Impressum axpy.prf
Sprache: Lisp
(axpynil 3320508538"" (skeep)"" (typepred "x1" ) (("" (rewrite "FcanonicBounded" ) nil nil ))nil ))nil )"Format" axpy nil )"bool" float nil )nil float nil )nil float nil )"above(1)" axpy nil )"bool" reals nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil numbers nil )nil integers nil )nil naturalnumbers nil )NOT const-decl "[bool -> bool]" booleans nil )nil booleans nil )nil booleans nil )nil float nil ))nil ))nil 3320496512"" (skeep)"" (expand "MinOrMax?" )"" (split)"1" (rewrite "RoundedModeUlp" :subst ("b" "b" "P" "isMin?" ))"1" (rewrite "isMin_RoundedMode" ) nil nil )) nil )"2" (rewrite "RoundedModeUlp" :subst ("b" "b" "P" "isMax?" ))"2" (rewrite "isMax_RoundedMode" ) nil nil )) nil ))nil ))nil ))nil )"bool" axpy nil )"bool" float nil )nil float nil )nil float nil )nil float nil )nil float nil )nil naturalnumbers nil )"bool" float nil )nil float nil )"bool" float nil )"Format" axpy nil )nil numbers nil )nil booleans nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )nil booleans nil )"bool" reals nil )nil integers nil )"above(1)" axpy nil )"(strict_total_order?[real])" real_props nil )nil float nil )"real" reals nil ))nil 3320496511"" (skeep) (("" (rewrite "FoppBounded" ) nil nil )) nil )nil float nil )nil naturalnumbers nil )nil float nil )"Format" axpy nil )nil float nil )nil numbers nil )nil booleans nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )nil booleans nil )"bool" reals nil )nil integers nil )"above(1)" axpy nil ))nil ))nil 3320496618"" (skeep)"" (expand "MinOrMax?" )"" (split)"1" (skosimp*) (("1" (rewrite "MinOppMax" ) nil nil )) nil )"2" (skosimp*) (("2" (rewrite "MaxOppMin" ) nil nil )) nil ))nil ))nil ))nil )"bool" axpy nil )nil float nil )"above(1)" axpy nil )nil integers nil )"bool" reals nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )nil float nil )"Format" axpy nil )nil float nil )nil naturalnumbers nil )nil float nil ))nil 3320496511"" (skeep)"" (rewrite "FcanonicBounded" )"" (rewrite "FpredCanonic" ) nil nil )) nil ))nil )nil float nil )nil naturalnumbers nil )nil float nil )"Format" axpy nil )nil float nil )"float" float nil )nil numbers nil )nil booleans nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )nil booleans nil )"bool" reals nil )nil integers nil )"above(1)" axpy nil )"(strict_total_order?[real])" real_props nil )nil float nil ))nil ))nil 3320504768"" (skeep) (("" (rewrite "FcanonicBounded" ) nil nil )) nil )nil float nil )nil naturalnumbers nil )nil float nil )"Format" axpy nil )nil float nil )nil numbers nil )nil booleans nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )nil booleans nil )"bool" reals nil )nil integers nil )"above(1)" axpy nil )"real" reals nil ))nil ))nil 3320504301"" (skeep)"" (case "z <= FtoR(f)" )"1" (expand "abs" -4)"1" (grind-reals)"1" (expand "MinOrMax?" )"1" (skosimp*)"1" (hide 1)"1" (expand "isMax?" )"1" (skosimp*)"1" (lemma "FsuccFpred" :subst ("f" "f" "b" "b" ))"1" (split)"1" (replace -1 1 :dir rl)"1" "FtoR(f!1)=FtoR(Fnormalize(b)(f!1))" )"1" (rewrite "FsuccProp" )"1" "z" :strict 1)"1" " FtoR[radix](f)-Fulp(b)(Fpred(b)(f))" "1" rewrite "FpredDiff" :dir rl)"1" (assert ) nil nil ))nil )"2" (assert ) nil nil ))nil ))nil )"2" (rewrite "FpredCanonic" ) nil nil ))nil )"2" (assert ) nil nil ))nil ))nil )"2" (propax) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (flip-ineq 1)"2" (expand "abs" -4)"2" (grind-reals)"2" (expand "MinOrMax?" )"2" (skosimp*)"2" (hide 2)"2" (expand "isMin?" )"2" (skosimp*)"2" "FpredFsucc" :subst ("f" "f" "b" "b" ))"2" (split)"1" (replace -1 1 :dir rl)"1" "FtoR(f!1)=FtoR(Fnormalize(b)(f!1))" )"1" rewrite "FpredProp" )"1" "z" :strict 2)"1" "FtoR(f)+Fulp(b)(Fpred(b)(f))" "1" (assert ) nil nil )"2" "FtoR(f)+Fulp(b)(f)" )"1" assert )"1" rewrite "FulpFpred1" )nil nil ))nil )"2" rewrite "FsuccDiff" :dir rl)"2" (assert ) nil nil ))nil )"3" "3" rewrite "FcanonicBounded" )nil nil ))nil )"4" rewrite "FcanonicBounded" )nil nil ))nil ))nil ))nil )"2" rewrite "FsuccCanonic" )nil nil ))nil )"2" (assert ) nil nil ))nil ))nil )"2" (propax) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"real" float nil )nil float nil )"above(1)" axpy nil )nil integers nil )"bool" reals nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil integers nil )"bool" reals nil )nil booleans nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil )"bool" float nil )nil float nil )nil naturalnumbers nil )nil float nil )"Format" axpy nil )"float" float nil )nil float nil )"real" float nil )"[numfield, numfield -> numfield]" number_fields nil )nil number_fields nil )nil float nil )"bool" reals nil )nil float nil )"[T, T -> boolean]" equalities nil )"bool" float nil )"bool" float nil )AND const-decl "[bool, bool -> bool]" booleans nil )"{x: (Fcanonic?(b)) | FtoR(x) = FtoR(f):: real AND Fexp(x) <= Fexp(f)}" nil )"bool" axpy nil )"{n: nonneg_real | n >= m AND n >= -m}" real_defsnil )nil float nil )nil real_props nil )"[bool, bool -> bool]" booleans nil )nil float nil )nil float nil )nil float nil )"real" reals nil )"[numfield, numfield -> numfield]" number_fields nil )nil float nil )"float" float nil )nil float nil )"bool" float nil )"real" reals nil )"(strict_total_order?[real])" real_props nil )"real" reals nil ))nil 3320505812"" (skeep) (("" (rewrite "FcanonicBounded" ) nil nil )) nil )nil float nil )nil naturalnumbers nil )nil float nil )"Format" axpy nil )nil float nil )nil numbers nil )nil booleans nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )nil booleans nil )"bool" reals nil )nil integers nil )"above(1)" axpy nil ))nil ))nil 3320505496"" (skeep)"" (expand * "MinOrMax?" "isMin?" )"" (skosimp*)"" (split)"1" (propax) nil nil )"2" (skosimp*)"2" (hide 2)"2" (lemma "FpredFsucc" :subst ("f" "f" "b" "b" ))"2" "FtoR(f!1)=FtoR(Fnormalize(b)(f!1))" )"1" (split)"1" (replace -1 1 :dir rl)"1" (rewrite "FpredProp" )"1" (hide 2 -1 -2)"1" (trans-ineq 1 "z" :strict 2)"1" (expand "abs" -4)"1" "FtoR(f)+Fulp(b)(f)" "1" (assert ) nil nil )"2" rewrite "FsuccDiff" :dir rl)"2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil )"2" (rewrite "FsuccCanonic" ) nil nil ))nil ))nil )"2" (propax) nil nil ))nil )"2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"bool" float nil )"bool" axpy nil )"[T, T -> boolean]" equalities nil )"real" float nil )"bool" float nil )"bool" 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 )nil float nil )"{n: nonneg_real | n >= m AND n >= -m}" real_defsnil )nil float nil )"real" reals nil )nil number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )"real" float nil ) (< const-decl "bool" reals nil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil )nil float nil )"float" float nil )nil float nil )nil naturalnumbers nil )nil float nil )"Format" axpy nil )nil float nil )nil numbers nil )nil booleans nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )nil booleans nil )"bool" reals nil )nil integers nil )"above(1)" axpy nil )"real" reals nil ))nil 3320506118"" (skeep) (("" (rewrite "FcanonicBounded" ) nil nil )) nil )nil float nil )nil naturalnumbers nil )nil float nil )"Format" axpy nil )nil float nil )nil numbers nil )nil booleans nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )nil booleans nil )"bool" reals nil )nil integers nil )"above(1)" axpy nil ))nil ))nil 3320505840"" (skeep)"" (case "z <= FtoR(f)" )"1" (expand "abs" -4)"1" (grind-reals)"1" (expand * "MinOrMax?" "isMax?" )"1" (skosimp*)"1" (hide 1)"1" (lemma "FsuccFpred" :subst ("f" "f" "b" "b" ))"1" (split)"1" (replace -1 1 :dir rl)"1" "FtoR(f!1)=FtoR(Fnormalize(b)(f!1))" )"1" (rewrite "FsuccProp" )"1" (hide 2 -1 -2)"1" (trans-ineq 1 "z" :strict 1)"1" "FtoR[radix](f) -Fulp(b)(f)" "1" rewrite "FulpCanonic" )"1" "f=(# Fnum:=0, Fexp:=-dExp(b) #)" )"1" "1" expand * "Fpred" "FtoR" )"1" "1" "radix_less_vNum" )"1" "b" )"1" "1" nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" rewrite "FcanonicUnique" "b" "b" "p" "f" "q" "(# Fnum := 0, Fexp := -dExp(b) #)" ))"1" rewrite -3 :dir rl)"1" expand "FtoR" 1)"1" (assert ) nil nil ))nil ))nil )"2" "2" expand *"Fcanonic?" "Fsubnormal?" "Fbounded?" )"2" "2" expand "abs" )"2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (assert ) nil nil ))nil ))nil ))nil )"2" (rewrite "FpredCanonic" ) nil nil ))nil )"2" (assert ) nil nil ))nil ))nil )"2" (propax) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (flip-ineq 1)"2" (expand * "MinOrMax?" "isMin?" )"2" (skosimp*)"2" (split)"1" (assert ) nil nil )"2" (skosimp*)"2" (expand "abs" -5)"2" (grind-reals)"2" (hide 2)"2" "FpredFsucc" :subst ("f" "f" "b" "b" ))"2" (split)"1" (replace -1 1 :dir rl)"1" "FtoR(f!1)=FtoR(Fnormalize(b)(f!1))" )"1" rewrite "FpredProp" )"1" "1" "z" :strict 2)"1" "FtoR(f)+Fulp(b)(f)" "1" (assert ) nil nil )"2" rewrite "FulpCanonic" )"2" "f=(# Fnum:=0, Fexp:=-dExp(b) #)" )"1" "1" expand * "Fsucc" "FtoR" )"1" "1" "1" "radix_less_vNum" )"1" "1" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" rewrite "FcanonicUnique" "b" "b" "p" "f" "q" "(# Fnum := 0, Fexp := -dExp(b) #)" ))"1" rewrite -4 :dir rl)"1" expand "FtoR" 1)"1" assert )nil nil ))nil ))nil )"2" "2" expand *"Fcanonic?" "Fsubnormal?" "Fbounded?" )"2" expand "abs" )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" rewrite "FsuccCanonic" )nil nil ))nil )"2" (assert ) nil nil ))nil ))nil )"2" (propax) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"real" float nil )nil float nil )"above(1)" axpy nil )nil integers nil )"bool" reals nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil integers nil )"bool" reals nil )nil booleans nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil )nil float nil )nil naturalnumbers nil )nil float nil )"Format" axpy nil )"float" float nil )nil float nil )"bool" reals nil )nil float nil )"posrat" exponentiation nil )"nnrat" exponentiation nil )nil float nil )"nonneg_rat" nil )nil real_props nil )"odd_int" integers nil )"bool" float nil )"(divides(m))" divides nil )"(divides(n))" divides nil )"{j: nonneg_int | j >= i}" nil )nil real_props nil )"nzrat" rationalsnil )"nzint" integers nil )"nzrat" rationalsnil )nil real_props nil )"posrat" nil )"int" integers nil )"(total_order?[real])" nil )nil float nil )"int" integers nil )"int" integers nil )"[numfield -> numfield]" number_fields nil )nil number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )"real" float nil )nil float nil )"[T, T -> boolean]" equalities nil )"bool" float nil )"bool" float nil )AND const-decl "[bool, bool -> bool]" booleans nil )"{x: (Fcanonic?(b)) | FtoR(x) = FtoR(f):: real AND Fexp(x) <= Fexp(f)}" nil )"bool" axpy nil )"bool" float nil )"{n: nonneg_real | n >= m AND n >= -m}" real_defsnil )"bool" float nil )nil float nil )"[numfield, numfield -> numfield]" number_fields nil )"real" reals nil )nil real_props nil )"posrat" nil )nil float nil )"float" float nil )nil float nil )"real" reals nil )"(strict_total_order?[real])" real_props nil )"real" reals nil ))nil 3320498426"" (skeep) (("" (rewrite "FcanonicBounded" ) nil nil )) nil )nil float nil )nil naturalnumbers nil )nil float nil )"Format" axpy nil )nil float nil )nil numbers nil )nil booleans nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )nil booleans nil )"bool" reals nil )nil integers nil )"above(1)" axpy nil ))nil ))nil 3320498426"" (skeep)"" (expand * "FtoR" "abs" ) (("" (grind-reals) nil nil )) nil )) nil )"{n: nonneg_real | n >= m AND n >= -m}" real_defsnil )"real" float nil )nil real_props nil )nil real_props nil )"rat" rationals nil )"nnrat" exponentiation nil )"even_int" integersnil )"(divides(n))" divides nil )"(divides(m))" divides nil )"int" integers nil )"(strict_total_order?[real])" real_props nil ))nil ))nil 3320498426"" (skosimp*) (("" (field -3) (("" (grind) nil nil )) nil )) nil )"rat" rationals nil )"nonneg_int" nil )"even_int" integersnil )"(divides(n))" divides nil )"(divides(m))" divides nil )nil booleans nil )nil booleans nil )NOT const-decl "[bool -> bool]" booleans nil )nil numbers nil )"boolean" notequal nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[numfield, numfield -> numfield]" number_fields 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 )"[numfield -> numfield]" number_fields nil )"{n: nonneg_real | n >= m AND n >= -m}" real_defsnil )nil integers nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )"bool" reals nil )nil integers nil )"above(1)" axpy nil )nil float nil )"nzrat" rationalsnil )"{j: nonneg_int | j >= i}" nil )"[T, T -> boolean]" equalities nil )nil extra_real_props nil )"[numfield, numfield -> numfield]" number_fields nil )nil number_fields nil )"[numfield, nznum -> numfield]" number_fields nil )"bool" functions nil )"(bijective?[T, T])" identity nil )"bool" booleans nil )"rat" rationals nil )"odd_int" integersnil )"int" integers nil )"(strict_total_order?[real])" real_props nil )"real" reals nil ))nil ))nil 3320498427"" (assert )"" (skeep)"" (assert )"" (case "0 < abs(Fnum(f))" )"1" (case "0 < 1 - 1 / (2 * abs(Fnum(f)))" )"1" (mult-by 2 "(1 - 1 / (2 * abs(Fnum(f))))" )"1" " abs(FtoR[radix](f)) * (1 - 1 / (2 * abs(Fnum(f)))) = ") "1" (move-terms 2 l 2)"1" (move-terms 2 r 1)"1" (trans-ineq 2 "abs(FtoR(f)-z)" )"1" (expand "abs" 1)"1" (grind-reals) nil nil )) nil )"2" (rewrite "ClosestUlp" )"2" (rewrite "FcanonicBounded" ) nil nil ))nil ))nil ))nil ))nil )"2" (hide 2 3)"2" (field 1)"2" (rewrite "FulpCanonic" )"2" (expand "FtoR" 1)"2" (rewrite "abs_mult" )"2" (expand "abs" 1 2)"2" (assert ) nil nil )) nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (case "1 <= abs(Fnum(f))" )"1" (field 1) nil nil )"2" (hide-all-but (-1 1)) (("2" (grind) nil nil )) nil ))nil ))nil )"2" (hide-all-but (1 2))"2" (expand * "abs" "FtoR" ) (("2" (grind-reals) nil nil ))nil ))nil ))nil ))nil ))nil ))nil )nil float nil )"above(1)" axpy nil )nil integers nil )"bool" reals nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil integers nil )"{n: nonneg_real | n >= m AND n >= -m}" real_defsnil )"[numfield -> numfield]" number_fields nil )nil number_fields nil )AND const-decl "[bool, bool -> bool]" booleans nil )nil real_types nil )"bool" reals nil ) (< const-decl "bool" reals nil )nil booleans nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )"odd_int" integersnil )nil extra_real_propsnil )"int" integers nil )nil real_props nil )nil real_types nil )"real" float nil )"float[radix]" axpy nil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil )"(strict_total_order?[real])" real_props nil )nil real_props nil )nil reals nil )nil float nil )"posrat" exponentiation nil )"nnrat" exponentiation nil )"real" exponentiation nil )OR const-decl "[bool, bool -> bool]" booleans nil )nil real_props nil )"{r: nonneg_rat | r >= q}" nil )"{r: posrat | r >= q}" nil )"nonneg_rat" nil )"nzint" integers nil )"even_int" integers nil )"rat" rationals nil )"odd_int" integers nil )"bool" booleans nil )"(bijective?[T, T])" identity nil )"bool" functions nil )nil real_props nil )nil rationals nil )NOT const-decl "[bool -> bool]" booleans nil )"[numfield, numfield -> numfield]" number_fields nil )"bool" reals nil )"[bool, bool -> bool]" booleans nil )"real" reals nil )"nnreal" nil )nil real_props nil )nil real_props nil )nil real_props nil )"real" reals nil )nil float nil )nil float nil )"real" reals nil )"real" reals nil )"[T, T -> boolean]" equalities nil )nil naturalnumbers nil )nil float nil )"bool" float nil )"real" float nil )"Format" axpy nil )"[numfield, numfield -> numfield]" number_fields nil )"[numfield, nznum -> numfield]" number_fields nil )nil number_fields nil )"boolean" notequal nil )"[numfield, numfield -> numfield]" number_fields nil )nil real_props nil )"(total_order?[real])" nil )"real" reals nil )"{j: nonneg_int | j >= i}" nil )"nonneg_int" nil )"even_int" integersnil )"(divides(n))" divides nil )"(divides(m))" divides nil )"nzrat" rationalsnil )"rat" rationals nil ))nil 3320499635"" (skeep) (("" (field -3) nil nil )) nil )"{j: nonneg_int | j >= i}" nil )"nzrat" rationalsnil )"rat" rationals nil )"nonneg_int" nil )"even_int" integersnil )"(divides(n))" divides nil )"(divides(m))" divides nil )nil booleans nil )nil booleans nil )NOT const-decl "[bool -> bool]" booleans nil )nil numbers nil )"boolean" notequal nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[numfield, numfield -> numfield]" number_fields 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 )"[numfield -> numfield]" number_fields nil )"{n: nonneg_real | n >= m AND n >= -m}" real_defsnil )nil integers nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )"bool" reals nil )nil integers nil )"above(1)" axpy nil )nil float nil ))nil ))nil 3320499635"" (skeep)"" (case "0 < abs(Fnum(f))" )"1" (case "0 < 1 + 1 / (2 * abs(Fnum(f)))" )"1" (mult-by 2 "(1 + 1 / (2 * abs(Fnum(f))))" )"1" (assert )"1" " abs(FtoR[radix](f)) * (1 / (2 * abs(Fnum(f)))) = ") "1" (move-terms 2 r 1)"1" (trans-ineq 2 "abs(FtoR(f)-z)" )"1" (hide-all-but 1)"1" (expand "abs" ) (("1" (grind-reals) nil nil ))nil ))nil )"2" (rewrite "ClosestUlp" )"2" (rewrite "FcanonicBounded" ) nil nil )) nil ))nil ))nil )"2" (field 1)"2" (rewrite "FulpCanonic" )"2" (hide-all-but 1)"2" (expand * "abs" "FtoR" )"2" (grind-reals) nil nil )) nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (field 1) nil nil ))nil )"2" (hide-all-but (1 2))"2" (expand * "abs" "FtoR" ) (("2" (grind-reals) nil nil ))nil ))nil ))nil ))nil )nil float nil )"above(1)" axpy nil )nil integers nil )"bool" reals nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil integers nil )"{n: nonneg_real | n >= m AND n >= -m}" real_defsnil )"[numfield -> numfield]" number_fields nil )nil number_fields nil )AND const-decl "[bool, bool -> bool]" booleans nil )nil real_types nil )"bool" reals nil ) (< const-decl "bool" reals nil )nil booleans nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )nil extra_real_propsnil )"real" reals nil )nil real_props nil )nil real_types nil )"real" float nil )"float[radix]" axpy nil )"(total_order?[real])" nil )"nzrat" rationalsnil )"{j: nonneg_int | j >= i}" nil )"(strict_total_order?[real])" real_props nil )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil )nil real_props nil )nil reals nil )"Format" axpy nil )"real" float nil )"bool" float nil )nil float nil )nil naturalnumbers nil )"[T, T -> boolean]" equalities nil )"real" reals nil )nil float nil )nil float nil )"real" reals nil )"real" reals nil )"[bool, bool -> bool]" booleans nil )"bool" reals nil )"[numfield, numfield -> numfield]" number_fields nil )nil float nil )"posrat" exponentiation nil )"nnrat" exponentiation nil )nil real_props nil )"rat" rationals nil )"rat" rationals nil )"int" integers nil )"bool" booleans nil )"(bijective?[T, T])" identity nil )"bool" functions nil )nil real_props nil )NOT const-decl "[bool -> bool]" booleans nil )"real" reals nil )"nonneg_int" nil )"even_int" integersnil )"(divides(n))" divides nil )"(divides(m))" divides nil )"rat" rationals nil )"[numfield, numfield -> numfield]" number_fields nil )"boolean" notequal nil )nil number_fields nil )"[numfield, nznum -> numfield]" number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )nil real_props nil ))nil 3320510642 ("" (subtype-tcc) nil nil )"(strict_total_order?[real])" real_props nil )"{j: nonneg_int | j >= i}" nil )"nat" exponentiation nil )"(total_order?[real])" nil )nil exponent_props "reals/" )"above(1)" axpy nil )nil integers nil )"bool" reals nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )"posnat" float nil )"bool" float nil )"real" float nil )"bool" float nil )"boolean" notequal nil )"nnrat" exponentiation nil )"posrat" exponentiation nil )"rat" rationals nil )"rat" rationals nil )"real" reals nil ))nil ))nil 3320510940"" (skeep)"" (lemma "errorBoundedPlus" )"" (inst -1 "b" "f" "p" "q" )"" (split)"1" (skosimp*)"1" (case "FtoR(e!1)=0" )"1" (assert ) nil nil )"2" (hide 2)"2" (typepred "e!1" )"2" (expand "Fbounded?" -1)"2" (flatten)"2" (expand "FtoR" 1)"2" (case "abs(Fnum(e!1)) < 1" )"1" (grind-reals) nil nil )"2" (hide 2)"2" (mult-by 1 "radix ^ (-dExp(b))" )"2" "abs(FtoR[radix](f) - (FtoR[radix](p) + FtoR[radix](q)))" "1" "abs(FtoR(e!1))" )"1" expand "FtoR" 1)"1" rewrite "abs_mult" )"1" expand "abs" 1 3)"1" "1" rewrite "Exp_increq_1" )nil nil ))nil ))nil ))nil ))nil )"2" rewrite -3)"2" expand "abs" 1)"2" (grind-reals) nil nil ))nil ))nil ))nil )"2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (propax) nil nil ) ("3" (propax) nil nil )"4" (propax) nil nil ) ("5" (propax) nil nil ))nil ))nil ))nil ))nil )"above(1)" axpy nil )nil integers nil )"bool" reals nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )nil float nil )"bool" float nil )"real" float nil )"[T, T -> boolean]" equalities nil )"real" reals nil )"{k: int | k <= i AND k <= j}" nil )"(strict_total_order?[real])" real_props nil )"nnrat" exponentiation nil )"posrat" exponentiation nil )"int" integers nil )NOT const-decl "[bool -> bool]" booleans nil )"{n: nonneg_real | n >= m AND n >= -m}" real_defsnil )"[numfield -> numfield]" number_fields nil )nil number_fields nil )AND const-decl "[bool, bool -> bool]" booleans nil )nil real_types nil )"bool" reals nil ) (< const-decl "bool" reals nil )"{j: nonneg_int | j >= i}" nil )"(divides(n))" divides nil )"(total_order?[real])" nil )"rat" rationals nil )nil real_props nil )nil extra_real_propsnil )OR const-decl "[bool, bool -> bool]" booleans nil )"boolean" notequal nil )"real" exponentiation nil )"(total_order?[real])" nil )"[numfield, numfield -> numfield]" number_fields nil )"posrat" nil )"nonneg_rat" nil )nil real_props nil )"{r: posrat | r >= q}" nil )"(strict_total_order?[real])" real_props nil )nil real_props nil )nil float nil )min const-decl "{p: real | p <= m AND p <= n}" real_defs nil )"bool" reals nil )"[numfield, numfield -> numfield]" number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )nil float nil )"Format" axpy nil )nil float nil )nil naturalnumbers nil )"real" reals nil ))nil 3320748868 ("" (subtype-tcc) nil nil )"even_int" integersnil )"above(1)" axpy nil )nil integers nil )"bool" reals nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )"posnat" float nil )"bool" float nil )"bool" float nil )"bool" float nil )"bool" float nil )"boolean" notequal nil )"int" integers nil )"(divides(n))" divides nil )"(divides(m))" divides nil ))nil ))nil 3320748869"" (skeep)"" (split)"1" (skosimp*)"1" (hide -2)"1" (expand * "Fnormal?" "Fbounded?" )"1" (flatten)"1" (case "radix /= 0" )"1" "radix ^ (Prec(b) - 1)*radix ^(-dExp(b))" "radix ^ (Prec(b) - 1)*radix ^(Fexp(f))" ))"1" (rewrite "expt_plus" :dir rl)"1" (assert ) nil nil )) nil )"2" (div-by 1 "radix ^ (Prec(b) - 1)" )"2" (rewrite "Exp_increq_1" ) nil nil )) nil )"3" (expand "FtoR" )"3" (rewrite "abs_mult" )"3" (expand "abs" 1 2)"3" (div-by 1 "radix ^ (Fexp(f))" )"3" (mult-by 1 "radix" )"3" (rewrite "abs_mult" )"3" expand "abs" -4 1)"3" "vNum(b)=radix ^ (Prec(b) - 1) * radix " )"1" (assert ) nil nil )"2" "vNum(b) = radix ^ (Prec(b) - 1) * radix^1" )"1" (rewrite "expt_x1" ) nil nil )"2" rewrite "expt_plus" :dir rl)"2" assert )"2" expand "vNum" )"2" (propax) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"4" (assert ) nil nil ) ("5" (assert ) nil nil )"6" (assert ) nil nil ) ("7" (assert ) nil nil )"8" (assert ) nil nil ) ("9" (assert ) nil nil )"10" (assert ) nil nil ))nil )"2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil )"2" (skosimp*)"2" (expand "Fcanonic?" )"2" (split)"1" (propax) nil nil )"2" (hide 1)"2" (flip-ineq -2)"2" (expand "Fsubnormal?" )"2" (flatten)"2" (expand "FtoR" )"2" (rewrite "abs_mult" )"2" (rewrite "abs_mult" )"2" (expand "abs" -3 1)"2" (expand "abs" 1 2)"2" "Prec(b) - 1 - dExp(b)=Prec(b) - 1 +(- dExp(b))" )"1" rewrite "expt_plus" )"1" "1" rewrite -4)"1" "radix ^ (-dExp(b))" )"1" "radix" )"1" "vNum(b) = radix ^ (Prec(b) - 1) * radix" )"1" (assert ) nil nil )"2" "vNum(b) = radix ^ (Prec(b) - 1) * radix^1" )"1" rewrite "expt_x1" )nil nil )"2" rewrite "expt_plus" "2" assert )"2" expand "vNum" )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"[bool, bool -> bool]" booleans nil )"(total_order?[real])" nil )"posrat" exponentiation nil )"nnrat" exponentiation nil )"(total_order?[real])" nil )"int" integers nil )"int" integers nil )"posrat" nil )AND const-decl "[bool, bool -> bool]" booleans nil )"bool" reals nil )OR const-decl "[bool, bool -> bool]" booleans nil )"bool" reals nil )"real" exponentiation nil )nil number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )nil naturalnumbers nil )nil float nil )"Format" axpy nil )"[numfield, numfield -> numfield]" number_fields nil )"[numfield -> numfield]" number_fields nil )nil float nil )nil real_types nil )"{n: nonneg_real | n >= m AND n >= -m}" real_defsnil )"real" float nil )"{j: nonneg_int | j >= i}" nil )"(strict_total_order?[real])" real_props nil )"int" integers nil )nil exponentiation nil )nil reals nil )nil float nil )nil reals nil )nil extra_real_props nil )nil real_types nil )nil real_props nil )nil real_props nil )"{r: posrat | r >= q}" nil )"nonneg_rat" nil )nil extra_real_props nil )"{j: posint | j >= i}" nil )"nonneg_int" nil )"posnat" float nil )nil integers nil )nil integers nil )"[T, T -> boolean]" equalities nil )"posint" exponentiation nil )nil exponentiation nil )nil real_props nil )"rat" rationals nil )nil numbers nil )nil booleans nil )"boolean" notequal nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )nil booleans nil )"bool" reals nil )nil integers nil )"above(1)" axpy nil )"bool" float nil )"(divides(m))" divides nil )"(divides(n))" divides nil )"bool" float nil )"bool" float nil )"bool" float nil )nil extra_real_propsnil )nil real_props nil )"[numfield, numfield -> numfield]" number_fields nil )"(strict_total_order?[real])" real_props nil ))nil 3320512021"" (skeep)"" (rewrite "FcanonicBounded" )"" (rewrite "FpredCanonic" ) nil nil )) nil ))nil )nil float nil )nil naturalnumbers nil )nil float nil )"Format" axpy nil )nil float nil )"float" float nil )"bool" float nil )nil numbers nil )nil booleans nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )nil booleans nil )"bool" reals nil )nil integers nil )"above(1)" axpy nil )"(strict_total_order?[real])" real_props nil )"real" reals nil )"real" reals nil )nil float nil ))nil ))nil 3320512021"" (skeep)"" case "abs(FtoR(u)-(y1+a1*x1)) < Fulp(b)(Fpred(b)(u)) / 2 + abs(FtoR(u)-(FtoR(t)+FtoR(y)))" )"1" (case "FtoR(t)+FtoR(y) <= FtoR(u)" )"1" (rewrite "MinOrMax1" )"1" (hide 2)"1" "abs(FtoR[radix](u) - FtoR[radix](t) - FtoR[radix](y)) + " :strict 1) "1" case "abs(FtoR[radix](u) - FtoR[radix](t) - FtoR[radix](y)) <= Fulp(b)(Fpred(b)(u)) / 2" )"1" (grind-reals) nil nil )"2" (hide 2)"2" (lemma "ClosestUlp2" )"2" (inst -1 "b" "u" "FtoR(t)+FtoR(y)" )"2" (split)"1" (case-replace "Fabs(u)=u" )"1" (assert ) nil nil )"2" (hide-all-but (1 -6))"2" (expand * "Fabs" "abs" "FtoR" )"2" "2" (decompose-equality 1) nil nil ))nil ))nil ))nil ))nil )"2" (trans-ineq 1 "FtoR[radix](u)" )"1" (hide 2)"1" (case "0 <= FtoR(t)+FtoR(y)" )"1" expand "abs" 1)"1" (grind-reals) nil nil ))nil )"2" "2" "2" "2" "RleRoundedLessR0" )"2" assert )"2" "Closest?" "b" "u" "FtoR(t)+FtoR(y)" )"2" "1" (assert ) nil nil )"2" (assert ) nil nil )"3" (propax) nil nil )"4" rewrite "Closest_RoundedMode" )nil nil )"5" "u" )"5" rewrite "FcanonicBounded" )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (hide-all-but (1 -5))"2" (expand "abs" )"2" expand "Fulp" )"2" (grind-reals) nil nil ))nil ))nil ))nil ))nil )"3" (propax) nil nil ) ("4" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (flip-ineq 1)"2" case "FtoR(t)+FtoR(y) <= FtoR(u)+ Fulp(b)(Fpred(b)(u)) / 2" )"1" (rewrite "MinOrMax1" )"1" "abs(-1 * FtoR(y) - FtoR(a) * FtoR(x) + a1 * x1 + y1)+abs(FtoR(t) - FtoR(a) * FtoR(x))+abs(FtoR(t)+FtoR(y)-FtoR(u))" "1" (hide-all-but 1)"1" (expand "abs" ) (("1" (grind-reals) nil nil ))nil ))nil )"2" (expand "abs" 1 3) (("2" (grind-reals) nil nil ))nil ))nil ))nil )"2" (flip-ineq 1)"2" (rewrite "MinOrMax2" )"2" (hide 2)"2" "abs(-1 * FtoR(y) - FtoR(a) * FtoR(x) + a1 * x1 + y1)+abs(FtoR(t) - FtoR(a) * FtoR(x))+abs(FtoR(t)+FtoR(y)-FtoR(u))" "1" (hide-all-but 1)"1" (expand "abs" )"1" (grind-reals) nil nil )) nil ))nil )"2" "Fulp(b)(Fpred(b)(u)) / 4 + Fulp(b)(Fpred(b)(u)) / 4 + " "1" (assert ) nil nil )"2" case " Fulp(b)(Fpred(b)(u)) <= Fulp(b)(u)" )"1" case "abs(FtoR[radix](t) + FtoR[radix](y) - FtoR[radix](u))<= Fulp(b)(u)/2" )"1" (assert ) nil nil )"2" (hide 2)"2" (lemma "ClosestUlp" )"2" "b" "u" "FtoR(t)+FtoR(y)" )"2" "1" "1" expand "abs" )"1" (grind-reals) nil nil ))nil ))nil )"2" (propax) nil nil )"3" rewrite "FcanonicBounded" )nil nil ))nil ))nil ))nil ))nil ))nil )"2" (rewrite "FulpFpred1" ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (hide 2)"2" "abs(y1-FtoR(y)+a1*x1-FtoR(a)*FtoR(x))+abs(FtoR(t)-FtoR(a)*FtoR(x))+abs(FtoR(u)-(FtoR(t)+FtoR(y)))" )"1" (hide-all-but 1)"1" (expand "abs" ) (("1" (grind-reals) nil nil )) nil ))nil )"2" (assert ) nil nil ))nil ))nil ))nil ))nil )"float" float nil )"real" float nil )"bool" float nil )"[numfield, nznum -> numfield]" number_fields nil )nil number_fields nil )"boolean" notequal nil )"[numfield, numfield -> numfield]" number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )"Format" axpy nil )"bool" float nil )nil float nil )nil naturalnumbers nil )"real" float nil )nil float nil )"above(1)" axpy nil )nil integers nil )"bool" reals nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil integers nil )"[numfield, numfield -> numfield]" number_fields nil )"{n: nonneg_real | n >= m AND n >= -m}" real_defsnil )"[numfield -> numfield]" number_fields nil )nil number_fields nil )AND const-decl "[bool, bool -> bool]" booleans nil )nil real_types nil )"bool" reals nil ) (< const-decl "bool" reals nil )nil booleans nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )nil float nil )nil float nil )nil axpy nil )"real" reals nil )"odd_int" integers nil )"nnreal" nil )nil axpy nil )"real" reals nil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil )nil float nil )"bool" float nil )nil float nil )NOT const-decl "[bool -> bool]" booleans nil )nil float nil )nil float nil )"(strict_total_order?[real])" real_props nil )"posrat" nil )"posrat" exponentiation nil )"float" float nil )"[T, T -> boolean]" equalities nil )"(total_order?[real])" nil )"int" integers nil )"nnrat" exponentiation nil )"rat" rationals nil )nil real_props nil )nil float nil )"bool" reals nil )"real" reals nil )"real" reals nil )"real" reals nil ))nil 3320575968"" (skeep)"" (trans-ineq 1 "Fulp(b)(t)/2" )"1" (rewrite "ClosestUlp" )"1" (rewrite "FcanonicBounded" ) nil nil )) nil )"2" (mult-by 1 "4" )"2" (field 1)"2" (trans-ineq 1 " radix* Fulp(b)(t)" )"1" (expand "Fulp" ) (("1" (grind-reals) nil nil )) nil )"2" case "Fcanonic?(b)((# Fnum:= Fnum(t), Fexp:=Fexp(t)+1 #))" )"1" "radix * Fulp(b)(t) =Fulp(b)( (# Fnum:= Fnum(t), Fexp:=Fexp(t)+1 #))" )"1" (rewrite "FulpMonotoneAbs" )"1" "abs(FtoR((# Fnum := Fnum(t), Fexp := 1 + Fexp(t) #)))=radix * abs(FtoR[radix](t))" )"1" (case "0 <= FtoR(Fpred(b)(u))" )"1" (expand "abs" 1 2) (("1" (propax) nil nil ))nil )"2" (rewrite "FpredPos" ) nil nil ))nil )"2" (hide-all-but 1)"2" (expand * "abs" "FtoR" )"2" (grind-reals)"1" (rewrite "expt_plus" )"1" (assert ) nil nil )) nil )"2" (rewrite "expt_plus" )"2" (assert ) nil nil )) nil ))nil ))nil ))nil ))nil )"2" (rewrite "FcanonicBounded" )"2" (rewrite "FpredCanonic" ) nil nil )) nil )"3" (rewrite "FcanonicBounded" ) nil nil ))nil )"2" (rewrite "FulpCanonic" )"2" (rewrite "FulpCanonic" )"2" (rewrite "expt_plus" ) nil nil )) nil ))nil )"3" (rewrite "FcanonicBounded" ) nil nil ))nil )"2" (hide-all-but (-4 1))"2" (expand * "Fcanonic?" "Fnormal?" "Fbounded?" )"2" (flatten)"2" (split)"1" (propax) nil nil ) ("2" (assert ) nil nil )"3" (propax) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"real" reals nil )nil booleans nil )nil booleans nil )AND const-decl "[bool, bool -> bool]" booleans nil )nil numbers nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"bool" reals nil ) (>= const-decl "bool" reals nil )nil real_types nil )nil number_fields nil )"[numfield -> numfield]" number_fields nil )"{n: nonneg_real | n >= m AND n >= -m}" real_defsnil )"[numfield, numfield -> numfield]" number_fields nil )nil integers nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )"bool" reals nil )nil integers nil )"above(1)" axpy nil )nil float nil )"real" float nil )nil naturalnumbers nil )nil float nil )"bool" float nil )"Format" axpy nil )"[numfield, numfield -> numfield]" number_fields nil )"boolean" notequal nil )nil number_fields nil )"[numfield, nznum -> numfield]" number_fields nil )"bool" float nil )"real" float nil )"float" float nil )"nnreal" nil )"(strict_total_order?[real])" real_props nil )"real" reals nil )"(total_order?[real])" nil )nil float nil )nil float nil )NOT const-decl "[bool -> bool]" booleans nil )"[T, T -> boolean]" equalities nil )"int" integers nil )"[numfield, numfield -> numfield]" number_fields nil )nil float nil )nil float nil )nil real_props nil )"rat" rationals nil )"rat" rationals nil )nil exponentiation nil )"posint" exponentiation nil )"int" integers nil )nil exponentiation nil )nil reals nil )"posrat" exponentiation nil )nil float nil )nil float nil )"bool" float nil )"(divides(m))" divides nil )"(divides(n))" divides nil )"int" integers nil )"{j: nonneg_int | j >= i}" nil )"posrat" nil )"nnrat" exponentiation nil )nil reals nil )nil real_props nil )nil real_types nil )nil real_props nil )"real" reals nil )"real" reals nil ))nil 3320577546"" (skeep)"" (trans-ineq 1 "Fulp(b)(t)/2" )"1" (rewrite "ClosestUlp" )"1" (rewrite "FcanonicBounded" ) nil nil )) nil )"2" (rewrite "FulpCanonic" )"2" (rewrite "FulpCanonic" )"1" (expand "Fsubnormal?" )"1" (flatten)"1" (rewrite -5)"1" (mult-by 1 "4" )"1" (field 1)"1" (trans-ineq 1 "radix*(radix ^ -dExp(b))" )"1" (grind-reals) nil nil )"2" " radix * (radix ^ -dExp(b))=radix ^(1 - dExp(b))" )"1" (rewrite "Exp_increq_1" ) nil nil )"2" (lemma "expt_plus" )"2" (inst -1 "1" "-dExp(b)" "radix" )"2" (rewrite "expt_x1" )"2" (assert ) nil nil )) nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (rewrite "FpredCanonic" ) nil nil ))nil ))nil ))nil ))nil )"real" reals nil )nil booleans nil )nil booleans nil )AND const-decl "[bool, bool -> bool]" booleans nil )nil numbers nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"bool" reals nil ) (>= const-decl "bool" reals nil )nil real_types nil )nil number_fields nil )"[numfield -> numfield]" number_fields nil )"{n: nonneg_real | n >= m AND n >= -m}" real_defsnil )"[numfield, numfield -> numfield]" number_fields nil )nil integers nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )"bool" reals nil )nil integers nil )"above(1)" axpy nil )nil float nil )"real" float nil )nil naturalnumbers nil )nil float nil )"bool" float nil )"Format" axpy nil )"[numfield, numfield -> numfield]" number_fields nil )"boolean" notequal nil )nil number_fields nil )"[numfield, nznum -> numfield]" number_fields nil )"bool" float nil )"real" float nil )"float" float nil )"int" integers nil )"(strict_total_order?[real])" real_props nil )"real" reals nil )"(total_order?[real])" nil )nil float nil )nil float nil )"posrat" nil )nil real_props nil )nil real_types nil )OR const-decl "[bool, bool -> bool]" booleans nil )"real" exponentiation nil )nil real_props nil )nil reals nil )nil real_props nil )nil exponentiation nil )nil exponentiation nil )"posint" exponentiation nil )"odd_int" integers nil )"int" integers nil )nil reals nil )nil float nil )"(total_order?[real])" nil )"[T, T -> boolean]" equalities nil )"posrat" nil )NOT const-decl "[bool -> bool]" booleans nil )"int" integers nil )"bool" float nil )"(divides(m))" divides nil )"(divides(n))" divides nil )nil float nil )"nnrat" exponentiation nil )"posrat" exponentiation nil )nil float nil )"real" reals nil )"real" reals nil ))nil 3320577865"" (skeep)"" (rewrite "MinOrMax1" )"" (hide 2)"" (copy -6)"" (rewrite -6)"" "abs(FtoR(t)-FtoR(a)*FtoR(x))+abs(-1 * FtoR(y) - FtoR(a) * FtoR(x) + a1 * x1 + y1)" "1" (hide-all-but 1)"1" (expand "abs" ) (("1" (grind-reals) nil nil ))nil ))nil )"2" "Fulp(b)(Fpred(b)(u)) / 2+ Fulp(b)(Fpred(b)(u)) / 4" "1" case "abs(FtoR[radix](t) - FtoR[radix](a) * FtoR[radix](x)) <= Fulp(b)(Fpred(b)(u)) / 2" )"1" (assert ) nil nil )"2" (hide 2)"2" (trans-ineq 1 "Fulp(b)(t)/2" )"1" (rewrite "ClosestUlp" )"1" (rewrite "FcanonicBounded" ) nil nil ))nil )"2" (field 1)"2" (case "Fcanonic?(b)(Fpred(b)(u))" )"1" (rewrite "FulpCanonic" )"1" (rewrite "FulpCanonic" )"1" rewrite "Exp_increq_1" )"1" "1" expand *"Fcanonic?" "Fnormal?" "Fsubnormal?" "Fbounded?" )"1" "1" "1" (assert ) nil nil ))nil )"2" "2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (rewrite "FpredCanonic" ) nil nil ))nil ))nil ))nil ))nil ))nil )"2" (expand "Fulp" 1) (("2" (grind-reals) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )nil axpy nil )nil integers nil )nil numbers nil )nil booleans nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil booleans nil )"bool" reals nil )nil integers nil )"above(1)" axpy nil )nil float nil )nil naturalnumbers nil )nil float nil )"bool" float nil )"Format" axpy nil )nil number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )"real" reals nil )"(strict_total_order?[real])" real_props nil )"nnreal" nil )AND const-decl "[bool, bool -> bool]" booleans nil )"bool" reals nil ) (>= const-decl "bool" reals nil )nil real_types nil )"[numfield -> numfield]" number_fields nil )"{n: nonneg_real | n >= m AND n >= -m}" real_defsnil )"[numfield, numfield -> numfield]" number_fields nil )"real" float nil ) (< const-decl "bool" reals nil )"bool" float nil )"real" float nil )"float" float nil )"(total_order?[real])" nil )"real" reals nil )"posrat" nil )nil float nil )nil float nil )"int" integers nil )"{j: nonneg_int | j >= i}" nil )"bool" float nil )"(divides(n))" divides nil )"(divides(m))" divides nil )"bool" float nil )"posrat" nil )nil float nil )"nnrat" exponentiation nil )"posrat" exponentiation nil )nil float nil )nil float nil )"bool" booleans nil )"(bijective?[T, T])" identity nil )"bool" functions nil )nil reals nil )nil real_props nil )nil real_types nil )nil real_props nil )"(strict_total_order?[real])" real_props nil )NOT const-decl "[bool -> bool]" booleans nil )"boolean" notequal nil )nil number_fields nil )"[numfield, nznum -> numfield]" number_fields nil )"odd_int" integers nil )"real" reals nil )"real" reals nil )"real" reals nil ))"(trans-ineq 1 \" Fulp(b)(t) / 2\") -> infinite loop" "" (skeep)"" (case "FtoR(u) <= a1*x1+y1" )"1" (rewrite "MinOrMax2" )"1" (hide 2)"1" "abs(FtoR(u)-(FtoR(t)+FtoR(y)))+abs(FtoR(t)-FtoR(a)*FtoR(x))+abs(-1 * FtoR(y) - FtoR(a) * FtoR(x) + a1 * x1 + y1)" "1" (hide-all-but 1)"1" (expand "abs" ) (("1" (grind-reals) nil nil )) nil ))nil )"2" case "abs(FtoR[radix](u) - (FtoR[radix](t) + FtoR[radix](y))) <= Fulp(b)(u)/2" )"1" case "abs(FtoR[radix](t) - FtoR[radix](a) * FtoR[radix](x)) <= Fulp(b)(u)/4" )"1" "Fulp(b)(u) / 4+ Fulp(b)(u) / 2+Fulp(b)(Fpred(b)(u)) / 4" "1" (assert ) nil nil )"2" (case "Fulp(b)(Fpred(b)(u)) <= Fulp(b)(u)" )"1" (grind-reals) nil nil )"2" (rewrite "FulpFpred1" ) nil nil ))nil ))nil )"2" (hide 2 -1)"2" (trans-ineq 1 "Fulp(b)(t)/2" )"1" (rewrite "ClosestUlp" )"1" (rewrite "FcanonicBounded" ) nil nil )) nil )"2" (field 1)"2" (rewrite "FulpCanonic" )"2" (rewrite "FulpCanonic" )"2" (trans-ineq 1 "radix^(1+Fexp(t))" )"1" (rewrite "expt_plus" )"1" (grind-reals) nil nil )) nil )"2" (rewrite "Exp_increq_1" )"2" expand "Fsubnormal?" )"2" (propax) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (rewrite "ClosestUlp" )"2" (rewrite "FcanonicBounded" ) nil nil )) nil ))nil ))nil ))nil ))nil )"2" (flip-ineq 1)"2" (case "Fabs(u)=u" )"1" (lemma "Axpy_aux2" )"1" (inst -1 "a" "a1" "t" "u" "x" "x1" "y" "y1" )"1" (split)"1" (propax) nil nil ) ("2" (propax) nil nil )"3" (rewrite "ExactSum_Near" )"1" "ClosestUlp2" :subst"b" "b" "p" "u" "r" "FtoR(t)+FtoR(y)" ))"1" (split)"1" "Fulp(b)(Fpred(b)(Fabs(u))) / 2" "1" (assert ) nil nil )"2" (rewrite "FulpCanonic" )"1" (grind-reals) nil nil )"2" (rewrite "FpredCanonic" ) nil nil ))nil ))nil )"2" (assert )"2" (flip-ineq 1)"2" (rewrite "MinOrMax1" )"2" (expand "abs" 1)"2" "abs(FtoR(t) + FtoR(y))- Fulp(b)(Fpred(b)(u)) / 2 - a1 * x1 - y1" "1" (assert ) nil nil )"2" expand "abs" 1)"2" "1" "1" "1" "RleRoundedLessR0" )"1" "Closest?" "b" "u" "FtoR(t)+FtoR(y)" )"1" "1" (assert ) nil nil )"2" (assert ) nil nil )"3" (propax) nil nil )"4" rewrite "Closest_RoundedMode" )nil nil )"5" rewrite "FcanonicBounded" )nil nil ))nil ))nil ))nil ))nil ))nil )"2" "2" "abs(FtoR(t)-FtoR(a)*FtoR(x)) + abs(-FtoR(y) - FtoR(a) * FtoR(x) + a1 * x1 + y1) - Fulp(b)(Fpred(b)(u)) / 2" )"1" "1" expand "abs" )"1" nil nil ))nil ))nil )"2" case "abs(FtoR[radix](t) - FtoR[radix](a) * FtoR[radix](x)) <= Fulp(b)(Fpred(b)(u)) / 2" )"1" (grind-reals) nil nil )"2" "2" "f2" "Fulp(b)(t) / 2" )"2" "f2" )"1" "1" rewrite "ClosestUlp" )"1" rewrite "FcanonicBounded" )nil nil ))nil ))nil )"2" "2" "2" )"2" rewrite "FulpCanonic" )"2" rewrite "FulpCanonic" )"1" expand "Fsubnormal?" )"1" "1" assert )nil nil ))nil ))nil )"2" rewrite "FpredCanonic" )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"3" (propax) nil nil )"4" (hide-all-but (-4 1))"4" (expand * "FtoR" "Fabs" "abs" )"4" (grind-reals) nil nil )) nil ))nil ))nil ))nil )"2" (decompose-equality)"2" (rewrite "FcanonicBounded" ) nil nil )) nil )"3" (rewrite "FcanonicBounded" ) nil nil )"4" (rewrite "FcanonicBounded" ) nil nil ))nil )"4" (propax) nil nil ) ("5" (propax) nil nil )"6" (propax) nil nil ) ("7" (propax) nil nil ))nil ))nil ))nil )"2" (hide-all-but (-4 1))"2" (expand * "FtoR" "Fabs" "abs" )"2" (grind-reals) (("2" (decompose-equality) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"[numfield, numfield -> numfield]" number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )nil number_fields nil )"Format" axpy nil )"bool" float nil )nil float nil )nil naturalnumbers nil )"real" float nil )nil float nil )"above(1)" axpy nil )nil integers nil )"bool" reals nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil integers nil )"bool" reals nil )nil booleans nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )"boolean" notequal nil )nil number_fields nil )"[numfield, nznum -> numfield]" number_fields nil )"(strict_total_order?[real])" real_props nil )nil real_props nil )nil real_types nil )nil real_props nil )nil reals nil )"bool" functions nil )"(bijective?[T, T])" identity nil )"bool" booleans nil )"[T, T -> boolean]" equalities nil )nil float nil )"bool" float nil )"(divides(m))" divides nil )"(divides(n))" divides nil )nil reals nil )nil exponentiation nil )"posint" exponentiation nil )nil exponentiation nil )nil real_props nil )"posrat" nil )"int" integers nil )OR const-decl "[bool, bool -> bool]" booleans nil )"real" exponentiation nil )"(total_order?[real])" nil )"[bool, bool -> bool]" booleans nil )"nnrat" exponentiation nil )"posrat" exponentiation nil )nil float nil )nil float nil )nil float nil )"float" float nil )nil float nil )"real" reals nil )"odd_int" integers nil )"nnreal" nil )AND const-decl "[bool, bool -> bool]" booleans nil )"bool" reals nil )nil real_types nil )"[numfield -> numfield]" number_fields nil )"{n: nonneg_real | n >= m AND n >= -m}" real_defsnil )"[numfield, numfield -> numfield]" number_fields nil )"bool" reals nil )"bool" float nil )"real" float nil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil )"int" integers nil )"int" integers nil )"real" reals nil )nil axpy nil )"float" float nil )nil axpy nil )"posrat" nil )nil float nil )nil real_props nil )"posreal" nil )nil real_props nil )nil float nil )nil float nil )nil float nil )"bool" float nil )"nnreal" nil )nil real_props nil )nil axpy nil )nil float nil )nil axpy nil )"rat" rationals nil )nil real_props nil )"real" reals nil )"real" reals nil )"real" reals nil ))nil 3320579990"" (skeep)"" (case "u <= a1*x1+y1" )"1" (rewrite "MinOrMax2" )"1" (hide 2)"1" "abs(u-(t+y))+abs(t-a*x)+abs(-1 * y - a * x + a1 * x1 + y1)" "1" (hide-all-but 1)"1" (expand "abs" ) (("1" (grind-reals) nil nil )) nil ))nil )"2" case "abs(FtoR[radix](u) - (FtoR[radix](t) + FtoR[radix](y))) <= Fulp(b)(u)/2" )"1" case "abs(FtoR[radix](t) - FtoR[radix](a) * FtoR[radix](x)) <= Fulp(b)(u)/4" )"1" "Fulp(b)(u) / 4+ Fulp(b)(u) / 2+Fulp(b)(Fpred(b)(u)) / 4" "1" (assert ) nil nil )"2" (case "Fulp(b)(Fpred(b)(u)) <= Fulp(b)(u)" )"1" (grind-reals) nil nil )"2" (rewrite "FulpFpred1" ) nil nil ))nil ))nil )"2" (hide 2 -1)"2" (trans-ineq 1 "Fulp(b)(t)/2" )"1" (rewrite "ClosestUlp" )"1" (rewrite "FcanonicBounded" ) nil nil )) nil )"2" (field 1)"2" (rewrite "FulpCanonic" )"2" (rewrite "FulpCanonic" )"2" (trans-ineq 1 "radix^(1+Fexp(t))" )"1" (rewrite "expt_plus" )"1" rewrite "expt_x1" )"1" (grind-reals) nil nil ))nil ))nil )"2" (rewrite "Exp_increq_1" )"2" expand "Fsubnormal?" )"2" (propax) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (rewrite "ClosestUlp" )"2" (rewrite "FcanonicBounded" ) nil nil )) nil ))nil ))nil ))nil ))nil )"2" (flip-ineq 1)"2" (case "Fabs(u)=u" )"1" (case "abs(t+y) <= u+Fulp(b)(Fpred(b)(u)) / 2" )"1" (lemma "Axpy_aux2" )"1" (inst -1 "a" "a1" "t" "u" "x" "x1" "y" "y1" )"1" (split)"1" (propax) nil nil ) ("2" (propax) nil nil )"3" (rewrite "ExactSum_Near" )"1" "ClosestUlp2" :subst"b" "b" "p" "u" "r" "t+y" ))"1" (split)"1" "Fulp(b)(Fpred(b)(Fabs(u))) / 2" :strict 2)"1" (assert ) nil nil )"2" (rewrite -3)"2" (rewrite "FulpCanonic" )"1" rewrite -8 :dir rl)"1" (grind-reals) nil nil ))nil )"2" (rewrite "FpredCanonic" ) nil nil ))nil ))nil ))nil )"2" (assert ) nil nil ) ("3" (propax) nil nil )"4" (assert ) nil nil ))nil ))nil )"2" (rewrite "FcanonicBounded" ) nil nil )"3" (rewrite "FcanonicBounded" ) nil nil )"4" (rewrite "FcanonicBounded" ) nil nil ))nil )"4" (propax) nil nil ) ("5" (propax) nil nil )"6" (propax) nil nil ) ("7" (propax) nil nil ))nil ))nil ))nil )"2" (flip-ineq 1)"2" (rewrite "MinOrMax1" )"2" (expand "abs" 1)"2" "abs(t + y)- Fulp(b)(Fpred(b)(u)) / 2 - a1 * x1 - y1" "1" (assert ) nil nil )"2" (expand "abs" 1)"2" (grind-reals)"1" (hide 1 2)"1" (flip-ineq -7)"1" (lemma "RleRoundedLessR0" )"1" "Closest?" "b" "u" "t+y" )"1" "1" (assert ) nil nil )"2" (assert ) nil nil )"3" (propax) nil nil )"4" rewrite "Closest_RoundedMode" )nil nil )"5" rewrite "FcanonicBounded" )nil nil ))nil ))nil ))nil ))nil ))nil )"2" (hide 1)"2" "abs(t-a*x) + abs(-y - a * x + a1 * x1 + y1) - Fulp(b)(Fpred(b)(u)) / 2" )"1" (hide-all-but 1)"1" expand "abs" )"1" (grind-reals) nil nil ))nil ))nil )"2" case "abs(FtoR[radix](t) - FtoR[radix](a) * FtoR[radix](x)) <= Fulp(b)(Fpred(b)(u)) / 2" )"1" (grind-reals) nil nil )"2" "2" " Fulp(b)(t) / 2" )"1" rewrite "ClosestUlp" )"1" rewrite "FcanonicBounded" )nil nil ))nil )"2" "2" )"2" rewrite "FulpCanonic" )"2" rewrite "FulpCanonic" )"1" expand "Fsubnormal?" )"1" "1" (assert ) nil nil ))nil ))nil )"2" rewrite "FpredCanonic" )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (hide-all-but (-4 1))"2" (expand * "FtoR" "Fabs" "abs" )"2" (grind-reals) (("2" (decompose-equality) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"bool" float nil )nil float nil )"real" float nil )nil float nil )nil float nil )"bool" float nil )nil float nil )nil float nil )nil float nil )"float" float nil )nil float nil )"real" float nil )"bool" float nil )"float" float nil )nil float nil )nil float nil )nil float nil )"bool" float nil )nil float nil )nil float nil ))nil 3320578694"" (skeep)"" (typepred "t" )"" (expand "Fcanonic?" )"" (split)"1" (lemma "Axpy_aux1" )"1" (inst -1 "a" "a1" "t" "u" "x" "x1" "y" "y1" )"1" (split)"1" (propax) nil nil ) ("2" (propax) nil nil )"3" (lemma "Axpy_aux1_aux1" )"3" (inst -1 "a" "t" "u" "x" "y" )"3" (split)"1" (propax) nil nil )"2" (split)"1" (propax) nil nil ) ("2" (propax) nil nil ))nil )"3" (propax) nil nil ) ("4" (propax) nil nil )"5" (propax) nil nil ) ("6" (propax) nil nil ))nil ))nil ))nil )"4" (propax) nil nil ) ("5" (propax) nil nil )"6" (propax) nil nil ))nil ))nil ))nil )"2" (case "1-dExp(b) <= Fexp(Fpred(b)(u))" )"1" (lemma "Axpy_aux1" )"1" (inst -1 "a" "a1" "t" "u" "x" "x1" "y" "y1" )"1" (split)"1" (propax) nil nil ) ("2" (propax) nil nil )"3" (lemma "Axpy_aux1_aux2" )"3" (inst -1 "a" "t" "u" "x" "y" )"3" (split)"1" (propax) nil nil ) ("2" (propax) nil nil )"3" (propax) nil nil ) ("4" (propax) nil nil )"5" (propax) nil nil ) ("6" (propax) nil nil ))nil ))nil ))nil )"4" (propax) nil nil ) ("5" (propax) nil nil )"6" (propax) nil nil ))nil ))nil ))nil )"2" (flip-ineq 1)"2" (case "1-dExp(b) <= Fexp(u)" )"1" (lemma "Axpy_aux3" )"1" (inst -1 "a" "a1" "t" "u" "x" "x1" "y" "y1" )"1" (split)"1" (propax) nil nil ) ("2" (propax) nil nil )"3" (propax) nil nil )"4" (case "Fbounded?(b)(Fpred(b)(u))" )"1" (expand "Fbounded?" )"1" (flatten) (("1" (assert ) nil nil ))nil ))nil )"2" (rewrite "FcanonicBounded" )"2" (rewrite "FpredCanonic" ) nil nil ))nil ))nil )"5" (propax) nil nil ) ("6" (propax) nil nil )"7" (propax) nil nil ) ("8" (propax) nil nil ))nil ))nil ))nil )"2" (flip-ineq 1)"2" (lemma "Axpy_aux2" )"2" (inst -1 "a" "a1" "t" "u" "x" "x1" "y" "y1" )"2" (split)"1" (propax) nil nil ) ("2" (propax) nil nil )"3" (rewrite "ExactSum_Near" )"1" "radix ^ (-dExp(b))/2" "1" "ClosestUlp" :subst"b" "b" "p" "u" "r" "FtoR(t)+FtoR(y)" ))"1" "1" "Fulp(b)(u) / 2" )"1" (assert ) nil nil )"2" "2" )"2" rewrite "FulpCanonic" )"2" rewrite "Exp_increq_1" )nil nil ))nil ))nil ))nil )"2" (propax) nil nil )"3" rewrite "FcanonicBounded" )nil nil ))nil ))nil )"2" (grind-reals) nil nil ))nil )"2" (rewrite "FcanonicBounded" ) nil nil )"3" (rewrite "FcanonicBounded" ) nil nil )"4" (rewrite "FcanonicBounded" ) nil nil ))nil )"4" (propax) nil nil ) ("5" (propax) nil nil )"6" (propax) nil nil ) ("7" (propax) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"Format" axpy nil )"bool" float nil )nil float nil )nil float nil )"above(1)" axpy nil )"bool" reals nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil numbers nil )nil integers nil )nil naturalnumbers nil )NOT const-decl "[bool -> bool]" booleans nil )nil booleans nil )nil booleans nil )nil axpy nil )nil axpy nil )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil )nil axpy nil )"posrat" exponentiation nil )"nnrat" exponentiation nil )nil float nil )"[numfield, numfield -> numfield]" number_fields nil )"real" float nil )nil float nil )nil float nil )nil reals nil )nil real_props nil )nil real_types nil )nil real_props nil )"posrat" nil )AND const-decl "[bool, bool -> bool]" booleans nil )"bool" reals nil )nil real_types nil )"[numfield -> numfield]" number_fields nil )"{n: nonneg_real | n >= m AND n >= -m}" real_defsnil )"real" float nil )"boolean" notequal nil )nil number_fields nil )"[numfield, nznum -> numfield]" number_fields nil )OR const-decl "[bool, bool -> bool]" booleans nil )"real" exponentiation nil )"bool" reals nil )nil axpy nil )nil axpy nil )nil float nil )"real" reals nil )nil float nil )"int" integers nil )"{j: nonneg_int | j >= i}" nil )"(strict_total_order?[real])" real_props nil )"bool" float nil )nil axpy nil )"float" float nil )"[numfield, numfield -> numfield]" number_fields nil )nil number_fields nil )"bool" reals nil )"int" integers nil )"real" reals nil )"real" reals nil )"real" reals nil ))nil 3320665877 ("" (subtype-tcc) nil nil )"Format" axpy nil )"bool" float nil )nil float nil )nil float nil )nil naturalnumbers nil )NOT const-decl "[bool -> bool]" booleans nil )"bool" float nil )"real" exponentiation nil )"even_int" integersnil )"nonneg_int" nil )"rat" rationals nil )"posnat" exponentiation nil )"int" integers nil )"(total_order?[real])" nil )"(total_order?[real])" nil )"posint" exponentiation nil )"(strict_total_order?[real])" real_props nil )"above(1)" axpy nil )nil integers nil )"bool" reals nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )"posnat" float nil )"bool" float nil )"bool" float nil )"real" float nil )"boolean" notequal nil )"nnrat" exponentiation nil )"posrat" exponentiation nil )"{j: nonneg_int | j >= i}" nil )"(divides(n))" divides nil )"(divides(m))" divides nil ))nil ))nil 3320665877"" (skeep)"" (expand "abs" -5)"" (grind-reals)"" (hide -1 -4)"" (expand * "FtoR" "Fnormal?" )"" (flatten)"" (expand "abs" )"" (grind-reals)"" (expand "Fpred" )"" (grind-reals)"1" (lemma "radix_less_vNum" )"1" (inst -1 "b" )"1" (grind-reals) nil nil )) nil ))nil )"2" (case-replace "Fnum(u)=1" )"1" (case-replace "vNum(b)=radix^(Prec(b))" )"1" (flip-ineq -4)"1" (assert )"1" case "radix^1 < radix ^ (Prec(b))" )"1" rewrite "expt_x1" )"1" (assert ) nil nil ))nil )"2" (rewrite "Exp_incr_1" ) nil nil ))nil ))nil ))nil )"2" (assert )"2" (expand "vNum" )"2" (propax) nil nil )) nil ))nil ))nil )"2" (assert ) nil nil ))nil )"3" (case-replace "Fnum(u)=1" )"1" (case "radix^1 < radix ^ (Prec(b))" )"1" (flip-ineq -4)"1" (assert )"1" "vNum(b)=radix^(Prec(b))" )"1" rewrite "expt_x1" )"1" (assert ) nil nil ))nil )"2" assert )"2" expand "vNum" )"2" (propax) nil nil ))nil ))nil ))nil ))nil ))nil )"2" (rewrite "Exp_incr_1" ) nil nil ))nil )"2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"{n: nonneg_real | n >= m AND n >= -m}" real_defsnil )nil real_props nil )nil real_props nil )"rat" rationals nil )"nnrat" exponentiation nil )"(total_order?[real])" nil )nil real_props nil )"posrat" nil )"(strict_total_order?[real])" real_props nil )"Format" axpy nil )nil float nil )nil naturalnumbers nil )"above(1)" axpy nil )nil integers nil )"bool" reals nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )nil float nil )"real" exponentiation nil )"boolean" notequal nil )OR const-decl "[bool, bool -> bool]" booleans nil )"posnat" float nil )nil integers nil )nil integers nil )"bool" reals nil )"posint" exponentiation nil )"even_int" integersnil )nil float nil )nil exponentiation nil )"bool" reals nil )"posint" nil )nil number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )"bool" float nil )nil float nil )"[T, T -> boolean]" equalities nil )"float" float nil )"int" integers nil )"real" float nil )"(divides(n))" divides nil )"(divides(m))" divides nil )"bool" float nil )nil real_props nil )"int" integers nil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil ))nil ))nil 3320665877 ("" (subtype-tcc) nil nil )"Format" axpy nil )"bool" float nil )nil float nil )nil float nil )nil naturalnumbers nil )NOT const-decl "[bool -> bool]" booleans nil )"bool" float nil )"real" exponentiation nil )"even_int" integersnil )"nonneg_int" nil )"rat" rationals nil )"posnat" exponentiation nil )"int" integers nil )"(total_order?[real])" nil )"(total_order?[real])" nil )"posint" exponentiation nil )"(strict_total_order?[real])" real_props nil )"above(1)" axpy nil )nil integers nil )"bool" reals nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )"posnat" float nil )"bool" float nil )"bool" float nil )"real" float nil )"boolean" notequal nil )"nnrat" exponentiation nil )"posrat" exponentiation nil )"{j: nonneg_int | j >= i}" nil )"(divides(n))" divides nil )"(divides(m))" divides nil ))nil ))nil 3320665877"" (skeep)"" (case "0 < (1 - 1 / (2 * abs(Fnum(t))))" )"1" (grind-reals) nil nil )"2" (hide -5)"2" (case "0 < abs(Fnum(t))" )"1" (case "1 <= abs(Fnum(t))" )"1" (grind-reals)"1" (move-terms 1 r 2) (("1" (assert ) nil nil )) nil ))nil )"2" (grind-reals)"2" (hide-all-but (-1 1)) (("2" (grind) nil nil )) nil ))nil ))nil )"2" (expand "Fnormal?" )"2" (flatten)"2" (hide-all-but (-2 1))"2" (expand "abs" ) (("2" (grind-reals) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"Format" axpy nil )"bool" float nil )nil float nil )nil naturalnumbers nil )nil float nil )"above(1)" axpy nil )nil integers nil )"bool" reals nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil integers nil )"{n: nonneg_real | n >= m AND n >= -m}" real_defsnil )"[numfield -> numfield]" number_fields nil )AND const-decl "[bool, bool -> bool]" booleans nil )nil real_types nil )"bool" reals nil )"[numfield, numfield -> numfield]" number_fields nil )"[numfield, nznum -> numfield]" number_fields nil )nil number_fields nil )"boolean" notequal nil )"[numfield, numfield -> numfield]" number_fields nil )nil number_fields nil )"bool" reals nil )nil booleans nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )"rat" rationals nil )"(divides(m))" divides nil )"(divides(n))" divides nil )"even_int" integersnil )"nonneg_int" nil )"(strict_total_order?[real])" real_props nil )"nzrat" rationalsnil )"{j: nonneg_int | j >= i}" nil )"(total_order?[real])" nil )"int" integers nil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil )"even_int" integersnil )nil real_props nil )"rat" rationals nil )"[bool, bool -> bool]" booleans nil )"[numfield, numfield -> numfield]" number_fields nil )"bool" reals nil )nil real_props nil )"bool" float nil ))nil ))nil 3320665877 ("" (subtype-tcc) nil nil )"Format" axpy nil )"bool" float nil )nil float nil )nil float nil )nil naturalnumbers nil )NOT const-decl "[bool -> bool]" booleans nil )"even_int" integersnil )"(strict_total_order?[real])" real_props nil )"{j: nonneg_int | j >= i}" nil )"posint" exponentiation nil )"(total_order?[real])" nil )"(total_order?[real])" nil )"int" integers nil )"posnat" exponentiation nil )"nat" exponentiation nil )"nat" exponentiation nil )"{n: nonneg_real | n >= m AND n >= -m}" real_defsnil )"real" exponentiation nil )"bool" float nil )"above(1)" axpy nil )nil integers nil )"bool" reals nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )"posnat" float nil )"bool" float nil )"bool" float nil )"real" float nil )"boolean" notequal nil )"nnrat" exponentiation nil )"posrat" exponentiation nil )"(divides(n))" divides nil )"(divides(m))" divides nil ))nil ))nil 3320665878"" case "FORALL (x:float): Fnormal?(b)(x) => radix^(Prec(b)-1) <= abs(Fnum(x))" )"1" (skeep)"1" (inst-cp -1 "t" )"1" (inst -1 "u" )"1" (split)"1" (split)"1" (case "2 <= radix ^ (Prec(b) - 1)" )"1" (mult-by 1 "(1 - 1 / (2 * abs(Fnum(t))))" )"1" "(1 + radix + radix ^ (4 - Prec(b))) * (1 - radix^(1-Prec(b))/2)" )"1" (move-terms 1 l 1)"1" (move-terms 1 l 4)"1" "radix/(radix^(Prec(b)-1)-1)+radix/(2*radix^(Prec(b)-1))+ ") "1" case "(abs(Fnum(Fpred(b)(u)))=radix ^ (Prec(b))-1 AND abs(Fnum(u))= radix ^ (Prec(b) - 1)) OR abs(Fnum(Fpred(b)(u)))=abs(Fnum(u))-1" )"1" (split)"1" "1" "1" replace -3)"1" "1" replace -2)"1" "1" case "radix ^ (Prec(b))=radix*radix ^ (Prec(b) - 1)" )"1" "AA" "radix ^ (Prec(b) - 1)" )"1" "1" replace -2)"1" "1" case "1 / (radix * AA - 1) <= 1 / (AA - 1)" )"1" case "(1 / (radix * AA - 1)) * (1 / (2 * AA)) * radix <= radix * 1 / (2 * AA) * 1 / (AA - 1)" )"1" case " (1 / (radix * AA - 1)) * radix <= radix / (AA - 1)" )"1" assert )nil nil )"2" "2" "radix" )"2" "radix / (AA - 1) / radix=1 / (AA - 1)" )"2" nil nil ))nil ))nil ))nil ))nil )"2" "2" "radix" )"2" " radix * 1 / (2 * AA) * 1 / (AA - 1) / radix = ") "1" "2*AA" )"1" "(1 / (radix * AA - 1)) * (1 / (2 * AA)) * (2 * AA)=1 / (radix * AA - 1)" )"1" "(1 / (AA - 1)) * (1 / (2 * AA)) * (2 * AA)=1 / (AA - 1)" )"1" nil nil ))nil )"2" nil nil ))nil ))nil )"2" nil nil ))nil ))nil ))nil ))nil )"2" "2" case "0 < radix * AA - 1" )"1" case "0 < AA - 1" )"1" "1" "AA" )"1" assert )nil nil ))nil ))nil )"2" nil nil ))nil )"2" "2" case "2*2 <= radix*AA" )"1" nil nil )"2" "2*AA" )"1" nil nil )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" rewrite "expt_div" "2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" replace -1)"2" "2" case "1 / (abs(Fnum(u)) - 1) <= 1 / (radix ^ (Prec(b) - 1) - 1)" )"1" "radix / (radix ^ (Prec(b) - 1) - 1) + 1 / (2 * abs(Fnum(u))) * radix + ") "1" (assert ) nil nil )"2" " radix / (radix ^ (Prec(b) - 1) - 1) = (1/ (radix ^ (Prec(b) - 1) - 1)) * radix" )"1" case "(1 / (abs(Fnum(u)) - 1)) * radix <= (1 / (radix ^ (Prec(b) - 1) - 1)) * radix" )"1" (assert ) nil nil )"2" "2" "radix" )nil nil ))nil ))nil )"2" (assert ) nil nil ))nil )"3" case " 1 / (2 * abs(Fnum(u))) * radix <= radix / (2 * radix ^ (Prec(b) - 1))" )"1" " radix / (radix ^ (Prec(b) - 1) - 1) + ") "1" (assert ) nil nil )"2" case "(1 / (abs(Fnum(u)) - 1)) * (1 / (2 * abs(Fnum(u)))) * radix <= radix * 1 / (2 * radix ^ (Prec(b) - 1)) * 1 / ") "1" (assert ) nil nil )"2" "2" "radix" )"2" " radix * 1 / (2 * radix ^ (Prec(b) - 1)) * 1 / ") "1" case "(1 / (2 * abs(Fnum(u)))) <= 1 / (2 * radix ^ (Prec(b) - 1))" )"1" " (1 / (abs(Fnum(u)) - 1)) * 1 / (2 * radix ^ (Prec(b) - 1))" )"1" "abs(Fnum(u)) - 1" )"1" "(1 / (abs(Fnum(u)) - 1)) * (1 / (2 * abs(Fnum(u)))) * ") "1" "(1 / (abs(Fnum(u)) - 1)) * 1 / (2 * radix ^ (Prec(b) - 1)) * ") "1" nil nil ))nil )"2" nil nil ))nil )"2" nil nil ))nil )"2" "2 * radix ^ (Prec(b) - 1)" )nil nil ))nil )"2" "2" nil nil ))nil ))nil )"2" nil nil ))nil ))nil ))nil ))nil ))nil )"2" "2" "radix" )"2" "radix / (2 * radix ^ (Prec(b) - 1)) / radix= 1 / (2*radix ^ (Prec(b) - 1))" )"1" "2" )"1" "1 / (2 * abs(Fnum(u))) * 2=1 /abs(Fnum(u))" )"1" "1 / (2 * radix ^ (Prec(b) - 1)) * 2=1/radix ^ (Prec(b) - 1)" )"1" nil nil )"2" nil nil ))nil )"2" nil nil )"3" nil nil ))nil ))nil )"2" (field 1) nil nil ))nil ))nil ))nil ))nil ))nil )"2" "2" case "0 < radix ^ (Prec(b) - 1) - 1" )"1" (grind-reals) nil nil )"2" case "radix^1 <= radix ^ (Prec(b) - 1)" )"1" rewrite "expt_x1" )"1" nil nil ))nil )"2" rewrite "Exp_increq_1" )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (hide 2)"2" case "0 < Fnum(u)" )"1" case "0 <= Fnum(Fpred(b)(u))" )"1" "abs(Fnum(Fpred(b)(u)))=Fnum(Fpred(b)(u))" )"1" "abs(Fnum(u))=Fnum(u)" )"1" "radix ^ (Prec(b))=vNum(b)" )"1" "radix ^ (Prec(b)-1)=vNum(b)/radix" )"1" expand "Fpred" 1)"1" nil nil ))nil )"2" rewrite -1 :dir rl)"2" rewrite "expt_div" nil nil ))nil ))nil )"2" expand "vNum" )"2" (propax) nil nil ))nil ))nil )"2" expand "abs" 1)"2" (grind-reals) nil nil ))nil ))nil )"2" expand "abs" 1)"2" (grind-reals) nil nil ))nil ))nil )"2" case "0 <= FtoR(Fpred(b)(u))" )"1" expand "FtoR" -1)"1" (grind-reals) nil nil ))nil )"2" rewrite "FpredPos" )nil nil ))nil ))nil )"2" expand "FtoR" -6)"2" (grind-reals) nil nil ))nil ))nil ))nil ))nil )"2" (hide -2 -3 -4 -5 -6)"2" "radix ^ (1 - Prec(b))=1/radix ^ (Prec(b) - 1)" )"1" " radix ^ (4 - Prec(b))=radix^3/radix ^ (Prec(b) - 1)" )"1" "1" "AA" "radix ^ (Prec(b) - 1)" )"1" case "0 < AA-1" )"1" case "0 < 2*AA" )"1" "AA-1" )"1" "2*AA" )"1" "3*AA*radix" )"1" (field 1) nil nil )"2" "2" "AA" )"2" "2" "2 * (AA * AA * radix ^ 3) / (2 * (AA * AA))=radix ^ 3" )"1" "2 * (AA * AA * radix) / (2 * AA)=AA * radix" )"1" "2 * (AA * AA * AA) / (2 * AA)=AA * AA" )"1" "2 * (AA * AA * AA * radix ^ 3) / (2 * (AA * AA))=AA * radix ^ 3" )"1" " 2 * (AA * AA * AA * radix) / (2 * AA)=AA * AA * radix" )"1" "1" assert )"1" "1" "1" "1" case "radix*radix <= AA" )"1" "1" rewrite "expt_x3" )"1" "1" " 2 * (AA * AA * radix * radix * radix)" )"1" "AA * AA + 3 * (AA * (AA * radix)) + 4 * (AA * AA * radix)" )"1" nil nil )"2" "AA" )"2" "AA" )"2" "1+7*radix" )"1" nil nil )"2" "2 * (AA * AA * radix * radix * radix) / AA / AA=2*radix*radix*radix" )"1" "1" "8*radix" )"1" nil nil )"2" "radix" )"2" "2" )"2" "8/2=4" )"1" "2*radix" )"1" nil nil )"2" nil nil ))nil )"2" nil nil ))nil ))nil ))nil ))nil ))nil )"2" nil nil ))nil ))nil ))nil ))nil ))nil )"2" nil nil ))nil ))nil ))nil ))nil )"2" "2" "2" rewrite "2" rewrite "expt_x2" "2" rewrite "Exp_increq_1" )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" nil nil ))nil )"2" nil nil ))nil )"2" nil nil ))nil )"2" nil nil ))nil )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" (grind-reals) nil nil ))nil ))nil )"2" "2" (grind-reals) nil nil ))nil ))nil ))nil ))nil )"2" "2" (rewrite "expt_div" ) nil nil ))nil ))nil )"2" "2" rewrite "expt_inverse" :dir rl)"2" (assert ) nil nil ))nil ))nil ))nil ))nil )"3" (hide 2)"3" (skosimp*)"3" "3" (grind-reals) nil nil ))nil ))nil ))nil )"4" (hide 2) (("4" (grind-reals) nil nil ))nil ))nil ))nil ))nil )"2" "1 + radix + radix ^ (4 - Prec(b))" )"2" "radix ^ (1 - Prec(b))=1/radix ^ (Prec(b) - 1)" )"1" (grind-reals) nil nil )"2" (rewrite "expt_inverse" :dir rl)"2" (assert ) nil nil )) nil ))nil ))nil ))nil )"2" (hide 2)"2" (split)"1" (field 1) nil nil ) ("2" (field 1) nil nil ))nil ))nil ))nil )"2" (trans-ineq 1 "radix ^1" )"1" (rewrite "expt_x1" ) (("1" (assert ) nil nil ))nil )"2" (rewrite "Exp_increq_1" ) nil nil ))nil ))nil )"2" (propax) nil nil ))nil )"2" (propax) nil nil ))nil ))nil ))nil ))nil )"2" (hide 2)"2" (skeep)"2" (expand "Fnormal?" )"2" (flatten)"2" (rewrite "abs_mult" )"2" (expand "abs" -2 1)"2" "vNum(b)= radix *radix ^ (Prec(b) - 1)" )"1" (grind-reals) nil nil )"2" (case-replace " vNum(b) =radix ^ (Prec(b))" )"1" (rewrite "expt_div" :dir rl)"1" (field 1) nil nil )) nil )"2" (assert )"2" (expand "vNum" ) (("2" (propax) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"3" (hide 2) (("3" (skeep) (("3" (assert ) nil nil )) nil )) nil ))nil )nil real_props nil )"{j: posint | j >= i}" nil )"rat" rationals nil )"rat" rationals nil )nil real_props nil )nil real_types nil )"[numfield, numfield -> numfield]" number_fields nil )"float" float nil )"{x: float | Fcanonic?(b)(x)}" axpy nil )"[numfield, numfield -> numfield]" number_fields nil )"[numfield, nznum -> numfield]" number_fields nil )nil number_fields nil )"nonneg_int" nil )"even_int" integersnil )"(divides(n))" divides nil )"(divides(m))" divides nil )"{j: nonneg_int | j >= i}" nil )"posrat" nil )"(total_order?[real])" nil )"nzrat" rationalsnil )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil )"nnrat" exponentiation nil )"posrat" exponentiation nil )"(strict_total_order?[real])" real_props nil )"rat" rationals nil )"rat" rationals nil )"nzrat" rationalsnil )"posint" nil )nil real_props nil )nil reals nil )nil real_props nil )"[bool, bool -> bool]" booleans nil )"posint" nil )nil real_props nil )"int" integers nil )nil real_props nil )nil integers nil )nil integers nil )"posnat" float nil )nil float nil )nil real_props nil )"real" float nil )nil exponentiation nil )nil exponentiation nil )nil reals nil )nil rationals nil )nil rationals nil )nil real_props nil )nil extra_real_props nil )NOT const-decl "[bool -> bool]" booleans nil )"bool" reals nil )nil real_props nil )nil real_props nil )nil real_props nil )nil real_props nil )"posint" exponentiation nil )nil float 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 )nil real_props nil )nil real_props nil )nil real_props nil )nil real_props nil )nil real_props nil )nil real_props nil )"even_int" integersnil )"nonneg_rat" nil )"even_int" integersnil )"{x: float | Fcanonic?(b)(x)}" axpy nil )"[T, T -> boolean]" equalities nil )"posrat" axpy nil )nil rationals nil )"bool" booleans nil )"(bijective?[T, T])" identity nil )"bool" functions nil )"even_int" integers nil )"nzint" integers nil )"odd_int" integers nil )nil exponentiation nil )nil real_props nil )nil extra_real_props nil )nil real_props nil )nil exponentiation nil )nil real_props nil )nil exponentiation nil )"posrat" nil )"posrat" nil )"odd_int" integersnil )nil real_props nil )"{n: nonneg_real | n >= Fnum(t) AND n >= -Fnum(t)}" axpy nil )nil extra_real_propsnil )"bool" float nil )"int" integers nil )nil integers nil )nil numbers nil )nil booleans nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil booleans nil )"bool" reals nil )nil integers nil )"above(1)" axpy nil )nil float nil )"[bool, bool -> bool]" booleans nil )nil naturalnumbers nil )nil float nil )"bool" float nil )"Format" axpy nil )"bool" reals nil )OR const-decl "[bool, bool -> bool]" booleans nil )"boolean" notequal nil )"bool" reals nil )"real" exponentiation nil )nil number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )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}" real_defsnil ))nil 3320593659 ("" (subtype-tcc) nil nil )"Format" axpy nil )"bool" float nil )nil float nil )nil float nil )nil naturalnumbers nil )NOT const-decl "[bool -> bool]" booleans nil )"even_int" integersnil )"(strict_total_order?[real])" real_props nil )"{j: nonneg_int | j >= i}" nil )"posint" exponentiation nil )"(total_order?[real])" nil )"(total_order?[real])" nil )"int" integers nil )"posnat" exponentiation nil )"(strict_total_order?[real])" real_props nil )"nat" exponentiation nil )"nat" exponentiation nil )"{n: nonneg_real | n >= m AND n >= -m}" real_defsnil )"real" exponentiation nil )"bool" float nil )"above(1)" axpy nil )nil integers nil )"bool" reals nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )"real" float nil )"bool" float nil )"posnat" float nil )"bool" float nil )"bool" float nil )"boolean" notequal nil )"nnrat" exponentiation nil )"posrat" exponentiation nil )"rat" rationals nil )"rat" rationals nil )"real" reals nil )"(divides(n))" divides nil )"(divides(m))" divides nil ))nil ))nil 3320593708"" (skeep)"" (case "Fnormal?(b)(u)" )"1" (case "0 < Fnum(Fpred(b)(u))" )"1" (case "0 < 1+1/Fnum(Fpred(b)(u))" )"1" (trans-ineq 1 "FtoR(u)/(1+1/Fnum(Fpred(b)(u)))" )"1" (mult-by 1 "1 + 1 / Fnum(Fpred(b)(u))" )"1" (case "0 < abs(Fnum(u))" )"1" (case "0 < 1+1/(2*abs(Fnum(u)))" )"1" "abs(FtoR(t)+FtoR(y))/(1+1/(2*abs(Fnum(u))))" )"1" (mult-by 1 "1 + 1 / (2 * abs(Fnum(u)))" )"1" "abs(FtoR(y)) - abs(FtoR(t))" )"1" (move-terms 1 r 2)"1" "abs(FtoR[radix](t))*(1+radix*((1 + 1 / Fnum(Fpred(b)(u)))*(1 + 1 / (2 * abs(Fnum(u))))))" )"1" (assert ) nil nil )"2" (case "0 < abs(Fnum(t))" )"1" case " 0 < (1 - 1 / (2 * abs(Fnum(t))))" )"1" case " abs(FtoR[radix](t)) <= ") "1" "(radix + 1 + radix ^ (4 - Prec(b))) * ") "1" "Axpy_opt_aux1_aux1" )"1" "t" "u" )"1" "1" "abs(FtoR(a)*FtoR(x))" )"1" "(1 + ") "1" "1" "abs(Fnum(Fpred(b)(u)))=Fnum(Fpred(b)(u))" )"1" "1" "G1" "(1 - 1 / (2 * abs(Fnum(t))))" )"1" "G2" "1 + 1 / (2 * abs(Fnum(u)))" )"1" "G3" "1 + 1 / Fnum(Fpred(b)(u))" )"1" "1 + radix * (G3 * G2)" )"1" "(1 + radix * G2 * G3) / G1 * abs(FtoR[radix](a) * FtoR[radix](x)) / ") "1" nil nil ))nil )"2" "2" case "radix * (G3 * G2) >= 0" )"1" "1" nil nil )"2" nil nil ))nil )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" expand "abs" 1)"2" nil nil ))nil ))nil ))nil ))nil )"2" (assert ) nil nil ))nil ))nil )"2" (propax) nil nil )"3" (propax) nil nil )"4" (propax) nil nil )"5" (propax) nil nil ))nil ))nil ))nil ))nil )"2" "2" rewrite "RoundLe" )"2" "2" expand * "abs" "FtoR" )"2" (grind-reals) nil nil ))nil ))nil ))nil ))nil )"3" (assert ) nil nil ))nil )"2" "2" "2" "2" assert )"2" case "1 <= abs(Fnum(t))" )"1" (grind-reals) nil nil )"2" "2" (grind) nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"3" (assert ) nil nil ))nil )"2" expand "Fnormal?" )"2" "2" "2" rewrite "abs_mult" )"2" expand "abs" -1 1)"2" (grind-reals) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (expand "abs" 1)"2" (grind-reals) nil nil )) nil ))nil ))nil )"2" (trans-ineq 1 "abs(FtoR(u))" )"1" (rewrite "RoundGe" ) nil nil )"2" (expand "abs" )"2" (grind-reals) nil nil )) nil ))nil )"3" (skosimp*) (("3" (grind-reals) nil nil )) nil )"4" (grind-reals) nil nil ))nil )"2" (case "0 <= 1 / (2 * abs(Fnum(u)))" )"1" (grind-reals) nil nil )"2" (grind-reals) nil nil ))nil )"3" (grind-reals) nil nil ))nil )"2" (expand "FtoR" -6)"2" (expand "abs" 1) (("2" (grind-reals) nil nil ))nil ))nil ))nil ))nil )"2" (mult-by 1 "1 + 1/Fnum(Fpred(b)(u))" )"2" (assert )"2" " FtoR[radix](Fpred(b)(u)) * (1 / Fnum(Fpred(b)(u))) = Fulp(b)(Fpred(b)(u))" )"1" (rewrite "FpredDiff" :dir rl)"1" (assert ) nil nil )) nil )"2" (hide 2)"2" (expand "FtoR" 1)"2" (rewrite "FulpCanonic" )"1" (field 1) nil nil )"2" (rewrite "FpredCanonic" ) nil nil ))nil ))nil ))nil )"3" (rewrite "FcanonicBounded" )"3" (rewrite "FpredCanonic" ) nil nil )) nil ))nil ))nil ))nil )"3" (hide 2)"3" (skosimp*) (("3" (grind-reals) nil nil )) nil )) nil )"4" (hide 2)"4" (skosimp*) (("4" (grind-reals) nil nil )) nil ))nil ))nil )"2" (hide-all-but (-1 1))"2" (trans-ineq 1 "1" :strict 1)"1" (assert ) nil nil )"2" (grind-reals)"2" (case "0 <= 1 / Fnum(Fpred(b)(u))" )"1" (grind-reals) nil nil )"2" (grind-reals) nil nil ))nil ))nil ))nil ))nil )"3" (assert ) nil nil ))nil )"2" (hide-all-but (-1 1 -4))"2" (expand * "FtoR" "Fnormal?" )"2" (flatten)"2" (expand "abs" -2)"2" (grind-reals)"2" (expand "Fpred" )"2" (grind-reals)"1" (lemma "radix_less_vNum" )"1" (inst -1 "b" )"1" (grind-reals) nil nil )) nil ))nil )"2" (mult-by 2 "radix" )"2" (assert )"2" (case "radix < vNum(b)" )"1" (grind-reals) nil nil )"2" "vNum(b)=radix^(Prec(b))" )"1" case "radix=radix^1" )"1" "radix^1" :strict 2)"1" (assert ) nil nil )"2" rewrite "Exp_incr_1" )nil nil ))nil )"2" (rewrite "expt_x1" ) nil nil ))nil )"2" assert )"2" expand "vNum" )"2" (propax) nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"3" (mult-by 2 "radix" )"3" (assert )"3" (case "radix < vNum(b)" )"1" (grind-reals) nil nil )"2" (trans-ineq 1 "radix^1" :strict 2)"1" rewrite "expt_x1" )"1" (assert ) nil nil ))nil )"2" "vNum(b)=radix^(Prec(b))" )"1" (rewrite "Exp_incr_1" ) nil nil )"2" assert )"2" expand "vNum" )"2" (propax) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (hide 2)"2" (lemma "Normal_iff" :subst ("f" "u" ))"2" (split)"1" (flatten)"1" case " radix ^ (Prec(b) - 1 - dExp(b)) <= abs(FtoR[radix](u))" )"1" (assert ) nil nil )"2" (trans-ineq 1 "abs(FtoR(t))" )"1" (lemma "Normal_iff" :subst ("f" "t" ))"1" (split)"1" (flatten) (("1" (assert ) nil nil )) nil )"2" (assert ) nil nil ))nil ))nil )"2" (hide 2 -1 -2)"2" (lemma "Closest_Monotone" )"2" (expand "Monotone?" )"2" "b" "abs(FtoR(t))" "abs(FtoR(t)+FtoR(y))" "Fabs(t)" "Fabs(u)" )"1" (split)"1" (rewrite "FabsCorrect" )"1" (rewrite "FabsCorrect" ) nil nil ))nil )"2" (rewrite "ClosestFabs" )"2" rewrite "FcanonicBounded" )nil nil ))nil )"3" (rewrite "ClosestFabs" )"1" rewrite "RoundedProjector" )"1" rewrite "Closest_RoundedMode" )nil nil )"2" rewrite "FcanonicBounded" )nil nil ))nil )"2" rewrite "FcanonicBounded" )nil nil ))nil )"4" (hide 2)"4" case "0 < 2*radix^(Prec(b)-1)" )"1" "abs(FtoR[radix](t))*((radix+1)*(1-1/(2*radix ^ (Prec(b) - 1)))-1) " "1" "abs(FtoR[radix](t))" )"1" case "radix*radix <=radix ^ (Prec(b) - 1)" )"1" "AA" "radix ^ (Prec(b) - 1)" )"1" "1" "1" "1" "1" "2*AA+AA" "1" "1" assert )"1" "radix*radix" )"1" "radix+radix" )"1" nil nil )"2" nil nil ))nil ))nil ))nil ))nil )"2" "AA" )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" rewrite "expt_x2" :dir rl)"2" rewrite "Exp_increq_1" )nil nil ))nil ))nil )"2" "2" expand * "Fnormal?" "FtoR" )"2" "2" rewrite "abs_mult" )"2" rewrite "abs_mult" )"2" expand "abs" -2 1)"2" expand "abs" 1 2)"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "abs(FtoR(y))-abs(FtoR(t))" )"1" "1" "(abs(FtoR[radix](t)) * ((radix + 1) * (1 - 1 / (2 * radix ^ (Prec(b) - 1)))))" )"1" (assert ) nil nil )"2" "abs(FtoR[radix](a) * FtoR[radix](x))*(radix + 1)" )"1" "rr" "radix+1" )"1" "rr" )"1" "1" "(abs(FtoR[radix](t)) *(1 - 1 / (2 * radix ^ (Prec(b) - 1))))" )"1" "G1" "1 / (2 * radix ^ (Prec(b) - 1))" )"1" nil nil ))nil )"2" "RoundLe" )"2" "t" "FtoR(a)*FtoR(x)" )"2" case " 0 < 2 * abs(Fnum(t))" )"1" case " 0 < 1 - 1 / (2 * abs(Fnum(t)))" )"1" "1" " 1 - 1 / (2 * abs(Fnum(t)))" )"1" "abs(FtoR[radix](t)) * (1 - 1 / (2 * abs(Fnum(t))))" )"1" "abs(FtoR[radix](t))" )"1" case " radix ^ (Prec(b) - 1) <= abs(Fnum(t))" )"1" nil nil )"2" "2" expand "Fnormal?" )"2" "2" "radix ^ (Prec(b) - 1)=vNum(b)/radix" )"1" "1" expand "abs" "1" assert )nil nil ))nil ))nil )"2" "vNum(b)=radix ^ (Prec(b))" )"1" rewrite "expt_div" nil nil )"2" assert )"2" expand "vNum" )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" nil nil ))nil ))nil )"2" nil nil )"3" "3" expand *"Fnormal?" "FtoR" )"3" "3" expand "abs" )"3" nil nil ))nil ))nil ))nil ))nil )"4" assert )nil nil ))nil )"2" "2" case "1 <= abs(Fnum(t))" )"1" nil nil )"2" "2" nil nil ))nil ))nil ))nil )"3" assert )nil nil ))nil )"2" "2" expand "Fnormal?" )"2" "2" expand "abs" )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "(radix + 1 + radix ^ (4 - Prec(b))) * ") "2" "abs(FtoR(a)*FtoR(x))" )"2" nil nil ))nil ))nil ))nil ))nil ))nil )"2" "2" expand "abs" )"2" (grind-reals) nil nil ))nil ))nil ))nil ))nil )"2" (grind-reals) nil nil )"3" (assert ) nil nil ))nil ))nil ))nil )"2" (rewrite "FabsBounded" )"2" (rewrite "FcanonicBounded" ) nil nil ))nil )"3" (rewrite "FabsBounded" )"3" (rewrite "FcanonicBounded" ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"3" (assert ) nil nil ))nil ))nil )"2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil )"bool" float nil )"Format" axpy nil )"bool" float nil )nil float nil )nil float nil )"above(1)" axpy nil )"bool" reals nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )nil integers nil )nil naturalnumbers nil )nil real_props nil )"posrat" nil )nil float nil )"posint" exponentiation nil )nil exponentiation nil )nil float nil )"posnat" float nil )nil integers nil )nil integers nil )nil extra_real_propsnil )"[numfield, nznum -> numfield]" number_fields nil )nil number_fields nil )"boolean" notequal nil )"[numfield, numfield -> numfield]" number_fields nil )nil number_fields nil )"rat" rationals nil )"real" float nil )"bool" float nil )nil float nil )nil float nil )nil float nil )nil float nil )nil real_props nil )nil real_types nil )"{x: float | Fcanonic?(b)(x)}" axpy nil )"(strict_total_order?[real])" real_props nil )nil real_props nil )nil reals nil )nil real_props nil )nil real_props nil )nil real_props nil )"nonneg_int" nil )"even_int" integersnil )"(divides(n))" divides nil )"(divides(m))" divides nil )"{j: nonneg_int | j >= i}" nil )nil axpy nil )"real" reals nil )"[bool, bool -> bool]" booleans nil )"even_int" integersnil )nil real_props nil )nil axpy nil )"[T, T -> boolean]" equalities nil )nil rationals nil )"posrat" exponentiation nil )"odd_int" integers nil )nil real_props nil )nil real_props nil )NOT const-decl "[bool -> bool]" booleans nil )nil extra_real_props nil )"rat" axpy nil )"rat" axpy nil )nil real_props nil )"int" integers nil )nil extra_real_propsnil )"rat" rationals nil )OR const-decl "[bool, bool -> bool]" booleans nil )"real" exponentiation nil )nil axpy nil )nil real_props nil )"rat" rationals nil )nil real_props nil )"{j: posint | j >= i}" nil )"rat" rationals nil )"nzrat" rationalsnil )"real" reals nil )"[numfield, numfield -> numfield]" number_fields nil )nil real_props nil )nil real_props nil )nil real_props nil )nil real_props nil )nil real_props nil )"(strict_total_order?[real])" real_props nil )"real" reals nil )"(total_order?[real])" nil )"posrat" nil )"posint" nil )"nnrat" exponentiation nil )"int" integers nil )"nnreal" nil )"real" float nil )"{n: nonneg_real | n >= m AND n >= -m}" real_defsnil )"[numfield -> numfield]" number_fields nil )nil real_types nil )"bool" reals nil )"[numfield, numfield -> numfield]" number_fields nil )"bool" reals nil )AND const-decl "[bool, bool -> bool]" booleans nil )"real" reals nil )"nnreal" nil )"(total_order?[real])" nil )"nzrat" rationalsnil )"[bool, bool -> bool]" booleans nil )nil real_props nil )nil real_props nil )"bool" reals nil )"float" float nil )nil axpy nil )nil float nil )"{x: float | Fcanonic?(b)(x)}" axpy nil )"float" float nil )"nonneg_rat" nil )"{r: posrat | r >= q}" nil )"posint" nil )nil real_props nil )"even_int" integers nil )"nzint" integers nil )"bool" booleans nil )"(bijective?[T, T])" identity nil )"bool" functions nil )nil rationals nil )nil rationals nil )nil float nil )nil exponentiation nil )nil extra_real_props nil )nil real_props nil )nil real_props nil )"posnat" exponentiation nil )"real" exponentiation nil )nil exponentiation nil )nil reals nil )nil real_props nil )nil real_props nil )nil real_props nil )nil real_props nil )"odd_int" integersnil )nil integers nil )"{x: float | Fcanonic?(b)(x)}" axpy nil )"{x: float | Fcanonic?(b)(x)}" axpy nil )"posrat" nil )nil float nil )nil float nil )nil float nil )"bool" float nil )nil float nil )nil float nil )nil float nil )"bool" float nil )"real" reals nil ))nil 3320755855 ("" (subtype-tcc) nil nil )"Format" axpy nil )"bool" float nil )nil float nil )nil float nil )nil naturalnumbers nil )NOT const-decl "[bool -> bool]" booleans nil )"even_int" integersnil )"(strict_total_order?[real])" real_props nil )"{j: nonneg_int | j >= i}" nil )"posint" exponentiation nil )"(total_order?[real])" nil )"(total_order?[real])" nil )"int" integers nil )"posnat" exponentiation nil )"(strict_total_order?[real])" real_props nil )"nat" exponentiation nil )"nat" exponentiation nil )"{n: nonneg_real | n >= m AND n >= -m}" real_defsnil )"real" exponentiation nil )"bool" float nil )"above(1)" axpy nil )nil integers nil )"bool" reals nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )"real" float nil )"bool" float nil )"posnat" float nil )"bool" float nil )"bool" float nil )"boolean" notequal nil )"nnrat" exponentiation nil )"posrat" exponentiation nil )"rat" rationals nil )"rat" rationals nil )"real" reals nil )"(divides(n))" divides nil )"(divides(m))" divides nil ))nil ))nil 3321271390 ("" (subtype-tcc) nil nil )"Format" axpy nil )"bool" float nil )nil float nil )nil float nil )nil naturalnumbers nil )NOT const-decl "[bool -> bool]" booleans nil )"even_int" integersnil )"(strict_total_order?[real])" real_props nil )"{j: nonneg_int | j >= i}" nil )"posint" exponentiation nil )"(total_order?[real])" nil )"(total_order?[real])" nil )"int" integers nil )"posnat" exponentiation nil )"(strict_total_order?[real])" real_props nil )"nat" exponentiation nil )"nat" exponentiation nil )"{n: nonneg_real | n >= m AND n >= -m}" real_defsnil )"real" exponentiation nil )"bool" float nil )"above(1)" axpy nil )nil integers nil )"bool" reals nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )"real" float nil )"bool" float nil )"posnat" float nil )"bool" float nil )"bool" float nil )"boolean" notequal nil )"nnrat" exponentiation nil )"posrat" exponentiation nil )"rat" rationals nil )"rat" rationals nil )"{r: nonneg_rat | r >= q}" nil )"nonneg_rat" nil )"nonneg_rat" nil )"real" reals nil )"(divides(n))" divides nil )"(divides(m))" divides nil ))nil ))nil 3320755856"" case "FORALL (x:float): Fnormal?(b)(x) => radix^(Prec(b)-1) <= abs(Fnum(x))" )"1" (skeep)"1" (case "Fnormal?(b)(Fpred(b)(u))" )"1" (case "Fnormal?(b)(u)" )"1" "abs(FtoR(u))*radix^(-Prec(b))" :strict 1)"1" (mult-by 1 "radix ^ (Prec(b))" )"1" "abs(FtoR[radix](y)) * radix/(radix+1)" "1" (rewrite "expt_div" :dir rl)"1" (field 1) nil nil )) nil )"2" (trans-ineq 1 "abs(FtoR(u))" :strict 1)"1" (case "0< 2*abs(Fnum(u))" )"1" (case "0< 1+1/(2*abs(Fnum(u)))" )"1" "abs(FtoR(t)+FtoR(y))/(1+1/(2*abs(Fnum(u))))" "1" (case "0 < 1+radix^(1-Prec(b))/2" )"1" " abs(FtoR[radix](t) + FtoR[radix](y)) / ( 1 + radix ^ (1 - Prec(b)) / 2)" "1" "1 + radix ^ (1 - Prec(b)) / 2" )"1" "abs(FtoR(y))-abs(FtoR(t))" "1" "1" "1" assert )"1" case "0 < 1+radix+radix^(4 - Prec(b))" )"1" case " 0 < 1 - radix ^ (1 - Prec(b)) / 2" )"1" "abs(FtoR(y))/(1 - radix ^ (1 - Prec(b)) / 2)*1/(1 + radix + radix ^ (4 - Prec(b)))" "1" case "0 < 2 * abs(Fnum(t))" )"1" case "0 < 1 - 1 / (2 * abs(Fnum(t)))" )"1" "abs(FtoR(a)*FtoR(x))/(1-1/(2*abs(Fnum(t))))" )"1" rewrite "RoundLe" )"1" "1" expand *"FtoR" "abs" )"1" nil nil ))nil ))nil ))nil )"2" "abs(FtoR[radix](a) * FtoR[radix](x)) / (1 - radix ^ (1 - Prec(b)) / 2)" )"1" "t" )"1" "1" "1" "abs(FtoR(a))" )"1" "abs(FtoR(x))" )"1" "1" "radix ^ (Prec(b) - 1)" )"1" "abs(Fnum(t))" )"1" assert )nil nil )"2" "abs(Fnum(t))" )"2" "2" rewrite "expt_plus" "2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" nil nil ))nil ))nil )"2" "(1 - radix ^ (1 - Prec(b)) / 2)" )"2" nil nil ))nil ))nil )"3" assert )nil nil )"4" assert )nil nil ))nil )"2" "2" assert )"2" "2" case "1 <= abs(Fnum(t))" )"1" nil nil )"2" "2" "gg" "abs(Fnum(t))" )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"3" assert )nil nil ))nil )"2" expand "Fnormal?" -13)"2" "2" "2" nil nil ))nil ))nil ))nil ))nil )"2" " radix ^ (4 - Prec(b))=radix^3*radix ^ (1 - Prec(b))" )"1" case "0 < radix ^ (1 - Prec(b))" )"1" case " radix ^ (1 - Prec(b)) <= radix^(-5)" )"1" "1" "AA" "radix ^ (1 - Prec(b))" )"1" "1" "1+radix" )"1" "abs(FtoR[radix](y))*(1+radix) - abs(FtoR[radix](y)) * radix - " "1" "abs(FtoR(y))" )"1" "1/ (1 - AA / 2) * 1 / (1 + radix + radix ^ 3 * AA) " "1" "1" " 1 - AA / 2" )"1" nil nil ))nil ))nil )"2" "( (1 + radix) - radix - " "1" "1-AA/2" )"1" "1 / (1 + radix + radix ^ 3 * AA) * (1 + radix)" "1" nil nil )"2" "1 + radix + radix ^ 3 * AA" )"2" "1+radix" "1" nil nil )"2" assert )"2" assert )"2" "2" "2" assert )"2" "2" "AA" )"2" "( 1 / 2 + radix ^ 3 * (AA / 2) + 1/ 2 * radix * radix + " "1" nil nil )"2" " (radix ^ 3 + AA / 4 * radix + " "1" "radix ^ 3" "1" "2" )"1" "1+radix^3*AA+radix*radix+radix^3*radix*AA+2*radix" )"1" nil nil )"2" "1 + radix ^ 3 * radix^(-5) + radix * radix + radix ^ 3 * radix * radix^(-5) + 2 * radix" )"1" "1" assert )"1" case "radix ^ 3 * AA <= radix ^ 3 * radix ^ (-5)" )"1" case "AA * radix ^ 3 * radix <= radix ^ 3 * radix ^ (-5) * radix" )"1" assert )nil nil )"2" nil nil ))nil )"2" nil nil ))nil ))nil ))nil )"2" rewrite "expt_plus" "2" "radix ^ 3 * radix * radix ^ (-5)=radix^(-1)" )"1" case "radix * radix <= radix ^ 3" )"1" case "1 + radix ^ -2 + radix ^ (-1) + 2 * radix <= ") "1" assert )nil nil )"2" "2" "radix^2+radix^2" )"1" case "1 + radix ^ -2 + radix ^ (-1) <= radix ^ 2" )"1" case " 2 * radix <= radix ^ 2" )"1" assert )nil nil )"2" rewrite "expt_x2" )"2" nil nil ))nil ))nil )"2" "2" "radix+1+1" )"1" case "1 <= radix" )"1" case "radix ^ -2 <= 1" )"1" case "radix ^ -1 <= 1" )"1" assert )nil nil )"2" "radix^0" )"1" rewrite "Exp_increq_1" )nil nil )"2" rewrite "expt_x0" )"2" assert )nil nil ))nil ))nil ))nil )"2" "radix^0" )"1" rewrite "Exp_increq_1" )nil nil )"2" rewrite "expt_x0" )"2" assert )nil nil ))nil ))nil ))nil )"2" assert )nil nil ))nil )"2" rewrite "expt_x2" )"2" "2" "radix+radix" )"1" nil nil )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "radix*radix^2" )"1" nil nil )"2" "expt_plus" )"2" "2" "1" "radix" )"2" rewrite "expt_x1" )"2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" rewrite "expt_x2" "2" rewrite "Exp_increq_1" )nil nil ))nil ))nil )"2" "expt_plus" )"2" "3" "1" "radix" )"2" rewrite "expt_x1" )"2" rewrite "2" rewrite "expt_plus" "2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" nil nil ))nil )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" nil nil ))nil ))nil )"2" "2" "2" case "0 < abs(FtoR(a)*FtoR(x))*(radix+1+radix^(4-Prec(b)))" )"1" nil nil )"2" " (radix + 1 + radix ^ (4 - Prec(b)))" )"2" "0" "1" nil nil )"2" case "0 <= abs(FtoR(a)*FtoR(x))" )"1" "1" case "FtoR[radix](t)=FtoR[radix]((#Fnum:=0, Fexp:=-dExp(b)#))" )"1" "1" expand *"Fnormal?" "FtoR" )"1" "1" nil nil ))nil ))nil ))nil )"2" "RoundedProjectorEq" )"2" "Closest?" "b" "(# Fnum := 0, Fexp := -dExp(b) #)" "t" )"2" "1" nil nil )"2" "FtoR((# Fnum := 0, Fexp := -dExp(b) #))=FtoR(a)*FtoR(x)" )"2" "2" expand "FtoR" "2" expand "abs" "2" nil nil ))nil ))nil ))nil ))nil )"3" rewrite "Closest_RoundedMode" )nil nil )"4" rewrite "FcanonicBounded" )nil nil )"5" "5" expand "Fbounded?" )"5" expand "abs" )"5" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" nil nil ))nil ))nil ))nil ))nil ))nil )"2" rewrite "Exp_increq_1" )nil nil ))nil )"2" assert )nil nil ))nil )"2" rewrite "expt_plus" nil nil ))nil )"3" (assert ) nil nil )"4" (assert ) nil nil ))nil )"2" "2" assert )"2" "2" "1/2" "1" "1" "radix^0" )"1" rewrite "Exp_increq_1" )nil nil )"2" rewrite "expt_x0" )"2" assert )nil nil ))nil ))nil ))nil )"2" nil nil ))nil ))nil ))nil ))nil ))nil )"2" "2" (grind-reals) nil nil ))nil ))nil ))nil ))nil ))nil )"2" "2" expand "abs" )"2" (grind-reals) nil nil ))nil ))nil ))nil ))nil )"2" (inst -6 "u" )"2" "1" "1" "abs(FtoR(t)+FtoR(y))" )"1" "1" case "1<=abs(Fnum(u)) * (radix ^ (1 - Prec(b)))" )"1" "abs(FtoR(t) + FtoR(y))" )"1" (assert ) nil nil ))nil )"2" "2" assert )"2" "radix ^ (Prec(b) - 1)" )"2" "abs(Fnum(u))" )"1" (assert ) nil nil )"2" "abs(Fnum(u))" )"2" "2" rewrite "expt_plus" "2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (propax) nil nil ))nil ))nil ))nil )"2" (trans-ineq 1 "1" :strict 1)"1" (assert ) nil nil )"2" (grind-reals) nil nil ))nil )"3" (assert ) nil nil ))nil )"2" (rewrite "RoundGe" ) nil nil )"3" (assert ) nil nil ) ("4" (assert ) nil nil ))nil )"2" (trans-ineq 1 "1" :strict 1)"1" (assert ) nil nil )"2" (move-terms 1 r 1)"2" (assert ) (("2" (grind-reals) nil nil ))nil ))nil ))nil )"3" (assert ) nil nil ))nil )"2" (expand "Fnormal?" -1)"2" (flatten)"2" (hide-all-but (-2 1))"2" (grind-reals) nil nil )) nil ))nil ))nil ))nil )"2" (rewrite "expt_inverse" )"2" (field 1) nil nil )) nil ))nil ))nil ))nil )"2" (rewrite "expt_inverse" )"2" (mult-by 1 "radix ^ Prec(b)" )"2" (trans-ineq 1 "abs(FtoR(u))" )"1" (field 1) nil nil )"2" " Fulp(b)(Fpred(b)(u))+ FtoR(Fpred(b)(u))" )"1" (rewrite "FpredDiff" :dir rl)"1" (assert )"1" (expand "abs" 1) (("1" (assert ) nil nil ))nil ))nil ))nil )"2" (expand "FtoR" 1)"2" (rewrite "FulpCanonic" )"1" (div-by 1 "radix ^ (Fexp(Fpred(b)(u)))" )"1" (trans-ineq 1 "Fnum(Fpred(b)(u))+1" )"1" (field 1) nil nil )"2" (hide-all-but (-2 1))"2" "ff" "Fpred(b)(u)" )"2" expand * "Fnormal?" "Fbounded?" )"2" "2" " radix ^ Prec(b)= vNum(b)" )"1" expand "abs" -2)"1" "1" case "FORALL (i,j:int): i<j => 1+i <= j" )"1" "1" (assert ) nil nil ))nil )"2" "2" (assert ) nil nil ))nil ))nil ))nil ))nil )"2" assert )"2" expand "vNum" )"2" (propax) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (rewrite "FpredCanonic" ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"3" (assert ) nil nil ) ("4" (assert ) nil nil ))nil )"2" (lemma "Normal_iff" )"2" (inst-cp -1 "u" )"2" (inst -1 "Fpred(b)(u)" )"2" (split)"1" (split)"1" (flatten)"1" (rewrite -2)"1" " abs(FtoR[radix](Fpred(b)(u)))" )"1" (case " 0 <= FtoR(Fpred(b)(u))" )"1" (expand "abs" 1)"1" "FpredLt" )"1" "b" "u" )"1" (assert ) nil nil ))nil ))nil ))nil )"2" (rewrite "FpredPos" ) nil nil ))nil ))nil ))nil ))nil )"2" (assert ) nil nil ))nil )"2" (assert )"2" (rewrite "FpredCanonic" ) nil nil )) nil ))nil ))nil ))nil ))nil ))nil )"2" (lemma "Normal_iff" )"2" (inst-cp -1 "Fpred(b)(u)" )"2" (inst -1 "t" )"2" (split)"1" (split)"1" (flatten)"1" (rewrite -2)"1" (trans-ineq 1 "abs(FtoR(t))" )"1" (trans-ineq 1 "radix*abs(FtoR(t))" )"1" (div-by 1 "abs(FtoR[radix](t))" )"1" (assert ) nil nil )) nil )"2" (expand "abs" 1 2)"2" (grind-reals)"1" "Axpy_opt_aux1" )"1" "a" "t" "u" "x" "y" )"1" "1" (assert ) nil nil )"2" (assert ) nil nil )"3" (propax) nil nil )"4" (assert ) nil nil )"5" (propax) nil nil )"6" (propax) nil nil )"7" (propax) nil nil ))nil ))nil ))nil )"2" "Axpy_opt_aux1" )"2" "a" "t" "u" "x" "y" )"2" "1" (propax) nil nil )"2" (assert ) nil nil )"3" (propax) nil nil )"4" (assert ) nil nil )"5" (propax) nil nil )"6" (propax) nil nil )"7" (propax) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (assert )"2" (rewrite "FpredCanonic" ) nil nil )) nil ))nil )"2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (hide 2)"2" (skeep)"2" (expand "Fnormal?" )"2" (flatten)"2" (case-replace "vNum(b)=radix ^ (Prec(b))" )"1" (rewrite "expt_div" :dir rl)"1" (rewrite "abs_mult" )"1" (expand "abs" -3 1)"1" (grind-reals) nil nil )) nil ))nil ))nil )"2" (assert )"2" (expand "vNum" ) (("2" (propax) nil nil )) nil ))nil )"3" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil )"3" (hide 2) (("3" (skosimp*) (("3" (assert ) nil nil )) nil ))nil ))nil )nil real_props nil )"real" reals nil )nil axpy nil )nil float nil )"posnat" float nil )nil integers nil )nil float nil )nil float nil )"posint" exponentiation nil )nil extra_real_propsnil )"rat" rationals nil )"{j: nonneg_int | j >= i}" nil )nil axpy nil )"rat" axpy nil )"posrat" axpy nil )"{n: nonneg_real | n >= Fnum(u) AND n >= -Fnum(u)}" axpy nil )"{x: float | Fcanonic?(b)(x)}" axpy nil )nil real_props nil )"{n: nonneg_real | n >= FtoR(t) + FtoR(y) AND n >= -(FtoR(t) + FtoR(y))}" nil )nil extra_real_propsnil )nil real_props nil )nil reals nil )"real" reals nil )"[bool, bool -> bool]" booleans nil )"rat" rationals nil )nil real_props nil )"{x: float | Fcanonic?(b)(x)}" axpy nil )nil real_props nil )nil real_props nil )nil real_props nil )"posrat" nil )"posint" nil )"even_int" integersnil )"odd_int" integers nil )nil exponentiation nil )nil float nil )nil real_props nil )nil integers nil )nil integers nil )nil real_props nil )nil real_props nil )nil real_props nil )nil real_props nil )nil real_props nil )"posrat" axpy nil )nil real_props nil )nil extra_real_props nil )nil real_props nil )"{j: posint | j >= i}" nil )"bool" float nil )nil float nil )nil float nil )nil float nil )nil float nil )"nonneg_rat" nil )nil real_props nil )"even_int" integersnil )nil real_props nil )nil real_props nil )"int" integers nil )"rat" rationals nil )nil axpy nil )nil rationals nil )nil rationals nil )"even_int" integersnil )"odd_int" integersnil )"[T, T -> boolean]" equalities nil )nil rationals nil )"posrat" exponentiation nil )"odd_int" integers nil )"(strict_total_order?[real])" real_props nil )nil real_props nil )nil real_props nil )nil real_props nil )nil real_props nil )nil real_types nil )"rat" axpy nil )"rat" axpy nil )"bool" functions nil )"(bijective?[T, T])" identity nil )"bool" booleans nil )"nzrat" rationalsnil )"{n: nonneg_real | n >= Fnum(t) AND n >= -Fnum(t)}" axpy nil )"{x: float | Fcanonic?(b)(x)}" axpy nil )"even_int" integers nil )nil real_props nil )nil real_props nil )nil exponentiation nil )"int" integers nil )nil exponentiation nil )"rat" rationals nil )"posrat" nil )"nonneg_rat" nil )nil extra_real_props nil )nil real_props nil )"real" reals nil )"real" reals nil )"nzrat" rationalsnil )nil real_props nil )"(divides(m))" divides nil )"(divides(n))" divides nil )"even_int" integersnil )"nonneg_int" nil )nil exponentiation nil )nil reals nil )nil exponentiation nil )"posrat" nil )nil exponentiation nil )NOT const-decl "[bool -> bool]" booleans nil )"(total_order?[real])" nil )"real" reals nil )"posrat" nil )"nnreal" nil )"real" float nil )"bool" float nil )"[numfield, numfield -> numfield]" number_fields nil )"real" float nil )"[numfield, numfield -> numfield]" number_fields nil )"[numfield, nznum -> numfield]" number_fields nil )nil number_fields nil )"bool" reals nil )"nnreal" nil )"nzint" integers nil )"nnreal" nil )"posint" nil )"(strict_total_order?[real])" real_props nil )"nnrat" exponentiation nil )"(total_order?[real])" nil )nil float nil )nil float nil )nil axpy nil )"bool" float nil )"float" float nil )"int" integers nil )nil integers nil )nil numbers nil )nil booleans nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil booleans nil )"bool" reals nil )nil integers nil )"above(1)" axpy nil )nil float nil )"[bool, bool -> bool]" booleans nil )nil naturalnumbers nil )nil float nil )"bool" float nil )"Format" axpy nil )"bool" reals nil )OR const-decl "[bool, bool -> bool]" booleans nil )"boolean" notequal nil )"bool" reals nil )"real" exponentiation nil )nil number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )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}" real_defsnil ))nil 3321283541 ("" (subtype-tcc) nil nil )"Format" axpy nil )"bool" float nil )nil float nil )nil float nil )nil naturalnumbers nil )NOT const-decl "[bool -> bool]" booleans nil )"even_int" integersnil )"(strict_total_order?[real])" real_props nil )"{j: nonneg_int | j >= i}" nil )"posint" exponentiation nil )"(total_order?[real])" nil )"(total_order?[real])" nil )"int" integers nil )"posnat" exponentiation nil )"(strict_total_order?[real])" real_props nil )"nat" exponentiation nil )"nat" exponentiation nil )"{n: nonneg_real | n >= m AND n >= -m}" real_defsnil )"real" exponentiation nil )"bool" float nil )"above(1)" axpy nil )nil integers nil )"bool" reals nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )"real" float nil )"bool" float nil )"posnat" float nil )"bool" float nil )"bool" float nil )"boolean" notequal nil )"nnrat" exponentiation nil )"posrat" exponentiation nil )"rat" rationals nil )"rat" rationals nil )"real" reals nil )"(divides(n))" divides nil )"(divides(m))" divides nil ))nil ))nil 3321283541 ("" (subtype-tcc) nil nil )"Format" axpy nil )"bool" float nil )nil float nil )nil float nil )nil naturalnumbers nil )NOT const-decl "[bool -> bool]" booleans nil )"even_int" integersnil )"(strict_total_order?[real])" real_props nil )"{j: nonneg_int | j >= i}" nil )"posint" exponentiation nil )"(total_order?[real])" nil )"(total_order?[real])" nil )"int" integers nil )"posnat" exponentiation nil )"(strict_total_order?[real])" real_props nil )"nat" exponentiation nil )"nat" exponentiation nil )"int" integers nil )"{n: nonneg_real | n >= m AND n >= -m}" real_defsnil )"real" exponentiation nil )"bool" float nil )"above(1)" axpy nil )nil integers nil )"bool" reals nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )"real" float nil )"bool" float nil )"posnat" float nil )"bool" float nil )"bool" float nil )"boolean" notequal nil )"nnrat" exponentiation nil )"posrat" exponentiation nil )"rat" rationals nil )"rat" rationals nil )"{r: nonneg_rat | r >= q}" nil )"nonneg_rat" nil )"nonneg_rat" nil )"real" reals nil )"(divides(n))" divides nil )"(divides(m))" divides nil ))nil ))nil 3321283542"" (skeep)"" (case "-dExp(b) <= Fexp(u)" )"1" (split-ineq -1)"1" (hide -2)"1" (mult-by 1 "radix/2+radix" )"1" (mult-by 1 "radix ^ (Prec(b)-1)" )"1" "abs(FtoR[radix](y)) * (radix ^ (1 - Prec(b)) * radix ^ (Prec(b) - 1))" "1" (assert ) nil nil )"2" (rewrite "expt_plus" :dir rl)"2" (case-replace "FtoR(y)=FtoR(u)-FtoR(t)" )"1" "abs(FtoR(u))+abs(FtoR(t))" :strict"1" (hide-all-but 1)"1" (expand "abs" )"1" (grind-reals) nil nil )) nil ))nil )"2" (rewrite "FulpCanonic" )"1" "Fexp(Fpred(b)(u))=-dExp(b)" )"1" (assert )"1" case "abs(FtoR[radix](t)) <= radix ^ (Prec(b) - 1) * radix ^ -dExp(b)*(radix/2)" )"1" case "abs(FtoR[radix](u)) < radix ^ (Prec(b) - 1) * radix ^ -dExp(b) * radix" )"1" (assert ) nil nil )"2" "2" rewrite "FabsCorrect" :dir rl)"2" expand "Fabs" )"2" expand "FtoR" 1)"2" rewrite -4 :dir rl)"2" "radix ^ -dExp(b)" )"2" "2" "vNum(b)" "1" case "Fbounded?(b)(u)" )"1" expand "Fbounded?" )"1" nil nil ))nil )"2" rewrite "FcanonicBounded" )nil nil ))nil )"2" "radix ^ (Prec(b))" )"1" assert )"1" expand "vNum" )"1" assert )nil nil ))nil ))nil )"2" "expt_plus" )"2" "1" "Prec(b)-1" "radix" )"2" rewrite "expt_x1" )"2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" expand * "Fsubnormal?" "FtoR" )"2" "2" "2" rewrite -3)"2" "radix ^ -dExp(b)" )"2" rewrite "abs_mult" )"2" rewrite "abs_mult" )"2" expand "abs" 1 2)"2" "2" " radix ^ (Prec(b) - 1)=vNum(b)/radix" )"1" "radix*abs(Fnum(t))" )"1" nil nil )"2" expand "abs" "2" nil nil ))nil ))nil )"2" rewrite "expt_div" "2" "2" expand "vNum" )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (hide-all-but (-2 -5 1))"2" (expand "Fpred" )"2" expand "FtoR" )"2" (grind-reals) nil nil ))nil ))nil ))nil ))nil )"2" (rewrite "FpredCanonic" ) nil nil ))nil ))nil )"2" (case "FtoR(u)=FtoR(t)+FtoR(y)" )"1" (assert ) nil nil )"2" (hide 2 3)"2" (lemma "ExactSum_Near" )"2" (inst?)"2" (split)"1" (propax) nil nil )"2" "radix ^ (-dExp(b))/2" "1" "Fulp(b)(u)/2" )"1" rewrite "ClosestUlp" )"1" rewrite "FcanonicBounded" )nil nil ))nil )"2" rewrite "FulpCanonic" )"2" (assert ) nil nil ))nil ))nil )"2" (grind-reals) nil nil ))nil )"3" (propax) nil nil )"4" rewrite "FcanonicBounded" )nil nil )"5" rewrite "FcanonicBounded" )nil nil )"6" rewrite "FcanonicBounded" )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (case "-dExp(b) < Fexp(u)" )"1" (hide -2 1)"1" (trans-ineq 1 "abs(FtoR(u))*radix^(-Prec(b)) " )"1" (mult-by 1 "radix ^ (Prec(b))" )"1" "abs(FtoR[radix](y)) / (radix/2 + radix)*(radix ^ (1 - Prec(b))*radix ^ (Prec(b)))" )"1" (assert ) nil nil )"2" (rewrite "expt_plus" :dir rl)"2" "abs(FtoR(u))*(radix ^ (-Prec(b)) * radix ^ (Prec(b)))" )"1" (rewrite "expt_plus" :dir rl)"1" (case-replace "(Prec(b)) + (-Prec(b))=0" )"1" (assert )"1" case "Fexp(u)=1-dExp(b) OR 2-dExp(b) <= Fexp(u)" )"1" "1" "1" "radix" )"1" "radix/2+radix" )"1" "abs(FtoR(u))+abs(FtoR(t))+abs(FtoR(u)-(FtoR(t)+FtoR(y)))" )"1" "1" expand "abs" )"1" nil nil ))nil ))nil )"2" "2" "abs(FtoR(u))+(radix^(Prec(b)-1)-1)*radix^(-dExp(b))+radix^(1-dExp(b))/2" )"1" case "abs(FtoR[radix](t)) <= (radix ^ (Prec(b) - 1) - 1) * radix ^ (-dExp(b))" )"1" case "abs(FtoR[radix](u) - (FtoR[radix](t) + FtoR[radix](y))) ") "1" (assert ) nil nil )"2" "2" "Fulp(b)(u)/2" )"1" rewrite "ClosestUlp" )"1" rewrite "FcanonicBounded" )nil nil ))nil )"2" rewrite "FulpCanonic" )"2" rewrite -2)"2" assert )nil nil ))nil ))nil ))nil ))nil ))nil )"2" expand "Fsubnormal?" )"2" "2" expand "FtoR" 1)"2" rewrite "abs_mult" )"2" expand "abs" "2" "2" rewrite "2" "2" " radix ^ -dExp(b)" )"2" "(radix ^ (Prec(b) - 1)-1)" )"1" rewrite "abs_mult" "1" expand "abs" "1" "vNum(b)=radix*(radix ^ (Prec(b) - 1))" )"1" "radix" )"1" case "FORALL (i,j:int):i<j=> i<= j-1" )"1" "1" assert )nil nil )"2" "2" nil nil ))nil ))nil )"2" "2" assert )nil nil ))nil ))nil ))nil )"2" rewrite "expt_div" "2" "2" expand "vNum" )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "radix" )"2" "abs(FtoR(u))*(radix/2+radix)" )"1" assert )"1" "1" assert )"1" "2" )"1" "1" "radix" )"1" "(2 * (radix ^ (Prec(b) - 1) * radix ^ (-dExp(b)) ) - ") "1" nil nil )"2" rewrite "FabsCorrect" "2" expand "FtoR" "2" expand "Fabs" )"2" rewrite "2" "radix ^ (1 - dExp(b))=radix*radix ^ (-dExp(b))" )"1" "radix ^ (-dExp(b))" )"1" "(radix -2 + ") "1" nil nil )"2" nil nil )"3" "3" "(radix ^ (Prec(b)))" )"1" case "2 <= radix" )"1" "1" rewrite "1" rewrite "1" assert )"1" "expt_plus" )"1" "1" "Prec(b)-1" "2" )"1" rewrite "expt_x1" )"1" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" case "2 < radix" )"1" "1" "3 * (radix ^ (Prec(b) - 1))" )"1" "1" assert )"1" "radix^1" )"1" rewrite "expt_x1" )"1" assert )nil nil ))nil )"2" rewrite "Exp_increq_1" )nil nil ))nil ))nil ))nil )"2" "radix * (radix ^ (Prec(b) - 1))" )"1" nil nil )"2" rewrite "expt_div" "2" nil nil ))nil ))nil ))nil ))nil )"2" assert )nil nil ))nil ))nil )"2" assert )nil nil ))nil )"2" "u" )"2" expand *"Fcanonic?" "Fsubnormal?" "Fnormal?" )"2" "2" rewrite "abs_mult" )"2" expand "abs" "2" "vNum(b)" )"2" assert )"2" expand "vNum" )"2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "expt_plus" )"2" "1" "-dExp(b)" "radix" )"2" rewrite "expt_x1" )"2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "radix" )"2" "radix / 2 + radix" )"2" case "0 < 1+radix^(1-Prec(b))/2" )"1" "abs(FtoR(u))*(1+radix^(1-Prec(b))/2+radix^(-2))" )"1" assert )"1" "1" "abs(FtoR[radix](y)) - abs(FtoR(t))" )"1" "1" "1" "1" assert )"1" "radix^(Prec(b)-1)*radix^(-dExp(b))" )"1" "1" rewrite "FabsCorrect" "1" expand *"FtoR" "Fabs" "Fsubnormal?" )"1" "1" "vNum(b)/radix*radix ^(-dExp(b))" )"1" rewrite "abs_mult" )"1" expand "abs" "1" "1" rewrite "1" "radix ^ -dExp(b) " )"1" nil nil ))nil ))nil ))nil ))nil ))nil )"2" "radix ^ -dExp(b) " )"2" rewrite "expt_div" "2" "2" expand "vNum" )"2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" rewrite "FabsCorrect" "2" expand "FtoR" "2" expand "Fabs" )"2" "abs(Fnum(u))*(radix ^ Fexp(u)*radix ^ (-2))" )"1" rewrite "expt_plus" "i" "Fexp(u)" ))"1" case "radix ^ (Prec(b) - 1) <= abs(Fnum(u))" )"1" case "radix ^ (-dExp(b)) <= radix ^ (Fexp(u) - 2)" )"1" nil nil )"2" rewrite "Exp_increq_1" )nil nil ))nil )"2" "2" "u" )"2" expand *"Fcanonic?" "Fnormal?" "Fsubnormal?" )"2" "2" rewrite "abs_mult" )"2" expand "abs" "2" " vNum(b)/radix" )"1" rewrite "expt_div" "1" "1" expand "vNum" )"1" assert )nil nil ))nil ))nil ))nil )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "abs(FtoR(t)+FtoR(y))" )"1" "1" expand "abs" )"1" nil nil ))nil ))nil )"2" case " 0 < 2*abs(Fnum(u))" )"1" case "0 < 1+1/(2*abs(Fnum(u)))" )"1" " abs(FtoR[radix](u))*(1+ 1 / (2 * abs(Fnum(u))))" )"1" "1 + 1 / (2 * abs(Fnum(u)))" )"1" rewrite "RoundGe" )nil nil ))nil )"2" assert )"2" "2" assert )"2" "abs(FtoR(u))" )"2" "2" )"2" "expt_inverse" )"2" "Prec(b)-1" "radix" )"2" "1 / (radix ^ (Prec(b) - 1))" )"1" "1" "1" "u" )"1" expand *"Fcanonic?" "Fnormal?" "Fsubnormal?" )"1" "1" rewrite "abs_mult" )"1" expand "abs" "1" "vNum(b)/radix" )"1" rewrite "expt_div" "1" "1" expand "vNum" )"1" assert )nil nil ))nil ))nil ))nil )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" "1" "1" assert )nil nil )"2" "2" assert )"2" nil nil ))nil ))nil ))nil ))nil )"3" assert )nil nil ))nil )"2" "2" expand *"FtoR" "abs" )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "abs(FtoR(u))" )"2" "2" "2" assert )"2" case "(2 * (radix ^ (-2) * radix)) <= 1" )"1" case "(radix * (radix ^ (1 - Prec(b)))) <= 1" )"1" assert )nil nil )"2" "2" "expt_plus" )"2" "1" "1-Prec(b)" "radix" )"2" rewrite "expt_x1" )"2" "radix ^ (2 - Prec(b))" )"1" assert )nil nil )"2" "radix ^ 0" )"1" rewrite "Exp_increq_1" )nil nil )"2" rewrite "expt_x0" )"2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" rewrite "expt_inverse" )"2" rewrite "expt_x2" )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (grind-reals) nil nil ))nil ))nil ))nil ))nil )"2" "2" "2" (grind-reals) nil nil ))nil ))nil ))nil ))nil )"2" (assert ) nil nil ))nil ))nil )"2" (assert ) nil nil ))nil ))nil ))nil ))nil )"2" (expand "abs" 1)"2" (rewrite "expt_inverse" )"2" (cross-mult 1)"2" (trans-ineq 1 "FtoR(u)" )"1" (assert ) nil nil )"2" "Fulp(b)(Fpred(b)(u))+FtoR(Fpred(b)(u))" )"1" (rewrite "FpredDiff" :dir rl)"1" (assert ) nil nil )) nil )"2" (rewrite "FulpCanonic" )"1" (expand "FtoR" 1)"1" "radix ^ (Fexp(Fpred(b)(u)))" )"1" "Fnum(Fpred(b)(u))+1" )"1" (field 1) nil nil )"2" case "Fbounded?(b)(Fpred(b)(u))" )"1" expand "Fbounded?" )"1" "1" "1" "radix ^ (Prec(b))=vNum(b)" )"1" expand "abs" )"1" "1" case "FORALL (i,j:int): i<j => 1+i <= j" )"1" nil nil )"2" nil nil ))nil ))nil ))nil )"2" assert )"2" expand "vNum" )"2" (propax) nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" rewrite "FcanonicBounded" )"2" rewrite "FpredCanonic" )nil nil ))nil ))nil ))nil ))nil ))nil )"2" (rewrite "FpredCanonic" ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (assert ) nil nil ))nil ))nil )"2" (case "Fbounded?(b)(u)" )"1" (expand "Fbounded?" ) (("1" (flatten) nil nil )) nil )"2" (rewrite "FcanonicBounded" ) nil nil ))nil ))nil ))nil )"bool" float nil )nil float nil )"Format" axpy nil )nil float nil )"above(1)" axpy nil )"bool" reals nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil integers nil )nil naturalnumbers nil )"[numfield -> numfield]" number_fields nil )nil number_fields nil )"bool" reals nil )nil booleans nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )"int" integers nil )"nzint" integers nil )nil exponentiation nil )nil real_props nil )"rat" rationals nil )nil real_props nil )"nzrat" rationalsnil )nil axpy nil )"{x: float | Fcanonic?(b)(x)}" axpy nil )nil exponentiation nil )"nzrat" rationalsnil )"{n: nonneg_real | n >= Fnum(u) AND n >= -Fnum(u)}" axpy nil )"posrat" axpy nil )nil real_props nil )nil real_props nil )nil real_props nil )"nzreal" nil )nil rationals nil )nil rationals nil )nil rationals nil )"even_int" integers nil )"bool" float nil )nil float nil )"rat" rationals nil )"[bool, bool -> bool]" booleans nil )"rat" rationals nil )nil extra_real_props nil )"real" exponentiation nil )"posnat" exponentiation nil )"rat" rationals nil )"posreal" nil )"posreal" nil )nil float nil )nil real_props nil )"posrat" exponentiation nil )nil reals nil )nil exponentiation nil )"int" integers nil )nil exponentiation nil )nil axpy nil )"real" reals nil )nil float nil )"real" reals nil )nil float nil )"float" float nil )"{j: nonneg_int | j >= i}" nil )NOT const-decl "[bool -> bool]" booleans nil )"[bool, bool -> bool]" booleans nil )"posint" exponentiation nil )nil exponentiation nil )nil float nil )nil integers nil )nil integers nil )"posnat" float nil )nil extra_real_props nil )nil real_props nil )"nonneg_rat" nil )"odd_int" integers nil )nil float nil )"rat" rationals nil )"(divides(n))" divides nil )"(divides(m))" divides nil )"bool" float nil )"{r: nonneg_rat | r >= q}" nil )nil real_props nil )"{r: posrat | r >= q}" nil )"nonneg_rat" nil )"(strict_total_order?[real])" real_props nil )"bool" functions nil )"(bijective?[T, T])" identity nil )"bool" booleans nil )"even_int" integersnil )nil exponentiation nil )"posint" nil )nil real_props nil )nil real_props nil )nil real_props nil )"even_int" integersnil )nil real_props nil )"nonneg_int" nil )"{j: posint | j >= i}" nil )nil real_props nil )nil real_props nil )nil float nil )"real" reals nil )"posrat" nil )"bool" reals nil )nil reals nil )nil real_props nil )"bool" reals nil )nil real_types nil )nil real_types nil )"[numfield, numfield -> numfield]" number_fields nil )"boolean" notequal nil )nil number_fields nil )"[numfield, nznum -> numfield]" number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )AND const-decl "[bool, bool -> bool]" booleans nil )"{n: nonneg_real | n >= m AND n >= -m}" real_defsnil )"real" float nil )OR const-decl "[bool, bool -> bool]" booleans nil )"real" exponentiation nil )"[numfield, numfield -> numfield]" number_fields nil )"bool" float nil )"real" float nil )"float" float nil )nil real_props nil )"posrat" nil )"nnreal" nil )"nnreal" nil )"int" integers nil )"nnrat" exponentiation nil )"posint" nil )"posrat" nil )"nnreal" nil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil )"real" reals nil )"(total_order?[real])" nil )"[T, T -> boolean]" equalities nil )"real" reals nil ))nil 3320768094 ("" (subtype-tcc) nil nil )"Format" axpy nil )"bool" float nil )nil float nil )nil float nil )nil naturalnumbers nil )NOT const-decl "[bool -> bool]" booleans nil )"(strict_total_order?[real])" real_props nil )"{j: nonneg_int | j >= i}" nil )"posint" exponentiation nil )"(total_order?[real])" nil )"(total_order?[real])" nil )"int" integers nil )"posnat" exponentiation nil )"(strict_total_order?[real])" real_props nil )"real" exponentiation nil )"bool" float nil )"bool" float nil )"bool" float nil )"posnat" float nil )"above(1)" axpy nil )nil integers nil )"bool" reals nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )"real" float nil )"bool" float nil )"boolean" notequal nil )"nnrat" exponentiation nil )"posrat" exponentiation nil )"rat" rationals nil )"rat" rationals nil )"real" reals nil )"(divides(n))" divides nil )"(divides(m))" divides nil ))nil ))nil 3320768094 ("" (subtype-tcc) nil nil )"Format" axpy nil )"bool" float nil )nil float nil )nil float nil )nil naturalnumbers nil )NOT const-decl "[bool -> bool]" booleans nil )"(strict_total_order?[real])" real_props nil )"{j: nonneg_int | j >= i}" nil )"posint" exponentiation nil )"(total_order?[real])" nil )"(total_order?[real])" nil )"int" integers nil )"posnat" exponentiation nil )"(strict_total_order?[real])" real_props nil )"nat" exponentiation nil )"int" integers nil )"nzrat" rationalsnil )"nat" exponentiation nil )"real" exponentiation nil )"bool" float nil )"bool" float nil )"bool" float nil )"posnat" float nil )"above(1)" axpy nil )nil integers nil )"bool" reals nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )"real" float nil )"bool" float nil )"boolean" notequal nil )"nnrat" exponentiation nil )"posrat" exponentiation nil )"rat" rationals nil )"rat" rationals nil )"{r: nonneg_rat | r >= q}" nil )"nonneg_rat" nil )"nonneg_rat" nil )"real" reals nil )"(divides(n))" divides nil )"(divides(m))" divides nil ))nil ))nil 3320768520"" (skeep)"" (lemma "AxpyPos" )"" (inst?)"" (inst -1 "t" "u" )"" (split)"1" (propax) nil nil )"2" "abs(FtoR[radix](y)) * radix ^ (1 - Prec(b)) / (6 * radix)" "2" (mult-by 1 "4" )"2" (typepred "t" )"2" (expand "Fcanonic?" )"2" (split)"1" "abs(FtoR(y))*radix^(1-Prec(b))/(radix+1)" "1" (hide-all-but 1)"1" (div-by 1 "abs(FtoR(y))" )"1" "radix ^ (1 - Prec(b)) / (6 * radix) * 4 " )"1" (field 1) nil nil )"2" "radix ^ (1 - Prec(b)) / (radix + 1)" )"1" "radix ^ (1 - Prec(b))" )"1" "4/(6*radix)" )"1" (field 1) nil nil )"2" "1/(radix + 1)" )"1" (grind-reals) nil nil )"2" (field 1) nil nil ))nil ))nil ))nil )"2" (field 1) nil nil ))nil ))nil ))nil ))nil )"2" (lemma "Axpy_opt_aux2" )"2" (inst?)"2" (inst -1 "t" "u" )"2" "1" (propax) nil nil )"2" (propax) nil nil )"3" (propax) nil nil )"4" (propax) nil nil )"5" (propax) nil nil )"6" (propax) nil nil )"7" (propax) nil nil ))nil ))nil ))nil ))nil ))nil )"2" "abs(FtoR(y))*radix^(1-Prec(b))/(radix+radix/2)" )"1" (hide-all-but 1)"1" (field 1) nil nil )) nil )"2" (lemma "Axpy_opt_aux3" )"2" (inst?)"2" (inst -1 "t" "u" )"2" "1" (propax) nil nil )"2" (propax) nil nil )"3" (propax) nil nil )"4" (assert ) nil nil )"5" (propax) nil nil )"6" (propax) nil nil )"7" (propax) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"3" (skosimp*)"3" (lemma "Axpy_opt_aux1" )"3" (inst?)"3" (inst -1 "t" "u" )"3" (split)"1" (propax) nil nil ) ("2" (propax) nil nil )"3" (propax) nil nil ) ("4" (assert ) nil nil )"5" (propax) nil nil ) ("6" (propax) nil nil )"7" (propax) nil nil ))nil ))nil ))nil ))nil ))nil )"4" (propax) nil nil ) ("5" (propax) nil nil )"6" (propax) nil nil ))nil ))nil ))nil ))nil ))nil )nil axpy nil )"odd_int" integersnil )"posrat" exponentiation nil )nil axpy nil )"int" integers nil )"posint" nil )"even_int" integersnil )"(divides(n))" divides nil )"(divides(m))" divides nil )"nnreal" nil )"real" reals nil )AND const-decl "[bool, bool -> bool]" booleans nil )"bool" reals nil ) (>= const-decl "bool" reals nil )nil real_types nil )nil number_fields nil )"[numfield -> numfield]" number_fields nil )"{n: nonneg_real | n >= m AND n >= -m}" real_defsnil )"[numfield, numfield -> numfield]" number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )"real" float nil )"[numfield, numfield -> numfield]" number_fields nil )"boolean" notequal nil )nil number_fields nil )"[numfield, nznum -> numfield]" number_fields nil )OR const-decl "[bool, bool -> bool]" booleans nil )"real" exponentiation nil )"bool" reals nil )"bool" float nil )"real" float nil )"float" float nil )"(total_order?[real])" nil )"nnrat" exponentiation nil )"nnreal" nil )"(strict_total_order?[real])" real_props nil )NOT const-decl "[bool -> bool]" booleans nil )nil axpy nil )"posrat" nil )"posrat" nil )nil real_props nil )nil real_props nil )nil real_props nil )"even_int" integersnil )"even_int" integersnil )"posrat" nil )nil real_props nil )nil real_props nil )nil real_props nil )"(strict_total_order?[real])" real_props nil )"{x: float | Fcanonic?(b)(x)}" axpy nil )nil real_props nil )"posint" nil )nil axpy nil )"(total_order?[real])" nil )nil reals nil )nil real_props nil )nil real_types nil )nil real_props nil )"Format" axpy nil )"bool" float nil )nil float nil )nil naturalnumbers nil )nil float nil )"above(1)" axpy nil )nil integers nil )"bool" reals nil )nil booleans nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )nil integers nil )"real" reals nil )"real" reals nil )"real" reals nil ))nil 3321008004 ("" (subtype-tcc) nil nil )"Format" axpy nil )"bool" float nil )nil float nil )nil float nil )nil naturalnumbers nil )NOT const-decl "[bool -> bool]" booleans nil )"rat" exponentiation nil )"(strict_total_order?[real])" real_props nil )"{j: nonneg_int | j >= i}" nil )"posint" exponentiation nil )"(total_order?[real])" nil )"(total_order?[real])" nil )"int" integers nil )"posnat" exponentiation nil )"real" exponentiation nil )"bool" float nil )"bool" float nil )"bool" float nil )"posnat" float nil )"above(1)" axpy nil )nil integers nil )"bool" reals nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )"real" float nil )"bool" float nil )"boolean" notequal nil )"nnrat" exponentiation nil )"posrat" exponentiation nil )"rat" rationals nil )"rat" rationals nil )"real" reals nil )"(divides(n))" divides nil )"(divides(m))" divides nil ))nil ))nil 3321008004 ("" (subtype-tcc) nil nil )"Format" axpy nil )"bool" float nil )nil float nil )nil float nil )nil naturalnumbers nil )NOT const-decl "[bool -> bool]" booleans nil )"rat" exponentiation nil )"(strict_total_order?[real])" real_props nil )"{j: nonneg_int | j >= i}" nil )"posint" exponentiation nil )"(total_order?[real])" nil )"(total_order?[real])" nil )"int" integers nil )"posnat" exponentiation nil )"rat" rationals nil )"int" integers nil )"real" exponentiation nil )"bool" float nil )"bool" float nil )"bool" float nil )"posnat" float nil )"above(1)" axpy nil )nil integers nil )"bool" reals nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )"real" float nil )"bool" float nil )"boolean" notequal nil )"nnrat" exponentiation nil )"posrat" exponentiation nil )"rat" rationals nil )"rat" rationals nil )"{r: nonneg_rat | r >= q}" nil )"nonneg_rat" nil )"nonneg_rat" nil )"real" reals nil )"(divides(n))" divides nil )"(divides(m))" divides nil ))nil ))nil 3321008612"" (skeep)"" (case "u=(# Fnum:= 0, Fexp:=-dExp(b) #)" )"1" (case "FtoR(u)=FtoR(t)+FtoR(y)" )"1" (case "FtoR(y)=0" )"1" (hide-all-but (-1 -8))"1" (flip-ineq -2)"1" (rewrite -1)"1" (expand "abs" ) (("1" (grind-reals) nil nil ))nil ))nil ))nil ))nil )"2" (case "radix*abs(FtoR(y)) <= (radix+1)*Fulp(b)(y)/2" )"1" (hide-all-but (-1 1))"1" (rewrite "FabsCorrect" :dir rl)"1" (rewrite "FulpCanonic" )"1" (expand * "Fabs" "FtoR" )"1" (div-by -1 "radix ^ (Fexp(y))" )"1" " radix * (abs(Fnum(y)) * radix ^ Fexp(y)) / radix ^ (Fexp(y))= radix * abs(Fnum(y))" )"1" "(radix ^ (Fexp(y)) + radix ^ (Fexp(y)) * radix) / 2 / ") "1" (hide -1 -2)"1" (div-by -1 "radix" )"1" case "abs(Fnum(y)) < 1" )"1" expand "abs" -1)"1" (grind-reals) nil nil ))nil )"2" "(radix + 1) / 2 / radix" "2" (grind-reals) nil nil ))nil ))nil ))nil ))nil )"2" (field 1) nil nil ))nil )"2" (field 1) nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" case "(radix+1) * abs(FtoR[radix](y)) <= abs(FtoR(y))+(radix + 1) * Fulp(b)(y) / 2" )"1" (grind-reals) nil nil )"2" (hide 2 4)"2" (div-by 1 "radix+1" )"2" "abs(FtoR[radix](y))/(radix+1)+ Fulp(b)(y) / 2" )"1" (move-terms 1 r 2)"1" (trans-ineq 1 "abs(FtoR(a)*FtoR(x))" )"1" (move-terms 1 l 2)"1" (move-terms 1 r 1)"1" (case-replace "y=Fopp(t)" )"1" rewrite "FulpOpp" )"1" rewrite "FoppCorrect" )"1" "abs(-FtoR(t))=abs(FtoR(t))" )"1" "abs(FtoR(t)-FtoR(a)*FtoR(x))" )"1" "1" case "FORALL (v,w:real): (v>=0 => w>=0)=> (v<=0 => w<=0) => abs(w)-abs(v)<=abs(w-v)" )"1" "FtoR(a)*FtoR(x)" "FtoR(t)" )"1" "1" (propax) nil nil )"2" "2" "RleRoundedLessR0" )"2" "Closest?" "b" "t" "FtoR(a)*FtoR(x)" )"2" "1" nil nil )"2" nil nil )"3" nil nil )"4" rewrite "Closest_RoundedMode" )nil nil )"5" rewrite "FcanonicBounded" )nil nil ))nil ))nil ))nil ))nil )"3" "3" "RleRoundedR0" )"3" "Closest?" "b" "t" "FtoR(a)*FtoR(x)" )"3" "1" assert )nil nil )"2" assert )nil nil )"3" nil nil )"4" rewrite "Closest_RoundedMode" )nil nil )"5" rewrite "FcanonicBounded" )nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" "2" expand "abs" )"2" nil nil ))nil ))nil ))nil ))nil ))nil )"2" rewrite "ClosestUlp" )"2" rewrite "FcanonicBounded" )nil nil ))nil ))nil )"2" "2" expand "abs" )"2" (grind-reals) nil nil ))nil ))nil ))nil ))nil )"2" rewrite "FcanonicBounded" )nil nil ))nil )"2" "FcanonicUnique" )"2" "b" "y" "Fopp(t)" )"2" "1" (propax) nil nil )"2" rewrite "FoppCorrect" )"2" (assert ) nil nil ))nil )"3" "FcanonicOpp" )"3" "b" "t" )"3" (assert ) nil nil ))nil ))nil )"4" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (mult-by 1 "radix+1" )"2" "(radix + 1 + radix ^ (4 - Prec(b))) * ") "2" (assert )"2" "2" "2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (field 1) nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (rewrite "ExactSum_Near" )"1" (trans-ineq 1 "Fulp(b)(u)/2" :strict 2)"1" (lemma "ClosestUlp" )"1" (inst -1 "b" "u" "FtoR(t)+FtoR(y)" )"1" (split)"1" (assert ) nil nil ) ("2" (propax) nil nil )"3" (rewrite "FcanonicBounded" ) nil nil ))nil ))nil ))nil )"2" (rewrite "FulpCanonic" )"2" (rewrite -1) (("2" (grind-reals) nil nil )) nil ))nil ))nil )"2" (rewrite "FcanonicBounded" ) nil nil )"3" (rewrite "FcanonicBounded" ) nil nil )"4" (rewrite "FcanonicBounded" ) nil nil ))nil ))nil )"2" (lemma "FcanonicUnique" )"2" (inst -1 "b" "u" "(# Fnum := 0, Fexp := -dExp(b) #)" )"2" (split)"1" (propax) nil nil )"2" (rewrite -3 :dir rl)"2" (expand "FtoR" 1) (("2" (assert ) nil nil )) nil ))nil )"3" (expand * "Fcanonic?" "Fsubnormal?" "Fbounded?" )"3" (flatten)"3" (split)"1" (expand "abs" 1) (("1" (grind-reals) nil nil ))nil )"2" (expand "abs" 1) (("2" (grind-reals) nil nil ))nil ))nil ))nil ))nil )"4" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil )
Messung V0.5 in Prozent C=100 H=100 G=100
¤ Die Informationen auf dieser Webseite wurden
nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit,
noch Qualität der bereit gestellten Informationen zugesichert.0.856Bemerkung:
(vorverarbeitet am 2026-04-27)
¤ *Bot Zugriff
2026-05-26
