src/HOL/Transfer.thy
 author huffman Fri Apr 20 22:54:13 2012 +0200 (2012-04-20) changeset 47636 b786388b4b3a parent 47635 ebb79474262c child 47637 7a34395441ff permissions -rw-r--r--
uniform naming scheme for transfer rules
```     1 (*  Title:      HOL/Transfer.thy
```
```     2     Author:     Brian Huffman, TU Muenchen
```
```     3 *)
```
```     4
```
```     5 header {* Generic theorem transfer using relations *}
```
```     6
```
```     7 theory Transfer
```
```     8 imports Plain Hilbert_Choice
```
```     9 uses ("Tools/transfer.ML")
```
```    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_relE:
```
```    30   assumes "(A ===> B) f g" and "A x y"
```
```    31   obtains "B (f x) (g y)"
```
```    32   using assms by (simp add: fun_rel_def)
```
```    33
```
```    34 lemma fun_rel_eq:
```
```    35   shows "((op =) ===> (op =)) = (op =)"
```
```    36   by (auto simp add: fun_eq_iff elim: fun_relE)
```
```    37
```
```    38 lemma fun_rel_eq_rel:
```
```    39   shows "((op =) ===> R) = (\<lambda>f g. \<forall>x. R (f x) (g x))"
```
```    40   by (simp add: fun_rel_def)
```
```    41
```
```    42
```
```    43 subsection {* Transfer method *}
```
```    44
```
```    45 text {* Explicit tags for application, abstraction, and relation
```
```    46 membership allow for backward proof methods. *}
```
```    47
```
```    48 definition App :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
```
```    49   where "App f \<equiv> f"
```
```    50
```
```    51 definition Abs :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
```
```    52   where "Abs f \<equiv> f"
```
```    53
```
```    54 definition Rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"
```
```    55   where "Rel r \<equiv> r"
```
```    56
```
```    57 text {* Handling of meta-logic connectives *}
```
```    58
```
```    59 definition transfer_forall where
```
```    60   "transfer_forall \<equiv> All"
```
```    61
```
```    62 definition transfer_implies where
```
```    63   "transfer_implies \<equiv> op \<longrightarrow>"
```
```    64
```
```    65 definition transfer_bforall :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
```
```    66   where "transfer_bforall \<equiv> (\<lambda>P Q. \<forall>x. P x \<longrightarrow> Q x)"
```
```    67
```
```    68 lemma transfer_forall_eq: "(\<And>x. P x) \<equiv> Trueprop (transfer_forall (\<lambda>x. P x))"
```
```    69   unfolding atomize_all transfer_forall_def ..
```
```    70
```
```    71 lemma transfer_implies_eq: "(A \<Longrightarrow> B) \<equiv> Trueprop (transfer_implies A B)"
```
```    72   unfolding atomize_imp transfer_implies_def ..
```
```    73
```
```    74 lemma transfer_bforall_unfold:
```
```    75   "Trueprop (transfer_bforall P (\<lambda>x. Q x)) \<equiv> (\<And>x. P x \<Longrightarrow> Q x)"
```
```    76   unfolding transfer_bforall_def atomize_imp atomize_all ..
```
```    77
```
```    78 lemma transfer_start: "\<lbrakk>Rel (op =) P Q; P\<rbrakk> \<Longrightarrow> Q"
```
```    79   unfolding Rel_def by simp
```
```    80
```
```    81 lemma transfer_start': "\<lbrakk>Rel (op \<longrightarrow>) P Q; P\<rbrakk> \<Longrightarrow> Q"
```
```    82   unfolding Rel_def by simp
```
```    83
```
```    84 lemma transfer_prover_start: "\<lbrakk>x = x'; Rel R x' y\<rbrakk> \<Longrightarrow> Rel R x y"
```
```    85   by simp
```
```    86
```
```    87 lemma Rel_eq_refl: "Rel (op =) x x"
```
```    88   unfolding Rel_def ..
```
```    89
```
```    90 lemma Rel_App:
```
```    91   assumes "Rel (A ===> B) f g" and "Rel A x y"
```
```    92   shows "Rel B (App f x) (App g y)"
```
```    93   using assms unfolding Rel_def App_def fun_rel_def by fast
```
```    94
```
```    95 lemma Rel_Abs:
```
```    96   assumes "\<And>x y. Rel A x y \<Longrightarrow> Rel B (f x) (g y)"
```
```    97   shows "Rel (A ===> B) (Abs (\<lambda>x. f x)) (Abs (\<lambda>y. g y))"
```
```    98   using assms unfolding Rel_def Abs_def fun_rel_def by fast
```
```    99
```
```   100 use "Tools/transfer.ML"
```
```   101
```
```   102 setup Transfer.setup
```
```   103
```
```   104 declare fun_rel_eq [relator_eq]
```
```   105
```
```   106 hide_const (open) App Abs Rel
```
```   107
```
```   108
```
```   109 subsection {* Predicates on relations, i.e. ``class constraints'' *}
```
```   110
```
```   111 definition right_total :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
```
```   112   where "right_total R \<longleftrightarrow> (\<forall>y. \<exists>x. R x y)"
```
```   113
```
```   114 definition right_unique :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
```
```   115   where "right_unique R \<longleftrightarrow> (\<forall>x y z. R x y \<longrightarrow> R x z \<longrightarrow> y = z)"
```
```   116
```
```   117 definition bi_total :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
```
```   118   where "bi_total R \<longleftrightarrow> (\<forall>x. \<exists>y. R x y) \<and> (\<forall>y. \<exists>x. R x y)"
```
```   119
```
```   120 definition bi_unique :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
```
```   121   where "bi_unique R \<longleftrightarrow>
```
```   122     (\<forall>x y z. R x y \<longrightarrow> R x z \<longrightarrow> y = z) \<and>
```
```   123     (\<forall>x y z. R x z \<longrightarrow> R y z \<longrightarrow> x = y)"
```
```   124
```
```   125 lemma right_total_alt_def:
```
```   126   "right_total R \<longleftrightarrow> ((R ===> op \<longrightarrow>) ===> op \<longrightarrow>) All All"
```
```   127   unfolding right_total_def fun_rel_def
```
```   128   apply (rule iffI, fast)
```
```   129   apply (rule allI)
```
```   130   apply (drule_tac x="\<lambda>x. True" in spec)
```
```   131   apply (drule_tac x="\<lambda>y. \<exists>x. R x y" in spec)
```
```   132   apply fast
```
```   133   done
```
```   134
```
```   135 lemma right_unique_alt_def:
```
```   136   "right_unique R \<longleftrightarrow> (R ===> R ===> op \<longrightarrow>) (op =) (op =)"
```
```   137   unfolding right_unique_def fun_rel_def by auto
```
```   138
```
```   139 lemma bi_total_alt_def:
```
```   140   "bi_total R \<longleftrightarrow> ((R ===> op =) ===> op =) All All"
```
```   141   unfolding bi_total_def fun_rel_def
```
```   142   apply (rule iffI, fast)
```
```   143   apply safe
```
```   144   apply (drule_tac x="\<lambda>x. \<exists>y. R x y" in spec)
```
```   145   apply (drule_tac x="\<lambda>y. True" in spec)
```
```   146   apply fast
```
```   147   apply (drule_tac x="\<lambda>x. True" in spec)
```
```   148   apply (drule_tac x="\<lambda>y. \<exists>x. R x y" in spec)
```
```   149   apply fast
```
```   150   done
```
```   151
```
```   152 lemma bi_unique_alt_def:
```
```   153   "bi_unique R \<longleftrightarrow> (R ===> R ===> op =) (op =) (op =)"
```
```   154   unfolding bi_unique_def fun_rel_def by auto
```
```   155
```
```   156
```
```   157 subsection {* Properties of relators *}
```
```   158
```
```   159 lemma right_total_eq [transfer_rule]: "right_total (op =)"
```
```   160   unfolding right_total_def by simp
```
```   161
```
```   162 lemma right_unique_eq [transfer_rule]: "right_unique (op =)"
```
```   163   unfolding right_unique_def by simp
```
```   164
```
```   165 lemma bi_total_eq [transfer_rule]: "bi_total (op =)"
```
```   166   unfolding bi_total_def by simp
```
```   167
```
```   168 lemma bi_unique_eq [transfer_rule]: "bi_unique (op =)"
```
```   169   unfolding bi_unique_def by simp
```
```   170
```
```   171 lemma right_total_fun [transfer_rule]:
```
```   172   "\<lbrakk>right_unique A; right_total B\<rbrakk> \<Longrightarrow> right_total (A ===> B)"
```
```   173   unfolding right_total_def fun_rel_def
```
```   174   apply (rule allI, rename_tac g)
```
```   175   apply (rule_tac x="\<lambda>x. SOME z. B z (g (THE y. A x y))" in exI)
```
```   176   apply clarify
```
```   177   apply (subgoal_tac "(THE y. A x y) = y", simp)
```
```   178   apply (rule someI_ex)
```
```   179   apply (simp)
```
```   180   apply (rule the_equality)
```
```   181   apply assumption
```
```   182   apply (simp add: right_unique_def)
```
```   183   done
```
```   184
```
```   185 lemma right_unique_fun [transfer_rule]:
```
```   186   "\<lbrakk>right_total A; right_unique B\<rbrakk> \<Longrightarrow> right_unique (A ===> B)"
```
```   187   unfolding right_total_def right_unique_def fun_rel_def
```
```   188   by (clarify, rule ext, fast)
```
```   189
```
```   190 lemma bi_total_fun [transfer_rule]:
```
```   191   "\<lbrakk>bi_unique A; bi_total B\<rbrakk> \<Longrightarrow> bi_total (A ===> B)"
```
```   192   unfolding bi_total_def fun_rel_def
```
```   193   apply safe
```
```   194   apply (rename_tac f)
```
```   195   apply (rule_tac x="\<lambda>y. SOME z. B (f (THE x. A x y)) z" in exI)
```
```   196   apply clarify
```
```   197   apply (subgoal_tac "(THE x. A x y) = x", simp)
```
```   198   apply (rule someI_ex)
```
```   199   apply (simp)
```
```   200   apply (rule the_equality)
```
```   201   apply assumption
```
```   202   apply (simp add: bi_unique_def)
```
```   203   apply (rename_tac g)
```
```   204   apply (rule_tac x="\<lambda>x. SOME z. B z (g (THE y. A x y))" in exI)
```
```   205   apply clarify
```
```   206   apply (subgoal_tac "(THE y. A x y) = y", simp)
```
```   207   apply (rule someI_ex)
```
```   208   apply (simp)
```
```   209   apply (rule the_equality)
```
```   210   apply assumption
```
```   211   apply (simp add: bi_unique_def)
```
```   212   done
```
```   213
```
```   214 lemma bi_unique_fun [transfer_rule]:
```
```   215   "\<lbrakk>bi_total A; bi_unique B\<rbrakk> \<Longrightarrow> bi_unique (A ===> B)"
```
```   216   unfolding bi_total_def bi_unique_def fun_rel_def fun_eq_iff
```
```   217   by (safe, metis, fast)
```
```   218
```
```   219
```
```   220 subsection {* Transfer rules *}
```
```   221
```
```   222 lemma eq_transfer [transfer_rule]:
```
```   223   assumes "bi_unique A"
```
```   224   shows "(A ===> A ===> op =) (op =) (op =)"
```
```   225   using assms unfolding bi_unique_def fun_rel_def by auto
```
```   226
```
```   227 lemma All_transfer [transfer_rule]:
```
```   228   assumes "bi_total A"
```
```   229   shows "((A ===> op =) ===> op =) All All"
```
```   230   using assms unfolding bi_total_def fun_rel_def by fast
```
```   231
```
```   232 lemma Ex_transfer [transfer_rule]:
```
```   233   assumes "bi_total A"
```
```   234   shows "((A ===> op =) ===> op =) Ex Ex"
```
```   235   using assms unfolding bi_total_def fun_rel_def by fast
```
```   236
```
```   237 lemma If_transfer [transfer_rule]: "(op = ===> A ===> A ===> A) If If"
```
```   238   unfolding fun_rel_def by simp
```
```   239
```
```   240 lemma Let_transfer [transfer_rule]: "(A ===> (A ===> B) ===> B) Let Let"
```
```   241   unfolding fun_rel_def by simp
```
```   242
```
```   243 lemma id_transfer [transfer_rule]: "(A ===> A) id id"
```
```   244   unfolding fun_rel_def by simp
```
```   245
```
```   246 lemma comp_transfer [transfer_rule]:
```
```   247   "((B ===> C) ===> (A ===> B) ===> (A ===> C)) (op \<circ>) (op \<circ>)"
```
```   248   unfolding fun_rel_def by simp
```
```   249
```
```   250 lemma fun_upd_transfer [transfer_rule]:
```
```   251   assumes [transfer_rule]: "bi_unique A"
```
```   252   shows "((A ===> B) ===> A ===> B ===> A ===> B) fun_upd fun_upd"
```
```   253   unfolding fun_upd_def [abs_def] by transfer_prover
```
```   254
```
```   255 lemma Domainp_iff: "Domainp T x \<longleftrightarrow> (\<exists>y. T x y)"
```
```   256   by auto
```
```   257
```
```   258 text {* Fallback rule for transferring universal quantifiers over
```
```   259   correspondence relations that are not bi-total, and do not have
```
```   260   custom transfer rules (e.g. relations between function types). *}
```
```   261
```
```   262 lemma Domainp_forall_transfer [transfer_rule]:
```
```   263   assumes "right_total A"
```
```   264   shows "((A ===> op =) ===> op =)
```
```   265     (transfer_bforall (Domainp A)) transfer_forall"
```
```   266   using assms unfolding right_total_def
```
```   267   unfolding transfer_forall_def transfer_bforall_def fun_rel_def Domainp_iff
```
```   268   by metis
```
```   269
```
```   270 text {* Preferred rule for transferring universal quantifiers over
```
```   271   bi-total correspondence relations (later rules are tried first). *}
```
```   272
```
```   273 lemma forall_transfer [transfer_rule]:
```
```   274   "bi_total A \<Longrightarrow> ((A ===> op =) ===> op =) transfer_forall transfer_forall"
```
```   275   unfolding transfer_forall_def by (rule All_transfer)
```
```   276
```
```   277 end
```