(*  Title:   HOL/LOrder.thy

     ID:      $Id$

     Author:  Steven Obua, TU Muenchen

     License: GPL (GNU GENERAL PUBLIC LICENSE)

 *)

 header {* Lattice Orders *}

 theory LOrder = HOL:

 text {*

   The theory of lattices developed here is taken from the book:

   \begin{itemize}

   \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979. 

   \end{itemize}

 *}

 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

 axclass join_semilorder < order

   join_exists: "? j. is_join j"

 axclass meet_semilorder < order

   meet_exists: "? m. is_meet m"

 axclass lorder < join_semilorder, meet_semilorder

 constdefs

   meet :: "('a::meet_semilorder) \<Rightarrow> 'a \<Rightarrow> 'a"

   "meet == THE m. is_meet m"

   join :: "('a::join_semilorder) \<Rightarrow> 'a \<Rightarrow> 'a"

   "join ==  THE j. is_join j"

 lemma is_meet_meet: "is_meet (meet::'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: meet_def theI[of is_meet])    

 qed

 lemma meet_unique: "(is_meet m) = (m = meet)" 

 by (insert is_meet_meet, auto simp add: is_meet_unique)

 lemma is_join_join: "is_join (join::'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: join_def theI[of is_join])    

 qed

 lemma join_unique: "(is_join j) = (j = join)"

 by (insert is_join_join, auto simp add: is_join_unique)

 lemma meet_left_le: "meet a b \<le> (a::'a::meet_semilorder)"

 by (insert is_meet_meet, auto simp add: is_meet_def)

 lemma meet_right_le: "meet a b \<le> (b::'a::meet_semilorder)"

 by (insert is_meet_meet, auto simp add: is_meet_def)

 lemma meet_imp_le: "x \<le> a \<Longrightarrow> x \<le> b \<Longrightarrow> x \<le> meet a (b::'a::meet_semilorder)"

 by (insert is_meet_meet, auto simp add: is_meet_def)

 lemma join_left_le: "a \<le> join a (b::'a::join_semilorder)"

 by (insert is_join_join, auto simp add: is_join_def)

 lemma join_right_le: "b \<le> join a (b::'a::join_semilorder)"

 by (insert is_join_join, auto simp add: is_join_def)

 lemma join_imp_le: "a \<le> x \<Longrightarrow> b \<le> x \<Longrightarrow> join a b \<le> (x::'a::join_semilorder)"

 by (insert is_join_join, auto simp add: is_join_def)

 lemmas meet_join_le = meet_left_le meet_right_le join_left_le join_right_le

 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> meet_semilorder

 proof

   from is_meet_min show "? (m::'a\<Rightarrow>'a\<Rightarrow>('a::linorder)). is_meet m" by auto

 qed

 instance linorder \<subseteq> join_semilorder

 proof

   from is_join_max show "? (j::'a\<Rightarrow>'a\<Rightarrow>('a::linorder)). is_join j" by auto 

 qed

 instance linorder \<subseteq> lorder ..

 lemma meet_min: "meet = (min :: 'a\<Rightarrow>'a\<Rightarrow>('a::linorder))" 

 by (simp add: is_meet_meet is_meet_min is_meet_unique)

 lemma join_max: "join = (max :: 'a\<Rightarrow>'a\<Rightarrow>('a::linorder))"

 by (simp add: is_join_join is_join_max is_join_unique)

 lemma meet_idempotent[simp]: "meet x x = x"

 by (rule order_antisym, simp_all add: meet_left_le meet_imp_le)

 lemma join_idempotent[simp]: "join x x = x"

 by (rule order_antisym, simp_all add: join_left_le join_imp_le)

 lemma meet_comm: "meet x y = meet y x" 

 by (rule order_antisym, (simp add: meet_left_le meet_right_le meet_imp_le)+)

 lemma join_comm: "join x y = join y x"

 by (rule order_antisym, (simp add: join_right_le join_left_le join_imp_le)+)

 lemma meet_assoc: "meet (meet x y) z = meet x (meet y z)" (is "?l=?r")

 proof - 

   have "?l <= meet x y & meet x y <= x & ?l <= z & meet x y <= y" by (simp add: meet_left_le meet_right_le)

   hence "?l <= x & ?l <= y & ?l <= z" by auto

   hence "?l <= ?r" by (simp add: meet_imp_le)

   hence a:"?l <= meet x (meet y z)" by (simp add: meet_imp_le)

   have "?r <= meet y z & meet y z <= y & meet y z <= z & ?r <= x" by (simp add: meet_left_le meet_right_le)  

   hence "?r <= x & ?r <= y & ?r <= z" by (auto) 

   hence "?r <= meet x y & ?r <= z" by (simp add: meet_imp_le)

   hence b:"?r <= ?l" by (simp add: meet_imp_le)

   from a b show "?l = ?r" by auto

 qed

 lemma join_assoc: "join (join x y) z = join x (join y z)" (is "?l=?r")

 proof -

   have "join x y <= ?l & x <= join x y & z <= ?l & y <= join x y" by (simp add: join_left_le join_right_le)

   hence "x <= ?l & y <= ?l & z <= ?l" by auto

   hence "join y z <= ?l & x <= ?l" by (simp add: join_imp_le)

   hence a:"?r <= ?l" by (simp add: join_imp_le)

   have "join y z <= ?r & y <= join y z & z <= join y z & x <= ?r" by (simp add: join_left_le join_right_le)

   hence "y <= ?r & z <= ?r & x <= ?r" by auto

   hence "join x y <= ?r & z <= ?r" by (simp add: join_imp_le)

   hence b:"?l <= ?r" by (simp add: join_imp_le)

   from a b show "?l = ?r" by auto

 qed

 lemma meet_left_comm: "meet a (meet b c) = meet b (meet a c)"

 by (simp add: meet_assoc[symmetric, of a b c], simp add: meet_comm[of a b], simp add: meet_assoc)

 lemma meet_left_idempotent: "meet y (meet y x) = meet y x"

 by (simp add: meet_assoc meet_comm meet_left_comm)

 lemma join_left_comm: "join a (join b c) = join b (join a c)"

 by (simp add: join_assoc[symmetric, of a b c], simp add: join_comm[of a b], simp add: join_assoc)

 lemma join_left_idempotent: "join y (join y x) = join y x"

 by (simp add: join_assoc join_comm join_left_comm)

 lemmas meet_aci = meet_assoc meet_comm meet_left_comm meet_left_idempotent

 lemmas join_aci = join_assoc join_comm join_left_comm join_left_idempotent

 lemma le_def_meet: "(x <= y) = (meet x y = x)" 

 proof -

   have u: "x <= y \<longrightarrow> meet x y = x"

   proof 

     assume "x <= y"

     hence "x <= meet x y & meet x y <= x" by (simp add: meet_imp_le meet_left_le)

     thus "meet x y = x" by auto

   qed

   have v:"meet x y = x \<longrightarrow> x <= y" 

   proof 

     have a:"meet x y <= y" by (simp add: meet_right_le)

     assume "meet x y = x"

     hence "x = meet x y" by auto

     with a show "x <= y" by (auto)

   qed

   from u v show ?thesis by blast

 qed

 lemma le_def_join: "(x <= y) = (join x y = y)" 

 proof -

   have u: "x <= y \<longrightarrow> join x y = y"

   proof 

     assume "x <= y"

     hence "join x y <= y & y <= join x y" by (simp add: join_imp_le join_right_le)

     thus "join x y = y" by auto

   qed

   have v:"join x y = y \<longrightarrow> x <= y" 

   proof 

     have a:"x <= join x y" by (simp add: join_left_le)

     assume "join x y = y"

     hence "y = join x y" by auto

     with a show "x <= y" by (auto)

   qed

   from u v show ?thesis by blast

 qed

 lemma meet_join_absorp: "meet x (join x y) = x"

 proof -

   have a:"meet x (join x y) <= x" by (simp add: meet_left_le)

   have b:"x <= meet x (join x y)" by (rule meet_imp_le, simp_all add: join_left_le)

   from a b show ?thesis by auto

 qed

 lemma join_meet_absorp: "join x (meet x y) = x"

 proof - 

   have a:"x <= join x (meet x y)" by (simp add: join_left_le)

   have b:"join x (meet x y) <= x" by (rule join_imp_le, simp_all add: meet_left_le)

   from a b show ?thesis by auto

 qed

 lemma meet_mono: "y \<le> z \<Longrightarrow> meet x y \<le> meet x z"

 proof -

   assume a: "y <= z"

   have "meet x y <= x & meet x y <= y" by (simp add: meet_left_le meet_right_le)

   with a have "meet x y <= x & meet x y <= z" by auto 

   thus "meet x y <= meet x z" by (simp add: meet_imp_le)

 qed

 lemma join_mono: "y \<le> z \<Longrightarrow> join x y \<le> join x z"

 proof -

   assume a: "y \<le> z"

   have "x <= join x z & z <= join x z" by (simp add: join_left_le join_right_le)

   with a have "x <= join x z & y <= join x z" by auto

   thus "join x y <= join x z" by (simp add: join_imp_le)

 qed

 lemma distrib_join_le: "join x (meet y z) \<le> meet (join x y) (join x z)" (is "_ <= ?r")

 proof -

   have a: "x <= ?r" by (rule meet_imp_le, simp_all add: join_left_le)

   from meet_join_le have b: "meet y z <= ?r" 

     by (rule_tac meet_imp_le, (blast intro: order_trans)+)

   from a b show ?thesis by (simp add: join_imp_le)

 qed

 lemma distrib_meet_le: "join (meet x y) (meet x z) \<le> meet x (join y z)" (is "?l <= _") 

 proof -

   have a: "?l <= x" by (rule join_imp_le, simp_all add: meet_left_le)

   from meet_join_le have b: "?l <= join y z" 

     by (rule_tac join_imp_le, (blast intro: order_trans)+)

   from a b show ?thesis by (simp add: meet_imp_le)

 qed

 lemma meet_join_eq_imp_le: "a = c \<or> a = d \<or> b = c \<or> b = d \<Longrightarrow> meet a b \<le> join c d"

 by (insert meet_join_le, blast intro: order_trans)

 lemma modular_le: "x \<le> z \<Longrightarrow> join x (meet y z) \<le> meet (join x y) z" (is "_ \<Longrightarrow> ?t <= _")

 proof -

   assume a: "x <= z"

   have b: "?t <= join x y" by (rule join_imp_le, simp_all add: meet_join_le meet_join_eq_imp_le)

   have c: "?t <= z" by (rule join_imp_le, simp_all add: meet_join_le a)

   from b c show ?thesis by (simp add: meet_imp_le)

 qed

 ML {*

 val is_meet_unique = thm "is_meet_unique";

 val is_join_unique = thm "is_join_unique";

 val join_exists = thm "join_exists";

 val meet_exists = thm "meet_exists";

 val is_meet_meet = thm "is_meet_meet";

 val meet_unique = thm "meet_unique";

 val is_join_join = thm "is_join_join";

 val join_unique = thm "join_unique";

 val meet_left_le = thm "meet_left_le";

 val meet_right_le = thm "meet_right_le";

 val meet_imp_le = thm "meet_imp_le";

 val join_left_le = thm "join_left_le";

 val join_right_le = thm "join_right_le";

 val join_imp_le = thm "join_imp_le";

 val meet_join_le = thms "meet_join_le";

 val is_meet_min = thm "is_meet_min";

 val is_join_max = thm "is_join_max";

 val meet_min = thm "meet_min";

 val join_max = thm "join_max";

 val meet_idempotent = thm "meet_idempotent";

 val join_idempotent = thm "join_idempotent";

 val meet_comm = thm "meet_comm";

 val join_comm = thm "join_comm";

 val meet_assoc = thm "meet_assoc";

 val join_assoc = thm "join_assoc";

 val meet_left_comm = thm "meet_left_comm";

 val meet_left_idempotent = thm "meet_left_idempotent";

 val join_left_comm = thm "join_left_comm";

 val join_left_idempotent = thm "join_left_idempotent";

 val meet_aci = thms "meet_aci";

 val join_aci = thms "join_aci";

 val le_def_meet = thm "le_def_meet";

 val le_def_join = thm "le_def_join";

 val meet_join_absorp = thm "meet_join_absorp";

 val join_meet_absorp = thm "join_meet_absorp";

 val meet_mono = thm "meet_mono";

 val join_mono = thm "join_mono";

 val distrib_join_le = thm "distrib_join_le";

 val distrib_meet_le = thm "distrib_meet_le";

 val meet_join_eq_imp_le = thm "meet_join_eq_imp_le";

 val modular_le = thm "modular_le";

 *}

 end