src/HOL/Tools/Qelim/langford.ML

-(*  Title:      HOL/Tools/Qelim/langford.ML
-    Author:     Amine Chaieb, TU Muenchen
-*)
-
-signature LANGFORD_QE = 
-sig
-  val dlo_tac : Proof.context -> int -> tactic
-  val dlo_conv : Proof.context -> cterm -> thm
-end
-
-structure LangfordQE :LANGFORD_QE = 
-struct
-
-val dest_set =
- let 
-  fun h acc ct = 
-   case term_of ct of
-     Const (@{const_name Orderings.bot}, _) => acc
-   | Const (@{const_name insert}, _) $ _ $ t => h (Thm.dest_arg1 ct :: acc) (Thm.dest_arg ct)
-in h [] end;
-
-fun prove_finite cT u = 
-let val [th0,th1] = map (instantiate' [SOME cT] []) @{thms "finite.intros"}
-    fun ins x th =
-     Thm.implies_elim (instantiate' [] [(SOME o Thm.dest_arg o Thm.dest_arg)
-                                     (Thm.cprop_of th), SOME x] th1) th
-in fold ins u th0 end;
-
-val simp_rule = Conv.fconv_rule (Conv.arg_conv (Simplifier.rewrite (HOL_basic_ss addsimps (@{thms "ball_simps"} @ simp_thms))));
-
-fun basic_dloqe ctxt stupid dlo_qeth dlo_qeth_nolb dlo_qeth_noub gather ep = 
- case term_of ep of 
-  Const("Ex",_)$_ => 
-   let 
-     val p = Thm.dest_arg ep
-     val ths = simplify (HOL_basic_ss addsimps gather) (instantiate' [] [SOME p] stupid)
-     val (L,U) = 
-       let 
-         val (x,q) = Thm.dest_abs NONE (Thm.dest_arg (Thm.rhs_of ths))
-       in (Thm.dest_arg1 q |> Thm.dest_arg1, Thm.dest_arg q |> Thm.dest_arg1)
-       end
-     fun proveneF S =         
-       let val (a,A) = Thm.dest_comb S |>> Thm.dest_arg
-           val cT = ctyp_of_term a
-           val ne = instantiate' [SOME cT] [SOME a, SOME A] 
-                    @{thm insert_not_empty}
-           val f = prove_finite cT (dest_set S)
-       in (ne, f) end
-
-     val qe = case (term_of L, term_of U) of 
-      (Const (@{const_name Orderings.bot}, _),_) =>  
-        let
-          val (neU,fU) = proveneF U 
-        in simp_rule (Thm.transitive ths (dlo_qeth_nolb OF [neU, fU])) end
-    | (_,Const (@{const_name Orderings.bot}, _)) =>  
-        let
-          val (neL,fL) = proveneF L
-        in simp_rule (Thm.transitive ths (dlo_qeth_noub OF [neL, fL])) end
-
-    | (_,_) =>  
-      let 
-       val (neL,fL) = proveneF L
-       val (neU,fU) = proveneF U
-      in simp_rule (Thm.transitive ths (dlo_qeth OF [neL, neU, fL, fU])) 
-      end
-   in qe end 
- | _ => error "dlo_qe : Not an existential formula";
-
-val all_conjuncts = 
- let fun h acc ct = 
-  case term_of ct of
-   @{term "op &"}$_$_ => h (h acc (Thm.dest_arg ct)) (Thm.dest_arg1 ct)
-  | _ => ct::acc
-in h [] end;
-
-fun conjuncts ct =
- case term_of ct of
-  @{term "op &"}$_$_ => (Thm.dest_arg1 ct)::(conjuncts (Thm.dest_arg ct))
-| _ => [ct];
-
-fun fold1 f = foldr1 (uncurry f);
-
-val list_conj = fold1 (fn c => fn c' => Thm.capply (Thm.capply @{cterm "op &"} c) c') ;
-
-fun mk_conj_tab th = 
- let fun h acc th = 
-   case prop_of th of
-   @{term "Trueprop"}$(@{term "op &"}$p$q) => 
-     h (h acc (th RS conjunct2)) (th RS conjunct1)
-  | @{term "Trueprop"}$p => (p,th)::acc
-in fold (Termtab.insert Thm.eq_thm) (h [] th) Termtab.empty end;
-
-fun is_conj (@{term "op &"}$_$_) = true
-  | is_conj _ = false;
-
-fun prove_conj tab cjs = 
- case cjs of 
-   [c] => if is_conj (term_of c) then prove_conj tab (conjuncts c) else tab c
- | c::cs => conjI OF [prove_conj tab [c], prove_conj tab cs];
-
-fun conj_aci_rule eq = 
- let 
-  val (l,r) = Thm.dest_equals eq
-  fun tabl c = the (Termtab.lookup (mk_conj_tab (Thm.assume l)) (term_of c))
-  fun tabr c = the (Termtab.lookup (mk_conj_tab (Thm.assume r)) (term_of c))
-  val ll = Thm.dest_arg l
-  val rr = Thm.dest_arg r
-  
-  val thl  = prove_conj tabl (conjuncts rr) 
-                |> Drule.implies_intr_hyps
-  val thr  = prove_conj tabr (conjuncts ll) 
-                |> Drule.implies_intr_hyps
-  val eqI = instantiate' [] [SOME ll, SOME rr] @{thm iffI}
- in Thm.implies_elim (Thm.implies_elim eqI thl) thr |> mk_meta_eq end;
-
-fun contains x ct = member (op aconv) (OldTerm.term_frees (term_of ct)) (term_of x);
-
-fun is_eqx x eq = case term_of eq of
-   Const("op =",_)$l$r => l aconv term_of x orelse r aconv term_of x
- | _ => false ;
-
-local 
-fun proc ct = 
- case term_of ct of
-  Const("Ex",_)$Abs (xn,_,_) => 
-   let 
-    val e = Thm.dest_fun ct
-    val (x,p) = Thm.dest_abs (SOME xn) (Thm.dest_arg ct)
-    val Pp = Thm.capply @{cterm "Trueprop"} p 
-    val (eqs,neqs) = List.partition (is_eqx x) (all_conjuncts p)
-   in case eqs of
-      [] => 
-        let 
-         val (dx,ndx) = List.partition (contains x) neqs
-         in case ndx of [] => NONE
-                      | _ =>
-            conj_aci_rule (Thm.mk_binop @{cterm "op == :: prop => _"} Pp 
-                 (Thm.capply @{cterm Trueprop} (list_conj (ndx @dx))))
-           |> Thm.abstract_rule xn x |> Drule.arg_cong_rule e 
-           |> Conv.fconv_rule (Conv.arg_conv 
-                   (Simplifier.rewrite 
-                      (HOL_basic_ss addsimps (simp_thms @ ex_simps))))
-           |> SOME
-          end
-    | _ => conj_aci_rule (Thm.mk_binop @{cterm "op == :: prop => _"} Pp 
-                 (Thm.capply @{cterm Trueprop} (list_conj (eqs@neqs))))
-           |> Thm.abstract_rule xn x |> Drule.arg_cong_rule e 
-           |> Conv.fconv_rule (Conv.arg_conv 
-                   (Simplifier.rewrite 
-                (HOL_basic_ss addsimps (simp_thms @ ex_simps)))) 
-           |> SOME
-   end
- | _ => NONE;
-in val reduce_ex_simproc = 
-  Simplifier.make_simproc 
-  {lhss = [@{cpat "EX x. ?P x"}] , name = "reduce_ex_simproc",
-   proc = K (K proc) , identifier = []}
-end;
-
-fun raw_dlo_conv dlo_ss 
-          ({qe_bnds, qe_nolb, qe_noub, gst, gs, atoms}:Langford_Data.entry) = 
- let 
-  val ss = dlo_ss addsimps @{thms "dnf_simps"} addsimprocs [reduce_ex_simproc]
-  val dnfex_conv = Simplifier.rewrite ss
-   val pcv = Simplifier.rewrite
-               (dlo_ss addsimps (simp_thms @ ex_simps @ all_simps)
-                @ [@{thm "all_not_ex"}, not_all, ex_disj_distrib])
- in fn p => 
-   Qelim.gen_qelim_conv pcv pcv dnfex_conv cons 
-                  (Thm.add_cterm_frees p [])  (K Thm.reflexive) (K Thm.reflexive)
-                  (K (basic_dloqe () gst qe_bnds qe_nolb qe_noub gs)) p
- end;
-
-
-val grab_atom_bop =
- let
-  fun h bounds tm =
-   (case term_of tm of
-     Const ("op =", T) $ _ $ _ =>
-       if domain_type T = HOLogic.boolT then find_args bounds tm
-       else Thm.dest_fun2 tm
-   | Const ("Not", _) $ _ => h bounds (Thm.dest_arg tm)
-   | Const ("All", _) $ _ => find_body bounds (Thm.dest_arg tm)
-   | Const ("all", _) $ _ => find_body bounds (Thm.dest_arg tm)
-   | Const ("Ex", _) $ _ => find_body bounds (Thm.dest_arg tm)
-   | Const ("op &", _) $ _ $ _ => find_args bounds tm
-   | Const ("op |", _) $ _ $ _ => find_args bounds tm
-   | Const ("op -->", _) $ _ $ _ => find_args bounds tm
-   | Const ("==>", _) $ _ $ _ => find_args bounds tm
-   | Const ("==", _) $ _ $ _ => find_args bounds tm
-   | Const ("Trueprop", _) $ _ => h bounds (Thm.dest_arg tm)
-   | _ => Thm.dest_fun2 tm)
-  and find_args bounds tm =
-    (h bounds (Thm.dest_arg tm) handle CTERM _ => h bounds (Thm.dest_arg1 tm))
- and find_body bounds b =
-   let val (_, b') = Thm.dest_abs (SOME (Name.bound bounds)) b
-   in h (bounds + 1) b' end;
-in h end;
-
-fun dlo_instance ctxt tm =
-  (fst (Langford_Data.get ctxt), 
-   Langford_Data.match ctxt (grab_atom_bop 0 tm));
-
-fun dlo_conv ctxt tm =
-  (case dlo_instance ctxt tm of
-    (_, NONE) => raise CTERM ("dlo_conv (langford): no corresponding instance in context!", [tm])
-  | (ss, SOME instance) => raw_dlo_conv ss instance tm);
-
-fun generalize_tac f = CSUBGOAL (fn (p, i) => PRIMITIVE (fn st =>
- let 
-   fun all T = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat "all"}
-   fun gen x t = Thm.capply (all (ctyp_of_term x)) (Thm.cabs x t)
-   val ts = sort (fn (a,b) => Term_Ord.fast_term_ord (term_of a, term_of b)) (f p)
-   val p' = fold_rev gen ts p
- in Thm.implies_intr p' (Thm.implies_elim st (fold Thm.forall_elim ts (Thm.assume p'))) end));
-
-
-fun cfrees ats ct =
- let 
-  val ins = insert (op aconvc)
-  fun h acc t = 
-   case (term_of t) of
-    b$_$_ => if member (op aconvc) ats (Thm.dest_fun2 t) 
-                then ins (Thm.dest_arg t) (ins (Thm.dest_arg1 t) acc) 
-                else h (h acc (Thm.dest_arg t)) (Thm.dest_fun t)
-  | _$_ => h (h acc (Thm.dest_arg t)) (Thm.dest_fun t)
-  | Abs(_,_,_) => Thm.dest_abs NONE t ||> h acc |> uncurry (remove (op aconvc))
-  | Free _ => if member (op aconvc) ats t then acc else ins t acc
-  | Var _ => if member (op aconvc) ats t then acc else ins t acc
-  | _ => acc
- in h [] ct end
-
-fun dlo_tac ctxt = CSUBGOAL (fn (p, i) =>
-  (case dlo_instance ctxt p of
-    (ss, NONE) => simp_tac ss i
-  | (ss,  SOME instance) =>
-      Object_Logic.full_atomize_tac i THEN
-      simp_tac ss i
-      THEN (CONVERSION Thm.eta_long_conversion) i
-      THEN (TRY o generalize_tac (cfrees (#atoms instance))) i
-      THEN Object_Logic.full_atomize_tac i
-      THEN CONVERSION (Object_Logic.judgment_conv (raw_dlo_conv ss instance)) i
-      THEN (simp_tac ss i)));  
-end;