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|
(* $Id$ *)
open Pp
open Util
open Names
open Generic
open Term
open Inductive
open Environ
open Reduction
open Typing
let whd_betadeltaiota_empty env = whd_betadeltaiota env Evd.empty
let make_lambda_string s t c = DOP2(Lambda,t,DLAM(Name(id_of_string s),c))
let make_prod_string s t c = DOP2(Prod,t,DLAM(Name(id_of_string s),c))
let lift_context n l =
let k = List.length l in
list_map_i (fun i (name,c) -> (name,liftn n (k-i) c)) 0 l
(*******************************************)
(* Building curryfied elimination *)
(*******************************************)
(*********************************************)
(* lc is the list of the constructors of ind *)
(*********************************************)
let mis_make_case_com depopt sigma mispec kinds =
let sp = mispec.mis_sp in
let tyi = mispec.mis_tyi in
let cl = mispec.mis_args in
let nparams = mis_nparams mispec in
let mip = mispec.mis_mip in
let mind = DOPN(MutInd(sp,tyi),cl) in
let kd = mis_kd mispec in
let kn = mis_kn mispec in
let t = mis_arity mispec in
let (lc,lct) = mis_type_mconstructs mispec in
let lnames,sort = splay_prod sigma t in
let nconstr = Array.length lc in
let dep = match depopt with
| None -> (sort<>DOP0(Sort(Prop Null)))
| Some d -> d
in
let _ =
if dep then begin
if not (List.exists (sort_cmp CONV_X kinds) kd) then
let pm = pTERM mind in
let ps = pTERM (DOP0(Sort kinds)) in
errorlabstrm "Case analysis"
[< 'sTR "Dependent case analysis on sort: "; ps; 'fNL;
'sTR "is not allowed for inductive definition: "; pm >]
end else if not (List.exists (sort_cmp CONV_X kinds) kn) then
let pm = pTERM mind in
let ps = pTERM (DOP0(Sort kinds)) in
errorlabstrm "Case analysis"
[< 'sTR "Non Dependent case analysis on sort: "; ps; 'fNL;
'sTR "is not allowed for inductive definition: "; pm >]
in
let lnamesar,lnamespar = chop_list (List.length lnames - nparams) lnames in
let lgar = List.length lnamesar in
let ar = hnf_prod_appvect sigma "make_case_dep" t (rel_vect 0 nparams) in
let typP =
if dep then
make_arity_dep (DOP0(Sort kinds)) ar
(appvect (mind,rel_vect 0 nparams))
else
make_arity_nodep (DOP0(Sort kinds)) ar
in
let rec add_branch k =
if k = nconstr then
it_lambda_name
(lambda_create
(appvect (mind,
(Array.append (rel_vect (nconstr+lgar+1) nparams)
(rel_vect 0 lgar))),
mkMutCaseA (ci_of_mind mind)
(Rel (nconstr+lgar+2))
(Rel 1)
(* (appvect (mind,
(Array.append (rel_vect (nconstr+lgar+2) nparams)
(rel_vect 1 lgar)))) *)
(rel_vect (lgar+1) nconstr)))
(lift_context (nconstr+1) lnamesar)
else
make_lambda_string "f"
(if dep then
type_one_branch_dep
(sigma,nparams,(rel_list (k+1) nparams),Rel (k+1)) lc.(k) lct.(k)
else
type_one_branch_nodep
(sigma,nparams,(rel_list (k+1) nparams),Rel (k+1)) lct.(k))
(add_branch (k+1))
in
it_lambda_name (make_lambda_string "P" typP (add_branch 0)) lnamespar
let make_case_com depopt sigma mind kinds =
let ity = mrectype_spec sigma mind in
let (sp,tyi,cl) = destMutInd ity in
let mispec = mind_specif_of_mind ity in
mis_make_case_com depopt sigma mispec kinds
let make_case_dep sigma = make_case_com (Some true) sigma
let make_case_nodep sigma = make_case_com (Some false) sigma
let make_case_gen sigma = make_case_com None sigma
(* check if the type depends recursively on one of the inductive scheme *)
(* Building the recursive elimination *)
(***********************************************************************
* t is the type of the constructor co and recargs is the information on
* the recursive calls.
* build the type of the corresponding branch of the recurrence principle
* assuming f has this type, branch_rec gives also the term
* [x1]..[xk](f xi (F xi) ...) to be put in the corresponding branch of
* the case operation
* FPvect gives for each inductive definition if we want an elimination
* on it with which predicate and which recursive function.
************************************************************************)
let simple_prod (n,t,c) = DOP2(Prod,t,DLAM(n,c))
let make_prod_dep dep = if dep then prod_name else simple_prod
let type_rec_branch dep (sigma,vargs,depPvect,decP) co t recargs =
let make_prod = make_prod_dep dep in
let nparams = Array.length vargs in
let st = hnf_prod_appvect sigma "type_rec_branch" t vargs in
let process_pos depK pk =
let rec prec i p =
match whd_betadeltaiota_stack sigma p [] with
| (DOP2(Prod,t,DLAM(n,c))),[] -> make_prod (n,t,prec (i+1) c)
| (DOPN(MutInd _,_),largs) ->
let (_,realargs) = chop_list nparams largs in
let base = applist (lift i pk,realargs) in
if depK then
mkAppList base [appvect (Rel (i+1),rel_vect 0 i)]
else
base
| _ -> assert false
in
prec 0
in
let rec process_constr i c recargs co =
match whd_betadeltaiota_stack sigma c [] with
| (DOP2(Prod,t,DLAM(n,c_0)),[]) ->
let (optionpos,rest) =
match recargs with
| [] -> None,[]
| (Param(_)::rest) -> (None,rest)
| (Norec::rest) -> (None,rest)
| (Imbr _::rest) ->
warning "Ignoring recursive call"; (None,rest)
|(Mrec j::rest) -> (depPvect.(j),rest)
in
(match optionpos with
| None ->
make_prod (n,t,process_constr (i+1) c_0 rest
(mkAppList (lift 1 co) [Rel 1]))
| Some(dep',p) ->
let nP = lift (i+1+decP) p in
let t_0 = process_pos dep' nP (lift 1 t) in
make_prod_dep (dep or dep')
(n,t,mkArrow t_0 (process_constr (i+2) (lift 1 c_0) rest
(mkAppList (lift 2 co) [Rel 2]))))
| (DOPN(MutInd(_,tyi),_),largs) ->
let nP = match depPvect.(tyi) with
| Some(_,p) -> lift (i+decP) p
| _ -> assert false in
let (_,realargs) = chop_list nparams largs in
let base = applist (nP,realargs) in
if dep then mkAppList base [co] else base
| _ -> assert false
in
process_constr 0 st recargs (appvect(co,vargs))
let rec_branch_arg (sigma,vargs,fvect,decF) f t recargs =
let nparams = Array.length vargs in
let st = hnf_prod_appvect sigma "type_rec_branch" t vargs in
let process_pos fk =
let rec prec i p =
(match whd_betadeltaiota_stack sigma p [] with
| (DOP2(Prod,t,DLAM(n,c))),[] -> lambda_name (n,t,prec (i+1) c)
| (DOPN(MutInd _,_),largs) ->
let (_,realargs) = chop_list nparams largs
and arg = appvect (Rel (i+1),rel_vect 0 i) in
applist(lift i fk,realargs@[arg])
| _ -> assert false)
in
prec 0
in
let rec process_constr i c f recargs =
match whd_betadeltaiota_stack sigma c [] with
| (DOP2(Prod,t,DLAM(n,c_0)),[]) ->
let (optionpos,rest) =
match recargs with
| [] -> None,[]
| (Param(i)::rest) -> None,rest
| (Norec::rest) -> None,rest
| (Imbr _::rest) -> None,rest
| (Mrec i::rest) -> fvect.(i),rest
in
(match optionpos with
| None ->
lambda_name
(n,t,process_constr (i+1) c_0
(applist(whd_beta_stack (lift 1 f) [(Rel 1)])) rest)
| Some(_,f_0) ->
let nF = lift (i+1+decF) f_0 in
let arg = process_pos nF (lift 1 t) in
lambda_name
(n,t,process_constr (i+1) c_0
(applist(whd_beta_stack (lift 1 f) [(Rel 1); arg]))
rest))
| (DOPN(MutInd(_,tyi),_),largs) -> f
| _ -> assert false
in
process_constr 0 st f recargs
let mis_make_indrec sigma listdepkind mispec =
let nparams = mis_nparams mispec in
let recargsvec = mis_recargs mispec in
let ntypes = mis_ntypes mispec in
let mind_arity = mis_arity mispec in
let (lnames, ckind) = splay_prod sigma mind_arity in
let kind = destSort ckind in
let lnamespar = lastn nparams lnames in
let listdepkind =
if listdepkind = [] then
let dep = kind <> Prop Null in [(mispec,dep,kind)]
else
listdepkind
in
let nrec = List.length listdepkind in
let depPvec = Array.create ntypes (None : (bool * constr) option) in
let _ =
let rec
assign k = function
| [] -> ()
| (mispeci,dep,_)::rest ->
(Array.set depPvec mispeci.tyi (Some(dep,Rel k));
assign (k-1) rest)
in
assign nrec listdepkind
in
let make_one_rec p =
let makefix nbconstruct =
let rec mrec i ln ltyp ldef = function
| (mispeci,dep,_)::rest ->
let tyi = mispeci.tyi in
let mind = DOPN(MutInd (mispeci.sp,tyi),mispeci.args) in
let (_,lct) = mis_type_mconstructs mispeci in
let nctyi = Array.length lct in (* nb constructeurs du type *)
let realar = hnf_prod_appvect sigma "make_branch"
(mis_arity mispeci)
(rel_vect (nrec+nbconstruct) nparams) in
(* arity in the contexte P1..Prec f1..f_nbconstruct *)
let lnames,_ = splay_prod sigma realar in
let nar = List.length lnames in
let decf = nar+nrec+nbconstruct+nrec in
let dect = nar+nrec+nbconstruct in
let vecfi = rel_vect (dect+1-i-nctyi) nctyi in
let branches =
map3_vect
(rec_branch_arg (sigma,rel_vect (decf+1) nparams,
depPvec,nar+1))
vecfi lct recargsvec.(tyi) in
let j = (match depPvec.(tyi) with
| Some (_,Rel j) -> j
| _ -> assert false) in
let deftyi =
it_lambda_name
(lambda_create
(appvect (mind,(Array.append (rel_vect decf nparams)
(rel_vect 0 nar))),
mkMutCaseA (ci_of_mind mind)
(Rel (decf-nrec+j+1)) (Rel 1) branches))
(lift_context nrec lnames)
in
let typtyi =
it_prod_name
(prod_create
(appvect (mind,(Array.append (rel_vect dect nparams)
(rel_vect 0 nar))),
(if dep then
appvect (Rel (dect-nrec+j+1), rel_vect 0 (nar+1))
else
appvect (Rel (dect-nrec+j+1),rel_vect 1 nar))))
lnames
in
mrec (i+nctyi) (nar::ln) (typtyi::ltyp) (deftyi::ldef) rest
| [] ->
let fixn = Array.of_list (List.rev ln) in
let fixtyi = Array.of_list (List.rev ltyp) in
let fixdef = Array.of_list (List.rev ldef) in
let makefixdef =
put_DLAMSV
(tabulate_list (fun _ -> Name(id_of_string "F")) nrec) fixdef
in
let fixspec = Array.append fixtyi [|makefixdef|] in
DOPN(Fix(fixn,p),fixspec)
in
mrec 0 [] [] []
in
let rec make_branch i = function
| (mispeci,dep,_)::rest ->
let tyi = mispeci.tyi in
let (lc,lct) = mis_type_mconstructs mispeci in
let rec onerec j =
if j = Array.length lc then
make_branch (i+j) rest
else
let co = lc.(j) in
let t = lct.(j) in
let recarg = recargsvec.(tyi).(j) in
let vargs = rel_vect (nrec+i+j) nparams in
let p_0 =
type_rec_branch dep (sigma,vargs,depPvec,i+j) co t recarg
in
DOP2(Lambda,p_0,DLAM(Name (id_of_string "f"),onerec (j+1)))
in onerec 0
| [] ->
makefix i listdepkind
in
let rec put_arity i = function
| ((mispeci,dep,kinds)::rest) ->
let mind = DOPN(MutInd (mispeci.sp,mispeci.tyi),mispeci.args) in
let arity = mis_arity mispeci in
let ar =
hnf_prod_appvect sigma "put_arity" arity (rel_vect i nparams)
in
let typP =
if dep then
make_arity_dep (DOP0(Sort kinds)) ar
(appvect (mind,rel_vect i nparams))
else
make_arity_nodep (DOP0(Sort kinds)) ar
in
DOP2(Lambda,typP,DLAM(Name(id_of_string "P"),put_arity (i+1) rest))
| [] ->
make_branch 0 listdepkind
in
let (mispeci,dep,kind) = List.nth listdepkind p in
if is_recursive (List.map (fun (mispec,_,_) -> mispec.tyi) listdepkind)
recargsvec.(mispeci.tyi) then
it_lambda_name (put_arity 0 listdepkind) lnamespar
else
mis_make_case_com (Some dep) sigma mispeci kind
in
tabulate_vect make_one_rec nrec
let make_indrec sigma listdepkind mind =
let ity = minductype_spec sigma mind in
let (sp,tyi,largs) = destMutInd ity in
let mispec = mind_specif_of_mind ity in
mis_make_indrec sigma listdepkind mispec
let change_sort_arity sort =
let rec drec = function
| (DOP2(Cast,c,t)) -> drec c
| (DOP2(Prod,t,DLAM(n,c))) -> DOP2(Prod,t,DLAM(n,drec c))
| (DOP0(Sort(_))) -> DOP0(Sort(sort))
| _ -> assert false
in
drec
let instanciate_indrec_scheme sort =
let rec drec npar elim =
let (n,t,c) = destLambda (strip_outer_cast elim) in
if npar = 0 then
mkLambda n (change_sort_arity sort t) c
else
mkLambda n t (drec (npar-1) c)
in
drec
let check_arities listdepkind =
List.iter
(function (mispeci,dep,kinds) ->
let mip = mispeci.mip in
if dep then
let kd = mis_kd mispeci in
if List.exists (sort_cmp CONV_X kinds) kd then
()
else
errorlabstrm "Bad Induction"
[<'sTR "Dependent induction for type ";
print_id mip.mINDTYPENAME;
'sTR " and sort "; pTERM (DOP0(Sort kinds));
'sTR "is not allowed">]
else
let kn = mis_kn mispeci in
if List.exists (sort_cmp CONV_X kinds) kn then
()
else
errorlabstrm "Bad Induction"
[<'sTR "Non dependent induction for type ";
print_id mip.mINDTYPENAME;
'sTR " and sort "; pTERM (DOP0(Sort kinds));
'sTR "is not allowed">])
listdepkind
let build_indrec sigma = function
| ((mind,dep,s)::lrecspec) ->
let redind = minductype_spec sigma mind in
let (sp,tyi,_) = destMutInd redind in
let listdepkind =
(mind_specif_of_mind redind, dep,s)::
(List.map (function (mind',dep',s') ->
let redind' = minductype_spec sigma mind' in
let (sp',_,_) = destMutInd redind' in
if sp=sp' then
(mind_specif_of_mind redind',dep',s')
else
error
"Induction schemes concern mutually inductive types")
lrecspec)
in
let _ = check_arities listdepkind in
make_indrec sigma listdepkind mind
| _ -> assert false
(* In order to interpret the Match operator *)
let type_mutind_rec env sigma ct pt p c =
let (mI,largs as mind) = find_minductype sigma ct in
let mispec = mind_specif_of_mind mI in
let recargs = mis_recarg mispec in
if is_recursive [mispec.tyi] recargs then
let dep = find_case_dep_mis env sigma mispec (c,p) mind pt in
let ntypes = mis_nconstr mispec
and tyi = mispec.tyi
and nparams = mis_nparams mispec in
let depPvec = Array.create ntypes (None : (bool * constr) option) in
let _ = Array.set depPvec mispec.tyi (Some(dep,p)) in
let (pargs,realargs) = chop_list nparams largs in
let vargs = Array.of_list pargs in
let (constrvec,typeconstrvec) = mis_type_mconstructs mispec in
let lft = map3_vect (type_rec_branch dep (sigma,vargs,depPvec,0))
constrvec typeconstrvec recargs in
(mI, lft,
if dep then applist(p,realargs@[c])
else applist(p,realargs) )
else
type_case_branches env sigma ct pt p c
let is_mutind sigma ct =
try let _ = find_minductype sigma ct in true with Induc -> false
let type_rec_branches recursive sigma env ct pt p c =
match whd_betadeltaiota_stack sigma ct [] with
| (DOPN(MutInd _,_),_) ->
if recursive then
type_mutind_rec env sigma ct pt p c
else
type_case_branches env sigma ct pt p c
| _ -> error"Elimination on a non-inductive type 1"
(* Awful special reduction function which skips abstraction on Xtra in order to
be safe for Program ... *)
let stacklamxtra recfun =
let rec lamrec sigma p_0 p_1 = match p_0,p_1 with
| (stack, (DOP2(Lambda,DOP1(XTRA("COMMENT",[]),_),DLAM(_,c)) as t)) ->
recfun stack (substl sigma t)
| ((h::t), (DOP2(Lambda,_,DLAM(_,c)))) -> lamrec (h::sigma) t c
| (stack, t) -> recfun stack (substl sigma t)
in
lamrec
let rec whrec x stack =
match x with
| DOP2(Lambda,DOP1(XTRA("COMMENT",[]),c),DLAM(name,t)) ->
let t' = applist (whrec t (List.map (lift 1) stack)) in
DOP2(Lambda,DOP1(XTRA("COMMENT",[]),c),DLAM(name,t')),[]
| DOP2(Lambda,c1,DLAM(name,c2)) ->
(match stack with
| [] -> (DOP2(Lambda,c1,DLAM(name,whd_betaxtra c2)),[])
| a1::rest -> stacklamxtra (fun l x -> whrec x l) [a1] rest c2)
| DOPN(AppL,cl) -> whrec (hd_vect cl) (app_tl_vect cl stack)
| DOP2(Cast,c,_) -> whrec c stack
| x -> x,stack
and whd_betaxtra x = applist(whrec x [])
let transform_rec env sigma cl (ct,pt) =
let (mI,largs as mind) = find_minductype sigma ct in
let p = cl.(0)
and c = cl.(1)
and lf = Array.sub cl 2 ((Array.length cl) - 2) in
let mispec = mind_specif_of_mind mI in
let recargs = mis_recarg mispec in
let expn = Array.length recargs in
if Array.length lf <> expn then error_number_branches CCI env c ct expn;
if is_recursive [mispec.tyi] recargs then
let dep = find_case_dep_mis env sigma mispec (c,p) mind pt in
let ntypes = mis_nconstr mispec
and tyi = mispec.tyi
and nparams = mis_nparams mispec in
let depFvec = Array.create ntypes (None : (bool * constr) option) in
let _ = Array.set depFvec mispec.tyi (Some(dep,Rel 1)) in
let (pargs,realargs) = chop_list nparams largs in
let vargs = Array.of_list pargs in
let (_,typeconstrvec) = mis_type_mconstructs mispec in
(* build now the fixpoint *)
let realar =
hnf_prod_appvect sigma "make_branch" (mis_arity mispec) vargs in
let lnames,_ = splay_prod sigma realar in
let nar = List.length lnames in
let branches =
map3_vect
(fun f t reca ->
whd_betaxtra
(rec_branch_arg
(sigma,(Array.map (lift (nar+2)) vargs),depFvec,nar+1)
f t reca))
(Array.map (lift (nar+2)) lf) typeconstrvec recargs
in
let deffix =
it_lambda_name
(lambda_create
(appvect (mI,Array.append (Array.map (lift (nar+1)) vargs)
(rel_vect 0 nar)),
mkMutCaseA (ci_of_mind mI)
(lift (nar+2) p) (Rel 1) branches))
(lift_context 1 lnames)
in
if noccurn 1 deffix then
whd_beta (applist (pop deffix,realargs@[c]))
else
let typPfix =
it_prod_name
(prod_create (appvect (mI,(Array.append
(Array.map (lift nar) vargs)
(rel_vect 0 nar))),
(if dep then applist (whd_beta_stack (lift (nar+1) p)
(rel_list 0 (nar+1)))
else applist (whd_beta_stack (lift (nar+1) p)
(rel_list 1 nar)))))
lnames
in
let fix = DOPN(Fix([|nar|],0),
[|typPfix;
DLAMV(Name(id_of_string "F"),[|deffix|])|])
in
applist (fix,realargs@[c])
else
mkMutCaseA (ci_of_mind mI) p c lf
(*** Building ML like case expressions without types ***)
let concl_n sigma =
let rec decrec m c = if m = 0 then c else
match whd_betadeltaiota sigma c with
| DOP2(Prod,_,DLAM(n,c_0)) -> decrec (m-1) c_0
| _ -> failwith "Typing.concl_n"
in
decrec
let count_rec_arg j =
let rec crec i = function
| [] -> i
| (Mrec k::l) -> crec (if k=j then (i+1) else i) l
| (_::l) -> crec i l
in
crec 0
let norec_branch_scheme sigma typc =
let rec crec typc = match whd_betadeltaiota sigma typc with
| DOP2(Prod,c,DLAM(name,t)) -> DOP2(Prod,c,DLAM(name,crec t))
| _ -> mkExistential
in
crec typc
let rec_branch_scheme sigma j typc recargs =
let rec crec (typc,recargs) =
match whd_betadeltaiota sigma typc, recargs with
| (DOP2(Prod,c,DLAM(name,t)),(ra::reca)) ->
DOP2(Prod,c,
match ra with
| Mrec k ->
if k=j then
DLAM(name,mkArrow mkExistential
(crec (lift 1 t,reca)))
else
DLAM(name,crec (t,reca))
| _ -> DLAM(name,crec (t,reca)))
| (_,_) -> mkExistential
in
crec (typc,recargs)
let branch_scheme sigma isrec i mind =
let typc = type_inst_construct sigma i mind in
if isrec then
let (mI,_) = find_mrectype sigma mind in
let (_,j,_) = destMutInd mI in
let mispec = mind_specif_of_mind mI in
let recarg = (mis_recarg mispec).(i-1) in
rec_branch_scheme sigma j typc recarg
else
norec_branch_scheme sigma typc
(* if arity of mispec is (p_bar:P_bar)(a_bar:A_bar)s where p_bar are the
* K parameters. Then then build_notdep builds the predicate
* [a_bar:A'_bar](lift k pred)
* where A'_bar = A_bar[p_bar <- globargs] *)
let build_notdep_pred mispec nparams globargs pred =
let arity = mis_arity mispec in
let lamarity = to_lambda nparams arity in
let inst_arity = whd_beta (appvect (lamarity,Array.of_list globargs)) in
let k = nb_prod inst_arity in
let env,_,npredlist = push_and_liftl k [] inst_arity [insert_lifted pred] in
let npred = (match npredlist with [npred] -> npred
| _ -> anomaly "push_and_lift should not behave this way") in
let _,finalpred,_ = lam_and_popl k env (extract_lifted npred) []
in
finalpred
let pred_case_ml_fail sigma isrec ct (i,ft) =
try
let (mI,largs) = find_mrectype sigma ct in
let (_,j,_) = destMutInd mI in
let mispec = mind_specif_of_mind mI in
let nparams = mis_nparams mispec in
let (globargs,la) = chop_list nparams largs in
let pred =
let recargs = (mis_recarg mispec) in
assert (Array.length recargs <> 0);
let recargi = recargs.(i-1) in
let nbrec = if isrec then count_rec_arg j recargi else 0 in
let nb_arg = List.length (recargs.(i-1)) + nbrec in
let pred = concl_n sigma nb_arg ft in
if noccur_bet 1 nb_arg pred then
lift (-nb_arg) pred
else
failwith "Dependent"
in
if la = [] then
pred
else (* we try with [_:T1]..[_:Tn](lift n pred) *)
build_notdep_pred mispec nparams globargs pred
with Induc ->
failwith "Inductive"
let pred_case_ml env sigma isrec (c,ct) lf (i,ft) =
try
pred_case_ml_fail sigma isrec ct (i,ft)
with Failure mes ->
error_ml_case mes env c ct lf.(i-1) ft
(* similar to pred_case_ml but does not expect the list lf of braches *)
let pred_case_ml_onebranch env sigma isrec (c,ct) (i,f,ft) =
try
pred_case_ml_fail sigma isrec ct (i,ft)
with Failure mes ->
error_ml_case mes env c ct f ft
let make_case_ml isrec pred c ci lf =
if isrec then
DOPN(XTRA("REC",[]),Array.append [|pred;c|] lf)
else
mkMutCaseA ci pred c lf
|