# HG changeset patch
# User haftmann
# Date 1163011714 -3600
# Node ID d594c58e24ed9e95b56dda3de7df6c7952886a40
# Parent  3fd22b0939ffdf898ae87886a69143a4aa4222f2
renamed Lattice_Locales to Lattices

diff -r 3fd22b0939ff -r d594c58e24ed src/HOL/Finite_Set.thy
--- a/src/HOL/Finite_Set.thy	Wed Nov 08 19:46:10 2006 +0100
+++ b/src/HOL/Finite_Set.thy	Wed Nov 08 19:48:34 2006 +0100
@@ -7,7 +7,7 @@
 header {* Finite sets *}
 
 theory Finite_Set
-imports Power Inductive Lattice_Locales
+imports Power Inductive Lattices
 begin
 
 subsection {* Definition and basic properties *}
diff -r 3fd22b0939ff -r d594c58e24ed src/HOL/IsaMakefile
--- a/src/HOL/IsaMakefile	Wed Nov 08 19:46:10 2006 +0100
+++ b/src/HOL/IsaMakefile	Wed Nov 08 19:48:34 2006 +0100
@@ -94,7 +94,7 @@
   Integ/cooper_proof.ML Integ/reflected_presburger.ML				\
   Integ/reflected_cooper.ML Integ/int_arith1.ML Integ/int_factor_simprocs.ML	\
   Integ/nat_simprocs.ML Integ/presburger.ML Integ/qelim.ML LOrder.thy		\
-  Lattice_Locales.thy List.ML List.thy Main.thy Map.thy			\
+  Lattices.thy List.ML List.thy Main.thy Map.thy			\
   Nat.ML Nat.thy OrderedGroup.ML OrderedGroup.thy	\
   Orderings.ML Orderings.thy Power.thy PreList.thy Product_Type.thy		\
   ROOT.ML Recdef.thy Reconstruction.thy Record.thy Refute.thy			\
diff -r 3fd22b0939ff -r d594c58e24ed src/HOL/LOrder.thy
--- a/src/HOL/LOrder.thy	Wed Nov 08 19:46:10 2006 +0100
+++ b/src/HOL/LOrder.thy	Wed Nov 08 19:48:34 2006 +0100
@@ -6,7 +6,7 @@
 header {* Lattice Orders *}
 
 theory LOrder
-imports Lattice_Locales
+imports Lattices
 begin
 
 text {*
diff -r 3fd22b0939ff -r d594c58e24ed src/HOL/Lattice_Locales.thy
--- a/src/HOL/Lattice_Locales.thy	Wed Nov 08 19:46:10 2006 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,336 +0,0 @@
-(*  Title:      HOL/Lattices.thy
-    ID:         $Id$
-    Author:     Tobias Nipkow
-*)
-
-header {* Lattices via Locales *}
-
-theory Lattice_Locales
-imports Orderings
-begin
-
-subsection{* Lattices *}
-
-text{* This theory of lattice locales only defines binary sup and inf
-operations. The extension to finite sets is done in theory @{text
-Finite_Set}. In the longer term it may be better to define arbitrary
-sups and infs via @{text THE}. *}
-
-locale lower_semilattice = partial_order +
-  fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<sqinter>" 70)
-  assumes inf_le1: "x \<sqinter> y \<sqsubseteq> x" and inf_le2: "x \<sqinter> y \<sqsubseteq> y"
-  and inf_least: "x \<sqsubseteq> y \<Longrightarrow> x \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> y \<sqinter> z"
-
-locale upper_semilattice = partial_order +
-  fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<squnion>" 65)
-  assumes sup_ge1: "x \<sqsubseteq> x \<squnion> y" and sup_ge2: "y \<sqsubseteq> x \<squnion> y"
-  and sup_greatest: "y \<sqsubseteq> x \<Longrightarrow> z \<sqsubseteq> x \<Longrightarrow> y \<squnion> z \<sqsubseteq> x"
-
-locale lattice = lower_semilattice + upper_semilattice
-
-lemma (in lower_semilattice) inf_commute: "(x \<sqinter> y) = (y \<sqinter> x)"
-by(blast intro: antisym inf_le1 inf_le2 inf_least)
-
-lemma (in upper_semilattice) sup_commute: "(x \<squnion> y) = (y \<squnion> x)"
-by(blast intro: antisym sup_ge1 sup_ge2 sup_greatest)
-
-lemma (in lower_semilattice) inf_assoc: "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
-by(blast intro: antisym inf_le1 inf_le2 inf_least trans del:refl)
-
-lemma (in upper_semilattice) sup_assoc: "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
-by(blast intro!: antisym sup_ge1 sup_ge2 intro: sup_greatest trans del:refl)
-
-lemma (in lower_semilattice) inf_idem[simp]: "x \<sqinter> x = x"
-by(blast intro: antisym inf_le1 inf_le2 inf_least refl)
-
-lemma (in upper_semilattice) sup_idem[simp]: "x \<squnion> x = x"
-by(blast intro: antisym sup_ge1 sup_ge2 sup_greatest refl)
-
-lemma (in lower_semilattice) inf_left_idem[simp]: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"
-by (simp add: inf_assoc[symmetric])
-
-lemma (in upper_semilattice) sup_left_idem[simp]: "x \<squnion> (x \<squnion> y) = x \<squnion> y"
-by (simp add: sup_assoc[symmetric])
-
-lemma (in lattice) inf_sup_absorb: "x \<sqinter> (x \<squnion> y) = x"
-by(blast intro: antisym inf_le1 inf_least sup_ge1)
-
-lemma (in lattice) sup_inf_absorb: "x \<squnion> (x \<sqinter> y) = x"
-by(blast intro: antisym sup_ge1 sup_greatest inf_le1)
-
-lemma (in lower_semilattice) inf_absorb: "x \<sqsubseteq> y \<Longrightarrow> x \<sqinter> y = x"
-by(blast intro: antisym inf_le1 inf_least refl)
-
-lemma (in upper_semilattice) sup_absorb: "x \<sqsubseteq> y \<Longrightarrow> x \<squnion> y = y"
-by(blast intro: antisym sup_ge2 sup_greatest refl)
-
-
-lemma (in lower_semilattice) less_eq_inf_conv [simp]:
- "x \<sqsubseteq> y \<sqinter> z = (x \<sqsubseteq> y \<and> x \<sqsubseteq> z)"
-by(blast intro: antisym inf_le1 inf_le2 inf_least refl trans)
-
-lemmas (in lower_semilattice) below_inf_conv = less_eq_inf_conv
-  -- {* a duplicate for backward compatibility *}
-
-lemma (in upper_semilattice) above_sup_conv[simp]:
- "x \<squnion> y \<sqsubseteq> z = (x \<sqsubseteq> z \<and> y \<sqsubseteq> z)"
-by(blast intro: antisym sup_ge1 sup_ge2 sup_greatest refl trans)
-
-
-text{* Towards distributivity: if you have one of them, you have them all. *}
-
-lemma (in lattice) distrib_imp1:
-assumes D: "!!x y z. x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
-shows "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
-proof-
-  have "x \<squnion> (y \<sqinter> z) = (x \<squnion> (x \<sqinter> z)) \<squnion> (y \<sqinter> z)" by(simp add:sup_inf_absorb)
-  also have "\<dots> = x \<squnion> (z \<sqinter> (x \<squnion> y))" by(simp add:D inf_commute sup_assoc)
-  also have "\<dots> = ((x \<squnion> y) \<sqinter> x) \<squnion> ((x \<squnion> y) \<sqinter> z)"
-    by(simp add:inf_sup_absorb inf_commute)
-  also have "\<dots> = (x \<squnion> y) \<sqinter> (x \<squnion> z)" by(simp add:D)
-  finally show ?thesis .
-qed
-
-lemma (in lattice) distrib_imp2:
-assumes D: "!!x y z. x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
-shows "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
-proof-
-  have "x \<sqinter> (y \<squnion> z) = (x \<sqinter> (x \<squnion> z)) \<sqinter> (y \<squnion> z)" by(simp add:inf_sup_absorb)
-  also have "\<dots> = x \<sqinter> (z \<squnion> (x \<sqinter> y))" by(simp add:D sup_commute inf_assoc)
-  also have "\<dots> = ((x \<sqinter> y) \<squnion> x) \<sqinter> ((x \<sqinter> y) \<squnion> z)"
-    by(simp add:sup_inf_absorb sup_commute)
-  also have "\<dots> = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" by(simp add:D)
-  finally show ?thesis .
-qed
-
-text{* A package of rewrite rules for deciding equivalence wrt ACI: *}
-
-lemma (in lower_semilattice) inf_left_commute: "x \<sqinter> (y \<sqinter> z) = y \<sqinter> (x \<sqinter> z)"
-proof -
-  have "x \<sqinter> (y \<sqinter> z) = (y \<sqinter> z) \<sqinter> x" by (simp only: inf_commute)
-  also have "... = y \<sqinter> (z \<sqinter> x)" by (simp only: inf_assoc)
-  also have "z \<sqinter> x = x \<sqinter> z" by (simp only: inf_commute)
-  finally(back_subst) show ?thesis .
-qed
-
-lemma (in upper_semilattice) sup_left_commute: "x \<squnion> (y \<squnion> z) = y \<squnion> (x \<squnion> z)"
-proof -
-  have "x \<squnion> (y \<squnion> z) = (y \<squnion> z) \<squnion> x" by (simp only: sup_commute)
-  also have "... = y \<squnion> (z \<squnion> x)" by (simp only: sup_assoc)
-  also have "z \<squnion> x = x \<squnion> z" by (simp only: sup_commute)
-  finally(back_subst) show ?thesis .
-qed
-
-lemma (in lower_semilattice) inf_left_idem: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"
-proof -
-  have "x \<sqinter> (x \<sqinter> y) = (x \<sqinter> x) \<sqinter> y" by(simp only:inf_assoc)
-  also have "\<dots> = x \<sqinter> y" by(simp)
-  finally show ?thesis .
-qed
-
-lemma (in upper_semilattice) sup_left_idem: "x \<squnion> (x \<squnion> y) = x \<squnion> y"
-proof -
-  have "x \<squnion> (x \<squnion> y) = (x \<squnion> x) \<squnion> y" by(simp only:sup_assoc)
-  also have "\<dots> = x \<squnion> y" by(simp)
-  finally show ?thesis .
-qed
-
-
-lemmas (in lower_semilattice) inf_ACI =
- inf_commute inf_assoc inf_left_commute inf_left_idem
-
-lemmas (in upper_semilattice) sup_ACI =
- sup_commute sup_assoc sup_left_commute sup_left_idem
-
-lemmas (in lattice) ACI = inf_ACI sup_ACI
-
-
-subsection{* Distributive lattices *}
-
-locale distrib_lattice = lattice +
-  assumes sup_inf_distrib1: "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
-
-lemma (in distrib_lattice) sup_inf_distrib2:
- "(y \<sqinter> z) \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)"
-by(simp add:ACI sup_inf_distrib1)
-
-lemma (in distrib_lattice) inf_sup_distrib1:
- "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
-by(rule distrib_imp2[OF sup_inf_distrib1])
-
-lemma (in distrib_lattice) inf_sup_distrib2:
- "(y \<squnion> z) \<sqinter> x = (y \<sqinter> x) \<squnion> (z \<sqinter> x)"
-by(simp add:ACI inf_sup_distrib1)
-
-lemmas (in distrib_lattice) distrib =
-  sup_inf_distrib1 sup_inf_distrib2 inf_sup_distrib1 inf_sup_distrib2
-
-
-subsection {* Least value operator and min/max -- properties *}
- 
-(*FIXME: derive more of the min/max laws generically via semilattices*)
-
-lemma LeastI2_order:
-  "[| P (x::'a::order);
-      !!y. P y ==> x <= y;
-      !!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |]
-   ==> Q (Least P)"
-  apply (unfold Least_def)
-  apply (rule theI2)
-    apply (blast intro: order_antisym)+
-  done
-
-lemma Least_equality:
-    "[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k"
-  apply (simp add: Least_def)
-  apply (rule the_equality)
-  apply (auto intro!: order_antisym)
-  done
-
-lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
-  by (simp add: min_def)
-
-lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
-  by (simp add: max_def)
-
-lemma min_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> min x least = least"
-  apply (simp add: min_def)
-  apply (blast intro: order_antisym)
-  done
-
-lemma max_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> max x least = x"
-  apply (simp add: max_def)
-  apply (blast intro: order_antisym)
-  done
-
-lemma min_of_mono:
-    "(!!x y. (f x <= f y) = (x <= y)) ==> min (f m) (f n) = f (min m n)"
-  by (simp add: min_def)
-
-lemma max_of_mono:
-    "(!!x y. (f x <= f y) = (x <= y)) ==> max (f m) (f n) = f (max m n)"
-  by (simp add: max_def)
-
-text{* Instantiate locales: *}
-
-interpretation min_max:
-  lower_semilattice["op \<le>" "op <" "min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a"]
-apply unfold_locales
-apply(simp add:min_def linorder_not_le order_less_imp_le)
-apply(simp add:min_def linorder_not_le order_less_imp_le)
-apply(simp add:min_def linorder_not_le order_less_imp_le)
-done
-
-interpretation min_max:
-  upper_semilattice["op \<le>" "op <" "max :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a"]
-apply unfold_locales
-apply(simp add: max_def linorder_not_le order_less_imp_le)
-apply(simp add: max_def linorder_not_le order_less_imp_le)
-apply(simp add: max_def linorder_not_le order_less_imp_le)
-done
-
-interpretation min_max:
-  lattice["op \<le>" "op <" "min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a" "max"]
-  by unfold_locales
-
-interpretation min_max:
-  distrib_lattice["op \<le>" "op <" "min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a" "max"]
-apply unfold_locales
-apply(rule_tac x=x and y=y in linorder_le_cases)
-apply(rule_tac x=x and y=z in linorder_le_cases)
-apply(rule_tac x=y and y=z in linorder_le_cases)
-apply(simp add:min_def max_def)
-apply(simp add:min_def max_def)
-apply(rule_tac x=y and y=z in linorder_le_cases)
-apply(simp add:min_def max_def)
-apply(simp add:min_def max_def)
-apply(rule_tac x=x and y=z in linorder_le_cases)
-apply(rule_tac x=y and y=z in linorder_le_cases)
-apply(simp add:min_def max_def)
-apply(simp add:min_def max_def)
-apply(rule_tac x=y and y=z in linorder_le_cases)
-apply(simp add:min_def max_def)
-apply(simp add:min_def max_def)
-done
-
-lemma le_max_iff_disj: "!!z::'a::linorder. (z <= max x y) = (z <= x | z <= y)"
-  apply(simp add:max_def)
-  apply (insert linorder_linear)
-  apply (blast intro: order_trans)
-  done
-
-lemmas le_maxI1 = min_max.sup_ge1
-lemmas le_maxI2 = min_max.sup_ge2
-
-lemma less_max_iff_disj: "!!z::'a::linorder. (z < max x y) = (z < x | z < y)"
-  apply (simp add: max_def order_le_less)
-  apply (insert linorder_less_linear)
-  apply (blast intro: order_less_trans)
-  done
-
-lemma max_less_iff_conj [simp]:
-    "!!z::'a::linorder. (max x y < z) = (x < z & y < z)"
-  apply (simp add: order_le_less max_def)
-  apply (insert linorder_less_linear)
-  apply (blast intro: order_less_trans)
-  done
-
-lemma min_less_iff_conj [simp]:
-    "!!z::'a::linorder. (z < min x y) = (z < x & z < y)"
-  apply (simp add: order_le_less min_def)
-  apply (insert linorder_less_linear)
-  apply (blast intro: order_less_trans)
-  done
-
-lemma min_le_iff_disj: "!!z::'a::linorder. (min x y <= z) = (x <= z | y <= z)"
-  apply (simp add: min_def)
-  apply (insert linorder_linear)
-  apply (blast intro: order_trans)
-  done
-
-lemma min_less_iff_disj: "!!z::'a::linorder. (min x y < z) = (x < z | y < z)"
-  apply (simp add: min_def order_le_less)
-  apply (insert linorder_less_linear)
-  apply (blast intro: order_less_trans)
-  done
-
-lemmas max_ac = min_max.sup_assoc min_max.sup_commute
-               mk_left_commute[of max,OF min_max.sup_assoc min_max.sup_commute]
-
-lemmas min_ac = min_max.inf_assoc min_max.inf_commute
-               mk_left_commute[of min,OF min_max.inf_assoc min_max.inf_commute]
-
-lemma split_min:
-    "P (min (i::'a::linorder) j) = ((i <= j --> P(i)) & (~ i <= j --> P(j)))"
-  by (simp add: min_def)
-
-lemma split_max:
-    "P (max (i::'a::linorder) j) = ((i <= j --> P(j)) & (~ i <= j --> P(i)))"
-  by (simp add: max_def)
-
-text {* ML legacy bindings *}
-
-ML {*
-val Least_def = thm "Least_def";
-val Least_equality = thm "Least_equality";
-val min_def = thm "min_def";
-val min_of_mono = thm "min_of_mono";
-val max_def = thm "max_def";
-val max_of_mono = thm "max_of_mono";
-val min_leastL = thm "min_leastL";
-val max_leastL = thm "max_leastL";
-val min_leastR = thm "min_leastR";
-val max_leastR = thm "max_leastR";
-val le_max_iff_disj = thm "le_max_iff_disj";
-val le_maxI1 = thm "le_maxI1";
-val le_maxI2 = thm "le_maxI2";
-val less_max_iff_disj = thm "less_max_iff_disj";
-val max_less_iff_conj = thm "max_less_iff_conj";
-val min_less_iff_conj = thm "min_less_iff_conj";
-val min_le_iff_disj = thm "min_le_iff_disj";
-val min_less_iff_disj = thm "min_less_iff_disj";
-val split_min = thm "split_min";
-val split_max = thm "split_max";
-*}
-
-end
diff -r 3fd22b0939ff -r d594c58e24ed src/HOL/Lattices.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Lattices.thy	Wed Nov 08 19:48:34 2006 +0100
@@ -0,0 +1,336 @@
+(*  Title:      HOL/Lattices.thy
+    ID:         $Id$
+    Author:     Tobias Nipkow
+*)
+
+header {* Lattices via Locales *}
+
+theory Lattices
+imports Orderings
+begin
+
+subsection{* Lattices *}
+
+text{* This theory of lattice locales only defines binary sup and inf
+operations. The extension to finite sets is done in theory @{text
+Finite_Set}. In the longer term it may be better to define arbitrary
+sups and infs via @{text THE}. *}
+
+locale lower_semilattice = partial_order +
+  fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<sqinter>" 70)
+  assumes inf_le1: "x \<sqinter> y \<sqsubseteq> x" and inf_le2: "x \<sqinter> y \<sqsubseteq> y"
+  and inf_least: "x \<sqsubseteq> y \<Longrightarrow> x \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> y \<sqinter> z"
+
+locale upper_semilattice = partial_order +
+  fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<squnion>" 65)
+  assumes sup_ge1: "x \<sqsubseteq> x \<squnion> y" and sup_ge2: "y \<sqsubseteq> x \<squnion> y"
+  and sup_greatest: "y \<sqsubseteq> x \<Longrightarrow> z \<sqsubseteq> x \<Longrightarrow> y \<squnion> z \<sqsubseteq> x"
+
+locale lattice = lower_semilattice + upper_semilattice
+
+lemma (in lower_semilattice) inf_commute: "(x \<sqinter> y) = (y \<sqinter> x)"
+by(blast intro: antisym inf_le1 inf_le2 inf_least)
+
+lemma (in upper_semilattice) sup_commute: "(x \<squnion> y) = (y \<squnion> x)"
+by(blast intro: antisym sup_ge1 sup_ge2 sup_greatest)
+
+lemma (in lower_semilattice) inf_assoc: "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
+by(blast intro: antisym inf_le1 inf_le2 inf_least trans del:refl)
+
+lemma (in upper_semilattice) sup_assoc: "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
+by(blast intro!: antisym sup_ge1 sup_ge2 intro: sup_greatest trans del:refl)
+
+lemma (in lower_semilattice) inf_idem[simp]: "x \<sqinter> x = x"
+by(blast intro: antisym inf_le1 inf_le2 inf_least refl)
+
+lemma (in upper_semilattice) sup_idem[simp]: "x \<squnion> x = x"
+by(blast intro: antisym sup_ge1 sup_ge2 sup_greatest refl)
+
+lemma (in lower_semilattice) inf_left_idem[simp]: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"
+by (simp add: inf_assoc[symmetric])
+
+lemma (in upper_semilattice) sup_left_idem[simp]: "x \<squnion> (x \<squnion> y) = x \<squnion> y"
+by (simp add: sup_assoc[symmetric])
+
+lemma (in lattice) inf_sup_absorb: "x \<sqinter> (x \<squnion> y) = x"
+by(blast intro: antisym inf_le1 inf_least sup_ge1)
+
+lemma (in lattice) sup_inf_absorb: "x \<squnion> (x \<sqinter> y) = x"
+by(blast intro: antisym sup_ge1 sup_greatest inf_le1)
+
+lemma (in lower_semilattice) inf_absorb: "x \<sqsubseteq> y \<Longrightarrow> x \<sqinter> y = x"
+by(blast intro: antisym inf_le1 inf_least refl)
+
+lemma (in upper_semilattice) sup_absorb: "x \<sqsubseteq> y \<Longrightarrow> x \<squnion> y = y"
+by(blast intro: antisym sup_ge2 sup_greatest refl)
+
+
+lemma (in lower_semilattice) less_eq_inf_conv [simp]:
+ "x \<sqsubseteq> y \<sqinter> z = (x \<sqsubseteq> y \<and> x \<sqsubseteq> z)"
+by(blast intro: antisym inf_le1 inf_le2 inf_least refl trans)
+
+lemmas (in lower_semilattice) below_inf_conv = less_eq_inf_conv
+  -- {* a duplicate for backward compatibility *}
+
+lemma (in upper_semilattice) above_sup_conv[simp]:
+ "x \<squnion> y \<sqsubseteq> z = (x \<sqsubseteq> z \<and> y \<sqsubseteq> z)"
+by(blast intro: antisym sup_ge1 sup_ge2 sup_greatest refl trans)
+
+
+text{* Towards distributivity: if you have one of them, you have them all. *}
+
+lemma (in lattice) distrib_imp1:
+assumes D: "!!x y z. x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
+shows "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
+proof-
+  have "x \<squnion> (y \<sqinter> z) = (x \<squnion> (x \<sqinter> z)) \<squnion> (y \<sqinter> z)" by(simp add:sup_inf_absorb)
+  also have "\<dots> = x \<squnion> (z \<sqinter> (x \<squnion> y))" by(simp add:D inf_commute sup_assoc)
+  also have "\<dots> = ((x \<squnion> y) \<sqinter> x) \<squnion> ((x \<squnion> y) \<sqinter> z)"
+    by(simp add:inf_sup_absorb inf_commute)
+  also have "\<dots> = (x \<squnion> y) \<sqinter> (x \<squnion> z)" by(simp add:D)
+  finally show ?thesis .
+qed
+
+lemma (in lattice) distrib_imp2:
+assumes D: "!!x y z. x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
+shows "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
+proof-
+  have "x \<sqinter> (y \<squnion> z) = (x \<sqinter> (x \<squnion> z)) \<sqinter> (y \<squnion> z)" by(simp add:inf_sup_absorb)
+  also have "\<dots> = x \<sqinter> (z \<squnion> (x \<sqinter> y))" by(simp add:D sup_commute inf_assoc)
+  also have "\<dots> = ((x \<sqinter> y) \<squnion> x) \<sqinter> ((x \<sqinter> y) \<squnion> z)"
+    by(simp add:sup_inf_absorb sup_commute)
+  also have "\<dots> = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" by(simp add:D)
+  finally show ?thesis .
+qed
+
+text{* A package of rewrite rules for deciding equivalence wrt ACI: *}
+
+lemma (in lower_semilattice) inf_left_commute: "x \<sqinter> (y \<sqinter> z) = y \<sqinter> (x \<sqinter> z)"
+proof -
+  have "x \<sqinter> (y \<sqinter> z) = (y \<sqinter> z) \<sqinter> x" by (simp only: inf_commute)
+  also have "... = y \<sqinter> (z \<sqinter> x)" by (simp only: inf_assoc)
+  also have "z \<sqinter> x = x \<sqinter> z" by (simp only: inf_commute)
+  finally(back_subst) show ?thesis .
+qed
+
+lemma (in upper_semilattice) sup_left_commute: "x \<squnion> (y \<squnion> z) = y \<squnion> (x \<squnion> z)"
+proof -
+  have "x \<squnion> (y \<squnion> z) = (y \<squnion> z) \<squnion> x" by (simp only: sup_commute)
+  also have "... = y \<squnion> (z \<squnion> x)" by (simp only: sup_assoc)
+  also have "z \<squnion> x = x \<squnion> z" by (simp only: sup_commute)
+  finally(back_subst) show ?thesis .
+qed
+
+lemma (in lower_semilattice) inf_left_idem: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"
+proof -
+  have "x \<sqinter> (x \<sqinter> y) = (x \<sqinter> x) \<sqinter> y" by(simp only:inf_assoc)
+  also have "\<dots> = x \<sqinter> y" by(simp)
+  finally show ?thesis .
+qed
+
+lemma (in upper_semilattice) sup_left_idem: "x \<squnion> (x \<squnion> y) = x \<squnion> y"
+proof -
+  have "x \<squnion> (x \<squnion> y) = (x \<squnion> x) \<squnion> y" by(simp only:sup_assoc)
+  also have "\<dots> = x \<squnion> y" by(simp)
+  finally show ?thesis .
+qed
+
+
+lemmas (in lower_semilattice) inf_ACI =
+ inf_commute inf_assoc inf_left_commute inf_left_idem
+
+lemmas (in upper_semilattice) sup_ACI =
+ sup_commute sup_assoc sup_left_commute sup_left_idem
+
+lemmas (in lattice) ACI = inf_ACI sup_ACI
+
+
+subsection{* Distributive lattices *}
+
+locale distrib_lattice = lattice +
+  assumes sup_inf_distrib1: "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
+
+lemma (in distrib_lattice) sup_inf_distrib2:
+ "(y \<sqinter> z) \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)"
+by(simp add:ACI sup_inf_distrib1)
+
+lemma (in distrib_lattice) inf_sup_distrib1:
+ "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
+by(rule distrib_imp2[OF sup_inf_distrib1])
+
+lemma (in distrib_lattice) inf_sup_distrib2:
+ "(y \<squnion> z) \<sqinter> x = (y \<sqinter> x) \<squnion> (z \<sqinter> x)"
+by(simp add:ACI inf_sup_distrib1)
+
+lemmas (in distrib_lattice) distrib =
+  sup_inf_distrib1 sup_inf_distrib2 inf_sup_distrib1 inf_sup_distrib2
+
+
+subsection {* Least value operator and min/max -- properties *}
+ 
+(*FIXME: derive more of the min/max laws generically via semilattices*)
+
+lemma LeastI2_order:
+  "[| P (x::'a::order);
+      !!y. P y ==> x <= y;
+      !!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |]
+   ==> Q (Least P)"
+  apply (unfold Least_def)
+  apply (rule theI2)
+    apply (blast intro: order_antisym)+
+  done
+
+lemma Least_equality:
+    "[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k"
+  apply (simp add: Least_def)
+  apply (rule the_equality)
+  apply (auto intro!: order_antisym)
+  done
+
+lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
+  by (simp add: min_def)
+
+lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
+  by (simp add: max_def)
+
+lemma min_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> min x least = least"
+  apply (simp add: min_def)
+  apply (blast intro: order_antisym)
+  done
+
+lemma max_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> max x least = x"
+  apply (simp add: max_def)
+  apply (blast intro: order_antisym)
+  done
+
+lemma min_of_mono:
+    "(!!x y. (f x <= f y) = (x <= y)) ==> min (f m) (f n) = f (min m n)"
+  by (simp add: min_def)
+
+lemma max_of_mono:
+    "(!!x y. (f x <= f y) = (x <= y)) ==> max (f m) (f n) = f (max m n)"
+  by (simp add: max_def)
+
+text{* Instantiate locales: *}
+
+interpretation min_max:
+  lower_semilattice["op \<le>" "op <" "min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a"]
+apply unfold_locales
+apply(simp add:min_def linorder_not_le order_less_imp_le)
+apply(simp add:min_def linorder_not_le order_less_imp_le)
+apply(simp add:min_def linorder_not_le order_less_imp_le)
+done
+
+interpretation min_max:
+  upper_semilattice["op \<le>" "op <" "max :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a"]
+apply unfold_locales
+apply(simp add: max_def linorder_not_le order_less_imp_le)
+apply(simp add: max_def linorder_not_le order_less_imp_le)
+apply(simp add: max_def linorder_not_le order_less_imp_le)
+done
+
+interpretation min_max:
+  lattice["op \<le>" "op <" "min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a" "max"]
+  by unfold_locales
+
+interpretation min_max:
+  distrib_lattice["op \<le>" "op <" "min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a" "max"]
+apply unfold_locales
+apply(rule_tac x=x and y=y in linorder_le_cases)
+apply(rule_tac x=x and y=z in linorder_le_cases)
+apply(rule_tac x=y and y=z in linorder_le_cases)
+apply(simp add:min_def max_def)
+apply(simp add:min_def max_def)
+apply(rule_tac x=y and y=z in linorder_le_cases)
+apply(simp add:min_def max_def)
+apply(simp add:min_def max_def)
+apply(rule_tac x=x and y=z in linorder_le_cases)
+apply(rule_tac x=y and y=z in linorder_le_cases)
+apply(simp add:min_def max_def)
+apply(simp add:min_def max_def)
+apply(rule_tac x=y and y=z in linorder_le_cases)
+apply(simp add:min_def max_def)
+apply(simp add:min_def max_def)
+done
+
+lemma le_max_iff_disj: "!!z::'a::linorder. (z <= max x y) = (z <= x | z <= y)"
+  apply(simp add:max_def)
+  apply (insert linorder_linear)
+  apply (blast intro: order_trans)
+  done
+
+lemmas le_maxI1 = min_max.sup_ge1
+lemmas le_maxI2 = min_max.sup_ge2
+
+lemma less_max_iff_disj: "!!z::'a::linorder. (z < max x y) = (z < x | z < y)"
+  apply (simp add: max_def order_le_less)
+  apply (insert linorder_less_linear)
+  apply (blast intro: order_less_trans)
+  done
+
+lemma max_less_iff_conj [simp]:
+    "!!z::'a::linorder. (max x y < z) = (x < z & y < z)"
+  apply (simp add: order_le_less max_def)
+  apply (insert linorder_less_linear)
+  apply (blast intro: order_less_trans)
+  done
+
+lemma min_less_iff_conj [simp]:
+    "!!z::'a::linorder. (z < min x y) = (z < x & z < y)"
+  apply (simp add: order_le_less min_def)
+  apply (insert linorder_less_linear)
+  apply (blast intro: order_less_trans)
+  done
+
+lemma min_le_iff_disj: "!!z::'a::linorder. (min x y <= z) = (x <= z | y <= z)"
+  apply (simp add: min_def)
+  apply (insert linorder_linear)
+  apply (blast intro: order_trans)
+  done
+
+lemma min_less_iff_disj: "!!z::'a::linorder. (min x y < z) = (x < z | y < z)"
+  apply (simp add: min_def order_le_less)
+  apply (insert linorder_less_linear)
+  apply (blast intro: order_less_trans)
+  done
+
+lemmas max_ac = min_max.sup_assoc min_max.sup_commute
+               mk_left_commute[of max,OF min_max.sup_assoc min_max.sup_commute]
+
+lemmas min_ac = min_max.inf_assoc min_max.inf_commute
+               mk_left_commute[of min,OF min_max.inf_assoc min_max.inf_commute]
+
+lemma split_min:
+    "P (min (i::'a::linorder) j) = ((i <= j --> P(i)) & (~ i <= j --> P(j)))"
+  by (simp add: min_def)
+
+lemma split_max:
+    "P (max (i::'a::linorder) j) = ((i <= j --> P(j)) & (~ i <= j --> P(i)))"
+  by (simp add: max_def)
+
+text {* ML legacy bindings *}
+
+ML {*
+val Least_def = thm "Least_def";
+val Least_equality = thm "Least_equality";
+val min_def = thm "min_def";
+val min_of_mono = thm "min_of_mono";
+val max_def = thm "max_def";
+val max_of_mono = thm "max_of_mono";
+val min_leastL = thm "min_leastL";
+val max_leastL = thm "max_leastL";
+val min_leastR = thm "min_leastR";
+val max_leastR = thm "max_leastR";
+val le_max_iff_disj = thm "le_max_iff_disj";
+val le_maxI1 = thm "le_maxI1";
+val le_maxI2 = thm "le_maxI2";
+val less_max_iff_disj = thm "less_max_iff_disj";
+val max_less_iff_conj = thm "max_less_iff_conj";
+val min_less_iff_conj = thm "min_less_iff_conj";
+val min_le_iff_disj = thm "min_le_iff_disj";
+val min_less_iff_disj = thm "min_less_iff_disj";
+val split_min = thm "split_min";
+val split_max = thm "split_max";
+*}
+
+end