path: root/tactics/equality.ml

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(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(*   \VV/  **************************************************************)
(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
(************************************************************************)

(* $Id: equality.ml 8878 2006-05-30 16:44:25Z herbelin $ *)

open Pp
open Util
open Names
open Nameops
open Univ
open Term
open Termops
open Inductive
open Inductiveops
open Environ
open Libnames
open Reductionops
open Typeops
open Typing
open Retyping
open Tacmach
open Proof_type
open Logic
open Evar_refiner
open Pattern
open Matching
open Hipattern
open Tacexpr
open Tacticals
open Tactics
open Tacred
open Rawterm
open Coqlib
open Vernacexpr
open Setoid_replace
open Declarations
open Indrec

(* Rewriting tactics *)

(* Warning : rewriting from left to right only works
   if there exists in the context a theorem named <eqname>_<suffsort>_r
   with type (A:<sort>)(x:A)(P:A->Prop)(P x)->(y:A)(eqname A y x)->(P y).
   If another equality myeq is introduced, then corresponding theorems
   myeq_ind_r, myeq_rec_r and myeq_rect_r have to be proven. See below.
   -- Eduardo (19/8/97)
*)

let general_s_rewrite_clause = function
  | None -> general_s_rewrite
  | Some id -> general_s_rewrite_in id

(* Ad hoc asymmetric general_elim_clause *)
let general_elim_clause cls c elim = match cls with
  | None ->
      (* was tclWEAK_PROGRESS which only fails for tactics generating one 
         subgoal and did not fail for useless conditional rewritings generating
         an extra condition *)
      tclNOTSAMEGOAL (general_elim c elim ~allow_K:false)
  | Some id ->
      general_elim_in id c elim 

let elimination_sort_of_clause = function
  | None -> elimination_sort_of_goal
  | Some id -> elimination_sort_of_hyp id

(* The next function decides in particular whether to try a regular 
  rewrite or a setoid rewrite. 

  Old approach was: 
   break everything, if [eq] appears in head position 
   then regular rewrite else try setoid rewrite
 
  New approach is: 
   if head position is a known setoid relation then setoid rewrite
   else back to the old approach
*)

let general_rewrite_bindings_clause cls lft2rgt (c,l) gl =
  let ctype = pf_type_of gl c in 
  (* A delta-reduction would be here too strong, since it would 
     break search for a defined setoid relation in head position. *)
  let t = snd (decompose_prod (whd_betaiotazeta ctype)) in 
  let head = if isApp t then fst (destApp t) else t in 
  if relation_table_mem head && l = NoBindings then 
    general_s_rewrite_clause cls lft2rgt c [] gl
  else 
    (* Original code. In particular, [splay_prod] performs delta-reduction. *)
    let env = pf_env gl in
    let sigma = project gl in 
    let _,t = splay_prod env sigma t in
    match match_with_equation t with
      | None -> 
	  if l = NoBindings
	  then general_s_rewrite_clause cls lft2rgt c [] gl
	  else error "The term provided does not end with an equation" 
      | Some (hdcncl,_) -> 
          let hdcncls = string_of_inductive hdcncl in 
	  let suffix = elimination_suffix (elimination_sort_of_clause cls gl) in
          let dir = if cls=None then lft2rgt else not lft2rgt in
          let rwr_thm = if dir then hdcncls^suffix^"_r" else hdcncls^suffix in
          let elim =
	    try pf_global gl (id_of_string rwr_thm)
            with Not_found ->
              error ("Cannot find rewrite principle "^rwr_thm)
          in 
	  general_elim_clause cls (c,l) (elim,NoBindings) gl

let general_rewrite_bindings = general_rewrite_bindings_clause None
let general_rewrite l2r c = general_rewrite_bindings l2r (c,NoBindings)

let general_rewrite_bindings_in l2r id =
  general_rewrite_bindings_clause (Some id) l2r
let general_rewrite_in l2r id c = 
  general_rewrite_bindings_clause (Some id) l2r (c,NoBindings)


let general_multi_rewrite l2r c cl = 
  let rec do_hyps = function 
    | [] -> tclIDTAC
    | ((_,id),_) :: l -> 
	tclTHENFIRST (general_rewrite_bindings_in l2r id c) (do_hyps l)
  in 
  let rec try_do_hyps = function 
    | [] -> tclIDTAC 
    | id :: l -> 
	tclTHENFIRST 
	  (tclTRY (general_rewrite_bindings_in l2r id c)) 
	  (try_do_hyps l)
  in 
  if cl.concl_occs <> [] then 
    error "The \"at\" syntax isn't available yet for the rewrite tactic"
  else 
    tclTHENFIRST 
      (if cl.onconcl then general_rewrite_bindings l2r c else tclIDTAC)
      (match cl.onhyps with 
	 | Some l -> do_hyps l
	 | None -> 
	     fun gl -> 
	       (* try to rewrite in all hypothesis 
		  (except maybe the rewritten one) *)
	       let ids = match kind_of_term (fst c) with 
		 | Var id -> list_remove id (pf_ids_of_hyps gl)
		 | _ -> pf_ids_of_hyps gl
	       in try_do_hyps ids gl)


(* Conditional rewriting, the success of a rewriting is related 
   to the resolution of the conditions by a given tactic *)

let conditional_rewrite lft2rgt tac (c,bl) = 
  tclTHENSFIRSTn (general_rewrite_bindings lft2rgt (c,bl))
    [|tclIDTAC|] (tclCOMPLETE tac)

let rewriteLR_bindings = general_rewrite_bindings true
let rewriteRL_bindings = general_rewrite_bindings false

let rewriteLR = general_rewrite true
let rewriteRL = general_rewrite false

let rewriteLRin_bindings = general_rewrite_bindings_in true
let rewriteRLin_bindings = general_rewrite_bindings_in false

let conditional_rewrite_in lft2rgt id tac (c,bl) = 
  tclTHENSFIRSTn (general_rewrite_bindings_in lft2rgt id (c,bl))
    [|tclIDTAC|] (tclCOMPLETE tac)

let rewriteRL_clause = function
  | None -> rewriteRL_bindings
  | Some id -> rewriteRLin_bindings id

(* Replacing tactics *)

(* eq,sym_eq : equality on Type and its symmetry theorem
   c2 c1 : c1 is to be replaced by c2
   unsafe : If true, do not check that c1 and c2 are convertible
   tac : Used to prove the equality c1 = c2 
   gl : goal *)

let abstract_replace clause c2 c1 unsafe tac gl =
  let t1 = pf_type_of gl c1 
  and t2 = pf_type_of gl c2 in
  if unsafe or (pf_conv_x gl t1 t2) then
    let e = build_coq_eq () in
    let sym = build_coq_sym_eq () in
    let eq = applist (e, [t1;c1;c2]) in
    tclTHENS (assert_tac false Anonymous eq)
      [onLastHyp (fun id -> 
	tclTHEN 
	  (tclTRY (rewriteRL_clause clause (mkVar id,NoBindings)))
	  (clear [id]));
       tclFIRST
	 [assumption;
	  tclTHEN (apply sym) assumption;
	  tclTRY (tclCOMPLETE tac)
	 ]
      ] gl
  else
    error "terms does not have convertible types"


let replace c2 c1 gl = abstract_replace None c2 c1 false tclIDTAC gl

let replace_in id c2 c1 gl = abstract_replace (Some id) c2 c1 false tclIDTAC gl

let replace_by c2 c1 tac gl = abstract_replace None c2 c1 false tac gl

let replace_in_by id c2 c1 tac gl = abstract_replace (Some id) c2 c1 false tac gl


let new_replace c2 c1 id tac_opt gl = 
  let tac = 
    match tac_opt with 
      | Some tac -> tac 
      | _ -> tclIDTAC
  in
  abstract_replace id c2 c1 false tac gl

(* End of Eduardo's code. The rest of this file could be improved
   using the functions match_with_equation, etc that I defined
   in Pattern.ml.
   -- Eduardo (19/8/97)
*)

(* Tactics for equality reasoning with the "eq" relation. This code
   will work with any equivalence relation which is substitutive *)

(* [find_positions t1 t2]

   will find the positions in the two terms which are suitable for
   discrimination, or for injection.  Obviously, if there is a
   position which is suitable for discrimination, then we want to
   exploit it, and not bother with injection.  So when we find a
   position which is suitable for discrimination, we will just raise
   an exception with that position.

   So the algorithm goes like this:

   if [t1] and [t2] start with the same constructor, then we can
   continue to try to find positions in the arguments of [t1] and
   [t2].

   if [t1] and [t2] do not start with the same constructor, then we
   have found a discrimination position

   if one [t1] or [t2] do not start with a constructor and the two
   terms are not already convertible, then we have found an injection
   position.

   A discriminating position consists of a constructor-path and a pair
   of operators.  The constructor-path tells us how to get down to the
   place where the two operators, which must differ, can be found.

   An injecting position has two terms instead of the two operators,
   since these terms are different, but not manifestly so.

   A constructor-path is a list of pairs of (operator * int), where
   the int (based at 0) tells us which argument of the operator we
   descended into.

 *)

exception DiscrFound of
  (constructor * int) list * constructor * constructor

let find_positions env sigma t1 t2 =
  let rec findrec posn t1 t2 =
    let hd1,args1 = whd_betadeltaiota_stack env sigma t1 in
    let hd2,args2 = whd_betadeltaiota_stack env sigma t2 in
    match (kind_of_term hd1, kind_of_term hd2) with
  	
      | Construct sp1, Construct sp2 
          when List.length args1 = mis_constructor_nargs_env env sp1
          ->
        (* both sides are fully applied constructors, so either we descend,
           or we can discriminate here. *)
	  if sp1 = sp2 then
            List.flatten
	      (list_map2_i
		 (fun i arg1 arg2 ->
		    findrec ((sp1,i)::posn) arg1 arg2)
		 0 args1 args2)
	  else
	    raise (DiscrFound(List.rev posn,sp1,sp2))

      | _ ->
	  let t1_0 = applist (hd1,args1) 
          and t2_0 = applist (hd2,args2) in
          if is_conv env sigma t1_0 t2_0 then 
	    []
          else
	    let ty1_0 = get_type_of env sigma t1_0 in
	    match get_sort_family_of env sigma ty1_0 with
	      | InSet | InType -> [(List.rev posn,t1_0,t2_0)]
	      | InProp -> []
	  in 
	  (try 
	     Inr(findrec [] t1 t2)
	   with DiscrFound (path,c1,c2) -> 
	     Inl (path,c1,c2))

let discriminable env sigma t1 t2 =
  match find_positions env sigma t1 t2 with
    | Inl _ -> true
    | _ -> false

(* Once we have found a position, we need to project down to it.  If
   we are discriminating, then we need to produce False on one of the
   branches of the discriminator, and True on the other one.  So the
   result type of the case-expressions is always Prop.

   If we are injecting, then we need to discover the result-type.
   This can be difficult, since the type of the two terms at the
   injection-position can be different, and we need to find a
   dependent sigma-type which generalizes them both.

   We can get an approximation to the right type to choose by:

   (0) Before beginning, we reserve a patvar for the default
   value of the match, to be used in all the bogus branches.

   (1) perform the case-splits, down to the site of the injection.  At
   each step, we have a term which is the "head" of the next
   case-split.  At the point when we actually reach the end of our
   path, the "head" is the term to return.  We compute its type, and
   then, backwards, make a sigma-type with every free debruijn
   reference in that type.  We can be finer, and first do a S(TRONG)NF
   on the type, so that we get the fewest number of references
   possible.

   (2) This gives us a closed type for the head, which we use for the
   types of all the case-splits.

   (3) Now, we can compute the type of one of T1, T2, and then unify
   it with the type of the last component of the result-type, and this
   will give us the bindings for the other arguments of the tuple.

 *)

(* The algorithm, then is to perform successive case-splits.  We have
   the result-type of the case-split, and also the type of that
   result-type.  We have a "direction" we want to follow, i.e. a
   constructor-number, and in all other "directions", we want to juse
   use the default-value.

   After doing the case-split, we call the afterfun, with the updated
   environment, to produce the term for the desired "direction".

   The assumption is made here that the result-type is not manifestly
   functional, so we can just use the length of the branch-type to
   know how many lambda's to stick in.

 *)

(* [descend_then sigma env head dirn]

   returns the number of products introduced, and the environment
   which is active, in the body of the case-branch given by [dirn],
   along with a continuation, which expects to be fed:

    (1) the value of the body of the branch given by [dirn]
    (2) the default-value

    (3) the type of the default-value, which must also be the type of
        the body of the [dirn] branch

   the continuation then constructs the case-split.
 *)
let descend_then sigma env head dirn =
  let IndType (indf,_) =
    try find_rectype env sigma (get_type_of env sigma head)
    with Not_found -> assert false in
  let ind,_ = dest_ind_family indf in
  let (mib,mip) = lookup_mind_specif env ind in
  let cstr = get_constructors env indf in
  let dirn_nlams = cstr.(dirn-1).cs_nargs in
  let dirn_env = push_rel_context cstr.(dirn-1).cs_args env in
  (dirn_nlams,
   dirn_env,
   (fun dirnval (dfltval,resty) ->
      let arsign,_ = get_arity env indf in
      let depind = build_dependent_inductive env indf in
      let deparsign = (Anonymous,None,depind)::arsign in
      let p =
	it_mkLambda_or_LetIn (lift (mip.mind_nrealargs+1) resty) deparsign in
      let build_branch i =
	let result = if i = dirn then dirnval else dfltval in
	it_mkLambda_or_LetIn_name env result cstr.(i-1).cs_args in
      let brl =
        List.map build_branch
          (interval 1 (Array.length mip.mind_consnames)) in
      let ci = make_default_case_info env RegularStyle ind in
      mkCase (ci, p, head, Array.of_list brl)))
  
(* Now we need to construct the discriminator, given a discriminable
   position.  This boils down to:

   (1) If the position is directly beneath us, then we need to do a
   case-split, with result-type Prop, and stick True and False into
   the branches, as is convenient.

   (2) If the position is not directly beneath us, then we need to
   call descend_then, to descend one step, and then recursively
   construct the discriminator.

 *)

(* [construct_discriminator env dirn headval]
   constructs a case-split on [headval], with the [dirn]-th branch
   giving [True], and all the rest giving False. *)

let construct_discriminator sigma env dirn c sort =
  let IndType(indf,_) =
    try find_rectype env sigma (type_of env sigma c)
    with Not_found ->
       (* one can find Rel(k) in case of dependent constructors 
          like T := c : (A:Set)A->T and a discrimination 
          on (c bool true) = (c bool false)
          CP : changed assert false in a more informative error
       *)
      errorlabstrm "Equality.construct_discriminator"
	(str "Cannot discriminate on inductive constructors with 
		 dependent types") in
  let (ind,_) = dest_ind_family indf in
  let (mib,mip) = lookup_mind_specif env ind in
  let arsign,arsort = get_arity env indf in
  let (true_0,false_0,sort_0) = build_coq_True(),build_coq_False(),Prop Null in
  let depind = build_dependent_inductive env indf in
  let deparsign = (Anonymous,None,depind)::arsign in
  let p = it_mkLambda_or_LetIn (mkSort sort_0) deparsign in
  let cstrs = get_constructors env indf in
  let build_branch i =
    let endpt = if i = dirn then true_0 else false_0 in
    it_mkLambda_or_LetIn endpt cstrs.(i-1).cs_args in
  let brl =
    List.map build_branch(interval 1 (Array.length mip.mind_consnames)) in
  let ci = make_default_case_info env RegularStyle ind in
  mkCase (ci, p, c, Array.of_list brl)
    
let rec build_discriminator sigma env dirn c sort = function
  | [] -> construct_discriminator sigma env dirn c sort
  | ((sp,cnum),argnum)::l ->
      let cty = type_of env sigma c in
      let IndType (indf,_) =
	try find_rectype env sigma cty with Not_found -> assert false in
      let (ind,_) = dest_ind_family indf in
      let (mib,mip) = lookup_mind_specif env ind in
      let nparams = mib.mind_nparams  in
      let (cnum_nlams,cnum_env,kont) = descend_then sigma env c cnum in
      let newc = mkRel(cnum_nlams-(argnum-nparams)) in
      let subval = build_discriminator sigma cnum_env dirn newc sort l  in
      kont subval (build_coq_False (),mkSort (Prop Null))

let gen_absurdity id gl =
  if is_empty_type (clause_type (onHyp id) gl)
  then
    simplest_elim (mkVar id) gl
  else
    errorlabstrm "Equality.gen_absurdity" 
      (str "Not the negation of an equality")

(* Precondition: eq is leibniz equality
 
  returns ((eq_elim t t1 P i t2), absurd_term)
  where  P=[e:t]discriminator 
         absurd_term=False
*)

let discrimination_pf e (t,t1,t2) discriminator lbeq gls =
  let i           = build_coq_I () in
  let absurd_term = build_coq_False () in
  let eq_elim     = lbeq.ind in
  (applist (eq_elim, [t;t1;mkNamedLambda e t discriminator;i;t2]), absurd_term)

exception NotDiscriminable

let discrEq (lbeq,(t,t1,t2)) id gls =
  let sort = pf_type_of gls (pf_concl gls) in 
  let sigma = project gls in
  let env = pf_env gls in
  (match find_positions env sigma t1 t2 with
     | Inr _ ->
	 errorlabstrm "discr" (str" Not a discriminable equality")
     | Inl (cpath, (_,dirn), _) ->
	 let e = pf_get_new_id (id_of_string "ee") gls in
	 let e_env = push_named (e,None,t) env in
	 let discriminator =
	   build_discriminator sigma e_env dirn (mkVar e) sort cpath in
	 let (pf, absurd_term) =
	   discrimination_pf e (t,t1,t2) discriminator lbeq gls 
	 in
	 tclCOMPLETE((tclTHENS (cut_intro absurd_term)
			([onLastHyp gen_absurdity;
			  refine (mkApp (pf, [| mkVar id |]))]))) gls)

let not_found_message id =
  (str "The variable" ++ spc () ++ str (string_of_id id) ++ spc () ++
    str" was not found in the current environment")

let onEquality tac id gls =
  let eqn = pf_whd_betadeltaiota gls (pf_get_hyp_typ gls id) in
  let eq =
    try find_eq_data_decompose eqn
    with PatternMatchingFailure ->
      errorlabstrm "" (pr_id id ++ str": not a primitive equality") 
  in tac eq id gls

let onNegatedEquality tac gls =
  let ccl = pf_concl gls in
  let eq =
    try match kind_of_term (hnf_constr (pf_env gls) (project gls) ccl) with
      | Prod (_,t,u) when is_empty_type u ->
          find_eq_data_decompose (pf_whd_betadeltaiota gls t)
      | _ -> raise PatternMatchingFailure
    with PatternMatchingFailure ->
      errorlabstrm "" (str "Not a negated primitive equality")
  in tclTHEN introf (onLastHyp (tac eq)) gls

let discrSimpleClause = function
  | None -> onNegatedEquality discrEq
  | Some ((_,id),_) -> onEquality discrEq id

let discr = onEquality discrEq

let discrClause = onClauses discrSimpleClause

let discrEverywhere = 
  tclORELSE
    (Tacticals.tryAllClauses discrSimpleClause)
    (fun gls -> 
       errorlabstrm "DiscrEverywhere" (str" No discriminable equalities"))

let discr_tac = function
  | None -> discrEverywhere
  | Some id -> try_intros_until discr id

let discrConcl gls  = discrClause onConcl gls
let discrHyp id gls = discrClause (onHyp id) gls

(* returns the sigma type (sigS, sigT) with the respective
    constructor depending on the sort *)

let find_sigma_data s =
  match s with  
    | Prop Pos  | Type _    -> build_sigma_type ()          (* Set/Type *)
    | Prop Null -> error "find_sigma_data"

(* [make_tuple env sigma (rterm,rty) lind] assumes [lind] is the lesser
   index bound in [rty]

   Then we build the term

     [(existT A P (mkRel lind) rterm)] of type [(sigS A P)]

   where [A] is the type of [mkRel lind] and [P] is [\na:A.rty{1/lind}]
 *)

let make_tuple env sigma (rterm,rty) lind =
  assert (dependent (mkRel lind) rty);
  let {intro = exist_term; typ = sig_term} =
    find_sigma_data (get_sort_of env sigma rty) in
  let a = type_of env sigma (mkRel lind) in
  let (na,_,_) = lookup_rel lind env in
  (* We move [lind] to [1] and lift other rels > [lind] by 1 *)
  let rty = lift (1-lind) (liftn lind (lind+1) rty) in
  (* Now [lind] is [mkRel 1] and we abstract on (na:a) *)
  let p = mkLambda (na, a, rty) in
  (applist(exist_term,[a;p;(mkRel lind);rterm]),
   applist(sig_term,[a;p]))

(* check that the free-references of the type of [c] are contained in
   the free-references of the normal-form of that type.  If the normal
   form of the type contains fewer references, we want to return that
   instead. *)

let minimal_free_rels env sigma (c,cty) =
  let cty_rels = free_rels cty in
  let nf_cty = nf_betadeltaiota env sigma cty in
  let nf_rels = free_rels nf_cty in
  if Intset.subset cty_rels nf_rels then
    (cty,cty_rels)
  else
    (nf_cty,nf_rels)

(* [sig_clausal_form siglen ty]
    
   Will explode [siglen] [sigS,sigT ]'s on [ty] (depending on the 
   type of ty), and return:

   (1) a pattern, with meta-variables in it for various arguments,
       which, when the metavariables are replaced with appropriate
       terms, will have type [ty]

   (2) an integer, which is the last argument - the one which we just
       returned.

   (3) a pattern, for the type of that last meta

   (4) a typing for each patvar

   WARNING: No checking is done to make sure that the 
            sigS(or sigT)'s are actually there.
          - Only homogenious pairs are built i.e. pairs where all the 
   dependencies are of the same sort

   [sig_clausal_form] proceed as follows: the default tuple is
   constructed by taking the tuple-type, exploding the first [tuplen]
   [sigS]'s, and replacing at each step the binder in the
   right-hand-type by a fresh metavariable. In addition, on the way
   back out, we will construct the pattern for the tuple which uses
   these meta-vars.

   This gives us a pattern, which we use to match against the type of
   [dflt]; if that fails, then against the S(TRONG)NF of that type.  If
   both fail, then we just cannot construct our tuple.  If one of
   those succeed, then we can construct our value easily - we just use
   the tuple-pattern.

 *)

let sig_clausal_form env sigma sort_of_ty siglen ty dflt =
  let { intro = exist_term } = find_sigma_data sort_of_ty in 
  let isevars = ref (Evd.create_evar_defs sigma) in
  let rec sigrec_clausal_form siglen p_i =
    if siglen = 0 then
      (* is the default value typable with the expected type *)
      let dflt_typ = type_of env sigma dflt in
      if Evarconv.e_cumul env isevars dflt_typ p_i then
	(* the_conv_x had a side-effect on isevars *)
	dflt
      else
	error "Cannot solve an unification problem"
    else
      let (a,p_i_minus_1) = match whd_beta_stack p_i with
	| (_sigS,[a;p]) -> (a,p)
 	| _ -> anomaly "sig_clausal_form: should be a sigma type" in
      let ev = Evarutil.e_new_evar isevars env a in
      let rty = beta_applist(p_i_minus_1,[ev]) in
      let tuple_tail = sigrec_clausal_form (siglen-1) rty in
      match
	Evd.existential_opt_value (Evd.evars_of !isevars) 
	(destEvar ev)
      with
	| Some w -> applist(exist_term,[a;p_i_minus_1;w;tuple_tail])
	| None -> anomaly "Not enough components to build the dependent tuple"
  in
  let scf = sigrec_clausal_form siglen ty in
  Evarutil.nf_evar (Evd.evars_of !isevars) scf

(* The problem is to build a destructor (a generalization of the
   predecessor) which, when applied to a term made of constructors
   (say [Ci(e1,Cj(e2,Ck(...,term,...),...),...)]), returns a given
   subterm of the term (say [term]).

   Let [typ] be the type of [term]. If [term] has no dependencies in
   the [e1], [e2], etc, then all is simple. If not, then we need to
   encapsulated the dependencies into a dependent tuple in such a way
   that the destructor has not a dependent type and rewriting can then
   be applied. The destructor has the form

   [e]Cases e of
      | ...
      | Ci (x1,x2,...) =>
          Cases x2 of
          | ...
          | Cj (y1,y2,...) =>
              Cases y2 of
              | ...
              | Ck (...,z,...) => z
              | ... end
          | ... end
      | ... end

   and the dependencies is expressed by the fact that [z] has a type
   dependent in the x1, y1, ...

   Assume [z] is typed as follows: env |- z:zty

   If [zty] has no dependencies, this is simple. Otherwise, assume
   [zty] has free (de Bruijn) variables in,...i1 then the role of
   [make_iterated_tuple sigma env (term,typ) (z,zty)] is to build the
   tuple

   [existT [xn]Pn Rel(in) .. (existT [x2]P2 Rel(i2) (existT [x1]P1 Rel(i1) z))]

   where P1 is zty[i1/x1], P2 is {x1 | P1[i2/x2]} etc.

   To do this, we find the free (relative) references of the strong NF
   of [z]'s type, gather them together in left-to-right order
   (i.e. highest-numbered is farthest-left), and construct a big
   iterated pair out of it. This only works when the references are
   all themselves to members of [Set]s, because we use [sigS] to
   construct the tuple.

   Suppose now that our constructed tuple is of length [tuplen]. We
   need also to construct a default value for the other branches of
   the destructor. As default value, we take a tuple of the form

    [existT [xn]Pn ?n (... existT [x2]P2 ?2 (existT [x1]P1 ?1 term))]

   but for this we have to solve the following unification problem:

     typ = zty[i1/?1;...;in/?n]

   This is done by [sig_clausal_form].
 *)

let make_iterated_tuple env sigma dflt (z,zty) =
  let (zty,rels) = minimal_free_rels env sigma (z,zty) in
  let sort_of_zty = get_sort_of env sigma zty in
  let sorted_rels = Sort.list (<) (Intset.elements rels) in
  let (tuple,tuplety) =
    List.fold_left (make_tuple env sigma) (z,zty) sorted_rels 
  in
  assert (closed0 tuplety);
  let n = List.length sorted_rels in
  let dfltval = sig_clausal_form env sigma sort_of_zty n tuplety dflt in
  (tuple,tuplety,dfltval)

let rec build_injrec sigma env dflt c = function
  | [] -> make_iterated_tuple env sigma dflt (c,type_of env sigma c)
  | ((sp,cnum),argnum)::l ->
      let cty = type_of env sigma c in
      let (ity,_) = find_mrectype env sigma cty in
      let (mib,mip) = lookup_mind_specif env ity in
      let nparams = mib.mind_nparams in
      let (cnum_nlams,cnum_env,kont) = descend_then sigma env c cnum in
      let newc = mkRel(cnum_nlams-(argnum-nparams)) in
      let (subval,tuplety,dfltval) =
      	build_injrec sigma cnum_env dflt newc l
      in
      (kont subval (dfltval,tuplety),
       tuplety,dfltval)

let build_injector sigma env dflt c cpath =
  let (injcode,resty,_) = build_injrec sigma env dflt c cpath in
  (injcode,resty)

let try_delta_expand env sigma t =
  let whdt = whd_betadeltaiota env sigma t  in 
  let rec hd_rec c  =
    match kind_of_term c with
      | Construct _ -> whdt
      | App (f,_)  -> hd_rec f
      | Cast (c,_,_) -> hd_rec c
      | _  -> t
  in 
  hd_rec whdt 

(* Given t1=t2 Inj calculates the whd normal forms of t1 and t2 and it 
   expands then only when the whdnf has a constructor of an inductive type
   in hd position, otherwise delta expansion is not done *)

let injEq (eq,(t,t1,t2)) id gls =
  let sigma = project gls in
  let env = pf_env gls in
  match find_positions env sigma t1 t2 with
    | Inl _ ->
	errorlabstrm "Inj"
	  (str (string_of_id id) ++
	    str" is not a projectable equality but a discriminable one")
    | Inr [] ->
	errorlabstrm "Equality.inj" 
	   (str"Nothing to do, it is an equality between convertible terms")
    | Inr posns ->
	let e = pf_get_new_id (id_of_string "e") gls in
	let e_env = push_named (e,None,t) env in
	let injectors =
	  map_succeed
	    (fun (cpath,t1_0,t2_0) ->
	      try 
		let (injbody,resty) =
                  (* take arbitrarily t1_0 as the injector default value *)
		  build_injector sigma e_env t1_0 (mkVar e) cpath in
		let injfun = mkNamedLambda e t injbody in
		let _ = type_of env sigma injfun in (injfun,resty)
	      with e when catchable_exception e ->
		(* may fail because ill-typed or because of a Prop argument *)
		(* error "find_sigma_data" *)
		failwith "caught")
            posns
	in
	if injectors = [] then
	  errorlabstrm "Equality.inj" 
	    (str "Failed to decompose the equality");
	tclMAP 
	  (fun (injfun,resty) ->
	     let pf = applist(eq.congr,
			      [t;resty;injfun;
			       try_delta_expand env sigma t1;
			       try_delta_expand env sigma t2;
			       mkVar id]) 
	     in
	     let ty =
	       try pf_nf gls (pf_type_of gls pf)
               with
		 | UserError("refiner__fail",_) -> 
		     errorlabstrm "InjClause" 
		(str (string_of_id id) ++ str" Not a projectable equality")
	     in ((tclTHENS  (cut  ty) ([tclIDTAC;refine pf]))))
	  injectors
	  gls

let inj = onEquality injEq

let injClause = function
  | None -> onNegatedEquality injEq
  | Some id -> try_intros_until inj id

let injConcl gls  = injClause None gls
let injHyp id gls = injClause (Some id) gls

let decompEqThen ntac (lbeq,(t,t1,t2)) id gls =
  let sort = pf_type_of gls (pf_concl gls) in 
  let sigma = project gls in
  let env = pf_env gls in 
  (match find_positions env sigma t1 t2 with
     | Inl (cpath, (_,dirn), _) ->
	 let e = pf_get_new_id (id_of_string "e") gls in
	 let e_env = push_named (e,None,t) env in
	 let discriminator =
	   build_discriminator sigma e_env dirn (mkVar e) sort cpath in
	 let (pf, absurd_term) =
	   discrimination_pf e (t,t1,t2) discriminator lbeq gls in
	 tclCOMPLETE
	   ((tclTHENS (cut_intro absurd_term)
	       ([onLastHyp gen_absurdity;
		 refine (mkApp (pf, [| mkVar id |]))]))) gls
     | Inr [] -> (* Change: do not fail, simplify clear this trivial hyp *)
         ntac 0 gls
     | Inr posns ->
	 (let e = pf_get_new_id (id_of_string "e") gls in
	  let e_env = push_named (e,None,t) env in
	  let injectors =
	    map_succeed
	      (fun (cpath,t1_0,t2_0) ->
		 let (injbody,resty) =
                  (* take arbitrarily t1_0 as the injector default value *)
		   build_injector sigma e_env t1_0 (mkVar e) cpath in
		 let injfun = mkNamedLambda e t injbody in
		 try 
		   let _ = type_of env sigma injfun in (injfun,resty)
		 with e when catchable_exception e -> failwith "caught")
	      posns 
	  in
	  if injectors = [] then
	    errorlabstrm "Equality.decompEqThen" 
              (str "Discriminate failed to decompose the equality");
	  (tclTHEN
	      (tclMAP (fun (injfun,resty) ->
			 let pf = applist(lbeq.congr,
					  [t;resty;injfun;t1;t2;
					   mkVar id]) in
			 let ty = pf_nf gls (pf_type_of gls pf) in
			 ((tclTHENS (cut ty) 
			     ([tclIDTAC;refine pf]))))
		 (List.rev injectors))
	      (ntac (List.length injectors)))
	  gls))

let dEqThen ntac = function
  | None -> onNegatedEquality (decompEqThen ntac)
  | Some id -> try_intros_until (onEquality (decompEqThen ntac)) id

let dEq = dEqThen (fun x -> tclIDTAC)

let rewrite_msg = function 
  | None -> str "passed term is not a primitive equality" 
  | Some id -> pr_id id ++ str "does not satisfy preconditions "

let swap_equands gls eqn =
  let (lbeq,(t,e1,e2)) = find_eq_data_decompose eqn in 
  applist(lbeq.eq,[t;e2;e1])

let swapEquandsInConcl gls =
  let (lbeq,(t,e1,e2)) = find_eq_data_decompose (pf_concl gls) in
  let sym_equal = lbeq.sym in
  refine (applist(sym_equal,[t;e2;e1;Evarutil.mk_new_meta()])) gls

let swapEquandsInHyp id gls =
  cut_replacing id (swap_equands gls (pf_get_hyp_typ gls id))
    (tclTHEN swapEquandsInConcl) gls

(* find_elim determines which elimination principle is necessary to
   eliminate lbeq on sort_of_gl.
   This is somehow an artificial choice as we could take eq_rect in
   all cases (eq_ind - and eq_rec - are instances of eq_rect) [HH 2/4/06].
*)

let find_elim sort_of_gl lbeq =
  match kind_of_term sort_of_gl  with
    | Sort(Prop Null)  (* Prop *)  -> lbeq.ind
    | _ (* Set/Type *) -> 
        (match lbeq.rect with
           | Some eq_rect -> eq_rect
           | None -> errorlabstrm "find_elim"
		 (str "this type of substitution is not allowed"))

(* Refine from [|- P e2] to [|- P e1] and [|- e1=e2:>t] (body is P (Rel 1)) *)

let bareRevSubstInConcl lbeq body (t,e1,e2) gls =
  (* find substitution scheme *)
  let eq_elim = find_elim (pf_type_of gls (pf_concl gls)) lbeq in
  (* build substitution predicate *)
  let p = lambda_create (pf_env gls) (t,body) in
  (* apply substitution scheme *)
  refine (applist(eq_elim,[t;e1;p;Evarutil.mk_new_meta();
                           e2;Evarutil.mk_new_meta()])) gls

(* [subst_tuple_term dep_pair B]

   Given that dep_pair looks like:

   (existT e1 (existT e2 ... (existT en en+1) ... ))

   and B might contain instances of the ei, we will return the term:

   ([x1:ty(e1)]...[xn:ty(en)]B
    (projS1 (mkRel 1))
    (projS1 (projS2 (mkRel 1)))
    ... etc ...)

   That is, we will abstract out the terms e1...en+1 as usual, but
   will then produce a term in which the abstraction is on a single
   term - the debruijn index [mkRel 1], which will be of the same type
   as dep_pair.

   ALGORITHM for abstraction:

   We have a list of terms, [e1]...[en+1], which we want to abstract
   out of [B].  For each term [ei], going backwards from [n+1], we
   just do a [subst_term], and then do a lambda-abstraction to the
   type of the [ei].

 *)

let decomp_tuple_term env c t = 
  let rec decomprec inner_code ex exty =
    try
      let {proj1=p1; proj2=p2},(a,p,car,cdr) = find_sigma_data_decompose ex in
      let car_code = applist (p1,[a;p;inner_code])
      and cdr_code = applist (p2,[a;p;inner_code]) in
      let cdrtyp = beta_applist (p,[car]) in
      ((car,a),car_code)::(decomprec cdr_code cdr cdrtyp)
    with PatternMatchingFailure ->
      [((ex,exty),inner_code)]
  in
  List.split (decomprec (mkRel 1) c t)

let subst_tuple_term env sigma dep_pair b =
  let typ = get_type_of env sigma dep_pair in
  let e_list,proj_list = decomp_tuple_term env dep_pair typ in
  let abst_B =
    List.fold_right
      (fun (e,t) body -> lambda_create env (t,subst_term e body)) e_list b in
  applist(abst_B,proj_list)
    
(* Comme "replace" mais decompose les egalites dependantes *)

let cutSubstInConcl_RL eqn gls =
  let (lbeq,(t,e1,e2 as eq)) = find_eq_data_decompose eqn in
  let body = pf_apply subst_tuple_term gls e2 (pf_concl gls) in
  assert (dependent (mkRel 1) body);
  bareRevSubstInConcl lbeq body eq gls

(* |- (P e1)
     BY CutSubstInConcl_LR (eq T e1 e2)
     |- (P e2)
     |- (eq T e1 e2)
 *)
let cutSubstInConcl_LR eqn gls =
  (tclTHENS (cutSubstInConcl_RL (swap_equands gls eqn))
     ([tclIDTAC;
       swapEquandsInConcl])) gls

let cutSubstInConcl l2r =if l2r then cutSubstInConcl_LR else cutSubstInConcl_RL

let cutSubstInHyp_LR eqn id gls =
  let (lbeq,(t,e1,e2 as eq)) = find_eq_data_decompose eqn in 
  let body = pf_apply subst_tuple_term gls e1 (pf_get_hyp_typ gls id) in
  assert (dependent (mkRel 1) body);
  cut_replacing id (subst1 e2 body)
    (tclTHENFIRST (bareRevSubstInConcl lbeq body eq)) gls

let cutSubstInHyp_RL eqn id gls =
  (tclTHENS (cutSubstInHyp_LR (swap_equands gls eqn) id)
     ([tclIDTAC;
       swapEquandsInConcl])) gls

let cutSubstInHyp l2r = if l2r then cutSubstInHyp_LR else cutSubstInHyp_RL

let try_rewrite tac gls =
  try 
    tac gls
  with 
    | PatternMatchingFailure ->
	errorlabstrm "try_rewrite" (str "Not a primitive equality here")
    | e when catchable_exception e -> 
	errorlabstrm "try_rewrite"
          (str "Cannot find a well-typed generalization of the goal that" ++
             str " makes the proof progress")

let cutSubstClause l2r eqn cls gls =
  match cls with
    | None ->    cutSubstInConcl l2r eqn gls
    | Some id -> cutSubstInHyp l2r eqn id gls

let cutRewriteClause l2r eqn cls = try_rewrite (cutSubstClause l2r eqn cls)
let cutRewriteInHyp l2r eqn id = cutRewriteClause l2r eqn (Some id)
let cutRewriteInConcl l2r eqn = cutRewriteClause l2r eqn None

let substClause l2r c cls gls =
  let eq = pf_type_of gls c in
  tclTHENS (cutSubstClause l2r eq cls) [tclIDTAC; exact_no_check c] gls

let rewriteClause l2r c cls = try_rewrite (substClause l2r c cls)
let rewriteInHyp l2r c id = rewriteClause l2r c (Some id)
let rewriteInConcl l2r c = rewriteClause l2r c None

(* Renaming scheme correspondence new name (old name)

      give equality                    give proof of equality

    / cutSubstClause (subst)          substClause (HypSubst on hyp)
raw | cutSubstInHyp (substInHyp)      substInHyp (none)
    \ cutSubstInConcl (substInConcl)  substInConcl (none) 

    / cutRewriteClause (none)         rewriteClause (none)
user| cutRewriteInHyp (substHyp)      rewriteInHyp (none)
    \ cutRewriteInConcl (substConcl)  rewriteInConcl (substHypInConcl on hyp)

raw = raise typing error or PatternMatchingFailure
user = raise user error specific to rewrite
*)

(* Summary of obsolete forms
let substInConcl = cutSubstInConcl
let substInHyp = cutSubstInHyp
let hypSubst l2r id = substClause l2r (mkVar id)
let hypSubst_LR = hypSubst true
let hypSubst_RL = hypSubst false
let substHypInConcl l2r id = rewriteInConcl l2r (mkVar id)
let substConcl = cutRewriteInConcl
let substHyp = cutRewriteInHyp
*)

(**********************************************************************)
(* Substitutions tactics (JCF) *)

let unfold_body x gl =
  let hyps = pf_hyps gl in
  let xval =
    match Sign.lookup_named x hyps with
        (_,Some xval,_) -> xval
      | _ -> errorlabstrm "unfold_body"
          (pr_id x ++ str" is not a defined hypothesis") in
  let aft = afterHyp x gl in
  let hl = List.fold_right (fun (y,yval,_) cl -> (([],y),InHyp) :: cl) aft [] in
  let xvar = mkVar x in
  let rfun _ _ c = replace_term xvar xval c in
  tclTHENLIST
    [tclMAP (fun h -> reduct_in_hyp rfun h) hl;
     reduct_in_concl (rfun,DEFAULTcast)] gl




exception FoundHyp of (identifier * constr * bool)

(* tests whether hyp [c] is [x = t] or [t = x], [x] not occuring in [t] *)
let is_eq_x x (id,_,c) =
  try
    let (_,lhs,rhs) = snd (find_eq_data_decompose c) in
    if (x = lhs) && not (occur_term x rhs) then raise (FoundHyp (id,rhs,true));
    if (x = rhs) && not (occur_term x lhs) then raise (FoundHyp (id,lhs,false))
  with PatternMatchingFailure ->
    ()

let subst_one x gl = 
  let hyps = pf_hyps gl in
  let (_,xval,_) = pf_get_hyp gl x in
  (* If x has a body, simply replace x with body and clear x *)
  if xval <> None then tclTHEN (unfold_body x) (clear [x]) gl else
  (* x is a variable: *)
  let varx = mkVar x in
  (* Find a non-recursive definition for x *)
  let (hyp,rhs,dir) = 
    try
      let test hyp _ = is_eq_x varx hyp in
      Sign.fold_named_context test ~init:() hyps;
      errorlabstrm "Subst"
        (str "cannot find any non-recursive equality over " ++ pr_id x)
    with FoundHyp res -> res
  in
  (* The set of hypotheses using x *)
  let depdecls = 
    let test (id,_,c as dcl) = 
      if id <> hyp && occur_var_in_decl (pf_env gl) x dcl then dcl
      else failwith "caught" in
    List.rev (map_succeed test hyps) in
  let dephyps = List.map (fun (id,_,_) -> id) depdecls in
  (* Decides if x appears in conclusion *)
  let depconcl = occur_var (pf_env gl) x (pf_concl gl) in
  (* The set of non-defined hypothesis: they must be abstracted,
     rewritten and reintroduced *)
  let abshyps =
    map_succeed
      (fun (id,v,_) -> if v=None then mkVar id else failwith "caught")
      depdecls in
  (* a tactic that either introduce an abstracted and rewritten hyp,
     or introduce a definition where x was replaced *)
  let introtac = function
      (id,None,_) -> intro_using id
    | (id,Some hval,htyp) ->
        letin_tac true (Name id)
	  (mkCast(replace_term varx rhs hval,DEFAULTcast,
	          replace_term varx rhs htyp)) nowhere
  in
  let need_rewrite = dephyps <> [] || depconcl in
  tclTHENLIST 
    ((if need_rewrite then
      [generalize abshyps;
       (if dir then rewriteLR else rewriteRL) (mkVar hyp);
       thin dephyps;
       tclMAP introtac depdecls]
    else
      [thin dephyps;
       tclMAP introtac depdecls]) @
     [tclTRY (clear [x;hyp])]) gl

let subst = tclMAP subst_one

let subst_all gl =
  let test (_,c) =
    try
      let (_,x,y) = snd (find_eq_data_decompose c) in
      (* J.F.: added to prevent failure on goal containing x=x as an hyp *)
      if eq_constr x y then failwith "caught";
      match kind_of_term x with Var x -> x | _ -> 
      match kind_of_term y with Var y -> y | _ -> failwith "caught"
    with PatternMatchingFailure -> failwith "caught"
  in
  let ids = map_succeed test (pf_hyps_types gl) in
  let ids = list_uniquize ids in
  subst ids gl

(* Rewrite the first assumption for which the condition faildir does not fail 
   and gives the direction of the rewrite *)

let rewrite_assumption_cond faildir gl =
   let rec arec = function
      | [] -> error "No such assumption"
      | (id,_,t)::rest -> 
	  (try let dir = faildir t gl in 
	       general_rewrite dir (mkVar id) gl
	   with Failure _ | UserError _  ->  arec rest)
   in arec (pf_hyps gl)


let rewrite_assumption_cond_in faildir hyp gl =
   let rec arec = function
      | [] -> error "No such assumption"
      | (id,_,t)::rest -> 
	  (try let dir = faildir t gl in 
	       general_rewrite_in dir hyp (mkVar id) gl
	   with Failure _ | UserError _ ->  arec rest)
   in arec (pf_hyps gl)

let cond_eq_term_left c t gl =
  try
    let (_,x,_) = snd (find_eq_data_decompose t) in
    if pf_conv_x gl c x then true else failwith "not convertible"
  with PatternMatchingFailure -> failwith "not an equality"

let cond_eq_term_right c t gl = 
  try
    let (_,_,x) = snd (find_eq_data_decompose t) in
    if pf_conv_x gl c x then false else failwith "not convertible"
  with PatternMatchingFailure -> failwith "not an equality"

let cond_eq_term c t gl = 
  try
    let (_,x,y) = snd (find_eq_data_decompose t) in
    if pf_conv_x gl c x then true 
    else if pf_conv_x gl c y then false
    else failwith "not convertible"
  with PatternMatchingFailure -> failwith "not an equality"

let replace_term_left t = rewrite_assumption_cond (cond_eq_term_left t)

let replace_term_right t = rewrite_assumption_cond (cond_eq_term_right t)
   
let replace_term t = rewrite_assumption_cond (cond_eq_term t)
   
let replace_term_in_left t = rewrite_assumption_cond_in (cond_eq_term_left t)

let replace_term_in_right t = rewrite_assumption_cond_in (cond_eq_term_right t)
   
let replace_term_in t = rewrite_assumption_cond_in (cond_eq_term t)

let _ = Setoid_replace.register_replace replace
let _ = Setoid_replace.register_general_rewrite general_rewrite