(weierstrass_approximation
(IMP_real_fun_on_compact_sets_TCC1 0
(IMP_real_fun_on_compact_sets_TCC1-1 nil 3479658253
("" (lemma "real_metric_space" ) (("" (inst?) nil nil )) nil )
((fullset const-decl "set" sets nil ) (set type-eq-decl nil sets nil )
(bool nonempty-type-eq-decl nil booleans 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_metric_space formula-decl nil real_metric_space nil ))
nil ))
(Weierstrass_approx_combin1_TCC1 0
(Weierstrass_approx_combin1_TCC1-1 nil 3479744181
("" (subtype-tcc) nil nil ) nil nil ))
(Weierstrass_approx_combin1_TCC2 0
(Weierstrass_approx_combin1_TCC2-1 nil 3479744181
("" (subtype-tcc) nil nil ) ((/= const-decl "boolean" notequal nil ))
nil ))
(Weierstrass_approx_combin1_TCC3 0
(Weierstrass_approx_combin1_TCC3-1 nil 3479744181
("" (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 )
(> 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 )
(>= const-decl "bool" reals nil )
(nonneg_int nonempty-type-eq-decl nil integers nil )
(posnat 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 )
(real_gt_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 ))
nil ))
(Weierstrass_approx_combin1 0
(Weierstrass_approx_combin1-1 nil 3479744182
("" (skeep)
((""
(case "FORALL (pp: nat): FORALL (kz: nat): pp<=2 AND kz+1 > pp IMPLIES
sigma(0,kz+1,LAMBDA (i:nat): IF i > kz+1 THEN 0 ELSE i^pp*C(kz+1,i)*(-1)^(kz+1-i) ENDIF) = 0")
(("1" (inst - "p" )
(("1" (inst - "k-1" ) (("1" (assert ) nil nil )) nil )) nil )
("2" (hide 2)
(("2" (hide -1)
(("2" (hide -1)
(("2" (induct "pp" )
(("1" (skosimp*)
(("1" (assert )
(("1" (lemma "binomial_theorem" )
(("1" (inst - "kz!1+1" "1" "-1" )
(("1" (assert )
(("1" (expand "^" -1 1)
(("1" (expand "expt" -1 1)
(("1" (lemma "sigma_restrict_eq" )
(("1"
(inst
-
"LAMBDA (i: nat):
IF i > kz!1+1 THEN 0 ELSE C(kz!1+1, i) * 1 ^ i * (-1) ^ (kz!1+1 - i) ENDIF"
"LAMBDA (i: nat):
IF i > kz!1+1 THEN 0 ELSE C(kz!1+1, i) * i ^ 0 * (-1) ^ (kz!1+1 - i) ENDIF"
"kz!1+1"
"0" )
(("1"
(assert )
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(decompose-equality)
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(expand "restrict" )
(("1"
(lift-if)
(("1"
(ground)
(("1"
(rewrite "expt_1i" )
(("1"
(expand "^" +)
(("1"
(expand "expt" +)
(("1"
(propax)
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
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(skosimp*)
(("2" (assert ) nil nil ))
nil ))
nil ))
nil ))
nil )
("2"
(skosimp*)
(("2" (assert ) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (skosimp*)
(("2" (lemma "sigma_split" )
(("2"
(inst - "LAMBDA (i: nat):
IF i > kz!1 + 1 THEN 0
ELSE (i ^ (j!1 + 1)) * C(kz!1 + 1, i) * (-1) ^ (kz!1 + 1 - i)
ENDIF" " kz!1 + 1" " 0" " 0")
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(("1" (hide -1)
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(("1" (expand "sigma" 1 1)
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(case "0^(1+j!1) = 0" )
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(replace -1)
(("1"
(hide -1)
(("1"
(assert )
(("1"
(case
"NOT (LAMBDA (i: nat):
IF i > 1 + kz!1 THEN 0
ELSE C(1 + kz!1, i) * (i ^ (1 + j!1)) * (-1) ^ (1 - i + kz!1)
ENDIF) = (LAMBDA (i: nat):
-(1+kz!1)*IF i > 1 + kz!1 or i = 0 THEN 0
ELSE C(kz!1, i-1) * (i ^ j!1) * (-1) ^ (kz!1-i)
ENDIF)")
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(hide 2)
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(hide -1)
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(ground)
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(replace -1)
(("1"
(case
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(("1"
(assert )
nil
nil )
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(expand
"^"
1)
(("2"
(expand
"expt"
1)
(("2"
(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil )
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(expand "C" )
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(case
"(-1) ^ (1 - x!1 + kz!1) = -(-1)^(kz!1-x!1)" )
(("1"
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-1)
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(hide
-1)
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(expand
"factorial"
2
1)
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(expand
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(case
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(hide
-1)
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nil
nil ))
nil ))
nil )
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"^"
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"expt"
1
1)
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(propax)
nil
nil ))
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nil ))
nil ))
nil ))
nil ))
nil ))
nil )
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(case
"kz!1 = x!1" )
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(replace
-1)
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(assert )
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(expand
"^"
1)
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(expand
"expt" )
(("1"
(expand
"expt" )
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(assert )
nil
nil ))
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nil ))
nil ))
nil ))
nil )
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(case
"kz!1 = x!1 -1" )
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-1)
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(assert )
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(expand
"^"
2)
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"expt"
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2)
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nil ))
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"^"
3)
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(assert )
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(expand
"expt"
3
1)
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nil )
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(replace -1)
(("2"
(hide -1)
(("2"
(lemma "sigma_shift_T2" )
(("2"
(inst
-
"(LAMBDA (i: nat):
-(1 + kz!1) *
IF i > 1 + kz!1 OR i = 0 THEN 0
ELSE C(kz!1, i - 1) * (i ^ j!1) * (-1) ^ (kz!1 - i)
ENDIF)"
"kz!1"
"0"
"1" )
(("1"
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(replace -1)
(("1"
(hide -1)
(("1"
(lemma
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(("1"
(inst
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"LAMBDA (i_1: nat):
IF 1 + i_1 > 1 + kz!1 THEN 0
ELSE (C(kz!1, i_1) * (-1) ^ (- i_1 + kz!1)) *
((1 + i_1) ^ j!1)
ENDIF"
"(1+kz!1)"
"kz!1"
"0" )
(("1"
(lemma
"sigma_restrict_eq" )
(("1"
(inst
-
"LAMBDA (i: nat):
(1 + kz!1) *
IF 1 + i > 1 + kz!1 THEN 0
ELSE (C(kz!1, i) * (-1) ^ (-i + kz!1)) * ((1 + i) ^ j!1)
ENDIF"
"LAMBDA (i_1: nat):
-(1 + kz!1) *
IF 1 + i_1 > 1 + kz!1 THEN 0
ELSE (C(kz!1, i_1) * (-1) ^ (-1 - i_1 + kz!1)) *
((1 + i_1) ^ j!1)
ENDIF"
"kz!1"
"0" )
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-1)
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-1)
(("1"
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-1)
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-1)
(("1"
(inst
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"kz!1-1" )
(("1"
(name
"AA"
"LAMBDA (i_1: nat):
IF 1 + i_1 > 1 + kz!1 THEN 0
ELSE (C(kz!1, i_1) * (-1) ^ (-i_1 + kz!1)) *
((1 + i_1) ^ j!1)
ENDIF")
(("1"
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-1)
(("1"
(case
"sigma(0,kz!1,AA) = 0" )
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nil )
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2)
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(name
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IF i > kz!1 THEN 0
ELSE C(kz!1, i) * i ^ j!1 * (-1) ^ (kz!1 - i)
ENDIF")
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(case
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IF 1 + i_1 > 1 + kz!1 THEN 0
ELSE C(kz!1, i_1) * (-1) ^ (-i_1 + kz!1)
ENDIF)")
(("1"
(case
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(replace
-1)
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(("1"
(lemma
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"CC"
"kz!1"
"0" )
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(expand
"+"
+)
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rl)
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(assert )
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nil )
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(hide
2)
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(replace
-1
+
:dir
rl)
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(hide
-1)
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(hide
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(hide
-1)
(("2"
(lemma
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(inst
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"kz!1"
"1"
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(assert )
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(expand
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-1
1)
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(case
"(LAMBDA (i: nat):
IF i > kz!1 THEN 0
ELSE C(kz!1, i) * 1 ^ i * (-1) ^ (kz!1 - i)
ENDIF) = (LAMBDA (i_1: nat):
IF 1 + i_1 > 1 + kz!1 THEN 0
ELSE C(kz!1, i_1) * (-1) ^ (-i_1 + kz!1)
ENDIF)")
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"^"
1)
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1)
(("1"
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"^"
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nil )
((- const-decl "[numfield, numfield -> numfield]" number_fields nil )
(- const-decl "[numfield -> numfield]" number_fields nil )
(C const-decl "posnat" binomial "reals/" )
(posnat nonempty-type-eq-decl nil integers nil )
(nonneg_int nonempty-type-eq-decl nil integers nil )
(^ const-decl "real" exponentiation nil )
(/= const-decl "boolean" notequal nil )
(* const-decl "[numfield, numfield -> numfield]" number_fields nil )
(IF const-decl "[boolean, T, T -> T]" if_def nil )
(sigma def-decl "real" sigma "reals/" )
(T_high type-eq-decl nil sigma "reals/" )
(T_low type-eq-decl nil sigma "reals/" )
(OR const-decl "[bool, bool -> bool]" booleans nil )
(= const-decl "[T, T -> boolean]" equalities nil )
(+ const-decl "[numfield, numfield -> numfield]" number_fields nil )
(numfield nonempty-type-eq-decl nil number_fields nil )
(> const-decl "bool" reals nil ) (<= const-decl "bool" reals 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 )
(int_minus_int_is_int application-judgement "int" integers nil )
(minus_odd_is_odd application-judgement "odd_int" integers nil )
(rat_times_rat_is_rat application-judgement "rat" rationals nil )
(nnint_plus_posint_is_posint application-judgement "posint"
integers nil )
(NOT const-decl "[bool -> bool]" booleans nil )
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil )
(real_le_is_total_order name-judgement "(total_order?[real])"
real_props nil )
(real_gt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil )
(real_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil )
(int_plus_int_is_int application-judgement "int" integers nil )
(pred type-eq-decl nil defined_types nil )
(nat_induction formula-decl nil naturalnumbers nil )
(posint_plus_nnint_is_posint application-judgement "posint"
integers nil )
(sigma_restrict_eq formula-decl nil sigma "reals/" )
(nat_expt application-judgement "nat" exponentiation nil )
(mult_divides2 application-judgement "(divides(m))" divides nil )
(mult_divides1 application-judgement "(divides(n))" divides nil )
(even_times_int_is_even application-judgement "even_int" integers
nil )
(nnint_times_nnint_is_nnint application-judgement "nonneg_int"
integers nil )
(restrict const-decl "[T -> real]" sigma "reals/" )
(expt_1i formula-decl nil exponentiation nil )
(int_expt application-judgement "int" exponentiation nil )
(nzreal_expt application-judgement "nzreal" exponentiation nil )
(nat_exp application-judgement "nat" exponentiation nil )
(nzreal_exp application-judgement "nzreal" exponentiation nil )
(rat_exp application-judgement "rat" exponentiation nil )
(posint_exp application-judgement "posint" exponentiation nil )
(posint_times_posint_is_posint application-judgement "posint"
integers nil )
(kz!1 skolem-const-decl "nat" weierstrass_approximation nil )
(expt def-decl "real" exponentiation nil )
(int_exp application-judgement "int" exponentiation nil )
(odd_plus_odd_is_even application-judgement "even_int" integers
nil )
(nzrat_times_nzrat_is_nzrat application-judgement "nzrat" rationals
nil )
(binomial_theorem formula-decl nil polynomials "reals/" )
(sigma_split formula-decl nil sigma "reals/" )
(even_minus_odd_is_odd application-judgement "odd_int" integers
nil )
(int_times_even_is_even application-judgement "even_int" integers
nil )
(real_plus_real_is_real application-judgement "real" reals nil )
(sigma_shift_T2 formula-decl nil sigma "reals/" )
(even_plus_odd_is_odd application-judgement "odd_int" integers nil )
(rat nonempty-type-eq-decl nil rationals nil )
(odd_plus_even_is_odd application-judgement "odd_int" integers nil )
(AA skolem-const-decl "[nat -> rational]" weierstrass_approximation
nil )
(BB skolem-const-decl "[nat -> rat]" weierstrass_approximation nil )
(+ const-decl "[T -> real]" real_fun_ops "reals/" )
(rat_plus_rat_is_rat application-judgement "rat" rationals nil )
(sigma_sum formula-decl nil sigma "reals/" )
(posnat_expt application-judgement "posnat" exponentiation nil )
(real_times_real_is_real application-judgement "real" reals nil )
(sigma_scal formula-decl nil sigma "reals/" )
(minus_nzint_is_nzint application-judgement "nzint" integers nil )
(factorial_1 formula-decl nil factorial "ints/" )
(nzint_times_nzint_is_nzint application-judgement "nzint" integers
nil )
(factorial_0 formula-decl nil factorial "ints/" )
(minus_int_is_int application-judgement "int" integers nil )
(posrat_div_posrat_is_posrat application-judgement "posrat"
rationals nil )
(factorial def-decl "posnat" factorial "ints/" )
(nzrat_div_nzrat_is_nzrat application-judgement "nzrat" rationals
nil )
(minus_nzrat_is_nzrat application-judgement "nzrat" rationals nil )
(odd_minus_even_is_odd application-judgement "odd_int" integers
nil )
(kz!1 skolem-const-decl "nat" weierstrass_approximation nil ))
shostak))
(Weierstrass_approximation_0_1 0
(Weierstrass_approximation_0_1-1 nil 3479658386
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(("" (lemma "closed_intervals_compact" )
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nil ))
nil ))
nil )
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(("2"
(case "NOT EXISTS (M: posreal): FORALL (x: (closed_intv(0,1))): abs(f(x)) < M" )
(("1" (hide "approx" )
(("1" (lemma "cont_on_compact_max" )
(("1" (inst?)
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(inst?)
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(assert )
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(skosimp*)
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(name
"Mz"
"max(abs(f(s!1))+1,abs(f(s!2))+1)" )
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(inst + "Mz" )
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(skosimp*)
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(expand "abs" +)
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(lift-if)
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(expand "abs" -2 1)
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nil ))
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nil )
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(inst - "x!1" )
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(inst - "x!1" )
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(expand
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2)
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nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (lemma "Heine" )
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(("1"
(label
"fbsname"
-1)
(("1"
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(("1"
(label
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-1)
(("1"
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(("1"
(lemma
"Bern_poly_inverse_def" )
(("1"
(inst
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"fbseq" )
(("1"
(inst
+
"Bern_poly_inverse(fbseq)"
"fbseq`index" )
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nil )
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-1)
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(case
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nil )
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(cross-mult
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nil ))
nil ))
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nil ))
nil ))
nil ))
nil ))
nil ))
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((2 * M!1) / sq(delta!1)) * (1 / N!1)")
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nil
nil ))
nil )
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(hide-all-but
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1)
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nil ))
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(label
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(label
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-1)
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(name
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-1)
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(("1"
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-1
-2)
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-1)
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(case
"((2 * M!1) / sq(delta!1)) *
(sq(x!1) + (1 / N!1) * (x!1 - sq(x!1)) - 2 * y!1 * x!1 + sq(y!1))
+ epsilon!1 / 2 = (epsilon!1 / 2) + ((2 * M!1) / sq(delta!1)) * sq(x!1 - y!1) +
((2 * M!1) / sq(delta!1)) * (1 / N!1) * (x!1 - sq(x!1))")
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nil )
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(hide
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1)
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(field
1)
nil
nil ))
nil ))
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nil )
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(case
"NOT (LAMBDA (i_1: nat):
IF i_1 > N!1 THEN 0
ELSE sigma(0, i_1,
LAMBDA (i: nat):
IF i > i_1 THEN 0
ELSE sq((i / N!1) - y!1) * C(i_1, i) *
C(N!1, i_1)
* (-1) ^ (i_1 - i)
ENDIF)
ENDIF
* (IF i_1 = 0 THEN 1 ELSE x!1 ^ i_1 ENDIF)) = (LAMBDA (i_1: nat):
IF i_1 > 3 THEN 0
ELSE sigma(0, i_1,
LAMBDA (i: nat):
IF i > i_1 THEN 0
ELSE sq((i / N!1) - y!1) * C(i_1, i) *
C(N!1, i_1)
* (-1) ^ (i_1 - i)
ENDIF)
ENDIF
* (IF i_1 = 0 THEN 1 ELSE x!1 ^ i_1 ENDIF))")
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IF i > x!2 THEN 0
ELSE sq((i / N!1) - y!1) * C(N!1, x!2) * C(x!2, i) *
(-1) ^ (x!2 - i)
ENDIF")
(("1"
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-1)
(("1"
(name
"BB"
"LAMBDA (i: nat):
IF i > x!2 THEN 0
ELSE sq(i / N!1) * C(N!1, x!2) * C(x!2, i) *
(-1) ^ (x!2 - i)
ENDIF")
(("1"
(name
"CC"
"LAMBDA (i: nat):
IF i > x!2 THEN 0
ELSE -2*(i / N!1)*y!1 * C(N!1, x!2) * C(x!2, i) *
(-1) ^ (x!2 - i)
ENDIF")
(("1"
(name
"DD"
"LAMBDA (i: nat):
IF i > x!2 THEN 0
ELSE sq(y!1) * C(N!1, x!2) * C(x!2, i) *
(-1) ^ (x!2 - i)
ENDIF")
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(expand
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:dir
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nil ))
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(expand
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+)
(("2"
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IF i > x!2 THEN 0
ELSE C(x!2, i) * i ^ 0 * (-1) ^ (x!2 - i)
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IF i_1 > x!2 THEN 0
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ENDIF) = (LAMBDA (i: nat):
IF i > x!2 THEN 0
ELSE sq(y!1) * C(N!1, x!2) * C(x!2, i) * (-1) ^ (x!2 - i)
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IF i > x!2 THEN 0
ELSE C(x!2, i) * i ^ 2 * (-1) ^ (x!2 - i)
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ELSE C(x!2, i_1) * i_1 ^ 2 * (-1) ^ (x!2 - i_1)
ENDIF) = (LAMBDA (i: nat):
IF i > x!2 THEN 0
ELSE sq(i / N!1) * C(N!1, x!2) * C(x!2, i) * (-1) ^ (x!2 - i)
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(("3"
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(hide
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(("3"
(hide
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(("3"
(hide
-1)
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(inst
-
"x!2"
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(assert )
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(mult-by
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(lemma
"sigma_scal" )
(("1"
(inst
-
"(LAMBDA (i: nat):
IF i > x!2 THEN 0
ELSE C(x!2, i) * i ^ 1 * (-1) ^ (x!2 - i)
ENDIF)"
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(("1"
(case
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ELSE C(x!2, i_1) * i_1 ^ 1 * (-1) ^ (x!2 - i_1)
ENDIF) = (LAMBDA (i: nat):
IF i > x!2 THEN 0
ELSE -2 *
(C(N!1, x!2) * C(x!2, i) * (-1) ^ (x!2 - i) * (i / N!1)
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nil
nil ))
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(skosimp*)
(("2"
(assert )
nil
nil ))
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(skosimp*)
(("3"
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nil
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(("4"
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nil
nil ))
nil )
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(skosimp*)
(("5"
(assert )
nil
nil ))
nil ))
nil ))
nil )
("2"
(replace
-1)
(("2"
(hide
-1)
(("2"
(hide
"sqdiffseqdef" )
(("2"
(name
"AA"
"(LAMBDA (i_1: nat):
IF i_1 > 3 THEN 0
ELSE sigma(0, i_1,
LAMBDA (i: nat):
IF i > i_1 THEN 0
ELSE sq((i / N!1) - y!1) * C(i_1, i) *
C(N!1, i_1)
* (-1) ^ (i_1 - i)
ENDIF)
ENDIF
* (IF i_1 = 0 THEN 1 ELSE x!1 ^ i_1 ENDIF))")
(("2"
(replace
-1)
(("2"
(lemma
"sigma_split" )
(("2"
(inst
-
"AA"
"N!1"
"0"
"3" )
(("2"
(assert )
(("2"
(replace
-1)
(("2"
(hide
-1)
(("2"
(case
"sigma(4,N!1,AA) = 0" )
(("1"
(replace
-1)
(("1"
(assert )
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(hide
-1)
(("1"
(replace
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:dir
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(("1"
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-1)
(("1"
(expand
"sigma" )
(("1"
(expand
"sigma" )
(("1"
(expand
"sigma" )
(("1"
(expand
"sigma" )
(("1"
(assert )
(("1"
(case
"C(N!1,0) = 1 AND C(N!1,1) = N!1 AND C(N!1,2) = N!1*(N!1-1)/2 AND C(N!1,3) = N!1*(N!1-1)*(N!1-2)/6" )
(("1"
(flatten)
(("1"
(replace
-1)
(("1"
(hide
-1)
(("1"
(replace
-1)
(("1"
(hide
-1)
(("1"
(replace
-1)
(("1"
(hide
-1)
(("1"
(replace
-1)
(("1"
(hide
-1)
(("1"
(mult-by
1
"sq(N!1)" )
(("1"
(grind)
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(hide
2)
(("2"
(split
+)
(("1"
(expand
"C" )
(("1"
(assert )
nil
nil ))
nil )
("2"
(expand
"C" )
(("2"
(expand
"factorial"
1
1)
(("2"
(assert )
nil
nil ))
nil ))
nil )
("3"
(expand
"C" )
(("3"
(expand
"factorial"
1
1)
(("3"
(expand
"factorial"
1
1)
(("3"
(expand
"factorial"
1
3)
(("3"
(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil )
("4"
(expand
"C" )
(("4"
(expand
"factorial"
1
1)
(("4"
(expand
"factorial"
1
1)
(("4"
(expand
"factorial"
1
1)
(("4"
(expand
"factorial"
1
2)
(("4"
(assert )
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(case
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(("1"
(replaces
-1)
(("1"
(assert )
nil
nil ))
nil )
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(hide
2)
(("2"
(grind)
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(lemma
"sigma_restrict_eq" )
(("2"
(inst
-
"AA"
"LAMBDA (i:nat): 0"
"N!1"
"4" )
(("2"
(split
-1)
(("1"
(lemma
"sigma_zero" )
(("1"
(inst
-
"N!1"
"4" )
(("1"
(assert )
nil
nil ))
nil ))
nil )
("2"
(hide
3)
(("2"
(decompose-equality)
(("2"
(expand
"restrict" )
(("2"
(lift-if)
(("2"
(hide
2)
(("2"
(replace
-1
+
:dir
rl)
(("2"
(hide
-1)
(("2"
(ground)
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("3"
(skosimp*)
(("3"
(assert )
nil
nil ))
nil )
("4"
(skosimp*)
(("4"
(assert )
nil
nil ))
nil )
("5"
(skosimp*)
(("5"
(assert )
nil
nil ))
nil )
("6"
(skosimp*)
(("6"
(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(name
"sqdiffzzz"
"(# index := N!1,bern_seq := (LAMBDA (pj: upto(N!1)): 2*M!1*sq(((pj/N!1)-y!1)/delta!1) + epsilon!1/2) #)" )
(("2"
(label
"sqdiffzzzname"
-1)
(("2"
(name
"BNzzz"
"Bern_poly(sqdiffzzz)" )
(("1"
(label
"BNzzzname"
-1)
(("1"
(case
"abs(BNf(x!1)-f(y!1)) <= BNzzz(x!1)" )
(("1"
(case
"BNzzz(x!1) = ((2 * M!1) / sq(delta!1)) * BNdiff(x!1) + epsilon!1 / 2" )
(("1"
(assert )
(("1"
(replace
-1)
(("1"
(propax)
nil
nil ))
nil ))
nil )
("2"
(assert )
(("2"
(hide
2)
(("2"
(hide
"bapprox2" )
(("2"
(hide
-1)
(("2"
(replace
"BNzzzname"
1
:dir
rl)
(("2"
(replace
"sqdiffzzzname"
1
:dir
rl)
(("2"
(hide
-1)
(("2"
(hide
-1)
(("2"
(replace
"BNdiffname"
:dir
rl)
(("2"
(replace
"sqdiffseqdef"
1
:dir
rl)
(("2"
(hide
"BNdiffname" )
(("2"
(hide
"sqdiffseqdef" )
(("2"
(hide
"BNfname" )
(("2"
(hide
"fbsname" )
(("2"
(hide
"fcont" )
(("2"
(hide
"intcompact" )
(("2"
(hide
"funif" )
(("2"
(hide
"nonempty" )
(("2"
(expand
"Bern_poly" )
(("2"
(lemma
"sigma_scal" )
(("2"
(inst
-
"LAMBDA (i: nat):
IF i > N!1 THEN 0
ELSE sq((i / N!1) - y!1) * Bern(i, N!1)(x!1)
ENDIF"
"((2 * M!1) / sq(delta!1))"
"N!1"
"0" )
(("1"
(replace
-1
:dir
rl)
(("1"
(hide
-1)
(("1"
(lemma
"Bernstein_partition_of_unity" )
(("1"
(inst
-
"N!1"
"x!1" )
(("1"
(mult-by
-1
"epsilon!1/2" )
(("1"
(lemma
"sigma_scal" )
(("1"
(inst
-
"LAMBDA (i: nat):
IF i > N!1 THEN 0 ELSE Bern(i, N!1)(x!1) ENDIF"
"epsilon!1/2"
"N!1"
"0" )
(("1"
(replace
-1
:dir
rl)
(("1"
(hide
-1)
(("1"
(name
"AA"
"LAMBDA (i: nat):
IF i > N!1 THEN 0
ELSE Bern(i, N!1)(x!1) * (epsilon!1 / 2) +
2 *
(Bern(i, N!1)(x!1) * sq(((i / N!1) - y!1) / delta!1) *
M!1)
ENDIF")
(("1"
(replace
-1)
(("1"
(name
"BB"
"LAMBDA (i_1: nat):
epsilon!1 / 2 *
IF i_1 > N!1 THEN 0 ELSE Bern(i_1, N!1)(x!1) ENDIF")
(("1"
(replace
-1)
(("1"
(hide
-1)
(("1"
(name
"CC"
"LAMBDA (i_1: nat):
((2 * M!1) / sq(delta!1)) *
IF i_1 > N!1 THEN 0
ELSE sq((i_1 / N!1) - y!1) * Bern(i_1, N!1)(x!1)
ENDIF")
(("1"
(replace
-1)
(("1"
(replace
-3
:dir
rl)
(("1"
(case
"AA = BB + CC" )
(("1"
(replace
-1)
(("1"
(hide
-1)
(("1"
(expand
"+"
1
1)
(("1"
(lemma
"sigma_sum" )
(("1"
(inst?)
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nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
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(hide
2)
(("2"
(decompose-equality)
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(expand
"+"
1)
(("2"
(expand
"AA"
+)
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(expand
"BB"
+)
(("2"
(expand
"CC"
+)
(("2"
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(ground)
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-1)
(("2"
(hide
-1)
(("2"
(hide
-1)
(("2"
(hide
"funif2" )
(("2"
(hide
"Mdef" )
(("2"
(grind
:exclude
"Bern" )
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 ))
nil ))
nil ))
nil ))
nil )
("2"
(skosimp*)
(("2"
(assert )
nil
nil ))
nil ))
nil ))
nil )
("2"
(skosimp*)
(("2"
(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(skosimp*)
(("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 )
("2"
(hide
2)
(("2"
(name
"BNfdiffseq"
"(# index:= N!1, bern_seq := (LAMBDA (pj:upto(N!1)): f(pj/N!1) - f(y!1)) #)" )
(("2"
(label
"BNfdiffseq"
-1)
(("2"
(name
"BNfdiff"
"Bern_poly(BNfdiffseq)" )
(("1"
(label
"BNfdiffname"
-1)
(("1"
(label
"BNfdiffseqname"
-2)
(("1"
(case
"abs(BNf(x!1)-f(y!1)) = abs(BNfdiff(x!1))" )
(("1"
(case
"abs(BNfdiff(x!1)) <= BNzzz(x!1)" )
(("1"
(assert )
nil
nil )
("2"
(hide
2)
(("2"
(hide
-1)
(("2"
(replace
"BNfdiffname"
1
:dir
rl)
(("2"
(replace
"BNzzzname"
1
:dir
rl)
(("2"
(expand
"BNfdiffseq"
1)
(("2"
(expand
"sqdiffzzz"
1)
(("2"
(expand
"Bern_poly"
1)
(("2"
(hide
"bapprox2" )
(("2"
(hide
"nonempty" )
(("2"
(hide
"BNfname" )
(("2"
(hide
"fbsname" )
(("2"
(hide
"sqdiffseqdef" )
(("2"
(hide
"sqdiffzzzname" )
(("2"
(lemma
"sigma_abs" )
(("2"
(inst
-
"(LAMBDA (i: nat):
IF i > N!1 THEN 0
ELSE Bern(i, N!1)(x!1) * f(i / N!1) -
Bern(i, N!1)(x!1) * f(y!1)
ENDIF)"
"N!1"
"0" )
(("1"
(lemma
"sigma_le" )
(("1"
(inst
-
"LAMBDA (n: nat):
abs(IF n > N!1 THEN 0
ELSE Bern(n, N!1)(x!1) * f(n / N!1) -
Bern(n, N!1)(x!1) * f(y!1)
ENDIF)"
"LAMBDA (i: nat):
IF i > N!1 THEN 0
ELSE Bern(i, N!1)(x!1) * (epsilon!1 / 2) +
2 *
(Bern(i, N!1)(x!1) * sq(((i / N!1) - y!1) / delta!1)
* M!1)
ENDIF"
"N!1"
"0" )
(("1"
(assert )
(("1"
(hide
2)
(("1"
(hide
-1)
(("1"
(skosimp*)
(("1"
(case
"FORALL (x1,y1: (closed_intv(0,1))): abs(f(x1)-f(y1)) <= 2*M!1*sq((x1-y1)/delta!1) + epsilon!1/2" )
(("1"
(inst
-
"n!1/N!1"
"y!1" )
(("1"
(lemma
"Bern_nonnegative" )
(("1"
(inst
-
"x!1" )
(("1"
(assert )
(("1"
(inst
-
"n!1"
"N!1" )
(("1"
(mult-by
-2
"Bern(n!1,N!1)(x!1)" )
(("1"
(case
"abs(Bern(n!1,N!1)(x!1)) = Bern(n!1,N!1)(x!1)" )
(("1"
(replace
-1
-2
:dir
rl)
(("1"
(hide
-1)
(("1"
(rewrite
"abs_mult"
:dir
rl)
(("1"
(expand
"abs"
-1
2)
(("1"
(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(expand
"abs"
1)
(("2"
(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(assert )
(("2"
(cross-mult
1)
nil
nil ))
nil ))
nil )
("2"
(assert )
(("2"
(hide
2)
(("2"
(skosimp*)
(("2"
(case
"abs(x1!1-y1!1) < delta!1" )
(("1"
(inst
"funif2"
"x1!1"
"y1!1" )
(("1"
(assert )
nil
nil ))
nil )
("2"
(case
"sq((x1!1 - y1!1)/delta!1) >= 1" )
(("1"
(mult-by
-1
"2*M!1" )
(("1"
(case
"abs(f(x1!1)-f(y1!1)) <= 2*M!1" )
(("1"
(assert )
nil
nil )
("2"
(lemma
"triangle" )
(("2"
(inst
-
"f(x1!1)"
"-f(y1!1)" )
(("2"
(rewrite
"abs_neg" )
(("2"
(copy
"Mdef" )
(("2"
(inst-cp
-
"x1!1" )
(("2"
(inst
-
"y1!1" )
(("2"
(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(lemma
"sq_lt" )
(("2"
(inst
-
"abs(x1!1-y1!1)"
"delta!1" )
(("2"
(rewrite
"sq_abs" )
(("2"
(assert )
(("2"
(rewrite
"sq_div" )
(("2"
(cross-mult
2)
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(assert )
(("2"
(skosimp*)
(("2"
(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(skosimp*)
(("2"
(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(case
"BNf(x!1) - f(y!1) = BNfdiff(x!1)" )
(("1"
(assert )
nil
nil )
("2"
(hide
2)
(("2"
(hide
2)
(("2"
(hide
2)
(("2"
(hide
2)
(("2"
(replace
"BNfname"
1
:dir
rl)
(("2"
(replace
"BNfdiffname"
1
:dir
rl)
(("2"
(replace
"fbsname"
1
:dir
rl)
(("2"
(replace
"BNfdiffseqname"
1
:dir
rl)
(("2"
(expand
"Bern_poly"
1)
(("2"
(hide
"fcont" )
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(hide
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(("2"
(hide
"Mdef" )
(("2"
(hide
"funif" )
(("2"
(hide
"funif2" )
(("2"
(lemma
"Bernstein_partition_of_unity" )
(("2"
(inst
-
"N!1"
"x!1" )
(("2"
(mult-by
-1
"-f(y!1)" )
(("1"
(lemma
"sigma_scal" )
(("1"
(inst
-
"LAMBDA (i: nat):
IF i > N!1 THEN 0 ELSE Bern(i, N!1)(x!1) ENDIF"
"-f(y!1)"
"N!1"
"0" )
(("1"
(replace
-1
:dir
rl)
(("1"
(hide
-1)
(("1"
(hide
"sqdiffzzzname" )
(("1"
(hide
"BNfdiffseqname" )
(("1"
(name
"AA"
"LAMBDA (i: nat):
IF i > N!1 THEN 0
ELSE Bern(i, N!1)(x!1) * f(i / N!1) -
Bern(i, N!1)(x!1) * f(y!1)
ENDIF")
(("1"
(replace
-1)
(("1"
(name
"BB"
"LAMBDA (i: nat):
IF i > N!1 THEN 0 ELSE f(i / N!1) * Bern(i, N!1)(x!1) ENDIF")
(("1"
(replace
-1)
(("1"
(name
"CC"
"LAMBDA (i_1: nat):
-f(y!1) * IF i_1 > N!1 THEN 0 ELSE Bern(i_1, N!1)(x!1) ENDIF")
(("1"
(replace
-1)
(("1"
(case
"AA = BB + CC" )
(("1"
(lemma
"sigma_sum" )
(("1"
(replace
-2
1)
(("1"
(expand
"+"
+)
(("1"
(inst
-
"BB"
"CC"
"N!1"
"0" )
(("1"
(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(hide
2)
(("2"
(decompose-equality)
(("2"
(expand
"+"
1)
(("2"
(expand
"AA"
+)
(("2"
(expand
"BB"
+)
(("2"
(expand
"CC"
+)
(("2"
(lift-if)
(("2"
(ground)
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(skosimp*)
(("2"
(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(skosimp*)
(("2"
(assert )
nil
nil ))
nil ))
nil ))
nil )
("2"
(skosimp*)
(("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"
(ground)
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(skosimp*)
(("2"
(hide
2)
(("2"
(hide
2)
(("2"
(hide
2)
(("2"
(assert )
(("2"
(split
+)
(("1"
(skosimp*)
(("1"
(ground)
nil
nil ))
nil )
("2"
(skosimp*)
(("2"
(assert )
nil
nil ))
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nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(ground)
(("2"
(skosimp*)
(("2"
(ground)
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(lemma
"archimedean" )
(("2"
(inst
-
"1/max(5,(M!1 / (delta!1 ^ 2 * epsilon!1)))" )
(("2"
(skosimp*)
(("2"
(cross-mult
-1)
(("2"
(expand
"max" )
(("2"
(lift-if)
(("2"
(ground)
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(lemma "archimedean" )
(("2"
(inst
-
"1/max(5,(M!1 / (delta!1 ^ 2 * epsilon!1)))" )
(("2"
(skosimp*)
(("2"
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(("1"
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(("1"
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("2"
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(("2"
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((closed_intervals_compact formula-decl nil real_metric_space nil )
(Heine formula-decl nil uniform_continuity nil )
(posreal_times_posreal_is_posreal application-judgement "posreal"
real_types 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 )
(above nonempty-type-eq-decl nil integers nil )
(* const-decl "[numfield, numfield -> numfield]" number_fields nil )
(OR const-decl "[bool, bool -> bool]" booleans nil )
(^ const-decl "real" exponentiation nil )
(archimedean formula-decl nil real_props nil )
(odd_times_odd_is_odd application-judgement "odd_int" integers nil )
(real_plus_real_is_real application-judgement "real" reals nil )
(both_sides_times_pos_le1_imp formula-decl nil extra_real_props
nil )
(div_mult_pos_le2 formula-decl nil real_props nil )
(sq_0 formula-decl nil sq "reals/" )
(nnreal_times_nnreal_is_nnreal application-judgement "nnreal"
real_types nil )
(real_times_real_is_real application-judgement "real" reals nil )
(polynomial const-decl "[real -> real]" polynomials "reals/" )
(int_exp application-judgement "int" exponentiation nil )
(sigma_0_neg formula-decl nil sigma_nat "reals/" )
(factorial_1 formula-decl nil factorial "ints/" )
(factorial_0 formula-decl nil factorial "ints/" )
(factorial def-decl "posnat" factorial "ints/" )
(posrat_times_posrat_is_posrat application-judgement "posrat"
rationals nil )
(int_plus_int_is_int application-judgement "int" integers nil )
(even_times_int_is_even application-judgement "even_int" integers
nil )
(both_sides_times1 formula-decl nil real_props nil )
(rat_div_nzrat_is_rat application-judgement "rat" rationals nil )
(restrict const-decl "[T -> real]" sigma "reals/" )
(sigma_zero formula-decl nil sigma "reals/" )
(sigma_nat application-judgement "nat" sigma_nat "reals/" )
(sigma_nnreal application-judgement "nnreal" sigma_nat "reals/" )
(sigma_restrict_eq formula-decl nil sigma "reals/" )
(even_minus_odd_is_odd application-judgement "odd_int" integers
nil )
(sigma_split formula-decl nil sigma "reals/" )
(Weierstrass_approx_combin1 formula-decl nil
weierstrass_approximation nil )
(+ const-decl "[T -> real]" real_fun_ops "reals/" )
(nzreal_expt application-judgement "nzreal" exponentiation nil )
(int_expt application-judgement "int" exponentiation nil )
(nzrat_div_nzrat_is_nzrat application-judgement "nzrat" rationals
nil )
(nzint_times_nzint_is_nzint application-judgement "nzint" integers
nil )
(nnrat_times_nnrat_is_nnrat application-judgement "nonneg_rat"
rationals nil )
(posint_times_posint_is_posint application-judgement "posint"
integers nil )
(mult_divides1 application-judgement "(divides(n))" divides nil )
(mult_divides2 application-judgement "(divides(m))" divides nil )
(rat_exp application-judgement "rat" exponentiation nil )
(nzrat_times_nzrat_is_nzrat application-judgement "nzrat" rationals
nil )
(rat_times_rat_is_rat application-judgement "rat" rationals nil )
(sigma_sum formula-decl nil sigma "reals/" )
(minus_nzint_is_nzint application-judgement "nzint" integers nil )
(minus_even_is_even application-judgement "even_int" integers nil )
(nat_exp application-judgement "nat" exponentiation nil )
(nnint_times_nnint_is_nnint application-judgement "nonneg_int"
integers nil )
(real_div_nzreal_is_real application-judgement "real" reals nil )
(times_div1 formula-decl nil real_props nil )
(zero_times1 formula-decl nil real_props nil )
(nat_expt application-judgement "nat" exponentiation nil )
(sigma_scal formula-decl nil sigma "reals/" )
(x!2 skolem-const-decl "nat" weierstrass_approximation nil )
(both_sides_times1_imp formula-decl nil extra_real_props nil )
(N!1 skolem-const-decl "above(5)" weierstrass_approximation nil )
(IMPLIES const-decl "[bool, bool -> bool]" booleans nil )
(C const-decl "posnat" binomial "reals/" )
(posnat nonempty-type-eq-decl nil integers nil )
(nonneg_int nonempty-type-eq-decl nil integers nil )
(sigma def-decl "real" sigma "reals/" )
(T_high type-eq-decl nil sigma "reals/" )
(T_low type-eq-decl nil sigma "reals/" )
(IF const-decl "[boolean, T, T -> T]" if_def nil )
(int_minus_int_is_int application-judgement "int" integers nil )
(minus_odd_is_odd application-judgement "odd_int" integers nil )
(nnreal_plus_nnreal_is_nnreal application-judgement "nnreal"
real_types nil )
(BNfdiffseq skolem-const-decl
"[# bern_seq: [upto(N!1) -> real], index: above(5) #]"
weierstrass_approximation nil )
(abs_neg formula-decl nil abs_lems "reals/" )
(triangle formula-decl nil real_props nil )
(both_sides_times_pos_ge1_imp formula-decl nil extra_real_props
nil )
(div_mult_pos_ge1 formula-decl nil real_props nil )
(nnreal_div_posreal_is_nnreal application-judgement "nnreal"
real_types nil )
(sq_div formula-decl nil sq "reals/" )
(sq_abs formula-decl nil sq "reals/" )
(sq_lt formula-decl nil sq "reals/" )
(subrange type-eq-decl nil integers nil )
(n!1 skolem-const-decl "subrange(0, N!1)" weierstrass_approximation
nil )
(abs_mult formula-decl nil real_props nil )
(x!1 skolem-const-decl "(LAMBDA (x: real): 0 <= x AND x <= 1)"
weierstrass_approximation nil )
(Bern_nonnegative formula-decl nil bernstein_polynomials "reals/" )
(div_mult_pos_le1 formula-decl nil real_props nil )
(abs_nat formula-decl nil abs_lems "reals/" )
(sigma_le formula-decl nil sigma "reals/" )
(sigma_abs formula-decl nil sigma "reals/" )
(sqdiffzzz skolem-const-decl
"[# bern_seq: [upto(N!1) -> posreal], index: above(5) #]"
weierstrass_approximation nil )
(BB skolem-const-decl "[nat -> numfield]" weierstrass_approximation
nil )
(CC skolem-const-decl "[nat -> real]" weierstrass_approximation
nil )
(AA skolem-const-decl "[nat -> numfield]" weierstrass_approximation
nil )
(Bernstein_partition_of_unity formula-decl nil
bernstein_polynomials "reals/" )
(posreal_exp application-judgement "posreal" exponentiation nil )
(BB skolem-const-decl "[nat -> real]" weierstrass_approximation
nil )
(CC skolem-const-decl "[nat -> real]" weierstrass_approximation
nil )
(AA skolem-const-decl "[nat -> numfield]" weierstrass_approximation
nil )
(Bern const-decl "real" bernstein_polynomials "reals/" )
(Bern_poly_inverse_def formula-decl nil bernstein_polynomials
"reals/" )
(Bern_poly_inverse const-decl "sequence[real]"
bernstein_polynomials "reals/" )
(sequence type-eq-decl nil sequences nil )
(div_mult_pos_lt1 formula-decl nil real_props nil )
(times_div2 formula-decl nil real_props nil )
(nonzero_real nonempty-type-eq-decl nil reals nil )
(div_mult_pos_lt2 formula-decl nil real_props nil )
(div_mult_pos_gt2 formula-decl nil extra_real_props nil )
(expt def-decl "real" exponentiation nil )
(both_sides_times_pos_gt1 formula-decl nil real_props nil )
(nzreal_exp application-judgement "nzreal" exponentiation nil )
(real_gt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil )
(sq_nz_pos application-judgement "posreal" sq "reals/" )
(sq const-decl "nonneg_real" sq "reals/" )
(- const-decl "[numfield, numfield -> numfield]" number_fields nil )
(posreal_plus_nnreal_is_posreal application-judgement "posreal"
real_types nil )
(IFF const-decl "[bool, bool -> bool]" booleans nil )
(Bern_poly const-decl "[real -> real]" bernstein_polynomials
"reals/" )
(Bernstein_Polynomial type-eq-decl nil bernstein_polynomials
"reals/" )
(upto nonempty-type-eq-decl nil naturalnumbers nil )
(nat nonempty-type-eq-decl nil naturalnumbers nil )
(nnrat_div_posrat_is_nnrat application-judgement "nonneg_rat"
rationals nil )
(n!1 skolem-const-decl "posnat" weierstrass_approximation nil )
(posreal_div_posreal_is_posreal application-judgement "posreal"
real_types nil )
(uniformly_continuous? const-decl "bool" uniform_continuity nil )
(cont_on_compact_min formula-decl nil real_fun_on_compact_sets nil )
(nnreal_plus_posreal_is_posreal application-judgement "posreal"
real_types nil )
(posreal_max application-judgement
"{z: posreal | z >= x AND z >= y}" real_defs nil )
(= const-decl "[T, T -> boolean]" equalities nil )
(max const-decl "{p: real | p >= m AND p >= n}" real_defs nil )
(+ const-decl "[numfield, numfield -> numfield]" number_fields nil )
(real_le_is_total_order name-judgement "(total_order?[real])"
real_props nil )
(minus_real_is_real application-judgement "real" reals nil )
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil )
(real_dist const-decl "nnreal" real_metric_space nil )
(nnreal type-eq-decl nil real_types nil )
(cont_on_compact_max formula-decl nil real_fun_on_compact_sets nil )
(NOT const-decl "[bool -> bool]" booleans nil )
(>= const-decl "bool" reals nil )
(nonneg_real nonempty-type-eq-decl nil real_types nil )
(> const-decl "bool" reals nil )
(posreal nonempty-type-eq-decl nil real_types nil )
(- const-decl "[numfield -> numfield]" number_fields nil )
(abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
nil )
(member const-decl "bool" sets nil )
(numfield nonempty-type-eq-decl nil number_fields nil )
(/= const-decl "boolean" notequal nil )
(nznum nonempty-type-eq-decl nil number_fields nil )
(/ const-decl "[numfield, nznum -> numfield]" number_fields nil )
(posrat_div_posrat_is_posrat application-judgement "posrat"
rationals nil )
(set type-eq-decl nil sets nil ) (empty? const-decl "bool" sets nil )
(AND const-decl "[bool, bool -> bool]" booleans nil )
(<= const-decl "bool" reals nil ) (< const-decl "bool" reals nil )
(bool nonempty-type-eq-decl nil booleans 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_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil )
(real_minus_real_is_real application-judgement "real" reals nil ))
shostak)))
Messung V0.5 in Prozent C=100 H=99 G=99
¤ Dauer der Verarbeitung: 0.637 Sekunden
(vorverarbeitet am 2026-05-05)
¤
*© Formatika GbR, Deutschland