src/HOL/Univ.thy
 author paulson Mon, 11 Mar 1996 14:05:45 +0100 changeset 1562 e98c7f6165c9 parent 1531 e5eb247ad13c child 3947 eb707467f8c5 permissions -rw-r--r--
deleted obsolete comment
```
(*  Title:      HOL/Univ.thy
ID:         \$Id\$
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

Declares the type 'a node, a subtype of (nat=>nat) * ('a+nat)

Defines "Cartesian Product" and "Disjoint Sum" as set operations.
Could <*> be generalized to a general summation (Sigma)?
*)

Univ = Arith + Sum +

(** lists, trees will be sets of nodes **)

typedef (Node)
'a node = "{p. EX f x k. p = (f::nat=>nat, x::'a+nat) & f(k)=0}"

types
'a item = 'a node set

consts
apfst     :: "['a=>'c, 'a*'b] => 'c*'b"
Push      :: [nat, nat=>nat] => (nat=>nat)

Push_Node :: [nat, 'a node] => 'a node
ndepth    :: 'a node => nat

Atom      :: "('a+nat) => 'a item"
Leaf      :: 'a => 'a item
Numb      :: nat => 'a item
"\$"       :: ['a item, 'a item]=> 'a item   (infixr 60)
In0,In1   :: 'a item => 'a item

ntrunc    :: [nat, 'a item] => 'a item

"<*>"  :: ['a item set, 'a item set]=> 'a item set (infixr 80)
"<+>"  :: ['a item set, 'a item set]=> 'a item set (infixr 70)

Split  :: [['a item, 'a item]=>'b, 'a item] => 'b
Case   :: [['a item]=>'b, ['a item]=>'b, 'a item] => 'b

diag   :: "'a set => ('a * 'a)set"
"<**>" :: "[('a item * 'a item)set, ('a item * 'a item)set]
=> ('a item * 'a item)set" (infixr 80)
"<++>" :: "[('a item * 'a item)set, ('a item * 'a item)set]
=> ('a item * 'a item)set" (infixr 70)

defs

Push_Node_def  "Push_Node == (%n x. Abs_Node (apfst (Push n) (Rep_Node x)))"

(*crude "lists" of nats -- needed for the constructions*)
apfst_def  "apfst == (%f (x,y). (f(x),y))"
Push_def   "Push == (%b h. nat_case (Suc b) h)"

(** operations on S-expressions -- sets of nodes **)

(*S-expression constructors*)
Atom_def   "Atom == (%x. {Abs_Node((%k.0, x))})"
Scons_def  "M\$N == (Push_Node(0) `` M) Un (Push_Node(Suc(0)) `` N)"

(*Leaf nodes, with arbitrary or nat labels*)
Leaf_def   "Leaf == Atom o Inl"
Numb_def   "Numb == Atom o Inr"

(*Injections of the "disjoint sum"*)
In0_def    "In0(M) == Numb(0) \$ M"
In1_def    "In1(M) == Numb(Suc(0)) \$ M"

(*the set of nodes with depth less than k*)
ndepth_def "ndepth(n) == (%(f,x). LEAST k. f(k)=0) (Rep_Node n)"
ntrunc_def "ntrunc k N == {n. n:N & ndepth(n)<k}"

(*products and sums for the "universe"*)
uprod_def  "A<*>B == UN x:A. UN y:B. { (x\$y) }"
usum_def   "A<+>B == In0``A Un In1``B"

(*the corresponding eliminators*)
Split_def  "Split c M == @u. ? x y. M = x\$y & u = c x y"

Case_def   "Case c d M == @u.  (? x . M = In0(x) & u = c(x))
| (? y . M = In1(y) & u = d(y))"

(** diagonal sets and equality for the "universe" **)

diag_def   "diag(A) == UN x:A. {(x,x)}"

dprod_def  "r<**>s == UN (x,x'):r. UN (y,y'):s. {(x\$y,x'\$y')}"

dsum_def   "r<++>s == (UN (x,x'):r. {(In0(x),In0(x'))}) Un
(UN (y,y'):s. {(In1(y),In1(y'))})"

end
```