(* Title: Pure/goal.ML
Author: Makarius
Goals in tactical theorem proving.
*)
signature BASIC_GOAL =
sig
val parallel_proofs: bool ref
val SELECT_GOAL: tactic -> int -> tactic
val CONJUNCTS: tactic -> int -> tactic
val PRECISE_CONJUNCTS: int -> tactic -> int -> tactic
end;
signature GOAL =
sig
include BASIC_GOAL
val init: cterm -> thm
val protect: thm -> thm
val conclude: thm -> thm
val finish: thm -> thm
val norm_result: thm -> thm
val future_enabled: unit -> bool
val future_result: Proof.context -> thm future -> term -> thm
val prove_internal: cterm list -> cterm -> (thm list -> tactic) -> thm
val prove_multi: Proof.context -> string list -> term list -> term list ->
({prems: thm list, context: Proof.context} -> tactic) -> thm list
val prove_future: Proof.context -> string list -> term list -> term ->
({prems: thm list, context: Proof.context} -> tactic) -> thm
val prove: Proof.context -> string list -> term list -> term ->
({prems: thm list, context: Proof.context} -> tactic) -> thm
val prove_global: theory -> string list -> term list -> term ->
({prems: thm list, context: Proof.context} -> tactic) -> thm
val extract: int -> int -> thm -> thm Seq.seq
val retrofit: int -> int -> thm -> thm -> thm Seq.seq
val conjunction_tac: int -> tactic
val precise_conjunction_tac: int -> int -> tactic
val recover_conjunction_tac: tactic
val norm_hhf_tac: int -> tactic
val compose_hhf_tac: thm -> int -> tactic
val assume_rule_tac: Proof.context -> int -> tactic
end;
structure Goal: GOAL =
struct
(** goals **)
(*
-------- (init)
C ==> #C
*)
val init =
let val A = #1 (Thm.dest_implies (Thm.cprop_of Drule.protectI))
in fn C => Thm.instantiate ([], [(A, C)]) Drule.protectI end;
(*
C
--- (protect)
#C
*)
fun protect th = Drule.comp_no_flatten (th, 0) 1 Drule.protectI;
(*
A ==> ... ==> #C
---------------- (conclude)
A ==> ... ==> C
*)
fun conclude th = Drule.comp_no_flatten (th, Thm.nprems_of th) 1 Drule.protectD;
(*
#C
--- (finish)
C
*)
fun finish th =
(case Thm.nprems_of th of
0 => conclude th
| n => raise THM ("Proof failed.\n" ^
Pretty.string_of (Pretty.chunks (Display.pretty_goals n th)) ^
("\n" ^ string_of_int n ^ " unsolved goal(s)!"), 0, [th]));
(** results **)
(* normal form *)
val norm_result =
Drule.flexflex_unique
#> MetaSimplifier.norm_hhf_protect
#> Thm.strip_shyps
#> Drule.zero_var_indexes;
(* future_enabled *)
val parallel_proofs = ref true;
fun future_enabled () =
Future.enabled () andalso ! parallel_proofs andalso is_some (Future.thread_data ());
(* future_result *)
fun future_result ctxt result prop =
let
val thy = ProofContext.theory_of ctxt;
val _ = Context.reject_draft thy;
val cert = Thm.cterm_of thy;
val certT = Thm.ctyp_of thy;
val assms = Assumption.assms_of ctxt;
val As = map Thm.term_of assms;
val xs = map Free (fold Term.add_frees (prop :: As) []);
val fixes = map cert xs;
val tfrees = fold Term.add_tfrees (prop :: As) [];
val instT = map (fn (a, S) => (certT (TVar ((a, 0), S)), certT (TFree (a, S)))) tfrees;
val global_prop =
Term.map_types Logic.varifyT (fold_rev Logic.all xs (Logic.list_implies (As, prop)));
val global_result = result |> Future.map
(Thm.adjust_maxidx_thm ~1 #>
Drule.implies_intr_list assms #>
Drule.forall_intr_list fixes #>
Thm.generalize (map #1 tfrees, []) 0);
val local_result =
Thm.future global_result (cert global_prop)
|> Thm.instantiate (instT, [])
|> Drule.forall_elim_list fixes
|> fold (Thm.elim_implies o Thm.assume) assms;
in local_result end;
(** tactical theorem proving **)
(* prove_internal -- minimal checks, no normalization of result! *)
fun prove_internal casms cprop tac =
(case SINGLE (tac (map Assumption.assume casms)) (init cprop) of
SOME th => Drule.implies_intr_list casms (finish th)
| NONE => error "Tactic failed.");
(* prove_common etc. *)
fun prove_common immediate ctxt xs asms props tac =
let
val thy = ProofContext.theory_of ctxt;
val string_of_term = Syntax.string_of_term ctxt;
val pos = Position.thread_data ();
fun err msg = cat_error msg
("The error(s) above occurred for the goal statement:\n" ^
string_of_term (Logic.list_implies (asms, Logic.mk_conjunction_list props)) ^
(case Position.str_of pos of "" => "" | s => "\n" ^ s));
fun cert_safe t = Thm.cterm_of thy (Envir.beta_norm (Term.no_dummy_patterns t))
handle TERM (msg, _) => err msg | TYPE (msg, _, _) => err msg;
val casms = map cert_safe asms;
val cprops = map cert_safe props;
val (prems, ctxt') = ctxt
|> Variable.add_fixes_direct xs
|> fold Variable.declare_term (asms @ props)
|> Assumption.add_assumes casms
||> Variable.set_body true;
val sorts = Variable.sorts_of ctxt';
val stmt = Thm.weaken_sorts sorts (Conjunction.mk_conjunction_balanced cprops);
fun result () =
(case SINGLE (tac {prems = prems, context = ctxt'}) (init stmt) of
NONE => err "Tactic failed."
| SOME st =>
let val res = finish st handle THM (msg, _, _) => err msg in
if Unify.matches_list thy [Thm.term_of stmt] [Thm.prop_of res]
then Thm.check_shyps sorts res
else err ("Proved a different theorem: " ^ string_of_term (Thm.prop_of res))
end);
val res =
if immediate orelse #maxidx (Thm.rep_cterm stmt) >= 0 orelse not (future_enabled ())
then result ()
else future_result ctxt' (Future.fork_pri ~1 result) (Thm.term_of stmt);
in
Conjunction.elim_balanced (length props) res
|> map (Assumption.export false ctxt' ctxt)
|> Variable.export ctxt' ctxt
|> map Drule.zero_var_indexes
end;
val prove_multi = prove_common true;
fun prove_future ctxt xs asms prop tac = hd (prove_common false ctxt xs asms [prop] tac);
fun prove ctxt xs asms prop tac = hd (prove_common true ctxt xs asms [prop] tac);
fun prove_global thy xs asms prop tac =
Drule.standard (prove (ProofContext.init thy) xs asms prop tac);
(** goal structure **)
(* nested goals *)
fun extract i n st =
(if i < 1 orelse n < 1 orelse i + n - 1 > Thm.nprems_of st then Seq.empty
else if n = 1 then Seq.single (Thm.cprem_of st i)
else
Seq.single (Conjunction.mk_conjunction_balanced (map (Thm.cprem_of st) (i upto i + n - 1))))
|> Seq.map (Thm.adjust_maxidx_cterm ~1 #> init);
fun retrofit i n st' st =
(if n = 1 then st
else st |> Drule.with_subgoal i (Conjunction.uncurry_balanced n))
|> Thm.compose_no_flatten false (conclude st', Thm.nprems_of st') i;
fun SELECT_GOAL tac i st =
if Thm.nprems_of st = 1 andalso i = 1 then tac st
else Seq.lifts (retrofit i 1) (Seq.maps tac (extract i 1 st)) st;
(* multiple goals *)
fun precise_conjunction_tac 0 i = eq_assume_tac i
| precise_conjunction_tac 1 i = SUBGOAL (K all_tac) i
| precise_conjunction_tac n i = PRIMITIVE (Drule.with_subgoal i (Conjunction.curry_balanced n));
val adhoc_conjunction_tac = REPEAT_ALL_NEW
(SUBGOAL (fn (goal, i) =>
if can Logic.dest_conjunction goal then rtac Conjunction.conjunctionI i
else no_tac));
val conjunction_tac = SUBGOAL (fn (goal, i) =>
precise_conjunction_tac (length (Logic.dest_conjunctions goal)) i ORELSE
TRY (adhoc_conjunction_tac i));
val recover_conjunction_tac = PRIMITIVE (fn th =>
Conjunction.uncurry_balanced (Thm.nprems_of th) th);
fun PRECISE_CONJUNCTS n tac =
SELECT_GOAL (precise_conjunction_tac n 1
THEN tac
THEN recover_conjunction_tac);
fun CONJUNCTS tac =
SELECT_GOAL (conjunction_tac 1
THEN tac
THEN recover_conjunction_tac);
(* hhf normal form *)
val norm_hhf_tac =
rtac Drule.asm_rl (*cheap approximation -- thanks to builtin Logic.flatten_params*)
THEN' SUBGOAL (fn (t, i) =>
if Drule.is_norm_hhf t then all_tac
else MetaSimplifier.rewrite_goal_tac Drule.norm_hhf_eqs i);
fun compose_hhf_tac th i st =
PRIMSEQ (Thm.bicompose false (false, Drule.lift_all (Thm.cprem_of st i) th, 0) i) st;
(* non-atomic goal assumptions *)
fun non_atomic (Const ("==>", _) $ _ $ _) = true
| non_atomic (Const ("all", _) $ _) = true
| non_atomic _ = false;
fun assume_rule_tac ctxt = norm_hhf_tac THEN' CSUBGOAL (fn (goal, i) =>
let
val ((_, goal'), ctxt') = Variable.focus goal ctxt;
val goal'' = Drule.cterm_rule (singleton (Variable.export ctxt' ctxt)) goal';
val Rs = filter (non_atomic o Thm.term_of) (Drule.strip_imp_prems goal'');
val tacs = Rs |> map (fn R =>
Tactic.etac (MetaSimplifier.norm_hhf (Thm.trivial R)) THEN_ALL_NEW assume_tac);
in fold_rev (curry op APPEND') tacs (K no_tac) i end);
end;
structure BasicGoal: BASIC_GOAL = Goal;
open BasicGoal;