1 theory QuotScript

     2 imports Plain ATP_Linkup Predicate

     3 begin

     4 

     5 definition

     6   "equivp E \<equiv> \<forall>x y. E x y = (E x = E y)"

     7 

     8 definition

     9   "reflp E \<equiv> \<forall>x. E x x"

    10 

    11 definition

    12   "symp E \<equiv> \<forall>x y. E x y \<longrightarrow> E y x"

    13 

    14 definition

    15   "transp E \<equiv> \<forall>x y z. E x y \<and> E y z \<longrightarrow> E x z"

    16 

    17 lemma equivp_reflp_symp_transp:

    18   shows "equivp E = (reflp E \<and> symp E \<and> transp E)"

    19   unfolding equivp_def reflp_def symp_def transp_def expand_fun_eq

    20   by (blast)

    21 

    22 lemma equivp_reflp:

    23   shows "equivp E \<Longrightarrow> (\<And>x. E x x)"

    24   by (simp only: equivp_reflp_symp_transp reflp_def)

    25 

    26 lemma equivp_symp:

    27   shows "equivp E \<Longrightarrow> (\<And>x y. E x y \<Longrightarrow> E y x)"

    28   by (metis equivp_reflp_symp_transp symp_def)

    29 

    30 lemma equivp_transp:

    31   shows "equivp E \<Longrightarrow> (\<And>x y z. E x y \<Longrightarrow> E y z \<Longrightarrow> E x z)"

    32   by (metis equivp_reflp_symp_transp transp_def)

    33 

    34 lemma equivpI:

    35   assumes "reflp R" "symp R" "transp R"

    36   shows "equivp R"

    37   using assms by (simp add: equivp_reflp_symp_transp)

    38 

    39 lemma eq_imp_rel:  

    40   shows "equivp R \<Longrightarrow> a = b \<longrightarrow> R a b" 

    41 by (simp add: equivp_reflp)

    42 

    43 definition

    44   "part_equivp E \<equiv> (\<exists>x. E x x) \<and> (\<forall>x y. E x y = (E x x \<and> E y y \<and> (E x = E y)))"

    45 

    46 lemma equivp_IMP_part_equivp:

    47   assumes a: "equivp E"

    48   shows "part_equivp E"

    49   using a unfolding equivp_def part_equivp_def

    50   by auto

    51 

    52 

    53 abbreviation 

    54   rel_conj (infixr "OOO" 75)

    55 where

    56   "r1 OOO r2 \<equiv> r1 OO r2 OO r1"

    57 

    58 definition

    59   "Quotient E Abs Rep \<equiv> (\<forall>a. Abs (Rep a) = a) \<and>

    60                         (\<forall>a. E (Rep a) (Rep a)) \<and>

    61                         (\<forall>r s. E r s = (E r r \<and> E s s \<and> (Abs r = Abs s)))"

    62 

    63 (* TEST

    64 lemma 

    65   fixes Abs1::"'b \<Rightarrow> 'c"

    66   and   Abs2::"'a \<Rightarrow> 'b"

    67   and   Rep1::"'c \<Rightarrow> 'b"

    68   and   Rep2::"'b \<Rightarrow> 'a"

    69   assumes "Quotient R1 Abs1 Rep1"

    70   and     "Quotient R2 Abs2 Rep2"

    71   shows "Quotient (f R2 R1) (Abs1 \<circ> Abs2) (Rep2 \<circ> Rep1)"

    72 *)

    73 

    74 lemma Quotient_abs_rep:

    75   assumes a: "Quotient E Abs Rep"

    76   shows "Abs (Rep a) \<equiv> a"

    77   using a unfolding Quotient_def

    78   by simp

    79 

    80 lemma Quotient_rep_reflp:

    81   assumes a: "Quotient E Abs Rep"

    82   shows "E (Rep a) (Rep a)"

    83   using a unfolding Quotient_def

    84   by blast

    85 

    86 lemma Quotient_rel:

    87   assumes a: "Quotient E Abs Rep"

    88   shows " E r s = (E r r \<and> E s s \<and> (Abs r = Abs s))"

    89   using a unfolding Quotient_def

    90   by blast

    91 

    92 lemma Quotient_rel_rep:

    93   assumes a: "Quotient R Abs Rep"

    94   shows "R (Rep a) (Rep b) \<equiv> (a = b)"

    95   apply (rule eq_reflection)

    96   using a unfolding Quotient_def

    97   by metis

    98 

    99 lemma Quotient_rep_abs:

   100   assumes a: "Quotient R Abs Rep"

   101   shows "R r r \<Longrightarrow> R (Rep (Abs r)) r"

   102   using a unfolding Quotient_def

   103   by blast

   104 

   105 lemma Quotient_rel_abs:

   106   assumes a: "Quotient E Abs Rep"

   107   shows "E r s \<Longrightarrow> Abs r = Abs s"

   108   using a unfolding Quotient_def

   109   by blast

   110 

   111 lemma identity_equivp:

   112   shows "equivp (op =)"

   113   unfolding equivp_def

   114   by auto

   115 

   116 lemma identity_quotient:

   117   shows "Quotient (op =) id id"

   118   unfolding Quotient_def id_def

   119   by blast

   120 

   121 lemma Quotient_symp:

   122   assumes a: "Quotient E Abs Rep"

   123   shows "symp E"

   124   using a unfolding Quotient_def symp_def

   125   by metis

   126 

   127 lemma Quotient_transp:

   128   assumes a: "Quotient E Abs Rep"

   129   shows "transp E"

   130   using a unfolding Quotient_def transp_def

   131   by metis

   132 

   133 definition

   134   fun_map (infixr "--->" 55)

   135 where

   136 [simp]: "fun_map f g h x = g (h (f x))"

   137 

   138 lemma fun_map_id:

   139   shows "(id ---> id) = id"

   140   by (simp add: expand_fun_eq id_def)

   141 

   142 definition

   143   fun_rel (infixr "===>" 55)

   144 where

   145 [simp]: "fun_rel E1 E2 f g = (\<forall>x y. E1 x y \<longrightarrow> E2 (f x) (g y))"

   146 

   147 lemma fun_rel_eq:

   148   "(op =) ===> (op =) \<equiv> (op =)"

   149   by (rule eq_reflection) (simp add: expand_fun_eq)

   150 

   151 lemma fun_quotient:

   152   assumes q1: "Quotient R1 abs1 rep1"

   153   and     q2: "Quotient R2 abs2 rep2"

   154   shows "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"

   155 proof -

   156   have "\<forall>a. (rep1 ---> abs2) ((abs1 ---> rep2) a) = a"

   157     apply(simp add: expand_fun_eq)

   158     using q1 q2

   159     apply(simp add: Quotient_def)

   160     done

   161   moreover

   162   have "\<forall>a. (R1 ===> R2) ((abs1 ---> rep2) a) ((abs1 ---> rep2) a)"

   163     apply(auto)

   164     using q1 q2 unfolding Quotient_def

   165     apply(metis)

   166     done

   167   moreover

   168   have "\<forall>r s. (R1 ===> R2) r s = ((R1 ===> R2) r r \<and> (R1 ===> R2) s s \<and> 

   169         (rep1 ---> abs2) r  = (rep1 ---> abs2) s)"

   170     apply(auto simp add: expand_fun_eq)

   171     using q1 q2 unfolding Quotient_def

   172     apply(metis)

   173     using q1 q2 unfolding Quotient_def

   174     apply(metis)

   175     using q1 q2 unfolding Quotient_def

   176     apply(metis)

   177     using q1 q2 unfolding Quotient_def

   178     apply(metis)

   179     done

   180   ultimately

   181   show "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"

   182     unfolding Quotient_def by blast

   183 qed

   184 

   185 definition

   186   Respects

   187 where

   188   "Respects R x \<equiv> (R x x)"

   189 

   190 lemma in_respects:

   191   shows "(x \<in> Respects R) = R x x"

   192   unfolding mem_def Respects_def by simp

   193 

   194 lemma equals_rsp:

   195   assumes q: "Quotient R Abs Rep"

   196   and     a: "R xa xb" "R ya yb"

   197   shows "R xa ya = R xb yb"

   198   using Quotient_symp[OF q] Quotient_transp[OF q] unfolding symp_def transp_def

   199   using a by blast

   200 

   201 lemma lambda_prs:

   202   assumes q1: "Quotient R1 Abs1 Rep1"

   203   and     q2: "Quotient R2 Abs2 Rep2"

   204   shows "(Rep1 ---> Abs2) (\<lambda>x. Rep2 (f (Abs1 x))) = (\<lambda>x. f x)"

   205   unfolding expand_fun_eq

   206   using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]

   207   by simp

   208 

   209 lemma lambda_prs1:

   210   assumes q1: "Quotient R1 Abs1 Rep1"

   211   and     q2: "Quotient R2 Abs2 Rep2"

   212   shows "(Rep1 ---> Abs2) (\<lambda>x. (Abs1 ---> Rep2) f x) = (\<lambda>x. f x)"

   213   unfolding expand_fun_eq

   214   using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]

   215   by simp

   216 

   217 lemma rep_abs_rsp:

   218   assumes q: "Quotient R Abs Rep"

   219   and     a: "R x1 x2"

   220   shows "R x1 (Rep (Abs x2))"

   221   using q a by (metis Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q])

   222 

   223 lemma rep_abs_rsp_left:

   224   assumes q: "Quotient R Abs Rep"

   225   and     a: "R x1 x2"

   226   shows "R (Rep (Abs x1)) x2"

   227 using q a by (metis Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q])

   228 

   229 (* In the following theorem R1 can be instantiated with anything,

   230    but we know some of the types of the Rep and Abs functions;

   231    so by solving Quotient assumptions we can get a unique R1 that

   232    will be provable; which is why we need to use apply_rsp and

   233    not the primed version *)

   234 lemma apply_rsp:

   235   fixes f g::"'a \<Rightarrow> 'c"

   236   assumes q: "Quotient R1 Abs1 Rep1"

   237   and     a: "(R1 ===> R2) f g" "R1 x y"

   238   shows "R2 (f x) (g y)"

   239   using a by simp

   240 

   241 lemma apply_rsp':

   242   assumes a: "(R1 ===> R2) f g" "R1 x y"

   243   shows "R2 (f x) (g y)"

   244   using a by simp

   245 

   246 (* Set of lemmas for regularisation of ball and bex *)

   247 

   248 lemma ball_reg_eqv:

   249   fixes P :: "'a \<Rightarrow> bool"

   250   assumes a: "equivp R"

   251   shows "Ball (Respects R) P = (All P)"

   252   by (metis equivp_def in_respects a)

   253 

   254 lemma bex_reg_eqv:

   255   fixes P :: "'a \<Rightarrow> bool"

   256   assumes a: "equivp R"

   257   shows "Bex (Respects R) P = (Ex P)"

   258   by (metis equivp_def in_respects a)

   259 

   260 lemma ball_reg_right:

   261   assumes a: "\<And>x. R x \<Longrightarrow> P x \<longrightarrow> Q x"

   262   shows "All P \<longrightarrow> Ball R Q"

   263   by (metis COMBC_def Collect_def Collect_mem_eq a)

   264 

   265 lemma bex_reg_left:

   266   assumes a: "\<And>x. R x \<Longrightarrow> Q x \<longrightarrow> P x"

   267   shows "Bex R Q \<longrightarrow> Ex P"

   268   by (metis COMBC_def Collect_def Collect_mem_eq a)

   269 

   270 lemma ball_reg_left:

   271   assumes a: "equivp R"

   272   shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ball (Respects R) Q \<longrightarrow> All P"

   273   by (metis equivp_reflp in_respects a)

   274 

   275 lemma bex_reg_right:

   276   assumes a: "equivp R"

   277   shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ex Q \<longrightarrow> Bex (Respects R) P"

   278   by (metis equivp_reflp in_respects a)

   279 

   280 lemma ball_reg_eqv_range:

   281   fixes P::"'a \<Rightarrow> bool"

   282   and x::"'a"

   283   assumes a: "equivp R2"

   284   shows   "(Ball (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = All (\<lambda>f. P (f x)))"

   285   apply(rule iffI)

   286   apply(rule allI)

   287   apply(drule_tac x="\<lambda>y. f x" in bspec)

   288   apply(simp add: in_respects)

   289   apply(rule impI)

   290   using a equivp_reflp_symp_transp[of "R2"]

   291   apply(simp add: reflp_def)

   292   apply(simp)

   293   apply(simp)

   294   done

   295 

   296 lemma bex_reg_eqv_range:

   297   assumes a: "equivp R2"

   298   shows   "(Bex (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = Ex (\<lambda>f. P (f x)))"

   299   apply(auto)

   300   apply(rule_tac x="\<lambda>y. f x" in bexI)

   301   apply(simp)

   302   apply(simp add: Respects_def in_respects)

   303   apply(rule impI)

   304   using a equivp_reflp_symp_transp[of "R2"]

   305   apply(simp add: reflp_def)

   306   done

   307 

   308 lemma all_reg:

   309   assumes a: "!x :: 'a. (P x --> Q x)"

   310   and     b: "All P"

   311   shows "All Q"

   312   using a b by (metis)

   313 

   314 lemma ex_reg:

   315   assumes a: "!x :: 'a. (P x --> Q x)"

   316   and     b: "Ex P"

   317   shows "Ex Q"

   318   using a b by (metis)

   319 

   320 lemma ball_reg:

   321   assumes a: "!x :: 'a. (R x --> P x --> Q x)"

   322   and     b: "Ball R P"

   323   shows "Ball R Q"

   324   using a b by (metis COMBC_def Collect_def Collect_mem_eq)

   325 

   326 lemma bex_reg:

   327   assumes a: "!x :: 'a. (R x --> P x --> Q x)"

   328   and     b: "Bex R P"

   329   shows "Bex R Q"

   330   using a b by (metis COMBC_def Collect_def Collect_mem_eq)

   331 

   332 lemma ball_all_comm:

   333   "(\<And>y. (\<forall>x\<in>P. A x y) \<longrightarrow> (\<forall>x. B x y)) \<Longrightarrow> ((\<forall>x\<in>P. \<forall>y. A x y) \<longrightarrow> (\<forall>x. \<forall>y. B x y))"

   334   by auto

   335 

   336 lemma bex_ex_comm:

   337   "((\<exists>y. \<exists>x. A x y) \<longrightarrow> (\<exists>y. \<exists>x\<in>P. B x y)) \<Longrightarrow> ((\<exists>x. \<exists>y. A x y) \<longrightarrow> (\<exists>x\<in>P. \<exists>y. B x y))"

   338   by auto

   339 

   340 (* Bounded abstraction *)

   341 definition

   342   Babs :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"

   343 where

   344   "(x \<in> p) \<Longrightarrow> (Babs p m x = m x)"

   345 

   346 definition

   347   Bexeq :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"

   348 where

   349   "Bexeq R P \<equiv> (\<exists>x \<in> Respects R. P x) \<and> (\<forall>x \<in> Respects R. \<forall>y \<in> Respects R. ((P x \<and> P y) \<longrightarrow> (R x y)))"

   350 

   351 (* 3 lemmas needed for proving repabs_inj *)

   352 lemma ball_rsp:

   353   assumes a: "(R ===> (op =)) f g"

   354   shows "Ball (Respects R) f = Ball (Respects R) g"

   355   using a by (simp add: Ball_def in_respects)

   356 

   357 lemma bex_rsp:

   358   assumes a: "(R ===> (op =)) f g"

   359   shows "(Bex (Respects R) f = Bex (Respects R) g)"

   360   using a by (simp add: Bex_def in_respects)

   361 

   362 lemma bex1_rsp:

   363   assumes a: "(R ===> (op =)) f g"

   364   shows "(Bex1 (Respects R) f = Bex1 (Respects R) g)"

   365   using a 

   366 by (simp add: Ex1_def Bex1_def in_respects) auto

   367 

   368 (* TODO/FIXME: simplify the repetitions in this proof *)

   369 lemma bexeq_rsp:

   370 assumes a: "Quotient R absf repf"

   371 shows "((R ===> op =) ===> op =) (Bexeq R) (Bexeq R)"

   372 apply simp

   373 unfolding Bexeq_def

   374 apply rule

   375 apply rule

   376 apply rule

   377 apply rule

   378 apply (erule conjE)+

   379 apply (erule bexE)

   380 apply rule

   381 apply (rule_tac x="xa" in bexI)

   382 apply metis

   383 apply metis

   384 apply rule+

   385 apply (erule_tac x="xb" in ballE)

   386 prefer 2

   387 apply (metis)

   388 apply (erule_tac x="ya" in ballE)

   389 prefer 2

   390 apply (metis)

   391 apply (metis in_respects)

   392 apply (erule conjE)+

   393 apply (erule bexE)

   394 apply rule

   395 apply (rule_tac x="xa" in bexI)

   396 apply metis

   397 apply metis

   398 apply rule+

   399 apply (erule_tac x="xb" in ballE)

   400 prefer 2

   401 apply (metis)

   402 apply (erule_tac x="ya" in ballE)

   403 prefer 2

   404 apply (metis)

   405 apply (metis in_respects)

   406 done

   407 

   408 lemma babs_rsp:

   409   assumes q: "Quotient R1 Abs1 Rep1"

   410   and     a: "(R1 ===> R2) f g"

   411   shows      "(R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)"

   412   apply (auto simp add: Babs_def)

   413   apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")

   414   using a apply (simp add: Babs_def)

   415   apply (simp add: in_respects)

   416   using Quotient_rel[OF q]

   417   by metis

   418 

   419 lemma babs_prs:

   420   assumes q1: "Quotient R1 Abs1 Rep1"

   421   and     q2: "Quotient R2 Abs2 Rep2"

   422   shows "(Rep1 ---> Abs2) (Babs (Respects R1) ((Abs1 ---> Rep2) f)) \<equiv> f"

   423   apply(rule eq_reflection)

   424   apply(rule ext)

   425   apply simp

   426   apply (subgoal_tac "Rep1 x \<in> Respects R1")

   427   apply (simp add: Babs_def Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2])

   428   apply (simp add: in_respects Quotient_rel_rep[OF q1])

   429   done

   430 

   431 lemma babs_simp:

   432   assumes q: "Quotient R1 Abs Rep"

   433   shows "((R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)) = ((R1 ===> R2) f g)"

   434   apply(rule iffI)

   435   apply(simp_all only: babs_rsp[OF q])

   436   apply(auto simp add: Babs_def)

   437   apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")

   438   apply(metis Babs_def)

   439   apply (simp add: in_respects)

   440   using Quotient_rel[OF q]

   441   by metis

   442 

   443 (* If a user proves that a particular functional relation 

   444    is an equivalence this may be useful in regularising *)

   445 lemma babs_reg_eqv:

   446   shows "equivp R \<Longrightarrow> Babs (Respects R) P = P"

   447   by (simp add: expand_fun_eq Babs_def in_respects equivp_reflp)

   448 

   449 (* 3 lemmas needed for cleaning of quantifiers *)

   450 lemma all_prs:

   451   assumes a: "Quotient R absf repf"

   452   shows "Ball (Respects R) ((absf ---> id) f) = All f"

   453   using a unfolding Quotient_def Ball_def in_respects fun_map_def id_apply

   454 by metis

   455 

   456 lemma ex_prs:

   457   assumes a: "Quotient R absf repf"

   458   shows "Bex (Respects R) ((absf ---> id) f) = Ex f"

   459   using a unfolding Quotient_def Bex_def in_respects fun_map_def id_apply

   460   by metis

   461 

   462 lemma ex1_prs:

   463   assumes a: "Quotient R absf repf"

   464   shows "((absf ---> id) ---> id) (Bexeq R) f = Ex1 f"

   465 apply simp

   466 apply (subst Bexeq_def)

   467 apply (subst Bex_def)

   468 apply (subst Ex1_def)

   469 apply simp

   470 apply rule

   471  apply (erule conjE)+

   472  apply (erule_tac exE)

   473  apply (erule conjE)

   474  apply (subgoal_tac "\<forall>y. R y y \<longrightarrow> f (absf y) \<longrightarrow> R x y")

   475   apply (rule_tac x="absf x" in exI)

   476   apply (thin_tac "\<forall>x\<in>Respects R. \<forall>y\<in>Respects R. f (absf x) \<and> f (absf y) \<longrightarrow> R x y")

   477   apply (simp)

   478   apply rule+

   479   using a unfolding Quotient_def

   480   apply metis

   481  apply rule+

   482  apply (erule_tac x="x" in ballE)

   483   apply (erule_tac x="y" in ballE)

   484    apply simp

   485   apply (simp add: in_respects)

   486  apply (simp add: in_respects)

   487 apply (erule_tac exE)

   488  apply rule

   489  apply (rule_tac x="repf x" in exI)

   490  apply (simp only: in_respects)

   491   apply rule

   492  apply (metis Quotient_rel_rep[OF a])

   493 using a unfolding Quotient_def apply (simp)

   494 apply rule+

   495 using a unfolding Quotient_def in_respects

   496 apply metis

   497 done

   498 

   499 lemma fun_rel_id:

   500   assumes a: "\<And>x y. R1 x y \<Longrightarrow> R2 (f x) (g y)"

   501   shows "(R1 ===> R2) f g"

   502   using a by simp

   503 

   504 lemma fun_rel_id_asm:

   505   assumes a: "\<And>x y. R1 x y \<Longrightarrow> (A \<longrightarrow> R2 (f x) (g y))"

   506   shows "A \<longrightarrow> (R1 ===> R2) f g"

   507   using a by auto

   508 

   509 lemma quot_rel_rsp:

   510   assumes a: "Quotient R Abs Rep"

   511   shows "(R ===> R ===> op =) R R"

   512   apply(rule fun_rel_id)+

   513   apply(rule equals_rsp[OF a])

   514   apply(assumption)+

   515   done

   516 

   517 lemma o_prs:

   518   assumes q1: "Quotient R1 Abs1 Rep1"

   519   and     q2: "Quotient R2 Abs2 Rep2"

   520   and     q3: "Quotient R3 Abs3 Rep3"

   521   shows "(Rep1 ---> Abs3) (((Abs2 ---> Rep3) f) o ((Abs1 ---> Rep2) g)) = f o g"

   522   using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2] Quotient_abs_rep[OF q3]

   523   unfolding o_def expand_fun_eq by simp

   524 

   525 lemma o_rsp:

   526   assumes q1: "Quotient R1 Abs1 Rep1"

   527   and     q2: "Quotient R2 Abs2 Rep2"

   528   and     q3: "Quotient R3 Abs3 Rep3"

   529   and     a1: "(R2 ===> R3) f1 f2"

   530   and     a2: "(R1 ===> R2) g1 g2"

   531   shows "(R1 ===> R3) (f1 o g1) (f2 o g2)"

   532   using a1 a2 unfolding o_def expand_fun_eq

   533   by (auto)

   534 

   535 lemma cond_prs:

   536   assumes a: "Quotient R absf repf"

   537   shows "absf (if a then repf b else repf c) = (if a then b else c)"

   538   using a unfolding Quotient_def by auto

   539 

   540 lemma if_prs:

   541   assumes q: "Quotient R Abs Rep"

   542   shows "Abs (If a (Rep b) (Rep c)) = If a b c"

   543 using Quotient_abs_rep[OF q] by auto

   544 

   545 (* q not used *)

   546 lemma if_rsp:

   547   assumes q: "Quotient R Abs Rep"

   548   and     a: "a1 = a2" "R b1 b2" "R c1 c2"

   549   shows "R (If a1 b1 c1) (If a2 b2 c2)"

   550 using a by auto

   551 

   552 lemma let_prs:

   553   assumes q1: "Quotient R1 Abs1 Rep1"

   554   and     q2: "Quotient R2 Abs2 Rep2"

   555   shows "Abs2 (Let (Rep1 x) ((Abs1 ---> Rep2) f)) = Let x f"

   556   using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2] by auto

   557 

   558 lemma let_rsp:

   559   assumes q1: "Quotient R1 Abs1 Rep1"

   560   and     a1: "(R1 ===> R2) f g"

   561   and     a2: "R1 x y"

   562   shows "R2 ((Let x f)::'c) ((Let y g)::'c)"

   563   using apply_rsp[OF q1 a1] a2 by auto

   564 

   565 

   566 

   567 

   568 (******************************************)

   569 (* REST OF THE FILE IS UNUSED (until now) *)

   570 (******************************************)

   571 

   572 lemma in_fun:

   573   shows "x \<in> ((f ---> g) s) = g (f x \<in> s)"

   574   by (simp add: mem_def)

   575 

   576 lemma respects_thm:

   577   shows "Respects (R1 ===> R2) f = (\<forall>x y. R1 x y \<longrightarrow> R2 (f x) (f y))"

   578   unfolding Respects_def

   579   by (simp add: expand_fun_eq)

   580 

   581 lemma respects_rep_abs:

   582   assumes a: "Quotient R1 Abs1 Rep1"

   583   and     b: "Respects (R1 ===> R2) f"

   584   and     c: "R1 x x"

   585   shows "R2 (f (Rep1 (Abs1 x))) (f x)"

   586   using a b[simplified respects_thm] c unfolding Quotient_def

   587   by blast

   588 

   589 lemma respects_mp:

   590   assumes a: "Respects (R1 ===> R2) f"

   591   and     b: "R1 x y"

   592   shows "R2 (f x) (f y)"

   593   using a b unfolding Respects_def

   594   by simp

   595 

   596 lemma respects_o:

   597   assumes a: "Respects (R2 ===> R3) f"

   598   and     b: "Respects (R1 ===> R2) g"

   599   shows "Respects (R1 ===> R3) (f o g)"

   600   using a b unfolding Respects_def

   601   by simp

   602 

   603 lemma abs_o_rep:

   604   assumes a: "Quotient r absf repf"

   605   shows "absf o repf = id"

   606   apply(rule ext)

   607   apply(simp add: Quotient_abs_rep[OF a])

   608   done

   609 

   610 lemma eq_comp_r: "op = OO R OO op = \<equiv> R"

   611   apply (rule eq_reflection)

   612   apply (rule ext)+

   613   apply auto

   614   done

   615 

   616 lemma fun_rel_eq_rel:

   617   assumes q1: "Quotient R1 Abs1 Rep1"

   618   and     q2: "Quotient R2 Abs2 Rep2"

   619   shows "(R1 ===> R2) f g = ((Respects (R1 ===> R2) f) \<and> (Respects (R1 ===> R2) g)

   620                              \<and> ((Rep1 ---> Abs2) f = (Rep1 ---> Abs2) g))"

   621   using fun_quotient[OF q1 q2] unfolding Respects_def Quotient_def expand_fun_eq

   622   by blast

   623 

   624 lemma let_babs:

   625   "v \<in> r \<Longrightarrow> Let v (Babs r lam) = Let v lam"

   626   by (simp add: Babs_def)

   627 

   628 lemma fun_rel_equals:

   629   assumes q1: "Quotient R1 Abs1 Rep1"

   630   and     q2: "Quotient R2 Abs2 Rep2"

   631   and     r1: "Respects (R1 ===> R2) f"

   632   and     r2: "Respects (R1 ===> R2) g" 

   633   shows "((Rep1 ---> Abs2) f = (Rep1 ---> Abs2) g) = (\<forall>x y. R1 x y \<longrightarrow> R2 (f x) (g y))"

   634   apply(rule_tac iffI)

   635   apply(rule)+

   636   apply (rule apply_rsp'[of "R1" "R2"])

   637   apply(subst Quotient_rel[OF fun_quotient[OF q1 q2]])

   638   apply auto

   639   using fun_quotient[OF q1 q2] r1 r2 unfolding Quotient_def Respects_def

   640   apply (metis let_rsp q1)

   641   apply (metis fun_rel_eq_rel let_rsp q1 q2 r2)

   642   using r1 unfolding Respects_def expand_fun_eq

   643   apply(simp (no_asm_use))

   644   apply(metis Quotient_rel[OF q2] Quotient_rel_rep[OF q1])

   645   done

   646 

   647 (* ask Peter: fun_rel_IMP used twice *) 

   648 lemma fun_rel_IMP2:

   649   assumes q1: "Quotient R1 Abs1 Rep1"

   650   and     q2: "Quotient R2 Abs2 Rep2"

   651   and     r1: "Respects (R1 ===> R2) f"

   652   and     r2: "Respects (R1 ===> R2) g" 

   653   and     a:  "(Rep1 ---> Abs2) f = (Rep1 ---> Abs2) g"

   654   shows "R1 x y \<Longrightarrow> R2 (f x) (g y)"

   655   using q1 q2 r1 r2 a

   656   by (simp add: fun_rel_equals)

   657 

   658 lemma lambda_rep_abs_rsp:

   659   assumes r1: "\<And>r r'. R1 r r' \<Longrightarrow>R1 r (Rep1 (Abs1 r'))"

   660   and     r2: "\<And>r r'. R2 r r' \<Longrightarrow>R2 r (Rep2 (Abs2 r'))"

   661   shows "(R1 ===> R2) f1 f2 \<Longrightarrow> (R1 ===> R2) f1 ((Abs1 ---> Rep2) ((Rep1 ---> Abs2) f2))"

   662   using r1 r2 by auto

   663 

   664 (* ask peter what are literal_case *)

   665 (* literal_case_PRS *)

   666 (* literal_case_RSP *)

   667 (* Cez: !f x. literal_case f x = f x *)

   668 

   669 (* We use id_simps which includes id_apply; so these 2 theorems can be removed *)

   670 lemma id_prs:

   671   assumes q: "Quotient R Abs Rep"

   672   shows "Abs (id (Rep e)) = id e"

   673   using Quotient_abs_rep[OF q] by auto

   674 

   675 lemma id_rsp:

   676   assumes q: "Quotient R Abs Rep"

   677   and     a: "R e1 e2"

   678   shows "R (id e1) (id e2)"

   679   using a by auto

   680 

   681 end

   682