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nil ))
nil )
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shostak))
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nil ))
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shostak))
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nil )
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shostak))
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nil )
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nil ))
(powerseries_deriv 0
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k * a!1(k) * x!1 ^ (k - 1)
ENDIF)
(n!1)) =
IF n!1 < 2 THEN abs(n!1 * a!1(n!1))
ELSE abs(n!1) * abs(a!1(n!1)) * abs(GET_tk(x!1, h!1, n!1) ^ (n!1 - 1) - x!1 ^ (n!1 - 1))
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nil
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nil )
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(hide 2)
(("2"
(hide -)
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(inst
-
"x!1" )
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(inst
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"x!1 + delta!1/2" )
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*
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inf_sum(1,(LAMBDA k:
a!1(k)*(((x!1+x!2)^k - x!1^k)/x!2)))")
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IF k < 2 THEN k* a!1(k)
ELSE k* a!1(k)* GET_tk(x!1, x!2,k) ^ (k - 1)
ENDIF)
- deriv_powerseq(a!1, x!1),
1) = series((LAMBDA k:
IF k < 2 THEN k* a!1(k)
ELSE k * a!1(k)*GET_tk(x!1, x!2,k) ^ (k - 1)
ENDIF)
- deriv_powerseq(a!1, x!1),
2)")
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2)
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ENDIF
- IF x = 0 THEN a!1(0) ELSE a!1(x) * x!1 ^ x ENDIF) =
(LAMBDA (x: nat):
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a!1(x_1) * x!1 ^ x_1
ENDIF) = (LAMBDA k:
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-2
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((not_one_element formula-decl nil power_series_deriv nil )
(T formal-subtype-decl nil power_series_deriv nil )
(T_pred const-decl "[real -> boolean]" power_series_deriv nil )
(real nonempty-type-from-decl nil reals nil )
(real_pred const-decl "[number_field -> boolean]" reals nil )
(number_field nonempty-type-from-decl nil number_fields nil )
(number_field_pred const-decl "[number -> boolean]" number_fields
nil )
(boolean nonempty-type-decl nil booleans nil )
(number nonempty-type-decl nil numbers nil )
(deriv_powerseries_conv formula-decl nil power_series_derivseq nil )
(deriv_fun_def formula-decl nil derivatives "analysis/" )
(deriv_domain? const-decl "bool" deriv_domain_def "analysis/" )
(not_one_element? const-decl "bool" deriv_domain_def "analysis/" )
(series const-decl "sequence[real]" series nil )
(deriv_powerseq const-decl "sequence[real]" power_series_derivseq
nil )
(= const-decl "[T, T -> boolean]" equalities nil )
(real_minus_real_is_real application-judgement "real" reals nil )
(derivative_squeeze_delta formula-decl nil derivatives_alt
"analysis/" )
(Gseq const-decl "real" power_series_deriv_scaf nil )
(inf_sum const-decl "real" series nil )
(conv_series? const-decl "bool" series nil )
(NQ const-decl "real" derivatives_def "analysis/" )
(- const-decl "[numfield, numfield -> numfield]" number_fields nil )
(numfield nonempty-type-eq-decl nil number_fields nil )
(A const-decl "setof[nzreal]" derivatives_def "analysis/" )
(setof type-eq-decl nil defined_types nil )
(nzreal nonempty-type-eq-decl nil reals nil )
(/= const-decl "boolean" notequal nil )
(posreal_div_posreal_is_posreal application-judgement "posreal"
real_types nil )
(delta!1 skolem-const-decl "posreal" power_series_deriv nil )
(nzreal_times_nzreal_is_nzreal application-judgement "nzreal"
real_types nil )
(delta!1 skolem-const-decl "posreal" power_series_deriv nil )
(real_times_real_is_real application-judgement "real" reals nil )
(x!1 skolem-const-decl "T" power_series_deriv nil )
(xp!1 skolem-const-decl "T" power_series_deriv nil )
(A2seq const-decl "real" power_series_deriv_scaf nil )
(* const-decl "[numfield, numfield -> numfield]" number_fields nil )
(Gseq_conv formula-decl nil power_series_deriv_scaf nil )
(real_le_is_total_order name-judgement "(total_order?[real])"
real_props nil )
(inf_sum_le formula-decl nil series nil )
(inf_sum_Gseq_abs formula-decl nil power_series_deriv_scaf nil )
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil )
(nnreal type-eq-decl nil real_types nil )
(upfrom nonempty-type-eq-decl nil integers nil )
(GET_tk const-decl
"{tk: between[T](x, x + h) | ((x + h) ^ k - x ^ k) / h = k * tk ^ (k - 1)}"
power_series_deriv_scaf nil )
(/ const-decl "[numfield, nznum -> numfield]" number_fields nil )
(nznum nonempty-type-eq-decl nil number_fields nil )
(posnat nonempty-type-eq-decl nil integers nil )
(nonneg_int nonempty-type-eq-decl nil integers nil )
(^ const-decl "real" exponentiation nil )
(OR const-decl "[bool, bool -> bool]" booleans nil )
(< const-decl "bool" reals nil )
(IF const-decl "[boolean, T, T -> T]" if_def nil )
(int_minus_int_is_int application-judgement "int" integers nil )
(nnreal_times_nnreal_is_nnreal application-judgement "nnreal"
real_types nil )
(nzint_abs_is_pos application-judgement "{j: posint | j >= i}"
real_defs nil )
(max const-decl "{p: real | p >= m AND p >= n}" real_defs nil )
(abs_mult formula-decl nil real_props nil )
(nonneg_real_max application-judgement
"{z: nonneg_real | z >= x AND z >= y}" real_defs nil )
(abs_0 formula-decl nil abs_lems "reals/" )
(open formula-decl nil power_series_deriv nil )
(derivable? const-decl "bool" derivatives "analysis/" )
(deriv const-decl "[T -> real]" derivatives "analysis/" )
(minus_real_is_real application-judgement "real" reals nil )
(expt def-decl "real" exponentiation nil )
(real_div_nzreal_is_real application-judgement "real" reals nil )
(le_times_le_any1 formula-decl nil extra_real_props nil )
(nnreal_expt application-judgement "nnreal" exponentiation nil )
(even_minus_odd_is_odd application-judgement "odd_int" integers
nil )
(even_minus_even_is_even application-judgement "even_int" integers
nil )
(expt_x0 formula-decl nil exponentiation nil )
(expt_x1 formula-decl nil exponentiation nil )
(MAX skolem-const-decl
"{z: nonneg_real | z >= abs(x!1) AND z >= abs(xp!1)}"
power_series_deriv nil )
(c!1 skolem-const-decl "T" power_series_deriv nil )
(abs_hat formula-decl nil exponent_props "reals/" )
(abs_eq_0 formula-decl nil abs_lems "reals/" )
(zero_hat formula-decl nil exponent_props "reals/" )
(n!1 skolem-const-decl "upfrom(2)" power_series_deriv nil )
(nnrat_exp application-judgement "nnrat" exponentiation nil )
(both_sides_expt_pos_ge formula-decl nil exponentiation nil )
(minus_odd_is_odd application-judgement "odd_int" integers nil )
(min const-decl "{p: real | p <= m AND p <= n}" real_defs nil )
(abs_neg formula-decl nil abs_lems "reals/" )
(derivable? const-decl "bool" derivatives_def "analysis/" )
(deriv const-decl "real" derivatives_def "analysis/" )
(deriv_x_to_n formula-decl nil derivatives "analysis/" )
(connected? const-decl "bool" deriv_domain_def "analysis/" )
(mean_value_abs formula-decl nil derivative_props "analysis/" )
(both_sides_times_pos_le1_imp formula-decl nil extra_real_props
nil )
(<= const-decl "bool" reals nil )
(int_abs_is_nonneg application-judgement "{j: nonneg_int | j >= i}"
real_defs nil )
(abs_abs formula-decl nil real_props nil )
(convergent_scal formula-decl nil convergence_ops "analysis/" )
(series_m_scal formula-decl nil series nil )
(series const-decl "sequence[real]" series nil )
(abs const-decl "sequence[real]" series nil )
(inf_sum_scal formula-decl nil series nil )
(IMPLIES const-decl "[bool, bool -> bool]" booleans nil )
(nonzero_abs_is_pos application-judgement "{y: posreal | y >= x}"
real_defs nil )
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nil )
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k * a!1(k) * x!1 ^ (k - 1)
ENDIF)
(n!1)) =
IF n!1 < 2 THEN abs(n!1 * a!1(n!1))
ELSE abs(n!1) * abs(a!1(n!1)) * abs(GET_tk(x!1, h!1, n!1) ^ (n!1 - 1) - x!1 ^ (n!1 - 1))
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IF k < 2 THEN k* a!1(k)
ELSE k* a!1(k)* GET_tk(x!1, x!2,k) ^ (k - 1)
ENDIF)
- deriv_powerseq(a!1, x!1),
1) = series((LAMBDA k:
IF k < 2 THEN k* a!1(k)
ELSE k * a!1(k)*GET_tk(x!1, x!2,k) ^ (k - 1)
ENDIF)
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2)")
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nil )
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(hide
2)
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nil
nil ))
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nil ))
nil ))
nil )
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(case-replace
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nil ))
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2)
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(inst?)
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(assert )
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nil ))
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("4"
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2)
(("4"
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(inst?)
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nil ))
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("2"
(hide 2)
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-3
*
rl)
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"-" )
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IF x = 0 THEN a!1(0)
ELSE a!1(x) * (x!1 + x!2) ^ x
ENDIF
- IF x = 0 THEN a!1(0) ELSE a!1(x) * x!1 ^ x ENDIF) =
(LAMBDA (x: nat):
IF x = 0 THEN 0
ELSE a!1(x) * (x!1 + x!2) ^ x - a!1(x) * x!1 ^ x ENDIF)")
(("1"
(hide
-1)
(("1"
(hide
-)
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(lemma
"limit_scal" )
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(inst?)
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(inst
-
"1/x!2" )
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(replace
-1
*
rl)
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-1)
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nil
nil )
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(hide
2)
(("2"
(case-replace
"(LAMBDA (x_1: nat):
1 / x!2 *
IF x_1 = 0 THEN 0
ELSE a!1(x_1) * (x!1 + x!2) ^ x_1 -
a!1(x_1) * x!1 ^ x_1
ENDIF) = (LAMBDA k:
a!1(k) * (((x!1 + x!2) ^ k - x!1 ^ k) / x!2))")
(("1"
(hide
-1)
(("1"
(lemma
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(inst?)
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"x!1" )
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2)
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(apply-extensionality
1
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t)
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t)
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("3"
(hide 2)
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-1
-2
-3)
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(expand
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(inst?)
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(inst?)
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nil
nil ))
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(lemma
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(skosimp*)
nil
nil ))
nil )
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(skosimp*)
nil
nil ))
nil ))
nil )
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(skosimp*)
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(inst?)
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(assert )
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(inst?)
nil
nil ))
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(lemma
"connected_domain" )
(("5"
(skosimp*)
nil
nil ))
nil ))
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nil ))
nil ))
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nil ))
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nil ))
nil ))
nil ))
nil ))
nil ))
nil )
((limit const-decl "real" convergence_sequences "analysis/" )
(powerseq const-decl "sequence[real]" power_series nil )
(series_scal formula-decl nil series nil )
(series_first formula-decl nil series nil )
(limit_scal formula-decl nil convergence_ops "analysis/" )
(series_diff formula-decl nil series nil )
(inf_sum_eq formula-decl nil series nil )
(conv_scaf3 formula-decl nil power_series_deriv_scaf nil )
(limit_diff formula-decl nil convergence_ops "analysis/" )
(limit_eq_rew formula-decl nil power_series_deriv_scaf nil )
(convergent_diff formula-decl nil convergence_ops "analysis/" )
(end_series_conv formula-decl nil series nil )
(conv_scaf2 formula-decl nil power_series_deriv_scaf nil )
(series_m_eq formula-decl nil series nil )
(sigma_first formula-decl nil sigma "reals/" )
(sigma def-decl "real" sigma "reals/" )
(T_high type-eq-decl nil sigma "reals/" )
(T_low type-eq-decl nil sigma "reals/" )
(series_m_diff formula-decl nil series nil )
(lim_deriv_alt formula-decl nil power_series_derivseq nil )
(A2_conv formula-decl nil power_series_deriv_scaf nil )
(cv_scal formula-decl nil lim_of_functions "analysis/" )
(cv_abs formula-decl nil lim_of_functions "analysis/" )
(adh_A_lem formula-decl nil derivatives_def "analysis/" )
(adh const-decl "setof[real]" convergence_functions "analysis/" )
(inf_sum_scal formula-decl nil series nil )
(abs const-decl "sequence[real]" series nil )
(series_m_scal formula-decl nil series nil )
(convergent_scal formula-decl nil convergence_ops "analysis/" )
(mean_value_abs formula-decl nil derivative_props "analysis/" )
(deriv_x_to_n formula-decl nil derivatives "analysis/" )
(deriv const-decl "real" derivatives_def "analysis/" )
(abs_neg formula-decl nil abs_lems "reals/" )
(zero_hat formula-decl nil exponent_props "reals/" )
(abs_eq_0 formula-decl nil abs_lems "reals/" )
(abs_hat formula-decl nil exponent_props "reals/" )
(deriv const-decl "[T -> real]" derivatives "analysis/" )
(abs_0 formula-decl nil abs_lems "reals/" )
(GET_tk const-decl
"{tk: between[T](x, x + h) | ((x + h) ^ k - x ^ k) / h = k * tk ^ (k - 1)}"
power_series_deriv_scaf nil )
(inf_sum_Gseq_abs formula-decl nil power_series_deriv_scaf nil )
(inf_sum_le formula-decl nil series nil )
(Gseq_conv formula-decl nil power_series_deriv_scaf nil )
(A2seq const-decl "real" power_series_deriv_scaf nil )
(A const-decl "setof[nzreal]" derivatives_def "analysis/" )
(NQ const-decl "real" derivatives_def "analysis/" )
(Gseq const-decl "real" power_series_deriv_scaf nil )
(deriv_powerseq const-decl "sequence[real]" power_series_derivseq
nil )
(deriv_fun_def formula-decl nil derivatives "analysis/" )
(deriv_powerseries_conv formula-decl nil power_series_derivseq
nil ))
nil )
(powerseries_deriv-9 nil 3299837690
("" (skosimp*)
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(("" (lemma "deriv_fun_def[T]" )
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(inst -
"(LAMBDA x: limit(series(deriv_powerseq(a!1, x))))" )
(("1" (split -1)
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(name "FA"
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(name
"FG"
"(LAMBDA x: limit(series(deriv_powerseq(a!1, x))))" )
(("1"
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(("1"
(lemma
"derivative_squeeze_delta[T]" )
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(inst?)
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(assert )
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(hide 2)
(("1"
(case-replace
"(LAMBDA (h: (A[T](x!1))): (NQ(FA, x!1)(h) - FG(x!1))) =
(LAMBDA (h: (A[T](x!1))): inf_sum(2,Gseq(a!1,x!1,h)))")
(("1"
(case
"EXISTS xp: abs(xp) > abs(x!1) AND (xp >= 0 IFF x!1 >= 0)" )
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(("1"
(hide -2 -3)
(("1"
(case
"xp!1 /= 0" )
(("1"
(flatten)
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(inst
+
"(LAMBDA (h: (A[T](x!1))): abs(h)*inf_sum(2,A2seq(a!1,x!1,xp!1)))" )
(("1"
(split +)
(("1"
(name-replace
"LL"
"inf_sum(2, A2seq(a!1, x!1, xp!1))" )
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(lemma
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(inst
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"0"
"LL"
"(LAMBDA (h: (A[T](x!1))): abs(h))"
"0" )
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2)
(("1"
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1)
(("1"
(lemma
"cv_abs[(A[T](x!1))]" )
(("1"
(inst?)
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(expand
"restrict" )
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(expand
"abs" )
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(propax)
nil
nil ))
nil ))
nil )
("2"
(lemma
"adh_A_lem[T]" )
(("2"
(inst?)
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(skosimp*)
nil
nil )
("3"
(skosimp*)
nil
nil ))
nil )
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(hide
2)
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(expand
"conv_series?" )
(("2"
(lemma
"A2_conv[T]" )
(("2"
(inst
-
"a!1"
"x!1"
"xp!1" )
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(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil )
("3"
(assert )
nil
nil ))
nil )
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(inst
+
"abs(xp!1 - x!1)" )
(("1"
(skosimp*)
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-4
-5
-6
-7
-10)
(("1"
(case
"(LAMBDA (h: (A[T](x!1))): (NQ(FA, x!1)(h) - FG(x!1)))(h!1) =
(LAMBDA (h: (A[T](x!1))): inf_sum(2, Gseq(a!1, x!1, h)))(h!1)")
(("1"
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-1)
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-1)
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(("1"
(inst?)
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-2)
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-2)
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(lemma
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"abs(Gseq(a!1, x!1, h!1))"
"abs(h!1)*A2seq(a!1, x!1, xp!1)"
"2" )
(("1"
(split
-1)
(("1"
(lemma
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(inst?)
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(inst
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"x!1" )
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(inst?)
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(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
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2)
(("2"
(skosimp*)
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(expand
"Gseq" )
(("2"
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"abs(abs(LAMBDA (k):
IF k < 2 THEN k * a!1(k)
ELSE k * a!1(k) * GET_tk(x!1, h!1, k) ^ (k - 1) -
k * a!1(k) * x!1 ^ (k - 1)
ENDIF)
(n!1)) =
IF n!1 < 2 THEN abs(n!1 * a!1(n!1))
ELSE abs(n!1) * abs(a!1(n!1)) * abs(GET_tk(x!1, h!1, n!1) ^ (n!1 - 1) - x!1 ^ (n!1 - 1))
ENDIF")
(("1"
(expand
"A2seq" )
(("1"
(assert )
(("1"
(hide
-1
-2
-5)
(("1"
(expand
"*" )
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(name-replace
"TK"
"GET_tk(x!1, h!1, n!1)" )
(("1"
(rewrite
"abs_mult" )
(("1"
(case-replace
"abs(TK ^ (n!1 - 1) - x!1 ^ (n!1 - 1)) <=
abs(max (abs(x!1), abs(xp!1)) ^ (n!1 - 2)) * abs(n!1 - 1) * abs(h!1)")
(("1"
(mult-by
-1
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(assert )
nil
nil ))
nil )
("2"
(hide
2)
(("2"
(name
"MAX"
"max(abs(x!1), abs(xp!1))" )
(("2"
(replace
-1)
(("2"
(lemma
"mean_value_abs[T]" )
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(inst
-
"TK"
"x!1"
"(LAMBDA (x:T): x^(n!1-1))" )
(("2"
(lemma
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"(n!1-1)"
"1" )
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(("1"
(hide
-1)
(("1"
(assert )
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(case-replace
"TK = x!1" )
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(assert )
nil
nil ))
nil ))
nil )
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(assert )
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(skosimp*)
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(case-replace
"deriv((LAMBDA (x: T): x ^ (n!1 - 1)), c!1) =
(n!1-1)*c!1^(n!1-2)")
(("1"
(hide
-1
-2
-3)
(("1"
(lemma
"abs_neg" )
(("1"
(inst
-
"x!1 ^ (n!1 - 1) - TK ^ (n!1 - 1)" )
(("1"
(replace
-1
-
rl)
(("1"
(replace
-4
+
rl)
(("1"
(rewrite
"abs_mult" )
(("1"
(name-replace
"NM1"
"abs((n!1 - 1))" )
(("1"
(hide
-1
-4
-8
-9)
(("1"
(case-replace
"abs(c!1 ^ (n!1 - 2)) <= abs(MAX ^ (n!1 - 2))" )
(("1"
(case-replace
"abs(x!1 - TK) <= abs(h!1)" )
(("1"
(mult-ineq
-1
-2)
(("1"
(mult-by
-1
"NM1" )
(("1"
(assert )
nil
nil ))
nil )
("2"
(prop)
(("1"
(hide-all-but
(-1
1))
(("1"
(grind)
nil
nil ))
nil )
("2"
(hide-all-but
(-2
1))
(("2"
(grind)
nil
nil ))
nil ))
nil ))
nil )
("2"
(hide
-1
-2
-3
3)
(("2"
(reveal
-16)
(("2"
(typepred
"GET_tk(x!1, h!1, n!1)" )
(("2"
(grind)
--> --------------------
--> maximum size reached
--> --------------------
Messung V0.5 in Prozent C=100 H=99 G=99
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