Quelle fermats_little_theorem.prf
Sprache: Lisp
(fermats_little_theoremnil 3564153926"" (subtype-tcc) nil nil )nil booleans nil )nil booleans nil )NOT const-decl "[bool -> bool]" booleans nil )nil numbers nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"bool" reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )"bool" reals nil )nil integers nil )nil integers nil )nil naturalnumbers nil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil )"boolean" notequal nil )"bool" divides nil )"bool" primes "ints/" ))nil ))nil 3564153932"" (skeep)"" (case "divides(p,factorial(p))" )"1" (case "divides(p,(factorial(np)*factorial(p-np))*C(p,np))" )"1" (lemma "prime_divides_prod" )"1" (inst - "p" )"1" (assert )"1" (inst?)"1" (assert )"1" (lemma "prime_divides_prod" )"1" (hide -3)"1" (inst - "p" )"1" (assert )"1" (inst?)"1" (assert )"1" "prime_divides_factorial" )"1" "1" assert )"1" "prime_divides_factorial" )"1" "np" "p" )"1" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (hide 2)"2" (expand "C" 1) (("2" (assert ) nil nil )) nil )) nil )"3" (assert ) nil nil ))nil )"2" (lemma "prime_divides_factorial" )"2" (inst - "p" "p" ) (("2" (assert ) nil nil )) nil )) nil ))nil ))nil )"posnat" factorial "ints/" )nil integers nil )"bool" reals nil )nil integers nil )nil naturalnumbers nil )"bool" reals nil )"bool" divides nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )"posrat" nil )"posrat" nil )nil primes "ints/" )"(strict_total_order?[real])" real_props nil )nil primes "ints/" )"(total_order?[real])" nil )"int" integers nil )"posint" nil )"(divides(n))" divides nil )"(divides(m))" divides nil )nil number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )"bool" reals nil )"posnat" binomial "reals/" ))nil 3411733664 ("" (subtype-tcc) nil nil )"boolean" notequal nil )"bool" divides nil )"bool" primes "ints/" ))nil ))nil 3411733696"" (induct "a" )"1" (assert )"1" (skeep)"1" (expand "^" )"1" (expand "expt" )"1" (assert )"1" (expand "eq_mod" )"1" (inst + "0" ) (("1" (assert ) nil nil )) nil )) nil ))nil ))nil ))nil ))nil ))nil )"2" (skeep)"2" (name "A" "j" )"2" (replace -1)"2" (assert )"2" (skeep)"2" (inst - "p" )"2" (assert )"2" (hide -1)"2" (lemma "binomial_theorem" )"2" (inst?)"2" (replace -1)"2" (hide -1)"2" (invoke (name "FF" "%1" ) (! 1 1 3))"1" replace -1)"1" expand "eq_mod" )"1" case "EXISTS (GG:[nat->nat]): FORALL (ii:nat): 1<=ii AND ii<=p-1 IMPLIES p*GG(ii)=FF(ii)" )"1" "1" "sigma_split" )"1" "FF" _ _ _)"1" "p" "0" "0" )"1" assert )"1" replace -2)"1" expand "sigma" + 1)"1" expand "sigma" "1" expand "FF" + 1)"1" expand "^" "1" expand "expt" "1" "p" "1" "p-1" )"1" assert )"1" replace "1" "1" expand "sigma" "1" expand "sigma" "1" case "FF(p) = 1" )"1" replace "1" assert )"1" case "divides(p,sigma(1, p - 1, FF))" )"1" expand "divides" "1" "kp" )"1" replace "1" "rj" )"1" replace "1" case "C(p,0) = 1" )"1" replace "1" assert )"1" "kp+rj" )"1" assert )nil nil ))nil ))nil ))nil )"2" "2" expand "C" )"2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" "2" expand "divides" )"2" "sigma(1,p-1,GG)" )"2" rewrite "sigma_scal" "2" rewrite "sigma_eq" )"2" "2" "2" "n!1" )"2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"3" "3" case "FORALL (TJ:nat): EXISTS (kr:nat): kr = sigma[nat](1,TJ,FF)" )"1" "1" "1" assert )nil nil ))nil ))nil )"2" "2" case "FORALL (TJ:nat): EXISTS (kr:nat): kr = FF(TJ)" )"1" "1" "TJ" )"1" expand "sigma" "1" nil nil ))nil )"2" "2" assert )"2" "2" expand "sigma" "2" replace "2" "1+j!1" )"2" "2" replace "2" "kr!1+kr" )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" "2" "2" expand "FF" )"2" assert )"2" case "TJ>p" )"1" assert )"1" "0" )nil nil ))nil )"2" assert )"2" "A" )"2" case "A = 0" )"1" assert )"1" replace "1" case "TJ = p" )"1" replace "1" case "C(p,p) = 1" )"1" replace "1" assert )"1" "expt_1i" )"1" "1" replace "1" assert )"1" expand "^" )"1" expand "expt" )"1" "1" )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" expand "C" )"2" assert )nil nil ))nil ))nil ))nil ))nil )"2" assert )"2" "0" )"2" expand "^" "2" expand "expt" "2" "2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" assert )"2" rewrite "expt_1i" )"2" assert )"2" "int_times_int_is_int" )"2" "1" "C(p, TJ) * A ^ (p - TJ)" )"1" assert )nil nil ))nil )"2" assert )"2" "int_expt" )"2" expand "^" "2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" expand "FF" )"2" assert )"2" rewrite "expt_1i" )"2" rewrite "expt_x0" )"2" expand "C" )"2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" case "FORALL (ii:nat): 1 <= ii AND ii <= p - 1 IMPLIES EXISTS (kk:nat): p*kk = FF(ii)" )"1" "LAMBDA (ii:nat): choose({kk:nat | 1 <= ii AND ii <= p - 1 IMPLIES p*kk = FF(ii)})" )"1" assert )"1" "1" assert )"1" "PK" "%1" )"1" replace -1)"1" assert )nil nil ))nil )"2" "2" expand "nonempty?" )"2" expand "empty?" )"2" "ii!1" )"2" assert )"2" "2" "kk!1" )"2" expand "member" )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" "ii!1" )"2" assert )"2" expand "nonempty?" )"2" expand "empty?" )"2" case "%1" )"1" "1" assert )"1" "1" "kk!1" )"1" expand "member" )"1" nil nil ))nil ))nil ))nil ))nil ))nil )"2" assert )"2" "0" )"2" expand "member" )"2" replace "2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" "2" expand "FF" )"2" "expt_1i" )"2" rewrite -1)"2" "2" assert )"2" "prime_divides_choose" )"2" "ii" "p" )"2" assert )"2" expand "divides" )"2" "2" replace "2" "x!1*A^(p-ii)" )"1" assert )nil nil )"2" assert )"2" case "x!1 > 0" )"1" "posreal_times_posreal_is_posreal" )"1" "x!1" "A^(p-ii)" )"1" assert )"1" "int_times_int_is_int" )"1" "1" assert )"1" case "FORALL (krp:nat): integer_pred(A^krp)" )"1" nil nil )"2" "2" "2" "krp" )"1" nil nil )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" assert )nil nil )"3" assert )nil nil ))nil ))nil )"2" "nnreal_times_nnreal_is_nnreal" )"2" "p" "-x!1" )"1" assert )nil nil )"2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" (grind) nil nil ))nil )"3" "3" (grind) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"posint" exponentiation nil )"posint" nil )nil polynomials "reals/" )NOT const-decl "[bool -> bool]" booleans nil )"[numfield, numfield -> numfield]" number_fields nil )"posnat" binomial "reals/" )nil integers nil )nil integers nil )"bool" reals nil )"[numfield, numfield -> numfield]" number_fields nil )"bool" reals nil )IF const-decl "[boolean, T, T -> T]" if_def nil )nil number_fields nil )"nonneg_rat" nil )nil real_types nil )"[numfield -> numfield]" number_fields nil )"int" integers nil )nil real_types nil )"posnat" factorial "ints/" )"even_int" integersnil )"nzrat" rationalsnil )nil real_types nil )nil real_types nil )"nat" fermats_little_theorem nil )"int" fermats_little_theorem nil )"rat" rationals nil )nil fermats_little_theorem nil )set type-eq-decl nil sets nil )"bool" sets nil )"(p)" sets nil )"bool" sets nil )"bool" sets nil )"odd_int" integersnil )"(total_order?[real])" nil )"real" reals nil )"real" sigma "reals/" )"[nat -> rational]" fermats_little_theoremnil )"even_int" integersnil )nil exponentiation nil )"bool" divides nil )"[numfield, numfield -> numfield]" number_fields nil )"int" integers nil )"posrat" nil )nil factorial "ints/" )"odd_int" integers nil )"nat" sigma_nat "reals/" )"nnreal" sigma_nat "reals/" )nil sigma "reals/" )nil integers nil )nil sigma "reals/" )"nonneg_int" nil )"(strict_total_order?[real])" real_props nil )"nat" fermats_little_theorem nil )"nat" fermats_little_theorem nil )"nat" fermats_little_theorem nil )nil exponentiation nil )nil integers nil )nil exponentiation nil )"posnat" exponentiation nil )"nnrat" exponentiation nil )"posint" nil )nil sigma "reals/" )nil sigma "reals/" )"(total_order?[real])" nil )nil sigma "reals/" )"[T, T -> boolean]" equalities nil )"int" integers nil )"nat" exponentiation nil )"(divides(m))" divides nil )"(divides(n))" divides nil )"even_int" integersnil )"nonneg_int" nil )"(strict_total_order?[real])" real_props nil )"real" exponentiation nil )nil naturalnumbers nil )"real" exponentiation nil )"boolean" notequal nil )OR const-decl "[bool, bool -> bool]" booleans nil )"bool" eq_mod nil )"bool" reals nil )"bool" primes "ints/" )AND const-decl "[bool, bool -> bool]" booleans nil )"[bool, bool -> bool]" booleans nil )nil defined_types nil )nil naturalnumbers nil )"bool" reals nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )"nat" exponentiation nil ))nil 3564160286"" (subtype-tcc) nil nil )"boolean" notequal nil )"bool" divides nil )"bool" primes "ints/" ))nil ))nil 3564160294"" (lemma "Fermats_Little" )"" (skeep)"" (inst - "mod(k,p)" "p" )"1" (assert )"1" (typepred "mod(k,p)" )"1" (case "NOT mod(k,p)>=0" )"1" (hide-all-but 1)"1" (expand "mod" )"1" (both-sides "+" "p*floor(k/p)" 1)"1" (assert )"1" (typepred "floor(k/p)" )"1" (mult-by -1 "p" ) (("1" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (expand "abs" -2)"2" (assert )"2" (expand "eq_mod" )"2" (expand "mod" )"2" (name "K" "floor(k/p)" )"2" (replace -1)"2" (skolem -4 "D" )"2" (assert )"2" "binomial_theorem" )"2" "p" "k" "-p*K" )"2" replace -1)"2" "2" assert )"2" "FF" "%1" )"1" replace -1)"1" expand "sigma" )"1" expand "FF" -5 1)"1" case "(-p * K)^0 = 1" )"1" replace -1)"1" case "C(p,p) = 1" )"1" "1" assert )"1" case "EXISTS (G:int): p*G = sigma(0,p-1,FF)" )"1" "1" replace "1" "D-K-G!1" )"1" assert )nil nil ))nil ))nil ))nil )"2" "2" "KR" " (LAMBDA (i: nat): IF i >= p THEN 0 ELSE -C(p, i) * (-p)^(p-i-1) * K ^ (p - i) * k ^ i ENDIF)") "1" "sigma(0,p-1,KR)" )"1" rewrite "sigma_scal" "1" rewrite "sigma_eq" )"1" "1" "1" "n" )"1" expand "KR" "1" expand "FF" "1" "expt_plus" )"1" "-1-n+p" "1" "-p" )"1" assert )"1" case "(-p)^1 = -p" )"1" assert )"1" "1" "mult_expt" )"1" "p-n" "K" "-p" )"1" assert )nil nil )"2" assert )"2" "2" replace "2" assert )"2" expand "^" "2" expand "expt" "2" assert )"2" "2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" case "FORALL (jj:nat): jj<=p-1 IMPLIES rational_pred(sigma[nat](0, jj, KR)) AND ") "1" "1" assert )nil nil ))nil )"2" "2" "jj" )"1" "1" "1" expand "sigma" )"1" expand "sigma" )"1" assert )"1" expand "KR" )"1" assert )"1" rewrite "int_times_int_is_int" )"1" rewrite "int_times_int_is_int" )"1" rewrite "int_times_int_is_int" )"1" "int_expt" )"1" expand "^" )"1" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" "2" assert )"2" "2" expand "sigma" "2" assert )"2" "1" rewrite "rat_plus_rat_is_rat" )nil nil )"2" rewrite "int_plus_int_is_int" )"2" expand "KR" "2" expand "^" )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" assert )"2" "2" assert )nil nil ))nil ))nil )"3" "3" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" expand "C" )"2" assert )nil nil ))nil ))nil ))nil )"2" "2" nil nil ))nil ))nil ))nil ))nil ))nil )"2" "2" (assert ) nil nil ))nil )"3" "3" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (assert )"2" (flatten)"2" (expand "prime?" ) (("2" (propax) nil nil )) nil ))nil ))nil ))nil ))nil )"2" (assert )"2" (hide 2)"2" (expand "mod" )"2" (both-sides "+" "p*floor(k/p)" 1)"2" (assert )"2" (typepred "floor(k/p)" )"2" (mult-by -1 "p" ) (("2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"3" (expand "prime?" )"3" (flatten) (("3" (assert ) nil nil )) nil )) nil ))nil ))nil ))nil )"(strict_total_order?[real])" real_props nil )"int" exponentiation nil )"bool" primes "ints/" )"(divides(n))" divides nil )"(divides(m))" divides nil )nil extra_real_propsnil )nil real_props nil )nil reals nil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil )nil real_props nil )"bool" divides nil )"[numfield, nznum -> numfield]" number_fields nil )nil number_fields nil )"{i | i <= x & x < i + 1}" floor_ceil nil )"[numfield, numfield -> numfield]" number_fields nil )"bool" reals nil )nil integers nil )"[numfield, numfield -> numfield]" number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )"[T, T -> boolean]" equalities nil )"int" integers nil )"rat" rationals nil )"int" integers nil )"int" integers nil )"[bool, bool -> bool]" booleans nil )nil rationals nil )"real" exponentiation nil )OR const-decl "[bool, bool -> bool]" booleans nil )"posnat" binomial "reals/" )nil integers nil )nil integers nil )"bool" reals nil )IF const-decl "[boolean, T, T -> T]" if_def nil )"real" sigma "reals/" )"real" reals nil )"[nat -> rat]" fermats_little_theorem nil )nil sigma "reals/" )nil exponentiation nil )"odd_int" integers nil )"nzrat" rationalsnil )nil exponentiation nil )"even_int" integersnil )"int" exponentiation nil )"real" exponentiation nil )"nat" exponentiation nil )"even_int" integersnil )"nonneg_int" nil )"{i | i <= k / p & k / p < 1 + i}" nil )nil reals nil )nil integers nil )"nzint" integers nil )nil sigma "reals/" )"posint" nil )"posint" nil )nil rationals nil )nil integers nil )nil exponentiation nil )nil integers nil )nil naturalnumbers nil )nil defined_types nil )nil sigma "reals/" )nil sigma "reals/" )"posint" nil )nil factorial "ints/" )"posrat" nil )"rat" exponentiation nil )"[nat -> rat]" fermats_little_theorem nil )"rat" rationals nil )nil polynomials "reals/" )"bool" eq_mod nil )NOT const-decl "[bool -> bool]" booleans nil )nil integers nil )"bool" reals nil )nil real_types nil )AND const-decl "[bool, bool -> bool]" booleans nil )nil number_fields nil )"[numfield -> numfield]" number_fields nil )"{n: nonneg_real | n >= m AND n >= -m}" real_defsnil )"{k | abs(k) < abs(j)}" mod nil )"int" fermats_little_theorem nil )nil numbers nil )nil booleans nil )"boolean" notequal nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )nil booleans nil )"bool" reals nil )nil naturalnumbers nil )"nat" fermats_little_theorem nil )"(total_order?[real])" nil )nil fermats_little_theorem nil ))
Messung V0.5 in Prozent C=100 H=100 G=100
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2026-05-26
