TFL/casesplit.ML
author wenzelm
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accomodate identification of type Sign.sg and theory;
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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 * Term.type -> int -> Thm.thm -> Thm.thm Seq.seq
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    casesplit_name : 
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      string -> int -> Thm.thm -> 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.thm list -> Thm.thm -> 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 -> Term.term
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  val mk_Trueprop : Term.term -> Term.term
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  val read_cterm : Sign.sg -> string -> Thm.cterm
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  val localize : Thm.thm list
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  val local_impI : Thm.thm
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  val atomize : Thm.thm list
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  val rulify1 : Thm.thm list
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  val rulify2 : Thm.thm list
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end;
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(* for HOL *)
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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 read_cterm = HOLogic.read_cterm;
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val localize = [Thm.symmetric (thm "induct_implies_def")];
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val local_impI = thm "induct_impliesI";
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val atomize = thms "induct_atomize";
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val rulify1 = thms "induct_rulify1";
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val rulify2 = thms "induct_rulify2";
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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 : Sign.sg -> Term.typ -> Thm.thm
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  val cases_thm_of_induct_thm : Thm.thm -> Thm.thm
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  (* case split tactics *)
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  val casesplit_free :
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      string * Term.typ -> int -> Thm.thm -> Thm.thm Seq.seq
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  val casesplit_name : string -> int -> Thm.thm -> 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 * Term.term -> (string * Term.typ) option
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  val find_thm_split :
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      Thm.thm -> int -> Thm.thm -> (string * Term.typ) option
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  val find_thms_split :
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      Thm.thm list -> int -> Thm.thm -> (string * Term.typ) option
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  (* try to recursively split conjectured thm to given list of thms *)
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  val splitto : Thm.thm list -> Thm.thm -> Thm.thm
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  (* for use with the recdef package *)
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  val derive_init_eqs :
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      Sign.sg ->
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      (Thm.thm * int) list -> Term.term list -> (Thm.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.rulify1;
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val atomize_goals = Tactic.rewrite_goals_rule Data.atomize;
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(* 
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val localize = Tactic.norm_hhf_rule o Tactic.simplify false Data.localize;
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fun atomize_term sg =
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  ObjectLogic.drop_judgment sg o MetaSimplifier.rewrite_term sg Data.atomize [];
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val rulify_tac =  Tactic.rewrite_goal_tac Data.rulify1;
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val atomize_tac =  Tactic.rewrite_goal_tac Data.atomize;
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Tactic.simplify_goal
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val rulify_tac =
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  Tactic.rewrite_goal_tac Data.rulify1 THEN'
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  Tactic.rewrite_goal_tac Data.rulify2 THEN'
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  Tactic.norm_hhf_tac;
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val atomize = Tactic.norm_hhf_rule o Tactic.simplify true Data.atomize;
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*)
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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,_)  => raise ERROR_MESSAGE 
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                                            ("Free type: " ^ s)   
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                   | TVar((s,i),_) => raise ERROR_MESSAGE 
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                                            ("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 => raise ERROR_MESSAGE ("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 tsig = Sign.tsig_of sgn;
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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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             Type.typ_match tsig (Vartab.empty, (Dty, ty)))
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          | _ => raise ERROR_MESSAGE ("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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                 | _ => raise ERROR_MESSAGE ("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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              raise ERROR_MESSAGE "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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local
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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 solve_eq (th, [], i) = 
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      raise ERROR_MESSAGE "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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in
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fun derive_init_eqs sgn rules eqs = 
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    let 
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      val eqths = map (Thm.trivial o (Thm.cterm_of sgn) o Data.mk_Trueprop) 
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                      eqs
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    in
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      (rev o map solve_eq)
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        (Library.foldln 
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           (fn (e,i) => 
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               (curry (op ::)) (e, (get_related_thms (i - 1) rules), i - 1)) 
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           eqths [])
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    end;
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end;
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(* 
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    val (rs_hwfc, unhidefs) = Library.split_list (map hide_prems rules)
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    (map2 (op |>) (ths, expfs))
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*)
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end;
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structure CaseSplit = CaseSplitFUN(CaseSplitData_HOL);