(* Title: Pure/tctical.ML
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Tacticals.
*)
infix 1 THEN THEN' THEN_ALL_NEW;
infix 0 ORELSE APPEND INTLEAVE ORELSE' APPEND' INTLEAVE';
infix 0 THEN_ELSE;
signature TACTICAL =
sig
type tactic = thm -> thm Seq.seq
val THEN: tactic * tactic -> tactic
val ORELSE: tactic * tactic -> tactic
val APPEND: tactic * tactic -> tactic
val INTLEAVE: tactic * tactic -> tactic
val THEN_ELSE: tactic * (tactic*tactic) -> tactic
val THEN': ('a -> tactic) * ('a -> tactic) -> 'a -> tactic
val ORELSE': ('a -> tactic) * ('a -> tactic) -> 'a -> tactic
val APPEND': ('a -> tactic) * ('a -> tactic) -> 'a -> tactic
val INTLEAVE': ('a -> tactic) * ('a -> tactic) -> 'a -> tactic
val all_tac: tactic
val no_tac: tactic
val DETERM: tactic -> tactic
val COND: (thm -> bool) -> tactic -> tactic -> tactic
val TRY: tactic -> tactic
val EVERY: tactic list -> tactic
val EVERY': ('a -> tactic) list -> 'a -> tactic
val EVERY1: (int -> tactic) list -> tactic
val FIRST: tactic list -> tactic
val FIRST': ('a -> tactic) list -> 'a -> tactic
val FIRST1: (int -> tactic) list -> tactic
val RANGE: (int -> tactic) list -> int -> tactic
val print_tac: string -> tactic
val pause_tac: tactic
val trace_REPEAT: bool ref
val suppress_tracing: bool ref
val tracify: bool ref -> tactic -> tactic
val traced_tac: (thm -> (thm * thm Seq.seq) option) -> tactic
val DETERM_UNTIL: (thm -> bool) -> tactic -> tactic
val REPEAT_DETERM_N: int -> tactic -> tactic
val REPEAT_DETERM: tactic -> tactic
val REPEAT: tactic -> tactic
val REPEAT_DETERM1: tactic -> tactic
val REPEAT1: tactic -> tactic
val FILTER: (thm -> bool) -> tactic -> tactic
val CHANGED: tactic -> tactic
val CHANGED_PROP: tactic -> tactic
val ALLGOALS: (int -> tactic) -> tactic
val SOMEGOAL: (int -> tactic) -> tactic
val FIRSTGOAL: (int -> tactic) -> tactic
val REPEAT_SOME: (int -> tactic) -> tactic
val REPEAT_DETERM_SOME: (int -> tactic) -> tactic
val REPEAT_FIRST: (int -> tactic) -> tactic
val REPEAT_DETERM_FIRST: (int -> tactic) -> tactic
val TRYALL: (int -> tactic) -> tactic
val CSUBGOAL: ((cterm * int) -> tactic) -> int -> tactic
val SUBGOAL: ((term * int) -> tactic) -> int -> tactic
val CHANGED_GOAL: (int -> tactic) -> int -> tactic
val THEN_ALL_NEW: (int -> tactic) * (int -> tactic) -> int -> tactic
val REPEAT_ALL_NEW: (int -> tactic) -> int -> tactic
val strip_context: term -> (string * typ) list * term list * term
val metahyps_thms: int -> thm -> thm list option
val METAHYPS: (thm list -> tactic) -> int -> tactic
val PRIMSEQ: (thm -> thm Seq.seq) -> tactic
val PRIMITIVE: (thm -> thm) -> tactic
val SINGLE: tactic -> thm -> thm option
val CONVERSION: conv -> int -> tactic
end;
structure Tactical : TACTICAL =
struct
(**** Tactics ****)
(*A tactic maps a proof tree to a sequence of proof trees:
if length of sequence = 0 then the tactic does not apply;
if length > 1 then backtracking on the alternatives can occur.*)
type tactic = thm -> thm Seq.seq;
(*** LCF-style tacticals ***)
(*the tactical THEN performs one tactic followed by another*)
fun (tac1 THEN tac2) st = Seq.maps tac2 (tac1 st);
(*The tactical ORELSE uses the first tactic that returns a nonempty sequence.
Like in LCF, ORELSE commits to either tac1 or tac2 immediately.
Does not backtrack to tac2 if tac1 was initially chosen. *)
fun (tac1 ORELSE tac2) st =
case Seq.pull(tac1 st) of
NONE => tac2 st
| sequencecell => Seq.make(fn()=> sequencecell);
(*The tactical APPEND combines the results of two tactics.
Like ORELSE, but allows backtracking on both tac1 and tac2.
The tactic tac2 is not applied until needed.*)
fun (tac1 APPEND tac2) st =
Seq.append (tac1 st) (Seq.make(fn()=> Seq.pull (tac2 st)));
(*Like APPEND, but interleaves results of tac1 and tac2.*)
fun (tac1 INTLEAVE tac2) st =
Seq.interleave(tac1 st,
Seq.make(fn()=> Seq.pull (tac2 st)));
(*Conditional tactic.
tac1 ORELSE tac2 = tac1 THEN_ELSE (all_tac, tac2)
tac1 THEN tac2 = tac1 THEN_ELSE (tac2, no_tac)
*)
fun (tac THEN_ELSE (tac1, tac2)) st =
case Seq.pull(tac st) of
NONE => tac2 st (*failed; try tactic 2*)
| seqcell => Seq.maps tac1 (Seq.make(fn()=> seqcell)); (*succeeded; use tactic 1*)
(*Versions for combining tactic-valued functions, as in
SOMEGOAL (resolve_tac rls THEN' assume_tac) *)
fun (tac1 THEN' tac2) x = tac1 x THEN tac2 x;
fun (tac1 ORELSE' tac2) x = tac1 x ORELSE tac2 x;
fun (tac1 APPEND' tac2) x = tac1 x APPEND tac2 x;
fun (tac1 INTLEAVE' tac2) x = tac1 x INTLEAVE tac2 x;
(*passes all proofs through unchanged; identity of THEN*)
fun all_tac st = Seq.single st;
(*passes no proofs through; identity of ORELSE and APPEND*)
fun no_tac st = Seq.empty;
(*Make a tactic deterministic by chopping the tail of the proof sequence*)
fun DETERM tac = Seq.DETERM tac;
(*Conditional tactical: testfun controls which tactic to use next.
Beware: due to eager evaluation, both thentac and elsetac are evaluated.*)
fun COND testfun thenf elsef = (fn prf =>
if testfun prf then thenf prf else elsef prf);
(*Do the tactic or else do nothing*)
fun TRY tac = tac ORELSE all_tac;
(*** List-oriented tactics ***)
local
(*This version of EVERY avoids backtracking over repeated states*)
fun EVY (trail, []) st =
Seq.make (fn()=> SOME(st,
Seq.make (fn()=> Seq.pull (evyBack trail))))
| EVY (trail, tac::tacs) st =
case Seq.pull(tac st) of
NONE => evyBack trail (*failed: backtrack*)
| SOME(st',q) => EVY ((st',q,tacs)::trail, tacs) st'
and evyBack [] = Seq.empty (*no alternatives*)
| evyBack ((st',q,tacs)::trail) =
case Seq.pull q of
NONE => evyBack trail
| SOME(st,q') => if Thm.eq_thm (st',st)
then evyBack ((st',q',tacs)::trail)
else EVY ((st,q',tacs)::trail, tacs) st
in
(* EVERY [tac1,...,tacn] equals tac1 THEN ... THEN tacn *)
fun EVERY tacs = EVY ([], tacs);
end;
(* EVERY' [tac1,...,tacn] i equals tac1 i THEN ... THEN tacn i *)
fun EVERY' tacs i = EVERY (map (fn f => f i) tacs);
(*Apply every tactic to 1*)
fun EVERY1 tacs = EVERY' tacs 1;
(* FIRST [tac1,...,tacn] equals tac1 ORELSE ... ORELSE tacn *)
fun FIRST tacs = fold_rev (curry op ORELSE) tacs no_tac;
(* FIRST' [tac1,...,tacn] i equals tac1 i ORELSE ... ORELSE tacn i *)
fun FIRST' tacs = fold_rev (curry op ORELSE') tacs (K no_tac);
(*Apply first tactic to 1*)
fun FIRST1 tacs = FIRST' tacs 1;
(*Apply tactics on consecutive subgoals*)
fun RANGE [] _ = all_tac
| RANGE (tac :: tacs) i = RANGE tacs (i + 1) THEN tac i;
(*** Tracing tactics ***)
(*Print the current proof state and pass it on.*)
fun print_tac msg =
(fn st =>
(tracing msg;
tracing ((Pretty.string_of o Pretty.chunks o
Display.pretty_goals (! Display.goals_limit)) st);
Seq.single st));
(*Pause until a line is typed -- if non-empty then fail. *)
fun pause_tac st =
(tracing "** Press RETURN to continue:";
if TextIO.inputLine TextIO.stdIn = SOME "\n" then Seq.single st
else (tracing "Goodbye"; Seq.empty));
exception TRACE_EXIT of thm
and TRACE_QUIT;
(*Tracing flags*)
val trace_REPEAT= ref false
and suppress_tracing = ref false;
(*Handle all tracing commands for current state and tactic *)
fun exec_trace_command flag (tac, st) =
case TextIO.inputLine TextIO.stdIn of
SOME "\n" => tac st
| SOME "f\n" => Seq.empty
| SOME "o\n" => (flag:=false; tac st)
| SOME "s\n" => (suppress_tracing:=true; tac st)
| SOME "x\n" => (tracing "Exiting now"; raise (TRACE_EXIT st))
| SOME "quit\n" => raise TRACE_QUIT
| _ => (tracing
"Type RETURN to continue or...\n\
\ f - to fail here\n\
\ o - to switch tracing off\n\
\ s - to suppress tracing until next entry to a tactical\n\
\ x - to exit at this point\n\
\ quit - to abort this tracing run\n\
\** Well? " ; exec_trace_command flag (tac, st));
(*Extract from a tactic, a thm->thm seq function that handles tracing*)
fun tracify flag tac st =
if !flag andalso not (!suppress_tracing)
then (Display.print_goals (! Display.goals_limit) st;
tracing "** Press RETURN to continue:";
exec_trace_command flag (tac,st))
else tac st;
(*Create a tactic whose outcome is given by seqf, handling TRACE_EXIT*)
fun traced_tac seqf st =
(suppress_tracing := false;
Seq.make (fn()=> seqf st
handle TRACE_EXIT st' => SOME(st', Seq.empty)));
(*Deterministic DO..UNTIL: only retains the first outcome; tail recursive.
Forces repitition until predicate on state is fulfilled.*)
fun DETERM_UNTIL p tac =
let val tac = tracify trace_REPEAT tac
fun drep st = if p st then SOME (st, Seq.empty)
else (case Seq.pull(tac st) of
NONE => NONE
| SOME(st',_) => drep st')
in traced_tac drep end;
(*Deterministic REPEAT: only retains the first outcome;
uses less space than REPEAT; tail recursive.
If non-negative, n bounds the number of repetitions.*)
fun REPEAT_DETERM_N n tac =
let val tac = tracify trace_REPEAT tac
fun drep 0 st = SOME(st, Seq.empty)
| drep n st =
(case Seq.pull(tac st) of
NONE => SOME(st, Seq.empty)
| SOME(st',_) => drep (n-1) st')
in traced_tac (drep n) end;
(*Allows any number of repetitions*)
val REPEAT_DETERM = REPEAT_DETERM_N ~1;
(*General REPEAT: maintains a stack of alternatives; tail recursive*)
fun REPEAT tac =
let val tac = tracify trace_REPEAT tac
fun rep qs st =
case Seq.pull(tac st) of
NONE => SOME(st, Seq.make(fn()=> repq qs))
| SOME(st',q) => rep (q::qs) st'
and repq [] = NONE
| repq(q::qs) = case Seq.pull q of
NONE => repq qs
| SOME(st,q) => rep (q::qs) st
in traced_tac (rep []) end;
(*Repeat 1 or more times*)
fun REPEAT_DETERM1 tac = DETERM tac THEN REPEAT_DETERM tac;
fun REPEAT1 tac = tac THEN REPEAT tac;
(** Filtering tacticals **)
fun FILTER pred tac st = Seq.filter pred (tac st);
(*Accept only next states that change the theorem somehow*)
fun CHANGED tac st =
let fun diff st' = not (Thm.eq_thm (st, st'));
in Seq.filter diff (tac st) end;
(*Accept only next states that change the theorem's prop field
(changes to signature, hyps, etc. don't count)*)
fun CHANGED_PROP tac st =
let fun diff st' = not (Thm.eq_thm_prop (st, st'));
in Seq.filter diff (tac st) end;
(*** Tacticals based on subgoal numbering ***)
(*For n subgoals, performs tac(n) THEN ... THEN tac(1)
Essential to work backwards since tac(i) may add/delete subgoals at i. *)
fun ALLGOALS tac st =
let fun doall 0 = all_tac
| doall n = tac(n) THEN doall(n-1)
in doall(nprems_of st)st end;
(*For n subgoals, performs tac(n) ORELSE ... ORELSE tac(1) *)
fun SOMEGOAL tac st =
let fun find 0 = no_tac
| find n = tac(n) ORELSE find(n-1)
in find(nprems_of st)st end;
(*For n subgoals, performs tac(1) ORELSE ... ORELSE tac(n).
More appropriate than SOMEGOAL in some cases.*)
fun FIRSTGOAL tac st =
let fun find (i,n) = if i>n then no_tac else tac(i) ORELSE find (i+1,n)
in find(1, nprems_of st)st end;
(*Repeatedly solve some using tac. *)
fun REPEAT_SOME tac = REPEAT1 (SOMEGOAL (REPEAT1 o tac));
fun REPEAT_DETERM_SOME tac = REPEAT_DETERM1 (SOMEGOAL (REPEAT_DETERM1 o tac));
(*Repeatedly solve the first possible subgoal using tac. *)
fun REPEAT_FIRST tac = REPEAT1 (FIRSTGOAL (REPEAT1 o tac));
fun REPEAT_DETERM_FIRST tac = REPEAT_DETERM1 (FIRSTGOAL (REPEAT_DETERM1 o tac));
(*For n subgoals, tries to apply tac to n,...1 *)
fun TRYALL tac = ALLGOALS (TRY o tac);
(*Make a tactic for subgoal i, if there is one. *)
fun CSUBGOAL goalfun i st =
(case SOME (Thm.cprem_of st i) handle THM _ => NONE of
SOME goal => goalfun (goal, i) st
| NONE => Seq.empty);
fun SUBGOAL goalfun =
CSUBGOAL (fn (goal, i) => goalfun (Thm.term_of goal, i));
(*Returns all states that have changed in subgoal i, counted from the LAST
subgoal. For stac, for example.*)
fun CHANGED_GOAL tac i st =
let val np = nprems_of st
val d = np-i (*distance from END*)
val t = List.nth(prems_of st, i-1)
fun diff st' =
nprems_of st' - d <= 0 (*the subgoal no longer exists*)
orelse
not (Pattern.aeconv (t,
List.nth(prems_of st',
nprems_of st' - d - 1)))
in Seq.filter diff (tac i st) end
handle Subscript => Seq.empty (*no subgoal i*);
fun (tac1 THEN_ALL_NEW tac2) i st =
st |> (tac1 i THEN (fn st' => Seq.INTERVAL tac2 i (i + nprems_of st' - nprems_of st) st'));
(*repeatedly dig into any emerging subgoals*)
fun REPEAT_ALL_NEW tac =
tac THEN_ALL_NEW (TRY o (fn i => REPEAT_ALL_NEW tac i));
(*Strips assumptions in goal yielding ( [x1,...,xm], [H1,...,Hn], B )
H1,...,Hn are the hypotheses; x1...xm are variants of the parameters.
Main difference from strip_assums concerns parameters:
it replaces the bound variables by free variables. *)
fun strip_context_aux (params, Hs, Const("==>", _) $ H $ B) =
strip_context_aux (params, H::Hs, B)
| strip_context_aux (params, Hs, Const("all",_)$Abs(a,T,t)) =
let val (b,u) = Syntax.variant_abs(a,T,t)
in strip_context_aux ((b,T)::params, Hs, u) end
| strip_context_aux (params, Hs, B) = (rev params, rev Hs, B);
fun strip_context A = strip_context_aux ([],[],A);
(**** METAHYPS -- tactical for using hypotheses as meta-level assumptions
METAHYPS (fn prems => tac prems) i
converts subgoal i, of the form !!x1...xm. [| A1;...;An] ==> A into a new
proof state A==>A, supplying A1,...,An as meta-level assumptions (in
"prems"). The parameters x1,...,xm become free variables. If the
resulting proof state is [| B1;...;Bk] ==> C (possibly assuming A1,...,An)
then it is lifted back into the original context, yielding k subgoals.
Replaces unknowns in the context by Frees having the prefix METAHYP_
New unknowns in [| B1;...;Bk] ==> C are lifted over x1,...,xm.
DOES NOT HANDLE TYPE UNKNOWNS.
****)
local
(*Left-to-right replacements: ctpairs = [...,(vi,ti),...].
Instantiates distinct free variables by terms of same type.*)
fun free_instantiate ctpairs =
forall_elim_list (map snd ctpairs) o forall_intr_list (map fst ctpairs);
fun free_of s ((a, i), T) =
Free (s ^ (case i of 0 => a | _ => a ^ "_" ^ string_of_int i), T)
fun mk_inst v = (Var v, free_of "METAHYP1_" v)
in
(*Common code for METAHYPS and metahyps_thms*)
fun metahyps_split_prem prem =
let (*find all vars in the hyps -- should find tvars also!*)
val hyps_vars = fold Term.add_vars (Logic.strip_assums_hyp prem) []
val insts = map mk_inst hyps_vars
(*replace the hyps_vars by Frees*)
val prem' = subst_atomic insts prem
val (params,hyps,concl) = strip_context prem'
in (insts,params,hyps,concl) end;
fun metahyps_aux_tac tacf (prem,gno) state =
let val (insts,params,hyps,concl) = metahyps_split_prem prem
val maxidx = Thm.maxidx_of state
val cterm = Thm.cterm_of (Thm.theory_of_thm state)
val chyps = map cterm hyps
val hypths = map assume chyps
val subprems = map (Thm.forall_elim_vars 0) hypths
val fparams = map Free params
val cparams = map cterm fparams
fun swap_ctpair (t,u) = (cterm u, cterm t)
(*Subgoal variables: make Free; lift type over params*)
fun mk_subgoal_inst concl_vars (v, T) =
if member (op =) concl_vars (v, T)
then ((v, T), true, free_of "METAHYP2_" (v, T))
else ((v, T), false, free_of "METAHYP2_" (v, map #2 params ---> T))
(*Instantiate subgoal vars by Free applied to params*)
fun mk_ctpair (v, in_concl, u) =
if in_concl then (cterm (Var v), cterm u)
else (cterm (Var v), cterm (list_comb (u, fparams)))
(*Restore Vars with higher type and index*)
fun mk_subgoal_swap_ctpair (((a, i), T), in_concl, u as Free (_, U)) =
if in_concl then (cterm u, cterm (Var ((a, i), T)))
else (cterm u, cterm (Var ((a, i + maxidx), U)))
(*Embed B in the original context of params and hyps*)
fun embed B = list_all_free (params, Logic.list_implies (hyps, B))
(*Strip the context using elimination rules*)
fun elim Bhyp = implies_elim_list (forall_elim_list cparams Bhyp) hypths
(*A form of lifting that discharges assumptions.*)
fun relift st =
let val prop = Thm.prop_of st
val subgoal_vars = (*Vars introduced in the subgoals*)
fold Term.add_vars (Logic.strip_imp_prems prop) []
and concl_vars = Term.add_vars (Logic.strip_imp_concl prop) []
val subgoal_insts = map (mk_subgoal_inst concl_vars) subgoal_vars
val st' = Thm.instantiate ([], map mk_ctpair subgoal_insts) st
val emBs = map (cterm o embed) (prems_of st')
val Cth = implies_elim_list st' (map (elim o assume) emBs)
in (*restore the unknowns to the hypotheses*)
free_instantiate (map swap_ctpair insts @
map mk_subgoal_swap_ctpair subgoal_insts)
(*discharge assumptions from state in same order*)
(implies_intr_list emBs
(forall_intr_list cparams (implies_intr_list chyps Cth)))
end
(*function to replace the current subgoal*)
fun next st = bicompose false (false, relift st, nprems_of st)
gno state
in Seq.maps next (tacf subprems (trivial (cterm concl))) end;
end;
(*Returns the theorem list that METAHYPS would supply to its tactic*)
fun metahyps_thms i state =
let val prem = Logic.nth_prem (i, Thm.prop_of state)
and cterm = cterm_of (Thm.theory_of_thm state)
val (_,_,hyps,_) = metahyps_split_prem prem
in SOME (map (Thm.forall_elim_vars 0 o Thm.assume o cterm) hyps) end
handle TERM ("nth_prem", [A]) => NONE;
local
fun print_vars_terms thy (n,thm) =
let
fun typed ty = " has type: " ^ Syntax.string_of_typ_global thy ty;
fun find_vars thy (Const (c, ty)) =
if null (Term.add_tvarsT ty []) then I
else insert (op =) (c ^ typed ty)
| find_vars thy (Var (xi, ty)) = insert (op =) (Term.string_of_vname xi ^ typed ty)
| find_vars _ (Free _) = I
| find_vars _ (Bound _) = I
| find_vars thy (Abs (_, _, t)) = find_vars thy t
| find_vars thy (t1 $ t2) =
find_vars thy t1 #> find_vars thy t1;
val prem = Logic.nth_prem (n, Thm.prop_of thm)
val tms = find_vars thy prem []
in
(warning "Found schematic vars in assumptions:"; warning (cat_lines tms))
end;
in
fun METAHYPS tacf n thm = SUBGOAL (metahyps_aux_tac tacf) n thm
handle THM("assume: variables",_,_) => (print_vars_terms (theory_of_thm thm) (n,thm); Seq.empty)
end;
(*Makes a tactic whose effect on a state is given by thmfun: thm->thm seq.*)
fun PRIMSEQ thmfun st = thmfun st handle THM _ => Seq.empty;
(*Makes a tactic whose effect on a state is given by thmfun: thm->thm.*)
fun PRIMITIVE thmfun = PRIMSEQ (Seq.single o thmfun);
(*Inverse (more or less) of PRIMITIVE*)
fun SINGLE tacf = Option.map fst o Seq.pull o tacf
(*Conversions as tactics*)
fun CONVERSION cv i st = Seq.single (Conv.gconv_rule cv i st)
handle THM _ => Seq.empty
| CTERM _ => Seq.empty
| TERM _ => Seq.empty
| TYPE _ => Seq.empty;
end;
open Tactical;