(derivative_inversenil 3472399224"" (lemma "connected_deriv_domain[T1]" )"" (lemma "connected_domain1" )"" (lemma not_one_element1) (("" (assert ) nil nil )) nil )) nil ))nil )nil derivative_inverse nil )nil derivative_inverse nil )nil deriv_domain_def nil )nil numbers nil )nil booleans nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" derivative_inverse nil )nil derivative_inverse nil ))nil )nil 3471606970"" (skosimp*)"" (lemma "connected_domain1" )"" (lemma "connected_deriv_domain[T1]" )"1" (replace -2) (("1" (inst?) nil nil )) nil )"2" (lemma "not_one_element1" ) (("2" (propax) nil nil )) nil ))nil ))nil ))nil )nil shostak))nil 3472399302"" (lemma "connected_deriv_domain[T2]" )"" (lemma "connected_domain2" )"" (lemma "not_one_element2" ) (("" (assert ) nil nil )) nil ))nil ))nil )nil derivative_inverse nil )nil derivative_inverse nil )nil deriv_domain_def nil )nil numbers nil )nil booleans nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" derivative_inverse nil )nil derivative_inverse nil ))nil )nil 3472399245"" (lemma "connected_domain2" )"" (lemma "connected_deriv_domain[T2]" )"" (replace -2)"" (lemma "not_one_element2" )"" (replace -1)"" (hide -1)"" (expand "deriv_domain?" ) (("" (propax) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"bool" deriv_domain_def nil )nil deriv_domain_def nil ))nil )nil 3471607027"" (skosimp*)"" (lemma "connected_domain2" )"" (lemma "connected_deriv_domain[T2]" )"1" (replace -2) (("1" (inst?) nil nil )) nil )"2" (lemma "not_one_element2" ) (("2" (propax) nil nil )) nil ))nil ))nil ))nil )nil nil )nil 3471607020";;; Proof deriv_domain1-1 for formula derivative_inverse.deriv_domain1" ";;; Proof deriv_domain1-1 for formula derivative_inverse.deriv_domain1" "connected_domain2" )";;; Proof deriv_domain1-1 for formula derivative_inverse.deriv_domain1" "connected_deriv_domain[T2]" )"1" (replace -2) (("1" (inst?) nil )))"2" (lemma "not_one_element1" ) (("2" (propax) nil ))))))))";;; developed with shostak decision procedures") nil nil ))nil 3262438616"" (lemma "deriv_domain1" ) (("" (propax) nil nil )) nil )nil derivative_inverse nil )) shostak))nil 3262438682"" (lemma "not_one_element1" ) (("" (propax) nil nil )) nil )nil derivative_inverse nil ))nil 3262438701"" (skosimp*)"" (lemma "deriv_domain2" )"" (expand "deriv_domain?" ) (("" (inst?) nil nil )) nil )) nil ))nil )nil derivative_inverse nil )) shostak))nil 3262438892"" (skolem 1 ("F" "G" "f" ))"" (flatten)"" (lemma "not_one_element2" )"" (expand "not_one_element?" ) (("" (propax) nil nil )) nil ))nil ))nil ))nil )nil derivative_inverse nil ))"Alternative" 3477737338"" (skolem 1 ("F" "G" "f" "Y0" ))"" (flatten)"" "derivative_equivalence3" "f" "F" "D" "f(G(Y0))" "x" "G(Y0)" ))"1" (name "X0" "G(Y0)" )"1" (replace -1)"1" (case "deriv(F, X0) = f(X0)" )"1" (case "derivable?(F, X0)" )"1" (assert )"1" (skolem! -4)"1" (flatten -4)"1" "derivative_equivalence3" "f" "G" "x" "Y0" "D" "1/f(G(Y0))" ))"1" (flatten -1)"1" (hide -1)"1" (split -1)"1" (flatten -1) nil nil )"2" "2" case "FORALL (y: T2): psi!1(G(y)) /= 0" )"1" "LAMBDA (y:T2): 1/psi!1(G(y))" )"1" "1" "cv_in_domain" "f" "psi!1" "x" "X0" "l" "f(X0)" ))"1" assert )"1" replace -5 1)"1" replace -1 1)"1" replace -5 1 rl)"1" case "continuous?(LAMBDA (y: T2): 1 / psi!1(G(y)), Y0)" )"1" rewrite nil nil )"2" "2" "inv_continuous[T2]" "g" "LAMBDA (y: T2): psi!1(G(y))" "x0" "Y0" ))"2" "1" expand "/" "1" nil nil ))nil )"2" "2" "inverse_continuous[T1,T2]" "g" "F" ))"1" replace "1" case "inverse(F)=LAMBDA (y:T2): G(y)" )"1" replace "1" case "continuous?[T1](psi!1,X0)" )"1" expand "continuous?" "1" "Y0" )"1" "composition_cont[T2,T1]" "f" "G" "g" "psi!1" "x0" "Y0" ))"1" expand "o" "1" replace "1" replace "1" "1" nil nil )"2" "2" expand "continuous?" )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" expand "convergence" )"2" expand "convergence" )"2" "2" "2" "continuity_def2[T1]" "f" "psi!1" "x0" "X0" ))"2" expand "convergent?" )"2" replace "2" "f(X0)" )"2" expand "continuous?" )"2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" "2" "extensionality" "f" "inverse(F)" "g" "(LAMBDA (y: T2): G(y))" ))"2" "1" nil nil )"2" "2" "2" "bijective_inverse" "f" "F" "y" "x!1" "x" "G(x!1)" ))"2" "comp_inverse_right_alt" "f" "F" "g" "G" "r" "x!1" ))"2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "derivable_cont_fun[T1]" "f" "F" ))"2" assert )nil nil ))nil )"3" "Y0" )nil nil )"4" "X0" )nil nil )"5" "connected_domain1" )"5" expand "connected?" )"5" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"3" nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "Y1" ))"2" "G(Y1)" )"2" "comp_inverse_right_alt" "f" "F" "g" "G" "r" "Y1" ))"2" replace -1)"2" "comp_inverse_right_alt" "f" "F" "g" "G" "r" "Y0" ))"2" replace -6 -8 rl)"2" replace -1 -8)"2" "div_cancel4" "y" "G(Y1) - G(Y0)" "x" "Y1 - Y0" "n0z" "psi!1(G(Y1))" ))"2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" "Y1" ))"2" "G(Y1)" )"2" "comp_inverse_right_alt" "f" "F" "g" "G" "r" "Y1" ))"2" "comp_inverse_right_alt" "f" "F" "g" "G" "r" "Y0" ))"2" replace -2 -7)"2" replace -5 -7 rl)"2" replace -1)"2" "2" expand "bijective?" "2" "2" expand "injective?" )"2" case "Y1 = Y0" )"1" "cv_in_domain" "f" "psi!1" "x" "X0" "l" "f(X0)" ))"1" replace "1" replace "1" assert )nil nil ))nil ))nil ))nil )"2" "G(Y1)" "G(Y0)" )"2" replace "2" replace "2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (use "not_one_element2" )"2" (expand "not_one_element?" )"2" (propax) nil nil )) nil ))nil )"3" (use "deriv_domain2" ) nil nil ))nil ))nil ))nil ))nil )"2" (expand "derivable?" -4)"2" (inst -4 "X0" ) nil nil )) nil ))nil )"2" (expand "deriv" -5)"2" (replace -5 1 rl) (("2" (assert ) nil nil )) nil ))nil )"3" (lemma "not_one_element1" ) (("3" (propax) nil nil ))nil ))nil ))nil ))nil )"2" (use "not_one_element1" )"2" (expand "not_one_element?" ) (("2" (propax) nil nil ))nil ))nil )"3" (use "deriv_domain1" ) nil nil ))nil ))nil ))nil )nil derivative_inverse nil )nil derivative_inverse nil )"[T, T -> boolean]" equalities nil )"real" derivatives_def nil )"bool" derivatives_def nil )"bool" derivatives nil )"real" reals nil )nil derivative_inverse nil )nil derivative_inverse nil )"bool" continuous_functions nil )nil continuous_functions nil )nil continuous_functions nil )nil derivative_inverse nil )"bool" deriv_domain_def nil )nil derivatives nil )nil functions nil )nil function_inverse nil )"bool" functions nil )"bool" function_inverse_def nil )nil function_inverse_def nil )"bool" lim_of_functions nil )nil continuous_functions nil )"bool" lim_of_functions nil )"bool" convergence_functions nil )nil composition_continuous nil )"T3" function_props nil )"D" function_inverse nil )"bool" booleans nil )"[bool, bool -> bool]" booleans nil )AND const-decl "[bool, bool -> bool]" booleans nil )"bool" reals nil )"bool" continuous_functions nil )nil inverse_continuous_functionsnil )"[T -> real]" real_fun_ops "reals/" )nil lim_of_functions nil )"odd_int" integers nil )nil real_props nil )nil reals nil )"[numfield, numfield -> numfield]" number_fields nil )"[T2 -> T1]" derivative_inverse nil )"[T1 -> real]" derivative_inverse nil )"bool" functions nil )"[numfield, nznum -> numfield]" number_fields nil )nil number_fields nil )nil number_fields nil )"nzreal" nil )"[T -> real]" derivatives nil )"real" reals nil )nil booleans nil )"bool" deriv_domain_def nil )"bool" deriv_domain_def nil )nil derivatives_alt nil )"boolean" notequal nil )nil reals nil )"[real -> boolean]" derivative_inverse nil )nil derivative_inverse nil )nil numbers nil )nil booleans nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" derivative_inverse nil )nil derivative_inverse nil ))nil )"Alternative" 3473181730"" (skolem 1 ("F" "G" "f" "Y0" ))"" (flatten)"" "derivative_equivalence3" "f" "F" "D" "f(G(Y0))" "x" "G(Y0)" ))"1" (name "X0" "G(Y0)" )"1" (replace -1)"1" (case "deriv(F, X0) = f(X0)" )"1" (case "derivable?(F, X0)" )"1" (assert )"1" (skolem! -4)"1" (flatten -4)"1" "derivative_equivalence3" "f" "G" "x" "Y0" "D" "1/f(G(Y0))" ))"1" (flatten -1)"1" (hide -1)"1" (split -1)"1" (flatten -1) nil nil )"2" "2" case "FORALL (y: T2): psi!1(G(y)) /= 0" )"1" "LAMBDA (y:T2): 1/psi!1(G(y))" )"1" "1" "cv_in_domain" "f" "psi!1" "x" "X0" "l" "f(X0)" ))"1" assert )"1" replace -5 1)"1" replace -1 1)"1" replace -5 1 rl)"1" case "continuous?(LAMBDA (y: T2): 1 / psi!1(G(y)), Y0)" )"1" rewrite nil nil )"2" "2" "inv_continuous[T2]" "g" "LAMBDA (y: T2): psi!1(G(y))" "x0" "Y0" ))"2" "1" expand "/" "1" nil nil ))nil )"2" "2" "inverse_continuous[T1,T2]" "g" "F" ))"1" replace "1" case "inverse(F)=LAMBDA (y:T2): G(y)" )"1" replace "1" case "continuous?[T1](psi!1,X0)" )"1" expand "continuous?" "1" "Y0" )"1" "composition_cont[T2,T1]" "f" "G" "g" "psi!1" "x0" "Y0" ))"1" expand "o" "1" replace "1" replace "1" "1" nil nil )"2" "2" expand "continuous?" )"2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" expand "convergence" )"2" expand "convergence" )"2" "2" "2" "continuity_def2[T1]" "f" "psi!1" "x0" "X0" ))"2" expand "convergent?" )"2" replace "2" "f(X0)" )"2" expand "continuous?" )"2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" "2" "extensionality" "f" "inverse(F)" "g" "(LAMBDA (y: T2): G(y))" ))"2" "1" nil nil )"2" "2" "2" "bijective_inverse" "f" "F" "y" "x!1" "x" "G(x!1)" ))"2" "comp_inverse_right_alt" "f" "F" "g" "G" "r" "x!1" ))"2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "derivable_cont_fun[T1]" "f" "F" ))"2" assert )nil nil ))nil )"3" "Y0" )nil nil )"4" "X0" )nil nil )"5" "connected_domain1" )"5" nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"3" nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "Y1" ))"2" "G(Y1)" )"2" "comp_inverse_right_alt" "f" "F" "g" "G" "r" "Y1" ))"2" replace -1)"2" "comp_inverse_right_alt" "f" "F" "g" "G" "r" "Y0" ))"2" replace -6 -8 rl)"2" replace -1 -8)"2" "div_cancel4" "y" "G(Y1) - G(Y0)" "x" "Y1 - Y0" "n0z" "psi!1(G(Y1))" ))"2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" quality 100%
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