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(in-package "PROOF-CHECKER-ARRAY")
(include-book "sb60")
;; ===================================================================
;; ============================ VARIABLES ============================
(defconst *2^58* (expt 2 58))
(defconst *2^59* (expt 2 59))
(defconst *2^60* (expt 2 60))
;; (signed-byte-p 60 x) --> (- *2^59*) <= x < *2^59*
;; *2^59* even, last literal should be odd because of pairing 0 0 1 -1 2 -2
;; (literalp x) --> 2 <= x <= (1- *2^59*) <--> < *2^59*
;; (encode x) = (if (< x 0) (1+ (* 2 (abs x))) (* 2 x))
;; (variablep x) --> 1 <= x <= (- *2^58* 2) <--> < (1- *2^58*)
(defun variablep (x)
(declare (xargs :guard t))
(and (integerp x)
(< 0 x)
(< x (1- *2^58*))))
;; ===================================================================
;; ============================ LITERALS =============================
(defun literalp (x)
(declare (xargs :guard t))
(and (integerp x)
(< 1 x)
(< x *2^59*)))
(defthm literalp-implies-eqlablep
(implies (literalp x)
(eqlablep x)))
(defthm literalp-implies-sb60p
(implies (literalp lit)
(sb60p lit))
:rule-classes :forward-chaining)
(defthm literalp-implies-<-2
(implies (literalp lit)
(< 1 lit))
:rule-classes :forward-chaining)
;; ===================================================================
;; ========================== LITERAL-LIST ===========================
(defun literal-listp (x)
(declare (xargs :guard t))
(if (atom x)
(null x)
(and (literalp (car x))
(literal-listp (cdr x)))))
(defthm literal-listp-implies-eqlable-listp
(implies (literal-listp x)
(eqlable-listp x)))
(defthm literal-listp-implies-literalp-member
(implies (and (literal-listp x)
(member e x))
(literalp e)))
(defthm literal-listp-append
(implies (and (literal-listp x)
(literal-listp y))
(literal-listp (append x y))))
;; ===================================================================
;; ============================ NEGATION =============================
(defun negatedp (x)
(declare (xargs :guard (literalp x)))
(logand x 1))
(defun negate (x)
(declare (xargs :guard (literalp x)))
(logxor x 1))
(encapsulate
()
(local (include-book "arithmetic-5/top" :dir :system))
(defthm literalp-negate
(equal (literalp (negate x))
(literalp x)))
(defthm literalp-implies-not-equal-negate
(implies (literalp x)
(not (equal (negate x) x))))
(defthm negate-negate
(implies (literalp x)
(equal (negate (negate x))
x))))
(encapsulate
()
(local (include-book "arithmetic-5/top" :dir :system))
(defthm literalp-implies-sb60p-negate
(implies (literalp lit)
(sb60p (negate lit)))))
;; ===================================================================
(encapsulate
()
(local (include-book "arithmetic-5/top" :dir :system))
(defthm equal-negate-1+
(implies (and (literalp lit)
(evenp lit))
(equal (negate lit)
(1+ lit))))
(in-theory (disable equal-negate-1+))
(defthm equal-1+-negate
(implies (and (literalp lit)
(evenp lit))
(equal (1+ lit)
(negate lit))))
(in-theory (disable equal-1+-negate))
(defthm equal-negate-1-
(implies (and (literalp lit)
(oddp lit))
(equal (negate lit)
(1- lit))))
(in-theory (disable equal-negate-1-))
(defthm evenp-implies-<-negate
(implies (and (literalp lit)
(integerp x)
(evenp x)
(< lit x))
(< (negate lit) x))
:hints (("Goal"
:cases ((evenp lit)))))
(defthm <=-0-negate
(implies (literalp lit)
(<= 0 (negate lit)))))
;; ===================================================================
;; ============================ DISABLES =============================
(in-theory (disable variablep
literalp
negatedp
negate
))
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