src/HOL/Library/Product_Order.thy
author haftmann
Thu Feb 14 14:14:55 2013 +0100 (2013-02-14)
changeset 51115 7dbd6832a689
parent 50573 src/HOL/Library/Product_Lattice.thy@765c22baa1c9
child 51542 738598beeb26
permissions -rw-r--r--
consolidation of library theories on product orders
     1 (*  Title:      HOL/Library/Product_Order.thy
     2     Author:     Brian Huffman
     3 *)
     4 
     5 header {* Pointwise order on product types *}
     6 
     7 theory Product_Order
     8 imports "~~/src/HOL/Library/Product_plus"
     9 begin
    10 
    11 subsection {* Pointwise ordering *}
    12 
    13 instantiation prod :: (ord, ord) ord
    14 begin
    15 
    16 definition
    17   "x \<le> y \<longleftrightarrow> fst x \<le> fst y \<and> snd x \<le> snd y"
    18 
    19 definition
    20   "(x::'a \<times> 'b) < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
    21 
    22 instance ..
    23 
    24 end
    25 
    26 lemma fst_mono: "x \<le> y \<Longrightarrow> fst x \<le> fst y"
    27   unfolding less_eq_prod_def by simp
    28 
    29 lemma snd_mono: "x \<le> y \<Longrightarrow> snd x \<le> snd y"
    30   unfolding less_eq_prod_def by simp
    31 
    32 lemma Pair_mono: "x \<le> x' \<Longrightarrow> y \<le> y' \<Longrightarrow> (x, y) \<le> (x', y')"
    33   unfolding less_eq_prod_def by simp
    34 
    35 lemma Pair_le [simp]: "(a, b) \<le> (c, d) \<longleftrightarrow> a \<le> c \<and> b \<le> d"
    36   unfolding less_eq_prod_def by simp
    37 
    38 instance prod :: (preorder, preorder) preorder
    39 proof
    40   fix x y z :: "'a \<times> 'b"
    41   show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
    42     by (rule less_prod_def)
    43   show "x \<le> x"
    44     unfolding less_eq_prod_def
    45     by fast
    46   assume "x \<le> y" and "y \<le> z" thus "x \<le> z"
    47     unfolding less_eq_prod_def
    48     by (fast elim: order_trans)
    49 qed
    50 
    51 instance prod :: (order, order) order
    52   by default auto
    53 
    54 
    55 subsection {* Binary infimum and supremum *}
    56 
    57 instantiation prod :: (semilattice_inf, semilattice_inf) semilattice_inf
    58 begin
    59 
    60 definition
    61   "inf x y = (inf (fst x) (fst y), inf (snd x) (snd y))"
    62 
    63 lemma inf_Pair_Pair [simp]: "inf (a, b) (c, d) = (inf a c, inf b d)"
    64   unfolding inf_prod_def by simp
    65 
    66 lemma fst_inf [simp]: "fst (inf x y) = inf (fst x) (fst y)"
    67   unfolding inf_prod_def by simp
    68 
    69 lemma snd_inf [simp]: "snd (inf x y) = inf (snd x) (snd y)"
    70   unfolding inf_prod_def by simp
    71 
    72 instance
    73   by default auto
    74 
    75 end
    76 
    77 instantiation prod :: (semilattice_sup, semilattice_sup) semilattice_sup
    78 begin
    79 
    80 definition
    81   "sup x y = (sup (fst x) (fst y), sup (snd x) (snd y))"
    82 
    83 lemma sup_Pair_Pair [simp]: "sup (a, b) (c, d) = (sup a c, sup b d)"
    84   unfolding sup_prod_def by simp
    85 
    86 lemma fst_sup [simp]: "fst (sup x y) = sup (fst x) (fst y)"
    87   unfolding sup_prod_def by simp
    88 
    89 lemma snd_sup [simp]: "snd (sup x y) = sup (snd x) (snd y)"
    90   unfolding sup_prod_def by simp
    91 
    92 instance
    93   by default auto
    94 
    95 end
    96 
    97 instance prod :: (lattice, lattice) lattice ..
    98 
    99 instance prod :: (distrib_lattice, distrib_lattice) distrib_lattice
   100   by default (auto simp add: sup_inf_distrib1)
   101 
   102 
   103 subsection {* Top and bottom elements *}
   104 
   105 instantiation prod :: (top, top) top
   106 begin
   107 
   108 definition
   109   "top = (top, top)"
   110 
   111 lemma fst_top [simp]: "fst top = top"
   112   unfolding top_prod_def by simp
   113 
   114 lemma snd_top [simp]: "snd top = top"
   115   unfolding top_prod_def by simp
   116 
   117 lemma Pair_top_top: "(top, top) = top"
   118   unfolding top_prod_def by simp
   119 
   120 instance
   121   by default (auto simp add: top_prod_def)
   122 
   123 end
   124 
   125 instantiation prod :: (bot, bot) bot
   126 begin
   127 
   128 definition
   129   "bot = (bot, bot)"
   130 
   131 lemma fst_bot [simp]: "fst bot = bot"
   132   unfolding bot_prod_def by simp
   133 
   134 lemma snd_bot [simp]: "snd bot = bot"
   135   unfolding bot_prod_def by simp
   136 
   137 lemma Pair_bot_bot: "(bot, bot) = bot"
   138   unfolding bot_prod_def by simp
   139 
   140 instance
   141   by default (auto simp add: bot_prod_def)
   142 
   143 end
   144 
   145 instance prod :: (bounded_lattice, bounded_lattice) bounded_lattice ..
   146 
   147 instance prod :: (boolean_algebra, boolean_algebra) boolean_algebra
   148   by default (auto simp add: prod_eqI inf_compl_bot sup_compl_top diff_eq)
   149 
   150 
   151 subsection {* Complete lattice operations *}
   152 
   153 instantiation prod :: (complete_lattice, complete_lattice) complete_lattice
   154 begin
   155 
   156 definition
   157   "Sup A = (SUP x:A. fst x, SUP x:A. snd x)"
   158 
   159 definition
   160   "Inf A = (INF x:A. fst x, INF x:A. snd x)"
   161 
   162 instance
   163   by default (simp_all add: less_eq_prod_def Inf_prod_def Sup_prod_def
   164     INF_lower SUP_upper le_INF_iff SUP_le_iff)
   165 
   166 end
   167 
   168 lemma fst_Sup: "fst (Sup A) = (SUP x:A. fst x)"
   169   unfolding Sup_prod_def by simp
   170 
   171 lemma snd_Sup: "snd (Sup A) = (SUP x:A. snd x)"
   172   unfolding Sup_prod_def by simp
   173 
   174 lemma fst_Inf: "fst (Inf A) = (INF x:A. fst x)"
   175   unfolding Inf_prod_def by simp
   176 
   177 lemma snd_Inf: "snd (Inf A) = (INF x:A. snd x)"
   178   unfolding Inf_prod_def by simp
   179 
   180 lemma fst_SUP: "fst (SUP x:A. f x) = (SUP x:A. fst (f x))"
   181   by (simp add: SUP_def fst_Sup image_image)
   182 
   183 lemma snd_SUP: "snd (SUP x:A. f x) = (SUP x:A. snd (f x))"
   184   by (simp add: SUP_def snd_Sup image_image)
   185 
   186 lemma fst_INF: "fst (INF x:A. f x) = (INF x:A. fst (f x))"
   187   by (simp add: INF_def fst_Inf image_image)
   188 
   189 lemma snd_INF: "snd (INF x:A. f x) = (INF x:A. snd (f x))"
   190   by (simp add: INF_def snd_Inf image_image)
   191 
   192 lemma SUP_Pair: "(SUP x:A. (f x, g x)) = (SUP x:A. f x, SUP x:A. g x)"
   193   by (simp add: SUP_def Sup_prod_def image_image)
   194 
   195 lemma INF_Pair: "(INF x:A. (f x, g x)) = (INF x:A. f x, INF x:A. g x)"
   196   by (simp add: INF_def Inf_prod_def image_image)
   197 
   198 
   199 text {* Alternative formulations for set infima and suprema over the product
   200 of two complete lattices: *}
   201 
   202 lemma Inf_prod_alt_def: "Inf A = (Inf (fst ` A), Inf (snd ` A))"
   203 by (auto simp: Inf_prod_def INF_def)
   204 
   205 lemma Sup_prod_alt_def: "Sup A = (Sup (fst ` A), Sup (snd ` A))"
   206 by (auto simp: Sup_prod_def SUP_def)
   207 
   208 lemma INFI_prod_alt_def: "INFI A f = (INFI A (fst o f), INFI A (snd o f))"
   209 by (auto simp: INF_def Inf_prod_def image_compose)
   210 
   211 lemma SUPR_prod_alt_def: "SUPR A f = (SUPR A (fst o f), SUPR A (snd o f))"
   212 by (auto simp: SUP_def Sup_prod_def image_compose)
   213 
   214 lemma INF_prod_alt_def:
   215   "(INF x:A. f x) = (INF x:A. fst (f x), INF x:A. snd (f x))"
   216 by (metis fst_INF snd_INF surjective_pairing)
   217 
   218 lemma SUP_prod_alt_def:
   219   "(SUP x:A. f x) = (SUP x:A. fst (f x), SUP x:A. snd (f x))"
   220 by (metis fst_SUP snd_SUP surjective_pairing)
   221 
   222 
   223 subsection {* Complete distributive lattices *}
   224 
   225 (* Contribution: Alessandro Coglio *)
   226 
   227 instance prod ::
   228   (complete_distrib_lattice, complete_distrib_lattice) complete_distrib_lattice
   229 proof
   230   case goal1 thus ?case
   231     by (auto simp: sup_prod_def Inf_prod_def INF_prod_alt_def sup_Inf sup_INF)
   232 next
   233   case goal2 thus ?case
   234     by (auto simp: inf_prod_def Sup_prod_def SUP_prod_alt_def inf_Sup inf_SUP)
   235 qed
   236 
   237 end
   238