(exp_termnil 3320227930"" (skosimp) (("" (assert ) nil nil )) nil )"(total_order?[real])" nil ))nil 3320473217"" (skosimp) (("" (assert ) nil nil )) nil )"(total_order?[real])" nil ))nil 3320469433"" (induct "m" )"1" (skosimp)"1" (rewrite "expt_x0" )"1" (expand "exp_term" )"1" (expand "abs" ) (("1" (assert ) nil nil )) nil )) nil ))nil ))nil )"2" (skosimp*)"2" (inst - "a!1" "n!1" )"2" (assert )"2" (case "abs(exp_term(a!1)(n!1)) /= 0" )"1" (rewrite "expt_plus" 2)"1" (rewrite "expt_x1" 2)"1" case "abs(exp_term(a!1)(j!1 * n!1 + n!1)) <= abs(exp_term(a!1)(n!1)) * abs(exp_term(a!1)(j!1 * n!1))" )"1" (assert )"1" "both_sides_times_pos_le1" "pz" "abs(exp_term(a!1)(n!1))" "x" "abs(exp_term(a!1)(j!1 * n!1))" "y" "abs(exp_term(a!1)(n!1)) ^ j!1" ))"1" (assert ) nil nil )) nil ))nil )"2" (hide 3 -2)"2" (expand "exp_term" )"2" (case-replace "n!1=0" )"1" (assert )"1" (expand "abs" ) (("1" (assert ) nil nil ))nil ))nil )"2" (assert )"2" (case-replace "j!1=0" )"1" (assert )"1" expand "abs" 2 3)"1" (assert ) nil nil ))nil ))nil )"2" (assert )"2" rewrite "abs_mult" 3 :dir rl)"2" rewrite "div_times" 3)"2" rewrite "expt_plus" 3 :dir rl)"2" rewrite "abs_div" 3)"2" rewrite "abs_div" 3)"2" expand "abs" 3 4)"2" expand "abs" 3 2)"2" rewrite "abs_expt" "2" case "abs(a!1)>0" )"1" "expt_pos" "px" "abs(a!1)" "i" "(j!1+1)*n!1" ))"1" "posreal_times_posreal_is_posreal" "px" "factorial(n!1)" "py" "factorial(j!1 * n!1)" ))"1" "both_sides_div_pos_lt2" "px" "factorial(j!1 * n!1 + n!1)" "pz" "abs(a!1) ^ (j!1 * n!1 + n!1)" "py" "factorial(n!1) * factorial(j!1 * n!1)" ))"1" assert )"1" "1" case "forall (pn,pm:posnat): factorial(pm)*factorial(pn) < factorial(pn+pm)" )"1" "n!1" "j!1*n!1" )"1" assert )nil nil ))nil )"2" "2" "pn" )"1" assert )nil nil )"2" assert )nil nil )"3" "3" "j!2=0" )"1" expand "factorial" "1" expand "factorial" "1" expand "factorial" "1" assert )nil nil ))nil ))nil ))nil )"2" assert )"2" "pm!1" )"2" expand "factorial" "2" expand "factorial" "2" "lt_times_lt_pos1" "px" "factorial(pm!1) * factorial(j!2)" "y" "factorial(pm!1 + j!2)" "nnz" "1+j!2" "w" "1+j!2+pm!1" ))"2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" assert )nil nil ))nil ))nil )"2" nil nil ))nil )"2" expand "abs" )"2" "a!1<0" )"1" assert )nil nil )"2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (hide-all-but (1 2))"2" (expand "exp_term" )"2" (case-replace "n!1=0" )"1" (expand "abs" ) (("1" (assert ) nil nil )) nil )"2" (assert )"2" (rewrite "abs_div" )"2" (expand "abs" -1 2)"2" (rewrite "abs_expt" -1 :dir rl)"2" "expt_nonzero" "n0x" "abs(a!1)" "i" "n!1" ))"1" (assert ) nil nil )"2" (expand "abs" )"2" "a!1<0" )"1" (assert ) nil nil )"2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"nnreal" exponentiation nil )"posint" nil )nil exponentiation nil )nil reals nil )nil exponentiation nil )"int" integers nil )"nnreal" nil )"nonneg_int" nil )"[numfield, numfield -> numfield]" number_fields nil )nil real_props nil )"bool" reals nil )nil real_types nil )"(strict_total_order?[real])" real_props nil )"real" reals nil )nil reals nil )nil real_props nil )"posint" nil )nil real_props nil )"nnreal" nil )nil exponentiation nil )"real" reals nil )nil exponentiation nil )nil real_props nil )nil real_props nil )"odd_int" integers nil )nil factorial "ints/" )"bool" reals nil )"posint" nil )"(strict_total_order?[real])" real_props nil )nil real_types nil )nil real_props nil )nil number_fields nil )"[numfield, nznum -> numfield]" number_fields nil )nil integers nil )nil integers nil )"posnat" factorial "ints/" )"even_int" integersnil )"real" reals nil )"even_int" integersnil )"even_int" integersnil )"{r: posrat | r >= q}" nil )"{j: nonneg_int | j >= i}" nil )"{j: posint | j >= i}" nil )"[T, T -> boolean]" equalities nil )nil exponentiation nil )"(total_order?[real])" nil )nil exponentiation nil )nil naturalnumbers nil )"real" exponentiation nil )OR const-decl "[bool, bool -> bool]" booleans nil )"[numfield, numfield -> numfield]" number_fields nil )"real" exp_term 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 )"boolean" notequal nil )"[bool, bool -> bool]" booleans nil )nil defined_types nil )nil naturalnumbers nil )"bool" reals nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )"nonneg_int" nil )"(divides(n))" divides nil )"(divides(m))" divides nil ))nil 3320472607"" (skosimp)"" (expand "exp_term" )"" (case-replace "n!1=0" )"1" (expand "factorial" )"1" (expand "factorial" )"1" (rewrite "expt_x1" )"1" (expand "abs" 2 2) (("1" (assert ) nil nil )) nil ))nil ))nil ))nil )"2" (assert )"2" (rewrite "abs_div" )"2" (rewrite "abs_div" )"2" (expand "abs" 3 4)"2" (expand "abs" 3 2)"2" (rewrite "abs_expt" 3 :dir rl)"2" (rewrite "abs_expt" 3 :dir rl)"2" (case "0 < abs(a!1)" )"1" (rewrite "expt_plus" 3)"1" (rewrite "expt_x1" )"1" (expand "factorial" 3 1)"1" "both_sides_times_pos_lt1" "x" "abs(a!1)/(1+n!1)" "y" "1" "pz" "abs(a!1) ^ n!1 / factorial(n!1)" ))"1" rewrite "div_times" -1)"1" assert )"1" "1" rewrite "div_mult_pos_lt1" 1)nil nil ))nil ))nil ))nil )"2" "expt_pos" "px" "abs(a!1)" "i" "n!1" ))"1" rewrite "div_mult_pos_gt1" 1)"1" (assert ) nil nil ))nil )"2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil )"2" (expand "abs" )"2" (case "a!1<0" )"1" (assert ) nil nil )"2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"real" reals nil )"real" exp_term nil )"{j: posint | j >= i}" nil )"{j: nonneg_int | j >= i}" nil )"{r: posrat | r >= q}" nil )"posint" nil )"real" reals nil )nil exponentiation nil )"nnreal" nil )nil exponentiation nil )"even_int" integersnil )nil extra_real_props nil )nil real_props nil )"nonneg_int" nil )nil real_props nil )"(strict_total_order?[real])" real_props nil )"[numfield, nznum -> numfield]" number_fields nil )nil number_fields nil )nil real_types nil )nil real_props nil )"[numfield -> numfield]" number_fields nil )AND const-decl "[bool, bool -> bool]" booleans nil )nil real_types nil )"bool" reals nil )"nnreal" exponentiation nil )nil exponentiation nil )nil reals nil )"nnreal" nil )"int" integers nil )"real" exponentiation nil )OR const-decl "[bool, bool -> bool]" booleans nil )"[numfield, numfield -> numfield]" number_fields nil )nil number_fields nil )nil integers nil )"bool" reals nil )nil integers nil )nil reals nil )"boolean" notequal nil )nil real_props nil )"posint" nil )"(divides(n))" divides nil )"(divides(m))" divides nil )"posnat" factorial "ints/" )nil exponentiation nil )"odd_int" integers nil )"posint" nil )"(strict_total_order?[real])" real_props nil )"{n: nonneg_real | n >= m AND n >= -m}" real_defsnil )nil naturalnumbers nil )"bool" reals nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )"[T, T -> boolean]" equalities nil )nil booleans nil )nil numbers nil ))nil 3320227935"" (skosimp*) (("" (assert ) nil nil )) nil )"(total_order?[real])" nil ))nil 3320228740"" (skosimp)"" (expand "tends_to_0?" )"" (skosimp)"" (case "EXISTS n: abs(exp_term(n0a!1)(n)) < 1" )"1" (skosimp)"1" (name "N" "floor(abs(n0a!1)+1)" )"1" (lemma "floor_def" ("x" "abs(n0a!1)+1" ))"1" (replace -2)"1" (flatten)"1" (case "N>=1" )"1" case "FORALL (px: posreal): ") "1" (case "0<abs(n0a!1)" )"1" (case "0<abs(exp_term(n0a!1)(n!1))" )"1" (inst - "abs(exp_term(n0a!1)(n!1))" )"1" assert )"1" "epsilon!1" )"1" "1" "M" "max(N,n!2)*n!1" )"1" "M" )"1" "1" case "forall (n:nat): abs(exp_term(n0a!1)(M+n)) <= abs(exp_term(n0a!1)(M))" )"1" "i!1-M" )"1" assert )"1" "exp_term_power" "a" "n0a!1" "n" "n!1" "m" "max(N,n!2)" ))"1" replace -3)"1" "both_sides_expt_lt1_le" "lt1x" "abs(exp_term(n0a!1)(n!1))" "i" "max(N,n!2)" "j" "n!2" ))"1" assert )nil nil ))nil ))nil ))nil ))nil )"2" expand "M" )"2" expand "max" )"2" "N<=n!2" )"1" assert )"1" "N<n!2" )"1" assert )nil nil )"2" assert )nil nil ))nil ))nil )"2" assert )nil nil ))nil ))nil ))nil ))nil )"2" "2" "n" )"1" (assert ) nil nil )"2" "2" expand "exp_term" )"2" "expt_plus" "n0x" "n0a!1" "i" "1" "j" "M+j!1" ))"2" rewrite "expt_x1" )"2" replace -1)"2" "2" expand "factorial" "2" assert )"2" case "M >=1" )"1" assert )"1" rewrite "abs_div" "1" rewrite "abs_div" "1" rewrite "abs_div" "1" expand "abs" "1" expand "abs" "1" expand "abs" "1" "posreal_times_posreal_is_posreal" "px" "1+M+j!1" "py" "factorial(M + j!1)" ))"1" expand "abs" "1" assert )"1" rewrite "abs_mult" "1" "le_times_le_pos" "nnx" "abs(n0a!1)/(1+M+j!1)" "y" "1" "nnz" "abs(n0a!1 ^ (M + j!1)) / factorial(M + j!1)" "w" "abs(n0a!1 ^ M) / factorial(M)" ))"1" assert )"1" "1" assert )"1" rewrite "div_times" "1" assert )nil nil ))nil ))nil )"2" "2" rewrite "div_mult_pos_le1" "2" expand "M" )"2" expand "max" )"2" "N<=n!2" )"1" assert )"1" "le_times_le_pos" "nnx" "1" "y" "n!1" "nnz" "N" "w" "n!2" ))"1" assert )"1" "N<n!2" )"1" assert )nil nil )"2" assert )nil nil ))nil ))nil ))nil ))nil )"2" assert )"2" "both_sides_times_pos_le1" "pz" "N" "x" "1" "y" "n!1" ))"2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" expand "M" )"2" expand "max" )"2" "N<=n!2" )"1" "N<n!2" )"1" assert )"1" "n!1=0" )"1" assert )nil nil )"2" "posreal_times_posreal_is_posreal" "px" "n!2" "py" "n!1" ))"1" assert )nil nil )"2" assert )nil nil ))nil ))nil ))nil )"2" assert )"2" "n!1=0" )"1" assert )nil nil )"2" assert )"2" "posreal_times_posreal_is_posreal" "px" "N" "py" "n!1" ))"2" assert )nil nil ))nil ))nil ))nil ))nil ))nil )"2" assert )"2" "posreal_times_posreal_is_posreal" "px" "N" "py" "n!1" ))"1" assert )nil nil )"2" "n!1=0" )"1" assert )nil nil )"2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (assert ) nil nil ))nil )"2" (hide-all-but (-1 1))"2" expand "exp_term" )"2" rewrite "abs_div" )"2" "n!1=0" )"1" expand "abs" )"1" (assert ) nil nil ))nil )"2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil )"2" (hide-all-but 1)"2" (expand "abs" )"2" "n0a!1 < 0" )"1" (assert ) nil nil )"2" (assert ) nil nil ))nil ))nil ))nil ))nil )"2" (hide-all-but 1)"2" (skosimp)"2" (lemma "small_expt" ("px" "px!1" ))"2" assert )"2" "2" "py!1" )"2" "2" "n!2=0" )"1" assert )"1" expand "expt" )"1" "1" )"1" rewrite "expt_x1" )"1" assert )nil nil ))nil ))nil ))nil ))nil )"2" "n!2" )"1" expand "^" )"1" (propax) nil nil ))nil )"2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (hide 2)"2" case "FORALL (n: nat, pn: posnat): pn ^ n <= factorial(pn + n)" )"1" (lemma "floor_def" ("x" "abs(n0a!1) + 1" ))"1" (flatten)"1" (expand "exp_term" )"1" (name-replace "N" "floor(abs(n0a!1) + 1)" )"1" (case "N>=1" )"1" (assert )"1" (inst + "2*N*N" )"1" "both_sides_times_pos_le1" "pz" "N" "x" "1" "y" "N" ))"1" assert )"1" rewrite "abs_div" 1)"1" expand "abs" 1 2)"1" rewrite "div_mult_pos_lt1" 1)"1" rewrite "abs_expt" 1 :dir rl)"1" rewrite "expt_times" 1)"1" "N*N" "N*N" )"1" "both_sides_expt_pos_lt" "px" "abs(n0a!1)^2" "py" "N*N" "pm" "N*N" ))"1" assert )"1" "1" expand "^" )"1" expand "expt" )"1" expand "expt" )"1" expand "expt" )"1" "lt_times_lt_pos1" "px" "abs(n0a!1)" "y" "N" "nnz" "abs(n0a!1)" "w" "N" ))"1" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil )"2" (hide 2)"2" (induct "n" )"1" (skosimp)"1" (rewrite "expt_x0" ) (("1" (assert ) nil nil ))nil ))nil )"2" (skosimp*)"2" (inst - "pn!1" )"2" (rewrite "expt_plus" )"2" (expand "factorial" 1)"2" "lt_times_lt_pos1" "px" "pn!1 ^ j!1" "y" "factorial(pn!1 + j!1)" "nnz" "pn!1" "w" "1+j!1+pn!1" ))"2" (rewrite "expt_x1" 1)"2" (assert ) nil nil )) nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"bool" exp_term nil )nil reals nil )"boolean" notequal nil )"real" exp_term 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 )nil naturalnumbers nil )"bool" reals nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )"{y: posreal | y >= x}" nil )"posreal" nil )"nat" floor_ceil nil )"[T, T -> boolean]" equalities nil )nil integers nil )"bool" reals nil )"[numfield, numfield -> numfield]" number_fields nil )"{i | i <= x & x < i + 1}" floor_ceil nil )nil exponentiation nil )"nat" exp_term nil )"posreal" exponentiation nil )"nzreal" exponentiation nil )"real" exponentiation nil )"(strict_total_order?[real])" real_props nil )"nzreal" exp_term nil )"nat" exp_term nil )"nonneg_int" nil )"(divides(n))" divides nil )"(divides(m))" divides nil )"[numfield, numfield -> numfield]" number_fields nil )max const-decl "{p: real | p >= m AND p >= n}" real_defs nil )nil exponentiation nil )"odd_int" integersnil )"nzreal" nil )"posnat" factorial "ints/" )nil reals nil )nil real_props nil )"posreal" nil )nil real_types nil )"posint" nil )nil real_props nil )nil number_fields nil )"[numfield, nznum -> numfield]" number_fields nil )nil real_props nil )nil real_props nil )nil real_props nil )"nnreal" nil )"real" reals nil )"posreal" nil )nil real_props nil )"even_int" integersnil )"real" reals nil )"{r: posrat | r >= q}" nil )"{j: nonneg_int | j >= i}" nil )"{j: posint | j >= i}" nil )nil exponentiation nil )"nzreal" nil )"nzreal" exponentiation nil )nil naturalnumbers nil )nil defined_types nil )"posint" nil )"int" integers nil )"[numfield, numfield -> numfield]" number_fields nil )"int" exp_term nil )"nonneg_int" exp_term nil )nil exp_term nil )nil exponentiation nil )"int" integers nil )"{k: int | i <= k AND j <= k}" nil )"{s: rat | s >= q AND s >= r}" nil )"nnreal" exponentiation nil )"real" reals nil )"real" reals nil )"nonneg_int" nil )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil )"(total_order?[real])" nil )"posint" nil )"nzreal" real_typesnil )"real" exponentiation nil )OR const-decl "[bool, bool -> bool]" booleans nil )nil integers nil )nil integers nil )"[bool, bool -> bool]" booleans nil )nil real_types nil )"bool" reals nil )nil floor_ceil nil )nil real_props nil )nil exponentiation nil )nil exponentiation nil )nil real_props nil )"nat" exponentiation nil )"posreal" exponentiation nil )nil exponentiation nil )"even_int" integersnil )nil exponentiation nil )"posint" exponentiation nil ))
Messung V0.5 in Prozent C=100 H=100 G=100
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