author  haftmann 
Thu, 14 Feb 2013 14:14:55 +0100  
changeset 51115  7dbd6832a689 
parent 50573  src/HOL/Library/Product_Lattice.thy@765c22baa1c9 
child 51542  738598beeb26 
permissions  rwrr 
51115
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(* Title: HOL/Library/Product_Order.thy 
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Author: Brian Huffman 
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*) 

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header {* Pointwise order on product types *} 
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theory Product_Order 
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imports "~~/src/HOL/Library/Product_plus" 
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begin 

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subsection {* Pointwise ordering *} 

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instantiation prod :: (ord, ord) ord 

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begin 

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definition 

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

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definition 

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

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instance .. 

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end 

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

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unfolding less_eq_prod_def by simp 

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

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unfolding less_eq_prod_def by simp 

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

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unfolding less_eq_prod_def by simp 

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

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unfolding less_eq_prod_def by simp 

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instance prod :: (preorder, preorder) preorder 

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proof 

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

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

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by (rule less_prod_def) 

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show "x \<le> x" 

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unfolding less_eq_prod_def 

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by fast 

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

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unfolding less_eq_prod_def 

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by (fast elim: order_trans) 

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qed 

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instance prod :: (order, order) order 

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by default auto 

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subsection {* Binary infimum and supremum *} 

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instantiation prod :: (semilattice_inf, semilattice_inf) semilattice_inf 

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begin 

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definition 

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

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

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unfolding inf_prod_def by simp 

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

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unfolding inf_prod_def by simp 

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

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unfolding inf_prod_def by simp 

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instance 

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by default auto 

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end 

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instantiation prod :: (semilattice_sup, semilattice_sup) semilattice_sup 

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begin 

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definition 

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

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

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unfolding sup_prod_def by simp 

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

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unfolding sup_prod_def by simp 

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

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unfolding sup_prod_def by simp 

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instance 

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by default auto 

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end 

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

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instance prod :: (distrib_lattice, distrib_lattice) distrib_lattice 

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by default (auto simp add: sup_inf_distrib1) 

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subsection {* Top and bottom elements *} 

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instantiation prod :: (top, top) top 

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begin 

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definition 

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"top = (top, top)" 

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

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unfolding top_prod_def by simp 

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

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unfolding top_prod_def by simp 

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

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unfolding top_prod_def by simp 

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instance 

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by default (auto simp add: top_prod_def) 

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end 

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instantiation prod :: (bot, bot) bot 

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begin 

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definition 

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"bot = (bot, bot)" 

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

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unfolding bot_prod_def by simp 

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

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unfolding bot_prod_def by simp 

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

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unfolding bot_prod_def by simp 

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instance 

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by default (auto simp add: bot_prod_def) 

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end 

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

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instance prod :: (boolean_algebra, boolean_algebra) boolean_algebra 

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

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subsection {* Complete lattice operations *} 

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instantiation prod :: (complete_lattice, complete_lattice) complete_lattice 

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begin 

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definition 

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

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definition 

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

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instance 

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by default (simp_all add: less_eq_prod_def Inf_prod_def Sup_prod_def 

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INF_lower SUP_upper le_INF_iff SUP_le_iff) 
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end 

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

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unfolding Sup_prod_def by simp 

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

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unfolding Sup_prod_def by simp 

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

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unfolding Inf_prod_def by simp 

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

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unfolding Inf_prod_def by simp 

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

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by (simp add: SUP_def fst_Sup image_image) 
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lemma snd_SUP: "snd (SUP x:A. f x) = (SUP x:A. snd (f x))" 

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by (simp add: SUP_def snd_Sup image_image) 
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lemma fst_INF: "fst (INF x:A. f x) = (INF x:A. fst (f x))" 

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by (simp add: INF_def fst_Inf image_image) 
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lemma snd_INF: "snd (INF x:A. f x) = (INF x:A. snd (f x))" 

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by (simp add: INF_def snd_Inf image_image) 
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lemma SUP_Pair: "(SUP x:A. (f x, g x)) = (SUP x:A. f x, SUP x:A. g x)" 

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by (simp add: SUP_def Sup_prod_def image_image) 
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lemma INF_Pair: "(INF x:A. (f x, g x)) = (INF x:A. f x, INF x:A. g x)" 

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by (simp add: INF_def Inf_prod_def image_image) 
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text {* Alternative formulations for set infima and suprema over the product 

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of two complete lattices: *} 

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lemma Inf_prod_alt_def: "Inf A = (Inf (fst ` A), Inf (snd ` A))" 

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by (auto simp: Inf_prod_def INF_def) 

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lemma Sup_prod_alt_def: "Sup A = (Sup (fst ` A), Sup (snd ` A))" 

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by (auto simp: Sup_prod_def SUP_def) 

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lemma INFI_prod_alt_def: "INFI A f = (INFI A (fst o f), INFI A (snd o f))" 

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by (auto simp: INF_def Inf_prod_def image_compose) 

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lemma SUPR_prod_alt_def: "SUPR A f = (SUPR A (fst o f), SUPR A (snd o f))" 

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by (auto simp: SUP_def Sup_prod_def image_compose) 

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lemma INF_prod_alt_def: 

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

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by (metis fst_INF snd_INF surjective_pairing) 

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lemma SUP_prod_alt_def: 

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

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by (metis fst_SUP snd_SUP surjective_pairing) 

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subsection {* Complete distributive lattices *} 

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(* Contribution: Alessandro Coglio *) 
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instance prod :: 

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(complete_distrib_lattice, complete_distrib_lattice) complete_distrib_lattice 

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proof 

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case goal1 thus ?case 

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by (auto simp: sup_prod_def Inf_prod_def INF_prod_alt_def sup_Inf sup_INF) 

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next 

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case goal2 thus ?case 

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by (auto simp: inf_prod_def Sup_prod_def SUP_prod_alt_def inf_Sup inf_SUP) 

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qed 

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end 
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