path: root/pretyping/evarconv.ml

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

open Errors
open Util
open Names
open Term
open Vars
open Closure
open Reduction
open Reductionops
open Termops
open Environ
open Recordops
open Evarutil
open Evarsolve
open Globnames
open Evd
open Pretype_errors

type flex_kind_of_term =
  | Rigid
  | MaybeFlexible (* approx'ed as reducible but not necessarily so *)
  | Flexible of existential

let flex_kind_of_term c sk =
  match kind_of_term c with
    | Rel _ | Const _ | Var _ -> MaybeFlexible
    | Lambda _ when not (Option.is_empty (decomp_stack sk)) -> MaybeFlexible
    | LetIn _  -> MaybeFlexible
    | Evar ev -> Flexible ev
    | Lambda _ | Prod _ | Sort _ | Ind _ | Construct _ | CoFix _ -> Rigid
    | Meta _ -> Rigid
    | Fix _ -> Rigid (* happens when the fixpoint is partially applied *)
    | Cast _ | App _ | Case _ -> assert false

let not_purely_applicative_stack args =
  List.exists (function (Zfix _ | Zcase _) -> true | _ -> false) args

let eval_flexible_term ts env c =
  match kind_of_term c with
  | Const c ->
      if is_transparent_constant ts c
      then constant_opt_value env c
      else None
  | Rel n ->
      (try let (_,v,_) = lookup_rel n env in Option.map (lift n) v
      with Not_found -> None)
  | Var id ->
      (try
	 if is_transparent_variable ts id then
	   let (_,v,_) = lookup_named id env in v
	 else None
       with Not_found -> None)
  | LetIn (_,b,_,c) -> Some (subst1 b c)
  | Lambda _ -> Some c
  | _ -> assert false

let apprec_nohdbeta ts env evd c =
  let (t,sk as appr) = Reductionops.whd_nored_state evd (c, []) in
  if not_purely_applicative_stack (snd (Reductionops.strip_app sk))
  then zip (fst (whd_betaiota_deltazeta_for_iota_state
		   ts env evd Cst_stack.empty appr))
  else c

let position_problem l2r = function
  | CONV -> None
  | CUMUL -> Some l2r

(* [check_conv_record (t1,l1) (t2,l2)] tries to decompose the problem
   (t1 l1) = (t2 l2) into a problem

     l1 = params1@c1::extra_args1
     l2 = us2@extra_args2
     (t1 params1 c1) = (proji params (c xs))
     (t2 us2) = (cstr us)
     extra_args1 = extra_args2

   by finding a record R and an object c := [xs:bs](Build_R params v1..vn)
   with vi = (cstr us), for which we know that the i-th projection proji
   satisfies

      (proji params (c xs)) = (cstr us)

   Rem: such objects, usable for conversion, are defined in the objdef
   table; practically, it amounts to "canonically" equip t2 into a
   object c in structure R (since, if c1 were not an evar, the
   projection would have been reduced) *)

let check_conv_record (t1,sk1) (t2,sk2) =
  try
    let proji = global_of_constr t1 in
    let canon_s,sk2_effective =
      try
	match kind_of_term t2 with
	    Prod (_,a,b) -> (* assert (l2=[]); *)
      	      if dependent (mkRel 1) b then raise Not_found
	      else lookup_canonical_conversion (proji, Prod_cs),[Zapp [a;pop b]]
	  | Sort s ->
	      lookup_canonical_conversion
		(proji, Sort_cs (family_of_sort s)),[]
	  | _ ->
	      let c2 = global_of_constr t2 in
		lookup_canonical_conversion (proji, Const_cs c2),sk2
      with Not_found ->
	lookup_canonical_conversion (proji,Default_cs),[]
    in
    let { o_DEF = c; o_INJ=n; o_TABS = bs;
          o_TPARAMS = params; o_NPARAMS = nparams; o_TCOMPS = us } = canon_s in
    let params1, c1, extra_args1 =
      match strip_n_app nparams sk1 with
	| Some (params1, c1,extra_args1) -> params1, c1, extra_args1
	| _ -> raise Not_found in
    let us2,extra_args2 =
      let l',s' = strip_app sk2_effective in
      let bef,aft = List.chop (List.length us) l' in
      (bef, append_stack_app_list aft s') in
    c,bs,(params,params1),(us,us2),(extra_args1,extra_args2),c1,
    (n,zip(t2,sk2))
  with Failure _ | Not_found ->
    raise Not_found

(* Precondition: one of the terms of the pb is an uninstantiated evar,
 * possibly applied to arguments. *)

let rec ise_try evd = function
    [] -> assert false
  | [f] -> f evd
  | f1::l ->
      match f1 evd with
      | Success _ as x -> x
      | UnifFailure _ -> ise_try evd l

let ise_and evd l =
  let rec ise_and i = function
      [] -> assert false
    | [f] -> f i
    | f1::l ->
        match f1 i with
	| Success i' -> ise_and i' l
	| UnifFailure _ as x -> x in
  ise_and evd l

(* This function requires to get the outermost arguments first.  It is
   a fold_right for backward compatibility.

   It tries to unify the suffix of 2 lists element by element and if
   it reaches the end of a list, it returns the remaining elements in
   the other list if there are some.
*)
let ise_exact ise x1 x2 =
  match ise x1 x2 with
  | None, out -> out
  | _, (UnifFailure _ as out) -> out
  | Some _, Success i -> UnifFailure (i,NotSameArgSize)

let generic_ise_list2 i f l1 l2 =
  let rec aux i l1 l2 =
    match l1,l2 with
    | [], [] -> (None, Success i)
    | l, [] -> (Some (Inl (List.rev l)), Success i)
    | [], l -> (Some (Inr (List.rev l)), Success i)
    | x::l1, y::l2 ->
        (match aux i l1 l2 with
        | aa, Success i' -> (aa, f i' x y)
        | _, (UnifFailure _ as x) -> None, x)
  in aux i (List.rev l1) (List.rev l2)

(* Same but the 2 lists must have the same length *)
let ise_list2 evd f l1 l2 =
  ise_exact (generic_ise_list2 evd f) l1 l2

let ise_array2 evd f v1 v2 =
  let rec allrec i = function
    | -1 -> Success i
    | n ->
        match f i v1.(n) v2.(n) with
	| Success i' -> allrec i' (n-1)
	| UnifFailure _ as x -> x in
  let lv1 = Array.length v1 in
  if Int.equal lv1 (Array.length v2) then allrec evd (pred lv1)
  else UnifFailure (evd,NotSameArgSize)

(* This function tries to unify 2 stacks element by element. It works
   from the end to the beginning. If it unifies a non empty suffix of
   stacks but not the entire stacks, the first part of the answer is
   Some(the remaining prefixes to tackle)) *)
let ise_stack2 no_app env evd f sk1 sk2 =
  let rec ise_stack2 deep i sk1 sk2 =
    let fail x = if deep then Some (List.rev sk1, List.rev sk2), Success i
      else None, x in
    match sk1, sk2 with
    | [], [] -> None, Success i
    | Zcase (_,t1,c1,_)::q1, Zcase (_,t2,c2,_)::q2 ->
      (match f env i CONV t1 t2 with
      | Success i' ->
	(match ise_array2 i' (fun ii -> f env ii CONV) c1 c2 with
	| Success i'' -> ise_stack2 true i'' q1 q2
        | UnifFailure _ as x -> fail x)
      | UnifFailure _ as x -> fail x)
    | Zfix (((li1, i1),(_,tys1,bds1 as recdef1)),a1,_)::q1, Zfix (((li2, i2),(_,tys2,bds2)),a2,_)::q2 ->
      if Int.equal i1 i2 && Array.equal Int.equal li1 li2 then
        match ise_and i [
	  (fun i -> ise_array2 i (fun ii -> f env ii CONV) tys1 tys2);
	  (fun i -> ise_array2 i (fun ii -> f (push_rec_types recdef1 env) ii CONV) bds1 bds2);
	  (fun i -> ise_exact (ise_stack2 false i) a1 a2)] with
        | Success i' -> ise_stack2 true i' q1 q2
        | UnifFailure _ as x -> fail x
      else fail (UnifFailure (i,NotSameHead))
    | Zupdate _ :: _, _ | Zshift _ :: _, _
    | _, Zupdate _ :: _ | _, Zshift _ :: _ -> assert false
    | Zapp l1 :: q1, Zapp l2 :: q2 ->
      if no_app&&deep then fail ((*dummy*)UnifFailure(i,NotSameHead)) else begin
      (* Is requiring to match on all the shorter list a restriction
	 here ? we could imagine a generalization of
	 generic_ise_list2 that succeed when it matches only a strict
	 non empty suffix of both lists and returns in this case the 2
	 prefixes *)
      match generic_ise_list2 i (fun ii -> f env ii CONV) l1 l2 with
      |_,(UnifFailure _ as x) -> fail x
      |None,Success i' -> ise_stack2 true i' q1 q2
      |Some (Inl r),Success i' -> ise_stack2 true i' (Zapp r :: q1) q2
      |Some (Inr r),Success i' -> ise_stack2 true i' q1 (Zapp r :: q2)
    end
    |_, _ -> fail (UnifFailure (i,(* Maybe improve: *) NotSameHead))
  in ise_stack2 false evd (List.rev sk1) (List.rev sk2)

(* Make sure that the matching suffix is the all stack *)
let exact_ise_stack2 env evd f sk1 sk2 =
  if Reductionops.compare_stack_shape sk1 sk2 then
    ise_exact (ise_stack2 false env evd f) sk1 sk2
  else UnifFailure (evd, (* Dummy *) NotSameHead)

let rec evar_conv_x ts env evd pbty term1 term2 =
  let term1 = whd_head_evar evd term1 in
  let term2 = whd_head_evar evd term2 in
  (* Maybe convertible but since reducing can erase evars which [evar_apprec]
     could have found, we do it only if the terms are free of evar.
     Note: incomplete heuristic... *)
  let ground_test =
    if is_ground_term evd term1 && is_ground_term evd term2 then
      if is_trans_fconv pbty ts env evd term1 term2 then
        Some true
      else if is_ground_env evd env then Some false
      else None
    else None in
  match ground_test with
    | Some true -> Success evd
    | Some false -> UnifFailure (evd,ConversionFailed (env,term1,term2))
    | None ->
	(* Until pattern-unification is used consistently, use nohdbeta to not
	   destroy beta-redexes that can be used for 1st-order unification *)
        let term1 = apprec_nohdbeta ts env evd term1 in
        let term2 = apprec_nohdbeta ts env evd term2 in
        begin match kind_of_term term1, kind_of_term term2 with
        | Evar ev, _ when Evd.is_undefined evd (fst ev) ->
          solve_simple_eqn (evar_conv_x ts) env evd
            (position_problem true pbty,ev,term2)
        | _, Evar ev when Evd.is_undefined evd (fst ev) ->
          solve_simple_eqn (evar_conv_x ts) env evd
            (position_problem false pbty,ev,term1)
        | _ ->
          evar_eqappr_x ts env evd pbty
            (whd_nored_state evd (term1,empty_stack), Cst_stack.empty)
            (whd_nored_state evd (term2,empty_stack), Cst_stack.empty)
        end

and evar_eqappr_x ?(rhs_is_already_stuck = false) ts env evd pbty
    ((term1,sk1 as appr1),csts1) ((term2,sk2 as appr2),csts2) =
  let default_fail i = (* costly *)
    UnifFailure (i,ConversionFailed (env, zip appr1, zip appr2)) in
  let miller_pfenning on_left fallback ev (_,skF) apprM evd =
    let tM = zip apprM in
    match list_of_app_stack skF with
    | None -> default_fail evd
    | Some lF -> match is_unification_pattern_evar env evd ev lF tM with
      | None -> fallback ()
      | Some l1' -> (* Miller-Pfenning's patterns unification *)
	let t2 = nf_evar evd tM in
	let t2 = solve_pattern_eqn env l1' t2 in
	solve_simple_eqn (evar_conv_x ts) env evd
	  (position_problem on_left pbty,ev,t2) in
  let flex_maybeflex on_left ev ((termF,skF as apprF),cstsF) ((termM, skM as apprM),cstsM) =
    let switch f a b = if on_left then f a b else f b a in
    let not_only_app = not_purely_applicative_stack skM in
    let f1 i = miller_pfenning on_left
      (fun () -> if not_only_app then (* Postpone the use of an heuristic *)
	switch (fun x y -> Success (add_conv_pb (pbty,env,zip x,zip y) i)) apprF apprM
       else default_fail i)
      ev apprF apprM i
    and f2 i =
      match switch (ise_stack2 not_only_app env i (evar_conv_x ts)) skF skM with
      |Some (l,r), Success i' when on_left && (not_only_app || List.is_empty l) ->
	switch (evar_conv_x ts env i' pbty) (zip(termF,l)) (zip(termM,r))
      |Some (r,l), Success i' when not on_left && (not_only_app || List.is_empty l) ->
	switch (evar_conv_x ts env i' pbty) (zip(termF,l)) (zip(termM,r))
      |None, Success i' -> switch (evar_conv_x ts env i' pbty) termF termM
      |_, (UnifFailure _ as x) -> x
      |Some _, _ -> UnifFailure (i,NotSameArgSize)
    and f3 i =
      match eval_flexible_term ts env termM with
      | Some vM ->
	switch (evar_eqappr_x ts env i pbty) (apprF,cstsF)
	  (whd_betaiota_deltazeta_for_iota_state ts env i cstsM (vM, skM))
      | None -> UnifFailure (i,NotSameHead)
    in
    ise_try evd [f1; f2; f3] in
  let eta env evd onleft sk term sk' term' =
    assert (match sk with [] -> true | _ -> false);
    let (na,c1,c'1) = destLambda term in
    let c = nf_evar evd c1 in
    let env' = push_rel (na,None,c) env in
    let out1 = whd_betaiota_deltazeta_for_iota_state
      ts env' evd Cst_stack.empty (c'1, empty_stack) in
    let out2 = whd_nored_state evd
      (zip (term', sk' @ [Zshift 1]), [Zapp [mkRel 1]]), Cst_stack.empty in
    if onleft then evar_eqappr_x ts env' evd CONV out1 out2
    else evar_eqappr_x ts env' evd CONV out2 out1
  in

  let app_empty = match sk1, sk2 with [], [] -> true | _ -> false in
  (* Evar must be undefined since we have flushed evars *)
  match (flex_kind_of_term term1 sk1, flex_kind_of_term term2 sk2) with
    | Flexible (sp1,al1 as ev1), Flexible (sp2,al2 as ev2) ->
	let f1 i =
	  match ise_stack2 false env i (evar_conv_x ts) sk1 sk2 with
	  |None, Success i' -> solve_simple_eqn (evar_conv_x ts) env i'
	    (position_problem true pbty,ev1,term2)
	  |Some (r,[]), Success i' -> solve_simple_eqn (evar_conv_x ts) env i'
	    (position_problem false pbty,ev2,zip(term1,r))
	  |Some ([],r), Success i' -> solve_simple_eqn (evar_conv_x ts) env i'
	          (position_problem true pbty,ev1,zip(term2,r))
	  |_, (UnifFailure _ as x) -> x
	  |Some _, _ -> UnifFailure (i,NotSameArgSize)
	and f2 i =
          if Evar.equal sp1 sp2 then
	    match ise_stack2 false env i (evar_conv_x ts) sk1 sk2 with
	    |None, Success i' ->
              Success (solve_refl (fun env i pbty a1 a2 ->
                is_success (evar_conv_x ts env i pbty a1 a2))
                env i' sp1 al1 al2)
	    |_, (UnifFailure _ as x) -> x
            |Some _, _ -> UnifFailure (i,NotSameArgSize)
          else UnifFailure (i,NotSameHead)
	in
	ise_try evd [f1; f2]

    | Flexible ev1, MaybeFlexible -> flex_maybeflex true ev1 (appr1,csts1) (appr2,csts2)

    | MaybeFlexible, Flexible ev2 -> flex_maybeflex false ev2 (appr2,csts2) (appr1,csts1)

    | MaybeFlexible, MaybeFlexible -> begin
        match kind_of_term term1, kind_of_term term2 with
        | LetIn (na,b1,t1,c'1), LetIn (_,b2,_,c'2) ->
        let f1 i =
          ise_and i
	    [(fun i -> evar_conv_x ts env i CONV b1 b2);
	     (fun i ->
	       let b = nf_evar i b1 in
	       let t = nf_evar i t1 in
	       evar_conv_x ts (push_rel (na,Some b,t) env) i pbty c'1 c'2);
	     (fun i -> exact_ise_stack2 env i (evar_conv_x ts) sk1 sk2)]
	and f2 i =
          let out1 = whd_betaiota_deltazeta_for_iota_state ts env i csts1 ((subst1 b1 c'1),sk1)
          and out2 = whd_betaiota_deltazeta_for_iota_state ts env i csts2 ((subst1 b2 c'2),sk2)
	  in evar_eqappr_x ts env i pbty out1 out2
	in
	ise_try evd [f1; f2]

	| _, _ ->
	let f1 i =
	  if eq_constr term1 term2 then
	    exact_ise_stack2 env i (evar_conv_x ts) sk1 sk2
	  else
	     UnifFailure (i,NotSameHead)
	and f2 i =
	  (try conv_record ts env i
             (try check_conv_record appr1 appr2
	      with Not_found -> check_conv_record appr2 appr1)
           with Not_found -> UnifFailure (i,NoCanonicalStructure))
	and f3 i =
          (* heuristic: unfold second argument first, exception made
             if the first argument is a beta-redex (expand a constant
             only if necessary) or the second argument is potentially
             usable as a canonical projection or canonical value *)
          let rec is_unnamed (hd, args) = match kind_of_term hd with
            | (Var _|Construct _|Ind _|Const _|Prod _|Sort _) ->
	      not_purely_applicative_stack args
            | (CoFix _|Meta _|Rel _)-> true
            | Evar _ -> not_purely_applicative_stack args
	    (* false (* immediate solution without Canon Struct *)*)
            | Lambda _ -> assert (match args with [] -> true | _ -> false); true
            | LetIn (_,b,_,c) -> is_unnamed
	      (fst (whd_betaiota_deltazeta_for_iota_state
		      ts env i Cst_stack.empty (subst1 b c, args)))
            | Case _| Fix _| App _| Cast _ -> assert false in
          let rhs_is_stuck_and_unnamed () =
            match eval_flexible_term ts env term2 with
            | None -> false
            | Some v2 ->
	      let applicative_stack = append_stack_app_list (fst (strip_app sk2)) empty_stack in
	      is_unnamed
		(fst (whd_betaiota_deltazeta_for_iota_state
			ts env i Cst_stack.empty (v2, applicative_stack))) in
          let rhs_is_already_stuck =
            rhs_is_already_stuck || rhs_is_stuck_and_unnamed () in
	  if (isLambda term1 || rhs_is_already_stuck)
	    && (not (not_purely_applicative_stack sk1)) then
	    match eval_flexible_term ts env term1 with
	    | Some v1 ->
	      evar_eqappr_x ~rhs_is_already_stuck ts env i pbty
		(whd_betaiota_deltazeta_for_iota_state
		   ts env i (Cst_stack.add_cst term1 csts1) (v1,sk1))
		(appr2,csts2)
	    | None ->
		match eval_flexible_term ts env term2 with
		| Some v2 ->
		    evar_eqappr_x ts env i pbty (appr1,csts1)
		      (whd_betaiota_deltazeta_for_iota_state
			 ts env i (Cst_stack.add_cst term2 csts2) (v2,sk2))
		| None -> UnifFailure (i,NotSameHead)
	  else
	    match eval_flexible_term ts env term2 with
	    | Some v2 ->
	      evar_eqappr_x ts env i pbty (appr1,csts1)
		(whd_betaiota_deltazeta_for_iota_state
		   ts env i (Cst_stack.add_cst term2 csts2) (v2,sk2))
	    | None ->
		match eval_flexible_term ts env term1 with
		| Some v1 ->
		  evar_eqappr_x ts env i pbty
		    (whd_betaiota_deltazeta_for_iota_state
		       ts env i (Cst_stack.add_cst term1 csts1) (v1,sk1))
		    (appr2,csts2)
		| None -> UnifFailure (i,NotSameHead)
	in
	ise_try evd [f1; f2; f3]
      end

    | Rigid, Rigid when isLambda term1 && isLambda term2 ->
        let (na,c1,c'1) = destLambda term1 in
        let (_,c2,c'2) = destLambda term2 in
        assert app_empty;
        ise_and evd
          [(fun i -> evar_conv_x ts env i CONV c1 c2);
           (fun i ->
	     let c = nf_evar i c1 in
	     evar_conv_x ts (push_rel (na,None,c) env) i CONV c'1 c'2)]

    | Flexible ev1, Rigid ->
      let f1 evd =
	miller_pfenning true
	(fun () -> Success ((* Postpone the use of an heuristic *)
	  add_conv_pb (pbty,env,zip appr1,zip appr2) evd))
	ev1 appr1 appr2 evd
      and f2 evd =
	if isLambda term2 then
	  eta env evd false sk2 term2 sk1 term1
	else UnifFailure (evd,NotSameHead)
      in ise_try evd [f1;f2]

    | Rigid, Flexible ev2 ->
      let f1 evd =
	miller_pfenning false
	  (fun () -> Success ((* Postpone the use of an heuristic *)
	     add_conv_pb (pbty,env,zip appr1,zip appr2) evd))
	ev2 appr2 appr1 evd
      and f2 evd =
	if isLambda term1 then
	  eta env evd true sk1 term1 sk2 term2
	else UnifFailure (evd,NotSameHead)
      in ise_try evd [f1;f2]

    | MaybeFlexible, Rigid ->
	let f3 i =
	  (try conv_record ts env i (check_conv_record appr1 appr2)
           with Not_found -> UnifFailure (i,NoCanonicalStructure))
	and f4 i =
	  match eval_flexible_term ts env term1 with
	    | Some v1 ->
		evar_eqappr_x ts env i pbty
		  (whd_betaiota_deltazeta_for_iota_state
		     ts env i (Cst_stack.add_cst term1 csts1) (v1,sk1))
		  (appr2,csts2)
	    | None ->
	      if isLambda term2 then eta env evd false sk2 term2 sk1 term1
	      else UnifFailure (i,NotSameHead)
	in
	  ise_try evd [f3; f4]

    | Rigid, MaybeFlexible ->
	let f3 i =
	  (try conv_record ts env i (check_conv_record appr2 appr1)
           with Not_found -> UnifFailure (i,NoCanonicalStructure))
	and f4 i =
	  match eval_flexible_term ts env term2 with
	    | Some v2 ->
		evar_eqappr_x ts env i pbty (appr1,csts1)
		  (whd_betaiota_deltazeta_for_iota_state
		     ts env i (Cst_stack.add_cst term2 csts2) (v2,sk2))
	    | None ->
	      if isLambda term1 then eta env evd true sk1 term1 sk2 term2
	      else UnifFailure (i,NotSameHead)
	in
	  ise_try evd [f3; f4]

    (* Eta-expansion *)
    | Rigid, _ when isLambda term1 ->
        eta env evd true sk1 term1 sk2 term2

    | _, Rigid when isLambda term2 ->
        eta env evd false sk2 term2 sk1 term1

    | Rigid, Rigid -> begin
        match kind_of_term term1, kind_of_term term2 with

	| Sort s1, Sort s2 when app_empty ->
	    (try
	       let evd' =
		 if pbty == CONV
		 then Evd.set_eq_sort evd s1 s2
		 else Evd.set_leq_sort evd s1 s2
	       in Success evd'
	     with Univ.UniverseInconsistency _ ->
               UnifFailure (evd,UnifUnivInconsistency)
	     | e when Errors.noncritical e -> UnifFailure (evd,NotSameHead))

	| Prod (n,c1,c'1), Prod (_,c2,c'2) when app_empty ->
            ise_and evd
              [(fun i -> evar_conv_x ts env i CONV c1 c2);
               (fun i ->
 	         let c = nf_evar i c1 in
	         evar_conv_x ts (push_rel (n,None,c) env) i pbty c'1 c'2)]

	| Ind sp1, Ind sp2 ->
	    if eq_ind sp1 sp2 then
              exact_ise_stack2 env evd (evar_conv_x ts) sk1 sk2
            else UnifFailure (evd,NotSameHead)

	| Construct sp1, Construct sp2 ->
	    if eq_constructor sp1 sp2 then
              exact_ise_stack2 env evd (evar_conv_x ts) sk1 sk2
            else UnifFailure (evd,NotSameHead)

	| Fix ((li1, i1),(_,tys1,bds1 as recdef1)), Fix ((li2, i2),(_,tys2,bds2)) -> (* Partially applied fixs *)
	  if Int.equal i1 i2 && Array.equal Int.equal li1 li2 then
            ise_and evd [
	      (fun i -> ise_array2 i (fun i' -> evar_conv_x ts env i' CONV) tys1 tys2);
	      (fun i -> ise_array2 i (fun i' -> evar_conv_x ts (push_rec_types recdef1 env) i' CONV) bds1 bds2);
	      (fun i -> exact_ise_stack2 env i (evar_conv_x ts) sk1 sk2)]
	  else UnifFailure (evd, NotSameHead)

	| CoFix (i1,(_,tys1,bds1 as recdef1)), CoFix (i2,(_,tys2,bds2)) ->
            if Int.equal i1 i2  then
              ise_and evd
                [(fun i -> ise_array2 i
                    (fun i -> evar_conv_x ts env i CONV) tys1 tys2);
                 (fun i -> ise_array2 i
		     (fun i -> evar_conv_x ts (push_rec_types recdef1 env) i CONV)
		     bds1 bds2);
                 (fun i -> exact_ise_stack2 env i
                     (evar_conv_x ts) sk1 sk2)]
            else UnifFailure (evd,NotSameHead)

	| (Meta _, _) | (_, Meta _) ->
	  begin match ise_stack2 true env evd (evar_conv_x ts) sk1 sk2 with
	  |_, (UnifFailure _ as x) -> x
	  |None, Success i' -> evar_conv_x ts env i' CONV term1 term2
	  |Some (sk1',sk2'), Success i' -> evar_conv_x ts env i' CONV (zip (term1,sk1')) (zip (term2,sk2'))
	  end

	| (Ind _ | Construct _ | Sort _ | Prod _ | CoFix _ | Fix _), _ -> UnifFailure (evd,NotSameHead)
	| _, (Ind _ | Construct _ | Sort _ | Prod _ | CoFix _ | Fix _) -> UnifFailure (evd,NotSameHead)

	| (App _ | Cast _ | Case _), _ -> assert false
	| (LetIn _ | Rel _ | Var _ | Const _ | Evar _), _ -> assert false
	| (Lambda _), _ -> assert false

      end

and conv_record trs env evd (c,bs,(params,params1),(us,us2),(ts,ts1),c1,(n,t2)) =
  if Reductionops.compare_stack_shape ts ts1 then
    let (evd',ks,_) =
      List.fold_left
	(fun (i,ks,m) b ->
	  if Int.equal m n then (i,t2::ks, m-1) else
	    let dloc = (Loc.ghost,Evar_kinds.InternalHole) in
            let (i',ev) = new_evar i env ~src:dloc (substl ks b) in
	    (i', ev :: ks, m - 1))
	(evd,[],List.length bs - 1) bs
    in
    let app = mkApp (c, Array.rev_of_list ks) in
    ise_and evd'
      [(fun i ->
	ise_list2 i
          (fun i' x1 x -> evar_conv_x trs env i' CONV x1 (substl ks x))
          params1 params);
       (fun i ->
	 ise_list2 i
           (fun i' u1 u -> evar_conv_x trs env i' CONV u1 (substl ks u))
           us2 us);
       (fun i -> evar_conv_x trs env i CONV c1 app);
       (fun i -> exact_ise_stack2 env i (evar_conv_x trs) ts ts1)]
  else UnifFailure(evd,(*dummy*)NotSameHead)

(* We assume here |l1| <= |l2| *)

let first_order_unification ts env evd (ev1,l1) (term2,l2) =
  let (deb2,rest2) = Array.chop (Array.length l2-Array.length l1) l2 in
  ise_and evd
    (* First compare extra args for better failure message *)
    [(fun i -> ise_array2 i (fun i -> evar_conv_x ts env i CONV) rest2 l1);
    (fun i ->
      (* Then instantiate evar unless already done by unifying args *)
      let t2 = mkApp(term2,deb2) in
      if is_defined i (fst ev1) then
	evar_conv_x ts env i CONV t2 (mkEvar ev1)
      else
	solve_simple_eqn ~choose:true (evar_conv_x ts) env i (None,ev1,t2))]

let choose_less_dependent_instance evk evd term args =
  let evi = Evd.find_undefined evd evk in
  let subst = make_pure_subst evi args in
  let subst' = List.filter (fun (id,c) -> eq_constr c term) subst in
  match subst' with
  | [] -> None
  | (id, _) :: _ -> Some (Evd.define evk (mkVar id) evd)

let apply_on_subterm evdref f c t =
  let rec applyrec (k,c as kc) t =
    (* By using eq_constr, we make an approximation, for instance, we *)
    (* could also be interested in finding a term u convertible to t *)
    (* such that c occurs in u *)
    if eq_constr c t then f k
    else
      match kind_of_term t with
      | Evar (evk,args) when Evd.is_undefined !evdref evk ->
          let ctx = evar_filtered_context (Evd.find_undefined !evdref evk) in
          let g (_,b,_) a = if Option.is_empty b then applyrec kc a else a in
          mkEvar (evk, Array.of_list (List.map2 g ctx (Array.to_list args)))
      | _ ->
        map_constr_with_binders_left_to_right (fun d (k,c) -> (k+1,lift 1 c))
	  applyrec kc t
  in
  applyrec (0,c) t

let filter_possible_projections c ty ctxt args =
  let fv1 = free_rels c in
  let fv2 = collect_vars c in
  let len = Array.length args in
  let tyvars = collect_vars ty in
  List.map_i (fun i (id,b,_) ->
    let () = assert (i < len) in
    let a = Array.unsafe_get args i in
    not (Option.is_empty b) ||
    a == c ||
    (* Here we make an approximation, for instance, we could also be *)
    (* interested in finding a term u convertible to c such that a occurs *)
    (* in u *)
    isRel a && Int.Set.mem (destRel a) fv1 ||
    isVar a && Id.Set.mem (destVar a) fv2 ||
    Id.Set.mem id tyvars)
    0 ctxt

let solve_evars = ref (fun _ -> failwith "solve_evars not installed")
let set_solve_evars f = solve_evars := f

(* We solve the problem env_rhs |- ?e[u1..un] = rhs knowing
 * x1:T1 .. xn:Tn |- ev : ty
 * by looking for a maximal well-typed abtraction over u1..un in rhs
 *
 * We first build C[e11..e1p1,..,en1..enpn] obtained from rhs by replacing
 * all occurrences of u1..un by evars eij of type Ti' where itself Ti' has
 * been obtained from the type of ui by also replacing all occurrences of
 * u1..ui-1 by evars.
 *
 * Then, we use typing to infer the relations between the different
 * occurrences. If some occurrence is still unconstrained after typing,
 * we instantiate successively the unresolved occurrences of un by xn,
 * of un-1 by xn-1, etc [the idea comes from Chung-Kil Hur, that he
 * used for his Heq plugin; extensions to several arguments based on a
 * proposition from Dan Grayson]
 *)

exception TypingFailed of evar_map

let second_order_matching ts env_rhs evd (evk,args) argoccs rhs =
  try
  let evi = Evd.find_undefined evd evk in
  let env_evar = evar_filtered_env evi in
  let sign = named_context_val env_evar in
  let ctxt = evar_filtered_context evi in
  let filter = evar_filter evi in
  let instance = List.map mkVar (List.map pi1 ctxt) in

  let rec make_subst = function
  | (id,_,t)::ctxt', c::l, occs::occsl when isVarId id c ->
      begin match occs with
      | Some _ ->
        error "Cannot force abstraction on identity instance."
      | None ->
        make_subst (ctxt',l,occsl)
      end
  | (id,_,t)::ctxt', c::l, occs::occsl ->
      let evs = ref [] in
      let ty = Retyping.get_type_of env_rhs evd c in
      let filter' = filter_possible_projections c ty ctxt args in
      let filter = List.map2 (&&) filter filter' in
      (id,t,c,ty,evs,filter,occs) :: make_subst (ctxt',l,occsl)
  | [], [], [] -> []
  | _ -> anomaly (Pp.str "Signature, instance and occurrences list do not match") in

  let rec set_holes evdref rhs = function
  | (id,_,c,cty,evsref,filter,occs)::subst ->
      let set_var k =
        match occs with
        | Some Locus.AllOccurrences -> mkVar id
        | Some _ -> error "Selection of specific occurrences not supported"
        | None ->
        let evty = set_holes evdref cty subst in
        let instance = List.filter_with filter instance in
        let evd,ev = new_evar_instance sign !evdref evty ~filter instance in
        evdref := evd;
        evsref := (fst (destEvar ev),evty)::!evsref;
        ev in
      set_holes evdref (apply_on_subterm evdref set_var c rhs) subst
  | [] -> rhs in

  let subst = make_subst (ctxt,Array.to_list args,argoccs) in

  let evdref = ref evd in
  let rhs = set_holes evdref rhs subst in
  let evd = !evdref in

  (* We instantiate the evars of which the value is forced by typing *)
  let evd,rhs =
    let evdref = ref evd in
    try let c = !solve_evars env_evar evdref rhs in !evdref,c
    with e when Pretype_errors.precatchable_exception e ->
      (* Could not revert all subterms *)
      raise (TypingFailed !evdref) in

  let rec abstract_free_holes evd = function
  | (id,idty,c,_,evsref,_,_)::l ->
      let rec force_instantiation evd = function
      | (evk,evty)::evs ->
          let evd =
            if is_undefined evd evk then
              (* We force abstraction over this unconstrained occurrence *)
              (* and we use typing to propagate this instantiation *)
              (* This is an arbitrary choice *)
              let evd = Evd.define evk (mkVar id) evd in
              match evar_conv_x ts env_evar evd CUMUL idty evty with
              | UnifFailure _ -> error "Cannot find an instance"
              | Success evd ->
              match reconsider_conv_pbs (evar_conv_x ts) evd with
              | UnifFailure _ -> error "Cannot find an instance"
              | Success evd ->
              evd
            else
              evd
          in
          force_instantiation evd evs
      | [] ->
          abstract_free_holes evd l
      in
      force_instantiation evd !evsref
  | [] ->
      Evd.define evk rhs evd in

  abstract_free_holes evd subst, true
  with TypingFailed evd -> Evd.define evk rhs evd, false

let second_order_matching_with_args ts env evd ev l t =
(*
  let evd,ev = evar_absorb_arguments env evd ev l in
  let argoccs = Array.map_to_list (fun _ -> None) (snd ev) in
  let evd, b = second_order_matching ts env evd ev argoccs t in
  if b then Success evd else
*)
  UnifFailure (evd, ConversionFailed (env,mkApp(mkEvar ev,l),t))

let apply_conversion_problem_heuristic ts env evd pbty t1 t2 =
  let t1 = apprec_nohdbeta ts env evd (whd_head_evar evd t1) in
  let t2 = apprec_nohdbeta ts env evd (whd_head_evar evd t2) in
  let (term1,l1 as appr1) = try destApp t1 with DestKO -> (t1, [||]) in
  let (term2,l2 as appr2) = try destApp t2 with DestKO -> (t2, [||]) in
  let app_empty = Array.is_empty l1 && Array.is_empty l2 in
  match kind_of_term term1, kind_of_term term2 with
  | Evar (evk1,args1), (Rel _|Var _) when app_empty
      && List.for_all (fun a -> eq_constr a term2 || isEvar a)
        (remove_instance_local_defs evd evk1 args1) ->
      (* The typical kind of constraint coming from pattern-matching return
         type inference *)
      (match choose_less_dependent_instance evk1 evd term2 args1 with
      | Some evd -> Success evd
      | None -> UnifFailure (evd, ConversionFailed (env,term1,term2)))
  | (Rel _|Var _), Evar (evk2,args2) when app_empty
      && List.for_all (fun a -> eq_constr a term1 || isEvar a)
        (remove_instance_local_defs evd evk2 args2) ->
      (* The typical kind of constraint coming from pattern-matching return
         type inference *)
      (match choose_less_dependent_instance evk2 evd term1 args2 with
      | Some evd -> Success evd
      | None -> UnifFailure (evd, ConversionFailed (env,term1,term2)))
  | Evar (evk1,args1), Evar (evk2,args2) when Evar.equal evk1 evk2 ->
      let f env evd pbty x y = is_trans_fconv pbty ts env evd x y in
      Success (solve_refl ~can_drop:true f env evd evk1 args1 args2)
  | Evar ev1, Evar ev2 ->
      Success (solve_evar_evar ~force:true
        (evar_define (evar_conv_x ts)) (evar_conv_x ts) env evd ev1 ev2)
  | Evar ev1,_ when Array.length l1 <= Array.length l2 ->
      (* On "?n t1 .. tn = u u1 .. u(n+p)", try first-order unification *)
      (* and otherwise second-order matching *)
      ise_try evd
        [(fun evd -> first_order_unification ts env evd (ev1,l1) appr2);
         (fun evd ->
           second_order_matching_with_args ts env evd ev1 l1 t2)]
  | _,Evar ev2 when Array.length l2 <= Array.length l1 ->
      (* On "u u1 .. u(n+p) = ?n t1 .. tn", try first-order unification *)
      (* and otherwise second-order matching *)
      ise_try evd
        [(fun evd -> first_order_unification ts env evd (ev2,l2) appr1);
         (fun evd ->
           second_order_matching_with_args ts env evd ev2 l2 t1)]
  | Evar ev1,_ ->
      (* Try second-order pattern-matching *)
      second_order_matching_with_args ts env evd ev1 l1 t2
  | _,Evar ev2 ->
      (* Try second-order pattern-matching *)
      second_order_matching_with_args ts env evd ev2 l2 t1
  | _ ->
      (* Some head evar have been instantiated, or unknown kind of problem *)
      evar_conv_x ts env evd pbty t1 t2

let check_problems_are_solved evd =
  match snd (extract_all_conv_pbs evd) with
  | (pbty,env,t1,t2)::_ -> Pretype_errors.error_cannot_unify env evd (t1, t2)
  | _ -> ()

let max_undefined_with_candidates evd =
  (* If evar were ordered with highest index first, fold_undefined
     would be going decreasingly and we could use fold_undefined to
     find the undefined evar of maximum index (alternatively,
     max_bindings from ocaml 3.12 could be used); instead we traverse
     the whole map *)
  let l = Evd.fold_undefined
    (fun evk ev_info evars ->
        match ev_info.evar_candidates with
        | None -> evars
        | Some l -> (evk,ev_info,l)::evars) evd [] in
  match l with
  | [] -> None
  | a::l -> Some (List.last (a::l))

let rec solve_unconstrained_evars_with_canditates evd =
  (* max_undefined is supposed to return the most recent, hence
     possibly most dependent evar *)
  match max_undefined_with_candidates evd with
  | None -> evd
  | Some (evk,ev_info,l) ->
      let rec aux = function
      | [] -> error "Unsolvable existential variables."
      | a::l ->
          try
            let conv_algo = evar_conv_x full_transparent_state in
            let evd = check_evar_instance evd evk a conv_algo in
            let evd = Evd.define evk a evd in
            match reconsider_conv_pbs conv_algo evd with
            | Success evd -> solve_unconstrained_evars_with_canditates evd
            | UnifFailure _ -> aux l
          with
          | IllTypedInstance _ -> aux l
          | e when Pretype_errors.precatchable_exception e -> aux l in
      (* List.rev is there to favor most dependent solutions *)
      (* and favor progress when used with the refine tactics *)
      let evd = aux (List.rev l) in
      solve_unconstrained_evars_with_canditates evd

let solve_unconstrained_impossible_cases evd =
  Evd.fold_undefined (fun evk ev_info evd' ->
    match ev_info.evar_source with
    | _,Evar_kinds.ImpossibleCase -> Evd.define evk (j_type (coq_unit_judge ())) evd'
    | _ -> evd') evd evd

let consider_remaining_unif_problems ?(ts=full_transparent_state) env evd =
  let evd = solve_unconstrained_evars_with_canditates evd in
  let rec aux evd pbs progress stuck =
    match pbs with
    | (pbty,env,t1,t2 as pb) :: pbs ->
        (match apply_conversion_problem_heuristic ts env evd pbty t1 t2 with
	| Success evd' ->
	    let (evd', rest) = extract_all_conv_pbs evd' in
            begin match rest with
            | [] -> aux evd' pbs true stuck
            | _ -> (* Unification got actually stuck, postpone *)
	      aux evd pbs progress (pb :: stuck)
            end
        | UnifFailure (evd,reason) ->
	    Pretype_errors.error_cannot_unify_loc (loc_of_conv_pb evd pb)
              env evd ~reason (t1, t2))
    | _ -> 
	if progress then aux evd stuck false []
	else 
	  match stuck with
	  | [] -> (* We're finished *) evd
	  | (pbty,env,t1,t2 as pb) :: _ ->
	      (* There remains stuck problems *)
	      Pretype_errors.error_cannot_unify_loc (loc_of_conv_pb evd pb)
                env evd (t1, t2)
  in
  let (evd,pbs) = extract_all_conv_pbs evd in
  let heuristic_solved_evd = aux evd pbs false [] in
  check_problems_are_solved heuristic_solved_evd;
  solve_unconstrained_impossible_cases heuristic_solved_evd

(* Main entry points *)

exception UnableToUnify of evar_map * unification_error

let the_conv_x ?(ts=full_transparent_state) env t1 t2 evd =
  match evar_conv_x ts env evd CONV  t1 t2 with
  | Success evd' -> evd'
  | UnifFailure (evd',e) -> raise (UnableToUnify (evd',e))

let the_conv_x_leq ?(ts=full_transparent_state) env t1 t2 evd =
  match evar_conv_x ts env evd CUMUL t1 t2 with
  | Success evd' -> evd'
  | UnifFailure (evd',e) -> raise (UnableToUnify (evd',e))

let e_conv ?(ts=full_transparent_state) env evdref t1 t2 =
  match evar_conv_x ts env !evdref CONV t1 t2 with
  | Success evd' -> evdref := evd'; true
  | _ -> false

let e_cumul ?(ts=full_transparent_state) env evdref t1 t2 =
  match evar_conv_x ts env !evdref CUMUL t1 t2 with
  | Success evd' -> evdref := evd'; true
  | _ -> false