Impressum bernstein_polynomials.prf
Interaktion und
(bernstein_polynomialsnil 3479200111 ("" (subtype-tcc) nil nil ) nil nil ))nil 3479200111 ("" (subtype-tcc) nil nil ) nil nil ))nil 3479200111 ("" (subtype-tcc) nil nil )"boolean" notequal nil )) nil ))nil 3479200335 ("" (subtype-tcc) nil nil )"boolean" notequal nil )) nil ))nil 3479206792 ("" (subtype-tcc) nil nil ) nil nil ))nil 3479206793"" (skeep)"" (expand "Bern" )"" (expand "C" )"" (expand "^" 1 1)"" (expand "expt" ) (("" (assert ) nil nil )) nil )) nil ))nil ))nil ))nil )"real" bernstein_polynomials nil )"real" exponentiation nil )nil factorial "ints/" )"posrat" nil )"real" reals nil )"real" exponentiation nil )"posnat" binomial nil )"(divides(m))" divides nil )"(divides(n))" divides nil )"posint" nil ))nil 3479206860 ("" (subtype-tcc) nil nil ) nil nil ))nil 3479206860 ("" (subtype-tcc) nil nil )"boolean" notequal nil )) nil ))nil 3479206860"" (skeep)"" (expand "Bern" )"" (expand "C" )"" (expand "^" 1 2)"" (expand "expt" ) (("" (assert ) nil nil )) nil )) nil ))nil ))nil ))nil )"real" bernstein_polynomials nil )"real" exponentiation nil )nil factorial "ints/" )"posrat" nil )"real" reals nil )"real" reals nil )"real" exponentiation nil )"posnat" binomial nil )"(divides(m))" divides nil )"(divides(n))" divides nil )"posint" nil ))nil 3480419658"" (skeep)"" (expand "Bern" )"" (case "0^i = 0" )"1" (assert ) nil nil )"2" (expand "^" 1)"2" (expand "expt" ) (("2" (assert ) nil nil )) nil )) nil ))nil ))nil ))nil )"real" bernstein_polynomials nil )"nat" exponentiation nil )"even_int" integersnil )"real" exponentiation nil )"nnrat" exponentiation nil )"(divides(m))" divides nil )"(divides(n))" divides nil )"nonneg_int" nil )"nonneg_rat" nil )"int" integers nil )"nat" exponentiation nil )nil numbers nil )nil booleans nil )"[T, T -> boolean]" equalities nil )nil reals nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )nil booleans nil )OR const-decl "[bool, bool -> bool]" booleans nil )"boolean" notequal nil )"bool" reals nil )"real" exponentiation nil )nil naturalnumbers nil ))nil 3480419912"" (skeep)"" (expand "Bern" )"" (case "0^(nn-i) = 0" )"1" (assert ) nil nil )"2" (expand "^" 1)"2" (expand "expt" 1) (("2" (assert ) nil nil )) nil )) nil ))nil ))nil ))nil )"real" bernstein_polynomials nil )"int" integers nil )"nat" exponentiation nil )"even_int" integersnil )"real" exponentiation nil )"posint" exponentiation nil )"(divides(m))" divides nil )"(divides(n))" divides nil )"nonneg_int" nil )"(total_order?[real])" nil )"int" integers nil )"nat" exponentiation nil )nil numbers nil )nil booleans nil )"[T, T -> boolean]" equalities nil )nil reals nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )nil booleans nil )OR const-decl "[bool, bool -> bool]" booleans nil )"boolean" notequal nil )"bool" reals nil )"real" exponentiation nil )nil number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )"bool" reals nil )nil naturalnumbers nil )nil integers nil ))nil 3479653703 ("" (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 )nil naturalnumbers nil )"bool" reals nil )nil number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )nil integers nil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil )"int" integers nil )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil ))nil ))nil 3479653703 ("" (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 )nil naturalnumbers nil )"bool" reals nil )nil number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )nil integers nil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil )"int" integers nil )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil ))nil ))nil 3479653689"" (skeep)"" (decompose-equality)"1" (expand "Bern" )"1" (lemma "neg_power_polynomial" )"1" case "FORALL (n: nat): LAMBDA (x: real): (1 - x) ^ n) = polynomial(neg_power_fs(n), n)") "1" (hide -2)"1" (case "nn>=i" )"1" (inst - "nn-i" )"1" (decompose-equality)"1" (inst - "x!1" )"1" (replace -1)"1" (hide -1)"1" (expand "polynomial" )"1" (rewrite "sigma_scal" :dir rl)"1" "Bfun" "LAMBDA (i_1: nat): IF i_1 = 0 THEN 1 ELSE x!1 ^ i_1 ENDIF)") "1" replace -1)"1" "Afun" "LAMBDA (i_1: nat): IF i_1 < i OR i_1 > nn THEN 0IF i_1 = 0 THEN 1 ELSE x!1 ^ i_1 ENDIF)") "1" replace -1)"1" "sigma_shift_T2" )"1" "lambda (iz:nat):IF iz < i THEN 0 ELSE Bfun(iz-i) ENDIF" "nn-i" "0" "i" )"1" assert )"1" case "(LAMBDA (i_1: nat): Bfun(i_1)) = Bfun" )"1" replace -1)"1" "1" replace "1" "1" "sigma_split" )"1" "Afun" "nn" "0" "i-1" )"1" assert )"1" replace "1" "1" "Bfun2" "LAMBDA (iz: nat): IF iz < i THEN 0 ELSE Bfun(iz - i) ENDIF" )"1" replace "1" case "sigma(0,i-1,Afun) = 0" )"1" replace "1" "sigma_restrict_eq" )"1" "Bfun2" "Afun" "nn" "i" )"1" assert )"1" "1" expand "restrict" )"1" "1" "1" "1" replace "1" replace "1" replace "1" assert )"1" "1" "1" assert )"1" "1" replace "1" case "i = 0" )"1" replace "1" assert )"1" expand "neg_power_fs" )"1" expand "C" )"1" assert )"1" expand "^" )"1" expand "expt" )"1" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" assert )nil nil ))nil ))nil )"2" replace "2" case "x!2 = i" )"1" replace "1" expand "neg_power_fs" )"1" expand "C" )"1" assert )nil nil ))nil ))nil ))nil )"2" assert )nil nil ))nil ))nil )"3" expand "neg_power_fs" )"3" expand "C" )"3" "3" "expt_plus" )"3" "x!2-i" "i" "x!1" )"1" assert )nil nil )"2" "2" replace "2" expand "^" )"2" expand "expt" )"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 ))nil ))nil )"2" "sigma_zero" )"2" "i-1" "0" )"2" "sigma_restrict_eq" )"2" "(LAMBDA (i: nat): 0)" "Afun" "i-1" "0" )"2" assert )"2" "2" "2" expand "restrict" )"2" "2" "2" replace "2" assert )"2" "2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" nil nil ))nil ))nil )"2" "2" (assert ) 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 )"2" (assert ) nil nil ))nil ))nil )"2" (hide 2)"2" (skeep)"2" (case "n>0" )"1" (inst - "n" ) nil nil )"2" (case "n = 0" )"1" (replace -1)"1" (hide-all-but 2)"1" (decompose-equality +)"1" (expand "^" )"1" (expand "expt" )"1" expand "polynomial" )"1" expand "sigma" )"1" assert )"1" expand "neg_power_fs" )"1" expand "C" )"1" assert )"1" expand "^" )"1" expand "expt" )"1" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (skeep) (("2" (assert ) nil nil )) nil )"3" (skeep) (("3" (assert ) nil nil )) nil ))nil ))nil )"(total_order?[real])" nil )"nzrat" rationalsnil )"odd_int" integers nil )"int" integers nil )"bool" reals nil )"above(i - 1)" bernstein_polynomials nil )nil integers nil )"[numfield, numfield -> numfield]" number_fields nil )nil number_fields nil )"bool" reals nil )"nat" bernstein_polynomials nil )"bool" reals nil )NOT const-decl "[bool -> bool]" booleans nil )AND const-decl "[bool, bool -> bool]" booleans nil )"[bool, bool -> bool]" booleans nil )nil naturalnumbers nil )"bool" reals nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )"real" bernstein_polynomials nil )nil sequences nil )"[real -> real]" polynomials nil )IF const-decl "[boolean, T, T -> T]" if_def nil )OR const-decl "[bool, bool -> bool]" booleans nil )"[numfield, numfield -> numfield]" number_fields nil )"boolean" notequal nil )"real" exponentiation nil )"[numfield -> numfield]" number_fields nil )nil integers nil )nil integers nil )"posnat" binomial nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )nil polynomials nil )"nat" bernstein_polynomials nil )nil sigma nil )nil sigma nil )nil sigma nil )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil )"real" sigma nil )nil sigma nil )"real" reals nil )"[T -> real]" sigma nil )"real" bernstein_polynomials nil )nil reals nil )"nonneg_rat" nil )nil exponentiation nil )nil factorial "ints/" )"nzint" integersnil )"real" exponentiation nil )"posrat" nil )"nzreal" exponentiation nil )"int" exponentiation nil )"posrat" nil )"even_int" integersnil )"int" exponentiation nil )nil sigma_nat nil )"(strict_total_order?[real])" real_props nil )"nonneg_int" nil )"even_int" integersnil )"nnreal" sigma_nat nil )"nat" sigma_nat nil )nil sigma nil )"odd_int" integersnil )nil sigma nil )"nonneg_int" nil )"int" integers nil )nil sigma nil )"nzreal" exponentiation nil )"rat" exponentiation nil )"(divides(m))" divides nil )"(divides(n))" divides nil )"posint" nil )"real" reals nil )"[nat -> int]" polynomials nil )"[T, T -> boolean]" equalities nil )"real" reals nil ))nil ))nil 3479211177 ("" (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 )"int" integers nil )"(strict_total_order?[real])" real_props nil ))nil ))nil 3479211177 ("" (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 )"int" integers nil )"(strict_total_order?[real])" real_props nil ))nil ))nil 3479211177 ("" (subtype-tcc) nil nil )nil nil ))nil 3479211177 ("" (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 )"int" integers nil )"(strict_total_order?[real])" real_props nil ))nil ))nil 3479211177"" (skeep)"" (case "C(k,r) = C(k-1,r) + C(k-1,r-1)" )"1" (mult-by -1 "x^r*(1-x)^(k-r)" )"1" (expand "Bern" )"1" (case "x^r = x^(r-1)*x" )"1" (case "(1-x)^(k-r) = (1-x)*(1-x)^(k-r-1)" )"1" (assert ) nil nil )"2" (expand "^" 1)"2" (assert )"2" (expand "expt" 1 1) (("2" (assert ) nil nil ))nil ))nil ))nil )"3" (assert ) nil nil ))nil )"2" (expand "^" 1)"2" (expand "expt" 1 1) (("2" (assert ) nil nil )) nil ))nil ))nil ))nil ))nil )"2" (hide 2)"2" (lemma "C_n_plus_1" )"2" (inst - "k-1" "r" ) (("2" (assert ) nil nil )) nil )) nil ))nil )"3" (assert ) nil nil ) ("4" (assert ) nil nil )"5" (assert ) nil nil ))nil ))nil )"[numfield, numfield -> numfield]" number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )nil number_fields nil )"posnat" binomial nil )nil integers nil )"bool" reals nil )nil integers nil )"bool" reals nil ) (>= const-decl "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 naturalnumbers nil )"[T, T -> boolean]" equalities nil )nil booleans nil )nil numbers nil )"int" integers nil )"posint" nil )"(total_order?[real])" nil )"(total_order?[real])" nil )"real" bernstein_polynomials nil )"int" integers nil )"real" exponentiation nil )"real" reals nil )"(strict_total_order?[real])" real_props nil )"real" reals nil )OR const-decl "[bool, bool -> bool]" booleans nil )"boolean" notequal nil )"real" bernstein_polynomials nil )"posnat" bernstein_polynomials nil )"posnat" bernstein_polynomials nil )"real" exponentiation nil )"[numfield, numfield -> numfield]" number_fields nil )nil extra_real_props nil )"real" reals nil )nil binomial nil ))nil 3479211589"" (skeep)"" (skeep)"" (expand "Bern" )"" (typepred "C(nn,i)" )"" (lemma "nnreal_expt" )"" (inst-cp - "i" "x" )"1" (inst - "nn-i" "1-x" )"1" (expand "^" )"1" (assert )"1" (lemma "nnreal_times_nnreal_is_nnreal" )"1" (inst?)"1" (lemma "nnreal_times_nnreal_is_nnreal" )"1" (inst?) nil nil )) nil ))nil ))nil ))nil ))nil )"2" (assert ) nil nil ))nil )"2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil )nil integers nil )"[numfield, numfield -> numfield]" number_fields nil )nil number_fields nil )"posnat" binomial nil )nil integers nil )nil integers nil )"bool" reals nil ) (>= const-decl "bool" reals nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil naturalnumbers nil )"bool" reals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil numbers nil )NOT const-decl "[bool -> bool]" booleans nil )nil booleans nil )nil booleans nil )"int" integers nil )"(total_order?[real])" nil )"real" bernstein_polynomials nil )nil real_types nil )"real" exponentiation nil )nil real_types nil )"above(i - 1)" bernstein_polynomials nil )"nat" bernstein_polynomials nil )nil real_types nil )"[numfield, numfield -> numfield]" number_fields nil )"real" exponentiation nil )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil )"real" reals nil )"real" reals nil )nil exponentiation nil )"real" bernstein_polynomials nil ))nil 3479211969"" (skeep)"" (skeep)"" (expand "Bern" )"" (typepred "C(nn,i)" )"" (lemma "posreal_expt" )"" (inst-cp - "i" "x" )"1" (inst - "nn-i" "1-x" )"1" (expand "^" )"1" (assert )"1" (lemma "posreal_times_posreal_is_posreal" )"1" (inst?)"1" case "C(nn, i) * expt((1 - x), (nn - i)) > 0" )"1" (assert ) nil nil )"2" (hide 2)"2" "posreal_times_posreal_is_posreal" )"2" (inst?) nil nil )) nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (assert ) nil nil ))nil )"2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil )nil integers nil )"[numfield, numfield -> numfield]" number_fields nil )nil number_fields nil )"posnat" binomial nil )nil integers nil )nil integers nil )"bool" reals nil ) (>= const-decl "bool" reals nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil naturalnumbers nil )"bool" reals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil numbers nil )NOT const-decl "[bool -> bool]" booleans nil )nil booleans nil )nil booleans nil )"int" integers nil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil )AND const-decl "[bool, bool -> bool]" booleans nil )"real" bernstein_polynomials nil )nil real_types nil )nil real_types nil )"real" exponentiation nil )nil real_types nil )"above(i - 1)" bernstein_polynomials nil )"nat" bernstein_polynomials nil )"[numfield, numfield -> numfield]" number_fields nil )"real" exponentiation nil )"(strict_total_order?[real])" real_props nil )"real" reals nil )"real" reals nil )nil exponentiation nil )"real" bernstein_polynomials nil ))nil ))nil 3479477346"" (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 )"int" integers nil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil ))nil ))nil 3479477346"" (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 )"int" integers nil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil ))nil ))nil 3479213227"" (skeep)"" case "C(k-1,p) = ((k-p)/k)*C(k,p) AND ((k-p)/k)*C(k,p) = ((p+1)/k)*C(k,p+1)" )"1" (flatten)"1" (mult-by -1 "x^p*(1-x)^(k-p-1)" )"1" (mult-by -2 "x^(p+1)*(1-x)^(k-p-1)" )"1" (expand "Bern" )"1" (case "(1-x)^(k-p) = (1-x)*(1-x)^(k-p-1)" )"1" (replace -1 1)"1" (hide -1)"1" (case "x^(p+1) = x*x^p" )"1" (assert ) nil nil )"2" (expand "^" 1)"2" (expand "expt" 1 1)"2" (propax) nil nil )) nil ))nil ))nil ))nil ))nil )"2" (expand "^" 1)"2" (assert )"2" (expand "expt" 1 1) (("2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (hide 2)"2" (split)"1" (expand "C" )"1" (expand "factorial" 1 4)"1" (expand "factorial" 1 6)"1" (assert ) (("1" (cross-mult 1) nil nil )) nil ))nil ))nil ))nil )"2" (expand "C" )"2" (expand "factorial" 1 3)"2" (assert )"2" (expand "factorial" 1 7)"2" (cross-mult 1) nil nil )) nil ))nil ))nil ))nil ))nil ))nil )"3" (assert ) nil nil ) ("4" (assert ) nil nil )"5" (assert ) nil nil ) ("6" (assert ) nil nil ))nil ))nil )"[numfield, numfield -> numfield]" number_fields nil )"[numfield, nznum -> numfield]" number_fields nil )nil number_fields nil )"boolean" notequal nil )"[numfield, numfield -> numfield]" number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )nil number_fields nil )"posnat" binomial nil )nil integers nil )"bool" reals nil )nil integers nil )"bool" reals nil ) (>= const-decl "bool" reals 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 naturalnumbers nil )"[T, T -> boolean]" equalities nil )nil numbers nil )AND const-decl "[bool, bool -> bool]" booleans nil )nil booleans nil )nil booleans nil )"posint" nil )"posrat" nil )"int" integers nil )"rat" rationals nil )"rat" rationals nil )"(total_order?[real])" nil )"(total_order?[real])" nil )"[bool, bool -> bool]" booleans nil )"real" reals nil )nil extra_real_props nil )"real" exponentiation nil )"posnat" bernstein_polynomials nil )"nat" bernstein_polynomials nil )"real" bernstein_polynomials nil )OR const-decl "[bool, bool -> bool]" booleans nil )"real" reals nil )"posrat" nil )"posint" nil )"(strict_total_order?[real])" real_props nil )"real" reals nil )"real" bernstein_polynomials nil )"int" integers nil )"real" exponentiation nil )"posnat" factorial "ints/" )"nzrat" rationalsnil )"posint" nil )nil real_props nil )nil real_props nil )nil real_props nil )nil real_props nil )nil reals nil )"(divides(m))" divides nil )"(divides(n))" divides nil )"nonneg_int" nil )"nonneg_rat" nil ))nil 3480405695 ("" (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 )nil naturalnumbers nil )"bool" reals nil )nil number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )nil naturalnumbers nil )"(total_order?[real])" nil )"posint" nil )"(total_order?[real])" nil ))nil ))nil 3480405695 ("" (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 )nil naturalnumbers nil )"bool" reals nil )nil number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )nil naturalnumbers nil )"int" integers nil )"(total_order?[real])" nil )"posint" nil )"(total_order?[real])" nil ))nil ))nil 3480405695 ("" (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 )nil naturalnumbers nil )"bool" reals nil )nil number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )nil naturalnumbers nil )"int" integers nil )"odd_int" integersnil )"(total_order?[real])" nil )"posint" nil )"(total_order?[real])" nil ))nil ))nil 3480405695 ("" (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 )nil naturalnumbers nil )"bool" reals nil )nil number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )nil naturalnumbers nil )"odd_int" integersnil )"(total_order?[real])" nil )"posint" nil )"(total_order?[real])" nil )"int" integers nil ))nil ))nil 3480410715 ("" (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 )nil naturalnumbers nil )"int" integers nil )"(total_order?[real])" nil ))nil ))nil 3480410715"" (termination-tcc) nil nil ) nil nil ))nil 3479472990"" (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 )"int" integers nil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil ))nil ))nil 3479476400"" (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 )"posint" nil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil ))nil ))"RWB - (field 1) on leaf removed to prevent infinite loop" "" (skeep)"" (case "k > p" )"1" (hide -2)"1" (lemma "Bernstein_Recursion" )"1" (inst - "k+1" "p+1" "x" )"1" (name "A" "1-x" )"1" (replace -1)"1" (assert )"1" case "A*Bern(1+p,k)(x) = (C(k,p+1)/C(k+1,p+1))*Bern(1+p,1+k)(x)" )"1" (replace -1)"1" (hide -1)"1" (expand "C" -2)"1" (expand "factorial" -2 4)"1" (expand "factorial" -2 7)"1" "const1" "Bern(p,k)(x)" )"1" replace -1)"1" "const2" "Bern(1+p,1+k)(x)" )"1" replace -1)"1" "const3" "factorial(k)" )"1" replace -1)"1" "const4" "factorial(1+p)" )"1" replace -1)"1" "const5" "factorial(-1-p+k)" )"1" replace -1)"1" "1" "1" "1" "1" "1" case "(const3 / (const4 * const5) / ") "1" replace "1" "1" "-" "((k-p)/(1+k))*const2" "1" assert )"1" replace "1" nil nil ))nil ))nil ))nil ))nil ))nil )"2" "2" "apz1" "(const3 / (const4 * const5) / ") "2" replace "2" "const4*const5" )"2" case "apz1*(const4*const5) = 1/((1 + k) / ") "1" replace "1" "1" "1" nil nil ))nil ))nil ))nil )"2" "2" "2" replace "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 ))nil ))nil ))nil ))nil ))nil )"2" (hide (-2 2))"2" (expand "Bern" )"2" (case "A*(1-x)^(k-p-1) = (1-x)^(k-p)" )"1" (assert ) nil nil )"2" (replace -1 :dir rl)"2" (expand "^" )"2" expand "expt" 1 3)"2" (propax) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (case "k = p" )"1" (replace -1)"1" (hide -1)"1" (hide -1)"1" (hide 1)"1" (lemma "Bern_top" )"1" (inst-cp - "p" "x" )"1" (inst - "p+1" "x" )"1" (replace -1)"1" (hide -1)"1" (replace -1)"1" (hide -1)"1" expand "^" )"1" expand "expt" 1 2)"1" (field) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (assert ) nil nil ))nil ))nil ))nil )nil naturalnumbers nil )nil integers nil )nil integers nil )"bool" reals nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )"bool" reals nil )nil booleans nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )nil bernstein_polynomials nil )"real" reals nil )"[T, T -> boolean]" equalities nil )"[numfield, numfield -> numfield]" number_fields nil )"(strict_total_order?[real])" real_props nil )OR const-decl "[bool, bool -> bool]" booleans nil )"real" exponentiation nil )"real" exponentiation nil )"int" integers nil )"posint" nil )"(divides(n))" divides nil )"(divides(m))" divides nil )"nonneg_int" nil )"nzrat" rationalsnil )"int" integers nil )"rat" rationals nil )"rat" rationals nil )nil integers nil )nil real_props nil )nil reals nil )nil real_props nil )"bool" functions nil )"(bijective?[T, T])" identity nil )"bool" booleans nil )"real" reals nil )nil real_props nil )NOT const-decl "[bool -> bool]" booleans nil )"real" reals nil )nil rationals nil )nil real_props nil )nil real_props nil )"rat" rationals nil )nil real_props nil )nil real_props nil )nil real_props nil )"rat" rationals nil )nil real_props nil )"nzrat" rationalsnil )"(total_order?[real])" nil )"[numfield -> numfield]" number_fields nil )"posnat" factorial "ints/" )"(total_order?[real])" nil )"[numfield, numfield -> numfield]" number_fields nil )nil integers nil )"real" bernstein_polynomials nil )"boolean" notequal nil )nil number_fields nil )"[numfield, nznum -> numfield]" number_fields nil )"bool" reals nil )"posnat" binomial nil )"posrat" nil )"real" reals nil )"odd_int" integers nil )nil number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )"posint" nil )"posint" nil )nil bernstein_polynomials nil )nil real_props nil ))nil 3479472990"" (skeep)"" (case "k > p" )"1" (hide -2)"1" (lemma "Bernstein_Recursion" )"1" (inst - "k+1" "p+1" "x" )"1" (name "A" "1-x" )"1" (replace -1)"1" (assert )"1" case "A*Bern(1+p,k)(x) = (C(k,p+1)/C(k+1,p+1))*Bern(1+p,1+k)(x)" )"1" (replace -1)"1" (hide -1)"1" (expand "C" -2)"1" (expand "factorial" -2 4)"1" (expand "factorial" -2 7)"1" "const1" "Bern(p,k)(x)" )"1" replace -1)"1" "const2" "Bern(1+p,1+k)(x)" )"1" replace -1)"1" "const3" "factorial(k)" )"1" replace -1)"1" "const4" "factorial(1+p)" )"1" replace -1)"1" "const5" "factorial(-1-p+k)" )"1" replace -1)"1" "1" "1" "1" "1" "1" case "(const3 / (const4 * const5) / ") "1" replace "1" "1" "-" "((k-p)/(1+k))*const2" "1" assert )"1" replace "1" nil nil ))nil ))nil ))nil ))nil ))nil )"2" "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 )"2" (hide (-2 2))"2" (expand "Bern" )"2" (case "A*(1-x)^(k-p-1) = (1-x)^(k-p)" )"1" (assert ) nil nil )"2" (replace -1 :dir rl)"2" (expand "^" )"2" expand "expt" 1 3)"2" (propax) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (case "k = p" )"1" (replace -1)"1" (hide -1)"1" (hide -1)"1" (hide 1)"1" (lemma "Bern_top" )"1" (inst-cp - "p" "x" )"1" (inst - "p+1" "x" )"1" (replace -1)"1" (hide -1)"1" (replace -1)"1" (hide -1)"1" expand "^" )"1" expand "expt" 1 2)"1" (field) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (assert ) nil nil ))nil ))nil ))nil )"posnat" factorial "ints/" )"posnat" binomial nil ))nil 3479214531"" (subtype-tcc) nil nil ) nil nil ))nil 3479214531"" (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 )nil naturalnumbers nil )"int" integers nil )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil ))nil ))nil 3479214531"" (skeep)"" (expand "Bern" )"" (lemma "binomial_theorem" )"" (inst - "p" "x" "(1-x)" )"" (replace -1 :dir rl)"" (hide -1)"" (assert ) (("" (rewrite "expt_1i" ) nil nil )) nil ))nil ))nil ))nil ))nil ))nil ))nil )"real" bernstein_polynomials nil )"real" reals nil )nil numbers nil )nil booleans 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 )nil number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )nil exponentiation nil )"real" reals nil )"real" reals nil )nil polynomials nil ))nil 3479546101"" (skeep)"" (skeep)"" (lemma "Bernstein_partition_of_unity" )"" (inst - "nn" "x" )"" "AA" "LAMBDA (i: nat): IF i > nn THEN 0 ELSE Bern(i, nn)(x) ENDIF" )"1" (case "FORALL (ijk:nat): AA(ijk)>=0" )"1" (replace -2)"1" (lemma "sigma_split" )"1" (inst - "AA" "nn" "0" "i-1" )"1" (assert )"1" (replace -1)"1" (hide -1)"1" (lemma "sigma_split" )"1" (inst - "AA" "nn" "i" "i" )"1" assert )"1" replace -1)"1" "1" "Ann" -1)"1" "Ann" )"1" expand "sigma" -2 3)"1" "sigma_le" )"1" "lambda (ij: nat): 0" "AA" "i-1" "0" )"1" "1" "sigma_le" )"1" "lambda (ij: nat): 0" "AA" "nn" "1+i" )"1" "1" "sigma_zero" )"1" "nn" "1+i" )"1" "sigma_zero" )"1" "i-1" "0" )"1" expand "sigma" "1" replace "1" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "Ann" )"2" "2" "n!1" )"2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "Ann" )"2" "2" "n!1" )"2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (skeep)"2" (replace -1 :dir rl)"2" (assert )"2" (lift-if)"2" (ground)"2" (lemma "Bern_nonnegative" )"2" (inst - "x" )"2" (assert )"2" (inst - "ijk" "nn" ) nil nil )) nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (assert )"2" (skosimp*) (("2" (assert ) nil nil )) nil )) nil ))nil ))nil ))nil ))nil ))nil )"(total_order?[real])" nil )"int" integers nil )nil numbers nil )nil booleans 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 )"bool" reals nil )nil number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )nil integers nil )nil sigma nil )"odd_int" integersnil )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil )"real" sigma nil )"nnreal" sigma_nat nil )"nat" sigma_nat nil )"(strict_total_order?[real])" real_props nil )nil sigma nil )AND const-decl "[bool, bool -> bool]" booleans nil )nil integers nil )"[numfield, numfield -> numfield]" number_fields nil )"posint" nil )nil sigma nil )"real" reals nil )OR const-decl "[bool, bool -> bool]" booleans nil )"bool" reals nil )nil sigma nil )nil sigma nil )nil bernstein_polynomials nil )"[bool, bool -> bool]" booleans nil )NOT const-decl "[bool -> bool]" booleans nil )"[T, T -> boolean]" equalities nil )IF const-decl "[boolean, T, T -> T]" if_def nil )"real" bernstein_polynomials nil )nil nil ))nil 3479457319"" (subtype-tcc) nil nil ) nil nil ))nil 3479457319"" (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 )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil ))nil ))nil 3479457319"" (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 )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil ))nil ))nil 3479487738"" (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 )"(strict_total_order?[real])" real_props nil )"int" integers nil ))nil ))nil 3479457319"" (skeep)"" case "FORALL (nn: nat): x ^ nn = LAMBDA (i: nat):IF i < nn OR i > k-p+nn THEN 0") "1" (inst - "p" ) (("1" (assert ) nil nil )) nil )"2" (hide 2)"2" (induct "nn" )"1" (expand "^" )"1" (expand "expt" )"1" (case "k = p" )"1" (replace -1)"1" (assert )"1" (expand "sigma" )"1" (assert )"1" (expand "C" )"1" (expand "Bern" )"1" (expand "^" )"1" expand "expt" )"1" expand "C" )"1" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (lemma "Bernstein_partition_of_unity" )"2" (inst - "k-p" "x" )"1" (lemma "sigma_restrict_eq" )"1" "LAMBDA (i: nat): IF i > k - p THEN 0" " LAMBDA (i: nat):IF i > k - p THEN 0 ELSE Bern(i, k - p)(x) ENDIF" "k-p" "0" )"1" (assert )"1" (hide 3)"1" (hide -1)"1" "1" expand "restrict" )"1" "1" "1" expand "C" )"1" (assert ) nil nil ))nil ))nil ))nil ))nil )"2" "2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil )"2" (skeep) (("2" (assert ) nil nil )) nil )"3" (skeep) (("3" (assert ) nil nil )) nil )"4" (skeep) (("4" (assert ) nil nil )) nil ))nil ))nil )"2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil )"2" (skeep)"2" "F" "LAMBDA (i: nat): IF i < j OR i > k - p + j THEN 0") "1" (replace -1)"1" "G" "LAMBDA (i: nat): IF i < j + 1 OR i > k - p + (j + 1) THEN 0") "1" (replace -1)"1" (mult-by -3 "x" )"1" (expand "^" )"1" (expand "expt" +)"1" (replace -1)"1" (hide -1)"1" "sigma_scal" )"1" "F" "x" "j-p+k" "j" )"1" replace -1 :dir rl)"1" "1" "H" "LAMBDA (i: nat): x * F(i)" )"1" replace -1)"1" "sigma_shift_T2" )"1" "H" "j-p+k+1" "1+j" "-1" )"1" assert )"1" replace -1)"1" "sigma_restrict_eq" )"1" "1" assert )"1" "1" "1" expand "restrict" )"1" "1" "1" "1" replace "1" "1" assert )"1" replace "1" assert )"1" "1" replace "1" assert )"1" "1" "Bernstein_index_degree_raise" )"1" "j-p+k" "x!1-1" "x" )"1" assert )"1" "(C(x!1 - 1, j) / C(j - p + k, j))" )"1" replace "1" "1" case "(x!1 / (1 - p + j + k)) * ") "1" "Bern(x!1,1-p+j+k)(x)" )"1" assert )nil nil ))nil )"2" "2" expand "C" )"2" expand "factorial" "2" "2" "2" expand "factorial" "2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "x!1" )"2" case "NOT j = 0" )"1" assert )nil nil )"2" replace "2" assert )"2" case "NOT x!1 = 0" )"1" case "k = p" )"1" replace "1" assert )"1" case "x!1 = 1" )"1" replace "1" assert )"1" "Bern_top" )"1" "0" "x" )"1" "1" "x" )"1" replace "1" replace "1" expand "^" )"1" expand "expt" )"1" expand "expt" )"1" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" assert )nil nil ))nil ))nil ))nil )"2" assert )nil 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 ))nil ))nil ))nil ))nil )"2" "2" "0" )"2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (skeep) (("2" (assert ) nil nil )) nil )"3" (skeep) (("3" (assert ) nil nil )) nil )"4" (skeep) (("4" (assert ) nil nil )) nil )"5" (skeep) (("5" (assert ) nil nil )) nil ))nil ))nil )"2" (skeep) (("2" (assert ) nil nil )) nil )"3" (skeep) (("3" (assert ) nil nil )) nil )"4" (skeep) (("4" (assert ) nil nil )) nil )"5" (skeep) (("5" (assert ) nil nil )) nil ))nil ))nil )"3" (skeep) (("3" (assert ) nil nil )) nil )"4" (skeep) (("4" (assert ) nil nil )) nil )"5" (skeep) (("5" (assert ) nil nil )) nil )"6" (skeep) (("6" (assert ) nil nil )) nil )"7" (skeep) (("7" (assert ) nil nil )) nil ))nil ))nil )"3" (skeep) (("3" (assert ) nil nil )) nil )"4" (skeep) (("4" (assert ) nil nil )) nil )"5" (skeep) (("5" (assert ) nil nil )) nil )"6" (skeep) (("6" (assert ) nil nil )) nil )"7" (skeep) (("7" (assert ) nil nil )) nil ))nil ))nil )"real" bernstein_polynomials nil )nil integers nil )"posnat" binomial nil )"[numfield, nznum -> numfield]" number_fields nil )nil number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )"bool" reals nil )IF const-decl "[boolean, T, T -> T]" if_def nil )nil integers nil )"bool" reals nil )nil integers nil )"[numfield, numfield -> numfield]" number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )nil number_fields nil )"real" sigma nil )nil sigma nil )nil sigma nil ) (<= const-decl "bool" reals nil )"real" exponentiation nil )"boolean" notequal nil )OR const-decl "[bool, bool -> bool]" booleans nil )"[T, T -> boolean]" equalities 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 )"int" integers nil )"real" reals nil )"int" integers nil )NOT const-decl "[bool -> bool]" booleans nil )AND const-decl "[bool, bool -> bool]" booleans nil )"[bool, bool -> bool]" booleans nil )"(total_order?[real])" nil )"nat" bernstein_polynomials nil )"posnat" bernstein_polynomials nil )nil defined_types nil )nil naturalnumbers nil )"real" exponentiation nil )nil nil )nil sigma nil )"(strict_total_order?[real])" real_props nil )"(divides(m))" divides nil )"[T -> real]" sigma nil )"odd_int" integers nil )"posrat" nil )"posrat" nil )"posint" nil )"(divides(n))" divides nil )nil factorial "ints/" )nil sigma_nat nil )"real" reals nil )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil )"posint" nil )nil extra_real_props nil )"posint" nil )"nat" bernstein_polynomials nil )"[numfield -> numfield]" number_fields nil )"posnat" factorial "ints/" )"bool" functions nil )"(bijective?[T, T])" identity nil )"bool" booleans nil )"nonneg_rat" nil )nil real_props nil )"rat" rationals nil )"rat" rationals nil )"nonneg_rat" nil )nil real_props nil )nil real_props nil )nil reals nil )nil real_props nil )"nonneg_int" nil )"nonneg_rat" nil )"rat" rationals nil )"rat" rationals nil )"odd_int" integers nil )nil bernstein_polynomials nil )"even_int" integersnil )nil nil )nil sigma nil )nil sigma nil ))nil 3479543185"" (skeep)"" (lemma "Bernstein_conversion" )"" (inst - "k" "p" "x" )"" (assert )"" (case "p = 0" )"1" (assert ) nil nil )"2" (lemma "sigma_split" )"2" "LAMBDA (i: nat): IF i < p OR i > k THEN 0" " k" " 0" " p-1") "1" (assert )"1" (replace -1)"1" (hide -1)"1" (lemma "sigma_restrict_eq" )"1" "LAMBDA (i: nat): IF i < p OR i > k THEN 0" " LAMBDA (i: nat):" " p-1" " 0") "1" (lemma "sigma_zero" )"1" (inst - "p-1" "0" )"1" assert )"1" "1" expand "restrict" )"1" (propax) nil nil ))nil )"2" "2" (assert ) nil nil ))nil )"3" "3" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil )"2" (skosimp*) (("2" (assert ) nil nil ))nil )"3" (skosimp*) (("3" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (skosimp*) (("2" (assert ) nil nil )) nil )"3" (skosimp*) (("3" (assert ) nil nil )) nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )nil bernstein_polynomials nil )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil )"(total_order?[real])" nil )nil sigma nil )"odd_int" integersnil )"[T -> real]" sigma nil )"(strict_total_order?[real])" real_props nil )"nnreal" sigma_nat nil )"nat" sigma_nat nil )"real" reals nil )nil sigma nil )nil sigma nil )"posrat" nil )nil sigma nil )nil sigma nil )"real" bernstein_polynomials nil )nil integers nil )"posnat" binomial nil )"[numfield, nznum -> numfield]" number_fields nil )nil number_fields nil )"boolean" notequal nil )"[numfield, numfield -> numfield]" number_fields nil )OR const-decl "[bool, bool -> bool]" booleans nil )IF const-decl "[boolean, T, T -> T]" if_def nil )nil number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )"bool" reals nil )"posnat" bernstein_polynomials nil )"nat" bernstein_polynomials nil )"bool" reals nil )NOT const-decl "[bool -> bool]" booleans nil )AND const-decl "[bool, bool -> bool]" booleans nil )"[bool, bool -> bool]" booleans nil )"real" reals nil )"int" integers nil )"[T, T -> boolean]" equalities nil )nil naturalnumbers nil )nil integers nil )"bool" reals nil )nil integers 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 ))nil 3479481006 ("" (subtype-tcc) nil nil )"(total_order?[real])" nil ))nil ))nil 3479481006 ("" (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 )nil naturalnumbers nil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil ))nil ))nil 3479481006 ("" (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 )nil naturalnumbers nil )"int" integers nil )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil ))nil ))nil 3480405695"" (skeep)"" (decompose-equality)"" (expand "Bern_poly" )"" (name "mm" "b`index" )"" (replace -1)"" (case "degree_raise(b)`index = mm+1" )"1" (replace -1)"1" "AA" "LAMBDA (i: nat): IF i > mm THEN 0 ELSE b`bern_seq(i) * Bern(i, mm)(x!1) ENDIF") "1" (replace -1)"1" "BB" "LAMBDA (i: nat): IF i > mm + 1 THEN 0") "1" (replace -1)"1" "Aleft" "LAMBDA (i:nat): IF i>mm THEN 0 ELSE b`bern_seq(i)*((mm+1-i)/(mm+1))*Bern(i,mm+1)(x!1) ENDIF" )"1" "Aright" "LAMBDA (i:nat): IF i > mm+1 OR i = 0 THEN 0 ELSE b`bern_seq(i-1)*(i/(mm+1))*Bern(i,mm+1)(x!1) ENDIF" )"1" (case "NOT BB = Aleft+Aright" )"1" "1" "1" "1" expand "+" +)"1" expand "BB" +)"1" expand "Aleft" +)"1" expand "Aright" +)"1" "1" "1" "1" "1" assert )"1" "1" expand "degree_raise" "1" "1" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" replace -1)"2" "2" "2" expand "+" +)"2" rewrite "sigma_sum" :dir rl)"2" "sigma_split" )"2" "Aleft" "mm+1" "0" "mm" )"1" assert )"1" replace -1)"1" "1" case "NOT sigma(1+mm,1+mm,Aleft) = 0" )"1" "1" expand "sigma" "1" expand "sigma" "1" expand "Aleft" "1" nil nil ))nil ))nil ))nil ))nil )"2" replace -1)"2" "2" assert )"2" "sigma_split" )"2" "Aright" "mm+1" "0" "0" )"2" assert )"2" replace "2" "2" case "NOT sigma(0,0,Aright) = 0" )"1" "1" expand "sigma" )"1" expand "sigma" )"1" expand "Aright" "1" nil nil ))nil ))nil ))nil ))nil )"2" replace "2" "2" assert )"2" "sigma_shift_T2" )"2" "Aright" "mm" "0" "1" )"2" assert )"2" replace "2" "2" "DD" "LAMBDA (i: nat): Aright(1 + i)" )"2" replace "2" case "NOT AA = Aleft + DD" )"1" "1" "1" expand "+" "1" expand "AA" "1" expand "Aleft" "1" expand "DD" "1" "1" assert )"1" "1" expand "Aright" "1" nil nil ))nil )"2" expand "Aright" "2" "Bernstein_degree_raise" )"2" "1+mm" "x!2" "x!1" )"2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" replace "2" "2" "sigma_sum" )"2" "Aleft" "DD" "mm" "0" )"2" expand "+" "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 ))nil ))nil ))nil ))nil ))nil )"2" "2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (skosimp*) (("2" (assert ) nil nil ))nil )"3" (skosimp*) (("3" (assert ) nil nil ))nil ))nil )"2" (skosimp*) (("2" (assert ) nil nil ))nil ))nil ))nil )"2" (skosimp*) (("2" (assert ) nil nil )) nil )"3" (skosimp*) (("3" (assert ) nil nil )) nil ))nil ))nil )"2" (skosimp*) (("2" (assert ) nil nil )) nil )"3" (skosimp*) (("3" (assert ) nil nil )) nil ))nil ))nil )"2" (hide 2)"2" (expand "degree_raise" ) (("2" (propax) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )nil naturalnumbers nil )nil naturalnumbers nil )nil bernstein_polynomials nil )"[real -> real]" bernstein_polynomials nil )"Bernstein_Polynomial" nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )"bool" reals nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )"[T, T -> boolean]" equalities nil )"[numfield, numfield -> numfield]" number_fields nil )nil number_fields nil )"posint" nil )"real" bernstein_polynomials nil )nil integers nil )"[numfield, numfield -> numfield]" number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )"bool" reals nil )IF const-decl "[boolean, T, T -> T]" if_def nil )"real" reals nil )"bool" reals nil )NOT const-decl "[bool -> bool]" booleans nil )"[bool, bool -> bool]" booleans nil )"boolean" notequal nil )nil number_fields nil )"[numfield, nznum -> numfield]" number_fields nil )"[T -> real]" real_fun_ops nil )"[nat -> real]" bernstein_polynomials nil )"(strict_total_order?[real])" real_props nil )"real" reals nil )"real" reals nil )"nonneg_rat" nil )"rat" rationals nil )"real" reals nil )"[nat -> real]" bernstein_polynomialsnil )"[nat -> real]" bernstein_polynomials nil )"posint" nil )"int" integers nil )nil sigma nil )"odd_int" integersnil )nil sigma nil )"odd_int" integers nil )nil bernstein_polynomials nil )nil integers nil )nil integers nil )"posrat" nil )"[nat -> real]" bernstein_polynomials nil )"[nat -> real]" bernstein_polynomials nil )"real" sigma nil )"(total_order?[real])" nil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil )nil sigma nil )nil sigma nil )nil sigma nil )AND const-decl "[bool, bool -> bool]" booleans nil )OR const-decl "[bool, bool -> bool]" booleans nil )"int" integers nil ))nil 3480666697"" (skeep)"" case "FORALL (jj:nat): Bern_poly(b) = Bern_poly(degree_raise_power(b, jj))" )"1" (inst - "j" ) nil nil )"2" (hide 2)"2" (induct "jj" )"1" (expand "degree_raise_power" ) (("1" (propax) nil nil ))nil )"2" (skosimp*)"2" (expand "degree_raise_power" +)"2" (lemma "degree_raise_id" )"2" (inst - "degree_raise_power(b,j!1)" )"2" (assert ) nil nil )) nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"Bernstein_Polynomial" nil )"[real -> real]" bernstein_polynomials nil )nil bernstein_polynomials nil )nil naturalnumbers nil )"[T, T -> boolean]" equalities 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 )nil defined_types nil )nil naturalnumbers nil )nil bernstein_polynomials nil ))nil ))nil 3485881402 ("" (subtype-tcc) nil nil )nil nil ))nil 3485881746"" (skeep)"" (expand "Bern_poly" )"" (case "Bern_poly_reverse(b)`index = b`index" )"1" (replace -1)"1" (lemma "sigma_reverse" )"1" (inst?)"1" (replace -1)"1" (hide -1)"1" (assert )"1" (lemma "sigma_restrict_eq" )"1" (inst?)"1" (assert )"1" (hide 2)"1" (decompose-equality 1)"1" expand "restrict" )"1" "1" "1" "1" expand "Bern_poly_reverse" )"1" case "Bern(b`index - x!1, b`index)(1 / 2 + x) = ") "1" (assert ) nil nil )"2" "2" expand "Bern" )"2" "C_symmetry" )"2" "b`index" "x!1" )"2" replace "2" "2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" (assert ) nil nil ))nil )"3" "3" (assert ) nil nil ))nil )"4" "4" (assert ) nil nil ))nil )"5" "5" (assert ) nil nil ))nil ))nil ))nil ))nil )"2" (skosimp*) (("2" (assert ) nil nil )) nil )"3" (skosimp*) (("3" (assert ) nil nil )) nil )"4" (skosimp*) (("4" (assert ) nil nil )) nil )"5" (skosimp*) (("5" (assert ) nil nil )) nil ))nil ))nil ))nil ))nil ))nil )"2" (skosimp*) (("2" (assert ) nil nil )) nil )"3" (skosimp*) (("3" (assert ) nil nil )) nil ))nil ))nil ))nil )"2" (hide 2)"2" (expand "Bern_poly_reverse" ) (("2" (propax) nil nil ))nil ))nil ))nil ))nil ))nil )"[real -> real]" bernstein_polynomials nil )"real" reals nil )"real" reals nil )"int" integers nil )"bool" reals nil )"Bernstein_Polynomial" bernstein_polynomialsnil )"bool" reals nil )NOT const-decl "[bool -> bool]" booleans nil )"[bool, bool -> bool]" booleans nil )"bool" reals nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )"[numfield, numfield -> numfield]" number_fields nil )nil number_fields nil )IF const-decl "[boolean, T, T -> T]" if_def nil )"[numfield, numfield -> numfield]" number_fields nil )nil integers nil )"real" bernstein_polynomials nil )"[numfield, numfield -> numfield]" number_fields nil )nil sigma nil )"[T -> real]" sigma nil )"boolean" notequal nil )nil number_fields nil )"[numfield, nznum -> numfield]" number_fields nil )"(total_order?[real])" nil )"posrat" nil )nil binomial nil )"(strict_total_order?[real])" real_props nil )OR const-decl "[bool, bool -> bool]" booleans nil )nil sigma nil )nil sigma nil )"int" integers nil )"real" reals nil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil )nil sigma_nat nil )nil numbers nil )nil booleans nil )"[T, T -> boolean]" equalities nil )nil naturalnumbers nil )nil naturalnumbers nil )nil reals nil )nil bernstein_polynomials nil )"Bernstein_Polynomial" nil ))nil ))nil 3485881402"" (skeep)"" (lemma "Bern_poly_reverse_sym_half" )"" (inst - "b" "x-1/2" ) (("" (assert ) nil nil )) nil )) nil ))nil )nil bernstein_polynomialsnil )"posrat" nil )"real" reals nil )"[numfield, nznum -> numfield]" number_fields nil )nil number_fields nil )"boolean" notequal nil )"[numfield, numfield -> numfield]" number_fields nil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )nil bernstein_polynomials nil )nil reals nil )nil naturalnumbers nil )nil naturalnumbers nil )"real" reals nil ))nil 3479635199 ("" (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 )nil naturalnumbers nil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil ))nil ))nil 3479644701 ("" (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 )nil naturalnumbers nil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil ))nil ))nil 3479644701 ("" (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 )nil naturalnumbers nil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil ))nil ))nil 3480943595"" (subtype-tcc) nil nil )nil bernstein_polynomials nil )nil naturalnumbers nil )nil naturalnumbers nil )"bool" reals nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )"bool" reals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil numbers nil )NOT const-decl "[bool -> bool]" booleans nil )nil booleans nil )nil booleans nil ))nil ))nil 3480943595"" (skeep)"" case "FORALL (mm: nat): (FORALL (bb:Bernstein_Polynomial): ((bb`index >= mm AND (FORALL (j:upto(bb`index)): j>mm IMPLIES bb`bern_seq(j) = 0)) IMPLIES P(bb)))" )"1" (skeep)"1" (inst - "b`index" )"1" (inst - "b" ) (("1" (assert ) nil nil )) nil )) nil ))nil )"2" (hide 2)"2" (induct "mm" )"1" (skeep)"1" (inst -4 "0" "bb`index" "bb`bern_seq(0)" )"1" (assert )"1" case "bb = (# index := bb`index, LAMBDA (j: upto(bb`index)):IF j = 0 THEN bb`bern_seq(0) ELSE 0 ENDIF #)") "1" (assert ) nil nil )"2" (hide 2)"2" (decompose-equality)"2" (decompose-equality)"2" (lift-if)"2" (ground)"2" (inst - "x!1" )"2" (assert ) nil nil )) nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (assert ) nil nil ))nil ))nil )"2" (skosimp*)"2" "bb1" "(# index := bb!1`index, bern_seq := (LAMBDA (i:upto(bb!1`index)): IF i <= j!1 THEN bb!1`bern_seq(i) ELSE 0 ENDIF) #)" )"2" "bb2" "(# index := bb!1`index, bern_seq := (LAMBDA (i:upto(bb!1`index)): IF i = 1+j!1 THEN bb!1`bern_seq(1+j!1) ELSE 0 ENDIF) #)" )"1" (inst - "bb1" )"1" (assert )"1" (split -3)"1" (inst -6 "bb1" "bb2" )"1" "1+j!1" "bb!1`index" "bb!1`bern_seq(1+j!1)" )"1" (assert )"1" (replace -2)"1" assert )"1" case "bb!1 = (# index := bb1`index, LAMBDA (i: upto(bb1`index)):") "1" (assert ) nil nil )"2" "2" "2" "1" expand "bb1" +)"1" expand "bb2" +)"1" "1" "1" "1" "x!1" )"1" "x!1" )"1" assert )"1" "1" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" (ground) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (skeep) (("2" (ground) nil nil )) nil ))nil )"2" (skeep)"2" (expand "bb1" 1)"2" (lift-if) (("2" (ground) nil nil ))nil ))nil ))nil ))nil ))nil )"2" (skeep) (("2" (ground) nil nil )) nil ))nil )"2" (skeep) (("2" (ground) nil nil )) nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"[T, T -> boolean]" equalities nil )"bool" reals nil ) (<= const-decl "bool" reals nil )AND const-decl "[bool, bool -> bool]" booleans nil )"[bool, bool -> bool]" booleans nil )nil bernstein_polynomials nil )nil naturalnumbers 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 )"(total_order?[real])" nil )nil defined_types nil )nil naturalnumbers nil )"(total_order?[real])" nil )"Bernstein_Polynomial" bernstein_polynomialsnil )IF const-decl "[boolean, T, T -> T]" if_def nil )"(strict_total_order?[real])" real_props nil )"[bool, bool -> bool]" booleans nil )"Bernstein_Polynomial" nil )"[# bern_seq: [upto(bb!1`index) -> real], index: nat #]" nil )"real" reals nil )NOT const-decl "[bool -> bool]" booleans nil )"[# bern_seq: [upto(bb!1`index) -> real], index: nat #]" nil )"posint" nil )"[numfield, numfield -> numfield]" number_fields nil )nil number_fields nil )"posint" nil ))nil 3479481006"" (subtype-tcc) nil nil ) nil nil ))nil 3479481006"" (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 )nil naturalnumbers nil )"bool" reals nil )nil naturalnumbers nil )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil )"(total_order?[real])" nil ))nil ))nil 3479486699"" (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 )nil naturalnumbers nil )"bool" reals nil )nil naturalnumbers nil )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil )"(total_order?[real])" nil ))nil ))nil 3479481006"" (skeep)"" case "FORALL (pp: nat): m>=pp IMPLIES polynomial(a, pp) = Bern_poly(Bernstein_polynomial(a, pp, m))" )"1" (inst - "p" ) (("1" (assert ) nil nil )) nil )"2" (hide 2)"2" (lemma "NAT_induction" )"2" (inst?)"2" (skeep)"2" (split -)"1" (inst - "pp" ) (("1" (assert ) nil nil )) nil )"2" (hide 2)"2" (skeep)"2" (case "j = 0" )"1" (replace -1)"1" (hide -1)"1" (hide -1)"1" (decompose-equality)"1" expand "polynomial" )"1" expand "sigma" )"1" assert )"1" expand "Bern_poly" )"1" expand "Bernstein_polynomial" )"1" "AA" "LAMBDA (i: nat): IF i > m THEN 0LAMBDA (i_1: nat):IF i_1 > i OR i_1 > 0 THEN 0") "1" replace -1)"1" case "AA = LAMBDA (i: nat): IF i>m THEN 0 ELSE a(0)*Bern(i,m)(x!1) ENDIF" )"1" replace -1)"1" "sigma_scal" )"1" "BB" "LAMBDA (i: nat): IF i > m THEN 0 ELSE Bern(i, m)(x!1) ENDIF") "1" "BB" "a(0)" "m" "0" )"1" case "(LAMBDA (i: nat): a(0) * BB(i)) = AA" )"1" replace -1)"1" replace "1" replace "1" case "sigma(0,m,BB) = 1" )"1" assert )nil nil )"2" "2" "2" "Bernstein_partition_of_unity" )"2" "m" "x!1" )"2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" replace "2" "1" replace "1" assert )"1" "1" nil nil ))nil ))nil ))nil )"2" "2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" replace -1 1 :dir rl)"2" "1" "1" "1" "sigma_split" )"1" "LAMBDA (i_1: nat): IF i_1 > x!2 OR i_1 > 0 THEN 0" "x!2" "0" "0" )"1" assert )"1" replace "1" "1" expand "sigma" "1" assert )"1" "sigma_restrict_eq" )"1" "LAMBDA (i_1: nat): IF i_1 > x!2 OR i_1 > 0 THEN 0" "LAMBDA (i_1: nat): " "x!2" "1" )"1" "1" "sigma_zero" )"1" "x!2" "1" )"1" replace "1" "Bern(x!2, m)(x!1)" )"1" replace "1" expand "C" )"1" assert )nil nil ))nil ))nil )"2" "2" assert )nil nil ))nil )"3" "3" assert )nil nil ))nil ))nil ))nil ))nil ))nil )"2" "1" expand "restrict" )"1" nil nil ))nil )"2" "2" assert )nil nil ))nil )"3" "3" assert )nil nil ))nil ))nil ))nil )"2" "2" assert )nil nil ))nil )"3" "3" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" assert )nil nil ))nil )"3" "3" assert )nil nil ))nil ))nil ))nil ))nil ))nil )"2" "2" assert )nil nil ))nil )"3" "3" assert )nil nil ))nil )"4" "4" assert )nil nil ))nil ))nil ))nil ))nil ))nil )"2" "2" (assert ) nil nil ))nil )"3" "3" (assert ) nil nil ))nil )"4" "4" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (inst - "j-1" )"1" (assert )"1" "aa" "LAMBDA (kz:nat): IF kz/=j THEN 0 ELSE a(j) ENDIF" )"1" case "polynomial(a,j) = polynomial(a,j-1) + polynomial(aa,j)" )"1" case "Bern_poly(Bernstein_polynomial(a, j, m))=Bern_poly(Bernstein_polynomial(a, j-1, m))+Bern_poly(Bernstein_polynomial(aa, j, m))" )"1" case "polynomial(aa,j) = Bern_poly(Bernstein_polynomial(aa,j,m))" )"1" (assert ) nil nil )"2" "2" "2" "2" "2" "2" expand "polynomial" )"2" expand "Bernstein_polynomial" )"2" expand "Bern_poly" )"2" assert )"2" "sigma_split" )"2" "LAMBDA (i: nat): aa(i) * (IF i = 0 THEN 1 ELSE x!1 ^ i ENDIF)" "j" "0" "j-1" )"2" assert )"2" replace "2" "2" "sigma_restrict_eq" )"2" "LAMBDA (i: nat): aa(i) * (IF i = 0 THEN 1 ELSE x!1 ^ i ENDIF)" "LAMBDA (i: nat): 0" "j-1" "0" )"2" "1" "sigma_zero" )"1" "1" replace "1" "1" replace "1" "1" assert )"1" expand "sigma" "1" assert )"1" expand "sigma" "1" case "aa(j) = a(j)" )"1" replace "1" "1" "Bernstein_conversion_2" )"1" "m" "j" "x!1" )"1" assert )"1" "a(j)" )"1" "sigma_scal" )"1" "1" "a(j)" )"1" replace "1" "1" "sigma_restrict_eq" )"1" "LAMBDA (i_1: nat): IF i_1 < j OR i_1 > m THEN 0" "LAMBDA (i_1: nat): IF i_1 > m THEN 0LAMBDA (i: nat):IF i > i_1 OR i > j THEN 0" "m" "0" )"1" assert )"1" "1" "1" "1" expand "restrict" )"1" "1" "1" "sigma_restrict_eq" )"1" "LAMBDA (i: nat): IF i > x!2 OR i > j THEN 0" "LAMBDA (i: nat): 0" "x!2" "0" )"1" "1" "sigma_zero" )"1" "x!2" "0" )"1" assert )nil nil ))nil ))nil )"2" "2" expand "restrict" )"2" "1" "1" "1" replace "1" assert )nil nil ))nil ))nil ))nil )"2" "2" assert )nil nil ))nil )"3" "3" assert )nil nil ))nil ))nil ))nil ))nil ))nil )"2" "2" assert )nil nil ))nil )"3" "3" assert )nil nil ))nil ))nil ))nil )"2" "BB" "LAMBDA (i: nat): IF i > x!2 OR i > j THEN 0") "1" replace "1" "sigma_split" )"1" "BB" "x!2" "0" "j-1" )"1" assert )"1" replace "1" "1" case "sigma(0,j-1,BB) = 0" )"1" replace "1" "1" assert )"1" case "x!2 = j" )"1" replace "1" expand "sigma" "1" expand "sigma" "1" replace "1" assert )"1" replace "1" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "sigma_split" )"2" "BB" "x!2" "j" "j" )"2" assert )"2" case "sigma(1+j,x!2,BB) = 0" )"1" replace "1" assert )"1" replace "1" expand "sigma" "1" expand "sigma" "1" replace "1" assert )"1" replace "1" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "sigma_restrict_eq" )"2" "BB" "LAMBDA (i:nat): 0" "x!2" "1+j" )"2" "1" "sigma_zero" )"1" "x!2" "1+j" )"1" assert )nil nil ))nil ))nil )"2" "2" expand "restrict" )"2" "2" "2" replace "2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "sigma_restrict_eq" )"2" "BB" "LAMBDA (i:nat): 0" "j-1" "0" )"2" "1" "sigma_zero" )"1" "j-1" "0" )"1" assert )nil nil ))nil ))nil )"2" "2" expand "restrict" )"2" "2" "2" replace "2" assert )"2" replace "2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" assert )nil nil ))nil )"3" "3" assert )nil nil ))nil ))nil ))nil ))nil ))nil )"2" "2" assert )nil nil ))nil )"3" "3" assert )nil nil ))nil )"4" "4" assert )nil nil ))nil )"5" "5" assert )nil nil ))nil )"6" "6" assert )nil nil ))nil ))nil ))nil ))nil ))nil )"2" "2" assert )nil nil ))nil )"3" "3" assert )nil nil ))nil )"4" "4" assert )nil nil ))nil )"5" "5" assert )nil nil ))nil )"6" "6" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" assert )nil nil ))nil )"3" "3" assert )nil nil ))nil ))nil ))nil )"2" "2" assert )nil nil ))nil )"3" "3" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" replace "2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" expand "restrict" )"2" "2" "1" replace "1" assert )nil nil ))nil )"2" replace "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" expand "+" 1)"2" expand "Bern_poly" 1)"2" expand "Bernstein_polynomial" "2" "AA" "LAMBDA (i: nat): IF i > m THEN 0LAMBDA (i_1: nat):IF i_1 > i OR i_1 > j THEN 0") "1" replace -1)"1" "BB" "LAMBDA (i: nat): IF i > m THEN 0LAMBDA (i_1: nat):IF i_1 > i OR i_1 > j THEN 0") "1" replace -1)"1" "CC" "LAMBDA (i: nat): IF i > m THEN 0LAMBDA (i_1: nat):IF i_1 > i OR i_1 > j - 1 THEN 0") "1" replace -1)"1" case "AA = (LAMBDA (i:nat): BB(i)+CC(i))" )"1" "sigma_sum" )"1" "1" assert )nil nil ))nil ))nil )"2" "2" "2" "2" replace "2" assert )"2" replace "2" assert )"2" replace "2" assert )"2" "2" "2" "2" "2" "2" "DD" "LAMBDA (i_1: nat): IF i_1 > x!2 OR i_1 > j THEN 0") "1" replace "1" "EE" "LAMBDA (i_1: nat): IF i_1 > x!2 OR i_1 > j - 1 THEN 0") "1" replace "1" "FF" "LAMBDA (i_1: nat): IF i_1 > x!2 OR i_1 > j THEN 0") "1" replace "1" case "DD = (LAMBDA (i:nat): EE(i)+FF(i))" )"1" "sigma_sum" )"1" "EE" "FF" "x!2" "0" )"1" assert )nil nil ))nil ))nil )"2" "2" "2" replace "2" "2" replace "2" "2" replace "2" "2" assert )"2" "2" "2" assert )"2" "2" "2" assert )"2" "1" replace "1" assert )nil nil ))nil )"2" replace "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 )"2" "2" assert )nil nil ))nil )"3" "3" assert )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 ))nil ))nil )"2" "2" assert )nil nil ))nil )"3" "3" assert )nil nil ))nil ))nil ))nil ))nil ))nil )"2" "2" (assert ) nil nil ))nil )"3" "3" (assert ) nil nil ))nil )"4" "4" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" expand "+" 1)"2" expand "polynomial" 1)"2" expand "sigma" 1 1)"2" "sigma_split" )"2" "LAMBDA (i: nat): aa(i) * (IF i = 0 THEN 1 ELSE x!1 ^ i ENDIF)" "j" "0" "j-1" )"2" assert )"2" replace -1)"2" "2" expand "sigma" 1 2)"2" expand "sigma" "2" "sigma_restrict_eq" )"2" "LAMBDA (i: nat): aa(i) * (IF i = 0 THEN 1 ELSE x!1 ^ i ENDIF)" "LAMBDA (i: nat): 0" "j-1" "0" )"2" "sigma_zero" )"2" "j-1" "0" )"2" assert )"2" replace "2" assert )"2" "2" expand "restrict" )"2" "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" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"Bernstein_Polynomial" nil )"[real -> real]" bernstein_polynomials nil )nil bernstein_polynomials nil )nil naturalnumbers nil )"[real -> real]" polynomials nil )nil sequences nil )"[T, T -> boolean]" equalities nil )"[bool, bool -> bool]" booleans 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 )"(total_order?[real])" nil )nil naturalnumbers nil )"real" sigma nil )"real" bernstein_polynomials nil )nil integers nil )"[numfield, numfield -> numfield]" number_fields nil )"posnat" binomial nil )nil integers nil )nil integers nil )"[numfield, nznum -> numfield]" number_fields nil )nil number_fields nil )"boolean" notequal nil )nil sigma nil )nil sigma nil ) (<= const-decl "bool" reals nil )OR const-decl "[bool, bool -> bool]" booleans nil )"[numfield, numfield -> numfield]" number_fields nil )"bool" reals nil )IF const-decl "[boolean, T, T -> T]" if_def nil )nil number_fields nil )AND const-decl "[bool, bool -> bool]" booleans nil )NOT const-decl "[bool -> bool]" booleans nil )nil sigma nil )"(strict_total_order?[real])" real_props nil )"nat" bernstein_polynomials nil )"int" integers nil )"posrat" nil )nil nil )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil )"nat" bernstein_polynomials nil )nil sigma nil )nil extra_real_props nil )"nnreal" sigma_nat nil )"nat" sigma_nat nil )"posint" nil )"(divides(n))" divides nil )"(divides(m))" divides nil )nil factorial "ints/" )nil sigma nil )"[T -> real]" sigma nil )"odd_int" integersnil )nil sigma nil )nil sigma_nat nil )"real" reals nil )"real" reals nil )"[T -> real]" real_fun_ops nil )nil sigma nil )"nonneg_int" nil )"even_int" integersnil )"bool" reals nil )"nat" bernstein_polynomials nil )"posint" nil )"[numfield, numfield -> numfield]" number_fields nil )nil bernstein_polynomials nil )"real" exponentiation nil )"nonneg_int" nil )"even_int" integersnil )"nat" bernstein_polynomials nil )nil defined_types nil ))nil 3479635199"" (skeep)"" (decompose-equality)"" (expand "polynomial" )"" case "NOT (LAMBDA (i: nat): IF i = 0 THEN 1 ELSE x!1 ^ i ENDIF)) = (LAMBDA (i: nat):") "1" (hide 2)"1" (decompose-equality)"1" (lift-if)"1" (ground)"1" (replace -1)"1" (expand "^" )"1" (expand "expt" ) (("1" (propax) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (replace -1)"2" (hide -1)"2" (expand "Bern_poly" )"2" (lemma "Bern_Polynomial" )"2" (expand "polynomial" )"2" case "NOT (LAMBDA (i: nat): IF i > b`index THEN 0LAMBDA (i: nat):LAMBDA (j:nat): IF j < i OR j > b`index OR i>b`index THEN 0 ELSE b`bern_seq(i)*(-1)^(j-i)*(C(b`index,j)*C(j,i))*x!1^j ENDIF)))") "1" (hide 2)"1" (decompose-equality)"1" (lift-if)"1" (ground)"1" "sigma_zero" )"1" "1" (assert ) nil nil ))nil ))nil )"2" assert )"2" "x!2" "b`index" )"2" replace -1)"2" "2" assert )"2" "AA" "LAMBDA (i_1: nat): IF i_1 < x!2 OR i_1 > b`index THEN 0IF i_1 = 0 THEN 1 ELSE x!1 ^ i_1 ENDIF)") "1" "BB" "LAMBDA (j: nat): IF j < x!2 OR j > b`index THEN 0") "1" replace -1)"1" replace -2)"1" "sigma_split" )"1" "AA" "b`index" "0" "x!2-1" )"1" assert )"1" replace -1)"1" "1" case "sigma(0,x!2-1,AA) = 0" )"1" replace "1" assert )"1" "1" rewrite "sigma_scal" "1" "sigma_restrict_eq" )"1" "1" assert )"1" "1" "1" expand "restrict" )"1" "1" "1" replace "1" replace "1" assert )"1" case "IF x!3 = 0 THEN 1 ELSE x!1 ^ x!3 ENDIF = x!1^x!3" )"1" assert )"1" replace "1" assert )nil nil ))nil ))nil )"2" "2" "2" "2" replace "2" expand "^" )"2" expand "expt" )"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 )"2" "sigma_zero" )"2" "x!2-1" "0" )"2" "sigma_restrict_eq" )"2" "AA" "LAMBDA (i:nat): 0" "x!2-1" "0" )"2" assert )"2" "2" expand "restrict" )"2" "2" "2" replace "2" assert )"2" "2" 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" (assert ) nil nil ))nil )"3" "3" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (skosimp*) (("2" (assert ) nil nil ))nil )"3" (skosimp*) (("3" (assert ) nil nil ))nil )"4" (skosimp*) (("4" (assert ) nil nil ))nil )"5" (skosimp*) (("5" (assert ) nil nil ))nil )"6" (skosimp*) (("6" (assert ) nil nil ))nil ))nil ))nil )"2" (replace -1)"2" (hide -1)"2" "F" "(LAMBDA (i,j:nat): IF j < i OR j > b`index OR i > b`index THEN 0 ") "2" (lemma "sigma_swap_triangle" )"2" "F" "b`index" "b`index" "0" )"2" assert )"2" case "(LAMBDA (i: nat): LAMBDA (j: nat):IF j < i OR j > b`index OR i > b`index THEN 0LAMBDA (i: nat): sigma(i, b`index, LAMBDA (j: nat): F(i, j)))") "1" replace -1)"1" "1" replace -1)"1" "sigma_restrict_eq" )"1" "(LAMBDA (i: nat): Bern_poly_inverse(b)(i) * x!1 ^ i)" "(LAMBDA (j: nat): sigma(0, j, LAMBDA (i: nat): F(i, j)))" "b`index" "0" )"1" assert )"1" "1" "1" expand "restrict" )"1" "1" "1" expand "Bern_poly_inverse" )"1" "CC" "LAMBDA (i: nat): IF i > x!2 THEN 0") "1" replace "1" "sigma_scal" )"1" "CC" "x!1^x!2" "x!2" "0" )"1" replace "1" "1" "sigma_restrict_eq" )"1" "1" assert )"1" "1" "1" expand "restrict" )"1" "1" "1" expand "CC" "1" expand "F" "1" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))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 )"2" assert )"2" "2" "1" "sigma_eq" )"1" "LAMBDA (j: nat): IF j < x!2 OR j > b`index OR x!2 > b`index THEN 0" "LAMBDA (j: nat): F(x!2, j)" "b`index" "x!2" )"1" assert )"1" "1" "1" expand "F" +)"1" nil nil ))nil ))nil ))nil ))nil )"2" "2" assert )nil nil ))nil )"3" "3" assert )nil nil ))nil )"4" "4" assert )nil nil ))nil ))nil ))nil )"2" "2" (assert ) nil nil ))nil )"3" "3" (assert ) nil nil ))nil )"4" "4" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"3" (skosimp*) (("3" (assert ) nil nil )) nil )"4" (skosimp*) (("4" (assert ) nil nil )) nil )"5" (skosimp*) (("5" (assert ) nil nil )) nil )"6" (skosimp*) (("6" (assert ) nil nil )) nil )"7" (skosimp*) (("7" (assert ) nil nil )) nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )nil booleans nil )"bool" reals nil )nil naturalnumbers nil )nil sequences nil )"[real -> real]" polynomials nil )nil naturalnumbers nil )nil bernstein_polynomials nil )"sequence[real]" nil )"[real -> real]" bernstein_polynomials nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )"real" exponentiation nil )"boolean" notequal nil )OR const-decl "[bool, bool -> bool]" booleans nil )IF const-decl "[boolean, T, T -> T]" if_def nil )"[numfield, numfield -> numfield]" number_fields nil )"[T, T -> boolean]" equalities nil )nil number_fields nil )NOT const-decl "[bool -> bool]" booleans nil )"real" reals nil )"[bool, bool -> bool]" booleans nil )"real" exponentiation nil )nil bernstein_polynomials nil )"posnat" binomial nil )nil integers nil )nil integers nil )"[numfield -> numfield]" number_fields nil )"bool" reals nil ) (sigma def-decl "real" sigma nil )nil sigma nil )nil sigma nil ) (<= const-decl "bool" reals nil )"real" bernstein_polynomials nil )nil integers nil )"[numfield, numfield -> numfield]" number_fields nil )"bool" reals nil )"int" integers nil )"odd_int" integers nil )AND const-decl "[bool, bool -> bool]" booleans nil )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil )"Bernstein_Polynomial" bernstein_polynomialsnil )"nnreal" sigma_nat nil )"nat" sigma_nat nil )nil sigma nil )"nzrat" rationalsnil )"rat" exponentiation nil )"nzreal" exponentiation nil )"posint" nil )"(divides(n))" divides nil )"(divides(m))" divides nil )nil sigma nil )"odd_int" integersnil )"nonneg_int" nil )"even_int" integersnil )"real" reals nil )nil sigma nil )"[T -> real]" sigma nil )"nzreal" exponentiation nil )"int" exponentiation nil )nil sigma nil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil )nil sigma_swap nil )"nat" bernstein_polynomials nil )nil sigma nil )"[nat -> real]" bernstein_polynomials nil )"[[nat, nat] -> real]" bernstein_polynomialsnil ))nil 3485777030 ("" (subtype-tcc) nil nil )nil nil ))nil 3485777030 ("" (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 naturalnumbers nil )nil naturalnumbers nil )nil bernstein_polynomials nil )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil )"(total_order?[real])" nil ))nil ))nil 3485777030 ("" (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 naturalnumbers nil )nil naturalnumbers nil )nil bernstein_polynomials nil )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil )"(total_order?[real])" nil ))nil ))nil 3485768080 ("" (subtype-tcc) nil nil )nil nil ))nil 3485768080 ("" (subtype-tcc) nil nil )nil nil ))nil 3485768080 ("" (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 naturalnumbers nil )nil naturalnumbers nil )nil bernstein_polynomials nil )"int" integers nil )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil )"(total_order?[real])" nil ))nil ))nil 3485777030 ("" (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 naturalnumbers nil )nil naturalnumbers nil )nil bernstein_polynomials nil )"int" integers nil )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil )"(total_order?[real])" nil ))nil ))nil 3485777078"" (skeep)"" case "FORALL (xx :real): Bern_poly(Bern_subdiv_left(b))(xx) = LAMBDA (x: real): (1 / 2) * x))(xx)") "1" (decompose-equality 1) nil nil )"2" (hide 2)"2" (skeep 1)"2" (name "aa" "Bern_poly_inverse(b)" )"2" (lemma "Bern_poly_inverse_def" )"2" (inst - "b" )"2" (replace -2)"2" (replace -1 :dir rl)"2" (expand "o" )"2" (name "nn" "b`index" )"2" (replace -1)"2" "az" "(LAMBDA (i:nat): (1/2^i)*aa(i))" )"2" case "NOT polynomial(az,nn)(xx) = polynomial(aa,nn)((1/2)*xx)" )"1" "1" expand "polynomial" +)"1" "sigma_restrict_eq" )"1" "1" assert )"1" "1" "1" expand "restrict" )"1" "1" "1" replace -1)"1" expand "az" +)"1" expand "^" +)"1" expand "expt" "1" assert )nil nil ))nil ))nil ))nil ))nil )"2" "mult_expt" )"2" "x!1" "1/2" "xx" )"1" replace -1)"1" "1" expand "az" "1" "div_expt" )"1" "x!1" "1" "2" )"1" replace "1" "1" case "1^x!1 = 1" )"1" assert )nil nil )"2" case "FORALL (jjj:nat): 1^jjj = 1" )"1" "x!1" )nil nil )"2" "jjj" )"1" expand "^" "1" expand "expt" "1" nil nil ))nil ))nil )"2" "2" expand "^" )"2" expand "expt" "2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" replace -1)"2" expand "^" "2" expand "expt" "2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" replace -1 :dir rl)"2" "2" "Bernstein_equivalence" )"2" "az" "nn" "nn" )"2" assert )"2" replace -1)"2" "2" expand "Bernstein_polynomial" )"2" assert )"2" expand "az" 1)"2" expand "aa" 1)"2" expand "Bern_poly_inverse" "2" case "Bern_subdiv_left(b) = (# index := nn, LAMBDA (s: upto(nn)):LAMBDA (i: nat):IF i > s OR i > nn THEN 0LAMBDA IF i_1 > i") "1" nil nil )"2" "2" expand "Bern_subdiv_left" )"2" replace "2" "1" "sigma_swap_triangle" )"1" "F" "LAMBDA (j,i:nat): IF i > x!1 OR i > nn OR j > i THEN 0 ELSE (1/2^i)*(C(x!1, i) / C(nn, i))*b`bern_seq(j) * C(i, j) * C(nn, i) * ") "1" case "NOT sigma(0, x!1, LAMBDA (i: nat):IF i > x!1 OR i > nn THEN 0LAMBDA (i_1: nat):IF i_1 > i THEN 0LAMBDA (i:nat): sigma(0,i,(LAMBDA (j:nat):F(j,i)))))") "1" "1" "sigma_restrict_eq" )"1" "1" assert )"1" "1" "1" expand "restrict" )"1" "1" "1" "sigma_scal" )"1" "LAMBDA (i_2: nat): IF i_2 > x!2 THEN 0" "(1 / 2 ^ x!2) " "x!2" "0" )"1" replace "1" "1" "sigma_restrict_eq" )"1" "1" assert )"1" "1" "1" expand "restrict" )"1" "1" "1" expand "F" "1" nil nil ))nil ))nil ))nil ))nil )"2" "2" assert )nil nil ))nil )"3" "3" assert )nil nil ))nil ))nil ))nil ))nil )"2" "2" assert )nil nil ))nil )"3" "3" assert )nil nil ))nil ))nil ))nil ))nil ))nil )"2" "2" assert )nil nil ))nil )"3" "3" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" assert )nil nil ))nil )"3" "3" assert )nil nil ))nil )"4" "4" assert )nil nil ))nil )"5" "5" assert )nil nil ))nil )"6" "6" assert )nil nil ))nil ))nil ))nil ))nil )"2" "2" assert )nil nil ))nil )"3" "3" assert )nil nil ))nil )"4" "4" assert )nil nil ))nil )"5" "5" assert )nil nil ))nil )"6" "6" assert )nil nil ))nil ))nil ))nil ))nil )"2" replace "2" "2" "F" "x!1" "x!1" "0" )"2" assert )"2" case "(1 / 2 ^ x!1) * LAMBDA (j: nat):IF j > x!1 THEN 0 ELSE C(x!1, j) * b`bern_seq(j) ENDIF) = sigma(0, x!1,LAMBDA (j: nat): sigma(j, x!1, LAMBDA (i: nat): F(j, i)))") "1" assert )nil nil )"2" "2" "sigma_scal" )"2" "LAMBDA (j: nat): IF j > x!1 THEN 0 ELSE C(x!1, j) * b`bern_seq(j) ENDIF" "1/2^x!1" "x!1" "0" )"1" replace "1" "1" "sigma_restrict_eq" )"1" "1" assert )"1" "1" "1" expand "restrict" "1" "1" "1" expand "F" "1" "1" "1" case "C(x!1, x!2) = LAMBDA (i: nat):IF i > x!1 OR i > nn OR x!2 > i THEN 0") "1" "1/2^x!1 * b`bern_seq(x!2)" )"1" replace "1" "1" "sigma_scal" )"1" "LAMBDA (i: nat): IF i > x!1 OR i > nn OR x!2 > i THEN 0" "(1 / 2 ^ x!1 * b`bern_seq(x!2))" "x!1" "x!2" )"1" replace "1" "1" "sigma_restrict_eq" )"1" "1" assert )"1" "1" "1" expand "restrict" )"1" "1" "1" case " (1 / 2 ^ (x!3 - x!1)) ") "1" replace "1" assert )nil nil ))nil )"2" "2" case "2^(x!3-x!1)*2^x!1 = 2^x!3" )"1" assert )nil nil )"2" "2" "expt_plus" )"2" "2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" assert )nil nil ))nil )"3" "3" assert )nil nil ))nil )"4" "4" assert )nil nil ))nil ))nil ))nil ))nil )"2" "2" assert )nil nil ))nil )"3" "3" assert )nil nil ))nil )"4" "4" assert )nil nil ))nil ))nil ))nil ))nil ))nil )"2" "2" assert )nil nil ))nil )"3" "3" assert )nil nil ))nil )"4" "4" assert )nil nil ))nil ))nil ))nil ))nil ))nil )"2" "2" assert )nil nil ))nil )"3" "3" assert )nil nil ))nil )"4" "4" assert )nil nil ))nil ))nil )"2" "2" case "C(x!1, x!2) = LAMBDA (i: nat):IF i > x!1 OR i > nn OR x!2 > i THEN 0") "1" assert )"1" replace "1" "sigma_restrict_eq" )"1" "1" assert )"1" "1" "1" expand "restrict" )"1" "1" nil nil ))nil ))nil )"2" "2" assert )nil nil ))nil )"3" "3" assert )nil nil ))nil )"4" "4" assert )nil nil ))nil ))nil ))nil ))nil )"2" "2" assert )nil nil ))nil )"3"
Messung V0.5 in Prozent C=100 H=99 G=99
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2026-05-26
