src/HOL/Lifting.thy
 author wenzelm Tue Sep 03 01:12:40 2013 +0200 (2013-09-03) changeset 53374 a14d2a854c02 parent 53219 ca237b9e4542 child 53651 ee90c67502c9 permissions -rw-r--r--
tuned proofs -- clarified flow of facts wrt. calculation;
```     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" :: 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 {* Other predicates on relations *}
```
```    28
```
```    29 definition left_total :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
```
```    30   where "left_total R \<longleftrightarrow> (\<forall>x. \<exists>y. R x y)"
```
```    31
```
```    32 lemma left_totalI:
```
```    33   "(\<And>x. \<exists>y. R x y) \<Longrightarrow> left_total R"
```
```    34 unfolding left_total_def by blast
```
```    35
```
```    36 lemma left_totalE:
```
```    37   assumes "left_total R"
```
```    38   obtains "(\<And>x. \<exists>y. R x y)"
```
```    39 using assms unfolding left_total_def by blast
```
```    40
```
```    41 definition left_unique :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
```
```    42   where "left_unique R \<longleftrightarrow> (\<forall>x y z. R x z \<longrightarrow> R y z \<longrightarrow> x = y)"
```
```    43
```
```    44 lemma left_total_fun:
```
```    45   "\<lbrakk>left_unique A; left_total B\<rbrakk> \<Longrightarrow> left_total (A ===> B)"
```
```    46   unfolding left_total_def fun_rel_def
```
```    47   apply (rule allI, rename_tac f)
```
```    48   apply (rule_tac x="\<lambda>y. SOME z. B (f (THE x. A x y)) z" in exI)
```
```    49   apply clarify
```
```    50   apply (subgoal_tac "(THE x. A x y) = x", simp)
```
```    51   apply (rule someI_ex)
```
```    52   apply (simp)
```
```    53   apply (rule the_equality)
```
```    54   apply assumption
```
```    55   apply (simp add: left_unique_def)
```
```    56   done
```
```    57
```
```    58 lemma left_unique_fun:
```
```    59   "\<lbrakk>left_total A; left_unique B\<rbrakk> \<Longrightarrow> left_unique (A ===> B)"
```
```    60   unfolding left_total_def left_unique_def fun_rel_def
```
```    61   by (clarify, rule ext, fast)
```
```    62
```
```    63 lemma left_total_eq: "left_total op=" unfolding left_total_def by blast
```
```    64
```
```    65 lemma left_unique_eq: "left_unique op=" unfolding left_unique_def by blast
```
```    66
```
```    67 subsection {* Quotient Predicate *}
```
```    68
```
```    69 definition
```
```    70   "Quotient R Abs Rep T \<longleftrightarrow>
```
```    71      (\<forall>a. Abs (Rep a) = a) \<and>
```
```    72      (\<forall>a. R (Rep a) (Rep a)) \<and>
```
```    73      (\<forall>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s) \<and>
```
```    74      T = (\<lambda>x y. R x x \<and> Abs x = y)"
```
```    75
```
```    76 lemma QuotientI:
```
```    77   assumes "\<And>a. Abs (Rep a) = a"
```
```    78     and "\<And>a. R (Rep a) (Rep a)"
```
```    79     and "\<And>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s"
```
```    80     and "T = (\<lambda>x y. R x x \<and> Abs x = y)"
```
```    81   shows "Quotient R Abs Rep T"
```
```    82   using assms unfolding Quotient_def by blast
```
```    83
```
```    84 context
```
```    85   fixes R Abs Rep T
```
```    86   assumes a: "Quotient R Abs Rep T"
```
```    87 begin
```
```    88
```
```    89 lemma Quotient_abs_rep: "Abs (Rep a) = a"
```
```    90   using a unfolding Quotient_def
```
```    91   by simp
```
```    92
```
```    93 lemma Quotient_rep_reflp: "R (Rep a) (Rep a)"
```
```    94   using a unfolding Quotient_def
```
```    95   by blast
```
```    96
```
```    97 lemma Quotient_rel:
```
```    98   "R r r \<and> R s s \<and> Abs r = Abs s \<longleftrightarrow> R r s" -- {* orientation does not loop on rewriting *}
```
```    99   using a unfolding Quotient_def
```
```   100   by blast
```
```   101
```
```   102 lemma Quotient_cr_rel: "T = (\<lambda>x y. R x x \<and> Abs x = y)"
```
```   103   using a unfolding Quotient_def
```
```   104   by blast
```
```   105
```
```   106 lemma Quotient_refl1: "R r s \<Longrightarrow> R r r"
```
```   107   using a unfolding Quotient_def
```
```   108   by fast
```
```   109
```
```   110 lemma Quotient_refl2: "R r s \<Longrightarrow> R s s"
```
```   111   using a unfolding Quotient_def
```
```   112   by fast
```
```   113
```
```   114 lemma Quotient_rel_rep: "R (Rep a) (Rep b) \<longleftrightarrow> a = b"
```
```   115   using a unfolding Quotient_def
```
```   116   by metis
```
```   117
```
```   118 lemma Quotient_rep_abs: "R r r \<Longrightarrow> R (Rep (Abs r)) r"
```
```   119   using a unfolding Quotient_def
```
```   120   by blast
```
```   121
```
```   122 lemma Quotient_rep_abs_fold_unmap:
```
```   123   assumes "x' \<equiv> Abs x" and "R x x" and "Rep x' \<equiv> Rep' x'"
```
```   124   shows "R (Rep' x') x"
```
```   125 proof -
```
```   126   have "R (Rep x') x" using assms(1-2) Quotient_rep_abs by auto
```
```   127   then show ?thesis using assms(3) by simp
```
```   128 qed
```
```   129
```
```   130 lemma Quotient_Rep_eq:
```
```   131   assumes "x' \<equiv> Abs x"
```
```   132   shows "Rep x' \<equiv> Rep x'"
```
```   133 by simp
```
```   134
```
```   135 lemma Quotient_rel_abs: "R r s \<Longrightarrow> Abs r = Abs s"
```
```   136   using a unfolding Quotient_def
```
```   137   by blast
```
```   138
```
```   139 lemma Quotient_rel_abs2:
```
```   140   assumes "R (Rep x) y"
```
```   141   shows "x = Abs y"
```
```   142 proof -
```
```   143   from assms have "Abs (Rep x) = Abs y" by (auto intro: Quotient_rel_abs)
```
```   144   then show ?thesis using assms(1) by (simp add: Quotient_abs_rep)
```
```   145 qed
```
```   146
```
```   147 lemma Quotient_symp: "symp R"
```
```   148   using a unfolding Quotient_def using sympI by (metis (full_types))
```
```   149
```
```   150 lemma Quotient_transp: "transp R"
```
```   151   using a unfolding Quotient_def using transpI by (metis (full_types))
```
```   152
```
```   153 lemma Quotient_part_equivp: "part_equivp R"
```
```   154 by (metis Quotient_rep_reflp Quotient_symp Quotient_transp part_equivpI)
```
```   155
```
```   156 end
```
```   157
```
```   158 lemma identity_quotient: "Quotient (op =) id id (op =)"
```
```   159 unfolding Quotient_def by simp
```
```   160
```
```   161 text {* TODO: Use one of these alternatives as the real definition. *}
```
```   162
```
```   163 lemma Quotient_alt_def:
```
```   164   "Quotient R Abs Rep T \<longleftrightarrow>
```
```   165     (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and>
```
```   166     (\<forall>b. T (Rep b) b) \<and>
```
```   167     (\<forall>x y. R x y \<longleftrightarrow> T x (Abs x) \<and> T y (Abs y) \<and> Abs x = Abs y)"
```
```   168 apply safe
```
```   169 apply (simp (no_asm_use) only: Quotient_def, fast)
```
```   170 apply (simp (no_asm_use) only: Quotient_def, fast)
```
```   171 apply (simp (no_asm_use) only: Quotient_def, fast)
```
```   172 apply (simp (no_asm_use) only: Quotient_def, fast)
```
```   173 apply (simp (no_asm_use) only: Quotient_def, fast)
```
```   174 apply (simp (no_asm_use) only: Quotient_def, fast)
```
```   175 apply (rule QuotientI)
```
```   176 apply simp
```
```   177 apply metis
```
```   178 apply simp
```
```   179 apply (rule ext, rule ext, metis)
```
```   180 done
```
```   181
```
```   182 lemma Quotient_alt_def2:
```
```   183   "Quotient R Abs Rep T \<longleftrightarrow>
```
```   184     (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and>
```
```   185     (\<forall>b. T (Rep b) b) \<and>
```
```   186     (\<forall>x y. R x y \<longleftrightarrow> T x (Abs y) \<and> T y (Abs x))"
```
```   187   unfolding Quotient_alt_def by (safe, metis+)
```
```   188
```
```   189 lemma Quotient_alt_def3:
```
```   190   "Quotient R Abs Rep T \<longleftrightarrow>
```
```   191     (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and> (\<forall>b. T (Rep b) b) \<and>
```
```   192     (\<forall>x y. R x y \<longleftrightarrow> (\<exists>z. T x z \<and> T y z))"
```
```   193   unfolding Quotient_alt_def2 by (safe, metis+)
```
```   194
```
```   195 lemma Quotient_alt_def4:
```
```   196   "Quotient R Abs Rep T \<longleftrightarrow>
```
```   197     (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and> (\<forall>b. T (Rep b) b) \<and> R = T OO conversep T"
```
```   198   unfolding Quotient_alt_def3 fun_eq_iff by auto
```
```   199
```
```   200 lemma fun_quotient:
```
```   201   assumes 1: "Quotient R1 abs1 rep1 T1"
```
```   202   assumes 2: "Quotient R2 abs2 rep2 T2"
```
```   203   shows "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2) (T1 ===> T2)"
```
```   204   using assms unfolding Quotient_alt_def2
```
```   205   unfolding fun_rel_def fun_eq_iff map_fun_apply
```
```   206   by (safe, metis+)
```
```   207
```
```   208 lemma apply_rsp:
```
```   209   fixes f g::"'a \<Rightarrow> 'c"
```
```   210   assumes q: "Quotient R1 Abs1 Rep1 T1"
```
```   211   and     a: "(R1 ===> R2) f g" "R1 x y"
```
```   212   shows "R2 (f x) (g y)"
```
```   213   using a by (auto elim: fun_relE)
```
```   214
```
```   215 lemma apply_rsp':
```
```   216   assumes a: "(R1 ===> R2) f g" "R1 x y"
```
```   217   shows "R2 (f x) (g y)"
```
```   218   using a by (auto elim: fun_relE)
```
```   219
```
```   220 lemma apply_rsp'':
```
```   221   assumes "Quotient R Abs Rep T"
```
```   222   and "(R ===> S) f f"
```
```   223   shows "S (f (Rep x)) (f (Rep x))"
```
```   224 proof -
```
```   225   from assms(1) have "R (Rep x) (Rep x)" by (rule Quotient_rep_reflp)
```
```   226   then show ?thesis using assms(2) by (auto intro: apply_rsp')
```
```   227 qed
```
```   228
```
```   229 subsection {* Quotient composition *}
```
```   230
```
```   231 lemma Quotient_compose:
```
```   232   assumes 1: "Quotient R1 Abs1 Rep1 T1"
```
```   233   assumes 2: "Quotient R2 Abs2 Rep2 T2"
```
```   234   shows "Quotient (T1 OO R2 OO conversep T1) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2) (T1 OO T2)"
```
```   235   using assms unfolding Quotient_alt_def4 by fastforce
```
```   236
```
```   237 lemma equivp_reflp2:
```
```   238   "equivp R \<Longrightarrow> reflp R"
```
```   239   by (erule equivpE)
```
```   240
```
```   241 subsection {* Respects predicate *}
```
```   242
```
```   243 definition Respects :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set"
```
```   244   where "Respects R = {x. R x x}"
```
```   245
```
```   246 lemma in_respects: "x \<in> Respects R \<longleftrightarrow> R x x"
```
```   247   unfolding Respects_def by simp
```
```   248
```
```   249 subsection {* Invariant *}
```
```   250
```
```   251 definition invariant :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
```
```   252   where "invariant R = (\<lambda>x y. R x \<and> x = y)"
```
```   253
```
```   254 lemma invariant_to_eq:
```
```   255   assumes "invariant P x y"
```
```   256   shows "x = y"
```
```   257 using assms by (simp add: invariant_def)
```
```   258
```
```   259 lemma fun_rel_eq_invariant:
```
```   260   shows "((invariant R) ===> S) = (\<lambda>f g. \<forall>x. R x \<longrightarrow> S (f x) (g x))"
```
```   261 by (auto simp add: invariant_def fun_rel_def)
```
```   262
```
```   263 lemma invariant_same_args:
```
```   264   shows "invariant P x x \<equiv> P x"
```
```   265 using assms by (auto simp add: invariant_def)
```
```   266
```
```   267 lemma UNIV_typedef_to_Quotient:
```
```   268   assumes "type_definition Rep Abs UNIV"
```
```   269   and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"
```
```   270   shows "Quotient (op =) Abs Rep T"
```
```   271 proof -
```
```   272   interpret type_definition Rep Abs UNIV by fact
```
```   273   from Abs_inject Rep_inverse Abs_inverse T_def show ?thesis
```
```   274     by (fastforce intro!: QuotientI fun_eq_iff)
```
```   275 qed
```
```   276
```
```   277 lemma UNIV_typedef_to_equivp:
```
```   278   fixes Abs :: "'a \<Rightarrow> 'b"
```
```   279   and Rep :: "'b \<Rightarrow> 'a"
```
```   280   assumes "type_definition Rep Abs (UNIV::'a set)"
```
```   281   shows "equivp (op=::'a\<Rightarrow>'a\<Rightarrow>bool)"
```
```   282 by (rule identity_equivp)
```
```   283
```
```   284 lemma typedef_to_Quotient:
```
```   285   assumes "type_definition Rep Abs S"
```
```   286   and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"
```
```   287   shows "Quotient (invariant (\<lambda>x. x \<in> S)) Abs Rep T"
```
```   288 proof -
```
```   289   interpret type_definition Rep Abs S by fact
```
```   290   from Rep Abs_inject Rep_inverse Abs_inverse T_def show ?thesis
```
```   291     by (auto intro!: QuotientI simp: invariant_def fun_eq_iff)
```
```   292 qed
```
```   293
```
```   294 lemma typedef_to_part_equivp:
```
```   295   assumes "type_definition Rep Abs S"
```
```   296   shows "part_equivp (invariant (\<lambda>x. x \<in> S))"
```
```   297 proof (intro part_equivpI)
```
```   298   interpret type_definition Rep Abs S by fact
```
```   299   show "\<exists>x. invariant (\<lambda>x. x \<in> S) x x" using Rep by (auto simp: invariant_def)
```
```   300 next
```
```   301   show "symp (invariant (\<lambda>x. x \<in> S))" by (auto intro: sympI simp: invariant_def)
```
```   302 next
```
```   303   show "transp (invariant (\<lambda>x. x \<in> S))" by (auto intro: transpI simp: invariant_def)
```
```   304 qed
```
```   305
```
```   306 lemma open_typedef_to_Quotient:
```
```   307   assumes "type_definition Rep Abs {x. P x}"
```
```   308   and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"
```
```   309   shows "Quotient (invariant P) Abs Rep T"
```
```   310   using typedef_to_Quotient [OF assms] by simp
```
```   311
```
```   312 lemma open_typedef_to_part_equivp:
```
```   313   assumes "type_definition Rep Abs {x. P x}"
```
```   314   shows "part_equivp (invariant P)"
```
```   315   using typedef_to_part_equivp [OF assms] by simp
```
```   316
```
```   317 text {* Generating transfer rules for quotients. *}
```
```   318
```
```   319 context
```
```   320   fixes R Abs Rep T
```
```   321   assumes 1: "Quotient R Abs Rep T"
```
```   322 begin
```
```   323
```
```   324 lemma Quotient_right_unique: "right_unique T"
```
```   325   using 1 unfolding Quotient_alt_def right_unique_def by metis
```
```   326
```
```   327 lemma Quotient_right_total: "right_total T"
```
```   328   using 1 unfolding Quotient_alt_def right_total_def by metis
```
```   329
```
```   330 lemma Quotient_rel_eq_transfer: "(T ===> T ===> op =) R (op =)"
```
```   331   using 1 unfolding Quotient_alt_def fun_rel_def by simp
```
```   332
```
```   333 lemma Quotient_abs_induct:
```
```   334   assumes "\<And>y. R y y \<Longrightarrow> P (Abs y)" shows "P x"
```
```   335   using 1 assms unfolding Quotient_def by metis
```
```   336
```
```   337 end
```
```   338
```
```   339 text {* Generating transfer rules for total quotients. *}
```
```   340
```
```   341 context
```
```   342   fixes R Abs Rep T
```
```   343   assumes 1: "Quotient R Abs Rep T" and 2: "reflp R"
```
```   344 begin
```
```   345
```
```   346 lemma Quotient_bi_total: "bi_total T"
```
```   347   using 1 2 unfolding Quotient_alt_def bi_total_def reflp_def by auto
```
```   348
```
```   349 lemma Quotient_id_abs_transfer: "(op = ===> T) (\<lambda>x. x) Abs"
```
```   350   using 1 2 unfolding Quotient_alt_def reflp_def fun_rel_def by simp
```
```   351
```
```   352 lemma Quotient_total_abs_induct: "(\<And>y. P (Abs y)) \<Longrightarrow> P x"
```
```   353   using 1 2 assms unfolding Quotient_alt_def reflp_def by metis
```
```   354
```
```   355 lemma Quotient_total_abs_eq_iff: "Abs x = Abs y \<longleftrightarrow> R x y"
```
```   356   using Quotient_rel [OF 1] 2 unfolding reflp_def by simp
```
```   357
```
```   358 end
```
```   359
```
```   360 text {* Generating transfer rules for a type defined with @{text "typedef"}. *}
```
```   361
```
```   362 context
```
```   363   fixes Rep Abs A T
```
```   364   assumes type: "type_definition Rep Abs A"
```
```   365   assumes T_def: "T \<equiv> (\<lambda>(x::'a) (y::'b). x = Rep y)"
```
```   366 begin
```
```   367
```
```   368 lemma typedef_left_unique: "left_unique T"
```
```   369   unfolding left_unique_def T_def
```
```   370   by (simp add: type_definition.Rep_inject [OF type])
```
```   371
```
```   372 lemma typedef_bi_unique: "bi_unique T"
```
```   373   unfolding bi_unique_def T_def
```
```   374   by (simp add: type_definition.Rep_inject [OF type])
```
```   375
```
```   376 (* the following two theorems are here only for convinience *)
```
```   377
```
```   378 lemma typedef_right_unique: "right_unique T"
```
```   379   using T_def type Quotient_right_unique typedef_to_Quotient
```
```   380   by blast
```
```   381
```
```   382 lemma typedef_right_total: "right_total T"
```
```   383   using T_def type Quotient_right_total typedef_to_Quotient
```
```   384   by blast
```
```   385
```
```   386 lemma typedef_rep_transfer: "(T ===> op =) (\<lambda>x. x) Rep"
```
```   387   unfolding fun_rel_def T_def by simp
```
```   388
```
```   389 end
```
```   390
```
```   391 text {* Generating the correspondence rule for a constant defined with
```
```   392   @{text "lift_definition"}. *}
```
```   393
```
```   394 lemma Quotient_to_transfer:
```
```   395   assumes "Quotient R Abs Rep T" and "R c c" and "c' \<equiv> Abs c"
```
```   396   shows "T c c'"
```
```   397   using assms by (auto dest: Quotient_cr_rel)
```
```   398
```
```   399 text {* Proving reflexivity *}
```
```   400
```
```   401 definition reflp' :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where "reflp' R \<equiv> reflp R"
```
```   402
```
```   403 lemma Quotient_to_left_total:
```
```   404   assumes q: "Quotient R Abs Rep T"
```
```   405   and r_R: "reflp R"
```
```   406   shows "left_total T"
```
```   407 using r_R Quotient_cr_rel[OF q] unfolding left_total_def by (auto elim: reflpE)
```
```   408
```
```   409 lemma reflp_Quotient_composition:
```
```   410   assumes "left_total R"
```
```   411   assumes "reflp T"
```
```   412   shows "reflp (R OO T OO R\<inverse>\<inverse>)"
```
```   413 using assms unfolding reflp_def left_total_def by fast
```
```   414
```
```   415 lemma reflp_fun1:
```
```   416   assumes "is_equality R"
```
```   417   assumes "reflp' S"
```
```   418   shows "reflp (R ===> S)"
```
```   419 using assms unfolding is_equality_def reflp'_def reflp_def fun_rel_def by blast
```
```   420
```
```   421 lemma reflp_fun2:
```
```   422   assumes "is_equality R"
```
```   423   assumes "is_equality S"
```
```   424   shows "reflp (R ===> S)"
```
```   425 using assms unfolding is_equality_def reflp_def fun_rel_def by blast
```
```   426
```
```   427 lemma is_equality_Quotient_composition:
```
```   428   assumes "is_equality T"
```
```   429   assumes "left_total R"
```
```   430   assumes "left_unique R"
```
```   431   shows "is_equality (R OO T OO R\<inverse>\<inverse>)"
```
```   432 using assms unfolding is_equality_def left_total_def left_unique_def OO_def conversep_iff
```
```   433 by fastforce
```
```   434
```
```   435 lemma left_total_composition: "left_total R \<Longrightarrow> left_total S \<Longrightarrow> left_total (R OO S)"
```
```   436 unfolding left_total_def OO_def by fast
```
```   437
```
```   438 lemma left_unique_composition: "left_unique R \<Longrightarrow> left_unique S \<Longrightarrow> left_unique (R OO S)"
```
```   439 unfolding left_unique_def OO_def by fast
```
```   440
```
```   441 lemma reflp_equality: "reflp (op =)"
```
```   442 by (auto intro: reflpI)
```
```   443
```
```   444 text {* Proving a parametrized correspondence relation *}
```
```   445
```
```   446 lemma eq_OO: "op= OO R = R"
```
```   447 unfolding OO_def by metis
```
```   448
```
```   449 definition POS :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where
```
```   450 "POS A B \<equiv> A \<le> B"
```
```   451
```
```   452 definition  NEG :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where
```
```   453 "NEG A B \<equiv> B \<le> A"
```
```   454
```
```   455 (*
```
```   456   The following two rules are here because we don't have any proper
```
```   457   left-unique ant left-total relations. Left-unique and left-total
```
```   458   assumptions show up in distributivity rules for the function type.
```
```   459 *)
```
```   460
```
```   461 lemma bi_unique_left_unique[transfer_rule]: "bi_unique R \<Longrightarrow> left_unique R"
```
```   462 unfolding bi_unique_def left_unique_def by blast
```
```   463
```
```   464 lemma bi_total_left_total[transfer_rule]: "bi_total R \<Longrightarrow> left_total R"
```
```   465 unfolding bi_total_def left_total_def by blast
```
```   466
```
```   467 lemma pos_OO_eq:
```
```   468   shows "POS (A OO op=) A"
```
```   469 unfolding POS_def OO_def by blast
```
```   470
```
```   471 lemma pos_eq_OO:
```
```   472   shows "POS (op= OO A) A"
```
```   473 unfolding POS_def OO_def by blast
```
```   474
```
```   475 lemma neg_OO_eq:
```
```   476   shows "NEG (A OO op=) A"
```
```   477 unfolding NEG_def OO_def by auto
```
```   478
```
```   479 lemma neg_eq_OO:
```
```   480   shows "NEG (op= OO A) A"
```
```   481 unfolding NEG_def OO_def by blast
```
```   482
```
```   483 lemma POS_trans:
```
```   484   assumes "POS A B"
```
```   485   assumes "POS B C"
```
```   486   shows "POS A C"
```
```   487 using assms unfolding POS_def by auto
```
```   488
```
```   489 lemma NEG_trans:
```
```   490   assumes "NEG A B"
```
```   491   assumes "NEG B C"
```
```   492   shows "NEG A C"
```
```   493 using assms unfolding NEG_def by auto
```
```   494
```
```   495 lemma POS_NEG:
```
```   496   "POS A B \<equiv> NEG B A"
```
```   497   unfolding POS_def NEG_def by auto
```
```   498
```
```   499 lemma NEG_POS:
```
```   500   "NEG A B \<equiv> POS B A"
```
```   501   unfolding POS_def NEG_def by auto
```
```   502
```
```   503 lemma POS_pcr_rule:
```
```   504   assumes "POS (A OO B) C"
```
```   505   shows "POS (A OO B OO X) (C OO X)"
```
```   506 using assms unfolding POS_def OO_def by blast
```
```   507
```
```   508 lemma NEG_pcr_rule:
```
```   509   assumes "NEG (A OO B) C"
```
```   510   shows "NEG (A OO B OO X) (C OO X)"
```
```   511 using assms unfolding NEG_def OO_def by blast
```
```   512
```
```   513 lemma POS_apply:
```
```   514   assumes "POS R R'"
```
```   515   assumes "R f g"
```
```   516   shows "R' f g"
```
```   517 using assms unfolding POS_def by auto
```
```   518
```
```   519 text {* Proving a parametrized correspondence relation *}
```
```   520
```
```   521 lemma fun_mono:
```
```   522   assumes "A \<ge> C"
```
```   523   assumes "B \<le> D"
```
```   524   shows   "(A ===> B) \<le> (C ===> D)"
```
```   525 using assms unfolding fun_rel_def by blast
```
```   526
```
```   527 lemma pos_fun_distr: "((R ===> S) OO (R' ===> S')) \<le> ((R OO R') ===> (S OO S'))"
```
```   528 unfolding OO_def fun_rel_def by blast
```
```   529
```
```   530 lemma functional_relation: "right_unique R \<Longrightarrow> left_total R \<Longrightarrow> \<forall>x. \<exists>!y. R x y"
```
```   531 unfolding right_unique_def left_total_def by blast
```
```   532
```
```   533 lemma functional_converse_relation: "left_unique R \<Longrightarrow> right_total R \<Longrightarrow> \<forall>y. \<exists>!x. R x y"
```
```   534 unfolding left_unique_def right_total_def by blast
```
```   535
```
```   536 lemma neg_fun_distr1:
```
```   537 assumes 1: "left_unique R" "right_total R"
```
```   538 assumes 2: "right_unique R'" "left_total R'"
```
```   539 shows "(R OO R' ===> S OO S') \<le> ((R ===> S) OO (R' ===> S')) "
```
```   540   using functional_relation[OF 2] functional_converse_relation[OF 1]
```
```   541   unfolding fun_rel_def OO_def
```
```   542   apply clarify
```
```   543   apply (subst all_comm)
```
```   544   apply (subst all_conj_distrib[symmetric])
```
```   545   apply (intro choice)
```
```   546   by metis
```
```   547
```
```   548 lemma neg_fun_distr2:
```
```   549 assumes 1: "right_unique R'" "left_total R'"
```
```   550 assumes 2: "left_unique S'" "right_total S'"
```
```   551 shows "(R OO R' ===> S OO S') \<le> ((R ===> S) OO (R' ===> S'))"
```
```   552   using functional_converse_relation[OF 2] functional_relation[OF 1]
```
```   553   unfolding fun_rel_def OO_def
```
```   554   apply clarify
```
```   555   apply (subst all_comm)
```
```   556   apply (subst all_conj_distrib[symmetric])
```
```   557   apply (intro choice)
```
```   558   by metis
```
```   559
```
```   560 subsection {* Domains *}
```
```   561
```
```   562 lemma pcr_Domainp_par_left_total:
```
```   563   assumes "Domainp B = P"
```
```   564   assumes "left_total A"
```
```   565   assumes "(A ===> op=) P' P"
```
```   566   shows "Domainp (A OO B) = P'"
```
```   567 using assms
```
```   568 unfolding Domainp_iff[abs_def] OO_def bi_unique_def left_total_def fun_rel_def
```
```   569 by (fast intro: fun_eq_iff)
```
```   570
```
```   571 lemma pcr_Domainp_par:
```
```   572 assumes "Domainp B = P2"
```
```   573 assumes "Domainp A = P1"
```
```   574 assumes "(A ===> op=) P2' P2"
```
```   575 shows "Domainp (A OO B) = (inf P1 P2')"
```
```   576 using assms unfolding fun_rel_def Domainp_iff[abs_def] OO_def
```
```   577 by (fast intro: fun_eq_iff)
```
```   578
```
```   579 definition rel_pred_comp :: "('a => 'b => bool) => ('b => bool) => 'a => bool"
```
```   580 where "rel_pred_comp R P \<equiv> \<lambda>x. \<exists>y. R x y \<and> P y"
```
```   581
```
```   582 lemma pcr_Domainp:
```
```   583 assumes "Domainp B = P"
```
```   584 shows "Domainp (A OO B) = (\<lambda>x. \<exists>y. A x y \<and> P y)"
```
```   585 using assms by blast
```
```   586
```
```   587 lemma pcr_Domainp_total:
```
```   588   assumes "bi_total B"
```
```   589   assumes "Domainp A = P"
```
```   590   shows "Domainp (A OO B) = P"
```
```   591 using assms unfolding bi_total_def
```
```   592 by fast
```
```   593
```
```   594 lemma Quotient_to_Domainp:
```
```   595   assumes "Quotient R Abs Rep T"
```
```   596   shows "Domainp T = (\<lambda>x. R x x)"
```
```   597 by (simp add: Domainp_iff[abs_def] Quotient_cr_rel[OF assms])
```
```   598
```
```   599 lemma invariant_to_Domainp:
```
```   600   assumes "Quotient (Lifting.invariant P) Abs Rep T"
```
```   601   shows "Domainp T = P"
```
```   602 by (simp add: invariant_def Domainp_iff[abs_def] Quotient_cr_rel[OF assms])
```
```   603
```
```   604 end
```
```   605
```
```   606 subsection {* ML setup *}
```
```   607
```
```   608 ML_file "Tools/Lifting/lifting_util.ML"
```
```   609
```
```   610 ML_file "Tools/Lifting/lifting_info.ML"
```
```   611 setup Lifting_Info.setup
```
```   612
```
```   613 lemmas [reflexivity_rule] =
```
```   614   reflp_equality reflp_Quotient_composition is_equality_Quotient_composition
```
```   615   left_total_fun left_unique_fun left_total_eq left_unique_eq left_total_composition
```
```   616   left_unique_composition
```
```   617
```
```   618 text {* add @{thm reflp_fun1} and @{thm reflp_fun2} manually through ML
```
```   619   because we don't want to get reflp' variant of these theorems *}
```
```   620
```
```   621 setup{*
```
```   622 Context.theory_map
```
```   623   (fold
```
```   624     (snd oo (Thm.apply_attribute Lifting_Info.add_reflexivity_rule_raw_attribute))
```
```   625       [@{thm reflp_fun1}, @{thm reflp_fun2}])
```
```   626 *}
```
```   627
```
```   628 (* setup for the function type *)
```
```   629 declare fun_quotient[quot_map]
```
```   630 declare fun_mono[relator_mono]
```
```   631 lemmas [relator_distr] = pos_fun_distr neg_fun_distr1 neg_fun_distr2
```
```   632
```
```   633 ML_file "Tools/Lifting/lifting_term.ML"
```
```   634
```
```   635 ML_file "Tools/Lifting/lifting_def.ML"
```
```   636
```
```   637 ML_file "Tools/Lifting/lifting_setup.ML"
```
```   638
```
```   639 hide_const (open) invariant POS NEG reflp'
```
```   640
```
```   641 end
```