(* Title: Provers/splitter
ID: $Id$
Author: Tobias Nipkow
Copyright 1995 TU Munich
Generic case-splitter, suitable for most logics.
Use:
val split_tac = mk_case_split_tac iffD;
by(case_split_tac splits i);
where splits = [P(elim(...)) == rhs, ...]
iffD = [| P <-> Q; Q |] ==> P (* is called iffD2 in HOL *)
*)
fun mk_case_split_tac iffD =
let
val lift =
let val ct = read_cterm (#sign(rep_thm iffD))
("[| !!x::'b::logic. Q(x) == R(x) |] ==> \
\P(%x.Q(x)) == P(%x.R(x))::'a::logic",propT)
in prove_goalw_cterm [] ct
(fn [prem] => [rewtac prem, rtac reflexive_thm 1])
end;
val trlift = lift RS transitive_thm;
val _ $ (Var(P,PT)$_) $ _ = concl_of trlift;
fun mk_cntxt Ts t pos T maxi =
let fun var (t,i) = Var(("X",i),type_of1(Ts,t));
fun down [] t i = Bound 0
| down (p::ps) t i =
let val (h,ts) = strip_comb t
val v1 = map var (take(p,ts) ~~ (i upto (i+p-1)))
val u::us = drop(p,ts)
val v2 = map var (us ~~ ((i+p) upto (i+length(ts)-2)))
in list_comb(h,v1@[down ps u (i+length ts)]@v2) end;
in Abs("", T, down (rev pos) t maxi) end;
fun add_lbnos(is,t) = add_loose_bnos(t,0,is);
(* check if the innermost quantifier that needs to be removed
has a body of type T; otherwise the expansion thm will fail later on
*)
fun type_test(T,lbnos,apsns) =
let val (_,U,_) = nth_elem(min lbnos,apsns)
in T=U end;
fun mk_split_pack(thm,T,n,ts,apsns) =
if n > length ts then []
else let val lev = length apsns
val lbnos = foldl add_lbnos ([],take(n,ts))
val flbnos = filter (fn i => i < lev) lbnos
in if null flbnos then [(thm,[])]
else if type_test(T,flbnos,apsns) then [(thm, rev apsns)] else []
end;
fun split_posns cmap Ts t =
let fun posns Ts pos apsns (Abs(_,T,t)) =
let val U = fastype_of1(T::Ts,t)
in posns (T::Ts) (0::pos) ((T,U,pos)::apsns) t end
| posns Ts pos apsns t =
let val (h,ts) = strip_comb t
fun iter((i,a),t) = (i+1, (posns Ts (i::pos) apsns t) @ a);
val a = case h of
Const(c,_) =>
(case assoc(cmap,c) of
Some(thm,T,n) => mk_split_pack(thm,T,n,ts,apsns)
| None => [])
| _ => []
in snd(foldl iter ((0,a),ts)) end
in posns Ts [] [] t end;
fun nth_subgoal i thm = nth_elem(i-1,prems_of thm);
fun shorter((_,ps),(_,qs)) = length ps <= length qs;
fun select cmap state i =
let val goali = nth_subgoal i state
val Ts = rev(map #2 (Logic.strip_params goali))
val _ $ t $ _ = Logic.strip_assums_concl goali;
in (Ts,t,sort shorter (split_posns cmap Ts t)) end;
fun inst_lift Ts t (T,U,pos) state lift i =
let val sg = #sign(rep_thm state)
val tsig = #tsig(Sign.rep_sg sg)
val cntxt = mk_cntxt Ts t pos (T-->U) (#maxidx(rep_thm lift))
val cu = cterm_of sg cntxt
val uT = #T(rep_cterm cu)
val cP' = cterm_of sg (Var(P,uT))
val ixnTs = Type.typ_match tsig ([],(PT,uT));
val ixncTs = map (fn (x,y) => (x,ctyp_of sg y)) ixnTs;
in instantiate (ixncTs, [(cP',cu)]) lift end;
fun split_tac [] i = no_tac
| split_tac splits i =
let fun const(thm) = let val _$(t as _$lhs)$_ = concl_of thm
val (Const(a,_),args) = strip_comb lhs
in (a,(thm,fastype_of t,length args)) end
val cmap = map const splits;
fun lift Ts t p state = rtac (inst_lift Ts t p state trlift i) i
fun lift_split state =
let val (Ts,t,splits) = select cmap state i
in case splits of
[] => no_tac
| (thm,apsns)::_ =>
(case apsns of
[] => rtac thm i
| p::_ => EVERY[STATE(lift Ts t p),
rtac reflexive_thm (i+1),
STATE lift_split])
end
in STATE(fn thm =>
if i <= nprems_of thm then rtac iffD i THEN STATE lift_split
else no_tac)
end;
in split_tac end;