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ELSE g!1(P!1`seq(n)) * f!1(xis!1`seq(n - 1)) -
g!1(P!1`seq(n - 1)) * f!1(xis!1`seq(n - 1))
ENDIF)")
(("1" (inst?)
(("1" (assert)
(("1" (replace -1)
(("1" (replace -1 :dir rl)
(("1" (propax) nil nil)) nil))
nil))
nil))
nil)
("2" (hide 2)
(("2"
(induct "NN" 1 "below_induction[length(P!1)-1]")
(("1" (assert)
(("1" (expand "sigma")
(("1" (expand "sigma") (("1" (propax) nil nil))
nil))
nil))
nil)
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(("2" (expand "sigma" 1) (("2" (assert) nil nil))
nil))
nil)
("3" (hide 2)
(("3" (skosimp*) (("3" (assert) nil nil)) nil))
nil))
nil))
nil)
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(("3" (skosimp*) (("3" (assert) nil nil)) nil)) nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
((real_times_real_is_real application-judgement "real" reals nil)
(Rie_sum const-decl "real" rs_integral_def nil)
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(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)" rs_integral_def
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(b!1 skolem-const-decl "T" rs_integral_def nil)
(a!1 skolem-const-decl "T" rs_integral_def nil)
(int_plus_int_is_int application-judgement "int" integers nil)
(IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(IF const-decl "[boolean, T, T -> T]" if_def nil)
(upto nonempty-type-eq-decl nil naturalnumbers nil)
(xis? type-eq-decl nil rs_integral_def nil)
(+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
(* const-decl "[numfield, numfield -> numfield]" number_fields 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)
(nnint_plus_posint_is_posint application-judgement "posint"
integers nil)
(posint_plus_nnint_is_posint application-judgement "posint"
integers nil)
(Rie_sec const-decl "real" rs_integral_def nil)
(real_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
(int_minus_int_is_int application-judgement "int" integers nil)
(real_minus_real_is_real application-judgement "real" reals 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)
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(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)
(increasing? const-decl "bool" sort_fseq "structures/")
(<= const-decl "bool" reals nil)
(restrict const-decl "R" restrict nil)
(= const-decl "[T, T -> boolean]" equalities nil)
(> const-decl "bool" reals nil)
(fseq type-eq-decl nil fseqs "structures/")
(barray type-eq-decl nil fseqs "structures/")
(T formal-nonempty-subtype-decl nil rs_integral_def nil)
(T_pred const-decl "[real -> boolean]" rs_integral_def nil)
(nat nonempty-type-eq-decl nil naturalnumbers nil)
(- const-decl "[numfield, numfield -> numfield]" number_fields nil)
(numfield nonempty-type-eq-decl nil number_fields nil)
(>= const-decl "bool" reals 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)
(number nonempty-type-decl nil numbers nil)
(AND const-decl "[bool, bool -> bool]" booleans nil)
(bool nonempty-type-eq-decl nil booleans nil)
(boolean nonempty-type-decl nil booleans nil))
nil)
(Rie_sum_alt_lem-2 nil 3280252800
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(0, NN,
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)))
=
sigma[upto(length(P!1) - 1)]
(1, NN+1,
LAMBDA (n: upto(length(P!1) - 1)):
IF n = 0 THEN 0
ELSE P!1`seq(n) * f!1(xis!1`seq(n - 1)) -
P!1`seq(n - 1) * f!1(xis!1`seq(n - 1))
ENDIF)")
(("1" (inst?) (("1" (assert) nil)))
("2" (hide 2)
(("2"
(induct "NN" 1 "below_induction[length(P!1)-1]")
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(("7" (assert)
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(("7" (assert) nil)))))))))))))
("3" (hide 2)
(("3" (skosimp*) (("3" (assert) nil)))))
("4" (hide 2)
(("4" (skosimp*) (("4" (assert) nil)))))
("5" (hide 2)
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(("5" (assert)
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(("5" (assert) nil)))))))))
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("7" (hide 2)
(("7" (skosimp*)
(("7" (assert)
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(("7" (assert) nil))))))))))))))))))))))))
nil)
nil nil)
(Rie_sum_alt_lem-1 nil 3278324183
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(0, NN,
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)))
=
sigma[upto(length(P!1) - 1)]
(1, NN+1,
LAMBDA (n: upto(length(P!1) - 1)):
IF n = 0 THEN 0
ELSE P!1`seq(n) * f!1(xis!1`seq(n - 1)) -
P!1`seq(n - 1) * f!1(xis!1`seq(n - 1))
ENDIF)")
(("1" (inst?) (("1" (assert) nil nil)) nil)
("2" (hide 2)
(("2"
(induct "NN" 1 "below_induction[length(P!1)-1]")
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nil))
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nil)
("3" (hide 2)
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nil)
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nil)
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(("5" (skosimp*)
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nil))
nil))
nil)
("6" (hide 2)
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nil)
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(("7" (assert) nil nil)) nil))
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nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
((sigma def-decl "real" sigma "reals/")) shostak))
(Riemann_sum_strictly_sort_TCC1 0
(Riemann_sum_strictly_sort_TCC1-1 nil 3492270809
("" (lemma "partition_strictly_sort")
(("" (assert)
(("" (skeep)
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(("" (inst - "a" "b")
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(partition type-eq-decl nil rs_partition nil)
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(integer_pred const-decl "[rational -> boolean]" integers nil)
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(rational_pred const-decl "[real -> boolean]" rationals nil)
(increasing? const-decl "bool" sort_fseq "structures/")
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(- const-decl "[numfield, numfield -> numfield]" number_fields nil)
(numfield nonempty-type-eq-decl nil number_fields nil)
(= const-decl "[T, T -> boolean]" equalities nil)
(> const-decl "bool" reals nil)
(AND const-decl "[bool, bool -> bool]" booleans nil)
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(partition_strictly_sort formula-decl nil rs_partition 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]" rs_integral_def nil)
(T formal-nonempty-subtype-decl nil rs_integral_def nil))
nil))
(Riemann_sum_strictly_sort 0
(Riemann_sum_strictly_sort-3 nil 3495206538
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(("" (skeep)
(("" (skeep)
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(("1" (label "XISfunname" -1)
(("1"
(case "FORALL (xis:xis?(a,b,P)): Rie_sum(a,b,g,P,xis,f) = Rie_sum(a,b,g,strictly_sort(P),XISfun(xis),f)")
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nil))
nil)
("2" (hide -1)
(("2" (hide -1)
(("2" (hide 2)
(("2" (skeep)
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(("1"
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--> --------------------
--> maximum size reached
--> --------------------
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