(************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) (* InProp | Prop Pos -> InSet | Type _ -> InType (********************************************************************) (* Constructions as implemented *) (********************************************************************) type cast_kind = VMcast | DEFAULTcast (* [constr array] is an instance matching definitional [named_context] in the same order (i.e. last argument first) *) type 'constr pexistential = existential_key * 'constr array type ('constr, 'types) prec_declaration = name array * 'types array * 'constr array type ('constr, 'types) pfixpoint = (int array * int) * ('constr, 'types) prec_declaration type ('constr, 'types) pcofixpoint = int * ('constr, 'types) prec_declaration (* [Var] is used for named variables and [Rel] for variables as de Bruijn indices. *) type ('constr, 'types) kind_of_term = | Rel of int | Var of identifier | Meta of metavariable | Evar of 'constr pexistential | Sort of sorts | Cast of 'constr * cast_kind * 'types | Prod of name * 'types * 'types | Lambda of name * 'types * 'constr | LetIn of name * 'constr * 'types * 'constr | App of 'constr * 'constr array | Const of constant | Ind of inductive | Construct of constructor | Case of case_info * 'constr * 'constr * 'constr array | Fix of ('constr, 'types) pfixpoint | CoFix of ('constr, 'types) pcofixpoint (* Experimental, used in Presburger contrib *) type ('constr, 'types) kind_of_type = | SortType of sorts | CastType of 'types * 'types | ProdType of name * 'types * 'types | LetInType of name * 'constr * 'types * 'types | AtomicType of 'constr * 'constr array let kind_of_type = function | Sort s -> SortType s | Cast (c,_,t) -> CastType (c, t) | Prod (na,t,c) -> ProdType (na, t, c) | LetIn (na,b,t,c) -> LetInType (na, b, t, c) | App (c,l) -> AtomicType (c, l) | (Rel _ | Meta _ | Var _ | Evar _ | Const _ | Case _ | Fix _ | CoFix _ | Ind _ as c) -> AtomicType (c,[||]) | (Lambda _ | Construct _) -> failwith "Not a type" (* constr is the fixpoint of the previous type. Requires option -rectypes of the Caml compiler to be set *) type constr = (constr,constr) kind_of_term type existential = existential_key * constr array type rec_declaration = name array * constr array * constr array type fixpoint = (int array * int) * rec_declaration type cofixpoint = int * rec_declaration (***************************) (* Hash-consing functions *) (***************************) (* Hash-consing of [constr] does not use the module [Hashcons] because [Hashcons] is not efficient on deep tree-like data structures. Indeed, [Hashcons] is based the (very efficient) generic hash function [Hashtbl.hash], which computes the hash key through a depth bounded traversal of the data structure to be hashed. As a consequence, for a deep [constr] like the natural number 1000 (S (S (... (S O)))), the same hash is assigned to all the sub [constr]s greater than the maximal depth handled by [Hashtbl.hash]. This entails a huge number of collisions in the hash table and leads to cubic hash-consing in this worst-case. In order to compute a hash key that is independent of the data structure depth while being constant-time, an incremental hashing function must be devised. A standard implementation creates a cache of the hashing function by decorating each node of the hash-consed data structure with its hash key. In that case, the hash function can deduce the hash key of a toplevel data structure by a local computation based on the cache held on its substructures. Unfortunately, this simple implementation introduces a space overhead that is damageable for the hash-consing of small [constr]s (the most common case). One can think of an heterogeneous distribution of caches on smartly chosen nodes, but this is forbidden by the use of generic equality in Coq source code. (Indeed, this forces each [constr] to have a unique canonical representation.) Given that hash-consing proceeds inductively, we can nonetheless computes the hash key incrementally during hash-consing by changing a little the signature of the hash-consing function: it now returns both the hash-consed term and its hash key. This simple solution is implemented in the following code: it does not introduce a space overhead in [constr], that's why the efficiency is unchanged for small [constr]s. Besides, it does handle deep [constr]s without introducing an unreasonable number of collisions in the hash table. Some benchmarks make us think that this implementation of hash-consing is linear in the size of the hash-consed data structure for our daily use of Coq. *) let array_eqeq t1 t2 = Array.length t1 = Array.length t2 && let rec aux i = (i = Array.length t1) || (t1.(i) == t2.(i) && aux (i + 1)) in aux 0 let comp_term t1 t2 = match t1, t2 with | Rel n1, Rel n2 -> n1 == n2 | Meta m1, Meta m2 -> m1 == m2 | Var id1, Var id2 -> id1 == id2 | Sort s1, Sort s2 -> s1 == s2 | Cast (c1,k1,t1), Cast (c2,k2,t2) -> c1 == c2 & k1 == k2 & t1 == t2 | Prod (n1,t1,c1), Prod (n2,t2,c2) -> n1 == n2 & t1 == t2 & c1 == c2 | Lambda (n1,t1,c1), Lambda (n2,t2,c2) -> n1 == n2 & t1 == t2 & c1 == c2 | LetIn (n1,b1,t1,c1), LetIn (n2,b2,t2,c2) -> n1 == n2 & b1 == b2 & t1 == t2 & c1 == c2 | App (c1,l1), App (c2,l2) -> c1 == c2 & array_eqeq l1 l2 | Evar (e1,l1), Evar (e2,l2) -> e1 = e2 & array_eqeq l1 l2 | Const c1, Const c2 -> c1 == c2 | Ind (sp1,i1), Ind (sp2,i2) -> sp1 == sp2 & i1 = i2 | Construct ((sp1,i1),j1), Construct ((sp2,i2),j2) -> sp1 == sp2 & i1 = i2 & j1 = j2 | Case (ci1,p1,c1,bl1), Case (ci2,p2,c2,bl2) -> ci1 == ci2 & p1 == p2 & c1 == c2 & array_eqeq bl1 bl2 | Fix (ln1,(lna1,tl1,bl1)), Fix (ln2,(lna2,tl2,bl2)) -> ln1 = ln2 & array_eqeq lna1 lna2 & array_eqeq tl1 tl2 & array_eqeq bl1 bl2 | CoFix(ln1,(lna1,tl1,bl1)), CoFix(ln2,(lna2,tl2,bl2)) -> ln1 = ln2 & array_eqeq lna1 lna2 & array_eqeq tl1 tl2 & array_eqeq bl1 bl2 | _ -> false (* The following module is a specialized version of [Hashtbl] that is a better space saver. Actually, [Hashcons] instanciates [Hashtbl.t] with [constr] used both as a key and as an image. Thus, in each cell of the internal bucketlist, there are two representations of the same value. In this implementation, there is only one. Besides, the responsibility of computing the hash function is now given to the caller, which makes possible the interleaving of the hash key computation and the hash-consing. *) module H : sig (* [may_add_and_get key constr] uses [key] to look for [constr] in the hash table [H]. If [constr] is [H], returns the specific representation that is stored in [H]. Otherwise, [constr] is stored in [H] and will be used as the canonical representation of this value in the future. *) val may_add_and_get : int -> constr -> constr end = struct type bucketlist = Empty | Cons of constr * int * bucketlist let initial_size = 19991 let table_data = ref (Array.make initial_size Empty) let table_size = ref 0 let resize () = let odata = !table_data in let osize = Array.length odata in let nsize = min (2 * osize + 1) Sys.max_array_length in if nsize <> osize then begin let ndata = Array.create nsize Empty in let rec insert_bucket = function | Empty -> () | Cons (key, hash, rest) -> let nidx = hash mod nsize in ndata.(nidx) <- Cons (key, hash, ndata.(nidx)); insert_bucket rest in for i = 0 to osize - 1 do insert_bucket odata.(i) done; table_data := ndata end let add hash key = let odata = !table_data in let osize = Array.length odata in let i = hash mod osize in odata.(i) <- Cons (key, hash, odata.(i)); incr table_size; if !table_size > osize lsl 1 then resize () let find_rec hash key bucket = let rec aux = function | Empty -> add hash key; key | Cons (k, h, rest) -> if hash == h && comp_term key k then k else aux rest in aux bucket let may_add_and_get hash key = let odata = !table_data in match odata.(hash mod (Array.length odata)) with | Empty -> add hash key; key | Cons (k1, h1, rest1) -> if hash == h1 && comp_term key k1 then k1 else match rest1 with | Empty -> add hash key; key | Cons (k2, h2, rest2) -> if hash == h2 && comp_term key k2 then k2 else match rest2 with | Empty -> add hash key; key | Cons (k3, h3, rest3) -> if hash == h2 && comp_term key k3 then k3 else find_rec hash key rest3 end (* These are helper functions to combine the hash keys in a similar way as [Hashtbl.hash] does. The constants [alpha] and [beta] must be prime numbers. There were chosen empirically. Notice that the problem of hashing trees is hard and there are plenty of study on this topic. Therefore, there must be room for improvement here. *) let ghash = Hashtbl.hash let alpha = 65599 let beta = 7 let combine2 x y = x * alpha + y let combine3 x y z = combine2 x (combine2 y z) let combine4 x y z t = combine2 x (combine3 y z t) let combine = combine2 let combinesmall x y = beta * x + y (* [hcons_term hash_consing_functions constr] computes an hash-consed representation for [constr] using [hash_consing_functions] on leaves. *) let hcons_term (sh_sort,sh_con,sh_kn,sh_na,sh_id) = let rec hash_term_array t = let accu = ref 0 in for i = 0 to Array.length t - 1 do let x, h = sh_rec t.(i) in accu := combine !accu h; t.(i) <- x done; (t, !accu) and hash_term t = match t with | Var i -> (Var (sh_id i), combinesmall 1 (ghash i)) | Sort s -> (Sort (sh_sort s), combinesmall 2 (ghash s)) | Cast (c, k, t) -> let c, hc = sh_rec c in let t, ht = sh_rec t in (Cast (c, k, t), combinesmall 3 (combine3 hc (ghash k) ht)) | Prod (na,t,c) -> let t, ht = sh_rec t and c, hc = sh_rec c in (Prod (sh_na na, t, c), combinesmall 4 (combine3 (ghash na) ht hc)) | Lambda (na,t,c) -> let t, ht = sh_rec t and c, hc = sh_rec c in (Lambda (sh_na na, t, c), combinesmall 5 (combine3 (ghash na) ht hc)) | LetIn (na,b,t,c) -> let b, hb = sh_rec b in let t, ht = sh_rec t in let c, hc = sh_rec c in (LetIn (sh_na na, b, t, c), combinesmall 6 (combine4 (ghash na) hb ht hc)) | App (c,l) -> let c, hc = sh_rec c in let l, hl = hash_term_array l in (App (c, l), combinesmall 7 (combine hl hc)) | Evar (e,l) -> let l, hl = hash_term_array l in (Evar (e, l), combinesmall 8 (combine (ghash e) hl)) | Const c -> (Const (sh_con c), combinesmall 9 (ghash c)) | Ind (kn,i) -> (Ind (sh_kn kn, i), combinesmall 9 (combine (ghash kn) i)) | Construct ((kn,i),j) -> (Construct ((sh_kn kn, i), j), combinesmall 10 (combine3 (ghash kn) i j)) | Case (ci,p,c,bl) -> (* TO DO: extract ind_kn *) let p, hp = sh_rec p and c, hc = sh_rec c in let bl, hbl = hash_term_array bl in let hbl = combine (combine hc hp) hbl in (Case (ci, p, c, bl), combinesmall 11 hbl) | Fix (ln,(lna,tl,bl)) -> let bl, hbl = hash_term_array bl in let tl, htl = hash_term_array tl in let h = combine hbl htl in Array.iteri (fun i x -> lna.(i) <- sh_na x) lna; (Fix (ln,(lna, tl, bl)), combinesmall 13 (combine (ghash lna) h)) | CoFix(ln,(lna,tl,bl)) -> let bl, hbl = hash_term_array bl in let tl, htl = hash_term_array tl in let h = combine hbl htl in Array.iteri (fun i x -> lna.(i) <- sh_na x) lna; (CoFix (ln, (lna, tl, bl)), combinesmall 14 (combine (ghash lna) h)) | Meta n -> (Meta n, combinesmall 15 n) | Rel n -> (Rel n, combinesmall 16 n) and sh_rec t = let (y, h) = hash_term t in (* [h] must be positive. *) let h = h land 0x3FFFFFFF in (H.may_add_and_get h y, h) in fun t -> fst (sh_rec t) (* Constructs a DeBrujin index with number n *) let rels = [|Rel 1;Rel 2;Rel 3;Rel 4;Rel 5;Rel 6;Rel 7; Rel 8; Rel 9;Rel 10;Rel 11;Rel 12;Rel 13;Rel 14;Rel 15; Rel 16|] let mkRel n = if 0 Cast (c,k1,t2) | _ -> Cast (t1,k2,t2) (* Constructs the product (x:t1)t2 *) let mkProd (x,t1,t2) = Prod (x,t1,t2) (* Constructs the abstraction [x:t1]t2 *) let mkLambda (x,t1,t2) = Lambda (x,t1,t2) (* Constructs [x=c_1:t]c_2 *) let mkLetIn (x,c1,t,c2) = LetIn (x,c1,t,c2) (* If lt = [t1; ...; tn], constructs the application (t1 ... tn) *) (* We ensure applicative terms have at least one argument and the function is not itself an applicative term *) let mkApp (f, a) = if Array.length a = 0 then f else match f with | App (g, cl) -> App (g, Array.append cl a) | _ -> App (f, a) (* Constructs a constant *) (* The array of terms correspond to the variables introduced in the section *) let mkConst c = Const c (* Constructs an existential variable *) let mkEvar e = Evar e (* Constructs the ith (co)inductive type of the block named kn *) (* The array of terms correspond to the variables introduced in the section *) let mkInd m = Ind m (* Constructs the jth constructor of the ith (co)inductive type of the block named kn. The array of terms correspond to the variables introduced in the section *) let mkConstruct c = Construct c (* Constructs the term

Case c of c1 | c2 .. | cn end *) let mkCase (ci, p, c, ac) = Case (ci, p, c, ac) let mkFix fix = Fix fix let mkCoFix cofix = CoFix cofix let kind_of_term c = c (************************************************************************) (* kind_of_term = constructions as seen by the user *) (************************************************************************) (* User view of [constr]. For [App], it is ensured there is at least one argument and the function is not itself an applicative term *) let kind_of_term = kind_of_term (**********************************************************************) (* Non primitive term destructors *) (**********************************************************************) (* Destructor operations : partial functions Raise invalid_arg "dest*" if the const has not the expected form *) (* Destructs a DeBrujin index *) let destRel c = match kind_of_term c with | Rel n -> n | _ -> invalid_arg "destRel" (* Destructs an existential variable *) let destMeta c = match kind_of_term c with | Meta n -> n | _ -> invalid_arg "destMeta" let isMeta c = match kind_of_term c with Meta _ -> true | _ -> false (* Destructs a variable *) let destVar c = match kind_of_term c with | Var id -> id | _ -> invalid_arg "destVar" (* Destructs a type *) let isSort c = match kind_of_term c with | Sort s -> true | _ -> false let destSort c = match kind_of_term c with | Sort s -> s | _ -> invalid_arg "destSort" let rec isprop c = match kind_of_term c with | Sort (Prop _) -> true | Cast (c,_,_) -> isprop c | _ -> false let rec is_Prop c = match kind_of_term c with | Sort (Prop Null) -> true | Cast (c,_,_) -> is_Prop c | _ -> false let rec is_Set c = match kind_of_term c with | Sort (Prop Pos) -> true | Cast (c,_,_) -> is_Set c | _ -> false let rec is_Type c = match kind_of_term c with | Sort (Type _) -> true | Cast (c,_,_) -> is_Type c | _ -> false let is_small = function | Prop _ -> true | _ -> false let iskind c = isprop c or is_Type c (* Tests if an evar *) let isEvar c = match kind_of_term c with Evar _ -> true | _ -> false let isEvar_or_Meta c = match kind_of_term c with | Evar _ | Meta _ -> true | _ -> false (* Destructs a casted term *) let destCast c = match kind_of_term c with | Cast (t1,k,t2) -> (t1,k,t2) | _ -> invalid_arg "destCast" let isCast c = match kind_of_term c with Cast _ -> true | _ -> false (* Tests if a de Bruijn index *) let isRel c = match kind_of_term c with Rel _ -> true | _ -> false (* Tests if a variable *) let isVar c = match kind_of_term c with Var _ -> true | _ -> false (* Tests if an inductive *) let isInd c = match kind_of_term c with Ind _ -> true | _ -> false (* Destructs the product (x:t1)t2 *) let destProd c = match kind_of_term c with | Prod (x,t1,t2) -> (x,t1,t2) | _ -> invalid_arg "destProd" let isProd c = match kind_of_term c with | Prod _ -> true | _ -> false (* Destructs the abstraction [x:t1]t2 *) let destLambda c = match kind_of_term c with | Lambda (x,t1,t2) -> (x,t1,t2) | _ -> invalid_arg "destLambda" let isLambda c = match kind_of_term c with | Lambda _ -> true | _ -> false (* Destructs the let [x:=b:t1]t2 *) let destLetIn c = match kind_of_term c with | LetIn (x,b,t1,t2) -> (x,b,t1,t2) | _ -> invalid_arg "destProd" let isLetIn c = match kind_of_term c with LetIn _ -> true | _ -> false (* Destructs an application *) let destApp c = match kind_of_term c with | App (f,a) -> (f, a) | _ -> invalid_arg "destApplication" let destApplication = destApp let isApp c = match kind_of_term c with App _ -> true | _ -> false (* Destructs a constant *) let destConst c = match kind_of_term c with | Const kn -> kn | _ -> invalid_arg "destConst" let isConst c = match kind_of_term c with Const _ -> true | _ -> false (* Destructs an existential variable *) let destEvar c = match kind_of_term c with | Evar (kn, a as r) -> r | _ -> invalid_arg "destEvar" (* Destructs a (co)inductive type named kn *) let destInd c = match kind_of_term c with | Ind (kn, a as r) -> r | _ -> invalid_arg "destInd" (* Destructs a constructor *) let destConstruct c = match kind_of_term c with | Construct (kn, a as r) -> r | _ -> invalid_arg "dest" let isConstruct c = match kind_of_term c with Construct _ -> true | _ -> false (* Destructs a term

Case c of lc1 | lc2 .. | lcn end *) let destCase c = match kind_of_term c with | Case (ci,p,c,v) -> (ci,p,c,v) | _ -> anomaly "destCase" let isCase c = match kind_of_term c with Case _ -> true | _ -> false let destFix c = match kind_of_term c with | Fix fix -> fix | _ -> invalid_arg "destFix" let isFix c = match kind_of_term c with Fix _ -> true | _ -> false let destCoFix c = match kind_of_term c with | CoFix cofix -> cofix | _ -> invalid_arg "destCoFix" let isCoFix c = match kind_of_term c with CoFix _ -> true | _ -> false (******************************************************************) (* Cast management *) (******************************************************************) let rec strip_outer_cast c = match kind_of_term c with | Cast (c,_,_) -> strip_outer_cast c | _ -> c (* Fonction spéciale qui laisse les cast clés sous les Fix ou les Case *) let under_outer_cast f c = match kind_of_term c with | Cast (b,k,t) -> mkCast (f b, k, f t) | _ -> f c let rec under_casts f c = match kind_of_term c with | Cast (c,k,t) -> mkCast (under_casts f c, k, t) | _ -> f c (******************************************************************) (* Flattening and unflattening of embedded applications and casts *) (******************************************************************) (* flattens application lists throwing casts in-between *) let rec collapse_appl c = match kind_of_term c with | App (f,cl) -> let rec collapse_rec f cl2 = match kind_of_term (strip_outer_cast f) with | App (g,cl1) -> collapse_rec g (Array.append cl1 cl2) | _ -> mkApp (f,cl2) in collapse_rec f cl | _ -> c let decompose_app c = match kind_of_term c with | App (f,cl) -> (f, Array.to_list cl) | _ -> (c,[]) (****************************************************************************) (* Functions to recur through subterms *) (****************************************************************************) (* [fold_constr f acc c] folds [f] on the immediate subterms of [c] starting from [acc] and proceeding from left to right according to the usual representation of the constructions; it is not recursive *) let fold_constr f acc c = match kind_of_term c with | (Rel _ | Meta _ | Var _ | Sort _ | Const _ | Ind _ | Construct _) -> acc | Cast (c,_,t) -> f (f acc c) t | Prod (_,t,c) -> f (f acc t) c | Lambda (_,t,c) -> f (f acc t) c | LetIn (_,b,t,c) -> f (f (f acc b) t) c | App (c,l) -> Array.fold_left f (f acc c) l | Evar (_,l) -> Array.fold_left f acc l | Case (_,p,c,bl) -> Array.fold_left f (f (f acc p) c) bl | Fix (_,(lna,tl,bl)) -> let fd = array_map3 (fun na t b -> (na,t,b)) lna tl bl in Array.fold_left (fun acc (na,t,b) -> f (f acc t) b) acc fd | CoFix (_,(lna,tl,bl)) -> let fd = array_map3 (fun na t b -> (na,t,b)) lna tl bl in Array.fold_left (fun acc (na,t,b) -> f (f acc t) b) acc fd (* [iter_constr f c] iters [f] on the immediate subterms of [c]; it is not recursive and the order with which subterms are processed is not specified *) let iter_constr f c = match kind_of_term c with | (Rel _ | Meta _ | Var _ | Sort _ | Const _ | Ind _ | Construct _) -> () | Cast (c,_,t) -> f c; f t | Prod (_,t,c) -> f t; f c | Lambda (_,t,c) -> f t; f c | LetIn (_,b,t,c) -> f b; f t; f c | App (c,l) -> f c; Array.iter f l | Evar (_,l) -> Array.iter f l | Case (_,p,c,bl) -> f p; f c; Array.iter f bl | Fix (_,(_,tl,bl)) -> Array.iter f tl; Array.iter f bl | CoFix (_,(_,tl,bl)) -> Array.iter f tl; Array.iter f bl (* [iter_constr_with_binders g f n c] iters [f n] on the immediate subterms of [c]; it carries an extra data [n] (typically a lift index) which is processed by [g] (which typically add 1 to [n]) at each binder traversal; it is not recursive and the order with which subterms are processed is not specified *) let iter_constr_with_binders g f n c = match kind_of_term c with | (Rel _ | Meta _ | Var _ | Sort _ | Const _ | Ind _ | Construct _) -> () | Cast (c,_,t) -> f n c; f n t | Prod (_,t,c) -> f n t; f (g n) c | Lambda (_,t,c) -> f n t; f (g n) c | LetIn (_,b,t,c) -> f n b; f n t; f (g n) c | App (c,l) -> f n c; Array.iter (f n) l | Evar (_,l) -> Array.iter (f n) l | Case (_,p,c,bl) -> f n p; f n c; Array.iter (f n) bl | Fix (_,(_,tl,bl)) -> Array.iter (f n) tl; Array.iter (f (iterate g (Array.length tl) n)) bl | CoFix (_,(_,tl,bl)) -> Array.iter (f n) tl; Array.iter (f (iterate g (Array.length tl) n)) bl (* [map_constr f c] maps [f] on the immediate subterms of [c]; it is not recursive and the order with which subterms are processed is not specified *) let map_constr f c = match kind_of_term c with | (Rel _ | Meta _ | Var _ | Sort _ | Const _ | Ind _ | Construct _) -> c | Cast (c,k,t) -> mkCast (f c, k, f t) | Prod (na,t,c) -> mkProd (na, f t, f c) | Lambda (na,t,c) -> mkLambda (na, f t, f c) | LetIn (na,b,t,c) -> mkLetIn (na, f b, f t, f c) | App (c,l) -> mkApp (f c, Array.map f l) | Evar (e,l) -> mkEvar (e, Array.map f l) | Case (ci,p,c,bl) -> mkCase (ci, f p, f c, Array.map f bl) | Fix (ln,(lna,tl,bl)) -> mkFix (ln,(lna,Array.map f tl,Array.map f bl)) | CoFix(ln,(lna,tl,bl)) -> mkCoFix (ln,(lna,Array.map f tl,Array.map f bl)) (* [map_constr_with_binders g f n c] maps [f n] on the immediate subterms of [c]; it carries an extra data [n] (typically a lift index) which is processed by [g] (which typically add 1 to [n]) at each binder traversal; it is not recursive and the order with which subterms are processed is not specified *) let map_constr_with_binders g f l c = match kind_of_term c with | (Rel _ | Meta _ | Var _ | Sort _ | Const _ | Ind _ | Construct _) -> c | Cast (c,k,t) -> mkCast (f l c, k, f l t) | Prod (na,t,c) -> mkProd (na, f l t, f (g l) c) | Lambda (na,t,c) -> mkLambda (na, f l t, f (g l) c) | LetIn (na,b,t,c) -> mkLetIn (na, f l b, f l t, f (g l) c) | App (c,al) -> mkApp (f l c, Array.map (f l) al) | Evar (e,al) -> mkEvar (e, Array.map (f l) al) | Case (ci,p,c,bl) -> mkCase (ci, f l p, f l c, Array.map (f l) bl) | Fix (ln,(lna,tl,bl)) -> let l' = iterate g (Array.length tl) l in mkFix (ln,(lna,Array.map (f l) tl,Array.map (f l') bl)) | CoFix(ln,(lna,tl,bl)) -> let l' = iterate g (Array.length tl) l in mkCoFix (ln,(lna,Array.map (f l) tl,Array.map (f l') bl)) (* [compare_constr f c1 c2] compare [c1] and [c2] using [f] to compare the immediate subterms of [c1] of [c2] if needed; Cast's, application associativity, binders name and Cases annotations are not taken into account *) let compare_constr f t1 t2 = match kind_of_term t1, kind_of_term t2 with | Rel n1, Rel n2 -> n1 = n2 | Meta m1, Meta m2 -> m1 = m2 | Var id1, Var id2 -> id1 = id2 | Sort s1, Sort s2 -> s1 = s2 | Cast (c1,_,_), _ -> f c1 t2 | _, Cast (c2,_,_) -> f t1 c2 | Prod (_,t1,c1), Prod (_,t2,c2) -> f t1 t2 & f c1 c2 | Lambda (_,t1,c1), Lambda (_,t2,c2) -> f t1 t2 & f c1 c2 | LetIn (_,b1,t1,c1), LetIn (_,b2,t2,c2) -> f b1 b2 & f t1 t2 & f c1 c2 | App (c1,l1), App (c2,l2) -> if Array.length l1 = Array.length l2 then f c1 c2 & array_for_all2 f l1 l2 else let (h1,l1) = decompose_app t1 in let (h2,l2) = decompose_app t2 in if List.length l1 = List.length l2 then f h1 h2 & List.for_all2 f l1 l2 else false | Evar (e1,l1), Evar (e2,l2) -> e1 = e2 & array_for_all2 f l1 l2 | Const c1, Const c2 -> eq_constant c1 c2 | Ind c1, Ind c2 -> eq_ind c1 c2 | Construct c1, Construct c2 -> eq_constructor c1 c2 | Case (_,p1,c1,bl1), Case (_,p2,c2,bl2) -> f p1 p2 & f c1 c2 & array_for_all2 f bl1 bl2 | Fix (ln1,(_,tl1,bl1)), Fix (ln2,(_,tl2,bl2)) -> ln1 = ln2 & array_for_all2 f tl1 tl2 & array_for_all2 f bl1 bl2 | CoFix(ln1,(_,tl1,bl1)), CoFix(ln2,(_,tl2,bl2)) -> ln1 = ln2 & array_for_all2 f tl1 tl2 & array_for_all2 f bl1 bl2 | _ -> false (***************************************************************************) (* Type of assumptions *) (***************************************************************************) type types = constr type strategy = types option type named_declaration = identifier * constr option * types type rel_declaration = name * constr option * types let map_named_declaration f (id, v, ty) = (id, Option.map f v, f ty) let map_rel_declaration = map_named_declaration let fold_named_declaration f (_, v, ty) a = f ty (Option.fold_right f v a) let fold_rel_declaration = fold_named_declaration let exists_named_declaration f (_, v, ty) = Option.cata f false v || f ty let exists_rel_declaration f (_, v, ty) = Option.cata f false v || f ty let for_all_named_declaration f (_, v, ty) = Option.cata f true v && f ty let for_all_rel_declaration f (_, v, ty) = Option.cata f true v && f ty (***************************************************************************) (* Type of local contexts (telescopes) *) (***************************************************************************) (*s Signatures of ordered optionally named variables, intended to be accessed by de Bruijn indices (to represent bound variables) *) type rel_context = rel_declaration list let empty_rel_context = [] let add_rel_decl d ctxt = d::ctxt let rec lookup_rel n sign = match n, sign with | 1, decl :: _ -> decl | n, _ :: sign -> lookup_rel (n-1) sign | _, [] -> raise Not_found let rel_context_length = List.length let rel_context_nhyps hyps = let rec nhyps acc = function | [] -> acc | (_,None,_)::hyps -> nhyps (1+acc) hyps | (_,Some _,_)::hyps -> nhyps acc hyps in nhyps 0 hyps (****************************************************************************) (* Functions for dealing with constr terms *) (****************************************************************************) (*********************) (* Occurring *) (*********************) exception LocalOccur (* (closedn n M) raises FreeVar if a variable of height greater than n occurs in M, returns () otherwise *) let closedn n c = let rec closed_rec n c = match kind_of_term c with | Rel m -> if m>n then raise LocalOccur | _ -> iter_constr_with_binders succ closed_rec n c in try closed_rec n c; true with LocalOccur -> false (* [closed0 M] is true iff [M] is a (deBruijn) closed term *) let closed0 = closedn 0 (* (noccurn n M) returns true iff (Rel n) does NOT occur in term M *) let noccurn n term = let rec occur_rec n c = match kind_of_term c with | Rel m -> if m = n then raise LocalOccur | _ -> iter_constr_with_binders succ occur_rec n c in try occur_rec n term; true with LocalOccur -> false (* (noccur_between n m M) returns true iff (Rel p) does NOT occur in term M for n <= p < n+m *) let noccur_between 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 try occur_rec n term; true with LocalOccur -> false (* Checking function for terms containing existential variables. The function [noccur_with_meta] considers the fact that each existential variable (as well as each isevar) in the term appears applied to its local context, which may contain the CoFix variables. These occurrences of CoFix variables are not considered *) let noccur_with_meta n m term = let rec occur_rec n c = match kind_of_term c with | Rel p -> if n<=p & p (match kind_of_term f with | Cast (c,_,_) when isMeta c -> () | Meta _ -> () | _ -> iter_constr_with_binders succ occur_rec n c) | Evar (_, _) -> () | _ -> iter_constr_with_binders succ occur_rec n c in try (occur_rec n term; true) with LocalOccur -> false (*********************) (* Lifting *) (*********************) (* The generic lifting function *) let rec exliftn el c = match kind_of_term c with | Rel i -> mkRel(reloc_rel i el) | _ -> map_constr_with_binders el_lift exliftn el c (* Lifting the binding depth across k bindings *) let liftn k n = match el_liftn (pred n) (el_shft k ELID) with | ELID -> (fun c -> c) | el -> exliftn el let lift k = liftn k 1 (*********************) (* Substituting *) (*********************) (* (subst1 M c) substitutes M for Rel(1) in c we generalise it to (substl [M1,...,Mn] c) which substitutes in parallel M1,...,Mn for respectively Rel(1),...,Rel(n) in c *) (* 1st : general case *) type info = Closed | Open | Unknown type 'a substituend = { mutable sinfo: info; sit: 'a } let rec lift_substituend depth s = match s.sinfo with | Closed -> s.sit | Open -> lift depth s.sit | Unknown -> s.sinfo <- if closed0 s.sit then Closed else Open; lift_substituend depth s let make_substituend c = { sinfo=Unknown; sit=c } let substn_many lamv n c = let lv = Array.length lamv in if lv = 0 then c else let rec substrec depth c = match kind_of_term c with | Rel k -> if k<=depth then c else if k-depth <= lv then lift_substituend depth lamv.(k-depth-1) else mkRel (k-lv) | _ -> map_constr_with_binders succ substrec depth c in substrec n c (* let substkey = Profile.declare_profile "substn_many";; let substn_many lamv n c = Profile.profile3 substkey substn_many lamv n c;; *) let substnl laml n = substn_many (Array.map make_substituend (Array.of_list laml)) n let substl laml = substnl laml 0 let subst1 lam = substl [lam] let substnl_decl laml k = map_rel_declaration (substnl laml k) let substl_decl laml = substnl_decl laml 0 let subst1_decl lam = substl_decl [lam] let substnl_named laml k = map_named_declaration (substnl laml k) let substl_named_decl = substl_decl let subst1_named_decl = subst1_decl (* (thin_val sigma) removes identity substitutions from sigma *) let rec thin_val = function | [] -> [] | (((id,{ sit = v }) as s)::tl) when isVar v -> if id = destVar v then thin_val tl else s::(thin_val tl) | h::tl -> h::(thin_val tl) (* (replace_vars sigma M) applies substitution sigma to term M *) let replace_vars var_alist = let var_alist = List.map (fun (str,c) -> (str,make_substituend c)) var_alist in let var_alist = thin_val var_alist in let rec substrec n c = match kind_of_term c with | Var x -> (try lift_substituend n (List.assoc x var_alist) with Not_found -> c) | _ -> map_constr_with_binders succ substrec n c in if var_alist = [] then (function x -> x) else substrec 0 (* let repvarkey = Profile.declare_profile "replace_vars";; let replace_vars vl c = Profile.profile2 repvarkey replace_vars vl c ;; *) (* (subst_var str t) substitute (VAR str) by (Rel 1) in t *) let subst_var str = replace_vars [(str, mkRel 1)] (* (subst_vars [id1;...;idn] t) substitute (VAR idj) by (Rel j) in t *) let substn_vars p vars = let _,subst = List.fold_left (fun (n,l) var -> ((n+1),(var,mkRel n)::l)) (p,[]) vars in replace_vars (List.rev subst) let subst_vars = substn_vars 1 (*********************) (* Term constructors *) (*********************) (* Constructs a DeBrujin index with number n *) let mkRel = mkRel (* Constructs an existential variable named "?n" *) let mkMeta = mkMeta (* Constructs a Variable named id *) let mkVar = mkVar (* Construct a type *) let mkProp = mkSort prop_sort let mkSet = mkSort set_sort let mkType u = mkSort (Type u) let mkSort = function | Prop Null -> mkProp (* Easy sharing *) | Prop Pos -> mkSet | s -> mkSort s (* Constructs the term t1::t2, i.e. the term t1 casted with the type t2 *) (* (that means t2 is declared as the type of t1) *) let mkCast = mkCast (* Constructs the product (x:t1)t2 *) let mkProd = mkProd let mkNamedProd id typ c = mkProd (Name id, typ, subst_var id c) (* Constructs the abstraction [x:t1]t2 *) let mkLambda = mkLambda let mkNamedLambda id typ c = mkLambda (Name id, typ, subst_var id c) (* Constructs [x=c_1:t]c_2 *) let mkLetIn = mkLetIn let mkNamedLetIn id c1 t c2 = mkLetIn (Name id, c1, t, subst_var id c2) (* Constructs either [(x:t)c] or [[x=b:t]c] *) let mkProd_or_LetIn (na,body,t) c = match body with | None -> mkProd (na, t, c) | Some b -> mkLetIn (na, b, t, c) let mkNamedProd_or_LetIn (id,body,t) c = match body with | None -> mkNamedProd id t c | Some b -> mkNamedLetIn id b t c (* Constructs either [[x:t]c] or [[x=b:t]c] *) let mkLambda_or_LetIn (na,body,t) c = match body with | None -> mkLambda (na, t, c) | Some b -> mkLetIn (na, b, t, c) let mkNamedLambda_or_LetIn (id,body,t) c = match body with | None -> mkNamedLambda id t c | Some b -> mkNamedLetIn id b t c (* Constructs either [(x:t)c] or [c] where [x] is replaced by [b] *) let mkProd_wo_LetIn (na,body,t) c = match body with | None -> mkProd (na, t, c) | Some b -> subst1 b c let mkNamedProd_wo_LetIn (id,body,t) c = match body with | None -> mkNamedProd id t c | Some b -> subst1 b (subst_var id c) (* non-dependent product t1 -> t2 *) let mkArrow t1 t2 = mkProd (Anonymous, t1, t2) (* If lt = [t1; ...; tn], constructs the application (t1 ... tn) *) (* We ensure applicative terms have at most one arguments and the function is not itself an applicative term *) let mkApp = mkApp (* Constructs a constant *) (* The array of terms correspond to the variables introduced in the section *) let mkConst = mkConst (* Constructs an existential variable *) let mkEvar = mkEvar (* Constructs the ith (co)inductive type of the block named kn *) (* The array of terms correspond to the variables introduced in the section *) let mkInd = mkInd (* Constructs the jth constructor of the ith (co)inductive type of the block named kn. The array of terms correspond to the variables introduced in the section *) let mkConstruct = mkConstruct (* Constructs the term

Case c of c1 | c2 .. | cn end *) let mkCase = mkCase (* If recindxs = [|i1,...in|] funnames = [|f1,...fn|] typarray = [|t1,...tn|] bodies = [|b1,...bn|] then mkFix ((recindxs,i),(funnames,typarray,bodies)) constructs the ith function of the block Fixpoint f1 [ctx1] : t1 := b1 with f2 [ctx2] : t2 := b2 ... with fn [ctxn] : tn := bn. where the lenght of the jth context is ij. *) let mkFix = mkFix (* If funnames = [|f1,...fn|] typarray = [|t1,...tn|] bodies = [|b1,...bn|] then mkCoFix (i,(funnames,typsarray,bodies)) constructs the ith function of the block CoFixpoint f1 : t1 := b1 with f2 : t2 := b2 ... with fn : tn := bn. *) let mkCoFix = mkCoFix (***************************) (* Other term constructors *) (***************************) (* prodn n [xn:Tn;..;x1:T1;Gamma] b = (x1:T1)..(xn:Tn)b *) let prodn n env b = let rec prodrec = function | (0, env, b) -> b | (n, ((v,t)::l), b) -> prodrec (n-1, l, mkProd (v,t,b)) | _ -> assert false in prodrec (n,env,b) (* compose_prod [xn:Tn;..;x1:T1] b = (x1:T1)..(xn:Tn)b *) let compose_prod l b = prodn (List.length l) l b (* lamn n [xn:Tn;..;x1:T1;Gamma] b = [x1:T1]..[xn:Tn]b *) let lamn n env b = let rec lamrec = function | (0, env, b) -> b | (n, ((v,t)::l), b) -> lamrec (n-1, l, mkLambda (v,t,b)) | _ -> assert false in lamrec (n,env,b) (* compose_lam [xn:Tn;..;x1:T1] b = [x1:T1]..[xn:Tn]b *) let compose_lam l b = lamn (List.length l) l b let applist (f,l) = mkApp (f, Array.of_list l) let applistc f l = mkApp (f, Array.of_list l) let appvect = mkApp let appvectc f l = mkApp (f,l) (* to_lambda n (x1:T1)...(xn:Tn)T = * [x1:T1]...[xn:Tn]T *) let rec to_lambda n prod = if n = 0 then prod else match kind_of_term prod with | Prod (na,ty,bd) -> mkLambda (na,ty,to_lambda (n-1) bd) | Cast (c,_,_) -> to_lambda n c | _ -> errorlabstrm "to_lambda" (mt ()) let rec to_prod n lam = if n=0 then lam else match kind_of_term lam with | Lambda (na,ty,bd) -> mkProd (na,ty,to_prod (n-1) bd) | Cast (c,_,_) -> to_prod n c | _ -> errorlabstrm "to_prod" (mt ()) (* pseudo-reduction rule: * [prod_app s (Prod(_,B)) N --> B[N] * with an strip_outer_cast on the first argument to produce a product *) let prod_app t n = match kind_of_term (strip_outer_cast t) with | Prod (_,_,b) -> subst1 n b | _ -> errorlabstrm "prod_app" (str"Needed a product, but didn't find one" ++ fnl ()) (* prod_appvect T [| a1 ; ... ; an |] -> (T a1 ... an) *) let prod_appvect t nL = Array.fold_left prod_app t nL (* prod_applist T [ a1 ; ... ; an ] -> (T a1 ... an) *) let prod_applist t nL = List.fold_left prod_app t nL let it_mkProd_or_LetIn = List.fold_left (fun c d -> mkProd_or_LetIn d c) let it_mkLambda_or_LetIn = List.fold_left (fun c d -> mkLambda_or_LetIn d c) (*********************************) (* Other term destructors *) (*********************************) (* Transforms a product term (x1:T1)..(xn:Tn)T into the pair ([(xn,Tn);...;(x1,T1)],T), where T is not a product *) let decompose_prod = let rec prodec_rec l c = match kind_of_term c with | Prod (x,t,c) -> prodec_rec ((x,t)::l) c | Cast (c,_,_) -> prodec_rec l c | _ -> l,c in prodec_rec [] (* Transforms a lambda term [x1:T1]..[xn:Tn]T into the pair ([(xn,Tn);...;(x1,T1)],T), where T is not a lambda *) let decompose_lam = let rec lamdec_rec l c = match kind_of_term c with | Lambda (x,t,c) -> lamdec_rec ((x,t)::l) c | Cast (c,_,_) -> lamdec_rec l c | _ -> l,c in lamdec_rec [] (* Given a positive integer n, transforms a product term (x1:T1)..(xn:Tn)T into the pair ([(xn,Tn);...;(x1,T1)],T) *) let decompose_prod_n n = if n < 0 then error "decompose_prod_n: integer parameter must be positive"; let rec prodec_rec l n c = if n=0 then l,c else match kind_of_term c with | Prod (x,t,c) -> prodec_rec ((x,t)::l) (n-1) c | Cast (c,_,_) -> prodec_rec l n c | _ -> error "decompose_prod_n: not enough products" in prodec_rec [] n (* Given a positive integer n, transforms a lambda term [x1:T1]..[xn:Tn]T into the pair ([(xn,Tn);...;(x1,T1)],T) *) let decompose_lam_n n = if n < 0 then error "decompose_lam_n: integer parameter must be positive"; let rec lamdec_rec l n c = if n=0 then l,c else match kind_of_term c with | Lambda (x,t,c) -> lamdec_rec ((x,t)::l) (n-1) c | Cast (c,_,_) -> lamdec_rec l n c | _ -> error "decompose_lam_n: not enough abstractions" in lamdec_rec [] n (* Transforms a product term (x1:T1)..(xn:Tn)T into the pair ([(xn,Tn);...;(x1,T1)],T), where T is not a product *) let decompose_prod_assum = let rec prodec_rec l c = match kind_of_term c with | Prod (x,t,c) -> prodec_rec (add_rel_decl (x,None,t) l) c | LetIn (x,b,t,c) -> prodec_rec (add_rel_decl (x,Some b,t) l) c | Cast (c,_,_) -> prodec_rec l c | _ -> l,c in prodec_rec empty_rel_context (* Transforms a lambda term [x1:T1]..[xn:Tn]T into the pair ([(xn,Tn);...;(x1,T1)],T), where T is not a lambda *) let decompose_lam_assum = let rec lamdec_rec l c = match kind_of_term c with | Lambda (x,t,c) -> lamdec_rec (add_rel_decl (x,None,t) l) c | LetIn (x,b,t,c) -> lamdec_rec (add_rel_decl (x,Some b,t) l) c | Cast (c,_,_) -> lamdec_rec l c | _ -> l,c in lamdec_rec empty_rel_context (* Given a positive integer n, transforms a product term (x1:T1)..(xn:Tn)T into the pair ([(xn,Tn);...;(x1,T1)],T) *) let decompose_prod_n_assum n = if n < 0 then error "decompose_prod_n_assum: integer parameter must be positive"; let rec prodec_rec l n c = if n=0 then l,c else match kind_of_term c with | Prod (x,t,c) -> prodec_rec (add_rel_decl (x,None,t) l) (n-1) c | LetIn (x,b,t,c) -> prodec_rec (add_rel_decl (x,Some b,t) l) (n-1) c | Cast (c,_,_) -> prodec_rec l n c | c -> error "decompose_prod_n_assum: not enough assumptions" in prodec_rec empty_rel_context n (* Given a positive integer n, transforms a lambda term [x1:T1]..[xn:Tn]T into the pair ([(xn,Tn);...;(x1,T1)],T) Lets in between are not expanded but turn into local definitions, but n is the actual number of destructurated lambdas. *) let decompose_lam_n_assum n = if n < 0 then error "decompose_lam_n_assum: integer parameter must be positive"; let rec lamdec_rec l n c = if n=0 then l,c else match kind_of_term c with | Lambda (x,t,c) -> lamdec_rec (add_rel_decl (x,None,t) l) (n-1) c | LetIn (x,b,t,c) -> lamdec_rec (add_rel_decl (x,Some b,t) l) n c | Cast (c,_,_) -> lamdec_rec l n c | c -> error "decompose_lam_n_assum: not enough abstractions" in lamdec_rec empty_rel_context n (* (nb_lam [na1:T1]...[nan:Tan]c) where c is not an abstraction * gives n (casts are ignored) *) let nb_lam = let rec nbrec n c = match kind_of_term c with | Lambda (_,_,c) -> nbrec (n+1) c | Cast (c,_,_) -> nbrec n c | _ -> n in nbrec 0 (* similar to nb_lam, but gives the number of products instead *) let nb_prod = let rec nbrec n c = match kind_of_term c with | Prod (_,_,c) -> nbrec (n+1) c | Cast (c,_,_) -> nbrec n c | _ -> n in nbrec 0 let prod_assum t = fst (decompose_prod_assum t) let prod_n_assum n t = fst (decompose_prod_n_assum n t) let strip_prod_assum t = snd (decompose_prod_assum t) let strip_prod t = snd (decompose_prod t) let strip_prod_n n t = snd (decompose_prod_n n t) let lam_assum t = fst (decompose_lam_assum t) let lam_n_assum n t = fst (decompose_lam_n_assum n t) let strip_lam_assum t = snd (decompose_lam_assum t) let strip_lam t = snd (decompose_lam t) let strip_lam_n n t = snd (decompose_lam_n n t) (***************************) (* Arities *) (***************************) (* An "arity" is a term of the form [[x1:T1]...[xn:Tn]s] with [s] a sort. Such a term can canonically be seen as the pair of a context of types and of a sort *) type arity = rel_context * sorts let destArity = let rec prodec_rec l c = match kind_of_term c with | Prod (x,t,c) -> prodec_rec ((x,None,t)::l) c | LetIn (x,b,t,c) -> prodec_rec ((x,Some b,t)::l) c | Cast (c,_,_) -> prodec_rec l c | Sort s -> l,s | _ -> anomaly "destArity: not an arity" in prodec_rec [] let mkArity (sign,s) = it_mkProd_or_LetIn (mkSort s) sign let rec isArity c = match kind_of_term c with | Prod (_,_,c) -> isArity c | LetIn (_,b,_,c) -> isArity (subst1 b c) | Cast (c,_,_) -> isArity c | Sort _ -> true | _ -> false (*******************************) (* alpha conversion functions *) (*******************************) (* alpha conversion : ignore print names and casts *) let rec eq_constr m n = (m==n) or compare_constr eq_constr m n let eq_constr m n = eq_constr m n (* to avoid tracing a recursive fun *) (*******************) (* hash-consing *) (*******************) module Htype = Hashcons.Make( struct type t = types type u = (constr -> constr) * (sorts -> sorts) (* let hash_sub (hc,hs) j = {body=hc j.body; typ=hs j.typ} let equal j1 j2 = j1.body==j2.body & j1.typ==j2.typ *) (**) let hash_sub (hc,hs) j = hc j let equal j1 j2 = j1==j2 (**) let hash = Hashtbl.hash end) module Hsorts = Hashcons.Make( struct type t = sorts type u = universe -> universe let hash_sub huniv = function Prop c -> Prop c | Type u -> Type (huniv u) let equal s1 s2 = match (s1,s2) with (Prop c1, Prop c2) -> c1=c2 | (Type u1, Type u2) -> u1 == u2 |_ -> false let hash = Hashtbl.hash end) let hsort = Hsorts.f let hcons_constr (hcon,hkn,hdir,hname,hident,hstr) = let hsortscci = Hashcons.simple_hcons hsort hcons1_univ in let hcci = hcons_term (hsortscci,hcon,hkn,hname,hident) in let htcci = Hashcons.simple_hcons Htype.f (hcci,hsortscci) in (hcci,htcci) let (hcons1_constr, hcons1_types) = hcons_constr (hcons_names()) (*******) (* Type of abstract machine values *) type values