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|
; cert_param: (uses-acl2r)
(in-package "ACL2")
(include-book "norm")
(include-book "std/util/define-sk" :dir :system)
(encapsulate
nil
(defthm unit-vector-is-unit
(implies (and (not (zvecp vec)) (real-listp vec))
(equal (eu-norm (scalar-* (/ (eu-norm vec)) vec)) 1))
:hints (("GOAL" :do-not-induct t
:in-theory (disable sqrt-* sqrt-/)
:use ((:instance sqrt-/ (x (norm^2 vec)))
(:instance norm-is-real)
(:instance zvecp-iff-zp-vec)))))
;; || u - v ||^2 = <u - v, u - v>
(local (defthm lemma-1
(implies (and (real-listp u) (real-listp v) (= (len u) (len v)))
(equal (norm^2 (vec-- u v))
(dot (vec-- u v) (vec-- u v))))
:hints (("GOAL" :do-not-induct t
:use ((:instance norm-inner-product-equivalence (vec (vec-- u v))))))))
;; || u - v ||^2 = <u, u-v> - <v, u-v>
(local (defthm lemma-2
(implies (and (real-listp u) (real-listp v) (= (len u) (len v)))
(equal (norm^2 (vec-- u v))
(+ (dot u (vec-- u v)) (- (dot v (vec-- u v))))))
:hints (("GOAL" :use ((:instance scalar-*-closure (a -1) (vec v))
(:instance dot-linear-first-coordinate-2
(vec1 u)
(vec2 (scalar-* -1 v))
(vec3 (vec-- u v))))))))
;; || u - v ||^2 = <u, u> - <u, v> - <v, u> + <v, v>
(local (defthm lemma-3
(implies (and (real-listp u) (real-listp v) (= (len u) (len v)))
(equal (norm^2 (vec-- u v))
(+ (dot u u) (- (dot u v)) (- (dot v u)) (dot v v))))
:hints (("GOAL" :use ((:instance scalar-*-closure (a -1) (vec v))
(:instance dot-linear-second-coordinate-2
(vec1 v)
(vec2 u)
(vec3 (scalar-* -1 v)))
(:instance dot-linear-second-coordinate-2
(vec1 u)
(vec2 u)
(vec3 (scalar-* -1 v))))))))
;; < u - v, u - v > = < u, u > - 2 < u , v > + < v, v >
(local (defthm lemma-4
(implies (and (real-listp u) (real-listp v) (= (len u) (len v)))
(equal (norm^2 (vec-- u v))
(+ (dot u u) (- (* 2 (dot u v))) (dot v v))))
:hints (("GOAL" :use ((:instance dot-commutativity (vec1 u) (vec2 v)))))))
;; 0 <= < u, u > - 2 < u , v > + < v, v >
(local (defthm lemma-6
(implies (and (real-listp u) (real-listp v) (= (len u) (len v)))
(equal (<= 0 (norm^2 (vec-- u v)))
(<= 0 (+ (dot u u) (- (* 2 (dot u v))) (dot v v)))))))
;; let v = (scalar-* (* (/ (dot v v)) (dot u v)) v)
(local (defthm lemma-7
(implies (and (real-listp u) (real-listp v) (= (len u) (len v)))
(equal (<= 0 (norm^2 (vec-- u
(scalar-* (* (/ (dot v v)) (dot u v)) v))))
(<= 0
(+ (dot u u)
(- (* 2 (dot u (scalar-* (* (/ (dot v v)) (dot u v)) v))))
(dot (scalar-* (* (/ (dot v v)) (dot u v)) v)
(scalar-* (* (/ (dot v v)) (dot u v)) v))))))
:hints (("GOAL" :use ((:instance scalar-*-closure (a (* (/ (dot v v)) (dot u v))) (vec v))
(:instance lemma-6 (v (scalar-* (* (/ (dot v v)) (dot u v)) v))))))))
;; 0 <= ||u||^2 - 2 <u, W> + (||v||^{-2})^2<u, v>^2||v||^2
(local (defthm lemma-8
(implies (and (real-listp u) (real-listp v) (= (len u) (len v)))
(equal (<= 0 (norm^2 (vec-- u
(scalar-* (* (/ (dot v v)) (dot u v)) v))))
(<= 0
(+ (dot u u)
(- (* (* 2 (* (/ (dot v v)) (dot u v)))
(dot u v)))
(* (/ (dot v v))
(dot u v)
(/ (dot v v))
(dot u v)
(dot v v))))))
:hints (("GOAL" :do-not-induct t
:use ((:instance scalar-*-closure (a (* (/ (dot v v)) (dot u v))) (vec v))
(:instance lemma-7))))))
;;
;; case when v is not zero
;;
(local (defthm lemma-9
(implies (and (realp x) (not (= x 0))
(realp y))
(equal (* (/ x) y (/ x) y x)
(* (/ x) y y)))))
(local (defthm lemma-10
(implies (and (real-listp v) (not (zvecp v)))
(and (not (equal (dot v v) 0))
(< 0 (dot v v))))
:hints (("GOAL" :use ((:instance norm-inner-product-equivalence (vec v))
(:instance zvecp-iff-zp-vec (vec v)))))))
;; 0 <= ||u||^2 - 2/||v||^2 <u, v>^2 + ||v||^{-2} <u, v>^2
(local (defthm lemma-11
(implies (and (real-listp u) (real-listp v) (= (len u) (len v)) (not (zvecp v)))
(equal (<= 0 (norm^2 (vec-- u
(scalar-* (* (/ (dot v v)) (dot u v)) v))))
(<= 0
(+ (dot u u)
(* -2 (* (/ (dot v v)) (dot u v) (dot u v)))
(* (/ (dot v v)) (dot u v) (dot u v))))))
:hints (("GOAL" :use ((:instance scalar-*-closure (a (* (/ (dot v v)) (dot u v))) (vec v))
(:instance lemma-9 (x (dot v v)) (y (dot u v)))
(:instance lemma-8))))))
;; ||v||^{-2}<u, v>^2 <= ||u||^2
(local (defthm lemma-13
(implies (and (real-listp u) (real-listp v) (= (len u) (len v)) (not (zvecp v)))
(equal (<= 0 (norm^2 (vec-- u
(scalar-* (* (/ (dot v v)) (dot u v)) v))))
(<= (* (/ (dot v v)) (dot u v) (dot u v))
(dot u u))))
:hints (("GOAL" :use ((:instance lemma-11))))))
(local (defthm lemma-14
(implies (and (realp x) (realp y) (realp z) (not (= x 0)) (< 0 x) (<= (* (/ x) y y) z))
(<= (* y y) (* z x)))))
(local (defthm lemma-15
(implies (and (real-listp u) (real-listp v) (= (len u) (len v)) (not (zvecp v)))
(equal (<= 0 (norm^2 (vec-- u
(scalar-* (* (/ (dot v v)) (dot u v)) v))))
(<= (* (dot u v) (dot u v))
(* (dot u u) (dot v v)))))
:hints (("GOAL" :do-not-induct t
:use ((:instance lemma-11)
(:instance lemma-13)
(:instance lemma-14 (x (dot v v)) (y (dot u v)) (z (dot u u))))))))
(defthm cauchy-schwarz-1
(implies (and (real-listp u) (real-listp v) (= (len u) (len v)))
(<= (* (dot u v) (dot u v))
(* (dot u u) (dot v v))))
:hints (("GOAL" :use ((:instance lemma-15))
:cases ((zvecp v)(not (zvecp v))))))
;;
;; Proving norm version of cauchy-schwarz by taking square roots
;;
(local (defthm lemma-16
(implies (and (real-listp u) (real-listp v) (= (len u) (len v)))
(and (equal (acl2-sqrt (* (dot u v) (dot u v)))
(abs (dot u v)))
(equal (acl2-sqrt (dot u u)) (eu-norm u))
(equal (acl2-sqrt (dot v v)) (eu-norm v))
(equal (acl2-sqrt (* (dot u u) (dot v v)))
(* (eu-norm u) (eu-norm v)))))
:hints (("GOAL" :use ((:instance norm-inner-product-equivalence (vec u))
(:instance norm-inner-product-equivalence (vec v)))))))
(local (defthm lemma-18
(implies (and (realp a) (realp b) (<= 0 a) (<= 0 b) (<= a b))
(<= (acl2-sqrt a) (acl2-sqrt b)))))
(defthm cauchy-schwarz-2
(implies (and (real-listp u) (real-listp v) (= (len u) (len v)))
(<= (abs (dot u v))
(* (eu-norm u) (eu-norm v))))
:hints (("GOAL" :do-not-induct t
:in-theory (disable eu-norm-*-eu-norm eu-norm abs)
:use ((:instance cauchy-schwarz-1)
(:instance norm-inner-product-equivalence (vec v))
(:instance norm-inner-product-equivalence (vec u))
(:instance lemma-18
(a (* (dot u v) (dot u v)))
(b (* (dot u u) (dot v v))))))))
;;
;; cs-2-implies-cs-1
;;
(local (defthm lemma-19
(implies (and (realp a) (realp b) (realp c) (= (abs a) (* b c)))
(= (* (abs a) (abs a)) (* (* b c) (* b c))))))
(local (defthm lemma-20
(implies (and (realp b) (realp c))
(= (* (* b c) (* b c)) (* (* b b) (* c c))))))
(local (defthm lemma-21
(implies (and (realp a) (realp b) (realp c) (= (abs a) (* b c)))
(= (* (abs a) (abs a)) (* (* b b) (* c c))))
:hints (("GOAL" :use ((:instance lemma-20))))))
;; |< u , v >| = || u || || v || implies
;; |< u , v >| |< u , v >| = < u , u > < v , v >
(local (defthm lemma-22
(implies (and (real-listp u) (real-listp v) (= (len u) (len v))
(= (abs (dot u v)) (* (eu-norm u) (eu-norm v))))
(= (* (abs (dot u v)) (abs (dot u v)))
(* (dot u u) (dot v v))))
:hints (("GOAL" :do-not-induct t
:in-theory (disable eu-norm eu-norm-*-eu-norm lemma-20 abs)
:use ((:instance lemma-21
(a (dot u v))
(b (eu-norm u))
(c (eu-norm v)))
(:instance dot-eu-norm (vec u))
(:instance dot-eu-norm (vec v)))))))
(local (defthm lemma-23
(implies (realp a)
(= (* (abs a) (abs a)) (* a a)))))
;; |< u , v >| = || u || || v || implies
;; |< u , v > < u , v >| = < u , u > < v , v >
(defthm cs-2-implies-cs-1
(implies (and (real-listp u) (real-listp v) (= (len u) (len v))
(= (abs (dot u v)) (* (eu-norm u) (eu-norm v))))
(= (* (dot u v) (dot u v))
(* (dot u u) (dot v v))))
:hints (("GOAL" :do-not-induct t
:in-theory (disable eu-norm eu-norm-*-eu-norm lemma-20 abs)
:use ((:instance lemma-22))))
:rule-classes nil)
)
;;
;; Now to show the conditions for equality
;;
(encapsulate
nil
;; \| u - v \|^2 = <u - v, u - v>
(local (defthm lemma-1
(implies (and (real-listp u) (real-listp v) (= (len u) (len v)))
(equal (norm^2 (vec-- u v))
(dot (vec-- u v) (vec-- u v))))
:hints (("GOAL" :use ((:instance norm-inner-product-equivalence (vec (vec-- u v))))))))
(local (defthm lemma-2
(implies (and (real-listp u) (real-listp v) (= (len u) (len v)))
(equal (norm^2 (vec-- u v))
(+ (dot u (vec-- u v)) (- (dot v (vec-- u v))))))
:hints (("GOAL" :use ((:instance scalar-*-closure (a -1) (vec v))
(:instance dot-linear-first-coordinate-2
(vec1 u)
(vec2 (scalar-* -1 v))
(vec3 (vec-- u v))))))))
(local (defthm lemma-3
(implies (and (real-listp u) (real-listp v) (= (len u) (len v)))
(equal (norm^2 (vec-- u v))
(+ (dot u u) (- (dot u v)) (- (dot v u)) (dot v v))))
:hints (("GOAL" :use ((:instance lemma-2)
(:instance scalar-*-closure (a -1) (vec v))
(:instance dot-linear-second-coordinate-2
(vec1 v)
(vec2 u)
(vec3 (scalar-* -1 v)))
(:instance dot-linear-second-coordinate-2
(vec1 u)
(vec2 u)
(vec3 (scalar-* -1 v))))))))
;; < u - v, u - v > = < u, u > - 2 < u , v > + < v, v >
(local (defthm lemma-4
(implies (and (real-listp u) (real-listp v) (= (len u) (len v)))
(equal (norm^2 (vec-- u v))
(+ (dot u u) (- (* 2 (dot u v))) (dot v v))))
:hints (("GOAL" :do-not-induct t
;:in-theory (disable <-+-negative-0-2)
:use ((:instance lemma-3)
(:instance dot-commutativity (vec1 u) (vec2 v)))))))
;; 0 <= \| u - v \|^2
(local (defthm lemma-5
(implies (and (real-listp u) (real-listp v) (= (len u) (len v)))
(<= 0 (norm^2 (vec-- u v))))))
;; 0 <= < u, u > - 2 < u , v > + < v, v >
(local (defthm lemma-6
(implies (and (real-listp u) (real-listp v) (= (len u) (len v)))
(equal (= 0 (norm^2 (vec-- u v)))
(= 0 (+ (dot u u) (- (* 2 (dot u v))) (dot v v)))))
:hints (("GOAL" :use ((:instance lemma-5)
(:instance lemma-4))))))
;; replacing v with (scalar-* (* (/ (dot v v)) (dot u v)) v)
(local (defthm lemma-7
(implies (and (real-listp u) (real-listp v) (= (len u) (len v)))
(equal (= 0 (norm^2 (vec-- u
(scalar-* (* (/ (dot v v)) (dot u v)) v))))
(= 0
(+ (dot u u)
(- (* 2 (dot u (scalar-* (* (/ (dot v v)) (dot u v)) v))))
(dot (scalar-* (* (/ (dot v v)) (dot u v)) v)
(scalar-* (* (/ (dot v v)) (dot u v)) v))))))
:hints (("GOAL" :do-not-induct t
:use ((:instance scalar-*-closure (a (* (/ (dot v v)) (dot u v))) (vec v))
(:instance lemma-6 (v (scalar-* (* (/ (dot v v)) (dot u v)) v))))))))
(local (defthm lemma-8
(implies (and (real-listp u) (real-listp v) (= (len u) (len v)))
(equal (= 0 (norm^2 (vec-- u
(scalar-* (* (/ (dot v v)) (dot u v)) v))))
(= 0
(+ (dot u u)
(- (* (* 2 (* (/ (dot v v)) (dot u v)))
(dot u v)))
(* (/ (dot v v))
(dot u v)
(/ (dot v v))
(dot u v)
(dot v v))))))
:hints (("GOAL" :do-not-induct t
:use ((:instance scalar-*-closure (a (* (/ (dot v v)) (dot u v))) (vec v))
(:instance lemma-7))))))
;;
;; case when v is not zero
;;
(local (defthm lemma-9
(implies (and (realp x) (not (= x 0))
(realp y))
(equal (* (/ x) y (/ x) y x)
(* (/ x) y y)))))
(local (defthm lemma-10
(implies (and (real-listp v) (not (zvecp v)))
(and (not (equal (dot v v) 0))
(< 0 (dot v v))))
:hints (("GOAL" :use ((:instance norm-inner-product-equivalence (vec v))
(:instance zvecp-iff-zp-vec (vec v)))))))
(local (defthm lemma-11
(implies (and (real-listp u) (real-listp v) (= (len u) (len v)) (not (zvecp v)))
(equal (= 0 (norm^2 (vec-- u
(scalar-* (* (/ (dot v v)) (dot u v)) v))))
(= 0
(+ (dot u u)
(* -2 (* (/ (dot v v)) (dot u v) (dot u v)))
(* (/ (dot v v)) (dot u v) (dot u v))))))
:hints (("GOAL" :do-not-induct t
:use ((:instance scalar-*-closure (a (* (/ (dot v v)) (dot u v))) (vec v))
(:instance lemma-9 (x (dot v v)) (y (dot u v)))
(:instance lemma-8))))))
(local (defthm lemma-12
(implies (and (real-listp u) (real-listp v) (= (len u) (len v)) (not (zvecp v)))
(equal (= 0 (norm^2 (vec-- u
(scalar-* (* (/ (dot v v)) (dot u v)) v))))
(= 0
(+ (dot u u)
(- (* (/ (dot v v)) (dot u v) (dot u v)))))))
:hints (("GOAL" :do-not-induct t
:use ((:instance lemma-11))
:in-theory (disable <-+-negative-0-2)))))
(local (defthm lemma-13
(implies (and (real-listp u) (real-listp v) (= (len u) (len v)) (not (zvecp v)))
(equal (= 0 (norm^2 (vec-- u
(scalar-* (* (/ (dot v v)) (dot u v)) v))))
(= (* (/ (dot v v)) (dot u v) (dot u v))
(dot u u))))
:hints (("GOAL" :do-not-induct t
:use ((:instance lemma-11))))))
(local (defthm lemma-14
(implies (and (realp x) (realp y) (realp z) (not (= x 0)))
(equal (= (* (/ x) y y) z)
(= (* y y) (* z x))))))
(local (defthm lemma-15
(implies (and (realp a) (realp b))
(= (* a b) (* b a)))
:rule-classes nil))
(local (defthm lemma-16
(implies (and (real-listp u) (real-listp v) (= (len u) (len v)) (not (zvecp v)))
(equal (= 0 (norm^2 (vec-- u
(scalar-* (* (/ (dot v v)) (dot u v)) v))))
(= (* (dot u v) (dot u v))
(* (dot u u) (dot v v)))))
:hints (("GOAL" :do-not-induct t
:in-theory (disable not commutativity-2-of-* commutativity-of-*)
:use ((:instance lemma-11)
(:instance lemma-13)
(:instance lemma-14 (x (dot v v)) (y (dot u v)) (z (dot u u))))))))
;; equality implies norm thing is zero
(local (defthm lemma-17
(implies (and (real-listp u) (real-listp v) (= (len u) (len v)) (not (zvecp v)))
(implies (= (* (dot u v) (dot u v))
(* (dot u u) (dot v v)))
(= 0 (norm^2 (vec-- u
(scalar-* (* (/ (dot v v)) (dot u v)) v))))))
:hints (("GOAL" :use ((:instance lemma-11)
(:instance lemma-13)
(:instance lemma-14 (x (dot v v)) (y (dot u v)) (z (dot u u))))))
:rule-classes nil))
;; equality implies the difference is zero
(local (defthm lemma-18
(implies (and (real-listp u) (real-listp v) (= (len u) (len v)) (not (zvecp v)))
(implies (= (* (dot u v) (dot u v))
(* (dot u u) (dot v v)))
(zvecp (vec-- u
(scalar-* (* (/ (dot v v)) (dot u v)) v)))))
:hints (("GOAL" :do-not-induct t
:use ((:instance lemma-17)
(:instance zvecp-iff-zp-vec
(vec (vec-- u (scalar-* (* (/ (dot v v)) (dot u v)) v)))))))))
;; equality means the two vectors are linear
(local (defthm lemma-19
(implies (and (real-listp u) (real-listp v) (= (len u) (len v)) (not (zvecp v))
(= (* (dot u v) (dot u v))
(* (dot u u) (dot v v))))
(equal u
(scalar-* (* (/ (dot v v)) (dot u v)) v)))
:rule-classes nil
:hints (("GOAL" :do-not-induct t
:use ((:instance scalar-*-closure (a (* (/ (dot v v)) (dot u v))) (vec v))
(:instance lemma-18)
(:instance vec---zero-inverse
(vec1 u)
(vec2 (scalar-* (* (/ (dot v v)) (dot u v)) v))))))))
(defun-sk linear-dependence-nz (u v)
(exists a
(equal u (scalar-* a v))))
(defun linear-dependence (u v)
(or (zvecp u) (zvecp v) (linear-dependence-nz u v)))
(local (defthm lemma-20
(implies (and (real-listp u) (real-listp v) (= (len u) (len v)) (not (zvecp v))
(= (* (dot u v) (dot u v))
(* (dot u u) (dot v v))))
(linear-dependence-nz u v))
:hints (("GOAL" :do-not-induct t
:cases (zvecp v)
:use ((:instance lemma-19))))))
;; CS1 equality implies v = 0 or u = av
(local (defthm lemma-21
(implies (and (real-listp u) (real-listp v) (= (len u) (len v))
(= (* (dot u v) (dot u v))
(* (dot u u) (dot v v))))
(or (zvecp v) (linear-dependence-nz u v)))
:rule-classes nil))
;; CS2 equality implies v = 0 or u = av
(local (defthm lemma-22
(implies (and (real-listp u) (real-listp v) (= (len u) (len v))
(= (abs (dot u v)) (* (eu-norm u) (eu-norm v))))
(or (zvecp v) (linear-dependence-nz u v)))
:hints (("GOAL" :use ((:instance cs-2-implies-cs-1))))
:rule-classes nil))
;; v = 0 implies CS1 equality
(local (defthm lemma-23
(implies (and (real-listp u) (real-listp v) (= (len u) (len v)) (zvecp v))
(= (* (dot u v) (dot u v))
(* (dot u u) (dot v v))))))
;; u = 0 implies CS1 equality
(local (defthm lemma-24
(implies (and (real-listp u) (real-listp v) (= (len u) (len v)) (zvecp u))
(= (* (dot u v) (dot u v))
(* (dot u u) (dot v v))))))
;; v = 0 implies CS2 equality
(local (defthm lemma-25
(implies (and (real-listp u) (real-listp v) (= (len u) (len v)) (zvecp v))
(= (abs (dot u v)) (* (eu-norm u) (eu-norm v))))
:hints (("GOAL" :use ((:instance eu-norm-zero-iff-zvecp (vec v)))))))
;; u = 0 implies CS2 equality
(local (defthm lemma-26
(implies (and (real-listp u) (real-listp v) (= (len u) (len v)) (zvecp u))
(= (abs (dot u v)) (* (eu-norm u) (eu-norm v))))
:hints (("GOAL" :use ((:instance eu-norm-zero-iff-zvecp (vec u)))))))
;; u = av implies CS1 equality
(local (defthm lemma-27
(implies (and (real-listp u) (real-listp v) (= (len u) (len v))
(linear-dependence-nz u v))
(= (* (dot u v) (dot u v))
(* (dot u u) (dot v v))))
:hints (("GOAL" ;:do-not-induct t
:use ((:instance scalar-*-unclosure (a (linear-dependence-nz-witness u v)))
(:instance dot-linear-first-coordinate-1
(a (linear-dependence-nz-witness u v))
(vec1 v)
(vec2 v)))
:cases ((realp (linear-dependence-nz-witness u v))))
("Subgoal 1''" :use ((:instance dot-linear-first-coordinate-1
(a (linear-dependence-nz-witness u v))
(vec1 v)
(vec2 v))
(:instance dot-linear-second-coordinate-1
(a (linear-dependence-nz-witness u v))
(vec2 v)
(vec1 (scalar-* (linear-dependence-nz-witness u v) v))))))))
(local (Defthm lemma-28
(implies (and (real-listp u) (real-listp v) (= (len u) (len v)) (realp a)
(equal u (scalar-* a v)))
(equal (eu-norm u)
(* (abs a) (eu-norm v))))
:hints (("GOAL" :use ((:instance eu-norm-scalar-* (vec v)))))))
(local (defthm lemma-29
(implies (and (realp a) (realp b) (<= 0 b))
(equal (abs (* a b))
(* (abs a) b)))))
(local (defthm lemma-30
(implies (and (real-listp u) (real-listp v) (= (len u) (len v)) (realp a)
(equal u (scalar-* a v)))
(equal (abs (dot u v))
(* (* (eu-norm v) (eu-norm v))
(abs a))))
:hints (("GOAL" :in-theory (disable abs eu-norm eu-norm-*-eu-norm)
:do-not-induct t
:use ((:instance dot-eu-norm (vec v))
(:instance dot-linear-first-coordinate-1
(a a)
(vec1 v)
(vec2 v)))))))
(local (defthm lemma-31
(implies (real-listp v)
(equal (norm^2 v)
(* (eu-norm v) (eu-norm v))))
:hints (("GOAL" :in-theory (disable abs eu-norm eu-norm-*-eu-norm)
:use ((:instance eu-norm-*-eu-norm (vec v)))
:do-not-induct t))))
(local (defthm lemma-32
(implies (and (real-listp u) (real-listp v) (= (len u) (len v))
(linear-dependence-nz u v))
(= (abs (dot u v)) (* (eu-norm u) (eu-norm v))))
:hints (("GOAL" :cases ((realp (linear-dependence-nz-witness u v)))
:in-theory (disable abs eu-norm eu-norm-*-eu-norm)
:use ((:instance scalar-*-unclosure (a (linear-dependence-nz-witness u v)))
(:instance eu-norm-zero-iff-zvecp (vec u)))))))
;; conditions for CS1 equality
(defthm cauchy-schwarz-3
(implies (and (real-listp u) (real-listp v) (= (len u) (len v)))
(equal (= (* (dot u v) (dot u v))
(* (dot u u) (dot v v)))
(or (zvecp u) (zvecp v) (linear-dependence-nz u v))))
:hints (("GOAL" :use ((:instance lemma-21)))))
; CS2 equality implies v = 0 or u = av
(defthm cauchy-schwarz-4
(implies (and (real-listp u) (real-listp v) (= (len u) (len v)))
(equal (= (abs (dot u v)) (* (eu-norm u) (eu-norm v)))
(or (zvecp u) (zvecp v) (linear-dependence-nz u v))))
:hints (("GOAL" :in-theory (disable abs lemma-31 eu-norm eu-norm-*-eu-norm lemma-27)
:do-not-induct t
:use ((:instance eu-norm-zero-iff-zvecp (vec u))
(:instance eu-norm-zero-iff-zvecp (vec v))
(:instance lemma-22)))))
)
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