Source file telescope.ml
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open Common
open Kernel
open Unchecked_types.Unchecked_types (Coh)
let middle_associator k =
let ps = Builtin.ps_comp (2 * k) in
let src = Coh (Builtin.comp_n (2 * k), Unchecked.(identity_ps ps)) in
let tgt =
let sub_assoc_middle =
let rec compute_sub i =
match i with
| i when i <= 0 -> [ (Var (Db 0), false) ]
| i when i < k ->
(Var (Db (2 * i)), true)
:: (Var (Db ((2 * i) - 1)), false)
:: compute_sub (i - 1)
| i when i = k ->
let sub_comp =
[
(Var (Db ((2 * k) + 2)), true);
(Var (Db ((2 * k) + 1)), false);
(Var (Db (2 * k)), true);
(Var (Db ((2 * k) - 1)), false);
(Var (Db ((2 * k) - 3)), false);
]
in
let comp = Coh (Builtin.comp_n 2, sub_comp) in
(comp, true)
:: (Var (Db ((2 * k) + 1)), false)
:: compute_sub (k - 1)
| i ->
(Var (Db ((2 * i) + 2)), true)
:: (Var (Db ((2 * i) + 1)), false)
:: compute_sub (i - 1)
in
compute_sub ((2 * k) - 1)
in
Coh (Builtin.comp_n ((2 * k) - 1), sub_assoc_middle)
in
Coh.check_inv ps src tgt ("focus", 0, [])
let middle_unitor k =
let ps = Builtin.ps_comp (2 * k) in
let src =
let sub_id_middle =
let rec compute_sub i =
match i with
| i when i <= 0 -> [ (Var (Db 0), false) ]
| i when i <= k ->
(Var (Db (2 * i)), true)
:: (Var (Db ((2 * i) - 1)), false)
:: compute_sub (i - 1)
| i when i = k + 1 ->
let id = Coh (Builtin.id (), [ (Var (Db ((2 * k) - 1)), false) ]) in
(id, true) :: (Var (Db ((2 * k) - 1)), false) :: compute_sub k
| i ->
(Var (Db ((2 * i) - 2)), true)
:: (Var (Db ((2 * i) - 3)), false)
:: compute_sub (i - 1)
in
compute_sub ((2 * k) + 1)
in
Coh (Builtin.comp_n ((2 * k) + 1), sub_id_middle)
in
let tgt = Coh (Builtin.comp_n (2 * k), Unchecked.(identity_ps ps)) in
Coh.check_inv ps src tgt ("unit", 0, [])
let middle_rewrite k =
let comp = Builtin.comp_n ((2 * k) + 1) in
let func_data = [ Var.Db ((2 * k) + 2) ] in
Functorialisation.coh_depth0 comp func_data
let tdb i = Var (Db i)
let cell_max k = Var.Db (4 * k)
let cell_forward k = Var.Db ((4 * k) - 2)
let cell_backward k = Var.Db ((4 * k) - 1)
let obj k = if k <= 0 then Var.Db 0 else Var.Db ((4 * k) - 3)
let type_cell_forward k = Arr (Obj, Var (obj (k - 1)), Var (obj k))
let type_cell_backward k = Arr (Obj, Var (obj k), Var (obj (k - 1)))
let type_cell_max k =
Arr
( Arr (Obj, Var (obj (k - 1)), Var (obj (k - 1))),
Coh
( Builtin.comp_n 2,
[
(Var (cell_backward k), true);
(Var (obj (k - 1)), false);
(Var (cell_forward k), true);
(Var (obj k), false);
(Var (obj (k - 1)), false);
] ),
Coh (Builtin.id (), [ (Var (obj (k - 1)), true) ]) )
let rec ctx k =
match k with
| k when k < 0 -> Error.fatal "cannot build context for the telescope"
| k when k = 0 -> [ (Var.Db 0, (Obj, false)) ]
| k ->
(cell_max k, (type_cell_max k, true))
:: (cell_backward k, (type_cell_backward k, false))
:: (cell_forward k, (type_cell_forward k, false))
:: (obj k, (Obj, false))
:: ctx (k - 1)
let rec subs_telescope_bdry ?(whisk = false) k =
match k with
| k when k <= 0 -> Error.fatal "telescopes must have positive length"
| k when k = 1 ->
( [ (tdb 2, true); (tdb 1, false); (tdb 0, false) ],
[ (tdb 3, true); (tdb 0, false) ] )
| k ->
let right, left = subs_telescope_bdry ~whisk:false (k - 1) in
if whisk then
let src_max_var =
Coh
( Builtin.comp_n 2,
[
(Var (cell_backward k), true);
(Var (obj (k - 1)), false);
(Var (cell_forward k), true);
(Var (obj k), false);
(Var (obj (k - 1)), false);
] )
in
( right,
List.append left
[
(Var (cell_max k), true);
(Coh (Builtin.id (), [ (Var (obj (k - 1)), true) ]), false);
(src_max_var, false);
(Var (obj (k - 1)), false);
] )
else
( (Var (cell_forward k), true) :: (Var (obj k), false) :: right,
List.append left
[ (Var (cell_backward k), true); (Var (obj (k - 1)), false) ] )
let sub_ps_telescope_bdry ?(whisk = false) k =
let right, left = subs_telescope_bdry ~whisk k in
List.append left right
let rec telescope k =
match k with
| k when k <= 0 -> Error.fatal "telescopes must have positive length"
| k when k = 1 -> tdb 4
| k ->
let comp = Suspension.coh (Some 1) (Builtin.comp_n 4) in
let tm_src_tgt coh sub_ps =
match Coh.forget coh with
| _, Arr (_, t, u), _ ->
let sub = Unchecked.sub_ps_to_sub sub_ps in
( Coh (coh, sub_ps),
Unchecked.tm_apply_sub t sub,
Unchecked.tm_apply_sub u sub )
| _ -> Error.fatal "coherence must be of an arrow type"
in
let m3, src_m3, tgt_m3 =
tm_src_tgt (middle_unitor (k - 1)) (sub_ps_telescope_bdry (k - 1))
in
let m2 =
Coh (middle_rewrite (k - 1), sub_ps_telescope_bdry ~whisk:true k)
in
let m1, src_m1, tgt_m1 =
tm_src_tgt (middle_associator k) (sub_ps_telescope_bdry k)
in
let sub_telescope =
[
(telescope (k - 1), true);
(Coh (Builtin.id (), [ (tdb 0, true) ]), false);
(m3, true);
(tgt_m3, false);
(m2, true);
(src_m3, false);
(m1, true);
(tgt_m1, false);
(src_m1, false);
(tdb 0, false);
(tdb 0, false);
]
in
Coh (comp, sub_telescope)
let checked k = check_term (Ctx.check (ctx k)) (telescope k)