src/HOL/Lattices.thy
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explicit checks stable_finished_theory/stable_command allow parallel asynchronous command transactions; tuned;
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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 {* Syntactic infimum and supremum operations *}
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class inf =
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class sup = 
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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)
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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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  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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  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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   163
  fix a b c
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   164
  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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   165
    by (rule antisym) (auto intro: le_infI1 le_infI2)
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
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parents: 34209
diff changeset
   166
  show "a \<sqinter> b = b \<sqinter> a"
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
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   167
    by (rule antisym) auto
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents: 34209
diff changeset
   168
  show "a \<sqinter> a = a"
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents: 34209
diff changeset
   169
    by (rule antisym) auto
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   170
qed
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haftmann
parents: 34209
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   171
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   172
context semilattice_inf
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   173
begin
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   174
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   175
lemma inf_assoc: "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
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   176
  by (fact inf.assoc)
21733
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   177
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   178
lemma inf_commute: "(x \<sqinter> y) = (y \<sqinter> x)"
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   179
  by (fact inf.commute)
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   180
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   181
lemma inf_left_commute: "x \<sqinter> (y \<sqinter> z) = y \<sqinter> (x \<sqinter> z)"
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   182
  by (fact inf.left_commute)
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   183
44921
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   184
lemma inf_idem: "x \<sqinter> x = x"
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   185
  by (fact inf.idem) (* already simp *)
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   186
44918
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   187
lemma inf_left_idem [simp]: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"
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   188
  by (fact inf.left_idem)
21733
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   189
32642
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   190
lemma inf_absorb1: "x \<sqsubseteq> y \<Longrightarrow> x \<sqinter> y = x"
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   191
  by (rule antisym) auto
21733
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diff changeset
   192
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   193
lemma inf_absorb2: "y \<sqsubseteq> x \<Longrightarrow> x \<sqinter> y = y"
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   194
  by (rule antisym) auto
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   195
 
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   196
lemmas inf_aci = inf_commute inf_assoc inf_left_commute inf_left_idem
21733
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   197
131dd2a27137 Modified lattice locale
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   198
end
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   199
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   200
sublocale semilattice_sup < sup!: semilattice sup
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   201
proof
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   202
  fix a b c
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   203
  show "(a \<squnion> b) \<squnion> c = a \<squnion> (b \<squnion> c)"
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
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   204
    by (rule antisym) (auto intro: le_supI1 le_supI2)
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents: 34209
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   205
  show "a \<squnion> b = b \<squnion> a"
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents: 34209
diff changeset
   206
    by (rule antisym) auto
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents: 34209
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   207
  show "a \<squnion> a = a"
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents: 34209
diff changeset
   208
    by (rule antisym) auto
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
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   209
qed
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents: 34209
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   210
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   211
context semilattice_sup
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   212
begin
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haftmann
parents:
diff changeset
   213
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   214
lemma sup_assoc: "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents: 34209
diff changeset
   215
  by (fact sup.assoc)
21733
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nipkow
parents: 21619
diff changeset
   216
34973
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diff changeset
   217
lemma sup_commute: "(x \<squnion> y) = (y \<squnion> x)"
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
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   218
  by (fact sup.commute)
21733
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nipkow
parents: 21619
diff changeset
   219
34973
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haftmann
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diff changeset
   220
lemma sup_left_commute: "x \<squnion> (y \<squnion> z) = y \<squnion> (x \<squnion> z)"
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents: 34209
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   221
  by (fact sup.left_commute)
21733
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diff changeset
   222
44921
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   223
lemma sup_idem: "x \<squnion> x = x"
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   224
  by (fact sup.idem) (* already simp *)
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diff changeset
   225
44918
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noschinl
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   226
lemma sup_left_idem [simp]: "x \<squnion> (x \<squnion> y) = x \<squnion> y"
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diff changeset
   227
  by (fact sup.left_idem)
21733
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parents: 21619
diff changeset
   228
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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   229
lemma sup_absorb1: "y \<sqsubseteq> x \<Longrightarrow> x \<squnion> y = x"
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   230
  by (rule antisym) auto
21733
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parents: 21619
diff changeset
   231
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
   232
lemma sup_absorb2: "x \<sqsubseteq> y \<Longrightarrow> x \<squnion> y = y"
32064
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diff changeset
   233
  by (rule antisym) auto
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   234
32064
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   235
lemmas sup_aci = sup_commute sup_assoc sup_left_commute sup_left_idem
21733
131dd2a27137 Modified lattice locale
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diff changeset
   236
131dd2a27137 Modified lattice locale
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diff changeset
   237
end
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parents:
diff changeset
   238
21733
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   239
context lattice
131dd2a27137 Modified lattice locale
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   240
begin
131dd2a27137 Modified lattice locale
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diff changeset
   241
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   242
lemma dual_lattice:
44845
5e51075cbd97 added syntactic classes for "inf" and "sup"
krauss
parents: 44085
diff changeset
   243
  "class.lattice sup (op \<ge>) (op >) inf"
36635
080b755377c0 locale predicates of classes carry a mandatory "class" prefix
haftmann
parents: 36352
diff changeset
   244
  by (rule class.lattice.intro, rule dual_semilattice, rule class.semilattice_sup.intro, rule dual_order)
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   245
    (unfold_locales, auto)
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   246
44918
6a80fbc4e72c tune simpset for Complete_Lattices
noschinl
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diff changeset
   247
lemma inf_sup_absorb [simp]: "x \<sqinter> (x \<squnion> y) = x"
25102
db3e412c4cb1 antisymmetry not a default intro rule any longer
haftmann
parents: 25062
diff changeset
   248
  by (blast intro: antisym inf_le1 inf_greatest sup_ge1)
21733
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diff changeset
   249
44918
6a80fbc4e72c tune simpset for Complete_Lattices
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diff changeset
   250
lemma sup_inf_absorb [simp]: "x \<squnion> (x \<sqinter> y) = x"
25102
db3e412c4cb1 antisymmetry not a default intro rule any longer
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parents: 25062
diff changeset
   251
  by (blast intro: antisym sup_ge1 sup_least inf_le1)
21733
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diff changeset
   252
32064
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   253
lemmas inf_sup_aci = inf_aci sup_aci
21734
283461c15fa7 renaming
nipkow
parents: 21733
diff changeset
   254
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   255
lemmas inf_sup_ord = inf_le1 inf_le2 sup_ge1 sup_ge2
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   256
21734
283461c15fa7 renaming
nipkow
parents: 21733
diff changeset
   257
text{* Towards distributivity *}
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   258
21734
283461c15fa7 renaming
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diff changeset
   259
lemma distrib_sup_le: "x \<squnion> (y \<sqinter> z) \<sqsubseteq> (x \<squnion> y) \<sqinter> (x \<squnion> z)"
32064
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haftmann
parents: 32063
diff changeset
   260
  by (auto intro: le_infI1 le_infI2 le_supI1 le_supI2)
21734
283461c15fa7 renaming
nipkow
parents: 21733
diff changeset
   261
283461c15fa7 renaming
nipkow
parents: 21733
diff changeset
   262
lemma distrib_inf_le: "(x \<sqinter> y) \<squnion> (x \<sqinter> z) \<sqsubseteq> x \<sqinter> (y \<squnion> z)"
32064
53ca12ff305d refinement of lattice classes
haftmann
parents: 32063
diff changeset
   263
  by (auto intro: le_infI1 le_infI2 le_supI1 le_supI2)
21734
283461c15fa7 renaming
nipkow
parents: 21733
diff changeset
   264
283461c15fa7 renaming
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diff changeset
   265
text{* If you have one of them, you have them all. *}
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d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   266
21733
131dd2a27137 Modified lattice locale
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diff changeset
   267
lemma distrib_imp1:
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   268
assumes D: "!!x y z. x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   269
shows "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   270
proof-
44918
6a80fbc4e72c tune simpset for Complete_Lattices
noschinl
parents: 44845
diff changeset
   271
  have "x \<squnion> (y \<sqinter> z) = (x \<squnion> (x \<sqinter> z)) \<squnion> (y \<sqinter> z)" by simp
6a80fbc4e72c tune simpset for Complete_Lattices
noschinl
parents: 44845
diff changeset
   272
  also have "\<dots> = x \<squnion> (z \<sqinter> (x \<squnion> y))"
6a80fbc4e72c tune simpset for Complete_Lattices
noschinl
parents: 44845
diff changeset
   273
    by (simp add: D inf_commute sup_assoc del: sup_inf_absorb)
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   274
  also have "\<dots> = ((x \<squnion> y) \<sqinter> x) \<squnion> ((x \<squnion> y) \<sqinter> z)"
44919
482f1807976e tune proofs
noschinl
parents: 44918
diff changeset
   275
    by(simp add: inf_commute)
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   276
  also have "\<dots> = (x \<squnion> y) \<sqinter> (x \<squnion> z)" by(simp add:D)
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   277
  finally show ?thesis .
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   278
qed
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   279
21733
131dd2a27137 Modified lattice locale
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diff changeset
   280
lemma distrib_imp2:
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   281
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
   282
shows "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   283
proof-
44918
6a80fbc4e72c tune simpset for Complete_Lattices
noschinl
parents: 44845
diff changeset
   284
  have "x \<sqinter> (y \<squnion> z) = (x \<sqinter> (x \<squnion> z)) \<sqinter> (y \<squnion> z)" by simp
6a80fbc4e72c tune simpset for Complete_Lattices
noschinl
parents: 44845
diff changeset
   285
  also have "\<dots> = x \<sqinter> (z \<squnion> (x \<sqinter> y))"
6a80fbc4e72c tune simpset for Complete_Lattices
noschinl
parents: 44845
diff changeset
   286
    by (simp add: D sup_commute inf_assoc del: inf_sup_absorb)
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   287
  also have "\<dots> = ((x \<sqinter> y) \<squnion> x) \<sqinter> ((x \<sqinter> y) \<squnion> z)"
44919
482f1807976e tune proofs
noschinl
parents: 44918
diff changeset
   288
    by(simp add: sup_commute)
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   289
  also have "\<dots> = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" by(simp add:D)
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   290
  finally show ?thesis .
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   291
qed
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   292
21733
131dd2a27137 Modified lattice locale
nipkow
parents: 21619
diff changeset
   293
end
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   294
32568
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   295
subsubsection {* Strict order *}
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   296
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34973
diff changeset
   297
context semilattice_inf
32568
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   298
begin
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   299
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   300
lemma less_infI1:
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   301
  "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
   302
  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
   303
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   304
lemma less_infI2:
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   305
  "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
   306
  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
   307
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   308
end
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   309
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34973
diff changeset
   310
context semilattice_sup
32568
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   311
begin
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   312
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   313
lemma less_supI1:
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   314
  "x \<sqsubset> a \<Longrightarrow> x \<sqsubset> a \<squnion> b"
44921
58eef4843641 tuned proofs
huffman
parents: 44919
diff changeset
   315
  using dual_semilattice
58eef4843641 tuned proofs
huffman
parents: 44919
diff changeset
   316
  by (rule semilattice_inf.less_infI1)
32568
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   317
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   318
lemma less_supI2:
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   319
  "x \<sqsubset> b \<Longrightarrow> x \<sqsubset> a \<squnion> b"
44921
58eef4843641 tuned proofs
huffman
parents: 44919
diff changeset
   320
  using dual_semilattice
58eef4843641 tuned proofs
huffman
parents: 44919
diff changeset
   321
  by (rule semilattice_inf.less_infI2)
32568
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   322
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   323
end
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   324
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   325
24164
haftmann
parents: 23948
diff changeset
   326
subsection {* Distributive lattices *}
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   327
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   328
class distrib_lattice = lattice +
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   329
  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
   330
21733
131dd2a27137 Modified lattice locale
nipkow
parents: 21619
diff changeset
   331
context distrib_lattice
131dd2a27137 Modified lattice locale
nipkow
parents: 21619
diff changeset
   332
begin
131dd2a27137 Modified lattice locale
nipkow
parents: 21619
diff changeset
   333
131dd2a27137 Modified lattice locale
nipkow
parents: 21619
diff changeset
   334
lemma sup_inf_distrib2:
44921
58eef4843641 tuned proofs
huffman
parents: 44919
diff changeset
   335
  "(y \<sqinter> z) \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)"
58eef4843641 tuned proofs
huffman
parents: 44919
diff changeset
   336
  by (simp add: sup_commute sup_inf_distrib1)
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   337
21733
131dd2a27137 Modified lattice locale
nipkow
parents: 21619
diff changeset
   338
lemma inf_sup_distrib1:
44921
58eef4843641 tuned proofs
huffman
parents: 44919
diff changeset
   339
  "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
58eef4843641 tuned proofs
huffman
parents: 44919
diff changeset
   340
  by (rule distrib_imp2 [OF sup_inf_distrib1])
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   341
21733
131dd2a27137 Modified lattice locale
nipkow
parents: 21619
diff changeset
   342
lemma inf_sup_distrib2:
44921
58eef4843641 tuned proofs
huffman
parents: 44919
diff changeset
   343
  "(y \<squnion> z) \<sqinter> x = (y \<sqinter> x) \<squnion> (z \<sqinter> x)"
58eef4843641 tuned proofs
huffman
parents: 44919
diff changeset
   344
  by (simp add: inf_commute inf_sup_distrib1)
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   345
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   346
lemma dual_distrib_lattice:
44845
5e51075cbd97 added syntactic classes for "inf" and "sup"
krauss
parents: 44085
diff changeset
   347
  "class.distrib_lattice sup (op \<ge>) (op >) inf"
36635
080b755377c0 locale predicates of classes carry a mandatory "class" prefix
haftmann
parents: 36352
diff changeset
   348
  by (rule class.distrib_lattice.intro, rule dual_lattice)
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   349
    (unfold_locales, fact inf_sup_distrib1)
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   350
36008
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   351
lemmas sup_inf_distrib =
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   352
  sup_inf_distrib1 sup_inf_distrib2
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   353
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   354
lemmas inf_sup_distrib =
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   355
  inf_sup_distrib1 inf_sup_distrib2
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   356
21733
131dd2a27137 Modified lattice locale
nipkow
parents: 21619
diff changeset
   357
lemmas distrib =
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   358
  sup_inf_distrib1 sup_inf_distrib2 inf_sup_distrib1 inf_sup_distrib2
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   359
21733
131dd2a27137 Modified lattice locale
nipkow
parents: 21619
diff changeset
   360
end
131dd2a27137 Modified lattice locale
nipkow
parents: 21619
diff changeset
   361
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   362
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   363
subsection {* Bounded lattices and boolean algebras *}
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   364
36352
f71978e47cd5 add bounded_lattice_bot and bounded_lattice_top type classes
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36096
diff changeset
   365
class bounded_lattice_bot = lattice + bot
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   366
begin
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_left [simp]:
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   369
  "\<bottom> \<sqinter> x = \<bottom>"
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   370
  by (rule inf_absorb1) 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 inf_bot_right [simp]:
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   373
  "x \<sqinter> \<bottom> = \<bottom>"
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   374
  by (rule inf_absorb2) simp
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   375
36352
f71978e47cd5 add bounded_lattice_bot and bounded_lattice_top type classes
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36096
diff changeset
   376
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
   377
  "\<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
   378
  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
   379
f71978e47cd5 add bounded_lattice_bot and bounded_lattice_top type classes
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36096
diff changeset
   380
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
   381
  "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
   382
  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
   383
f71978e47cd5 add bounded_lattice_bot and bounded_lattice_top type classes
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36096
diff changeset
   384
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
   385
  "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
   386
  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
   387
f71978e47cd5 add bounded_lattice_bot and bounded_lattice_top type classes
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36096
diff changeset
   388
end
f71978e47cd5 add bounded_lattice_bot and bounded_lattice_top type classes
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36096
diff changeset
   389
f71978e47cd5 add bounded_lattice_bot and bounded_lattice_top type classes
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36096
diff changeset
   390
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
   391
begin
f71978e47cd5 add bounded_lattice_bot and bounded_lattice_top type classes
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36096
diff changeset
   392
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   393
lemma sup_top_left [simp]:
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   394
  "\<top> \<squnion> x = \<top>"
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   395
  by (rule sup_absorb1) simp
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   396
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   397
lemma sup_top_right [simp]:
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   398
  "x \<squnion> \<top> = \<top>"
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   399
  by (rule sup_absorb2) simp
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   400
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   401
lemma inf_top_left [simp]:
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   402
  "\<top> \<sqinter> x = x"
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   403
  by (rule inf_absorb2) simp
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   404
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   405
lemma inf_top_right [simp]:
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   406
  "x \<sqinter> \<top> = x"
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   407
  by (rule inf_absorb1) simp
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   408
36008
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   409
lemma inf_eq_top_iff [simp]:
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   410
  "x \<sqinter> y = \<top> \<longleftrightarrow> x = \<top> \<and> y = \<top>"
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   411
  by (simp add: eq_iff)
32568
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   412
36352
f71978e47cd5 add bounded_lattice_bot and bounded_lattice_top type classes
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36096
diff changeset
   413
end
f71978e47cd5 add bounded_lattice_bot and bounded_lattice_top type classes
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36096
diff changeset
   414
f71978e47cd5 add bounded_lattice_bot and bounded_lattice_top type classes
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36096
diff changeset
   415
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
   416
begin
f71978e47cd5 add bounded_lattice_bot and bounded_lattice_top type classes
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36096
diff changeset
   417
f71978e47cd5 add bounded_lattice_bot and bounded_lattice_top type classes
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36096
diff changeset
   418
lemma dual_bounded_lattice:
44845
5e51075cbd97 added syntactic classes for "inf" and "sup"
krauss
parents: 44085
diff changeset
   419
  "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
   420
  by unfold_locales (auto simp add: less_le_not_le)
32568
89518a3074e1 some lemmas about strict order in lattices
haftmann
parents: 32512
diff changeset
   421
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   422
end
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   423
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   424
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
   425
  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
   426
    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
   427
  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
   428
begin
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   429
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   430
lemma dual_boolean_algebra:
44845
5e51075cbd97 added syntactic classes for "inf" and "sup"
krauss
parents: 44085
diff changeset
   431
  "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
   432
  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
   433
    (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
   434
44918
6a80fbc4e72c tune simpset for Complete_Lattices
noschinl
parents: 44845
diff changeset
   435
lemma compl_inf_bot [simp]:
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   436
  "- x \<sqinter> x = \<bottom>"
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   437
  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
   438
44918
6a80fbc4e72c tune simpset for Complete_Lattices
noschinl
parents: 44845
diff changeset
   439
lemma compl_sup_top [simp]:
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   440
  "- x \<squnion> x = \<top>"
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   441
  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
   442
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   443
lemma compl_unique:
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   444
  assumes "x \<sqinter> y = \<bottom>"
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   445
    and "x \<squnion> y = \<top>"
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   446
  shows "- x = y"
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   447
proof -
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   448
  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
   449
    using inf_compl_bot assms(1) by simp
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   450
  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
   451
    by (simp add: inf_commute)
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   452
  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
   453
    by (simp add: inf_sup_distrib1)
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   454
  then have "- x \<sqinter> \<top> = y \<sqinter> \<top>"
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   455
    using sup_compl_top assms(2) by simp
34209
c7f621786035 killed a few warnings
krauss
parents: 34007
diff changeset
   456
  then show "- x = y" by simp
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   457
qed
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   458
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   459
lemma double_compl [simp]:
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   460
  "- (- x) = x"
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   461
  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
   462
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   463
lemma compl_eq_compl_iff [simp]:
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   464
  "- x = - y \<longleftrightarrow> x = y"
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   465
proof
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   466
  assume "- x = - y"
36008
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   467
  then have "- (- x) = - (- y)" by (rule arg_cong)
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   468
  then show "x = y" by simp
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   469
next
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   470
  assume "x = y"
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   471
  then show "- x = - y" by simp
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   472
qed
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   473
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   474
lemma compl_bot_eq [simp]:
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   475
  "- \<bottom> = \<top>"
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   476
proof -
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   477
  from sup_compl_top have "\<bottom> \<squnion> - \<bottom> = \<top>" .
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   478
  then show ?thesis 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_top_eq [simp]:
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   482
  "- \<top> = \<bottom>"
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 inf_compl_bot have "\<top> \<sqinter> - \<top> = \<bottom>" .
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_inf [simp]:
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   489
  "- (x \<sqinter> y) = - x \<squnion> - y"
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   490
proof (rule compl_unique)
36008
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   491
  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
   492
    by (simp only: inf_sup_distrib inf_aci)
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   493
  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
   494
    by (simp add: inf_compl_bot)
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   495
next
36008
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   496
  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
   497
    by (simp only: sup_inf_distrib sup_aci)
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   498
  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
   499
    by (simp add: sup_compl_top)
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   500
qed
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   501
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   502
lemma compl_sup [simp]:
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   503
  "- (x \<squnion> y) = - x \<sqinter> - y"
44921
58eef4843641 tuned proofs
huffman
parents: 44919
diff changeset
   504
  using dual_boolean_algebra
58eef4843641 tuned proofs
huffman
parents: 44919
diff changeset
   505
  by (rule boolean_algebra.compl_inf)
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   506
36008
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   507
lemma compl_mono:
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   508
  "x \<sqsubseteq> y \<Longrightarrow> - y \<sqsubseteq> - x"
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   509
proof -
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   510
  assume "x \<sqsubseteq> y"
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   511
  then have "x \<squnion> y = y" by (simp only: le_iff_sup)
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   512
  then have "- (x \<squnion> y) = - y" by simp
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   513
  then have "- x \<sqinter> - y = - y" by simp
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   514
  then have "- y \<sqinter> - x = - y" by (simp only: inf_commute)
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   515
  then show "- y \<sqsubseteq> - x" by (simp only: le_iff_inf)
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   516
qed
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   517
44918
6a80fbc4e72c tune simpset for Complete_Lattices
noschinl
parents: 44845
diff changeset
   518
lemma compl_le_compl_iff [simp]:
43753
fe5e846c0839 tuned notation
haftmann
parents: 41082
diff changeset
   519
  "- x \<sqsubseteq> - y \<longleftrightarrow> y \<sqsubseteq> x"
43873
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   520
  by (auto dest: compl_mono)
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   521
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   522
lemma compl_le_swap1:
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   523
  assumes "y \<sqsubseteq> - x" shows "x \<sqsubseteq> -y"
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   524
proof -
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   525
  from assms have "- (- x) \<sqsubseteq> - y" by (simp only: compl_le_compl_iff)
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   526
  then show ?thesis by simp
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   527
qed
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   528
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   529
lemma compl_le_swap2:
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   530
  assumes "- y \<sqsubseteq> x" shows "- x \<sqsubseteq> y"
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   531
proof -
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   532
  from assms have "- x \<sqsubseteq> - (- y)" by (simp only: compl_le_compl_iff)
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   533
  then show ?thesis by simp
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   534
qed
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   535
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   536
lemma compl_less_compl_iff: (* TODO: declare [simp] ? *)
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   537
  "- x \<sqsubset> - y \<longleftrightarrow> y \<sqsubset> x"
44919
482f1807976e tune proofs
noschinl
parents: 44918
diff changeset
   538
  by (auto simp add: less_le)
43873
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   539
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   540
lemma compl_less_swap1:
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   541
  assumes "y \<sqsubset> - x" shows "x \<sqsubset> - y"
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   542
proof -
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   543
  from assms have "- (- x) \<sqsubset> - y" by (simp only: compl_less_compl_iff)
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   544
  then show ?thesis by simp
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   545
qed
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   546
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   547
lemma compl_less_swap2:
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   548
  assumes "- y \<sqsubset> x" shows "- x \<sqsubset> y"
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   549
proof -
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   550
  from assms have "- x \<sqsubset> - (- y)" by (simp only: compl_less_compl_iff)
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   551
  then show ?thesis by simp
8a2f339641c1 more on complement
haftmann
parents: 43753
diff changeset
   552
qed
36008
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   553
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   554
end
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   555
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   556
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   557
subsection {* Uniqueness of inf and sup *}
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   558
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34973
diff changeset
   559
lemma (in semilattice_inf) inf_unique:
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   560
  fixes f (infixl "\<triangle>" 70)
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   561
  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
   562
  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
   563
  shows "x \<sqinter> y = x \<triangle> y"
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   564
proof (rule antisym)
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   565
  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
   566
next
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   567
  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
   568
  show "x \<sqinter> y \<sqsubseteq> x \<triangle> y" by (rule leI) simp_all
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   569
qed
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   570
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34973
diff changeset
   571
lemma (in semilattice_sup) sup_unique:
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   572
  fixes f (infixl "\<nabla>" 70)
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   573
  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
   574
  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
   575
  shows "x \<squnion> y = x \<nabla> y"
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   576
proof (rule antisym)
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   577
  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
   578
next
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   579
  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
   580
  show "x \<nabla> y \<sqsubseteq> x \<squnion> y" by (rule leI) simp_all
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   581
qed
36008
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 35724
diff changeset
   582
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   583
22916
haftmann
parents: 22737
diff changeset
   584
subsection {* @{const min}/@{const max} on linear orders as
haftmann
parents: 22737
diff changeset
   585
  special case of @{const inf}/@{const sup} *}
haftmann
parents: 22737
diff changeset
   586
44845
5e51075cbd97 added syntactic classes for "inf" and "sup"
krauss
parents: 44085
diff changeset
   587
sublocale linorder < min_max!: distrib_lattice min less_eq less max
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28692
diff changeset
   588
proof
22916
haftmann
parents: 22737
diff changeset
   589
  fix x y z
32512
d14762642cdd proper class syntax for sublocale class < expr
haftmann
parents: 32436
diff changeset
   590
  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
   591
    by (auto simp add: min_def max_def)
22916
haftmann
parents: 22737
diff changeset
   592
qed (auto simp add: min_def max_def not_le less_imp_le)
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   593
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34973
diff changeset
   594
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
   595
  by (rule ext)+ (auto intro: antisym)
21733
131dd2a27137 Modified lattice locale
nipkow
parents: 21619
diff changeset
   596
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34973
diff changeset
   597
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
   598
  by (rule ext)+ (auto intro: antisym)
21733
131dd2a27137 Modified lattice locale
nipkow
parents: 21619
diff changeset
   599
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   600
lemmas le_maxI1 = min_max.sup_ge1
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   601
lemmas le_maxI2 = min_max.sup_ge2
21381
79e065f2be95 reworking of min/max lemmas
haftmann
parents: 21312
diff changeset
   602
 
34973
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents: 34209
diff changeset
   603
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
   604
  min_max.inf.left_commute
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   605
34973
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents: 34209
diff changeset
   606
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
   607
  min_max.sup.left_commute
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents: 34209
diff changeset
   608
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   609
46631
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   610
subsection {* Lattice on @{typ bool} *}
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   611
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   612
instantiation bool :: boolean_algebra
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   613
begin
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   614
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   615
definition
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   616
  bool_Compl_def [simp]: "uminus = Not"
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   617
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   618
definition
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   619
  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
   620
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   621
definition
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   622
  [simp]: "P \<sqinter> Q \<longleftrightarrow> P \<and> Q"
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   623
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   624
definition
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   625
  [simp]: "P \<squnion> Q \<longleftrightarrow> P \<or> Q"
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   626
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   627
instance proof
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   628
qed auto
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   629
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   630
end
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   631
32781
19c01bd7f6ae moved lemmas about sup on bool to Lattices.thy
haftmann
parents: 32780
diff changeset
   632
lemma sup_boolI1:
19c01bd7f6ae moved lemmas about sup on bool to Lattices.thy
haftmann
parents: 32780
diff changeset
   633
  "P \<Longrightarrow> P \<squnion> Q"
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   634
  by simp
32781
19c01bd7f6ae moved lemmas about sup on bool to Lattices.thy
haftmann
parents: 32780
diff changeset
   635
19c01bd7f6ae moved lemmas about sup on bool to Lattices.thy
haftmann
parents: 32780
diff changeset
   636
lemma sup_boolI2:
19c01bd7f6ae moved lemmas about sup on bool to Lattices.thy
haftmann
parents: 32780
diff changeset
   637
  "Q \<Longrightarrow> P \<squnion> Q"
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   638
  by simp
32781
19c01bd7f6ae moved lemmas about sup on bool to Lattices.thy
haftmann
parents: 32780
diff changeset
   639
19c01bd7f6ae moved lemmas about sup on bool to Lattices.thy
haftmann
parents: 32780
diff changeset
   640
lemma sup_boolE:
19c01bd7f6ae moved lemmas about sup on bool to Lattices.thy
haftmann
parents: 32780
diff changeset
   641
  "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
   642
  by auto
32781
19c01bd7f6ae moved lemmas about sup on bool to Lattices.thy
haftmann
parents: 32780
diff changeset
   643
23878
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   644
46631
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   645
subsection {* Lattice on @{typ "_ \<Rightarrow> _"} *}
23878
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   646
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   647
instantiation "fun" :: (type, lattice) lattice
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   648
begin
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   649
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   650
definition
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   651
  "f \<sqinter> g = (\<lambda>x. f x \<sqinter> g x)"
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   652
46882
6242b4bc05bc tuned simpset
noschinl
parents: 46691
diff changeset
   653
lemma inf_apply [simp] (* CANDIDATE [code] *):
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   654
  "(f \<sqinter> g) x = f x \<sqinter> g x"
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   655
  by (simp add: inf_fun_def)
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   656
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   657
definition
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   658
  "f \<squnion> g = (\<lambda>x. f x \<squnion> g x)"
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   659
46882
6242b4bc05bc tuned simpset
noschinl
parents: 46691
diff changeset
   660
lemma sup_apply [simp] (* CANDIDATE [code] *):
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   661
  "(f \<squnion> g) x = f x \<squnion> g x"
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   662
  by (simp add: sup_fun_def)
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   663
32780
be337ec31268 tuned proofs
haftmann
parents: 32642
diff changeset
   664
instance proof
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46882
diff changeset
   665
qed (simp_all add: le_fun_def)
23878
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   666
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25482
diff changeset
   667
end
23878
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   668
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   669
instance "fun" :: (type, distrib_lattice) distrib_lattice proof
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46882
diff changeset
   670
qed (rule ext, simp add: sup_inf_distrib1)
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   671
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   672
instance "fun" :: (type, bounded_lattice) bounded_lattice ..
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 32781
diff changeset
   673
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   674
instantiation "fun" :: (type, uminus) uminus
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   675
begin
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   676
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   677
definition
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   678
  fun_Compl_def: "- A = (\<lambda>x. - A x)"
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   679
46882
6242b4bc05bc tuned simpset
noschinl
parents: 46691
diff changeset
   680
lemma uminus_apply [simp] (* CANDIDATE [code] *):
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   681
  "(- A) x = - (A x)"
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   682
  by (simp add: fun_Compl_def)
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   683
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   684
instance ..
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   685
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   686
end
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   687
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   688
instantiation "fun" :: (type, minus) minus
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   689
begin
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   690
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   691
definition
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   692
  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
   693
46882
6242b4bc05bc tuned simpset
noschinl
parents: 46691
diff changeset
   694
lemma minus_apply [simp] (* CANDIDATE [code] *):
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   695
  "(A - B) x = A x - B x"
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   696
  by (simp add: fun_diff_def)
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   697
31991
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   698
instance ..
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   699
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   700
end
37390299214a added boolean_algebra type class; tuned lattice duals
haftmann
parents: 30729
diff changeset
   701
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   702
instance "fun" :: (type, boolean_algebra) boolean_algebra proof
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46882
diff changeset
   703
qed (rule ext, simp_all add: inf_compl_bot sup_compl_top diff_eq)+
26794
354c3844dfde - Now imports Fun rather than Orderings
berghofe
parents: 26233
diff changeset
   704
46631
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   705
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   706
subsection {* Lattice on unary and binary predicates *}
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   707
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   708
lemma inf1I: "A x \<Longrightarrow> B x \<Longrightarrow> (A \<sqinter> B) x"
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   709
  by (simp add: inf_fun_def)
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   710
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   711
lemma inf2I: "A x y \<Longrightarrow> B x y \<Longrightarrow> (A \<sqinter> B) x y"
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   712
  by (simp add: inf_fun_def)
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   713
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   714
lemma inf1E: "(A \<sqinter> B) x \<Longrightarrow> (A x \<Longrightarrow> B x \<Longrightarrow> P) \<Longrightarrow> P"
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   715
  by (simp add: inf_fun_def)
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   716
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   717
lemma inf2E: "(A \<sqinter> B) x y \<Longrightarrow> (A x y \<Longrightarrow> B x y \<Longrightarrow> P) \<Longrightarrow> P"
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   718
  by (simp add: inf_fun_def)
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   719
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   720
lemma inf1D1: "(A \<sqinter> B) x \<Longrightarrow> A x"
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   721
  by (simp add: inf_fun_def)
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   722
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   723
lemma inf2D1: "(A \<sqinter> B) x y \<Longrightarrow> A x y"
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   724
  by (simp add: inf_fun_def)
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   725
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   726
lemma inf1D2: "(A \<sqinter> B) x \<Longrightarrow> B x"
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   727
  by (simp add: inf_fun_def)
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   728
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   729
lemma inf2D2: "(A \<sqinter> B) x y \<Longrightarrow> B x y"
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   730
  by (simp add: inf_fun_def)
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   731
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   732
lemma sup1I1: "A x \<Longrightarrow> (A \<squnion> B) x"
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   733
  by (simp add: sup_fun_def)
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   734
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   735
lemma sup2I1: "A x y \<Longrightarrow> (A \<squnion> B) x y"
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   736
  by (simp add: sup_fun_def)
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   737
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   738
lemma sup1I2: "B x \<Longrightarrow> (A \<squnion> B) x"
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   739
  by (simp add: sup_fun_def)
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   740
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   741
lemma sup2I2: "B x y \<Longrightarrow> (A \<squnion> B) x y"
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   742
  by (simp add: sup_fun_def)
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   743
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   744
lemma sup1E: "(A \<squnion> B) x \<Longrightarrow> (A x \<Longrightarrow> P) \<Longrightarrow> (B x \<Longrightarrow> P) \<Longrightarrow> P"
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   745
  by (simp add: sup_fun_def) iprover
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   746
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   747
lemma sup2E: "(A \<squnion> B) x y \<Longrightarrow> (A x y \<Longrightarrow> P) \<Longrightarrow> (B x y \<Longrightarrow> P) \<Longrightarrow> P"
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   748
  by (simp add: sup_fun_def) iprover
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   749
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   750
text {*
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   751
  \medskip Classical introduction rule: no commitment to @{text A} vs
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   752
  @{text B}.
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   753
*}
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   754
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   755
lemma sup1CI: "(\<not> B x \<Longrightarrow> A x) \<Longrightarrow> (A \<squnion> B) x"
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   756
  by (auto simp add: sup_fun_def)
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   757
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   758
lemma sup2CI: "(\<not> B x y \<Longrightarrow> A x y) \<Longrightarrow> (A \<squnion> B) x y"
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   759
  by (auto simp add: sup_fun_def)
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   760
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   761
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24749
diff changeset
   762
no_notation
46691
72d81e789106 tuned syntax declarations; tuned structure
haftmann
parents: 46689
diff changeset
   763
  less_eq (infix "\<sqsubseteq>" 50) and
72d81e789106 tuned syntax declarations; tuned structure
haftmann
parents: 46689
diff changeset
   764
  less (infix "\<sqsubset>" 50)
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24749
diff changeset
   765
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   766
end
46631
2c5c003cee35 moved lemmas for orderings and lattices on predicates to corresponding theories, retaining declaration order of classical rules; tuned headings; tuned syntax
haftmann
parents: 46557
diff changeset
   767