|
(deriv_sigma
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(IMP_derivatives_TCC2-1 nil 3514223777
("" (lemma "not_one_element") (("" (propax) nil nil)) nil)
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nil))
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(real_minus_real_is_real application-judgement "real" reals nil))
nil))
(sigma_derivable_TCC3 0
(sigma_derivable_TCC3-1 nil 3460460862 ("" (subtype-tcc) nil nil)
((boolean nonempty-type-decl nil booleans nil)
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(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)
(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)
(>= const-decl "bool" reals nil)
(nonneg_int nonempty-type-eq-decl nil integers nil)
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(T_pred const-decl "[real -> boolean]" deriv_sigma nil)
(T formal-subtype-decl nil deriv_sigma nil)
(real_lt_is_strict_total_order name-judgement
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(real_gt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props 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)
(< const-decl "bool" reals nil)
(nat nonempty-type-eq-decl nil naturalnumbers nil)
(Index type-eq-decl nil vectors "vectors/")
(integer nonempty-type-from-decl nil integers nil)
(fullset const-decl "set" sets nil)
(adh const-decl "setof[real]" convergence_functions "analysis/")
(NQ const-decl "real" derivatives_def "analysis/")
(convergence const-decl "bool" convergence_functions "analysis/")
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(convergent? const-decl "bool" lim_of_functions "analysis/")
(derivable? const-decl "bool" derivatives_def "analysis/")
(real_minus_real_is_real application-judgement "real" reals nil)
(real_div_nzreal_is_real application-judgement "real" reals nil))
nil))
(sigma_derivable 0
(sigma_derivable-1 nil 3460463493
("" (induct "m")
(("1" (assert) nil nil) ("2" (assert) nil nil)
("3" (skosimp*)
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(("1" (assert)
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(case-replace "(LAMBDA (x: T):
sigma(0, j!1, LAMBDA (i: Index[1 + j!1]): h!1(x)(i))) =
(LAMBDA (x: T):
sigma(0, j!1-1, LAMBDA (i: Index[1 + j!1]): h!1(x)(i)) + h!1(x)(j!1))")
(("1" (hide -1)
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(inst - "y!1"
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"(LAMBDA (x: T):
sigma(0, j!1 - 1,
LAMBDA (i: Index[1 + j!1]): h!1(x)(i)))"
"y!1")
(("1" (assert)
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(expand "+
")
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("3" (hide 2)
(("3"
(case-replace
"(LAMBDA (x_1: T):
sigma(0, j!1 - 1,
LAMBDA (i_1: Index[j!1]): h!1(x_1)(i_1))) =
(LAMBDA (x: T):
sigma(0, j!1 - 1,
LAMBDA (i: Index[1 + j!1]): h!1(x)(i)))
")
(("1"
(hide 2)
(("1"
(apply-extensionality 1 :hide? t)
(("1"
(hide -1)
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(lemma
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(inst?)
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nil))
nil))
nil)
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(skosimp*)
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nil)
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(skosimp*)
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nil))
nil))
nil)
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(skosimp*)
(("2" (assert) nil nil))
nil)
("3"
(skosimp*)
(("3" (assert) nil nil))
nil))
nil))
nil))
nil))
nil)
("2" (skosimp*) (("2" (assert) nil nil)) nil))
nil))
nil)
("2" (skosimp*) (("2" (inst?) nil nil)) nil))
nil))
nil))
nil))
nil)
("2" (hide -2 2)
(("2" (apply-extensionality 1 :hide? t)
(("1" (lemma "sigma_last[Index[j!1+1]]")
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("2" (skosimp*) (("2" (assert) nil nil)) nil))
nil)
("2" (skosimp*) (("2" (assert) nil nil)) nil))
nil))
nil)
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nil))
nil)
("2" (hide -1)
(("2" (assert)
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nil))
nil))
nil))
nil)
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("6" (skosimp*) (("6" (assert) nil nil)) nil))
nil)
((posint_plus_nnint_is_posint application-judgement "posint"
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real_props nil)
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real_props nil)
(real_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
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(j!1 skolem-const-decl "nat" deriv_sigma nil)
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(= const-decl "[T, T -> boolean]" equalities nil)
(+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
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(posnat nonempty-type-eq-decl nil integers nil)
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(nat_induction formula-decl nil naturalnumbers nil)
(sigma def-decl "real" sigma "reals/")
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(T_low type-eq-decl nil sigma "reals/")
(pred type-eq-decl nil defined_types nil)
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(AND const-decl "[bool, bool -> bool]" booleans nil)
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(number nonempty-type-decl nil numbers nil)
(boolean nonempty-type-decl nil booleans nil)
(number_field_pred const-decl "[number -> boolean]" number_fields
nil)
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(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)
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(integer_pred const-decl "[rational -> boolean]" integers nil)
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(int_minus_int_is_int application-judgement "int" integers nil))
shostak))
(deriv_sigma_TCC1 0
(deriv_sigma_TCC1-1 nil 3460386482 ("" (subtype-tcc) nil nil)
((boolean nonempty-type-decl nil booleans nil)
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(integer_pred const-decl "[rational -> boolean]" integers nil)
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(nat nonempty-type-eq-decl nil naturalnumbers nil)
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(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)
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(NQ const-decl "real" derivatives_def "analysis/")
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(real_div_nzreal_is_real application-judgement "real" reals nil))
nil))
(deriv_sigma_TCC2 0
(deriv_sigma_TCC2-1 nil 3460461487 ("" (subtype-tcc) nil nil)
((boolean nonempty-type-decl nil booleans nil)
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nil)
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(rational nonempty-type-from-decl nil rationals nil)
(integer_pred const-decl "[rational -> boolean]" integers nil)
(int nonempty-type-eq-decl nil integers nil)
(nat nonempty-type-eq-decl nil naturalnumbers nil)
(T_pred const-decl "[real -> boolean]" deriv_sigma nil)
(T formal-subtype-decl nil deriv_sigma nil)
(int_minus_int_is_int application-judgement "int" integers 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)
(fullset const-decl "set" sets nil)
(adh const-decl "setof[real]" convergence_functions "analysis/")
(NQ const-decl "real" derivatives_def "analysis/")
(convergence const-decl "bool" convergence_functions "analysis/")
(convergence const-decl "bool" lim_of_functions "analysis/")
(convergent? const-decl "bool" lim_of_functions "analysis/")
(derivable? const-decl "bool" derivatives_def "analysis/")
(real_minus_real_is_real application-judgement "real" reals nil)
(real_div_nzreal_is_real application-judgement "real" reals nil))
nil))
(deriv_sigma_TCC3 0
(deriv_sigma_TCC3-1 nil 3460461487 ("" (subtype-tcc) nil 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)
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(integer_pred const-decl "[rational -> boolean]" integers nil)
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(nat nonempty-type-eq-decl nil naturalnumbers nil)
(T_pred const-decl "[real -> boolean]" deriv_sigma nil)
(T formal-subtype-decl nil deriv_sigma nil)
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(Index type-eq-decl nil vectors "vectors/")
(< const-decl "bool" reals nil)
(real_le_is_total_order name-judgement "(total_order?[real])"
real_props nil)
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"(strict_total_order?[real])" real_props nil)
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil)
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"(strict_total_order?[real])" real_props nil)
(fullset const-decl "set" sets nil)
(adh const-decl "setof[real]" convergence_functions "analysis/")
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(derivable? const-decl "bool" derivatives_def "analysis/")
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(real_div_nzreal_is_real application-judgement "real" reals nil))
nil))
(deriv_sigma_TCC4 0
(deriv_sigma_TCC4-1 nil 3460461487
("" (skosimp*)
(("" (lemma "sigma_derivable")
(("" (inst - "n!1" "y!1" "h!1") (("" (assert) nil nil)) nil))
nil))
nil)
((sigma_derivable formula-decl nil deriv_sigma nil)
(int_minus_int_is_int application-judgement "int" integers nil)
(Index type-eq-decl nil vectors "vectors/")
(< const-decl "bool" reals nil)
(T formal-subtype-decl nil deriv_sigma nil)
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(posnat nonempty-type-eq-decl nil integers nil)
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(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)
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(bool nonempty-type-eq-decl nil 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)
(>= const-decl "bool" reals nil)
(nat nonempty-type-eq-decl nil naturalnumbers nil)
(n!1 skolem-const-decl "nat" deriv_sigma nil)
(real_gt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil))
nil))
(deriv_sigma_TCC5 0
(deriv_sigma_TCC5-1 nil 3460462472 ("" (subtype-tcc) nil 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)
(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)
(nat nonempty-type-eq-decl nil naturalnumbers nil)
(T_pred const-decl "[real -> boolean]" deriv_sigma nil)
(T formal-subtype-decl nil deriv_sigma 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_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
(fullset const-decl "set" sets nil)
(adh const-decl "setof[real]" convergence_functions "analysis/")
(NQ const-decl "real" derivatives_def "analysis/")
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(convergent? const-decl "bool" lim_of_functions "analysis/")
(derivable? const-decl "bool" derivatives_def "analysis/")
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(real_div_nzreal_is_real application-judgement "real" reals nil))
nil))
(deriv_sigma_TCC6 0
(deriv_sigma_TCC6-1 nil 3460463371 ("" (subtype-tcc) nil 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)
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(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)
(nat nonempty-type-eq-decl nil naturalnumbers nil)
(T_pred const-decl "[real -> boolean]" deriv_sigma nil)
(T formal-subtype-decl nil deriv_sigma nil)
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real_props nil)
(real_gt_is_strict_total_order name-judgement
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(real_div_nzreal_is_real application-judgement "real" reals nil))
nil))
(deriv_sigma_TCC7 0
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((boolean nonempty-type-decl nil booleans nil)
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nil))
(deriv_sigma_TCC8 0
(deriv_sigma_TCC8-1 nil 3460463371
("" (skosimp*) (("" (assert) nil nil)) 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)
(real_gt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil))
nil))
(deriv_sigma 0
(deriv_sigma-1 nil 3460387081
("" (induct "n")
(("1" (skosimp*)
(("1" (expand "sigma")
(("1" (lemma "deriv_const[T]")
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(("1" (expand "const_fun") (("1" (inst?) nil nil)) nil))
nil))
nil))
nil))
nil)
("2" (skosimp*)
(("2" (case "j!1 = 0")
(("1" (replace -1)
(("1" (hide -2)
(("1" (expand "sigma")
(("1" (expand "sigma")
(("1" (expand "deriv" 1 2) (("1" (propax) nil nil))
nil))
nil))
nil))
nil))
nil)
("2" (inst - "y!1")
(("2" (assert)
(("1"
(inst -
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(("1" (split -1)
(("1" (lemma "sigma_last[below(j!1+1)]")
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(rational_pred const-decl "[real -> boolean]" rationals nil)
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(integer_pred const-decl "[rational -> boolean]" integers nil)
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(> const-decl "bool" reals nil) (< const-decl "bool" reals nil)
(Index type-eq-decl nil vectors "vectors/")
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(derivable? const-decl "bool" derivatives_def "analysis/")
(OR const-decl "[bool, bool -> bool]" booleans nil)
(<= const-decl "bool" reals nil)
(int_minus_int_is_int application-judgement "int" integers nil))
shostak)))
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