src/HOL/Old_Datatype.thy
author blanchet
Mon Sep 01 16:17:46 2014 +0200 (2014-09-01)
changeset 58112 8081087096ad
parent 55642 src/HOL/Datatype.thy@63beb38e9258
child 58157 c376c43c346c
permissions -rw-r--r--
renamed modules defining old datatypes, as a step towards having 'datatype_new' take 'datatype's place
blanchet@58112
     1
(*  Title:      HOL/Old_Datatype.thy
wenzelm@20819
     2
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
wenzelm@11954
     3
    Author:     Stefan Berghofer and Markus Wenzel, TU Muenchen
berghofe@5181
     4
*)
berghofe@5181
     5
blanchet@58112
     6
header {* Old Datatype package: constructing datatypes from Cartesian Products and Disjoint Sums *}
wenzelm@11954
     7
blanchet@58112
     8
theory Old_Datatype
haftmann@33959
     9
imports Product_Type Sum_Type Nat
wenzelm@46950
    10
keywords "datatype" :: thy_decl
nipkow@15131
    11
begin
wenzelm@11954
    12
haftmann@33959
    13
subsection {* The datatype universe *}
haftmann@33959
    14
wenzelm@45694
    15
definition "Node = {p. EX f x k. p = (f :: nat => 'b + nat, x ::'a + nat) & f k = Inr 0}"
wenzelm@45694
    16
wenzelm@49834
    17
typedef ('a, 'b) node = "Node :: ((nat => 'b + nat) * ('a + nat)) set"
wenzelm@45694
    18
  morphisms Rep_Node Abs_Node
wenzelm@45694
    19
  unfolding Node_def by auto
wenzelm@20819
    20
wenzelm@20819
    21
text{*Datatypes will be represented by sets of type @{text node}*}
wenzelm@20819
    22
bulwahn@42163
    23
type_synonym 'a item        = "('a, unit) node set"
bulwahn@42163
    24
type_synonym ('a, 'b) dtree = "('a, 'b) node set"
wenzelm@20819
    25
wenzelm@20819
    26
consts
wenzelm@20819
    27
  Push      :: "[('b + nat), nat => ('b + nat)] => (nat => ('b + nat))"
wenzelm@20819
    28
wenzelm@20819
    29
  Push_Node :: "[('b + nat), ('a, 'b) node] => ('a, 'b) node"
wenzelm@20819
    30
  ndepth    :: "('a, 'b) node => nat"
wenzelm@20819
    31
wenzelm@20819
    32
  Atom      :: "('a + nat) => ('a, 'b) dtree"
wenzelm@20819
    33
  Leaf      :: "'a => ('a, 'b) dtree"
wenzelm@20819
    34
  Numb      :: "nat => ('a, 'b) dtree"
wenzelm@20819
    35
  Scons     :: "[('a, 'b) dtree, ('a, 'b) dtree] => ('a, 'b) dtree"
wenzelm@20819
    36
  In0       :: "('a, 'b) dtree => ('a, 'b) dtree"
wenzelm@20819
    37
  In1       :: "('a, 'b) dtree => ('a, 'b) dtree"
wenzelm@20819
    38
  Lim       :: "('b => ('a, 'b) dtree) => ('a, 'b) dtree"
wenzelm@20819
    39
wenzelm@20819
    40
  ntrunc    :: "[nat, ('a, 'b) dtree] => ('a, 'b) dtree"
wenzelm@20819
    41
wenzelm@20819
    42
  uprod     :: "[('a, 'b) dtree set, ('a, 'b) dtree set]=> ('a, 'b) dtree set"
wenzelm@20819
    43
  usum      :: "[('a, 'b) dtree set, ('a, 'b) dtree set]=> ('a, 'b) dtree set"
wenzelm@20819
    44
wenzelm@20819
    45
  Split     :: "[[('a, 'b) dtree, ('a, 'b) dtree]=>'c, ('a, 'b) dtree] => 'c"
wenzelm@20819
    46
  Case      :: "[[('a, 'b) dtree]=>'c, [('a, 'b) dtree]=>'c, ('a, 'b) dtree] => 'c"
wenzelm@20819
    47
wenzelm@20819
    48
  dprod     :: "[(('a, 'b) dtree * ('a, 'b) dtree)set, (('a, 'b) dtree * ('a, 'b) dtree)set]
wenzelm@20819
    49
                => (('a, 'b) dtree * ('a, 'b) dtree)set"
wenzelm@20819
    50
  dsum      :: "[(('a, 'b) dtree * ('a, 'b) dtree)set, (('a, 'b) dtree * ('a, 'b) dtree)set]
wenzelm@20819
    51
                => (('a, 'b) dtree * ('a, 'b) dtree)set"
wenzelm@20819
    52
wenzelm@20819
    53
wenzelm@20819
    54
defs
wenzelm@20819
    55
wenzelm@20819
    56
  Push_Node_def:  "Push_Node == (%n x. Abs_Node (apfst (Push n) (Rep_Node x)))"
wenzelm@20819
    57
wenzelm@20819
    58
  (*crude "lists" of nats -- needed for the constructions*)
blanchet@55415
    59
  Push_def:   "Push == (%b h. case_nat b h)"
wenzelm@20819
    60
wenzelm@20819
    61
  (** operations on S-expressions -- sets of nodes **)
wenzelm@20819
    62
wenzelm@20819
    63
  (*S-expression constructors*)
wenzelm@20819
    64
  Atom_def:   "Atom == (%x. {Abs_Node((%k. Inr 0, x))})"
wenzelm@20819
    65
  Scons_def:  "Scons M N == (Push_Node (Inr 1) ` M) Un (Push_Node (Inr (Suc 1)) ` N)"
wenzelm@20819
    66
wenzelm@20819
    67
  (*Leaf nodes, with arbitrary or nat labels*)
wenzelm@20819
    68
  Leaf_def:   "Leaf == Atom o Inl"
wenzelm@20819
    69
  Numb_def:   "Numb == Atom o Inr"
wenzelm@20819
    70
wenzelm@20819
    71
  (*Injections of the "disjoint sum"*)
wenzelm@20819
    72
  In0_def:    "In0(M) == Scons (Numb 0) M"
wenzelm@20819
    73
  In1_def:    "In1(M) == Scons (Numb 1) M"
wenzelm@20819
    74
wenzelm@20819
    75
  (*Function spaces*)
wenzelm@20819
    76
  Lim_def: "Lim f == Union {z. ? x. z = Push_Node (Inl x) ` (f x)}"
wenzelm@20819
    77
wenzelm@20819
    78
  (*the set of nodes with depth less than k*)
wenzelm@20819
    79
  ndepth_def: "ndepth(n) == (%(f,x). LEAST k. f k = Inr 0) (Rep_Node n)"
wenzelm@20819
    80
  ntrunc_def: "ntrunc k N == {n. n:N & ndepth(n)<k}"
wenzelm@20819
    81
wenzelm@20819
    82
  (*products and sums for the "universe"*)
wenzelm@20819
    83
  uprod_def:  "uprod A B == UN x:A. UN y:B. { Scons x y }"
wenzelm@20819
    84
  usum_def:   "usum A B == In0`A Un In1`B"
wenzelm@20819
    85
wenzelm@20819
    86
  (*the corresponding eliminators*)
wenzelm@20819
    87
  Split_def:  "Split c M == THE u. EX x y. M = Scons x y & u = c x y"
wenzelm@20819
    88
wenzelm@20819
    89
  Case_def:   "Case c d M == THE u.  (EX x . M = In0(x) & u = c(x))
wenzelm@20819
    90
                                  | (EX y . M = In1(y) & u = d(y))"
wenzelm@20819
    91
wenzelm@20819
    92
wenzelm@20819
    93
  (** equality for the "universe" **)
wenzelm@20819
    94
wenzelm@20819
    95
  dprod_def:  "dprod r s == UN (x,x'):r. UN (y,y'):s. {(Scons x y, Scons x' y')}"
wenzelm@20819
    96
wenzelm@20819
    97
  dsum_def:   "dsum r s == (UN (x,x'):r. {(In0(x),In0(x'))}) Un
wenzelm@20819
    98
                          (UN (y,y'):s. {(In1(y),In1(y'))})"
wenzelm@20819
    99
wenzelm@20819
   100
wenzelm@20819
   101
wenzelm@20819
   102
lemma apfst_convE: 
wenzelm@20819
   103
    "[| q = apfst f p;  !!x y. [| p = (x,y);  q = (f(x),y) |] ==> R  
wenzelm@20819
   104
     |] ==> R"
wenzelm@20819
   105
by (force simp add: apfst_def)
wenzelm@20819
   106
wenzelm@20819
   107
(** Push -- an injection, analogous to Cons on lists **)
wenzelm@20819
   108
wenzelm@20819
   109
lemma Push_inject1: "Push i f = Push j g  ==> i=j"
nipkow@39302
   110
apply (simp add: Push_def fun_eq_iff) 
wenzelm@20819
   111
apply (drule_tac x=0 in spec, simp) 
wenzelm@20819
   112
done
wenzelm@20819
   113
wenzelm@20819
   114
lemma Push_inject2: "Push i f = Push j g  ==> f=g"
nipkow@39302
   115
apply (auto simp add: Push_def fun_eq_iff) 
wenzelm@20819
   116
apply (drule_tac x="Suc x" in spec, simp) 
wenzelm@20819
   117
done
wenzelm@20819
   118
wenzelm@20819
   119
lemma Push_inject:
wenzelm@20819
   120
    "[| Push i f =Push j g;  [| i=j;  f=g |] ==> P |] ==> P"
wenzelm@20819
   121
by (blast dest: Push_inject1 Push_inject2) 
wenzelm@20819
   122
wenzelm@20819
   123
lemma Push_neq_K0: "Push (Inr (Suc k)) f = (%z. Inr 0) ==> P"
nipkow@39302
   124
by (auto simp add: Push_def fun_eq_iff split: nat.split_asm)
wenzelm@20819
   125
wenzelm@45607
   126
lemmas Abs_Node_inj = Abs_Node_inject [THEN [2] rev_iffD1]
wenzelm@20819
   127
wenzelm@20819
   128
wenzelm@20819
   129
(*** Introduction rules for Node ***)
wenzelm@20819
   130
wenzelm@20819
   131
lemma Node_K0_I: "(%k. Inr 0, a) : Node"
wenzelm@20819
   132
by (simp add: Node_def)
wenzelm@20819
   133
wenzelm@20819
   134
lemma Node_Push_I: "p: Node ==> apfst (Push i) p : Node"
wenzelm@20819
   135
apply (simp add: Node_def Push_def) 
blanchet@55642
   136
apply (fast intro!: apfst_conv nat.case(2)[THEN trans])
wenzelm@20819
   137
done
wenzelm@20819
   138
wenzelm@20819
   139
wenzelm@20819
   140
subsection{*Freeness: Distinctness of Constructors*}
wenzelm@20819
   141
wenzelm@20819
   142
(** Scons vs Atom **)
wenzelm@20819
   143
wenzelm@20819
   144
lemma Scons_not_Atom [iff]: "Scons M N \<noteq> Atom(a)"
huffman@35216
   145
unfolding Atom_def Scons_def Push_Node_def One_nat_def
huffman@35216
   146
by (blast intro: Node_K0_I Rep_Node [THEN Node_Push_I] 
wenzelm@20819
   147
         dest!: Abs_Node_inj 
wenzelm@20819
   148
         elim!: apfst_convE sym [THEN Push_neq_K0])  
wenzelm@20819
   149
wenzelm@45607
   150
lemmas Atom_not_Scons [iff] = Scons_not_Atom [THEN not_sym]
haftmann@21407
   151
wenzelm@20819
   152
wenzelm@20819
   153
(*** Injectiveness ***)
wenzelm@20819
   154
wenzelm@20819
   155
(** Atomic nodes **)
wenzelm@20819
   156
wenzelm@20819
   157
lemma inj_Atom: "inj(Atom)"
wenzelm@20819
   158
apply (simp add: Atom_def)
wenzelm@20819
   159
apply (blast intro!: inj_onI Node_K0_I dest!: Abs_Node_inj)
wenzelm@20819
   160
done
wenzelm@45607
   161
lemmas Atom_inject = inj_Atom [THEN injD]
wenzelm@20819
   162
wenzelm@20819
   163
lemma Atom_Atom_eq [iff]: "(Atom(a)=Atom(b)) = (a=b)"
wenzelm@20819
   164
by (blast dest!: Atom_inject)
wenzelm@20819
   165
wenzelm@20819
   166
lemma inj_Leaf: "inj(Leaf)"
wenzelm@20819
   167
apply (simp add: Leaf_def o_def)
wenzelm@20819
   168
apply (rule inj_onI)
wenzelm@20819
   169
apply (erule Atom_inject [THEN Inl_inject])
wenzelm@20819
   170
done
wenzelm@20819
   171
wenzelm@45607
   172
lemmas Leaf_inject [dest!] = inj_Leaf [THEN injD]
wenzelm@20819
   173
wenzelm@20819
   174
lemma inj_Numb: "inj(Numb)"
wenzelm@20819
   175
apply (simp add: Numb_def o_def)
wenzelm@20819
   176
apply (rule inj_onI)
wenzelm@20819
   177
apply (erule Atom_inject [THEN Inr_inject])
wenzelm@20819
   178
done
wenzelm@20819
   179
wenzelm@45607
   180
lemmas Numb_inject [dest!] = inj_Numb [THEN injD]
wenzelm@20819
   181
wenzelm@20819
   182
wenzelm@20819
   183
(** Injectiveness of Push_Node **)
wenzelm@20819
   184
wenzelm@20819
   185
lemma Push_Node_inject:
wenzelm@20819
   186
    "[| Push_Node i m =Push_Node j n;  [| i=j;  m=n |] ==> P  
wenzelm@20819
   187
     |] ==> P"
wenzelm@20819
   188
apply (simp add: Push_Node_def)
wenzelm@20819
   189
apply (erule Abs_Node_inj [THEN apfst_convE])
wenzelm@20819
   190
apply (rule Rep_Node [THEN Node_Push_I])+
wenzelm@20819
   191
apply (erule sym [THEN apfst_convE]) 
wenzelm@20819
   192
apply (blast intro: Rep_Node_inject [THEN iffD1] trans sym elim!: Push_inject)
wenzelm@20819
   193
done
wenzelm@20819
   194
wenzelm@20819
   195
wenzelm@20819
   196
(** Injectiveness of Scons **)
wenzelm@20819
   197
wenzelm@20819
   198
lemma Scons_inject_lemma1: "Scons M N <= Scons M' N' ==> M<=M'"
huffman@35216
   199
unfolding Scons_def One_nat_def
huffman@35216
   200
by (blast dest!: Push_Node_inject)
wenzelm@20819
   201
wenzelm@20819
   202
lemma Scons_inject_lemma2: "Scons M N <= Scons M' N' ==> N<=N'"
huffman@35216
   203
unfolding Scons_def One_nat_def
huffman@35216
   204
by (blast dest!: Push_Node_inject)
wenzelm@20819
   205
wenzelm@20819
   206
lemma Scons_inject1: "Scons M N = Scons M' N' ==> M=M'"
wenzelm@20819
   207
apply (erule equalityE)
wenzelm@20819
   208
apply (iprover intro: equalityI Scons_inject_lemma1)
wenzelm@20819
   209
done
wenzelm@20819
   210
wenzelm@20819
   211
lemma Scons_inject2: "Scons M N = Scons M' N' ==> N=N'"
wenzelm@20819
   212
apply (erule equalityE)
wenzelm@20819
   213
apply (iprover intro: equalityI Scons_inject_lemma2)
wenzelm@20819
   214
done
wenzelm@20819
   215
wenzelm@20819
   216
lemma Scons_inject:
wenzelm@20819
   217
    "[| Scons M N = Scons M' N';  [| M=M';  N=N' |] ==> P |] ==> P"
wenzelm@20819
   218
by (iprover dest: Scons_inject1 Scons_inject2)
wenzelm@20819
   219
wenzelm@20819
   220
lemma Scons_Scons_eq [iff]: "(Scons M N = Scons M' N') = (M=M' & N=N')"
wenzelm@20819
   221
by (blast elim!: Scons_inject)
wenzelm@20819
   222
wenzelm@20819
   223
(*** Distinctness involving Leaf and Numb ***)
wenzelm@20819
   224
wenzelm@20819
   225
(** Scons vs Leaf **)
wenzelm@20819
   226
wenzelm@20819
   227
lemma Scons_not_Leaf [iff]: "Scons M N \<noteq> Leaf(a)"
huffman@35216
   228
unfolding Leaf_def o_def by (rule Scons_not_Atom)
wenzelm@20819
   229
wenzelm@45607
   230
lemmas Leaf_not_Scons  [iff] = Scons_not_Leaf [THEN not_sym]
wenzelm@20819
   231
wenzelm@20819
   232
(** Scons vs Numb **)
wenzelm@20819
   233
wenzelm@20819
   234
lemma Scons_not_Numb [iff]: "Scons M N \<noteq> Numb(k)"
huffman@35216
   235
unfolding Numb_def o_def by (rule Scons_not_Atom)
wenzelm@20819
   236
wenzelm@45607
   237
lemmas Numb_not_Scons [iff] = Scons_not_Numb [THEN not_sym]
wenzelm@20819
   238
wenzelm@20819
   239
wenzelm@20819
   240
(** Leaf vs Numb **)
wenzelm@20819
   241
wenzelm@20819
   242
lemma Leaf_not_Numb [iff]: "Leaf(a) \<noteq> Numb(k)"
wenzelm@20819
   243
by (simp add: Leaf_def Numb_def)
wenzelm@20819
   244
wenzelm@45607
   245
lemmas Numb_not_Leaf [iff] = Leaf_not_Numb [THEN not_sym]
wenzelm@20819
   246
wenzelm@20819
   247
wenzelm@20819
   248
(*** ndepth -- the depth of a node ***)
wenzelm@20819
   249
wenzelm@20819
   250
lemma ndepth_K0: "ndepth (Abs_Node(%k. Inr 0, x)) = 0"
wenzelm@20819
   251
by (simp add: ndepth_def  Node_K0_I [THEN Abs_Node_inverse] Least_equality)
wenzelm@20819
   252
wenzelm@20819
   253
lemma ndepth_Push_Node_aux:
blanchet@55415
   254
     "case_nat (Inr (Suc i)) f k = Inr 0 --> Suc(LEAST x. f x = Inr 0) <= k"
wenzelm@20819
   255
apply (induct_tac "k", auto)
wenzelm@20819
   256
apply (erule Least_le)
wenzelm@20819
   257
done
wenzelm@20819
   258
wenzelm@20819
   259
lemma ndepth_Push_Node: 
wenzelm@20819
   260
    "ndepth (Push_Node (Inr (Suc i)) n) = Suc(ndepth(n))"
wenzelm@20819
   261
apply (insert Rep_Node [of n, unfolded Node_def])
wenzelm@20819
   262
apply (auto simp add: ndepth_def Push_Node_def
wenzelm@20819
   263
                 Rep_Node [THEN Node_Push_I, THEN Abs_Node_inverse])
wenzelm@20819
   264
apply (rule Least_equality)
wenzelm@20819
   265
apply (auto simp add: Push_def ndepth_Push_Node_aux)
wenzelm@20819
   266
apply (erule LeastI)
wenzelm@20819
   267
done
wenzelm@20819
   268
wenzelm@20819
   269
wenzelm@20819
   270
(*** ntrunc applied to the various node sets ***)
wenzelm@20819
   271
wenzelm@20819
   272
lemma ntrunc_0 [simp]: "ntrunc 0 M = {}"
wenzelm@20819
   273
by (simp add: ntrunc_def)
wenzelm@20819
   274
wenzelm@20819
   275
lemma ntrunc_Atom [simp]: "ntrunc (Suc k) (Atom a) = Atom(a)"
wenzelm@20819
   276
by (auto simp add: Atom_def ntrunc_def ndepth_K0)
wenzelm@20819
   277
wenzelm@20819
   278
lemma ntrunc_Leaf [simp]: "ntrunc (Suc k) (Leaf a) = Leaf(a)"
huffman@35216
   279
unfolding Leaf_def o_def by (rule ntrunc_Atom)
wenzelm@20819
   280
wenzelm@20819
   281
lemma ntrunc_Numb [simp]: "ntrunc (Suc k) (Numb i) = Numb(i)"
huffman@35216
   282
unfolding Numb_def o_def by (rule ntrunc_Atom)
wenzelm@20819
   283
wenzelm@20819
   284
lemma ntrunc_Scons [simp]: 
wenzelm@20819
   285
    "ntrunc (Suc k) (Scons M N) = Scons (ntrunc k M) (ntrunc k N)"
huffman@35216
   286
unfolding Scons_def ntrunc_def One_nat_def
huffman@35216
   287
by (auto simp add: ndepth_Push_Node)
wenzelm@20819
   288
wenzelm@20819
   289
wenzelm@20819
   290
wenzelm@20819
   291
(** Injection nodes **)
wenzelm@20819
   292
wenzelm@20819
   293
lemma ntrunc_one_In0 [simp]: "ntrunc (Suc 0) (In0 M) = {}"
wenzelm@20819
   294
apply (simp add: In0_def)
wenzelm@20819
   295
apply (simp add: Scons_def)
wenzelm@20819
   296
done
wenzelm@20819
   297
wenzelm@20819
   298
lemma ntrunc_In0 [simp]: "ntrunc (Suc(Suc k)) (In0 M) = In0 (ntrunc (Suc k) M)"
wenzelm@20819
   299
by (simp add: In0_def)
wenzelm@20819
   300
wenzelm@20819
   301
lemma ntrunc_one_In1 [simp]: "ntrunc (Suc 0) (In1 M) = {}"
wenzelm@20819
   302
apply (simp add: In1_def)
wenzelm@20819
   303
apply (simp add: Scons_def)
wenzelm@20819
   304
done
wenzelm@20819
   305
wenzelm@20819
   306
lemma ntrunc_In1 [simp]: "ntrunc (Suc(Suc k)) (In1 M) = In1 (ntrunc (Suc k) M)"
wenzelm@20819
   307
by (simp add: In1_def)
wenzelm@20819
   308
wenzelm@20819
   309
wenzelm@20819
   310
subsection{*Set Constructions*}
wenzelm@20819
   311
wenzelm@20819
   312
wenzelm@20819
   313
(*** Cartesian Product ***)
wenzelm@20819
   314
wenzelm@20819
   315
lemma uprodI [intro!]: "[| M:A;  N:B |] ==> Scons M N : uprod A B"
wenzelm@20819
   316
by (simp add: uprod_def)
wenzelm@20819
   317
wenzelm@20819
   318
(*The general elimination rule*)
wenzelm@20819
   319
lemma uprodE [elim!]:
wenzelm@20819
   320
    "[| c : uprod A B;   
wenzelm@20819
   321
        !!x y. [| x:A;  y:B;  c = Scons x y |] ==> P  
wenzelm@20819
   322
     |] ==> P"
wenzelm@20819
   323
by (auto simp add: uprod_def) 
wenzelm@20819
   324
wenzelm@20819
   325
wenzelm@20819
   326
(*Elimination of a pair -- introduces no eigenvariables*)
wenzelm@20819
   327
lemma uprodE2: "[| Scons M N : uprod A B;  [| M:A;  N:B |] ==> P |] ==> P"
wenzelm@20819
   328
by (auto simp add: uprod_def)
wenzelm@20819
   329
wenzelm@20819
   330
wenzelm@20819
   331
(*** Disjoint Sum ***)
wenzelm@20819
   332
wenzelm@20819
   333
lemma usum_In0I [intro]: "M:A ==> In0(M) : usum A B"
wenzelm@20819
   334
by (simp add: usum_def)
wenzelm@20819
   335
wenzelm@20819
   336
lemma usum_In1I [intro]: "N:B ==> In1(N) : usum A B"
wenzelm@20819
   337
by (simp add: usum_def)
wenzelm@20819
   338
wenzelm@20819
   339
lemma usumE [elim!]: 
wenzelm@20819
   340
    "[| u : usum A B;   
wenzelm@20819
   341
        !!x. [| x:A;  u=In0(x) |] ==> P;  
wenzelm@20819
   342
        !!y. [| y:B;  u=In1(y) |] ==> P  
wenzelm@20819
   343
     |] ==> P"
wenzelm@20819
   344
by (auto simp add: usum_def)
wenzelm@20819
   345
wenzelm@20819
   346
wenzelm@20819
   347
(** Injection **)
wenzelm@20819
   348
wenzelm@20819
   349
lemma In0_not_In1 [iff]: "In0(M) \<noteq> In1(N)"
huffman@35216
   350
unfolding In0_def In1_def One_nat_def by auto
wenzelm@20819
   351
wenzelm@45607
   352
lemmas In1_not_In0 [iff] = In0_not_In1 [THEN not_sym]
wenzelm@20819
   353
wenzelm@20819
   354
lemma In0_inject: "In0(M) = In0(N) ==>  M=N"
wenzelm@20819
   355
by (simp add: In0_def)
wenzelm@20819
   356
wenzelm@20819
   357
lemma In1_inject: "In1(M) = In1(N) ==>  M=N"
wenzelm@20819
   358
by (simp add: In1_def)
wenzelm@20819
   359
wenzelm@20819
   360
lemma In0_eq [iff]: "(In0 M = In0 N) = (M=N)"
wenzelm@20819
   361
by (blast dest!: In0_inject)
wenzelm@20819
   362
wenzelm@20819
   363
lemma In1_eq [iff]: "(In1 M = In1 N) = (M=N)"
wenzelm@20819
   364
by (blast dest!: In1_inject)
wenzelm@20819
   365
wenzelm@20819
   366
lemma inj_In0: "inj In0"
wenzelm@20819
   367
by (blast intro!: inj_onI)
wenzelm@20819
   368
wenzelm@20819
   369
lemma inj_In1: "inj In1"
wenzelm@20819
   370
by (blast intro!: inj_onI)
wenzelm@20819
   371
wenzelm@20819
   372
wenzelm@20819
   373
(*** Function spaces ***)
wenzelm@20819
   374
wenzelm@20819
   375
lemma Lim_inject: "Lim f = Lim g ==> f = g"
wenzelm@20819
   376
apply (simp add: Lim_def)
wenzelm@20819
   377
apply (rule ext)
wenzelm@20819
   378
apply (blast elim!: Push_Node_inject)
wenzelm@20819
   379
done
wenzelm@20819
   380
wenzelm@20819
   381
wenzelm@20819
   382
(*** proving equality of sets and functions using ntrunc ***)
wenzelm@20819
   383
wenzelm@20819
   384
lemma ntrunc_subsetI: "ntrunc k M <= M"
wenzelm@20819
   385
by (auto simp add: ntrunc_def)
wenzelm@20819
   386
wenzelm@20819
   387
lemma ntrunc_subsetD: "(!!k. ntrunc k M <= N) ==> M<=N"
wenzelm@20819
   388
by (auto simp add: ntrunc_def)
wenzelm@20819
   389
wenzelm@20819
   390
(*A generalized form of the take-lemma*)
wenzelm@20819
   391
lemma ntrunc_equality: "(!!k. ntrunc k M = ntrunc k N) ==> M=N"
wenzelm@20819
   392
apply (rule equalityI)
wenzelm@20819
   393
apply (rule_tac [!] ntrunc_subsetD)
wenzelm@20819
   394
apply (rule_tac [!] ntrunc_subsetI [THEN [2] subset_trans], auto) 
wenzelm@20819
   395
done
wenzelm@20819
   396
wenzelm@20819
   397
lemma ntrunc_o_equality: 
wenzelm@20819
   398
    "[| !!k. (ntrunc(k) o h1) = (ntrunc(k) o h2) |] ==> h1=h2"
wenzelm@20819
   399
apply (rule ntrunc_equality [THEN ext])
nipkow@39302
   400
apply (simp add: fun_eq_iff) 
wenzelm@20819
   401
done
wenzelm@20819
   402
wenzelm@20819
   403
wenzelm@20819
   404
(*** Monotonicity ***)
wenzelm@20819
   405
wenzelm@20819
   406
lemma uprod_mono: "[| A<=A';  B<=B' |] ==> uprod A B <= uprod A' B'"
wenzelm@20819
   407
by (simp add: uprod_def, blast)
wenzelm@20819
   408
wenzelm@20819
   409
lemma usum_mono: "[| A<=A';  B<=B' |] ==> usum A B <= usum A' B'"
wenzelm@20819
   410
by (simp add: usum_def, blast)
wenzelm@20819
   411
wenzelm@20819
   412
lemma Scons_mono: "[| M<=M';  N<=N' |] ==> Scons M N <= Scons M' N'"
wenzelm@20819
   413
by (simp add: Scons_def, blast)
wenzelm@20819
   414
wenzelm@20819
   415
lemma In0_mono: "M<=N ==> In0(M) <= In0(N)"
huffman@35216
   416
by (simp add: In0_def Scons_mono)
wenzelm@20819
   417
wenzelm@20819
   418
lemma In1_mono: "M<=N ==> In1(M) <= In1(N)"
huffman@35216
   419
by (simp add: In1_def Scons_mono)
wenzelm@20819
   420
wenzelm@20819
   421
wenzelm@20819
   422
(*** Split and Case ***)
wenzelm@20819
   423
wenzelm@20819
   424
lemma Split [simp]: "Split c (Scons M N) = c M N"
wenzelm@20819
   425
by (simp add: Split_def)
wenzelm@20819
   426
wenzelm@20819
   427
lemma Case_In0 [simp]: "Case c d (In0 M) = c(M)"
wenzelm@20819
   428
by (simp add: Case_def)
wenzelm@20819
   429
wenzelm@20819
   430
lemma Case_In1 [simp]: "Case c d (In1 N) = d(N)"
wenzelm@20819
   431
by (simp add: Case_def)
wenzelm@20819
   432
wenzelm@20819
   433
wenzelm@20819
   434
wenzelm@20819
   435
(**** UN x. B(x) rules ****)
wenzelm@20819
   436
wenzelm@20819
   437
lemma ntrunc_UN1: "ntrunc k (UN x. f(x)) = (UN x. ntrunc k (f x))"
wenzelm@20819
   438
by (simp add: ntrunc_def, blast)
wenzelm@20819
   439
wenzelm@20819
   440
lemma Scons_UN1_x: "Scons (UN x. f x) M = (UN x. Scons (f x) M)"
wenzelm@20819
   441
by (simp add: Scons_def, blast)
wenzelm@20819
   442
wenzelm@20819
   443
lemma Scons_UN1_y: "Scons M (UN x. f x) = (UN x. Scons M (f x))"
wenzelm@20819
   444
by (simp add: Scons_def, blast)
wenzelm@20819
   445
wenzelm@20819
   446
lemma In0_UN1: "In0(UN x. f(x)) = (UN x. In0(f(x)))"
wenzelm@20819
   447
by (simp add: In0_def Scons_UN1_y)
wenzelm@20819
   448
wenzelm@20819
   449
lemma In1_UN1: "In1(UN x. f(x)) = (UN x. In1(f(x)))"
wenzelm@20819
   450
by (simp add: In1_def Scons_UN1_y)
wenzelm@20819
   451
wenzelm@20819
   452
wenzelm@20819
   453
(*** Equality for Cartesian Product ***)
wenzelm@20819
   454
wenzelm@20819
   455
lemma dprodI [intro!]: 
wenzelm@20819
   456
    "[| (M,M'):r;  (N,N'):s |] ==> (Scons M N, Scons M' N') : dprod r s"
wenzelm@20819
   457
by (auto simp add: dprod_def)
wenzelm@20819
   458
wenzelm@20819
   459
(*The general elimination rule*)
wenzelm@20819
   460
lemma dprodE [elim!]: 
wenzelm@20819
   461
    "[| c : dprod r s;   
wenzelm@20819
   462
        !!x y x' y'. [| (x,x') : r;  (y,y') : s;  
wenzelm@20819
   463
                        c = (Scons x y, Scons x' y') |] ==> P  
wenzelm@20819
   464
     |] ==> P"
wenzelm@20819
   465
by (auto simp add: dprod_def)
wenzelm@20819
   466
wenzelm@20819
   467
wenzelm@20819
   468
(*** Equality for Disjoint Sum ***)
wenzelm@20819
   469
wenzelm@20819
   470
lemma dsum_In0I [intro]: "(M,M'):r ==> (In0(M), In0(M')) : dsum r s"
wenzelm@20819
   471
by (auto simp add: dsum_def)
wenzelm@20819
   472
wenzelm@20819
   473
lemma dsum_In1I [intro]: "(N,N'):s ==> (In1(N), In1(N')) : dsum r s"
wenzelm@20819
   474
by (auto simp add: dsum_def)
wenzelm@20819
   475
wenzelm@20819
   476
lemma dsumE [elim!]: 
wenzelm@20819
   477
    "[| w : dsum r s;   
wenzelm@20819
   478
        !!x x'. [| (x,x') : r;  w = (In0(x), In0(x')) |] ==> P;  
wenzelm@20819
   479
        !!y y'. [| (y,y') : s;  w = (In1(y), In1(y')) |] ==> P  
wenzelm@20819
   480
     |] ==> P"
wenzelm@20819
   481
by (auto simp add: dsum_def)
wenzelm@20819
   482
wenzelm@20819
   483
wenzelm@20819
   484
(*** Monotonicity ***)
wenzelm@20819
   485
wenzelm@20819
   486
lemma dprod_mono: "[| r<=r';  s<=s' |] ==> dprod r s <= dprod r' s'"
wenzelm@20819
   487
by blast
wenzelm@20819
   488
wenzelm@20819
   489
lemma dsum_mono: "[| r<=r';  s<=s' |] ==> dsum r s <= dsum r' s'"
wenzelm@20819
   490
by blast
wenzelm@20819
   491
wenzelm@20819
   492
wenzelm@20819
   493
(*** Bounding theorems ***)
wenzelm@20819
   494
wenzelm@20819
   495
lemma dprod_Sigma: "(dprod (A <*> B) (C <*> D)) <= (uprod A C) <*> (uprod B D)"
wenzelm@20819
   496
by blast
wenzelm@20819
   497
wenzelm@45607
   498
lemmas dprod_subset_Sigma = subset_trans [OF dprod_mono dprod_Sigma]
wenzelm@20819
   499
wenzelm@20819
   500
(*Dependent version*)
wenzelm@20819
   501
lemma dprod_subset_Sigma2:
blanchet@58112
   502
    "(dprod (Sigma A B) (Sigma C D)) <= Sigma (uprod A C) (Split (%x y. uprod (B x) (D y)))"
wenzelm@20819
   503
by auto
wenzelm@20819
   504
wenzelm@20819
   505
lemma dsum_Sigma: "(dsum (A <*> B) (C <*> D)) <= (usum A C) <*> (usum B D)"
wenzelm@20819
   506
by blast
wenzelm@20819
   507
wenzelm@45607
   508
lemmas dsum_subset_Sigma = subset_trans [OF dsum_mono dsum_Sigma]
wenzelm@20819
   509
wenzelm@20819
   510
haftmann@24162
   511
text {* hides popular names *}
wenzelm@36176
   512
hide_type (open) node item
wenzelm@36176
   513
hide_const (open) Push Node Atom Leaf Numb Lim Split Case
wenzelm@20819
   514
blanchet@58112
   515
ML_file "Tools/Old_Datatype/old_datatype.ML"
wenzelm@12918
   516
wenzelm@48891
   517
ML_file "Tools/inductive_realizer.ML"
haftmann@33959
   518
setup InductiveRealizer.setup
berghofe@13635
   519
blanchet@58112
   520
ML_file "Tools/Old_Datatype/old_datatype_realizer.ML"
blanchet@58112
   521
setup Old_Datatype_Realizer.setup
berghofe@13635
   522
berghofe@5181
   523
end