author | paulson |
Thu, 19 Jun 1997 11:31:14 +0200 | |
changeset 3451 | d10f100676d8 |
parent 3244 | 71b760618f30 |
child 3533 | b976967a92fc |
permissions | -rw-r--r-- |
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(* Title: Provers/blast |
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ID: $Id$ |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Copyright 1997 University of Cambridge |
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Generic tableau prover with proof reconstruction |
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SKOLEMIZES ReplaceI WRONGLY: allow new vars in prems, or forbid such rules?? |
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Needs explicit instantiation of assumptions? |
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Blast_tac is often more powerful than fast_tac, but has some limitations. |
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Blast_tac... |
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* ignores addss, addbefore, addafter; this restriction is intrinsic |
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* ignores elimination rules that don't have the correct format |
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(conclusion must be a formula variable) |
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* ignores types, which can make HOL proofs fail |
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* rules must not require higher-order unification, e.g. apply_type in ZF |
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+ message "Function Var's argument not a bound variable" relates to this |
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* its proof strategy is more general but can actually be slower |
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Known problems: |
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"Recursive" chains of rules can sometimes exclude other unsafe formulae |
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from expansion. This happens because newly-created formulae always |
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have priority over existing ones. But obviously recursive rules |
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such as transitivity are treated specially to prevent this. |
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With overloading: |
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Overloading can't follow all chains of type dependencies. Cannot |
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prove "In1 x ~: Part A In0" because PartE introducees the polymorphic |
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equality In1 x = In0 y, when the corresponding rule uses the (distinct) |
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set equality. Workaround: supply a type instance of the rule that |
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creates new equalities (e.g. PartE in HOL/ex/Simult) |
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==> affects freeness reasoning about Sexp constants (since they have |
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type ... set) |
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With substition for equalities (hyp_subst_tac): |
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When substitution affects a haz formula or literal, it is moved |
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back to the list of safe formulae. |
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But there's no way of putting it in the right place. A "moved" or |
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"no DETERM" flag would prevent proofs failing here. |
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*) |
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(*Should be a type abbreviation?*) |
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type netpair = (int*(bool*thm)) Net.net * (int*(bool*thm)) Net.net; |
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(*Assumptions about constants: |
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--The negation symbol is "Not" |
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--The equality symbol is "op =" |
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--The is-true judgement symbol is "Trueprop" |
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--There are no constants named "*Goal* or "*False*" |
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*) |
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signature BLAST_DATA = |
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sig |
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type claset |
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val notE : thm (* [| ~P; P |] ==> R *) |
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val ccontr : thm |
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val contr_tac : int -> tactic |
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val dup_intr : thm -> thm |
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val vars_gen_hyp_subst_tac : bool -> int -> tactic |
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val claset : claset ref |
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val rep_claset : |
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claset -> {safeIs: thm list, safeEs: thm list, |
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hazIs: thm list, hazEs: thm list, |
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uwrapper: (int -> tactic) -> (int -> tactic), |
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swrapper: (int -> tactic) -> (int -> tactic), |
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safe0_netpair: netpair, safep_netpair: netpair, |
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haz_netpair: netpair, dup_netpair: netpair} |
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end; |
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signature BLAST = |
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sig |
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type claset |
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datatype term = |
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Const of string |
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| OConst of string * int |
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| Skolem of string * term option ref list |
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| Free of string |
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| Var of term option ref |
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| Bound of int |
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| Abs of string*term |
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| $ of term*term; |
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type branch |
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val depth_tac : claset -> int -> int -> tactic |
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val blast_tac : claset -> int -> tactic |
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val Blast_tac : int -> tactic |
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val declConsts : string list * thm list -> unit |
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(*debugging tools*) |
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val trace : bool ref |
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val fullTrace : branch list list ref |
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val fromTerm : Type.type_sig -> Term.term -> term |
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val fromSubgoal : Type.type_sig -> Term.term -> term |
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val toTerm : int -> term -> Term.term |
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val readGoal : Sign.sg -> string -> term |
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val tryInThy : theory -> int -> string -> |
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(int->tactic) list * branch list list * (int*int*exn) list |
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val trygl : claset -> int -> int -> |
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(int->tactic) list * branch list list * (int*int*exn) list |
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val Trygl : int -> int -> |
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(int->tactic) list * branch list list * (int*int*exn) list |
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val normBr : branch -> branch |
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end; |
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functor BlastFun(Data: BLAST_DATA) : BLAST = |
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struct |
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type claset = Data.claset; |
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val trace = ref false; |
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datatype term = |
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Const of string |
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| OConst of string * int |
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| Skolem of string * term option ref list |
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| Free of string |
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| Var of term option ref |
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| Bound of int |
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| Abs of string*term |
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| op $ of term*term; |
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(** Basic syntactic operations **) |
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fun is_Var (Var _) = true |
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| is_Var _ = false; |
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fun dest_Var (Var x) = x; |
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fun rand (f$x) = x; |
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(* maps (f, [t1,...,tn]) to f(t1,...,tn) *) |
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val list_comb : term * term list -> term = foldl (op $); |
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(* maps f(t1,...,tn) to (f, [t1,...,tn]) ; naturally tail-recursive*) |
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fun strip_comb u : term * term list = |
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let fun stripc (f$t, ts) = stripc (f, t::ts) |
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| stripc x = x |
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in stripc(u,[]) end; |
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(* maps f(t1,...,tn) to f , which is never a combination *) |
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fun head_of (f$t) = head_of f |
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| head_of u = u; |
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(** Particular constants **) |
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fun negate P = Const"Not" $ P; |
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fun mkGoal P = Const"*Goal*" $ P; |
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fun isGoal (Const"*Goal*" $ _) = true |
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| isGoal _ = false; |
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val Trueprop = Term.Const("Trueprop", Type("o",[])-->propT); |
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fun mk_tprop P = Term.$ (Trueprop, P); |
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fun isTrueprop (Term.Const("Trueprop",_)) = true |
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| isTrueprop _ = false; |
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(** Dealing with overloaded constants **) |
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(*Result is a symbol table, indexed by names of overloaded constants. |
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Each constant maps to a list of (pattern,Blast.Const) pairs. |
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Any Term.Const that matches a pattern gets replaced by the Blast.Const. |
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*) |
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fun addConsts (t as Term.Const(a,_), tab) = |
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(case Symtab.lookup (tab,a) of |
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None => tab (*ignore: not a constant that we are looking for*) |
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| Some patList => |
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(case gen_assoc (op aconv) (patList, t) of |
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None => Symtab.update |
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((a, (t, OConst (a, length patList)) :: patList), |
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tab) |
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| _ => tab)) |
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| addConsts (Term.Abs(_,_,body), tab) = addConsts (body, tab) |
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| addConsts (Term.$ (t,u), tab) = addConsts (t, addConsts (u, tab)) |
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| addConsts (_, tab) = tab (*ignore others*); |
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fun addRules (rls,tab) = foldr addConsts (map (#prop o rep_thm) rls, tab); |
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fun declConst (a,tab) = |
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case Symtab.lookup (tab,a) of |
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None => Symtab.update((a,[]), tab) (*create a brand new entry*) |
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| Some _ => tab (*preserve old entry*); |
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(*maps the name of each overloaded constant to a list of archetypal constants, |
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which may be polymorphic.*) |
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local |
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val overLoadTab = ref (Symtab.null : (Term.term * term) list Symtab.table) |
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(*The alists in this table should only be increased*) |
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in |
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fun declConsts (names, rls) = |
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overLoadTab := addRules (rls, foldr declConst (names, !overLoadTab)); |
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(*Convert a possibly overloaded Term.Const to a Blast.Const*) |
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fun fromConst tsig (t as Term.Const (a,_)) = |
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let fun find [] = Const a |
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| find ((pat,t')::patList) = |
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if Pattern.matches tsig (pat,t) then t' |
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else find patList |
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in case Symtab.lookup(!overLoadTab, a) of |
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None => Const a |
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| Some patList => find patList |
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end; |
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end; |
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(*Tests whether 2 terms are alpha-convertible; chases instantiations*) |
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fun (Const a) aconv (Const b) = a=b |
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| (OConst ai) aconv (OConst bj) = ai=bj |
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| (Skolem (a,_)) aconv (Skolem (b,_)) = a=b (*arglists must then be equal*) |
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| (Free a) aconv (Free b) = a=b |
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| (Var(ref(Some t))) aconv u = t aconv u |
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| t aconv (Var(ref(Some u))) = t aconv u |
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| (Var v) aconv (Var w) = v=w (*both Vars are un-assigned*) |
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| (Bound i) aconv (Bound j) = i=j |
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| (Abs(_,t)) aconv (Abs(_,u)) = t aconv u |
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| (f$t) aconv (g$u) = (f aconv g) andalso (t aconv u) |
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| _ aconv _ = false; |
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fun mem_term (_, []) = false |
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| mem_term (t, t'::ts) = t aconv t' orelse mem_term(t,ts); |
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fun ins_term(t,ts) = if mem_term(t,ts) then ts else t :: ts; |
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fun mem_var (v: term option ref, []) = false |
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| mem_var (v, v'::vs) = v=v' orelse mem_var(v,vs); |
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fun ins_var(v,vs) = if mem_var(v,vs) then vs else v :: vs; |
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(** Vars **) |
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(*Accumulates the Vars in the term, suppressing duplicates*) |
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fun add_term_vars (Skolem(a,args), vars) = add_vars_vars(args,vars) |
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| add_term_vars (Var (v as ref None), vars) = ins_var (v, vars) |
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| add_term_vars (Var (ref (Some u)), vars) = add_term_vars(u,vars) |
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| add_term_vars (Abs (_,body), vars) = add_term_vars(body,vars) |
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| add_term_vars (f$t, vars) = add_term_vars (f, add_term_vars(t, vars)) |
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| add_term_vars (_, vars) = vars |
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(*Term list version. [The fold functionals are slow]*) |
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and add_terms_vars ([], vars) = vars |
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| add_terms_vars (t::ts, vars) = add_terms_vars (ts, add_term_vars(t,vars)) |
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(*Var list version.*) |
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and add_vars_vars ([], vars) = vars |
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| add_vars_vars (ref (Some u) :: vs, vars) = |
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add_vars_vars (vs, add_term_vars(u,vars)) |
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| add_vars_vars (v::vs, vars) = (*v must be a ref None*) |
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add_vars_vars (vs, ins_var (v, vars)); |
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(*Chase assignments in "vars"; return a list of unassigned variables*) |
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fun vars_in_vars vars = add_vars_vars(vars,[]); |
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(*increment a term's non-local bound variables |
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inc is increment for bound variables |
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lev is level at which a bound variable is considered 'loose'*) |
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fun incr_bv (inc, lev, u as Bound i) = if i>=lev then Bound(i+inc) else u |
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| incr_bv (inc, lev, Abs(a,body)) = Abs(a, incr_bv(inc,lev+1,body)) |
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| incr_bv (inc, lev, f$t) = incr_bv(inc,lev,f) $ incr_bv(inc,lev,t) |
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| incr_bv (inc, lev, u) = u; |
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fun incr_boundvars 0 t = t |
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| incr_boundvars inc t = incr_bv(inc,0,t); |
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(*Accumulate all 'loose' bound vars referring to level 'lev' or beyond. |
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(Bound 0) is loose at level 0 *) |
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fun add_loose_bnos (Bound i, lev, js) = if i<lev then js |
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else (i-lev) ins_int js |
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| add_loose_bnos (Abs (_,t), lev, js) = add_loose_bnos (t, lev+1, js) |
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| add_loose_bnos (f$t, lev, js) = |
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add_loose_bnos (f, lev, add_loose_bnos (t, lev, js)) |
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| add_loose_bnos (_, _, js) = js; |
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fun loose_bnos t = add_loose_bnos (t, 0, []); |
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fun subst_bound (arg, t) : term = |
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let fun subst (t as Bound i, lev) = |
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if i<lev then t (*var is locally bound*) |
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else if i=lev then incr_boundvars lev arg |
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else Bound(i-1) (*loose: change it*) |
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| subst (Abs(a,body), lev) = Abs(a, subst(body,lev+1)) |
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| subst (f$t, lev) = subst(f,lev) $ subst(t,lev) |
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| subst (t,lev) = t |
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in subst (t,0) end; |
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(*Normalize...but not the bodies of ABSTRACTIONS*) |
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fun norm t = case t of |
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Skolem (a,args) => Skolem(a, vars_in_vars args) |
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| (Var (ref None)) => t |
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| (Var (ref (Some u))) => norm u |
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| (f $ u) => (case norm f of |
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Abs(_,body) => norm (subst_bound (u, body)) |
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| nf => nf $ norm u) |
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| _ => t; |
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(*Weak (one-level) normalize for use in unification*) |
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fun wkNormAux t = case t of |
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(Var v) => (case !v of |
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Some u => wkNorm u |
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| None => t) |
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| (f $ u) => (case wkNormAux f of |
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Abs(_,body) => wkNorm (subst_bound (u, body)) |
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| nf => nf $ u) |
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| Abs (a,body) => (*eta-contract if possible*) |
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(case wkNormAux body of |
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nb as (f $ t) => |
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if (0 mem_int loose_bnos f) orelse wkNorm t <> Bound 0 |
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then Abs(a,nb) |
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else wkNorm (incr_boundvars ~1 f) |
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| nb => Abs (a,nb)) |
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| _ => t |
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and wkNorm t = case head_of t of |
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Const _ => t |
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| OConst _ => t |
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| Skolem(a,args) => t |
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| Free _ => t |
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| _ => wkNormAux t; |
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(*Does variable v occur in u? For unification.*) |
|
339 |
fun varOccur v = |
|
340 |
let fun occL [] = false (*same as (exists occ), but faster*) |
|
341 |
| occL (u::us) = occ u orelse occL us |
|
342 |
and occ (Var w) = |
|
343 |
v=w orelse |
|
344 |
(case !w of None => false |
|
345 |
| Some u => occ u) |
|
346 |
| occ (Skolem(_,args)) = occL (map Var args) |
|
347 |
| occ (Abs(_,u)) = occ u |
|
348 |
| occ (f$u) = occ u orelse occ f |
|
349 |
| occ (_) = false; |
|
350 |
in occ end; |
|
351 |
||
352 |
exception UNIFY; |
|
353 |
||
354 |
val trail = ref [] : term option ref list ref; |
|
355 |
val ntrail = ref 0; |
|
356 |
||
357 |
||
358 |
(*Restore the trail to some previous state: for backtracking*) |
|
359 |
fun clearTo n = |
|
3083 | 360 |
while !ntrail<>n do |
2854 | 361 |
(hd(!trail) := None; |
362 |
trail := tl (!trail); |
|
363 |
ntrail := !ntrail - 1); |
|
364 |
||
365 |
||
366 |
(*First-order unification with bound variables. |
|
367 |
"vars" is a list of variables local to the rule and NOT to be put |
|
368 |
on the trail (no point in doing so) |
|
369 |
*) |
|
2999
fad7cc426242
Penalty for branching instantiations reduced from log3 to log4.
paulson
parents:
2952
diff
changeset
|
370 |
fun unify(allowClash,vars,t,u) = |
2854 | 371 |
let val n = !ntrail |
372 |
fun update (t as Var v, u) = |
|
373 |
if t aconv u then () |
|
374 |
else if varOccur v u then raise UNIFY |
|
375 |
else if mem_var(v, vars) then v := Some u |
|
376 |
else (*avoid updating Vars in the branch if possible!*) |
|
377 |
if is_Var u andalso mem_var(dest_Var u, vars) |
|
378 |
then dest_Var u := Some t |
|
379 |
else (v := Some u; |
|
380 |
trail := v :: !trail; ntrail := !ntrail + 1) |
|
381 |
fun unifyAux (t,u) = |
|
382 |
case (wkNorm t, wkNorm u) of |
|
383 |
(nt as Var v, nu) => update(nt,nu) |
|
384 |
| (nu, nt as Var v) => update(nt,nu) |
|
2999
fad7cc426242
Penalty for branching instantiations reduced from log3 to log4.
paulson
parents:
2952
diff
changeset
|
385 |
| (Const a, OConst(b,_)) => if allowClash andalso a=b then () |
fad7cc426242
Penalty for branching instantiations reduced from log3 to log4.
paulson
parents:
2952
diff
changeset
|
386 |
else raise UNIFY |
fad7cc426242
Penalty for branching instantiations reduced from log3 to log4.
paulson
parents:
2952
diff
changeset
|
387 |
| (OConst(a,_), Const b) => if allowClash andalso a=b then () |
fad7cc426242
Penalty for branching instantiations reduced from log3 to log4.
paulson
parents:
2952
diff
changeset
|
388 |
else raise UNIFY |
2854 | 389 |
| (Abs(_,t'), Abs(_,u')) => unifyAux(t',u') |
390 |
(*NB: can yield unifiers having dangling Bound vars!*) |
|
391 |
| (f$t', g$u') => (unifyAux(f,g); unifyAux(t',u')) |
|
392 |
| (nt, nu) => if nt aconv nu then () else raise UNIFY |
|
3083 | 393 |
in (unifyAux(t,u); true) handle UNIFY => (clearTo n; false) |
2854 | 394 |
end; |
395 |
||
396 |
||
397 |
(*Convert from "real" terms to prototerms; eta-contract*) |
|
398 |
fun fromTerm tsig t = |
|
399 |
let val alist = ref [] |
|
400 |
fun from (t as Term.Const _) = fromConst tsig t |
|
401 |
| from (Term.Free (a,_)) = Free a |
|
402 |
| from (Term.Bound i) = Bound i |
|
403 |
| from (Term.Var (ixn,T)) = |
|
404 |
(case (assoc_string_int(!alist,ixn)) of |
|
405 |
None => let val t' = Var(ref None) |
|
406 |
in alist := (ixn, (t', T)) :: !alist; t' |
|
407 |
end |
|
408 |
| Some (v,_) => v) |
|
409 |
| from (Term.Abs (a,_,u)) = |
|
410 |
(case from u of |
|
411 |
u' as (f $ Bound 0) => |
|
412 |
if (0 mem_int loose_bnos f) then Abs(a,u') |
|
413 |
else incr_boundvars ~1 f |
|
414 |
| u' => Abs(a,u')) |
|
415 |
| from (Term.$ (f,u)) = from f $ from u |
|
416 |
in from t end; |
|
417 |
||
418 |
(* A1==>...An==>B goes to [A1,...,An], where B is not an implication *) |
|
419 |
fun strip_imp_prems (Const"==>" $ (Const"Trueprop" $ A) $ B) = |
|
420 |
A :: strip_imp_prems B |
|
421 |
| strip_imp_prems (Const"==>" $ A $ B) = A :: strip_imp_prems B |
|
422 |
| strip_imp_prems _ = []; |
|
423 |
||
424 |
(* A1==>...An==>B goes to B, where B is not an implication *) |
|
425 |
fun strip_imp_concl (Const"==>" $ A $ B) = strip_imp_concl B |
|
426 |
| strip_imp_concl (Const"Trueprop" $ A) = A |
|
427 |
| strip_imp_concl A = A : term; |
|
428 |
||
429 |
||
430 |
(*** Conversion of Elimination Rules to Tableau Operations ***) |
|
431 |
||
432 |
(*The conclusion becomes the goal/negated assumption *False*: delete it!*) |
|
433 |
fun squash_nots [] = [] |
|
434 |
| squash_nots (Const "*Goal*" $ (Var (ref (Some (Const"*False*")))) :: Ps) = |
|
435 |
squash_nots Ps |
|
436 |
| squash_nots (Const "Not" $ (Var (ref (Some (Const"*False*")))) :: Ps) = |
|
437 |
squash_nots Ps |
|
438 |
| squash_nots (P::Ps) = P :: squash_nots Ps; |
|
439 |
||
440 |
fun skoPrem vars (Const "all" $ Abs (_, P)) = |
|
441 |
skoPrem vars (subst_bound (Skolem (gensym "S_", vars), P)) |
|
442 |
| skoPrem vars P = P; |
|
443 |
||
444 |
fun convertPrem t = |
|
445 |
squash_nots (mkGoal (strip_imp_concl t) :: strip_imp_prems t); |
|
446 |
||
447 |
(*Expects elimination rules to have a formula variable as conclusion*) |
|
448 |
fun convertRule vars t = |
|
449 |
let val (P::Ps) = strip_imp_prems t |
|
450 |
val Var v = strip_imp_concl t |
|
451 |
in v := Some (Const"*False*"); |
|
452 |
(P, map (convertPrem o skoPrem vars) Ps) |
|
453 |
end; |
|
454 |
||
455 |
||
456 |
(*Like dup_elim, but puts the duplicated major premise FIRST*) |
|
457 |
fun rev_dup_elim th = th RSN (2, revcut_rl) |> assumption 2 |> Sequence.hd; |
|
458 |
||
459 |
||
460 |
(*Count new hyps so that they can be rotated*) |
|
461 |
fun nNewHyps [] = 0 |
|
462 |
| nNewHyps (Const "*Goal*" $ _ :: Ps) = nNewHyps Ps |
|
463 |
| nNewHyps (P::Ps) = 1 + nNewHyps Ps; |
|
464 |
||
465 |
fun rot_subgoals_tac [] i st = Sequence.single st |
|
466 |
| rot_subgoals_tac (k::ks) i st = |
|
467 |
rot_subgoals_tac ks (i+1) (Sequence.hd (rotate_tac (~k) i st)) |
|
468 |
handle OPTION _ => Sequence.null; |
|
469 |
||
2999
fad7cc426242
Penalty for branching instantiations reduced from log3 to log4.
paulson
parents:
2952
diff
changeset
|
470 |
fun TRACE rl tac st i = if !trace then (prth rl; tac st i) else tac st i; |
2854 | 471 |
|
472 |
(*Tableau rule from elimination rule. Flag "dup" requests duplication of the |
|
473 |
affected formula.*) |
|
474 |
fun fromRule vars rl = |
|
475 |
let val {tsig,...} = Sign.rep_sg (#sign (rep_thm rl)) |
|
476 |
val trl = rl |> rep_thm |> #prop |> fromTerm tsig |> convertRule vars |
|
477 |
fun tac dup i = |
|
2999
fad7cc426242
Penalty for branching instantiations reduced from log3 to log4.
paulson
parents:
2952
diff
changeset
|
478 |
TRACE rl (etac (if dup then rev_dup_elim rl else rl)) i |
2854 | 479 |
THEN rot_subgoals_tac (map nNewHyps (#2 trl)) i |
480 |
||
3244
71b760618f30
Basis library version of type "option" now resides in its own structure Option
paulson
parents:
3101
diff
changeset
|
481 |
in Option.SOME (trl, tac) end |
2924
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
482 |
handle Bind => (*reject: conclusion is not just a variable*) |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
483 |
(if !trace then (writeln"Warning: ignoring ill-formed elimination rule"; |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
484 |
print_thm rl) |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
485 |
else (); |
3244
71b760618f30
Basis library version of type "option" now resides in its own structure Option
paulson
parents:
3101
diff
changeset
|
486 |
Option.NONE); |
2854 | 487 |
|
488 |
||
3101
e8a92f497295
Again "norm" DOES NOT normalize bodies of abstractions
paulson
parents:
3092
diff
changeset
|
489 |
(*** Conversion of Introduction Rules ***) |
2854 | 490 |
|
491 |
fun convertIntrPrem t = mkGoal (strip_imp_concl t) :: strip_imp_prems t; |
|
492 |
||
493 |
fun convertIntrRule vars t = |
|
494 |
let val Ps = strip_imp_prems t |
|
495 |
val P = strip_imp_concl t |
|
496 |
in (mkGoal P, map (convertIntrPrem o skoPrem vars) Ps) |
|
497 |
end; |
|
498 |
||
499 |
(*Tableau rule from introduction rule. Since haz rules are now delayed, |
|
500 |
"dup" is always FALSE for introduction rules.*) |
|
501 |
fun fromIntrRule vars rl = |
|
502 |
let val {tsig,...} = Sign.rep_sg (#sign (rep_thm rl)) |
|
503 |
val trl = rl |> rep_thm |> #prop |> fromTerm tsig |> convertIntrRule vars |
|
504 |
fun tac dup i = |
|
2999
fad7cc426242
Penalty for branching instantiations reduced from log3 to log4.
paulson
parents:
2952
diff
changeset
|
505 |
TRACE rl (DETERM o (rtac (if dup then Data.dup_intr rl else rl))) i |
2854 | 506 |
THEN rot_subgoals_tac (map nNewHyps (#2 trl)) i |
507 |
in (trl, tac) end; |
|
508 |
||
509 |
||
3030 | 510 |
val dummyVar = Term.Var (("etc",0), dummyT); |
2854 | 511 |
|
512 |
(*Convert from prototerms to ordinary terms with dummy types |
|
2924
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
513 |
Ignore abstractions; identify all Vars; STOP at given depth*) |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
514 |
fun toTerm 0 _ = dummyVar |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
515 |
| toTerm d (Const a) = Term.Const (a,dummyT) |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
516 |
| toTerm d (OConst(a,_)) = Term.Const (a,dummyT) |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
517 |
| toTerm d (Skolem(a,_)) = Term.Const (a,dummyT) |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
518 |
| toTerm d (Free a) = Term.Free (a,dummyT) |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
519 |
| toTerm d (Bound i) = Term.Bound i |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
520 |
| toTerm d (Var _) = dummyVar |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
521 |
| toTerm d (Abs(a,_)) = dummyVar |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
522 |
| toTerm d (f $ u) = Term.$ (toTerm d f, toTerm (d-1) u); |
2854 | 523 |
|
524 |
||
525 |
fun netMkRules P vars (nps: netpair list) = |
|
526 |
case P of |
|
527 |
(Const "*Goal*" $ G) => |
|
2924
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
528 |
let val pG = mk_tprop (toTerm 2 G) |
2854 | 529 |
val intrs = List.concat |
530 |
(map (fn (inet,_) => Net.unify_term inet pG) |
|
531 |
nps) |
|
532 |
in map (fromIntrRule vars o #2) (orderlist intrs) end |
|
533 |
| _ => |
|
2924
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
534 |
let val pP = mk_tprop (toTerm 3 P) |
2854 | 535 |
val elims = List.concat |
536 |
(map (fn (_,enet) => Net.unify_term enet pP) |
|
537 |
nps) |
|
538 |
in List.mapPartial (fromRule vars o #2) (orderlist elims) end; |
|
539 |
||
540 |
(** |
|
541 |
end; |
|
542 |
**) |
|
543 |
||
3092 | 544 |
|
545 |
(*Pending formulae carry md (may duplicate) flags*) |
|
546 |
type branch = ((term*bool) list * (*safe formulae on this level*) |
|
547 |
(term*bool) list) list * (*haz formulae on this level*) |
|
548 |
term list * (*literals: irreducible formulae*) |
|
549 |
term option ref list * (*variables occurring in branch*) |
|
550 |
int; (*resource limit*) |
|
551 |
||
552 |
val fullTrace = ref[] : branch list list ref; |
|
553 |
||
554 |
(*Normalize a branch--for tracing*) |
|
555 |
fun norm2 (G,md) = (norm G, md); |
|
556 |
||
557 |
fun normLev (Gs,Hs) = (map norm2 Gs, map norm2 Hs); |
|
558 |
||
559 |
fun normBr (br, lits, vars, lim) = |
|
560 |
(map normLev br, map norm lits, vars, lim); |
|
561 |
||
562 |
||
563 |
val dummyVar2 = Term.Var(("var",0), dummyT); |
|
564 |
||
565 |
(*Convert from prototerms to ordinary terms with dummy types for tracing*) |
|
566 |
fun showTerm d (Const a) = Term.Const (a,dummyT) |
|
567 |
| showTerm d (OConst(a,_)) = Term.Const (a,dummyT) |
|
568 |
| showTerm d (Skolem(a,_)) = Term.Const (a,dummyT) |
|
569 |
| showTerm d (Free a) = Term.Free (a,dummyT) |
|
570 |
| showTerm d (Bound i) = Term.Bound i |
|
3101
e8a92f497295
Again "norm" DOES NOT normalize bodies of abstractions
paulson
parents:
3092
diff
changeset
|
571 |
| showTerm d (Var(ref(Some u))) = showTerm d u |
e8a92f497295
Again "norm" DOES NOT normalize bodies of abstractions
paulson
parents:
3092
diff
changeset
|
572 |
| showTerm d (Var(ref None)) = dummyVar2 |
3092 | 573 |
| showTerm d (Abs(a,t)) = if d=0 then dummyVar |
574 |
else Term.Abs(a, dummyT, showTerm (d-1) t) |
|
575 |
| showTerm d (f $ u) = if d=0 then dummyVar |
|
576 |
else Term.$ (showTerm d f, showTerm (d-1) u); |
|
577 |
||
578 |
||
579 |
fun traceTerm sign t = Sign.string_of_term sign (showTerm 8 (norm t)); |
|
580 |
||
581 |
||
582 |
(*Print tracing information at each iteration of prover*) |
|
583 |
fun tracing sign brs = |
|
584 |
let fun printPairs (((G,_)::_,_)::_) = prs(traceTerm sign G) |
|
585 |
| printPairs (([],(H,_)::_)::_) = prs(traceTerm sign H ^ "\t (Unsafe)") |
|
586 |
| printPairs _ = () |
|
587 |
fun printBrs (brs0 as (pairs, lits, _, lim) :: brs) = |
|
588 |
(fullTrace := brs0 :: !fullTrace; |
|
589 |
seq (fn _ => prs "+") brs; |
|
590 |
prs (" [" ^ Int.toString lim ^ "] "); |
|
591 |
printPairs pairs; |
|
592 |
writeln"") |
|
593 |
in if !trace then printBrs (map normBr brs) else () |
|
594 |
end; |
|
595 |
||
596 |
(*Tracing: variables updated in the last branch operation?*) |
|
597 |
fun traceVars ntrl = |
|
598 |
if !trace then |
|
599 |
(case !ntrail-ntrl of |
|
600 |
0 => writeln"" |
|
601 |
| 1 => writeln"\t1 variable updated" |
|
602 |
| n => writeln("\t" ^ Int.toString n ^ " variables updated")) |
|
603 |
else (); |
|
604 |
||
605 |
(*Tracing: how many new branches are created?*) |
|
606 |
fun traceNew prems = |
|
607 |
if !trace then |
|
608 |
case length prems of |
|
609 |
0 => prs"branch closed by rule" |
|
610 |
| 1 => prs"branch extended (1 new subgoal)" |
|
611 |
| n => prs("branch split: "^ Int.toString n ^ " new subgoals") |
|
612 |
else (); |
|
613 |
||
614 |
||
615 |
||
2854 | 616 |
(*** Code for handling equality: naive substitution, like hyp_subst_tac ***) |
617 |
||
618 |
(*Replace the ATOMIC term "old" by "new" in t*) |
|
619 |
fun subst_atomic (old,new) t = |
|
620 |
let fun subst (Var(ref(Some u))) = subst u |
|
621 |
| subst (Abs(a,body)) = Abs(a, subst body) |
|
622 |
| subst (f$t) = subst f $ subst t |
|
623 |
| subst t = if t aconv old then new else t |
|
624 |
in subst t end; |
|
625 |
||
626 |
(*Eta-contract a term from outside: just enough to reduce it to an atom*) |
|
627 |
fun eta_contract_atom (t0 as Abs(a, body)) = |
|
628 |
(case eta_contract2 body of |
|
629 |
f $ Bound 0 => if (0 mem_int loose_bnos f) then t0 |
|
630 |
else eta_contract_atom (incr_boundvars ~1 f) |
|
631 |
| _ => t0) |
|
632 |
| eta_contract_atom t = t |
|
633 |
and eta_contract2 (f$t) = f $ eta_contract_atom t |
|
634 |
| eta_contract2 t = eta_contract_atom t; |
|
635 |
||
636 |
||
637 |
(*When can we safely delete the equality? |
|
638 |
Not if it equates two constants; consider 0=1. |
|
639 |
Not if it resembles x=t[x], since substitution does not eliminate x. |
|
640 |
Not if it resembles ?x=0; another goal could instantiate ?x to Suc(i) |
|
641 |
Prefer to eliminate Bound variables if possible. |
|
642 |
Result: true = use as is, false = reorient first *) |
|
643 |
||
644 |
(*Does t occur in u? For substitution. |
|
645 |
Does NOT check args of Skolem terms: substitution does not affect them. |
|
646 |
NOT reflexive since hyp_subst_tac fails on x=x.*) |
|
647 |
fun substOccur t = |
|
648 |
let fun occEq u = (t aconv u) orelse occ u |
|
649 |
and occ (Var(ref None)) = false |
|
650 |
| occ (Var(ref(Some u))) = occEq u |
|
651 |
| occ (Abs(_,u)) = occEq u |
|
652 |
| occ (f$u) = occEq u orelse occEq f |
|
653 |
| occ (_) = false; |
|
654 |
in occEq end; |
|
655 |
||
3092 | 656 |
exception DEST_EQ; |
657 |
||
658 |
(*Take apart an equality (plain or overloaded). NO constant Trueprop*) |
|
659 |
fun dest_eq (Const "op =" $ t $ u) = |
|
660 |
(eta_contract_atom t, eta_contract_atom u) |
|
661 |
| dest_eq (OConst("op =",_) $ t $ u) = |
|
662 |
(eta_contract_atom t, eta_contract_atom u) |
|
663 |
| dest_eq _ = raise DEST_EQ; |
|
664 |
||
2854 | 665 |
fun check (t,u,v) = if substOccur t u then raise DEST_EQ else v; |
666 |
||
667 |
(*IF the goal is an equality with a substitutable variable |
|
668 |
THEN orient that equality ELSE raise exception DEST_EQ*) |
|
3092 | 669 |
fun orientGoal (t,u) = case (t,u) of |
2854 | 670 |
(Skolem _, _) => check(t,u,(t,u)) (*eliminates t*) |
671 |
| (_, Skolem _) => check(u,t,(u,t)) (*eliminates u*) |
|
672 |
| (Free _, _) => check(t,u,(t,u)) (*eliminates t*) |
|
673 |
| (_, Free _) => check(u,t,(u,t)) (*eliminates u*) |
|
674 |
| _ => raise DEST_EQ; |
|
675 |
||
2894 | 676 |
(*Substitute through the branch if an equality goal (else raise DEST_EQ). |
677 |
Moves affected literals back into the branch, but it is not clear where |
|
678 |
they should go: this could make proofs fail. ??? *) |
|
3092 | 679 |
fun equalSubst sign (G, pairs, lits, vars, lim) = |
680 |
let val (t,u) = orientGoal(dest_eq G) |
|
681 |
val subst = subst_atomic (t,u) |
|
2854 | 682 |
fun subst2(G,md) = (subst G, md) |
2924
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
683 |
(*substitute throughout Haz list; move affected ones to Safe part*) |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
684 |
fun subHazs ([], Gs, nHs) = (Gs,nHs) |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
685 |
| subHazs ((H,md)::Hs, Gs, nHs) = |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
686 |
let val nH = subst H |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
687 |
in if nH aconv H then subHazs (Hs, Gs, (H,md)::nHs) |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
688 |
else subHazs (Hs, (nH,md)::Gs, nHs) |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
689 |
end |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
690 |
val pairs' = map (fn(Gs,Hs) => subHazs(rev Hs, map subst2 Gs, [])) pairs |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
691 |
(*substitute throughout literals; move affected ones to Safe part*) |
2894 | 692 |
fun subLits ([], [], nlits) = (pairs', nlits, vars, lim) |
693 |
| subLits ([], Gs, nlits) = ((Gs,[])::pairs', nlits, vars, lim) |
|
694 |
| subLits (lit::lits, Gs, nlits) = |
|
2854 | 695 |
let val nlit = subst lit |
2894 | 696 |
in if nlit aconv lit then subLits (lits, Gs, nlit::nlits) |
697 |
else subLits (lits, (nlit,true)::Gs, nlits) |
|
2854 | 698 |
end |
3092 | 699 |
in if !trace then writeln ("Substituting " ^ traceTerm sign u ^ |
700 |
" for " ^ traceTerm sign t ^ " in branch" ) |
|
701 |
else (); |
|
3083 | 702 |
subLits (rev lits, [], []) |
2854 | 703 |
end; |
704 |
||
705 |
||
706 |
exception NEWBRANCHES and CLOSEF; |
|
707 |
||
708 |
exception PROVE; |
|
709 |
||
710 |
val eq_contr_tac = eresolve_tac [Data.notE] THEN' eq_assume_tac; |
|
711 |
||
712 |
val eContr_tac = TRACE Data.notE (eq_contr_tac ORELSE' Data.contr_tac); |
|
713 |
val eAssume_tac = TRACE asm_rl (eq_assume_tac ORELSE' assume_tac); |
|
714 |
||
715 |
(*Try to unify complementary literals and return the corresponding tactic. *) |
|
3083 | 716 |
fun tryClose (Const"*Goal*" $ G, L) = |
717 |
if unify(true,[],G,L) then Some eAssume_tac else None |
|
718 |
| tryClose (G, Const"*Goal*" $ L) = |
|
719 |
if unify(true,[],G,L) then Some eAssume_tac else None |
|
720 |
| tryClose (Const"Not" $ G, L) = |
|
721 |
if unify(true,[],G,L) then Some eContr_tac else None |
|
722 |
| tryClose (G, Const"Not" $ L) = |
|
723 |
if unify(true,[],G,L) then Some eContr_tac else None |
|
724 |
| tryClose _ = None; |
|
2854 | 725 |
|
726 |
||
727 |
(*Were there Skolem terms in the premise? Must NOT chase Vars*) |
|
728 |
fun hasSkolem (Skolem _) = true |
|
729 |
| hasSkolem (Abs (_,body)) = hasSkolem body |
|
730 |
| hasSkolem (f$t) = hasSkolem f orelse hasSkolem t |
|
731 |
| hasSkolem _ = false; |
|
732 |
||
733 |
(*Attach the right "may duplicate" flag to new formulae: if they contain |
|
734 |
Skolem terms then allow duplication.*) |
|
735 |
fun joinMd md [] = [] |
|
736 |
| joinMd md (G::Gs) = (G, hasSkolem G orelse md) :: joinMd md Gs; |
|
737 |
||
2894 | 738 |
(*Convert a Goal to an ordinary Not. Used also in dup_intr, where a goal like |
739 |
Ex(P) is duplicated as the assumption ~Ex(P). *) |
|
740 |
fun negOfGoal (Const"*Goal*" $ G) = negate G |
|
741 |
| negOfGoal G = G; |
|
742 |
||
743 |
fun negOfGoal2 (G,md) = (negOfGoal G, md); |
|
744 |
||
745 |
(*Converts all Goals to Nots in the safe parts of a branch. They could |
|
746 |
have been moved there from the literals list after substitution (equalSubst). |
|
747 |
There can be at most one--this function could be made more efficient.*) |
|
748 |
fun negOfGoals pairs = map (fn (Gs,haz) => (map negOfGoal2 Gs, haz)) pairs; |
|
749 |
||
750 |
(*Tactic. Convert *Goal* to negated assumption in FIRST position*) |
|
751 |
val negOfGoal_tac = rtac Data.ccontr THEN' rotate_tac ~1; |
|
752 |
||
2854 | 753 |
|
754 |
(** Backtracking and Pruning **) |
|
755 |
||
756 |
(*clashVar vars (n,trail) determines whether any of the last n elements |
|
757 |
of "trail" occur in "vars" OR in their instantiations*) |
|
758 |
fun clashVar [] = (fn _ => false) |
|
759 |
| clashVar vars = |
|
760 |
let fun clash (0, _) = false |
|
761 |
| clash (_, []) = false |
|
762 |
| clash (n, v::vs) = exists (varOccur v) vars orelse clash(n-1,vs) |
|
763 |
in clash end; |
|
764 |
||
765 |
||
766 |
(*nbrs = # of branches just prior to closing this one. Delete choice points |
|
767 |
for goals proved by the latest inference, provided NO variables in the |
|
768 |
next branch have been updated.*) |
|
769 |
fun prune (1, nxtVars, choices) = choices (*DON'T prune at very end: allow |
|
770 |
backtracking over bad proofs*) |
|
771 |
| prune (nbrs, nxtVars, choices) = |
|
772 |
let fun traceIt last = |
|
773 |
let val ll = length last |
|
774 |
and lc = length choices |
|
775 |
in if !trace andalso ll<lc then |
|
3083 | 776 |
(writeln("Pruning " ^ Int.toString(lc-ll) ^ " levels"); |
2854 | 777 |
last) |
778 |
else last |
|
779 |
end |
|
780 |
fun pruneAux (last, _, _, []) = last |
|
3083 | 781 |
| pruneAux (last, ntrl, trl, (ntrl',nbrs',exn) :: choices) = |
2854 | 782 |
if nbrs' < nbrs |
783 |
then last (*don't backtrack beyond first solution of goal*) |
|
784 |
else if nbrs' > nbrs then pruneAux (last, ntrl, trl, choices) |
|
785 |
else (* nbrs'=nbrs *) |
|
786 |
if clashVar nxtVars (ntrl-ntrl', trl) then last |
|
787 |
else (*no clashes: can go back at least this far!*) |
|
788 |
pruneAux(choices, ntrl', List.drop(trl, ntrl-ntrl'), |
|
789 |
choices) |
|
790 |
in traceIt (pruneAux (choices, !ntrail, !trail, choices)) end; |
|
791 |
||
2894 | 792 |
fun nextVars ((br, lits, vars, lim) :: _) = map Var vars |
3083 | 793 |
| nextVars [] = []; |
2854 | 794 |
|
3083 | 795 |
fun backtrack (choices as (ntrl, nbrs, exn)::_) = |
796 |
(if !trace then (writeln ("Backtracking; now there are " ^ |
|
797 |
Int.toString nbrs ^ " branches")) |
|
798 |
else (); |
|
799 |
raise exn) |
|
800 |
| backtrack _ = raise PROVE; |
|
2854 | 801 |
|
2894 | 802 |
(*Add the literal G, handling *Goal* and detecting duplicates.*) |
803 |
fun addLit (Const "*Goal*" $ G, lits) = |
|
804 |
(*New literal is a *Goal*, so change all other Goals to Nots*) |
|
2854 | 805 |
let fun bad (Const"*Goal*" $ _) = true |
806 |
| bad (Const"Not" $ G') = G aconv G' |
|
807 |
| bad _ = false; |
|
808 |
fun change [] = [] |
|
809 |
| change (Const"*Goal*" $ G' :: lits) = |
|
810 |
if G aconv G' then change lits |
|
811 |
else Const"Not" $ G' :: change lits |
|
812 |
| change (Const"Not" $ G' :: lits) = |
|
813 |
if G aconv G' then change lits |
|
814 |
else Const"Not" $ G' :: change lits |
|
815 |
| change (lit::lits) = lit :: change lits |
|
816 |
in |
|
817 |
Const "*Goal*" $ G :: (if exists bad lits then change lits else lits) |
|
818 |
end |
|
819 |
| addLit (G,lits) = ins_term(G, lits) |
|
820 |
||
821 |
||
2952 | 822 |
(*For calculating the "penalty" to assess on a branching factor of n |
823 |
log2 seems a little too severe*) |
|
3083 | 824 |
fun log n = if n<4 then 0 else 1 + log(n div 4); |
2924
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
825 |
|
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
826 |
|
3021
39806db47be9
Loop detection: before expanding a haz formula, see whether it is a duplicate
paulson
parents:
2999
diff
changeset
|
827 |
(*match(t,u) says whether the term u might be an instance of the pattern t |
39806db47be9
Loop detection: before expanding a haz formula, see whether it is a duplicate
paulson
parents:
2999
diff
changeset
|
828 |
Used to detect "recursive" rules such as transitivity*) |
39806db47be9
Loop detection: before expanding a haz formula, see whether it is a duplicate
paulson
parents:
2999
diff
changeset
|
829 |
fun match (Var _) u = true |
39806db47be9
Loop detection: before expanding a haz formula, see whether it is a duplicate
paulson
parents:
2999
diff
changeset
|
830 |
| match (Const"*Goal*") (Const"Not") = true |
39806db47be9
Loop detection: before expanding a haz formula, see whether it is a duplicate
paulson
parents:
2999
diff
changeset
|
831 |
| match (Const"Not") (Const"*Goal*") = true |
39806db47be9
Loop detection: before expanding a haz formula, see whether it is a duplicate
paulson
parents:
2999
diff
changeset
|
832 |
| match (Const a) (Const b) = (a=b) |
39806db47be9
Loop detection: before expanding a haz formula, see whether it is a duplicate
paulson
parents:
2999
diff
changeset
|
833 |
| match (OConst ai) (OConst bj) = (ai=bj) |
39806db47be9
Loop detection: before expanding a haz formula, see whether it is a duplicate
paulson
parents:
2999
diff
changeset
|
834 |
| match (Free a) (Free b) = (a=b) |
39806db47be9
Loop detection: before expanding a haz formula, see whether it is a duplicate
paulson
parents:
2999
diff
changeset
|
835 |
| match (Bound i) (Bound j) = (i=j) |
39806db47be9
Loop detection: before expanding a haz formula, see whether it is a duplicate
paulson
parents:
2999
diff
changeset
|
836 |
| match (Abs(_,t)) (Abs(_,u)) = match t u |
39806db47be9
Loop detection: before expanding a haz formula, see whether it is a duplicate
paulson
parents:
2999
diff
changeset
|
837 |
| match (f$t) (g$u) = match f g andalso match t u |
39806db47be9
Loop detection: before expanding a haz formula, see whether it is a duplicate
paulson
parents:
2999
diff
changeset
|
838 |
| match t u = false; |
39806db47be9
Loop detection: before expanding a haz formula, see whether it is a duplicate
paulson
parents:
2999
diff
changeset
|
839 |
|
39806db47be9
Loop detection: before expanding a haz formula, see whether it is a duplicate
paulson
parents:
2999
diff
changeset
|
840 |
|
2854 | 841 |
(*Tableau prover based on leanTaP. Argument is a list of branches. Each |
842 |
branch contains a list of unexpanded formulae, a list of literals, and a |
|
843 |
bound on unsafe expansions.*) |
|
3030 | 844 |
fun prove (sign, cs, brs, cont) = |
2854 | 845 |
let val {safe0_netpair, safep_netpair, haz_netpair, ...} = Data.rep_claset cs |
846 |
val safeList = [safe0_netpair, safep_netpair] |
|
847 |
and hazList = [haz_netpair] |
|
3083 | 848 |
fun prv (tacs, trs, choices, []) = cont (tacs, trs, choices) |
2854 | 849 |
| prv (tacs, trs, choices, |
2894 | 850 |
brs0 as (((G,md)::br, haz)::pairs, lits, vars, lim) :: brs) = |
2854 | 851 |
let exception PRV (*backtrack to precisely this recursion!*) |
852 |
val ntrl = !ntrail |
|
853 |
val nbrs = length brs0 |
|
854 |
val nxtVars = nextVars brs |
|
855 |
val G = norm G |
|
2924
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
856 |
val rules = netMkRules G vars safeList |
2854 | 857 |
(*Make a new branch, decrementing "lim" if instantiations occur*) |
2894 | 858 |
fun newBr (vars',lim') prems = |
859 |
map (fn prem => |
|
860 |
if (exists isGoal prem) |
|
861 |
then (((joinMd md prem, []) :: |
|
862 |
negOfGoals ((br, haz)::pairs)), |
|
863 |
map negOfGoal lits, |
|
864 |
vars', lim') |
|
865 |
else (((joinMd md prem, []) :: (br, haz) :: pairs), |
|
866 |
lits, vars', lim')) |
|
2854 | 867 |
prems @ |
868 |
brs |
|
869 |
(*Seek a matching rule. If unifiable then add new premises |
|
870 |
to branch.*) |
|
871 |
fun deeper [] = raise NEWBRANCHES |
|
872 |
| deeper (((P,prems),tac)::grls) = |
|
3083 | 873 |
if unify(false, add_term_vars(P,[]), P, G) |
874 |
then (*P comes from the rule; G comes from the branch.*) |
|
875 |
let val ntrl' = !ntrail |
|
876 |
val lim' = if ntrl < !ntrail |
|
877 |
then lim - (1+log(length rules)) |
|
878 |
else lim (*discourage branching updates*) |
|
879 |
val vars = vars_in_vars vars |
|
880 |
val vars' = foldr add_terms_vars (prems, vars) |
|
881 |
val choices' = (ntrl, nbrs, PRV) :: choices |
|
882 |
val tacs' = (DETERM o (tac false)) :: tacs |
|
883 |
(*no duplication*) |
|
884 |
in |
|
3092 | 885 |
traceNew prems; traceVars ntrl; |
3083 | 886 |
(if null prems then (*closed the branch: prune!*) |
887 |
prv(tacs', brs0::trs, |
|
888 |
prune (nbrs, nxtVars, choices'), |
|
889 |
brs) |
|
890 |
else |
|
891 |
if lim'<0 (*faster to kill ALL the alternatives*) |
|
892 |
then raise NEWBRANCHES |
|
893 |
else |
|
894 |
prv(tacs', brs0::trs, choices', |
|
895 |
newBr (vars',lim') prems)) |
|
896 |
handle PRV => |
|
897 |
if ntrl < ntrl' (*Vars have been updated*) |
|
898 |
then |
|
899 |
(*Backtrack at this level. |
|
900 |
Reset Vars and try another rule*) |
|
901 |
(clearTo ntrl; deeper grls) |
|
902 |
else (*backtrack to previous level*) |
|
903 |
backtrack choices |
|
904 |
end |
|
905 |
else deeper grls |
|
2854 | 906 |
(*Try to close branch by unifying with head goal*) |
907 |
fun closeF [] = raise CLOSEF |
|
908 |
| closeF (L::Ls) = |
|
3083 | 909 |
case tryClose(G,L) of |
910 |
None => closeF Ls |
|
911 |
| Some tac => |
|
912 |
let val choices' = |
|
3092 | 913 |
(if !trace then (prs"branch closed"; |
914 |
traceVars ntrl) |
|
3083 | 915 |
else (); |
916 |
prune (nbrs, nxtVars, |
|
917 |
(ntrl, nbrs, PRV) :: choices)) |
|
918 |
in prv (tac::tacs, brs0::trs, choices', brs) |
|
919 |
handle PRV => |
|
920 |
(*reset Vars and try another literal |
|
921 |
[this handler is pruned if possible!]*) |
|
922 |
(clearTo ntrl; closeF Ls) |
|
923 |
end |
|
2894 | 924 |
fun closeFl [] = raise CLOSEF |
925 |
| closeFl ((br, haz)::pairs) = |
|
926 |
closeF (map fst br) |
|
927 |
handle CLOSEF => closeF (map fst haz) |
|
928 |
handle CLOSEF => closeFl pairs |
|
3083 | 929 |
in tracing sign brs0; |
2854 | 930 |
if lim<0 then backtrack choices |
931 |
else |
|
3092 | 932 |
prv ((TRY o Data.vars_gen_hyp_subst_tac false) :: tacs, |
933 |
(*The TRY above allows the substitution to fail, e.g. if |
|
934 |
the real version is z = f(?x z). Rest of proof might |
|
935 |
still succeed!*) |
|
2854 | 936 |
brs0::trs, choices, |
3092 | 937 |
equalSubst sign (G, (br,haz)::pairs, lits, vars, lim) :: brs) |
2854 | 938 |
handle DEST_EQ => closeF lits |
2894 | 939 |
handle CLOSEF => closeFl ((br,haz)::pairs) |
2854 | 940 |
handle CLOSEF => |
2924
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
941 |
deeper rules |
2894 | 942 |
handle NEWBRANCHES => |
943 |
(case netMkRules G vars hazList of |
|
944 |
[] => (*no plausible rules: move G to literals*) |
|
945 |
prv (tacs, brs0::trs, choices, |
|
946 |
((br,haz)::pairs, addLit(G,lits), vars, lim) |
|
947 |
::brs) |
|
948 |
| _ => (*G admits some haz rules: try later*) |
|
949 |
prv (if isGoal G then negOfGoal_tac :: tacs |
|
950 |
else tacs, |
|
951 |
brs0::trs, choices, |
|
952 |
((br, haz@[(negOfGoal G, md)])::pairs, |
|
953 |
lits, vars, lim) :: brs)) |
|
2854 | 954 |
end |
2924
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
955 |
| prv (tacs, trs, choices, |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
956 |
(([],haz)::(Gs,haz')::pairs, lits, vars, lim) :: brs) = |
2894 | 957 |
(*no more "safe" formulae: transfer haz down a level*) |
2924
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
958 |
prv (tacs, trs, choices, |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
959 |
((Gs,haz@haz')::pairs, lits, vars, lim) :: brs) |
2854 | 960 |
| prv (tacs, trs, choices, |
2894 | 961 |
brs0 as ([([], (H,md)::Hs)], lits, vars, lim) :: brs) = |
962 |
(*no safe steps possible at any level: apply a haz rule*) |
|
2854 | 963 |
let exception PRV (*backtrack to precisely this recursion!*) |
2894 | 964 |
val H = norm H |
2854 | 965 |
val ntrl = !ntrail |
2924
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
966 |
val rules = netMkRules H vars hazList |
3021
39806db47be9
Loop detection: before expanding a haz formula, see whether it is a duplicate
paulson
parents:
2999
diff
changeset
|
967 |
(*new premises of haz rules may NOT be duplicated*) |
39806db47be9
Loop detection: before expanding a haz formula, see whether it is a duplicate
paulson
parents:
2999
diff
changeset
|
968 |
fun newPrem (vars,recur,dup,lim') prem = |
39806db47be9
Loop detection: before expanding a haz formula, see whether it is a duplicate
paulson
parents:
2999
diff
changeset
|
969 |
let val Gs' = map (fn P => (P,false)) prem |
39806db47be9
Loop detection: before expanding a haz formula, see whether it is a duplicate
paulson
parents:
2999
diff
changeset
|
970 |
and Hs' = if dup then Hs @ [(negOfGoal H, md)] else Hs |
39806db47be9
Loop detection: before expanding a haz formula, see whether it is a duplicate
paulson
parents:
2999
diff
changeset
|
971 |
in (if recur then [(Gs',Hs')] else [(Gs',[]), ([],Hs')], |
39806db47be9
Loop detection: before expanding a haz formula, see whether it is a duplicate
paulson
parents:
2999
diff
changeset
|
972 |
lits, vars, lim') |
39806db47be9
Loop detection: before expanding a haz formula, see whether it is a duplicate
paulson
parents:
2999
diff
changeset
|
973 |
end |
2854 | 974 |
fun newBr x prems = map (newPrem x) prems @ brs |
975 |
(*Seek a matching rule. If unifiable then add new premises |
|
976 |
to branch.*) |
|
977 |
fun deeper [] = raise NEWBRANCHES |
|
978 |
| deeper (((P,prems),tac)::grls) = |
|
3083 | 979 |
if unify(false, add_term_vars(P,[]), P, H) |
980 |
then |
|
981 |
let val ntrl' = !ntrail |
|
982 |
val vars = vars_in_vars vars |
|
983 |
val vars' = foldr add_terms_vars (prems, vars) |
|
984 |
(*duplicate H if md and the subgoal has new vars*) |
|
985 |
val dup = md andalso vars' <> vars |
|
986 |
(*any instances of P in the subgoals?*) |
|
987 |
and recur = exists (exists (match P)) prems |
|
988 |
val lim' = (*Decrement "lim" extra if updates occur*) |
|
989 |
if ntrl < !ntrail then lim - (1+log(length rules)) |
|
990 |
else lim-1 |
|
991 |
(*It is tempting to leave "lim" UNCHANGED if |
|
992 |
both dup and recur are false. Proofs are |
|
993 |
found at shallower depths, but looping |
|
994 |
occurs too often...*) |
|
995 |
val mayUndo = not(null grls) (*other rules to try?*) |
|
996 |
orelse ntrl < ntrl' (*variables updated?*) |
|
997 |
orelse vars=vars' (*no new Vars?*) |
|
998 |
val tac' = if mayUndo then tac dup |
|
999 |
else DETERM o (tac dup) |
|
1000 |
in |
|
1001 |
if lim'<0 andalso not (null prems) |
|
1002 |
then (*it's faster to kill ALL the alternatives*) |
|
1003 |
raise NEWBRANCHES |
|
1004 |
else |
|
3092 | 1005 |
traceNew prems; traceVars ntrl; |
3083 | 1006 |
prv(tac dup :: tacs, |
3101
e8a92f497295
Again "norm" DOES NOT normalize bodies of abstractions
paulson
parents:
3092
diff
changeset
|
1007 |
(*FIXME: if recur then the tactic should not |
e8a92f497295
Again "norm" DOES NOT normalize bodies of abstractions
paulson
parents:
3092
diff
changeset
|
1008 |
call rotate_tac: new formulae should be last*) |
3083 | 1009 |
brs0::trs, |
1010 |
(ntrl, length brs0, PRV) :: choices, |
|
1011 |
newBr (vars', recur, dup, lim') prems) |
|
1012 |
handle PRV => |
|
1013 |
if mayUndo |
|
1014 |
then (*reset Vars and try another rule*) |
|
1015 |
(clearTo ntrl; deeper grls) |
|
1016 |
else (*backtrack to previous level*) |
|
1017 |
backtrack choices |
|
1018 |
end |
|
1019 |
else deeper grls |
|
1020 |
in tracing sign brs0; |
|
2854 | 1021 |
if lim<1 then backtrack choices |
2924
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
1022 |
else deeper rules |
2854 | 1023 |
handle NEWBRANCHES => |
2894 | 1024 |
(*cannot close branch: move H to literals*) |
2854 | 1025 |
prv (tacs, brs0::trs, choices, |
2894 | 1026 |
([([], Hs)], H::lits, vars, lim)::brs) |
2854 | 1027 |
end |
1028 |
| prv (tacs, trs, choices, _ :: brs) = backtrack choices |
|
1029 |
in prv ([], [], [(!ntrail, length brs, PROVE)], brs) end; |
|
1030 |
||
1031 |
||
2883 | 1032 |
(*Construct an initial branch.*) |
2854 | 1033 |
fun initBranch (ts,lim) = |
2894 | 1034 |
([(map (fn t => (t,true)) ts, [])], |
1035 |
[], add_terms_vars (ts,[]), lim); |
|
2854 | 1036 |
|
1037 |
||
1038 |
(*** Conversion & Skolemization of the Isabelle proof state ***) |
|
1039 |
||
1040 |
(*Make a list of all the parameters in a subgoal, even if nested*) |
|
1041 |
local open Term |
|
1042 |
in |
|
1043 |
fun discard_foralls (Const("all",_)$Abs(a,T,t)) = discard_foralls t |
|
1044 |
| discard_foralls t = t; |
|
1045 |
end; |
|
1046 |
||
1047 |
||
1048 |
(*List of variables not appearing as arguments to the given parameter*) |
|
1049 |
fun getVars [] i = [] |
|
1050 |
| getVars ((_,(v,is))::alist) i = |
|
1051 |
if i mem is then getVars alist i |
|
1052 |
else v :: getVars alist i; |
|
1053 |
||
1054 |
||
1055 |
(*Conversion of a subgoal: Skolemize all parameters*) |
|
1056 |
fun fromSubgoal tsig t = |
|
1057 |
let val alist = ref [] |
|
1058 |
fun hdvar ((ix,(v,is))::_) = v |
|
1059 |
fun from lev t = |
|
1060 |
let val (ht,ts) = Term.strip_comb t |
|
1061 |
fun apply u = list_comb (u, map (from lev) ts) |
|
1062 |
fun bounds [] = [] |
|
1063 |
| bounds (Term.Bound i::ts) = |
|
1064 |
if i<lev then error"Function Var's argument not a parameter" |
|
1065 |
else i-lev :: bounds ts |
|
1066 |
| bounds ts = error"Function Var's argument not a bound variable" |
|
1067 |
in |
|
1068 |
case ht of |
|
1069 |
t as Term.Const _ => apply (fromConst tsig t) |
|
1070 |
| Term.Free (a,_) => apply (Free a) |
|
1071 |
| Term.Bound i => apply (Bound i) |
|
1072 |
| Term.Var (ix,_) => |
|
1073 |
(case (assoc_string_int(!alist,ix)) of |
|
1074 |
None => (alist := (ix, (ref None, bounds ts)) |
|
1075 |
:: !alist; |
|
1076 |
Var (hdvar(!alist))) |
|
1077 |
| Some(v,is) => if is=bounds ts then Var v |
|
1078 |
else error("Discrepancy among occurrences of ?" |
|
1079 |
^ Syntax.string_of_vname ix)) |
|
1080 |
| Term.Abs (a,_,body) => |
|
1081 |
if null ts then Abs(a, from (lev+1) body) |
|
1082 |
else error "fromSubgoal: argument not in normal form" |
|
1083 |
end |
|
1084 |
||
1085 |
val npars = length (Logic.strip_params t) |
|
1086 |
||
1087 |
(*Skolemize a subgoal from a proof state*) |
|
1088 |
fun skoSubgoal i t = |
|
1089 |
if i<npars then |
|
1090 |
skoSubgoal (i+1) |
|
3092 | 1091 |
(subst_bound (Skolem (gensym "T_", getVars (!alist) i), |
2854 | 1092 |
t)) |
1093 |
else t |
|
1094 |
||
1095 |
in skoSubgoal 0 (from 0 (discard_foralls t)) end; |
|
1096 |
||
1097 |
||
1098 |
(*Tactic using tableau engine and proof reconstruction. |
|
1099 |
"lim" is depth limit.*) |
|
1100 |
fun depth_tac cs lim i st = |
|
1101 |
(fullTrace:=[]; trail := []; ntrail := 0; |
|
3030 | 1102 |
let val {sign,...} = rep_thm st |
1103 |
val {tsig,...} = Sign.rep_sg sign |
|
2854 | 1104 |
val skoprem = fromSubgoal tsig (List.nth(prems_of st, i-1)) |
1105 |
val hyps = strip_imp_prems skoprem |
|
1106 |
and concl = strip_imp_concl skoprem |
|
3083 | 1107 |
fun cont (tacs,_,choices) = |
2854 | 1108 |
(case Sequence.pull(EVERY' (rev tacs) i st) of |
1109 |
None => (writeln ("PROOF FAILED for depth " ^ |
|
1110 |
Int.toString lim); |
|
1111 |
backtrack choices) |
|
1112 |
| cell => Sequence.seqof(fn()=> cell)) |
|
3030 | 1113 |
in prove (sign, cs, [initBranch (mkGoal concl :: hyps, lim)], cont) |
2854 | 1114 |
end |
3451
d10f100676d8
Made proofs more concise by replacing calls to spy_analz_tac by uses of
paulson
parents:
3244
diff
changeset
|
1115 |
handle PROVE => Sequence.null); |
2854 | 1116 |
|
1117 |
fun blast_tac cs = (DEEPEN (1,20) (depth_tac cs) 0); |
|
1118 |
||
1119 |
fun Blast_tac i = blast_tac (!Data.claset) i; |
|
1120 |
||
2924
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
1121 |
|
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
1122 |
(*** For debugging: these apply the prover to a subgoal and return |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
1123 |
the resulting tactics, trace, etc. ***) |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
1124 |
|
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
1125 |
(*Translate subgoal i from a proof state*) |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
1126 |
fun trygl cs lim i = |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
1127 |
(fullTrace:=[]; trail := []; ntrail := 0; |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
1128 |
let val st = topthm() |
3030 | 1129 |
val {sign,...} = rep_thm st |
1130 |
val {tsig,...} = Sign.rep_sg sign |
|
2924
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
1131 |
val skoprem = fromSubgoal tsig (List.nth(prems_of st, i-1)) |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
1132 |
val hyps = strip_imp_prems skoprem |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
1133 |
and concl = strip_imp_concl skoprem |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
1134 |
in timeap prove |
3030 | 1135 |
(sign, cs, [initBranch (mkGoal concl :: hyps, lim)], I) |
2924
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
1136 |
end |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
1137 |
handle Subscript => error("There is no subgoal " ^ Int.toString i)); |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
1138 |
|
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
1139 |
fun Trygl lim i = trygl (!Data.claset) lim i; |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
1140 |
|
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
1141 |
(*Read a string to make an initial, singleton branch*) |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
1142 |
fun readGoal sign s = read_cterm sign (s,propT) |> |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
1143 |
term_of |> fromTerm (#tsig(Sign.rep_sg sign)) |> |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
1144 |
rand |> mkGoal; |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
1145 |
|
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
1146 |
fun tryInThy thy lim s = |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
1147 |
(fullTrace:=[]; trail := []; ntrail := 0; |
3030 | 1148 |
timeap prove (sign_of thy, |
1149 |
!Data.claset, |
|
2924
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
1150 |
[initBranch ([readGoal (sign_of thy) s], lim)], |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
1151 |
I)); |
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
1152 |
|
af506c35b4ed
Control over excessive branching by applying a log2 penalty
paulson
parents:
2894
diff
changeset
|
1153 |
|
2854 | 1154 |
end; |
1155 |