Created
July 17, 2018 12:04
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module Peano = struct | |
type z = Zero | |
type 'a s = Succ | |
type 'a t = | |
| Z : z t | |
| S : 'a t -> 'a s t | |
and zero = z t | |
let zero : z t = Z | |
let one : z s t = S Z | |
end | |
module Plus = struct | |
type ('a, 'b, 'r) t = | |
| Z : (Peano.z, 'a, 'a) t | |
| S : ('p, 'q, 'r) t -> ('p Peano.s, 'q, 'r Peano.s) t | |
end | |
module Eq = struct | |
type ('a, 'b) refl = Refl : ('a, 'a) refl | |
end | |
module Ordinal = struct | |
type u = unit | |
type ('a, 'n) t = | |
| Z : (u, Peano.z) t | |
| S : ('e, 'n) t -> ('e * u, 'n Peano.s) t | |
module N = struct | |
type ('a, 'n) t = | |
| Z : ('e * u, Peano.z) t | |
| S : ('e, 'n) t -> ('e * u, 'n Peano.s) t | |
end | |
let rec in_range : type e n m. (e, n) N.t -> (e, m) t -> (e, e) Eq.refl | |
= fun v r -> match v, r with | |
| N.Z, S _ -> Eq.Refl | |
| N.S v, S r -> match in_range v r with | |
| Eq.Refl -> Eq.Refl | |
end | |
type ('e, 'l, 'h, 'r) t = | |
| Z : ('e, 'h) Ordinal.t -> ('e, Peano.z, 'h, 'h) t | |
| S : ('e, 'l, 'h, 'r) t -> ('e * Ordinal.u, 'l Peano.s, 'h, 'r Peano.s) t | |
module N = struct | |
type ('e, 'l, 'm, 'h) t = | |
| Z : ('e, 'm) Ordinal.N.t -> ('e, Peano.z, 'm, 'h) t | |
| S : ('e, 'l, 'm, 'h) t -> ('e * Ordinal.u, 'l Peano.s, 'm Peano.s, 'h) t | |
end | |
let rec in_range | |
: type e h0 h1 l m n. (e, l, m, h0) N.t -> (e, l, n, h1) t -> (e, e) Eq.refl | |
= fun v t -> match v, t with | |
| N.Z v, Z o -> Ordinal.in_range v o | |
| N.S v, S o -> match in_range v o with | |
| Eq.Refl -> Eq.Refl | |
let range_2_4 = S (S (Z Ordinal.(S (S Z)))) | |
let _ = in_range N.(S (S (Z Ordinal.N.(S Z)))) range_2_4 |
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