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(* Title: Provers/splitter.ML
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.
*)
signature SPLITTER_DATA =
sig
val context : Proof.context
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 (* "[| EX 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" *)
val safe_tac : Proof.context -> tactic
end
signature SPLITTER =
sig
(* somewhat more internal functions *)
val cmap_of_split_thms: thm list -> (string * (typ * term * thm * typ * int) list) list
val split_posns: (string * (typ * term * thm * typ * int) list) list ->
theory -> typ list -> term -> (thm * (typ * typ * int list) list * int list * typ * term) list
(* first argument is a "cmap", returns a list of "split packs" *)
(* the "real" interface, providing a number of tactics *)
val split_tac : Proof.context -> thm list -> int -> tactic
val split_inside_tac: Proof.context -> thm list -> int -> tactic
val split_asm_tac : Proof.context -> thm list -> int -> tactic
val add_split: thm -> Proof.context -> Proof.context
val add_split_bang: thm -> Proof.context -> Proof.context
val del_split: thm -> Proof.context -> Proof.context
val split_modifiers : Method.modifier parser list
end;
functor Splitter(Data: SPLITTER_DATA): SPLITTER =
struct
val Const (const_not, _) $ _ =
Object_Logic.drop_judgment Data.context
(#1 (Logic.dest_implies (Thm.prop_of Data.notnotD)));
val Const (const_or , _) $ _ $ _ =
Object_Logic.drop_judgment Data.context
(#1 (Logic.dest_implies (Thm.prop_of Data.disjE)));
val const_Trueprop = Object_Logic.judgment_name Data.context;
fun split_format_err () = error "Wrong format for split rule";
fun split_thm_info thm =
(case Thm.concl_of (Data.mk_eq thm) of
Const(\<^const_name>\<open>Pure.eq\<close>, _) $ (Var _ $ t) $ c =>
(case strip_comb t of
(Const p, _) => (p, case c of (Const (s, _) $ _) => s = const_not | _ => false)
| _ => split_format_err ())
| _ => split_format_err ());
fun cmap_of_split_thms thms =
let
val splits = map Data.mk_eq thms
fun add_thm thm cmap =
(case Thm.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 AList.lookup (op =) cmap a of
SOME infos => AList.update (op =) (a, info::infos) cmap
| NONE => (a,[info])::cmap
end
| _ => split_format_err())
| _ => split_format_err())
in
fold add_thm splits []
end;
val abss = fold (Term.abs o pair "");
(* ------------------------------------------------------------------------- *)
(* mk_case_split_tac *)
(* ------------------------------------------------------------------------- *)
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; (* (P == Q) ==> Q ==> P *)
val lift = Goal.prove_global \<^theory> ["P", "Q", "R"]
[Syntax.read_prop_global \<^theory>\<open>Pure\<close> "!!x :: 'b. Q(x) == R(x) :: 'c"]
(Syntax.read_prop_global \<^theory>\<open>Pure\<close> "P(%x. Q(x)) == P(%x. R(x))")
(fn {context = ctxt, prems} =>
rewrite_goals_tac ctxt prems THEN resolve_tac ctxt [reflexive_thm] 1)
val _ $ _ $ (_ $ (_ $ abs_lift) $ _) = Thm.prop_of lift;
val trlift = lift RS transitive_thm;
(************************************************************************
Set up term for instantiation of P in the lift-theorem
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(...)
*************************************************************************)
fun mk_cntxt t pos T =
let
fun down [] t = (Bound 0, t)
| down (p :: ps) t =
let
val (h, ts) = strip_comb t
val (ts1, u :: ts2) = chop p ts
val (u1, u2) = down ps u
in
(list_comb (incr_boundvars 1 h,
map (incr_boundvars 1) ts1 @ u1 ::
map (incr_boundvars 1) ts2),
u2)
end;
val (u1, u2) = down (rev pos) t
in (Abs ("", T, u1), u2) 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 Envir.aeconv(incr_boundvars lev tt, 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 t is = 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: typ, _) = nth apsns (foldl1 Int.min lbnos)
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: typ, T', n, ts, apsns, pos, TB, t) =
if n > length ts then []
else let val lev = length apsns
val lbnos = fold 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 occurrences of Const(key,...) $ ...
Returns a list of "split-packs" (one for each occurrence 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 thy (tyenv, TU) = Sign.typ_match thy TU tyenv
handle Type.TYPE_MATCH => raise MATCH;
fun fomatch thy args =
let
fun mtch tyinsts = fn
(Ts, Var(_,T), t) =>
typ_match thy (tyinsts, (T, fastype_of1(Ts,t)))
| (_, Free (a,T), Free (b,U)) =>
if a=b then typ_match thy (tyinsts,(T,U)) else raise MATCH
| (_, Const (a,T), Const (b,U)) =>
if a=b then typ_match thy (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 thy (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 : (string * (typ * term * thm * typ * int) list) list) thy 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 t (i, a) = (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 thy (cT, gcT) andalso fomatch thy (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 (these (AList.lookup (op =) cmap c)) end
| _ => []
in snd (fold iter ts (0, a)) end
in posns Ts [] [] t end;
fun shorter ((_,ps,pos,_,_), (_,qs,qos,_,_)) =
prod_ord (int_ord o apply2 length) (order o apply2 length)
((ps, pos), (qs, qos));
(************************************************************
call split_posns with appropriate parameters
*************************************************************)
fun select thy cmap state i =
let
val goal = Thm.term_of (Thm.cprem_of state i);
val Ts = rev (map #2 (Logic.strip_params goal));
val _ $ t $ _ = Logic.strip_assums_concl goal;
in (Ts, t, sort shorter (split_posns cmap thy Ts t)) end;
fun exported_split_posns cmap thy Ts t =
sort shorter (split_posns cmap thy Ts t);
(*************************************************************
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
i : no. of subgoal
**************************************************************)
fun inst_lift ctxt Ts t (T, U, pos) state i =
let
val (cntxt, u) = mk_cntxt t pos (T --> U);
val trlift' = Thm.lift_rule (Thm.cprem_of state i)
(Thm.rename_boundvars abs_lift u trlift);
val (Var (P, _), _) =
strip_comb (fst (Logic.dest_equals
(Logic.strip_assums_concl (Thm.prop_of trlift'))));
in infer_instantiate ctxt [(P, Thm.cterm_of ctxt (abss Ts 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 ctxt Ts t tt thm TB state i =
let
val thm' = Thm.lift_rule (Thm.cprem_of state i) thm;
val (Var (P, _), _) =
strip_comb (fst (Logic.dest_equals
(Logic.strip_assums_concl (Thm.prop_of thm'))));
val cntxt = mk_cntxt_splitthm t tt TB;
in infer_instantiate ctxt [(P, Thm.cterm_of ctxt (abss Ts cntxt))] 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 ctxt splits i =
let val cmap = cmap_of_split_thms splits
fun lift_tac Ts t p st = compose_tac ctxt (false, inst_lift ctxt Ts t p st i, 2) i st
fun lift_split_tac state =
let val (Ts, t, splits) = select (Proof_Context.theory_of ctxt) cmap state i
in case splits of
[] => no_tac state
| (thm, apsns, pos, TB, tt)::_ =>
(case apsns of
[] =>
compose_tac ctxt (false, inst_split ctxt Ts t tt thm TB state i, 0) i state
| p::_ => EVERY [lift_tac Ts t p,
resolve_tac ctxt [reflexive_thm] (i+1),
lift_split_tac] state)
end
in COND (has_fewer_prems i) no_tac
(resolve_tac ctxt [meta_iffD] i THEN lift_split_tac)
end;
in (split_tac, exported_split_posns) end; (* mk_case_split_tac *)
val (split_tac, split_posns) = 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 ctxt splits =
let val cname_list = map (fst o fst o split_thm_info) splits;
fun tac (t,i) =
let val n = find_index (exists_Const (member (op =) cname_list o #1))
(Logic.strip_assums_hyp t);
fun first_prem_is_disj (Const (\<^const_name>\<open>Pure.imp\<close>, _) $ (Const (c, _)
$ (Const (s, _) $ _ $ _ )) $ _ ) = c = const_Trueprop andalso s = const_or
| first_prem_is_disj (Const(\<^const_name>\<open>Pure.all\<close>,_)$Abs(_,_,t)) =
first_prem_is_disj t
| first_prem_is_disj _ = false;
(* does not work properly if the split variable is bound by a quantifier *)
fun flat_prems_tac i = SUBGOAL (fn (t,i) =>
(if first_prem_is_disj t
then EVERY[eresolve_tac ctxt [Data.disjE] i, rotate_tac ~1 i,
rotate_tac ~1 (i+1),
flat_prems_tac (i+1)]
else all_tac)
THEN REPEAT (eresolve_tac ctxt [Data.conjE,Data.exE] i)
THEN REPEAT (dresolve_tac ctxt [Data.notnotD] i)) i;
in if n<0 then no_tac else (DETERM (EVERY'
[rotate_tac n, eresolve_tac ctxt [Data.contrapos2],
split_tac ctxt splits,
rotate_tac ~1, eresolve_tac ctxt [Data.contrapos], rotate_tac ~1,
flat_prems_tac] i))
end;
in SUBGOAL tac
end;
fun gen_split_tac _ [] = K no_tac
| gen_split_tac ctxt (split::splits) =
let val (_,asm) = split_thm_info split
in (if asm then split_asm_tac else split_tac) ctxt [split] ORELSE'
gen_split_tac ctxt splits
end;
(** declare split rules **)
(* add_split / del_split *)
fun string_of_typ (Type (s, Ts)) =
(if null Ts then "" else enclose "(" ")" (commas (map string_of_typ Ts))) ^ s
| string_of_typ _ = "_";
fun split_name (name, T) asm = "split " ^
(if asm then "asm " else "") ^ name ^ " :: " ^ string_of_typ T;
fun gen_add_split bang split ctxt =
let
val (name, asm) = split_thm_info split
val split0 = Thm.trim_context split
fun tac ctxt' =
let val split' = Thm.transfer' ctxt' split0 in
(if asm then split_asm_tac ctxt' [split']
else if bang
then split_tac ctxt' [split'] THEN_ALL_NEW
TRY o (SELECT_GOAL (Data.safe_tac ctxt'))
else split_tac ctxt' [split'])
end
in Simplifier.addloop (ctxt, (split_name name asm, tac)) end;
val add_split = gen_add_split false;
val add_split_bang = gen_add_split true;
fun del_split split ctxt =
let val (name, asm) = split_thm_info split
in Simplifier.delloop (ctxt, split_name name asm) end;
(* attributes *)
val splitN = "split";
fun split_add bang = Simplifier.attrib (gen_add_split bang);
val split_del = Simplifier.attrib del_split;
val add_del = Scan.lift
(Args.bang >> K (split_add true)
|| Args.del >> K split_del
|| Scan.succeed (split_add false));
val _ = Theory.setup
(Attrib.setup \<^binding>\<open>split\<close> add_del "declare case split rule");
(* methods *)
val split_modifiers =
[Args.$$$ splitN -- Args.colon >> K (Method.modifier (split_add false) \<^here>),
Args.$$$ splitN -- Args.bang_colon >> K (Method.modifier (split_add true) \<^here>),
Args.$$$ splitN -- Args.del -- Args.colon >> K (Method.modifier split_del \<^here>)];
val _ =
Theory.setup
(Method.setup \<^binding>\<open>split\<close>
(Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD' (CHANGED_PROP o gen_split_tac ctxt ths)))
"apply case split rule");
end;
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