src/HOL/Lifting_Set.thy
 author hoelzl Fri May 30 14:55:10 2014 +0200 (2014-05-30) changeset 57129 7edb7550663e parent 56524 f4ba736040fa child 57599 7ef939f89776 permissions -rw-r--r--
introduce more powerful reindexing rules for big operators
```     1 (*  Title:      HOL/Lifting_Set.thy
```
```     2     Author:     Brian Huffman and Ondrej Kuncar
```
```     3 *)
```
```     4
```
```     5 header {* Setup for Lifting/Transfer for the set type *}
```
```     6
```
```     7 theory Lifting_Set
```
```     8 imports Lifting
```
```     9 begin
```
```    10
```
```    11 subsection {* Relator and predicator properties *}
```
```    12
```
```    13 definition rel_set :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> bool"
```
```    14   where "rel_set R = (\<lambda>A B. (\<forall>x\<in>A. \<exists>y\<in>B. R x y) \<and> (\<forall>y\<in>B. \<exists>x\<in>A. R x y))"
```
```    15
```
```    16 lemma rel_setI:
```
```    17   assumes "\<And>x. x \<in> A \<Longrightarrow> \<exists>y\<in>B. R x y"
```
```    18   assumes "\<And>y. y \<in> B \<Longrightarrow> \<exists>x\<in>A. R x y"
```
```    19   shows "rel_set R A B"
```
```    20   using assms unfolding rel_set_def by simp
```
```    21
```
```    22 lemma rel_setD1: "\<lbrakk> rel_set R A B; x \<in> A \<rbrakk> \<Longrightarrow> \<exists>y \<in> B. R x y"
```
```    23   and rel_setD2: "\<lbrakk> rel_set R A B; y \<in> B \<rbrakk> \<Longrightarrow> \<exists>x \<in> A. R x y"
```
```    24 by(simp_all add: rel_set_def)
```
```    25
```
```    26 lemma rel_set_conversep [simp]: "rel_set A\<inverse>\<inverse> = (rel_set A)\<inverse>\<inverse>"
```
```    27   unfolding rel_set_def by auto
```
```    28
```
```    29 lemma rel_set_eq [relator_eq]: "rel_set (op =) = (op =)"
```
```    30   unfolding rel_set_def fun_eq_iff by auto
```
```    31
```
```    32 lemma rel_set_mono[relator_mono]:
```
```    33   assumes "A \<le> B"
```
```    34   shows "rel_set A \<le> rel_set B"
```
```    35 using assms unfolding rel_set_def by blast
```
```    36
```
```    37 lemma rel_set_OO[relator_distr]: "rel_set R OO rel_set S = rel_set (R OO S)"
```
```    38   apply (rule sym)
```
```    39   apply (intro ext, rename_tac X Z)
```
```    40   apply (rule iffI)
```
```    41   apply (rule_tac b="{y. (\<exists>x\<in>X. R x y) \<and> (\<exists>z\<in>Z. S y z)}" in relcomppI)
```
```    42   apply (simp add: rel_set_def, fast)
```
```    43   apply (simp add: rel_set_def, fast)
```
```    44   apply (simp add: rel_set_def, fast)
```
```    45   done
```
```    46
```
```    47 lemma Domainp_set[relator_domain]:
```
```    48   "Domainp (rel_set T) = (\<lambda>A. Ball A (Domainp T))"
```
```    49 unfolding rel_set_def Domainp_iff[abs_def]
```
```    50 apply (intro ext)
```
```    51 apply (rule iffI)
```
```    52 apply blast
```
```    53 apply (rename_tac A, rule_tac x="{y. \<exists>x\<in>A. T x y}" in exI, fast)
```
```    54 done
```
```    55
```
```    56 lemma left_total_rel_set[transfer_rule]:
```
```    57   "left_total A \<Longrightarrow> left_total (rel_set A)"
```
```    58   unfolding left_total_def rel_set_def
```
```    59   apply safe
```
```    60   apply (rename_tac X, rule_tac x="{y. \<exists>x\<in>X. A x y}" in exI, fast)
```
```    61 done
```
```    62
```
```    63 lemma left_unique_rel_set[transfer_rule]:
```
```    64   "left_unique A \<Longrightarrow> left_unique (rel_set A)"
```
```    65   unfolding left_unique_def rel_set_def
```
```    66   by fast
```
```    67
```
```    68 lemma right_total_rel_set [transfer_rule]:
```
```    69   "right_total A \<Longrightarrow> right_total (rel_set A)"
```
```    70 using left_total_rel_set[of "A\<inverse>\<inverse>"] by simp
```
```    71
```
```    72 lemma right_unique_rel_set [transfer_rule]:
```
```    73   "right_unique A \<Longrightarrow> right_unique (rel_set A)"
```
```    74   unfolding right_unique_def rel_set_def by fast
```
```    75
```
```    76 lemma bi_total_rel_set [transfer_rule]:
```
```    77   "bi_total A \<Longrightarrow> bi_total (rel_set A)"
```
```    78 by(simp add: bi_total_alt_def left_total_rel_set right_total_rel_set)
```
```    79
```
```    80 lemma bi_unique_rel_set [transfer_rule]:
```
```    81   "bi_unique A \<Longrightarrow> bi_unique (rel_set A)"
```
```    82   unfolding bi_unique_def rel_set_def by fast
```
```    83
```
```    84 lemma set_relator_eq_onp [relator_eq_onp]:
```
```    85   "rel_set (eq_onp P) = eq_onp (\<lambda>A. Ball A P)"
```
```    86   unfolding fun_eq_iff rel_set_def eq_onp_def Ball_def by fast
```
```    87
```
```    88 lemma bi_unique_rel_set_lemma:
```
```    89   assumes "bi_unique R" and "rel_set R X Y"
```
```    90   obtains f where "Y = image f X" and "inj_on f X" and "\<forall>x\<in>X. R x (f x)"
```
```    91 proof
```
```    92   def f \<equiv> "\<lambda>x. THE y. R x y"
```
```    93   { fix x assume "x \<in> X"
```
```    94     with `rel_set R X Y` `bi_unique R` have "R x (f x)"
```
```    95       by (simp add: bi_unique_def rel_set_def f_def) (metis theI)
```
```    96     with assms `x \<in> X`
```
```    97     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"
```
```    98       by (fastforce simp add: bi_unique_def rel_set_def)+ }
```
```    99   note * = this
```
```   100   moreover
```
```   101   { fix y assume "y \<in> Y"
```
```   102     with `rel_set R X Y` *(3) `y \<in> Y` have "\<exists>x\<in>X. y = f x"
```
```   103       by (fastforce simp: rel_set_def) }
```
```   104   ultimately show "\<forall>x\<in>X. R x (f x)" "Y = image f X" "inj_on f X"
```
```   105     by (auto simp: inj_on_def image_iff)
```
```   106 qed
```
```   107
```
```   108 subsection {* Quotient theorem for the Lifting package *}
```
```   109
```
```   110 lemma Quotient_set[quot_map]:
```
```   111   assumes "Quotient R Abs Rep T"
```
```   112   shows "Quotient (rel_set R) (image Abs) (image Rep) (rel_set T)"
```
```   113   using assms unfolding Quotient_alt_def4
```
```   114   apply (simp add: rel_set_OO[symmetric])
```
```   115   apply (simp add: rel_set_def, fast)
```
```   116   done
```
```   117
```
```   118 subsection {* Transfer rules for the Transfer package *}
```
```   119
```
```   120 subsubsection {* Unconditional transfer rules *}
```
```   121
```
```   122 context
```
```   123 begin
```
```   124 interpretation lifting_syntax .
```
```   125
```
```   126 lemma empty_transfer [transfer_rule]: "(rel_set A) {} {}"
```
```   127   unfolding rel_set_def by simp
```
```   128
```
```   129 lemma insert_transfer [transfer_rule]:
```
```   130   "(A ===> rel_set A ===> rel_set A) insert insert"
```
```   131   unfolding rel_fun_def rel_set_def by auto
```
```   132
```
```   133 lemma union_transfer [transfer_rule]:
```
```   134   "(rel_set A ===> rel_set A ===> rel_set A) union union"
```
```   135   unfolding rel_fun_def rel_set_def by auto
```
```   136
```
```   137 lemma Union_transfer [transfer_rule]:
```
```   138   "(rel_set (rel_set A) ===> rel_set A) Union Union"
```
```   139   unfolding rel_fun_def rel_set_def by simp fast
```
```   140
```
```   141 lemma image_transfer [transfer_rule]:
```
```   142   "((A ===> B) ===> rel_set A ===> rel_set B) image image"
```
```   143   unfolding rel_fun_def rel_set_def by simp fast
```
```   144
```
```   145 lemma UNION_transfer [transfer_rule]:
```
```   146   "(rel_set A ===> (A ===> rel_set B) ===> rel_set B) UNION UNION"
```
```   147   unfolding Union_image_eq [symmetric, abs_def] by transfer_prover
```
```   148
```
```   149 lemma Ball_transfer [transfer_rule]:
```
```   150   "(rel_set A ===> (A ===> op =) ===> op =) Ball Ball"
```
```   151   unfolding rel_set_def rel_fun_def by fast
```
```   152
```
```   153 lemma Bex_transfer [transfer_rule]:
```
```   154   "(rel_set A ===> (A ===> op =) ===> op =) Bex Bex"
```
```   155   unfolding rel_set_def rel_fun_def by fast
```
```   156
```
```   157 lemma Pow_transfer [transfer_rule]:
```
```   158   "(rel_set A ===> rel_set (rel_set A)) Pow Pow"
```
```   159   apply (rule rel_funI, rename_tac X Y, rule rel_setI)
```
```   160   apply (rename_tac X', rule_tac x="{y\<in>Y. \<exists>x\<in>X'. A x y}" in rev_bexI, clarsimp)
```
```   161   apply (simp add: rel_set_def, fast)
```
```   162   apply (rename_tac Y', rule_tac x="{x\<in>X. \<exists>y\<in>Y'. A x y}" in rev_bexI, clarsimp)
```
```   163   apply (simp add: rel_set_def, fast)
```
```   164   done
```
```   165
```
```   166 lemma rel_set_transfer [transfer_rule]:
```
```   167   "((A ===> B ===> op =) ===> rel_set A ===> rel_set B ===> op =) rel_set rel_set"
```
```   168   unfolding rel_fun_def rel_set_def by fast
```
```   169
```
```   170 lemma bind_transfer [transfer_rule]:
```
```   171   "(rel_set A ===> (A ===> rel_set B) ===> rel_set B) Set.bind Set.bind"
```
```   172   unfolding bind_UNION [abs_def] by transfer_prover
```
```   173
```
```   174 lemma INF_parametric [transfer_rule]:
```
```   175   "(rel_set A ===> (A ===> HOL.eq) ===> HOL.eq) INFIMUM INFIMUM"
```
```   176   unfolding INF_def [abs_def] by transfer_prover
```
```   177
```
```   178 lemma SUP_parametric [transfer_rule]:
```
```   179   "(rel_set R ===> (R ===> HOL.eq) ===> HOL.eq) SUPREMUM SUPREMUM"
```
```   180   unfolding SUP_def [abs_def] by transfer_prover
```
```   181
```
```   182
```
```   183 subsubsection {* Rules requiring bi-unique, bi-total or right-total relations *}
```
```   184
```
```   185 lemma member_transfer [transfer_rule]:
```
```   186   assumes "bi_unique A"
```
```   187   shows "(A ===> rel_set A ===> op =) (op \<in>) (op \<in>)"
```
```   188   using assms unfolding rel_fun_def rel_set_def bi_unique_def by fast
```
```   189
```
```   190 lemma right_total_Collect_transfer[transfer_rule]:
```
```   191   assumes "right_total A"
```
```   192   shows "((A ===> op =) ===> rel_set A) (\<lambda>P. Collect (\<lambda>x. P x \<and> Domainp A x)) Collect"
```
```   193   using assms unfolding right_total_def rel_set_def rel_fun_def Domainp_iff by fast
```
```   194
```
```   195 lemma Collect_transfer [transfer_rule]:
```
```   196   assumes "bi_total A"
```
```   197   shows "((A ===> op =) ===> rel_set A) Collect Collect"
```
```   198   using assms unfolding rel_fun_def rel_set_def bi_total_def by fast
```
```   199
```
```   200 lemma inter_transfer [transfer_rule]:
```
```   201   assumes "bi_unique A"
```
```   202   shows "(rel_set A ===> rel_set A ===> rel_set A) inter inter"
```
```   203   using assms unfolding rel_fun_def rel_set_def bi_unique_def by fast
```
```   204
```
```   205 lemma Diff_transfer [transfer_rule]:
```
```   206   assumes "bi_unique A"
```
```   207   shows "(rel_set A ===> rel_set A ===> rel_set A) (op -) (op -)"
```
```   208   using assms unfolding rel_fun_def rel_set_def bi_unique_def
```
```   209   unfolding Ball_def Bex_def Diff_eq
```
```   210   by (safe, simp, metis, simp, metis)
```
```   211
```
```   212 lemma subset_transfer [transfer_rule]:
```
```   213   assumes [transfer_rule]: "bi_unique A"
```
```   214   shows "(rel_set A ===> rel_set A ===> op =) (op \<subseteq>) (op \<subseteq>)"
```
```   215   unfolding subset_eq [abs_def] by transfer_prover
```
```   216
```
```   217 lemma right_total_UNIV_transfer[transfer_rule]:
```
```   218   assumes "right_total A"
```
```   219   shows "(rel_set A) (Collect (Domainp A)) UNIV"
```
```   220   using assms unfolding right_total_def rel_set_def Domainp_iff by blast
```
```   221
```
```   222 lemma UNIV_transfer [transfer_rule]:
```
```   223   assumes "bi_total A"
```
```   224   shows "(rel_set A) UNIV UNIV"
```
```   225   using assms unfolding rel_set_def bi_total_def by simp
```
```   226
```
```   227 lemma right_total_Compl_transfer [transfer_rule]:
```
```   228   assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "right_total A"
```
```   229   shows "(rel_set A ===> rel_set A) (\<lambda>S. uminus S \<inter> Collect (Domainp A)) uminus"
```
```   230   unfolding Compl_eq [abs_def]
```
```   231   by (subst Collect_conj_eq[symmetric]) transfer_prover
```
```   232
```
```   233 lemma Compl_transfer [transfer_rule]:
```
```   234   assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A"
```
```   235   shows "(rel_set A ===> rel_set A) uminus uminus"
```
```   236   unfolding Compl_eq [abs_def] by transfer_prover
```
```   237
```
```   238 lemma right_total_Inter_transfer [transfer_rule]:
```
```   239   assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "right_total A"
```
```   240   shows "(rel_set (rel_set A) ===> rel_set A) (\<lambda>S. Inter S \<inter> Collect (Domainp A)) Inter"
```
```   241   unfolding Inter_eq[abs_def]
```
```   242   by (subst Collect_conj_eq[symmetric]) transfer_prover
```
```   243
```
```   244 lemma Inter_transfer [transfer_rule]:
```
```   245   assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A"
```
```   246   shows "(rel_set (rel_set A) ===> rel_set A) Inter Inter"
```
```   247   unfolding Inter_eq [abs_def] by transfer_prover
```
```   248
```
```   249 lemma filter_transfer [transfer_rule]:
```
```   250   assumes [transfer_rule]: "bi_unique A"
```
```   251   shows "((A ===> op=) ===> rel_set A ===> rel_set A) Set.filter Set.filter"
```
```   252   unfolding Set.filter_def[abs_def] rel_fun_def rel_set_def by blast
```
```   253
```
```   254 lemma finite_transfer [transfer_rule]:
```
```   255   "bi_unique A \<Longrightarrow> (rel_set A ===> op =) finite finite"
```
```   256   by (rule rel_funI, erule (1) bi_unique_rel_set_lemma)
```
```   257      (auto dest: finite_imageD)
```
```   258
```
```   259 lemma card_transfer [transfer_rule]:
```
```   260   "bi_unique A \<Longrightarrow> (rel_set A ===> op =) card card"
```
```   261   by (rule rel_funI, erule (1) bi_unique_rel_set_lemma)
```
```   262      (simp add: card_image)
```
```   263
```
```   264 lemma vimage_parametric [transfer_rule]:
```
```   265   assumes [transfer_rule]: "bi_total A" "bi_unique B"
```
```   266   shows "((A ===> B) ===> rel_set B ===> rel_set A) vimage vimage"
```
```   267   unfolding vimage_def[abs_def] by transfer_prover
```
```   268
```
```   269 end
```
```   270
```
```   271 lemma (in comm_monoid_set) F_parametric [transfer_rule]:
```
```   272   fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"
```
```   273   assumes "bi_unique A"
```
```   274   shows "rel_fun (rel_fun A (op =)) (rel_fun (rel_set A) (op =)) F F"
```
```   275 proof(rule rel_funI)+
```
```   276   fix f :: "'b \<Rightarrow> 'a" and g S T
```
```   277   assume "rel_fun A (op =) f g" "rel_set A S T"
```
```   278   with `bi_unique A` obtain i where "bij_betw i S T" "\<And>x. x \<in> S \<Longrightarrow> f x = g (i x)"
```
```   279     by (auto elim: bi_unique_rel_set_lemma simp: rel_fun_def bij_betw_def)
```
```   280   then show "F f S = F g T"
```
```   281     by (simp add: reindex_bij_betw)
```
```   282 qed
```
```   283
```
```   284 lemmas setsum_parametric = setsum.F_parametric
```
```   285 lemmas setprod_parametric = setprod.F_parametric
```
```   286
```
```   287 end
```