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(hide 2)
(("1"
(inst
+
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"(LAMBDA (ii: below(length(Pb!1) - 1)):
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1)):
UUPart(a!1, b!1, c!1, Pa!1, Pb!1)`seq(1 + n) *
f!1((xisa!1 o xisb!1)(n))
-
UUPart(a!1, b!1, c!1, Pa!1, Pb!1)`seq(n) *
f!1((xisa!1 o xisb!1)(n))) = sigma[below(length(Pa!1) - 1)]
(0, length(Pa!1) - 2,
LAMBDA (n: below(length(Pa!1) - 1)):
Pa!1`seq(1 + n) * f!1(xisa!1(n)) -
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]")
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"(LAMBDA (n:
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((partition type-eq-decl nil integral_def nil )
(xis? const-decl "bool" integral_def nil )
(UUPart const-decl "partition[T](a, c)" step_fun_props nil )
(sigma_split formula-decl nil sigma "reals/" )
(sigma_diff_eq formula-decl nil sigma_below_sub "reals/" )
(sigma_diff_shift formula-decl nil sigma_below_sub "reals/" )
(sigma def-decl "real" sigma "reals/" )
(Rie_sum const-decl "real" integral_def nil )
(max const-decl
"{a: T | SS(a) AND (FORALL (x: T): SS(x) IMPLIES x <= a)}"
finite_sets_minmax "finite_sets/" )
(width const-decl "posreal" integral_def nil ))
shostak))
(iss_prep_TCC1 0
(iss_prep_TCC1-1 nil 3282395953
("" (skosimp*) (("" (assert ) nil nil )) nil )
((real_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil ))
shostak))
(iss_prep 0
(OWRE "FIX" 3399972591
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((""
(case "(EXISTS (jj: below(length(P!1))): seq(P!1)(jj) = b!1)" )
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(("1" (assert ) (("1" (inst?) nil nil )) nil )) nil ))
nil )
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(name "ab"
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(("1"
(inst 2 " LET N = length(P!1)+1 IN
(# length := N,
seq := (LAMBDA (ii: below(N)):
IF ii < ab THEN P!1`seq(ii)
ELSIF ii = ab THEN b!1
ELSE P!1`seq(ii-1)
ENDIF)
#)")
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((real_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil )
(min const-decl "{a | S(a) AND (FORALL x: S(x) IMPLIES a <= x)}"
min_nat nil )
(IMPLIES const-decl "[bool, bool -> bool]" booleans nil )
(nonempty? const-decl "bool" sets nil )
(set type-eq-decl nil sets nil )
(step_function_on? const-decl "bool" step_fun_def nil )
(posint_plus_nnint_is_posint application-judgement "posint"
integers nil )
(open_interval type-eq-decl nil intervals_real "reals/" )
(ii!1 skolem-const-decl "below(length(P!1))" integral_split nil )
(x!1 skolem-const-decl "open_interval
[T](IF ii!1 < ab THEN P!1`seq(ii!1)
ELSIF ii!1 = ab THEN b!1
ELSE P!1`seq(ii!1 - 1)
ENDIF,
IF 1 + ii!1 < ab THEN P!1`seq(1 + ii!1)
ELSIF 1 + ii!1 = ab THEN b!1
ELSE P!1`seq(ii!1)
ENDIF)" integral_split nil)
(x!1 skolem-const-decl "open_interval
[T](IF ii!1 < ab THEN P!1`seq(ii!1)
ELSIF ii!1 = ab THEN b!1
ELSE P!1`seq(ii!1 - 1)
ENDIF,
IF 1 + ii!1 < ab THEN P!1`seq(1 + ii!1)
ELSIF 1 + ii!1 = ab THEN b!1
ELSE P!1`seq(ii!1)
ENDIF)" integral_split nil)
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil )
(x!1 skolem-const-decl "open_interval
[T](IF ii!1 < ab THEN P!1`seq(ii!1)
ELSIF ii!1 = ab THEN b!1
ELSE P!1`seq(ii!1 - 1)
ENDIF,
IF 1 + ii!1 < ab THEN P!1`seq(1 + ii!1)
ELSIF 1 + ii!1 = ab THEN b!1
ELSE P!1`seq(ii!1)
ENDIF)" integral_split nil)
(int_plus_int_is_int application-judgement "int" integers nil )
(real_gt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil )
(IF const-decl "[boolean, T, T -> T]" if_def nil )
(NOT const-decl "[bool -> bool]" booleans nil )
(nonneg_int nonempty-type-eq-decl nil integers nil )
(posint nonempty-type-eq-decl nil integers nil )
(a!1 skolem-const-decl "T" integral_split nil )
(c!1 skolem-const-decl "T" integral_split nil )
(P!1 skolem-const-decl "partition[T](a!1, c!1)" integral_split nil )
(b!1 skolem-const-decl "T" integral_split nil )
(ab skolem-const-decl "{a |
seq(P!1)(a) > b!1 AND
(FORALL (x: below(length(P!1))):
seq(P!1)(x) > b!1 IMPLIES a <= x)}" integral_split nil)
(nnint_plus_posint_is_posint application-judgement "posint"
integers nil )
(int_minus_int_is_int application-judgement "int" integers nil )
(member const-decl "bool" sets nil )
(empty? const-decl "bool" sets nil )
(number nonempty-type-decl nil numbers nil )
(boolean nonempty-type-decl nil booleans nil )
(number_field_pred const-decl "[number -> boolean]" number_fields
nil )
(number_field nonempty-type-from-decl nil number_fields nil )
(real_pred const-decl "[number_field -> boolean]" reals nil )
(real nonempty-type-from-decl nil reals nil )
(rational_pred const-decl "[real -> boolean]" rationals nil )
(rational nonempty-type-from-decl nil rationals nil )
(integer_pred const-decl "[rational -> boolean]" integers nil )
(int nonempty-type-eq-decl nil integers nil )
(bool nonempty-type-eq-decl nil booleans nil )
(>= const-decl "bool" reals nil )
(nat nonempty-type-eq-decl nil naturalnumbers nil )
(< const-decl "bool" reals nil )
(below type-eq-decl nil nat_types nil )
(T_pred const-decl "[real -> boolean]" integral_split nil )
(T formal-nonempty-subtype-decl nil integral_split nil )
(AND const-decl "[bool, bool -> bool]" booleans nil )
(<= const-decl "bool" reals nil )
(closed_interval type-eq-decl nil intervals_real "reals/" )
(finite_sequence type-eq-decl nil finite_sequences nil )
(> const-decl "bool" reals nil )
(= const-decl "[T, T -> boolean]" equalities nil )
(numfield nonempty-type-eq-decl nil number_fields nil )
(- const-decl "[numfield, numfield -> numfield]" number_fields nil )
(below type-eq-decl nil naturalnumbers nil )
(+ const-decl "[numfield, numfield -> numfield]" number_fields nil )
(partition type-eq-decl nil integral_def nil ))
shostak)
(iss_prep-2 nil 3399972563
(";;; Proof iss_prep-2 for formula integral_split.iss_prep"
(skosimp*)
((";;; Proof iss_prep-2 for formula integral_split.iss_prep"
(assert )
((";;; Proof iss_prep-2 for formula integral_split.iss_prep"
(case "(EXISTS (jj: below(length(P!1))): seq(P!1)(jj) = b!1)" )
(("1" (skosimp*)
(("1" (inst + "P!1" ) (("1" (assert ) (("1" (inst?) nil )))))))
("2"
(name "ab"
"min_nat.min({ii: below(length(P!1)) | seq(P!1)(ii) > b!1})" )
(("1"
(inst 2 " LET N = length(P!1)+1 IN
(# length := N,
seq := (LAMBDA (ii: below(N)):
IF ii < ab THEN P!1`seq(ii)
ELSIF ii = ab THEN b!1
ELSE P!1`seq(ii-1)
ENDIF)
#)")
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nil )))))))))))))))))))))))))))))
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";;; developed with shostak decision procedures")
((partition type-eq-decl nil integral_def nil )
(step_function_on? const-decl "bool" step_fun_def nil ))
nil )
(iss_prep-1 nil 3282319775
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nil )
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(("1"
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LET N = length(P!1)+1 IN
(# length := N,
seq := (LAMBDA (ii: below(N)):
IF ii < ab THEN P!1`seq(ii)
ELSIF ii = ab THEN b!1
ELSE P!1`seq(ii-1)
ENDIF)
#)")
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((step_function_on? const-decl "bool" step_fun_def nil )
(partition type-eq-decl nil integral_def nil ))
shostak))
(integrable_split_step_TCC1 0
(integrable_split_step_TCC1-1 nil 3276447547
("" (skosimp*) (("" (assert ) nil nil )) nil )
((real_lt_is_strict_total_order name-judgement
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shostak))
(integrable_split_step 0
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(0, length(LP) - 2,
LAMBDA (ii: below(length(LP) - 1)):
LP`seq(1 + ii) *
integral_step[T].val_in(a!1, b!1, LP, ii, f!1)
-
LP`seq(ii) * integral_step[T].val_in(a!1, b!1, LP, ii, f!1)) = sigma[below[length(PP!1) - 1]].sigma
(0, ib!1 - 1,
LAMBDA (ii: below(length(PP!1) - 1)):
PP!1`seq(1 + ii) *
integral_step[T].val_in(a!1, c!1, PP!1, ii, f!1)
-
PP!1`seq(ii) *
integral_step[T].val_in(a!1, c!1, PP!1, ii, f!1))")
(("1"
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-1)
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-
"(LAMBDA (ii: below(length(PP!1) - 1)):
PP!1`seq(1 + ii) * val_in(a!1, c!1, PP!1, ii, f!1) -
PP!1`seq(ii) * val_in(a!1, c!1, PP!1, ii, f!1))"
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nil ))
nil ))
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nil ))
nil )
("2" (hide 2) (("2" (postpone) nil nil )) nil )
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nil shostak))
(integrable_split 0
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(";;; Proof integrable_split-12 for formula integral_split.integrable_split"
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((";;; Proof integrable_split-12 for formula integral_split.integrable_split"
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ELSIF x <= b!1 THEN f2!2(x)
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(inst
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"a!1"
"b!1"
"c!1"
"FF2 - FF1" )
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rl)
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-7)
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nil )
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(assert )
nil )))))))))))))))))))
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-1
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rl)
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-3)
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(rewrite
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";;; developed with shostak decision procedures")
((= const-decl "[T, T -> boolean]" equalities nil )
(IF const-decl "[boolean, T, T -> T]" if_def nil )
(OR const-decl "[bool, bool -> bool]" booleans nil )
(< const-decl "bool" reals nil ) (<= const-decl "bool" reals nil )
(real_gt_is_strict_total_order name-judgement
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(real_le_is_total_order name-judgement "(total_order?[real])"
real_props nil )
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(AND const-decl "[bool, bool -> bool]" booleans nil )
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(step_function_integrable? formula-decl nil integral_step nil )
(- const-decl "[T -> real]" real_fun_ops "reals/" )
(integrable_split_step formula-decl nil integral_split nil )
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(integral const-decl "{S: real | integral?(a, b, ff, S)}"
integral_def nil )
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(integrable? const-decl "bool" integral_def nil )
(split_step_is_step formula-decl nil step_fun_props nil )
(step_function? const-decl "bool" step_fun_def nil )
(/ const-decl "[numfield, nznum -> numfield]" number_fields nil )
(nznum nonempty-type-eq-decl nil number_fields nil )
(/= const-decl "boolean" notequal nil )
(numfield nonempty-type-eq-decl nil number_fields nil )
(posreal nonempty-type-eq-decl nil real_types nil )
(> const-decl "bool" reals nil )
(nonneg_real nonempty-type-eq-decl nil real_types nil )
(>= const-decl "bool" reals nil )
(posreal_div_posreal_is_posreal application-judgement "posreal"
real_types nil )
(step_to_integrable formula-decl nil integral_step nil )
(real_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil )
(bool nonempty-type-eq-decl nil booleans nil )
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(number nonempty-type-decl nil numbers nil )
(boolean nonempty-type-decl nil booleans nil )
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nil )
(number_field nonempty-type-from-decl nil number_fields nil )
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(real nonempty-type-from-decl nil reals nil )
(T_pred const-decl "[real -> boolean]" integral_split nil )
(T formal-nonempty-subtype-decl nil integral_split nil ))
nil )
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(("1"
(name
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ELSIF x <= b!1 THEN f2!2(x)
ELSE f2!1(x) ENDIF)")
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(lemma
"diff_step_is_step[T]" )
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(lemma
"step_function_integrable?[T]" )
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nil ))
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(inst
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"FF2 - FF1" )
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nil ))
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*
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(inst?)
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-1
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-4
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"integrable?(b!1, c!1, FF2 - FF1) AND integral(b!1, c!1, FF2 - FF1) = integral(b!1, c!1, f2!1 - f1!1)" )
(("1"
(flatten)
(("1"
(assert )
nil
nil ))
nil )
("2"
(hide
2)
(("2"
(replace
-7
*
rl)
(("2"
(hide
-7)
(("2"
(replace
-7
*
rl)
(("2"
(hide
-7)
(("2"
(hide
-1
-2
-3
-4
-5
-6
-7
-8)
(("2"
(lemma
"integral_restr_eq" )
(("2"
(inst?)
(("2"
(assert )
(("2"
(split
-1)
(("1"
(propax)
nil
nil )
("2"
(assert )
(("2"
(hide
2)
(("2"
(skosimp*)
(("2"
(expand
"-" )
(("2"
(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("3"
--> --------------------
--> maximum size reached
--> --------------------
Messung V0.5 in Prozent C=100 H=100 G=100
¤ Dauer der Verarbeitung: 0.985 Sekunden
(vorverarbeitet am 2026-04-29)
¤
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