src/HOL/Transfer.thy
 author blanchet Thu Mar 06 15:40:33 2014 +0100 (2014-03-06) changeset 55945 e96383acecf9 parent 55811 aa1acc25126b child 56085 3d11892ea537 permissions -rw-r--r--
renamed 'fun_rel' to 'rel_fun'
```     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 right_total :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
```
```   139   where "right_total R \<longleftrightarrow> (\<forall>y. \<exists>x. R x y)"
```
```   140
```
```   141 definition right_unique :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
```
```   142   where "right_unique R \<longleftrightarrow> (\<forall>x y z. R x y \<longrightarrow> R x z \<longrightarrow> y = z)"
```
```   143
```
```   144 definition bi_total :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
```
```   145   where "bi_total R \<longleftrightarrow> (\<forall>x. \<exists>y. R x y) \<and> (\<forall>y. \<exists>x. R x y)"
```
```   146
```
```   147 definition bi_unique :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
```
```   148   where "bi_unique R \<longleftrightarrow>
```
```   149     (\<forall>x y z. R x y \<longrightarrow> R x z \<longrightarrow> y = z) \<and>
```
```   150     (\<forall>x y z. R x z \<longrightarrow> R y z \<longrightarrow> x = y)"
```
```   151
```
```   152 lemma bi_uniqueDr: "\<lbrakk> bi_unique A; A x y; A x z \<rbrakk> \<Longrightarrow> y = z"
```
```   153 by(simp add: bi_unique_def)
```
```   154
```
```   155 lemma bi_uniqueDl: "\<lbrakk> bi_unique A; A x y; A z y \<rbrakk> \<Longrightarrow> x = z"
```
```   156 by(simp add: bi_unique_def)
```
```   157
```
```   158 lemma right_uniqueI: "(\<And>x y z. \<lbrakk> A x y; A x z \<rbrakk> \<Longrightarrow> y = z) \<Longrightarrow> right_unique A"
```
```   159 unfolding right_unique_def by blast
```
```   160
```
```   161 lemma right_uniqueD: "\<lbrakk> right_unique A; A x y; A x z \<rbrakk> \<Longrightarrow> y = z"
```
```   162 unfolding right_unique_def by blast
```
```   163
```
```   164 lemma right_total_alt_def:
```
```   165   "right_total R \<longleftrightarrow> ((R ===> op \<longrightarrow>) ===> op \<longrightarrow>) All All"
```
```   166   unfolding right_total_def rel_fun_def
```
```   167   apply (rule iffI, fast)
```
```   168   apply (rule allI)
```
```   169   apply (drule_tac x="\<lambda>x. True" in spec)
```
```   170   apply (drule_tac x="\<lambda>y. \<exists>x. R x y" in spec)
```
```   171   apply fast
```
```   172   done
```
```   173
```
```   174 lemma right_unique_alt_def:
```
```   175   "right_unique R \<longleftrightarrow> (R ===> R ===> op \<longrightarrow>) (op =) (op =)"
```
```   176   unfolding right_unique_def rel_fun_def by auto
```
```   177
```
```   178 lemma bi_total_alt_def:
```
```   179   "bi_total R \<longleftrightarrow> ((R ===> op =) ===> op =) All All"
```
```   180   unfolding bi_total_def rel_fun_def
```
```   181   apply (rule iffI, fast)
```
```   182   apply safe
```
```   183   apply (drule_tac x="\<lambda>x. \<exists>y. R x y" in spec)
```
```   184   apply (drule_tac x="\<lambda>y. True" in spec)
```
```   185   apply fast
```
```   186   apply (drule_tac x="\<lambda>x. True" in spec)
```
```   187   apply (drule_tac x="\<lambda>y. \<exists>x. R x y" in spec)
```
```   188   apply fast
```
```   189   done
```
```   190
```
```   191 lemma bi_unique_alt_def:
```
```   192   "bi_unique R \<longleftrightarrow> (R ===> R ===> op =) (op =) (op =)"
```
```   193   unfolding bi_unique_def rel_fun_def by auto
```
```   194
```
```   195 lemma bi_unique_conversep [simp]: "bi_unique R\<inverse>\<inverse> = bi_unique R"
```
```   196 by(auto simp add: bi_unique_def)
```
```   197
```
```   198 lemma bi_total_conversep [simp]: "bi_total R\<inverse>\<inverse> = bi_total R"
```
```   199 by(auto simp add: bi_total_def)
```
```   200
```
```   201 text {* Properties are preserved by relation composition. *}
```
```   202
```
```   203 lemma OO_def: "R OO S = (\<lambda>x z. \<exists>y. R x y \<and> S y z)"
```
```   204   by auto
```
```   205
```
```   206 lemma bi_total_OO: "\<lbrakk>bi_total A; bi_total B\<rbrakk> \<Longrightarrow> bi_total (A OO B)"
```
```   207   unfolding bi_total_def OO_def by metis
```
```   208
```
```   209 lemma bi_unique_OO: "\<lbrakk>bi_unique A; bi_unique B\<rbrakk> \<Longrightarrow> bi_unique (A OO B)"
```
```   210   unfolding bi_unique_def OO_def by metis
```
```   211
```
```   212 lemma right_total_OO:
```
```   213   "\<lbrakk>right_total A; right_total B\<rbrakk> \<Longrightarrow> right_total (A OO B)"
```
```   214   unfolding right_total_def OO_def by metis
```
```   215
```
```   216 lemma right_unique_OO:
```
```   217   "\<lbrakk>right_unique A; right_unique B\<rbrakk> \<Longrightarrow> right_unique (A OO B)"
```
```   218   unfolding right_unique_def OO_def by metis
```
```   219
```
```   220
```
```   221 subsection {* Properties of relators *}
```
```   222
```
```   223 lemma right_total_eq [transfer_rule]: "right_total (op =)"
```
```   224   unfolding right_total_def by simp
```
```   225
```
```   226 lemma right_unique_eq [transfer_rule]: "right_unique (op =)"
```
```   227   unfolding right_unique_def by simp
```
```   228
```
```   229 lemma bi_total_eq [transfer_rule]: "bi_total (op =)"
```
```   230   unfolding bi_total_def by simp
```
```   231
```
```   232 lemma bi_unique_eq [transfer_rule]: "bi_unique (op =)"
```
```   233   unfolding bi_unique_def by simp
```
```   234
```
```   235 lemma right_total_fun [transfer_rule]:
```
```   236   "\<lbrakk>right_unique A; right_total B\<rbrakk> \<Longrightarrow> right_total (A ===> B)"
```
```   237   unfolding right_total_def rel_fun_def
```
```   238   apply (rule allI, rename_tac g)
```
```   239   apply (rule_tac x="\<lambda>x. SOME z. B z (g (THE y. A x y))" in exI)
```
```   240   apply clarify
```
```   241   apply (subgoal_tac "(THE y. A x y) = y", simp)
```
```   242   apply (rule someI_ex)
```
```   243   apply (simp)
```
```   244   apply (rule the_equality)
```
```   245   apply assumption
```
```   246   apply (simp add: right_unique_def)
```
```   247   done
```
```   248
```
```   249 lemma right_unique_fun [transfer_rule]:
```
```   250   "\<lbrakk>right_total A; right_unique B\<rbrakk> \<Longrightarrow> right_unique (A ===> B)"
```
```   251   unfolding right_total_def right_unique_def rel_fun_def
```
```   252   by (clarify, rule ext, fast)
```
```   253
```
```   254 lemma bi_total_fun [transfer_rule]:
```
```   255   "\<lbrakk>bi_unique A; bi_total B\<rbrakk> \<Longrightarrow> bi_total (A ===> B)"
```
```   256   unfolding bi_total_def rel_fun_def
```
```   257   apply safe
```
```   258   apply (rename_tac f)
```
```   259   apply (rule_tac x="\<lambda>y. SOME z. B (f (THE x. A x y)) z" in exI)
```
```   260   apply clarify
```
```   261   apply (subgoal_tac "(THE x. A x y) = x", simp)
```
```   262   apply (rule someI_ex)
```
```   263   apply (simp)
```
```   264   apply (rule the_equality)
```
```   265   apply assumption
```
```   266   apply (simp add: bi_unique_def)
```
```   267   apply (rename_tac g)
```
```   268   apply (rule_tac x="\<lambda>x. SOME z. B z (g (THE y. A x y))" in exI)
```
```   269   apply clarify
```
```   270   apply (subgoal_tac "(THE y. A x y) = y", simp)
```
```   271   apply (rule someI_ex)
```
```   272   apply (simp)
```
```   273   apply (rule the_equality)
```
```   274   apply assumption
```
```   275   apply (simp add: bi_unique_def)
```
```   276   done
```
```   277
```
```   278 lemma bi_unique_fun [transfer_rule]:
```
```   279   "\<lbrakk>bi_total A; bi_unique B\<rbrakk> \<Longrightarrow> bi_unique (A ===> B)"
```
```   280   unfolding bi_total_def bi_unique_def rel_fun_def fun_eq_iff
```
```   281   by (safe, metis, fast)
```
```   282
```
```   283
```
```   284 subsection {* Transfer rules *}
```
```   285
```
```   286 lemma Domainp_forall_transfer [transfer_rule]:
```
```   287   assumes "right_total A"
```
```   288   shows "((A ===> op =) ===> op =)
```
```   289     (transfer_bforall (Domainp A)) transfer_forall"
```
```   290   using assms unfolding right_total_def
```
```   291   unfolding transfer_forall_def transfer_bforall_def rel_fun_def Domainp_iff
```
```   292   by metis
```
```   293
```
```   294 text {* Transfer rules using implication instead of equality on booleans. *}
```
```   295
```
```   296 lemma transfer_forall_transfer [transfer_rule]:
```
```   297   "bi_total A \<Longrightarrow> ((A ===> op =) ===> op =) transfer_forall transfer_forall"
```
```   298   "right_total A \<Longrightarrow> ((A ===> op =) ===> implies) transfer_forall transfer_forall"
```
```   299   "right_total A \<Longrightarrow> ((A ===> implies) ===> implies) transfer_forall transfer_forall"
```
```   300   "bi_total A \<Longrightarrow> ((A ===> op =) ===> rev_implies) transfer_forall transfer_forall"
```
```   301   "bi_total A \<Longrightarrow> ((A ===> rev_implies) ===> rev_implies) transfer_forall transfer_forall"
```
```   302   unfolding transfer_forall_def rev_implies_def rel_fun_def right_total_def bi_total_def
```
```   303   by metis+
```
```   304
```
```   305 lemma transfer_implies_transfer [transfer_rule]:
```
```   306   "(op =        ===> op =        ===> op =       ) transfer_implies transfer_implies"
```
```   307   "(rev_implies ===> implies     ===> implies    ) transfer_implies transfer_implies"
```
```   308   "(rev_implies ===> op =        ===> implies    ) transfer_implies transfer_implies"
```
```   309   "(op =        ===> implies     ===> implies    ) transfer_implies transfer_implies"
```
```   310   "(op =        ===> op =        ===> implies    ) transfer_implies transfer_implies"
```
```   311   "(implies     ===> rev_implies ===> rev_implies) transfer_implies transfer_implies"
```
```   312   "(implies     ===> op =        ===> rev_implies) transfer_implies transfer_implies"
```
```   313   "(op =        ===> rev_implies ===> rev_implies) transfer_implies transfer_implies"
```
```   314   "(op =        ===> op =        ===> rev_implies) transfer_implies transfer_implies"
```
```   315   unfolding transfer_implies_def rev_implies_def rel_fun_def by auto
```
```   316
```
```   317 lemma eq_imp_transfer [transfer_rule]:
```
```   318   "right_unique A \<Longrightarrow> (A ===> A ===> op \<longrightarrow>) (op =) (op =)"
```
```   319   unfolding right_unique_alt_def .
```
```   320
```
```   321 lemma eq_transfer [transfer_rule]:
```
```   322   assumes "bi_unique A"
```
```   323   shows "(A ===> A ===> op =) (op =) (op =)"
```
```   324   using assms unfolding bi_unique_def rel_fun_def by auto
```
```   325
```
```   326 lemma right_total_Ex_transfer[transfer_rule]:
```
```   327   assumes "right_total A"
```
```   328   shows "((A ===> op=) ===> op=) (Bex (Collect (Domainp A))) Ex"
```
```   329 using assms unfolding right_total_def Bex_def rel_fun_def Domainp_iff[abs_def]
```
```   330 by blast
```
```   331
```
```   332 lemma right_total_All_transfer[transfer_rule]:
```
```   333   assumes "right_total A"
```
```   334   shows "((A ===> op =) ===> op =) (Ball (Collect (Domainp A))) All"
```
```   335 using assms unfolding right_total_def Ball_def rel_fun_def Domainp_iff[abs_def]
```
```   336 by blast
```
```   337
```
```   338 lemma All_transfer [transfer_rule]:
```
```   339   assumes "bi_total A"
```
```   340   shows "((A ===> op =) ===> op =) All All"
```
```   341   using assms unfolding bi_total_def rel_fun_def by fast
```
```   342
```
```   343 lemma Ex_transfer [transfer_rule]:
```
```   344   assumes "bi_total A"
```
```   345   shows "((A ===> op =) ===> op =) Ex Ex"
```
```   346   using assms unfolding bi_total_def rel_fun_def by fast
```
```   347
```
```   348 lemma If_transfer [transfer_rule]: "(op = ===> A ===> A ===> A) If If"
```
```   349   unfolding rel_fun_def by simp
```
```   350
```
```   351 lemma Let_transfer [transfer_rule]: "(A ===> (A ===> B) ===> B) Let Let"
```
```   352   unfolding rel_fun_def by simp
```
```   353
```
```   354 lemma id_transfer [transfer_rule]: "(A ===> A) id id"
```
```   355   unfolding rel_fun_def by simp
```
```   356
```
```   357 lemma comp_transfer [transfer_rule]:
```
```   358   "((B ===> C) ===> (A ===> B) ===> (A ===> C)) (op \<circ>) (op \<circ>)"
```
```   359   unfolding rel_fun_def by simp
```
```   360
```
```   361 lemma fun_upd_transfer [transfer_rule]:
```
```   362   assumes [transfer_rule]: "bi_unique A"
```
```   363   shows "((A ===> B) ===> A ===> B ===> A ===> B) fun_upd fun_upd"
```
```   364   unfolding fun_upd_def [abs_def] by transfer_prover
```
```   365
```
```   366 lemma case_nat_transfer [transfer_rule]:
```
```   367   "(A ===> (op = ===> A) ===> op = ===> A) case_nat case_nat"
```
```   368   unfolding rel_fun_def by (simp split: nat.split)
```
```   369
```
```   370 lemma rec_nat_transfer [transfer_rule]:
```
```   371   "(A ===> (op = ===> A ===> A) ===> op = ===> A) rec_nat rec_nat"
```
```   372   unfolding rel_fun_def by (clarsimp, rename_tac n, induct_tac n, simp_all)
```
```   373
```
```   374 lemma funpow_transfer [transfer_rule]:
```
```   375   "(op = ===> (A ===> A) ===> (A ===> A)) compow compow"
```
```   376   unfolding funpow_def by transfer_prover
```
```   377
```
```   378 lemma mono_transfer[transfer_rule]:
```
```   379   assumes [transfer_rule]: "bi_total A"
```
```   380   assumes [transfer_rule]: "(A ===> A ===> op=) op\<le> op\<le>"
```
```   381   assumes [transfer_rule]: "(B ===> B ===> op=) op\<le> op\<le>"
```
```   382   shows "((A ===> B) ===> op=) mono mono"
```
```   383 unfolding mono_def[abs_def] by transfer_prover
```
```   384
```
```   385 lemma right_total_relcompp_transfer[transfer_rule]:
```
```   386   assumes [transfer_rule]: "right_total B"
```
```   387   shows "((A ===> B ===> op=) ===> (B ===> C ===> op=) ===> A ===> C ===> op=)
```
```   388     (\<lambda>R S x z. \<exists>y\<in>Collect (Domainp B). R x y \<and> S y z) op OO"
```
```   389 unfolding OO_def[abs_def] by transfer_prover
```
```   390
```
```   391 lemma relcompp_transfer[transfer_rule]:
```
```   392   assumes [transfer_rule]: "bi_total B"
```
```   393   shows "((A ===> B ===> op=) ===> (B ===> C ===> op=) ===> A ===> C ===> op=) op OO op OO"
```
```   394 unfolding OO_def[abs_def] by transfer_prover
```
```   395
```
```   396 lemma right_total_Domainp_transfer[transfer_rule]:
```
```   397   assumes [transfer_rule]: "right_total B"
```
```   398   shows "((A ===> B ===> op=) ===> A ===> op=) (\<lambda>T x. \<exists>y\<in>Collect(Domainp B). T x y) Domainp"
```
```   399 apply(subst(2) Domainp_iff[abs_def]) by transfer_prover
```
```   400
```
```   401 lemma Domainp_transfer[transfer_rule]:
```
```   402   assumes [transfer_rule]: "bi_total B"
```
```   403   shows "((A ===> B ===> op=) ===> A ===> op=) Domainp Domainp"
```
```   404 unfolding Domainp_iff[abs_def] by transfer_prover
```
```   405
```
```   406 lemma reflp_transfer[transfer_rule]:
```
```   407   "bi_total A \<Longrightarrow> ((A ===> A ===> op=) ===> op=) reflp reflp"
```
```   408   "right_total A \<Longrightarrow> ((A ===> A ===> implies) ===> implies) reflp reflp"
```
```   409   "right_total A \<Longrightarrow> ((A ===> A ===> op=) ===> implies) reflp reflp"
```
```   410   "bi_total A \<Longrightarrow> ((A ===> A ===> rev_implies) ===> rev_implies) reflp reflp"
```
```   411   "bi_total A \<Longrightarrow> ((A ===> A ===> op=) ===> rev_implies) reflp reflp"
```
```   412 using assms unfolding reflp_def[abs_def] rev_implies_def bi_total_def right_total_def rel_fun_def
```
```   413 by fast+
```
```   414
```
```   415 lemma right_unique_transfer [transfer_rule]:
```
```   416   assumes [transfer_rule]: "right_total A"
```
```   417   assumes [transfer_rule]: "right_total B"
```
```   418   assumes [transfer_rule]: "bi_unique B"
```
```   419   shows "((A ===> B ===> op=) ===> implies) right_unique right_unique"
```
```   420 using assms unfolding right_unique_def[abs_def] right_total_def bi_unique_def rel_fun_def
```
```   421 by metis
```
```   422
```
```   423 end
```
```   424
```
```   425 end
```