(* Title: TFL/post.ML
ID: $Id$
Author: Konrad Slind, Cambridge University Computer Laboratory
Copyright 1997 University of Cambridge
Second part of main module (postprocessing of TFL definitions).
*)
signature TFL =
sig
val trace: bool ref
val quiet_mode: bool ref
val message: string -> unit
val tgoalw: theory -> thm list -> thm list -> thm list
val tgoal: theory -> thm list -> thm list
val define_i: bool -> theory -> claset -> simpset -> thm list -> thm list -> xstring ->
term -> term list -> theory * {rules: (thm * int) list, induct: thm, tcs: term list}
val define: bool -> theory -> claset -> simpset -> thm list -> thm list -> xstring ->
string -> string list -> theory * {rules: (thm * int) list, induct: thm, tcs: term list}
val defer_i: theory -> thm list -> xstring -> term list -> theory * thm
val defer: theory -> thm list -> xstring -> string list -> theory * thm
end;
structure Tfl: TFL =
struct
structure S = USyntax
(* messages *)
val trace = Prim.trace
val quiet_mode = ref false;
fun message s = if ! quiet_mode then () else writeln s;
(* misc *)
(*---------------------------------------------------------------------------
* Extract termination goals so that they can be put it into a goalstack, or
* have a tactic directly applied to them.
*--------------------------------------------------------------------------*)
fun termination_goals rules =
map (Type.freeze o HOLogic.dest_Trueprop)
(foldr (fn (th,A) => gen_union (op aconv) (prems_of th, A)) [] rules);
(*---------------------------------------------------------------------------
* Finds the termination conditions in (highly massaged) definition and
* puts them into a goalstack.
*--------------------------------------------------------------------------*)
fun tgoalw thy defs rules =
case termination_goals rules of
[] => error "tgoalw: no termination conditions to prove"
| L => OldGoals.goalw_cterm defs
(Thm.cterm_of (Theory.sign_of thy)
(HOLogic.mk_Trueprop(USyntax.list_mk_conj L)));
fun tgoal thy = tgoalw thy [];
(*---------------------------------------------------------------------------
* Three postprocessors are applied to the definition. It
* attempts to prove wellfoundedness of the given relation, simplifies the
* non-proved termination conditions, and finally attempts to prove the
* simplified termination conditions.
*--------------------------------------------------------------------------*)
fun std_postprocessor strict cs ss wfs =
Prim.postprocess strict
{wf_tac = REPEAT (ares_tac wfs 1),
terminator = asm_simp_tac ss 1
THEN TRY (silent_arith_tac 1 ORELSE
fast_tac (cs addSDs [not0_implies_Suc] addss ss) 1),
simplifier = Rules.simpl_conv ss []};
val concl = #2 o Rules.dest_thm;
(*---------------------------------------------------------------------------
* Postprocess a definition made by "define". This is a separate stage of
* processing from the definition stage.
*---------------------------------------------------------------------------*)
local
structure R = Rules
structure U = Utils
(* The rest of these local definitions are for the tricky nested case *)
val solved = not o can S.dest_eq o #2 o S.strip_forall o concl
fun id_thm th =
let val {lhs,rhs} = S.dest_eq (#2 (S.strip_forall (#2 (R.dest_thm th))));
in lhs aconv rhs end
handle U.ERR _ => false;
fun prover s = prove_goal HOL.thy s (fn _ => [fast_tac HOL_cs 1]);
val P_imp_P_iff_True = prover "P --> (P= True)" RS mp;
val P_imp_P_eq_True = P_imp_P_iff_True RS eq_reflection;
fun mk_meta_eq r = case concl_of r of
Const("==",_)$_$_ => r
| _ $(Const("op =",_)$_$_) => r RS eq_reflection
| _ => r RS P_imp_P_eq_True
(*Is this the best way to invoke the simplifier??*)
fun rewrite L = rewrite_rule (map mk_meta_eq (List.filter(not o id_thm) L))
fun join_assums th =
let val {sign,...} = rep_thm th
val tych = cterm_of sign
val {lhs,rhs} = S.dest_eq(#2 (S.strip_forall (concl th)))
val cntxtl = (#1 o S.strip_imp) lhs (* cntxtl should = cntxtr *)
val cntxtr = (#1 o S.strip_imp) rhs (* but union is solider *)
val cntxt = gen_union (op aconv) (cntxtl, cntxtr)
in
R.GEN_ALL
(R.DISCH_ALL
(rewrite (map (R.ASSUME o tych) cntxt) (R.SPEC_ALL th)))
end
val gen_all = S.gen_all
in
fun proof_stage strict cs ss wfs theory {f, R, rules, full_pats_TCs, TCs} =
let
val _ = message "Proving induction theorem ..."
val ind = Prim.mk_induction theory {fconst=f, R=R, SV=[], pat_TCs_list=full_pats_TCs}
val _ = message "Postprocessing ...";
val {rules, induction, nested_tcs} =
std_postprocessor strict cs ss wfs theory {rules=rules, induction=ind, TCs=TCs}
in
case nested_tcs
of [] => {induction=induction, rules=rules,tcs=[]}
| L => let val dummy = message "Simplifying nested TCs ..."
val (solved,simplified,stubborn) =
fold_rev (fn th => fn (So,Si,St) =>
if (id_thm th) then (So, Si, th::St) else
if (solved th) then (th::So, Si, St)
else (So, th::Si, St)) nested_tcs ([],[],[])
val simplified' = map join_assums simplified
val dummy = (Prim.trace_thms "solved =" solved;
Prim.trace_thms "simplified' =" simplified')
val rewr = full_simplify (ss addsimps (solved @ simplified'));
val dummy = Prim.trace_thms "Simplifying the induction rule..."
[induction]
val induction' = rewr induction
val dummy = Prim.trace_thms "Simplifying the recursion rules..."
[rules]
val rules' = rewr rules
val _ = message "... Postprocessing finished";
in
{induction = induction',
rules = rules',
tcs = map (gen_all o S.rhs o #2 o S.strip_forall o concl)
(simplified@stubborn)}
end
end;
(*lcp: curry the predicate of the induction rule*)
fun curry_rule rl =
SplitRule.split_rule_var (Term.head_of (HOLogic.dest_Trueprop (concl_of rl))) rl;
(*lcp: put a theorem into Isabelle form, using meta-level connectives*)
val meta_outer =
curry_rule o standard o
rule_by_tactic (REPEAT (FIRSTGOAL (resolve_tac [allI, impI, conjI] ORELSE' etac conjE)));
(*Strip off the outer !P*)
val spec'= read_instantiate [("x","P::?'b=>bool")] spec;
fun tracing true _ = ()
| tracing false msg = writeln msg;
fun simplify_defn strict thy cs ss congs wfs id pats def0 =
let val def = Thm.freezeT def0 RS meta_eq_to_obj_eq
val {rules,rows,TCs,full_pats_TCs} =
Prim.post_definition congs (thy, (def,pats))
val {lhs=f,rhs} = S.dest_eq (concl def)
val (_,[R,_]) = S.strip_comb rhs
val dummy = Prim.trace_thms "congs =" congs
(*the next step has caused simplifier looping in some cases*)
val {induction, rules, tcs} =
proof_stage strict cs ss wfs thy
{f = f, R = R, rules = rules,
full_pats_TCs = full_pats_TCs,
TCs = TCs}
val rules' = map (standard o ObjectLogic.rulify_no_asm)
(R.CONJUNCTS rules)
in {induct = meta_outer (ObjectLogic.rulify_no_asm (induction RS spec')),
rules = ListPair.zip(rules', rows),
tcs = (termination_goals rules') @ tcs}
end
handle U.ERR {mesg,func,module} =>
error (mesg ^
"\n (In TFL function " ^ module ^ "." ^ func ^ ")");
(* Derive the initial equations from the case-split rules to meet the
users specification of the recursive function.
Note: We don't do this if the wf conditions fail to be solved, as each
case may have a different wf condition. We could group the conditions
together 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, for now, I
prefer leaving more fine-grain control to the user.
-- Lucas Dixon, Aug 2004 *)
local
fun get_related_thms i =
List.mapPartial ((fn (r,x) => if x = i then SOME r else NONE));
fun solve_eq (th, [], i) =
error "derive_init_eqs: missing rules"
| solve_eq (th, [a], i) = [(a, i)]
| solve_eq (th, splitths as (_ :: _), i) =
(writeln "Proving unsplit equation...";
[((standard o ObjectLogic.rulify_no_asm)
(CaseSplit.splitto splitths th), i)])
(* if there's an error, pretend nothing happened with this definition
We should probably print something out so that the user knows...? *)
handle ERROR s =>
(warning ("recdef (solve_eq): " ^ s); map (fn x => (x,i)) splitths);
in
fun derive_init_eqs sgn rules eqs =
let
val eqths = map (Thm.trivial o (Thm.cterm_of sgn) o HOLogic.mk_Trueprop)
eqs
fun countlist l =
(rev o snd o (Library.foldl (fn ((i,L), e) => (i + 1,(e,i) :: L)))) ((0,[]), l)
in
List.concat (map (fn (e,i) => solve_eq (e, (get_related_thms i rules), i))
(countlist eqths))
end;
end;
(*---------------------------------------------------------------------------
* Defining a function with an associated termination relation.
*---------------------------------------------------------------------------*)
fun define_i strict thy cs ss congs wfs fid R eqs =
let val {functional,pats} = Prim.mk_functional thy eqs
val (thy, def) = Prim.wfrec_definition0 thy (Sign.base_name fid) R functional
val {induct, rules, tcs} =
simplify_defn strict thy cs ss congs wfs fid pats def
val rules' =
if strict then derive_init_eqs (Theory.sign_of thy) rules eqs
else rules
in (thy, {rules = rules', induct = induct, tcs = tcs}) end;
fun define strict thy cs ss congs wfs fid R seqs =
define_i strict thy cs ss congs wfs fid (Sign.read_term thy R) (map (Sign.read_term thy) seqs)
handle U.ERR {mesg,...} => error mesg;
(*---------------------------------------------------------------------------
*
* Definitions with synthesized termination relation
*
*---------------------------------------------------------------------------*)
fun func_of_cond_eqn tm =
#1 (S.strip_comb (#lhs (S.dest_eq (#2 (S.strip_forall (#2 (S.strip_imp tm)))))));
fun defer_i thy congs fid eqs =
let val {rules,R,theory,full_pats_TCs,SV,...} =
Prim.lazyR_def thy (Sign.base_name fid) congs eqs
val f = func_of_cond_eqn (concl (R.CONJUNCT1 rules handle U.ERR _ => rules));
val dummy = message "Proving induction theorem ...";
val induction = Prim.mk_induction theory
{fconst=f, R=R, SV=SV, pat_TCs_list=full_pats_TCs}
in (theory,
(*return the conjoined induction rule and recursion equations,
with assumptions remaining to discharge*)
standard (induction RS (rules RS conjI)))
end
fun defer thy congs fid seqs =
defer_i thy congs fid (map (Sign.read_term thy) seqs)
handle U.ERR {mesg,...} => error mesg;
end;
end;