Added mk_int and mk_list.
(* Title: HOL/Datatype_Universe.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
Declares the type ('a, 'b) node, a subtype of (nat=>'b+nat) * ('a+nat)
Defines "Cartesian Product" and "Disjoint Sum" as set operations.
Could <*> be generalized to a general summation (Sigma)?
*)
Datatype_Universe = NatArith + Sum_Type +
(** lists, trees will be sets of nodes **)
typedef (Node)
('a,'b) node = "{p. EX f x k. p = (f::nat=>'b+nat, x::'a+nat) & f k = Inr 0}"
types
'a item = ('a, unit) node set
('a, 'b) dtree = ('a, 'b) node set
consts
apfst :: "['a=>'c, 'a*'b] => 'c*'b"
Push :: "[('b + nat), nat => ('b + nat)] => (nat => ('b + nat))"
Push_Node :: "[('b + nat), ('a, 'b) node] => ('a, 'b) node"
ndepth :: ('a, 'b) node => nat
Atom :: "('a + nat) => ('a, 'b) dtree"
Leaf :: 'a => ('a, 'b) dtree
Numb :: nat => ('a, 'b) dtree
Scons :: [('a, 'b) dtree, ('a, 'b) dtree] => ('a, 'b) dtree
In0,In1 :: ('a, 'b) dtree => ('a, 'b) dtree
Lim :: ('b => ('a, 'b) dtree) => ('a, 'b) dtree
ntrunc :: [nat, ('a, 'b) dtree] => ('a, 'b) dtree
uprod :: [('a, 'b) dtree set, ('a, 'b) dtree set]=> ('a, 'b) dtree set
usum :: [('a, 'b) dtree set, ('a, 'b) dtree set]=> ('a, 'b) dtree set
Split :: [[('a, 'b) dtree, ('a, 'b) dtree]=>'c, ('a, 'b) dtree] => 'c
Case :: [[('a, 'b) dtree]=>'c, [('a, 'b) dtree]=>'c, ('a, 'b) dtree] => 'c
dprod :: "[(('a, 'b) dtree * ('a, 'b) dtree)set, (('a, 'b) dtree * ('a, 'b) dtree)set]
=> (('a, 'b) dtree * ('a, 'b) dtree)set"
dsum :: "[(('a, 'b) dtree * ('a, 'b) dtree)set, (('a, 'b) dtree * ('a, 'b) dtree)set]
=> (('a, 'b) dtree * ('a, 'b) dtree)set"
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 b h)"
(** operations on S-expressions -- sets of nodes **)
(*S-expression constructors*)
Atom_def "Atom == (%x. {Abs_Node((%k. Inr 0, x))})"
Scons_def "Scons M N == (Push_Node (Inr 1) ` M) Un (Push_Node (Inr (Suc 1)) ` 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) == Scons (Numb 0) M"
In1_def "In1(M) == Scons (Numb 1) M"
(*Function spaces*)
Lim_def "Lim f == Union {z. ? x. z = Push_Node (Inl x) ` (f x)}"
(*the set of nodes with depth less than k*)
ndepth_def "ndepth(n) == (%(f,x). LEAST k. f k = Inr 0) (Rep_Node n)"
ntrunc_def "ntrunc k N == {n. n:N & ndepth(n)<k}"
(*products and sums for the "universe"*)
uprod_def "uprod A B == UN x:A. UN y:B. { Scons x y }"
usum_def "usum A B == In0`A Un In1`B"
(*the corresponding eliminators*)
Split_def "Split c M == THE u. EX x y. M = Scons x y & u = c x y"
Case_def "Case c d M == THE u. (EX x . M = In0(x) & u = c(x))
| (EX y . M = In1(y) & u = d(y))"
(** equality for the "universe" **)
dprod_def "dprod r s == UN (x,x'):r. UN (y,y'):s. {(Scons x y, Scons x' y')}"
dsum_def "dsum r s == (UN (x,x'):r. {(In0(x),In0(x'))}) Un
(UN (y,y'):s. {(In1(y),In1(y'))})"
end