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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 {* Lattices via Locales *}


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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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text{* This theory of lattice locales only defines binary sup and inf


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operations. The extension to finite sets is done in theory @{text


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Finite_Set}. In the longer term it may be better to define arbitrary


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sups and infs via @{text THE}. *}


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locale lower_semilattice = partial_order +


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fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<sqinter>" 70)


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assumes inf_le1: "x \<sqinter> y \<sqsubseteq> x" and inf_le2: "x \<sqinter> y \<sqsubseteq> y"


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and inf_least: "x \<sqsubseteq> y \<Longrightarrow> x \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> y \<sqinter> z"


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locale upper_semilattice = partial_order +


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fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<squnion>" 65)


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assumes sup_ge1: "x \<sqsubseteq> x \<squnion> y" and sup_ge2: "y \<sqsubseteq> x \<squnion> y"


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and sup_greatest: "y \<sqsubseteq> x \<Longrightarrow> z \<sqsubseteq> x \<Longrightarrow> y \<squnion> z \<sqsubseteq> x"


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locale lattice = lower_semilattice + upper_semilattice


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lemma (in lower_semilattice) inf_commute: "(x \<sqinter> y) = (y \<sqinter> x)"


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by(blast intro: antisym inf_le1 inf_le2 inf_least)


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lemma (in upper_semilattice) sup_commute: "(x \<squnion> y) = (y \<squnion> x)"


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by(blast intro: antisym sup_ge1 sup_ge2 sup_greatest)


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lemma (in lower_semilattice) inf_assoc: "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"


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by(blast intro: antisym inf_le1 inf_le2 inf_least trans del:refl)


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lemma (in upper_semilattice) sup_assoc: "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"


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by(blast intro!: antisym sup_ge1 sup_ge2 intro: sup_greatest trans del:refl)


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lemma (in lower_semilattice) inf_idem[simp]: "x \<sqinter> x = x"


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by(blast intro: antisym inf_le1 inf_le2 inf_least refl)


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lemma (in upper_semilattice) sup_idem[simp]: "x \<squnion> x = x"


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by(blast intro: antisym sup_ge1 sup_ge2 sup_greatest refl)


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lemma (in lower_semilattice) inf_left_idem[simp]: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"


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by (simp add: inf_assoc[symmetric])


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lemma (in upper_semilattice) sup_left_idem[simp]: "x \<squnion> (x \<squnion> y) = x \<squnion> y"


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by (simp add: sup_assoc[symmetric])


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lemma (in lattice) inf_sup_absorb: "x \<sqinter> (x \<squnion> y) = x"


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by(blast intro: antisym inf_le1 inf_least sup_ge1)


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lemma (in lattice) sup_inf_absorb: "x \<squnion> (x \<sqinter> y) = x"


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by(blast intro: antisym sup_ge1 sup_greatest inf_le1)


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lemma (in lower_semilattice) inf_absorb: "x \<sqsubseteq> y \<Longrightarrow> x \<sqinter> y = x"


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by(blast intro: antisym inf_le1 inf_least refl)


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lemma (in upper_semilattice) sup_absorb: "x \<sqsubseteq> y \<Longrightarrow> x \<squnion> y = y"


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by(blast intro: antisym sup_ge2 sup_greatest refl)


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lemma (in lower_semilattice) less_eq_inf_conv [simp]:


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"x \<sqsubseteq> y \<sqinter> z = (x \<sqsubseteq> y \<and> x \<sqsubseteq> z)"


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by(blast intro: antisym inf_le1 inf_le2 inf_least refl trans)


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lemmas (in lower_semilattice) below_inf_conv = less_eq_inf_conv


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 {* a duplicate for backward compatibility *}


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lemma (in upper_semilattice) above_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 intro: antisym sup_ge1 sup_ge2 sup_greatest refl trans)


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text{* Towards distributivity: if you have one of them, you have them all. *}


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lemma (in lattice) distrib_imp1:


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assumes D: "!!x y z. x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"


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shows "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"


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proof


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have "x \<squnion> (y \<sqinter> z) = (x \<squnion> (x \<sqinter> z)) \<squnion> (y \<sqinter> z)" by(simp add:sup_inf_absorb)


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also have "\<dots> = x \<squnion> (z \<sqinter> (x \<squnion> y))" by(simp add:D inf_commute sup_assoc)


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also have "\<dots> = ((x \<squnion> y) \<sqinter> x) \<squnion> ((x \<squnion> y) \<sqinter> z)"


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by(simp add:inf_sup_absorb inf_commute)


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also have "\<dots> = (x \<squnion> y) \<sqinter> (x \<squnion> z)" by(simp add:D)


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finally show ?thesis .


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qed


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lemma (in lattice) distrib_imp2:


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assumes D: "!!x y z. x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"


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shows "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"


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proof


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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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also have "\<dots> = x \<sqinter> (z \<squnion> (x \<sqinter> y))" by(simp add:D sup_commute inf_assoc)


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also have "\<dots> = ((x \<sqinter> y) \<squnion> x) \<sqinter> ((x \<sqinter> y) \<squnion> z)"


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by(simp add:sup_inf_absorb sup_commute)


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also have "\<dots> = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" by(simp add:D)


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finally show ?thesis .


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qed


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text{* A package of rewrite rules for deciding equivalence wrt ACI: *}


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lemma (in lower_semilattice) inf_left_commute: "x \<sqinter> (y \<sqinter> z) = y \<sqinter> (x \<sqinter> z)"


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proof 


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have "x \<sqinter> (y \<sqinter> z) = (y \<sqinter> z) \<sqinter> x" by (simp only: inf_commute)


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also have "... = y \<sqinter> (z \<sqinter> x)" by (simp only: inf_assoc)


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also have "z \<sqinter> x = x \<sqinter> z" by (simp only: inf_commute)


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finally(back_subst) show ?thesis .


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qed


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lemma (in upper_semilattice) sup_left_commute: "x \<squnion> (y \<squnion> z) = y \<squnion> (x \<squnion> z)"


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proof 


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have "x \<squnion> (y \<squnion> z) = (y \<squnion> z) \<squnion> x" by (simp only: sup_commute)


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also have "... = y \<squnion> (z \<squnion> x)" by (simp only: sup_assoc)


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also have "z \<squnion> x = x \<squnion> z" by (simp only: sup_commute)


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finally(back_subst) show ?thesis .


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qed


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lemma (in lower_semilattice) inf_left_idem: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"


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proof 


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have "x \<sqinter> (x \<sqinter> y) = (x \<sqinter> x) \<sqinter> y" by(simp only:inf_assoc)


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also have "\<dots> = x \<sqinter> y" by(simp)


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finally show ?thesis .


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qed


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lemma (in upper_semilattice) sup_left_idem: "x \<squnion> (x \<squnion> y) = x \<squnion> y"


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proof 


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have "x \<squnion> (x \<squnion> y) = (x \<squnion> x) \<squnion> y" by(simp only:sup_assoc)


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also have "\<dots> = x \<squnion> y" by(simp)


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finally show ?thesis .


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qed


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lemmas (in lower_semilattice) inf_ACI =


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inf_commute inf_assoc inf_left_commute inf_left_idem


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lemmas (in upper_semilattice) sup_ACI =


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sup_commute sup_assoc sup_left_commute sup_left_idem


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lemmas (in lattice) ACI = inf_ACI sup_ACI


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subsection{* Distributive lattices *}


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locale distrib_lattice = lattice +


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assumes sup_inf_distrib1: "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"


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lemma (in distrib_lattice) sup_inf_distrib2:


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"(y \<sqinter> z) \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)"


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by(simp add:ACI sup_inf_distrib1)


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lemma (in distrib_lattice) inf_sup_distrib1:


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"x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"


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by(rule distrib_imp2[OF sup_inf_distrib1])


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lemma (in distrib_lattice) inf_sup_distrib2:


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"(y \<squnion> z) \<sqinter> x = (y \<sqinter> x) \<squnion> (z \<sqinter> x)"


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by(simp add:ACI inf_sup_distrib1)


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lemmas (in distrib_lattice) distrib =


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sup_inf_distrib1 sup_inf_distrib2 inf_sup_distrib1 inf_sup_distrib2


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subsection {* Least value operator and min/max  properties *}


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(*FIXME: derive more of the min/max laws generically via semilattices*)


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lemma LeastI2_order:


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"[ P (x::'a::order);


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!!y. P y ==> x <= y;


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!!x. [ P x; ALL y. P y > x \<le> y ] ==> Q x ]


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==> Q (Least P)"


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apply (unfold Least_def)


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apply (rule theI2)


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apply (blast intro: order_antisym)+


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done


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lemma Least_equality:


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"[ P (k::'a::order); !!x. P x ==> k <= x ] ==> (LEAST x. P x) = k"


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apply (simp add: Least_def)


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apply (rule the_equality)


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apply (auto intro!: order_antisym)


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done


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lemma min_leastL: "(!!x. least <= x) ==> min least x = least"


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by (simp add: min_def)


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lemma max_leastL: "(!!x. least <= x) ==> max least x = x"


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by (simp add: max_def)


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lemma min_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> min x least = least"


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apply (simp add: min_def)


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apply (blast intro: order_antisym)


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done


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lemma max_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> max x least = x"


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apply (simp add: max_def)


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apply (blast intro: order_antisym)


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done


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lemma min_of_mono:


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"(!!x y. (f x <= f y) = (x <= y)) ==> min (f m) (f n) = f (min m n)"


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by (simp add: min_def)


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lemma max_of_mono:


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"(!!x y. (f x <= f y) = (x <= y)) ==> max (f m) (f n) = f (max m n)"


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by (simp add: max_def)


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text{* Instantiate locales: *}


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interpretation min_max:


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lower_semilattice["op \<le>" "op <" "min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a"]


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apply unfold_locales


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apply(simp add:min_def linorder_not_le order_less_imp_le)


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apply(simp add:min_def linorder_not_le order_less_imp_le)


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apply(simp add:min_def linorder_not_le order_less_imp_le)


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done


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interpretation min_max:


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upper_semilattice["op \<le>" "op <" "max :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a"]


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apply unfold_locales


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apply(simp add: max_def linorder_not_le order_less_imp_le)


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apply(simp add: max_def linorder_not_le order_less_imp_le)


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apply(simp add: max_def linorder_not_le order_less_imp_le)


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done


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interpretation min_max:


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lattice["op \<le>" "op <" "min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a" "max"]


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by unfold_locales


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interpretation min_max:


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distrib_lattice["op \<le>" "op <" "min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a" "max"]


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apply unfold_locales


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apply(rule_tac x=x and y=y in linorder_le_cases)


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apply(rule_tac x=x and y=z in linorder_le_cases)


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apply(rule_tac x=y and y=z in linorder_le_cases)


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apply(simp add:min_def max_def)


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apply(simp add:min_def max_def)


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apply(rule_tac x=y and y=z in linorder_le_cases)


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apply(simp add:min_def max_def)


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apply(simp add:min_def max_def)


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apply(rule_tac x=x and y=z in linorder_le_cases)


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apply(rule_tac x=y and y=z in linorder_le_cases)


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apply(simp add:min_def max_def)


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apply(simp add:min_def max_def)


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apply(rule_tac x=y and y=z in linorder_le_cases)


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apply(simp add:min_def max_def)


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apply(simp add:min_def max_def)


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done


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lemma le_max_iff_disj: "!!z::'a::linorder. (z <= max x y) = (z <= x  z <= y)"


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apply(simp add:max_def)


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apply (insert linorder_linear)


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apply (blast intro: order_trans)


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done


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lemmas le_maxI1 = min_max.sup_ge1


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lemmas le_maxI2 = min_max.sup_ge2


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lemma less_max_iff_disj: "!!z::'a::linorder. (z < max x y) = (z < x  z < y)"


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apply (simp add: max_def order_le_less)


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apply (insert linorder_less_linear)


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apply (blast intro: order_less_trans)


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done


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lemma max_less_iff_conj [simp]:


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"!!z::'a::linorder. (max x y < z) = (x < z & y < z)"


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apply (simp add: order_le_less max_def)


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apply (insert linorder_less_linear)


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apply (blast intro: order_less_trans)


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done


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lemma min_less_iff_conj [simp]:


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"!!z::'a::linorder. (z < min x y) = (z < x & z < y)"


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apply (simp add: order_le_less min_def)


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apply (insert linorder_less_linear)


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apply (blast intro: order_less_trans)


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done


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lemma min_le_iff_disj: "!!z::'a::linorder. (min x y <= z) = (x <= z  y <= z)"


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apply (simp add: min_def)


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apply (insert linorder_linear)


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apply (blast intro: order_trans)


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done


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lemma min_less_iff_disj: "!!z::'a::linorder. (min x y < z) = (x < z  y < z)"


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apply (simp add: min_def order_le_less)


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apply (insert linorder_less_linear)


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apply (blast intro: order_less_trans)


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done


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lemmas max_ac = min_max.sup_assoc min_max.sup_commute


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mk_left_commute[of max,OF min_max.sup_assoc min_max.sup_commute]


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lemmas min_ac = min_max.inf_assoc min_max.inf_commute


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mk_left_commute[of min,OF min_max.inf_assoc min_max.inf_commute]


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lemma split_min:


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"P (min (i::'a::linorder) j) = ((i <= j > P(i)) & (~ i <= j > P(j)))"


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by (simp add: min_def)


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lemma split_max:


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"P (max (i::'a::linorder) j) = ((i <= j > P(j)) & (~ i <= j > P(i)))"


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by (simp add: max_def)


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text {* ML legacy bindings *}


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ML {*


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val Least_def = thm "Least_def";


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val Least_equality = thm "Least_equality";


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val min_def = thm "min_def";


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val min_of_mono = thm "min_of_mono";


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val max_def = thm "max_def";


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val max_of_mono = thm "max_of_mono";


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val min_leastL = thm "min_leastL";


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val max_leastL = thm "max_leastL";


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val min_leastR = thm "min_leastR";


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val max_leastR = thm "max_leastR";


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val le_max_iff_disj = thm "le_max_iff_disj";


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val le_maxI1 = thm "le_maxI1";


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val le_maxI2 = thm "le_maxI2";


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val less_max_iff_disj = thm "less_max_iff_disj";


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val max_less_iff_conj = thm "max_less_iff_conj";


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val min_less_iff_conj = thm "min_less_iff_conj";


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val min_le_iff_disj = thm "min_le_iff_disj";


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val min_less_iff_disj = thm "min_less_iff_disj";


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val split_min = thm "split_min";


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val split_max = thm "split_max";


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*}


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end
