src/HOL/Transfer.thy
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```     1 (*  Title:      HOL/Transfer.thy
```
```     2     Author:     Brian Huffman, TU Muenchen
```
```     3     Author:     Ondrej Kuncar, TU Muenchen
```
```     4 *)
```
```     5
```
```     6 section \<open>Generic theorem transfer using relations\<close>
```
```     7
```
```     8 theory Transfer
```
```     9 imports Basic_BNF_LFPs Hilbert_Choice Metis
```
```    10 begin
```
```    11
```
```    12 subsection \<open>Relator for function space\<close>
```
```    13
```
```    14 bundle lifting_syntax
```
```    15 begin
```
```    16   notation rel_fun  (infixr "===>" 55)
```
```    17   notation map_fun  (infixr "--->" 55)
```
```    18 end
```
```    19
```
```    20 context includes lifting_syntax
```
```    21 begin
```
```    22
```
```    23 lemma rel_funD2:
```
```    24   assumes "rel_fun A B f g" and "A x x"
```
```    25   shows "B (f x) (g x)"
```
```    26   using assms by (rule rel_funD)
```
```    27
```
```    28 lemma rel_funE:
```
```    29   assumes "rel_fun A B f g" and "A x y"
```
```    30   obtains "B (f x) (g y)"
```
```    31   using assms by (simp add: rel_fun_def)
```
```    32
```
```    33 lemmas rel_fun_eq = fun.rel_eq
```
```    34
```
```    35 lemma rel_fun_eq_rel:
```
```    36 shows "rel_fun (=) R = (\<lambda>f g. \<forall>x. R (f x) (g x))"
```
```    37   by (simp add: rel_fun_def)
```
```    38
```
```    39
```
```    40 subsection \<open>Transfer method\<close>
```
```    41
```
```    42 text \<open>Explicit tag for relation membership allows for
```
```    43   backward proof methods.\<close>
```
```    44
```
```    45 definition Rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"
```
```    46   where "Rel r \<equiv> r"
```
```    47
```
```    48 text \<open>Handling of equality relations\<close>
```
```    49
```
```    50 definition is_equality :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
```
```    51   where "is_equality R \<longleftrightarrow> R = (=)"
```
```    52
```
```    53 lemma is_equality_eq: "is_equality (=)"
```
```    54   unfolding is_equality_def by simp
```
```    55
```
```    56 text \<open>Reverse implication for monotonicity rules\<close>
```
```    57
```
```    58 definition rev_implies where
```
```    59   "rev_implies x y \<longleftrightarrow> (y \<longrightarrow> x)"
```
```    60
```
```    61 text \<open>Handling of meta-logic connectives\<close>
```
```    62
```
```    63 definition transfer_forall where
```
```    64   "transfer_forall \<equiv> All"
```
```    65
```
```    66 definition transfer_implies where
```
```    67   "transfer_implies \<equiv> (\<longrightarrow>)"
```
```    68
```
```    69 definition transfer_bforall :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
```
```    70   where "transfer_bforall \<equiv> (\<lambda>P Q. \<forall>x. P x \<longrightarrow> Q x)"
```
```    71
```
```    72 lemma transfer_forall_eq: "(\<And>x. P x) \<equiv> Trueprop (transfer_forall (\<lambda>x. P x))"
```
```    73   unfolding atomize_all transfer_forall_def ..
```
```    74
```
```    75 lemma transfer_implies_eq: "(A \<Longrightarrow> B) \<equiv> Trueprop (transfer_implies A B)"
```
```    76   unfolding atomize_imp transfer_implies_def ..
```
```    77
```
```    78 lemma transfer_bforall_unfold:
```
```    79   "Trueprop (transfer_bforall P (\<lambda>x. Q x)) \<equiv> (\<And>x. P x \<Longrightarrow> Q x)"
```
```    80   unfolding transfer_bforall_def atomize_imp atomize_all ..
```
```    81
```
```    82 lemma transfer_start: "\<lbrakk>P; Rel (=) P Q\<rbrakk> \<Longrightarrow> Q"
```
```    83   unfolding Rel_def by simp
```
```    84
```
```    85 lemma transfer_start': "\<lbrakk>P; Rel (\<longrightarrow>) P Q\<rbrakk> \<Longrightarrow> Q"
```
```    86   unfolding Rel_def by simp
```
```    87
```
```    88 lemma transfer_prover_start: "\<lbrakk>x = x'; Rel R x' y\<rbrakk> \<Longrightarrow> Rel R x y"
```
```    89   by simp
```
```    90
```
```    91 lemma untransfer_start: "\<lbrakk>Q; Rel (=) P Q\<rbrakk> \<Longrightarrow> P"
```
```    92   unfolding Rel_def by simp
```
```    93
```
```    94 lemma Rel_eq_refl: "Rel (=) x x"
```
```    95   unfolding Rel_def ..
```
```    96
```
```    97 lemma Rel_app:
```
```    98   assumes "Rel (A ===> B) f g" and "Rel A x y"
```
```    99   shows "Rel B (f x) (g y)"
```
```   100   using assms unfolding Rel_def rel_fun_def by fast
```
```   101
```
```   102 lemma Rel_abs:
```
```   103   assumes "\<And>x y. Rel A x y \<Longrightarrow> Rel B (f x) (g y)"
```
```   104   shows "Rel (A ===> B) (\<lambda>x. f x) (\<lambda>y. g y)"
```
```   105   using assms unfolding Rel_def rel_fun_def by fast
```
```   106
```
```   107 subsection \<open>Predicates on relations, i.e. ``class constraints''\<close>
```
```   108
```
```   109 definition left_total :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
```
```   110   where "left_total R \<longleftrightarrow> (\<forall>x. \<exists>y. R x y)"
```
```   111
```
```   112 definition left_unique :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
```
```   113   where "left_unique R \<longleftrightarrow> (\<forall>x y z. R x z \<longrightarrow> R y z \<longrightarrow> x = y)"
```
```   114
```
```   115 definition right_total :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
```
```   116   where "right_total R \<longleftrightarrow> (\<forall>y. \<exists>x. R x y)"
```
```   117
```
```   118 definition right_unique :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
```
```   119   where "right_unique R \<longleftrightarrow> (\<forall>x y z. R x y \<longrightarrow> R x z \<longrightarrow> y = z)"
```
```   120
```
```   121 definition bi_total :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
```
```   122   where "bi_total R \<longleftrightarrow> (\<forall>x. \<exists>y. R x y) \<and> (\<forall>y. \<exists>x. R x y)"
```
```   123
```
```   124 definition bi_unique :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
```
```   125   where "bi_unique R \<longleftrightarrow>
```
```   126     (\<forall>x y z. R x y \<longrightarrow> R x z \<longrightarrow> y = z) \<and>
```
```   127     (\<forall>x y z. R x z \<longrightarrow> R y z \<longrightarrow> x = y)"
```
```   128
```
```   129 lemma left_uniqueI: "(\<And>x y z. \<lbrakk> A x z; A y z \<rbrakk> \<Longrightarrow> x = y) \<Longrightarrow> left_unique A"
```
```   130 unfolding left_unique_def by blast
```
```   131
```
```   132 lemma left_uniqueD: "\<lbrakk> left_unique A; A x z; A y z \<rbrakk> \<Longrightarrow> x = y"
```
```   133 unfolding left_unique_def by blast
```
```   134
```
```   135 lemma left_totalI:
```
```   136   "(\<And>x. \<exists>y. R x y) \<Longrightarrow> left_total R"
```
```   137 unfolding left_total_def by blast
```
```   138
```
```   139 lemma left_totalE:
```
```   140   assumes "left_total R"
```
```   141   obtains "(\<And>x. \<exists>y. R x y)"
```
```   142 using assms unfolding left_total_def by blast
```
```   143
```
```   144 lemma bi_uniqueDr: "\<lbrakk> bi_unique A; A x y; A x z \<rbrakk> \<Longrightarrow> y = z"
```
```   145 by(simp add: bi_unique_def)
```
```   146
```
```   147 lemma bi_uniqueDl: "\<lbrakk> bi_unique A; A x y; A z y \<rbrakk> \<Longrightarrow> x = z"
```
```   148 by(simp add: bi_unique_def)
```
```   149
```
```   150 lemma right_uniqueI: "(\<And>x y z. \<lbrakk> A x y; A x z \<rbrakk> \<Longrightarrow> y = z) \<Longrightarrow> right_unique A"
```
```   151 unfolding right_unique_def by fast
```
```   152
```
```   153 lemma right_uniqueD: "\<lbrakk> right_unique A; A x y; A x z \<rbrakk> \<Longrightarrow> y = z"
```
```   154 unfolding right_unique_def by fast
```
```   155
```
```   156 lemma right_totalI: "(\<And>y. \<exists>x. A x y) \<Longrightarrow> right_total A"
```
```   157 by(simp add: right_total_def)
```
```   158
```
```   159 lemma right_totalE:
```
```   160   assumes "right_total A"
```
```   161   obtains x where "A x y"
```
```   162 using assms by(auto simp add: right_total_def)
```
```   163
```
```   164 lemma right_total_alt_def2:
```
```   165   "right_total R \<longleftrightarrow> ((R ===> (\<longrightarrow>)) ===> (\<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_def2:
```
```   175   "right_unique R \<longleftrightarrow> (R ===> R ===> (\<longrightarrow>)) (=) (=)"
```
```   176   unfolding right_unique_def rel_fun_def by auto
```
```   177
```
```   178 lemma bi_total_alt_def2:
```
```   179   "bi_total R \<longleftrightarrow> ((R ===> (=)) ===> (=)) 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_def2:
```
```   192   "bi_unique R \<longleftrightarrow> (R ===> R ===> (=)) (=) (=)"
```
```   193   unfolding bi_unique_def rel_fun_def by auto
```
```   194
```
```   195 lemma [simp]:
```
```   196   shows left_unique_conversep: "left_unique A\<inverse>\<inverse> \<longleftrightarrow> right_unique A"
```
```   197   and right_unique_conversep: "right_unique A\<inverse>\<inverse> \<longleftrightarrow> left_unique A"
```
```   198 by(auto simp add: left_unique_def right_unique_def)
```
```   199
```
```   200 lemma [simp]:
```
```   201   shows left_total_conversep: "left_total A\<inverse>\<inverse> \<longleftrightarrow> right_total A"
```
```   202   and right_total_conversep: "right_total A\<inverse>\<inverse> \<longleftrightarrow> left_total A"
```
```   203 by(simp_all add: left_total_def right_total_def)
```
```   204
```
```   205 lemma bi_unique_conversep [simp]: "bi_unique R\<inverse>\<inverse> = bi_unique R"
```
```   206 by(auto simp add: bi_unique_def)
```
```   207
```
```   208 lemma bi_total_conversep [simp]: "bi_total R\<inverse>\<inverse> = bi_total R"
```
```   209 by(auto simp add: bi_total_def)
```
```   210
```
```   211 lemma right_unique_alt_def: "right_unique R = (conversep R OO R \<le> (=))" unfolding right_unique_def by blast
```
```   212 lemma left_unique_alt_def: "left_unique R = (R OO (conversep R) \<le> (=))" unfolding left_unique_def by blast
```
```   213
```
```   214 lemma right_total_alt_def: "right_total R = (conversep R OO R \<ge> (=))" unfolding right_total_def by blast
```
```   215 lemma left_total_alt_def: "left_total R = (R OO conversep R \<ge> (=))" unfolding left_total_def by blast
```
```   216
```
```   217 lemma bi_total_alt_def: "bi_total A = (left_total A \<and> right_total A)"
```
```   218 unfolding left_total_def right_total_def bi_total_def by blast
```
```   219
```
```   220 lemma bi_unique_alt_def: "bi_unique A = (left_unique A \<and> right_unique A)"
```
```   221 unfolding left_unique_def right_unique_def bi_unique_def by blast
```
```   222
```
```   223 lemma bi_totalI: "left_total R \<Longrightarrow> right_total R \<Longrightarrow> bi_total R"
```
```   224 unfolding bi_total_alt_def ..
```
```   225
```
```   226 lemma bi_uniqueI: "left_unique R \<Longrightarrow> right_unique R \<Longrightarrow> bi_unique R"
```
```   227 unfolding bi_unique_alt_def ..
```
```   228
```
```   229 end
```
```   230
```
```   231
```
```   232
```
```   233 ML_file "Tools/Transfer/transfer.ML"
```
```   234 declare refl [transfer_rule]
```
```   235
```
```   236 hide_const (open) Rel
```
```   237
```
```   238 context includes lifting_syntax
```
```   239 begin
```
```   240
```
```   241 text \<open>Handling of domains\<close>
```
```   242
```
```   243 lemma Domainp_iff: "Domainp T x \<longleftrightarrow> (\<exists>y. T x y)"
```
```   244   by auto
```
```   245
```
```   246 lemma Domainp_refl[transfer_domain_rule]:
```
```   247   "Domainp T = Domainp T" ..
```
```   248
```
```   249 lemma Domain_eq_top[transfer_domain_rule]: "Domainp (=) = top" by auto
```
```   250
```
```   251 lemma Domainp_pred_fun_eq[relator_domain]:
```
```   252   assumes "left_unique T"
```
```   253   shows "Domainp (T ===> S) = pred_fun (Domainp T) (Domainp S)"
```
```   254   using assms unfolding rel_fun_def Domainp_iff[abs_def] left_unique_def fun_eq_iff pred_fun_def
```
```   255   apply safe
```
```   256    apply blast
```
```   257   apply (subst all_comm)
```
```   258   apply (rule choice)
```
```   259   apply blast
```
```   260   done
```
```   261
```
```   262 text \<open>Properties are preserved by relation composition.\<close>
```
```   263
```
```   264 lemma OO_def: "R OO S = (\<lambda>x z. \<exists>y. R x y \<and> S y z)"
```
```   265   by auto
```
```   266
```
```   267 lemma bi_total_OO: "\<lbrakk>bi_total A; bi_total B\<rbrakk> \<Longrightarrow> bi_total (A OO B)"
```
```   268   unfolding bi_total_def OO_def by fast
```
```   269
```
```   270 lemma bi_unique_OO: "\<lbrakk>bi_unique A; bi_unique B\<rbrakk> \<Longrightarrow> bi_unique (A OO B)"
```
```   271   unfolding bi_unique_def OO_def by blast
```
```   272
```
```   273 lemma right_total_OO:
```
```   274   "\<lbrakk>right_total A; right_total B\<rbrakk> \<Longrightarrow> right_total (A OO B)"
```
```   275   unfolding right_total_def OO_def by fast
```
```   276
```
```   277 lemma right_unique_OO:
```
```   278   "\<lbrakk>right_unique A; right_unique B\<rbrakk> \<Longrightarrow> right_unique (A OO B)"
```
```   279   unfolding right_unique_def OO_def by fast
```
```   280
```
```   281 lemma left_total_OO: "left_total R \<Longrightarrow> left_total S \<Longrightarrow> left_total (R OO S)"
```
```   282 unfolding left_total_def OO_def by fast
```
```   283
```
```   284 lemma left_unique_OO: "left_unique R \<Longrightarrow> left_unique S \<Longrightarrow> left_unique (R OO S)"
```
```   285 unfolding left_unique_def OO_def by blast
```
```   286
```
```   287
```
```   288 subsection \<open>Properties of relators\<close>
```
```   289
```
```   290 lemma left_total_eq[transfer_rule]: "left_total (=)"
```
```   291   unfolding left_total_def by blast
```
```   292
```
```   293 lemma left_unique_eq[transfer_rule]: "left_unique (=)"
```
```   294   unfolding left_unique_def by blast
```
```   295
```
```   296 lemma right_total_eq [transfer_rule]: "right_total (=)"
```
```   297   unfolding right_total_def by simp
```
```   298
```
```   299 lemma right_unique_eq [transfer_rule]: "right_unique (=)"
```
```   300   unfolding right_unique_def by simp
```
```   301
```
```   302 lemma bi_total_eq[transfer_rule]: "bi_total (=)"
```
```   303   unfolding bi_total_def by simp
```
```   304
```
```   305 lemma bi_unique_eq[transfer_rule]: "bi_unique (=)"
```
```   306   unfolding bi_unique_def by simp
```
```   307
```
```   308 lemma left_total_fun[transfer_rule]:
```
```   309   "\<lbrakk>left_unique A; left_total B\<rbrakk> \<Longrightarrow> left_total (A ===> B)"
```
```   310   unfolding left_total_def rel_fun_def
```
```   311   apply (rule allI, rename_tac f)
```
```   312   apply (rule_tac x="\<lambda>y. SOME z. B (f (THE x. A x y)) z" in exI)
```
```   313   apply clarify
```
```   314   apply (subgoal_tac "(THE x. A x y) = x", simp)
```
```   315   apply (rule someI_ex)
```
```   316   apply (simp)
```
```   317   apply (rule the_equality)
```
```   318   apply assumption
```
```   319   apply (simp add: left_unique_def)
```
```   320   done
```
```   321
```
```   322 lemma left_unique_fun[transfer_rule]:
```
```   323   "\<lbrakk>left_total A; left_unique B\<rbrakk> \<Longrightarrow> left_unique (A ===> B)"
```
```   324   unfolding left_total_def left_unique_def rel_fun_def
```
```   325   by (clarify, rule ext, fast)
```
```   326
```
```   327 lemma right_total_fun [transfer_rule]:
```
```   328   "\<lbrakk>right_unique A; right_total B\<rbrakk> \<Longrightarrow> right_total (A ===> B)"
```
```   329   unfolding right_total_def rel_fun_def
```
```   330   apply (rule allI, rename_tac g)
```
```   331   apply (rule_tac x="\<lambda>x. SOME z. B z (g (THE y. A x y))" in exI)
```
```   332   apply clarify
```
```   333   apply (subgoal_tac "(THE y. A x y) = y", simp)
```
```   334   apply (rule someI_ex)
```
```   335   apply (simp)
```
```   336   apply (rule the_equality)
```
```   337   apply assumption
```
```   338   apply (simp add: right_unique_def)
```
```   339   done
```
```   340
```
```   341 lemma right_unique_fun [transfer_rule]:
```
```   342   "\<lbrakk>right_total A; right_unique B\<rbrakk> \<Longrightarrow> right_unique (A ===> B)"
```
```   343   unfolding right_total_def right_unique_def rel_fun_def
```
```   344   by (clarify, rule ext, fast)
```
```   345
```
```   346 lemma bi_total_fun[transfer_rule]:
```
```   347   "\<lbrakk>bi_unique A; bi_total B\<rbrakk> \<Longrightarrow> bi_total (A ===> B)"
```
```   348   unfolding bi_unique_alt_def bi_total_alt_def
```
```   349   by (blast intro: right_total_fun left_total_fun)
```
```   350
```
```   351 lemma bi_unique_fun[transfer_rule]:
```
```   352   "\<lbrakk>bi_total A; bi_unique B\<rbrakk> \<Longrightarrow> bi_unique (A ===> B)"
```
```   353   unfolding bi_unique_alt_def bi_total_alt_def
```
```   354   by (blast intro: right_unique_fun left_unique_fun)
```
```   355
```
```   356 end
```
```   357
```
```   358 lemma if_conn:
```
```   359   "(if P \<and> Q then t else e) = (if P then if Q then t else e else e)"
```
```   360   "(if P \<or> Q then t else e) = (if P then t else if Q then t else e)"
```
```   361   "(if P \<longrightarrow> Q then t else e) = (if P then if Q then t else e else t)"
```
```   362   "(if \<not> P then t else e) = (if P then e else t)"
```
```   363 by auto
```
```   364
```
```   365 ML_file "Tools/Transfer/transfer_bnf.ML"
```
```   366 ML_file "Tools/BNF/bnf_fp_rec_sugar_transfer.ML"
```
```   367
```
```   368 declare pred_fun_def [simp]
```
```   369 declare rel_fun_eq [relator_eq]
```
```   370
```
```   371 (* Delete the automated generated rule from the bnf command;
```
```   372   we have a more general rule (Domainp_pred_fun_eq) that subsumes it. *)
```
```   373 declare fun.Domainp_rel[relator_domain del]
```
```   374
```
```   375 subsection \<open>Transfer rules\<close>
```
```   376
```
```   377 context includes lifting_syntax
```
```   378 begin
```
```   379
```
```   380 lemma Domainp_forall_transfer [transfer_rule]:
```
```   381   assumes "right_total A"
```
```   382   shows "((A ===> (=)) ===> (=))
```
```   383     (transfer_bforall (Domainp A)) transfer_forall"
```
```   384   using assms unfolding right_total_def
```
```   385   unfolding transfer_forall_def transfer_bforall_def rel_fun_def Domainp_iff
```
```   386   by fast
```
```   387
```
```   388 text \<open>Transfer rules using implication instead of equality on booleans.\<close>
```
```   389
```
```   390 lemma transfer_forall_transfer [transfer_rule]:
```
```   391   "bi_total A \<Longrightarrow> ((A ===> (=)) ===> (=)) transfer_forall transfer_forall"
```
```   392   "right_total A \<Longrightarrow> ((A ===> (=)) ===> implies) transfer_forall transfer_forall"
```
```   393   "right_total A \<Longrightarrow> ((A ===> implies) ===> implies) transfer_forall transfer_forall"
```
```   394   "bi_total A \<Longrightarrow> ((A ===> (=)) ===> rev_implies) transfer_forall transfer_forall"
```
```   395   "bi_total A \<Longrightarrow> ((A ===> rev_implies) ===> rev_implies) transfer_forall transfer_forall"
```
```   396   unfolding transfer_forall_def rev_implies_def rel_fun_def right_total_def bi_total_def
```
```   397   by fast+
```
```   398
```
```   399 lemma transfer_implies_transfer [transfer_rule]:
```
```   400   "((=)        ===> (=)        ===> (=)       ) transfer_implies transfer_implies"
```
```   401   "(rev_implies ===> implies     ===> implies    ) transfer_implies transfer_implies"
```
```   402   "(rev_implies ===> (=)        ===> implies    ) transfer_implies transfer_implies"
```
```   403   "((=)        ===> implies     ===> implies    ) transfer_implies transfer_implies"
```
```   404   "((=)        ===> (=)        ===> implies    ) transfer_implies transfer_implies"
```
```   405   "(implies     ===> rev_implies ===> rev_implies) transfer_implies transfer_implies"
```
```   406   "(implies     ===> (=)        ===> rev_implies) transfer_implies transfer_implies"
```
```   407   "((=)        ===> rev_implies ===> rev_implies) transfer_implies transfer_implies"
```
```   408   "((=)        ===> (=)        ===> rev_implies) transfer_implies transfer_implies"
```
```   409   unfolding transfer_implies_def rev_implies_def rel_fun_def by auto
```
```   410
```
```   411 lemma eq_imp_transfer [transfer_rule]:
```
```   412   "right_unique A \<Longrightarrow> (A ===> A ===> (\<longrightarrow>)) (=) (=)"
```
```   413   unfolding right_unique_alt_def2 .
```
```   414
```
```   415 text \<open>Transfer rules using equality.\<close>
```
```   416
```
```   417 lemma left_unique_transfer [transfer_rule]:
```
```   418   assumes "right_total A"
```
```   419   assumes "right_total B"
```
```   420   assumes "bi_unique A"
```
```   421   shows "((A ===> B ===> (=)) ===> implies) left_unique left_unique"
```
```   422 using assms unfolding left_unique_def[abs_def] right_total_def bi_unique_def rel_fun_def
```
```   423 by metis
```
```   424
```
```   425 lemma eq_transfer [transfer_rule]:
```
```   426   assumes "bi_unique A"
```
```   427   shows "(A ===> A ===> (=)) (=) (=)"
```
```   428   using assms unfolding bi_unique_def rel_fun_def by auto
```
```   429
```
```   430 lemma right_total_Ex_transfer[transfer_rule]:
```
```   431   assumes "right_total A"
```
```   432   shows "((A ===> (=)) ===> (=)) (Bex (Collect (Domainp A))) Ex"
```
```   433 using assms unfolding right_total_def Bex_def rel_fun_def Domainp_iff[abs_def]
```
```   434 by fast
```
```   435
```
```   436 lemma right_total_All_transfer[transfer_rule]:
```
```   437   assumes "right_total A"
```
```   438   shows "((A ===> (=)) ===> (=)) (Ball (Collect (Domainp A))) All"
```
```   439 using assms unfolding right_total_def Ball_def rel_fun_def Domainp_iff[abs_def]
```
```   440 by fast
```
```   441
```
```   442 lemma All_transfer [transfer_rule]:
```
```   443   assumes "bi_total A"
```
```   444   shows "((A ===> (=)) ===> (=)) All All"
```
```   445   using assms unfolding bi_total_def rel_fun_def by fast
```
```   446
```
```   447 lemma Ex_transfer [transfer_rule]:
```
```   448   assumes "bi_total A"
```
```   449   shows "((A ===> (=)) ===> (=)) Ex Ex"
```
```   450   using assms unfolding bi_total_def rel_fun_def by fast
```
```   451
```
```   452 lemma Ex1_parametric [transfer_rule]:
```
```   453   assumes [transfer_rule]: "bi_unique A" "bi_total A"
```
```   454   shows "((A ===> (=)) ===> (=)) Ex1 Ex1"
```
```   455 unfolding Ex1_def[abs_def] by transfer_prover
```
```   456
```
```   457 declare If_transfer [transfer_rule]
```
```   458
```
```   459 lemma Let_transfer [transfer_rule]: "(A ===> (A ===> B) ===> B) Let Let"
```
```   460   unfolding rel_fun_def by simp
```
```   461
```
```   462 declare id_transfer [transfer_rule]
```
```   463
```
```   464 declare comp_transfer [transfer_rule]
```
```   465
```
```   466 lemma curry_transfer [transfer_rule]:
```
```   467   "((rel_prod A B ===> C) ===> A ===> B ===> C) curry curry"
```
```   468   unfolding curry_def by transfer_prover
```
```   469
```
```   470 lemma fun_upd_transfer [transfer_rule]:
```
```   471   assumes [transfer_rule]: "bi_unique A"
```
```   472   shows "((A ===> B) ===> A ===> B ===> A ===> B) fun_upd fun_upd"
```
```   473   unfolding fun_upd_def [abs_def] by transfer_prover
```
```   474
```
```   475 lemma case_nat_transfer [transfer_rule]:
```
```   476   "(A ===> ((=) ===> A) ===> (=) ===> A) case_nat case_nat"
```
```   477   unfolding rel_fun_def by (simp split: nat.split)
```
```   478
```
```   479 lemma rec_nat_transfer [transfer_rule]:
```
```   480   "(A ===> ((=) ===> A ===> A) ===> (=) ===> A) rec_nat rec_nat"
```
```   481   unfolding rel_fun_def by (clarsimp, rename_tac n, induct_tac n, simp_all)
```
```   482
```
```   483 lemma funpow_transfer [transfer_rule]:
```
```   484   "((=) ===> (A ===> A) ===> (A ===> A)) compow compow"
```
```   485   unfolding funpow_def by transfer_prover
```
```   486
```
```   487 lemma mono_transfer[transfer_rule]:
```
```   488   assumes [transfer_rule]: "bi_total A"
```
```   489   assumes [transfer_rule]: "(A ===> A ===> (=)) (\<le>) (\<le>)"
```
```   490   assumes [transfer_rule]: "(B ===> B ===> (=)) (\<le>) (\<le>)"
```
```   491   shows "((A ===> B) ===> (=)) mono mono"
```
```   492 unfolding mono_def[abs_def] by transfer_prover
```
```   493
```
```   494 lemma right_total_relcompp_transfer[transfer_rule]:
```
```   495   assumes [transfer_rule]: "right_total B"
```
```   496   shows "((A ===> B ===> (=)) ===> (B ===> C ===> (=)) ===> A ===> C ===> (=))
```
```   497     (\<lambda>R S x z. \<exists>y\<in>Collect (Domainp B). R x y \<and> S y z) (OO)"
```
```   498 unfolding OO_def[abs_def] by transfer_prover
```
```   499
```
```   500 lemma relcompp_transfer[transfer_rule]:
```
```   501   assumes [transfer_rule]: "bi_total B"
```
```   502   shows "((A ===> B ===> (=)) ===> (B ===> C ===> (=)) ===> A ===> C ===> (=)) (OO) (OO)"
```
```   503 unfolding OO_def[abs_def] by transfer_prover
```
```   504
```
```   505 lemma right_total_Domainp_transfer[transfer_rule]:
```
```   506   assumes [transfer_rule]: "right_total B"
```
```   507   shows "((A ===> B ===> (=)) ===> A ===> (=)) (\<lambda>T x. \<exists>y\<in>Collect(Domainp B). T x y) Domainp"
```
```   508 apply(subst(2) Domainp_iff[abs_def]) by transfer_prover
```
```   509
```
```   510 lemma Domainp_transfer[transfer_rule]:
```
```   511   assumes [transfer_rule]: "bi_total B"
```
```   512   shows "((A ===> B ===> (=)) ===> A ===> (=)) Domainp Domainp"
```
```   513 unfolding Domainp_iff[abs_def] by transfer_prover
```
```   514
```
```   515 lemma reflp_transfer[transfer_rule]:
```
```   516   "bi_total A \<Longrightarrow> ((A ===> A ===> (=)) ===> (=)) reflp reflp"
```
```   517   "right_total A \<Longrightarrow> ((A ===> A ===> implies) ===> implies) reflp reflp"
```
```   518   "right_total A \<Longrightarrow> ((A ===> A ===> (=)) ===> implies) reflp reflp"
```
```   519   "bi_total A \<Longrightarrow> ((A ===> A ===> rev_implies) ===> rev_implies) reflp reflp"
```
```   520   "bi_total A \<Longrightarrow> ((A ===> A ===> (=)) ===> rev_implies) reflp reflp"
```
```   521 unfolding reflp_def[abs_def] rev_implies_def bi_total_def right_total_def rel_fun_def
```
```   522 by fast+
```
```   523
```
```   524 lemma right_unique_transfer [transfer_rule]:
```
```   525   "\<lbrakk> right_total A; right_total B; bi_unique B \<rbrakk>
```
```   526   \<Longrightarrow> ((A ===> B ===> (=)) ===> implies) right_unique right_unique"
```
```   527 unfolding right_unique_def[abs_def] right_total_def bi_unique_def rel_fun_def
```
```   528 by metis
```
```   529
```
```   530 lemma left_total_parametric [transfer_rule]:
```
```   531   assumes [transfer_rule]: "bi_total A" "bi_total B"
```
```   532   shows "((A ===> B ===> (=)) ===> (=)) left_total left_total"
```
```   533 unfolding left_total_def[abs_def] by transfer_prover
```
```   534
```
```   535 lemma right_total_parametric [transfer_rule]:
```
```   536   assumes [transfer_rule]: "bi_total A" "bi_total B"
```
```   537   shows "((A ===> B ===> (=)) ===> (=)) right_total right_total"
```
```   538 unfolding right_total_def[abs_def] by transfer_prover
```
```   539
```
```   540 lemma left_unique_parametric [transfer_rule]:
```
```   541   assumes [transfer_rule]: "bi_unique A" "bi_total A" "bi_total B"
```
```   542   shows "((A ===> B ===> (=)) ===> (=)) left_unique left_unique"
```
```   543 unfolding left_unique_def[abs_def] by transfer_prover
```
```   544
```
```   545 lemma prod_pred_parametric [transfer_rule]:
```
```   546   "((A ===> (=)) ===> (B ===> (=)) ===> rel_prod A B ===> (=)) pred_prod pred_prod"
```
```   547 unfolding prod.pred_set[abs_def] Basic_BNFs.fsts_def Basic_BNFs.snds_def fstsp.simps sndsp.simps
```
```   548 by simp transfer_prover
```
```   549
```
```   550 lemma apfst_parametric [transfer_rule]:
```
```   551   "((A ===> B) ===> rel_prod A C ===> rel_prod B C) apfst apfst"
```
```   552 unfolding apfst_def[abs_def] by transfer_prover
```
```   553
```
```   554 lemma rel_fun_eq_eq_onp: "((=) ===> eq_onp P) = eq_onp (\<lambda>f. \<forall>x. P(f x))"
```
```   555 unfolding eq_onp_def rel_fun_def by auto
```
```   556
```
```   557 lemma rel_fun_eq_onp_rel:
```
```   558   shows "((eq_onp R) ===> S) = (\<lambda>f g. \<forall>x. R x \<longrightarrow> S (f x) (g x))"
```
```   559 by (auto simp add: eq_onp_def rel_fun_def)
```
```   560
```
```   561 lemma eq_onp_transfer [transfer_rule]:
```
```   562   assumes [transfer_rule]: "bi_unique A"
```
```   563   shows "((A ===> (=)) ===> A ===> A ===> (=)) eq_onp eq_onp"
```
```   564 unfolding eq_onp_def[abs_def] by transfer_prover
```
```   565
```
```   566 lemma rtranclp_parametric [transfer_rule]:
```
```   567   assumes "bi_unique A" "bi_total A"
```
```   568   shows "((A ===> A ===> (=)) ===> A ===> A ===> (=)) rtranclp rtranclp"
```
```   569 proof(rule rel_funI iffI)+
```
```   570   fix R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and R' x y x' y'
```
```   571   assume R: "(A ===> A ===> (=)) R R'" and "A x x'"
```
```   572   {
```
```   573     assume "R\<^sup>*\<^sup>* x y" "A y y'"
```
```   574     thus "R'\<^sup>*\<^sup>* x' y'"
```
```   575     proof(induction arbitrary: y')
```
```   576       case base
```
```   577       with \<open>bi_unique A\<close> \<open>A x x'\<close> have "x' = y'" by(rule bi_uniqueDr)
```
```   578       thus ?case by simp
```
```   579     next
```
```   580       case (step y z z')
```
```   581       from \<open>bi_total A\<close> obtain y' where "A y y'" unfolding bi_total_def by blast
```
```   582       hence "R'\<^sup>*\<^sup>* x' y'" by(rule step.IH)
```
```   583       moreover from R \<open>A y y'\<close> \<open>A z z'\<close> \<open>R y z\<close>
```
```   584       have "R' y' z'" by(auto dest: rel_funD)
```
```   585       ultimately show ?case ..
```
```   586     qed
```
```   587   next
```
```   588     assume "R'\<^sup>*\<^sup>* x' y'" "A y y'"
```
```   589     thus "R\<^sup>*\<^sup>* x y"
```
```   590     proof(induction arbitrary: y)
```
```   591       case base
```
```   592       with \<open>bi_unique A\<close> \<open>A x x'\<close> have "x = y" by(rule bi_uniqueDl)
```
```   593       thus ?case by simp
```
```   594     next
```
```   595       case (step y' z' z)
```
```   596       from \<open>bi_total A\<close> obtain y where "A y y'" unfolding bi_total_def by blast
```
```   597       hence "R\<^sup>*\<^sup>* x y" by(rule step.IH)
```
```   598       moreover from R \<open>A y y'\<close> \<open>A z z'\<close> \<open>R' y' z'\<close>
```
```   599       have "R y z" by(auto dest: rel_funD)
```
```   600       ultimately show ?case ..
```
```   601     qed
```
```   602   }
```
```   603 qed
```
```   604
```
```   605 lemma right_unique_parametric [transfer_rule]:
```
```   606   assumes [transfer_rule]: "bi_total A" "bi_unique B" "bi_total B"
```
```   607   shows "((A ===> B ===> (=)) ===> (=)) right_unique right_unique"
```
```   608 unfolding right_unique_def[abs_def] by transfer_prover
```
```   609
```
```   610 lemma map_fun_parametric [transfer_rule]:
```
```   611   "((A ===> B) ===> (C ===> D) ===> (B ===> C) ===> A ===> D) map_fun map_fun"
```
```   612 unfolding map_fun_def[abs_def] by transfer_prover
```
```   613
```
```   614 end
```
```   615
```
```   616
```
```   617 subsection \<open>@{const of_nat}\<close>
```
```   618
```
```   619 lemma transfer_rule_of_nat:
```
```   620   fixes R :: "'a::semiring_1 \<Rightarrow> 'b::semiring_1 \<Rightarrow> bool"
```
```   621   assumes [transfer_rule]: "R 0 0" "R 1 1"
```
```   622     "rel_fun R (rel_fun R R) plus plus"
```
```   623   shows "rel_fun HOL.eq R of_nat of_nat"
```
```   624   by (unfold of_nat_def [abs_def]) transfer_prover
```
```   625
```
```   626 end
```