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
|
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(** Type of regular tree with nodes labelled by values of type 'a
The implementation uses de Bruijn indices, so binding capture
is avoided by the lift operator (see example below) *)
type 'a t
(** Building trees *)
(** build a node given a label and the vector of sons *)
val mk_node : 'a -> 'a t array -> 'a t
(** Build mutually recursive trees:
X_1 = f_1(X_1,..,X_n) ... X_n = f_n(X_1,..,X_n)
is obtained by the following pseudo-code
let vx = mk_rec_calls n in
let [|x_1;..;x_n|] =
mk_rec[|f_1(vx.(0),..,vx.(n-1);..;f_n(vx.(0),..,vx.(n-1))|]
First example: build rec X = a(X,Y) and Y = b(X,Y,Y)
let [|vx;vy|] = mk_rec_calls 2 in
let [|x;y|] = mk_rec [|mk_node a [|vx;vy|]; mk_node b [|vx;vy;vy|]|]
Another example: nested recursive trees rec Y = b(rec X = a(X,Y),Y,Y)
let [|vy|] = mk_rec_calls 1 in
let [|vx|] = mk_rec_calls 1 in
let [|x|] = mk_rec[|mk_node a vx;lift 1 vy|]
let [|y|] = mk_rec[|mk_node b x;vy;vy|]
(note the lift to avoid
*)
val mk_rec_calls : int -> 'a t array
val mk_rec : 'a t array -> 'a t array
(** [lift k t] increases of [k] the free parameters of [t]. Needed
to avoid captures when a tree appears under [mk_rec] *)
val lift : int -> 'a t -> 'a t
val is_node : 'a t -> bool
(** Destructors (recursive calls are expanded) *)
val dest_node : 'a t -> 'a * 'a t array
(** dest_param is not needed for closed trees (i.e. with no free variable) *)
val dest_param : 'a t -> int * int
(** Tells if a tree has an infinite branch. The first arg is a comparison
used to detect already seen elements, hence loops *)
val is_infinite : ('a -> 'a -> bool) -> 'a t -> bool
(** [Rtree.equiv eq eqlab t1 t2] compares t1 t2 (top-down).
If t1 and t2 are both nodes, [eqlab] is called on their labels,
in case of success deeper nodes are examined.
In case of loop (detected via structural equality parametrized
by [eq]), then the comparison is successful. *)
val equiv :
('a -> 'a -> bool) -> ('a -> 'a -> bool) -> 'a t -> 'a t -> bool
(** [Rtree.equal eq t1 t2] compares t1 and t2, first via physical
equality, then by structural equality (using [eq] on elements),
then by logical equivalence [Rtree.equiv eq eq] *)
val equal : ('a -> 'a -> bool) -> 'a t -> 'a t -> bool
(** Iterators *)
val map : ('a -> 'b) -> 'a t -> 'b t
(** [(smartmap f t) == t] if [(f a) ==a ] for all nodes *)
val smartmap : ('a -> 'a) -> 'a t -> 'a t
(** A rather simple minded pretty-printer *)
val pp_tree : ('a -> Pp.std_ppcmds) -> 'a t -> Pp.std_ppcmds
val eq_rtree : ('a -> 'a -> bool) -> 'a t -> 'a t -> bool
(** @deprecated Same as [Rtree.equal] *)
|
