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nil)
nil shostak)
(deriv_polynomial-1 nil 3259333426
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nil))
nil))
nil))
nil)
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(("1" (expand "sigma" 1)
(("1" (inst - "a")
(("1" (assert)
(("1"
(lemma "deriv_sum_fun"
("ff1"
"LAMBDA (x: real): sigma(0, j, LAMBDA (i: nat): a(i) * (IF i = 0 THEN 1 ELSE x ^ i ENDIF))"
"ff2"
"LAMBDA (x: real): a(1 + j) * x ^ (1 + j)"))
(("1" (expand "+" -1)
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(("1" (replace -3 1)
(("1"
(hide -1 -3)
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(lemma
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(lemma "deriv_id_fun[real]")
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(expand "const_fun")
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(lemma
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"n"
"1+j"))
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(expand "^" -1)
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(flatten -1)
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(lemma
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"a(1+j)"
"ff"
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(replace -3 -1)
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(expand "*" -1)
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1)
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(propax)
nil
nil))
nil))
nil))
nil))
nil)
("2"
(propax)
nil
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
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(assert)
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nil))
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nil))
nil))
nil))
nil))
nil))
nil))
nil)
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(lemma "derivable_polynomial"
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(("3" (propax) nil nil)) nil))
nil))
nil))
nil))
nil))
nil))
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(lemma
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("b" "a(0)"))
(("2"
(expand "I")
(("2"
(expand "const_fun")
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(lemma
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"a(1)"
"ff"
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(expand "*" -1)
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"ff2"
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nil))
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(lemma
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(expand "I")
(("2" (propax) nil nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("3" (assert) nil nil))
nil))
nil))
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nil))
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nil)
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("4" (hide 2)
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(("4" (lemma "derivable_polynomial" ("a" "a" "n" "n"))
(("4" (propax) nil nil)) nil))
nil))
nil))
nil)
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(const_derivable_fun formula-decl nil derivatives nil))
shostak))
(derivable_n_times_polynomial 0
(derivable_n_times_polynomial-1 nil 3261819507
("" (induct "n")
(("1" (skosimp*)
(("1" (expand "derivable_n_times?") (("1" (propax) nil nil))
nil))
nil)
("2" (skosimp*)
(("2" (expand "derivable_n_times?" 1)
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nil))
nil))
nil))
nil))
nil))
nil)
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(("2" (replace -1 2)
(("2"
(inst -
"LAMBDA (i: nat): b!1(1 + i) + i * b!1(1 + i)"
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nil nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
((derivable_polynomial formula-decl nil polynomial_deriv nil)
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(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(- const-decl "[numfield, numfield -> numfield]" number_fields nil)
(* const-decl "[numfield, numfield -> numfield]" number_fields nil)
(+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
(numfield nonempty-type-eq-decl nil number_fields nil)
(sigma def-decl "real" sigma "reals/")
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(real_pred const-decl "[number_field -> boolean]" reals nil)
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(number_field_pred const-decl "[number -> boolean]" number_fields
nil)
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(number nonempty-type-decl nil numbers nil))
shostak))
(nderiv_polynomial_TCC1 0
(nderiv_polynomial_TCC1-1 nil 3260944881
("" (skosimp*)
((""
(lemma "derivable_n_times_polynomial"
("n" "n!1" "b" "b!1" "m" "m!1"))
(("" (propax) nil nil)) nil))
nil)
((sequence type-eq-decl nil sequences nil)
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(>= const-decl "bool" reals nil)
(bool nonempty-type-eq-decl nil booleans nil)
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(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)
(derivable_n_times_polynomial formula-decl nil polynomial_deriv
nil))
shostak))
(nderiv_polynomial_TCC2 0
(nderiv_polynomial_TCC2-1 nil 3260944938 ("" (grind) nil nil) nil
shostak))
(nderiv_polynomial_TCC3 0
(nderiv_polynomial_TCC3-1 nil 3260944949 ("" (grind) nil nil)
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(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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(>= 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)
(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))
shostak))
(nderiv_polynomial 0
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(("1" (skosimp*)
(("1" (assert)
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(lemma "extensionality"
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nil)
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(("2"
(expand "sigma")
(("2"
(expand "sigma")
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nil))
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nil))
nil))
nil))
nil))
nil)
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(("1" (assert) nil nil)
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(assert)
(("2"
(lemma
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("f"
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factorial(j) * b(1 + i_1 + j) * C(i_1 + j, j) +
i_1 * factorial(j) * b(1 + i_1 + j) * C(i_1 + j, j)
+ factorial(j) * b(1 + i_1 + j) * C(i_1 + j, j) * j"
"g"
"LAMBDA (i: nat):
factorial(1 + j) * b(1 + i + j) * C(1 + i + j,1+j)"))
(("2"
(split -1)
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(("1" (propax) nil nil))
nil)
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(hide 4)
(("2"
(skolem 1 ("i"))
(("2"
(name-replace
"K1"
"b(1 + i + j)")
(("2"
--> --------------------
--> maximum size reached
--> --------------------
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