(piecewise_continuous
(IMP_fundamental_theorem_TCC1 0
(IMP_fundamental_theorem_TCC1-1 nil 3612529043
("" (lemma "connected_domain" ) (("" (propax) nil nil )) nil )
((connected_domain formula-decl nil piecewise_continuous nil )) nil ))
(IMP_fundamental_theorem_TCC2 0
(IMP_fundamental_theorem_TCC2-1 nil 3612529043
("" (lemma "not_one_element" ) (("" (propax) nil nil )) nil )
((not_one_element formula-decl nil piecewise_continuous nil )) nil ))
(piecewise_continuous?_TCC1 0
(piecewise_continuous?_TCC1-1 nil 3611578171
("" (skosimp*) (("" (assert ) nil nil )) nil )
((real_le_is_total_order name-judgement "(total_order?[real])"
real_props nil ))
nil ))
(piecewise_continuous?_TCC2 0
(piecewise_continuous?_TCC2-1 nil 3611578171
("" (skosimp*) (("" (assert ) nil nil )) nil )
((nnint_plus_posint_is_posint application-judgement "posint"
integers nil )
(real_le_is_total_order name-judgement "(total_order?[real])"
real_props nil ))
nil ))
(piecewise_continuous?_TCC3 0
(piecewise_continuous?_TCC3-1 nil 3611578171
("" (subtype-tcc) nil nil )
((strict_increasing? const-decl "bool" real_fun_preds "reals/" )
(continuous_on? const-decl "bool" continuous_functions nil ) nil
(real_minus_real_is_real application-judgement "real" reals nil ))
nil ))
(piecewise_continuous?_TCC4 0
(piecewise_continuous?_TCC4-1 nil 3611578171
("" (skeep) (("" (assert ) nil nil )) nil )
((real_le_is_total_order name-judgement "(total_order?[real])"
real_props nil ))
nil ))
(piecewise_continuous?_TCC5 0
(piecewise_continuous?_TCC5-1 nil 3611591182 ("" (grind) nil nil )
((strict_increasing? const-decl "bool" real_fun_preds "reals/" )
(abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
nil )
(continuous_on? const-decl "bool" continuous_functions nil ) nil
(real_minus_real_is_real application-judgement "real" reals nil ))
nil ))
(piecewise_continuous?_TCC6 0
(piecewise_continuous?_TCC6-1 nil 3611591182 ("" (grind) nil nil )
((strict_increasing? const-decl "bool" real_fun_preds "reals/" )
(abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
nil )
(continuous_on? const-decl "bool" continuous_functions nil ) nil
(real_minus_real_is_real application-judgement "real" reals nil ))
nil ))
(continuous_on_integrable 0
(continuous_on_integrable-1 nil 3611681282
("" (skosimp*)
(("" (lemma "unif_cont_interval[closed_interval(a!1,b!1)]" )
(("1" (inst?)
(("1" (inst -1 "f!1" )
(("1" (assert )
(("1" (expand "restrict" )
(("1"
(case " continuous_on?(LAMBDA (s: closed_interval[T](a!1, b!1)): f!1(s),
{xx: closed_interval[T](a!1, b!1) | TRUE})")
(("1" (assert )
(("1" (hide -1)
(("1" (rewrite "step_to_integrable" )
(("1" (skosimp*)
(("1" (expand "uniformly_continuous?" )
(("1" (inst -1 "(eps!1/2)/(b!1-a!1)" )
(("1" (skosimp*)
(("1"
(name
"PP"
"eq_partition(a!1,b!1,floor((b!1-a!1)/delta!1)+2)" )
(("1"
(case
"step_function?(a!1, b!1, fmin(a!1, b!1, PP, f!1))" )
(("1"
(case
"step_function?(a!1, b!1, fmax(a!1, b!1, PP, f!1))" )
(("1"
(inst
+
"fmin(a!1,b!1,PP,f!1)"
"fmax(a!1,b!1,PP,f!1)" )
(("1"
(assert )
(("1"
(case
"integrable?(a!1, b!1, fmax(a!1, b!1, PP, f!1) - fmin(a!1, b!1, PP, f!1))" )
(("1"
(assert )
(("1"
(case
" integral(a!1, b!1, fmax(a!1, b!1, PP, f!1) - fmin(a!1, b!1, PP, f!1)) < eps!1" )
(("1"
(assert )
(("1"
(skosimp*)
(("1"
(hide-all-but
(-7 1))
(("1"
(split 1)
(("1"
(typepred
"fmin(a!1, b!1, PP, f!1)" )
(("1"
(lemma
"part_induction" )
(("1"
(inst
-1
"(LAMBDA x: fmin(a!1, b!1, PP, f!1)(x) <= f!1(x))"
"a!1"
"b!1"
"PP"
"xx!1" )
(("1"
(assert )
(("1"
(assert )
(("1"
(skosimp*)
(("1"
(assert )
(("1"
(inst
-
"ii!1" )
(("1"
(inst
-
"xx!1" )
(("1"
(flatten)
(("1"
(split
-4)
(("1"
(assert )
nil
nil )
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(flatten)
(("2"
(assert )
(("2"
(replace
-3)
(("2"
(hide
-3)
(("2"
(assert )
(("2"
(typepred
"min_x(seq(PP)(ii!1), seq(PP)(1 + ii!1), f!1)" )
(("2"
(hide
-1
-2
-3)
(("2"
(inst
-1
"xx!1" )
(("2"
(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(typepred
"fmax(a!1, b!1, PP, f!1)" )
(("2"
(lemma
"part_induction" )
(("2"
(inst
-1
"(LAMBDA x: fmax(a!1, b!1, PP, f!1)(x) >= f!1(x))"
"a!1"
"b!1"
"PP"
"xx!1" )
(("2"
(assert )
(("2"
(assert )
(("2"
(skosimp*)
(("2"
(inst
-
"ii!1" )
(("2"
(inst
-
"xx!1" )
(("2"
(flatten)
(("2"
(split
-4)
(("1"
(assert )
nil
nil )
("2"
(assert )
(("2"
(flatten)
(("2"
(assert )
(("2"
(replace
-3)
(("2"
(hide
-3)
(("2"
(typepred
"max_x(seq(PP)(ii!1), seq(PP)(1 + ii!1), f!1)" )
(("2"
(hide
-1
-2
-3)
(("2"
(inst
-
"xx!1" )
(("2"
(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(hide 2)
(("2"
(lemma
"width_eq_part" )
(("2"
(inst?)
(("2"
(assert )
(("2"
(replace -5)
(("2"
(case
"width(a!1,b!1,PP) < delta!1" )
(("1"
(hide
-2
-6)
(("1"
(lemma
"integral_bound_abs" )
(("1"
(inst
-1
"eps!1/(2*(b!1-a!1))"
"a!1"
"b!1"
"fmax(a!1, b!1, PP, f!1) - fmin(a!1, b!1, PP, f!1)" )
(("1"
(name-replace
"BMA"
"b!1-a!1" )
(("1"
(assert )
(("1"
(split
-1)
(("1"
(name
"III"
"integral(a!1, b!1, fmax(a!1, b!1, PP, f!1) - fmin(a!1, b!1, PP, f!1))" )
(("1"
(case
"III >= 0" )
(("1"
(replace
-2)
(("1"
(expand
"abs"
-3)
(("1"
(assert )
nil
nil ))
nil ))
nil )
("2"
(assert )
nil
nil ))
nil ))
nil )
("2"
(hide
2)
(("2"
(skosimp*)
(("2"
(expand
"-" )
(("2"
(lemma
"part_induction" )
(("2"
(inst
-1
"(LAMBDA x: (fmax(a!1, b!1, PP, f!1) - fmin(a!1, b!1, PP, f!1))(x) <= eps!1 / (2 * BMA))"
"a!1"
"b!1"
"PP"
"x!1" )
(("2"
(assert )
(("2"
(expand
"-" )
(("2"
(case
"fmax(a!1, b!1, PP, f!1)(x!1) - fmin(a!1, b!1, PP, f!1)(x!1) >= 0" )
(("1"
(expand
"abs"
1)
(("1"
(assert )
(("1"
(assert )
(("1"
(skosimp*)
(("1"
(typepred
"fmax(a!1, b!1, PP, f!1)" )
(("1"
(typepred
"fmin(a!1, b!1, PP, f!1)" )
(("1"
(inst
-1
"ii!1" )
(("1"
(inst
-2
"ii!1" )
(("1"
(inst
-1
"x!1" )
(("1"
(inst
-2
"x!1" )
(("1"
(flatten)
(("1"
(case
"(seq(PP)(ii!1) = x!1 OR x!1 = seq(PP)(1 + ii!1))" )
(("1"
(assert )
(("1"
(split
-1)
(("1"
(assert )
(("1"
(hide
-2
-4)
(("1"
(replace
-2)
(("1"
(replace
-3)
(("1"
(assert )
(("1"
(hide-all-but
1)
(("1"
(reveal
-21)
(("1"
(mult-by
1
"(2*BMA)" )
(("1"
(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(assert )
(("2"
(hide
-2
-4)
(("2"
(replace
-2)
(("2"
(replace
-3)
(("2"
(assert )
(("2"
(hide-all-but
1)
(("2"
(reveal
-21)
(("2"
(mult-by
1
"(2*BMA)" )
(("2"
(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(flatten)
(("2"
(assert )
(("2"
(hide
-2
-4)
(("2"
(replace
-1)
(("2"
(hide
-1)
(("2"
(replace
-1)
(("2"
(hide
-1)
(("2"
(name
"MIN_x"
"min_x(seq(PP)(ii!1), seq(PP)(1 + ii!1), f!1)" )
(("2"
(replace
-1)
(("2"
(name
"MAX_x"
"max_x(seq(PP)(ii!1), seq(PP)(1 + ii!1), f!1)" )
(("2"
(replace
-1)
(("2"
(case
"(FORALL (x,y: closed_interval(seq(PP)(ii!1),seq(PP)(ii!1+1))): abs(x-y) <= seq(PP)(1 + ii!1) - seq(PP)(ii!1))" )
(("1"
(case
"abs(MAX_x - MIN_x) <= seq(PP)(1+ii!1) - seq(PP)(ii!1)" )
(("1"
(inst
-12
"MAX_x"
"MIN_x" )
(("1"
(assert )
(("1"
(lemma
"width_lem" )
(("1"
(expand
"finseq_appl" )
(("1"
(inst?)
(("1"
(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(ground)
nil
nil )
("3"
(ground)
nil
nil ))
nil )
("2"
(assert )
(("2"
(hide-all-but
(-2
-3
-16
1))
(("2"
(typepred
"MAX_x" )
(("2"
(typepred
"MIN_x" )
(("2"
(hide
-1
-4
-5
-8
-9
-10)
(("2"
(expand
"abs" )
(("2"
(lift-if)
(("2"
(ground)
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(assert )
(("2"
(hide-all-but
1)
(("2"
(skeep)
(("2"
(typepred
"x" )
(("2"
(typepred
"y" )
(("2"
(expand
"abs" )
(("2"
(lift-if)
(("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 ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(hide
2)
(("2"
(rewrite
"fmax_ge" )
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(hide
-3
-4
-6
-8
2)
(("2"
(replace
-1)
(("2"
(hide
-1
-2
-3)
(("2"
(name-replace
"BMA"
"b!1-a!1" )
(("2"
(mult-by
1
"(1 + floor(BMA / delta!1))" )
(("1"
(div-by
1
"delta!1" )
(("1"
(assert )
nil
nil ))
nil )
("2"
(hide
2)
(("2"
(reveal
-3)
(("2"
(assert )
(("2"
(case-replace
"BMA / delta!1 >= 0" )
(("1"
(assert )
nil
nil )
("2"
(hide
2)
(("2"
(mult-by
1
"delta!1" )
(("2"
(assert )
(("2"
(reveal
-4)
(("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 )
("2"
(hide 2)
(("2"
(lemma "integral_diff" )
(("2"
(inst
-1
"a!1"
"b!1"
"fmax(a!1, b!1, PP, f!1)"
"fmin(a!1, b!1, PP, f!1)" )
(("2"
(assert )
(("2"
(split -1)
(("1"
(flatten)
(("1"
(assert )
(("1"
(expand "-" )
(("1"
(propax)
nil
nil ))
nil ))
nil ))
nil )
("2"
(hide 2)
(("2"
(rewrite
"step_function_integrable?" )
nil
nil ))
nil )
("3"
(hide 2)
(("3"
(rewrite
"step_function_integrable?" )
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(hide 2)
(("2"
(rewrite "fmax_step" )
nil
nil ))
nil ))
nil )
("2"
(hide 2)
(("2"
(rewrite "fmin_step" )
nil
nil ))
nil )
("3"
(hide 2)
(("3"
(skosimp*)
(("3"
(hide -1 -2 -3)
(("3"
(expand "continuous_on?" )
(("3"
(expand "continuous?" )
(("3"
(inst - "x!1" )
(("3"
(skeep)
(("3"
(inst - "epsilon" )
(("3"
(skeep)
(("3"
(inst + "delta" )
(("3"
(skeep)
(("3"
(inst - "x" )
(("3"
(expand
"restrict" )
(("3"
(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(assert )
(("2"
(mult-by 1 "delta!1" )
(("2"
(assert )
(("2"
(case
"floor((b!1 - a!1) / delta!1)>=0" )
(("1"
(mult-by -1 "delta!1" )
(("1" (assert ) nil nil ))
nil )
("2"
(assert )
(("2"
(mult-by -2 "1/delta!1" )
(("2" (assert ) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (hide 2)
(("2"
(split +)
(("1"
(mult-by 1 "b!1-a!1" )
(("1" (assert ) nil nil ))
nil )
("2"
(mult-by 1 "b!1-a!1" )
(("2" (assert ) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (hide -1 2)
(("2" (expand "continuous_on?" )
(("2" (propax) nil nil )) nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (assert ) nil nil ) ("3" (assert ) nil nil ))
nil )
("2" (expand "not_one_element?" )
(("2" (skosimp*)
(("2" (typepred "x!1" )
(("2" (inst-cp 1 "a!1" )
(("2" (inst-cp 1 "b!1" ) (("2" (assert ) nil nil )) nil ))
nil ))
nil ))
nil ))
nil )
("3" (lemma "connected_domain" )
(("3" (expand "connected?" )
(("3" (assert )
(("3" (skosimp*)
(("3" (lemma "connected_domain" )
(("3" (expand "connected?" )
(("3" (inst -1 "a!1" "b!1" "z!1" )
(("3" (assert ) nil nil )) nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
((closed_interval type-eq-decl nil intervals_real "reals/" )
(<= const-decl "bool" reals nil )
(AND const-decl "[bool, bool -> bool]" booleans nil )
(bool nonempty-type-eq-decl nil booleans nil )
(T formal-nonempty-subtype-decl nil piecewise_continuous nil )
(T_pred const-decl "[real -> boolean]" piecewise_continuous 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 )
(unif_cont_interval formula-decl nil unif_cont_fun nil )
(connected? const-decl "bool" deriv_domain_def nil )
(not_one_element? const-decl "bool" deriv_domain_def nil )
(restrict const-decl "R" restrict nil )
(step_to_integrable formula-decl nil integral_step nil )
(real_minus_real_is_real application-judgement "real" reals nil )
(uniformly_continuous? const-decl "bool" unif_cont_fun nil )
(real_plus_real_is_real application-judgement "real" reals nil )
(both_sides_times_pos_lt1 formula-decl nil real_props nil )
(posreal_times_posreal_is_posreal application-judgement "posreal"
real_types nil )
(both_sides_times_pos_le1_imp formula-decl nil extra_real_props
nil )
(step_function? const-decl "bool" step_fun_def nil )
(continuous? const-decl "bool" continuous_functions nil )
(fun_cont_on type-eq-decl nil integral_cont_scaf nil )
(IMPLIES const-decl "[bool, bool -> bool]" booleans nil )
(fun_cont_on type-eq-decl nil interval_minmax nil )
(min_x const-decl "{mx: T |
a <= mx AND
mx <= b AND
(FORALL (x: T): a <= x AND x <= b IMPLIES f(mx) <= f(x))}"
interval_minmax nil )
(OR const-decl "[bool, bool -> bool]" booleans nil )
(fmin const-decl "{ff: [T -> real] |
LET xx = seq(P) IN
FORALL (ii: below(length(P) - 1)):
FORALL (x: T):
(xx(ii) < x AND x < xx(ii + 1) IMPLIES
ff(x) = f(min_x(xx(ii), xx(ii + 1), f)))
AND ((xx(ii) = x OR x = xx(ii + 1)) IMPLIES ff(x) = f(x))}"
integral_cont_scaf nil )
(fmax_step formula-decl nil integral_cont_scaf nil )
(integrable? const-decl "bool" integral_def nil )
(- const-decl "[T -> real]" real_fun_ops "reals/" )
(integral? const-decl "bool" integral_def nil )
(integral const-decl "{S: real | integral?(a, b, ff, S)}"
integral_def nil )
(part_induction formula-decl nil integral_def nil )
(posint_plus_nnint_is_posint application-judgement "posint"
integers nil )
(NOT const-decl "[bool -> bool]" booleans nil )
(width_eq_part formula-decl nil integral_def nil )
(width const-decl "posreal" integral_def nil )
(integral_bound_abs formula-decl nil integral_prep nil )
(minus_odd_is_odd application-judgement "odd_int" integers nil )
(abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
nil )
(both_sides_times_pos_le1 formula-decl nil real_props nil )
(BMA skolem-const-decl "real" piecewise_continuous nil )
(div_cancel2 formula-decl nil real_props nil )
(nonzero_real nonempty-type-eq-decl nil reals nil )
(- const-decl "[numfield -> numfield]" number_fields nil )
(minus_real_is_real application-judgement "real" reals nil )
(PP skolem-const-decl
"{fs: finite_sequence[{x | a!1 <= x AND x <= b!1}] |
length(fs) > 1 AND
seq(fs)(0) = a!1 AND
seq(fs)(length(fs) - 1) = b!1 AND
(FORALL (ii: below(length(fs) - 1)):
seq(fs)(ii) < seq(fs)(1 + ii))}" piecewise_continuous
nil )
(ii!1 skolem-const-decl "below(length(PP) - 1)"
piecewise_continuous nil )
(f!1 skolem-const-decl "[T -> real]" piecewise_continuous nil )
(MAX_x skolem-const-decl "{mx: T |
seq(PP)(ii!1) <= mx AND
mx <= seq(PP)(1 + ii!1) AND
(FORALL (x: T):
seq(PP)(ii!1) <= x AND x <= seq(PP)(1 + ii!1) IMPLIES
f!1(mx) >= f!1(x))}" piecewise_continuous nil)
(MIN_x skolem-const-decl "{mx: T |
seq(PP)(ii!1) <= mx AND
mx <= seq(PP)(1 + ii!1) AND
(FORALL (x: T):
seq(PP)(ii!1) <= x AND x <= seq(PP)(1 + ii!1) IMPLIES
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Messung V0.5 in Prozent C=100 H=100 G=100
¤ Dauer der Verarbeitung: 0.455 Sekunden
(vorverarbeitet am 2026-04-27)
¤
*© Formatika GbR, Deutschland