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(* Title: HOL/univ.thy
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ID: $Id$
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory
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Copyright 1993 University of Cambridge
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Move LEAST to nat.thy??? Could it be defined for all types 'a::ord?
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Declares the type 'a node, a subtype of (nat=>nat) * ('a+nat)
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Defines "Cartesian Product" and "Disjoint Sum" as set operations.
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Could <*> be generalized to a general summation (Sigma)?
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*)
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Univ = Arith +
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types node 1
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arities node :: (term)term
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consts
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Least :: "(nat=>bool) => nat" (binder "LEAST " 10)
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apfst :: "['a=>'c, 'a*'b] => 'c*'b"
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Push :: "[nat, nat=>nat] => (nat=>nat)"
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Node :: "((nat=>nat) * ('a+nat)) set"
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Rep_Node :: "'a node => (nat=>nat) * ('a+nat)"
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Abs_Node :: "(nat=>nat) * ('a+nat) => 'a node"
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Push_Node :: "[nat, 'a node] => 'a node"
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ndepth :: "'a node => nat"
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Atom :: "('a+nat) => 'a node set"
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Leaf :: "'a => 'a node set"
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Numb :: "nat => 'a node set"
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"." :: "['a node set, 'a node set]=> 'a node set" (infixr 60)
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In0,In1 :: "'a node set => 'a node set"
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ntrunc :: "[nat, 'a node set] => 'a node set"
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"<*>" :: "['a node set set, 'a node set set]=> 'a node set set" (infixr 80)
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"<+>" :: "['a node set set, 'a node set set]=> 'a node set set" (infixr 70)
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Split :: "['a node set, ['a node set, 'a node set]=>'b] => 'b"
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Case :: "['a node set, ['a node set]=>'b, ['a node set]=>'b] => 'b"
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diag :: "'a set => ('a * 'a)set"
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"<**>" :: "[('a node set * 'a node set)set, ('a node set * 'a node set)set] \
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\ => ('a node set * 'a node set)set" (infixr 80)
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"<++>" :: "[('a node set * 'a node set)set, ('a node set * 'a node set)set] \
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\ => ('a node set * 'a node set)set" (infixr 70)
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rules
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(*least number operator*)
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Least_def "Least(P) == @k. P(k) & (ALL j. j<k --> ~P(j))"
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(** lists, trees will be sets of nodes **)
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Node_def "Node == {p. EX f x k. p = <f,x> & f(k)=0}"
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(*faking the type definition 'a node == (nat=>nat) * ('a+nat) *)
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Rep_Node "Rep_Node(n): Node"
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Rep_Node_inverse "Abs_Node(Rep_Node(n)) = n"
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Abs_Node_inverse "p: Node ==> Rep_Node(Abs_Node(p)) = p"
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Push_Node_def "Push_Node == (%n x. Abs_Node (apfst(Push(n),Rep_Node(x))))"
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(*crude "lists" of nats -- needed for the constructions*)
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apfst_def "apfst == (%f p.split(p, %x y. <f(x),y>))"
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Push_def "Push == (%b h n. nat_case(n,Suc(b),h))"
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(** operations on S-expressions -- sets of nodes **)
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(*S-expression constructors*)
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Atom_def "Atom == (%x. {Abs_Node(<%k.0, x>)})"
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Scons_def "M.N == (Push_Node(0) `` M) Un (Push_Node(Suc(0)) `` N)"
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(*Leaf nodes, with arbitrary or nat labels*)
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Leaf_def "Leaf == Atom o Inl"
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Numb_def "Numb == Atom o Inr"
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(*Injections of the "disjoint sum"*)
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In0_def "In0(M) == Numb(0) . M"
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In1_def "In1(M) == Numb(Suc(0)) . M"
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(*the set of nodes with depth less than k*)
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ndepth_def "ndepth(n) == split(Rep_Node(n), %f x. LEAST k. f(k)=0)"
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ntrunc_def "ntrunc(k,N) == {n. n:N & ndepth(n)<k}"
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(*products and sums for the "universe"*)
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uprod_def "A<*>B == UN x:A. UN y:B. { (x.y) }"
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usum_def "A<+>B == In0``A Un In1``B"
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(*the corresponding eliminators*)
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Split_def "Split(M,c) == @u. ? x y. M = x.y & u = c(x,y)"
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Case_def "Case(M,c,d) == @u. (? x . M = In0(x) & u = c(x)) \
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\ | (? y . M = In1(y) & u = d(y))"
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(** diagonal sets and equality for the "universe" **)
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diag_def "diag(A) == UN x:A. {<x,x>}"
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dprod_def "r<**>s == UN u:r. UN v:s. \
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\ split(u, %x x'. split(v, %y y'. {<x.y,x'.y'>}))"
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dsum_def "r<++>s == (UN u:r. split(u, %x x'. {<In0(x),In0(x')>})) Un \
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\ (UN v:s. split(v, %y y'. {<In1(y),In1(y')>}))"
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end
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