Coq Index Relation - coq

Question

I'm defining an indexed inductive type in Coq:
Module Typ.
(* My index -- a t is a `t H` or a `t Z`. *)
Inductive hz : Set := H | Z .
(* I'd like to use this relation to constrain Cursor and Arrow. *)
(* E.g. one specialized Cursor has type `t H -> t Z -> t Z` *)
Inductive maxHZ : hz -> hz -> hz -> Type :=
| HZ : maxHZ H Z Z
| ZH : maxHZ Z H Z
| HH : maxHZ H H H
.
Inductive t : hz -> Type :=
| Num : t H
| Hole : t H
| Cursor : t H -> t Z
| Arrow : forall a b c, t a -> t b -> t c
| Sum : forall a b c, t a -> t b -> t c
.
End Typ.
How can I constrain my Arrow / Sum indices to be the same shape as the maxHZ relation (short of creating more constructors, like ArrowHZ : t H -> t Z -> t Z).

Answer 1

One approach:
(* Bring the coercion is_true into scope *)
From Coq Require Import ssreflect ssrfun ssrbool.
Module Typ.
(* My index -- a t is a `t H` or a `t Z`. *)
Inductive hz : Set := H | Z .
(* I'd like to use this relation to constrain Cursor and Arrow. *)
(* E.g. one specialized Cursor has type `t H -> t Z -> t Z` *)
Definition maxHZ (x y z : hz) : bool :=
match x, y, z with
| H, Z, Z
| Z, H, Z
| H, H, H => true
| _, _, _ => false
end.
Inductive t : hz -> Type :=
| Num : t H
| Hole : t H
| Cursor : t H -> t Z
| Arrow : forall a b c, maxHZ a b c -> t a -> t b -> t c
| Sum : forall a b c, maxHZ a b c -> t a -> t b -> t c
.
End Typ.
the other:
Inductive t : hz -> Type :=
| Num : t H
| Hole : t H
| Cursor : t H -> t Z
| Arrow : forall a b c, t a -> t b -> t c
| Sum : forall a b c, t a -> t b -> t c
.
Definition t_wf x (m : t x) : bool :=
match m with
| Arrow a b c _ _ => maxHZ a b c
| Sum a b c _ _ => maxHZ a b c
| _ => true
end.
Definition t' x := { m : t x | t_wf x m }.
(* This is automatically a subtype. *)