(power_series_derivseq
(IMP_power_series_conv_TCC1 0
(IMP_power_series_conv_TCC1-1 nil 3297687390
("" (lemma "not_one_element" ) (("" (propax) nil nil )) nil )
((not_one_element formula-decl nil power_series_derivseq nil ))
shostak))
(IMP_power_series_conv_TCC2 0
(IMP_power_series_conv_TCC2-1 nil 3297687390
("" (lemma "connected_domain" )
(("" (lemma "connected_deriv_domain[T]" )
(("" (lemma "not_one_element" ) (("" (assert ) nil nil )) nil ))
nil ))
nil )
((T formal-subtype-decl nil power_series_derivseq nil )
(T_pred const-decl "[real -> boolean]" power_series_derivseq nil )
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(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 )
(connected_deriv_domain formula-decl nil deriv_domain_def
"analysis/" )
(not_one_element formula-decl nil power_series_derivseq nil )
(connected_domain formula-decl nil power_series_derivseq nil ))
shostak))
(IMP_power_series_conv_TCC3 0
(IMP_power_series_conv_TCC3-1 nil 3297687390
("" (lemma "open" ) (("" (propax) nil nil )) nil )
((open formula-decl nil power_series_derivseq nil )) shostak))
(deriv_powerseq_TCC1 0
(deriv_powerseq_TCC1-1 nil 3279280814 ("" (subtype-tcc) nil nil )
((boolean nonempty-type-decl nil booleans nil )
(bool nonempty-type-eq-decl nil booleans nil )
(NOT const-decl "[bool -> bool]" booleans nil )
(number nonempty-type-decl nil numbers nil )
(number_field_pred const-decl "[number -> boolean]" number_fields
nil )
(number_field nonempty-type-from-decl nil number_fields nil )
(real_pred const-decl "[number_field -> boolean]" reals nil )
(real nonempty-type-from-decl nil reals nil )
(T_pred const-decl "[real -> boolean]" power_series_derivseq nil )
(T formal-subtype-decl nil power_series_derivseq nil )
(>= const-decl "bool" reals nil )
(rational_pred const-decl "[real -> boolean]" rationals nil )
(rational nonempty-type-from-decl nil rationals nil )
(integer_pred const-decl "[rational -> boolean]" integers nil )
(int nonempty-type-eq-decl nil integers nil )
(nat nonempty-type-eq-decl nil naturalnumbers nil )
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil )
(int_minus_int_is_int application-judgement "int" integers nil )
(/= const-decl "boolean" notequal nil ))
nil ))
(deriv_powerseries_conv 0
(deriv_powerseries_conv-6 nil 3320063098
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(("1" (rewrite "zero_series_conv" ) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
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(("2" (expand "deriv_powerseq" )
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(< const-decl "bool" reals nil ) (> const-decl "bool" reals nil )
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real_defs nil )
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real_defs nil )
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(nnreal_div_posreal_is_nnreal application-judgement "nnreal"
real_types nil )
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nil )
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nil )
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nil
nil )
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nil
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*
rl)
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nil ))
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*
rl)
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nil
nil ))
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3)
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1)
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-6
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*
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nil
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nil )
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("2"
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1
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(lemma
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(inst?)
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(assert )
nil
nil ))
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nil ))
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nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(hide -1 -2 -5 3)
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(case
"EXISTS (NBIG: posnat): NBIG > abs(x!1)/(y!1 - abs(x!1))" )
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(skosimp*)
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(inst + "NBIG!1" )
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1
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(("1"
(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(hide 2)
(("2"
(inst
+
"ceiling(abs(x!1) / (y!1 - abs(x!1)))+1" )
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nil
nil )
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(assert )
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(case-replace
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(assert )
nil
nil )
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(hide 2)
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(cross-mult
1)
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(assert )
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(lemma
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(inst?)
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(assert )
nil
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nil ))
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nil ))
nil ))
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(replace -1 * rl)
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nil
nil ))
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(inst
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"1" )
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nil
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(inst - "n!1" )
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(lemma
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(inst
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"x!1" )
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(assert )
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(case-replace
"abs(a!1(n!1) * x!1 ^ (n!1 - 1) * n!1) =
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-1)
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(expand
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-1
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-1
"n!1" )
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nil
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4)
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1
3)
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(("1"
(rewrite
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1)
nil
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(hide
2)
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(case-replace
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nil
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(hide-all-but
1)
(("2"
(grind)
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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"
(skosimp*)
(("2" (assert ) nil nil ))
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nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
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nil ))
nil )
("2" (hide -1 3)
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nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
((powerseries_bounded formula-decl nil power_series nil )
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(powerseq const-decl "sequence[real]" power_series nil )
(tail_series_conv2 formula-decl nil series nil )
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(zero_hat formula-decl nil exponent_props "reals/" ))
nil )
(deriv_powerseries_conv-4 nil 3297690436
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(("1"
(case-replace "(LAMBDA n:
2 * (a!1(2 + n) * 0 ^ (1 + n)) +
n * a!1(2 + n) * 0 ^ (1 + n)) = (LAMBDA n: 0)")
(("1" (rewrite "zero_series_conv" ) nil nil )
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(assert )
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(inst - "x!1" )
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(hide 4)
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(inst?)
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(replace -1)
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(hide -1)
(("1"
(case-replace
"(LAMBDA n:
x!1 *
IF n = 0 THEN 0
ELSE n * a!1(n) * x!1 ^ (n - 1)
ENDIF) = (LAMBDA n:
IF n = 0 THEN 0
ELSE n * a!1(n) * x!1 ^ n
ENDIF)")
(("1"
(hide -1)
(("1"
(lemma
"comparison_test_gen" )
(("1"
(inst
-1
"(LAMBDA (k: nat):
IF k = 0
THEN 0
ELSE k * a!1(k) * x!1 ^ k
ENDIF)"
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(inst?)
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(inst
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"r!1/abs(x!1) - 1" )
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"n!1" )
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1)
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(expand
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-2
-6)
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(case
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-2)
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-1)
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(case-replace
"a!1(n!2) = 0" )
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nil )
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-5
3)
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(inst?)
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(lemma
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(inst?)
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(rewrite
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(inst?)
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(inst
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"n!2" )
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"^" )
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nil ))
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nil ))
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nil )
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(lemma
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(inst?)
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nil
nil ))
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nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
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-5
3)
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(inst
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"n2n" )
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nil )
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nil ))
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nil )
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(lemma
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(expand
"increasing?" )
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(inst
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"exp((1 / n!2) * ln(n!2))"
"1" )
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(rewrite
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(inst
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"1"
"n!2" )
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(lemma
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(assert )
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(lemma
"ln_bij" )
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(expand
"bijective?" )
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(flatten)
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(expand
"injective?" )
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(inst
-
"1"
"n!2" )
(("2"
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nil
nil ))
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nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(case
"r!1 / abs(x!1) > 1" )
(("1"
(assert )
nil
nil )
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(hide
2
3)
(("2"
(cross-mult
1)
(("2"
(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(hide 2)
(("2"
(apply-extensionality
1
:hide?
t)
(("2"
(lift-if)
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(ground)
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(expand "^" )
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(expand
expt
2
2)
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(assert )
nil
nil ))
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nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (hide 4)
(("2" (lemma "open" )
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(("2"
(case "x!1 >= 0" )
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(inst - "x!1" )
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(skosimp*)
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(inst + "x!1 + delta!1/2" )
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nil )
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(inst - "x!1+delta!1/2" )
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(assert )
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(("2" (assert ) nil nil ))
nil ))
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nil ))
nil ))
nil ))
nil )
("2"
(inst - "-x!1" )
(("1"
(skosimp*)
(("1"
(inst + "-x!1+delta!1/2" )
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(expand "abs" )
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nil )
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(inst - "-x!1+delta!1/2" )
(("2"
(assert )
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(expand "abs" )
(("2" (assert ) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(lemma "ball" )
(("2"
(inst?)
(("2" (assert ) nil nil ))
nil ))
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nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
((zero_series_conv formula-decl nil series nil )
(tail_series_conv2 formula-decl nil series nil )
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((powerseq const-decl "sequence[real]" power_series nil )
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(deriv_series_shift_TCC1 0
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nil ))
nil )
((deriv_powerseries_conv formula-decl nil power_series_derivseq nil )
(deriv_powerseq const-decl "sequence[real]" power_series_derivseq
nil )
(T formal-subtype-decl nil power_series_derivseq nil )
(T_pred const-decl "[real -> boolean]" power_series_derivseq nil )
(sequence type-eq-decl nil sequences nil )
(nat nonempty-type-eq-decl nil naturalnumbers nil )
(>= const-decl "bool" reals nil )
(bool nonempty-type-eq-decl nil booleans nil )
(int nonempty-type-eq-decl nil integers nil )
(integer_pred const-decl "[rational -> boolean]" integers nil )
(rational nonempty-type-from-decl nil rationals nil )
(rational_pred const-decl "[real -> boolean]" rationals 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 ))
nil ))
(deriv_series_shift_TCC2 0
(deriv_series_shift_TCC2-1 nil 3279280814
("" (skosimp*) (("" (rewrite "derivseq_conv" ) nil nil )) nil )
((derivseq_conv formula-decl nil power_series_derivseq nil )
(number nonempty-type-decl nil numbers nil )
(boolean nonempty-type-decl nil booleans nil )
(number_field_pred const-decl "[number -> boolean]" number_fields
nil )
(number_field nonempty-type-from-decl nil number_fields nil )
(real_pred const-decl "[number_field -> boolean]" reals nil )
(real nonempty-type-from-decl nil reals nil )
(rational_pred const-decl "[real -> boolean]" rationals nil )
(rational nonempty-type-from-decl nil rationals nil )
(integer_pred const-decl "[rational -> boolean]" integers nil )
(int nonempty-type-eq-decl nil integers nil )
(bool nonempty-type-eq-decl nil booleans nil )
(>= const-decl "bool" reals nil )
(nat nonempty-type-eq-decl nil naturalnumbers nil )
(sequence type-eq-decl nil sequences nil )
(T_pred const-decl "[real -> boolean]" power_series_derivseq nil )
(T formal-subtype-decl nil power_series_derivseq nil ))
nil ))
(deriv_series_shift 0
(deriv_series_shift-3 "we" 3403525340
("" (skosimp*)
(("" (expand "powerseq" )
(("" (expand "derivseq" )
(("" (expand "deriv_powerseq" )
(("" (lemma "limit_series_shift" )
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(("1" (split -1)
(("1" (rewrite "sigma_rew" )
(("1" (assert )
(("1"
(case-replace "(LAMBDA (k: nat):
IF k = 0 THEN 1 * a!1(1)
ELSE a!1(1 + k) * x!1 ^ k + k * a!1(1 + k) * x!1 ^ k
ENDIF) = ((LAMBDA n:
a!1(1 + n) * x!1 ^ n + n * a!1(1 + n) * x!1 ^ n))")
(("1" (assert )
(("1" (hide 2)
(("1" (apply-extensionality 1 :hide? t)
(("1"
(lift-if)
(("1"
(ground)
(("1"
(hide -2 -3)
(("1"
(replace -1)
(("1" (grind) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (hide 2)
(("2" (assert )
(("2" (lemma "deriv_powerseries_conv" )
(("2" (skosimp*) (("2" (assert ) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (hide 2)
(("2" (assert )
(("2" (lemma "deriv_powerseries_conv" )
(("2" (inst?)
(("2" (replace -2)
(("2" (assert )
(("2"
(inst - "x!1" )
(("2"
(expand "deriv_powerseq" )
(("2" (propax) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (skosimp*) (("2" (assert ) nil nil )) nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
((powerseq const-decl "sequence[real]" power_series nil )
(deriv_powerseq const-decl "sequence[real]" power_series_derivseq
nil )
(int_minus_int_is_int application-judgement "int" integers nil )
(- const-decl "[numfield, numfield -> numfield]" number_fields nil )
(numfield nonempty-type-eq-decl nil number_fields nil )
(x!1 skolem-const-decl "T" power_series_derivseq nil )
(T formal-subtype-decl nil power_series_derivseq nil )
(T_pred const-decl "[real -> boolean]" power_series_derivseq nil )
(/= const-decl "boolean" notequal nil )
(OR const-decl "[bool, bool -> bool]" booleans nil )
(= const-decl "[T, T -> boolean]" equalities nil )
(NOT const-decl "[bool -> bool]" booleans nil )
(AND const-decl "[bool, bool -> bool]" booleans nil )
(IMPLIES const-decl "[bool, bool -> bool]" booleans nil )
(nat nonempty-type-eq-decl nil naturalnumbers nil )
(>= const-decl "bool" reals nil )
(bool nonempty-type-eq-decl nil booleans nil )
(int nonempty-type-eq-decl nil integers nil )
(integer_pred const-decl "[rational -> boolean]" integers nil )
(rational nonempty-type-from-decl nil rationals nil )
(rational_pred const-decl "[real -> boolean]" rationals 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 )
(sequence type-eq-decl nil sequences nil )
(IF const-decl "[boolean, T, T -> T]" if_def nil )
(* const-decl "[numfield, numfield -> numfield]" number_fields nil )
(^ const-decl "real" exponentiation nil )
(deriv_powerseries_conv formula-decl nil power_series_derivseq nil )
(odd_plus_even_is_odd application-judgement "odd_int" integers nil )
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil )
(expt def-decl "real" exponentiation nil )
(posint_plus_nnint_is_posint application-judgement "posint"
integers nil )
(+ const-decl "[numfield, numfield -> numfield]" number_fields nil )
(real_gt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil )
(nnint_plus_posint_is_posint application-judgement "posint"
integers nil )
(real_le_is_total_order name-judgement "(total_order?[real])"
real_props nil )
(real_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil )
(real_plus_real_is_real application-judgement "real" reals nil )
(odd_minus_odd_is_even application-judgement "even_int" integers
nil )
(sigma_rew formula-decl nil sigma "reals/" )
(<= const-decl "bool" reals nil )
(T_high type-eq-decl nil sigma "reals/" )
(T_low type-eq-decl nil sigma "reals/" )
(nonneg_int nonempty-type-eq-decl nil integers nil )
(> const-decl "bool" reals nil )
(posnat nonempty-type-eq-decl nil integers nil )
(limit_series_shift formula-decl nil series nil )
(derivseq const-decl "sequence[real]" power_series_derivseq nil )
(real_times_real_is_real application-judgement "real" reals nil ))
nil )
(deriv_series_shift-2 "we" 3297672214
("" (skosimp*)
(("" (expand "powerseq" )
(("" (expand "derivseq" )
(("" (expand "deriv_powerseq" )
(("" (lemma "limit_series_shift" )
(("" (inst?)
(("1" (inst -1 "1" )
(("1" (split -1)
(("1" (expand "sigma" )
(("1"
(case-replace "(LAMBDA (k: nat):
IF k = 0 THEN 1 * a!1(1)
ELSE a!1(1 + k) * x!1 ^ k + k * a!1(1 + k) * x!1 ^ k
ENDIF) = ((LAMBDA n:
a!1(1 + n) * x!1 ^ n + n * a!1(1 + n) * x!1 ^ n))")
(("1" (assert )
(("1" (hide 2)
(("1" (apply-extensionality 1 :hide? t)
(("1" (lift-if)
(("1"
(ground)
(("1"
(hide -2 -3)
(("1"
(replace -1)
(("1" (grind) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (hide 2)
(("2" (assert )
(("2" (lemma "deriv_powerseries_conv" )
(("2" (inst?)
(("2" (replace -2)
(("2" (assert )
(("2"
(inst - "x!1" )
(("2"
(expand "deriv_powerseq" )
(("2" (propax) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (skosimp*) (("2" (assert ) nil nil )) nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
((powerseq const-decl "sequence[real]" power_series nil )
(sigma def-decl "real" sigma "reals/" )
(limit_series_shift formula-decl nil series nil ))
nil )
(as "we" 3297671925
("" (skosimp*)
(("" (expand "powerseq" )
(("" (expand "derivseq" )
(("" (expand "deriv_powerseq" )
(("" (lemma "limit_series_shift" )
(("" (inst?)
(("1" (inst -1 "1" )
(("1" (split -1)
(("1" (expand "sigma" ) (("1" (propax) nil nil )) nil )
("2" (assert )
(("2" (hide 2)
(("2" (lemma "deriv_powerseries_conv" )
(("2" (inst?)
(("2" (replace -2)
(("2" (inst?)
(("1"
(expand "deriv_powerseq" )
(("1"
(assert )
(("1" (postpone) nil nil ))
nil ))
nil )
("2" (postpone) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (postpone) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
((limit_series_shift formula-decl nil series nil )
(sigma def-decl "real" sigma "reals/" )
(powerseq const-decl "sequence[real]" power_series nil ))
shostak)
(deriv_series_shift-1 nil 3279280814
("" (skosimp*)
(("" (expand "powerseq" )
(("" (expand "derivseq" )
(("" (expand "deriv_powerseq" )
(("" (lemma "limit_series_shift" )
(("" (inst?)
(("1" (inst -1 "1" )
(("1" (split -1)
(("1" (expand "sigma" ) (("1" (propax) nil nil )) nil )
("2" (assert )
(("2" (hide 2)
(("2" (lemma "deriv_powerseries_conv" )
(("2" (inst?)
(("2" (replace -2)
(("2" (assert ) (("2" (inst?) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (hide 2)
(("2" (skosimp*) (("2" (assert ) nil nil )) nil )) nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
((powerseq const-decl "sequence[real]" power_series nil )
(sigma def-decl "real" sigma "reals/" )
(limit_series_shift formula-decl nil series nil ))
nil ))
(conv_derivseq 0
(conv_derivseq-1 nil 3298199522
("" (skosimp*)
(("" (expand "conv_powerseries?" )
(("" (skosimp*)
(("" (lemma "derivseq_conv" )
(("" (inst?)
(("" (expand "conv_powerseries?" )
(("" (inst?)
(("" (replace -2)
(("" (expand "powerseries" ) (("" (propax) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
((conv_powerseries? const-decl "bool" power_series_conv nil )
(derivseq_conv formula-decl nil power_series_derivseq nil )
(powerseries const-decl "sequence[real]" power_series nil )
(T_pred const-decl "[real -> boolean]" power_series_derivseq nil )
(T formal-subtype-decl nil power_series_derivseq nil )
(sequence type-eq-decl nil sequences nil )
(nat nonempty-type-eq-decl nil naturalnumbers nil )
(>= const-decl "bool" reals nil )
(bool nonempty-type-eq-decl nil booleans nil )
(int nonempty-type-eq-decl nil integers nil )
(integer_pred const-decl "[rational -> boolean]" integers nil )
(rational nonempty-type-from-decl nil rationals nil )
(rational_pred const-decl "[real -> boolean]" rationals 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 ))
shostak))
(deriv_powerseq_lem 0
(deriv_powerseq_lem-1 nil 3279280814
("" (skosimp*)
(("" (expand "powerseq" )
(("" (case-replace "n!1 = 0" )
(("1" (expand "deriv_powerseq" )
(("1" (split +)
(("1" (lemma "const_derivable_fun[T]" )
(("1" (expand "const_fun" )
(("1" (assert ) (("1" (inst?) nil nil )) nil )) nil ))
nil )
("2" (assert )
(("2" (lemma "deriv_const_fun[T]" )
(("2" (expand "const_fun" ) (("2" (inst?) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (lemma "deriv_exp_fun[T]" )
(("2" (inst -1 "(LAMBDA x: x)" "n!1" )
(("2" (assert )
(("2" (split -1)
(("1" (flatten)
(("1" (split)
(("1" (expand "^" -1)
(("1" (hide -2)
(("1" (lemma "scal_derivable_fun[T]" )
(("1"
(inst -1 "a!1(n!1)" "(LAMBDA t: t^n!1)" )
(("1" (expand "*" )
(("1"
(assert )
(("1"
(lemma "deriv_scal_fun[T]" )
(("1"
(inst
-1
"a!1(n!1)"
"(LAMBDA x: x^n!1)" )
(("1"
(expand "*" -1)
(("1"
(hide -1)
(("1"
(expand "^" -2)
(("1"
(hide -2)
(("1"
(case-replace
"deriv((LAMBDA x: x)) = const_fun(1)" )
(("1"
(expand "restrict" )
(("1"
(propax)
nil
nil ))
nil )
("2"
(expand "restrict" )
(("2"
(propax)
nil
nil ))
nil )
("3"
(expand "restrict" )
(("3"
(propax)
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (expand "deriv_powerseq" )
(("2" (lemma "deriv_scal_fun[T]" )
(("2" (inst -1 "a!1(n!1)" "(LAMBDA x: x^n!1)" )
(("1" (expand "*" -1)
(("1" (replace -1)
(("1"
(hide -1)
(("1"
(assert )
(("1"
(expand "^" -2)
(("1"
(replace -2)
(("1"
(hide -2)
(("1"
(expand "*" )
(("1"
(case-replace
"deriv((LAMBDA x: x)) = const_fun(1)" )
(("1"
(expand "const_fun" )
(("1"
(hide -1 -2)
(("1"
(ground)
(("1"
(apply-extensionality
1
:hide?
t)
(("1"
(lift-if)
(("1"
(ground)
(("1"
(replace -1)
(("1"
(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(hide -1 2 3)
(("2"
(lemma
"deriv_id_fun[T]" )
(("2"
(expand "const_fun" )
(("2"
(propax)
nil
nil ))
nil ))
nil ))
nil )
("3"
(hide 2)
(("3"
(lemma
"identity_derivable_fun[T]" )
(("3"
(expand "I" )
(("3"
(propax)
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (hide 2)
(("2" (lemma "identity_derivable_fun[T]" )
(("2"
(expand "I" )
(("2"
(expand "^" -2)
(("2" (propax) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (hide 3)
(("2" (lemma "identity_derivable_fun[T]" )
(("2" (expand "I" ) (("2" (propax) nil nil )) nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
((real_times_real_is_real application-judgement "real" reals nil )
(powerseq const-decl "sequence[real]" power_series nil )
(deriv_exp_fun formula-decl nil derivatives "analysis/" )
(real_gt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil )
(n!1 skolem-const-decl "nat" power_series_derivseq nil )
(int_minus_int_is_int application-judgement "int" integers nil )
(* const-decl "[T -> real]" real_fun_ops "reals/" )
(I const-decl "(bijective?[T, T])" identity nil )
(identity_derivable_fun formula-decl nil derivatives "analysis/" )
(deriv_id_fun formula-decl nil derivatives "analysis/" )
(odd_minus_odd_is_even application-judgement "even_int" integers
nil )
(IF const-decl "[boolean, T, T -> T]" if_def nil )
(- const-decl "[numfield, numfield -> numfield]" number_fields nil )
(* const-decl "[numfield, numfield -> numfield]" number_fields nil )
(numfield nonempty-type-eq-decl nil number_fields nil )
(^ const-decl "[T -> real]" real_fun_ops "reals/" )
(scal_derivable_fun formula-decl nil derivatives "analysis/" )
(* const-decl "[T -> real]" real_fun_ops "reals/" )
(deriv_scal_fun formula-decl nil derivatives "analysis/" )
(const_fun const-decl "[T -> real]" real_fun_ops "reals/" )
(deriv const-decl "[T -> real]" derivatives "analysis/" )
(constant_seq2 application-judgement "(convergent_nz?)"
convergence_ops "analysis/" )
(const_fun_continuous application-judgement "continuous_fun"
continuous_functions "analysis/" )
(derivable_const application-judgement "deriv_fun" derivatives
"analysis/" )
(derivable? const-decl "bool" derivatives "analysis/" )
(deriv_fun type-eq-decl nil derivatives "analysis/" )
(restrict const-decl "R" restrict nil )
(OR const-decl "[bool, bool -> bool]" booleans nil )
(/= const-decl "boolean" notequal nil )
(^ const-decl "real" exponentiation nil )
(deriv_powerseq const-decl "sequence[real]" power_series_derivseq
nil )
(deriv_const_fun formula-decl nil derivatives "analysis/" )
(const_derivable_fun formula-decl nil derivatives "analysis/" )
(T_pred const-decl "[real -> boolean]" power_series_derivseq nil )
(T formal-subtype-decl nil power_series_derivseq nil )
(sequence type-eq-decl nil sequences nil )
(expt_x0 formula-decl nil exponentiation nil )
(nat nonempty-type-eq-decl nil naturalnumbers nil )
(>= const-decl "bool" reals nil )
(bool nonempty-type-eq-decl nil booleans nil )
(int nonempty-type-eq-decl nil integers nil )
(integer_pred const-decl "[rational -> boolean]" integers nil )
(rational nonempty-type-from-decl nil rationals nil )
(rational_pred const-decl "[real -> boolean]" rationals 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 )
(= const-decl "[T, T -> boolean]" equalities nil )
(boolean nonempty-type-decl nil booleans nil )
(number nonempty-type-decl nil numbers nil ))
nil ))
(sigma_power_derivable_TCC1 0
(sigma_power_derivable_TCC1-1 nil 3403524860
("" (subtype-tcc) nil nil ) nil nil ))
(sigma_power_derivable 0
(sigma_power_derivable-3 nil 3403525533
("" (induct "n" )
(("1" (skosimp*)
(("1" (expand "sigma" )
(("1" (expand "sigma" )
(("1" (expand "powerseq" )
(("1" (expand "^" )
(("1" (expand "expt" )
(("1" (lemma "const_derivable_fun[T]" )
(("1" (inst -1 "a!1(0)" )
(("1" (expand "const_fun" ) (("1" (propax) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (skosimp*)
(("2" (expand "sigma" 1)
(("2" (lemma "sum_derivable_fun[T]" )
(("2"
(inst -1
"(LAMBDA x: powerseq(a!1, x)(1 + j!1))"
"(LAMBDA x: sigma(0, j!1, powerseq(a!1, x)))" )
(("2" (expand "+" -1)
(("2" (assert )
(("2" (hide 2)
(("2" (inst?)
(("2" (assert )
(("2" (hide -1)
(("2" (lemma "deriv_powerseq_lem" )
(("2" (inst?) (("2" (flatten) nil nil )) nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
((sum_derivable_fun formula-decl nil derivatives "analysis/" )
(+ const-decl "[T -> real]" real_fun_ops "reals/" )
(deriv_powerseq_lem formula-decl nil power_series_derivseq nil )
(numfield nonempty-type-eq-decl nil number_fields nil )
(+ const-decl "[numfield, numfield -> numfield]" number_fields nil )
(posint_plus_nnint_is_posint application-judgement "posint"
integers nil )
(^ const-decl "real" exponentiation nil )
(const_derivable_fun formula-decl nil derivatives "analysis/" )
(expt def-decl "real" exponentiation nil )
(real_times_real_is_real application-judgement "real" reals nil )
(nat_induction formula-decl nil naturalnumbers nil )
(powerseq const-decl "sequence[real]" power_series nil )
(sigma def-decl "real" sigma "reals/" )
(T_high type-eq-decl nil sigma "reals/" )
(T_low type-eq-decl nil sigma "reals/" )
(<= const-decl "bool" reals nil )
(OR const-decl "[bool, bool -> bool]" booleans nil )
(derivable? const-decl "bool" derivatives "analysis/" )
(T formal-subtype-decl nil power_series_derivseq nil )
(T_pred const-decl "[real -> boolean]" power_series_derivseq nil )
(sequence type-eq-decl nil sequences nil )
(pred type-eq-decl nil defined_types nil )
(nat nonempty-type-eq-decl nil naturalnumbers nil )
(>= const-decl "bool" reals nil )
(bool nonempty-type-eq-decl nil booleans nil )
(int nonempty-type-eq-decl nil integers nil )
(integer_pred const-decl "[rational -> boolean]" integers nil )
(rational nonempty-type-from-decl nil rationals nil )
(rational_pred const-decl "[real -> boolean]" rationals 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 ))
nil )
(sigma_power_derivable-2 nil 3297176600
("" (induct "n" )
(("1" (skosimp*)
(("1" (expand "sigma" )
(("1" (expand "powerseq" )
(("1" (expand "^" )
(("1" (expand "expt" )
(("1" (lemma "const_derivable_fun[T]" )
(("1" (inst -1 "a!1(0)" )
(("1" (expand "const_fun" ) (("1" (propax) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (skosimp*)
(("2" (expand "sigma" 1)
(("2" (lemma "sum_derivable_fun[T]" )
(("2"
(inst -1
"(LAMBDA x: powerseq(a!1, x)(1 + j!1))"
"(LAMBDA x: sigma(0, j!1, powerseq(a!1, x)))" )
(("2" (expand "+" -1)
(("2" (assert )
(("2" (hide 2)
(("2" (inst?)
(("2" (assert )
(("2" (hide -1)
(("2" (lemma "deriv_powerseq_lem" )
(("2" (inst?) (("2" (flatten) nil nil )) nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
((sum_derivable_fun formula-decl nil derivatives "analysis/" )
(+ const-decl "[T -> real]" real_fun_ops "reals/" )
(const_fun const-decl "[T -> real]" real_fun_ops "reals/" )
(const_derivable_fun formula-decl nil derivatives "analysis/" )
(powerseq const-decl "sequence[real]" power_series nil )
(sigma def-decl "real" sigma "reals/" ))
nil )
(sigma_power_derivable-1 nil 3279280814
("" (induct "n" )
(("1" (skosimp*)
(("1" (expand "sigma" )
(("1" (expand "powerseq" )
(("1" (expand "^" )
(("1" (expand "expt" )
(("1" (lemma "const_derivable_fun[real]" )
(("1" (inst -1 "a!1(0)" )
(("1" (expand "const_fun" ) (("1" (propax) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (skosimp*)
(("2" (expand "sigma" 1)
(("2" (lemma "sum_derivable_fun[real]" )
(("2"
(inst -1
"(LAMBDA x: powerseq(a!1, x)(1 + j!1))"
"(LAMBDA x: sigma(0, j!1, powerseq(a!1, x)))" )
(("2" (expand "+" -1)
(("2" (assert )
(("2" (hide 2)
(("2" (inst?)
(("2" (assert )
(("2" (hide -1)
(("2" (lemma "deriv_powerseq_lem" )
(("2" (inst?) (("2" (flatten) nil nil )) nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
((sum_derivable_fun formula-decl nil derivatives "analysis/" )
(+ const-decl "[T -> real]" real_fun_ops "reals/" )
(const_fun const-decl "[T -> real]" real_fun_ops "reals/" )
(const_derivable_fun formula-decl nil derivatives "analysis/" )
(powerseq const-decl "sequence[real]" power_series nil )
(sigma def-decl "real" sigma "reals/" ))
nil ))
(deriv_sigma_power_TCC1 0
(deriv_sigma_power_TCC1-1 nil 3279280814
("" (skosimp*)
(("" (lemma "sigma_power_derivable" ) (("" (inst?) nil nil )) nil ))
nil )
((sigma_power_derivable formula-decl nil power_series_derivseq nil )
(number nonempty-type-decl nil numbers nil )
(boolean nonempty-type-decl nil booleans nil )
(number_field_pred const-decl "[number -> boolean]" number_fields
nil )
(number_field nonempty-type-from-decl nil number_fields nil )
(real_pred const-decl "[number_field -> boolean]" reals nil )
(real nonempty-type-from-decl nil reals nil )
(rational_pred const-decl "[real -> boolean]" rationals nil )
(rational nonempty-type-from-decl nil rationals nil )
(integer_pred const-decl "[rational -> boolean]" integers nil )
(int nonempty-type-eq-decl nil integers nil )
(bool nonempty-type-eq-decl nil booleans nil )
(>= const-decl "bool" reals nil )
(nat nonempty-type-eq-decl nil naturalnumbers nil )
(sequence type-eq-decl nil sequences nil ))
nil ))
(deriv_sigma_power 0
(deriv_sigma_power-2 nil 3297176628
("" (induct "n" 1)
(("1" (skosimp*)
(("1" (expand "sigma" )
(("1" (expand "sigma" )
(("1" (expand "powerseq" )
(("1" (expand "deriv_powerseq" )
(("1" (expand "^" )
(("1" (expand "expt" )
(("1" (lemma "deriv_const_fun[T]" )
(("1" (expand "const_fun" ) (("1" (inst?) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (skosimp*)
(("2" (expand "sigma" 1)
(("2" (lemma "deriv_sum_fun[T]" )
(("2"
(inst -1
"(LAMBDA x: powerseq(a!1, x)(1 + j!1))"
"(LAMBDA x: sigma(0, j!1, powerseq(a!1, x)))" )
(("1" (expand "+" -1)
(("1" (replace -1)
(("1" (hide -1)
(("1" (apply-extensionality 1 :hide? t)
(("1" (inst?)
(("1" (replace -1)
(("1" (hide -1)
(("1" (lemma "deriv_powerseq_lem" )
(("1" (inst?)
(("1"
(flatten)
(("1"
(replace -2)
(("1"
(hide -2)
(("1"
(expand "deriv_powerseq" )
(("1" (propax) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (hide -1)
(("2" (skosimp*)
(("2" (rewrite "sigma_power_derivable" ) nil
nil ))
nil ))
nil )
("3" (skosimp*)
(("3" (hide -1)
(("3" (lemma "deriv_powerseq_lem" )
(("3" (inst?) (("3" (flatten) nil nil )) nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (hide -1 2)
(("2" (rewrite "sigma_power_derivable" ) nil nil )) nil )
("3" (hide -1 2)
(("3" (lemma "deriv_powerseq_lem" )
(("3" (inst?) (("3" (flatten) nil nil )) nil )) nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("3" (hide 2)
(("3" (skosimp*)
(("3" (rewrite "sigma_power_derivable" ) nil nil )) nil ))
nil ))
nil )
((deriv_sum_fun formula-decl nil derivatives "analysis/" )
(+ const-decl "[T -> real]" real_fun_ops "reals/" )
(sigma_power_derivable formula-decl nil power_series_derivseq nil )
(real_times_real_is_real application-judgement "real" reals nil )
(deriv_powerseq_lem formula-decl nil power_series_derivseq nil )
(real_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil )
(real_le_is_total_order name-judgement "(total_order?[real])"
real_props nil )
(real_plus_real_is_real application-judgement "real" reals nil )
(j!1 skolem-const-decl "nat" power_series_derivseq nil )
(+ const-decl "[numfield, numfield -> numfield]" number_fields nil )
(numfield nonempty-type-eq-decl nil number_fields nil )
(a!1 skolem-const-decl "sequence[real]" power_series_derivseq nil )
(posint_plus_nnint_is_posint application-judgement "posint"
integers nil )
(expt def-decl "real" exponentiation nil )
(deriv_const_fun formula-decl nil derivatives "analysis/" )
(^ const-decl "real" exponentiation nil )
(nat_induction formula-decl nil naturalnumbers nil )
(deriv_powerseq const-decl "sequence[real]" power_series_derivseq
nil )
(deriv const-decl "[T -> real]" derivatives "analysis/" )
(deriv_fun type-eq-decl nil derivatives "analysis/" )
(= const-decl "[T, T -> boolean]" equalities nil )
(pred type-eq-decl nil defined_types nil )
(number nonempty-type-decl nil numbers nil )
(boolean nonempty-type-decl nil booleans nil )
(number_field_pred const-decl "[number -> boolean]" number_fields
nil )
(number_field nonempty-type-from-decl nil number_fields nil )
(real_pred const-decl "[number_field -> boolean]" reals nil )
(real nonempty-type-from-decl nil reals nil )
(rational_pred const-decl "[real -> boolean]" rationals nil )
(rational nonempty-type-from-decl nil rationals nil )
(integer_pred const-decl "[rational -> boolean]" integers nil )
(int nonempty-type-eq-decl nil integers nil )
(bool nonempty-type-eq-decl nil booleans nil )
(>= const-decl "bool" reals nil )
(nat nonempty-type-eq-decl nil naturalnumbers nil )
(sequence type-eq-decl nil sequences nil )
(T_pred const-decl "[real -> boolean]" power_series_derivseq nil )
(T formal-subtype-decl nil power_series_derivseq nil )
(derivable? const-decl "bool" derivatives "analysis/" )
(OR const-decl "[bool, bool -> bool]" booleans nil )
(<= const-decl "bool" reals nil )
(T_low type-eq-decl nil sigma "reals/" )
(T_high type-eq-decl nil sigma "reals/" )
(sigma def-decl "real" sigma "reals/" )
(powerseq const-decl "sequence[real]" power_series nil ))
nil )
(deriv_sigma_power-1 nil 3279280814
("" (induct "n" 1)
(("1" (skosimp*)
(("1" (expand "sigma" )
(("1" (expand "powerseq" )
(("1" (expand "deriv_powerseq" )
(("1" (expand "^" )
(("1" (expand "expt" )
(("1" (lemma "deriv_const_fun[real]" )
(("1" (expand "const_fun" ) (("1" (inst?) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (skosimp*)
(("2" (expand "sigma" 1)
(("2" (lemma "deriv_sum_fun[real]" )
(("2"
(inst -1
"(LAMBDA x: powerseq(a!1, x)(1 + j!1))"
"(LAMBDA x: sigma(0, j!1, powerseq(a!1, x)))" )
(("1" (expand "+" -1)
(("1" (replace -1)
(("1" (hide -1)
(("1" (apply-extensionality 1 :hide? t)
(("1" (inst?)
(("1" (replace -1)
(("1" (hide -1)
(("1" (lemma "deriv_powerseq_lem" )
(("1" (inst?)
(("1"
(flatten)
(("1"
(replace -2)
(("1"
(hide -2)
(("1"
(expand "deriv_powerseq" )
(("1" (propax) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (hide -1)
(("2" (rewrite "sigma_power_derivable" ) nil nil ))
nil )
("3" (lemma "deriv_powerseq_lem" )
(("3" (inst?) (("3" (flatten) nil nil )) nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (hide -1 2)
(("2" (rewrite "sigma_power_derivable" ) nil nil )) nil )
("3" (hide -1 2)
(("3" (lemma "deriv_powerseq_lem" )
(("3" (inst?) (("3" (flatten) nil nil )) nil )) nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("3" (hide 2)
(("3" (skosimp*)
(("3" (rewrite "sigma_power_derivable" ) nil nil )) nil ))
nil ))
nil )
((deriv_sum_fun formula-decl nil derivatives "analysis/" )
(+ const-decl "[T -> real]" real_fun_ops "reals/" )
(deriv_const_fun formula-decl nil derivatives "analysis/" )
(const_fun const-decl "[T -> real]" real_fun_ops "reals/" )
(deriv const-decl "[T -> real]" derivatives "analysis/" )
(deriv_fun type-eq-decl nil derivatives "analysis/" )
(sigma def-decl "real" sigma "reals/" )
(powerseq const-decl "sequence[real]" power_series nil ))
nil ))
(deriv_sigma_power_conv 0
(deriv_sigma_power_conv-1 nil 3279280814
("" (skosimp*)
(("" (lemma "deriv_powerseries_conv" )
(("" (inst?)
(("" (replace -2)
(("" (assert )
(("" (inst -1 "x1!1" )
(("" (lemma "deriv_sigma_power" )
(("" (inst?)
((""
(case-replace "(LAMBDA (n: nat):
deriv(LAMBDA x: sigma(0, n, powerseq(a!1, x)))(x1!1)) =
(LAMBDA (n: nat):
(LAMBDA x: sigma(0, n, deriv_powerseq(a!1, x)))(x1!1))")
(("1" (hide -1 -2)
(("1" (expand "deriv_powerseq" )
(("1" (assert )
(("1" (expand "series" )
(("1" (propax) nil nil )) nil ))
nil ))
nil ))
nil )
("2" (hide 2)
(("2" (apply-extensionality 1 :hide? t)
(("1" (inst?)
(("1" (assert )
(("1" (replace -1)
(("1"
(hide -1 -2)
(("1"
(expand "sigma" )
(("1" (propax) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (hide -1 -2 2)
(("2" (skosimp*)
(("2" (rewrite "sigma_power_derivable" ) nil
nil ))
nil ))
nil ))
nil ))
nil )
("3" (hide -1 -2 2)
(("3" (skosimp*)
(("3" (rewrite "sigma_power_derivable" ) nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
((deriv_powerseries_conv formula-decl nil power_series_derivseq nil )
(T formal-subtype-decl nil power_series_derivseq nil )
(T_pred const-decl "[real -> boolean]" power_series_derivseq nil )
(sigma_power_derivable formula-decl nil power_series_derivseq nil )
(real_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil )
(real_le_is_total_order name-judgement "(total_order?[real])"
real_props nil )
(a!1 skolem-const-decl "sequence[real]" power_series_derivseq nil )
(series const-decl "sequence[real]" series nil )
(deriv_powerseq const-decl "sequence[real]" power_series_derivseq
nil )
(powerseq const-decl "sequence[real]" power_series nil )
(sigma def-decl "real" sigma "reals/" )
(T_high type-eq-decl nil sigma "reals/" )
(T_low type-eq-decl nil sigma "reals/" )
(<= const-decl "bool" reals nil )
(OR const-decl "[bool, bool -> bool]" booleans nil )
(deriv const-decl "[T -> real]" derivatives "analysis/" )
(deriv_fun type-eq-decl nil derivatives "analysis/" )
(derivable? const-decl "bool" derivatives "analysis/" )
(= const-decl "[T, T -> boolean]" equalities nil )
(IMPLIES const-decl "[bool, bool -> bool]" booleans nil )
(deriv_sigma_power formula-decl nil power_series_derivseq nil )
(sequence type-eq-decl nil sequences nil )
(nat nonempty-type-eq-decl nil naturalnumbers nil )
(>= const-decl "bool" reals nil )
(bool nonempty-type-eq-decl nil booleans nil )
(int nonempty-type-eq-decl nil integers nil )
(integer_pred const-decl "[rational -> boolean]" integers nil )
(rational nonempty-type-from-decl nil rationals nil )
(rational_pred const-decl "[real -> boolean]" rationals 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 ))
nil ))
(deriv_conv_prep_TCC1 0
(deriv_conv_prep_TCC1-1 nil 3403524860 ("" (subtype-tcc) nil nil )
((powerseries const-decl "sequence[real]" power_series nil )
(series const-decl "sequence[real]" series nil )
(convergence const-decl "bool" convergence_sequences "analysis/" )
(convergent? const-decl "bool" convergence_sequences "analysis/" )
(T formal-subtype-decl nil power_series_derivseq nil )
(T_pred const-decl "[real -> boolean]" power_series_derivseq 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 )
(conv_powerseries? const-decl "bool" power_series_conv nil )
(real_minus_real_is_real application-judgement "real" reals nil ))
nil ))
(deriv_conv_prep 0
(deriv_conv_prep-1 nil 3279280814
("" (skosimp*)
(("" (lemma "deriv_powerseries_conv" )
(("" (inst?)
(("" (replace -2)
(("" (assert )
(("" (expand "deriv_powerseq" )
(("" (inst?)
(("" (expand "series" ) (("" (propax) nil nil )) nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
((deriv_powerseries_conv formula-decl nil power_series_derivseq nil )
(deriv_powerseq const-decl "sequence[real]" power_series_derivseq
nil )
(series const-decl "sequence[real]" series nil )
(T_pred const-decl "[real -> boolean]" power_series_derivseq nil )
(T formal-subtype-decl nil power_series_derivseq nil )
(sequence type-eq-decl nil sequences nil )
(nat nonempty-type-eq-decl nil naturalnumbers nil )
(>= const-decl "bool" reals nil )
(bool nonempty-type-eq-decl nil booleans nil )
(int nonempty-type-eq-decl nil integers nil )
(integer_pred const-decl "[rational -> boolean]" integers nil )
(rational nonempty-type-from-decl nil rationals nil )
(rational_pred const-decl "[real -> boolean]" rationals 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 ))
nil ))
(deriv_powerseq_cont 0
(deriv_powerseq_cont-2 nil 3297176902
("" (induct "n" )
(("1" (expand "sigma" )
(("1" (expand "deriv_powerseq" )
(("1" (expand "sigma" )
(("1" (skosimp*)
(("1" (lemma "const_fun_continuous[T]" )
(("1" (expand "const_fun" ) (("1" (inst?) nil nil )) nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (skosimp*)
(("2" (expand "sigma" 1)
(("2" (lemma "sum_fun_continuous[T]" )
(("2"
(inst -1
"(LAMBDA x: deriv_powerseq(a!1, x)(1 + j!1))"
"(LAMBDA x: sigma(0, j!1, deriv_powerseq(a!1, x)))" )
(("1" (expand "+" -1) (("1" (propax) nil nil )) nil )
("2" (hide 2) (("2" (inst?) nil nil )) nil )
("3" (hide 2)
(("3" (hide -1)
(("3" (name-replace "J1" "j!1+1" )
(("3" (expand "deriv_powerseq" )
(("3" (case "J1 > 0" )
(("1" (assert )
(("1" (lemma "prod_fun_continuous[T]" )
(("1"
(inst -1 "(LAMBDA x: a!1(J1)*J1)"
"(LAMBDA x: x ^ (J1-1))" )
(("1" (ground)
(("1"
(lemma "const_fun_continuous[T]" )
(("1"
(expand "const_fun" )
(("1"
(inst?)
(("1"
(expand "*" )
(("1"
(case-replace
"(LAMBDA x:
IF J1 = 1 THEN a!1(1) ELSE a!1(J1) * x ^ (J1 - 1) * J1 ENDIF) = (LAMBDA (x_1: T): a!1(J1) * x_1 ^ (J1 - 1) * J1)")
(("1"
(hide 2)
(("1"
(apply-extensionality
1
:hide?
t)
(("1"
(lift-if)
(("1"
(ground)
(("1"
(replace -1)
(("1"
(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (hide 2)
(("2"
(lemma "x_to_n_continuous[T]" )
(("2" (inst?) nil nil ))
nil ))
nil )
("3" (lemma "const_fun_continuous[T]" )
(("3"
(expand "const_fun" )
(("3" (inst?) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (hide 2)
(("2" (reveal -1) (("2" (assert ) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
((sum_fun_continuous judgement-tcc nil continuous_functions
"analysis/" )
(posint nonempty-type-eq-decl nil integers nil )
(> const-decl "bool" reals nil )
(nonneg_int nonempty-type-eq-decl nil integers nil )
(= const-decl "[T, T -> boolean]" equalities nil )
(nnint_plus_posint_is_posint application-judgement "posint"
integers nil )
(int_minus_int_is_int application-judgement "int" integers nil )
(real_minus_real_is_real application-judgement "real" reals nil )
(prod_fun_continuous judgement-tcc nil continuous_functions
"analysis/" )
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil )
(x_to_n_continuous formula-decl nil continuous_functions
"analysis/" )
(* const-decl "[T -> real]" real_fun_ops "reals/" )
(odd_minus_odd_is_even application-judgement "even_int" integers
nil )
(expt_x0 formula-decl nil exponentiation nil )
(IF const-decl "[boolean, T, T -> T]" if_def nil )
(NOT const-decl "[bool -> bool]" booleans nil )
(IMPLIES const-decl "[bool, bool -> bool]" booleans nil )
(- const-decl "[numfield, numfield -> numfield]" number_fields nil )
(^ const-decl "real" exponentiation nil )
(/= const-decl "boolean" notequal nil )
(J1 skolem-const-decl "posint" power_series_derivseq nil )
(* const-decl "[numfield, numfield -> numfield]" number_fields nil )
(real_times_real_is_real application-judgement "real" reals nil )
(real_gt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil )
(+ const-decl "[T -> real]" real_fun_ops "reals/" )
(continuous_fun nonempty-type-eq-decl nil continuous_functions
"analysis/" )
(j!1 skolem-const-decl "nat" power_series_derivseq nil )
(+ const-decl "[numfield, numfield -> numfield]" number_fields nil )
(numfield nonempty-type-eq-decl nil number_fields nil )
(a!1 skolem-const-decl "sequence[real]" power_series_derivseq nil )
(posint_plus_nnint_is_posint application-judgement "posint"
integers nil )
(const_fun_continuous judgement-tcc nil continuous_functions
"analysis/" )
(const_fun const-decl "[T -> real]" real_fun_ops "reals/" )
(nat_induction formula-decl nil naturalnumbers nil )
(deriv_powerseq const-decl "sequence[real]" power_series_derivseq
nil )
(sigma def-decl "real" sigma "reals/" )
(T_high type-eq-decl nil sigma "reals/" )
(T_low type-eq-decl nil sigma "reals/" )
(<= const-decl "bool" reals nil )
(OR const-decl "[bool, bool -> bool]" booleans nil )
(continuous? const-decl "bool" continuous_functions "analysis/" )
(T formal-subtype-decl nil power_series_derivseq nil )
(T_pred const-decl "[real -> boolean]" power_series_derivseq nil )
(sequence type-eq-decl nil sequences nil )
(pred type-eq-decl nil defined_types nil )
(nat nonempty-type-eq-decl nil naturalnumbers nil )
(>= const-decl "bool" reals nil )
(bool nonempty-type-eq-decl nil booleans nil )
(int nonempty-type-eq-decl nil integers nil )
(integer_pred const-decl "[rational -> boolean]" integers nil )
(rational nonempty-type-from-decl nil rationals nil )
(rational_pred const-decl "[real -> boolean]" rationals 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 ))
nil )
(deriv_powerseq_cont-1 nil 3279280814
("" (induct "n" )
(("1" (expand "sigma" )
(("1" (expand "deriv_powerseq" )
(("1" (skosimp*)
(("1" (lemma "const_fun_continuous[real]" )
(("1" (expand "const_fun" ) (("1" (inst?) nil nil )) nil ))
nil ))
nil ))
nil ))
nil )
("2" (skosimp*)
(("2" (expand "sigma" 1)
(("2" (lemma "sum_fun_continuous[real]" )
(("2"
(inst -1
"(LAMBDA x: deriv_powerseq(a!1, x)(1 + j!1))"
"(LAMBDA x: sigma(0, j!1, deriv_powerseq(a!1, x)))" )
(("1" (expand "+" -1) (("1" (propax) nil nil )) nil )
("2" (hide 2) (("2" (inst?) nil nil )) nil )
("3" (hide 2)
(("3" (hide -1)
(("3" (name-replace "J1" "j!1+1" )
(("3" (expand "deriv_powerseq" )
(("3" (case "J1 > 0" )
(("1" (assert )
(("1" (lemma "prod_fun_continuous[real]" )
(("1"
(inst -1 "(LAMBDA x: a!1(J1)*J1)"
"(LAMBDA x: x ^ (J1-1))" )
(("1" (expand "*" ) (("1" (propax) nil nil ))
nil )
("2" (hide 2)
(("2"
(lemma "x_to_n_continuous" )
(("2" (inst?) nil nil ))
nil ))
nil )
("3" (lemma "const_fun_continuous[real]" )
(("3"
(expand "const_fun" )
(("3" (inst?) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (hide 2)
(("2" (reveal -1) (("2" (assert ) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
((sum_fun_continuous judgement-tcc nil continuous_functions
"analysis/" )
(prod_fun_continuous judgement-tcc nil continuous_functions
"analysis/" )
(+ const-decl "[T -> real]" real_fun_ops "reals/" )
(continuous_fun nonempty-type-eq-decl nil continuous_functions
"analysis/" )
(const_fun const-decl "[T -> real]" real_fun_ops "reals/" )
(const_fun_continuous judgement-tcc nil continuous_functions
"analysis/" )
(sigma def-decl "real" sigma "reals/" ))
nil ))
(lim_deriv_alt_TCC1 0
(lim_deriv_alt_TCC1-1 nil 3299583090
("" (skosimp*)
(("" (lemma "deriv_powerseries_conv" )
(("" (inst?)
(("" (assert )
(("" (inst?)
(("" (lemma "end_series_conv" )
(("" (inst?) (("" (assert ) nil nil )) nil )) nil ))
nil ))
nil ))
nil ))
nil ))
nil )
((deriv_powerseries_conv formula-decl nil power_series_derivseq nil )
(end_series_conv formula-decl nil series nil )
(deriv_powerseq const-decl "sequence[real]" power_series_derivseq
nil )
(T_pred const-decl "[real -> boolean]" power_series_derivseq nil )
(T formal-subtype-decl nil power_series_derivseq nil )
(sequence type-eq-decl nil sequences nil )
(nat nonempty-type-eq-decl nil naturalnumbers nil )
(>= const-decl "bool" reals nil )
(bool nonempty-type-eq-decl nil booleans nil )
(int nonempty-type-eq-decl nil integers nil )
(integer_pred const-decl "[rational -> boolean]" integers nil )
(rational nonempty-type-from-decl nil rationals nil )
(rational_pred const-decl "[real -> boolean]" rationals 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 ))
shostak))
(lim_deriv_alt 0
(lim_deriv_alt-1 nil 3299582635
("" (skosimp*)
(("" (expand "series" )
((""
(case-replace
"(LAMBDA (n: nat): sigma(0, n, deriv_powerseq(a!1, x!1))) =
(LAMBDA (n: nat): sigma(1, n, deriv_powerseq(a!1, x!1)))")
(("" (hide 2)
(("" (apply-extensionality 1 :hide? t)
(("" (lemma "sigma_first" )
(("" (inst?)
(("" (assert )
(("" (case "x!2 = 0" )
(("1" (replace -1)
(("1" (expand "sigma" )
(("1" (expand "deriv_powerseq" )
(("1" (propax) nil nil )) nil ))
nil ))
nil )
("2" (assert )
(("2" (replace -1)
(("2" (hide -1)
(("2" (expand "deriv_powerseq" )
(("2" (propax) nil nil )) nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
((series const-decl "sequence[real]" series nil )
(series const-decl "sequence[real]" series nil )
(sigma_first formula-decl nil sigma "reals/" )
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil )
(real_plus_real_is_real application-judgement "real" reals nil )
(real_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil )
(real_le_is_total_order name-judgement "(total_order?[real])"
real_props nil )
(deriv_powerseq const-decl "sequence[real]" power_series_derivseq
nil )
(T formal-subtype-decl nil power_series_derivseq nil )
(T_pred const-decl "[real -> boolean]" power_series_derivseq nil )
(sequence type-eq-decl nil sequences nil )
(sigma def-decl "real" sigma "reals/" )
(T_high type-eq-decl nil sigma "reals/" )
(T_low type-eq-decl nil sigma "reals/" )
(<= const-decl "bool" reals nil )
(OR const-decl "[bool, bool -> bool]" booleans nil )
(= const-decl "[T, T -> boolean]" equalities nil )
(nat nonempty-type-eq-decl nil naturalnumbers nil )
(>= const-decl "bool" reals nil )
(bool nonempty-type-eq-decl nil booleans nil )
(int nonempty-type-eq-decl nil integers nil )
(integer_pred const-decl "[rational -> boolean]" integers nil )
(rational nonempty-type-from-decl nil rationals nil )
(rational_pred const-decl "[real -> boolean]" rationals 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 ))
nil )))
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