src/HOL/Library/Product_Order.thy

+(*  Title:      HOL/Library/Product_Order.thy
+    Author:     Brian Huffman
+*)
+
+header {* Pointwise order on product types *}
+
+theory Product_Order
+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_lower SUP_upper 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: SUP_def fst_Sup image_image)
+
+lemma snd_SUP: "snd (SUP x:A. f x) = (SUP x:A. snd (f x))"
+  by (simp add: SUP_def snd_Sup image_image)
+
+lemma fst_INF: "fst (INF x:A. f x) = (INF x:A. fst (f x))"
+  by (simp add: INF_def fst_Inf image_image)
+
+lemma snd_INF: "snd (INF x:A. f x) = (INF x:A. snd (f x))"
+  by (simp add: INF_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: SUP_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: INF_def Inf_prod_def image_image)
+
+
+text {* Alternative formulations for set infima and suprema over the product
+of two complete lattices: *}
+
+lemma Inf_prod_alt_def: "Inf A = (Inf (fst ` A), Inf (snd ` A))"
+by (auto simp: Inf_prod_def INF_def)
+
+lemma Sup_prod_alt_def: "Sup A = (Sup (fst ` A), Sup (snd ` A))"
+by (auto simp: Sup_prod_def SUP_def)
+
+lemma INFI_prod_alt_def: "INFI A f = (INFI A (fst o f), INFI A (snd o f))"
+by (auto simp: INF_def Inf_prod_def image_compose)
+
+lemma SUPR_prod_alt_def: "SUPR A f = (SUPR A (fst o f), SUPR A (snd o f))"
+by (auto simp: SUP_def Sup_prod_def image_compose)
+
+lemma INF_prod_alt_def:
+  "(INF x:A. f x) = (INF x:A. fst (f x), INF x:A. snd (f x))"
+by (metis fst_INF snd_INF surjective_pairing)
+
+lemma SUP_prod_alt_def:
+  "(SUP x:A. f x) = (SUP x:A. fst (f x), SUP x:A. snd (f x))"
+by (metis fst_SUP snd_SUP surjective_pairing)
+
+
+subsection {* Complete distributive lattices *}
+
+(* Contribution: Alessandro Coglio *)
+
+instance prod ::
+  (complete_distrib_lattice, complete_distrib_lattice) complete_distrib_lattice
+proof
+  case goal1 thus ?case
+    by (auto simp: sup_prod_def Inf_prod_def INF_prod_alt_def sup_Inf sup_INF)
+next
+  case goal2 thus ?case
+    by (auto simp: inf_prod_def Sup_prod_def SUP_prod_alt_def inf_Sup inf_SUP)
+qed
+
+end
+