## Stream: Coq users

### Topic: Operation on Functions

#### Zach Flores (Jul 18 2022 at 15:24):

Hello all, this is my first post on the Zulip (not sure if I really need to announce that, but would like everyone to know I'm new to this). Let me first describe what we would really like to implement mathematically:

Let's assume all the $c_i, b_j,$ and $d$ live in Set.

$f : c_1 \rightarrow \ldots \rightarrow c_n \rightarrow d$

$g : b_1 \rightarrow \ldots \rightarrow b_m \rightarrow c_i$

We want to construct a function in Coq with all this information, called $f \circ_i g$ such that

$f\circ_i g : c_1 \rightarrow\cdots c_{i-1} \rightarrow b_1\rightarrow\cdots \rightarrow b_m\rightarrow c_{i+1}\rightarrow\cdots\rightarrow c_n\rightarrow d$

and

$z_1 : c_1,\ldots, z_{i-1}: c_{i-1}, z_i : b_1,\ldots, z_{i+m-1} : b_m, z_{i+m} : c_{i+1},\ldots, z_{n+m-1} : c_n$

And

$(f\circ_i g) (z_1,\ldots, z_{i+m-1}) = f(z_1,\ldots, z_{i-1}, g (z_i,\ldots, z_{i+m-1}), z_{i+m},\ldots , z_{n+m-1})$

So our current attempt to do this was to do what I thought was the obvious thing: Define heterogeneous lists in Coq, and then make a definition of this that involved a bunch of removing, inserting and keeping the correct members of the heterogeneous list. This turns out to be a type casting nightmare, and is generally really ugly. It also needs to come with with several hypotheses about the size of the lists involved.

Another unfortunate consequence of choices that were made in this direction requires us to define hom sets in this setting as:

Definition homSet (d : Set)(c : list Set) : Type :=
hList c -> d.


Where hList c is the type of all heterogeneous lists that can be made from the list c. This carries with it the difficulty in defining the identity function, among other problems.

In order to remedy the situation, and wanting to avoid defining new types, we'd really like to perform operations analogous to some of the stuff in the n-ary library, though probably a little different. We start with an analogue of nfun for lists:

Fixpoint listFun (B:Set)(l: list Set) :=
match l with
| [] => B
| h::t => h -> listFun B t
end.

Notation " l --> B " := (listFun B l)
(at level 50) : type_scope.


Now remember our goal is, given

Definition lookupSet (i: nat)(xs : list Set)  : Set :=
nth i xs unit.

Definition insert{A : Type} (i:nat) (old new : list A) : list A :=
firstn i old ++ new ++ skipn (i+ 1) old.

f: c --> d
g: b--> ci
lookupSet i c  = ci


we want to obtain a function

circ i f g : insert i c b --> d


Currently trying to understand the Coq n-ary library in hopes that it gives us a direction on how to do this, but don't have a lot of ideas at the moment, so any help (or ideas!) would be greatly appreciated.

#### James Wood (Jul 19 2022 at 09:26):

My intuition would be to define a family Elem {A : Type} : A -> list A -> Set of proof-relevant locations in a list, and use a bit of automation to generate inhabitants of it from the list and a natural number (essentially a beefed-up version of how you can generate Fin.t ns by providing an m and checking that ltb m n evaluates to true). From an Elem X old, you should be able to read off the relevant portions of old and splice new in in a controlled way.

#### Li-yao (Jul 19 2022 at 18:57):

The Haskell way (with type classes):

(* f : c1 -> ... -> cn -> d
g : b1 -> ... -> bm -> ci

f .i g : c1 -> ... -> c(i-1)
-> b1 -> ... -> bm
-> ci+1 -> ... -> cn -> d *)

Class Compose (i : nat) (d c r : Type) : Type :=
compose : d -> c -> r.

#[global]
Instance Compose_S n c0 d c r {Compose n d c r}
: Compose (S n) (c0 -> d) c (c0 -> r) :=
fun f g x0 => compose (f x0) g.

#[global]
Instance Compose_0_c c0 d b c r {Compose 0 (c0 -> d) c r}
: Compose 0 (c0 -> d) (b -> c) (b -> r) :=
fun f g x0 => compose f (g x0).

#[global]
Instance Compose_0 d c : Compose 0 (c -> d) c d :=
fun f g => f g.

Arguments compose : clear implicits.
Arguments compose i {d c r _} _ _.

Parameter c : nat -> Type.
Parameter b : nat -> Type.

Parameter f : c 1 -> c 2 -> c 3 -> c 4 -> nat.
Parameter g : b 1 -> b 2 -> b 3 -> c 2.

Check compose 1 f g.
(* compose 1 f g : c 1 -> b 1 -> b 2 -> b 3 -> c 3 -> c 4 -> nat *)


#### Zach Flores (Jul 29 2022 at 15:34):

Li-yao said:

The Haskell way (with type classes):

(* f : c1 -> ... -> cn -> d
g : b1 -> ... -> bm -> ci

f .i g : c1 -> ... -> c(i-1)
-> b1 -> ... -> bm
-> ci+1 -> ... -> cn -> d *)

Class Compose (i : nat) (d c r : Type) : Type :=
compose : d -> c -> r.

#[global]
Instance Compose_S n c0 d c r {Compose n d c r}
: Compose (S n) (c0 -> d) c (c0 -> r) :=
fun f g x0 => compose (f x0) g.

#[global]
Instance Compose_0_c c0 d b c r {Compose 0 (c0 -> d) c r}
: Compose 0 (c0 -> d) (b -> c) (b -> r) :=
fun f g x0 => compose f (g x0).

#[global]
Instance Compose_0 d c : Compose 0 (c -> d) c d :=
fun f g => f g.

Arguments compose : clear implicits.
Arguments compose i {d c r _} _ _.

Parameter c : nat -> Type.
Parameter b : nat -> Type.

Parameter f : c 1 -> c 2 -> c 3 -> c 4 -> nat.
Parameter g : b 1 -> b 2 -> b 3 -> c 2.

Check compose 1 f g.
(* compose 1 f g : c 1 -> b 1 -> b 2 -> b 3 -> c 3 -> c 4 -> nat *)


Oh, wow. That is exactly what I wanted. Several of us were really struggling with this for a long time. Would you mind walking me through how this works? I am mathematician and not a functional programmer.

(deleted)

#### Li-yao (Jul 29 2022 at 16:07):

The idea is that the Class declares a function with a very (overly) general signature compose : d -> c -> r, and Instances define inference rules such that Compose i d c r is derivable exactly when the type d -> c -> r has a shape matching your desired .i operator.

#### Paolo Giarrusso (Jul 29 2022 at 18:50):

I was wondering: this is useful, but it does not implement \ring_i as a function of i, only concrete instances for the indexes needed in your program

#### Paolo Giarrusso (Jul 29 2022 at 18:51):

Is this sufficient for @Zach Flores 's original request? That's not obvious...

#### Zach Flores (Aug 01 2022 at 14:53):

Paolo Giarrusso said:

I was wondering: this is useful, but it does not implement \ring_i as a function of i, only concrete instances for the indexes needed in your program

I will say that it definitely seems a step in the right direction. I'm going to say this definitely seems to help, and going to see how sufficient it is. I'll keep everyone posted!

#### Zach Flores (Sep 23 2022 at 15:35):

I have been curious if this can be made into something using inductive types. It's not hard to make a function that does this for types (this is just some list insertion), but we don't know how to do this without making some kind of crazy thing using heterogeneous lists. Even then, it's a mess of parameters and I haven't gone back and tried to create a concrete example.

Any thoughts on this would be really appreciated.

#### Zach Flores (Sep 23 2022 at 15:37):

Li-yao said:

The idea is that the Class declares a function with a very (overly) general signature compose : d -> c -> r, and Instances define inference rules such that Compose i d c r is derivable exactly when the type d -> c -> r has a shape matching your desired .i operator.

Is there any way to do this without type classes?

#### Li-yao (Sep 23 2022 at 22:19):

Lucky for you, I've been practicing Agda. Note there is a slight simplification you can do, which removes the parameter n: use (ci+1 -> ... -> cn -> d) as d. In hindsight, this simplification probably allows this implementation to be rewritten with lists instead of vectors.

From Equations Require Import Equations.

From Coq Require Import Vector.

Notation "[]" := (Vector.nil _) : vector_scope.
Notation "t :: ts" := (Vector.cons _ t _ ts) : vector_scope.
Infix "^" := Vector.t.

#[local] Open Scope vector_scope.

Equations arr {n : nat} (ts : Type ^ n) (t' : Type) : Type :=
arr [] t' := t' ;
arr (t :: ts) t' := t -> arr ts t'.

Equations compose' {m : nat} (bs : Type ^ m) {ci d : Type} : (ci -> d) -> arr bs ci -> arr bs d :=
compose' [] f g := f g ;
compose' (b :: bs) f g := fun x => compose' bs f (g x).

(* f : c0 -> ... -> cn-1 -> cn -> d   (* NB: d itself may be a function type *)
g : b0 -> ... -> bm -> cn
compose cs bs f g : c0 ... cn-1 b0 ... bm -> d
*)
Equations compose {n m : nat} (cs : Type ^ n) {ci d : Type} (bs : Type ^ m)
: arr cs (ci -> d) -> arr bs ci -> arr cs (arr bs d) :=
compose [] bs f g := compose' bs f g ;
compose (c :: cs) bs f g := fun x => compose cs bs (f x) g.

(* Infer the lists from the types *)

Class Split (T : Type) (i : nat) (Args : Type ^ i) (d : Type) : Prop :=
Split_eq : arr Args d = T.

#[global] Instance Split_S {B n A As d} {e : Split B n As d}
: Split (A -> B) (S n) (A :: As) d :=
match e with
| eq_refl => eq_refl
end.

#[global] Instance Split_O A : Split A O [] A := eq_refl.

Definition compose_ (i : nat) {cs : Type ^ i} {m} {bs : Type ^ m} {ci d cd bc : Type}
{ec : Split cd i cs (ci -> d)} {eb : Split bc m bs ci} : cd -> bc -> arr cs (arr bs d) :=
let 'eq_refl := ec in
let 'eq_refl := eb in compose cs bs.

Section Example.

Context (c : nat -> Type) (b : nat -> Type) (d : Type).
Context (f : c 0 -> c 1 -> c 2 -> d).
Context (g : b 0 -> b 1 -> c 1).

(* Without the class, we must provide the list of types. *)
Definition fg := compose (c 0 :: []) (b 0 :: b 1 :: []) f g.
Definition type_of {A : Type} (_ : A) : Type := A.
Compute (type_of fg).

(* With the class, the lists of types can be inferred. *)
Definition fg' := compose_ 1 f g.
Compute (type_of fg').

End Example.


Last updated: Jun 24 2024 at 12:02 UTC