(holder_scaf
(mu_TCC1 0
(mu_TCC1-1 nil 3477456171
("" (typepred "S" )
(("" (expand "sigma_algebra?" )
(("" (flatten)
(("" (expand "subset_algebra_empty?" ) (("" (assert ) nil nil ))
nil ))
nil ))
nil ))
nil )
((subset_algebra_empty? const-decl "bool" subset_algebra_def
"measure_integration/" )
(finite_emptyset name-judgement "finite_set" finite_sets nil )
(finite_emptyset name-judgement "finite_set[T]" countable_props
"sets_aux/" )
(finite_emptyset name-judgement "finite_set[T]" countable_setofsets
"sets_aux/" )
(member const-decl "bool" sets nil )
(boolean nonempty-type-decl nil booleans nil )
(bool nonempty-type-eq-decl nil booleans nil )
(NOT const-decl "[bool -> bool]" booleans nil )
(T formal-type-decl nil holder_scaf nil )
(setof type-eq-decl nil defined_types nil )
(setofsets type-eq-decl nil sets nil )
(sigma_algebra? const-decl "bool" subset_algebra_def
"measure_integration/" )
(sigma_algebra nonempty-type-eq-decl nil subset_algebra_def
"measure_integration/" )
(S formal-const-decl "sigma_algebra[T]" holder_scaf nil ))
nil ))
(q_TCC1 0 (q_TCC1-1 nil 3477456171 ("" (assert ) nil nil ) nil nil ))
(f_TCC1 0
(f_TCC1-1 nil 3477456171 ("" (assert ) nil nil )
((real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil ))
nil ))
(g_TCC1 0
(g_TCC1-1 nil 3477456171
("" (typepred "q" )
(("" (typepred "p" )
(("" (lemma "div_cancel3" ("x" "1" "n0z" "q" "y" "1-1/p" ))
(("" (assert )
(("" (flatten)
(("" (hide -2)
(("" (split)
(("1" (replace -1 1)
(("1"
(lemma "both_sides_times_pos_ge1"
("pz" "q" "x" "1" "y" "1-1/p" ))
(("1" (assert )
(("1" (hide 2)
(("1"
(lemma "posreal_div_posreal_is_posreal"
("px" "1" "py" "p" ))
(("1" (assert ) nil nil )) nil ))
nil ))
nil ))
nil ))
nil )
("2" (assert ) nil nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
((nzreal_times_nzreal_is_nzreal application-judgement "nzreal"
real_types nil )
(real_times_real_is_real application-judgement "real" reals nil )
(real_plus_real_is_real application-judgement "real" reals nil )
(nzreal_div_nzreal_is_nzreal application-judgement "nzreal"
real_types nil )
(posreal_div_posreal_is_posreal application-judgement "posreal"
real_types 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 )
(posreal_times_posreal_is_posreal application-judgement "posreal"
real_types nil )
(posreal_div_posreal_is_posreal judgement-tcc nil real_types nil )
(AND const-decl "[bool, bool -> bool]" booleans nil )
(both_sides_times_pos_ge1 formula-decl nil real_props nil )
(real_minus_real_is_real application-judgement "real" reals nil )
(div_cancel3 formula-decl nil real_props nil )
(nonzero_real nonempty-type-eq-decl nil reals nil )
(- const-decl "[numfield, numfield -> numfield]" number_fields 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 )
(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 "[T, T -> boolean]" equalities nil )
(numfield nonempty-type-eq-decl nil number_fields nil )
(+ const-decl "[numfield, numfield -> numfield]" 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 )
(p formal-const-decl "{a: real | a > 1}" holder_scaf nil )
(q formal-const-decl "{a: posreal | 1 / p + 1 / a = 1}" holder_scaf
nil ))
nil ))
(holder_aux 0
(holder_aux-1 nil 3477461478
("" (skosimp)
(("" (typepred "f!1" )
(("" (typepred "g!1" )
(("" (case "p>1" )
(("1" (case "q>1" )
(("1" (typepred "norm[T, S, mu, q](g!1)" )
(("1" (typepred "norm[T, S, mu, p](f!1)" )
(("1" (expand ">=" )
(("1" (expand "<=" -1)
(("1" (split)
(("1" (expand "<=" -2)
(("1" (split)
(("1" (expand "p_integrable?" )
(("1" (flatten)
(("1"
(rewrite
"prod_complex_measurable[T,S]" )
(("1"
(name
"ALPHA"
"norm[T, S, mu, p](f!1)" )
(("1"
(replace -1)
(("1"
(name
"BETA"
"norm[T, S, mu, q](g!1)" )
(("1"
(replace -1)
(("1"
(lemma
"posreal_times_posreal_is_posreal"
("px" "ALPHA" "py" "BETA" ))
(("1"
(lemma
"real_expt_pos"
("px" "ALPHA" "a" "p" ))
(("1"
(lemma
"real_expt_pos"
("px" "BETA" "a" "q" ))
(("1"
(case
"FORALL (x: T):
abs(f!1(x) * g!1(x)) / (ALPHA * BETA) <=
(1 / p) * (abs(f!1(x)) ^ p / (ALPHA ^ p)) +
(1 / q) * (abs(g!1(x)) ^ q / (BETA ^ q))")
(("1"
(case-replace
"abs(f!1 * g!1) ^ 1=abs(f!1 * g!1)" )
(("1"
(hide -1)
(("1"
(expand "norm" )
(("1"
(case-replace
"abs(f!1 * g!1) ^ 1=abs(f!1 * g!1)" )
(("1"
(hide -1)
(("1"
(hide
-11
-13)
(("1"
(lemma
"both_sides_real_expt"
("x"
"integral.integral(abs(g!1) ^ q) ^ (1 / q)"
"y"
"BETA"
"pa"
"q" ))
(("1"
(rewrite
"real_expt_times"
-1
:dir
rl)
(("1"
(assert )
(("1"
(lemma
"both_sides_real_expt"
("x"
"integral.integral(abs(f!1)^p)^(1/p)"
"y"
"ALPHA"
"pa"
"p" ))
(("1"
(rewrite
"real_expt_times"
-1
:dir
rl)
(("1"
(assert )
(("1"
(rewrite
"real_expt_x1"
-1)
(("1"
(rewrite
"real_expt_x1"
-2)
(("1"
(lemma
"integral.integrable_is_measurable" )
(("1"
(inst-cp
-
"abs(f!1) ^ p" )
(("1"
(inst
-
"abs(g!1) ^ q" )
(("1"
(lemma
"expt_measurable"
("g"
"abs(f!1) ^ p"
"a"
"1/p" ))
(("1"
(lemma
"expt_measurable"
("g"
"abs(g!1) ^ q"
"a"
"1/q" ))
(("1"
(case-replace
"(^[T](abs(g!1) ^ q, 1 / q))=abs(g!1)" )
(("1"
(case-replace
"(^[T](abs(f!1)^p, 1 / p))=abs(f!1)" )
(("1"
(hide
-1
-2
-5
-6)
(("1"
(case
"measurable_function?[T,S](abs(f!1*g!1))" )
(("1"
(lemma
"scal_measurable"
("c"
"1/(ALPHA * BETA)"
"g"
"abs(f!1 * g!1)" ))
(("1"
(hide
-2
-3
-4)
(("1"
(lemma
"integral.integrable_scal"
("c"
"(1/p)*(1/ALPHA ^ p)"
"f"
"abs(f!1) ^ p" ))
(("1"
(lemma
"integral.integrable_scal"
("c"
"(1/q)*(1/BETA ^ q)"
"f"
"abs(g!1) ^ q" ))
(("1"
(lemma
"integral.integrable_add"
("f1"
"*[T]((1 / p) * (1 / ALPHA ^ p), abs(f!1) ^ p)"
"f2"
"*[T]((1 / q) * (1 / BETA ^ q), abs(g!1) ^ q)" ))
(("1"
(hide
-2
-3)
(("1"
(lemma
"integral_ae_abs"
("h"
"*[T](1 / (ALPHA * BETA), abs(f!1 * g!1))"
"f"
"(+[T])
(*[T]((1 / p) * (1 / ALPHA ^ p), abs(f!1) ^ p),
*[T]((1 / q) * (1 / BETA ^ q), abs(g!1) ^ q))"))
(("1"
(split
-1)
(("1"
(flatten
-1)
(("1"
(lemma
"integral.integrable_scal"
("c"
"ALPHA*BETA"
"f"
"*[T](1 / (ALPHA * BETA), abs(f!1 * g!1))" ))
(("1"
(case-replace
"(*[T]
(ALPHA * BETA,
*[T](1 / (ALPHA * BETA), abs(f!1 * g!1))))=abs(f!1 * g!1)")
(("1"
(rewrite
"real_expt_x1" )
(("1"
(lemma
"integral.integral_scal"
("c"
"1/(ALPHA*BETA)"
"f"
"abs(f!1*g!1)" ))
(("1"
(replace
-1)
(("1"
(lemma
"integral_nonneg"
("f"
"abs(f!1*g!1)" ))
(("1"
(split
-1)
(("1"
(rewrite
"real_props.abs_mult"
-6)
(("1"
(lemma
"posreal_div_posreal_is_posreal"
("px"
"1"
"py"
"ALPHA*BETA" ))
(("1"
(expand
"abs"
-7
1)
(("1"
(assert )
(("1"
(expand
"abs"
-7
1)
(("1"
(case
"integral(abs((+[T])
(*[T]((1 / p) * (1 / ALPHA ^ p), abs(f!1) ^ p),
*[T]((1 / q) * (1 / BETA ^ q), abs(g!1) ^ q)))) <=1")
(("1"
(assert )
(("1"
(lemma
"div_mult_pos_le1"
("z"
"integral.integral(abs(f!1 * g!1))"
"py"
"ALPHA * BETA"
"x"
"1" ))
(("1"
(assert )
nil
nil ))
nil ))
nil )
("2"
(hide
2
-1
-2
-3
-4
-5
-6
-7)
(("2"
(case-replace
"abs((+[T])
(*[T]((1 / p) * (1 / ALPHA ^ p), abs(f!1) ^ p),
*[T]((1 / q) * (1 / BETA ^ q), abs(g!1) ^ q)))= ((1/p)*1/(ALPHA^p))* abs(f!1)^p + ((1/q)*1/(BETA^q))* abs(g!1)^q")
(("1"
(hide
-1)
(("1"
(lemma
"posreal_times_posreal_is_posreal"
("px"
"1/p"
"py"
"1/(ALPHA^p)" ))
(("1"
(lemma
"posreal_times_posreal_is_posreal"
("px"
"1/q"
"py"
"1/(BETA^q)" ))
(("1"
(name-replace
"AA"
"1 / p * (1 / (ALPHA ^ p))" )
(("1"
(name-replace
"BB"
"1 / q * (1 / (BETA ^ q))" )
(("1"
(lemma
"integral.integral_scal"
("c"
"AA"
"f"
"abs(f!1)^p" ))
(("1"
(lemma
"integral.integral_scal"
("c"
"BB"
"f"
"abs(g!1)^q" ))
(("1"
(replace
-7)
(("1"
(replace
-8)
(("1"
(lemma
"integral.integral_add"
("f1"
"AA * abs(f!1) ^ p"
"f2"
"BB * abs(g!1) ^ q" ))
(("1"
(replace
-2)
(("1"
(replace
-3)
(("1"
(replace
-1)
(("1"
(hide
-1
-2
-3
-20
-21
-8
-9
-6
-7
-14
-15
-10)
(("1"
(expand
"AA" )
(("1"
(expand
"BB" )
(("1"
(name-replace
"AP"
"ALPHA ^ p" )
(("1"
(rewrite
"div_cancel1" )
(("1"
(typepred
"q" )
(("1"
(assert )
(("1"
(assert )
(("1"
(rewrite
"one_times" )
(("1"
(replace
-3)
(("1"
(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(assert )
(("2"
(lemma
"posreal_div_posreal_is_posreal"
("px"
"1"
"py"
"BETA^q" ))
(("2"
(assert )
nil
nil ))
nil ))
nil ))
nil )
("2"
(lemma
"posreal_div_posreal_is_posreal"
("px"
"1"
"py"
"ALPHA^p" ))
(("2"
(assert )
nil
nil ))
nil ))
nil ))
nil )
("2"
(hide
2)
(("2"
(apply-extensionality
:hide?
t)
(("1"
(expand
"abs"
+)
(("1"
(expand
"+"
+)
(("1"
(expand
"*"
+)
(("1"
(expand
"^"
1
1)
(("1"
(expand
"^"
1
1)
(("1"
(expand
"^"
1
6)
--> --------------------
--> maximum size reached
--> --------------------
quality 91%
¤ Dauer der Verarbeitung: 0.30 Sekunden
(vorverarbeitet)
¤
*© Formatika GbR, Deutschland