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|
(in-package "ACL2")
(local (include-book "arithmetic/idiv" :dir :system))
(local (include-book "arithmetic/realp" :dir :system))
(include-book "inverse-monotone")
(include-book "exp")
(include-book "complex-polar")
; Added by Matt K. for v2-7.
(add-match-free-override :once t)
(set-match-free-default :once)
(in-theory (enable acl2-exp))
(defthm real-sumlist-exp-list
(implies (realp x)
(realp (sumlist (taylor-exp-list n i x)))))
(defthm realp-standard-part
(implies (realp x)
(realp (standard-part x))))
(defthm realp-standard-part
(implies (realp x)
(realp (standard-part x))))
(defthm-std realp-exp
(implies (realp x)
(realp (acl2-exp x))))
(defthm-std standard-exp
(implies (standardp x)
(standardp (acl2-exp x))))
(defthm expt-is-non-decreasing-for-non-negatives
(implies (and (realp x)
(realp y)
(<= 0 x)
(<= x y)
(integerp n)
(<= 0 n))
(<= (expt x n) (expt y n))))
(encapsulate
()
(local
(defthm lemma-1
(implies (and (realp x1)
(realp x2)
(realp y1)
(realp y2)
(<= x1 x2)
(<= y1 y2))
(<= (+ x1 y1) (+ x2 y2)))))
(defthm taylor-exp-is-non-decreasing-for-non-negatives
(implies (and (realp x)
(realp y)
(<= 0 x)
(< x y))
(<= (sumlist (taylor-exp-list n i x))
(sumlist (taylor-exp-list n i y)))))
)
(defthm-std acl2-exp-is-non-decreasing-for-non-negatives
(implies (and (realp x)
(realp y)
(<= 0 x)
(< x y))
(<= (acl2-exp x) (acl2-exp y))))
(encapsulate
()
(local
(defthm taylor-exp-list-0
(equal (sumlist (taylor-exp-list nterms counter 0))
(if (or (zp nterms)
(not (equal counter 0)))
0
1))))
(defthm exp-0
(equal (acl2-exp 0) 1)))
(defthm acl2-exp---x
(equal (acl2-exp (- x))
(/ (acl2-exp (fix x))))
:hints (("Goal"
:use ((:instance exp-sum
(x x)
(y (- x)))
(:instance exp-0)
(:instance Uniqueness-of-*-inverses
(x (acl2-exp x))
(y (acl2-exp (- x)))))
:in-theory (disable exp-sum
exp-0
Uniqueness-of-*-inverses))))
(encapsulate
()
(local
(defthm taylor-exp->=-0-for-non-negatives
(implies (and (realp x)
(<= 0 x))
(<= 0 (sumlist (taylor-exp-list n i x))))))
(local
(defthm taylor-exp->-1-for-non-negatives-lemma
(implies (and (realp x)
(<= 0 x)
(integerp n)
(< 1 n))
(<= (+ 1 x) (sumlist (taylor-exp-list n 0 x))))
:hints (("Goal"
:do-not-induct t
:expand ((taylor-exp-list n 0 x)))
("Goal'"
:do-not-induct t
:expand ((taylor-exp-list (+ -1 n) 1 x)))
("Goal''"
:do-not-induct t
:use ((:instance taylor-exp->=-0-for-non-negatives
(n (+ -2 n))
(i 2)))
:in-theory (disable taylor-exp->=-0-for-non-negatives)))))
(defthm-std acl2-exp->-1-for-non-negatives-lemma
(implies (and (realp x)
(<= 0 x))
(<= (+ 1 x) (acl2-exp x)))
:hints (("Goal"
:use ((:instance standard-part-<=
(x (+ 1 x))
(y (sumlist (taylor-exp-list (i-large-integer) 0 x))))
(:instance taylor-exp->-1-for-non-negatives-lemma
(n (i-large-integer)))
(:instance large->-non-large
(x (i-large-integer))
(y 1)))
:in-theory (disable standard-part-<=
taylor-exp->-1-for-non-negatives-lemma
large->-non-large))))
(defthm acl2-exp->-1-for-positives
(implies (and (realp x)
(< 0 x))
(< 1 (acl2-exp x)))
:hints (("Goal"
:use ((:instance acl2-exp->-1-for-non-negatives-lemma))
:in-theory (disable acl2-exp->-1-for-non-negatives-lemma
acl2-exp))))
)
(defthm acl2-exp->-0-for-reals
(implies (realp x)
(< 0 (acl2-exp x)))
:hints (("Goal"
:cases ((< 0 x) (= 0 x) (< x 0)))
("Subgoal 3"
:use ((:instance acl2-exp->-1-for-positives))
:in-theory (disable acl2-exp->-1-for-positives))
("Subgoal 1"
:use ((:instance acl2-exp---x)
(:instance exp-sum (x x) (y (- x)))
(:instance acl2-exp->-1-for-positives (x (- x)))
(:instance REALP-EXP (x (- x)))
(:instance
(:theorem (implies (and (equal 1 (* a b))
(< 0 b)
(realp a)
(realp b))
(< 0 a)))
(a (acl2-exp x))
(b (acl2-exp (- x)))))
:in-theory (disable acl2-exp---x
REALP-EXP
exp-sum
acl2-exp->-1-for-positives
(:type-prescription acl2-exp)))))
(defthm acl2-exp-x-never-zero
(implies (acl2-numberp x)
(not (equal (acl2-exp x) 0)))
:hints (("Goal"
:use ((:instance exp-sum
(x (- x))
(y x))
(:instance exp-0)
(:instance Uniqueness-of-*-inverses
(x (acl2-exp x))
(y (acl2-exp (- x)))))
:in-theory (disable exp-sum
exp-0
Uniqueness-of-*-inverses))))
(defthm acl2-exp-is-non-decreasing-for-non-positives
(implies (and (realp x)
(realp y)
(< x y)
(<= y 0))
(<= (acl2-exp x) (acl2-exp y)))
:hints (("Goal"
:use ((:instance acl2-exp-is-non-decreasing-for-non-negatives
(x (- y))
(y (- x)))
(:instance /-inverts-weak-order
(x (acl2-exp (- y)))
(y (acl2-exp (- x)))))
:in-theory (disable acl2-exp-is-non-decreasing-for-non-negatives
/-inverts-weak-order
equal-/))))
(defthm acl2-exp-is-non-decreasing
(implies (and (realp x)
(realp y)
(< x y))
(<= (acl2-exp x) (acl2-exp y)))
:hints (("Goal"
:cases ((<= y 0)
(and (< x 0) (< 0 y))
(<= 0 x)))
("Subgoal 2"
:use ((:instance acl2-exp-is-non-decreasing-for-non-positives
(x x)
(y 0))
(:instance acl2-exp-is-non-decreasing-for-non-negatives
(x 0)
(y y)))
:in-theory (disable acl2-exp-is-non-decreasing-for-non-positives
acl2-exp-is-non-decreasing-for-non-negatives))))
(encapsulate
()
(local
(defthm lemma-1
(implies (and (realp x)
(realp y)
(< x y))
(not (equal (acl2-exp x)
(acl2-exp y))))
:hints (("Goal"
:use ((:instance exp-sum
(x x)
(y (- y x)))
(:instance equal-*-x-y-x
(x (acl2-exp x))
(y (acl2-exp (- y x))))
(:instance acl2-exp->-1-for-positives
(x (- y x))))
:in-theory (disable exp-sum
equal-*-x-y-x
acl2-exp->-0-for-reals)))))
(defthm acl2-exp-is-1-1
(implies (and (realp x)
(realp y)
(not (equal x y)))
(not (equal (acl2-exp x)
(acl2-exp y))))
:hints (("Goal"
:cases ((< x y) (= x y) (< y x)))
("Subgoal 1"
:use ((:instance lemma-1 (x y) (y x)))
:in-theory (disable lemma-1))))
)
(defthm acl2-exp-is-increasing
(implies (and (realp x)
(realp y)
(< x y))
(< (acl2-exp x) (acl2-exp y)))
:hints (("Goal"
:use ((:instance acl2-exp-is-non-decreasing)
(:instance acl2-exp-is-1-1))
:in-theory (disable acl2-exp-is-non-decreasing
acl2-exp-is-1-1))))
(defthm acl2-exp->=-1-for-non-negatives
(implies (and (realp x)
(<= 0 x))
(<= 1 (acl2-exp x)))
:hints (("Goal"
:use ((:instance acl2-exp->-1-for-positives)
(:instance exp-0))
:in-theory (disable acl2-exp->-1-for-positives
exp-0))))
(defthm acl2-exp-x->=-x-for-x->=-1
(implies (and (realp x)
(<= 1 x))
(<= x (acl2-exp x)))
:hints (("Goal"
:use ((:instance acl2-exp->-1-for-non-negatives-lemma))
:in-theory (disable acl2-exp->-1-for-non-negatives-lemma))))
(in-theory (disable acl2-exp))
(defun pos-exp (x)
(acl2-exp (realfix x)))
(defun real->=-1 (x)
(and (realp x)
(<= 1 x)))
(defun exp-interval (x)
(interval 0 x))
(local
(defthm exp-lemma-1
(IMPLIES (INSIDE-INTERVAL-P Y (INTERVAL 1 NIL))
(INTERVAL-P (INTERVAL 0 Y)))
:hints (("Goal"
:in-theory (enable interval-definition-theory)))))
(local
(defthm exp-lemma-2
(IMPLIES (INSIDE-INTERVAL-P Y (INTERVAL 1 NIL))
(SUBINTERVAL-P (INTERVAL 0 Y)
(INTERVAL 0 NIL)))
:hints (("Goal"
:in-theory (enable interval-definition-theory)))))
(local
(defthm exp-lemma-3
(IMPLIES (AND (INSIDE-INTERVAL-P Y (INTERVAL 1 NIL))
(< Y 1))
(<= (ACL2-EXP Y) Y))
:hints (("Goal"
:in-theory (enable interval-definition-theory)))))
(local
(defthm exp-lemma-4
(IMPLIES (INSIDE-INTERVAL-P X (INTERVAL 0 NIL))
(INSIDE-INTERVAL-P (ACL2-EXP X)
(INTERVAL 1 NIL)))
:hints (("Goal"
:in-theory (enable interval-definition-theory)))))
(definv pos-exp
:domain (interval 0 nil)
:range (interval 1 nil)
:inverse-interval exp-interval)
(defun acl2-ln-for-positive (y)
(if (< y 1)
(- (pos-exp-inverse (/ y)))
(pos-exp-inverse y)))
(defthm acl2-exp-ln-for-positive
(implies (and (realp y)
(< 0 y))
(and (realp (acl2-ln-for-positive y))
(equal (acl2-exp (acl2-ln-for-positive y)) y)))
:hints (("Goal"
:use ((:instance pos-exp-inverse-exists (y (/ y)))
(:instance pos-exp-inverse-exists (y y)))
:in-theory (enable-disable (interval-definition-theory) (pos-exp-inverse-exists))))
)
(defthm realp-ln-for-positive
(implies (and (realp x)
(< 0 x))
(realp (acl2-ln-for-positive x)))
:hints (("Goal"
:use ((:instance pos-exp-inverse-exists (y x))
(:instance pos-exp-inverse-exists (y (/ x))))
:in-theory (enable-disable (interval-definition-theory) (pos-exp-inverse-exists)))))
(defthm ln-for-positive-exp
(implies (realp x)
(equal (acl2-ln-for-positive (acl2-exp x))
x))
:hints (("Goal"
:use ((:instance acl2-exp-is-1-1
(x x)
(y (acl2-ln-for-positive (acl2-exp x))))
(:instance acl2-exp-ln-for-positive
(y (acl2-exp x))))
:in-theory (disable acl2-exp-is-1-1
acl2-exp-ln-for-positive
acl2-ln-for-positive))))
(defthm uniqueness-of-ln-for-positive
(implies (and (realp x)
(equal (acl2-exp x) y))
(equal (acl2-ln-for-positive y) x)))
(in-theory (disable acl2-ln-for-positive (acl2-ln-for-positive)))
(defun acl2-ln (y)
(complex (acl2-ln-for-positive (radiuspart y))
(anglepart y)))
(defthm realp-ln
(implies (and (realp y)
(< 0 y))
(realp (acl2-ln y))))
(defthm complex-0-b
(implies (realp b)
(equal (complex 0 b)
(* #c(0 1) b)))
:hints (("Goal"
:use ((:instance complex-definition
(x 0)
(y b))))))
(defthm exp-complex
(implies (and (realp a)
(realp b))
(equal (acl2-exp (complex a b))
(* (acl2-exp a) (acl2-exp (complex 0 b)))))
:hints (("Goal"
:use ((:instance complex-definition (x a) (y b))
(:instance exp-sum
(x a)
(y (* #c(0 1) b))))
:in-theory (disable exp-sum
e^ix-cos-i-sin))))
(defthm radiuspart-real-non-zero
(implies (not (equal (fix y) 0))
(and (realp (radiuspart y))
(< 0 (radiuspart y))))
:hints (("Goal"
:use ((:instance radiuspart-is-zero-only-for-zero
(x y)))
:in-theory (disable radiuspart
radiuspart-is-zero-only-for-zero)))
:rule-classes (:type-prescription :rewrite))
(defthm real-ln-radiuspart
(implies (not (equal (fix y) 0))
(realp (acl2-ln-for-positive (radiuspart y))))
:hints (("Goal"
:in-theory (disable radiuspart))))
(defthm exp-ln
(implies (not (equal (fix y) 0))
(equal (acl2-exp (acl2-ln y)) y))
:hints (("Goal"
:in-theory (disable anglepart
radiuspart
e^ix-cos-i-sin))))
(local
(defthm +-complex
(implies (and (realp i) (realp j) (realp r) (realp s))
(equal (+ (complex i j) (complex r s))
(complex (+ i r) (+ j s))))
:hints (("Goal"
:use ((:instance complex-definition (x i) (y j))
(:instance complex-definition (x r) (y s))
(:instance complex-definition (x (+ i r)) (y (+ j s))))))))
(local
(encapsulate
()
(local
(defthm *-complex-lemma-1
(implies (and (realp a) (realp b) (realp r) (realp s))
(equal (* (complex a b) (complex r s))
(* (+ a (* #C(0 1) b)) (+ R (* #C(0 1) S)))))
:hints (("Goal"
:use ((:instance complex-definition (x a) (y b))
(:instance complex-definition (x r) (y s)))))))
;; ...the next step is to factor everything out, remembering that
;; i^2=-1....
(local
(defthm *-complex-lemma-2
(implies (and (realp a) (realp b) (realp r) (realp s))
(equal (complex (- (* a r) (* b s))
(+ (* a s) (* b r)))
(+ (+ (* a R) (- (* b S)))
(* #C(0 1) (+ (* a S) (* b R))))))
:hints (("Goal"
:use ((:instance complex-definition
(x (- (* a r) (* b s)))
(y (+ (* a s) (* b r)))))))))
;; And so now we can get a formula for the product of two complex
;; numbers.
(defthm *-complex
(implies (and (realp i) (realp j) (realp r) (realp s))
(equal (* (complex i j) (complex r s))
(complex (- (* i r) (* j s))
(+ (* i s) (* j r)))))))
)
(defthm radiuspart-*
(equal (radiuspart (* x y))
(* (radiuspart x) (radiuspart y)))
)
(defthm abs-exp
(implies (realp x)
(equal (abs (acl2-exp x))
(acl2-exp x)))
:hints (("Goal"
:use ((:instance acl2-exp->-0-for-reals))
:in-theory (enable-disable (abs) (acl2-exp->-0-for-reals)))))
(defthm imagpart-+
(equal (imagpart (+ x y))
(+ (imagpart x) (imagpart y))))
(defthm realpart-+
(equal (realpart (+ x y))
(+ (realpart x) (realpart y))))
(defthm realpart-of-real
(implies (realp x)
(equal (realpart x) x)))
(defthm realpart-of-imaginary
(implies (realp x)
(equal (realpart (* #c(0 1) x)) 0))
:hints (("Goal"
:use ((:instance complex-0-b
(b x)))
:in-theory (disable complex-0-b))))
(defthm imagpart-of-imaginary
(implies (realp x)
(equal (imagpart (* #c(0 1) x)) x))
:hints (("Goal"
:use ((:instance complex-0-b
(b x)))
:in-theory (disable complex-0-b))))
(defthm realpart-sine
(implies (realp x)
(equal (realpart (acl2-sine x))
(acl2-sine x))))
(defthm imagpart-sine
(implies (realp x)
(equal (imagpart (acl2-sine x))
0)))
(defthm realpart-cosine
(implies (realp x)
(equal (realpart (acl2-cosine x))
(acl2-cosine x))))
(defthm imagpart-cosine
(implies (realp x)
(equal (imagpart (acl2-cosine x))
0)))
(defthm radiuspart-e-i-theta
(implies (realp theta)
(equal (radiuspart (acl2-exp (* #c(0 1) theta)))
1))
:hints (("Goal"
:use ((:instance sin**2+cos**2
(x theta)))
:in-theory (disable sin**2+cos**2)))
)
(defthm sine->=-0-=>-x-<=-pi
(implies (and (realp x)
(<= 0 x)
(< x (* (acl2-pi) 2))
(<= 0 (acl2-sine x))
)
(<= x (acl2-pi)))
:hints (("Goal"
:use ((:instance sine-negative-in-pi-3pi/2)
(:instance sine-3pi/2)
(:instance sine-negative-in-3pi/2-2pi))
:in-theory (disable sine-negative-in-pi-3pi/2
sine-3pi/2
sine-negative-in-3pi/2-2pi)))
)
(defthm sine-<-0-=>-x->-pi
(implies (and (realp x)
(<= 0 x)
(< (acl2-sine x) 0)
)
(< (acl2-pi) x))
:hints (("Goal"
:use ((:instance sine-0)
(:instance sine-positive-in-0-pi/2)
(:instance sine-pi/2)
(:instance sine-positive-in-pi/2-pi))
:in-theory (disable sine-0
sine-positive-in-0-pi/2
sine-pi/2
sine-positive-in-pi/2-pi
sine-2x
sine-3x
<-*-/-LEFT
<-*-/-right))))
(local
(defthm realp-cosine-inverse
(implies (realp theta)
(realp (acl2-acos (acl2-cosine theta))))
:hints (("Goal"
:use ((:instance acl2-acos-exists (y (acl2-cosine theta))))
:in-theory (enable-disable (interval-definition-theory)
(acl2-acos-exists))))))
(defthm anglepart-e-i-theta
(implies (and (realp theta)
(<= 0 theta)
(< theta (* (acl2-pi) 2)))
(equal (anglepart (acl2-exp (* #c(0 1) theta)))
theta))
:hints (("Goal"
:use ((:instance sin**2+cos**2
(x theta)))
:in-theory (disable sin**2+cos**2))
("Subgoal 3"
:use ((:instance complex-definition
(x (acl2-cosine theta))
(y (acl2-sine theta)))
(:instance complex-equal
(x1 (acl2-cosine theta))
(y1 (acl2-sine theta))
(x2 0)
(y2 0)))
:in-theory (disable complex-equal
sin**2+cos**2))
("Subgoal 2"
:use ((:instance sine->=-0-=>-x-<=-pi
(x theta))
(:instance acl2-acos-exists
(y (acl2-cosine theta)))
(:instance cosine-is-1-1-on-domain
(x1 theta)
(x2 (ACL2-ACOS (ACL2-COSINE THETA))))
)
:in-theory (enable-disable (interval-definition-theory)
(sine->=-0-=>-x-<=-pi
acl2-acos-exists)))
("Subgoal 1"
:use ((:instance sine-<-0-=>-x->-pi
(x theta))
(:instance acl2-acos-exists
(y (acl2-cosine (+ (* (acl2-pi) 2) (- theta)))))
(:instance cosine-is-1-1-on-domain
(x1 (+ (* (acl2-pi) 2) (- theta)))
(x2 (ACL2-ACOS (ACL2-COSINE (+ (* (acl2-pi) 2) (- theta)))))))
:in-theory (enable-disable (interval-definition-theory)
(sine-<-0-=>-x->-pi
acl2-acos-exists
COS-2PI-X
COS-2PI+X)))))
(encapsulate
()
(local
(defthm realpart-*-real
(implies (realp x)
(equal (realpart (* x y))
(* x (realpart y))))
:hints (("Goal"
:use ((:instance *-complex
(i x)
(j 0)
(r (realpart y))
(s (imagpart y)))
(:instance realpart-complex
(x (* (realpart y) x))
(y (* (imagpart y) x))))
:in-theory (disable *-complex
realpart-complex)))))
(local
(defthm *-real-complex
(implies (and (realp x)
(realp r)
(realp s))
(equal (* x (complex r s))
(complex (* r x) (* s x))))
:hints (("Goal"
:use ((:instance *-complex
(i x)
(j 0)))))))
(local
(defthm imagpart-*-real
(implies (realp x)
(equal (imagpart (* x y))
(* x (imagpart y))))
:hints (("Goal"
:use ((:instance *-complex
(i x)
(j 0)
(r (realpart y))
(s (imagpart y)))
(:instance IMAGPART-COMPLEX
(x (* (realpart y) x))
(y (* (imagpart y) x))))
:in-theory (disable *-complex
imagpart-complex)))))
(defthm anglepart-*-real
(implies (and (realp x)
(< 0 x))
(equal (anglepart (* x y))
(anglepart y)))))
(defthm acl2-ln-for-positive-product
(implies (and (realp x)
(realp y)
(< 0 x)
(< 0 y))
(equal (acl2-ln-for-positive (* x y))
(+ (acl2-ln-for-positive x)
(acl2-ln-for-positive y))))
:hints (("Goal"
:use ((:instance uniqueness-of-ln-for-positive
(x (+ (acl2-ln-for-positive x)
(acl2-ln-for-positive y)))
(y (* x y)))
(:instance exp-sum
(x (acl2-ln-for-positive x))
(y (acl2-ln-for-positive y)))
(:instance acl2-exp-ln-for-positive (y x))
(:instance acl2-exp-ln-for-positive (y y))
)
:in-theory (disable uniqueness-of-ln-for-positive exp-sum acl2-exp-ln-for-positive ACL2-LN-FOR-POSITIVE))))
(defthm ln-exp
(implies (and (acl2-numberp x)
(<= 0 (imagpart x))
(< (imagpart x) (* 2 (acl2-pi))))
(equal (acl2-ln (acl2-exp x))
x))
:hints (("Goal"
:use ((:instance radiuspart-e-i-theta (theta (imagpart x))))
:in-theory (disable anglepart
radiuspart
radiuspart-e-i-theta
e^ix-cos-i-sin))))
(defthm uniqueness-of-ln
(implies (and (acl2-numberp x)
(<= 0 (imagpart x))
(< (imagpart x) (* 2 (acl2-pi)))
(equal (acl2-exp x) y))
(equal (acl2-ln y) x)))
(in-theory (disable acl2-ln (acl2-ln)))
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