(* Title: HOL/LOrder.thy ID: $Id: LOrder.thy,v 1.6 2005/09/20 12:03:37 wenzelm Exp $ Author: Steven Obua, TU Muenchen *) header {* Lattice Orders *} theory LOrder imports Orderings begin 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) => 'a => 'a) => bool" "is_meet m == ! a b x. m a b ≤ a ∧ m a b ≤ b ∧ (x ≤ a ∧ x ≤ b --> x ≤ m a b)" is_join :: "(('a::order) => 'a => 'a) => bool" "is_join j == ! a b x. a ≤ j a b ∧ b ≤ j a b ∧ (a ≤ x ∧ b ≤ x --> j a b ≤ x)" lemma is_meet_unique: assumes "is_meet u" "is_meet v" shows "u = v" proof - { fix a b :: "'a => 'a => '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 => 'a => '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) => 'a => 'a" "meet == THE m. is_meet m" join :: "('a::join_semilorder) => 'a => 'a" "join == THE j. is_join j" lemma is_meet_meet: "is_meet (meet::'a => 'a => ('a::meet_semilorder))" proof - from meet_exists obtain k::"'a => 'a => '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 => 'a => ('a::join_semilorder))" proof - from join_exists obtain k::"'a => 'a => '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 ≤ (a::'a::meet_semilorder)" by (insert is_meet_meet, auto simp add: is_meet_def) lemma meet_right_le: "meet a b ≤ (b::'a::meet_semilorder)" by (insert is_meet_meet, auto simp add: is_meet_def) lemma meet_imp_le: "x ≤ a ==> x ≤ b ==> x ≤ meet a (b::'a::meet_semilorder)" by (insert is_meet_meet, auto simp add: is_meet_def) lemma join_left_le: "a ≤ join a (b::'a::join_semilorder)" by (insert is_join_join, auto simp add: is_join_def) lemma join_right_le: "b ≤ join a (b::'a::join_semilorder)" by (insert is_join_join, auto simp add: is_join_def) lemma join_imp_le: "a ≤ x ==> b ≤ x ==> join a b ≤ (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 => 'a => ('a::linorder))" by (auto simp add: is_meet_def min_def) lemma is_join_max: "is_join (max::'a => 'a => ('a::linorder))" by (auto simp add: is_join_def max_def) instance linorder ⊆ meet_semilorder proof from is_meet_min show "? (m::'a=>'a=>('a::linorder)). is_meet m" by auto qed instance linorder ⊆ join_semilorder proof from is_join_max show "? (j::'a=>'a=>('a::linorder)). is_join j" by auto qed instance linorder ⊆ lorder .. lemma meet_min: "meet = (min :: 'a=>'a=>('a::linorder))" by (simp add: is_meet_meet is_meet_min is_meet_unique) lemma join_max: "join = (max :: 'a=>'a=>('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 --> 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 --> 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 --> 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 --> 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 ≤ z ==> meet x y ≤ 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 ≤ z ==> join x y ≤ join x z" proof - assume a: "y ≤ 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) ≤ 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) ≤ 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 ∨ a = d ∨ b = c ∨ b = d ==> meet a b ≤ join c d" by (insert meet_join_le, blast intro: order_trans) lemma modular_le: "x ≤ z ==> join x (meet y z) ≤ meet (join x y) z" (is "_ ==> ?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 end
lemma is_meet_unique:
[| is_meet u; is_meet v |] ==> u = v
lemma is_join_unique:
[| is_join u; is_join v |] ==> u = v
lemma is_meet_meet:
is_meet meet
lemma meet_unique:
is_meet m = (m = meet)
lemma is_join_join:
is_join join
lemma join_unique:
is_join j = (j = join)
lemma meet_left_le:
meet a b ≤ a
lemma meet_right_le:
meet a b ≤ b
lemma meet_imp_le:
[| x ≤ a; x ≤ b |] ==> x ≤ meet a b
lemma join_left_le:
a ≤ join a b
lemma join_right_le:
b ≤ join a b
lemma join_imp_le:
[| a ≤ x; b ≤ x |] ==> join a b ≤ x
lemmas meet_join_le:
meet a b ≤ a
meet a b ≤ b
a ≤ join a b
b ≤ join a b
lemmas meet_join_le:
meet a b ≤ a
meet a b ≤ b
a ≤ join a b
b ≤ join a b
lemma is_meet_min:
is_meet min
lemma is_join_max:
is_join max
lemma meet_min:
meet = min
lemma join_max:
join = max
lemma meet_idempotent:
meet x x = x
lemma join_idempotent:
join x x = x
lemma meet_comm:
meet x y = meet y x
lemma join_comm:
join x y = join y x
lemma meet_assoc:
meet (meet x y) z = meet x (meet y z)
lemma join_assoc:
join (join x y) z = join x (join y z)
lemma meet_left_comm:
meet a (meet b c) = meet b (meet a c)
lemma meet_left_idempotent:
meet y (meet y x) = meet y x
lemma join_left_comm:
join a (join b c) = join b (join a c)
lemma join_left_idempotent:
join y (join y x) = join y x
lemmas meet_aci:
meet (meet x y) z = meet x (meet y z)
meet x y = meet y x
meet a (meet b c) = meet b (meet a c)
meet y (meet y x) = meet y x
lemmas meet_aci:
meet (meet x y) z = meet x (meet y z)
meet x y = meet y x
meet a (meet b c) = meet b (meet a c)
meet y (meet y x) = meet y x
lemmas join_aci:
join (join x y) z = join x (join y z)
join x y = join y x
join a (join b c) = join b (join a c)
join y (join y x) = join y x
lemmas join_aci:
join (join x y) z = join x (join y z)
join x y = join y x
join a (join b c) = join b (join a c)
join y (join y x) = join y x
lemma le_def_meet:
(x ≤ y) = (meet x y = x)
lemma le_def_join:
(x ≤ y) = (join x y = y)
lemma meet_join_absorp:
meet x (join x y) = x
lemma join_meet_absorp:
join x (meet x y) = x
lemma meet_mono:
y ≤ z ==> meet x y ≤ meet x z
lemma join_mono:
y ≤ z ==> join x y ≤ join x z
lemma distrib_join_le:
join x (meet y z) ≤ meet (join x y) (join x z)
lemma distrib_meet_le:
join (meet x y) (meet x z) ≤ meet x (join y z)
lemma meet_join_eq_imp_le:
a = c ∨ a = d ∨ b = c ∨ b = d ==> meet a b ≤ join c d
lemma modular_le:
x ≤ z ==> join x (meet y z) ≤ meet (join x y) z