Impressum float.prf
Interaktion und
(float (vNum_TCC1 0nil 3321636413 ("" (subtype-tcc) nil nil )"boolean" notequal nil )) nil ))nil 3319366647"" (judgement-tcc)"" "expt(radix, Prec(b!1)) / radix=expt(radix, Prec(b!1)-1)" )"1" (grind-reals) nil nil )"2" (hide 2)"2" (rewrite "expt_minus_aux" )"2" (rewrite "expt_x1_aux" ) nil nil )) nil ))nil ))nil ))nil )"int" integersnil )nil numbers nil )nil booleans nil )"[T, T -> boolean]" equalities nil )"[number -> boolean]" nil )nil number_fields nil )nil number_fields nil )"boolean" notequal nil )nil number_fields nil )"[numfield, nznum -> numfield]" number_fieldsnil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integersnil )nil integers nil )nil booleans nil )"bool" reals nil )nil naturalnumbers nil )"real" exponentiation nil )"bool" reals nil )nil integers nil )"above(1)" float nil )nil float nil )"[numfield, numfield -> numfield]" nil )nil reals nil )nil exponentiation nil )nil exponentiation nil )"posnat" exponentiationnil )"real" exponentiation nil )"(total_order?[real])" nil )"posrat" nil )"posint" exponentiationnil )"posnat" float nil ))nil ))nil 3319366647"" (judgement-tcc)"" (lemma "both_sides_expt_gt1_ge_aux" )"" (inst -1 "radix" "Prec(b!1)-1" "0" )"" (grind-reals)"" (rewrite "expt_x1_aux" -1) (("" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil )nil exponentiationnil )"int" integersnil )"posint" nil )"odd_int" nil )nil exponentiation nil )nil float nil )"[numfield, numfield -> numfield]" nil )nil number_fields nil )nil naturalnumbers nil )"above(1)" float nil )nil integers nil )nil integers nil )"[rational -> boolean]" integersnil )nil rationals nil )"[real -> boolean]" rationals nil )nil real_types nil )"bool" reals nil )nil real_types nil )"bool" reals nil )nil booleans nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" nil )nil booleans nil )nil numbers nil )"int" integersnil )"(strict_total_order?[real])" real_props nil )"posnat" exponentiationnil )"real" exponentiation nil )"(total_order?[real])" nil )"posint" exponentiationnil )"(total_order?[real])" nil )"posnat" float nil ))nil ))nil 3318615387 ("" (subtype-tcc) nil nil )"boolean" notequal nil )) nil ))nil 3544780855 ("" (grind) nil nil )"float" float nil )"real" exponentiation nil )"real" exponentiation nil )"real" float nil ))nil 3319889173 ("" (subtype-tcc) nil nil )"boolean" notequal nil )"{k: int | k <= i AND k <= j}" nil ))nil ))nil 3319889173 ("" (subtype-tcc) nil nil )"boolean" notequal nil )"{k: int | k <= i AND k <= j}" nil ))nil ))nil 3319889173 ("" (subtype-tcc) nil nil )"real" exponentiation nil )"int" integersnil )"posnat" exponentiationnil )"int" integersnil )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil )"{k: int | k <= i AND k <= j}" nil )"nnrat" exponentiation nil )"posrat" exponentiationnil )"rat" rationalsnil )"rat" rationalsnil ))nil ))nil 3319889173 ("" (subtype-tcc) nil nil )"real" exponentiation nil )"(divides(m))" dividesnil )"(divides(n))" dividesnil )"posnat" exponentiationnil )"int" integersnil )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil )"{k: int | k <= i AND k <= j}" nil )"nnrat" exponentiation nil )"posrat" exponentiationnil )"rat" rationalsnil )"rat" rationalsnil ))nil ))nil 3545039871 ("" (grind) nil nil )"(divides(m))" dividesnil )"(divides(n))" dividesnil )"posint" nil )"posnat" exponentiationnil )"int" integersnil )"(strict_total_order?[real])" real_props nil )"real" exponentiation nil )min const-decl "{p: real | p <= m AND p <= n}" real_defsnil )"real" exponentiation nil )"float" float nil )"float" float nil ))nil 3545039902 ("" (grind) nil nil )"float" float nil )"float" float nil ))nil 3545396627 ("" (subtype-tcc) nil nil )"nnrat" exponentiation nil )"posrat" exponentiationnil )"real" float nil ))nil ))nil 3545396632"" (skeep)"" (expand "^" )"" (expand "Fexpt" )"" (expand "FtoR" )"" (lemma "mult_expt" )"" (case "n = 0" )"1" (replace -1)"1" (hide -) (("1" (grind) nil nil )) nil )) nil )"2" (inst - "n" _ _)"2" (expand "^" - 1)"2" (rewrite -1)"1" (lemma "expt_times" )"1" (inst - "Fexp(f)" "n" "radix" )"1" replace -1 :dir rl)"1" "1" expand "^" + 2)"1" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil )"2" (flatten)"2" (replace -1)"2" expand "expt" )"2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"real" exponentiation nil )"float" float nil )"real" float nil )nil naturalnumbers nil )"bool" reals nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integersnil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" nil )"[T, T -> boolean]" equalities nil )nil booleans nil )nil numbers nil )"rat" rationalsnil )"even_int" nil )"(divides(n))" dividesnil )"(divides(m))" dividesnil )"posnat" exponentiationnil )"posrat" nil )"(total_order?[real])" nil )"rat" exponentiation nil )"int" integers nil )"real" exponentiation nil )"posrat" exponentiationnil )"nnrat" exponentiation nil )"int" exponentiation nil )"nzreal" exponentiationnil )"nonneg_rat" rationals nil )"nonneg_int" integers nil )"nat" exponentiation nil )"int" integersnil )nil exponentiation nil )nil reals nil )OR const-decl "[bool, bool -> bool]" booleans nil )"bool" reals nil )nil integers nil )"above(1)" float nil )"float" float nil )nil float nil )"boolean" notequal nil )"int" exponentiation nil )nil exponentiation nil )"float" float nil ))nil ))nil 3544869888 ("" (subtype-tcc) nil nil )nil booleans nil )nil booleans nil )NOT const-decl "[bool -> bool]" booleans nil )nil numbers nil )"[number -> boolean]" nil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"bool" reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integersnil )nil integers nil )nil naturalnumbers nil )"int" integersnil )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil ))nil ))nil 3544869888 ("" (termination-tcc) nil nil ) nil nil ))nil 3544783712"" (skeep)"" (typepred "i" )"" (case "radix = mod(radix,i) + i * floor(radix / i)" )"1" (replace -3)"1" (assert )"1" (replace -1)"1" (assert )"1" case "FORALL (ii,jj:int): integer_pred(ii*jj)" )"1" (inst - "Fnum(f)" "floor(radix/i)" )"1" (assert )"1" (hide-all-but (-1 1))"1" (grind :exclude "floor" ) nil nil ))nil ))nil ))nil )"2" (hide-all-but 1) (("2" (grind) nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (expand "mod" ) (("2" (propax) nil nil )) nil ))nil ))nil ))nil )"above(1)" float nil )nil integers nil )"bool" reals nil )"{k | abs(k) < abs(j)}" mod nil )"{n: nonneg_real | n >= m AND n >= -m}" nil )"[numfield -> numfield]" number_fields nil )nil number_fields nil )AND const-decl "[bool, bool -> bool]" booleans nil )nil real_types nil )"bool" reals nil )"bool" reals nil )nil integers nil )"[T, T -> boolean]" equalities nil )nil integers nil )nil integers nil )"[rational -> boolean]" integersnil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" nil )"boolean" notequal nil )nil numbers nil )NOT const-decl "[bool -> bool]" booleans nil )nil booleans nil )nil booleans nil )"(divides(n))" dividesnil )"(divides(m))" dividesnil )nil float nil )"rat" rationalsnil )"rat" rationalsnil )"int" integersnil )"nzrat" nil )"[numfield, numfield -> numfield]" nil )"[numfield, numfield -> numfield]" nil )nil integers nil )"bool" reals nil )"{i | i <= x & x < i + 1}" floor_ceil nil )nil number_fields nil )"[numfield, nznum -> numfield]" number_fieldsnil ))nil ))nil 3544783715"" (skeep)"" (expand "mod" )"" (name "k" "floor(radix/i)" )"" (case "k = radix/i" )"1" (expand "FDivInt" )"1" (replace -1 :dir rl)"1" (expand "FtoR" )"1" (lemma "expt_plus" )"1" (inst - "Fexp(f)-1" "1" "radix" )"1" (assert ) nil nil )) nil ))nil ))nil ))nil ))nil )"2" (assert )"2" (replace -1) (("2" (assert ) nil nil )) nil ))nil ))nil ))nil ))nil ))nil )"(divides(n))" dividesnil )"(divides(m))" dividesnil )"{k | abs(k) < abs(j)}" mod nil )nil exponentiation nil )nil exponentiation nil )"rat" rationalsnil )"rat" rationalsnil )"posint" exponentiationnil )"posrat" nil )"int" integersnil )"posrat" exponentiationnil )"nnrat" exponentiation nil )"[numfield, numfield -> numfield]" nil )nil float nil )nil reals nil )"int" integersnil )"real" float nil )"float" float nil )"real" realsnil )"nzrat" nil )nil numbers nil )nil booleans nil )"[T, T -> boolean]" equalities nil )"[number -> boolean]" nil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integersnil )nil integers nil )nil booleans nil )AND const-decl "[bool, bool -> bool]" booleans nil )"bool" reals nil )"bool" reals nil )nil number_fields nil )"[numfield, numfield -> numfield]" nil )"{i | i <= x & x < i + 1}" floor_ceil nil )"boolean" notequal nil )nil number_fields nil )"[numfield, nznum -> numfield]" number_fieldsnil )nil integers nil )"bool" reals nil )nil integers nil )"above(1)" float nil )nil integers nil ))nil 3318954543"" (grind-reals)"" (wrap-formula -1 "exp" )"1" (rewrite "exp_ln" -1) (("1" (grind-reals) nil nil ))nil )"2" (grind-reals) nil nil ))nil ))nil )nil numbers nil )"[T, T -> boolean]" equalities nil )"[number -> boolean]" nil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"bool" reals nil )nil real_types nil )"bool" reals nil )nil real_types nil )"real" ln_exp "lnexp_fnd/" )"{py | x = ln(py)}" ln_exp "lnexp_fnd/" )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integersnil )nil integers nil )nil integers nil )"above(1)" float nil )"bool" booleans nil )"(bijective?[T, T])" identity nil )"bool" functions nil )nil booleans nil )nil booleans nil )nil ln_exp "lnexp_fnd/" )nil ln_exp "lnexp_fnd/" ))nil ))nil 3318954543 ("" (subtype-tcc) nil nil )nil booleans nil )nil booleans nil )NOT const-decl "[bool -> bool]" booleans nil )nil numbers nil )"[number -> boolean]" nil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"bool" reals nil )nil real_types nil )"bool" reals nil )nil real_types nil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil )"boolean" notequal nil )"real" ln_exp "lnexp_fnd/" )"real" realsnil ))nil ))nil 3318954554"" (skeep)"" (expand "^^" )"" (case "radix=0" )"1" (assert ) nil nil )"2" (assert )"2" "-(ln(r) / ln(radix)) * ln(radix) = -ln(r)" )"1" (hide -1)"1" (rewrite "exp_neg" )"1" (rewrite "exp_ln" ) nil nil )) nil ))nil )"2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil )"real" realsnil )"real" reals nil )"nnreal" expt "lnexp_fnd/" )"posreal" real_types nil )"real" realsnil )nil ln_exp "lnexp_fnd/" )nil ln_exp "lnexp_fnd/" )"real" ln_exp "lnexp_fnd/" )nil real_types nil )nil real_types nil )"bool" reals nil )"[numfield, nznum -> numfield]" number_fieldsnil )nil number_fields nil )"boolean" notequal nil )"[numfield -> numfield]" number_fields nil )"[numfield, numfield -> numfield]" nil )nil number_fields nil )nil numbers nil )nil booleans nil )"[T, T -> boolean]" equalities nil )"[number -> boolean]" nil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integersnil )nil integers nil )nil booleans nil )"bool" reals nil )nil integers nil )"above(1)" float nil ))nil 3319195930 ("" (subtype-tcc) nil nil )nil booleans nil )nil booleans nil )NOT const-decl "[bool -> bool]" booleans nil )nil numbers nil )"[number -> boolean]" nil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integersnil )nil integers nil )"boolean" notequal nil ))nil ))nil 3545651010 ("" (subtype-tcc) nil nil )nil booleans nil )nil booleans nil )NOT const-decl "[bool -> bool]" booleans nil )nil numbers nil )"[number -> boolean]" nil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integersnil )nil integers nil )"boolean" notequal nil ))nil ))nil 3319197088"" (skeep)"" (expand "^" )"" (case "e1 >= 0" )"1" (grind-reals)"1" (rewrite "hathat_nat" )"1" (expand "^" ) (("1" (propax) nil nil )) nil ))nil ))nil )"2" (grind-reals)"2" (lemma "hathat_diff" )"2" (inst -1 "radix" "-e1" "0" )"2" (case-replace "0--e1=e1" )"1" (rewrite -2)"1" (rewrite "hathat_nat" )"1" (expand "^" )"1" (grind-reals) nil nil )) nil ))nil ))nil )"2" (hide -1 2 3) (("2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"real" exponentiation nil )nil real_props nil )"posrat" nil )"int" integers nil )"nnreal" nil )nil number_fields nil )"[numfield -> numfield]" number_fields nil )nil expt "lnexp_fnd/" )"real" realsnil )"nnreal" nil )"posrat" nil )nil real_props nil )"nzreal" nil )"int" integersnil )"[T, T -> boolean]" equalities nil )"[numfield, numfield -> numfield]" nil )nil expt "lnexp_fnd/" )"(total_order?[real])" nil )"posnat" exponentiationnil )"nnrat" exponentiation nil )"posrat" exponentiationnil )nil naturalnumbers nil )"above(1)" float nil )nil integers nil )nil real_types nil )"bool" reals nil )nil real_types nil )nil expt "lnexp_fnd/" )nil numbers nil )nil booleans nil )"[number -> boolean]" nil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )nil booleans nil )"bool" reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integersnil )nil integers nil ))nil 3321636413"" (skeep)"" (lemma "radix_div_vNum" ) (("" (inst?) nil nil )) nil ))nil )nil float nil )nil naturalnumbers nil )nil integers nil )nil float nil ))nil ))nil 3319211903"" (skeep)"" (case-replace "-vNum(b) / radix = -(vNum(b)/radix)" )"1" (hide -1)"1" (lemma "integers.closed_neg" )"1" (inst?)"1" (hide 2)"1" (lemma "radix_div_vNum" )"1" (inst -1 "b" ) nil nil )) nil ))nil ))nil ))nil ))nil )"2" (hide 2) (("2" (assert ) nil nil )) nil ))nil ))nil )"nzrat" nil )"nzrat" rationalsnil )nil numbers nil )nil booleans nil )"[T, T -> boolean]" equalities nil )"[number -> boolean]" nil )nil number_fields nil )nil number_fields nil )"boolean" notequal nil )nil number_fields nil )"[numfield, nznum -> numfield]" number_fieldsnil )"[numfield -> numfield]" number_fields nil )nil naturalnumbers nil )nil integers nil )nil float nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integersnil )nil integers nil )nil booleans nil )"bool" reals nil )nil integers nil )"bool" reals nil )nil integers nil )"posnat" float nil )"above(1)" float nil )"int" integersnil )"nzint" integersnil )nil integers nil )nil float nil )"Format" float nil )"posrat" nil ))nil ))nil 3318871237 ("" (subtype-tcc) nil nil )nil booleans nil )nil booleans nil )NOT const-decl "[bool -> bool]" booleans nil )nil naturalnumbers nil )nil integers nil )nil float nil )nil integers nil )nil float nil )"bool" float nil )"real" exponentiation nil )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil )"(total_order?[real])" nil )"int" integers nil )"int" integersnil )"posnat" exponentiationnil )"posnat" float nil )"{j: nonneg_int | j >= i}" real_defs nil )"posint" exponentiationnil ))nil ))nil 3318871237 ("" (subtype-tcc) nil nil )nil booleans nil )nil booleans nil )NOT const-decl "[bool -> bool]" booleans nil )nil naturalnumbers nil )nil integers nil )nil float nil )nil integers nil )nil float nil )"real" exponentiation nil )"(strict_total_order?[real])" real_props nil )"{j: nonneg_int | j >= i}" real_defs nil )"posint" exponentiationnil )"(total_order?[real])" nil )"(total_order?[real])" nil )"int" integers nil )"posnat" exponentiationnil )"{n: nonneg_real | n >= m AND n >= -m}" nil )"posnat" float nil )"bool" float nil )"bool" float nil )"bool" float nil )"bool" float nil )"real" float nil )"nnrat" exponentiation nil )"posrat" exponentiationnil ))nil ))nil 3318871237 ("" (subtype-tcc) nil nil )nil booleans nil )nil booleans nil )NOT const-decl "[bool -> bool]" booleans nil )nil naturalnumbers nil )nil integers nil )nil float nil )nil integers nil )nil float nil )"bool" float nil )"real" exponentiation nil )"posnat" exponentiationnil )"int" integers nil )"(total_order?[real])" nil )"(total_order?[real])" nil )"posint" exponentiationnil )"{j: nonneg_int | j >= i}" real_defs nil )"(strict_total_order?[real])" real_props nil )"posnat" float nil )"bool" float nil )"bool" float nil )"bool" float nil )"(divides(n))" dividesnil )"(divides(m))" dividesnil ))nil ))nil 3318871237 ("" (subtype-tcc) nil nil )nil booleans nil )nil booleans nil )NOT const-decl "[bool -> bool]" booleans nil )nil naturalnumbers nil )nil integers nil )nil float nil )nil integers nil )nil float nil )"posnat" exponentiationnil )"int" integers nil )"(total_order?[real])" nil )"(total_order?[real])" nil )"posint" exponentiationnil )"{j: nonneg_int | j >= i}" real_defs nil )"(strict_total_order?[real])" real_props nil )"real" exponentiation nil )"posnat" float nil )"bool" float nil )"(divides(n))" dividesnil )"(divides(m))" dividesnil ))nil ))nil 3318871660"" (skosimp)"" (grind-reals)"" (expand "abs" 4 2)"" (wrap-formula 2 "abs" )"" (expand "abs" 1 2)"" (cancel-by 4 "abs(Fnum(f!1))" )"" (hide-all-but (-1 3)) (("" (grind) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )nil real_props nil )nil real_props nil )"int" integers nil )"(total_order?[real])" nil )"{j: nonneg_int | j >= i}" real_defs nil )"nonneg_int" integers nil )"{j: posint | j >= i}" real_defs nil )"(strict_total_order?[real])" real_props nil )"int" integersnil )"(total_order?[real])" nil )NOT const-decl "[bool -> bool]" booleans nil )nil numbers nil )"[number -> boolean]" nil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals 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 )"[numfield, numfield -> numfield]" nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integersnil )nil integers nil )"bool" reals nil )nil integers nil )"above(1)" float nil )nil float nil )nil naturalnumbers nil )nil float nil )"bool" float nil )nil integers nil )nil integers nil )"posnat" float nil )nil booleans nil )nil booleans nil )"bool" functions nil )"(bijective?[T, T])" identity nil )"bool" booleans nil )nil real_props nil )nil real_props nil )"rat" rationalsnil )"bool" reals nil )nil real_props nil )"(strict_total_order?[real])" real_props nil )"boolean" notequal nil )"Format" float nil )"(Fbounded?(b!1))" float nil )"{n: nonneg_real | n >= Fnum(f!1) AND n >= -Fnum(f!1)}" nil )nil number_fields nil )"[numfield, nznum -> numfield]" number_fieldsnil )nil nil )"nzrat" nil )"[T, T -> boolean]" equalities nil )"{n: nonneg_real | n >= m AND n >= -m}" nil )"(divides(m))" dividesnil )"(divides(n))" dividesnil ))nil ))nil 3319287157"" (skosimp)"" (split)"1" "v!1(b!1)((# Fnum := radix * Fnum(f!1), Fexp := Fexp(f!1) - 1 #))" )"1" (replaces -2)"1" (expand "FtoR" )"1" (rewrite "expt_div" :dir rl)"1" (rewrite "expt_x1" ) (("1" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil )"2" "v!1(b!1)((# Fnum := radix * Fnum(f!1), Fexp := Fexp(f!1) - 1 #))" )"2" (assert ) nil nil )) nil ))nil ))nil )nil reals nil )"boolean" notequal nil )nil exponentiation nil )"posrat" exponentiationnil )"rat" rationalsnil )"posint" exponentiationnil )"nnrat" exponentiation nil )"posrat" nil )nil exponentiation nil )"(total_order?[real])" nil )"int" integersnil )nil booleans nil )nil booleans nil )NOT const-decl "[bool -> bool]" booleans nil )nil naturalnumbers nil )nil integers nil )nil float nil )nil integers nil )nil float nil )"bool" float nil )"bool" float nil )nil numbers nil )"[number -> boolean]" nil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"bool" reals nil )nil number_fields nil )"[numfield, numfield -> numfield]" nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integersnil )"bool" reals nil )nil integers nil )"bool" reals nil )nil integers nil )"posnat" float nil )nil real_types nil )AND const-decl "[bool, bool -> bool]" booleans nil )"[numfield -> numfield]" number_fields nil )"{n: nonneg_real | n >= m AND n >= -m}" nil )"[T, T -> boolean]" equalities nil )"real" float nil )"bool" reals nil )"[numfield, numfield -> numfield]" nil )"above(1)" float nil )"(divides(m))" dividesnil )"(divides(n))" dividesnil )"{j: nonneg_int | j >= i}" real_defs nil ))nil ))nil 3318887371 ("" (subtype-tcc) nil nil )nil booleans nil )nil booleans nil )NOT const-decl "[bool -> bool]" booleans nil )nil naturalnumbers nil )nil integers nil )nil float nil )nil integers nil )nil float nil )"bool" float nil )"posnat" float nil )"int" integers nil )"(total_order?[real])" nil )"(total_order?[real])" nil )"posint" exponentiationnil )"{j: nonneg_int | j >= i}" real_defs nil )"(strict_total_order?[real])" real_props nil )"boolean" notequal nil ))nil ))nil 3319221457"" (skeep)"" (split)"1" (skosimp*)"1" (expand "Fcanonic?" )"1" (split)"1" (case "Fnormal?(b)(Fopp(f))" )"1" (assert ) nil nil )"2" (hide 2)"2" (expand * "Fnormal?" "Fbounded?" "Fopp" )"2" (flatten)"2" (split)"1" "abs(-(Fnum(f)))=abs(Fnum(f))" )"1" "1" expand "abs" )"1" (grind-reals) nil nil ))nil ))nil ))nil )"2" (propax) nil nil )"3" "abs(radix * -(Fnum(f)))=abs(radix * Fnum(f))" )"3" "3" expand "abs" )"3" "1" case "-(radix * (Fnum(f))) > 0" )"1" (grind-reals) nil nil )"2" "2" "radix" )nil nil ))nil ))nil )"2" "2" "2" "radix" )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (case "Fsubnormal?(b)(Fopp(f))" )"1" (assert ) nil nil )"2" (hide 2)"2" (expand * "Fopp" "Fsubnormal?" "Fbounded?" )"2" (flatten)"2" (split)"1" "abs(-(Fnum(f)))=abs(Fnum(f))" )"1" "1" expand "abs" )"1" (grind-reals) nil nil ))nil ))nil ))nil )"2" (propax) nil nil )"3" (propax) nil nil )"4" "abs(radix * -(Fnum(f)))=abs(radix * Fnum(f))" )"4" "4" expand "abs" )"4" "1" "1" "radix" )nil nil ))nil )"2" "2" "2" "radix" )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (skosimp*)"2" (expand "Fcanonic?" )"2" (split)"1" (case "Fnormal?(b)(f)" )"1" (assert ) nil nil )"2" (hide 2)"2" (expand * "Fnormal?" "Fopp" "Fbounded?" )"2" (flatten)"2" (split)"1" "abs(Fnum(f))=abs(-(Fnum(f)))" )"1" "1" expand "abs" )"1" (grind-reals) nil nil ))nil ))nil ))nil )"2" (propax) nil nil )"3" "abs(radix * (Fnum(f)))=abs(radix * -(Fnum(f)))" )"3" "3" expand "abs" )"3" "1" "1" "radix" )nil nil ))nil )"2" "2" "radix" )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (case " Fsubnormal?(b)(f)" )"1" (assert ) nil nil )"2" (hide 2)"2" (expand * "Fsubnormal?" "Fopp" "Fbounded?" )"2" (flatten)"2" (split)"1" "abs(Fnum(f))=abs(-(Fnum(f)))" )"1" "1" expand "abs" )"1" (grind-reals) nil nil ))nil ))nil ))nil )"2" (propax) nil nil )"3" (propax) nil nil )"4" "abs(radix * (Fnum(f)))=abs(radix * -(Fnum(f)))" )"4" "4" expand "abs" )"4" "1" "1" "radix" )nil nil ))nil )"2" "2" "radix" )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"bool" float nil )"bool" float nil )"float" float nil )"bool" float nil )nil booleans nil )nil booleans nil )nil float nil )nil integers nil )nil float nil )nil integers nil )nil naturalnumbers nil )"bool" float nil )"int" integers nil )"(divides(n))" dividesnil )"(divides(m))" dividesnil )"(strict_total_order?[real])" real_props nil )"{n: nonneg_real | n >= m AND n >= -m}" nil )"[numfield -> numfield]" number_fields nil )nil number_fields nil )AND const-decl "[bool, bool -> bool]" booleans nil )nil real_types nil )"bool" reals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" nil )"[T, T -> boolean]" equalities nil )nil numbers nil )"{j: nonneg_int | j >= i}" real_defs nil )nil real_props nil )nil real_props nil )"above(1)" float nil )"bool" reals nil )"[rational -> boolean]" integersnil )nil rationals nil )"[real -> boolean]" rationals nil )"[numfield, numfield -> numfield]" nil ))nil 3319221336"" (skeep)"" (expand * "Fcanonic?" "Fnormal?" "Fsubnormal?" )"" (split)"1" (flatten) nil nil ) ("2" (flatten) nil nil )) nil ))nil ))nil )"bool" float nil )"(divides(m))" dividesnil )"(divides(n))" dividesnil )"bool" float nil )"bool" float nil ))nil 3319212045"" (skeep)"" (expand "Fpred" )"" (grind-reals)"1" (expand "Fcanonic?" 1 1)"1" case "Fnormal?(b)((# Fnum := -(vNum(b) / radix), Fexp := 1 + Fexp(f) #))" )"1" (assert ) nil nil )"2" (hide 2)"2" (expand * "Fnormal?" "Fbounded?" )"2" (split)"1" (expand "abs" 1 1)"1" (lift-if)"1" (split 1)"1" "1" "1" "vNum(b)" )nil nil ))nil ))nil )"2" "2" (field 2) nil nil ))nil ))nil ))nil ))nil )"2" (grind) nil nil ) ("3" (grind) nil nil ))nil ))nil ))nil )"3" (hide -1 -2 2)"3" (rewrite "integers.closed_neg" )"3" (rewrite "radix_div_vNum" ) nil nil )) nil ))nil ))nil ))nil )"2" (hide -1 -2 1)"2" case "Fnormal?(b)((# Fnum := vNum(b) - 1, Fexp := Fexp(f) - 1 #))" )"1" (expand "Fcanonic?" ) (("1" (assert ) nil nil ))nil )"2" (hide 2)"2" (expand * "Fnormal?" "Fbounded?" )"2" (split)"1" (expand "abs" 1 1)"1" (grind-reals) nil nil )) nil )"2" (case "vNum(b) * radix - radix >=0" )"1" (expand "abs" 1 1)"1" (grind-reals)"1" "1" case "vNum(b) >= radix" )"1" case "vNum(b) + radix <= vNum(b) * 2" )"1" case "vNum(b) * 2 <= vNum(b) * radix" )"1" (grind-reals) nil nil )"2" "2" "vNum(b)" )nil nil ))nil ))nil )"2" "2" (grind-reals) nil nil ))nil ))nil )"2" "2" case "integer_pred(vNum(b) / radix)" )"1" case "vNum(b)>=1" )"1" "1" "div_simple" )"1" "vNum(b)" "radix" )"1" "1" "radix" )"1" "1" "1" case "k!1 >= 1 OR k!1 <=0" )"1" "1" nil nil )"2" case "k!1 * radix <= 0" )"1" nil nil )"2" nil nil ))nil ))nil )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (assert ) nil nil ))nil )"2" assert )"2" rewrite "radix_div_vNum" )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (hide -1 2)"2" (cancel-by 1 "radix" ) nil nil )) nil ))nil ))nil ))nil ))nil ))nil ))nil )"3" (flip-ineq 2)"3" case "abs(radix*(Fnum(f) - 1))< vNum(b) OR abs(radix*(Fnum(f) - 1)) >= vNum(b)" )"1" (split)"1" case "Fsubnormal?(b)((# Fnum := Fnum(f) - 1, Fexp := Fexp(f) #))" )"1" (expand "Fcanonic?" )"1" (assert ) nil nil )) nil )"2" (hide 2 3)"2" (expand * "Fsubnormal?" "Fbounded?" )"2" (split)"1" (mult-by 1 "radix" )"1" "abs(Fnum(f) - 1) * radix=abs(radix * (Fnum(f) - 1))" )"1" case " vNum(b) <=vNum(b) * radix" )"1" (grind-reals) nil nil )"2" "2" "vNum(b)" )"2" (assert ) nil nil ))nil ))nil ))nil )"2" "2" rewrite "abs_mult" )"2" expand "abs" 1 2)"2" (grind-reals) nil nil ))nil ))nil ))nil ))nil ))nil )"2" (case "Fbounded?(b)(f)" )"1" expand "Fbounded?" )"1" (flatten) nil nil ))nil )"2" rewrite "FcanonicBounded" )nil nil ))nil )"3" (case "Fbounded?(b)(f)" )"1" expand "Fbounded?" )"1" "1" (grind-reals) nil nil ))nil ))nil )"2" rewrite "FcanonicBounded" )nil nil ))nil )"4" (grind-reals) nil nil ))nil ))nil ))nil ))nil )"2" case "Fnormal?(b)((# Fnum := Fnum(f) - 1, Fexp := Fexp(f) #))" )"1" (expand "Fcanonic?" )"1" (assert ) nil nil )) nil )"2" (hide 3)"2" (expand * "Fnormal?" "Fbounded?" )"2" (split)"1" (case "Fbounded?(b)(f)" )"1" expand "Fbounded?" )"1" "1" "1" expand * "abs" )"1" (grind-reals) nil nil ))nil ))nil ))nil ))nil )"2" rewrite "FcanonicBounded" )nil nil ))nil )"2" (case "Fbounded?(b)(f)" )"1" expand "Fbounded?" )"1" (flatten) nil nil ))nil )"2" rewrite "FcanonicBounded" )nil nil ))nil )"3" (hide -2 -3 2)"3" (grind-reals) nil nil )) nil ))nil ))nil ))nil ))nil ))nil )"2" (hide -1 -2 2 3) (("2" (grind-reals) nil nil ))nil ))nil ))nil )"4" (case "Fbounded?(b)(f)" )"1" (expand "Fbounded?" )"1" (flatten)"1" (split-ineq -2)"1" case "abs(radix*(Fnum(f) - 1))< vNum(b) OR abs(radix*(Fnum(f) - 1)) >= vNum(b)" )"1" (split)"1" case "Fsubnormal?(b)((# Fnum := Fnum(f) - 1, Fexp := Fexp(f) #))" )"1" (expand "Fcanonic?" )"1" (assert ) nil nil )) nil )"2" (hide 4)"2" expand * "Fsubnormal?" "Fbounded?" )"2" "1" case " abs(Fnum(f) - 1) <= abs(radix * (Fnum(f) - 1))" )"1" (grind-reals) nil nil )"2" rewrite "abs_mult" )"2" expand "abs" 1 2)"2" "abs(Fnum(f) - 1)" )"2" (assert ) nil nil ))nil ))nil ))nil ))nil )"2" (grind-reals) nil nil ))nil ))nil ))nil ))nil )"2" case "Fnormal?(b)((# Fnum := Fnum(f) - 1, Fexp := Fexp(f) #))" )"1" (expand "Fcanonic?" )"1" (assert ) nil nil )) nil )"2" (hide 4)"2" expand * "Fnormal?" "Fbounded?" )"2" "1" "1" expand * "abs" )"1" (grind-reals) nil nil ))nil ))nil )"2" (grind-reals) nil nil ))nil ))nil ))nil ))nil ))nil )"2" (grind-reals) nil nil ))nil )"2" case "Fnormal?(b)((# Fnum := Fnum(f) - 1, Fexp := Fexp(f) #))" )"1" (expand "Fcanonic?" )"1" (assert ) nil nil )) nil )"2" (hide 5)"2" (case "Fnormal?(b)(f)" )"1" (expand * "Fnormal?" "Fbounded?" )"1" "1" "1" expand * "abs" )"1" (grind-reals) nil nil ))nil ))nil )"2" "2" expand * "abs" )"2" "1" case "Fnum(f) <=0" )"1" (grind-reals) nil nil )"2" case "Fnum(f) < 1" )"1" (grind-reals) nil nil )"2" "radix" )"2" "2" nil nil ))nil ))nil ))nil ))nil )"2" case "vNum(b) = radix * Fnum(f) OR vNum(b) < radix * Fnum(f)" )"1" "1" (assert ) nil nil )"2" "2" "radix" )"2" "radix" )"2" "(Fnum(f) * radix - radix) / radix=Fnum(f)-1" )"1" "1" "div_simple" )"1" "vNum(b)" "radix" )"1" case "(EXISTS (k:int): vNum(b) = k * radix)" )"1" "1" "vNum(b) / radix=k!1" )"1" "1" nil nil ))nil )"2" nil nil ))nil ))nil )"2" "2" rewrite "radix_div_vNum" )nil nil ))nil ))nil ))nil ))nil ))nil )"2" nil nil ))nil ))nil ))nil ))nil ))nil )"2" (grind-reals) nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (hide 2 4 5)"2" expand "Fcanonic?" )"2" "1" (propax) nil nil )"2" expand "Fsubnormal?" )"2" (propax) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (rewrite "FcanonicBounded" ) nil nil ))nil ))nil ))nil ))nil )"int" integersnil )"float" float nil )nil real_props nil )nil real_props nil )nil real_props nil )nil extra_real_props nil )"float" float nil )nil extra_real_props nil )nil reals nil )nil real_props nil )nil real_props nil )nil real_props nil )"{j: posint | j >= i}" real_defs nil )nil real_props nil )nil float nil )AND const-decl "[bool, bool -> bool]" booleans nil )"nonneg_rat" rationals nil )"bool" reals nil )"[bool, bool -> bool]" booleans nil )"posint" nil )nil integer_props nil )NOT const-decl "[bool -> bool]" booleans nil )OR const-decl "[bool, bool -> bool]" booleans nil )nil real_props nil )nil real_props nil )nil real_props nil )nil integers nil )"even_int" nil )nil real_props nil )nil real_types nil )nil real_props nil )"[numfield, numfield -> numfield]" nil )"bool" float nil )nil float nil )nil integers nil )"bool" booleans nil )"(bijective?[T, T])" identity nil )"bool" functions nil )nil real_props nil )nil real_props nil )nil real_props nil )nil real_props nil )nil real_props nil )"posrat" nil )"bool" reals nil )"(total_order?[real])" nil )"[numfield, numfield -> numfield]" nil )nil real_types nil )nil nil )"[T, T -> boolean]" equalities nil )"posint" nil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil )nil extra_tegies nil )nil real_props nil )"{n: nonneg_real | n >= m AND n >= -m}" nil )"{j: nonneg_int | j >= i}" real_defs nil )"posint" exponentiationnil )"posnat" exponentiationnil )"bool" float nil )nil real_props nil )"real" exponentiation nil )"real" exponentiation nil )nil real_props nil )nil real_props nil )"{r: posrat | r >= q}" real_defs nil )"rat" rationals nil )"rat" rationalsnil )"rat" rationalsnil )"nonneg_int" integers nil )"even_int" nil )"nzrat" nil )"bool" float nil )"nzrat" rationalsnil )nil naturalnumbers nil )nil integers nil )nil float nil )nil integers nil )nil float nil )nil booleans nil )nil booleans nil )"bool" float nil )nil numbers nil )"[number -> boolean]" nil )nil number_fields nil )"[numfield -> numfield]" number_fields nil )nil number_fields nil )"boolean" notequal nil )nil number_fields nil )"[numfield, nznum -> numfield]" number_fieldsnil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integersnil )"bool" reals nil )nil integers nil )"bool" reals nil )nil integers nil )"posnat" float nil )"above(1)" float nil )"[numfield, numfield -> numfield]" nil )"int" integersnil )"nzint" integersnil )"nzrat" nil )"(strict_total_order?[real])" real_props nil )"(divides(m))" dividesnil )"(divides(n))" dividesnil )"posrat" nil )"int" integers nil )nil real_props nil ))nil 3543227658"" (subtype-tcc) nil nil )"boolean" notequal nil )"int" integers nil ))nil ))nil 3543227658"" (skeep)"" (assert )"" (case "x*radix/vNum(b) > 0" )"1" (assert ) nil nil )"2" (hide 2)"2" (cross-mult 1)"2" (lemma "posreal_times_posreal_is_posreal" )"2" (inst?) (("2" (assert ) nil nil )) nil )) nil ))nil ))nil ))nil ))nil ))nil )"real" realsnil )"real" realsnil )"int" integers nil )"int" integersnil )"nnrat" exponentiation nil )"posrat" nil )"(total_order?[real])" nil )nil nil )"(divides(m))" dividesnil )"(divides(n))" dividesnil )"even_int" nil )"nonneg_int" integers nil )nil real_types nil )nil real_types nil )nil extra_real_props nil )"(strict_total_order?[real])" real_props nil )nil numbers nil )nil booleans nil )"[number -> boolean]" nil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )nil booleans nil )"bool" reals nil )nil number_fields nil )"boolean" notequal nil )nil number_fields nil )"[numfield, nznum -> numfield]" number_fieldsnil )"[numfield, numfield -> numfield]" nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integersnil )nil integers nil )nil integers nil )"above(1)" float nil )nil naturalnumbers nil )nil float nil )"bool" reals nil )nil integers nil )nil integers nil )"posnat" float nil ))nil ))nil 3543227658"" (skosimp*) (("" (assert ) nil nil )) nil )"int" integers nil )"int" integersnil )"nnrat" exponentiation nil )"posrat" nil )"(total_order?[real])" nil ))nil ))nil 3543227658"" (skeep)"" (expand "vNum" )"" (rewrite "expt_plus" :dir rl)"" (mult-by -1 "radix ^ (1 + dExp(b) - Prec(b))" )"" (rewrite "expt_plus" :dir rl) nil nil )) nil ))nil ))nil ))nil )"posnat" float nil )nil nil )"bool" reals nil )nil real_types nil )OR const-decl "[bool, bool -> bool]" booleans nil )"real" exponentiation nil )"[numfield, numfield -> numfield]" nil )nil exponentiation nil )"nnrat" exponentiation nil )"posrat" exponentiationnil )"int" integersnil )"int" integers nil )"posint" nil )"posint" exponentiationnil )"above(1)" float nil )"bool" reals nil )nil booleans nil )nil reals nil )"boolean" notequal nil )nil float nil )nil integers nil )nil naturalnumbers nil )"[numfield -> numfield]" number_fields nil )"[numfield, numfield -> numfield]" nil )nil number_fields nil )nil integers nil )"[rational -> boolean]" integersnil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" nil )nil booleans nil )nil numbers nil )nil exponentiation nil )"int" integersnil ))nil ))nil 3543227662"" case "FORALL (b: Format, x: nnreal): ") "1" (skeep)"1" (inst - "b" "x" )"1" (assert ) nil nil )"2" (expand "vNum" )"2" (typepred "radix^(Prec(b)-dExp(b)-1)" )"1" (assert ) nil nil ) ("2" (assert ) nil nil )) nil ))nil ))nil ))nil )"2" (hide 2)"2" (skeep)"2" (expand "vNum" )"2" (rewrite "expt_plus" :dir rl)"2" (name "yy" "x*radix^(dExp(b) - Prec(b) + 1)" )"2" (case "NOT yy>=1" )"1" (hide 2)"1" (expand "yy" +)"1" "radix ^ (Prec(b) - 1 + -dExp(b))" )"1" (lemma "expt_plus" )"1" "1 + dExp(b) - Prec(b)" "Prec(b) - 1 + -dExp(b)" "radix" )"1" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil )"2" case "NOT x = yy*radix ^ (Prec(b) - 1 + -dExp(b))" )"1" (hide 2)"1" (expand "yy" +)"1" (rewrite "expt_plus" :dir rl)"1" (assert ) nil nil )) nil ))nil ))nil )"2" (replace -1 +)"2" (rewrite "ln_div" )"1" case "radix ^ (Prec(b) - 1 + -dExp(b)) * radix = radix ^ (Prec(b) + -dExp(b))" )"1" replace -1)"1" "1" rewrite "ln_mult" )"1" rewrite "expt_plus" +)"1" rewrite "ln_mult" )"1" assert )"1" case "(ln(radix ^ -dExp(b)) + ln(yy)) / ln(radix) = (ln(yy)/ln(radix))-dExp(b)" )"1" replace -1)"1" "1" "floor_eq_log_nat_ge_1" )"1" "yy" "radix" )"1" case "floor((ln(yy) / ln(radix)) - dExp(b)) = floor(ln(yy) / ln(radix)) - dExp(b)" )"1" assert )nil nil )"2" "2" "floor_plus_int" )"2" "-dExp(b)" "ln(yy)/ln(radix)" )"2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" case "ln(radix^(-dExp(b)))/ln(radix) = -dExp(b)" )"1" (assert ) nil nil )"2" "2" "2" rewrite "ln_expt" )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" "expt_plus" )"2" "Prec(b)-1+-dExp(b)" "1" "radix" )"2" expand "^" - 3)"2" expand "expt" )"2" expand "expt" )"2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (hide 2)"2" "posreal_times_posreal_is_posreal" )"2" (inst?) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"3" (hide 2)"3" (skeep)"3" (expand "vNum" )"3" (mult-by 1 "radix ^ (Prec(b) - 1 + -dExp(b))" )"3" (lemma "expt_plus" )"3" "1 + dExp(b) - Prec(b)" "Prec(b) - 1 + -dExp(b)" "radix" )"3" (assert )"3" (replace -1 :dir rl)"3" (lemma "expt_plus" )"3" "Prec(b)" "-dExp(b)-1" "radix" )"3" (assert ) nil nil )) nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"4" (hide 2) (("4" (skeep) (("4" (assert ) nil nil )) nil ))nil )"5" (hide 2)"5" (skeep)"5" (assert )"5" (lemma "exp_ln" )"5" (inst?)"5" (replace -3) (("5" (assert ) nil nil )) nil ))nil ))nil ))nil ))nil ))nil )"6" (hide 2)"6" (skeep)"6" (cross-mult 1)"6" (assert )"6" (lemma "posreal_times_posreal_is_posreal" )"6" (inst?) (("6" (assert ) nil nil )) nil )) nil ))nil ))nil ))nil ))nil )"7" (hide 2) (("7" (skeep) (("7" (assert ) nil nil )) nil ))nil ))nil )nil extra_real_props nil )"nonneg_int" integers nil )"even_int" nil )"(divides(n))" dividesnil )"(divides(m))" dividesnil )nil ln_exp "lnexp_fnd/" )nil ln_exp "lnexp_fnd/" )"Format" float nil )nil ln_exp "lnexp_fnd/" )"real" realsnil )"real" exponentiation nil )nil ln_exp "lnexp_fnd/" )"odd_int" integersnil )"real" realsnil )nil floor_ceil nil )nil ln_approx"lnexp_fnd/" )nil ln_exp "lnexp_fnd/" )nil reals nil )nil real_props nil )nil nil )nil real_props nil )nil exponentiation nil )"nnreal" float nil )"int" integersnil )nil exponentiation nil )nil reals nil )"real" realsnil )"(strict_total_order?[real])" real_props nil )NOT const-decl "[bool -> bool]" booleans nil )"real" float nil )"(total_order?[real])" nil )"int" integers nil )"nnreal" nil )"posint" nil )"nnrat" exponentiation nil )"posrat" exponentiationnil )"posint" exponentiationnil )"int" integersnil )"posrat" nil )"nnreal" nil )"real" realsnil )nil naturalnumbers nil )nil integers nil )nil float nil )nil numbers nil )nil booleans nil )"[number -> boolean]" nil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )nil booleans nil )"bool" reals nil )nil real_types nil )"[bool, bool -> bool]" booleans nil )nil number_fields nil )"[numfield, numfield -> numfield]" nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integersnil )nil integers nil )OR const-decl "[bool, bool -> bool]" booleans nil )"boolean" notequal nil )"real" exponentiation nil )"bool" reals nil )"above(1)" float nil )"[numfield, numfield -> numfield]" nil )"[numfield -> numfield]" number_fields nil )nil integers nil )nil integers nil )"posnat" float nil )"[T, T -> boolean]" equalities nil )nil integers nil )AND const-decl "[bool, bool -> bool]" booleans nil )"bool" reals nil )"bool" reals nil )"[numfield, numfield -> numfield]" nil )"{i | i <= x & x < i + 1}" floor_ceil nil )nil number_fields nil )"[numfield, nznum -> numfield]" number_fieldsnil )nil real_types nil )nil real_types nil )"real" ln_exp "lnexp_fnd/" )"[n: nat, {y | y < p AND x = p ^ n * y}]" "reals/" ))nil ))nil 3318940757 ("" (subtype-tcc) nil nil )nil booleans nil )nil booleans nil )NOT const-decl "[bool -> bool]" booleans nil )nil numbers nil )"[number -> boolean]" nil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"bool" reals nil )nil real_types nil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil )"nonneg_int" integers nil )"(divides(n))" dividesnil )"(divides(m))" dividesnil )"nat" exponentiation nil )"int" integersnil )"(strict_total_order?[real])" real_props nil )nil power_series "series/" )"boolean" notequal nil )"posnat" float nil )"int" integers nil )"nnrat" exponentiation nil )"posrat" exponentiationnil )"posint" exponentiationnil ))nil ))nil 3318940757"" (skeep)"" (expand "Fcanonic?" )"" case "Fsubnormal?(b) ") "1" (assert ) nil nil )"2" (hide 2)"2" (expand * "Fsubnormal?" )"2" case "abs(radix * floor(x * radix ^ (dExp(b)))) < vNum(b)" )"1" (expand "Fbounded?" )"1" (split)"1" (rewrite "abs_mult" -1)"1" (expand "abs" -1 1)"1" (mult-by 1 "radix" )"1" "abs(floor(x * radix ^ (dExp(b))))" "radix" )"1" case "vNum(b)<=vNum(b) * radix" )"1" (grind-reals) nil nil )"2" "2" "vNum(b)" )"2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (propax) nil nil ))nil ))nil )"2" (hide 2)"2" (rewrite "abs_mult" )"2" (expand "abs" 1 1)"2" (div-by 1 "radix" )"2" case "floor(x * radix ^ (dExp(b))) >=0" )"1" expand "abs" 1 1)"1" case "x * radix ^ (dExp(b)) < vNum(b) / radix" )"1" (grind-reals) nil nil )"2" "2" "expt_inverse" )"2" "dExp(b)+1" "radix" )"2" "-(dExp(b) + 1)=-dExp(b) - 1" )"1" rewrite -2 -3)"1" rewrite "expt_plus" -3)"1" nil nil ))nil ))nil )"2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" (grind-reals) nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"3" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"bool" float nil )"{n: nonneg_real | n >= m AND n >= -m}" nil )nil integers nil )nil integers nil )"posnat" float nil )"(total_order?[real])" nil )"[T, T -> boolean]" equalities nil )nil extra_real_props nil )nil reals nil )nil real_props nil )nil real_types nil )nil real_props nil )"(total_order?[real])" nil )"posint" nil )nil nil )"{j: posint | j >= i}" real_defs nil )"{j: nonneg_int | j >= i}" real_defs nil )nil real_props nil )"bool" float nil )nil real_props nil )"posint" nil )nil reals nil )nil real_props nil )nil real_props nil )nil real_props nil )nil real_props nil )"posint" nil )nil exponentiation nil )nil exponentiation nil )"nzint" integersnil )"[numfield, numfield -> numfield]" nil )nil exponentiation nil )"[numfield, nznum -> numfield]" number_fieldsnil )nil number_fields nil )"posrat" nil )"(divides(m))" dividesnil )"(divides(n))" dividesnil )"nonneg_int" integers nil )"int" integersnil )"nnrat" exponentiation nil )"posrat" nil )"(strict_total_order?[real])" real_props nil )"nat" floor_ceilnil )"int" integers nil )"nnreal" nil )nil naturalnumbers nil )nil integers nil )nil float nil )nil integers nil )nil float nil )nil booleans nil )nil booleans nil )"bool" float nil )nil numbers nil )"[number -> boolean]" nil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integersnil )nil integers nil )AND const-decl "[bool, bool -> bool]" booleans nil )"bool" reals nil )"bool" reals nil )nil number_fields nil )"[numfield, numfield -> numfield]" nil )"{i | i <= x & x < i + 1}" floor_ceil nil )"[numfield, numfield -> numfield]" nil )"bool" reals nil )nil real_types nil )OR const-decl "[bool, bool -> bool]" booleans nil )"boolean" notequal nil )"real" exponentiation nil )"bool" reals nil )"above(1)" float nil )"[numfield -> numfield]" number_fields nil )"posint" exponentiationnil ))nil ))"" 3318950531"" (skeep) (("" (assert ) nil nil )) nil )"int" integers nil )"int" integersnil )"nnrat" exponentiation nil )"posrat" nil )"(strict_total_order?[real])" real_props nil ))nil 3318940757"" (skeep)"" (expand "Fcanonic?" )"" case "Fsubnormal?(b) ") "1" (assert ) nil nil )"2" (hide 2)"2" (expand * "Fsubnormal?" )"2" case "abs(radix * floor(x * radix ^ (dExp(b)))) < vNum(b)" )"1" (expand "Fbounded?" )"1" (split)"1" (rewrite "abs_mult" -1)"1" (expand "abs" -1 1)"1" (mult-by 1 "radix" )"1" "abs(floor(x * radix ^ (dExp(b))))" "radix" )"1" case "vNum(b)<=vNum(b) * radix" )"1" (grind-reals) nil nil )"2" "2" "vNum(b)" )"2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (propax) nil nil ))nil ))nil )"2" (hide 2)"2" (rewrite "abs_mult" )"2" (expand "abs" 1 1)"2" (expand "abs" 1 1)"2" (div-by 1 "radix" )"2" case "x * radix ^ (dExp(b)) < vNum(b) / radix" )"1" "1" (postpone) nil nil )"2" (postpone) nil nil ))nil )"2" "2" "radix ^ (dExp(b))" )"2" case "radix ^ (-dExp(b) - 1)=1/radix ^ (dExp(b)+1)" )"1" rewrite -1 -2)"1" rewrite "expt_plus" )"1" rewrite "expt_x1" )"1" nil nil ))nil ))nil ))nil )"2" "2" (grind) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"3" (grind-reals) nil nil ))nil ))nil ))nil ))nil ))nil ))nil )nil nil ))nil 3318940757"" (skeep)"" (expand "vNum" )"" (rewrite "expt_plus" :dir rl)"" (mult-by 2 "radix ^ (Prec(b) - 1 + -dExp(b))" )"" (assert )"" (rewrite "expt_plus" :dir rl)"" (assert ) nil nil )) nil ))nil ))nil ))nil ))nil ))nil )"posnat" float nil )"nnreal" nil )nil real_props nil )"bool" reals nil )nil real_types nil )nil real_types nil )OR const-decl "[bool, bool -> bool]" booleans nil )"real" exponentiation nil )"[numfield, numfield -> numfield]" nil )"[numfield, numfield -> numfield]" nil )nil exponentiation nil )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil )"posrat" nil )"nnrat" exponentiation nil )"posrat" exponentiationnil )"int" integersnil )"int" integers nil )"posint" nil )"posint" exponentiationnil )"above(1)" float nil )"bool" reals nil )nil booleans nil )nil reals nil )"boolean" notequal nil )nil float nil )nil integers nil )nil naturalnumbers nil )"[numfield -> numfield]" number_fields nil )"[numfield, numfield -> numfield]" nil )nil number_fields nil )nil integers nil )"[rational -> boolean]" integersnil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" nil )nil booleans nil )nil numbers nil )nil exponentiation nil )"int" integersnil ))nil ))nil 3318940757"" (skeep)"" (expand "vNum" )"" (rewrite "expt_plus" :dir rl)"" (mult-by 1 "radix ^ (1 + dExp(b) - Prec(b))" )"" (rewrite "expt_plus" :dir rl)"" (assert ) nil nil )) nil ))nil ))nil ))nil ))nil )"posnat" float nil )nil nil )"bool" reals nil )nil real_types nil )OR const-decl "[bool, bool -> bool]" booleans nil )"real" exponentiation nil )"[numfield, numfield -> numfield]" nil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil )"[numfield, numfield -> numfield]" nil )"bool" reals nil )"posrat" nil )"nnreal" nil )nil exponentiation nil )"int" integersnil )"int" integers nil )"above(1)" float nil )"bool" reals nil )nil booleans nil )nil reals nil )"boolean" notequal nil )nil float nil )nil integers nil )nil naturalnumbers nil )"[numfield -> numfield]" number_fields nil )"[numfield, numfield -> numfield]" nil )nil number_fields nil )nil integers nil )"[rational -> boolean]" integersnil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" nil )nil booleans nil )nil numbers nil )nil exponentiation nil )"int" integersnil )"posint" nil )"nnrat" exponentiation nil )"posrat" exponentiationnil )"posint" exponentiationnil ))nil ))"fix" 3399983082"" (skeep)"" (lemma "RND_log_compute" )"" (inst?)"" (assert )"" (replace -1 :dir rl)"" (hide -1)"" (skosimp)"" (expand "Fcanonic?" )"" (assert )"" (flatten)"" (typepred "e!1" )"" replace -2 :hide? t)"" "AA" "floor(ln(x * radix / vNum(b)) / ln(radix))" )"" replace -1)"" expand "Fnormal?" )"" expand "Fbounded?" )"" "1" case "0<= floor(x * radix ^ (-AA))" )"1" expand "abs" 1 1)"1" "1" case "x * radix ^ (-AA) < vNum(b)" )"1" nil nil )"2" "2" case "x * radix ^ (-AA)=x * radix ^^ (-AA)" )"1" replace -1)"1" case "x * radix ^^ (-AA) < x * radix ^^ (-(ln(x * radix / vNum(b)) / ln(radix)-1))" )"1" case "x * radix ^^ (-(ln(x * radix / vNum(b)) / ln(radix) - 1)) <= vNum(b)" )"1" "1" "WW" "x * radix ^^ (-(ln(x * radix / vNum(b)) / ln(radix) - 1))" )"1" assert )nil nil ))nil ))nil )"2" "2" "(ln(x * radix / vNum(b)) / ln(radix) - 1)=ln(x / vNum(b)) / ln(radix)" )"1" "1" rewrite "hathatln" )"1" nil nil ))nil ))nil )"2" "2" rewrite "ln_div" )"2" rewrite "ln_div" )"2" rewrite "ln_mult" )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" "x" )"2" "2" expand "^^" )"2" "exp_strict_increasing" )"2" expand "strict_increasing?" )"2" rewrite "2" "2" "2" case "(ln(x * radix / vNum(b)) / ln(radix) - 1) < AA" )"1" rewrite "ln_div" )"1" rewrite "ln_mult" )"1" "1" "ln(radix)" )"1" nil nil )"2" "2" case "ln(radix) > 0" )"1" "1" nil nil )"2" nil nil ))nil )"2" "2" "ln_strict_increasing" )"2" expand "strict_increasing?" )"2" "1" "radix" )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" case "x>0" )"1" nil nil )"2" "2" nil nil ))nil ))nil ))nil ))nil )"2" "2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" rewrite "hathat_int" )nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" case "0 <= x * radix ^ (-AA)" )"1" "1" nil nil ))nil )"2" "2" nil nil ))nil ))nil ))nil )"2" case "-dExp(b) <= ln(x * radix / vNum(b)) / ln(radix)" )"1" nil nil )"2" "2" "ln(radix)" "1" "exp" )"1" "exp(-dExp(b) * ln(radix))=radix^^(-dExp(b))" )"1" rewrite "exp_ln" )"1" "radix" )"1" "vNum(b)" )"1" " radix ^ (-dExp(b)) / radix=radix ^ (-dExp(b) - 1)" )"1" "1" rewrite "hathat_int" )"1" "radix" )"1" "1" "(vNum(b) * radix ^ (-dExp(b))) / radix =radix ^ (-dExp(b) - 1) * vNum(b)" )"1" "1" nil nil ))nil )"2" "2" "expt_plus" )"2" "-dExp(b)" "-1" "radix" )"2" "2" expand "^" "2" "2" rewrite "expt_x1_aux" )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" "2" case "radix^(-dExp(b)-1)*radix = radix^(-dExp(b)-1)*radix^1" )"1" replace "1" "1" rewrite "expt_plus" nil nil ))nil ))nil )"2" expand "^" "2" expand "expt" "2" expand "expt" "2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" expand "^^" )"2" nil nil ))nil ))nil ))nil )"2" "ln_increasing" )"2" expand "increasing?" )"2" "ln_exp" )"2" nil nil ))nil ))nil ))nil ))nil )"2" "2" case "ln(radix) > 0" )"1" "1" assert )nil nil )"2" nil nil ))nil )"2" "2" "ln_strict_increasing" )"2" expand "strict_increasing?" )"2" "1" "radix" )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"3" expand "abs" 1 1)"3" "3" "radix" )"3" "3" case "vNum(b) / radix <= x * radix ^ (-AA)" )"1" case "integer_pred(vNum(b) / radix)" )"1" "1" "div_simple" )"1" "vNum(b)" "radix" )"1" case "(EXISTS (k:int): vNum(b) = k * radix)" )"1" "1" " vNum(b) / radix=k!1" )"1" nil nil )"2" "2" nil nil ))nil ))nil ))nil )"2" "2" assert )nil nil ))nil ))nil ))nil ))nil ))nil )"2" assert )"2" "2" rewrite "radix_div_vNum" )nil nil ))nil ))nil ))nil )"2" "2" case "vNum(b) / radix <= x * radix ^^ (-(ln(x * radix / vNum(b)) / ln(radix)))" )"1" case "x * radix ^^ (-(ln(x * radix / vNum(b)) / ln(radix))) <= x * radix ^ (-AA)" )"1" "A1" "x * radix ^^ (-(ln(x * radix / vNum(b)) / ln(radix)))" )"1" "A2" "vNum(b) / radix" )"1" assert )nil nil ))nil ))nil )"2" "2" "x" )"2" "2" "radix ^ (-AA)=radix ^^ (-AA)" )"1" expand "^^" )"1" "exp_increasing" )"1" expand "increasing?" )"1" rewrite "1" "1" "-ln(radix)" "1" nil nil )"2" case "ln(radix)>0" )"1" assert )nil nil )"2" "2" "ln_strict_increasing" )"2" expand "strict_increasing?" )"2" "1" "radix" )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" rewrite "hathat_int" )nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" rewrite "hathatln" )"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 float nil )"(total_order?[real])" nil )"int" integersnil )"posrat" nil )"(strict_total_order?[real])" real_props nil )"bool" float nil )"bool" float nil )nil real_props nil )"posint" nil )"nonneg_rat" rationals nil )"nonneg_rat" nil )nil float nil )nil integers nil )nil integer_props nil )nil real_props nil )nil real_props nil )nil real_props nil )nil real_types nil )nil real_types nil )nil real_props nil )nil ln_exp "lnexp_fnd/" )"nzreal" nil )"nonneg_real" float nil )nil real_props nil )nil real_props nil )nil rationals nil )nil rationals nil )nil real_props nil )nil real_props nil )nil real_props nil )nil real_props nil )nil real_props nil )nil real_props nil )nil reals nil )nil ln_exp "lnexp_fnd/" )nil ln_exp "lnexp_fnd/" )"bool" real_fun_preds "reals/" )"posrat" nil )nil reals nil )"posnat" exponentiationnil )nil exponentiation nil )nil real_props nil )nil real_props nil )"int" integersnil )nil real_props nil )nil exponentiation nil )nil extra_real_props nil )"real" exponentiation nil )nil ln_exp "lnexp_fnd/" )"{py | x = ln(py)}" ln_exp "lnexp_fnd/" )OR const-decl "[bool, bool -> bool]" booleans nil )"real" exponentiation nil )"[numfield -> numfield]" number_fields nil )"(total_order?[real])" nil )nil float nil )nil ln_exp "lnexp_fnd/" )nil real_props nil )nil ln_exp "lnexp_fnd/" )nil ln_exp "lnexp_fnd/" )"bool" real_fun_preds"reals/" )nil real_props nil )nil real_props nil )nil real_props nil )nil nil )"nonneg_real" float nil )nil real_props nil )nil real_props nil )nil real_props nil )nil real_props nil )nil real_props nil )"bool" functions nil )"(bijective?[T, T])" identity nil )"bool" booleans nil )nil real_props nil )"(strict_total_order?[real])" real_props nil )nil float nil )"nzreal" nil )nil real_props nil )nil ln_exp "lnexp_fnd/" )nil ln_exp "lnexp_fnd/" )"odd_int" integersnil )"real" realsnil )nil real_props nil )"real" realsnil )"[numfield, numfield -> numfield]" nil )"real" reals nil )"nnreal" expt "lnexp_fnd/" )nil real_types nil )"{n: nonneg_real | n >= m AND n >= -m}" nil )"bool" float nil )"(divides(m))" dividesnil )"(divides(n))" dividesnil )"nonneg_int" integers nil )"[T, T -> boolean]" equalities nil )nil integers nil )AND const-decl "[bool, bool -> bool]" booleans nil )"bool" reals nil )"bool" reals nil )nil number_fields nil )"[numfield, numfield -> numfield]" nil )"{i | i <= x & x < i + 1}" floor_ceil nil )"boolean" notequal nil )nil number_fields nil )"[numfield, nznum -> numfield]" number_fieldsnil )"bool" reals nil )nil real_types nil )"real" ln_exp "lnexp_fnd/" )"[numfield, numfield -> numfield]" nil )"above(1)" float nil )nil integers nil )nil integers nil )"posnat" float nil )nil integers nil )"[rational -> boolean]" integersnil )nil rationals nil )"[real -> boolean]" rationals nil )NOT const-decl "[bool -> bool]" booleans nil )"real" realsnil )"nnreal" nil )"nat" floor_ceilnil )nil real_types nil )"bool" reals nil )nil booleans nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" nil )nil booleans nil )nil numbers nil )nil float nil )nil integers nil )nil naturalnumbers nil )"real" realsnil )"int" integers nil )"nnreal" nil )"posint" nil )"nnrat" exponentiation nil )"posrat" exponentiationnil )"posint" exponentiationnil ))nil 3399983047";;; Proof RND_aux_TCC6-2 for formula float.RND_aux_TCC6" ";;; Proof RND_aux_TCC6-2 for formula float.RND_aux_TCC6" ";;; Proof RND_aux_TCC6-2 for formula float.RND_aux_TCC6" expand "Fcanonic?" )";;; Proof RND_aux_TCC6-2 for formula float.RND_aux_TCC6" assert )";;; Proof RND_aux_TCC6-2 for formula float.RND_aux_TCC6" ";;; Proof RND_aux_TCC6-2 for formula float.RND_aux_TCC6" "e!1" )";;; Proof RND_aux_TCC6-2 for formula float.RND_aux_TCC6" replace -3 :hide? t)";;; Proof RND_aux_TCC6-2 for formula float.RND_aux_TCC6" "AA" "floor(ln(x * radix / vNum(b)) / ln(radix))" )";;; Proof RND_aux_TCC6-2 for formula float.RND_aux_TCC6" replace -1)";;; Proof RND_aux_TCC6-2 for formula float.RND_aux_TCC6" expand "Fnormal?" )";;; Proof RND_aux_TCC6-2 for formula float.RND_aux_TCC6" expand "Fbounded?" )";;; Proof RND_aux_TCC6-2 for formula float.RND_aux_TCC6" "1" case "0<= floor(x * radix ^ (-AA))" )"1" expand "abs" 1 1)"1" "1" case "x * radix ^ (-AA) < vNum(b)" )"1" (grind-reals) nil )"2" "2" case "x * radix ^ (-AA)=x * radix ^^ (-AA)" )"1" replace -1)"1" case "x * radix ^^ (-AA) < x * radix ^^ (-(ln(x * radix / vNum(b)) / ln(radix)-1))" )"1" case "x * radix ^^ (-(ln(x * radix / vNum(b)) / ln(radix) - 1)) <= vNum(b)" )"1" "1" "WW" "x * radix ^^ (-(ln(x * radix / vNum(b)) / ln(radix) - 1))" )"1" assert )nil )))))"2" "2" "(ln(x * radix / vNum(b)) / ln(radix) - 1)=ln(x / vNum(b)) / ln(radix)" )"1" "1" rewrite "hathatln" )"1" nil )))))"2" "2" rewrite "ln_div" )"2" rewrite "ln_div" )"2" rewrite "ln_mult" )"2" nil )))))))))))))))"2" "2" "x" )"2" "2" expand "^^" )"2" "exp_strict_increasing" )"2" expand "strict_increasing?" )"2" rewrite "2" "2" "2" case "(ln(x * radix / vNum(b)) / ln(radix) - 1) < AA" )"1" rewrite "ln_div" )"1" rewrite "ln_mult" )"1" "1" "ln(radix)" )"1" nil )"2" "2" case "ln(radix) > 0" )"1" "1" nil )"2" nil )))"2" "2" "ln_strict_increasing" )"2" expand "strict_increasing?" )"2" "1" "radix" )"2" nil )))))))))))))))))))"2" "2" case "x>0" )"1" nil )"2" "2" nil )))))))))"2" "2" nil )))))))))))))))))))))))))))"2" "2" rewrite "hathat_int" )nil )))))))))))))"2" case "0 <= x * radix ^ (-AA)" )"1" "1" (grind-reals) nil )))"2" "2" (grind-reals) nil )))))))"2" case "-dExp(b) <= ln(x * radix / vNum(b)) / ln(radix)" )"1" (grind-reals) nil )"2" "2" "ln(radix)" +)"1" "exp" )"1" "exp(-dExp(b) * ln(radix))=radix^^(-dExp(b))" )"1" rewrite "exp_ln" )"1" "radix" )"1" "vNum(b)" )"1" " radix ^ (-dExp(b)) / radix=radix ^ (-dExp(b) - 1)" )"1" "1" rewrite "hathat_int" )"1" "radix" )"1" "1" "(vNum(b) * radix ^ (-dExp(b))) / radix =radix ^ (-dExp(b) - 1) * vNum(b)" )"1" "1" nil )))"2" "2" "expt_plus" )"2" "-dExp(b)" "-1" "radix" )"2" "2" expand "^" "2" "2" rewrite "expt_x1_aux" )nil )))))))))))))))))))))))"2" "2" "expt_plus" )"2" "-dExp(b)" "-1" "radix" )"2" "2" expand "^" "2" "2" rewrite "expt_x1_aux" )nil )))))))))))))))))))))"2" "2" expand "^^" )"2" (propax) nil )))))))"2" "ln_increasing" )"2" expand "increasing?" )"2" "ln_exp" )"2" nil )))))))))"2" "2" case "ln(radix) > 0" )"1" "1" (assert ) nil )"2" (propax) nil )))"2" "2" "ln_strict_increasing" )"2" expand "strict_increasing?" )"2" "1" "radix" )"2" nil )))))))))))))))))))"3" expand "abs" 1 1)"3" "3" "radix" )"3" "3" case "vNum(b) / radix <= x * radix ^ (-AA)" )"1" case "integer_pred(vNum(b) / radix)" )"1" "1" "div_simple" )"1" "vNum(b)" "radix" )"1" case "(EXISTS (k:int): vNum(b) = k * radix)" )"1" "1" " vNum(b) / radix=k!1" )"1" nil )"2" "2" nil )))))))"2" "2" assert )nil )))))))))))"2" assert )"2" "2" rewrite "radix_div_vNum" )nil )))))))"2" "2" case "vNum(b) / radix <= x * radix ^^ (-(ln(x * radix / vNum(b)) / ln(radix)))" )"1" case "x * radix ^^ (-(ln(x * radix / vNum(b)) / ln(radix))) <= x * radix ^ (-AA)" )"1" (assert ) nil )"2" "2" "x" )"2" "2" "radix ^ (-AA)=radix ^^ (-AA)" )"1" expand "^^" )"1" "exp_increasing" )"1" expand "increasing?" )"1" rewrite "1" "1" "-ln(radix)" "1" nil )"2" case "ln(radix)>0" )"1" assert )nil )"2" "2" "ln_strict_increasing" )"2" expand "strict_increasing?" )"2" "1" "radix" )"2" nil )))))))))))))))))))))))"2" "2" rewrite "hathat_int" )nil )))))))))))))"2" "2" rewrite "hathatln" )"2" nil ))))))))))))))))))))))))))))))))))))))))))";;; developed with shostak decision procedures") "bool" real_fun_preds "reals/" )"bool" real_fun_preds"reals/" ))nil )nil 3318940757"" (skeep)"" case "Fnormal?(b) ") "1" (expand "Fcanonic?" ) (("1" (assert ) nil nil )) nil )"2" (hide 3)"2" "AA" "floor(ln(x * radix / vNum(b)) / ln(radix))" )"2" (replace -1 1)"2" (case "radix ^ (-dExp(b) - 1) * vNum(b) <= x" )"1" (hide 2)"1" (expand * "Fnormal?" "Fbounded?" )"1" (split)"1" (case "0<= floor(x * radix ^ (-AA))" )"1" (expand "abs" 1 1)"1" "1" case "x * radix ^ (-AA) < vNum(b)" )"1" (grind-reals) nil nil )"2" "2" case "x * radix ^ (-AA)=x * radix ^^ (-AA)" )"1" replace -1)"1" case "x * radix ^^ (-AA) < x * radix ^^ (-(ln(x * radix / vNum(b)) / ln(radix)-1))" )"1" case "x * radix ^^ (-(ln(x * radix / vNum(b)) / ln(radix) - 1)) <= vNum(b)" )"1" "1" "WW" "x * radix ^^ (-(ln(x * radix / vNum(b)) / ln(radix) - 1))" )"1" assert )nil nil ))nil ))nil )"2" "2" "(ln(x * radix / vNum(b)) / ln(radix) - 1)=ln(x / vNum(b)) / ln(radix)" )"1" "1" rewrite "hathatln" )"1" nil nil ))nil ))nil )"2" "2" rewrite "ln_div" )"2" rewrite "ln_div" )"2" rewrite "ln_mult" )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" "x" )"2" "2" expand "^^" )"2" "exp_strict_increasing" )"2" expand "strict_increasing?" )"2" rewrite -1)"2" "2" "2" case "(ln(x * radix / vNum(b)) / ln(radix) - 1) < AA" )"1" rewrite "ln_div" )"1" rewrite "ln_mult" )"1" "1" "ln(radix)" )"1" nil nil )"2" "2" case "ln(radix) > 0" )"1" "1" nil nil )"2" nil nil ))nil )"2" "2" "ln_strict_increasing" )"2" expand "strict_increasing?" )"2" "1" "radix" )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" case "x>0" )"1" nil nil )"2" "2" nil nil ))nil ))nil ))nil ))nil )"2" "2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" rewrite "hathat_int" )nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (case "0 <= x * radix ^ (-AA)" )"1" "1" (grind-reals) nil nil ))nil )"2" "2" (grind-reals) nil nil ))nil ))nil ))nil )"2" case "-dExp(b) <= ln(x * radix / vNum(b)) / ln(radix)" )"1" (grind-reals) nil nil )"2" (hide -2 2)"2" "ln(radix)" +)"1" "exp" )"1" "exp(-dExp(b) * ln(radix))=radix^^(-dExp(b))" )"1" rewrite "exp_ln" )"1" "radix" )"1" "vNum(b)" )"1" " radix ^ (-dExp(b)) / radix=radix ^ (-dExp(b) - 1)" )"1" "1" rewrite "hathat_int" )"1" "radix" )"1" "1" "(vNum(b) * radix ^ (-dExp(b))) / radix =radix ^ (-dExp(b) - 1) * vNum(b)" )"1" "1" nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" "expt_plus" )"2" "-dExp(b)" "-1" "radix" )"2" "2" expand "^" 1 2)"2" "2" rewrite "expt_x1_aux" )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" expand "^^" )"2" (propax) nil nil ))nil ))nil ))nil )"2" "ln_increasing" )"2" expand "increasing?" )"2" "ln_exp" )"2" (grind-reals) nil nil ))nil ))nil ))nil ))nil )"2" "2" case "ln(radix) > 0" )"1" "1" (assert ) nil nil )"2" (propax) nil nil ))nil )"2" "2" "ln_strict_increasing" )"2" expand "strict_increasing?" )"2" "1" "radix" )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"3" (expand "abs" 1 1)"3" (grind-reals)"3" "radix" )"3" "3" case "vNum(b) / radix <= x * radix ^ (-AA)" )"1" case "integer_pred(vNum(b) / radix)" )"1" "1" "div_simple" )"1" "vNum(b)" "radix" )"1" case "(EXISTS (k:int): vNum(b) = k * radix)" )"1" "1" " vNum(b) / radix=k!1" )"1" nil nil )"2" "2" nil nil ))nil ))nil ))nil )"2" "2" assert )nil nil ))nil ))nil ))nil ))nil ))nil )"2" assert )"2" "2" rewrite "radix_div_vNum" )nil nil ))nil ))nil ))nil )"2" "2" case "vNum(b) / radix <= x * radix ^^ (-(ln(x * radix / vNum(b)) / ln(radix)))" )"1" case "x * radix ^^ (-(ln(x * radix / vNum(b)) / ln(radix))) <= x * radix ^ (-AA)" )"1" (assert ) nil nil )"2" "2" "x" )"2" "2" "radix ^ (-AA)=radix ^^ (-AA)" )"1" expand "^^" )"1" "exp_increasing" )"1" expand "increasing?" )"1" rewrite -1)"1" "1" "-ln(radix)" "1" nil nil )"2" case "ln(radix)>0" )"1" assert )nil nil )"2" "2" "ln_strict_increasing" )"2" expand "strict_increasing?" )"2" "1" "radix" )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" rewrite "hathat_int" )nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" rewrite "hathatln" )"2" (field) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (assert ) nil nil ))nil ))nil ))nil ))nil )"3" (hide 2 3) (("3" (grind-reals) nil nil )) nil ))nil ))nil )"bool" real_fun_preds "reals/" )"bool" real_fun_preds"reals/" ))nil ))nil 3544343150"" (skeep)"" (case "x>=0" )"1" (split-ineq -1) nil nil ) ("2" (assert ) nil nil ))nil ))nil )nil real_types nil )"bool" reals nil )nil booleans nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" nil )nil booleans nil )nil numbers nil )"[T, T -> boolean]" equalities nil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil )"posrat" nil )"nnrat" exponentiation nil )"int" integersnil )"int" integers nil )"(strict_total_order?[real])" real_props nil )"nnreal" nil )"nnreal" nil ))nil ))nil 3544343150"" (skosimp*)"" (case "radix = 1" )"1" (replace -1) (("1" (assert ) nil nil )) nil )"2" (assert )"2" (lemma "ln_eq_0" )"2" (inst?) (("2" (assert ) nil nil )) nil )) nil ))nil ))nil ))nil )"above(1)" float nil )nil integers nil )"bool" reals nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integersnil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" nil )"[T, T -> boolean]" equalities nil )nil booleans nil )nil numbers nil )"posrat" exponentiationnil )"nnrat" exponentiation nil )"int" integers nil )"(strict_total_order?[real])" real_props nil )"posrat" nil )"int" integersnil )"real" realsnil )"nnreal" nil )"nnreal" nil ))nil ))nil 3544343150"" (skeep)"" (lemma "RND_aux_TCC6" )"" (inst?)"" (assert )"" (skosimp*)"" (inst - "e!1" )"" (ground)"" (lemma "RND_log_compute" )"" (inst?) (("" (assert ) nil nil )) nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )nil float nil )"int" integersnil )"posrat" nil )"(strict_total_order?[real])" real_props nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integersnil )nil integers nil )nil integers nil )AND const-decl "[bool, bool -> bool]" booleans nil )"bool" reals nil )nil number_fields nil )"boolean" notequal nil )nil number_fields nil )"[numfield, nznum -> numfield]" number_fieldsnil )"bool" reals nil )nil real_types nil )"real" ln_exp "lnexp_fnd/" )"[numfield, numfield -> numfield]" nil )"above(1)" float nil )nil integers nil )nil integers nil )"posnat" float nil )"bool" reals nil )"[numfield, numfield -> numfield]" nil )nil float nil )"(total_order?[real])" nil )"nat" floor_ceilnil )"nnreal" nil )"real" realsnil )nil real_types nil )"bool" reals nil )nil booleans nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" nil )nil booleans nil )nil numbers nil )nil float nil )nil integers nil )nil naturalnumbers nil )"posint" nil )"int" integers nil )"posint" exponentiationnil )"nnreal" nil )"nnrat" exponentiation nil )"posrat" exponentiationnil ))nil ))nil 3544343151"" (decompose-equality)"" (decompose-equality)"" (expand "RND_aux" )"" (expand "RND_aux_alt" )"" (lift-if)"" (ground)"" (lemma "RND_log_compute" )"" (inst - "x!1" "x!2" ) (("" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"nnreal" nil )"(strict_total_order?[real])" real_props nil )"posrat" nil )"nat" floor_ceilnil )"real" realsnil )"nnreal" nil )"(total_order?[real])" nil )nil float nil )"int" integers nil )"int" integersnil )"posrat" exponentiationnil )"nnrat" exponentiation nil )"posint" nil )"posint" exponentiationnil )nil naturalnumbers nil )nil integers nil )nil float nil )nil numbers nil )nil booleans nil )"[number -> boolean]" nil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )nil booleans nil )"bool" reals nil )nil real_types nil )nil integers nil )nil float nil )"bool" float nil )"(Fcanonic?(b))" float nil )"(Fcanonic?(b))" float nil ))nil 3319384410 ("" (subtype-tcc) nil nil )nil booleans nil )nil booleans nil )NOT const-decl "[bool -> bool]" booleans nil )nil numbers nil )"[number -> boolean]" nil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"(total_order?[real])" nil )"(total_order?[real])" nil ))nil ))nil 3319384410 ("" (subtype-tcc) nil nil )nil booleans nil )nil booleans nil )NOT const-decl "[bool -> bool]" booleans nil )nil numbers nil )"[number -> boolean]" nil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"real" reals nil )"(total_order?[real])" nil )"(total_order?[real])" nil ))nil ))nil 3319384410"" (skeep)"" (lemma "FcanonicOpp" )"" (inst -1 "b" "RND_aux(b)(-x)" )"1" (assert ) nil nil ) ("2" (assert ) nil nil )) nil ))nil ))nil )nil float nil )"(total_order?[real])" nil )"(Fcanonic?(b))" float nil )"bool" float nil )nil real_types nil )nil float nil )nil integers nil )nil float nil )nil integers nil )nil naturalnumbers nil )nil numbers nil )nil booleans nil )"[number -> boolean]" nil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )nil booleans nil )"bool" reals nil )nil number_fields nil )"[numfield -> numfield]" number_fields nil )"real" float nil )"real" reals nil )"(total_order?[real])" nil ))nil ))nil 3319384410"" (skeep)"" (rewrite "FpredCanonic" )"" (lemma "FcanonicOpp" )"" (inst -1 "b" "RND_aux(b)(-x)" )"" (assert ) nil nil )) nil ))nil ))nil ))nil )"(total_order?[real])" nil )"real" reals nil )nil float nil )nil naturalnumbers nil )nil integers nil )nil float nil )nil integers nil )nil float nil )"float" float nil )nil numbers nil )nil booleans nil )"[number -> boolean]" nil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )nil booleans nil )"bool" reals nil )nil real_types nil )"bool" float nil )"(Fcanonic?(b))" float nil )nil number_fields nil )"[numfield -> numfield]" number_fields nil )"(total_order?[real])" nil )nil float nil ))nil ))nil 3319894713"" (skeep)"" (lemma "FcanonicOpp" )"" (inst -1 "b" "RND_Min(b)(-x)" ) (("" (assert ) nil nil ))nil ))nil ))nil )nil float nil )"[numfield -> numfield]" number_fields nil )nil number_fields nil )"(Fcanonic?(b))" float nil )"bool" float nil )nil booleans nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" nil )nil booleans nil )nil numbers nil )nil float nil )nil integers nil )nil float nil )nil integers nil )nil naturalnumbers nil )"real" reals nil ))nil ))nil 3318763952 ("" (subtype-tcc) nil nil )nil booleans nil )nil booleans nil )NOT const-decl "[bool -> bool]" booleans nil )nil numbers nil )"[number -> boolean]" nil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integersnil )nil integers nil )"(strict_total_order?[real])" real_props nil )"boolean" notequal nil ))nil ))nil 3318763991"" (skeep)"" (use "both_sides_expt_gt1_lt" )"1" (assert ) nil nil ) ("2" (assert ) nil nil )) nil ))nil )nil exponentiation nil )nil real_types nil )nil real_types nil )"bool" reals nil )nil numbers nil )nil booleans nil )"[number -> boolean]" nil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )nil booleans nil )"bool" reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integersnil )nil integers nil )nil integers nil )"above(1)" float nil )"(strict_total_order?[real])" real_props nil )"(strict_total_order?[real])" real_props nil )"nnrat" exponentiation nil )"posrat" exponentiationnil ))nil 3318865674"" (skeep)"" (use " both_sides_expt_gt1_lt" )"1" (assert ) nil nil ) ("2" (assert ) nil nil )) nil ))nil )nil exponentiation nil )nil real_types nil )nil real_types nil )"bool" reals nil )nil numbers nil )nil booleans nil )"[number -> boolean]" nil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )nil booleans nil )"bool" reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integersnil )nil integers nil )nil integers nil )"above(1)" float nil )"(strict_total_order?[real])" real_props nil )"(strict_total_order?[real])" real_props nil )"nnrat" exponentiation nil )"posrat" exponentiationnil ))nil 3318783146 ("" (subtype-tcc) nil nil )nil booleans nil )nil booleans nil )NOT const-decl "[bool -> bool]" booleans nil )nil numbers nil )"[number -> boolean]" nil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integersnil )nil integers nil )"(total_order?[real])" nil )"boolean" notequal nil ))nil ))nil 3318865769"" (skeep)"" (use "both_sides_expt_gt1_le" )"1" (assert ) nil nil ) ("2" (assert ) nil nil )) nil ))nil )nil exponentiation nil )nil real_types nil )nil real_types nil )"bool" reals nil )nil numbers nil )nil booleans nil )"[number -> boolean]" nil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )nil booleans nil )"bool" reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integersnil )nil integers nil )nil integers nil )"above(1)" float nil )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil )"nnrat" exponentiation nil )"posrat" exponentiationnil ))nil 3318865818"" (skeep)"" (use "both_sides_expt_gt1_le" )"1" (assert ) nil nil ) ("2" (assert ) nil nil )) nil ))nil )nil exponentiation nil )nil real_types nil )nil real_types nil )"bool" reals nil )nil numbers nil )nil booleans nil )"[number -> boolean]" nil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )nil booleans nil )"bool" reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integersnil )nil integers nil )nil integers nil )"above(1)" float nil )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil )"nnrat" exponentiation nil )"posrat" exponentiationnil ))nil 3318698220"" (skeep)"" (case "0 <= e1" )"1" (case "e1 <= 0" )"1" (grind) nil nil )"2" (rewrite "Exp_increq_2" )"2" (rewrite "expt_x0" ) (("2" (assert ) nil nil ))nil ))nil ))nil )"2" (rewrite "Exp_increq_2" )"2" (rewrite "expt_x0" ) (("2" (assert ) nil nil )) nil ))nil ))nil ))nil )nil integers nil )"[rational -> boolean]" integersnil )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]" nil )nil booleans nil )nil numbers nil )nil exponentiation nil )"posint" exponentiationnil )nil float nil )"real" exponentiation nil )"real" exponentiation nil )"(total_order?[real])" nil )"posrat" nil )"posnat" exponentiationnil )"posrat" exponentiationnil )"nnrat" exponentiation nil )"(total_order?[real])" nil ))nil 3318696278"" (skeep)"" (case "e1 <= e2" )"1" (case "e2 <= e1" )"1" (grind) nil nil )"2" (rewrite "Exp_increq_2" ) nil nil ))nil )"2" (rewrite "Exp_increq_2" ) nil nil ))nil ))nil )nil integers nil )"[rational -> boolean]" integersnil )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]" nil )nil booleans nil )nil numbers nil )nil float nil )"real" exponentiation nil )"real" exponentiation nil )"int" integers nil )"(total_order?[real])" nil )"posrat" nil )"posnat" exponentiationnil )"posrat" exponentiationnil )"nnrat" exponentiation nil )"(total_order?[real])" nil ))nil 3319544683 ("" (subtype-tcc) nil nil )"boolean" notequal nil )) nil ))nil 3319544683"" (induct "n" )"1" (grind-reals) (("1" (rewrite "expt_x1" ) nil nil )) nil )"2" (rewrite "expt_x1" ) (("2" (grind-reals) nil nil )) nil )"3" (skeep)"3" (split 1)"1" (skosimp*)"1" (rewrite "expt_plus" )"1" (rewrite "expt_x1" )"1" (grind-reals)"1" (rewrite "odd_times_odd_is_odd" ) nil nil ))nil ))nil ))nil ))nil )"2" (skosimp*)"2" (rewrite "expt_plus" )"2" (rewrite "expt_x1" )"2" (lemma "even_or_odd" )"2" (inst -1 "radix" )"2" (flatten)"2" (case "even?(radix)" )"1" (hide -2 -3 1)"1" "even_iff_not_odd" )"1" "(radix ^ (1 + j)) * radix" )"1" "1" case "even?((radix ^ (1 + j)) * radix)" )"1" (assert ) nil nil )"2" rewrite "int_times_even_is_even" )nil nil ))nil ))nil ))nil ))nil ))nil )"2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )nil naturalnumbers nil )"[numfield, numfield -> numfield]" nil )nil integers nil )nil integers nil )nil naturalnumbers nil )"bool" integers nil )"posint" nil )nil integers nil )nil integers nil )"posint" nil )"(divides(n))" dividesnil )"(divides(m))" dividesnil )"int" integersnil )nil exponentiation nil )nil reals nil )"odd_int" nil )nil exponentiation nil )nil naturalnumbers nil )"[numfield, numfield -> numfield]" nil )nil number_fields nil )"real" exponentiation nil )"boolean" notequal nil )OR const-decl "[bool, bool -> bool]" booleans nil )"above(1)" float nil )nil integers nil )"bool" reals nil )"bool" integers nil )"[bool, bool -> bool]" booleans nil )nil defined_types nil )nil naturalnumbers nil )"bool" reals nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integersnil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" nil )nil booleans nil )nil numbers nil )"posint" nil )"posint" exponentiationnil ))nil 3319562624"" (induct "n" )"1" (grind-reals) (("1" (rewrite "expt_x1" ) nil nil )) nil )"2" (grind-reals) (("2" (rewrite "expt_x1" ) nil nil )) nil )"3" (skeep)"3" (rewrite 1 "expt_plus" )"1" rewrite "expt_plus" :subst"n0x" "radix" "i" "j+1" "j" "1" ))"1" (rewrite "expt_x1" )"1" (split 1)"1" (skosimp*)"1" (split -2)"1" (rewrite "even_times_int_is_even" ) nil nil )"2" (propax) nil nil ))nil ))nil )"2" (skosimp*)"2" (case-replace "1+j=j+1" )"1" (hide -1)"1" "even?((radix ^ (j + 1)) * radix)=NOT odd?((radix ^ (1 + j)) * radix)" )"1" "even?(radix)=NOT odd?(radix)" )"1" rewrite "odd_times_odd_is_odd" )"1" "odd_or_even_int" "x" "radix ^ (1 + j)" ))"1" (assert ) nil nil ))nil ))nil )"2" "even_iff_not_odd" "x" "radix" ))"2" (grind-reals) nil nil ))nil ))nil )"2" (case-replace "1+j=j+1" )"1" "even_iff_not_odd" "x" "((radix ^ (j + 1)) * radix)" ))"1" (grind-reals) nil nil ))nil )"2" (assert ) nil nil ))nil ))nil ))nil )"2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil )"2" (skosimp*)"2" (hide 2)"2" (case-replace "even?(radix)=NOT odd?(radix)" )"1" "even?(radix^(1+j))=NOT odd?(radix^(1+j))" )"1" (case "odd?(radix ^ (1 + j))" )"1" "even?(radix^(2+j))=NOT odd?(radix^(2+j))" )"1" (case-replace "2+j=1+(1+j)" )"1" (rewrite "expt_plus" )"1" rewrite "expt_x1" )"1" rewrite "odd_times_odd_is_odd" )nil nil ))nil ))nil )"2" (assert ) nil nil ))nil )"2" "even_iff_not_odd" :subst"x" "radix ^ (2 + j)" ))"2" (grind-reals) nil nil )) nil ))nil )"2" (assert ) nil nil ))nil )"2" "even_iff_not_odd" :subst"x" "radix ^ (1 + j)" ))"2" (grind-reals) nil nil )) nil ))nil )"2" "even_iff_not_odd" :subst ("x" "radix" ))"2" (grind-reals) nil nil )) nil ))nil ))nil ))nil ))nil ))nil ))nil )nil reals nil )nil exponentiation nil )"(divides(m))" dividesnil )"(divides(n))" dividesnil )"posint" nil )nil naturalnumbers nil )nil integers nil )nil integers nil )nil naturalnumbers nil )"[numfield, numfield -> numfield]" nil )NOT const-decl "[bool -> bool]" booleans nil )"bool" integers nil )"(total_order?[real])" nil )"[T, T -> boolean]" equalities nil )nil integers nil )nil integers nil )"int" integersnil )"posint" nil )"odd_int" nil )nil exponentiation nil )nil naturalnumbers nil )"[numfield, numfield -> numfield]" nil )nil number_fields nil )"real" exponentiation nil )"boolean" notequal nil )OR const-decl "[bool, bool -> bool]" booleans nil )"above(1)" float nil )nil integers nil )"bool" reals nil )"bool" integers nil )"[bool, bool -> bool]" booleans nil )nil defined_types nil )nil naturalnumbers nil )"bool" reals nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integersnil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" nil )nil booleans nil )nil numbers nil )"posint" nil )"posint" exponentiationnil ))nil 3319368570"" (skeep)"" (expand * "FtoR" "Fopp" ) (("" (grind-reals) nil nil ))nil ))nil )"float" float nil )"real" float nil )"rat" rationalsnil )"int" integers nil )"nnrat" exponentiation nil )"rat" rationals nil ))nil 3545043329"" (grind) (("" (decompose-equality) nil nil )) nil )nil integers nil )"[rational -> boolean]" integersnil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" nil )nil booleans nil )nil numbers nil )nil float nil )"[numfield -> numfield]" number_fields nil )nil number_fields nil )"int" integers nil )"float" float nil ))nil 3545043536"" (skeep) (("" (grind) nil nil )) nil )"int" integers nil )"(divides(n))" dividesnil )"(divides(m))" dividesnil )"int" integersnil )"float" float nil )"float" float nil )"float" float nil )"float" float nil ))nil 3545043574 ("" (grind) nil nil )"float" float nil )"float" float nil )"float" float nil )"float" float nil )"(divides(m))" dividesnil )"(divides(n))" dividesnil )"int" integers nil ))nil 3319889181"" (skeep)"" (expand * "Fabs" "FtoR" )"" (rewrite "abs_mult" )"" (expand "abs" 1 3) (("" (propax) nil nil )) nil ))nil ))nil ))nil )"real" float nil )"rat" rationalsnil )"posrat" exponentiationnil )"nnrat" exponentiation nil )"float" float nil )"{n: nonneg_real | n >= m AND n >= -m}" nil )"nonneg_rat" rationals nil )"{j: nonneg_int | j >= i}" real_defs nil )"{r: posrat | r >= q}" real_defs nil )"above(1)" float nil )nil integers nil )"bool" reals nil )"real" exponentiation nil )"bool" reals nil )"boolean" notequal nil )OR const-decl "[bool, bool -> bool]" booleans nil )nil booleans nil )"[rational -> boolean]" integersnil )nil rationals nil )"[real -> boolean]" rationals nil )nil float nil )nil integers nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" nil )nil booleans nil )nil numbers nil )nil real_props nil ))nil 3319889234"" (skeep)"" (expand * "Fplus" "FtoR" )"" (rewrite "expt_div" :dir rl)"" (rewrite "expt_div" :dir rl) (("" (field) nil nil ))nil ))nil ))nil ))nil )"real" float nil )"rat" rationalsnil )"{k: int | k <= i AND k <= j}" nil )"nnrat" exponentiation nil )"posrat" exponentiationnil )"float" float nil )NOT const-decl "[bool -> bool]" booleans nil )OR const-decl "[bool, bool -> bool]" booleans nil )"bool" reals nil )"rat" rationalsnil )"posrat" nil )"odd_int" integersnil )"above(1)" float nil )nil integers nil )"bool" reals nil )nil reals nil )"boolean" notequal nil )min const-decl "{p: real | p <= m AND p <= n}" real_defsnil )"bool" reals nil )AND const-decl "[bool, bool -> bool]" booleans nil )nil booleans nil )nil float nil )nil integers nil )"[rational -> boolean]" integersnil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" nil )nil booleans nil )nil numbers nil )nil exponentiation nil ))nil 3545564973"" (expand "associative?" )"" (skeep)"" (expand "Fplus" + 2)"" (assert )"" (expand "min" )"" (assert )"" (lift-if)"" (ground)"1" (expand "Fplus" + 3)"1" (expand "min" )"1" (lift-if)"1" "1" "1" assert )"1" expand "Fplus" + 1)"1" expand "min" )"1" assert )"1" expand "Fplus" + 1)"1" expand "min" )"1" assert )"1" "expt_plus" )"1" "Fexp(x)-Fexp(y)" "Fexp(y)-Fexp(z)" "radix" )"1" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" assert )"2" expand "Fplus" 2 1)"2" expand "min" )"2" assert )"2" expand "Fplus" 2 1)"2" expand "min" )"2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (expand "Fplus" 2 3)"2" (expand "min" )"2" (lift-if)"2" "1" expand "Fplus" 1 1)"1" expand "min" )"1" "1" "1" expand "Fplus" )"1" assert )"1" expand "min" )"1" assert )"1" "expt_plus" )"1" "Fexp(x)-Fexp(z)" "Fexp(y)-Fexp(x)" "radix" )"1" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" assert )"2" expand "Fplus" )"2" expand "min" )"2" assert )"2" "expt_plus" )"2" "Fexp(y)-Fexp(z)" "Fexp(z)-Fexp(x)" "radix" )"2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" expand "Fplus" 2 1)"2" expand "min" )"2" assert )"2" expand "Fplus" 2 1)"2" expand "min" )"2" assert )"2" "expt_plus" )"2" "Fexp(y)-Fexp(x)" "Fexp(z)-Fexp(y)" "radix" )"2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"int" integersnil )"rat" rationalsnil )"rat" rationalsnil )"(strict_total_order?[real])" real_props nil )"real" realsnil )nil exponentiation nil )"posint" exponentiationnil )nil numbers nil )nil booleans nil )"[number -> boolean]" nil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integersnil )nil integers nil )nil number_fields nil )"[numfield, numfield -> numfield]" nil )nil float nil )"boolean" notequal nil )nil reals nil )nil booleans nil )"bool" reals nil )nil integers nil )"above(1)" float nil )"int" integersnil )"posrat" nil )nil exponentiation nil )"(divides(n))" dividesnil )"(divides(m))" dividesnil )min const-decl "{p: real | p <= m AND p <= n}" real_defsnil )"float" float nil )"posrat" exponentiationnil )"nnrat" exponentiation nil )"{k: int | k <= i AND k <= j}" nil )"bool" operator_defs nil ))nil 3545572260 ("" (grind) nil nil )"(divides(n))" dividesnil )"(divides(m))" dividesnil )"int" integersnil )"bool" operator_defs nil )"float" float nil ))nil 3319889348"" (skeep)"" (expand * "Fminus" "FtoR" )"" (rewrite "expt_div" :dir rl)"" (rewrite "expt_div" :dir rl) (("" (field) nil nil ))nil ))nil ))nil ))nil )"real" float nil )"{k: int | k <= i AND k <= j}" nil )"nnrat" exponentiation nil )"posrat" exponentiationnil )"rat" rationalsnil )"float" float nil )NOT const-decl "[bool -> bool]" booleans nil )OR const-decl "[bool, bool -> bool]" booleans nil )"bool" reals nil )"rat" rationalsnil )"posrat" nil )"odd_int" integersnil )"above(1)" float nil )nil integers nil )"bool" reals nil )nil reals nil )"boolean" notequal nil )min const-decl "{p: real | p <= m AND p <= n}" real_defsnil )"bool" reals nil )AND const-decl "[bool, bool -> bool]" booleans nil )nil booleans nil )nil float nil )nil integers nil )"[rational -> boolean]" integersnil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" nil )nil booleans nil )nil numbers nil )nil exponentiation nil ))nil 3319889392"" (skeep)"" (expand * "Fmult" "FtoR" )"" (rewrite "expt_plus" ) (("" (field) nil nil )) nil ))nil ))nil )"real" float nil )"float" float nil )"bool" reals nil )OR const-decl "[bool, bool -> bool]" booleans nil )NOT const-decl "[bool -> bool]" booleans nil )"posrat" nil )"nnrat" exponentiation nil )"rat" rationalsnil )"posrat" exponentiationnil )"(divides(m))" dividesnil )"(divides(n))" dividesnil )"above(1)" float nil )nil integers nil )"bool" reals nil )nil booleans nil )nil reals nil )"boolean" notequal nil )nil float nil )nil integers nil )"[rational -> boolean]" integersnil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" nil )nil booleans nil )nil numbers nil )nil exponentiation nil ))nil 3545401225"" (grind)"" (decompose-equality)"" (expand "divides" )"" (expand "divides" )"" (inst + "1" ) (("" (assert ) nil nil )) nil )) nil ))nil ))nil ))nil )"float" float nil )nil float nil )"bool" divides nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integersnil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" nil )nil booleans nil )nil numbers nil )"bool" divides nil )"(divides(m))" dividesnil )"(divides(n))" dividesnil )"float" float nil )"float" float nil )"float" float nil ))nil 3545401264"" (grind) (("" (decompose-equality) nil nil )) nil )"bool" divides nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integersnil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" nil )nil booleans nil )nil numbers nil )nil number_fields nil )"[numfield, numfield -> numfield]" nil )nil float nil )"(divides(m))" dividesnil )"(divides(n))" dividesnil )"float" float nil )"float" float nil )"float" float nil ))nil ))nil 3545401333 ("" (grind) nil nil )"float" float nil )"float" float nil )"float" float nil )min const-decl "{p: real | p <= m AND p <= n}" real_defsnil )"real" exponentiation nil )"real" exponentiation nil )"float" float nil )"float" float nil ))nil 3545401340 ("" (grind) nil nil )"float" float nil )"float" float nil )"float" float nil )min const-decl "{p: real | p <= m AND p <= n}" real_defsnil )"real" exponentiation nil )"real" exponentiation nil )"float" float nil )"float" float nil ))nil 3545654182 ("" (subtype-tcc) nil nil )"boolean" notequal nil )) nil ))nil 3545654182 ("" (subtype-tcc) nil nil ) nil nil ))nil 3545654182"" (skeep)"" (case "FORALL (ik,jk:int): integer_pred(ik*jk)" )"1" (inst?)"1" (hide 2)"1" case "FORALL (rp:nat): integer_pred((radix^rp)/(k^rp))" )"1" (inst?) nil nil )"2" (hide 2)"2" (induct "rp" )"1" (grind) nil nil )"2" (skeep)"2" case "FORALL (ik,jk:int): integer_pred(ik*jk)" )"1" case "(radix^(j+1))/(k^(j+1)) = (radix/k)*((radix^j/k^j))" )"1" (replaces -1)"1" "1" "1" "k" )"1" expand "mod" )"1" case "floor(radix/k) = radix/k" )"1" replace -1 :dir rl)"1" (assert ) nil nil ))nil )"2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (expand "^" +)"2" expand "expt" + 1)"2" expand "expt" + 2)"2" (assert ) nil nil ))nil ))nil ))nil ))nil )"2" (hide-all-but 1)"2" (grind) nil nil )) nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (hide-all-but 1) (("2" (grind) nil nil )) nil ))nil ))nil )"[numfield, numfield -> numfield]" nil )nil number_fields nil )nil integers nil )"[rational -> boolean]" integersnil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" nil )nil booleans nil )nil numbers nil )"(divides(m))" dividesnil )"(divides(n))" dividesnil )"nzrat" nil )"posint" nil )"[numfield, numfield -> numfield]" nil )"nat" float nil )NOT const-decl "[bool -> bool]" booleans nil )nil integers nil )"bool" reals nil )"{i | i <= x & x < i + 1}" floor_ceil nil )"int" integersnil )"nzint" nil )"posint" nil )"real" exponentiation nil )"int" exponentiation nil )"nzreal" exponentiationnil )"(total_order?[real])" nil )"posrat" nil )"posnat" exponentiationnil )nil naturalnumbers nil )nil defined_types nil )"nat" float nil )"{k: nzint | mod(radix, k) = 0}" floatnil )"{k | abs(k) < abs(j)}" mod nil )"{n: nonneg_real | n >= m AND n >= -m}" nil )"[numfield -> numfield]" number_fields nil )AND const-decl "[bool, bool -> bool]" booleans nil )nil real_types nil )"bool" reals nil )nil integers nil )"[T, T -> boolean]" equalities nil )nil integers nil )nil naturalnumbers nil )"above(1)" float nil )nil integers nil )"bool" reals nil )"real" exponentiation nil )"bool" reals nil )OR const-decl "[bool, bool -> bool]" booleans nil )nil booleans nil )"[numfield, nznum -> numfield]" number_fieldsnil )nil number_fields nil )"boolean" notequal nil )nil float nil )"int" exponentiation nil )"nzreal" exponentiationnil )"nzrat" nil )"posint" exponentiationnil ))nil ))nil 3545651846"" (induct "i" )"1" (assert )"1" (skeep)"1" (expand "FDivKexpt" )"1" (assert )"1" (assert )"1" (expand "FtoR" ) (("1" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (skeep)"2" (skeep)"2" case "NOT FORALL (zd:nat): integer_pred((radix ^ (zd)) / (k ^ (zd)))" )"1" (hide (-1 2))"1" (induct "zd" )"1" (grind) nil nil )"2" (skolem 1 "rk" )"2" (flatten)"2" case "FORALL (ik,jk:int): integer_pred(ik*jk)" )"1" "(radix ^ rk) / (k ^ rk)" "radix/k" )"1" (expand "^" )"1" expand "expt" +)"1" (assert ) nil nil ))nil ))nil )"2" (hide-all-but 1)"2" "k" )"2" expand "mod" )"2" case "floor(radix/k) = radix/k" )"1" assert )"1" replace -1 :dir rl)"1" (assert ) nil nil ))nil ))nil )"2" "2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (hide-all-but 1)"2" (grind) nil nil )) nil ))nil ))nil ))nil ))nil ))nil )"2" (label "hyp1" -1)"2" (hide "hyp1" )"2" (inst - "f" "k" )"2" (expand "FDivKexpt" )"2" (mult-by 1 "k" )"1" case "FtoR(f) / k ^ (1 + j) * k = FtoR(f)/k^j" )"1" (replace -1)"1" "1" replace -1 :dir rl)"1" "1" "float_int_def" )"1" "k" )"1" case "FORALL (aa:real): aa*k = aa*FtoR(float_int(k))" )"1" rewrite -1)"1" "1" "FmultCorrect" )"1" rewrite -1 :dir rl)"1" "1" assert )"1" expand "FtoR" )"1" expand "Fmult" )"1" "1" assert )"1" expand "float_int" )"1" case "k^(1+j) = k*(k^j)" )"1" replace "1" "1" "expt_plus" )"1" "1+j" "Fexp(f)-1-j" "radix" )"1" assert )"1" "Fnum(f) * (k ^ j)" )"1" replace "1" "1" assert )"1" "Fexp(f)-j" "j" "radix" )"1" "Fnum(f) * (k ^ j)" )"1" replace "1" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" expand "^" )"2" expand "expt" "2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "hyp1" )"2" "2" assert )"2" case "FORALL (ik,jk:int): integer_pred(ik*jk)" )"1" nil nil )"2" "2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "hyp1" )"2" "2" assert )"2" case "FORALL (ik,jk:int): integer_pred(ik*jk)" )"1" nil nil )"2" "2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" "float_int_def" )"2" "2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (expand "^" )"2" expand "expt" + 1)"2" (assert ) nil nil ))nil ))nil ))nil )"2" (reveal "hyp1" )"2" (inst?)"2" assert )"2" case "FORALL (ik,jk:int): integer_pred(ik*jk)" )"1" (inst?) nil nil )"2" "2" (grind) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )NOT const-decl "[bool -> bool]" booleans nil )"posnat" exponentiationnil )"posrat" nil )"(total_order?[real])" nil )"nzreal" exponentiationnil )"int" exponentiation nil )"real" exponentiation nil )"nat" float nil )"{k: nzint | mod(radix, k) = 0}" floatnil )"nzrat" nil )"posint" nil )"nzint" nil )nil integers nil )"bool" reals nil )"[numfield, numfield -> numfield]" nil )"{i | i <= x & x < i + 1}" floor_ceil nil )"int" integersnil )"[numfield, numfield -> numfield]" nil )"(divides(m))" dividesnil )"(divides(n))" dividesnil )"real" realsnil )nil float nil )nil real_props nil )nil real_props nil )"posrat" exponentiationnil )nil exponentiation nil )"posrat" nil )nil extra_real_props nil )nil reals nil )"float" float nil )"int" integersnil )"float" float nil )nil float nil )"posint" nil )"float" float nil )"nat" float nil )"[numfield, numfield -> numfield]" nil )nil reals nil )nil real_props nil )nil exponentiation nil )"rat" rationalsnil )"nnrat" exponentiation nil )"rat" rationalsnil )"nzrat" nil )"posint" exponentiationnil )"real" realsnil )nil naturalnumbers nil )"real" exponentiation nil )OR const-decl "[bool, bool -> bool]" booleans nil )"[numfield, nznum -> numfield]" number_fieldsnil )nil number_fields nil )"float" float nil )"real" float nil )"above(1)" float nil )nil integers nil )"bool" reals nil )"{k | abs(k) < abs(j)}" mod nil )"{n: nonneg_real | n >= m AND n >= -m}" nil )"[numfield -> numfield]" number_fields nil )nil number_fields nil )AND const-decl "[bool, bool -> bool]" booleans nil )nil real_types nil )"bool" reals nil )nil integers nil )"[T, T -> boolean]" equalities nil )nil integers nil )"boolean" notequal nil )nil float nil )nil defined_types nil )nil naturalnumbers nil )"bool" reals nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integersnil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" nil )nil booleans nil )nil numbers nil )"int" exponentiation nil )"nzreal" exponentiationnil ))nil ))nil 3319899604"" (skeep)"" (expand * "Fbounded?" "Fopp" )"" (flatten)"" (split)"1" (expand "abs" ) (("1" (grind-reals) nil nil )) nil )"2" (propax) nil nil ))nil ))nil ))nil ))nil )"float" float nil )"int" integers nil )"bool" float nil )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil )"{n: nonneg_real | n >= m AND n >= -m}" nil ))nil 3320082424"" (skeep)"" (expand * "Fbounded?" "Fabs" )"" (flatten)"" (split)"1" (grind-reals) nil nil ) ("2" (propax) nil nil ))nil ))nil ))nil ))nil )"float" float nil )"{j: nonneg_int | j >= i}" real_defs nil )"bool" float nil )"(strict_total_order?[real])" real_props nil )nil real_props nil ))nil 3320082549"" (skeep)"" (expand "Fcanonic?" )"" (split)"1" (case "Fnormal?(b)(Fabs(f))" )"1" (assert ) nil nil )"2" (hide 2)"2" (expand "Fnormal?" )"2" (flatten)"2" (split)"1" (rewrite "FabsBounded" ) nil nil )"2" (expand "Fabs" )"2" (expand "abs" )"2" (grind-reals) nil nil )) nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (case "Fsubnormal?(b)(Fabs(f))" )"1" (assert ) nil nil )"2" (hide 2)"2" (expand "Fsubnormal?" )"2" (flatten)"2" (split)"1" (rewrite "FabsBounded" ) nil nil )"2" (expand "Fabs" ) (("2" (propax) nil nil ))nil )"3" (expand "Fabs" )"3" (expand "abs" )"3" (grind-reals) nil nil )) nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"bool" float nil )"bool" float nil )"float" float nil )"bool" float nil )nil booleans nil )nil booleans nil )nil float nil )nil integers nil )nil float nil )nil integers nil )nil naturalnumbers nil )"(divides(n))" dividesnil )"(divides(m))" dividesnil )nil float nil )"{n: nonneg_real | n >= m AND n >= -m}" nil )nil real_props nil )"(total_order?[real])" nil )"int" integers nil )"(strict_total_order?[real])" real_props nil )"nonneg_int" integers nil )"{j: nonneg_int | j >= i}" real_defs nil ))nil 3318679671"" (skeep)"" (expand "FtoR" ) (("" (grind-reals) nil nil )) nil )) nil )"real" float nil )nil real_props nil )"(total_order?[real])" nil )"rat" rationalsnil )"nnrat" exponentiation nil ))nil 3318680805"" (skosimp*)"" (expand "FtoR" ) (("" (grind-reals) nil nil )) nil )) nil )"real" float nil )nil real_props nil )"(total_order?[real])" nil )"rat" rationalsnil )"nnrat" exponentiation nil ))nil 3319904574"" (skeep)"" (split)"1" (skosimp*)"1" (expand "Fle?" )"1" (case "FtoR(p) - FtoR(q) <= 0" )"1" (assert ) nil nil )"2" (rewrite "FminusCorrect" :dir rl)"2" (expand "FtoR" ) (("2" (grind-reals) nil nil ))nil ))nil ))nil ))nil ))nil )"2" (skosimp*)"2" (expand "Fle?" )"2" (case "FtoR(Fminus(p, q)) <= 0" )"1" (expand "FtoR" ) (("1" (grind-reals) nil nil ))nil )"2" (rewrite "FminusCorrect" )"2" (assert ) nil nil )) nil ))nil ))nil ))nil ))nil ))nil )"bool" float nil )nil float nil )"odd_int" integersnil )"nnrat" exponentiation nil )"rat" rationalsnil )nil real_props nil )"(total_order?[real])" nil )"real" realsnil )nil numbers nil )nil booleans nil )"[number -> boolean]" nil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )nil booleans nil )"bool" reals nil )nil number_fields nil )"[numfield, numfield -> numfield]" nil )nil integers nil )nil float nil )"real" float nil )"float" float nil ))nil 3319904922"" (skeep)"" (split)"1" (skosimp*)"1" (expand "Flt?" )"1" (case "FtoR(Fminus(p, q)) < 0" )"1" (rewrite "FminusCorrect" )"1" (assert ) nil nil )) nil )"2" (expand "FtoR" ) (("2" (grind-reals) nil nil ))nil ))nil ))nil ))nil )"2" (skosimp*)"2" (case "FtoR(Fminus(p, q)) < 0" )"1" (expand "FtoR" )"1" (expand "Flt?" ) (("1" (grind-reals) nil nil ))nil ))nil )"2" (rewrite "FminusCorrect" )"2" (assert ) nil nil )) nil ))nil ))nil ))nil ))nil )"bool" float nil )nil real_props nil )"rat" rationalsnil )"nnrat" exponentiation nil )"(total_order?[real])" nil )nil float nil )"real" realsnil )"(strict_total_order?[real])" real_props nil )nil numbers nil )nil booleans nil )"[number -> boolean]" nil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )nil booleans nil )"bool" reals nil )nil integers nil )nil float nil )"real" float nil )"float" float nil ))nil 3320169103"" (skeep)"" (expand "Fmin" )"" (lemma "FleCorrect" )"" (inst -1 "p" "q" )"" (rewrite -1)"" (expand "min" ) (("" (grind-reals) nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"float" float nil )nil integers nil )nil float nil )min const-decl "{p: real | p <= m AND p <= n}" real_defsnil )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil )nil float nil ))nil 3320169183"" (skeep)"" (expand "Fmax" )"" (lemma "FleCorrect" :subst ("p" "p" "q" "q" ))"" (rewrite -1)"" (expand "max" ) (("" (grind-reals) nil nil )) nil ))nil ))nil ))nil ))nil )"float" float nil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil )max const-decl "{p: real | p >= m AND p >= n}" real_defsnil )nil float nil )nil integers nil )nil float nil ))nil 3318681540"" (skosimp*)"" (expand "FtoR" )"" (expand "Fsubnormal?" )"" (flatten)"" (decompose-equality) (("" (grind-reals) nil nil ))nil ))nil ))nil ))nil ))nil )"real" float nil )"{j: posint | j >= i}" real_defs nil )"nonneg_int" integers nil )nil real_props nil )nil real_props nil )"posrat" exponentiationnil )nil float nil )nil integers nil )"int" integers nil )"(strict_total_order?[real])" real_props nil )"{j: nonneg_int | j >= i}" real_defs nil )"rat" rationalsnil )"nnrat" exponentiation nil )"bool" float nil )"(divides(m))" dividesnil )"(divides(n))" dividesnil ))nil 3318683813"" (skeep)"" (expand * "Fnormal?" "Fbounded?" "FtoR" )"" (flatten)"" (case-replace "Fexp(p)=Fexp(q)" )"1" (decompose-equality)"1" (grind-reals) nil nil )) nil )"2" (hide 2)"2" (case "Fexp(p)<Fexp(q)+1" )"1" (case "Fexp(q)<Fexp(p)+1" )"1" (hide -3 -4 -5 -6 -7 -8 -9)"1" (grind) nil nil )) nil )"2" (hide -1 2)"2" (rewrite "Exp_incr_2" )"2" (hide 2)"2" (mult-by 1 "abs(Fnum(q))" )"2" "abs(Fnum(q)) * radix ^ Fexp(q)=abs(Fnum(p)) * radix ^ Fexp(p)" )"1" rewrite "expt_plus" )"1" "radix ^ Fexp(p)" )"1" rewrite "abs_mult" -8)"1" expand "abs" -8 1)"1" (assert ) nil nil ))nil ))nil ))nil ))nil )"2" "2" "abs" )"2" rewrite * ("abs_mult" ))"2" expand "abs" -1 (2 4))"2" rewrite * ("abs_mult" ))"2" expand "abs" -1 3)"2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (hide 2)"2" (rewrite "Exp_incr_2" )"2" (hide 2)"2" (mult-by 1 "abs(Fnum(p))" )"2" "abs(Fnum(p)) * radix ^ Fexp(p)=abs(Fnum(q)) * radix ^ Fexp(q)" )"1" rewrite "expt_plus" )"1" "radix ^ Fexp(q)" )"1" rewrite "abs_mult" -5)"1" expand "abs" -5 1)"1" (assert ) nil nil ))nil ))nil ))nil ))nil )"2" "2" "abs" )"2" rewrite * ("abs_mult" ))"2" expand "abs" -1 (2 4))"2" rewrite * ("abs_mult" ))"2" expand "abs" -1 3)"2" (propax) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"bool" float nil )"real" float nil )"(divides(n))" dividesnil )"(divides(m))" dividesnil )"bool" float nil )nil numbers nil )nil booleans nil )"[T, T -> boolean]" equalities nil )nil integers nil )nil float nil )"posrat" exponentiationnil )"nnrat" exponentiation nil )"nonneg_int" integers nil )"{j: posint | j >= i}" real_defs nil )nil real_props nil )nil real_props nil )"(strict_total_order?[real])" real_props nil )"{j: nonneg_int | j >= i}" real_defs nil )"(total_order?[real])" nil )"int" integers nil )"rat" rationalsnil )"int" integersnil )"[number -> boolean]" nil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )nil booleans nil )"bool" reals nil )nil number_fields nil )"[numfield, numfield -> numfield]" nil )NOT const-decl "[bool -> bool]" booleans nil )nil rationals nil )nil rationals nil )"(strict_total_order?[real])" real_props nil )"bool" booleans nil )"(bijective?[T, T])" identity nil )"bool" functions nil )"[numfield, nznum -> numfield]" number_fieldsnil )nil number_fields nil )"posrat" nil )nil exponentiation nil )"posrat" nil )"posint" exponentiationnil )"int" integersnil )nil exponentiation nil )nil reals nil )"{r: nonneg_rat | r >= q}" real_defs nil )"{r: posrat | r >= q}" real_defs nil )"nonneg_rat" rationals nil )"[numfield, numfield -> numfield]" nil )"(total_order?[real])" nil )"above(1)" float nil )nil integers nil )"bool" reals nil )"real" exponentiation nil )"boolean" notequal nil )OR const-decl "[bool, bool -> bool]" booleans nil )"{n: nonneg_real | n >= m AND n >= -m}" nil )"[numfield -> numfield]" number_fields nil )AND const-decl "[bool, bool -> bool]" booleans nil )nil real_types nil )"bool" reals nil )nil nil )nil float nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integersnil ))nil 3318856796"" (skeep)"" (case "abs(FtoR(q)) < abs (FtoR(p))" )"1" (assert ) nil nil )"2" (expand * "FtoR" "Fnormal?" "Fsubnormal?" )"2" (flatten)"2" (hide -6)"2" (rewrite "abs_mult" )"2" (expand "abs" 1 2)"2" (rewrite "abs_mult" )"2" (expand "abs" 1 3)"2" (mult-by 1 "radix" )"2" "abs(Fnum(q))" *"radix ^ (Fexp(q))" "radix" r)"2" "abs(Fnum(p))" "radix ^ (Fexp(p))" "radix" "2" case "abs(Fnum(q)) * radix < abs(Fnum(p)) * radix" "radix ^ Fexp(q) <= radix ^ Fexp(p)" )"1" (grind-reals) nil nil )"2" "2" expand "Fbounded?" )"2" "2" rewrite -6)"2" rewrite "Exp_increq_1" )nil nil ))nil ))nil ))nil ))nil )"3" "3" rewrite "abs_mult" -2)"3" rewrite "abs_mult" -5)"3" expand "abs" -2 1)"3" expand "abs" -5 1)"3" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"real" float nil )nil float nil )nil integers nil )"{n: nonneg_real | n >= m AND n >= -m}" nil )"[numfield -> numfield]" number_fields nil )nil number_fields nil )AND const-decl "[bool, bool -> bool]" booleans nil )nil real_types nil )"bool" reals nil )"bool" reals nil )nil booleans nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" nil )nil booleans nil )nil numbers nil )"(strict_total_order?[real])" real_props nil )"above(1)" float nil )nil integers nil )"bool" reals nil )"real" exponentiation nil )"boolean" notequal nil )OR const-decl "[bool, bool -> bool]" booleans nil )"[rational -> boolean]" integersnil )nil rationals nil )"[real -> boolean]" rationals nil )nil real_props nil )"{r: posrat | r >= q}" real_defs nil )"{j: nonneg_int | j >= i}" real_defs nil )"nonneg_rat" rationals nil )nil nil )"[numfield, numfield -> numfield]" nil )"(total_order?[real])" nil )"{j: posint | j >= i}" real_defs nil )nil float nil )nil naturalnumbers nil )nil float nil )"int" integers nil )"bool" float nil )nil real_props nil )nil real_props nil )"(total_order?[real])" nil )"bool" reals nil )"nonneg_int" integers nil )"[T, T -> boolean]" equalities nil )"rat" rationalsnil )"posrat" exponentiationnil )"nnrat" exponentiation nil )"bool" float nil )"(divides(n))" dividesnil )"(divides(m))" dividesnil )"bool" float nil ))nil 3318616280"" (skeep)"" (expand "Fcanonic?" )"" (split)"1" (split)"1" (use "FnormalUnique" ) (("1" (assert ) nil nil ))nil )"2" (use "NormalAndSubNormalNotEq" )"2" (assert ) nil nil )) nil ))nil )"2" (split)"1" (lemma "NormalAndSubNormalNotEq" )"1" (inst -1 "b" "q" "p" ) (("1" (assert ) nil nil ))nil ))nil )"2" (use "FsubnormalUnique" ) (("2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil )"bool" float nil )nil float nil )nil float nil )nil integers nil )nil naturalnumbers nil )nil integers nil )nil float nil )nil float nil )nil float nil ))nil 3319283991"" (skeep) (("" (assert ) nil nil )) nil ) nil shostak))nil 3319375218 ("" (subtype-tcc) nil nil )"posnat" float nil )"bool" float nil )"bool" float nil )"bool" float nil )"bool" float nil )"(divides(n))" dividesnil )"(divides(m))" dividesnil ))nil ))nil 3319375218"" (skeep)"" (expand "Fulp" )"" (case-replace "Fnormalize(b)(p)=p" )"" (hide 2)"" (lemma "FcanonicUnique" )"" (inst -1 "b" "Fnormalize(b)(p)" "p" )"" (rewrite -1) nil nil )) nil ))nil ))nil ))nil ))nil ))nil )"real" float nil )nil float nil )"{x: (Fcanonic?(b)) | FtoR(x) = FtoR(f):: real AND Fexp(x) <= Fexp(f)}" nil )"bool" reals nil )"real" float nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" nil )nil numbers nil )AND const-decl "[bool, bool -> bool]" booleans nil )"bool" float nil )"bool" float nil )nil booleans nil )nil float nil )nil integers nil )nil naturalnumbers nil )"[T, T -> boolean]" equalities nil )nil booleans nil )nil float nil )nil integers nil ))nil 3319234645"" (skeep)"" (case "0<= Fnum(p)" )"1" (case "0<= Fnum(q)" )"1" (expand "Fcanonic?" )"1" (split)"1" (split)"1" (case "Fexp(p) < Fexp(q)+1" )"1" (grind-reals) nil nil )"2" (hide 2)"2" (expand * "Fnormal?" "Fbounded?" )"2" (flatten)"2" (hide -2 -3 -4 -5)"2" rewrite "abs_mult" -2)"2" expand * "abs" )"2" "2" rewrite "Exp_incr_2" )"2" "2" "Fnum(p)" )"2" "radix ^ Fexp(p) * Fnum(p)= FtoR(p)" )"1" case "FtoR(q)<radix ^ (1 + Fexp(q)) * Fnum(p)" )"1" nil nil )"2" "2" expand "FtoR" )"2" rewrite "expt_plus" )"2" rewrite "expt_x1" )"2" "radix ^ (Fexp(q))" )"2" "radix * radix ^ Fexp(q) * Fnum(p) / radix ^ (Fexp(q))=radix * Fnum(p)" )"1" nil nil )"2" "2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" expand "FtoR" )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" expand * "Fnormal?" "Fsubnormal?" "Fbounded?" )"2" (flatten)"2" (rewrite "abs_mult" )"2" (rewrite "abs_mult" )"2" (expand * "abs" )"2" "2" "2" "2" case "FtoR(q) < FtoR(p)" )"1" (grind-reals) nil nil )"2" "2" expand "FtoR" )"2" case "Fnum(q) * radix ^ (Fexp(q)) <= Fnum(q) * radix ^ (Fexp(p))" )"1" case "Fnum(q) * radix ^ (Fexp(p)) < Fnum(p) * radix ^ (Fexp(p))" )"1" "1" "AA" "Fnum(q) * radix ^ (Fexp(q))" )"1" "BB" "Fnum(q) * radix ^ (Fexp(p))" )"1" nil nil ))nil ))nil ))nil )"2" "2" "radix ^ (Fexp(p))" )"2" "radix" )"2" nil nil ))nil ))nil ))nil ))nil )"2" "Fnum(q)" )"2" rewrite "Exp_increq_1" )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (expand * "Fsubnormal?" "Fnormal?" "Fbounded?" )"2" (flatten) (("2" (grind-reals) nil nil )) nil ))nil ))nil ))nil )"2" (hide -1 -2 -3 2)"2" (mult-by 1 "radix^Fexp(q)" )"1" (expand "FtoR" ) (("1" (grind-reals) nil nil ))nil )"2" (grind-reals) nil nil ))nil ))nil ))nil )"2" (hide -1 -2 -4 2)"2" (expand "FtoR" ) (("2" (grind-reals) nil nil )) nil ))nil ))nil ))nil )nil float nil )nil integers nil )"bool" reals nil )nil booleans nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" nil )nil booleans nil )nil numbers nil )"nonneg_rat" rationals nil )"float" float nil )"bool" float nil )"above(1)" float nil )nil integers nil )"bool" reals nil )"[rational -> boolean]" integersnil )nil rationals nil )"[real -> boolean]" rationals nil )nil real_props nil )"{j: nonneg_int | j >= i}" real_defs nil )"{j: posint | j >= i}" real_defs nil )"nonneg_int" integers nil )"int" integers nil )"real" float nil )"[T, T -> boolean]" equalities nil )nil reals nil )nil exponentiation nil )"int" integersnil )"posint" exponentiationnil )"posrat" nil )nil exponentiation nil )"[numfield, nznum -> numfield]" number_fieldsnil )nil number_fields nil )"rat" rationalsnil )nil real_props nil )NOT const-decl "[bool -> bool]" booleans nil )nil reals nil )nil extra_real_props nil )nil real_types nil )nil real_props nil )nil real_props nil )"(strict_total_order?[real])" real_props nil )"rat" rationalsnil )"[numfield, numfield -> numfield]" nil )"(total_order?[real])" nil )"real" exponentiation nil )"boolean" notequal nil )OR const-decl "[bool, bool -> bool]" booleans nil )nil real_types nil )"bool" reals nil )nil nil )"nnrat" exponentiation nil )"posrat" exponentiationnil )nil float nil )"{n: nonneg_real | n >= m AND n >= -m}" nil )"bool" float nil )"(divides(m))" dividesnil )"(divides(n))" dividesnil )"bool" float nil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil )"int" integersnil )"bool" reals nil )nil number_fields nil )"[numfield, numfield -> numfield]" nil )"posrat" nil )nil real_props nil )nil real_props nil )nil real_props nil )nil real_props nil )nil real_props nil )nil real_props nil )nil real_props nil )"bool" functions nil )"(bijective?[T, T])" identity nil )"bool" booleans nil )nil rationals nil )nil rationals nil )nil rationals nil )nil float nil )"nzrat" nil )AND const-decl "[bool, bool -> bool]" booleans nil )"int" float nil )"even_int" nil )nil real_props nil )nil real_props nil )nil real_props nil )nil real_props nil )"bool" float nil )nil real_props nil ))nil 3319282878"" (skeep)"" (case-replace "p=Fnormalize(b)(q)" )"1" (hide -1 -2 -4) (("1" (assert ) nil nil )) nil )"2" (hide 2)"2" (lemma "FcanonicUnique" )"2" (inst -1 "b" "p" "Fnormalize(b)(q)" )"2" (rewrite -1) nil nil )) nil ))nil ))nil ))nil ))nil )nil integers nil )nil float nil )nil booleans nil )"[T, T -> boolean]" equalities nil )nil naturalnumbers nil )nil integers nil )nil float nil )nil booleans nil )"bool" float nil )"bool" float nil )AND const-decl "[bool, bool -> bool]" booleans nil )nil numbers nil )"[number -> boolean]" nil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"real" float nil )"bool" reals nil )"{x: (Fcanonic?(b)) | FtoR(x) = FtoR(f):: real AND Fexp(x) <= Fexp(f)}" nil )"(total_order?[real])" nil )nil float nil ))nil 3319890253"" (skeep)"" (split)"1" (case "vNum(b) <= abs(radix * Fnum(p))" )"1" (expand * "Fcanonic?" "Fnormal?" )"1" (assert ) nil nil )) nil )"2" (expand * "Fcanonic?" "Fsubnormal?" )"2" (assert ) nil nil )) nil ))nil )"2" (expand * "Fcanonic?" "Fnormal?" )"2" (assert ) nil nil )) nil ))nil ))nil )"bool" float nil )"bool" float nil )"bool" float nil )"(total_order?[real])" nil )"{j: nonneg_int | j >= i}" real_defs nil )"int" integers nil )nil numbers nil )nil booleans nil )"[number -> boolean]" nil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )nil booleans nil )"bool" reals nil )nil naturalnumbers nil )nil integers nil )nil float nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integersnil )nil integers nil )"bool" reals nil )nil integers nil )"bool" reals nil )nil integers nil )"posnat" float 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}" nil )"[numfield, numfield -> numfield]" nil )"above(1)" float nil )nil float nil )"(divides(n))" dividesnil )"(divides(m))" dividesnil ))nil 3319817879"" (skeep) (("" (rewrite "FoppBounded" ) nil nil )) nil )nil float nil )nil naturalnumbers nil )nil integers nil )nil float nil )nil integers nil )nil float nil ))nil ))nil 3319817880"" (skeep)"" (expand "Fulp" )"" (case "Fnormalize(b)(Fopp(f))=Fopp(Fnormalize(b)(f))" )"1" (rewrite -1)"1" (expand "Fopp" ) (("1" (propax) nil nil )) nil ))nil )"2" (rewrite "FcanonicUnique" :subst ("b" "b" ))"1" case "FtoR(Fnormalize(b)(Fopp(f)))= FtoR(Fopp(f))" )"1" (rewrite -1)"1" (rewrite "FoppCorrect" )"1" (rewrite "FoppCorrect" )"1" (assert ) nil nil )) nil ))nil ))nil )"2" (assert ) nil nil ))nil )"2" (lemma "FcanonicOpp" )"2" (inst -1 "b" "Fnormalize(b)(f)" )"2" (assert ) nil nil )) nil ))nil ))nil ))nil ))nil ))nil )"real" float nil )nil float nil )"posrat" exponentiationnil )"nnrat" exponentiation nil )"real" reals nil )nil float nil )nil float nil )nil integers nil )nil float nil )nil booleans nil )"[T, T -> boolean]" equalities nil )nil naturalnumbers nil )nil integers nil )nil float nil )nil booleans nil )"bool" float nil )"bool" float nil )AND const-decl "[bool, bool -> bool]" booleans nil )nil numbers nil )"[number -> boolean]" nil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"real" float nil )"bool" reals nil )"{x: (Fcanonic?(b)) | FtoR(x) = FtoR(f):: real AND Fexp(x) <= Fexp(f)}" nil )"float" float nil ))nil 3320082088"" (skeep) (("" (rewrite "FabsBounded" ) nil nil )) nil )nil float nil )nil naturalnumbers nil )nil integers nil )nil float nil )nil integers nil )nil float nil ))nil ))nil 3320082088"" (skeep)"" (expand "Fabs" )"" (expand "abs" )"" (grind-reals)"1" "(# Fnum := -Fnum(f), Fexp := Fexp(f) #)=Fopp(f)" )"1" (rewrite "FulpOpp" ) nil nil )"2" (expand "Fopp" ) (("2" (propax) nil nil )) nil ))nil )"2" "(# Fnum := Fnum(f), Fexp := Fexp(f) #)=f" )"2" (decompose-equality) nil nil )) nil ))nil ))nil ))nil ))nil )"float" float nil )"int" integers nil )"(strict_total_order?[real])" real_props nil )nil float nil )nil integers nil )nil naturalnumbers nil )nil float nil )"float" float nil )nil float nil )"[numfield -> numfield]" number_fields nil )"[T, T -> boolean]" equalities nil )nil number_fields nil )nil integers nil )"[rational -> boolean]" integersnil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" nil )nil booleans nil )nil numbers nil )"{n: nonneg_real | n >= m AND n >= -m}" nil ))nil 3320081916"" (skeep)"" (expand "Fulp" )"" (rewrite "Exp_increq_1" )"" (rewrite "Lexico" :subst ("b" "b" )) nil nil )) nil ))nil ))nil )"real" float nil )nil float nil )"(total_order?[real])" nil )"nnrat" exponentiation nil )"posrat" exponentiationnil )"{x: (Fcanonic?(b)) | FtoR(x) = FtoR(f):: real AND Fexp(x) <= Fexp(f)}" nil )"bool" reals nil )"real" float nil )"[T, T -> boolean]" equalities nil )AND const-decl "[bool, bool -> bool]" booleans nil )"bool" float nil )"bool" float nil )nil booleans nil )nil float nil )nil float nil )nil integers nil )nil naturalnumbers nil )nil integers nil )"[rational -> boolean]" integersnil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" nil )nil booleans nil )nil numbers nil )nil float nil ))nil 3320082046"" (skeep)"" (rewrite "FulpAbs" :dir rl)"" (rewrite "FulpAbs" :dir rl :subst ("f" "q" ))"" (rewrite "FulpMonotone" )"1" (rewrite "FabsCorrect" )"1" (rewrite "FabsCorrect" ) nil nil )) nil )"2" (rewrite "FabsCorrect" ) (("2" (assert ) nil nil ))nil )"3" (rewrite "FabsBounded" ) nil nil )"4" (rewrite "FabsBounded" ) nil nil ))nil ))nil ))nil ))nil )nil float nil )nil naturalnumbers nil )nil integers nil )nil float nil )nil integers nil )nil float nil )nil float nil )"float" float nil )"(total_order?[real])" nil )nil float nil )nil float nil ))nil 3319810465"" (skeep)"" case "FORALL (b: Format, p: float): ") "1" (inst -1 "b" "Fnormalize(b)(p)" )"1" (split)"1" (skosimp*)"1" (inst 1 "f!1" )"1" (case "FtoR(Fnormalize(b)(p))=FtoR(p)" )"1" case " Fulp(b)(Fnormalize(b)(p))=Fulp(b)(p)" )"1" (assert ) nil nil )"2" (hide -1 -2 2)"2" (expand "Fulp" )"2" rewrite "FcanonicUnique" :subst"b" "b" "p" "Fnormalize(b)(Fnormalize(b)(p))" "q" "Fnormalize(b)(p)" ))nil nil ))nil ))nil ))nil )"2" (assert ) nil nil ))nil ))nil ))nil )"2" (assert ) nil nil ))nil ))nil )"2" (hide 2 -1)"2" (skosimp*)"2" case "Fnum(p!1) = vNum(b!1)-1 OR Fnum(p!1) <= vNum(b!1)-2" )"1" (split)"1" "(# Fnum:=vNum(b!1)/radix, Fexp:=Fexp(p!1)+1 #)" )"1" (rewrite "FulpCanonic" )"1" (expand "FtoR" )"1" (rewrite "expt_plus" )"1" (rewrite "expt_x1" )"1" (field) nil nil )) nil ))nil ))nil ))nil )"2" (expand "Fbounded?" )"2" (case "Fbounded?(b!1)(p!1)" )"1" (expand "Fbounded?" )"1" (flatten)"1" "1" expand "abs" )"1" "1" "vNum(b!1)" )"1" (assert ) nil nil ))nil ))nil ))nil )"2" (assert ) nil nil ))nil ))nil ))nil )"2" (rewrite "FcanonicBounded" ) nil nil ))nil ))nil )"3" (lemma "radix_div_vNum" )"3" (inst -1 "b!1" ) nil nil )) nil ))nil )"2" "(# Fnum:=Fnum(p!1)+1, Fexp:=Fexp(p!1) #)" )"1" (rewrite "FulpCanonic" )"1" (expand "FtoR" ) (("1" (propax) nil nil ))nil ))nil )"2" (expand "Fbounded?" )"2" (case "Fbounded?(b!1)(p!1)" )"1" (expand "Fbounded?" )"1" (flatten)"1" "1" expand "abs" )"1" (grind-reals) nil nil ))nil )"2" (propax) nil nil ))nil ))nil ))nil )"2" (rewrite "FcanonicBounded" ) nil nil ))nil ))nil ))nil ))nil )"2" (case "Fbounded?(b!1)(p!1)" )"1" (expand "Fbounded?" )"1" (flatten) (("1" (grind-reals) nil nil ))nil ))nil )"2" (rewrite "FcanonicBounded" ) nil nil ))nil ))nil ))nil ))nil )"3" (skosimp*)"3" (rewrite "FcanonicBounded" ) nil nil )) nil ))nil ))nil )"real" float nil )"[numfield, numfield -> numfield]" nil )nil number_fields nil )"real" float nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" nil )"[T, T -> boolean]" equalities nil )nil numbers nil )"bool" float nil )"bool" float nil )"[bool, bool -> bool]" booleans nil )nil booleans nil )nil booleans nil )nil float nil )nil integers nil )nil float nil )nil integers nil )nil naturalnumbers nil )"real" realsnil )nil float nil )AND const-decl "[bool, bool -> bool]" booleans nil )"bool" reals nil )"{x: (Fcanonic?(b)) | FtoR(x) = FtoR(f):: real AND Fexp(x) <= Fexp(f)}" nil )"float" float nil )"Format" float nil )nil float nil )nil float nil )nil real_props nil )nil real_props nil )"(total_order?[real])" nil )"int" integers nil )"(strict_total_order?[real])" real_props nil )"nzrat" rationalsnil )"nonneg_int" integers nil )"even_int" nil )"(divides(n))" dividesnil )"(divides(m))" dividesnil )"posint" nil )nil extra_real_props nil )nil reals nil )nil real_props nil )nil real_types nil )nil real_types nil )nil real_props nil )"{n: nonneg_real | n >= m AND n >= -m}" nil )"{j: nonneg_int | j >= i}" real_defs nil )nil float nil )"posrat" exponentiationnil )"nnrat" exponentiation nil )nil reals nil )nil exponentiation nil )"rat" rationalsnil )"posint" exponentiationnil )"posrat" nil )nil exponentiation nil )NOT const-decl "[bool -> bool]" booleans nil )"rat" rationalsnil )"[numfield, numfield -> numfield]" nil )"real" exponentiation nil )"float" float nil )"above(1)" float nil )"Format" float nil )"[numfield, nznum -> numfield]" number_fieldsnil )nil number_fields nil )"boolean" notequal nil )"int" integersnil )"posrat" nil )"int" integersnil )OR const-decl "[bool, bool -> bool]" booleans nil )"[numfield, numfield -> numfield]" nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integersnil )"bool" reals nil )nil integers nil )"bool" reals nil )nil integers nil )"posnat" float nil ))nil 3319814672"" (skeep)"" (lemma "FloatPlusUlpBounded" )"" (inst -1 "b" "Fopp(p)" )"" (split)"1" (skosimp*)"1" (inst 1 "Fopp(f!1)" )"1" (rewrite "FoppCorrect" )"1" (rewrite "FoppCorrect" )"1" (rewrite -1)"1" (case "Fulp(b)(Fopp(p))=Fulp(b)(p)" )"1" (assert ) nil nil )"2" (hide -1 2)"2" expand "Fulp" )"2" case "Fnormalize(b)(Fopp(p))=Fopp(Fnormalize(b)(p))" )"1" rewrite -1)"1" expand "Fopp" )"1" (propax) nil nil ))nil ))nil )"2" "2" rewrite "FcanonicUnique" "b" "b" ))"1" case "FtoR(Fnormalize(b)(Fopp(p)))=FtoR(Fopp(p))" )"1" rewrite -1)"1" rewrite "FoppCorrect" )"1" rewrite "FoppCorrect" )"1" (assert ) nil nil ))nil ))nil ))nil )"2" (assert ) nil nil ))nil )"2" "FcanonicOpp" )"2" "b" "Fnormalize(b)(p)" )"2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (rewrite "FoppBounded" ) nil nil ))nil ))nil )"2" (rewrite "FoppBounded" ) nil nil ))nil ))nil ))nil ))nil )nil float nil )"(Fbounded?(b))" float nil )"Format" float nil )"bool" float nil )nil booleans nil )nil booleans nil )"real" float nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" nil )"[T, T -> boolean]" equalities nil )nil numbers nil )"real" realsnil )nil float nil )nil float nil )"{x: (Fcanonic?(b)) | FtoR(x) = FtoR(f):: real AND Fexp(x) <= Fexp(f)}" nil )"bool" reals nil )"real" float nil )AND const-decl "[bool, bool -> bool]" booleans nil )"bool" float nil )"real" realsnil )"real" reals nil )nil float nil )nil float nil )"float" float nil )nil float nil )nil integers nil )nil float nil )nil integers nil )nil naturalnumbers nil ))nil 3319211967"" (skeep)"" (expand * "Fpred" "Fopp" "Fsucc" )"" (grind-reals) nil nil )) nil ))nil )"float" float nil )"nzint" integersnil )"float" float nil )"int" integersnil )"float" float nil )"(divides(n))" dividesnil )"(divides(m))" dividesnil )"(strict_total_order?[real])" real_props nil )nil real_props nil )"int" integers nil )"nzrat" rationalsnil )"posrat" nil )"int" integersnil ))nil 3319212016"" (skeep)"" (expand * "Fsucc" "Fopp" "Fpred" )"" (grind-reals) nil nil )) nil ))nil )"float" float nil )"int" integersnil )"float" float nil )"nzint" integersnil )"float" float nil )"(divides(n))" dividesnil )"(divides(m))" dividesnil )"(strict_total_order?[real])" real_props nil )nil real_props nil )"int" integers nil )"nzrat" rationalsnil )"nzrat" nil )"int" integersnil ))nil 3319298166"" (skeep)"" (expand * "Fpred" "Fsucc" )"" (grind-reals)"1" (lemma "radix_less_vNum" :subst ("b" "b" ))"1" (hide -2 -3) (("1" (field -2) nil nil )) nil ))nil )"2" (apply-extensionality :hide? t) nil nil )"3" expand * "Fcanonic?" "Fnormal?" "Fsubnormal?" "Fbounded?" )nil nil )"4" (decompose-equality 2)"1" (rewrite "div_cancel4" ) (("1" (grind) nil nil ))nil )"2" (rewrite "div_cancel4" )"2" (replace -2 :dir rl :hide? t)"2" (grind) nil nil )) nil ))nil ))nil )"5" expand * "Fcanonic?" "Fnormal?" "Fsubnormal?" "Fbounded?" )"5" (rewrite -2) (("5" (grind-reals) nil nil )) nil ))nil )"6" (decompose-equality 3) nil nil )"7" expand * "Fcanonic?" "Fnormal?" "Fsubnormal?" "Fbounded?" )"7" (rewrite -2) (("7" (grind-reals) nil nil )) nil ))nil )"8" expand * "Fcanonic?" "Fnormal?" "Fsubnormal?" "Fbounded?" )"8" (flatten)"8" (case-replace "Fnum(f)=1-(vNum(b) / radix)" )"1" (hide-all-but -3)"1" (rewrite "abs_mult" )"1" (expand "abs" )"1" (grind-reals)"1" "radix_less_vNum" :subst"b" "b" ))"1" (grind-reals) nil nil )) nil ))nil ))nil ))nil ))nil )"2" (assert ) nil nil ))nil ))nil ))nil )"9" (decompose-equality 3) nil nil ))nil ))nil ))nil )"float" float nil )"nzint" integersnil )"int" integersnil )"float" float nil )"rat" rationalsnil )"[T, T -> boolean]" equalities nil )"posrat" nil )nil real_props nil )"{r: nonneg_rat | r >= q}" real_defs nil )"nonneg_rat" rationals nil )"rat" rationalsnil )"rat" rationals nil )nil real_props nil )"{j: posint | j >= i}" real_defs nil )"(total_order?[real])" nil )AND const-decl "[bool, bool -> bool]" booleans nil )"float" float nil )"above(1)" float nil )"[numfield, nznum -> numfield]" number_fieldsnil )nil number_fields nil )"boolean" notequal nil )nil rationals nil )nil rationals nil )nil real_props nil )nil real_props nil )"{n: nonneg_real | n >= m AND n >= -m}" nil )"real" exponentiation nil )"real" exponentiation nil )"posint" exponentiationnil )"posnat" exponentiationnil )"(strict_total_order?[real])" real_props nil )"{j: nonneg_int | j >= i}" real_defs nil )"bool" float nil )"bool" float nil )"bool" float nil )"bool" float nil )nil float nil )"posnat" float nil )nil integers nil )"bool" reals nil )nil integers nil )"bool" reals nil )nil booleans nil )"[numfield, numfield -> numfield]" nil )"[numfield -> numfield]" number_fields nil )nil number_fields nil )nil integers nil )"[rational -> boolean]" integersnil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" nil )nil booleans nil )nil numbers nil )nil float nil )nil naturalnumbers nil )nil integers nil )nil float nil )"(total_order?[real])" nil )"int" integersnil )"posrat" nil )"(strict_total_order?[real])" real_props nil )"nzrat" nil )"(divides(m))" dividesnil )"(divides(n))" dividesnil )"posint" nil )"int" integers nil )nil real_props nil )nil real_props nil ))nil 3319299927"" (skeep)"" (case "f=Fopp(Fopp(f))" )"1" (rewrite -1 :target-fnums 1)"1" (rewrite "FsuccFoppFpred" )"1" rewrite "FpredFoppFsucc" :subst"f" "(Fpred(b)(Fopp(f)))" ))"1" (rewrite "FsuccFpred" )"1" (lemma "FcanonicOpp" )"1" (grind-reals) nil nil )) nil ))nil ))nil ))nil ))nil )"2" (hide -1 2)"2" (expand "Fopp" )"2" (decompose-equality) nil nil )) nil ))nil ))nil ))nil )"float" float nil )"[T, T -> boolean]" equalities nil )nil booleans nil )nil float nil )nil integers nil )nil float nil )nil naturalnumbers nil )nil integers nil )nil float nil )nil float nil )nil float nil )nil float nil )"float" float nil )"int" integers nil )nil numbers nil )"[number -> boolean]" nil )nil number_fields nil )"[numfield -> numfield]" number_fields nil ))nil 3320428258"" (skeep)"" (expand * "Fbounded?" "Fpred" )"" (grind-reals)"1" (expand "abs" 1)"1" (field 1)"1" (div-by 1 "vNum(b)" ) (("1" (assert ) nil nil ))nil ))nil ))nil )"2" (expand "abs" 2) (("2" (assert ) nil nil )) nil )"3" (expand "abs" ) (("3" (grind-reals) nil nil )) nil )"4" (expand "abs" ) (("4" (grind-reals) nil nil )) nil ))nil ))nil ))nil )"float" float nil )"nzrat" nil )"nzint" integersnil )"int" integersnil )"bool" float nil )"{n: nonneg_real | n >= m AND n >= -m}" nil )nil real_props nil )"above(1)" float nil )"[numfield, numfield -> numfield]" nil )"[numfield -> numfield]" number_fields nil )nil number_fields nil )"posnat" float nil )nil integers nil )nil integers nil )nil integers nil )"[rational -> boolean]" integersnil )nil rationals nil )"[real -> boolean]" rationals nil )nil float nil )nil integers nil )nil naturalnumbers nil )nil real_types nil )"bool" reals nil )nil real_types nil )"bool" reals nil )nil booleans nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" nil )nil booleans nil )nil numbers nil )nil real_props nil )"boolean" notequal nil )nil reals nil )nil extra_real_props nil )"posint" nil )"(strict_total_order?[real])" real_props nil )"(divides(m))" dividesnil )"(divides(n))" dividesnil )"{j: posint | j >= i}" real_defs nil )nil real_props nil )"posrat" nil )"{r: posrat | r >= q}" real_defs nil )nil real_props nil )"int" integersnil )"int" integers nil )"(total_order?[real])" nil )"{j: nonneg_int | j >= i}" real_defs nil )"(strict_total_order?[real])" real_props nil )nil real_props nil ))nil 3320428394"" (skeep)"" (case-replace "Fsucc(b)(f)=Fopp(Fopp(Fsucc(b)(f)))" )"1" (rewrite "FoppBounded" )"1" (rewrite "FpredFoppFsucc" :dir rl)"1" (rewrite "FpredBounded" )"1" (rewrite "FoppBounded" ) nil nil )) nil ))nil ))nil )"2" (expand "Fopp" )"2" (decompose-equality 1) nil nil )) nil ))nil ))nil )nil integers nil )nil float nil )nil booleans nil )"[T, T -> boolean]" equalities nil )nil naturalnumbers nil )nil integers nil )nil float nil )"float" float nil )"float" float nil )nil float nil )nil float nil )nil float nil )"[numfield -> numfield]" number_fields nil )nil number_fields nil )"[number -> boolean]" nil )nil numbers nil )"int" integers nil ))nil 3319233185"" (skeep)"" (lemma "FcanonicOpp" )"" (inst-cp -1 "b" "Fsucc(b)(f)" )"" (flatten)"" (rewrite -3)"" (hide -2 -3 2)"" (lemma "FpredFoppFsucc" )"" (inst -1 "b" "f" )"" (case "Fcanonic?(b)(Fpred(b)(Fopp(f)))" )"1" (assert ) nil nil )"2" (rewrite "FpredCanonic" )"2" (hide 2 3 -1)"2" "b" "f" )"2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )nil float nil )nil float nil )nil booleans nil )nil booleans nil )"bool" float nil )"float" float nil )"float" float nil )nil float nil )"float" float nil )nil float nil )nil integers nil )nil float nil )nil integers nil )nil naturalnumbers nil ))nil 3319306101"" (skeep)"" (expand "FtoR" )"" (div-by -2 "radix ^ (Fexp(p))" )"" (case-replace "0 / radix ^ (Fexp(p))=0" )"1" (hide -1)"1" (expand "Fpred" ) (("1" (grind-reals) nil nil ))nil ))nil )"2" (grind-reals) nil nil ))nil ))nil ))nil ))nil )"real" float nil )"[numfield, nznum -> numfield]" number_fieldsnil )nil number_fields nil )"[T, T -> boolean]" equalities nil )"nonneg_rat" nil )"int" integersnil )"float" float nil )"(strict_total_order?[real])" real_props nil )"(divides(m))" dividesnil )"(divides(n))" dividesnil )nil real_props nil )nil real_props nil )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil )"nzrat" nil )"int" integers nil )"int" integersnil )nil reals nil )nil extra_real_props nil )"rat" rationalsnil )nil numbers nil )nil booleans nil )"[number -> boolean]" nil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )nil booleans nil )"bool" reals nil )nil real_types nil )"bool" reals nil )nil real_types nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integersnil )nil integers nil )OR const-decl "[bool, bool -> bool]" booleans nil )"boolean" notequal nil )"real" exponentiation nil )nil integers nil )"above(1)" float nil )nil float nil )nil number_fields nil )"[numfield, numfield -> numfield]" nil )nil real_props nil )"nnrat" exponentiation nil )"posrat" exponentiationnil ))nil 3319462089"" (skeep)"" (case "0 <=Fnum(p)" )"1" (case "0 < Fnum(Fsucc(b)(p))" )"1" (expand "FtoR" 1) (("1" (grind-reals) nil nil ))nil )"2" (hide 2)"2" (expand "Fsucc" )"2" (lift-if)"2" (split +)"1" (flatten) (("1" (cross-mult 1) nil nil ))nil )"2" (flatten)"2" (split 2)"1" (flatten) (("1" (grind) nil nil )) nil )"2" (flatten) (("2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (expand "FtoR" -2) (("2" (grind-reals) nil nil ))nil ))nil ))nil )nil float nil )nil integers nil )"bool" reals nil )nil booleans nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" nil )nil booleans nil )nil numbers nil )"int" integersnil )"rat" rationalsnil )"bool" float nil )"bool" float nil )"bool" float nil )"bool" float nil )"{n: nonneg_real | n >= m AND n >= -m}" nil )"real" exponentiation nil )"real" exponentiation nil )"nzrat" nil )"posint" exponentiationnil )"posnat" exponentiationnil )"posrat" nil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil )"int" integers nil )"int" integersnil )"posint" nil )"{j: nonneg_int | j >= i}" real_defs nil )"bool" reals nil )nil real_types nil )"bool" reals nil )nil real_types nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integersnil )"above(1)" float nil )nil integers nil )nil integers nil )"posnat" float nil )"(divides(m))" dividesnil )"(divides(n))" dividesnil )"even_int" nil )"nonneg_int" integers nil )nil real_props nil )"nzint" integersnil )"real" float nil )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil )"rat" rationalsnil )"nnrat" exponentiation nil )nil real_props nil )"bool" reals nil )nil naturalnumbers nil )nil integers nil )nil float nil )"float" float nil )nil real_props nil ))nil 3319375218"" (skeep)"" (rewrite "FcanonicBounded" )"" (rewrite "FpredCanonic" ) nil nil )) nil ))nil )nil float nil )nil naturalnumbers nil )nil integers nil )nil float nil )nil integers nil )nil float nil )"float" float nil )"(strict_total_order?[real])" real_props nil )nil float nil ))nil ))nil 3319375376"" (skeep)"" (rewrite "FulpCanonic" )"1" (expand * "FtoR" "Fpred" )"1" (grind-reals)"1" (rewrite "expt_div" :dir rl)"1" (rewrite "expt_x1" )"1" (grind-reals) nil nil )) nil ))nil ))nil ))nil )"2" (rewrite "FpredCanonic" ) nil nil ))nil ))nil )nil float nil )nil naturalnumbers nil )nil integers nil )nil float nil )nil integers nil )nil float nil )"float" float nil )"posrat" exponentiationnil )"nnrat" exponentiation nil )"real" realsnil )"(strict_total_order?[real])" real_props nil )nil real_props nil )nil real_props nil )"rat" rationalsnil )"int" integers nil )"int" integersnil )"(divides(n))" dividesnil )"(divides(m))" dividesnil )"(strict_total_order?[real])" real_props nil )"posrat" nil )"rat" rationalsnil )nil real_props nil )nil exponentiation nil )"posrat" nil )"posint" exponentiationnil )nil exponentiation nil )nil numbers nil )nil booleans nil )"[number -> boolean]" nil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integersnil )"boolean" notequal nil )nil reals nil )nil booleans nil )"bool" reals nil )"above(1)" float nil )"real" float nil )"rat" rationalsnil )"int" integersnil )"nzint" integersnil )"nzrat" nil )nil float nil ))nil 3319461787"" (skeep)"" (case "f=Fpred(b)(Fsucc(b)(f))" )"1" (case-replace "FtoR(f)=FtoR(Fpred(b)(Fsucc(b)(f)))" )"1" "Fulp(b)(f)=Fulp(b)(Fpred(b)(Fsucc(b)(f)))" )"1" (rewrite "FpredDiff" )"1" (hide -1 -2 2)"1" (rewrite "FsuccPos" ) nil nil )) nil )"2" (rewrite "FsuccCanonic" ) nil nil ))nil )"2" (grind-reals) nil nil )"3" (rewrite -2 :dir rl)"3" (rewrite "FcanonicBounded" ) nil nil )) nil ))nil )"2" (grind-reals) nil nil ))nil )"2" (rewrite "FpredFsucc" ) nil nil ))nil ))nil )"float" float nil )"float" float nil )nil float nil )nil integers nil )nil naturalnumbers nil )"[T, T -> boolean]" equalities nil )nil booleans nil )nil float nil )nil integers nil )nil booleans nil )"bool" float nil )"real" float nil )nil float nil )nil float nil )"(total_order?[real])" nil )"real" realsnil )"(strict_total_order?[real])" real_props nil )nil float nil )nil float nil )"real" float nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" nil )nil numbers nil )nil float nil ))nil 3319233508"" (skeep)"" (expand "Fpred" )"" (grind-reals)"1" (expand "FtoR" )"1" (rewrite "expt_plus" )"1" (rewrite "expt_x1" )"1" (rewrite -1) (("1" (field) nil nil )) nil ))nil ))nil ))nil )"2" (expand "FtoR" )"2" "radix ^ (Fexp(f) - 1)=radix ^ (Fexp(f))/radix" )"1" (hide -1 -3 1)"1" (field) (("1" (grind-reals) nil nil )) nil ))nil )"2" (hide -1 -2 2 3)"2" (lemma "expt_div" )"2" (inst -1 "Fexp(f)" "1" "radix" )"2" (rewrite "expt_x1" -1)"2" (grind-reals) nil nil )) nil ))nil ))nil ))nil ))nil ))nil )"3" (hide 1 2)"3" (expand "FtoR" ) (("3" (grind-reals) nil nil ))nil ))nil )"4" (expand "FtoR" )"4" (hide 1 2) (("4" (grind-reals) nil nil )) nil ))nil ))nil ))nil ))nil )"int" integersnil )"float" float nil )nil real_props nil )nil exponentiation nil )"rat" rationalsnil )"rat" rationalsnil )nil real_props nil )nil real_props nil )"[numfield, nznum -> numfield]" number_fieldsnil )nil number_fields nil )"[numfield, numfield -> numfield]" nil )nil number_fields nil )"real" exponentiation nil )"[T, T -> boolean]" equalities nil )"posrat" nil )"real" float nil )"nzrat" nil )"rat" rationalsnil )NOT const-decl "[bool -> bool]" booleans nil )OR const-decl "[bool, bool -> bool]" booleans nil )"bool" reals nil )"(total_order?[real])" nil )nil real_props nil )nil real_props nil )nil exponentiation nil )"posrat" nil )"posint" exponentiationnil )"nnrat" exponentiation nil )"posrat" exponentiationnil )"above(1)" float nil )nil integers nil )"bool" reals nil )nil booleans nil )nil reals nil )"boolean" notequal nil )nil float nil )nil integers nil )"[rational -> boolean]" integersnil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" nil )nil booleans nil )nil numbers nil )nil exponentiation nil )"int" integersnil )"nzint" integersnil )"nzrat" nil )"(strict_total_order?[real])" real_props nil )"(divides(m))" dividesnil )"(divides(n))" dividesnil )"int" integers nil )"(strict_total_order?[real])" real_props nil )nil real_props nil ))nil 3319391212"" (skeep)"" "FtoR(Fpred(b)(p))=FtoR(p)-Fulp(b)(Fpred(b)(p))" )"1" "FtoR(Fpred(b)(q))=FtoR(q)-Fulp(b)(Fpred(b)(q))" )"1" (rewrite "FulpCanonic" )"1" (rewrite "FulpCanonic" )"1" (hide -1 -2)"1" (case "0<Fnum(p)" )"1" (case "0<Fnum(q)" )"1" (case "Fexp(p) <= Fexp(q)" )"1" (expand "Fpred" )"1" (grind-reals)"1" expand * "FtoR" )"1" rewrite "expt_div" :dir rl)"1" rewrite "expt_div" :dir rl)"1" rewrite "expt_x1" )"1" "1" "vNum(b)-1" )"1" "Fnum(p)=vNum(b)/radix" )"1" "Fnum(q)=vNum(b)/radix" )"1" "(vNum(b) / radix * (radix ^ Fexp(p)) * radix - (radix ^ Fexp(p))) / ") "1" "(vNum(b) / radix * (radix ^ Fexp(q)) * radix - (radix ^ Fexp(q))) / ") "1" rewrite "Exp_increq_1" )nil nil )"2" nil nil ))nil )"2" nil nil ))nil )"2" "2" nil nil ))nil ))nil )"2" expand "Fcanonic?" )"2" "1" expand "Fnormal?" )"1" "1" rewrite "abs_mult" "1" expand "abs" "1" "radix" )"1" nil nil ))nil ))nil ))nil ))nil ))nil )"2" " Fnum(p) * (radix ^ Fexp(p)) * radix - (radix ^ Fexp(p)) ") "1" case "(vNum(b) - 1) * radix ^ Fexp(p) <= (vNum(b) - 1) * radix ^ Fexp(q)" )"1" case "(vNum(b) - 1) * radix ^ Fexp(q) <= Fnum(q) * radix ^ (Fexp(q)) * radix - radix ^ Fexp(q) * radix" )"1" nil nil )"2" "radix ^ Fexp(q)" )nil nil ))nil )"2" "2" case "vNum(b) = radix * Fnum(q) OR vNum(b) < radix * Fnum(q)" )"1" "1" nil nil )"2" "radix_div_vNum" )"2" "b" )"2" rewrite "div_simple" "2" "2" "radix" )"2" "vNum(b) / radix=k!1" )"1" case "k!1 <= Fnum(q)-1" )"1" "radix" )"1" nil nil ))nil )"2" nil nil ))nil )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" nil nil ))nil ))nil ))nil )"2" "vNum(b) - 1" )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" expand "Fcanonic?" -5)"2" "1" expand "Fnormal?" )"1" "1" rewrite "abs_mult" -2)"1" expand "abs" -2)"1" expand "FtoR" 2)"1" rewrite "expt_div" "1" rewrite "expt_x1" )"1" "radix" )"1" "1" " Fnum(p) * (radix ^ Fexp(p)) * radix - (radix ^ Fexp(p)) ") "1" case "(vNum(b) - 1) * radix ^ Fexp(p) <= (vNum(b) - 1) * radix ^ Fexp(q)" )"1" case "(vNum(b) - 1) * radix ^ Fexp(q) <= Fnum(q) * radix ^ (Fexp(q)) * radix - radix ^ Fexp(q) * radix" )"1" nil nil )"2" "radix ^ Fexp(q)" )"2" "2" case "vNum(b) = radix * Fnum(q) OR vNum(b) < radix * Fnum(q)" )"1" "1" nil nil )"2" "radix_div_vNum" )"2" "b" )"2" rewrite "div_simple" "2" "2" "radix" )"2" "vNum(b) / radix=k!1" )"1" case "k!1 <= Fnum(q)-1" )"1" "radix" )"1" nil nil ))nil )"2" nil nil ))nil )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" nil nil ))nil ))nil ))nil ))nil )"2" "vNum(b) - 1" )"2" rewrite "Exp_increq_1" )nil nil ))nil ))nil )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" expand "Fsubnormal?" )"2" (propax) nil nil ))nil ))nil ))nil )"3" case "Fbounded?(b)(p)" )"1" expand "Fbounded?" )"1" "1" case "-dExp(b) = Fexp(p)" )"1" expand "abs" -2)"1" expand "FtoR" 2)"1" case "(vNum(b)-2)*radix ^ Fexp(p) <= Fnum(q) * radix ^ (Fexp(q)) - radix ^ (Fexp(q) - 1)" )"1" case "Fnum(p) * radix ^ (Fexp(p)) - radix ^ Fexp(p) <= (vNum(b) - 2) * radix ^ Fexp(p)" )"1" nil nil )"2" "radix ^ (Fexp(p))" )"2" "(Fnum(p) * radix ^ (Fexp(p)) - radix ^ Fexp(p)) / radix ^ (Fexp(p)) ") "1" nil nil )"2" nil nil ))nil ))nil ))nil )"2" "2" case
Messung V0.5 in Prozent C=100 H=100 G=100
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2026-05-26
