(*  Title:      Product_Lattice.thy

     Author:     Brian Huffman

 *)

 header {* Lattice operations on product types *}

 theory Product_Lattice

 imports "~~/src/HOL/Library/Product_plus"

 begin

 subsection {* Pointwise ordering *}

 instantiation prod :: (ord, ord) ord

 begin

 definition

   "x \<le> y \<longleftrightarrow> fst x \<le> fst y \<and> snd x \<le> snd y"

 definition

   "(x::'a \<times> 'b) < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"

 instance ..

 end

 lemma fst_mono: "x \<le> y \<Longrightarrow> fst x \<le> fst y"

   unfolding less_eq_prod_def by simp

 lemma snd_mono: "x \<le> y \<Longrightarrow> snd x \<le> snd y"

   unfolding less_eq_prod_def by simp

 lemma Pair_mono: "x \<le> x' \<Longrightarrow> y \<le> y' \<Longrightarrow> (x, y) \<le> (x', y')"

   unfolding less_eq_prod_def by simp

 lemma Pair_le [simp]: "(a, b) \<le> (c, d) \<longleftrightarrow> a \<le> c \<and> b \<le> d"

   unfolding less_eq_prod_def by simp

 instance prod :: (preorder, preorder) preorder

 proof

   fix x y z :: "'a \<times> 'b"

   show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"

     by (rule less_prod_def)

   show "x \<le> x"

     unfolding less_eq_prod_def

     by fast

   assume "x \<le> y" and "y \<le> z" thus "x \<le> z"

     unfolding less_eq_prod_def

     by (fast elim: order_trans)

 qed

 instance prod :: (order, order) order

   by default auto

 subsection {* Binary infimum and supremum *}

 instantiation prod :: (semilattice_inf, semilattice_inf) semilattice_inf

 begin

 definition

   "inf x y = (inf (fst x) (fst y), inf (snd x) (snd y))"

 lemma inf_Pair_Pair [simp]: "inf (a, b) (c, d) = (inf a c, inf b d)"

   unfolding inf_prod_def by simp

 lemma fst_inf [simp]: "fst (inf x y) = inf (fst x) (fst y)"

   unfolding inf_prod_def by simp

 lemma snd_inf [simp]: "snd (inf x y) = inf (snd x) (snd y)"

   unfolding inf_prod_def by simp

 instance

   by default auto

 end

 instantiation prod :: (semilattice_sup, semilattice_sup) semilattice_sup

 begin

 definition

   "sup x y = (sup (fst x) (fst y), sup (snd x) (snd y))"

 lemma sup_Pair_Pair [simp]: "sup (a, b) (c, d) = (sup a c, sup b d)"

   unfolding sup_prod_def by simp

 lemma fst_sup [simp]: "fst (sup x y) = sup (fst x) (fst y)"

   unfolding sup_prod_def by simp

 lemma snd_sup [simp]: "snd (sup x y) = sup (snd x) (snd y)"

   unfolding sup_prod_def by simp

 instance

   by default auto

 end

 instance prod :: (lattice, lattice) lattice ..

 instance prod :: (distrib_lattice, distrib_lattice) distrib_lattice

   by default (auto simp add: sup_inf_distrib1)

 subsection {* Top and bottom elements *}

 instantiation prod :: (top, top) top

 begin

 definition

   "top = (top, top)"

 lemma fst_top [simp]: "fst top = top"

   unfolding top_prod_def by simp

 lemma snd_top [simp]: "snd top = top"

   unfolding top_prod_def by simp

 lemma Pair_top_top: "(top, top) = top"

   unfolding top_prod_def by simp

 instance

   by default (auto simp add: top_prod_def)

 end

 instantiation prod :: (bot, bot) bot

 begin

 definition

   "bot = (bot, bot)"

 lemma fst_bot [simp]: "fst bot = bot"

   unfolding bot_prod_def by simp

 lemma snd_bot [simp]: "snd bot = bot"

   unfolding bot_prod_def by simp

 lemma Pair_bot_bot: "(bot, bot) = bot"

   unfolding bot_prod_def by simp

 instance

   by default (auto simp add: bot_prod_def)

 end

 instance prod :: (bounded_lattice, bounded_lattice) bounded_lattice ..

 instance prod :: (boolean_algebra, boolean_algebra) boolean_algebra

   by default (auto simp add: prod_eqI inf_compl_bot sup_compl_top diff_eq)

 subsection {* Complete lattice operations *}

 instantiation prod :: (complete_lattice, complete_lattice) complete_lattice

 begin

 definition

   "Sup A = (SUP x:A. fst x, SUP x:A. snd x)"

 definition

   "Inf A = (INF x:A. fst x, INF x:A. snd x)"

 instance

   by default (simp_all add: less_eq_prod_def Inf_prod_def Sup_prod_def

     INF_leI le_SUPI le_INF_iff SUP_le_iff)

 end

 lemma fst_Sup: "fst (Sup A) = (SUP x:A. fst x)"

   unfolding Sup_prod_def by simp

 lemma snd_Sup: "snd (Sup A) = (SUP x:A. snd x)"

   unfolding Sup_prod_def by simp

 lemma fst_Inf: "fst (Inf A) = (INF x:A. fst x)"

   unfolding Inf_prod_def by simp

 lemma snd_Inf: "snd (Inf A) = (INF x:A. snd x)"

   unfolding Inf_prod_def by simp

 lemma fst_SUP: "fst (SUP x:A. f x) = (SUP x:A. fst (f x))"

   by (simp add: SUPR_def fst_Sup image_image)

 lemma snd_SUP: "snd (SUP x:A. f x) = (SUP x:A. snd (f x))"

   by (simp add: SUPR_def snd_Sup image_image)

 lemma fst_INF: "fst (INF x:A. f x) = (INF x:A. fst (f x))"

   by (simp add: INFI_def fst_Inf image_image)

 lemma snd_INF: "snd (INF x:A. f x) = (INF x:A. snd (f x))"

   by (simp add: INFI_def snd_Inf image_image)

 lemma SUP_Pair: "(SUP x:A. (f x, g x)) = (SUP x:A. f x, SUP x:A. g x)"

   by (simp add: SUPR_def Sup_prod_def image_image)

 lemma INF_Pair: "(INF x:A. (f x, g x)) = (INF x:A. f x, INF x:A. g x)"

   by (simp add: INFI_def Inf_prod_def image_image)

 end