual_semilattice)
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   325
  assume "x \<sqsubset> b"
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   326
  then show "x \<sqsubset> a \<squnion> b"
32568
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   327
    by (fact dual.less_infI2)
89518a3074e1 some lemmas about strict order in lattices
haftmann
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diff changeset
   328
qed
89518a3074e1 some lemmas about strict order in lattices
haftmann
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diff changeset
   329
89518a3074e1 some lemmas about strict order in lattices
haftmann
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diff changeset
   330
end
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   331
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
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   332
24164
haftmann
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   333
subsection {* Distributive lattices *}
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d594c58e24ed renamed Lattice_Locales to Lattices
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parents:
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   334
22454
c3654ba76a09 integrated with LOrder.thy
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   335
class distrib_lattice = lattice +
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parents:
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   336
  assumes sup_inf_distrib1: "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
d594c58e24ed renamed Lattice_Locales to Lattices
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parents:
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   337
21733
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   338
context distrib_lattice
131dd2a27137 Modified lattice locale
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   339
begin
131dd2a27137 Modified lattice locale
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   340
131dd2a27137 Modified lattice locale
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   341
lemma sup_inf_distrib2:
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parents:
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   342
 "(y \<sqinter> z) \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)"
32064
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   343
by(simp add: inf_sup_aci sup_inf_distrib1)
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parents:
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   344
21733
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   345
lemma inf_sup_distrib1:
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parents:
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   346
 "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
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   347
by(rule distrib_imp2[OF sup_inf_distrib1])
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   348
21733
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   349
lemma inf_sup_distrib2:
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haftmann
parents:
diff changeset
   350
 "(y \<squnion> z) \<sqinter> x = (y \<sqinter> x) \<squnion> (z \<sqinter> x)"
32064
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haftmann
parents: 32063
diff changeset
   351
by(simp add: inf_sup_aci inf_sup_distrib1)
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   352
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   353
lemma dual_distrib_lattice:
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   354
  "class.distrib_lattice sup (op \<ge>) (op >) inf"
36635
080b755377c0 locale predicates of classes carry a mandatory "class" prefix
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   355
  by (rule class.distrib_lattice.intro, rule dual_lattice)
31991
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haftmann
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diff changeset
   356
    (unfold_locales, fact inf_sup_distrib1)
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diff changeset
   357
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   358
lemmas sup_inf_distrib =
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   359
  sup_inf_distrib1 sup_inf_distrib2
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   360
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   361
lemmas inf_sup_distrib =
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   362
  inf_sup_distrib1 inf_sup_distrib2
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   363
21733
131dd2a27137 Modified lattice locale
nipkow
parents: 21619
diff changeset
   364
lemmas distrib =
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   365
  sup_inf_distrib1 sup_inf_distrib2 inf_sup_distrib1 inf_sup_distrib2
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   366
21733
131dd2a27137 Modified lattice locale
nipkow
parents: 21619
diff changeset
   367
end
131dd2a27137 Modified lattice locale
nipkow
parents: 21619
diff changeset
   368
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   369
34007
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   370
subsection {* Bounded lattices and boolean algebras *}
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   371
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Cezary Kaliszyk <kaliszyk@in.tum.de>
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   372
class bounded_lattice_bot = lattice + bot
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haftmann
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   373
begin
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   374
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
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diff changeset
   375
lemma inf_bot_left [simp]:
34007
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   376
  "\<bottom> \<sqinter> x = \<bottom>"
31991
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haftmann
parents: 30729
diff changeset
   377
  by (rule inf_absorb1) simp
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haftmann
parents: 30729
diff changeset
   378
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
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diff changeset
   379
lemma inf_bot_right [simp]:
34007
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haftmann
parents: 32781
diff changeset
   380
  "x \<sqinter> \<bottom> = \<bottom>"
31991
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haftmann
parents: 30729
diff changeset
   381
  by (rule inf_absorb2) simp
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   382
36352
f71978e47cd5 add bounded_lattice_bot and bounded_lattice_top type classes
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36096
diff changeset
   383
lemma sup_bot_left [simp]:
f71978e47cd5 add bounded_lattice_bot and bounded_lattice_top type classes
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36096
diff changeset
   384
  "\<bottom> \<squnion> x = x"
f71978e47cd5 add bounded_lattice_bot and bounded_lattice_top type classes
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36096
diff changeset
   385
  by (rule sup_absorb2) simp
f71978e47cd5 add bounded_lattice_bot and bounded_lattice_top type classes
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36096
diff changeset
   386
f71978e47cd5 add bounded_lattice_bot and bounded_lattice_top type classes
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36096
diff changeset
   387
lemma sup_bot_right [simp]:
f71978e47cd5 add bounded_lattice_bot and bounded_lattice_top type classes
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36096
diff changeset
   388
  "x \<squnion> \<bottom> = x"
f71978e47cd5 add bounded_lattice_bot and bounded_lattice_top type classes
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36096
diff changeset
   389
  by (rule sup_absorb1) simp
f71978e47cd5 add bounded_lattice_bot and bounded_lattice_top type classes
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36096
diff changeset
   390
f71978e47cd5 add bounded_lattice_bot and bounded_lattice_top type classes
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36096
diff changeset
   391
lemma sup_eq_bot_iff [simp]:
f71978e47cd5 add bounded_lattice_bot and bounded_lattice_top type classes
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36096
diff changeset
   392
  "x \<squnion> y = \<bottom> \<longleftrightarrow> x = \<bottom> \<and> y = \<bottom>"
f71978e47cd5 add bounded_lattice_bot and bounded_lattice_top type classes
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36096
diff changeset
   393
  by (simp add: eq_iff)
f71978e47cd5 add bounded_lattice_bot and bounded_lattice_top type classes
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36096
diff changeset
   394
f71978e47cd5 add bounded_lattice_bot and bounded_lattice_top type classes
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36096
diff changeset
   395
end
f71978e47cd5 add bounded_lattice_bot and bounded_lattice_top type classes
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36096
diff changeset
   396
f71978e47cd5 add bounded_lattice_bot and bounded_lattice_top type classes
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36096
diff changeset
   397
class bounded_lattice_top = lattice + top
f71978e47cd5 add bounded_lattice_bot and bounded_lattice_top type classes
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36096
diff changeset
   398
begin
f71978e47cd5 add bounded_lattice_bot and bounded_lattice_top type classes
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36096
diff changeset
   399
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   400
lemma sup_top_left [simp]:
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   401
  "\<top> \<squnion> x = \<top>"
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   402
  by (rule sup_absorb1) simp
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   403
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   404
lemma sup_top_right [simp]:
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   405
  "x \<squnion> \<top> = \<top>"
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   406
  by (rule sup_absorb2) simp
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   407
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   408
lemma inf_top_left [simp]:
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   409
  "\<top> \<sqinter> x = x"
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   410
  by (rule inf_absorb2) simp
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   411
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   412
lemma inf_top_right [simp]:
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   413
  "x \<sqinter> \<top> = x"
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   414
  by (rule inf_absorb1) simp
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   415
36008
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   416
lemma inf_eq_top_iff [simp]:
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   417
  "x \<sqinter> y = \<top> \<longleftrightarrow> x = \<top> \<and> y = \<top>"
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   418
  by (simp add: eq_iff)
32568
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   419
36352
f71978e47cd5 add bounded_lattice_bot and bounded_lattice_top type classes
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36096
diff changeset
   420
end
f71978e47cd5 add bounded_lattice_bot and bounded_lattice_top type classes
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36096
diff changeset
   421
f71978e47cd5 add bounded_lattice_bot and bounded_lattice_top type classes
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36096
diff changeset
   422
class bounded_lattice = bounded_lattice_bot + bounded_lattice_top
f71978e47cd5 add bounded_lattice_bot and bounded_lattice_top type classes
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36096
diff changeset
   423
begin
f71978e47cd5 add bounded_lattice_bot and bounded_lattice_top type classes
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36096
diff changeset
   424
f71978e47cd5 add bounded_lattice_bot and bounded_lattice_top type classes
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36096
diff changeset
   425
lemma dual_bounded_lattice:
44845
5e51075cbd97 added syntactic classes for "inf" and "sup"
krauss
parents: 44085
diff changeset
   426
  "class.bounded_lattice sup greater_eq greater inf \<top> \<bottom>"
36352
f71978e47cd5 add bounded_lattice_bot and bounded_lattice_top type classes
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36096
diff changeset
   427
  by unfold_locales (auto simp add: less_le_not_le)
32568
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   428
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   429
end
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   430
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   431
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
   432
  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
   433
    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
   434
  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
   435
begin
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   436
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   437
lemma dual_boolean_algebra:
44845
5e51075cbd97 added syntactic classes for "inf" and "sup"
krauss
parents: 44085
diff changeset
   438
  "class.boolean_algebra (\<lambda>x y. x \<squnion> - y) uminus sup greater_eq greater inf \<top> \<bottom>"
36635
080b755377c0 locale predicates of classes carry a mandatory "class" prefix
haftmann
parents: 36352
diff changeset
   439
  by (rule class.boolean_algebra.intro, rule dual_bounded_lattice, rule dual_distrib_lattice)
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   440
    (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
   441
44085
a65e26f1427b move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents: 43873
diff changeset
   442
lemma compl_inf_bot (*[simp]*):
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   443
  "- x \<sqinter> x = \<bottom>"
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   444
  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
   445
44085
a65e26f1427b move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents: 43873
diff changeset
   446
lemma compl_sup_top (*[simp]*):
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   447
  "- x \<squnion> x = \<top>"
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   448
  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
   449
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   450
lemma compl_unique:
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   451
  assumes "x \<sqinter> y = \<bottom>"
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   452
    and "x \<squnion> y = \<top>"
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   453
  shows "- x = y"
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   454
proof -
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   455
  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
   456
    using inf_compl_bot assms(1) by simp
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   457
  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
   458
    by (simp add: inf_commute)
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   459
  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
   460
    by (simp add: inf_sup_distrib1)
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   461
  then have "- x \<sqinter> \<top> = y \<sqinter> \<top>"
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   462
    using sup_compl_top assms(2) by simp
34209
c7f621786035 killed a few warnings
krauss
parents: 34007
diff changeset
   463
  then show "- x = y" by simp
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   464
qed
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   465
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   466
lemma double_compl [simp]:
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   467
  "- (- x) = x"
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   468
  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
   469
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   470
lemma compl_eq_compl_iff [simp]:
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   471
  "- x = - y \<longleftrightarrow> x = y"
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   472
proof
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   473
  assume "- x = - y"
36008
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   474
  then have "- (- x) = - (- y)" by (rule arg_cong)
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   475
  then show "x = y" by simp
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   476
next
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   477
  assume "x = y"
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   478
  then show "- x = - y" by simp
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   479
qed
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   480
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   481
lemma compl_bot_eq [simp]:
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   482
  "- \<bottom> = \<top>"
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   483
proof -
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   484
  from sup_compl_top have "\<bottom> \<squnion> - \<bottom> = \<top>" .
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   485
  then show ?thesis by simp
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   486
qed
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   487
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   488
lemma compl_top_eq [simp]:
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   489
  "- \<top> = \<bottom>"
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   490
proof -
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   491
  from inf_compl_bot have "\<top> \<sqinter> - \<top> = \<bottom>" .
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   492
  then show ?thesis by simp
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   493
qed
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   494
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   495
lemma compl_inf [simp]:
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   496
  "- (x \<sqinter> y) = - x \<squnion> - y"
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   497
proof (rule compl_unique)
36008
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   498
  have "(x \<sqinter> y) \<sqinter> (- x \<squnion> - y) = (y \<sqinter> (x \<sqinter> - x)) \<squnion> (x \<sqinter> (y \<sqinter> - y))"
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   499
    by (simp only: inf_sup_distrib inf_aci)
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   500
  then show "(x \<sqinter> y) \<sqinter> (- x \<squnion> - y) = \<bottom>"
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   501
    by (simp add: inf_compl_bot)
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   502
next
36008
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   503
  have "(x \<sqinter> y) \<squnion> (- x \<squnion> - y) = (- y \<squnion> (x \<squnion> - x)) \<sqinter> (- x \<squnion> (y \<squnion> - y))"
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   504
    by (simp only: sup_inf_distrib sup_aci)
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   505
  then show "(x \<sqinter> y) \<squnion> (- x \<squnion> - y) = \<top>"
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   506
    by (simp add: sup_compl_top)
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   507
qed
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   508
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   509
lemma compl_sup [simp]:
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   510
  "- (x \<squnion> y) = - x \<sqinter> - y"
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   511
proof -
44845
5e51075cbd97 added syntactic classes for "inf" and "sup"
krauss
parents: 44085
diff changeset
   512
  interpret boolean_algebra "\<lambda>x y. x \<squnion> - y" uminus sup greater_eq greater inf \<top> \<bottom>
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   513
    by (rule dual_boolean_algebra)
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   514
  then show ?thesis by simp
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
36008
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   517
lemma compl_mono:
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   518
  "x \<sqsubseteq> y \<Longrightarrow> - y \<sqsubseteq> - x"
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   519
proof -
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   520
  assume "x \<sqsubseteq> y"
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   521
  then have "x \<squnion> y = y" by (simp only: le_iff_sup)
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   522
  then have "- (x \<squnion> y) = - y" by simp
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   523
  then have "- x \<sqinter> - y = - y" by simp
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   524
  then have "- y \<sqinter> - x = - y" by (simp only: inf_commute)
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   525
  then show "- y \<sqsubseteq> - x" by (simp only: le_iff_inf)
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   526
qed
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   527
44085
a65e26f1427b move legacy candiates to bottom; marked candidates for default simp rules
haftmann
parents: 43873
diff changeset
   528
lemma compl_le_compl_iff (*[simp]*):
43753
fe5e846c0839 tuned notation
haftmann
parents: 41082
diff changeset
   529
  "- x \<sqsubseteq> - y \<longleftrightarrow> y \<sqsubseteq> x"
43873
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   530
  by (auto dest: compl_mono)
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   531
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   532
lemma compl_le_swap1:
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   533
  assumes "y \<sqsubseteq> - x" shows "x \<sqsubseteq> -y"
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   534
proof -
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   535
  from assms have "- (- x) \<sqsubseteq> - y" by (simp only: compl_le_compl_iff)
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   536
  then show ?thesis by simp
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   537
qed
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   538
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   539
lemma compl_le_swap2:
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   540
  assumes "- y \<sqsubseteq> x" shows "- x \<sqsubseteq> y"
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   541
proof -
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   542
  from assms have "- x \<sqsubseteq> - (- y)" by (simp only: compl_le_compl_iff)
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   543
  then show ?thesis by simp
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   544
qed
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   545
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   546
lemma compl_less_compl_iff: (* TODO: declare [simp] ? *)
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   547
  "- x \<sqsubset> - y \<longleftrightarrow> y \<sqsubset> x"
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   548
  by (auto simp add: less_le compl_le_compl_iff)
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   549
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   550
lemma compl_less_swap1:
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   551
  assumes "y \<sqsubset> - x" shows "x \<sqsubset> - y"
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   552
proof -
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   553
  from assms have "- (- x) \<sqsubset> - y" by (simp only: compl_less_compl_iff)
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   554
  then show ?thesis by simp
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   555
qed
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   556
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   557
lemma compl_less_swap2:
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   558
  assumes "- y \<sqsubset> x" shows "- x \<sqsubset> y"
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   559
proof -
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   560
  from assms have "- x \<sqsubset> - (- y)" by (simp only: compl_less_compl_iff)
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   561
  then show ?thesis by simp
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   562
qed
36008
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   563
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   564
end
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   565
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   566
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   567
subsection {* Uniqueness of inf and sup *}
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   568
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34973
diff changeset
   569
lemma (in semilattice_inf) inf_unique:
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   570
  fixes f (infixl "\<triangle>" 70)
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   571
  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
   572
  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
   573
  shows "x \<sqinter> y = x \<triangle> y"
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   574
proof (rule antisym)
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   575
  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
   576
next
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   577
  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
   578
  show "x \<sqinter> y \<sqsubseteq> x \<triangle> y" by (rule leI) simp_all
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   579
qed
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   580
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34973
diff changeset
   581
lemma (in semilattice_sup) sup_unique:
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   582
  fixes f (infixl "\<nabla>" 70)
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   583
  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
   584
  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
   585
  shows "x \<squnion> y = x \<nabla> y"
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   586
proof (rule antisym)
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   587
  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
   588
next
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   589
  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
   590
  show "x \<nabla> y \<sqsubseteq> x \<squnion> y" by (rule leI) simp_all
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   591
qed
36008
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   592
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   593
22916
haftmann
parents: 22737
diff changeset
   594
subsection {* @{const min}/@{const max} on linear orders as
haftmann
parents: 22737
diff changeset
   595
  special case of @{const inf}/@{const sup} *}
haftmann
parents: 22737
diff changeset
   596
44845
5e51075cbd97 added syntactic classes for "inf" and "sup"
krauss
parents: 44085
diff changeset
   597
sublocale linorder < min_max!: distrib_lattice min less_eq less max
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28692
diff changeset
   598
proof
22916
haftmann
parents: 22737
diff changeset
   599
  fix x y z
32512
d14762642cdd proper class syntax for sublocale class < expr
haftmann
parents: 32436
diff changeset
   600
  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
   601
    by (auto simp add: min_def max_def)
22916
haftmann
parents: 22737
diff changeset
   602
qed (auto simp add: min_def max_def not_le less_imp_le)
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   603
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34973
diff changeset
   604
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
   605
  by (rule ext)+ (auto intro: antisym)
21733
131dd2a27137 Modified lattice locale
nipkow
parents: 21619
diff changeset
   606
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34973
diff changeset
   607
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
   608
  by (rule ext)+ (auto intro: antisym)
21733
131dd2a27137 Modified lattice locale
nipkow
parents: 21619
diff changeset
   609
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   610
lemmas le_maxI1 = min_max.sup_ge1
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   611
lemmas le_maxI2 = min_max.sup_ge2
21381
79e065f2be95 reworking of min/max lemmas
haftmann
parents: 21312
diff changeset
   612
 
34973
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents: 34209
diff changeset
   613
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
   614
  min_max.inf.left_commute
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   615
34973
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents: 34209
diff changeset
   616
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
   617
  min_max.sup.left_commute
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents: 34209
diff changeset
   618
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   619
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   620
subsection {* Bool as lattice *}
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   621
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   622
instantiation bool :: boolean_algebra
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   623
begin
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   624
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   625
definition
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   626
  bool_Compl_def [simp]: "uminus = Not"
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   627
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   628
definition
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   629
  bool_diff_def [simp]: "A - B \<longleftrightarrow> A \<and> \<not> B"
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   630
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   631
definition
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   632
  [simp]: "P \<sqinter> Q \<longleftrightarrow> P \<and> Q"
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   633
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   634
definition
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   635
  [simp]: "P \<squnion> Q \<longleftrightarrow> P \<or> Q"
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   636
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   637
instance proof
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   638
qed auto
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   639
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   640
end
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   641
32781
19c01bd7f6ae moved lemmas about sup on bool to Lattices.thy
haftmann
parents: 32780
diff changeset
   642
lemma sup_boolI1:
19c01bd7f6ae moved lemmas about sup on bool to Lattices.thy
haftmann
parents: 32780
diff changeset
   643
  "P \<Longrightarrow> P \<squnion> Q"
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   644
  by simp
32781
19c01bd7f6ae moved lemmas about sup on bool to Lattices.thy
haftmann
parents: 32780
diff changeset
   645
19c01bd7f6ae moved lemmas about sup on bool to Lattices.thy
haftmann
parents: 32780
diff changeset
   646
lemma sup_boolI2:
19c01bd7f6ae moved lemmas about sup on bool to Lattices.thy
haftmann
parents: 32780
diff changeset
   647
  "Q \<Longrightarrow> P \<squnion> Q"
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   648
  by simp
32781
19c01bd7f6ae moved lemmas about sup on bool to Lattices.thy
haftmann
parents: 32780
diff changeset
   649
19c01bd7f6ae moved lemmas about sup on bool to Lattices.thy
haftmann
parents: 32780
diff changeset
   650
lemma sup_boolE:
19c01bd7f6ae moved lemmas about sup on bool to Lattices.thy
haftmann
parents: 32780
diff changeset
   651
  "P \<squnion> Q \<Longrightarrow> (P \<Longrightarrow> R) \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   652
  by auto
32781
19c01bd7f6ae moved lemmas about sup on bool to Lattices.thy
haftmann
parents: 32780
diff changeset
   653
23878
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   654
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   655
subsection {* Fun as lattice *}
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   656
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   657
instantiation "fun" :: (type, lattice) lattice
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   658
begin
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   659
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   660
definition
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   661
  "f \<sqinter> g = (\<lambda>x. f x \<sqinter> g x)"
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   662
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   663
lemma inf_apply:
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   664
  "(f \<sqinter> g) x = f x \<sqinter> g x"
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   665
  by (simp add: inf_fun_def)
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   666
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   667
definition
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   668
  "f \<squnion> g = (\<lambda>x. f x \<squnion> g x)"
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   669
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   670
lemma sup_apply:
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   671
  "(f \<squnion> g) x = f x \<squnion> g x"
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   672
  by (simp add: sup_fun_def)
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   673
32780
be337ec31268 tuned proofs
haftmann
parents: 32642
diff changeset
   674
instance proof
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   675
qed (simp_all add: le_fun_def inf_apply sup_apply)
23878
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   676
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   677
end
23878
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   678
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   679
instance "fun" :: (type, distrib_lattice) distrib_lattice proof
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   680
qed (rule ext, simp add: sup_inf_distrib1 inf_apply sup_apply)
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   681
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   682
instance "fun" :: (type, bounded_lattice) bounded_lattice ..
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   683
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   684
instantiation "fun" :: (type, uminus) uminus
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   685
begin
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   686
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   687
definition
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   688
  fun_Compl_def: "- A = (\<lambda>x. - A x)"
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   689
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   690
lemma uminus_apply:
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   691
  "(- A) x = - (A x)"
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   692
  by (simp add: fun_Compl_def)
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   693
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   694
instance ..
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   695
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   696
end
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   697
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   698
instantiation "fun" :: (type, minus) minus
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   699
begin
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   700
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   701
definition
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   702
  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
   703
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   704
lemma minus_apply:
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   705
  "(A - B) x = A x - B x"
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   706
  by (simp add: fun_diff_def)
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   707
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   708
instance ..
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   709
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   710
end
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   711
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   712
instance "fun" :: (type, boolean_algebra) boolean_algebra proof
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   713
qed (rule ext, simp_all add: inf_apply sup_apply bot_apply top_apply uminus_apply minus_apply inf_compl_bot sup_compl_top diff_eq)+
26794
354c3844dfde - Now imports Fun rather than Orderings
berghofe
parents: 26233
diff changeset
   714
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24749
diff changeset
   715
no_notation
25382
72cfe89f7b21 tuned specifications of 'notation';
wenzelm
parents: 25206
diff changeset
   716
  less_eq  (infix "\<sqsubseteq>" 50) and
72cfe89f7b21 tuned specifications of 'notation';
wenzelm
parents: 25206
diff changeset
   717
  less (infix "\<sqsubset>" 50) and
72cfe89f7b21 tuned specifications of 'notation';
wenzelm
parents: 25206
diff changeset
   718
  inf  (infixl "\<sqinter>" 70) and
32568
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   719
  sup  (infixl "\<squnion>" 65) and
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   720
  top ("\<top>") and
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   721
  bot ("\<bottom>")
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24749
diff changeset
   722
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   723
end