(* ID: $Id$
use "/homes/lcp/Isa/new/blast.ML";
SKOLEMIZES ReplaceI WRONGLY: allow new vars in prems, or forbid such rules??
Needs explicit instantiation of assumptions? (#55 takes 32s)
*)
proof_timing:=true;
print_depth 20;
structure List = List_;
(*Should be a type abbreviation?*)
type netpair = (int*(bool*thm)) Net.net * (int*(bool*thm)) Net.net;
(*Assumptions about constants:
--The negation symbol is "Not"
--The equality symbol is "op ="
--The is-true judgement symbol is "Trueprop"
--There are no constants named "*Goal* or "*False*"
*)
signature BLAST_DATA =
sig
type claset
val notE : thm (* [| ~P; P |] ==> R *)
val ccontr : thm
val contr_tac : int -> tactic
val dup_intr : thm -> thm
val vars_gen_hyp_subst_tac : bool -> int -> tactic
val claset : claset ref
val rep_claset :
claset -> {safeIs: thm list, safeEs: thm list,
hazIs: thm list, hazEs: thm list,
uwrapper: (int -> tactic) -> (int -> tactic),
swrapper: (int -> tactic) -> (int -> tactic),
safe0_netpair: netpair, safep_netpair: netpair,
haz_netpair: netpair, dup_netpair: netpair}
end;
signature BLAST =
sig
type claset
val depth_tac : claset -> int -> int -> tactic
val blast_tac : claset -> int -> tactic
val Blast_tac : int -> tactic
end;
functor BlastFun(Data: BLAST_DATA): BLAST =
struct
type claset = Data.claset;
val trace = ref false;
datatype term =
Const of string
| OConst of string * int
| Skolem of string * term option ref list
| Free of string
| Var of term option ref
| Bound of int
| Abs of string*term
| op $ of term*term;
exception DEST_EQ;
(*Take apart an equality (plain or overloaded). NO constant Trueprop*)
fun dest_eq (Const "op =" $ t $ u) = (t,u)
| dest_eq (OConst("op =",_) $ t $ u) = (t,u)
| dest_eq _ = raise DEST_EQ;
(** Basic syntactic operations **)
fun is_Var (Var _) = true
| is_Var _ = false;
fun dest_Var (Var x) = x;
fun rand (f$x) = x;
(* maps (f, [t1,...,tn]) to f(t1,...,tn) *)
val list_comb : term * term list -> term = foldl (op $);
(* maps f(t1,...,tn) to (f, [t1,...,tn]) ; naturally tail-recursive*)
fun strip_comb u : term * term list =
let fun stripc (f$t, ts) = stripc (f, t::ts)
| stripc x = x
in stripc(u,[]) end;
(* maps f(t1,...,tn) to f , which is never a combination *)
fun head_of (f$t) = head_of f
| head_of u = u;
(** Particular constants **)
fun negate P = Const"Not" $ P;
fun mkGoal P = Const"*Goal*" $ P;
fun isGoal (Const"*Goal*" $ _) = true
| isGoal _ = false;
val Trueprop = Term.Const("Trueprop", Type("o",[])-->propT);
fun mk_tprop P = Term.$ (Trueprop, P);
fun isTrueprop (Term.Const("Trueprop",_)) = true
| isTrueprop _ = false;
(** Dealing with overloaded constants **)
(*Result is a symbol table, indexed by names of overloaded constants.
Each constant maps to a list of (pattern,Blast.Const) pairs.
Any Term.Const that matches a pattern gets replaced by the Blast.Const.
*)
fun addConsts (t as Term.Const(a,_), tab) =
(case Symtab.lookup (tab,a) of
None => tab (*ignore: not a constant that we are looking for*)
| Some patList =>
(case gen_assoc (op aconv) (patList, t) of
None => Symtab.update
((a, (t, OConst (a, length patList)) :: patList),
tab)
| _ => tab))
| addConsts (Term.Abs(_,_,body), tab) = addConsts (body, tab)
| addConsts (Term.$ (t,u), tab) = addConsts (t, addConsts (u, tab))
| addConsts (_, tab) = tab (*ignore others*);
fun addRules (rls,tab) = foldr addConsts (map (#prop o rep_thm) rls, tab);
fun declConst (a,tab) = Symtab.update((a,[]), tab);
(*maps the name of each overloaded constant to a list of archetypal constants,
which may be polymorphic.*)
local
val overLoadTab = ref (Symtab.null : (Term.term * term) list Symtab.table)
(*The alists in this table should only be increased*)
in
fun declConsts (names, rls) =
overLoadTab := addRules (rls, foldr declConst (names, !overLoadTab));
(*Convert a possibly overloaded Term.Const to a Blast.Const*)
fun fromConst tsig (t as Term.Const (a,_)) =
let fun find [] = Const a
| find ((pat,t')::patList) =
if Pattern.matches tsig (pat,t) then t'
else find patList
in case Symtab.lookup(!overLoadTab, a) of
None => Const a
| Some patList => find patList
end;
end;
(*Tests whether 2 terms are alpha-convertible; chases instantiations*)
fun (Const a) aconv (Const b) = a=b
| (OConst ai) aconv (OConst bj) = ai=bj
| (Skolem (a,_)) aconv (Skolem (b,_)) = a=b (*arglists must then be equal*)
| (Free a) aconv (Free b) = a=b
| (Var(ref(Some t))) aconv u = t aconv u
| t aconv (Var(ref(Some u))) = t aconv u
| (Var v) aconv (Var w) = v=w (*both Vars are un-assigned*)
| (Bound i) aconv (Bound j) = i=j
| (Abs(_,t)) aconv (Abs(_,u)) = t aconv u
| (f$t) aconv (g$u) = (f aconv g) andalso (t aconv u)
| _ aconv _ = false;
fun mem_term (_, []) = false
| mem_term (t, t'::ts) = t aconv t' orelse mem_term(t,ts);
fun ins_term(t,ts) = if mem_term(t,ts) then ts else t :: ts;
fun mem_var (v: term option ref, []) = false
| mem_var (v, v'::vs) = v=v' orelse mem_var(v,vs);
fun ins_var(v,vs) = if mem_var(v,vs) then vs else v :: vs;
(** Vars **)
(*Accumulates the Vars in the term, suppressing duplicates*)
fun add_term_vars (Skolem(a,args), vars) = add_vars_vars(args,vars)
| add_term_vars (Var (v as ref None), vars) = ins_var (v, vars)
| add_term_vars (Var (ref (Some u)), vars) = add_term_vars(u,vars)
| add_term_vars (Abs (_,body), vars) = add_term_vars(body,vars)
| add_term_vars (f$t, vars) = add_term_vars (f, add_term_vars(t, vars))
| add_term_vars (_, vars) = vars
(*Term list version. [The fold functionals are slow]*)
and add_terms_vars ([], vars) = vars
| add_terms_vars (t::ts, vars) = add_terms_vars (ts, add_term_vars(t,vars))
(*Var list version.*)
and add_vars_vars ([], vars) = vars
| add_vars_vars (ref (Some u) :: vs, vars) =
add_vars_vars (vs, add_term_vars(u,vars))
| add_vars_vars (v::vs, vars) = (*v must be a ref None*)
add_vars_vars (vs, ins_var (v, vars));
(*Chase assignments in "vars"; return a list of unassigned variables*)
fun vars_in_vars vars = add_vars_vars(vars,[]);
(*increment a term's non-local bound variables
inc is increment for bound variables
lev is level at which a bound variable is considered 'loose'*)
fun incr_bv (inc, lev, u as Bound i) = if i>=lev then Bound(i+inc) else u
| incr_bv (inc, lev, Abs(a,body)) = Abs(a, incr_bv(inc,lev+1,body))
| incr_bv (inc, lev, f$t) = incr_bv(inc,lev,f) $ incr_bv(inc,lev,t)
| incr_bv (inc, lev, u) = u;
fun incr_boundvars 0 t = t
| incr_boundvars inc t = incr_bv(inc,0,t);
(*Accumulate all 'loose' bound vars referring to level 'lev' or beyond.
(Bound 0) is loose at level 0 *)
fun add_loose_bnos (Bound i, lev, js) = if i<lev then js
else (i-lev) ins_int js
| add_loose_bnos (Abs (_,t), lev, js) = add_loose_bnos (t, lev+1, js)
| add_loose_bnos (f$t, lev, js) =
add_loose_bnos (f, lev, add_loose_bnos (t, lev, js))
| add_loose_bnos (_, _, js) = js;
fun loose_bnos t = add_loose_bnos (t, 0, []);
fun subst_bound (arg, t) : term =
let fun subst (t as Bound i, lev) =
if i<lev then t (*var is locally bound*)
else if i=lev then incr_boundvars lev arg
else Bound(i-1) (*loose: change it*)
| subst (Abs(a,body), lev) = Abs(a, subst(body,lev+1))
| subst (f$t, lev) = subst(f,lev) $ subst(t,lev)
| subst (t,lev) = t
in subst (t,0) end;
(*Normalize...but not the bodies of ABSTRACTIONS*)
fun norm t = case t of
Skolem(a,args) => Skolem(a, vars_in_vars args)
| (Var (ref None)) => t
| (Var (ref (Some u))) => norm u
| (f $ u) => (case norm f of
Abs(_,body) => norm (subst_bound (u, body))
| nf => nf $ norm u)
| _ => t;
(*Weak (one-level) normalize for use in unification*)
fun wkNormAux t = case t of
(Var v) => (case !v of
Some u => wkNorm u
| None => t)
| (f $ u) => (case wkNormAux f of
Abs(_,body) => wkNorm (subst_bound (u, body))
| nf => nf $ u)
| _ => t
and wkNorm t = case head_of t of
Const _ => t
| OConst _ => t
| Skolem(a,args) => t
| Free _ => t
| _ => wkNormAux t;
(*Does variable v occur in u? For unification.*)
fun varOccur v =
let fun occL [] = false (*same as (exists occ), but faster*)
| occL (u::us) = occ u orelse occL us
and occ (Var w) =
v=w orelse
(case !w of None => false
| Some u => occ u)
| occ (Skolem(_,args)) = occL (map Var args)
| occ (Abs(_,u)) = occ u
| occ (f$u) = occ u orelse occ f
| occ (_) = false;
in occ end;
exception UNIFY;
val trail = ref [] : term option ref list ref;
val ntrail = ref 0;
(*Restore the trail to some previous state: for backtracking*)
fun clearTo n =
while !ntrail>n do
(hd(!trail) := None;
trail := tl (!trail);
ntrail := !ntrail - 1);
(*First-order unification with bound variables.
"vars" is a list of variables local to the rule and NOT to be put
on the trail (no point in doing so)
*)
fun unify(vars,t,u) =
let val n = !ntrail
fun update (t as Var v, u) =
if t aconv u then ()
else if varOccur v u then raise UNIFY
else if mem_var(v, vars) then v := Some u
else (*avoid updating Vars in the branch if possible!*)
if is_Var u andalso mem_var(dest_Var u, vars)
then dest_Var u := Some t
else (v := Some u;
trail := v :: !trail; ntrail := !ntrail + 1)
fun unifyAux (t,u) =
case (wkNorm t, wkNorm u) of
(nt as Var v, nu) => update(nt,nu)
| (nu, nt as Var v) => update(nt,nu)
| (Abs(_,t'), Abs(_,u')) => unifyAux(t',u')
(*NB: can yield unifiers having dangling Bound vars!*)
| (f$t', g$u') => (unifyAux(f,g); unifyAux(t',u'))
| (nt, nu) => if nt aconv nu then () else raise UNIFY
in unifyAux(t,u) handle UNIFY => (clearTo n; raise UNIFY)
end;
(*Convert from "real" terms to prototerms; eta-contract*)
fun fromTerm tsig t =
let val alist = ref []
fun from (t as Term.Const _) = fromConst tsig t
| from (Term.Free (a,_)) = Free a
| from (Term.Bound i) = Bound i
| from (Term.Var (ixn,T)) =
(case (assoc_string_int(!alist,ixn)) of
None => let val t' = Var(ref None)
in alist := (ixn, (t', T)) :: !alist; t'
end
| Some (v,_) => v)
| from (Term.Abs (a,_,u)) =
(case from u of
u' as (f $ Bound 0) =>
if (0 mem_int loose_bnos f) then Abs(a,u')
else incr_boundvars ~1 f
| u' => Abs(a,u'))
| from (Term.$ (f,u)) = from f $ from u
in from t end;
(* A1==>...An==>B goes to [A1,...,An], where B is not an implication *)
fun strip_imp_prems (Const"==>" $ (Const"Trueprop" $ A) $ B) =
A :: strip_imp_prems B
| strip_imp_prems (Const"==>" $ A $ B) = A :: strip_imp_prems B
| strip_imp_prems _ = [];
(* A1==>...An==>B goes to B, where B is not an implication *)
fun strip_imp_concl (Const"==>" $ A $ B) = strip_imp_concl B
| strip_imp_concl (Const"Trueprop" $ A) = A
| strip_imp_concl A = A : term;
(*** Conversion of Elimination Rules to Tableau Operations ***)
(*The conclusion becomes the goal/negated assumption *False*: delete it!*)
fun squash_nots [] = []
| squash_nots (Const "*Goal*" $ (Var (ref (Some (Const"*False*")))) :: Ps) =
squash_nots Ps
| squash_nots (Const "Not" $ (Var (ref (Some (Const"*False*")))) :: Ps) =
squash_nots Ps
| squash_nots (P::Ps) = P :: squash_nots Ps;
fun skoPrem vars (Const "all" $ Abs (_, P)) =
skoPrem vars (subst_bound (Skolem (gensym "S_", vars), P))
| skoPrem vars P = P;
fun convertPrem t =
squash_nots (mkGoal (strip_imp_concl t) :: strip_imp_prems t);
(*Expects elimination rules to have a formula variable as conclusion*)
fun convertRule vars t =
let val (P::Ps) = strip_imp_prems t
val Var v = strip_imp_concl t
in v := Some (Const"*False*");
(P, map (convertPrem o skoPrem vars) Ps)
end;
(*Like dup_elim, but puts the duplicated major premise FIRST*)
fun rev_dup_elim th = th RSN (2, revcut_rl) |> assumption 2 |> Sequence.hd;
(*Count new hyps so that they can be rotated*)
fun nNewHyps [] = 0
| nNewHyps (Const "*Goal*" $ _ :: Ps) = nNewHyps Ps
| nNewHyps (P::Ps) = 1 + nNewHyps Ps;
fun rot_subgoals_tac [] i st = Sequence.single st
| rot_subgoals_tac (k::ks) i st =
rot_subgoals_tac ks (i+1) (Sequence.hd (rotate_tac (~k) i st))
handle OPTION _ => Sequence.null;
fun TRACE rl tac st = if !trace then (prth rl; tac st) else tac st;
(*Tableau rule from elimination rule. Flag "dup" requests duplication of the
affected formula.*)
fun fromRule vars rl =
let val {tsig,...} = Sign.rep_sg (#sign (rep_thm rl))
val trl = rl |> rep_thm |> #prop |> fromTerm tsig |> convertRule vars
fun tac dup i =
TRACE rl
(DETERM (etac (if dup then rev_dup_elim rl else rl) i))
THEN rot_subgoals_tac (map nNewHyps (#2 trl)) i
in General.SOME (trl, tac) end
handle Bind => General.NONE (*reject: conclusion is not just a variable*);
(*** Conversion of Introduction Rules (needed for efficiency in
proof reconstruction) ***)
fun convertIntrPrem t = mkGoal (strip_imp_concl t) :: strip_imp_prems t;
fun convertIntrRule vars t =
let val Ps = strip_imp_prems t
val P = strip_imp_concl t
in (mkGoal P, map (convertIntrPrem o skoPrem vars) Ps)
end;
(*Tableau rule from introduction rule. Since haz rules are now delayed,
"dup" is always FALSE for introduction rules.*)
fun fromIntrRule vars rl =
let val {tsig,...} = Sign.rep_sg (#sign (rep_thm rl))
val trl = rl |> rep_thm |> #prop |> fromTerm tsig |> convertIntrRule vars
fun tac dup i =
TRACE rl (DETERM (rtac (if dup then Data.dup_intr rl else rl) i))
THEN rot_subgoals_tac (map nNewHyps (#2 trl)) i
in (trl, tac) end;
val dummyVar = Term.Var (("Doom",666), dummyT);
(*Convert from prototerms to ordinary terms with dummy types
Ignore abstractions; identify all Vars*)
fun dummyTerm 0 _ = dummyVar
| dummyTerm d (Const a) = Term.Const (a,dummyT)
| dummyTerm d (OConst(a,_)) = Term.Const (a,dummyT)
| dummyTerm d (Skolem(a,_)) = Term.Const (a,dummyT)
| dummyTerm d (Free a) = Term.Free (a,dummyT)
| dummyTerm d (Bound i) = Term.Bound i
| dummyTerm d (Var _) = dummyVar
| dummyTerm d (Abs(a,_)) = dummyVar
| dummyTerm d (f $ u) = Term.$ (dummyTerm d f, dummyTerm (d-1) u);
fun netMkRules P vars (nps: netpair list) =
case P of
(Const "*Goal*" $ G) =>
let val pG = mk_tprop (dummyTerm 2 G)
val intrs = List.concat
(map (fn (inet,_) => Net.unify_term inet pG)
nps)
in map (fromIntrRule vars o #2) (orderlist intrs) end
| _ =>
let val pP = mk_tprop (dummyTerm 3 P)
val elims = List.concat
(map (fn (_,enet) => Net.unify_term enet pP)
nps)
in List.mapPartial (fromRule vars o #2) (orderlist elims) end;
(**
end;
**)
(*** Code for handling equality: naive substitution, like hyp_subst_tac ***)
(*Replace the ATOMIC term "old" by "new" in t*)
fun subst_atomic (old,new) t =
let fun subst (Var(ref(Some u))) = subst u
| subst (Abs(a,body)) = Abs(a, subst body)
| subst (f$t) = subst f $ subst t
| subst t = if t aconv old then new else t
in subst t end;
(*Eta-contract a term from outside: just enough to reduce it to an atom*)
fun eta_contract_atom (t0 as Abs(a, body)) =
(case eta_contract2 body of
f $ Bound 0 => if (0 mem_int loose_bnos f) then t0
else eta_contract_atom (incr_boundvars ~1 f)
| _ => t0)
| eta_contract_atom t = t
and eta_contract2 (f$t) = f $ eta_contract_atom t
| eta_contract2 t = eta_contract_atom t;
(*When can we safely delete the equality?
Not if it equates two constants; consider 0=1.
Not if it resembles x=t[x], since substitution does not eliminate x.
Not if it resembles ?x=0; another goal could instantiate ?x to Suc(i)
Prefer to eliminate Bound variables if possible.
Result: true = use as is, false = reorient first *)
(*Does t occur in u? For substitution.
Does NOT check args of Skolem terms: substitution does not affect them.
NOT reflexive since hyp_subst_tac fails on x=x.*)
fun substOccur t =
let fun occEq u = (t aconv u) orelse occ u
and occ (Var(ref None)) = false
| occ (Var(ref(Some u))) = occEq u
| occ (Abs(_,u)) = occEq u
| occ (f$u) = occEq u orelse occEq f
| occ (_) = false;
in occEq end;
fun check (t,u,v) = if substOccur t u then raise DEST_EQ else v;
(*IF the goal is an equality with a substitutable variable
THEN orient that equality ELSE raise exception DEST_EQ*)
fun orientGoal (t,u) =
case (eta_contract_atom t, eta_contract_atom u) of
(Skolem _, _) => check(t,u,(t,u)) (*eliminates t*)
| (_, Skolem _) => check(u,t,(u,t)) (*eliminates u*)
| (Free _, _) => check(t,u,(t,u)) (*eliminates t*)
| (_, Free _) => check(u,t,(u,t)) (*eliminates u*)
| _ => raise DEST_EQ;
(*Convert a Goal to an ordinary Not. Used also in dup_intr, where a goal like
Ex(P) is duplicated as the assumption ~Ex(P). *)
fun negOfGoal (Const"*Goal*" $ G, md) = (negate G, md)
| negOfGoal G = G;
(*Substitute through the branch if an equality goal (else raise DEST_EQ)*)
fun equalSubst (G, br, hazs, lits, vars, lim) =
let val subst = subst_atomic (orientGoal(dest_eq G))
fun subst2(G,md) = (subst G, md)
fun subLits ([], br, nlits) =
(br, map (map subst2) hazs, nlits, vars, lim)
| subLits (lit::lits, br, nlits) =
let val nlit = subst lit
in if nlit aconv lit then subLits (lits, br, nlit::nlits)
else subLits (lits, (nlit,true)::br, nlits)
end
in subLits (rev lits, map subst2 br, [])
end;
exception NEWBRANCHES and CLOSEF;
type branch = (term*bool) list * (*pending formulae with md flags*)
(term*bool) list list * (*stack of haz formulae*)
term list * (*literals: irreducible formulae*)
term option ref list * (*variables occurring in branch*)
int; (*resource limit*)
val fullTrace = ref[] : branch list list ref;
exception PROVE;
val eq_contr_tac = eresolve_tac [Data.notE] THEN' eq_assume_tac;
val eContr_tac = TRACE Data.notE (eq_contr_tac ORELSE' Data.contr_tac);
val eAssume_tac = TRACE asm_rl (eq_assume_tac ORELSE' assume_tac);
(*Try to unify complementary literals and return the corresponding tactic. *)
fun tryClose (Const"*Goal*" $ G, L) = (unify([],G,L); eAssume_tac)
| tryClose (G, Const"*Goal*" $ L) = (unify([],G,L); eAssume_tac)
| tryClose (Const"Not" $ G, L) = (unify([],G,L); eContr_tac)
| tryClose (G, Const"Not" $ L) = (unify([],G,L); eContr_tac)
| tryClose _ = raise UNIFY;
(*hazs is a list of lists of unsafe formulae. This "stack" keeps them
in the right relative order: they must go after *all* safe formulae,
with newly introduced ones coming before older ones.*)
(*Add an empty "stack frame" unless there's already one there*)
fun nilHaz hazs =
case hazs of []::_ => hazs
| _ => []::hazs;
fun addHaz (G, haz::hazs) = (haz@[negOfGoal G]) :: hazs;
(*Convert *Goal* to negated assumption in FIRST position*)
val negOfGoal_tac = rtac Data.ccontr THEN' rotate_tac ~1;
(*Were there Skolem terms in the premise? Must NOT chase Vars*)
fun hasSkolem (Skolem _) = true
| hasSkolem (Abs (_,body)) = hasSkolem body
| hasSkolem (f$t) = hasSkolem f orelse hasSkolem t
| hasSkolem _ = false;
(*Attach the right "may duplicate" flag to new formulae: if they contain
Skolem terms then allow duplication.*)
fun joinMd md [] = []
| joinMd md (G::Gs) = (G, hasSkolem G orelse md) :: joinMd md Gs;
(*Join new formulae to a branch.*)
fun appendBr md (ts,us) =
if (exists isGoal ts) then joinMd md ts @ map negOfGoal us
else joinMd md ts @ us;
(** Backtracking and Pruning **)
(*clashVar vars (n,trail) determines whether any of the last n elements
of "trail" occur in "vars" OR in their instantiations*)
fun clashVar [] = (fn _ => false)
| clashVar vars =
let fun clash (0, _) = false
| clash (_, []) = false
| clash (n, v::vs) = exists (varOccur v) vars orelse clash(n-1,vs)
in clash end;
(*nbrs = # of branches just prior to closing this one. Delete choice points
for goals proved by the latest inference, provided NO variables in the
next branch have been updated.*)
fun prune (1, nxtVars, choices) = choices (*DON'T prune at very end: allow
backtracking over bad proofs*)
| prune (nbrs, nxtVars, choices) =
let fun traceIt last =
let val ll = length last
and lc = length choices
in if !trace andalso ll<lc then
(writeln("PRUNING " ^ Int.toString(lc-ll) ^ " LEVELS");
last)
else last
end
fun pruneAux (last, _, _, []) = last
| pruneAux (last, ntrl, trl, ch' as (ntrl',nbrs',exn) :: choices) =
if nbrs' < nbrs
then last (*don't backtrack beyond first solution of goal*)
else if nbrs' > nbrs then pruneAux (last, ntrl, trl, choices)
else (* nbrs'=nbrs *)
if clashVar nxtVars (ntrl-ntrl', trl) then last
else (*no clashes: can go back at least this far!*)
pruneAux(choices, ntrl', List.drop(trl, ntrl-ntrl'),
choices)
in traceIt (pruneAux (choices, !ntrail, !trail, choices)) end;
fun nextVars ((br, hazs, lits, vars, lim) :: _) = map Var vars
| nextVars [] = [];
fun backtrack ((_, _, exn)::_) = raise exn
| backtrack _ = raise PROVE;
(*Change all *Goal* literals to Not. Also delete all those identical to G.*)
fun addLit (Const "*Goal*" $ G,lits) =
let fun bad (Const"*Goal*" $ _) = true
| bad (Const"Not" $ G') = G aconv G'
| bad _ = false;
fun change [] = []
| change (Const"*Goal*" $ G' :: lits) =
if G aconv G' then change lits
else Const"Not" $ G' :: change lits
| change (Const"Not" $ G' :: lits) =
if G aconv G' then change lits
else Const"Not" $ G' :: change lits
| change (lit::lits) = lit :: change lits
in
Const "*Goal*" $ G :: (if exists bad lits then change lits else lits)
end
| addLit (G,lits) = ins_term(G, lits)
(*Tableau prover based on leanTaP. Argument is a list of branches. Each
branch contains a list of unexpanded formulae, a list of literals, and a
bound on unsafe expansions.*)
fun prove (cs, brs, cont) =
let val {safe0_netpair, safep_netpair, haz_netpair, ...} = Data.rep_claset cs
val safeList = [safe0_netpair, safep_netpair]
and hazList = [haz_netpair]
fun prv (tacs, trs, choices, []) = (cont (trs,choices,tacs))
| prv (tacs, trs, choices,
brs0 as ((G,md)::br, hazs, lits, vars, lim) :: brs) =
let exception PRV (*backtrack to precisely this recursion!*)
val ntrl = !ntrail
val nbrs = length brs0
val nxtVars = nextVars brs
val G = norm G
(*Make a new branch, decrementing "lim" if instantiations occur*)
fun newBr vars prems =
map (fn prem => (appendBr md (prem, br),
nilHaz hazs, lits,
add_terms_vars (prem,vars),
if ntrl < !ntrail then lim-3 else lim))
prems @
brs
(*Seek a matching rule. If unifiable then add new premises
to branch.*)
fun deeper [] = raise NEWBRANCHES
| deeper (((P,prems),tac)::grls) =
let val dummy = unify(add_term_vars(P,[]), P, G)
(*P comes from the rule; G comes from the branch.*)
val ntrl' = !ntrail
val choices' = (ntrl, nbrs, PRV) :: choices
in
if null prems then (*closed the branch: prune!*)
prv(tac false :: tacs, (*no duplication*)
brs0::trs,
prune (nbrs, nxtVars, choices'),
brs)
handle PRV =>
(*reset Vars and try another rule*)
(clearTo ntrl; deeper grls)
else
prv(tac false :: tacs, (*no duplication*)
brs0::trs, choices',
newBr (vars_in_vars vars) prems)
handle PRV =>
if ntrl < ntrl' then
(*Vars have been updated: must backtrack
even if not mentioned in other goals!
Reset Vars and try another rule*)
(clearTo ntrl; deeper grls)
else (*backtrack to previous level*)
backtrack choices
end
handle UNIFY => deeper grls
(*Try to close branch by unifying with head goal*)
fun closeF [] = raise CLOSEF
| closeF (L::Ls) =
let val tacs' = tryClose(G,L)::tacs
val choices' = prune (nbrs, nxtVars,
(ntrl, nbrs, PRV) :: choices)
in prv (tacs', brs0::trs, choices', brs)
handle PRV =>
(*reset Vars and try another literal
[this handler is pruned if possible!]*)
(clearTo ntrl; closeF Ls)
end
handle UNIFY => closeF Ls
in if !trace then fullTrace := brs0 :: !fullTrace else ();
if lim<0 then backtrack choices
else
prv (Data.vars_gen_hyp_subst_tac false :: tacs,
brs0::trs, choices,
equalSubst (G, br, hazs, lits, vars, lim) :: brs)
handle DEST_EQ => closeF lits
handle CLOSEF => closeF (map #1 br)
handle CLOSEF => closeF (map #1 (List.concat hazs))
handle CLOSEF =>
(deeper (netMkRules G vars safeList)
handle NEWBRANCHES =>
(case netMkRules G vars hazList of
[] => (*no plausible rules: move G to literals*)
prv (tacs, trs, choices,
(br, hazs, addLit(G,lits), vars, lim)::brs)
| _ => (*G admits some haz rules: try later*)
prv (if isGoal G then negOfGoal_tac :: tacs
else tacs,
trs, choices,
(br, addHaz((G,md),hazs), lits, vars, lim)
::brs)))
end
| prv (tacs, trs, choices, ([], []::hazs, lits, vars, lim) :: brs) =
(*removal of empty list from hazs*)
prv (tacs, trs, choices, ([], hazs, lits, vars, lim) :: brs)
| prv (tacs, trs, choices,
brs0 as ([], ((G,md)::Gs)::hazs, lits, vars, lim) :: brs) =
(*application of haz rule*)
let exception PRV (*backtrack to precisely this recursion!*)
val G = norm G
val ntrl = !ntrail
fun newPrem (vars,dup) prem =
(map (fn P => (P,false)) prem,
nilHaz (if dup then Gs :: hazs @ [[negOfGoal (G,md)]]
else Gs :: hazs),
lits,
vars,
(*Decrement "lim" if instantiations occur or the
formula is duplicated*)
if ntrl < !ntrail then lim-3
else if dup then lim-1 else lim)
fun newBr x prems = map (newPrem x) prems @ brs
(*Seek a matching rule. If unifiable then add new premises
to branch.*)
fun deeper [] = raise NEWBRANCHES
| deeper (((P,prems),tac)::grls) =
let val dummy = unify(add_term_vars(P,[]), P, G)
val ntrl' = !ntrail
val vars = vars_in_vars vars
val vars' = foldr add_terms_vars (prems, vars)
val dup = md andalso vars' <> vars
(*duplicate G only if md and the premise has new vars*)
in
prv(tac dup :: tacs,
brs0::trs,
(ntrl, length brs0, PRV) :: choices,
newBr (vars', dup) prems)
handle PRV =>
if ntrl < ntrl' (*variables updated?*)
orelse vars=vars' (*pseudo-unsafe: no new Vars?*)
then (*reset Vars and try another rule*)
(clearTo ntrl; deeper grls)
else (*backtrack to previous level*)
backtrack choices
end
handle UNIFY => deeper grls
in if !trace then fullTrace := brs0 :: !fullTrace else ();
if lim<1 then backtrack choices
else
deeper (netMkRules G vars hazList)
handle NEWBRANCHES =>
(*cannot close branch: move G to literals*)
prv (tacs, brs0::trs, choices,
([], Gs::hazs, G::lits, vars, lim)::brs)
end
| prv (tacs, trs, choices, _ :: brs) = backtrack choices
in prv ([], [], [(!ntrail, length brs, PROVE)], brs) end;
fun initBranch (ts,lim) =
(map (fn t => (t,true)) ts,
[[]], [], add_terms_vars (ts,[]), lim);
(*** Conversion & Skolemization of the Isabelle proof state ***)
(*Make a list of all the parameters in a subgoal, even if nested*)
local open Term
in
fun discard_foralls (Const("all",_)$Abs(a,T,t)) = discard_foralls t
| discard_foralls t = t;
end;
(*List of variables not appearing as arguments to the given parameter*)
fun getVars [] i = []
| getVars ((_,(v,is))::alist) i =
if i mem is then getVars alist i
else v :: getVars alist i;
(*Conversion of a subgoal: Skolemize all parameters*)
fun fromSubgoal tsig t =
let val alist = ref []
fun hdvar ((ix,(v,is))::_) = v
fun from lev t =
let val (ht,ts) = Term.strip_comb t
fun apply u = list_comb (u, map (from lev) ts)
fun bounds [] = []
| bounds (Term.Bound i::ts) =
if i<lev then error"Function Var's argument not a parameter"
else i-lev :: bounds ts
| bounds ts = error"Function Var's argument not a bound variable"
in
case ht of
t as Term.Const _ => apply (fromConst tsig t)
| Term.Free (a,_) => apply (Free a)
| Term.Bound i => apply (Bound i)
| Term.Var (ix,_) =>
(case (assoc_string_int(!alist,ix)) of
None => (alist := (ix, (ref None, bounds ts))
:: !alist;
Var (hdvar(!alist)))
| Some(v,is) => if is=bounds ts then Var v
else error("Discrepancy among occurrences of ?"
^ Syntax.string_of_vname ix))
| Term.Abs (a,_,body) =>
if null ts then Abs(a, from (lev+1) body)
else error "fromSubgoal: argument not in normal form"
end
val npars = length (Logic.strip_params t)
(*Skolemize a subgoal from a proof state*)
fun skoSubgoal i t =
if i<npars then
skoSubgoal (i+1)
(subst_bound (Skolem (gensym "SG_", getVars (!alist) i),
t))
else t
in skoSubgoal 0 (from 0 (discard_foralls t)) end;
(*Tactic using tableau engine and proof reconstruction.
"lim" is depth limit.*)
fun depth_tac cs lim i st =
(fullTrace:=[]; trail := []; ntrail := 0;
let val {tsig,...} = Sign.rep_sg (#sign (rep_thm st))
val skoprem = fromSubgoal tsig (List.nth(prems_of st, i-1))
val hyps = strip_imp_prems skoprem
and concl = strip_imp_concl skoprem
fun cont (_,choices,tacs) =
(case Sequence.pull(EVERY' (rev tacs) i st) of
None => (writeln ("PROOF FAILED for depth " ^
Int.toString lim);
backtrack choices)
| cell => Sequence.seqof(fn()=> cell))
in prove (cs, [initBranch (mkGoal concl :: hyps, lim)], cont)
end
handle Subscript => Sequence.null
| PROVE => Sequence.null);
fun blast_tac cs = (DEEPEN (1,20) (depth_tac cs) 0);
fun Blast_tac i = blast_tac (!Data.claset) i;
end;