(* Title: ZF/ex/Comb.thy
ID: $Id$
Author: Lawrence C Paulson
Copyright 1994 University of Cambridge
Combinatory Logic example: the Church-Rosser Theorem
Curiously, combinators do not include free variables.
Example taken from
J. Camilleri and T. F. Melham.
Reasoning with Inductively Defined Relations in the HOL Theorem Prover.
Report 265, University of Cambridge Computer Laboratory, 1992.
*)
Comb = Datatype +
(** Datatype definition of combinators S and K, with infixed application **)
consts comb :: "i"
datatype (* <= "univ(0)" *)
"comb" = K
| S
| "#" ("p: comb", "q: comb") (infixl 90)
(** Inductive definition of contractions, -1->
and (multi-step) reductions, --->
**)
consts
contract :: "i"
"-1->" :: "[i,i] => o" (infixl 50)
"--->" :: "[i,i] => o" (infixl 50)
translations
"p -1-> q" == "<p,q> : contract"
"p ---> q" == "<p,q> : contract^*"
inductive
domains "contract" <= "comb*comb"
intrs
K "[| p:comb; q:comb |] ==> K#p#q -1-> p"
S "[| p:comb; q:comb; r:comb |] ==> S#p#q#r -1-> (p#r)#(q#r)"
Ap1 "[| p-1->q; r:comb |] ==> p#r -1-> q#r"
Ap2 "[| p-1->q; r:comb |] ==> r#p -1-> r#q"
type_intrs "comb.intrs"
(** Inductive definition of parallel contractions, =1=>
and (multi-step) parallel reductions, ===>
**)
consts
parcontract :: "i"
"=1=>" :: "[i,i] => o" (infixl 50)
"===>" :: "[i,i] => o" (infixl 50)
translations
"p =1=> q" == "<p,q> : parcontract"
"p ===> q" == "<p,q> : parcontract^+"
inductive
domains "parcontract" <= "comb*comb"
intrs
refl "[| p:comb |] ==> p =1=> p"
K "[| p:comb; q:comb |] ==> K#p#q =1=> p"
S "[| p:comb; q:comb; r:comb |] ==> S#p#q#r =1=> (p#r)#(q#r)"
Ap "[| p=1=>q; r=1=>s |] ==> p#r =1=> q#s"
type_intrs "comb.intrs"
(*Misc definitions*)
consts
diamond :: "i => o"
I :: "i"
defs
diamond_def "diamond(r) == ALL x y. <x,y>:r --> \
\ (ALL y'. <x,y'>:r --> \
\ (EX z. <y,z>:r & <y',z> : r))"
I_def "I == S#K#K"
end