src/HOL/Lifting_Set.thy
 author haftmann Sun Mar 16 18:09:04 2014 +0100 (2014-03-16) changeset 56166 9a241bc276cd parent 55945 e96383acecf9 child 56212 3253aaf73a01 permissions -rw-r--r--
normalising simp rules for compound 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   assumes "Domainp T = R"
```
```    49   shows "Domainp (rel_set T) = (\<lambda>A. Ball A R)"
```
```    50 using assms unfolding rel_set_def Domainp_iff[abs_def]
```
```    51 apply (intro ext)
```
```    52 apply (rule iffI)
```
```    53 apply blast
```
```    54 apply (rename_tac A, rule_tac x="{y. \<exists>x\<in>A. T x y}" in exI, fast)
```
```    55 done
```
```    56
```
```    57 lemma left_total_rel_set[reflexivity_rule]:
```
```    58   "left_total A \<Longrightarrow> left_total (rel_set A)"
```
```    59   unfolding left_total_def rel_set_def
```
```    60   apply safe
```
```    61   apply (rename_tac X, rule_tac x="{y. \<exists>x\<in>X. A x y}" in exI, fast)
```
```    62 done
```
```    63
```
```    64 lemma left_unique_rel_set[reflexivity_rule]:
```
```    65   "left_unique A \<Longrightarrow> left_unique (rel_set A)"
```
```    66   unfolding left_unique_def rel_set_def
```
```    67   by fast
```
```    68
```
```    69 lemma right_total_rel_set [transfer_rule]:
```
```    70   "right_total A \<Longrightarrow> right_total (rel_set A)"
```
```    71 using left_total_rel_set[of "A\<inverse>\<inverse>"] by simp
```
```    72
```
```    73 lemma right_unique_rel_set [transfer_rule]:
```
```    74   "right_unique A \<Longrightarrow> right_unique (rel_set A)"
```
```    75   unfolding right_unique_def rel_set_def by fast
```
```    76
```
```    77 lemma bi_total_rel_set [transfer_rule]:
```
```    78   "bi_total A \<Longrightarrow> bi_total (rel_set A)"
```
```    79 by(simp add: bi_total_conv_left_right left_total_rel_set right_total_rel_set)
```
```    80
```
```    81 lemma bi_unique_rel_set [transfer_rule]:
```
```    82   "bi_unique A \<Longrightarrow> bi_unique (rel_set A)"
```
```    83   unfolding bi_unique_def rel_set_def by fast
```
```    84
```
```    85 lemma set_invariant_commute [invariant_commute]:
```
```    86   "rel_set (Lifting.invariant P) = Lifting.invariant (\<lambda>A. Ball A P)"
```
```    87   unfolding fun_eq_iff rel_set_def Lifting.invariant_def Ball_def by fast
```
```    88
```
```    89 subsection {* Quotient theorem for the Lifting package *}
```
```    90
```
```    91 lemma Quotient_set[quot_map]:
```
```    92   assumes "Quotient R Abs Rep T"
```
```    93   shows "Quotient (rel_set R) (image Abs) (image Rep) (rel_set T)"
```
```    94   using assms unfolding Quotient_alt_def4
```
```    95   apply (simp add: rel_set_OO[symmetric])
```
```    96   apply (simp add: rel_set_def, fast)
```
```    97   done
```
```    98
```
```    99 subsection {* Transfer rules for the Transfer package *}
```
```   100
```
```   101 subsubsection {* Unconditional transfer rules *}
```
```   102
```
```   103 context
```
```   104 begin
```
```   105 interpretation lifting_syntax .
```
```   106
```
```   107 lemma empty_transfer [transfer_rule]: "(rel_set A) {} {}"
```
```   108   unfolding rel_set_def by simp
```
```   109
```
```   110 lemma insert_transfer [transfer_rule]:
```
```   111   "(A ===> rel_set A ===> rel_set A) insert insert"
```
```   112   unfolding rel_fun_def rel_set_def by auto
```
```   113
```
```   114 lemma union_transfer [transfer_rule]:
```
```   115   "(rel_set A ===> rel_set A ===> rel_set A) union union"
```
```   116   unfolding rel_fun_def rel_set_def by auto
```
```   117
```
```   118 lemma Union_transfer [transfer_rule]:
```
```   119   "(rel_set (rel_set A) ===> rel_set A) Union Union"
```
```   120   unfolding rel_fun_def rel_set_def by simp fast
```
```   121
```
```   122 lemma image_transfer [transfer_rule]:
```
```   123   "((A ===> B) ===> rel_set A ===> rel_set B) image image"
```
```   124   unfolding rel_fun_def rel_set_def by simp fast
```
```   125
```
```   126 lemma UNION_transfer [transfer_rule]:
```
```   127   "(rel_set A ===> (A ===> rel_set B) ===> rel_set B) UNION UNION"
```
```   128   unfolding Union_image_eq [symmetric, abs_def] by transfer_prover
```
```   129
```
```   130 lemma Ball_transfer [transfer_rule]:
```
```   131   "(rel_set A ===> (A ===> op =) ===> op =) Ball Ball"
```
```   132   unfolding rel_set_def rel_fun_def by fast
```
```   133
```
```   134 lemma Bex_transfer [transfer_rule]:
```
```   135   "(rel_set A ===> (A ===> op =) ===> op =) Bex Bex"
```
```   136   unfolding rel_set_def rel_fun_def by fast
```
```   137
```
```   138 lemma Pow_transfer [transfer_rule]:
```
```   139   "(rel_set A ===> rel_set (rel_set A)) Pow Pow"
```
```   140   apply (rule rel_funI, rename_tac X Y, rule rel_setI)
```
```   141   apply (rename_tac X', rule_tac x="{y\<in>Y. \<exists>x\<in>X'. A x y}" in rev_bexI, clarsimp)
```
```   142   apply (simp add: rel_set_def, fast)
```
```   143   apply (rename_tac Y', rule_tac x="{x\<in>X. \<exists>y\<in>Y'. A x y}" in rev_bexI, clarsimp)
```
```   144   apply (simp add: rel_set_def, fast)
```
```   145   done
```
```   146
```
```   147 lemma rel_set_transfer [transfer_rule]:
```
```   148   "((A ===> B ===> op =) ===> rel_set A ===> rel_set B ===> op =)
```
```   149     rel_set rel_set"
```
```   150   unfolding rel_fun_def rel_set_def by fast
```
```   151
```
```   152 lemma SUPR_parametric [transfer_rule]:
```
```   153   "(rel_set R ===> (R ===> op =) ===> op =) SUPR (SUPR :: _ \<Rightarrow> _ \<Rightarrow> _::complete_lattice)"
```
```   154 proof(rule rel_funI)+
```
```   155   fix A B f and g :: "'b \<Rightarrow> 'c"
```
```   156   assume AB: "rel_set R A B"
```
```   157     and fg: "(R ===> op =) f g"
```
```   158   show "SUPR A f = SUPR B g"
```
```   159     by(rule SUPR_eq)(auto 4 4 dest: rel_setD1[OF AB] rel_setD2[OF AB] rel_funD[OF fg] intro: rev_bexI)
```
```   160 qed
```
```   161
```
```   162 lemma bind_transfer [transfer_rule]:
```
```   163   "(rel_set A ===> (A ===> rel_set B) ===> rel_set B) Set.bind Set.bind"
```
```   164 unfolding bind_UNION[abs_def] by transfer_prover
```
```   165
```
```   166 subsubsection {* Rules requiring bi-unique, bi-total or right-total relations *}
```
```   167
```
```   168 lemma member_transfer [transfer_rule]:
```
```   169   assumes "bi_unique A"
```
```   170   shows "(A ===> rel_set A ===> op =) (op \<in>) (op \<in>)"
```
```   171   using assms unfolding rel_fun_def rel_set_def bi_unique_def by fast
```
```   172
```
```   173 lemma right_total_Collect_transfer[transfer_rule]:
```
```   174   assumes "right_total A"
```
```   175   shows "((A ===> op =) ===> rel_set A) (\<lambda>P. Collect (\<lambda>x. P x \<and> Domainp A x)) Collect"
```
```   176   using assms unfolding right_total_def rel_set_def rel_fun_def Domainp_iff by fast
```
```   177
```
```   178 lemma Collect_transfer [transfer_rule]:
```
```   179   assumes "bi_total A"
```
```   180   shows "((A ===> op =) ===> rel_set A) Collect Collect"
```
```   181   using assms unfolding rel_fun_def rel_set_def bi_total_def by fast
```
```   182
```
```   183 lemma inter_transfer [transfer_rule]:
```
```   184   assumes "bi_unique A"
```
```   185   shows "(rel_set A ===> rel_set A ===> rel_set A) inter inter"
```
```   186   using assms unfolding rel_fun_def rel_set_def bi_unique_def by fast
```
```   187
```
```   188 lemma Diff_transfer [transfer_rule]:
```
```   189   assumes "bi_unique A"
```
```   190   shows "(rel_set A ===> rel_set A ===> rel_set A) (op -) (op -)"
```
```   191   using assms unfolding rel_fun_def rel_set_def bi_unique_def
```
```   192   unfolding Ball_def Bex_def Diff_eq
```
```   193   by (safe, simp, metis, simp, metis)
```
```   194
```
```   195 lemma subset_transfer [transfer_rule]:
```
```   196   assumes [transfer_rule]: "bi_unique A"
```
```   197   shows "(rel_set A ===> rel_set A ===> op =) (op \<subseteq>) (op \<subseteq>)"
```
```   198   unfolding subset_eq [abs_def] by transfer_prover
```
```   199
```
```   200 lemma right_total_UNIV_transfer[transfer_rule]:
```
```   201   assumes "right_total A"
```
```   202   shows "(rel_set A) (Collect (Domainp A)) UNIV"
```
```   203   using assms unfolding right_total_def rel_set_def Domainp_iff by blast
```
```   204
```
```   205 lemma UNIV_transfer [transfer_rule]:
```
```   206   assumes "bi_total A"
```
```   207   shows "(rel_set A) UNIV UNIV"
```
```   208   using assms unfolding rel_set_def bi_total_def by simp
```
```   209
```
```   210 lemma right_total_Compl_transfer [transfer_rule]:
```
```   211   assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "right_total A"
```
```   212   shows "(rel_set A ===> rel_set A) (\<lambda>S. uminus S \<inter> Collect (Domainp A)) uminus"
```
```   213   unfolding Compl_eq [abs_def]
```
```   214   by (subst Collect_conj_eq[symmetric]) transfer_prover
```
```   215
```
```   216 lemma Compl_transfer [transfer_rule]:
```
```   217   assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A"
```
```   218   shows "(rel_set A ===> rel_set A) uminus uminus"
```
```   219   unfolding Compl_eq [abs_def] by transfer_prover
```
```   220
```
```   221 lemma right_total_Inter_transfer [transfer_rule]:
```
```   222   assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "right_total A"
```
```   223   shows "(rel_set (rel_set A) ===> rel_set A) (\<lambda>S. Inter S \<inter> Collect (Domainp A)) Inter"
```
```   224   unfolding Inter_eq[abs_def]
```
```   225   by (subst Collect_conj_eq[symmetric]) transfer_prover
```
```   226
```
```   227 lemma Inter_transfer [transfer_rule]:
```
```   228   assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A"
```
```   229   shows "(rel_set (rel_set A) ===> rel_set A) Inter Inter"
```
```   230   unfolding Inter_eq [abs_def] by transfer_prover
```
```   231
```
```   232 lemma filter_transfer [transfer_rule]:
```
```   233   assumes [transfer_rule]: "bi_unique A"
```
```   234   shows "((A ===> op=) ===> rel_set A ===> rel_set A) Set.filter Set.filter"
```
```   235   unfolding Set.filter_def[abs_def] rel_fun_def rel_set_def by blast
```
```   236
```
```   237 lemma bi_unique_rel_set_lemma:
```
```   238   assumes "bi_unique R" and "rel_set R X Y"
```
```   239   obtains f where "Y = image f X" and "inj_on f X" and "\<forall>x\<in>X. R x (f x)"
```
```   240 proof
```
```   241   let ?f = "\<lambda>x. THE y. R x y"
```
```   242   from assms show f: "\<forall>x\<in>X. R x (?f x)"
```
```   243     apply (clarsimp simp add: rel_set_def)
```
```   244     apply (drule (1) bspec, clarify)
```
```   245     apply (rule theI2, assumption)
```
```   246     apply (simp add: bi_unique_def)
```
```   247     apply assumption
```
```   248     done
```
```   249   from assms show "Y = image ?f X"
```
```   250     apply safe
```
```   251     apply (clarsimp simp add: rel_set_def)
```
```   252     apply (drule (1) bspec, clarify)
```
```   253     apply (rule image_eqI)
```
```   254     apply (rule the_equality [symmetric], assumption)
```
```   255     apply (simp add: bi_unique_def)
```
```   256     apply assumption
```
```   257     apply (clarsimp simp add: rel_set_def)
```
```   258     apply (frule (1) bspec, clarify)
```
```   259     apply (rule theI2, assumption)
```
```   260     apply (clarsimp simp add: bi_unique_def)
```
```   261     apply (simp add: bi_unique_def, metis)
```
```   262     done
```
```   263   show "inj_on ?f X"
```
```   264     apply (rule inj_onI)
```
```   265     apply (drule f [rule_format])
```
```   266     apply (drule f [rule_format])
```
```   267     apply (simp add: assms(1) [unfolded bi_unique_def])
```
```   268     done
```
```   269 qed
```
```   270
```
```   271 lemma finite_transfer [transfer_rule]:
```
```   272   "bi_unique A \<Longrightarrow> (rel_set A ===> op =) finite finite"
```
```   273   by (rule rel_funI, erule (1) bi_unique_rel_set_lemma,
```
```   274     auto dest: finite_imageD)
```
```   275
```
```   276 lemma card_transfer [transfer_rule]:
```
```   277   "bi_unique A \<Longrightarrow> (rel_set A ===> op =) card card"
```
```   278   by (rule rel_funI, erule (1) bi_unique_rel_set_lemma, simp add: card_image)
```
```   279
```
```   280 lemma vimage_parametric [transfer_rule]:
```
```   281   assumes [transfer_rule]: "bi_total A" "bi_unique B"
```
```   282   shows "((A ===> B) ===> rel_set B ===> rel_set A) vimage vimage"
```
```   283 unfolding vimage_def[abs_def] by transfer_prover
```
```   284
```
```   285 lemma setsum_parametric [transfer_rule]:
```
```   286   assumes "bi_unique A"
```
```   287   shows "((A ===> op =) ===> rel_set A ===> op =) setsum setsum"
```
```   288 proof(rule rel_funI)+
```
```   289   fix f :: "'a \<Rightarrow> 'c" and g S T
```
```   290   assume fg: "(A ===> op =) f g"
```
```   291     and ST: "rel_set A S T"
```
```   292   show "setsum f S = setsum g T"
```
```   293   proof(rule setsum_reindex_cong)
```
```   294     let ?f = "\<lambda>t. THE s. A s t"
```
```   295     show "S = ?f ` T"
```
```   296       by(blast dest: rel_setD1[OF ST] rel_setD2[OF ST] bi_uniqueDl[OF assms]
```
```   297            intro: rev_image_eqI the_equality[symmetric] subst[rotated, where P="\<lambda>x. x \<in> S"])
```
```   298
```
```   299     show "inj_on ?f T"
```
```   300     proof(rule inj_onI)
```
```   301       fix t1 t2
```
```   302       assume "t1 \<in> T" "t2 \<in> T" "?f t1 = ?f t2"
```
```   303       from ST `t1 \<in> T` obtain s1 where "A s1 t1" "s1 \<in> S" by(auto dest: rel_setD2)
```
```   304       hence "?f t1 = s1" by(auto dest: bi_uniqueDl[OF assms])
```
```   305       moreover
```
```   306       from ST `t2 \<in> T` obtain s2 where "A s2 t2" "s2 \<in> S" by(auto dest: rel_setD2)
```
```   307       hence "?f t2 = s2" by(auto dest: bi_uniqueDl[OF assms])
```
```   308       ultimately have "s1 = s2" using `?f t1 = ?f t2` by simp
```
```   309       with `A s1 t1` `A s2 t2` show "t1 = t2" by(auto dest: bi_uniqueDr[OF assms])
```
```   310     qed
```
```   311
```
```   312     fix t
```
```   313     assume "t \<in> T"
```
```   314     with ST obtain s where "A s t" "s \<in> S" by(auto dest: rel_setD2)
```
```   315     hence "?f t = s" by(auto dest: bi_uniqueDl[OF assms])
```
```   316     moreover from fg `A s t` have "f s = g t" by(rule rel_funD)
```
```   317     ultimately show "g t = f (?f t)" by simp
```
```   318   qed
```
```   319 qed
```
```   320
```
```   321 end
```
```   322
```
```   323 end
```