(* Title: Pure/term.ML
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Author: Makarius
Simply typed lambda-calculus: types, terms, and basic operations.
*)
infix 9 $;
infixr 5 -->;
infixr --->;
infix aconv;
signature BASIC_TERM =
sig
type indexname = string * int
type class = string
type sort = class list
type arity = string * sort list * sort
datatype typ =
Type of string * typ list |
TFree of string * sort |
TVar of indexname * sort
datatype term =
Const of string * typ |
Free of string * typ |
Var of indexname * typ |
Bound of int |
Abs of string * typ * term |
$ of term * term
exception TYPE of string * typ list * term list
exception TERM of string * term list
val dummyS: sort
val dummyT: typ
val no_dummyT: typ -> typ
val --> : typ * typ -> typ
val ---> : typ list * typ -> typ
val dest_Type: typ -> string * typ list
val dest_TVar: typ -> indexname * sort
val dest_TFree: typ -> string * sort
val is_Bound: term -> bool
val is_Const: term -> bool
val is_Free: term -> bool
val is_Var: term -> bool
val is_TVar: typ -> bool
val dest_Const: term -> string * typ
val dest_Free: term -> string * typ
val dest_Var: term -> indexname * typ
val dest_comb: term -> term * term
val domain_type: typ -> typ
val range_type: typ -> typ
val dest_funT: typ -> typ * typ
val binder_types: typ -> typ list
val body_type: typ -> typ
val strip_type: typ -> typ list * typ
val type_of1: typ list * term -> typ
val type_of: term -> typ
val fastype_of1: typ list * term -> typ
val fastype_of: term -> typ
val strip_abs: term -> (string * typ) list * term
val strip_abs_body: term -> term
val strip_abs_vars: term -> (string * typ) list
val strip_qnt_body: string -> term -> term
val strip_qnt_vars: string -> term -> (string * typ) list
val list_comb: term * term list -> term
val strip_comb: term -> term * term list
val head_of: term -> term
val size_of_term: term -> int
val size_of_typ: typ -> int
val map_atyps: (typ -> typ) -> typ -> typ
val map_aterms: (term -> term) -> term -> term
val map_type_tvar: (indexname * sort -> typ) -> typ -> typ
val map_type_tfree: (string * sort -> typ) -> typ -> typ
val map_types: (typ -> typ) -> term -> term
val fold_atyps: (typ -> 'a -> 'a) -> typ -> 'a -> 'a
val fold_atyps_sorts: (typ * sort -> 'a -> 'a) -> typ -> 'a -> 'a
val fold_aterms: (term -> 'a -> 'a) -> term -> 'a -> 'a
val fold_term_types: (term -> typ -> 'a -> 'a) -> term -> 'a -> 'a
val fold_types: (typ -> 'a -> 'a) -> term -> 'a -> 'a
val burrow_types: (typ list -> typ list) -> term list -> term list
val aconv: term * term -> bool
val propT: typ
val strip_all_body: term -> term
val strip_all_vars: term -> (string * typ) list
val incr_bv: int * int * term -> term
val incr_boundvars: int -> term -> term
val add_loose_bnos: term * int * int list -> int list
val loose_bnos: term -> int list
val loose_bvar: term * int -> bool
val loose_bvar1: term * int -> bool
val subst_bounds: term list * term -> term
val subst_bound: term * term -> term
val betapply: term * term -> term
val betapplys: term * term list -> term
val subst_free: (term * term) list -> term -> term
val abstract_over: term * term -> term
val lambda: term -> term -> term
val absfree: string * typ -> term -> term
val absdummy: typ -> term -> term
val subst_atomic: (term * term) list -> term -> term
val typ_subst_atomic: (typ * typ) list -> typ -> typ
val subst_atomic_types: (typ * typ) list -> term -> term
val typ_subst_TVars: (indexname * typ) list -> typ -> typ
val subst_TVars: (indexname * typ) list -> term -> term
val subst_Vars: (indexname * term) list -> term -> term
val subst_vars: (indexname * typ) list * (indexname * term) list -> term -> term
val is_first_order: string list -> term -> bool
val maxidx_of_typ: typ -> int
val maxidx_of_typs: typ list -> int
val maxidx_of_term: term -> int
val exists_subtype: (typ -> bool) -> typ -> bool
val exists_type: (typ -> bool) -> term -> bool
val exists_subterm: (term -> bool) -> term -> bool
val exists_Const: (string * typ -> bool) -> term -> bool
end;
signature TERM =
sig
include BASIC_TERM
val aT: sort -> typ
val itselfT: typ -> typ
val a_itselfT: typ
val argument_type_of: term -> int -> typ
val abs: string * typ -> term -> term
val add_tvar_namesT: typ -> indexname list -> indexname list
val add_tvar_names: term -> indexname list -> indexname list
val add_tvarsT: typ -> (indexname * sort) list -> (indexname * sort) list
val add_tvars: term -> (indexname * sort) list -> (indexname * sort) list
val add_var_names: term -> indexname list -> indexname list
val add_vars: term -> (indexname * typ) list -> (indexname * typ) list
val add_tfree_namesT: typ -> string list -> string list
val add_tfree_names: term -> string list -> string list
val add_tfreesT: typ -> (string * sort) list -> (string * sort) list
val add_tfrees: term -> (string * sort) list -> (string * sort) list
val add_free_names: term -> string list -> string list
val add_frees: term -> (string * typ) list -> (string * typ) list
val add_const_names: term -> string list -> string list
val add_consts: term -> (string * typ) list -> (string * typ) list
val hidden_polymorphism: term -> (indexname * sort) list
val declare_typ_names: typ -> Name.context -> Name.context
val declare_term_names: term -> Name.context -> Name.context
val declare_term_frees: term -> Name.context -> Name.context
val variant_frees: term -> (string * 'a) list -> (string * 'a) list
val rename_wrt_term: term -> (string * 'a) list -> (string * 'a) list
val eq_ix: indexname * indexname -> bool
val eq_tvar: (indexname * sort) * (indexname * sort) -> bool
val eq_var: (indexname * typ) * (indexname * typ) -> bool
val aconv_untyped: term * term -> bool
val could_unify: term * term -> bool
val strip_abs_eta: int -> term -> (string * typ) list * term
val match_bvars: (term * term) -> (string * string) list -> (string * string) list
val map_abs_vars: (string -> string) -> term -> term
val rename_abs: term -> term -> term -> term option
val is_open: term -> bool
val is_dependent: term -> bool
val lambda_name: string * term -> term -> term
val close_schematic_term: term -> term
val maxidx_typ: typ -> int -> int
val maxidx_typs: typ list -> int -> int
val maxidx_term: term -> int -> int
val has_abs: term -> bool
val dest_abs: string * typ * term -> string * term
val dummy_pattern: typ -> term
val dummy: term
val dummy_prop: term
val is_dummy_pattern: term -> bool
val free_dummy_patterns: term -> Name.context -> term * Name.context
val no_dummy_patterns: term -> term
val replace_dummy_patterns: term -> int -> term * int
val is_replaced_dummy_pattern: indexname -> bool
val show_dummy_patterns: term -> term
val string_of_vname: indexname -> string
val string_of_vname': indexname -> string
end;
structure Term: TERM =
struct
(*Indexnames can be quickly renamed by adding an offset to the integer part,
for resolution.*)
type indexname = string * int;
(* Types are classified by sorts. *)
type class = string;
type sort = class list;
type arity = string * sort list * sort;
(* The sorts attached to TFrees and TVars specify the sort of that variable *)
datatype typ = Type of string * typ list
| TFree of string * sort
| TVar of indexname * sort;
(*Terms. Bound variables are indicated by depth number.
Free variables, (scheme) variables and constants have names.
An term is "closed" if every bound variable of level "lev"
is enclosed by at least "lev" abstractions.
It is possible to create meaningless terms containing loose bound vars
or type mismatches. But such terms are not allowed in rules. *)
datatype term =
Const of string * typ
| Free of string * typ
| Var of indexname * typ
| Bound of int
| Abs of string*typ*term
| op $ of term*term;
(*Errors involving type mismatches*)
exception TYPE of string * typ list * term list;
(*Errors errors involving terms*)
exception TERM of string * term list;
(*Note variable naming conventions!
a,b,c: string
f,g,h: functions (including terms of function type)
i,j,m,n: int
t,u: term
v,w: indexnames
x,y: any
A,B,C: term (denoting formulae)
T,U: typ
*)
(** Types **)
(*dummies for type-inference etc.*)
val dummyS = [""];
val dummyT = Type ("dummy", []);
fun no_dummyT typ =
let
fun check (T as Type ("dummy", _)) =
raise TYPE ("Illegal occurrence of '_' dummy type", [T], [])
| check (Type (_, Ts)) = List.app check Ts
| check _ = ();
in check typ; typ end;
fun S --> T = Type("fun",[S,T]);
(*handy for multiple args: [T1,...,Tn]--->T gives T1-->(T2--> ... -->T)*)
val op ---> = Library.foldr (op -->);
fun dest_Type (Type x) = x
| dest_Type T = raise TYPE ("dest_Type", [T], []);
fun dest_TVar (TVar x) = x
| dest_TVar T = raise TYPE ("dest_TVar", [T], []);
fun dest_TFree (TFree x) = x
| dest_TFree T = raise TYPE ("dest_TFree", [T], []);
(** Discriminators **)
fun is_Bound (Bound _) = true
| is_Bound _ = false;
fun is_Const (Const _) = true
| is_Const _ = false;
fun is_Free (Free _) = true
| is_Free _ = false;
fun is_Var (Var _) = true
| is_Var _ = false;
fun is_TVar (TVar _) = true
| is_TVar _ = false;
(** Destructors **)
fun dest_Const (Const x) = x
| dest_Const t = raise TERM("dest_Const", [t]);
fun dest_Free (Free x) = x
| dest_Free t = raise TERM("dest_Free", [t]);
fun dest_Var (Var x) = x
| dest_Var t = raise TERM("dest_Var", [t]);
fun dest_comb (t1 $ t2) = (t1, t2)
| dest_comb t = raise TERM("dest_comb", [t]);
fun domain_type (Type ("fun", [T, _])) = T;
fun range_type (Type ("fun", [_, U])) = U;
fun dest_funT (Type ("fun", [T, U])) = (T, U)
| dest_funT T = raise TYPE ("dest_funT", [T], []);
(* maps [T1,...,Tn]--->T to the list [T1,T2,...,Tn]*)
fun binder_types (Type ("fun", [T, U])) = T :: binder_types U
| binder_types _ = [];
(* maps [T1,...,Tn]--->T to T*)
fun body_type (Type ("fun", [_, U])) = body_type U
| body_type T = T;
(* maps [T1,...,Tn]--->T to ([T1,T2,...,Tn], T) *)
fun strip_type T = (binder_types T, body_type T);
(*Compute the type of the term, checking that combinations are well-typed
Ts = [T0,T1,...] holds types of bound variables 0, 1, ...*)
fun type_of1 (Ts, Const (_,T)) = T
| type_of1 (Ts, Free (_,T)) = T
| type_of1 (Ts, Bound i) = (nth Ts i
handle General.Subscript => raise TYPE("type_of: bound variable", [], [Bound i]))
| type_of1 (Ts, Var (_,T)) = T
| type_of1 (Ts, Abs (_,T,body)) = T --> type_of1(T::Ts, body)
| type_of1 (Ts, f$u) =
let val U = type_of1(Ts,u)
and T = type_of1(Ts,f)
in case T of
Type("fun",[T1,T2]) =>
if T1=U then T2 else raise TYPE
("type_of: type mismatch in application", [T1,U], [f$u])
| _ => raise TYPE
("type_of: function type is expected in application",
[T,U], [f$u])
end;
fun type_of t : typ = type_of1 ([],t);
(*Determines the type of a term, with minimal checking*)
fun fastype_of1 (Ts, f$u) =
(case fastype_of1 (Ts,f) of
Type("fun",[_,T]) => T
| _ => raise TERM("fastype_of: expected function type", [f$u]))
| fastype_of1 (_, Const (_,T)) = T
| fastype_of1 (_, Free (_,T)) = T
| fastype_of1 (Ts, Bound i) = (nth Ts i
handle General.Subscript => raise TERM("fastype_of: Bound", [Bound i]))
| fastype_of1 (_, Var (_,T)) = T
| fastype_of1 (Ts, Abs (_,T,u)) = T --> fastype_of1 (T::Ts, u);
fun fastype_of t : typ = fastype_of1 ([],t);
(*Determine the argument type of a function*)
fun argument_type_of tm k =
let
fun argT i (Type ("fun", [T, U])) = if i = 0 then T else argT (i - 1) U
| argT _ T = raise TYPE ("argument_type_of", [T], []);
fun arg 0 _ (Abs (_, T, _)) = T
| arg i Ts (Abs (_, T, t)) = arg (i - 1) (T :: Ts) t
| arg i Ts (t $ _) = arg (i + 1) Ts t
| arg i Ts a = argT i (fastype_of1 (Ts, a));
in arg k [] tm end;
fun abs (x, T) t = Abs (x, T, t);
fun strip_abs (Abs (a, T, t)) =
let val (a', t') = strip_abs t
in ((a, T) :: a', t') end
| strip_abs t = ([], t);
(* maps (x1,...,xn)t to t *)
fun strip_abs_body (Abs(_,_,t)) = strip_abs_body t
| strip_abs_body u = u;
(* maps (x1,...,xn)t to [x1, ..., xn] *)
fun strip_abs_vars (Abs(a,T,t)) = (a,T) :: strip_abs_vars t
| strip_abs_vars u = [] : (string*typ) list;
fun strip_qnt_body qnt =
let fun strip(tm as Const(c,_)$Abs(_,_,t)) = if c=qnt then strip t else tm
| strip t = t
in strip end;
fun strip_qnt_vars qnt =
let fun strip(Const(c,_)$Abs(a,T,t)) = if c=qnt then (a,T)::strip t else []
| strip t = [] : (string*typ) list
in strip end;
(* maps (f, [t1,...,tn]) to f(t1,...,tn) *)
val list_comb : term * term list -> term = Library.foldl (op $);
(* maps f(t1,...,tn) to (f, [t1,...,tn]) ; naturally tail-recursive*)
fun strip_comb u : term * term list =
let fun stripc (f$t, ts) = stripc (f, t::ts)
| stripc x = x
in stripc(u,[]) end;
(* maps f(t1,...,tn) to f , which is never a combination *)
fun head_of (f$t) = head_of f
| head_of u = u;
(*number of atoms and abstractions in a term*)
fun size_of_term tm =
let
fun add_size (t $ u) n = add_size t (add_size u n)
| add_size (Abs (_ ,_, t)) n = add_size t (n + 1)
| add_size _ n = n + 1;
in add_size tm 0 end;
(*number of atoms and constructors in a type*)
fun size_of_typ ty =
let
fun add_size (Type (_, tys)) n = fold add_size tys (n + 1)
| add_size _ n = n + 1;
in add_size ty 0 end;
fun map_atyps f (Type (a, Ts)) = Type (a, map (map_atyps f) Ts)
| map_atyps f T = f T;
fun map_aterms f (t $ u) = map_aterms f t $ map_aterms f u
| map_aterms f (Abs (a, T, t)) = Abs (a, T, map_aterms f t)
| map_aterms f t = f t;
fun map_type_tvar f = map_atyps (fn TVar x => f x | T => T);
fun map_type_tfree f = map_atyps (fn TFree x => f x | T => T);
fun map_types f =
let
fun map_aux (Const (a, T)) = Const (a, f T)
| map_aux (Free (a, T)) = Free (a, f T)
| map_aux (Var (v, T)) = Var (v, f T)
| map_aux (Bound i) = Bound i
| map_aux (Abs (a, T, t)) = Abs (a, f T, map_aux t)
| map_aux (t $ u) = map_aux t $ map_aux u;
in map_aux end;
(* fold types and terms *)
fun fold_atyps f (Type (_, Ts)) = fold (fold_atyps f) Ts
| fold_atyps f T = f T;
fun fold_atyps_sorts f =
fold_atyps (fn T as TFree (_, S) => f (T, S) | T as TVar (_, S) => f (T, S));
fun fold_aterms f (t $ u) = fold_aterms f t #> fold_aterms f u
| fold_aterms f (Abs (_, _, t)) = fold_aterms f t
| fold_aterms f a = f a;
fun fold_term_types f (t as Const (_, T)) = f t T
| fold_term_types f (t as Free (_, T)) = f t T
| fold_term_types f (t as Var (_, T)) = f t T
| fold_term_types f (Bound _) = I
| fold_term_types f (t as Abs (_, T, b)) = f t T #> fold_term_types f b
| fold_term_types f (t $ u) = fold_term_types f t #> fold_term_types f u;
fun fold_types f = fold_term_types (K f);
fun replace_types (Const (c, _)) (T :: Ts) = (Const (c, T), Ts)
| replace_types (Free (x, _)) (T :: Ts) = (Free (x, T), Ts)
| replace_types (Var (xi, _)) (T :: Ts) = (Var (xi, T), Ts)
| replace_types (Bound i) Ts = (Bound i, Ts)
| replace_types (Abs (x, _, b)) (T :: Ts) =
let val (b', Ts') = replace_types b Ts
in (Abs (x, T, b'), Ts') end
| replace_types (t $ u) Ts =
let
val (t', Ts') = replace_types t Ts;
val (u', Ts'') = replace_types u Ts';
in (t' $ u', Ts'') end;
fun burrow_types f ts =
let
val Ts = rev ((fold o fold_types) cons ts []);
val Ts' = f Ts;
val (ts', []) = fold_map replace_types ts Ts';
in ts' end;
(*collect variables*)
val add_tvar_namesT = fold_atyps (fn TVar (xi, _) => insert (op =) xi | _ => I);
val add_tvar_names = fold_types add_tvar_namesT;
val add_tvarsT = fold_atyps (fn TVar v => insert (op =) v | _ => I);
val add_tvars = fold_types add_tvarsT;
val add_var_names = fold_aterms (fn Var (xi, _) => insert (op =) xi | _ => I);
val add_vars = fold_aterms (fn Var v => insert (op =) v | _ => I);
val add_tfree_namesT = fold_atyps (fn TFree (a, _) => insert (op =) a | _ => I);
val add_tfree_names = fold_types add_tfree_namesT;
val add_tfreesT = fold_atyps (fn TFree v => insert (op =) v | _ => I);
val add_tfrees = fold_types add_tfreesT;
val add_free_names = fold_aterms (fn Free (x, _) => insert (op =) x | _ => I);
val add_frees = fold_aterms (fn Free v => insert (op =) v | _ => I);
val add_const_names = fold_aterms (fn Const (c, _) => insert (op =) c | _ => I);
val add_consts = fold_aterms (fn Const c => insert (op =) c | _ => I);
(*extra type variables in a term, not covered by its type*)
fun hidden_polymorphism t =
let
val T = fastype_of t;
val tvarsT = add_tvarsT T [];
val extra_tvars = fold_types (fold_atyps
(fn TVar v => if member (op =) tvarsT v then I else insert (op =) v | _ => I)) t [];
in extra_tvars end;
(* renaming variables *)
val declare_typ_names = fold_atyps (fn TFree (a, _) => Name.declare a | _ => I);
fun declare_term_names tm =
fold_aterms
(fn Const (a, _) => Name.declare (Long_Name.base_name a)
| Free (a, _) => Name.declare a
| _ => I) tm #>
fold_types declare_typ_names tm;
val declare_term_frees = fold_aterms (fn Free (x, _) => Name.declare x | _ => I);
fun variant_frees t frees =
fst (fold_map Name.variant (map fst frees) (declare_term_names t Name.context)) ~~
map snd frees;
fun rename_wrt_term t frees = rev (variant_frees t frees); (*reversed result!*)
(** Comparing terms **)
(* variables *)
fun eq_ix ((x, i): indexname, (y, j)) = i = j andalso x = y;
fun eq_tvar ((xi, S: sort), (xi', S')) = eq_ix (xi, xi') andalso S = S';
fun eq_var ((xi, T: typ), (xi', T')) = eq_ix (xi, xi') andalso T = T';
(* alpha equivalence *)
fun tm1 aconv tm2 =
pointer_eq (tm1, tm2) orelse
(case (tm1, tm2) of
(t1 $ u1, t2 $ u2) => t1 aconv t2 andalso u1 aconv u2
| (Abs (_, T1, t1), Abs (_, T2, t2)) => t1 aconv t2 andalso T1 = T2
| (a1, a2) => a1 = a2);
fun aconv_untyped (tm1, tm2) =
pointer_eq (tm1, tm2) orelse
(case (tm1, tm2) of
(t1 $ u1, t2 $ u2) => aconv_untyped (t1, t2) andalso aconv_untyped (u1, u2)
| (Abs (_, _, t1), Abs (_, _, t2)) => aconv_untyped (t1, t2)
| (Const (a, _), Const (b, _)) => a = b
| (Free (x, _), Free (y, _)) => x = y
| (Var (xi, _), Var (yj, _)) => xi = yj
| (Bound i, Bound j) => i = j
| _ => false);
(*A fast unification filter: true unless the two terms cannot be unified.
Terms must be NORMAL. Treats all Vars as distinct. *)
fun could_unify (t, u) =
let
fun matchrands (f $ t) (g $ u) = could_unify (t, u) andalso matchrands f g
| matchrands _ _ = true;
in
case (head_of t, head_of u) of
(_, Var _) => true
| (Var _, _) => true
| (Const (a, _), Const (b, _)) => a = b andalso matchrands t u
| (Free (a, _), Free (b, _)) => a = b andalso matchrands t u
| (Bound i, Bound j) => i = j andalso matchrands t u
| (Abs _, _) => true (*because of possible eta equality*)
| (_, Abs _) => true
| _ => false
end;
(** Connectives of higher order logic **)
fun aT S = TFree (Name.aT, S);
fun itselfT ty = Type ("itself", [ty]);
val a_itselfT = itselfT (TFree (Name.aT, []));
val propT : typ = Type ("prop",[]);
(* maps !!x1...xn. t to t *)
fun strip_all_body (Const("all",_)$Abs(_,_,t)) = strip_all_body t
| strip_all_body t = t;
(* maps !!x1...xn. t to [x1, ..., xn] *)
fun strip_all_vars (Const("all",_)$Abs(a,T,t)) =
(a,T) :: strip_all_vars t
| strip_all_vars t = [] : (string*typ) list;
(*increments a term's non-local bound variables
required when moving a term within abstractions
inc is increment for bound variables
lev is level at which a bound variable is considered 'loose'*)
fun incr_bv (inc, lev, u as Bound i) = if i>=lev then Bound(i+inc) else u
| incr_bv (inc, lev, Abs(a,T,body)) =
Abs(a, T, incr_bv(inc,lev+1,body))
| incr_bv (inc, lev, f$t) =
incr_bv(inc,lev,f) $ incr_bv(inc,lev,t)
| incr_bv (inc, lev, u) = u;
fun incr_boundvars 0 t = t
| incr_boundvars inc t = incr_bv(inc,0,t);
(*Scan a pair of terms; while they are similar,
accumulate corresponding bound vars in "al"*)
fun match_bvs(Abs(x,_,s),Abs(y,_,t), al) =
match_bvs(s, t, if x="" orelse y="" then al
else (x,y)::al)
| match_bvs(f$s, g$t, al) = match_bvs(f,g,match_bvs(s,t,al))
| match_bvs(_,_,al) = al;
(* strip abstractions created by parameters *)
fun match_bvars (s,t) al = match_bvs(strip_abs_body s, strip_abs_body t, al);
fun map_abs_vars f (t $ u) = map_abs_vars f t $ map_abs_vars f u
| map_abs_vars f (Abs (a, T, t)) = Abs (f a, T, map_abs_vars f t)
| map_abs_vars f t = t;
fun rename_abs pat obj t =
let
val ren = match_bvs (pat, obj, []);
fun ren_abs (Abs (x, T, b)) =
Abs (the_default x (AList.lookup (op =) ren x), T, ren_abs b)
| ren_abs (f $ t) = ren_abs f $ ren_abs t
| ren_abs t = t
in if null ren then NONE else SOME (ren_abs t) end;
(*Accumulate all 'loose' bound vars referring to level 'lev' or beyond.
(Bound 0) is loose at level 0 *)
fun add_loose_bnos (Bound i, lev, js) =
if i<lev then js else insert (op =) (i - lev) js
| add_loose_bnos (Abs (_,_,t), lev, js) = add_loose_bnos (t, lev+1, js)
| add_loose_bnos (f$t, lev, js) =
add_loose_bnos (f, lev, add_loose_bnos (t, lev, js))
| add_loose_bnos (_, _, js) = js;
fun loose_bnos t = add_loose_bnos (t, 0, []);
(* loose_bvar(t,k) iff t contains a 'loose' bound variable referring to
level k or beyond. *)
fun loose_bvar(Bound i,k) = i >= k
| loose_bvar(f$t, k) = loose_bvar(f,k) orelse loose_bvar(t,k)
| loose_bvar(Abs(_,_,t),k) = loose_bvar(t,k+1)
| loose_bvar _ = false;
fun loose_bvar1(Bound i,k) = i = k
| loose_bvar1(f$t, k) = loose_bvar1(f,k) orelse loose_bvar1(t,k)
| loose_bvar1(Abs(_,_,t),k) = loose_bvar1(t,k+1)
| loose_bvar1 _ = false;
fun is_open t = loose_bvar (t, 0);
fun is_dependent t = loose_bvar1 (t, 0);
(*Substitute arguments for loose bound variables.
Beta-reduction of arg(n-1)...arg0 into t replacing (Bound i) with (argi).
Note that for ((%x y. c) a b), the bound vars in c are x=1 and y=0
and the appropriate call is subst_bounds([b,a], c) .
Loose bound variables >=n are reduced by "n" to
compensate for the disappearance of lambdas.
*)
fun subst_bounds (args: term list, t) : term =
let
val n = length args;
fun subst (t as Bound i, lev) =
(if i < lev then raise Same.SAME (*var is locally bound*)
else incr_boundvars lev (nth args (i - lev))
handle General.Subscript => Bound (i - n)) (*loose: change it*)
| subst (Abs (a, T, body), lev) = Abs (a, T, subst (body, lev + 1))
| subst (f $ t, lev) =
(subst (f, lev) $ (subst (t, lev) handle Same.SAME => t)
handle Same.SAME => f $ subst (t, lev))
| subst _ = raise Same.SAME;
in case args of [] => t | _ => (subst (t, 0) handle Same.SAME => t) end;
(*Special case: one argument*)
fun subst_bound (arg, t) : term =
let
fun subst (Bound i, lev) =
if i < lev then raise Same.SAME (*var is locally bound*)
else if i = lev then incr_boundvars lev arg
else Bound (i - 1) (*loose: change it*)
| subst (Abs (a, T, body), lev) = Abs (a, T, subst (body, lev + 1))
| subst (f $ t, lev) =
(subst (f, lev) $ (subst (t, lev) handle Same.SAME => t)
handle Same.SAME => f $ subst (t, lev))
| subst _ = raise Same.SAME;
in subst (t, 0) handle Same.SAME => t end;
(*beta-reduce if possible, else form application*)
fun betapply (Abs(_,_,t), u) = subst_bound (u,t)
| betapply (f,u) = f$u;
val betapplys = Library.foldl betapply;
(*unfolding abstractions with substitution
of bound variables and implicit eta-expansion*)
fun strip_abs_eta k t =
let
val used = fold_aterms declare_term_frees t Name.context;
fun strip_abs t (0, used) = (([], t), (0, used))
| strip_abs (Abs (v, T, t)) (k, used) =
let
val (v', used') = Name.variant v used;
val t' = subst_bound (Free (v', T), t);
val ((vs, t''), (k', used'')) = strip_abs t' (k - 1, used');
in (((v', T) :: vs, t''), (k', used'')) end
| strip_abs t (k, used) = (([], t), (k, used));
fun expand_eta [] t _ = ([], t)
| expand_eta (T::Ts) t used =
let
val (v, used') = Name.variant "" used;
val (vs, t') = expand_eta Ts (t $ Free (v, T)) used';
in ((v, T) :: vs, t') end;
val ((vs1, t'), (k', used')) = strip_abs t (k, used);
val Ts = fst (chop k' (binder_types (fastype_of t')));
val (vs2, t'') = expand_eta Ts t' used';
in (vs1 @ vs2, t'') end;
(*Substitute new for free occurrences of old in a term*)
fun subst_free [] = I
| subst_free pairs =
let fun substf u =
case AList.lookup (op aconv) pairs u of
SOME u' => u'
| NONE => (case u of Abs(a,T,t) => Abs(a, T, substf t)
| t$u' => substf t $ substf u'
| _ => u)
in substf end;
(*Abstraction of the term "body" over its occurrences of v,
which must contain no loose bound variables.
The resulting term is ready to become the body of an Abs.*)
fun abstract_over (v, body) =
let
fun abs lev tm =
if v aconv tm then Bound lev
else
(case tm of
Abs (a, T, t) => Abs (a, T, abs (lev + 1) t)
| t $ u =>
(abs lev t $ (abs lev u handle Same.SAME => u)
handle Same.SAME => t $ abs lev u)
| _ => raise Same.SAME);
in abs 0 body handle Same.SAME => body end;
fun term_name (Const (x, _)) = Long_Name.base_name x
| term_name (Free (x, _)) = x
| term_name (Var ((x, _), _)) = x
| term_name _ = Name.uu;
fun lambda_name (x, v) t =
Abs (if x = "" then term_name v else x, fastype_of v, abstract_over (v, t));
fun lambda v t = lambda_name ("", v) t;
fun absfree (a, T) body = Abs (a, T, abstract_over (Free (a, T), body));
fun absdummy T body = Abs (Name.uu_, T, body);
(*Replace the ATOMIC term ti by ui; inst = [(t1,u1), ..., (tn,un)].
A simultaneous substitution: [ (a,b), (b,a) ] swaps a and b. *)
fun subst_atomic [] tm = tm
| subst_atomic inst tm =
let
fun subst (Abs (a, T, body)) = Abs (a, T, subst body)
| subst (t $ u) = subst t $ subst u
| subst t = the_default t (AList.lookup (op aconv) inst t);
in subst tm end;
(*Replace the ATOMIC type Ti by Ui; inst = [(T1,U1), ..., (Tn,Un)].*)
fun typ_subst_atomic [] ty = ty
| typ_subst_atomic inst ty =
let
fun subst (Type (a, Ts)) = Type (a, map subst Ts)
| subst T = the_default T (AList.lookup (op = : typ * typ -> bool) inst T);
in subst ty end;
fun subst_atomic_types [] tm = tm
| subst_atomic_types inst tm = map_types (typ_subst_atomic inst) tm;
fun typ_subst_TVars [] ty = ty
| typ_subst_TVars inst ty =
let
fun subst (Type (a, Ts)) = Type (a, map subst Ts)
| subst (T as TVar (xi, _)) = the_default T (AList.lookup (op =) inst xi)
| subst T = T;
in subst ty end;
fun subst_TVars [] tm = tm
| subst_TVars inst tm = map_types (typ_subst_TVars inst) tm;
fun subst_Vars [] tm = tm
| subst_Vars inst tm =
let
fun subst (t as Var (xi, _)) = the_default t (AList.lookup (op =) inst xi)
| subst (Abs (a, T, t)) = Abs (a, T, subst t)
| subst (t $ u) = subst t $ subst u
| subst t = t;
in subst tm end;
fun subst_vars ([], []) tm = tm
| subst_vars ([], inst) tm = subst_Vars inst tm
| subst_vars (instT, inst) tm =
let
fun subst (Const (a, T)) = Const (a, typ_subst_TVars instT T)
| subst (Free (a, T)) = Free (a, typ_subst_TVars instT T)
| subst (Var (xi, T)) =
(case AList.lookup (op =) inst xi of
NONE => Var (xi, typ_subst_TVars instT T)
| SOME t => t)
| subst (t as Bound _) = t
| subst (Abs (a, T, t)) = Abs (a, typ_subst_TVars instT T, subst t)
| subst (t $ u) = subst t $ subst u;
in subst tm end;
fun close_schematic_term t =
let
val extra_types = map (fn v => Const ("Pure.type", itselfT (TVar v))) (hidden_polymorphism t);
val extra_terms = map Var (add_vars t []);
in fold lambda (extra_terms @ extra_types) t end;
(** Identifying first-order terms **)
(*Differs from proofterm/is_fun in its treatment of TVar*)
fun is_funtype (Type ("fun", [_, _])) = true
| is_funtype _ = false;
(*Argument Ts is a reverse list of binder types, needed if term t contains Bound vars*)
fun has_not_funtype Ts t = not (is_funtype (fastype_of1 (Ts, t)));
(*First order means in all terms of the form f(t1,...,tn) no argument has a
function type. The supplied quantifiers are excluded: their argument always
has a function type through a recursive call into its body.*)
fun is_first_order quants =
let fun first_order1 Ts (Abs (_,T,body)) = first_order1 (T::Ts) body
| first_order1 Ts (Const(q,_) $ Abs(a,T,body)) =
member (op =) quants q andalso (*it is a known quantifier*)
not (is_funtype T) andalso first_order1 (T::Ts) body
| first_order1 Ts t =
case strip_comb t of
(Var _, ts) => forall (first_order1 Ts andf has_not_funtype Ts) ts
| (Free _, ts) => forall (first_order1 Ts andf has_not_funtype Ts) ts
| (Const _, ts) => forall (first_order1 Ts andf has_not_funtype Ts) ts
| (Bound _, ts) => forall (first_order1 Ts andf has_not_funtype Ts) ts
| (Abs _, ts) => false (*not in beta-normal form*)
| _ => error "first_order: unexpected case"
in first_order1 [] end;
(* maximum index of typs and terms *)
fun maxidx_typ (TVar ((_, j), _)) i = Int.max (i, j)
| maxidx_typ (Type (_, Ts)) i = maxidx_typs Ts i
| maxidx_typ (TFree _) i = i
and maxidx_typs [] i = i
| maxidx_typs (T :: Ts) i = maxidx_typs Ts (maxidx_typ T i);
fun maxidx_term (Var ((_, j), T)) i = maxidx_typ T (Int.max (i, j))
| maxidx_term (Const (_, T)) i = maxidx_typ T i
| maxidx_term (Free (_, T)) i = maxidx_typ T i
| maxidx_term (Bound _) i = i
| maxidx_term (Abs (_, T, t)) i = maxidx_term t (maxidx_typ T i)
| maxidx_term (t $ u) i = maxidx_term u (maxidx_term t i);
fun maxidx_of_typ T = maxidx_typ T ~1;
fun maxidx_of_typs Ts = maxidx_typs Ts ~1;
fun maxidx_of_term t = maxidx_term t ~1;
(** misc syntax operations **)
(* substructure *)
fun exists_subtype P =
let
fun ex ty = P ty orelse
(case ty of Type (_, Ts) => exists ex Ts | _ => false);
in ex end;
fun exists_type P =
let
fun ex (Const (_, T)) = P T
| ex (Free (_, T)) = P T
| ex (Var (_, T)) = P T
| ex (Bound _) = false
| ex (Abs (_, T, t)) = P T orelse ex t
| ex (t $ u) = ex t orelse ex u;
in ex end;
fun exists_subterm P =
let
fun ex tm = P tm orelse
(case tm of
t $ u => ex t orelse ex u
| Abs (_, _, t) => ex t
| _ => false);
in ex end;
fun exists_Const P = exists_subterm (fn Const c => P c | _ => false);
fun has_abs (Abs _) = true
| has_abs (t $ u) = has_abs t orelse has_abs u
| has_abs _ = false;
(* dest abstraction *)
fun dest_abs (x, T, body) =
let
fun name_clash (Free (y, _)) = (x = y)
| name_clash (t $ u) = name_clash t orelse name_clash u
| name_clash (Abs (_, _, t)) = name_clash t
| name_clash _ = false;
in
if name_clash body then
dest_abs (singleton (Name.variant_list [x]) x, T, body) (*potentially slow*)
else (x, subst_bound (Free (x, T), body))
end;
(* dummy patterns *)
fun dummy_pattern T = Const ("Pure.dummy_pattern", T);
val dummy = dummy_pattern dummyT;
val dummy_prop = dummy_pattern propT;
fun is_dummy_pattern (Const ("Pure.dummy_pattern", _)) = true
| is_dummy_pattern _ = false;
fun no_dummy_patterns tm =
if not (fold_aterms (fn t => fn b => b orelse is_dummy_pattern t) tm false) then tm
else raise TERM ("Illegal occurrence of '_' dummy pattern", [tm]);
fun free_dummy_patterns (Const ("Pure.dummy_pattern", T)) used =
let val [x] = Name.invent used Name.uu 1
in (Free (Name.internal x, T), Name.declare x used) end
| free_dummy_patterns (Abs (x, T, b)) used =
let val (b', used') = free_dummy_patterns b used
in (Abs (x, T, b'), used') end
| free_dummy_patterns (t $ u) used =
let
val (t', used') = free_dummy_patterns t used;
val (u', used'') = free_dummy_patterns u used';
in (t' $ u', used'') end
| free_dummy_patterns a used = (a, used);
fun replace_dummy Ts (Const ("Pure.dummy_pattern", T)) i =
(list_comb (Var (("_dummy_", i), Ts ---> T), map_range Bound (length Ts)), i + 1)
| replace_dummy Ts (Abs (x, T, t)) i =
let val (t', i') = replace_dummy (T :: Ts) t i
in (Abs (x, T, t'), i') end
| replace_dummy Ts (t $ u) i =
let
val (t', i') = replace_dummy Ts t i;
val (u', i'') = replace_dummy Ts u i';
in (t' $ u', i'') end
| replace_dummy _ a i = (a, i);
val replace_dummy_patterns = replace_dummy [];
fun is_replaced_dummy_pattern ("_dummy_", _) = true
| is_replaced_dummy_pattern _ = false;
fun show_dummy_patterns (Var (("_dummy_", _), T)) = dummy_pattern T
| show_dummy_patterns (t $ u) = show_dummy_patterns t $ show_dummy_patterns u
| show_dummy_patterns (Abs (x, T, t)) = Abs (x, T, show_dummy_patterns t)
| show_dummy_patterns a = a;
(* display variables *)
fun string_of_vname (x, i) =
let
val idx = string_of_int i;
val dot =
(case rev (Symbol.explode x) of
_ :: "\\<^sub>" :: _ => false
| c :: _ => Symbol.is_digit c
| _ => true);
in
if dot then "?" ^ x ^ "." ^ idx
else if i <> 0 then "?" ^ x ^ idx
else "?" ^ x
end;
fun string_of_vname' (x, ~1) = x
| string_of_vname' xi = string_of_vname xi;
end;
structure Basic_Term: BASIC_TERM = Term;
open Basic_Term;