(* Title: HOL/univ.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
Move LEAST to nat.thy??? Could it be defined for all types 'a::ord?
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 +
types
'a node
arities
node :: (term)term
consts
Least :: "(nat=>bool) => nat" (binder "LEAST " 10)
apfst :: "['a=>'c, 'a*'b] => 'c*'b"
Push :: "[nat, nat=>nat] => (nat=>nat)"
Node :: "((nat=>nat) * ('a+nat)) set"
Rep_Node :: "'a node => (nat=>nat) * ('a+nat)"
Abs_Node :: "(nat=>nat) * ('a+nat) => 'a node"
Push_Node :: "[nat, 'a node] => 'a node"
ndepth :: "'a node => nat"
Atom :: "('a+nat) => 'a node set"
Leaf :: "'a => 'a node set"
Numb :: "nat => 'a node set"
"$" :: "['a node set, 'a node set]=> 'a node set" (infixr 60)
In0,In1 :: "'a node set => 'a node set"
ntrunc :: "[nat, 'a node set] => 'a node set"
"<*>" :: "['a node set set, 'a node set set]=> 'a node set set" (infixr 80)
"<+>" :: "['a node set set, 'a node set set]=> 'a node set set" (infixr 70)
Split :: "['a node set, ['a node set, 'a node set]=>'b] => 'b"
Case :: "['a node set, ['a node set]=>'b, ['a node set]=>'b] => 'b"
diag :: "'a set => ('a * 'a)set"
"<**>" :: "[('a node set * 'a node set)set, ('a node set * 'a node set)set] \
\ => ('a node set * 'a node set)set" (infixr 80)
"<++>" :: "[('a node set * 'a node set)set, ('a node set * 'a node set)set] \
\ => ('a node set * 'a node set)set" (infixr 70)
rules
(*least number operator*)
Least_def "Least(P) == @k. P(k) & (ALL j. j<k --> ~P(j))"
(** lists, trees will be sets of nodes **)
Node_def "Node == {p. EX f x k. p = <f,x> & f(k)=0}"
(*faking the type definition 'a node == (nat=>nat) * ('a+nat) *)
Rep_Node "Rep_Node(n): Node"
Rep_Node_inverse "Abs_Node(Rep_Node(n)) = n"
Abs_Node_inverse "p: Node ==> Rep_Node(Abs_Node(p)) = p"
Push_Node_def "Push_Node == (%n x. Abs_Node (apfst(Push(n),Rep_Node(x))))"
(*crude "lists" of nats -- needed for the constructions*)
apfst_def "apfst == (%f p.split(p, %x y. <f(x),y>))"
Push_def "Push == (%b h n. nat_case(n,Suc(b),h))"
(** operations on S-expressions -- sets of nodes **)
(*S-expression constructors*)
Atom_def "Atom == (%x. {Abs_Node(<%k.0, x>)})"
Scons_def "M$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) == split(Rep_Node(n), %f x. LEAST k. f(k)=0)"
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$y) }"
usum_def "A<+>B == In0``A Un In1``B"
(*the corresponding eliminators*)
Split_def "Split(M,c) == @u. ? x y. M = x$y & u = c(x,y)"
Case_def "Case(M,c,d) == @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 u:r. UN v:s. \
\ split(u, %x x'. split(v, %y y'. {<x$y,x'$y'>}))"
dsum_def "r<++>s == (UN u:r. split(u, %x x'. {<In0(x),In0(x')>})) Un \
\ (UN v:s. split(v, %y y'. {<In1(y),In1(y')>}))"
end