1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
|
(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(** Explicit substitutions *)
(** {6 Explicit substitutions } *)
(** Explicit substitutions of type ['a].
- ESID(n) = %n END bounded identity
- CONS([|t1..tn|],S) = (S.t1...tn) parallel substitution
(beware of the order: indice 1 is substituted by tn)
- SHIFT(n,S) = (^n o S) terms in S are relocated with n vars
- LIFT(n,S) = (%n S) stands for ((^n o S).n...1)
(corresponds to S crossing n binders) *)
type 'a subs = private
| ESID of int
| CONS of 'a array * 'a subs
| SHIFT of int * 'a subs
| LIFT of int * 'a subs
(** Derived constructors granting basic invariants *)
val subs_id : int -> 'a subs
val subs_cons: 'a array * 'a subs -> 'a subs
val subs_shft: int * 'a subs -> 'a subs
val subs_lift: 'a subs -> 'a subs
val subs_liftn: int -> 'a subs -> 'a subs
(** [subs_shift_cons(k,s,[|t1..tn|])] builds (^k s).t1..tn *)
val subs_shift_cons: int * 'a subs * 'a array -> 'a subs
(** [expand_rel k subs] expands de Bruijn [k] in the explicit substitution
[subs]. The result is either (Inl(lams,v)) when the variable is
substituted by value [v] under lams binders (i.e. v *has* to be
shifted by lams), or (Inr (k',p)) when the variable k is just relocated
as k'; p is None if the variable points inside subs and Some(k) if the
variable points k bindings beyond subs (cf argument of ESID).
*)
val expand_rel: int -> 'a subs -> (int * 'a, int * int option) Util.union
(** Tests whether a substitution behaves like the identity *)
val is_subs_id: 'a subs -> bool
(** Composition of substitutions: [comp mk_clos s1 s2] computes a
substitution equivalent to applying s2 then s1. Argument
mk_clos is used when a closure has to be created, i.e. when
s1 is applied on an element of s2.
*)
val comp : ('a subs * 'a -> 'a) -> 'a subs -> 'a subs -> 'a subs
(** {6 Compact representation } *)
(** Compact representation of explicit relocations
- [ELSHFT(l,n)] == lift of [n], then apply [lift l].
- [ELLFT(n,l)] == apply [l] to de Bruijn > [n] i.e under n binders.
Invariant ensured by the private flag: no lift contains two consecutive
[ELSHFT] nor two consecutive [ELLFT].
*)
type lift = private
| ELID
| ELSHFT of lift * int
| ELLFT of int * lift
val el_id : lift
val el_shft : int -> lift -> lift
val el_liftn : int -> lift -> lift
val el_lift : lift -> lift
val reloc_rel : int -> lift -> int
val is_lift_id : lift -> bool
(** Lift applied to substitution: [lift_subst mk_clos el s] computes a
substitution equivalent to applying el then s. Argument
mk_clos is used when a closure has to be created, i.e. when
el is applied on an element of s.
*)
val lift_subst : (lift -> 'a -> 'b) -> lift -> 'a subs -> 'b subs
|
