(* Title: HOL/hologic.ML
Author: Lawrence C Paulson and Markus Wenzel
Abstract syntax operations for HOL.
*)
signature HOLOGIC =
sig
val typeS: sort
val typeT: typ
val boolN: string
val boolT: typ
val Trueprop: term
val mk_Trueprop: term -> term
val dest_Trueprop: term -> term
val true_const: term
val false_const: term
val mk_setT: typ -> typ
val dest_setT: typ -> typ
val Collect_const: typ -> term
val mk_Collect: string * typ * term -> term
val mk_mem: term * term -> term
val dest_mem: term -> term * term
val mk_set: typ -> term list -> term
val dest_set: term -> term list
val mk_UNIV: typ -> term
val conj_intr: thm -> thm -> thm
val conj_elim: thm -> thm * thm
val conj_elims: thm -> thm list
val conj: term
val disj: term
val imp: term
val Not: term
val mk_conj: term * term -> term
val mk_disj: term * term -> term
val mk_imp: term * term -> term
val mk_not: term -> term
val dest_conj: term -> term list
val dest_disj: term -> term list
val disjuncts: term -> term list
val dest_imp: term -> term * term
val dest_not: term -> term
val eq_const: typ -> term
val mk_eq: term * term -> term
val dest_eq: term -> term * term
val all_const: typ -> term
val mk_all: string * typ * term -> term
val list_all: (string * typ) list * term -> term
val exists_const: typ -> term
val mk_exists: string * typ * term -> term
val choice_const: typ -> term
val class_eq: string
val mk_binop: string -> term * term -> term
val mk_binrel: string -> term * term -> term
val dest_bin: string -> typ -> term -> term * term
val unitT: typ
val is_unitT: typ -> bool
val unit: term
val is_unit: term -> bool
val mk_prodT: typ * typ -> typ
val dest_prodT: typ -> typ * typ
val pair_const: typ -> typ -> term
val mk_prod: term * term -> term
val dest_prod: term -> term * term
val mk_fst: term -> term
val mk_snd: term -> term
val split_const: typ * typ * typ -> term
val mk_split: term -> term
val flatten_tupleT: typ -> typ list
val mk_tupleT: typ list -> typ
val strip_tupleT: typ -> typ list
val mk_tuple: term list -> term
val strip_tuple: term -> term list
val mk_ptupleT: int list list -> typ list -> typ
val strip_ptupleT: int list list -> typ -> typ list
val flat_tupleT_paths: typ -> int list list
val mk_ptuple: int list list -> typ -> term list -> term
val strip_ptuple: int list list -> term -> term list
val flat_tuple_paths: term -> int list list
val mk_psplits: int list list -> typ -> typ -> term -> term
val strip_psplits: term -> term * typ list * int list list
val natT: typ
val zero: term
val is_zero: term -> bool
val mk_Suc: term -> term
val dest_Suc: term -> term
val Suc_zero: term
val mk_nat: int -> term
val dest_nat: term -> int
val class_size: string
val size_const: typ -> term
val code_numeralT: typ
val intT: typ
val pls_const: term
val min_const: term
val bit0_const: term
val bit1_const: term
val mk_bit: int -> term
val dest_bit: term -> int
val mk_numeral: int -> term
val dest_numeral: term -> int
val number_of_const: typ -> term
val add_numerals: term -> (term * typ) list -> (term * typ) list
val mk_number: typ -> int -> term
val dest_number: term -> typ * int
val realT: typ
val nibbleT: typ
val mk_nibble: int -> term
val dest_nibble: term -> int
val charT: typ
val mk_char: int -> term
val dest_char: term -> int
val listT: typ -> typ
val nil_const: typ -> term
val cons_const: typ -> term
val mk_list: typ -> term list -> term
val dest_list: term -> term list
val stringT: typ
val mk_string: string -> term
val dest_string: term -> string
val literalT: typ
val mk_literal: string -> term
val dest_literal: term -> string
val mk_typerep: typ -> term
val termT: typ
val term_of_const: typ -> term
val mk_term_of: typ -> term -> term
val reflect_term: term -> term
val mk_valtermify_app: string -> (string * typ) list -> typ -> term
val mk_return: typ -> typ -> term -> term
val mk_ST: ((term * typ) * (string * typ) option) list -> term -> typ -> typ option * typ -> term
val mk_random: typ -> term -> term
end;
structure HOLogic: HOLOGIC =
struct
(* HOL syntax *)
val typeS: sort = ["HOL.type"];
val typeT = TypeInfer.anyT typeS;
(* bool and set *)
val boolN = "bool";
val boolT = Type (boolN, []);
val true_const = Const ("True", boolT);
val false_const = Const ("False", boolT);
fun mk_setT T = T --> boolT;
fun dest_setT (Type ("fun", [T, Type ("bool", [])])) = T
| dest_setT T = raise TYPE ("dest_setT: set type expected", [T], []);
fun mk_set T ts =
let
val sT = mk_setT T;
val empty = Const ("Orderings.bot_class.bot", sT);
fun insert t u = Const ("Set.insert", T --> sT --> sT) $ t $ u;
in fold_rev insert ts empty end;
fun mk_UNIV T = Const ("Orderings.top_class.top", mk_setT T);
fun dest_set (Const ("Orderings.bot_class.bot", _)) = []
| dest_set (Const ("Set.insert", _) $ t $ u) = t :: dest_set u
| dest_set t = raise TERM ("dest_set", [t]);
fun Collect_const T = Const ("Collect", (T --> boolT) --> mk_setT T);
fun mk_Collect (a, T, t) = Collect_const T $ absfree (a, T, t);
fun mk_mem (x, A) =
let val setT = fastype_of A in
Const ("op :", dest_setT setT --> setT --> boolT) $ x $ A
end;
fun dest_mem (Const ("op :", _) $ x $ A) = (x, A)
| dest_mem t = raise TERM ("dest_mem", [t]);
(* logic *)
val Trueprop = Const ("Trueprop", boolT --> propT);
fun mk_Trueprop P = Trueprop $ P;
fun dest_Trueprop (Const ("Trueprop", _) $ P) = P
| dest_Trueprop t = raise TERM ("dest_Trueprop", [t]);
fun conj_intr thP thQ =
let
val (P, Q) = pairself (ObjectLogic.dest_judgment o Thm.cprop_of) (thP, thQ)
handle CTERM (msg, _) => raise THM (msg, 0, [thP, thQ]);
val inst = Thm.instantiate ([], [(@{cpat "?P::bool"}, P), (@{cpat "?Q::bool"}, Q)]);
in Drule.implies_elim_list (inst @{thm conjI}) [thP, thQ] end;
fun conj_elim thPQ =
let
val (P, Q) = Thm.dest_binop (ObjectLogic.dest_judgment (Thm.cprop_of thPQ))
handle CTERM (msg, _) => raise THM (msg, 0, [thPQ]);
val inst = Thm.instantiate ([], [(@{cpat "?P::bool"}, P), (@{cpat "?Q::bool"}, Q)]);
val thP = Thm.implies_elim (inst @{thm conjunct1}) thPQ;
val thQ = Thm.implies_elim (inst @{thm conjunct2}) thPQ;
in (thP, thQ) end;
fun conj_elims th =
let val (th1, th2) = conj_elim th
in conj_elims th1 @ conj_elims th2 end handle THM _ => [th];
val conj = @{term "op &"}
and disj = @{term "op |"}
and imp = @{term "op -->"}
and Not = @{term "Not"};
fun mk_conj (t1, t2) = conj $ t1 $ t2
and mk_disj (t1, t2) = disj $ t1 $ t2
and mk_imp (t1, t2) = imp $ t1 $ t2
and mk_not t = Not $ t;
fun dest_conj (Const ("op &", _) $ t $ t') = t :: dest_conj t'
| dest_conj t = [t];
fun dest_disj (Const ("op |", _) $ t $ t') = t :: dest_disj t'
| dest_disj t = [t];
(*Like dest_disj, but flattens disjunctions however nested*)
fun disjuncts_aux (Const ("op |", _) $ t $ t') disjs = disjuncts_aux t (disjuncts_aux t' disjs)
| disjuncts_aux t disjs = t::disjs;
fun disjuncts t = disjuncts_aux t [];
fun dest_imp (Const("op -->",_) $ A $ B) = (A, B)
| dest_imp t = raise TERM ("dest_imp", [t]);
fun dest_not (Const ("Not", _) $ t) = t
| dest_not t = raise TERM ("dest_not", [t]);
fun eq_const T = Const ("op =", [T, T] ---> boolT);
fun mk_eq (t, u) = eq_const (fastype_of t) $ t $ u;
fun dest_eq (Const ("op =", _) $ lhs $ rhs) = (lhs, rhs)
| dest_eq t = raise TERM ("dest_eq", [t])
fun all_const T = Const ("All", [T --> boolT] ---> boolT);
fun mk_all (x, T, P) = all_const T $ absfree (x, T, P);
fun list_all (xs, t) = fold_rev (fn (x, T) => fn P => all_const T $ Abs (x, T, P)) xs t;
fun exists_const T = Const ("Ex", [T --> boolT] ---> boolT);
fun mk_exists (x, T, P) = exists_const T $ absfree (x, T, P);
fun choice_const T = Const("Hilbert_Choice.Eps", (T --> boolT) --> T);
val class_eq = "HOL.eq";
(* binary operations and relations *)
fun mk_binop c (t, u) =
let val T = fastype_of t in
Const (c, [T, T] ---> T) $ t $ u
end;
fun mk_binrel c (t, u) =
let val T = fastype_of t in
Const (c, [T, T] ---> boolT) $ t $ u
end;
(*destruct the application of a binary operator. The dummyT case is a crude
way of handling polymorphic operators.*)
fun dest_bin c T (tm as Const (c', Type ("fun", [T', _])) $ t $ u) =
if c = c' andalso (T=T' orelse T=dummyT) then (t, u)
else raise TERM ("dest_bin " ^ c, [tm])
| dest_bin c _ tm = raise TERM ("dest_bin " ^ c, [tm]);
(* unit *)
val unitT = Type ("Product_Type.unit", []);
fun is_unitT (Type ("Product_Type.unit", [])) = true
| is_unitT _ = false;
val unit = Const ("Product_Type.Unity", unitT);
fun is_unit (Const ("Product_Type.Unity", _)) = true
| is_unit _ = false;
(* prod *)
fun mk_prodT (T1, T2) = Type ("*", [T1, T2]);
fun dest_prodT (Type ("*", [T1, T2])) = (T1, T2)
| dest_prodT T = raise TYPE ("dest_prodT", [T], []);
fun pair_const T1 T2 = Const ("Pair", [T1, T2] ---> mk_prodT (T1, T2));
fun mk_prod (t1, t2) =
let val T1 = fastype_of t1 and T2 = fastype_of t2 in
pair_const T1 T2 $ t1 $ t2
end;
fun dest_prod (Const ("Pair", _) $ t1 $ t2) = (t1, t2)
| dest_prod t = raise TERM ("dest_prod", [t]);
fun mk_fst p =
let val pT = fastype_of p in
Const ("fst", pT --> fst (dest_prodT pT)) $ p
end;
fun mk_snd p =
let val pT = fastype_of p in
Const ("snd", pT --> snd (dest_prodT pT)) $ p
end;
fun split_const (A, B, C) =
Const ("split", (A --> B --> C) --> mk_prodT (A, B) --> C);
fun mk_split t =
(case Term.fastype_of t of
T as (Type ("fun", [A, Type ("fun", [B, C])])) =>
Const ("split", T --> mk_prodT (A, B) --> C) $ t
| _ => raise TERM ("mk_split: bad body type", [t]));
(*Maps the type T1 * ... * Tn to [T1, ..., Tn], however nested*)
fun flatten_tupleT (Type ("*", [T1, T2])) = flatten_tupleT T1 @ flatten_tupleT T2
| flatten_tupleT T = [T];
(* tuples with right-fold structure *)
fun mk_tupleT [] = unitT
| mk_tupleT Ts = foldr1 mk_prodT Ts;
fun strip_tupleT (Type ("Product_Type.unit", [])) = []
| strip_tupleT (Type ("*", [T1, T2])) = T1 :: strip_tupleT T2
| strip_tupleT T = [T];
fun mk_tuple [] = unit
| mk_tuple ts = foldr1 mk_prod ts;
fun strip_tuple (Const ("Product_Type.Unity", _)) = []
| strip_tuple (Const ("Pair", _) $ t1 $ t2) = t1 :: strip_tuple t2
| strip_tuple t = [t];
(* tuples with specific arities
an "arity" of a tuple is a list of lists of integers,
denoting paths to subterms that are pairs
*)
fun ptuple_err s = raise TERM (s ^ ": inconsistent use of nested products", []);
fun mk_ptupleT ps =
let
fun mk p Ts =
if p mem ps then
let
val (T, Ts') = mk (1::p) Ts;
val (U, Ts'') = mk (2::p) Ts'
in (mk_prodT (T, U), Ts'') end
else (hd Ts, tl Ts)
in fst o mk [] end;
fun strip_ptupleT ps =
let
fun factors p T = if p mem ps then (case T of
Type ("*", [T1, T2]) =>
factors (1::p) T1 @ factors (2::p) T2
| _ => ptuple_err "strip_ptupleT") else [T]
in factors [] end;
val flat_tupleT_paths =
let
fun factors p (Type ("*", [T1, T2])) =
p :: factors (1::p) T1 @ factors (2::p) T2
| factors p _ = []
in factors [] end;
fun mk_ptuple ps =
let
fun mk p T ts =
if p mem ps then (case T of
Type ("*", [T1, T2]) =>
let
val (t, ts') = mk (1::p) T1 ts;
val (u, ts'') = mk (2::p) T2 ts'
in (pair_const T1 T2 $ t $ u, ts'') end
| _ => ptuple_err "mk_ptuple")
else (hd ts, tl ts)
in fst oo mk [] end;
fun strip_ptuple ps =
let
fun dest p t = if p mem ps then (case t of
Const ("Pair", _) $ t $ u =>
dest (1::p) t @ dest (2::p) u
| _ => ptuple_err "strip_ptuple") else [t]
in dest [] end;
val flat_tuple_paths =
let
fun factors p (Const ("Pair", _) $ t $ u) =
p :: factors (1::p) t @ factors (2::p) u
| factors p _ = []
in factors [] end;
(*In mk_psplits ps S T u, term u expects separate arguments for the factors of S,
with result type T. The call creates a new term expecting one argument
of type S.*)
fun mk_psplits ps T T3 u =
let
fun ap ((p, T) :: pTs) =
if p mem ps then (case T of
Type ("*", [T1, T2]) =>
split_const (T1, T2, map snd pTs ---> T3) $
ap ((1::p, T1) :: (2::p, T2) :: pTs)
| _ => ptuple_err "mk_psplits")
else Abs ("x", T, ap pTs)
| ap [] =
let val k = length ps
in list_comb (incr_boundvars (k + 1) u, map Bound (k downto 0)) end
in ap [([], T)] end;
val strip_psplits =
let
fun strip [] qs Ts t = (t, rev Ts, qs)
| strip (p :: ps) qs Ts (Const ("split", _) $ t) =
strip ((1 :: p) :: (2 :: p) :: ps) (p :: qs) Ts t
| strip (p :: ps) qs Ts (Abs (s, T, t)) = strip ps qs (T :: Ts) t
| strip (p :: ps) qs Ts t = strip ps qs
(hd (binder_types (fastype_of1 (Ts, t))) :: Ts)
(incr_boundvars 1 t $ Bound 0)
in strip [[]] [] [] end;
(* nat *)
val natT = Type ("nat", []);
val zero = Const ("Algebras.zero_class.zero", natT);
fun is_zero (Const ("Algebras.zero_class.zero", _)) = true
| is_zero _ = false;
fun mk_Suc t = Const ("Suc", natT --> natT) $ t;
fun dest_Suc (Const ("Suc", _) $ t) = t
| dest_Suc t = raise TERM ("dest_Suc", [t]);
val Suc_zero = mk_Suc zero;
fun mk_nat n =
let
fun mk 0 = zero
| mk n = mk_Suc (mk (n - 1));
in if n < 0 then raise TERM ("mk_nat: negative number", []) else mk n end;
fun dest_nat (Const ("Algebras.zero_class.zero", _)) = 0
| dest_nat (Const ("Suc", _) $ t) = dest_nat t + 1
| dest_nat t = raise TERM ("dest_nat", [t]);
val class_size = "Nat.size";
fun size_const T = Const ("Nat.size_class.size", T --> natT);
(* code numeral *)
val code_numeralT = Type ("Code_Numeral.code_numeral", []);
(* binary numerals and int -- non-unique representation due to leading zeros/ones! *)
val intT = Type ("Int.int", []);
val pls_const = Const ("Int.Pls", intT)
and min_const = Const ("Int.Min", intT)
and bit0_const = Const ("Int.Bit0", intT --> intT)
and bit1_const = Const ("Int.Bit1", intT --> intT);
fun mk_bit 0 = bit0_const
| mk_bit 1 = bit1_const
| mk_bit _ = raise TERM ("mk_bit", []);
fun dest_bit (Const ("Int.Bit0", _)) = 0
| dest_bit (Const ("Int.Bit1", _)) = 1
| dest_bit t = raise TERM ("dest_bit", [t]);
fun mk_numeral 0 = pls_const
| mk_numeral ~1 = min_const
| mk_numeral i =
let val (q, r) = Integer.div_mod i 2;
in mk_bit r $ mk_numeral q end;
fun dest_numeral (Const ("Int.Pls", _)) = 0
| dest_numeral (Const ("Int.Min", _)) = ~1
| dest_numeral (Const ("Int.Bit0", _) $ bs) = 2 * dest_numeral bs
| dest_numeral (Const ("Int.Bit1", _) $ bs) = 2 * dest_numeral bs + 1
| dest_numeral t = raise TERM ("dest_numeral", [t]);
fun number_of_const T = Const ("Int.number_class.number_of", intT --> T);
fun add_numerals (Const ("Int.number_class.number_of", Type (_, [_, T])) $ t) = cons (t, T)
| add_numerals (t $ u) = add_numerals t #> add_numerals u
| add_numerals (Abs (_, _, t)) = add_numerals t
| add_numerals _ = I;
fun mk_number T 0 = Const ("Algebras.zero_class.zero", T)
| mk_number T 1 = Const ("Algebras.one_class.one", T)
| mk_number T i = number_of_const T $ mk_numeral i;
fun dest_number (Const ("Algebras.zero_class.zero", T)) = (T, 0)
| dest_number (Const ("Algebras.one_class.one", T)) = (T, 1)
| dest_number (Const ("Int.number_class.number_of", Type ("fun", [_, T])) $ t) =
(T, dest_numeral t)
| dest_number t = raise TERM ("dest_number", [t]);
(* real *)
val realT = Type ("RealDef.real", []);
(* list *)
fun listT T = Type ("List.list", [T]);
fun nil_const T = Const ("List.list.Nil", listT T);
fun cons_const T =
let val lT = listT T
in Const ("List.list.Cons", T --> lT --> lT) end;
fun mk_list T ts =
let
val lT = listT T;
val Nil = Const ("List.list.Nil", lT);
fun Cons t u = Const ("List.list.Cons", T --> lT --> lT) $ t $ u;
in fold_rev Cons ts Nil end;
fun dest_list (Const ("List.list.Nil", _)) = []
| dest_list (Const ("List.list.Cons", _) $ t $ u) = t :: dest_list u
| dest_list t = raise TERM ("dest_list", [t]);
(* nibble *)
val nibbleT = Type ("String.nibble", []);
fun mk_nibble n =
let val s =
if 0 <= n andalso n <= 9 then chr (n + ord "0")
else if 10 <= n andalso n <= 15 then chr (n + ord "A" - 10)
else raise TERM ("mk_nibble", [])
in Const ("String.nibble.Nibble" ^ s, nibbleT) end;
fun dest_nibble t =
let fun err () = raise TERM ("dest_nibble", [t]) in
(case try (unprefix "String.nibble.Nibble" o fst o Term.dest_Const) t of
NONE => err ()
| SOME c =>
if size c <> 1 then err ()
else if "0" <= c andalso c <= "9" then ord c - ord "0"
else if "A" <= c andalso c <= "F" then ord c - ord "A" + 10
else err ())
end;
(* char *)
val charT = Type ("String.char", []);
fun mk_char n =
if 0 <= n andalso n <= 255 then
Const ("String.char.Char", nibbleT --> nibbleT --> charT) $
mk_nibble (n div 16) $ mk_nibble (n mod 16)
else raise TERM ("mk_char", []);
fun dest_char (Const ("String.char.Char", _) $ t $ u) =
dest_nibble t * 16 + dest_nibble u
| dest_char t = raise TERM ("dest_char", [t]);
(* string *)
val stringT = listT charT;
val mk_string = mk_list charT o map (mk_char o ord) o explode;
val dest_string = implode o map (chr o dest_char) o dest_list;
(* literal *)
val literalT = Type ("String.literal", []);
fun mk_literal s = Const ("String.literal.STR", stringT --> literalT)
$ mk_string s;
fun dest_literal (Const ("String.literal.STR", _) $ t) =
dest_string t
| dest_literal t = raise TERM ("dest_literal", [t]);
(* typerep and term *)
val typerepT = Type ("Typerep.typerep", []);
fun mk_typerep (Type (tyco, Ts)) = Const ("Typerep.typerep.Typerep",
literalT --> listT typerepT --> typerepT) $ mk_literal tyco
$ mk_list typerepT (map mk_typerep Ts)
| mk_typerep (T as TFree _) = Const ("Typerep.typerep_class.typerep",
Term.itselfT T --> typerepT) $ Logic.mk_type T;
val termT = Type ("Code_Evaluation.term", []);
fun term_of_const T = Const ("Code_Evaluation.term_of_class.term_of", T --> termT);
fun mk_term_of T t = term_of_const T $ t;
fun reflect_term (Const (c, T)) =
Const ("Code_Evaluation.Const", literalT --> typerepT --> termT)
$ mk_literal c $ mk_typerep T
| reflect_term (t1 $ t2) =
Const ("Code_Evaluation.App", termT --> termT --> termT)
$ reflect_term t1 $ reflect_term t2
| reflect_term (Abs (v, _, t)) = Abs (v, termT, reflect_term t)
| reflect_term t = t;
fun mk_valtermify_app c vs T =
let
fun termifyT T = mk_prodT (T, unitT --> termT);
fun valapp T T' = Const ("Code_Evaluation.valapp",
termifyT (T --> T') --> termifyT T --> termifyT T');
fun mk_fTs [] _ = []
| mk_fTs (_ :: Ts) T = (Ts ---> T) :: mk_fTs Ts T;
val Ts = map snd vs;
val t = Const (c, Ts ---> T);
val tt = mk_prod (t, Abs ("u", unitT, reflect_term t));
fun app (fT, (v, T)) t = valapp T fT $ t $ Free (v, termifyT T);
in fold app (mk_fTs Ts T ~~ vs) tt end;
(* open state monads *)
fun mk_return T U x = pair_const T U $ x;
fun mk_ST clauses t U (someT, V) =
let
val R = case someT of SOME T => mk_prodT (T, V) | NONE => V
fun mk_clause ((t, U), SOME (v, T)) (t', U') =
(Const ("Product_Type.scomp", (U --> mk_prodT (T, U')) --> (T --> U' --> R) --> U --> R)
$ t $ lambda (Free (v, T)) t', U)
| mk_clause ((t, U), NONE) (t', U') =
(Const ("Product_Type.fcomp", (U --> U') --> (U' --> R) --> U --> R)
$ t $ t', U)
in fold_rev mk_clause clauses (t, U) |> fst end;
val code_numeralT = Type ("Code_Numeral.code_numeral", []);
val random_seedT = mk_prodT (code_numeralT, code_numeralT);
fun mk_random T t = Const ("Quickcheck.random_class.random", code_numeralT
--> random_seedT --> mk_prodT (mk_prodT (T, unitT --> termT), random_seedT)) $ t;
end;