Those who have dealt with proof assistants based on type theories (Such as Agda and Coq) might have noticed that we often require types to have a positivity restriction. This restriction essentially states that you can't have non-positive occurrences of the recursive type variable. As I described earlier, System F avoids this whole positivity restriction by forcing the programmer to demonstrate that constructors themselves (as a Church encoding) can be implemented, avoiding the problem of uninhabited types being inadvertently

*asserted*.

So this means that in some sense positivity is probably the same thing as saying that there is an algorithmic method of translation of a data-type into System-F. I suspect (but can not yet prove) it also means that we should be allowed to write down datatypes as long as we can show that the inductive or coinductive types are inhabited by a Church encoding! If we could also do this trick for the Calculus of Constructions it might give a tricky way to increase the number of (co)inductive types we are allowed to write down.

Now, I'll write down an example of what I'm talking about in System-F, embedded in the propositional fragment of Coq (which is suitably impredicative), so that you can see a very straightforwardly non-positive type which has a Church encoding. This example of the LamMu type came from Andreas Abel's paper on sized types (which one I forget!) where he demonstrates that his system allows the definition.

In the very final entry we attempt to enter LamMu into the Coq Inductive type framework and watch it fail (for reasons of positivity violation). The original type definition gives us three constructors, each of which is proved in turn below by a "constructor".

`Definition LamMu := forall (X : Prop), `

(Nat -> X) ->

(Nat -> List X -> X) ->

((forall (Y : Prop), ((X -> Y) -> Y)) -> X) -> X.

Definition var : Nat -> LamMu :=

fun (n : Nat) =>

fun (X : Prop)

(v : Nat -> X)

(f : Nat -> List X -> X)

(m : (forall (Y : Prop), ((X -> Y) -> Y)) -> X) =>

v n.

Definition func : Nat -> List LamMu -> LamMu :=

fun (n : Nat) =>

fun (t : List LamMu) =>

fun (X : Prop)

(v : Nat -> X)

(f : Nat -> List X -> X)

(m : (forall (Y : Prop), ((X -> Y) -> Y)) -> X) =>

f n (t (List X) (nil X) (fun (x : LamMu) (y : List X) => cons (x X v f m) y)).

Definition mu : (forall (Y : Prop), (LamMu -> Y) -> Y) -> LamMu.

Proof.

unfold LamMu.

refine

(fun (zi : (forall (Y : Prop), (LamMu -> Y) -> Y)) =>

fun (X : Prop)

(v : Nat -> X)

(f : Nat -> List X -> X)

(m : (forall (Y : Prop), ((X -> Y) -> Y)) -> X) =>

m (fun (Y : Prop) (g : X -> Y) =>

g (zi X (fun e : LamMu => e X v f m)))).

Defined.

Inductive LamMu2 : Type :=

| Var : Nat -> LamMu2

| Fun : Nat -> LamMu2 -> LamMu2

| Mu : ((forall Y, ((LamMu2 -> Y) -> Y)) -> LamMu2) -> LamMu2

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