(************************************************************************)
   v       *    The Coq Proof Assistant   /   The Coq Development Team     
  < O _ _ _ , , *    INRIA - CNRS - LIX - LRI - PPS - Copyright 1999 - 2010     
    \VV/   * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
(************************************************************************)

  File initially created by   and   in 1984
  Extension to inductive constructions by   for Coq V5.6
  Extension to mutual inductive constructions by   for 
    Coq V5.10.2
   Coq V5.10.2 *)
  Extension to co - inductive constructions by  
  Optimization of substitution functions by  
  Optimization of lifting functions by   , Mar 1997
  Hash - consing by   in Feb 1998
  Restructuration of Coq of the type - checking kernel by      , 1999
   Filliâtre, 1999 *)
  Abstraction of the syntax of terms and iterators by   , 2000
  Cleaning and lightening of the kernel by   , Nov 2001

  This file defines the internal syntax of the Calculus of 
    Inductive Constructions ( CIC ) terms together with constructors , 
    destructors , iterators and basic functions
   Inductive Constructions (CIC) terms together with constructors,
   destructors, iterators and basic functions *)

open Util
open Pp
open Names
open Univ
open Esubst


type existential_key = int
type metavariable = int

(* This defines the strategy to use for verifiying a Cast *)

(* This defines Cases annotations *)
type case_style = LetStyle | IfStyle | LetPatternStyle | MatchStyle | RegularStyle
type case_printing =
  { ind_nargs : int; (* length of the arity of the inductive type *)
    style     : case_style }
type case_info =
  { ci_ind        : inductive;
    ci_npar       : int;
    ci_cstr_ndecls : int array; (* number of pattern vars of each constructor *)
    ci_pp_info    : case_printing (* not interpreted by the kernel *)
  }

(* Sorts. *)

type contents = Pos | Null

type sorts =
  | Prop of contents                      (* proposition types *)
  | Type of universe

let prop_sort = Prop Null
let set_sort = Prop Pos
let type1_sort = Type type1_univ

type sorts_family = InProp | InSet | InType

let family_of_sort = function
  | Prop Null -> InProp
  | Prop Pos -> InSet
  | Type _ -> InType

(********************************************************************)
(*       Constructions as implemented                               *)
(********************************************************************)

type cast_kind = VMcast | DEFAULTcast

  [ constr array ] is an instance matching definitional [ named_context ] in 
    the same order ( i.e. last argument first )
   the same order (i.e. last argument first) *)
type 'constr pexistential = existential_key * 'constr array
type ('constr, 'types) prec_declaration =
    name array * 'types array * 'constr array
type ('constr, 'types) pfixpoint =
    (int array * int) * ('constr, 'types) prec_declaration
type ('constr, 'types) pcofixpoint =
    int * ('constr, 'types) prec_declaration

  [  ] is used for named variables and [ Rel ] for variables as 
    de Bruijn indices .
   de Bruijn indices. *)
type ('constr, 'types) kind_of_term =
  | Rel       of int
  | Var       of identifier
  | Meta      of metavariable
  | Evar      of 'constr pexistential
  | Sort      of sorts
  | Cast      of 'constr * cast_kind * 'types
  | Prod      of name * 'types * 'types
  | Lambda    of name * 'types * 'constr
  | LetIn     of name * 'constr * 'types * 'constr
  | App       of 'constr * 'constr array
  | Const     of constant
  | Ind       of inductive
  | Construct of constructor
  | Case      of case_info * 'constr * 'constr * 'constr array
  | Fix       of ('constr, 'types) pfixpoint
  | CoFix     of ('constr, 'types) pcofixpoint

  Experimental , used in Presburger contrib
type ('constr, 'types) kind_of_type =
  | SortType   of sorts
  | CastType   of 'types * 'types
  | ProdType   of name * 'types * 'types
  | LetInType  of name * 'constr * 'types * 'types
  | AtomicType of 'constr * 'constr array

let kind_of_type = function
  | Sort s -> SortType s
  | Cast (c,_,t) -> CastType (c, t)
  | Prod (na,t,c) -> ProdType (na, t, c)
  | LetIn (na,b,t,c) -> LetInType (na, b, t, c)
  | App (c,l) -> AtomicType (c, l)
  | (Rel _ | Meta _ | Var _ | Evar _ | Const _ | Case _ | Fix _ | CoFix _ | Ind _ as c)
    -> AtomicType (c,[||])
  | (Lambda _ | Construct _) -> failwith "Not a type"

  constr is the fixpoint of the previous type . Requires option 
    -rectypes of the  compiler to be set
   -rectypes of the Caml compiler to be set *)
type constr = (constr,constr) kind_of_term

type existential = existential_key * constr array
type rec_declaration = name array * constr array * constr array
type fixpoint = (int array * int) * rec_declaration
type cofixpoint = int * rec_declaration

(***************************)
(* hash-consing functions  *)
(***************************)

let comp_term t1 t2 =
  match t1, t2 with
  | Rel n1, Rel n2 -> n1 = n2
  | Meta m1, Meta m2 -> m1 == m2
  | Var id1, Var id2 -> id1 == id2
  | Sort s1, Sort s2 -> s1 == s2
  | Cast (c1,_,t1), Cast (c2,_,t2) -> c1 == c2 & t1 == t2
  | Prod (n1,t1,c1), Prod (n2,t2,c2) -> n1 == n2 & t1 == t2 & c1 == c2
  | Lambda (n1,t1,c1), Lambda (n2,t2,c2) -> n1 == n2 & t1 == t2 & c1 == c2
  | LetIn (n1,b1,t1,c1), LetIn (n2,b2,t2,c2) ->
      n1 == n2 & b1 == b2 & t1 == t2 & c1 == c2
  | App (c1,l1), App (c2,l2) -> c1 == c2 & array_for_all2 (==) l1 l2
  | Evar (e1,l1), Evar (e2,l2) -> e1 = e2 & array_for_all2 (==) l1 l2
  | Const c1, Const c2 -> c1 == c2
  | Ind (sp1,i1), Ind (sp2,i2) -> sp1 == sp2 & i1 = i2
  | Construct ((sp1,i1),j1), Construct ((sp2,i2),j2) ->
      sp1 == sp2 & i1 = i2 & j1 = j2
  | Case (ci1,p1,c1,bl1), Case (ci2,p2,c2,bl2) ->
      ci1 == ci2 & p1 == p2 & c1 == c2 & array_for_all2 (==) bl1 bl2
  | Fix (ln1,(lna1,tl1,bl1)), Fix (ln2,(lna2,tl2,bl2)) ->
      ln1 = ln2
      & array_for_all2 (==) lna1 lna2
      & array_for_all2 (==) tl1 tl2
      & array_for_all2 (==) bl1 bl2
  | CoFix(ln1,(lna1,tl1,bl1)), CoFix(ln2,(lna2,tl2,bl2)) ->
      ln1 = ln2
      & array_for_all2 (==) lna1 lna2
      & array_for_all2 (==) tl1 tl2
      & array_for_all2 (==) bl1 bl2
  | _ -> false

let hash_term (sh_rec,(sh_sort,sh_con,sh_kn,sh_na,sh_id)) t =
  match t with
  | Rel _ -> t
  | Meta x -> Meta x
  | Var x -> Var (sh_id x)
  | Sort s -> Sort (sh_sort s)
  | Cast (c, k, t) -> Cast (sh_rec c, k, (sh_rec t))
  | Prod (na,t,c) -> Prod (sh_na na, sh_rec t, sh_rec c)
  | Lambda (na,t,c) -> Lambda (sh_na na, sh_rec t, sh_rec c)
  | LetIn (na,b,t,c) -> LetIn (sh_na na, sh_rec b, sh_rec t, sh_rec c)
  | App (c,l) -> App (sh_rec c, Array.map sh_rec l)
  | Evar (e,l) -> Evar (e, Array.map sh_rec l)
  | Const c -> Const (sh_con c)
  | Ind (kn,i) -> Ind (sh_kn kn,i)
  | Construct ((kn,i),j) -> Construct ((sh_kn kn,i),j)
  TO DO : extract ind_kn
      Case (ci, sh_rec p, sh_rec c, Array.map sh_rec bl)
  | Fix (ln,(lna,tl,bl)) ->
      Fix (ln,(Array.map sh_na lna,
                 Array.map sh_rec tl,
                 Array.map sh_rec bl))
  | CoFix(ln,(lna,tl,bl)) ->
      CoFix (ln,(Array.map sh_na lna,
                   Array.map sh_rec tl,
                   Array.map sh_rec bl))

module Hconstr =
  Hashcons.Make(
    struct
      type t = constr
      type u = (constr -> constr) *
               ((sorts -> sorts) * (constant -> constant) *
               (mutual_inductive -> mutual_inductive) * (name -> name) *
               (identifier -> identifier))
      let hash_sub = hash_term
      let equal = comp_term
      let hash = Hashtbl.hash
    end)

let hcons_term (hsorts,hcon,hkn,hname,hident) =
  Hashcons.recursive_hcons Hconstr.f (hsorts,hcon,hkn,hname,hident)

  Constructs a  index with number n
let rels =
  [|Rel  1;Rel  2;Rel  3;Rel  4;Rel  5;Rel  6;Rel  7; Rel  8;
    Rel  9;Rel 10;Rel 11;Rel 12;Rel 13;Rel 14;Rel 15; Rel 16|]

let mkRel n = if 0<n & n<=16 then rels.(n-1) else Rel n

(* Constructs an existential variable named "?n" *)
let mkMeta  n =  Meta n

(* Constructs a Variable named id *)
let mkVar id = Var id

  Construct a type
let mkSort s = Sort s

(* Constructs the term t1::t2, i.e. the term t1 casted with the type t2 *)
(* (that means t2 is declared as the type of t1)
   [s] is the strategy to use when *)
let mkCast (t1,k2,t2) =
  match t1 with
  | Cast (c,k1, _) when k1 = k2 -> Cast (c,k1,t2)
  | _ -> Cast (t1,k2,t2)

(* Constructs the product (x:t1)t2 *)
let mkProd (x,t1,t2) = Prod (x,t1,t2)

(* Constructs the abstraction [x:t1]t2 *)
let mkLambda (x,t1,t2) = Lambda (x,t1,t2)

(* Constructs [x=c_1:t]c_2 *)
let mkLetIn (x,c1,t,c2) = LetIn (x,c1,t,c2)

(* If lt = [t1; ...; tn], constructs the application (t1 ... tn) *)
  We ensure applicative terms have at least one argument and the 
    function is not itself an applicative term
   function is not itself an applicative term *)
let mkApp (f, a) =
  if Array.length a = 0 then f else
    match f with
      | App (g, cl) -> App (g, Array.append cl a)
      | _ -> App (f, a)


  Constructs a constant
(* The array of terms correspond to the variables introduced in the section *)
let mkConst c = Const c

(* Constructs an existential variable *)
let mkEvar e = Evar e

(* Constructs the ith (co)inductive type of the block named kn *)
(* The array of terms correspond to the variables introduced in the section *)
let mkInd m = Ind m

  Constructs the jth constructor of the ith ( co)inductive type of the 
    block named kn . The array of terms correspond to the variables 
    introduced in the section
   block named kn. The array of terms correspond to the variables
   introduced in the section *)
let mkConstruct c = Construct c

  Constructs the term < p > Case c of c1 | c2 .. | cn end
let mkCase (ci, p, c, ac) = Case (ci, p, c, ac)

let mkFix fix = Fix fix

let mkCoFix cofix = CoFix cofix

let kind_of_term c = c

(************************************************************************)
(*    kind_of_term = constructions as seen by the user                 *)
(************************************************************************)

  User view of [ constr ] . For [ App ] , it is ensured there is at 
    least one argument and the function is not itself an applicative 
    term
   least one argument and the function is not itself an applicative
   term *)

let kind_of_term = kind_of_term

(**********************************************************************)
(*          Non primitive term destructors                            *)
(**********************************************************************)

(* Destructor operations : partial functions
   Raise invalid_arg "dest*" if the const has not the expected form *)

  Destructs a  index
let destRel c = match kind_of_term c with
  | Rel n -> n
  | _ -> invalid_arg "destRel"

(* Destructs an existential variable *)
let destMeta c = match kind_of_term c with
  | Meta n -> n
  | _ -> invalid_arg "destMeta"

let isMeta c = match kind_of_term c with Meta _ -> true | _ -> false

(* Destructs a variable *)
let destVar c = match kind_of_term c with
  | Var id -> id
  | _ -> invalid_arg "destVar"

  Destructs a type
let isSort c = match kind_of_term c with
  | Sort s -> true
  | _ -> false

let destSort c = match kind_of_term c with
  | Sort s -> s
  | _ -> invalid_arg "destSort"

let rec isprop c = match kind_of_term c with
  | Sort (Prop _) -> true
  | Cast (c,_,_) -> isprop c
  | _ -> false

let rec is_Prop c = match kind_of_term c with
  | Sort (Prop Null) -> true
  | Cast (c,_,_) -> is_Prop c
  | _ -> false

let rec is_Set c = match kind_of_term c with
  | Sort (Prop Pos) -> true
  | Cast (c,_,_) -> is_Set c
  | _ -> false

let rec is_Type c = match kind_of_term c with
  | Sort (Type _) -> true
  | Cast (c,_,_) -> is_Type c
  | _ -> false

let is_small = function
  | Prop _ -> true
  | _ -> false

let iskind c = isprop c or is_Type c

(* Tests if an evar *)
let isEvar c = match kind_of_term c with Evar _ -> true | _ -> false

let isEvar_or_Meta c = match kind_of_term c with
  | Evar _ | Meta _ -> true
  | _ -> false

  Destructs a casted term
let destCast c = match kind_of_term c with
  | Cast (t1,k,t2) -> (t1,k,t2)
  | _ -> invalid_arg "destCast"

let isCast c = match kind_of_term c with Cast _ -> true | _ -> false


(* Tests if a de Bruijn index *)
let isRel c = match kind_of_term c with Rel _ -> true | _ -> false

(* Tests if a variable *)
let isVar c = match kind_of_term c with Var _ -> true | _ -> false

(* Tests if an inductive *)
let isInd c = match kind_of_term c with Ind _ -> true | _ -> false

(* Destructs the product (x:t1)t2 *)
let destProd c = match kind_of_term c with
  | Prod (x,t1,t2) -> (x,t1,t2)
  | _ -> invalid_arg "destProd"

let isProd c = match kind_of_term c with | Prod _ -> true | _ -> false

(* Destructs the abstraction [x:t1]t2 *)
let destLambda c = match kind_of_term c with
  | Lambda (x,t1,t2) -> (x,t1,t2)
  | _ -> invalid_arg "destLambda"

let isLambda c = match kind_of_term c with | Lambda _ -> true | _ -> false

(* Destructs the let [x:=b:t1]t2 *)
let destLetIn c = match kind_of_term c with
  | LetIn (x,b,t1,t2) -> (x,b,t1,t2)
  | _ -> invalid_arg "destProd"

let isLetIn c =  match kind_of_term c with LetIn _ -> true | _ -> false

  Destructs an application
let destApp c = match kind_of_term c with
  | App (f,a) -> (f, a)
  | _ -> invalid_arg "destApplication"

let destApplication = destApp

let isApp c = match kind_of_term c with App _ -> true | _ -> false

(* Destructs a constant *)
let destConst c = match kind_of_term c with
  | Const kn -> kn
  | _ -> invalid_arg "destConst"

let isConst c = match kind_of_term c with Const _ -> true | _ -> false

(* Destructs an existential variable *)
let destEvar c = match kind_of_term c with
  | Evar (kn, a as r) -> r
  | _ -> invalid_arg "destEvar"

(* Destructs a (co)inductive type named kn *)
let destInd c = match kind_of_term c with
  | Ind (kn, a as r) -> r
  | _ -> invalid_arg "destInd"

  Destructs a constructor
let destConstruct c = match kind_of_term c with
  | Construct (kn, a as r) -> r
  | _ -> invalid_arg "dest"

let isConstruct c = match kind_of_term c with Construct _ -> true | _ -> false

  Destructs a term < p > Case c of lc1 | lc2 .. | lcn end
let destCase c = match kind_of_term c with
  | Case (ci,p,c,v) -> (ci,p,c,v)
  | _ -> anomaly "destCase"

let isCase c =  match kind_of_term c with Case _ -> true | _ -> false

let destFix c = match kind_of_term c with
  | Fix fix -> fix
  | _ -> invalid_arg "destFix"

let isFix c =  match kind_of_term c with Fix _ -> true | _ -> false

let destCoFix c = match kind_of_term c with
  | CoFix cofix -> cofix
  | _ -> invalid_arg "destCoFix"

let isCoFix c =  match kind_of_term c with CoFix _ -> true | _ -> false

(******************************************************************)
(* Cast management                                                *)
(******************************************************************)

let rec strip_outer_cast c = match kind_of_term c with
  | Cast (c,_,_) -> strip_outer_cast c
  | _ -> c

  Fonction spéciale qui laisse les cast       Case

let under_outer_cast f c =  match kind_of_term c with
  | Cast (b,k,t) -> mkCast (f b, k, f t)
  | _ -> f c

let rec under_casts f c = match kind_of_term c with
  | Cast (c,k,t) -> mkCast (under_casts f c, k, t)
  | _            -> f c

(******************************************************************)
(* Flattening and unflattening of embedded applications and casts *)
(******************************************************************)

(* flattens application lists throwing casts in-between *)
let rec collapse_appl c = match kind_of_term c with
  | App (f,cl) ->
      let rec collapse_rec f cl2 =
        match kind_of_term (strip_outer_cast f) with
	| App (g,cl1) -> collapse_rec g (Array.append cl1 cl2)
	| _ -> mkApp (f,cl2)
      in
      collapse_rec f cl
  | _ -> c

let decompose_app c =
  match kind_of_term c with
    | App (f,cl) -> (f, Array.to_list cl)
    | _ -> (c,[])

(****************************************************************************)
(*              Functions to recur through subterms                         *)
(****************************************************************************)

  [ fold_constr f acc c ] folds [ f ] on the immediate subterms of [ c ] 
    starting from [ acc ] and proceeding from left to right according to 
    the usual representation of the constructions ; it is not recursive
   starting from [acc] and proceeding from left to right according to
   the usual representation of the constructions; it is not recursive *)

let fold_constr f acc c = match kind_of_term c with
  | (Rel _ | Meta _ | Var _   | Sort _ | Const _ | Ind _
    | Construct _) -> acc
  | Cast (c,_,t) -> f (f acc c) t
  | Prod (_,t,c) -> f (f acc t) c
  | Lambda (_,t,c) -> f (f acc t) c
  | LetIn (_,b,t,c) -> f (f (f acc b) t) c
  | App (c,l) -> Array.fold_left f (f acc c) l
  | Evar (_,l) -> Array.fold_left f acc l
  | Case (_,p,c,bl) -> Array.fold_left f (f (f acc p) c) bl
  | Fix (_,(lna,tl,bl)) ->
      let fd = array_map3 (fun na t b -> (na,t,b)) lna tl bl in
      Array.fold_left (fun acc (na,t,b) -> f (f acc t) b) acc fd
  | CoFix (_,(lna,tl,bl)) ->
      let fd = array_map3 (fun na t b -> (na,t,b)) lna tl bl in
      Array.fold_left (fun acc (na,t,b) -> f (f acc t) b) acc fd

(* [iter_constr f c] iters [f] on the immediate subterms of [c]; it is
   not recursive and the order with which subterms are processed is
   not specified *)

let iter_constr f c = match kind_of_term c with
  | (Rel _ | Meta _ | Var _   | Sort _ | Const _ | Ind _
    | Construct _) -> ()
  | Cast (c,_,t) -> f c; f t
  | Prod (_,t,c) -> f t; f c
  | Lambda (_,t,c) -> f t; f c
  | LetIn (_,b,t,c) -> f b; f t; f c
  | App (c,l) -> f c; Array.iter f l
  | Evar (_,l) -> Array.iter f l
  | Case (_,p,c,bl) -> f p; f c; Array.iter f bl
  | Fix (_,(_,tl,bl)) -> Array.iter f tl; Array.iter f bl
  | CoFix (_,(_,tl,bl)) -> Array.iter f tl; Array.iter f bl

  [ iter_constr_with_binders g f n c ] iters [ f n ] on the immediate 
    subterms of [ c ] ; it carries an extra data [ n ] ( typically a lift 
    index ) which is processed by [ g ] ( which typically add 1 to [ n ] ) at 
    each binder traversal ; it is not recursive and the order with which 
    subterms are processed is not specified
   subterms of [c]; it carries an extra data [n] (typically a lift
   index) which is processed by [g] (which typically add 1 to [n]) at
   each binder traversal; it is not recursive and the order with which
   subterms are processed is not specified *)

let iter_constr_with_binders g f n c = match kind_of_term c with
  | (Rel _ | Meta _ | Var _   | Sort _ | Const _ | Ind _
    | Construct _) -> ()
  | Cast (c,_,t) -> f n c; f n t
  | Prod (_,t,c) -> f n t; f (g n) c
  | Lambda (_,t,c) -> f n t; f (g n) c
  | LetIn (_,b,t,c) -> f n b; f n t; f (g n) c
  | App (c,l) -> f n c; Array.iter (f n) l
  | Evar (_,l) -> Array.iter (f n) l
  | Case (_,p,c,bl) -> f n p; f n c; Array.iter (f n) bl
  | Fix (_,(_,tl,bl)) ->
      Array.iter (f n) tl;
      Array.iter (f (iterate g (Array.length tl) n)) bl
  | CoFix (_,(_,tl,bl)) ->
      Array.iter (f n) tl;
      Array.iter (f (iterate g (Array.length tl) n)) bl

(* [map_constr f c] maps [f] on the immediate subterms of [c]; it is
   not recursive and the order with which subterms are processed is
   not specified *)

let map_constr f c = match kind_of_term c with
  | (Rel _ | Meta _ | Var _   | Sort _ | Const _ | Ind _
    | Construct _) -> c
  | Cast (c,k,t) -> mkCast (f c, k, f t)
  | Prod (na,t,c) -> mkProd (na, f t, f c)
  | Lambda (na,t,c) -> mkLambda (na, f t, f c)
  | LetIn (na,b,t,c) -> mkLetIn (na, f b, f t, f c)
  | App (c,l) -> mkApp (f c, Array.map f l)
  | Evar (e,l) -> mkEvar (e, Array.map f l)
  | Case (ci,p,c,bl) -> mkCase (ci, f p, f c, Array.map f bl)
  | Fix (ln,(lna,tl,bl)) ->
      mkFix (ln,(lna,Array.map f tl,Array.map f bl))
  | CoFix(ln,(lna,tl,bl)) ->
      mkCoFix (ln,(lna,Array.map f tl,Array.map f bl))

  [ map_constr_with_binders g f n c ] maps [ f n ] on the immediate 
    subterms of [ c ] ; it carries an extra data [ n ] ( typically a lift 
    index ) which is processed by [ g ] ( which typically add 1 to [ n ] ) at 
    each binder traversal ; it is not recursive and the order with which 
    subterms are processed is not specified
   subterms of [c]; it carries an extra data [n] (typically a lift
   index) which is processed by [g] (which typically add 1 to [n]) at
   each binder traversal; it is not recursive and the order with which
   subterms are processed is not specified *)

let map_constr_with_binders g f l c = match kind_of_term c with
  | (Rel _ | Meta _ | Var _   | Sort _ | Const _ | Ind _
    | Construct _) -> c
  | Cast (c,k,t) -> mkCast (f l c, k, f l t)
  | Prod (na,t,c) -> mkProd (na, f l t, f (g l) c)
  | Lambda (na,t,c) -> mkLambda (na, f l t, f (g l) c)
  | LetIn (na,b,t,c) -> mkLetIn (na, f l b, f l t, f (g l) c)
  | App (c,al) -> mkApp (f l c, Array.map (f l) al)
  | Evar (e,al) -> mkEvar (e, Array.map (f l) al)
  | Case (ci,p,c,bl) -> mkCase (ci, f l p, f l c, Array.map (f l) bl)
  | Fix (ln,(lna,tl,bl)) ->
      let l' = iterate g (Array.length tl) l in
      mkFix (ln,(lna,Array.map (f l) tl,Array.map (f l') bl))
  | CoFix(ln,(lna,tl,bl)) ->
      let l' = iterate g (Array.length tl) l in
      mkCoFix (ln,(lna,Array.map (f l) tl,Array.map (f l') bl))

  [ compare_constr f c1 c2 ] compare [ c1 ] and [ c2 ] using [ f ] to compare 
    the immediate subterms of [ c1 ] of [ c2 ] if needed ; Cast 's , 
    application associativity , binders name and Cases annotations are 
    not taken into account
   the immediate subterms of [c1] of [c2] if needed; Cast's,
   application associativity, binders name and Cases annotations are
   not taken into account *)


let compare_constr f t1 t2 =
  match kind_of_term t1, kind_of_term t2 with
  | Rel n1, Rel n2 -> n1 = n2
  | Meta m1, Meta m2 -> m1 = m2
  | Var id1, Var id2 -> id1 = id2
  | Sort s1, Sort s2 -> s1 = s2
  | Cast (c1,_,_), _ -> f c1 t2
  | _, Cast (c2,_,_) -> f t1 c2
  | Prod (_,t1,c1), Prod (_,t2,c2) -> f t1 t2 & f c1 c2
  | Lambda (_,t1,c1), Lambda (_,t2,c2) -> f t1 t2 & f c1 c2
  | LetIn (_,b1,t1,c1), LetIn (_,b2,t2,c2) -> f b1 b2 & f t1 t2 & f c1 c2
  | App (c1,l1), App (c2,l2) ->
      if Array.length l1 = Array.length l2 then
        f c1 c2 & array_for_all2 f l1 l2
      else
        let (h1,l1) = decompose_app t1 in
        let (h2,l2) = decompose_app t2 in
        if List.length l1 = List.length l2 then
          f h1 h2 & List.for_all2 f l1 l2
        else false
  | Evar (e1,l1), Evar (e2,l2) -> e1 = e2 & array_for_all2 f l1 l2
  | Const c1, Const c2 -> eq_constant c1 c2
  | Ind c1, Ind c2 -> eq_ind c1 c2
  | Construct c1, Construct c2 -> eq_constructor c1 c2
  | Case (_,p1,c1,bl1), Case (_,p2,c2,bl2) ->
      f p1 p2 & f c1 c2 & array_for_all2 f bl1 bl2
  | Fix (ln1,(_,tl1,bl1)), Fix (ln2,(_,tl2,bl2)) ->
      ln1 = ln2 & array_for_all2 f tl1 tl2 & array_for_all2 f bl1 bl2
  | CoFix(ln1,(_,tl1,bl1)), CoFix(ln2,(_,tl2,bl2)) ->
      ln1 = ln2 & array_for_all2 f tl1 tl2 & array_for_all2 f bl1 bl2
  | _ -> false

(***************************************************************************)
(*     Type of assumptions                                                 *)
(***************************************************************************)

type types = constr

type strategy = types option

type named_declaration = identifier * constr option * types
type rel_declaration = name * constr option * types

let map_named_declaration f (id, v, ty) = (id, Option.map f v, f ty)
let map_rel_declaration = map_named_declaration

let fold_named_declaration f (_, v, ty) a = f ty (Option.fold_right f v a)
let fold_rel_declaration = fold_named_declaration

(***************************************************************************)
(*     Type of local contexts (telescopes)                                 *)
(***************************************************************************)

s Signatures of ordered optionally named variables , intended to be 
    accessed by   indices ( to represent bound variables )
   accessed by de Bruijn indices (to represent bound variables) *)

type rel_context = rel_declaration list

let empty_rel_context = []

let add_rel_decl d ctxt = d::ctxt

let rec lookup_rel n sign =
  match n, sign with
  | 1, decl :: _ -> decl
  | n, _ :: sign -> lookup_rel (n-1) sign
  | _, []        -> raise Not_found

let rel_context_length = List.length

let rel_context_nhyps hyps =
  let rec nhyps acc = function
    | [] -> acc
    | (_,None,_)::hyps -> nhyps (1+acc) hyps
    | (_,Some _,_)::hyps -> nhyps acc hyps in
  nhyps 0 hyps

(****************************************************************************)
(*              Functions for dealing with constr terms                     *)
(****************************************************************************)

(*********************)
(*     Occurring     *)
(*********************)

exception LocalOccur

  ( closedn n M ) raises FreeVar if a variable of height greater than n 
    occurs in M , returns ( ) otherwise
   occurs in M, returns () otherwise *)

let closedn n c =
  let rec closed_rec n c = match kind_of_term c with
    | Rel m -> if m>n then raise LocalOccur
    | _ -> iter_constr_with_binders succ closed_rec n c
  in
  try closed_rec n c; true with LocalOccur -> false

  [ closed0 M ] is true iff [ M ] is a ( deBruijn ) closed term

let closed0 = closedn 0

  ( noccurn n M ) returns true iff ( Rel n ) does NOT occur in term M  

let noccurn n term =
  let rec occur_rec n c = match kind_of_term c with
    | Rel m -> if m = n then raise LocalOccur
    | _ -> iter_constr_with_binders succ occur_rec n c
  in
  try occur_rec n term; true with LocalOccur -> false

  ( noccur_between n m M ) returns true iff ( Rel p ) does NOT occur in term M 
   for n < = p < n+m
  for n <= p < n+m *)

let noccur_between n m term =
  let rec occur_rec n c = match kind_of_term c with
    | Rel(p) -> if n<=p && p<n+m then raise LocalOccur
    | _        -> iter_constr_with_binders succ occur_rec n c
  in
  try occur_rec n term; true with LocalOccur -> false

  Checking function for terms containing existential variables . 
  The function [ noccur_with_meta ] considers the fact that 
  each existential variable ( as well as each isevar ) 
  in the term appears applied to its local context , 
  which may contain the CoFix variables . These occurrences of CoFix variables 
  are not considered
 The function [noccur_with_meta] considers the fact that
 each existential variable (as well as each isevar)
 in the term appears applied to its local context,
 which may contain the CoFix variables. These occurrences of CoFix variables
 are not considered *)

let noccur_with_meta n m term =
  let rec occur_rec n c = match kind_of_term c with
    | Rel p -> if n<=p & p<n+m then raise LocalOccur
    | App(f,cl) ->
	(match kind_of_term f with
           | Cast (c,_,_) when isMeta c -> ()
           | Meta _ -> ()
	   | _ -> iter_constr_with_binders succ occur_rec n c)
    | Evar (_, _) -> ()
    | _ -> iter_constr_with_binders succ occur_rec n c
  in
  try (occur_rec n term; true) with LocalOccur -> false


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

(* The generic lifting function *)
let rec exliftn el c = match kind_of_term c with
  | Rel i -> mkRel(reloc_rel i el)
  | _ -> map_constr_with_binders el_lift exliftn el c

(* Lifting the binding depth across k bindings *)

let liftn k n =
  match el_liftn (pred n) (el_shft k ELID) with
    | ELID -> (fun c -> c)
    | el -> exliftn el

let lift k = liftn k 1

(*********************)
(*   Substituting    *)
(*********************)

  ( subst1 M c ) substitutes M for Rel(1 ) in c 
    we generalise it to ( substl [ M1, ... ,Mn ] c ) which substitutes in parallel 
    M1, ... ,Mn for respectively Rel(1), ... ,Rel(n ) in c
   we generalise it to (substl [M1,...,Mn] c) which substitutes in parallel
   M1,...,Mn for respectively Rel(1),...,Rel(n) in c *)

(* 1st : general case *)

type info = Closed | Open | Unknown
type 'a substituend = { mutable sinfo: info; sit: 'a }

let rec lift_substituend depth s =
  match s.sinfo with
    | Closed -> s.sit
    | Open -> lift depth s.sit
    | Unknown ->
        s.sinfo <- if closed0 s.sit then Closed else Open;
        lift_substituend depth s

let make_substituend c = { sinfo=Unknown; sit=c }

let substn_many lamv n c =
  let lv = Array.length lamv in
  if lv = 0 then c
  else
    let rec substrec depth c = match kind_of_term c with
      | Rel k     ->
          if k<=depth then c
          else if k-depth <= lv then lift_substituend depth lamv.(k-depth-1)
          else mkRel (k-lv)
      | _ -> map_constr_with_binders succ substrec depth c in
    substrec n c

(*
let substkey = Profile.declare_profile "substn_many";;
let substn_many lamv n c = Profile.profile3 substkey substn_many lamv n c;;
*)

let substnl laml n =
  substn_many (Array.map make_substituend (Array.of_list laml)) n
let substl laml = substnl laml 0
let subst1 lam = substl [lam]

let substnl_decl laml k (id,bodyopt,typ) =
  (id,Option.map (substnl laml k) bodyopt,substnl laml k typ)
let substl_decl laml = substnl_decl laml 0
let subst1_decl lam = substl_decl [lam]
let subst1_named_decl = subst1_decl
let substl_named_decl = substl_decl

(* (thin_val sigma) removes identity substitutions from sigma *)

let rec thin_val = function
  | [] -> []
  | (((id,{ sit = v }) as s)::tl) when isVar v ->
      if id = destVar v then thin_val tl else s::(thin_val tl)
  | h::tl -> h::(thin_val tl)

(* (replace_vars sigma M) applies substitution sigma to term M *)
let replace_vars var_alist =
  let var_alist =
    List.map (fun (str,c) -> (str,make_substituend c)) var_alist in
  let var_alist = thin_val var_alist in
  let rec substrec n c = match kind_of_term c with
    | Var x ->
        (try lift_substituend n (List.assoc x var_alist)
         with Not_found -> c)
    | _ -> map_constr_with_binders succ substrec n c
  in
  if var_alist = [] then (function x -> x) else substrec 0

(*
let repvarkey = Profile.declare_profile "replace_vars";;
let replace_vars vl c = Profile.profile2 repvarkey replace_vars vl c ;;
*)

  ( subst_var str t ) substitute (   ) by ( Rel 1 ) in t
let subst_var str = replace_vars [(str, mkRel 1)]

  ( subst_vars [ id1; ... ;idn ] t ) substitute ( VAR idj ) by ( Rel j ) in t
let substn_vars p vars =
  let _,subst =
    List.fold_left (fun (n,l) var -> ((n+1),(var,mkRel n)::l)) (p,[]) vars
  in replace_vars (List.rev subst)

let subst_vars = substn_vars 1

(*********************)
(* Term constructors *)
(*********************)

  Constructs a  index with number n
let mkRel = mkRel

(* Constructs an existential variable named "?n" *)
let mkMeta = mkMeta

(* Constructs a Variable named id *)
let mkVar = mkVar

  Construct a type
let mkProp   = mkSort prop_sort
let mkSet    = mkSort set_sort
let mkType u = mkSort (Type u)
let mkSort   = function
  | Prop Null -> mkProp (* Easy sharing *)
  | Prop Pos -> mkSet
  | s -> mkSort s

(* Constructs the term t1::t2, i.e. the term t1 casted with the type t2 *)
(* (that means t2 is declared as the type of t1) *)
let mkCast = mkCast

(* Constructs the product (x:t1)t2 *)
let mkProd = mkProd
let mkNamedProd id typ c = mkProd (Name id, typ, subst_var id c)

(* Constructs the abstraction [x:t1]t2 *)
let mkLambda = mkLambda
let mkNamedLambda id typ c = mkLambda (Name id, typ, subst_var id c)

(* Constructs [x=c_1:t]c_2 *)
let mkLetIn = mkLetIn
let mkNamedLetIn id c1 t c2 = mkLetIn (Name id, c1, t, subst_var id c2)

  Constructs either [ ( x : t)c ] or [ [ x = b : t]c ]
let mkProd_or_LetIn (na,body,t) c =
  match body with
    | None -> mkProd (na, t, c)
    | Some b -> mkLetIn (na, b, t, c)

let mkNamedProd_or_LetIn (id,body,t) c =
  match body with
    | None -> mkNamedProd id t c
    | Some b -> mkNamedLetIn id b t c

  Constructs either [ [ x : t]c ] or [ [ x = b : t]c ]
let mkLambda_or_LetIn (na,body,t) c =
  match body with
    | None -> mkLambda (na, t, c)
    | Some b -> mkLetIn (na, b, t, c)

let mkNamedLambda_or_LetIn (id,body,t) c =
  match body with
    | None -> mkNamedLambda id t c
    | Some b -> mkNamedLetIn id b t c

  Constructs either [ ( x : t)c ] or [ c ] where [ x ] is replaced by [ b ]
let mkProd_wo_LetIn (na,body,t) c =
  match body with
    | None -> mkProd (na,  t, c)
    | Some b -> subst1 b c

let mkNamedProd_wo_LetIn (id,body,t) c =
  match body with
    | None -> mkNamedProd id t c
    | Some b -> subst1 b (subst_var id c)

(* non-dependent product t1 -> t2 *)
let mkArrow t1 t2 = mkProd (Anonymous, t1, t2)

(* If lt = [t1; ...; tn], constructs the application (t1 ... tn) *)
  We ensure applicative terms have at most one arguments and the 
    function is not itself an applicative term
   function is not itself an applicative term *)
let mkApp = mkApp

  Constructs a constant
(* The array of terms correspond to the variables introduced in the section *)
let mkConst = mkConst

(* Constructs an existential variable *)
let mkEvar = mkEvar

(* Constructs the ith (co)inductive type of the block named kn *)
(* The array of terms correspond to the variables introduced in the section *)
let mkInd = mkInd

  Constructs the jth constructor of the ith ( co)inductive type of the 
    block named kn . The array of terms correspond to the variables 
    introduced in the section
   block named kn. The array of terms correspond to the variables
   introduced in the section *)
let mkConstruct = mkConstruct

  Constructs the term < p > Case c of c1 | c2 .. | cn end
let mkCase = mkCase

  If recindxs = [ |i1, ... in| ] 
       funnames = [ |f1, ... fn| ] 
       typarray = [ |t1, ...  ] 
       bodies    = [ |b1, ... bn| ] 
    then 

        ( ( recindxs , i),(funnames , typarray , bodies ) ) 

    constructs the ith function of the block 

     Fixpoint f1 [ ctx1 ] : t1 : = b1 
     with      f2 [ ctx2 ] : t2 : = b2 
     ... 
     with      fn [ ctxn ] : tn : = bn . 

    where the lenght of the jth context is ij . 

      funnames = [|f1,...fn|]
      typarray = [|t1,...tn|]
      bodies   = [|b1,...bn|]
   then

      mkFix ((recindxs,i),(funnames,typarray,bodies))

   constructs the ith function of the block

    Fixpoint f1 [ctx1] : t1 := b1
    with     f2 [ctx2] : t2 := b2
    ...
    with     fn [ctxn] : tn := bn.

   where the lenght of the jth context is ij.
*)

let mkFix = mkFix

  If funnames = [ |f1, ... fn| ] 
       typarray = [ |t1, ...  ] 
       bodies    = [ |b1, ... bn| ] 
    then 

       mkCoFix ( i,(funnames , typsarray , bodies ) ) 

    constructs the ith function of the block 

     CoFixpoint f1 : t1 : = b1 
     with        f2 : t2 : = b2 
     ... 
     with        fn : tn : = bn . 

      typarray = [|t1,...tn|]
      bodies   = [|b1,...bn|]
   then

      mkCoFix (i,(funnames,typsarray,bodies))

   constructs the ith function of the block

    CoFixpoint f1 : t1 := b1
    with       f2 : t2 := b2
    ...
    with       fn : tn := bn.
*)
let mkCoFix = mkCoFix

  Construct an implicit
let implicit_sort = Type (make_univ(make_dirpath[id_of_string"implicit"],0))
let mkImplicit = mkSort implicit_sort

(***************************)
(* Other term constructors *)
(***************************)

let abs_implicit c = mkLambda (Anonymous, mkImplicit, c)
let lambda_implicit a = mkLambda (Name(id_of_string"y"), mkImplicit, a)
let lambda_implicit_lift n a = iterate lambda_implicit n (lift n a)

(* prodn n [xn:Tn;..;x1:T1;Gamma] b = (x1:T1)..(xn:Tn)b *)
let prodn n env b =
  let rec prodrec = function
    | (0, env, b)        -> b
    | (n, ((v,t)::l), b) -> prodrec (n-1,  l, mkProd (v,t,b))
    | _ -> assert false
  in
  prodrec (n,env,b)

(* compose_prod [xn:Tn;..;x1:T1] b = (x1:T1)..(xn:Tn)b *)
let compose_prod l b = prodn (List.length l) l b

(* lamn n [xn:Tn;..;x1:T1;Gamma] b = [x1:T1]..[xn:Tn]b *)
let lamn n env b =
  let rec lamrec = function
    | (0, env, b)        -> b
    | (n, ((v,t)::l), b) -> lamrec (n-1,  l, mkLambda (v,t,b))
    | _ -> assert false
  in
  lamrec (n,env,b)

(* compose_lam [xn:Tn;..;x1:T1] b = [x1:T1]..[xn:Tn]b *)
let compose_lam l b = lamn (List.length l) l b

let applist (f,l) = mkApp (f, Array.of_list l)

let applistc f l = mkApp (f, Array.of_list l)

let appvect = mkApp

let appvectc f l = mkApp (f,l)

(* to_lambda n (x1:T1)...(xn:Tn)T =
 * [x1:T1]...[xn:Tn]T *)
let rec to_lambda n prod =
  if n = 0 then
    prod
  else
    match kind_of_term prod with
      | Prod (na,ty,bd) -> mkLambda (na,ty,to_lambda (n-1) bd)
      | Cast (c,_,_) -> to_lambda n c
      | _   -> errorlabstrm "to_lambda" (mt ())

let rec to_prod n lam =
  if n=0 then
    lam
  else
    match kind_of_term lam with
      | Lambda (na,ty,bd) -> mkProd (na,ty,to_prod (n-1) bd)
      | Cast (c,_,_) -> to_prod n c
      | _   -> errorlabstrm "to_prod" (mt ())

  pseudo - reduction rule : 
  * [ prod_app   s ( Prod(_,B ) ) N -- > B[N ] 
  * with an strip_outer_cast on the first argument to produce a product
 * [prod_app  s (Prod(_,B)) N --> B[N]
 * with an strip_outer_cast on the first argument to produce a product *)

let prod_app t n =
  match kind_of_term (strip_outer_cast t) with
    | Prod (_,_,b) -> subst1 n b
    | _ ->
	errorlabstrm "prod_app"
	  (str"Needed a product, but didn't find one" ++ fnl ())


(* prod_appvect T [| a1 ; ... ; an |] -> (T a1 ... an) *)
let prod_appvect t nL = Array.fold_left prod_app t nL

  prod_applist T [ a1 ; ... ; an ] - > ( T a1 ... an )
let prod_applist t nL = List.fold_left prod_app t nL

let it_mkProd_or_LetIn   = List.fold_left (fun c d -> mkProd_or_LetIn d c)
let it_mkLambda_or_LetIn = List.fold_left (fun c d -> mkLambda_or_LetIn d c)

(*********************************)
(* Other term destructors        *)
(*********************************)

(* Transforms a product term (x1:T1)..(xn:Tn)T into the pair
   ([(xn,Tn);...;(x1,T1)],T), where T is not a product *)
let decompose_prod =
  let rec prodec_rec l c = match kind_of_term c with
    | Prod (x,t,c) -> prodec_rec ((x,t)::l) c
    | Cast (c,_,_)   -> prodec_rec l c
    | _              -> l,c
  in
  prodec_rec []

(* Transforms a lambda term [x1:T1]..[xn:Tn]T into the pair
   ([(xn,Tn);...;(x1,T1)],T), where T is not a lambda *)
let decompose_lam =
  let rec lamdec_rec l c = match kind_of_term c with
    | Lambda (x,t,c) -> lamdec_rec ((x,t)::l) c
    | Cast (c,_,_)     -> lamdec_rec l c
    | _                -> l,c
  in
  lamdec_rec []

(* Given a positive integer n, transforms a product term (x1:T1)..(xn:Tn)T
   into the pair ([(xn,Tn);...;(x1,T1)],T) *)
let decompose_prod_n n =
  if n < 0 then error "decompose_prod_n: integer parameter must be positive";
  let rec prodec_rec l n c =
    if n=0 then l,c
    else match kind_of_term c with
      | Prod (x,t,c) -> prodec_rec ((x,t)::l) (n-1) c
      | Cast (c,_,_)   -> prodec_rec l n c
      | _ -> error "decompose_prod_n: not enough products"
  in
  prodec_rec [] n

(* Given a positive integer n, transforms a lambda term [x1:T1]..[xn:Tn]T
   into the pair ([(xn,Tn);...;(x1,T1)],T) *)
let decompose_lam_n n =
  if n < 0 then error "decompose_lam_n: integer parameter must be positive";
  let rec lamdec_rec l n c =
    if n=0 then l,c
    else match kind_of_term c with
      | Lambda (x,t,c) -> lamdec_rec ((x,t)::l) (n-1) c
      | Cast (c,_,_)     -> lamdec_rec l n c
      | _ -> error "decompose_lam_n: not enough abstractions"
  in
  lamdec_rec [] n

(* Transforms a product term (x1:T1)..(xn:Tn)T into the pair
   ([(xn,Tn);...;(x1,T1)],T), where T is not a product *)
let decompose_prod_assum =
  let rec prodec_rec l c =
    match kind_of_term c with
    | Prod (x,t,c)    -> prodec_rec (add_rel_decl (x,None,t) l) c
    | LetIn (x,b,t,c) -> prodec_rec (add_rel_decl (x,Some b,t) l) c
    | Cast (c,_,_)      -> prodec_rec l c
    | _               -> l,c
  in
  prodec_rec empty_rel_context

(* Transforms a lambda term [x1:T1]..[xn:Tn]T into the pair
   ([(xn,Tn);...;(x1,T1)],T), where T is not a lambda *)
let decompose_lam_assum =
  let rec lamdec_rec l c =
    match kind_of_term c with
    | Lambda (x,t,c)  -> lamdec_rec (add_rel_decl (x,None,t) l) c
    | LetIn (x,b,t,c) -> lamdec_rec (add_rel_decl (x,Some b,t) l) c
    | Cast (c,_,_)      -> lamdec_rec l c
    | _               -> l,c
  in
  lamdec_rec empty_rel_context

(* Given a positive integer n, transforms a product term (x1:T1)..(xn:Tn)T
   into the pair ([(xn,Tn);...;(x1,T1)],T) *)
let decompose_prod_n_assum n =
  if n < 0 then
    error "decompose_prod_n_assum: integer parameter must be positive";
  let rec prodec_rec l n c =
    if n=0 then l,c
    else match kind_of_term c with
    | Prod (x,t,c)    -> prodec_rec (add_rel_decl (x,None,t) l) (n-1) c
    | LetIn (x,b,t,c) -> prodec_rec (add_rel_decl (x,Some b,t) l) (n-1) c
    | Cast (c,_,_)      -> prodec_rec l n c
    | c -> error "decompose_prod_n_assum: not enough assumptions"
  in
  prodec_rec empty_rel_context n

(* Given a positive integer n, transforms a lambda term [x1:T1]..[xn:Tn]T
   into the pair ([(xn,Tn);...;(x1,T1)],T)
   Lets in between are not expanded but turn into local definitions,
   but n is the actual number of destructurated lambdas.  *)
let decompose_lam_n_assum n =
  if n < 0 then
    error "decompose_lam_n_assum: integer parameter must be positive";
  let rec lamdec_rec l n c =
    if n=0 then l,c
    else match kind_of_term c with
    | Lambda (x,t,c)  -> lamdec_rec (add_rel_decl (x,None,t) l) (n-1) c
    | LetIn (x,b,t,c) -> lamdec_rec (add_rel_decl (x,Some b,t) l) n c
    | Cast (c,_,_)      -> lamdec_rec l n c
    | c -> error "decompose_lam_n_assum: not enough abstractions"
  in
  lamdec_rec empty_rel_context n

  ( nb_lam [ na1 : T1] ... [nan :  ) where c is not an abstraction 
  * gives n ( casts are ignored )
 * gives n (casts are ignored) *)
let nb_lam =
  let rec nbrec n c = match kind_of_term c with
    | Lambda (_,_,c) -> nbrec (n+1) c
    | Cast (c,_,_) -> nbrec n c
    | _ -> n
  in
  nbrec 0

(* similar to nb_lam, but gives the number of products instead *)
let nb_prod =
  let rec nbrec n c = match kind_of_term c with
    | Prod (_,_,c) -> nbrec (n+1) c
    | Cast (c,_,_) -> nbrec n c
    | _ -> n
  in
  nbrec 0

let prod_assum t = fst (decompose_prod_assum t)
let prod_n_assum n t = fst (decompose_prod_n_assum n t)
let strip_prod_assum t = snd (decompose_prod_assum t)
let strip_prod t = snd (decompose_prod t)
let strip_prod_n n t = snd (decompose_prod_n n t)
let lam_assum t = fst (decompose_lam_assum t)
let lam_n_assum n t = fst (decompose_lam_n_assum n t)
let strip_lam_assum t = snd (decompose_lam_assum t)
let strip_lam t = snd (decompose_lam t)
let strip_lam_n n t = snd (decompose_lam_n n t)

(***************************)
(* Arities                 *)
(***************************)

(* An "arity" is a term of the form [[x1:T1]...[xn:Tn]s] with [s] a sort.
   Such a term can canonically be seen as the pair of a context of types
   and of a sort *)

type arity = rel_context * sorts

let destArity =
  let rec prodec_rec l c =
    match kind_of_term c with
    | Prod (x,t,c)    -> prodec_rec ((x,None,t)::l) c
    | LetIn (x,b,t,c) -> prodec_rec ((x,Some b,t)::l) c
    | Cast (c,_,_)      -> prodec_rec l c
    | Sort s          -> l,s
    | _               -> anomaly "destArity: not an arity"
  in
  prodec_rec []

let mkArity (sign,s) = it_mkProd_or_LetIn (mkSort s) sign

let rec isArity c =
  match kind_of_term c with
  | Prod (_,_,c)    -> isArity c
  | LetIn (_,b,_,c) -> isArity (subst1 b c)
  | Cast (c,_,_)      -> isArity c
  | Sort _          -> true
  | _               -> false

(*******************************)
(*  alpha conversion functions *)
(*******************************)

(* alpha conversion : ignore print names and casts *)

let rec eq_constr m n =
  (m==n) or
  compare_constr eq_constr m n

let eq_constr m n = eq_constr m n (* to avoid tracing a recursive fun *)

(*******************)
(*  hash-consing   *)
(*******************)

module Htype =
  Hashcons.Make(
    struct
      type t = types
      type u = (constr -> constr) * (sorts -> sorts)

       let  ( hc , hs ) j = { body = hc j.body ; typ = hs j.typ } 
       let equal j1 j2 = j1.body==j2.body & j1.typ==j2.typ 

      let hash_sub (hc,hs) j = {body=hc j.body; typ=hs j.typ}
      let equal j1 j2 = j1.body==j2.body & j1.typ==j2.typ
*)
(**)
      let hash_sub (hc,hs) j = hc j
      let equal j1 j2 = j1==j2
(**)
      let hash = Hashtbl.hash
    end)

module Hsorts =
  Hashcons.Make(
    struct
      type t = sorts
      type u = universe -> universe
      let hash_sub huniv = function
          Prop c -> Prop c
        | Type u -> Type (huniv u)
      let equal s1 s2 =
        match (s1,s2) with
            (Prop c1, Prop c2) -> c1=c2
          | (Type u1, Type u2) -> u1 == u2
          |_ -> false
      let hash = Hashtbl.hash
    end)

let hsort = Hsorts.f

let hcons_constr (hcon,hkn,hdir,hname,hident,hstr) =
  let hsortscci = Hashcons.simple_hcons hsort hcons1_univ in
  let hcci = hcons_term (hsortscci,hcon,hkn,hname,hident) in
  let htcci = Hashcons.simple_hcons Htype.f (hcci,hsortscci) in
  (hcci,htcci)

let (hcons1_constr, hcons1_types) = hcons_constr (hcons_names())


(*******)
(* Type of abstract machine values *)
type values

**********************************************************************
    //   *      This file is distributed under the terms of the       
         *       GNU Lesser General Public License Version 2.1        
**********************************************************************
 This defines the strategy to use for verifiying a Cast 
 This defines Cases annotations 
 length of the arity of the inductive type 
 number of pattern vars of each constructor 
 not interpreted by the kernel 
 Sorts. 
 proposition types 
******************************************************************
       Constructions as implemented                               
******************************************************************
*************************
 hash-consing functions  
*************************
 Constructs an existential variable named "?n" 
 Constructs a Variable named id 
 Constructs the term t1::t2, i.e. the term t1 casted with the type t2 
 (that means t2 is declared as the type of t1)
   [s] is the strategy to use when 
 Constructs the product (x:t1)t2 
 Constructs the abstraction [x:t1]t2 
 Constructs [x=c_1:t]c_2 
 If lt = [t1; ...; tn], constructs the application (t1 ... tn) 
 The array of terms correspond to the variables introduced in the section 
 Constructs an existential variable 
 Constructs the ith (co)inductive type of the block named kn 
 The array of terms correspond to the variables introduced in the section 
**********************************************************************
    kind_of_term = constructions as seen by the user                 
**********************************************************************
********************************************************************
          Non primitive term destructors                            
********************************************************************
 Destructor operations : partial functions
   Raise invalid_arg "dest*" if the const has not the expected form 
 Destructs an existential variable 
 Destructs a variable 
 Tests if an evar 
 Tests if a de Bruijn index 
 Tests if a variable 
 Tests if an inductive 
 Destructs the product (x:t1)t2 
 Destructs the abstraction [x:t1]t2 
 Destructs the let [x:=b:t1]t2 
 Destructs a constant 
 Destructs an existential variable 
 Destructs a (co)inductive type named kn 
****************************************************************
 Cast management                                                
****************************************************************
****************************************************************
 Flattening and unflattening of embedded applications and casts 
****************************************************************
 flattens application lists throwing casts in-between 
**************************************************************************
              Functions to recur through subterms                         
**************************************************************************
 [iter_constr f c] iters [f] on the immediate subterms of [c]; it is
   not recursive and the order with which subterms are processed is
   not specified 
 [map_constr f c] maps [f] on the immediate subterms of [c]; it is
   not recursive and the order with which subterms are processed is
   not specified 
*************************************************************************
     Type of assumptions                                                 
*************************************************************************
*************************************************************************
     Type of local contexts (telescopes)                                 
*************************************************************************
**************************************************************************
              Functions for dealing with constr terms                     
**************************************************************************
*******************
     Occurring     
*******************
*******************
      Lifting      
*******************
 The generic lifting function 
 Lifting the binding depth across k bindings 
*******************
   Substituting    
*******************
 1st : general case 

let substkey = Profile.declare_profile "substn_many";;
let substn_many lamv n c = Profile.profile3 substkey substn_many lamv n c;;

 (thin_val sigma) removes identity substitutions from sigma 
 (replace_vars sigma M) applies substitution sigma to term M 

let repvarkey = Profile.declare_profile "replace_vars";;
let replace_vars vl c = Profile.profile2 repvarkey replace_vars vl c ;;

*******************
 Term constructors 
*******************
 Constructs an existential variable named "?n" 
 Constructs a Variable named id 
 Easy sharing 
 Constructs the term t1::t2, i.e. the term t1 casted with the type t2 
 (that means t2 is declared as the type of t1) 
 Constructs the product (x:t1)t2 
 Constructs the abstraction [x:t1]t2 
 Constructs [x=c_1:t]c_2 
 non-dependent product t1 -> t2 
 If lt = [t1; ...; tn], constructs the application (t1 ... tn) 
 The array of terms correspond to the variables introduced in the section 
 Constructs an existential variable 
 Constructs the ith (co)inductive type of the block named kn 
 The array of terms correspond to the variables introduced in the section 
*************************
 Other term constructors 
*************************
 prodn n [xn:Tn;..;x1:T1;Gamma] b = (x1:T1)..(xn:Tn)b 
 compose_prod [xn:Tn;..;x1:T1] b = (x1:T1)..(xn:Tn)b 
 lamn n [xn:Tn;..;x1:T1;Gamma] b = [x1:T1]..[xn:Tn]b 
 compose_lam [xn:Tn;..;x1:T1] b = [x1:T1]..[xn:Tn]b 
 to_lambda n (x1:T1)...(xn:Tn)T =
 * [x1:T1]...[xn:Tn]T 
 prod_appvect T [| a1 ; ... ; an |] -> (T a1 ... an) 
*******************************
 Other term destructors        
*******************************
 Transforms a product term (x1:T1)..(xn:Tn)T into the pair
   ([(xn,Tn);...;(x1,T1)],T), where T is not a product 
 Transforms a lambda term [x1:T1]..[xn:Tn]T into the pair
   ([(xn,Tn);...;(x1,T1)],T), where T is not a lambda 
 Given a positive integer n, transforms a product term (x1:T1)..(xn:Tn)T
   into the pair ([(xn,Tn);...;(x1,T1)],T) 
 Given a positive integer n, transforms a lambda term [x1:T1]..[xn:Tn]T
   into the pair ([(xn,Tn);...;(x1,T1)],T) 
 Transforms a product term (x1:T1)..(xn:Tn)T into the pair
   ([(xn,Tn);...;(x1,T1)],T), where T is not a product 
 Transforms a lambda term [x1:T1]..[xn:Tn]T into the pair
   ([(xn,Tn);...;(x1,T1)],T), where T is not a lambda 
 Given a positive integer n, transforms a product term (x1:T1)..(xn:Tn)T
   into the pair ([(xn,Tn);...;(x1,T1)],T) 
 Given a positive integer n, transforms a lambda term [x1:T1]..[xn:Tn]T
   into the pair ([(xn,Tn);...;(x1,T1)],T)
   Lets in between are not expanded but turn into local definitions,
   but n is the actual number of destructurated lambdas.  
 similar to nb_lam, but gives the number of products instead 
*************************
 Arities                 
*************************
 An "arity" is a term of the form [[x1:T1]...[xn:Tn]s] with [s] a sort.
   Such a term can canonically be seen as the pair of a context of types
   and of a sort 
*****************************
  alpha conversion functions 
*****************************
 alpha conversion : ignore print names and casts 
 to avoid tracing a recursive fun 
*****************
  hash-consing   
*****************


*****
 Type of abstract machine values 
   v       *    The Coq Proof Assistant   /   The Coq Development Team     
  < O _ _ _ , , *    INRIA - CNRS - LIX - LRI - PPS - Copyright 1999 - 2010     
    \VV/   * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

  File initially created by   and   in 1984
  Extension to inductive constructions by   for Coq V5.6
  Extension to mutual inductive constructions by   for 
    Coq V5.10.2
   Coq V5.10.2 *)
  Extension to co - inductive constructions by  
  Optimization of substitution functions by  
  Optimization of lifting functions by   , Mar 1997
  Hash - consing by   in Feb 1998
  Restructuration of Coq of the type - checking kernel by      , 1999
   Filliâtre, 1999 *)
  Abstraction of the syntax of terms and iterators by   , 2000
  Cleaning and lightening of the kernel by   , Nov 2001

  This file defines the internal syntax of the Calculus of 
    Inductive Constructions ( CIC ) terms together with constructors , 
    destructors , iterators and basic functions
   Inductive Constructions (CIC) terms together with constructors,
   destructors, iterators and basic functions *)

open Util
open Pp
open Names
open Univ
open Esubst


type existential_key = int
type metavariable = int


type case_style = LetStyle | IfStyle | LetPatternStyle | MatchStyle | RegularStyle
type case_printing =
    style     : case_style }
type case_info =
  { ci_ind        : inductive;
    ci_npar       : int;
  }


type contents = Pos | Null

type sorts =
  | Type of universe

let prop_sort = Prop Null
let set_sort = Prop Pos
let type1_sort = Type type1_univ

type sorts_family = InProp | InSet | InType

let family_of_sort = function
  | Prop Null -> InProp
  | Prop Pos -> InSet
  | Type _ -> InType


type cast_kind = VMcast | DEFAULTcast

  [ constr array ] is an instance matching definitional [ named_context ] in 
    the same order ( i.e. last argument first )
   the same order (i.e. last argument first) *)
type 'constr pexistential = existential_key * 'constr array
type ('constr, 'types) prec_declaration =
    name array * 'types array * 'constr array
type ('constr, 'types) pfixpoint =
    (int array * int) * ('constr, 'types) prec_declaration
type ('constr, 'types) pcofixpoint =
    int * ('constr, 'types) prec_declaration

  [  ] is used for named variables and [ Rel ] for variables as 
    de Bruijn indices .
   de Bruijn indices. *)
type ('constr, 'types) kind_of_term =
  | Rel       of int
  | Var       of identifier
  | Meta      of metavariable
  | Evar      of 'constr pexistential
  | Sort      of sorts
  | Cast      of 'constr * cast_kind * 'types
  | Prod      of name * 'types * 'types
  | Lambda    of name * 'types * 'constr
  | LetIn     of name * 'constr * 'types * 'constr
  | App       of 'constr * 'constr array
  | Const     of constant
  | Ind       of inductive
  | Construct of constructor
  | Case      of case_info * 'constr * 'constr * 'constr array
  | Fix       of ('constr, 'types) pfixpoint
  | CoFix     of ('constr, 'types) pcofixpoint

  Experimental , used in Presburger contrib
type ('constr, 'types) kind_of_type =
  | SortType   of sorts
  | CastType   of 'types * 'types
  | ProdType   of name * 'types * 'types
  | LetInType  of name * 'constr * 'types * 'types
  | AtomicType of 'constr * 'constr array

let kind_of_type = function
  | Sort s -> SortType s
  | Cast (c,_,t) -> CastType (c, t)
  | Prod (na,t,c) -> ProdType (na, t, c)
  | LetIn (na,b,t,c) -> LetInType (na, b, t, c)
  | App (c,l) -> AtomicType (c, l)
  | (Rel _ | Meta _ | Var _ | Evar _ | Const _ | Case _ | Fix _ | CoFix _ | Ind _ as c)
    -> AtomicType (c,[||])
  | (Lambda _ | Construct _) -> failwith "Not a type"

  constr is the fixpoint of the previous type . Requires option 
    -rectypes of the  compiler to be set
   -rectypes of the Caml compiler to be set *)
type constr = (constr,constr) kind_of_term

type existential = existential_key * constr array
type rec_declaration = name array * constr array * constr array
type fixpoint = (int array * int) * rec_declaration
type cofixpoint = int * rec_declaration


let comp_term t1 t2 =
  match t1, t2 with
  | Rel n1, Rel n2 -> n1 = n2
  | Meta m1, Meta m2 -> m1 == m2
  | Var id1, Var id2 -> id1 == id2
  | Sort s1, Sort s2 -> s1 == s2
  | Cast (c1,_,t1), Cast (c2,_,t2) -> c1 == c2 & t1 == t2
  | Prod (n1,t1,c1), Prod (n2,t2,c2) -> n1 == n2 & t1 == t2 & c1 == c2
  | Lambda (n1,t1,c1), Lambda (n2,t2,c2) -> n1 == n2 & t1 == t2 & c1 == c2
  | LetIn (n1,b1,t1,c1), LetIn (n2,b2,t2,c2) ->
      n1 == n2 & b1 == b2 & t1 == t2 & c1 == c2
  | App (c1,l1), App (c2,l2) -> c1 == c2 & array_for_all2 (==) l1 l2
  | Evar (e1,l1), Evar (e2,l2) -> e1 = e2 & array_for_all2 (==) l1 l2
  | Const c1, Const c2 -> c1 == c2
  | Ind (sp1,i1), Ind (sp2,i2) -> sp1 == sp2 & i1 = i2
  | Construct ((sp1,i1),j1), Construct ((sp2,i2),j2) ->
      sp1 == sp2 & i1 = i2 & j1 = j2
  | Case (ci1,p1,c1,bl1), Case (ci2,p2,c2,bl2) ->
      ci1 == ci2 & p1 == p2 & c1 == c2 & array_for_all2 (==) bl1 bl2
  | Fix (ln1,(lna1,tl1,bl1)), Fix (ln2,(lna2,tl2,bl2)) ->
      ln1 = ln2
      & array_for_all2 (==) lna1 lna2
      & array_for_all2 (==) tl1 tl2
      & array_for_all2 (==) bl1 bl2
  | CoFix(ln1,(lna1,tl1,bl1)), CoFix(ln2,(lna2,tl2,bl2)) ->
      ln1 = ln2
      & array_for_all2 (==) lna1 lna2
      & array_for_all2 (==) tl1 tl2
      & array_for_all2 (==) bl1 bl2
  | _ -> false

let hash_term (sh_rec,(sh_sort,sh_con,sh_kn,sh_na,sh_id)) t =
  match t with
  | Rel _ -> t
  | Meta x -> Meta x
  | Var x -> Var (sh_id x)
  | Sort s -> Sort (sh_sort s)
  | Cast (c, k, t) -> Cast (sh_rec c, k, (sh_rec t))
  | Prod (na,t,c) -> Prod (sh_na na, sh_rec t, sh_rec c)
  | Lambda (na,t,c) -> Lambda (sh_na na, sh_rec t, sh_rec c)
  | LetIn (na,b,t,c) -> LetIn (sh_na na, sh_rec b, sh_rec t, sh_rec c)
  | App (c,l) -> App (sh_rec c, Array.map sh_rec l)
  | Evar (e,l) -> Evar (e, Array.map sh_rec l)
  | Const c -> Const (sh_con c)
  | Ind (kn,i) -> Ind (sh_kn kn,i)
  | Construct ((kn,i),j) -> Construct ((sh_kn kn,i),j)
  TO DO : extract ind_kn
      Case (ci, sh_rec p, sh_rec c, Array.map sh_rec bl)
  | Fix (ln,(lna,tl,bl)) ->
      Fix (ln,(Array.map sh_na lna,
                 Array.map sh_rec tl,
                 Array.map sh_rec bl))
  | CoFix(ln,(lna,tl,bl)) ->
      CoFix (ln,(Array.map sh_na lna,
                   Array.map sh_rec tl,
                   Array.map sh_rec bl))

module Hconstr =
  Hashcons.Make(
    struct
      type t = constr
      type u = (constr -> constr) *
               ((sorts -> sorts) * (constant -> constant) *
               (mutual_inductive -> mutual_inductive) * (name -> name) *
               (identifier -> identifier))
      let hash_sub = hash_term
      let equal = comp_term
      let hash = Hashtbl.hash
    end)

let hcons_term (hsorts,hcon,hkn,hname,hident) =
  Hashcons.recursive_hcons Hconstr.f (hsorts,hcon,hkn,hname,hident)

  Constructs a  index with number n
let rels =
  [|Rel  1;Rel  2;Rel  3;Rel  4;Rel  5;Rel  6;Rel  7; Rel  8;
    Rel  9;Rel 10;Rel 11;Rel 12;Rel 13;Rel 14;Rel 15; Rel 16|]

let mkRel n = if 0<n & n<=16 then rels.(n-1) else Rel n

let mkMeta  n =  Meta n

let mkVar id = Var id

  Construct a type
let mkSort s = Sort s

let mkCast (t1,k2,t2) =
  match t1 with
  | Cast (c,k1, _) when k1 = k2 -> Cast (c,k1,t2)
  | _ -> Cast (t1,k2,t2)

let mkProd (x,t1,t2) = Prod (x,t1,t2)

let mkLambda (x,t1,t2) = Lambda (x,t1,t2)

let mkLetIn (x,c1,t,c2) = LetIn (x,c1,t,c2)

  We ensure applicative terms have at least one argument and the 
    function is not itself an applicative term
   function is not itself an applicative term *)
let mkApp (f, a) =
  if Array.length a = 0 then f else
    match f with
      | App (g, cl) -> App (g, Array.append cl a)
      | _ -> App (f, a)


  Constructs a constant
let mkConst c = Const c

let mkEvar e = Evar e

let mkInd m = Ind m

  Constructs the jth constructor of the ith ( co)inductive type of the 
    block named kn . The array of terms correspond to the variables 
    introduced in the section
   block named kn. The array of terms correspond to the variables
   introduced in the section *)
let mkConstruct c = Construct c

  Constructs the term < p > Case c of c1 | c2 .. | cn end
let mkCase (ci, p, c, ac) = Case (ci, p, c, ac)

let mkFix fix = Fix fix

let mkCoFix cofix = CoFix cofix

let kind_of_term c = c


  User view of [ constr ] . For [ App ] , it is ensured there is at 
    least one argument and the function is not itself an applicative 
    term
   least one argument and the function is not itself an applicative
   term *)

let kind_of_term = kind_of_term



  Destructs a  index
let destRel c = match kind_of_term c with
  | Rel n -> n
  | _ -> invalid_arg "destRel"

let destMeta c = match kind_of_term c with
  | Meta n -> n
  | _ -> invalid_arg "destMeta"

let isMeta c = match kind_of_term c with Meta _ -> true | _ -> false

let destVar c = match kind_of_term c with
  | Var id -> id
  | _ -> invalid_arg "destVar"

  Destructs a type
let isSort c = match kind_of_term c with
  | Sort s -> true
  | _ -> false

let destSort c = match kind_of_term c with
  | Sort s -> s
  | _ -> invalid_arg "destSort"

let rec isprop c = match kind_of_term c with
  | Sort (Prop _) -> true
  | Cast (c,_,_) -> isprop c
  | _ -> false

let rec is_Prop c = match kind_of_term c with
  | Sort (Prop Null) -> true
  | Cast (c,_,_) -> is_Prop c
  | _ -> false

let rec is_Set c = match kind_of_term c with
  | Sort (Prop Pos) -> true
  | Cast (c,_,_) -> is_Set c
  | _ -> false

let rec is_Type c = match kind_of_term c with
  | Sort (Type _) -> true
  | Cast (c,_,_) -> is_Type c
  | _ -> false

let is_small = function
  | Prop _ -> true
  | _ -> false

let iskind c = isprop c or is_Type c

let isEvar c = match kind_of_term c with Evar _ -> true | _ -> false

let isEvar_or_Meta c = match kind_of_term c with
  | Evar _ | Meta _ -> true
  | _ -> false

  Destructs a casted term
let destCast c = match kind_of_term c with
  | Cast (t1,k,t2) -> (t1,k,t2)
  | _ -> invalid_arg "destCast"

let isCast c = match kind_of_term c with Cast _ -> true | _ -> false


let isRel c = match kind_of_term c with Rel _ -> true | _ -> false

let isVar c = match kind_of_term c with Var _ -> true | _ -> false

let isInd c = match kind_of_term c with Ind _ -> true | _ -> false

let destProd c = match kind_of_term c with
  | Prod (x,t1,t2) -> (x,t1,t2)
  | _ -> invalid_arg "destProd"

let isProd c = match kind_of_term c with | Prod _ -> true | _ -> false

let destLambda c = match kind_of_term c with
  | Lambda (x,t1,t2) -> (x,t1,t2)
  | _ -> invalid_arg "destLambda"

let isLambda c = match kind_of_term c with | Lambda _ -> true | _ -> false

let destLetIn c = match kind_of_term c with
  | LetIn (x,b,t1,t2) -> (x,b,t1,t2)
  | _ -> invalid_arg "destProd"

let isLetIn c =  match kind_of_term c with LetIn _ -> true | _ -> false

  Destructs an application
let destApp c = match kind_of_term c with
  | App (f,a) -> (f, a)
  | _ -> invalid_arg "destApplication"

let destApplication = destApp

let isApp c = match kind_of_term c with App _ -> true | _ -> false

let destConst c = match kind_of_term c with
  | Const kn -> kn
  | _ -> invalid_arg "destConst"

let isConst c = match kind_of_term c with Const _ -> true | _ -> false

let destEvar c = match kind_of_term c with
  | Evar (kn, a as r) -> r
  | _ -> invalid_arg "destEvar"

let destInd c = match kind_of_term c with
  | Ind (kn, a as r) -> r
  | _ -> invalid_arg "destInd"

  Destructs a constructor
let destConstruct c = match kind_of_term c with
  | Construct (kn, a as r) -> r
  | _ -> invalid_arg "dest"

let isConstruct c = match kind_of_term c with Construct _ -> true | _ -> false

  Destructs a term < p > Case c of lc1 | lc2 .. | lcn end
let destCase c = match kind_of_term c with
  | Case (ci,p,c,v) -> (ci,p,c,v)
  | _ -> anomaly "destCase"

let isCase c =  match kind_of_term c with Case _ -> true | _ -> false

let destFix c = match kind_of_term c with
  | Fix fix -> fix
  | _ -> invalid_arg "destFix"

let isFix c =  match kind_of_term c with Fix _ -> true | _ -> false

let destCoFix c = match kind_of_term c with
  | CoFix cofix -> cofix
  | _ -> invalid_arg "destCoFix"

let isCoFix c =  match kind_of_term c with CoFix _ -> true | _ -> false


let rec strip_outer_cast c = match kind_of_term c with
  | Cast (c,_,_) -> strip_outer_cast c
  | _ -> c

  Fonction spéciale qui laisse les cast       Case

let under_outer_cast f c =  match kind_of_term c with
  | Cast (b,k,t) -> mkCast (f b, k, f t)
  | _ -> f c

let rec under_casts f c = match kind_of_term c with
  | Cast (c,k,t) -> mkCast (under_casts f c, k, t)
  | _            -> f c


let rec collapse_appl c = match kind_of_term c with
  | App (f,cl) ->
      let rec collapse_rec f cl2 =
        match kind_of_term (strip_outer_cast f) with
	| App (g,cl1) -> collapse_rec g (Array.append cl1 cl2)
	| _ -> mkApp (f,cl2)
      in
      collapse_rec f cl
  | _ -> c

let decompose_app c =
  match kind_of_term c with
    | App (f,cl) -> (f, Array.to_list cl)
    | _ -> (c,[])


  [ fold_constr f acc c ] folds [ f ] on the immediate subterms of [ c ] 
    starting from [ acc ] and proceeding from left to right according to 
    the usual representation of the constructions ; it is not recursive
   starting from [acc] and proceeding from left to right according to
   the usual representation of the constructions; it is not recursive *)

let fold_constr f acc c = match kind_of_term c with
  | (Rel _ | Meta _ | Var _   | Sort _ | Const _ | Ind _
    | Construct _) -> acc
  | Cast (c,_,t) -> f (f acc c) t
  | Prod (_,t,c) -> f (f acc t) c
  | Lambda (_,t,c) -> f (f acc t) c
  | LetIn (_,b,t,c) -> f (f (f acc b) t) c
  | App (c,l) -> Array.fold_left f (f acc c) l
  | Evar (_,l) -> Array.fold_left f acc l
  | Case (_,p,c,bl) -> Array.fold_left f (f (f acc p) c) bl
  | Fix (_,(lna,tl,bl)) ->
      let fd = array_map3 (fun na t b -> (na,t,b)) lna tl bl in
      Array.fold_left (fun acc (na,t,b) -> f (f acc t) b) acc fd
  | CoFix (_,(lna,tl,bl)) ->
      let fd = array_map3 (fun na t b -> (na,t,b)) lna tl bl in
      Array.fold_left (fun acc (na,t,b) -> f (f acc t) b) acc fd


let iter_constr f c = match kind_of_term c with
  | (Rel _ | Meta _ | Var _   | Sort _ | Const _ | Ind _
    | Construct _) -> ()
  | Cast (c,_,t) -> f c; f t
  | Prod (_,t,c) -> f t; f c
  | Lambda (_,t,c) -> f t; f c
  | LetIn (_,b,t,c) -> f b; f t; f c
  | App (c,l) -> f c; Array.iter f l
  | Evar (_,l) -> Array.iter f l
  | Case (_,p,c,bl) -> f p; f c; Array.iter f bl
  | Fix (_,(_,tl,bl)) -> Array.iter f tl; Array.iter f bl
  | CoFix (_,(_,tl,bl)) -> Array.iter f tl; Array.iter f bl

  [ iter_constr_with_binders g f n c ] iters [ f n ] on the immediate 
    subterms of [ c ] ; it carries an extra data [ n ] ( typically a lift 
    index ) which is processed by [ g ] ( which typically add 1 to [ n ] ) at 
    each binder traversal ; it is not recursive and the order with which 
    subterms are processed is not specified
   subterms of [c]; it carries an extra data [n] (typically a lift
   index) which is processed by [g] (which typically add 1 to [n]) at
   each binder traversal; it is not recursive and the order with which
   subterms are processed is not specified *)

let iter_constr_with_binders g f n c = match kind_of_term c with
  | (Rel _ | Meta _ | Var _   | Sort _ | Const _ | Ind _
    | Construct _) -> ()
  | Cast (c,_,t) -> f n c; f n t
  | Prod (_,t,c) -> f n t; f (g n) c
  | Lambda (_,t,c) -> f n t; f (g n) c
  | LetIn (_,b,t,c) -> f n b; f n t; f (g n) c
  | App (c,l) -> f n c; Array.iter (f n) l
  | Evar (_,l) -> Array.iter (f n) l
  | Case (_,p,c,bl) -> f n p; f n c; Array.iter (f n) bl
  | Fix (_,(_,tl,bl)) ->
      Array.iter (f n) tl;
      Array.iter (f (iterate g (Array.length tl) n)) bl
  | CoFix (_,(_,tl,bl)) ->
      Array.iter (f n) tl;
      Array.iter (f (iterate g (Array.length tl) n)) bl


let map_constr f c = match kind_of_term c with
  | (Rel _ | Meta _ | Var _   | Sort _ | Const _ | Ind _
    | Construct _) -> c
  | Cast (c,k,t) -> mkCast (f c, k, f t)
  | Prod (na,t,c) -> mkProd (na, f t, f c)
  | Lambda (na,t,c) -> mkLambda (na, f t, f c)
  | LetIn (na,b,t,c) -> mkLetIn (na, f b, f t, f c)
  | App (c,l) -> mkApp (f c, Array.map f l)
  | Evar (e,l) -> mkEvar (e, Array.map f l)
  | Case (ci,p,c,bl) -> mkCase (ci, f p, f c, Array.map f bl)
  | Fix (ln,(lna,tl,bl)) ->
      mkFix (ln,(lna,Array.map f tl,Array.map f bl))
  | CoFix(ln,(lna,tl,bl)) ->
      mkCoFix (ln,(lna,Array.map f tl,Array.map f bl))

  [ map_constr_with_binders g f n c ] maps [ f n ] on the immediate 
    subterms of [ c ] ; it carries an extra data [ n ] ( typically a lift 
    index ) which is processed by [ g ] ( which typically add 1 to [ n ] ) at 
    each binder traversal ; it is not recursive and the order with which 
    subterms are processed is not specified
   subterms of [c]; it carries an extra data [n] (typically a lift
   index) which is processed by [g] (which typically add 1 to [n]) at
   each binder traversal; it is not recursive and the order with which
   subterms are processed is not specified *)

let map_constr_with_binders g f l c = match kind_of_term c with
  | (Rel _ | Meta _ | Var _   | Sort _ | Const _ | Ind _
    | Construct _) -> c
  | Cast (c,k,t) -> mkCast (f l c, k, f l t)
  | Prod (na,t,c) -> mkProd (na, f l t, f (g l) c)
  | Lambda (na,t,c) -> mkLambda (na, f l t, f (g l) c)
  | LetIn (na,b,t,c) -> mkLetIn (na, f l b, f l t, f (g l) c)
  | App (c,al) -> mkApp (f l c, Array.map (f l) al)
  | Evar (e,al) -> mkEvar (e, Array.map (f l) al)
  | Case (ci,p,c,bl) -> mkCase (ci, f l p, f l c, Array.map (f l) bl)
  | Fix (ln,(lna,tl,bl)) ->
      let l' = iterate g (Array.length tl) l in
      mkFix (ln,(lna,Array.map (f l) tl,Array.map (f l') bl))
  | CoFix(ln,(lna,tl,bl)) ->
      let l' = iterate g (Array.length tl) l in
      mkCoFix (ln,(lna,Array.map (f l) tl,Array.map (f l') bl))

  [ compare_constr f c1 c2 ] compare [ c1 ] and [ c2 ] using [ f ] to compare 
    the immediate subterms of [ c1 ] of [ c2 ] if needed ; Cast 's , 
    application associativity , binders name and Cases annotations are 
    not taken into account
   the immediate subterms of [c1] of [c2] if needed; Cast's,
   application associativity, binders name and Cases annotations are
   not taken into account *)


let compare_constr f t1 t2 =
  match kind_of_term t1, kind_of_term t2 with
  | Rel n1, Rel n2 -> n1 = n2
  | Meta m1, Meta m2 -> m1 = m2
  | Var id1, Var id2 -> id1 = id2
  | Sort s1, Sort s2 -> s1 = s2
  | Cast (c1,_,_), _ -> f c1 t2
  | _, Cast (c2,_,_) -> f t1 c2
  | Prod (_,t1,c1), Prod (_,t2,c2) -> f t1 t2 & f c1 c2
  | Lambda (_,t1,c1), Lambda (_,t2,c2) -> f t1 t2 & f c1 c2
  | LetIn (_,b1,t1,c1), LetIn (_,b2,t2,c2) -> f b1 b2 & f t1 t2 & f c1 c2
  | App (c1,l1), App (c2,l2) ->
      if Array.length l1 = Array.length l2 then
        f c1 c2 & array_for_all2 f l1 l2
      else
        let (h1,l1) = decompose_app t1 in
        let (h2,l2) = decompose_app t2 in
        if List.length l1 = List.length l2 then
          f h1 h2 & List.for_all2 f l1 l2
        else false
  | Evar (e1,l1), Evar (e2,l2) -> e1 = e2 & array_for_all2 f l1 l2
  | Const c1, Const c2 -> eq_constant c1 c2
  | Ind c1, Ind c2 -> eq_ind c1 c2
  | Construct c1, Construct c2 -> eq_constructor c1 c2
  | Case (_,p1,c1,bl1), Case (_,p2,c2,bl2) ->
      f p1 p2 & f c1 c2 & array_for_all2 f bl1 bl2
  | Fix (ln1,(_,tl1,bl1)), Fix (ln2,(_,tl2,bl2)) ->
      ln1 = ln2 & array_for_all2 f tl1 tl2 & array_for_all2 f bl1 bl2
  | CoFix(ln1,(_,tl1,bl1)), CoFix(ln2,(_,tl2,bl2)) ->
      ln1 = ln2 & array_for_all2 f tl1 tl2 & array_for_all2 f bl1 bl2
  | _ -> false


type types = constr

type strategy = types option

type named_declaration = identifier * constr option * types
type rel_declaration = name * constr option * types

let map_named_declaration f (id, v, ty) = (id, Option.map f v, f ty)
let map_rel_declaration = map_named_declaration

let fold_named_declaration f (_, v, ty) a = f ty (Option.fold_right f v a)
let fold_rel_declaration = fold_named_declaration


s Signatures of ordered optionally named variables , intended to be 
    accessed by   indices ( to represent bound variables )
   accessed by de Bruijn indices (to represent bound variables) *)

type rel_context = rel_declaration list

let empty_rel_context = []

let add_rel_decl d ctxt = d::ctxt

let rec lookup_rel n sign =
  match n, sign with
  | 1, decl :: _ -> decl
  | n, _ :: sign -> lookup_rel (n-1) sign
  | _, []        -> raise Not_found

let rel_context_length = List.length

let rel_context_nhyps hyps =
  let rec nhyps acc = function
    | [] -> acc
    | (_,None,_)::hyps -> nhyps (1+acc) hyps
    | (_,Some _,_)::hyps -> nhyps acc hyps in
  nhyps 0 hyps



exception LocalOccur

  ( closedn n M ) raises FreeVar if a variable of height greater than n 
    occurs in M , returns ( ) otherwise
   occurs in M, returns () otherwise *)

let closedn n c =
  let rec closed_rec n c = match kind_of_term c with
    | Rel m -> if m>n then raise LocalOccur
    | _ -> iter_constr_with_binders succ closed_rec n c
  in
  try closed_rec n c; true with LocalOccur -> false

  [ closed0 M ] is true iff [ M ] is a ( deBruijn ) closed term

let closed0 = closedn 0

  ( noccurn n M ) returns true iff ( Rel n ) does NOT occur in term M  

let noccurn n term =
  let rec occur_rec n c = match kind_of_term c with
    | Rel m -> if m = n then raise LocalOccur
    | _ -> iter_constr_with_binders succ occur_rec n c
  in
  try occur_rec n term; true with LocalOccur -> false

  ( noccur_between n m M ) returns true iff ( Rel p ) does NOT occur in term M 
   for n < = p < n+m
  for n <= p < n+m *)

let noccur_between n m term =
  let rec occur_rec n c = match kind_of_term c with
    | Rel(p) -> if n<=p && p<n+m then raise LocalOccur
    | _        -> iter_constr_with_binders succ occur_rec n c
  in
  try occur_rec n term; true with LocalOccur -> false

  Checking function for terms containing existential variables . 
  The function [ noccur_with_meta ] considers the fact that 
  each existential variable ( as well as each isevar ) 
  in the term appears applied to its local context , 
  which may contain the CoFix variables . These occurrences of CoFix variables 
  are not considered
 The function [noccur_with_meta] considers the fact that
 each existential variable (as well as each isevar)
 in the term appears applied to its local context,
 which may contain the CoFix variables. These occurrences of CoFix variables
 are not considered *)

let noccur_with_meta n m term =
  let rec occur_rec n c = match kind_of_term c with
    | Rel p -> if n<=p & p<n+m then raise LocalOccur
    | App(f,cl) ->
	(match kind_of_term f with
           | Cast (c,_,_) when isMeta c -> ()
           | Meta _ -> ()
	   | _ -> iter_constr_with_binders succ occur_rec n c)
    | Evar (_, _) -> ()
    | _ -> iter_constr_with_binders succ occur_rec n c
  in
  try (occur_rec n term; true) with LocalOccur -> false



let rec exliftn el c = match kind_of_term c with
  | Rel i -> mkRel(reloc_rel i el)
  | _ -> map_constr_with_binders el_lift exliftn el c


let liftn k n =
  match el_liftn (pred n) (el_shft k ELID) with
    | ELID -> (fun c -> c)
    | el -> exliftn el

let lift k = liftn k 1


  ( subst1 M c ) substitutes M for Rel(1 ) in c 
    we generalise it to ( substl [ M1, ... ,Mn ] c ) which substitutes in parallel 
    M1, ... ,Mn for respectively Rel(1), ... ,Rel(n ) in c
   we generalise it to (substl [M1,...,Mn] c) which substitutes in parallel
   M1,...,Mn for respectively Rel(1),...,Rel(n) in c *)


type info = Closed | Open | Unknown
type 'a substituend = { mutable sinfo: info; sit: 'a }

let rec lift_substituend depth s =
  match s.sinfo with
    | Closed -> s.sit
    | Open -> lift depth s.sit
    | Unknown ->
        s.sinfo <- if closed0 s.sit then Closed else Open;
        lift_substituend depth s

let make_substituend c = { sinfo=Unknown; sit=c }

let substn_many lamv n c =
  let lv = Array.length lamv in
  if lv = 0 then c
  else
    let rec substrec depth c = match kind_of_term c with
      | Rel k     ->
          if k<=depth then c
          else if k-depth <= lv then lift_substituend depth lamv.(k-depth-1)
          else mkRel (k-lv)
      | _ -> map_constr_with_binders succ substrec depth c in
    substrec n c


let substnl laml n =
  substn_many (Array.map make_substituend (Array.of_list laml)) n
let substl laml = substnl laml 0
let subst1 lam = substl [lam]

let substnl_decl laml k (id,bodyopt,typ) =
  (id,Option.map (substnl laml k) bodyopt,substnl laml k typ)
let substl_decl laml = substnl_decl laml 0
let subst1_decl lam = substl_decl [lam]
let subst1_named_decl = subst1_decl
let substl_named_decl = substl_decl


let rec thin_val = function
  | [] -> []
  | (((id,{ sit = v }) as s)::tl) when isVar v ->
      if id = destVar v then thin_val tl else s::(thin_val tl)
  | h::tl -> h::(thin_val tl)

let replace_vars var_alist =
  let var_alist =
    List.map (fun (str,c) -> (str,make_substituend c)) var_alist in
  let var_alist = thin_val var_alist in
  let rec substrec n c = match kind_of_term c with
    | Var x ->
        (try lift_substituend n (List.assoc x var_alist)
         with Not_found -> c)
    | _ -> map_constr_with_binders succ substrec n c
  in
  if var_alist = [] then (function x -> x) else substrec 0


  ( subst_var str t ) substitute (   ) by ( Rel 1 ) in t
let subst_var str = replace_vars [(str, mkRel 1)]

  ( subst_vars [ id1; ... ;idn ] t ) substitute ( VAR idj ) by ( Rel j ) in t
let substn_vars p vars =
  let _,subst =
    List.fold_left (fun (n,l) var -> ((n+1),(var,mkRel n)::l)) (p,[]) vars
  in replace_vars (List.rev subst)

let subst_vars = substn_vars 1


  Constructs a  index with number n
let mkRel = mkRel

let mkMeta = mkMeta

let mkVar = mkVar

  Construct a type
let mkProp   = mkSort prop_sort
let mkSet    = mkSort set_sort
let mkType u = mkSort (Type u)
let mkSort   = function
  | Prop Pos -> mkSet
  | s -> mkSort s

let mkCast = mkCast

let mkProd = mkProd
let mkNamedProd id typ c = mkProd (Name id, typ, subst_var id c)

let mkLambda = mkLambda
let mkNamedLambda id typ c = mkLambda (Name id, typ, subst_var id c)

let mkLetIn = mkLetIn
let mkNamedLetIn id c1 t c2 = mkLetIn (Name id, c1, t, subst_var id c2)

  Constructs either [ ( x : t)c ] or [ [ x = b : t]c ]
let mkProd_or_LetIn (na,body,t) c =
  match body with
    | None -> mkProd (na, t, c)
    | Some b -> mkLetIn (na, b, t, c)

let mkNamedProd_or_LetIn (id,body,t) c =
  match body with
    | None -> mkNamedProd id t c
    | Some b -> mkNamedLetIn id b t c

  Constructs either [ [ x : t]c ] or [ [ x = b : t]c ]
let mkLambda_or_LetIn (na,body,t) c =
  match body with
    | None -> mkLambda (na, t, c)
    | Some b -> mkLetIn (na, b, t, c)

let mkNamedLambda_or_LetIn (id,body,t) c =
  match body with
    | None -> mkNamedLambda id t c
    | Some b -> mkNamedLetIn id b t c

  Constructs either [ ( x : t)c ] or [ c ] where [ x ] is replaced by [ b ]
let mkProd_wo_LetIn (na,body,t) c =
  match body with
    | None -> mkProd (na,  t, c)
    | Some b -> subst1 b c

let mkNamedProd_wo_LetIn (id,body,t) c =
  match body with
    | None -> mkNamedProd id t c
    | Some b -> subst1 b (subst_var id c)

let mkArrow t1 t2 = mkProd (Anonymous, t1, t2)

  We ensure applicative terms have at most one arguments and the 
    function is not itself an applicative term
   function is not itself an applicative term *)
let mkApp = mkApp

  Constructs a constant
let mkConst = mkConst

let mkEvar = mkEvar

let mkInd = mkInd

  Constructs the jth constructor of the ith ( co)inductive type of the 
    block named kn . The array of terms correspond to the variables 
    introduced in the section
   block named kn. The array of terms correspond to the variables
   introduced in the section *)
let mkConstruct = mkConstruct

  Constructs the term < p > Case c of c1 | c2 .. | cn end
let mkCase = mkCase

  If recindxs = [ |i1, ... in| ] 
       funnames = [ |f1, ... fn| ] 
       typarray = [ |t1, ...  ] 
       bodies    = [ |b1, ... bn| ] 
    then 

        ( ( recindxs , i),(funnames , typarray , bodies ) ) 

    constructs the ith function of the block 

     Fixpoint f1 [ ctx1 ] : t1 : = b1 
     with      f2 [ ctx2 ] : t2 : = b2 
     ... 
     with      fn [ ctxn ] : tn : = bn . 

    where the lenght of the jth context is ij . 

      funnames = [|f1,...fn|]
      typarray = [|t1,...tn|]
      bodies   = [|b1,...bn|]
   then

      mkFix ((recindxs,i),(funnames,typarray,bodies))

   constructs the ith function of the block

    Fixpoint f1 [ctx1] : t1 := b1
    with     f2 [ctx2] : t2 := b2
    ...
    with     fn [ctxn] : tn := bn.

   where the lenght of the jth context is ij.
*)

let mkFix = mkFix

  If funnames = [ |f1, ... fn| ] 
       typarray = [ |t1, ...  ] 
       bodies    = [ |b1, ... bn| ] 
    then 

       mkCoFix ( i,(funnames , typsarray , bodies ) ) 

    constructs the ith function of the block 

     CoFixpoint f1 : t1 : = b1 
     with        f2 : t2 : = b2 
     ... 
     with        fn : tn : = bn . 

      typarray = [|t1,...tn|]
      bodies   = [|b1,...bn|]
   then

      mkCoFix (i,(funnames,typsarray,bodies))

   constructs the ith function of the block

    CoFixpoint f1 : t1 := b1
    with       f2 : t2 := b2
    ...
    with       fn : tn := bn.
*)
let mkCoFix = mkCoFix

  Construct an implicit
let implicit_sort = Type (make_univ(make_dirpath[id_of_string"implicit"],0))
let mkImplicit = mkSort implicit_sort


let abs_implicit c = mkLambda (Anonymous, mkImplicit, c)
let lambda_implicit a = mkLambda (Name(id_of_string"y"), mkImplicit, a)
let lambda_implicit_lift n a = iterate lambda_implicit n (lift n a)

let prodn n env b =
  let rec prodrec = function
    | (0, env, b)        -> b
    | (n, ((v,t)::l), b) -> prodrec (n-1,  l, mkProd (v,t,b))
    | _ -> assert false
  in
  prodrec (n,env,b)

let compose_prod l b = prodn (List.length l) l b

let lamn n env b =
  let rec lamrec = function
    | (0, env, b)        -> b
    | (n, ((v,t)::l), b) -> lamrec (n-1,  l, mkLambda (v,t,b))
    | _ -> assert false
  in
  lamrec (n,env,b)

let compose_lam l b = lamn (List.length l) l b

let applist (f,l) = mkApp (f, Array.of_list l)

let applistc f l = mkApp (f, Array.of_list l)

let appvect = mkApp

let appvectc f l = mkApp (f,l)

let rec to_lambda n prod =
  if n = 0 then
    prod
  else
    match kind_of_term prod with
      | Prod (na,ty,bd) -> mkLambda (na,ty,to_lambda (n-1) bd)
      | Cast (c,_,_) -> to_lambda n c
      | _   -> errorlabstrm "to_lambda" (mt ())

let rec to_prod n lam =
  if n=0 then
    lam
  else
    match kind_of_term lam with
      | Lambda (na,ty,bd) -> mkProd (na,ty,to_prod (n-1) bd)
      | Cast (c,_,_) -> to_prod n c
      | _   -> errorlabstrm "to_prod" (mt ())

  pseudo - reduction rule : 
  * [ prod_app   s ( Prod(_,B ) ) N -- > B[N ] 
  * with an strip_outer_cast on the first argument to produce a product
 * [prod_app  s (Prod(_,B)) N --> B[N]
 * with an strip_outer_cast on the first argument to produce a product *)

let prod_app t n =
  match kind_of_term (strip_outer_cast t) with
    | Prod (_,_,b) -> subst1 n b
    | _ ->
	errorlabstrm "prod_app"
	  (str"Needed a product, but didn't find one" ++ fnl ())


let prod_appvect t nL = Array.fold_left prod_app t nL

  prod_applist T [ a1 ; ... ; an ] - > ( T a1 ... an )
let prod_applist t nL = List.fold_left prod_app t nL

let it_mkProd_or_LetIn   = List.fold_left (fun c d -> mkProd_or_LetIn d c)
let it_mkLambda_or_LetIn = List.fold_left (fun c d -> mkLambda_or_LetIn d c)


let decompose_prod =
  let rec prodec_rec l c = match kind_of_term c with
    | Prod (x,t,c) -> prodec_rec ((x,t)::l) c
    | Cast (c,_,_)   -> prodec_rec l c
    | _              -> l,c
  in
  prodec_rec []

let decompose_lam =
  let rec lamdec_rec l c = match kind_of_term c with
    | Lambda (x,t,c) -> lamdec_rec ((x,t)::l) c
    | Cast (c,_,_)     -> lamdec_rec l c
    | _                -> l,c
  in
  lamdec_rec []

let decompose_prod_n n =
  if n < 0 then error "decompose_prod_n: integer parameter must be positive";
  let rec prodec_rec l n c =
    if n=0 then l,c
    else match kind_of_term c with
      | Prod (x,t,c) -> prodec_rec ((x,t)::l) (n-1) c
      | Cast (c,_,_)   -> prodec_rec l n c
      | _ -> error "decompose_prod_n: not enough products"
  in
  prodec_rec [] n

let decompose_lam_n n =
  if n < 0 then error "decompose_lam_n: integer parameter must be positive";
  let rec lamdec_rec l n c =
    if n=0 then l,c
    else match kind_of_term c with
      | Lambda (x,t,c) -> lamdec_rec ((x,t)::l) (n-1) c
      | Cast (c,_,_)     -> lamdec_rec l n c
      | _ -> error "decompose_lam_n: not enough abstractions"
  in
  lamdec_rec [] n

let decompose_prod_assum =
  let rec prodec_rec l c =
    match kind_of_term c with
    | Prod (x,t,c)    -> prodec_rec (add_rel_decl (x,None,t) l) c
    | LetIn (x,b,t,c) -> prodec_rec (add_rel_decl (x,Some b,t) l) c
    | Cast (c,_,_)      -> prodec_rec l c
    | _               -> l,c
  in
  prodec_rec empty_rel_context

let decompose_lam_assum =
  let rec lamdec_rec l c =
    match kind_of_term c with
    | Lambda (x,t,c)  -> lamdec_rec (add_rel_decl (x,None,t) l) c
    | LetIn (x,b,t,c) -> lamdec_rec (add_rel_decl (x,Some b,t) l) c
    | Cast (c,_,_)      -> lamdec_rec l c
    | _               -> l,c
  in
  lamdec_rec empty_rel_context

let decompose_prod_n_assum n =
  if n < 0 then
    error "decompose_prod_n_assum: integer parameter must be positive";
  let rec prodec_rec l n c =
    if n=0 then l,c
    else match kind_of_term c with
    | Prod (x,t,c)    -> prodec_rec (add_rel_decl (x,None,t) l) (n-1) c
    | LetIn (x,b,t,c) -> prodec_rec (add_rel_decl (x,Some b,t) l) (n-1) c
    | Cast (c,_,_)      -> prodec_rec l n c
    | c -> error "decompose_prod_n_assum: not enough assumptions"
  in
  prodec_rec empty_rel_context n

let decompose_lam_n_assum n =
  if n < 0 then
    error "decompose_lam_n_assum: integer parameter must be positive";
  let rec lamdec_rec l n c =
    if n=0 then l,c
    else match kind_of_term c with
    | Lambda (x,t,c)  -> lamdec_rec (add_rel_decl (x,None,t) l) (n-1) c
    | LetIn (x,b,t,c) -> lamdec_rec (add_rel_decl (x,Some b,t) l) n c
    | Cast (c,_,_)      -> lamdec_rec l n c
    | c -> error "decompose_lam_n_assum: not enough abstractions"
  in
  lamdec_rec empty_rel_context n

  ( nb_lam [ na1 : T1] ... [nan :  ) where c is not an abstraction 
  * gives n ( casts are ignored )
 * gives n (casts are ignored) *)
let nb_lam =
  let rec nbrec n c = match kind_of_term c with
    | Lambda (_,_,c) -> nbrec (n+1) c
    | Cast (c,_,_) -> nbrec n c
    | _ -> n
  in
  nbrec 0

let nb_prod =
  let rec nbrec n c = match kind_of_term c with
    | Prod (_,_,c) -> nbrec (n+1) c
    | Cast (c,_,_) -> nbrec n c
    | _ -> n
  in
  nbrec 0

let prod_assum t = fst (decompose_prod_assum t)
let prod_n_assum n t = fst (decompose_prod_n_assum n t)
let strip_prod_assum t = snd (decompose_prod_assum t)
let strip_prod t = snd (decompose_prod t)
let strip_prod_n n t = snd (decompose_prod_n n t)
let lam_assum t = fst (decompose_lam_assum t)
let lam_n_assum n t = fst (decompose_lam_n_assum n t)
let strip_lam_assum t = snd (decompose_lam_assum t)
let strip_lam t = snd (decompose_lam t)
let strip_lam_n n t = snd (decompose_lam_n n t)



type arity = rel_context * sorts

let destArity =
  let rec prodec_rec l c =
    match kind_of_term c with
    | Prod (x,t,c)    -> prodec_rec ((x,None,t)::l) c
    | LetIn (x,b,t,c) -> prodec_rec ((x,Some b,t)::l) c
    | Cast (c,_,_)      -> prodec_rec l c
    | Sort s          -> l,s
    | _               -> anomaly "destArity: not an arity"
  in
  prodec_rec []

let mkArity (sign,s) = it_mkProd_or_LetIn (mkSort s) sign

let rec isArity c =
  match kind_of_term c with
  | Prod (_,_,c)    -> isArity c
  | LetIn (_,b,_,c) -> isArity (subst1 b c)
  | Cast (c,_,_)      -> isArity c
  | Sort _          -> true
  | _               -> false



let rec eq_constr m n =
  (m==n) or
  compare_constr eq_constr m n



module Htype =
  Hashcons.Make(
    struct
      type t = types
      type u = (constr -> constr) * (sorts -> sorts)

       let  ( hc , hs ) j = { body = hc j.body ; typ = hs j.typ } 
       let equal j1 j2 = j1.body==j2.body & j1.typ==j2.typ 

      let hash_sub (hc,hs) j = {body=hc j.body; typ=hs j.typ}
      let equal j1 j2 = j1.body==j2.body & j1.typ==j2.typ
*)
      let hash_sub (hc,hs) j = hc j
      let equal j1 j2 = j1==j2
      let hash = Hashtbl.hash
    end)

module Hsorts =
  Hashcons.Make(
    struct
      type t = sorts
      type u = universe -> universe
      let hash_sub huniv = function
          Prop c -> Prop c
        | Type u -> Type (huniv u)
      let equal s1 s2 =
        match (s1,s2) with
            (Prop c1, Prop c2) -> c1=c2
          | (Type u1, Type u2) -> u1 == u2
          |_ -> false
      let hash = Hashtbl.hash
    end)

let hsort = Hsorts.f

let hcons_constr (hcon,hkn,hdir,hname,hident,hstr) =
  let hsortscci = Hashcons.simple_hcons hsort hcons1_univ in
  let hcci = hcons_term (hsortscci,hcon,hkn,hname,hident) in
  let htcci = Hashcons.simple_hcons Htype.f (hcci,hsortscci) in
  (hcci,htcci)

let (hcons1_constr, hcons1_types) = hcons_constr (hcons_names())


type values
