src/HOL/Library/reflection.ML
 author huffman Fri Mar 30 12:32:35 2012 +0200 (2012-03-30) changeset 47220 52426c62b5d0 parent 46763 aa9f5c3bcd4c child 51717 9e7d1c139569 permissions -rw-r--r--
replace lemmas eval_nat_numeral with a simpler reformulation
```     1 (*  Title:      HOL/Library/reflection.ML
```
```     2     Author:     Amine Chaieb, TU Muenchen
```
```     3
```
```     4 A trial for automatical reification.
```
```     5 *)
```
```     6
```
```     7 signature REFLECTION =
```
```     8 sig
```
```     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
```
```    13   val genreif : Proof.context -> thm list -> term -> thm
```
```    14 end;
```
```    15
```
```    16 structure Reflection : REFLECTION =
```
```    17 struct
```
```    18
```
```    19   (* Make a congruence rule out of a defining equation for the interpretation *)
```
```    20   (* th is one defining equation of f, i.e.
```
```    21      th is "f (Cp ?t1 ... ?tn) = P(f ?t1, .., f ?tn)" *)
```
```    22   (* Cp is a constructor pattern and P is a pattern *)
```
```    23
```
```    24   (* The result is:
```
```    25       [|?A1 = f ?t1 ; .. ; ?An= f ?tn |] ==> P (?A1, .., ?An) = f (Cp ?t1 .. ?tn) *)
```
```    26   (*  + the a list of names of the A1 .. An, Those are fresh in the ctxt*)
```
```    27
```
```    28 fun mk_congeq ctxt fs th =
```
```    29   let
```
```    30    val Const (fN, _) = th |> prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq
```
```    31      |> fst |> strip_comb |> fst
```
```    32    val thy = Proof_Context.theory_of ctxt
```
```    33    val cert = Thm.cterm_of thy
```
```    34    val (((_,_),[th']), ctxt') = Variable.import true [th] ctxt
```
```    35    val (lhs, rhs) = HOLogic.dest_eq (HOLogic.dest_Trueprop (Thm.prop_of th'))
```
```    36    fun add_fterms (t as t1 \$ t2) =
```
```    37        if exists (fn f => Term.could_unify (t |> strip_comb |> fst, f)) fs then insert (op aconv) t
```
```    38        else add_fterms t1 #> add_fterms t2
```
```    39      | add_fterms (t as Abs _) =
```
```    40        if exists_Const (fn (c, _) => c = fN) t then (fn _ => [t]) else (fn _ => [])
```
```    41      | add_fterms _ = I
```
```    42    val fterms = add_fterms rhs []
```
```    43    val (xs, ctxt'') = Variable.variant_fixes (replicate (length fterms) "x") ctxt'
```
```    44    val tys = map fastype_of fterms
```
```    45    val vs = map Free (xs ~~ tys)
```
```    46    val env = fterms ~~ vs
```
```    47                     (* FIXME!!!!*)
```
```    48    fun replace_fterms (t as t1 \$ t2) =
```
```    49        (case AList.lookup (op aconv) env t of
```
```    50             SOME v => v
```
```    51           | NONE => replace_fterms t1 \$ replace_fterms t2)
```
```    52      | replace_fterms t = (case AList.lookup (op aconv) env t of
```
```    53                                SOME v => v
```
```    54                              | NONE => t)
```
```    55
```
```    56    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)))
```
```    57      | mk_def (t, v) = HOLogic.mk_Trueprop (HOLogic.mk_eq (v, t))
```
```    58    fun tryext x = (x RS @{lemma "(\<forall>x. f x = g x) \<Longrightarrow> f = g" by blast} handle THM _ =>  x)
```
```    59    val cong =
```
```    60     (Goal.prove ctxt'' [] (map mk_def env)
```
```    61       (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, replace_fterms rhs)))
```
```    62       (fn {context, prems, ...} =>
```
```    63         Local_Defs.unfold_tac context (map tryext prems) THEN rtac th' 1)) RS sym
```
```    64
```
```    65    val (cong' :: vars') =
```
```    66        Variable.export ctxt'' ctxt (cong :: map (Drule.mk_term o cert) vs)
```
```    67    val vs' = map (fst o fst o Term.dest_Var o Thm.term_of o Drule.dest_term) vars'
```
```    68
```
```    69   in  (vs', cong') end;
```
```    70  (* congs is a list of pairs (P,th) where th is a theorem for *)
```
```    71         (* [| f p1 = A1; ...; f pn = An|] ==> f (C p1 .. pn) = P *)
```
```    72 val FWD = curry (op OF);
```
```    73
```
```    74
```
```    75 exception REIF of string;
```
```    76
```
```    77 fun dest_listT (Type (@{type_name "list"}, [T])) = T;
```
```    78
```
```    79 fun rearrange congs =
```
```    80   let
```
```    81     fun P (_, th) =
```
```    82       let val @{term "Trueprop"}\$(Const (@{const_name HOL.eq},_) \$l\$_) = concl_of th
```
```    83       in can dest_Var l end
```
```    84     val (yes,no) = List.partition P congs
```
```    85   in no @ yes end
```
```    86
```
```    87 fun genreif ctxt raw_eqs t =
```
```    88   let
```
```    89     fun index_of t bds =
```
```    90       let
```
```    91         val tt = HOLogic.listT (fastype_of t)
```
```    92       in
```
```    93        (case AList.lookup Type.could_unify bds tt of
```
```    94           NONE => error "index_of : type not found in environements!"
```
```    95         | SOME (tbs,tats) =>
```
```    96           let
```
```    97             val i = find_index (fn t' => t' = t) tats
```
```    98             val j = find_index (fn t' => t' = t) tbs
```
```    99           in (if j = ~1 then
```
```   100               if i = ~1
```
```   101               then (length tbs + length tats,
```
```   102                     AList.update Type.could_unify (tt,(tbs,tats@[t])) bds)
```
```   103               else (i, bds) else (j, bds))
```
```   104           end)
```
```   105       end;
```
```   106
```
```   107     (* Generic decomp for reification : matches the actual term with the
```
```   108        rhs of one cong rule. The result of the matching guides the
```
```   109        proof synthesis: The matches of the introduced Variables A1 .. An are
```
```   110        processed recursively
```
```   111        The rest is instantiated in the cong rule,i.e. no reification is needed *)
```
```   112
```
```   113     (* da is the decomposition for atoms, ie. it returns ([],g) where g
```
```   114        returns the right instance f (AtC n) = t , where AtC is the Atoms
```
```   115        constructor and n is the number of the atom corresponding to t *)
```
```   116     fun decomp_genreif da cgns (t,ctxt) bds =
```
```   117       let
```
```   118         val thy = Proof_Context.theory_of ctxt
```
```   119         val cert = cterm_of thy
```
```   120         fun tryabsdecomp (s,ctxt) bds =
```
```   121           (case s of
```
```   122              Abs(_, xT, ta) => (
```
```   123                let
```
```   124                  val ([xn],ctxt') = Variable.variant_fixes ["x"] ctxt
```
```   125                  val (xn,ta) = Syntax_Trans.variant_abs (xn,xT,ta)  (* FIXME !? *)
```
```   126                  val x = Free(xn,xT)
```
```   127                  val bds = (case AList.lookup Type.could_unify bds (HOLogic.listT xT)
```
```   128                           of NONE => error "tryabsdecomp: Type not found in the Environement"
```
```   129                            | SOME (bsT,atsT) =>
```
```   130                              (AList.update Type.could_unify (HOLogic.listT xT, ((x::bsT), atsT)) bds))
```
```   131                in (([(ta, ctxt')],
```
```   132                     fn ([th], bds) =>
```
```   133                       (hd (Variable.export ctxt' ctxt [(Thm.forall_intr (cert x) th) COMP allI]),
```
```   134                        let val (bsT,asT) = the(AList.lookup Type.could_unify bds (HOLogic.listT xT))
```
```   135                        in AList.update Type.could_unify (HOLogic.listT xT,(tl bsT,asT)) bds
```
```   136                        end)),
```
```   137                    bds)
```
```   138                end)
```
```   139            | _ => da (s,ctxt) bds)
```
```   140       in
```
```   141         (case cgns of
```
```   142           [] => tryabsdecomp (t,ctxt) bds
```
```   143         | ((vns,cong)::congs) =>
```
```   144             (let
```
```   145               val cert = cterm_of thy
```
```   146               val certy = ctyp_of thy
```
```   147               val (tyenv, tmenv) =
```
```   148                 Pattern.match thy
```
```   149                   ((fst o HOLogic.dest_eq o HOLogic.dest_Trueprop) (concl_of cong), t)
```
```   150                   (Vartab.empty, Vartab.empty)
```
```   151               val (fnvs,invs) = List.partition (fn ((vn,_),_) => member (op =) vns vn) (Vartab.dest tmenv)
```
```   152               val (fts,its) =
```
```   153                 (map (snd o snd) fnvs,
```
```   154                  map (fn ((vn,vi),(tT,t)) => (cert(Var ((vn,vi),tT)), cert t)) invs)
```
```   155               val ctyenv = map (fn ((vn,vi),(s,ty)) => (certy (TVar((vn,vi),s)), certy ty)) (Vartab.dest tyenv)
```
```   156             in ((fts ~~ (replicate (length fts) ctxt),
```
```   157                  Library.apfst (FWD (Drule.instantiate_normalize (ctyenv, its) cong))), bds)
```
```   158             end handle Pattern.MATCH => decomp_genreif da congs (t,ctxt) bds))
```
```   159       end;
```
```   160
```
```   161  (* looks for the atoms equation and instantiates it with the right number *)
```
```   162     fun mk_decompatom eqs (t,ctxt) bds = (([], fn (_, bds) =>
```
```   163       let
```
```   164         val tT = fastype_of t
```
```   165         fun isat eq =
```
```   166           let
```
```   167             val rhs = eq |> prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> snd
```
```   168           in exists_Const
```
```   169             (fn (n,ty) => n = @{const_name "List.nth"}
```
```   170                           andalso
```
```   171                           AList.defined Type.could_unify bds (domain_type ty)) rhs
```
```   172             andalso Type.could_unify (fastype_of rhs, tT)
```
```   173           end
```
```   174
```
```   175         fun get_nths t acc =
```
```   176           case t of
```
```   177             Const(@{const_name "List.nth"},_)\$vs\$n => insert (fn ((a,_),(b,_)) => a aconv b) (t,(vs,n)) acc
```
```   178           | t1\$t2 => get_nths t1 (get_nths t2 acc)
```
```   179           | Abs(_,_,t') => get_nths t'  acc
```
```   180           | _ => acc
```
```   181
```
```   182         fun
```
```   183            tryeqs [] bds = error "Can not find the atoms equation"
```
```   184          | tryeqs (eq::eqs) bds = ((
```
```   185           let
```
```   186             val rhs = eq |> prop_of |> HOLogic.dest_Trueprop  |> HOLogic.dest_eq |> snd
```
```   187             val nths = get_nths rhs []
```
```   188             val (vss,_ ) = fold_rev (fn (_, (vs, n)) => fn (vss, ns) =>
```
```   189               (insert (op aconv) vs vss, insert (op aconv) n ns)) nths ([], [])
```
```   190             val (vsns, ctxt') = Variable.variant_fixes (replicate (length vss) "vs") ctxt
```
```   191             val (xns, ctxt'') = Variable.variant_fixes (replicate (length nths) "x") ctxt'
```
```   192             val thy = Proof_Context.theory_of ctxt''
```
```   193             val cert = cterm_of thy
```
```   194             val certT = ctyp_of thy
```
```   195             val vsns_map = vss ~~ vsns
```
```   196             val xns_map = (fst (split_list nths)) ~~ xns
```
```   197             val subst = map (fn (nt, xn) => (nt, Var ((xn,0), fastype_of nt))) xns_map
```
```   198             val rhs_P = subst_free subst rhs
```
```   199             val (tyenv, tmenv) = Pattern.match thy (rhs_P, t) (Vartab.empty, Vartab.empty)
```
```   200             val sbst = Envir.subst_term (tyenv, tmenv)
```
```   201             val sbsT = Envir.subst_type tyenv
```
```   202             val subst_ty = map (fn (n,(s,t)) => (certT (TVar (n, s)), certT t))
```
```   203                                (Vartab.dest tyenv)
```
```   204             val tml = Vartab.dest tmenv
```
```   205             val (subst_ns, bds) = fold_map
```
```   206                 (fn (Const _ \$ _ \$ n, Var (xn0, _)) => fn bds =>
```
```   207                   let
```
```   208                     val name = snd (the (AList.lookup (op =) tml xn0))
```
```   209                     val (idx, bds) = index_of name bds
```
```   210                   in ((cert n, idx |> (HOLogic.mk_nat #> cert)), bds) end) subst bds
```
```   211             val subst_vs =
```
```   212               let
```
```   213                 fun h (Const _ \$ (vs as Var (_, lT)) \$ _, Var (_, T)) =
```
```   214                   let
```
```   215                     val cns = sbst (Const(@{const_name "List.Cons"}, T --> lT --> lT))
```
```   216                     val lT' = sbsT lT
```
```   217                     val (bsT, _) = the (AList.lookup Type.could_unify bds lT)
```
```   218                     val vsn = the (AList.lookup (op =) vsns_map vs)
```
```   219                     val cvs = cert (fold_rev (fn x => fn xs => cns\$x\$xs) bsT (Free (vsn, lT')))
```
```   220                   in (cert vs, cvs) end
```
```   221               in map h subst end
```
```   222             val cts = map (fn ((vn,vi),(tT,t)) => (cert(Var ((vn,vi),tT)), cert t))
```
```   223                           (fold (AList.delete (fn (((a: string),_),(b,_)) => a = b))
```
```   224                                 (map (fn n => (n,0)) xns) tml)
```
```   225             val substt =
```
```   226               let val ih = Drule.cterm_rule (Thm.instantiate (subst_ty,[]))
```
```   227               in map (fn (v,t) => (ih v, ih t)) (subst_ns@subst_vs@cts)  end
```
```   228             val th = (Drule.instantiate_normalize (subst_ty, substt)  eq) RS sym
```
```   229           in (hd (Variable.export ctxt'' ctxt [th]), bds) end)
```
```   230           handle Pattern.MATCH => tryeqs eqs bds)
```
```   231       in tryeqs (filter isat eqs) bds end), bds);
```
```   232
```
```   233   (* Generic reification procedure: *)
```
```   234   (* creates all needed cong rules and then just uses the theorem synthesis *)
```
```   235
```
```   236     fun mk_congs ctxt raw_eqs =
```
```   237       let
```
```   238         val fs = fold_rev (fn eq =>
```
```   239                            insert (op =) (eq |> prop_of |> HOLogic.dest_Trueprop
```
```   240                            |> HOLogic.dest_eq |> fst |> strip_comb
```
```   241                            |> fst)) raw_eqs []
```
```   242         val tys = fold_rev (fn f => fold (insert (op =)) (f |> fastype_of |> binder_types |> tl)
```
```   243                             ) fs []
```
```   244         val (vs, ctxt') = Variable.variant_fixes (replicate (length tys) "vs") ctxt
```
```   245         val thy = Proof_Context.theory_of ctxt'
```
```   246         val cert = cterm_of thy
```
```   247         val vstys = map (fn (t,v) => (t,SOME (cert (Free(v,t)))))
```
```   248                     (tys ~~ vs)
```
```   249         val is_Var = can dest_Var
```
```   250         fun insteq eq vs =
```
```   251           let
```
```   252             val subst = map (fn (v as Var(_, t)) => (cert v, (the o the) (AList.lookup (op =) vstys t)))
```
```   253                         (filter is_Var vs)
```
```   254           in Thm.instantiate ([],subst) eq
```
```   255           end
```
```   256
```
```   257         val bds = AList.make (fn _ => ([],[])) tys
```
```   258         val eqs = map (fn eq => eq |> prop_of |> HOLogic.dest_Trueprop
```
```   259                                    |> HOLogic.dest_eq |> fst |> strip_comb |> snd |> tl
```
```   260                                    |> (insteq eq)) raw_eqs
```
```   261         val (ps,congs) = split_list (map (mk_congeq ctxt' fs) eqs)
```
```   262       in (ps ~~ (Variable.export ctxt' ctxt congs), bds)
```
```   263       end
```
```   264
```
```   265     val (congs, bds) = mk_congs ctxt raw_eqs
```
```   266     val congs = rearrange congs
```
```   267     val (th, bds) = divide_and_conquer' (decomp_genreif (mk_decompatom raw_eqs) congs) (t,ctxt) bds
```
```   268     fun is_listVar (Var (_,t)) = can dest_listT t
```
```   269          | is_listVar _ = false
```
```   270     val vars = th |> prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> snd
```
```   271                   |> strip_comb |> snd |> filter is_listVar
```
```   272     val cert = cterm_of (Proof_Context.theory_of ctxt)
```
```   273     val cvs = map (fn (v as Var(_, t)) => (cert v,
```
```   274                   the (AList.lookup Type.could_unify bds t) |> snd |> HOLogic.mk_list (dest_listT t) |> cert)) vars
```
```   275     val th' = Drule.instantiate_normalize ([], cvs) th
```
```   276     val t' = (fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) th'
```
```   277     val th'' = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq (t, t')))
```
```   278                (fn _ => simp_tac (simpset_of ctxt) 1)
```
```   279   in FWD trans [th'',th']
```
```   280   end
```
```   281
```
```   282
```
```   283 fun genreflect ctxt conv corr_thms raw_eqs t =
```
```   284   let
```
```   285     val reifth = genreif ctxt raw_eqs t
```
```   286     fun trytrans [] = error "No suitable correctness theorem found"
```
```   287       | trytrans (th::ths) =
```
```   288            (FWD trans [reifth, th RS sym] handle THM _ => trytrans ths)
```
```   289     val th = trytrans corr_thms
```
```   290     val ft = (Thm.dest_arg1 o Thm.dest_arg o Thm.dest_arg o cprop_of) th
```
```   291     val rth = conv ft
```
```   292   in
```
```   293     simplify (HOL_basic_ss addsimps raw_eqs addsimps @{thms nth_Cons_0 nth_Cons_Suc})
```
```   294              (simplify (HOL_basic_ss addsimps [rth]) th)
```
```   295   end
```
```   296
```
```   297 fun genreify_tac ctxt eqs to = SUBGOAL (fn (goal, i) =>
```
```   298   let
```
```   299     val t = (case to of NONE => HOLogic.dest_Trueprop goal | SOME x => x)
```
```   300     val th = genreif ctxt eqs t RS ssubst
```
```   301   in rtac th i end);
```
```   302
```
```   303     (* Reflection calls reification and uses the correctness *)
```
```   304         (* theorem assumed to be the head of the list *)
```
```   305 fun gen_reflection_tac ctxt conv corr_thms raw_eqs to = SUBGOAL (fn (goal, i) =>
```
```   306   let
```
```   307     val t = (case to of NONE => HOLogic.dest_Trueprop goal | SOME x => x)
```
```   308     val th = genreflect ctxt conv corr_thms raw_eqs t RS ssubst
```
```   309   in rtac th i THEN TRY (rtac TrueI i) end);  (* FIXME THEN_ALL_NEW !? *)
```
```   310
```
```   311 fun reflection_tac ctxt = gen_reflection_tac ctxt
```
```   312   (Code_Evaluation.dynamic_conv (Proof_Context.theory_of ctxt));
```
```   313   (*FIXME why Code_Evaluation.dynamic_conv?  very specific...*)
```
```   314
```
```   315 end
```