|
(integral_prep
(IMP_integral_def_TCC1 0
(IMP_integral_def_TCC1-1 nil 3253532115
("" (skosimp*)
(("" (lemma "connected_domain")
(("" (inst?)
(("" (assert)
(("" (inst -1 "y!1") (("" (assert) nil nil)) nil)) nil))
nil))
nil))
nil)
((connected_domain formula-decl nil integral_prep nil)) nil))
(IMP_integral_def_TCC2 0
(IMP_integral_def_TCC2-1 nil 3253532115
("" (skosimp*)
(("" (lemma "not_one_element") (("" (inst?) nil nil)) nil)) nil)
((not_one_element formula-decl nil integral_prep nil)) nil))
(integral_const_fun 0
(integral_const_fun-1 nil 3253532115
("" (skosimp*)
(("" (rewrite "integral_def")
(("" (expand "integral?")
(("" (skosimp*)
(("" (auto-rewrite "Rie_sum")
(("" (inst + "1")
(("" (skosimp*)
(("" (case-replace "R!1 = D!1*(b!1-a!1)")
(("1" (assert)
(("1" (expand "abs") (("1" (assert) nil nil)) nil))
nil)
("2" (hide 2)
(("2" (typepred "R!1")
(("2" (expand "Riemann_sum?")
(("2" (skosimp*)
(("2" (replace -1)
(("2" (hide -1)
(("2"
(assert)
(("1"
(expand "const_fun")
(("1"
(case
"FORALL (n: below(length(P!1))): n > 0 IMPLIES sigma[below(length(P!1) - 1)] (0, n-1, LAMBDA (n: below(length(P!1) - 1)): seq(P!1)(1 + n) * D!1 - seq(P!1)(n) * D!1) = D!1*seq(P!1)(n) - D!1*seq(P!1)(0)")
(("1"
(inst -1 "length(P!1)-1")
(("1" (assert) nil nil))
nil)
("2"
(hide 2)
(("2"
(induct
"n"
1
"below_induction[length(P!1)]")
(("1" (assert) nil nil)
("2"
(skosimp*)
(("2"
(assert)
(("1"
(case-replace "jb!1 = 0")
(("1"
(hide -3)
(("1"
(expand "sigma")
(("1"
(assert)
nil
nil))
nil))
nil)
("2"
(assert)
(("1"
(expand "sigma" 2)
(("1"
(replace -2)
(("1"
(assert)
nil
nil))
nil))
nil)
("2"
(skosimp*)
(("2"
(assert)
nil
nil))
nil))
nil))
nil)
("2"
(skosimp*)
(("2" (assert) nil nil))
nil))
nil))
nil)
("3"
(hide 2)
(("3"
(skosimp*)
(("3" (assert) nil nil))
nil))
nil)
("4"
(hide 2)
(("4"
(skosimp*)
(("4" (assert) nil nil))
nil))
nil))
nil))
nil)
("3"
(hide 2)
(("3"
(skosimp*)
(("3" (assert) nil nil))
nil))
nil)
("4"
(hide 2)
(("4"
(skosimp*)
(("4" (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)
((real_times_real_is_real application-judgement "real" reals nil)
(integral_def formula-decl nil integral_def nil)
(const_fun const-decl "[T -> real]" real_fun_ops "reals/")
(numfield nonempty-type-eq-decl nil number_fields nil)
(* const-decl "[numfield, numfield -> numfield]" number_fields nil)
(- const-decl "[numfield, numfield -> numfield]" number_fields 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)
(T_pred const-decl "[real -> boolean]" integral_prep nil)
(T formal-subtype-decl nil integral_prep nil)
(real_minus_real_is_real application-judgement "real" reals 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)
(bool nonempty-type-eq-decl nil booleans nil)
(= const-decl "[T, T -> boolean]" equalities nil)
(< const-decl "bool" reals nil)
(nat nonempty-type-eq-decl nil naturalnumbers nil)
(below type-eq-decl nil nat_types nil)
(AND const-decl "[bool, bool -> bool]" booleans 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)
(below type-eq-decl nil naturalnumbers nil)
(+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
(Riemann_sum? const-decl "bool" integral_def nil)
(closed_interval type-eq-decl nil intervals_real "reals/")
(finite_sequence type-eq-decl nil finite_sequences nil)
(partition type-eq-decl nil integral_def nil)
(abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
nil)
(real_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(odd_plus_even_is_odd application-judgement "odd_int" integers nil)
(real_plus_real_is_real application-judgement "real" reals nil)
(minus_odd_is_odd application-judgement "odd_int" integers nil)
(nnint_plus_posint_is_posint application-judgement "posint"
integers nil)
(sigma_0_neg formula-decl nil sigma_below "reals/")
(below_induction formula-decl nil bounded_nat_inductions nil)
(pred type-eq-decl nil defined_types nil)
(P!1 skolem-const-decl "partition[T](a!1, b!1)" integral_prep nil)
(b!1 skolem-const-decl "T" integral_prep nil)
(a!1 skolem-const-decl "T" integral_prep nil)
(integer nonempty-type-from-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/")
(OR const-decl "[bool, bool -> bool]" booleans nil)
(IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
(posint_plus_nnint_is_posint application-judgement "posint"
integers nil)
(Rie_sum const-decl "real" integral_def nil)
(finseq_appl const-decl "[below[length(fs)] -> T]" finite_sequences
nil)
(real_le_is_total_order name-judgement "(total_order?[real])"
real_props 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)
(int_minus_int_is_int application-judgement "int" integers nil)
(integral? const-decl "bool" integral_def nil))
nil))
(integral_const_restrict 0
(integral_const_restrict-3 nil 3278176127
("" (skosimp*)
(("" (lemma "integral_const_fun")
(("" (inst?)
(("" (assert)
(("" (flatten)
(("" (typepred "integral(a!1, b!1, const_fun[T](D!1))")
(("1" (expand "integral?")
(("1" (skosimp*)
(("1" (inst -1 "epsi!1")
(("1" (skosimp*)
(("1" (inst + "delta!1")
(("1" (skosimp*)
(("1" (inst?)
(("1" (assert)
(("1"
(replace -3)
(("1"
(typepred "R!1")
(("1"
(inst -2 "R!1")
(("1" (assert) nil nil)
("2"
(hide 2)
(("2"
(expand "Riemann_sum?")
(("2"
(auto-rewrite "Rie_sum")
(("2"
(skosimp*)
(("2"
(assert)
(("2"
(inst + "xis!1")
(("2"
(replace -1)
(("2"
(hide -1)
(("2"
(assert)
(("2"
(expand
"const_fun")
(("2"
(rewrite
"sigma_restrict_eq[below(length(P!1) - 1)]")
(("1"
(hide 2)
(("1"
(expand
"restrict")
(("1"
(apply-extensionality
1
:hide?
t)
(("1"
(inst?)
(("1"
(assert)
nil
nil)
("2"
(hide
2)
(("2"
(typepred
"P!1")
(("2"
(typepred
"xis!1")
(("2"
(assert)
(("2"
(flatten)
(("2"
(assert)
nil
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2"
(skosimp*)
(("2"
(typepred
"xis!1")
(("2"
(assert)
(("2"
(flatten)
(("2"
(assert)
nil
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2"
(skosimp*)
(("2"
(hide 2)
(("2"
(typepred
"xis!1")
(("2"
(assert)
(("2"
(flatten)
(("2"
(assert)
nil
nil))
nil))
nil))
nil))
nil))
nil)
("3"
(hide 2)
(("3"
(skosimp*)
(("3"
(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" (propax) nil nil))
nil))
nil))
nil))
nil))
nil))
nil)
((integral_const_fun formula-decl nil integral_prep nil)
(real_times_real_is_real application-judgement "real" reals nil)
(real_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
(integral const-decl "{S: real | integral?(a, b, ff, S)}"
integral_def nil)
(integrable? const-decl "bool" integral_def nil)
(const_fun const-decl "[T -> real]" real_fun_ops "reals/")
(integral? const-decl "bool" integral_def nil)
(< const-decl "bool" reals nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(bool nonempty-type-eq-decl nil booleans nil)
(Riemann_sum? const-decl "bool" integral_def nil)
(real_gt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props 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)
(finseq_appl const-decl "[below[length(fs)] -> T]" finite_sequences
nil)
(Rie_sum const-decl "real" integral_def nil)
(integer nonempty-type-from-decl nil integers nil)
(IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
(T_low type-eq-decl nil sigma "reals/")
(T_high type-eq-decl nil sigma "reals/")
(OR const-decl "[bool, bool -> bool]" booleans nil)
(* const-decl "[numfield, numfield -> numfield]" number_fields nil)
(sigma_restrict_eq formula-decl nil sigma "reals/")
(posint_plus_nnint_is_posint application-judgement "posint"
integers nil)
(restrict const-decl "[T -> real]" sigma "reals/")
(xis? const-decl "bool" integral_def nil)
(int_minus_int_is_int application-judgement "int" integers nil)
(R!1 skolem-const-decl "(Riemann_sum?(a!1, b!1, P!1, f!1))"
integral_prep nil)
(f!1 skolem-const-decl "[T -> real]" integral_prep nil)
(D!1 skolem-const-decl "real" integral_prep nil)
(P!1 skolem-const-decl "partition[T](a!1, b!1)" integral_prep nil)
(b!1 skolem-const-decl "T" integral_prep nil)
(a!1 skolem-const-decl "T" integral_prep nil)
(minus_odd_is_odd application-judgement "odd_int" integers nil)
(partition type-eq-decl nil integral_def nil)
(+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
(below type-eq-decl nil naturalnumbers 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)
(- const-decl "[numfield, numfield -> numfield]" number_fields nil)
(numfield nonempty-type-eq-decl nil number_fields nil)
(= const-decl "[T, T -> boolean]" equalities nil)
(finite_sequence type-eq-decl nil finite_sequences nil)
(closed_interval type-eq-decl nil intervals_real "reals/")
(<= const-decl "bool" reals nil)
(AND const-decl "[bool, bool -> bool]" booleans nil)
(below type-eq-decl nil nat_types nil)
(nat nonempty-type-eq-decl nil naturalnumbers 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)
(T formal-subtype-decl nil integral_prep nil)
(T_pred const-decl "[real -> boolean]" integral_prep 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_minus_real_is_real application-judgement "real" reals nil))
nil)
(integral_const_restrict-2 nil 3277741047
("" (skosimp*)
(("" (lemma "integral_const_fun")
(("" (inst?)
(("" (assert)
(("" (flatten)
(("" (typepred "integral(a!1, b!1, const_fun[T](D!1))")
(("1" (expand "integral?")
(("1" (skosimp*)
(("1" (inst -1 "epsi!1")
(("1" (skosimp*)
(("1" (inst + "delta!1")
(("1" (skosimp*)
(("1" (inst?)
(("1" (assert)
(("1"
(replace -3)
(("1"
(typepred "R!1")
(("1"
(inst -2 "R!1")
(("1" (assert) nil nil)
("2"
(hide 2)
(("2"
(expand "Riemann_sum?")
(("2"
(skosimp*)
(("2"
(assert)
(("2"
(inst + "xis!1")
(("2"
(replace -1)
(("2"
(hide -1)
(("2"
(assert)
(("2"
(expand
"const_fun")
(("2"
(rewrite
"sigma_restrict_eq[below(length(P!1) - 1)]")
(("1"
(hide 2)
(("1"
(expand
"restrict")
(("1"
(apply-extensionality
1
:hide?
t)
(("1"
(inst?)
(("1"
(assert)
nil
nil)
("2"
(hide
2)
(("2"
(typepred
"P!1")
(("2"
(typepred
"xis!1")
(("2"
(assert)
(("2"
(flatten)
(("2"
(assert)
nil
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2"
(skosimp*)
(("2"
(typepred
"xis!1")
(("2"
(assert)
(("2"
(flatten)
(("2"
(assert)
nil
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2"
(skosimp*)
(("2"
(hide 2)
(("2"
(typepred
"xis!1")
(("2"
(assert)
(("2"
(flatten)
(("2"
(assert)
nil
nil))
nil))
nil))
nil))
nil))
nil)
("3"
(hide 2)
(("3"
(skosimp*)
(("3"
(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" (propax) nil nil))
nil))
nil))
nil))
nil))
nil))
nil)
((x_in const-decl "{t: T | aa <= t AND t <= bb}" integral_def nil)
(integral const-decl "{S: real | integral?(a, b, ff, S)}"
integral_def nil)
(integrable? const-decl "bool" integral_def nil)
(const_fun const-decl "[T -> real]" real_fun_ops "reals/")
(integral? const-decl "bool" integral_def nil)
(Riemann_sum? const-decl "bool" integral_def nil)
(xis? const-decl "bool" integral_def nil)
(restrict const-decl "[T -> real]" sigma "reals/")
(sigma_restrict_eq formula-decl nil sigma "reals/")
(partition type-eq-decl nil integral_def nil))
nil)
(integral_const_restrict-1 nil 3253532115
("" (skosimp*)
(("" (lemma "integral_const_fun")
(("" (inst?)
(("" (assert)
(("" (flatten)
(("" (typepred "integral(a!1, b!1, const_fun[T](D!1))")
(("1" (expand "is_integral")
(("1" (skosimp*)
(("1" (inst -1 "epsi!1")
(("1" (skosimp*)
(("1" (inst + "delta!1")
(("1" (skosimp*)
(("1" (inst?)
(("1" (assert)
(("1"
(replace -3)
(("1"
(hide -3)
(("1"
(case-replace
"Riemann_sum(a!1, b!1, P!1, f!1) = Riemann_sum(a!1, b!1, P!1, const_fun[T](D!1))")
(("1" (assert) nil nil)
("2"
(hide -1 2)
(("2"
(expand "Riemann_sum")
(("2"
(rewrite "sigma_restrict_eq")
(("1"
(hide 2)
(("1"
(expand "restrict")
(("1"
(apply-extensionality
1
:hide?
t)
(("1"
(lift-if)
(("1"
(ground)
(("1"
(expand
"const_fun")
(("1"
(inst?)
(("1"
(replace -3)
(("1"
(propax)
nil
nil))
nil))
nil))
nil))
nil))
nil)
("2"
(skosimp*)
(("2"
(typepred "P!1")
(("2"
(inst - "i!1-1")
(("1"
(assert)
nil
nil)
("2"
(assert)
nil
nil))
nil))
nil))
nil)
("3"
(skosimp*)
(("3"
(assert)
nil
nil))
nil))
nil))
nil))
nil)
("2"
(hide 2)
(("2"
(skosimp*)
(("2"
(assert)
(("2"
(typepred "P!1")
(("2"
(inst - "n!1-1")
(("2"
(assert)
nil
nil))
nil))
nil))
nil))
nil))
nil)
("3"
(hide-all-but 1)
(("3"
(skosimp*)
(("3" (assert) nil nil))
nil))
nil)
("4"
(hide-all-but 1)
(("4"
(skosimp*)
(("4" (ground) nil nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2" (propax) nil nil))
nil))
nil))
nil))
nil))
nil))
nil)
((integral const-decl "{S: real | integral?(a, b, ff, S)}"
integral_def nil)
(integrable? const-decl "bool" integral_def nil)
(const_fun const-decl "[T -> real]" real_fun_ops "reals/")
(x_in const-decl "{t: T | aa <= t AND t <= bb}" integral_def nil)
(sigma_restrict_eq formula-decl nil sigma "reals/")
(restrict const-decl "[T -> real]" sigma "reals/")
(partition type-eq-decl nil integral_def nil))
nil))
(integral_scal 0
(integral_scal-3 nil 3280253820
("" (skosimp*)
(("" (case-replace "D!1 = 0")
(("1" (assert)
(("1" (expand "*")
(("1" (lemma "integral_const_fun")
(("1" (inst - "0" "a!1" "b!1")
(("1" (expand "const_fun")
(("1" (flatten) (("1" (assert) nil nil)) nil)) nil))
nil))
nil))
nil))
nil)
("2" (rewrite "integral_def")
(("2" (typepred "integral(a!1, b!1, f!1)")
(("1" (expand "integral?")
(("1" (skosimp*)
(("1" (inst -1 "epsi!1/abs(D!1)")
(("1" (skosimp*)
(("1" (inst + "delta!1")
(("1" (skosimp*)
(("1" (inst?)
(("1" (assert)
(("1" (typepred "R!1")
(("1" (expand "Riemann_sum?")
(("1"
(auto-rewrite "Rie_sum")
(("1"
(assert)
(("1"
(skosimp*)
(("1"
(replace -1)
(("1"
(hide -1)
(("1"
(case-replace
"Rie_sum(a!1, b!1, P!1, xis!1, D!1 * f!1) = D!1*Rie_sum(a!1, b!1, P!1, xis!1, f!1)")
(("1"
(expand "Rie_sum")
(("1"
(assert)
(("1"
(replace -1)
(("1"
(hide -1)
(("1"
(inst
-
"sigma[below(length(P!1) - 1)]
(0, length(P!1) - 2,
LAMBDA (n: below(length(P!1) - 1)):
P!1`seq(1 + n) * f!1(xis!1(n)) -
P!1`seq(n) * f!1(xis!1(n)))")
(("1"
(cross-mult -1)
(("1"
(assert)
(("1"
(split -1)
(("1"
(flatten)
(("1"
(rewrite
"abs_mult "
:dir
rl)
(("1"
(assert)
nil
nil)
("2"
(hide
3)
(("2"
(skosimp*)
(("2"
(assert)
nil
nil))
nil))
nil))
nil))
nil)
("2"
(flatten)
(("2"
(hide-all-but
(1 2))
(("2"
(expand
"abs")
(("2"
(lift-if)
(("2"
(ground)
nil
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2"
(skosimp*)
(("2"
(assert)
nil
nil))
nil)
("3"
(hide-all-but
(1 2))
(("3"
(grind)
nil
nil))
nil))
nil)
("2"
(hide 3)
(("2"
(expand
"Riemann_sum?")
(("2"
(assert)
(("2"
(inst
+
"xis!1")
nil
nil))
nil))
nil))
nil)
("3"
(skosimp*)
(("3"
(assert)
nil
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2"
(hide 2)
(("2"
(hide 2)
(("2"
(expand "Rie_sum")
(("2"
(assert)
(("2"
(lemma
"sigma_scal[below(length(P!1) - 1)]")
(("1"
(inst?)
(("1"
(assert)
(("1"
(expand "*")
(("1"
(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)
("2" (hide 3)
(("2" (assert)
(("2" (case "epsi!1 / abs(D!1) > 0")
(("1" (assert) nil nil)
("2" (hide 2)
(("2" (hide -2)
(("2" (cross-mult 1)
(("1" (grind) nil nil)
("2" (assert)
(("2"
(hide 2)
(("2" (grind) nil nil))
nil))
nil))
nil))
nil))
nil)
("3" (hide 2) (("3" (grind) nil nil)) nil))
nil))
nil))
nil)
("3" (hide -2 3) (("3" (grind) nil nil)) nil))
nil))
nil))
nil)
("2" (propax) nil nil) ("3" (propax) nil nil))
nil))
nil))
nil))
nil)
((number nonempty-type-decl nil numbers nil)
(boolean nonempty-type-decl nil booleans nil)
(= const-decl "[T, T -> boolean]" equalities 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 "[T -> real]" real_fun_ops "reals/")
(T formal-subtype-decl nil integral_prep nil)
(T_pred const-decl "[real -> boolean]" integral_prep nil)
(real_times_real_is_real application-judgement "real" reals nil)
(const_fun const-decl "[T -> real]" real_fun_ops "reals/")
(integral_const_fun formula-decl nil integral_prep nil)
(real_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(div_mult_pos_gt1 formula-decl nil extra_real_props nil)
(int_minus_int_is_int application-judgement "int" integers nil)
(finseq_appl const-decl "[below[length(fs)] -> T]" finite_sequences
nil)
(Rie_sum const-decl "real" integral_def nil)
(xis? const-decl "bool" integral_def nil)
(real_le_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(div_mult_pos_neg_lt2 formula-decl nil extra_real_props nil)
(nonzero_real nonempty-type-eq-decl nil reals nil)
(abs_mult formula-decl nil real_props nil)
(nnreal_times_nnreal_is_nnreal application-judgement "nnreal"
real_types nil)
(xis!1 skolem-const-decl "(xis?(a!1, b!1, P!1))" integral_prep 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)
(f!1 skolem-const-decl "[T -> real]" integral_prep nil)
(IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
(integer nonempty-type-from-decl nil integers nil)
(P!1 skolem-const-decl "partition[T](a!1, b!1)" integral_prep nil)
(b!1 skolem-const-decl "T" integral_prep nil)
(a!1 skolem-const-decl "T" integral_prep nil)
(posint_plus_nnint_is_posint application-judgement "posint"
integers nil)
(sigma_scal formula-decl nil sigma "reals/")
(Riemann_sum? const-decl "bool" integral_def nil)
(nat nonempty-type-eq-decl nil naturalnumbers nil)
(below type-eq-decl nil nat_types 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 "[numfield, numfield -> numfield]" number_fields 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)
(below type-eq-decl nil naturalnumbers nil)
(+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
(partition type-eq-decl nil integral_def nil)
(epsi!1 skolem-const-decl "posreal" integral_prep nil)
(posreal nonempty-type-eq-decl nil real_types nil)
(> const-decl "bool" reals nil)
(/ const-decl "[numfield, nznum -> numfield]" number_fields nil)
(nznum nonempty-type-eq-decl nil number_fields nil)
(D!1 skolem-const-decl "real" integral_prep nil)
(abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
nil)
(- const-decl "[numfield -> numfield]" number_fields nil)
(AND const-decl "[bool, bool -> bool]" booleans nil)
(nonneg_real nonempty-type-eq-decl nil real_types nil)
(>= const-decl "bool" reals nil)
(/= const-decl "boolean" notequal nil)
(nzreal_div_nzreal_is_nzreal application-judgement "nzreal"
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)
(real_minus_real_is_real application-judgement "real" reals nil)
(integral const-decl "{S: real | integral?(a, b, ff, S)}"
integral_def nil)
(integral? const-decl "bool" integral_def nil)
(integrable? const-decl "bool" integral_def nil)
(< const-decl "bool" reals nil)
(bool nonempty-type-eq-decl nil booleans nil)
(* const-decl "[numfield, numfield -> numfield]" number_fields nil)
(numfield nonempty-type-eq-decl nil number_fields nil)
(integral_def formula-decl nil integral_def nil))
nil)
(integral_scal-2 nil 3278176311
("" (skosimp*)
(("" (case-replace "D!1 = 0")
(("1" (assert)
(("1" (expand "*")
(("1" (lemma "integral_const_fun")
(("1" (inst - "0" "a!1" "b!1")
(("1" (expand "const_fun")
(("1" (flatten) (("1" (assert) nil nil)) nil)) nil))
nil))
nil))
nil))
nil)
("2" (rewrite "integral_def")
(("2" (typepred "integral(a!1, b!1, f!1)")
(("1" (expand "integral?")
(("1" (skosimp*)
(("1" (inst -1 "epsi!1/abs(D!1)")
(("1" (skosimp*)
(("1" (inst + "delta!1")
(("1" (skosimp*)
(("1" (inst?)
(("1" (assert)
(("1" (typepred "R!1")
(("1" (expand "Riemann_sum?")
(("1"
(auto-rewrite "Rie_sum")
(("1"
(assert)
(("1"
(skosimp*)
(("1"
(replace -1)
(("1"
(hide -1)
(("1"
(case-replace
"Rie_sum(a!1, b!1, P!1, xis!1, D!1 * f!1) = D!1*Rie_sum(a!1, b!1, P!1, xis!1, f!1)")
(("1"
(expand "Rie_sum")
(("1"
(assert)
(("1"
(replace -1)
(("1"
(hide -1)
(("1"
(inst
-
"sigma[below(length(P!1) - 1)]
(0, length(P!1) - 2,
LAMBDA (n: below(length(P!1) - 1)):
P!1`seq(1 + n) * f!1(xis!1`seq(n)) -
P!1`seq(n) * f!1(xis!1`seq(n)))")
(("1"
(cross-mult -1)
(("1"
(assert)
(("1"
(split -1)
(("1"
(flatten)
(("1"
(rewrite
"abs_mult "
:dir
rl)
(("1"
(assert)
nil
nil)
("2"
(hide
3)
(("2"
(skosimp*)
(("2"
(assert)
nil
nil))
nil))
nil)
("3"
(skosimp*)
(("3"
(assert)
(("3"
(typepred
"xis!1")
(("3"
(assert)
nil
nil))
nil))
nil))
nil))
nil))
nil)
("2"
(flatten)
(("2"
(hide-all-but
(1 2))
(("2"
(expand
"abs")
--> --------------------
--> maximum size reached
--> --------------------
¤ Dauer der Verarbeitung: 0.123 Sekunden
(vorverarbeitet)
¤
|
Haftungshinweis
Die Informationen auf dieser Webseite wurden
nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit,
noch Qualität der bereit gestellten Informationen zugesichert.
Bemerkung:
Die farbliche Syntaxdarstellung ist noch experimentell.
|
