TFL/casesplit.ML

added transitivity rules, reworking of min/max lemmas

+
−(* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- *)
+
−(*  Title:      TFL/casesplit.ML
+
−    Author:     Lucas Dixon, University of Edinburgh
+
−                [e-mail]
+
−    Date:       17 Aug 2004
+
−*)
+
−(* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- *)
+
−(*  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
+
−
+
−    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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−
+
−(* 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
+
−end;
+
−
+
−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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−
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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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−
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−
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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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−
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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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−
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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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−
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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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−
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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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−
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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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−
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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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−
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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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−
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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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−
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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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−
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−      val abs_ct = ctermify abst;
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−      val free_ct = ctermify x;
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−
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−      val casethm_vars = rev (Term.term_vars (Thm.concl_of case_thm));
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−
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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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−
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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 (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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−
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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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−
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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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−
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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 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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−
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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 = Name.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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−
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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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−
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−      val split_goal_th = splitter_thm RS subgoalth';
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−
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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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−
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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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−
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−by (casesplit_name "y" 2);
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−
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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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−
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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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−
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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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−
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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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−
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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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−
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−
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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. *)
+
−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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−
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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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−
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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)
+
−    in
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−      recsplitf genth
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−    end;
+
−
+
−
+
−(* 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,
+
−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. *)
+
−
+
−(* derive eqs, assuming strict, ie the rules have no assumptions = all
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−   the well-foundness conditions have been solved. *)
+
−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;
+
−  in
+
−    []
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−    |> fold_index add_eq eqths 
+
−    |> map solve_eq
+
−    |> rev
+
−  end;
+
−
+
−end;
+
−
+
−
+
−structure CaseSplit = CaseSplitFUN(CaseSplitData_HOL);