(* Title: Pure/term_ord.ML
Author: Tobias Nipkow and Makarius, TU Muenchen
Term orderings.
*)
signature BASIC_TERM_ORD =
sig
structure Vartab: TABLE
structure Sorttab: TABLE
structure Typtab: TABLE
structure Termtab: TABLE
end;
signature TERM_ORD =
sig
include BASIC_TERM_ORD
val fast_indexname_ord: indexname * indexname -> order
val sort_ord: sort * sort -> order
val typ_ord: typ * typ -> order
val fast_term_ord: term * term -> order
val syntax_term_ord: term * term -> order
val indexname_ord: indexname * indexname -> order
val tvar_ord: (indexname * sort) * (indexname * sort) -> order
val var_ord: (indexname * typ) * (indexname * typ) -> order
val term_ord: term * term -> order
val hd_ord: term * term -> order
val termless: term * term -> bool
val term_lpo: (term -> int) -> term * term -> order
val term_cache: (term -> 'a) -> term -> 'a
end;
structure Term_Ord: TERM_ORD =
struct
(* fast syntactic ordering -- tuned for inequalities *)
fun fast_indexname_ord ((x, i), (y, j)) =
(case int_ord (i, j) of EQUAL => fast_string_ord (x, y) | ord => ord);
fun sort_ord SS =
if pointer_eq SS then EQUAL
else dict_ord fast_string_ord SS;
local
fun cons_nr (TVar _) = 0
| cons_nr (TFree _) = 1
| cons_nr (Type _) = 2;
in
fun typ_ord TU =
if pointer_eq TU then EQUAL
else
(case TU of
(Type (a, Ts), Type (b, Us)) =>
(case fast_string_ord (a, b) of EQUAL => dict_ord typ_ord (Ts, Us) | ord => ord)
| (TFree (a, S), TFree (b, S')) =>
(case fast_string_ord (a, b) of EQUAL => sort_ord (S, S') | ord => ord)
| (TVar (xi, S), TVar (yj, S')) =>
(case fast_indexname_ord (xi, yj) of EQUAL => sort_ord (S, S') | ord => ord)
| (T, U) => int_ord (cons_nr T, cons_nr U));
end;
local
fun cons_nr (Const _) = 0
| cons_nr (Free _) = 1
| cons_nr (Var _) = 2
| cons_nr (Bound _) = 3
| cons_nr (Abs _) = 4
| cons_nr (_ $ _) = 5;
fun struct_ord (Abs (_, _, t), Abs (_, _, u)) = struct_ord (t, u)
| struct_ord (t1 $ t2, u1 $ u2) =
(case struct_ord (t1, u1) of EQUAL => struct_ord (t2, u2) | ord => ord)
| struct_ord (t, u) = int_ord (cons_nr t, cons_nr u);
fun atoms_ord (Abs (_, _, t), Abs (_, _, u)) = atoms_ord (t, u)
| atoms_ord (t1 $ t2, u1 $ u2) =
(case atoms_ord (t1, u1) of EQUAL => atoms_ord (t2, u2) | ord => ord)
| atoms_ord (Const (a, _), Const (b, _)) = fast_string_ord (a, b)
| atoms_ord (Free (x, _), Free (y, _)) = fast_string_ord (x, y)
| atoms_ord (Var (xi, _), Var (yj, _)) = fast_indexname_ord (xi, yj)
| atoms_ord (Bound i, Bound j) = int_ord (i, j)
| atoms_ord _ = EQUAL;
fun types_ord (Abs (_, T, t), Abs (_, U, u)) =
(case typ_ord (T, U) of EQUAL => types_ord (t, u) | ord => ord)
| types_ord (t1 $ t2, u1 $ u2) =
(case types_ord (t1, u1) of EQUAL => types_ord (t2, u2) | ord => ord)
| types_ord (Const (_, T), Const (_, U)) = typ_ord (T, U)
| types_ord (Free (_, T), Free (_, U)) = typ_ord (T, U)
| types_ord (Var (_, T), Var (_, U)) = typ_ord (T, U)
| types_ord _ = EQUAL;
fun comments_ord (Abs (x, _, t), Abs (y, _, u)) =
(case fast_string_ord (x, y) of EQUAL => comments_ord (t, u) | ord => ord)
| comments_ord (t1 $ t2, u1 $ u2) =
(case comments_ord (t1, u1) of EQUAL => comments_ord (t2, u2) | ord => ord)
| comments_ord _ = EQUAL;
in
fun fast_term_ord tu =
if pointer_eq tu then EQUAL
else
(case struct_ord tu of
EQUAL => (case atoms_ord tu of EQUAL => types_ord tu | ord => ord)
| ord => ord);
fun syntax_term_ord tu =
(case fast_term_ord tu of EQUAL => comments_ord tu | ord => ord);
end;
(* term_ord *)
(*a linear well-founded AC-compatible ordering for terms:
s < t <=> 1. size(s) < size(t) or
2. size(s) = size(t) and s=f(...) and t=g(...) and f<g or
3. size(s) = size(t) and s=f(s1..sn) and t=f(t1..tn) and
(s1..sn) < (t1..tn) (lexicographically)*)
fun indexname_ord ((x, i), (y, j)) =
(case int_ord (i, j) of EQUAL => string_ord (x, y) | ord => ord);
val tvar_ord = prod_ord indexname_ord sort_ord;
val var_ord = prod_ord indexname_ord typ_ord;
local
fun hd_depth (t $ _, n) = hd_depth (t, n + 1)
| hd_depth p = p;
fun dest_hd (Const (a, T)) = (((a, 0), T), 0)
| dest_hd (Free (a, T)) = (((a, 0), T), 1)
| dest_hd (Var v) = (v, 2)
| dest_hd (Bound i) = ((("", i), dummyT), 3)
| dest_hd (Abs (_, T, _)) = ((("", 0), T), 4);
in
fun term_ord tu =
if pointer_eq tu then EQUAL
else
(case tu of
(Abs (_, T, t), Abs(_, U, u)) =>
(case term_ord (t, u) of EQUAL => typ_ord (T, U) | ord => ord)
| (t, u) =>
(case int_ord (size_of_term t, size_of_term u) of
EQUAL =>
(case prod_ord hd_ord int_ord (hd_depth (t, 0), hd_depth (u, 0)) of
EQUAL => args_ord (t, u) | ord => ord)
| ord => ord))
and hd_ord (f, g) =
prod_ord (prod_ord indexname_ord typ_ord) int_ord (dest_hd f, dest_hd g)
and args_ord (f $ t, g $ u) =
(case args_ord (f, g) of EQUAL => term_ord (t, u) | ord => ord)
| args_ord _ = EQUAL;
fun termless tu = (term_ord tu = LESS);
end;
(* Lexicographic path order on terms *)
(*
See Baader & Nipkow, Term rewriting, CUP 1998.
Without variables. Const, Var, Bound, Free and Abs are treated all as
constants.
f_ord maps terms to integers and serves two purposes:
- Predicate on constant symbols. Those that are not recognised by f_ord
must be mapped to ~1.
- Order on the recognised symbols. These must be mapped to distinct
integers >= 0.
The argument of f_ord is never an application.
*)
local
fun unrecognized (Const (a, T)) = ((1, ((a, 0), T)), 0)
| unrecognized (Free (a, T)) = ((1, ((a, 0), T)), 0)
| unrecognized (Var v) = ((1, v), 1)
| unrecognized (Bound i) = ((1, (("", i), dummyT)), 2)
| unrecognized (Abs (_, T, _)) = ((1, (("", 0), T)), 3);
fun dest_hd f_ord t =
let val ord = f_ord t
in if ord = ~1 then unrecognized t else ((0, (("", ord), fastype_of t)), 0) end;
fun term_lpo f_ord (s, t) =
let val (f, ss) = strip_comb s and (g, ts) = strip_comb t in
if forall (fn si => term_lpo f_ord (si, t) = LESS) ss
then case hd_ord f_ord (f, g) of
GREATER =>
if forall (fn ti => term_lpo f_ord (s, ti) = GREATER) ts
then GREATER else LESS
| EQUAL =>
if forall (fn ti => term_lpo f_ord (s, ti) = GREATER) ts
then list_ord (term_lpo f_ord) (ss, ts)
else LESS
| LESS => LESS
else GREATER
end
and hd_ord f_ord (f, g) = case (f, g) of
(Abs (_, T, t), Abs (_, U, u)) =>
(case term_lpo f_ord (t, u) of EQUAL => typ_ord (T, U) | ord => ord)
| (_, _) => prod_ord (prod_ord int_ord
(prod_ord indexname_ord typ_ord)) int_ord
(dest_hd f_ord f, dest_hd f_ord g);
in
val term_lpo = term_lpo
end;
(* tables and caches *)
structure Vartab = Table(type key = indexname val ord = fast_indexname_ord);
structure Sorttab = Table(type key = sort val ord = sort_ord);
structure Typtab = Table(type key = typ val ord = typ_ord);
structure Termtab = Table(type key = term val ord = fast_term_ord);
fun term_cache f = Cache.create Termtab.empty Termtab.lookup Termtab.update f;
end;
structure Basic_Term_Ord: BASIC_TERM_ORD = Term_Ord;
open Basic_Term_Ord;
structure Var_Graph = Graph(type key = indexname val ord = Term_Ord.fast_indexname_ord);
structure Sort_Graph = Graph(type key = sort val ord = Term_Ord.sort_ord);
structure Typ_Graph = Graph(type key = typ val ord = Term_Ord.typ_ord);
structure Term_Graph = Graph(type key = term val ord = Term_Ord.fast_term_ord);