| Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/PVS/lnexp_fnd/ (Beweissystem der NASA Version 6.0.9©) Datei vom 28.9.2014 mit Größe 149 kB |
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Quelle ln_exp.prfSprache: Lisp |
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(ln_exp (conn_posreal 0 (conn_posreal-1 nil 3477844344 ("" (expand "connected?") (("" (skosimp*) (("" (assert) nil nil)) nil)) 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) (connected? const-decl "bool" deriv_domain_def "analysis/")) shostak)) (one_over_t_cont 0 (one_over_t_cont-1 nil 3253532189 ("" (lemma "div_fun_continuous[posreal]") (("" (inst -1 "(LAMBDA (t: posreal): 1)" "(LAMBDA (t: posreal): t)") (("1" (expand "/") (("1" (propax) nil nil)) nil) ("2" (assert) (("2" (hide 2) (("2" (lemma "id_fun_continuous[posreal]") (("2" (expand "I") (("2" (propax) nil nil)) nil)) nil)) nil)) nil) ("3" (hide 2) (("3" (lemma "const_fun_continuous[posreal]") (("3" (inst?) (("3" (expand "const_fun") (("3" (propax) nil nil)) nil)) nil)) nil)) nil)) nil)) nil) ((continuous? const-decl "bool" continuous_functions "analysis/") (continuous_fun nonempty-type-eq-decl nil continuous_functions "analysis/") (/= const-decl "boolean" notequal nil) (nzreal nonempty-type-eq-decl nil reals nil) (nz_continuous_fun type-eq-decl nil continuous_functions "analysis/") (/ const-decl "[T -> real]" real_fun_ops "reals/") (I const-decl "(bijective?[T, T])" identity nil) (id_fun_continuous judgement-tcc nil continuous_functions "analysis/") (posreal_div_posreal_is_posreal application-judgement "posreal" real_types nil) (const_fun_continuous judgement-tcc nil continuous_functions "analysis/") (const_fun const-decl "[T -> real]" real_fun_ops "reals/") (div_fun_continuous judgement-tcc nil continuous_functions "analysis/") (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) (bool nonempty-type-eq-decl nil booleans 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)) nil)) (ln_prep_TCC1 0 (ln_prep_TCC1-1 nil 3253532189 ("" (assuming-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) (nonneg_real nonempty-type-eq-decl nil real_types nil) (> const-decl "bool" reals nil) (posreal nonempty-type-eq-decl nil real_types 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) (real_ge_is_total_order name-judgement "(total_order?[real])" real_props nil) (connected? const-decl "bool" deriv_domain_def "analysis/")) nil)) (ln_prep_TCC2 0 (ln_prep_TCC2-1 nil 3253532189 ("" (expand "not_one_element?") (("" (skosimp*) (("" (inst + "x!1+1") (("" (assert) nil nil)) nil)) nil)) nil) ((+ const-decl "[numfield, numfield -> numfield]" number_fields nil) (numfield nonempty-type-eq-decl nil number_fields 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) (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) (posreal_plus_nnreal_is_posreal application-judgement "posreal" real_types nil) (not_one_element? const-decl "bool" deriv_domain_def "analysis/")) nil)) (ln_prep 0 (ln_prep-1 nil 3253532189 ("" (skosimp*) (("" (lemma "continuous_Integrable?[posreal]") (("1" (inst?) (("1" (assert) (("1" (hide 2) (("1" (skosimp*) (("1" (lemma "one_over_t_cont") (("1" (expand "continuous?" -1) (("1" (inst?) nil nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (hide 2) (("2" (ground) (("2" (expand "connected?") (("2" (skosimp*) (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) 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) (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) (continuous_Integrable? formula-decl nil integral "analysis/") (connected? const-decl "bool" deriv_domain_def "analysis/") (not_one_element? const-decl "bool" deriv_domain_def "analysis/") (continuous? const-decl "bool" continuous_functions "analysis/") (Closed_interval type-eq-decl nil intervals_real "reals/") (< const-decl "bool" reals nil) (<= const-decl "bool" reals nil) (IMPLIES const-decl "[bool, bool -> bool]" booleans nil) (AND const-decl "[bool, bool -> bool]" booleans nil) (one_over_t_cont formula-decl nil ln_exp nil) (/ const-decl "[numfield, nznum -> numfield]" number_fields nil) (nznum nonempty-type-eq-decl nil number_fields nil) (/= const-decl "boolean" notequal nil) (numfield nonempty-type-eq-decl nil number_fields nil) (posreal_div_posreal_is_posreal application-judgement "posreal" real_types nil)) nil)) (ln_TCC1 0 (ln_TCC1-1 nil 3253532189 ("" (skosimp*) (("" (rewrite "ln_prep") nil nil)) nil) ((ln_prep formula-decl nil ln_exp 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) (bool nonempty-type-eq-decl nil booleans 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)) nil)) (ln_derivable_TCC1 0 (ln_derivable_TCC1-1 nil 3471688875 ("" (lemma "deriv_domain_posreal") (("" (propax) nil nil)) nil) ((deriv_domain_posreal formula-decl nil deriv_domain "analysis/")) nil)) (ln_derivable 0 (ln_derivable-1 nil 3253532189 ("" (lemma "fundamental[posreal]") (("1" (inst -1 "ln" "1" "(LAMBDA (t: posreal): 1 / t)") (("1" (split -1) (("1" (propax) nil nil) ("2" (hide 2) (("2" (rewrite "one_over_t_cont") nil nil)) nil) ("3" (hide 2) (("3" (skosimp*) (("3" (expand "ln") (("3" (propax) nil nil)) nil)) nil)) nil)) nil)) nil) ("2" (assert) (("2" (expand "not_one_element?") (("2" (skosimp*) (("2" (inst + "x!1+1") (("2" (assert) nil nil)) nil)) nil)) nil)) nil) ("3" (assert) (("3" (expand "connected?") (("3" (skosimp*) (("3" (assert) nil nil)) nil)) nil)) nil)) nil) ((posreal_div_posreal_is_posreal application-judgement "posreal" real_types nil) (ln const-decl "real" ln_exp nil) (numfield nonempty-type-eq-decl nil number_fields nil) (/= const-decl "boolean" notequal nil) (nznum nonempty-type-eq-decl nil number_fields nil) (/ const-decl "[numfield, nznum -> numfield]" number_fields nil) (one_over_t_cont formula-decl nil ln_exp nil) (not_one_element? const-decl "bool" deriv_domain_def "analysis/") (connected? const-decl "bool" deriv_domain_def "analysis/") (fundamental formula-decl nil fundamental_theorem "analysis/") (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) (bool nonempty-type-eq-decl nil booleans 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)) nil)) (ln_continuous 0 (ln_continuous-2 nil 3445353494 ("" (lemma "ln_derivable") (("" (lemma "derivable_continuous[posreal]") (("1" (inst?) (("1" (flatten) (("1" (expand "continuous?" 1) (("1" (skosimp*) (("1" (inst?) (("1" (assert) (("1" (expand "derivable?" -1) (("1" (inst?) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (skosimp*) (("2" (inst-cp + "1") (("2" (assert) (("2" (inst + "2") (("2" (assert) nil nil)) nil)) nil)) nil)) nil) ("3" (skosimp*) (("3" (assert) (("3" (lemma "deriv_domain_posreal") (("3" (expand "deriv_domain?") (("3" (inst?) nil nil)) nil)) nil)) nil)) nil)) nil)) 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) (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) (derivable_continuous formula-decl nil derivatives_def "analysis/") (deriv_domain? const-decl "bool" deriv_domain_def "analysis/") (not_one_element? const-decl "bool" deriv_domain_def "analysis/") (posreal_div_posreal_is_posreal application-judgement "posreal" real_types nil) (derivable? const-decl "bool" derivatives "analysis/") (continuous? const-decl "bool" continuous_functions "analysis/") (ln const-decl "real" ln_exp nil) (ln_derivable formula-decl nil ln_exp nil)) nil) (ln_continuous-1 nil 3253532189 ("" (lemma "ln_derivable") (("" (lemma "derivable_continuous[posreal]") (("1" (inst?) (("1" (flatten) (("1" (expand "continuous?" 1) (("1" (skosimp*) (("1" (inst?) (("1" (assert) (("1" (expand "derivable" -1) (("1" (inst?) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (skosimp*) (("2" (inst-cp + "1") (("2" (assert) (("2" (inst + "2") (("2" (assert) nil nil)) nil)) nil)) nil)) nil) ("3" (skosimp*) (("3" (assert) nil nil)) nil)) nil)) nil) ((derivable_continuous formula-decl nil derivatives_def "analysis/")) nil)) (ln_1 0 (ln_1-1 nil 3253532189 ("" (expand "ln") (("" (expand "Integral") (("" (propax) nil nil)) nil)) nil) ((Integral const-decl "real" integral_def "analysis/") (ln const-decl "real" ln_exp nil)) nil)) (ln_mult 0 (ln_mult-3 nil 3445353567 ("" (skosimp*) (("" (case "derivable?[posreal](LAMBDA (x: posreal): ln(py!1 * x))") (("1" (lemma "ln_derivable") (("1" (flatten) (("1" (case "deriv(LAMBDA (x: posreal): ln(py!1*x)) = deriv(LAMBDA (x: posreal): ln(x))") (("1" (lemma "derivs_eq[posreal]") (("1" (inst?) (("1" (assert) (("1" (case-replace "(LAMBDA (x: posreal): ln(x)) = ln") (("1" (assert) (("1" (hide -1) (("1" (replace -3) (("1" (assert) (("1" (skosimp*) (("1" (expand "+ ") (("1" (expand "const_fun") (("1" (case "(LAMBDA (x: posreal): ln(py!1 * x))(1) = (LAMBDA (x_824: posreal): ln(x_824) + c!1)(1)") (("1" (assert) (("1" (hide -2) (("1" (rewrite "ln_1") (("1" (assert) (("1" (reveal -1) (("1" (replace -2 * rl) (("1" (case-replace "(LAMBDA (x: posreal): ln(py!1 * x))(px!1) = (LAMBDA (x_824: posreal): ln(x_824) + ln(py!1)) (("1" (hide -2) (("1" (assert) nil nil)) nil) ("2" (replace -1) (("2" (propax) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (replace -1) (("2" (propax) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (apply-extensionality 1 :hide? t) nil nil)) nil)) nil)) nil) ("2" (assert) (("2" (expand "not_one_element?") (("2" (skosimp*) (("2" (assert) (("2" (inst + "x!1+1") (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil) ("3" (assert) (("3" (expand "connected?") (("3" (skosimp*) (("3" (assert) nil nil)) nil)) nil)) nil)) nil) ("2" (hide 2) (("2" (lemma "deriv_composition[posreal,posreal]") (("1" (case-replace "(LAMBDA (x: posreal): ln(x)) = ln") (("1" (hide -1) (("1" (replace -3) (("1" (apply-extensionality 1 :hide? t) (("1" (inst - "(LAMBDA (t: posreal): py!1*t)" "ln" "x!1") (("1" (assert) (("1" (case "derivable?((LAMBDA (t: posreal): py!1 * t), x!1) ") (("1" (assert) (("1" (expand "derivable?" -3) (("1" (inst -3 "py!1*x!1") (("1" (assert) (("1" (expand "o ") (("1" (expand "deriv" 1) (("1" (case "deriv(ln)(py!1*x!1) = (LAMBDA (t: posreal): 1 / t)(py!1*x!1)") (("1" (expand "deriv" -1) (("1" (replace -1) (("1" (hide -1) (("1" (case-replace "deriv[posreal]((LAMBDA (t: posreal): py!1 * t), x!1) = py!1") (("1" (assert) nil nil) ("2" (hide 2) (("2" (lemma "deriv_scal[posreal]") (("2" (inst -1 "py!1" "I[posreal]" "x!1") (("2" (split -1) (("1" (expand "I") (("1" (expand "*") (("1" (replace -1) (("1" (lemma "deriv_identity[posreal]") (("1" (expand "I") (("1" (inst?) (("1" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (hide-all-but 1) (("2" (lemma "identity_derivable[posreal]") (("2" (inst?) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (assert) (("2" (hide 2) (("2" (replace -4) (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (hide-all-but 1) (("2" (lemma "scal_derivable[posreal]") (("2" (inst -1 "py!1" "I" "x!1") (("2" (lemma "identity_derivable[posreal]") (("2" (inst?) (("2" (split -2) (("1" (expand "I") (("1" (expand "*") (("1" (propax) nil nil)) nil)) nil) ("2" (propax) nil nil)) nil)) nil)) nil)) nil)) nil)) nil) ("3" (lemma "deriv_domain_posreal") (("3" (expand "deriv_domain?") (("3" (propax) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (apply-extensionality 1 :hide? t) nil nil)) nil) ("2" (skosimp*) (("2" (inst + "x!1+1") (("2" (assert) nil nil)) nil)) nil) ("3" (assert) (("3" (lemma "deriv_domain_posreal") (("3" (propax) nil nil)) nil)) nil)) nil)) nil) ("3" (case-replace "(LAMBDA (x: posreal): ln(x)) = ln") (("3" (apply-extensionality 1 :hide? t) nil nil)) nil) ("4" (propax) nil nil)) nil)) nil)) nil) ("2" (hide 2) (("2" (assert) (("2" (lemma "ln_derivable") (("2" (flatten) (("2" (assert) (("2" (lemma "scal_derivable[posreal]") (("1" (expand "derivable?" 1) (("1" (skosimp*) (("1" (inst -1 "py!1" "I" "x!1") (("1" (lemma "identity_derivable[posreal]") (("1" (inst?) (("1" (lemma "composition_derivable[posreal,posreal]") (("1" (inst -1 "(LAMBDA (t:posreal): py!1*t)" "ln" "x!1") (("1" (expand "o ") (("1" (split -1) (("1" (propax) nil nil) ("2" (expand "derivable?" -3) (("2" (inst -3 "py!1*x!1") (("2" (hide 2) (("2" (expand "I") (("2" (expand "*") (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil) ("3" (hide 2) (("3" (expand "derivable?" -3) (("3" (inst -3 "py!1*x!1") nil nil)) nil)) nil)) nil)) nil)) nil) ("2" (lemma "deriv_domain_posreal") (("2" (propax) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (skosimp*) (("2" (inst-cp + "1") (("2" (inst + "2") (("2" (assert) nil nil)) nil)) nil)) nil) ("3" (skosimp*) (("3" (assert) (("3" (lemma "deriv_domain_posreal") (("3" (expand "deriv_domain?") (("3" (inst?) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ((* const-decl "[numfield, numfield -> numfield]" number_fields nil) (numfield nonempty-type-eq-decl nil number_fields nil) (ln const-decl "real" ln_exp nil) (derivable? const-decl "bool" derivatives "analysis/") (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) (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) (posreal_times_posreal_is_posreal application-judgement "posreal" real_types nil) (derivable? const-decl "bool" derivatives_def "analysis/") (I const-decl "(bijective?[T, T])" identity nil) (bijective? const-decl "bool" functions nil) (identity_derivable formula-decl nil derivatives_def "analysis/") (real_times_real_is_real application-judgement "real" reals nil) (deriv_identity formula-decl nil derivatives_def "analysis/") (* const-decl "[T -> real]" real_fun_ops "reals/") (deriv_scal formula-decl nil derivatives_def "analysis/") (deriv const-decl "real" derivatives_def "analysis/") (O const-decl "T3" function_props nil) (scal_derivable formula-decl nil derivatives_def "analysis/") (py!1 skolem-const-decl "posreal" ln_exp nil) (/ const-decl "[numfield, nznum -> numfield]" number_fields nil) (nznum nonempty-type-eq-decl nil number_fields nil) (/= const-decl "boolean" notequal nil) (deriv_domain? const-decl "bool" deriv_domain_def "analysis/") (deriv_composition formula-decl nil chain_rule "analysis/") (derivs_eq formula-decl nil indefinite_integral "analysis/") (connected? const-decl "bool" deriv_domain_def "analysis/") (not_one_element? const-decl "bool" deriv_domain_def "analysis/") (real_plus_real_is_real application-judgement "real" reals nil) (posreal_div_posreal_is_posreal application-judgement "posreal" real_types nil) (const_fun const-decl "[T -> real]" real_fun_ops "reals/") (ln_1 formula-decl nil ln_exp nil) (+ const-decl "[numfield, numfield -> numfield]" number_fields nil) (+ const-decl "[T -> real]" real_fun_ops "reals/") (= const-decl "[T, T -> boolean]" equalities nil) (deriv_fun type-eq-decl nil derivatives "analysis/") (deriv const-decl "[T -> real]" derivatives "analysis/") (ln_derivable formula-decl nil ln_exp nil) (composition_derivable formula-decl nil chain_rule "analysis/")) nil) (ln_mult-2 nil 3445353520 (";;; Proof ln_mult-1 for formula ln_exp.ln_mult" (skosimp*) ((";;; Proof ln_mult-1 for formula ln_exp.ln_mult" (case "derivable?[posreal](LAMBDA (x: posreal): ln(py!1 * x))") (("1" (lemma "ln_derivable") (("1" (flatten) (("1" (case "deriv(LAMBDA (x: posreal): ln(py!1*x)) = deriv(LAMBDA (x: posreal): ln(x))") (("1" (lemma "derivs_eq[posreal]") (("1" (inst?) 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(("" (prop) (("1" (expand "injective?") (("1" (skosimp*) (("1" (expand "strict_increasing?") (("1" (inst-cp - "x1!1" "x2!1") (("1" (inst - "x2!1" "x1!1") (("1" (ground) nil nil)) nil)) nil)) nil)) nil)) nil) ("2" (expand "surjective?") (("2" (skosimp*) (("2" (hide -1) (("2" (rewrite "ln_image_all") nil nil)) nil)) nil)) nil)) nil)) nil)) nil) ((bijective? const-decl "bool" functions nil) (surjective? const-decl "bool" functions nil) (ln_image_all formula-decl nil ln_exp nil) (injective? const-decl "bool" functions nil) (strict_increasing? const-decl "bool" real_fun_preds "reals/") (real_lt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props 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) (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) (ln_strict_increasing formula-decl nil ln_exp nil)) nil)) (large_ln 0 (large_ln-1 nil 3253532189 ("" (skosimp*) (("" (lemma "ln_strict_increasing") (("" (expand "strict_increasing?") (("" (lemma "ln_bij") (("" (expand "bijective?") (("" (flatten) (("" (hide -1) (("" (expand "surjective?") (("" (inst -1 "a!1") (("" (skosimp*) (("" (replace -1 * rl) (("" (hide -1) (("" (inst -1 "x!1" "x!1+1") (("" (assert) (("" (inst + "1+x!1") nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ((ln_strict_increasing formula-decl nil ln_exp nil) (ln_bij formula-decl nil ln_exp nil) (surjective? const-decl "bool" functions nil) (real_lt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (+ const-decl "[numfield, numfield -> numfield]" number_fields nil) (numfield nonempty-type-eq-decl nil number_fields 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) (posreal_plus_nnreal_is_posreal application-judgement "posreal" real_types 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) (bijective? const-decl "bool" functions nil) (strict_increasing? const-decl "bool" real_fun_preds "reals/")) nil)) (ln_eq_0 0 (ln_eq_0-1 nil 3297441392 ("" (skosimp*) (("" (lemma "ln_bij") (("" (expand "bijective?") (("" (flatten) (("" (expand "injective?") (("" (inst - "px!1" "1") (("" (lemma "ln_1") (("" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ((ln_bij formula-decl nil ln_exp 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) (bool nonempty-type-eq-decl nil booleans 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) (ln_1 formula-decl nil ln_exp nil) (injective? const-decl "bool" functions nil) (bijective? const-decl "bool" functions nil)) shostak)) (ln_ge_0 0 (ln_ge_0-1 nil 3297691676 ("" (skosimp*) (("" (lemma "ln_increasing") (("" (expand "increasing?") (("" (inst - "1" "px!1") (("" (rewrite "ln_1") (("" (assert) nil nil)) nil)) nil)) nil)) nil)) nil) ((ln_increasing formula-decl nil ln_exp 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) (bool nonempty-type-eq-decl nil booleans 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) (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) (ln_1 formula-decl nil ln_exp nil) (increasing? const-decl "bool" real_fun_preds "reals/")) shostak)) (ln_gt_0 0 (ln_gt_0-1 nil 3297691602 ("" (skosimp*) (("" (lemma "ln_strict_increasing") (("" (expand "strict_increasing?") (("" (inst - "1" "px!1") (("" (rewrite "ln_1") (("" (assert) nil nil)) nil)) nil)) nil)) nil)) nil) ((ln_strict_increasing formula-decl nil ln_exp 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) (bool nonempty-type-eq-decl nil booleans 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) (real_lt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (real_gt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (ln_1 formula-decl nil ln_exp nil) (strict_increasing? const-decl "bool" real_fun_preds "reals/")) shostak)) (exp_TCC1 0 (exp_TCC1-1 nil 3253532189 ("" (inst + "(LAMBDA x: inverse(ln)(x))") (("" (lemma "bijective_inverse[posreal,real]") (("" (skosimp*) (("" (inst?) (("1" (assert) nil nil) ("2" (hide 2) (("2" (rewrite "ln_bij") nil nil)) nil)) nil)) nil)) nil)) nil) ((bijective_inverse formula-decl nil function_inverse nil) (bijective? const-decl "bool" functions nil) (ln_bij formula-decl nil ln_exp nil) (inverse const-decl "D" function_inverse nil) (ln const-decl "real" ln_exp nil) (= const-decl "[T, T -> boolean]" equalities 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) (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)) nil)) (exp_def 0 (exp_def-1 nil 3253532189 ("" (apply-extensionality 1 :hide? t) (("1" (typepred "exp(x!1)") (("1" (lemma "bijective_inverse[posreal,real]") (("1" (inst?) (("1" (ground) nil nil) ("2" (rewrite "ln_bij") nil nil)) nil)) nil)) nil) ("2" (skosimp*) (("2" (lemma "bijective_inverse[posreal,real]") (("2" (inst?) (("1" (ground) nil nil) ("2" (hide 2) (("2" (rewrite "ln_bij") nil nil)) nil)) nil)) nil)) nil)) nil) ((NOT const-decl "[bool -> bool]" booleans nil) (bijective? const-decl "bool" functions nil) (ln_bij formula-decl nil ln_exp nil) (bijective_inverse formula-decl nil function_inverse nil) (inverse const-decl "D" function_inverse nil) (exp const-decl "{py | x = ln(py)}" ln_exp nil) (ln const-decl "real" ln_exp nil) (= const-decl "[T, T -> boolean]" equalities 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) (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)) nil)) (exp_bij 0 (exp_bij-1 nil 3253532189 ("" (rewrite "exp_def") (("" (assert) (("" (lemma "bijective_inverse_is_bijective[posreal,real]") (("" (inst?) (("" (hide 2) (("" (rewrite "ln_bij") nil nil)) nil)) nil)) nil)) nil)) nil) ((bijective? const-decl "bool" functions nil) (ln const-decl "real" ln_exp nil) (ln_bij formula-decl nil ln_exp nil) (bijective_inverse_is_bijective judgement-tcc nil function_inverse 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) (bool nonempty-type-eq-decl nil booleans 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) (exp_def formula-decl nil ln_exp nil)) nil)) (ln_exp 0 (ln_exp-1 nil 3253532189 ("" (skosimp*) (("" (assert) nil nil)) nil) nil nil)) (exp_ln 0 (exp_ln-1 nil 3253532189 ("" (skosimp*) (("" (assert) (("" (typepred "exp(ln(py!1))") (("" (lemma "ln_bij") (("" (expand "bijective?") (("" (flatten) (("" (expand "injective?") (("" (inst -1 "py!1" "exp(ln(py!1))") (("" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ((ln_bij formula-decl nil ln_exp 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) (injective? const-decl "bool" functions nil) (bijective? const-decl "bool" functions 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) (nonneg_real nonempty-type-eq-decl nil real_types nil) (> const-decl "bool" reals nil) (posreal nonempty-type-eq-decl nil real_types nil) (= const-decl "[T, T -> boolean]" equalities nil) (ln const-decl "real" ln_exp nil) (exp const-decl "{py | x = ln(py)}" ln_exp nil)) nil)) (ln_e 0 (ln_e-1 nil 3253532189 ("" (assert) (("" (expand "e") (("" (assert) nil nil)) nil)) nil) ((e const-decl "posreal" ln_exp nil)) nil)) (ln_2_bnds 0 (ln_2_bnds-1 nil 3253532189 ("" (lemma "Integral_bound[posreal]") (("1" (inst -1 "1" "1" "2" "(LAMBDA (t: posreal): 1/t)" "1/2") (("1" (assert) (("1" (case-replace "Integral(1, 2, (LAMBDA (t: posreal): 1 / t)) = ln(2)") (("1" (rewrite "ln_prep") (("1" (assert) (("1" (split -2) (("1" (propax) nil nil) ("2" (skosimp*) (("2" (hide-all-but 1) (("2" (prop) (("1" (mult-by 1 "x!1") (("1" (assert) nil nil)) nil) ("2" (mult-by 1 "x!1") (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (expand "ln") (("2" (propax) nil nil)) nil) ("3" (rewrite "ln_prep") nil nil)) nil)) nil)) nil) ("2" (expand "not_one_element?") (("2" (skosimp*) (("2" (inst + "x!1+1") (("2" (assert) nil nil)) nil)) nil)) nil) ("3" (assert) (("3" (expand "connected?") (("3" (skosimp*) (("3" (assert) nil nil)) nil)) nil)) nil)) nil) ((posrat_div_posrat_is_posrat application-judgement "posrat" rationals nil) (posreal_div_posreal_is_posreal application-judgement "posreal" real_types nil) (numfield nonempty-type-eq-decl nil number_fields nil) (/= const-decl "boolean" notequal nil) (nznum nonempty-type-eq-decl nil number_fields nil) (/ const-decl "[numfield, nznum -> numfield]" number_fields nil) (= const-decl "[T, T -> boolean]" equalities nil) (Integrable? const-decl "bool" integral_def "analysis/") (Integrable_funs type-eq-decl nil integral_def "analysis/") (Integral const-decl "real" integral_def "analysis/") (ln const-decl "real" ln_exp nil) (posreal_times_posreal_is_posreal application-judgement "posreal" real_types nil) (nonzero_real nonempty-type-eq-decl nil reals nil) (div_cancel2 formula-decl nil real_props nil) (Closed_interval type-eq-decl nil intervals_real "reals/") (<= const-decl "bool" reals nil) (AND const-decl "[bool, bool -> bool]" booleans nil) (both_sides_times_pos_le1 formula-decl nil real_props nil) (ln_prep formula-decl nil ln_exp nil) (real_lt_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) (not_one_element? const-decl "bool" deriv_domain_def "analysis/") (connected? const-decl "bool" deriv_domain_def "analysis/") (Integral_bound formula-decl nil integral "analysis/") (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) (bool nonempty-type-eq-decl nil booleans 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)) nil)) (exp_int_TCC1 0 (exp_int_TCC1-1 nil 3253532189 ("" (skosimp*) (("" (assert) nil nil)) nil) nil nil)) (exp_int 0 (exp_int-1 nil 3253532189 ("" (skosimp*) (("" (typepred "exp(i!1)") (("" (lemma "ln_expt") (("" (inst?) 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(("2" (flatten) (("2" (expand "injective?") 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(("2" (propax) nil nil)) nil)) nil)) nil) ((= const-decl "[T, T -> boolean]" equalities nil) (ln const-decl "real" ln_exp nil) (exp const-decl "{py | x = ln(py)}" ln_exp nil) (deriv_inverse_fun formula-decl nil derivative_inverse "analysis/") (ln_derivable formula-decl nil ln_exp nil) (ln_exp formula-decl nil ln_exp nil) (inverse? const-decl "bool" function_inverse_def nil) (ln_bij formula-decl nil ln_exp nil) (/ const-decl "[numfield, nznum -> numfield]" number_fields nil) (nznum nonempty-type-eq-decl nil number_fields nil) (numfield nonempty-type-eq-decl nil number_fields nil) (nzreal nonempty-type-eq-decl nil reals nil) (/= const-decl "boolean" notequal nil) (posreal_div_posreal_is_posreal application-judgement "posreal" real_types nil) (not_one_element? const-decl "bool" deriv_domain_def "analysis/") (connected? const-decl "bool" deriv_domain_def "analysis/") (inverse_derivable_fun formula-decl nil derivative_inverse "analysis/") (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) (bool nonempty-type-eq-decl nil booleans 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)) shostak)) (e_bnds 0 (e_bnds-1 nil 3253532189 ("" (expand "e") (("" (lemma "ln_2_bnds") (("" (flatten) (("" (mult-by -1 "2") (("" (lemma "ln_expt") (("" (inst -1 "2" "2") (("" (replace -1 * rl) (("" (hide -1) (("" (expand "^") (("" (expand "expt") (("" (expand "expt") (("" (expand "expt") (("" (typepred "exp(1)") (("" (hide -1 -2) (("" (replace -1 (-2 -3)) (("" (hide -1) (("" (lemma "exp_increasing") (("" (expand "increasing?") (("" (inst -1 "ln(exp(1))" "ln(4)") (("" (assert) (("" (rewrite "exp_ln") (("" (rewrite "exp_ln") (("" (assert) (("" (lemma "exp_increasing") (("" (expand "increasing?") 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¤ Dauer der Verarbeitung: 0.122 Sekunden (vorverarbeitet am 2026-05-01) ¤*© Formatika GbR, Deutschland |
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2026-05-26