(* Title: tactic
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1991 University of Cambridge
Tactics
*)
signature TACTIC =
sig
structure Tactical: TACTICAL and Net: NET
local open Tactical Tactical.Thm Net
in
val ares_tac: thm list -> int -> tactic
val asm_rewrite_goal_tac:
(meta_simpset -> tactic) -> meta_simpset -> int -> tactic
val assume_tac: int -> tactic
val atac: int ->tactic
val bimatch_from_nets_tac: (int*(bool*thm)) net * (int*(bool*thm)) net -> int -> tactic
val bimatch_tac: (bool*thm)list -> int -> tactic
val biresolve_from_nets_tac: (int*(bool*thm)) net * (int*(bool*thm)) net -> int -> tactic
val biresolve_tac: (bool*thm)list -> int -> tactic
val build_net: thm list -> (int*thm) net
val build_netpair: (bool*thm)list -> (int*(bool*thm)) net * (int*(bool*thm)) net
val compose_inst_tac: (string*string)list -> (bool*thm*int) -> int -> tactic
val compose_tac: (bool * thm * int) -> int -> tactic
val cut_facts_tac: thm list -> int -> tactic
val dmatch_tac: thm list -> int -> tactic
val dresolve_tac: thm list -> int -> tactic
val dres_inst_tac: (string*string)list -> thm -> int -> tactic
val dtac: thm -> int ->tactic
val etac: thm -> int ->tactic
val eq_assume_tac: int -> tactic
val ematch_tac: thm list -> int -> tactic
val eresolve_tac: thm list -> int -> tactic
val eres_inst_tac: (string*string)list -> thm -> int -> tactic
val filter_thms: (term*term->bool) -> int*term*thm list -> thm list
val filt_resolve_tac: thm list -> int -> int -> tactic
val flexflex_tac: tactic
val fold_goals_tac: thm list -> tactic
val fold_tac: thm list -> tactic
val forward_tac: thm list -> int -> tactic
val forw_inst_tac: (string*string)list -> thm -> int -> tactic
val is_fact: thm -> bool
val lessb: (bool * thm) * (bool * thm) -> bool
val lift_inst_rule: thm * int * (string*string)list * thm -> thm
val make_elim: thm -> thm
val match_from_net_tac: (int*thm) net -> int -> tactic
val match_tac: thm list -> int -> tactic
val metacut_tac: thm -> int -> tactic
val net_bimatch_tac: (bool*thm) list -> int -> tactic
val net_biresolve_tac: (bool*thm) list -> int -> tactic
val net_match_tac: thm list -> int -> tactic
val net_resolve_tac: thm list -> int -> tactic
val PRIMITIVE: (thm -> thm) -> tactic
val PRIMSEQ: (thm -> thm Sequence.seq) -> tactic
val prune_params_tac: tactic
val rename_tac: string -> int -> tactic
val rename_last_tac: string -> string list -> int -> tactic
val resolve_from_net_tac: (int*thm) net -> int -> tactic
val resolve_tac: thm list -> int -> tactic
val res_inst_tac: (string*string)list -> thm -> int -> tactic
val rewrite_goals_tac: thm list -> tactic
val rewrite_tac: thm list -> tactic
val rewtac: thm -> tactic
val rtac: thm -> int -> tactic
val rule_by_tactic: tactic -> thm -> thm
val subgoals_of_brl: bool * thm -> int
val subgoal_tac: string -> int -> tactic
val trace_goalno_tac: (int -> tactic) -> int -> tactic
end
end;
functor TacticFun (structure Logic: LOGIC and Drule: DRULE and
Tactical: TACTICAL and Net: NET
sharing Drule.Thm = Tactical.Thm) : TACTIC =
struct
structure Tactical = Tactical;
structure Thm = Tactical.Thm;
structure Net = Net;
structure Sequence = Thm.Sequence;
structure Sign = Thm.Sign;
local open Tactical Tactical.Thm Drule
in
(*Discover what goal is chosen: SOMEGOAL(trace_goalno_tac tac) *)
fun trace_goalno_tac tf i = Tactic (fn state =>
case Sequence.pull(tapply(tf i, state)) of
None => Sequence.null
| seqcell => (prs("Subgoal " ^ string_of_int i ^ " selected\n");
Sequence.seqof(fn()=> seqcell)));
fun string_of (a,0) = a
| string_of (a,i) = a ^ "_" ^ string_of_int i;
(*convert all Vars in a theorem to Frees -- export??*)
fun freeze th =
let val fth = freezeT th
val {prop,sign,...} = rep_thm fth
fun mk_inst (Var(v,T)) =
(Sign.cterm_of sign (Var(v,T)),
Sign.cterm_of sign (Free(string_of v, T)))
val insts = map mk_inst (term_vars prop)
in instantiate ([],insts) fth end;
(*Makes a rule by applying a tactic to an existing rule*)
fun rule_by_tactic (Tactic tf) rl =
case Sequence.pull(tf (freeze (standard rl))) of
None => raise THM("rule_by_tactic", 0, [rl])
| Some(rl',_) => standard rl';
(*** Basic tactics ***)
(*Makes a tactic whose effect on a state is given by thmfun: thm->thm seq.*)
fun PRIMSEQ thmfun = Tactic (fn state => thmfun state
handle THM _ => Sequence.null);
(*Makes a tactic whose effect on a state is given by thmfun: thm->thm.*)
fun PRIMITIVE thmfun = PRIMSEQ (Sequence.single o thmfun);
(*** 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 (assumption i);
(*Solve subgoal i by assumption, using no unification*)
fun eq_assume_tac i = PRIMITIVE (eq_assumption i);
(** Resolution/matching tactics **)
(*The composition rule/state: no lifting or var renaming.
The arg = (bires_flg, orule, m) ; see bicompose for explanation.*)
fun compose_tac arg i = PRIMSEQ (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 (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*)
fun rtac rl = resolve_tac [rl];
fun etac rl = eresolve_tac [rl];
fun dtac rl = dresolve_tac [rl];
val atac = assume_tac;
(*Use an assumption or some rules ... A popular combination!*)
fun ares_tac rules = assume_tac ORELSE' resolve_tac rules;
(*Matching tactics -- as above, but forbid updating of state*)
fun bimatch_tac brules i = PRIMSEQ (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 flexflex_rule;
(*Lift and instantiate a rule wrt the given state and subgoal number *)
fun lift_inst_rule (state, i, sinsts, rule) =
let val {maxidx,sign,...} = rep_thm state
val (_, _, Bi, _) = dest_state(state,i)
val params = Logic.strip_params Bi (*params of subgoal i*)
val params = rev(rename_wrt_term Bi params) (*as they are printed*)
val paramTs = map #2 params
and inc = maxidx+1
fun liftvar (Var ((a,j), T)) = Var((a, j+inc), paramTs---> incr_tvar inc T)
| liftvar t = raise TERM("Variable expected", [t]);
fun liftterm t = list_abs_free (params,
Logic.incr_indexes(paramTs,inc) t)
(*Lifts instantiation pair over params*)
fun liftpair (cv,ct) = (Sign.cfun liftvar cv, Sign.cfun liftterm ct)
fun lifttvar((a,i),ctyp) =
let val {T,sign} = Sign.rep_ctyp ctyp
in ((a,i+inc), Sign.ctyp_of sign (incr_tvar inc T)) end
val rts = types_sorts rule and (types,sorts) = types_sorts state
fun types'(a,~1) = (case assoc(params,a) of None => types(a,~1) | sm => sm)
| types'(ixn) = types ixn;
val (Tinsts,insts) = Sign.read_insts sign rts (types',sorts) sinsts
in instantiate (map lifttvar Tinsts, map liftpair insts)
(lift_rule (state,i) rule)
end;
(*** Resolve after lifting and instantation; may refer to parameters of the
subgoal. Fails if "i" is out of range. ***)
(*compose version: arguments are as for bicompose.*)
fun compose_inst_tac sinsts (bires_flg, rule, nsubgoal) i =
STATE ( fn state =>
compose_tac (bires_flg, lift_inst_rule (state, i, sinsts, rule),
nsubgoal) i
handle TERM (msg,_) => (writeln msg; no_tac)
| THM _ => no_tac );
(*Resolve version*)
fun res_inst_tac sinsts rule i =
compose_inst_tac sinsts (false, rule, nprems_of rule) i;
(*eresolve (elimination) version*)
fun eres_inst_tac sinsts rule i =
compose_inst_tac sinsts (true, rule, nprems_of rule) i;
(*For forw_inst_tac and dres_inst_tac: preserve Var indexes of rl.
Fails if rl's major premise contains !! or ==> ; it should not anyway!*)
fun make_elim_preserve rl =
let val revcut_rl' = lift_rule (rl,1) revcut_rl
val arg = (false, rl, nprems_of rl)
val [th] = Sequence.list_of_s (bicompose false arg 1 revcut_rl')
in th end
handle Bind => raise THM("make_elim_preserve", 1, [rl]);
(*forward version*)
fun forw_inst_tac sinsts rule =
res_inst_tac sinsts (make_elim_preserve rule) THEN' assume_tac;
(*dresolve version*)
fun dres_inst_tac sinsts rule = eres_inst_tac sinsts (make_elim_preserve rule);
(*** Applications of cut_rl -- forward reasoning ***)
(*Used by metacut_tac*)
fun bires_cut_tac arg i =
resolve_tac [cut_rl] i THEN biresolve_tac arg (i+1) ;
(*The conclusion of the rule gets assumed in subgoal i,
while subgoal i+1,... are the premises of the rule.*)
fun metacut_tac rule = bires_cut_tac [(false,rule)];
(*Recognizes theorems that are not rules, but simple propositions*)
fun is_fact rl =
case prems_of rl of
[] => true | _::_ => false;
(*"Cut" all facts from theorem list into the goal as assumptions. *)
fun cut_facts_tac ths i =
EVERY (map (fn th => metacut_tac th i) (filter is_fact ths));
(*Introduce the given proposition as a lemma and subgoal*)
fun subgoal_tac sprop = res_inst_tac [("psi", sprop)] cut_rl;
(**** 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 tags*)
fun taglist k [] = []
| taglist k (x::xs) = (k,x) :: taglist (k+1) xs;
(*remove tags and suppress duplicates -- list is assumed sorted!*)
fun untaglist [] = []
| untaglist [(k:int,x)] = [x]
| untaglist ((k,x) :: (rest as (k',x')::_)) =
if k=k' then untaglist rest
else x :: untaglist rest;
(*return list elements in original order*)
val orderlist = untaglist o sort (fn(x,y)=> #1 x < #1 y);
(*insert one tagged brl into the pair of nets*)
fun insert_kbrl (kbrl as (k,(eres,th)), (inet,enet)) =
if eres then
case prems_of th of
prem::_ => (inet, Net.insert_term ((prem,kbrl), enet, K false))
| [] => error"insert_kbrl: elimination rule with no premises"
else (Net.insert_term ((concl_of th, kbrl), inet, K false), enet);
(*build a pair of nets for biresolution*)
fun build_netpair brls =
foldr insert_kbrl (taglist 1 brls, (Net.empty,Net.empty));
(*biresolution using a pair of nets rather than rules*)
fun biresolution_from_nets_tac 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 @
flat (map (Net.unify_term enet) hyps)
in PRIMSEQ (biresolution match (orderlist kbrls) i) end);
(*versions taking pre-built nets*)
val biresolve_from_nets_tac = biresolution_from_nets_tac false;
val bimatch_from_nets_tac = biresolution_from_nets_tac true;
(*fast versions using nets internally*)
val net_biresolve_tac = biresolve_from_nets_tac o build_netpair;
val net_bimatch_tac = bimatch_from_nets_tac o build_netpair;
(*** 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) =
Net.insert_term ((concl_of th, krl), net, K false);
(*build a net of rules for resolution*)
fun build_net rls =
foldr insert_krl (taglist 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
(biresolution match (map (pair false) (orderlist 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) = length (prems_of rule) - 1
| subgoals_of_brl (false,rule) = length (prems_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;
(*** Meta-Rewriting Tactics ***)
fun result1 tacf mss thm =
case Sequence.pull(tapply(tacf mss,thm)) of
None => None
| Some(thm,_) => Some(thm);
(*Rewrite subgoal i only *)
fun asm_rewrite_goal_tac prover_tac mss i =
PRIMITIVE(rewrite_goal_rule (result1 prover_tac) mss i);
(*Rewrite or fold throughout proof state. *)
fun rewrite_tac thms = PRIMITIVE(rewrite_rule thms);
fun fold_tac rths = rewrite_tac (map symmetric rths);
(*Rewrite subgoals only, not main goal. *)
fun rewrite_goals_tac thms = PRIMITIVE (rewrite_goals_rule thms);
fun fold_goals_tac rths = rewrite_goals_tac (map symmetric rths);
fun rewtac rth = rewrite_goals_tac [rth];
(** Renaming of parameters in a subgoal
Names may contain letters, digits or primes and must be
separated by blanks **)
(*Calling this will generate the warning "Same as previous level" since
it affects nothing but the names of bound variables!*)
fun rename_tac str i =
let val cs = explode str
in
if !Logic.auto_rename
then (writeln"Note: setting Logic.auto_rename := false";
Logic.auto_rename := false)
else ();
case #2 (take_prefix (is_letdig orf is_blank) cs) of
[] => PRIMITIVE (rename_params_rule (scanwords is_letdig cs, i))
| c::_ => error ("Illegal character: " ^ c)
end;
(*Rename recent parameters using names generated from (a) and the suffixes,
provided the string (a), which represents a term, is an identifier. *)
fun rename_last_tac a sufs i =
let val names = map (curry op^ a) sufs
in if Syntax.is_identifier a
then PRIMITIVE (rename_params_rule (names,i))
else all_tac
end;
(*Prunes all redundant parameters from the proof state by rewriting*)
val prune_params_tac = rewrite_tac [triv_forall_equality];
end;
end;