Removing the datatype declaration of "order" allows the standard General.order
to be used. Thus we can use Int.compare and String.compare instead of the
slower home-grown versions.
(* Title: Pure/Proof/extraction.ML
ID: $Id$
Author: Stefan Berghofer, TU Muenchen
License: GPL (GNU GENERAL PUBLIC LICENSE)
Extraction of programs from proofs.
*)
signature EXTRACTION =
sig
val set_preprocessor : (Sign.sg -> Proofterm.proof -> Proofterm.proof) -> theory -> theory
val add_realizes_eqns_i : ((term * term) list * (term * term)) list -> theory -> theory
val add_realizes_eqns : string list -> theory -> theory
val add_typeof_eqns_i : ((term * term) list * (term * term)) list -> theory -> theory
val add_typeof_eqns : string list -> theory -> theory
val add_realizers_i : (string * (string list * term * Proofterm.proof)) list
-> theory -> theory
val add_realizers : (thm * (string list * string * string)) list
-> theory -> theory
val add_expand_thms : thm list -> theory -> theory
val add_types : (xstring * ((term -> term option) list *
(term -> typ -> term -> typ -> term) option)) list -> theory -> theory
val extract : (thm * string list) list -> theory -> theory
val nullT : typ
val nullt : term
val mk_typ : typ -> term
val etype_of : theory -> string list -> typ list -> term -> typ
val realizes_of: theory -> string list -> term -> term -> term
val parsers: OuterSyntax.parser list
val setup: (theory -> theory) list
end;
structure Extraction : EXTRACTION =
struct
open Proofterm;
(**** tools ****)
fun add_syntax thy =
thy
|> Theory.copy
|> Theory.root_path
|> Theory.add_types [("Type", 0, NoSyn), ("Null", 0, NoSyn)]
|> Theory.add_arities [("Type", [], "logic"), ("Null", [], "logic")]
|> Theory.add_consts
[("typeof", "'b::logic => Type", NoSyn),
("Type", "'a::logic itself => Type", NoSyn),
("Null", "Null", NoSyn),
("realizes", "'a::logic => 'b::logic => 'b", NoSyn)];
val nullT = Type ("Null", []);
val nullt = Const ("Null", nullT);
fun mk_typ T =
Const ("Type", itselfT T --> Type ("Type", [])) $ Logic.mk_type T;
fun typeof_proc defaultS vs (Const ("typeof", _) $ u) =
Some (mk_typ (case strip_comb u of
(Var ((a, i), _), _) =>
if a mem vs then TFree ("'" ^ a ^ ":" ^ string_of_int i, defaultS)
else nullT
| (Free (a, _), _) =>
if a mem vs then TFree ("'" ^ a, defaultS) else nullT
| _ => nullT))
| typeof_proc _ _ _ = None;
fun rlz_proc (Const ("realizes", Type (_, [Type ("Null", []), _])) $ r $ t) = Some t
| rlz_proc (Const ("realizes", Type (_, [T, _])) $ r $ t) =
(case strip_comb t of
(Var (ixn, U), ts) => Some (list_comb (Var (ixn, T --> U), r :: ts))
| (Free (s, U), ts) => Some (list_comb (Free (s, T --> U), r :: ts))
| _ => None)
| rlz_proc _ = None;
val unpack_ixn = apfst implode o apsnd (fst o read_int o tl) o
take_prefix (not o equal ":") o explode;
type rules =
{next: int, rs: ((term * term) list * (term * term)) list,
net: (int * ((term * term) list * (term * term))) Net.net};
val empty_rules : rules = {next = 0, rs = [], net = Net.empty};
fun add_rule (r as (_, (lhs, _)), {next, rs, net} : rules) =
{next = next - 1, rs = r :: rs, net = Net.insert_term
((Pattern.eta_contract lhs, (next, r)), net, K false)};
fun merge_rules
({next, rs = rs1, net} : rules) ({next = next2, rs = rs2, ...} : rules) =
foldr add_rule (rs2 \\ rs1, {next = next, rs = rs1, net = net});
fun condrew sign rules procs =
let
val tsig = Sign.tsig_of sign;
fun rew tm =
Pattern.rewrite_term tsig [] (condrew' :: procs) tm
and condrew' tm = get_first (fn (_, (prems, (tm1, tm2))) =>
let
fun ren t = if_none (Term.rename_abs tm1 tm t) t;
val inc = Logic.incr_indexes ([], maxidx_of_term tm + 1);
val env as (Tenv, tenv) = Pattern.match tsig (inc tm1, tm);
val prems' = map (pairself (subst_vars env o inc o ren)) prems;
val env' = Envir.Envir
{maxidx = foldl Int.max
(~1, map (Int.max o pairself maxidx_of_term) prems'),
iTs = Vartab.make Tenv, asol = Vartab.make tenv};
val env'' = foldl (fn (env, p) =>
Pattern.unify (sign, env, [pairself rew p])) (env', prems')
in Some (Envir.norm_term env'' (inc (ren tm2)))
end handle Pattern.MATCH => None | Pattern.Unif => None)
(sort (Int.compare o pairself fst)
(Net.match_term rules (Pattern.eta_contract tm)));
in rew end;
val chtype = change_type o Some;
fun add_prefix a b = NameSpace.pack (a :: NameSpace.unpack b);
fun corr_name s vs =
add_prefix "extr" (space_implode "_" (s :: vs)) ^ "_correctness";
fun extr_name s vs = add_prefix "extr" (space_implode "_" (s :: vs));
fun msg d s = priority (implode (replicate d " ") ^ s);
fun vars_of t = rev (foldl_aterms
(fn (vs, v as Var _) => v ins vs | (vs, _) => vs) ([], t));
fun vfs_of t = vars_of t @ sort (make_ord atless) (term_frees t);
fun forall_intr (t, prop) =
let val (a, T) = (case t of Var ((a, _), T) => (a, T) | Free p => p)
in all T $ Abs (a, T, abstract_over (t, prop)) end;
fun forall_intr_prf (t, prf) =
let val (a, T) = (case t of Var ((a, _), T) => (a, T) | Free p => p)
in Abst (a, Some T, prf_abstract_over t prf) end;
val mkabs = foldr (fn (v, t) => Abs ("x", fastype_of v, abstract_over (v, t)));
fun strip_abs 0 t = t
| strip_abs n (Abs (_, _, t)) = strip_abs (n-1) t
| strip_abs _ _ = error "strip_abs: not an abstraction";
fun prf_subst_TVars tye =
map_proof_terms (subst_TVars tye) (typ_subst_TVars tye);
fun relevant_vars types prop = foldr (fn
(Var ((a, i), T), vs) => (case strip_type T of
(_, Type (s, _)) => if s mem types then a :: vs else vs
| _ => vs)
| (_, vs) => vs) (vars_of prop, []);
fun tname_of (Type (s, _)) = s
| tname_of _ = "";
fun get_var_type t =
let
val vs = Term.add_vars ([], t);
val fs = Term.add_frees ([], t)
in fn
Var (ixn, _) => (case assoc (Term.add_vars ([], t), ixn) of
None => error "get_var_type: no such variable in term"
| Some T => Var (ixn, T))
| Free (s, _) => (case assoc (Term.add_frees ([], t), s) of
None => error "get_var_type: no such variable in term"
| Some T => Free (s, T))
| _ => error "get_var_type: not a variable"
end;
(**** theory data ****)
(* data kind 'Pure/extraction' *)
structure ExtractionArgs =
struct
val name = "Pure/extraction";
type T =
{realizes_eqns : rules,
typeof_eqns : rules,
types : (string * ((term -> term option) list *
(term -> typ -> term -> typ -> term) option)) list,
realizers : (string list * (term * proof)) list Symtab.table,
defs : thm list,
expand : (string * term) list,
prep : (Sign.sg -> proof -> proof) option}
val empty =
{realizes_eqns = empty_rules,
typeof_eqns = empty_rules,
types = [],
realizers = Symtab.empty,
defs = [],
expand = [],
prep = None};
val copy = I;
val prep_ext = I;
fun merge
(({realizes_eqns = realizes_eqns1, typeof_eqns = typeof_eqns1, types = types1,
realizers = realizers1, defs = defs1, expand = expand1, prep = prep1},
{realizes_eqns = realizes_eqns2, typeof_eqns = typeof_eqns2, types = types2,
realizers = realizers2, defs = defs2, expand = expand2, prep = prep2}) : T * T) =
{realizes_eqns = merge_rules realizes_eqns1 realizes_eqns2,
typeof_eqns = merge_rules typeof_eqns1 typeof_eqns2,
types = merge_alists types1 types2,
realizers = Symtab.merge_multi' (eq_set o pairself #1)
(realizers1, realizers2),
defs = gen_merge_lists eq_thm defs1 defs2,
expand = merge_lists expand1 expand2,
prep = (case prep1 of None => prep2 | _ => prep1)};
fun print sg (x : T) = ();
end;
structure ExtractionData = TheoryDataFun(ExtractionArgs);
fun read_condeq thy =
let val sg = sign_of (add_syntax thy)
in fn s =>
let val t = Logic.varify (term_of (read_cterm sg (s, propT)))
in (map Logic.dest_equals (Logic.strip_imp_prems t),
Logic.dest_equals (Logic.strip_imp_concl t))
end handle TERM _ => error ("Not a (conditional) meta equality:\n" ^ s)
end;
(** preprocessor **)
fun set_preprocessor prep thy =
let val {realizes_eqns, typeof_eqns, types, realizers,
defs, expand, ...} = ExtractionData.get thy
in
ExtractionData.put
{realizes_eqns = realizes_eqns, typeof_eqns = typeof_eqns, types = types,
realizers = realizers, defs = defs, expand = expand, prep = Some prep} thy
end;
(** equations characterizing realizability **)
fun gen_add_realizes_eqns prep_eq eqns thy =
let val {realizes_eqns, typeof_eqns, types, realizers,
defs, expand, prep} = ExtractionData.get thy;
in
ExtractionData.put
{realizes_eqns = foldr add_rule (map (prep_eq thy) eqns, realizes_eqns),
typeof_eqns = typeof_eqns, types = types, realizers = realizers,
defs = defs, expand = expand, prep = prep} thy
end
val add_realizes_eqns_i = gen_add_realizes_eqns (K I);
val add_realizes_eqns = gen_add_realizes_eqns read_condeq;
(** equations characterizing type of extracted program **)
fun gen_add_typeof_eqns prep_eq eqns thy =
let
val {realizes_eqns, typeof_eqns, types, realizers,
defs, expand, prep} = ExtractionData.get thy;
val eqns' = map (prep_eq thy) eqns
in
ExtractionData.put
{realizes_eqns = realizes_eqns, realizers = realizers,
typeof_eqns = foldr add_rule (eqns', typeof_eqns),
types = types, defs = defs, expand = expand, prep = prep} thy
end
val add_typeof_eqns_i = gen_add_typeof_eqns (K I);
val add_typeof_eqns = gen_add_typeof_eqns read_condeq;
fun thaw (T as TFree (a, S)) =
if ":" mem explode a then TVar (unpack_ixn a, S) else T
| thaw (Type (a, Ts)) = Type (a, map thaw Ts)
| thaw T = T;
fun freeze (TVar ((a, i), S)) = TFree (a ^ ":" ^ string_of_int i, S)
| freeze (Type (a, Ts)) = Type (a, map freeze Ts)
| freeze T = T;
fun freeze_thaw f x =
map_term_types thaw (f (map_term_types freeze x));
fun etype_of sg vs Ts t =
let
val {typeof_eqns, ...} = ExtractionData.get_sg sg;
fun err () = error ("Unable to determine type of extracted program for\n" ^
Sign.string_of_term sg t)
in case strip_abs_body (freeze_thaw (condrew sg (#net typeof_eqns)
[typeof_proc (Sign.defaultS sg) vs]) (list_abs (map (pair "x") (rev Ts),
Const ("typeof", fastype_of1 (Ts, t) --> Type ("Type", [])) $ t))) of
Const ("Type", _) $ u => (Logic.dest_type u handle TERM _ => err ())
| _ => err ()
end;
(** realizers for axioms / theorems, together with correctness proofs **)
fun gen_add_realizers prep_rlz rs thy =
let val {realizes_eqns, typeof_eqns, types, realizers,
defs, expand, prep} = ExtractionData.get thy
in
ExtractionData.put
{realizes_eqns = realizes_eqns, typeof_eqns = typeof_eqns, types = types,
realizers = foldr Symtab.update_multi
(map (prep_rlz thy) (rev rs), realizers),
defs = defs, expand = expand, prep = prep} thy
end
fun prep_realizer thy =
let
val {realizes_eqns, typeof_eqns, defs, types, ...} =
ExtractionData.get thy;
val procs = flat (map (fst o snd) types);
val rtypes = map fst types;
val eqns = Net.merge (#net realizes_eqns, #net typeof_eqns, K false);
val thy' = add_syntax thy;
val sign = sign_of thy';
val tsg = Sign.tsig_of sign;
val rd = ProofSyntax.read_proof thy' false
in fn (thm, (vs, s1, s2)) =>
let
val name = Thm.name_of_thm thm;
val _ = assert (name <> "") "add_realizers: unnamed theorem";
val prop = Pattern.rewrite_term tsg
(map (Logic.dest_equals o prop_of) defs) [] (prop_of thm);
val vars = vars_of prop;
val vars' = filter_out (fn v =>
tname_of (body_type (fastype_of v)) mem rtypes) vars;
val T = etype_of sign vs [] prop;
val (T', thw) = Type.freeze_thaw_type
(if T = nullT then nullT else map fastype_of vars' ---> T);
val t = map_term_types thw (term_of (read_cterm sign (s1, T')));
val r' = freeze_thaw (condrew sign eqns
(procs @ [typeof_proc (Sign.defaultS sign) vs, rlz_proc]))
(Const ("realizes", T --> propT --> propT) $
(if T = nullT then t else list_comb (t, vars')) $ prop);
val r = foldr forall_intr (map (get_var_type r') vars, r');
val prf = Reconstruct.reconstruct_proof sign r (rd s2);
in (name, (vs, (t, prf))) end
end;
val add_realizers_i = gen_add_realizers
(fn _ => fn (name, (vs, t, prf)) => (name, (vs, (t, prf))));
val add_realizers = gen_add_realizers prep_realizer;
fun realizes_of thy vs t prop =
let
val thy' = add_syntax thy;
val sign = sign_of thy';
val {realizes_eqns, typeof_eqns, defs, types, ...} =
ExtractionData.get thy';
val procs = flat (map (fst o snd) types);
val eqns = Net.merge (#net realizes_eqns, #net typeof_eqns, K false);
val prop' = Pattern.rewrite_term (Sign.tsig_of sign)
(map (Logic.dest_equals o prop_of) defs) [] prop;
in freeze_thaw (condrew sign eqns
(procs @ [typeof_proc (Sign.defaultS sign) vs, rlz_proc]))
(Const ("realizes", fastype_of t --> propT --> propT) $ t $ prop')
end;
(** expanding theorems / definitions **)
fun add_expand_thm (thy, thm) =
let
val {realizes_eqns, typeof_eqns, types, realizers,
defs, expand, prep} = ExtractionData.get thy;
val name = Thm.name_of_thm thm;
val _ = assert (name <> "") "add_expand_thms: unnamed theorem";
val is_def =
(case strip_comb (fst (Logic.dest_equals (prop_of thm))) of
(Const _, ts) => forall is_Var ts andalso null (duplicates ts)
andalso exists (fn thy =>
is_some (Symtab.lookup (#axioms (rep_theory thy), name)))
(thy :: ancestors_of thy)
| _ => false) handle TERM _ => false;
val name = Thm.name_of_thm thm;
val _ = assert (name <> "") "add_expand_thms: unnamed theorem";
in
(ExtractionData.put (if is_def then
{realizes_eqns = realizes_eqns,
typeof_eqns = add_rule (([],
Logic.dest_equals (prop_of (Drule.abs_def thm))), typeof_eqns),
types = types,
realizers = realizers, defs = gen_ins eq_thm (thm, defs),
expand = expand, prep = prep}
else
{realizes_eqns = realizes_eqns, typeof_eqns = typeof_eqns, types = types,
realizers = realizers, defs = defs,
expand = (name, prop_of thm) ins expand, prep = prep}) thy, thm)
end;
fun add_expand_thms thms thy = foldl (fst o add_expand_thm) (thy, thms);
(** types with computational content **)
fun add_types tys thy =
let val {realizes_eqns, typeof_eqns, types, realizers,
defs, expand, prep} = ExtractionData.get thy;
in
ExtractionData.put
{realizes_eqns = realizes_eqns, typeof_eqns = typeof_eqns,
types = map (apfst (Sign.intern_tycon (sign_of thy))) tys @ types,
realizers = realizers, defs = defs, expand = expand, prep = prep} thy
end;
(**** extract program ****)
val dummyt = Const ("dummy", dummyT);
fun extract thms thy =
let
val sg = sign_of (add_syntax thy);
val tsg = Sign.tsig_of sg;
val {realizes_eqns, typeof_eqns, types, realizers, defs, expand, prep} =
ExtractionData.get thy;
val procs = flat (map (fst o snd) types);
val rtypes = map fst types;
val typroc = typeof_proc (Sign.defaultS sg);
val prep = if_none prep (K I) sg o ProofRewriteRules.elim_defs sg false defs o
Reconstruct.expand_proof sg (("", None) :: map (apsnd Some) expand);
val rrews = Net.merge (#net realizes_eqns, #net typeof_eqns, K false);
fun find_inst prop Ts ts vs =
let
val rvs = relevant_vars rtypes prop;
val vars = vars_of prop;
val n = Int.min (length vars, length ts);
fun add_args ((Var ((a, i), _), t), (vs', tye)) =
if a mem rvs then
let val T = etype_of sg vs Ts t
in if T = nullT then (vs', tye)
else (a :: vs', (("'" ^ a, i), T) :: tye)
end
else (vs', tye)
in foldr add_args (take (n, vars) ~~ take (n, ts), ([], [])) end;
fun find vs = apsome snd o find_first (curry eq_set vs o fst);
fun find' s = map snd o filter (equal s o fst)
fun app_rlz_rews Ts vs t = strip_abs (length Ts) (freeze_thaw
(condrew sg rrews (procs @ [typroc vs, rlz_proc])) (list_abs
(map (pair "x") (rev Ts), t)));
fun realizes_null vs prop = app_rlz_rews [] vs
(Const ("realizes", nullT --> propT --> propT) $ nullt $ prop);
fun corr d defs vs ts Ts hs (PBound i) _ _ = (defs, PBound i)
| corr d defs vs ts Ts hs (Abst (s, Some T, prf)) (Abst (_, _, prf')) t =
let val (defs', corr_prf) = corr d defs vs [] (T :: Ts)
(dummyt :: hs) prf (incr_pboundvars 1 0 prf')
(case t of Some (Abs (_, _, u)) => Some u | _ => None)
in (defs', Abst (s, Some T, corr_prf)) end
| corr d defs vs ts Ts hs (AbsP (s, Some prop, prf)) (AbsP (_, _, prf')) t =
let
val T = etype_of sg vs Ts prop;
val u = if T = nullT then
(case t of Some u => Some (incr_boundvars 1 u) | None => None)
else (case t of Some (Abs (_, _, u)) => Some u | _ => None);
val (defs', corr_prf) = corr d defs vs [] (T :: Ts) (prop :: hs)
(incr_pboundvars 0 1 prf) (incr_pboundvars 0 1 prf') u;
val rlz = Const ("realizes", T --> propT --> propT)
in (defs',
if T = nullT then AbsP ("R",
Some (app_rlz_rews Ts vs (rlz $ nullt $ prop)),
prf_subst_bounds [nullt] corr_prf)
else Abst (s, Some T, AbsP ("R",
Some (app_rlz_rews (T :: Ts) vs
(rlz $ Bound 0 $ incr_boundvars 1 prop)), corr_prf)))
end
| corr d defs vs ts Ts hs (prf % Some t) (prf' % _) t' =
let
val (Us, T) = strip_type (fastype_of1 (Ts, t));
val (defs', corr_prf) = corr d defs vs (t :: ts) Ts hs prf prf'
(if tname_of T mem rtypes then t'
else (case t' of Some (u $ _) => Some u | _ => None));
val u = if not (tname_of T mem rtypes) then t else
let
val eT = etype_of sg vs Ts t;
val (r, Us') = if eT = nullT then (nullt, Us) else
(Bound (length Us), eT :: Us);
val u = list_comb (incr_boundvars (length Us') t,
map Bound (length Us - 1 downto 0));
val u' = (case assoc (types, tname_of T) of
Some ((_, Some f)) => f r eT u T
| _ => Const ("realizes", eT --> T --> T) $ r $ u)
in app_rlz_rews Ts vs (list_abs (map (pair "x") Us', u')) end
in (defs', corr_prf % Some u) end
| corr d defs vs ts Ts hs (prf1 %% prf2) (prf1' %% prf2') t =
let
val prop = Reconstruct.prop_of' hs prf2';
val T = etype_of sg vs Ts prop;
val (defs1, f, u) = if T = nullT then (defs, t, None) else
(case t of
Some (f $ u) => (defs, Some f, Some u)
| _ =>
let val (defs1, u) = extr d defs vs [] Ts hs prf2'
in (defs1, None, Some u) end)
val (defs2, corr_prf1) = corr d defs1 vs [] Ts hs prf1 prf1' f;
val (defs3, corr_prf2) = corr d defs2 vs [] Ts hs prf2 prf2' u;
in
if T = nullT then (defs3, corr_prf1 %% corr_prf2) else
(defs3, corr_prf1 % u %% corr_prf2)
end
| corr d defs vs ts Ts hs (prf0 as PThm ((name, _), prf, prop, Some Ts')) _ _ =
let
val (vs', tye) = find_inst prop Ts ts vs;
val tye' = (map fst (term_tvars prop) ~~ Ts') @ tye;
val T = etype_of sg vs' [] prop;
val defs' = if T = nullT then defs
else fst (extr d defs vs ts Ts hs prf0)
in
if T = nullT andalso realizes_null vs' prop aconv prop then (defs, prf0)
else case Symtab.lookup (realizers, name) of
None => (case find vs' (find' name defs') of
None =>
let
val _ = assert (T = nullT) "corr: internal error";
val _ = msg d ("Building correctness proof for " ^ quote name ^
(if null vs' then ""
else " (relevant variables: " ^ commas_quote vs' ^ ")"));
val prf' = prep (Reconstruct.reconstruct_proof sg prop prf);
val (defs'', corr_prf) =
corr (d + 1) defs' vs' [] [] [] prf' prf' None;
val corr_prop = Reconstruct.prop_of corr_prf;
val corr_prf' = foldr forall_intr_prf
(map (get_var_type corr_prop) (vfs_of prop), proof_combt
(PThm ((corr_name name vs', []), corr_prf, corr_prop,
Some (map TVar (term_tvars corr_prop))), vfs_of corr_prop))
in
((name, (vs', ((nullt, nullt), (corr_prf, corr_prf')))) :: defs'',
prf_subst_TVars tye' corr_prf')
end
| Some (_, (_, prf')) => (defs', prf_subst_TVars tye' prf'))
| Some rs => (case find vs' rs of
Some (_, prf') => (defs', prf_subst_TVars tye' prf')
| None => error ("corr: no realizer for instance of theorem " ^
quote name ^ ":\n" ^ Sign.string_of_term sg (Envir.beta_norm
(Reconstruct.prop_of (proof_combt (prf0, ts))))))
end
| corr d defs vs ts Ts hs (prf0 as PAxm (s, prop, Some Ts')) _ _ =
let
val (vs', tye) = find_inst prop Ts ts vs;
val tye' = (map fst (term_tvars prop) ~~ Ts') @ tye
in
if etype_of sg vs' [] prop = nullT andalso
realizes_null vs' prop aconv prop then (defs, prf0)
else case find vs' (Symtab.lookup_multi (realizers, s)) of
Some (_, prf) => (defs, prf_subst_TVars tye' prf)
| None => error ("corr: no realizer for instance of axiom " ^
quote s ^ ":\n" ^ Sign.string_of_term sg (Envir.beta_norm
(Reconstruct.prop_of (proof_combt (prf0, ts)))))
end
| corr d defs vs ts Ts hs _ _ _ = error "corr: bad proof"
and extr d defs vs ts Ts hs (PBound i) = (defs, Bound i)
| extr d defs vs ts Ts hs (Abst (s, Some T, prf)) =
let val (defs', t) = extr d defs vs []
(T :: Ts) (dummyt :: hs) (incr_pboundvars 1 0 prf)
in (defs', Abs (s, T, t)) end
| extr d defs vs ts Ts hs (AbsP (s, Some t, prf)) =
let
val T = etype_of sg vs Ts t;
val (defs', t) = extr d defs vs [] (T :: Ts) (t :: hs)
(incr_pboundvars 0 1 prf)
in (defs',
if T = nullT then subst_bound (nullt, t) else Abs (s, T, t))
end
| extr d defs vs ts Ts hs (prf % Some t) =
let val (defs', u) = extr d defs vs (t :: ts) Ts hs prf
in (defs',
if tname_of (body_type (fastype_of1 (Ts, t))) mem rtypes then u
else u $ t)
end
| extr d defs vs ts Ts hs (prf1 %% prf2) =
let
val (defs', f) = extr d defs vs [] Ts hs prf1;
val prop = Reconstruct.prop_of' hs prf2;
val T = etype_of sg vs Ts prop
in
if T = nullT then (defs', f) else
let val (defs'', t) = extr d defs' vs [] Ts hs prf2
in (defs'', f $ t) end
end
| extr d defs vs ts Ts hs (prf0 as PThm ((s, _), prf, prop, Some Ts')) =
let
val (vs', tye) = find_inst prop Ts ts vs;
val tye' = (map fst (term_tvars prop) ~~ Ts') @ tye
in
case Symtab.lookup (realizers, s) of
None => (case find vs' (find' s defs) of
None =>
let
val _ = msg d ("Extracting " ^ quote s ^
(if null vs' then ""
else " (relevant variables: " ^ commas_quote vs' ^ ")"));
val prf' = prep (Reconstruct.reconstruct_proof sg prop prf);
val (defs', t) = extr (d + 1) defs vs' [] [] [] prf';
val (defs'', corr_prf) =
corr (d + 1) defs' vs' [] [] [] prf' prf' (Some t);
val nt = Envir.beta_norm t;
val args = filter_out (fn v => tname_of (body_type
(fastype_of v)) mem rtypes) (vfs_of prop);
val args' = filter (fn v => Logic.occs (v, nt)) args;
val t' = mkabs (args', nt);
val T = fastype_of t';
val cname = extr_name s vs';
val c = Const (cname, T);
val u = mkabs (args, list_comb (c, args'));
val eqn = Logic.mk_equals (c, t');
val rlz =
Const ("realizes", fastype_of nt --> propT --> propT);
val lhs = app_rlz_rews [] vs' (rlz $ nt $ prop);
val rhs = app_rlz_rews [] vs' (rlz $ list_comb (c, args') $ prop);
val f = app_rlz_rews [] vs'
(Abs ("x", T, rlz $ list_comb (Bound 0, args') $ prop));
val corr_prf' =
chtype [] equal_elim_axm %> lhs %> rhs %%
(chtype [propT] symmetric_axm %> rhs %> lhs %%
(chtype [propT, T] combination_axm %> f %> f %> c %> t' %%
(chtype [T --> propT] reflexive_axm %> f) %%
PAxm (cname ^ "_def", eqn,
Some (map TVar (term_tvars eqn))))) %% corr_prf;
val corr_prop = Reconstruct.prop_of corr_prf';
val corr_prf'' = foldr forall_intr_prf
(map (get_var_type corr_prop) (vfs_of prop), proof_combt
(PThm ((corr_name s vs', []), corr_prf', corr_prop,
Some (map TVar (term_tvars corr_prop))), vfs_of corr_prop));
in
((s, (vs', ((t', u), (corr_prf', corr_prf'')))) :: defs'',
subst_TVars tye' u)
end
| Some ((_, u), _) => (defs, subst_TVars tye' u))
| Some rs => (case find vs' rs of
Some (t, _) => (defs, subst_TVars tye' t)
| None => error ("extr: no realizer for instance of theorem " ^
quote s ^ ":\n" ^ Sign.string_of_term sg (Envir.beta_norm
(Reconstruct.prop_of (proof_combt (prf0, ts))))))
end
| extr d defs vs ts Ts hs (prf0 as PAxm (s, prop, Some Ts')) =
let
val (vs', tye) = find_inst prop Ts ts vs;
val tye' = (map fst (term_tvars prop) ~~ Ts') @ tye
in
case find vs' (Symtab.lookup_multi (realizers, s)) of
Some (t, _) => (defs, subst_TVars tye' t)
| None => error ("extr: no realizer for instance of axiom " ^
quote s ^ ":\n" ^ Sign.string_of_term sg (Envir.beta_norm
(Reconstruct.prop_of (proof_combt (prf0, ts)))))
end
| extr d defs vs ts Ts hs _ = error "extr: bad proof";
fun prep_thm (thm, vs) =
let
val {prop, der = (_, prf), sign, ...} = rep_thm thm;
val name = Thm.name_of_thm thm;
val _ = assert (name <> "") "extraction: unnamed theorem";
val _ = assert (etype_of sg vs [] prop <> nullT) ("theorem " ^
quote name ^ " has no computational content")
in (Reconstruct.reconstruct_proof sign prop prf, vs) end;
val defs = foldl (fn (defs, (prf, vs)) =>
fst (extr 0 defs vs [] [] [] prf)) ([], map prep_thm thms);
val {path, ...} = Sign.rep_sg sg;
fun add_def ((s, (vs, ((t, u), (prf, _)))), thy) =
(case Sign.const_type (sign_of thy) (extr_name s vs) of
None =>
let
val corr_prop = Reconstruct.prop_of prf;
val ft = fst (Type.freeze_thaw t);
val fu = fst (Type.freeze_thaw u);
val thy' = if t = nullt then thy else thy |>
Theory.add_consts_i [(extr_name s vs, fastype_of ft, NoSyn)] |>
fst o PureThy.add_defs_i false [((extr_name s vs ^ "_def",
Logic.mk_equals (head_of (strip_abs_body fu), ft)), [])];
in
fst (PureThy.store_thm ((corr_name s vs,
Thm.varifyT (funpow (length (term_vars corr_prop))
(forall_elim_var 0) (forall_intr_frees
(ProofChecker.thm_of_proof thy'
(fst (Proofterm.freeze_thaw_prf prf)))))), []) thy')
end
| Some _ => thy);
in thy |>
Theory.absolute_path |>
curry (foldr add_def) defs |>
Theory.add_path (NameSpace.pack (if_none path []))
end;
(**** interface ****)
structure P = OuterParse and K = OuterSyntax.Keyword;
val parse_vars = Scan.optional (P.$$$ "(" |-- P.list1 P.name --| P.$$$ ")") [];
val realizersP =
OuterSyntax.command "realizers"
"specify realizers for primitive axioms / theorems, together with correctness proof"
K.thy_decl
(Scan.repeat1 (P.xname -- parse_vars --| P.$$$ ":" -- P.string -- P.string) >>
(fn xs => Toplevel.theory (fn thy => add_realizers
(map (fn (((a, vs), s1), s2) =>
(PureThy.get_thm thy a, (vs, s1, s2))) xs) thy)));
val realizabilityP =
OuterSyntax.command "realizability"
"add equations characterizing realizability" K.thy_decl
(Scan.repeat1 P.string >> (Toplevel.theory o add_realizes_eqns));
val typeofP =
OuterSyntax.command "extract_type"
"add equations characterizing type of extracted program" K.thy_decl
(Scan.repeat1 P.string >> (Toplevel.theory o add_typeof_eqns));
val extractP =
OuterSyntax.command "extract" "extract terms from proofs" K.thy_decl
(Scan.repeat1 (P.xname -- parse_vars) >> (fn xs => Toplevel.theory
(fn thy => extract (map (apfst (PureThy.get_thm thy)) xs) thy)));
val parsers = [realizersP, realizabilityP, typeofP, extractP];
val setup =
[ExtractionData.init,
add_types [("prop", ([], None))],
add_typeof_eqns
["(typeof (PROP P)) == (Type (TYPE(Null))) ==> \
\ (typeof (PROP Q)) == (Type (TYPE('Q))) ==> \
\ (typeof (PROP P ==> PROP Q)) == (Type (TYPE('Q)))",
"(typeof (PROP Q)) == (Type (TYPE(Null))) ==> \
\ (typeof (PROP P ==> PROP Q)) == (Type (TYPE(Null)))",
"(typeof (PROP P)) == (Type (TYPE('P))) ==> \
\ (typeof (PROP Q)) == (Type (TYPE('Q))) ==> \
\ (typeof (PROP P ==> PROP Q)) == (Type (TYPE('P => 'Q)))",
"(%x. typeof (PROP P (x))) == (%x. Type (TYPE(Null))) ==> \
\ (typeof (!!x. PROP P (x))) == (Type (TYPE(Null)))",
"(%x. typeof (PROP P (x))) == (%x. Type (TYPE('P))) ==> \
\ (typeof (!!x::'a. PROP P (x))) == (Type (TYPE('a => 'P)))",
"(%x. typeof (f (x))) == (%x. Type (TYPE('f))) ==> \
\ (typeof (f)) == (Type (TYPE('f)))"],
add_realizes_eqns
["(typeof (PROP P)) == (Type (TYPE(Null))) ==> \
\ (realizes (r) (PROP P ==> PROP Q)) == \
\ (PROP realizes (Null) (PROP P) ==> PROP realizes (r) (PROP Q))",
"(typeof (PROP P)) == (Type (TYPE('P))) ==> \
\ (typeof (PROP Q)) == (Type (TYPE(Null))) ==> \
\ (realizes (r) (PROP P ==> PROP Q)) == \
\ (!!x::'P. PROP realizes (x) (PROP P) ==> PROP realizes (Null) (PROP Q))",
"(realizes (r) (PROP P ==> PROP Q)) == \
\ (!!x. PROP realizes (x) (PROP P) ==> PROP realizes (r (x)) (PROP Q))",
"(%x. typeof (PROP P (x))) == (%x. Type (TYPE(Null))) ==> \
\ (realizes (r) (!!x. PROP P (x))) == \
\ (!!x. PROP realizes (Null) (PROP P (x)))",
"(realizes (r) (!!x. PROP P (x))) == \
\ (!!x. PROP realizes (r (x)) (PROP P (x)))"],
Attrib.add_attributes
[("extraction_expand",
(Attrib.no_args add_expand_thm, K Attrib.undef_local_attribute),
"specify theorems / definitions to be expanded during extraction")]];
val etype_of = etype_of o sign_of o add_syntax;
end;
OuterSyntax.add_parsers Extraction.parsers;