(************************************************************************)
(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
(*   \VV/  **************************************************************)
(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
(************************************************************************)

open Errors
open Util
open Names
open Term
open Vars
open Context
open Termops
open Univ
open Evd
open Environ

exception Elimconst

(** This module implements a call by name reduction used by (at
    least) evarconv unification and cbn tactic.

    It has an ability to "refold" constants by storing constants and
    their parameters in its stack.
*)

(** Machinery to custom the behavior of the reduction *)
module ReductionBehaviour = struct
  open Globnames
  open Libobject

  type t = {
    b_nargs: int;
    b_recargs: int list;
    b_dont_expose_case: bool;
  }

  let table =
    Summary.ref (Refmap.empty : t Refmap.t) ~name:"reductionbehaviour"

  type flag = [ `ReductionDontExposeCase | `ReductionNeverUnfold ]
  type req =
    | ReqLocal
    | ReqGlobal of global_reference * (int list * int * flag list)

  let load _ (_,(_,(r, b))) =
    table := Refmap.add r b !table

  let cache o = load 1 o

  let classify = function
    | ReqLocal, _ -> Dispose
    | ReqGlobal _, _ as o -> Substitute o

  let subst (subst, (_, (r,o as orig))) =
    ReqLocal,
    let r' = fst (subst_global subst r) in if r==r' then orig else (r',o)

  let discharge = function
    | _,(ReqGlobal (ConstRef c, req), (_, b)) ->
       let c' = pop_con c in
       let vars, _subst, _ctx = Lib.section_segment_of_constant c in
       let extra = List.length vars in
       let nargs' = if b.b_nargs < 0 then b.b_nargs else b.b_nargs + extra in
       let recargs' = List.map ((+) extra) b.b_recargs in
       let b' = { b with b_nargs = nargs'; b_recargs = recargs' } in
       Some (ReqGlobal (ConstRef c', req), (ConstRef c', b'))
    | _ -> None

  let rebuild = function
    | req, (ConstRef c, _ as x) -> req, x
    | _ -> assert false

  let inRedBehaviour = declare_object {
			(default_object "REDUCTIONBEHAVIOUR") with
			load_function = load;
			cache_function = cache;
			classify_function = classify;
			subst_function = subst;
			discharge_function = discharge;
			rebuild_function = rebuild;
		      }

  let set local r (recargs, nargs, flags as req) =
    let nargs = if List.mem `ReductionNeverUnfold flags then max_int else nargs in
    let behaviour = {
      b_nargs = nargs; b_recargs = recargs;
      b_dont_expose_case = List.mem `ReductionDontExposeCase flags } in
    let req = if local then ReqLocal else ReqGlobal (r, req) in
    Lib.add_anonymous_leaf (inRedBehaviour (req, (r, behaviour)))
  ;;

  let get r =
    try
      let b = Refmap.find r !table in
      let flags =
	if Int.equal b.b_nargs max_int then [`ReductionNeverUnfold]
	else if b.b_dont_expose_case then [`ReductionDontExposeCase] else [] in
      Some (b.b_recargs, (if Int.equal b.b_nargs max_int then -1 else b.b_nargs), flags)
    with Not_found -> None

  let print ref =
    let open Pp in
    let pr_global = Nametab.pr_global_env Id.Set.empty in
    match get ref with
    | None -> mt ()
    | Some (recargs, nargs, flags) ->
       let never = List.mem `ReductionNeverUnfold flags in
       let nomatch = List.mem `ReductionDontExposeCase flags in
       let pp_nomatch = spc() ++ if nomatch then
				   str "avoiding to expose match constructs" else str"" in
       let pp_recargs = spc() ++ str "when the " ++
			  pr_enum (fun x -> pr_nth (x+1)) recargs ++ str (String.plural (List.length recargs) " argument") ++
			  str (String.plural (if List.length recargs >= 2 then 1 else 2) " evaluate") ++
			  str " to a constructor" in
       let pp_nargs =
	 spc() ++ str "when applied to " ++ int nargs ++
	   str (String.plural nargs " argument") in
       hov 2 (str "The reduction tactics " ++
		 match recargs, nargs, never with
		 | _,_, true -> str "never unfold " ++ pr_global ref
		 | [], 0, _ -> str "always unfold " ++ pr_global ref
		 | _::_, n, _ when n < 0 ->
		    str "unfold " ++ pr_global ref ++ pp_recargs ++ pp_nomatch
		 | _::_, n, _ when n > List.fold_left max 0 recargs ->
		    str "unfold " ++ pr_global ref ++ pp_recargs ++
		      str " and" ++ pp_nargs ++ pp_nomatch
		 | _::_, _, _ ->
		    str "unfold " ++ pr_global ref ++ pp_recargs ++ pp_nomatch
		 | [], n, _ when n > 0 ->
		    str "unfold " ++ pr_global ref ++ pp_nargs ++ pp_nomatch
		 | _ -> str "unfold " ++ pr_global ref ++ pp_nomatch )
end

(** Machinery about stack of unfolded constants *)
module Cst_stack = struct
(** constant * params * args

- constant applied to params = term in head applied to args
- there is at most one arguments with an empty list of args, it must be the first.
- in args, the int represents the indice of the first arg to consider *)
  type t = (constr * constr list * (int * constr array) list)  list

  let empty = []

  let sanity x y =
    assert(Term.eq_constr x y)

  let drop_useless = function
    | _ :: ((_,_,[])::_ as q) -> q
    | l -> l

  let add_param h cst_l =
    let append2cst = function
      | (c,params,[]) -> (c, h::params, [])
      | (c,params,((i,t)::q)) when i = pred (Array.length t) ->
	 let () = sanity h t.(i) in (c, params, q)
      | (c,params,(i,t)::q) ->
	let () = sanity h t.(i) in (c, params, (succ i,t)::q)
    in
      drop_useless (List.map append2cst cst_l)

  let add_args cl =
    List.map (fun (a,b,args) -> (a,b,(0,cl)::args))

  let add_cst cst = function
    | (_,_,[]) :: q as l -> l
    | l -> (cst,[],[])::l

  let best_cst = function
    | (cst,params,[])::_ -> Some(cst,params)
    | _ -> None

  (** [best_replace d cst_l c] makes the best replacement for [d]
      by [cst_l] in [c] *)
  let best_replace d cst_l c =
    let reconstruct_head = List.fold_left
      (fun t (i,args) -> mkApp (t,Array.sub args i (Array.length args - i))) in
    List.fold_right
      (fun (cst,params,args) t -> Termops.replace_term
	(reconstruct_head d args)
	(applist (cst, List.rev params))
	t) cst_l c

  let pr l =
    let open Pp in
    let p_c = Termops.print_constr in
    prlist_with_sep pr_semicolon
      (fun (c,params,args) ->
	hov 1 (p_c c ++ spc () ++ pr_sequence p_c params ++ spc () ++ str "(args:" ++
		 pr_sequence (fun (i,el) -> prvect_with_sep spc p_c (Array.sub el i (Array.length el - i))) args ++
		 str ")")) l
end


(** The type of (machine) stacks (= lambda-bar-calculus' contexts) *)
module Stack :
sig
  type 'a app_node
  val pr_app_node : ('a -> Pp.std_ppcmds) -> 'a app_node -> Pp.std_ppcmds

  type 'a cst_member = 
    | Cst_const of pconstant
    | Cst_proj of projection * 'a

  type 'a member =
  | App of 'a app_node
  | Case of case_info * 'a * 'a array * Cst_stack.t
  | Proj of int * int * projection
  | Fix of fixpoint * 'a t * Cst_stack.t
  | Cst of 'a cst_member * int * int list * 'a t * Cst_stack.t
  | Shift of int
  | Update of 'a
  and 'a t = 'a member list
  val pr : ('a -> Pp.std_ppcmds) -> 'a t -> Pp.std_ppcmds
  val empty : 'a t
  val is_empty : 'a t -> bool
  val append_app : 'a array -> 'a t -> 'a t
  val decomp : 'a t -> ('a * 'a t) option
  val decomp_node_last : 'a app_node -> 'a t -> ('a * 'a t)
  val equal : ('a * int -> 'a * int -> bool) -> (fixpoint * int -> fixpoint * int -> bool)
    -> 'a t -> 'a t -> (int * int) option
  val compare_shape : 'a t -> 'a t -> bool
  val map : (constr -> constr) -> constr t -> constr t
  val fold2 : ('a -> constr -> constr -> 'a) -> 'a ->
    constr t -> constr t -> 'a * int * int
  val append_app_list : 'a list -> 'a t -> 'a t
  val strip_app : 'a t -> 'a t * 'a t
  val strip_n_app : int -> 'a t -> ('a t * 'a * 'a t) option
  val not_purely_applicative : 'a t -> bool
  val list_of_app_stack : constr t -> constr list option
  val assign : 'a t -> int -> 'a -> 'a t
  val args_size : 'a t -> int
  val tail : int -> 'a t -> 'a t
  val nth : 'a t -> int -> 'a
  val best_state : constr * constr t -> Cst_stack.t -> constr * constr t
  val zip : ?refold:bool -> constr * constr t -> constr
end =
struct
  type 'a app_node = int * 'a array * int
  (* first releavnt position, arguments, last relevant position *)

  (*
     Invariant that this module must ensure :
     (behare of direct access to app_node by the rest of Reductionops)
     - in app_node (i,_,j) i <= j
     - There is no array realocation (outside of debug printing)
   *)

  let pr_app_node pr (i,a,j) =
    let open Pp in surround (
		     prvect_with_sep pr_comma pr (Array.sub a i (j - i + 1))
		     )


  type 'a cst_member = 
    | Cst_const of pconstant
    | Cst_proj of projection * 'a

  type 'a member =
  | App of 'a app_node
  | Case of Term.case_info * 'a * 'a array * Cst_stack.t
  | Proj of int * int * projection
  | Fix of fixpoint * 'a t * Cst_stack.t
  | Cst of 'a cst_member * int * int list * 'a t * Cst_stack.t
  | Shift of int
  | Update of 'a
  and 'a t = 'a member list

  let rec pr_member pr_c member =
    let open Pp in
    let pr_c x = hov 1 (pr_c x) in
    match member with
    | App app -> str "ZApp" ++ pr_app_node pr_c app
    | Case (_,_,br,cst) ->
       str "ZCase(" ++
	 prvect_with_sep (pr_bar) pr_c br
       ++ str ")"
    | Proj (n,m,p) ->
      str "ZProj(" ++ int n ++ pr_comma () ++ int m ++
	pr_comma () ++ pr_con p ++ str ")"
    | Fix (f,args,cst) ->
       str "ZFix(" ++ Termops.pr_fix Termops.print_constr f
       ++ pr_comma () ++ pr pr_c args ++ str ")"
    | Cst (mem,curr,remains,params,cst_l) ->
      str "ZCst(" ++ pr_cst_member pr_c mem ++ pr_comma () ++ int curr
      ++ pr_comma () ++
	prlist_with_sep pr_semicolon int remains ++
	pr_comma () ++ pr pr_c params ++ str ")"
    | Shift i -> str "ZShift(" ++ int i ++ str ")"
    | Update t -> str "ZUpdate(" ++ pr_c t ++ str ")"
  and pr pr_c l =
    let open Pp in
    prlist_with_sep pr_semicolon (fun x -> hov 1 (pr_member pr_c x)) l

  and pr_cst_member pr_c c =
    let open Pp in 
      match c with
      | Cst_const (c, u) -> 
	if Univ.Instance.is_empty u then Constant.print c
	else str"(" ++ Constant.print c ++ str ", " ++ Univ.Instance.pr u ++ str")"
      | Cst_proj (p, c) ->
	pr_c c ++ str".(" ++ Constant.print p ++ str")"

  let empty = []
  let is_empty = CList.is_empty

  let append_app v s =
    let le = Array.length v in
    if Int.equal le 0 then s else App (0,v,pred le) :: s

  let decomp_node (i,l,j) sk =
    if i < j then (l.(i), App (succ i,l,j) :: sk)
    else (l.(i), sk)

  let decomp = function
    | App node::s -> Some (decomp_node node s)
    | _ -> None

  let decomp_node_last (i,l,j) sk =
    if i < j then (l.(j), App (i,l,pred j) :: sk)
    else (l.(j), sk)

  let equal f f_fix sk1 sk2 =
    let equal_cst_member x lft1 y lft2 = 
      match x, y with
      | Cst_const (c1,u1), Cst_const (c2, u2) ->
	Constant.equal c1 c2 && Univ.Instance.equal u1 u2
      | Cst_proj (p1,c1), Cst_proj (p2,c2) ->
	Constant.equal p1 p2 && f (c1,lft1) (c2,lft2)
      | _, _ -> false
    in
    let rec equal_rec sk1 lft1 sk2 lft2  =
      match sk1,sk2 with
      | [],[] -> Some (lft1,lft2)
      | (Update _ :: _, _ | _, Update _ :: _) -> assert false
      | Shift k :: s1, _ -> equal_rec  s1  (lft1+k)  sk2 lft2
      | _, Shift k :: s2 -> equal_rec sk1 lft1 s2 (lft2+k)
      | App a1 :: s1, App a2 :: s2 ->
	let t1,s1' = decomp_node_last a1 s1 in
	let t2,s2' = decomp_node_last a2 s2 in
	if f (t1,lft1) (t2,lft2) then equal_rec s1' lft1 s2' lft2 else None
      | Case (_,t1,a1,_) :: s1, Case (_,t2,a2,_) :: s2 ->
	if f (t1,lft1) (t2,lft2) && CArray.equal (fun x y -> f (x,lft1) (y,lft2)) a1 a2
	then equal_rec s1 lft1 s2 lft2
	else None
      | (Proj (n1,m1,p)::s1, Proj(n2,m2,p2)::s2) ->
	if Int.equal n1 n2 && Int.equal m1 m2 && Names.Constant.CanOrd.equal p p2
	then equal_rec s1 lft1 s2 lft2
	else None
      | Fix (f1,s1,_) :: s1', Fix (f2,s2,_) :: s2' ->
	if f_fix (f1,lft1) (f2,lft2) then
	  match equal_rec (List.rev s1) lft1 (List.rev s2) lft2 with
	  | None -> None
	  | Some (lft1',lft2') -> equal_rec s1' lft1' s2' lft2'
	else None
      | Cst (c1,curr1,remains1,params1,_)::s1', Cst (c2,curr2,remains2,params2,_)::s2' ->
	if equal_cst_member c1 lft1 c2 lft2 then
	  match equal_rec (List.rev params1) lft1 (List.rev params2) lft2 with
	  | Some (lft1',lft2') -> equal_rec s1' lft1' s2' lft2'
	  | None -> None
	else None
      | ((App _|Case _|Proj _|Fix _|Cst _)::_|[]), _ -> None
    in equal_rec (List.rev sk1) 0 (List.rev sk2) 0

  let compare_shape stk1 stk2 =
    let rec compare_rec bal stk1 stk2 =
      match (stk1,stk2) with
	([],[]) -> Int.equal bal 0
      | ((Update _|Shift _)::s1, _) -> compare_rec bal s1 stk2
      | (_, (Update _|Shift _)::s2) -> compare_rec bal stk1 s2
      | (App (i,_,j)::s1, _) -> compare_rec (bal + j + 1 - i) s1 stk2
      | (_, App (i,_,j)::s2) -> compare_rec (bal - j - 1 + i) stk1 s2
      | (Case(c1,_,_,_)::s1, Case(c2,_,_,_)::s2) ->
        Int.equal bal 0 (* && c1.ci_ind  = c2.ci_ind *) && compare_rec 0 s1 s2
      | (Proj (n1,m1,p)::s1, Proj(n2,m2,p2)::s2) ->
	Int.equal bal 0 && compare_rec 0 s1 s2
      | (Fix(_,a1,_)::s1, Fix(_,a2,_)::s2) ->
        Int.equal bal 0 && compare_rec 0 a1 a2 && compare_rec 0 s1 s2
      | (Cst (_,_,_,p1,_)::s1, Cst (_,_,_,p2,_)::s2) ->
	Int.equal bal 0 && compare_rec 0 p1 p2 && compare_rec 0 s1 s2
      | (_,_) -> false in
    compare_rec 0 stk1 stk2

  let fold2 f o sk1 sk2 =
    let rec aux o lft1 sk1 lft2 sk2 =
      let fold_array =
	Array.fold_left2 (fun a x y -> f a (Vars.lift lft1 x) (Vars.lift lft2 y))
      in
      match sk1,sk2 with
      | [], [] -> o,lft1,lft2
      | Shift n :: q1, _ -> aux o (lft1+n) q1 lft2 sk2
      | _, Shift n :: q2 -> aux o lft1 sk1 (lft2+n) q2
      | App n1 :: q1, App n2 :: q2 ->
	 let t1,l1 = decomp_node_last n1 q1 in
	 let t2,l2 = decomp_node_last n2 q2 in
	 aux (f o (Vars.lift lft1 t1) (Vars.lift lft2 t2))
	     lft1 l1 lft2 l2
      | Case (_,t1,a1,_) :: q1, Case (_,t2,a2,_) :: q2 ->
	aux (fold_array
	       (f o (Vars.lift lft1 t1) (Vars.lift lft2 t2))
	       a1 a2) lft1 q1 lft2 q2
      | Proj (n1,m1,p1) :: q1, Proj (n2,m2,p2) :: q2 ->
	aux o lft1 q1 lft2 q2
      | Fix ((_,(_,a1,b1)),s1,_) :: q1, Fix ((_,(_,a2,b2)),s2,_) :: q2 ->
	let (o',lft1',lft2') = aux (fold_array (fold_array o b1 b2) a1 a2)
	  lft1 (List.rev s1) lft2 (List.rev s2) in
	aux o' lft1' q1 lft2' q2
      | Cst (cst1,_,_,params1,_) :: q1, Cst (cst2,_,_,params2,_) :: q2 ->
	let (o',lft1',lft2') =
	  aux o lft1 (List.rev params1) lft2 (List.rev params2)
	in aux o' lft1' q1 lft2' q2
      | (((Update _|App _|Case _|Proj _|Fix _| Cst _) :: _|[]), _) ->
	raise (Invalid_argument "Reductionops.Stack.fold2")
    in aux o 0 (List.rev sk1) 0 (List.rev sk2)

  let rec map f x = List.map (function
			       | Update _ -> assert false
			       | (Proj (_,_,_) | Shift _) as e -> e
			       | App (i,a,j) ->
				  let le = j - i + 1 in
				  App (0,Array.map f (Array.sub a i le), le-1)
			       | Case (info,ty,br,alt) -> Case (info, f ty, Array.map f br, alt)
			       | Fix ((r,(na,ty,bo)),arg,alt) ->
				  Fix ((r,(na,Array.map f ty, Array.map f bo)),map f arg,alt)
			       | Cst (cst,curr,remains,params,alt) ->
				 Cst (cst,curr,remains,map f params,alt)
  ) x

  let append_app_list l s =
    let a = Array.of_list l in
    append_app a s

  let rec args_size = function
    | App (i,_,j)::s -> j + 1 - i + args_size s
    | Shift(_)::s -> args_size s
    | Update(_)::s -> args_size s
    | (Case _|Fix _|Proj _|Cst _)::_ | [] -> 0

  let strip_app s =
    let rec aux out = function
      | ( App _ | Shift _ as e) :: s -> aux (e :: out) s
      | s -> List.rev out,s
    in aux [] s
  let strip_n_app n s =
    let rec aux n out = function
      | Shift k as e :: s -> aux n (e :: out) s
      | App (i,a,j) as e :: s ->
	 let nb = j  - i + 1 in
	 if n >= nb then
	   aux (n - nb) (e::out) s
	 else
	   let p = i+n in
	   Some (CList.rev
	      (if Int.equal n 0 then out else App (i,a,p-1) :: out),
	    a.(p),
	    if j > p then App(succ p,a,j)::s else s)
      | s -> None
    in aux n [] s

  let not_purely_applicative args =
    List.exists (function (Fix _ | Case _ | Proj _ | Cst _) -> true | _ -> false) args
  let list_of_app_stack s =
    let rec aux = function
      | App (i,a,j) :: s ->
	 let (k,(args',s')) = aux s in
	 let a' = Array.map (Vars.lift k) (Array.sub a i (j - i + 1)) in
	 k,(Array.fold_right (fun x y -> x::y) a' args', s')
      | Shift n :: s ->
	 let (k,(args',s')) = aux s in (k+n,(args', s'))
      | s -> (0,([],s)) in
    let (lft,(out,s')) = aux s in
    let init = match s' with [] when Int.equal lft 0 -> true | _ -> false in
    Option.init init out

  let assign s p c =
    match strip_n_app p s with
    | Some (pre,_,sk) -> pre @ (App (0,[|c|],0)::sk)
    | None -> assert false

  let tail n0 s0 =
    let rec aux lft n s =
      let out s = if Int.equal lft 0 then s else Shift lft :: s in
      if Int.equal n 0 then out s else
	match s with
      | App (i,a,j) :: s ->
	 let nb = j  - i + 1 in
	 if n >= nb then
	   aux lft (n - nb) s
	 else
	   let p = i+n in
	   if j >= p then App(p,a,j)::s else s
	| Shift k :: s' -> aux (lft+k) n s'
	| _ -> raise (Invalid_argument "Reductionops.Stack.tail")
    in aux 0 n0 s0

  let nth s p =
    match strip_n_app p s with
    | Some (_,el,_) -> el
    | None -> raise Not_found

  (** This function breaks the abstraction of Cst_stack ! *)
  let best_state (_,sk as s) l =
    let rec aux sk def = function
      |(cst, params, []) -> (cst, append_app_list (List.rev params) sk)
      |(cst, params, (i,t)::q) -> match decomp sk with
	| Some (el,sk') when Constr.equal el t.(i) ->
	  if i = pred (Array.length t)
	  then aux sk' def (cst, params, q)
	  else aux sk' def (cst, params, (succ i,t)::q)
	| _ -> def
    in List.fold_left (aux sk) s l

  let constr_of_cst_member f sk = 
    match f with
    | Cst_const (c, u) -> mkConstU (c,u), sk
    | Cst_proj (p, c) -> mkProj (p, c), tail 1 sk

  let rec zip ?(refold=false) = function
    | f, [] -> f
    | f, (App (i,a,j) :: s) ->
       let a' = if Int.equal i 0 && Int.equal j (Array.length a - 1)
		then a
		else Array.sub a i (j - i + 1) in
       zip ~refold (mkApp (f, a'), s)
    | f, (Case (ci,rt,br,cst_l)::s) when refold ->
      zip ~refold (best_state (mkCase (ci,rt,f,br), s) cst_l)
    | f, (Case (ci,rt,br,_)::s) -> zip ~refold (mkCase (ci,rt,f,br), s)
    | f, (Fix (fix,st,cst_l)::s) when refold ->
      zip ~refold (best_state (mkFix fix, st @ (append_app [|f|] s)) cst_l)
  | f, (Fix (fix,st,_)::s) -> zip ~refold
    (mkFix fix, st @ (append_app [|f|] s))
  | f, (Cst (cst,_,_,params,cst_l)::s) when refold ->
    zip ~refold (best_state (constr_of_cst_member cst (params @ (append_app [|f|] s))) cst_l)
  | f, (Cst (cst,_,_,params,_)::s) ->
    zip ~refold (constr_of_cst_member cst (params @ (append_app [|f|] s)))
  | f, (Shift n::s) -> zip ~refold (lift n f, s)
  | f, (Proj (n,m,p)::s) -> zip ~refold (mkProj (p,f),s)
  | _, (Update _::_) -> assert false
end

(** The type of (machine) states (= lambda-bar-calculus' cuts) *)
type state = constr * constr Stack.t

type  contextual_reduction_function = env ->  evar_map -> constr -> constr
type  reduction_function =  contextual_reduction_function
type local_reduction_function = evar_map -> constr -> constr
type e_reduction_function = env -> evar_map -> constr -> evar_map * constr

type  contextual_stack_reduction_function =
    env ->  evar_map -> constr -> constr * constr list
type  stack_reduction_function =  contextual_stack_reduction_function
type local_stack_reduction_function =
    evar_map -> constr -> constr * constr list

type  contextual_state_reduction_function =
    env ->  evar_map -> state -> state
type  state_reduction_function =  contextual_state_reduction_function
type local_state_reduction_function = evar_map -> state -> state

let pr_state (tm,sk) =
  let open Pp in
  h 0 (Termops.print_constr tm ++ str "|" ++ cut () ++ Stack.pr Termops.print_constr sk)

(*************************************)
(*** Reduction Functions Operators ***)
(*************************************)

let safe_evar_value sigma ev =
  try Some (Evd.existential_value sigma ev)
  with NotInstantiatedEvar | Not_found -> None

let safe_meta_value sigma ev =
  try Some (Evd.meta_value sigma ev)
  with Not_found -> None

let strong whdfun env sigma t =
  let rec strongrec env t =
    map_constr_with_full_binders push_rel strongrec env (whdfun env sigma t) in
  strongrec env t

let local_strong whdfun sigma =
  let rec strongrec t = map_constr strongrec (whdfun sigma t) in
  strongrec

let rec strong_prodspine redfun sigma c =
  let x = redfun sigma c in
  match kind_of_term x with
    | Prod (na,a,b) -> mkProd (na,a,strong_prodspine redfun sigma b)
    | _ -> x

(*************************************)
(*** Reduction using bindingss ***)
(*************************************)

(* Local *)
let nored = Closure.RedFlags.no_red
let beta = Closure.beta
let eta = Closure.RedFlags.mkflags [Closure.RedFlags.fETA]
let zeta = Closure.RedFlags.mkflags [Closure.RedFlags.fZETA]
let betaiota = Closure.betaiota
let betaiotazeta = Closure.betaiotazeta

(* Contextual *)
let delta = Closure.RedFlags.mkflags [Closure.RedFlags.fDELTA]
let betalet = Closure.RedFlags.mkflags [Closure.RedFlags.fBETA;Closure.RedFlags.fZETA]
let betaetalet = Closure.RedFlags.red_add betalet Closure.RedFlags.fETA
let betadelta = Closure.RedFlags.red_add betalet Closure.RedFlags.fDELTA
let betadeltaeta = Closure.RedFlags.red_add betadelta Closure.RedFlags.fETA
let betadeltaiota = Closure.RedFlags.red_add betadelta Closure.RedFlags.fIOTA
let betadeltaiota_nolet = Closure.betadeltaiotanolet
let betadeltaiotaeta = Closure.RedFlags.red_add betadeltaiota Closure.RedFlags.fETA

(* Beta Reduction tools *)

let apply_subst recfun env cst_l t stack =
  let rec aux env cst_l t stack =
    match (Stack.decomp stack,kind_of_term t) with
    | Some (h,stacktl), Lambda (_,_,c) ->
      aux (h::env) (Cst_stack.add_param h cst_l) c stacktl
    | _ -> recfun cst_l (substl env t, stack)
  in aux env cst_l t stack
    
let stacklam recfun env t stack =
  apply_subst (fun _ -> recfun) env Cst_stack.empty t stack

let beta_applist (c,l) =
  stacklam Stack.zip [] c (Stack.append_app_list l Stack.empty)

(* Iota reduction tools *)

type 'a miota_args = {
  mP      : constr;     (* the result type *)
  mconstr : constr;     (* the constructor *)
  mci     : case_info;  (* special info to re-build pattern *)
  mcargs  : 'a list;    (* the constructor's arguments *)
  mlf     : 'a array }  (* the branch code vector *)

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

(** @return c if there is a constant c whose body is bd
    @return bd else.

    It has only a meaning because internal representation of "Fixpoint f x
    := t" is Definition f := fix f x => t

    Even more fragile that we could hope because do Module M. Fixpoint
    f x := t. End M. Definition f := u. and say goodbye to any hope
    of refolding M.f this way ...
*)
let magicaly_constant_of_fixbody env bd = function
  | Name.Anonymous -> bd
  | Name.Name id ->
    try
      let cst = Nametab.locate_constant
    	(Libnames.make_qualid DirPath.empty id) in
      let (cst, u), ctx = Universes.fresh_constant_instance env cst in
      match constant_opt_value_in env (cst,u) with
      | None -> bd
      | Some t ->
        let b, csts = Universes.eq_constr_universes t bd in
    	let subst = Universes.Constraints.fold (fun (l,d,r) acc ->
    	  Univ.LMap.add (Option.get (Universe.level l)) (Option.get (Universe.level r)) acc)
    	  csts Univ.LMap.empty
    	in
    	let inst = Instance.subst_fn (fun u -> Univ.LMap.find u subst) u in
          if b then mkConstU (cst,inst) else bd
    with
    | Not_found -> bd

let contract_cofix ?env (bodynum,(names,types,bodies as typedbodies)) =
  let nbodies = Array.length bodies in
  let make_Fi j =
    let ind = nbodies-j-1 in
    if Int.equal bodynum ind then mkCoFix (ind,typedbodies)
    else
      let bd = mkCoFix (ind,typedbodies) in
      match env with
      | None -> bd
      | Some e -> magicaly_constant_of_fixbody e bd names.(ind) in
  let closure = List.init nbodies make_Fi in
  substl closure bodies.(bodynum)

(** Similar to the "fix" case below *)
let reduce_and_refold_cofix recfun env cst_l cofix sk =
  let raw_answer = contract_cofix ~env cofix in
  apply_subst
    (fun x (t,sk') -> recfun x (Cst_stack.best_replace (mkCoFix cofix) cst_l t,sk'))
    [] Cst_stack.empty raw_answer sk

let reduce_mind_case mia =
  match kind_of_term mia.mconstr with
    | Construct ((ind_sp,i),u) ->
(*	let ncargs = (fst mia.mci).(i-1) in*)
	let real_cargs = List.skipn mia.mci.ci_npar mia.mcargs in
        applist (mia.mlf.(i-1),real_cargs)
    | CoFix cofix ->
	let cofix_def = contract_cofix cofix in
	mkCase (mia.mci, mia.mP, applist(cofix_def,mia.mcargs), mia.mlf)
    | _ -> assert false

(* contracts fix==FIX[nl;i](A1...Ak;[F1...Fk]{B1....Bk}) to produce
   Bi[Fj --> FIX[nl;j](A1...Ak;[F1...Fk]{B1...Bk})] *)

let contract_fix ?env ((recindices,bodynum),(names,types,bodies as typedbodies)) =
    let nbodies = Array.length recindices in
    let make_Fi j =
      let ind = nbodies-j-1 in
      if Int.equal bodynum ind then mkFix ((recindices,ind),typedbodies)
      else
	let bd = mkFix ((recindices,ind),typedbodies) in
	match env with
	| None -> bd
	| Some e -> magicaly_constant_of_fixbody e bd names.(ind) in
    let closure = List.init nbodies make_Fi in
    substl closure bodies.(bodynum)

(** First we substitute the Rel bodynum by the fixpoint and then we try to
    replace the fixpoint by the best constant from [cst_l]
    Other rels are directly substituted by constants "magically found from the
    context" in contract_fix *)
let reduce_and_refold_fix recfun env cst_l fix sk =
  let raw_answer = contract_fix ~env fix in
  apply_subst
    (fun x (t,sk') -> recfun x (Cst_stack.best_replace (mkFix fix) cst_l t,sk'))
    [] Cst_stack.empty raw_answer sk

let fix_recarg ((recindices,bodynum),_) stack =
  assert (0 <= bodynum && bodynum < Array.length recindices);
  let recargnum = Array.get recindices bodynum in
  try
    Some (recargnum, Stack.nth stack recargnum)
  with Not_found ->
    None

(** Generic reduction function with environment

    Here is where unfolded constant are stored in order to be
    eventualy refolded.

    If tactic_mode is true, it uses ReductionBehaviour, prefers
    refold constant instead of value and tries to infer constants
    fix and cofix came from.

    It substitutes fix and cofix by the constant they come from in
    contract_* in any case .
*)
let rec whd_state_gen ?csts tactic_mode flags env sigma =
  let rec whrec cst_l (x, stack as s) =
    let fold () =
      if tactic_mode then (Stack.best_state s cst_l,Cst_stack.empty) else (s,cst_l)
    in
    match kind_of_term x with
    | Rel n when Closure.RedFlags.red_set flags Closure.RedFlags.fDELTA ->
      (match lookup_rel n env with
      | (_,Some body,_) -> whrec Cst_stack.empty (lift n body, stack)
      | _ -> fold ())
    | Var id when Closure.RedFlags.red_set flags (Closure.RedFlags.fVAR id) ->
      (match lookup_named id env with
      | (_,Some body,_) -> whrec (Cst_stack.add_cst (mkVar id) cst_l) (body, stack)
      | _ -> fold ())
    | Evar ev ->
      (match safe_evar_value sigma ev with
      | Some body -> whrec cst_l (body, stack)
      | None -> fold ())
    | Meta ev ->
      (match safe_meta_value sigma ev with
      | Some body -> whrec cst_l (body, stack)
      | None -> fold ())
    | Const (c,u as const) when Closure.RedFlags.red_set flags (Closure.RedFlags.fCONST c) ->
       (match constant_opt_value_in env const with
	| None -> fold ()
	| Some body ->
	   if not tactic_mode
	   then whrec (Cst_stack.add_cst (mkConstU const) cst_l) (body, stack)
	   else (* Looks for ReductionBehaviour *)
	     match ReductionBehaviour.get (Globnames.ConstRef c) with
	     | None -> whrec (Cst_stack.add_cst (mkConstU const) cst_l) (body, stack)
	     | Some (recargs, nargs, flags) ->
		if (List.mem `ReductionNeverUnfold flags
		    || (nargs > 0 && Stack.args_size stack < nargs))
		then fold ()
		else (* maybe unfolds *)
		  if List.mem `ReductionDontExposeCase flags then
		    let app_sk,sk = Stack.strip_app stack in
		    let (tm',sk'),cst_l' =
		      whrec (Cst_stack.add_cst (mkConstU const) cst_l) (body, app_sk) in
		    let f_equal (x,lft1) (y,lft2) = Constr.equal (Vars.lift lft1 x) (Vars.lift lft2 y) in
		    if
		      (match Stack.equal f_equal
			  (fun (a,b) (c,d) -> f_equal (Constr.mkFix a, b) (Constr.mkFix c, d))
			  app_sk sk' with
			  | None -> false
			  | Some (lft1,lft2) -> f_equal (x, lft1) (tm', lft2)
		      ) || Stack.not_purely_applicative sk'
		    then fold ()
		    else whrec cst_l' (tm', sk' @ sk)
		  else match recargs with
		  |[] -> (* if nargs has been specified *)
			 (* CAUTION : the constant is NEVER refold
                            (even when it hides a (co)fix) *)
		    whrec cst_l (body, stack)
		  |curr::remains -> match Stack.strip_n_app curr stack with
		    | None -> fold ()
		    | Some (bef,arg,s') ->
		      whrec Cst_stack.empty (arg,Stack.Cst(Stack.Cst_const const,curr,remains,bef,cst_l)::s')
       )
    | Proj (p, c) when Closure.RedFlags.red_set flags (Closure.RedFlags.fCONST p) ->
      (match (lookup_constant p env).Declarations.const_proj with
      | None -> assert false
      | Some pb -> begin
	let npars = pb.Declarations.proj_npars 
	and arg = pb.Declarations.proj_arg in
	  match ReductionBehaviour.get (Globnames.ConstRef p) with
	  | None ->
	    let stack' = (c, Stack.Proj (npars, arg, p) :: stack) in
	      whrec Cst_stack.empty(* cst_l *) stack'
	  | Some (recargs, nargs, flags) ->
	    if (List.mem `ReductionNeverUnfold flags
		|| (nargs > 0 && Stack.args_size stack < (nargs - (npars + 1))))
	    then fold ()
	    else
	      let recargs = List.map_filter (fun x -> 
		let idx = x - npars in 
		  if idx < 0 then None else Some idx) recargs
	      in
	      match recargs with
	      |[] -> (* if nargs has been specified *)
	      (* CAUTION : the constant is NEVER refold
                 (even when it hides a (co)fix) *)
		let stack' = (c, Stack.Proj (npars, arg, p) :: stack) in
		  whrec Cst_stack.empty(* cst_l *) stack'
	      | curr::remains -> 
		if curr == 0 then (* Try to reduce the record argument *)
		  whrec Cst_stack.empty 
		    (c, Stack.Cst(Stack.Cst_proj (p, c),curr,remains,Stack.empty,cst_l)::stack)
		else
		  match Stack.strip_n_app curr stack with
		  | None -> fold ()
		  | Some (bef,arg,s') ->
		    whrec Cst_stack.empty 
		      (arg,Stack.Cst(Stack.Cst_proj (p, c),curr,remains,bef,cst_l)::s')
      end)
    | LetIn (_,b,_,c) when Closure.RedFlags.red_set flags Closure.RedFlags.fZETA ->
      apply_subst whrec [b] cst_l c stack
    | Cast (c,_,_) -> whrec cst_l (c, stack)
    | App (f,cl)  ->
      whrec
	(Cst_stack.add_args cl cst_l)
	(f, Stack.append_app cl stack)
    | Lambda (na,t,c) ->
      (match Stack.decomp stack with
      | Some _ when Closure.RedFlags.red_set flags Closure.RedFlags.fBETA ->
	apply_subst whrec [] cst_l x stack
      | None when Closure.RedFlags.red_set flags Closure.RedFlags.fETA ->
	let env' = push_rel (na,None,t) env in
	let whrec' = whd_state_gen tactic_mode flags env' sigma in
        (match kind_of_term (Stack.zip ~refold:true (fst (whrec' (c, Stack.empty)))) with
        | App (f,cl) ->
	  let napp = Array.length cl in
	  if napp > 0 then
	    let (x', l'),_ = whrec' (Array.last cl, Stack.empty) in
            match kind_of_term x', l' with
            | Rel 1, [] ->
	      let lc = Array.sub cl 0 (napp-1) in
	      let u = if Int.equal napp 1 then f else appvect (f,lc) in
	      if noccurn 1 u then (pop u,Stack.empty),Cst_stack.empty else fold ()
            | _ -> fold ()
	  else fold ()
	| _ -> fold ())
      | _ -> fold ())

    | Case (ci,p,d,lf) ->
      whrec Cst_stack.empty (d, Stack.Case (ci,p,lf,cst_l) :: stack)

    | Fix ((ri,n),_ as f) ->
      (match Stack.strip_n_app ri.(n) stack with
      |None -> fold ()
      |Some (bef,arg,s') ->
	whrec Cst_stack.empty (arg, Stack.Fix(f,bef,cst_l)::s'))

    | Construct ((ind,c),u) ->
      if Closure.RedFlags.red_set flags Closure.RedFlags.fIOTA then
	match Stack.strip_app stack with
	|args, (Stack.Case(ci, _, lf,_)::s') ->
	  whrec Cst_stack.empty (lf.(c-1), (Stack.tail ci.ci_npar args) @ s')
	|args, (Stack.Proj (n,m,p)::s') ->
	  whrec Cst_stack.empty (Stack.nth args (n+m), s')
	|args, (Stack.Fix (f,s',cst_l)::s'') ->
	  let x' = Stack.zip(x,args) in
	  let out_sk = s' @ (Stack.append_app [|x'|] s'') in
	  if tactic_mode
	  then reduce_and_refold_fix whrec env cst_l f out_sk
	  else whrec Cst_stack.empty (contract_fix f, out_sk)
	|args, (Stack.Cst (const,curr,remains,s',cst_l) :: s'') ->
	  let x' = Stack.zip(x,args) in
	  begin match remains with
	  | [] -> 
	    (match const with
	    | Stack.Cst_const const ->
	      (match constant_opt_value_in env const with
	      | None -> fold ()
	      | Some body ->
		whrec (Cst_stack.add_cst (mkConstU const) cst_l)
		  (body, s' @ (Stack.append_app [|x'|] s'')))
	    | Stack.Cst_proj (p, c) ->
	      let pb = lookup_projection p env in
	      let npars = pb.Declarations.proj_npars in
	      let narg = pb.Declarations.proj_arg in
	      let stack = s' @ (Stack.append_app [|x'|] s'') in
		match Stack.strip_n_app 0 stack with
		| None -> assert false
		| Some (_,arg,s'') ->
		  whrec Cst_stack.empty (arg, Stack.Proj (npars,narg,p) :: s''))
	  | next :: remains' -> match Stack.strip_n_app (next-curr-1) s'' with
	    | None -> fold ()
	    | Some (bef,arg,s''') ->
	      whrec Cst_stack.empty
		(arg,
		 Stack.Cst (const,next,remains',s' @ (Stack.append_app [|x'|] bef),cst_l) :: s''')
	  end
	|_, (Stack.App _|Stack.Update _|Stack.Shift _)::_ -> assert false
	|_, [] -> fold ()
      else fold ()

    | CoFix cofix ->
      if Closure.RedFlags.red_set flags Closure.RedFlags.fIOTA then
	match Stack.strip_app stack with
	|args, (Stack.Case(ci, _, lf,_)::s') ->
	  if tactic_mode
	  then reduce_and_refold_cofix whrec env cst_l cofix stack
	  else whrec Cst_stack.empty (contract_cofix cofix, stack)
	|_ -> fold ()
      else fold ()

    | Rel _ | Var _ | Const _ | LetIn _ | Proj _ -> fold ()
    | Sort _ | Ind _ | Prod _ -> fold ()
  in
  whrec (Option.default Cst_stack.empty csts)

(** reduction machine without global env and refold machinery *)
let local_whd_state_gen flags sigma =
  let rec whrec (x, stack as s) =
    match kind_of_term x with
    | LetIn (_,b,_,c) when Closure.RedFlags.red_set flags Closure.RedFlags.fZETA ->
      stacklam whrec [b] c stack
    | Cast (c,_,_) -> whrec (c, stack)
    | App (f,cl)  -> whrec (f, Stack.append_app cl stack)
    | Lambda (_,_,c) ->
      (match Stack.decomp stack with
      | Some (a,m) when Closure.RedFlags.red_set flags Closure.RedFlags.fBETA ->
	stacklam whrec [a] c m
      | None when Closure.RedFlags.red_set flags Closure.RedFlags.fETA ->
        (match kind_of_term (Stack.zip (whrec (c, Stack.empty))) with
        | App (f,cl) ->
	  let napp = Array.length cl in
	  if napp > 0 then
	    let x', l' = whrec (Array.last cl, Stack.empty) in
            match kind_of_term x', l' with
            | Rel 1, [] ->
	      let lc = Array.sub cl 0 (napp-1) in
	      let u = if Int.equal napp 1 then f else appvect (f,lc) in
	      if noccurn 1 u then (pop u,Stack.empty) else s
            | _ -> s
	  else s
	| _ -> s)
      | _ -> s)

    | Proj (p,c) when Closure.RedFlags.red_set flags (Closure.RedFlags.fCONST p) ->
      (match (lookup_constant p (Global.env ())).Declarations.const_proj with
      | None -> assert false
      | Some pb -> whrec (c, Stack.Proj (pb.Declarations.proj_npars, pb.Declarations.proj_arg, p)
        :: stack))

    | Case (ci,p,d,lf) ->
      whrec (d, Stack.Case (ci,p,lf,Cst_stack.empty) :: stack)

    | Fix ((ri,n),_ as f) ->
      (match Stack.strip_n_app ri.(n) stack with
      |None -> s
      |Some (bef,arg,s') -> whrec (arg, Stack.Fix(f,bef,Cst_stack.empty)::s'))

    | Evar ev ->
      (match safe_evar_value sigma ev with
        Some c -> whrec (c,stack)
      | None -> s)

    | Meta ev ->
      (match safe_meta_value sigma ev with
        Some c -> whrec (c,stack)
      | None -> s)

    | Construct ((ind,c),u) ->
      if Closure.RedFlags.red_set flags Closure.RedFlags.fIOTA then
	match Stack.strip_app stack with
	|args, (Stack.Case(ci, _, lf,_)::s') ->
	  whrec (lf.(c-1), (Stack.tail ci.ci_npar args) @ s')
	|args, (Stack.Proj (n,m,p) :: s') ->
	  whrec (Stack.nth args (n+m), s')
	|args, (Stack.Fix (f,s',cst)::s'') ->
	  let x' = Stack.zip(x,args) in
	  whrec (contract_fix f, s' @ (Stack.append_app [|x'|] s''))
	|_, (Stack.App _|Stack.Update _|Stack.Shift _|Stack.Cst _)::_ -> assert false
	|_, [] -> s
      else s

    | CoFix cofix ->
      if Closure.RedFlags.red_set flags Closure.RedFlags.fIOTA then
	match Stack.strip_app stack with
	|args, (Stack.Case(ci, _, lf,_)::s') ->
	  whrec (contract_cofix cofix, stack)
	|_ -> s
      else s

    | x -> s
  in
  whrec

let raw_whd_state_gen flags env =
  let f sigma s = fst (whd_state_gen false flags env sigma s) in
  f

let stack_red_of_state_red f =
  let f sigma x = decompose_app (Stack.zip (f sigma (x, Stack.empty))) in
  f

(* Drops the Cst_stack *)
let iterate_whd_gen refold flags env sigma s =
  let rec aux t =
  let (hd,sk),_ = whd_state_gen refold flags env sigma (t,Stack.empty) in
  let whd_sk = Stack.map aux sk in
  Stack.zip ~refold (hd,whd_sk)
  in aux s

let red_of_state_red f sigma x =
  Stack.zip (f sigma (x,Stack.empty))

(* 0. No Reduction Functions *)

let whd_nored_state = local_whd_state_gen nored
let whd_nored_stack = stack_red_of_state_red whd_nored_state
let whd_nored = red_of_state_red whd_nored_state

(* 1. Beta Reduction Functions *)

let whd_beta_state = local_whd_state_gen beta
let whd_beta_stack = stack_red_of_state_red whd_beta_state
let whd_beta = red_of_state_red whd_beta_state

(* Nouveau ! *)
let whd_betaetalet_state = local_whd_state_gen betaetalet
let whd_betaetalet_stack = stack_red_of_state_red whd_betaetalet_state
let whd_betaetalet = red_of_state_red whd_betaetalet_state

let whd_betalet_state = local_whd_state_gen betalet
let whd_betalet_stack = stack_red_of_state_red whd_betalet_state
let whd_betalet = red_of_state_red whd_betalet_state

(* 2. Delta Reduction Functions *)

let whd_delta_state e = raw_whd_state_gen delta e
let whd_delta_stack env = stack_red_of_state_red (whd_delta_state env)
let whd_delta env = red_of_state_red  (whd_delta_state env)

let whd_betadelta_state e = raw_whd_state_gen betadelta e
let whd_betadelta_stack env =
  stack_red_of_state_red (whd_betadelta_state env)
let whd_betadelta env =
  red_of_state_red (whd_betadelta_state env)


let whd_betadeltaeta_state e = raw_whd_state_gen betadeltaeta e
let whd_betadeltaeta_stack env =
  stack_red_of_state_red (whd_betadeltaeta_state env)
let whd_betadeltaeta env =
  red_of_state_red (whd_betadeltaeta_state env)

(* 3. Iota reduction Functions *)

let whd_betaiota_state = local_whd_state_gen betaiota
let whd_betaiota_stack = stack_red_of_state_red whd_betaiota_state
let whd_betaiota = red_of_state_red whd_betaiota_state

let whd_betaiotazeta_state = local_whd_state_gen betaiotazeta
let whd_betaiotazeta_stack = stack_red_of_state_red whd_betaiotazeta_state
let whd_betaiotazeta = red_of_state_red whd_betaiotazeta_state

let whd_betadeltaiota_state env = raw_whd_state_gen betadeltaiota env
let whd_betadeltaiota_stack env =
  stack_red_of_state_red (whd_betadeltaiota_state env)
let whd_betadeltaiota env =
  red_of_state_red (whd_betadeltaiota_state env)

let whd_betadeltaiotaeta_state env = raw_whd_state_gen betadeltaiotaeta env
let whd_betadeltaiotaeta_stack env =
  stack_red_of_state_red (whd_betadeltaiotaeta_state env)
let whd_betadeltaiotaeta env =
  red_of_state_red (whd_betadeltaiotaeta_state env)

let whd_betadeltaiota_nolet_state env = raw_whd_state_gen betadeltaiota_nolet env
let whd_betadeltaiota_nolet_stack env =
  stack_red_of_state_red (whd_betadeltaiota_nolet_state env)
let whd_betadeltaiota_nolet env =
  red_of_state_red (whd_betadeltaiota_nolet_state env)

(* 4. Eta reduction Functions *)

let whd_eta c = Stack.zip (local_whd_state_gen eta Evd.empty (c,Stack.empty))

(* 5. Zeta Reduction Functions *)

let whd_zeta c = Stack.zip (local_whd_state_gen zeta Evd.empty (c,Stack.empty))

(****************************************************************************)
(*                   Reduction Functions                                    *)
(****************************************************************************)

(* Replacing defined evars for error messages *)
let rec whd_evar sigma c =
  match kind_of_term c with
    | Evar ev ->
        (match safe_evar_value sigma ev with
            Some c -> whd_evar sigma c
          | None -> c)
    | Sort (Type u) -> 
      let u' = Evd.normalize_universe sigma u in
	if u' == u then c else mkSort (Sorts.sort_of_univ u')
    | Const (c', u) when not (Univ.Instance.is_empty u) -> 
      let u' = Evd.normalize_universe_instance sigma u in
	if u' == u then c else mkConstU (c', u')
    | Ind (i, u) when not (Univ.Instance.is_empty u) -> 
      let u' = Evd.normalize_universe_instance sigma u in
	if u' == u then c else mkIndU (i, u')
    | Construct (co, u) when not (Univ.Instance.is_empty u) ->
      let u' = Evd.normalize_universe_instance sigma u in
	if u' == u then c else mkConstructU (co, u')
    | _ -> c

let nf_evar =
  local_strong whd_evar

(* lazy reduction functions. The infos must be created for each term *)
(* Note by HH [oct 08] : why would it be the job of clos_norm_flags to add
   a [nf_evar] here *)
let clos_norm_flags flgs env sigma t =
  try
    let evars ev = safe_evar_value sigma ev in
    Closure.norm_val
      (Closure.create_clos_infos ~evars flgs env)
      (Closure.inject t)
  with e when is_anomaly e -> error "Tried to normalize ill-typed term"

let nf_beta = clos_norm_flags Closure.beta (Global.env ())
let nf_betaiota = clos_norm_flags Closure.betaiota (Global.env ())
let nf_betaiotazeta = clos_norm_flags Closure.betaiotazeta (Global.env ())
let nf_betadeltaiota env sigma =
  clos_norm_flags Closure.betadeltaiota env sigma


(********************************************************************)
(*                         Conversion                               *)
(********************************************************************)
(*
let fkey = Profile.declare_profile "fhnf";;
let fhnf info v = Profile.profile2 fkey fhnf info v;;

let fakey = Profile.declare_profile "fhnf_apply";;
let fhnf_apply info k h a = Profile.profile4 fakey fhnf_apply info k h a;;
*)

let is_transparent e k =
  match Conv_oracle.get_strategy (Environ.oracle e) k with
  | Conv_oracle.Opaque -> false
  | _ -> true

(* Conversion utility functions *)

type conversion_test = constraints -> constraints

let pb_is_equal pb = pb == Reduction.CONV

let pb_equal = function
  | Reduction.CUMUL -> Reduction.CONV
  | Reduction.CONV -> Reduction.CONV

let sort_cmp cv_pb s1 s2 u = 
  Reduction.check_sort_cmp_universes cv_pb s1 s2 u

let test_trans_conversion (f: ?l2r:bool-> ?evars:'a->'b) reds env sigma x y =
  try
    let evars ev = safe_evar_value sigma ev in
    let _ = f ~evars reds env (Evd.universes sigma) x y in
    true
  with Reduction.NotConvertible -> false
    | e when is_anomaly e -> error "Conversion test raised an anomaly"

let is_trans_conv reds env sigma = test_trans_conversion Reduction.trans_conv_universes reds env sigma
let is_trans_conv_leq reds env sigma = test_trans_conversion Reduction.trans_conv_leq_universes reds env sigma
let is_trans_fconv = function Reduction.CONV -> is_trans_conv | Reduction.CUMUL -> is_trans_conv_leq

let is_conv = is_trans_conv full_transparent_state
let is_conv_leq = is_trans_conv_leq full_transparent_state
let is_fconv = function | Reduction.CONV -> is_conv | Reduction.CUMUL -> is_conv_leq

let check_conv ?(pb=Reduction.CUMUL) ?(ts=full_transparent_state) env sigma x y = 
  let f = match pb with
    | Reduction.CONV -> Reduction.trans_conv_universes
    | Reduction.CUMUL -> Reduction.trans_conv_leq_universes 
  in
    try f ~evars:(safe_evar_value sigma) ts env (Evd.universes sigma) x y; true
    with Reduction.NotConvertible -> false
    | Univ.UniverseInconsistency _ -> false
    | e when is_anomaly e -> error "Conversion test raised an anomaly"

let sigma_compare_sorts pb s0 s1 sigma =
  match pb with
  | Reduction.CONV -> Evd.set_eq_sort sigma s0 s1
  | Reduction.CUMUL -> Evd.set_leq_sort sigma s0 s1
    
let sigma_compare_instances flex i0 i1 sigma =
  try Evd.set_eq_instances ~flex sigma i0 i1
  with Evd.UniversesDiffer -> raise Reduction.NotConvertible

let sigma_univ_state = 
  { Reduction.compare = sigma_compare_sorts;
    Reduction.compare_instances = sigma_compare_instances }

let infer_conv ?(pb=Reduction.CUMUL) ?(ts=full_transparent_state) env sigma x y = 
  try 
    let b, sigma = 
      let b, cstrs = 
	if pb == Reduction.CUMUL then 
	  Universes.leq_constr_univs_infer (Evd.universes sigma) x y 
	else
	  Universes.eq_constr_univs_infer (Evd.universes sigma) x y 
      in
	if b then 
	  try true, Evd.add_universe_constraints sigma cstrs
	  with Univ.UniverseInconsistency _ | Evd.UniversesDiffer -> false, sigma
	else false, sigma
    in
      if b then sigma, true
      else
	let sigma' = 
	  Reduction.generic_conv pb false (safe_evar_value sigma) ts 
	    env (sigma, sigma_univ_state) x y in
	  sigma', true
  with
  | Reduction.NotConvertible -> sigma, false
  | Univ.UniverseInconsistency _ -> sigma, false
  | e when is_anomaly e -> error "Conversion test raised an anomaly"
    
(********************************************************************)
(*             Special-Purpose Reduction                            *)
(********************************************************************)

let whd_meta sigma c = match kind_of_term c with
  | Meta p -> (try meta_value sigma p with Not_found -> c)
  | _ -> c

let default_plain_instance_ident = Id.of_string "H"

(* Try to replace all metas. Does not replace metas in the metas' values
 * Differs from (strong whd_meta). *)
let plain_instance s c =
  let rec irec n u = match kind_of_term u with
    | Meta p -> (try lift n (Metamap.find p s) with Not_found -> u)
    | App (f,l) when isCast f ->
        let (f,_,t) = destCast f in
        let l' = CArray.Fun1.smartmap irec n l in
        (match kind_of_term f with
        | Meta p ->
	    (* Don't flatten application nodes: this is used to extract a
               proof-term from a proof-tree and we want to keep the structure
               of the proof-tree *)
	    (try let g = Metamap.find p s in
	    match kind_of_term g with
            | App _ ->
                let l' = CArray.Fun1.smartmap lift 1 l' in
                mkLetIn (Name default_plain_instance_ident,g,t,mkApp(mkRel 1, l'))
            | _ -> mkApp (g,l')
	    with Not_found -> mkApp (f,l'))
        | _ -> mkApp (irec n f,l'))
    | Cast (m,_,_) when isMeta m ->
	(try lift n (Metamap.find (destMeta m) s) with Not_found -> u)
    | _ ->
	map_constr_with_binders succ irec n u
  in
  if Metamap.is_empty s then c
  else irec 0 c

(* [instance] is used for [res_pf]; the call to [local_strong whd_betaiota]
   has (unfortunately) different subtle side effects:

   - ** Order of subgoals **
     If the lemma is a case analysis with parameters, it will move the
     parameters as first subgoals (e.g. "case H" applied on
     "H:D->A/\B|-C" will present the subgoal |-D first while w/o
     betaiota the subgoal |-D would have come last).

   - ** Betaiota-contraction in statement **
     If the lemma has a parameter which is a function and this
     function is applied in the lemma, then the _strong_ betaiota will
     contract the application of the function to its argument (e.g.
     "apply (H (fun x => x))" in "H:forall f, f 0 = 0 |- 0=0" will
     result in applying the lemma 0=0 in which "(fun x => x) 0" has
     been contracted). A goal to rewrite may then fail or succeed
     differently.

   - ** Naming of hypotheses **
     If a lemma is a function of the form "fun H:(forall a:A, P a)
     => .. F H .." where the expected type of H is "forall b:A, P b",
     then, without reduction, the application of the lemma will
     generate a subgoal "forall a:A, P a" (and intro will use name
     "a"), while with reduction, it will generate a subgoal "forall
     b:A, P b" (and intro will use name "b").

   - ** First-order pattern-matching **
     If a lemma has the type "(fun x => p) t" then rewriting t may fail
     if the type of the lemma is first beta-reduced (this typically happens
     when rewriting a single variable and the type of the lemma is obtained
     by meta_instance (with empty map) which itself calls instance with this
     empty map).
 *)

let instance sigma s c =
  (* if s = [] then c else *)
  local_strong whd_betaiota sigma (plain_instance s c)

(* pseudo-reduction rule:
 * [hnf_prod_app env s (Prod(_,B)) N --> B[N]
 * with an HNF on the first argument to produce a product.
 * if this does not work, then we use the string S as part of our
 * error message. *)

let hnf_prod_app env sigma t n =
  match kind_of_term (whd_betadeltaiota env sigma t) with
    | Prod (_,_,b) -> subst1 n b
    | _ -> anomaly ~label:"hnf_prod_app" (Pp.str "Need a product")

let hnf_prod_appvect env sigma t nl =
  Array.fold_left (hnf_prod_app env sigma) t nl

let hnf_prod_applist env sigma t nl =
  List.fold_left (hnf_prod_app env sigma) t nl

let hnf_lam_app env sigma t n =
  match kind_of_term (whd_betadeltaiota env sigma t) with
    | Lambda (_,_,b) -> subst1 n b
    | _ -> anomaly ~label:"hnf_lam_app" (Pp.str "Need an abstraction")

let hnf_lam_appvect env sigma t nl =
  Array.fold_left (hnf_lam_app env sigma) t nl

let hnf_lam_applist env sigma t nl =
  List.fold_left (hnf_lam_app env sigma) t nl

let splay_prod env sigma =
  let rec decrec env m c =
    let t = whd_betadeltaiota env sigma c in
    match kind_of_term t with
      | Prod (n,a,c0) ->
	  decrec (push_rel (n,None,a) env)
	    ((n,a)::m) c0
      | _ -> m,t
  in
  decrec env []

let splay_lam env sigma =
  let rec decrec env m c =
    let t = whd_betadeltaiota env sigma c in
    match kind_of_term t with
      | Lambda (n,a,c0) ->
	  decrec (push_rel (n,None,a) env)
	    ((n,a)::m) c0
      | _ -> m,t
  in
  decrec env []

let splay_prod_assum env sigma =
  let rec prodec_rec env l c =
    let t = whd_betadeltaiota_nolet env sigma c in
    match kind_of_term t with
    | Prod (x,t,c)  ->
	prodec_rec (push_rel (x,None,t) env)
	  (add_rel_decl (x, None, t) l) c
    | LetIn (x,b,t,c) ->
	prodec_rec (push_rel (x, Some b, t) env)
	  (add_rel_decl (x, Some b, t) l) c
    | Cast (c,_,_)    -> prodec_rec env l c
    | _               -> 
      let t' = whd_betadeltaiota env sigma t in
	if Term.eq_constr t t' then l,t
	else prodec_rec env l t'
  in
  prodec_rec env empty_rel_context

let splay_arity env sigma c =
  let l, c = splay_prod env sigma c in
  match kind_of_term c with
    | Sort s -> l,s
    | _ -> invalid_arg "splay_arity"

let sort_of_arity env sigma c = snd (splay_arity env sigma c)

let splay_prod_n env sigma n =
  let rec decrec env m ln c = if Int.equal m 0 then (ln,c) else
    match kind_of_term (whd_betadeltaiota env sigma c) with
      | Prod (n,a,c0) ->
	  decrec (push_rel (n,None,a) env)
	    (m-1) (add_rel_decl (n,None,a) ln) c0
      | _                      -> invalid_arg "splay_prod_n"
  in
  decrec env n empty_rel_context

let splay_lam_n env sigma n =
  let rec decrec env m ln c = if Int.equal m 0 then (ln,c) else
    match kind_of_term (whd_betadeltaiota env sigma c) with
      | Lambda (n,a,c0) ->
	  decrec (push_rel (n,None,a) env)
	    (m-1) (add_rel_decl (n,None,a) ln) c0
      | _                      -> invalid_arg "splay_lam_n"
  in
  decrec env n empty_rel_context

let is_sort env sigma t =
  match kind_of_term (whd_betadeltaiota env sigma t) with
  | Sort s -> true
  | _ -> false

(* reduction to head-normal-form allowing delta/zeta only in argument
   of case/fix (heuristic used by evar_conv) *)

let whd_betaiota_deltazeta_for_iota_state ts env sigma csts s =
  let rec whrec csts s =
    let (t, stack as s),csts' = whd_state_gen ~csts false betaiota env sigma s in
    match Stack.strip_app stack with
      |args, (Stack.Case _ :: _ as stack') ->
	let (t_o,stack_o),csts_o = whd_state_gen ~csts:csts' false
	  (Closure.RedFlags.red_add_transparent betadeltaiota ts) env sigma (t,args) in
	if reducible_mind_case t_o then whrec csts_o (t_o, stack_o@stack') else s,csts'
      |args, (Stack.Fix _ :: _ as stack') ->
	let (t_o,stack_o),csts_o = whd_state_gen ~csts:csts' false
	  (Closure.RedFlags.red_add_transparent betadeltaiota ts) env sigma (t,args) in
	if isConstruct t_o then whrec csts_o (t_o, stack_o@stack') else s,csts'
      |args, (Stack.Proj (n,m,p) :: stack'' as stack') ->
	let (t_o,stack_o),csts_o = whd_state_gen ~csts:csts' false
	  (Closure.RedFlags.red_add_transparent betadeltaiota ts) env sigma (t,args) in
	if isConstruct t_o then
	  if Closure.is_transparent_constant ts p then
	    whrec Cst_stack.empty (* csts_o *) (Stack.nth stack_o (n+m), stack'')
	  else (* Won't unfold *) (whd_betaiota_state sigma (t_o, stack_o@stack'),csts')
	else s,csts'
      |_, ((Stack.App _| Stack.Shift _|Stack.Update _|Stack.Cst _) :: _|[]) -> s,csts'
  in whrec csts s

let find_conclusion env sigma =
  let rec decrec env c =
    let t = whd_betadeltaiota env sigma c in
    match kind_of_term t with
      | Prod (x,t,c0) -> decrec (push_rel (x,None,t) env) c0
      | Lambda (x,t,c0) -> decrec (push_rel (x,None,t) env) c0
      | t -> t
  in
  decrec env

let is_arity env sigma c =
  match find_conclusion env sigma c with
    | Sort _ -> true
    | _ -> false

(*************************************)
(* Metas *)

let meta_value evd mv =
  let rec valrec mv =
    match meta_opt_fvalue evd mv with
    | Some (b,_) ->
      let metas = Metamap.bind valrec b.freemetas in
      instance evd metas b.rebus
    | None -> mkMeta mv
  in
  valrec mv

let meta_instance sigma b =
  let fm = b.freemetas in
  if Metaset.is_empty fm then b.rebus
  else
    let c_sigma = Metamap.bind (fun mv -> meta_value sigma mv) fm in
    instance sigma c_sigma b.rebus

let nf_meta sigma c = meta_instance sigma (mk_freelisted c)

(* Instantiate metas that create beta/iota redexes *)

let meta_reducible_instance evd b =
  let fm = b.freemetas in
  let fold mv accu =
    let fvalue = try meta_opt_fvalue evd mv with Not_found -> None in
    match fvalue with
    | None -> accu
    | Some (g, (_, s)) -> Metamap.add mv (g.rebus, s) accu
  in
  let metas = Metaset.fold fold fm Metamap.empty in
  let rec irec u =
    let u = whd_betaiota Evd.empty u in
    match kind_of_term u with
    | Case (ci,p,c,bl) when isMeta (strip_outer_cast c) ->
	let m = destMeta (strip_outer_cast c) in
	(match
	  try
	    let g, s = Metamap.find m metas in
            let is_coerce = match s with CoerceToType -> true | _ -> false in
	    if isConstruct g || not is_coerce then Some g else None
	  with Not_found -> None
	  with
	    | Some g -> irec (mkCase (ci,p,g,bl))
	    | None -> mkCase (ci,irec p,c,Array.map irec bl))
    | App (f,l) when isMeta (strip_outer_cast f) ->
	let m = destMeta (strip_outer_cast f) in
	(match
	  try
	    let g, s = Metamap.find m metas in
            let is_coerce = match s with CoerceToType -> true | _ -> false in
	    if isLambda g || not is_coerce then Some g else None
	  with Not_found -> None
	 with
	   | Some g -> irec (mkApp (g,l))
	   | None -> mkApp (f,Array.map irec l))
    | Meta m ->
	(try let g, s = Metamap.find m metas in
          let is_coerce = match s with CoerceToType -> true | _ -> false in
          if not is_coerce then irec g else u
	 with Not_found -> u)
    | Proj (p,c) when isMeta c || isCast c && isMeta (pi1 (destCast c)) ->
	let m = try destMeta c with _ -> destMeta (pi1 (destCast c)) in
	  (match
	  try
	    let g, s = Metamap.find m metas in
            let is_coerce = match s with CoerceToType -> true | _ -> false in
	    if isConstruct g || not is_coerce then Some g else None
	  with Not_found -> None
	  with
	    | Some g -> irec (mkProj (p,g))
	    | None -> mkProj (p,c))
    | _ -> map_constr irec u
  in
  if Metaset.is_empty fm then (* nf_betaiota? *) b.rebus
  else irec b.rebus


let head_unfold_under_prod ts env _ c =
  let unfold (cst,u as cstu) =
    if Cpred.mem cst (snd ts) then
      match constant_opt_value_in env cstu with
	| Some c -> c
	| None -> mkConstU cstu
    else mkConstU cstu in
  let rec aux c =
    match kind_of_term c with
      | Prod (n,t,c) -> mkProd (n,aux t, aux c)
      | _ ->
	  let (h,l) = decompose_app c in
	  match kind_of_term h with
	    | Const cst -> beta_applist (unfold cst,l)
	    | _ -> c in
  aux c