formal/hol/Boyer_Moore/induction.ml

(******************************************************************************)
(* FILE          : induction.ml                                               *)
(* DESCRIPTION   : Induction.                                                 *)
(*                                                                            *)
(* READS FILES   : <none>                                                     *)
(* WRITES FILES  : <none>                                                     *)
(*                                                                            *)
(* AUTHOR        : R.J.Boulton                                                *)
(* DATE          : 26th June 1991                                             *)
(*                                                                            *)
(* LAST MODIFIED : P. Papapanagiotou (University of Edinburgh)                *)
(* DATE          : 2008                                                       *)
(******************************************************************************)

let (CONV_OF_RCONV: conv -> conv) =
  let rec get_bv tm =
    if is_abs tm then bndvar tm
    else if is_comb tm then
         try get_bv (rand tm) with Failure _ -> get_bv (rator tm)
    else failwith "" in
  fun conv tm ->
  let v = get_bv tm in
  let th1 = conv tm in
  let th2 = ONCE_DEPTH_CONV (GEN_ALPHA_CONV v) (rhs(concl th1)) in
  TRANS th1 th2;;

let (CONV_OF_THM: thm -> conv) =
  CONV_OF_RCONV o REWR_CONV;;

let RIGHT_IMP_FORALL_CONV = CONV_OF_THM RIGHT_IMP_FORALL_THM;;
(* Does this work?? *)

(*----------------------------------------------------------------------------*)
(* is_rec_const_app : term -> bool                                            *)
(*                                                                            *)
(* This function returns true if the term it is given is an application of a  *)
(* currently known recursive function constant.                               *)
(*----------------------------------------------------------------------------*)

let is_rec_const_app tm =
try (let (f,args) = strip_comb tm
  in  let (n,defs) = (get_def o fst o dest_const) f
  in  (n > 0) &&
      ((length o snd o strip_comb o lhs o concl o snd o hd) defs = length args)
 ) with Failure _ -> false;;


(*----------------------------------------------------------------------------*)
(* possible_inductions : term -> (term list # term list)                      *)
(*                                                                            *)
(* Function to compute two lists of variables on which induction could be     *)
(* performed. The first list of variables for which the induction is unflawed *)
(* and the second is of variables for which the induction is flawed.          *)
(*                                                                            *)
(* From a list of applications of recursive functions, the arguments are      *)
(* split into those that are in a recursive argument position and those that  *)
(* are not. Possible inductions are on the variables in the recursive         *)
(* argument positions, but if the variable also appears in a non-recursive    *)
(* argument position then the induction is flawed.                            *)
(*----------------------------------------------------------------------------*)

let possible_inductions tm =
   let apps = find_bm_terms is_rec_const_app tm
   in  let (rec_args,other_args) =
          List.split (map (fun app -> let (f,args) = strip_comb app
                            in  let name = fst (dest_const f)
                            in  let n = (fst o get_def) name
                            in  remove_el n args) apps)
   in  let vars = setify (filter is_var rec_args)
   in  let others = setify (flat other_args)
   in  partition (fun v -> not (mem v others)) vars;;

(*----------------------------------------------------------------------------*)
(* DEPTH_FORALL_CONV : conv -> conv                                           *)
(*                                                                            *)
(* Given a term of the form "!x1 ... xn. t", this function applies the        *)
(* argument conversion to "t".                                                *)
(*----------------------------------------------------------------------------*)

let rec DEPTH_FORALL_CONV conv tm =
   if (is_forall tm)
   then RAND_CONV (ABS_CONV (DEPTH_FORALL_CONV conv)) tm
   else conv tm;;

(*----------------------------------------------------------------------------*)
(* induction_heuristic : (term # bool) -> ((term # bool) list # proof)        *)
(*                                                                            *)
(* Heuristic for induction. It performs one of the possible unflawed          *)
(* inductions on a clause, or failing that, one of the flawed inductions.     *)
(* The heuristic fails if no inductions are possible.                         *)
(*                                                                            *)
(* Having obtained a variable on which to perform induction, the function     *)
(* computes the name of the top-level type constructor in the type of the     *)
(* variable. The appropriate induction theorem is then obtained from the      *)
(* shell environment. The theorem is specialised for the argument clause and  *)
(* beta-reduction is performed at the appropriate places.                     *)
(*                                                                            *)
(* The resulting theorem will be of the form:                                 *)
(*                                                                            *)
(*    |- (case1 /\ ... /\ casen) ==> (!x. f[x])                         ( * ) *)
(*                                                                            *)
(* So, if we can establish casei for each i, we shall have |- !x. f[x]. When  *)
(* specialised with the induction variable, this theorem has the original     *)
(* clause as its conclusion. Each casei is of one of these forms:             *)
(*                                                                            *)
(*    !x1 ... xn. s ==> (!y1 ... ym. t)                                       *)
(*    !x1 ... xn. t                                                           *)
(*                                                                            *)
(* where the yi's do not appear in s. We simplify the casei's that have the   *)
(* first form by proving theorems like:                                       *)
(*                                                                            *)
(*    |- (!x1 ... xn. s ==> (!y1 ... ym. t)) =                                *)
(*       (!x1 ... xn y1 ... ym. s ==> t)                                      *)
(*                                                                            *)
(* For consistency, theorems of the form |- (!x1 ... xn. t) = (!x1 ... xn. t) *)
(* are proved for the casei's that have the second form. The bodies of the    *)
(* right-hand sides of these equations are returned as the new goal clauses.  *)
(* A body that is an implication is taken to be an inductive step and so is   *)
(* returned paired with true. Bodies that are not implications are paired     *)
(* with false.                                                                *)
(*                                                                            *)
(* The proof of the original clause from these new clauses proceeds as        *)
(* follows. The universal quantifications that were stripped from the         *)
(* right-hand sides are restored by generalizing the theorems. From the       *)
(* equations we can then obtain theorems for the left-hand sides. These are   *)
(* conjoined and used to satisfy the antecedant of the theorem ( * ). As      *)
(* described above, specialising the resulting theorem gives a theorem for    *)
(* the original clause.                                                       *)
(*----------------------------------------------------------------------------*)

let induction_heuristic (tm,(ind:bool)) =
try
 (let (unflawed,flawed) = possible_inductions tm
  in  let var = try (hd unflawed) with Failure _ -> (hd flawed)
  in  let ty_name = fst (dest_type (type_of var))
  in  let induct_thm = (sys_shell_info ty_name).induct
  in  let P = mk_abs (var,tm)
  in  let th1 = ISPEC P induct_thm
  in  let th2 =
         CONV_RULE
            (ONCE_DEPTH_CONV
                (fun tm -> if (rator tm = P) then BETA_CONV tm else failwith "")) th1
  in  let new_goals = conj_list (rand (rator (concl th2)))
  in  let ths =
         map (REPEATC (DEPTH_FORALL_CONV RIGHT_IMP_FORALL_CONV)) new_goals
  in  let (varsl,tml) = List.split (map (strip_forall o rhs o concl) ths)
  in  let proof thl =
         let thl' = map (uncurry GENL) (lcombinep (varsl,thl))
         in  let thl'' = map (fun (eq,th) -> EQ_MP (SYM eq) th) (lcombinep (ths,thl'))
         in  SPEC var (MP th2 (LIST_CONJ thl''))
  in  (map (fun tm -> (tm,((is_imp tm) && (not (is_neg tm))))) tml,
       apply_fproof "induction_heuristic" proof tml)
 ) with Failure _ -> failwith "induction_heuristic";;