src/HOL/Transfer.thy
author kuncar
Thu Apr 10 17:48:14 2014 +0200 (2014-04-10)
changeset 56518 beb3b6851665
parent 56085 3d11892ea537
child 56520 3373f5d1e074
permissions -rw-r--r--
left_total and left_unique rules are now transfer rules (cleaner solution, reflexvity_rule attribute not needed anymore)
     1 (*  Title:      HOL/Transfer.thy
     2     Author:     Brian Huffman, TU Muenchen
     3     Author:     Ondrej Kuncar, TU Muenchen
     4 *)
     5 
     6 header {* Generic theorem transfer using relations *}
     7 
     8 theory Transfer
     9 imports Hilbert_Choice Basic_BNFs Metis
    10 begin
    11 
    12 subsection {* Relator for function space *}
    13 
    14 locale lifting_syntax
    15 begin
    16   notation rel_fun (infixr "===>" 55)
    17   notation map_fun (infixr "--->" 55)
    18 end
    19 
    20 context
    21 begin
    22 interpretation lifting_syntax .
    23 
    24 lemma rel_funD2:
    25   assumes "rel_fun A B f g" and "A x x"
    26   shows "B (f x) (g x)"
    27   using assms by (rule rel_funD)
    28 
    29 lemma rel_funE:
    30   assumes "rel_fun A B f g" and "A x y"
    31   obtains "B (f x) (g y)"
    32   using assms by (simp add: rel_fun_def)
    33 
    34 lemmas rel_fun_eq = fun.rel_eq
    35 
    36 lemma rel_fun_eq_rel:
    37 shows "rel_fun (op =) R = (\<lambda>f g. \<forall>x. R (f x) (g x))"
    38   by (simp add: rel_fun_def)
    39 
    40 
    41 subsection {* Transfer method *}
    42 
    43 text {* Explicit tag for relation membership allows for
    44   backward proof methods. *}
    45 
    46 definition Rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"
    47   where "Rel r \<equiv> r"
    48 
    49 text {* Handling of equality relations *}
    50 
    51 definition is_equality :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
    52   where "is_equality R \<longleftrightarrow> R = (op =)"
    53 
    54 lemma is_equality_eq: "is_equality (op =)"
    55   unfolding is_equality_def by simp
    56 
    57 text {* Reverse implication for monotonicity rules *}
    58 
    59 definition rev_implies where
    60   "rev_implies x y \<longleftrightarrow> (y \<longrightarrow> x)"
    61 
    62 text {* Handling of meta-logic connectives *}
    63 
    64 definition transfer_forall where
    65   "transfer_forall \<equiv> All"
    66 
    67 definition transfer_implies where
    68   "transfer_implies \<equiv> op \<longrightarrow>"
    69 
    70 definition transfer_bforall :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
    71   where "transfer_bforall \<equiv> (\<lambda>P Q. \<forall>x. P x \<longrightarrow> Q x)"
    72 
    73 lemma transfer_forall_eq: "(\<And>x. P x) \<equiv> Trueprop (transfer_forall (\<lambda>x. P x))"
    74   unfolding atomize_all transfer_forall_def ..
    75 
    76 lemma transfer_implies_eq: "(A \<Longrightarrow> B) \<equiv> Trueprop (transfer_implies A B)"
    77   unfolding atomize_imp transfer_implies_def ..
    78 
    79 lemma transfer_bforall_unfold:
    80   "Trueprop (transfer_bforall P (\<lambda>x. Q x)) \<equiv> (\<And>x. P x \<Longrightarrow> Q x)"
    81   unfolding transfer_bforall_def atomize_imp atomize_all ..
    82 
    83 lemma transfer_start: "\<lbrakk>P; Rel (op =) P Q\<rbrakk> \<Longrightarrow> Q"
    84   unfolding Rel_def by simp
    85 
    86 lemma transfer_start': "\<lbrakk>P; Rel (op \<longrightarrow>) P Q\<rbrakk> \<Longrightarrow> Q"
    87   unfolding Rel_def by simp
    88 
    89 lemma transfer_prover_start: "\<lbrakk>x = x'; Rel R x' y\<rbrakk> \<Longrightarrow> Rel R x y"
    90   by simp
    91 
    92 lemma untransfer_start: "\<lbrakk>Q; Rel (op =) P Q\<rbrakk> \<Longrightarrow> P"
    93   unfolding Rel_def by simp
    94 
    95 lemma Rel_eq_refl: "Rel (op =) x x"
    96   unfolding Rel_def ..
    97 
    98 lemma Rel_app:
    99   assumes "Rel (A ===> B) f g" and "Rel A x y"
   100   shows "Rel B (f x) (g y)"
   101   using assms unfolding Rel_def rel_fun_def by fast
   102 
   103 lemma Rel_abs:
   104   assumes "\<And>x y. Rel A x y \<Longrightarrow> Rel B (f x) (g y)"
   105   shows "Rel (A ===> B) (\<lambda>x. f x) (\<lambda>y. g y)"
   106   using assms unfolding Rel_def rel_fun_def by fast
   107 
   108 end
   109 
   110 ML_file "Tools/transfer.ML"
   111 setup Transfer.setup
   112 
   113 declare refl [transfer_rule]
   114 
   115 declare rel_fun_eq [relator_eq]
   116 
   117 hide_const (open) Rel
   118 
   119 context
   120 begin
   121 interpretation lifting_syntax .
   122 
   123 text {* Handling of domains *}
   124 
   125 lemma Domainp_iff: "Domainp T x \<longleftrightarrow> (\<exists>y. T x y)"
   126   by auto
   127 
   128 lemma Domaimp_refl[transfer_domain_rule]:
   129   "Domainp T = Domainp T" ..
   130 
   131 lemma Domainp_prod_fun_eq[transfer_domain_rule]:
   132   assumes "Domainp T = P"
   133   shows "Domainp (op= ===> T) = (\<lambda>f. \<forall>x. P (f x))"
   134 by (auto intro: choice simp: assms[symmetric] Domainp_iff rel_fun_def fun_eq_iff)
   135 
   136 subsection {* Predicates on relations, i.e. ``class constraints'' *}
   137 
   138 definition left_total :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
   139   where "left_total R \<longleftrightarrow> (\<forall>x. \<exists>y. R x y)"
   140 
   141 definition left_unique :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
   142   where "left_unique R \<longleftrightarrow> (\<forall>x y z. R x z \<longrightarrow> R y z \<longrightarrow> x = y)"
   143 
   144 definition right_total :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
   145   where "right_total R \<longleftrightarrow> (\<forall>y. \<exists>x. R x y)"
   146 
   147 definition right_unique :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
   148   where "right_unique R \<longleftrightarrow> (\<forall>x y z. R x y \<longrightarrow> R x z \<longrightarrow> y = z)"
   149 
   150 definition bi_total :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
   151   where "bi_total R \<longleftrightarrow> (\<forall>x. \<exists>y. R x y) \<and> (\<forall>y. \<exists>x. R x y)"
   152 
   153 definition bi_unique :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
   154   where "bi_unique R \<longleftrightarrow>
   155     (\<forall>x y z. R x y \<longrightarrow> R x z \<longrightarrow> y = z) \<and>
   156     (\<forall>x y z. R x z \<longrightarrow> R y z \<longrightarrow> x = y)"
   157 
   158 lemma left_uniqueI: "(\<And>x y z. \<lbrakk> A x z; A y z \<rbrakk> \<Longrightarrow> x = y) \<Longrightarrow> left_unique A"
   159 unfolding left_unique_def by blast
   160 
   161 lemma left_uniqueD: "\<lbrakk> left_unique A; A x z; A y z \<rbrakk> \<Longrightarrow> x = y"
   162 unfolding left_unique_def by blast
   163 
   164 lemma left_totalI:
   165   "(\<And>x. \<exists>y. R x y) \<Longrightarrow> left_total R"
   166 unfolding left_total_def by blast
   167 
   168 lemma left_totalE:
   169   assumes "left_total R"
   170   obtains "(\<And>x. \<exists>y. R x y)"
   171 using assms unfolding left_total_def by blast
   172 
   173 lemma bi_uniqueDr: "\<lbrakk> bi_unique A; A x y; A x z \<rbrakk> \<Longrightarrow> y = z"
   174 by(simp add: bi_unique_def)
   175 
   176 lemma bi_uniqueDl: "\<lbrakk> bi_unique A; A x y; A z y \<rbrakk> \<Longrightarrow> x = z"
   177 by(simp add: bi_unique_def)
   178 
   179 lemma right_uniqueI: "(\<And>x y z. \<lbrakk> A x y; A x z \<rbrakk> \<Longrightarrow> y = z) \<Longrightarrow> right_unique A"
   180 unfolding right_unique_def by fast
   181 
   182 lemma right_uniqueD: "\<lbrakk> right_unique A; A x y; A x z \<rbrakk> \<Longrightarrow> y = z"
   183 unfolding right_unique_def by fast
   184 
   185 lemma right_total_alt_def:
   186   "right_total R \<longleftrightarrow> ((R ===> op \<longrightarrow>) ===> op \<longrightarrow>) All All"
   187   unfolding right_total_def rel_fun_def
   188   apply (rule iffI, fast)
   189   apply (rule allI)
   190   apply (drule_tac x="\<lambda>x. True" in spec)
   191   apply (drule_tac x="\<lambda>y. \<exists>x. R x y" in spec)
   192   apply fast
   193   done
   194 
   195 lemma right_unique_alt_def:
   196   "right_unique R \<longleftrightarrow> (R ===> R ===> op \<longrightarrow>) (op =) (op =)"
   197   unfolding right_unique_def rel_fun_def by auto
   198 
   199 lemma bi_total_alt_def:
   200   "bi_total R \<longleftrightarrow> ((R ===> op =) ===> op =) All All"
   201   unfolding bi_total_def rel_fun_def
   202   apply (rule iffI, fast)
   203   apply safe
   204   apply (drule_tac x="\<lambda>x. \<exists>y. R x y" in spec)
   205   apply (drule_tac x="\<lambda>y. True" in spec)
   206   apply fast
   207   apply (drule_tac x="\<lambda>x. True" in spec)
   208   apply (drule_tac x="\<lambda>y. \<exists>x. R x y" in spec)
   209   apply fast
   210   done
   211 
   212 lemma bi_unique_alt_def:
   213   "bi_unique R \<longleftrightarrow> (R ===> R ===> op =) (op =) (op =)"
   214   unfolding bi_unique_def rel_fun_def by auto
   215 
   216 lemma [simp]:
   217   shows left_unique_conversep: "left_unique A\<inverse>\<inverse> \<longleftrightarrow> right_unique A"
   218   and right_unique_conversep: "right_unique A\<inverse>\<inverse> \<longleftrightarrow> left_unique A"
   219 by(auto simp add: left_unique_def right_unique_def)
   220 
   221 lemma [simp]:
   222   shows left_total_conversep: "left_total A\<inverse>\<inverse> \<longleftrightarrow> right_total A"
   223   and right_total_conversep: "right_total A\<inverse>\<inverse> \<longleftrightarrow> left_total A"
   224 by(simp_all add: left_total_def right_total_def)
   225 
   226 lemma bi_unique_conversep [simp]: "bi_unique R\<inverse>\<inverse> = bi_unique R"
   227 by(auto simp add: bi_unique_def)
   228 
   229 lemma bi_total_conversep [simp]: "bi_total R\<inverse>\<inverse> = bi_total R"
   230 by(auto simp add: bi_total_def)
   231 
   232 lemma bi_total_iff: "bi_total A = (right_total A \<and> left_total A)"
   233 unfolding left_total_def right_total_def bi_total_def by blast
   234 
   235 lemma bi_total_conv_left_right: "bi_total R \<longleftrightarrow> left_total R \<and> right_total R"
   236 by(simp add: left_total_def right_total_def bi_total_def)
   237 
   238 lemma bi_unique_iff: "bi_unique A  \<longleftrightarrow> right_unique A \<and> left_unique A"
   239 unfolding left_unique_def right_unique_def bi_unique_def by blast
   240 
   241 lemma bi_unique_conv_left_right: "bi_unique R \<longleftrightarrow> left_unique R \<and> right_unique R"
   242 by(auto simp add: left_unique_def right_unique_def bi_unique_def)
   243 
   244 lemma bi_totalI: "left_total R \<Longrightarrow> right_total R \<Longrightarrow> bi_total R"
   245 unfolding bi_total_iff ..
   246 
   247 lemma bi_uniqueI: "left_unique R \<Longrightarrow> right_unique R \<Longrightarrow> bi_unique R"
   248 unfolding bi_unique_iff ..
   249 
   250 
   251 text {* Properties are preserved by relation composition. *}
   252 
   253 lemma OO_def: "R OO S = (\<lambda>x z. \<exists>y. R x y \<and> S y z)"
   254   by auto
   255 
   256 lemma bi_total_OO: "\<lbrakk>bi_total A; bi_total B\<rbrakk> \<Longrightarrow> bi_total (A OO B)"
   257   unfolding bi_total_def OO_def by fast
   258 
   259 lemma bi_unique_OO: "\<lbrakk>bi_unique A; bi_unique B\<rbrakk> \<Longrightarrow> bi_unique (A OO B)"
   260   unfolding bi_unique_def OO_def by blast
   261 
   262 lemma right_total_OO:
   263   "\<lbrakk>right_total A; right_total B\<rbrakk> \<Longrightarrow> right_total (A OO B)"
   264   unfolding right_total_def OO_def by fast
   265 
   266 lemma right_unique_OO:
   267   "\<lbrakk>right_unique A; right_unique B\<rbrakk> \<Longrightarrow> right_unique (A OO B)"
   268   unfolding right_unique_def OO_def by fast
   269 
   270 lemma left_total_OO: "left_total R \<Longrightarrow> left_total S \<Longrightarrow> left_total (R OO S)"
   271 unfolding left_total_def OO_def by fast
   272 
   273 lemma left_unique_OO: "left_unique R \<Longrightarrow> left_unique S \<Longrightarrow> left_unique (R OO S)"
   274 unfolding left_unique_def OO_def by blast
   275 
   276 
   277 subsection {* Properties of relators *}
   278 
   279 lemma left_total_eq[transfer_rule]: "left_total op=" 
   280   unfolding left_total_def by blast
   281 
   282 lemma left_unique_eq[transfer_rule]: "left_unique op=" 
   283   unfolding left_unique_def by blast
   284 
   285 lemma right_total_eq [transfer_rule]: "right_total op="
   286   unfolding right_total_def by simp
   287 
   288 lemma right_unique_eq [transfer_rule]: "right_unique op="
   289   unfolding right_unique_def by simp
   290 
   291 lemma bi_total_eq[transfer_rule]: "bi_total (op =)"
   292   unfolding bi_total_def by simp
   293 
   294 lemma bi_unique_eq[transfer_rule]: "bi_unique (op =)"
   295   unfolding bi_unique_def by simp
   296 
   297 lemma left_total_fun[transfer_rule]:
   298   "\<lbrakk>left_unique A; left_total B\<rbrakk> \<Longrightarrow> left_total (A ===> B)"
   299   unfolding left_total_def rel_fun_def
   300   apply (rule allI, rename_tac f)
   301   apply (rule_tac x="\<lambda>y. SOME z. B (f (THE x. A x y)) z" in exI)
   302   apply clarify
   303   apply (subgoal_tac "(THE x. A x y) = x", simp)
   304   apply (rule someI_ex)
   305   apply (simp)
   306   apply (rule the_equality)
   307   apply assumption
   308   apply (simp add: left_unique_def)
   309   done
   310 
   311 lemma left_unique_fun[transfer_rule]:
   312   "\<lbrakk>left_total A; left_unique B\<rbrakk> \<Longrightarrow> left_unique (A ===> B)"
   313   unfolding left_total_def left_unique_def rel_fun_def
   314   by (clarify, rule ext, fast)
   315 
   316 lemma right_total_fun [transfer_rule]:
   317   "\<lbrakk>right_unique A; right_total B\<rbrakk> \<Longrightarrow> right_total (A ===> B)"
   318   unfolding right_total_def rel_fun_def
   319   apply (rule allI, rename_tac g)
   320   apply (rule_tac x="\<lambda>x. SOME z. B z (g (THE y. A x y))" in exI)
   321   apply clarify
   322   apply (subgoal_tac "(THE y. A x y) = y", simp)
   323   apply (rule someI_ex)
   324   apply (simp)
   325   apply (rule the_equality)
   326   apply assumption
   327   apply (simp add: right_unique_def)
   328   done
   329 
   330 lemma right_unique_fun [transfer_rule]:
   331   "\<lbrakk>right_total A; right_unique B\<rbrakk> \<Longrightarrow> right_unique (A ===> B)"
   332   unfolding right_total_def right_unique_def rel_fun_def
   333   by (clarify, rule ext, fast)
   334 
   335 lemma bi_total_fun[transfer_rule]:
   336   "\<lbrakk>bi_unique A; bi_total B\<rbrakk> \<Longrightarrow> bi_total (A ===> B)"
   337   unfolding bi_unique_iff bi_total_iff
   338   by (blast intro: right_total_fun left_total_fun)
   339 
   340 lemma bi_unique_fun[transfer_rule]:
   341   "\<lbrakk>bi_total A; bi_unique B\<rbrakk> \<Longrightarrow> bi_unique (A ===> B)"
   342   unfolding bi_unique_iff bi_total_iff
   343   by (blast intro: right_unique_fun left_unique_fun)
   344 
   345 subsection {* Transfer rules *}
   346 
   347 lemma Domainp_forall_transfer [transfer_rule]:
   348   assumes "right_total A"
   349   shows "((A ===> op =) ===> op =)
   350     (transfer_bforall (Domainp A)) transfer_forall"
   351   using assms unfolding right_total_def
   352   unfolding transfer_forall_def transfer_bforall_def rel_fun_def Domainp_iff
   353   by fast
   354 
   355 text {* Transfer rules using implication instead of equality on booleans. *}
   356 
   357 lemma transfer_forall_transfer [transfer_rule]:
   358   "bi_total A \<Longrightarrow> ((A ===> op =) ===> op =) transfer_forall transfer_forall"
   359   "right_total A \<Longrightarrow> ((A ===> op =) ===> implies) transfer_forall transfer_forall"
   360   "right_total A \<Longrightarrow> ((A ===> implies) ===> implies) transfer_forall transfer_forall"
   361   "bi_total A \<Longrightarrow> ((A ===> op =) ===> rev_implies) transfer_forall transfer_forall"
   362   "bi_total A \<Longrightarrow> ((A ===> rev_implies) ===> rev_implies) transfer_forall transfer_forall"
   363   unfolding transfer_forall_def rev_implies_def rel_fun_def right_total_def bi_total_def
   364   by fast+
   365 
   366 lemma transfer_implies_transfer [transfer_rule]:
   367   "(op =        ===> op =        ===> op =       ) transfer_implies transfer_implies"
   368   "(rev_implies ===> implies     ===> implies    ) transfer_implies transfer_implies"
   369   "(rev_implies ===> op =        ===> implies    ) transfer_implies transfer_implies"
   370   "(op =        ===> implies     ===> implies    ) transfer_implies transfer_implies"
   371   "(op =        ===> op =        ===> implies    ) transfer_implies transfer_implies"
   372   "(implies     ===> rev_implies ===> rev_implies) transfer_implies transfer_implies"
   373   "(implies     ===> op =        ===> rev_implies) transfer_implies transfer_implies"
   374   "(op =        ===> rev_implies ===> rev_implies) transfer_implies transfer_implies"
   375   "(op =        ===> op =        ===> rev_implies) transfer_implies transfer_implies"
   376   unfolding transfer_implies_def rev_implies_def rel_fun_def by auto
   377 
   378 lemma eq_imp_transfer [transfer_rule]:
   379   "right_unique A \<Longrightarrow> (A ===> A ===> op \<longrightarrow>) (op =) (op =)"
   380   unfolding right_unique_alt_def .
   381 
   382 text {* Transfer rules using equality. *}
   383 
   384 lemma left_unique_transfer [transfer_rule]:
   385   assumes "right_total A"
   386   assumes "right_total B"
   387   assumes "bi_unique A"
   388   shows "((A ===> B ===> op=) ===> implies) left_unique left_unique"
   389 using assms unfolding left_unique_def[abs_def] right_total_def bi_unique_def rel_fun_def
   390 by metis
   391 
   392 lemma eq_transfer [transfer_rule]:
   393   assumes "bi_unique A"
   394   shows "(A ===> A ===> op =) (op =) (op =)"
   395   using assms unfolding bi_unique_def rel_fun_def by auto
   396 
   397 lemma right_total_Ex_transfer[transfer_rule]:
   398   assumes "right_total A"
   399   shows "((A ===> op=) ===> op=) (Bex (Collect (Domainp A))) Ex"
   400 using assms unfolding right_total_def Bex_def rel_fun_def Domainp_iff[abs_def]
   401 by fast
   402 
   403 lemma right_total_All_transfer[transfer_rule]:
   404   assumes "right_total A"
   405   shows "((A ===> op =) ===> op =) (Ball (Collect (Domainp A))) All"
   406 using assms unfolding right_total_def Ball_def rel_fun_def Domainp_iff[abs_def]
   407 by fast
   408 
   409 lemma All_transfer [transfer_rule]:
   410   assumes "bi_total A"
   411   shows "((A ===> op =) ===> op =) All All"
   412   using assms unfolding bi_total_def rel_fun_def by fast
   413 
   414 lemma Ex_transfer [transfer_rule]:
   415   assumes "bi_total A"
   416   shows "((A ===> op =) ===> op =) Ex Ex"
   417   using assms unfolding bi_total_def rel_fun_def by fast
   418 
   419 lemma If_transfer [transfer_rule]: "(op = ===> A ===> A ===> A) If If"
   420   unfolding rel_fun_def by simp
   421 
   422 lemma Let_transfer [transfer_rule]: "(A ===> (A ===> B) ===> B) Let Let"
   423   unfolding rel_fun_def by simp
   424 
   425 lemma id_transfer [transfer_rule]: "(A ===> A) id id"
   426   unfolding rel_fun_def by simp
   427 
   428 lemma comp_transfer [transfer_rule]:
   429   "((B ===> C) ===> (A ===> B) ===> (A ===> C)) (op \<circ>) (op \<circ>)"
   430   unfolding rel_fun_def by simp
   431 
   432 lemma fun_upd_transfer [transfer_rule]:
   433   assumes [transfer_rule]: "bi_unique A"
   434   shows "((A ===> B) ===> A ===> B ===> A ===> B) fun_upd fun_upd"
   435   unfolding fun_upd_def [abs_def] by transfer_prover
   436 
   437 lemma case_nat_transfer [transfer_rule]:
   438   "(A ===> (op = ===> A) ===> op = ===> A) case_nat case_nat"
   439   unfolding rel_fun_def by (simp split: nat.split)
   440 
   441 lemma rec_nat_transfer [transfer_rule]:
   442   "(A ===> (op = ===> A ===> A) ===> op = ===> A) rec_nat rec_nat"
   443   unfolding rel_fun_def by (clarsimp, rename_tac n, induct_tac n, simp_all)
   444 
   445 lemma funpow_transfer [transfer_rule]:
   446   "(op = ===> (A ===> A) ===> (A ===> A)) compow compow"
   447   unfolding funpow_def by transfer_prover
   448 
   449 lemma mono_transfer[transfer_rule]:
   450   assumes [transfer_rule]: "bi_total A"
   451   assumes [transfer_rule]: "(A ===> A ===> op=) op\<le> op\<le>"
   452   assumes [transfer_rule]: "(B ===> B ===> op=) op\<le> op\<le>"
   453   shows "((A ===> B) ===> op=) mono mono"
   454 unfolding mono_def[abs_def] by transfer_prover
   455 
   456 lemma right_total_relcompp_transfer[transfer_rule]: 
   457   assumes [transfer_rule]: "right_total B"
   458   shows "((A ===> B ===> op=) ===> (B ===> C ===> op=) ===> A ===> C ===> op=) 
   459     (\<lambda>R S x z. \<exists>y\<in>Collect (Domainp B). R x y \<and> S y z) op OO"
   460 unfolding OO_def[abs_def] by transfer_prover
   461 
   462 lemma relcompp_transfer[transfer_rule]: 
   463   assumes [transfer_rule]: "bi_total B"
   464   shows "((A ===> B ===> op=) ===> (B ===> C ===> op=) ===> A ===> C ===> op=) op OO op OO"
   465 unfolding OO_def[abs_def] by transfer_prover
   466 
   467 lemma right_total_Domainp_transfer[transfer_rule]:
   468   assumes [transfer_rule]: "right_total B"
   469   shows "((A ===> B ===> op=) ===> A ===> op=) (\<lambda>T x. \<exists>y\<in>Collect(Domainp B). T x y) Domainp"
   470 apply(subst(2) Domainp_iff[abs_def]) by transfer_prover
   471 
   472 lemma Domainp_transfer[transfer_rule]:
   473   assumes [transfer_rule]: "bi_total B"
   474   shows "((A ===> B ===> op=) ===> A ===> op=) Domainp Domainp"
   475 unfolding Domainp_iff[abs_def] by transfer_prover
   476 
   477 lemma reflp_transfer[transfer_rule]: 
   478   "bi_total A \<Longrightarrow> ((A ===> A ===> op=) ===> op=) reflp reflp"
   479   "right_total A \<Longrightarrow> ((A ===> A ===> implies) ===> implies) reflp reflp"
   480   "right_total A \<Longrightarrow> ((A ===> A ===> op=) ===> implies) reflp reflp"
   481   "bi_total A \<Longrightarrow> ((A ===> A ===> rev_implies) ===> rev_implies) reflp reflp"
   482   "bi_total A \<Longrightarrow> ((A ===> A ===> op=) ===> rev_implies) reflp reflp"
   483 using assms unfolding reflp_def[abs_def] rev_implies_def bi_total_def right_total_def rel_fun_def 
   484 by fast+
   485 
   486 lemma right_unique_transfer [transfer_rule]:
   487   assumes [transfer_rule]: "right_total A"
   488   assumes [transfer_rule]: "right_total B"
   489   assumes [transfer_rule]: "bi_unique B"
   490   shows "((A ===> B ===> op=) ===> implies) right_unique right_unique"
   491 using assms unfolding right_unique_def[abs_def] right_total_def bi_unique_def rel_fun_def
   492 by metis
   493 
   494 end
   495 
   496 end