(* Title: HOL/Library/reflection.ML
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 -> thm list -> term option -> int -> tactic
val gen_reflection_tac: Proof.context -> (cterm -> thm)
-> thm list -> thm list -> term option -> int -> tactic
val genreif : Proof.context -> thm list -> term -> thm
end;
structure Reflection : REFLECTION =
struct
val ext2 = @{thm ext2};
val nth_Cons_0 = @{thm nth_Cons_0};
val nth_Cons_Suc = @{thm nth_Cons_Suc};
(* 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 => 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(xn,xT,t')) =
if exists_Const (fn (c, _) => c = fN) t then (fn _ => [t]) else (fn _ => [])
| 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$(Bound 0), t)))
| mk_def (t, v) = HOLogic.mk_Trueprop (HOLogic.mk_eq (v, t))
fun tryext x = (x RS ext2 handle THM _ => x)
val cong = (Goal.prove ctxt'' [] (map mk_def env)
(HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, replace_fterms rhs)))
(fn x => Local_Defs.unfold_tac (#context x) (map tryext (#prems x))
THEN rtac th' 1)) RS sym
val (cong' :: vars') =
Variable.export ctxt'' ctxt (cong :: map (Drule.mk_term o cert) vs)
val vs' = map (fst o fst o Term.dest_Var o Thm.term_of o Drule.dest_term) vars'
in (vs', cong') end;
(* congs is a list of pairs (P,th) where th is a theorem for *)
(* [| f p1 = A1; ...; f pn = An|] ==> f (C p1 .. pn) = P *)
val FWD = curry (op OF);
exception REIF of string;
fun dest_listT (Type (@{type_name "list"}, [T])) = T;
fun rearrange congs =
let
fun P (_, th) =
let val @{term "Trueprop"}$(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 genreif ctxt raw_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 = ProofContext.theory_of ctxt
val cert = cterm_of thy
fun tryabsdecomp (s,ctxt) bds =
(case s of
Abs(xn,xT,ta) => (
let
val ([xn],ctxt') = Variable.variant_fixes ["x"] ctxt
val (xn,ta) = variant_abs (xn,xT,ta)
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),
Library.apfst (FWD (instantiate (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,ns) = 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 = ProofContext.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 t's = map (fn xn => snd (the (AList.lookup (op =) tml (xn,0)))) xns (* FIXME : Express with sbst*)
val (subst_ns, bds) = fold_map
(fn (Const _ $ vs $ n, Var (xn0,T)) => 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 ty (Const _ $ (vs as Var (vsn,lT)) $ n, Var (xn0,T)) = (certT T, certT (sbsT T))
fun h (Const _ $ (vs as Var (vsn,lT)) $ n, Var (xn0,T)) =
let
val cns = sbst (Const(@{const_name "List.Cons"}, T --> lT --> lT))
val lT' = sbsT lT
val (bsT,asT) = 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 = (instantiate (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 raw_eqs =
let
val fs = fold_rev (fn eq =>
insert (op =) (eq |> prop_of |> HOLogic.dest_Trueprop
|> HOLogic.dest_eq |> fst |> strip_comb
|> fst)) raw_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 = ProofContext.theory_of ctxt'
val cert = cterm_of thy
val vstys = map (fn (t,v) => (t,SOME (cert (Free(v,t)))))
(tys ~~ vs)
val is_Var = can dest_Var
fun insteq eq vs =
let
val subst = map (fn (v as Var(n,t)) => (cert v, (the o the) (AList.lookup (op =) vstys t)))
(filter is_Var vs)
in Thm.instantiate ([],subst) eq
end
val bds = AList.make (fn _ => ([],[])) tys
val eqs = map (fn eq => eq |> prop_of |> HOLogic.dest_Trueprop
|> HOLogic.dest_eq |> fst |> strip_comb |> snd |> tl
|> (insteq eq)) raw_eqs
val (ps,congs) = split_list (map (mk_congeq ctxt' fs) eqs)
in (ps ~~ (Variable.export ctxt' ctxt congs), bds)
end
val (congs, bds) = mk_congs ctxt raw_eqs
val congs = rearrange congs
val (th, bds) = divide_and_conquer' (decomp_genreif (mk_decompatom raw_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 (ProofContext.theory_of ctxt)
val cvs = map (fn (v as Var(n,t)) => (cert v,
the (AList.lookup Type.could_unify bds t) |> snd |> HOLogic.mk_list (dest_listT t) |> cert)) vars
val th' = instantiate ([], 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 (simpset_of ctxt) 1)
in FWD trans [th'',th']
end
fun genreflect ctxt conv corr_thms raw_eqs t =
let
val reifth = genreif ctxt raw_eqs t
fun trytrans [] = error "No suitable correctness theorem found"
| trytrans (th::ths) =
(FWD trans [reifth, th RS sym] handle THM _ => trytrans ths)
val th = trytrans corr_thms
val ft = (Thm.dest_arg1 o Thm.dest_arg o Thm.dest_arg o cprop_of) th
val rth = 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
fun P () = HOLogic.dest_Trueprop (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 gen_reflection_tac ctxt conv corr_thms raw_eqs to i = (fn st =>
let
val P = HOLogic.dest_Trueprop (nth (prems_of st) (i - 1));
val t = the_default P to;
val th = genreflect ctxt conv corr_thms raw_eqs t
RS ssubst;
in (rtac th i THEN TRY(rtac TrueI i)) st end);
fun reflection_tac ctxt = gen_reflection_tac ctxt Codegen.evaluation_conv;
(*FIXME why Codegen.evaluation_conv? very specific...*)
end