(* Title: Provers/splitter
ID: $Id$
Author: Tobias Nipkow
Copyright 1995 TU Munich
Generic case-splitter, suitable for most logics.
Deals with equalities of the form ?P(f args) = ...
where "f args" must be a first-order term without duplicate variables.
*)
infix 4 addsplits delsplits;
signature SPLITTER_DATA =
sig
structure Simplifier: SIMPLIFIER
val mk_eq : thm -> thm
val meta_eq_to_iff: thm (* "x == y ==> x = y" *)
val iffD : thm (* "[| P = Q; Q |] ==> P" *)
val disjE : thm (* "[| P | Q; P ==> R; Q ==> R |] ==> R" *)
val conjE : thm (* "[| P & Q; [| P; Q |] ==> R |] ==> R" *)
val exE : thm (* "[| x. P x; !!x. P x ==> Q |] ==> Q" *)
val contrapos : thm (* "[| ~ Q; P ==> Q |] ==> ~ P" *)
val contrapos2 : thm (* "[| Q; ~ P ==> ~ Q |] ==> P" *)
val notnotD : thm (* "~ ~ P ==> P" *)
end
signature SPLITTER =
sig
type simpset
val split_tac : thm list -> int -> tactic
val split_inside_tac: thm list -> int -> tactic
val split_asm_tac : thm list -> int -> tactic
val addsplits : simpset * thm list -> simpset
val delsplits : simpset * thm list -> simpset
val Addsplits : thm list -> unit
val Delsplits : thm list -> unit
val split_add_global: theory attribute
val split_del_global: theory attribute
val split_add_local: Proof.context attribute
val split_del_local: Proof.context attribute
val split_modifiers : (Args.T list -> (Method.modifier * Args.T list)) list
val setup: (theory -> theory) list
end;
functor SplitterFun(Data: SPLITTER_DATA): SPLITTER =
struct
structure Simplifier = Data.Simplifier;
type simpset = Simplifier.simpset;
val Const ("==>", _) $ (Const ("Trueprop", _) $
(Const (const_not, _) $ _ )) $ _ = #prop (rep_thm(Data.notnotD));
val Const ("==>", _) $ (Const ("Trueprop", _) $
(Const (const_or , _) $ _ $ _)) $ _ = #prop (rep_thm(Data.disjE));
fun split_format_err() = error("Wrong format for split rule");
fun split_thm_info thm = case concl_of (Data.mk_eq thm) of
Const("==", _)$(Var _$t)$c =>
(case strip_comb t of
(Const(a,_),_) => (a,case c of (Const(s,_)$_)=>s=const_not|_=> false)
| _ => split_format_err())
| _ => split_format_err();
fun mk_case_split_tac order =
let
(************************************************************
Create lift-theorem "trlift" :
[| !!x. Q x == R x; P(%x. R x) == C |] ==> P (%x. Q x) == C
*************************************************************)
val meta_iffD = Data.meta_eq_to_iff RS Data.iffD;
val lift =
let val ct = read_cterm (#sign(rep_thm Data.iffD))
("[| !!x. (Q::('b::logic)=>('c::logic))(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 _ $ (P $ _) $ _ = concl_of trlift;
(************************************************************************
Set up term for instantiation of P in the lift-theorem
Ts : types of parameters (i.e. variables bound by meta-quantifiers)
t : lefthand side of meta-equality in subgoal
the lift theorem is applied to (see select)
pos : "path" leading to abstraction, coded as a list
T : type of body of P(...)
maxi : maximum index of Vars
*************************************************************************)
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 = ListPair.map var (take(p,ts), i upto (i+p-1))
val u::us = drop(p,ts)
val v2 = ListPair.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;
(************************************************************************
Set up term for instantiation of P in the split-theorem
P(...) == rhs
t : lefthand side of meta-equality in subgoal
the split theorem is applied to (see select)
T : type of body of P(...)
tt : the term Const(key,..) $ ...
*************************************************************************)
fun mk_cntxt_splitthm t tt T =
let fun repl lev t =
if incr_boundvars lev tt aconv t then Bound lev
else case t of
(Abs (v, T2, t)) => Abs (v, T2, repl (lev+1) t)
| (Bound i) => Bound (if i>=lev then i+1 else i)
| (t1 $ t2) => (repl lev t1) $ (repl lev t2)
| t => t
in Abs("", T, repl 0 t) end;
(* add all loose bound variables in t to list is *)
fun add_lbnos(is,t) = add_loose_bnos(t,0,is);
(* check if the innermost abstraction 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(foldl Int.min (hd lbnos, tl lbnos), apsns)
in T=U end;
(*************************************************************************
Create a "split_pack".
thm : the relevant split-theorem, i.e. P(...) == rhs , where P(...)
is of the form
P( Const(key,...) $ t_1 $ ... $ t_n ) (e.g. key = "if")
T : type of P(...)
T' : type of term to be scanned
n : number of arguments expected by Const(key,...)
ts : list of arguments actually found
apsns : list of tuples of the form (T,U,pos), one tuple for each
abstraction that is encountered on the way to the position where
Const(key, ...) $ ... occurs, where
T : type of the variable bound by the abstraction
U : type of the abstraction's body
pos : "path" leading to the body of the abstraction
pos : "path" leading to the position where Const(key, ...) $ ... occurs.
TB : type of Const(key,...) $ t_1 $ ... $ t_n
t : the term Const(key,...) $ t_1 $ ... $ t_n
A split pack is a tuple of the form
(thm, apsns, pos, TB, tt)
Note : apsns is reversed, so that the outermost quantifier's position
comes first ! If the terms in ts don't contain variables bound
by other than meta-quantifiers, apsns is empty, because no further
lifting is required before applying the split-theorem.
******************************************************************************)
fun mk_split_pack(thm, T, T', n, ts, apsns, pos, TB, t) =
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
val tt = incr_boundvars (~lev) t
in if null flbnos then
if T = T' then [(thm,[],pos,TB,tt)] else []
else if type_test(T,flbnos,apsns) then [(thm, rev apsns,pos,TB,tt)]
else []
end;
(****************************************************************************
Recursively scans term for occurences of Const(key,...) $ ...
Returns a list of "split-packs" (one for each occurence of Const(key,...) )
cmap : association list of split-theorems that should be tried.
The elements have the format (key,(thm,T,n)) , where
key : the theorem's key constant ( Const(key,...) $ ... )
thm : the theorem itself
T : type of P( Const(key,...) $ ... )
n : number of arguments expected by Const(key,...)
Ts : types of parameters
t : the term to be scanned
******************************************************************************)
(* Simplified first-order matching;
assumes that all Vars in the pattern are distinct;
see Pure/pattern.ML for the full version;
*)
local
exception MATCH
in
fun typ_match tsig args = (Type.typ_match tsig args)
handle Type.TYPE_MATCH => raise MATCH;
fun fomatch tsig args =
let
fun mtch tyinsts = fn
(Ts,Var(_,T), t) => typ_match tsig (tyinsts, (T, fastype_of1(Ts,t)))
| (_,Free (a,T), Free (b,U)) =>
if a=b then typ_match tsig (tyinsts,(T,U)) else raise MATCH
| (_,Const (a,T), Const (b,U)) =>
if a=b then typ_match tsig (tyinsts,(T,U)) else raise MATCH
| (_,Bound i, Bound j) => if i=j then tyinsts else raise MATCH
| (Ts,Abs(_,T,t), Abs(_,U,u)) =>
mtch (typ_match tsig (tyinsts,(T,U))) (U::Ts,t,u)
| (Ts, f$t, g$u) => mtch (mtch tyinsts (Ts,f,g)) (Ts, t, u)
| _ => raise MATCH
in (mtch Vartab.empty args; true) handle MATCH => false end;
end
fun split_posns cmap sg Ts t =
let
val T' = fastype_of1 (Ts, t);
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, cT) =>
let fun find [] = []
| find ((gcT, pat, thm, T, n)::tups) =
let val t2 = list_comb (h, take (n, ts))
in if Sign.typ_instance sg (cT, gcT)
andalso fomatch (Sign.tsig_of sg) (Ts,pat,t2)
then mk_split_pack(thm,T,T',n,ts,apsns,pos,type_of1(Ts,t2),t2)
else find tups
end
in find (assocs cmap c) end
| _ => []
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,pos,_,_),(_,qs,qos,_,_)) =
prod_ord (int_ord o pairself length) (order o pairself length)
((ps, pos), (qs, qos));
(************************************************************
call split_posns with appropriate parameters
*************************************************************)
fun select cmap state i =
let val sg = #sign(rep_thm state)
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 sg Ts t)) end;
(*************************************************************
instantiate lift theorem
if t is of the form
... ( Const(...,...) $ Abs( .... ) ) ...
then
P = %a. ... ( Const(...,...) $ a ) ...
where a has type T --> U
Ts : types of parameters
t : lefthand side of meta-equality in subgoal
the split theorem is applied to (see cmap)
T,U,pos : see mk_split_pack
state : current proof state
lift : the lift theorem
i : no. of subgoal
**************************************************************)
fun inst_lift Ts t (T, U, pos) state i =
let
val cert = cterm_of (sign_of_thm state);
val cntxt = mk_cntxt Ts t pos (T --> U) (#maxidx(rep_thm trlift));
in cterm_instantiate [(cert P, cert cntxt)] trlift
end;
(*************************************************************
instantiate split theorem
Ts : types of parameters
t : lefthand side of meta-equality in subgoal
the split theorem is applied to (see cmap)
tt : the term Const(key,..) $ ...
thm : the split theorem
TB : type of body of P(...)
state : current proof state
i : number of subgoal
**************************************************************)
fun inst_split Ts t tt thm TB state i =
let
val thm' = Thm.lift_rule (state, i) thm;
val (P, _) = strip_comb (fst (Logic.dest_equals
(Logic.strip_assums_concl (#prop (rep_thm thm')))));
val cert = cterm_of (sign_of_thm state);
val cntxt = mk_cntxt_splitthm t tt TB;
val abss = foldl (fn (t, T) => Abs ("", T, t));
in cterm_instantiate [(cert P, cert (abss (cntxt, Ts)))] thm'
end;
(*****************************************************************************
The split-tactic
splits : list of split-theorems to be tried
i : number of subgoal the tactic should be applied to
*****************************************************************************)
fun split_tac [] i = no_tac
| split_tac splits i =
let val splits = map Data.mk_eq splits;
fun add_thm(cmap,thm) =
(case concl_of thm of _$(t as _$lhs)$_ =>
(case strip_comb lhs of (Const(a,aT),args) =>
let val info = (aT,lhs,thm,fastype_of t,length args)
in case assoc(cmap,a) of
Some infos => overwrite(cmap,(a,info::infos))
| None => (a,[info])::cmap
end
| _ => split_format_err())
| _ => split_format_err())
val cmap = foldl add_thm ([],splits);
fun lift_tac Ts t p st = rtac (inst_lift Ts t p st i) i st
fun lift_split_tac state =
let val (Ts, t, splits) = select cmap state i
in case splits of
[] => no_tac state
| (thm, apsns, pos, TB, tt)::_ =>
(case apsns of
[] => compose_tac (false, inst_split Ts t tt thm TB state i, 0) i state
| p::_ => EVERY [lift_tac Ts t p,
rtac reflexive_thm (i+1),
lift_split_tac] state)
end
in COND (has_fewer_prems i) no_tac
(rtac meta_iffD i THEN lift_split_tac)
end;
in split_tac end;
val split_tac = mk_case_split_tac int_ord;
val split_inside_tac = mk_case_split_tac (rev_order o int_ord);
(*****************************************************************************
The split-tactic for premises
splits : list of split-theorems to be tried
****************************************************************************)
fun split_asm_tac [] = K no_tac
| split_asm_tac splits =
let val cname_list = map (fst o split_thm_info) splits;
fun is_case (a,_) = a mem cname_list;
fun tac (t,i) =
let val n = find_index (exists_Const is_case)
(Logic.strip_assums_hyp t);
fun first_prem_is_disj (Const ("==>", _) $ (Const ("Trueprop", _)
$ (Const (s, _) $ _ $ _ )) $ _ ) = (s=const_or)
| first_prem_is_disj (Const("all",_)$Abs(_,_,t)) =
first_prem_is_disj t
| first_prem_is_disj _ = false;
(* does not work properly if the split variable is bound by a quantfier *)
fun flat_prems_tac i = SUBGOAL (fn (t,i) =>
(if first_prem_is_disj t
then EVERY[etac Data.disjE i,rotate_tac ~1 i,
rotate_tac ~1 (i+1),
flat_prems_tac (i+1)]
else all_tac)
THEN REPEAT (eresolve_tac [Data.conjE,Data.exE] i)
THEN REPEAT (dresolve_tac [Data.notnotD] i)) i;
in if n<0 then no_tac else DETERM (EVERY'
[rotate_tac n, etac Data.contrapos2,
split_tac splits,
rotate_tac ~1, etac Data.contrapos, rotate_tac ~1,
flat_prems_tac] i)
end;
in SUBGOAL tac
end;
fun gen_split_tac [] = K no_tac
| gen_split_tac (split::splits) =
let val (_,asm) = split_thm_info split
in (if asm then split_asm_tac else split_tac) [split] ORELSE'
gen_split_tac splits
end;
(** declare split rules **)
(* addsplits / delsplits *)
fun split_name name asm = "split " ^ name ^ (if asm then " asm" else "");
fun ss addsplits splits =
let fun addsplit (ss,split) =
let val (name,asm) = split_thm_info split
in Simplifier.addloop(ss,(split_name name asm,
(if asm then split_asm_tac else split_tac) [split])) end
in foldl addsplit (ss,splits) end;
fun ss delsplits splits =
let fun delsplit(ss,split) =
let val (name,asm) = split_thm_info split
in Simplifier.delloop(ss,split_name name asm)
end in foldl delsplit (ss,splits) end;
fun Addsplits splits = (Simplifier.simpset_ref() :=
Simplifier.simpset() addsplits splits);
fun Delsplits splits = (Simplifier.simpset_ref() :=
Simplifier.simpset() delsplits splits);
(* attributes *)
val splitN = "split";
val split_add_global = Simplifier.change_global_ss (op addsplits);
val split_del_global = Simplifier.change_global_ss (op delsplits);
val split_add_local = Simplifier.change_local_ss (op addsplits);
val split_del_local = Simplifier.change_local_ss (op delsplits);
val split_attr =
(Attrib.add_del_args split_add_global split_del_global,
Attrib.add_del_args split_add_local split_del_local);
(* methods *)
val split_modifiers =
[Args.$$$ splitN -- Args.colon >> K ((I, split_add_local): Method.modifier),
Args.$$$ splitN -- Args.add -- Args.colon >> K (I, split_add_local),
Args.$$$ splitN -- Args.del -- Args.colon >> K (I, split_del_local)];
val split_args = #2 oo Method.syntax Attrib.local_thms;
fun split_meth ths = Method.SIMPLE_METHOD' HEADGOAL (CHANGED_PROP o gen_split_tac ths);
(** theory setup **)
val setup =
[Attrib.add_attributes [(splitN, split_attr, "declaration of case split rule")],
Method.add_methods [(splitN, split_meth oo split_args, "apply case split rule")]];
end;