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|
; Copyright (C) 2022, Regents of the University of Texas
; Written by Matt Kaufmann and J Moore
; License: A 3-clause BSD license. See the LICENSE file distributed with ACL2.
(include-book "projects/apply/top" :dir :system)
(defun rev (x)
(if (endp x)
nil
(append (rev (cdr x)) (list (car x)))))
(defthm assoc-of-append
(equal (append (append a b) c)
(append a (append b c))))
(thm
(implies (true-listp lst)
(equal (loop$ with ans = ans0
with lst = xxx
do
(if (consp lst)
(progn (setq ans (cons (car lst) ans))
(setq lst (cdr lst)))
(return ans)))
(append (rev xxx) ans0))))
; Note that we (unnecessarily) pick up (true-listp lst) as a Q here, and prove
; the formula.
(thm
(implies (true-listp lst)
(equal (loop$ with ans = ans0
with lst = lst
do
(if (consp lst)
(progn (setq ans (cons (car lst) ans))
(setq lst (cdr lst)))
(return ans)))
(append (rev lst) ans0)))
:hints (("Goal" :in-theory (disable (:induction rev) (:induction true-listp)))))
; This works. And we pick up (true-listp xxx) as Q.
(thm
(implies (true-listp xxx)
(equal (loop$ with ans = ans0
with lst = xxx
do
(if (consp lst)
(progn (setq ans (cons (car lst) ans))
(setq lst (cdr lst)))
(return ans)))
(append (rev xxx) ans0)))
:hints (("Goal" :in-theory (disable (:induction rev) (:induction true-listp)))))
; This proves. We pick (natp iii) as Q.
(thm
(implies (natp iii)
(equal (loop$ with iii = iii
do
(if (equal iii 0)
(return 'silly)
(setq iii (- iii 1))))
'silly)))
; This works. But we do not automatically choose Q when there is an induct hint.
; So we have to make sure the induct hint includes it, which means if the induct
; hint is a do loop$ we need to modify its body to include the q.
(thm
(implies (natp iii)
(equal (loop$ with iii = iii
do
(if (equal iii 0)
(return 'silly)
(setq iii (- iii 1))))
'silly))
:hints (("Goal" :induct (loop$ with iii = iii
do
(if (natp iii) ; <--- modified do loop as hint!
(if (equal iii 0)
(return 'silly)
(setq iii (- iii 1)))
(return 'dumb))))))
; This works. We choose Q correctly.
(thm
(implies (and (natp iii)
(natp nnn))
(equal (loop$ with i = iii
with n = nnn
do
:guard (and (natp i) (natp n))
:measure (nfix (- n i))
(if (< i n)
(setq i (+ i 1))
(return t)))
t))
:otf-flg t)
; This works and we choose Q correctly.
(thm
(implies (and (natp iii)
(natp nnn))
(equal (loop$ with i = iii
with n = nnn
do
:guard (and (natp i) (natp n))
:measure (nfix (- n i))
(if (< i n)
(setq i (+ i 1))
(return t)))
t))
:hints (("Goal" :induct t)))
; This works. We choose (natp iii) for Q.
(thm
(implies (natp iii)
(equal (loop$ with i = iii
with n = 10
do
:guard (and (natp n) (natp i))
:measure (nfix (- n i))
(if (< i n)
(setq i (+ i 1))
(return t)))
t)))
; This works. It tests the ability to do a true (:induction do$) induction
; rather than semi-concrete one.
(thm (implies (and (natp (cdr (assoc-eq-safe 'n alist)))
(natp (cdr (assoc-eq-safe 'm alist))))
(equal
(do$ (lambda$ (alist) (acl2-count (cdr (assoc-eq-safe 'n alist))))
alist
(lambda$ (alist)
(if (zp (cdr (assoc-eq-safe 'n alist)))
(list :return (cdr (assoc-eq-safe 'm alist)) alist)
(list nil nil
(list (cons 'n (- (cdr (assoc-eq-safe 'n alist)) 1))
(cons 'm (+ (cdr (assoc-eq-safe 'm alist)) 1))))))
(lambda$ (alist)
(list nil nil
(list (cons 'n (cdr (assoc-eq-safe 'n alist))))))
nil nil)
(+ (cdr (assoc-eq-safe 'n alist))
(cdr (assoc-eq-safe 'm alist))))))
(defun rev1 (x ac)
(if (endp x)
ac
(rev1 (cdr x) (cons (car x) ac))))
(thm
(equal (loop$ with x = lst
with ans = ans0
do
(if (consp x)
(progn (setq ans (cons (car x) ans))
(setq x (cdr x)))
(return ans)))
(rev1 lst ans0)))
; Write a do loop$ that computes (member e lst) and prove it correct.
(thm
(equal (loop$ with x = lst
do
(if (consp x)
(if (equal (car x) e)
(return x)
(setq x (cdr x)))
(return nil)))
(member e lst)))
(thm
(implies (natp ans0)
(equal (loop$ with x = lst
with ans = ans0
do
(if (consp x)
(progn (setq ans (+ 1 ans))
(setq x (cdr x)))
(return ans)))
(+ ans0 (len lst)))))
(thm
(implies (natp n)
(equal (loop$ with x = lst
with i = n
do
(cond ((endp x) (return nil))
((equal i 0) (return (car x)))
(t (progn (setq i (- i 1))
(setq x (cdr x))))))
(nth n lst))))
(thm
(implies (and (true-listp ans0)
(true-listp lst))
(equal (loop$ with x = lst
with ans = ans0
do
(if (consp x)
(progn (setq ans (append ans (list (car x))))
(setq x (cdr x)))
(return ans)))
(append ans0 lst)))
:hints (("Goal" :in-theory (disable (:induction binary-append)))))
(thm
(implies (and (natp m)
(natp n))
(equal (loop$ with i = m
with j = n
do
(if (integerp i)
(if (< 0 i)
(progn (setq i (- i 1))
(setq j (+ 1 j)))
(return j))
(return j)))
(+ m n))))
(defun fact (n)
(if (zp n)
1
(* n (fact (- n 1)))))
(defthm commutivity-2-of-*
(equal (* a (* b c)) (* b (* a c)))
:hints
(("Goal" :use ((:instance commutativity-of-*
(x a)(y (* b c)))))))
(thm
(implies (and (natp n)
(natp ans0))
(equal (loop$ with i = n
with ans = ans0
do
(if (integerp i)
(if (< 0 i)
(progn (setq ans (* i ans))
(setq i (- i 1)))
(return ans))
(return ans)))
(* ans0 (fact n)))))
(defun sq (x) (* x x))
(defwarrant sq)
(thm
(implies (and (warrant sq)
(integer-listp lst0)
(integerp u0)
(integerp v0))
(equal (loop$ with lst = lst0
with u = u0
with v = v0
do
(cond ((consp lst)
(progn (setq u (+ u (car lst)))
(setq v (+ v (* (car lst) (car lst))))
(setq lst (cdr lst))))
(t (return (cons u v)))))
(cons (+ u0 (loop$ for e in lst0 sum e))
(+ v0 (loop$ for e in lst0 sum (sq e)))))))
(thm
(equal (loop$ with lst = lst
with syms = syms
with non-syms = non-syms
do
(cond ((consp lst)
(cond
((symbolp (car lst))
(progn (setq syms (cons (car lst) syms))
(setq lst (cdr lst))))
(t
(progn (setq non-syms (cons (car lst) non-syms))
(setq lst (cdr lst))))))
(t (return (cons syms non-syms)))))
(cons (append (rev (loop$ for e in lst when (symbolp e) collect e))
syms)
(append (rev (loop$ for e in lst when (not (symbolp e)) collect e))
non-syms))))
(defun nats-ac-up (i n ans)
(declare (xargs :measure (nfix (- (+ 1 (nfix n)) (nfix i)))))
(let ((i (nfix i))
(n (nfix n)))
(if (> i n)
ans
(nats-ac-up (+ i 1) n (cons i ans)))))
(thm
(implies (and (natp n)
(natp i0))
(equal
(loop$ with i = i0
with ans = ans0
do
:measure (nfix (+ 1 (- i) n))
(if (< n i)
(return ans)
(progn (setq ans (cons i ans))
(setq i (+ 1 i)))))
(nats-ac-up i0 n ans0))))
(defun make-pair (i j)
(declare (xargs :guard t))
(cons i j))
(defwarrant make-pair)
(defun apdh2 (i j ans)
(cond
((natp j)
(cond ((< j 1) ans)
(t (apdh2 i (- j 1) (cons (make-pair i j) ans)))))
(t nil)))
(defun apdh1 (i jmax ans)
(cond
((natp i)
(cond ((< i 1) ans)
(t (apdh1 (- i 1) jmax (apdh2 i jmax ans)))))
(t nil)))
(defthm lemma1
(implies (natp jmax) ; <--- no warrant req'd
(equal ; because no make-
(loop$ with j = jmax ; pair anymore
with ans = ans0
do
:guard (natp j)
(cond
((< j 1)
(return ans))
(t (progn
(setq ans (cons (cons i j) ; <--- make-pair opened
ans))
(setq j (- j 1))))))
(apdh2 i jmax ans0))))
; Prove that the generalized outer loop$ is apdh1.
(thm
(implies
(and (warrant do$) ; <--- no warrant make-pair
(natp imax)
(natp jmax))
(equal
(loop$ with i = imax
with ans = ans0 ; <--- ans0 instead of nil
do
:guard (and (natp i) (natp jmax))
(cond
((< i 1)
(return ans))
(t (progn
(setq ans (loop$ with j = jmax ; <--- normalized inner
with ans = ans ; loop$
do
:guard (natp j)
(cond
((< j 1)
(return ans))
(t (progn
(setq ans (cons (cons i j) ans))
(setq j (- j 1)))))))
(setq i (- i 1))))))
(apdh1 imax jmax ans0))))
|
