src/HOL/Datatype_Universe.thy
 author obua Mon Apr 10 16:00:34 2006 +0200 (2006-04-10) changeset 19404 9bf2cdc9e8e8 parent 17589 58eeffd73be1 child 20799 46694b230cfb permissions -rw-r--r--
Moved stuff from Ring_and_Field to Matrix
```     1 (*  Title:      HOL/Datatype_Universe.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     4     Copyright   1993  University of Cambridge
```
```     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_Universe
```
```    12 imports NatArith Sum_Type
```
```    13 begin
```
```    14
```
```    15
```
```    16 typedef (Node)
```
```    17   ('a,'b) node = "{p. EX f x k. p = (f::nat=>'b+nat, x::'a+nat) & f k = Inr 0}"
```
```    18     --{*it is a subtype of @{text "(nat=>'b+nat) * ('a+nat)"}*}
```
```    19   by auto
```
```    20
```
```    21 text{*Datatypes will be represented by sets of type @{text node}*}
```
```    22
```
```    23 types 'a item        = "('a, unit) node set"
```
```    24       ('a, 'b) dtree = "('a, 'b) node set"
```
```    25
```
```    26 consts
```
```    27   apfst     :: "['a=>'c, 'a*'b] => 'c*'b"
```
```    28   Push      :: "[('b + nat), nat => ('b + nat)] => (nat => ('b + nat))"
```
```    29
```
```    30   Push_Node :: "[('b + nat), ('a, 'b) node] => ('a, 'b) node"
```
```    31   ndepth    :: "('a, 'b) node => nat"
```
```    32
```
```    33   Atom      :: "('a + nat) => ('a, 'b) dtree"
```
```    34   Leaf      :: "'a => ('a, 'b) dtree"
```
```    35   Numb      :: "nat => ('a, 'b) dtree"
```
```    36   Scons     :: "[('a, 'b) dtree, ('a, 'b) dtree] => ('a, 'b) dtree"
```
```    37   In0       :: "('a, 'b) dtree => ('a, 'b) dtree"
```
```    38   In1       :: "('a, 'b) dtree => ('a, 'b) dtree"
```
```    39   Lim       :: "('b => ('a, 'b) dtree) => ('a, 'b) dtree"
```
```    40
```
```    41   ntrunc    :: "[nat, ('a, 'b) dtree] => ('a, 'b) dtree"
```
```    42
```
```    43   uprod     :: "[('a, 'b) dtree set, ('a, 'b) dtree set]=> ('a, 'b) dtree set"
```
```    44   usum      :: "[('a, 'b) dtree set, ('a, 'b) dtree set]=> ('a, 'b) dtree set"
```
```    45
```
```    46   Split     :: "[[('a, 'b) dtree, ('a, 'b) dtree]=>'c, ('a, 'b) dtree] => 'c"
```
```    47   Case      :: "[[('a, 'b) dtree]=>'c, [('a, 'b) dtree]=>'c, ('a, 'b) dtree] => 'c"
```
```    48
```
```    49   dprod     :: "[(('a, 'b) dtree * ('a, 'b) dtree)set, (('a, 'b) dtree * ('a, 'b) dtree)set]
```
```    50                 => (('a, 'b) dtree * ('a, 'b) dtree)set"
```
```    51   dsum      :: "[(('a, 'b) dtree * ('a, 'b) dtree)set, (('a, 'b) dtree * ('a, 'b) dtree)set]
```
```    52                 => (('a, 'b) dtree * ('a, 'b) dtree)set"
```
```    53
```
```    54
```
```    55 defs
```
```    56
```
```    57   Push_Node_def:  "Push_Node == (%n x. Abs_Node (apfst (Push n) (Rep_Node x)))"
```
```    58
```
```    59   (*crude "lists" of nats -- needed for the constructions*)
```
```    60   apfst_def:  "apfst == (%f (x,y). (f(x),y))"
```
```    61   Push_def:   "Push == (%b h. nat_case b h)"
```
```    62
```
```    63   (** operations on S-expressions -- sets of nodes **)
```
```    64
```
```    65   (*S-expression constructors*)
```
```    66   Atom_def:   "Atom == (%x. {Abs_Node((%k. Inr 0, x))})"
```
```    67   Scons_def:  "Scons M N == (Push_Node (Inr 1) ` M) Un (Push_Node (Inr (Suc 1)) ` N)"
```
```    68
```
```    69   (*Leaf nodes, with arbitrary or nat labels*)
```
```    70   Leaf_def:   "Leaf == Atom o Inl"
```
```    71   Numb_def:   "Numb == Atom o Inr"
```
```    72
```
```    73   (*Injections of the "disjoint sum"*)
```
```    74   In0_def:    "In0(M) == Scons (Numb 0) M"
```
```    75   In1_def:    "In1(M) == Scons (Numb 1) M"
```
```    76
```
```    77   (*Function spaces*)
```
```    78   Lim_def: "Lim f == Union {z. ? x. z = Push_Node (Inl x) ` (f x)}"
```
```    79
```
```    80   (*the set of nodes with depth less than k*)
```
```    81   ndepth_def: "ndepth(n) == (%(f,x). LEAST k. f k = Inr 0) (Rep_Node n)"
```
```    82   ntrunc_def: "ntrunc k N == {n. n:N & ndepth(n)<k}"
```
```    83
```
```    84   (*products and sums for the "universe"*)
```
```    85   uprod_def:  "uprod A B == UN x:A. UN y:B. { Scons x y }"
```
```    86   usum_def:   "usum A B == In0`A Un In1`B"
```
```    87
```
```    88   (*the corresponding eliminators*)
```
```    89   Split_def:  "Split c M == THE u. EX x y. M = Scons x y & u = c x y"
```
```    90
```
```    91   Case_def:   "Case c d M == THE u.  (EX x . M = In0(x) & u = c(x))
```
```    92                                   | (EX y . M = In1(y) & u = d(y))"
```
```    93
```
```    94
```
```    95   (** equality for the "universe" **)
```
```    96
```
```    97   dprod_def:  "dprod r s == UN (x,x'):r. UN (y,y'):s. {(Scons x y, Scons x' y')}"
```
```    98
```
```    99   dsum_def:   "dsum r s == (UN (x,x'):r. {(In0(x),In0(x'))}) Un
```
```   100                           (UN (y,y'):s. {(In1(y),In1(y'))})"
```
```   101
```
```   102
```
```   103
```
```   104 (** apfst -- can be used in similar type definitions **)
```
```   105
```
```   106 lemma apfst_conv [simp]: "apfst f (a,b) = (f(a),b)"
```
```   107 by (simp add: apfst_def)
```
```   108
```
```   109
```
```   110 lemma apfst_convE:
```
```   111     "[| q = apfst f p;  !!x y. [| p = (x,y);  q = (f(x),y) |] ==> R
```
```   112      |] ==> R"
```
```   113 by (force simp add: apfst_def)
```
```   114
```
```   115 (** Push -- an injection, analogous to Cons on lists **)
```
```   116
```
```   117 lemma Push_inject1: "Push i f = Push j g  ==> i=j"
```
```   118 apply (simp add: Push_def expand_fun_eq)
```
```   119 apply (drule_tac x=0 in spec, simp)
```
```   120 done
```
```   121
```
```   122 lemma Push_inject2: "Push i f = Push j g  ==> f=g"
```
```   123 apply (auto simp add: Push_def expand_fun_eq)
```
```   124 apply (drule_tac x="Suc x" in spec, simp)
```
```   125 done
```
```   126
```
```   127 lemma Push_inject:
```
```   128     "[| Push i f =Push j g;  [| i=j;  f=g |] ==> P |] ==> P"
```
```   129 by (blast dest: Push_inject1 Push_inject2)
```
```   130
```
```   131 lemma Push_neq_K0: "Push (Inr (Suc k)) f = (%z. Inr 0) ==> P"
```
```   132 by (auto simp add: Push_def expand_fun_eq split: nat.split_asm)
```
```   133
```
```   134 lemmas Abs_Node_inj = Abs_Node_inject [THEN [2] rev_iffD1, standard]
```
```   135
```
```   136
```
```   137 (*** Introduction rules for Node ***)
```
```   138
```
```   139 lemma Node_K0_I: "(%k. Inr 0, a) : Node"
```
```   140 by (simp add: Node_def)
```
```   141
```
```   142 lemma Node_Push_I: "p: Node ==> apfst (Push i) p : Node"
```
```   143 apply (simp add: Node_def Push_def)
```
```   144 apply (fast intro!: apfst_conv nat_case_Suc [THEN trans])
```
```   145 done
```
```   146
```
```   147
```
```   148 subsection{*Freeness: Distinctness of Constructors*}
```
```   149
```
```   150 (** Scons vs Atom **)
```
```   151
```
```   152 lemma Scons_not_Atom [iff]: "Scons M N \<noteq> Atom(a)"
```
```   153 apply (simp add: Atom_def Scons_def Push_Node_def One_nat_def)
```
```   154 apply (blast intro: Node_K0_I Rep_Node [THEN Node_Push_I]
```
```   155          dest!: Abs_Node_inj
```
```   156          elim!: apfst_convE sym [THEN Push_neq_K0])
```
```   157 done
```
```   158
```
```   159 lemmas Atom_not_Scons = Scons_not_Atom [THEN not_sym, standard]
```
```   160 declare Atom_not_Scons [iff]
```
```   161
```
```   162 (*** Injectiveness ***)
```
```   163
```
```   164 (** Atomic nodes **)
```
```   165
```
```   166 lemma inj_Atom: "inj(Atom)"
```
```   167 apply (simp add: Atom_def)
```
```   168 apply (blast intro!: inj_onI Node_K0_I dest!: Abs_Node_inj)
```
```   169 done
```
```   170 lemmas Atom_inject = inj_Atom [THEN injD, standard]
```
```   171
```
```   172 lemma Atom_Atom_eq [iff]: "(Atom(a)=Atom(b)) = (a=b)"
```
```   173 by (blast dest!: Atom_inject)
```
```   174
```
```   175 lemma inj_Leaf: "inj(Leaf)"
```
```   176 apply (simp add: Leaf_def o_def)
```
```   177 apply (rule inj_onI)
```
```   178 apply (erule Atom_inject [THEN Inl_inject])
```
```   179 done
```
```   180
```
```   181 lemmas Leaf_inject = inj_Leaf [THEN injD, standard]
```
```   182 declare Leaf_inject [dest!]
```
```   183
```
```   184 lemma inj_Numb: "inj(Numb)"
```
```   185 apply (simp add: Numb_def o_def)
```
```   186 apply (rule inj_onI)
```
```   187 apply (erule Atom_inject [THEN Inr_inject])
```
```   188 done
```
```   189
```
```   190 lemmas Numb_inject = inj_Numb [THEN injD, standard]
```
```   191 declare Numb_inject [dest!]
```
```   192
```
```   193
```
```   194 (** Injectiveness of Push_Node **)
```
```   195
```
```   196 lemma Push_Node_inject:
```
```   197     "[| Push_Node i m =Push_Node j n;  [| i=j;  m=n |] ==> P
```
```   198      |] ==> P"
```
```   199 apply (simp add: Push_Node_def)
```
```   200 apply (erule Abs_Node_inj [THEN apfst_convE])
```
```   201 apply (rule Rep_Node [THEN Node_Push_I])+
```
```   202 apply (erule sym [THEN apfst_convE])
```
```   203 apply (blast intro: Rep_Node_inject [THEN iffD1] trans sym elim!: Push_inject)
```
```   204 done
```
```   205
```
```   206
```
```   207 (** Injectiveness of Scons **)
```
```   208
```
```   209 lemma Scons_inject_lemma1: "Scons M N <= Scons M' N' ==> M<=M'"
```
```   210 apply (simp add: Scons_def One_nat_def)
```
```   211 apply (blast dest!: Push_Node_inject)
```
```   212 done
```
```   213
```
```   214 lemma Scons_inject_lemma2: "Scons M N <= Scons M' N' ==> N<=N'"
```
```   215 apply (simp add: Scons_def One_nat_def)
```
```   216 apply (blast dest!: Push_Node_inject)
```
```   217 done
```
```   218
```
```   219 lemma Scons_inject1: "Scons M N = Scons M' N' ==> M=M'"
```
```   220 apply (erule equalityE)
```
```   221 apply (iprover intro: equalityI Scons_inject_lemma1)
```
```   222 done
```
```   223
```
```   224 lemma Scons_inject2: "Scons M N = Scons M' N' ==> N=N'"
```
```   225 apply (erule equalityE)
```
```   226 apply (iprover intro: equalityI Scons_inject_lemma2)
```
```   227 done
```
```   228
```
```   229 lemma Scons_inject:
```
```   230     "[| Scons M N = Scons M' N';  [| M=M';  N=N' |] ==> P |] ==> P"
```
```   231 by (iprover dest: Scons_inject1 Scons_inject2)
```
```   232
```
```   233 lemma Scons_Scons_eq [iff]: "(Scons M N = Scons M' N') = (M=M' & N=N')"
```
```   234 by (blast elim!: Scons_inject)
```
```   235
```
```   236 (*** Distinctness involving Leaf and Numb ***)
```
```   237
```
```   238 (** Scons vs Leaf **)
```
```   239
```
```   240 lemma Scons_not_Leaf [iff]: "Scons M N \<noteq> Leaf(a)"
```
```   241 by (simp add: Leaf_def o_def Scons_not_Atom)
```
```   242
```
```   243 lemmas Leaf_not_Scons = Scons_not_Leaf [THEN not_sym, standard]
```
```   244 declare Leaf_not_Scons [iff]
```
```   245
```
```   246 (** Scons vs Numb **)
```
```   247
```
```   248 lemma Scons_not_Numb [iff]: "Scons M N \<noteq> Numb(k)"
```
```   249 by (simp add: Numb_def o_def Scons_not_Atom)
```
```   250
```
```   251 lemmas Numb_not_Scons = Scons_not_Numb [THEN not_sym, standard]
```
```   252 declare Numb_not_Scons [iff]
```
```   253
```
```   254
```
```   255 (** Leaf vs Numb **)
```
```   256
```
```   257 lemma Leaf_not_Numb [iff]: "Leaf(a) \<noteq> Numb(k)"
```
```   258 by (simp add: Leaf_def Numb_def)
```
```   259
```
```   260 lemmas Numb_not_Leaf = Leaf_not_Numb [THEN not_sym, standard]
```
```   261 declare Numb_not_Leaf [iff]
```
```   262
```
```   263
```
```   264 (*** ndepth -- the depth of a node ***)
```
```   265
```
```   266 lemma ndepth_K0: "ndepth (Abs_Node(%k. Inr 0, x)) = 0"
```
```   267 by (simp add: ndepth_def  Node_K0_I [THEN Abs_Node_inverse] Least_equality)
```
```   268
```
```   269 lemma ndepth_Push_Node_aux:
```
```   270      "nat_case (Inr (Suc i)) f k = Inr 0 --> Suc(LEAST x. f x = Inr 0) <= k"
```
```   271 apply (induct_tac "k", auto)
```
```   272 apply (erule Least_le)
```
```   273 done
```
```   274
```
```   275 lemma ndepth_Push_Node:
```
```   276     "ndepth (Push_Node (Inr (Suc i)) n) = Suc(ndepth(n))"
```
```   277 apply (insert Rep_Node [of n, unfolded Node_def])
```
```   278 apply (auto simp add: ndepth_def Push_Node_def
```
```   279                  Rep_Node [THEN Node_Push_I, THEN Abs_Node_inverse])
```
```   280 apply (rule Least_equality)
```
```   281 apply (auto simp add: Push_def ndepth_Push_Node_aux)
```
```   282 apply (erule LeastI)
```
```   283 done
```
```   284
```
```   285
```
```   286 (*** ntrunc applied to the various node sets ***)
```
```   287
```
```   288 lemma ntrunc_0 [simp]: "ntrunc 0 M = {}"
```
```   289 by (simp add: ntrunc_def)
```
```   290
```
```   291 lemma ntrunc_Atom [simp]: "ntrunc (Suc k) (Atom a) = Atom(a)"
```
```   292 by (auto simp add: Atom_def ntrunc_def ndepth_K0)
```
```   293
```
```   294 lemma ntrunc_Leaf [simp]: "ntrunc (Suc k) (Leaf a) = Leaf(a)"
```
```   295 by (simp add: Leaf_def o_def ntrunc_Atom)
```
```   296
```
```   297 lemma ntrunc_Numb [simp]: "ntrunc (Suc k) (Numb i) = Numb(i)"
```
```   298 by (simp add: Numb_def o_def ntrunc_Atom)
```
```   299
```
```   300 lemma ntrunc_Scons [simp]:
```
```   301     "ntrunc (Suc k) (Scons M N) = Scons (ntrunc k M) (ntrunc k N)"
```
```   302 by (auto simp add: Scons_def ntrunc_def One_nat_def ndepth_Push_Node)
```
```   303
```
```   304
```
```   305
```
```   306 (** Injection nodes **)
```
```   307
```
```   308 lemma ntrunc_one_In0 [simp]: "ntrunc (Suc 0) (In0 M) = {}"
```
```   309 apply (simp add: In0_def)
```
```   310 apply (simp add: Scons_def)
```
```   311 done
```
```   312
```
```   313 lemma ntrunc_In0 [simp]: "ntrunc (Suc(Suc k)) (In0 M) = In0 (ntrunc (Suc k) M)"
```
```   314 by (simp add: In0_def)
```
```   315
```
```   316 lemma ntrunc_one_In1 [simp]: "ntrunc (Suc 0) (In1 M) = {}"
```
```   317 apply (simp add: In1_def)
```
```   318 apply (simp add: Scons_def)
```
```   319 done
```
```   320
```
```   321 lemma ntrunc_In1 [simp]: "ntrunc (Suc(Suc k)) (In1 M) = In1 (ntrunc (Suc k) M)"
```
```   322 by (simp add: In1_def)
```
```   323
```
```   324
```
```   325 subsection{*Set Constructions*}
```
```   326
```
```   327
```
```   328 (*** Cartesian Product ***)
```
```   329
```
```   330 lemma uprodI [intro!]: "[| M:A;  N:B |] ==> Scons M N : uprod A B"
```
```   331 by (simp add: uprod_def)
```
```   332
```
```   333 (*The general elimination rule*)
```
```   334 lemma uprodE [elim!]:
```
```   335     "[| c : uprod A B;
```
```   336         !!x y. [| x:A;  y:B;  c = Scons x y |] ==> P
```
```   337      |] ==> P"
```
```   338 by (auto simp add: uprod_def)
```
```   339
```
```   340
```
```   341 (*Elimination of a pair -- introduces no eigenvariables*)
```
```   342 lemma uprodE2: "[| Scons M N : uprod A B;  [| M:A;  N:B |] ==> P |] ==> P"
```
```   343 by (auto simp add: uprod_def)
```
```   344
```
```   345
```
```   346 (*** Disjoint Sum ***)
```
```   347
```
```   348 lemma usum_In0I [intro]: "M:A ==> In0(M) : usum A B"
```
```   349 by (simp add: usum_def)
```
```   350
```
```   351 lemma usum_In1I [intro]: "N:B ==> In1(N) : usum A B"
```
```   352 by (simp add: usum_def)
```
```   353
```
```   354 lemma usumE [elim!]:
```
```   355     "[| u : usum A B;
```
```   356         !!x. [| x:A;  u=In0(x) |] ==> P;
```
```   357         !!y. [| y:B;  u=In1(y) |] ==> P
```
```   358      |] ==> P"
```
```   359 by (auto simp add: usum_def)
```
```   360
```
```   361
```
```   362 (** Injection **)
```
```   363
```
```   364 lemma In0_not_In1 [iff]: "In0(M) \<noteq> In1(N)"
```
```   365 by (auto simp add: In0_def In1_def One_nat_def)
```
```   366
```
```   367 lemmas In1_not_In0 = In0_not_In1 [THEN not_sym, standard]
```
```   368 declare In1_not_In0 [iff]
```
```   369
```
```   370 lemma In0_inject: "In0(M) = In0(N) ==>  M=N"
```
```   371 by (simp add: In0_def)
```
```   372
```
```   373 lemma In1_inject: "In1(M) = In1(N) ==>  M=N"
```
```   374 by (simp add: In1_def)
```
```   375
```
```   376 lemma In0_eq [iff]: "(In0 M = In0 N) = (M=N)"
```
```   377 by (blast dest!: In0_inject)
```
```   378
```
```   379 lemma In1_eq [iff]: "(In1 M = In1 N) = (M=N)"
```
```   380 by (blast dest!: In1_inject)
```
```   381
```
```   382 lemma inj_In0: "inj In0"
```
```   383 by (blast intro!: inj_onI)
```
```   384
```
```   385 lemma inj_In1: "inj In1"
```
```   386 by (blast intro!: inj_onI)
```
```   387
```
```   388
```
```   389 (*** Function spaces ***)
```
```   390
```
```   391 lemma Lim_inject: "Lim f = Lim g ==> f = g"
```
```   392 apply (simp add: Lim_def)
```
```   393 apply (rule ext)
```
```   394 apply (blast elim!: Push_Node_inject)
```
```   395 done
```
```   396
```
```   397
```
```   398 (*** proving equality of sets and functions using ntrunc ***)
```
```   399
```
```   400 lemma ntrunc_subsetI: "ntrunc k M <= M"
```
```   401 by (auto simp add: ntrunc_def)
```
```   402
```
```   403 lemma ntrunc_subsetD: "(!!k. ntrunc k M <= N) ==> M<=N"
```
```   404 by (auto simp add: ntrunc_def)
```
```   405
```
```   406 (*A generalized form of the take-lemma*)
```
```   407 lemma ntrunc_equality: "(!!k. ntrunc k M = ntrunc k N) ==> M=N"
```
```   408 apply (rule equalityI)
```
```   409 apply (rule_tac [!] ntrunc_subsetD)
```
```   410 apply (rule_tac [!] ntrunc_subsetI [THEN [2] subset_trans], auto)
```
```   411 done
```
```   412
```
```   413 lemma ntrunc_o_equality:
```
```   414     "[| !!k. (ntrunc(k) o h1) = (ntrunc(k) o h2) |] ==> h1=h2"
```
```   415 apply (rule ntrunc_equality [THEN ext])
```
```   416 apply (simp add: expand_fun_eq)
```
```   417 done
```
```   418
```
```   419
```
```   420 (*** Monotonicity ***)
```
```   421
```
```   422 lemma uprod_mono: "[| A<=A';  B<=B' |] ==> uprod A B <= uprod A' B'"
```
```   423 by (simp add: uprod_def, blast)
```
```   424
```
```   425 lemma usum_mono: "[| A<=A';  B<=B' |] ==> usum A B <= usum A' B'"
```
```   426 by (simp add: usum_def, blast)
```
```   427
```
```   428 lemma Scons_mono: "[| M<=M';  N<=N' |] ==> Scons M N <= Scons M' N'"
```
```   429 by (simp add: Scons_def, blast)
```
```   430
```
```   431 lemma In0_mono: "M<=N ==> In0(M) <= In0(N)"
```
```   432 by (simp add: In0_def subset_refl Scons_mono)
```
```   433
```
```   434 lemma In1_mono: "M<=N ==> In1(M) <= In1(N)"
```
```   435 by (simp add: In1_def subset_refl Scons_mono)
```
```   436
```
```   437
```
```   438 (*** Split and Case ***)
```
```   439
```
```   440 lemma Split [simp]: "Split c (Scons M N) = c M N"
```
```   441 by (simp add: Split_def)
```
```   442
```
```   443 lemma Case_In0 [simp]: "Case c d (In0 M) = c(M)"
```
```   444 by (simp add: Case_def)
```
```   445
```
```   446 lemma Case_In1 [simp]: "Case c d (In1 N) = d(N)"
```
```   447 by (simp add: Case_def)
```
```   448
```
```   449
```
```   450
```
```   451 (**** UN x. B(x) rules ****)
```
```   452
```
```   453 lemma ntrunc_UN1: "ntrunc k (UN x. f(x)) = (UN x. ntrunc k (f x))"
```
```   454 by (simp add: ntrunc_def, blast)
```
```   455
```
```   456 lemma Scons_UN1_x: "Scons (UN x. f x) M = (UN x. Scons (f x) M)"
```
```   457 by (simp add: Scons_def, blast)
```
```   458
```
```   459 lemma Scons_UN1_y: "Scons M (UN x. f x) = (UN x. Scons M (f x))"
```
```   460 by (simp add: Scons_def, blast)
```
```   461
```
```   462 lemma In0_UN1: "In0(UN x. f(x)) = (UN x. In0(f(x)))"
```
```   463 by (simp add: In0_def Scons_UN1_y)
```
```   464
```
```   465 lemma In1_UN1: "In1(UN x. f(x)) = (UN x. In1(f(x)))"
```
```   466 by (simp add: In1_def Scons_UN1_y)
```
```   467
```
```   468
```
```   469 (*** Equality for Cartesian Product ***)
```
```   470
```
```   471 lemma dprodI [intro!]:
```
```   472     "[| (M,M'):r;  (N,N'):s |] ==> (Scons M N, Scons M' N') : dprod r s"
```
```   473 by (auto simp add: dprod_def)
```
```   474
```
```   475 (*The general elimination rule*)
```
```   476 lemma dprodE [elim!]:
```
```   477     "[| c : dprod r s;
```
```   478         !!x y x' y'. [| (x,x') : r;  (y,y') : s;
```
```   479                         c = (Scons x y, Scons x' y') |] ==> P
```
```   480      |] ==> P"
```
```   481 by (auto simp add: dprod_def)
```
```   482
```
```   483
```
```   484 (*** Equality for Disjoint Sum ***)
```
```   485
```
```   486 lemma dsum_In0I [intro]: "(M,M'):r ==> (In0(M), In0(M')) : dsum r s"
```
```   487 by (auto simp add: dsum_def)
```
```   488
```
```   489 lemma dsum_In1I [intro]: "(N,N'):s ==> (In1(N), In1(N')) : dsum r s"
```
```   490 by (auto simp add: dsum_def)
```
```   491
```
```   492 lemma dsumE [elim!]:
```
```   493     "[| w : dsum r s;
```
```   494         !!x x'. [| (x,x') : r;  w = (In0(x), In0(x')) |] ==> P;
```
```   495         !!y y'. [| (y,y') : s;  w = (In1(y), In1(y')) |] ==> P
```
```   496      |] ==> P"
```
```   497 by (auto simp add: dsum_def)
```
```   498
```
```   499
```
```   500 (*** Monotonicity ***)
```
```   501
```
```   502 lemma dprod_mono: "[| r<=r';  s<=s' |] ==> dprod r s <= dprod r' s'"
```
```   503 by blast
```
```   504
```
```   505 lemma dsum_mono: "[| r<=r';  s<=s' |] ==> dsum r s <= dsum r' s'"
```
```   506 by blast
```
```   507
```
```   508
```
```   509 (*** Bounding theorems ***)
```
```   510
```
```   511 lemma dprod_Sigma: "(dprod (A <*> B) (C <*> D)) <= (uprod A C) <*> (uprod B D)"
```
```   512 by blast
```
```   513
```
```   514 lemmas dprod_subset_Sigma = subset_trans [OF dprod_mono dprod_Sigma, standard]
```
```   515
```
```   516 (*Dependent version*)
```
```   517 lemma dprod_subset_Sigma2:
```
```   518      "(dprod (Sigma A B) (Sigma C D)) <=
```
```   519       Sigma (uprod A C) (Split (%x y. uprod (B x) (D y)))"
```
```   520 by auto
```
```   521
```
```   522 lemma dsum_Sigma: "(dsum (A <*> B) (C <*> D)) <= (usum A C) <*> (usum B D)"
```
```   523 by blast
```
```   524
```
```   525 lemmas dsum_subset_Sigma = subset_trans [OF dsum_mono dsum_Sigma, standard]
```
```   526
```
```   527
```
```   528 (*** Domain ***)
```
```   529
```
```   530 lemma Domain_dprod [simp]: "Domain (dprod r s) = uprod (Domain r) (Domain s)"
```
```   531 by auto
```
```   532
```
```   533 lemma Domain_dsum [simp]: "Domain (dsum r s) = usum (Domain r) (Domain s)"
```
```   534 by auto
```
```   535
```
```   536 ML
```
```   537 {*
```
```   538 val apfst_conv = thm "apfst_conv";
```
```   539 val apfst_convE = thm "apfst_convE";
```
```   540 val Push_inject1 = thm "Push_inject1";
```
```   541 val Push_inject2 = thm "Push_inject2";
```
```   542 val Push_inject = thm "Push_inject";
```
```   543 val Push_neq_K0 = thm "Push_neq_K0";
```
```   544 val Abs_Node_inj = thm "Abs_Node_inj";
```
```   545 val Node_K0_I = thm "Node_K0_I";
```
```   546 val Node_Push_I = thm "Node_Push_I";
```
```   547 val Scons_not_Atom = thm "Scons_not_Atom";
```
```   548 val Atom_not_Scons = thm "Atom_not_Scons";
```
```   549 val inj_Atom = thm "inj_Atom";
```
```   550 val Atom_inject = thm "Atom_inject";
```
```   551 val Atom_Atom_eq = thm "Atom_Atom_eq";
```
```   552 val inj_Leaf = thm "inj_Leaf";
```
```   553 val Leaf_inject = thm "Leaf_inject";
```
```   554 val inj_Numb = thm "inj_Numb";
```
```   555 val Numb_inject = thm "Numb_inject";
```
```   556 val Push_Node_inject = thm "Push_Node_inject";
```
```   557 val Scons_inject1 = thm "Scons_inject1";
```
```   558 val Scons_inject2 = thm "Scons_inject2";
```
```   559 val Scons_inject = thm "Scons_inject";
```
```   560 val Scons_Scons_eq = thm "Scons_Scons_eq";
```
```   561 val Scons_not_Leaf = thm "Scons_not_Leaf";
```
```   562 val Leaf_not_Scons = thm "Leaf_not_Scons";
```
```   563 val Scons_not_Numb = thm "Scons_not_Numb";
```
```   564 val Numb_not_Scons = thm "Numb_not_Scons";
```
```   565 val Leaf_not_Numb = thm "Leaf_not_Numb";
```
```   566 val Numb_not_Leaf = thm "Numb_not_Leaf";
```
```   567 val ndepth_K0 = thm "ndepth_K0";
```
```   568 val ndepth_Push_Node_aux = thm "ndepth_Push_Node_aux";
```
```   569 val ndepth_Push_Node = thm "ndepth_Push_Node";
```
```   570 val ntrunc_0 = thm "ntrunc_0";
```
```   571 val ntrunc_Atom = thm "ntrunc_Atom";
```
```   572 val ntrunc_Leaf = thm "ntrunc_Leaf";
```
```   573 val ntrunc_Numb = thm "ntrunc_Numb";
```
```   574 val ntrunc_Scons = thm "ntrunc_Scons";
```
```   575 val ntrunc_one_In0 = thm "ntrunc_one_In0";
```
```   576 val ntrunc_In0 = thm "ntrunc_In0";
```
```   577 val ntrunc_one_In1 = thm "ntrunc_one_In1";
```
```   578 val ntrunc_In1 = thm "ntrunc_In1";
```
```   579 val uprodI = thm "uprodI";
```
```   580 val uprodE = thm "uprodE";
```
```   581 val uprodE2 = thm "uprodE2";
```
```   582 val usum_In0I = thm "usum_In0I";
```
```   583 val usum_In1I = thm "usum_In1I";
```
```   584 val usumE = thm "usumE";
```
```   585 val In0_not_In1 = thm "In0_not_In1";
```
```   586 val In1_not_In0 = thm "In1_not_In0";
```
```   587 val In0_inject = thm "In0_inject";
```
```   588 val In1_inject = thm "In1_inject";
```
```   589 val In0_eq = thm "In0_eq";
```
```   590 val In1_eq = thm "In1_eq";
```
```   591 val inj_In0 = thm "inj_In0";
```
```   592 val inj_In1 = thm "inj_In1";
```
```   593 val Lim_inject = thm "Lim_inject";
```
```   594 val ntrunc_subsetI = thm "ntrunc_subsetI";
```
```   595 val ntrunc_subsetD = thm "ntrunc_subsetD";
```
```   596 val ntrunc_equality = thm "ntrunc_equality";
```
```   597 val ntrunc_o_equality = thm "ntrunc_o_equality";
```
```   598 val uprod_mono = thm "uprod_mono";
```
```   599 val usum_mono = thm "usum_mono";
```
```   600 val Scons_mono = thm "Scons_mono";
```
```   601 val In0_mono = thm "In0_mono";
```
```   602 val In1_mono = thm "In1_mono";
```
```   603 val Split = thm "Split";
```
```   604 val Case_In0 = thm "Case_In0";
```
```   605 val Case_In1 = thm "Case_In1";
```
```   606 val ntrunc_UN1 = thm "ntrunc_UN1";
```
```   607 val Scons_UN1_x = thm "Scons_UN1_x";
```
```   608 val Scons_UN1_y = thm "Scons_UN1_y";
```
```   609 val In0_UN1 = thm "In0_UN1";
```
```   610 val In1_UN1 = thm "In1_UN1";
```
```   611 val dprodI = thm "dprodI";
```
```   612 val dprodE = thm "dprodE";
```
```   613 val dsum_In0I = thm "dsum_In0I";
```
```   614 val dsum_In1I = thm "dsum_In1I";
```
```   615 val dsumE = thm "dsumE";
```
```   616 val dprod_mono = thm "dprod_mono";
```
```   617 val dsum_mono = thm "dsum_mono";
```
```   618 val dprod_Sigma = thm "dprod_Sigma";
```
```   619 val dprod_subset_Sigma = thm "dprod_subset_Sigma";
```
```   620 val dprod_subset_Sigma2 = thm "dprod_subset_Sigma2";
```
```   621 val dsum_Sigma = thm "dsum_Sigma";
```
```   622 val dsum_subset_Sigma = thm "dsum_subset_Sigma";
```
```   623 val Domain_dprod = thm "Domain_dprod";
```
```   624 val Domain_dsum = thm "Domain_dsum";
```
```   625 *}
```
```   626
```
```   627 end
```