(************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) (* match !impossible_default_case with | Some (id,type_of_id) -> make_judge id type_of_id | None -> (* In case the constants id/ID are not defined *) make_judge (mkLambda (na1,mkProp,mkLambda(na2,mkRel 1,mkRel 1))) (mkProd (na1,mkProp,mkArrow (mkRel 1) (mkRel 2))) (**********************************************************************) module type S = sig val compile_cases : loc -> case_style -> (type_constraint -> env -> evar_defs ref -> rawconstr -> unsafe_judgment) * evar_defs ref -> type_constraint -> env -> rawconstr option * tomatch_tuples * cases_clauses -> unsafe_judgment end let rec list_try_compile f = function | [a] -> f a | [] -> anomaly "try_find_f" | h::t -> try f h with UserError _ | TypeError _ | PretypeError _ | Stdpp.Exc_located (_,(UserError _ | TypeError _ | PretypeError _)) -> list_try_compile f t let force_name = let nx = Name (id_of_string "x") in function Anonymous -> nx | na -> na (************************************************************************) (* Pattern-matching compilation (Cases) *) (************************************************************************) (************************************************************************) (* Configuration, errors and warnings *) open Pp let msg_may_need_inversion () = strbrk "Found a matching with no clauses on a term unknown to have an empty inductive type." (* Utils *) let make_anonymous_patvars n = list_make n (PatVar (dummy_loc,Anonymous)) (* Environment management *) let push_rels vars env = List.fold_right push_rel vars env let push_rel_defs = List.fold_right (fun (x,d,t) e -> push_rel (x,Some d,t) e) (* We have x1:t1...xn:tn,xi':ti,y1..yk |- c and re-generalize over xi:ti to get x1:t1...xn:tn,xi':ti,y1..yk |- c[xi:=xi'] *) let regeneralize_rel i k j = if j = i+k then k else if j < i+k then j else j let rec regeneralize_index i k t = match kind_of_term t with | Rel j when j = i+k -> mkRel (k+1) | Rel j when j < i+k -> t | Rel j when j > i+k -> t | _ -> map_constr_with_binders succ (regeneralize_index i) k t type alias_constr = | DepAlias | NonDepAlias let mkSpecialLetInJudge j (na,(deppat,nondeppat,d,t)) = { uj_val = if isRel deppat or not (dependent (mkRel 1) j.uj_val) or d = NonDepAlias & not (dependent (mkRel 1) j.uj_type) then (* The body of pat is not needed to type j - see *) (* insert_aliases - and both deppat and nondeppat have the *) (* same type, then one can freely substitute one by the other *) subst1 nondeppat j.uj_val else (* The body of pat is not needed to type j but its value *) (* is dependent in the type of j; our choice is to *) (* enforce this dependency *) mkLetIn (na,deppat,t,j.uj_val); uj_type = subst1 deppat j.uj_type } (**********************************************************************) (* Structures used in compiling pattern-matching *) type 'a rhs = { rhs_env : env; rhs_vars : identifier list; avoid_ids : identifier list; it : 'a option} type 'a equation = { patterns : cases_pattern list; rhs : 'a rhs; alias_stack : name list; eqn_loc : loc; used : bool ref } type 'a matrix = 'a equation list type dep_status = KnownDep | KnownNotDep | DepUnknown (* 1st argument of IsInd is the original ind before extracting the summary *) type tomatch_type = | IsInd of types * inductive_type * name list | NotInd of constr option * types type tomatch_status = | Pushed of ((constr * tomatch_type) * int list * (name * dep_status)) | Alias of (constr * constr * alias_constr * constr) | Abstract of rel_declaration type tomatch_stack = tomatch_status list (* We keep a constr for aliases and a cases_pattern for error message *) type alias_builder = | AliasLeaf | AliasConstructor of constructor type pattern_history = | Top | MakeAlias of alias_builder * pattern_continuation and pattern_continuation = | Continuation of int * cases_pattern list * pattern_history | Result of cases_pattern list let start_history n = Continuation (n, [], Top) let initial_history = function Continuation (_,[],Top) -> true | _ -> false let feed_history arg = function | Continuation (n, l, h) when n>=1 -> Continuation (n-1, arg :: l, h) | Continuation (n, _, _) -> anomaly ("Bad number of expected remaining patterns: "^(string_of_int n)) | Result _ -> anomaly "Exhausted pattern history" (* This is for non exhaustive error message *) let rec rawpattern_of_partial_history args2 = function | Continuation (n, args1, h) -> let args3 = make_anonymous_patvars (n - (List.length args2)) in build_rawpattern (List.rev_append args1 (args2@args3)) h | Result pl -> pl and build_rawpattern args = function | Top -> args | MakeAlias (AliasLeaf, rh) -> assert (args = []); rawpattern_of_partial_history [PatVar (dummy_loc, Anonymous)] rh | MakeAlias (AliasConstructor pci, rh) -> rawpattern_of_partial_history [PatCstr (dummy_loc, pci, args, Anonymous)] rh let complete_history = rawpattern_of_partial_history [] (* This is to build glued pattern-matching history and alias bodies *) let rec simplify_history = function | Continuation (0, l, Top) -> Result (List.rev l) | Continuation (0, l, MakeAlias (f, rh)) -> let pargs = List.rev l in let pat = match f with | AliasConstructor pci -> PatCstr (dummy_loc,pci,pargs,Anonymous) | AliasLeaf -> assert (l = []); PatVar (dummy_loc, Anonymous) in feed_history pat rh | h -> h (* Builds a continuation expecting [n] arguments and building [ci] applied to this [n] arguments *) let push_history_pattern n current cont = Continuation (n, [], MakeAlias (current, cont)) (* A pattern-matching problem has the following form: env, evd |- Cases tomatch of mat end where tomatch is some sequence of "instructions" (t1 ... tn) and mat is some matrix (p11 ... p1n -> rhs1) ( ... ) (pm1 ... pmn -> rhsm) Terms to match: there are 3 kinds of instructions - "Pushed" terms to match are typed in [env]; these are usually just Rel(n) except for the initial terms given by user and typed in [env] - "Abstract" instructions means an abstraction has to be inserted in the current branch to build (this means a pattern has been detected dependent in another one and generalisation is necessary to ensure well-typing) - "Alias" instructions means an alias has to be inserted (this alias is usually removed at the end, except when its type is not the same as the type of the matched term from which it comes - typically because the inductive types are "real" parameters) Right-hand-sides: They consist of a raw term to type in an environment specific to the clause they belong to: the names of declarations are those of the variables present in the patterns. Therefore, they come with their own [rhs_env] (actually it is the same as [env] except for the names of variables). *) type 'a pattern_matching_problem = { env : env; evdref : evar_defs ref; pred : constr; tomatch : tomatch_stack; history : pattern_continuation; mat : 'a matrix; caseloc : loc; casestyle : case_style; typing_function: type_constraint -> env -> evar_defs ref -> 'a option -> unsafe_judgment } (*--------------------------------------------------------------------------* * A few functions to infer the inductive type from the patterns instead of * * checking that the patterns correspond to the ind. type of the * * destructurated object. Allows type inference of examples like * * match n with O => true | _ => false end * * match x in I with C => true | _ => false end * *--------------------------------------------------------------------------*) (* Computing the inductive type from the matrix of patterns *) (* We use the "in I" clause to coerce the terms to match and otherwise use the constructor to know in which type is the matching problem Note that insertion of coercions inside nested patterns is done each time the matrix is expanded *) let rec find_row_ind = function [] -> None | PatVar _ :: l -> find_row_ind l | PatCstr(loc,c,_,_) :: _ -> Some (loc,c) let inductive_template evdref env tmloc ind = let arsign = get_full_arity_sign env ind in let hole_source = match tmloc with | Some loc -> fun i -> (loc, TomatchTypeParameter (ind,i)) | None -> fun _ -> (dummy_loc, InternalHole) in let (_,evarl,_) = List.fold_right (fun (na,b,ty) (subst,evarl,n) -> match b with | None -> let ty' = substl subst ty in let e = e_new_evar evdref env ~src:(hole_source n) ty' in (e::subst,e::evarl,n+1) | Some b -> (b::subst,evarl,n+1)) arsign ([],[],1) in applist (mkInd ind,List.rev evarl) let try_find_ind env sigma typ realnames = let (IndType(_,realargs) as ind) = find_rectype env sigma typ in let names = match realnames with | Some names -> names | None -> list_make (List.length realargs) Anonymous in IsInd (typ,ind,names) let inh_coerce_to_ind evdref env ty tyi = let expected_typ = inductive_template evdref env None tyi in (* devrait être indifférent d'exiger leq ou pas puisque pour un inductif cela doit être égal *) let _ = e_cumul env evdref expected_typ ty in () let unify_tomatch_with_patterns evdref env loc typ pats realnames = match find_row_ind pats with | None -> NotInd (None,typ) | Some (_,(ind,_)) -> inh_coerce_to_ind evdref env typ ind; try try_find_ind env (evars_of !evdref) typ realnames with Not_found -> NotInd (None,typ) let find_tomatch_tycon evdref env loc = function (* Try if some 'in I ...' is present and can be used as a constraint *) | Some (_,ind,_,realnal) -> mk_tycon (inductive_template evdref env loc ind),Some (List.rev realnal) | None -> empty_tycon,None let coerce_row typing_fun evdref env pats (tomatch,(_,indopt)) = let loc = Some (loc_of_rawconstr tomatch) in let tycon,realnames = find_tomatch_tycon evdref env loc indopt in let j = typing_fun tycon env evdref tomatch in let typ = nf_evar (evars_of !evdref) j.uj_type in let t = try try_find_ind env (evars_of !evdref) typ realnames with Not_found -> unify_tomatch_with_patterns evdref env loc typ pats realnames in (j.uj_val,t) let coerce_to_indtype typing_fun evdref env matx tomatchl = let pats = List.map (fun r -> r.patterns) matx in let matx' = match matrix_transpose pats with | [] -> List.map (fun _ -> []) tomatchl (* no patterns at all *) | m -> m in List.map2 (coerce_row typing_fun evdref env) matx' tomatchl (************************************************************************) (* Utils *) let mkExistential env ?(src=(dummy_loc,InternalHole)) evdref = e_new_evar evdref env ~src:src (new_Type ()) let evd_comb2 f evdref x y = let (evd',y) = f !evdref x y in evdref := evd'; y module Cases_F(Coercion : Coercion.S) : S = struct let adjust_tomatch_to_pattern pb ((current,typ),deps,dep) = (* Ideally, we could find a common inductive type to which both the term to match and the patterns coerce *) (* In practice, we coerce the term to match if it is not already an inductive type and it is not dependent; moreover, we use only the first pattern type and forget about the others *) let typ,names = match typ with IsInd(t,_,names) -> t,Some names | NotInd(_,t) -> t,None in let typ = try try_find_ind pb.env (evars_of !(pb.evdref)) typ names with Not_found -> NotInd (None,typ) in let tomatch = ((current,typ),deps,dep) in match typ with | NotInd (None,typ) -> let tm1 = List.map (fun eqn -> List.hd eqn.patterns) pb.mat in (match find_row_ind tm1 with | None -> tomatch | Some (_,(ind,_)) -> let indt = inductive_template pb.evdref pb.env None ind in let current = if deps = [] & isEvar typ then (* Don't insert coercions if dependent; only solve evars *) let _ = e_cumul pb.env pb.evdref indt typ in current else (evd_comb2 (Coercion.inh_conv_coerce_to dummy_loc pb.env) pb.evdref (make_judge current typ) (mk_tycon_type indt)).uj_val in let sigma = evars_of !(pb.evdref) in let typ = try_find_ind pb.env sigma indt names in ((current,typ),deps,dep)) | _ -> tomatch let type_of_tomatch = function | IsInd (t,_,_) -> t | NotInd (_,t) -> t let mkDeclTomatch na = function | IsInd (t,_,_) -> (na,None,t) | NotInd (c,t) -> (na,c,t) let map_tomatch_type f = function | IsInd (t,ind,names) -> IsInd (f t,map_inductive_type f ind,names) | NotInd (c,t) -> NotInd (Option.map f c, f t) let liftn_tomatch_type n depth = map_tomatch_type (liftn n depth) let lift_tomatch_type n = liftn_tomatch_type n 1 let lift_tomatch n ((current,typ),info) = ((lift n current,lift_tomatch_type n typ),info) (**********************************************************************) (* Utilities on patterns *) let current_pattern eqn = match eqn.patterns with | pat::_ -> pat | [] -> anomaly "Empty list of patterns" let alias_of_pat = function | PatVar (_,name) -> name | PatCstr(_,_,_,name) -> name let unalias_pat = function | PatVar (c,name) as p -> if name = Anonymous then p else PatVar (c,Anonymous) | PatCstr(a,b,c,name) as p -> if name = Anonymous then p else PatCstr (a,b,c,Anonymous) let remove_current_pattern eqn = match eqn.patterns with | pat::pats -> { eqn with patterns = pats; alias_stack = alias_of_pat pat :: eqn.alias_stack } | [] -> anomaly "Empty list of patterns" let prepend_pattern tms eqn = {eqn with patterns = tms@eqn.patterns } (**********************************************************************) (* Well-formedness tests *) (* Partial check on patterns *) exception NotAdjustable let rec adjust_local_defs loc = function | (pat :: pats, (_,None,_) :: decls) -> pat :: adjust_local_defs loc (pats,decls) | (pats, (_,Some _,_) :: decls) -> PatVar (loc, Anonymous) :: adjust_local_defs loc (pats,decls) | [], [] -> [] | _ -> raise NotAdjustable let check_and_adjust_constructor env ind cstrs = function | PatVar _ as pat -> pat | PatCstr (loc,((_,i) as cstr),args,alias) as pat -> (* Check it is constructor of the right type *) let ind' = inductive_of_constructor cstr in if Closure.mind_equiv env ind' ind then (* Check the constructor has the right number of args *) let ci = cstrs.(i-1) in let nb_args_constr = ci.cs_nargs in if List.length args = nb_args_constr then pat else try let args' = adjust_local_defs loc (args, List.rev ci.cs_args) in PatCstr (loc, cstr, args', alias) with NotAdjustable -> error_wrong_numarg_constructor_loc loc (Global.env()) cstr nb_args_constr else (* Try to insert a coercion *) try Coercion.inh_pattern_coerce_to loc pat ind' ind with Not_found -> error_bad_constructor_loc loc cstr ind let check_all_variables typ mat = List.iter (fun eqn -> match current_pattern eqn with | PatVar (_,id) -> () | PatCstr (loc,cstr_sp,_,_) -> error_bad_pattern_loc loc cstr_sp typ) mat let check_unused_pattern env eqn = if not !(eqn.used) then raise_pattern_matching_error (eqn.eqn_loc, env, UnusedClause eqn.patterns) let set_used_pattern eqn = eqn.used := true let extract_rhs pb = match pb.mat with | [] -> errorlabstrm "build_leaf" (msg_may_need_inversion()) | eqn::_ -> set_used_pattern eqn; eqn.rhs (**********************************************************************) (* Functions to deal with matrix factorization *) let occur_in_rhs na rhs = match na with | Anonymous -> false | Name id -> List.mem id rhs.rhs_vars let is_dep_patt_in eqn = function | PatVar (_,name) -> occur_in_rhs name eqn.rhs | PatCstr _ -> true let mk_dep_patt_row (pats,eqn) = List.map (is_dep_patt_in eqn) pats let dependencies_in_pure_rhs nargs eqns = if eqns = [] then list_make nargs false (* Only "_" patts *) else let deps_rows = List.map mk_dep_patt_row eqns in let deps_columns = matrix_transpose deps_rows in List.map (List.exists ((=) true)) deps_columns let dependent_decl a = function | (na,None,t) -> dependent a t | (na,Some c,t) -> dependent a t || dependent a c let rec dep_in_tomatch n = function | (Pushed _ | Alias _) :: l -> dep_in_tomatch n l | Abstract d :: l -> dependent_decl (mkRel n) d or dep_in_tomatch (n+1) l | [] -> false let dependencies_in_rhs nargs current tms eqns = match kind_of_term current with | Rel n when dep_in_tomatch n tms -> list_make nargs true | _ -> dependencies_in_pure_rhs nargs eqns (* Computing the matrix of dependencies *) (* We are in context d1...dn |- and [find_dependencies k 1 nextlist] computes for declaration [k+1] in which of declarations in [nextlist] (which corresponds to d(k+2)...dn) it depends; declarations are expressed by index, e.g. in dependency list [n-2;1], [1] points to [dn] and [n-2] to [d3] *) let rec find_dependency_list k n = function | [] -> [] | (used,tdeps,d)::rest -> let deps = find_dependency_list k (n+1) rest in if used && dependent_decl (mkRel n) d then list_add_set (List.length rest + 1) (list_union deps tdeps) else deps let find_dependencies is_dep_or_cstr_in_rhs d (k,nextlist) = let deps = find_dependency_list k 1 nextlist in if is_dep_or_cstr_in_rhs || deps <> [] then (k-1,(true ,deps,d)::nextlist) else (k-1,(false,[] ,d)::nextlist) let find_dependencies_signature deps_in_rhs typs = let k = List.length deps_in_rhs in let _,l = List.fold_right2 find_dependencies deps_in_rhs typs (k,[]) in List.map (fun (_,deps,_) -> deps) l (******) (* A Pushed term to match has just been substituted by some constructor t = (ci x1...xn) and the terms x1 ... xn have been added to match - all terms to match and to push (dependent on t by definition) must have (Rel depth) substituted by t and Rel's>depth lifted by n - all pushed terms to match (non dependent on t by definition) must be lifted by n We start with depth=1 *) let regeneralize_index_tomatch n = let rec genrec depth = function | [] -> [] | Pushed ((c,tm),l,dep) :: rest -> let c = regeneralize_index n depth c in let tm = map_tomatch_type (regeneralize_index n depth) tm in let l = List.map (regeneralize_rel n depth) l in Pushed ((c,tm),l,dep) :: genrec depth rest | Alias (c1,c2,d,t) :: rest -> Alias (regeneralize_index n depth c1,c2,d,t) :: genrec depth rest | Abstract d :: rest -> Abstract (map_rel_declaration (regeneralize_index n depth) d) :: genrec (depth+1) rest in genrec 0 let rec replace_term n c k t = if t = mkRel (n+k) then lift k c else map_constr_with_binders succ (replace_term n c) k t let length_of_tomatch_type_sign (dep,_) = function | NotInd _ -> if dep<>Anonymous then 1 else 0 | IsInd (_,_,names) -> List.length names + if dep<>Anonymous then 1 else 0 let replace_tomatch n c = let rec replrec depth = function | [] -> [] | Pushed ((b,tm),l,dep) :: rest -> let b = replace_term n c depth b in let tm = map_tomatch_type (replace_term n c depth) tm in List.iter (fun i -> if i=n+depth then anomaly "replace_tomatch") l; Pushed ((b,tm),l,dep) :: replrec depth rest | Alias (c1,c2,d,t) :: rest -> Alias (replace_term n c depth c1,c2,d,t) :: replrec depth rest | Abstract d :: rest -> Abstract (map_rel_declaration (replace_term n c depth) d) :: replrec (depth+1) rest in replrec 0 let liftn_rel_declaration n k = map_rel_declaration (liftn n k) let substnl_rel_declaration sigma k = map_rel_declaration (substnl sigma k) let rec liftn_tomatch_stack n depth = function | [] -> [] | Pushed ((c,tm),l,dep)::rest -> let c = liftn n depth c in let tm = liftn_tomatch_type n depth tm in let l = List.map (fun i -> if i Alias (liftn n depth c1,liftn n depth c2,d,liftn n depth t) ::(liftn_tomatch_stack n depth rest) | Abstract d::rest -> Abstract (map_rel_declaration (liftn n depth) d) ::(liftn_tomatch_stack n (depth+1) rest) let lift_tomatch_stack n = liftn_tomatch_stack n 1 (* if [current] has type [I(p1...pn u1...um)] and we consider the case of constructor [ci] of type [I(p1...pn u'1...u'm)], then the default variable [name] is expected to have which type? Rem: [current] is [(Rel i)] except perhaps for initial terms to match *) (************************************************************************) (* Some heuristics to get names for variables pushed in pb environment *) (* Typical requirement: [match y with (S (S x)) => x | x => x end] should be compiled into [match y with O => y | (S n) => match n with O => y | (S x) => x end end] and [match y with (S (S n)) => n | n => n end] into [match y with O => y | (S n0) => match n0 with O => y | (S n) => n end end] i.e. user names should be preserved and created names should not interfere with user names *) let merge_name get_name obj = function | Anonymous -> get_name obj | na -> na let merge_names get_name = List.map2 (merge_name get_name) let get_names env sign eqns = let names1 = list_make (List.length sign) Anonymous in (* If any, we prefer names used in pats, from top to bottom *) let names2 = List.fold_right (fun (pats,eqn) names -> merge_names alias_of_pat pats names) eqns names1 in (* Otherwise, we take names from the parameters of the constructor but avoiding conflicts with user ids *) let allvars = List.fold_left (fun l (_,eqn) -> list_union l eqn.rhs.avoid_ids) [] eqns in let names4,_ = List.fold_left2 (fun (l,avoid) d na -> let na = merge_name (fun (na,_,t) -> Name (next_name_away (named_hd env t na) avoid)) d na in (na::l,(out_name na)::avoid)) ([],allvars) (List.rev sign) names2 in names4 (************************************************************************) (* Recovering names for variables pushed to the rhs' environment *) let recover_alias_names get_name = List.map2 (fun x (_,c,t) ->(get_name x,c,t)) let push_rels_eqn sign eqn = {eqn with rhs = {eqn.rhs with rhs_env = push_rels sign eqn.rhs.rhs_env} } let push_rels_eqn_with_names sign eqn = let pats = List.rev (list_firstn (List.length sign) eqn.patterns) in let sign = recover_alias_names alias_of_pat pats sign in push_rels_eqn sign eqn let build_aliases_context env sigma names allpats pats = (* pats is the list of bodies to push as an alias *) (* They all are defined in env and we turn them into a sign *) (* cuts in sign need to be done in allpats *) let rec insert env sign1 sign2 n newallpats oldallpats = function | (deppat,_,_,_)::pats, Anonymous::names when not (isRel deppat) -> (* Anonymous leaves must be considered named and treated in the *) (* next clause because they may occur in implicit arguments *) insert env sign1 sign2 n newallpats (List.map List.tl oldallpats) (pats,names) | (deppat,nondeppat,d,t)::pats, na::names -> let nondeppat = lift n nondeppat in let deppat = lift n deppat in let newallpats = List.map2 (fun l1 l2 -> List.hd l2::l1) newallpats oldallpats in let oldallpats = List.map List.tl oldallpats in let decl = (na,Some deppat,t) in let a = (deppat,nondeppat,d,t) in insert (push_rel decl env) (decl::sign1) ((na,a)::sign2) (n+1) newallpats oldallpats (pats,names) | [], [] -> newallpats, sign1, sign2, env | _ -> anomaly "Inconsistent alias and name lists" in let allpats = List.map (fun x -> [x]) allpats in insert env [] [] 0 (List.map (fun _ -> []) allpats) allpats (pats, names) let insert_aliases_eqn sign eqnnames alias_rest eqn = let thissign = List.map2 (fun na (_,c,t) -> (na,c,t)) eqnnames sign in { eqn with alias_stack = alias_rest; rhs = {eqn.rhs with rhs_env = push_rels thissign eqn.rhs.rhs_env } } let insert_aliases env sigma alias eqns = (* Là, y a une faiblesse, si un alias est utilisé dans un cas par *) (* défaut présent mais inutile, ce qui est le cas général, l'alias *) (* est introduit même s'il n'est pas utilisé dans les cas réguliers *) let eqnsnames = List.map (fun eqn -> List.hd eqn.alias_stack) eqns in let alias_rests = List.map (fun eqn -> List.tl eqn.alias_stack) eqns in (* name2 takes the meet of all needed aliases *) let name2 = List.fold_right (merge_name (fun x -> x)) eqnsnames Anonymous in (* Only needed aliases are kept by build_aliases_context *) let eqnsnames, sign1, sign2, env = build_aliases_context env sigma [name2] eqnsnames [alias] in let eqns = list_map3 (insert_aliases_eqn sign1) eqnsnames alias_rests eqns in sign2, env, eqns (**********************************************************************) (* Functions to deal with elimination predicate *) exception Occur let noccur_between_without_evar n m term = let rec occur_rec n c = match kind_of_term c with | Rel p -> if n<=p && p () | _ -> iter_constr_with_binders succ occur_rec n c in (m = 0) or (try occur_rec n term; true with Occur -> false) (* Infering the predicate *) (* The problem to solve is the following: We match Gamma |- t : I(u01..u0q) against the following constructors: Gamma, x11...x1p1 |- C1(x11..x1p1) : I(u11..u1q) ... Gamma, xn1...xnpn |- Cn(xn1..xnp1) : I(un1..unq) Assume the types in the branches are the following Gamma, x11...x1p1 |- branch1 : T1 ... Gamma, xn1...xnpn |- branchn : Tn Assume the type of the global case expression is Gamma |- T The predicate has the form phi = [y1..yq][z:I(y1..yq)]? and must satisfy the following n+1 equations: Gamma, x11...x1p1 |- (phi u11..u1q (C1 x11..x1p1)) = T1 ... Gamma, xn1...xnpn |- (phi un1..unq (Cn xn1..xnpn)) = Tn Gamma |- (phi u01..u0q t) = T Some hints: - Clearly, if xij occurs in Ti, then, a "match z with (Ci xi1..xipi) => ..." or a "psi(yk)", with psi extracting xij from uik, should be inserted somewhere in Ti. - If T is undefined, an easy solution is to insert a "match z with (Ci xi1..xipi) => ..." in front of each Ti - Otherwise, T1..Tn and T must be step by step unified, if some of them diverge, then try to replace the diverging subterm by one of y1..yq or z. - The main problem is what to do when an existential variables is encountered *) (* Propagation of user-provided predicate through compilation steps *) let rec map_predicate f k ccl = function | [] -> f k ccl | Pushed ((_,tm),_,dep) :: rest -> let k' = length_of_tomatch_type_sign dep tm in map_predicate f (k+k') ccl rest | Alias _ :: rest -> map_predicate f k ccl rest | Abstract _ :: rest -> map_predicate f (k+1) ccl rest let noccurn_predicate = map_predicate noccurn let liftn_predicate n = map_predicate (liftn n) let lift_predicate n = liftn_predicate n 1 let regeneralize_index_predicate n = map_predicate (regeneralize_index n) 0 let substnl_predicate sigma = map_predicate (substnl sigma) (* This is parallel bindings *) let subst_predicate (args,copt) ccl tms = let sigma = match copt with | None -> List.rev args | Some c -> c::(List.rev args) in substnl_predicate sigma 0 ccl tms let specialize_predicate_var (cur,typ,dep) tms ccl = let c = if dep<>Anonymous then Some cur else None in let l = match typ with | IsInd (_,IndType(_,realargs),names) -> if names<>[] then realargs else [] | NotInd _ -> [] in subst_predicate (l,c) ccl tms (*****************************************************************************) (* We have pred = [X:=realargs;x:=c]P typed in Gamma1, x:I(realargs), Gamma2 *) (* and we want to abstract P over y:t(x) typed in the same context to get *) (* *) (* pred' = [X:=realargs;x':=c](y':t(x'))P[y:=y'] *) (* *) (* We first need to lift t(x) s.t. it is typed in Gamma, X:=rargs, x' *) (* then we have to replace x by x' in t(x) and y by y' in P *) (*****************************************************************************) let generalize_predicate (names,(nadep,_)) ny d tms ccl = if nadep=Anonymous then anomaly "Undetected dependency"; let p = List.length names + 1 in let ccl = lift_predicate 1 ccl tms in regeneralize_index_predicate (ny+p+1) ccl tms let rec extract_predicate l ccl = function | Alias (deppat,nondeppat,_,_)::tms -> let tms' = match kind_of_term nondeppat with | Rel i -> replace_tomatch i deppat tms | _ -> (* initial terms are not dependent *) tms in extract_predicate l ccl tms' | Abstract d'::tms -> let d' = map_rel_declaration (lift (List.length l)) d' in substl l (mkProd_or_LetIn d' (extract_predicate [] ccl tms)) | Pushed ((cur,NotInd _),_,(dep,_))::tms -> extract_predicate (if dep<>Anonymous then cur::l else l) ccl tms | Pushed ((cur,IsInd (_,IndType(_,realargs),_)),_,(dep,_))::tms -> let l = List.rev realargs@l in extract_predicate (if dep<>Anonymous then cur::l else l) ccl tms | [] -> substl l ccl let abstract_predicate env sigma indf cur (names,(nadep,_)) tms ccl = let sign = make_arity_signature env true indf in (* n is the number of real args + 1 *) let n = List.length sign in let tms = lift_tomatch_stack n tms in let tms = match kind_of_term cur with | Rel i -> regeneralize_index_tomatch (i+n) tms | _ -> (* Initial case *) tms in let sign = List.map2 (fun na (_,c,t) -> (na,c,t)) (nadep::names) sign in let ccl = if nadep <> Anonymous then ccl else lift_predicate 1 ccl tms in let pred = extract_predicate [] ccl tms in it_mkLambda_or_LetIn_name env pred sign let known_dependent (_,dep) = (dep = KnownDep) (* [expand_arg] is used by [specialize_predicate] it replaces gamma, x1...xn, x1...xk |- pred by gamma, x1...xn, x1...xk-1 |- [X=realargs,xk=xk]pred (if dep) or by gamma, x1...xn, x1...xk-1 |- [X=realargs]pred (if not dep) *) let expand_arg tms ccl ((_,t),_,na) = let k = length_of_tomatch_type_sign na t in lift_predicate (k-1) ccl tms let adjust_impossible_cases pb pred tomatch submat = if submat = [] then match kind_of_term (whd_evar (evars_of !(pb.evdref)) pred) with | Evar (evk,_) when snd (evar_source evk !(pb.evdref)) = ImpossibleCase -> let default = (coq_unit_judge ()).uj_type in pb.evdref := Evd.evar_define evk default !(pb.evdref); (* we add an "assert false" case *) let pats = List.map (fun _ -> PatVar (dummy_loc,Anonymous)) tomatch in let aliasnames = map_succeed (function Alias _ -> Anonymous | _ -> failwith"") tomatch in [ { patterns = pats; rhs = { rhs_env = pb.env; rhs_vars = []; avoid_ids = []; it = None }; alias_stack = Anonymous::aliasnames; eqn_loc = dummy_loc; used = ref false } ] | _ -> submat else submat (*****************************************************************************) (* pred = [X:=realargs;x:=c]P types the following problem: *) (* *) (* Gamma |- match Pushed(c:I(realargs)) rest with...end: pred *) (* *) (* where the branch with constructor Ci:(x1:T1)...(xn:Tn)->I(realargsi) *) (* is considered. Assume each Ti is some Ii(argsi). *) (* We let e=Ci(x1,...,xn) and replace pred by *) (* *) (* pred' = [X1:=rargs1,x1:=x1']...[Xn:=rargsn,xn:=xn'](P[X:=realargsi;x:=e]) *) (* *) (* s.t Gamma,x1'..xn' |- match Pushed(x1')..Pushed(xn') rest with..end :pred'*) (* *) (*****************************************************************************) let specialize_predicate newtomatchs (names,(depna,_)) cs tms ccl = (* Assume some gamma st: gamma, (X,x:=realargs,copt), tms |- ccl *) let nrealargs = List.length names in let k = nrealargs + (if depna<>Anonymous then 1 else 0) in (* We adjust pred st: gamma, x1..xn, (X,x:=realargs,copt), tms |- ccl' *) let n = cs.cs_nargs in let ccl' = liftn_predicate n (k+1) ccl tms in let argsi = if nrealargs <> 0 then Array.to_list cs.cs_concl_realargs else [] in let copti = if depna<>Anonymous then Some (build_dependent_constructor cs) else None in (* The substituends argsi, copti are all defined in gamma, x1...xn *) (* We need _parallel_ bindings to get gamma, x1...xn, tms |- ccl'' *) let ccl'' = whd_betaiota (subst_predicate (argsi, copti) ccl' tms) in (* We adjust ccl st: gamma, x1..xn, x1..xn, tms |- ccl'' *) let ccl''' = liftn_predicate n (n+1) ccl'' tms in (* We finally get gamma,x1..xn |- [X1,x1:=R1,x1]..[Xn,xn:=Rn,xn]pred'''*) List.fold_left (expand_arg tms) ccl''' newtomatchs let find_predicate loc env evdref p current (IndType (indf,realargs)) dep tms = let pred= abstract_predicate env (evars_of !evdref) indf current dep tms p in (pred, whd_betaiota (applist (pred, realargs@[current])), new_Type ()) let adjust_predicate_from_tomatch ((_,oldtyp),_,(nadep,_)) typ pb = match typ, oldtyp with | IsInd (_,_,names), NotInd _ -> let k = if nadep <> Anonymous then 2 else 1 in let n = List.length names in { pb with pred = liftn_predicate n k pb.pred pb.tomatch } | _ -> pb (************************************************************************) (* Sorting equations by constructor *) type inversion_problem = (* the discriminating arg in some Ind and its order in Ind *) | Incompatible of int * (int * int) | Constraints of (int * constr) list let solve_constraints constr_info indt = (* TODO *) Constraints [] let rec irrefutable env = function | PatVar (_,name) -> true | PatCstr (_,cstr,args,_) -> let ind = inductive_of_constructor cstr in let (_,mip) = Inductive.lookup_mind_specif env ind in let one_constr = Array.length mip.mind_user_lc = 1 in one_constr & List.for_all (irrefutable env) args let first_clause_irrefutable env = function | eqn::mat -> List.for_all (irrefutable env) eqn.patterns | _ -> false let group_equations pb ind current cstrs mat = let mat = if first_clause_irrefutable pb.env mat then [List.hd mat] else mat in let brs = Array.create (Array.length cstrs) [] in let only_default = ref true in let _ = List.fold_right (* To be sure it's from bottom to top *) (fun eqn () -> let rest = remove_current_pattern eqn in let pat = current_pattern eqn in match check_and_adjust_constructor pb.env ind cstrs pat with | PatVar (_,name) -> (* This is a default clause that we expand *) for i=1 to Array.length cstrs do let args = make_anonymous_patvars cstrs.(i-1).cs_nargs in brs.(i-1) <- (args, rest) :: brs.(i-1) done | PatCstr (loc,((_,i)),args,_) -> (* This is a regular clause *) only_default := false; brs.(i-1) <- (args,rest) :: brs.(i-1)) mat () in (brs,!only_default) (************************************************************************) (* Here starts the pattern-matching compilation algorithm *) (* Abstracting over dependent subterms to match *) let rec generalize_problem names pb = function | [] -> pb | i::l -> let d = map_rel_declaration (lift i) (Environ.lookup_rel i pb.env) in let pb' = generalize_problem names pb l in let tomatch = lift_tomatch_stack 1 pb'.tomatch in let tomatch = regeneralize_index_tomatch (i+1) tomatch in { pb' with tomatch = Abstract d :: tomatch; pred = generalize_predicate names i d pb.tomatch pb'.pred } (* No more patterns: typing the right-hand-side of equations *) let build_leaf pb = let rhs = extract_rhs pb in let j = pb.typing_function (mk_tycon pb.pred) rhs.rhs_env pb.evdref rhs.it in j_nf_evar (evars_of !(pb.evdref)) j (* Building the sub-problem when all patterns are variables *) let shift_problem ((current,t),_,(nadep,_)) pb = {pb with tomatch = Alias (current,current,NonDepAlias,type_of_tomatch t)::pb.tomatch; pred = specialize_predicate_var (current,t,nadep) pb.tomatch pb.pred; history = push_history_pattern 0 AliasLeaf pb.history; mat = List.map remove_current_pattern pb.mat } (* Building the sub-pattern-matching problem for a given branch *) let build_branch current deps (realnames,dep) pb eqns const_info = (* We remember that we descend through a constructor *) let alias_type = if Array.length const_info.cs_concl_realargs = 0 & not (known_dependent dep) & deps = [] then NonDepAlias else DepAlias in let history = push_history_pattern const_info.cs_nargs (AliasConstructor const_info.cs_cstr) pb.history in (* We find matching clauses *) let cs_args = const_info.cs_args in let names = get_names pb.env cs_args eqns in let submat = List.map (fun (tms,eqn) -> prepend_pattern tms eqn) eqns in let typs = List.map2 (fun (_,c,t) na -> (na,c,t)) cs_args names in let _,typs',_ = List.fold_right (fun (na,c,t as d) (env,typs,tms) -> let tms = List.map List.tl tms in (push_rel d env, (na,NotInd(c,t))::typs,tms)) typs (pb.env,[],List.map fst eqns) in let dep_sign = find_dependencies_signature (dependencies_in_rhs const_info.cs_nargs current pb.tomatch eqns) (List.rev typs) in (* The dependent term to subst in the types of the remaining UnPushed terms is relative to the current context enriched by topushs *) let ci = build_dependent_constructor const_info in (* We replace [(mkRel 1)] by its expansion [ci] *) (* and context "Gamma = Gamma1, current, Gamma2" by "Gamma;typs;curalias" *) (* This is done in two steps : first from "Gamma |- tms" *) (* into "Gamma; typs; curalias |- tms" *) let tomatch = lift_tomatch_stack const_info.cs_nargs pb.tomatch in let typs'' = list_map2_i (fun i (na,t) deps -> let dep = match dep with | Name _ as na',k -> (if na <> Anonymous then na else na'),k | Anonymous,KnownNotDep -> if deps = [] && noccurn_predicate 1 pb.pred tomatch then (Anonymous,KnownNotDep) else (force_name na,KnownDep) | _,_ -> anomaly "Inconsistent dependency" in ((mkRel i, lift_tomatch_type i t),deps,dep)) 1 typs' (List.rev dep_sign) in let pred = specialize_predicate typs'' (realnames,dep) const_info tomatch pb.pred in let currents = List.map (fun x -> Pushed x) typs'' in let sign = List.map (fun (na,t) -> mkDeclTomatch na t) typs' in let ind = appvect ( applist (mkInd (inductive_of_constructor const_info.cs_cstr), List.map (lift const_info.cs_nargs) const_info.cs_params), const_info.cs_concl_realargs) in let cur_alias = lift (List.length sign) current in let currents = Alias (ci,cur_alias,alias_type,ind) :: currents in let tomatch = List.rev_append currents tomatch in let submat = adjust_impossible_cases pb pred tomatch submat in if submat = [] then raise_pattern_matching_error (dummy_loc, pb.env, NonExhaustive (complete_history history)); sign, { pb with env = push_rels sign pb.env; tomatch = tomatch; pred = pred; history = history; mat = List.map (push_rels_eqn_with_names sign) submat } (********************************************************************** INVARIANT: pb = { env, subst, tomatch, mat, ...} tomatch = list of Pushed (c:T) or Abstract (na:T) or Alias (c:T) "Pushed" terms and types are relative to env "Abstract" types are relative to env enriched by the previous terms to match *) (**********************************************************************) (* Main compiling descent *) let rec compile pb = match pb.tomatch with | (Pushed cur)::rest -> match_current { pb with tomatch = rest } cur | (Alias x)::rest -> compile_alias pb x rest | (Abstract d)::rest -> compile_generalization pb d rest | [] -> build_leaf pb and match_current pb tomatch = let ((current,typ),deps,dep as ct) = adjust_tomatch_to_pattern pb tomatch in let pb = adjust_predicate_from_tomatch tomatch typ pb in match typ with | NotInd (_,typ) -> check_all_variables typ pb.mat; compile (shift_problem ct pb) | IsInd (_,(IndType(indf,realargs) as indt),names) -> let mind,_ = dest_ind_family indf in let cstrs = get_constructors pb.env indf in let eqns,onlydflt = group_equations pb mind current cstrs pb.mat in if (Array.length cstrs <> 0 or pb.mat <> []) & onlydflt then compile (shift_problem ct pb) else let _constraints = Array.map (solve_constraints indt) cstrs in (* We generalize over terms depending on current term to match *) let pb = generalize_problem (names,dep) pb deps in (* We compile branches *) let brs = array_map2 (compile_branch current (names,dep) deps pb) eqns cstrs in (* We build the (elementary) case analysis *) let brvals = Array.map (fun (v,_) -> v) brs in let (pred,typ,s) = find_predicate pb.caseloc pb.env pb.evdref pb.pred current indt (names,dep) pb.tomatch in let ci = make_case_info pb.env mind pb.casestyle in let case = mkCase (ci,nf_betaiota pred,current,brvals) in let inst = List.map mkRel deps in { uj_val = applist (case, inst); uj_type = substl inst typ } and compile_branch current names deps pb eqn cstr = let sign, pb = build_branch current deps names pb eqn cstr in let j = compile pb in (it_mkLambda_or_LetIn j.uj_val sign, j.uj_type) and compile_generalization pb d rest = let pb = { pb with env = push_rel d pb.env; tomatch = rest; mat = List.map (push_rels_eqn [d]) pb.mat } in let j = compile pb in { uj_val = mkLambda_or_LetIn d j.uj_val; uj_type = mkProd_or_LetIn d j.uj_type } and compile_alias pb (deppat,nondeppat,d,t) rest = let history = simplify_history pb.history in let sign, newenv, mat = insert_aliases pb.env (evars_of !(pb.evdref)) (deppat,nondeppat,d,t) pb.mat in let n = List.length sign in (* We had Gamma1; x:current; Gamma2 |- tomatch(x) and we rebind x to get *) (* Gamma1; x:current; Gamma2; typs; x':=curalias |- tomatch(x') *) let tomatch = lift_tomatch_stack n rest in let tomatch = match kind_of_term nondeppat with | Rel i -> if n = 1 then regeneralize_index_tomatch (i+n) tomatch else replace_tomatch i deppat tomatch | _ -> (* initial terms are not dependent *) tomatch in let pb = {pb with env = newenv; tomatch = tomatch; pred = lift_predicate n pb.pred tomatch; history = history; mat = mat } in let j = compile pb in List.fold_left mkSpecialLetInJudge j sign (* pour les alias des initiaux, enrichir les env de ce qu'il faut et substituer après par les initiaux *) (**************************************************************************) (* Preparation of the pattern-matching problem *) (* builds the matrix of equations testing that each eqn has n patterns * and linearizing the _ patterns. * Syntactic correctness has already been done in astterm *) let matx_of_eqns env tomatchl eqns = let build_eqn (loc,ids,lpat,rhs) = let initial_lpat,initial_rhs = lpat,rhs in let initial_rhs = rhs in let rhs = { rhs_env = env; rhs_vars = free_rawvars initial_rhs; avoid_ids = ids@(ids_of_named_context (named_context env)); it = Some initial_rhs } in { patterns = initial_lpat; alias_stack = []; eqn_loc = loc; used = ref false; rhs = rhs } in List.map build_eqn eqns (************************************************************************) (* preparing the elimination predicate if any *) let build_expected_arity env evdref isdep tomatchl = let cook n = function | _,IsInd (_,IndType(indf,_),_) -> let indf' = lift_inductive_family n indf in Some (build_dependent_inductive env indf', fst (get_arity env indf')) | _,NotInd _ -> None in let rec buildrec n env = function | [] -> new_Type () | tm::ltm -> match cook n tm with | None -> buildrec n env ltm | Some (ty1,aritysign) -> let rec follow n env = function | d::sign -> mkProd_or_LetIn_name env (follow (n+1) (push_rel d env) sign) d | [] -> if isdep then mkProd (Anonymous, ty1, buildrec (n+1) (push_rel_assum (Anonymous, ty1) env) ltm) else buildrec n env ltm in follow n env (List.rev aritysign) in buildrec 0 env tomatchl let extract_predicate_conclusion isdep tomatchl pred = let cook = function | _,IsInd (_,IndType(_,args),_) -> Some (List.length args) | _,NotInd _ -> None in let rec decomp_lam_force n l p = if n=0 then (l,p) else match kind_of_term p with | Lambda (na,_,c) -> decomp_lam_force (n-1) (na::l) c | _ -> (* eta-expansion *) let na = Name (id_of_string "x") in decomp_lam_force (n-1) (na::l) (applist (lift 1 p, [mkRel 1])) in let rec buildrec allnames p = function | [] -> (List.rev allnames,p) | tm::ltm -> match cook tm with | None -> let p = (* adjust to a sign containing the NotInd's *) if isdep then lift 1 p else p in let names = if isdep then [Anonymous] else [] in buildrec (names::allnames) p ltm | Some n -> let n = if isdep then n+1 else n in let names,p = decomp_lam_force n [] p in buildrec (names::allnames) p ltm in buildrec [] pred tomatchl let set_arity_signature dep n arsign tomatchl pred x = (* avoid is not exhaustive ! *) let rec decomp_lam_force n avoid l p = if n = 0 then (List.rev l,p,avoid) else match p with | RLambda (_,(Name id as na),_,_,c) -> decomp_lam_force (n-1) (id::avoid) (na::l) c | RLambda (_,(Anonymous as na),_,_,c) -> decomp_lam_force (n-1) avoid (na::l) c | _ -> let x = next_ident_away (id_of_string "x") avoid in decomp_lam_force (n-1) (x::avoid) (Name x :: l) (* eta-expansion *) (let a = RVar (dummy_loc,x) in match p with | RApp (loc,p,l) -> RApp (loc,p,l@[a]) | _ -> (RApp (dummy_loc,p,[a]))) in let rec decomp_block avoid p = function | ([], _) -> x := Some p | ((_,IsInd (_,IndType(indf,realargs),_))::l),(y::l') -> let (ind,params) = dest_ind_family indf in let (nal,p,avoid') = decomp_lam_force (List.length realargs) avoid [] p in let na,p,avoid' = if dep then decomp_lam_force 1 avoid' [] p else [Anonymous],p,avoid' in y := (List.hd na, if List.for_all ((=) Anonymous) nal then None else Some (dummy_loc, ind, (List.map (fun _ -> Anonymous) params)@nal)); decomp_block avoid' p (l,l') | (_::l),(y::l') -> y := (Anonymous,None); decomp_block avoid p (l,l') | _ -> anomaly "set_arity_signature" in decomp_block [] pred (tomatchl,arsign) (***************** Building an inversion predicate ************************) (* Let "match t1 in I1 u11..u1n_1 ... tm in Im um1..umn_m with ... end : T" be a pattern-matching problem. We assume that the each uij can be decomposed under the form pij(vij1..vijq_ij) where pij(aij1..aijq_ij) is a pattern depending on some variables aijk and the vijk are instances of these variables. We also assume that each ti has the form of a pattern qi(wi1..wiq_i) where qi(bi1..biq_i) is a pattern depending on some variables bik and the wik are instances of these variables (in practice, there is no reason that ti is already constructed and the qi will be degenerated). We then look for a type U(..a1jk..b1 .. ..amjk..bm) so that T = U(..v1jk..t1 .. ..vmjk..tm). This a higher-order matching problem with a priori different solution (one of them if T itself!). We finally invert the uij and the ti and build the return clause phi(x11..x1n_1y1..xm1..xmn_mym) = match x11..x1n_1 y1 .. xm1..xmn_m ym with | p11..p1n_1 q1 .. pm1..pmn_m qm => U(..a1jk..b1 .. ..amjk..bm) | _ .. _ _ .. _ .. _ _ => True end so that "phi(u11..u1n_1t1..um1..umn_mtm) = T" (note that the clause returning True never happens and any inhabited type can be put instead). *) let adjust_to_extended_env_and_remove_deps env extenv subst t = let n = rel_context_length (rel_context env) in let n' = rel_context_length (rel_context extenv) in (* We first remove the bindings that are dependently typed (they are difficult to manage and it is not sure these are so useful in practice); Notes: - [subst] is made of pairs [(id,u)] where id is a name in [extenv] and [u] a term typed in [env]; - [subst0] is made of items [(p,u,(u,ty))] where [ty] is the type of [u] and both are adjusted to [extenv] while [p] is the index of [id] in [extenv] (after expansion of the aliases) *) let subst0 = map_succeed (fun (x,u) -> (* d1 ... dn dn+1 ... dn'-p+1 ... dn' *) (* \--env-/ (= x:ty) *) (* \--------------extenv------------/ *) let (p,_) = lookup_rel_id x (rel_context extenv) in let rec aux n (_,b,ty) = match b with | Some c -> assert (isRel c); let p = n + destRel c in aux p (lookup_rel p (rel_context extenv)) | None -> (n,ty) in let (p,ty) = aux p (lookup_rel p (rel_context extenv)) in if noccur_between_without_evar 1 (n'-p-n+1) ty then let u = lift (n'-n) u in (p,u,(expand_vars_in_term extenv u,lift p ty)) else failwith "") subst in let t0 = lift (n'-n) t in (subst0,t0) (* Let vijk and ti be a set of dependent terms and T a type, all * defined in some environment env. The vijk and ti are supposed to be * instances for variables aijk and bi. * * [abstract_tycon Gamma0 Sigma subst T Gamma] looks for U(..v1jk..t1 .. ..vmjk..tm) * defined in some extended context * "Gamma0, ..a1jk:V1jk.. b1:W1 .. ..amjk:Vmjk.. bm:Wm" * such that env |- T = U(..v1jk..t1 .. ..vmjk..tm). To not commit to * a particular solution, we replace each subterm t in T that unifies with * a subset u1..ul of the vijk and ti by a special evar * ?id(x=t;c1:=c1,..,cl=cl) defined in context Gamma0,x,c1,...,cl |- ?id * (where the c1..cl are the aijk and bi matching the u1..ul), and * similarly for each ti. *) let abstract_tycon loc env evdref subst _tycon extenv t = let t = nf_betaiota t in (* it helps in some cases to remove K-redex... *) let sigma = evars_of !evdref in let subst0,t0 = adjust_to_extended_env_and_remove_deps env extenv subst t in (* We traverse the type T of the original problem Xi looking for subterms that match the non-constructor part of the constraints (this part is in subst); these subterms are the "good" subterms and we replace them by an evar that may depend (and only depend) on the corresponding convertible subterms of the substitution *) let rec aux (k,env,subst as x) t = let good = List.filter (fun (_,u,_) -> is_conv_leq env sigma t u) subst in if good <> [] then let (u,ty) = pi3 (List.hd good) in let vl = List.map pi1 good in let inst = list_map_i (fun i _ -> if List.mem i vl then u else mkRel i) 1 (rel_context extenv) in let rel_filter = List.map (fun a -> not (isRel a) or dependent a u) inst in let named_filter = List.map (fun (id,_,_) -> dependent (mkVar id) u) (named_context extenv) in let filter = rel_filter@named_filter in let ev = e_new_evar evdref extenv ~src:(loc, CasesType) ~filter:filter ty in evdref := add_conv_pb (Reduction.CONV,extenv,substl inst ev,u) !evdref; lift k ev else map_constr_with_full_binders (fun d (k,env,subst) -> k+1, push_rel d env, List.map (fun (na,u,d) -> (na,lift 1 u,d)) subst) aux x t in aux (0,extenv,subst0) t0 let build_tycon loc env tycon_env subst tycon extenv evdref t = let t = match t with | None -> (* This is the situation we are building a return predicate and we are in an impossible branch *) let n = rel_context_length (rel_context env) in let n' = rel_context_length (rel_context tycon_env) in let impossible_case_type = e_new_evar evdref env ~src:(loc,ImpossibleCase) (new_Type ()) in lift (n'-n) impossible_case_type | Some t -> abstract_tycon loc tycon_env evdref subst tycon extenv t in get_judgment_of extenv (evars_of !evdref) t (* For a multiple pattern-matching problem Xi on t1..tn with return * type T, [build_inversion_problem Gamma Sigma (t1..tn) T] builds a return * predicate for Xi that is itself made by an auxiliary * pattern-matching problem of which the first clause reveals the * pattern structure of the constraints on the inductive types of the t1..tn, * and the second clause is a wildcard clause for catching the * impossible cases. See above "Building an inversion predicate" for * further explanations *) let build_inversion_problem loc env evdref tms t = let sigma = evars_of !evdref in let make_patvar t (subst,avoid) = let id = next_name_away (named_hd env t Anonymous) avoid in PatVar (dummy_loc,Name id), ((id,t)::subst, id::avoid) in let rec reveal_pattern t (subst,avoid as acc) = match kind_of_term (whd_betadeltaiota env sigma t) with | Construct cstr -> PatCstr (dummy_loc,cstr,[],Anonymous), acc | App (f,v) when isConstruct f -> let cstr = destConstruct f in let n = constructor_nrealargs env cstr in let l = list_lastn n (Array.to_list v) in let l,acc = list_fold_map' reveal_pattern l acc in PatCstr (dummy_loc,cstr,l,Anonymous), acc | _ -> make_patvar t acc in let rec aux n env acc_sign tms acc = match tms with | [] -> [], acc_sign, acc | (t, IsInd (_,IndType(indf,realargs),_)) :: tms -> let patl,acc = list_fold_map' reveal_pattern realargs acc in let pat,acc = make_patvar t acc in let indf' = lift_inductive_family n indf in let sign = make_arity_signature env true indf' in let p = List.length realargs in let env' = push_rels sign env in let patl',acc_sign,acc = aux (n+p+1) env' (sign@acc_sign) tms acc in patl@pat::patl',acc_sign,acc | (t, NotInd (bo,typ)) :: tms -> aux n env acc_sign tms acc in let avoid0 = ids_of_context env in (* [patl] is a list of patterns revealing the substructure of constructors present in the constraints on the type of the multiple terms t1..tn that are matched in the original problem; [subst] is the substitution of the free pattern variables in [patl] that returns the non-constructor parts of the constraints. Especially, if the ti has type I ui1..uin_i, and the patterns associated to ti are pi1..pin_i, then subst(pij) is uij; the substitution is useful to recognize which subterms of the whole type T of the original problem have to be abstracted *) let patl,sign,(subst,avoid) = aux 0 env [] tms ([],avoid0) in let n = List.length sign in let (pb_env,_),sub_tms = list_fold_map (fun (env,i) (na,b,t as d) -> let typ = if b<>None then NotInd(None,t) else try try_find_ind env sigma t None with Not_found -> NotInd (None,t) in let ty = lift_tomatch_type (n-i) typ in let tm = Pushed ((mkRel (n-i),ty),[],(Anonymous,KnownNotDep)) in ((push_rel d env,i+1),tm)) (env,0) (List.rev sign) in let subst = List.map (fun (na,t) -> (na,lift n t)) subst in (* [eqn1] is the first clause of the auxiliary pattern-matching that serves as skeleton for the return type: [patl] is the substructure of constructors extracted from the list of constraints on the inductive types of the multiple terms matched in the original pattern-matching problem Xi *) let eqn1 = { patterns = patl; alias_stack = []; eqn_loc = dummy_loc; used = ref false; rhs = { rhs_env = pb_env; (* we assume all vars are used; in practice we discard dependent vars so that the field rhs_vars is normally not used *) rhs_vars = List.map fst subst; avoid_ids = avoid; it = Some (lift n t) } } in (* [eqn2] is the default clause of the auxiliary pattern-matching: it will catch the clauses of the original pattern-matching problem Xi whose type constraints are incompatible with the constraints on the inductive types of the multiple terms matched in Xi *) let eqn2 = { patterns = List.map (fun _ -> PatVar (dummy_loc,Anonymous)) patl; alias_stack = []; eqn_loc = dummy_loc; used = ref false; rhs = { rhs_env = pb_env; rhs_vars = []; avoid_ids = avoid0; it = None } } in (* [pb] is the auxiliary pattern-matching serving as skeleton for the return type of the original problem Xi *) let pb = { env = pb_env; evdref = evdref; pred = new_Type(); tomatch = sub_tms; history = start_history n; mat = [eqn1;eqn2]; caseloc = loc; casestyle = RegularStyle; typing_function = build_tycon loc env pb_env subst} in (compile pb).uj_val let prepare_predicate_from_tycon loc dep env evdref tomatchs sign c = let cook (n, l, env, signs) = function | c,IsInd (_,IndType(indf,realargs),_) -> let indf' = lift_inductive_family n indf in let sign = make_arity_signature env dep indf' in let p = List.length realargs in if dep then (n + p + 1, c::(List.rev realargs)@l, push_rels sign env,sign::signs) else (n + p, (List.rev realargs)@l, push_rels sign env,sign::signs) | c,NotInd (bo,typ) -> let sign = [Anonymous,Option.map (lift n) bo,lift n typ] in let sign = name_context env sign in (n + 1, c::l, push_rels sign env, sign::signs) in let n,allargs,env',signs = List.fold_left cook (0, [], env, []) tomatchs in let names = List.rev (List.map (List.map pi1) signs) in names, build_inversion_problem loc env evdref tomatchs c (* Here, [pred] is assumed to be in the context built from all *) (* realargs and terms to match *) let build_initial_predicate knowndep allnames pred = let nar = List.fold_left (fun n names -> List.length names + n) 0 allnames in let rec buildrec n pred nal = function | [] -> List.rev nal,pred | names::lnames -> let names' = List.tl names in let n' = n + List.length names' in let pred, p = if dependent (mkRel (nar-n')) pred then pred, 1 else liftn (-1) (nar-n') pred, 0 in let na = if p=1 then let na = List.hd names in ((if na = Anonymous then (* can happen if evars occur in the return clause *) Name (id_of_string "x") (*Hum*) else na),knowndep) else (Anonymous,KnownNotDep) in buildrec (n'+1) pred (na::nal) lnames in buildrec 0 pred [] allnames let extract_arity_signature env0 tomatchl tmsign = let get_one_sign n tm (na,t) = match tm with | NotInd (bo,typ) -> (match t with | None -> [na,Option.map (lift n) bo,lift n typ] | Some (loc,_,_,_) -> user_err_loc (loc,"", str"Unexpected type annotation for a term of non inductive type.")) | IsInd (term,IndType(indf,realargs),_) -> let indf' = lift_inductive_family n indf in let (ind,params) = dest_ind_family indf' in let nrealargs = List.length realargs in let realnal = match t with | Some (loc,ind',nparams,realnal) -> if ind <> ind' then user_err_loc (loc,"",str "Wrong inductive type."); if List.length params <> nparams or nrealargs <> List.length realnal then anomaly "Ill-formed 'in' clause in cases"; List.rev realnal | None -> list_make nrealargs Anonymous in let arsign = fst (get_arity env0 indf') in (* let na = *) (* match na with *) (* | Name _ -> na *) (* | Anonymous -> *) (* match kind_of_term term with *) (* | Rel n -> pi1 (lookup_rel n (Environ.rel_context env0)) *) (* | _ -> Anonymous *) (* in *) (na,None,build_dependent_inductive env0 indf') ::(List.map2 (fun x (_,c,t) ->(x,c,t)) realnal arsign) in let rec buildrec n = function | [],[] -> [] | (_,tm)::ltm, x::tmsign -> let l = get_one_sign n tm x in l :: buildrec (n + List.length l) (ltm,tmsign) | _ -> assert false in List.rev (buildrec 0 (tomatchl,tmsign)) let inh_conv_coerce_to_tycon loc env evdref j tycon = match tycon with | Some p -> let (evd',j) = Coercion.inh_conv_coerce_to loc env !evdref j p in evdref := evd'; j | None -> j (* We put the tycon inside the arity signature, possibly discovering dependencies. *) let prepare_predicate_from_arsign_tycon loc env tomatchs sign arsign c = let nar = List.fold_left (fun n sign -> List.length sign + n) 0 arsign in let subst, len = List.fold_left2 (fun (subst, len) (tm, tmtype) sign -> let signlen = List.length sign in match kind_of_term tm with | Rel n when dependent tm c && signlen = 1 (* The term to match is not of a dependent type itself *) -> ((n, len) :: subst, len - signlen) | Rel _ when not (dependent tm c) && signlen > 1 (* The term is of a dependent type but does not appear in the tycon, maybe some variable in its type does. *) -> (match tmtype with NotInd _ -> (* len - signlen, subst*) assert false (* signlen > 1 *) | IsInd (_, IndType(indf,realargs),_) -> List.fold_left (fun (subst, len) arg -> match kind_of_term arg with | Rel n when dependent arg c -> ((n, len) :: subst, pred len) | _ -> (subst, pred len)) (subst, len) realargs) | _ -> (subst, len - signlen)) ([], nar) tomatchs arsign in let rec predicate lift c = match kind_of_term c with | Rel n when n > lift -> (try (* Make the predicate dependent on the matched variable *) let idx = List.assoc (n - lift) subst in mkRel (idx + lift) with Not_found -> (* A variable that is not matched, lift over the arsign. *) mkRel (n + nar)) | _ -> map_constr_with_binders succ predicate lift c in predicate 0 c (* Builds the predicate. If the predicate is dependent, its context is * made of 1+nrealargs assumptions for each matched term in an inductive * type and 1 assumption for each term not _syntactically_ in an * inductive type. * Each matched terms are independently considered dependent or not. * A type constraint but no annotation case: we try to specialize the * tycon to make the predicate if it is not closed. *) let is_dependent_on_rel x t = match kind_of_term x with Rel n -> not (noccur_with_meta n n t) | _ -> false let prepare_predicate loc typing_fun evdref env tomatchs sign tycon pred = match pred with (* No type annotation *) | None -> (match tycon with | Some (None, t) when not (noccur_with_meta 0 max_int t) -> (* If the tycon is not closed w.r.t real variables *) (* We try two different strategies *) let evdref2 = ref !evdref in let arsign = extract_arity_signature env tomatchs sign in let env' = List.fold_right push_rels arsign env in (* First strategy: we abstract the tycon wrt to the dependencies *) let names1 = List.rev (List.map (List.map pi1) arsign) in let pred1 = prepare_predicate_from_arsign_tycon loc env' tomatchs sign arsign t in let nal1,pred1 = build_initial_predicate KnownDep names1 pred1 in (* Second strategy: we build an "inversion" predicate *) let names2,pred2 = prepare_predicate_from_tycon loc true env evdref2 tomatchs sign t in let nal2,pred2 = build_initial_predicate DepUnknown names2 pred2 in [evdref, nal1, pred1; evdref2, nal2, pred2] | Some (None, t) -> (* Only one strategy: we build an "inversion" predicate *) let names,pred = prepare_predicate_from_tycon loc true env evdref tomatchs sign t in let nal,pred = build_initial_predicate DepUnknown names pred in [evdref, nal, pred] | _ -> (* No type constaints: we use two strategies *) let evdref2 = ref !evdref in let t1 = mkExistential env ~src:(loc, CasesType) evdref in (* First strategy: we pose a possibly dependent "inversion" evar *) let names1,pred1 = prepare_predicate_from_tycon loc true env evdref tomatchs sign t1 in let nal1,pred1 = build_initial_predicate DepUnknown names1 pred1 in (* Second strategy: we pose a non dependent evar *) let t2 = mkExistential env ~src:(loc, CasesType) evdref2 in let arsign = extract_arity_signature env tomatchs sign in let names2 = List.rev (List.map (List.map pi1) arsign) in let nal2,pred2 = build_initial_predicate KnownNotDep names2 t2 in [evdref, nal1, pred1; evdref2, nal2, pred2]) (* Some type annotation *) | Some rtntyp -> (* We extract the signature of the arity *) let arsign = extract_arity_signature env tomatchs sign in let env = List.fold_right push_rels arsign env in let allnames = List.rev (List.map (List.map pi1) arsign) in let predcclj = typing_fun (mk_tycon (new_Type ())) env evdref rtntyp in let _ = Option.map (fun tycon -> evdref := Coercion.inh_conv_coerces_to loc env !evdref predcclj.uj_val (lift_tycon_type (List.length arsign) tycon)) tycon in let predccl = (j_nf_isevar !evdref predcclj).uj_val in let nal,pred = build_initial_predicate KnownDep allnames predccl in [evdref, nal, pred] (**************************************************************************) (* Main entry of the matching compilation *) let compile_cases loc style (typing_fun, evdref) tycon env (predopt, tomatchl, eqns) = (* We build the matrix of patterns and right-hand-side *) let matx = matx_of_eqns env tomatchl eqns in (* We build the vector of terms to match consistently with the *) (* constructors found in patterns *) let tomatchs = coerce_to_indtype typing_fun evdref env matx tomatchl in (* If an elimination predicate is provided, we check it is compatible with the type of arguments to match; if none is provided, we build alternative possible predicates *) let sign = List.map snd tomatchl in let preds = prepare_predicate loc typing_fun evdref env tomatchs sign tycon predopt in let compile_for_one_predicate (myevdref,nal,pred) = (* We push the initial terms to match and push their alias to rhs' envs *) (* names of aliases will be recovered from patterns (hence Anonymous *) (* here) *) let initial_pushed = List.map2 (fun tm na -> Pushed(tm,[],na)) tomatchs nal in (* A typing function that provides with a canonical term for absurd cases*) let typing_fun tycon env evdref = function | Some t -> typing_fun tycon env evdref t | None -> coq_unit_judge () in let pb = { env = env; evdref = myevdref; pred = pred; tomatch = initial_pushed; history = start_history (List.length initial_pushed); mat = matx; caseloc = loc; casestyle = style; typing_function = typing_fun } in let j = compile pb in evdref := !myevdref; j in (* Return the term compiled with the first possible elimination *) (* predicate for which the compilation succeeds *) let j = list_try_compile compile_for_one_predicate preds in (* We check for unused patterns *) List.iter (check_unused_pattern env) matx; (* We coerce to the tycon (if an elim predicate was provided) *) inh_conv_coerce_to_tycon loc env evdref j tycon end