src/HOL/Induct/SList.thy
author wenzelm
Sun Nov 02 18:21:45 2014 +0100 (2014-11-02)
changeset 58889 5b7a9633cfa8
parent 58305 57752a91eec4
child 60530 44f9873d6f6f
permissions -rw-r--r--
modernized header uniformly as section;
     1 (*  Title:      HOL/Induct/SList.thy
     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     3 
     4 This theory is strictly of historical interest. It illustrates how
     5 recursive datatypes can be constructed in higher-order logic from
     6 first principles. Isabelle's datatype package automates a
     7 construction of this sort.
     8 
     9 Enriched theory of lists; mutual indirect recursive data-types.
    10 
    11 Definition of type 'a list (strict lists) by a least fixed point
    12 
    13 We use          list(A) == lfp(%Z. {NUMB(0)} <+> A <*> Z)
    14 and not         list    == lfp(%Z. {NUMB(0)} <+> range(Leaf) <*> Z)
    15 
    16 so that list can serve as a "functor" for defining other recursive types.
    17 
    18 This enables the conservative construction of mutual recursive datatypes
    19 such as
    20 
    21 datatype 'a m = Node 'a * 'a m list
    22 *)
    23 
    24 section {* Extended List Theory (old) *}
    25 
    26 theory SList
    27 imports Sexp
    28 begin
    29 
    30 (*Hilbert_Choice is needed for the function "inv"*)
    31 
    32 (* *********************************************************************** *)
    33 (*                                                                         *)
    34 (* Building up data type                                                   *)
    35 (*                                                                         *)
    36 (* *********************************************************************** *)
    37 
    38 
    39 (* Defining the Concrete Constructors *)
    40 definition
    41   NIL  :: "'a item" where
    42   "NIL = In0(Numb(0))"
    43 
    44 definition
    45   CONS :: "['a item, 'a item] => 'a item" where
    46   "CONS M N = In1(Scons M N)"
    47 
    48 inductive_set
    49   list :: "'a item set => 'a item set"
    50   for A :: "'a item set"
    51   where
    52     NIL_I:  "NIL: list A"
    53   | CONS_I: "[| a: A;  M: list A |] ==> CONS a M : list A"
    54 
    55 
    56 definition "List = list (range Leaf)"
    57 
    58 typedef 'a list = "List :: 'a item set"
    59   morphisms Rep_List Abs_List
    60   unfolding List_def by (blast intro: list.NIL_I)
    61 
    62 abbreviation "Case == Old_Datatype.Case"
    63 abbreviation "Split == Old_Datatype.Split"
    64 
    65 definition
    66   List_case :: "['b, ['a item, 'a item]=>'b, 'a item] => 'b" where
    67   "List_case c d = Case(%x. c)(Split(d))"
    68   
    69 definition
    70   List_rec  :: "['a item, 'b, ['a item, 'a item, 'b]=>'b] => 'b" where
    71   "List_rec M c d = wfrec (pred_sexp^+)
    72                            (%g. List_case c (%x y. d x y (g y))) M"
    73 
    74 
    75 (* *********************************************************************** *)
    76 (*                                                                         *)
    77 (* Abstracting data type                                                   *)
    78 (*                                                                         *)
    79 (* *********************************************************************** *)
    80 
    81 (*Declaring the abstract list constructors*)
    82 
    83 no_translations
    84   "[x, xs]" == "x#[xs]"
    85   "[x]" == "x#[]"
    86 no_notation
    87   Nil  ("[]") and
    88   Cons (infixr "#" 65)
    89 
    90 definition
    91   Nil       :: "'a list"                               ("[]") where
    92    "Nil  = Abs_List(NIL)"
    93 
    94 definition
    95   "Cons"       :: "['a, 'a list] => 'a list"           (infixr "#" 65) where
    96    "x#xs = Abs_List(CONS (Leaf x)(Rep_List xs))"
    97 
    98 definition
    99   (* list Recursion -- the trancl is Essential; see list.ML *)
   100   list_rec  :: "['a list, 'b, ['a, 'a list, 'b]=>'b] => 'b" where
   101    "list_rec l c d =
   102       List_rec(Rep_List l) c (%x y r. d(inv Leaf x)(Abs_List y) r)"
   103 
   104 definition
   105   list_case :: "['b, ['a, 'a list]=>'b, 'a list] => 'b" where
   106    "list_case a f xs = list_rec xs a (%x xs r. f x xs)"
   107 
   108 (* list Enumeration *)
   109 translations
   110   "[x, xs]" == "x#[xs]"
   111   "[x]"     == "x#[]"
   112 
   113   "case xs of [] => a | y#ys => b" == "CONST list_case(a, %y ys. b, xs)"
   114 
   115 
   116 (* *********************************************************************** *)
   117 (*                                                                         *)
   118 (* Generalized Map Functionals                                             *)
   119 (*                                                                         *)
   120 (* *********************************************************************** *)
   121 
   122   
   123 (* Generalized Map Functionals *)
   124 
   125 definition
   126   Rep_map   :: "('b => 'a item) => ('b list => 'a item)" where
   127    "Rep_map f xs = list_rec xs  NIL(%x l r. CONS(f x) r)"
   128 
   129 definition
   130   Abs_map   :: "('a item => 'b) => 'a item => 'b list" where
   131    "Abs_map g M  = List_rec M Nil (%N L r. g(N)#r)"
   132 
   133 
   134 definition
   135   map       :: "('a=>'b) => ('a list => 'b list)" where
   136   "map f xs = list_rec xs [] (%x l r. f(x)#r)"
   137 
   138 primrec take :: "['a list,nat] => 'a list" where
   139   take_0:  "take xs 0 = []"
   140 | take_Suc: "take xs (Suc n) = list_case [] (%x l. x # take l n) xs"
   141 
   142 lemma ListI: "x : list (range Leaf) ==> x : List"
   143 by (simp add: List_def)
   144 
   145 lemma ListD: "x : List ==> x : list (range Leaf)"
   146 by (simp add: List_def)
   147 
   148 lemma list_unfold: "list(A) = usum {Numb(0)} (uprod A (list(A)))"
   149   by (fast intro!: list.intros [unfolded NIL_def CONS_def]
   150            elim: list.cases [unfolded NIL_def CONS_def])
   151 
   152 (*This justifies using list in other recursive type definitions*)
   153 lemma list_mono: "A<=B ==> list(A) <= list(B)"
   154 apply (rule subsetI)
   155 apply (erule list.induct)
   156 apply (auto intro!: list.intros)
   157 done
   158 
   159 (*Type checking -- list creates well-founded sets*)
   160 lemma list_sexp: "list(sexp) <= sexp"
   161 apply (rule subsetI)
   162 apply (erule list.induct)
   163 apply (unfold NIL_def CONS_def)
   164 apply (auto intro: sexp.intros sexp_In0I sexp_In1I)
   165 done
   166 
   167 (* A <= sexp ==> list(A) <= sexp *)
   168 lemmas list_subset_sexp = subset_trans [OF list_mono list_sexp] 
   169 
   170 
   171 (*Induction for the type 'a list *)
   172 lemma list_induct:
   173     "[| P(Nil);    
   174         !!x xs. P(xs) ==> P(x # xs) |]  ==> P(l)"
   175 apply (unfold Nil_def Cons_def) 
   176 apply (rule Rep_List_inverse [THEN subst])
   177 (*types force good instantiation*)
   178 apply (rule Rep_List [unfolded List_def, THEN list.induct], simp)
   179 apply (erule Abs_List_inverse [unfolded List_def, THEN subst], blast) 
   180 done
   181 
   182 
   183 (*** Isomorphisms ***)
   184 
   185 lemma inj_on_Abs_list: "inj_on Abs_List (list(range Leaf))"
   186 apply (rule inj_on_inverseI)
   187 apply (erule Abs_List_inverse [unfolded List_def])
   188 done
   189 
   190 (** Distinctness of constructors **)
   191 
   192 lemma CONS_not_NIL [iff]: "CONS M N ~= NIL"
   193 by (simp add: NIL_def CONS_def)
   194 
   195 lemmas NIL_not_CONS [iff] = CONS_not_NIL [THEN not_sym]
   196 lemmas CONS_neq_NIL = CONS_not_NIL [THEN notE]
   197 lemmas NIL_neq_CONS = sym [THEN CONS_neq_NIL]
   198 
   199 lemma Cons_not_Nil [iff]: "x # xs ~= Nil"
   200 apply (unfold Nil_def Cons_def)
   201 apply (rule CONS_not_NIL [THEN inj_on_Abs_list [THEN inj_on_contraD]])
   202 apply (simp_all add: list.intros rangeI Rep_List [unfolded List_def])
   203 done
   204 
   205 lemmas Nil_not_Cons = Cons_not_Nil [THEN not_sym]
   206 declare Nil_not_Cons [iff]
   207 lemmas Cons_neq_Nil = Cons_not_Nil [THEN notE]
   208 lemmas Nil_neq_Cons = sym [THEN Cons_neq_Nil]
   209 
   210 (** Injectiveness of CONS and Cons **)
   211 
   212 lemma CONS_CONS_eq [iff]: "(CONS K M)=(CONS L N) = (K=L & M=N)"
   213 by (simp add: CONS_def)
   214 
   215 (*For reasoning about abstract list constructors*)
   216 declare Rep_List [THEN ListD, intro] ListI [intro]
   217 declare list.intros [intro,simp]
   218 declare Leaf_inject [dest!]
   219 
   220 lemma Cons_Cons_eq [iff]: "(x#xs=y#ys) = (x=y & xs=ys)"
   221 apply (simp add: Cons_def)
   222 apply (subst Abs_List_inject)
   223 apply (auto simp add: Rep_List_inject)
   224 done
   225 
   226 lemmas Cons_inject2 = Cons_Cons_eq [THEN iffD1, THEN conjE]
   227 
   228 lemma CONS_D: "CONS M N: list(A) ==> M: A & N: list(A)"
   229   by (induct L == "CONS M N" rule: list.induct) auto
   230 
   231 lemma sexp_CONS_D: "CONS M N: sexp ==> M: sexp & N: sexp"
   232 apply (simp add: CONS_def In1_def)
   233 apply (fast dest!: Scons_D)
   234 done
   235 
   236 
   237 (*Reasoning about constructors and their freeness*)
   238 
   239 
   240 lemma not_CONS_self: "N: list(A) ==> !M. N ~= CONS M N"
   241 apply (erule list.induct) apply simp_all done
   242 
   243 lemma not_Cons_self2: "\<forall>x. l ~= x#l"
   244 by (induct l rule: list_induct) simp_all
   245 
   246 
   247 lemma neq_Nil_conv2: "(xs ~= []) = (\<exists>y ys. xs = y#ys)"
   248 by (induct xs rule: list_induct) auto
   249 
   250 (** Conversion rules for List_case: case analysis operator **)
   251 
   252 lemma List_case_NIL [simp]: "List_case c h NIL = c"
   253 by (simp add: List_case_def NIL_def)
   254 
   255 lemma List_case_CONS [simp]: "List_case c h (CONS M N) = h M N"
   256 by (simp add: List_case_def CONS_def)
   257 
   258 
   259 (*** List_rec -- by wf recursion on pred_sexp ***)
   260 
   261 (* The trancl(pred_sexp) is essential because pred_sexp_CONS_I1,2 would not
   262    hold if pred_sexp^+ were changed to pred_sexp. *)
   263 
   264 lemma List_rec_unfold_lemma:
   265      "(%M. List_rec M c d) == 
   266       wfrec (pred_sexp^+) (%g. List_case c (%x y. d x y (g y)))"
   267 by (simp add: List_rec_def)
   268 
   269 lemmas List_rec_unfold = 
   270     def_wfrec [OF List_rec_unfold_lemma wf_pred_sexp [THEN wf_trancl]]
   271 
   272 
   273 (** pred_sexp lemmas **)
   274 
   275 lemma pred_sexp_CONS_I1: 
   276     "[| M: sexp;  N: sexp |] ==> (M, CONS M N) : pred_sexp^+"
   277 by (simp add: CONS_def In1_def)
   278 
   279 lemma pred_sexp_CONS_I2: 
   280     "[| M: sexp;  N: sexp |] ==> (N, CONS M N) : pred_sexp^+"
   281 by (simp add: CONS_def In1_def)
   282 
   283 lemma pred_sexp_CONS_D: 
   284     "(CONS M1 M2, N) : pred_sexp^+ ==>  
   285      (M1,N) : pred_sexp^+ & (M2,N) : pred_sexp^+"
   286 apply (frule pred_sexp_subset_Sigma [THEN trancl_subset_Sigma, THEN subsetD])
   287 apply (blast dest!: sexp_CONS_D intro: pred_sexp_CONS_I1 pred_sexp_CONS_I2 
   288                     trans_trancl [THEN transD])
   289 done
   290 
   291 
   292 (** Conversion rules for List_rec **)
   293 
   294 lemma List_rec_NIL [simp]: "List_rec NIL c h = c"
   295 apply (rule List_rec_unfold [THEN trans])
   296 apply (simp add: List_case_NIL)
   297 done
   298 
   299 lemma List_rec_CONS [simp]:
   300      "[| M: sexp;  N: sexp |] 
   301       ==> List_rec (CONS M N) c h = h M N (List_rec N c h)"
   302 apply (rule List_rec_unfold [THEN trans])
   303 apply (simp add: pred_sexp_CONS_I2)
   304 done
   305 
   306 
   307 (*** list_rec -- by List_rec ***)
   308 
   309 lemmas Rep_List_in_sexp =
   310     subsetD [OF range_Leaf_subset_sexp [THEN list_subset_sexp]
   311                 Rep_List [THEN ListD]] 
   312 
   313 
   314 lemma list_rec_Nil [simp]: "list_rec Nil c h = c"
   315 by (simp add: list_rec_def ListI [THEN Abs_List_inverse] Nil_def)
   316 
   317 
   318 lemma list_rec_Cons [simp]: "list_rec (a#l) c h = h a l (list_rec l c h)"
   319 by (simp add: list_rec_def ListI [THEN Abs_List_inverse] Cons_def
   320               Rep_List_inverse Rep_List [THEN ListD] inj_Leaf Rep_List_in_sexp)
   321 
   322 
   323 (*Type checking.  Useful?*)
   324 lemma List_rec_type:
   325      "[| M: list(A);      
   326          A<=sexp;         
   327          c: C(NIL);       
   328          !!x y r. [| x: A;  y: list(A);  r: C(y) |] ==> h x y r: C(CONS x y)  
   329       |] ==> List_rec M c h : C(M :: 'a item)"
   330 apply (erule list.induct, simp) 
   331 apply (insert list_subset_sexp) 
   332 apply (subst List_rec_CONS, blast+)
   333 done
   334 
   335 
   336 
   337 (** Generalized map functionals **)
   338 
   339 lemma Rep_map_Nil [simp]: "Rep_map f Nil = NIL"
   340 by (simp add: Rep_map_def)
   341 
   342 lemma Rep_map_Cons [simp]: 
   343     "Rep_map f(x#xs) = CONS(f x)(Rep_map f xs)"
   344 by (simp add: Rep_map_def)
   345 
   346 lemma Rep_map_type: "(!!x. f(x): A) ==> Rep_map f xs: list(A)"
   347 apply (simp add: Rep_map_def)
   348 apply (rule list_induct, auto)
   349 done
   350 
   351 lemma Abs_map_NIL [simp]: "Abs_map g NIL = Nil"
   352 by (simp add: Abs_map_def)
   353 
   354 lemma Abs_map_CONS [simp]: 
   355     "[| M: sexp;  N: sexp |] ==> Abs_map g (CONS M N) = g(M) # Abs_map g N"
   356 by (simp add: Abs_map_def)
   357 
   358 (*Eases the use of primitive recursion.*)
   359 lemma def_list_rec_NilCons:
   360     "[| !!xs. f(xs) = list_rec xs c h  |] 
   361      ==> f [] = c & f(x#xs) = h x xs (f xs)"
   362 by simp 
   363 
   364 lemma Abs_map_inverse:
   365      "[| M: list(A);  A<=sexp;  !!z. z: A ==> f(g(z)) = z |]  
   366       ==> Rep_map f (Abs_map g M) = M"
   367 apply (erule list.induct, simp_all)
   368 apply (insert list_subset_sexp) 
   369 apply (subst Abs_map_CONS, blast)
   370 apply blast 
   371 apply simp 
   372 done
   373 
   374 (*Rep_map_inverse is obtained via Abs_Rep_map and map_ident*)
   375 
   376 (** list_case **)
   377 
   378 (* setting up rewrite sets *)
   379 
   380 text{*Better to have a single theorem with a conjunctive conclusion.*}
   381 declare def_list_rec_NilCons [OF list_case_def, simp]
   382 
   383 (** list_case **)
   384 
   385 lemma expand_list_case: 
   386  "P(list_case a f xs) = ((xs=[] --> P a ) & (!y ys. xs=y#ys --> P(f y ys)))"
   387 by (induct xs rule: list_induct) simp_all
   388 
   389 
   390 (**** Function definitions ****)
   391 
   392 declare def_list_rec_NilCons [OF map_def, simp]
   393 
   394 (** The functional "map" and the generalized functionals **)
   395 
   396 lemma Abs_Rep_map: 
   397      "(!!x. f(x): sexp) ==>  
   398         Abs_map g (Rep_map f xs) = map (%t. g(f(t))) xs"
   399 apply (induct xs rule: list_induct)
   400 apply (simp_all add: Rep_map_type list_sexp [THEN subsetD])
   401 done
   402 
   403 
   404 (** Additional mapping lemmas **)
   405 
   406 lemma map_ident [simp]: "map(%x. x)(xs) = xs"
   407 by (induct xs rule: list_induct) simp_all
   408 
   409 lemma map_compose: "map(f o g)(xs) = map f (map g xs)"
   410 apply (simp add: o_def)
   411 apply (induct xs rule: list_induct)
   412 apply simp_all
   413 done
   414 
   415 
   416 (** take **)
   417 
   418 lemma take_Suc1 [simp]: "take [] (Suc x) = []"
   419 by simp
   420 
   421 lemma take_Suc2 [simp]: "take(a#xs)(Suc x) = a#take xs x"
   422 by simp
   423 
   424 lemma take_Nil [simp]: "take [] n = []"
   425 by (induct n) simp_all
   426 
   427 lemma take_take_eq [simp]: "\<forall>n. take (take xs n) n = take xs n"
   428 apply (induct xs rule: list_induct)
   429 apply simp_all
   430 apply (rule allI)
   431 apply (induct_tac n)
   432 apply auto
   433 done
   434 
   435 end