(* Title: Pure/tactic.ML
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Fundamental tactics.
*)
signature BASIC_TACTIC =
sig
val trace_goalno_tac: (int -> tactic) -> int -> tactic
val rule_by_tactic: Proof.context -> tactic -> thm -> thm
val assume_tac: int -> tactic
val eq_assume_tac: int -> tactic
val compose_tac: (bool * thm * int) -> int -> tactic
val make_elim: thm -> thm
val biresolve_tac: (bool * thm) list -> int -> tactic
val resolve_tac: thm list -> int -> tactic
val eresolve_tac: thm list -> int -> tactic
val forward_tac: thm list -> int -> tactic
val dresolve_tac: thm list -> int -> tactic
val atac: int -> tactic
val rtac: thm -> int -> tactic
val dtac: thm -> int -> tactic
val etac: thm -> int -> tactic
val ftac: thm -> int -> tactic
val ares_tac: thm list -> int -> tactic
val solve_tac: thm list -> int -> tactic
val bimatch_tac: (bool * thm) list -> int -> tactic
val match_tac: thm list -> int -> tactic
val ematch_tac: thm list -> int -> tactic
val dmatch_tac: thm list -> int -> tactic
val flexflex_tac: tactic
val distinct_subgoal_tac: int -> tactic
val distinct_subgoals_tac: tactic
val cut_tac: thm -> int -> tactic
val cut_rules_tac: thm list -> int -> tactic
val cut_facts_tac: thm list -> int -> tactic
val filter_thms: (term * term -> bool) -> int * term * thm list -> thm list
val biresolution_from_nets_tac: ('a list -> (bool * thm) list) ->
bool -> 'a Net.net * 'a Net.net -> int -> tactic
val biresolve_from_nets_tac: (int * (bool * thm)) Net.net * (int * (bool * thm)) Net.net ->
int -> tactic
val bimatch_from_nets_tac: (int * (bool * thm)) Net.net * (int * (bool * thm)) Net.net ->
int -> tactic
val net_biresolve_tac: (bool * thm) list -> int -> tactic
val net_bimatch_tac: (bool * thm) list -> int -> tactic
val filt_resolve_tac: thm list -> int -> int -> tactic
val resolve_from_net_tac: (int * thm) Net.net -> int -> tactic
val match_from_net_tac: (int * thm) Net.net -> int -> tactic
val net_resolve_tac: thm list -> int -> tactic
val net_match_tac: thm list -> int -> tactic
val subgoals_of_brl: bool * thm -> int
val lessb: (bool * thm) * (bool * thm) -> bool
val rename_tac: string list -> int -> tactic
val rotate_tac: int -> int -> tactic
val defer_tac: int -> tactic
val prefer_tac: int -> tactic
val filter_prems_tac: (term -> bool) -> int -> tactic
end;
signature TACTIC =
sig
include BASIC_TACTIC
val insert_tagged_brl: 'a * (bool * thm) ->
('a * (bool * thm)) Net.net * ('a * (bool * thm)) Net.net ->
('a * (bool * thm)) Net.net * ('a * (bool * thm)) Net.net
val build_netpair: (int * (bool * thm)) Net.net * (int * (bool * thm)) Net.net ->
(bool * thm) list -> (int * (bool * thm)) Net.net * (int * (bool * thm)) Net.net
val delete_tagged_brl: bool * thm ->
('a * (bool * thm)) Net.net * ('a * (bool * thm)) Net.net ->
('a * (bool * thm)) Net.net * ('a * (bool * thm)) Net.net
val eq_kbrl: ('a * (bool * thm)) * ('a * (bool * thm)) -> bool
val build_net: thm list -> (int * thm) Net.net
end;
structure Tactic: TACTIC =
struct
(*Discover which goal is chosen: SOMEGOAL(trace_goalno_tac tac) *)
fun trace_goalno_tac tac i st =
case Seq.pull(tac i st) of
NONE => Seq.empty
| seqcell => (tracing ("Subgoal " ^ string_of_int i ^ " selected");
Seq.make(fn()=> seqcell));
(*Makes a rule by applying a tactic to an existing rule*)
fun rule_by_tactic ctxt tac rl =
let
val ctxt' = Variable.declare_thm rl ctxt;
val ((_, [st]), ctxt'') = Variable.import true [rl] ctxt';
in
(case Seq.pull (tac st) of
NONE => raise THM ("rule_by_tactic", 0, [rl])
| SOME (st', _) => zero_var_indexes (singleton (Variable.export ctxt'' ctxt') st'))
end;
(*** Basic tactics ***)
(*** The following fail if the goal number is out of range:
thus (REPEAT (resolve_tac rules i)) stops once subgoal i disappears. *)
(*Solve subgoal i by assumption*)
fun assume_tac i = PRIMSEQ (Thm.assumption i);
(*Solve subgoal i by assumption, using no unification*)
fun eq_assume_tac i = PRIMITIVE (Thm.eq_assumption i);
(** Resolution/matching tactics **)
(*The composition rule/state: no lifting or var renaming.
The arg = (bires_flg, orule, m); see Thm.bicompose for explanation.*)
fun compose_tac arg i = PRIMSEQ (Thm.bicompose false arg i);
(*Converts a "destruct" rule like P&Q==>P to an "elimination" rule
like [| P&Q; P==>R |] ==> R *)
fun make_elim rl = zero_var_indexes (rl RS revcut_rl);
(*Attack subgoal i by resolution, using flags to indicate elimination rules*)
fun biresolve_tac brules i = PRIMSEQ (Thm.biresolution false brules i);
(*Resolution: the simple case, works for introduction rules*)
fun resolve_tac rules = biresolve_tac (map (pair false) rules);
(*Resolution with elimination rules only*)
fun eresolve_tac rules = biresolve_tac (map (pair true) rules);
(*Forward reasoning using destruction rules.*)
fun forward_tac rls = resolve_tac (map make_elim rls) THEN' assume_tac;
(*Like forward_tac, but deletes the assumption after use.*)
fun dresolve_tac rls = eresolve_tac (map make_elim rls);
(*Shorthand versions: for resolution with a single theorem*)
val atac = assume_tac;
fun rtac rl = resolve_tac [rl];
fun dtac rl = dresolve_tac [rl];
fun etac rl = eresolve_tac [rl];
fun ftac rl = forward_tac [rl];
(*Use an assumption or some rules ... A popular combination!*)
fun ares_tac rules = assume_tac ORELSE' resolve_tac rules;
fun solve_tac rules = resolve_tac rules THEN_ALL_NEW assume_tac;
(*Matching tactics -- as above, but forbid updating of state*)
fun bimatch_tac brules i = PRIMSEQ (Thm.biresolution true brules i);
fun match_tac rules = bimatch_tac (map (pair false) rules);
fun ematch_tac rules = bimatch_tac (map (pair true) rules);
fun dmatch_tac rls = ematch_tac (map make_elim rls);
(*Smash all flex-flex disagreement pairs in the proof state.*)
val flexflex_tac = PRIMSEQ Thm.flexflex_rule;
(*Remove duplicate subgoals.*)
val permute_tac = PRIMITIVE oo Thm.permute_prems;
fun distinct_tac (i, k) =
permute_tac 0 (i - 1) THEN
permute_tac 1 (k - 1) THEN
DETERM (PRIMSEQ (fn st =>
Thm.compose_no_flatten false (st, 0) 1
(Drule.incr_indexes st Drule.distinct_prems_rl))) THEN
permute_tac 1 (1 - k) THEN
permute_tac 0 (1 - i);
fun distinct_subgoal_tac i st =
(case drop (i - 1) (Thm.prems_of st) of
[] => no_tac st
| A :: Bs =>
st |> EVERY (fold (fn (B, k) =>
if A aconv B then cons (distinct_tac (i, k)) else I) (Bs ~~ (1 upto length Bs)) []));
fun distinct_subgoals_tac state =
let
val goals = Thm.prems_of state;
val dups = distinct (eq_fst (op aconv)) (goals ~~ (1 upto length goals));
in EVERY (rev (map (distinct_subgoal_tac o snd) dups)) state end;
(*** Applications of cut_rl ***)
(*The conclusion of the rule gets assumed in subgoal i,
while subgoal i+1,... are the premises of the rule.*)
fun cut_tac rule i = rtac cut_rl i THEN rtac rule (i + 1);
(*"Cut" a list of rules into the goal. Their premises will become new
subgoals.*)
fun cut_rules_tac ths i = EVERY (map (fn th => cut_tac th i) ths);
(*As above, but inserts only facts (unconditional theorems);
generates no additional subgoals. *)
fun cut_facts_tac ths = cut_rules_tac (filter Thm.no_prems ths);
(**** Indexing and filtering of theorems ****)
(*Returns the list of potentially resolvable theorems for the goal "prem",
using the predicate could(subgoal,concl).
Resulting list is no longer than "limit"*)
fun filter_thms could (limit, prem, ths) =
let val pb = Logic.strip_assums_concl prem; (*delete assumptions*)
fun filtr (limit, []) = []
| filtr (limit, th::ths) =
if limit=0 then []
else if could(pb, concl_of th) then th :: filtr(limit-1, ths)
else filtr(limit,ths)
in filtr(limit,ths) end;
(*** biresolution and resolution using nets ***)
(** To preserve the order of the rules, tag them with increasing integers **)
(*insert one tagged brl into the pair of nets*)
fun insert_tagged_brl (kbrl as (k, (eres, th))) (inet, enet) =
if eres then
(case try Thm.major_prem_of th of
SOME prem => (inet, Net.insert_term (K false) (prem, kbrl) enet)
| NONE => error "insert_tagged_brl: elimination rule with no premises")
else (Net.insert_term (K false) (concl_of th, kbrl) inet, enet);
(*build a pair of nets for biresolution*)
fun build_netpair netpair brls =
fold_rev insert_tagged_brl (tag_list 1 brls) netpair;
(*delete one kbrl from the pair of nets*)
fun eq_kbrl ((_, (_, th)), (_, (_, th'))) = Thm.eq_thm_prop (th, th')
fun delete_tagged_brl (brl as (eres, th)) (inet, enet) =
(if eres then
(case try Thm.major_prem_of th of
SOME prem => (inet, Net.delete_term eq_kbrl (prem, ((), brl)) enet)
| NONE => (inet, enet)) (*no major premise: ignore*)
else (Net.delete_term eq_kbrl (Thm.concl_of th, ((), brl)) inet, enet))
handle Net.DELETE => (inet,enet);
(*biresolution using a pair of nets rather than rules.
function "order" must sort and possibly filter the list of brls.
boolean "match" indicates matching or unification.*)
fun biresolution_from_nets_tac order match (inet,enet) =
SUBGOAL
(fn (prem,i) =>
let val hyps = Logic.strip_assums_hyp prem
and concl = Logic.strip_assums_concl prem
val kbrls = Net.unify_term inet concl @ maps (Net.unify_term enet) hyps
in PRIMSEQ (Thm.biresolution match (order kbrls) i) end);
(*versions taking pre-built nets. No filtering of brls*)
val biresolve_from_nets_tac = biresolution_from_nets_tac order_list false;
val bimatch_from_nets_tac = biresolution_from_nets_tac order_list true;
(*fast versions using nets internally*)
val net_biresolve_tac =
biresolve_from_nets_tac o build_netpair(Net.empty,Net.empty);
val net_bimatch_tac =
bimatch_from_nets_tac o build_netpair(Net.empty,Net.empty);
(*** Simpler version for resolve_tac -- only one net, and no hyps ***)
(*insert one tagged rl into the net*)
fun insert_krl (krl as (k,th)) =
Net.insert_term (K false) (concl_of th, krl);
(*build a net of rules for resolution*)
fun build_net rls =
fold_rev insert_krl (tag_list 1 rls) Net.empty;
(*resolution using a net rather than rules; pred supports filt_resolve_tac*)
fun filt_resolution_from_net_tac match pred net =
SUBGOAL
(fn (prem,i) =>
let val krls = Net.unify_term net (Logic.strip_assums_concl prem)
in
if pred krls
then PRIMSEQ
(Thm.biresolution match (map (pair false) (order_list krls)) i)
else no_tac
end);
(*Resolve the subgoal using the rules (making a net) unless too flexible,
which means more than maxr rules are unifiable. *)
fun filt_resolve_tac rules maxr =
let fun pred krls = length krls <= maxr
in filt_resolution_from_net_tac false pred (build_net rules) end;
(*versions taking pre-built nets*)
val resolve_from_net_tac = filt_resolution_from_net_tac false (K true);
val match_from_net_tac = filt_resolution_from_net_tac true (K true);
(*fast versions using nets internally*)
val net_resolve_tac = resolve_from_net_tac o build_net;
val net_match_tac = match_from_net_tac o build_net;
(*** For Natural Deduction using (bires_flg, rule) pairs ***)
(*The number of new subgoals produced by the brule*)
fun subgoals_of_brl (true,rule) = nprems_of rule - 1
| subgoals_of_brl (false,rule) = nprems_of rule;
(*Less-than test: for sorting to minimize number of new subgoals*)
fun lessb (brl1,brl2) = subgoals_of_brl brl1 < subgoals_of_brl brl2;
(*Renaming of parameters in a subgoal*)
fun rename_tac xs i =
case Library.find_first (not o Symbol_Pos.is_identifier) xs of
SOME x => error ("Not an identifier: " ^ x)
| NONE => PRIMITIVE (Thm.rename_params_rule (xs, i));
(*rotate_tac n i: rotate the assumptions of subgoal i by n positions, from
right to left if n is positive, and from left to right if n is negative.*)
fun rotate_tac 0 i = all_tac
| rotate_tac k i = PRIMITIVE (Thm.rotate_rule k i);
(*Rotates the given subgoal to be the last.*)
fun defer_tac i = PRIMITIVE (Thm.permute_prems (i - 1) 1);
(*Rotates the given subgoal to be the first.*)
fun prefer_tac i = PRIMITIVE (Thm.permute_prems (i - 1) 1 #> Thm.permute_prems 0 ~1);
(* remove premises that do not satisfy p; fails if all prems satisfy p *)
fun filter_prems_tac p =
let fun Then NONE tac = SOME tac
| Then (SOME tac) tac' = SOME(tac THEN' tac');
fun thins H (tac,n) =
if p H then (tac,n+1)
else (Then tac (rotate_tac n THEN' etac thin_rl),0);
in SUBGOAL(fn (subg,n) =>
let val Hs = Logic.strip_assums_hyp subg
in case fst(fold thins Hs (NONE,0)) of
NONE => no_tac | SOME tac => tac n
end)
end;
end;
structure Basic_Tactic: BASIC_TACTIC = Tactic;
open Basic_Tactic;