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(defun unchangeables (formals args quick-block-info subset ans)
; We compute the set of all variable names occurring in args in
; unchanging measured formal positions. We accumulate the answer onto
; ans.
(cond ((endp formals) ans)
((and (member-eq (car formals) subset)
(eq (car quick-block-info) 'unchanging))
(unchangeables (cdr formals) (cdr args) (cdr quick-block-info) subset
(all-vars1 (car args) ans)))
(t
(unchangeables (cdr formals) (cdr args) (cdr quick-block-info) subset
ans))))
(defun changeables (formals args quick-block-info subset ans)
; We compute the args in changing measured formal positions. We
; accumulate the answer onto ans.
(cond ((endp formals) ans)
((and (member-eq (car formals) subset)
(not (eq (car quick-block-info) 'unchanging)))
(changeables (cdr formals) (cdr args) (cdr quick-block-info)
subset
(cons (car args) ans)))
(t
(changeables (cdr formals) (cdr args) (cdr quick-block-info)
subset
ans))))
(defun sound-induction-principle-mask1 (formals args quick-block-info
subset
unchangeables
changeables)
; See sound-induction-principle-mask.
(cond
((endp formals) nil)
(t (let ((var (car formals))
(arg (car args))
(q (car quick-block-info)))
(mv-let (mask-ele new-unchangeables new-changeables)
(cond ((member-eq var subset)
(cond ((eq q 'unchanging)
(mv 'unchangeable unchangeables changeables))
(t (mv 'changeable unchangeables changeables))))
((and (variablep arg)
(eq q 'unchanging))
(cond ((member-eq arg changeables)
(mv nil unchangeables changeables))
(t (mv 'unchangeable
(add-to-set-eq arg unchangeables)
changeables))))
((and (variablep arg)
(not (member-eq arg changeables))
(not (member-eq arg unchangeables)))
(mv 'changeable
unchangeables
(cons arg changeables)))
(t (mv nil unchangeables changeables)))
(cons mask-ele
(sound-induction-principle-mask1 (cdr formals)
(cdr args)
(cdr quick-block-info)
subset
new-unchangeables
new-changeables)))))))
(defun sound-induction-principle-mask (term formals quick-block-info subset)
; Term is a call of some fn on some args. The formals and
; quick-block-info are those of fn, and subset is one of fn's measured
; subset (a subset of formals). We wish to determine, in the
; terminology of ACL, whether the induction applies to term. If so we
; return a mask indicating how to build the substitutions for the
; induction from args and the machine for fn. Otherwise we return
; nil.
; Let the changeables be those args that are in measured formal
; positions that sometimes change in the recursion. Let the
; unchangeables be all of the variables occurring in measured actuals
; that never change in recursion. The induction applies if
; changeables is a sequence of distinct variable names and has an
; empty intersection with unchangeables.
; If the induction is applicable then the substitutions should
; substitute for the changeables just as the recursion would, and hold
; each unchangeable fixed -- i.e., substitute each for itself. With
; such substitutions it is possible to prove the measure lemmas
; analogous to those proved when justification of subset was stored,
; except that the measure is obtained by instantiating the measure
; term used in the justification by the measured actuals in unchanging
; slots. Actual variables that are neither among the changeables or
; unchangeables may be substituted for arbitrarily.
; If the induction is applicable we return a mask with as many
; elements as there are args. For each arg the mask contains either
; 'changeable, 'unchangeable, or nil. 'Changeable means the arg
; should be instantiated as specified in the recursion. 'Unchangeable
; means each var in the actual should be held fixed. Nil means that
; the corresponding substitution pairs in the machine for the function
; should be ignored.
; Abstractly, this function builds the mask by first putting either
; 'changeable or 'unchangeable in each measured slot. It then fills
; in the remaining slots from the left so as to permit the actual to
; be instantiated or held fixed as desired by the recursion, provided
; that in so doing it does not permit substitutions for previously
; allocated actuals.
(let ((unchangeables
(unchangeables formals (fargs term) quick-block-info subset nil))
(changeables
(changeables formals (fargs term) quick-block-info subset nil)))
(cond ((or (not (no-duplicatesp-equal changeables))
(not (all-variablep changeables))
(intersectp-eq changeables unchangeables))
nil)
(t (sound-induction-principle-mask1 formals
(fargs term)
quick-block-info
subset
unchangeables
changeables)))))
(defrec candidate
(score controllers changed-vars unchangeable-vars
tests-and-alists-lst justification induction-term other-terms
xinduction-term xother-terms xancestry))
; This record is used to represent one of the plausible inductions that must be
; considered. The SCORE represents some fairly artificial estimation of how
; many terms in the conjecture want this induction. CONTROLLERS is the list of
; the controllers -- including unchanging controllers -- for all the inductions
; merged to form this one. The CHANGED-VARS is a list of all those variables
; that will be instantiated (non-identically) in some induction hypotheses.
; Thus, CHANGED-VARS include both variables that actually contribute to why
; some measure goes down and variables like accumulators that are just along
; for the ride. UNCHANGEABLE-VARS is a list of all those vars which may not be
; changed by any substitution if this induction is to be sound for the reasons
; specified. TESTS-AND-ALISTS-LST is a list of TESTS-AND-ALISTS which
; indicates the case analysis for the induction and how the various induction
; hypotheses are obtained via substitutions. JUSTIFICATION is the
; JUSTIFICATION that justifies this induction, and INDUCTION-TERM is the term
; that suggested this induction and from which you obtain the actuals to
; substitute into the template. OTHER-TERMS are the induction-terms of
; candidates that have been merged into this one for heuristic reasons.
; Because of induction rules we can think of some terms being aliases for
; others. We have to make a distinction between the terms in the conjecture
; that suggested an induction and the actual terms that suggested the
; induction. For example, if an induction rule maps (fn x y) to (recur x y),
; then (recur x y) will be the INDUCTION-TERM because it suggested the
; induction and we will, perhaps, have to recover its induction machine or
; quick block information to implement various heuristics. But we do not wish
; to report (recur x y) to the user, preferring instead to report (fn x y).
; Therefore, corresponding to INDUCTION-TERM and OTHER-TERMS we have
; XINDUCTION-TERM and XOTHER-TERMS, which are maintained in exactly the same
; way as their counterparts but which deal completely with the user-level view
; of the induction. In practice this means that we will initialize the
; XINDUCTION-TERM field of a candidate with the term from the conjecture,
; initialize the INDUCTION-TERM with its alias, and then merge the fields
; completely independently but analogously. XANCESTRY is a field maintained by
; merging to contain the user-level terms that caused us to change our
; tests-and-alists. (Some candidates may be flushed or merged into this one
; without changing it.)
(defun count-non-nils (lst)
(cond ((endp lst) 0)
((car lst) (1+ (count-non-nils (cdr lst))))
(t (count-non-nils (cdr lst)))))
(defun controllers (formals args subset ans)
(cond ((endp formals) ans)
((member (car formals) subset)
(controllers (cdr formals) (cdr args) subset
(all-vars1 (car args) ans)))
(t (controllers (cdr formals) (cdr args) subset ans))))
(defun changed/unchanged-vars (x args mask ans)
(cond ((endp mask) ans)
((eq (car mask) x)
(changed/unchanged-vars x (cdr args) (cdr mask)
(all-vars1 (car args) ans)))
(t (changed/unchanged-vars x (cdr args) (cdr mask) ans))))
(defrec tests-and-alists (tests alists))
(defun tests-and-alists/alist (alist args mask call-args)
; Alist is an alist that maps the formals of some fn to its actuals,
; args. Mask is a sound-induction-principle-mask indicating the args
; for which we will build substitution pairs. Call-args is the list of
; arguments to some recursive call of fn occurring in the induction
; machine for fn. We build an alist mapping the masked args to the
; instantiations (under alist) of the values in call-args.
(cond
((endp mask) nil)
((null (car mask))
(tests-and-alists/alist alist (cdr args) (cdr mask) (cdr call-args)))
((eq (car mask) 'changeable)
(cons (cons (car args)
(sublis-var alist (car call-args)))
(tests-and-alists/alist alist
(cdr args)
(cdr mask)
(cdr call-args))))
(t (let ((vars (all-vars (car args))))
(append (pairlis vars vars)
(tests-and-alists/alist alist
(cdr args)
(cdr mask)
(cdr call-args)))))))
(defun tests-and-alists/alists (alist args mask calls)
; Alist is an alist that maps the formals of some fn to its actuals,
; args. Mask is a sound-induction-principle-mask indicating the args
; for which we will build substitution pairs. Calls is the list of
; calls for a given case of the induction machine. We build the alists
; from those calls.
(cond
((endp calls) nil)
(t (cons (tests-and-alists/alist alist args mask (fargs (car calls)))
(tests-and-alists/alists alist args mask (cdr calls))))))
; The following record contains the tests leading to a collection of
; recursive calls at the end of a branch through a defun. In general,
; the calls may not be to the function being defuned but rather to
; some other function in the same mutually recursive clique, but in
; the context of induction we know that all the calls are to the same
; fn because we haven't implemented mutually recursive inductions yet.
; A list of these records constitute the induction machine for a function.
(defrec tests-and-calls (tests . calls))
; The justification record contains a subset of the formals of the function
; under which it is stored, the domain predicate, mp, and the relation, rel,
; which is well-founded on mp objects, and the mp-measure term which has been
; proved to decrease in the recursion. A list of such records is stored under
; each function symbol by the defun principle.
(defrec justification (subset mp rel measure))
(defun tests-and-alists (alist args mask tc)
; Alist is an alist that maps the formals of some fn to its actuals,
; args. Mask is a sound-induction-principle-mask indicating the args
; for which we will build substitution pairs. Tc is one of the
; tests-and-calls in the induction machine for the function. We make
; a tests-and-alists record containing the instantiated tests of tc
; and alists derived from the calls of tc.
(make tests-and-alists
:tests (sublis-var-lst alist (access tests-and-calls tc :tests))
:alists (tests-and-alists/alists alist
args
mask
(access tests-and-calls tc :calls))))
(defun tests-and-alists-lst (alist args mask machine)
; We build a list of tests-and-alists from machine, instantiating it
; with alist, which is a map from the formals of the function to the
; actuals, args. Mask is the sound-induction-principle-mask that
; indicates the args for which we substitute.
(cond
((endp machine) nil)
(t (cons (tests-and-alists alist args mask (car machine))
(tests-and-alists-lst alist args mask (cdr machine))))))
(defun flesh-out-induction-principle (term formals justification mask machine
xterm)
; Term is a call of some function the indicated formals and induction machine.
; Justification is its 'justification and mask is a sound-induction-
; principle-mask for the term. We build the induction candidate suggested by
; term.
(make candidate
:score
(/ (count-non-nils mask) (length mask))
:controllers
(controllers formals (fargs term)
(access justification justification :subset)
nil)
:changed-vars
(changed/unchanged-vars 'changeable (fargs term) mask nil)
:unchangeable-vars
(changed/unchanged-vars 'unchangeable (fargs term) mask nil)
:tests-and-alists-lst
(tests-and-alists-lst (pairlis formals (fargs term))
(fargs term) mask machine)
:justification justification
:induction-term term
:xinduction-term xterm
:other-terms nil
:xother-terms nil
:xancestry nil))
(defun intrinsic-suggested-induction-cand
(term formals quick-block-info justification machine xterm ens wrld)
; Note: An "intrinsically suggested" induction scheme is an induction scheme
; suggested by a justification of a recursive function. We
; distinguish between intrinsically suggested inductions and those suggested
; via records of induction-rule type. Intrinsic inductions have no embodiment
; as induction-rules but are, instead, the basis of the semantics of such
; rules. That is, the inductions suggested by (fn x y) is the union of those
; suggested by the terms to which it is linked via induction-rules together
; with the intrinsic suggestion for (fn x y).
; Term, above, is a call of some fn with the given formals, quick-block-info,
; justification and induction machine. We return a list of induction
; candidates, said list either being empty or the singleton list containing the
; induction candidate intrinsically suggested by term. Xterm is logically
; unrelated to term and is the term appearing in the original conjecture from
; which we (somehow) obtained term for consideration.
(cond
((enabled-numep (fn-nume :INDUCTION (ffn-symb term) wrld) ens)
(let ((mask (sound-induction-principle-mask term formals
quick-block-info
(access justification
justification
:subset))))
(cond
(mask
(list (flesh-out-induction-principle term formals
justification
mask
machine
xterm)))
(t nil))))
(t nil)))
(defrec induction-rule ((nume . pattern) condition . scheme))
; In ACL2 this nest does not actually terminate because the user may
; set up looping induction rules. Here is a trivial example that
; forces ACL2 Version 2.7 into a stack overflow upon attacking any
; inductive theorem involving append.
; (defthm looper t
; :rule-classes ((:induction :pattern (append a b)
; :scheme (append a b))))
; ACL2 does not detect such loops. Paco limits them: it doesn't allow
; more than nnn rules to be invoked on any path.
(acl2::set-well-founded-relation e0-ord-<)
(mutual-recursion
(defun apply-induction-rule (rule term type-alist xterm ens wrld nnn)
(declare (xargs :measure (cons (cons (+ 1 (nfix nnn)) 3) 0)
:hints (("Goal" :in-theory (disable type-set
one-way-unify1)))))
; We apply the induction-rule, rule, to term, and return a possibly
; empty list of suggested inductions. The basic idea is to check that
; the :pattern of the rule matches term. If so, we check that the
; :condition of the rule is true under the current type-alist. This
; check is heuristic only. If type-set says the :condition is
; non-nil, we fire the rule by instantiating the :scheme and
; recursively getting the suggested inductions for that term. Note
; that we are not recursively exploring the instantiated scheme but
; just getting the inductions suggested by its top-level function
; symbol.
(cond
((enabled-numep (access induction-rule rule :nume) ens)
(mv-let
(ans alist)
(one-way-unify (access induction-rule rule :pattern)
term)
(cond
(ans
(let ((ts
(type-set (sublis-var alist
(access induction-rule rule :condition))
type-alist nil ens wrld *type-set-nnn*)))
(cond
((ts-intersectp *ts-nil* ts) nil)
(t (let ((term1
(sublis-var alist (access induction-rule rule :scheme))))
(cond
((or (variablep term1)
(fquotep term1))
nil)
(t (suggested-induction-cands term1 type-alist
xterm ens wrld nnn))))))))
(t nil))))
(t nil)))
(defun suggested-induction-cands1 (induction-rules term type-alist
xterm ens wrld nnn)
(declare (xargs :measure (cons (cons (+ 1 (nfix nnn)) 1)
(acl2-count induction-rules))))
; We map down induction-rules and apply each enabled rule to term,
; which is known to be an application of the function symbol fn to
; some args. Each rule gives us a possibly empty list of suggested
; inductions. We append all these suggestions together. In addition,
; if fn is recursively defined, we collect the intrinsic suggestion
; for term as well.
(cond
((or (endp induction-rules)
(zp nnn))
(let* ((fn (ffn-symb term))
(machine (getprop fn 'induction-machine nil wrld)))
(cond
((null machine) nil)
(t
(intrinsic-suggested-induction-cand
term
(formals fn wrld)
(getprop fn 'quick-block-info nil wrld)
(getprop fn 'justification nil wrld)
machine
xterm
ens
wrld)))))
(t (append (apply-induction-rule (car induction-rules) term type-alist
xterm ens wrld (- nnn 1))
(suggested-induction-cands1 (cdr induction-rules)
term type-alist xterm
ens wrld nnn)))))
(defun suggested-induction-cands (term type-alist xterm ens wrld nnn)
(declare (xargs :measure (cons (cons (+ 1 (nfix nnn)) 2) 0)))
; Term is some fn applied to args. Xterm is some term occurring in the
; conjecture we are exploring and is the term upon which this induction
; suggestion will be "blamed" and from which we have obtained term via
; induction-rules. We return all of the induction candidates suggested by
; term. Lambda applications suggest no inductions.
(cond
((flambdap (ffn-symb term)) nil)
(t (suggested-induction-cands1
(getprop (ffn-symb term) 'induction-rules nil wrld)
term type-alist xterm ens wrld nnn))))
)
; Note: A reasonable value for nnn on the initial call is 3. This is
; the depth of rule nesting. Thus, a term could have an arbitrary
; number of associated induction-rules and each of them would be
; applied with a nesting depth of 3. But upon applying one of those
; rules, i.e., upon recurring into the :scheme of the rule, the
; nesting depth is decreased to 2. When the nesting depth reaches 0,
; we just use the intrinsic induction associated with the scheme,
; rather than any other rules.
(defconst *suggested-induction-cands-nnn* 3)
(mutual-recursion
(defun get-induction-cands (term type-alist ens wrld ans)
; We explore term and accumulate onto ans all of the induction
; candidates suggested by it.
(cond
((variablep term) ans)
((fquotep term) ans)
(t (get-induction-cands-lst
(fargs term)
type-alist ens wrld
(append (suggested-induction-cands term type-alist term ens wrld
*suggested-induction-cands-nnn*)
ans)))))
(defun get-induction-cands-lst (lst type-alist ens wrld ans)
; We explore the list of terms, lst, and accumulate onto ans all of
; the induction candidates.
(cond ((endp lst) ans)
(t (get-induction-cands
(car lst)
type-alist ens wrld
(get-induction-cands-lst
(cdr lst)
type-alist ens wrld ans)))))
)
(defun get-induction-cands-from-cl-set1 (cl-set ens wrld ans)
; We explore cl-set and accumulate onto ans all of the induction
; candidates.
(cond
((endp cl-set) ans)
(t (mv-let (contradictionp type-alist)
(type-alist-clause (car cl-set) nil ens wrld)
; We need a type-alist with which to apply induction rules. It is of
; heuristic use only. We assume no contradiction is found. If by
; chance one is, then type-alist is nil, which is an acceptable
; type-alist.
(declare (ignore contradictionp))
(get-induction-cands-lst
(car cl-set) type-alist ens wrld
(get-induction-cands-from-cl-set1 (cdr cl-set)
ens wrld ans))))))
(defun get-induction-cands-from-cl-set (cl-set pspv wrld)
; We explore cl-set and collect all induction candidates.
(get-induction-cands-from-cl-set1 cl-set
(access rewrite-constant
(access prove-spec-var
pspv
:rewrite-constant)
:ens)
wrld nil))
; That completes the development of the code for exploring a clause set
; and gathering the induction candidates suggested.
; Section: Pigeon-Holep
; We next develop pigeon-holep, which tries to fit some "pigeons" into
; some "holes" using a function to determine the sense of the word
; "fit". Since ACL2 is first-order we can't pass arbitrary functions
; and hence we pass symbols and define our own special-purpose "apply"
; that interprets the particular symbols passed to calls of
; pigeon-holep.
; However, it turns out that the applications of pigeon-holep require
; passing functions that themselves call pigeon-holep. And so
; pigeon-holep-apply is mutually recursive with pigeon-holep (but only
; because the application functions use pigeon-holep).
(defun pigeon-hole-fn-measure (fn)
(case fn (pair-fitp 1) (alist-fitp 2) (otherwise 3)))
(mutual-recursion
(defun pigeon-holep-apply (fn pigeon hole)
(declare (xargs :measure (lex4 (pigeon-hole-fn-measure fn) 0 0 0)))
; See pigeon-holep for the problem and terminology. This function
; takes a symbol denoting a predicate and two arguments. It applies
; the predicate to the arguments. When the predicate holds we say
; the pigeon argument "fits" into the hole argument.
(case fn
(pair-fitp
; This predicate is applied to two pairs, each taken from two substitutions.
; We say (v1 . term1) (the "pigeon") fits into (v2 . term2) (the "hole")
; if v1 is v2 and term1 occurs in term2.
(and (eq (car pigeon) (car hole))
(occur (cdr pigeon) (cdr hole))))
(alist-fitp
; This predicate is applied to two substitutions. We say the pigeon
; alist fits into the hole alist if each pair of the pigeon fits into
; a pair of the hole and no pair of the hole is used more than once.
(pigeon-holep pigeon hole nil 'pair-fitp))
(tests-and-alists-fitp
; This predicate is applied to two tests-and-alists records. The
; first fits into the second if the tests of the first are a subset
; of those of the second and either they are both base cases (i.e., have
; no alists) or each substitution of the first fits into a substitution of
; the second, using no substitution of the second more than once.
(and (subsetp-equal (access tests-and-alists pigeon :tests)
(access tests-and-alists hole :tests))
(or (and (null (access tests-and-alists pigeon :alists))
(null (access tests-and-alists hole :alists)))
(pigeon-holep (access tests-and-alists pigeon :alists)
(access tests-and-alists hole :alists)
nil
'alist-fitp))))))
(defun pigeon-holep (pigeons holes filled-holes fn)
(declare (xargs :measure (lex4 (pigeon-hole-fn-measure fn)
(acl2-count pigeons)
0
(acl2-count holes))))
; Both pigeons and holes are lists of arbitrary objects. The holes
; are implicitly enumerated left-to-right from 0. Filled-holes is a
; list of the indices of holes we consider "filled." Fn is a
; predicate known to pigeon-holep-apply. If fn applied to a pigeon and
; a hole is non-nil, then we say the pigeon "fits" into the hole. We
; can only "put" a pigeon into a hole if the hole is unfilled and the
; pigeon fits. The act of putting the pigeon into the hole causes the
; hole to become filled. We return t iff it is possible to put each
; pigeon into a hole under these rules.
(cond
((endp pigeons) t)
(t (pigeon-holep1 (car pigeons)
(cdr pigeons)
holes 0
holes filled-holes fn))))
(defun pigeon-holep1 (pigeon pigeons lst n holes filled-holes fn)
; Lst is a terminal sublist of holes, whose first element has index n.
; We map over lst looking for an unfilled hole h such that (a) we can
; put pigeon into h and (b) we can dispose of the rest of the pigeons
; after filling h.
(declare (xargs :measure (lex4 (pigeon-hole-fn-measure fn)
(acl2-count pigeons)
1
(acl2-count lst))))
(cond
((endp lst) nil)
((member n filled-holes)
(pigeon-holep1 pigeon pigeons (cdr lst) (1+ n) holes filled-holes fn))
((and (pigeon-holep-apply fn pigeon (car lst))
(pigeon-holep pigeons holes
(cons n filled-holes)
fn))
t)
(t (pigeon-holep1 pigeon pigeons (cdr lst) (1+ n)
holes filled-holes fn))))
)
(defun flush-cand1-down-cand2 (cand1 cand2)
; This function takes two induction candidates and determines whether
; the first is subsumed by the second. If so, it constructs a new
; candidate that is logically equivalent (vis a vis the induction
; suggested) to the second but which now carries with it the weight
; and heuristic burdens of the first.
(cond
((and (subsetp-eq (access candidate cand1 :changed-vars)
(access candidate cand2 :changed-vars))
(subsetp-eq (access candidate cand1 :unchangeable-vars)
(access candidate cand2 :unchangeable-vars))
(pigeon-holep (access candidate cand1 :tests-and-alists-lst)
(access candidate cand2 :tests-and-alists-lst)
nil
'tests-and-alists-fitp))
(change candidate cand2
:score (+ (access candidate cand1 :score)
(access candidate cand2 :score))
:controllers (union-eq (access candidate cand1 :controllers)
(access candidate cand2 :controllers))
:other-terms (add-to-set-equal
(access candidate cand1 :induction-term)
(union-equal
(access candidate cand1 :other-terms)
(access candidate cand2 :other-terms)))
:xother-terms (add-to-set-equal
(access candidate cand1 :xinduction-term)
(union-equal
(access candidate cand1 :xother-terms)
(access candidate cand2 :xother-terms)))))
(t nil)))
(defun flush-candidates (cand1 cand2)
; This function determines whether one of the two induction candidates
; given flushes down the other and if so returns the appropriate
; new candidate. This function is a mate and merge function used
; by m&m and is hence known to m&m-apply.
(or (flush-cand1-down-cand2 cand1 cand2)
(flush-cand1-down-cand2 cand2 cand1)))
; We now begin the development of the merging of two induction
; candidates. The basic idea is that if two functions both replace x
; by x', and one of them simultaneously replaces a by a' while the
; other replaces b by b', then we should consider inducting on x, a,
; and b, by x', a', and b'. Of course, by the time we get here, the
; recursion is coded into substitution alists: ((x . x') (a . a')) and
; ((x . x') (b . b')). We merge these two alists into ((x . x') (a .
; a') (b . b')). The merge of two sufficiently compatible alists is
; accomplished by just unioning them together.
; The key ideas are (1) recognizing when two alists are describing the
; "same" recursive step (i.e., they are both replacing x by x', where
; x is somehow a key variable); (2) observing that they do not
; explicitly disagree on what to do with the other variables.
(defun alists-agreep (alist1 alist2 vars)
; Two alists agree on vars iff for each var in vars the image of var under
; the first alist is equal to that under the second.
(cond ((endp vars) t)
((equal (let ((temp (assoc-eq (car vars) alist1)))
(cond (temp (cdr temp))(t (car vars))))
(let ((temp (assoc-eq (car vars) alist2)))
(cond (temp (cdr temp))(t (car vars)))))
(alists-agreep alist1 alist2 (cdr vars)))
(t nil)))
(defun irreconcilable-alistsp (alist1 alist2)
; Two alists are irreconcilable iff there is a var v that they both
; explicitly map to different values. Put another way, there exists a
; v such that (v . a) is a member of alist1 and (v . b) is a member of
; alist2, where a and b are different. If two substitutions are
; reconcilable then their union is a substitution.
; We rely on the fact that this function return t or nil.
(cond ((endp alist1) nil)
(t (let ((temp (assoc-eq (caar alist1) alist2)))
(cond ((null temp)
(irreconcilable-alistsp (cdr alist1) alist2))
((equal (cdar alist1) (cdr temp))
(irreconcilable-alistsp (cdr alist1) alist2))
(t t))))))
(defun affinity (aff alist1 alist2 vars)
; We say two alists that agree on vars but are irreconcilable are
; "antagonists". Two alists that agree on vars and are not irreconcilable
; are "mates".
; Aff is either 'antagonists or 'mates and denotes one of the two relations
; above. We return t iff the other args are in the given relation.
(and (alists-agreep alist1 alist2 vars)
(eq (irreconcilable-alistsp alist1 alist2)
(eq aff 'antagonists))))
(defun member-affinity (aff alist alist-lst vars)
; We determine if some member of alist-lst has the given affinity with alist.
(cond ((endp alist-lst) nil)
((affinity aff alist (car alist-lst) vars)
t)
(t (member-affinity aff alist (cdr alist-lst) vars))))
(defun occur-affinity (aff alist lst vars)
; Lst is a list of tests-and-alists. We determine whether alist has
; the given affinity with some alist in lst. We call this occur
; because we are looking inside the elements of lst. But it is
; technically a misnomer because we don't look inside recursively; we
; treat lst as though it were a list of lists.
(cond
((endp lst) nil)
((member-affinity aff alist
(access tests-and-alists (car lst) :alists)
vars)
t)
(t (occur-affinity aff alist (cdr lst) vars))))
(defun some-occur-affinity (aff alists lst vars)
; Lst is a list of tests-and-alists. We determine whether some alist
; in alists has the given affinity with some alist in lst.
(cond ((endp alists) nil)
(t (or (occur-affinity aff (car alists) lst vars)
(some-occur-affinity aff (cdr alists) lst vars)))))
(defun all-occur-affinity (aff alists lst vars)
; Lst is a list of tests-and-alists. We determine whether every alist
; in alists has the given affinity with some alist in lst.
(cond ((endp alists) t)
(t (and (occur-affinity aff (car alists) lst vars)
(all-occur-affinity aff (cdr alists) lst vars)))))
(defun contains-affinity (aff lst vars)
; We determine if two members of lst have the given affinity.
(cond ((endp lst) nil)
((member-affinity aff (car lst) (cdr lst) vars) t)
(t (contains-affinity aff (cdr lst) vars))))
; So much for general-purpose scanners. We now develop the predicates
; that are used to determine if we can merge lst1 into lst2 on vars.
; See merge-tests-and-alists-lsts for extensive comments on the ideas
; behind the following functions.
(defun antagonistic-tests-and-alists-lstp (lst vars)
; Lst is a list of tests-and-alists. Consider just the set of all
; alists in lst. We return t iff that set contains an antagonistic
; pair.
; We operate as follows. Each element of lst contains some alists.
; We take the first element and ask whether its alists contain an
; antagonistic pair. If so, we're done. Otherwise, we ask whether
; any alist in that first element is antagonistic with the alists in
; the rest of lst. If so, we're done. Otherwise, we recursively
; look at the rest of lst.
(cond
((endp lst) nil)
(t (or (contains-affinity
'antagonists
(access tests-and-alists (car lst) :alists)
vars)
(some-occur-affinity
'antagonists
(access tests-and-alists (car lst) :alists)
(cdr lst)
vars)
(antagonistic-tests-and-alists-lstp (cdr lst) vars)))))
(defun antagonistic-tests-and-alists-lstsp (lst1 lst2 vars)
; Both lst1 and lst2 are lists of tests-and-alists. We determine whether
; there exists an alist1 in lst1 and an alist2 in lst2 such that alist1
; and alist2 are antagonists.
(cond
((endp lst1) nil)
(t (or (some-occur-affinity
'antagonists
(access tests-and-alists (car lst1) :alists)
lst2
vars)
(antagonistic-tests-and-alists-lstsp (cdr lst1) lst2 vars)))))
(defun every-alist1-matedp (lst1 lst2 vars)
; Both lst1 and lst2 are lists of tests-and-alists. We determine for every
; alist1 in lst1 there exists an alist2 in lst2 that agrees with alist1 on
; vars and that is not irreconcilable.
(cond ((endp lst1) t)
(t (and (all-occur-affinity
'mates
(access tests-and-alists (car lst1) :alists)
lst2
vars)
(every-alist1-matedp (cdr lst1) lst2 vars)))))
; The functions above are used to determine that lst1 and lst2 contain
; no antagonistic pairs, that every alist in lst1 has a mate somewhere in
; lst2, and that the process of merging alists from lst1 with their
; mates will not produce alists that are antagonistic with other alists
; in lst1. We now develop the code for merging nonantagonistic alists
; and work up the structural hierarchy to merging lists of tests and alists.
(defun merge-alist1-into-alist2 (alist1 alist2 vars)
; We assume alist1 and alist2 are not antagonists. Thus, either they
; agree on vars and have no explicit disagreements, or they simply
; don't agree on vars. If they agree on vars, we merge alist1 into
; alist2 by just unioning them together. If they don't agree on vars,
; then we merge alist1 into alist2 by ignoring alist1.
(cond
((alists-agreep alist1 alist2 vars)
(union-equal alist1 alist2))
(t alist2)))
; Now we begin working up the structural hierarchy. Our strategy is
; to focus on a given alist2 and hit it with every alist1 we have.
; Then we'll use that to copy lst2 once, hitting each alist2 in it
; with everything we have. We could decompose the problem the other
; way: hit lst2 with successive alist1's. That suffers from forcing
; us to copy lst2 repeatedly, and there are parts of that structure
; (i.e., the :tests) that don't change.
(defun merge-alist1-lst-into-alist2 (alist1-lst alist2 vars)
; Alist1-lst is a list of alists and alist2 is an alist. We know that
; there is no antagonists between the elements of alist1-lst and in
; alist2. We merge each alist1 into alist2 and return
; the resulting extension of alist2.
(cond
((endp alist1-lst) alist2)
(t (merge-alist1-lst-into-alist2
(cdr alist1-lst)
(merge-alist1-into-alist2 (car alist1-lst) alist2 vars)
vars))))
(defun merge-lst1-into-alist2 (lst1 alist2 vars)
; Given a list of tests-and-alists, lst1, and an alist2, we hit alist2
; with every alist1 in lst1.
(cond ((endp lst1) alist2)
(t (merge-lst1-into-alist2
(cdr lst1)
(merge-alist1-lst-into-alist2
(access tests-and-alists (car lst1) :alists)
alist2
vars)
vars))))
; So now we write the code to copy lst2, hitting each alist in it with lst1.
(defun merge-lst1-into-alist2-lst (lst1 alist2-lst vars)
(cond ((endp alist2-lst) nil)
(t (cons (merge-lst1-into-alist2 lst1 (car alist2-lst) vars)
(merge-lst1-into-alist2-lst lst1 (cdr alist2-lst) vars)))))
(defun merge-lst1-into-lst2 (lst1 lst2 vars)
(cond ((endp lst2) nil)
(t (cons (change tests-and-alists (car lst2)
:alists
(merge-lst1-into-alist2-lst
lst1
(access tests-and-alists (car lst2) :alists)
vars))
(merge-lst1-into-lst2 lst1 (cdr lst2) vars)))))
(defun merge-tests-and-alists-lsts (lst1 lst2 vars)
; Lst1 and lst2 are each tests-and-alists-lsts from some induction
; candidates. Intuitively, we try to stuff the alists of lst1 into
; those of lst2 to construct a new lst2 that combines the induction
; schemes of both. If we fail we return nil. Otherwise we return the
; modified lst2. The modified lst2 has exactly the same tests as
; before; only its alists are different and they are different only by
; virtue of having been extended with some addition pairs. So the
; justification of the merged induction is the same as the
; justification of the original lst2.
; Given an alist1 from lst1, which alist2's of lst2 do you extend and
; how? Suppose alist1 maps x to x' and y to y'. Then intuitively we
; think "the first candidate is trying to keep x and y in step, so
; that when x goes to x' y goes to y'." So, if you see an alist in
; lst2 that is replacing x by x', one might be tempted to augment it
; by replacing y by y'. However, what if x is just an accumulator?
; The role of vars is to specify upon which variables two
; substitutions must agree in order to be merged. Usually, vars
; consists of the measured variables.
; So now we get a little more technical. We will try to "merge" each
; alist1 from lst1 into each alist2 from lst2 (preserving the case structure
; of lst2). If alist1 and alist2 do not agree on vars then their merge
; is just alist2. If they do agree on vars, then their merge is their
; union, provided that is a substitution. It may fail to be a substitution
; because the two alists disagree on some other variable. In that case
; we say the two are irreconcilable. We now give three simple examples:
; Let vars be {x}. Let alist2 be {(x . x') (z . z')}. If alist1 maps
; x to x'', then their merge is just alist2 because alist1 is
; addressing a different case of the induction. If alist1 maps x to x'
; and y to y', then their merge is {(x . x') (y . y') (z . z')}. If
; alist1 maps x to x' and z to z'', then the two are irreconcilable.
; Two irreconcilable alists that agree on vars are called "antagonists"
; because they "want" to merge but can't. We cannot merge lst1 into lst2
; if they have an antagonistic pair between them. If an antagonistic pair
; is discovered, the entire merge operation fails.
; Now we will successively consider each alist1 in lst1 and merge it
; into lst2, forming successive lst2's. We insist that each alist1 of
; lst1 have at least one mate in lst2 with which it agrees and is
; reconcilable. (Otherwise, we could merge completely disjoint
; substitutions.)
; Because we try the alist1's successively, each alist1 is actually
; merged into the lst2 produced by all the previous alist1's. That
; produces an apparent order dependence. However, this is avoided by
; the requirement that we never produce antagonistic pairs.
; For example, suppose that in one case of lst1, x is mapped to x' and
; y is mapped to y', but in another case x is mapped to x' and y is
; mapped to y''. Now imagine trying to merge that lst1 into a lst2 in
; which x is mapped to x' and z is mapped to z'. The first alist of
; lst1 extends lst2 to (((x . x') (y . y') (z . z'))). But the second
; alist is then antagonistic. The same thing happens if we tried the two
; alists of lst1 in the other order. Thus, the above lst1 cannot be merged
; into lst2. Note that they can be merged in the other order! That is,
; lst2 can be merged into lst1, because the case structure of lst1 is
; richer.
; We can detect the situation above without forming the intermediate
; lst2. In particular, if lst1 contains an antagonistic pair, then it
; cannot be merged with any lst2 and we can quit.
; Note: Once upon a time, indeed, for the first 20 years or so of the
; existence of the merge routine, we took the attitude that if
; irreconcilable but agreeable alists arose, then we just added to
; alist2 those pairs of alist1 that were reconcilable and we left out
; the irreconcilable pairs. This however resulted in the system often
; merging complicated accumulator using functions (like TAUTOLOGYP)
; into simpler functions (like NORMALIZEDP) by dropping the
; accumulators that got in the way. This idea of just not doing
; "hostile merges" is being tried out for the first time in ACL2.
(cond ((antagonistic-tests-and-alists-lstp lst1 vars) nil)
((antagonistic-tests-and-alists-lstsp lst1 lst2 vars) nil)
((not (every-alist1-matedp lst1 lst2 vars)) nil)
(t (merge-lst1-into-lst2 lst1 lst2 vars))))
(defun merge-cand1-into-cand2 (cand1 cand2)
; Can induction candidate cand1 be merged into cand2? If so, return
; their merge. The guts of this function is merge-tests-and-alists-
; lsts. The tests preceding it are heuristic only. If
; merge-tests-and-alists-lsts returns non-nil, then it returns a sound
; induction; indeed, it merely extends some of the substitutions in
; the second candidate.
(let ((vars (or (intersection-eq
(access candidate cand1 :controllers)
(intersection-eq
(access candidate cand2 :controllers)
(intersection-eq
(access candidate cand1 :changed-vars)
(access candidate cand2 :changed-vars))))
(intersection-eq
(access candidate cand1 :changed-vars)
(access candidate cand2 :changed-vars)))))
(cond
((and vars
(not (intersectp-eq (access candidate cand1 :unchangeable-vars)
(access candidate cand2 :changed-vars)))
(not (intersectp-eq (access candidate cand2 :unchangeable-vars)
(access candidate cand1 :changed-vars))))
(let ((temp (merge-tests-and-alists-lsts
(access candidate cand1 :tests-and-alists-lst)
(access candidate cand2 :tests-and-alists-lst)
vars)))
(cond (temp
(make candidate
:score (+ (access candidate cand1 :score)
(access candidate cand2 :score))
:controllers (union-eq
(access candidate cand1 :controllers)
(access candidate cand2 :controllers))
:changed-vars (union-eq
(access candidate cand1 :changed-vars)
(access candidate cand2 :changed-vars))
:unchangeable-vars (union-eq
(access candidate cand1
:unchangeable-vars)
(access candidate cand2
:unchangeable-vars))
:tests-and-alists-lst temp
:justification (access candidate cand2 :justification)
:induction-term (access candidate cand2 :induction-term)
:other-terms (add-to-set-equal
(access candidate cand1 :induction-term)
(union-equal
(access candidate cand1 :other-terms)
(access candidate cand2 :other-terms)))
:xinduction-term (access candidate cand2 :xinduction-term)
:xother-terms (add-to-set-equal
(access candidate cand1 :xinduction-term)
(union-equal
(access candidate cand1 :xother-terms)
(access candidate cand2 :xother-terms)))
:xancestry (cond
((equal temp
(access candidate cand2
:tests-and-alists-lst))
(access candidate cand2 :xancestry))
(t (add-to-set-equal
(access candidate cand1 :xinduction-term)
(union-equal
(access candidate cand1 :xancestry)
(access candidate cand2 :xancestry)))))))
(t nil))))
(t nil))))
(defun merge-candidates (cand1 cand2)
; This function determines whether one of the two induction candidates
; can be merged into the other. If so, it returns their merge. This
; function is a mate and merge function used by m&m and is hence known
; to m&m-apply.
(or (merge-cand1-into-cand2 cand1 cand2)
(merge-cand1-into-cand2 cand2 cand1)))
; We now move from merging to flawing. The idea is to determine which
; inductions get in the way of others.
(defun controller-variables1 (args controller-pocket)
; Controller-pocket is a list of t's and nil's in 1:1 correspondence with
; args, indicating which args are controllers. We collect those controller
; args that are variable symbols.
(declare (xargs :measure (acl2-count controller-pocket)))
(cond ((endp controller-pocket) nil)
((and (car controller-pocket)
(variablep (car args)))
(add-to-set-eq (car args)
(controller-variables1 (cdr args)
(cdr controller-pocket))))
(t (controller-variables1 (cdr args)
(cdr controller-pocket)))))
(defun controller-variables (term controller-alists)
; Controller-alists is the corresponding property of the function
; symbol, fn, of term. Recall that each element of controller-alists
; is an alist that associates with each function in fn's mutually
; recursive clique the controller pockets used in a given
; justification of the clique. In induction, as things stand today,
; we know that fn is singly recursive because we don't know how to
; handle mutual recursion yet. But no use is made of that here. We
; collect all the variables in controller slots of term.
(cond ((endp controller-alists) nil)
(t (union-eq
(controller-variables1 (fargs term)
(cdr (assoc-eq (ffn-symb term)
(car controller-alists))))
(controller-variables term (cdr controller-alists))))))
(defun induct-vars1 (lst wrld)
; Lst is a list of terms. We collect every variable symbol occuring in a
; controller slot of any term in lst.
(cond ((endp lst) nil)
(t (union-eq (controller-variables (car lst)
(getprop (ffn-symb (car lst))
'controller-alists
nil wrld))
(induct-vars1 (cdr lst) wrld)))))
(defun induct-vars (cand wrld)
(induct-vars1 (cons (access candidate cand :induction-term)
(access candidate cand :other-terms))
wrld))
(defun vetoedp (cand vars lst changed-vars-flg)
; Vars is a list of variables. We return t iff there exists a candidate
; in lst, other than cand, whose unchangeable-vars (or, if changed-vars-flg,
; changed-vars or unchangeable-vars) intersect with vars.
; This function is used both by compute-vetoes1, where flg is t and
; vars is the list of changing induction vars of cand, and by
; compute-vetoes2, where flg is nil and vars is the list of
; changed-vars cand. We combine these two into one function simply to
; eliminate a definition from the system.
(cond ((endp lst) nil)
((equal cand (car lst))
(vetoedp cand vars (cdr lst) changed-vars-flg))
((and changed-vars-flg
(intersectp-eq vars
(access candidate (car lst) :changed-vars)))
t)
((intersectp-eq vars
(access candidate (car lst) :unchangeable-vars))
t)
(t (vetoedp cand vars (cdr lst) changed-vars-flg))))
(defun compute-vetoes1 (lst cand-lst wrld)
; Lst is a tail of cand-lst. We throw out from lst any candidate
; whose changing induct-vars intersect the changing or unchanging vars
; of another candidate in cand-lst. We assume no two elements of
; cand-lst are equal, an invariant assured by the fact that we have
; done merging and flushing before this.
(cond ((endp lst) nil)
((vetoedp (car lst)
(intersection-eq
(access candidate (car lst) :changed-vars)
(induct-vars (car lst) wrld))
cand-lst
t)
(compute-vetoes1 (cdr lst) cand-lst wrld))
(t (cons (car lst)
(compute-vetoes1 (cdr lst) cand-lst wrld)))))
; If the first veto computation throws out all candidates, we revert to
; another heuristic.
(defun compute-vetoes2 (lst cand-lst)
; Lst is a tail of cand-lst. We throw out from lst any candidate
; whose changed-vars intersect the unchangeable-vars of another
; candidate in cand-lst. Again, we assume no two elements of cand-lst
; are equal.
(cond ((endp lst) nil)
((vetoedp (car lst)
(access candidate (car lst) :changed-vars)
cand-lst
nil)
(compute-vetoes2 (cdr lst) cand-lst))
(t (cons (car lst)
(compute-vetoes2 (cdr lst) cand-lst)))))
(defun compute-vetoes (cand-lst wrld)
; We try two different techniques for throwing out candidates. If the
; first throws out everything, we try the second. If the second throws
; out everything, we throw out nothing.
; The two are: (1) throw out a candidate if its changing induct-vars
; (the variables in control slots that change) intersect with either
; the changed-vars or the unchangeable-vars of another candidate. (2)
; throw out a candidate if its changed-vars intersect the
; unchangeable-vars of another candidate.
(or (compute-vetoes1 cand-lst cand-lst wrld)
(compute-vetoes2 cand-lst cand-lst)
cand-lst))
; The next heuristic is to select complicated candidates, based on
; support for non-primitive recursive function schemas.
(defun induction-complexity1 (lst wrld)
; The "function" induction-complexity does not exist. It is a symbol
; passed to maximal-elements-apply which calls this function on the list
; of terms supported by an induction candidate. We count the number of
; non pr fns supported.
(cond ((endp lst) 0)
((getprop (ffn-symb (car lst)) 'primitive-recursive-defunp nil wrld)
(induction-complexity1 (cdr lst) wrld))
(t (1+ (induction-complexity1 (cdr lst) wrld)))))
; We develop a general-purpose function for selecting maximal elements from
; a list under a measure. That function, maximal-elements, is then used
; with the induction-complexity measure to collect both the most complex
; inductions and then to select those with the highest scores.
(defun maximal-elements-apply (fn x wrld)
; This function must produce an integerp. This is just the apply function
; for maximal-elements.
(case fn
(induction-complexity
(induction-complexity1 (cons (access candidate x :induction-term)
(access candidate x :other-terms))
wrld))
(score (access candidate x :score))
(otherwise 0)))
(defun maximal-elements1 (lst winners maximum fn wrld)
; We are scanning down lst collecting into winners all those elements
; with maximal scores as computed by fn. Maximum is the maximal score seen
; so far and winners is the list of all the elements passed so far with
; that score.
(cond ((endp lst) winners)
(t (let ((temp (maximal-elements-apply fn (car lst) wrld)))
(cond ((> temp maximum)
(maximal-elements1 (cdr lst)
(list (car lst))
temp fn wrld))
((= temp maximum)
(maximal-elements1 (cdr lst)
(cons (car lst) winners)
maximum fn wrld))
(t
(maximal-elements1 (cdr lst)
winners
maximum fn wrld)))))))
(defun maximal-elements (lst fn wrld)
; Return the subset of lst that have the highest score as computed by
; fn. The functional parameter fn must be known to maximal-elements-apply.
; We reverse the accumulated elements to preserve the order used by
; nqthm.
(cond ((endp lst) nil)
((endp (cdr lst)) lst)
(t (reverse
(maximal-elements1 (cdr lst)
(list (car lst))
(maximal-elements-apply fn (car lst) wrld)
fn wrld)))))
; All that is left in the heuristic selection of the induction candidate is
; the function m&m that mates and merges arbitrary objects. We develop that
; now.
; The following three functions are not part of induction but are
; used by other callers of m&m and so have to be introduced now
; so we can define m&m-apply and get on with induct.
(defun intersectp-eq/union-equal (x y)
(cond ((intersectp-eq (car x) (car y))
(cons (union-eq (car x) (car y))
(union-equal (cdr x) (cdr y))))
(t nil)))
(defun equal/union-equal (x y)
(cond ((equal (car x) (car y))
(cons (car x)
(union-equal (cdr x) (cdr y))))
(t nil)))
(defun subsetp-equal/smaller (x y)
(cond ((subsetp-equal x y) x)
((subsetp-equal y x) y)
(t nil)))
(defun m&m-apply (fn x y)
; This is a first-order function that really just applies fn to x and
; y, but does so only for a fixed set of fns. In fact, this function
; handles exactly those functions that we give to m&m.
(case fn
(intersectp-eq/union-equal (intersectp-eq/union-equal x y))
(equal/union-equal (equal/union-equal x y))
(flush-candidates (flush-candidates x y))
(merge-candidates (merge-candidates x y))
(subsetp-equal/smaller (subsetp-equal/smaller x y))))
(defun count-off (n lst)
; Pair the elements of lst with successive integers starting at n.
(cond ((endp lst) nil)
(t (cons (cons n (car lst))
(count-off (1+ n) (cdr lst))))))
(defun m&m-search (x y-lst del fn)
; Y-lst is a list of pairs, (id . y). The ids are integers. If id is
; a member of del, we think of y as "deleted" from y-lst. That is,
; y-lst and del together characterize a list of precisely those y such
; that (id . y) is in y-lst and id is not in del.
; We search y-lst for the first y that is not deleted and that mates
; with x. We return two values, the merge of x and y and the integer
; id of y. If no such y exists, return two nils.
(cond ((endp y-lst) (mv nil nil))
((member (caar y-lst) del)
(m&m-search x (cdr y-lst) del fn))
(t (let ((z (m&m-apply fn x (cdar y-lst))))
(cond (z (mv z (caar y-lst)))
(t (m&m-search x (cdr y-lst) del fn)))))))
(defun count-undel (pairs del)
(cond ((endp pairs) 0)
((member (caar pairs) del)
(count-undel (cdr pairs) del))
(t (+ 1 (count-undel (cdr pairs) del)))))
(defun exists-pair-with-car (x lst)
(cond ((endp lst) nil)
((equal x (caar lst)) t)
(t (exists-pair-with-car x (cdr lst)))))
(defthm m&m-search-property1
(implies (car (m&m-search x y-lst del fn))
(exists-pair-with-car (mv-nth 1 (m&m-search x y-lst del fn))
y-lst))
:hints (("Goal" :in-theory (disable m&m-apply))))
(defthm m&m-search-property2
(implies (car (m&m-search x y-lst del fn))
(not (member (mv-nth 1 (m&m-search x y-lst del fn))
del)))
:hints (("Goal" :in-theory (disable m&m-apply))))
(defthm m&m1-admission-crux
(implies (and (exists-pair-with-car y-id pairs)
(not (member y-id del)))
(< (count-undel pairs (cons y-id del))
(count-undel pairs del)))
:rule-classes :linear)
(defun m&m1 (pairs del ans n fn)
; This is workhorse for m&m. See that fn for a general description of
; the problem and the terminology. Pairs is a list of pairs. The car
; of each pair is an integer and the cdr is a possible element of the
; bag we are closing under fn. Del is a list of the integers
; identifying all the elements of pairs that have already been
; deleted. Abstractly, pairs and del together represent a bag we call
; the "unprocessed bag". The elements of the unprocessed bag are
; precisely those ele such that (i . ele) is in pairs and i is not in
; del.
; Without assuming any properties of fn, this function can be
; specified as follows: If the first element, x, of the unprocessed
; bag, mates with some y in the rest of the uprocessed bag, then put
; the merge of x and the first such y in place of x, delete that y,
; and iterate. If the first element has no such mate, put it in the
; answer accumulator ans. N, by the way, is the next available unique
; identifier integer.
; If one is willing to make the assumptions that the mate and merge
; fns of fn are associative and commutative and have the distributive
; and non-preclusion properties, then it is possible to say more about
; this function. The rest of this comment makes those assumptions.
; Ans is a bag with the property that no element of ans mates with any
; other element of ans or with any element of the unprocessed bag. N
; is the next available unique identifier integer; it is always larger
; than any such integer in pairs or in del.
; Abstractly, this function closes B under fn, where B is the bag
; union of the unprocessed bag and ans.
(declare (xargs :measure (cons (+ 1 (len pairs))
(count-undel pairs del))
:hints (("Goal" :in-theory (disable m&m-search)))))
(cond
((endp pairs) ans)
((member (caar pairs) del)
(m&m1 (cdr pairs) del ans n fn))
(t (mv-let (mrg y-id)
(m&m-search (cdar pairs) (cdr pairs) del fn)
(cond
((null mrg)
(m&m1 (cdr pairs)
del
(cons (cdar pairs) ans)
n fn))
(t (m&m1 (cons (cons n mrg) (cdr pairs))
(cons y-id del)
ans
(1+ n)
fn)))))))
(defun m&m (bag fn)
; This function takes a bag and a symbol naming a dyadic function, fn,
; known to m&m-apply and about which we assume certain properties
; described below. Let z be (m&m-apply fn x y). Then we say x and y
; "mate" if z is non-nil. If x and y mate, we say z is the "merge" of
; x and y. The name of this function abbreviates the phrase "mate and
; merge".
; We consider each element, x, of bag in turn and seek the first
; successive element, y, that mates with it. If we find one, we throw
; out both, add their merge in place of x and iterate. If we find no
; mate for x, we deposit it in our answer accumulator.
; The specification above is explicit about the order in which we try
; the elements of the bag. If we try to loosen the specification so
; that order is unimportant, we must require that fn have certain
; properties. We discuss this below.
; First, note that we have implicitly assumed that mate and merge are
; commutative because we haven't said in which order we present the
; arguments.
; Second, note that if x doesn't mate with any y, we set it aside in
; our accumulating answer. We do not even try to mate such an x with
; the offspring of the y's it didn't like. This makes us order
; dependent. For example, consider the bag {x y1 y2}. Suppose x
; won't mate with either y1 or y2, but that y1 mates with y2 to
; produce y3 and x mates with y3 to produce y4. Then if we seek mates
; for x first we find none and it gets into our final answer. Then y1
; and y2 mate to form y3. The final answer is hence {x y3}. But if
; we seek mates for y1 first we find y2, produce y3, add it to the
; bag, forming {y3 x}, and then mate x with y3 to get the final answer
; {y4}. This order dependency cannot arise if fn has the property
; that if x mates with the merge of y and z then x mates with either y
; or z. This is called the "distributive" property of mate over merge.
; Third, note that if x does mate with y to produce z then we throw x
; out in favor of z. Thus, x is not mated against any but the first
; y. Thus, if we have {x y1 y2} and x mates with y1 to form z1 and x
; mates with y2 to form z2 and there are no other mates, then we can
; either get {z1 y2} or {z2 y1} as the final bag, depending on whether
; we mate x with y1 or y2. This order dependency cannot arise if fn
; has the property that if x mates with y1 and x mates with y2, then
; (a) the merge of x and y1 mates with y2, and (b) merge has the
; "commutativity-2" (merge (merge x y1) y2) = (merge (merge x y2) y1).
; We call property (a) "non-preclusion" property of mate and merge,
; i.e., merging doesn't preclude mating.
; The commutativity-2 property is implied by associativity and (the
; already assumed commutativity). Thus, another way to avoid the
; third order dependency is if legal merges are associative and have
; the non-preclusion property.
; Important Note: The commonly used fn of unioning together two alists
; that agree on the intersection of their domains, does not have the
; non-preclusion property! Suppose x, y1, and y2 are all alists and
; all map A to 0. Suppose in addition y1 maps B to 1 but y2 maps B to
; 2. Finally, suppose x maps C to 3. Then x mates with both y1 and
; y2. But merging y1 into x precludes mating with y2 and vice versa.
; We claim, but do not prove, that if the mate and merge functions for
; fn are commutative and associative, and have the distributive and
; non-preclusion properties, then m&m is order independent.
; For efficiency we have chosen to implement deletion by keeping a
; list of the deleted elements. But we cannot make a list of the
; deleted elements themselves because there may be duplicate elements
; in the bag and we need to be able to delete occurrences. Thus, the
; workhorse function actually operates on a list of pairs, (i . ele),
; where i is a unique identification integer and ele is an element of
; the bag. In fact we just assign the position of each occurrence to
; each element of the initial bag and thereafter count up as we
; generate new elements.
;
; See m&m1 for the details.
(m&m1 (count-off 0 bag) nil nil (length bag) fn))
; We now develop a much more powerful concept, that of mapping m&m over the
; powerset of a set. This is how we actually merge induction candidates.
; That is, we try to mash together every possible subset of the candidates,
; largest subsets first. See m&m-over-powerset for some implementation
; commentary before going on.
(defun cons-subset-tree (x y)
; We are representing full binary trees of t's and nil's and
; collapsing trees of all nil's to nil and trees of all t's to t. See
; the long comment in m&m-over-powerset. We avoid consing when
; convenient.
(if (eq x t)
(if (eq y t)
t
(if y (cons x y) '(t)))
(if x
(cons x y)
(if (eq y t)
'(nil . t)
(if y (cons x y) nil)))))
(defmacro car-subset-tree (x)
; See cons-subset-tree.
`(let ((x ,x))
(if (eq x t) t (car x))))
(defmacro cdr-subset-tree (x)
; See cons-subset-tree.
`(let ((x ,x))
(if (eq x t) t (cdr x))))
(defun or-subset-trees (tree1 tree2)
; We disjoin the tips of two binary t/nil trees. See cons-subset-tree.
(declare (xargs :measure
(+ (if (equal tree1 nil) 0 (+ 1 (acl2-count tree1)))
(if (equal tree2 nil) 0 (+ 1 (acl2-count tree2))))))
(cond ((or (eq tree1 t)(eq tree2 t)) t)
((null tree1) tree2)
((null tree2) tree1)
(t (cons-subset-tree (or-subset-trees (car-subset-tree tree1)
(car-subset-tree tree2))
(or-subset-trees (cdr-subset-tree tree1)
(cdr-subset-tree tree2))))))
(defun m&m-over-powerset1 (st subset stree ans fn)
; See m&m-over-powerset.
(cond
((eq stree t) (mv t ans))
((endp st)
(let ((z (m&m subset fn)))
(cond ((and z (null (cdr z)))
(mv t (cons (car z) ans)))
(t (mv nil ans)))))
(t
(mv-let (stree1 ans1)
(m&m-over-powerset1 (cdr st)
(cons (car st) subset)
(cdr-subset-tree stree)
ans fn)
(mv-let (stree2 ans2)
(m&m-over-powerset1 (cdr st)
subset
(or-subset-trees
(car-subset-tree stree)
stree1)
ans1 fn)
(mv (cons-subset-tree stree2 stree1) ans2))))))
(defun m&m-over-powerset (st fn)
; Fn is a function known to m&m-apply. Let (fn* s) be defined to be z,
; if (m&m s fn) = {z} and nil otherwise. Informally, (fn* s) is the
; result of somehow mating and merging all the elements of s into a single
; object, or nil if you can't.
; This function applies fn* to the powerset of st and collects all those
; non-nil values produced from maximal s's. I.e., we keep (fn* s) iff it
; is non-nil and no superset of s produces a non-nil value.
; We do this amazing feat (recall that the powerset of a set of n
; things contains 2**n subsets) by generating the powerset in order
; from largest to smallest subsets and don't generate or test any
; subset under a non-nil fn*. Nevertheless, if the size of set is
; very big, this function will get you.
; An informal specification of this function is that it is like m&m
; except that we permit an element to be merged into more than one
; other element (but an element can be used at most once per final
; element) and we try to maximize the amount of merging we can do.
; For example, if x mates with y1 to form z1, and x mates with y2 to
; form z2, and no other mates occur, then this function would
; transform {x y1 y2} into {z1 z2}. It searches by generate and test:
; s (fn* s)
; (x y1 y2) nil
; (x y1) z1
; (x y2) z2
; (x) subsumed
; (y1 y2) nil
; (y1) subsumed
; (y2) subsumed
; nil subsumed
; Here, s1 is "subsumed" by s2 means s1 is a subset of s2. (Just the
; opposite technical definition but exactly the same meaning as in the
; clausal sense.)
; The way we generate the powerset elements is suggested by the
; following trivial von Neumann function, ps, which, when called as in
; (ps set nil), calls PROCESS on each member of the powerset of set,
; in the order in which we generate them:
; (defun ps (set subset)
; (cond ((null set) (PROCESS subset))
; (t (ps (cdr set) (cons (car set) subset)) ;rhs
; (ps (cdr set) subset)))) ;lhs
; By generating larger subsets first we know that if a subset subsumes
; the set we are considering then that subset has already been
; considered. Therefore, we need a way to keep track of the subsets
; with non-nil values. We do this with a "subset tree". Let U be the
; universe of objects in some order. Then the full binary tree with
; depth |U| can be thought of as the powerset of U. In particular,
; any branch through the tree, from top-most node to tip, represents a
; subset of U by labelling the nodes at successive depth by the
; successive elements of U (the topmost node being labelled with the
; first element of U) and adopting the convention that taking a
; right-hand (cdr) branch at a node indicates that the label is in the
; subset and a left-hand (car) branch indicates that the label is not
; in the subset. At the tip of the tree we store a T indicating that
; the subset had a non-nil value or a NIL indicating that it had a nil
; value.
; For storage efficiency we let nil represent an arbitrarily deep full
; binary tree will nil at every tip and we let t represent the
; analogous trees with t at every tip. Car-subset-tree,
; cdr-subset-tree and cons-subset-tree implement these abstractions.
; Of course, we don't have the tree when we start and we generate it
; as we go. That is a really weird idea because generating the tree
; that tells us who was a subset of whom in the past seems to have little
; use as we move forward. But that is not true.
; Observe that there is a correspondence between these trees and the
; function ps above for generating the power set. The recursion
; labelled "rhs" above is going down the right-hand side of the tree
; and the "lhs" recursion is going down the left-hand side. Note that
; we go down the rhs first.
; The neat fact about these trees is that there is a close
; relationship between the right-hand subtree (rhs) and left-hand
; subtree (lhs) of any given node of the tree: lhs can be obtained
; from rhs by turning some nils into ts. The reason is that the tips
; of the lhs of a node labelled by x denote exactly the same subsets
; as the corresponding tips of the right-hand side, except that on the
; right x was present in the subset and on the left it is not. So
; when we do the right hand side we come back with a tree and if we
; used that very tree for the left hand side (interpreting nil as
; meaning "compute it and see" and t as meaning "a superset of this
; set has non-nil value") then it is correct. But we can do a little
; better than that because we might have come into this node with a
; tree (i.e., one to go into the right hand side with and another to go
; into the left hand side with) and so after we have gone into the
; right and come back with its new tree, we can disjoin the output of
; the right side with the input for the left side to form the tree we
; will actually use to explore the left side.
(mv-let (stree ans)
(m&m-over-powerset1 st nil nil nil fn)
(declare (ignore stree))
ans))
; Ok, so now we have finished the selection process and we begin the
; construction of the induction formula itself.
(defun all-picks2 (pocket pick ans)
; See all-picks.
(cond ((endp pocket) ans)
(t (cons (cons (car pocket) pick)
(all-picks2 (cdr pocket) pick ans)))))
(defun all-picks2r (pocket pick ans)
; See all-picks.
(cond ((endp pocket) ans)
(t (all-picks2r (cdr pocket) pick
(cons (cons (car pocket) pick) ans)))))
(defun all-picks1 (pocket picks ans rflg)
; See all-picks.
(cond ((endp picks) ans)
(t (all-picks1 pocket (cdr picks)
(if rflg
(all-picks2r pocket (car picks) ans)
(all-picks2 pocket (car picks) ans))
rflg))))
(defun all-picks (pockets rflg)
; Pockets is a list of pockets, each pocket containing 0 or more
; objects. We return a list of all the possible ways you can pick one
; thing from each pocket. If rflg is nil initially, then the order of
; the resulting list is exactly the same as it was in nqthm. There is
; not much else to recommend this particular choice of definition!
; Historical Plaque from Nqthm:
; (DEFUN ALL-PICKS (POCKET-LIST)
; (COND ((NULL POCKET-LIST) (LIST NIL))
; (T (ITERATE FOR PICK IN (ALL-PICKS (CDR POCKET-LIST))
; NCONC (ITERATE FOR CHOICE IN (CAR POCKET-LIST)
; COLLECT (CONS CHOICE PICK))))))
; Nqthm's construction is a very natural recursive one, except that it
; used nconc to join together the various segments of the answer. If
; we tried the analogous construction here we would have to append the
; segments together and copy a very long list. So we do it via an
; accumulator. The trouble however is that we reverse the order of
; the little buckets in our answer every time we process a pocket. We
; could avoid that if we wanted to recurse down the length of our
; answer on recursive calls, but we were afraid of running out of
; stack, and so we have coded this with tail recursion only. We do
; non-tail recursion only over short things like individual pockets or
; the list of pockets. And so to (a) avoid unnecessary copying, (b)
; non-tail recursion, and (c) constructing our answer in a different
; order, we introduced rflg. Rflg causes us either to reverse or not
; reverse a certain intermediate result every other recursion. It
; would be reassuring to see a mechanically checked proof that this
; definition of all-picks is equivalent to nqthm's.
(cond ((endp pockets) '(nil))
(t (all-picks1 (car pockets)
(all-picks (cdr pockets) (not rflg))
nil
rflg))))
(defun dumb-negate-lit-lst-lst (cl-set)
; We apply dumb-negate-lit-lst to every list in cl-set and collect the
; result. You can think of this as negating a clause set (i.e., an
; implicit conjunction of disjunctions), but you have to then imagine
; that the implicit "and" at the top has been turned into an "or" and
; vice versa at the lower level.
(cond ((endp cl-set) nil)
(t (cons (dumb-negate-lit-lst (car cl-set))
(dumb-negate-lit-lst-lst (cdr cl-set))))))
(defun induction-hyp-clause-segments2 (alists cl ans)
; See induction-hyp-clause-segments1.
(cond ((endp alists) ans)
(t (cons (sublis-var-lst (car alists) cl)
(induction-hyp-clause-segments2 (cdr alists) cl ans)))))
(defun induction-hyp-clause-segments1 (alists cl-set ans)
; This function applies all of the substitutions in alists to all of
; the clauses in cl-set and appends the result to ans to create one
; list of instantiated clauses.
(cond ((endp cl-set) ans)
(t (induction-hyp-clause-segments2
alists
(car cl-set)
(induction-hyp-clause-segments1 alists (cdr cl-set) ans)))))
(defun induction-hyp-clause-segments (alists cl-set)
; Cl-set is a set of clauses. We are trying to prove the conjunction
; over that set, i.e., cl1 & cl2 ... & clk, by induction. We are in a
; case in which we can assume every instance under alists of that
; conjunction. Thus, we can assume any lit from cl1, any lit from
; cl2, etc., instantiated via all of the alists. We wish to return a
; list of clause segments. Each segment will be spliced into the a
; clause we are trying to prove and together the resulting set of
; clauses is supposed to be equivalent to assuming all instances of
; the conjunction over cl-set.
; So one way to create the answer would be to first instantiate each
; of the k clauses with each of the n alists, getting a set of n*k
; clauses. Then we could run all-picks over that, selecting one
; literal from each of the instantiated clauses to assume. Then we'd
; negate each literal within each pick to create a clause hypothesis
; segment. That is nearly what we do, except that we do the negation
; first so as to share structure among the all-picks answers.
; Note: The code below calls (dumb-negate-lit lit) on each lit. Nqthm
; used (negate-lit lit nil ...) on each lit, employing
; negate-lit-lst-lst, which has since been deleted but was strictly
; analogous to the dumb version called below. But since the
; type-alist is nil in Nqthm's call, it seems unlikely that the
; literal will be decided by type-set. We changed to dumb-negate-lit
; to avoid having to deal both with ttrees and the enabled structure
; implicit in type-set.
(all-picks
(induction-hyp-clause-segments1 alists
(dumb-negate-lit-lst-lst cl-set)
nil)
nil))
(defun induction-formula3 (neg-tests hyp-segments cl ans)
; Neg-tests is the list of the negated tests of an induction
; tests-and-alists entry. hyp-segments is a list of hypothesis clause
; segments (i.e., more negated tests), and cl is a clause. For each
; hyp segment we create the clause obtained by disjoining the tests,
; the segment, and cl. We conjoin the resulting clauses to ans.
; See induction-formula for a comment about this iteration.
(cond
((endp hyp-segments) ans)
(t (induction-formula3 neg-tests
(cdr hyp-segments)
cl
(conjoin-clause-to-clause-set
; Historical Plaque from Nqthm:
; We once implemented the idea of "homographication" in which we combined
; both induction, opening up of the recursive fns in the conclusion, and
; generalizing away some recursive calls. This function did the expansion
; and generalization. If the idea is reconsidered the following theorems are
; worthy of consideration:
; (ORDERED (SORT X)),
; (IMPLIES (ORDERED X)
; (ORDERED (ADDTOLIST I X))),
; (IMPLIES (AND (NUMBER-LISTP X)
; (ORDERED X)
; (NUMBERP I)
; (NOT (< (CAR X) I)))
; (EQUAL (ADDTOLIST I X) (CONS I X))), and
; (IMPLIES (AND (NUMBER-LISTP X) (ORDERED X)) (EQUAL (SORT X) X)).
; Observe that we simply disjoin the negated tests, hyp segments, and clause.
; Homographication further manipulated the clause before adding it to the
; answer.
(disjoin-clauses
neg-tests
(disjoin-clauses (car hyp-segments) cl))
ans)))))
(defun induction-formula2 (cl cl-set ta-lst ans)
; Cl is a clause in cl-set, which is a set of clauses we are proving
; by induction. Ta-lst is the tests-and-alists-lst component of the
; induction candidate we are applying to prove cl-set. We are now
; focussed on the proof of cl, using the induction schema of ta-lst
; but getting to assume all the clauses in cl-set in our induction
; hypothesis. We will map across ta-lst, getting a set of tests and
; some alists at each stop, and for each stop add a bunch of clauses
; to ans.
(cond
((endp ta-lst) ans)
(t (induction-formula2 cl cl-set (cdr ta-lst)
(induction-formula3
(dumb-negate-lit-lst
(access tests-and-alists (car ta-lst) :tests))
(induction-hyp-clause-segments
(access tests-and-alists (car ta-lst) :alists)
cl-set)
cl
ans)))))
(defun induction-formula1 (lst cl-set ta-lst ans)
; Lst is a tail of cl-set. Cl-set is a set of clauses we are trying to prove.
; Ta-lst is the tests-and-alists-lst component of the induction candidate
; we wish to apply to cl-set. We map down lst forming a set of clauses
; for each cl in lst. Basically, the set we form for cl is of the form
; ... -> cl, where ... involves all the case analysis under the tests in
; ta-lst and all the induction hypotheses from cl-set under the alists in
; each test-and-alists. We add our clauses to ans.
(cond
((endp lst) ans)
(t (induction-formula1 (cdr lst) cl-set ta-lst
(induction-formula2 (car lst)
cl-set ta-lst ans)))))
(defun induction-formula (cl-set ta-lst)
; Cl-set is a set of clauses we are to try to prove by induction, applying
; the inductive scheme described by the tests-and-alists-lst, ta-lst,
; of some induction candidate. The following historical plaque tells all.
; Historical Plaque from Nqthm:
; TESTS-AND-ALISTS-LST is a such a list that the disjunction of the
; conjunctions of the TESTS components of the members is T. Furthermore,
; there exists a measure M, a well-founded relation R, and a sequence of
; variables x1, ..., xn such that for each T&Ai in TESTS-AND-ALISTS-LST, for
; each alist alst in the ALISTS component of T&Ai, the conjunction of the
; TESTS component, say qi, implies that (R (M x1 ... xn)/alst (M x1 ... xn)).
; To prove thm, the conjunction of the disjunctions of the members of CL-SET,
; it is sufficient, by the principle of induction, to prove instead the
; conjunction of the terms qi & thm' & thm'' ... -> thm, where the primed
; terms are the results of substituting the alists in the ALISTS field of the
; ith member of TESTS-AND-ALISTS-LST into thm.
; If thm1, thm2, ..., thmn are the disjunctions of the members of CL-SET,
; then it is sufficient to prove all of the formulas qi & thm' & thm'' ...
; -> thmj. This is a trivial proposition fact, to prove (IMPLIES A (AND B
; C)) it is sufficient to prove (IMPLIES A B) and (IMPLIES A C).
; The (ITERATE FOR PICK ...)* expression below returns a list of
; clauses whose conjunction propositionally implies qi & thm' &
; thm'' ... -> thmj, where TA is the ith member of
; TESTS-AND-ALISTS-LST and CL is the jth member of CL-SET. Proof:
; Let THM have the form:
;
; (AND (OR a1 ...)
; (OR b1 ...)
; ...
; (OR z1 ...)).
; Then qi & thm' & thm'' ... -> thmj has the form:
; (IMPLIES (AND qi
; (AND (OR a1 ... )
; (OR b1 ... )
; ...
; (OR z1 ... ))'
; (AND (OR a1 ... )
; (OR b1 ... )
; ...
; (OR z1 ... ))''
; ...
; (AND (OR a1 ... )
; (OR b1 ... )
; ...
; (OR z1 ... )))'''...'
; thmj).
;
; Suppose this formula is false for some values of the free variables. Then
; under those values, each disjunction in the hypothesis is true. Thus there
; exists a way of choosing one literal from each of the disjunctions, all of
; which are true. This choice is one of the PICKs below. But we prove that
; (IMPLIES (AND qi PICK) thmj).
; Note: The (ITERATE FOR PICK ...) expression mentioned above is the function
; induction-formula3 above.
(m&m (reverse (induction-formula1 cl-set cl-set ta-lst nil))
'subsetp-equal/smaller))
; Because the preceding computation is potentially explosive we will
; sometimes reduce its complexity by shrinking the given clause set to
; a singleton set containing a unit clause. To decide whether to do that
; we will use the following rough measures:
(defun all-picks-size (cl-set)
; This returns the size of the all-picks computed by induction-formula3.
(cond ((endp cl-set) 1)
(t (* (length (car cl-set)) (all-picks-size (cdr cl-set))))))
(defun induction-formula-size1 (hyps-size concl-size ta-lst)
; We determine roughly the number of clauses that ta-lst will generate when
; the number of all-picks through the hypotheses is hyps-size and the
; number of conclusion clauses is concl-size. The individual cases of
; the tests-and-alists combine additively. But we must pick our way through
; the hyps for each instantiation.
(cond ((endp ta-lst) 0)
(t
(+ (* concl-size (expt hyps-size
(length (access tests-and-alists
(car ta-lst)
:alists))))
(induction-formula-size1 hyps-size concl-size (cdr ta-lst))))))
(defun induction-formula-size (cl-set ta-lst)
; This function returns a rough upper bound on the number of clauses
; that will be generated by induction-formula on the given arguments.
; See the comment in that function.
(induction-formula-size1 (all-picks-size cl-set)
(length cl-set)
ta-lst))
; The following constant determines the limit on the estimated number of
; clauses induct, below, will return. When normal processing would exceed
; this number, we try to cut down the combinatorics by collapsing clauses
; back into terms.
(defconst *maximum-induct-size* 100)
; And here is how we convert a hairy set of clauses into a term when we
; have to.
(defun termify-clause-set (clauses)
; This function is similar to termify-clause except that it converts a
; set of clauses into an equivalent term. The set of clauses is
; understood to be implicitly conjoined and we therefore produce a
; conjunction expressed as (if cl1 cl2 nil).
(cond ((endp clauses) *t*)
((endp (cdr clauses)) (disjoin (car clauses)))
(t (fcons-term* 'if
(disjoin (car clauses))
(termify-clause-set (cdr clauses))
*nil*))))
; Once we have created the set of clauses to prove, we inform the
; simplifier of what to look out for during the early processing.
(defun inform-simplify3 (alist terms ans)
; Instantiate every term in terms with alist and add them to ans.
(cond ((endp terms) ans)
(t (inform-simplify3 alist (cdr terms)
(add-to-set-equal (sublis-var alist (car terms))
ans)))))
(defun inform-simplify2 (alists terms ans)
; Using every alist in alists, instantiate every term in terms and add
; them all to ans.
(cond ((endp alists) ans)
(t (inform-simplify2 (cdr alists) terms
(inform-simplify3 (car alists) terms ans)))))
(defun inform-simplify1 (ta-lst terms ans)
; Using every alist mentioned in any tests-and-alists record of ta-lst
; we instantiate every term in terms and add them all to ans.
(cond ((endp ta-lst) ans)
(t (inform-simplify1 (cdr ta-lst) terms
(inform-simplify2 (access tests-and-alists
(car ta-lst)
:alists)
terms
ans)))))
(defun inform-simplify (ta-lst terms pspv)
; Historical Plaque from Nqthm:
; Two of the variables effecting REWRITE are TERMS-TO-BE-IGNORED-BY-REWRITE
; and EXPAND-LST. When any term on the former is encountered REWRITE returns
; it without rewriting it. Terms on the latter must be calls of defined fns
; and when encountered are replaced by the rewritten body.
; We believe that the theorem prover will perform significantly faster on
; many theorems if, after an induction, it does not waste time (a) trying to
; simplify the recursive calls introduced in the induction hypotheses and (b)
; trying to decide whether to expand the terms inducted for in the induction
; conclusion. This suspicion is due to some testing done with the idea of
; "homographication" which was just a jokingly suggested name for the idea of
; generalizing the recursive calls away at INDUCT time after expanding the
; induction terms in the conclusion. Homographication speeded the
; theorem-prover on many theorems but lost on several others because of the
; premature generalization. See the comment in FORM-INDUCTION-CLAUSE.
; To avoid the generalization at INDUCT time we are going to try using
; TERMS-TO-BE-IGNORED-BY-REWRITE. The idea is this, during the initial
; simplification of a clause produced by INDUCT we will have the recursive
; terms on TERMS-TO-BE-IGNORED-BY-REWRITE. When the clause settles down --
; hopefully it will often be proved first -- we will restore
; TERMS-TO-BE-IGNORED-BY-REWRITE to its pre-INDUCT value. Note however that
; we have to mess with TERMS-TO-BE-IGNORED-BY-REWRITE on a clause by clause
; basis, not just once in INDUCT.
; So here is the plan. INDUCT will set INDUCTION-HYP-TERMS to the list of
; instances of the induction terms, and will set INDUCTION-CONCL-TERMS to the
; induction terms themselves. SIMPLIFY-CLAUSE will look at the history of
; the clause to determine whether it has settled down since induction. If
; not it will bind TERMS-TO-BE-IGNORED-BY-REWRITE to the concatenation of
; INDUCTION-HYP-TERMS and its old value and will analogously bind EXPAND-LST.
; A new process, called SETTLED-DOWN-SENT, will be used to mark when in the
; history the clause settled down.
; In addition, induct-clause resets the :ens of the rcnst in pspv so
; that it it the initial global ens. This way inductive proofs that
; work on the outside will work from within a larger proof.
(change prove-spec-var pspv
:rewrite-constant
(change rewrite-constant
(access prove-spec-var pspv :rewrite-constant)
:ens (access prove-spec-var pspv :global-ens))
:induction-concl-terms terms
:induction-hyp-terms (inform-simplify1 ta-lst terms nil)))
(defun remove-trivial-clauses (clauses ens wrld)
(cond
((endp clauses) nil)
((trivial-clause-p (car clauses) ens wrld)
(remove-trivial-clauses (cdr clauses) ens wrld))
(t (cons (car clauses)
(remove-trivial-clauses (cdr clauses) ens wrld)))))
(defun induct-clause (id cl hist pspv wrld)
; This is a standard waterfall processor. We return four results.
; The first is either the signal 'MISS (meaning we could find no
; induction to do) or 'HIT, meaning we're going to use induction. The
; second value is a list of clauses, representing the induction base
; cases and steps. The third is the hist-alist entry describing the
; induction. It contains just one pair (:SUGGESTORS . terms), where
; terms are the terms that suggested or played a role in shaping the
; induction chosen. The fourth is a new pspv. We modify pspv to
; store the induction-hyp-terms and induction-concl-terms for the
; simplifier.
(declare (ignore id hist))
; Some functions below operate on clause sets, so we turn cl into such
; a set.
(let ((cl-set (list cl))
(induct-hint-val (access prove-spec-var pspv :induct-hint-val)))
; Induct-hint-val is nil, :do-not-induct, t, or a term. Nil is
; treated like t and means we should choose an induction from those
; suggested by cl. (The difference between nil and t is felt
; elsewhere: nil means let the waterfall get to induction in the
; natural course of events and non-nil means go to induction.)
; :do-not-induct means we MISS (and hence fail). Otherwise, we choose
; an induction from those suggested by the induct-hint-val.
(cond
((eq induct-hint-val :DO-NOT-INDUCT)
(mv 'MISS nil nil nil))
(t
(let* ((pspv (change prove-spec-var pspv :induct-hint-val nil))
(candidates
(get-induction-cands-from-cl-set
(if (or (eq induct-hint-val t)
(eq induct-hint-val nil))
cl-set
(list (list induct-hint-val)))
pspv wrld))
(flushed-candidates
(m&m candidates 'flush-candidates))
(merged-candidates
(cond ((< (length flushed-candidates) 10)
(m&m-over-powerset flushed-candidates 'merge-candidates))
(t (m&m flushed-candidates 'merge-candidates))))
; Note: We really respect powerset. If the set we're trying to merge
; has more than 10 things in it -- an arbitrary choice at the time of
; this writing -- we just do the m&m instead, which causes us to miss
; some merges because we only use each candidate once and merging
; early merges can prevent later ones.
(unvetoed-candidates
(compute-vetoes merged-candidates wrld))
(complicated-candidates
(maximal-elements unvetoed-candidates 'induction-complexity wrld))
(high-scoring-candidates
(maximal-elements complicated-candidates 'score wrld))
(winning-candidate (car high-scoring-candidates)))
(cond
(winning-candidate
(<induct-clause-id>
(let* (
; First, we estimate the size of the answer if we persist in using cl-set.
(estimated-size
(induction-formula-size cl-set
(access candidate
winning-candidate
:tests-and-alists-lst)))
; Next we create clauses, the set of clauses we wish to prove.
; Observe that if the estimated size is greater than
; *maximum-induct-size* we squeeze the cl-set into the form {{p}},
; where p is a single term. This eliminates the combinatoric
; explosion at the expense of making the rest of the theorem prover
; suffer through opening things back up. The idea, however, is that
; it is better to give the user something to look at, so he sees the
; problem blowing up in front of him in rewrite, than to just
; disappear into induction and never come out.
(clauses0
(induction-formula
(cond ((> estimated-size *maximum-induct-size*)
(list (list (termify-clause-set cl-set))))
(t cl-set))
(access candidate winning-candidate
:tests-and-alists-lst)))
(clauses
(cond ((> estimated-size *maximum-induct-size*)
clauses0)
(t (remove-trivial-clauses
clauses0
(access rewrite-constant
(access prove-spec-var
pspv
:rewrite-constant)
:ens)
wrld))))
; Now we inform the simplifier of this induction and store the ttree of
; the winning candidate into the tag-tree of the pspv.
(new-pspv
(inform-simplify
(access candidate winning-candidate
:tests-and-alists-lst)
(cons (access candidate winning-candidate
:xinduction-term)
(access candidate winning-candidate
:xother-terms))
pspv)))
(mv 'HIT
clauses
(list
(cons :suggestors
(cons
(access candidate winning-candidate
:xinduction-term)
(access candidate winning-candidate
:xother-terms))))
new-pspv))))
(t
; Otherwise, we return the 'lose signal.
(mv 'MISS nil nil nil))))))))
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