src/HOL/Datatype.thy
author haftmann
Tue Oct 31 14:58:14 2006 +0100 (2006-10-31)
changeset 21126 4dbc3ccbaab0
parent 21111 624ed9c7c4fe
child 21243 afffe1f72143
permissions -rw-r--r--
adapted seralizer syntax
     1 (*  Title:      HOL/Datatype.thy
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Author:     Stefan Berghofer and Markus Wenzel, TU Muenchen
     5 
     6 Could <*> be generalized to a general summation (Sigma)?
     7 *)
     8 
     9 header{*Analogues of the Cartesian Product and Disjoint Sum for Datatypes*}
    10 
    11 theory Datatype
    12 imports NatArith Sum_Type
    13 begin
    14 
    15 typedef (Node)
    16   ('a,'b) node = "{p. EX f x k. p = (f::nat=>'b+nat, x::'a+nat) & f k = Inr 0}"
    17     --{*it is a subtype of @{text "(nat=>'b+nat) * ('a+nat)"}*}
    18   by auto
    19 
    20 text{*Datatypes will be represented by sets of type @{text node}*}
    21 
    22 types 'a item        = "('a, unit) node set"
    23       ('a, 'b) dtree = "('a, 'b) node set"
    24 
    25 consts
    26   apfst     :: "['a=>'c, 'a*'b] => 'c*'b"
    27   Push      :: "[('b + nat), nat => ('b + nat)] => (nat => ('b + nat))"
    28 
    29   Push_Node :: "[('b + nat), ('a, 'b) node] => ('a, 'b) node"
    30   ndepth    :: "('a, 'b) node => nat"
    31 
    32   Atom      :: "('a + nat) => ('a, 'b) dtree"
    33   Leaf      :: "'a => ('a, 'b) dtree"
    34   Numb      :: "nat => ('a, 'b) dtree"
    35   Scons     :: "[('a, 'b) dtree, ('a, 'b) dtree] => ('a, 'b) dtree"
    36   In0       :: "('a, 'b) dtree => ('a, 'b) dtree"
    37   In1       :: "('a, 'b) dtree => ('a, 'b) dtree"
    38   Lim       :: "('b => ('a, 'b) dtree) => ('a, 'b) dtree"
    39 
    40   ntrunc    :: "[nat, ('a, 'b) dtree] => ('a, 'b) dtree"
    41 
    42   uprod     :: "[('a, 'b) dtree set, ('a, 'b) dtree set]=> ('a, 'b) dtree set"
    43   usum      :: "[('a, 'b) dtree set, ('a, 'b) dtree set]=> ('a, 'b) dtree set"
    44 
    45   Split     :: "[[('a, 'b) dtree, ('a, 'b) dtree]=>'c, ('a, 'b) dtree] => 'c"
    46   Case      :: "[[('a, 'b) dtree]=>'c, [('a, 'b) dtree]=>'c, ('a, 'b) dtree] => 'c"
    47 
    48   dprod     :: "[(('a, 'b) dtree * ('a, 'b) dtree)set, (('a, 'b) dtree * ('a, 'b) dtree)set]
    49                 => (('a, 'b) dtree * ('a, 'b) dtree)set"
    50   dsum      :: "[(('a, 'b) dtree * ('a, 'b) dtree)set, (('a, 'b) dtree * ('a, 'b) dtree)set]
    51                 => (('a, 'b) dtree * ('a, 'b) dtree)set"
    52 
    53 
    54 defs
    55 
    56   Push_Node_def:  "Push_Node == (%n x. Abs_Node (apfst (Push n) (Rep_Node x)))"
    57 
    58   (*crude "lists" of nats -- needed for the constructions*)
    59   apfst_def:  "apfst == (%f (x,y). (f(x),y))"
    60   Push_def:   "Push == (%b h. nat_case b h)"
    61 
    62   (** operations on S-expressions -- sets of nodes **)
    63 
    64   (*S-expression constructors*)
    65   Atom_def:   "Atom == (%x. {Abs_Node((%k. Inr 0, x))})"
    66   Scons_def:  "Scons M N == (Push_Node (Inr 1) ` M) Un (Push_Node (Inr (Suc 1)) ` N)"
    67 
    68   (*Leaf nodes, with arbitrary or nat labels*)
    69   Leaf_def:   "Leaf == Atom o Inl"
    70   Numb_def:   "Numb == Atom o Inr"
    71 
    72   (*Injections of the "disjoint sum"*)
    73   In0_def:    "In0(M) == Scons (Numb 0) M"
    74   In1_def:    "In1(M) == Scons (Numb 1) M"
    75 
    76   (*Function spaces*)
    77   Lim_def: "Lim f == Union {z. ? x. z = Push_Node (Inl x) ` (f x)}"
    78 
    79   (*the set of nodes with depth less than k*)
    80   ndepth_def: "ndepth(n) == (%(f,x). LEAST k. f k = Inr 0) (Rep_Node n)"
    81   ntrunc_def: "ntrunc k N == {n. n:N & ndepth(n)<k}"
    82 
    83   (*products and sums for the "universe"*)
    84   uprod_def:  "uprod A B == UN x:A. UN y:B. { Scons x y }"
    85   usum_def:   "usum A B == In0`A Un In1`B"
    86 
    87   (*the corresponding eliminators*)
    88   Split_def:  "Split c M == THE u. EX x y. M = Scons x y & u = c x y"
    89 
    90   Case_def:   "Case c d M == THE u.  (EX x . M = In0(x) & u = c(x))
    91                                   | (EX y . M = In1(y) & u = d(y))"
    92 
    93 
    94   (** equality for the "universe" **)
    95 
    96   dprod_def:  "dprod r s == UN (x,x'):r. UN (y,y'):s. {(Scons x y, Scons x' y')}"
    97 
    98   dsum_def:   "dsum r s == (UN (x,x'):r. {(In0(x),In0(x'))}) Un
    99                           (UN (y,y'):s. {(In1(y),In1(y'))})"
   100 
   101 
   102 
   103 (** apfst -- can be used in similar type definitions **)
   104 
   105 lemma apfst_conv [simp]: "apfst f (a,b) = (f(a),b)"
   106 by (simp add: apfst_def)
   107 
   108 
   109 lemma apfst_convE: 
   110     "[| q = apfst f p;  !!x y. [| p = (x,y);  q = (f(x),y) |] ==> R  
   111      |] ==> R"
   112 by (force simp add: apfst_def)
   113 
   114 (** Push -- an injection, analogous to Cons on lists **)
   115 
   116 lemma Push_inject1: "Push i f = Push j g  ==> i=j"
   117 apply (simp add: Push_def expand_fun_eq) 
   118 apply (drule_tac x=0 in spec, simp) 
   119 done
   120 
   121 lemma Push_inject2: "Push i f = Push j g  ==> f=g"
   122 apply (auto simp add: Push_def expand_fun_eq) 
   123 apply (drule_tac x="Suc x" in spec, simp) 
   124 done
   125 
   126 lemma Push_inject:
   127     "[| Push i f =Push j g;  [| i=j;  f=g |] ==> P |] ==> P"
   128 by (blast dest: Push_inject1 Push_inject2) 
   129 
   130 lemma Push_neq_K0: "Push (Inr (Suc k)) f = (%z. Inr 0) ==> P"
   131 by (auto simp add: Push_def expand_fun_eq split: nat.split_asm)
   132 
   133 lemmas Abs_Node_inj = Abs_Node_inject [THEN [2] rev_iffD1, standard]
   134 
   135 
   136 (*** Introduction rules for Node ***)
   137 
   138 lemma Node_K0_I: "(%k. Inr 0, a) : Node"
   139 by (simp add: Node_def)
   140 
   141 lemma Node_Push_I: "p: Node ==> apfst (Push i) p : Node"
   142 apply (simp add: Node_def Push_def) 
   143 apply (fast intro!: apfst_conv nat_case_Suc [THEN trans])
   144 done
   145 
   146 
   147 subsection{*Freeness: Distinctness of Constructors*}
   148 
   149 (** Scons vs Atom **)
   150 
   151 lemma Scons_not_Atom [iff]: "Scons M N \<noteq> Atom(a)"
   152 apply (simp add: Atom_def Scons_def Push_Node_def One_nat_def)
   153 apply (blast intro: Node_K0_I Rep_Node [THEN Node_Push_I] 
   154          dest!: Abs_Node_inj 
   155          elim!: apfst_convE sym [THEN Push_neq_K0])  
   156 done
   157 
   158 lemmas Atom_not_Scons = Scons_not_Atom [THEN not_sym, standard]
   159 declare Atom_not_Scons [iff]
   160 
   161 (*** Injectiveness ***)
   162 
   163 (** Atomic nodes **)
   164 
   165 lemma inj_Atom: "inj(Atom)"
   166 apply (simp add: Atom_def)
   167 apply (blast intro!: inj_onI Node_K0_I dest!: Abs_Node_inj)
   168 done
   169 lemmas Atom_inject = inj_Atom [THEN injD, standard]
   170 
   171 lemma Atom_Atom_eq [iff]: "(Atom(a)=Atom(b)) = (a=b)"
   172 by (blast dest!: Atom_inject)
   173 
   174 lemma inj_Leaf: "inj(Leaf)"
   175 apply (simp add: Leaf_def o_def)
   176 apply (rule inj_onI)
   177 apply (erule Atom_inject [THEN Inl_inject])
   178 done
   179 
   180 lemmas Leaf_inject = inj_Leaf [THEN injD, standard]
   181 declare Leaf_inject [dest!]
   182 
   183 lemma inj_Numb: "inj(Numb)"
   184 apply (simp add: Numb_def o_def)
   185 apply (rule inj_onI)
   186 apply (erule Atom_inject [THEN Inr_inject])
   187 done
   188 
   189 lemmas Numb_inject = inj_Numb [THEN injD, standard]
   190 declare Numb_inject [dest!]
   191 
   192 
   193 (** Injectiveness of Push_Node **)
   194 
   195 lemma Push_Node_inject:
   196     "[| Push_Node i m =Push_Node j n;  [| i=j;  m=n |] ==> P  
   197      |] ==> P"
   198 apply (simp add: Push_Node_def)
   199 apply (erule Abs_Node_inj [THEN apfst_convE])
   200 apply (rule Rep_Node [THEN Node_Push_I])+
   201 apply (erule sym [THEN apfst_convE]) 
   202 apply (blast intro: Rep_Node_inject [THEN iffD1] trans sym elim!: Push_inject)
   203 done
   204 
   205 
   206 (** Injectiveness of Scons **)
   207 
   208 lemma Scons_inject_lemma1: "Scons M N <= Scons M' N' ==> M<=M'"
   209 apply (simp add: Scons_def One_nat_def)
   210 apply (blast dest!: Push_Node_inject)
   211 done
   212 
   213 lemma Scons_inject_lemma2: "Scons M N <= Scons M' N' ==> N<=N'"
   214 apply (simp add: Scons_def One_nat_def)
   215 apply (blast dest!: Push_Node_inject)
   216 done
   217 
   218 lemma Scons_inject1: "Scons M N = Scons M' N' ==> M=M'"
   219 apply (erule equalityE)
   220 apply (iprover intro: equalityI Scons_inject_lemma1)
   221 done
   222 
   223 lemma Scons_inject2: "Scons M N = Scons M' N' ==> N=N'"
   224 apply (erule equalityE)
   225 apply (iprover intro: equalityI Scons_inject_lemma2)
   226 done
   227 
   228 lemma Scons_inject:
   229     "[| Scons M N = Scons M' N';  [| M=M';  N=N' |] ==> P |] ==> P"
   230 by (iprover dest: Scons_inject1 Scons_inject2)
   231 
   232 lemma Scons_Scons_eq [iff]: "(Scons M N = Scons M' N') = (M=M' & N=N')"
   233 by (blast elim!: Scons_inject)
   234 
   235 (*** Distinctness involving Leaf and Numb ***)
   236 
   237 (** Scons vs Leaf **)
   238 
   239 lemma Scons_not_Leaf [iff]: "Scons M N \<noteq> Leaf(a)"
   240 by (simp add: Leaf_def o_def Scons_not_Atom)
   241 
   242 lemmas Leaf_not_Scons = Scons_not_Leaf [THEN not_sym, standard]
   243 declare Leaf_not_Scons [iff]
   244 
   245 (** Scons vs Numb **)
   246 
   247 lemma Scons_not_Numb [iff]: "Scons M N \<noteq> Numb(k)"
   248 by (simp add: Numb_def o_def Scons_not_Atom)
   249 
   250 lemmas Numb_not_Scons = Scons_not_Numb [THEN not_sym, standard]
   251 declare Numb_not_Scons [iff]
   252 
   253 
   254 (** Leaf vs Numb **)
   255 
   256 lemma Leaf_not_Numb [iff]: "Leaf(a) \<noteq> Numb(k)"
   257 by (simp add: Leaf_def Numb_def)
   258 
   259 lemmas Numb_not_Leaf = Leaf_not_Numb [THEN not_sym, standard]
   260 declare Numb_not_Leaf [iff]
   261 
   262 
   263 (*** ndepth -- the depth of a node ***)
   264 
   265 lemma ndepth_K0: "ndepth (Abs_Node(%k. Inr 0, x)) = 0"
   266 by (simp add: ndepth_def  Node_K0_I [THEN Abs_Node_inverse] Least_equality)
   267 
   268 lemma ndepth_Push_Node_aux:
   269      "nat_case (Inr (Suc i)) f k = Inr 0 --> Suc(LEAST x. f x = Inr 0) <= k"
   270 apply (induct_tac "k", auto)
   271 apply (erule Least_le)
   272 done
   273 
   274 lemma ndepth_Push_Node: 
   275     "ndepth (Push_Node (Inr (Suc i)) n) = Suc(ndepth(n))"
   276 apply (insert Rep_Node [of n, unfolded Node_def])
   277 apply (auto simp add: ndepth_def Push_Node_def
   278                  Rep_Node [THEN Node_Push_I, THEN Abs_Node_inverse])
   279 apply (rule Least_equality)
   280 apply (auto simp add: Push_def ndepth_Push_Node_aux)
   281 apply (erule LeastI)
   282 done
   283 
   284 
   285 (*** ntrunc applied to the various node sets ***)
   286 
   287 lemma ntrunc_0 [simp]: "ntrunc 0 M = {}"
   288 by (simp add: ntrunc_def)
   289 
   290 lemma ntrunc_Atom [simp]: "ntrunc (Suc k) (Atom a) = Atom(a)"
   291 by (auto simp add: Atom_def ntrunc_def ndepth_K0)
   292 
   293 lemma ntrunc_Leaf [simp]: "ntrunc (Suc k) (Leaf a) = Leaf(a)"
   294 by (simp add: Leaf_def o_def ntrunc_Atom)
   295 
   296 lemma ntrunc_Numb [simp]: "ntrunc (Suc k) (Numb i) = Numb(i)"
   297 by (simp add: Numb_def o_def ntrunc_Atom)
   298 
   299 lemma ntrunc_Scons [simp]: 
   300     "ntrunc (Suc k) (Scons M N) = Scons (ntrunc k M) (ntrunc k N)"
   301 by (auto simp add: Scons_def ntrunc_def One_nat_def ndepth_Push_Node) 
   302 
   303 
   304 
   305 (** Injection nodes **)
   306 
   307 lemma ntrunc_one_In0 [simp]: "ntrunc (Suc 0) (In0 M) = {}"
   308 apply (simp add: In0_def)
   309 apply (simp add: Scons_def)
   310 done
   311 
   312 lemma ntrunc_In0 [simp]: "ntrunc (Suc(Suc k)) (In0 M) = In0 (ntrunc (Suc k) M)"
   313 by (simp add: In0_def)
   314 
   315 lemma ntrunc_one_In1 [simp]: "ntrunc (Suc 0) (In1 M) = {}"
   316 apply (simp add: In1_def)
   317 apply (simp add: Scons_def)
   318 done
   319 
   320 lemma ntrunc_In1 [simp]: "ntrunc (Suc(Suc k)) (In1 M) = In1 (ntrunc (Suc k) M)"
   321 by (simp add: In1_def)
   322 
   323 
   324 subsection{*Set Constructions*}
   325 
   326 
   327 (*** Cartesian Product ***)
   328 
   329 lemma uprodI [intro!]: "[| M:A;  N:B |] ==> Scons M N : uprod A B"
   330 by (simp add: uprod_def)
   331 
   332 (*The general elimination rule*)
   333 lemma uprodE [elim!]:
   334     "[| c : uprod A B;   
   335         !!x y. [| x:A;  y:B;  c = Scons x y |] ==> P  
   336      |] ==> P"
   337 by (auto simp add: uprod_def) 
   338 
   339 
   340 (*Elimination of a pair -- introduces no eigenvariables*)
   341 lemma uprodE2: "[| Scons M N : uprod A B;  [| M:A;  N:B |] ==> P |] ==> P"
   342 by (auto simp add: uprod_def)
   343 
   344 
   345 (*** Disjoint Sum ***)
   346 
   347 lemma usum_In0I [intro]: "M:A ==> In0(M) : usum A B"
   348 by (simp add: usum_def)
   349 
   350 lemma usum_In1I [intro]: "N:B ==> In1(N) : usum A B"
   351 by (simp add: usum_def)
   352 
   353 lemma usumE [elim!]: 
   354     "[| u : usum A B;   
   355         !!x. [| x:A;  u=In0(x) |] ==> P;  
   356         !!y. [| y:B;  u=In1(y) |] ==> P  
   357      |] ==> P"
   358 by (auto simp add: usum_def)
   359 
   360 
   361 (** Injection **)
   362 
   363 lemma In0_not_In1 [iff]: "In0(M) \<noteq> In1(N)"
   364 by (auto simp add: In0_def In1_def One_nat_def)
   365 
   366 lemmas In1_not_In0 = In0_not_In1 [THEN not_sym, standard]
   367 declare In1_not_In0 [iff]
   368 
   369 lemma In0_inject: "In0(M) = In0(N) ==>  M=N"
   370 by (simp add: In0_def)
   371 
   372 lemma In1_inject: "In1(M) = In1(N) ==>  M=N"
   373 by (simp add: In1_def)
   374 
   375 lemma In0_eq [iff]: "(In0 M = In0 N) = (M=N)"
   376 by (blast dest!: In0_inject)
   377 
   378 lemma In1_eq [iff]: "(In1 M = In1 N) = (M=N)"
   379 by (blast dest!: In1_inject)
   380 
   381 lemma inj_In0: "inj In0"
   382 by (blast intro!: inj_onI)
   383 
   384 lemma inj_In1: "inj In1"
   385 by (blast intro!: inj_onI)
   386 
   387 
   388 (*** Function spaces ***)
   389 
   390 lemma Lim_inject: "Lim f = Lim g ==> f = g"
   391 apply (simp add: Lim_def)
   392 apply (rule ext)
   393 apply (blast elim!: Push_Node_inject)
   394 done
   395 
   396 
   397 (*** proving equality of sets and functions using ntrunc ***)
   398 
   399 lemma ntrunc_subsetI: "ntrunc k M <= M"
   400 by (auto simp add: ntrunc_def)
   401 
   402 lemma ntrunc_subsetD: "(!!k. ntrunc k M <= N) ==> M<=N"
   403 by (auto simp add: ntrunc_def)
   404 
   405 (*A generalized form of the take-lemma*)
   406 lemma ntrunc_equality: "(!!k. ntrunc k M = ntrunc k N) ==> M=N"
   407 apply (rule equalityI)
   408 apply (rule_tac [!] ntrunc_subsetD)
   409 apply (rule_tac [!] ntrunc_subsetI [THEN [2] subset_trans], auto) 
   410 done
   411 
   412 lemma ntrunc_o_equality: 
   413     "[| !!k. (ntrunc(k) o h1) = (ntrunc(k) o h2) |] ==> h1=h2"
   414 apply (rule ntrunc_equality [THEN ext])
   415 apply (simp add: expand_fun_eq) 
   416 done
   417 
   418 
   419 (*** Monotonicity ***)
   420 
   421 lemma uprod_mono: "[| A<=A';  B<=B' |] ==> uprod A B <= uprod A' B'"
   422 by (simp add: uprod_def, blast)
   423 
   424 lemma usum_mono: "[| A<=A';  B<=B' |] ==> usum A B <= usum A' B'"
   425 by (simp add: usum_def, blast)
   426 
   427 lemma Scons_mono: "[| M<=M';  N<=N' |] ==> Scons M N <= Scons M' N'"
   428 by (simp add: Scons_def, blast)
   429 
   430 lemma In0_mono: "M<=N ==> In0(M) <= In0(N)"
   431 by (simp add: In0_def subset_refl Scons_mono)
   432 
   433 lemma In1_mono: "M<=N ==> In1(M) <= In1(N)"
   434 by (simp add: In1_def subset_refl Scons_mono)
   435 
   436 
   437 (*** Split and Case ***)
   438 
   439 lemma Split [simp]: "Split c (Scons M N) = c M N"
   440 by (simp add: Split_def)
   441 
   442 lemma Case_In0 [simp]: "Case c d (In0 M) = c(M)"
   443 by (simp add: Case_def)
   444 
   445 lemma Case_In1 [simp]: "Case c d (In1 N) = d(N)"
   446 by (simp add: Case_def)
   447 
   448 
   449 
   450 (**** UN x. B(x) rules ****)
   451 
   452 lemma ntrunc_UN1: "ntrunc k (UN x. f(x)) = (UN x. ntrunc k (f x))"
   453 by (simp add: ntrunc_def, blast)
   454 
   455 lemma Scons_UN1_x: "Scons (UN x. f x) M = (UN x. Scons (f x) M)"
   456 by (simp add: Scons_def, blast)
   457 
   458 lemma Scons_UN1_y: "Scons M (UN x. f x) = (UN x. Scons M (f x))"
   459 by (simp add: Scons_def, blast)
   460 
   461 lemma In0_UN1: "In0(UN x. f(x)) = (UN x. In0(f(x)))"
   462 by (simp add: In0_def Scons_UN1_y)
   463 
   464 lemma In1_UN1: "In1(UN x. f(x)) = (UN x. In1(f(x)))"
   465 by (simp add: In1_def Scons_UN1_y)
   466 
   467 
   468 (*** Equality for Cartesian Product ***)
   469 
   470 lemma dprodI [intro!]: 
   471     "[| (M,M'):r;  (N,N'):s |] ==> (Scons M N, Scons M' N') : dprod r s"
   472 by (auto simp add: dprod_def)
   473 
   474 (*The general elimination rule*)
   475 lemma dprodE [elim!]: 
   476     "[| c : dprod r s;   
   477         !!x y x' y'. [| (x,x') : r;  (y,y') : s;  
   478                         c = (Scons x y, Scons x' y') |] ==> P  
   479      |] ==> P"
   480 by (auto simp add: dprod_def)
   481 
   482 
   483 (*** Equality for Disjoint Sum ***)
   484 
   485 lemma dsum_In0I [intro]: "(M,M'):r ==> (In0(M), In0(M')) : dsum r s"
   486 by (auto simp add: dsum_def)
   487 
   488 lemma dsum_In1I [intro]: "(N,N'):s ==> (In1(N), In1(N')) : dsum r s"
   489 by (auto simp add: dsum_def)
   490 
   491 lemma dsumE [elim!]: 
   492     "[| w : dsum r s;   
   493         !!x x'. [| (x,x') : r;  w = (In0(x), In0(x')) |] ==> P;  
   494         !!y y'. [| (y,y') : s;  w = (In1(y), In1(y')) |] ==> P  
   495      |] ==> P"
   496 by (auto simp add: dsum_def)
   497 
   498 
   499 (*** Monotonicity ***)
   500 
   501 lemma dprod_mono: "[| r<=r';  s<=s' |] ==> dprod r s <= dprod r' s'"
   502 by blast
   503 
   504 lemma dsum_mono: "[| r<=r';  s<=s' |] ==> dsum r s <= dsum r' s'"
   505 by blast
   506 
   507 
   508 (*** Bounding theorems ***)
   509 
   510 lemma dprod_Sigma: "(dprod (A <*> B) (C <*> D)) <= (uprod A C) <*> (uprod B D)"
   511 by blast
   512 
   513 lemmas dprod_subset_Sigma = subset_trans [OF dprod_mono dprod_Sigma, standard]
   514 
   515 (*Dependent version*)
   516 lemma dprod_subset_Sigma2:
   517      "(dprod (Sigma A B) (Sigma C D)) <= 
   518       Sigma (uprod A C) (Split (%x y. uprod (B x) (D y)))"
   519 by auto
   520 
   521 lemma dsum_Sigma: "(dsum (A <*> B) (C <*> D)) <= (usum A C) <*> (usum B D)"
   522 by blast
   523 
   524 lemmas dsum_subset_Sigma = subset_trans [OF dsum_mono dsum_Sigma, standard]
   525 
   526 
   527 (*** Domain ***)
   528 
   529 lemma Domain_dprod [simp]: "Domain (dprod r s) = uprod (Domain r) (Domain s)"
   530 by auto
   531 
   532 lemma Domain_dsum [simp]: "Domain (dsum r s) = usum (Domain r) (Domain s)"
   533 by auto
   534 
   535 
   536 subsection {* Finishing the datatype package setup *}
   537 
   538 text {* Belongs to theory @{text Datatype_Universe}; hides popular names. *}
   539 setup "DatatypeCodegen.setup_hooks"
   540 hide (open) const Push Node Atom Leaf Numb Lim Split Case
   541 hide (open) type node item
   542 
   543 
   544 section {* Datatypes *}
   545 
   546 subsection {* Representing primitive types *}
   547 
   548 rep_datatype bool
   549   distinct True_not_False False_not_True
   550   induction bool_induct
   551 
   552 declare case_split [cases type: bool]
   553   -- "prefer plain propositional version"
   554 
   555 rep_datatype unit
   556   induction unit_induct
   557 
   558 rep_datatype prod
   559   inject Pair_eq
   560   induction prod_induct
   561 
   562 rep_datatype sum
   563   distinct Inl_not_Inr Inr_not_Inl
   564   inject Inl_eq Inr_eq
   565   induction sum_induct
   566 
   567 lemma surjective_sum: "sum_case (%x::'a. f (Inl x)) (%y::'b. f (Inr y)) s = f(s)"
   568   apply (rule_tac s = s in sumE)
   569    apply (erule ssubst)
   570    apply (rule sum.cases(1))
   571   apply (erule ssubst)
   572   apply (rule sum.cases(2))
   573   done
   574 
   575 lemma sum_case_weak_cong: "s = t ==> sum_case f g s = sum_case f g t"
   576   -- {* Prevents simplification of @{text f} and @{text g}: much faster. *}
   577   by simp
   578 
   579 lemma sum_case_inject:
   580   "sum_case f1 f2 = sum_case g1 g2 ==> (f1 = g1 ==> f2 = g2 ==> P) ==> P"
   581 proof -
   582   assume a: "sum_case f1 f2 = sum_case g1 g2"
   583   assume r: "f1 = g1 ==> f2 = g2 ==> P"
   584   show P
   585     apply (rule r)
   586      apply (rule ext)
   587      apply (cut_tac x = "Inl x" in a [THEN fun_cong], simp)
   588     apply (rule ext)
   589     apply (cut_tac x = "Inr x" in a [THEN fun_cong], simp)
   590     done
   591 qed
   592 
   593 constdefs
   594   Suml :: "('a => 'c) => 'a + 'b => 'c"
   595   "Suml == (%f. sum_case f arbitrary)"
   596 
   597   Sumr :: "('b => 'c) => 'a + 'b => 'c"
   598   "Sumr == sum_case arbitrary"
   599 
   600 lemma Suml_inject: "Suml f = Suml g ==> f = g"
   601   by (unfold Suml_def) (erule sum_case_inject)
   602 
   603 lemma Sumr_inject: "Sumr f = Sumr g ==> f = g"
   604   by (unfold Sumr_def) (erule sum_case_inject)
   605 
   606 hide (open) const Suml Sumr
   607 
   608 
   609 subsection {* Further cases/induct rules for tuples *}
   610 
   611 lemma prod_cases3 [cases type]:
   612   obtains (fields) a b c where "y = (a, b, c)"
   613   by (cases y, case_tac b) blast
   614 
   615 lemma prod_induct3 [case_names fields, induct type]:
   616     "(!!a b c. P (a, b, c)) ==> P x"
   617   by (cases x) blast
   618 
   619 lemma prod_cases4 [cases type]:
   620   obtains (fields) a b c d where "y = (a, b, c, d)"
   621   by (cases y, case_tac c) blast
   622 
   623 lemma prod_induct4 [case_names fields, induct type]:
   624     "(!!a b c d. P (a, b, c, d)) ==> P x"
   625   by (cases x) blast
   626 
   627 lemma prod_cases5 [cases type]:
   628   obtains (fields) a b c d e where "y = (a, b, c, d, e)"
   629   by (cases y, case_tac d) blast
   630 
   631 lemma prod_induct5 [case_names fields, induct type]:
   632     "(!!a b c d e. P (a, b, c, d, e)) ==> P x"
   633   by (cases x) blast
   634 
   635 lemma prod_cases6 [cases type]:
   636   obtains (fields) a b c d e f where "y = (a, b, c, d, e, f)"
   637   by (cases y, case_tac e) blast
   638 
   639 lemma prod_induct6 [case_names fields, induct type]:
   640     "(!!a b c d e f. P (a, b, c, d, e, f)) ==> P x"
   641   by (cases x) blast
   642 
   643 lemma prod_cases7 [cases type]:
   644   obtains (fields) a b c d e f g where "y = (a, b, c, d, e, f, g)"
   645   by (cases y, case_tac f) blast
   646 
   647 lemma prod_induct7 [case_names fields, induct type]:
   648     "(!!a b c d e f g. P (a, b, c, d, e, f, g)) ==> P x"
   649   by (cases x) blast
   650 
   651 
   652 subsection {* The option type *}
   653 
   654 datatype 'a option = None | Some 'a
   655 
   656 lemma not_None_eq [iff]: "(x ~= None) = (EX y. x = Some y)"
   657   by (induct x) auto
   658 
   659 lemma not_Some_eq [iff]: "(ALL y. x ~= Some y) = (x = None)"
   660   by (induct x) auto
   661 
   662 text{*Although it may appear that both of these equalities are helpful
   663 only when applied to assumptions, in practice it seems better to give
   664 them the uniform iff attribute. *}
   665 
   666 lemma option_caseE:
   667   assumes c: "(case x of None => P | Some y => Q y)"
   668   obtains
   669     (None) "x = None" and P
   670   | (Some) y where "x = Some y" and "Q y"
   671   using c by (cases x) simp_all
   672 
   673 
   674 subsubsection {* Operations *}
   675 
   676 consts
   677   the :: "'a option => 'a"
   678 primrec
   679   "the (Some x) = x"
   680 
   681 consts
   682   o2s :: "'a option => 'a set"
   683 primrec
   684   "o2s None = {}"
   685   "o2s (Some x) = {x}"
   686 
   687 lemma ospec [dest]: "(ALL x:o2s A. P x) ==> A = Some x ==> P x"
   688   by simp
   689 
   690 ML_setup {* change_claset (fn cs => cs addSD2 ("ospec", thm "ospec")) *}
   691 
   692 lemma elem_o2s [iff]: "(x : o2s xo) = (xo = Some x)"
   693   by (cases xo) auto
   694 
   695 lemma o2s_empty_eq [simp]: "(o2s xo = {}) = (xo = None)"
   696   by (cases xo) auto
   697 
   698 
   699 constdefs
   700   option_map :: "('a => 'b) => ('a option => 'b option)"
   701   "option_map == %f y. case y of None => None | Some x => Some (f x)"
   702 
   703 lemma option_map_None [simp]: "option_map f None = None"
   704   by (simp add: option_map_def)
   705 
   706 lemma option_map_Some [simp]: "option_map f (Some x) = Some (f x)"
   707   by (simp add: option_map_def)
   708 
   709 lemma option_map_is_None [iff]:
   710     "(option_map f opt = None) = (opt = None)"
   711   by (simp add: option_map_def split add: option.split)
   712 
   713 lemma option_map_eq_Some [iff]:
   714     "(option_map f xo = Some y) = (EX z. xo = Some z & f z = y)"
   715   by (simp add: option_map_def split add: option.split)
   716 
   717 lemma option_map_comp:
   718     "option_map f (option_map g opt) = option_map (f o g) opt"
   719   by (simp add: option_map_def split add: option.split)
   720 
   721 lemma option_map_o_sum_case [simp]:
   722     "option_map f o sum_case g h = sum_case (option_map f o g) (option_map f o h)"
   723   by (rule ext) (simp split: sum.split)
   724 
   725 
   726 subsubsection {* Code generator setup *}
   727 
   728 definition
   729   is_none :: "'a option \<Rightarrow> bool"
   730   is_none_none [normal post, symmetric, code inline]: "is_none x \<longleftrightarrow> x = None"
   731 
   732 lemma is_none_code [code]:
   733   shows "is_none None \<longleftrightarrow> True"
   734     and "is_none (Some x) \<longleftrightarrow> False"
   735   unfolding is_none_none [symmetric] by simp_all
   736 
   737 hide (open) const is_none
   738 
   739 lemmas [code] = imp_conv_disj
   740 
   741 lemma [code func]:
   742   "(\<not> True) = False" by (rule HOL.simp_thms)
   743 
   744 lemma [code func]:
   745   "(\<not> False) = True" by (rule HOL.simp_thms)
   746 
   747 lemma [code func]:
   748   shows "(False \<and> x) = False"
   749     and "(True \<and> x) = x"
   750     and "(x \<and> False) = False"
   751     and "(x \<and> True) = x" by simp_all
   752 
   753 lemma [code func]:
   754   shows "(False \<or> x) = x"
   755     and "(True \<or> x) = True"
   756     and "(x \<or> False) = x"
   757     and "(x \<or> True) = True" by simp_all
   758 
   759 declare
   760   if_True [code func]
   761   if_False [code func]
   762   fst_conv [code]
   763   snd_conv [code]
   764 
   765 lemma split_is_prod_case [code inline]:
   766     "split = prod_case"
   767   by (simp add: expand_fun_eq split_def prod.cases)
   768 
   769 code_type bool
   770   (SML "bool")
   771   (Haskell "Bool")
   772 
   773 code_const True and False and Not and "op &" and "op |" and If
   774   (SML "true" and "false" and "not"
   775     and infixl 1 "andalso" and infixl 0 "orelse"
   776     and "!(if (_)/ then (_)/ else (_))")
   777   (Haskell "True" and "False" and "not"
   778     and infixl 3 "&&" and infixl 2 "||"
   779     and "!(if (_)/ then (_)/ else (_))")
   780 
   781 code_type *
   782   (SML infix 2 "*")
   783   (Haskell "!((_),/ (_))")
   784 
   785 code_const Pair
   786   (SML "!((_),/ (_))")
   787   (Haskell "!((_),/ (_))")
   788 
   789 code_type unit
   790   (SML "unit")
   791   (Haskell "()")
   792 
   793 code_const Unity
   794   (SML "()")
   795   (Haskell "()")
   796 
   797 code_type option
   798   (SML "_ option")
   799   (Haskell "Maybe _")
   800 
   801 code_const None and Some
   802   (SML "NONE" and "SOME")
   803   (Haskell "Nothing" and "Just")
   804 
   805 code_instance option :: eq
   806   (Haskell -)
   807 
   808 code_const "Code_Generator.eq \<Colon> 'a\<Colon>eq option \<Rightarrow> 'a option \<Rightarrow> bool"
   809   (Haskell infixl 4 "==")
   810 
   811 code_reserved SML
   812   bool true false not unit option NONE SOME
   813 
   814 code_reserved Haskell
   815   Bool True False not Maybe Nothing Just
   816 
   817 ML
   818 {*
   819 val apfst_conv = thm "apfst_conv";
   820 val apfst_convE = thm "apfst_convE";
   821 val Push_inject1 = thm "Push_inject1";
   822 val Push_inject2 = thm "Push_inject2";
   823 val Push_inject = thm "Push_inject";
   824 val Push_neq_K0 = thm "Push_neq_K0";
   825 val Abs_Node_inj = thm "Abs_Node_inj";
   826 val Node_K0_I = thm "Node_K0_I";
   827 val Node_Push_I = thm "Node_Push_I";
   828 val Scons_not_Atom = thm "Scons_not_Atom";
   829 val Atom_not_Scons = thm "Atom_not_Scons";
   830 val inj_Atom = thm "inj_Atom";
   831 val Atom_inject = thm "Atom_inject";
   832 val Atom_Atom_eq = thm "Atom_Atom_eq";
   833 val inj_Leaf = thm "inj_Leaf";
   834 val Leaf_inject = thm "Leaf_inject";
   835 val inj_Numb = thm "inj_Numb";
   836 val Numb_inject = thm "Numb_inject";
   837 val Push_Node_inject = thm "Push_Node_inject";
   838 val Scons_inject1 = thm "Scons_inject1";
   839 val Scons_inject2 = thm "Scons_inject2";
   840 val Scons_inject = thm "Scons_inject";
   841 val Scons_Scons_eq = thm "Scons_Scons_eq";
   842 val Scons_not_Leaf = thm "Scons_not_Leaf";
   843 val Leaf_not_Scons = thm "Leaf_not_Scons";
   844 val Scons_not_Numb = thm "Scons_not_Numb";
   845 val Numb_not_Scons = thm "Numb_not_Scons";
   846 val Leaf_not_Numb = thm "Leaf_not_Numb";
   847 val Numb_not_Leaf = thm "Numb_not_Leaf";
   848 val ndepth_K0 = thm "ndepth_K0";
   849 val ndepth_Push_Node_aux = thm "ndepth_Push_Node_aux";
   850 val ndepth_Push_Node = thm "ndepth_Push_Node";
   851 val ntrunc_0 = thm "ntrunc_0";
   852 val ntrunc_Atom = thm "ntrunc_Atom";
   853 val ntrunc_Leaf = thm "ntrunc_Leaf";
   854 val ntrunc_Numb = thm "ntrunc_Numb";
   855 val ntrunc_Scons = thm "ntrunc_Scons";
   856 val ntrunc_one_In0 = thm "ntrunc_one_In0";
   857 val ntrunc_In0 = thm "ntrunc_In0";
   858 val ntrunc_one_In1 = thm "ntrunc_one_In1";
   859 val ntrunc_In1 = thm "ntrunc_In1";
   860 val uprodI = thm "uprodI";
   861 val uprodE = thm "uprodE";
   862 val uprodE2 = thm "uprodE2";
   863 val usum_In0I = thm "usum_In0I";
   864 val usum_In1I = thm "usum_In1I";
   865 val usumE = thm "usumE";
   866 val In0_not_In1 = thm "In0_not_In1";
   867 val In1_not_In0 = thm "In1_not_In0";
   868 val In0_inject = thm "In0_inject";
   869 val In1_inject = thm "In1_inject";
   870 val In0_eq = thm "In0_eq";
   871 val In1_eq = thm "In1_eq";
   872 val inj_In0 = thm "inj_In0";
   873 val inj_In1 = thm "inj_In1";
   874 val Lim_inject = thm "Lim_inject";
   875 val ntrunc_subsetI = thm "ntrunc_subsetI";
   876 val ntrunc_subsetD = thm "ntrunc_subsetD";
   877 val ntrunc_equality = thm "ntrunc_equality";
   878 val ntrunc_o_equality = thm "ntrunc_o_equality";
   879 val uprod_mono = thm "uprod_mono";
   880 val usum_mono = thm "usum_mono";
   881 val Scons_mono = thm "Scons_mono";
   882 val In0_mono = thm "In0_mono";
   883 val In1_mono = thm "In1_mono";
   884 val Split = thm "Split";
   885 val Case_In0 = thm "Case_In0";
   886 val Case_In1 = thm "Case_In1";
   887 val ntrunc_UN1 = thm "ntrunc_UN1";
   888 val Scons_UN1_x = thm "Scons_UN1_x";
   889 val Scons_UN1_y = thm "Scons_UN1_y";
   890 val In0_UN1 = thm "In0_UN1";
   891 val In1_UN1 = thm "In1_UN1";
   892 val dprodI = thm "dprodI";
   893 val dprodE = thm "dprodE";
   894 val dsum_In0I = thm "dsum_In0I";
   895 val dsum_In1I = thm "dsum_In1I";
   896 val dsumE = thm "dsumE";
   897 val dprod_mono = thm "dprod_mono";
   898 val dsum_mono = thm "dsum_mono";
   899 val dprod_Sigma = thm "dprod_Sigma";
   900 val dprod_subset_Sigma = thm "dprod_subset_Sigma";
   901 val dprod_subset_Sigma2 = thm "dprod_subset_Sigma2";
   902 val dsum_Sigma = thm "dsum_Sigma";
   903 val dsum_subset_Sigma = thm "dsum_subset_Sigma";
   904 val Domain_dprod = thm "Domain_dprod";
   905 val Domain_dsum = thm "Domain_dsum";
   906 *}
   907 
   908 end