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;; David M. Russinoff
;; david@russinoff.com
;; http://www.russinoff.com
(in-package "RTL")
(local (include-book "support/fermat"))
(set-enforce-redundancy t)
(set-inhibit-warnings "theory") ; avoid warning in the next event
(local (in-theory nil))
;; The proof depends on Euclid's Theorem:
(include-book "euclid")
(defsection fermat
:parents (number-theory)
:short "This book contains a proof of Fermat's Theorem: if @('p') is a prime and @('m')
is not divisible by @('p'), then @('mod(m^(p-1),p) = 1')."
:long "The proof depends on Euclid's Theorem:"
"We shall construct two lists of integers, each of which is a permutation of the other.
@(def perm)"
(defun perm (a b)
(if (consp a)
(if (member (car a) b)
(perm (cdr a) (remove1 (car a) b))
())
(not (consp b))))
"The first list is @({positives(p-1) = (1, 2, ..., p-1)})
@(def positives)"
(defun positives (n)
(if (zp n)
()
(cons n (positives (1- n)))))
"The second list is @({mod-prods(p-1,a,p) = (mod(a,p), mod(2*a,p), ..., mod((p-1)*a,p))})
@(def mod-prods)"
(defun mod-prods (n m p)
(if (zp n)
()
(cons (mod (* m n) p)
(mod-prods (1- n) m p))))
"The proof is based on the pigeonhole principle, as stated below.
@(thm not-member-remove1)
@(thm perm-member)
@(def distinct-positives)
@(thm member-positives)
@(thm pigeonhole-principle)"
(defthm not-member-remove1
(implies (not (member x l))
(not (member x (remove1 y l)))))
(defthm perm-member
(implies (and (perm a b)
(member x a))
(member x b)))
(defun distinct-positives (l n)
(if (consp l)
(and (not (zp n))
(not (zp (car l)))
(<= (car l) n)
(not (member (car l) (cdr l)))
(distinct-positives (cdr l) n))
t))
(defthm member-positives
(iff (member x (positives n))
(and (not (zp n))
(not (zp x))
(<= x n))))
(defthm pigeonhole-principle
(implies (distinct-positives l (len l))
(perm (positives (len l)) l))
:rule-classes ())
"We must show that @('mod-prods(p-1,m,p)') is a list of length @('p-1') of distinct
integers between @('1') and @('p-1').
@(thm len-mod-prods)
@(thm mod-distinct)
@(thm mod-p-bnds)
@(thm mod-prods-distinct-positives)
@(thm perm-mod-prods)"
(defthm len-mod-prods
(implies (natp n)
(equal (len (mod-prods n m p)) n)))
(defthm mod-distinct
(implies (and (primep p)
(not (zp i))
(< i p)
(not (zp j))
(< j p)
(not (= j i))
(integerp m)
(not (divides p m)))
(not (equal (mod (* m i) p) (mod (* m j) p)))))
(defthm mod-p-bnds
(implies (and (primep p)
(not (zp i))
(< i p)
(integerp m)
(not (divides p m)))
(and (< 0 (mod (* m i) p))
(> p (mod (* m i) p))))
:rule-classes ())
(defthm mod-prods-distinct-positives
(implies (and (primep p)
(natp n)
(< n p)
(integerp m)
(not (divides p m)))
(distinct-positives (mod-prods n m p) (1- p)))
:rule-classes ())
(defthm perm-mod-prods
(implies (and (primep p)
(integerp m)
(not (divides p m)))
(perm (positives (1- p))
(mod-prods (1- p) m p)))
:rule-classes ())
"The product of the members of a list is invariant under permutation:
@(def times-list)
@(thm perm-times-list)"
(defun times-list (l)
(if (consp l)
(* (ifix (car l))
(times-list (cdr l)))
1))
(defthm perm-times-list
(implies (perm l1 l2)
(equal (times-list l1)
(times-list l2)))
:rule-classes ())
"It follows that the product of the members of @('mod-prods(p-1,m,p)') is @('(p-1)!').
@(thm times-list-positives)
@(thm times-list-equal-fact)
@(thm times-list-mod-prods)"
(defthm times-list-positives
(equal (times-list (positives n))
(fact n)))
(defthm times-list-equal-fact
(implies (perm (positives n) l)
(equal (times-list l) (fact n))))
(defthm times-list-mod-prods
(implies (and (primep p)
(integerp m)
(not (divides p m)))
(equal (times-list (mod-prods (1- p) m p))
(fact (1- p)))))
"On the other hand, the product modulo @('p') may be computed as follows.
@(thm mod-mod-prods)"
(defthm mod-mod-prods
(implies (and (not (zp p))
(integerp m)
(natp n))
(equal (mod (times-list (mod-prods n m p)) p)
(mod (* (fact n) (expt m n)) p)))
:rule-classes ())
"Fermat's theorem now follows easily.
@(thm not-divides-p-fact)
@(thm mod-times-prime)
@(thm fermat)"
(defthm not-divides-p-fact
(implies (and (primep p)
(natp n)
(< n p))
(not (divides p (fact n))))
:rule-classes ())
(defthm mod-times-prime
(implies (and (primep p)
(integerp a)
(integerp b)
(integerp c)
(not (divides p a))
(= (mod (* a b) p) (mod (* a c) p)))
(= (mod b p) (mod c p)))
:rule-classes ())
(defthm fermat
(implies (and (primep p)
(integerp m)
(not (divides p m)))
(equal (mod (expt m (1- p)) p)
1))
:rule-classes ())
)
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