src/HOL/Univ.thy
 changeset 923 ff1574a81019 child 972 e61b058d58d2
```     1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/src/HOL/Univ.thy	Fri Mar 03 12:02:25 1995 +0100
1.3 @@ -0,0 +1,103 @@
1.4 +(*  Title:      HOL/Univ.thy
1.5 +    ID:         \$Id\$
1.6 +    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
1.7 +    Copyright   1993  University of Cambridge
1.8 +
1.9 +Move LEAST to Nat.thy???  Could it be defined for all types 'a::ord?
1.10 +
1.11 +Declares the type 'a node, a subtype of (nat=>nat) * ('a+nat)
1.12 +
1.13 +Defines "Cartesian Product" and "Disjoint Sum" as set operations.
1.14 +Could <*> be generalized to a general summation (Sigma)?
1.15 +*)
1.16 +
1.17 +Univ = Arith + Sum +
1.18 +
1.19 +(** lists, trees will be sets of nodes **)
1.20 +
1.21 +subtype (Node)
1.22 +  'a node = "{p. EX f x k. p = <f::nat=>nat, x::'a+nat> & f(k)=0}"
1.23 +
1.24 +types
1.25 +  'a item = "'a node set"
1.26 +
1.27 +consts
1.28 +  Least     :: "(nat=>bool) => nat"    (binder "LEAST " 10)
1.29 +
1.30 +  apfst     :: "['a=>'c, 'a*'b] => 'c*'b"
1.31 +  Push      :: "[nat, nat=>nat] => (nat=>nat)"
1.32 +
1.33 +  Push_Node :: "[nat, 'a node] => 'a node"
1.34 +  ndepth    :: "'a node => nat"
1.35 +
1.36 +  Atom      :: "('a+nat) => 'a item"
1.37 +  Leaf      :: "'a => 'a item"
1.38 +  Numb      :: "nat => 'a item"
1.39 +  "\$"       :: "['a item, 'a item]=> 'a item"   (infixr 60)
1.40 +  In0,In1   :: "'a item => 'a item"
1.41 +
1.42 +  ntrunc    :: "[nat, 'a item] => 'a item"
1.43 +
1.44 +  "<*>"  :: "['a item set, 'a item set]=> 'a item set" (infixr 80)
1.45 +  "<+>"  :: "['a item set, 'a item set]=> 'a item set" (infixr 70)
1.46 +
1.47 +  Split  :: "[['a item, 'a item]=>'b, 'a item] => 'b"
1.48 +  Case   :: "[['a item]=>'b, ['a item]=>'b, 'a item] => 'b"
1.49 +
1.50 +  diag   :: "'a set => ('a * 'a)set"
1.51 +  "<**>" :: "[('a item * 'a item)set, ('a item * 'a item)set] \
1.52 +\           => ('a item * 'a item)set" (infixr 80)
1.53 +  "<++>" :: "[('a item * 'a item)set, ('a item * 'a item)set] \
1.54 +\           => ('a item * 'a item)set" (infixr 70)
1.55 +
1.56 +defs
1.57 +
1.58 +  (*least number operator*)
1.59 +  Least_def      "Least(P) == @k. P(k) & (ALL j. j<k --> ~P(j))"
1.60 +
1.61 +  Push_Node_def  "Push_Node == (%n x. Abs_Node (apfst (Push n) (Rep_Node x)))"
1.62 +
1.63 +  (*crude "lists" of nats -- needed for the constructions*)
1.64 +  apfst_def  "apfst == (%f. split(%x y. <f(x),y>))"
1.65 +  Push_def   "Push == (%b h. nat_case (Suc b) h)"
1.66 +
1.67 +  (** operations on S-expressions -- sets of nodes **)
1.68 +
1.69 +  (*S-expression constructors*)
1.70 +  Atom_def   "Atom == (%x. {Abs_Node(<%k.0, x>)})"
1.71 +  Scons_def  "M\$N == (Push_Node(0) `` M) Un (Push_Node(Suc(0)) `` N)"
1.72 +
1.73 +  (*Leaf nodes, with arbitrary or nat labels*)
1.74 +  Leaf_def   "Leaf == Atom o Inl"
1.75 +  Numb_def   "Numb == Atom o Inr"
1.76 +
1.77 +  (*Injections of the "disjoint sum"*)
1.78 +  In0_def    "In0(M) == Numb(0) \$ M"
1.79 +  In1_def    "In1(M) == Numb(Suc(0)) \$ M"
1.80 +
1.81 +  (*the set of nodes with depth less than k*)
1.82 +  ndepth_def "ndepth(n) == split (%f x. LEAST k. f(k)=0) (Rep_Node n)"
1.83 +  ntrunc_def "ntrunc k N == {n. n:N & ndepth(n)<k}"
1.84 +
1.85 +  (*products and sums for the "universe"*)
1.86 +  uprod_def  "A<*>B == UN x:A. UN y:B. { (x\$y) }"
1.87 +  usum_def   "A<+>B == In0``A Un In1``B"
1.88 +
1.89 +  (*the corresponding eliminators*)
1.90 +  Split_def  "Split c M == @u. ? x y. M = x\$y & u = c x y"
1.91 +
1.92 +  Case_def   "Case c d M == @u.  (? x . M = In0(x) & u = c(x)) \
1.93 +\                              | (? y . M = In1(y) & u = d(y))"
1.94 +
1.95 +
1.96 +  (** diagonal sets and equality for the "universe" **)
1.97 +
1.98 +  diag_def   "diag(A) == UN x:A. {<x,x>}"
1.99 +
1.100 +  dprod_def  "r<**>s == UN u:r. split (%x x'. \
1.101 +\                       UN v:s. split (%y y'. {<x\$y,x'\$y'>}) v) u"
1.102 +
1.103 +  dsum_def   "r<++>s == (UN u:r. split (%x x'. {<In0(x),In0(x')>}) u) Un \
1.104 +\                       (UN v:s. split (%y y'. {<In1(y),In1(y')>}) v)"
1.105 +
1.106 +end
```