src/HOL/Lifting.thy
 author blanchet Wed Sep 24 15:45:55 2014 +0200 (2014-09-24) changeset 58425 246985c6b20b parent 58186 a6c3962ea907 child 58821 11e226e8a095 permissions -rw-r--r--
simpler proof
```     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 Equiv_Relations Transfer
```
```    10 keywords
```
```    11   "parametric" and
```
```    12   "print_quot_maps" "print_quotients" :: diag and
```
```    13   "lift_definition" :: thy_goal and
```
```    14   "setup_lifting" "lifting_forget" "lifting_update" :: thy_decl
```
```    15 begin
```
```    16
```
```    17 subsection {* Function map *}
```
```    18
```
```    19 context
```
```    20 begin
```
```    21 interpretation lifting_syntax .
```
```    22
```
```    23 lemma map_fun_id:
```
```    24   "(id ---> id) = id"
```
```    25   by (simp add: fun_eq_iff)
```
```    26
```
```    27 subsection {* Quotient Predicate *}
```
```    28
```
```    29 definition
```
```    30   "Quotient R Abs Rep T \<longleftrightarrow>
```
```    31      (\<forall>a. Abs (Rep a) = a) \<and>
```
```    32      (\<forall>a. R (Rep a) (Rep a)) \<and>
```
```    33      (\<forall>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s) \<and>
```
```    34      T = (\<lambda>x y. R x x \<and> Abs x = y)"
```
```    35
```
```    36 lemma QuotientI:
```
```    37   assumes "\<And>a. Abs (Rep a) = a"
```
```    38     and "\<And>a. R (Rep a) (Rep a)"
```
```    39     and "\<And>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s"
```
```    40     and "T = (\<lambda>x y. R x x \<and> Abs x = y)"
```
```    41   shows "Quotient R Abs Rep T"
```
```    42   using assms unfolding Quotient_def by blast
```
```    43
```
```    44 context
```
```    45   fixes R Abs Rep T
```
```    46   assumes a: "Quotient R Abs Rep T"
```
```    47 begin
```
```    48
```
```    49 lemma Quotient_abs_rep: "Abs (Rep a) = a"
```
```    50   using a unfolding Quotient_def
```
```    51   by simp
```
```    52
```
```    53 lemma Quotient_rep_reflp: "R (Rep a) (Rep a)"
```
```    54   using a unfolding Quotient_def
```
```    55   by blast
```
```    56
```
```    57 lemma Quotient_rel:
```
```    58   "R r r \<and> R s s \<and> Abs r = Abs s \<longleftrightarrow> R r s" -- {* orientation does not loop on rewriting *}
```
```    59   using a unfolding Quotient_def
```
```    60   by blast
```
```    61
```
```    62 lemma Quotient_cr_rel: "T = (\<lambda>x y. R x x \<and> Abs x = y)"
```
```    63   using a unfolding Quotient_def
```
```    64   by blast
```
```    65
```
```    66 lemma Quotient_refl1: "R r s \<Longrightarrow> R r r"
```
```    67   using a unfolding Quotient_def
```
```    68   by fast
```
```    69
```
```    70 lemma Quotient_refl2: "R r s \<Longrightarrow> R s s"
```
```    71   using a unfolding Quotient_def
```
```    72   by fast
```
```    73
```
```    74 lemma Quotient_rel_rep: "R (Rep a) (Rep b) \<longleftrightarrow> a = b"
```
```    75   using a unfolding Quotient_def
```
```    76   by metis
```
```    77
```
```    78 lemma Quotient_rep_abs: "R r r \<Longrightarrow> R (Rep (Abs r)) r"
```
```    79   using a unfolding Quotient_def
```
```    80   by blast
```
```    81
```
```    82 lemma Quotient_rep_abs_eq: "R t t \<Longrightarrow> R \<le> op= \<Longrightarrow> Rep (Abs t) = t"
```
```    83   using a unfolding Quotient_def
```
```    84   by blast
```
```    85
```
```    86 lemma Quotient_rep_abs_fold_unmap:
```
```    87   assumes "x' \<equiv> Abs x" and "R x x" and "Rep x' \<equiv> Rep' x'"
```
```    88   shows "R (Rep' x') x"
```
```    89 proof -
```
```    90   have "R (Rep x') x" using assms(1-2) Quotient_rep_abs by auto
```
```    91   then show ?thesis using assms(3) by simp
```
```    92 qed
```
```    93
```
```    94 lemma Quotient_Rep_eq:
```
```    95   assumes "x' \<equiv> Abs x"
```
```    96   shows "Rep x' \<equiv> Rep x'"
```
```    97 by simp
```
```    98
```
```    99 lemma Quotient_rel_abs: "R r s \<Longrightarrow> Abs r = Abs s"
```
```   100   using a unfolding Quotient_def
```
```   101   by blast
```
```   102
```
```   103 lemma Quotient_rel_abs2:
```
```   104   assumes "R (Rep x) y"
```
```   105   shows "x = Abs y"
```
```   106 proof -
```
```   107   from assms have "Abs (Rep x) = Abs y" by (auto intro: Quotient_rel_abs)
```
```   108   then show ?thesis using assms(1) by (simp add: Quotient_abs_rep)
```
```   109 qed
```
```   110
```
```   111 lemma Quotient_symp: "symp R"
```
```   112   using a unfolding Quotient_def using sympI by (metis (full_types))
```
```   113
```
```   114 lemma Quotient_transp: "transp R"
```
```   115   using a unfolding Quotient_def using transpI by (metis (full_types))
```
```   116
```
```   117 lemma Quotient_part_equivp: "part_equivp R"
```
```   118 by (metis Quotient_rep_reflp Quotient_symp Quotient_transp part_equivpI)
```
```   119
```
```   120 end
```
```   121
```
```   122 lemma identity_quotient: "Quotient (op =) id id (op =)"
```
```   123 unfolding Quotient_def by simp
```
```   124
```
```   125 text {* TODO: Use one of these alternatives as the real definition. *}
```
```   126
```
```   127 lemma Quotient_alt_def:
```
```   128   "Quotient R Abs Rep T \<longleftrightarrow>
```
```   129     (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and>
```
```   130     (\<forall>b. T (Rep b) b) \<and>
```
```   131     (\<forall>x y. R x y \<longleftrightarrow> T x (Abs x) \<and> T y (Abs y) \<and> Abs x = Abs y)"
```
```   132 apply safe
```
```   133 apply (simp (no_asm_use) only: Quotient_def, fast)
```
```   134 apply (simp (no_asm_use) only: Quotient_def, fast)
```
```   135 apply (simp (no_asm_use) only: Quotient_def, fast)
```
```   136 apply (simp (no_asm_use) only: Quotient_def, fast)
```
```   137 apply (simp (no_asm_use) only: Quotient_def, fast)
```
```   138 apply (simp (no_asm_use) only: Quotient_def, fast)
```
```   139 apply (rule QuotientI)
```
```   140 apply simp
```
```   141 apply metis
```
```   142 apply simp
```
```   143 apply (rule ext, rule ext, metis)
```
```   144 done
```
```   145
```
```   146 lemma Quotient_alt_def2:
```
```   147   "Quotient R Abs Rep T \<longleftrightarrow>
```
```   148     (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and>
```
```   149     (\<forall>b. T (Rep b) b) \<and>
```
```   150     (\<forall>x y. R x y \<longleftrightarrow> T x (Abs y) \<and> T y (Abs x))"
```
```   151   unfolding Quotient_alt_def by (safe, metis+)
```
```   152
```
```   153 lemma Quotient_alt_def3:
```
```   154   "Quotient R Abs Rep T \<longleftrightarrow>
```
```   155     (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and> (\<forall>b. T (Rep b) b) \<and>
```
```   156     (\<forall>x y. R x y \<longleftrightarrow> (\<exists>z. T x z \<and> T y z))"
```
```   157   unfolding Quotient_alt_def2 by (safe, metis+)
```
```   158
```
```   159 lemma Quotient_alt_def4:
```
```   160   "Quotient R Abs Rep T \<longleftrightarrow>
```
```   161     (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and> (\<forall>b. T (Rep b) b) \<and> R = T OO conversep T"
```
```   162   unfolding Quotient_alt_def3 fun_eq_iff by auto
```
```   163
```
```   164 lemma Quotient_alt_def5:
```
```   165   "Quotient R Abs Rep T \<longleftrightarrow>
```
```   166     T \<le> BNF_Def.Grp UNIV Abs \<and> BNF_Def.Grp UNIV Rep \<le> T\<inverse>\<inverse> \<and> R = T OO T\<inverse>\<inverse>"
```
```   167   unfolding Quotient_alt_def4 Grp_def by blast
```
```   168
```
```   169 lemma fun_quotient:
```
```   170   assumes 1: "Quotient R1 abs1 rep1 T1"
```
```   171   assumes 2: "Quotient R2 abs2 rep2 T2"
```
```   172   shows "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2) (T1 ===> T2)"
```
```   173   using assms unfolding Quotient_alt_def2
```
```   174   unfolding rel_fun_def fun_eq_iff map_fun_apply
```
```   175   by (safe, metis+)
```
```   176
```
```   177 lemma apply_rsp:
```
```   178   fixes f g::"'a \<Rightarrow> 'c"
```
```   179   assumes q: "Quotient R1 Abs1 Rep1 T1"
```
```   180   and     a: "(R1 ===> R2) f g" "R1 x y"
```
```   181   shows "R2 (f x) (g y)"
```
```   182   using a by (auto elim: rel_funE)
```
```   183
```
```   184 lemma apply_rsp':
```
```   185   assumes a: "(R1 ===> R2) f g" "R1 x y"
```
```   186   shows "R2 (f x) (g y)"
```
```   187   using a by (auto elim: rel_funE)
```
```   188
```
```   189 lemma apply_rsp'':
```
```   190   assumes "Quotient R Abs Rep T"
```
```   191   and "(R ===> S) f f"
```
```   192   shows "S (f (Rep x)) (f (Rep x))"
```
```   193 proof -
```
```   194   from assms(1) have "R (Rep x) (Rep x)" by (rule Quotient_rep_reflp)
```
```   195   then show ?thesis using assms(2) by (auto intro: apply_rsp')
```
```   196 qed
```
```   197
```
```   198 subsection {* Quotient composition *}
```
```   199
```
```   200 lemma Quotient_compose:
```
```   201   assumes 1: "Quotient R1 Abs1 Rep1 T1"
```
```   202   assumes 2: "Quotient R2 Abs2 Rep2 T2"
```
```   203   shows "Quotient (T1 OO R2 OO conversep T1) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2) (T1 OO T2)"
```
```   204   using assms unfolding Quotient_alt_def4 by fastforce
```
```   205
```
```   206 lemma equivp_reflp2:
```
```   207   "equivp R \<Longrightarrow> reflp R"
```
```   208   by (erule equivpE)
```
```   209
```
```   210 subsection {* Respects predicate *}
```
```   211
```
```   212 definition Respects :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set"
```
```   213   where "Respects R = {x. R x x}"
```
```   214
```
```   215 lemma in_respects: "x \<in> Respects R \<longleftrightarrow> R x x"
```
```   216   unfolding Respects_def by simp
```
```   217
```
```   218 lemma UNIV_typedef_to_Quotient:
```
```   219   assumes "type_definition Rep Abs UNIV"
```
```   220   and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"
```
```   221   shows "Quotient (op =) Abs Rep T"
```
```   222 proof -
```
```   223   interpret type_definition Rep Abs UNIV by fact
```
```   224   from Abs_inject Rep_inverse Abs_inverse T_def show ?thesis
```
```   225     by (fastforce intro!: QuotientI fun_eq_iff)
```
```   226 qed
```
```   227
```
```   228 lemma UNIV_typedef_to_equivp:
```
```   229   fixes Abs :: "'a \<Rightarrow> 'b"
```
```   230   and Rep :: "'b \<Rightarrow> 'a"
```
```   231   assumes "type_definition Rep Abs (UNIV::'a set)"
```
```   232   shows "equivp (op=::'a\<Rightarrow>'a\<Rightarrow>bool)"
```
```   233 by (rule identity_equivp)
```
```   234
```
```   235 lemma typedef_to_Quotient:
```
```   236   assumes "type_definition Rep Abs S"
```
```   237   and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"
```
```   238   shows "Quotient (eq_onp (\<lambda>x. x \<in> S)) Abs Rep T"
```
```   239 proof -
```
```   240   interpret type_definition Rep Abs S by fact
```
```   241   from Rep Abs_inject Rep_inverse Abs_inverse T_def show ?thesis
```
```   242     by (auto intro!: QuotientI simp: eq_onp_def fun_eq_iff)
```
```   243 qed
```
```   244
```
```   245 lemma typedef_to_part_equivp:
```
```   246   assumes "type_definition Rep Abs S"
```
```   247   shows "part_equivp (eq_onp (\<lambda>x. x \<in> S))"
```
```   248 proof (intro part_equivpI)
```
```   249   interpret type_definition Rep Abs S by fact
```
```   250   show "\<exists>x. eq_onp (\<lambda>x. x \<in> S) x x" using Rep by (auto simp: eq_onp_def)
```
```   251 next
```
```   252   show "symp (eq_onp (\<lambda>x. x \<in> S))" by (auto intro: sympI simp: eq_onp_def)
```
```   253 next
```
```   254   show "transp (eq_onp (\<lambda>x. x \<in> S))" by (auto intro: transpI simp: eq_onp_def)
```
```   255 qed
```
```   256
```
```   257 lemma open_typedef_to_Quotient:
```
```   258   assumes "type_definition Rep Abs {x. P x}"
```
```   259   and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"
```
```   260   shows "Quotient (eq_onp P) Abs Rep T"
```
```   261   using typedef_to_Quotient [OF assms] by simp
```
```   262
```
```   263 lemma open_typedef_to_part_equivp:
```
```   264   assumes "type_definition Rep Abs {x. P x}"
```
```   265   shows "part_equivp (eq_onp P)"
```
```   266   using typedef_to_part_equivp [OF assms] by simp
```
```   267
```
```   268 text {* Generating transfer rules for quotients. *}
```
```   269
```
```   270 context
```
```   271   fixes R Abs Rep T
```
```   272   assumes 1: "Quotient R Abs Rep T"
```
```   273 begin
```
```   274
```
```   275 lemma Quotient_right_unique: "right_unique T"
```
```   276   using 1 unfolding Quotient_alt_def right_unique_def by metis
```
```   277
```
```   278 lemma Quotient_right_total: "right_total T"
```
```   279   using 1 unfolding Quotient_alt_def right_total_def by metis
```
```   280
```
```   281 lemma Quotient_rel_eq_transfer: "(T ===> T ===> op =) R (op =)"
```
```   282   using 1 unfolding Quotient_alt_def rel_fun_def by simp
```
```   283
```
```   284 lemma Quotient_abs_induct:
```
```   285   assumes "\<And>y. R y y \<Longrightarrow> P (Abs y)" shows "P x"
```
```   286   using 1 assms unfolding Quotient_def by metis
```
```   287
```
```   288 end
```
```   289
```
```   290 text {* Generating transfer rules for total quotients. *}
```
```   291
```
```   292 context
```
```   293   fixes R Abs Rep T
```
```   294   assumes 1: "Quotient R Abs Rep T" and 2: "reflp R"
```
```   295 begin
```
```   296
```
```   297 lemma Quotient_left_total: "left_total T"
```
```   298   using 1 2 unfolding Quotient_alt_def left_total_def reflp_def by auto
```
```   299
```
```   300 lemma Quotient_bi_total: "bi_total T"
```
```   301   using 1 2 unfolding Quotient_alt_def bi_total_def reflp_def by auto
```
```   302
```
```   303 lemma Quotient_id_abs_transfer: "(op = ===> T) (\<lambda>x. x) Abs"
```
```   304   using 1 2 unfolding Quotient_alt_def reflp_def rel_fun_def by simp
```
```   305
```
```   306 lemma Quotient_total_abs_induct: "(\<And>y. P (Abs y)) \<Longrightarrow> P x"
```
```   307   using 1 2 assms unfolding Quotient_alt_def reflp_def by metis
```
```   308
```
```   309 lemma Quotient_total_abs_eq_iff: "Abs x = Abs y \<longleftrightarrow> R x y"
```
```   310   using Quotient_rel [OF 1] 2 unfolding reflp_def by simp
```
```   311
```
```   312 end
```
```   313
```
```   314 text {* Generating transfer rules for a type defined with @{text "typedef"}. *}
```
```   315
```
```   316 context
```
```   317   fixes Rep Abs A T
```
```   318   assumes type: "type_definition Rep Abs A"
```
```   319   assumes T_def: "T \<equiv> (\<lambda>(x::'a) (y::'b). x = Rep y)"
```
```   320 begin
```
```   321
```
```   322 lemma typedef_left_unique: "left_unique T"
```
```   323   unfolding left_unique_def T_def
```
```   324   by (simp add: type_definition.Rep_inject [OF type])
```
```   325
```
```   326 lemma typedef_bi_unique: "bi_unique T"
```
```   327   unfolding bi_unique_def T_def
```
```   328   by (simp add: type_definition.Rep_inject [OF type])
```
```   329
```
```   330 (* the following two theorems are here only for convinience *)
```
```   331
```
```   332 lemma typedef_right_unique: "right_unique T"
```
```   333   using T_def type Quotient_right_unique typedef_to_Quotient
```
```   334   by blast
```
```   335
```
```   336 lemma typedef_right_total: "right_total T"
```
```   337   using T_def type Quotient_right_total typedef_to_Quotient
```
```   338   by blast
```
```   339
```
```   340 lemma typedef_rep_transfer: "(T ===> op =) (\<lambda>x. x) Rep"
```
```   341   unfolding rel_fun_def T_def by simp
```
```   342
```
```   343 end
```
```   344
```
```   345 text {* Generating the correspondence rule for a constant defined with
```
```   346   @{text "lift_definition"}. *}
```
```   347
```
```   348 lemma Quotient_to_transfer:
```
```   349   assumes "Quotient R Abs Rep T" and "R c c" and "c' \<equiv> Abs c"
```
```   350   shows "T c c'"
```
```   351   using assms by (auto dest: Quotient_cr_rel)
```
```   352
```
```   353 text {* Proving reflexivity *}
```
```   354
```
```   355 lemma Quotient_to_left_total:
```
```   356   assumes q: "Quotient R Abs Rep T"
```
```   357   and r_R: "reflp R"
```
```   358   shows "left_total T"
```
```   359 using r_R Quotient_cr_rel[OF q] unfolding left_total_def by (auto elim: reflpE)
```
```   360
```
```   361 lemma Quotient_composition_ge_eq:
```
```   362   assumes "left_total T"
```
```   363   assumes "R \<ge> op="
```
```   364   shows "(T OO R OO T\<inverse>\<inverse>) \<ge> op="
```
```   365 using assms unfolding left_total_def by fast
```
```   366
```
```   367 lemma Quotient_composition_le_eq:
```
```   368   assumes "left_unique T"
```
```   369   assumes "R \<le> op="
```
```   370   shows "(T OO R OO T\<inverse>\<inverse>) \<le> op="
```
```   371 using assms unfolding left_unique_def by blast
```
```   372
```
```   373 lemma eq_onp_le_eq:
```
```   374   "eq_onp P \<le> op=" unfolding eq_onp_def by blast
```
```   375
```
```   376 lemma reflp_ge_eq:
```
```   377   "reflp R \<Longrightarrow> R \<ge> op=" unfolding reflp_def by blast
```
```   378
```
```   379 lemma ge_eq_refl:
```
```   380   "R \<ge> op= \<Longrightarrow> R x x" by blast
```
```   381
```
```   382 text {* Proving a parametrized correspondence relation *}
```
```   383
```
```   384 definition POS :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where
```
```   385 "POS A B \<equiv> A \<le> B"
```
```   386
```
```   387 definition  NEG :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where
```
```   388 "NEG A B \<equiv> B \<le> A"
```
```   389
```
```   390 lemma pos_OO_eq:
```
```   391   shows "POS (A OO op=) A"
```
```   392 unfolding POS_def OO_def by blast
```
```   393
```
```   394 lemma pos_eq_OO:
```
```   395   shows "POS (op= OO A) A"
```
```   396 unfolding POS_def OO_def by blast
```
```   397
```
```   398 lemma neg_OO_eq:
```
```   399   shows "NEG (A OO op=) A"
```
```   400 unfolding NEG_def OO_def by auto
```
```   401
```
```   402 lemma neg_eq_OO:
```
```   403   shows "NEG (op= OO A) A"
```
```   404 unfolding NEG_def OO_def by blast
```
```   405
```
```   406 lemma POS_trans:
```
```   407   assumes "POS A B"
```
```   408   assumes "POS B C"
```
```   409   shows "POS A C"
```
```   410 using assms unfolding POS_def by auto
```
```   411
```
```   412 lemma NEG_trans:
```
```   413   assumes "NEG A B"
```
```   414   assumes "NEG B C"
```
```   415   shows "NEG A C"
```
```   416 using assms unfolding NEG_def by auto
```
```   417
```
```   418 lemma POS_NEG:
```
```   419   "POS A B \<equiv> NEG B A"
```
```   420   unfolding POS_def NEG_def by auto
```
```   421
```
```   422 lemma NEG_POS:
```
```   423   "NEG A B \<equiv> POS B A"
```
```   424   unfolding POS_def NEG_def by auto
```
```   425
```
```   426 lemma POS_pcr_rule:
```
```   427   assumes "POS (A OO B) C"
```
```   428   shows "POS (A OO B OO X) (C OO X)"
```
```   429 using assms unfolding POS_def OO_def by blast
```
```   430
```
```   431 lemma NEG_pcr_rule:
```
```   432   assumes "NEG (A OO B) C"
```
```   433   shows "NEG (A OO B OO X) (C OO X)"
```
```   434 using assms unfolding NEG_def OO_def by blast
```
```   435
```
```   436 lemma POS_apply:
```
```   437   assumes "POS R R'"
```
```   438   assumes "R f g"
```
```   439   shows "R' f g"
```
```   440 using assms unfolding POS_def by auto
```
```   441
```
```   442 text {* Proving a parametrized correspondence relation *}
```
```   443
```
```   444 lemma fun_mono:
```
```   445   assumes "A \<ge> C"
```
```   446   assumes "B \<le> D"
```
```   447   shows   "(A ===> B) \<le> (C ===> D)"
```
```   448 using assms unfolding rel_fun_def by blast
```
```   449
```
```   450 lemma pos_fun_distr: "((R ===> S) OO (R' ===> S')) \<le> ((R OO R') ===> (S OO S'))"
```
```   451 unfolding OO_def rel_fun_def by blast
```
```   452
```
```   453 lemma functional_relation: "right_unique R \<Longrightarrow> left_total R \<Longrightarrow> \<forall>x. \<exists>!y. R x y"
```
```   454 unfolding right_unique_def left_total_def by blast
```
```   455
```
```   456 lemma functional_converse_relation: "left_unique R \<Longrightarrow> right_total R \<Longrightarrow> \<forall>y. \<exists>!x. R x y"
```
```   457 unfolding left_unique_def right_total_def by blast
```
```   458
```
```   459 lemma neg_fun_distr1:
```
```   460 assumes 1: "left_unique R" "right_total R"
```
```   461 assumes 2: "right_unique R'" "left_total R'"
```
```   462 shows "(R OO R' ===> S OO S') \<le> ((R ===> S) OO (R' ===> S')) "
```
```   463   using functional_relation[OF 2] functional_converse_relation[OF 1]
```
```   464   unfolding rel_fun_def OO_def
```
```   465   apply clarify
```
```   466   apply (subst all_comm)
```
```   467   apply (subst all_conj_distrib[symmetric])
```
```   468   apply (intro choice)
```
```   469   by metis
```
```   470
```
```   471 lemma neg_fun_distr2:
```
```   472 assumes 1: "right_unique R'" "left_total R'"
```
```   473 assumes 2: "left_unique S'" "right_total S'"
```
```   474 shows "(R OO R' ===> S OO S') \<le> ((R ===> S) OO (R' ===> S'))"
```
```   475   using functional_converse_relation[OF 2] functional_relation[OF 1]
```
```   476   unfolding rel_fun_def OO_def
```
```   477   apply clarify
```
```   478   apply (subst all_comm)
```
```   479   apply (subst all_conj_distrib[symmetric])
```
```   480   apply (intro choice)
```
```   481   by metis
```
```   482
```
```   483 subsection {* Domains *}
```
```   484
```
```   485 lemma composed_equiv_rel_eq_onp:
```
```   486   assumes "left_unique R"
```
```   487   assumes "(R ===> op=) P P'"
```
```   488   assumes "Domainp R = P''"
```
```   489   shows "(R OO eq_onp P' OO R\<inverse>\<inverse>) = eq_onp (inf P'' P)"
```
```   490 using assms unfolding OO_def conversep_iff Domainp_iff[abs_def] left_unique_def rel_fun_def eq_onp_def
```
```   491 fun_eq_iff by blast
```
```   492
```
```   493 lemma composed_equiv_rel_eq_eq_onp:
```
```   494   assumes "left_unique R"
```
```   495   assumes "Domainp R = P"
```
```   496   shows "(R OO op= OO R\<inverse>\<inverse>) = eq_onp P"
```
```   497 using assms unfolding OO_def conversep_iff Domainp_iff[abs_def] left_unique_def eq_onp_def
```
```   498 fun_eq_iff is_equality_def by metis
```
```   499
```
```   500 lemma pcr_Domainp_par_left_total:
```
```   501   assumes "Domainp B = P"
```
```   502   assumes "left_total A"
```
```   503   assumes "(A ===> op=) P' P"
```
```   504   shows "Domainp (A OO B) = P'"
```
```   505 using assms
```
```   506 unfolding Domainp_iff[abs_def] OO_def bi_unique_def left_total_def rel_fun_def
```
```   507 by (fast intro: fun_eq_iff)
```
```   508
```
```   509 lemma pcr_Domainp_par:
```
```   510 assumes "Domainp B = P2"
```
```   511 assumes "Domainp A = P1"
```
```   512 assumes "(A ===> op=) P2' P2"
```
```   513 shows "Domainp (A OO B) = (inf P1 P2')"
```
```   514 using assms unfolding rel_fun_def Domainp_iff[abs_def] OO_def
```
```   515 by (fast intro: fun_eq_iff)
```
```   516
```
```   517 definition rel_pred_comp :: "('a => 'b => bool) => ('b => bool) => 'a => bool"
```
```   518 where "rel_pred_comp R P \<equiv> \<lambda>x. \<exists>y. R x y \<and> P y"
```
```   519
```
```   520 lemma pcr_Domainp:
```
```   521 assumes "Domainp B = P"
```
```   522 shows "Domainp (A OO B) = (\<lambda>x. \<exists>y. A x y \<and> P y)"
```
```   523 using assms by blast
```
```   524
```
```   525 lemma pcr_Domainp_total:
```
```   526   assumes "left_total B"
```
```   527   assumes "Domainp A = P"
```
```   528   shows "Domainp (A OO B) = P"
```
```   529 using assms unfolding left_total_def
```
```   530 by fast
```
```   531
```
```   532 lemma Quotient_to_Domainp:
```
```   533   assumes "Quotient R Abs Rep T"
```
```   534   shows "Domainp T = (\<lambda>x. R x x)"
```
```   535 by (simp add: Domainp_iff[abs_def] Quotient_cr_rel[OF assms])
```
```   536
```
```   537 lemma eq_onp_to_Domainp:
```
```   538   assumes "Quotient (eq_onp P) Abs Rep T"
```
```   539   shows "Domainp T = P"
```
```   540 by (simp add: eq_onp_def Domainp_iff[abs_def] Quotient_cr_rel[OF assms])
```
```   541
```
```   542 end
```
```   543
```
```   544 subsection {* ML setup *}
```
```   545
```
```   546 ML_file "Tools/Lifting/lifting_util.ML"
```
```   547
```
```   548 named_theorems relator_eq_onp
```
```   549   "theorems that a relator of an eq_onp is an eq_onp of the corresponding predicate"
```
```   550 ML_file "Tools/Lifting/lifting_info.ML"
```
```   551 setup Lifting_Info.setup
```
```   552
```
```   553 (* setup for the function type *)
```
```   554 declare fun_quotient[quot_map]
```
```   555 declare fun_mono[relator_mono]
```
```   556 lemmas [relator_distr] = pos_fun_distr neg_fun_distr1 neg_fun_distr2
```
```   557
```
```   558 ML_file "Tools/Lifting/lifting_bnf.ML"
```
```   559 ML_file "Tools/Lifting/lifting_term.ML"
```
```   560 ML_file "Tools/Lifting/lifting_def.ML"
```
```   561 ML_file "Tools/Lifting/lifting_setup.ML"
```
```   562
```
```   563 hide_const (open) POS NEG
```
```   564
```
```   565 end
```