File: esubst.ml

package info (click to toggle)
coq 8.9.0-1
  • links: PTS, VCS
  • area: main
  • in suites: bullseye, sid
  • size: 30,604 kB
  • sloc: ml: 192,230; sh: 2,585; python: 2,206; ansic: 1,878; makefile: 818; lisp: 202; xml: 24; sed: 2
file content (186 lines) | stat: -rw-r--r-- 6,859 bytes parent folder | download
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
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
(************************************************************************)
(*         *   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)         *)
(************************************************************************)

(* Created by Bruno Barras for Coq V7.0, Mar 2001 *)

(* Support for explicit substitutions *)

open Util

(*********************)
(*      Lifting      *)
(*********************)

(* Explicit lifts and basic operations *)
(* Invariant to preserve in this module: no lift contains two consecutive
    [ELSHFT] nor two consecutive [ELLFT]. *)
type lift =
  | ELID
  | ELSHFT of lift * int (* ELSHFT(l,n) == lift of n, then apply lift l *)
  | ELLFT of int * lift  (* ELLFT(n,l)  == apply l to de Bruijn > n *)
                         (*                 i.e under n binders *)

let el_id = ELID

(* compose a relocation of magnitude n *)
let el_shft_rec n = function
  | ELSHFT(el,k) -> ELSHFT(el,k+n)
  | el           -> ELSHFT(el,n)
let el_shft n el = if Int.equal n 0 then el else el_shft_rec n el

(* cross n binders *)
let el_liftn_rec n = function
  | ELID        -> ELID
  | ELLFT(k,el) -> ELLFT(n+k, el)
  | el          -> ELLFT(n, el)
let el_liftn n el = if Int.equal n 0 then el else el_liftn_rec n el

let el_lift el = el_liftn_rec 1 el

(* relocation of de Bruijn n in an explicit lift *)
let rec reloc_rel n = function
  | ELID -> n
  | ELLFT(k,el) ->
      if n <= k then n else (reloc_rel (n-k) el) + k
  | ELSHFT(el,k) -> (reloc_rel (n+k) el)

let rec is_lift_id = function
  | ELID -> true
  | ELSHFT(e,n) -> Int.equal n 0 && is_lift_id e
  | ELLFT (_,e) -> is_lift_id e

(*********************)
(*  Substitutions    *)
(*********************)

(* (bounded) explicit substitutions of type 'a *)
type 'a subs =
  | ESID of int            (* ESID(n)    = %n END   bounded identity *)
  | CONS of 'a array * 'a subs
                           (* CONS([|t1..tn|],S)  =
                              (S.t1...tn)    parallel substitution
                              beware of the order *)
  | SHIFT of int * 'a subs (* SHIFT(n,S) = (^n o S) terms in S are relocated *)
                           (*                        with n vars *)
  | LIFT of int * 'a subs  (* LIFT(n,S) = (%n S) stands for ((^n o S).n...1) *)

(* operations of subs: collapses constructors when possible.
 * Needn't be recursive if we always use these functions
 *)

let subs_id i = ESID i

let subs_cons(x,s) = if Int.equal (Array.length x) 0 then s else CONS(x,s)

let subs_liftn n = function
  | ESID p        -> ESID (p+n) (* bounded identity lifted extends by p *)
  | LIFT (p,lenv) -> LIFT (p+n, lenv)
  | lenv          -> LIFT (n,lenv)

let subs_lift a = subs_liftn 1 a
let subs_liftn n a = if Int.equal n 0 then a else subs_liftn n a

let subs_shft = function
  | (0, s)            -> s
  | (n, SHIFT (k,s1)) -> SHIFT (k+n, s1)
  | (n, s)            -> SHIFT (n,s)
let subs_shft s = if Int.equal (fst s) 0 then snd s else subs_shft s

let subs_shift_cons = function
  (0, s, t)           -> CONS(t,s)
| (k, SHIFT(n,s1), t) -> CONS(t,SHIFT(k+n, s1))
| (k, s, t)           -> CONS(t,SHIFT(k, s));;

(* Tests whether a substitution is equal to the identity *)
let rec is_subs_id = function
    ESID _     -> true
  | LIFT(_,s)  -> is_subs_id s
  | SHIFT(0,s) -> is_subs_id s
  | CONS(x,s)  -> Int.equal (Array.length x) 0 && is_subs_id s
  | _          -> false

(* Expands de Bruijn k in the explicit substitution subs
 * lams accumulates de shifts to perform when retrieving the i-th value
 * the rules used are the following:
 *
 *    [id]k       --> k
 *    [S.t]1      --> t
 *    [S.t]k      --> [S](k-1)  if k > 1
 *    [^n o S] k  --> [^n]([S]k)
 *    [(%n S)] k  --> k         if k <= n
 *    [(%n S)] k  --> [^n]([S](k-n))
 *
 * the result is (Inr (k+lams,p)) when the variable is just relocated
 * where p is None if the variable points inside subs and Some(k) if the
 * variable points k bindings beyond subs.
 *)
let rec exp_rel lams k subs =
  match subs with
    | CONS (def,_) when k <= Array.length def
                           -> Inl(lams,def.(Array.length def - k))
    | CONS (v,l)           -> exp_rel lams (k - Array.length v) l
    | LIFT (n,_) when k<=n -> Inr(lams+k,None)
    | LIFT (n,l)           -> exp_rel (n+lams) (k-n) l
    | SHIFT (n,s)          -> exp_rel (n+lams) k s
    | ESID n when k<=n     -> Inr(lams+k,None)
    | ESID n               -> Inr(lams+k,Some (k-n))

let expand_rel k subs = exp_rel 0 k subs

let rec subs_map f = function
| ESID _ as s -> s
| CONS (x, s) -> CONS (Array.map f x, subs_map f s)
| SHIFT (n, s) -> SHIFT (n, subs_map f s)
| LIFT (n, s) -> LIFT (n, subs_map f s)

let rec lift_subst mk_cl s1 s2 = match s1 with
| ELID -> subs_map (fun c -> mk_cl ELID c) s2
| ELSHFT(s, k) -> subs_shft(k, lift_subst mk_cl s s2)
| ELLFT (k, s) ->
  match s2 with
  | CONS(x,s') ->
      CONS(CArray.Fun1.map mk_cl s1 x, lift_subst mk_cl s1 s')
  | ESID n -> lift_subst mk_cl s (ESID (n + k))
  | SHIFT(k',s') ->
      if k<k'
      then subs_shft(k, lift_subst mk_cl s (subs_shft(k'-k, s')))
      else subs_shft(k', lift_subst mk_cl (el_liftn (k-k') s) s')
  | LIFT(k',s') ->
      if k<k'
      then subs_liftn k (lift_subst mk_cl s (subs_liftn (k'-k) s'))
      else subs_liftn k' (lift_subst mk_cl (el_liftn (k-k') s) s')

let rec comp mk_cl s1 s2 =
  match (s1, s2) with
    | _, ESID _ -> s1
    | ESID _, _ -> s2
    | SHIFT(k,s), _ -> subs_shft(k, comp mk_cl s s2)
    | _, CONS(x,s') ->
        CONS(Array.Fun1.map (fun s t -> mk_cl(s,t)) s1 x, comp mk_cl s1 s')
    | CONS(x,s), SHIFT(k,s') ->
        let lg = Array.length x in
        if k == lg then comp mk_cl s s'
        else if k > lg then comp mk_cl s (SHIFT(k-lg, s'))
        else comp mk_cl (CONS(Array.sub x 0 (lg-k), s)) s'
    | CONS(x,s), LIFT(k,s') ->
        let lg = Array.length x in
        if k == lg then CONS(x, comp mk_cl s s')
        else if k > lg then CONS(x, comp mk_cl s (LIFT(k-lg, s')))
        else
          CONS(Array.sub x (lg-k) k,
               comp mk_cl (CONS(Array.sub x 0 (lg-k),s)) s')
    | LIFT(k,s), SHIFT(k',s') ->
        if k<k'
        then subs_shft(k, comp mk_cl s (subs_shft(k'-k, s')))
        else subs_shft(k', comp mk_cl (subs_liftn (k-k') s) s')
    | LIFT(k,s), LIFT(k',s') ->
        if k<k'
        then subs_liftn k (comp mk_cl s (subs_liftn (k'-k) s'))
        else subs_liftn k' (comp mk_cl (subs_liftn (k-k') s) s')