(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
(************************************************************************)

module CVars = Vars

open Pp
open CErrors
open Util
open Names
open Nameops
open Constr
open Context
open Termops
open Environ
open EConstr
open Vars
open Namegen
open Declarations
open Inductiveops
open Reductionops
open Type_errors
open Glob_term
open Glob_ops
open Retyping
open Pretype_errors
open Evarutil
open Evardefine
open Evarsolve
open Evarconv
open Evd
open Context.Rel.Declaration
open GlobEnv

module RelDecl = Context.Rel.Declaration
module NamedDecl = Context.Named.Declaration

(* Pattern-matching errors *)

type pattern_matching_error =
  | BadPattern of constructor * constr
  | BadConstructor of constructor * inductive
  | WrongNumargConstructor of
      {cstr:constructor; expanded:bool; nargs:int; expected_nassums:int; expected_ndecls:int}
  | WrongNumargInductive of
      {ind:inductive; expanded:bool; nargs:int; expected_nassums:int; expected_ndecls:int}
  | UnusedClause of cases_pattern list
  | NonExhaustive of cases_pattern list
  | CannotInferPredicate of (constr * types) array

exception PatternMatchingError of env * evar_map * pattern_matching_error

let raise_pattern_matching_error ?loc (env,sigma,te) =
  Loc.raise ?loc (PatternMatchingError(env,sigma,te))

let error_bad_pattern ?loc env sigma cstr ind =
  raise_pattern_matching_error ?loc
    (env, sigma, BadPattern (cstr,ind))

let error_bad_constructor ?loc env cstr ind =
  raise_pattern_matching_error ?loc
    (env, Evd.empty, BadConstructor (cstr,ind))

let error_wrong_numarg_constructor ?loc env ~cstr ~expanded ~nargs ~expected_nassums ~expected_ndecls =
  raise_pattern_matching_error ?loc (env, Evd.empty,
    WrongNumargConstructor {cstr; expanded; nargs; expected_nassums; expected_ndecls})

let error_wrong_numarg_inductive ?loc env ~ind ~expanded ~nargs ~expected_nassums ~expected_ndecls =
  raise_pattern_matching_error ?loc (env, Evd.empty,
    WrongNumargInductive {ind; expanded; nargs; expected_nassums; expected_ndecls})

let list_try_compile f l =
  let rec aux errors = function
  | [] -> if errors = [] then anomaly (str "try_find_f.") else Exninfo.iraise (List.last errors)
  | h::t ->
      try f h
      with UserError _ | TypeError _ | PretypeError _ | PatternMatchingError _ as e ->
            let e = Exninfo.capture e in
            aux (e::errors) t in
  aux [] l

let force_name =
  let nx = Name default_dependent_ident in function Anonymous -> nx | na -> na

(************************************************************************)
(*            Pattern-matching compilation (Cases)                      *)
(************************************************************************)

(************************************************************************)
(* Configuration, errors and warnings *)

open Pp

let msg_may_need_inversion () =
  strbrk "Found a matching with no clauses on a term unknown to have an empty inductive type."

(* Utils *)
let make_anonymous_patvars n =
  List.make n (DAst.make @@ PatVar Anonymous)

(* We have x1:t1...xn:tn,xi':ti,y1..yk |- c and re-generalize
   over xi:ti to get x1:t1...xn:tn,xi':ti,y1..yk |- c[xi:=xi'] *)

let relocate_rel n1 n2 k j = if Int.equal j (n1 + k) then n2+k else j

let rec relocate_index sigma n1 n2 k t =
  match EConstr.kind sigma t with
  | Rel j when Int.equal j (n1 + k) -> mkRel (n2+k)
  | Rel j when j < n1+k -> t
  | Rel j when j > n1+k -> t
  | _ -> EConstr.map_with_binders sigma succ (relocate_index sigma n1 n2) k t

(**********************************************************************)
(* Structures used in compiling pattern-matching *)

let (!!) env = GlobEnv.env env

type 'a rhs =
    { rhs_env    : GlobEnv.t;
      rhs_vars   : Id.Set.t;
      avoid_ids  : Id.Set.t;
      it         : 'a option}

type 'a equation =
    { patterns     : cases_pattern list;
      rhs          : 'a rhs;
      alias_stack  : Name.t list;
      eqn_loc      : Loc.t option;
      orig         : int option;
      catch_all_vars : Id.t CAst.t list }

type 'a matrix = 'a equation list

(* 1st argument of IsInd is the original ind before extracting the summary *)
type tomatch_type =
  | IsInd of types * inductive_type * Name.t list
  | NotInd of constr option * types

(* spiwack: The first argument of [Pushed] is [true] for initial
   Pushed and [false] otherwise. Used to decide whether the term being
   matched on must be aliased in the variable case (only initial
   Pushed need to be aliased). The first argument of [Alias] is [true]
   if the alias was introduced by an initial pushed and [false]
   otherwise.*)
type tomatch_status =
  | Pushed of (bool*((constr * tomatch_type) * int list * Name.t))
  | Alias of (bool*(Name.t * constr * (constr * types)))
  | NonDepAlias
  | Abstract of int * rel_declaration

type tomatch_stack = tomatch_status list

(* We keep a constr for aliases and a cases_pattern for error message *)

type pattern_history =
  | Top
  | MakeConstructor of constructor * pattern_continuation

and pattern_continuation =
  | Continuation of int * cases_pattern list * pattern_history
  | Result of cases_pattern list

let start_history n = Continuation (n, [], Top)

let feed_history arg = function
  | Continuation (n, l, h) when n>=1 ->
      Continuation (n-1, arg :: l, h)
  | Continuation (n, _, _) ->
      anomaly (str "Bad number of expected remaining patterns: " ++ int n ++ str ".")
  | Result _ ->
      anomaly (Pp.str "Exhausted pattern history.")

(* This is for non exhaustive error message *)

let rec glob_pattern_of_partial_history args2 = function
  | Continuation (n, args1, h) ->
      let args3 = make_anonymous_patvars (n - (List.length args2)) in
      build_glob_pattern (List.rev_append args1 (args2@args3)) h
  | Result pl -> pl

and build_glob_pattern args = function
  | Top -> args
  | MakeConstructor (pci, rh) ->
      glob_pattern_of_partial_history
        [DAst.make @@ PatCstr (pci, args, Anonymous)] rh

let complete_history = glob_pattern_of_partial_history []

(* This is to build glued pattern-matching history and alias bodies *)

let pop_history_pattern = function
  | Continuation (0, l, Top) ->
      Result (List.rev l)
  | Continuation (0, l, MakeConstructor (pci, rh)) ->
      feed_history (DAst.make @@ PatCstr (pci,List.rev l,Anonymous)) rh
  | _ ->
      anomaly (Pp.str "Constructor not yet filled with its arguments.")

let pop_history h =
  feed_history (DAst.make @@ PatVar Anonymous) h

(* Builds a continuation expecting [n] arguments and building [ci] applied
   to this [n] arguments *)

let push_history_pattern n pci cont =
  Continuation (n, [], MakeConstructor (pci, cont))

(* A pattern-matching problem has the following form:

   env, evd |- match terms_to_tomatch return pred with mat end

  where terms_to_match is some sequence of "instructions" (t1 ... tp)

  and mat is some matrix

   (p11 ... p1n -> rhs1)
   (    ...            )
   (pm1 ... pmn -> rhsm)

  Terms to match: there are 3 kinds of instructions

  - "Pushed" terms to match are typed in [env]; these are usually just
    Rel(n) except for the initial terms given by user; in Pushed ((c,tm),deps,na),
    [c] is the reference to the term (which is a Rel or an initial term), [tm] is
    its type (telling whether we know if it is an inductive type or not),
    [deps] is the list of terms to abstract before matching on [c] (these are
    rels too)
  - "Abstract" instructions mean that an abstraction has to be inserted in the
    current branch to build (this means a pattern has been detected dependent
    in another one and a generalization is necessary to ensure well-typing)
    Abstract instructions extend the [env] in which the other instructions
    are typed
  - "Alias" instructions mean an alias has to be inserted (this alias
    is usually removed at the end, except when its type is not the
    same as the type of the matched term from which it comes -
    typically because the inductive types are "real" parameters)
  - "NonDepAlias" instructions mean the completion of a matching over
    a term to match as for Alias but without inserting this alias
    because there is no dependency in it

  Right-hand sides:

  They consist of a raw term to type in an environment specific to the
  clause they belong to: the names of declarations are those of the
  variables present in the patterns. Therefore, they come with their
  own [rhs_env] (actually it is the same as [env] except for the names
  of variables).

*)

type 'a pattern_matching_problem =
    { env       : GlobEnv.t;
      pred      : constr;
      tomatch   : tomatch_stack;
      history   : pattern_continuation;
      mat       : 'a matrix;
      caseloc   : Loc.t option;
      casestyle : case_style;
      typing_function: type_constraint -> GlobEnv.t -> evar_map -> 'a option -> evar_map * unsafe_judgment }

(*--------------------------------------------------------------------------*
 * A few functions to infer the inductive type from the patterns instead of *
 * checking that the patterns correspond to the ind. type of the            *
 * destructurated object. Allows type inference of examples like            *
 *  match n with O => true | _ => false end                                 *
 *  match x in I with C => true | _ => false end                            *
 *--------------------------------------------------------------------------*)

(* Computing the inductive type from the matrix of patterns *)

(* We use the "in I" clause to coerce the terms to match and otherwise
   use the constructor to know in which type is the matching problem

   Note that insertion of coercions inside nested patterns is done
   each time the matrix is expanded *)

let rec find_row_ind = function
    [] -> None
  | p :: l ->
    match DAst.get p with
    | PatVar _ -> find_row_ind l
    | PatCstr(c,_,_) -> Some (p.CAst.loc,c)

let inductive_template env sigma tmloc ind =
  let sigma, indu = Evd.fresh_inductive_instance env sigma ind in
  let arsign = inductive_alldecls env indu in
  let indu = on_snd EInstance.make indu in
  let hole_source i = match tmloc with
    | Some loc -> Loc.tag ~loc @@ Evar_kinds.TomatchTypeParameter (ind,i)
    | None     -> Loc.tag      @@ Evar_kinds.TomatchTypeParameter (ind,i) in
   let (sigma, _, evarl, _) =
    List.fold_right
      (fun decl (sigma, subst, evarl, n) ->
        match decl with
        | LocalAssum (na,ty) ->
            let ty = EConstr.of_constr ty in
            let ty' = substl subst ty in
            let sigma, e =
              Evarutil.new_evar env ~src:(hole_source n) sigma ty'
            in
            (sigma, e::subst,e::evarl,n+1)
        | LocalDef (na,b,ty) ->
            let b = EConstr.of_constr b in
            (sigma, substl subst b::subst,evarl,n+1))
      arsign (sigma, [], [], 1) in
   sigma, applist (mkIndU indu,List.rev evarl)

let try_find_ind env sigma typ realnames =
  let (IndType(indf,realargs) as ind) = find_rectype env sigma typ in
  let names =
    match realnames with
      | Some names -> names
      | None ->
          let ind = fst (fst (dest_ind_family indf)) in
          List.make (inductive_nrealdecls env ind) Anonymous in
  IsInd (typ,ind,names)

let inh_coerce_to_ind env sigma0 loc ty tyi =
  let sigma, expected_typ = inductive_template env sigma0 loc tyi in
  (* Try to refine the type with inductive information coming from the
     constructor and renounce if not able to give more information *)
  (* devrait être indifférent d'exiger leq ou pas puisque pour
     un inductif cela doit être égal *)
  match Evarconv.unify_leq_delay env sigma expected_typ ty with
  | sigma -> sigma
  | exception Evarconv.UnableToUnify _ -> sigma0

let binding_vars_of_inductive sigma = function
  | NotInd _ -> []
  | IsInd (_,IndType(_,realargs),_) -> List.filter (isRel sigma) realargs

let set_tomatch_realnames names = function
  | NotInd _ as t -> t
  | IsInd (typ,ind,_) -> IsInd (typ,ind,names)

let extract_inductive_data env sigma decl =
  match decl with
  | LocalAssum (_,t) ->
    let tmtyp =
      try try_find_ind env sigma t None
      with Not_found -> NotInd (None,t) in
    let tmtypvars = binding_vars_of_inductive sigma tmtyp in
    (tmtyp,tmtypvars)
  | LocalDef (_,_,t) ->
    (NotInd (None, t), [])

let unify_tomatch_with_patterns env sigma loc typ pats realnames =
  match find_row_ind pats with
    | None -> sigma, NotInd (None,typ)
    | Some (_,(ind,_)) ->
        let sigma = inh_coerce_to_ind env sigma loc typ ind in
        try sigma, try_find_ind env sigma typ realnames
        with Not_found -> sigma, NotInd (None,typ)

let find_tomatch_tycon env sigma loc = function
  (* Try if some 'in I ...' is present and can be used as a constraint *)
  | Some {CAst.v=(ind,realnal)} ->
      let sigma, tycon = inductive_template env sigma loc ind in
      sigma, mk_tycon tycon, Some (List.rev realnal)
  | None ->
      sigma, empty_tycon, None

let make_return_predicate_ltac_lvar env sigma na tm c =
  (* If we have an [x as x return ...] clause and [x] expands to [c],
     we have to update the status of [x] in the substitution:
     - if [c] is a variable [id'], then [x] should now become [id']
     - otherwise, [x] should be hidden *)
  match na, DAst.get tm with
  | Name id, (GVar id' | GRef (GlobRef.VarRef id', _)) when Id.equal id id' ->
     let expansion = match kind sigma c with
       | Var id' -> Name id'
       | _ -> Anonymous in
       GlobEnv.hide_variable env expansion id
  | _ -> env

let is_patvar pat =
  match DAst.get pat with
  | PatVar _ -> true
  | _ -> false

let coerce_row ~program_mode typing_fun env sigma pats (tomatch,(na,indopt)) =
  let loc = loc_of_glob_constr tomatch in
  let sigma, tycon, realnames = find_tomatch_tycon !!env sigma loc indopt in
  let sigma, j = typing_fun tycon env sigma tomatch in
  let sigma, j = Coercion.inh_coerce_to_base ?loc:(loc_of_glob_constr tomatch) ~program_mode !!env sigma j in
  let typ = nf_evar sigma j.uj_type in
  let env = make_return_predicate_ltac_lvar env sigma na tomatch j.uj_val in
  let sigma, t =
    if realnames = None && pats <> [] && List.for_all is_patvar pats then
      sigma, NotInd (None,typ)
    else
    try sigma, try_find_ind !!env sigma typ realnames
    with Not_found ->
      unify_tomatch_with_patterns !!env sigma loc typ pats realnames
  in
  ((env, sigma), (j.uj_val,t))

let coerce_to_indtype ~program_mode typing_fun env sigma matx tomatchl =
  let pats = List.map (fun r ->  r.patterns) matx in
  let matx' = match matrix_transpose pats with
    | [] -> List.map (fun _ -> []) tomatchl (* no patterns at all *)
    | m -> m in
  let (env, sigma), tms = List.fold_left2_map (fun (env, sigma) -> coerce_row ~program_mode typing_fun env sigma) (env, sigma) matx' tomatchl in
  env, sigma, tms

(************************************************************************)
(* Utils *)

let mkExistential ?(src=(Loc.tag Evar_kinds.InternalHole)) env sigma =
  let sigma, (e, u) = Evarutil.new_type_evar env sigma ~src:src univ_flexible_alg in
  sigma, e

let adjust_tomatch_to_pattern ~program_mode sigma pb ((current,typ),deps,dep) =
  (* Ideally, we could find a common inductive type to which both the
     term to match and the patterns coerce *)
  (* In practice, we coerce the term to match if it is not already an
     inductive type and it is not dependent; moreover, we use only
     the first pattern type and forget about the others *)
  let typ,names =
    match typ with IsInd(t,_,names) -> t,Some names | NotInd(_,t) -> t,None in
  let tmtyp =
    try try_find_ind !!(pb.env) sigma typ names
    with Not_found -> NotInd (None,typ) in
  match tmtyp with
  | NotInd (None,typ) ->
      let tm1 = List.map (fun eqn -> List.hd eqn.patterns) pb.mat in
      (match find_row_ind tm1 with
        | None -> sigma, (current, tmtyp)
        | Some (loc,(ind,_)) ->
            let sigma, indt = inductive_template !!(pb.env) sigma None ind in
            let sigma, current =
              if List.is_empty deps && isEvar sigma typ then
              (* Don't insert coercions if dependent; only solve evars *)
                match Evarconv.unify_leq_delay !!(pb.env) sigma indt typ with
                | exception Evarconv.UnableToUnify _ -> sigma, current
                | sigma -> sigma, current
              else
                let sigma, j, _trace = Coercion.inh_conv_coerce_to ?loc ~program_mode true !!(pb.env) sigma (make_judge current typ) indt in
                sigma, j.uj_val
            in
            sigma, (current, try_find_ind !!(pb.env) sigma indt names))
  | _ -> sigma, (current, tmtyp)

let type_of_tomatch = function
  | IsInd (t,_,_) -> t
  | NotInd (_,t) -> t

let map_tomatch_type f = function
  | IsInd (t,ind,names) -> IsInd (f t,map_inductive_type f ind,names)
  | NotInd (c,t) -> NotInd (Option.map f c, f t)

let liftn_tomatch_type n depth = map_tomatch_type (Vars.liftn n depth)
let lift_tomatch_type n = liftn_tomatch_type n 1

(**********************************************************************)
(* Utilities on patterns *)

let current_pattern eqn =
  match eqn.patterns with
    | pat::_ -> pat
    | [] -> anomaly (Pp.str "Empty list of patterns.")

let remove_current_pattern eqn =
  match eqn.patterns with
    | pat::pats ->
        { eqn with
            patterns = pats;
            alias_stack = alias_of_pat pat :: eqn.alias_stack }
    | [] -> anomaly (Pp.str "Empty list of patterns.")

let push_current_pattern ~program_mode sigma (cur,ty) eqn =
  let hypnaming = if program_mode then ProgramNaming else KeepUserNameAndRenameExistingButSectionNames in
  match eqn.patterns with
    | pat::pats ->
        let r = Sorts.Relevant in (* TODO relevance *)
        let _,rhs_env = push_rel ~hypnaming sigma (LocalDef (make_annot (alias_of_pat pat) r,cur,ty)) eqn.rhs.rhs_env in
        { eqn with
            rhs = { eqn.rhs with rhs_env = rhs_env };
            patterns = pats }
    | [] -> anomaly (Pp.str "Empty list of patterns.")

(* spiwack: like [push_current_pattern] but does not introduce an
   alias in rhs_env. Aliasing binders are only useful for variables at
   the root of a pattern matching problem (initial push), so we
   distinguish the cases. *)
let push_noalias_current_pattern eqn =
  match eqn.patterns with
  | _::pats ->
      { eqn with patterns = pats }
  | [] -> anomaly (Pp.str "push_noalias_current_pattern: Empty list of patterns.")



let prepend_pattern tms eqn = {eqn with patterns = tms@eqn.patterns }

(**********************************************************************)
(* Well-formedness tests *)
(* Partial check on patterns *)

exception NotAdjustable

let rec adjust_local_defs ?loc = function
  | (pat :: pats, LocalAssum _ :: decls) ->
      pat :: adjust_local_defs ?loc (pats,decls)
  | (pats, LocalDef _ :: decls) ->
      (DAst.make ?loc @@ PatVar Anonymous) :: adjust_local_defs ?loc (pats,decls)
  | [], [] -> []
  | _ -> raise NotAdjustable

let check_and_adjust_constructor env ind cstrs pat = match DAst.get pat with
  | PatVar _ -> pat
  | PatCstr (((_,i) as cstr),args,alias) ->
      let loc = pat.CAst.loc in
      (* Check it is constructor of the right type *)
      let ind' = inductive_of_constructor cstr in
      if Ind.CanOrd.equal ind' ind then
        (* Check the constructor has the right number of args *)
        let ci = cstrs.(i-1) in
        let nb_args_constr = ci.cs_nargs in
        let nargs = List.length args in
        if Int.equal nargs nb_args_constr then pat
        else
          try
            let args' = adjust_local_defs ?loc (args, List.rev ci.cs_args)
            in DAst.make ?loc @@ PatCstr (cstr, args', alias)
          with NotAdjustable ->
            let nlet = List.count (function LocalDef _ -> true | _ -> false) ci.cs_args in
            (* In practice, this is already checked at interning *)
            error_wrong_numarg_constructor ?loc env ~cstr
              (* as if not expanded: *) ~expanded:false ~nargs ~expected_nassums:nb_args_constr
              ~expected_ndecls:(nb_args_constr + nlet)
      else
        (* Try to insert a coercion *)
        try
          Coercion.inh_pattern_coerce_to ?loc env pat ind' ind
        with Not_found ->
          error_bad_constructor ?loc env cstr ind

let check_all_variables env sigma typ mat =
  List.iter
    (fun eqn ->
      let pat = current_pattern eqn in
      match DAst.get pat with
       | PatVar id -> ()
       | PatCstr (cstr_sp,_,_) ->
          let loc = pat.CAst.loc in
           error_bad_pattern ?loc env sigma cstr_sp typ)
    mat

let set_pattern_catch_all_var ?loc eqn = function
  | Name id when not (Id.Set.mem id eqn.rhs.rhs_vars) ->
    { eqn with catch_all_vars = CAst.make ?loc id :: eqn.catch_all_vars }
  | _ ->
    eqn

let warn_named_multi_catch_all =
  CWarnings.create ~name:"unused-pattern-matching-variable" ~category:"pattern-matching"
         (fun id ->
          strbrk "Unused variable " ++ Id.print id ++ strbrk " catches more than one case.")

let wildcard_id = Id.of_string "wildcard'"

let is_wildcard id =
  Id.equal (Id.of_string (Nameops.atompart_of_id id)) wildcard_id

let check_unused_pattern_eqn env vars eqn =
  match List.length vars with
  | 0 -> raise_pattern_matching_error ?loc:eqn.eqn_loc (env, Evd.empty, UnusedClause eqn.patterns)
  | 1 -> ()
  | _ ->
    let warn {CAst.v = id; loc} =
      (* Convention: Names starting with `_` and derivatives of Program's
         "wildcard'" internal name deactivate the warning *)
      if (Id.to_string id).[0] <> '_' && not (is_wildcard id)
      then warn_named_multi_catch_all ?loc id in
    List.iter warn (List.uniquize (List.flatten vars))

let check_unused_pattern env used matx =
  let result = Array.init (List.length matx) (fun _ -> []) in
  List.iter (function (Some n,vars) -> result.(n) <- vars :: result.(n) | _ -> ()) used;
  List.iter2 (check_unused_pattern_eqn env) (Array.to_list result) matx

let extract_rhs pb =
  match pb.mat with
    | [] -> user_err (msg_may_need_inversion())
    | eqn::_ -> ([eqn.orig,eqn.catch_all_vars], eqn.rhs)

(**********************************************************************)
(* Functions to deal with matrix factorization *)

let occur_in_rhs na rhs =
  match na with
    | Anonymous -> false
    | Name id -> Id.Set.mem id rhs.rhs_vars

let is_dep_patt_in eqn pat = match DAst.get pat with
  | PatVar name -> occur_in_rhs name eqn.rhs
  | PatCstr _ -> true

let mk_dep_patt_row ~program_mode (pats,_,eqn) =
  if program_mode then List.map (fun _ -> true) pats
  else List.map (is_dep_patt_in eqn) pats

let dependencies_in_pure_rhs ~program_mode nargs eqns =
  if List.is_empty eqns then
    List.make nargs (not program_mode)  (* Only "_" patts *) else
  let deps_rows = List.map (mk_dep_patt_row ~program_mode) eqns in
  let deps_columns = matrix_transpose deps_rows in
  List.map (List.exists (fun x -> x)) deps_columns

let dependent_decl sigma a =
  function
  | LocalAssum (na,t) -> dependent sigma a t
  | LocalDef (na,c,t) -> dependent sigma a t || dependent sigma a c

let rec dep_in_tomatch sigma n = function
  | (Pushed _ | Alias _ | NonDepAlias) :: l -> dep_in_tomatch sigma n l
  | Abstract (_,d) :: l -> RelDecl.exists (fun c -> not (noccurn sigma n c)) d || dep_in_tomatch sigma (n+1) l
  | [] -> false

let dependencies_in_rhs ~program_mode sigma nargs current tms eqns =
  match EConstr.kind sigma current with
  | Rel n when dep_in_tomatch sigma n tms -> List.make nargs true
  | _ -> dependencies_in_pure_rhs ~program_mode nargs eqns

(* Computing the matrix of dependencies *)

(* [find_dependency_list tmi [d(i+1);...;dn]] computes in which
   declarations [d(i+1);...;dn] the term [tmi] is dependent in.

   [find_dependencies_signature (used1,...,usedn) ((tm1,d1),...,(tmn,dn))]
   returns [(deps1,...,depsn)] where [depsi] is a subset of tm(i+1),..,tmn
   denoting in which of the d(i+1)...dn, the term tmi is dependent.
*)

let rec find_dependency_list sigma tmblock = function
  | [] -> []
  | (used,tdeps,tm,d)::rest ->
      let deps = find_dependency_list sigma tmblock rest in
      if used && List.exists (fun x -> dependent_decl sigma x d) tmblock
      then
        match EConstr.kind sigma tm with
        | Rel n -> List.add_set Int.equal n (List.union Int.equal deps tdeps)
        | _ -> List.union Int.equal deps tdeps
      else deps

let find_dependencies sigma is_dep_or_cstr_in_rhs (tm,(_,tmtypleaves),d) nextlist =
  let deps = find_dependency_list sigma (tm::tmtypleaves) nextlist in
  if is_dep_or_cstr_in_rhs || not (List.is_empty deps)
  then ((true ,deps,tm,d)::nextlist)
  else ((false,[]  ,tm,d)::nextlist)

let find_dependencies_signature sigma deps_in_rhs typs =
  let l = List.fold_right2 (find_dependencies sigma) deps_in_rhs typs [] in
  List.map (fun (_,deps,_,_) -> deps) l

(* Assume we had terms t1..tq to match in a context xp:Tp,...,x1:T1 |-
   and xn:Tn has just been regeneralized into x:Tn so that the terms
   to match are now to be considered in the context xp:Tp,...,x1:T1,x:Tn |-.

   [relocate_index_tomatch n 1 tomatch] updates t1..tq so that
   former references to xn1 are now references to x. Note that t1..tq
   are already adjusted to the context xp:Tp,...,x1:T1,x:Tn |-.

   [relocate_index_tomatch 1 n tomatch] will go the way back.
 *)

let relocate_index_tomatch sigma n1 n2 =
  let rec genrec depth = function
  | [] ->
      []
  | Pushed (b,((c,tm),l,na)) :: rest ->
      let c = relocate_index sigma n1 n2 depth c in
      let tm = map_tomatch_type (relocate_index sigma n1 n2 depth) tm in
      let l = List.map (relocate_rel n1 n2 depth) l in
      Pushed (b,((c,tm),l,na)) :: genrec depth rest
  | Alias (initial,(na,c,d)) :: rest ->
      (* [c] is out of relocation scope *)
      Alias (initial,(na,c,map_pair (relocate_index sigma n1 n2 depth) d)) :: genrec depth rest
  | NonDepAlias :: rest ->
      NonDepAlias :: genrec depth rest
  | Abstract (i,d) :: rest ->
      let i = relocate_rel n1 n2 depth i in
      Abstract (i, RelDecl.map_constr (fun c -> relocate_index sigma n1 n2 depth c) d)
      :: genrec (depth+1) rest in
  genrec 0

(* [replace_tomatch n c tomatch] replaces [Rel n] by [c] in [tomatch] *)

let rec replace_term sigma n c k t =
  if isRel sigma t && Int.equal (destRel sigma t) (n + k) then Vars.lift k c
  else EConstr.map_with_binders sigma succ (replace_term sigma n c) k t

let length_of_tomatch_type_sign na t =
  let l = match na with
  | Anonymous -> 0
  | Name _ -> 1
  in
  match t with
  | NotInd _ -> l
  | IsInd (_, _, names) -> List.length names + l

let replace_tomatch sigma n c =
  let rec replrec depth = function
  | [] -> []
  | Pushed (initial,((b,tm),l,na)) :: rest ->
      let b = replace_term sigma n c depth b in
      let tm = map_tomatch_type (replace_term sigma n c depth) tm in
      List.iter (fun i -> if Int.equal i (n + depth) then anomaly (Pp.str "replace_tomatch.")) l;
      Pushed (initial,((b,tm),l,na)) :: replrec depth rest
  | Alias (initial,(na,b,d)) :: rest ->
      (* [b] is out of replacement scope *)
      Alias (initial,(na,b,map_pair (replace_term sigma n c depth) d)) :: replrec depth rest
  | NonDepAlias  :: rest ->
      NonDepAlias :: replrec depth rest
  | Abstract (i,d) :: rest ->
      Abstract (i, RelDecl.map_constr (fun t -> replace_term sigma n c depth t) d)
      :: replrec (depth+1) rest in
  replrec 0

(* [liftn_tomatch_stack]: a term to match has just been substituted by
   some constructor t = (ci x1...xn) and the terms x1 ... xn have been
   added to match; all pushed terms to match must be lifted by n
   (knowing that [Abstract] introduces a binder in the list of pushed
   terms to match).
*)

let rec liftn_tomatch_stack n depth = function
  | [] -> []
  | Pushed (initial,((c,tm),l,na))::rest ->
      let c = liftn n depth c in
      let tm = liftn_tomatch_type n depth tm in
      let l = List.map (fun i -> if i<depth then i else i+n) l in
      Pushed (initial,((c,tm),l,na))::(liftn_tomatch_stack n depth rest)
  | Alias (initial,(na,c,d))::rest ->
      Alias (initial,(na,liftn n depth c,map_pair (liftn n depth) d))
      ::(liftn_tomatch_stack n depth rest)
  | NonDepAlias  :: rest ->
      NonDepAlias :: liftn_tomatch_stack n depth rest
  | Abstract (i,d)::rest ->
      let i = if i<depth then i else i+n in
      Abstract (i, RelDecl.map_constr (liftn n depth) d)
      ::(liftn_tomatch_stack n (depth+1) rest)

let lift_tomatch_stack n = liftn_tomatch_stack n 1

(* if [current] has type [I(p1...pn u1...um)] and we consider the case
   of constructor [ci] of type [I(p1...pn u'1...u'm)], then the
   default variable [name] is expected to have which type?
   Rem: [current] is [(Rel i)] except perhaps for initial terms to match *)

(************************************************************************)
(* Some heuristics to get names for variables pushed in pb environment *)
(* Typical requirement:

   [match y with (S (S x)) => x | x => x end] should be compiled into
   [match y with O => y | (S n) => match n with O => y | (S x) => x end end]

   and [match y with (S (S n)) => n | n => n end] into
   [match y with O => y | (S n0) => match n0 with O => y | (S n) => n end end]

   i.e. user names should be preserved and created names should not
   interfere with user names

   The exact names here are not important for typing (because they are
   put in pb.env and not in the rhs.rhs_env of branches. However,
   whether a name is Anonymous or not may have an effect on whether a
   generalization is done or not.
 *)

let merge_name get_name obj = function
  | Anonymous -> get_name obj
  | na -> na

let merge_names get_name = List.map2 (merge_name get_name)

let get_names avoid env sigma sign eqns =
  let names1 = List.make (Context.Rel.length sign) Anonymous in
  (* If any, we prefer names used in pats, from top to bottom *)
  let names2,aliasname =
    List.fold_right
      (fun (pats,pat_alias,eqn) (names,aliasname) ->
        (merge_names alias_of_pat pats names,
         merge_name (fun x -> x) pat_alias aliasname))
      eqns (names1,Anonymous) in
  (* Otherwise, we take names from the parameters of the constructor but
     avoiding conflicts with user ids *)
  let allvars =
    List.fold_left (fun l (_,_,eqn) -> Id.Set.union l eqn.rhs.avoid_ids)
      avoid eqns in
  let names3,_ =
    List.fold_left2
      (fun (l,avoid) d na ->
         let na =
           merge_name
             (fun decl ->
                let na = get_name decl in
                let t = get_type decl in
                Name (next_name_away (named_hd env sigma t na) avoid))
             d na
         in
         (na::l,Id.Set.add (Name.get_id na) avoid))
      ([],allvars) (List.rev sign) names2 in
  names3,aliasname

(*****************************************************************)
(* Recovering names for variables pushed to the rhs' environment *)
(* We just factorized a match over a matrix of equations         *)
(* "C xi1 .. xin as xi" as a single match over "C y1 .. yn as y" *)
(* We now replace the names y1 .. yn y by the actual names       *)
(* xi1 .. xin xi to be found in the i-th clause of the matrix    *)

let recover_initial_subpattern_names = List.map2 RelDecl.set_name

let recover_and_adjust_alias_names (_,avoid) names sign =
  let rec aux = function
  | [],[] ->
      []
  | x::names, LocalAssum (x',t)::sign ->
      (x, LocalAssum ({x' with binder_name=alias_of_pat x},t)) :: aux (names,sign)
  | names, (LocalDef (na,_,_) as decl)::sign ->
      (DAst.make @@ PatVar na.binder_name, decl) :: aux (names,sign)
  | _ -> assert false
  in
  List.split (aux (names,sign))

let push_rels_eqn ~hypnaming sigma sign eqn =
  {eqn with
     rhs = {eqn.rhs with rhs_env = snd (push_rel_context ~hypnaming sigma sign eqn.rhs.rhs_env) } }

let push_rels_eqn_with_names sigma sign eqn =
  let subpats = List.rev (List.firstn (List.length sign) eqn.patterns) in
  let subpatnames = List.map alias_of_pat subpats in
  let sign = recover_initial_subpattern_names subpatnames sign in
  push_rels_eqn sigma sign eqn

let push_generalized_decl_eqn ~hypnaming env sigma n decl eqn =
  match RelDecl.get_name decl with
  | Anonymous ->
      push_rels_eqn ~hypnaming sigma [decl] eqn
  | Name _ ->
      push_rels_eqn ~hypnaming sigma [RelDecl.set_name (RelDecl.get_name (Environ.lookup_rel n !!(eqn.rhs.rhs_env))) decl] eqn

let drop_alias_eqn eqn =
  { eqn with alias_stack = List.tl eqn.alias_stack }

let push_alias_eqn sigma alias eqn =
  let aliasname = List.hd eqn.alias_stack in
  let eqn = drop_alias_eqn eqn in
  let alias = RelDecl.set_name aliasname alias in
  push_rels_eqn sigma [alias] eqn

(**********************************************************************)
(* Functions to deal with elimination predicate *)

(* Inferring the predicate *)
(*
The problem to solve is the following:

We match Gamma |- t : I(u01..u0q) against the following constructors:

  Gamma, x11...x1p1 |- C1(x11..x1p1) : I(u11..u1q)
   ...
  Gamma, xn1...xnpn |- Cn(xn1..xnp1) : I(un1..unq)

Assume the types in the branches are the following

  Gamma, x11...x1p1 |- branch1 : T1
   ...
  Gamma, xn1...xnpn |- branchn : Tn

Assume the type of the global case expression is Gamma |- T

The predicate has the form phi = [y1..yq][z:I(y1..yq)]psi and it has to
satisfy the following n+1 equations:

  Gamma, x11...x1p1 |- (phi u11..u1q (C1 x11..x1p1))  =  T1
   ...
  Gamma, xn1...xnpn |- (phi un1..unq (Cn xn1..xnpn))  =  Tn
  Gamma             |- (phi u01..u0q t)               =  T

Some hints:

- Clearly, if xij occurs in Ti, then, a "match z with (Ci xi1..xipi)
  => ... end"  or a "psi(yk)", with psi extracting xij from uik, should be
  inserted somewhere in Ti.

- If T is undefined, an easy solution is to insert a "match z with
  (Ci xi1..xipi) => ... end" in front of each Ti

- Otherwise, T1..Tn and T must be step by step unified, if some of them
  diverge, then try to replace the diverging subterm by one of y1..yq or z.

- The main problem is what to do when an existential variables is encountered

*)

(* Propagation of user-provided predicate through compilation steps *)

let rec map_predicate f k ccl = function
  | [] -> f k ccl
  | Pushed (_,((_,tm),_,na)) :: rest ->
      let k' = length_of_tomatch_type_sign na tm in
      map_predicate f (k+k') ccl rest
  | (Alias _ | NonDepAlias) :: rest ->
      map_predicate f k ccl rest
  | Abstract _ :: rest ->
      map_predicate f (k+1) ccl rest

let noccur_predicate_between sigma n = map_predicate (noccur_between sigma n)

let liftn_predicate n = map_predicate (liftn n)

let lift_predicate n = liftn_predicate n 1

let regeneralize_index_predicate sigma n = map_predicate (relocate_index sigma n 1) 0

let substnl_predicate sigma = map_predicate (substnl sigma)

(* This is parallel bindings *)
let subst_predicate (subst,copt) ccl tms =
  let sigma = match copt with
    | None -> subst
    | Some c -> c::subst in
  substnl_predicate sigma 0 ccl tms

let specialize_predicate_var (cur,typ,dep) env tms ccl =
  let c = match dep with
  | Anonymous -> None
  | Name _ -> Some cur
  in
  let l =
    match typ with
    | IsInd (_, IndType (_, _), []) -> []
    | IsInd (_, IndType (indf, realargs), names) ->
       let arsign,_ = get_arity env indf in
       let arsign = List.map EConstr.of_rel_decl arsign in
       subst_of_rel_context_instance_list arsign realargs
    | NotInd _ -> [] in
  subst_predicate (l,c) ccl tms

(*****************************************************************************)
(* We have pred = [X:=realargs;x:=c]P typed in Gamma1, x:I(realargs), Gamma2 *)
(* and we want to abstract P over y:t(x) typed in the same context to get    *)
(*                                                                           *)
(*    pred' = [X:=realargs;x':=c](y':t(x'))P[y:=y']                          *)
(*                                                                           *)
(* We first need to lift t(x) s.t. it is typed in Gamma, X:=rargs, x'        *)
(* then we have to replace x by x' in t(x) and y by y' in P                  *)
(*****************************************************************************)
let generalize_predicate sigma (names,na) ny d tms ccl =
  let () = match na with
  | Anonymous -> anomaly (Pp.str "Undetected dependency.")
  | _ -> () in
  let p = List.length names + 1 in
  let ccl = lift_predicate 1 ccl tms in
  regeneralize_index_predicate sigma (ny+p+1) ccl tms

(*****************************************************************************)
(* We just matched over cur:ind(realargs) in the following matching problem  *)
(*                                                                           *)
(*   env |- match cur tms return ccl with ... end                            *)
(*                                                                           *)
(* and we want to build the predicate corresponding to the individual        *)
(* matching over cur                                                         *)
(*                                                                           *)
(*    pred = fun X:realargstyps x:ind(X)] PI tms.ccl                         *)
(*                                                                           *)
(* where pred is computed by abstract_predicate and PI tms.ccl by            *)
(* extract_predicate                                                         *)
(*****************************************************************************)
let rec extract_predicate ccl = function
  | (Alias _ | NonDepAlias)::tms ->
      (* substitution already done in build_branch *)
      extract_predicate ccl tms
  | Abstract (i,d)::tms ->
      mkProd_wo_LetIn d (extract_predicate ccl tms)
  | Pushed (_,((cur,NotInd _),_,na))::tms ->
      begin match na with
      | Anonymous -> extract_predicate ccl tms
      | Name _ ->
        let tms = lift_tomatch_stack 1 tms in
        let pred = extract_predicate ccl tms in
        subst1 cur pred
      end
  | Pushed (_,((cur,IsInd (_,IndType(_,realargs),_)),_,na))::tms ->
      let realargs = List.rev realargs in
      let k, nrealargs = match na with
      | Anonymous -> 0, realargs
      | Name _ -> 1, (cur :: realargs)
      in
      let tms = lift_tomatch_stack (List.length realargs + k) tms in
      let pred = extract_predicate ccl tms in
      substl nrealargs pred
  | [] ->
      ccl

let abstract_predicate env sigma indf cur realargs (names,na) tms ccl =
  let sign = make_arity_signature !!env sigma true indf in
  (* n is the number of real args + 1 (+ possible let-ins in sign) *)
  let n = List.length sign in
  (* Before abstracting we generalize over cur and on those realargs *)
  (* that are rels, consistently with the specialization made in     *)
  (* build_branch                                                    *)
  let tms = List.fold_right2 (fun par arg tomatch ->
    match EConstr.kind sigma par with
    | Rel i -> relocate_index_tomatch sigma (i+n) (destRel sigma arg) tomatch
    | _ -> tomatch) (realargs@[cur]) (Context.Rel.instance_list EConstr.mkRel 0 sign)
       (lift_tomatch_stack n tms) in
  (* Pred is already dependent in the current term to match (if      *)
  (* (na<>Anonymous) and its realargs; we just need to adjust it to  *)
  (* full sign if dep in cur is not taken into account               *)
  let ccl = match na with
  | Anonymous -> lift_predicate 1 ccl tms
  | Name _ -> ccl
  in
  let pred = extract_predicate ccl tms in
  (* Build the predicate properly speaking *)
  let sign = List.map2 set_name (na::names) sign in
  it_mkLambda_or_LetIn_name !!env sigma pred sign

(* [expand_arg] is used by [specialize_predicate]
   if Yk denotes [Xk;xk] or [Xk],
   it replaces gamma, x1...xn, x1...xk Yk+1...Yn |- pred
   by gamma, x1...xn, x1...xk-1 [Xk;xk] Yk+1...Yn |- pred (if dep) or
   by gamma, x1...xn, x1...xk-1 [Xk] Yk+1...Yn |- pred (if not dep) *)

let expand_arg tms (p,ccl) ((_,t),_,na) =
  let k = length_of_tomatch_type_sign na t in
  (p+k,liftn_predicate (k-1) (p+1) ccl tms)

let add_assert_false_case pb tomatch =
  let pats = List.map (fun _ -> DAst.make @@ PatVar Anonymous) tomatch in
  let aliasnames =
    List.map_filter (function Alias _ | NonDepAlias -> Some Anonymous | _ -> None) tomatch
  in
  [ { patterns = pats;
      rhs = { rhs_env = pb.env;
              rhs_vars = Id.Set.empty;
              avoid_ids = Id.Set.empty;
              it = None };
      alias_stack = Anonymous::aliasnames;
      eqn_loc = None;
      orig = None;
      catch_all_vars = [] } ]

let adjust_impossible_cases sigma pb pred tomatch submat =
  match submat with
  | [] ->
    (* FIXME: This breaks if using evar-insensitive primitives. In particular,
       this means that the Evd.define below may redefine an already defined
       evar. See e.g. first definition of test for bug #3388. *)
    let pred = EConstr.Unsafe.to_constr pred in
    begin match Constr.kind pred with
    | Evar (evk,_) when snd (evar_source evk sigma) == Evar_kinds.ImpossibleCase ->
        let sigma =
          if not (Evd.is_defined sigma evk) then
            let sigma, default = coq_unit_judge !!(pb.env) sigma in
            let sigma = Evd.define evk default.uj_type sigma in
            sigma
          else sigma
        in
        sigma, add_assert_false_case pb tomatch
    | _ ->
        sigma, submat
    end
  | _ ->
    sigma, submat

(*****************************************************************************)
(* Let  pred = PI [X;x:I(X)]. PI tms. P  be a typing predicate for the       *)
(* following pattern-matching problem:                                       *)
(*                                                                           *)
(*  Gamma |- match Pushed(c:I(V)) as x in I(X), tms return pred with...end   *)
(*                                                                           *)
(* where the branch with constructor Ci:(x1:T1)...(xn:Tn)->I(realargsi)      *)
(* is considered. Assume each Ti is some Ii(argsi) with Ti:PI Ui. sort_i     *)
(* We let subst = X:=realargsi;x:=Ci(x1,...,xn) and replace pred by          *)
(*                                                                           *)
(* pred' = PI [X1:Ui;x1:I1(X1)]...[Xn:Un;xn:In(Xn)]. (PI tms. P)[subst]      *)
(*                                                                           *)
(* s.t. the following well-typed sub-pattern-matching problem is obtained    *)
(*                                                                           *)
(* Gamma,x'1..x'n |-                                                         *)
(*   match                                                                   *)
(*      Pushed(x'1) as x1 in I(X1),                                          *)
(*      ..,                                                                  *)
(*      Pushed(x'n) as xn in I(Xn),                                          *)
(*      tms                                                                  *)
(*      return pred'                                                         *)
(*   with .. end                                                             *)
(*                                                                           *)
(*****************************************************************************)
let specialize_predicate env newtomatchs (names,depna) arsign cs tms ccl =
  (* Assume some gamma st: gamma |- PI [X,x:I(X)]. PI tms. ccl *)
  let nrealargs = List.length names in
  let l = match depna with Anonymous -> 0 | Name _ -> 1 in
  let k = nrealargs + l in
  (* We adjust pred st: gamma, x1..xn |- PI [X,x:I(X)]. PI tms. ccl' *)
  (* so that x can later be instantiated by Ci(x1..xn) *)
  (* and X by the realargs for Ci *)
  let n = cs.cs_nargs in
  let ccl' = liftn_predicate n (k+1) ccl tms in
  (* We prepare the substitution of X and x:I(X) *)
  let realargsi =
    if not (Int.equal nrealargs 0) then
      CVars.subst_of_rel_context_instance arsign cs.cs_concl_realargs
    else
      [] in
  let realargsi = List.map EConstr.of_constr realargsi in
  let copti = match depna with
  | Anonymous -> None
  | Name _ -> Some (EConstr.of_constr (build_dependent_constructor cs))
  in
  (* The substituends realargsi, copti are all defined in gamma, x1...xn *)
  (* We need _parallel_ bindings to get gamma, x1...xn |- PI tms. ccl'' *)
  (* Note: applying the substitution in tms is not important (is it sure?) *)
  let ccl'' =
    whd_betaiota env Evd.empty (subst_predicate (realargsi, copti) ccl' tms) in
  (* We adjust ccl st: gamma, x'1..x'n, x1..xn, tms |- ccl'' *)
  let ccl''' = liftn_predicate n (n+1) ccl'' tms in
  (* We finally get gamma,x'1..x'n,x |- [X1;x1:I(X1)]..[Xn;xn:I(Xn)]pred'''*)
  snd (List.fold_left (expand_arg tms) (1,ccl''') newtomatchs)

let find_predicate loc env sigma p current (IndType (indf,realargs)) dep tms =
  let pred = abstract_predicate env sigma indf current realargs dep tms p in
  (pred, whd_betaiota !!env sigma
           (applist (pred, realargs@[current])))

(* Take into account that a type has been discovered to be inductive, leading
   to more dependencies in the predicate if the type has indices *)
let adjust_predicate_from_tomatch tomatch (current,typ as ct) pb =
  let ((_,oldtyp),deps,na) = tomatch in
  match typ, oldtyp with
  | IsInd (_,_,names), NotInd _ ->
      let k = match na with
      | Anonymous -> 1
      | Name _ -> 2
      in
      let n = List.length names in
      { pb with pred = liftn_predicate n k pb.pred pb.tomatch },
      (ct,List.map (fun i -> if i >= k then i+n else i) deps,na)
  | _ ->
      pb, (ct,deps,na)

(* Remove commutative cuts that turn out to be non-dependent after
   some evars have been instantiated *)

let rec ungeneralize sigma n ng body =
  match EConstr.kind sigma body with
  | Lambda (_,_,c) when Int.equal ng 0 ->
      subst1 (mkRel n) c
  | Lambda (na,t,c) ->
      (* We traverse an inner generalization *)
      mkLambda (na,t,ungeneralize sigma (n+1) (ng-1) c)
  | LetIn (na,b,t,c) ->
      (* We traverse an alias *)
      mkLetIn (na,b,t,ungeneralize sigma (n+1) ng c)
  | Case (ci,u,pms,p,iv,c,brs) ->
      (* We traverse a split *)
      let p =
        let (nas, p) = p in
        let sign2,p = decompose_prod_n_assum sigma ng p in
        let p = prod_applist sigma p [mkRel (n+Array.length nas+ng)] in
        nas, it_mkProd_or_LetIn p sign2
      in
      let map (nas, br) = nas, ungeneralize sigma (n + Array.length nas) ng br in
      mkCase (ci, u, pms, p, iv, c, Array.map map brs)
  | App (f,args) ->
      (* We traverse an inner generalization *)
      assert (isCase sigma f);
      mkApp (ungeneralize sigma n (ng+Array.length args) f,args)
  | _ -> assert false

let ungeneralize_branch sigma n k (sign,body) cs =
  (sign,ungeneralize sigma (n+cs.cs_nargs) k body)

let rec is_dependent_generalization sigma ng body =
  match EConstr.kind sigma body with
  | Lambda (_,_,c) when Int.equal ng 0 ->
      not (noccurn sigma 1 c)
  | Lambda (na,t,c) ->
      (* We traverse an inner generalization *)
      is_dependent_generalization sigma (ng-1) c
  | LetIn (na,b,t,c) ->
      (* We traverse an alias *)
      is_dependent_generalization sigma ng c
  | Case (ci,u,pms,p,iv,c,brs) ->
      (* We traverse a split *)
      Array.exists (fun (_, b) -> is_dependent_generalization sigma ng b) brs
  | App (g,args) ->
      (* We traverse an inner generalization *)
      assert (isCase sigma g);
      is_dependent_generalization sigma (ng+Array.length args) g
  | _ -> assert false

let is_dependent_branch sigma k (_,br) =
  is_dependent_generalization sigma k br

let postprocess_dependencies evd tocheck brs tomatch pred deps cs =
  let rec aux k brs tomatch pred tocheck deps = match deps, tomatch with
  | [], _ -> brs,tomatch,pred,[]
  | n::deps, Abstract (i,d) :: tomatch ->
      let d = map_constr (fun c -> nf_evar evd c) d in
      let is_d = match d with LocalAssum _ -> false | LocalDef _ -> true in
      if is_d || List.exists (fun c -> dependent_decl evd (lift k c) d) tocheck
                 && Array.exists (is_dependent_branch evd k) brs then
        (* Dependency in the current term to match and its dependencies is real *)
        let brs,tomatch,pred,inst = aux (k+1) brs tomatch pred (mkRel n::tocheck) deps in
        let inst = match d with
        | LocalAssum _ -> mkRel n :: inst
        | _ -> inst
        in
        brs, Abstract (i,d) :: tomatch, pred, inst
      else
        (* Finally, no dependency remains, so, we can replace the generalized *)
        (* terms by its actual value in both the remaining terms to match and *)
        (* the bodies of the Case *)
        let pred = lift_predicate (-1) pred tomatch in
        let tomatch = relocate_index_tomatch evd 1 (n+1) tomatch in
        let tomatch = lift_tomatch_stack (-1) tomatch in
        let brs = Array.map2 (ungeneralize_branch evd n k) brs cs in
        aux k brs tomatch pred tocheck deps
  | _ -> assert false
  in aux 0 brs tomatch pred tocheck deps

(************************************************************************)
(* Sorting equations by constructor *)

let rec irrefutable env pat = match DAst.get pat with
  | PatVar name -> true
  | PatCstr (cstr,args,_) ->
      let ind = inductive_of_constructor cstr in
      let (_,mip) = Inductive.lookup_mind_specif env ind in
      let one_constr = Int.equal (Array.length mip.mind_user_lc) 1 in
      one_constr && List.for_all (irrefutable env) args

let first_clause_irrefutable env = function
  | eqn::mat -> List.for_all (irrefutable env) eqn.patterns
  | _ -> false

let group_equations pb ind current cstrs mat =
  let mat =
    if first_clause_irrefutable !!(pb.env) mat then [List.hd mat] else mat in
  let brs = Array.make (Array.length cstrs) [] in
  let only_default = ref None in
  let _ =
    List.fold_right (* To be sure it's from bottom to top *)
      (fun eqn () ->
         let rest = remove_current_pattern eqn in
         let pat = current_pattern eqn in
         match DAst.get (check_and_adjust_constructor !!(pb.env) ind cstrs pat) with
           | PatVar name ->
               (* This is a default clause that we expand *)
               let rest = set_pattern_catch_all_var ?loc:pat.CAst.loc rest name in
               for i=1 to Array.length cstrs do
                 let args = make_anonymous_patvars cstrs.(i-1).cs_nargs in
                 brs.(i-1) <- (args, name, rest) :: brs.(i-1)
               done;
               if !only_default == None then only_default := Some true
           | PatCstr (((_,i)),args,name) ->
               (* This is a regular clause *)
               only_default := Some false;
               brs.(i-1) <- (args, name, rest) :: brs.(i-1)) mat () in
  (brs,Option.default false !only_default)

(************************************************************************)
(* Here starts the pattern-matching compilation algorithm *)

(* Abstracting over dependent subterms to match *)
let rec generalize_problem names sigma pb = function
  | [] -> pb, []
  | i::l ->
      let pb',deps = generalize_problem names sigma pb l in
      let d = map_constr (lift i) (lookup_rel i !!(pb.env)) in
      begin match d with
      | LocalDef ({binder_name=Anonymous},_,_) -> pb', deps
      | _ ->
         (* for better rendering *)
        let d = RelDecl.map_type (fun c -> whd_betaiota !!(pb.env) sigma c) d in
        let tomatch = lift_tomatch_stack 1 pb'.tomatch in
        let tomatch = relocate_index_tomatch sigma (i+1) 1 tomatch in
        { pb' with
            tomatch = Abstract (i,d) :: tomatch;
            pred = generalize_predicate sigma names i d pb'.tomatch pb'.pred  },
        i::deps
      end

(* No more patterns: typing the right-hand side of equations *)
let build_leaf sigma pb =
  let used, rhs = extract_rhs pb in
  let sigma, j = pb.typing_function (mk_tycon pb.pred) rhs.rhs_env sigma rhs.it in
  used, sigma, j_nf_evar sigma j

(* Build the sub-pattern-matching problem for a given branch "C x1..xn as x" *)
(* spiwack: the [initial] argument keeps track whether the branch is a
   toplevel branch ([true]) or a deep one ([false]). *)
let build_branch ~program_mode initial current realargs deps (realnames,curname) sigma pb arsign eqns const_info =
  (* We remember that we descend through constructor C *)
  let history =
    push_history_pattern const_info.cs_nargs (fst const_info.cs_cstr) pb.history in

  (* We prepare the matching on x1:T1 .. xn:Tn using some heuristic to *)
  (* build the name x1..xn from the names present in the equations *)
  (* that had matched constructor C *)
  let cs_args = const_info.cs_args in
  let cs_args = List.map (fun d -> map_rel_decl EConstr.of_constr d) cs_args in
  let names,aliasname = get_names (GlobEnv.vars_of_env pb.env) !!(pb.env) sigma cs_args eqns in
  let typs = List.map2 RelDecl.set_name names cs_args
  in

  (* Beta-iota-normalize types to better compatibility of refine with 8.4 behavior *)
  (* This is a bit too strong I think, in the sense that what we would *)
  (* really like is to have beta-iota reduction only at the positions where *)
  (* parameters are substituted *)
  let typs = List.map (map_type (nf_betaiota !!(pb.env) sigma)) typs in

  (* We build the matrix obtained by expanding the matching on *)
  (* "C x1..xn as x" followed by a residual matching on eqn into *)
  (* a matching on "x1 .. xn eqn" *)
  let submat = List.map (fun (tms,_,eqn) -> prepend_pattern tms eqn) eqns in

  (* We adjust the terms to match in the context they will be once the *)
  (* context [x1:T1,..,xn:Tn] will have been pushed on the current env *)
  let typs' =
    List.map_i (fun i d -> (mkRel i, map_constr (lift i) d)) 1 typs in

  let hypnaming = if program_mode then ProgramNaming else KeepUserNameAndRenameExistingButSectionNames in
  let typs,extenv = push_rel_context ~hypnaming sigma typs pb.env in

  let typs' =
    List.map (fun (c,d) ->
      (c,extract_inductive_data !!extenv sigma d,d)) typs' in

  (* We compute over which of x(i+1)..xn and x matching on xi will need a *)
  (* generalization *)
  let dep_sign =
    find_dependencies_signature sigma
      (dependencies_in_rhs ~program_mode sigma const_info.cs_nargs current pb.tomatch eqns)
      (List.rev typs') in

  (* The dependent term to subst in the types of the remaining UnPushed
     terms is relative to the current context enriched by topushs *)
  let ci = EConstr.of_constr (build_dependent_constructor const_info) in

  (* Current context Gamma has the form Gamma1;cur:I(realargs);Gamma2 *)
  (* We go from Gamma |- PI tms. pred to                              *)
  (* Gamma;x1..xn;curalias:I(x1..xn) |- PI tms'. pred'                *)
  (* where, in tms and pred, those realargs that are vars are         *)
  (* replaced by the corresponding xi and cur replaced by curalias    *)
  let cirealargs = Array.map_to_list EConstr.of_constr const_info.cs_concl_realargs in

  (* Do the specialization for terms to match *)
  let tomatch = List.fold_right2 (fun par arg tomatch ->
    match EConstr.kind sigma par with
    | Rel i -> replace_tomatch sigma (i+const_info.cs_nargs) arg tomatch
    | _ -> tomatch) (current::realargs) (ci::cirealargs)
      (lift_tomatch_stack const_info.cs_nargs pb.tomatch) in

  let pred_is_not_dep =
    noccur_predicate_between sigma 1 (List.length realnames + 1) pb.pred tomatch in

  let typs' =
    List.map2
      (fun (tm, (tmtyp,_), decl) deps ->
        let na = RelDecl.get_name decl in
        let na = match curname, na with
        | Name _, Anonymous -> curname
        | Name _, Name _ -> na
        | Anonymous, _ ->
            if List.is_empty deps && pred_is_not_dep then Anonymous else force_name na in
        ((tm,tmtyp),deps,na))
      typs' (List.rev dep_sign) in

  (* Do the specialization for the predicate *)
  let pred =
    specialize_predicate !!(pb.env) typs' (realnames,curname) arsign const_info tomatch pb.pred in

  let currents = List.map (fun x -> Pushed (false,x)) typs' in

  let alias = match aliasname with
  | Anonymous ->
      NonDepAlias
  | Name _ ->
      let cur_alias = lift const_info.cs_nargs current in
      let ind =
        mkApp (
          applist (mkIndU (inductive_of_constructor (fst const_info.cs_cstr), EInstance.make (snd const_info.cs_cstr)),
                   List.map (EConstr.of_constr %> lift const_info.cs_nargs) const_info.cs_params),
          Array.map EConstr.of_constr const_info.cs_concl_realargs) in
      Alias (initial,(aliasname,cur_alias,(ci,ind))) in

  let tomatch = List.rev_append (alias :: currents) tomatch in

  let sigma, submat = adjust_impossible_cases sigma pb pred tomatch submat in
  let () = match submat with
  | [] ->
    raise_pattern_matching_error (!!(pb.env), Evd.empty, NonExhaustive (complete_history history))
  | _ -> ()
  in

  sigma, typs,
  { pb with
      env = extenv;
      tomatch = tomatch;
      pred = pred;
      history = history;
      mat = List.map (push_rels_eqn_with_names ~hypnaming sigma typs) submat }

(**********************************************************************
 INVARIANT:

  pb = { env, pred, tomatch, mat, ...}
  tomatch = list of Pushed (c:T), Abstract (na:T), Alias (c:T) or NonDepAlias

  all terms and types in Pushed, Abstract and Alias are relative to env
  enriched by the Abstract coming before

*)

(**********************************************************************)
(* Main compiling descent *)
let compile ~program_mode sigma pb =
  let rec compile sigma pb =
    match pb.tomatch with
      | Pushed cur :: rest -> match_current sigma { pb with tomatch = rest } cur
      | Alias (initial,x) :: rest -> compile_alias initial sigma pb x rest
      | NonDepAlias :: rest -> compile_non_dep_alias sigma pb rest
      | Abstract (i,d) :: rest -> compile_generalization sigma pb i d rest
      | [] -> build_leaf sigma pb

(* Case splitting *)
  and match_current sigma pb (initial,tomatch) =
    let sigma, tm = adjust_tomatch_to_pattern ~program_mode sigma pb tomatch in
    let pb,tomatch = adjust_predicate_from_tomatch tomatch tm pb in
    let ((current,typ),deps,dep) = tomatch in
    match typ with
      | NotInd (_,typ) ->
          check_all_variables !!(pb.env) sigma typ pb.mat;
          compile_all_variables initial tomatch sigma pb
      | IsInd (_,(IndType(indf,realargs) as indt),names) ->
        let mind,_ = dest_ind_family indf in
          let mind = Tacred.check_privacy !!(pb.env) mind in
          let cstrs = get_constructors !!(pb.env) indf in
          let arsign, _ = get_arity !!(pb.env) indf in
        let eqns,onlydflt = group_equations pb (fst mind) current cstrs pb.mat in
          let no_cstr = Int.equal (Array.length cstrs) 0 in
        if (not no_cstr || not (List.is_empty pb.mat)) && onlydflt then
            compile_all_variables initial tomatch sigma pb
        else
          (* We generalize over terms depending on current term to match *)
            let pb,deps = generalize_problem (names,dep) sigma pb deps in

          (* We compile branches *)
            let fold_br sigma eqn cstr =
              let used, sigma, j = compile_branch initial current realargs (names,dep) deps sigma pb arsign eqn cstr in
              sigma, (used, j)
            in
            let sigma, brvals = Array.fold_left2_map fold_br sigma eqns cstrs in
            let used, brvals = Array.split brvals in
          (* We build the (elementary) case analysis *)
            let depstocheck = current::binding_vars_of_inductive sigma typ in
            let brvals,tomatch,pred,inst =
              postprocess_dependencies sigma depstocheck
                brvals pb.tomatch pb.pred deps cstrs in
            let brvals = Array.map (fun (sign,body) ->
              it_mkLambda_or_LetIn body sign) brvals in
            let (pred,typ) =
              find_predicate pb.caseloc pb.env sigma
                pred current indt (names,dep) tomatch
            in
            let rci = Typing.check_allowed_sort !!(pb.env) sigma mind current pred in
            let ci = make_case_info !!(pb.env) (fst mind) rci pb.casestyle in
            let pred = nf_betaiota !!(pb.env) sigma pred in
            let case = make_case_or_project !!(pb.env) sigma indt ci pred current brvals in
            let sigma, _ = Typing.type_of !!(pb.env) sigma pred in
            let used = List.flatten (Array.to_list used) in
            used, sigma, { uj_val = applist (case, inst);
              uj_type = prod_applist sigma typ inst }


  (* Building the sub-problem when all patterns are variables. Case
     where [current] is an initially pushed term. *)
  and shift_problem ((current,t),_,na) sigma pb =
    let ty = type_of_tomatch t in
    let tomatch = lift_tomatch_stack 1 pb.tomatch in
    let pred = specialize_predicate_var (current,t,na) !!(pb.env) pb.tomatch pb.pred in
    let env = Name.fold_left (fun env id -> hide_variable env Anonymous id) pb.env na in
    let hypnaming = if program_mode then ProgramNaming else KeepUserNameAndRenameExistingButSectionNames in
    let pb =
      { pb with
         env = snd (push_rel ~hypnaming sigma (LocalDef (annotR na,current,ty)) env);
         tomatch = tomatch;
         pred = lift_predicate 1 pred tomatch;
         history = pop_history pb.history;
         mat = List.map (push_current_pattern ~program_mode sigma (current,ty)) pb.mat } in
    let used, sigma, j = compile sigma pb in
    used, sigma, { uj_val = subst1 current j.uj_val;
      uj_type = subst1 current j.uj_type }

  (* Building the sub-problem when all patterns are variables,
     non-initial case. Variables which appear as subterms of constructor
     are already introduced in the context, we avoid creating aliases to
     themselves by treating this case specially. *)
  and pop_problem ((current,t),_,na) sigma pb =
    let pred = specialize_predicate_var (current,t,na) !!(pb.env) pb.tomatch pb.pred in
    let pb =
      { pb with
         pred = pred;
         history = pop_history pb.history;
         mat = List.map push_noalias_current_pattern pb.mat } in
    compile sigma pb

  (* Building the sub-problem when all patterns are variables. *)
  and compile_all_variables initial cur sigma pb =
    if initial then shift_problem cur sigma pb
    else pop_problem cur sigma pb

  (* Building the sub-problem when all patterns are variables *)
  and compile_branch initial current realargs names deps sigma pb arsign eqns cstr =
    let sigma, sign, pb = build_branch ~program_mode initial current realargs deps names sigma pb arsign eqns cstr in
    let used, sigma, j = compile sigma pb in
    used, sigma, (sign, j.uj_val)

  (* Abstract over a declaration before continuing splitting *)
  and compile_generalization sigma pb i d rest =
    let hypnaming = if program_mode then ProgramNaming else KeepUserNameAndRenameExistingButSectionNames in
    let pb =
      { pb with
         env = snd (push_rel ~hypnaming sigma d pb.env);
         tomatch = rest;
         mat = List.map (push_generalized_decl_eqn ~hypnaming pb.env sigma i d) pb.mat } in
    let used, sigma, j = compile sigma pb in
    used, sigma, { uj_val = mkLambda_or_LetIn d j.uj_val;
      uj_type = mkProd_wo_LetIn d j.uj_type }

  (* spiwack: the [initial] argument keeps track whether the alias has
     been introduced by a toplevel branch ([true]) or a deep one
     ([false]). *)
  and compile_alias initial sigma pb (na,orig,(expanded,expanded_typ)) rest =
    let hypnaming = if program_mode then ProgramNaming else KeepUserNameAndRenameExistingButSectionNames in
    let f c t =
      let r = Retyping.relevance_of_type !!(pb.env) sigma t in
      let alias = LocalDef (make_annot na r,c,t) in
      let pb =
        { pb with
           env = snd (push_rel ~hypnaming sigma alias pb.env);
           tomatch = lift_tomatch_stack 1 rest;
           pred = lift_predicate 1 pb.pred pb.tomatch;
           history = pop_history_pattern pb.history;
           mat = List.map (push_alias_eqn ~hypnaming sigma alias) pb.mat } in
      let used, sigma, j = compile sigma pb in
      used, sigma, { uj_val =
          if isRel sigma c || isVar sigma c || count_occurrences sigma (mkRel 1) j.uj_val <= 1 then
            subst1 c j.uj_val
          else
            mkLetIn (make_annot na r,c,t,j.uj_val);
        uj_type = subst1 c j.uj_type } in
    (* spiwack: when an alias appears on a deep branch, its non-expanded
       form is automatically a variable of the same name. We avoid
       introducing such superfluous aliases so that refines are elegant. *)
    let just_pop sigma =
      let pb =
        { pb with
          tomatch = rest;
          history = pop_history_pattern pb.history;
          mat = List.map drop_alias_eqn pb.mat } in
      compile sigma pb
    in
    (* If the "match" was originally over a variable, as in "match x with
       O => true | n => n end", we give preference to non-expansion in
       the default clause (i.e. "match x with O => true | n => n end"
       rather than "match x with O => true | S p => S p end";
       computationally, this avoids reallocating constructors in cbv
       evaluation; the drawback is that it might duplicate the instances
       of the term to match when the corresponding variable is
       substituted by a non-evaluated expression *)
    if not program_mode && (isRel sigma orig || isVar sigma orig) then
      (* Try to compile first using non expanded alias *)
      try
        if initial then f orig (Retyping.get_type_of !!(pb.env) sigma orig)
        else just_pop sigma
      with e when precatchable_exception e ->
      (* Try then to compile using expanded alias *)
      (* Could be needed in case of dependent return clause *)
      f expanded expanded_typ
    else
      (* Try to compile first using expanded alias *)
      try f expanded expanded_typ
      with e when precatchable_exception e ->
      (* Try then to compile using non expanded alias *)
      (* Could be needed in case of a recursive call which requires to
         be on a variable for size reasons *)
      if initial then f orig (Retyping.get_type_of !!(pb.env) sigma orig)
      else just_pop sigma


  (* Remember that a non-trivial pattern has been consumed *)
  and compile_non_dep_alias sigma pb rest =
    let pb =
      { pb with
         tomatch = rest;
         history = pop_history_pattern pb.history;
         mat = List.map drop_alias_eqn pb.mat } in
    compile sigma pb
  in
  compile sigma pb

(* pour les alias des initiaux, enrichir les env de ce qu'il faut et
substituer après par les initiaux *)

(**************************************************************************)
(* Preparation of the pattern-matching problem                            *)

(* builds the matrix of equations testing that each eqn has n patterns
 * and linearizing the _ patterns.
 * Syntactic correctness has already been done in constrintern *)
let matx_of_eqns env eqns =
  let build_eqn i {CAst.loc;v=(ids,initial_lpat,initial_rhs)} =
    let avoid = ids_of_named_context_val (named_context_val !!env) in
    let avoid = List.fold_left (fun accu id -> Id.Set.add id accu) avoid ids in
    let rhs =
      { rhs_env = env;
        rhs_vars = free_glob_vars initial_rhs;
        avoid_ids = avoid;
        it = Some initial_rhs } in
    { patterns = initial_lpat;
      alias_stack = [];
      eqn_loc = loc;
      orig = Some i;
      catch_all_vars = [];
      rhs = rhs }
  in List.map_i build_eqn 0 eqns

(***************** Building an inversion predicate ************************)

(* Let "match t1 in I1 u11..u1n_1 ... tm in Im um1..umn_m with ... end : T"
   be a pattern-matching problem. We assume that each uij can be
   decomposed under the form pij(vij1..vijq_ij) where pij(aij1..aijq_ij)
   is a pattern depending on some variables aijk and the vijk are
   instances of these variables.  We also assume that each ti has the
   form of a pattern qi(wi1..wiq_i) where qi(bi1..biq_i) is a pattern
   depending on some variables bik and the wik are instances of these
   variables (in practice, there is no reason that ti is already
   constructed and the qi will be degenerated).

   We then look for a type U(..a1jk..b1 .. ..amjk..bm) so that
   T = U(..v1jk..t1 .. ..vmjk..tm). This a higher-order matching
   problem with a priori different solutions (one of them if T itself!).

   We finally invert the uij and the ti and build the return clause

   phi(x11..x1n_1y1..xm1..xmn_mym) =
     match x11..x1n_1 y1 .. xm1..xmn_m ym with
         | p11..p1n_1 q1 .. pm1..pmn_m qm => U(..a1jk..b1 .. ..amjk..bm)
         |  _ .. _    _  ..  _ .. _    _  => True
    end

   so that "phi(u11..u1n_1t1..um1..umn_mtm) = T" (note that the clause
   returning True never happens and any inhabited type can be put instead).
*)

let adjust_to_extended_env_and_remove_deps env extenv sigma subst t =
  let n = Context.Rel.length (rel_context !!env) in
  let n' = Context.Rel.length (rel_context !!extenv) in
  (* We first remove the bindings that are dependently typed (they are
     difficult to manage and it is not sure these are so useful in practice);
     Notes:
     - [subst] is made of pairs [(id,u)] where id is a name in [extenv] and
       [u] a term typed in [env];
     - [subst0] is made of items [(p,u,(u,ty))] where [ty] is the type of [u]
       and both are adjusted to [extenv] while [p] is the index of [id] in
       [extenv] (after expansion of the aliases) *)
  let map (x, u) =
    (* d1 ... dn dn+1 ... dn'-p+1 ... dn' *)
    (* \--env-/          (= x:ty)         *)
    (* \--------------extenv------------/ *)
    let (p, _, _) = lookup_rel_id x (rel_context !!extenv) in
    let rec traverse_local_defs p =
      match lookup_rel p !!extenv with
      | LocalDef (_,c,_) -> assert (isRel sigma c); traverse_local_defs (p + destRel sigma c)
      | LocalAssum _ -> p in
    let p = traverse_local_defs p in
    let u = lift (n' - n) u in
    try Some (p, u, expand_vars_in_term !!extenv sigma u)
      (* pedrot: does this really happen to raise [Failure _]? *)
    with Failure _ -> None in
  let subst0 = List.map_filter map subst in
  let t0 = lift (n' - n) t in
  (subst0, t0)

let push_binder sigma d (k,env,subst) =
  (k+1,snd (push_rel ~hypnaming:KeepUserNameAndRenameExistingButSectionNames sigma d env),List.map (fun (na,u,d) -> (na,lift 1 u,d)) subst)

let rec list_assoc_in_triple x = function
    [] -> raise Not_found
  | (a, b, _)::l -> if Int.equal a x then b else list_assoc_in_triple x l

(* Let vijk and ti be a set of dependent terms and T a type, all
 * defined in some environment env. The vijk and ti are supposed to be
 * instances for variables aijk and bi.
 *
 * [abstract_tycon Gamma0 Sigma subst T Gamma] looks for U(..v1jk..t1 .. ..vmjk..tm)
 * defined in some extended context
 * "Gamma0, ..a1jk:V1jk.. b1:W1 .. ..amjk:Vmjk.. bm:Wm"
 * such that env |- T = U(..v1jk..t1 .. ..vmjk..tm). To not commit to
 * a particular solution, we replace each subterm t in T that unifies with
 * a subset u1..ul of the vijk and ti by a special evar
 * ?id(x=t;c1:=c1,..,cl=cl) defined in context Gamma0,x,c1,...,cl |- ?id
 * (where the c1..cl are the aijk and bi matching the u1..ul), and
 * similarly for each ti.
*)

let abstract_tycon ?loc env sigma subst tycon extenv t =
  let t = nf_betaiota !!env sigma t in (* it helps in some cases to remove K-redex*)
  let src = match EConstr.kind sigma t with
    | Evar (evk,_) -> (Loc.tag ?loc @@ Evar_kinds.SubEvar (None,evk))
    | _ -> (Loc.tag ?loc @@ Evar_kinds.CasesType true) in
  let subst0,t0 = adjust_to_extended_env_and_remove_deps env extenv sigma subst t in
  (* We traverse the type T of the original problem Xi looking for subterms
     that match the non-constructor part of the constraints (this part
     is in subst); these subterms are the "good" subterms and we replace them
     by an evar that may depend (and only depend) on the corresponding
     convertible subterms of the substitution *)
  let evdref = ref sigma in
  let rec aux (k,env,subst as x) t =
    (* Use a reference because the [map_constr_with_full_binders] does not
       allow threading a state. *)
    let sigma = !evdref in
    match EConstr.kind sigma t with
    | Rel n when is_local_def (lookup_rel n !!env) -> t
    | Evar ev ->
        let ty = get_type_of !!env sigma t in
        let sigma, ty = refresh_universes (Some false) !!env sigma ty in
        let inst =
          List.map_i
            (fun i _ ->
              try list_assoc_in_triple i subst0 with Not_found -> mkRel i)
              1 (rel_context !!env) in
        let sigma, ev' = Evarutil.new_evar ~src ~typeclass_candidate:false !!env sigma ty in
        begin
          let flags = (default_flags_of TransparentState.full) in
          match solve_simple_eqn evar_unify flags !!env sigma (None,ev,substl inst ev') with
          | Success evd -> evdref := evd
          | UnifFailure _ -> evdref := add_conv_pb (Reduction.CONV,!!env,substl inst ev',t) sigma
        end;
        ev'
    | _ ->
    let good = List.filter (fun (_,u,_) -> is_conv_leq !!env sigma t u) subst in
    match good with
    | [] ->
      map_constr_with_full_binders !!env sigma (push_binder sigma) aux x t
    | (_, _, u) :: _ -> (* u is in extenv *)
      let vl = List.map pi1 good in
      let ty =
        let ty = get_type_of !!env sigma t in
        let sigma, res = refresh_universes (Some false) !!env !evdref ty in
        evdref := sigma; res
      in
      let dummy_subst = List.init k (fun _ -> mkProp) in
      let ty = substl dummy_subst (aux x ty) in
      let sigma = !evdref in
      let depvl = free_rels sigma ty in
      let inst =
        List.map_i
          (fun i _ -> if Int.List.mem i vl then u else mkRel i) 1
          (rel_context !!extenv) in
      let map a = match EConstr.kind sigma a with
      | Rel n -> not (noccurn sigma n u) || Int.Set.mem n depvl
      | _ -> true
      in
      let rel_filter = List.map map inst in
      let named_filter =
        List.map (fun d -> local_occur_var sigma (NamedDecl.get_id d) u)
          (named_context !!extenv) in
      let filter = Filter.make (rel_filter @ named_filter) in
      let candidates = List.rev (u :: List.map mkRel vl) in
      let sigma, ev = Evarutil.new_evar !!extenv ~src ~filter ~candidates ~typeclass_candidate:false sigma ty in
      let () = evdref := sigma in
      lift k ev
  in
  let ans = aux (0,extenv,subst0) t0 in
  !evdref, ans

let build_tycon ?loc env tycon_env s subst tycon extenv sigma t =
  let s = mkSort s in
  match t with
    | None ->
        (* This is the situation we are building a return predicate and
           we are in an impossible branch *)
        let n = Context.Rel.length (rel_context !!env) in
        let n' = Context.Rel.length (rel_context !!tycon_env) in
        let src = Loc.tag ?loc Evar_kinds.ImpossibleCase in
        let sigma, impossible_case_type =
          Evarutil.new_evar (reset_context !!env) sigma ~src ~typeclass_candidate:false s in
        (sigma, { uj_val = lift (n'-n) impossible_case_type; uj_type = s })
    | Some t ->
        let sigma, t = abstract_tycon ?loc tycon_env sigma subst tycon extenv t in
        let sigma, tt = Typing.type_of !!extenv sigma t in
        match unify_leq_delay !!env sigma tt s with
        | exception Evarconv.UnableToUnify _ -> anomaly (Pp.str "Build_tycon: should be a type.");
        | sigma -> (sigma, { uj_val = t; uj_type = tt })


(* For a multiple pattern-matching problem Xi on t1..tn with return
 * type T, [build_inversion_problem Gamma Sigma (t1..tn) T] builds a return
 * predicate for Xi that is itself made by an auxiliary
 * pattern-matching problem of which the first clause reveals the
 * pattern structure of the constraints on the inductive types of the t1..tn,
 * and the second clause is a wildcard clause for catching the
 * impossible cases. See above "Building an inversion predicate" for
 * further explanations
 *)

let build_inversion_problem ~program_mode loc env sigma tms t =
  let make_patvar t (subst,avoid) =
    let id = next_name_away (named_hd !!env sigma t Anonymous) avoid in
    DAst.make @@ PatVar (Name id), ((id,t)::subst, Id.Set.add id avoid) in
  let rec reveal_pattern t (subst,avoid as acc) =
    match EConstr.kind sigma (whd_all !!env sigma t) with
    | Construct (cstr,u) -> DAst.make (PatCstr (cstr,[],Anonymous)), acc
    | App (f,v) when isConstruct sigma f ->
        let cstr,u = destConstruct sigma f in
        let n = constructor_nrealargs !!env cstr in
        let l = List.lastn n (Array.to_list v) in
        let l,acc = List.fold_right_map reveal_pattern l acc in
        DAst.make (PatCstr (cstr,l,Anonymous)), acc
    | _ -> make_patvar t acc in
  let rec aux n env acc_sign tms acc =
    match tms with
    | [] -> [], acc_sign, acc
    | (t, IsInd (_,IndType(indf,realargs),_)) :: tms ->
        let patl,acc = List.fold_right_map reveal_pattern realargs acc in
        let pat,acc = make_patvar t acc in
        let indf' = lift_inductive_family n indf in
        let sign = make_arity_signature !!env sigma true indf' in
        let patl = pat :: List.rev patl in
        let patl,sign = recover_and_adjust_alias_names acc patl sign in
        let p = List.length patl in
        let _,env' = push_rel_context ~hypnaming:KeepUserNameAndRenameExistingButSectionNames sigma sign env in
        let patl',acc_sign,acc = aux (n+p) env' (sign@acc_sign) tms acc in
        List.rev_append patl patl',acc_sign,acc
    | (t, NotInd (bo,typ)) :: tms ->
      let pat,acc = make_patvar t acc in
      let typ = lift n typ in
      let d = LocalAssum (annotR (alias_of_pat pat),typ) in
      let patl,acc_sign,acc = aux (n+1) (snd (push_rel ~hypnaming:KeepUserNameAndRenameExistingButSectionNames sigma d env)) (d::acc_sign) tms acc in
      pat::patl,acc_sign,acc in
  let avoid0 = GlobEnv.vars_of_env env in
  (* [patl] is a list of patterns revealing the substructure of
     constructors present in the constraints on the type of the
     multiple terms t1..tn that are matched in the original problem;
     [subst] is the substitution of the free pattern variables in
     [patl] that returns the non-constructor parts of the constraints.
     Especially, if the ti has type I ui1..uin_i, and the patterns associated
     to ti are pi1..pin_i, then subst(pij) is uij; the substitution is
     useful to recognize which subterms of the whole type T of the original
     problem have to be abstracted *)
  let patl,sign,(subst,avoid) = aux 0 env [] tms ([],avoid0) in
  let n = List.length sign in

  let decls =
    List.map_i (fun i d -> (mkRel i, map_constr (lift i) d)) 1 sign in

  let _,pb_env = push_rel_context ~hypnaming:KeepUserNameAndRenameExistingButSectionNames sigma sign env in
  let decls =
    List.map (fun (c,d) -> (c,extract_inductive_data !!(pb_env) sigma d,d)) decls in

  let decls = List.rev decls in
  let dep_sign = find_dependencies_signature sigma (List.make n true) decls in

  let sub_tms =
    List.map2 (fun deps (tm, (tmtyp,_), decl) ->
      let na = if List.is_empty deps then Anonymous else force_name (RelDecl.get_name decl) in
      Pushed (true,((tm,tmtyp),deps,na)))
      dep_sign decls in
  let subst = List.map (fun (na,t) -> (na,lift n t)) subst in
  (* [main_eqn] is the main clause of the auxiliary pattern-matching that
     serves as skeleton for the return type: [patl] is the
     substructure of constructors extracted from the list of
     constraints on the inductive types of the multiple terms matched
     in the original pattern-matching problem Xi *)
  let main_eqn =
    { patterns = patl;
      alias_stack = [];
      eqn_loc = None;
      orig = None;
      catch_all_vars = [];
      rhs = { rhs_env = pb_env;
              (* we assume all vars are used; in practice we discard dependent
                 vars so that the field rhs_vars is normally not used *)
              rhs_vars = List.fold_left (fun accu (id, _) -> Id.Set.add id accu) Id.Set.empty subst;
              avoid_ids = avoid;
              it = Some (lift n t) } } in
  (* [catch_all] is a catch-all default clause of the auxiliary
     pattern-matching, if needed: it will catch the clauses
     of the original pattern-matching problem Xi whose type
     constraints are incompatible with the constraints on the
     inductive types of the multiple terms matched in Xi *)
  let catch_all_eqn =
    if List.for_all (irrefutable !!env) patl then
      (* No need for a catch all clause *)
      []
    else
      [ { patterns = List.map (fun _ -> DAst.make @@ PatVar Anonymous) patl;
          alias_stack = [];
          eqn_loc = None;
          orig = None;
          catch_all_vars = [];
          rhs = { rhs_env = pb_env;
                  rhs_vars = Id.Set.empty;
                  avoid_ids = avoid0;
                  it = None } } ] in
  (* [pb] is the auxiliary pattern-matching serving as skeleton for the
      return type of the original problem Xi *)
  let s = Retyping.get_sort_of !!env sigma t in
  let sigma, s = Sorts.(match s with
  | SProp | Prop | Set ->
    (* To anticipate a possible restriction on an elimination from
       SProp, Prop or (impredicative) Set we preserve the sort of the
       main branch, knowing that the default impossible case shall
       always be coercible to one of those *)
      sigma, s
  | Type _ ->
    (* If the sort has algebraic universes, we cannot use this sort a
       type constraint for the impossible case; especially if the
       default case is not the canonical one provided in Prop by Coq
       but one given by the user, which may be in either sort (an
       example is in Vector.caseS', even if this one can probably be
       put in Prop too with some care) *)
    let sigma, s' = Evd.new_sort_variable univ_flexible sigma in
    let sigma = Evd.set_leq_sort !!env sigma s s' in
    sigma, s') in
  let pb =
    { env       = pb_env;
      pred      = (*ty *) mkSort s;
      tomatch   = sub_tms;
      history   = start_history n;
      mat       = main_eqn :: catch_all_eqn;
      caseloc   = loc;
      casestyle = RegularStyle;
      typing_function = build_tycon ?loc env pb_env s subst} in
  let _used, sigma, j = compile ~program_mode sigma pb in
  (sigma, j.uj_val)

(* Here, [pred] is assumed to be in the context built from all *)
(* realargs and terms to match *)
let build_initial_predicate arsign pred =
  let rec buildrec pred tmnames = function
    | [] -> List.rev tmnames,pred
    | (decl::realdecls)::lnames ->
        let na = RelDecl.get_name decl in
        let realnames = List.map RelDecl.get_name realdecls in
        buildrec pred ((force_name na,realnames)::tmnames) lnames
    | _ -> assert false
  in buildrec pred [] (List.rev arsign)

let extract_arity_signature ?(dolift=true) env0 tomatchl tmsign =
  let lift = if dolift then lift else fun n t -> t in
  let get_one_sign n tm (na,t) =
    match tm with
      | NotInd (bo,typ) ->
          (match t with
            | None ->
              let r = Sorts.Relevant in (* TODO relevance *)
              let sign = match bo with
                       | None -> [LocalAssum (make_annot na r, lift n typ)]
                       | Some b -> [LocalDef (make_annot na r, lift n b, lift n typ)] in sign
            | Some {CAst.loc} ->
            user_err ?loc
                (str"Unexpected type annotation for a term of non inductive type."))
      | IsInd (term,IndType(indf,realargs),_) ->
          let indf' = if dolift then lift_inductive_family n indf else indf in
          let ((ind,u),_) = dest_ind_family indf' in
          let nrealargs_ctxt = inductive_nrealdecls env0 ind in
          let arsign, inds = get_arity env0 indf' in
          let arsign = List.map (fun d -> map_rel_decl EConstr.of_constr d) arsign in
          let realnal =
            match t with
              | Some {CAst.loc;v=(ind',realnal)} ->
                  if not (Ind.CanOrd.equal ind ind') then
                    user_err ?loc  (str "Wrong inductive type.");
                  if not (Int.equal nrealargs_ctxt (List.length realnal)) then
                      anomaly (Pp.str "Ill-formed 'in' clause in cases.");
                  List.rev realnal
              | None ->
                  List.make nrealargs_ctxt Anonymous in
          let r = Sorts.relevance_of_sort_family inds in
          let t = EConstr.of_constr (build_dependent_inductive env0 indf') in
          LocalAssum (make_annot na r, t) :: List.map2 RelDecl.set_name realnal arsign in
  let rec buildrec n = function
    | [],[] -> []
    | (_,tm)::ltm, (_,x)::tmsign ->
        let l = get_one_sign n tm x in
        l :: buildrec (n + List.length l) (ltm,tmsign)
    | _ -> assert false
  in List.rev (buildrec 0 (tomatchl,tmsign))

let inh_conv_coerce_to_tycon ?loc ~program_mode env sigma j tycon =
  match tycon with
    | Some p ->
      let (evd,v,_trace) =
        Coercion.inh_conv_coerce_to ?loc ~program_mode true env sigma
          ~flags:(default_flags_of TransparentState.full) j p
      in
      (evd,v)
    | None -> sigma, j

(* We put the tycon inside the arity signature, possibly discovering dependencies. *)

let add_subst sigma c len (rel_subst,var_subst) =
  match EConstr.kind sigma c with
  | Rel n -> (n,len) :: rel_subst, var_subst
  | Var id -> rel_subst, (id,len) :: var_subst
  | _ -> assert false

let dependent_rel_or_var sigma tm c =
  match EConstr.kind sigma tm with
  | Rel n -> not (noccurn sigma n c)
  | Var id -> Termops.local_occur_var sigma id c
  | _ -> assert false

let prepare_predicate_from_arsign_tycon ~program_mode env sigma loc tomatchs arsign c =
  let nar = List.fold_left (fun n sign -> Context.Rel.nhyps sign + n) 0 arsign in
  let (rel_subst,var_subst), len =
    List.fold_right2 (fun (tm, tmtype) sign (subst, len) ->
      let signlen = List.length sign in
        match EConstr.kind sigma tm with
          | Rel _ | Var _ when Int.equal signlen 1 && dependent_rel_or_var sigma tm c
            (* The term to match is not of a dependent type itself *) ->
              (add_subst sigma tm len subst, len - signlen)
          | Rel _ | Var _ when signlen > 1 (* The term is of a dependent type,
                                      maybe some variable in its type appears in the tycon. *) ->
              (match tmtype with
                  NotInd _ -> (subst, len - signlen)
                | IsInd (_, IndType(indf,realargs),_) ->
                    let subst, len =
                      List.fold_left
                        (fun (subst, len) arg ->
                          match EConstr.kind sigma arg with
                          | Rel _ | Var _ when dependent_rel_or_var sigma arg c ->
                              (add_subst sigma arg len subst, pred len)
                          | _ -> (subst, pred len))
                        (subst, len) realargs
                    in
                    let subst =
                      if dependent_rel_or_var sigma tm c && List.for_all (fun c -> isRel sigma c || isVar sigma c) realargs
                      then add_subst sigma tm len subst else subst
                    in (subst, pred len))
          | _ -> (subst, len - signlen))
      (List.rev tomatchs) arsign (([],[]), nar)
  in
  let rec predicate lift c =
    match EConstr.kind sigma c with
      | Rel n when n > lift ->
          (try
              (* Make the predicate dependent on the matched variable *)
              let idx = Int.List.assoc (n - lift) rel_subst in
                mkRel (idx + lift)
            with Not_found ->
              (* A variable that is not matched, lift over the arsign *)
              mkRel (n + nar))
      | Var id ->
          (try
              (* Make the predicate dependent on the matched variable *)
              let idx = Id.List.assoc id var_subst in
                mkRel (idx + lift)
            with Not_found ->
              (* A variable that is not matched *)
              c)
      | _ ->
          EConstr.map_with_binders sigma succ predicate lift c
  in
  assert (len == 0);
  let p = predicate 0 c in
  let hypnaming = if program_mode then ProgramNaming else KeepUserNameAndRenameExistingButSectionNames in
  let arsign,env' = List.fold_right_map (push_rel_context ~hypnaming sigma) arsign env in
  try let sigma' = fst (Typing.type_of !!env' sigma p) in
        Some (sigma', p, arsign)
  with e when precatchable_exception e -> None

let expected_elimination_sort env tomatchl =
  List.fold_right (fun (_,tm) s ->
      match tm with
      | IsInd (_,IndType(indf,_),_) ->
        (* Not a degenerated line, see coerce_to_indtype *)
        let s' = Inductive.elim_sort (Inductive.lookup_mind_specif env (fst (fst (dest_ind_family indf)))) in
        if Sorts.family_leq s s' then s else s'
      | NotInd _ -> s) tomatchl Sorts.InType

(* Builds the predicate. If the predicate is dependent, its context is
 * made of 1+nrealargs assumptions for each matched term in an inductive
 * type and 1 assumption for each term not _syntactically_ in an
 * inductive type.

 * Each matched term is independently considered dependent or not.
 *)

let prepare_predicate ?loc ~program_mode typing_fun env sigma tomatchs arsign tycon pred =
  let refresh_tycon sigma t =
    (* If we put the typing constraint in the term, it has to be
       refreshed to preserve the invariant that no algebraic universe
       can appear in the term.  *)
    refresh_universes ~status:Evd.univ_flexible ~onlyalg:true (Some true)
                      !!env sigma t
  in
  let preds =
    match pred with
    (* No return clause *)
    | None ->
        let sigma,t =
          match tycon with
          | Some t -> refresh_tycon sigma t
          | None ->
             (* No type constraint: we first create a generic evar type constraint *)
             let src = (loc, Evar_kinds.CasesType false) in
             let sigma, (t, _) = Evarutil.new_type_evar !!env sigma univ_flexible ~src in
             sigma, t in
        (* First strategy: we build an "inversion" predicate, also replacing the *)
        (* dependencies with existential variables *)
        let sigma1,pred1 = build_inversion_problem loc ~program_mode env sigma tomatchs t in
        (* Optional second strategy: we abstract the tycon wrt to the dependencies *)
        let p2 =
          prepare_predicate_from_arsign_tycon ~program_mode env sigma loc tomatchs arsign t in
        (* Third strategy: we take the type constraint as it is; of course we could *)
        (* need something in between, abstracting some but not all of the dependencies *)
        (* the "inversion" strategy deals with that but unification may not be *)
        (* powerful enough so strategy 2 and 3 helps; moreover, inverting does not *)
        (* work (yet) when a constructor has a type not precise enough for the inversion *)
        (* see log message for details *)
        let pred3 = lift (List.length (List.flatten arsign)) t in
        (match p2 with
         | Some (sigma2,pred2,arsign) when not (EConstr.eq_constr sigma pred2 pred3) ->
             [sigma1, pred1, arsign; sigma2, pred2, arsign; sigma, pred3, arsign]
         | _ ->
             [sigma1, pred1, arsign; sigma, pred3, arsign])
    (* Some type annotation *)
    | Some rtntyp ->
      (* We extract the signature of the arity *)
      let building_arsign,envar = List.fold_right_map (push_rel_context ~hypnaming:KeepUserNameAndRenameExistingButSectionNames sigma) arsign env in
      (* We put a type constraint on the predicate so that one
         branch type-checked first does not lead to a lower type than
         another branch; we take into account the possible elimination
         constraints on the predicate *)
      let sigma, rtnsort = fresh_sort_in_family sigma (expected_elimination_sort !!env tomatchs) in
      let sigma, predcclj = typing_fun (Some (mkSort rtnsort)) envar sigma rtntyp in
      let predccl = nf_evar sigma predcclj.uj_val in
      [sigma, predccl, building_arsign]
  in
  List.map
    (fun (sigma,pred,arsign) ->
      let (nal,pred) = build_initial_predicate arsign pred in
      sigma,nal,pred)
    preds

(** Program cases *)

open Program

let ($) f x = f x

let string_of_name name =
  match name with
    | Anonymous -> "anonymous"
    | Name n -> Id.to_string n

let make_prime_id name =
  let str = string_of_name name in
    Id.of_string str, Id.of_string (str ^ "'")

let prime avoid name =
  let previd, id = make_prime_id name in
    previd, next_ident_away id avoid

let make_prime avoid prevname =
  let previd, id = prime !avoid prevname in
    avoid := Id.Set.add id !avoid;
    previd, id

let eq_id avoid id =
  let hid = Id.of_string ("Heq_" ^ Id.to_string id) in
  let hid' = next_ident_away hid avoid in
    hid'

let mk_eq sigma typ x y = papp sigma coq_eq_ind [| typ; x ; y |]
let mk_eq_refl sigma typ x = papp sigma coq_eq_refl [| typ; x |]
let mk_JMeq sigma typ x typ' y =
  papp sigma coq_JMeq_ind [| typ; x ; typ'; y |]
let mk_JMeq_refl sigma typ x =
  papp sigma coq_JMeq_refl [| typ; x |]

let hole na = DAst.make @@
  GHole (Evar_kinds.QuestionMark {
      Evar_kinds.qm_obligation= Evar_kinds.Define false;
      Evar_kinds.qm_name=na;
      Evar_kinds.qm_record_field=None},
         IntroAnonymous, None)

let constr_of_pat env sigma arsign pat avoid =
  let rec typ env sigma (ty, realargs) pat avoid =
    let loc = pat.CAst.loc in
    match DAst.get pat with
    | PatVar name ->
        let name, avoid = match name with
            Name n -> name, avoid
          | Anonymous ->
              let id = next_ident_away wildcard_id avoid in
                Name id, Id.Set.add id avoid
        in
        let r = Sorts.Relevant in (* TODO relevance *)
          (sigma, (DAst.make ?loc @@ PatVar name), [LocalAssum (make_annot name r, ty)] @ realargs, mkRel 1, ty,
           (List.map (fun x -> mkRel 1) realargs), 1, avoid)
    | PatCstr (((_, i) as cstr),args,alias) ->
        let cind = inductive_of_constructor cstr in
        let IndType (indf, _) =
          try find_rectype env sigma (lift (-(List.length realargs)) ty)
          with Not_found -> error_case_not_inductive env sigma
            {uj_val = ty; uj_type = Retyping.get_type_of env sigma ty}
        in
        let (ind,u), params = dest_ind_family indf in
        let params = List.map EConstr.of_constr params in
        if not (Ind.CanOrd.equal ind cind) then error_bad_constructor ?loc env cstr ind;
        let cstrs = get_constructors env indf in
        let ci = cstrs.(i-1) in
        let nb_args_constr = ci.cs_nargs in
        assert (Int.equal nb_args_constr (List.length args));
        let sigma, patargs, args, sign, env, n, m, avoid =
          List.fold_right2
            (fun decl ua (sigma, patargs, args, sign, env, n, m, avoid)  ->
               let t = EConstr.of_constr (RelDecl.get_type decl) in
               let sigma, pat', sign', arg', typ', argtypargs, n', avoid =
                 let liftt = liftn (List.length sign) (succ (List.length args)) t in
                   typ env sigma (substl args liftt, []) ua avoid
               in
               let args' = arg' :: List.map (lift n') args in
               let env' = EConstr.push_rel_context sign' env in
                 (sigma, pat' :: patargs, args', sign' @ sign, env', n' + n, succ m, avoid))
            ci.cs_args (List.rev args) (sigma, [], [], [], env, 0, 0, avoid)
        in
        let args = List.rev args in
        let patargs = List.rev patargs in
        let pat' = DAst.make ?loc @@ PatCstr (cstr, patargs, alias) in
        let cstr = mkConstructU (on_snd EInstance.make ci.cs_cstr) in
        let app = applist (cstr, List.map (lift (List.length sign)) params) in
        let app = applist (app, args) in
        let apptype = Retyping.get_type_of env sigma app in
        let IndType (indf, realargs) = find_rectype env sigma apptype in
          match alias with
              Anonymous ->
                sigma, pat', sign, app, apptype, realargs, n, avoid
            | Name id ->
                let _, inds = get_arity env indf in
                let r = Sorts.relevance_of_sort_family inds in
                let sign = LocalAssum (make_annot alias r, lift m ty) :: sign in
                let avoid = Id.Set.add id avoid in
                let sigma, sign, i, avoid =
                  try
                    let env = EConstr.push_rel_context sign env in
                    let sigma = unify_leq_delay (EConstr.push_rel_context sign env) sigma
                      (lift (succ m) ty) (lift 1 apptype) in
                    let sigma, eq_t = mk_eq sigma (lift (succ m) ty)
                      (mkRel 1) (* alias *)
                      (lift 1 app) (* aliased term *)
                    in
                    let neq = eq_id avoid id in
                    (* if we ever allow using a SProp-typed coq_eq_ind this relevance will be wrong *)
                      sigma, LocalDef (nameR neq, mkRel 0, eq_t) :: sign, 2, Id.Set.add neq avoid
                  with Evarconv.UnableToUnify _ -> sigma, sign, 1, avoid
                in
                  (* Mark the equality as a hole *)
                  sigma, pat', sign, lift i app, lift i apptype, realargs, n + i, avoid
  in
  let sigma, pat', sign, patc, patty, args, z, avoid = typ env sigma (RelDecl.get_type (List.hd arsign), List.tl arsign) pat avoid in
    sigma, pat', (sign, patc, (RelDecl.get_type (List.hd arsign), args), pat'), avoid


(* shadows functional version *)
let eq_id avoid id =
  let hid = Id.of_string ("Heq_" ^ Id.to_string id) in
  let hid' = next_ident_away hid !avoid in
    avoid := Id.Set.add hid' !avoid;
    hid'

let is_topvar sigma t =
match EConstr.kind sigma t with
| Rel 0 -> true
| _ -> false

let rels_of_patsign sigma =
  List.map (fun decl ->
            match decl with
            | LocalDef (na,t',t) when is_topvar sigma t' -> LocalAssum (na,t)
            | _ -> decl)

let vars_of_ctx sigma ctx =
  let _, y =
    List.fold_right (fun decl (prev, vars) ->
      match decl with
        | LocalDef (na,t',t) when is_topvar sigma t' ->
            prev,
            (DAst.make @@ GApp (
                (DAst.make @@ GRef (delayed_force coq_eq_refl_ref, None)),
                   [hole na.binder_name; DAst.make @@ GVar prev])) :: vars
        | _ ->
            match RelDecl.get_name decl with
                Anonymous -> invalid_arg "vars_of_ctx"
              | Name n -> n, (DAst.make @@ GVar n) :: vars)
      ctx (Id.of_string "vars_of_ctx_error", [])
  in List.rev y

let rec is_included x y =
  match DAst.get x, DAst.get y with
    | PatVar _, _ -> true
    | _, PatVar _ -> true
    | PatCstr ((_, i), args, alias), PatCstr ((_, i'), args', alias')  ->
        if Int.equal i i' then List.for_all2 is_included args args'
        else false

(* liftsign is the current pattern's complete signature length.
   Hence pats is already typed in its
   full signature. However prevpatterns are in the original one signature per pattern form.
 *)
let build_ineqs sigma prevpatterns pats liftsign =
  let sigma, diffs =
    List.fold_left
      (fun (sigma, c) eqnpats ->
          let sigma, acc = List.fold_left2
            (* ppat is the pattern we are discriminating against, curpat is the current one. *)
            (fun (sigma, acc) (ppat_sign, ppat_c, (ppat_ty, ppat_tyargs), ppat)
              (curpat_sign, curpat_c, (curpat_ty, curpat_tyargs), curpat) ->
              match acc with
                  None -> sigma, None
                | Some (sign, len, n, c) -> (* FixMe: do not work with ppat_args *)
                    if is_included curpat ppat then
                      (* Length of previous pattern's signature *)
                      let lens = List.length ppat_sign in
                      (* Accumulated length of previous pattern's signatures *)
                      let len' = lens + len in
                      let sigma, c' =
                        papp sigma coq_eq_ind
                          [| lift (len' + liftsign) curpat_ty;
                            liftn (len + liftsign) (succ lens) ppat_c ;
                            lift len' curpat_c |]
                      in
                      let acc =
                        ((* Jump over previous prevpat signs *)
                          lift_rel_context len ppat_sign @ sign,
                          len',
                          succ n, (* nth pattern *)
                          c' :: List.map (lift lens (* Jump over this prevpat signature *)) c)
                      in sigma, Some acc
                    else sigma, None)
           (sigma, Some ([], 0, 0, [])) eqnpats pats
         in match acc with
             None -> sigma, c
            | Some (sign, len, _, c') ->
               let sigma, conj = mk_coq_and sigma c' in
               let sigma, neg = mk_coq_not sigma conj in
               let conj = it_mkProd_or_LetIn neg (lift_rel_context liftsign sign) in
               sigma, conj :: c)
      (sigma, []) prevpatterns
  in match diffs with [] -> sigma, None
    | _ -> let sigma, conj = mk_coq_and sigma diffs in sigma, Some conj

let constrs_of_pats typing_fun env sigma eqns tomatchs sign neqs arity =
  let i = ref 0 in
  let (sigma, x, y, z) =
    List.fold_left
      (fun (sigma, branches, eqns, prevpatterns) eqn ->
         let sigma, _, newpatterns, pats =
           List.fold_left2
             (fun (sigma, idents, newpatterns, pats) pat arsign ->
                let sigma, pat', cpat, idents = constr_of_pat !!env sigma arsign pat idents in
                  (sigma, idents, pat' :: newpatterns, cpat :: pats))
              (sigma, Id.Set.empty, [], []) eqn.patterns sign
         in
         let newpatterns = List.rev newpatterns and opats = List.rev pats in
         let rhs_rels, pats, signlen =
           List.fold_left
             (fun (renv, pats, n) (sign,c, (s, args), p) ->
               (* Recombine signatures and terms of all of the row's patterns *)
               let sign' = lift_rel_context n sign in
               let len = List.length sign' in
                 (sign' @ renv,
                 (* lift to get outside of previous pattern's signatures. *)
                 (sign', liftn n (succ len) c,
                  (s, List.map (liftn n (succ len)) args), p) :: pats,
                 len + n))
             ([], [], 0) opats in
         let pats, _ = List.fold_left
           (* lift to get outside of past patterns to get terms in the combined environment. *)
           (fun (pats, n) (sign, c, (s, args), p) ->
             let len = List.length sign in
               ((rels_of_patsign sigma sign, lift n c,
                 (s, List.map (lift n) args), p) :: pats, len + n))
           ([], 0) pats
         in
         let sigma, ineqs = build_ineqs sigma prevpatterns pats signlen in
         let rhs_rels' = rels_of_patsign sigma rhs_rels in
         let _signenv,_ = push_rel_context ~hypnaming:ProgramNaming sigma rhs_rels' env in
         let arity =
           let args, nargs =
             List.fold_right (fun (sign, c, (_, args), _) (allargs,n) ->
               (args @ c :: allargs, List.length args + succ n))
               pats ([], 0)
           in
           let args = List.rev args in
             substl args (liftn signlen (succ nargs) arity)
         in
         let r = Sorts.Relevant in (* TODO relevance *)
         let rhs_rels', tycon =
           let neqs_rels, arity =
             match ineqs with
             | None -> [], arity
             | Some ineqs ->
                 [LocalAssum (make_annot Anonymous r, ineqs)], lift 1 arity
           in
           let eqs_rels, arity = decompose_prod_n_assum sigma neqs arity in
             eqs_rels @ neqs_rels @ rhs_rels', arity
         in
         let _,rhs_env = push_rel_context ~hypnaming:ProgramNaming sigma rhs_rels' env in
         let sigma, j = typing_fun (mk_tycon tycon) rhs_env sigma eqn.rhs.it in
         let bbody = it_mkLambda_or_LetIn j.uj_val rhs_rels'
         and btype = it_mkProd_or_LetIn j.uj_type rhs_rels' in
         let sigma, _btype = Typing.type_of !!env sigma bbody in
         let branch_name = Id.of_string ("program_branch_" ^ (string_of_int !i)) in
         let branch_decl = LocalDef (make_annot (Name branch_name) r, lift !i bbody, lift !i btype) in
         let branch =
           let bref = DAst.make @@ GVar branch_name in
             match vars_of_ctx sigma rhs_rels with
                 [] -> bref
               | l  -> DAst.make @@ GApp (bref, l)
         in
         let branch = match ineqs with
             Some _ -> DAst.make @@ GApp (branch, [ hole Anonymous ])
           | None -> branch
         in
         incr i;
         let rhs = { eqn.rhs with it = Some branch } in
           (sigma, branch_decl :: branches,
           { eqn with patterns = newpatterns; rhs = rhs } :: eqns,
           opats :: prevpatterns))
      (sigma, [], [], []) eqns
  in
  sigma, x, y

(* Builds the predicate. If the predicate is dependent, its context is
 * made of 1+nrealargs assumptions for each matched term in an inductive
 * type and 1 assumption for each term not _syntactically_ in an
 * inductive type.

 * Each matched terms are independently considered dependent or not.

 * A type constraint but no annotation case: it is assumed non dependent.
 *)

let lift_ctx n ctx =
  let ctx', _ =
    List.fold_right (fun (c, t) (ctx, n') ->
                       (liftn n n' c, liftn_tomatch_type n n' t) :: ctx, succ n')
      ctx ([], 0)
  in ctx'

(* Turn matched terms into variables. *)
let abstract_tomatch env sigma tomatchs tycon =
  let prev, ctx, names, tycon =
    List.fold_left
      (fun (prev, ctx, names, tycon) (c, t) ->
         let lenctx =  List.length ctx in
         match EConstr.kind sigma c with
             Rel n -> (lift lenctx c, lift_tomatch_type lenctx t) :: prev, ctx, names, tycon
           | _ ->
               let tycon = Option.map
                 (fun t -> subst_term sigma (lift 1 c) (lift 1 t)) tycon in
               let name = next_ident_away (Id.of_string "filtered_var") names in
               let r = Sorts.Relevant in (* TODO relevance *)
                 (mkRel 1, lift_tomatch_type (succ lenctx) t) :: lift_ctx 1 prev,
               LocalDef (make_annot (Name name) r, lift lenctx c, lift lenctx $ type_of_tomatch t) :: ctx,
               Id.Set.add name names, tycon)
      ([], [], Id.Set.empty, tycon) tomatchs
  in List.rev prev, ctx, tycon

let build_dependent_signature env sigma avoid tomatchs arsign =
  let avoid = ref avoid in
  let arsign = List.rev arsign in
  let allnames = List.rev_map (List.map RelDecl.get_name) arsign in
  let nar = List.fold_left (fun n names -> List.length names + n) 0 allnames in
  let sigma, eqs, neqs, refls, slift, arsign' =
    List.fold_left2
      (fun (sigma, eqs, neqs, refl_args, slift, arsigns) (tm, ty) arsign ->
         (* The accumulator:
            previous eqs,
            number of previous eqs,
            lift to get outside eqs and in the introduced variables ('as' and 'in'),
            new arity signatures
         *)
         match ty with
         | IsInd (ty, IndType (indf, args), _) when List.length args > 0 ->
             (* Build the arity signature following the names in matched terms
                as much as possible *)
             let argsign = List.tl arsign in (* arguments in inverse application order *)
             let app_decl = List.hd arsign in (* The matched argument *)
             let appn = RelDecl.get_name app_decl in
             let appt = RelDecl.get_type app_decl in
             let argsign = List.rev argsign in (* arguments in application order *)
             let sigma, env', nargeqs, argeqs, refl_args, slift, argsign' =
               List.fold_left2
                 (fun (sigma, env, nargeqs, argeqs, refl_args, slift, argsign') arg decl ->
                    let name = RelDecl.get_name decl in
                    let t = RelDecl.get_type decl in
                    let argt = Retyping.get_type_of env sigma arg in
                    let sigma, eq, refl_arg =
                      if Reductionops.is_conv env sigma argt t then
                        let sigma, eq =
                          mk_eq sigma (lift (nargeqs + slift) argt)
                            (mkRel (nargeqs + slift))
                            (lift (nargeqs + nar) arg)
                        in
                        let sigma, refl = mk_eq_refl sigma argt arg in
                        sigma, eq, refl
                      else
                        let sigma, eq =
                          mk_JMeq sigma (lift (nargeqs + slift) t)
                            (mkRel (nargeqs + slift))
                            (lift (nargeqs + nar) argt)
                            (lift (nargeqs + nar) arg)
                        in
                        let sigma, refl = mk_JMeq_refl sigma argt arg in
                        (sigma, eq, refl)
                    in
                    let previd, id =
                      let name =
                        match EConstr.kind sigma arg with
                        Rel n -> RelDecl.get_name (lookup_rel n env)
                        | _ -> name
                      in
                        make_prime avoid name
                    in
                      (sigma, env, succ nargeqs,
                       (LocalAssum (make_annot (Name (eq_id avoid previd)) Sorts.Relevant, eq)) :: argeqs,
                       refl_arg :: refl_args,
                       pred slift,
                       RelDecl.set_name (Name id) decl :: argsign'))
                 (sigma, env, neqs, [], [], slift, []) args argsign
             in
              let sigma, eq =
                mk_JMeq sigma
                  (lift (nargeqs + slift) appt)
                  (mkRel (nargeqs + slift))
                  (lift (nargeqs + nar) ty)
                  (lift (nargeqs + nar) tm)
             in
             let sigma, refl_eq = mk_JMeq_refl sigma ty tm in
             let previd, id = make_prime avoid appn in
               (sigma, (LocalAssum (make_annot (Name (eq_id avoid previd)) Sorts.Relevant, eq) :: argeqs) :: eqs,
                succ nargeqs,
                refl_eq :: refl_args,
                pred slift,
                ((RelDecl.set_name (Name id) app_decl :: argsign') :: arsigns))

         | _ -> (* Non dependent inductive or not inductive, just use a regular equality *)
             let decl = match arsign with [x] -> x | _ -> assert(false) in
             let name = RelDecl.get_name decl in
             let previd, id = make_prime avoid name in
             let arsign' = RelDecl.set_name (Name id) decl in
             let tomatch_ty = type_of_tomatch ty in
            let sigma, eq =
              mk_eq sigma (lift nar tomatch_ty)
                (mkRel slift) (lift nar tm)
            in
            let sigma, refl = mk_eq_refl sigma tomatch_ty tm in
            let na = make_annot (Name (eq_id avoid previd)) Sorts.Relevant in
            (sigma,
            [LocalAssum (na, eq)] :: eqs, succ neqs,
            refl :: refl_args,
            pred slift, (arsign' :: []) :: arsigns))
      (sigma, [], 0, [], nar, []) tomatchs arsign
  in
  let arsign'' = List.rev arsign' in
    assert(Int.equal slift 0); (* we must have folded over all elements of the arity signature *)
    sigma, arsign'', allnames, nar, eqs, neqs, refls

let context_of_arsign l =
  let (x, _) = List.fold_right
    (fun c (x, n) ->
       (lift_rel_context n c @ x, List.length c + n))
    l ([], 0)
  in x

let compile_program_cases ?loc style (typing_function, sigma) tycon env
    (predopt, tomatchl, eqns) =
  let typing_fun tycon env sigma = function
    | Some t ->	typing_function tycon env sigma t
    | None -> coq_unit_judge !!env sigma in

  (* We build the matrix of patterns and right-hand side *)
  let matx = matx_of_eqns env eqns in

  (* We build the vector of terms to match consistently with the *)
  (* constructors found in patterns *)
  let env, sigma, tomatchs = coerce_to_indtype ~program_mode:true typing_function env sigma matx tomatchl in
  let tycon = valcon_of_tycon tycon in
  let tomatchs, tomatchs_lets, tycon' = abstract_tomatch env sigma tomatchs tycon in
  let _,env = push_rel_context ~hypnaming:ProgramNaming sigma tomatchs_lets env in
  let len = List.length eqns in
  let sigma, sign, allnames, signlen, eqs, neqs, args =
    (* The arity signature *)
    let arsign = extract_arity_signature ~dolift:false !!env tomatchs tomatchl in
      (* Build the dependent arity signature, the equalities which makes
         the first part of the predicate and their instantiations. *)
    let avoid = Id.Set.empty in
      build_dependent_signature !!env sigma avoid tomatchs arsign

  in
  let sigma, tycon, arity =
    let nar = List.fold_left (fun n sign -> List.length sign + n) 0 sign in
    match tycon' with
    | None ->
      let sigma, ev = mkExistential !!env sigma in
      sigma, ev, lift nar ev
    | Some t ->
        let sigma, pred =
          match prepare_predicate_from_arsign_tycon ~program_mode:true env sigma loc tomatchs sign t with
          | Some (evd, pred, arsign) -> evd, pred
          | None -> sigma, lift nar t
        in
        sigma, Option.get tycon, pred
  in
  let neqs, arity =
    let ctx = context_of_arsign eqs in
    let neqs = List.length ctx in
      neqs, it_mkProd_or_LetIn (lift neqs arity) ctx
  in
  let sigma, lets, matx =
    (* Type the rhs under the assumption of equations *)
    constrs_of_pats typing_fun env sigma matx tomatchs sign neqs arity
  in
  let matx = List.rev matx in
  let _ = assert (Int.equal len (List.length lets)) in
  let _,env = push_rel_context ~hypnaming:ProgramNaming sigma lets env in
  let matx = List.map (fun eqn -> { eqn with rhs = { eqn.rhs with rhs_env = env } }) matx in
  let tomatchs = List.map (fun (x, y) -> lift len x, lift_tomatch_type len y) tomatchs in
  let args = List.rev_map (lift len) args in
  let pred = liftn len (succ signlen) arity in
  let nal, pred = build_initial_predicate sign pred in

  (* We push the initial terms to match and push their alias to rhs' envs *)
  (* names of aliases will be recovered from patterns (hence Anonymous here) *)

  (* TODO relevance *)
  let out_tmt na = function NotInd (None,t) -> LocalAssum (make_annot na Sorts.Relevant,t)
                          | NotInd (Some b, t) -> LocalDef (make_annot na Sorts.Relevant,b,t)
                          | IsInd (typ,_,_) -> LocalAssum (make_annot na Sorts.Relevant,typ) in
  let typs = List.map2 (fun (na,_) (tm,tmt) -> (tm,out_tmt na tmt)) nal tomatchs in

  let typs =
    List.map (fun (c,d) -> (c,extract_inductive_data !!env sigma d,d)) typs in

  let dep_sign =
    find_dependencies_signature sigma
      (List.make (List.length typs) true)
      typs in

  let typs' =
    List.map3
      (fun (tm,tmt) deps (na,realnames) ->
         let deps = if not (isRel sigma tm) then [] else deps in
          let tmt = set_tomatch_realnames realnames tmt in
           ((tm,tmt),deps,na))
      tomatchs dep_sign nal in

  let initial_pushed = List.map (fun x -> Pushed (true,x)) typs' in

  let typing_function tycon env sigma = function
    | Some t -> typing_function tycon env sigma t
    | None -> coq_unit_judge !!env sigma in

  let pb =
    { env      = env;
      pred     = pred;
      tomatch  = initial_pushed;
      history  = start_history (List.length initial_pushed);
      mat      = matx;
      caseloc  = loc;
      casestyle= style;
      typing_function = typing_function } in

  let used, sigma, j = compile ~program_mode:true sigma pb in
    (* We check for unused patterns *)
    check_unused_pattern !!env used matx;
    let body = it_mkLambda_or_LetIn (applist (j.uj_val, args)) lets in
    let j =
      { uj_val = it_mkLambda_or_LetIn body tomatchs_lets;
        (* XXX: is this normalization needed? *)
        uj_type = Evarutil.nf_evar sigma tycon; }
    in sigma, j

(**************************************************************************)
(* Main entry of the matching compilation                                 *)

let compile_cases ?loc ~program_mode style (typing_fun, sigma) tycon env (predopt, tomatchl, eqns) =
  if predopt == None && program_mode && Program.is_program_cases () then
    compile_program_cases ?loc style (typing_fun, sigma)
      tycon env (predopt, tomatchl, eqns)
  else

  (* We build the matrix of patterns and right-hand side *)
  let matx = matx_of_eqns env eqns in

  (* We build the vector of terms to match consistently with the *)
  (* constructors found in patterns *)
  let predenv, sigma, tomatchs = coerce_to_indtype ~program_mode typing_fun env sigma matx tomatchl in

  (* If an elimination predicate is provided, we check it is compatible
     with the type of arguments to match; if none is provided, we
     build alternative possible predicates *)
  let arsign = extract_arity_signature !!env tomatchs tomatchl in
  let preds = prepare_predicate ?loc ~program_mode typing_fun predenv sigma tomatchs arsign tycon predopt in

  let compile_for_one_predicate (sigma,nal,pred) =
    (* We push the initial terms to match and push their alias to rhs' envs *)
    (* names of aliases will be recovered from patterns (hence Anonymous *)
    (* here) *)

    (* TODO relevance *)
    let out_tmt na = function NotInd (None,t) -> LocalAssum (na,t)
                            | NotInd (Some b,t) -> LocalDef (na,b,t)
                            | IsInd (typ,_,_) -> LocalAssum (na,typ) in
    let typs = List.map2 (fun (na,_) (tm,tmt) -> (tm,out_tmt (make_annot na Sorts.Relevant) tmt)) nal tomatchs in

    let typs =
      List.map (fun (c,d) -> (c,extract_inductive_data !!env sigma d,d)) typs in

    let dep_sign =
      find_dependencies_signature sigma
        (List.make (List.length typs) true)
        typs in

    let typs' =
      List.map3
        (fun (tm,tmt) deps (na,realnames) ->
          let deps = if not (isRel sigma tm) then [] else deps in
          let tmt = set_tomatch_realnames realnames tmt in
          ((tm,tmt),deps,na))
        tomatchs dep_sign nal in

    let initial_pushed = List.map (fun x -> Pushed (true,x)) typs' in

    (* A typing function that provides with a canonical term for absurd cases*)
    let typing_fun tycon env sigma = function
    | Some t -> typing_fun tycon env sigma t
    | None -> coq_unit_judge !!env sigma in

    let pb =
      { env       = env;
        pred      = pred;
        tomatch   = initial_pushed;
        history   = start_history (List.length initial_pushed);
        mat       = matx;
        caseloc   = loc;
        casestyle = style;
        typing_function = typing_fun } in

    let used, sigma, j = compile ~program_mode sigma pb in

    (* We coerce to the tycon (if an elim predicate was provided) *)
    let sigma, j = inh_conv_coerce_to_tycon ?loc ~program_mode !!env sigma j tycon in
    used, sigma, j
  in

  (* Return the term compiled with the first possible elimination  *)
  (* predicate for which the compilation succeeds *)
  let used, sigma, j = list_try_compile compile_for_one_predicate preds in

  (* We check for unused patterns *)
  check_unused_pattern !!env used matx;

  sigma, j
