src/HOL/Tools/TFL/casesplit.ML

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