path: root/pretyping/cases.ml

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

(* $Id: cases.ml 13329 2010-07-26 11:05:39Z herbelin $ *)

open Util
open Names
open Nameops
open Term
open Termops
open Namegen
open Declarations
open Inductiveops
open Environ
open Sign
open Reductionops
open Typeops
open Type_errors
open Rawterm
open Retyping
open Pretype_errors
open Evarutil
open Evarconv
open Evd

(* Pattern-matching errors *)

type pattern_matching_error =
  | BadPattern of constructor * constr
  | BadConstructor of constructor * inductive
  | WrongNumargConstructor of constructor * int
  | WrongNumargInductive of inductive * int
  | WrongPredicateArity of constr * constr * constr
  | NeedsInversion of constr * constr
  | UnusedClause of cases_pattern list
  | NonExhaustive of cases_pattern list
  | CannotInferPredicate of (constr * types) array

exception PatternMatchingError of env * pattern_matching_error

let raise_pattern_matching_error (loc,ctx,te) =
  Stdpp.raise_with_loc loc (PatternMatchingError(ctx,te))

let error_bad_pattern_loc loc cstr ind =
  raise_pattern_matching_error (loc, Global.env(), BadPattern (cstr,ind))

let error_bad_constructor_loc loc cstr ind =
  raise_pattern_matching_error (loc, Global.env(), BadConstructor (cstr,ind))

let error_wrong_numarg_constructor_loc loc env c n =
  raise_pattern_matching_error (loc, env, WrongNumargConstructor(c,n))

let error_wrong_numarg_inductive_loc loc env c n =
  raise_pattern_matching_error (loc, env, WrongNumargInductive(c,n))

let error_wrong_predicate_arity_loc loc env c n1 n2 =
  raise_pattern_matching_error (loc, env, WrongPredicateArity (c,n1,n2))

let error_needs_inversion env x t =
  raise (PatternMatchingError (env, NeedsInversion (x,t)))

(**********************************************************************)
(* Functions to deal with impossible cases *)

let impossible_default_case = ref None

let set_impossible_default_clause c = impossible_default_case := Some c

let coq_unit_judge =
  let na1 = Name (id_of_string "A") in
  let na2 = Name (id_of_string "H") in
  fun () ->
    match !impossible_default_case with
    | Some (id,type_of_id) ->
	make_judge id type_of_id
    | None ->
	(* In case the constants id/ID are not defined *)
	make_judge (mkLambda (na1,mkProp,mkLambda(na2,mkRel 1,mkRel 1)))
                 (mkProd (na1,mkProp,mkArrow (mkRel 1) (mkRel 2)))

(**********************************************************************)

module type S = sig
  val compile_cases :
    loc -> case_style ->
    (type_constraint -> env -> evar_map ref -> rawconstr -> unsafe_judgment) * evar_map ref ->
    type_constraint ->
    env -> rawconstr option * tomatch_tuples * cases_clauses ->
    unsafe_judgment
end

let rec list_try_compile f = function
  | [a] -> f a
  | [] -> anomaly "try_find_f"
  | h::t ->
      try f h
      with UserError _ | TypeError _ | PretypeError _
	| Stdpp.Exc_located (_,(UserError _ | TypeError _ | PretypeError _)) ->
	    list_try_compile f t

let force_name =
  let nx = Name (id_of_string "x") 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 (PatVar (dummy_loc,Anonymous))

(* Environment management *)
let push_rels vars env = List.fold_right push_rel vars env

(* 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 regeneralize_rel i k j = if j = i+k then k+1 else j

let rec regeneralize_index i k t = match kind_of_term t with
  | Rel j when j = i+k -> mkRel (k+1)
  | Rel j when j < i+k -> t
  | Rel j when j > i+k -> t
  | _ -> map_constr_with_binders succ (regeneralize_index i) k t

type alias_constr =
  | DepAlias
  | NonDepAlias

let mkSpecialLetInJudge j (na,(deppat,nondeppat,d,t)) =
  { uj_val =
    if
      isRel deppat or not (dependent (mkRel 1) j.uj_val) or
      d = NonDepAlias & not (dependent (mkRel 1) j.uj_type)
    then
      (* The body of pat is not needed to type j - see *)
      (* insert_aliases - and both deppat and nondeppat have the *)
      (* same type, then one can freely substitute one by the other. *)
      (* We use nondeppat only if it's a Rel to preserve sharing. *)
      if isRel nondeppat then
        subst1 nondeppat j.uj_val
      else subst1 deppat j.uj_val
    else
      (* The body of pat is not needed to type j but its value *)
      (* is dependent in the type of j; our choice is to *)
      (* enforce this dependency *)
      mkLetIn (na,deppat,t,j.uj_val);
    uj_type = subst1 deppat j.uj_type }

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

type 'a rhs =
    { rhs_env    : env;
      rhs_vars   : identifier list;
      avoid_ids  : identifier list;
      it         : 'a option}

type 'a equation =
    { patterns     : cases_pattern list;
      rhs          : 'a rhs;
      alias_stack  : name list;
      eqn_loc      : loc;
      used         : bool ref }

type 'a matrix = 'a equation list

type dep_status = KnownDep | KnownNotDep | DepUnknown

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

type tomatch_status =
  | Pushed of ((constr * tomatch_type) * int list * (name * dep_status))
  | Alias of (constr * constr * alias_constr * constr)
  | Abstract of rel_declaration

type tomatch_stack = tomatch_status list

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

type alias_builder =
  | AliasLeaf
  | AliasConstructor of constructor

type pattern_history =
  | Top
  | MakeAlias of alias_builder * 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 ("Bad number of expected remaining patterns: "^(string_of_int n))
  | Result _ ->
      anomaly "Exhausted pattern history"

(* This is for non exhaustive error message *)

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

and build_rawpattern args = function
  | Top -> args
  | MakeAlias (AliasLeaf, rh) ->
      assert (args = []);
      rawpattern_of_partial_history [PatVar (dummy_loc, Anonymous)] rh
  | MakeAlias (AliasConstructor pci, rh) ->
      rawpattern_of_partial_history
	[PatCstr (dummy_loc, pci, args, Anonymous)] rh

let complete_history = rawpattern_of_partial_history []

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

let rec simplify_history = function
  | Continuation (0, l, Top) -> Result (List.rev l)
  | Continuation (0, l, MakeAlias (f, rh)) ->
      let pargs = List.rev l in
      let pat = match f with
	| AliasConstructor pci ->
	    PatCstr (dummy_loc,pci,pargs,Anonymous)
	| AliasLeaf ->
	    assert (l = []);
	    PatVar (dummy_loc, Anonymous) in
      feed_history pat rh
  | h -> h

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

let push_history_pattern n current cont =
  Continuation (n, [], MakeAlias (current, cont))

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

   env, evd |- <pred> Cases tomatch of mat end

  where tomatch is some sequence of "instructions" (t1  ... tn)

  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 and typed in [env]
  - "Abstract" instructions means an abstraction has to be inserted in the
    current branch to build (this means a pattern has been detected dependent
    in another one and generalisation is necessary to ensure well-typing)
  - "Alias" instructions means 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)

  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       : env;
      evdref    : evar_map ref;
      pred      : constr;
      tomatch   : tomatch_stack;
      history   : pattern_continuation;
      mat       : 'a matrix;
      caseloc   : loc;
      casestyle : case_style;
      typing_function: type_constraint -> env -> evar_map ref -> 'a option -> 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
  | PatVar _ :: l -> find_row_ind l
  | PatCstr(loc,c,_,_) :: _ -> Some (loc,c)

let inductive_template evdref env tmloc ind =
  let arsign = get_full_arity_sign env ind in
  let hole_source = match tmloc with
    | Some loc -> fun i -> (loc, TomatchTypeParameter (ind,i))
    | None -> fun _ -> (dummy_loc, InternalHole) in
   let (_,evarl,_) =
    List.fold_right
      (fun (na,b,ty) (subst,evarl,n) ->
	match b with
        | None ->
	    let ty' = substl subst ty in
	    let e = e_new_evar evdref env ~src:(hole_source n) ty' in
	    (e::subst,e::evarl,n+1)
	| Some b ->
	    (b::subst,evarl,n+1))
      arsign ([],[],1) in
   applist (mkInd ind,List.rev evarl)

let try_find_ind env sigma typ realnames =
  let (IndType(_,realargs) as ind) = find_rectype env sigma typ in
  let names =
    match realnames with
      | Some names -> names
      | None -> list_make (List.length realargs) Anonymous in
  IsInd (typ,ind,names)


let inh_coerce_to_ind evdref env ty tyi =
  let expected_typ = inductive_template evdref env None tyi in
     (* devrait être indifférent d'exiger leq ou pas puisque pour
        un inductif cela doit être égal *)
  let _ = e_cumul env evdref expected_typ ty in ()

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

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

let coerce_row typing_fun evdref env pats (tomatch,(_,indopt)) =
  let loc = Some (loc_of_rawconstr tomatch) in
  let tycon,realnames = find_tomatch_tycon evdref env loc indopt in
  let j = typing_fun tycon env evdref tomatch in
  let typ = nf_evar !evdref j.uj_type in
  let t =
    try try_find_ind env !evdref typ realnames
    with Not_found ->
      unify_tomatch_with_patterns evdref env loc typ pats realnames in
  (j.uj_val,t)

let coerce_to_indtype typing_fun evdref env 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
  List.map2 (coerce_row typing_fun evdref env) matx' tomatchl

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

let mkExistential env ?(src=(dummy_loc,InternalHole)) evdref =
  e_new_evar evdref env ~src:src (new_Type ())

let evd_comb2 f evdref x y =
  let (evd',y) = f !evdref x y in
  evdref := evd';
  y


module Cases_F(Coercion : Coercion.S) : S = struct

let adjust_tomatch_to_pattern 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 typ =
    try try_find_ind pb.env !(pb.evdref) typ names
    with Not_found -> NotInd (None,typ) in
  let tomatch = ((current,typ),deps,dep) in
  match typ with
  | NotInd (None,typ) ->
      let tm1 = List.map (fun eqn -> List.hd eqn.patterns) pb.mat in
      (match find_row_ind tm1 with
	| None -> tomatch
	| Some (_,(ind,_)) ->
	    let indt = inductive_template pb.evdref pb.env None ind in
	    let current =
	      if deps = [] & isEvar typ then
	      (* Don't insert coercions if dependent; only solve evars *)
		let _ = e_cumul pb.env pb.evdref indt typ in
		current
	      else
		(evd_comb2 (Coercion.inh_conv_coerce_to dummy_loc pb.env)
		  pb.evdref (make_judge current typ) (mk_tycon_type indt)).uj_val in
	    let sigma =  !(pb.evdref) in
	    let typ = try_find_ind pb.env sigma indt names in
	    ((current,typ),deps,dep))
  | _ -> tomatch

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

let mkDeclTomatch na = function
  | IsInd (t,_,_) -> (na,None,t)
  | NotInd (c,t) -> (na,c,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 (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 "Empty list of patterns"

let alias_of_pat = function
  | PatVar (_,name) -> name
  | PatCstr(_,_,_,name) -> name

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 "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, (_,None,_) :: decls) ->
      pat :: adjust_local_defs loc (pats,decls)
  | (pats, (_,Some _,_) :: decls) ->
      PatVar (loc, Anonymous) :: adjust_local_defs loc (pats,decls)
  | [], [] -> []
  | _ -> raise NotAdjustable

let check_and_adjust_constructor env ind cstrs = function
  | PatVar _ as pat -> pat
  | PatCstr (loc,((_,i) as cstr),args,alias) as pat ->
      (* Check it is constructor of the right type *)
      let ind' = inductive_of_constructor cstr in
      if eq_ind 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
	if List.length args = nb_args_constr then pat
	else
	  try
	    let args' = adjust_local_defs loc (args, List.rev ci.cs_args)
	    in PatCstr (loc, cstr, args', alias)
	  with NotAdjustable ->
	    error_wrong_numarg_constructor_loc loc (Global.env())
	      cstr nb_args_constr
      else
	(* Try to insert a coercion *)
	try
	  Coercion.inh_pattern_coerce_to loc pat ind' ind
	with Not_found ->
	  error_bad_constructor_loc loc cstr ind

let check_all_variables typ mat =
  List.iter
    (fun eqn -> match current_pattern eqn with
       | PatVar (_,id) -> ()
       | PatCstr (loc,cstr_sp,_,_) ->
	   error_bad_pattern_loc loc cstr_sp typ)
    mat

let check_unused_pattern env eqn =
  if not !(eqn.used) then
    raise_pattern_matching_error
      (eqn.eqn_loc, env, UnusedClause eqn.patterns)

let set_used_pattern eqn = eqn.used := true

let extract_rhs pb =
  match pb.mat with
    | [] -> errorlabstrm "build_leaf" (msg_may_need_inversion())
    | eqn::_ ->
	set_used_pattern eqn;
        eqn.rhs

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

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

let is_dep_patt_in eqn = function
  | PatVar (_,name) -> occur_in_rhs name eqn.rhs
  | PatCstr _ -> true

let mk_dep_patt_row (pats,eqn) =
  List.map (is_dep_patt_in eqn) pats

let dependencies_in_pure_rhs nargs eqns =
  if eqns = [] then list_make nargs false (* Only "_" patts *) else
  let deps_rows = List.map mk_dep_patt_row eqns in
  let deps_columns = matrix_transpose deps_rows in
  List.map (List.exists ((=) true)) deps_columns

let dependent_decl a = function
  | (na,None,t) -> dependent a t
  | (na,Some c,t) -> dependent a t || dependent a c

let rec dep_in_tomatch n = function
  | (Pushed _ | Alias _) :: l -> dep_in_tomatch n l
  | Abstract d :: l -> dependent_decl (mkRel n) d or dep_in_tomatch (n+1) l
  | [] -> false

let dependencies_in_rhs nargs current tms eqns =
  match kind_of_term current with
  | Rel n when dep_in_tomatch n tms -> list_make nargs true
  | _ -> dependencies_in_pure_rhs nargs eqns

(* Computing the matrix of dependencies *)

(* We are in context d1...dn |- and [find_dependencies k 1 nextlist]
   computes for declaration [k+1] in which of declarations in
   [nextlist] (which corresponds to d(k+2)...dn) it depends;
   declarations are expressed by index, e.g. in dependency list
   [n-2;1], [1] points to [dn] and [n-2] to [d3] *)

let rec find_dependency_list k n = function
  | [] -> []
  | (used,tdeps,d)::rest ->
      let deps = find_dependency_list k (n+1) rest in
      if used && dependent_decl (mkRel n) d
      then list_add_set (List.length rest + 1) (list_union deps tdeps)
      else deps

let find_dependencies is_dep_or_cstr_in_rhs d (k,nextlist) =
  let deps = find_dependency_list k 1 nextlist in
  if is_dep_or_cstr_in_rhs || deps <> []
  then (k-1,(true ,deps,d)::nextlist)
  else (k-1,(false,[]  ,d)::nextlist)

let find_dependencies_signature deps_in_rhs typs =
  let k = List.length deps_in_rhs in
  let _,l = List.fold_right2 find_dependencies deps_in_rhs typs (k,[]) 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 |-.

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

let regeneralize_index_tomatch n =
  let rec genrec depth = function
  | [] ->
      []
  | Pushed ((c,tm),l,dep) :: rest ->
      let c = regeneralize_index n depth c in
      let tm = map_tomatch_type (regeneralize_index n depth) tm in
      let l = List.map (regeneralize_rel n depth) l in
      Pushed ((c,tm),l,dep) :: genrec depth rest
  | Alias (c1,c2,d,t) :: rest ->
      Alias (regeneralize_index n depth c1,c2,d,t) :: genrec depth rest
  | Abstract d :: rest ->
      Abstract (map_rel_declaration (regeneralize_index n depth) d)
      :: genrec (depth+1) rest in
  genrec 0

let rec replace_term n c k t =
  if t = mkRel (n+k) then lift k c
  else map_constr_with_binders succ (replace_term n c) k t

let length_of_tomatch_type_sign (dep,_) = function
  | NotInd _ -> if dep<>Anonymous then 1 else 0
  | IsInd (_,_,names) -> List.length names + if dep<>Anonymous then 1 else 0

let replace_tomatch n c =
  let rec replrec depth = function
  | [] -> []
  | Pushed ((b,tm),l,dep) :: rest ->
      let b = replace_term n c depth b in
      let tm = map_tomatch_type (replace_term n c depth) tm in
      List.iter (fun i -> if i=n+depth then anomaly "replace_tomatch") l;
      Pushed ((b,tm),l,dep) :: replrec depth rest
  | Alias (c1,c2,d,t) :: rest ->
      Alias (replace_term n c depth c1,c2,d,t) :: replrec depth rest
  | Abstract d :: rest ->
      Abstract (map_rel_declaration (replace_term n c depth) 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 ((c,tm),l,dep)::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 ((c,tm),l,dep)::(liftn_tomatch_stack n depth rest)
  | Alias (c1,c2,d,t)::rest ->
      Alias (liftn n depth c1,liftn n depth c2,d,liftn n depth t)
      ::(liftn_tomatch_stack n depth rest)
  | Abstract d::rest ->
      Abstract (map_rel_declaration (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 *)

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 env sign eqns =
  let names1 = list_make (List.length sign) Anonymous in
  (* If any, we prefer names used in pats, from top to bottom *)
  let names2 =
    List.fold_right
      (fun (pats,eqn) names -> merge_names alias_of_pat pats names)
      eqns names1 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) -> list_union l eqn.rhs.avoid_ids) [] eqns in
  let names4,_ =
    List.fold_left2
      (fun (l,avoid) d na ->
	 let na =
	   merge_name
	     (fun (na,_,t) -> Name (next_name_away (named_hd env t na) avoid))
	     d na
	 in
         (na::l,(out_name na)::avoid))
      ([],allvars) (List.rev sign) names2 in
  names4

(************************************************************************)
(* Recovering names for variables pushed to the rhs' environment *)

let recover_alias_names get_name = List.map2 (fun x (_,c,t) ->(get_name x,c,t))

let push_rels_eqn sign eqn =
  {eqn with rhs = {eqn.rhs with rhs_env = push_rels sign eqn.rhs.rhs_env} }

let push_rels_eqn_with_names sign eqn =
  let pats = List.rev (list_firstn (List.length sign) eqn.patterns) in
  let sign = recover_alias_names alias_of_pat pats sign in
  push_rels_eqn sign eqn

let build_aliases_context env sigma names allpats pats =
  (* pats is the list of bodies to push as an alias *)
  (* They all are defined in env and we turn them into a sign *)
  (* cuts in sign need to be done in allpats *)
  let rec insert env sign1 sign2 n newallpats oldallpats = function
    | (deppat,_,_,_)::pats, Anonymous::names when not (isRel deppat) ->
        (* Anonymous leaves must be considered named and treated in the *)
        (* next clause because they may occur in implicit arguments *)
	insert env sign1 sign2
	  n newallpats (List.map List.tl oldallpats) (pats,names)
    | (deppat,nondeppat,d,t)::pats, na::names ->
	let nondeppat = lift n nondeppat in
	let deppat = lift n deppat in
	let newallpats =
	  List.map2 (fun l1 l2 -> List.hd l2::l1) newallpats oldallpats in
	let oldallpats = List.map List.tl oldallpats in
	let decl = (na,Some deppat,t) in
	let a = (deppat,nondeppat,d,t) in
	insert (push_rel decl env) (decl::sign1) ((na,a)::sign2) (n+1)
	  newallpats oldallpats (pats,names)
    | [], [] -> newallpats, sign1, sign2, env
    | _ -> anomaly "Inconsistent alias and name lists" in
  let allpats = List.map (fun x -> [x]) allpats
  in insert env [] [] 0 (List.map (fun _ -> []) allpats) allpats (pats, names)

let insert_aliases_eqn sign eqnnames alias_rest eqn =
  let thissign = List.map2 (fun na (_,c,t) -> (na,c,t)) eqnnames sign in
  { eqn with
      alias_stack = alias_rest;
      rhs = {eqn.rhs with rhs_env = push_rels thissign eqn.rhs.rhs_env } }

let insert_aliases env sigma alias eqns =
  (* Là, y a une faiblesse, si un alias est utilisé dans un cas par *)
  (* défaut présent mais inutile, ce qui est le cas général, l'alias *)
  (* est introduit même s'il n'est pas utilisé dans les cas réguliers *)
  let eqnsnames = List.map (fun eqn -> List.hd eqn.alias_stack) eqns in
  let alias_rests = List.map (fun eqn -> List.tl eqn.alias_stack) eqns in
  (* name2 takes the meet of all needed aliases *)
  let name2 =
    List.fold_right (merge_name (fun x -> x)) eqnsnames Anonymous in
  (* Only needed aliases are kept by build_aliases_context *)
  let eqnsnames, sign1, sign2, env =
    build_aliases_context env sigma [name2] eqnsnames [alias] in
  let eqns = list_map3 (insert_aliases_eqn sign1) eqnsnames alias_rests eqns in
  sign2, env, eqns

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

exception Occur
let noccur_between_without_evar n m term =
  let rec occur_rec n c = match kind_of_term c with
    | Rel p       -> if n<=p && p<n+m then raise Occur
    | Evar (_,cl) -> ()
    | _             -> iter_constr_with_binders succ occur_rec n c
  in
  (m = 0) or (try occur_rec n term; true with Occur -> false)


(* Infering 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)]? and must 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)
  => ..."  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) => ..." 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),_,dep) :: rest ->
      let k' = length_of_tomatch_type_sign dep tm in
      map_predicate f (k+k') ccl rest
  | Alias _ :: rest ->
      map_predicate f k ccl rest
  | Abstract _ :: rest ->
      map_predicate f (k+1) ccl rest

let noccur_predicate_between n = map_predicate (noccur_between n)

let liftn_predicate n = map_predicate (liftn n)

let lift_predicate n = liftn_predicate n 1

let regeneralize_index_predicate n = map_predicate (regeneralize_index n) 0

let substnl_predicate sigma = map_predicate (substnl sigma)

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

let specialize_predicate_var (cur,typ,dep) tms ccl =
  let c = if dep<>Anonymous then Some cur else None in
  let l =
    match typ with
    | IsInd (_,IndType(_,realargs),names) -> if names<>[] then realargs else []
    | 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 (names,(nadep,_)) ny d tms ccl =
  if nadep=Anonymous then anomaly "Undetected dependency";
  let p = List.length names + 1 in
  let ccl = lift_predicate 1 ccl tms in
  regeneralize_index_predicate (ny+p+1) ccl tms

let rec extract_predicate l ccl = function
  | Alias (deppat,nondeppat,_,_)::tms ->
      let tms' = match kind_of_term nondeppat with
        | Rel i -> replace_tomatch i deppat tms
        | _ -> (* initial terms are not dependent *) tms in
      extract_predicate l ccl tms'
  | Abstract d'::tms ->
      let d' = map_rel_declaration (lift (List.length l)) d' in
      substl l (mkProd_or_LetIn d' (extract_predicate [] ccl tms))
  | Pushed ((cur,NotInd _),_,(dep,_))::tms ->
      extract_predicate (if dep<>Anonymous then cur::l else l) ccl tms
  | Pushed ((cur,IsInd (_,IndType(_,realargs),_)),_,(dep,_))::tms ->
      let l = List.rev realargs@l in
      extract_predicate (if dep<>Anonymous then cur::l else l) ccl tms
  | [] ->
      substl l ccl

let abstract_predicate env sigma indf cur (names,(nadep,_)) tms ccl =
  let sign = make_arity_signature env true indf in
  (* n is the number of real args + 1 *)
  let n = List.length sign in
  let tms = lift_tomatch_stack n tms in
  let tms =
    match kind_of_term cur with
      | Rel i -> regeneralize_index_tomatch (i+n) tms
      | _ -> (* Initial case *) tms in
  let sign = List.map2 (fun na (_,c,t) -> (na,c,t)) (nadep::names) sign in
  let ccl = if nadep <> Anonymous then ccl else lift_predicate 1 ccl tms in
  let pred = extract_predicate [] ccl tms in
  it_mkLambda_or_LetIn_name env pred sign

let known_dependent (_,dep) = (dep = KnownDep)

(* [expand_arg] is used by [specialize_predicate]
   it replaces gamma, x1...xn, x1...xk |- pred
   by gamma, x1...xn, x1...xk-1 |- [X=realargs,xk=xk]pred (if dep) or
   by gamma, x1...xn, x1...xk-1 |- [X=realargs]pred (if not dep) *)

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

let adjust_impossible_cases pb pred tomatch submat =
  if submat = [] then
    match kind_of_term (whd_evar !(pb.evdref) pred) with
    | Evar (evk,_) when snd (evar_source evk !(pb.evdref)) = ImpossibleCase ->
	let default = (coq_unit_judge ()).uj_type in
	pb.evdref := Evd.define evk default !(pb.evdref);
      (* we add an "assert false" case *)
      let pats = List.map (fun _ -> PatVar (dummy_loc,Anonymous)) tomatch in
      let aliasnames =
	map_succeed (function Alias _ -> Anonymous | _ -> failwith"") tomatch
      in
      [ { patterns = pats;
          rhs = { rhs_env = pb.env;
	          rhs_vars = [];
		  avoid_ids = [];
		  it = None };
	  alias_stack = Anonymous::aliasnames;
	  eqn_loc = dummy_loc;
	  used = ref false } ]
    | _ ->
	submat
  else
    submat

(*****************************************************************************)
(* pred = [X:=realargs;x:=c]P types the following problem:                   *)
(*                                                                           *)
(*  Gamma |- match Pushed(c:I(realargs)) rest with...end: pred               *)
(*                                                                           *)
(* where the branch with constructor Ci:(x1:T1)...(xn:Tn)->I(realargsi)      *)
(* is considered. Assume each Ti is some Ii(argsi).                          *)
(* We let e=Ci(x1,...,xn) and replace pred by                                *)
(*                                                                           *)
(* pred' = [X1:=rargs1,x1:=x1']...[Xn:=rargsn,xn:=xn'](P[X:=realargsi;x:=e]) *)
(*                                                                           *)
(* s.t Gamma,x1'..xn' |- match Pushed(x1')..Pushed(xn') rest with..end :pred'*)
(*                                                                           *)
(*****************************************************************************)
let specialize_predicate newtomatchs (names,(depna,_)) arsign cs tms ccl =
  (* Assume some gamma st: gamma, (X,x:=realargs,copt), tms |- ccl *)
  let nrealargs = List.length names in
  let k = nrealargs + (if depna<>Anonymous then 1 else 0) in
  (* We adjust pred st: gamma, x1..xn, (X,x:=realargs,copt), tms |- ccl' *)
  let n = cs.cs_nargs in
  let ccl' = liftn_predicate n (k+1) ccl tms in
  let argsi =
    if nrealargs <> 0 then
      adjust_subst_to_rel_context arsign (Array.to_list cs.cs_concl_realargs)
    else
      [] in
  let copti = if depna<>Anonymous then Some (build_dependent_constructor cs) else None in
  (* The substituends argsi, copti are all defined in gamma, x1...xn *)
  (* We need _parallel_ bindings to get gamma, x1...xn, tms |- ccl'' *)
  let ccl'' =
    whd_betaiota Evd.empty (subst_predicate (argsi, copti) ccl' tms) in
  (* We adjust ccl st: gamma, x1..xn, x1..xn, tms |- ccl'' *)
  let ccl''' = liftn_predicate n (n+1) ccl'' tms in
  (* We finally get gamma,x1..xn |- [X1,x1:=R1,x1]..[Xn,xn:=Rn,xn]pred'''*)
  List.fold_left (expand_arg tms) ccl''' newtomatchs

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

let adjust_predicate_from_tomatch ((_,oldtyp),_,(nadep,_)) typ pb =
  match typ, oldtyp with
  | IsInd (_,_,names), NotInd _ ->
      let k = if nadep <> Anonymous then 2 else 1 in
      let n = List.length names in
      { pb with	pred = liftn_predicate n k pb.pred pb.tomatch }
  | _ ->
      pb

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

type inversion_problem =
  (* the discriminating arg in some Ind and its order in Ind *)
  | Incompatible of int * (int * int)
  | Constraints of (int * constr) list

let solve_constraints constr_info indt =
  (* TODO *)
  Constraints []

let rec irrefutable env = function
  | 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 = 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.create (Array.length cstrs) [] in
  let only_default = ref true 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 check_and_adjust_constructor pb.env ind cstrs pat with
	   | PatVar (_,name) ->
	       (* This is a default clause that we expand *)
	       for i=1 to Array.length cstrs do
		 let args = make_anonymous_patvars cstrs.(i-1).cs_nargs in
		 brs.(i-1) <- (args, rest) :: brs.(i-1)
	       done
	   | PatCstr (loc,((_,i)),args,_) ->
	       (* This is a regular clause *)
	       only_default := false;
	       brs.(i-1) <- (args,rest) :: brs.(i-1)) mat () in
  (brs,!only_default)

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

(* Abstracting over dependent subterms to match *)
let rec generalize_problem names pb = function
  | [] -> pb
  | i::l ->
      let d = map_rel_declaration (lift i) (Environ.lookup_rel i pb.env) in
      let d = on_pi3 (whd_betaiota !(pb.evdref)) d in (* for better rendering *)
      let pb' = generalize_problem names pb l in
      let tomatch = lift_tomatch_stack 1 pb'.tomatch in
      let tomatch = regeneralize_index_tomatch (i+1) tomatch in
      { pb' with
	  tomatch = Abstract d :: tomatch;
          pred = generalize_predicate names i d pb.tomatch pb'.pred  }

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

(* Building the sub-problem when all patterns are variables *)
let shift_problem ((current,t),_,(nadep,_)) pb =
  {pb with
     tomatch = Alias (current,current,NonDepAlias,type_of_tomatch t)::pb.tomatch;
     pred = specialize_predicate_var (current,t,nadep) pb.tomatch pb.pred;
     history = push_history_pattern 0 AliasLeaf pb.history;
     mat = List.map remove_current_pattern pb.mat }

(* Building the sub-pattern-matching problem for a given branch *)
let build_branch current deps (realnames,dep) pb arsign eqns const_info =
  (* We remember that we descend through a constructor *)
  let alias_type =
    if Array.length const_info.cs_concl_realargs = 0
      & not (known_dependent dep) & deps = []
    then
      NonDepAlias
    else
      DepAlias
  in
  let history =
    push_history_pattern const_info.cs_nargs
      (AliasConstructor const_info.cs_cstr)
      pb.history in

  (* We find matching clauses *)
  let cs_args = const_info.cs_args in
  let names = get_names pb.env cs_args eqns in
  let submat = List.map (fun (tms,eqn) -> prepend_pattern tms eqn) eqns in
  let typs = List.map2 (fun (_,c,t) na -> (na,c,t)) cs_args names in
  let _,typs',_ =
    List.fold_right
      (fun (na,c,t as d) (env,typs,tms) ->
	 let tms = List.map List.tl tms in
 	 (push_rel d env, (na,NotInd(c,t))::typs,tms))
      typs (pb.env,[],List.map fst eqns) in

  let dep_sign =
    find_dependencies_signature
      (dependencies_in_rhs 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 = build_dependent_constructor const_info in

  (* We replace [(mkRel 1)] by its expansion [ci] *)
  (* and context "Gamma = Gamma1, current, Gamma2" by "Gamma;typs;curalias" *)
  (* This is done in two steps : first from "Gamma |- tms" *)
  (* into  "Gamma; typs; curalias |- tms" *)
  let tomatch = lift_tomatch_stack const_info.cs_nargs pb.tomatch in

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

  let typs'' =
    list_map2_i
      (fun i (na,t) deps ->
	let dep = match dep with
	  | Name _ as na',k -> (if na <> Anonymous then na else na'),k
	  | Anonymous,KnownNotDep ->
	      if deps = [] && pred_is_not_dep then
		(Anonymous,KnownNotDep)
	      else
		(force_name na,KnownDep)
	  | _,_ -> anomaly "Inconsistent dependency" in
	((mkRel i, lift_tomatch_type i t),deps,dep))
      1 typs' (List.rev dep_sign) in

  let pred =
    specialize_predicate typs'' (realnames,dep) arsign const_info tomatch pb.pred in

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

  let sign = List.map (fun (na,t) -> mkDeclTomatch na t) typs' in

  let ind =
    appvect (
      applist (mkInd (inductive_of_constructor const_info.cs_cstr),
      List.map (lift const_info.cs_nargs) const_info.cs_params),
      const_info.cs_concl_realargs) in

  let cur_alias = lift (List.length sign) current in
  let currents = Alias (ci,cur_alias,alias_type,ind) :: currents in
  let tomatch = List.rev_append currents tomatch in

  let submat = adjust_impossible_cases pb pred tomatch submat in
  if submat = [] then
    raise_pattern_matching_error
      (dummy_loc, pb.env, NonExhaustive (complete_history history));

  sign,
  { pb with
      env = push_rels sign pb.env;
      tomatch = tomatch;
      pred = pred;
      history = history;
      mat = List.map (push_rels_eqn_with_names sign) submat }

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

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

  "Pushed" terms and types are relative to env
  "Abstract" types are relative to env enriched by the previous terms to match

*)

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

and match_current pb tomatch =
  let ((current,typ),deps,dep as ct) = adjust_tomatch_to_pattern pb tomatch in
  let pb = adjust_predicate_from_tomatch tomatch typ pb in
  match typ with
    | NotInd (_,typ) ->
	check_all_variables typ pb.mat;
	compile (shift_problem ct pb)
    | IsInd (_,(IndType(indf,realargs) as indt),names) ->
	let mind,_ = dest_ind_family indf in
	let cstrs = get_constructors pb.env indf in
	let arsign, _ = get_arity pb.env indf in
	let eqns,onlydflt = group_equations pb mind current cstrs pb.mat in
	if (Array.length cstrs <> 0 or pb.mat <> []) & onlydflt  then
	  compile (shift_problem ct pb)
	else
          let _constraints = Array.map (solve_constraints indt) cstrs in

	  (* We generalize over terms depending on current term to match *)
	  let pb = generalize_problem (names,dep) pb deps in

	  (* We compile branches *)
	  let brs = array_map2 (compile_branch current (names,dep) deps pb arsign) eqns cstrs in

	  (* We build the (elementary) case analysis *)
	  let brvals = Array.map (fun (v,_) -> v) brs in
	  let (pred,typ,s) =
	    find_predicate pb.caseloc pb.env pb.evdref
	      pb.pred current indt (names,dep) pb.tomatch in
	  let ci = make_case_info pb.env mind pb.casestyle in
	  let case = mkCase (ci,nf_betaiota Evd.empty pred,current,brvals) in
	  let inst = List.map mkRel deps in
	  { uj_val = applist (case, inst);
	    uj_type = substl inst typ }

and compile_branch current names deps pb arsign eqn cstr =
  let sign, pb = build_branch current deps names pb arsign eqn cstr in
  let j = compile pb in
  (it_mkLambda_or_LetIn j.uj_val sign, j.uj_type)

and compile_generalization pb d rest =
  let pb =
    { pb with
       env = push_rel d pb.env;
       tomatch = rest;
       mat = List.map (push_rels_eqn [d]) pb.mat } in
  let j = compile pb in
  { uj_val = mkLambda_or_LetIn d j.uj_val;
    uj_type = mkProd_or_LetIn d j.uj_type }

and compile_alias pb (deppat,nondeppat,d,t) rest =
  let history = simplify_history pb.history in
  let sign, newenv, mat =
    insert_aliases pb.env !(pb.evdref) (deppat,nondeppat,d,t) pb.mat in
  let n = List.length sign in

  (* We had Gamma1; x:current; Gamma2 |- tomatch(x) and we rebind x to get *)
  (* Gamma1; x:current; Gamma2; typs; x':=curalias |- tomatch(x') *)
  let tomatch = lift_tomatch_stack n rest in
  let tomatch = match kind_of_term nondeppat with
    | Rel i ->
	if n = 1 then regeneralize_index_tomatch (i+n) tomatch
	else replace_tomatch i deppat tomatch
    | _ -> (* initial terms are not dependent *) tomatch in

  let pb =
    {pb with
       env = newenv;
       tomatch = tomatch;
       pred = lift_predicate n pb.pred tomatch;
       history = history;
       mat = mat } in
  let j = compile pb in
  List.fold_left mkSpecialLetInJudge j sign

(* 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 astterm *)
let matx_of_eqns env tomatchl eqns =
  let build_eqn (loc,ids,lpat,rhs) =
    let initial_lpat,initial_rhs = lpat,rhs in
    let initial_rhs = rhs in
    let rhs =
      { rhs_env = env;
        rhs_vars = free_rawvars initial_rhs;
	avoid_ids = ids@(ids_of_named_context (named_context env));
	it = Some initial_rhs } in
    { patterns = initial_lpat;
      alias_stack = [];
      eqn_loc = loc;
      used = ref false;
      rhs = rhs }
  in List.map build_eqn 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 the 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 subst t =
  let n = rel_context_length (rel_context env) in
  let n' = rel_context_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 subst0 = map_succeed (fun (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 aux n (_,b,ty) =
      match b with
      | Some c ->
	  assert (isRel c);
	  let p = n + destRel c in aux p (lookup_rel p extenv)
      | None ->
	  (n,ty) in
    let (p,ty) = aux p (lookup_rel p extenv) in
    if noccur_between_without_evar 1 (n'-p-n+1) ty
    then
      let u = lift (n'-n) u in
      (p,u,(expand_vars_in_term extenv u,lift p ty))
    else
      failwith "") subst in
  let t0 = lift (n'-n) t in
  (subst0,t0)

let push_binder d (k,env,subst) =
  (k+1,push_rel d env,List.map (fun (na,u,d) -> (na,lift 1 u,d)) subst)

(* 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 evdref subst _tycon extenv t =
  let sigma = !evdref in
  let t = nf_betaiota sigma t in (* it helps in some cases to remove K-redex *)
  let subst0,t0 = adjust_to_extended_env_and_remove_deps env extenv 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 rec aux (k,env,subst as x) t =
    if isRel t && pi2 (lookup_rel (destRel t) env) <> None then
      map_constr_with_full_binders push_binder aux x t
    else
    let good = List.filter (fun (_,u,_) -> is_conv_leq env sigma t u) subst in
    if good <> [] then
      let (u,ty) = pi3 (List.hd good) in
      let vl = List.map pi1 good in
      let inst =
	list_map_i
	  (fun i _ -> if List.mem i vl then u else mkRel i) 1
	  (rel_context extenv) in
      let rel_filter =
	List.map (fun a -> not (isRel a) or dependent a u) inst in
      let named_filter =
	List.map (fun (id,_,_) -> dependent (mkVar id) u)
	  (named_context extenv) in
      let filter = rel_filter@named_filter in
      let ev =
	e_new_evar evdref extenv ~src:(loc, CasesType) ~filter:filter ty in
      evdref := add_conv_pb (Reduction.CONV,extenv,substl inst ev,u) !evdref;
      lift k ev
    else
      map_constr_with_full_binders push_binder aux x t in
  aux (0,extenv,subst0) t0

let build_tycon loc env tycon_env subst tycon extenv evdref t =
  let t = match t with
    | None ->
	(* This is the situation we are building a return predicate and
           we are in an impossible branch *)
	let n = rel_context_length (rel_context env) in
	let n' = rel_context_length (rel_context tycon_env) in
	let impossible_case_type =
	  e_new_evar evdref env ~src:(loc,ImpossibleCase) (new_Type ()) in
	lift (n'-n) impossible_case_type
    | Some t -> abstract_tycon loc tycon_env evdref subst tycon extenv t in
  get_judgment_of extenv !evdref t

(* 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 loc env sigma tms t =
  let make_patvar t (subst,avoid) =
    let id = next_name_away (named_hd env t Anonymous) avoid in
    PatVar (dummy_loc,Name id), ((id,t)::subst, id::avoid) in
  let rec reveal_pattern t (subst,avoid as acc) =
    match kind_of_term (whd_betadeltaiota env sigma t) with
    | Construct cstr -> PatCstr (dummy_loc,cstr,[],Anonymous), acc
    | App (f,v) when isConstruct f ->
	let cstr = destConstruct f in
	let n = constructor_nrealargs env cstr in
	let l = list_lastn n (Array.to_list v) in
	let l,acc = list_fold_map' reveal_pattern l acc in
	PatCstr (dummy_loc,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_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 true indf' in
	let p = List.length realargs in
	let env' = push_rels sign env in
	let patl',acc_sign,acc = aux (n+p+1) env' (sign@acc_sign) tms acc in
	patl@pat::patl',acc_sign,acc
    | (t, NotInd (bo,typ)) :: tms ->
	aux n env acc_sign tms acc in
  let avoid0 = ids_of_context 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 (pb_env,_),sub_tms =
    list_fold_map (fun (env,i) (na,b,t as d) ->
      let typ =
	if b<>None then NotInd(None,t) else
	  try try_find_ind env sigma t None
	  with Not_found -> NotInd (None,t) in
      let ty = lift_tomatch_type (n-i) typ in
      let tm = Pushed ((mkRel (n-i),ty),[],(Anonymous,KnownNotDep)) in
      ((push_rel d env,i+1),tm))
      (env,0) (List.rev sign) in
  let subst = List.map (fun (na,t) -> (na,lift n t)) subst in
  (* [eqn1] is the first 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 eqn1 =
    { patterns = patl;
      alias_stack = [];
      eqn_loc = dummy_loc;
      used = ref false;
      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.map fst subst;
              avoid_ids = avoid;
	      it = Some (lift n t) } } in
  (* [eqn2] is the default clause of the auxiliary pattern-matching: 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 eqn2 =
    { patterns = List.map (fun _ -> PatVar (dummy_loc,Anonymous)) patl;
      alias_stack = [];
      eqn_loc = dummy_loc;
      used = ref false;
      rhs = { rhs_env = pb_env;
              rhs_vars = [];
	      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 evdref = ref sigma in
  let pb =
    { env       = pb_env;
      evdref    = evdref;
      pred      = new_Type();
      tomatch   = sub_tms;
      history   = start_history n;
      mat       = [eqn1;eqn2];
      caseloc   = loc;
      casestyle = RegularStyle;
      typing_function = build_tycon loc env pb_env subst} in
  let pred = (compile pb).uj_val in
  (!evdref,pred)

(* Here, [pred] is assumed to be in the context built from all *)
(* realargs and terms to match *)
let build_initial_predicate knowndep allnames pred =
  let nar = List.fold_left (fun n names -> List.length names + n) 0 allnames in
  let rec buildrec n pred nal = function
    | [] -> List.rev nal,pred
    | names::lnames ->
        let names' = List.tl names in
        let n' = n + List.length names' in
        let pred, p =
          if dependent (mkRel (nar-n')) pred then pred, 1
          else liftn (-1) (nar-n') pred, 0 in
        let na =
          if p=1 then
            let na = List.hd names in
            ((if na = Anonymous then
              (* can happen if evars occur in the return clause *)
              Name (id_of_string "x") (*Hum*)
            else na),knowndep)
          else (Anonymous,KnownNotDep) in
        buildrec (n'+1) pred (na::nal) lnames
  in buildrec 0 pred [] allnames

let extract_arity_signature env0 tomatchl tmsign =
  let get_one_sign n tm (na,t) =
    match tm with
      | NotInd (bo,typ) ->
	  (match t with
	    | None -> [na,Option.map (lift n) bo,lift n typ]
	    | Some (loc,_,_,_) ->
 	    user_err_loc (loc,"",
	    str"Unexpected type annotation for a term of non inductive type."))
      | IsInd (term,IndType(indf,realargs),_) ->
          let indf' = lift_inductive_family n indf in
	  let (ind,_) = dest_ind_family indf' in
	  let nparams_ctxt,nrealargs_ctxt = inductive_nargs env0 ind in
	  let arsign = fst (get_arity env0 indf') in
	  let realnal =
	    match t with
	      | Some (loc,ind',nparams,realnal) ->
		  if ind <> ind' then
		    user_err_loc (loc,"",str "Wrong inductive type.");
		  if nparams_ctxt <> nparams
		    or nrealargs_ctxt <> List.length realnal then
		      anomaly "Ill-formed 'in' clause in cases";
		  List.rev realnal
	      | None -> list_make nrealargs_ctxt Anonymous in
(* 	  let na =  *)
(* 	    match na with  *)
(* 	      | Name _ -> na *)
(* 	      | Anonymous -> *)
(* 		  match kind_of_term term with *)
(* 		    | Rel n -> pi1 (lookup_rel n (Environ.rel_context env0)) *)
(* 		    | _ -> Anonymous *)
(* 	  in *)
	  (na,None,build_dependent_inductive env0 indf')
	  ::(List.map2 (fun x (_,c,t) ->(x,c,t)) 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 env evdref j tycon =
  match tycon with
    | Some p ->
	let (evd',j) = Coercion.inh_conv_coerce_to loc env !evdref j p in
          evdref := evd';
          j
    | None -> j

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

let prepare_predicate_from_arsign_tycon loc tomatchs sign arsign c =
  let nar = List.fold_left (fun n sign -> List.length sign + n) 0 arsign in
  let subst, len =
    List.fold_left2 (fun (subst, len) (tm, tmtype) sign ->
      let signlen = List.length sign in
	match kind_of_term tm with
	  | Rel n when dependent tm c
		&& signlen = 1 (* The term to match is not of a dependent type itself *) ->
	      ((n, len) :: subst, len - signlen)
	  | Rel n when signlen > 1 (* The term is of a dependent type,
				      maybe some variable in its type appears in the tycon. *) ->
	      (match tmtype with
		  NotInd _ -> (* len - signlen, subst*) assert false (* signlen > 1 *)
		| IsInd (_, IndType(indf,realargs),_) ->
		    let subst =
		      if dependent tm c && List.for_all isRel realargs
		      then (n, 1) :: subst else subst
		    in
		      List.fold_left
			(fun (subst, len) arg ->
			  match kind_of_term arg with
			  | Rel n when dependent arg c ->
			      ((n, len) :: subst, pred len)
			  | _ -> (subst, pred len))
			(subst, len) realargs)
	  | _ -> (subst, len - signlen))
      ([], nar) tomatchs arsign
  in
  let rec predicate lift c =
    match kind_of_term c with
      | Rel n when n > lift ->
	  (try
	      (* Make the predicate dependent on the matched variable *)
	      let idx = List.assoc (n - lift) subst in
		mkRel (idx + lift)
	    with Not_found ->
	      (* A variable that is not matched, lift over the arsign. *)
	      mkRel (n + nar))
      | _ ->
	  map_constr_with_binders succ predicate lift c
  in predicate 0 c


(* 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: we try to specialize the
 * tycon to make the predicate if it is not closed.
 *)

let prepare_predicate loc typing_fun evdref env tomatchs sign tycon pred =
  let arsign = extract_arity_signature env tomatchs sign in
  let names = List.rev (List.map (List.map pi1) arsign) in
  let preds =
    match pred, tycon with
    (* No type annotation *)
    | None, Some (None, t) when not (noccur_with_meta 0 max_int t) ->
	(* If the tycon is not closed w.r.t real variables, we try *)
        (* two different strategies *)
	(* First strategy: we abstract the tycon wrt to the dependencies *)
        let pred1 =
          prepare_predicate_from_arsign_tycon loc tomatchs sign arsign t in
	(* Second strategy: we build an "inversion" predicate *)
	let sigma2,pred2 = build_inversion_problem loc env !evdref tomatchs t in
	[!evdref, KnownDep, pred1; sigma2, DepUnknown, pred2]
    | None, Some (None, t) ->
	(* Only one strategy: we build an "inversion" predicate *)
	let sigma,pred = build_inversion_problem loc env !evdref tomatchs t in
	[sigma, DepUnknown, pred]
    | None,  _ ->
	(* No type constaints: we use two strategies *)
	let t = mkExistential env ~src:(loc, CasesType) evdref in
	(* First strategy: we build an inversion problem *)
	let sigma1,pred1 = build_inversion_problem loc env !evdref tomatchs t in
	(* Second strategy: we directly use the evar as a non dependent pred *)
        let pred2 = lift (List.length names) t in
	[sigma1, DepUnknown, pred1; !evdref, KnownNotDep, pred2]
    (* Some type annotation *)
    | Some rtntyp, _ ->
      (* We extract the signature of the arity *)
      let env = List.fold_right push_rels arsign env in
      let predcclj = typing_fun (mk_tycon (new_Type ())) env evdref rtntyp in
      Option.iter (fun tycon ->
          let tycon = lift_tycon_type (List.length arsign) tycon in
	  evdref :=
            Coercion.inh_conv_coerces_to loc env !evdref predcclj.uj_val tycon)
        tycon;
      let predccl = (j_nf_evar !evdref predcclj).uj_val in
      [!evdref, KnownDep, predccl]
  in
  List.map
    (fun (sigma,dep,pred) ->
      let (nal,pred) = build_initial_predicate dep names pred in
      sigma,nal,pred)
    preds

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

let compile_cases loc style (typing_fun, evdref) tycon env (predopt, tomatchl, eqns) =

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

  (* We build the vector of terms to match consistently with the *)
  (* constructors found in patterns *)
  let tomatchs = coerce_to_indtype typing_fun evdref env 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 sign = List.map snd tomatchl in
  let preds = prepare_predicate loc typing_fun evdref env tomatchs sign 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) *)
    let initial_pushed = List.map2 (fun tm na -> Pushed(tm,[],na)) tomatchs nal in

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

    let myevdref = ref sigma in

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

    let j = compile pb in
    evdref := !myevdref;
    j in

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

  (* We check for unused patterns *)
  List.iter (check_unused_pattern env) matx;

  (* We coerce to the tycon (if an elim predicate was provided) *)
  inh_conv_coerce_to_tycon loc env evdref j tycon

end