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; Section: THE WATERFALL
; The waterfall is a simple finite state machine (whose individual
; state transitions are very complicated). Abstractly, each state
; contains a "processor" and transitions to one of two neighbor
; states, the "hit" state and the "miss" state. Roughly speaking,
; when we are in a state we apply its processor to the input clause
; and obtain either a "hit" signal (and some new clauses) or "miss"
; signal. We then transit to the appropriate state and continue.
; However, the "hit" state for every state is the top of the falls.
; +<---------------------------------------+
; | |
; apply-hints-clause ---------------------->|
; | |
; simplify-clause ------------------------->|
; | |
; settled-down-clause---------------------->|
; | |
; ... |
; | |
; induct-clause --------------------------->+
; |
; UNPROVED
; We therefore represent a state s of the waterfall as a pair whose car
; is the processor for s and whose cdr is the miss state for s. The hit
; state for every state is the constant state below, which includes, by
; successive cdrs, every state below it in the falls.
; Because the word "STATE" has a very different meaning in ACL2 than we have
; been using thus far in this discussion, we refer to the "states" of the
; waterfall as "ledges" and basically name them by the processors on each.
; Observe that the cdr of the 'simplify-clause ledge, for example, is the
; 'settled-down-clause ledge, etc. That is, each ledge contains the
; ones below it.
; Note: To add a new processor to the waterfall you must add the
; appropriate entry to the *waterfall* and redefine
; apply-waterfall-process below.
; The waterfall builds a proof attempt, represented by
(defrec proof-attempt (process depth-to-go clause-id
input-clause input-hist input-pspv
output-signal . children))
; A proof attempt is a tree of proot-attempt nodes. The process is
; one of the waterfall processes or else INDUCT. The depth-to-go
; records how much recursion depth is left in the waterfall -- there
; is no reason to believe that repeated simplification will ultimately
; terminate. Input-clause, input-hist and input-pspv record the input
; to the processor. The world is always constant and is not part of
; the proof-attempt. Output-signal records the output signal produced
; by the processor. Finally, children is the recursively obtained
; list of proof-attempts for each child produced by the processor.
; The legal signals are HIT, MISS, ABORT or anything else. The last
; indicates an error and is actually handled like an ABORT: further
; processing of the children is not pursued, but ongoing computations for the
; other higher- and peer-subgoals continues uninterrupted.
; About Hints: The pspv contains :hint-settings, which is an alist
; pairing clause ids with hint information. To apply hints, we first
; determine whether there is a pair for the given clause id. If there is,
; we may change the clause (i.e., adding a hypothesis as per a :USE
; hint). We may change the :rewrite-constant in the pspv (i.e., to
; install a new enabled structure). We also change the :hint-settings
; in the pspv, removing the hint just used. We also change the hist,
; adding an APPLY-HINTS-CLAUSE entry.
; Here is a generic hint-settings
; ((id . ((:USE . (term1 term2 ... termk)) ; a list of terms to add as hyps
; (:ENS . ens) ; a new enabled structure
; (:INDUCT . term) ; a term to suggest an induction
; ))
; ...)
(defun apply-hints-clause1 (alist cl pspv)
; Here, alist is the alist pairing :keys to values. That alist was
; paired in the hint-settings with the id of clause cl. We apply that
; alist to cl and pspv, obtaining (mv cl' pspv').
(cond
((endp alist) (mv cl pspv))
(t (let ((key (car (car alist)))
(val (cdr (car alist)))
(rcnst (access prove-spec-var pspv :rewrite-constant)))
(case key
(:USE
(apply-hints-clause1 (cdr alist)
(disjoin-clauses
(dumb-negate-lit-lst val)
cl)
pspv))
(:IN-THEORY
(apply-hints-clause1 (cdr alist)
cl
(change prove-spec-var pspv
:rewrite-constant
(change rewrite-constant
rcnst
:ens val))))
(:INDUCT
(apply-hints-clause1 (cdr alist)
cl
(change prove-spec-var pspv
:induct-hint-val
val)))
(:HANDS-OFF
(apply-hints-clause1 (cdr alist)
cl
(change prove-spec-var pspv
:rewrite-constant
(change rewrite-constant
rcnst
:fns-to-be-ignored val))))
(:EXPAND
(apply-hints-clause1 (cdr alist)
cl
(change prove-spec-var pspv
:rewrite-constant
(change rewrite-constant
rcnst
:expand-lst val))))
(:DO-NOT
(apply-hints-clause1 (cdr alist)
cl
(change prove-spec-var pspv
:do-not-processes
val)))
(otherwise
(apply-hints-clause1 (cdr alist) cl pspv)))))))
(defun apply-hints-clause (id cl hist pspv wrld)
; This is a standard waterfall processor. In fact, it is the first
; processor in the waterfall. As for all processors, the results are
; (mv signal cl-set hist-obj new-pspv).
; The idea here is to recognize whether id has an associated hint in
; the pspv and if so to apply the hint. Typically a hint might change
; the clause and the pspv, e.g., adding hyps to one and deleting the
; hint from the other after modifying the :rewrite-constant.
(declare (ignore hist wrld))
(let ((temp
(assoc-equal id (access prove-spec-var pspv :hint-settings))))
(cond
(temp
(mv-let (new-cl new-pspv)
(apply-hints-clause1
(cdr temp)
cl
(change prove-spec-var pspv
:hint-settings
(delete1-equal temp
(access prove-spec-var pspv
:hint-settings))))
(mv 'HIT (list new-cl) nil new-pspv)))
(t (mv 'MISS nil nil nil)))))
; Here is the function that applies a waterfall processor. Note that
; if the pspv contains an :induct-hint-val other than :DO-NOT-INDUCT,
; then every processor except induct-clause is a no-op. If the hint
; is :DO-NOT-INDUCT we just take the natural course through the
; waterfall and fail when (if) we get to induction. If an induction
; hint of T or a term was provided, we skip other processes and go
; straight to induction.
; This is another identity function used for nume tracking. But it is
; not cut from the standard mold because it also prints a message
; about the just-completed waterfall process.
(defun apply-waterfall-process (process id cl hist pspv wrld)
(cond
((member-eq process (access prove-spec-var pspv :do-not-processes))
(mv 'miss nil nil nil))
((and (access prove-spec-var pspv :induct-hint-val)
(not (equal (access prove-spec-var pspv :induct-hint-val)
:do-not-induct))
(not (equal process 'induct-clause)))
(mv 'miss nil nil nil))
(t
; This is a weird call of my macro. Push a new empty frame. If the
; result is HIT, then print the stuff indicated and pop the frame with flg t
; else don't print and pop the frame with flg nil.
; The problem is that I don't have any assurance that the "printing" code
; prints instead of doing some arbitrary unsound raw lisp thing.
(<apply-waterfall-process-id>
(case process
(apply-hints-clause
(apply-hints-clause id cl hist pspv wrld))
(simplify-clause
(simplify-clause id cl hist pspv wrld))
(settled-down-clause
(settled-down-clause id cl hist pspv wrld))
(eliminate-destructors-clause
(eliminate-destructors-clause id cl hist pspv wrld))
(induct-clause
(induct-clause id cl hist pspv wrld))
(otherwise
(mv 'abort nil (list (cons :UNKNOWN-PROCESS process)) pspv)))))))
(defconst *waterfall*
'(apply-hints-clause
simplify-clause
settled-down-clause
eliminate-destructors-clause
induct-clause
))
; Guide to the Waterfall Code: The top-level function is waterfall.
; It trickles down the ledges, trying each process until it comes to
; one that HITs. When that happens, it calls waterfall-lst on the
; children. That function computes the new clause id for each child
; and calls waterfall on each one. The lex4 measure is used, but only
; the first 3 components are relevant. The last is always 0; I didn't
; have lex3.
(acl2::set-well-founded-relation e0-ord-<)
(mutual-recursion
(defun waterfall (ledge id cl hist pspv wrld nnn)
(declare (xargs :measure (lex4 (nfix nnn) (acl2-count ledge) 0 0)
:hints (("Goal"
:in-theory (disable apply-waterfall-process)))))
(cond
((or (endp ledge) (zp nnn))
(make proof-attempt
:process 'UNPROVED
:depth-to-go nnn
:clause-id id
:input-clause cl
:input-hist hist
:input-pspv pspv
:output-signal (if (zp nnn) 'TIMEOUT 'UNPROVED)
:children nil))
(t (mv-let (signal cl-set hist-alist new-pspv)
(apply-waterfall-process (car ledge) id cl hist pspv wrld)
(case signal
(HIT
(let ((new-hist
(cons (make history-entry
:processor (car ledge)
:clause cl
:alist hist-alist)
hist))
; Some hints are not inherited from their parents. This is the place
; we reset the un-inherited hints. At the moment, the only such hint
; is the :do-not one. If that were inherited and the user, say,
; turned off simplify-clause for a given goal, it would stay off for
; all children until turned back on.
(new-pspv (change prove-spec-var new-pspv
:do-not-processes nil)))
(make proof-attempt
:process (car ledge)
:depth-to-go nnn
:clause-id id
:input-clause cl
:input-hist hist
:input-pspv pspv
:output-signal 'HIT
:children (waterfall-lst id
cl-set
new-hist
new-pspv
wrld
(- nnn 1)))))
(MISS (waterfall (cdr ledge) id cl hist pspv wrld nnn))
(otherwise
; Typically the only other signal is ABORT, which means the proof has
; failed and the input-clause is an unproved subgoal. But the signal
; might be an arbitrary "error message".
(let ((new-hist (cons (make history-entry
:processor (car ledge)
:clause cl
:alist hist-alist)
hist)))
(make proof-attempt
:process (car ledge)
:depth-to-go nnn
:clause-id id
:input-clause cl
:input-hist new-hist
:input-pspv pspv
:output-signal signal
:children nil))))))))
(defun waterfall-lst (id0 clauses hist pspv wrld nnn)
(declare (xargs :measure
(lex4 (nfix nnn)
(acl2-count *waterfall*)
(acl2-count clauses)
0)))
(cond
((endp clauses) nil)
(t (cons (waterfall *waterfall*
(append id0 (list (len clauses)))
(car clauses) hist pspv wrld nnn)
(waterfall-lst id0 (cdr clauses) hist pspv wrld nnn)))))
)
(mutual-recursion
(defun scan-proof-attempt1 (p errors subgoals)
; We explore a proof-attempt p and collect into errors every error signal
; or other anomaly and collect into subgoals every unproved
; subgoal. Each new element is collected in a keyword pair:
; (:SUBGOAL id clause-as-formula)
; (:ERROR signal)
; We return (mv errors subgoals).
(cond ((endp p)
(mv (cons '(:ERROR ILL-FORMED) errors)
subgoals))
((eq (access proof-attempt p :process) 'UNPROVED)
(mv errors
(cons (list :SUBGOAL (access proof-attempt p :clause-id)
(prettyify-clause
(access proof-attempt p :input-clause)))
subgoals)))
((member (access proof-attempt p :process) *waterfall*)
(cond ((eq (access proof-attempt p :output-signal) 'HIT)
(scan-proof-attempt1-lst (access proof-attempt p :children)
errors subgoals))
(t (mv (cons (list :ERROR 'UDS
(access proof-attempt p :output-signal))
errors)
subgoals))))
(t (mv (cons (list :ERROR 'UDF
(access proof-attempt p :process))
errors)
subgoals))))
(defun scan-proof-attempt1-lst (plst errors subgoals)
(cond ((endp plst) (mv errors subgoals))
(t (mv-let (errors subgoals)
(scan-proof-attempt1 (car plst) errors subgoals)
(scan-proof-attempt1-lst (cdr plst) errors subgoals)))))
)
(defun scan-proof-attempt (p)
(mv-let (errors subgoals)
(scan-proof-attempt1 p nil nil)
(cond ((and (null errors) (null subgoals)) :QED)
(t (list :FAILURE
(cons :ERRORS errors)
(cons :SUBGOALS subgoals))))))
(mutual-recursion
(defun condense-proof-attempt1 (p)
; We assume there are no errors in the scanned proof attempt.
(cond
((consp p)
(list* (access proof-attempt p :process)
(list :SUBGOAL (access proof-attempt p :clause-id)
(prettyify-clause
(access proof-attempt p :input-clause)))
(condense-proof-attempt1-lst (access proof-attempt p :children))))
(t nil)))
(defun condense-proof-attempt1-lst (plst)
(cond
((endp plst) nil)
(t (cons (condense-proof-attempt1 (car plst))
(condense-proof-attempt1-lst (cdr plst))))))
)
(defun condense-proof-attempt (p)
(let ((temp (scan-proof-attempt p)))
(cond ((eq temp :QED)
(condense-proof-attempt1 p))
((and (consp temp)
(null (cdr (assoc-eq :ERRORS (cdr temp)))))
(condense-proof-attempt1 p))
(t temp))))
(defun describe-proof-attempt (p d-level)
(case d-level
(0 (scan-proof-attempt p))
(1 (condense-proof-attempt p))
(otherwise p)))
; THE PROVER
(defun prove1 (term pspv wrld waterfall-depth)
(waterfall *waterfall* nil (list term) nil pspv wrld waterfall-depth))
(defun make-initial-pspv (ens hint-settings)
(make prove-spec-var
:rewrite-constant
(make rewrite-constant
:expand-lst nil
:terms-to-be-ignored-by-rewrite nil
:top-clause nil
:current-clause nil
:ens ens
:current-literal nil)
:induct-hint-val nil
:induction-hyp-terms nil
:induction-concl-terms nil
:do-not-processes nil
:hint-settings hint-settings
:global-ens ens))
; The following function is still the identity function. But it is
; not cut from the standard mold. After doing the standard nume
; tracking and evaluating body, it prints all the numes in the
; top-most frame of the nume stack. It first converts them to runes.
(defun prove (term ens wrld hint-settings waterfall-depth)
(<prove-id>
(prove1 term
(make-initial-pspv ens hint-settings)
wrld waterfall-depth)))
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