TFL/casesplit.ML
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Sign.typ_equiv;
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(* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- *)
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(*  Title:      TFL/casesplit.ML
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    Author:     Lucas Dixon, University of Edinburgh
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                lucas.dixon@ed.ac.uk
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    Date:       17 Aug 2004
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*)
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(* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- *)
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(*  DESCRIPTION:
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    A structure that defines a tactic to program case splits.
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    casesplit_free :
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      string * typ -> int -> thm -> thm Seq.seq
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    casesplit_name :
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      string -> int -> thm -> thm Seq.seq
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    These use the induction theorem associated with the recursive data
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    type to be split.
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    The structure includes a function to try and recursively split a
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    conjecture into a list sub-theorems:
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    splitto : thm list -> thm -> thm
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*)
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(* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- *)
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(* logic-specific *)
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signature CASE_SPLIT_DATA =
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sig
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  val dest_Trueprop : term -> term
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  val mk_Trueprop : term -> term
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  val atomize : thm list
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  val rulify : thm list
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end;
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structure CaseSplitData_HOL : CASE_SPLIT_DATA =
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struct
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val dest_Trueprop = HOLogic.dest_Trueprop;
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val mk_Trueprop = HOLogic.mk_Trueprop;
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val atomize = thms "induct_atomize";
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val rulify = thms "induct_rulify";
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val rulify_fallback = thms "induct_rulify_fallback";
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end;
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signature CASE_SPLIT =
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sig
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  (* failure to find a free to split on *)
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  exception find_split_exp of string
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  (* getting a case split thm from the induction thm *)
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  val case_thm_of_ty : theory -> typ -> thm
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  val cases_thm_of_induct_thm : thm -> thm
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  (* case split tactics *)
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  val casesplit_free :
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      string * typ -> int -> thm -> thm Seq.seq
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  val casesplit_name : string -> int -> thm -> thm Seq.seq
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  (* finding a free var to split *)
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  val find_term_split :
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      term * term -> (string * typ) option
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  val find_thm_split :
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      thm -> int -> thm -> (string * typ) option
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  val find_thms_split :
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      thm list -> int -> thm -> (string * typ) option
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  (* try to recursively split conjectured thm to given list of thms *)
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  val splitto : thm list -> thm -> thm
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  (* for use with the recdef package *)
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  val derive_init_eqs :
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      theory ->
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      (thm * int) list -> term list -> (thm * int) list
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end;
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functor CaseSplitFUN(Data : CASE_SPLIT_DATA) =
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struct
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val rulify_goals = Tactic.rewrite_goals_rule Data.rulify;
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val atomize_goals = Tactic.rewrite_goals_rule Data.atomize;
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(* beta-eta contract the theorem *)
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fun beta_eta_contract thm =
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    let
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      val thm2 = equal_elim (Thm.beta_conversion true (Thm.cprop_of thm)) thm
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      val thm3 = equal_elim (Thm.eta_conversion (Thm.cprop_of thm2)) thm2
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    in thm3 end;
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(* make a casethm from an induction thm *)
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val cases_thm_of_induct_thm =
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     Seq.hd o (ALLGOALS (fn i => REPEAT (etac Drule.thin_rl i)));
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(* get the case_thm (my version) from a type *)
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fun case_thm_of_ty sgn ty  =
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    let
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      val dtypestab = DatatypePackage.get_datatypes sgn;
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      val ty_str = case ty of
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                     Type(ty_str, _) => ty_str
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                   | TFree(s,_)  => error ("Free type: " ^ s)
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                   | TVar((s,i),_) => error ("Free variable: " ^ s)
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      val dt = case Symtab.lookup dtypestab ty_str
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                of SOME dt => dt
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                 | NONE => error ("Not a Datatype: " ^ ty_str)
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    in
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      cases_thm_of_induct_thm (#induction dt)
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    end;
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(*
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 val ty = (snd o hd o map Term.dest_Free o Term.term_frees) t;
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*)
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(* for use when there are no prems to the subgoal *)
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(* does a case split on the given variable *)
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fun mk_casesplit_goal_thm sgn (vstr,ty) gt =
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    let
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      val x = Free(vstr,ty)
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      val abst = Abs(vstr, ty, Term.abstract_over (x, gt));
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      val ctermify = Thm.cterm_of sgn;
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      val ctypify = Thm.ctyp_of sgn;
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      val case_thm = case_thm_of_ty sgn ty;
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      val abs_ct = ctermify abst;
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      val free_ct = ctermify x;
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      val casethm_vars = rev (Term.term_vars (Thm.concl_of case_thm));
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      val casethm_tvars = Term.term_tvars (Thm.concl_of case_thm);
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      val (Pv, Dv, type_insts) =
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          case (Thm.concl_of case_thm) of
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            (_ $ ((Pv as Var(P,Pty)) $ (Dv as Var(D, Dty)))) =>
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            (Pv, Dv,
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             Sign.typ_match sgn (Dty, ty) Vartab.empty)
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          | _ => error "not a valid case thm";
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      val type_cinsts = map (fn (ixn, (S, T)) => (ctypify (TVar (ixn, S)), ctypify T))
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        (Vartab.dest type_insts);
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      val cPv = ctermify (Envir.subst_TVars type_insts Pv);
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      val cDv = ctermify (Envir.subst_TVars type_insts Dv);
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    in
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      (beta_eta_contract
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         (case_thm
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            |> Thm.instantiate (type_cinsts, [])
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            |> Thm.instantiate ([], [(cPv, abs_ct), (cDv, free_ct)])))
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    end;
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(* for use when there are no prems to the subgoal *)
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(* does a case split on the given variable (Free fv) *)
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fun casesplit_free fv i th =
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    let
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      val (subgoalth, exp) = IsaND.fix_alls i th;
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      val subgoalth' = atomize_goals subgoalth;
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      val gt = Data.dest_Trueprop (Logic.get_goal (Thm.prop_of subgoalth') 1);
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      val sgn = Thm.sign_of_thm th;
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      val splitter_thm = mk_casesplit_goal_thm sgn fv gt;
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      val nsplits = Thm.nprems_of splitter_thm;
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      val split_goal_th = splitter_thm RS subgoalth';
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      val rulified_split_goal_th = rulify_goals split_goal_th;
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    in
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      IsaND.export_back exp rulified_split_goal_th
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    end;
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(* for use when there are no prems to the subgoal *)
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(* does a case split on the given variable *)
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fun casesplit_name vstr i th =
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    let
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      val (subgoalth, exp) = IsaND.fix_alls i th;
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      val subgoalth' = atomize_goals subgoalth;
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      val gt = Data.dest_Trueprop (Logic.get_goal (Thm.prop_of subgoalth') 1);
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      val freets = Term.term_frees gt;
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      fun getter x =
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          let val (n,ty) = Term.dest_Free x in
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            (if vstr = n orelse vstr = Syntax.dest_skolem n
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             then SOME (n,ty) else NONE )
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            handle Fail _ => NONE (* dest_skolem *)
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          end;
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      val (n,ty) = case Library.get_first getter freets
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                of SOME (n, ty) => (n, ty)
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                 | _ => error ("no such variable " ^ vstr);
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      val sgn = Thm.sign_of_thm th;
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      val splitter_thm = mk_casesplit_goal_thm sgn (n,ty) gt;
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      val nsplits = Thm.nprems_of splitter_thm;
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      val split_goal_th = splitter_thm RS subgoalth';
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      val rulified_split_goal_th = rulify_goals split_goal_th;
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    in
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      IsaND.export_back exp rulified_split_goal_th
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    end;
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(* small example:
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Goal "P (x :: nat) & (C y --> Q (y :: nat))";
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by (rtac (thm "conjI") 1);
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val th = topthm();
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val i = 2;
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val vstr = "y";
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by (casesplit_name "y" 2);
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val th = topthm();
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val i = 1;
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val th' = casesplit_name "x" i th;
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*)
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(* the find_XXX_split functions are simply doing a lightwieght (I
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think) term matching equivalent to find where to do the next split *)
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(* assuming two twems are identical except for a free in one at a
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subterm, or constant in another, ie assume that one term is a plit of
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another, then gives back the free variable that has been split. *)
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exception find_split_exp of string
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fun find_term_split (Free v, _ $ _) = SOME v
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  | find_term_split (Free v, Const _) = SOME v
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  | find_term_split (Free v, Abs _) = SOME v (* do we really want this case? *)
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  | find_term_split (Free v, Var _) = NONE (* keep searching *)
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  | find_term_split (a $ b, a2 $ b2) =
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    (case find_term_split (a, a2) of
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       NONE => find_term_split (b,b2)
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     | vopt => vopt)
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  | find_term_split (Abs(_,ty,t1), Abs(_,ty2,t2)) =
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    find_term_split (t1, t2)
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  | find_term_split (Const (x,ty), Const(x2,ty2)) =
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    if x = x2 then NONE else (* keep searching *)
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    raise find_split_exp (* stop now *)
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            "Terms are not identical upto a free varaible! (Consts)"
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  | find_term_split (Bound i, Bound j) =
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    if i = j then NONE else (* keep searching *)
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    raise find_split_exp (* stop now *)
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            "Terms are not identical upto a free varaible! (Bound)"
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  | find_term_split (a, b) =
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    raise find_split_exp (* stop now *)
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            "Terms are not identical upto a free varaible! (Other)";
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(* assume that "splitth" is a case split form of subgoal i of "genth",
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then look for a free variable to split, breaking the subgoal closer to
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splitth. *)
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fun find_thm_split splitth i genth =
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    find_term_split (Logic.get_goal (Thm.prop_of genth) i,
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                     Thm.concl_of splitth) handle find_split_exp _ => NONE;
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(* as above but searches "splitths" for a theorem that suggest a case split *)
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fun find_thms_split splitths i genth =
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    Library.get_first (fn sth => find_thm_split sth i genth) splitths;
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(* split the subgoal i of "genth" until we get to a member of
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splitths. Assumes that genth will be a general form of splitths, that
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can be case-split, as needed. Otherwise fails. Note: We assume that
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all of "splitths" are split to the same level, and thus it doesn't
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matter which one we choose to look for the next split. Simply add
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search on splitthms and split variable, to change this.  *)
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(* Note: possible efficiency measure: when a case theorem is no longer
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useful, drop it? *)
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(* Note: This should not be a separate tactic but integrated into the
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case split done during recdef's case analysis, this would avoid us
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having to (re)search for variables to split. *)
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fun splitto splitths genth =
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    let
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      val _ = assert (not (null splitths)) "splitto: no given splitths";
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      val sgn = Thm.sign_of_thm genth;
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      (* check if we are a member of splitths - FIXME: quicker and
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      more flexible with discrim net. *)
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      fun solve_by_splitth th split =
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          Thm.biresolution false [(false,split)] 1 th;
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      fun split th =
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          (case find_thms_split splitths 1 th of
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             NONE =>
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             (writeln "th:";
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              Display.print_thm th; writeln "split ths:";
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              Display.print_thms splitths; writeln "\n--";
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              error "splitto: cannot find variable to split on")
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            | SOME v =>
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             let
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               val gt = Data.dest_Trueprop (List.nth(Thm.prems_of th, 0));
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               val split_thm = mk_casesplit_goal_thm sgn v gt;
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               val (subthms, expf) = IsaND.fixed_subgoal_thms split_thm;
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             in
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               expf (map recsplitf subthms)
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             end)
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      and recsplitf th =
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          (* note: multiple unifiers! we only take the first element,
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             probably fine -- there is probably only one anyway. *)
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          (case Library.get_first (Seq.pull o solve_by_splitth th) splitths of
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             NONE => split th
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           | SOME (solved_th, more) => solved_th)
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    in
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      recsplitf genth
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    end;
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(* Note: We dont do this if wf conditions fail to be solved, as each
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case may have a different wf condition - we could group the conditions
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togeather and say that they must be true to solve the general case,
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but that would hide from the user which sub-case they were related
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to. Probably this is not important, and it would work fine, but I
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prefer leaving more fine grain control to the user. *)
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(* derive eqs, assuming strict, ie the rules have no assumptions = all
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   the well-foundness conditions have been solved. *)
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fun derive_init_eqs sgn rules eqs =
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  let
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    fun get_related_thms i =
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      List.mapPartial ((fn (r, x) => if x = i then SOME r else NONE));
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    fun add_eq (i, e) xs =
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      (e, (get_related_thms i rules), i) :: xs
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    fun solve_eq (th, [], i) =
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          error "derive_init_eqs: missing rules"
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      | solve_eq (th, [a], i) = (a, i)
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      | solve_eq (th, splitths as (_ :: _), i) = (splitto splitths th, i);
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    val eqths =
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      map (Thm.trivial o Thm.cterm_of sgn o Data.mk_Trueprop) eqs;
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  in
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    []
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    |> fold_index add_eq eqths 
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    |> map solve_eq
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    |> rev
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  end;
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end;
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structure CaseSplit = CaseSplitFUN(CaseSplitData_HOL);