(* Title: HOLCF/Tools/domain/domain_constructors.ML
Author: Brian Huffman
Defines constructor functions for a given domain isomorphism
and proves related theorems.
*)
signature DOMAIN_CONSTRUCTORS =
sig
val add_domain_constructors :
string
-> typ * (binding * (bool * binding option * typ) list * mixfix) list
-> term * term
-> thm * thm
-> theory
-> { con_consts : term list,
con_defs : thm list,
con_compacts : thm list,
sel_rews : thm list }
* theory;
end;
structure Domain_Constructors :> DOMAIN_CONSTRUCTORS =
struct
(******************************************************************************)
(************************** building types and terms **************************)
(******************************************************************************)
(*** Operations from Isabelle/HOL ***)
val boolT = HOLogic.boolT;
val mk_equals = Logic.mk_equals;
val mk_eq = HOLogic.mk_eq;
val mk_trp = HOLogic.mk_Trueprop;
val mk_fst = HOLogic.mk_fst;
val mk_snd = HOLogic.mk_snd;
val mk_not = HOLogic.mk_not;
(*** Continuous function space ***)
(* ->> is taken from holcf_logic.ML *)
fun mk_cfunT (T, U) = Type(@{type_name "->"}, [T, U]);
infixr 6 ->>; val (op ->>) = mk_cfunT;
infix -->>; val (op -->>) = Library.foldr mk_cfunT;
fun dest_cfunT (Type(@{type_name "->"}, [T, U])) = (T, U)
| dest_cfunT T = raise TYPE ("dest_cfunT", [T], []);
fun capply_const (S, T) =
Const(@{const_name Rep_CFun}, (S ->> T) --> (S --> T));
fun cabs_const (S, T) =
Const(@{const_name Abs_CFun}, (S --> T) --> (S ->> T));
fun mk_cabs t =
let val T = Term.fastype_of t
in cabs_const (Term.domain_type T, Term.range_type T) $ t end
(* builds the expression (LAM v. rhs) *)
fun big_lambda v rhs =
cabs_const (Term.fastype_of v, Term.fastype_of rhs) $ Term.lambda v rhs;
(* builds the expression (LAM v1 v2 .. vn. rhs) *)
fun big_lambdas [] rhs = rhs
| big_lambdas (v::vs) rhs = big_lambda v (big_lambdas vs rhs);
fun mk_capply (t, u) =
let val (S, T) =
case Term.fastype_of t of
Type(@{type_name "->"}, [S, T]) => (S, T)
| _ => raise TERM ("mk_capply " ^ ML_Syntax.print_list ML_Syntax.print_term [t, u], [t, u]);
in capply_const (S, T) $ t $ u end;
infix 9 ` ; val (op `) = mk_capply;
val list_ccomb : term * term list -> term = Library.foldl mk_capply;
fun mk_ID T = Const (@{const_name ID}, T ->> T);
fun cfcomp_const (T, U, V) =
Const (@{const_name cfcomp}, (U ->> V) ->> (T ->> U) ->> (T ->> V));
fun mk_cfcomp (f, g) =
let
val (U, V) = dest_cfunT (Term.fastype_of f);
val (T, U') = dest_cfunT (Term.fastype_of g);
in
if U = U'
then mk_capply (mk_capply (cfcomp_const (T, U, V), f), g)
else raise TYPE ("mk_cfcomp", [U, U'], [f, g])
end;
fun mk_bottom T = Const (@{const_name UU}, T);
(*** Product type ***)
fun mk_tupleT [] = HOLogic.unitT
| mk_tupleT [T] = T
| mk_tupleT (T :: Ts) = HOLogic.mk_prodT (T, mk_tupleT Ts);
(* builds the expression (v1,v2,..,vn) *)
fun mk_tuple [] = HOLogic.unit
| mk_tuple (t::[]) = t
| mk_tuple (t::ts) = HOLogic.mk_prod (t, mk_tuple ts);
(* builds the expression (%(v1,v2,..,vn). rhs) *)
fun lambda_tuple [] rhs = Term.lambda (Free("unit", HOLogic.unitT)) rhs
| lambda_tuple (v::[]) rhs = Term.lambda v rhs
| lambda_tuple (v::vs) rhs =
HOLogic.mk_split (Term.lambda v (lambda_tuple vs rhs));
(*** Lifted cpo type ***)
fun mk_upT T = Type(@{type_name "u"}, [T]);
fun dest_upT (Type(@{type_name "u"}, [T])) = T
| dest_upT T = raise TYPE ("dest_upT", [T], []);
fun up_const T = Const(@{const_name up}, T ->> mk_upT T);
fun mk_up t = up_const (Term.fastype_of t) ` t;
fun fup_const (T, U) =
Const(@{const_name fup}, (T ->> U) ->> mk_upT T ->> U);
fun from_up T = fup_const (T, T) ` mk_ID T;
(*** Strict product type ***)
val oneT = @{typ "one"};
fun mk_sprodT (T, U) = Type(@{type_name "**"}, [T, U]);
fun dest_sprodT (Type(@{type_name "**"}, [T, U])) = (T, U)
| dest_sprodT T = raise TYPE ("dest_sprodT", [T], []);
fun spair_const (T, U) =
Const(@{const_name spair}, T ->> U ->> mk_sprodT (T, U));
(* builds the expression (:t, u:) *)
fun mk_spair (t, u) =
spair_const (Term.fastype_of t, Term.fastype_of u) ` t ` u;
(* builds the expression (:t1,t2,..,tn:) *)
fun mk_stuple [] = @{term "ONE"}
| mk_stuple (t::[]) = t
| mk_stuple (t::ts) = mk_spair (t, mk_stuple ts);
fun sfst_const (T, U) =
Const(@{const_name sfst}, mk_sprodT (T, U) ->> T);
fun ssnd_const (T, U) =
Const(@{const_name ssnd}, mk_sprodT (T, U) ->> U);
(*** Strict sum type ***)
fun mk_ssumT (T, U) = Type(@{type_name "++"}, [T, U]);
fun dest_ssumT (Type(@{type_name "++"}, [T, U])) = (T, U)
| dest_ssumT T = raise TYPE ("dest_ssumT", [T], []);
fun sinl_const (T, U) = Const(@{const_name sinl}, T ->> mk_ssumT (T, U));
fun sinr_const (T, U) = Const(@{const_name sinr}, U ->> mk_ssumT (T, U));
(* builds the list [sinl(t1), sinl(sinr(t2)), ... sinr(...sinr(tn))] *)
fun mk_sinjects ts =
let
val Ts = map Term.fastype_of ts;
fun combine (t, T) (us, U) =
let
val v = sinl_const (T, U) ` t;
val vs = map (fn u => sinr_const (T, U) ` u) us;
in
(v::vs, mk_ssumT (T, U))
end
fun inj [] = error "mk_sinjects: empty list"
| inj ((t, T)::[]) = ([t], T)
| inj ((t, T)::ts) = combine (t, T) (inj ts);
in
fst (inj (ts ~~ Ts))
end;
fun sscase_const (T, U, V) =
Const(@{const_name sscase},
(T ->> V) ->> (U ->> V) ->> mk_ssumT (T, U) ->> V);
fun from_sinl (T, U) =
sscase_const (T, U, T) ` mk_ID T ` mk_bottom (U ->> T);
fun from_sinr (T, U) =
sscase_const (T, U, U) ` mk_bottom (T ->> U) ` mk_ID U;
(*** miscellaneous constructions ***)
val trT = @{typ "tr"};
val deflT = @{typ "udom alg_defl"};
fun mapT T =
let
fun argTs (Type (_, Ts)) = Ts | argTs _ = [];
fun auto T = T ->> T;
in
map auto (argTs T) -->> auto T
end;
fun mk_strict t =
let val (T, U) = dest_cfunT (Term.fastype_of t);
in mk_eq (t ` mk_bottom T, mk_bottom U) end;
fun mk_defined t = mk_not (mk_eq (t, mk_bottom (Term.fastype_of t)));
fun mk_compact t =
Const (@{const_name compact}, Term.fastype_of t --> boolT) $ t;
fun mk_cont t =
Const (@{const_name cont}, Term.fastype_of t --> boolT) $ t;
fun mk_fix t =
let val (T, _) = dest_cfunT (Term.fastype_of t)
in mk_capply (Const(@{const_name fix}, (T ->> T) ->> T), t) end;
fun mk_Rep_of T =
Const (@{const_name Rep_of}, Term.itselfT T --> deflT) $ Logic.mk_type T;
fun coerce_const T = Const (@{const_name coerce}, T);
fun isodefl_const T =
Const (@{const_name isodefl}, (T ->> T) --> deflT --> boolT);
(* splits a cterm into the right and lefthand sides of equality *)
fun dest_eqs t = HOLogic.dest_eq (HOLogic.dest_Trueprop t);
fun mk_eqs (t, u) = HOLogic.mk_Trueprop (HOLogic.mk_eq (t, u));
(************************** miscellaneous functions ***************************)
fun define_consts
(specs : (binding * term * mixfix) list)
(thy : theory)
: (term list * thm list) * theory =
let
fun mk_decl (b, t, mx) = (b, Term.fastype_of t, mx);
val decls = map mk_decl specs;
val thy = Cont_Consts.add_consts decls thy;
fun mk_const (b, T, mx) = Const (Sign.full_name thy b, T);
val consts = map mk_const decls;
fun mk_def c (b, t, mx) =
(Binding.suffix_name "_def" b, Logic.mk_equals (c, t));
val defs = map2 mk_def consts specs;
val (def_thms, thy) =
PureThy.add_defs false (map Thm.no_attributes defs) thy;
in
((consts, def_thms), thy)
end;
fun prove
(thy : theory)
(defs : thm list)
(goal : term)
(tacs : {prems: thm list, context: Proof.context} -> tactic list)
: thm =
let
fun tac {prems, context} =
rewrite_goals_tac defs THEN
EVERY (tacs {prems = map (rewrite_rule defs) prems, context = context})
in
Goal.prove_global thy [] [] goal tac
end;
(************** generating beta reduction rules from definitions **************)
local
fun arglist (Const _ $ Abs (s, T, t)) =
let
val arg = Free (s, T);
val (args, body) = arglist (subst_bound (arg, t));
in (arg :: args, body) end
| arglist t = ([], t);
in
fun beta_of_def thy def_thm =
let
val (con, lam) = Logic.dest_equals (concl_of def_thm);
val (args, rhs) = arglist lam;
val lhs = list_ccomb (con, args);
val goal = mk_equals (lhs, rhs);
val cs = ContProc.cont_thms lam;
val betas = map (fn c => mk_meta_eq (c RS @{thm beta_cfun})) cs;
in
prove thy (def_thm::betas) goal (K [rtac reflexive_thm 1])
end;
end;
(******************************************************************************)
(************** definitions and theorems for selector functions ***************)
(******************************************************************************)
fun add_selectors
(spec : (term * (bool * binding option * typ) list) list)
(rep_const : term)
(abs_inv : thm)
(rep_strict : thm)
(rep_strict_iff : thm)
(con_betas : thm list)
(thy : theory)
: thm list * theory =
let
(* define selector functions *)
val ((sel_consts, sel_defs), thy) =
let
fun rangeT s = snd (dest_cfunT (Term.fastype_of s));
fun mk_outl s = mk_cfcomp (from_sinl (dest_ssumT (rangeT s)), s);
fun mk_outr s = mk_cfcomp (from_sinr (dest_ssumT (rangeT s)), s);
fun mk_sfst s = mk_cfcomp (sfst_const (dest_sprodT (rangeT s)), s);
fun mk_ssnd s = mk_cfcomp (ssnd_const (dest_sprodT (rangeT s)), s);
fun mk_down s = mk_cfcomp (from_up (dest_upT (rangeT s)), s);
fun sels_of_arg s (lazy, NONE, T) = []
| sels_of_arg s (lazy, SOME b, T) =
[(b, if lazy then mk_down s else s, NoSyn)];
fun sels_of_args s [] = []
| sels_of_args s (v :: []) = sels_of_arg s v
| sels_of_args s (v :: vs) =
sels_of_arg (mk_sfst s) v @ sels_of_args (mk_ssnd s) vs;
fun sels_of_cons s [] = []
| sels_of_cons s ((con, args) :: []) = sels_of_args s args
| sels_of_cons s ((con, args) :: cs) =
sels_of_args (mk_outl s) args @ sels_of_cons (mk_outr s) cs;
val sel_eqns : (binding * term * mixfix) list =
sels_of_cons rep_const spec;
in
define_consts sel_eqns thy
end
(* replace bindings with terms in constructor spec *)
val spec2 : (term * (bool * term option * typ) list) list =
let
fun prep_arg (lazy, NONE, T) sels = ((lazy, NONE, T), sels)
| prep_arg (lazy, SOME _, T) sels =
((lazy, SOME (hd sels), T), tl sels);
fun prep_con (con, args) sels =
apfst (pair con) (fold_map prep_arg args sels);
in
fst (fold_map prep_con spec sel_consts)
end;
(* prove selector strictness rules *)
val sel_stricts : thm list =
let
val rules = rep_strict :: @{thms sel_strict_rules};
val tacs = [simp_tac (HOL_basic_ss addsimps rules) 1];
fun sel_strict sel =
let
val goal = mk_trp (mk_strict sel);
in
prove thy sel_defs goal (K tacs)
end
in
map sel_strict sel_consts
end
(* prove selector application rules *)
val sel_apps : thm list =
let
val defs = con_betas @ sel_defs;
val rules = @{thms sel_app_rules};
val simps = simp_thms @ [abs_inv] @ rules;
val tacs = [asm_simp_tac (HOL_basic_ss addsimps simps) 1];
fun sel_apps_of (i, (con, args)) =
let
val Ts : typ list = map #3 args;
val ns : string list = Datatype_Prop.make_tnames Ts;
val vs : term list = map Free (ns ~~ Ts);
val con_app : term = list_ccomb (con, vs);
val vs' : (bool * term) list = map #1 args ~~ vs;
fun one_same (n, sel, T) =
let
val xs = map snd (filter_out fst (nth_drop n vs'));
val assms = map (mk_trp o mk_defined) xs;
val concl = mk_trp (mk_eq (sel ` con_app, nth vs n));
val goal = Logic.list_implies (assms, concl);
in
prove thy defs goal (K tacs)
end;
fun one_diff (n, sel, T) =
let
val goal = mk_trp (mk_eq (sel ` con_app, mk_bottom T));
in
prove thy defs goal (K tacs)
end;
fun one_con (j, (_, args')) : thm list =
let
fun prep (i, (lazy, NONE, T)) = NONE
| prep (i, (lazy, SOME sel, T)) = SOME (i, sel, T);
val sels : (int * term * typ) list =
map_filter prep (map_index I args');
in
if i = j
then map one_same sels
else map one_diff sels
end
in
flat (map_index one_con spec2)
end
in
flat (map_index sel_apps_of spec2)
end
(* prove selector definedness rules *)
val sel_defins : thm list =
let
val rules = rep_strict_iff :: @{thms sel_defined_iff_rules};
val tacs = [simp_tac (HOL_basic_ss addsimps rules) 1];
fun sel_defin sel =
let
val (T, U) = dest_cfunT (Term.fastype_of sel);
val x = Free ("x", T);
val lhs = mk_eq (sel ` x, mk_bottom U);
val rhs = mk_eq (x, mk_bottom T);
val goal = mk_trp (mk_eq (lhs, rhs));
in
prove thy sel_defs goal (K tacs)
end
fun one_arg (false, SOME sel, T) = SOME (sel_defin sel)
| one_arg _ = NONE;
in
case spec2 of
[(con, args)] => map_filter one_arg args
| _ => []
end;
in
(sel_stricts @ sel_defins @ sel_apps, thy)
end
(******************************************************************************)
(************* definitions and theorems for constructor functions *************)
(******************************************************************************)
fun add_domain_constructors
(dname : string)
(lhsT : typ,
spec : (binding * (bool * binding option * typ) list * mixfix) list)
(rep_const : term, abs_const : term)
(rep_iso_thm : thm, abs_iso_thm : thm)
(thy : theory) =
let
(* prove rep/abs strictness rules *)
val iso_locale = @{thm iso.intro} OF [abs_iso_thm, rep_iso_thm];
val rep_strict = iso_locale RS @{thm iso.rep_strict};
val abs_strict = iso_locale RS @{thm iso.abs_strict};
val rep_defined_iff = iso_locale RS @{thm iso.rep_defined_iff};
val abs_defined_iff = iso_locale RS @{thm iso.abs_defined_iff};
(* define constructor functions *)
val ((con_consts, con_def_thms), thy) =
let
fun vars_of args =
let
val Ts = map (fn (lazy,sel,T) => T) args;
val ns = Datatype_Prop.make_tnames Ts;
in
map Free (ns ~~ Ts)
end;
fun one_arg (lazy,_,_) var = if lazy then mk_up var else var;
fun one_con (_,args,_) = mk_stuple (map2 one_arg args (vars_of args));
fun mk_abs t = abs_const ` t;
val rhss = map mk_abs (mk_sinjects (map one_con spec));
fun mk_def (bind, args, mx) rhs =
(bind, big_lambdas (vars_of args) rhs, mx);
in
define_consts (map2 mk_def spec rhss) thy
end;
(* prove beta reduction rules for constructors *)
val con_beta_thms = map (beta_of_def thy) con_def_thms;
(* TODO: enable this earlier *)
val thy = Sign.add_path dname thy;
(* replace bindings with terms in constructor spec *)
val con_spec : (term * (bool * typ) list) list =
let fun one_arg (lazy, sel, T) = (lazy, T);
fun one_con con (b, args, mx) = (con, map one_arg args);
in map2 one_con con_consts spec end;
(* prove compactness rules for constructors *)
val con_compacts =
let
val rules = @{thms compact_sinl compact_sinr compact_spair
compact_up compact_ONE};
val tacs =
[rtac (iso_locale RS @{thm iso.compact_abs}) 1,
REPEAT (resolve_tac rules 1 ORELSE atac 1)];
fun con_compact (con, args) =
let
val Ts = map snd args;
val ns = Datatype_Prop.make_tnames Ts;
val vs = map Free (ns ~~ Ts);
val con_app = list_ccomb (con, vs);
val concl = mk_trp (mk_compact con_app);
val assms = map (mk_trp o mk_compact) vs;
val goal = Logic.list_implies (assms, concl);
in
prove thy con_beta_thms goal (K tacs)
end;
in
map con_compact con_spec
end;
(* replace bindings with terms in constructor spec *)
val sel_spec : (term * (bool * binding option * typ) list) list =
map2 (fn con => fn (b, args, mx) => (con, args)) con_consts spec;
(* define and prove theorems for selector functions *)
val (sel_thms : thm list, thy : theory) =
add_selectors sel_spec rep_const
abs_iso_thm rep_strict rep_defined_iff con_beta_thms thy;
(* restore original signature path *)
val thy = Sign.parent_path thy;
val result =
{ con_consts = con_consts,
con_defs = con_def_thms,
con_compacts = con_compacts,
sel_rews = sel_thms };
in
(result, thy)
end;
end;