src/Provers/splitter.ML

Sign.print_data;

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−(*  Title:      Provers/splitter
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−    ID:         $Id$
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−    Author:     Tobias Nipkow
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−    Copyright   1995  TU Munich
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−
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−Generic case-splitter, suitable for most logics.
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−
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−Use:
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−
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−val split_tac = mk_case_split_tac iffD;
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−
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−by(case_split_tac splits i);
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−
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−where splits = [P(elim(...)) == rhs, ...]
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−      iffD  = [| P <-> Q; Q |] ==> P (* is called iffD2 in HOL *)
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−
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−*)
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−
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−local
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−
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−fun mk_case_split_tac_2 iffD order =
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−let
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−
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−
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−(************************************************************
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−   Create lift-theorem "trlift" :
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−
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−   [| !! x. Q(x)==R(x) ; P(R) == C |] ==> P(Q)==C
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−
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−*************************************************************)
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− 
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−val lift =
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−  let val ct = read_cterm (#sign(rep_thm iffD))
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−           ("[| !!x::'b::logic. Q(x) == R(x) |] ==> \
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−            \P(%x. Q(x)) == P(%x. R(x))::'a::logic",propT)
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−  in prove_goalw_cterm [] ct
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−     (fn [prem] => [rewtac prem, rtac reflexive_thm 1])
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−  end;
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−
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−val trlift = lift RS transitive_thm;
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−val _ $ (Var(P,PT)$_) $ _ = concl_of trlift;
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−
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−
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−(************************************************************************ 
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−   Set up term for instantiation of P in the lift-theorem
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−   
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−   Ts    : types of parameters (i.e. variables bound by meta-quantifiers)
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−   t     : lefthand side of meta-equality in subgoal
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−           the lift theorem is applied to (see select)
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−   pos   : "path" leading to abstraction, coded as a list
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−   T     : type of body of P(...)
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−   maxi  : maximum index of Vars
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−*************************************************************************)
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−
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−fun mk_cntxt Ts t pos T maxi =
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−  let fun var (t,i) = Var(("X",i),type_of1(Ts,t));
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−      fun down [] t i = Bound 0
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−        | down (p::ps) t i =
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−            let val (h,ts) = strip_comb t
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−                val v1 = ListPair.map var (take(p,ts), i upto (i+p-1))
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−                val u::us = drop(p,ts)
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−                val v2 = ListPair.map var (us, (i+p) upto (i+length(ts)-2))
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−      in list_comb(h,v1@[down ps u (i+length ts)]@v2) end;
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−  in Abs("", T, down (rev pos) t maxi) end;
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−
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−
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−(************************************************************************ 
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−   Set up term for instantiation of P in the split-theorem
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−   P(...) == rhs
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−
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−   Ts    : types of parameters (i.e. variables bound by meta-quantifiers)
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−   t     : lefthand side of meta-equality in subgoal
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−           the split theorem is applied to (see select)
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−   T     : type of body of P(...)
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−   tt    : the term  Const(..,..) $ ...
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−   maxi  : maximum index of Vars
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−
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−   lev   : abstraction level
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−*************************************************************************)
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−
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−fun mk_cntxt_splitthm Ts t tt T maxi =
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−  let fun down lev (Abs(v,T2,t)) = Abs(v,T2,down (lev+1) t)
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−        | down lev (Bound i) = if i >= lev
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−                               then Var(("X",maxi+i-lev),nth_elem(i-lev,Ts))
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−                               else Bound i 
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−        | down lev t = 
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−            let val (h,ts) = strip_comb t
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−                val h2 = (case h of Bound _ => down lev h | _ => h)
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−            in if incr_bv(lev,0,tt)=t 
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−               then
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−                 Bound (lev)
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−               else
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−                 list_comb(h2,map (down lev) ts)
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−            end;
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−  in Abs("",T,down 0 t) end;
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−
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−
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−(* add all loose bound variables in t to list is *)
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−fun add_lbnos(is,t) = add_loose_bnos(t,0,is);
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−
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−(* check if the innermost quantifier that needs to be removed
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−   has a body of type T; otherwise the expansion thm will fail later on
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−*)
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−fun type_test(T,lbnos,apsns) =
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−  let val (_,U,_) = nth_elem(foldl Int.min (hd lbnos, tl lbnos), apsns)
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−  in T=U end;
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−
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−(*************************************************************************
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−   Create a "split_pack".
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−
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−   thm   : the relevant split-theorem, i.e. P(...) == rhs , where P(...)
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−           is of the form
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−           P( Const(key,...) $ t_1 $ ... $ t_n )      (e.g. key = "if")
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−   T     : type of P(...)
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−   n     : number of arguments expected by Const(key,...)
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−   ts    : list of arguments actually found
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−   apsns : list of tuples of the form (T,U,pos), one tuple for each
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−           abstraction that is encountered on the way to the position where 
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−           Const(key, ...) $ ...  occurs, where
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−           T   : type of the variable bound by the abstraction
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−           U   : type of the abstraction's body
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−           pos : "path" leading to the body of the abstraction
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−   pos   : "path" leading to the position where Const(key, ...) $ ...  occurs.
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−   TB    : type of  Const(key,...) $ t_1 $ ... $ t_n
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−   t     : the term Const(key,...) $ t_1 $ ... $ t_n
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−
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−   A split pack is a tuple of the form
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−   (thm, apsns, pos, TB)
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−   Note : apsns is reversed, so that the outermost quantifier's position
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−          comes first ! If the terms in ts don't contain variables bound
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−          by other than meta-quantifiers, apsns is empty, because no further
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−          lifting is required before applying the split-theorem.
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−******************************************************************************) 
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−
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−fun mk_split_pack(thm,T,n,ts,apsns,pos,TB,t) =
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−  if n > length ts then []
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−  else let val lev = length apsns
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−           val lbnos = foldl add_lbnos ([],take(n,ts))
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−           val flbnos = filter (fn i => i < lev) lbnos
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−           val tt = incr_bv(~lev,0,t)
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−       in if null flbnos then [(thm,[],pos,TB,tt)]
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−          else if type_test(T,flbnos,apsns) then [(thm, rev apsns,pos,TB,tt)] 
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−               else []
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−       end;
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−
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−
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−(****************************************************************************
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−   Recursively scans term for occurences of Const(key,...) $ ...
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−   Returns a list of "split-packs" (one for each occurence of Const(key,...) )
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−
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−   cmap : association list of split-theorems that should be tried.
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−          The elements have the format (key,(thm,T,n)) , where
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−          key : the theorem's key constant ( Const(key,...) $ ... )
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−          thm : the theorem itself
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−          T   : type of P( Const(key,...) $ ... )
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−          n   : number of arguments expected by Const(key,...)
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−   Ts   : types of parameters
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−   t    : the term to be scanned
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−******************************************************************************)
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−
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−fun split_posns cmap Ts t =
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−  let fun posns Ts pos apsns (Abs(_,T,t)) =
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−            let val U = fastype_of1(T::Ts,t)
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−            in posns (T::Ts) (0::pos) ((T,U,pos)::apsns) t end
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−        | posns Ts pos apsns t =
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−            let val (h,ts) = strip_comb t
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−                fun iter((i,a),t) = (i+1, (posns Ts (i::pos) apsns t) @ a);
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−                val a = case h of
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−                  Const(c,_) =>
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−                    (case assoc(cmap,c) of
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−                       Some(thm,T,n) => mk_split_pack(thm,T,n,ts,apsns,pos,type_of1(Ts,t),t)
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−                     | None => [])
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−                | _ => []
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−             in snd(foldl iter ((0,a),ts)) end
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−  in posns Ts [] [] t end;
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−
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−
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−fun nth_subgoal i thm = nth_elem(i-1,prems_of thm);
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−
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−fun shorter((_,ps,pos,_,_),(_,qs,qos,_,_)) =
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−  let val ms = length ps and ns = length qs
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−  in ms < ns orelse (ms = ns andalso order(length pos,length qos)) end;
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−
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−
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−(************************************************************
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−   call split_posns with appropriate parameters
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−*************************************************************)
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−
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−fun select cmap state i =
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−  let val goali = nth_subgoal i state
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−      val Ts = rev(map #2 (Logic.strip_params goali))
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−      val _ $ t $ _ = Logic.strip_assums_concl goali;
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−  in (Ts,t,sort shorter (split_posns cmap Ts t)) end;
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−
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−
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−(*************************************************************
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−   instantiate lift theorem
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−
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−   if t is of the form
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−   ... ( Const(...,...) $ Abs( .... ) ) ...
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−   then
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−   P = %a.  ... ( Const(...,...) $ a ) ...
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−   where a has type T --> U
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−
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−   Ts      : types of parameters
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−   t       : lefthand side of meta-equality in subgoal
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−             the split theorem is applied to (see cmap)
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−   T,U,pos : see mk_split_pack
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−   state   : current proof state
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−   lift    : the lift theorem
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−   i       : no. of subgoal
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−**************************************************************)
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−
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−fun inst_lift Ts t (T,U,pos) state lift i =
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−  let val sg = #sign(rep_thm state)
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−      val tsig = #tsig(Sign.rep_sg sg)
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−      val cntxt = mk_cntxt Ts t pos (T-->U) (#maxidx(rep_thm lift))
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−      val cu = cterm_of sg cntxt
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−      val uT = #T(rep_cterm cu)
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−      val cP' = cterm_of sg (Var(P,uT))
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−      val ixnTs = Type.typ_match tsig ([],(PT,uT));
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−      val ixncTs = map (fn (x,y) => (x,ctyp_of sg y)) ixnTs;
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−  in instantiate (ixncTs, [(cP',cu)]) lift end;
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−
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−
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−(*************************************************************
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−   instantiate split theorem
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−
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−   Ts    : types of parameters
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−   t     : lefthand side of meta-equality in subgoal
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−           the split theorem is applied to (see cmap)
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−   pos   : "path" to the body of P(...)
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−   thm   : the split theorem
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−   TB    : type of body of P(...)
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−   state : current proof state
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−**************************************************************)
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−
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−fun inst_split Ts t tt thm TB state =
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−  let val _$((Var(P2,PT2))$_)$_ = concl_of thm
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−      val sg = #sign(rep_thm state)
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−      val tsig = #tsig(Sign.rep_sg sg)
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−      val cntxt = mk_cntxt_splitthm Ts t tt TB (#maxidx(rep_thm thm))
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−      val cu = cterm_of sg cntxt
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−      val uT = #T(rep_cterm cu)
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−      val cP' = cterm_of sg (Var(P2,uT))
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−      val ixnTs = Type.typ_match tsig ([],(PT2,uT));
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−      val ixncTs = map (fn (x,y) => (x,ctyp_of sg y)) ixnTs;
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−  in instantiate (ixncTs, [(cP',cu)]) thm end;
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−
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−
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−(*****************************************************************************
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−   The split-tactic
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−   
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−   splits : list of split-theorems to be tried
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−   i      : number of subgoal the tactic should be applied to
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−*****************************************************************************)
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−
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−fun split_tac [] i = no_tac
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−  | split_tac splits i =
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−  let fun const(thm) = let val _$(t as _$lhs)$_ = concl_of thm
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−                           val (Const(a,_),args) = strip_comb lhs
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−                       in (a,(thm,fastype_of t,length args)) end
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−      val cmap = map const splits;
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−      fun lift_tac Ts t p st = (rtac (inst_lift Ts t p st trlift i) i) st
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−      fun lift_split_tac st = st |>
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−            let val (Ts,t,splits) = select cmap st i
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−            in case splits of
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−                 [] => no_tac
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−               | (thm,apsns,pos,TB,tt)::_ =>
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−                   (case apsns of
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−                      [] => (fn state => state |>
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−			           rtac (inst_split Ts t tt thm TB state) i)
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−                    | p::_ => EVERY[lift_tac Ts t p,
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−                                    rtac reflexive_thm (i+1),
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−                                    lift_split_tac])
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−            end
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−  in COND (has_fewer_prems i) no_tac 
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−          (rtac iffD i THEN lift_split_tac)
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−  end;
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−
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−in split_tac end;
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−
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−in
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−
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−fun mk_case_split_tac iffD = mk_case_split_tac_2 iffD (op <=) ;
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−
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−fun mk_case_split_inside_tac iffD = mk_case_split_tac_2 iffD (op >=) ;
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−
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−end;