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shostak))
(noa_gt_m1 0
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real_props nil )
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real_props nil )
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real_props nil )
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shostak))
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nil )
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(< const-decl "bool" reals nil ) (>= const-decl "bool" reals nil )
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(abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
nil )
(posrat_div_posrat_is_posrat application-judgement "posrat"
rationals nil )
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"(strict_total_order?[real])" real_props nil )
(nzrat_abs_is_pos application-judgement "{r: posrat | r >= q}"
real_defs nil )
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(deriv_domain_gtm1 0
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(nderiv_ln_TCC1 0
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((const_derivable_fun formula-decl nil derivatives "analysis/" )
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(* const-decl "[T -> real]" real_fun_ops "reals/" )
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real_types nil )
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real_types nil )
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nil )
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integers nil )
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real_props nil )
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integers nil )
(I const-decl "(bijective?[T, T])" identity nil )
(deriv_id_fun formula-decl nil derivatives "analysis/" )
(identity_derivable_fun formula-decl nil derivatives "analysis/" )
(ln_derivable formula-decl nil ln_exp nil )
(extensionality formula-decl nil functions nil )
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(factorial_1 formula-decl nil factorial "ints/" )
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(factorial_0 formula-decl nil factorial "ints/" )
(posreal_div_posreal_is_posreal application-judgement "posreal"
real_types nil )
(nnint_plus_posint_is_posint application-judgement "posint"
integers nil )
(minus_nzint_is_nzint application-judgement "nzint" integers nil )
(nat_induction formula-decl nil naturalnumbers nil )
(pred type-eq-decl nil defined_types nil )
(IMPLIES const-decl "[bool, bool -> bool]" booleans nil )
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real_types nil )
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nil )
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nil )
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(real nonempty-type-from-decl nil reals nil )
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(ln const-decl "real" ln_exp nil )
(= const-decl "[T, T -> boolean]" equalities 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 3302280454
(""
(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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"n"
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"-factorial(j!1 - 1)" ))
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(replace
-4)
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(lemma
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("f"
"-factorial(j!1 - 1) * (LAMBDA (x: posreal): 1 / -x) ^ j!1"
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(replace
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"j"
"2" ))
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(lemma
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(replace
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(lemma
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"-factorial(j!1 - 1) * (LAMBDA (x: posreal): 1 / -x) ^ j!1"
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(split
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(lemma
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"-factorial(j!1 - 1) *
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(LAMBDA (x: posreal): --1 / (-x * -x)))"
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"i"
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"j"
"2" ))
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(lemma "Taylors[posreal]"
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(lemma "sigma_eq[nat]"
("low" "0" "high" "n!1" "F"
"LAMBDA (nn:nat):
IF nn > n!1 THEN 0
ELSIF nn = 0 THEN ln(1)
ELSE nderiv(nn, ln)(1) * xgm1!1 ^ nn / factorial(nn)
ENDIF" " G" " LAMBDA (nn: nat):
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(expand "factorial" 1 2)
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(name-replace
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(lemma
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"((-1/ (-c!1) ^ (1 + n!1)) * xgm1!1 ^ (1 + n!1)) / (1+n!1)"
"n0z"
"K2" ))
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(name-replace
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(inst?)
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(assert )
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(simplify 2)
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"-factorial(n!2 - 1) / (-1) ^ n!2 * xgm1!1 ^ n!2"
"n0x"
"factorial(n!2)"
"y"
"-(-xgm1!1) ^ n!2"
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"0"
"high"
"n!1" ))
(("2"
(lemma
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("low"
"0"
"high"
"n!1"
"F"
"LAMBDA (i: nat): 0"
"G"
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real_props nil )
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(/= const-decl "boolean" notequal nil )
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nil )
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nil )
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nil )
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(lemma "sigma_eq[nat]"
("low" "0" "high" "n!1" "F"
"LAMBDA (nn:nat):
IF nn > n!1 THEN 0
ELSIF nn = 0 THEN ln(1)
ELSE nderiv(nn, ln)(1) * xgm1!1 ^ nn / factorial(nn)
ENDIF" " G" " LAMBDA (nn: nat):
IF nn = 0 THEN 0 ELSE -(-xgm1!1) ^ nn / nn ENDIF"))
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(name-replace
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"((-1/ (-c!1) ^ (1 + n!1)) * xgm1!1 ^ (1 + n!1)) / (1+n!1)"
"n0z"
"K2" ))
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-1)
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(name-replace
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"-factorial(n!2 - 1) / (-1) ^ n!2 * xgm1!1 ^ n!2"
"n0x"
"factorial(n!2)"
"y"
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"n0y"
"n!2" ))
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(replace
-1
2)
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3)
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(expand
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2
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"xgm1!1"
"n0y"
"-1"
"i"
"n!2" ))
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nil )
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(assert )
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(hide-all-but 1)
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(lemma
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"0"
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(lemma
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("low"
"0"
"high"
"n!1"
"F"
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(expand
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(expand
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(assert )
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nil ))
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nil ))
nil ))
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nil ))
nil ))
nil ))
nil ))
nil )
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real_props nil )
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(nzreal_div_nzreal_is_nzreal application-judgement "nzreal"
real_types nil )
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real_types nil )
(< const-decl "bool" reals nil )
(AND const-decl "[bool, bool -> bool]" booleans nil )
(abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
nil )
(/= const-decl "boolean" notequal nil )
(nznum nonempty-type-eq-decl nil number_fields nil )
(/ const-decl "[numfield, nznum -> numfield]" number_fields nil )
(OR const-decl "[bool, bool -> bool]" booleans nil )
(^ const-decl "real" exponentiation nil )
(IMPLIES const-decl "[bool, bool -> bool]" booleans nil )
(+ const-decl "[numfield, numfield -> numfield]" number_fields nil )
(= const-decl "[T, T -> boolean]" equalities nil )
(between type-eq-decl nil taylors "analysis/" )
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(Gt_m1 type-eq-decl nil ln_series nil )
(- const-decl "[numfield -> numfield]" number_fields nil )
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(> const-decl "bool" reals nil )
(nat nonempty-type-eq-decl nil naturalnumbers nil )
(>= const-decl "bool" reals nil )
(bool nonempty-type-eq-decl nil booleans 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
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shostak))
(lnp1_TCC1 0
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(- const-decl "[numfield -> numfield]" number_fields nil )
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nil )
(abslt1 type-eq-decl nil ln_series nil )
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nil ))
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nil )
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(("2" (skosimp*)
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nil ))
nil )
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nil ))
nil ))
nil ))
nil ))
nil )
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(number_field_pred const-decl "[number -> boolean]" number_fields
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(x1!1 skolem-const-decl "abslt1" ln_series nil )
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real_props nil )
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real_types nil ))
nil ))
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real_props nil )
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(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil )
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nil ))
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(lnp1-2 nil 3321095882
("" (skosimp*) (("" (expand "ln" ) (("" (propax) nil nil )) nil )) nil )
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(ln_estimate_increasing_odd 0
(ln_estimate_increasing_odd-1 nil 3558194748
(""
(case "FORALL (m:nat): derivable?[real](LAMBDA (pr: real): ln_estimate(pr, m))" )
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(("1" (copy -1)
(("1" (label "derivtt2" -1)
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(("1" (skeep)
(("1" (lemma "nonneg_derivative[real]" )
(("1" (inst-cp -2 "2*n+1" )
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(("1" (assert )
(("1" (ground)
(("1" (expand "increasing?" )
(("1" (inst - "px" "py" )
(("1" (assert ) nil )))))
("2" (hide (2 3))
(("2"
(case "FORALL (m:nat): FORALL (x: real):
deriv(LAMBDA (pr: real): ln_estimate(pr, m), x) = sigma(0,m-1,(LAMBDA (nn:nat): (-x)^nn))")
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(hide-all-but 1)
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(expand "^" )
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(expand "expt" )
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(grind)
nil )))))))))))
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(replaces -1)
(("2"
(rewrite "sigma_ge_0" )
nil )))))
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(label "xtp" 1)
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(hide "xtp" )
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(case
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(lemma
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(inst
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"1+2*n"
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"x" )
(("1"
(assert )
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(("1"
(skeep
-1)
(("1"
(replaces
-1)
(("1"
(assert )
nil )))))
("2"
(inst
+
"x" )
nil )))
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(case
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(("1"
(inst
-
"2*n" )
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(hide-all-but
(-1
+))
(("1"
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"^" )
nil )))))
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(hide-all-but
(-1
+))
(("2"
(induct
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3)
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(("2"
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"^" )
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(rewrite
-1)
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(expand
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+)
(("2"
(case
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-1
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(("1"
(assert )
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(grind
:exclude
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nil )))))))))))))))))))))))
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(lemma
"both_sides_expt_pos_ge_aux" )
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(inst
-
"2*n"
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"1" )
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(assert )
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(("1"
(inst
-
"n" )
(("1"
(ground)
nil )))
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(hide-all-but
1)
(("2"
(induct
"nn" )
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(grind)
nil )
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(grind)
nil )
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(skeep)
(("3"
(split)
(("1"
(expand
"^" )
(("1"
(expand
"expt"
+)
(("1"
(expand
"expt"
+)
(("1"
(assert )
nil )))))))
("2"
(assert )
(("2"
(expand
"expt"
+)
(("2"
(expand
"expt"
+)
(("2"
(propax)
nil )))))))))))))))))))))))))))))))))
("2"
(hide-all-but
1)
(("2"
(induct
"m" )
(("1"
(grind)
nil )
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(skeep)
(("2"
(expand
"^" )
(("2"
(expand
"expt"
+)
(("2"
(expand
"expt"
+)
(("2"
(assert )
nil )))))))))))))))))))))
("2"
(assert )
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(flatten)
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(replaces
-1)
(("2"
(assert )
nil )))))))))))))))))
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(hide-all-but 1)
(("2"
(induct "kk" )
(("1" (grind) nil )
("2"
(skeep)
(("2"
(assert )
(("2"
(expand
"sigma"
+)
(("2"
(replaces -1)
(("2"
(assert )
(("2"
(expand
"^" )
(("2"
(expand
"expt"
+
3)
(("2"
(assert )
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(name
"yy"
"expt(-x,j)" )
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(replaces
-1)
(("2"
(case
"yy = (-(1-yy)+(1-(-x)*yy))/(1+x)" )
(("1"
(assert )
nil )
("2"
(hide
2)
(("2"
(cross-mult
1)
nil )))))))))))))))))))))))))
("3"
(reveal "xtp" )
(("3"
(ground)
nil )))
("4"
(hide 2)
(("4"
(skeep)
(("4"
(assert )
nil )))))))))
("3"
(reveal "xtp" )
(("3"
(ground)
nil )))))))))))))))))))))
("2" (induct "m" )
(("1"
(hide 2)
(("1"
(skeep)
(("1"
(assert )
(("1"
(lemma "deriv_const[real]" )
(("1"
(expand "ln_estimate" +)
(("1"
(expand "sigma" +)
(("1"
(expand "sigma" +)
(("1"
(expand "const_fun" )
(("1"
(rewrite -1)
nil )))))))))))))))))
("2"
(hide 2)
(("2"
(skeep)
(("2"
(case "j = 0" )
(("1"
(replaces -1)
(("1"
(assert )
(("1"
(skeep)
(("1"
(expand "ln_estimate" +)
(("1"
(expand "sigma" +)
(("1"
(expand "sigma" +)
(("1"
(expand "sigma" +)
(("1"
(expand "^" )
(("1"
(expand "expt" +)
(("1"
(expand
"expt"
+)
(("1"
(assert )
(("1"
(lemma
"deriv_identity[real]" )
(("1"
(expand
"I" )
(("1"
(rewrite
-1)
nil )))))))))))))))))))))))))))
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(label "jpos" 1)
(("2"
(hide 1)
(("2"
(skeep)
(("2"
(inst - "x" )
(("2"
(lemma
"deriv_sum[real]" )
(("2"
(inst
-
"LAMBDA (pr: real): ln_estimate(pr, j)"
"LAMBDA (pr:real): -(-pr) ^ (1+j) / (1+j)"
"x" )
(("2"
(assert )
(("2"
(split -)
(("1"
(expand
"sigma"
+)
(("1"
(assert )
(("1"
(invoke
(case
"%1 = %2 AND %3 = %4 AND %5 = %6" )
(! -1 1)
(! 1 1)
(! -1 2 1)
(! 1 2 2)
(! -1 2 2)
(! 1 2 1))
(("1"
(flatten)
(("1"
(assert )
nil )))
("2"
(split)
(("1"
(invoke
(case
"%1 = %2" )
(!
1
1
1)
(!
1
2
1))
(("1"
(assert )
nil )
("2"
(hide
(2
3))
(("2"
(decompose-equality)
(("2"
(expand
"ln_estimate"
+)
(("2"
(expand
"+"
+)
(("2"
(expand
"sigma"
+
2)
(("2"
(propax)
nil )))))))))))))
("2"
(propax)
nil )
("3"
(lemma
"deriv_neg[real]" )
(("3"
(hide
(-2
2))
(("3"
(case
"EXISTS (c:real): (c=1 OR c=-1) AND (-x)^j = -c*x^j AND (LAMBDA (pr: real): -(-pr) ^ (1 + j) / (1 + j)) = (LAMBDA (pr: real): -c*(pr) ^ (1 + j) / (1 + j))" )
(("1"
(skeep
-1)
(("1"
(replace
-3)
(("1"
(replace
-2)
(("1"
(lemma
"deriv_scal[real]" )
(("1"
(inst
-
"-c / (1+j)"
"LAMBDA (pr: real): (pr) ^ (1 + j)"
"x" )
(("1"
(split
-)
(("1"
(invoke
(case
"%1 = %2" )
(!
-1
1
1)
(!
1
1
1))
(("1"
(replaces
-1)
(("1"
(replaces
-1)
(("1"
(lemma
"deriv_x_n[real]" )
(("1"
(inst
-
"LAMBDA (pr: real): (pr) ^ (1 + j)"
"1+j" )
(("1"
(assert )
(("1"
(flatten)
(("1"
(decompose-equality
-2)
(("1"
(inst
-
"x" )
(("1"
(expand
"deriv"
-1)
(("1"
(replaces
-1)
(("1"
(cross-mult
1)
nil )))))))))))))))))))))
("2"
(hide-all-but
1)
(("2"
(decompose-equality)
(("2"
(expand
"*" )
(("2"
(assert )
nil )))))))))
("2"
(lemma
"deriv_x_n[real]" )
(("2"
(inst
-
"LAMBDA (pr: real): (pr) ^ (1 + j)"
"1+j" )
(("2"
(assert )
(("2"
(flatten)
(("2"
(expand
"derivable?"
-1)
(("2"
(inst
-
"x" )
nil )))))))))))))))))))))))
("2"
(hide-all-but
1)
(("2"
(name
"cc"
"IF even?(j) THEN -1 ELSE 1 ENDIF" )
(("2"
(inst
+
"cc" )
(("2"
(split)
(("1"
(replaces
-1
:dir
rl)
(("1"
(grind)
nil )))
("2"
(lemma
"mult_expt" )
(("2"
(inst
-
"j"
"-1"
"x" )
(("1"
(assert )
(("1"
(replace
-1)
(("1"
(lemma
"even_m1_pow" )
(("1"
(inst
-
"j" )
(("1"
(lemma
"not_even_m1_pow" )
(("1"
(inst
-
"j" )
(("1"
(lift-if)
(("1"
(ground)
nil )))))))))))))))
("2"
(flatten)
(("2"
(replaces
-1)
(("2"
(reveal
"jpos" )
(("2"
(expand
"^" )
(("2"
(expand
"expt" )
(("2"
(assert )
nil )))))))))))))))
("3"
(decompose-equality)
(("3"
(lemma
"mult_expt" )
(("3"
(inst
-
"1+j"
"-1"
"x!1" )
(("1"
(replaces
-1)
(("1"
(lemma
"not_even_m1_pow" )
(("1"
(inst
-
"1+j" )
(("1"
(case
"even?(1+j) IFF (NOT even?(j))" )
(("1"
(replace
-1)
(("1"
(assert )
(("1"
(lemma
"even_m1_pow" )
(("1"
(inst
-
"1+j" )
(("1"
(replace
-2)
(("1"
(lift-if)
(("1"
(ground)
nil )))))))))))))
("2"
(assert )
(("2"
(hide-all-but
1)
(("2"
(expand
"even?" )
(("2"
(ground)
(("1"
(skosimp*)
(("1"
(assert )
(("1"
(replace
-2)
(("1"
(assert )
(("1"
(case
"EXISTS (kk:nat): 1= 2*kk" )
(("1"
(skosimp*)
(("1"
(assert )
nil )))
("2"
(inst
+
"j!1-j!2" )
(("2"
(assert )
nil )))))))))))))
("2"
(lemma
"even_or_odd" )
(("2"
(inst
-
"j" )
(("2"
(expand
"even?" )
(("2"
(expand
"odd?" )
(("2"
(ground)
(("2"
(skosimp*)
(("2"
(inst
3
"j!1+1" )
(("2"
(assert )
nil )))))))))))))))))))))))))))))))
("2"
(flatten)
(("2"
(replaces
-1)
(("2"
(reveal
"jpos" )
(("2"
(expand
"^" )
(("2"
(expand
"expt" )
(("2"
(assert )
nil )))))))))))))))))))))))))))))))))))))))
("2"
(inst -2 "j" )
(("2"
(expand
"derivable?"
-2)
(("2"
(inst
-
"x" )
nil )))))
("3"
(case
"EXISTS (c:real): (c=1 OR c=-1) AND (-x)^j = -c*x^j AND (LAMBDA (pr: real): -(-pr) ^ (1 + j) / (1 + j)) = (LAMBDA (pr: real): -c*(pr) ^ (1 + j) / (1 + j))" )
(("1"
(skeep -1)
(("1"
(replace
-3)
(("1"
(lemma
"derivable_scal[real]" )
(("1"
(inst
-
"-c / (1+j)"
"LAMBDA (pr: real): (pr) ^ (1 + j)" )
(("1"
(expand
"*"
-1)
(("1"
(expand
"derivable?"
-1)
(("1"
(inst
-
"x" )
(("1"
(assert )
nil )))))))
("2"
(lemma
"deriv_x_n[real]" )
(("2"
(inst
-
"LAMBDA (pr: real): (pr) ^ (1 + j)"
"1+j" )
(("2"
(assert )
nil )))))))))))))
("2"
(hide-all-but
1)
(("2"
(name
"cc"
"IF even?(j) THEN -1 ELSE 1 ENDIF" )
(("2"
(inst
+
"cc" )
(("2"
(split)
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(replaces
-1
:dir
rl)
(("1"
(grind)
nil )))
("2"
(lemma
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(inst
-
"j"
"-1"
"x" )
(("1"
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-1)
(("1"
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(("1"
(inst
-
"j" )
(("1"
(lemma
"not_even_m1_pow" )
(("1"
(inst
-
"j" )
(("1"
(lift-if)
(("1"
(ground)
nil )))))))))))))))
("2"
(reveal
"jpos" )
(("2"
(flatten)
(("2"
(replaces
-1)
(("2"
(expand
"^" )
(("2"
(expand
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(("2"
(assert )
nil )))))))))))))))
("3"
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(("3"
(lemma
"mult_expt" )
(("3"
(inst
-
"1+j"
"-1"
"x!1" )
(("1"
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-1)
(("1"
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(("1"
(inst
-
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(("1"
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(("1"
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(("1"
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(("1"
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(ground)
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(assert )
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1)
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(skosimp*)
(("1"
(assert )
nil )))
("2"
(inst
+
"j!1-j!2" )
(("2"
(assert )
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("2"
(lemma
"even_or_odd" )
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(inst
-
"j" )
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(expand
"odd?" )
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nil )))))))))))))))))))))))))))))))
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(flatten)
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(replaces
-1)
(("2"
(assert )
nil )))))))))))))))))))))))))))))))))))))))))))))
("3" (skosimp*) (("3" (assert ) nil )))
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(reveal "derivtt2" )
(("4"
(skeep)
(("4"
(inst - "m" )
(("4"
(expand "derivable?" -1)
(("4" (inst - "x" ) nil )))))))))))
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(("4"
(inst - "m" )
(("4"
(expand "derivable?" -1)
(("4" (inst - "x" ) nil )))))))))))))))
("2" (inst - "1+2*n" ) nil )))))
("2" (lemma "connected_real" )
(("2" (propax) nil )))))))))))))))
("2" (hide 2)
(("2" (induct "m" )
(("1" (expand "ln_estimate" )
(("1" (expand "sigma" )
(("1" (expand "sigma" )
(("1" (lemma "derivable_const[real]" )
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(("1" (expand "expt" )
(("1" (expand "expt" )
(("1"
(lemma "identity_derivable[real]" )
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(expand "derivable?" +)
(("1"
(skeep)
(("1"
(inst - "x" )
(("1"
(expand "I" )
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(assert )
nil )))))))))))))))))))))))))))))
("2" (label "jpos" 1)
(("2" (hide 1)
(("2" (expand "ln_estimate" )
(("2" (expand "sigma" +)
(("2" (lemma "derivable_sum[real]" )
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(inst -
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"LAMBDA (pr: real): -(-pr) ^ (1 + j) / (1 + j)" )
(("1"
(invoke (case "%1 = %2" ) (! -1 1) (! 1 1))
(("1" (assert ) nil )
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(expand "+" )
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(expand "ln_estimate" )
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(hide 2)
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(inst
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"LAMBDA (pr: real): (pr) ^ (1 + j)"
"1+j" )
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(assert )
nil )))))))))))))))))
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(hide-all-but 1)
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(name
"cc"
"IF even?(j) THEN -1 ELSE 1 ENDIF" )
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(assert )
nil )))
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nil )
((derivable_sum judgement-tcc nil derivatives "analysis/" )
(x!1 skolem-const-decl "real" ln_series nil )
(j skolem-const-decl "nat" ln_series nil )
(identity_derivable formula-decl nil derivatives_def "analysis/" )
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(connected_real formula-decl nil deriv_domain "analysis/" )
(even_plus_odd_is_odd application-judgement "odd_int" integers nil )
(nnint_plus_posint_is_posint application-judgement "posint"
integers 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 )
(real_le_is_total_order name-judgement "(total_order?[real])"
real_props nil )
(not_one_element_real formula-decl nil deriv_domain "analysis/" )
(deriv_domain_real formula-decl nil deriv_domain "analysis/" )
(deriv_const formula-decl nil derivatives_def "analysis/" )
(const_fun const-decl "[T -> real]" real_fun_ops "reals/" )
(sigma_0_neg formula-decl nil sigma_nat "reals/" )
(x!1 skolem-const-decl "real" ln_series nil )
(IFF const-decl "[bool, bool -> bool]" booleans nil )
(int_plus_int_is_int application-judgement "int" integers nil )
(even_or_odd formula-decl nil naturalnumbers nil )
(nnrat_div_posrat_is_nnrat application-judgement "nonneg_rat"
rationals nil )
(even_m1_pow formula-decl nil exponentiation nil )
(not_even_m1_pow formula-decl nil exponentiation nil )
(x skolem-const-decl "real" ln_series nil )
(odd_int nonempty-type-eq-decl nil integers nil )
(odd? const-decl "bool" integers nil )
(even? const-decl "bool" integers nil )
(IF const-decl "[boolean, T, T -> T]" if_def nil )
(* const-decl "[T -> real]" real_fun_ops "reals/" )
(div_cancel3 formula-decl nil real_props nil )
(times_div2 formula-decl nil real_props nil )
(deriv const-decl "[T -> real]" derivatives "analysis/" )
(deriv_x_n formula-decl nil derivatives "analysis/" )
(deriv_scal formula-decl nil derivatives_def "analysis/" )
(deriv_neg formula-decl nil derivatives_def "analysis/" )
(+ const-decl "[T -> real]" real_fun_ops "reals/" )
(IMPLIES const-decl "[bool, bool -> bool]" booleans nil )
(x!1 skolem-const-decl "real" ln_series nil )
(minus_int_is_int application-judgement "int" integers nil )
(nat_exp application-judgement "nat" exponentiation nil )
(derivable_scal judgement-tcc nil derivatives "analysis/" )
(j skolem-const-decl "nat" ln_series nil )
(deriv_sum formula-decl nil derivatives_def "analysis/" )
(I const-decl "(bijective?[T, T])" identity nil )
(deriv_identity formula-decl nil derivatives_def "analysis/" )
(real_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil )
(odd_minus_odd_is_even application-judgement "even_int" integers
nil )
(real_div_nzreal_is_real application-judgement "real" reals nil )
(real_plus_real_is_real application-judgement "real" reals nil )
(nznum nonempty-type-eq-decl nil number_fields nil )
(/ const-decl "[numfield, nznum -> numfield]" number_fields nil )
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(mult_expt formula-decl nil exponentiation nil )
(rat_div_nzrat_is_rat application-judgement "rat" rationals nil )
(hat_02n formula-decl nil power_series "series/" )
(real_times_real_is_real application-judgement "real" reals nil )
(nzint_times_nzint_is_nzint application-judgement "nzint" integers
nil )
(nzrat_div_nzrat_is_nzrat application-judgement "nzrat" rationals
nil )
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(< const-decl "bool" reals nil )
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(abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
nil )
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(> const-decl "bool" reals nil )
(nnreal_times_nnreal_is_nnreal application-judgement "nnreal"
real_types nil )
(nnreal type-eq-decl nil real_types nil )
(posreal_plus_nnreal_is_posreal application-judgement "posreal"
real_types nil )
(nnreal_exp application-judgement "nnreal" exponentiation nil )
(posint_times_posint_is_posint application-judgement "posint"
integers nil )
(posnat_expt application-judgement "posnat" exponentiation nil )
(both_sides_expt_pos_ge_aux formula-decl nil exponentiation nil )
(real_gt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil )
(nonzero_real nonempty-type-eq-decl nil reals nil )
(div_mult_pos_neg_ge1 formula-decl nil extra_real_props nil )
(nzreal nonempty-type-eq-decl nil reals nil )
(x skolem-const-decl "real" ln_series nil )
(even_minus_odd_is_odd application-judgement "odd_int" integers
nil )
(div_cancel4 formula-decl nil real_props nil )
(sigma_ge_0 formula-decl nil sigma "reals/" )
(minus_even_is_even application-judgement "even_int" integers nil )
(int_exp application-judgement "int" exponentiation nil )
(int_expt application-judgement "int" exponentiation nil )
(NOT const-decl "[bool -> bool]" booleans nil )
(AND const-decl "[bool, bool -> bool]" booleans nil )
(subrange type-eq-decl nil integers nil )
(expt def-decl "real" exponentiation nil )
(nzreal_exp application-judgement "nzreal" exponentiation nil )
(expt_1i formula-decl nil exponentiation nil )
(minus_odd_is_odd application-judgement "odd_int" integers nil )
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil )
(- const-decl "[numfield -> numfield]" number_fields nil )
(^ const-decl "real" exponentiation nil )
(/= const-decl "boolean" notequal nil )
(- const-decl "[numfield, numfield -> numfield]" number_fields 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 "real" derivatives_def "analysis/" )
(derivable? const-decl "bool" derivatives_def "analysis/" )
(= const-decl "[T, T -> boolean]" equalities nil )
(minus_real_is_real application-judgement "real" reals nil )
(int_minus_int_is_int application-judgement "int" integers nil )
(increasing? const-decl "bool" real_fun_preds "reals/" )
(odd_plus_even_is_odd application-judgement "odd_int" integers nil )
(posint_plus_nnint_is_posint application-judgement "posint"
integers nil )
(deriv_fun type-eq-decl nil derivatives "analysis/" )
(n skolem-const-decl "nat" ln_series 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 )
(connected? const-decl "bool" deriv_domain_def "analysis/" )
(nonneg_derivative formula-decl nil derivative_props "analysis/" )
(not_one_element? const-decl "bool" deriv_domain_def "analysis/" )
(deriv_domain? const-decl "bool" deriv_domain_def "analysis/" )
(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 )
(derivable? const-decl "bool" derivatives "analysis/" )
(ln_estimate const-decl "real" ln_series nil ))
nil ))
(ln_estimate_increasing_even 0
(ln_estimate_increasing_even-1 nil 3558194834
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nil )
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(("2"
(case "-(-x1) ^ (1 + 2 * j) / (1 + 2 * j)
+ -(-x1) ^ (2 + 2 * j) / (2 + 2 * j)
<=
-(-y1) ^ (1 + 2 * j) / (1 + 2 * j)
+ -(-y1) ^ (2 + 2 * j) / (2 + 2 * j)")
(("1" (assert ) nil nil )
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nil ))
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nil ))
nil )
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1)
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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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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 ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(hide
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(("5" (inst - "(LAMBDA (x: {x: real | x > -1}): x+1)" )
(("1" (expand "/" )
(("1" (lemma "continuous_Integrable?[abslt1]" )
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((nz_continuous_fun type-eq-decl nil continuous_functions
"analysis/" )
(continuous_Integrable? formula-decl nil integral "analysis/" )
(linear_fun_cont formula-decl nil continuous_functions "analysis/" )
(inv_fun_continuous formula-decl nil continuous_functions
"analysis/" )
(Int_diff_dom formula-decl nil integral_diff_doms "analysis/" )
(limit_series_shift formula-decl nil series "series/" )
(sigma def-decl "real" sigma "reals/" )
(integ_powerseq const-decl "sequence[real]" power_series_integ
"series/" )
(limit const-decl "real" convergence_sequences "analysis/" )
(m1_conv formula-decl nil power_series_conv "series/" )
(powerseq const-decl "sequence[real]" power_series "series/" )
(integral_powerseries formula-decl nil power_series_integ
"series/" )
(convergence const-decl "bool" convergence_sequences "analysis/" )
(Integral const-decl "real" integral_def "analysis/" )
(Integrable_funs type-eq-decl nil integral_def "analysis/" )
(Integrable? const-decl "bool" integral_def "analysis/" ))
shostak)))
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