src/HOL/Lifting.thy
 author paulson Tue Apr 25 16:39:54 2017 +0100 (2017-04-25) changeset 65578 e4997c181cce parent 63343 fb5d8a50c641 child 67229 4ecf0ef70efb permissions -rw-r--r--
New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
```     1 (*  Title:      HOL/Lifting.thy
```
```     2     Author:     Brian Huffman and Ondrej Kuncar
```
```     3     Author:     Cezary Kaliszyk and Christian Urban
```
```     4 *)
```
```     5
```
```     6 section \<open>Lifting package\<close>
```
```     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 \<open>Function map\<close>
```
```    18
```
```    19 context includes lifting_syntax
```
```    20 begin
```
```    21
```
```    22 lemma map_fun_id:
```
```    23   "(id ---> id) = id"
```
```    24   by (simp add: fun_eq_iff)
```
```    25
```
```    26 subsection \<open>Quotient Predicate\<close>
```
```    27
```
```    28 definition
```
```    29   "Quotient R Abs Rep T \<longleftrightarrow>
```
```    30      (\<forall>a. Abs (Rep a) = a) \<and>
```
```    31      (\<forall>a. R (Rep a) (Rep a)) \<and>
```
```    32      (\<forall>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s) \<and>
```
```    33      T = (\<lambda>x y. R x x \<and> Abs x = y)"
```
```    34
```
```    35 lemma QuotientI:
```
```    36   assumes "\<And>a. Abs (Rep a) = a"
```
```    37     and "\<And>a. R (Rep a) (Rep a)"
```
```    38     and "\<And>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s"
```
```    39     and "T = (\<lambda>x y. R x x \<and> Abs x = y)"
```
```    40   shows "Quotient R Abs Rep T"
```
```    41   using assms unfolding Quotient_def by blast
```
```    42
```
```    43 context
```
```    44   fixes R Abs Rep T
```
```    45   assumes a: "Quotient R Abs Rep T"
```
```    46 begin
```
```    47
```
```    48 lemma Quotient_abs_rep: "Abs (Rep a) = a"
```
```    49   using a unfolding Quotient_def
```
```    50   by simp
```
```    51
```
```    52 lemma Quotient_rep_reflp: "R (Rep a) (Rep a)"
```
```    53   using a unfolding Quotient_def
```
```    54   by blast
```
```    55
```
```    56 lemma Quotient_rel:
```
```    57   "R r r \<and> R s s \<and> Abs r = Abs s \<longleftrightarrow> R r s" \<comment> \<open>orientation does not loop on rewriting\<close>
```
```    58   using a unfolding Quotient_def
```
```    59   by blast
```
```    60
```
```    61 lemma Quotient_cr_rel: "T = (\<lambda>x y. R x x \<and> Abs x = y)"
```
```    62   using a unfolding Quotient_def
```
```    63   by blast
```
```    64
```
```    65 lemma Quotient_refl1: "R r s \<Longrightarrow> R r r"
```
```    66   using a unfolding Quotient_def
```
```    67   by fast
```
```    68
```
```    69 lemma Quotient_refl2: "R r s \<Longrightarrow> R s s"
```
```    70   using a unfolding Quotient_def
```
```    71   by fast
```
```    72
```
```    73 lemma Quotient_rel_rep: "R (Rep a) (Rep b) \<longleftrightarrow> a = b"
```
```    74   using a unfolding Quotient_def
```
```    75   by metis
```
```    76
```
```    77 lemma Quotient_rep_abs: "R r r \<Longrightarrow> R (Rep (Abs r)) r"
```
```    78   using a unfolding Quotient_def
```
```    79   by blast
```
```    80
```
```    81 lemma Quotient_rep_abs_eq: "R t t \<Longrightarrow> R \<le> op= \<Longrightarrow> Rep (Abs t) = t"
```
```    82   using a unfolding Quotient_def
```
```    83   by blast
```
```    84
```
```    85 lemma Quotient_rep_abs_fold_unmap:
```
```    86   assumes "x' \<equiv> Abs x" and "R x x" and "Rep x' \<equiv> Rep' x'"
```
```    87   shows "R (Rep' x') x"
```
```    88 proof -
```
```    89   have "R (Rep x') x" using assms(1-2) Quotient_rep_abs by auto
```
```    90   then show ?thesis using assms(3) by simp
```
```    91 qed
```
```    92
```
```    93 lemma Quotient_Rep_eq:
```
```    94   assumes "x' \<equiv> Abs x"
```
```    95   shows "Rep x' \<equiv> Rep x'"
```
```    96 by simp
```
```    97
```
```    98 lemma Quotient_rel_abs: "R r s \<Longrightarrow> Abs r = Abs s"
```
```    99   using a unfolding Quotient_def
```
```   100   by blast
```
```   101
```
```   102 lemma Quotient_rel_abs2:
```
```   103   assumes "R (Rep x) y"
```
```   104   shows "x = Abs y"
```
```   105 proof -
```
```   106   from assms have "Abs (Rep x) = Abs y" by (auto intro: Quotient_rel_abs)
```
```   107   then show ?thesis using assms(1) by (simp add: Quotient_abs_rep)
```
```   108 qed
```
```   109
```
```   110 lemma Quotient_symp: "symp R"
```
```   111   using a unfolding Quotient_def using sympI by (metis (full_types))
```
```   112
```
```   113 lemma Quotient_transp: "transp R"
```
```   114   using a unfolding Quotient_def using transpI by (metis (full_types))
```
```   115
```
```   116 lemma Quotient_part_equivp: "part_equivp R"
```
```   117 by (metis Quotient_rep_reflp Quotient_symp Quotient_transp part_equivpI)
```
```   118
```
```   119 end
```
```   120
```
```   121 lemma identity_quotient: "Quotient (op =) id id (op =)"
```
```   122 unfolding Quotient_def by simp
```
```   123
```
```   124 text \<open>TODO: Use one of these alternatives as the real definition.\<close>
```
```   125
```
```   126 lemma Quotient_alt_def:
```
```   127   "Quotient R Abs Rep T \<longleftrightarrow>
```
```   128     (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and>
```
```   129     (\<forall>b. T (Rep b) b) \<and>
```
```   130     (\<forall>x y. R x y \<longleftrightarrow> T x (Abs x) \<and> T y (Abs y) \<and> Abs x = Abs y)"
```
```   131 apply safe
```
```   132 apply (simp (no_asm_use) only: Quotient_def, fast)
```
```   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 (rule QuotientI)
```
```   139 apply simp
```
```   140 apply metis
```
```   141 apply simp
```
```   142 apply (rule ext, rule ext, metis)
```
```   143 done
```
```   144
```
```   145 lemma Quotient_alt_def2:
```
```   146   "Quotient R Abs Rep T \<longleftrightarrow>
```
```   147     (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and>
```
```   148     (\<forall>b. T (Rep b) b) \<and>
```
```   149     (\<forall>x y. R x y \<longleftrightarrow> T x (Abs y) \<and> T y (Abs x))"
```
```   150   unfolding Quotient_alt_def by (safe, metis+)
```
```   151
```
```   152 lemma Quotient_alt_def3:
```
```   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>
```
```   155     (\<forall>x y. R x y \<longleftrightarrow> (\<exists>z. T x z \<and> T y z))"
```
```   156   unfolding Quotient_alt_def2 by (safe, metis+)
```
```   157
```
```   158 lemma Quotient_alt_def4:
```
```   159   "Quotient R Abs Rep T \<longleftrightarrow>
```
```   160     (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and> (\<forall>b. T (Rep b) b) \<and> R = T OO conversep T"
```
```   161   unfolding Quotient_alt_def3 fun_eq_iff by auto
```
```   162
```
```   163 lemma Quotient_alt_def5:
```
```   164   "Quotient R Abs Rep T \<longleftrightarrow>
```
```   165     T \<le> BNF_Def.Grp UNIV Abs \<and> BNF_Def.Grp UNIV Rep \<le> T\<inverse>\<inverse> \<and> R = T OO T\<inverse>\<inverse>"
```
```   166   unfolding Quotient_alt_def4 Grp_def by blast
```
```   167
```
```   168 lemma fun_quotient:
```
```   169   assumes 1: "Quotient R1 abs1 rep1 T1"
```
```   170   assumes 2: "Quotient R2 abs2 rep2 T2"
```
```   171   shows "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2) (T1 ===> T2)"
```
```   172   using assms unfolding Quotient_alt_def2
```
```   173   unfolding rel_fun_def fun_eq_iff map_fun_apply
```
```   174   by (safe, metis+)
```
```   175
```
```   176 lemma apply_rsp:
```
```   177   fixes f g::"'a \<Rightarrow> 'c"
```
```   178   assumes q: "Quotient R1 Abs1 Rep1 T1"
```
```   179   and     a: "(R1 ===> R2) f g" "R1 x y"
```
```   180   shows "R2 (f x) (g y)"
```
```   181   using a by (auto elim: rel_funE)
```
```   182
```
```   183 lemma apply_rsp':
```
```   184   assumes a: "(R1 ===> R2) f g" "R1 x y"
```
```   185   shows "R2 (f x) (g y)"
```
```   186   using a by (auto elim: rel_funE)
```
```   187
```
```   188 lemma apply_rsp'':
```
```   189   assumes "Quotient R Abs Rep T"
```
```   190   and "(R ===> S) f f"
```
```   191   shows "S (f (Rep x)) (f (Rep x))"
```
```   192 proof -
```
```   193   from assms(1) have "R (Rep x) (Rep x)" by (rule Quotient_rep_reflp)
```
```   194   then show ?thesis using assms(2) by (auto intro: apply_rsp')
```
```   195 qed
```
```   196
```
```   197 subsection \<open>Quotient composition\<close>
```
```   198
```
```   199 lemma Quotient_compose:
```
```   200   assumes 1: "Quotient R1 Abs1 Rep1 T1"
```
```   201   assumes 2: "Quotient R2 Abs2 Rep2 T2"
```
```   202   shows "Quotient (T1 OO R2 OO conversep T1) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2) (T1 OO T2)"
```
```   203   using assms unfolding Quotient_alt_def4 by fastforce
```
```   204
```
```   205 lemma equivp_reflp2:
```
```   206   "equivp R \<Longrightarrow> reflp R"
```
```   207   by (erule equivpE)
```
```   208
```
```   209 subsection \<open>Respects predicate\<close>
```
```   210
```
```   211 definition Respects :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set"
```
```   212   where "Respects R = {x. R x x}"
```
```   213
```
```   214 lemma in_respects: "x \<in> Respects R \<longleftrightarrow> R x x"
```
```   215   unfolding Respects_def by simp
```
```   216
```
```   217 lemma UNIV_typedef_to_Quotient:
```
```   218   assumes "type_definition Rep Abs UNIV"
```
```   219   and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"
```
```   220   shows "Quotient (op =) Abs Rep T"
```
```   221 proof -
```
```   222   interpret type_definition Rep Abs UNIV by fact
```
```   223   from Abs_inject Rep_inverse Abs_inverse T_def show ?thesis
```
```   224     by (fastforce intro!: QuotientI fun_eq_iff)
```
```   225 qed
```
```   226
```
```   227 lemma UNIV_typedef_to_equivp:
```
```   228   fixes Abs :: "'a \<Rightarrow> 'b"
```
```   229   and Rep :: "'b \<Rightarrow> 'a"
```
```   230   assumes "type_definition Rep Abs (UNIV::'a set)"
```
```   231   shows "equivp (op=::'a\<Rightarrow>'a\<Rightarrow>bool)"
```
```   232 by (rule identity_equivp)
```
```   233
```
```   234 lemma typedef_to_Quotient:
```
```   235   assumes "type_definition Rep Abs S"
```
```   236   and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"
```
```   237   shows "Quotient (eq_onp (\<lambda>x. x \<in> S)) Abs Rep T"
```
```   238 proof -
```
```   239   interpret type_definition Rep Abs S by fact
```
```   240   from Rep Abs_inject Rep_inverse Abs_inverse T_def show ?thesis
```
```   241     by (auto intro!: QuotientI simp: eq_onp_def fun_eq_iff)
```
```   242 qed
```
```   243
```
```   244 lemma typedef_to_part_equivp:
```
```   245   assumes "type_definition Rep Abs S"
```
```   246   shows "part_equivp (eq_onp (\<lambda>x. x \<in> S))"
```
```   247 proof (intro part_equivpI)
```
```   248   interpret type_definition Rep Abs S by fact
```
```   249   show "\<exists>x. eq_onp (\<lambda>x. x \<in> S) x x" using Rep by (auto simp: eq_onp_def)
```
```   250 next
```
```   251   show "symp (eq_onp (\<lambda>x. x \<in> S))" by (auto intro: sympI simp: eq_onp_def)
```
```   252 next
```
```   253   show "transp (eq_onp (\<lambda>x. x \<in> S))" by (auto intro: transpI simp: eq_onp_def)
```
```   254 qed
```
```   255
```
```   256 lemma open_typedef_to_Quotient:
```
```   257   assumes "type_definition Rep Abs {x. P x}"
```
```   258   and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"
```
```   259   shows "Quotient (eq_onp P) Abs Rep T"
```
```   260   using typedef_to_Quotient [OF assms] by simp
```
```   261
```
```   262 lemma open_typedef_to_part_equivp:
```
```   263   assumes "type_definition Rep Abs {x. P x}"
```
```   264   shows "part_equivp (eq_onp P)"
```
```   265   using typedef_to_part_equivp [OF assms] by simp
```
```   266
```
```   267 lemma type_definition_Quotient_not_empty: "Quotient (eq_onp P) Abs Rep T \<Longrightarrow> \<exists>x. P x"
```
```   268 unfolding eq_onp_def by (drule Quotient_rep_reflp) blast
```
```   269
```
```   270 lemma type_definition_Quotient_not_empty_witness: "Quotient (eq_onp P) Abs Rep T \<Longrightarrow> P (Rep undefined)"
```
```   271 unfolding eq_onp_def by (drule Quotient_rep_reflp) blast
```
```   272
```
```   273
```
```   274 text \<open>Generating transfer rules for quotients.\<close>
```
```   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 rel_fun_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 end
```
```   295
```
```   296 text \<open>Generating transfer rules for total quotients.\<close>
```
```   297
```
```   298 context
```
```   299   fixes R Abs Rep T
```
```   300   assumes 1: "Quotient R Abs Rep T" and 2: "reflp R"
```
```   301 begin
```
```   302
```
```   303 lemma Quotient_left_total: "left_total T"
```
```   304   using 1 2 unfolding Quotient_alt_def left_total_def reflp_def by auto
```
```   305
```
```   306 lemma Quotient_bi_total: "bi_total T"
```
```   307   using 1 2 unfolding Quotient_alt_def bi_total_def reflp_def by auto
```
```   308
```
```   309 lemma Quotient_id_abs_transfer: "(op = ===> T) (\<lambda>x. x) Abs"
```
```   310   using 1 2 unfolding Quotient_alt_def reflp_def rel_fun_def by simp
```
```   311
```
```   312 lemma Quotient_total_abs_induct: "(\<And>y. P (Abs y)) \<Longrightarrow> P x"
```
```   313   using 1 2 unfolding Quotient_alt_def reflp_def by metis
```
```   314
```
```   315 lemma Quotient_total_abs_eq_iff: "Abs x = Abs y \<longleftrightarrow> R x y"
```
```   316   using Quotient_rel [OF 1] 2 unfolding reflp_def by simp
```
```   317
```
```   318 end
```
```   319
```
```   320 text \<open>Generating transfer rules for a type defined with \<open>typedef\<close>.\<close>
```
```   321
```
```   322 context
```
```   323   fixes Rep Abs A T
```
```   324   assumes type: "type_definition Rep Abs A"
```
```   325   assumes T_def: "T \<equiv> (\<lambda>(x::'a) (y::'b). x = Rep y)"
```
```   326 begin
```
```   327
```
```   328 lemma typedef_left_unique: "left_unique T"
```
```   329   unfolding left_unique_def T_def
```
```   330   by (simp add: type_definition.Rep_inject [OF type])
```
```   331
```
```   332 lemma typedef_bi_unique: "bi_unique T"
```
```   333   unfolding bi_unique_def T_def
```
```   334   by (simp add: type_definition.Rep_inject [OF type])
```
```   335
```
```   336 (* the following two theorems are here only for convinience *)
```
```   337
```
```   338 lemma typedef_right_unique: "right_unique T"
```
```   339   using T_def type Quotient_right_unique typedef_to_Quotient
```
```   340   by blast
```
```   341
```
```   342 lemma typedef_right_total: "right_total T"
```
```   343   using T_def type Quotient_right_total typedef_to_Quotient
```
```   344   by blast
```
```   345
```
```   346 lemma typedef_rep_transfer: "(T ===> op =) (\<lambda>x. x) Rep"
```
```   347   unfolding rel_fun_def T_def by simp
```
```   348
```
```   349 end
```
```   350
```
```   351 text \<open>Generating the correspondence rule for a constant defined with
```
```   352   \<open>lift_definition\<close>.\<close>
```
```   353
```
```   354 lemma Quotient_to_transfer:
```
```   355   assumes "Quotient R Abs Rep T" and "R c c" and "c' \<equiv> Abs c"
```
```   356   shows "T c c'"
```
```   357   using assms by (auto dest: Quotient_cr_rel)
```
```   358
```
```   359 text \<open>Proving reflexivity\<close>
```
```   360
```
```   361 lemma Quotient_to_left_total:
```
```   362   assumes q: "Quotient R Abs Rep T"
```
```   363   and r_R: "reflp R"
```
```   364   shows "left_total T"
```
```   365 using r_R Quotient_cr_rel[OF q] unfolding left_total_def by (auto elim: reflpE)
```
```   366
```
```   367 lemma Quotient_composition_ge_eq:
```
```   368   assumes "left_total T"
```
```   369   assumes "R \<ge> op="
```
```   370   shows "(T OO R OO T\<inverse>\<inverse>) \<ge> op="
```
```   371 using assms unfolding left_total_def by fast
```
```   372
```
```   373 lemma Quotient_composition_le_eq:
```
```   374   assumes "left_unique T"
```
```   375   assumes "R \<le> op="
```
```   376   shows "(T OO R OO T\<inverse>\<inverse>) \<le> op="
```
```   377 using assms unfolding left_unique_def by blast
```
```   378
```
```   379 lemma eq_onp_le_eq:
```
```   380   "eq_onp P \<le> op=" unfolding eq_onp_def by blast
```
```   381
```
```   382 lemma reflp_ge_eq:
```
```   383   "reflp R \<Longrightarrow> R \<ge> op=" unfolding reflp_def by blast
```
```   384
```
```   385 text \<open>Proving a parametrized correspondence relation\<close>
```
```   386
```
```   387 definition POS :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where
```
```   388 "POS A B \<equiv> A \<le> B"
```
```   389
```
```   390 definition  NEG :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where
```
```   391 "NEG A B \<equiv> B \<le> A"
```
```   392
```
```   393 lemma pos_OO_eq:
```
```   394   shows "POS (A OO op=) A"
```
```   395 unfolding POS_def OO_def by blast
```
```   396
```
```   397 lemma pos_eq_OO:
```
```   398   shows "POS (op= OO A) A"
```
```   399 unfolding POS_def OO_def by blast
```
```   400
```
```   401 lemma neg_OO_eq:
```
```   402   shows "NEG (A OO op=) A"
```
```   403 unfolding NEG_def OO_def by auto
```
```   404
```
```   405 lemma neg_eq_OO:
```
```   406   shows "NEG (op= OO A) A"
```
```   407 unfolding NEG_def OO_def by blast
```
```   408
```
```   409 lemma POS_trans:
```
```   410   assumes "POS A B"
```
```   411   assumes "POS B C"
```
```   412   shows "POS A C"
```
```   413 using assms unfolding POS_def by auto
```
```   414
```
```   415 lemma NEG_trans:
```
```   416   assumes "NEG A B"
```
```   417   assumes "NEG B C"
```
```   418   shows "NEG A C"
```
```   419 using assms unfolding NEG_def by auto
```
```   420
```
```   421 lemma POS_NEG:
```
```   422   "POS A B \<equiv> NEG B A"
```
```   423   unfolding POS_def NEG_def by auto
```
```   424
```
```   425 lemma NEG_POS:
```
```   426   "NEG A B \<equiv> POS B A"
```
```   427   unfolding POS_def NEG_def by auto
```
```   428
```
```   429 lemma POS_pcr_rule:
```
```   430   assumes "POS (A OO B) C"
```
```   431   shows "POS (A OO B OO X) (C OO X)"
```
```   432 using assms unfolding POS_def OO_def by blast
```
```   433
```
```   434 lemma NEG_pcr_rule:
```
```   435   assumes "NEG (A OO B) C"
```
```   436   shows "NEG (A OO B OO X) (C OO X)"
```
```   437 using assms unfolding NEG_def OO_def by blast
```
```   438
```
```   439 lemma POS_apply:
```
```   440   assumes "POS R R'"
```
```   441   assumes "R f g"
```
```   442   shows "R' f g"
```
```   443 using assms unfolding POS_def by auto
```
```   444
```
```   445 text \<open>Proving a parametrized correspondence relation\<close>
```
```   446
```
```   447 lemma fun_mono:
```
```   448   assumes "A \<ge> C"
```
```   449   assumes "B \<le> D"
```
```   450   shows   "(A ===> B) \<le> (C ===> D)"
```
```   451 using assms unfolding rel_fun_def by blast
```
```   452
```
```   453 lemma pos_fun_distr: "((R ===> S) OO (R' ===> S')) \<le> ((R OO R') ===> (S OO S'))"
```
```   454 unfolding OO_def rel_fun_def by blast
```
```   455
```
```   456 lemma functional_relation: "right_unique R \<Longrightarrow> left_total R \<Longrightarrow> \<forall>x. \<exists>!y. R x y"
```
```   457 unfolding right_unique_def left_total_def by blast
```
```   458
```
```   459 lemma functional_converse_relation: "left_unique R \<Longrightarrow> right_total R \<Longrightarrow> \<forall>y. \<exists>!x. R x y"
```
```   460 unfolding left_unique_def right_total_def by blast
```
```   461
```
```   462 lemma neg_fun_distr1:
```
```   463 assumes 1: "left_unique R" "right_total R"
```
```   464 assumes 2: "right_unique R'" "left_total R'"
```
```   465 shows "(R OO R' ===> S OO S') \<le> ((R ===> S) OO (R' ===> S')) "
```
```   466   using functional_relation[OF 2] functional_converse_relation[OF 1]
```
```   467   unfolding rel_fun_def OO_def
```
```   468   apply clarify
```
```   469   apply (subst all_comm)
```
```   470   apply (subst all_conj_distrib[symmetric])
```
```   471   apply (intro choice)
```
```   472   by metis
```
```   473
```
```   474 lemma neg_fun_distr2:
```
```   475 assumes 1: "right_unique R'" "left_total R'"
```
```   476 assumes 2: "left_unique S'" "right_total S'"
```
```   477 shows "(R OO R' ===> S OO S') \<le> ((R ===> S) OO (R' ===> S'))"
```
```   478   using functional_converse_relation[OF 2] functional_relation[OF 1]
```
```   479   unfolding rel_fun_def OO_def
```
```   480   apply clarify
```
```   481   apply (subst all_comm)
```
```   482   apply (subst all_conj_distrib[symmetric])
```
```   483   apply (intro choice)
```
```   484   by metis
```
```   485
```
```   486 subsection \<open>Domains\<close>
```
```   487
```
```   488 lemma composed_equiv_rel_eq_onp:
```
```   489   assumes "left_unique R"
```
```   490   assumes "(R ===> op=) P P'"
```
```   491   assumes "Domainp R = P''"
```
```   492   shows "(R OO eq_onp P' OO R\<inverse>\<inverse>) = eq_onp (inf P'' P)"
```
```   493 using assms unfolding OO_def conversep_iff Domainp_iff[abs_def] left_unique_def rel_fun_def eq_onp_def
```
```   494 fun_eq_iff by blast
```
```   495
```
```   496 lemma composed_equiv_rel_eq_eq_onp:
```
```   497   assumes "left_unique R"
```
```   498   assumes "Domainp R = P"
```
```   499   shows "(R OO op= OO R\<inverse>\<inverse>) = eq_onp P"
```
```   500 using assms unfolding OO_def conversep_iff Domainp_iff[abs_def] left_unique_def eq_onp_def
```
```   501 fun_eq_iff is_equality_def by metis
```
```   502
```
```   503 lemma pcr_Domainp_par_left_total:
```
```   504   assumes "Domainp B = P"
```
```   505   assumes "left_total A"
```
```   506   assumes "(A ===> op=) P' P"
```
```   507   shows "Domainp (A OO B) = P'"
```
```   508 using assms
```
```   509 unfolding Domainp_iff[abs_def] OO_def bi_unique_def left_total_def rel_fun_def
```
```   510 by (fast intro: fun_eq_iff)
```
```   511
```
```   512 lemma pcr_Domainp_par:
```
```   513 assumes "Domainp B = P2"
```
```   514 assumes "Domainp A = P1"
```
```   515 assumes "(A ===> op=) P2' P2"
```
```   516 shows "Domainp (A OO B) = (inf P1 P2')"
```
```   517 using assms unfolding rel_fun_def Domainp_iff[abs_def] OO_def
```
```   518 by (fast intro: fun_eq_iff)
```
```   519
```
```   520 definition rel_pred_comp :: "('a => 'b => bool) => ('b => bool) => 'a => bool"
```
```   521 where "rel_pred_comp R P \<equiv> \<lambda>x. \<exists>y. R x y \<and> P y"
```
```   522
```
```   523 lemma pcr_Domainp:
```
```   524 assumes "Domainp B = P"
```
```   525 shows "Domainp (A OO B) = (\<lambda>x. \<exists>y. A x y \<and> P y)"
```
```   526 using assms by blast
```
```   527
```
```   528 lemma pcr_Domainp_total:
```
```   529   assumes "left_total B"
```
```   530   assumes "Domainp A = P"
```
```   531   shows "Domainp (A OO B) = P"
```
```   532 using assms unfolding left_total_def
```
```   533 by fast
```
```   534
```
```   535 lemma Quotient_to_Domainp:
```
```   536   assumes "Quotient R Abs Rep T"
```
```   537   shows "Domainp T = (\<lambda>x. R x x)"
```
```   538 by (simp add: Domainp_iff[abs_def] Quotient_cr_rel[OF assms])
```
```   539
```
```   540 lemma eq_onp_to_Domainp:
```
```   541   assumes "Quotient (eq_onp P) Abs Rep T"
```
```   542   shows "Domainp T = P"
```
```   543 by (simp add: eq_onp_def Domainp_iff[abs_def] Quotient_cr_rel[OF assms])
```
```   544
```
```   545 end
```
```   546
```
```   547 (* needed for lifting_def_code_dt.ML (moved from Lifting_Set) *)
```
```   548 lemma right_total_UNIV_transfer:
```
```   549   assumes "right_total A"
```
```   550   shows "(rel_set A) (Collect (Domainp A)) UNIV"
```
```   551   using assms unfolding right_total_def rel_set_def Domainp_iff by blast
```
```   552
```
```   553 subsection \<open>ML setup\<close>
```
```   554
```
```   555 ML_file "Tools/Lifting/lifting_util.ML"
```
```   556
```
```   557 named_theorems relator_eq_onp
```
```   558   "theorems that a relator of an eq_onp is an eq_onp of the corresponding predicate"
```
```   559 ML_file "Tools/Lifting/lifting_info.ML"
```
```   560
```
```   561 (* setup for the function type *)
```
```   562 declare fun_quotient[quot_map]
```
```   563 declare fun_mono[relator_mono]
```
```   564 lemmas [relator_distr] = pos_fun_distr neg_fun_distr1 neg_fun_distr2
```
```   565
```
```   566 ML_file "Tools/Lifting/lifting_bnf.ML"
```
```   567 ML_file "Tools/Lifting/lifting_term.ML"
```
```   568 ML_file "Tools/Lifting/lifting_def.ML"
```
```   569 ML_file "Tools/Lifting/lifting_setup.ML"
```
```   570 ML_file "Tools/Lifting/lifting_def_code_dt.ML"
```
```   571
```
```   572 lemma pred_prod_beta: "pred_prod P Q xy \<longleftrightarrow> P (fst xy) \<and> Q (snd xy)"
```
```   573 by(cases xy) simp
```
```   574
```
```   575 lemma pred_prod_split: "P (pred_prod Q R xy) \<longleftrightarrow> (\<forall>x y. xy = (x, y) \<longrightarrow> P (Q x \<and> R y))"
```
```   576 by(cases xy) simp
```
```   577
```
```   578 hide_const (open) POS NEG
```
```   579
```
```   580 end
```