src/HOL/Lifting_Set.thy
author wenzelm
Wed Jun 22 10:09:20 2016 +0200 (2016-06-22)
changeset 63343 fb5d8a50c641
parent 63040 eb4ddd18d635
child 64267 b9a1486e79be
permissions -rw-r--r--
bundle lifting_syntax;
     1 (*  Title:      HOL/Lifting_Set.thy
     2     Author:     Brian Huffman and Ondrej Kuncar
     3 *)
     4 
     5 section \<open>Setup for Lifting/Transfer for the set type\<close>
     6 
     7 theory Lifting_Set
     8 imports Lifting
     9 begin
    10 
    11 subsection \<open>Relator and predicator properties\<close>
    12 
    13 lemma rel_setD1: "\<lbrakk> rel_set R A B; x \<in> A \<rbrakk> \<Longrightarrow> \<exists>y \<in> B. R x y"
    14   and rel_setD2: "\<lbrakk> rel_set R A B; y \<in> B \<rbrakk> \<Longrightarrow> \<exists>x \<in> A. R x y"
    15   by (simp_all add: rel_set_def)
    16 
    17 lemma rel_set_conversep [simp]: "rel_set A\<inverse>\<inverse> = (rel_set A)\<inverse>\<inverse>"
    18   unfolding rel_set_def by auto
    19 
    20 lemma rel_set_eq [relator_eq]: "rel_set (op =) = (op =)"
    21   unfolding rel_set_def fun_eq_iff by auto
    22 
    23 lemma rel_set_mono[relator_mono]:
    24   assumes "A \<le> B"
    25   shows "rel_set A \<le> rel_set B"
    26   using assms unfolding rel_set_def by blast
    27 
    28 lemma rel_set_OO[relator_distr]: "rel_set R OO rel_set S = rel_set (R OO S)"
    29   apply (rule sym)
    30   apply (intro ext)
    31   subgoal for X Z
    32     apply (rule iffI)
    33     apply (rule relcomppI [where b="{y. (\<exists>x\<in>X. R x y) \<and> (\<exists>z\<in>Z. S y z)}"])
    34     apply (simp add: rel_set_def, fast)+
    35     done
    36   done
    37 
    38 lemma Domainp_set[relator_domain]:
    39   "Domainp (rel_set T) = (\<lambda>A. Ball A (Domainp T))"
    40   unfolding rel_set_def Domainp_iff[abs_def]
    41   apply (intro ext)
    42   apply (rule iffI) 
    43   apply blast
    44   subgoal for A by (rule exI [where x="{y. \<exists>x\<in>A. T x y}"]) fast
    45   done
    46 
    47 lemma left_total_rel_set[transfer_rule]: 
    48   "left_total A \<Longrightarrow> left_total (rel_set A)"
    49   unfolding left_total_def rel_set_def
    50   apply safe
    51   subgoal for X by (rule exI [where x="{y. \<exists>x\<in>X. A x y}"]) fast
    52   done
    53 
    54 lemma left_unique_rel_set[transfer_rule]: 
    55   "left_unique A \<Longrightarrow> left_unique (rel_set A)"
    56   unfolding left_unique_def rel_set_def
    57   by fast
    58 
    59 lemma right_total_rel_set [transfer_rule]:
    60   "right_total A \<Longrightarrow> right_total (rel_set A)"
    61   using left_total_rel_set[of "A\<inverse>\<inverse>"] by simp
    62 
    63 lemma right_unique_rel_set [transfer_rule]:
    64   "right_unique A \<Longrightarrow> right_unique (rel_set A)"
    65   unfolding right_unique_def rel_set_def by fast
    66 
    67 lemma bi_total_rel_set [transfer_rule]:
    68   "bi_total A \<Longrightarrow> bi_total (rel_set A)"
    69   by(simp add: bi_total_alt_def left_total_rel_set right_total_rel_set)
    70 
    71 lemma bi_unique_rel_set [transfer_rule]:
    72   "bi_unique A \<Longrightarrow> bi_unique (rel_set A)"
    73   unfolding bi_unique_def rel_set_def by fast
    74 
    75 lemma set_relator_eq_onp [relator_eq_onp]:
    76   "rel_set (eq_onp P) = eq_onp (\<lambda>A. Ball A P)"
    77   unfolding fun_eq_iff rel_set_def eq_onp_def Ball_def by fast
    78 
    79 lemma bi_unique_rel_set_lemma:
    80   assumes "bi_unique R" and "rel_set R X Y"
    81   obtains f where "Y = image f X" and "inj_on f X" and "\<forall>x\<in>X. R x (f x)"
    82 proof
    83   define f where "f x = (THE y. R x y)" for x
    84   { fix x assume "x \<in> X"
    85     with \<open>rel_set R X Y\<close> \<open>bi_unique R\<close> have "R x (f x)"
    86       by (simp add: bi_unique_def rel_set_def f_def) (metis theI)
    87     with assms \<open>x \<in> X\<close> 
    88     have  "R x (f x)" "\<forall>x'\<in>X. R x' (f x) \<longrightarrow> x = x'" "\<forall>y\<in>Y. R x y \<longrightarrow> y = f x" "f x \<in> Y"
    89       by (fastforce simp add: bi_unique_def rel_set_def)+ }
    90   note * = this
    91   moreover
    92   { fix y assume "y \<in> Y"
    93     with \<open>rel_set R X Y\<close> *(3) \<open>y \<in> Y\<close> have "\<exists>x\<in>X. y = f x"
    94       by (fastforce simp: rel_set_def) }
    95   ultimately show "\<forall>x\<in>X. R x (f x)" "Y = image f X" "inj_on f X"
    96     by (auto simp: inj_on_def image_iff)
    97 qed
    98 
    99 subsection \<open>Quotient theorem for the Lifting package\<close>
   100 
   101 lemma Quotient_set[quot_map]:
   102   assumes "Quotient R Abs Rep T"
   103   shows "Quotient (rel_set R) (image Abs) (image Rep) (rel_set T)"
   104   using assms unfolding Quotient_alt_def4
   105   apply (simp add: rel_set_OO[symmetric])
   106   apply (simp add: rel_set_def)
   107   apply fast
   108   done
   109 
   110 
   111 subsection \<open>Transfer rules for the Transfer package\<close>
   112 
   113 subsubsection \<open>Unconditional transfer rules\<close>
   114 
   115 context includes lifting_syntax
   116 begin
   117 
   118 lemma empty_transfer [transfer_rule]: "(rel_set A) {} {}"
   119   unfolding rel_set_def by simp
   120 
   121 lemma insert_transfer [transfer_rule]:
   122   "(A ===> rel_set A ===> rel_set A) insert insert"
   123   unfolding rel_fun_def rel_set_def by auto
   124 
   125 lemma union_transfer [transfer_rule]:
   126   "(rel_set A ===> rel_set A ===> rel_set A) union union"
   127   unfolding rel_fun_def rel_set_def by auto
   128 
   129 lemma Union_transfer [transfer_rule]:
   130   "(rel_set (rel_set A) ===> rel_set A) Union Union"
   131   unfolding rel_fun_def rel_set_def by simp fast
   132 
   133 lemma image_transfer [transfer_rule]:
   134   "((A ===> B) ===> rel_set A ===> rel_set B) image image"
   135   unfolding rel_fun_def rel_set_def by simp fast
   136 
   137 lemma UNION_transfer [transfer_rule]:
   138   "(rel_set A ===> (A ===> rel_set B) ===> rel_set B) UNION UNION"
   139   by transfer_prover
   140 
   141 lemma Ball_transfer [transfer_rule]:
   142   "(rel_set A ===> (A ===> op =) ===> op =) Ball Ball"
   143   unfolding rel_set_def rel_fun_def by fast
   144 
   145 lemma Bex_transfer [transfer_rule]:
   146   "(rel_set A ===> (A ===> op =) ===> op =) Bex Bex"
   147   unfolding rel_set_def rel_fun_def by fast
   148 
   149 lemma Pow_transfer [transfer_rule]:
   150   "(rel_set A ===> rel_set (rel_set A)) Pow Pow"
   151   apply (rule rel_funI)
   152   apply (rule rel_setI)
   153   subgoal for X Y X'
   154     apply (rule rev_bexI [where x="{y\<in>Y. \<exists>x\<in>X'. A x y}"])
   155     apply clarsimp
   156     apply (simp add: rel_set_def)
   157     apply fast
   158     done
   159   subgoal for X Y Y'
   160     apply (rule rev_bexI [where x="{x\<in>X. \<exists>y\<in>Y'. A x y}"])
   161     apply clarsimp
   162     apply (simp add: rel_set_def)
   163     apply fast
   164     done
   165   done
   166 
   167 lemma rel_set_transfer [transfer_rule]:
   168   "((A ===> B ===> op =) ===> rel_set A ===> rel_set B ===> op =) rel_set rel_set"
   169   unfolding rel_fun_def rel_set_def by fast
   170 
   171 lemma bind_transfer [transfer_rule]:
   172   "(rel_set A ===> (A ===> rel_set B) ===> rel_set B) Set.bind Set.bind"
   173   unfolding bind_UNION [abs_def] by transfer_prover
   174 
   175 lemma INF_parametric [transfer_rule]:
   176   "(rel_set A ===> (A ===> HOL.eq) ===> HOL.eq) INFIMUM INFIMUM"
   177   by transfer_prover
   178 
   179 lemma SUP_parametric [transfer_rule]:
   180   "(rel_set R ===> (R ===> HOL.eq) ===> HOL.eq) SUPREMUM SUPREMUM"
   181   by transfer_prover
   182 
   183 
   184 subsubsection \<open>Rules requiring bi-unique, bi-total or right-total relations\<close>
   185 
   186 lemma member_transfer [transfer_rule]:
   187   assumes "bi_unique A"
   188   shows "(A ===> rel_set A ===> op =) (op \<in>) (op \<in>)"
   189   using assms unfolding rel_fun_def rel_set_def bi_unique_def by fast
   190 
   191 lemma right_total_Collect_transfer[transfer_rule]:
   192   assumes "right_total A"
   193   shows "((A ===> op =) ===> rel_set A) (\<lambda>P. Collect (\<lambda>x. P x \<and> Domainp A x)) Collect"
   194   using assms unfolding right_total_def rel_set_def rel_fun_def Domainp_iff by fast
   195 
   196 lemma Collect_transfer [transfer_rule]:
   197   assumes "bi_total A"
   198   shows "((A ===> op =) ===> rel_set A) Collect Collect"
   199   using assms unfolding rel_fun_def rel_set_def bi_total_def by fast
   200 
   201 lemma inter_transfer [transfer_rule]:
   202   assumes "bi_unique A"
   203   shows "(rel_set A ===> rel_set A ===> rel_set A) inter inter"
   204   using assms unfolding rel_fun_def rel_set_def bi_unique_def by fast
   205 
   206 lemma Diff_transfer [transfer_rule]:
   207   assumes "bi_unique A"
   208   shows "(rel_set A ===> rel_set A ===> rel_set A) (op -) (op -)"
   209   using assms unfolding rel_fun_def rel_set_def bi_unique_def
   210   unfolding Ball_def Bex_def Diff_eq
   211   by (safe, simp, metis, simp, metis)
   212 
   213 lemma subset_transfer [transfer_rule]:
   214   assumes [transfer_rule]: "bi_unique A"
   215   shows "(rel_set A ===> rel_set A ===> op =) (op \<subseteq>) (op \<subseteq>)"
   216   unfolding subset_eq [abs_def] by transfer_prover
   217 
   218 declare right_total_UNIV_transfer[transfer_rule]
   219 
   220 lemma UNIV_transfer [transfer_rule]:
   221   assumes "bi_total A"
   222   shows "(rel_set A) UNIV UNIV"
   223   using assms unfolding rel_set_def bi_total_def by simp
   224 
   225 lemma right_total_Compl_transfer [transfer_rule]:
   226   assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "right_total A"
   227   shows "(rel_set A ===> rel_set A) (\<lambda>S. uminus S \<inter> Collect (Domainp A)) uminus"
   228   unfolding Compl_eq [abs_def]
   229   by (subst Collect_conj_eq[symmetric]) transfer_prover
   230 
   231 lemma Compl_transfer [transfer_rule]:
   232   assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A"
   233   shows "(rel_set A ===> rel_set A) uminus uminus"
   234   unfolding Compl_eq [abs_def] by transfer_prover
   235 
   236 lemma right_total_Inter_transfer [transfer_rule]:
   237   assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "right_total A"
   238   shows "(rel_set (rel_set A) ===> rel_set A) (\<lambda>S. \<Inter>S \<inter> Collect (Domainp A)) Inter"
   239   unfolding Inter_eq[abs_def]
   240   by (subst Collect_conj_eq[symmetric]) transfer_prover
   241 
   242 lemma Inter_transfer [transfer_rule]:
   243   assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A"
   244   shows "(rel_set (rel_set A) ===> rel_set A) Inter Inter"
   245   unfolding Inter_eq [abs_def] by transfer_prover
   246 
   247 lemma filter_transfer [transfer_rule]:
   248   assumes [transfer_rule]: "bi_unique A"
   249   shows "((A ===> op=) ===> rel_set A ===> rel_set A) Set.filter Set.filter"
   250   unfolding Set.filter_def[abs_def] rel_fun_def rel_set_def by blast
   251 
   252 lemma finite_transfer [transfer_rule]:
   253   "bi_unique A \<Longrightarrow> (rel_set A ===> op =) finite finite"
   254   by (rule rel_funI, erule (1) bi_unique_rel_set_lemma)
   255      (auto dest: finite_imageD)
   256 
   257 lemma card_transfer [transfer_rule]:
   258   "bi_unique A \<Longrightarrow> (rel_set A ===> op =) card card"
   259   by (rule rel_funI, erule (1) bi_unique_rel_set_lemma)
   260      (simp add: card_image)
   261 
   262 lemma vimage_parametric [transfer_rule]:
   263   assumes [transfer_rule]: "bi_total A" "bi_unique B"
   264   shows "((A ===> B) ===> rel_set B ===> rel_set A) vimage vimage"
   265   unfolding vimage_def[abs_def] by transfer_prover
   266 
   267 lemma Image_parametric [transfer_rule]:
   268   assumes "bi_unique A"
   269   shows "(rel_set (rel_prod A B) ===> rel_set A ===> rel_set B) op `` op ``"
   270   by (intro rel_funI rel_setI)
   271     (force dest: rel_setD1 bi_uniqueDr[OF assms], force dest: rel_setD2 bi_uniqueDl[OF assms])
   272 
   273 end
   274 
   275 lemma (in comm_monoid_set) F_parametric [transfer_rule]:
   276   fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"
   277   assumes "bi_unique A"
   278   shows "rel_fun (rel_fun A (op =)) (rel_fun (rel_set A) (op =)) F F"
   279 proof (rule rel_funI)+
   280   fix f :: "'b \<Rightarrow> 'a" and g S T
   281   assume "rel_fun A (op =) f g" "rel_set A S T"
   282   with \<open>bi_unique A\<close> obtain i where "bij_betw i S T" "\<And>x. x \<in> S \<Longrightarrow> f x = g (i x)"
   283     by (auto elim: bi_unique_rel_set_lemma simp: rel_fun_def bij_betw_def)
   284   then show "F f S = F g T"
   285     by (simp add: reindex_bij_betw)
   286 qed
   287 
   288 lemmas setsum_parametric = setsum.F_parametric
   289 lemmas setprod_parametric = setprod.F_parametric
   290 
   291 lemma rel_set_UNION:
   292   assumes [transfer_rule]: "rel_set Q A B" "rel_fun Q (rel_set R) f g"
   293   shows "rel_set R (UNION A f) (UNION B g)"
   294   by transfer_prover
   295 
   296 end