(* Title: HOL/Tools/TFL/post.ML
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 define_i: bool -> Proof.context -> thm list -> thm list -> xstring -> term -> term list ->
theory -> {lhs: term, rules: (thm * int) list, induct: thm, tcs: term list} * theory
val define: bool -> Proof.context -> thm list -> thm list -> xstring -> string -> string list ->
theory -> {lhs: term, rules: (thm * int) list, induct: thm, tcs: term list} * theory
val defer_i: thm list -> xstring -> term list -> theory -> thm * theory
val defer: thm list -> xstring -> string list -> theory -> thm * theory
end;
structure Tfl: TFL =
struct
(* 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.legacy_freeze o HOLogic.dest_Trueprop)
(fold_rev (union (op aconv) o prems_of) rules []);
(*---------------------------------------------------------------------------
* 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 ctxt wfs =
Prim.postprocess strict
{wf_tac = REPEAT (ares_tac wfs 1),
terminator =
asm_simp_tac ctxt 1
THEN TRY (Arith_Data.arith_tac ctxt 1 ORELSE
fast_force_tac (ctxt addSDs [@{thm not0_implies_Suc}]) 1),
simplifier = Rules.simpl_conv ctxt []};
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
(* The rest of these local definitions are for the tricky nested case *)
val solved = not o can USyntax.dest_eq o #2 o USyntax.strip_forall o concl
fun id_thm th =
let val {lhs,rhs} = USyntax.dest_eq (#2 (USyntax.strip_forall (#2 (Rules.dest_thm th))));
in lhs aconv rhs end
handle Utils.ERR _ => false;
val P_imp_P_eq_True = @{thm eqTrueI} RS eq_reflection;
fun mk_meta_eq r = case concl_of r of
Const("==",_)$_$_ => r
| _ $(Const(@{const_name HOL.eq},_)$_$_) => r RS eq_reflection
| _ => r RS P_imp_P_eq_True
(*Is this the best way to invoke the simplifier??*)
fun rewrite ctxt L = rewrite_rule ctxt (map mk_meta_eq (filter_out id_thm L))
fun join_assums ctxt th =
let val thy = Thm.theory_of_thm th
val tych = cterm_of thy
val {lhs,rhs} = USyntax.dest_eq(#2 (USyntax.strip_forall (concl th)))
val cntxtl = (#1 o USyntax.strip_imp) lhs (* cntxtl should = cntxtr *)
val cntxtr = (#1 o USyntax.strip_imp) rhs (* but union is solider *)
val cntxt = union (op aconv) cntxtl cntxtr
in
Rules.GEN_ALL
(Rules.DISCH_ALL
(rewrite ctxt (map (Rules.ASSUME o tych) cntxt) (Rules.SPEC_ALL th)))
end
val gen_all = USyntax.gen_all
in
fun proof_stage strict ctxt wfs theory {f, R, rules, full_pats_TCs, TCs} =
let
val _ = writeln "Proving induction theorem ..."
val ind = Prim.mk_induction theory {fconst=f, R=R, SV=[], pat_TCs_list=full_pats_TCs}
val _ = writeln "Postprocessing ...";
val {rules, induction, nested_tcs} =
std_postprocessor strict ctxt wfs theory {rules=rules, induction=ind, TCs=TCs}
in
case nested_tcs
of [] => {induction=induction, rules=rules,tcs=[]}
| L => let val dummy = writeln "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 ctxt) simplified
val dummy = (Prim.trace_thms "solved =" solved;
Prim.trace_thms "simplified' =" simplified')
val rewr = full_simplify (ctxt 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 _ = writeln "... Postprocessing finished";
in
{induction = induction',
rules = rules',
tcs = map (gen_all o USyntax.rhs o #2 o USyntax.strip_forall o concl)
(simplified@stubborn)}
end
end;
(*lcp: curry the predicate of the induction rule*)
fun curry_rule ctxt rl =
Split_Rule.split_rule_var ctxt (Term.head_of (HOLogic.dest_Trueprop (concl_of rl))) rl;
(*lcp: put a theorem into Isabelle form, using meta-level connectives*)
fun meta_outer ctxt =
curry_rule ctxt o Drule.export_without_context o
rule_by_tactic ctxt (REPEAT (FIRSTGOAL (resolve_tac [allI, impI, conjI] ORELSE' etac conjE)));
(*Strip off the outer !P*)
val spec'= Rule_Insts.read_instantiate @{context} [(("x", 0), "P::'b=>bool")] [] spec;
fun tracing true _ = ()
| tracing false msg = writeln msg;
fun simplify_defn strict thy ctxt congs wfs id pats def0 =
let
val def = Thm.unvarify_global def0 RS meta_eq_to_obj_eq
val {rules,rows,TCs,full_pats_TCs} =
Prim.post_definition congs thy ctxt (def, pats)
val {lhs=f,rhs} = USyntax.dest_eq (concl def)
val (_,[R,_]) = USyntax.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 ctxt wfs thy
{f = f, R = R, rules = rules,
full_pats_TCs = full_pats_TCs,
TCs = TCs}
val rules' = map (Drule.export_without_context o Object_Logic.rulify_no_asm ctxt)
(Rules.CONJUNCTS rules)
in {induct = meta_outer ctxt (Object_Logic.rulify_no_asm ctxt (induction RS spec')),
rules = ListPair.zip(rules', rows),
tcs = (termination_goals rules') @ tcs}
end
handle Utils.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. *)
local
fun get_related_thms i =
map_filter ((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 ctxt (th, splitths, i) =
(writeln "Proving unsplit equation...";
[((Drule.export_without_context o Object_Logic.rulify_no_asm ctxt)
(CaseSplit.splitto ctxt splitths th), i)])
handle ERROR s =>
(warning ("recdef (solve_eq): " ^ s); map (fn x => (x,i)) splitths);
in
fun derive_init_eqs ctxt rules eqs =
map (Thm.trivial o Thm.cterm_of (Proof_Context.theory_of ctxt) o HOLogic.mk_Trueprop) eqs
|> map_index (fn (i, e) => solve_eq ctxt (e, (get_related_thms i rules), i))
|> flat;
end;
(*---------------------------------------------------------------------------
* Defining a function with an associated termination relation.
*---------------------------------------------------------------------------*)
fun define_i strict ctxt congs wfs fid R eqs thy =
let val {functional,pats} = Prim.mk_functional thy eqs
val (thy, def) = Prim.wfrec_definition0 thy fid R functional
val ctxt = Proof_Context.transfer thy ctxt
val (lhs, _) = Logic.dest_equals (prop_of def)
val {induct, rules, tcs} = simplify_defn strict thy ctxt congs wfs fid pats def
val rules' =
if strict then derive_init_eqs ctxt rules eqs
else rules
in ({lhs = lhs, rules = rules', induct = induct, tcs = tcs}, thy) end;
fun define strict ctxt congs wfs fid R seqs thy =
define_i strict ctxt congs wfs fid
(Syntax.read_term ctxt R) (map (Syntax.read_term ctxt) seqs) thy
handle Utils.ERR {mesg,...} => error mesg;
(*---------------------------------------------------------------------------
*
* Definitions with synthesized termination relation
*
*---------------------------------------------------------------------------*)
fun func_of_cond_eqn tm =
#1 (USyntax.strip_comb (#lhs (USyntax.dest_eq (#2 (USyntax.strip_forall (#2 (USyntax.strip_imp tm)))))));
fun defer_i congs fid eqs thy =
let val {rules,R,theory,full_pats_TCs,SV,...} = Prim.lazyR_def thy fid congs eqs
val f = func_of_cond_eqn (concl (Rules.CONJUNCT1 rules handle Utils.ERR _ => rules));
val dummy = writeln "Proving induction theorem ...";
val induction = Prim.mk_induction theory
{fconst=f, R=R, SV=SV, pat_TCs_list=full_pats_TCs}
in
(*return the conjoined induction rule and recursion equations,
with assumptions remaining to discharge*)
(Drule.export_without_context (induction RS (rules RS conjI)), theory)
end
fun defer congs fid seqs thy =
defer_i congs fid (map (Syntax.read_term_global thy) seqs) thy
handle Utils.ERR {mesg,...} => error mesg;
end;
end;