src/HOL/Transfer.thy
 author wenzelm Tue Sep 01 22:32:58 2015 +0200 (2015-09-01) changeset 61076 bdc1e2f0a86a parent 60758 d8d85a8172b5 child 61630 608520e0e8e2 permissions -rw-r--r--
eliminated \<Colon>;
```     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 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 \<open>Transfer method\<close>
```
```    42
```
```    43 text \<open>Explicit tag for relation membership allows for
```
```    44   backward proof methods.\<close>
```
```    45
```
```    46 definition Rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"
```
```    47   where "Rel r \<equiv> r"
```
```    48
```
```    49 text \<open>Handling of equality relations\<close>
```
```    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 \<open>Reverse implication for monotonicity rules\<close>
```
```    58
```
```    59 definition rev_implies where
```
```    60   "rev_implies x y \<longleftrightarrow> (y \<longrightarrow> x)"
```
```    61
```
```    62 text \<open>Handling of meta-logic connectives\<close>
```
```    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 subsection \<open>Predicates on relations, i.e. ``class constraints''\<close>
```
```   109
```
```   110 definition left_total :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
```
```   111   where "left_total R \<longleftrightarrow> (\<forall>x. \<exists>y. R x y)"
```
```   112
```
```   113 definition left_unique :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
```
```   114   where "left_unique R \<longleftrightarrow> (\<forall>x y z. R x z \<longrightarrow> R y z \<longrightarrow> x = y)"
```
```   115
```
```   116 definition right_total :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
```
```   117   where "right_total R \<longleftrightarrow> (\<forall>y. \<exists>x. R x y)"
```
```   118
```
```   119 definition right_unique :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
```
```   120   where "right_unique R \<longleftrightarrow> (\<forall>x y z. R x y \<longrightarrow> R x z \<longrightarrow> y = z)"
```
```   121
```
```   122 definition bi_total :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
```
```   123   where "bi_total R \<longleftrightarrow> (\<forall>x. \<exists>y. R x y) \<and> (\<forall>y. \<exists>x. R x y)"
```
```   124
```
```   125 definition bi_unique :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
```
```   126   where "bi_unique R \<longleftrightarrow>
```
```   127     (\<forall>x y z. R x y \<longrightarrow> R x z \<longrightarrow> y = z) \<and>
```
```   128     (\<forall>x y z. R x z \<longrightarrow> R y z \<longrightarrow> x = y)"
```
```   129
```
```   130 lemma left_uniqueI: "(\<And>x y z. \<lbrakk> A x z; A y z \<rbrakk> \<Longrightarrow> x = y) \<Longrightarrow> left_unique A"
```
```   131 unfolding left_unique_def by blast
```
```   132
```
```   133 lemma left_uniqueD: "\<lbrakk> left_unique A; A x z; A y z \<rbrakk> \<Longrightarrow> x = y"
```
```   134 unfolding left_unique_def by blast
```
```   135
```
```   136 lemma left_totalI:
```
```   137   "(\<And>x. \<exists>y. R x y) \<Longrightarrow> left_total R"
```
```   138 unfolding left_total_def by blast
```
```   139
```
```   140 lemma left_totalE:
```
```   141   assumes "left_total R"
```
```   142   obtains "(\<And>x. \<exists>y. R x y)"
```
```   143 using assms unfolding left_total_def by blast
```
```   144
```
```   145 lemma bi_uniqueDr: "\<lbrakk> bi_unique A; A x y; A x z \<rbrakk> \<Longrightarrow> y = z"
```
```   146 by(simp add: bi_unique_def)
```
```   147
```
```   148 lemma bi_uniqueDl: "\<lbrakk> bi_unique A; A x y; A z y \<rbrakk> \<Longrightarrow> x = z"
```
```   149 by(simp add: bi_unique_def)
```
```   150
```
```   151 lemma right_uniqueI: "(\<And>x y z. \<lbrakk> A x y; A x z \<rbrakk> \<Longrightarrow> y = z) \<Longrightarrow> right_unique A"
```
```   152 unfolding right_unique_def by fast
```
```   153
```
```   154 lemma right_uniqueD: "\<lbrakk> right_unique A; A x y; A x z \<rbrakk> \<Longrightarrow> y = z"
```
```   155 unfolding right_unique_def by fast
```
```   156
```
```   157 lemma right_totalI: "(\<And>y. \<exists>x. A x y) \<Longrightarrow> right_total A"
```
```   158 by(simp add: right_total_def)
```
```   159
```
```   160 lemma right_totalE:
```
```   161   assumes "right_total A"
```
```   162   obtains x where "A x y"
```
```   163 using assms by(auto simp add: right_total_def)
```
```   164
```
```   165 lemma right_total_alt_def2:
```
```   166   "right_total R \<longleftrightarrow> ((R ===> op \<longrightarrow>) ===> op \<longrightarrow>) All All"
```
```   167   unfolding right_total_def rel_fun_def
```
```   168   apply (rule iffI, fast)
```
```   169   apply (rule allI)
```
```   170   apply (drule_tac x="\<lambda>x. True" in spec)
```
```   171   apply (drule_tac x="\<lambda>y. \<exists>x. R x y" in spec)
```
```   172   apply fast
```
```   173   done
```
```   174
```
```   175 lemma right_unique_alt_def2:
```
```   176   "right_unique R \<longleftrightarrow> (R ===> R ===> op \<longrightarrow>) (op =) (op =)"
```
```   177   unfolding right_unique_def rel_fun_def by auto
```
```   178
```
```   179 lemma bi_total_alt_def2:
```
```   180   "bi_total R \<longleftrightarrow> ((R ===> op =) ===> op =) All All"
```
```   181   unfolding bi_total_def rel_fun_def
```
```   182   apply (rule iffI, fast)
```
```   183   apply safe
```
```   184   apply (drule_tac x="\<lambda>x. \<exists>y. R x y" in spec)
```
```   185   apply (drule_tac x="\<lambda>y. True" in spec)
```
```   186   apply fast
```
```   187   apply (drule_tac x="\<lambda>x. True" in spec)
```
```   188   apply (drule_tac x="\<lambda>y. \<exists>x. R x y" in spec)
```
```   189   apply fast
```
```   190   done
```
```   191
```
```   192 lemma bi_unique_alt_def2:
```
```   193   "bi_unique R \<longleftrightarrow> (R ===> R ===> op =) (op =) (op =)"
```
```   194   unfolding bi_unique_def rel_fun_def by auto
```
```   195
```
```   196 lemma [simp]:
```
```   197   shows left_unique_conversep: "left_unique A\<inverse>\<inverse> \<longleftrightarrow> right_unique A"
```
```   198   and right_unique_conversep: "right_unique A\<inverse>\<inverse> \<longleftrightarrow> left_unique A"
```
```   199 by(auto simp add: left_unique_def right_unique_def)
```
```   200
```
```   201 lemma [simp]:
```
```   202   shows left_total_conversep: "left_total A\<inverse>\<inverse> \<longleftrightarrow> right_total A"
```
```   203   and right_total_conversep: "right_total A\<inverse>\<inverse> \<longleftrightarrow> left_total A"
```
```   204 by(simp_all add: left_total_def right_total_def)
```
```   205
```
```   206 lemma bi_unique_conversep [simp]: "bi_unique R\<inverse>\<inverse> = bi_unique R"
```
```   207 by(auto simp add: bi_unique_def)
```
```   208
```
```   209 lemma bi_total_conversep [simp]: "bi_total R\<inverse>\<inverse> = bi_total R"
```
```   210 by(auto simp add: bi_total_def)
```
```   211
```
```   212 lemma right_unique_alt_def: "right_unique R = (conversep R OO R \<le> op=)" unfolding right_unique_def by blast
```
```   213 lemma left_unique_alt_def: "left_unique R = (R OO (conversep R) \<le> op=)" unfolding left_unique_def by blast
```
```   214
```
```   215 lemma right_total_alt_def: "right_total R = (conversep R OO R \<ge> op=)" unfolding right_total_def by blast
```
```   216 lemma left_total_alt_def: "left_total R = (R OO conversep R \<ge> op=)" unfolding left_total_def by blast
```
```   217
```
```   218 lemma bi_total_alt_def: "bi_total A = (left_total A \<and> right_total A)"
```
```   219 unfolding left_total_def right_total_def bi_total_def by blast
```
```   220
```
```   221 lemma bi_unique_alt_def: "bi_unique A = (left_unique A \<and> right_unique A)"
```
```   222 unfolding left_unique_def right_unique_def bi_unique_def by blast
```
```   223
```
```   224 lemma bi_totalI: "left_total R \<Longrightarrow> right_total R \<Longrightarrow> bi_total R"
```
```   225 unfolding bi_total_alt_def ..
```
```   226
```
```   227 lemma bi_uniqueI: "left_unique R \<Longrightarrow> right_unique R \<Longrightarrow> bi_unique R"
```
```   228 unfolding bi_unique_alt_def ..
```
```   229
```
```   230 end
```
```   231
```
```   232 subsection \<open>Equality restricted by a predicate\<close>
```
```   233
```
```   234 definition eq_onp :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
```
```   235   where "eq_onp R = (\<lambda>x y. R x \<and> x = y)"
```
```   236
```
```   237 lemma eq_onp_Grp: "eq_onp P = BNF_Def.Grp (Collect P) id"
```
```   238 unfolding eq_onp_def Grp_def by auto
```
```   239
```
```   240 lemma eq_onp_to_eq:
```
```   241   assumes "eq_onp P x y"
```
```   242   shows "x = y"
```
```   243 using assms by (simp add: eq_onp_def)
```
```   244
```
```   245 lemma eq_onp_top_eq_eq: "eq_onp top = op="
```
```   246 by (simp add: eq_onp_def)
```
```   247
```
```   248 lemma eq_onp_same_args:
```
```   249   shows "eq_onp P x x = P x"
```
```   250 using assms by (auto simp add: eq_onp_def)
```
```   251
```
```   252 lemma Ball_Collect: "Ball A P = (A \<subseteq> (Collect P))"
```
```   253 by auto
```
```   254
```
```   255 ML_file "Tools/Transfer/transfer.ML"
```
```   256 declare refl [transfer_rule]
```
```   257
```
```   258 hide_const (open) Rel
```
```   259
```
```   260 context
```
```   261 begin
```
```   262 interpretation lifting_syntax .
```
```   263
```
```   264 text \<open>Handling of domains\<close>
```
```   265
```
```   266 lemma Domainp_iff: "Domainp T x \<longleftrightarrow> (\<exists>y. T x y)"
```
```   267   by auto
```
```   268
```
```   269 lemma Domainp_refl[transfer_domain_rule]:
```
```   270   "Domainp T = Domainp T" ..
```
```   271
```
```   272 lemma Domain_eq_top: "Domainp op= = top" by auto
```
```   273
```
```   274 lemma Domainp_prod_fun_eq[relator_domain]:
```
```   275   "Domainp (op= ===> T) = (\<lambda>f. \<forall>x. (Domainp T) (f x))"
```
```   276 by (auto intro: choice simp: Domainp_iff rel_fun_def fun_eq_iff)
```
```   277
```
```   278 text \<open>Properties are preserved by relation composition.\<close>
```
```   279
```
```   280 lemma OO_def: "R OO S = (\<lambda>x z. \<exists>y. R x y \<and> S y z)"
```
```   281   by auto
```
```   282
```
```   283 lemma bi_total_OO: "\<lbrakk>bi_total A; bi_total B\<rbrakk> \<Longrightarrow> bi_total (A OO B)"
```
```   284   unfolding bi_total_def OO_def by fast
```
```   285
```
```   286 lemma bi_unique_OO: "\<lbrakk>bi_unique A; bi_unique B\<rbrakk> \<Longrightarrow> bi_unique (A OO B)"
```
```   287   unfolding bi_unique_def OO_def by blast
```
```   288
```
```   289 lemma right_total_OO:
```
```   290   "\<lbrakk>right_total A; right_total B\<rbrakk> \<Longrightarrow> right_total (A OO B)"
```
```   291   unfolding right_total_def OO_def by fast
```
```   292
```
```   293 lemma right_unique_OO:
```
```   294   "\<lbrakk>right_unique A; right_unique B\<rbrakk> \<Longrightarrow> right_unique (A OO B)"
```
```   295   unfolding right_unique_def OO_def by fast
```
```   296
```
```   297 lemma left_total_OO: "left_total R \<Longrightarrow> left_total S \<Longrightarrow> left_total (R OO S)"
```
```   298 unfolding left_total_def OO_def by fast
```
```   299
```
```   300 lemma left_unique_OO: "left_unique R \<Longrightarrow> left_unique S \<Longrightarrow> left_unique (R OO S)"
```
```   301 unfolding left_unique_def OO_def by blast
```
```   302
```
```   303
```
```   304 subsection \<open>Properties of relators\<close>
```
```   305
```
```   306 lemma left_total_eq[transfer_rule]: "left_total op="
```
```   307   unfolding left_total_def by blast
```
```   308
```
```   309 lemma left_unique_eq[transfer_rule]: "left_unique op="
```
```   310   unfolding left_unique_def by blast
```
```   311
```
```   312 lemma right_total_eq [transfer_rule]: "right_total op="
```
```   313   unfolding right_total_def by simp
```
```   314
```
```   315 lemma right_unique_eq [transfer_rule]: "right_unique op="
```
```   316   unfolding right_unique_def by simp
```
```   317
```
```   318 lemma bi_total_eq[transfer_rule]: "bi_total (op =)"
```
```   319   unfolding bi_total_def by simp
```
```   320
```
```   321 lemma bi_unique_eq[transfer_rule]: "bi_unique (op =)"
```
```   322   unfolding bi_unique_def by simp
```
```   323
```
```   324 lemma left_total_fun[transfer_rule]:
```
```   325   "\<lbrakk>left_unique A; left_total B\<rbrakk> \<Longrightarrow> left_total (A ===> B)"
```
```   326   unfolding left_total_def rel_fun_def
```
```   327   apply (rule allI, rename_tac f)
```
```   328   apply (rule_tac x="\<lambda>y. SOME z. B (f (THE x. A x y)) z" in exI)
```
```   329   apply clarify
```
```   330   apply (subgoal_tac "(THE x. A x y) = x", simp)
```
```   331   apply (rule someI_ex)
```
```   332   apply (simp)
```
```   333   apply (rule the_equality)
```
```   334   apply assumption
```
```   335   apply (simp add: left_unique_def)
```
```   336   done
```
```   337
```
```   338 lemma left_unique_fun[transfer_rule]:
```
```   339   "\<lbrakk>left_total A; left_unique B\<rbrakk> \<Longrightarrow> left_unique (A ===> B)"
```
```   340   unfolding left_total_def left_unique_def rel_fun_def
```
```   341   by (clarify, rule ext, fast)
```
```   342
```
```   343 lemma right_total_fun [transfer_rule]:
```
```   344   "\<lbrakk>right_unique A; right_total B\<rbrakk> \<Longrightarrow> right_total (A ===> B)"
```
```   345   unfolding right_total_def rel_fun_def
```
```   346   apply (rule allI, rename_tac g)
```
```   347   apply (rule_tac x="\<lambda>x. SOME z. B z (g (THE y. A x y))" in exI)
```
```   348   apply clarify
```
```   349   apply (subgoal_tac "(THE y. A x y) = y", simp)
```
```   350   apply (rule someI_ex)
```
```   351   apply (simp)
```
```   352   apply (rule the_equality)
```
```   353   apply assumption
```
```   354   apply (simp add: right_unique_def)
```
```   355   done
```
```   356
```
```   357 lemma right_unique_fun [transfer_rule]:
```
```   358   "\<lbrakk>right_total A; right_unique B\<rbrakk> \<Longrightarrow> right_unique (A ===> B)"
```
```   359   unfolding right_total_def right_unique_def rel_fun_def
```
```   360   by (clarify, rule ext, fast)
```
```   361
```
```   362 lemma bi_total_fun[transfer_rule]:
```
```   363   "\<lbrakk>bi_unique A; bi_total B\<rbrakk> \<Longrightarrow> bi_total (A ===> B)"
```
```   364   unfolding bi_unique_alt_def bi_total_alt_def
```
```   365   by (blast intro: right_total_fun left_total_fun)
```
```   366
```
```   367 lemma bi_unique_fun[transfer_rule]:
```
```   368   "\<lbrakk>bi_total A; bi_unique B\<rbrakk> \<Longrightarrow> bi_unique (A ===> B)"
```
```   369   unfolding bi_unique_alt_def bi_total_alt_def
```
```   370   by (blast intro: right_unique_fun left_unique_fun)
```
```   371
```
```   372 end
```
```   373
```
```   374 lemma if_conn:
```
```   375   "(if P \<and> Q then t else e) = (if P then if Q then t else e else e)"
```
```   376   "(if P \<or> Q then t else e) = (if P then t else if Q then t else e)"
```
```   377   "(if P \<longrightarrow> Q then t else e) = (if P then if Q then t else e else t)"
```
```   378   "(if \<not> P then t else e) = (if P then e else t)"
```
```   379 by auto
```
```   380
```
```   381 ML_file "Tools/Transfer/transfer_bnf.ML"
```
```   382 ML_file "Tools/BNF/bnf_fp_rec_sugar_transfer.ML"
```
```   383
```
```   384 declare pred_fun_def [simp]
```
```   385 declare rel_fun_eq [relator_eq]
```
```   386
```
```   387 subsection \<open>Transfer rules\<close>
```
```   388
```
```   389 context
```
```   390 begin
```
```   391 interpretation lifting_syntax .
```
```   392
```
```   393 lemma Domainp_forall_transfer [transfer_rule]:
```
```   394   assumes "right_total A"
```
```   395   shows "((A ===> op =) ===> op =)
```
```   396     (transfer_bforall (Domainp A)) transfer_forall"
```
```   397   using assms unfolding right_total_def
```
```   398   unfolding transfer_forall_def transfer_bforall_def rel_fun_def Domainp_iff
```
```   399   by fast
```
```   400
```
```   401 text \<open>Transfer rules using implication instead of equality on booleans.\<close>
```
```   402
```
```   403 lemma transfer_forall_transfer [transfer_rule]:
```
```   404   "bi_total A \<Longrightarrow> ((A ===> op =) ===> op =) transfer_forall transfer_forall"
```
```   405   "right_total A \<Longrightarrow> ((A ===> op =) ===> implies) transfer_forall transfer_forall"
```
```   406   "right_total A \<Longrightarrow> ((A ===> implies) ===> implies) transfer_forall transfer_forall"
```
```   407   "bi_total A \<Longrightarrow> ((A ===> op =) ===> rev_implies) transfer_forall transfer_forall"
```
```   408   "bi_total A \<Longrightarrow> ((A ===> rev_implies) ===> rev_implies) transfer_forall transfer_forall"
```
```   409   unfolding transfer_forall_def rev_implies_def rel_fun_def right_total_def bi_total_def
```
```   410   by fast+
```
```   411
```
```   412 lemma transfer_implies_transfer [transfer_rule]:
```
```   413   "(op =        ===> op =        ===> op =       ) transfer_implies transfer_implies"
```
```   414   "(rev_implies ===> implies     ===> implies    ) transfer_implies transfer_implies"
```
```   415   "(rev_implies ===> op =        ===> implies    ) transfer_implies transfer_implies"
```
```   416   "(op =        ===> implies     ===> implies    ) transfer_implies transfer_implies"
```
```   417   "(op =        ===> op =        ===> implies    ) transfer_implies transfer_implies"
```
```   418   "(implies     ===> rev_implies ===> rev_implies) transfer_implies transfer_implies"
```
```   419   "(implies     ===> op =        ===> rev_implies) transfer_implies transfer_implies"
```
```   420   "(op =        ===> rev_implies ===> rev_implies) transfer_implies transfer_implies"
```
```   421   "(op =        ===> op =        ===> rev_implies) transfer_implies transfer_implies"
```
```   422   unfolding transfer_implies_def rev_implies_def rel_fun_def by auto
```
```   423
```
```   424 lemma eq_imp_transfer [transfer_rule]:
```
```   425   "right_unique A \<Longrightarrow> (A ===> A ===> op \<longrightarrow>) (op =) (op =)"
```
```   426   unfolding right_unique_alt_def2 .
```
```   427
```
```   428 text \<open>Transfer rules using equality.\<close>
```
```   429
```
```   430 lemma left_unique_transfer [transfer_rule]:
```
```   431   assumes "right_total A"
```
```   432   assumes "right_total B"
```
```   433   assumes "bi_unique A"
```
```   434   shows "((A ===> B ===> op=) ===> implies) left_unique left_unique"
```
```   435 using assms unfolding left_unique_def[abs_def] right_total_def bi_unique_def rel_fun_def
```
```   436 by metis
```
```   437
```
```   438 lemma eq_transfer [transfer_rule]:
```
```   439   assumes "bi_unique A"
```
```   440   shows "(A ===> A ===> op =) (op =) (op =)"
```
```   441   using assms unfolding bi_unique_def rel_fun_def by auto
```
```   442
```
```   443 lemma right_total_Ex_transfer[transfer_rule]:
```
```   444   assumes "right_total A"
```
```   445   shows "((A ===> op=) ===> op=) (Bex (Collect (Domainp A))) Ex"
```
```   446 using assms unfolding right_total_def Bex_def rel_fun_def Domainp_iff[abs_def]
```
```   447 by fast
```
```   448
```
```   449 lemma right_total_All_transfer[transfer_rule]:
```
```   450   assumes "right_total A"
```
```   451   shows "((A ===> op =) ===> op =) (Ball (Collect (Domainp A))) All"
```
```   452 using assms unfolding right_total_def Ball_def rel_fun_def Domainp_iff[abs_def]
```
```   453 by fast
```
```   454
```
```   455 lemma All_transfer [transfer_rule]:
```
```   456   assumes "bi_total A"
```
```   457   shows "((A ===> op =) ===> op =) All All"
```
```   458   using assms unfolding bi_total_def rel_fun_def by fast
```
```   459
```
```   460 lemma Ex_transfer [transfer_rule]:
```
```   461   assumes "bi_total A"
```
```   462   shows "((A ===> op =) ===> op =) Ex Ex"
```
```   463   using assms unfolding bi_total_def rel_fun_def by fast
```
```   464
```
```   465 lemma Ex1_parametric [transfer_rule]:
```
```   466   assumes [transfer_rule]: "bi_unique A" "bi_total A"
```
```   467   shows "((A ===> op =) ===> op =) Ex1 Ex1"
```
```   468 unfolding Ex1_def[abs_def] by transfer_prover
```
```   469
```
```   470 declare If_transfer [transfer_rule]
```
```   471
```
```   472 lemma Let_transfer [transfer_rule]: "(A ===> (A ===> B) ===> B) Let Let"
```
```   473   unfolding rel_fun_def by simp
```
```   474
```
```   475 declare id_transfer [transfer_rule]
```
```   476
```
```   477 declare comp_transfer [transfer_rule]
```
```   478
```
```   479 lemma curry_transfer [transfer_rule]:
```
```   480   "((rel_prod A B ===> C) ===> A ===> B ===> C) curry curry"
```
```   481   unfolding curry_def by transfer_prover
```
```   482
```
```   483 lemma fun_upd_transfer [transfer_rule]:
```
```   484   assumes [transfer_rule]: "bi_unique A"
```
```   485   shows "((A ===> B) ===> A ===> B ===> A ===> B) fun_upd fun_upd"
```
```   486   unfolding fun_upd_def [abs_def] by transfer_prover
```
```   487
```
```   488 lemma case_nat_transfer [transfer_rule]:
```
```   489   "(A ===> (op = ===> A) ===> op = ===> A) case_nat case_nat"
```
```   490   unfolding rel_fun_def by (simp split: nat.split)
```
```   491
```
```   492 lemma rec_nat_transfer [transfer_rule]:
```
```   493   "(A ===> (op = ===> A ===> A) ===> op = ===> A) rec_nat rec_nat"
```
```   494   unfolding rel_fun_def by (clarsimp, rename_tac n, induct_tac n, simp_all)
```
```   495
```
```   496 lemma funpow_transfer [transfer_rule]:
```
```   497   "(op = ===> (A ===> A) ===> (A ===> A)) compow compow"
```
```   498   unfolding funpow_def by transfer_prover
```
```   499
```
```   500 lemma mono_transfer[transfer_rule]:
```
```   501   assumes [transfer_rule]: "bi_total A"
```
```   502   assumes [transfer_rule]: "(A ===> A ===> op=) op\<le> op\<le>"
```
```   503   assumes [transfer_rule]: "(B ===> B ===> op=) op\<le> op\<le>"
```
```   504   shows "((A ===> B) ===> op=) mono mono"
```
```   505 unfolding mono_def[abs_def] by transfer_prover
```
```   506
```
```   507 lemma right_total_relcompp_transfer[transfer_rule]:
```
```   508   assumes [transfer_rule]: "right_total B"
```
```   509   shows "((A ===> B ===> op=) ===> (B ===> C ===> op=) ===> A ===> C ===> op=)
```
```   510     (\<lambda>R S x z. \<exists>y\<in>Collect (Domainp B). R x y \<and> S y z) op OO"
```
```   511 unfolding OO_def[abs_def] by transfer_prover
```
```   512
```
```   513 lemma relcompp_transfer[transfer_rule]:
```
```   514   assumes [transfer_rule]: "bi_total B"
```
```   515   shows "((A ===> B ===> op=) ===> (B ===> C ===> op=) ===> A ===> C ===> op=) op OO op OO"
```
```   516 unfolding OO_def[abs_def] by transfer_prover
```
```   517
```
```   518 lemma right_total_Domainp_transfer[transfer_rule]:
```
```   519   assumes [transfer_rule]: "right_total B"
```
```   520   shows "((A ===> B ===> op=) ===> A ===> op=) (\<lambda>T x. \<exists>y\<in>Collect(Domainp B). T x y) Domainp"
```
```   521 apply(subst(2) Domainp_iff[abs_def]) by transfer_prover
```
```   522
```
```   523 lemma Domainp_transfer[transfer_rule]:
```
```   524   assumes [transfer_rule]: "bi_total B"
```
```   525   shows "((A ===> B ===> op=) ===> A ===> op=) Domainp Domainp"
```
```   526 unfolding Domainp_iff[abs_def] by transfer_prover
```
```   527
```
```   528 lemma reflp_transfer[transfer_rule]:
```
```   529   "bi_total A \<Longrightarrow> ((A ===> A ===> op=) ===> op=) reflp reflp"
```
```   530   "right_total A \<Longrightarrow> ((A ===> A ===> implies) ===> implies) reflp reflp"
```
```   531   "right_total A \<Longrightarrow> ((A ===> A ===> op=) ===> implies) reflp reflp"
```
```   532   "bi_total A \<Longrightarrow> ((A ===> A ===> rev_implies) ===> rev_implies) reflp reflp"
```
```   533   "bi_total A \<Longrightarrow> ((A ===> A ===> op=) ===> rev_implies) reflp reflp"
```
```   534 using assms unfolding reflp_def[abs_def] rev_implies_def bi_total_def right_total_def rel_fun_def
```
```   535 by fast+
```
```   536
```
```   537 lemma right_unique_transfer [transfer_rule]:
```
```   538   "\<lbrakk> right_total A; right_total B; bi_unique B \<rbrakk>
```
```   539   \<Longrightarrow> ((A ===> B ===> op=) ===> implies) right_unique right_unique"
```
```   540 unfolding right_unique_def[abs_def] right_total_def bi_unique_def rel_fun_def
```
```   541 by metis
```
```   542
```
```   543 lemma left_total_parametric [transfer_rule]:
```
```   544   assumes [transfer_rule]: "bi_total A" "bi_total B"
```
```   545   shows "((A ===> B ===> op =) ===> op =) left_total left_total"
```
```   546 unfolding left_total_def[abs_def] by transfer_prover
```
```   547
```
```   548 lemma right_total_parametric [transfer_rule]:
```
```   549   assumes [transfer_rule]: "bi_total A" "bi_total B"
```
```   550   shows "((A ===> B ===> op =) ===> op =) right_total right_total"
```
```   551 unfolding right_total_def[abs_def] by transfer_prover
```
```   552
```
```   553 lemma left_unique_parametric [transfer_rule]:
```
```   554   assumes [transfer_rule]: "bi_unique A" "bi_total A" "bi_total B"
```
```   555   shows "((A ===> B ===> op =) ===> op =) left_unique left_unique"
```
```   556 unfolding left_unique_def[abs_def] by transfer_prover
```
```   557
```
```   558 lemma prod_pred_parametric [transfer_rule]:
```
```   559   "((A ===> op =) ===> (B ===> op =) ===> rel_prod A B ===> op =) pred_prod pred_prod"
```
```   560 unfolding pred_prod_def[abs_def] Basic_BNFs.fsts_def Basic_BNFs.snds_def fstsp.simps sndsp.simps
```
```   561 by simp transfer_prover
```
```   562
```
```   563 lemma apfst_parametric [transfer_rule]:
```
```   564   "((A ===> B) ===> rel_prod A C ===> rel_prod B C) apfst apfst"
```
```   565 unfolding apfst_def[abs_def] by transfer_prover
```
```   566
```
```   567 lemma rel_fun_eq_eq_onp: "(op= ===> eq_onp P) = eq_onp (\<lambda>f. \<forall>x. P(f x))"
```
```   568 unfolding eq_onp_def rel_fun_def by auto
```
```   569
```
```   570 lemma rel_fun_eq_onp_rel:
```
```   571   shows "((eq_onp R) ===> S) = (\<lambda>f g. \<forall>x. R x \<longrightarrow> S (f x) (g x))"
```
```   572 by (auto simp add: eq_onp_def rel_fun_def)
```
```   573
```
```   574 lemma eq_onp_transfer [transfer_rule]:
```
```   575   assumes [transfer_rule]: "bi_unique A"
```
```   576   shows "((A ===> op=) ===> A ===> A ===> op=) eq_onp eq_onp"
```
```   577 unfolding eq_onp_def[abs_def] by transfer_prover
```
```   578
```
```   579 lemma rtranclp_parametric [transfer_rule]:
```
```   580   assumes "bi_unique A" "bi_total A"
```
```   581   shows "((A ===> A ===> op =) ===> A ===> A ===> op =) rtranclp rtranclp"
```
```   582 proof(rule rel_funI iffI)+
```
```   583   fix R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and R' x y x' y'
```
```   584   assume R: "(A ===> A ===> op =) R R'" and "A x x'"
```
```   585   {
```
```   586     assume "R\<^sup>*\<^sup>* x y" "A y y'"
```
```   587     thus "R'\<^sup>*\<^sup>* x' y'"
```
```   588     proof(induction arbitrary: y')
```
```   589       case base
```
```   590       with \<open>bi_unique A\<close> \<open>A x x'\<close> have "x' = y'" by(rule bi_uniqueDr)
```
```   591       thus ?case by simp
```
```   592     next
```
```   593       case (step y z z')
```
```   594       from \<open>bi_total A\<close> obtain y' where "A y y'" unfolding bi_total_def by blast
```
```   595       hence "R'\<^sup>*\<^sup>* x' y'" by(rule step.IH)
```
```   596       moreover from R \<open>A y y'\<close> \<open>A z z'\<close> \<open>R y z\<close>
```
```   597       have "R' y' z'" by(auto dest: rel_funD)
```
```   598       ultimately show ?case ..
```
```   599     qed
```
```   600   next
```
```   601     assume "R'\<^sup>*\<^sup>* x' y'" "A y y'"
```
```   602     thus "R\<^sup>*\<^sup>* x y"
```
```   603     proof(induction arbitrary: y)
```
```   604       case base
```
```   605       with \<open>bi_unique A\<close> \<open>A x x'\<close> have "x = y" by(rule bi_uniqueDl)
```
```   606       thus ?case by simp
```
```   607     next
```
```   608       case (step y' z' z)
```
```   609       from \<open>bi_total A\<close> obtain y where "A y y'" unfolding bi_total_def by blast
```
```   610       hence "R\<^sup>*\<^sup>* x y" by(rule step.IH)
```
```   611       moreover from R \<open>A y y'\<close> \<open>A z z'\<close> \<open>R' y' z'\<close>
```
```   612       have "R y z" by(auto dest: rel_funD)
```
```   613       ultimately show ?case ..
```
```   614     qed
```
```   615   }
```
```   616 qed
```
```   617
```
```   618 lemma right_unique_parametric [transfer_rule]:
```
```   619   assumes [transfer_rule]: "bi_total A" "bi_unique B" "bi_total B"
```
```   620   shows "((A ===> B ===> op =) ===> op =) right_unique right_unique"
```
```   621 unfolding right_unique_def[abs_def] by transfer_prover
```
```   622
```
```   623 end
```
```   624
```
```   625 end
```