(* 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: Proof.context -> int -> tactic
val eq_assume_tac: int -> tactic
val compose_tac: Proof.context -> (bool * thm * int) -> int -> tactic
val make_elim: thm -> thm
val biresolve0_tac: (bool * thm) list -> int -> tactic
val biresolve_tac: Proof.context -> (bool * thm) list -> int -> tactic
val resolve0_tac: thm list -> int -> tactic
val resolve_tac: Proof.context -> thm list -> int -> tactic
val eresolve0_tac: thm list -> int -> tactic
val eresolve_tac: Proof.context -> thm list -> int -> tactic
val forward_tac: Proof.context -> thm list -> int -> tactic
val dresolve0_tac: thm list -> int -> tactic
val dresolve_tac: Proof.context -> thm list -> int -> tactic
val ares_tac: Proof.context -> thm list -> int -> tactic
val solve_tac: Proof.context -> thm list -> int -> tactic
val bimatch_tac: Proof.context -> (bool * thm) list -> int -> tactic
val match_tac: Proof.context -> thm list -> int -> tactic
val ematch_tac: Proof.context -> thm list -> int -> tactic
val dmatch_tac: Proof.context -> thm list -> int -> tactic
val flexflex_tac: Proof.context -> 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: Proof.context ->
('a list -> (bool * thm) list) -> bool -> 'a Net.net * 'a Net.net -> int -> tactic
val biresolve_from_nets_tac: Proof.context ->
(int * (bool * thm)) Net.net * (int * (bool * thm)) Net.net -> int -> tactic
val bimatch_from_nets_tac: Proof.context ->
(int * (bool * thm)) Net.net * (int * (bool * thm)) Net.net -> int -> tactic
val filt_resolve_from_net_tac: Proof.context -> int -> (int * thm) Net.net -> int -> tactic
val resolve_from_net_tac: Proof.context -> (int * thm) Net.net -> int -> tactic
val match_from_net_tac: Proof.context -> (int * thm) Net.net -> 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: Proof.context -> (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 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 thy = Proof_Context.theory_of ctxt;
val ctxt' = Variable.declare_thm rl ctxt;
val ((_, [st]), ctxt'') = Variable.import true [Thm.transfer thy 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 ctxt i = PRIMSEQ (Thm.assumption (SOME ctxt) 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 ctxt arg i =
PRIMSEQ (Thm.bicompose (SOME ctxt) {flatten = true, match = false, incremented = false} arg i);
(*Converts a "destruct" rule like P \<and> Q \<Longrightarrow> P to an "elimination" rule
like \<lbrakk>P \<and> Q; P \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R *)
fun make_elim rl = zero_var_indexes (rl RS revcut_rl);
(*Attack subgoal i by resolution, using flags to indicate elimination rules*)
fun biresolve0_tac brules i = PRIMSEQ (Thm.biresolution NONE false brules i);
fun biresolve_tac ctxt brules i = PRIMSEQ (Thm.biresolution (SOME ctxt) false brules i);
(*Resolution: the simple case, works for introduction rules*)
fun resolve0_tac rules = biresolve0_tac (map (pair false) rules);
fun resolve_tac ctxt rules = biresolve_tac ctxt (map (pair false) rules);
(*Resolution with elimination rules only*)
fun eresolve0_tac rules = biresolve0_tac (map (pair true) rules);
fun eresolve_tac ctxt rules = biresolve_tac ctxt (map (pair true) rules);
(*Forward reasoning using destruction rules.*)
fun forward_tac ctxt rls = resolve_tac ctxt (map make_elim rls) THEN' assume_tac ctxt;
(*Like forward_tac, but deletes the assumption after use.*)
fun dresolve0_tac rls = eresolve0_tac (map make_elim rls);
fun dresolve_tac ctxt rls = eresolve_tac ctxt (map make_elim rls);
(*Use an assumption or some rules*)
fun ares_tac ctxt rules = assume_tac ctxt ORELSE' resolve_tac ctxt rules;
fun solve_tac ctxt rules = resolve_tac ctxt rules THEN_ALL_NEW assume_tac ctxt;
(*Matching tactics -- as above, but forbid updating of state*)
fun bimatch_tac ctxt brules i = PRIMSEQ (Thm.biresolution (SOME ctxt) true brules i);
fun match_tac ctxt rules = bimatch_tac ctxt (map (pair false) rules);
fun ematch_tac ctxt rules = bimatch_tac ctxt (map (pair true) rules);
fun dmatch_tac ctxt rls = ematch_tac ctxt (map make_elim rls);
(*Smash all flex-flex disagreement pairs in the proof state.*)
fun flexflex_tac ctxt = PRIMSEQ (Thm.flexflex_rule (SOME ctxt));
(*Remove duplicate subgoals.*)
fun distinct_subgoals_tac st =
let
val subgoals = Thm.cprems_of st;
val (tab, n) =
(subgoals, (Ctermtab.empty, 0)) |-> fold (fn ct => fn (tab, i) =>
if Ctermtab.defined tab ct then (tab, i)
else (Ctermtab.update (ct, i) tab, i + 1));
val st' =
if n = length subgoals then st
else
let
val thy = Thm.theory_of_thm st;
fun cert_prop i = Thm.global_cterm_of thy (Free (Name.bound i, propT));
val As = map (cert_prop o the o Ctermtab.lookup tab) subgoals;
val As' = map cert_prop (0 upto (n - 1));
val C = cert_prop n;
val template = Drule.list_implies (As, C);
val inst =
(dest_Free (Thm.term_of C), Thm.cconcl_of st) ::
Ctermtab.fold (fn (ct, i) => cons ((Name.bound i, propT), ct)) tab [];
in
Thm.assume template
|> fold (Thm.elim_implies o Thm.assume) As
|> fold_rev Thm.implies_intr As'
|> Thm.implies_intr template
|> Thm.instantiate_frees ([], inst)
|> Thm.elim_implies st
end;
in Seq.single st' 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 = resolve0_tac [cut_rl] i THEN resolve0_tac [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, Thm.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) (Thm.concl_of th, kbrl) inet, enet);
(*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 ctxt order match (inet, enet) =
SUBGOAL
(fn (prem, i) =>
let
val hyps = Logic.strip_assums_hyp prem;
val concl = Logic.strip_assums_concl prem;
val kbrls = Net.unify_term inet concl @ maps (Net.unify_term enet) hyps;
in PRIMSEQ (Thm.biresolution (SOME ctxt) match (order kbrls) i) end);
(*versions taking pre-built nets. No filtering of brls*)
fun biresolve_from_nets_tac ctxt = biresolution_from_nets_tac ctxt order_list false;
fun bimatch_from_nets_tac ctxt = biresolution_from_nets_tac ctxt order_list true;
(*** 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) (Thm.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 ctxt 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 (SOME ctxt) 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_from_net_tac ctxt maxr net =
let fun pred krls = length krls <= maxr
in filt_resolution_from_net_tac ctxt false pred net end;
(*versions taking pre-built nets*)
fun resolve_from_net_tac ctxt = filt_resolution_from_net_tac ctxt false (K true);
fun match_from_net_tac ctxt = filt_resolution_from_net_tac ctxt true (K true);
(*** For Natural Deduction using (bires_flg, rule) pairs ***)
(*The number of new subgoals produced by the brule*)
fun subgoals_of_brl (true, rule) = Thm.nprems_of rule - 1
| subgoals_of_brl (false, rule) = Thm.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 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);
(*Rotate the given subgoal to be the last.*)
fun defer_tac i = PRIMITIVE (Thm.permute_prems (i - 1) 1);
(*Rotate 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 pred; fails if all prems satisfy pred.*)
fun filter_prems_tac ctxt pred =
let
fun Then NONE tac = SOME tac
| Then (SOME tac) tac' = SOME (tac THEN' tac');
fun thins H (tac, n) =
if pred H then (tac, n + 1)
else (Then tac (rotate_tac n THEN' eresolve_tac ctxt [thin_rl]), 0);
in
SUBGOAL (fn (goal, i) =>
let val Hs = Logic.strip_assums_hyp goal in
(case fst (fold thins Hs (NONE, 0)) of
NONE => no_tac
| SOME tac => tac i)
end)
end;
end;
structure Basic_Tactic: BASIC_TACTIC = Tactic;
open Basic_Tactic;