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|
; Copyright (C) 2023, ForrestHunt, Inc.
; Written by J Strother Moore
; License: A 3-clause BSD license. See the LICENSE file distributed with ACL2.
; Release approved by DARPA with "DISTRIBUTION STATEMENT A. Approved
; for public release. Distribution is unlimited."
; Seven Proofs that PPR1 returns a PPR-TUPLE-P
; J Moore
; October, 2023
; This book illustrates the ideas discussed in
; :DOC induction-coarse-v-fine-grained
; -----------------------------------------------------------------
; 0 Setup
; We inherit lemmas from fmt-support and disable the lemma we'll repeatedly
; prove.
(in-package "ACL2")
(include-book "std/testing/must-fail" :dir :system)
(include-book "system/fmt-support" :dir :system)
; Matt K. mod, 2/2/2024 to avoid failure apparently due to ACL2(p) parallelism
; issues:
(set-waterfall-parallelism nil)
(in-theory (disable fix ppr-tuple-p-ppr1 ppr-tuple-lst-p-ppr1-lst))
; We will create two induction schemes, a ``coarse'' one and a ``fine'' one.
; We'll repeatedly prove the ppr-tuple-p-ppr1 theorem (and its counterpart
; about ppr1-lst) using both schemes and various hints.
(make-flag coarse-induction-scheme ppr1
:ruler-extenders :basic)
(make-flag fine-induction-scheme ppr1
:ruler-extenders :lambdas)
; This function sketches an induction scheme. We illustrate below.
(defun about-imach (fn wrld)
(cons (len (getpropc fn 'induction-machine nil wrld))
(merge-sort-lexorder
(loop$ with imach = (getpropc fn 'induction-machine nil wrld)
with alist = nil
do
(cond
((consp imach)
(progn
(setq alist
(let ((n (len (access tests-and-calls (car imach) :calls))))
(put-assoc-equal n
(+ 1
(or (cdr (assoc-equal n alist)) 0))
alist)))
(setq imach (cdr imach))))
(t (return alist)))))))
(assert-event
(equal (about-imach 'coarse-induction-scheme (w state))
'(76 ;; 76 cases in the induction scheme
(0 . 6) ;; no induction hyps in 6 cases, i.e., Base Cases
(1 . 8) ;; 1 induction hyp in 8 cases
(2 . 2) ;; 2 induction hyps in 2 cases
(3 . 16) ;; ...
(4 . 32)
(8 . 4)
(9 . 8) ;; 9 induction hyps in 8 cases
)))
(assert-event
(equal (about-imach 'fine-induction-scheme (w state))
'(256 ;; 256 cases in the induction scheme
(0 . 6) ;; no induction hyps in 6 cases, i.e., Base Cases
(1 . 8) ;; 1 induction hyp in 8 cases
(2 . 82) ;; 2 induction hyps in 82 cases
(3 . 80) ;; 3 induction hyps in 80 cases
(4 . 80) ;; 4 induction hyps in 80 cases
)))
; The following are used in our computed hint.
(defun find-all-calls-of-any (fns term ans)
(cond
((endp fns) ans)
(t (find-all-calls-of-any (cdr fns)
term
(find-all-calls (car fns) term ans)))))
(defun all-fnnames-intersect-fns (fns clause)
; We wish to compute the intersection of all the fnnames used in clause with
; the set of names listed in fns. But rather than collect all the fnnames in
; clause we just ask of each fn in fns whether it is used in clause.
(cond ((endp fns) nil)
((ffnnamep-lst (car fns) clause)
(cons (car fns)
(all-fnnames-intersect-fns (cdr fns) clause)))
(t (all-fnnames-intersect-fns (cdr fns) clause))))
(set-state-ok t)
(defun computed-hint
(fns opened-conclp enabledp clause stable-under-simplificationp state)
; The idea here is that we'll start with ppr1/ppr1-lst disabled. The flagged
; induction will produce a bunch of hideous induction steps which we'll simplify
; without expanding any ppr1/ppr1-lst terms. When the subgoal becomes stable,
; we'll explicitly :expand only the ppr1/ppr1-lst in the concluding literal.
; If it becomes stable again and ppr1/ppr1-lst still occur in the subgoal,
; we'll enable them and let the prover just do its job.
; Fns is the list of functions whose opening we're interested in, opened-conclp
; is a flag that tells us that we've already :expanded the calls in the
; conclusion, and enabledp is a flag that tells us that we've enabled the fns
; in question.
(cond
(stable-under-simplificationp
(cond
((not opened-conclp)
(let ((calls-in-concl (find-all-calls-of-any fns (car (last clause)) nil)))
(cond (calls-in-concl
(value
`(:computed-hint-replacement
((computed-hint
',fns
t ,enabledp clause stable-under-simplificationp state))
:expand ,calls-in-concl)))
((not enabledp)
(let ((fns-to-enable (all-fnnames-intersect-fns fns clause)))
(cond (fns-to-enable
(value `(:in-theory (enable ,fns-to-enable))))
(t (value nil)))))
(t (value nil)))))
((not enabledp)
(let ((fns-to-enable (all-fnnames-intersect-fns fns clause)))
(cond (fns-to-enable
(value `(:in-theory (enable ,fns-to-enable))))
(t (value nil)))))
(t (value nil))))
(t (value nil))))
; Note that ppr1 and ppr1-lst will be globally disabled here, but the first two
; will eventually be enabled by the computed hint below, if an unproved subgoal
; involving ppr1 or ppr1-lst becomes stable after we've opened up ppr1 and/or
; ppr1-lst in the conclusion.
; Now we begin the tests.
; -----------------------------------------------------------------
; Scenario 1
(in-theory (enable ppr1 ppr1-lst))
; The following proof takes about 2200 seconds if allowed to proceed to
; completion. But we don't want the ordinary user, who will probably do make
; regression at some point, to have to wait an extra 30 minutes to see that
; this proof takes a long time. So in this version of the book, we just abort
; this proof attempt after 350,000 steps. The successful proof takes on the
; order of 38 million steps. To see the successful proof, certify
; ppr1-experiments-thm-1-ppr1.lisp.
; In September, 2023, it took 2165.19 seconds running the proto-Version 8.6 in
; CCL on a MacBook Pro.
(must-fail
(with-prover-step-limit 350000
(defthm-coarse-induction-scheme ; coarse, enabled, :expand
(defthm thm-1-ppr1
(implies (and
(unsigned-byte-p #.*small-nat-bits* width)
(unsigned-byte-p #.*small-nat-bits* rpc)
(symbol-to-fixnat-alistp
(table-alist 'ppr-special-syms (w state))))
(ppr-tuple-p (ppr1 x print-base print-radix width rpc state eviscp)))
:flag ppr1
:rule-classes nil)
(defthm thm-1-ppr1-lst
(implies (and
(unsigned-byte-p #.*small-nat-bits* width)
(unsigned-byte-p #.*small-nat-bits* rpc)
(symbol-to-fixnat-alistp
(table-alist 'ppr-special-syms (w state))))
(ppr-tuple-lst-p (ppr1-lst lst print-base print-radix width rpc
pair-keywords-p state eviscp)))
:flag ppr1-lst
:rule-classes nil)
:hints (("Goal"
:expand ((:free (print-base print-radix width rpc state eviscp)
(ppr1 x print-base print-radix width rpc state eviscp))
(:free (print-base print-radix width rpc
pair-keywords-p state eviscp)
(ppr1-lst lst print-base print-radix width rpc
pair-keywords-p state eviscp))))))
))
; This proof times out after about 11 seconds. The last few subgoals enumerated
; look like:
; Subgoal *1/63.9
; Subgoal *1/63.9'
; Subgoal *1/63.9.2
; Subgoal *1/63.9.1
; Subgoal *1/63.8
; Subgoal *1/63.8'
; Subgoal *1/63.8.4
; Subgoal *1/63.8.4.2
; Subgoal *1/63.8.4.2.3
; Subgoal *1/63.8.4.2.3.2
; Subgoal *1/63.8.4.2.3.1
; Subgoal *1/63.8.4.2.2
; Subgoal *1/63.8.4.2.2.3
; Subgoal *1/63.8.4.2.2.2
; -----------------------------------------------------------------
; Scenario 2
(in-theory (disable ppr1 ppr1-lst))
(defthm-coarse-induction-scheme ; coarse, disabled, computed
(defthm thm-2-ppr1
(implies (and
(unsigned-byte-p #.*small-nat-bits* width)
(unsigned-byte-p #.*small-nat-bits* rpc)
(symbol-to-fixnat-alistp
(table-alist 'ppr-special-syms (w state))))
(ppr-tuple-p (ppr1 x print-base print-radix width rpc state eviscp)))
:flag ppr1
:rule-classes nil)
(defthm thm-2-ppr1-lst
(implies (and
(unsigned-byte-p #.*small-nat-bits* width)
(unsigned-byte-p #.*small-nat-bits* rpc)
(symbol-to-fixnat-alistp
(table-alist 'ppr-special-syms (w state))))
(ppr-tuple-lst-p (ppr1-lst lst print-base print-radix width rpc
pair-keywords-p state eviscp)))
:flag ppr1-lst
:rule-classes nil)
:hints ((computed-hint '(ppr1 ppr1-lst) nil nil
clause stable-under-simplificationp state)))
; -----------------------------------------------------------------
; Scenario 3
(in-theory (enable ppr1 ppr1-lst))
(defthm-fine-induction-scheme ; fine, enabled, :expand
(defthm thm-3-ppr1
(implies (and
(unsigned-byte-p #.*small-nat-bits* width)
(unsigned-byte-p #.*small-nat-bits* rpc)
(symbol-to-fixnat-alistp
(table-alist 'ppr-special-syms (w state))))
(ppr-tuple-p (ppr1 x print-base print-radix width rpc state eviscp)))
:flag ppr1
:rule-classes nil)
(defthm thm-3-ppr1-lst
(implies (and
(unsigned-byte-p #.*small-nat-bits* width)
(unsigned-byte-p #.*small-nat-bits* rpc)
(symbol-to-fixnat-alistp
(table-alist 'ppr-special-syms (w state))))
(ppr-tuple-lst-p (ppr1-lst lst print-base print-radix width rpc
pair-keywords-p state eviscp)))
:flag ppr1-lst
:rule-classes nil)
:hints (("Goal"
:expand ((:free (print-base print-radix width rpc state eviscp)
(ppr1 x print-base print-radix width rpc state eviscp))
(:free (print-base print-radix width rpc
pair-keywords-p state eviscp)
(ppr1-lst lst print-base print-radix width rpc
pair-keywords-p state eviscp))))))
; -----------------------------------------------------------------
; Scenario 4
(in-theory (disable ppr1 ppr1-lst))
(defthm-fine-induction-scheme ; fine, disabled, computed hint + :expand
(defthm thm-4-ppr1
(implies (and
(unsigned-byte-p #.*small-nat-bits* width)
(unsigned-byte-p #.*small-nat-bits* rpc)
(symbol-to-fixnat-alistp
(table-alist 'ppr-special-syms (w state))))
(ppr-tuple-p (ppr1 x print-base print-radix width rpc state eviscp)))
:flag ppr1
:rule-classes nil)
(defthm thm-4-ppr1-lst
(implies (and
(unsigned-byte-p #.*small-nat-bits* width)
(unsigned-byte-p #.*small-nat-bits* rpc)
(symbol-to-fixnat-alistp
(table-alist 'ppr-special-syms (w state))))
(ppr-tuple-lst-p (ppr1-lst lst print-base print-radix width rpc
pair-keywords-p state eviscp)))
:flag ppr1-lst
:rule-classes nil)
:hints ((computed-hint '(ppr1 ppr1-lst) nil nil
clause stable-under-simplificationp state)
("Goal"
:expand ((:free (print-base print-radix width rpc state eviscp)
(ppr1 x print-base print-radix width rpc state eviscp))
(:free (print-base print-radix width rpc
pair-keywords-p state eviscp)
(ppr1-lst lst print-base print-radix width rpc
pair-keywords-p state eviscp))))
))
; -----------------------------------------------------------------
; Scenario 5
(in-theory (disable ppr1 ppr1-lst))
; Subgoal *1/255''' of the thm-5-ppr1-lst proof below fails because (PPR1-LST
; LST ...) in the conclusion becomes (PPR1-LST NIL ...), the :EXPAND hint
; doesn't catch that, and the TWEAK prevents the computed hint from detecting
; it. But the following trivial lemma does the job.
(defthm ppr1-lst-nil
(equal (PPR1-LST NIL PRINT-BASE PRINT-RADIX WIDTH
RPC PAIR-KEYWORDS-P STATE EVISCP)
nil)
:hints (("Goal"
:expand (PPR1-LST NIL PRINT-BASE PRINT-RADIX WIDTH
RPC PAIR-KEYWORDS-P STATE EVISCP))))
(defthm-fine-induction-scheme ; fine, disabled, computed' + :expand + NIL lem
(defthm thm-5-ppr1
(implies (and
(unsigned-byte-p #.*small-nat-bits* width)
(unsigned-byte-p #.*small-nat-bits* rpc)
(symbol-to-fixnat-alistp
(table-alist 'ppr-special-syms (w state))))
(ppr-tuple-p (ppr1 x print-base print-radix width rpc state eviscp)))
:flag ppr1
:rule-classes nil)
(defthm thm-5-ppr1-lst
(implies (and
(unsigned-byte-p #.*small-nat-bits* width)
(unsigned-byte-p #.*small-nat-bits* rpc)
(symbol-to-fixnat-alistp
(table-alist 'ppr-special-syms (w state))))
(ppr-tuple-lst-p (ppr1-lst lst print-base print-radix width rpc
pair-keywords-p state eviscp)))
:flag ppr1-lst
:rule-classes nil)
:hints ((computed-hint '(ppr1 ppr1-lst) t nil ; <--- TWEAK!
clause stable-under-simplificationp state)
("Goal"
:expand ((:free (print-base print-radix width rpc state eviscp)
(ppr1 x print-base print-radix width rpc state eviscp))
(:free (print-base print-radix width rpc
pair-keywords-p state eviscp)
(ppr1-lst lst print-base print-radix width rpc
pair-keywords-p state eviscp))))
))
(in-theory (disable ppr1-lst-nil))
; -----------------------------------------------------------------
; Scenario 6
(in-theory (disable ppr1 ppr1-lst))
(defthm-fine-induction-scheme ; fine, disabled, computed
(defthm thm-6-ppr1
(implies (and
(unsigned-byte-p #.*small-nat-bits* width)
(unsigned-byte-p #.*small-nat-bits* rpc)
(symbol-to-fixnat-alistp
(table-alist 'ppr-special-syms (w state))))
(ppr-tuple-p (ppr1 x print-base print-radix width rpc state eviscp)))
:flag ppr1
:rule-classes nil)
(defthm thm-6-ppr1-lst
(implies (and
(unsigned-byte-p #.*small-nat-bits* width)
(unsigned-byte-p #.*small-nat-bits* rpc)
(symbol-to-fixnat-alistp
(table-alist 'ppr-special-syms (w state))))
(ppr-tuple-lst-p (ppr1-lst lst print-base print-radix width rpc
pair-keywords-p state eviscp)))
:flag ppr1-lst
:rule-classes nil)
:hints ((computed-hint '(ppr1 ppr1-lst) nil nil
clause stable-under-simplificationp state)))
; -----------------------------------------------------------------
; Scenario 7
; Now we look at one other avenue: disabling MAX after proving a couple of
; useful lemmas about it. We otherwise just use the fastest setup we've found
; above, namely a fine induction with the fns disabled and both the computed
; hint and the :expand hint. Chances are that these MAX tweaks would help all
; the scenarios we've tried.
(defthm integerp-max
(implies (and (integerp x)
(integerp y))
(integerp (max x y))))
(defthm nonneg-max
(implies (and (<= 0 x)
(<= 0 y))
(<= 0 (max x y)))
:rule-classes :linear)
(defthm max-upper-bound
(and (<= x (max x y))
(<= y (max x y)))
:rule-classes :linear)
(in-theory (disable max))
(in-theory (disable ppr1 ppr1-lst))
(defthm-fine-induction-scheme ; fine, disabled, computed + :expand + MAX
(defthm thm-7-ppr1
(implies (and
(unsigned-byte-p #.*small-nat-bits* width)
(unsigned-byte-p #.*small-nat-bits* rpc)
(symbol-to-fixnat-alistp
(table-alist 'ppr-special-syms (w state))))
(ppr-tuple-p (ppr1 x print-base print-radix width rpc state eviscp)))
:flag ppr1
:rule-classes nil)
(defthm thm-7-ppr1-lst
(implies (and
(unsigned-byte-p #.*small-nat-bits* width)
(unsigned-byte-p #.*small-nat-bits* rpc)
(symbol-to-fixnat-alistp
(table-alist 'ppr-special-syms (w state))))
(ppr-tuple-lst-p (ppr1-lst lst print-base print-radix width rpc
pair-keywords-p state eviscp)))
:flag ppr1-lst
:rule-classes nil)
:hints ((computed-hint '(ppr1 ppr1-lst) nil nil
clause stable-under-simplificationp state)
("Goal"
:expand ((:free (print-base print-radix width rpc state eviscp)
(ppr1 x print-base print-radix width rpc state eviscp))
(:free (print-base print-radix width rpc
pair-keywords-p state eviscp)
(ppr1-lst lst print-base print-radix width rpc
pair-keywords-p state eviscp))))))
; We now disable the lemmas we just proved about max and re-enable max to
; return the system to the state it was in before Scenario 7, in case we add
; more scenarios.
(in-theory (e/d (max)
(integerp-max nonneg-max max-upper-bound)))
|
