Converted reification to use fold_map instead of Library.foldl_map. Use antiquotations.
1 (* Title: HOL/Library/reflection.ML
2 Author: Amine Chaieb, TU Muenchen
4 A trial for automatical reification.
9 val genreify_tac: Proof.context -> thm list -> term option -> int -> tactic
10 val reflection_tac: Proof.context -> thm list -> thm list -> term option -> int -> tactic
11 val gen_reflection_tac: Proof.context -> (cterm -> thm)
12 -> thm list -> thm list -> term option -> int -> tactic
15 structure Reflection : REFLECTION =
18 val ext2 = @{thm ext2};
19 val nth_Cons_0 = @{thm nth_Cons_0};
20 val nth_Cons_Suc = @{thm nth_Cons_Suc};
22 (* Make a congruence rule out of a defining equation for the interpretation *)
23 (* th is one defining equation of f, i.e.
24 th is "f (Cp ?t1 ... ?tn) = P(f ?t1, .., f ?tn)" *)
25 (* Cp is a constructor pattern and P is a pattern *)
28 [|?A1 = f ?t1 ; .. ; ?An= f ?tn |] ==> P (?A1, .., ?An) = f (Cp ?t1 .. ?tn) *)
29 (* + the a list of names of the A1 .. An, Those are fresh in the ctxt*)
31 fun mk_congeq ctxt fs th =
33 val (f as Const(fN,fT)) = th |> prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq
34 |> fst |> strip_comb |> fst
35 val thy = ProofContext.theory_of ctxt
36 val cert = Thm.cterm_of thy
37 val (((_,_),[th']), ctxt') = Variable.import_thms true [th] ctxt
38 val (lhs, rhs) = HOLogic.dest_eq (HOLogic.dest_Trueprop (Thm.prop_of th'))
39 fun add_fterms (t as t1 $ t2) =
40 if exists (fn f => Term.could_unify (t |> strip_comb |> fst, f)) fs then insert (op aconv) t
41 else add_fterms t1 #> add_fterms t2
42 | add_fterms (t as Abs(xn,xT,t')) =
43 if exists_Const (fn (c, _) => c = fN) t then (fn _ => [t]) else (fn _ => [])
45 val fterms = add_fterms rhs []
46 val (xs, ctxt'') = Variable.variant_fixes (replicate (length fterms) "x") ctxt'
47 val tys = map fastype_of fterms
48 val vs = map Free (xs ~~ tys)
49 val env = fterms ~~ vs
51 fun replace_fterms (t as t1 $ t2) =
52 (case AList.lookup (op aconv) env t of
54 | NONE => replace_fterms t1 $ replace_fterms t2)
55 | replace_fterms t = (case AList.lookup (op aconv) env t of
59 fun mk_def (Abs(x,xT,t),v) = HOLogic.mk_Trueprop ((HOLogic.all_const xT)$ Abs(x,xT,HOLogic.mk_eq(v$(Bound 0), t)))
60 | mk_def (t, v) = HOLogic.mk_Trueprop (HOLogic.mk_eq (v, t))
61 fun tryext x = (x RS ext2 handle THM _ => x)
62 val cong = (Goal.prove ctxt'' [] (map mk_def env)
63 (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, replace_fterms rhs)))
64 (fn x => LocalDefs.unfold_tac (#context x) (map tryext (#prems x))
65 THEN rtac th' 1)) RS sym
67 val (cong' :: vars') =
68 Variable.export ctxt'' ctxt (cong :: map (Drule.mk_term o cert) vs)
69 val vs' = map (fst o fst o Term.dest_Var o Thm.term_of o Drule.dest_term) vars'
72 (* congs is a list of pairs (P,th) where th is a theorem for *)
73 (* [| f p1 = A1; ...; f pn = An|] ==> f (C p1 .. pn) = P *)
74 val FWD = curry (op OF);
77 exception REIF of string;
79 fun dest_listT (Type (@{type_name "list"}, [T])) = T;
81 (* This modified version of divide_and_conquer allows the threading
82 of a state variable *)
83 fun divide_and_conquer' decomp s x =
84 let val ((ys, recomb), s') = decomp s x
85 in recomb (fold_map (divide_and_conquer' decomp) ys s') end;
90 let val @{term "Trueprop"}$(Const ("op =",_) $l$_) = concl_of th
92 val (yes,no) = List.partition P congs
95 fun genreif ctxt raw_eqs t =
99 val tt = HOLogic.listT (fastype_of t)
101 (case AList.lookup Type.could_unify bds tt of
102 NONE => error "index_of : type not found in environements!"
105 val i = find_index_eq t tats
106 val j = find_index_eq t tbs
109 then (length tbs + length tats,
110 AList.update Type.could_unify (tt,(tbs,tats@[t])) bds)
111 else (i, bds) else (j, bds))
115 (* Generic decomp for reification : matches the actual term with the
116 rhs of one cong rule. The result of the matching guides the
117 proof synthesis: The matches of the introduced Variables A1 .. An are
118 processed recursively
119 The rest is instantiated in the cong rule,i.e. no reification is needed *)
121 (* da is the decomposition for atoms, ie. it returns ([],g) where g
122 returns the right instance f (AtC n) = t , where AtC is the Atoms
123 constructor and n is the number of the atom corresponding to t *)
124 fun decomp_genreif da cgns (t,ctxt) bds =
126 val thy = ProofContext.theory_of ctxt
127 val cert = cterm_of thy
128 fun tryabsdecomp (s,ctxt) bds =
132 val ([xn],ctxt') = Variable.variant_fixes ["x"] ctxt
133 val (xn,ta) = variant_abs (xn,xT,ta)
135 val bds = (case AList.lookup Type.could_unify bds (HOLogic.listT xT)
136 of NONE => error "tryabsdecomp: Type not found in the Environement"
138 (AList.update Type.could_unify (HOLogic.listT xT, ((x::bsT), atsT)) bds))
141 (hd (Variable.export ctxt' ctxt [(forall_intr (cert x) th) COMP allI]),
142 let val (bsT,asT) = the(AList.lookup Type.could_unify bds (HOLogic.listT xT))
143 in AList.update Type.could_unify (HOLogic.listT xT,(tl bsT,asT)) bds
147 | _ => da (s,ctxt) bds)
149 [] => tryabsdecomp (t,ctxt) bds
150 | ((vns,cong)::congs) => ((let
151 val cert = cterm_of thy
152 val certy = ctyp_of thy
155 ((fst o HOLogic.dest_eq o HOLogic.dest_Trueprop) (concl_of cong), t)
156 (Envir.type_env (Envir.empty 0), Vartab.empty)
157 val (fnvs,invs) = List.partition (fn ((vn,_),_) => vn mem vns) (Vartab.dest tmenv)
159 (map (snd o snd) fnvs,
160 map (fn ((vn,vi),(tT,t)) => (cert(Var ((vn,vi),tT)), cert t)) invs)
161 val ctyenv = map (fn ((vn,vi),(s,ty)) => (certy (TVar((vn,vi),s)), certy ty)) (Vartab.dest tyenv)
162 in ((fts ~~ (replicate (length fts) ctxt),
163 Library.apfst (FWD (instantiate (ctyenv, its) cong))), bds)
165 handle MATCH => decomp_genreif da congs (t,ctxt) bds))
168 (* looks for the atoms equation and instantiates it with the right number *)
169 fun mk_decompatom eqs (t,ctxt) bds = (([], fn (_, bds) =>
171 val tT = fastype_of t
174 val rhs = eq |> prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> snd
176 (fn (n,ty) => n = @{const_name "List.nth"}
178 AList.defined Type.could_unify bds (domain_type ty)) rhs
179 andalso Type.could_unify (fastype_of rhs, tT)
184 Const(@{const_name "List.nth"},_)$vs$n => insert (fn ((a,_),(b,_)) => a aconv b) (t,(vs,n)) acc
185 | t1$t2 => get_nths t1 (get_nths t2 acc)
186 | Abs(_,_,t') => get_nths t' acc
190 tryeqs [] bds = error "Can not find the atoms equation"
191 | tryeqs (eq::eqs) bds = ((
193 val rhs = eq |> prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> snd
194 val nths = get_nths rhs []
195 val (vss,ns) = fold_rev (fn (_,(vs,n)) => fn (vss,ns) =>
196 (insert (op aconv) vs vss, insert (op aconv) n ns)) nths ([],[])
197 val (vsns, ctxt') = Variable.variant_fixes (replicate (length vss) "vs") ctxt
198 val (xns, ctxt'') = Variable.variant_fixes (replicate (length nths) "x") ctxt'
199 val thy = ProofContext.theory_of ctxt''
200 val cert = cterm_of thy
201 val certT = ctyp_of thy
202 val vsns_map = vss ~~ vsns
203 val xns_map = (fst (split_list nths)) ~~ xns
204 val subst = map (fn (nt, xn) => (nt, Var ((xn,0), fastype_of nt))) xns_map
205 val rhs_P = subst_free subst rhs
206 val (tyenv, tmenv) = Pattern.match
208 (Envir.type_env (Envir.empty 0), Vartab.empty)
209 val sbst = Envir.subst_vars (tyenv, tmenv)
210 val sbsT = Envir.typ_subst_TVars tyenv
211 val subst_ty = map (fn (n,(s,t)) => (certT (TVar (n, s)), certT t))
213 val tml = Vartab.dest tmenv
214 val t's = map (fn xn => snd (the (AList.lookup (op =) tml (xn,0)))) xns (* FIXME : Express with sbst*)
215 val (subst_ns, bds) = fold_map
216 (fn (Const _ $ vs $ n, Var (xn0,T)) => fn bds =>
218 val name = snd (the (AList.lookup (op =) tml xn0))
219 val (idx, bds) = index_of name bds
220 in ((cert n, idx |> (HOLogic.mk_nat #> cert)), bds) end) subst bds
223 fun ty (Const _ $ (vs as Var (vsn,lT)) $ n, Var (xn0,T)) = (certT T, certT (sbsT T))
224 fun h (Const _ $ (vs as Var (vsn,lT)) $ n, Var (xn0,T)) =
226 val cns = sbst (Const(@{const_name "List.Cons"}, T --> lT --> lT))
228 val (bsT,asT) = the (AList.lookup Type.could_unify bds lT)
229 val vsn = the (AList.lookup (op =) vsns_map vs)
230 val cvs = cert (fold_rev (fn x => fn xs => cns$x$xs) bsT (Free (vsn, lT')))
231 in (cert vs, cvs) end
233 val cts = map (fn ((vn,vi),(tT,t)) => (cert(Var ((vn,vi),tT)), cert t))
234 (fold (AList.delete (fn (((a: string),_),(b,_)) => a = b))
235 (map (fn n => (n,0)) xns) tml)
237 let val ih = Drule.cterm_rule (Thm.instantiate (subst_ty,[]))
238 in map (fn (v,t) => (ih v, ih t)) (subst_ns@subst_vs@cts) end
239 val th = (instantiate (subst_ty, substt) eq) RS sym
240 in (hd (Variable.export ctxt'' ctxt [th]), bds) end)
241 handle MATCH => tryeqs eqs bds)
242 in tryeqs (filter isat eqs) bds end), bds);
244 (* Generic reification procedure: *)
245 (* creates all needed cong rules and then just uses the theorem synthesis *)
247 fun mk_congs ctxt raw_eqs =
249 val fs = fold_rev (fn eq =>
250 insert (op =) (eq |> prop_of |> HOLogic.dest_Trueprop
251 |> HOLogic.dest_eq |> fst |> strip_comb
253 val tys = fold_rev (fn f => fold (insert (op =)) (f |> fastype_of |> binder_types |> tl)
255 val (vs, ctxt') = Variable.variant_fixes (replicate (length tys) "vs") ctxt
256 val thy = ProofContext.theory_of ctxt'
257 val cert = cterm_of thy
258 val vstys = map (fn (t,v) => (t,SOME (cert (Free(v,t)))))
260 val is_Var = can dest_Var
263 val subst = map (fn (v as Var(n,t)) => (cert v, (the o the) (AList.lookup (op =) vstys t)))
265 in Thm.instantiate ([],subst) eq
268 val bds = AList.make (fn _ => ([],[])) tys
269 val eqs = map (fn eq => eq |> prop_of |> HOLogic.dest_Trueprop
270 |> HOLogic.dest_eq |> fst |> strip_comb |> snd |> tl
271 |> (insteq eq)) raw_eqs
272 val (ps,congs) = split_list (map (mk_congeq ctxt' fs) eqs)
273 in (ps ~~ (Variable.export ctxt' ctxt congs), bds)
276 val (congs, bds) = mk_congs ctxt raw_eqs
277 val congs = rearrange congs
278 val (th, bds) = divide_and_conquer' (decomp_genreif (mk_decompatom raw_eqs) congs) (t,ctxt) bds
279 fun is_listVar (Var (_,t)) = can dest_listT t
280 | is_listVar _ = false
281 val vars = th |> prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> snd
282 |> strip_comb |> snd |> filter is_listVar
283 val cert = cterm_of (ProofContext.theory_of ctxt)
284 val cvs = map (fn (v as Var(n,t)) => (cert v,
285 the (AList.lookup Type.could_unify bds t) |> snd |> HOLogic.mk_list (dest_listT t) |> cert)) vars
286 val th' = instantiate ([], cvs) th
287 val t' = (fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) th'
288 val th'' = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq (t, t')))
289 (fn _ => simp_tac (local_simpset_of ctxt) 1)
290 in FWD trans [th'',th']
294 fun genreflect ctxt conv corr_thms raw_eqs t =
296 val reifth = genreif ctxt raw_eqs t
297 fun trytrans [] = error "No suitable correctness theorem found"
298 | trytrans (th::ths) =
299 (FWD trans [reifth, th RS sym] handle THM _ => trytrans ths)
300 val th = trytrans corr_thms
301 val ft = (Thm.dest_arg1 o Thm.dest_arg o Thm.dest_arg o cprop_of) th
303 in simplify (HOL_basic_ss addsimps raw_eqs addsimps [nth_Cons_0, nth_Cons_Suc])
304 (simplify (HOL_basic_ss addsimps [rth]) th)
307 fun genreify_tac ctxt eqs to i = (fn st =>
309 fun P () = HOLogic.dest_Trueprop (nth (prems_of st) (i - 1))
310 val t = (case to of NONE => P () | SOME x => x)
311 val th = (genreif ctxt eqs t) RS ssubst
315 (* Reflection calls reification and uses the correctness *)
316 (* theorem assumed to be the dead of the list *)
317 fun gen_reflection_tac ctxt conv corr_thms raw_eqs to i = (fn st =>
319 val P = HOLogic.dest_Trueprop (nth (prems_of st) (i - 1));
320 val t = the_default P to;
321 val th = genreflect ctxt conv corr_thms raw_eqs t
323 in (rtac th i THEN TRY(rtac TrueI i)) st end);
325 fun reflection_tac ctxt = gen_reflection_tac ctxt Codegen.evaluation_conv;
326 (*FIXME why Codegen.evaluation_conv? very specific...*)