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

(*i camlp4deps: "parsing/grammar.cma parsing/q_constr.cmo" i*)

open Pp
open Util
open Names
open Nameops
open Term
open Sign
open Termops
open Reductionops
open Inductiveops
open Evd
open Environ
open Clenv
open Pattern
open Matching
open Coqlib
open Declarations

(* I implemented the following functions which test whether a term t
   is an inductive but non-recursive type, a general conjuction, a
   general disjunction, or a type with no constructors.

   They are more general than matching with or_term, and_term, etc,
   since they do not depend on the name of the type. Hence, they
   also work on ad-hoc disjunctions introduced by the user.

  -- Eduardo (6/8/97). *)

type 'a matching_function = constr -> 'a option

type testing_function  = constr -> bool

let mkmeta n = Nameops.make_ident "X" (Some n)
let meta1 = mkmeta 1
let meta2 = mkmeta 2
let meta3 = mkmeta 3
let meta4 = mkmeta 4

let op2bool = function Some _ -> true | None -> false

let match_with_non_recursive_type t =
  match kind_of_term t with
    | App _ ->
        let (hdapp,args) = decompose_app t in
        (match kind_of_term hdapp with
           | Ind ind ->
               if not (Global.lookup_mind (fst ind)).mind_finite then
		 Some (hdapp,args)
	       else
		 None
           | _ -> None)
    | _ -> None

let is_non_recursive_type t = op2bool (match_with_non_recursive_type t)

(* Test dependencies *)

(* NB: we consider also the let-in case in the following function,
   since they may appear in types of inductive constructors (see #2629) *)

let rec has_nodep_prod_after n c =
  match kind_of_term c with
    | Prod (_,_,b) | LetIn (_,_,_,b) ->
	( n>0 || not (dependent (mkRel 1) b))
	&& (has_nodep_prod_after (n-1) b)
    | _            -> true

let has_nodep_prod = has_nodep_prod_after 0

(* A general conjunctive type is a non-recursive with-no-indices inductive
   type with only one constructor and no dependencies between argument;
   it is strict if it has the form
   "Inductive I A1 ... An := C (_:A1) ... (_:An)" *)

(* style: None = record; Some false = conjunction; Some true = strict conj *)

let match_with_one_constructor style allow_rec t =
  let (hdapp,args) = decompose_app t in
  match kind_of_term hdapp with
  | Ind ind ->
      let (mib,mip) = Global.lookup_inductive ind in
      if (Array.length mip.mind_consnames = 1)
	&& (allow_rec or not (mis_is_recursive (ind,mib,mip)))
        && (mip.mind_nrealargs = 0)
      then
	if style = Some true (* strict conjunction *) then
	  let ctx =
	    (prod_assum (snd
	      (decompose_prod_n_assum mib.mind_nparams mip.mind_nf_lc.(0)))) in
	  if
	    List.for_all
	      (fun (_,b,c) -> b=None && isRel c && destRel c = mib.mind_nparams) ctx
	  then
	    Some (hdapp,args)
	  else None
	else
	  let ctyp = prod_applist mip.mind_nf_lc.(0) args in
	  let cargs = List.map pi3 ((prod_assum ctyp)) in
	  if style <> Some false || has_nodep_prod ctyp then
	    (* Record or non strict conjunction *)
	    Some (hdapp,List.rev cargs)
	  else
	      None
      else
	None
  | _ -> None

let match_with_conjunction ?(strict=false) t =
  match_with_one_constructor (Some strict) false t

let match_with_record t =
  match_with_one_constructor None false t

let is_conjunction ?(strict=false) t =
  op2bool (match_with_conjunction ~strict t)

let is_record t =
  op2bool (match_with_record t)

let match_with_tuple t =
  let t = match_with_one_constructor None true t in
  Option.map (fun (hd,l) ->
    let ind = destInd hd in
    let (mib,mip) = Global.lookup_inductive ind in
    let isrec = mis_is_recursive (ind,mib,mip) in
    (hd,l,isrec)) t

let is_tuple t =
  op2bool (match_with_tuple t)

(* A general disjunction type is a non-recursive with-no-indices inductive
   type with of which all constructors have a single argument;
   it is strict if it has the form
   "Inductive I A1 ... An := C1 (_:A1) | ... | Cn : (_:An)" *)

let test_strict_disjunction n lc =
  array_for_all_i (fun i c ->
    match (prod_assum (snd (decompose_prod_n_assum n c))) with
    | [_,None,c] -> isRel c && destRel c = (n - i)
    | _ -> false) 0 lc

let match_with_disjunction ?(strict=false) t =
  let (hdapp,args) = decompose_app t in
  match kind_of_term hdapp with
  | Ind ind  ->
      let car = mis_constr_nargs ind in
      let (mib,mip) = Global.lookup_inductive ind in
      if array_for_all (fun ar -> ar = 1) car
        && not (mis_is_recursive (ind,mib,mip))
        && (mip.mind_nrealargs = 0)
      then
	if strict then
	  if test_strict_disjunction mib.mind_nparams mip.mind_nf_lc then
	    Some (hdapp,args)
	  else
	    None
	else
	  let cargs =
	    Array.map (fun ar -> pi2 (destProd (prod_applist ar args)))
	      mip.mind_nf_lc in
	  Some (hdapp,Array.to_list cargs)
      else
	None
  | _ -> None

let is_disjunction ?(strict=false) t =
  op2bool (match_with_disjunction ~strict t)

(* An empty type is an inductive type, possible with indices, that has no
   constructors *)

let match_with_empty_type t =
  let (hdapp,args) = decompose_app t in
  match (kind_of_term hdapp) with
    | Ind ind ->
        let (mib,mip) = Global.lookup_inductive ind in
        let nconstr = Array.length mip.mind_consnames in
	if nconstr = 0 then Some hdapp else None
    | _ ->  None

let is_empty_type t = op2bool (match_with_empty_type t)

(* This filters inductive types with one constructor with no arguments;
   Parameters and indices are allowed *)

let match_with_unit_or_eq_type t =
  let (hdapp,args) = decompose_app t in
  match (kind_of_term hdapp) with
    | Ind ind  ->
        let (mib,mip) = Global.lookup_inductive ind in
        let constr_types = mip.mind_nf_lc in
        let nconstr = Array.length mip.mind_consnames in
        let zero_args c = nb_prod c = mib.mind_nparams in
	if nconstr = 1 && zero_args constr_types.(0) then
	  Some hdapp
	else
	  None
    | _ -> None

let is_unit_or_eq_type t = op2bool (match_with_unit_or_eq_type t)

(* A unit type is an inductive type with no indices but possibly
   (useless) parameters, and that has no arguments in its unique
   constructor *)

let is_unit_type t =
  match match_with_conjunction t with
  | Some (_,t) when List.length t = 0 -> true
  | _ -> false

(* Checks if a given term is an application of an
   inductive binary relation R, so that R has only one constructor
   establishing its reflexivity.  *)

type equation_kind =
  | MonomorphicLeibnizEq of constr * constr
  | PolymorphicLeibnizEq of constr * constr * constr
  | HeterogenousEq of constr * constr * constr * constr

exception NoEquationFound

let coq_refl_leibniz1_pattern = PATTERN [ forall x:_, _ x x ]
let coq_refl_leibniz2_pattern = PATTERN [ forall A:_, forall x:A, _ A x x ]
let coq_refl_jm_pattern       = PATTERN [ forall A:_, forall x:A, _ A x A x ]

open Libnames

let match_with_equation t =
  if not (isApp t) then raise NoEquationFound;
  let (hdapp,args) = destApp t in
  match kind_of_term hdapp with
  | Ind ind ->
      if IndRef ind = glob_eq then
	Some (build_coq_eq_data()),hdapp,
	PolymorphicLeibnizEq(args.(0),args.(1),args.(2))
      else if IndRef ind = glob_identity then
	Some (build_coq_identity_data()),hdapp,
	PolymorphicLeibnizEq(args.(0),args.(1),args.(2))
      else if IndRef ind = glob_jmeq then
	Some (build_coq_jmeq_data()),hdapp,
	HeterogenousEq(args.(0),args.(1),args.(2),args.(3))
      else
        let (mib,mip) = Global.lookup_inductive ind in
        let constr_types = mip.mind_nf_lc in
        let nconstr = Array.length mip.mind_consnames in
	if nconstr = 1 then
          if is_matching coq_refl_leibniz1_pattern constr_types.(0) then
	    None, hdapp, MonomorphicLeibnizEq(args.(0),args.(1))
	  else if is_matching coq_refl_leibniz2_pattern constr_types.(0) then
	    None, hdapp, PolymorphicLeibnizEq(args.(0),args.(1),args.(2))
	  else if is_matching coq_refl_jm_pattern constr_types.(0) then
	    None, hdapp, HeterogenousEq(args.(0),args.(1),args.(2),args.(3))
	  else raise NoEquationFound
        else raise NoEquationFound
    | _ -> raise NoEquationFound

let is_inductive_equality ind =
  let (mib,mip) = Global.lookup_inductive ind in
  let nconstr = Array.length mip.mind_consnames in
  nconstr = 1 && constructor_nrealargs (Global.env()) (ind,1) = 0

let match_with_equality_type t =
  let (hdapp,args) = decompose_app t in
  match (kind_of_term hdapp) with
  | Ind ind when is_inductive_equality ind -> Some (hdapp,args)
  | _ -> None

let is_equality_type t = op2bool (match_with_equality_type t)

let coq_arrow_pattern = PATTERN [ ?X1 -> ?X2 ]

let match_arrow_pattern t =
  match matches coq_arrow_pattern t with
    | [(m1,arg);(m2,mind)] -> assert (m1=meta1 & m2=meta2); (arg, mind)
    | _ -> anomaly "Incorrect pattern matching"

let match_with_nottype t =
  try
    let (arg,mind) = match_arrow_pattern t in
    if is_empty_type mind then Some (mind,arg) else None
  with PatternMatchingFailure -> None

let is_nottype t = op2bool (match_with_nottype t)

let match_with_forall_term c=
  match kind_of_term c with
    | Prod (nam,a,b) -> Some (nam,a,b)
    | _            -> None

let is_forall_term c = op2bool (match_with_forall_term c)

let match_with_imp_term c=
  match kind_of_term c with
    | Prod (_,a,b) when not (dependent (mkRel 1) b) ->Some (a,b)
    | _              -> None

let is_imp_term c = op2bool (match_with_imp_term c)

let match_with_nodep_ind t =
  let (hdapp,args) = decompose_app t in
    match (kind_of_term hdapp) with
      | Ind ind  ->
          let (mib,mip) = Global.lookup_inductive ind in
	    if Array.length (mib.mind_packets)>1 then None else
	      let nodep_constr = has_nodep_prod_after mib.mind_nparams in
		if array_for_all nodep_constr mip.mind_nf_lc then
		  let params=
		    if mip.mind_nrealargs=0 then args else
		      fst (list_chop mib.mind_nparams args) in
		    Some (hdapp,params,mip.mind_nrealargs)
		else
		  None
      | _ -> None

let is_nodep_ind t=op2bool (match_with_nodep_ind t)

let match_with_sigma_type t=
  let (hdapp,args) = decompose_app t in
  match (kind_of_term hdapp) with
    | Ind ind  ->
        let (mib,mip) = Global.lookup_inductive ind in
          if (Array.length (mib.mind_packets)=1) &&
	    (mip.mind_nrealargs=0) &&
	    (Array.length mip.mind_consnames=1) &&
	    has_nodep_prod_after (mib.mind_nparams+1) mip.mind_nf_lc.(0) then
	      (*allowing only 1 existential*)
	      Some (hdapp,args)
	  else
	    None
    | _ -> None

let is_sigma_type t=op2bool (match_with_sigma_type t)

(***** Destructing patterns bound to some theory *)

let rec first_match matcher = function
  | [] -> raise PatternMatchingFailure
  | (pat,build_set)::l ->
      try (build_set (),matcher pat)
      with PatternMatchingFailure -> first_match matcher l

(*** Equality *)

(* Patterns "(eq ?1 ?2 ?3)" and "(identity ?1 ?2 ?3)" *)
let coq_eq_pattern_gen eq = lazy PATTERN [ %eq ?X1 ?X2 ?X3 ]
let coq_eq_pattern = coq_eq_pattern_gen coq_eq_ref
let coq_identity_pattern = coq_eq_pattern_gen coq_identity_ref
let coq_jmeq_pattern = lazy PATTERN [ %coq_jmeq_ref ?X1 ?X2 ?X3 ?X4 ]
let coq_eq_true_pattern = lazy PATTERN [ %coq_eq_true_ref ?X1 ]

let match_eq eqn eq_pat =
  let pat =
    try Lazy.force eq_pat
    with e when Errors.noncritical e -> raise PatternMatchingFailure
  in
  match matches pat eqn with
    | [(m1,t);(m2,x);(m3,y)] ->
	assert (m1 = meta1 & m2 = meta2 & m3 = meta3);
	PolymorphicLeibnizEq (t,x,y)
    | [(m1,t);(m2,x);(m3,t');(m4,x')] ->
	assert (m1 = meta1 & m2 = meta2 & m3 = meta3 & m4 = meta4);
	HeterogenousEq (t,x,t',x')
    | _ -> anomaly "match_eq: an eq pattern should match 3 or 4 terms"

let equalities =
  [coq_eq_pattern, build_coq_eq_data;
   coq_jmeq_pattern, build_coq_jmeq_data;
   coq_identity_pattern, build_coq_identity_data]

let find_eq_data eqn = (* fails with PatternMatchingFailure *)
  first_match (match_eq eqn) equalities

let extract_eq_args gl = function
  | MonomorphicLeibnizEq (e1,e2) ->
      let t = Tacmach.pf_type_of gl e1 in (t,e1,e2)
  | PolymorphicLeibnizEq (t,e1,e2) -> (t,e1,e2)
  | HeterogenousEq (t1,e1,t2,e2) ->
      if Tacmach.pf_conv_x gl t1 t2 then (t1,e1,e2)
      else raise PatternMatchingFailure

let find_eq_data_decompose gl eqn =
  let (lbeq,eq_args) = find_eq_data eqn in
  (lbeq,extract_eq_args gl eq_args)

let inversible_equalities =
  [coq_eq_pattern, build_coq_inversion_eq_data;
   coq_jmeq_pattern, build_coq_inversion_jmeq_data;
   coq_identity_pattern, build_coq_inversion_identity_data;
   coq_eq_true_pattern, build_coq_inversion_eq_true_data]

let find_this_eq_data_decompose gl eqn =
  let (lbeq,eq_args) =
    try (*first_match (match_eq eqn) inversible_equalities*)
      find_eq_data eqn
    with PatternMatchingFailure ->
      errorlabstrm "" (str "No primitive equality found.") in
  let eq_args =
    try extract_eq_args gl eq_args
    with PatternMatchingFailure ->
      error "Don't know what to do with JMeq on arguments not of same type." in
  (lbeq,eq_args)

open Tacmach
open Tacticals

let match_eq_nf gls eqn eq_pat =
  match pf_matches gls (Lazy.force eq_pat) eqn with
    | [(m1,t);(m2,x);(m3,y)] ->
	assert (m1 = meta1 & m2 = meta2 & m3 = meta3);
	(t,pf_whd_betadeltaiota gls x,pf_whd_betadeltaiota gls y)
    | _ -> anomaly "match_eq: an eq pattern should match 3 terms"

let dest_nf_eq gls eqn =
  try
    snd (first_match (match_eq_nf gls eqn) equalities)
  with PatternMatchingFailure ->
    error "Not an equality."

(*** Sigma-types *)

(* Patterns "(existS ?1 ?2 ?3 ?4)" and "(existT ?1 ?2 ?3 ?4)" *)
let coq_ex_pattern_gen ex = lazy PATTERN [ %ex ?X1 ?X2 ?X3 ?X4 ]
let coq_existT_pattern = coq_ex_pattern_gen coq_existT_ref
let coq_exist_pattern = coq_ex_pattern_gen coq_exist_ref

let match_sigma ex ex_pat =
  match matches (Lazy.force ex_pat) ex with
    | [(m1,a);(m2,p);(m3,car);(m4,cdr)] ->
	assert (m1=meta1 & m2=meta2 & m3=meta3 & m4=meta4);
	(a,p,car,cdr)
    | _ ->
	anomaly "match_sigma: a successful sigma pattern should match 4 terms"

let find_sigma_data_decompose ex = (* fails with PatternMatchingFailure *)
  first_match (match_sigma ex)
    [coq_existT_pattern, build_sigma_type;
     coq_exist_pattern, build_sigma]

(* Pattern "(sig ?1 ?2)" *)
let coq_sig_pattern = lazy PATTERN [ %coq_sig_ref ?X1 ?X2 ]

let match_sigma t =
  match matches (Lazy.force coq_sig_pattern) t with
    | [(_,a); (_,p)] -> (a,p)
    | _ -> anomaly "Unexpected pattern"

let is_matching_sigma t = is_matching (Lazy.force coq_sig_pattern) t

(*** Decidable equalities *)

(* The expected form of the goal for the tactic Decide Equality *)

(* Pattern "{<?1>x=y}+{~(<?1>x=y)}" *)
(* i.e. "(sumbool (eq ?1 x y) ~(eq ?1 x y))" *)

let coq_eqdec_inf_pattern =
 lazy PATTERN [ { ?X2 = ?X3 :> ?X1 } + { ~ ?X2 = ?X3 :> ?X1 } ]

let coq_eqdec_inf_rev_pattern =
 lazy PATTERN [ { ~ ?X2 = ?X3 :> ?X1 } + { ?X2 = ?X3 :> ?X1 } ]

let coq_eqdec_pattern =
 lazy PATTERN [ %coq_or_ref (?X2 = ?X3 :> ?X1) (~ ?X2 = ?X3 :> ?X1) ]

let coq_eqdec_rev_pattern =
 lazy PATTERN [ %coq_or_ref (~ ?X2 = ?X3 :> ?X1) (?X2 = ?X3 :> ?X1) ]

let op_or = coq_or_ref
let op_sum = coq_sumbool_ref

let match_eqdec t =
  let eqonleft,op,subst =
    try true,op_sum,matches (Lazy.force coq_eqdec_inf_pattern) t
    with PatternMatchingFailure ->
    try false,op_sum,matches (Lazy.force coq_eqdec_inf_rev_pattern) t
    with PatternMatchingFailure ->
    try true,op_or,matches (Lazy.force coq_eqdec_pattern) t
    with PatternMatchingFailure ->
        false,op_or,matches (Lazy.force coq_eqdec_rev_pattern) t in
  match subst with
  | [(_,typ);(_,c1);(_,c2)] ->
      eqonleft, Libnames.constr_of_global (Lazy.force op), c1, c2, typ
  | _ -> anomaly "Unexpected pattern"

(* Patterns "~ ?" and "? -> False" *)
let coq_not_pattern = lazy PATTERN [ ~ _ ]
let coq_imp_False_pattern = lazy PATTERN [ _ -> %coq_False_ref ]

let is_matching_not t = is_matching (Lazy.force coq_not_pattern) t
let is_matching_imp_False t = is_matching (Lazy.force coq_imp_False_pattern) t

(* Remark: patterns that have references to the standard library must
   be evaluated lazily (i.e. at the time they are used, not a the time
   coqtop starts) *)
