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+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
+(*   \VV/  **************************************************************)
+(*    //   *      This file is distributed under the terms of the       *)
+(*         *       GNU Lesser General Public License Version 2.1        *)
+(************************************************************************)
+
+(* $Id: esubst.ml,v 1.4.2.1 2004/07/16 19:30:25 herbelin Exp $ *)
+
+open Util
+
+(*********************)
+(*      Lifting      *)
+(*********************)
+
+(* Explicit lifts and basic operations *)
+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 *)
+
+(* compose a relocation of magnitude n *)
+let rec el_shft_rec n = function
+  | ELSHFT(el,k) -> el_shft_rec (k+n) el
+  | el           -> ELSHFT(el,n)
+let el_shft n el = if n = 0 then el else el_shft_rec n el
+
+(* cross n binders *)
+let rec el_liftn_rec n = function
+  | ELID        -> ELID
+  | ELLFT(k,el) -> el_liftn_rec (n+k) el
+  | el          -> ELLFT(n, el)
+let el_liftn n el = if 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) -> 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 * 'a subs   (* CONS(t,S)  = (S.t)    parallel substitution *)
+  | 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_cons(x,s) = 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 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 (n,a) = if n = 0 then a else subs_shft(n,a)
+
+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 extensionnaly 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
+  | _          -> 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 (k,subs) with
+    | (1, CONS (def,_))         -> Inl(lams,def)
+    | (_, CONS (_,l))           -> exp_rel lams (pred k) 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 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(mk_cl(s1,x), comp mk_cl s1 s')
+    | CONS(x,s), SHIFT(k,s') -> comp mk_cl s (subs_shft(k-1, s'))
+    | CONS(x,s), LIFT(k,s') -> CONS(x,comp mk_cl s (subs_liftn (k-1) 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')