(*
ID: $Id$
Author: Amine Chaieb, TU Muenchen
A trial for automatical reification.
*)
signature REFLECTION = sig
val genreify_tac: Proof.context -> thm list -> term option -> int -> tactic
val reflection_tac: Proof.context -> thm list -> term option -> int -> tactic
end;
structure Reflection : REFLECTION
= struct
val ext2 = thm "ext2";
(* 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 (f as Const(fN,fT)) = th |> prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq
|> fst |> strip_comb |> fst
val thy = ProofContext.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 => (t |> strip_comb |> fst) aconv f) fs then insert (op aconv) t
else add_fterms t1 #> add_fterms t2
| add_fterms (t as Abs(xn,xT,t')) =
if (fN mem (term_consts t)) then (fn _ => [t]) else (fn _ => [])
| add_fterms _ = I
val fterms = add_fterms rhs []
val (xs, ctxt'') = Variable.invent_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 ext2 handle _ => x)
val cong = (Goal.prove ctxt'' [] (map mk_def env)
(HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, replace_fterms rhs)))
(fn x => LocalDefs.unfold_tac (#context x) (map tryext (#prems x))
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 *)
val FWD = curry (op OF);
(* 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 *)
(* 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 *)
exception REIF of string;
val bds = ref ([]: (typ * ((term list) * (term list))) list);
fun index_of t =
let
val tt = HOLogic.listT (fastype_of t)
in
(case AList.lookup (op =) (!bds) tt of
NONE => error "index_of : type not found in environements!"
| SOME (tbs,tats) =>
let
val i = find_index_eq t tats
val j = find_index_eq t tbs
in (if j= ~1 then
if i= ~1
then (bds := AList.update (op =) (tt,(tbs,tats@[t])) (!bds) ;
length tbs + length tats)
else i else j)
end)
end;
fun dest_listT (Type ("List.list", [T])) = T;
fun decomp_genreif da cgns (t,ctxt) =
let
val thy = ProofContext.theory_of ctxt
val cert = cterm_of thy
fun tryabsdecomp (s,ctxt) =
(case s of
Abs(xn,xT,ta) =>
(let
val ([xn],ctxt') = Variable.invent_fixes ["x"] ctxt
val (xn,ta) = variant_abs (xn,xT,ta)
val x = Free(xn,xT)
val _ = (case AList.lookup (op =) (!bds) (HOLogic.listT xT)
of NONE => error "tryabsdecomp: Type not found in the Environement"
| SOME (bsT,atsT) =>
(bds := AList.update (op =) (HOLogic.listT xT, ((x::bsT), atsT)) (!bds)))
in ([(ta, ctxt')] ,
fn [th] => ((let val (bsT,asT) = the(AList.lookup (op =) (!bds) (HOLogic.listT xT))
in (bds := AList.update (op =) (HOLogic.listT xT,(tl bsT,asT)) (!bds))
end) ;
hd (Variable.export ctxt' ctxt [(forall_intr (cert x) th) COMP allI])))
end)
| _ => da (s,ctxt))
in
(case cgns of
[] => tryabsdecomp (t,ctxt)
| ((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)
(Envir.type_env (Envir.empty 0),Term.Vartab.empty)
val (fnvs,invs) = List.partition (fn ((vn,_),_) => vn mem vns) (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), FWD (instantiate (ctyenv, its) cong))
end)
handle MATCH => decomp_genreif da congs (t,ctxt)))
end;
(* looks for the atoms equation and instantiates it with the right number *)
fun mk_decompatom eqs (t,ctxt) =
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="List.nth"
andalso
AList.defined (op =) (!bds) (domain_type ty)) rhs
andalso fastype_of rhs = tT
end
fun get_nth t =
case t of
Const("List.nth",_)$vs$n => (t,vs,n)
| t1$t2 => (get_nth t1 handle REIF "get_nth" => get_nth t2)
| Abs(_,_,t') => get_nth t'
| _ => raise REIF "get_nth"
val ([xn,vsn],ctxt') = Variable.invent_fixes ["x","vs"] ctxt
val thy = ProofContext.theory_of ctxt'
val cert = cterm_of thy
fun tryeqs [] = raise REIF "Can not find the atoms equation"
| tryeqs (eq::eqs) = ((
let
val rhs = eq |> prop_of |> HOLogic.dest_Trueprop
|> HOLogic.dest_eq |> snd
val (nt,vs,n) = get_nth rhs
val ntT = fastype_of nt
val ntlT = HOLogic.listT ntT
val (bsT,asT) = the (AList.lookup (op =) (!bds) ntlT)
val x = Var ((xn,0),ntT)
val rhs_P = subst_free [(nt,x)] rhs
val (_, tmenv) = Pattern.match
thy (rhs_P, t)
(Envir.type_env (Envir.empty 0),Term.Vartab.empty)
val tml = Vartab.dest tmenv
val SOME (_,t') = AList.lookup (op =) tml (xn,0)
val cvs =
cert (foldr (fn (x,xs) => Const("List.list.Cons", ntT --> ntlT --> ntlT)$x$xs)
(Free(vsn,ntlT)) bsT)
val cts = map (fn ((vn,vi),(tT,t)) => (cert(Var ((vn,vi),tT)), cert t))
(AList.delete (op =) (xn,0) tml)
val th = (instantiate
([],
[(cert vs, cvs),(cert n, t' |> index_of |> HOLogic.mk_nat |> cert)]
@cts) eq) RS sym
in hd (Variable.export ctxt' ctxt [th])
end)
handle MATCH => tryeqs eqs)
in ([], fn _ => tryeqs (filter isat eqs))
end;
(*
fun mk_decompatom eqs (t,ctxt) =
let
val tT = fastype_of t
val tlT = HOLogic.listT tT
val (bsT,asT) = (the (AList.lookup (op =) (!bds) tlT)
handle Option => error "mk_decompatom: Type not found in the env.")
fun isateq (_$_$(Const("List.nth",_)$vs$_)) = (fastype_of vs = tlT)
| isateq _ = false
in case List.find (isateq o HOLogic.dest_Trueprop o prop_of) eqs of
NONE => raise REIF "Can not find the atoms equation"
| SOME th =>
([],
fn ths =>
let
val ([x], ctxt') = Variable.invent_fixes ["vs"] ctxt
val cert = cterm_of (ProofContext.theory_of ctxt')
val (Const("List.nth",_)$(vs as Var((vsn,vsi),_))$n) =
(snd o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) th
val cvs =
cert (foldr (fn (x,xs) => Const("List.list.Cons", tT --> tlT --> tlT)$x$xs)
(Free(x,tlT)) bsT)
val th' = (instantiate ([],
[(cert vs, cvs),
(cert n, cert (HOLogic.mk_nat(index_of t)))]) th)
RS sym
in hd (Variable.export ctxt' ctxt [th']) end)
end;
*)
(* Generic reification procedure: *)
(* creates all needed cong rules and then just uses the theorem synthesis *)
fun mk_congs ctxt raw_eqs =
let
val fs = foldr (fn (eq,fns) =>
(eq |> prop_of |> HOLogic.dest_Trueprop
|> HOLogic.dest_eq |> fst |> strip_comb
|> fst) ins fns) [] raw_eqs
val tys = foldr (fn (f,ts) => (f |> fastype_of |> binder_types |> split_last |> fst)
union ts) [] fs
val _ = bds := AList.make (fn _ => ([],[])) tys
val (vs, ctxt') = Variable.invent_fixes (replicate (length tys) "vs") ctxt
val thy = ProofContext.theory_of ctxt'
val cert = cterm_of thy
val vstys = map (fn (t,v) => (t,SOME (cert (Free(v,t)))))
(tys ~~ vs)
fun insteq eq ts =
let val itms = map (fn t => t|> (AList.lookup (op =) vstys) |> the) ts
in instantiate' [] itms eq
end
val eqs = map (fn eq => eq |> prop_of |> HOLogic.dest_Trueprop
|> HOLogic.dest_eq |> fst |> strip_comb |> fst |> fastype_of
|> binder_types |> split_last |> fst
|> (insteq eq)) raw_eqs
val (ps,congs) = split_list (map (mk_congeq ctxt' fs) eqs)
in ps ~~ (Variable.export ctxt' ctxt congs)
end;
fun genreif ctxt raw_eqs t =
let
val _ = bds := []
val congs = mk_congs ctxt raw_eqs
val th = divide_and_conquer (decomp_genreif (mk_decompatom raw_eqs) congs) (t,ctxt)
val tys = th |> prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> snd
|> strip_comb |> fst |> fastype_of |> strip_type |> fst
|> split_last |> fst
val cert = cterm_of (ProofContext.theory_of ctxt)
val cvs = map (fn t => t |> (AList.lookup (op =) (!bds)) |> the |> snd
|> HOLogic.mk_list I (dest_listT t) |> cert |> SOME)
tys
val th' = (instantiate' [] cvs (th RS sym)) RS sym
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 1)
val _ = bds := []
in FWD trans [th'',th']
end;
fun genreflect ctxt corr_thm raw_eqs t =
let val th = FWD trans [genreif ctxt raw_eqs t, corr_thm RS sym]
val ft = (snd o Thm.dest_comb o snd o Thm.dest_comb o snd o Thm.dest_comb o cprop_of) th
val rth = NBE.normalization_conv ft
in simplify (HOL_basic_ss addsimps raw_eqs addsimps [nth_Cons_0, nth_Cons_Suc])
(simplify (HOL_basic_ss addsimps [rth]) th)
end
fun genreify_tac ctxt eqs to i = (fn st =>
let
val P = HOLogic.dest_Trueprop (List.nth (prems_of st, i - 1))
val t = (case to of NONE => P | SOME x => x)
val th = (genreif ctxt eqs t) RS ssubst
in rtac th i st
end);
(* Reflection calls reification and uses the correctness *)
(* theorem assumed to be the dead of the list *)
fun reflection_tac ctxt (corr_thm::raw_eqs) to i =
(fn st =>
let val P = (HOLogic.dest_Trueprop (List.nth (prems_of st, i - 1)))
val t = (case to of NONE => P | SOME x => x)
val th = (genreflect ctxt corr_thm raw_eqs t) RS ssubst
in rtac th i st end);
end