Built-in axioms and theorems
of the
Definition:
(defaxiom booleanp-bad-atom<= (or (equal (bad-atom<= x y) t) (equal (bad-atom<= x y) nil)) :rule-classes :type-prescription)
Definition:
(defaxiom code-char-type (characterp (code-char n)) :rule-classes :type-prescription)
Definition:
(defaxiom symbolp-pkg-witness (symbolp (pkg-witness x)) :rule-classes :type-prescription)
Definition:
(defaxiom symbolp-intern-in-package-of-symbol (symbolp (intern-in-package-of-symbol x y)) :rule-classes :type-prescription)
Definition:
(defaxiom stringp-symbol-package-name (stringp (symbol-package-name x)) :rule-classes :type-prescription)
Definition:
(defaxiom nonnegative-product (implies (real/rationalp x) (and (real/rationalp (* x x)) (<= 0 (* x x)))) :rule-classes ((:type-prescription :typed-term (* x x))))
Theorem:
(defthm natp-from-to-by-measure (natp (from-to-by-measure i j)) :rule-classes :type-prescription)
Theorem:
(defthm stringp-df-string (stringp (df-string x)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-df-rationalize (rationalp (df-rationalize x)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-df-pi (rationalp (df-pi)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-df-atanh-fn (rationalp (df-atanh-fn x)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-df-acosh-fn (rationalp (df-acosh-fn x)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-df-asinh-fn (rationalp (df-asinh-fn x)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-df-tanh-fn (rationalp (df-tanh-fn x)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-df-cosh-fn (rationalp (df-cosh-fn x)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-df-sinh-fn (rationalp (df-sinh-fn x)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-df-atan-fn (rationalp (df-atan-fn x)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-df-acos-fn (rationalp (df-acos-fn x)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-df-asin-fn (rationalp (df-asin-fn x)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-df-tan-fn (rationalp (df-tan-fn x)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-df-cos-fn (rationalp (df-cos-fn x)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-df-sin-fn (rationalp (df-sin-fn x)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-df-abs-fn (rationalp (df-abs-fn x)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-unary-df-log (rationalp (unary-df-log x)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-binary-df-log (rationalp (binary-df-log x y)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-df-sqrt-fn (rationalp (df-sqrt-fn x)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-df-exp-fn (rationalp (df-exp-fn x)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-df-expt-fn (rationalp (df-expt-fn x y)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-to-df (rationalp (to-df x)) :rule-classes :type-prescription)
Theorem:
(defthm true-listp-explode-atom (true-listp (explode-atom n print-base)) :rule-classes :type-prescription)
Theorem:
(defthm true-listp-chars-for-tilde-@-clause-id-phrase/periods (true-listp (chars-for-tilde-@-clause-id-phrase/periods lst)) :rule-classes :type-prescription)
Theorem:
(defthm fn-count-evg-rec-type-prescription (implies (natp acc) (natp (fn-count-evg-rec evg acc calls))) :rule-classes :type-prescription)
Theorem:
(defthm canonical-pathname-type (or (equal (canonical-pathname x dir-p state) nil) (stringp (canonical-pathname x dir-p state))) :rule-classes :type-prescription)
Theorem:
(defthm true-listp-merge-sort-lexorder (implies (and (true-listp l1) (true-listp l2)) (true-listp (merge-lexorder l1 l2 acc))) :rule-classes :type-prescription)
Theorem:
(defthm nth-0-read-run-time-type-prescription (implies (state-p1 state) (rationalp (nth 0 (read-run-time state)))) :rule-classes ((:type-prescription :typed-term (nth 0 (read-run-time state)))))
Theorem:
(defthm natp-random$ (natp (car (random$ n state))) :rule-classes :type-prescription)
Theorem:
(defthm true-listp-subseq-type-prescription (implies (not (stringp seq)) (true-listp (subseq seq start end))) :rule-classes :type-prescription)
Theorem:
(defthm stringp-subseq-type-prescription (implies (stringp seq) (stringp (subseq seq start end))) :rule-classes :type-prescription)
Theorem:
(defthm true-listp-update-nth (implies (true-listp l) (true-listp (update-nth key val l))) :rule-classes :type-prescription)
Theorem:
(defthm characterp-char-upcase (characterp (char-upcase x)) :rule-classes :type-prescription)
Theorem:
(defthm characterp-char-downcase (characterp (char-downcase x)) :rule-classes :type-prescription)
Theorem:
(defthm natp-position-ac (implies (and (integerp acc) (<= 0 acc)) (or (equal (position-ac item lst acc) nil) (and (integerp (position-ac item lst acc)) (<= 0 (position-ac item lst acc))))) :rule-classes :type-prescription)
Theorem:
(defthm true-list-listp-forward-to-true-listp-assoc-equal (implies (true-list-listp l) (true-listp (assoc-equal key l))) :rule-classes (:type-prescription (:forward-chaining :trigger-terms ((assoc-equal key l)))))
Theorem:
(defthm true-listp-explode-nonnegative-integer (implies (true-listp ans) (true-listp (explode-nonnegative-integer n print-base ans))) :rule-classes :type-prescription)
Theorem:
(defthm consp-assoc-equal (implies (alistp l) (or (consp (assoc-equal name l)) (equal (assoc-equal name l) nil))) :rule-classes (:type-prescription (:forward-chaining :trigger-terms ((assoc-equal name l)))))
Theorem:
(defthm true-listp-substitute-type-prescription (implies (not (stringp seq)) (true-listp (substitute new old seq))) :rule-classes :type-prescription)
Theorem:
(defthm stringp-substitute-type-prescription (implies (stringp seq) (stringp (substitute new old seq))) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-expt-type-prescription (implies (rationalp r) (rationalp (expt r i))) :rule-classes :type-prescription)
Theorem:
(defthm expt-type-prescription-non-zero-base (implies (and (acl2-numberp r) (not (equal r 0))) (not (equal (expt r i) 0))) :rule-classes :type-prescription)
Theorem:
(defthm natp-position-equal-ac (implies (natp acc) (or (natp (position-equal-ac item lst acc)) (equal (position-equal-ac item lst acc) nil))) :rule-classes :type-prescription)
Theorem:
(defthm natp-position-ac-eql-exec (implies (natp acc) (or (natp (position-ac-eql-exec item lst acc)) (equal (position-ac-eql-exec item lst acc) nil))) :rule-classes :type-prescription)
Theorem:
(defthm natp-position-ac-eq-exec (implies (natp acc) (or (natp (position-ac-eq-exec item lst acc)) (equal (position-ac-eq-exec item lst acc) nil))) :rule-classes :type-prescription)
Theorem:
(defthm true-listp-nthcdr-type-prescription (implies (true-listp x) (true-listp (nthcdr n x))) :rule-classes :type-prescription)
Theorem:
(defthm true-listp-first-n-ac-type-prescription (true-listp (first-n-ac i l ac)) :rule-classes :type-prescription)
Theorem:
(defthm true-listp-revappend-type-prescription (implies (true-listp y) (true-listp (revappend x y))) :rule-classes :type-prescription)
Theorem:
(defthm true-listp-append (implies (true-listp b) (true-listp (append a b))) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-constrained-df-pi (rationalp (constrained-df-pi)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-constrained-df-atanh-fn (rationalp (constrained-df-atanh-fn x)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-constrained-df-acosh-fn (rationalp (constrained-df-acosh-fn x)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-constrained-df-asinh-fn (rationalp (constrained-df-asinh-fn x)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-constrained-df-tanh-fn (rationalp (constrained-df-tanh-fn x)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-constrained-df-cosh-fn (rationalp (constrained-df-cosh-fn x)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-constrained-df-sinh-fn (rationalp (constrained-df-sinh-fn x)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-constrained-df-atan-fn (rationalp (constrained-df-atan-fn x)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-constrained-df-acos-fn (rationalp (constrained-df-acos-fn x)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-constrained-df-asin-fn (rationalp (constrained-df-asin-fn x)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-constrained-df-tan-fn (rationalp (constrained-df-tan-fn x)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-constrained-df-cos-fn (rationalp (constrained-df-cos-fn x)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-constrained-df-sin-fn (rationalp (constrained-df-sin-fn x)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-constrained-df-abs-fn (rationalp (constrained-df-abs-fn x)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-constrained-unary-df-log (rationalp (constrained-unary-df-log x)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-constrained-binary-df-log (rationalp (constrained-binary-df-log x y)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-constrained-df-sqrt-fn (rationalp (constrained-df-sqrt-fn x)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-constrained-df-exp-fn (rationalp (constrained-df-exp-fn x)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-constrained-df-expt-fn (rationalp (constrained-df-expt-fn x y)) :rule-classes :type-prescription)
Theorem:
(defthm natp-conjoin-clause-sets-bound (natp (conjoin-clause-sets-bound)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-constrained-df-rationalize (rationalp (constrained-df-rationalize x)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-df-round (rationalp (df-round x)) :rule-classes :type-prescription)
Theorem:
(defthm rationalp-constrained-to-df (rationalp (constrained-to-df x)) :rule-classes :type-prescription)
Theorem:
(defthm stringp-constrained-df-string (stringp (constrained-df-string x)) :rule-classes :type-prescription)
Theorem:
(defthm characterp-char-downcase-non-standard (characterp (char-downcase-non-standard x)) :rule-classes :type-prescription)
Theorem:
(defthm characterp-char-upcase-non-standard (characterp (char-upcase-non-standard x)) :rule-classes :type-prescription)
Theorem:
(defthm booleanp-lower-case-p-non-standard (booleanp (lower-case-p-non-standard x)) :rule-classes :type-prescription)
Theorem:
(defthm booleanp-upper-case-p-non-standard (booleanp (upper-case-p-non-standard x)) :rule-classes :type-prescription)
Theorem:
(defthm booleanp-alpha-char-p-non-standard (booleanp (alpha-char-p-non-standard x)) :rule-classes :type-prescription)