src/HOL/Transfer.thy
 author kuncar Mon May 13 13:59:04 2013 +0200 (2013-05-13) changeset 51956 a4d81cdebf8b parent 51955 04d9381bebff child 52354 acb4f932dd24 permissions -rw-r--r--
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
```     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
```
```    10 begin
```
```    11
```
```    12 subsection {* Relator for function space *}
```
```    13
```
```    14 definition
```
```    15   fun_rel :: "('a \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool" (infixr "===>" 55)
```
```    16 where
```
```    17   "fun_rel A B = (\<lambda>f g. \<forall>x y. A x y \<longrightarrow> B (f x) (g y))"
```
```    18
```
```    19 lemma fun_relI [intro]:
```
```    20   assumes "\<And>x y. A x y \<Longrightarrow> B (f x) (g y)"
```
```    21   shows "(A ===> B) f g"
```
```    22   using assms by (simp add: fun_rel_def)
```
```    23
```
```    24 lemma fun_relD:
```
```    25   assumes "(A ===> B) f g" and "A x y"
```
```    26   shows "B (f x) (g y)"
```
```    27   using assms by (simp add: fun_rel_def)
```
```    28
```
```    29 lemma fun_relD2:
```
```    30   assumes "(A ===> B) f g" and "A x x"
```
```    31   shows "B (f x) (g x)"
```
```    32   using assms unfolding fun_rel_def by auto
```
```    33
```
```    34 lemma fun_relE:
```
```    35   assumes "(A ===> B) f g" and "A x y"
```
```    36   obtains "B (f x) (g y)"
```
```    37   using assms by (simp add: fun_rel_def)
```
```    38
```
```    39 lemma fun_rel_eq:
```
```    40   shows "((op =) ===> (op =)) = (op =)"
```
```    41   by (auto simp add: fun_eq_iff elim: fun_relE)
```
```    42
```
```    43 lemma fun_rel_eq_rel:
```
```    44   shows "((op =) ===> R) = (\<lambda>f g. \<forall>x. R (f x) (g x))"
```
```    45   by (simp add: fun_rel_def)
```
```    46
```
```    47
```
```    48 subsection {* Transfer method *}
```
```    49
```
```    50 text {* Explicit tag for relation membership allows for
```
```    51   backward proof methods. *}
```
```    52
```
```    53 definition Rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"
```
```    54   where "Rel r \<equiv> r"
```
```    55
```
```    56 text {* Handling of equality relations *}
```
```    57
```
```    58 definition is_equality :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
```
```    59   where "is_equality R \<longleftrightarrow> R = (op =)"
```
```    60
```
```    61 lemma is_equality_eq: "is_equality (op =)"
```
```    62   unfolding is_equality_def by simp
```
```    63
```
```    64 text {* Handling of meta-logic connectives *}
```
```    65
```
```    66 definition transfer_forall where
```
```    67   "transfer_forall \<equiv> All"
```
```    68
```
```    69 definition transfer_implies where
```
```    70   "transfer_implies \<equiv> op \<longrightarrow>"
```
```    71
```
```    72 definition transfer_bforall :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
```
```    73   where "transfer_bforall \<equiv> (\<lambda>P Q. \<forall>x. P x \<longrightarrow> Q x)"
```
```    74
```
```    75 lemma transfer_forall_eq: "(\<And>x. P x) \<equiv> Trueprop (transfer_forall (\<lambda>x. P x))"
```
```    76   unfolding atomize_all transfer_forall_def ..
```
```    77
```
```    78 lemma transfer_implies_eq: "(A \<Longrightarrow> B) \<equiv> Trueprop (transfer_implies A B)"
```
```    79   unfolding atomize_imp transfer_implies_def ..
```
```    80
```
```    81 lemma transfer_bforall_unfold:
```
```    82   "Trueprop (transfer_bforall P (\<lambda>x. Q x)) \<equiv> (\<And>x. P x \<Longrightarrow> Q x)"
```
```    83   unfolding transfer_bforall_def atomize_imp atomize_all ..
```
```    84
```
```    85 lemma transfer_start: "\<lbrakk>P; Rel (op =) P Q\<rbrakk> \<Longrightarrow> Q"
```
```    86   unfolding Rel_def by simp
```
```    87
```
```    88 lemma transfer_start': "\<lbrakk>P; Rel (op \<longrightarrow>) P Q\<rbrakk> \<Longrightarrow> Q"
```
```    89   unfolding Rel_def by simp
```
```    90
```
```    91 lemma transfer_prover_start: "\<lbrakk>x = x'; Rel R x' y\<rbrakk> \<Longrightarrow> Rel R x y"
```
```    92   by simp
```
```    93
```
```    94 lemma Rel_eq_refl: "Rel (op =) 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 fun_rel_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 fun_rel_def by fast
```
```   106
```
```   107 ML_file "Tools/transfer.ML"
```
```   108 setup Transfer.setup
```
```   109
```
```   110 declare refl [transfer_rule]
```
```   111
```
```   112 declare fun_rel_eq [relator_eq]
```
```   113
```
```   114 hide_const (open) Rel
```
```   115
```
```   116 text {* Handling of domains *}
```
```   117
```
```   118 lemma Domaimp_refl[transfer_domain_rule]:
```
```   119   "Domainp T = Domainp T" ..
```
```   120
```
```   121 subsection {* Predicates on relations, i.e. ``class constraints'' *}
```
```   122
```
```   123 definition right_total :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
```
```   124   where "right_total R \<longleftrightarrow> (\<forall>y. \<exists>x. R x y)"
```
```   125
```
```   126 definition right_unique :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
```
```   127   where "right_unique R \<longleftrightarrow> (\<forall>x y z. R x y \<longrightarrow> R x z \<longrightarrow> y = z)"
```
```   128
```
```   129 definition bi_total :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
```
```   130   where "bi_total R \<longleftrightarrow> (\<forall>x. \<exists>y. R x y) \<and> (\<forall>y. \<exists>x. R x y)"
```
```   131
```
```   132 definition bi_unique :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
```
```   133   where "bi_unique R \<longleftrightarrow>
```
```   134     (\<forall>x y z. R x y \<longrightarrow> R x z \<longrightarrow> y = z) \<and>
```
```   135     (\<forall>x y z. R x z \<longrightarrow> R y z \<longrightarrow> x = y)"
```
```   136
```
```   137 lemma right_total_alt_def:
```
```   138   "right_total R \<longleftrightarrow> ((R ===> op \<longrightarrow>) ===> op \<longrightarrow>) All All"
```
```   139   unfolding right_total_def fun_rel_def
```
```   140   apply (rule iffI, fast)
```
```   141   apply (rule allI)
```
```   142   apply (drule_tac x="\<lambda>x. True" in spec)
```
```   143   apply (drule_tac x="\<lambda>y. \<exists>x. R x y" in spec)
```
```   144   apply fast
```
```   145   done
```
```   146
```
```   147 lemma right_unique_alt_def:
```
```   148   "right_unique R \<longleftrightarrow> (R ===> R ===> op \<longrightarrow>) (op =) (op =)"
```
```   149   unfolding right_unique_def fun_rel_def by auto
```
```   150
```
```   151 lemma bi_total_alt_def:
```
```   152   "bi_total R \<longleftrightarrow> ((R ===> op =) ===> op =) All All"
```
```   153   unfolding bi_total_def fun_rel_def
```
```   154   apply (rule iffI, fast)
```
```   155   apply safe
```
```   156   apply (drule_tac x="\<lambda>x. \<exists>y. R x y" in spec)
```
```   157   apply (drule_tac x="\<lambda>y. True" in spec)
```
```   158   apply fast
```
```   159   apply (drule_tac x="\<lambda>x. True" in spec)
```
```   160   apply (drule_tac x="\<lambda>y. \<exists>x. R x y" in spec)
```
```   161   apply fast
```
```   162   done
```
```   163
```
```   164 lemma bi_unique_alt_def:
```
```   165   "bi_unique R \<longleftrightarrow> (R ===> R ===> op =) (op =) (op =)"
```
```   166   unfolding bi_unique_def fun_rel_def by auto
```
```   167
```
```   168 text {* Properties are preserved by relation composition. *}
```
```   169
```
```   170 lemma OO_def: "R OO S = (\<lambda>x z. \<exists>y. R x y \<and> S y z)"
```
```   171   by auto
```
```   172
```
```   173 lemma bi_total_OO: "\<lbrakk>bi_total A; bi_total B\<rbrakk> \<Longrightarrow> bi_total (A OO B)"
```
```   174   unfolding bi_total_def OO_def by metis
```
```   175
```
```   176 lemma bi_unique_OO: "\<lbrakk>bi_unique A; bi_unique B\<rbrakk> \<Longrightarrow> bi_unique (A OO B)"
```
```   177   unfolding bi_unique_def OO_def by metis
```
```   178
```
```   179 lemma right_total_OO:
```
```   180   "\<lbrakk>right_total A; right_total B\<rbrakk> \<Longrightarrow> right_total (A OO B)"
```
```   181   unfolding right_total_def OO_def by metis
```
```   182
```
```   183 lemma right_unique_OO:
```
```   184   "\<lbrakk>right_unique A; right_unique B\<rbrakk> \<Longrightarrow> right_unique (A OO B)"
```
```   185   unfolding right_unique_def OO_def by metis
```
```   186
```
```   187
```
```   188 subsection {* Properties of relators *}
```
```   189
```
```   190 lemma right_total_eq [transfer_rule]: "right_total (op =)"
```
```   191   unfolding right_total_def by simp
```
```   192
```
```   193 lemma right_unique_eq [transfer_rule]: "right_unique (op =)"
```
```   194   unfolding right_unique_def by simp
```
```   195
```
```   196 lemma bi_total_eq [transfer_rule]: "bi_total (op =)"
```
```   197   unfolding bi_total_def by simp
```
```   198
```
```   199 lemma bi_unique_eq [transfer_rule]: "bi_unique (op =)"
```
```   200   unfolding bi_unique_def by simp
```
```   201
```
```   202 lemma right_total_fun [transfer_rule]:
```
```   203   "\<lbrakk>right_unique A; right_total B\<rbrakk> \<Longrightarrow> right_total (A ===> B)"
```
```   204   unfolding right_total_def fun_rel_def
```
```   205   apply (rule allI, rename_tac g)
```
```   206   apply (rule_tac x="\<lambda>x. SOME z. B z (g (THE y. A x y))" in exI)
```
```   207   apply clarify
```
```   208   apply (subgoal_tac "(THE y. A x y) = y", simp)
```
```   209   apply (rule someI_ex)
```
```   210   apply (simp)
```
```   211   apply (rule the_equality)
```
```   212   apply assumption
```
```   213   apply (simp add: right_unique_def)
```
```   214   done
```
```   215
```
```   216 lemma right_unique_fun [transfer_rule]:
```
```   217   "\<lbrakk>right_total A; right_unique B\<rbrakk> \<Longrightarrow> right_unique (A ===> B)"
```
```   218   unfolding right_total_def right_unique_def fun_rel_def
```
```   219   by (clarify, rule ext, fast)
```
```   220
```
```   221 lemma bi_total_fun [transfer_rule]:
```
```   222   "\<lbrakk>bi_unique A; bi_total B\<rbrakk> \<Longrightarrow> bi_total (A ===> B)"
```
```   223   unfolding bi_total_def fun_rel_def
```
```   224   apply safe
```
```   225   apply (rename_tac f)
```
```   226   apply (rule_tac x="\<lambda>y. SOME z. B (f (THE x. A x y)) z" in exI)
```
```   227   apply clarify
```
```   228   apply (subgoal_tac "(THE x. A x y) = x", simp)
```
```   229   apply (rule someI_ex)
```
```   230   apply (simp)
```
```   231   apply (rule the_equality)
```
```   232   apply assumption
```
```   233   apply (simp add: bi_unique_def)
```
```   234   apply (rename_tac g)
```
```   235   apply (rule_tac x="\<lambda>x. SOME z. B z (g (THE y. A x y))" in exI)
```
```   236   apply clarify
```
```   237   apply (subgoal_tac "(THE y. A x y) = y", simp)
```
```   238   apply (rule someI_ex)
```
```   239   apply (simp)
```
```   240   apply (rule the_equality)
```
```   241   apply assumption
```
```   242   apply (simp add: bi_unique_def)
```
```   243   done
```
```   244
```
```   245 lemma bi_unique_fun [transfer_rule]:
```
```   246   "\<lbrakk>bi_total A; bi_unique B\<rbrakk> \<Longrightarrow> bi_unique (A ===> B)"
```
```   247   unfolding bi_total_def bi_unique_def fun_rel_def fun_eq_iff
```
```   248   by (safe, metis, fast)
```
```   249
```
```   250
```
```   251 subsection {* Transfer rules *}
```
```   252
```
```   253 text {* Transfer rules using implication instead of equality on booleans. *}
```
```   254
```
```   255 lemma eq_imp_transfer [transfer_rule]:
```
```   256   "right_unique A \<Longrightarrow> (A ===> A ===> op \<longrightarrow>) (op =) (op =)"
```
```   257   unfolding right_unique_alt_def .
```
```   258
```
```   259 lemma forall_imp_transfer [transfer_rule]:
```
```   260   "right_total A \<Longrightarrow> ((A ===> op \<longrightarrow>) ===> op \<longrightarrow>) transfer_forall transfer_forall"
```
```   261   unfolding right_total_alt_def transfer_forall_def .
```
```   262
```
```   263 lemma eq_transfer [transfer_rule]:
```
```   264   assumes "bi_unique A"
```
```   265   shows "(A ===> A ===> op =) (op =) (op =)"
```
```   266   using assms unfolding bi_unique_def fun_rel_def by auto
```
```   267
```
```   268 lemma Domainp_iff: "Domainp T x \<longleftrightarrow> (\<exists>y. T x y)"
```
```   269   by auto
```
```   270
```
```   271 lemma right_total_Ex_transfer[transfer_rule]:
```
```   272   assumes "right_total A"
```
```   273   shows "((A ===> op=) ===> op=) (Bex (Collect (Domainp A))) Ex"
```
```   274 using assms unfolding right_total_def Bex_def fun_rel_def Domainp_iff[abs_def]
```
```   275 by blast
```
```   276
```
```   277 lemma right_total_All_transfer[transfer_rule]:
```
```   278   assumes "right_total A"
```
```   279   shows "((A ===> op =) ===> op =) (Ball (Collect (Domainp A))) All"
```
```   280 using assms unfolding right_total_def Ball_def fun_rel_def Domainp_iff[abs_def]
```
```   281 by blast
```
```   282
```
```   283 lemma All_transfer [transfer_rule]:
```
```   284   assumes "bi_total A"
```
```   285   shows "((A ===> op =) ===> op =) All All"
```
```   286   using assms unfolding bi_total_def fun_rel_def by fast
```
```   287
```
```   288 lemma Ex_transfer [transfer_rule]:
```
```   289   assumes "bi_total A"
```
```   290   shows "((A ===> op =) ===> op =) Ex Ex"
```
```   291   using assms unfolding bi_total_def fun_rel_def by fast
```
```   292
```
```   293 lemma If_transfer [transfer_rule]: "(op = ===> A ===> A ===> A) If If"
```
```   294   unfolding fun_rel_def by simp
```
```   295
```
```   296 lemma Let_transfer [transfer_rule]: "(A ===> (A ===> B) ===> B) Let Let"
```
```   297   unfolding fun_rel_def by simp
```
```   298
```
```   299 lemma id_transfer [transfer_rule]: "(A ===> A) id id"
```
```   300   unfolding fun_rel_def by simp
```
```   301
```
```   302 lemma comp_transfer [transfer_rule]:
```
```   303   "((B ===> C) ===> (A ===> B) ===> (A ===> C)) (op \<circ>) (op \<circ>)"
```
```   304   unfolding fun_rel_def by simp
```
```   305
```
```   306 lemma fun_upd_transfer [transfer_rule]:
```
```   307   assumes [transfer_rule]: "bi_unique A"
```
```   308   shows "((A ===> B) ===> A ===> B ===> A ===> B) fun_upd fun_upd"
```
```   309   unfolding fun_upd_def [abs_def] by transfer_prover
```
```   310
```
```   311 lemma nat_case_transfer [transfer_rule]:
```
```   312   "(A ===> (op = ===> A) ===> op = ===> A) nat_case nat_case"
```
```   313   unfolding fun_rel_def by (simp split: nat.split)
```
```   314
```
```   315 lemma nat_rec_transfer [transfer_rule]:
```
```   316   "(A ===> (op = ===> A ===> A) ===> op = ===> A) nat_rec nat_rec"
```
```   317   unfolding fun_rel_def by (clarsimp, rename_tac n, induct_tac n, simp_all)
```
```   318
```
```   319 lemma funpow_transfer [transfer_rule]:
```
```   320   "(op = ===> (A ===> A) ===> (A ===> A)) compow compow"
```
```   321   unfolding funpow_def by transfer_prover
```
```   322
```
```   323 lemma Domainp_forall_transfer [transfer_rule]:
```
```   324   assumes "right_total A"
```
```   325   shows "((A ===> op =) ===> op =)
```
```   326     (transfer_bforall (Domainp A)) transfer_forall"
```
```   327   using assms unfolding right_total_def
```
```   328   unfolding transfer_forall_def transfer_bforall_def fun_rel_def Domainp_iff
```
```   329   by metis
```
```   330
```
```   331 lemma forall_transfer [transfer_rule]:
```
```   332   "bi_total A \<Longrightarrow> ((A ===> op =) ===> op =) transfer_forall transfer_forall"
```
```   333   unfolding transfer_forall_def by (rule All_transfer)
```
```   334
```
```   335 end
```