--- a/src/HOL/LOrder.thy Fri Mar 16 21:33:31 2007 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,194 +0,0 @@
-(* Title: HOL/LOrder.thy
- ID: $Id$
- Author: Steven Obua, TU Muenchen
-
-FIXME integrate properly with lattice locales
-- get rid of the implicit there-is-a-meet/join but require explicit ops.
-- abandone semilorder axclasses, instead turn lattice locales into classes
-- rename meet/join to inf/sup in all theorems
-*)
-
-header "Lattice Orders"
-
-theory LOrder
-imports Lattices
-begin
-
-text {* The theory of lattices developed here is taken from
-\cite{Birkhoff79}. *}
-
-constdefs
- is_meet :: "(('a::order) \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> bool"
- "is_meet m == ! a b x. m a b \<le> a \<and> m a b \<le> b \<and> (x \<le> a \<and> x \<le> b \<longrightarrow> x \<le> m a b)"
- is_join :: "(('a::order) \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> bool"
- "is_join j == ! a b x. a \<le> j a b \<and> b \<le> j a b \<and> (a \<le> x \<and> b \<le> x \<longrightarrow> j a b \<le> x)"
-
-lemma is_meet_unique:
- assumes "is_meet u" "is_meet v" shows "u = v"
-proof -
- {
- fix a b :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
- assume a: "is_meet a"
- assume b: "is_meet b"
- {
- fix x y
- let ?za = "a x y"
- let ?zb = "b x y"
- from a have za_le: "?za <= x & ?za <= y" by (auto simp add: is_meet_def)
- with b have "?za <= ?zb" by (auto simp add: is_meet_def)
- }
- }
- note f_le = this
- show "u = v" by ((rule ext)+, simp_all add: order_antisym prems f_le)
-qed
-
-lemma is_join_unique:
- assumes "is_join u" "is_join v" shows "u = v"
-proof -
- {
- fix a b :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
- assume a: "is_join a"
- assume b: "is_join b"
- {
- fix x y
- let ?za = "a x y"
- let ?zb = "b x y"
- from a have za_le: "x <= ?za & y <= ?za" by (auto simp add: is_join_def)
- with b have "?zb <= ?za" by (auto simp add: is_join_def)
- }
- }
- note f_le = this
- show "u = v" by ((rule ext)+, simp_all add: order_antisym prems f_le)
-qed
-
-lemma is_meet_inf: "is_meet (inf \<Colon> 'a\<Colon>lower_semilattice \<Rightarrow> 'a \<Rightarrow> 'a)"
-unfolding is_meet_def by auto
-
-lemma is_join_sup: "is_join (sup \<Colon> 'a\<Colon>upper_semilattice \<Rightarrow> 'a \<Rightarrow> 'a)"
-unfolding is_join_def by auto
-
-axclass meet_semilorder < order
- meet_exists: "? m. is_meet m"
-
-axclass join_semilorder < order
- join_exists: "? j. is_join j"
-
-axclass lorder < meet_semilorder, join_semilorder
-
-definition
- inf :: "('a::meet_semilorder) \<Rightarrow> 'a \<Rightarrow> 'a" where
- "inf = (THE m. is_meet m)"
-
-definition
- sup :: "('a::join_semilorder) \<Rightarrow> 'a \<Rightarrow> 'a" where
- "sup = (THE j. is_join j)"
-
-lemma is_meet_meet: "is_meet (inf::'a \<Rightarrow> 'a \<Rightarrow> ('a::meet_semilorder))"
-proof -
- from meet_exists obtain k::"'a \<Rightarrow> 'a \<Rightarrow> 'a" where "is_meet k" ..
- with is_meet_unique[of _ k] show ?thesis
- by (simp add: inf_def theI [of is_meet])
-qed
-
-lemma is_join_join: "is_join (sup::'a \<Rightarrow> 'a \<Rightarrow> ('a::join_semilorder))"
-proof -
- from join_exists obtain k::"'a \<Rightarrow> 'a \<Rightarrow> 'a" where "is_join k" ..
- with is_join_unique[of _ k] show ?thesis
- by (simp add: sup_def theI[of is_join])
-qed
-
-lemma meet_unique: "(is_meet m) = (m = inf)"
-by (insert is_meet_meet, auto simp add: is_meet_unique)
-
-lemma join_unique: "(is_join j) = (j = sup)"
-by (insert is_join_join, auto simp add: is_join_unique)
-
-lemma inf_unique: "(is_meet m) = (m = (Lattices.inf \<Colon> 'a \<Rightarrow> 'a \<Rightarrow> 'a\<Colon>lower_semilattice))"
-by (insert is_meet_inf, auto simp add: is_meet_unique)
-
-lemma sup_unique: "(is_join j) = (j = (Lattices.sup \<Colon> 'a \<Rightarrow> 'a \<Rightarrow> 'a\<Colon>upper_semilattice))"
-by (insert is_join_sup, auto simp add: is_join_unique)
-
-interpretation inf_semilat:
- lower_semilattice ["op \<le> \<Colon> 'a\<Colon>meet_semilorder \<Rightarrow> 'a \<Rightarrow> bool" "op <" inf]
-proof unfold_locales
- fix x y z :: "'a\<Colon>meet_semilorder"
- from is_meet_meet have "is_meet inf" by blast
- note meet = this is_meet_def
- from meet show "inf x y \<le> x" by blast
- from meet show "inf x y \<le> y" by blast
- from meet show "x \<le> y \<Longrightarrow> x \<le> z \<Longrightarrow> x \<le> inf y z" by blast
-qed
-
-interpretation sup_semilat:
- upper_semilattice ["op \<le> \<Colon> 'a\<Colon>join_semilorder \<Rightarrow> 'a \<Rightarrow> bool" "op <" sup]
-proof unfold_locales
- fix x y z :: "'a\<Colon>join_semilorder"
- from is_join_join have "is_join sup" by blast
- note join = this is_join_def
- from join show "x \<le> sup x y" by blast
- from join show "y \<le> sup x y" by blast
- from join show "x \<le> z \<Longrightarrow> y \<le> z \<Longrightarrow> sup x y \<le> z" by blast
-qed
-
-interpretation inf_sup_lat:
- lattice ["op \<le> \<Colon> 'a\<Colon>lorder \<Rightarrow> 'a \<Rightarrow> bool" "op <" inf sup]
- by unfold_locales
-
-lemma is_meet_min: "is_meet (min::'a \<Rightarrow> 'a \<Rightarrow> ('a::linorder))"
- by (auto simp add: is_meet_def min_def)
- lemma is_join_max: "is_join (max::'a \<Rightarrow> 'a \<Rightarrow> ('a::linorder))"
- by (auto simp add: is_join_def max_def)
-
-instance linorder \<subseteq> lorder
-proof
- from is_meet_min show "? (m::'a\<Rightarrow>'a\<Rightarrow>('a::linorder)). is_meet m" by auto
- from is_join_max show "? (j::'a\<Rightarrow>'a\<Rightarrow>('a::linorder)). is_join j" by auto
-qed
-
-lemma meet_min: "inf = (min \<Colon> 'a\<Colon>{linorder} \<Rightarrow> 'a \<Rightarrow> 'a)"
- by (simp add: is_meet_meet is_meet_min is_meet_unique)
-lemma join_max: "sup = (max \<Colon> 'a\<Colon>{linorder} \<Rightarrow> 'a \<Rightarrow> 'a)"
- by (simp add: is_join_join is_join_max is_join_unique)
-
-lemmas [rule del] = sup_semilat.antisym_intro inf_semilat.antisym_intro
- sup_semilat.le_supE inf_semilat.le_infE
-
-lemmas inf_le1 = inf_semilat.inf_le1
-lemmas inf_le2 = inf_semilat.inf_le2
-lemmas le_infI [rule del] = inf_semilat.le_infI
- (*intro! breaks a proof in Hyperreal/SEQ and NumberTheory/IntPrimes*)
-lemmas sup_ge1 = sup_semilat.sup_ge1
-lemmas sup_ge2 = sup_semilat.sup_ge2
-lemmas le_supI [rule del] = sup_semilat.le_supI
-lemmas inf_sup_ord = inf_le1 inf_le2 sup_ge1 sup_ge2
-lemmas le_inf_iff = inf_semilat.le_inf_iff
-lemmas ge_sup_conv = sup_semilat.ge_sup_conv
-lemmas inf_idem = inf_semilat.inf_idem
-lemmas sup_idem = sup_semilat.sup_idem
-lemmas inf_commute = inf_semilat.inf_commute
-lemmas sup_commute = sup_semilat.sup_commute
-lemmas le_infI1 [rule del] = inf_semilat.le_infI1
-lemmas le_infI2 [rule del] = inf_semilat.le_infI2
-lemmas le_supI1 [rule del] = sup_semilat.le_supI1
-lemmas le_supI2 [rule del] = sup_semilat.le_supI2
-lemmas inf_assoc = inf_semilat.inf_assoc
-lemmas sup_assoc = sup_semilat.sup_assoc
-lemmas inf_left_commute = inf_semilat.inf_left_commute
-lemmas inf_left_idem = inf_semilat.inf_left_idem
-lemmas sup_left_commute = sup_semilat.sup_left_commute
-lemmas sup_left_idem= sup_semilat.sup_left_idem
-lemmas inf_aci = inf_assoc inf_commute inf_left_commute inf_left_idem
-lemmas sup_aci = sup_assoc sup_commute sup_left_commute sup_left_idem
-lemmas le_iff_inf = inf_semilat.le_iff_inf
-lemmas le_iff_sup = sup_semilat.le_iff_sup
-lemmas sup_absorb2 = sup_semilat.sup_absorb2
-lemmas sup_absorb1 = sup_semilat.sup_absorb1
-lemmas inf_absorb1 = inf_semilat.inf_absorb1
-lemmas inf_absorb2 = inf_semilat.inf_absorb2
-lemmas inf_sup_absorb = inf_sup_lat.inf_sup_absorb
-lemmas sup_inf_absorb = inf_sup_lat.sup_inf_absorb
-lemmas distrib_sup_le = inf_sup_lat.distrib_sup_le
-lemmas distrib_inf_le = inf_sup_lat.distrib_inf_le
-
-end