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(deriv_domain_gtm1 0
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("" (expand "deriv_domain?" )
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real_defs nil )
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(>= const-decl "bool" reals nil )
(/ const-decl "[numfield, nznum -> numfield]" number_fields nil )
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(- const-decl "[numfield -> numfield]" number_fields nil )
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shostak))
(nderiv_ln_TCC1 0
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(nderiv_ln_TCC2 0
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nil ))
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(nderiv_ln 0
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nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
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nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
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nil ))
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nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
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nil ))
nil ))
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nil ))
nil )
((const_derivable_fun formula-decl nil derivatives "analysis/" )
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integers nil )
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(factorial_1 formula-decl nil factorial "ints/" )
(factorial_0 formula-decl nil factorial "ints/" )
(posreal_div_posreal_is_posreal application-judgement "posreal"
real_types nil )
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integers nil )
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real_types nil )
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nil )
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nil )
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(real_pred const-decl "[number_field -> boolean]" reals nil )
(real nonempty-type-from-decl nil reals nil )
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(= const-decl "[T, T -> boolean]" equalities nil )
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(/ const-decl "[numfield, nznum -> numfield]" number_fields nil )
(- const-decl "[numfield -> numfield]" number_fields nil )
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(- const-decl "[numfield, numfield -> numfield]" number_fields nil )
(OR const-decl "[bool, bool -> bool]" booleans nil )
(^ const-decl "real" exponentiation nil ))
nil )
(nderiv_ln-1 nil 3270048547
(""
(case "FORALL (n: nat): derivable_n_times[posreal](ln, n) & nderiv[posreal](n,ln) = IF n = 0 THEN ln ELSE (LAMBDA (x:posreal): -factorial(n-1)/(-x)^n) ENDIF" )
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nil )
("2" (expand "nderiv" ) (("2" (propax) nil nil )) nil )
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(lemma "extensionality"
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(("1" (assert ) nil nil ))
nil ))
nil )
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(replace -2)
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(skosimp*)
(("2"
(expand "factorial" )
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(expand "factorial" )
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nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
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(("2"
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(("2" (assert ) nil nil ))
nil ))
nil ))
nil )
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nil )
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nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
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(+ const-decl "[numfield, numfield -> numfield]" number_fields nil )
(real_plus_real_is_real application-judgement "real" reals nil )
(both_sides_div1 formula-decl nil real_props nil )
(* const-decl "[numfield, numfield -> numfield]" number_fields nil )
(minus_odd_is_odd application-judgement "odd_int" integers nil )
(posint_plus_nnint_is_posint application-judgement "posint"
integers nil )
(expt def-decl "real" exponentiation nil )
(sigma_0_neg formula-decl nil sigma_nat "reals/" )
(nat_induction formula-decl nil naturalnumbers nil )
(pred type-eq-decl nil defined_types nil )
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil )
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(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_div_nzreal_is_real application-judgement "real" reals nil )
(real_minus_real_is_real application-judgement "real" reals nil ))
shostak))
(ln_estimate_scaf5_TCC1 0
(ln_estimate_scaf5_TCC1-1 nil 3309550054
("" (skosimp) (("" (assert ) nil nil )) nil )
((real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil )
(real_minus_real_is_real application-judgement "real" reals nil ))
shostak))
(ln_estimate_scaf5_TCC2 0
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real_props nil )
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shostak))
(ln_estimate_scaf5 0
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(("" (assert ) nil nil )) nil ))
nil )
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(numfield nonempty-type-eq-decl nil number_fields 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 )
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(nzreal_div_nzreal_is_nzreal application-judgement "nzreal"
real_types nil )
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(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_div_nzreal_is_real application-judgement "real" reals nil )
(nnint_plus_posint_is_posint application-judgement "posint"
integers nil ))
shostak))
(ln_estimate_scaf6 0
(ln_estimate_scaf6-2 nil 3352449131
("" (induct "n" )
(("1" (expand "ln_estimate" )
(("1" (expand "sigma" )
(("1" (assert )
(("1" (lemma "const_derivable_fun[posreal]" ("b" "0" ))
(("1" (expand "const_fun" ) (("1" (propax) nil nil )) nil ))
nil ))
nil ))
nil ))
nil )
("2" (skosimp*)
(("2" (expand "ln_estimate" )
(("2" (expand "sigma" 1)
(("2"
(name "F" "(LAMBDA px:
sigma(0, j!1,
LAMBDA (nn: nat):
IF nn = 0 THEN 0 ELSE -(-(px - 1)) ^ nn / nn ENDIF))")
(("1"
(lemma "extensionality"
("f" "(LAMBDA px:
sigma(0, j!1,
LAMBDA (nn: nat):
IF nn = 0 THEN 0 ELSE -(-(px - 1)) ^ nn / nn ENDIF)
+ -(-(px - 1)) ^ (1 + j!1) / (1 + j!1))"
"g" "F-(LAMBDA px: (1-px)^(1+j!1)/(1+j!1))" ))
(("1" (split -1)
(("1" (replace -1)
(("1" (replace -2)
(("1" (hide -1 -2)
(("1" (lemma "identity_derivable_fun[posreal]" )
(("1" (expand "I" )
(("1"
(lemma "const_derivable_fun[posreal]"
("b" "1" ))
(("1"
(lemma "diff_derivable_fun[posreal]"
("f1"
"LAMBDA (x: posreal): 1"
"f2"
"LAMBDA (x: posreal): x" ))
(("1"
(expand "const_fun" )
(("1"
(assert )
(("1"
(expand "-" -1)
(("1"
(lemma
"deriv_exp_fun[posreal]"
("f"
"LAMBDA (x:posreal): 1-x"
"n"
"1+j!1" ))
(("1"
(assert )
(("1"
(flatten)
(("1"
(hide -2)
(("1"
(expand "^" -1)
(("1"
(lemma
"scal_derivable_fun[posreal]"
("b"
"1/(1+j!1)"
"f"
"LAMBDA (t: posreal): (1 - t) ^ (1 + j!1)" ))
(("1"
(assert )
(("1"
(lemma
"extensionality"
("f"
"1 / (1 + j!1) * (LAMBDA (t: posreal): (1 - t) ^ (1 + j!1))"
"g"
"(LAMBDA px: (1 - px) ^ (1 + j!1) / (1 + j!1))" ))
(("1"
(split -1)
(("1"
(replace -1)
(("1"
(lemma
"diff_derivable_fun[posreal]"
("f1"
"F"
"f2"
"(LAMBDA px: (1 - px) ^ (1 + j!1) / (1 + j!1))" ))
(("1"
(assert )
nil
nil ))
nil ))
nil )
("2"
(hide-all-but
1)
(("2"
(skosimp)
(("2"
(expand
"*" )
(("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 )
("2" (hide-all-but 1)
(("2" (skosimp)
(("2" (expand "F" )
(("2" (expand "-" )
(("2" (case-replace "-(x!1 - 1) = 1-x!1" )
(("1" (assert ) nil nil )
("2" (assert ) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (skosimp*) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil )
((minus_real_is_real application-judgement "real" reals nil )
(posint_plus_nnint_is_posint application-judgement "posint"
integers nil )
(extensionality formula-decl nil functions nil )
(+ const-decl "[numfield, numfield -> numfield]" number_fields nil )
(- const-decl "[T -> real]" real_fun_ops "reals/" )
(F skolem-const-decl "[posreal -> real]" ln_exp_series_alt nil )
(I const-decl "(bijective?[T, T])" identity nil )
(diff_derivable_fun formula-decl nil derivatives "analysis/" )
(real_gt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil )
(posrat_div_posrat_is_posrat application-judgement "posrat"
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(* const-decl "[T -> real]" real_fun_ops "reals/" )
(real_times_real_is_real application-judgement "real" reals nil )
(^ const-decl "[T -> real]" real_fun_ops "reals/" )
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(IMPLIES const-decl "[bool, bool -> bool]" booleans nil )
(NOT const-decl "[bool -> bool]" booleans nil )
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real_props nil )
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(= const-decl "[T, T -> boolean]" equalities nil )
(OR const-decl "[bool, bool -> bool]" booleans nil )
(<= const-decl "bool" reals nil )
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(T_high type-eq-decl nil sigma "reals/" )
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(/= const-decl "boolean" notequal nil )
(nznum nonempty-type-eq-decl nil number_fields nil )
(/ const-decl "[numfield, nznum -> numfield]" number_fields nil )
(- const-decl "[numfield -> numfield]" number_fields nil )
(^ const-decl "real" exponentiation nil )
(sigma_0_neg formula-decl nil sigma_nat "reals/" )
(const_derivable_fun formula-decl nil derivatives "analysis/" )
(sigma def-decl "real" sigma "reals/" )
(nat_induction formula-decl nil naturalnumbers nil )
(- const-decl "[numfield, numfield -> numfield]" number_fields nil )
(numfield nonempty-type-eq-decl nil number_fields nil )
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(derivable? const-decl "bool" derivatives "analysis/" )
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(> const-decl "bool" reals nil )
(nonneg_real nonempty-type-eq-decl nil real_types 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 )
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(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 )
(real_minus_real_is_real application-judgement "real" reals nil ))
nil )
(ln_estimate_scaf6-1 nil 3309551069
("" (induct "n" )
(("1" (expand "ln_estimate" )
(("1" (expand "sigma" )
(("1" (lemma "const_derivable_fun[posreal]" ("b" "0" ))
(("1" (expand "const_fun" ) (("1" (propax) nil nil )) nil ))
nil ))
nil ))
nil )
("2" (skosimp*)
(("2" (expand "ln_estimate" )
(("2" (expand "sigma" 1)
(("2"
(name "F" "(LAMBDA px:
sigma(0, j!1,
LAMBDA (nn: nat):
IF nn = 0 THEN 0 ELSE -(-(px - 1)) ^ nn / nn ENDIF))")
(("1"
(lemma "extensionality"
("f" "(LAMBDA px:
sigma(0, j!1,
LAMBDA (nn: nat):
IF nn = 0 THEN 0 ELSE -(-(px - 1)) ^ nn / nn ENDIF)
+ -(-(px - 1)) ^ (1 + j!1) / (1 + j!1))" " g"
"F-(LAMBDA px: (1-px)^(1+j!1)/(1+j!1))" ))
(("1" (split -1)
(("1" (replace -1)
(("1" (replace -2)
(("1" (hide -1 -2)
(("1" (lemma "identity_derivable_fun[posreal]" )
(("1" (expand "I" )
(("1"
(lemma "const_derivable_fun[posreal]"
("b" "1" ))
(("1"
(lemma "diff_derivable_fun"
("f1"
"LAMBDA (x: posreal): 1"
"f2"
"LAMBDA (x: posreal): x" ))
(("1"
(expand "const_fun" )
(("1"
(assert )
(("1"
(expand "-" -1)
(("1"
(lemma
"deriv_exp_fun"
("f"
"LAMBDA (x:posreal): 1-x"
"n"
"1+j!1" ))
(("1"
(assert )
(("1"
(flatten)
(("1"
(hide -2)
(("1"
(expand "^" -1)
(("1"
(lemma
"scal_derivable_fun"
("b"
"1/(1+j!1)"
"f"
"LAMBDA (t: posreal): (1 - t) ^ (1 + j!1)" ))
(("1"
(assert )
(("1"
(lemma
"extensionality"
("f"
"1 / (1 + j!1) * (LAMBDA (t: posreal): (1 - t) ^ (1 + j!1))"
"g"
"(LAMBDA px: (1 - px) ^ (1 + j!1) / (1 + j!1))" ))
(("1"
(split -1)
(("1"
(replace -1)
(("1"
(lemma
"diff_derivable_fun"
("f1"
"F"
"f2"
"(LAMBDA px: (1 - px) ^ (1 + j!1) / (1 + j!1))" ))
(("1"
(assert )
nil
nil ))
nil ))
nil )
("2"
(hide-all-but
1)
(("2"
(skosimp)
(("2"
(expand
"*" )
(("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 )
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(("2" (skosimp)
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("2" (assert ) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (skosimp*) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil )
((diff_derivable_fun formula-decl nil derivatives "analysis/" )
(deriv_exp_fun formula-decl nil derivatives "analysis/" )
(scal_derivable_fun formula-decl nil derivatives "analysis/" )
(identity_derivable_fun formula-decl nil derivatives "analysis/" )
(const_derivable_fun formula-decl nil derivatives "analysis/" )
(const_fun const-decl "[T -> real]" real_fun_ops "reals/" )
(sigma def-decl "real" sigma "reals/" ))
shostak))
(ln_estimate_scaf7_TCC1 0
(ln_estimate_scaf7_TCC1-1 nil 3309557879
("" (lemma "ln_estimate_scaf6" )
(("" (skosimp) (("" (inst - "n!1+1" ) nil nil )) nil )) nil )
((+ const-decl "[numfield, numfield -> numfield]" number_fields nil )
(numfield nonempty-type-eq-decl nil number_fields nil )
(nat nonempty-type-eq-decl nil naturalnumbers nil )
(>= const-decl "bool" reals nil )
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(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 )
(nnint_plus_posint_is_posint application-judgement "posint"
integers nil )
(ln_estimate_scaf6 formula-decl nil ln_exp_series_alt nil ))
shostak))
(ln_estimate_scaf7 0
(ln_estimate_scaf7-2 nil 3352449229
("" (induct "n" )
(("1" (expand "ln_estimate" )
(("1" (expand "sigma" )
(("1" (expand "sigma" )
(("1" (lemma "identity_derivable_fun[posreal]" )
(("1" (lemma "deriv_id_fun[posreal]" )
(("1" (expand "I" )
(("1" (lemma "const_derivable_fun[posreal]" ("b" "1" ))
(("1" (lemma "deriv_const_fun[posreal]" ("b" "1" ))
(("1" (expand "const_fun" )
(("1"
(lemma "diff_derivable_fun[posreal]"
("f1" "LAMBDA (x: posreal): x" "f2"
"LAMBDA (x: posreal): 1" ))
(("1" (assert )
(("1"
(lemma "deriv_diff_fun[posreal]"
("ff1" "LAMBDA (x: posreal): x" "ff2"
"LAMBDA (x: posreal): 1" ))
(("1" (replace -3)
(("1"
(replace -5)
(("1"
(expand "-" )
(("1"
(lemma
"extensionality"
("f"
"LAMBDA px: -(-(px - 1)) ^ 1 / 1"
"g"
"lambda px: px-1" ))
(("1"
(split -1)
(("1"
(replace -1)
(("1"
(replace -2)
(("1"
(apply-extensionality
1
:hide?
t)
(("1"
(expand "^" 1)
(("1"
(expand "expt" 1)
(("1"
(propax)
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(hide-all-but 1)
(("2"
(skosimp)
(("2"
(expand "^" )
(("2"
(expand "expt" )
(("2"
(expand "expt" )
(("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 )
("2" (skosimp*)
(("2" (expand "ln_estimate" )
(("2" (expand "sigma" 1)
(("2"
(name "F" "(LAMBDA px:
-sigma(0, 1 + j!1,
LAMBDA (nn: nat):
IF nn = 0 THEN 0 ELSE -(-(1 - px)) ^ nn / nn ENDIF))")
(("1"
(lemma "extensionality"
("f" "F+ (lambda px: (px-1) ^ (2 + j!1) / (2 + j!1))"
"g" "(LAMBDA px:
-(sigma(0, 1 + j!1,
LAMBDA (nn: nat):
IF nn = 0 THEN 0 ELSE -(-(1 - px)) ^ nn / nn ENDIF)
+ -(-(1 - px)) ^ (2 + j!1) / (2 + j!1)))"))
(("1" (split -1)
(("1" (hide -1 -2)
(("1"
(name "FF" "(LAMBDA px:
sigma(0, 1 + j!1,
LAMBDA (nn: nat):
IF nn = 0 THEN 0 ELSE -(-(px - 1)) ^ nn / nn ENDIF))")
(("1" (replace -1)
(("1"
(lemma "extensionality"
("f" "(LAMBDA px:
sigma(0, 1 + j!1,
LAMBDA (nn: nat):
IF nn = 0 THEN 0 ELSE -(-(px - 1)) ^ nn / nn ENDIF)
+ -(-(px - 1)) ^ (2 + j!1) / (2 + j!1))" " g"
"FF-(lambda px: (1-px)^(2+j!1)/(2+j!1))" ))
(("1" (split -1)
(("1" (replace -1)
(("1"
(lemma "ln_estimate_scaf6" ("n" "1+j!1" ))
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(expand "ln_estimate" )
(("1"
(replace -3)
(("1"
(hide -2 -3)
(("1"
(name
"GG"
"(LAMBDA px: sigma(0, j!1, (LAMBDA (i: nat): (1 - px) ^ i)))" )
(("1"
(lemma
"extensionality"
("f"
"(LAMBDA px:
(1 - px) ^ (1 + j!1) +
sigma(0, j!1, (LAMBDA (i: nat): (1 - px) ^ i)))"
"g"
"GG+(LAMBDA px:
(1 - px) ^ (1 + j!1))"))
(("1"
(split -1)
(("1"
(replace -1)
(("1"
(replace -2)
(("1"
(hide -1 -2)
(("1"
(lemma
"identity_derivable_fun[posreal]" )
(("1"
(lemma
"deriv_id_fun[posreal]" )
(("1"
(expand "I" )
(("1"
(lemma
"const_derivable_fun[posreal]"
("b" "1" ))
(("1"
(lemma
"deriv_const_fun[posreal]"
("b" "1" ))
(("1"
(expand
"const_fun" )
(("1"
(lemma
"diff_derivable_fun[posreal]"
("f1"
"LAMBDA (x: posreal): 1"
"f2"
"LAMBDA (x: posreal): x" ))
(("1"
(assert )
(("1"
(lemma
"deriv_diff_fun[posreal]"
("ff1"
"LAMBDA (x: posreal): 1"
"ff2"
"LAMBDA (x: posreal): x" ))
(("1"
(replace
-3)
(("1"
(replace
-5)
(("1"
(expand
"-" )
(("1"
(lemma
"deriv_exp_fun[posreal]"
("f"
"LAMBDA (x:posreal): 1-x"
"n"
"2+j!1" ))
(("1"
(assert )
(("1"
(flatten)
(("1"
(expand
"^"
(-1
-2))
(("1"
(replace
-3)
(("1"
(lemma
"scal_derivable_fun[posreal]"
("b"
"1/(2+j!1)"
"f"
"LAMBDA (t: posreal): (1 - t) ^ (2 + j!1)" ))
(("1"
(assert )
(("1"
(lemma
"deriv_scal_fun[posreal]"
("b"
"1/(2+j!1)"
"ff"
"LAMBDA (t: posreal): (1 - t) ^ (2 + j!1)" ))
(("1"
(replace
-4)
(("1"
(lemma
"extensionality"
("f"
"1 / (2 + j!1) *
(((2 + j!1) * (LAMBDA (t: posreal): (1 - t) ^ (1 + j!1))) *
(LAMBDA (x_1: posreal): -1))"
"g"
"LAMBDA (t: posreal): - ((1 - t) ^ (1 + j!1))" ))
(("1"
(split
-1)
(("1"
(replace
-1)
(("1"
(lemma
"extensionality"
("f"
"(1 / (2 + j!1) * (LAMBDA (t: posreal): (1 - t) ^ (2 + j!1)))"
"g"
"(LAMBDA (t: posreal): (1 - t) ^ (2 + j!1)/(2+j!1))" ))
(("1"
(split
-1)
(("1"
(replace
-1)
(("1"
(hide
-1
-2
-5
-6
-7
-8
-9
-10
-11
-12)
(("1"
(lemma
"diff_derivable_fun[posreal]"
("f1"
"FF"
"f2"
"(LAMBDA (t: posreal): (1 - t) ^ (2 + j!1) / (2 + j!1))" ))
(("1"
(assert )
(("1"
(lemma
"deriv_diff_fun[posreal]"
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(ln_estimate_scaf8 0
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Integral(1, px, LAMBDA (t: posreal): ((1 - t) ^ (1 + n)) / t))"
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(lemma "extensionality_postulate"
("f" "ln" "g" "lambda px: ln_estimate(px - 1, n + 1) +
(-1) ^ (n + 1) *
Integral(1, px, LAMBDA (t: posreal): (1 - t) ^ (n + 1) / t)"))
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(name "II"
"LAMBDA (px: posreal): Integral(1, px, LAMBDA (t: posreal): ((1 - t) ^ (1 + n)) / t)" )
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(lemma "extensionality"
("f" "(LAMBDA (px: posreal):
ln_estimate(px - 1, 1 + n) +
Integral(1, px, LAMBDA (t: posreal): ((1 - t) ^ (1 + n)) / t))"
"g" "L+II" ))
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(lemma
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"LAMBDA (t: posreal): ((1 - t) ^ (1 + n)) / t"
"F"
"II"
"a"
"1" ))
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(lemma
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"L"
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(lemma
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(Integral_a_to_a formula-decl nil integral "analysis/" )
(+ const-decl "[T -> real]" real_fun_ops "reals/" ))
shostak))
(ln_estimate_scaf9_TCC1 0
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nil ))
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nil ))
nil )
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nil ))
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nil )
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(/ const-decl "[numfield, nznum -> numfield]" number_fields nil )
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(posreal_div_posreal_is_posreal application-judgement "posreal"
real_types nil )
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(ln_estimate const-decl "real" ln_exp_series_alt nil )
(int_plus_int_is_int application-judgement "int" integers nil )
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil )
(int_minus_int_is_int application-judgement "int" integers nil )
(ln_estimate_scaf8 formula-decl nil ln_exp_series_alt nil ))
shostak))
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(ln_estimate_scaf10_TCC1-1 nil 3309628050
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integers nil )
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real_props nil )
(real_minus_real_is_real application-judgement "real" reals nil )
(real_le_is_total_order name-judgement "(total_order?[real])"
real_props nil ))
shostak))
(ln_estimate_scaf10 0
(ln_estimate_scaf10-6 nil 3445353682
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(lemma "continuous_Integrable?[posreal]"
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"LAMBDA (t: posreal): (1 - t) ^ n!1" ))
(("1" (split -1)
(("1" (lemma "const_derivable_fun[posreal]" ("b" "1" ))
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(lemma "diff_derivable_fun[posreal]"
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"LAMBDA (x: posreal): x" ))
(("1" (assert )
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(lemma "deriv_diff_fun[posreal]"
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"LAMBDA (x: posreal): 1"
"ff2"
"LAMBDA (x: posreal): x" ))
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(replace -3)
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(replace -5)
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(expand "-" )
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(lemma
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"1+n!1"
"f"
"LAMBDA (x_1: posreal): 1 - x_1" ))
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(expand "^" -1)
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-5
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(lemma
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"-1/(1+n!1)"
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(assert )
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(lemma
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(expand "*" )
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-4
-1)
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(lemma
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("f"
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-1 * ((1 - x) ^ n!1 * (-1 / (1 + n!1))) -
(1 - x) ^ n!1 * (-1 / (1 + n!1)) * n!1)"
"g"
"LAMBDA (t: posreal): (1 - t) ^ n!1" ))
(("1"
(split
-1)
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(replace
-1)
(("1"
(hide
-1)
(("1"
(lemma
"fundamental3[posreal]"
("f"
"LAMBDA (t: posreal): (1 - t) ^ n!1"
"F"
"LAMBDA (x: posreal): -1 / (1 + n!1) * (1 - x) ^ (1 + n!1)"
"a"
"1"
"b"
"px!1" ))
(("1"
(replace
-2
-1)
(("1"
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(("1"
(replace
-2
-1)
(("1"
(name-replace
"DRL100"
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(hide-all-but
(-1
-9
-10
1))
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(rewrite
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-1)
(("1"
(rewrite
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-1)
(("1"
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1)
(("1"
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1)
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-2
:dir
rl)
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(expand
"abs" )
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(assert )
nil
nil ))
nil ))
nil )
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(replace
-1
*
rl)
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(expand
"^" )
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(expand
"expt" )
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(rewrite
"zero_times1" )
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(expand
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(propax)
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(hide
2)
(("2"
(hide
-2
-3
-4
-5)
(("2"
(hide
-1)
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(expand
"<="
-3)
(("2"
(split
-3)
(("1"
(lemma
"ln_estimate_scaf2"
("x"
"px!1"
"n"
"n!1" ))
(("1"
(name-replace
"F"
"LAMBDA (t: posreal): (1 - t) ^ n!1 / t" )
(("1"
(name-replace
"G"
"LAMBDA (t: posreal): (1 - t) ^ n!1" )
(("1"
(case
"even?(n!1)" )
(("1"
(lemma
"Integral_ge_0_open[posreal]"
("a"
"1"
"b"
"px!1"
"f"
"F" ))
(("1"
(lemma
"Integral_ge_0_open[posreal]"
("a"
"1"
"b"
"px!1"
"f"
"G" ))
(("1"
(assert )
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(split
-1)
(("1"
(split
-2)
(("1"
(expand
"abs" )
(("1"
(assert )
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(lemma
"Integral_diff[posreal]"
("a"
"1"
"b"
"px!1"
"f"
"G"
"g"
"F" ))
(("1"
(assert )
(("1"
(flatten)
(("1"
(lemma
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("a"
"1"
"b"
"px!1"
"f"
"LAMBDA (x: posreal): G(x) - F(x)" ))
(("1"
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(hide-all-but
(1
-7
-5
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(expand
"even?" )
(("1"
(skosimp*)
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(expand
"F" )
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(expand
"G" )
(("1"
(lemma
"posreal_times_posreal_is_posreal"
("px"
"(1 - x!1) ^ n!1"
"py"
"1-1/x!1" ))
(("1"
(assert )
nil
nil )
("2"
(assert )
(("2"
(hide
2)
(("2"
(typepred
"x!1" )
(("2"
(lemma
"div_mult_pos_lt1"
("py"
"x!1"
"x"
"1"
"z"
"1" ))
(("2"
(assert )
(("2"
(flatten)
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(typepred
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(assert )
nil
nil ))
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nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("3"
(hide
2)
(("3"
(replace
-1)
(("3"
(rewrite
"expt_times"
1)
(("1"
(lemma
"sq_nz_pos"
("nz"
"1-x!1" ))
(("1"
(expand
"sq" )
(("1"
(expand
"^"
1
1)
(("1"
(expand
"^"
1
2)
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(expand
"expt" )
(("1"
(expand
"expt" )
(("1"
(expand
"expt" )
(("1"
(lemma
"expt_pos"
("px"
"1 - 2 * x!1 + x!1 * x!1"
"i"
"j!1" ))
(("1"
(assert )
nil
nil )
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(assert )
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nil ))
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nil ))
nil ))
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nil ))
nil )
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(flatten)
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nil ))
nil ))
nil ))
nil )
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nil ))
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nil ))
nil ))
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nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
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nil ))
nil ))
nil ))
nil )
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(hide
-1
2)
(("2"
(skosimp)
(("2"
(expand
"F"
1)
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(typepred
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(rewrite
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1)
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(expand
"even?" )
(("2"
(skosimp)
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(replace
-5
1)
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(rewrite
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1)
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(lemma
"sq_nz_pos"
("nz"
"1-x!1" ))
(("1"
(expand
"sq" )
(("1"
(expand
"^"
1
1)
(("1"
(expand
"expt"
1)
(("1"
(expand
"expt"
1)
(("1"
(expand
"expt"
1)
(("1"
(lemma
"expt_pos"
("px"
"1 - 2 * x!1 + x!1 * x!1"
"i"
"j!1" ))
(("1"
(assert )
nil
nil )
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(assert )
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nil ))
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nil ))
nil ))
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(flatten)
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nil ))
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nil )
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(hide
-1
2)
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"even?" )
(("2"
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(("2"
(expand
"G" )
(("2"
(hide
-2
-4
-5)
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-1)
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(rewrite
"expt_times"
1)
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(lemma
"sq_nz_pos"
("nz"
"1-x!1" ))
(("1"
(expand
"sq" )
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(expand
"^"
1
1)
(("1"
(expand
"expt"
1)
(("1"
(expand
"expt"
1)
(("1"
(expand
"expt"
1)
(("1"
(lemma
"expt_pos"
("px"
"1 - 2 * x!1 + x!1 * x!1"
"i"
"j!1" ))
(("1"
(assert )
nil
nil )
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(assert )
nil
nil ))
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nil ))
nil ))
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nil ))
nil )
("2"
(flatten)
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nil ))
nil ))
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nil )
("2"
(flatten)
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(typepred
"x!1" )
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nil
nil ))
nil ))
nil ))
nil ))
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nil )
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(rewrite
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(("2"
(case
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(("1"
(case
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(("1"
(case
"forall (x:posreal): 1 < x & x <= px!1 => G(x) < F(x)" )
(("1"
(lemma
"Integral_ge_0_open[posreal]"
("a"
"1"
"b"
"px!1"
"f"
"-F" ))
(("1"
(lemma
"Integral_ge_0_open[posreal]"
("a"
"1"
"b"
"px!1"
"f"
"-G" ))
(("1"
(lemma
"Integral_neg[posreal]"
("a"
"1"
"b"
"px!1"
"f"
"F" ))
(("1"
(lemma
"Integral_neg[posreal]"
("a"
"1"
"b"
"px!1"
"f"
"G" ))
(("1"
(assert )
(("1"
(flatten)
(("1"
(assert )
(("1"
(split
-5)
(("1"
(split
-6)
(("1"
(expand
">="
(-1
-2))
(("1"
(expand
"<="
(-1
-2))
(("1"
(split
-1)
(("1"
(expand
"abs"
1)
(("1"
(assert )
(("1"
(split
-2)
(("1"
(assert )
(("1"
(lemma
"Integral_diff[posreal]"
("a"
"1"
"b"
"px!1"
"f"
"-G"
"g"
"-F" ))
(("1"
(assert )
(("1"
(flatten)
(("1"
(lemma
"Integral_ge_0_open[posreal]"
("a"
"1"
"b"
"px!1"
"f"
"LAMBDA (x: posreal): (-G)(x) - (-F)(x)" ))
(("1"
(replace
-2)
(("1"
(split
-1)
(("1"
(assert )
nil
nil )
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(hide
2)
(("2"
(assert )
nil
nil ))
nil )
("3"
(skosimp)
(("3"
(expand
"-"
1)
(("3"
(inst
-9
"x!1" )
(("3"
(assert )
(("3"
(typepred
"x!1" )
(("3"
(assert )
(("3"
(ground)
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(replace
-1
*
rl)
(("2"
(assert )
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(case-replace
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(lemma
"Integral_diff[posreal]"
("a"
"1"
"b"
"px!1"
"f"
"-G"
"g"
"-F" ))
(("1"
(assert )
(("1"
(replace
-3
-1
rl)
(("1"
(flatten)
(("1"
(lemma
"Integral_ge_0_open[posreal]"
("a"
"1"
"b"
"px!1"
"f"
"LAMBDA (x: posreal): (-G)(x) - (-F)(x)" ))
(("1"
(replace
-2)
(("1"
(split
-1)
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(assert )
nil
nil )
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(assert )
nil
nil )
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(skosimp)
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(expand
"-"
1)
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"x!1" )
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(assert )
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(typepred
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(ground)
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(hide
2)
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(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
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(expand
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(assert )
nil
nil ))
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nil )
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(skosimp)
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(expand
"-"
1)
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(inst
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"x!1" )
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(typepred
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(ground)
nil
nil ))
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nil )
("2"
(skosimp)
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(expand
"-"
1)
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"x!1" )
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(ground)
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
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nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(hide
2)
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(skosimp)
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(expand
"odd?" )
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(skosimp*)
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(expand
"G" )
(("2"
(expand
"F" )
(("2"
(rewrite
"div_mult_pos_lt2"
1)
(("2"
(lemma
"both_sides_times_neg_lt1"
("nz"
"(1 - x!1) ^ n!1"
"y"
"1"
"x"
"x!1" ))
(("1"
(assert )
nil
nil )
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(hide-all-but
(-1
-2
-7
-10
1
-5))
(("2"
(replace
-3)
(("2"
(rewrite
"expt_plus"
1)
(("2"
(rewrite
"expt_x1" )
(("2"
(lemma
"both_sides_times_pos_lt1"
("pz"
"(1 - x!1) ^ (2 * j!1)"
"x"
"1-x!1"
"y"
"0" ))
(("1"
(assert )
nil
nil )
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(hide
2)
(("2"
(rewrite
"expt_times" )
(("2"
(lemma
"sq_nz_pos"
("nz"
"1-x!1" ))
(("1"
(lemma
"expt_pos"
("px"
"sq(1 - x!1)"
"i"
"j!1" ))
(("1"
(expand
"sq" )
(("1"
(expand
"^"
1
1)
(("1"
(expand
"^"
1
2)
(("1"
(expand
"expt" )
(("1"
(expand
"expt" )
(("1"
(expand
"expt" )
(("1"
(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(propax)
nil
nil ))
nil )
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(assert )
nil
nil ))
nil ))
nil ))
nil )
("3"
(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(hide
2
-3
-5
-6)
(("2"
(skosimp)
(("2"
(inst
-
"x!1" )
(("2"
(assert )
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(expand
"F" )
(("2"
(expand
"G" )
(("2"
(rewrite
"div_mult_pos_lt1"
1)
(("2"
(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(hide
-2
-4
-5
2)
(("2"
(expand
"odd?" )
(("2"
(skosimp*)
(("2"
(expand
"G" )
(("2"
(replace
-3)
(("2"
(rewrite
"expt_plus" )
(("2"
(rewrite
"expt_x1" )
(("2"
(rewrite
"expt_times" )
(("2"
(lemma
"sq_nz_pos"
("nz"
"1-x!1" ))
(("1"
(expand
"sq" )
(("1"
(expand
"^"
1
1)
(("1"
(expand
"expt"
1)
(("1"
(expand
"expt"
1)
(("1"
(expand
"expt"
1)
(("1"
(lemma
"expt_pos"
("px"
"1 - 2 * x!1 + x!1 * x!1"
"i"
"j!1" ))
(("1"
(lemma
"both_sides_times_pos_lt1"
("pz"
"(1 - 2 * x!1 + x!1 * x!1) ^ j!1"
"x"
"1-x!1"
"y"
"0" ))
(("1"
(assert )
nil
nil )
("2"
(assert )
nil
nil ))
nil )
("2"
(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(replace
-1
*
rl)
(("2"
(lemma
"Integral_a_to_a[posreal]"
("a"
"1"
"f"
"LAMBDA (t: posreal): (1 - t) ^ n!1" ))
(("2"
(replace
-1)
(("2"
(lemma
"Integral_a_to_a[posreal]"
("a"
"1"
"f"
"LAMBDA (t: posreal): (1 - t) ^ n!1/t" ))
(("2"
(replace
-1)
(("2"
(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("3"
(lemma
"ln_estimate_scaf2"
("x"
"px!1"
"n"
"n!1" ))
(("3"
(propax)
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(hide-all-but
1)
(("2"
(skosimp)
(("2"
(lemma
"div_cancel1"
("x"
"(1 - x!1) ^ n!1"
"n0z"
"-1-n!1" ))
(("2"
(case-replace
"-1 / (1 + n!1) = 1/(-1-n!1)" )
(("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 ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (hide 2)
(("2" (skosimp)
(("2" (expand "continuous?" -1)
(("2" (inst - "x!1" ) nil nil )) nil ))
nil ))
nil ))
nil )
("2" (assert )
(("2" (expand "not_one_element?" )
(("2" (skosimp*)
(("2" (inst + "x!1+1" ) (("2" (assert ) nil nil )) nil ))
nil ))
nil ))
nil )
("3" (assert )
(("3" (expand "connected?" )
(("3" (skosimp*) (("3" (assert ) nil nil )) nil )) nil ))
nil ))
nil ))
nil ))
nil )
("2" (hide 2)
(("2" (lemma "const_fun_continuous[posreal]" ("u" "1" ))
(("2" (lemma "id_fun_continuous[posreal]" )
(("2" (expand "I" )
(("2" (expand "const_fun" )
(("2"
(lemma "diff_fun_continuous[posreal]"
("h1" "LAMBDA (x: posreal): 1" "h2"
"LAMBDA (x: posreal): x" ))
(("1" (expand "-" )
(("1" (induct "n" )
(("1"
(lemma "extensionality"
("f" "LAMBDA (t: posreal): (1 - t) ^ 0" "g"
"LAMBDA (x: posreal): 1" ))
(("1" (split -1)
(("1" (assert ) nil nil )
("2" (hide-all-but 1)
(("2" (skosimp)
(("2"
(expand "^" )
(("2"
(expand "expt" )
(("2" (propax) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (skosimp*)
(("2"
(lemma "prod_fun_continuous[posreal]"
("h1" "LAMBDA (x_1: posreal): 1 - x_1" "h2"
"LAMBDA (t: posreal): (1 - t) ^ j!1" ))
(("1"
(lemma "extensionality"
("f" "((LAMBDA (x_1: posreal): 1 - x_1) *
(LAMBDA (t: posreal): (1 - t) ^ j!1))"
"g"
"(LAMBDA (t: posreal): (1 - t) ^ (j!1 + 1))" ))
(("1" (split -1)
(("1" (assert ) nil nil )
("2"
(hide-all-but 1)
(("2"
(skosimp)
(("2"
(expand "*" )
(("2"
(expand "^" )
(("2"
(expand "expt" 1 3)
(("2" (propax) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (propax) nil nil )
("3" (propax) nil nil ))
nil ))
nil ))
nil ))
nil )
("2" (propax) nil nil ) ("3" (propax) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
((- const-decl "[numfield, numfield -> numfield]" number_fields nil )
(numfield nonempty-type-eq-decl nil number_fields nil )
(^ const-decl "real" exponentiation nil )
(/= const-decl "boolean" notequal nil )
(OR const-decl "[bool, bool -> bool]" booleans nil )
(continuous? const-decl "bool" continuous_functions "analysis/" )
(posreal nonempty-type-eq-decl nil real_types nil )
(> const-decl "bool" reals nil )
(nonneg_real nonempty-type-eq-decl nil real_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 )
(real_minus_real_is_real application-judgement "real" reals nil )
(deriv_const_fun formula-decl nil derivatives "analysis/" )
(deriv_id_fun formula-decl nil derivatives "analysis/" )
(diff_derivable_fun formula-decl nil derivatives "analysis/" )
(deriv_diff_fun formula-decl nil derivatives "analysis/" )
(derivable? const-decl "bool" derivatives "analysis/" )
(deriv_fun type-eq-decl nil derivatives "analysis/" )
(deriv_exp_fun formula-decl nil derivatives "analysis/" )
(+ const-decl "[numfield, numfield -> numfield]" number_fields nil )
(^ const-decl "[T -> real]" real_fun_ops "reals/" )
(nzrat_div_nzrat_is_nzrat application-judgement "nzrat" rationals
nil )
(scal_derivable_fun formula-decl nil derivatives "analysis/" )
(nznum nonempty-type-eq-decl nil number_fields nil )
(/ const-decl "[numfield, nznum -> numfield]" number_fields nil )
(- const-decl "[numfield -> numfield]" number_fields nil )
(deriv_scal_fun formula-decl nil derivatives "analysis/" )
(extensionality formula-decl nil functions nil )
(* const-decl "[numfield, numfield -> numfield]" number_fields nil )
(int_minus_int_is_int application-judgement "int" integers nil )
(div_cancel1 formula-decl nil real_props nil )
(fundamental3 formula-decl nil fundamental_theorem "analysis/" )
(nat_exp application-judgement "nat" exponentiation nil )
(rat_times_rat_is_rat application-judgement "rat" rationals nil )
(expt def-decl "real" exponentiation nil )
(even? const-decl "bool" integers nil )
(div_mult_pos_ge1 formula-decl nil real_props nil )
(Integral_diff formula-decl nil integral "analysis/" )
(F skolem-const-decl "[posreal -> real]" ln_exp_series_alt nil )
(posreal_times_posreal_is_posreal judgement-tcc nil real_types nil )
(IMPLIES const-decl "[bool, bool -> bool]" booleans nil )
(< const-decl "bool" reals nil )
(Open_interval type-eq-decl nil intervals_real "reals/" )
(posreal_div_posreal_is_posreal application-judgement "posreal"
real_types nil )
(div_mult_pos_lt1 formula-decl nil real_props nil )
(posreal_times_posreal_is_posreal application-judgement "posreal"
real_types nil )
(NOT const-decl "[bool -> bool]" booleans nil )
(sq_nz_pos judgement-tcc nil sq "reals/" )
(expt_pos formula-decl nil exponentiation nil )
(real_plus_real_is_real application-judgement "real" reals nil )
(sq const-decl "nonneg_real" sq "reals/" )
(expt_times formula-decl nil exponentiation nil )
(G skolem-const-decl "[posreal -> real]" ln_exp_series_alt nil )
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil )
(Integral_ge_0_open formula-decl nil integral "analysis/" )
(minus_even_is_even application-judgement "even_int" integers nil )
(Integral_neg formula-decl nil integral "analysis/" )
(- const-decl "[T -> real]" real_fun_ops "reals/" )
(both_sides_times_neg_lt1 formula-decl nil real_props nil )
(nonpos_real nonempty-type-eq-decl nil real_types nil )
(negreal nonempty-type-eq-decl nil real_types nil )
(odd_plus_even_is_odd application-judgement "odd_int" integers nil )
(expt_x1 formula-decl nil exponentiation nil )
(both_sides_times_pos_lt1 formula-decl nil real_props nil )
(expt_plus formula-decl nil exponentiation nil )
(div_mult_pos_lt2 formula-decl nil real_props nil )
(odd? const-decl "bool" integers nil )
(even_or_odd formula-decl nil naturalnumbers nil )
(ln_estimate_scaf2 formula-decl nil ln_exp_series_alt nil )
(Integral_a_to_a formula-decl nil integral "analysis/" )
(abs_0 formula-decl nil abs_lems "reals/" )
(nonzero_real nonempty-type-eq-decl nil reals nil )
(abs_div formula-decl nil real_props nil )
(nzint_abs_is_pos application-judgement "{j: posint | j >= i}"
real_defs nil )
(posrat_div_posrat_is_posrat application-judgement "posrat"
rationals nil )
(minus_real_is_real application-judgement "real" reals nil )
(nnreal_exp application-judgement "nnreal" exponentiation nil )
(real_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil )
(abs_expt formula-decl nil exponentiation nil )
(nzreal nonempty-type-eq-decl nil reals nil )
(zero_times1 formula-decl nil real_props nil )
(int_abs_is_nonneg application-judgement "{j: nonneg_int | j >= i}"
real_defs nil )
(nnint_times_nnint_is_nnint application-judgement "nonneg_int"
integers nil )
(even_times_int_is_even application-judgement "even_int" integers
nil )
(mult_divides1 application-judgement "(divides(n))" divides nil )
(mult_divides2 application-judgement "(divides(m))" divides nil )
(nnreal_times_nnreal_is_nnreal application-judgement "nnreal"
real_types nil )
(nzrat_abs_is_pos application-judgement "{r: posrat | r >= q}"
real_defs nil )
(abs_mult formula-decl nil real_props nil )
(= const-decl "[T, T -> boolean]" equalities nil )
(Integral const-decl "real" integral_def "analysis/" )
(Integrable_funs type-eq-decl nil integral_def "analysis/" )
(Integrable? const-decl "bool" integral_def "analysis/" )
(abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
nil )
(AND const-decl "[bool, bool -> bool]" booleans nil )
(<= const-decl "bool" reals nil )
(nat_expt application-judgement "nat" exponentiation nil )
(* const-decl "[T -> real]" real_fun_ops "reals/" )
(* const-decl "[T -> real]" real_fun_ops "reals/" )
(real_times_real_is_real application-judgement "real" reals nil )
(minus_odd_is_odd application-judgement "odd_int" integers nil )
(real_gt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil )
(- const-decl "[T -> real]" real_fun_ops "reals/" )
(real_le_is_total_order name-judgement "(total_order?[real])"
real_props nil )
(real_div_nzreal_is_real application-judgement "real" reals nil )
(nnint_plus_posint_is_posint application-judgement "posint"
integers nil )
(posint_plus_nnint_is_posint application-judgement "posint"
integers nil )
(I const-decl "(bijective?[T, T])" identity nil )
(identity_derivable_fun formula-decl nil derivatives "analysis/" )
(const_derivable_fun formula-decl nil derivatives "analysis/" )
(Closed_interval type-eq-decl nil intervals_real "reals/" )
(not_one_element? const-decl "bool" deriv_domain_def "analysis/" )
(connected? const-decl "bool" deriv_domain_def "analysis/" )
(continuous_Integrable? formula-decl nil integral "analysis/" )
(const_fun_continuous judgement-tcc nil continuous_functions
"analysis/" )
(diff_fun_continuous judgement-tcc nil continuous_functions
"analysis/" )
(continuous_fun nonempty-type-eq-decl nil continuous_functions
"analysis/" )
(pred type-eq-decl nil defined_types nil )
(nat_induction formula-decl nil naturalnumbers nil )
(prod_fun_continuous judgement-tcc nil continuous_functions
"analysis/" )
(const_fun const-decl "[T -> real]" real_fun_ops "reals/" )
(id_fun_continuous judgement-tcc nil continuous_functions
"analysis/" ))
nil )
(ln_estimate_scaf10-5 nil 3413217110
("" (stop-rewrite "abs_nat" )
((""
(case "forall (n:nat): continuous[posreal](LAMBDA (t: posreal): (1 - t) ^ n)" )
(("1" (skosimp)
(("1" (inst - "n!1" )
(("1"
(lemma "continuous_Integrable?[posreal]"
("a" "1" "b" "px!1" "f"
"LAMBDA (t: posreal): (1 - t) ^ n!1" ))
(("1" (split -1)
(("1" (lemma "const_derivable_fun[posreal]" ("b" "1" ))
(("1" (lemma "deriv_const_fun[posreal]" ("b" "1" ))
(("1" (lemma "identity_derivable_fun[posreal]" )
(("1" (lemma "deriv_id_fun[posreal]" )
(("1" (expand "I" )
(("1" (expand "const_fun" )
(("1"
(lemma "diff_derivable_fun[posreal]"
("f1" "LAMBDA (x: posreal): 1" "f2"
"LAMBDA (x: posreal): x" ))
(("1" (assert )
(("1"
(lemma
"deriv_diff_fun[posreal]"
("ff1"
"LAMBDA (x: posreal): 1"
"ff2"
"LAMBDA (x: posreal): x" ))
(("1"
(replace -3)
(("1"
(replace -5)
(("1"
(expand "-" )
(("1"
(lemma
"deriv_exp_fun[posreal]"
("n"
"1+n!1"
"f"
"LAMBDA (x_1: posreal): 1 - x_1" ))
(("1"
(assert )
(("1"
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(("1"
(flatten)
(("1"
(replace -3)
(("1"
(hide
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-4
-5
-6
-7
-8)
(("1"
(lemma
"scal_derivable_fun[posreal]"
("b"
"-1/(1+n!1)"
"f"
"LAMBDA (t: posreal): (1 - t) ^ (1 + n!1)" ))
(("1"
(assert )
(("1"
(lemma
"deriv_scal_fun[posreal]"
("b"
"-1/(1+n!1)"
"ff"
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(("1"
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(("1"
(replace
-4
-1)
(("1"
(assert )
(("1"
(lemma
"extensionality"
("f"
"(LAMBDA (x: posreal):
-1 * ((1 - x) ^ n!1 * (-1 / (1 + n!1))) -
(1 - x) ^ n!1 * (-1 / (1 + n!1)) * n!1)"
"g"
"LAMBDA (t: posreal): (1 - t) ^ n!1" ))
(("1"
(split
-1)
(("1"
(replace
-1)
(("1"
(hide
-1)
(("1"
(lemma
"fundamental3[posreal]"
("f"
"LAMBDA (t: posreal): (1 - t) ^ n!1"
"F"
"LAMBDA (x: posreal): -1 / (1 + n!1) * (1 - x) ^ (1 + n!1)"
"a"
"1"
"b"
"px!1" ))
(("1"
(replace
-2
-1)
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(replace
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(("1"
(replace
-2
-1)
(("1"
(name-replace
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(("1"
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(("1"
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(("1"
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(("1"
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(split
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(("1"
(rewrite
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-2
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rl)
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(expand
"abs" )
(("1"
(assert )
nil
nil ))
nil ))
nil )
("2"
(replace
-1
*
rl)
(("2"
(expand
"^" )
(("2"
(expand
"expt" )
(("2"
(rewrite
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(("2"
(expand
"abs" )
(("2"
(propax)
nil
nil ))
--> --------------------
--> maximum size reached
--> --------------------
Messung V0.5 in Prozent C=100 H=99 G=99
¤ Dauer der Verarbeitung: 0.657 Sekunden
(vorverarbeitet am 2026-04-30)
¤
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