wenzelm@9536: (* Title: Pure/term.ML clasohm@0: ID: $Id$ wenzelm@9536: Author: Lawrence C Paulson, Cambridge University Computer Laboratory clasohm@0: Copyright Cambridge University 1992 clasohm@1364: wenzelm@4444: Simply typed lambda-calculus: types, terms, and basic operations. clasohm@0: *) clasohm@0: clasohm@1364: infix 9 $; clasohm@1364: infixr 5 -->; wenzelm@4444: infixr --->; wenzelm@4444: infix aconv; clasohm@1364: wenzelm@4444: signature BASIC_TERM = wenzelm@4444: sig wenzelm@4444: type indexname wenzelm@4444: type class wenzelm@4444: type sort wenzelm@14829: type arity wenzelm@4444: datatype typ = wenzelm@4444: Type of string * typ list | wenzelm@4444: TFree of string * sort | wenzelm@4444: TVar of indexname * sort wenzelm@16537: datatype term = wenzelm@16537: Const of string * typ | wenzelm@16537: Free of string * typ | wenzelm@16537: Var of indexname * typ | wenzelm@16537: Bound of int | wenzelm@16537: Abs of string * typ * term | wenzelm@17756: $ of term * term wenzelm@16537: exception TYPE of string * typ list * term list wenzelm@16537: exception TERM of string * term list wenzelm@16710: val dummyT: typ wenzelm@16710: val no_dummyT: typ -> typ wenzelm@4444: val --> : typ * typ -> typ wenzelm@4444: val ---> : typ list * typ -> typ wenzelm@16710: val dest_Type: typ -> string * typ list wenzelm@16710: val dest_TVar: typ -> indexname * sort wenzelm@16710: val dest_TFree: typ -> string * sort wenzelm@16710: val is_Bound: term -> bool wenzelm@16710: val is_Const: term -> bool wenzelm@16710: val is_Free: term -> bool wenzelm@16710: val is_Var: term -> bool wenzelm@4444: val is_TVar: typ -> bool paulson@15573: val is_funtype: typ -> bool wenzelm@16710: val dest_Const: term -> string * typ wenzelm@16710: val dest_Free: term -> string * typ wenzelm@16710: val dest_Var: term -> indexname * typ wenzelm@4444: val domain_type: typ -> typ wenzelm@4480: val range_type: typ -> typ wenzelm@4444: val binder_types: typ -> typ list wenzelm@4444: val body_type: typ -> typ wenzelm@4444: val strip_type: typ -> typ list * typ wenzelm@16710: val type_of1: typ list * term -> typ wenzelm@4444: val type_of: term -> typ wenzelm@16710: val fastype_of1: typ list * term -> typ wenzelm@4444: val fastype_of: term -> typ wenzelm@10806: val list_abs: (string * typ) list * term -> term haftmann@18927: val strip_abs: term -> (string * typ) list * term wenzelm@4444: val strip_abs_body: term -> term wenzelm@4444: val strip_abs_vars: term -> (string * typ) list wenzelm@4444: val strip_qnt_body: string -> term -> term wenzelm@4444: val strip_qnt_vars: string -> term -> (string * typ) list wenzelm@4444: val list_comb: term * term list -> term wenzelm@4444: val strip_comb: term -> term * term list wenzelm@4444: val head_of: term -> term wenzelm@4444: val size_of_term: term -> int haftmann@18847: val map_atyps: (typ -> typ) -> typ -> typ haftmann@18847: val map_aterms: (term -> term) -> term -> term wenzelm@4444: val map_type_tvar: (indexname * sort -> typ) -> typ -> typ wenzelm@4444: val map_type_tfree: (string * sort -> typ) -> typ -> typ wenzelm@4444: val map_term_types: (typ -> typ) -> term -> term wenzelm@16943: val fold_atyps: (typ -> 'a -> 'a) -> typ -> 'a -> 'a wenzelm@16943: val fold_aterms: (term -> 'a -> 'a) -> term -> 'a -> 'a wenzelm@16943: val fold_term_types: (term -> typ -> 'a -> 'a) -> term -> 'a -> 'a wenzelm@16943: val fold_types: (typ -> 'a -> 'a) -> term -> 'a -> 'a haftmann@18927: val it_term_types: (typ * 'a -> 'a) -> term * 'a -> 'a wenzelm@16943: val add_term_names: term * string list -> string list wenzelm@16943: val add_term_varnames: term -> indexname list -> indexname list wenzelm@16943: val term_varnames: term -> indexname list wenzelm@18149: val find_free: term -> string -> term option wenzelm@16710: val aconv: term * term -> bool wenzelm@16710: val aconvs: term list * term list -> bool wenzelm@16537: structure Vartab: TABLE wenzelm@16537: structure Typtab: TABLE wenzelm@16537: structure Termtab: TABLE wenzelm@4444: val itselfT: typ -> typ wenzelm@4444: val a_itselfT: typ wenzelm@4444: val propT: typ wenzelm@4444: val implies: term wenzelm@4444: val all: typ -> term wenzelm@4444: val equals: typ -> term wenzelm@4444: val strip_all_body: term -> term wenzelm@4444: val strip_all_vars: term -> (string * typ) list wenzelm@4444: val incr_bv: int * int * term -> term wenzelm@4444: val incr_boundvars: int -> term -> term wenzelm@4444: val add_loose_bnos: term * int * int list -> int list wenzelm@4444: val loose_bnos: term -> int list wenzelm@4444: val loose_bvar: term * int -> bool wenzelm@4444: val loose_bvar1: term * int -> bool wenzelm@4444: val subst_bounds: term list * term -> term wenzelm@4444: val subst_bound: term * term -> term wenzelm@4444: val betapply: term * term -> term wenzelm@18183: val betapplys: term * term list -> term wenzelm@4444: val eq_ix: indexname * indexname -> bool wenzelm@4444: val ins_ix: indexname * indexname list -> indexname list wenzelm@4444: val mem_ix: indexname * indexname list -> bool wenzelm@4444: val mem_term: term * term list -> bool wenzelm@4444: val ins_term: term * term list -> term list wenzelm@4444: val could_unify: term * term -> bool wenzelm@4444: val subst_free: (term * term) list -> term -> term wenzelm@4444: val xless: (string * int) * indexname -> bool wenzelm@4444: val abstract_over: term * term -> term wenzelm@11922: val lambda: term -> term -> term wenzelm@4444: val absfree: string * typ * term -> term wenzelm@17786: val absdummy: typ * term -> term wenzelm@4444: val list_abs_free: (string * typ) list * term -> term wenzelm@4444: val list_all_free: (string * typ) list * term -> term wenzelm@4444: val list_all: (string * typ) list * term -> term wenzelm@16710: val subst_atomic: (term * term) list -> term -> term wenzelm@16710: val typ_subst_atomic: (typ * typ) list -> typ -> typ wenzelm@16710: val subst_atomic_types: (typ * typ) list -> term -> term wenzelm@16710: val typ_subst_TVars: (indexname * typ) list -> typ -> typ wenzelm@16710: val subst_TVars: (indexname * typ) list -> term -> term wenzelm@16710: val subst_Vars: (indexname * term) list -> term -> term wenzelm@16710: val subst_vars: (indexname * typ) list * (indexname * term) list -> term -> term wenzelm@16710: val is_first_order: string list -> term -> bool wenzelm@4444: val maxidx_of_typ: typ -> int wenzelm@4444: val maxidx_of_typs: typ list -> int wenzelm@4444: val maxidx_of_term: term -> int wenzelm@4444: val variant: string list -> string -> string wenzelm@4444: val variantlist: string list * string list -> string list wenzelm@16537: (*note reversed order of args wrt. variant!*) wenzelm@4444: val add_typ_classes: typ * class list -> class list wenzelm@4444: val add_typ_tycons: typ * string list -> string list wenzelm@4444: val add_term_classes: term * class list -> class list wenzelm@16710: val add_term_tycons: term * string list -> string list wenzelm@4444: val add_term_consts: term * string list -> string list nipkow@13646: val term_consts: term -> string list wenzelm@16943: val exists_subterm: (term -> bool) -> term -> bool wenzelm@16710: val exists_Const: (string * typ -> bool) -> term -> bool wenzelm@16710: val add_term_free_names: term * string list -> string list wenzelm@16710: val add_typ_tvars: typ * (indexname * sort) list -> (indexname * sort) list wenzelm@16710: val add_typ_tfree_names: typ * string list -> string list wenzelm@16710: val add_typ_tfrees: typ * (string * sort) list -> (string * sort) list wenzelm@16710: val add_typ_varnames: typ * string list -> string list wenzelm@16710: val add_term_tvars: term * (indexname * sort) list -> (indexname * sort) list wenzelm@16710: val add_term_tfrees: term * (string * sort) list -> (string * sort) list wenzelm@16710: val add_term_tfree_names: term * string list -> string list wenzelm@16710: val add_term_tvarnames: term * string list -> string list wenzelm@16710: val typ_tfrees: typ -> (string * sort) list wenzelm@16710: val typ_tvars: typ -> (indexname * sort) list wenzelm@16710: val term_tfrees: term -> (string * sort) list wenzelm@16710: val term_tvars: term -> (indexname * sort) list wenzelm@16710: val add_typ_ixns: indexname list * typ -> indexname list wenzelm@16710: val add_term_tvar_ixns: term * indexname list -> indexname list wenzelm@16710: val add_term_vars: term * term list -> term list wenzelm@16710: val term_vars: term -> term list wenzelm@4444: val add_term_frees: term * term list -> term list wenzelm@4444: val term_frees: term -> term list wenzelm@16710: val variant_abs: string * typ * term -> string * term wenzelm@16710: val rename_wrt_term: term -> (string * typ) list -> (string * typ) list wenzelm@15986: val show_question_marks: bool ref wenzelm@4444: end; clasohm@0: wenzelm@4444: signature TERM = wenzelm@4444: sig wenzelm@4444: include BASIC_TERM wenzelm@16710: val argument_type_of: term -> typ wenzelm@16943: val add_tvarsT: typ -> (indexname * sort) list -> (indexname * sort) list wenzelm@16943: val add_tvars: term -> (indexname * sort) list -> (indexname * sort) list wenzelm@16943: val add_vars: term -> (indexname * typ) list -> (indexname * typ) list wenzelm@16943: val add_tfreesT: typ -> (string * sort) list -> (string * sort) list wenzelm@16943: val add_tfrees: term -> (string * sort) list -> (string * sort) list wenzelm@16943: val add_frees: term -> (string * typ) list -> (string * typ) list wenzelm@16678: val fast_indexname_ord: indexname * indexname -> order wenzelm@16537: val indexname_ord: indexname * indexname -> order wenzelm@16537: val sort_ord: sort * sort -> order wenzelm@16537: val typ_ord: typ * typ -> order wenzelm@16678: val fast_term_ord: term * term -> order wenzelm@16537: val term_ord: term * term -> order wenzelm@16537: val hd_ord: term * term -> order wenzelm@16537: val termless: term * term -> bool nipkow@16570: val term_lpo: (string -> int) -> term * term -> order wenzelm@12981: val match_bvars: (term * term) * (string * string) list -> (string * string) list wenzelm@12981: val rename_abs: term -> term -> term -> term option wenzelm@16943: val eq_tvar: (indexname * sort) * (indexname * sort) -> bool wenzelm@16882: val eq_var: (indexname * typ) * (indexname * typ) -> bool wenzelm@16943: val tvar_ord: (indexname * sort) * (indexname * sort) -> order wenzelm@16943: val var_ord: (indexname * typ) * (indexname * typ) -> order wenzelm@16882: val instantiate: ((indexname * sort) * typ) list * ((indexname * typ) * term) list wenzelm@16882: -> term -> term wenzelm@16882: val instantiateT: ((indexname * sort) * typ) list -> typ -> typ wenzelm@16710: val maxidx_typ: typ -> int -> int wenzelm@16710: val maxidx_typs: typ list -> int -> int wenzelm@16710: val maxidx_term: term -> int -> int wenzelm@12499: val invent_names: string list -> string -> int -> string list wenzelm@16710: val dest_abs: string * typ * term -> string * term wenzelm@16990: val bound: int -> string wenzelm@16990: val is_bound: string -> bool wenzelm@16943: val zero_var_indexesT: typ -> typ wenzelm@16943: val zero_var_indexes: term -> term wenzelm@16943: val zero_var_indexes_inst: term -> wenzelm@16943: ((indexname * sort) * typ) list * ((indexname * typ) * term) list wenzelm@9536: val dummy_patternN: string wenzelm@18253: val dummy_pattern: typ -> term wenzelm@9536: val no_dummy_patterns: term -> term wenzelm@9536: val replace_dummy_patterns: int * term -> int * term wenzelm@10552: val is_replaced_dummy_pattern: indexname -> bool wenzelm@16035: val show_dummy_patterns: term -> term wenzelm@13484: val adhoc_freeze_vars: term -> term * string list wenzelm@14786: val string_of_vname: indexname -> string wenzelm@14786: val string_of_vname': indexname -> string wenzelm@17777: val str_of_term: term -> string wenzelm@4444: end; wenzelm@4444: wenzelm@4444: structure Term: TERM = clasohm@1364: struct clasohm@0: clasohm@0: (*Indexnames can be quickly renamed by adding an offset to the integer part, clasohm@0: for resolution.*) wenzelm@16537: type indexname = string * int; clasohm@0: wenzelm@4626: (* Types are classified by sorts. *) clasohm@0: type class = string; clasohm@0: type sort = class list; wenzelm@14829: type arity = string * sort list * sort; clasohm@0: clasohm@0: (* The sorts attached to TFrees and TVars specify the sort of that variable *) clasohm@0: datatype typ = Type of string * typ list clasohm@0: | TFree of string * sort wenzelm@9536: | TVar of indexname * sort; clasohm@0: paulson@6033: (*Terms. Bound variables are indicated by depth number. clasohm@0: Free variables, (scheme) variables and constants have names. wenzelm@4626: An term is "closed" if every bound variable of level "lev" wenzelm@13000: is enclosed by at least "lev" abstractions. clasohm@0: clasohm@0: It is possible to create meaningless terms containing loose bound vars clasohm@0: or type mismatches. But such terms are not allowed in rules. *) clasohm@0: wenzelm@13000: datatype term = clasohm@0: Const of string * typ wenzelm@13000: | Free of string * typ clasohm@0: | Var of indexname * typ clasohm@0: | Bound of int clasohm@0: | Abs of string*typ*term wenzelm@3965: | op $ of term*term; clasohm@0: wenzelm@16537: (*Errors involving type mismatches*) clasohm@0: exception TYPE of string * typ list * term list; clasohm@0: wenzelm@16537: (*Errors errors involving terms*) clasohm@0: exception TERM of string * term list; clasohm@0: clasohm@0: (*Note variable naming conventions! clasohm@0: a,b,c: string clasohm@0: f,g,h: functions (including terms of function type) clasohm@0: i,j,m,n: int clasohm@0: t,u: term clasohm@0: v,w: indexnames clasohm@0: x,y: any clasohm@0: A,B,C: term (denoting formulae) clasohm@0: T,U: typ clasohm@0: *) clasohm@0: clasohm@0: paulson@6033: (** Types **) paulson@6033: wenzelm@16537: (*dummy type for parsing and printing etc.*) wenzelm@16537: val dummyT = Type ("dummy", []); wenzelm@16537: wenzelm@16537: fun no_dummyT typ = wenzelm@16537: let wenzelm@16537: fun check (T as Type ("dummy", _)) = wenzelm@16537: raise TYPE ("Illegal occurrence of '_' dummy type", [T], []) wenzelm@16537: | check (Type (_, Ts)) = List.app check Ts wenzelm@16537: | check _ = (); wenzelm@16537: in check typ; typ end; wenzelm@16537: paulson@6033: fun S --> T = Type("fun",[S,T]); paulson@6033: paulson@6033: (*handy for multiple args: [T1,...,Tn]--->T gives T1-->(T2--> ... -->T)*) skalberg@15570: val op ---> = Library.foldr (op -->); paulson@6033: paulson@6033: fun dest_Type (Type x) = x paulson@6033: | dest_Type T = raise TYPE ("dest_Type", [T], []); dixon@15914: fun dest_TVar (TVar x) = x dixon@15914: | dest_TVar T = raise TYPE ("dest_TVar", [T], []); dixon@15914: fun dest_TFree (TFree x) = x dixon@15914: | dest_TFree T = raise TYPE ("dest_TFree", [T], []); paulson@6033: wenzelm@16537: clasohm@0: (** Discriminators **) clasohm@0: nipkow@7318: fun is_Bound (Bound _) = true nipkow@7318: | is_Bound _ = false; nipkow@7318: clasohm@0: fun is_Const (Const _) = true clasohm@0: | is_Const _ = false; clasohm@0: clasohm@0: fun is_Free (Free _) = true clasohm@0: | is_Free _ = false; clasohm@0: clasohm@0: fun is_Var (Var _) = true clasohm@0: | is_Var _ = false; clasohm@0: clasohm@0: fun is_TVar (TVar _) = true clasohm@0: | is_TVar _ = false; clasohm@0: paulson@15573: (*Differs from proofterm/is_fun in its treatment of TVar*) paulson@15573: fun is_funtype (Type("fun",[_,_])) = true paulson@15573: | is_funtype _ = false; paulson@15573: wenzelm@16537: clasohm@0: (** Destructors **) clasohm@0: clasohm@0: fun dest_Const (Const x) = x clasohm@0: | dest_Const t = raise TERM("dest_Const", [t]); clasohm@0: clasohm@0: fun dest_Free (Free x) = x clasohm@0: | dest_Free t = raise TERM("dest_Free", [t]); clasohm@0: clasohm@0: fun dest_Var (Var x) = x clasohm@0: | dest_Var t = raise TERM("dest_Var", [t]); clasohm@0: clasohm@0: paulson@4464: fun domain_type (Type("fun", [T,_])) = T paulson@4464: and range_type (Type("fun", [_,T])) = T; paulson@4064: clasohm@0: (* maps [T1,...,Tn]--->T to the list [T1,T2,...,Tn]*) clasohm@0: fun binder_types (Type("fun",[S,T])) = S :: binder_types T clasohm@0: | binder_types _ = []; clasohm@0: clasohm@0: (* maps [T1,...,Tn]--->T to T*) clasohm@0: fun body_type (Type("fun",[S,T])) = body_type T clasohm@0: | body_type T = T; clasohm@0: clasohm@0: (* maps [T1,...,Tn]--->T to ([T1,T2,...,Tn], T) *) clasohm@0: fun strip_type T : typ list * typ = clasohm@0: (binder_types T, body_type T); clasohm@0: clasohm@0: clasohm@0: (*Compute the type of the term, checking that combinations are well-typed clasohm@0: Ts = [T0,T1,...] holds types of bound variables 0, 1, ...*) clasohm@0: fun type_of1 (Ts, Const (_,T)) = T clasohm@0: | type_of1 (Ts, Free (_,T)) = T skalberg@15570: | type_of1 (Ts, Bound i) = (List.nth (Ts,i) skalberg@15570: handle Subscript => raise TYPE("type_of: bound variable", [], [Bound i])) clasohm@0: | type_of1 (Ts, Var (_,T)) = T clasohm@0: | type_of1 (Ts, Abs (_,T,body)) = T --> type_of1(T::Ts, body) wenzelm@13000: | type_of1 (Ts, f$u) = clasohm@0: let val U = type_of1(Ts,u) clasohm@0: and T = type_of1(Ts,f) clasohm@0: in case T of wenzelm@9536: Type("fun",[T1,T2]) => wenzelm@9536: if T1=U then T2 else raise TYPE wenzelm@9536: ("type_of: type mismatch in application", [T1,U], [f$u]) wenzelm@13000: | _ => raise TYPE wenzelm@9536: ("type_of: function type is expected in application", wenzelm@9536: [T,U], [f$u]) clasohm@0: end; clasohm@0: clasohm@0: fun type_of t : typ = type_of1 ([],t); clasohm@0: clasohm@0: (*Determines the type of a term, with minimal checking*) wenzelm@13000: fun fastype_of1 (Ts, f$u) = lcp@61: (case fastype_of1 (Ts,f) of wenzelm@9536: Type("fun",[_,T]) => T wenzelm@9536: | _ => raise TERM("fastype_of: expected function type", [f$u])) lcp@61: | fastype_of1 (_, Const (_,T)) = T lcp@61: | fastype_of1 (_, Free (_,T)) = T skalberg@15570: | fastype_of1 (Ts, Bound i) = (List.nth(Ts,i) skalberg@15570: handle Subscript => raise TERM("fastype_of: Bound", [Bound i])) wenzelm@13000: | fastype_of1 (_, Var (_,T)) = T lcp@61: | fastype_of1 (Ts, Abs (_,T,u)) = T --> fastype_of1 (T::Ts, u); lcp@61: lcp@61: fun fastype_of t : typ = fastype_of1 ([],t); clasohm@0: wenzelm@16678: (*Determine the argument type of a function*) wenzelm@16678: fun argument_type_of tm = wenzelm@16678: let wenzelm@16678: fun argT i (Type ("fun", [T, U])) = if i = 0 then T else argT (i - 1) U wenzelm@16678: | argT _ T = raise TYPE ("argument_type_of", [T], []); wenzelm@16678: wenzelm@16678: fun arg 0 _ (Abs (_, T, _)) = T wenzelm@16678: | arg i Ts (Abs (_, T, t)) = arg (i - 1) (T :: Ts) t wenzelm@16678: | arg i Ts (t $ _) = arg (i + 1) Ts t wenzelm@16678: | arg i Ts a = argT i (fastype_of1 (Ts, a)); wenzelm@16678: in arg 0 [] tm end; wenzelm@16678: clasohm@0: skalberg@15570: val list_abs = Library.foldr (fn ((x, T), t) => Abs (x, T, t)); wenzelm@10806: haftmann@18927: fun strip_abs (Abs (a, T, t)) = haftmann@18927: let val (a', t') = strip_abs t haftmann@18927: in ((a, T) :: a', t') end haftmann@18927: | strip_abs t = ([], t); haftmann@18927: clasohm@0: (* maps (x1,...,xn)t to t *) wenzelm@13000: fun strip_abs_body (Abs(_,_,t)) = strip_abs_body t clasohm@0: | strip_abs_body u = u; clasohm@0: clasohm@0: (* maps (x1,...,xn)t to [x1, ..., xn] *) wenzelm@13000: fun strip_abs_vars (Abs(a,T,t)) = (a,T) :: strip_abs_vars t clasohm@0: | strip_abs_vars u = [] : (string*typ) list; clasohm@0: clasohm@0: clasohm@0: fun strip_qnt_body qnt = clasohm@0: let fun strip(tm as Const(c,_)$Abs(_,_,t)) = if c=qnt then strip t else tm clasohm@0: | strip t = t clasohm@0: in strip end; clasohm@0: clasohm@0: fun strip_qnt_vars qnt = clasohm@0: let fun strip(Const(c,_)$Abs(a,T,t)) = if c=qnt then (a,T)::strip t else [] clasohm@0: | strip t = [] : (string*typ) list clasohm@0: in strip end; clasohm@0: clasohm@0: clasohm@0: (* maps (f, [t1,...,tn]) to f(t1,...,tn) *) skalberg@15570: val list_comb : term * term list -> term = Library.foldl (op $); clasohm@0: clasohm@0: clasohm@0: (* maps f(t1,...,tn) to (f, [t1,...,tn]) ; naturally tail-recursive*) wenzelm@13000: fun strip_comb u : term * term list = clasohm@0: let fun stripc (f$t, ts) = stripc (f, t::ts) wenzelm@13000: | stripc x = x clasohm@0: in stripc(u,[]) end; clasohm@0: clasohm@0: clasohm@0: (* maps f(t1,...,tn) to f , which is never a combination *) clasohm@0: fun head_of (f$t) = head_of f clasohm@0: | head_of u = u; clasohm@0: clasohm@0: wenzelm@16599: (*number of atoms and abstractions in a term*) wenzelm@16599: fun size_of_term tm = wenzelm@16599: let wenzelm@16678: fun add_size (t $ u, n) = add_size (t, add_size (u, n)) wenzelm@16678: | add_size (Abs (_ ,_, t), n) = add_size (t, n + 1) wenzelm@16678: | add_size (_, n) = n + 1; wenzelm@16678: in add_size (tm, 0) end; clasohm@0: haftmann@18847: fun map_atyps f (Type (a, Ts)) = Type (a, map (map_atyps f) Ts) haftmann@18847: | map_atyps f T = T; haftmann@18847: haftmann@18847: fun map_aterms f (t $ u) = map_aterms f t $ map_aterms f u haftmann@18847: | map_aterms f (Abs (a, T, t)) = Abs (a, T, map_aterms f t) haftmann@18847: | map_aterms f t = f t; haftmann@18847: wenzelm@16678: fun map_type_tvar f = wenzelm@16678: let wenzelm@16678: fun map_aux (Type (a, Ts)) = Type (a, map map_aux Ts) wenzelm@16678: | map_aux (TVar x) = f x wenzelm@16678: | map_aux T = T; wenzelm@16678: in map_aux end; nipkow@949: wenzelm@16678: fun map_type_tfree f = wenzelm@16678: let wenzelm@16678: fun map_aux (Type (a, Ts)) = Type (a, map map_aux Ts) wenzelm@16678: | map_aux (TFree x) = f x wenzelm@16678: | map_aux T = T; wenzelm@16678: in map_aux end; nipkow@949: clasohm@0: fun map_term_types f = wenzelm@16678: let wenzelm@16678: fun map_aux (Const (a, T)) = Const (a, f T) wenzelm@16678: | map_aux (Free (a, T)) = Free (a, f T) wenzelm@16678: | map_aux (Var (v, T)) = Var (v, f T) wenzelm@16678: | map_aux (t as Bound _) = t wenzelm@16678: | map_aux (Abs (a, T, t)) = Abs (a, f T, map_aux t) wenzelm@16678: | map_aux (t $ u) = map_aux t $ map_aux u; wenzelm@16678: in map_aux end; clasohm@0: clasohm@0: (* iterate a function over all types in a term *) clasohm@0: fun it_term_types f = clasohm@0: let fun iter(Const(_,T), a) = f(T,a) clasohm@0: | iter(Free(_,T), a) = f(T,a) clasohm@0: | iter(Var(_,T), a) = f(T,a) clasohm@0: | iter(Abs(_,T,t), a) = iter(t,f(T,a)) clasohm@0: | iter(f$u, a) = iter(f, iter(u, a)) clasohm@0: | iter(Bound _, a) = a clasohm@0: in iter end clasohm@0: clasohm@0: wenzelm@16943: (* fold types and terms *) wenzelm@16943: wenzelm@16943: (*fold atoms of type*) wenzelm@16943: fun fold_atyps f (Type (_, Ts)) = fold (fold_atyps f) Ts wenzelm@16943: | fold_atyps f T = f T; wenzelm@16943: wenzelm@16943: (*fold atoms of term*) wenzelm@16943: fun fold_aterms f (t $ u) = fold_aterms f t #> fold_aterms f u wenzelm@16943: | fold_aterms f (Abs (_, _, t)) = fold_aterms f t wenzelm@16943: | fold_aterms f a = f a; wenzelm@16943: wenzelm@16943: (*fold types of term*) wenzelm@16943: fun fold_term_types f (t as Const (_, T)) = f t T wenzelm@16943: | fold_term_types f (t as Free (_, T)) = f t T wenzelm@16943: | fold_term_types f (t as Var (_, T)) = f t T wenzelm@16943: | fold_term_types f (Bound _) = I wenzelm@16943: | fold_term_types f (t as Abs (_, T, b)) = f t T #> fold_term_types f b wenzelm@16943: | fold_term_types f (t $ u) = fold_term_types f t #> fold_term_types f u; wenzelm@16943: wenzelm@16943: fun fold_types f = fold_term_types (K f); wenzelm@16943: wenzelm@16943: (*collect variables*) wenzelm@16943: val add_tvarsT = fold_atyps (fn TVar v => insert (op =) v | _ => I); wenzelm@16943: val add_tvars = fold_types add_tvarsT; wenzelm@16943: val add_vars = fold_aterms (fn Var v => insert (op =) v | _ => I); wenzelm@16943: val add_tfreesT = fold_atyps (fn TFree v => insert (op =) v | _ => I); wenzelm@16943: val add_tfrees = fold_types add_tfreesT; wenzelm@16943: val add_frees = fold_aterms (fn Free v => insert (op =) v | _ => I); wenzelm@16943: wenzelm@16943: (*collect variable names*) wenzelm@16943: val add_term_varnames = fold_aterms (fn Var (xi, _) => insert (op =) xi | _ => I); wenzelm@16943: fun term_varnames t = add_term_varnames t []; wenzelm@16943: wenzelm@18149: fun find_free t x = wenzelm@18149: let wenzelm@18149: exception Found of term; wenzelm@18149: fun find (t as Free (x', _)) = if x = x' then raise Found t else I wenzelm@18149: | find _ = I; wenzelm@18149: in (fold_aterms find t (); NONE) handle Found v => SOME v end; wenzelm@18149: wenzelm@18149: wenzelm@16943: wenzelm@16678: (** Comparing terms, types, sorts etc. **) wenzelm@16537: wenzelm@16678: (* fast syntactic comparison *) wenzelm@16678: wenzelm@16678: fun fast_indexname_ord ((x, i), (y, j)) = wenzelm@16678: (case int_ord (i, j) of EQUAL => fast_string_ord (x, y) | ord => ord); wenzelm@16537: wenzelm@16599: fun sort_ord SS = wenzelm@16599: if pointer_eq SS then EQUAL wenzelm@16990: else dict_ord fast_string_ord SS; wenzelm@16678: wenzelm@16678: local wenzelm@16537: wenzelm@16678: fun cons_nr (TVar _) = 0 wenzelm@16678: | cons_nr (TFree _) = 1 wenzelm@16678: | cons_nr (Type _) = 2; wenzelm@16537: wenzelm@16678: in wenzelm@16537: wenzelm@16537: fun typ_ord TU = wenzelm@16537: if pointer_eq TU then EQUAL wenzelm@16537: else wenzelm@16537: (case TU of wenzelm@16678: (Type (a, Ts), Type (b, Us)) => wenzelm@16990: (case fast_string_ord (a, b) of EQUAL => dict_ord typ_ord (Ts, Us) | ord => ord) wenzelm@16678: | (TFree (a, S), TFree (b, S')) => wenzelm@16678: (case fast_string_ord (a, b) of EQUAL => sort_ord (S, S') | ord => ord) wenzelm@16678: | (TVar (xi, S), TVar (yj, S')) => wenzelm@16678: (case fast_indexname_ord (xi, yj) of EQUAL => sort_ord (S, S') | ord => ord) wenzelm@16678: | (T, U) => int_ord (cons_nr T, cons_nr U)); wenzelm@16678: wenzelm@16678: end; wenzelm@16678: wenzelm@16678: local wenzelm@16678: wenzelm@16678: fun cons_nr (Const _) = 0 wenzelm@16678: | cons_nr (Free _) = 1 wenzelm@16678: | cons_nr (Var _) = 2 wenzelm@16678: | cons_nr (Bound _) = 3 wenzelm@16678: | cons_nr (Abs _) = 4 wenzelm@16678: | cons_nr (_ $ _) = 5; wenzelm@16678: wenzelm@16678: fun struct_ord (Abs (_, _, t), Abs (_, _, u)) = struct_ord (t, u) wenzelm@16678: | struct_ord (t1 $ t2, u1 $ u2) = wenzelm@16678: (case struct_ord (t1, u1) of EQUAL => struct_ord (t2, u2) | ord => ord) wenzelm@16678: | struct_ord (t, u) = int_ord (cons_nr t, cons_nr u); wenzelm@16678: wenzelm@16678: fun atoms_ord (Abs (_, _, t), Abs (_, _, u)) = atoms_ord (t, u) wenzelm@16678: | atoms_ord (t1 $ t2, u1 $ u2) = wenzelm@16678: (case atoms_ord (t1, u1) of EQUAL => atoms_ord (t2, u2) | ord => ord) wenzelm@16678: | atoms_ord (Const (a, _), Const (b, _)) = fast_string_ord (a, b) wenzelm@16678: | atoms_ord (Free (x, _), Free (y, _)) = fast_string_ord (x, y) wenzelm@16678: | atoms_ord (Var (xi, _), Var (yj, _)) = fast_indexname_ord (xi, yj) wenzelm@16678: | atoms_ord (Bound i, Bound j) = int_ord (i, j) wenzelm@16678: | atoms_ord _ = sys_error "atoms_ord"; wenzelm@16678: wenzelm@16678: fun types_ord (Abs (_, T, t), Abs (_, U, u)) = wenzelm@16678: (case typ_ord (T, U) of EQUAL => types_ord (t, u) | ord => ord) wenzelm@16678: | types_ord (t1 $ t2, u1 $ u2) = wenzelm@16678: (case types_ord (t1, u1) of EQUAL => types_ord (t2, u2) | ord => ord) wenzelm@16678: | types_ord (Const (_, T), Const (_, U)) = typ_ord (T, U) wenzelm@16678: | types_ord (Free (_, T), Free (_, U)) = typ_ord (T, U) wenzelm@16678: | types_ord (Var (_, T), Var (_, U)) = typ_ord (T, U) wenzelm@16678: | types_ord (Bound _, Bound _) = EQUAL wenzelm@16678: | types_ord _ = sys_error "types_ord"; wenzelm@16678: wenzelm@16678: in wenzelm@16678: wenzelm@16678: fun fast_term_ord tu = wenzelm@16678: if pointer_eq tu then EQUAL wenzelm@16678: else wenzelm@16678: (case struct_ord tu of wenzelm@16678: EQUAL => (case atoms_ord tu of EQUAL => types_ord tu | ord => ord) wenzelm@16678: | ord => ord); wenzelm@16678: wenzelm@16678: fun op aconv tu = (fast_term_ord tu = EQUAL); wenzelm@16678: fun aconvs ts_us = (list_ord fast_term_ord ts_us = EQUAL); wenzelm@16678: wenzelm@16678: structure Vartab = TableFun(type key = indexname val ord = fast_indexname_ord); wenzelm@16678: structure Typtab = TableFun(type key = typ val ord = typ_ord); wenzelm@16678: structure Termtab = TableFun(type key = term val ord = fast_term_ord); wenzelm@16678: wenzelm@16678: end; wenzelm@16537: wenzelm@16537: wenzelm@16537: (* term_ord *) wenzelm@16537: wenzelm@16537: (*a linear well-founded AC-compatible ordering for terms: wenzelm@16537: s < t <=> 1. size(s) < size(t) or wenzelm@16537: 2. size(s) = size(t) and s=f(...) and t=g(...) and f string_ord (x, y) | ord => ord); wenzelm@16678: wenzelm@16667: local wenzelm@16667: wenzelm@16667: fun hd_depth (t $ _, n) = hd_depth (t, n + 1) wenzelm@16667: | hd_depth p = p; wenzelm@16537: wenzelm@16537: fun dest_hd (Const (a, T)) = (((a, 0), T), 0) wenzelm@16537: | dest_hd (Free (a, T)) = (((a, 0), T), 1) wenzelm@16537: | dest_hd (Var v) = (v, 2) wenzelm@16537: | dest_hd (Bound i) = ((("", i), dummyT), 3) wenzelm@16537: | dest_hd (Abs (_, T, _)) = ((("", 0), T), 4); wenzelm@16537: wenzelm@16667: in wenzelm@16667: wenzelm@16537: fun term_ord tu = wenzelm@16537: if pointer_eq tu then EQUAL wenzelm@16537: else wenzelm@16537: (case tu of wenzelm@16537: (Abs (_, T, t), Abs(_, U, u)) => wenzelm@16537: (case term_ord (t, u) of EQUAL => typ_ord (T, U) | ord => ord) wenzelm@16667: | (t, u) => wenzelm@16537: (case int_ord (size_of_term t, size_of_term u) of wenzelm@16537: EQUAL => wenzelm@16943: (case prod_ord hd_ord int_ord (hd_depth (t, 0), hd_depth (u, 0)) of wenzelm@16943: EQUAL => args_ord (t, u) | ord => ord) wenzelm@16537: | ord => ord)) wenzelm@16537: and hd_ord (f, g) = wenzelm@16537: prod_ord (prod_ord indexname_ord typ_ord) int_ord (dest_hd f, dest_hd g) wenzelm@16667: and args_ord (f $ t, g $ u) = wenzelm@16667: (case args_ord (f, g) of EQUAL => term_ord (t, u) | ord => ord) wenzelm@16667: | args_ord _ = EQUAL; wenzelm@16537: wenzelm@16537: fun termless tu = (term_ord tu = LESS); wenzelm@16537: wenzelm@16667: end; wenzelm@16667: wenzelm@16667: wenzelm@16667: (** Lexicographic path order on terms **) wenzelm@16667: wenzelm@16667: (* nipkow@16570: See Baader & Nipkow, Term rewriting, CUP 1998. nipkow@16570: Without variables. Const, Var, Bound, Free and Abs are treated all as nipkow@16570: constants. nipkow@16570: nipkow@16570: f_ord maps strings to integers and serves two purposes: nipkow@16570: - Predicate on constant symbols. Those that are not recognised by f_ord nipkow@16570: must be mapped to ~1. nipkow@16570: - Order on the recognised symbols. These must be mapped to distinct nipkow@16570: integers >= 0. nipkow@16570: wenzelm@16667: *) nipkow@16570: nipkow@16570: local wenzelm@16667: fun dest_hd f_ord (Const (a, T)) = nipkow@16570: let val ord = f_ord a in nipkow@16570: if ord = ~1 then ((1, ((a, 0), T)), 0) else ((0, (("", ord), T)), 0) nipkow@16570: end nipkow@16570: | dest_hd _ (Free (a, T)) = ((1, ((a, 0), T)), 0) nipkow@16570: | dest_hd _ (Var v) = ((1, v), 1) nipkow@16570: | dest_hd _ (Bound i) = ((1, (("", i), dummyT)), 2) nipkow@16570: | dest_hd _ (Abs (_, T, _)) = ((1, (("", 0), T)), 3); nipkow@16570: nipkow@16570: fun term_lpo f_ord (s, t) = nipkow@16570: let val (f, ss) = strip_comb s and (g, ts) = strip_comb t in nipkow@16570: if forall (fn si => term_lpo f_ord (si, t) = LESS) ss nipkow@16570: then case hd_ord f_ord (f, g) of wenzelm@16667: GREATER => wenzelm@16667: if forall (fn ti => term_lpo f_ord (s, ti) = GREATER) ts wenzelm@16667: then GREATER else LESS nipkow@16570: | EQUAL => wenzelm@16667: if forall (fn ti => term_lpo f_ord (s, ti) = GREATER) ts wenzelm@16667: then list_ord (term_lpo f_ord) (ss, ts) wenzelm@16667: else LESS nipkow@16570: | LESS => LESS nipkow@16570: else GREATER nipkow@16570: end nipkow@16570: and hd_ord f_ord (f, g) = case (f, g) of nipkow@16570: (Abs (_, T, t), Abs (_, U, u)) => nipkow@16570: (case term_lpo f_ord (t, u) of EQUAL => typ_ord (T, U) | ord => ord) nipkow@16570: | (_, _) => prod_ord (prod_ord int_ord nipkow@16570: (prod_ord indexname_ord typ_ord)) int_ord nipkow@16570: (dest_hd f_ord f, dest_hd f_ord g) nipkow@16570: in nipkow@16570: val term_lpo = term_lpo nipkow@16570: end; nipkow@16570: wenzelm@16537: clasohm@0: (** Connectives of higher order logic **) clasohm@0: wenzelm@375: fun itselfT ty = Type ("itself", [ty]); wenzelm@14854: val a_itselfT = itselfT (TFree ("'a", [])); wenzelm@375: clasohm@0: val propT : typ = Type("prop",[]); clasohm@0: clasohm@0: val implies = Const("==>", propT-->propT-->propT); clasohm@0: clasohm@0: fun all T = Const("all", (T-->propT)-->propT); clasohm@0: clasohm@0: fun equals T = Const("==", T-->T-->propT); clasohm@0: clasohm@0: (* maps !!x1...xn. t to t *) wenzelm@13000: fun strip_all_body (Const("all",_)$Abs(_,_,t)) = strip_all_body t clasohm@0: | strip_all_body t = t; clasohm@0: clasohm@0: (* maps !!x1...xn. t to [x1, ..., xn] *) clasohm@0: fun strip_all_vars (Const("all",_)$Abs(a,T,t)) = wenzelm@13000: (a,T) :: strip_all_vars t clasohm@0: | strip_all_vars t = [] : (string*typ) list; clasohm@0: clasohm@0: (*increments a term's non-local bound variables clasohm@0: required when moving a term within abstractions clasohm@0: inc is increment for bound variables clasohm@0: lev is level at which a bound variable is considered 'loose'*) wenzelm@13000: fun incr_bv (inc, lev, u as Bound i) = if i>=lev then Bound(i+inc) else u clasohm@0: | incr_bv (inc, lev, Abs(a,T,body)) = wenzelm@9536: Abs(a, T, incr_bv(inc,lev+1,body)) wenzelm@13000: | incr_bv (inc, lev, f$t) = clasohm@0: incr_bv(inc,lev,f) $ incr_bv(inc,lev,t) clasohm@0: | incr_bv (inc, lev, u) = u; clasohm@0: clasohm@0: fun incr_boundvars 0 t = t clasohm@0: | incr_boundvars inc t = incr_bv(inc,0,t); clasohm@0: wenzelm@12981: (*Scan a pair of terms; while they are similar, wenzelm@12981: accumulate corresponding bound vars in "al"*) wenzelm@12981: fun match_bvs(Abs(x,_,s),Abs(y,_,t), al) = wenzelm@12981: match_bvs(s, t, if x="" orelse y="" then al wenzelm@12981: else (x,y)::al) wenzelm@12981: | match_bvs(f$s, g$t, al) = match_bvs(f,g,match_bvs(s,t,al)) wenzelm@12981: | match_bvs(_,_,al) = al; wenzelm@12981: wenzelm@12981: (* strip abstractions created by parameters *) wenzelm@12981: fun match_bvars((s,t),al) = match_bvs(strip_abs_body s, strip_abs_body t, al); wenzelm@12981: wenzelm@12981: fun rename_abs pat obj t = wenzelm@12981: let wenzelm@12981: val ren = match_bvs (pat, obj, []); wenzelm@12981: fun ren_abs (Abs (x, T, b)) = wenzelm@18942: Abs (the_default x (AList.lookup (op =) ren x), T, ren_abs b) wenzelm@12981: | ren_abs (f $ t) = ren_abs f $ ren_abs t wenzelm@12981: | ren_abs t = t skalberg@15531: in if null ren then NONE else SOME (ren_abs t) end; clasohm@0: clasohm@0: (*Accumulate all 'loose' bound vars referring to level 'lev' or beyond. clasohm@0: (Bound 0) is loose at level 0 *) wenzelm@13000: fun add_loose_bnos (Bound i, lev, js) = wenzelm@9536: if i= k clasohm@0: | loose_bvar(f$t, k) = loose_bvar(f,k) orelse loose_bvar(t,k) clasohm@0: | loose_bvar(Abs(_,_,t),k) = loose_bvar(t,k+1) clasohm@0: | loose_bvar _ = false; clasohm@0: nipkow@2792: fun loose_bvar1(Bound i,k) = i = k nipkow@2792: | loose_bvar1(f$t, k) = loose_bvar1(f,k) orelse loose_bvar1(t,k) nipkow@2792: | loose_bvar1(Abs(_,_,t),k) = loose_bvar1(t,k+1) nipkow@2792: | loose_bvar1 _ = false; clasohm@0: clasohm@0: (*Substitute arguments for loose bound variables. clasohm@0: Beta-reduction of arg(n-1)...arg0 into t replacing (Bound i) with (argi). wenzelm@4626: Note that for ((%x y. c) a b), the bound vars in c are x=1 and y=0 wenzelm@9536: and the appropriate call is subst_bounds([b,a], c) . clasohm@0: Loose bound variables >=n are reduced by "n" to clasohm@0: compensate for the disappearance of lambdas. clasohm@0: *) wenzelm@13000: fun subst_bounds (args: term list, t) : term = clasohm@0: let val n = length args; clasohm@0: fun subst (t as Bound i, lev) = wenzelm@9536: (if i Bound(i-n) (*loose: change it*)) wenzelm@9536: | subst (Abs(a,T,body), lev) = Abs(a, T, subst(body,lev+1)) wenzelm@9536: | subst (f$t, lev) = subst(f,lev) $ subst(t,lev) wenzelm@9536: | subst (t,lev) = t clasohm@0: in case args of [] => t | _ => subst (t,0) end; clasohm@0: paulson@2192: (*Special case: one argument*) wenzelm@13000: fun subst_bound (arg, t) : term = paulson@2192: let fun subst (t as Bound i, lev) = wenzelm@9536: if i true clasohm@0: | (Var _, _) => true clasohm@0: | (Const(a,_), Const(b,_)) => a=b andalso matchrands(t,u) clasohm@0: | (Free(a,_), Free(b,_)) => a=b andalso matchrands(t,u) clasohm@0: | (Bound i, Bound j) => i=j andalso matchrands(t,u) clasohm@0: | (Abs _, _) => true (*because of possible eta equality*) clasohm@0: | (_, Abs _) => true clasohm@0: | _ => false clasohm@0: end; clasohm@0: clasohm@0: (*Substitute new for free occurrences of old in a term*) clasohm@0: fun subst_free [] = (fn t=>t) clasohm@0: | subst_free pairs = wenzelm@13000: let fun substf u = haftmann@17314: case AList.lookup (op aconv) pairs u of skalberg@15531: SOME u' => u' skalberg@15531: | NONE => (case u of Abs(a,T,t) => Abs(a, T, substf t) wenzelm@9536: | t$u' => substf t $ substf u' wenzelm@9536: | _ => u) clasohm@0: in substf end; clasohm@0: clasohm@0: (*a total, irreflexive ordering on index names*) clasohm@0: fun xless ((a,i), (b,j): indexname) = i Abs (a, T, abs (lev + 1) t) wenzelm@16990: | t $ u => (abs lev t $ (abs lev u handle SAME => u) handle SAME => t $ abs lev u) wenzelm@16990: | _ => raise SAME); wenzelm@16990: in abs 0 body handle SAME => body end; clasohm@0: wenzelm@18942: fun lambda (v as Const (x, T)) t = Abs (x, T, abstract_over (v, t)) wenzelm@18942: | lambda (v as Free (x, T)) t = Abs (x, T, abstract_over (v, t)) berghofe@13665: | lambda (v as Var ((x, _), T)) t = Abs (x, T, abstract_over (v, t)) berghofe@13665: | lambda v t = raise TERM ("lambda", [v, t]); clasohm@0: clasohm@0: (*Form an abstraction over a free variable.*) clasohm@0: fun absfree (a,T,body) = Abs(a, T, abstract_over (Free(a,T), body)); wenzelm@17786: fun absdummy (T, body) = Abs ("uu", T, body); clasohm@0: clasohm@0: (*Abstraction over a list of free variables*) clasohm@0: fun list_abs_free ([ ] , t) = t wenzelm@13000: | list_abs_free ((a,T)::vars, t) = clasohm@0: absfree(a, T, list_abs_free(vars,t)); clasohm@0: clasohm@0: (*Quantification over a list of free variables*) clasohm@0: fun list_all_free ([], t: term) = t wenzelm@13000: | list_all_free ((a,T)::vars, t) = clasohm@0: (all T) $ (absfree(a, T, list_all_free(vars,t))); clasohm@0: clasohm@0: (*Quantification over a list of variables (already bound in body) *) clasohm@0: fun list_all ([], t) = t wenzelm@13000: | list_all ((a,T)::vars, t) = clasohm@0: (all T) $ (Abs(a, T, list_all(vars,t))); clasohm@0: wenzelm@16678: (*Replace the ATOMIC term ti by ui; inst = [(t1,u1), ..., (tn,un)]. clasohm@0: A simultaneous substitution: [ (a,b), (b,a) ] swaps a and b. *) wenzelm@16678: fun subst_atomic [] tm = tm wenzelm@16678: | subst_atomic inst tm = wenzelm@16678: let wenzelm@16678: fun subst (Abs (a, T, body)) = Abs (a, T, subst body) wenzelm@16678: | subst (t $ u) = subst t $ subst u wenzelm@18942: | subst t = the_default t (AList.lookup (op aconv) inst t); wenzelm@16678: in subst tm end; clasohm@0: wenzelm@16678: (*Replace the ATOMIC type Ti by Ui; inst = [(T1,U1), ..., (Tn,Un)].*) wenzelm@16678: fun typ_subst_atomic [] ty = ty wenzelm@16678: | typ_subst_atomic inst ty = wenzelm@16678: let wenzelm@16678: fun subst (Type (a, Ts)) = Type (a, map subst Ts) wenzelm@18942: | subst T = the_default T (AList.lookup (op = : typ * typ -> bool) inst T); wenzelm@16678: in subst ty end; berghofe@15797: wenzelm@16678: fun subst_atomic_types [] tm = tm wenzelm@16678: | subst_atomic_types inst tm = map_term_types (typ_subst_atomic inst) tm; wenzelm@16678: wenzelm@16678: fun typ_subst_TVars [] ty = ty wenzelm@16678: | typ_subst_TVars inst ty = wenzelm@16678: let wenzelm@16678: fun subst (Type (a, Ts)) = Type (a, map subst Ts) wenzelm@18942: | subst (T as TVar (xi, _)) = the_default T (AList.lookup (op =) inst xi) wenzelm@16678: | subst T = T; wenzelm@16678: in subst ty end; clasohm@0: wenzelm@16678: fun subst_TVars [] tm = tm wenzelm@16678: | subst_TVars inst tm = map_term_types (typ_subst_TVars inst) tm; clasohm@0: wenzelm@16678: (*see also Envir.norm_term*) wenzelm@16678: fun subst_Vars [] tm = tm wenzelm@16678: | subst_Vars inst tm = wenzelm@16678: let wenzelm@18942: fun subst (t as Var (xi, _)) = the_default t (AList.lookup (op =) inst xi) wenzelm@16678: | subst (Abs (a, T, t)) = Abs (a, T, subst t) wenzelm@16678: | subst (t $ u) = subst t $ subst u wenzelm@16678: | subst t = t; wenzelm@16678: in subst tm end; clasohm@0: wenzelm@16678: (*see also Envir.norm_term*) wenzelm@16678: fun subst_vars ([], []) tm = tm wenzelm@16678: | subst_vars ([], inst) tm = subst_Vars inst tm wenzelm@16678: | subst_vars (instT, inst) tm = wenzelm@16678: let wenzelm@16678: fun subst (Const (a, T)) = Const (a, typ_subst_TVars instT T) wenzelm@16678: | subst (Free (a, T)) = Free (a, typ_subst_TVars instT T) wenzelm@16678: | subst (t as Var (xi, T)) = haftmann@17271: (case AList.lookup (op =) inst xi of wenzelm@16678: NONE => Var (xi, typ_subst_TVars instT T) wenzelm@16678: | SOME t => t) wenzelm@16678: | subst (t as Bound _) = t wenzelm@16678: | subst (Abs (a, T, t)) = Abs (a, typ_subst_TVars instT T, subst t) wenzelm@16678: | subst (t $ u) = subst t $ subst u; wenzelm@16678: in subst tm end; clasohm@0: clasohm@0: wenzelm@16943: (* instantiation of schematic variables (types before terms) *) wenzelm@16882: wenzelm@16882: local exception SAME in wenzelm@16882: wenzelm@16943: fun instantiateT_same [] _ = raise SAME wenzelm@16943: | instantiateT_same instT ty = wenzelm@16882: let wenzelm@16882: fun subst_typ (Type (a, Ts)) = Type (a, subst_typs Ts) wenzelm@16882: | subst_typ (TVar v) = haftmann@17314: (case AList.lookup eq_tvar instT v of wenzelm@16882: SOME T => T wenzelm@16882: | NONE => raise SAME) wenzelm@16882: | subst_typ _ = raise SAME wenzelm@16882: and subst_typs (T :: Ts) = wenzelm@16882: (subst_typ T :: (subst_typs Ts handle SAME => Ts) wenzelm@16882: handle SAME => T :: subst_typs Ts) wenzelm@16882: | subst_typs [] = raise SAME; wenzelm@16882: in subst_typ ty end; wenzelm@16882: wenzelm@16882: fun instantiate ([], []) tm = tm wenzelm@16882: | instantiate (instT, inst) tm = wenzelm@16882: let wenzelm@16943: val substT = instantiateT_same instT; wenzelm@16943: fun subst (Const (c, T)) = Const (c, substT T) wenzelm@16943: | subst (Free (x, T)) = Free (x, substT T) wenzelm@16882: | subst (Var (xi, T)) = wenzelm@16943: let val (T', same) = (substT T, false) handle SAME => (T, true) in haftmann@17314: (case AList.lookup eq_var inst (xi, T') of wenzelm@16882: SOME t => t wenzelm@16882: | NONE => if same then raise SAME else Var (xi, T')) wenzelm@16882: end wenzelm@16882: | subst (Bound _) = raise SAME wenzelm@16882: | subst (Abs (x, T, t)) = wenzelm@16943: (Abs (x, substT T, subst t handle SAME => t) wenzelm@16882: handle SAME => Abs (x, T, subst t)) wenzelm@16882: | subst (t $ u) = (subst t $ (subst u handle SAME => u) handle SAME => t $ subst u); wenzelm@16882: in subst tm handle SAME => tm end; wenzelm@16882: wenzelm@16882: fun instantiateT instT ty = wenzelm@16943: instantiateT_same instT ty handle SAME => ty; wenzelm@16882: wenzelm@16882: end; wenzelm@16882: wenzelm@16882: paulson@15573: (** Identifying first-order terms **) paulson@15573: paulson@15573: (*Argument Ts is a reverse list of binder types, needed if term t contains Bound vars*) paulson@15573: fun has_not_funtype Ts t = not (is_funtype (fastype_of1 (Ts,t))); paulson@15573: wenzelm@16537: (*First order means in all terms of the form f(t1,...,tn) no argument has a paulson@16589: function type. The supplied quantifiers are excluded: their argument always paulson@16589: has a function type through a recursive call into its body.*) wenzelm@16667: fun is_first_order quants = paulson@16589: let fun first_order1 Ts (Abs (_,T,body)) = first_order1 (T::Ts) body wenzelm@16667: | first_order1 Ts (Const(q,_) $ Abs(a,T,body)) = wenzelm@16667: q mem_string quants andalso (*it is a known quantifier*) paulson@16589: not (is_funtype T) andalso first_order1 (T::Ts) body wenzelm@16667: | first_order1 Ts t = wenzelm@16667: case strip_comb t of wenzelm@16667: (Var _, ts) => forall (first_order1 Ts andf has_not_funtype Ts) ts wenzelm@16667: | (Free _, ts) => forall (first_order1 Ts andf has_not_funtype Ts) ts wenzelm@16667: | (Const _, ts) => forall (first_order1 Ts andf has_not_funtype Ts) ts wenzelm@16667: | (Bound _, ts) => forall (first_order1 Ts andf has_not_funtype Ts) ts wenzelm@16667: | (Abs _, ts) => false (*not in beta-normal form*) wenzelm@16667: | _ => error "first_order: unexpected case" paulson@16589: in first_order1 [] end; paulson@15573: wenzelm@16710: wenzelm@16990: (* maximum index of typs and terms *) clasohm@0: wenzelm@16710: fun maxidx_typ (TVar ((_, j), _)) i = Int.max (i, j) wenzelm@16710: | maxidx_typ (Type (_, Ts)) i = maxidx_typs Ts i wenzelm@16710: | maxidx_typ (TFree _) i = i wenzelm@16710: and maxidx_typs [] i = i wenzelm@16710: | maxidx_typs (T :: Ts) i = maxidx_typs Ts (maxidx_typ T i); clasohm@0: wenzelm@16710: fun maxidx_term (Var ((_, j), T)) i = maxidx_typ T (Int.max (i, j)) wenzelm@16710: | maxidx_term (Const (_, T)) i = maxidx_typ T i wenzelm@16710: | maxidx_term (Free (_, T)) i = maxidx_typ T i wenzelm@16710: | maxidx_term (Bound _) i = i wenzelm@16710: | maxidx_term (Abs (_, T, t)) i = maxidx_term t (maxidx_typ T i) wenzelm@16710: | maxidx_term (t $ u) i = maxidx_term u (maxidx_term t i); clasohm@0: wenzelm@16710: fun maxidx_of_typ T = maxidx_typ T ~1; wenzelm@16710: fun maxidx_of_typs Ts = maxidx_typs Ts ~1; wenzelm@16710: fun maxidx_of_term t = maxidx_term t ~1; berghofe@13665: clasohm@0: clasohm@0: clasohm@0: (**** Syntax-related declarations ****) clasohm@0: clasohm@0: (*** Printing ***) clasohm@0: wenzelm@16943: (*Makes a variant of a name distinct from the names in 'used'. wenzelm@14676: First attaches the suffix and then increments this; wenzelm@12306: preserves a suffix of underscores "_". *) wenzelm@16943: fun variant used name = wenzelm@12306: let wenzelm@12306: val (c, u) = pairself implode (Library.take_suffix (equal "_") (Symbol.explode name)); wenzelm@16943: fun vary2 c = if ((c ^ u) mem_string used) then vary2 (Symbol.bump_string c) else c; wenzelm@16943: fun vary1 c = if ((c ^ u) mem_string used) then vary2 (Symbol.bump_init c) else c; wenzelm@12306: in vary1 (if c = "" then "u" else c) ^ u end; clasohm@0: clasohm@0: (*Create variants of the list of names, with priority to the first ones*) clasohm@0: fun variantlist ([], used) = [] wenzelm@13000: | variantlist(b::bs, used) = clasohm@0: let val b' = variant used b clasohm@0: in b' :: variantlist (bs, b'::used) end; clasohm@0: wenzelm@14695: (*Invent fresh names*) wenzelm@14695: fun invent_names _ _ 0 = [] wenzelm@14695: | invent_names used a n = wenzelm@14695: let val b = Symbol.bump_string a in wenzelm@14695: if a mem_string used then invent_names used b n wenzelm@14695: else a :: invent_names used b (n - 1) wenzelm@14695: end; wenzelm@11353: wenzelm@16537: wenzelm@4017: (** Consts etc. **) wenzelm@4017: skalberg@15574: fun add_typ_classes (Type (_, Ts), cs) = foldr add_typ_classes cs Ts wenzelm@4017: | add_typ_classes (TFree (_, S), cs) = S union cs wenzelm@4017: | add_typ_classes (TVar (_, S), cs) = S union cs; wenzelm@4017: wenzelm@16294: fun add_typ_tycons (Type (c, Ts), cs) = foldr add_typ_tycons (c ins_string cs) Ts wenzelm@4017: | add_typ_tycons (_, cs) = cs; wenzelm@4017: wenzelm@4017: val add_term_classes = it_term_types add_typ_classes; wenzelm@4017: val add_term_tycons = it_term_types add_typ_tycons; wenzelm@4017: wenzelm@9319: fun add_term_consts (Const (c, _), cs) = c ins_string cs wenzelm@4017: | add_term_consts (t $ u, cs) = add_term_consts (t, add_term_consts (u, cs)) wenzelm@4017: | add_term_consts (Abs (_, _, t), cs) = add_term_consts (t, cs) wenzelm@4017: | add_term_consts (_, cs) = cs; wenzelm@4017: nipkow@13646: fun term_consts t = add_term_consts(t,[]); nipkow@13646: wenzelm@16943: fun exists_subterm P = wenzelm@16943: let wenzelm@16943: fun ex tm = P tm orelse wenzelm@16943: (case tm of wenzelm@16943: t $ u => ex t orelse ex u wenzelm@16943: | Abs (_, _, t) => ex t wenzelm@16943: | _ => false); wenzelm@16943: in ex end; obua@16108: wenzelm@16943: fun exists_Const P = exists_subterm (fn Const c => P c | _ => false); nipkow@4631: wenzelm@4017: clasohm@0: (** TFrees and TVars **) clasohm@0: wenzelm@12802: (*Accumulates the names of Frees in the term, suppressing duplicates.*) wenzelm@12802: fun add_term_free_names (Free(a,_), bs) = a ins_string bs wenzelm@12802: | add_term_free_names (f$u, bs) = add_term_free_names (f, add_term_free_names(u, bs)) wenzelm@12802: | add_term_free_names (Abs(_,_,t), bs) = add_term_free_names(t,bs) wenzelm@12802: | add_term_free_names (_, bs) = bs; wenzelm@12802: clasohm@0: (*Accumulates the names in the term, suppressing duplicates. clasohm@0: Includes Frees and Consts. For choosing unambiguous bound var names.*) wenzelm@10666: fun add_term_names (Const(a,_), bs) = NameSpace.base a ins_string bs paulson@2176: | add_term_names (Free(a,_), bs) = a ins_string bs clasohm@0: | add_term_names (f$u, bs) = add_term_names (f, add_term_names(u, bs)) clasohm@0: | add_term_names (Abs(_,_,t), bs) = add_term_names(t,bs) clasohm@0: | add_term_names (_, bs) = bs; clasohm@0: clasohm@0: (*Accumulates the TVars in a type, suppressing duplicates. *) skalberg@15574: fun add_typ_tvars(Type(_,Ts),vs) = foldr add_typ_tvars vs Ts clasohm@0: | add_typ_tvars(TFree(_),vs) = vs wenzelm@16294: | add_typ_tvars(TVar(v),vs) = insert (op =) v vs; clasohm@0: clasohm@0: (*Accumulates the TFrees in a type, suppressing duplicates. *) skalberg@15574: fun add_typ_tfree_names(Type(_,Ts),fs) = foldr add_typ_tfree_names fs Ts paulson@2176: | add_typ_tfree_names(TFree(f,_),fs) = f ins_string fs clasohm@0: | add_typ_tfree_names(TVar(_),fs) = fs; clasohm@0: skalberg@15574: fun add_typ_tfrees(Type(_,Ts),fs) = foldr add_typ_tfrees fs Ts wenzelm@16294: | add_typ_tfrees(TFree(f),fs) = insert (op =) f fs clasohm@0: | add_typ_tfrees(TVar(_),fs) = fs; clasohm@0: skalberg@15574: fun add_typ_varnames(Type(_,Ts),nms) = foldr add_typ_varnames nms Ts paulson@2176: | add_typ_varnames(TFree(nm,_),nms) = nm ins_string nms paulson@2176: | add_typ_varnames(TVar((nm,_),_),nms) = nm ins_string nms; nipkow@949: clasohm@0: (*Accumulates the TVars in a term, suppressing duplicates. *) clasohm@0: val add_term_tvars = it_term_types add_typ_tvars; clasohm@0: clasohm@0: (*Accumulates the TFrees in a term, suppressing duplicates. *) clasohm@0: val add_term_tfrees = it_term_types add_typ_tfrees; clasohm@0: val add_term_tfree_names = it_term_types add_typ_tfree_names; clasohm@0: nipkow@949: val add_term_tvarnames = it_term_types add_typ_varnames; nipkow@949: clasohm@0: (*Non-list versions*) clasohm@0: fun typ_tfrees T = add_typ_tfrees(T,[]); clasohm@0: fun typ_tvars T = add_typ_tvars(T,[]); clasohm@0: fun term_tfrees t = add_term_tfrees(t,[]); clasohm@0: fun term_tvars t = add_term_tvars(t,[]); clasohm@0: nipkow@949: (*special code to enforce left-to-right collection of TVar-indexnames*) nipkow@949: skalberg@15570: fun add_typ_ixns(ixns,Type(_,Ts)) = Library.foldl add_typ_ixns (ixns,Ts) wenzelm@13000: | add_typ_ixns(ixns,TVar(ixn,_)) = if mem_ix (ixn, ixns) then ixns wenzelm@9536: else ixns@[ixn] nipkow@949: | add_typ_ixns(ixns,TFree(_)) = ixns; nipkow@949: nipkow@949: fun add_term_tvar_ixns(Const(_,T),ixns) = add_typ_ixns(ixns,T) nipkow@949: | add_term_tvar_ixns(Free(_,T),ixns) = add_typ_ixns(ixns,T) nipkow@949: | add_term_tvar_ixns(Var(_,T),ixns) = add_typ_ixns(ixns,T) nipkow@949: | add_term_tvar_ixns(Bound _,ixns) = ixns nipkow@949: | add_term_tvar_ixns(Abs(_,T,t),ixns) = nipkow@949: add_term_tvar_ixns(t,add_typ_ixns(ixns,T)) nipkow@949: | add_term_tvar_ixns(f$t,ixns) = nipkow@949: add_term_tvar_ixns(t,add_term_tvar_ixns(f,ixns)); nipkow@949: wenzelm@16537: clasohm@0: (** Frees and Vars **) clasohm@0: clasohm@0: (*Accumulates the Vars in the term, suppressing duplicates*) clasohm@0: fun add_term_vars (t, vars: term list) = case t of wenzelm@16990: Var _ => OrdList.insert term_ord t vars clasohm@0: | Abs (_,_,body) => add_term_vars(body,vars) clasohm@0: | f$t => add_term_vars (f, add_term_vars(t, vars)) clasohm@0: | _ => vars; clasohm@0: clasohm@0: fun term_vars t = add_term_vars(t,[]); clasohm@0: clasohm@0: (*Accumulates the Frees in the term, suppressing duplicates*) clasohm@0: fun add_term_frees (t, frees: term list) = case t of wenzelm@16990: Free _ => OrdList.insert term_ord t frees clasohm@0: | Abs (_,_,body) => add_term_frees(body,frees) clasohm@0: | f$t => add_term_frees (f, add_term_frees(t, frees)) clasohm@0: | _ => frees; clasohm@0: clasohm@0: fun term_frees t = add_term_frees(t,[]); clasohm@0: clasohm@0: (*Given an abstraction over P, replaces the bound variable by a Free variable wenzelm@17851: having a unique name -- SLOW!*) clasohm@0: fun variant_abs (a,T,P) = clasohm@0: let val b = variant (add_term_names(P,[])) a paulson@2192: in (b, subst_bound (Free(b,T), P)) end; clasohm@0: wenzelm@16678: fun dest_abs (x, T, body) = wenzelm@16678: let wenzelm@16678: fun name_clash (Free (y, _)) = (x = y) wenzelm@16678: | name_clash (t $ u) = name_clash t orelse name_clash u wenzelm@16678: | name_clash (Abs (_, _, t)) = name_clash t wenzelm@16678: | name_clash _ = false; wenzelm@16678: in wenzelm@16678: if name_clash body then wenzelm@16678: dest_abs (variant [x] x, T, body) (*potentially slow, but rarely happens*) wenzelm@16678: else (x, subst_bound (Free (x, T), body)) wenzelm@16678: end; wenzelm@16678: wenzelm@16990: (*names for numbered variables -- wenzelm@16990: preserves order wrt. int_ord vs. string_ord, avoids allocating new strings*) wenzelm@16990: local wenzelm@16990: val small_int = Vector.tabulate (1000, fn i => wenzelm@16990: let val leading = if i < 10 then "00" else if i < 100 then "0" else "" wenzelm@16990: in ":" ^ leading ^ string_of_int i end); wenzelm@16990: in wenzelm@16990: fun bound n = wenzelm@16990: if n < 1000 then Vector.sub (small_int, n) wenzelm@16990: else ":" ^ bound (n div 1000) ^ Vector.sub (small_int, n mod 1000); wenzelm@16990: end; wenzelm@16990: wenzelm@16990: val is_bound = String.isPrefix ":"; wenzelm@16943: clasohm@0: (* renames and reverses the strings in vars away from names *) clasohm@0: fun rename_aTs names vars : (string*typ)list = clasohm@0: let fun rename_aT (vars,(a,T)) = wenzelm@9536: (variant (map #1 vars @ names) a, T) :: vars skalberg@15570: in Library.foldl rename_aT ([],vars) end; clasohm@0: clasohm@0: fun rename_wrt_term t = rename_aTs (add_term_names(t,[])); clasohm@1364: paulson@1417: wenzelm@16943: (* zero var indexes *) wenzelm@4286: wenzelm@16943: fun zero_var_inst vars = wenzelm@16943: fold (fn v as ((x, i), X) => fn (used, inst) => wenzelm@16943: let wenzelm@17642: val x' = variant used (if is_bound x then "u" else x); wenzelm@16943: val used' = x' :: used; wenzelm@16943: in if x = x' andalso i = 0 then (used', inst) else (used', (v, ((x', 0), X)) :: inst) end) wenzelm@16943: vars ([], []) |> #2; wenzelm@4286: wenzelm@16943: fun zero_var_indexesT ty = wenzelm@16943: instantiateT (map (apsnd TVar) (zero_var_inst (sort tvar_ord (add_tvarsT ty [])))) ty; haftmann@16790: wenzelm@16943: fun zero_var_indexes_inst tm = wenzelm@16943: let wenzelm@16943: val instT = map (apsnd TVar) (zero_var_inst (sort tvar_ord (fold_types add_tvarsT tm []))); wenzelm@16943: val inst = wenzelm@16943: add_vars tm [] |> map (apsnd (instantiateT instT)) wenzelm@16943: |> sort var_ord |> zero_var_inst |> map (apsnd Var); wenzelm@16943: in (instT, inst) end; wenzelm@12499: wenzelm@16943: fun zero_var_indexes tm = instantiate (zero_var_indexes_inst tm) tm; wenzelm@4286: paulson@1417: wenzelm@9536: (* dummy patterns *) wenzelm@9536: wenzelm@9536: val dummy_patternN = "dummy_pattern"; wenzelm@9536: wenzelm@18253: fun dummy_pattern T = Const (dummy_patternN, T); wenzelm@18253: wenzelm@9536: fun is_dummy_pattern (Const ("dummy_pattern", _)) = true wenzelm@9536: | is_dummy_pattern _ = false; wenzelm@9536: wenzelm@9536: fun no_dummy_patterns tm = haftmann@16787: if not (fold_aterms (fn t => fn b => b orelse is_dummy_pattern t) tm false) then tm wenzelm@9536: else raise TERM ("Illegal occurrence of '_' dummy pattern", [tm]); wenzelm@9536: wenzelm@11903: fun replace_dummy Ts (i, Const ("dummy_pattern", T)) = wenzelm@11903: (i + 1, list_comb (Var (("_dummy_", i), Ts ---> T), map Bound (0 upto length Ts - 1))) wenzelm@11903: | replace_dummy Ts (i, Abs (x, T, t)) = wenzelm@11903: let val (i', t') = replace_dummy (T :: Ts) (i, t) wenzelm@11903: in (i', Abs (x, T, t')) end wenzelm@11903: | replace_dummy Ts (i, t $ u) = wenzelm@11903: let val (i', t') = replace_dummy Ts (i, t); val (i'', u') = replace_dummy Ts (i', u) wenzelm@11903: in (i'', t' $ u') end wenzelm@11903: | replace_dummy _ (i, a) = (i, a); wenzelm@11903: wenzelm@11903: val replace_dummy_patterns = replace_dummy []; wenzelm@9536: wenzelm@10552: fun is_replaced_dummy_pattern ("_dummy_", _) = true wenzelm@10552: | is_replaced_dummy_pattern _ = false; wenzelm@9536: wenzelm@16035: fun show_dummy_patterns (Var (("_dummy_", _), T)) = Const ("dummy_pattern", T) wenzelm@16035: | show_dummy_patterns (t $ u) = show_dummy_patterns t $ show_dummy_patterns u wenzelm@16035: | show_dummy_patterns (Abs (x, T, t)) = Abs (x, T, show_dummy_patterns t) wenzelm@16035: | show_dummy_patterns a = a; wenzelm@16035: wenzelm@13484: wenzelm@13484: (* adhoc freezing *) wenzelm@13484: wenzelm@13484: fun adhoc_freeze_vars tm = wenzelm@13484: let wenzelm@13484: fun mk_inst (var as Var ((a, i), T)) = wenzelm@13484: let val x = a ^ Library.gensym "_" ^ string_of_int i wenzelm@13484: in ((var, Free(x, T)), x) end; wenzelm@13484: val (insts, xs) = split_list (map mk_inst (term_vars tm)); wenzelm@13484: in (subst_atomic insts tm, xs) end; wenzelm@13484: wenzelm@13484: wenzelm@14786: (* string_of_vname *) wenzelm@14786: wenzelm@15986: val show_question_marks = ref true; berghofe@15472: wenzelm@14786: fun string_of_vname (x, i) = wenzelm@14786: let wenzelm@15986: val question_mark = if ! show_question_marks then "?" else ""; wenzelm@15986: val idx = string_of_int i; wenzelm@15986: val dot = wenzelm@15986: (case rev (Symbol.explode x) of wenzelm@15986: _ :: "\\<^isub>" :: _ => false wenzelm@15986: | _ :: "\\<^isup>" :: _ => false wenzelm@15986: | c :: _ => Symbol.is_digit c wenzelm@15986: | _ => true); wenzelm@14786: in wenzelm@15986: if dot then question_mark ^ x ^ "." ^ idx wenzelm@15986: else if i <> 0 then question_mark ^ x ^ idx wenzelm@15986: else question_mark ^ x wenzelm@14786: end; wenzelm@14786: wenzelm@14786: fun string_of_vname' (x, ~1) = x wenzelm@14786: | string_of_vname' xi = string_of_vname xi; wenzelm@14786: wenzelm@17777: wenzelm@17777: (* str_of_term *) wenzelm@17777: wenzelm@17777: fun str_of_term (Const (c, _)) = c wenzelm@17777: | str_of_term (Free (x, _)) = x wenzelm@17777: | str_of_term (Var (xi, _)) = string_of_vname xi wenzelm@17777: | str_of_term (Bound i) = string_of_int i wenzelm@17777: | str_of_term (Abs (x, _, t)) = "%" ^ x ^ ". " ^ str_of_term t wenzelm@17777: | str_of_term (t $ u) = "(" ^ str_of_term t ^ " " ^ str_of_term u ^ ")"; wenzelm@17777: clasohm@1364: end; clasohm@1364: wenzelm@4444: structure BasicTerm: BASIC_TERM = Term; wenzelm@4444: open BasicTerm;