src/HOL/Induct/Comb.thy
 author berghofe Fri, 24 Jul 1998 13:39:47 +0200 changeset 5191 8ceaa19f7717 parent 5184 9b8547a9496a child 9101 b643f4d7b9e9 permissions -rw-r--r--
Renamed '\$' to 'Scons' because of clashes with constants of the same name in theories using datatypes.
```
(*  Title:      HOL/ex/Comb.thy
ID:         \$Id\$
Author:     Lawrence C Paulson
Copyright   1996  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 **)
datatype comb = K
| S
| "#" comb comb (infixl 90)

(** Inductive definition of contractions, -1->
and (multi-step) reductions, --->
**)
consts
contract  :: "(comb*comb) set"
"-1->"    :: [comb,comb] => bool   (infixl 50)
"--->"    :: [comb,comb] => bool   (infixl 50)

translations
"x -1-> y" == "(x,y) : contract"
"x ---> y" == "(x,y) : contract^*"

inductive contract
intrs
K     "K#x#y -1-> x"
S     "S#x#y#z -1-> (x#z)#(y#z)"
Ap1   "x-1->y ==> x#z -1-> y#z"
Ap2   "x-1->y ==> z#x -1-> z#y"

(** Inductive definition of parallel contractions, =1=>
and (multi-step) parallel reductions, ===>
**)
consts
parcontract :: "(comb*comb) set"
"=1=>"    :: [comb,comb] => bool   (infixl 50)
"===>"    :: [comb,comb] => bool   (infixl 50)

translations
"x =1=> y" == "(x,y) : parcontract"
"x ===> y" == "(x,y) : parcontract^*"

inductive parcontract
intrs
refl  "x =1=> x"
K     "K#x#y =1=> x"
S     "S#x#y#z =1=> (x#z)#(y#z)"
Ap    "[| x=1=>y;  z=1=>w |] ==> x#z =1=> y#w"

(*Misc definitions*)
constdefs
I :: comb
"I == S#K#K"

(*confluence; Lambda/Commutation treats this more abstractly*)
diamond   :: "('a * 'a)set => bool"
"diamond(r) == ALL x y. (x,y):r -->
(ALL y'. (x,y'):r -->
(EX z. (y,z):r & (y',z) : r))"

end
```