Adding rules to the database
For an introduction to rule-classes, see rule-classes-introduction.
Example Form (from community book finite-set-theory/total-ordering.lisp): (defthm <<-trichotomy (implies (and (ordinaryp x) (ordinaryp y)) (or (<< x y) (equal x y) (<< y x))) :rule-classes ((:rewrite :corollary (implies (and (ordinaryp x) (ordinaryp y) (not (<< x y)) (not (equal x y))) (<< y x))))) General Form: a true list of rule class objects as defined below Special Cases: a symbol abbreviating a single rule class object
When defthm is used to prove a named theorem, rules may be derived
from the proved formula and stored in the database. The user specifies which
kinds of rules are to be built, by providing a list of rule class names
or, more generally, rule class objects, which name the kind of rule to
build and optionally specify various attributes of the desired rule. The rule
class names are
Note that not all events generate rules. For example, a defthm event that specifies
A rule class object is either one of the
(:class :COROLLARY term :TRIGGER-FNS (fn1 ... fnk) ; provided :class = :META (!) :WELL-FORMEDNESS-GUARANTEE x ; provided :class = :META or :class = :CLAUSE-PROCESSOR :TRIGGER-TERMS (t1 ... tk) ; provided :class = :FORWARD-CHAINING ; or :class = :LINEAR :TYPE-SET n ; provided :class = :TYPE-SET-INVERTER :TYPED-TERM term ; provided :class = :TYPE-PRESCRIPTION :CLIQUE (fn1 ... fnk) ; provided :class = :DEFINITION :CONTROLLER-ALIST alist ; provided :class = :DEFINITION :INSTALL-BODY directive ; provided :class = :DEFINITION :LOOP-STOPPER alist ; provided :class = :REWRITE or :class = :REWRITE-QUOTED-CONSTANT :PATTERN term ; provided :class = :INDUCTION (!) :CONDITION term ; provided :class = :INDUCTION :SCHEME term ; provided :class = :INDUCTION (!) :MATCH-FREE all-or-once ; provided :class = :REWRITE ; or :class = :LINEAR ; or :class = :FORWARD-CHAINING :BACKCHAIN-LIMIT-LST limit ; provided :class = :REWRITE ; or :class = :META ; or :class = :LINEAR ; or :class = :TYPE-PRESCRIPTION :HINTS hints ; provided instrs = nil :INSTRUCTIONS instrs ; provided hints = nil :OTF-FLG flg)
When rule class objects are provided by the user, most of the fields are
optional and their values are computed in a context sensitive way. When a
Before we get into the discussion of rule classes, let us return to an
important point. In spite of the large variety of rule classes available, at
present we recommend that new ACL2 users rely almost exclusively on
(conditional) rewrite rules. A reasonable but slightly bolder approach is to
We therefore now consider
Each element of the expanded value of
:Corollary — its value, term, must be a term. If omitted, this field defaults to thm. The :corollary of a rule class object is the formula actually used to justify the rule created and thus determines the form of the rule. Nqthm provided no similar capability: each rule was determined by thm, the theorem or axiom added. ACL2 permits thmto be stated ``elegantly'' and then allows the :corollary of a rule class object to specify how that elegant statement is to be interpreted as a rule. For the rule class object to be well-formed, its (defaulted) :corollary, term, must follow from thm. Unless termfollows trivially from thmusing little more than propositional logic, the formula (implies thm term)is submitted to the theorem prover and the proof attempt must be successful. During that proof attempt the values of :hints, :instructions, and :otf-flg, as provided in the rule class object, are provided as arguments to the prover. Such auxiliary proofs give the sort of output that one expects from the prover. However, as noted above, corollaries that follow trivially are not submitted to the prover; thus, such corollaries cause no prover output. Note that no rule is stored for the theorem until all corollaries have been proved.
Note that before
termis stored, all calls of macros in it are expanded away. See trans.
:Hints, :instructions, :otf-flg — the values of these fields must satisfy the same restrictions placed on the fields of the same names in defthm. These values are passed to the recursive call of the prover used to establish that the :corollary of the rule class object follows from the theorem or axiom thm.
:Type-set — this field may be supplied only if the :classis :type-set-inverter. When provided, the value must be a type-set, an integer in a certain range. If not provided, an attempt is made to compute it from the corollary. See type-set-inverter.
:Typed-term— this field may be supplied only if the :classis :type-prescription. When provided, the value is the term for which the :corollary is a type-prescription lemma. If no :typed-termis provided in a :type-prescription rule class object, we try to compute heuristically an acceptable term. See type-prescription.
:Trigger-terms— this field may be supplied only if the :classis :forward-chaining or :linear. When provided, the value is a list of terms, each of which is to trigger the attempted application of the rule. If no :trigger-termsis provided, we attempt to compute heuristically an appropriate set of triggers. See forward-chaining or see linear.
:Trigger-fns— this field must (and may only) be supplied if the :classis :meta. Its value must be a list of function symbols (except that a macro alias can stand in for a function symbol; see add-macro-alias). Terms with these symbols trigger the application of the rule. See meta. thm-name1 ; :META or :CLAUSE-PROCESSOR rules  (thm-name1) ; :META rules  (thm-name1 thm-name2) ; :META rules
thm-name1and thm-name2are the names of previously proved theorems establishing that the results of applying the metafunction(s) or clause-processor will be syntactically well-formed. See :meta and :clause-processor for details of the required forms of these well-formedness theorems. Forms  and  may be used for :metarules where no hypothesis metafunction is involved. Form  must be used for :metarules with hypothesis metafunctions; that is, if you provide a well-formedness guarantee for a metatheorem with a hypothesis metafunction you must guarantee the well-formedness of both the metafunction (with thm-name1) and the hypothesis metafunction (with thm-name2). Form  must be used for :clause-processorrules. In the absence of a proper :well-formedness-guaranteethe well-formedness of the output of a both kinds of rules is checked every time the rule is fired. These checks are skipped when a proper :well-formedness-guaranteeis provided or when overridden as described in set-skip-meta-termp-checks.
:Cliqueand :controller-alist— these two fields may only be supplied if the :classis :definition. If they are omitted, then ACL2 will attempt to guess them. Suppose the :corollary of the rule is (implies hyp (equiv (fn a1 ... an) body)). The value of the :cliquefield should be a true list of function symbols, and if non- nilmust include fn. These symbols are all the members of the mutually recursive clique containing this definition of fn. That is, a call of any function in :cliqueis considered a ``recursive call'' for purposes of the expansion heuristics. The value of the :controller-alistfield should be an alist that maps each function symbol in the :cliqueto a list of t's and nil's of length equal to the arity of the function. For example, if :cliqueconsists of just two symbols, fn1and fn2, of arities 2and 3respectively, then ((fn1 t nil) (fn2 nil t t))is a legal value of :controller-alist. The value associated with a function symbol in this alist is a ``mask'' specifying which argument slots of the function ``control'' the recursion for heuristic purposes. Sloppy choice of :cliqueor :controller-alistcan result in infinite expansion and stack overflow.
:Install-body— this field may only be supplied if the :classis :definition. Its value must be t, nil, or the default, :normalize. A value of tor :normalizewill cause ACL2 to install this rule as the new body of the function being ``defined'' ( fnin the paragraph just above); hence this definition will be installed for future :expandhints. Furthermore, if this field is omitted or the value is :normalize, then this definition will be simplified with the normalization procedure that is used by default when processing definitions made with defun. You must explicitly specify :install-body nilin the following cases: the arguments are not a list of distinct variables, equiv(as above) is not equal, or there are free variables in the hypotheses or right-hand side (see free-variables). However, supplying :install-body nilwill not affect the rewriter's application of the :definitionrule, other than to avoid using the rule to apply :expandhints. If a definition rule equates (f a1 ... ak)with bodybut there are hypotheses, hyps, then :expandhints will replace terms (f term1 ... termk)by corresponding terms (if hyps body (hide (f term1 ... termk))).
:Loop-stopper — this field may only be supplied if the class is :rewrite or :rewrite-quoted-constant. Its value must be a list of entries each consisting of two variables followed by a (possibly empty) list of function symbols, for example ((x y binary-+) (u v foo bar)). It will be used to restrict application of rewrite rules by requiring that the list of instances of the second variables must be ``smaller'' than the list of instances of the first variables in a sense related to the corresponding functions listed; see loop-stopper. The list as a whole is allowed to be nil, indicating that no such restriction shall be made. Note that any such entry that contains a variable not being instantiated, i.e., not occurring on the left side of the rewrite rule, will be ignored. However, for simplicity we merely require that every variable mentioned should appear somewhere in the corresponding :corollary formula.
:Pattern, :Condition, :Scheme— the first and last of these fields must (and may only) be supplied if the class is :induction. :Conditionis optional but may only be supplied if the class is :induction. The values must all be terms and indicate, respectively, the pattern to which a new induction scheme is to be attached, the condition under which the suggestion is to be made, and a term which suggests the new scheme. See induction.
:Match-free— this field must be :allor :onceand may be supplied only if the :classis either :rewrite, :linear, or :forward-chaining. (This field is not implemented for other rule classes, including the :type-prescription rule class.) See free-variables for a description of this field. Note: Although this field is intended to be used for controlling retries of matching free variables in hypotheses, it is legal to supply it even if there are no such free variables. This can simplify the automated generation of rules, but note that when :match-freeis supplied, the warning otherwise provided for the presence of free variables in hypotheses will be suppressed.
:Backchain-limit-lst— this field may be supplied only if the :classis either :rewrite, :meta, :linear, or :type-prescription. It is further required that either only one rule is generated from the formula or, at least, every such rule has the same list of hypotheses. The value for :backchain-limit-lstmust be nil; a non-negative integer; or, except in the case of :meta rules, a true list each element of which is either nilor a non-negative integer. If it is a list, its length must be equal to the number of hypotheses of the rule and each item in the list is the ``backchain limit'' associated with the corresponding hypothesis. If backchain-limit-lstis a non-negative integer, it is defaulted to a list of the appropriate number of repetitions of that integer. The backchain limit of a hypothesis is used to limit the effort that ACL2 will expend when relieving the hypothesis. If it is NIL, no new limits are imposed; if it is an integer, the hypothesis will be limited to backchaining at most that many times. Note that backchaining may be further limited by a global backchain-limit; see backchain-limit for details. For different ways to reign in the rewriter, see rewrite-stack-limit and see set-prover-step-limit. Jared Davis has pointed out that you can set the :backchain-limit-lstto 0 to avoid any attempt to relieve forced hypotheses, which can lead to a significant speed-up in some cases.