(* Title: HOL/Library/reflection.ML
Author: Amine Chaieb, TU Muenchen
A trial for automatical reification.
*)
signature REFLECTION =
sig
val gen_reify: Proof.context -> thm list -> term -> thm
val gen_reify_tac: Proof.context -> thm list -> term option -> int -> tactic
val gen_reflection_tac: Proof.context -> (cterm -> thm)
-> thm list -> thm list -> term option -> int -> tactic
val get_default: Proof.context -> { reification_eqs: thm list, correctness_thms: thm list }
val add_reification_eq: attribute
val del_reification_eq: attribute
val add_correctness_thm: attribute
val del_correctness_thm: attribute
val default_reify_tac: Proof.context -> thm list -> term option -> int -> tactic
val default_reflection_tac: Proof.context -> thm list -> thm list -> term option -> int -> tactic
end;
structure Reflection : REFLECTION =
struct
val FWD = curry (op OF);
fun dest_listT (Type (@{type_name "list"}, [T])) = T;
(* Make a congruence rule out of a defining equation for the interpretation
th is one defining equation of f,
i.e. th is "f (Cp ?t1 ... ?tn) = P(f ?t1, .., f ?tn)"
Cp is a constructor pattern and P is a pattern
The result is:
[|?A1 = f ?t1 ; .. ; ?An= f ?tn |] ==> P (?A1, .., ?An) = f (Cp ?t1 .. ?tn)
+ the a list of names of the A1 .. An, Those are fresh in the ctxt *)
fun mk_congeq ctxt fs th =
let
val Const (fN, _) = th |> prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq
|> fst |> strip_comb |> fst;
val thy = Proof_Context.theory_of ctxt;
val cert = Thm.cterm_of thy;
val ((_, [th']), ctxt') = Variable.import true [th] ctxt;
val (lhs, rhs) = HOLogic.dest_eq (HOLogic.dest_Trueprop (Thm.prop_of th'));
fun add_fterms (t as t1 $ t2) =
if exists (fn f => Term.could_unify (t |> strip_comb |> fst, f)) fs
then insert (op aconv) t
else add_fterms t1 #> add_fterms t2
| add_fterms (t as Abs _) =
if exists_Const (fn (c, _) => c = fN) t
then K [t]
else K []
| add_fterms _ = I;
val fterms = add_fterms rhs [];
val (xs, ctxt'') = Variable.variant_fixes (replicate (length fterms) "x") ctxt';
val tys = map fastype_of fterms;
val vs = map Free (xs ~~ tys);
val env = fterms ~~ vs; (*FIXME*)
fun replace_fterms (t as t1 $ t2) =
(case AList.lookup (op aconv) env t of
SOME v => v
| NONE => replace_fterms t1 $ replace_fterms t2)
| replace_fterms t =
(case AList.lookup (op aconv) env t of
SOME v => v
| NONE => t);
fun mk_def (Abs (x, xT, t), v) =
HOLogic.mk_Trueprop (HOLogic.all_const xT $ Abs (x, xT, HOLogic.mk_eq (v $ Bound 0, t)))
| mk_def (t, v) = HOLogic.mk_Trueprop (HOLogic.mk_eq (v, t));
fun tryext x =
(x RS @{lemma "(\<forall>x. f x = g x) \<Longrightarrow> f = g" by blast} handle THM _ => x);
val cong =
(Goal.prove ctxt'' [] (map mk_def env)
(HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, replace_fterms rhs)))
(fn {context, prems, ...} =>
Local_Defs.unfold_tac context (map tryext prems) THEN rtac th' 1)) RS sym;
val (cong' :: vars') =
Variable.export ctxt'' ctxt (cong :: map (Drule.mk_term o cert) vs);
val vs' = map (fst o fst o Term.dest_Var o Thm.term_of o Drule.dest_term) vars';
in (vs', cong') end;
(* congs is a list of pairs (P,th) where th is a theorem for
[| f p1 = A1; ...; f pn = An|] ==> f (C p1 .. pn) = P *)
fun rearrange congs =
let
fun P (_, th) =
let val @{term "Trueprop"} $ (Const (@{const_name HOL.eq}, _) $ l $ _) = concl_of th
in can dest_Var l end;
val (yes, no) = List.partition P congs;
in no @ yes end;
fun gen_reify ctxt eqs t =
let
fun index_of t bds =
let
val tt = HOLogic.listT (fastype_of t);
in
(case AList.lookup Type.could_unify bds tt of
NONE => error "index_of : type not found in environements!"
| SOME (tbs, tats) =>
let
val i = find_index (fn t' => t' = t) tats;
val j = find_index (fn t' => t' = t) tbs;
in
if j = ~1 then
if i = ~1
then (length tbs + length tats, AList.update Type.could_unify (tt, (tbs, tats @ [t])) bds)
else (i, bds)
else (j, bds)
end)
end;
(* Generic decomp for reification : matches the actual term with the
rhs of one cong rule. The result of the matching guides the
proof synthesis: The matches of the introduced Variables A1 .. An are
processed recursively
The rest is instantiated in the cong rule,i.e. no reification is needed *)
(* da is the decomposition for atoms, ie. it returns ([],g) where g
returns the right instance f (AtC n) = t , where AtC is the Atoms
constructor and n is the number of the atom corresponding to t *)
fun decomp_genreif da cgns (t, ctxt) bds =
let
val thy = Proof_Context.theory_of ctxt;
val cert = cterm_of thy;
fun tryabsdecomp (s, ctxt) bds =
(case s of
Abs (_, xT, ta) =>
let
val ([raw_xn], ctxt') = Variable.variant_fixes ["x"] ctxt;
val (xn, ta) = Syntax_Trans.variant_abs (raw_xn, xT, ta); (* FIXME !? *)
val x = Free(xn, xT);
val bds = (case AList.lookup Type.could_unify bds (HOLogic.listT xT) of
NONE => error "tryabsdecomp: Type not found in the Environement"
| SOME (bsT, atsT) => AList.update Type.could_unify (HOLogic.listT xT, (x :: bsT, atsT)) bds);
in (([(ta, ctxt')],
fn ([th], bds) =>
(hd (Variable.export ctxt' ctxt [(Thm.forall_intr (cert x) th) COMP allI]),
let
val (bsT, asT) = the (AList.lookup Type.could_unify bds (HOLogic.listT xT));
in
AList.update Type.could_unify (HOLogic.listT xT,(tl bsT, asT)) bds
end)),
bds)
end
| _ => da (s, ctxt) bds)
in
(case cgns of
[] => tryabsdecomp (t, ctxt) bds
| ((vns, cong) :: congs) =>
(let
val cert = cterm_of thy;
val certy = ctyp_of thy;
val (tyenv, tmenv) =
Pattern.match thy
((fst o HOLogic.dest_eq o HOLogic.dest_Trueprop) (concl_of cong), t)
(Vartab.empty, Vartab.empty);
val (fnvs, invs) = List.partition (fn ((vn, _),_) => member (op =) vns vn) (Vartab.dest tmenv);
val (fts, its) =
(map (snd o snd) fnvs,
map (fn ((vn, vi), (tT, t)) => (cert (Var ((vn, vi), tT)), cert t)) invs);
val ctyenv = map (fn ((vn, vi), (s, ty)) => (certy (TVar((vn, vi), s)), certy ty)) (Vartab.dest tyenv);
in
((fts ~~ replicate (length fts) ctxt,
apfst (FWD (Drule.instantiate_normalize (ctyenv, its) cong))), bds)
end handle Pattern.MATCH => decomp_genreif da congs (t,ctxt) bds))
end;
(* looks for the atoms equation and instantiates it with the right number *)
fun mk_decompatom eqs (t, ctxt) bds = (([], fn (_, bds) =>
let
val tT = fastype_of t;
fun isat eq =
let
val rhs = eq |> prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> snd;
in exists_Const
(fn (n, ty) => n = @{const_name "List.nth"}
andalso AList.defined Type.could_unify bds (domain_type ty)) rhs
andalso Type.could_unify (fastype_of rhs, tT)
end;
fun get_nths t acc =
case t of
Const(@{const_name "List.nth"}, _) $ vs $ n => insert (fn ((a, _), (b, _)) => a aconv b) (t, (vs, n)) acc
| t1 $ t2 => get_nths t1 (get_nths t2 acc)
| Abs (_ ,_ ,t') => get_nths t' acc
| _ => acc;
fun tryeqs [] bds = error "Can not find the atoms equation"
| tryeqs (eq :: eqs) bds = ((
let
val rhs = eq |> prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> snd;
val nths = get_nths rhs [];
val (vss, _) = fold_rev (fn (_, (vs, n)) => fn (vss, ns) =>
(insert (op aconv) vs vss, insert (op aconv) n ns)) nths ([], []);
val (vsns, ctxt') = Variable.variant_fixes (replicate (length vss) "vs") ctxt;
val (xns, ctxt'') = Variable.variant_fixes (replicate (length nths) "x") ctxt';
val thy = Proof_Context.theory_of ctxt'';
val cert = cterm_of thy;
val certT = ctyp_of thy;
val vsns_map = vss ~~ vsns;
val xns_map = fst (split_list nths) ~~ xns;
val subst = map (fn (nt, xn) => (nt, Var ((xn, 0), fastype_of nt))) xns_map;
val rhs_P = subst_free subst rhs;
val (tyenv, tmenv) = Pattern.match thy (rhs_P, t) (Vartab.empty, Vartab.empty);
val sbst = Envir.subst_term (tyenv, tmenv);
val sbsT = Envir.subst_type tyenv;
val subst_ty = map (fn (n, (s, t)) =>
(certT (TVar (n, s)), certT t)) (Vartab.dest tyenv)
val tml = Vartab.dest tmenv;
val (subst_ns, bds) = fold_map
(fn (Const _ $ _ $ n, Var (xn0, _)) => fn bds =>
let
val name = snd (the (AList.lookup (op =) tml xn0));
val (idx, bds) = index_of name bds;
in ((cert n, idx |> (HOLogic.mk_nat #> cert)), bds) end) subst bds;
val subst_vs =
let
fun h (Const _ $ (vs as Var (_, lT)) $ _, Var (_, T)) =
let
val cns = sbst (Const (@{const_name "List.Cons"}, T --> lT --> lT));
val lT' = sbsT lT;
val (bsT, _) = the (AList.lookup Type.could_unify bds lT);
val vsn = the (AList.lookup (op =) vsns_map vs);
val cvs = cert (fold_rev (fn x => fn xs => cns $ x $xs) bsT (Free (vsn, lT')));
in (cert vs, cvs) end;
in map h subst end;
val cts = map (fn ((vn, vi), (tT, t)) => (cert (Var ((vn, vi), tT)), cert t))
(fold (AList.delete (fn (((a : string), _), (b, _)) => a = b))
(map (fn n => (n, 0)) xns) tml);
val substt =
let
val ih = Drule.cterm_rule (Thm.instantiate (subst_ty, []));
in map (fn (v, t) => (ih v, ih t)) (subst_ns @ subst_vs @ cts) end;
val th = (Drule.instantiate_normalize (subst_ty, substt) eq) RS sym;
in (hd (Variable.export ctxt'' ctxt [th]), bds) end)
handle Pattern.MATCH => tryeqs eqs bds)
in tryeqs (filter isat eqs) bds end), bds);
(* Generic reification procedure: *)
(* creates all needed cong rules and then just uses the theorem synthesis *)
fun mk_congs ctxt eqs =
let
val fs = fold_rev (fn eq => insert (op =) (eq |> prop_of |> HOLogic.dest_Trueprop
|> HOLogic.dest_eq |> fst |> strip_comb
|> fst)) eqs [];
val tys = fold_rev (fn f => fold (insert (op =)) (f |> fastype_of |> binder_types |> tl)) fs [];
val (vs, ctxt') = Variable.variant_fixes (replicate (length tys) "vs") ctxt;
val thy = Proof_Context.theory_of ctxt';
val cert = cterm_of thy;
val vstys = map (fn (t, v) => (t, SOME (cert (Free (v, t))))) (tys ~~ vs);
fun prep_eq eq =
let
val (_, _ :: vs) = eq |> prop_of |> HOLogic.dest_Trueprop
|> HOLogic.dest_eq |> fst |> strip_comb;
val subst = map (fn (v as Var (_, t)) =>
(cert v, (the o the) (AList.lookup (op =) vstys t))) (filter is_Var vs);
in Thm.instantiate ([], subst) eq end;
val (ps, congs) = map_split (mk_congeq ctxt' fs o prep_eq) eqs;
val bds = AList.make (K ([], [])) tys;
in (ps ~~ Variable.export ctxt' ctxt congs, bds) end
val (congs, bds) = mk_congs ctxt eqs;
val congs = rearrange congs;
val (th, bds) = divide_and_conquer' (decomp_genreif (mk_decompatom eqs) congs) (t,ctxt) bds;
fun is_listVar (Var (_, t)) = can dest_listT t
| is_listVar _ = false;
val vars = th |> prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> snd
|> strip_comb |> snd |> filter is_listVar;
val cert = cterm_of (Proof_Context.theory_of ctxt);
val cvs = map (fn (v as Var(_, t)) => (cert v,
the (AList.lookup Type.could_unify bds t) |> snd |> HOLogic.mk_list (dest_listT t) |> cert)) vars;
val th' = Drule.instantiate_normalize ([], cvs) th;
val t' = (fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) th';
val th'' = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq (t, t')))
(fn _ => simp_tac ctxt 1)
in FWD trans [th'',th'] end;
fun gen_reflect ctxt conv corr_thms eqs t =
let
val reify_thm = gen_reify ctxt eqs t;
fun try_corr thm =
SOME (FWD trans [reify_thm, thm RS sym]) handle THM _ => NONE;
val thm = case get_first try_corr corr_thms
of NONE => error "No suitable correctness theorem found"
| SOME thm => thm;
val ft = (Thm.dest_arg1 o Thm.dest_arg o Thm.dest_arg o cprop_of) thm;
val rth = conv ft;
in
thm
|> simplify (put_simpset HOL_basic_ss ctxt addsimps [rth])
|> simplify (put_simpset HOL_basic_ss ctxt addsimps eqs addsimps @{thms nth_Cons_0 nth_Cons_Suc})
end;
fun tac_of_thm mk_thm to = SUBGOAL (fn (goal, i) =>
let
val t = (case to of NONE => HOLogic.dest_Trueprop goal | SOME t => t)
val thm = mk_thm t RS ssubst;
in rtac thm i end);
fun gen_reify_tac ctxt eqs = tac_of_thm (gen_reify ctxt eqs);
(*Reflection calls reification and uses the correctness theorem assumed to be the head of the list*)
fun gen_reflection_tac ctxt conv corr_thms eqs =
tac_of_thm (gen_reflect ctxt conv corr_thms eqs);
structure Data = Generic_Data
(
type T = thm list * thm list;
val empty = ([], []);
val extend = I;
fun merge ((ths1, rths1), (ths2, rths2)) =
(Thm.merge_thms (ths1, ths2), Thm.merge_thms (rths1, rths2));
);
fun get_default ctxt =
let
val (reification_eqs, correctness_thms) = Data.get (Context.Proof ctxt);
in { reification_eqs = reification_eqs, correctness_thms = correctness_thms } end;
val add_reification_eq = Thm.declaration_attribute (Data.map o apfst o Thm.add_thm);
val del_reification_eq = Thm.declaration_attribute (Data.map o apfst o Thm.del_thm);
val add_correctness_thm = Thm.declaration_attribute (Data.map o apsnd o Thm.add_thm);
val del_correctness_thm = Thm.declaration_attribute (Data.map o apsnd o Thm.del_thm);
val _ = Context.>> (Context.map_theory
(Attrib.setup @{binding reify}
(Attrib.add_del add_reification_eq del_reification_eq) "declare reification equations" #>
Attrib.setup @{binding reflection}
(Attrib.add_del add_correctness_thm del_correctness_thm) "declare reflection correctness theorems"));
fun default_reify_tac ctxt user_eqs =
let
val { reification_eqs = default_eqs, correctness_thms = _ } =
get_default ctxt;
val eqs = fold Thm.add_thm user_eqs default_eqs;
in gen_reify_tac ctxt eqs end;
fun default_reflection_tac ctxt user_thms user_eqs =
let
val { reification_eqs = default_eqs, correctness_thms = default_thms } =
get_default ctxt;
val corr_thms = fold Thm.add_thm user_thms default_thms;
val eqs = fold Thm.add_thm user_eqs default_eqs;
val conv = Code_Evaluation.dynamic_conv (Proof_Context.theory_of ctxt);
(*FIXME why Code_Evaluation.dynamic_conv? very specific*)
in gen_reflection_tac ctxt conv corr_thms eqs end;
end