## Stream: new members

### Topic: mutual inductive definition

#### Shaun Steenkamp (Nov 16 2018 at 10:35):

can one of you explain why I cannot get this mutual inductive definition to work?

mutual inductive Foo, Bar
with Foo : Type
| bizz : Foo
| buzz {f : Foo} : Bar f → Foo
with Bar : Foo → Type
| baz (f : Foo) : Bar f


For reference, the following works in Agda

mutual
data Foo : Set where
bizz : Foo
buzz : {f : Foo} → Bar f → Foo

data Bar : Foo → Set where
baz : (f : Foo) → Bar f


#### Mario Carneiro (Nov 16 2018 at 10:39):

this is called an inductive-recursive definition, and it is stronger than lean's axiomatics

#### Mario Carneiro (Nov 16 2018 at 10:40):

There may be another way to encode the type you want, but inductive recursive definitions can be used to prove DTT is consistent, so lean can't do it

#### Keeley Hoek (Nov 16 2018 at 10:51):

Incidentally, Mario, were you working on a Lean consistency project at some point?

What is DTT?

#### Patrick Massot (Nov 16 2018 at 11:02):

dependent type theory

#### Patrick Massot (Nov 16 2018 at 11:03):

That's the name of the game we are playing

#### Mario Carneiro (Nov 16 2018 at 11:30):

yes, I'll be defending my masters thesis in a few weeks and lean consistency is in it

#### Mario Carneiro (Nov 16 2018 at 11:30):

https://github.com/digama0/lean-type-theory/releases/tag/v0.21

#### Kenny Lau (Nov 16 2018 at 11:31):

so, is lean consistent?

#### Abhimanyu Pallavi Sudhir (Nov 16 2018 at 11:38):

@Kenny Lau How is that not independent?

#### Mario Carneiro (Nov 16 2018 at 12:02):

yes, assuming some large cardinal assumptions

#### Mario Carneiro (Nov 16 2018 at 12:03):

which is the usual story with consistency proofs

#### Johannes Hölzl (Nov 16 2018 at 12:25):

I think this shape is called inductive-inductive: mutual inductive types where at least one is indexed by the other one, inductive-recursive means to mutually define inductive types and recursive definitions where at least one inductive type depends on the recursive definition.
I think inductive-inductive can be reduced to inductive, inductive-recursive cannot

#### Mario Carneiro (Nov 16 2018 at 12:30):

you might be right about the name, but I'm pretty sure inductive-inductive is also axiomatically strong

#### Mario Carneiro (Nov 16 2018 at 12:32):

because you can define the well typed terms of DTT by induction-induction with the set of well formed contexts and the set of types in a context

#### Mario Carneiro (Nov 16 2018 at 12:34):

Inductive recursive can be reduced to inductive inductive, I think, where the recursive function becomes a functional relation

#### Shaun Steenkamp (Nov 16 2018 at 13:45):

There may be another way to encode the type you want, but inductive recursive definitions can be used to prove DTT is consistent, so lean can't do it

Consistent with respect to what?

#### Johannes Hölzl (Nov 16 2018 at 14:54):

with respect to DTT with a larger type universe and inductive recursive definitions (I guess)

#### Floris van Doorn (Nov 16 2018 at 15:05):

This is indeed an inductive-inductive definition. I think adding these won't increase the proof stength of the system. In extensional type theory, inductive-inductive definitions can be defined from indexed inductive types, but I don't think it's known in whether this is the case in intensional type theory.
Inductive-recursive definitions indeed do increase the proof stength of the system: you can build a (Tarski) universe using induction-recursion.