src/HOL/Lifting.thy
 author Christian Sternagel Thu Aug 30 15:44:03 2012 +0900 (2012-08-30) changeset 49093 fdc301f592c4 parent 48891 c0eafbd55de3 child 51112 da97167e03f7 permissions -rw-r--r--
```     1 (*  Title:      HOL/Lifting.thy
```
```     2     Author:     Brian Huffman and Ondrej Kuncar
```
```     3     Author:     Cezary Kaliszyk and Christian Urban
```
```     4 *)
```
```     5
```
```     6 header {* Lifting package *}
```
```     7
```
```     8 theory Lifting
```
```     9 imports Plain Equiv_Relations Transfer
```
```    10 keywords
```
```    11   "print_quotmaps" "print_quotients" :: diag and
```
```    12   "lift_definition" :: thy_goal and
```
```    13   "setup_lifting" :: thy_decl
```
```    14 begin
```
```    15
```
```    16 subsection {* Function map *}
```
```    17
```
```    18 notation map_fun (infixr "--->" 55)
```
```    19
```
```    20 lemma map_fun_id:
```
```    21   "(id ---> id) = id"
```
```    22   by (simp add: fun_eq_iff)
```
```    23
```
```    24 subsection {* Quotient Predicate *}
```
```    25
```
```    26 definition
```
```    27   "Quotient R Abs Rep T \<longleftrightarrow>
```
```    28      (\<forall>a. Abs (Rep a) = a) \<and>
```
```    29      (\<forall>a. R (Rep a) (Rep a)) \<and>
```
```    30      (\<forall>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s) \<and>
```
```    31      T = (\<lambda>x y. R x x \<and> Abs x = y)"
```
```    32
```
```    33 lemma QuotientI:
```
```    34   assumes "\<And>a. Abs (Rep a) = a"
```
```    35     and "\<And>a. R (Rep a) (Rep a)"
```
```    36     and "\<And>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s"
```
```    37     and "T = (\<lambda>x y. R x x \<and> Abs x = y)"
```
```    38   shows "Quotient R Abs Rep T"
```
```    39   using assms unfolding Quotient_def by blast
```
```    40
```
```    41 context
```
```    42   fixes R Abs Rep T
```
```    43   assumes a: "Quotient R Abs Rep T"
```
```    44 begin
```
```    45
```
```    46 lemma Quotient_abs_rep: "Abs (Rep a) = a"
```
```    47   using a unfolding Quotient_def
```
```    48   by simp
```
```    49
```
```    50 lemma Quotient_rep_reflp: "R (Rep a) (Rep a)"
```
```    51   using a unfolding Quotient_def
```
```    52   by blast
```
```    53
```
```    54 lemma Quotient_rel:
```
```    55   "R r r \<and> R s s \<and> Abs r = Abs s \<longleftrightarrow> R r s" -- {* orientation does not loop on rewriting *}
```
```    56   using a unfolding Quotient_def
```
```    57   by blast
```
```    58
```
```    59 lemma Quotient_cr_rel: "T = (\<lambda>x y. R x x \<and> Abs x = y)"
```
```    60   using a unfolding Quotient_def
```
```    61   by blast
```
```    62
```
```    63 lemma Quotient_refl1: "R r s \<Longrightarrow> R r r"
```
```    64   using a unfolding Quotient_def
```
```    65   by fast
```
```    66
```
```    67 lemma Quotient_refl2: "R r s \<Longrightarrow> R s s"
```
```    68   using a unfolding Quotient_def
```
```    69   by fast
```
```    70
```
```    71 lemma Quotient_rel_rep: "R (Rep a) (Rep b) \<longleftrightarrow> a = b"
```
```    72   using a unfolding Quotient_def
```
```    73   by metis
```
```    74
```
```    75 lemma Quotient_rep_abs: "R r r \<Longrightarrow> R (Rep (Abs r)) r"
```
```    76   using a unfolding Quotient_def
```
```    77   by blast
```
```    78
```
```    79 lemma Quotient_rep_abs_fold_unmap:
```
```    80   assumes "x' \<equiv> Abs x" and "R x x" and "Rep x' \<equiv> Rep' x'"
```
```    81   shows "R (Rep' x') x"
```
```    82 proof -
```
```    83   have "R (Rep x') x" using assms(1-2) Quotient_rep_abs by auto
```
```    84   then show ?thesis using assms(3) by simp
```
```    85 qed
```
```    86
```
```    87 lemma Quotient_Rep_eq:
```
```    88   assumes "x' \<equiv> Abs x"
```
```    89   shows "Rep x' \<equiv> Rep x'"
```
```    90 by simp
```
```    91
```
```    92 lemma Quotient_rel_abs: "R r s \<Longrightarrow> Abs r = Abs s"
```
```    93   using a unfolding Quotient_def
```
```    94   by blast
```
```    95
```
```    96 lemma Quotient_rel_abs2:
```
```    97   assumes "R (Rep x) y"
```
```    98   shows "x = Abs y"
```
```    99 proof -
```
```   100   from assms have "Abs (Rep x) = Abs y" by (auto intro: Quotient_rel_abs)
```
```   101   then show ?thesis using assms(1) by (simp add: Quotient_abs_rep)
```
```   102 qed
```
```   103
```
```   104 lemma Quotient_symp: "symp R"
```
```   105   using a unfolding Quotient_def using sympI by (metis (full_types))
```
```   106
```
```   107 lemma Quotient_transp: "transp R"
```
```   108   using a unfolding Quotient_def using transpI by (metis (full_types))
```
```   109
```
```   110 lemma Quotient_part_equivp: "part_equivp R"
```
```   111 by (metis Quotient_rep_reflp Quotient_symp Quotient_transp part_equivpI)
```
```   112
```
```   113 end
```
```   114
```
```   115 lemma identity_quotient: "Quotient (op =) id id (op =)"
```
```   116 unfolding Quotient_def by simp
```
```   117
```
```   118 text {* TODO: Use one of these alternatives as the real definition. *}
```
```   119
```
```   120 lemma Quotient_alt_def:
```
```   121   "Quotient R Abs Rep T \<longleftrightarrow>
```
```   122     (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and>
```
```   123     (\<forall>b. T (Rep b) b) \<and>
```
```   124     (\<forall>x y. R x y \<longleftrightarrow> T x (Abs x) \<and> T y (Abs y) \<and> Abs x = Abs y)"
```
```   125 apply safe
```
```   126 apply (simp (no_asm_use) only: Quotient_def, fast)
```
```   127 apply (simp (no_asm_use) only: Quotient_def, fast)
```
```   128 apply (simp (no_asm_use) only: Quotient_def, fast)
```
```   129 apply (simp (no_asm_use) only: Quotient_def, fast)
```
```   130 apply (simp (no_asm_use) only: Quotient_def, fast)
```
```   131 apply (simp (no_asm_use) only: Quotient_def, fast)
```
```   132 apply (rule QuotientI)
```
```   133 apply simp
```
```   134 apply metis
```
```   135 apply simp
```
```   136 apply (rule ext, rule ext, metis)
```
```   137 done
```
```   138
```
```   139 lemma Quotient_alt_def2:
```
```   140   "Quotient R Abs Rep T \<longleftrightarrow>
```
```   141     (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and>
```
```   142     (\<forall>b. T (Rep b) b) \<and>
```
```   143     (\<forall>x y. R x y \<longleftrightarrow> T x (Abs y) \<and> T y (Abs x))"
```
```   144   unfolding Quotient_alt_def by (safe, metis+)
```
```   145
```
```   146 lemma Quotient_alt_def3:
```
```   147   "Quotient R Abs Rep T \<longleftrightarrow>
```
```   148     (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and> (\<forall>b. T (Rep b) b) \<and>
```
```   149     (\<forall>x y. R x y \<longleftrightarrow> (\<exists>z. T x z \<and> T y z))"
```
```   150   unfolding Quotient_alt_def2 by (safe, metis+)
```
```   151
```
```   152 lemma Quotient_alt_def4:
```
```   153   "Quotient R Abs Rep T \<longleftrightarrow>
```
```   154     (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and> (\<forall>b. T (Rep b) b) \<and> R = T OO conversep T"
```
```   155   unfolding Quotient_alt_def3 fun_eq_iff by auto
```
```   156
```
```   157 lemma fun_quotient:
```
```   158   assumes 1: "Quotient R1 abs1 rep1 T1"
```
```   159   assumes 2: "Quotient R2 abs2 rep2 T2"
```
```   160   shows "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2) (T1 ===> T2)"
```
```   161   using assms unfolding Quotient_alt_def2
```
```   162   unfolding fun_rel_def fun_eq_iff map_fun_apply
```
```   163   by (safe, metis+)
```
```   164
```
```   165 lemma apply_rsp:
```
```   166   fixes f g::"'a \<Rightarrow> 'c"
```
```   167   assumes q: "Quotient R1 Abs1 Rep1 T1"
```
```   168   and     a: "(R1 ===> R2) f g" "R1 x y"
```
```   169   shows "R2 (f x) (g y)"
```
```   170   using a by (auto elim: fun_relE)
```
```   171
```
```   172 lemma apply_rsp':
```
```   173   assumes a: "(R1 ===> R2) f g" "R1 x y"
```
```   174   shows "R2 (f x) (g y)"
```
```   175   using a by (auto elim: fun_relE)
```
```   176
```
```   177 lemma apply_rsp'':
```
```   178   assumes "Quotient R Abs Rep T"
```
```   179   and "(R ===> S) f f"
```
```   180   shows "S (f (Rep x)) (f (Rep x))"
```
```   181 proof -
```
```   182   from assms(1) have "R (Rep x) (Rep x)" by (rule Quotient_rep_reflp)
```
```   183   then show ?thesis using assms(2) by (auto intro: apply_rsp')
```
```   184 qed
```
```   185
```
```   186 subsection {* Quotient composition *}
```
```   187
```
```   188 lemma Quotient_compose:
```
```   189   assumes 1: "Quotient R1 Abs1 Rep1 T1"
```
```   190   assumes 2: "Quotient R2 Abs2 Rep2 T2"
```
```   191   shows "Quotient (T1 OO R2 OO conversep T1) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2) (T1 OO T2)"
```
```   192   using assms unfolding Quotient_alt_def4 by (auto intro!: ext)
```
```   193
```
```   194 lemma equivp_reflp2:
```
```   195   "equivp R \<Longrightarrow> reflp R"
```
```   196   by (erule equivpE)
```
```   197
```
```   198 subsection {* Respects predicate *}
```
```   199
```
```   200 definition Respects :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set"
```
```   201   where "Respects R = {x. R x x}"
```
```   202
```
```   203 lemma in_respects: "x \<in> Respects R \<longleftrightarrow> R x x"
```
```   204   unfolding Respects_def by simp
```
```   205
```
```   206 subsection {* Invariant *}
```
```   207
```
```   208 definition invariant :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
```
```   209   where "invariant R = (\<lambda>x y. R x \<and> x = y)"
```
```   210
```
```   211 lemma invariant_to_eq:
```
```   212   assumes "invariant P x y"
```
```   213   shows "x = y"
```
```   214 using assms by (simp add: invariant_def)
```
```   215
```
```   216 lemma fun_rel_eq_invariant:
```
```   217   shows "((invariant R) ===> S) = (\<lambda>f g. \<forall>x. R x \<longrightarrow> S (f x) (g x))"
```
```   218 by (auto simp add: invariant_def fun_rel_def)
```
```   219
```
```   220 lemma invariant_same_args:
```
```   221   shows "invariant P x x \<equiv> P x"
```
```   222 using assms by (auto simp add: invariant_def)
```
```   223
```
```   224 lemma UNIV_typedef_to_Quotient:
```
```   225   assumes "type_definition Rep Abs UNIV"
```
```   226   and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"
```
```   227   shows "Quotient (op =) Abs Rep T"
```
```   228 proof -
```
```   229   interpret type_definition Rep Abs UNIV by fact
```
```   230   from Abs_inject Rep_inverse Abs_inverse T_def show ?thesis
```
```   231     by (fastforce intro!: QuotientI fun_eq_iff)
```
```   232 qed
```
```   233
```
```   234 lemma UNIV_typedef_to_equivp:
```
```   235   fixes Abs :: "'a \<Rightarrow> 'b"
```
```   236   and Rep :: "'b \<Rightarrow> 'a"
```
```   237   assumes "type_definition Rep Abs (UNIV::'a set)"
```
```   238   shows "equivp (op=::'a\<Rightarrow>'a\<Rightarrow>bool)"
```
```   239 by (rule identity_equivp)
```
```   240
```
```   241 lemma typedef_to_Quotient:
```
```   242   assumes "type_definition Rep Abs S"
```
```   243   and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"
```
```   244   shows "Quotient (invariant (\<lambda>x. x \<in> S)) Abs Rep T"
```
```   245 proof -
```
```   246   interpret type_definition Rep Abs S by fact
```
```   247   from Rep Abs_inject Rep_inverse Abs_inverse T_def show ?thesis
```
```   248     by (auto intro!: QuotientI simp: invariant_def fun_eq_iff)
```
```   249 qed
```
```   250
```
```   251 lemma typedef_to_part_equivp:
```
```   252   assumes "type_definition Rep Abs S"
```
```   253   shows "part_equivp (invariant (\<lambda>x. x \<in> S))"
```
```   254 proof (intro part_equivpI)
```
```   255   interpret type_definition Rep Abs S by fact
```
```   256   show "\<exists>x. invariant (\<lambda>x. x \<in> S) x x" using Rep by (auto simp: invariant_def)
```
```   257 next
```
```   258   show "symp (invariant (\<lambda>x. x \<in> S))" by (auto intro: sympI simp: invariant_def)
```
```   259 next
```
```   260   show "transp (invariant (\<lambda>x. x \<in> S))" by (auto intro: transpI simp: invariant_def)
```
```   261 qed
```
```   262
```
```   263 lemma open_typedef_to_Quotient:
```
```   264   assumes "type_definition Rep Abs {x. P x}"
```
```   265   and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"
```
```   266   shows "Quotient (invariant P) Abs Rep T"
```
```   267   using typedef_to_Quotient [OF assms] by simp
```
```   268
```
```   269 lemma open_typedef_to_part_equivp:
```
```   270   assumes "type_definition Rep Abs {x. P x}"
```
```   271   shows "part_equivp (invariant P)"
```
```   272   using typedef_to_part_equivp [OF assms] by simp
```
```   273
```
```   274 text {* Generating transfer rules for quotients. *}
```
```   275
```
```   276 context
```
```   277   fixes R Abs Rep T
```
```   278   assumes 1: "Quotient R Abs Rep T"
```
```   279 begin
```
```   280
```
```   281 lemma Quotient_right_unique: "right_unique T"
```
```   282   using 1 unfolding Quotient_alt_def right_unique_def by metis
```
```   283
```
```   284 lemma Quotient_right_total: "right_total T"
```
```   285   using 1 unfolding Quotient_alt_def right_total_def by metis
```
```   286
```
```   287 lemma Quotient_rel_eq_transfer: "(T ===> T ===> op =) R (op =)"
```
```   288   using 1 unfolding Quotient_alt_def fun_rel_def by simp
```
```   289
```
```   290 lemma Quotient_abs_induct:
```
```   291   assumes "\<And>y. R y y \<Longrightarrow> P (Abs y)" shows "P x"
```
```   292   using 1 assms unfolding Quotient_def by metis
```
```   293
```
```   294 lemma Quotient_All_transfer:
```
```   295   "((T ===> op =) ===> op =) (Ball (Respects R)) All"
```
```   296   unfolding fun_rel_def Respects_def Quotient_cr_rel [OF 1]
```
```   297   by (auto, metis Quotient_abs_induct)
```
```   298
```
```   299 lemma Quotient_Ex_transfer:
```
```   300   "((T ===> op =) ===> op =) (Bex (Respects R)) Ex"
```
```   301   unfolding fun_rel_def Respects_def Quotient_cr_rel [OF 1]
```
```   302   by (auto, metis Quotient_rep_reflp [OF 1] Quotient_abs_rep [OF 1])
```
```   303
```
```   304 lemma Quotient_forall_transfer:
```
```   305   "((T ===> op =) ===> op =) (transfer_bforall (\<lambda>x. R x x)) transfer_forall"
```
```   306   using Quotient_All_transfer
```
```   307   unfolding transfer_forall_def transfer_bforall_def
```
```   308     Ball_def [abs_def] in_respects .
```
```   309
```
```   310 end
```
```   311
```
```   312 text {* Generating transfer rules for total quotients. *}
```
```   313
```
```   314 context
```
```   315   fixes R Abs Rep T
```
```   316   assumes 1: "Quotient R Abs Rep T" and 2: "reflp R"
```
```   317 begin
```
```   318
```
```   319 lemma Quotient_bi_total: "bi_total T"
```
```   320   using 1 2 unfolding Quotient_alt_def bi_total_def reflp_def by auto
```
```   321
```
```   322 lemma Quotient_id_abs_transfer: "(op = ===> T) (\<lambda>x. x) Abs"
```
```   323   using 1 2 unfolding Quotient_alt_def reflp_def fun_rel_def by simp
```
```   324
```
```   325 lemma Quotient_total_abs_induct: "(\<And>y. P (Abs y)) \<Longrightarrow> P x"
```
```   326   using 1 2 assms unfolding Quotient_alt_def reflp_def by metis
```
```   327
```
```   328 lemma Quotient_total_abs_eq_iff: "Abs x = Abs y \<longleftrightarrow> R x y"
```
```   329   using Quotient_rel [OF 1] 2 unfolding reflp_def by simp
```
```   330
```
```   331 end
```
```   332
```
```   333 text {* Generating transfer rules for a type defined with @{text "typedef"}. *}
```
```   334
```
```   335 context
```
```   336   fixes Rep Abs A T
```
```   337   assumes type: "type_definition Rep Abs A"
```
```   338   assumes T_def: "T \<equiv> (\<lambda>(x::'a) (y::'b). x = Rep y)"
```
```   339 begin
```
```   340
```
```   341 lemma typedef_bi_unique: "bi_unique T"
```
```   342   unfolding bi_unique_def T_def
```
```   343   by (simp add: type_definition.Rep_inject [OF type])
```
```   344
```
```   345 lemma typedef_rep_transfer: "(T ===> op =) (\<lambda>x. x) Rep"
```
```   346   unfolding fun_rel_def T_def by simp
```
```   347
```
```   348 lemma typedef_All_transfer: "((T ===> op =) ===> op =) (Ball A) All"
```
```   349   unfolding T_def fun_rel_def
```
```   350   by (metis type_definition.Rep [OF type]
```
```   351     type_definition.Abs_inverse [OF type])
```
```   352
```
```   353 lemma typedef_Ex_transfer: "((T ===> op =) ===> op =) (Bex A) Ex"
```
```   354   unfolding T_def fun_rel_def
```
```   355   by (metis type_definition.Rep [OF type]
```
```   356     type_definition.Abs_inverse [OF type])
```
```   357
```
```   358 lemma typedef_forall_transfer:
```
```   359   "((T ===> op =) ===> op =)
```
```   360     (transfer_bforall (\<lambda>x. x \<in> A)) transfer_forall"
```
```   361   unfolding transfer_bforall_def transfer_forall_def Ball_def [symmetric]
```
```   362   by (rule typedef_All_transfer)
```
```   363
```
```   364 end
```
```   365
```
```   366 text {* Generating the correspondence rule for a constant defined with
```
```   367   @{text "lift_definition"}. *}
```
```   368
```
```   369 lemma Quotient_to_transfer:
```
```   370   assumes "Quotient R Abs Rep T" and "R c c" and "c' \<equiv> Abs c"
```
```   371   shows "T c c'"
```
```   372   using assms by (auto dest: Quotient_cr_rel)
```
```   373
```
```   374 text {* Proving reflexivity *}
```
```   375
```
```   376 definition left_total :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
```
```   377   where "left_total R \<longleftrightarrow> (\<forall>x. \<exists>y. R x y)"
```
```   378
```
```   379 lemma left_totalI:
```
```   380   "(\<And>x. \<exists>y. R x y) \<Longrightarrow> left_total R"
```
```   381 unfolding left_total_def by blast
```
```   382
```
```   383 lemma left_totalE:
```
```   384   assumes "left_total R"
```
```   385   obtains "(\<And>x. \<exists>y. R x y)"
```
```   386 using assms unfolding left_total_def by blast
```
```   387
```
```   388 lemma Quotient_to_left_total:
```
```   389   assumes q: "Quotient R Abs Rep T"
```
```   390   and r_R: "reflp R"
```
```   391   shows "left_total T"
```
```   392 using r_R Quotient_cr_rel[OF q] unfolding left_total_def by (auto elim: reflpE)
```
```   393
```
```   394 lemma reflp_Quotient_composition:
```
```   395   assumes lt_R1: "left_total R1"
```
```   396   and r_R2: "reflp R2"
```
```   397   shows "reflp (R1 OO R2 OO R1\<inverse>\<inverse>)"
```
```   398 using assms
```
```   399 proof -
```
```   400   {
```
```   401     fix x
```
```   402     from lt_R1 obtain y where "R1 x y" unfolding left_total_def by blast
```
```   403     moreover then have "R1\<inverse>\<inverse> y x" by simp
```
```   404     moreover have "R2 y y" using r_R2 by (auto elim: reflpE)
```
```   405     ultimately have "(R1 OO R2 OO R1\<inverse>\<inverse>) x x" by auto
```
```   406   }
```
```   407   then show ?thesis by (auto intro: reflpI)
```
```   408 qed
```
```   409
```
```   410 lemma reflp_equality: "reflp (op =)"
```
```   411 by (auto intro: reflpI)
```
```   412
```
```   413 subsection {* ML setup *}
```
```   414
```
```   415 ML_file "Tools/Lifting/lifting_util.ML"
```
```   416
```
```   417 ML_file "Tools/Lifting/lifting_info.ML"
```
```   418 setup Lifting_Info.setup
```
```   419
```
```   420 declare fun_quotient[quot_map]
```
```   421 lemmas [reflexivity_rule] = reflp_equality reflp_Quotient_composition
```
```   422
```
```   423 ML_file "Tools/Lifting/lifting_term.ML"
```
```   424
```
```   425 ML_file "Tools/Lifting/lifting_def.ML"
```
```   426
```
```   427 ML_file "Tools/Lifting/lifting_setup.ML"
```
```   428
```
```   429 hide_const (open) invariant
```
```   430
```
```   431 end
```