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[if] [then] [rec1 [[if] [then] [rec1] [rec2] genrec] rec2] ifte
From "Recursion Theory and Joy" (j05cmp.html) by Manfred von Thun: "The genrec combinator takes four program parameters in addition to whatever data parameters it needs. Fourth from the top is an if-part, followed by a then-part. If the if-part yields true, then the then-part is executed and the combinator terminates. The other two parameters are the rec1-part and the rec2-part. If the if-part yields false, the rec1-part is executed. Following that the four program parameters and the combinator are again pushed onto the stack bundled up in a quoted form. Then the rec2-part is executed, where it will find the bundled form. Typically it will then execute the bundled form, either with i or with app2, or some other combinator."
The way to design one of these is to fix your base case [then] and the test [if], and then treat rec1 and rec2 as an else-part "sandwiching" a quotation of the whole function.
For example, given a (general recursive) function 'F': :
F == [I] [T] [R1] [R2] genrec
If the [I] if-part fails you must derive R1 and R2 from: :
... R1 [F] R2
Just set the stack arguments in front, and figure out what R1 and R2 have to do to apply the quoted [F] in the proper way. In effect, the genrec combinator turns into an ifte combinator with a quoted copy of the original definition in the else-part: :
F == [I] [T] [R1] [R2] genrec
== [I] [T] [R1 [F] R2] ifte
Primitive recursive functions are those where R2 == i. :
P == [I] [T] [R] tailrec
== [I] [T] [R [P] i] ifte
== [I] [T] [R P] ifte
Gentzen diagram.
Definition
if not basis.
Derivation
if not basis.
Source
if basis
Discussion
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