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; Copyright (C) 2014, Regents of the University of Texas
; Written by J Moore, 6/13/01
; License: A 3-clause BSD license. See the LICENSE file distributed with ACL2.
(in-package "ACL2")
; J Moore, 6/13/01
(include-book "dft")
(dft comm2-test-1
(equal (* a (* b c)) (* b (* a c)))
:rule-classes nil
:otf-flg nil
:proof
((consider (* a (* b c)))
(= (* (* a b) c))
(= (* (* b a) c) :disable (associativity-of-*))
(= (* b (* a c)))))
(include-book "arithmetic/top-with-meta" :dir :system)
; Here I prove Euclid's theorem, that p|ab implies p|a or p|b, for prime p. I
; defaxiom a few "basic" facts. My point is to illustrate dft.
(progn
(defstub primep (x) t)
(defstub divides (x y) t)
(defstub quotient (x y) t)
(defstub my-gcd (x y) t)
(defaxiom fact0
(implies (and (integerp x)
(integerp y))
(integerp (quotient x y))))
(defaxiom fact1
(implies (and (integerp x)
(integerp y))
(integerp (my-gcd x y))))
(defaxiom fact2
(implies (and (integerp x)
(integerp y)
(divides x y))
(equal (* x (quotient y x)) y)))
(defaxiom fact3
(implies (and (integerp x)
(integerp y)
(primep x)
(not (divides x y)))
(equal (my-gcd x y) 1)))
(defaxiom fact4
(implies (and (integerp x)
(integerp y)
(integerp z))
(equal (my-gcd (* x y) (* x z))
(* x (my-gcd y z)))))
(defaxiom fact5
(implies (and (integerp x)
(integerp y))
(divides x (* x y))))
(dft prime-key
(implies (and (integerp a)
(integerp b)
(integerp p)
(primep p)
(divides p (* a b)))
(or (divides p a)
(divides p b)))
:rule-classes nil
:Proof
((Observe (equal (* p (quotient (* a b) p)) (* a b)))
(Generalize (quotient (* a b) p) to i where (integerp i))
(case (not (divides p a))
(observe (equal 1 (my-gcd p a)))
(consider b)
(= (* b (my-gcd p a)))
(= (my-gcd (* b p) (* b a)) :by fact4)
(= (my-gcd (* p b) (* p i)))
(= (* p (my-gcd b i)) :by fact4)
(so-it-suffices-to-prove
(implies (and (integerp a)
(integerp b)
(integerp p)
(integerp i))
(divides p (* p (my-gcd b i)))))
(observe (divides p (* p (my-gcd b i))))))))
; This is a theorem similar to one I had a hard time proving during the
; K5 FDIV proof.
(dft abs-chain-proof-1
(implies (and (rationalp x)
(rationalp y)
(rationalp c)
(<= c (+ x y))
(integerp i))
(<= (* (expt 2 i) c)
(abs (+ (* (expt 2 i) x) (* (expt 2 i) y)))))
:rule-classes nil
:proof
((let e be (expt 2 i))
(consider (* e c))
(<= (* e (+ x y)))
(= (+ (* e x) (* e y)))
(<= (abs (+ (* e x) (* e y))) :enable abs)))
(dft abs-chain-proof-2
(implies (and (rationalp x)
(rationalp y)
(rationalp c)
(<= c (+ x y))
(integerp i))
(<= (* (expt 2 i) c)
(abs (+ (* (expt 2 i) x) (* (expt 2 i) y)))))
:rule-classes nil
:proof
((let e be (expt 2 i))
(Observe (<= c (abs (+ x y))) :enable abs)
(Consider (abs (+ (* e x) (* e y))))
(= (abs (* e (+ x y))))
(= (* e (abs (+ x y)))
:proof
((observe (rationalp e))
(observe (equal (abs e) e))
(Theorem (implies (and (rationalp x)
(rationalp y))
(equal (abs (* x y)) (* (abs x) (abs y))))
:enable abs)
(Instantiate (x e) (y (+ x y)))))
(Observe (<= (* e c)
(abs (+ (* e x) (* e y)))))))
(dft abs-chain-proof-3
(implies (and (rationalp x)
(rationalp y)
(rationalp c)
(<= c (+ x y))
(integerp i))
(<= (* (expt 2 i) c)
(abs (+ (* (expt 2 i) x) (* (expt 2 i) y)))))
:rule-classes nil
:proof
((let e be (expt 2 i))
(let rhs be (abs (+ (* (expt 2 i) x) (* (expt 2 i) y))))
(Observe (<= c (abs (+ x y))) :enable abs)
(Consider rhs)
(= (abs (* e (+ x y))))
(= (* e (abs (+ x y)))
:proof
((generalize (+ x y) to z where (rationalp z))
(observe (equal (abs (* e z)) (* e (abs z))) :enable abs)))
(observe (<= (* e c) rhs))))
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