(************************************************************************)
(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014     *)
(*   \VV/  **************************************************************)
(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
(************************************************************************)

(* File initially created by Christine Paulin, 1996 *)

(* This file builds various inductive schemes *)

open Pp
open Util
open Names
open Libnames
open Nameops
open Term
open Namegen
open Declarations
open Entries
open Inductive
open Inductiveops
open Environ
open Reductionops
open Typeops
open Type_errors
open Safe_typing
open Nametab
open Sign

type dep_flag = bool

(* Errors related to recursors building *)
type recursion_scheme_error =
  | NotAllowedCaseAnalysis of (*isrec:*) bool * sorts * inductive
  | NotMutualInScheme of inductive * inductive

exception RecursionSchemeError of recursion_scheme_error

let make_prod_dep dep env = if dep then mkProd_name env else mkProd
let mkLambda_string s t c = mkLambda (Name (id_of_string s), t, c)

(*******************************************)
(* Building curryfied elimination          *)
(*******************************************)

(**********************************************************************)
(* Building case analysis schemes *)
(* Christine Paulin, 1996 *)

let mis_make_case_com dep env sigma ind (mib,mip as specif) kind =
  let lnamespar = List.map
    (fun (n, c, t) -> (n, c, Termops.refresh_universes t))
    mib.mind_params_ctxt
  in

  if not (List.mem kind (elim_sorts specif)) then
    raise
      (RecursionSchemeError
	 (NotAllowedCaseAnalysis (false, Termops.new_sort_in_family kind, ind)));

  let ndepar = mip.mind_nrealargs_ctxt + 1 in

  (* Pas génant car env ne sert pas à typer mais juste à renommer les Anonym *)
  (* mais pas très joli ... (mais manque get_sort_of à ce niveau) *)
  let env' = push_rel_context lnamespar env in

  let indf = make_ind_family(ind, Termops.extended_rel_list 0 lnamespar) in
  let constrs = get_constructors env indf in

  let rec add_branch env k =
    if k = Array.length mip.mind_consnames then
      let nbprod = k+1 in

      let indf' = lift_inductive_family nbprod indf in
      let arsign,_ = get_arity env indf' in
      let depind = build_dependent_inductive env indf' in
      let deparsign = (Anonymous,None,depind)::arsign in

      let ci = make_case_info env ind RegularStyle in
      let pbody =
        appvect
          (mkRel (ndepar + nbprod),
           if dep then Termops.extended_rel_vect 0 deparsign
           else Termops.extended_rel_vect 1 arsign) in
      let p =
	it_mkLambda_or_LetIn_name env'
	  ((if dep then mkLambda_name env' else mkLambda)
	   (Anonymous,depind,pbody))
          arsign
      in
      it_mkLambda_or_LetIn_name env'
       	(mkCase (ci, lift ndepar p,
		     mkRel 1,
		     Termops.rel_vect ndepar k))
       	deparsign
    else
      let cs = lift_constructor (k+1) constrs.(k) in
      let t = build_branch_type env dep (mkRel (k+1)) cs in
      mkLambda_string "f" t
	(add_branch (push_rel (Anonymous, None, t) env) (k+1))
  in
  let typP = make_arity env' dep indf (Termops.new_sort_in_family kind) in
  it_mkLambda_or_LetIn_name env
    (mkLambda_string "P" typP
       (add_branch (push_rel (Anonymous,None,typP) env') 0)) lnamespar

(* check if the type depends recursively on one of the inductive scheme *)

(**********************************************************************)
(* Building the recursive elimination *)
(* Christine Paulin, 1996 *)

(*
 * t is the type of the constructor co and recargs is the information on
 * the recursive calls. (It is assumed to be in form given by the user).
 * build the type of the corresponding branch of the recurrence principle
 * assuming f has this type, branch_rec gives also the term
 *   [x1]..[xk](f xi (F xi) ...) to be put in the corresponding branch of
 * the case operation
 * FPvect gives for each inductive definition if we want an elimination
 * on it with which predicate and which recursive function.
 *)

let type_rec_branch is_rec dep env sigma (vargs,depPvect,decP) tyi cs recargs =
  let make_prod = make_prod_dep dep in
  let nparams = List.length vargs in
  let process_pos env depK pk =
    let rec prec env i sign p =
      let p',largs = whd_betadeltaiota_nolet_stack env sigma p in
      match kind_of_term p' with
	| Prod (n,t,c) ->
	    let d = (n,None,t) in
	    make_prod env (n,t,prec (push_rel d env) (i+1) (d::sign) c)
	| LetIn (n,b,t,c) ->
	    let d = (n,Some b,t) in
	    mkLetIn (n,b,t,prec (push_rel d env) (i+1) (d::sign) c)
     	| Ind (_,_) ->
	    let realargs = list_skipn nparams largs in
	    let base = applist (lift i pk,realargs) in
            if depK then
	      Reduction.beta_appvect
                base [|applist (mkRel (i+1), Termops.extended_rel_list 0 sign)|]
            else
	      base
      	| _ -> assert false
    in
    prec env 0 []
  in
  let rec process_constr env i c recargs nhyps li =
    if nhyps > 0 then match kind_of_term c with
      | Prod (n,t,c_0) ->
          let (optionpos,rest) =
	    match recargs with
	      | [] -> None,[]
              | ra::rest ->
                  (match dest_recarg ra with
	            | Mrec (_,j) when is_rec -> (depPvect.(j),rest)
	            | Imbr _  ->
		        Flags.if_warn msg_warning (str "Ignoring recursive call");
		        (None,rest)
                    | _ -> (None, rest))
	  in
          (match optionpos with
	     | None ->
		 make_prod env
		   (n,t,
		    process_constr (push_rel (n,None,t) env) (i+1) c_0 rest
		      (nhyps-1) (i::li))
             | Some(dep',p) ->
		 let nP = lift (i+1+decP) p in
                 let env' = push_rel (n,None,t) env in
		 let t_0 = process_pos env' dep' nP (lift 1 t) in
		 make_prod_dep (dep or dep') env
                   (n,t,
		    mkArrow t_0
		      (process_constr
			(push_rel (Anonymous,None,t_0) env')
			 (i+2) (lift 1 c_0) rest (nhyps-1) (i::li))))
      | LetIn (n,b,t,c_0) ->
	  mkLetIn (n,b,t,
		   process_constr
		     (push_rel (n,Some b,t) env)
		     (i+1) c_0 recargs (nhyps-1) li)
      | _ -> assert false
    else
      if dep then
	let realargs = List.map (fun k -> mkRel (i-k)) (List.rev li) in
        let params = List.map (lift i) vargs in
        let co = applist (mkConstruct cs.cs_cstr,params@realargs) in
	Reduction.beta_appvect c [|co|]
      else c
  in
  let nhyps = List.length cs.cs_args in
  let nP = match depPvect.(tyi) with
    | Some(_,p) -> lift (nhyps+decP) p
    | _ -> assert false in
  let base = appvect (nP,cs.cs_concl_realargs) in
  let c = it_mkProd_or_LetIn base cs.cs_args in
  process_constr env 0 c recargs nhyps []

let make_rec_branch_arg env sigma (nparrec,fvect,decF) f cstr recargs =
  let process_pos env fk  =
    let rec prec env i hyps p =
      let p',largs = whd_betadeltaiota_nolet_stack env sigma p in
      match kind_of_term p' with
	| Prod (n,t,c) ->
	    let d = (n,None,t) in
	    mkLambda_name env (n,t,prec (push_rel d env) (i+1) (d::hyps) c)
	| LetIn (n,b,t,c) ->
	    let d = (n,Some b,t) in
	    mkLetIn (n,b,t,prec (push_rel d env) (i+1) (d::hyps) c)
     	| Ind _ ->
            let realargs = list_skipn nparrec largs
            and arg = appvect (mkRel (i+1), Termops.extended_rel_vect 0 hyps) in
            applist(lift i fk,realargs@[arg])
     	| _ -> assert false
    in
    prec env 0 []
  in
  (* ici, cstrprods est la liste des produits du constructeur instantié *)
  let rec process_constr env i f = function
    | (n,None,t as d)::cprest, recarg::rest ->
        let optionpos =
	  match dest_recarg recarg with
            | Norec   -> None
            | Imbr _  -> None
            | Mrec (_,i)  -> fvect.(i)
	in
        (match optionpos with
           | None ->
	       mkLambda_name env
		 (n,t,process_constr (push_rel d env) (i+1)
		    (whd_beta Evd.empty (applist (lift 1 f, [(mkRel 1)])))
		    (cprest,rest))
           | Some(_,f_0) ->
	       let nF = lift (i+1+decF) f_0 in
               let env' = push_rel d env in
	       let arg = process_pos env' nF (lift 1 t) in
               mkLambda_name env
		 (n,t,process_constr env' (i+1)
		    (whd_beta Evd.empty (applist (lift 1 f, [(mkRel 1); arg])))
		    (cprest,rest)))
    | (n,Some c,t as d)::cprest, rest ->
	mkLetIn
	  (n,c,t,
	   process_constr (push_rel d env) (i+1) (lift 1 f)
	     (cprest,rest))
    | [],[] -> f
    | _,[] | [],_ -> anomaly "process_constr"

  in
  process_constr env 0 f (List.rev cstr.cs_args, recargs)


(* Cut a context ctx in 2 parts (ctx1,ctx2) with ctx1 containing k
   variables *)
let context_chop k ctx =
  let rec chop_aux acc = function
    | (0, l2) -> (List.rev acc, l2)
    | (n, ((_,Some _,_ as h)::t)) -> chop_aux (h::acc) (n, t)
    | (n, (h::t)) -> chop_aux (h::acc) (pred n, t)
    | (_, []) -> failwith "context_chop"
  in chop_aux [] (k,ctx)


(* Main function *)
let mis_make_indrec env sigma listdepkind mib =
  let nparams = mib.mind_nparams in
  let nparrec = mib. mind_nparams_rec in
  let lnonparrec,lnamesparrec =
    context_chop (nparams-nparrec) mib.mind_params_ctxt in
  let nrec = List.length listdepkind in
  let depPvec =
    Array.create mib.mind_ntypes (None : (bool * constr) option) in
  let _ =
    let rec
	assign k = function
	  | [] -> ()
          | (indi,mibi,mipi,dep,_)::rest ->
              (Array.set depPvec (snd indi) (Some(dep,mkRel k));
               assign (k-1) rest)
    in
      assign nrec listdepkind in
  let recargsvec =
    Array.map (fun mip -> mip.mind_recargs) mib.mind_packets in
  (* recarg information for non recursive parameters *)
  let rec recargparn l n =
    if n = 0 then l else recargparn (mk_norec::l) (n-1) in
  let recargpar = recargparn [] (nparams-nparrec) in
  let make_one_rec p =
    let makefix nbconstruct =
      let rec mrec i ln ltyp ldef = function
	| (indi,mibi,mipi,dep,_)::rest ->
	    let tyi = snd indi in
	    let nctyi =
              Array.length mipi.mind_consnames in (* nb constructeurs du type*)

            (* arity in the context of the fixpoint, i.e.
               P1..P_nrec f1..f_nbconstruct *)
	    let args = Termops.extended_rel_list (nrec+nbconstruct) lnamesparrec in
	    let indf = make_ind_family(indi,args) in

	    let arsign,_ = get_arity env indf in
	    let depind = build_dependent_inductive env indf in
	    let deparsign = (Anonymous,None,depind)::arsign in

	    let nonrecpar = rel_context_length lnonparrec in
	    let larsign = rel_context_length deparsign in
	    let ndepar = larsign - nonrecpar in
	    let dect = larsign+nrec+nbconstruct in

            (* constructors in context of the Cases expr, i.e.
               P1..P_nrec f1..f_nbconstruct F_1..F_nrec a_1..a_nar x:I *)
	    let args' = Termops.extended_rel_list (dect+nrec) lnamesparrec in
	    let args'' = Termops.extended_rel_list ndepar lnonparrec in
            let indf' = make_ind_family(indi,args'@args'') in

	    let branches =
	      let constrs = get_constructors env indf' in
	      let fi = Termops.rel_vect (dect-i-nctyi) nctyi in
	      let vecfi = Array.map
		(fun f -> appvect (f, Termops.extended_rel_vect ndepar lnonparrec))
		fi
	      in
		array_map3
		  (make_rec_branch_arg env sigma
		      (nparrec,depPvec,larsign))
                  vecfi constrs (dest_subterms recargsvec.(tyi))
	    in

	    let j = (match depPvec.(tyi) with
	      | Some (_,c) when isRel c -> destRel c
	      | _ -> assert false)
	    in

	    (* Predicate in the context of the case *)

	    let depind' = build_dependent_inductive env indf' in
	    let arsign',_ = get_arity env indf' in
	    let deparsign' = (Anonymous,None,depind')::arsign' in

	    let pargs =
	      let nrpar = Termops.extended_rel_list (2*ndepar) lnonparrec
	      and nrar = if dep then Termops.extended_rel_list 0 deparsign'
		else Termops.extended_rel_list 1 arsign'
	      in nrpar@nrar

	    in

	    (* body of i-th component of the mutual fixpoint *)
	    let deftyi =
	      let ci = make_case_info env indi RegularStyle in
	      let concl = applist (mkRel (dect+j+ndepar),pargs) in
	      let pred =
		it_mkLambda_or_LetIn_name env
		  ((if dep then mkLambda_name env else mkLambda)
		      (Anonymous,depind',concl))
		  arsign'
	      in
		it_mkLambda_or_LetIn_name env
		  (mkCase (ci, pred,
		          mkRel 1,
			  branches))
		  (Termops.lift_rel_context nrec deparsign)
	    in

	    (* type of i-th component of the mutual fixpoint *)

	    let typtyi =
	      let concl =
		let pargs = if dep then Termops.extended_rel_vect 0 deparsign
		  else Termops.extended_rel_vect 1 arsign
		in appvect (mkRel (nbconstruct+ndepar+nonrecpar+j),pargs)
	      in it_mkProd_or_LetIn_name env
		concl
		deparsign
	    in
	      mrec (i+nctyi) (rel_context_nhyps arsign ::ln) (typtyi::ltyp)
                (deftyi::ldef) rest
        | [] ->
	    let fixn = Array.of_list (List.rev ln) in
            let fixtyi = Array.of_list (List.rev ltyp) in
            let fixdef = Array.of_list (List.rev ldef) in
            let names = Array.create nrec (Name(id_of_string "F")) in
	      mkFix ((fixn,p),(names,fixtyi,fixdef))
      in
	mrec 0 [] [] []
    in
    let rec make_branch env i = function
      | (indi,mibi,mipi,dep,_)::rest ->
          let tyi = snd indi in
	  let nconstr = Array.length mipi.mind_consnames in
	  let rec onerec env j =
	    if j = nconstr then
	      make_branch env (i+j) rest
	    else
	      let recarg = (dest_subterms recargsvec.(tyi)).(j) in
	      let recarg = recargpar@recarg in
	      let vargs = Termops.extended_rel_list (nrec+i+j) lnamesparrec in
	      let cs = get_constructor (indi,mibi,mipi,vargs) (j+1) in
	      let p_0 =
		type_rec_branch
                  true dep env sigma (vargs,depPvec,i+j) tyi cs recarg
	      in
		mkLambda_string "f" p_0
		  (onerec (push_rel (Anonymous,None,p_0) env) (j+1))
	  in onerec env 0
      | [] ->
	  makefix i listdepkind
    in
    let rec put_arity env i = function
      | (indi,_,_,dep,kinds)::rest ->
	  let indf = make_ind_family (indi, Termops.extended_rel_list i lnamesparrec) in
	  let typP = make_arity env dep indf (Termops.new_sort_in_family kinds) in
	    mkLambda_string "P" typP
	      (put_arity (push_rel (Anonymous,None,typP) env) (i+1) rest)
      | [] ->
	  make_branch env 0 listdepkind
    in

    (* Body on make_one_rec *)
    let (indi,mibi,mipi,dep,kind) = List.nth listdepkind p in

      if (mis_is_recursive_subset
	(List.map (fun (indi,_,_,_,_) -> snd indi) listdepkind)
	mipi.mind_recargs)
      then
	let env' = push_rel_context lnamesparrec env in
	  it_mkLambda_or_LetIn_name env (put_arity env' 0 listdepkind)
	    lnamesparrec
      else
	mis_make_case_com dep env sigma indi (mibi,mipi) kind
  in
    (* Body of mis_make_indrec *)
    list_tabulate make_one_rec nrec

(**********************************************************************)
(* This builds elimination predicate for Case tactic *)

let build_case_analysis_scheme env sigma ity dep kind =
  let (mib,mip) = lookup_mind_specif env ity in
  mis_make_case_com dep env sigma ity (mib,mip) kind

let build_case_analysis_scheme_default env sigma ity kind =
  let (mib,mip) = lookup_mind_specif env ity in
  let dep = inductive_sort_family mip <> InProp in
  mis_make_case_com dep env sigma ity (mib,mip) kind


(**********************************************************************)
(* [modify_sort_scheme s rec] replaces the sort of the scheme
   [rec] by [s] *)

let change_sort_arity sort =
  let rec drec a = match kind_of_term a with
    | Cast (c,_,_) -> drec c
    | Prod (n,t,c) -> mkProd (n, t, drec c)
    | LetIn (n,b,t,c) -> mkLetIn (n,b, t, drec c)
    | Sort _ -> mkSort sort
    | _ -> assert false
  in
    drec

(* [npar] is the number of expected arguments (then excluding letin's) *)
let modify_sort_scheme sort =
  let rec drec npar elim =
    match kind_of_term elim with
      | Lambda (n,t,c) ->
	  if npar = 0 then
	    mkLambda (n, change_sort_arity sort t, c)
	  else
	    mkLambda (n, t, drec (npar-1) c)
      | LetIn (n,b,t,c) -> mkLetIn (n,b,t,drec npar c)
      | _ -> anomaly "modify_sort_scheme: wrong elimination type"
  in
  drec

(* Change the sort in the type of an inductive definition, builds the
   corresponding eta-expanded term *)
let weaken_sort_scheme sort npars term =
  let rec drec np elim =
    match kind_of_term elim with
      | Prod (n,t,c) ->
	  if np = 0 then
            let t' = change_sort_arity sort t in
            mkProd (n, t', c),
            mkLambda (n, t', mkApp(term,Termops.rel_vect 0 (npars+1)))
	  else
            let c',term' = drec (np-1) c in
	    mkProd (n, t, c'), mkLambda (n, t, term')
      | LetIn (n,b,t,c) -> let c',term' = drec np c in
           mkLetIn (n,b,t,c'), mkLetIn (n,b,t,term')
      | _ -> anomaly "weaken_sort_scheme: wrong elimination type"
  in
  drec npars

(**********************************************************************)
(* Interface to build complex Scheme *)
(* Check inductive types only occurs once
(otherwise we obtain a meaning less scheme) *)

let check_arities listdepkind =
  let _ = List.fold_left
    (fun ln ((_,ni as mind),mibi,mipi,dep,kind) ->
       let kelim = elim_sorts (mibi,mipi) in
       if not (List.exists ((=) kind) kelim) then raise
	 (RecursionSchemeError
	   (NotAllowedCaseAnalysis (true, Termops.new_sort_in_family kind,mind)))
       else if List.mem ni ln then raise
	 (RecursionSchemeError (NotMutualInScheme (mind,mind)))
       else ni::ln)
	    [] listdepkind
  in true

let build_mutual_induction_scheme env sigma = function
  | (mind,dep,s)::lrecspec ->
      let (mib,mip) = Global.lookup_inductive mind in
      let (sp,tyi) = mind in
      let listdepkind =
	(mind,mib,mip,dep,s)::
    	(List.map
	   (function (mind',dep',s') ->
	      let (sp',_) = mind' in
	      if sp=sp' then
                let (mibi',mipi') = lookup_mind_specif env mind' in
		(mind',mibi',mipi',dep',s')
	      else
		raise (RecursionSchemeError (NotMutualInScheme (mind,mind'))))
	   lrecspec)
      in
      let _ = check_arities listdepkind in
      mis_make_indrec env sigma listdepkind mib
  | _ -> anomaly "build_induction_scheme expects a non empty list of inductive types"

let build_induction_scheme env sigma ind dep kind =
  let (mib,mip) = lookup_mind_specif env ind in
  List.hd (mis_make_indrec env sigma [(ind,mib,mip,dep,kind)] mib)

(*s Eliminations. *)

let elimination_suffix = function
  | InProp -> "_ind"
  | InSet  -> "_rec"
  | InType -> "_rect"

let case_suffix = "_case"

let make_elimination_ident id s = add_suffix id (elimination_suffix s)

(* Look up function for the default elimination constant *)

let lookup_eliminator ind_sp s =
  let kn,i = ind_sp in
  let mp,dp,l = repr_mind kn in
  let ind_id = (Global.lookup_mind kn).mind_packets.(i).mind_typename in
  let id = add_suffix ind_id (elimination_suffix s) in
  (* Try first to get an eliminator defined in the same section as the *)
  (* inductive type *)
  try
    let cst =Global.constant_of_delta_kn (make_kn mp dp (label_of_id id)) in
    let _ = Global.lookup_constant cst in
      mkConst cst
  with Not_found ->
  (* Then try to get a user-defined eliminator in some other places *)
  (* using short name (e.g. for "eq_rec") *)
  try constr_of_global (Nametab.locate (qualid_of_ident id))
  with Not_found ->
    errorlabstrm "default_elim"
      (strbrk "Cannot find the elimination combinator " ++
       pr_id id ++ strbrk ", the elimination of the inductive definition " ++
       pr_global_env Idset.empty (IndRef ind_sp) ++
       strbrk " on sort " ++ Termops.pr_sort_family s ++
       strbrk " is probably not allowed.")
