src/HOL/Univ.thy
 author clasohm Fri Mar 03 12:02:25 1995 +0100 (1995-03-03) changeset 923 ff1574a81019 child 972 e61b058d58d2 permissions -rw-r--r--
new version of HOL with curried function application
```     1 (*  Title:      HOL/Univ.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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```     4     Copyright   1993  University of Cambridge
```
```     5
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```     6 Move LEAST to Nat.thy???  Could it be defined for all types 'a::ord?
```
```     7
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```     8 Declares the type 'a node, a subtype of (nat=>nat) * ('a+nat)
```
```     9
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```    10 Defines "Cartesian Product" and "Disjoint Sum" as set operations.
```
```    11 Could <*> be generalized to a general summation (Sigma)?
```
```    12 *)
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```    13
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```    14 Univ = Arith + Sum +
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```    15
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```    16 (** lists, trees will be sets of nodes **)
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```    17
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```    18 subtype (Node)
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```    19   'a node = "{p. EX f x k. p = <f::nat=>nat, x::'a+nat> & f(k)=0}"
```
```    20
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```    21 types
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```    22   'a item = "'a node set"
```
```    23
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```    24 consts
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```    25   Least     :: "(nat=>bool) => nat"    (binder "LEAST " 10)
```
```    26
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```    27   apfst     :: "['a=>'c, 'a*'b] => 'c*'b"
```
```    28   Push      :: "[nat, nat=>nat] => (nat=>nat)"
```
```    29
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```    30   Push_Node :: "[nat, 'a node] => 'a node"
```
```    31   ndepth    :: "'a node => nat"
```
```    32
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```    33   Atom      :: "('a+nat) => 'a item"
```
```    34   Leaf      :: "'a => 'a item"
```
```    35   Numb      :: "nat => 'a item"
```
```    36   "\$"       :: "['a item, 'a item]=> 'a item"   (infixr 60)
```
```    37   In0,In1   :: "'a item => 'a item"
```
```    38
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```    39   ntrunc    :: "[nat, 'a item] => 'a item"
```
```    40
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```    41   "<*>"  :: "['a item set, 'a item set]=> 'a item set" (infixr 80)
```
```    42   "<+>"  :: "['a item set, 'a item set]=> 'a item set" (infixr 70)
```
```    43
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```    44   Split  :: "[['a item, 'a item]=>'b, 'a item] => 'b"
```
```    45   Case   :: "[['a item]=>'b, ['a item]=>'b, 'a item] => 'b"
```
```    46
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```    47   diag   :: "'a set => ('a * 'a)set"
```
```    48   "<**>" :: "[('a item * 'a item)set, ('a item * 'a item)set] \
```
```    49 \           => ('a item * 'a item)set" (infixr 80)
```
```    50   "<++>" :: "[('a item * 'a item)set, ('a item * 'a item)set] \
```
```    51 \           => ('a item * 'a item)set" (infixr 70)
```
```    52
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```    53 defs
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```    54
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```    55   (*least number operator*)
```
```    56   Least_def      "Least(P) == @k. P(k) & (ALL j. j<k --> ~P(j))"
```
```    57
```
```    58   Push_Node_def  "Push_Node == (%n x. Abs_Node (apfst (Push n) (Rep_Node x)))"
```
```    59
```
```    60   (*crude "lists" of nats -- needed for the constructions*)
```
```    61   apfst_def  "apfst == (%f. split(%x y. <f(x),y>))"
```
```    62   Push_def   "Push == (%b h. nat_case (Suc b) h)"
```
```    63
```
```    64   (** operations on S-expressions -- sets of nodes **)
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```    65
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```    66   (*S-expression constructors*)
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```    67   Atom_def   "Atom == (%x. {Abs_Node(<%k.0, x>)})"
```
```    68   Scons_def  "M\$N == (Push_Node(0) `` M) Un (Push_Node(Suc(0)) `` N)"
```
```    69
```
```    70   (*Leaf nodes, with arbitrary or nat labels*)
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```    71   Leaf_def   "Leaf == Atom o Inl"
```
```    72   Numb_def   "Numb == Atom o Inr"
```
```    73
```
```    74   (*Injections of the "disjoint sum"*)
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```    75   In0_def    "In0(M) == Numb(0) \$ M"
```
```    76   In1_def    "In1(M) == Numb(Suc(0)) \$ M"
```
```    77
```
```    78   (*the set of nodes with depth less than k*)
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```    79   ndepth_def "ndepth(n) == split (%f x. LEAST k. f(k)=0) (Rep_Node n)"
```
```    80   ntrunc_def "ntrunc k N == {n. n:N & ndepth(n)<k}"
```
```    81
```
```    82   (*products and sums for the "universe"*)
```
```    83   uprod_def  "A<*>B == UN x:A. UN y:B. { (x\$y) }"
```
```    84   usum_def   "A<+>B == In0``A Un In1``B"
```
```    85
```
```    86   (*the corresponding eliminators*)
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```    87   Split_def  "Split c M == @u. ? x y. M = x\$y & u = c x y"
```
```    88
```
```    89   Case_def   "Case c d M == @u.  (? x . M = In0(x) & u = c(x)) \
```
```    90 \                              | (? y . M = In1(y) & u = d(y))"
```
```    91
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```    92
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```    93   (** diagonal sets and equality for the "universe" **)
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```    94
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```    95   diag_def   "diag(A) == UN x:A. {<x,x>}"
```
```    96
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```    97   dprod_def  "r<**>s == UN u:r. split (%x x'. \
```
```    98 \                       UN v:s. split (%y y'. {<x\$y,x'\$y'>}) v) u"
```
```    99
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```   100   dsum_def   "r<++>s == (UN u:r. split (%x x'. {<In0(x),In0(x')>}) u) Un \
```
```   101 \                       (UN v:s. split (%y y'. {<In1(y),In1(y')>}) v)"
```
```   102
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```   103 end
```