src/HOL/Lattices.thy
author haftmann
Fri, 20 Jul 2007 14:27:56 +0200
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child 23948 261bd4678076
permissions -rw-r--r--
simplified HOL bootstrap
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(*  Title:      HOL/Lattices.thy
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    ID:         $Id$
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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
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begin
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subsection{* Lattices *}
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class lower_semilattice = order +
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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 upper_semilattice = order +
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  assumes sup_ge1 [simp]: "x \<sqsubseteq> x \<squnion> y"
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  and sup_ge2 [simp]: "y \<sqsubseteq> x \<squnion> y"
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  and sup_least: "y \<sqsubseteq> x \<Longrightarrow> z \<sqsubseteq> x \<Longrightarrow> y \<squnion> z \<sqsubseteq> x"
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class lattice = lower_semilattice + upper_semilattice
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subsubsection{* Intro and elim rules*}
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context lower_semilattice
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begin
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lemmas antisym_intro [intro!] = antisym
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lemmas (in -) [rule del] = antisym_intro
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lemma le_infI1[intro]: "a \<sqsubseteq> x \<Longrightarrow> a \<sqinter> b \<sqsubseteq> x"
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apply(subgoal_tac "a \<sqinter> b \<sqsubseteq> a")
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 apply(blast intro: order_trans)
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apply simp
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done
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lemmas (in -) [rule del] = le_infI1
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lemma le_infI2[intro]: "b \<sqsubseteq> x \<Longrightarrow> a \<sqinter> b \<sqsubseteq> x"
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apply(subgoal_tac "a \<sqinter> b \<sqsubseteq> b")
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 apply(blast intro: order_trans)
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apply simp
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done
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lemmas (in -) [rule del] = le_infI2
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lemma le_infI[intro!]: "x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> b \<Longrightarrow> x \<sqsubseteq> a \<sqinter> b"
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by(blast intro: inf_greatest)
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lemmas (in -) [rule del] = le_infI
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lemma le_infE [elim!]: "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)
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lemmas (in -) [rule del] = le_infE
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lemma le_inf_iff [simp]:
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 "x \<sqsubseteq> y \<sqinter> z = (x \<sqsubseteq> y \<and> x \<sqsubseteq> z)"
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by blast
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lemma le_iff_inf: "(x \<sqsubseteq> y) = (x \<sqinter> y = x)"
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by(blast dest:eq_iff[THEN iffD1])
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end
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lemma mono_inf: "mono f \<Longrightarrow> f (inf A B) \<le> inf (f A) (f B)"
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  by (auto simp add: mono_def)
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context upper_semilattice
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begin
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lemmas antisym_intro [intro!] = antisym
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lemmas (in -) [rule del] = antisym_intro
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lemma le_supI1[intro]: "x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> a \<squnion> b"
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apply(subgoal_tac "a \<sqsubseteq> a \<squnion> b")
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 apply(blast intro: order_trans)
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apply simp
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done
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lemmas (in -) [rule del] = le_supI1
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lemma le_supI2[intro]: "x \<sqsubseteq> b \<Longrightarrow> x \<sqsubseteq> a \<squnion> b"
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apply(subgoal_tac "b \<sqsubseteq> a \<squnion> b")
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 apply(blast intro: order_trans)
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apply simp
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done
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lemmas (in -) [rule del] = le_supI2
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lemma le_supI[intro!]: "a \<sqsubseteq> x \<Longrightarrow> b \<sqsubseteq> x \<Longrightarrow> a \<squnion> b \<sqsubseteq> x"
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by(blast intro: sup_least)
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lemmas (in -) [rule del] = le_supI
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lemma le_supE[elim!]: "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)
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lemmas (in -) [rule del] = le_supE
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lemma ge_sup_conv[simp]:
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 "x \<squnion> y \<sqsubseteq> z = (x \<sqsubseteq> z \<and> y \<sqsubseteq> z)"
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by blast
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lemma le_iff_sup: "(x \<sqsubseteq> y) = (x \<squnion> y = y)"
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by(blast dest:eq_iff[THEN iffD1])
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end
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lemma mono_sup: "mono f \<Longrightarrow> sup (f A) (f B) \<le> f (sup A B)"
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  by (auto simp add: mono_def)
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subsubsection{* Equational laws *}
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context lower_semilattice
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begin
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lemma inf_commute: "(x \<sqinter> y) = (y \<sqinter> x)"
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by blast
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lemma inf_assoc: "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
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by blast
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lemma inf_idem[simp]: "x \<sqinter> x = x"
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by blast
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lemma inf_left_idem[simp]: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"
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by blast
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lemma inf_absorb1: "x \<sqsubseteq> y \<Longrightarrow> x \<sqinter> y = x"
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by blast
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lemma inf_absorb2: "y \<sqsubseteq> x \<Longrightarrow> x \<sqinter> y = y"
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by blast
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lemma inf_left_commute: "x \<sqinter> (y \<sqinter> z) = y \<sqinter> (x \<sqinter> z)"
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by blast
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lemmas inf_ACI = inf_commute inf_assoc inf_left_commute inf_left_idem
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end
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context upper_semilattice
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begin
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lemma sup_commute: "(x \<squnion> y) = (y \<squnion> x)"
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by blast
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lemma sup_assoc: "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
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by blast
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lemma sup_idem[simp]: "x \<squnion> x = x"
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by blast
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lemma sup_left_idem[simp]: "x \<squnion> (x \<squnion> y) = x \<squnion> y"
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by blast
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lemma sup_absorb1: "y \<sqsubseteq> x \<Longrightarrow> x \<squnion> y = x"
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by blast
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lemma sup_absorb2: "x \<sqsubseteq> y \<Longrightarrow> x \<squnion> y = y"
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by blast
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lemma sup_left_commute: "x \<squnion> (y \<squnion> z) = y \<squnion> (x \<squnion> z)"
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by blast
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lemmas sup_ACI = sup_commute sup_assoc sup_left_commute sup_left_idem
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end
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context lattice
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begin
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lemma inf_sup_absorb: "x \<sqinter> (x \<squnion> y) = x"
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by(blast intro: antisym inf_le1 inf_greatest sup_ge1)
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lemma sup_inf_absorb: "x \<squnion> (x \<sqinter> y) = x"
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by(blast intro: antisym sup_ge1 sup_least inf_le1)
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lemmas ACI = inf_ACI sup_ACI
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   183
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lemmas inf_sup_ord = inf_le1 inf_le2 sup_ge1 sup_ge2
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text{* Towards distributivity *}
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   187
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lemma distrib_sup_le: "x \<squnion> (y \<sqinter> z) \<sqsubseteq> (x \<squnion> y) \<sqinter> (x \<squnion> z)"
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   189
by blast
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   190
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   191
lemma distrib_inf_le: "(x \<sqinter> y) \<squnion> (x \<sqinter> z) \<sqsubseteq> x \<sqinter> (y \<squnion> z)"
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   192
by blast
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   193
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   194
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text{* If you have one of them, you have them all. *}
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lemma distrib_imp1:
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   198
assumes D: "!!x y z. x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
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parents:
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   199
shows "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
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parents:
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   200
proof-
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parents:
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   201
  have "x \<squnion> (y \<sqinter> z) = (x \<squnion> (x \<sqinter> z)) \<squnion> (y \<sqinter> z)" by(simp add:sup_inf_absorb)
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parents:
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   202
  also have "\<dots> = x \<squnion> (z \<sqinter> (x \<squnion> y))" by(simp add:D inf_commute sup_assoc)
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haftmann
parents:
diff changeset
   203
  also have "\<dots> = ((x \<squnion> y) \<sqinter> x) \<squnion> ((x \<squnion> y) \<sqinter> z)"
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parents:
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   204
    by(simp add:inf_sup_absorb inf_commute)
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haftmann
parents:
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   205
  also have "\<dots> = (x \<squnion> y) \<sqinter> (x \<squnion> z)" by(simp add:D)
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parents:
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   206
  finally show ?thesis .
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parents:
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qed
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parents:
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   208
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   209
lemma distrib_imp2:
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   210
assumes D: "!!x y z. x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
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parents:
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   211
shows "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
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haftmann
parents:
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   212
proof-
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parents:
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   213
  have "x \<sqinter> (y \<squnion> z) = (x \<sqinter> (x \<squnion> z)) \<sqinter> (y \<squnion> z)" by(simp add:inf_sup_absorb)
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parents:
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   214
  also have "\<dots> = x \<sqinter> (z \<squnion> (x \<sqinter> y))" by(simp add:D sup_commute inf_assoc)
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parents:
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   215
  also have "\<dots> = ((x \<sqinter> y) \<squnion> x) \<sqinter> ((x \<sqinter> y) \<squnion> z)"
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parents:
diff changeset
   216
    by(simp add:sup_inf_absorb sup_commute)
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haftmann
parents:
diff changeset
   217
  also have "\<dots> = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" by(simp add:D)
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haftmann
parents:
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   218
  finally show ?thesis .
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haftmann
parents:
diff changeset
   219
qed
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   220
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   221
(* seems unused *)
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   222
lemma modular_le: "x \<sqsubseteq> z \<Longrightarrow> x \<squnion> (y \<sqinter> z) \<sqsubseteq> (x \<squnion> y) \<sqinter> z"
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   223
by blast
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diff changeset
   224
21733
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   225
end
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parents:
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   226
d594c58e24ed renamed Lattice_Locales to Lattices
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parents:
diff changeset
   227
d594c58e24ed renamed Lattice_Locales to Lattices
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parents:
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   228
subsection{* Distributive lattices *}
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   229
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   230
class distrib_lattice = lattice +
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   231
  assumes sup_inf_distrib1: "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
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parents:
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   232
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   233
context distrib_lattice
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   234
begin
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   235
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   236
lemma sup_inf_distrib2:
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parents:
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   237
 "(y \<sqinter> z) \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)"
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haftmann
parents:
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   238
by(simp add:ACI sup_inf_distrib1)
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haftmann
parents:
diff changeset
   239
21733
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   240
lemma inf_sup_distrib1:
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parents:
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   241
 "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
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parents:
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   242
by(rule distrib_imp2[OF sup_inf_distrib1])
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haftmann
parents:
diff changeset
   243
21733
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   244
lemma inf_sup_distrib2:
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parents:
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   245
 "(y \<squnion> z) \<sqinter> x = (y \<sqinter> x) \<squnion> (z \<sqinter> x)"
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haftmann
parents:
diff changeset
   246
by(simp add:ACI inf_sup_distrib1)
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   247
21733
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   248
lemmas distrib =
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parents:
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   249
  sup_inf_distrib1 sup_inf_distrib2 inf_sup_distrib1 inf_sup_distrib2
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haftmann
parents:
diff changeset
   250
21733
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   251
end
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   252
21249
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haftmann
parents:
diff changeset
   253
22454
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   254
subsection {* Uniqueness of inf and sup *}
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parents: 22422
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   255
22737
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parents: 22548
diff changeset
   256
lemma (in lower_semilattice) inf_unique:
22454
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parents: 22422
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   257
  fixes f (infixl "\<triangle>" 70)
22737
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parents: 22548
diff changeset
   258
  assumes le1: "\<And>x y. x \<triangle> y \<^loc>\<le> x" and le2: "\<And>x y. x \<triangle> y \<^loc>\<le> y"
haftmann
parents: 22548
diff changeset
   259
  and greatest: "\<And>x y z. x \<^loc>\<le> y \<Longrightarrow> x \<^loc>\<le> z \<Longrightarrow> x \<^loc>\<le> y \<triangle> z"
haftmann
parents: 22548
diff changeset
   260
  shows "x \<sqinter> y = x \<triangle> y"
22454
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haftmann
parents: 22422
diff changeset
   261
proof (rule antisym)
23389
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wenzelm
parents: 23087
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   262
  show "x \<triangle> y \<^loc>\<le> x \<sqinter> y" by (rule le_infI) (rule le1, rule le2)
22454
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haftmann
parents: 22422
diff changeset
   263
next
22737
haftmann
parents: 22548
diff changeset
   264
  have leI: "\<And>x y z. x \<^loc>\<le> y \<Longrightarrow> x \<^loc>\<le> z \<Longrightarrow> x \<^loc>\<le> y \<triangle> z" by (blast intro: greatest)
haftmann
parents: 22548
diff changeset
   265
  show "x \<sqinter> y \<^loc>\<le> x \<triangle> y" by (rule leI) simp_all
22454
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parents: 22422
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   266
qed
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   267
22737
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parents: 22548
diff changeset
   268
lemma (in upper_semilattice) sup_unique:
22454
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haftmann
parents: 22422
diff changeset
   269
  fixes f (infixl "\<nabla>" 70)
22737
haftmann
parents: 22548
diff changeset
   270
  assumes ge1 [simp]: "\<And>x y. x \<^loc>\<le> x \<nabla> y" and ge2: "\<And>x y. y \<^loc>\<le> x \<nabla> y"
haftmann
parents: 22548
diff changeset
   271
  and least: "\<And>x y z. y \<^loc>\<le> x \<Longrightarrow> z \<^loc>\<le> x \<Longrightarrow> y \<nabla> z \<^loc>\<le> x"
haftmann
parents: 22548
diff changeset
   272
  shows "x \<squnion> y = x \<nabla> y"
22454
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haftmann
parents: 22422
diff changeset
   273
proof (rule antisym)
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23087
diff changeset
   274
  show "x \<squnion> y \<^loc>\<le> x \<nabla> y" by (rule le_supI) (rule ge1, rule ge2)
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   275
next
22737
haftmann
parents: 22548
diff changeset
   276
  have leI: "\<And>x y z. x \<^loc>\<le> z \<Longrightarrow> y \<^loc>\<le> z \<Longrightarrow> x \<nabla> y \<^loc>\<le> z" by (blast intro: least)
haftmann
parents: 22548
diff changeset
   277
  show "x \<nabla> y \<^loc>\<le> x \<squnion> y" by (rule leI) simp_all
22454
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haftmann
parents: 22422
diff changeset
   278
qed
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   279
  
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haftmann
parents: 22422
diff changeset
   280
22916
haftmann
parents: 22737
diff changeset
   281
subsection {* @{const min}/@{const max} on linear orders as
haftmann
parents: 22737
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   282
  special case of @{const inf}/@{const sup} *}
haftmann
parents: 22737
diff changeset
   283
haftmann
parents: 22737
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   284
lemma (in linorder) distrib_lattice_min_max:
23018
1d29bc31b0cb no special treatment in naming of locale predicates stemming form classes
haftmann
parents: 22916
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   285
  "distrib_lattice (op \<^loc>\<le>) (op \<^loc><) min max"
22916
haftmann
parents: 22737
diff changeset
   286
proof unfold_locales
haftmann
parents: 22737
diff changeset
   287
  have aux: "\<And>x y \<Colon> 'a. x \<^loc>< y \<Longrightarrow> y \<^loc>\<le> x \<Longrightarrow> x = y"
haftmann
parents: 22737
diff changeset
   288
    by (auto simp add: less_le antisym)
haftmann
parents: 22737
diff changeset
   289
  fix x y z
haftmann
parents: 22737
diff changeset
   290
  show "max x (min y z) = min (max x y) (max x z)"
haftmann
parents: 22737
diff changeset
   291
  unfolding min_def max_def
haftmann
parents: 22737
diff changeset
   292
    by (auto simp add: intro: antisym, unfold not_le,
haftmann
parents: 22737
diff changeset
   293
      auto intro: less_trans le_less_trans aux)
haftmann
parents: 22737
diff changeset
   294
qed (auto simp add: min_def max_def not_le less_imp_le)
21249
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haftmann
parents:
diff changeset
   295
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   296
interpretation min_max:
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   297
  distrib_lattice ["op \<le> \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> bool" "op <" min max]
23087
ad7244663431 rudimentary class target implementation
haftmann
parents: 23018
diff changeset
   298
  by (rule distrib_lattice_min_max [folded ord_class.min ord_class.max])
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   299
22454
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haftmann
parents: 22422
diff changeset
   300
lemma inf_min: "inf = (min \<Colon> 'a\<Colon>{lower_semilattice, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)"
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haftmann
parents: 22422
diff changeset
   301
  by (rule ext)+ auto
21733
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nipkow
parents: 21619
diff changeset
   302
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   303
lemma sup_max: "sup = (max \<Colon> 'a\<Colon>{upper_semilattice, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)"
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   304
  by (rule ext)+ auto
21733
131dd2a27137 Modified lattice locale
nipkow
parents: 21619
diff changeset
   305
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   306
lemmas le_maxI1 = min_max.sup_ge1
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   307
lemmas le_maxI2 = min_max.sup_ge2
21381
79e065f2be95 reworking of min/max lemmas
haftmann
parents: 21312
diff changeset
   308
 
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   309
lemmas max_ac = min_max.sup_assoc min_max.sup_commute
22422
ee19cdb07528 stepping towards uniform lattice theory development in HOL
haftmann
parents: 22384
diff changeset
   310
  mk_left_commute [of max, OF min_max.sup_assoc min_max.sup_commute]
21249
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haftmann
parents:
diff changeset
   311
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   312
lemmas min_ac = min_max.inf_assoc min_max.inf_commute
22422
ee19cdb07528 stepping towards uniform lattice theory development in HOL
haftmann
parents: 22384
diff changeset
   313
  mk_left_commute [of min, OF min_max.inf_assoc min_max.inf_commute]
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   314
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   315
text {*
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   316
  Now we have inherited antisymmetry as an intro-rule on all
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   317
  linear orders. This is a problem because it applies to bool, which is
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   318
  undesirable.
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   319
*}
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   320
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   321
lemmas [rule del] = min_max.antisym_intro min_max.le_infI min_max.le_supI
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   322
  min_max.le_supE min_max.le_infE min_max.le_supI1 min_max.le_supI2
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   323
  min_max.le_infI1 min_max.le_infI2
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   324
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   325
23878
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   326
subsection {* Complete lattices *}
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   327
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   328
class complete_lattice = lattice +
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   329
  fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   330
  assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x"
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   331
  assumes Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A"
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   332
begin
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   333
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   334
definition
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   335
  Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   336
where
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   337
  "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<^loc>\<le> b}"
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   338
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   339
lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<^loc>\<le> a}"
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  unfolding Sup_def by (auto intro: Inf_greatest Inf_lower)
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   341
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   342
lemma Sup_upper: "x \<in> A \<Longrightarrow> x \<^loc>\<le> \<Squnion>A"
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  by (auto simp: Sup_def intro: Inf_greatest)
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   344
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   345
lemma Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<^loc>\<le> z) \<Longrightarrow> \<Squnion>A \<^loc>\<le> z"
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   346
  by (auto simp: Sup_def intro: Inf_lower)
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   347
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   348
lemma Inf_Univ: "\<Sqinter>UNIV = \<Squnion>{}"
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  unfolding Sup_def by auto
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   350
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   351
lemma Sup_Univ: "\<Squnion>UNIV = \<Sqinter>{}"
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  unfolding Inf_Sup by auto
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   353
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   354
lemma Inf_insert: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A"
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  apply (rule antisym)
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   356
  apply (rule le_infI)
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   357
  apply (rule Inf_lower)
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   358
  apply simp
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   359
  apply (rule Inf_greatest)
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   360
  apply (rule Inf_lower)
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   361
  apply simp
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   362
  apply (rule Inf_greatest)
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   363
  apply (erule insertE)
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   364
  apply (rule le_infI1)
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   365
  apply simp
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   366
  apply (rule le_infI2)
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   367
  apply (erule Inf_lower)
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   368
  done
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   369
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   370
lemma Sup_insert: "\<Squnion>insert a A = a \<squnion> \<Squnion>A"
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   371
  apply (rule antisym)
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   372
  apply (rule Sup_least)
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   373
  apply (erule insertE)
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   374
  apply (rule le_supI1)
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   375
  apply simp
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   376
  apply (rule le_supI2)
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   377
  apply (erule Sup_upper)
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   378
  apply (rule le_supI)
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   379
  apply (rule Sup_upper)
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diff changeset
   380
  apply simp
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   381
  apply (rule Sup_least)
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   382
  apply (rule Sup_upper)
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   383
  apply simp
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   384
  done
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   385
bd651ecd4b8a simplified HOL bootstrap
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   386
lemma Inf_singleton [simp]:
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   387
  "\<Sqinter>{a} = a"
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diff changeset
   388
  by (auto intro: antisym Inf_lower Inf_greatest)
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diff changeset
   389
bd651ecd4b8a simplified HOL bootstrap
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diff changeset
   390
lemma Sup_singleton [simp]:
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   391
  "\<Squnion>{a} = a"
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diff changeset
   392
  by (auto intro: antisym Sup_upper Sup_least)
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diff changeset
   393
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   394
lemma Inf_insert_simp:
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   395
  "\<Sqinter>insert a A = (if A = {} then a else a \<sqinter> \<Sqinter>A)"
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diff changeset
   396
  by (cases "A = {}") (simp_all, simp add: Inf_insert)
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diff changeset
   397
bd651ecd4b8a simplified HOL bootstrap
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diff changeset
   398
lemma Sup_insert_simp:
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diff changeset
   399
  "\<Squnion>insert a A = (if A = {} then a else a \<squnion> \<Squnion>A)"
bd651ecd4b8a simplified HOL bootstrap
haftmann
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diff changeset
   400
  by (cases "A = {}") (simp_all, simp add: Sup_insert)
bd651ecd4b8a simplified HOL bootstrap
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diff changeset
   401
bd651ecd4b8a simplified HOL bootstrap
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diff changeset
   402
lemma Inf_binary:
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diff changeset
   403
  "\<Sqinter>{a, b} = a \<sqinter> b"
bd651ecd4b8a simplified HOL bootstrap
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diff changeset
   404
  by (simp add: Inf_insert_simp)
bd651ecd4b8a simplified HOL bootstrap
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parents: 23389
diff changeset
   405
bd651ecd4b8a simplified HOL bootstrap
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diff changeset
   406
lemma Sup_binary:
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diff changeset
   407
  "\<Squnion>{a, b} = a \<squnion> b"
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diff changeset
   408
  by (simp add: Sup_insert_simp)
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diff changeset
   409
bd651ecd4b8a simplified HOL bootstrap
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diff changeset
   410
end
bd651ecd4b8a simplified HOL bootstrap
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diff changeset
   411
bd651ecd4b8a simplified HOL bootstrap
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   412
lemmas Sup_def = Sup_def [folded complete_lattice_class.Sup]
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   413
lemmas Sup_upper = Sup_upper [folded complete_lattice_class.Sup]
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   414
lemmas Sup_least = Sup_least [folded complete_lattice_class.Sup]
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diff changeset
   415
bd651ecd4b8a simplified HOL bootstrap
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diff changeset
   416
lemmas Sup_insert [code func] = Sup_insert [folded complete_lattice_class.Sup]
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diff changeset
   417
lemmas Sup_singleton [simp, code func] = Sup_singleton [folded complete_lattice_class.Sup]
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   418
lemmas Sup_insert_simp = Sup_insert_simp [folded complete_lattice_class.Sup]
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   419
lemmas Sup_binary = Sup_binary [folded complete_lattice_class.Sup]
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diff changeset
   420
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   421
definition
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   422
  top :: "'a::complete_lattice"
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   423
where
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   424
  "top = Inf {}"
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diff changeset
   425
bd651ecd4b8a simplified HOL bootstrap
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diff changeset
   426
definition
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diff changeset
   427
  bot :: "'a::complete_lattice"
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   428
where
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diff changeset
   429
  "bot = Sup {}"
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diff changeset
   430
bd651ecd4b8a simplified HOL bootstrap
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diff changeset
   431
lemma top_greatest [simp]: "x \<le> top"
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diff changeset
   432
  by (unfold top_def, rule Inf_greatest, simp)
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diff changeset
   433
bd651ecd4b8a simplified HOL bootstrap
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diff changeset
   434
lemma bot_least [simp]: "bot \<le> x"
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diff changeset
   435
  by (unfold bot_def, rule Sup_least, simp)
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diff changeset
   436
bd651ecd4b8a simplified HOL bootstrap
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diff changeset
   437
definition
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diff changeset
   438
  SUPR :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b::complete_lattice) \<Rightarrow> 'b"
bd651ecd4b8a simplified HOL bootstrap
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   439
where
bd651ecd4b8a simplified HOL bootstrap
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diff changeset
   440
  "SUPR A f == Sup (f ` A)"
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diff changeset
   441
bd651ecd4b8a simplified HOL bootstrap
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diff changeset
   442
definition
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diff changeset
   443
  INFI :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b::complete_lattice) \<Rightarrow> 'b"
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diff changeset
   444
where
bd651ecd4b8a simplified HOL bootstrap
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diff changeset
   445
  "INFI A f == Inf (f ` A)"
bd651ecd4b8a simplified HOL bootstrap
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diff changeset
   446
bd651ecd4b8a simplified HOL bootstrap
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diff changeset
   447
syntax
bd651ecd4b8a simplified HOL bootstrap
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diff changeset
   448
  "_SUP1"     :: "pttrns => 'b => 'b"           ("(3SUP _./ _)" [0, 10] 10)
bd651ecd4b8a simplified HOL bootstrap
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diff changeset
   449
  "_SUP"      :: "pttrn => 'a set => 'b => 'b"  ("(3SUP _:_./ _)" [0, 10] 10)
bd651ecd4b8a simplified HOL bootstrap
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diff changeset
   450
  "_INF1"     :: "pttrns => 'b => 'b"           ("(3INF _./ _)" [0, 10] 10)
bd651ecd4b8a simplified HOL bootstrap
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   451
  "_INF"      :: "pttrn => 'a set => 'b => 'b"  ("(3INF _:_./ _)" [0, 10] 10)
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diff changeset
   452
bd651ecd4b8a simplified HOL bootstrap
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diff changeset
   453
translations
bd651ecd4b8a simplified HOL bootstrap
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diff changeset
   454
  "SUP x y. B"   == "SUP x. SUP y. B"
bd651ecd4b8a simplified HOL bootstrap
haftmann
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diff changeset
   455
  "SUP x. B"     == "CONST SUPR UNIV (%x. B)"
bd651ecd4b8a simplified HOL bootstrap
haftmann
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diff changeset
   456
  "SUP x. B"     == "SUP x:UNIV. B"
bd651ecd4b8a simplified HOL bootstrap
haftmann
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diff changeset
   457
  "SUP x:A. B"   == "CONST SUPR A (%x. B)"
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   458
  "INF x y. B"   == "INF x. INF y. B"
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   459
  "INF x. B"     == "CONST INFI UNIV (%x. B)"
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   460
  "INF x. B"     == "INF x:UNIV. B"
bd651ecd4b8a simplified HOL bootstrap
haftmann
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diff changeset
   461
  "INF x:A. B"   == "CONST INFI A (%x. B)"
bd651ecd4b8a simplified HOL bootstrap
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diff changeset
   462
bd651ecd4b8a simplified HOL bootstrap
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diff changeset
   463
(* To avoid eta-contraction of body: *)
bd651ecd4b8a simplified HOL bootstrap
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   464
print_translation {*
bd651ecd4b8a simplified HOL bootstrap
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   465
let
bd651ecd4b8a simplified HOL bootstrap
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diff changeset
   466
  fun btr' syn (A :: Abs abs :: ts) =
bd651ecd4b8a simplified HOL bootstrap
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diff changeset
   467
    let val (x,t) = atomic_abs_tr' abs
bd651ecd4b8a simplified HOL bootstrap
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diff changeset
   468
    in list_comb (Syntax.const syn $ x $ A $ t, ts) end
bd651ecd4b8a simplified HOL bootstrap
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diff changeset
   469
  val const_syntax_name = Sign.const_syntax_name @{theory} o fst o dest_Const
bd651ecd4b8a simplified HOL bootstrap
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diff changeset
   470
in
bd651ecd4b8a simplified HOL bootstrap
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diff changeset
   471
[(const_syntax_name @{term SUPR}, btr' "_SUP"),(const_syntax_name @{term "INFI"}, btr' "_INF")]
bd651ecd4b8a simplified HOL bootstrap
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diff changeset
   472
end
bd651ecd4b8a simplified HOL bootstrap
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diff changeset
   473
*}
bd651ecd4b8a simplified HOL bootstrap
haftmann
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diff changeset
   474
bd651ecd4b8a simplified HOL bootstrap
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diff changeset
   475
lemma le_SUPI: "i : A \<Longrightarrow> M i \<le> (SUP i:A. M i)"
bd651ecd4b8a simplified HOL bootstrap
haftmann
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diff changeset
   476
  by (auto simp add: SUPR_def intro: Sup_upper)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   477
bd651ecd4b8a simplified HOL bootstrap
haftmann
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diff changeset
   478
lemma SUP_leI: "(\<And>i. i : A \<Longrightarrow> M i \<le> u) \<Longrightarrow> (SUP i:A. M i) \<le> u"
bd651ecd4b8a simplified HOL bootstrap
haftmann
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diff changeset
   479
  by (auto simp add: SUPR_def intro: Sup_least)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   480
bd651ecd4b8a simplified HOL bootstrap
haftmann
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diff changeset
   481
lemma INF_leI: "i : A \<Longrightarrow> (INF i:A. M i) \<le> M i"
bd651ecd4b8a simplified HOL bootstrap
haftmann
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diff changeset
   482
  by (auto simp add: INFI_def intro: Inf_lower)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   483
bd651ecd4b8a simplified HOL bootstrap
haftmann
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diff changeset
   484
lemma le_INFI: "(\<And>i. i : A \<Longrightarrow> u \<le> M i) \<Longrightarrow> u \<le> (INF i:A. M i)"
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   485
  by (auto simp add: INFI_def intro: Inf_greatest)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   486
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   487
lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (SUP i:A. M) = M"
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   488
  by (auto intro: order_antisym SUP_leI le_SUPI)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   489
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   490
lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (INF i:A. M) = M"
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   491
  by (auto intro: order_antisym INF_leI le_INFI)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   492
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   493
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   494
subsection {* Bool as lattice *}
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   495
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   496
instance bool :: distrib_lattice
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   497
  inf_bool_eq: "inf P Q \<equiv> P \<and> Q"
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   498
  sup_bool_eq: "sup P Q \<equiv> P \<or> Q"
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   499
  by intro_classes (auto simp add: inf_bool_eq sup_bool_eq le_bool_def)
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   500
23878
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   501
instance bool :: complete_lattice
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   502
  Inf_bool_def: "Inf A \<equiv> \<forall>x\<in>A. x"
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   503
  apply intro_classes
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   504
  apply (unfold Inf_bool_def)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   505
  apply (iprover intro!: le_boolI elim: ballE)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   506
  apply (iprover intro!: ballI le_boolI elim: ballE le_boolE)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   507
  done
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   508
23878
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   509
theorem Sup_bool_eq: "Sup A \<longleftrightarrow> (\<exists>x\<in>A. x)"
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   510
  apply (rule order_antisym)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   511
  apply (rule Sup_least)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   512
  apply (rule le_boolI)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   513
  apply (erule bexI, assumption)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   514
  apply (rule le_boolI)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   515
  apply (erule bexE)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   516
  apply (rule le_boolE)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   517
  apply (rule Sup_upper)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   518
  apply assumption+
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   519
  done
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   520
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   521
lemma Inf_empty_bool [simp]:
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   522
  "Inf {}"
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   523
  unfolding Inf_bool_def by auto
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   524
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   525
lemma not_Sup_empty_bool [simp]:
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   526
  "\<not> Sup {}"
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   527
  unfolding Sup_def Inf_bool_def by auto
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   528
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   529
lemma top_bool_eq: "top = True"
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   530
  by (iprover intro!: order_antisym le_boolI top_greatest)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   531
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   532
lemma bot_bool_eq: "bot = False"
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   533
  by (iprover intro!: order_antisym le_boolI bot_least)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   534
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   535
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   536
subsection {* Set as lattice *}
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   537
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   538
instance set :: (type) distrib_lattice
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   539
  inf_set_eq: "inf A B \<equiv> A \<inter> B"
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   540
  sup_set_eq: "sup A B \<equiv> A \<union> B"
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   541
  by intro_classes (auto simp add: inf_set_eq sup_set_eq)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   542
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   543
lemmas [code func del] = inf_set_eq sup_set_eq
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   544
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   545
lemmas mono_Int = mono_inf
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   546
  [where ?'a="?'a set", where ?'b="?'b set", unfolded inf_set_eq sup_set_eq]
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   547
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   548
lemmas mono_Un = mono_sup
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   549
  [where ?'a="?'a set", where ?'b="?'b set", unfolded inf_set_eq sup_set_eq]
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   550
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   551
instance set :: (type) complete_lattice
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   552
  Inf_set_def: "Inf S \<equiv> \<Inter>S"
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   553
  by intro_classes (auto simp add: Inf_set_def)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   554
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   555
lemmas [code func del] = Inf_set_def
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   556
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   557
theorem Sup_set_eq: "Sup S = \<Union>S"
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   558
  apply (rule subset_antisym)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   559
  apply (rule Sup_least)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   560
  apply (erule Union_upper)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   561
  apply (rule Union_least)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   562
  apply (erule Sup_upper)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   563
  done
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   564
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   565
lemma top_set_eq: "top = UNIV"
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   566
  by (iprover intro!: subset_antisym subset_UNIV top_greatest)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   567
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   568
lemma bot_set_eq: "bot = {}"
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   569
  by (iprover intro!: subset_antisym empty_subsetI bot_least)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   570
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   571
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   572
subsection {* Fun as lattice *}
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   573
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   574
instance "fun" :: (type, lattice) lattice
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   575
  inf_fun_eq: "inf f g \<equiv> (\<lambda>x. inf (f x) (g x))"
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   576
  sup_fun_eq: "sup f g \<equiv> (\<lambda>x. sup (f x) (g x))"
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   577
apply intro_classes
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   578
unfolding inf_fun_eq sup_fun_eq
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   579
apply (auto intro: le_funI)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   580
apply (rule le_funI)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   581
apply (auto dest: le_funD)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   582
apply (rule le_funI)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   583
apply (auto dest: le_funD)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   584
done
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   585
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   586
lemmas [code func del] = inf_fun_eq sup_fun_eq
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   587
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   588
instance "fun" :: (type, distrib_lattice) distrib_lattice
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   589
  by default (auto simp add: inf_fun_eq sup_fun_eq sup_inf_distrib1)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   590
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   591
instance "fun" :: (type, complete_lattice) complete_lattice
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   592
  Inf_fun_def: "Inf A \<equiv> (\<lambda>x. Inf {y. \<exists>f\<in>A. y = f x})"
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   593
  apply intro_classes
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   594
  apply (unfold Inf_fun_def)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   595
  apply (rule le_funI)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   596
  apply (rule Inf_lower)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   597
  apply (rule CollectI)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   598
  apply (rule bexI)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   599
  apply (rule refl)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   600
  apply assumption
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   601
  apply (rule le_funI)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   602
  apply (rule Inf_greatest)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   603
  apply (erule CollectE)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   604
  apply (erule bexE)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   605
  apply (iprover elim: le_funE)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   606
  done
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   607
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   608
lemmas [code func del] = Inf_fun_def
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   609
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   610
theorem Sup_fun_eq: "Sup A = (\<lambda>x. Sup {y. \<exists>f\<in>A. y = f x})"
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   611
  apply (rule order_antisym)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   612
  apply (rule Sup_least)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   613
  apply (rule le_funI)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   614
  apply (rule Sup_upper)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   615
  apply fast
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   616
  apply (rule le_funI)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   617
  apply (rule Sup_least)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   618
  apply (erule CollectE)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   619
  apply (erule bexE)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   620
  apply (drule le_funD [OF Sup_upper])
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   621
  apply simp
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   622
  done
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   623
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   624
lemma Inf_empty_fun:
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   625
  "Inf {} = (\<lambda>_. Inf {})"
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   626
  by rule (auto simp add: Inf_fun_def)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   627
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   628
lemma Sup_empty_fun:
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   629
  "Sup {} = (\<lambda>_. Sup {})"
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   630
proof -
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   631
  have aux: "\<And>x. {y. \<exists>f. y = f x} = UNIV" by auto
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   632
  show ?thesis
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   633
  by (auto simp add: Sup_def Inf_fun_def Inf_binary inf_bool_eq aux)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   634
qed
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   635
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   636
lemma top_fun_eq: "top = (\<lambda>x. top)"
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   637
  by (iprover intro!: order_antisym le_funI top_greatest)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   638
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   639
lemma bot_fun_eq: "bot = (\<lambda>x. bot)"
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   640
  by (iprover intro!: order_antisym le_funI bot_least)
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   641
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   642
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   643
text {* redundant bindings *}
22454
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   644
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   645
lemmas inf_aci = inf_ACI
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   646
lemmas sup_aci = sup_ACI
c3654ba76a09 integrated with LOrder.thy
haftmann
parents: 22422
diff changeset
   647
23878
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   648
ML {*
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   649
val sup_fun_eq = @{thm sup_fun_eq}
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   650
val sup_bool_eq = @{thm sup_bool_eq}
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   651
*}
bd651ecd4b8a simplified HOL bootstrap
haftmann
parents: 23389
diff changeset
   652
21249
d594c58e24ed renamed Lattice_Locales to Lattices
haftmann
parents:
diff changeset
   653
end