author  nipkow 
Fri, 23 May 1997 09:18:06 +0200  
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permissions  rwrr 
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(* Title: HOL/ex/Comb.thy 
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ID: $Id$ 
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Author: Lawrence C Paulson 
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Copyright 1996 University of Cambridge 
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Combinatory Logic example: the ChurchRosser Theorem 
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Curiously, combinators do not include free variables. 
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Example taken from 
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J. Camilleri and T. F. Melham. 
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Reasoning with Inductively Defined Relations in the HOL Theorem Prover. 
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Report 265, University of Cambridge Computer Laboratory, 1992. 
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*) 
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3309  16 
Comb = Arith + 
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(** Datatype definition of combinators S and K, with infixed application **) 
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datatype comb = K 
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 S 
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 "#" comb comb (infixl 90) 
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(** Inductive definition of contractions, 1> 
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and (multistep) reductions, > 
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**) 
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consts 
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contract :: "(comb*comb) set" 
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"1>" :: [comb,comb] => bool (infixl 50) 
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">" :: [comb,comb] => bool (infixl 50) 
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translations 
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"x 1> y" == "(x,y) : contract" 
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"x > y" == "(x,y) : contract^*" 
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inductive contract 
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intrs 
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K "K#x#y 1> x" 
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S "S#x#y#z 1> (x#z)#(y#z)" 
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Ap1 "x1>y ==> x#z 1> y#z" 
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Ap2 "x1>y ==> z#x 1> z#y" 
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(** Inductive definition of parallel contractions, =1=> 
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and (multistep) parallel reductions, ===> 
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**) 
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consts 
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parcontract :: "(comb*comb) set" 
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"=1=>" :: [comb,comb] => bool (infixl 50) 
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"===>" :: [comb,comb] => bool (infixl 50) 
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translations 
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"x =1=> y" == "(x,y) : parcontract" 
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"x ===> y" == "(x,y) : parcontract^*" 
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inductive parcontract 
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intrs 
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refl "x =1=> x" 
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K "K#x#y =1=> x" 
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S "S#x#y#z =1=> (x#z)#(y#z)" 
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Ap "[ x=1=>y; z=1=>w ] ==> x#z =1=> y#w" 
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(*Misc definitions*) 
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constdefs 
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I :: comb 
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"I == S#K#K" 
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(*confluence; Lambda/Commutation treats this more abstractly*) 
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diamond :: "('a * 'a)set => bool" 
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"diamond(r) == ALL x y. (x,y):r > 
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(ALL y'. (x,y'):r > 
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(EX z. (y,z):r & (y',z) : r))" 
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end 