src/HOL/Induct/Comb.thy
author berghofe
Fri, 24 Jul 1998 13:39:47 +0200
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Renamed '$' to 'Scons' because of clashes with constants of the same name in theories using datatypes.
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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 Church-Rosser 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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Comb = Datatype +
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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 (multi-step) 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   "x-1->y ==> x#z -1-> y#z"
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    Ap2   "x-1->y ==> z#x -1-> z#y"
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(** Inductive definition of parallel contractions, =1=>
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             and (multi-step) 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