Built-in axioms and theorems without rule classes.
Definition:
(defaxiom bad-atom<=-total (implies (and (bad-atom x) (bad-atom y)) (or (bad-atom<= x y) (bad-atom<= y x))) :rule-classes nil)
Definition:
(defaxiom bad-atom<=-antisymmetric (implies (and (bad-atom x) (bad-atom y) (bad-atom<= x y) (bad-atom<= y x)) (equal x y)) :rule-classes nil)
Definition:
(defaxiom completion-of-realpart (equal (realpart x) (if (acl2-numberp x) (realpart x) 0)) :rule-classes nil)
Definition:
(defaxiom completion-of-numerator (equal (numerator x) (if (rationalp x) (numerator x) 0)) :rule-classes nil)
Definition:
(defaxiom completion-of-intern-in-package-of-symbol (equal (intern-in-package-of-symbol x y) (if (and (stringp x) (symbolp y)) (intern-in-package-of-symbol x y) nil)) :rule-classes nil)
Definition:
(defaxiom completion-of-imagpart (equal (imagpart x) (if (acl2-numberp x) (imagpart x) 0)) :rule-classes nil)
Definition:
(defaxiom completion-of-denominator (equal (denominator x) (if (rationalp x) (denominator x) 1)) :rule-classes nil)
Definition:
(defaxiom completion-of-coerce (equal (coerce x y) (cond ((equal y 'list) (if (stringp x) (coerce x 'list) nil)) (t (coerce (make-character-list x) 'string)))) :rule-classes nil)
Definition:
(defaxiom completion-of-complex (equal (complex x y) (complex (if (real/rationalp x) x 0) (if (real/rationalp y) y 0))) :rule-classes nil)
Definition:
(defaxiom completion-of-code-char (equal (code-char x) (if (and (integerp x) (>= x 0) (< x 256)) (code-char x) *null-char*)) :rule-classes nil)
Definition:
(defaxiom completion-of-char-code (equal (char-code x) (if (characterp x) (char-code x) 0)) :rule-classes nil)
Definition:
(defaxiom completion-of-cdr (equal (cdr x) (cond ((consp x) (cdr x)) (t nil))) :rule-classes nil)
Definition:
(defaxiom completion-of-car (equal (car x) (cond ((consp x) (car x)) (t nil))) :rule-classes nil)
Definition:
(defaxiom completion-of-< (equal (< x y) (if (and (real/rationalp x) (real/rationalp y)) (< x y) (let ((x1 (if (acl2-numberp x) x 0)) (y1 (if (acl2-numberp y) y 0))) (or (< (realpart x1) (realpart y1)) (and (equal (realpart x1) (realpart y1)) (< (imagpart x1) (imagpart y1))))))) :rule-classes nil)
Definition:
(defaxiom completion-of-unary-/ (equal (/ x) (if (and (acl2-numberp x) (not (equal x 0))) (/ x) 0)) :rule-classes nil)
Definition:
(defaxiom completion-of-unary-minus (equal (- x) (if (acl2-numberp x) (- x) 0)) :rule-classes nil)
Definition:
(defaxiom completion-of-* (equal (* x y) (if (acl2-numberp x) (if (acl2-numberp y) (* x y) 0) 0)) :rule-classes nil)
Definition:
(defaxiom completion-of-+ (equal (+ x y) (if (acl2-numberp x) (if (acl2-numberp y) (+ x y) x) (if (acl2-numberp y) y 0))) :rule-classes nil)
Definition:
(defaxiom completion-of-symbol-package-name (equal (symbol-package-name x) (if (symbolp x) (symbol-package-name x) "")) :rule-classes nil)
Definition:
(defaxiom completion-of-symbol-name (equal (symbol-name x) (if (symbolp x) (symbol-name x) "")) :rule-classes nil)
Definition:
(defaxiom nil-is-not-circular (equal nil (intern-in-package-of-symbol (coerce (cons #\N (cons #\I (cons #\L 0))) 'string) 'string)) :rule-classes nil)
Definition:
(defaxiom string-is-not-circular (equal 'string (intern-in-package-of-symbol (coerce (cons #\S (cons #\T (cons #\R (cons #\I (cons #\N (cons #\G 0)))))) (cons #\S (cons #\T (cons #\R (cons #\I (cons #\N (cons #\G 0))))))) (intern-in-package-of-symbol 0 0))) :rule-classes nil)
Definition:
(defaxiom completion-of-pkg-imports (equal (pkg-imports x) (if (stringp x) (pkg-imports x) nil)) :rule-classes nil)
Definition:
(defaxiom characterp-return (characterp #\Return) :rule-classes nil)
Definition:
(defaxiom characterp-rubout (characterp #\Rubout) :rule-classes nil)
Definition:
(defaxiom characterp-tab (characterp #\Tab) :rule-classes nil)
Definition:
(defaxiom characterp-page (characterp #\Page) :rule-classes nil)
Definition:
(defaxiom booleanp-characterp (booleanp (characterp x)) :rule-classes nil)
Definition:
(defaxiom lowest-terms (implies (and (integerp n) (rationalp x) (integerp r) (integerp q) (< 0 n) (equal (numerator x) (* n r)) (equal (denominator x) (* n q))) (equal n 1)) :rule-classes nil)
Definition:
(defaxiom integer-step (implies (integerp x) (and (integerp (+ x 1)) (integerp (+ x -1)))) :rule-classes nil)
Definition:
(defaxiom integer-1 (integerp 1) :rule-classes nil)
Definition:
(defaxiom integer-0 (integerp 0) :rule-classes nil)
Definition:
(defaxiom nonzero-imagpart (implies (complex/complex-rationalp x) (not (equal 0 (imagpart x)))) :rule-classes nil)
Definition:
(defaxiom complex-implies1 (and (real/rationalp (realpart x)) (real/rationalp (imagpart x))) :rule-classes nil)
Definition:
(defaxiom integer-implies-rational (implies (integerp x) (rationalp x)) :rule-classes nil)
Definition:
(defaxiom rational-implies1 (implies (rationalp x) (and (integerp (denominator x)) (integerp (numerator x)) (< 0 (denominator x)))) :rule-classes nil)
Definition:
(defaxiom positive (and (implies (and (< 0 x) (< 0 y)) (< 0 (+ x y))) (implies (and (real/rationalp x) (real/rationalp y) (< 0 x) (< 0 y)) (< 0 (* x y)))) :rule-classes nil)
Definition:
(defaxiom trichotomy (and (implies (acl2-numberp x) (or (< 0 x) (equal x 0) (< 0 (- x)))) (or (not (< 0 x)) (not (< 0 (- x))))) :rule-classes nil)
Definition:
(defaxiom zero (not (< 0 0)) :rule-classes nil)
Definition:
(defaxiom <-on-others (equal (< x y) (< (+ x (- y)) 0)) :rule-classes nil)
Definition:
(defaxiom closure (and (acl2-numberp (+ x y)) (acl2-numberp (* x y)) (acl2-numberp (- x)) (acl2-numberp (/ x))) :rule-classes nil)
Theorem:
(defthm apply$-userfn-takes-arity-args (implies (badge-userfn fn) (equal (apply$-userfn fn args) (apply$-userfn fn (take (apply$-badge-arity (badge-userfn fn)) args)))) :rule-classes nil)
Theorem:
(defthm add-def-complex (equal (+ x y) (complex (+ (realpart x) (realpart y)) (+ (imagpart x) (imagpart y)))) :rule-classes nil)
Theorem:
(defthm symbol-equality (implies (and (or (symbolp s1) (symbolp s2)) (equal (symbol-name s1) (symbol-name s2)) (equal (symbol-package-name s1) (symbol-package-name s2))) (equal s1 s2)) :rule-classes nil)
Theorem:
(defthm all-boundp-preserves-assoc-equal (implies (and (all-boundp tbl1 tbl2) (assoc-equal x tbl1)) (assoc-equal x tbl2)) :rule-classes nil)
Theorem:
(defthm equal-char-code (implies (and (characterp x) (characterp y)) (implies (equal (char-code x) (char-code y)) (equal x y))) :rule-classes nil)
Theorem:
(defthm doppelganger-apply$-userfn-takes-arity-args (implies (doppelganger-badge-userfn fn) (equal (doppelganger-apply$-userfn fn args) (doppelganger-apply$-userfn fn (take (caddr (doppelganger-badge-userfn fn)) args)))) :rule-classes nil)
Theorem:
(defthm doppelganger-badge-userfn-type (or (null (doppelganger-badge-userfn fn)) (let ((x (doppelganger-badge-userfn fn))) (and (weak-apply$-badge-p x) (natp (access apply$-badge x :arity)) (natp (access apply$-badge x :out-arity)) (or (eq (access apply$-badge x :ilks) t) (and (true-listp (access apply$-badge x :ilks)) (equal (len (access apply$-badge x :ilks)) (access apply$-badge x :arity)) (not (all-nils (access apply$-badge x :ilks))) (subsetp (access apply$-badge x :ilks) '(nil :fn :expr))))))) :rule-classes nil)
Theorem:
(defthm function-symbolp-ev-fncall+-fns-strictp (let ((fn (ev-fncall+-fns fn args wrld big-n safe-mode gc-off t))) (and (symbolp fn) (or (null fn) (function-symbolp fn wrld)))) :rule-classes nil)
Theorem:
(defthm lower/upper-case-p-non-standard-disjointness (not (and (lower-case-p-non-standard x) (upper-case-p-non-standard x))) :rule-classes nil)