# Question 24247: Exact Finite Exponential Sums Are Not General Infinite-Value Certificates [Return to catalogue page 122](https://ocme.evemisslab.com/en/questions/page/122/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261113-r02-real-exponential-function-foundations/#q-20261015-advance-20261113-r02-real-exponential-function-0022) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0047-24247 · q-20261015-advance-20261113-r02-real-exponential-function-0022 ## Original question and givens Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=22/23, cutoff=26, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary). - G001 — Use the whole real series E(x)=sum over n≥0 of x^n/n!, with power-series coefficients1/n!. A finite sum is a separate object; whole-series identities require the declared convergence and rearrangement obligations. Use exact x=22/23, cutoff=26, including terms0 through cutoff. Give (x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-real-exponential-function](https://ocme.evemisslab.com/en/mko/mko-real-exponential-function/) ## Step-by-step derivation S001 · impliesBefore source roles = [exp-finite-partial-sum] After derived = [[x, cutoff, terms, partial_sums, finite_sum, infinite_value_certificate, boundary]] Conditions: none List the source field order without JavaScript.Gradient boundary uses its original mode to choose6,8 or7 fields;no padding or answer-derived mode. Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · impliesBefore source roles = [0] After derived = [1] Conditions: none The empty product0! is1,not0. Step identifiers and dependenciesRule: empty_factorial_is_multiplicative_unit · Uses: G001, S001 S003 · impliesBefore (22/23)^0 After Exp ORIGINAL x power0 = 1 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S002 S004 · calculationBefore Exp term0 power over factorial := div(1, 1) After Exp term0 power over factorial = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Exp cumulative sum through0 := sum(0, 1) After Exp cumulative sum through0 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Exp term1 factorial step1 := mul(1, 1) After Exp term1 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore (22/23)^1 After Exp ORIGINAL x power1 = 22/23 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S006 S008 · calculationBefore Exp term1 power over factorial := div(22/23, 1) After Exp term1 power over factorial = 22/23 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Exp cumulative sum through1 := sum(1, 22/23) After Exp cumulative sum through1 = 45/23 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore Exp term2 factorial step1 := mul(1, 1) After Exp term2 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore Exp term2 factorial step2 := mul(1, 2) After Exp term2 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · impliesBefore (22/23)^2 After Exp ORIGINAL x power2 = 484/529 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S011 S013 · calculationBefore Exp term2 power over factorial := div(484/529, 2) After Exp term2 power over factorial = 242/529 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore Exp cumulative sum through2 := sum(45/23, 242/529) After Exp cumulative sum through2 = 1277/529 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore Exp term3 factorial step1 := mul(1, 1) After Exp term3 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore Exp term3 factorial step2 := mul(1, 2) After Exp term3 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore Exp term3 factorial step3 := mul(2, 3) After Exp term3 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · impliesBefore (22/23)^3 After Exp ORIGINAL x power3 = 10648/12167 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S017 S019 · calculationBefore Exp term3 power over factorial := div(10648/12167, 6) After Exp term3 power over factorial = 5324/36501 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore Exp cumulative sum through3 := sum(1277/529, 5324/36501) After Exp cumulative sum through3 = 93437/36501 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore Exp term4 factorial step1 := mul(1, 1) After Exp term4 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore Exp term4 factorial step2 := mul(1, 2) After Exp term4 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore Exp term4 factorial step3 := mul(2, 3) After Exp term4 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore Exp term4 factorial step4 := mul(6, 4) After Exp term4 factorial step4 = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · impliesBefore (22/23)^4 After Exp ORIGINAL x power4 = 234256/279841 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S024 S026 · calculationBefore Exp term4 power over factorial := div(234256/279841, 24) After Exp term4 power over factorial = 29282/839523 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore Exp cumulative sum through4 := sum(93437/36501, 29282/839523) After Exp cumulative sum through4 = 726111/279841 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore Exp term5 factorial step1 := mul(1, 1) After Exp term5 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · calculationBefore Exp term5 factorial step2 := mul(1, 2) After Exp term5 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · calculationBefore Exp term5 factorial step3 := mul(2, 3) After Exp term5 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · calculationBefore Exp term5 factorial step4 := mul(6, 4) After Exp term5 factorial step4 = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · calculationBefore Exp term5 factorial step5 := mul(24, 5) After Exp term5 factorial step5 = 120 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · impliesBefore (22/23)^5 After Exp ORIGINAL x power5 = 5153632/6436343 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S032 S034 · calculationBefore Exp term5 power over factorial := div(5153632/6436343, 120) After Exp term5 power over factorial = 644204/96545145 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · calculationBefore Exp cumulative sum through5 := sum(726111/279841, 644204/96545145) After Exp cumulative sum through5 = 251152499/96545145 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034 S036 · calculationBefore Exp term6 factorial step1 := mul(1, 1) After Exp term6 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · calculationBefore Exp term6 factorial step2 := mul(1, 2) After Exp term6 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036 S038 · calculationBefore Exp term6 factorial step3 := mul(2, 3) After Exp term6 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S037 S039 · calculationBefore Exp term6 factorial step4 := mul(6, 4) After Exp term6 factorial step4 = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038 S040 · calculationBefore Exp term6 factorial step5 := mul(24, 5) After Exp term6 factorial step5 = 120 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S039 S041 · calculationBefore Exp term6 factorial step6 := mul(120, 6) After Exp term6 factorial step6 = 720 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S040 S042 · impliesBefore (22/23)^6 After Exp ORIGINAL x power6 = 113379904/148035889 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S041 S043 · calculationBefore Exp term6 power over factorial := div(113379904/148035889, 720) After Exp term6 power over factorial = 7086244/6661615005 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042 S044 · calculationBefore Exp cumulative sum through6 := sum(251152499/96545145, 7086244/6661615005) After Exp cumulative sum through6 = 3467321735/1332323001 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S043 S045 · calculationBefore Exp term7 factorial step1 := mul(1, 1) After Exp term7 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S044 S046 · calculationBefore Exp term7 factorial step2 := mul(1, 2) After Exp term7 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S045 S047 · calculationBefore Exp term7 factorial step3 := mul(2, 3) After Exp term7 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S046 S048 · calculationBefore Exp term7 factorial step4 := mul(6, 4) After Exp term7 factorial step4 = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S047 S049 · calculationBefore Exp term7 factorial step5 := mul(24, 5) After Exp term7 factorial step5 = 120 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S048 S050 · calculationBefore Exp term7 factorial step6 := mul(120, 6) After Exp term7 factorial step6 = 720 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S049 S051 · calculationBefore Exp term7 factorial step7 := mul(720, 7) After Exp term7 factorial step7 = 5040 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S050 S052 · impliesBefore (22/23)^7 After Exp ORIGINAL x power7 = 2494357888/3404825447 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S051 S053 · calculationBefore Exp term7 power over factorial := div(2494357888/3404825447, 5040) After Exp term7 power over factorial = 155897368/1072520015805 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S052 S054 · calculationBefore Exp cumulative sum through7 := sum(3467321735/1332323001, 155897368/1072520015805) After Exp cumulative sum through7 = 310149988227/119168890645 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S053 S055 · calculationBefore Exp term8 factorial step1 := mul(1, 1) After Exp term8 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S054 S056 · calculationBefore Exp term8 factorial step2 := mul(1, 2) After Exp term8 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S055 S057 · calculationBefore Exp term8 factorial step3 := mul(2, 3) After Exp term8 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S056 S058 · calculationBefore Exp term8 factorial step4 := mul(6, 4) After Exp term8 factorial step4 = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S057 S059 · calculationBefore Exp term8 factorial step5 := mul(24, 5) After Exp term8 factorial step5 = 120 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S058 S060 · calculationBefore Exp term8 factorial step6 := mul(120, 6) After Exp term8 factorial step6 = 720 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S059 S061 · calculationBefore Exp term8 factorial step7 := mul(720, 7) After Exp term8 factorial step7 = 5040 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S060 S062 · calculationBefore Exp term8 factorial step8 := mul(5040, 8) After Exp term8 factorial step8 = 40320 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S061 S063 · impliesBefore (22/23)^8 After Exp ORIGINAL x power8 = 54875873536/78310985281 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S062 S064 · calculationBefore Exp term8 power over factorial := div(54875873536/78310985281, 40320) After Exp term8 power over factorial = 428717762/24667960363515 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S063 S065 · calculationBefore Exp cumulative sum through8 := sum(310149988227/119168890645, 428717762/24667960363515) After Exp cumulative sum through8 = 64201476280751/24667960363515 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S064 S066 · calculationBefore Exp term9 factorial step1 := mul(1, 1) After Exp term9 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S065 S067 · calculationBefore Exp term9 factorial step2 := mul(1, 2) After Exp term9 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S066 S068 · calculationBefore Exp term9 factorial step3 := mul(2, 3) After Exp term9 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S067 S069 · calculationBefore Exp term9 factorial step4 := mul(6, 4) After Exp term9 factorial step4 = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S068 S070 · calculationBefore Exp term9 factorial step5 := mul(24, 5) After Exp term9 factorial step5 = 120 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S069 S071 · calculationBefore Exp term9 factorial step6 := mul(120, 6) After Exp term9 factorial step6 = 720 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S070 S072 · calculationBefore Exp term9 factorial step7 := mul(720, 7) After Exp term9 factorial step7 = 5040 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S071 S073 · calculationBefore Exp term9 factorial step8 := mul(5040, 8) After Exp term9 factorial step8 = 40320 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S072 S074 · calculationBefore Exp term9 factorial step9 := mul(40320, 9) After Exp term9 factorial step9 = 362880 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S073 S075 · impliesBefore (22/23)^9 After Exp ORIGINAL x power9 = 1207269217792/1801152661463 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S074 S076 · calculationBefore Exp term9 power over factorial := div(1207269217792/1801152661463, 362880) After Exp term9 power over factorial = 9431790764/5106267795247605 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S075 S077 · calculationBefore Exp cumulative sum through9 := sum(64201476280751/24667960363515, 9431790764/5106267795247605) After Exp cumulative sum through9 = 13289715021906221/5106267795247605 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S076 S078 · calculationBefore Exp term10 factorial step1 := mul(1, 1) After Exp term10 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S077 S079 · calculationBefore Exp term10 factorial step2 := mul(1, 2) After Exp term10 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S078 S080 · calculationBefore Exp term10 factorial step3 := mul(2, 3) After Exp term10 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S079 S081 · calculationBefore Exp term10 factorial step4 := mul(6, 4) After Exp term10 factorial step4 = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S080 S082 · calculationBefore Exp term10 factorial step5 := mul(24, 5) After Exp term10 factorial step5 = 120 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S081 S083 · calculationBefore Exp term10 factorial step6 := mul(120, 6) After Exp term10 factorial step6 = 720 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S082 S084 · calculationBefore Exp term10 factorial step7 := mul(720, 7) After Exp term10 factorial step7 = 5040 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S083 S085 · calculationBefore Exp term10 factorial step8 := mul(5040, 8) After Exp term10 factorial step8 = 40320 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S084 S086 · calculationBefore Exp term10 factorial step9 := mul(40320, 9) After Exp term10 factorial step9 = 362880 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S085 S087 · calculationBefore Exp term10 factorial step10 := mul(362880, 10) After Exp term10 factorial step10 = 3628800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S086 S088 · impliesBefore (22/23)^10 After Exp ORIGINAL x power10 = 26559922791424/41426511213649 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S087 S089 · calculationBefore Exp term10 power over factorial := div(26559922791424/41426511213649, 3628800) After Exp term10 power over factorial = 103749698404/587220796453474575 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S088 S090 · calculationBefore Exp cumulative sum through10 := sum(13289715021906221/5106267795247605, 103749698404/587220796453474575) After Exp cumulative sum through10 = 169813036807657091/65246755161497175 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S089 S091 · calculationBefore Exp term11 factorial step1 := mul(1, 1) After Exp term11 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S090 S092 · calculationBefore Exp term11 factorial step2 := mul(1, 2) After Exp term11 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S091 S093 · calculationBefore Exp term11 factorial step3 := mul(2, 3) After Exp term11 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S092 S094 · calculationBefore Exp term11 factorial step4 := mul(6, 4) After Exp term11 factorial step4 = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S093 S095 · calculationBefore Exp term11 factorial step5 := mul(24, 5) After Exp term11 factorial step5 = 120 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S094 S096 · calculationBefore Exp term11 factorial step6 := mul(120, 6) After Exp term11 factorial step6 = 720 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S095 S097 · calculationBefore Exp term11 factorial step7 := mul(720, 7) After Exp term11 factorial step7 = 5040 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S096 S098 · calculationBefore Exp term11 factorial step8 := mul(5040, 8) After Exp term11 factorial step8 = 40320 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S097 S099 · calculationBefore Exp term11 factorial step9 := mul(40320, 9) After Exp term11 factorial step9 = 362880 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S098 S100 · calculationBefore Exp term11 factorial step10 := mul(362880, 10) After Exp term11 factorial step10 = 3628800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S099 S101 · calculationBefore Exp term11 factorial step11 := mul(3628800, 11) After Exp term11 factorial step11 = 39916800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S100 S102 · impliesBefore (22/23)^11 After Exp ORIGINAL x power11 = 584318301411328/952809757913927 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S101 S103 · calculationBefore Exp term11 power over factorial := div(584318301411328/952809757913927, 39916800) After Exp term11 power over factorial = 207499396808/13506078318429915225 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S102 S104 · calculationBefore Exp cumulative sum through11 := sum(169813036807657091/65246755161497175, 207499396808/13506078318429915225) After Exp cumulative sum through11 = 7030259765336882929/2701215663685983045 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S103 S105 · calculationBefore Exp term12 factorial step1 := mul(1, 1) After Exp term12 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S104 S106 · calculationBefore Exp term12 factorial step2 := mul(1, 2) After Exp term12 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S105 S107 · calculationBefore Exp term12 factorial step3 := mul(2, 3) After Exp term12 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S106 S108 · calculationBefore Exp term12 factorial step4 := mul(6, 4) After Exp term12 factorial step4 = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S107 S109 · calculationBefore Exp term12 factorial step5 := mul(24, 5) After Exp term12 factorial step5 = 120 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S108 S110 · calculationBefore Exp term12 factorial step6 := mul(120, 6) After Exp term12 factorial step6 = 720 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S109 S111 · calculationBefore Exp term12 factorial step7 := mul(720, 7) After Exp term12 factorial step7 = 5040 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S110 S112 · calculationBefore Exp term12 factorial step8 := mul(5040, 8) After Exp term12 factorial step8 = 40320 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S111 S113 · calculationBefore Exp term12 factorial step9 := mul(40320, 9) After Exp term12 factorial step9 = 362880 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S112 S114 · calculationBefore Exp term12 factorial step10 := mul(362880, 10) After Exp term12 factorial step10 = 3628800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S113 S115 · calculationBefore Exp term12 factorial step11 := mul(3628800, 11) After Exp term12 factorial step11 = 39916800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S114 S116 · calculationBefore Exp term12 factorial step12 := mul(39916800, 12) After Exp term12 factorial step12 = 479001600 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S115 S117 · impliesBefore (22/23)^12 After Exp ORIGINAL x power12 = 12855002631049216/21914624432020321 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S116 S118 · calculationBefore Exp term12 power over factorial := div(12855002631049216/21914624432020321, 479001600) After Exp term12 power over factorial = 1141246682444/931919403971664150525 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S117 S119 · calculationBefore Exp cumulative sum through12 := sum(7030259765336882929/2701215663685983045, 1141246682444/931919403971664150525) After Exp cumulative sum through12 = 2425439620182471292949/931919403971664150525 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S118 S120 · calculationBefore Exp term13 factorial step1 := mul(1, 1) After Exp term13 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S119 S121 · calculationBefore Exp term13 factorial step2 := mul(1, 2) After Exp term13 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S120 S122 · calculationBefore Exp term13 factorial step3 := mul(2, 3) After Exp term13 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S121 S123 · calculationBefore Exp term13 factorial step4 := mul(6, 4) After Exp term13 factorial step4 = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S122 S124 · calculationBefore Exp term13 factorial step5 := mul(24, 5) After Exp term13 factorial step5 = 120 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S123 S125 · calculationBefore Exp term13 factorial step6 := mul(120, 6) After Exp term13 factorial step6 = 720 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S124 S126 · calculationBefore Exp term13 factorial step7 := mul(720, 7) After Exp term13 factorial step7 = 5040 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S125 S127 · calculationBefore Exp term13 factorial step8 := mul(5040, 8) After Exp term13 factorial step8 = 40320 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S126 S128 · calculationBefore Exp term13 factorial step9 := mul(40320, 9) After Exp term13 factorial step9 = 362880 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S127 S129 · calculationBefore Exp term13 factorial step10 := mul(362880, 10) After Exp term13 factorial step10 = 3628800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S128 S130 · calculationBefore Exp term13 factorial step11 := mul(3628800, 11) After Exp term13 factorial step11 = 39916800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S129 S131 · calculationBefore Exp term13 factorial step12 := mul(39916800, 12) After Exp term13 factorial step12 = 479001600 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S130 S132 · calculationBefore Exp term13 factorial step13 := mul(479001600, 13) After Exp term13 factorial step13 = 6227020800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S131 S133 · impliesBefore (22/23)^13 After Exp ORIGINAL x power13 = 282810057883082752/504036361936467383 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S132 S134 · calculationBefore Exp term13 power over factorial := div(282810057883082752/504036361936467383, 6227020800) After Exp term13 power over factorial = 25107427013768/278643901787527581006975 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S133 S135 · calculationBefore Exp cumulative sum through13 := sum(2425439620182471292949/931919403971664150525, 25107427013768/278643901787527581006975) After Exp cumulative sum through13 = 26859498017024679392797/10320144510649169666925 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S134 S136 · calculationBefore Exp term14 factorial step1 := mul(1, 1) After Exp term14 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S135 S137 · calculationBefore Exp term14 factorial step2 := mul(1, 2) After Exp term14 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S136 S138 · calculationBefore Exp term14 factorial step3 := mul(2, 3) After Exp term14 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S137 S139 · calculationBefore Exp term14 factorial step4 := mul(6, 4) After Exp term14 factorial step4 = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S138 S140 · calculationBefore Exp term14 factorial step5 := mul(24, 5) After Exp term14 factorial step5 = 120 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S139 S141 · calculationBefore Exp term14 factorial step6 := mul(120, 6) After Exp term14 factorial step6 = 720 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S140 S142 · calculationBefore Exp term14 factorial step7 := mul(720, 7) After Exp term14 factorial step7 = 5040 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S141 S143 · calculationBefore Exp term14 factorial step8 := mul(5040, 8) After Exp term14 factorial step8 = 40320 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S142 S144 · calculationBefore Exp term14 factorial step9 := mul(40320, 9) After Exp term14 factorial step9 = 362880 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S143 S145 · calculationBefore Exp term14 factorial step10 := mul(362880, 10) After Exp term14 factorial step10 = 3628800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S144 S146 · calculationBefore Exp term14 factorial step11 := mul(3628800, 11) After Exp term14 factorial step11 = 39916800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S145 S147 · calculationBefore Exp term14 factorial step12 := mul(39916800, 12) After Exp term14 factorial step12 = 479001600 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S146 S148 · calculationBefore Exp term14 factorial step13 := mul(479001600, 13) After Exp term14 factorial step13 = 6227020800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S147 S149 · calculationBefore Exp term14 factorial step14 := mul(6227020800, 14) After Exp term14 factorial step14 = 87178291200 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S148 S150 · impliesBefore (22/23)^14 After Exp ORIGINAL x power14 = 6221821273427820544/11592836324538749809 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S149 S151 · calculationBefore Exp term14 power over factorial := div(6221821273427820544/11592836324538749809, 87178291200) After Exp term14 power over factorial = 276181697151448/44861668187791940542122975 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S150 S152 · calculationBefore Exp cumulative sum through14 := sum(26859498017024679392797/10320144510649169666925, 276181697151448/44861668187791940542122975) After Exp cumulative sum through14 = 116758237880282463017640007/44861668187791940542122975 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S151 S153 · calculationBefore Exp term15 factorial step1 := mul(1, 1) After Exp term15 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S152 S154 · calculationBefore Exp term15 factorial step2 := mul(1, 2) After Exp term15 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S153 S155 · calculationBefore Exp term15 factorial step3 := mul(2, 3) After Exp term15 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S154 S156 · calculationBefore Exp term15 factorial step4 := mul(6, 4) After Exp term15 factorial step4 = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S155 S157 · calculationBefore Exp term15 factorial step5 := mul(24, 5) After Exp term15 factorial step5 = 120 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S156 S158 · calculationBefore Exp term15 factorial step6 := mul(120, 6) After Exp term15 factorial step6 = 720 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S157 S159 · calculationBefore Exp term15 factorial step7 := mul(720, 7) After Exp term15 factorial step7 = 5040 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S158 S160 · calculationBefore Exp term15 factorial step8 := mul(5040, 8) After Exp term15 factorial step8 = 40320 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S159 S161 · calculationBefore Exp term15 factorial step9 := mul(40320, 9) After Exp term15 factorial step9 = 362880 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S160 S162 · calculationBefore Exp term15 factorial step10 := mul(362880, 10) After Exp term15 factorial step10 = 3628800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S161 S163 · calculationBefore Exp term15 factorial step11 := mul(3628800, 11) After Exp term15 factorial step11 = 39916800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S162 S164 · calculationBefore Exp term15 factorial step12 := mul(39916800, 12) After Exp term15 factorial step12 = 479001600 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S163 S165 · calculationBefore Exp term15 factorial step13 := mul(479001600, 13) After Exp term15 factorial step13 = 6227020800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S164 S166 · calculationBefore Exp term15 factorial step14 := mul(6227020800, 14) After Exp term15 factorial step14 = 87178291200 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S165 S167 · calculationBefore Exp term15 factorial step15 := mul(87178291200, 15) After Exp term15 factorial step15 = 1307674368000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S166 S168 · impliesBefore (22/23)^15 After Exp ORIGINAL x power15 = 136880068015412051968/266635235464391245607 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S167 S169 · calculationBefore Exp term15 power over factorial := div(136880068015412051968/266635235464391245607, 1307674368000) After Exp term15 power over factorial = 6075997337331856/15477275524788219487032426375 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S168 S170 · calculationBefore Exp cumulative sum through15 := sum(116758237880282463017640007/44861668187791940542122975, 6075997337331856/15477275524788219487032426375) After Exp cumulative sum through15 = 40281592068703525738423134271/15477275524788219487032426375 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S169 S171 · calculationBefore Exp term16 factorial step1 := mul(1, 1) After Exp term16 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S170 S172 · calculationBefore Exp term16 factorial step2 := mul(1, 2) After Exp term16 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S171 S173 · calculationBefore Exp term16 factorial step3 := mul(2, 3) After Exp term16 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S172 S174 · calculationBefore Exp term16 factorial step4 := mul(6, 4) After Exp term16 factorial step4 = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S173 S175 · calculationBefore Exp term16 factorial step5 := mul(24, 5) After Exp term16 factorial step5 = 120 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S174 S176 · calculationBefore Exp term16 factorial step6 := mul(120, 6) After Exp term16 factorial step6 = 720 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S175 S177 · calculationBefore Exp term16 factorial step7 := mul(720, 7) After Exp term16 factorial step7 = 5040 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S176 S178 · calculationBefore Exp term16 factorial step8 := mul(5040, 8) After Exp term16 factorial step8 = 40320 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S177 S179 · calculationBefore Exp term16 factorial step9 := mul(40320, 9) After Exp term16 factorial step9 = 362880 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S178 S180 · calculationBefore Exp term16 factorial step10 := mul(362880, 10) After Exp term16 factorial step10 = 3628800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S179 S181 · calculationBefore Exp term16 factorial step11 := mul(3628800, 11) After Exp term16 factorial step11 = 39916800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S180 S182 · calculationBefore Exp term16 factorial step12 := mul(39916800, 12) After Exp term16 factorial step12 = 479001600 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S181 S183 · calculationBefore Exp term16 factorial step13 := mul(479001600, 13) After Exp term16 factorial step13 = 6227020800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S182 S184 · calculationBefore Exp term16 factorial step14 := mul(6227020800, 14) After Exp term16 factorial step14 = 87178291200 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S183 S185 · calculationBefore Exp term16 factorial step15 := mul(87178291200, 15) After Exp term16 factorial step15 = 1307674368000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S184 S186 · calculationBefore Exp term16 factorial step16 := mul(1307674368000, 16) After Exp term16 factorial step16 = 20922789888000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S185 S187 · impliesBefore (22/23)^16 After Exp ORIGINAL x power16 = 3011361496339065143296/6132610415680998648961 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S186 S188 · calculationBefore Exp term16 power over factorial := div(3011361496339065143296/6132610415680998648961, 20922789888000) After Exp term16 power over factorial = 8354496338831302/355977337070129048201745806625 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S187 S189 · calculationBefore Exp cumulative sum through16 := sum(40281592068703525738423134271/15477275524788219487032426375, 8354496338831302/355977337070129048201745806625) After Exp cumulative sum through16 = 6862789759853255159111636441/2636869163482437394087005975 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S188 S190 · calculationBefore Exp term17 factorial step1 := mul(1, 1) After Exp term17 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S189 S191 · calculationBefore Exp term17 factorial step2 := mul(1, 2) After Exp term17 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S190 S192 · calculationBefore Exp term17 factorial step3 := mul(2, 3) After Exp term17 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S191 S193 · calculationBefore Exp term17 factorial step4 := mul(6, 4) After Exp term17 factorial step4 = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S192 S194 · calculationBefore Exp term17 factorial step5 := mul(24, 5) After Exp term17 factorial step5 = 120 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S193 S195 · calculationBefore Exp term17 factorial step6 := mul(120, 6) After Exp term17 factorial step6 = 720 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S194 S196 · calculationBefore Exp term17 factorial step7 := mul(720, 7) After Exp term17 factorial step7 = 5040 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S195 S197 · calculationBefore Exp term17 factorial step8 := mul(5040, 8) After Exp term17 factorial step8 = 40320 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S196 S198 · calculationBefore Exp term17 factorial step9 := mul(40320, 9) After Exp term17 factorial step9 = 362880 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S197 S199 · calculationBefore Exp term17 factorial step10 := mul(362880, 10) After Exp term17 factorial step10 = 3628800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S198 S200 · calculationBefore Exp term17 factorial step11 := mul(3628800, 11) After Exp term17 factorial step11 = 39916800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S199 S201 · calculationBefore Exp term17 factorial step12 := mul(39916800, 12) After Exp term17 factorial step12 = 479001600 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S200 S202 · calculationBefore Exp term17 factorial step13 := mul(479001600, 13) After Exp term17 factorial step13 = 6227020800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S201 S203 · calculationBefore Exp term17 factorial step14 := mul(6227020800, 14) After Exp term17 factorial step14 = 87178291200 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S202 S204 · calculationBefore Exp term17 factorial step15 := mul(87178291200, 15) After Exp term17 factorial step15 = 1307674368000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S203 S205 · calculationBefore Exp term17 factorial step16 := mul(1307674368000, 16) After Exp term17 factorial step16 = 20922789888000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S204 S206 · calculationBefore Exp term17 factorial step17 := mul(20922789888000, 17) After Exp term17 factorial step17 = 355687428096000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S205 S207 · impliesBefore (22/23)^17 After Exp ORIGINAL x power17 = 66249952919459433152512/141050039560662968926103 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S206 S208 · calculationBefore Exp term17 power over factorial := div(66249952919459433152512/141050039560662968926103, 355687428096000) After Exp term17 power over factorial = 183798919454288644/139187138794420457846882610390375 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S207 S209 · calculationBefore Exp cumulative sum through17 := sum(6862789759853255159111636441/2636869163482437394087005975, 183798919454288644/139187138794420457846882610390375) After Exp cumulative sum through17 = 362252357473854257372627183826829/139187138794420457846882610390375 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S208 S210 · calculationBefore Exp term18 factorial step1 := mul(1, 1) After Exp term18 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S209 S211 · calculationBefore Exp term18 factorial step2 := mul(1, 2) After Exp term18 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S210 S212 · calculationBefore Exp term18 factorial step3 := mul(2, 3) After Exp term18 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S211 S213 · calculationBefore Exp term18 factorial step4 := mul(6, 4) After Exp term18 factorial step4 = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S212 S214 · calculationBefore Exp term18 factorial step5 := mul(24, 5) After Exp term18 factorial step5 = 120 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S213 S215 · calculationBefore Exp term18 factorial step6 := mul(120, 6) After Exp term18 factorial step6 = 720 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S214 S216 · calculationBefore Exp term18 factorial step7 := mul(720, 7) After Exp term18 factorial step7 = 5040 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S215 S217 · calculationBefore Exp term18 factorial step8 := mul(5040, 8) After Exp term18 factorial step8 = 40320 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S216 S218 · calculationBefore Exp term18 factorial step9 := mul(40320, 9) After Exp term18 factorial step9 = 362880 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S217 S219 · calculationBefore Exp term18 factorial step10 := mul(362880, 10) After Exp term18 factorial step10 = 3628800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S218 S220 · calculationBefore Exp term18 factorial step11 := mul(3628800, 11) After Exp term18 factorial step11 = 39916800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S219 S221 · calculationBefore Exp term18 factorial step12 := mul(39916800, 12) After Exp term18 factorial step12 = 479001600 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S220 S222 · calculationBefore Exp term18 factorial step13 := mul(479001600, 13) After Exp term18 factorial step13 = 6227020800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S221 S223 · calculationBefore Exp term18 factorial step14 := mul(6227020800, 14) After Exp term18 factorial step14 = 87178291200 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S222 S224 · calculationBefore Exp term18 factorial step15 := mul(87178291200, 15) After Exp term18 factorial step15 = 1307674368000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S223 S225 · calculationBefore Exp term18 factorial step16 := mul(1307674368000, 16) After Exp term18 factorial step16 = 20922789888000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S224 S226 · calculationBefore Exp term18 factorial step17 := mul(20922789888000, 17) After Exp term18 factorial step17 = 355687428096000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S225 S227 · calculationBefore Exp term18 factorial step18 := mul(355687428096000, 18) After Exp term18 factorial step18 = 6402373705728000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S226 S228 · impliesBefore (22/23)^18 After Exp ORIGINAL x power18 = 1457498964228107529355264/3244150909895248285300369 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S227 S229 · calculationBefore Exp term18 power over factorial := div(1457498964228107529355264/3244150909895248285300369, 6402373705728000) After Exp term18 power over factorial = 2021788113997175084/28811737730445034774304700350807625 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S228 S230 · calculationBefore Exp cumulative sum through18 := sum(362252357473854257372627183826829/139187138794420457846882610390375, 2021788113997175084/28811737730445034774304700350807625) After Exp cumulative sum through18 = 5768172153622141022917072388409899/2216287517726541136484976950062125 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S229 S231 · calculationBefore Exp term19 factorial step1 := mul(1, 1) After Exp term19 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S230 S232 · calculationBefore Exp term19 factorial step2 := mul(1, 2) After Exp term19 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S231 S233 · calculationBefore Exp term19 factorial step3 := mul(2, 3) After Exp term19 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S232 S234 · calculationBefore Exp term19 factorial step4 := mul(6, 4) After Exp term19 factorial step4 = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S233 S235 · calculationBefore Exp term19 factorial step5 := mul(24, 5) After Exp term19 factorial step5 = 120 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S234 S236 · calculationBefore Exp term19 factorial step6 := mul(120, 6) After Exp term19 factorial step6 = 720 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S235 S237 · calculationBefore Exp term19 factorial step7 := mul(720, 7) After Exp term19 factorial step7 = 5040 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S236 S238 · calculationBefore Exp term19 factorial step8 := mul(5040, 8) After Exp term19 factorial step8 = 40320 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S237 S239 · calculationBefore Exp term19 factorial step9 := mul(40320, 9) After Exp term19 factorial step9 = 362880 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S238 S240 · calculationBefore Exp term19 factorial step10 := mul(362880, 10) After Exp term19 factorial step10 = 3628800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S239 S241 · calculationBefore Exp term19 factorial step11 := mul(3628800, 11) After Exp term19 factorial step11 = 39916800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S240 S242 · calculationBefore Exp term19 factorial step12 := mul(39916800, 12) After Exp term19 factorial step12 = 479001600 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S241 S243 · calculationBefore Exp term19 factorial step13 := mul(479001600, 13) After Exp term19 factorial step13 = 6227020800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S242 S244 · calculationBefore Exp term19 factorial step14 := mul(6227020800, 14) After Exp term19 factorial step14 = 87178291200 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S243 S245 · calculationBefore Exp term19 factorial step15 := mul(87178291200, 15) After Exp term19 factorial step15 = 1307674368000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S244 S246 · calculationBefore Exp term19 factorial step16 := mul(1307674368000, 16) After Exp term19 factorial step16 = 20922789888000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S245 S247 · calculationBefore Exp term19 factorial step17 := mul(20922789888000, 17) After Exp term19 factorial step17 = 355687428096000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S246 S248 · calculationBefore Exp term19 factorial step18 := mul(355687428096000, 18) After Exp term19 factorial step18 = 6402373705728000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S247 S249 · calculationBefore Exp term19 factorial step19 := mul(6402373705728000, 19) After Exp term19 factorial step19 = 121645100408832000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S248 S250 · impliesBefore (22/23)^19 After Exp ORIGINAL x power19 = 32064977213018365645815808/74615470927590710561908487 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S249 S251 · calculationBefore Exp term19 power over factorial := div(32064977213018365645815808/74615470927590710561908487, 121645100408832000) After Exp term19 power over factorial = 44479338507937851848/12590729388204480196371154053302932125 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S250 S252 · calculationBefore Exp cumulative sum through19 := sum(5768172153622141022917072388409899/2216287517726541136484976950062125, 44479338507937851848/12590729388204480196371154053302932125) After Exp cumulative sum through19 = 3640998444969709243963469638499387563/1398969932022720021819017117033659125 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S251 S253 · calculationBefore Exp term20 factorial step1 := mul(1, 1) After Exp term20 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S252 S254 · calculationBefore Exp term20 factorial step2 := mul(1, 2) After Exp term20 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S253 S255 · calculationBefore Exp term20 factorial step3 := mul(2, 3) After Exp term20 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S254 S256 · calculationBefore Exp term20 factorial step4 := mul(6, 4) After Exp term20 factorial step4 = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S255 S257 · calculationBefore Exp term20 factorial step5 := mul(24, 5) After Exp term20 factorial step5 = 120 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S256 S258 · calculationBefore Exp term20 factorial step6 := mul(120, 6) After Exp term20 factorial step6 = 720 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S257 S259 · calculationBefore Exp term20 factorial step7 := mul(720, 7) After Exp term20 factorial step7 = 5040 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S258 S260 · calculationBefore Exp term20 factorial step8 := mul(5040, 8) After Exp term20 factorial step8 = 40320 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S259 S261 · calculationBefore Exp term20 factorial step9 := mul(40320, 9) After Exp term20 factorial step9 = 362880 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S260 S262 · calculationBefore Exp term20 factorial step10 := mul(362880, 10) After Exp term20 factorial step10 = 3628800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S261 S263 · calculationBefore Exp term20 factorial step11 := mul(3628800, 11) After Exp term20 factorial step11 = 39916800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S262 S264 · calculationBefore Exp term20 factorial step12 := mul(39916800, 12) After Exp term20 factorial step12 = 479001600 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S263 S265 · calculationBefore Exp term20 factorial step13 := mul(479001600, 13) After Exp term20 factorial step13 = 6227020800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S264 S266 · calculationBefore Exp term20 factorial step14 := mul(6227020800, 14) After Exp term20 factorial step14 = 87178291200 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S265 S267 · calculationBefore Exp term20 factorial step15 := mul(87178291200, 15) After Exp term20 factorial step15 = 1307674368000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S266 S268 · calculationBefore Exp term20 factorial step16 := mul(1307674368000, 16) After Exp term20 factorial step16 = 20922789888000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S267 S269 · calculationBefore Exp term20 factorial step17 := mul(20922789888000, 17) After Exp term20 factorial step17 = 355687428096000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S268 S270 · calculationBefore Exp term20 factorial step18 := mul(355687428096000, 18) After Exp term20 factorial step18 = 6402373705728000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S269 S271 · calculationBefore Exp term20 factorial step19 := mul(6402373705728000, 19) After Exp term20 factorial step19 = 121645100408832000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S270 S272 · calculationBefore Exp term20 factorial step20 := mul(121645100408832000, 20) After Exp term20 factorial step20 = 2432902008176640000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S271 S273 · impliesBefore (22/23)^20 After Exp ORIGINAL x power20 = 705429498686404044207947776/1716155831334586342923895201 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S272 S274 · calculationBefore Exp term20 power over factorial := div(705429498686404044207947776/1716155831334586342923895201, 2432902008176640000) After Exp term20 power over factorial = 244636361793658185164/1447933879643515222582682716129837194375 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S273 S275 · calculationBefore Exp cumulative sum through20 := sum(3640998444969709243963469638499387563/1398969932022720021819017117033659125, 244636361793658185164/1447933879643515222582682716129837194375) After Exp cumulative sum through20 = 3768433390543649067746827437640524312869/1447933879643515222582682716129837194375 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S274 S276 · calculationBefore Exp term21 factorial step1 := mul(1, 1) After Exp term21 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S275 S277 · calculationBefore Exp term21 factorial step2 := mul(1, 2) After Exp term21 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S276 S278 · calculationBefore Exp term21 factorial step3 := mul(2, 3) After Exp term21 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S277 S279 · calculationBefore Exp term21 factorial step4 := mul(6, 4) After Exp term21 factorial step4 = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S278 S280 · calculationBefore Exp term21 factorial step5 := mul(24, 5) After Exp term21 factorial step5 = 120 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S279 S281 · calculationBefore Exp term21 factorial step6 := mul(120, 6) After Exp term21 factorial step6 = 720 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S280 S282 · calculationBefore Exp term21 factorial step7 := mul(720, 7) After Exp term21 factorial step7 = 5040 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S281 S283 · calculationBefore Exp term21 factorial step8 := mul(5040, 8) After Exp term21 factorial step8 = 40320 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S282 S284 · calculationBefore Exp term21 factorial step9 := mul(40320, 9) After Exp term21 factorial step9 = 362880 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S283 S285 · calculationBefore Exp term21 factorial step10 := mul(362880, 10) After Exp term21 factorial step10 = 3628800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S284 S286 · calculationBefore Exp term21 factorial step11 := mul(3628800, 11) After Exp term21 factorial step11 = 39916800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S285 S287 · calculationBefore Exp term21 factorial step12 := mul(39916800, 12) After Exp term21 factorial step12 = 479001600 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S286 S288 · calculationBefore Exp term21 factorial step13 := mul(479001600, 13) After Exp term21 factorial step13 = 6227020800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S287 S289 · calculationBefore Exp term21 factorial step14 := mul(6227020800, 14) After Exp term21 factorial step14 = 87178291200 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S288 S290 · calculationBefore Exp term21 factorial step15 := mul(87178291200, 15) After Exp term21 factorial step15 = 1307674368000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S289 S291 · calculationBefore Exp term21 factorial step16 := mul(1307674368000, 16) After Exp term21 factorial step16 = 20922789888000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S290 S292 · calculationBefore Exp term21 factorial step17 := mul(20922789888000, 17) After Exp term21 factorial step17 = 355687428096000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S291 S293 · calculationBefore Exp term21 factorial step18 := mul(355687428096000, 18) After Exp term21 factorial step18 = 6402373705728000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S292 S294 · calculationBefore Exp term21 factorial step19 := mul(6402373705728000, 19) After Exp term21 factorial step19 = 121645100408832000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S293 S295 · calculationBefore Exp term21 factorial step20 := mul(121645100408832000, 20) After Exp term21 factorial step20 = 2432902008176640000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S294 S296 · calculationBefore Exp term21 factorial step21 := mul(2432902008176640000, 21) After Exp term21 factorial step21 = 51090942171709440000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S295 S297 · impliesBefore (22/23)^21 After Exp ORIGINAL x power21 = 15519448971100888972574851072/39471584120695485887249589623 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S296 S298 · calculationBefore Exp term21 power over factorial := div(15519448971100888972574851072/39471584120695485887249589623, 51090942171709440000) After Exp term21 power over factorial = 5381999959460480073608/699352063867817852507435751890711364883125 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S297 S299 · calculationBefore Exp cumulative sum through21 := sum(3768433390543649067746827437640524312869/1447933879643515222582682716129837194375, 5381999959460480073608/699352063867817852507435751890711364883125) After Exp cumulative sum through21 = 364030665526516499945419930467966744637867/139870412773563570501487150378142272976625 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S298 S300 · calculationBefore Exp term22 factorial step1 := mul(1, 1) After Exp term22 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S299 S301 · calculationBefore Exp term22 factorial step2 := mul(1, 2) After Exp term22 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S300 S302 · calculationBefore Exp term22 factorial step3 := mul(2, 3) After Exp term22 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S301 S303 · calculationBefore Exp term22 factorial step4 := mul(6, 4) After Exp term22 factorial step4 = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S302 S304 · calculationBefore Exp term22 factorial step5 := mul(24, 5) After Exp term22 factorial step5 = 120 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S303 S305 · calculationBefore Exp term22 factorial step6 := mul(120, 6) After Exp term22 factorial step6 = 720 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S304 S306 · calculationBefore Exp term22 factorial step7 := mul(720, 7) After Exp term22 factorial step7 = 5040 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S305 S307 · calculationBefore Exp term22 factorial step8 := mul(5040, 8) After Exp term22 factorial step8 = 40320 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S306 S308 · calculationBefore Exp term22 factorial step9 := mul(40320, 9) After Exp term22 factorial step9 = 362880 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S307 S309 · calculationBefore Exp term22 factorial step10 := mul(362880, 10) After Exp term22 factorial step10 = 3628800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S308 S310 · calculationBefore Exp term22 factorial step11 := mul(3628800, 11) After Exp term22 factorial step11 = 39916800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S309 S311 · calculationBefore Exp term22 factorial step12 := mul(39916800, 12) After Exp term22 factorial step12 = 479001600 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S310 S312 · calculationBefore Exp term22 factorial step13 := mul(479001600, 13) After Exp term22 factorial step13 = 6227020800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S311 S313 · calculationBefore Exp term22 factorial step14 := mul(6227020800, 14) After Exp term22 factorial step14 = 87178291200 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S312 S314 · calculationBefore Exp term22 factorial step15 := mul(87178291200, 15) After Exp term22 factorial step15 = 1307674368000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S313 S315 · calculationBefore Exp term22 factorial step16 := mul(1307674368000, 16) After Exp term22 factorial step16 = 20922789888000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S314 S316 · calculationBefore Exp term22 factorial step17 := mul(20922789888000, 17) After Exp term22 factorial step17 = 355687428096000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S315 S317 · calculationBefore Exp term22 factorial step18 := mul(355687428096000, 18) After Exp term22 factorial step18 = 6402373705728000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S316 S318 · calculationBefore Exp term22 factorial step19 := mul(6402373705728000, 19) After Exp term22 factorial step19 = 121645100408832000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S317 S319 · calculationBefore Exp term22 factorial step20 := mul(121645100408832000, 20) After Exp term22 factorial step20 = 2432902008176640000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S318 S320 · calculationBefore Exp term22 factorial step21 := mul(2432902008176640000, 21) After Exp term22 factorial step21 = 51090942171709440000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S319 S321 · calculationBefore Exp term22 factorial step22 := mul(51090942171709440000, 22) After Exp term22 factorial step22 = 1124000727777607680000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S320 S322 · impliesBefore (22/23)^22 After Exp ORIGINAL x power22 = 341427877364219557396646723584/907846434775996175406740561329 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S321 S323 · calculationBefore Exp term22 power over factorial := div(341427877364219557396646723584/907846434775996175406740561329, 1124000727777607680000) After Exp term22 power over factorial = 5381999959460480073608/16085097468959810607671022293486361392311875 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S322 S324 · calculationBefore Exp cumulative sum through22 := sum(364030665526516499945419930467966744637867/139870412773563570501487150378142272976625, 5381999959460480073608/16085097468959810607671022293486361392311875) After Exp cumulative sum through22 = 1550500982798125833101062000139838374571419/595744350702215207691519344203198570085625 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S323 S325 · calculationBefore Exp term23 factorial step1 := mul(1, 1) After Exp term23 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S324 S326 · calculationBefore Exp term23 factorial step2 := mul(1, 2) After Exp term23 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S325 S327 · calculationBefore Exp term23 factorial step3 := mul(2, 3) After Exp term23 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S326 S328 · calculationBefore Exp term23 factorial step4 := mul(6, 4) After Exp term23 factorial step4 = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S327 S329 · calculationBefore Exp term23 factorial step5 := mul(24, 5) After Exp term23 factorial step5 = 120 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S328 S330 · calculationBefore Exp term23 factorial step6 := mul(120, 6) After Exp term23 factorial step6 = 720 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S329 S331 · calculationBefore Exp term23 factorial step7 := mul(720, 7) After Exp term23 factorial step7 = 5040 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S330 S332 · calculationBefore Exp term23 factorial step8 := mul(5040, 8) After Exp term23 factorial step8 = 40320 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S331 S333 · calculationBefore Exp term23 factorial step9 := mul(40320, 9) After Exp term23 factorial step9 = 362880 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S332 S334 · calculationBefore Exp term23 factorial step10 := mul(362880, 10) After Exp term23 factorial step10 = 3628800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S333 S335 · calculationBefore Exp term23 factorial step11 := mul(3628800, 11) After Exp term23 factorial step11 = 39916800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S334 S336 · calculationBefore Exp term23 factorial step12 := mul(39916800, 12) After Exp term23 factorial step12 = 479001600 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S335 S337 · calculationBefore Exp term23 factorial step13 := mul(479001600, 13) After Exp term23 factorial step13 = 6227020800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S336 S338 · calculationBefore Exp term23 factorial step14 := mul(6227020800, 14) After Exp term23 factorial step14 = 87178291200 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S337 S339 · calculationBefore Exp term23 factorial step15 := mul(87178291200, 15) After Exp term23 factorial step15 = 1307674368000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S338 S340 · calculationBefore Exp term23 factorial step16 := mul(1307674368000, 16) After Exp term23 factorial step16 = 20922789888000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S339 S341 · calculationBefore Exp term23 factorial step17 := mul(20922789888000, 17) After Exp term23 factorial step17 = 355687428096000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S340 S342 · calculationBefore Exp term23 factorial step18 := mul(355687428096000, 18) After Exp term23 factorial step18 = 6402373705728000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S341 S343 · calculationBefore Exp term23 factorial step19 := mul(6402373705728000, 19) After Exp term23 factorial step19 = 121645100408832000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S342 S344 · calculationBefore Exp term23 factorial step20 := mul(121645100408832000, 20) After Exp term23 factorial step20 = 2432902008176640000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S343 S345 · calculationBefore Exp term23 factorial step21 := mul(2432902008176640000, 21) After Exp term23 factorial step21 = 51090942171709440000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S344 S346 · calculationBefore Exp term23 factorial step22 := mul(51090942171709440000, 22) After Exp term23 factorial step22 = 1124000727777607680000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S345 S347 · calculationBefore Exp term23 factorial step23 := mul(1124000727777607680000, 23) After Exp term23 factorial step23 = 25852016738884976640000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S346 S348 · impliesBefore (22/23)^23 After Exp ORIGINAL x power23 = 7511413302012830262726227918848/20880467999847912034355032910567 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S347 S349 · calculationBefore Exp term23 power over factorial := div(7511413302012830262726227918848/20880467999847912034355032910567, 25852016738884976640000) After Exp term23 power over factorial = 118403999108130561619376/8509016561079739811457970793254285176532981875 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S348 S350 · calculationBefore Exp cumulative sum through23 := sum(1550500982798125833101062000139838374571419/595744350702215207691519344203198570085625, 118403999108130561619376/8509016561079739811457970793254285176532981875) After Exp cumulative sum through23 = 22145805537305631274182586951996419634565196953/8509016561079739811457970793254285176532981875 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S349 S351 · calculationBefore Exp term24 factorial step1 := mul(1, 1) After Exp term24 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S350 S352 · calculationBefore Exp term24 factorial step2 := mul(1, 2) After Exp term24 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S351 S353 · calculationBefore Exp term24 factorial step3 := mul(2, 3) After Exp term24 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S352 S354 · calculationBefore Exp term24 factorial step4 := mul(6, 4) After Exp term24 factorial step4 = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S353 S355 · calculationBefore Exp term24 factorial step5 := mul(24, 5) After Exp term24 factorial step5 = 120 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S354 S356 · calculationBefore Exp term24 factorial step6 := mul(120, 6) After Exp term24 factorial step6 = 720 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S355 S357 · calculationBefore Exp term24 factorial step7 := mul(720, 7) After Exp term24 factorial step7 = 5040 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S356 S358 · calculationBefore Exp term24 factorial step8 := mul(5040, 8) After Exp term24 factorial step8 = 40320 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S357 S359 · calculationBefore Exp term24 factorial step9 := mul(40320, 9) After Exp term24 factorial step9 = 362880 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S358 S360 · calculationBefore Exp term24 factorial step10 := mul(362880, 10) After Exp term24 factorial step10 = 3628800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S359 S361 · calculationBefore Exp term24 factorial step11 := mul(3628800, 11) After Exp term24 factorial step11 = 39916800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S360 S362 · calculationBefore Exp term24 factorial step12 := mul(39916800, 12) After Exp term24 factorial step12 = 479001600 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S361 S363 · calculationBefore Exp term24 factorial step13 := mul(479001600, 13) After Exp term24 factorial step13 = 6227020800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S362 S364 · calculationBefore Exp term24 factorial step14 := mul(6227020800, 14) After Exp term24 factorial step14 = 87178291200 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S363 S365 · calculationBefore Exp term24 factorial step15 := mul(87178291200, 15) After Exp term24 factorial step15 = 1307674368000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S364 S366 · calculationBefore Exp term24 factorial step16 := mul(1307674368000, 16) After Exp term24 factorial step16 = 20922789888000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S365 S367 · calculationBefore Exp term24 factorial step17 := mul(20922789888000, 17) After Exp term24 factorial step17 = 355687428096000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S366 S368 · calculationBefore Exp term24 factorial step18 := mul(355687428096000, 18) After Exp term24 factorial step18 = 6402373705728000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S367 S369 · calculationBefore Exp term24 factorial step19 := mul(6402373705728000, 19) After Exp term24 factorial step19 = 121645100408832000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S368 S370 · calculationBefore Exp term24 factorial step20 := mul(121645100408832000, 20) After Exp term24 factorial step20 = 2432902008176640000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S369 S371 · calculationBefore Exp term24 factorial step21 := mul(2432902008176640000, 21) After Exp term24 factorial step21 = 51090942171709440000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S370 S372 · calculationBefore Exp term24 factorial step22 := mul(51090942171709440000, 22) After Exp term24 factorial step22 = 1124000727777607680000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S371 S373 · calculationBefore Exp term24 factorial step23 := mul(1124000727777607680000, 23) After Exp term24 factorial step23 = 25852016738884976640000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S372 S374 · calculationBefore Exp term24 factorial step24 := mul(25852016738884976640000, 24) After Exp term24 factorial step24 = 620448401733239439360000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S373 S375 · impliesBefore (22/23)^24 After Exp ORIGINAL x power24 = 165251092644282265779977014214656/480250763996501976790165756943041 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S374 S376 · calculationBefore Exp term24 power over factorial := div(165251092644282265779977014214656/480250763996501976790165756943041, 620448401733239439360000) After Exp term24 power over factorial = 325610997547359044453284/587122142714502046990599984734545677180775749375 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S375 S377 · calculationBefore Exp cumulative sum through24 := sum(22145805537305631274182586951996419634565196953/8509016561079739811457970793254285176532981875, 325610997547359044453284/587122142714502046990599984734545677180775749375) After Exp cumulative sum through24 = 1528060582074088557918598825298750502144043043041/587122142714502046990599984734545677180775749375 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S376 S378 · calculationBefore Exp term25 factorial step1 := mul(1, 1) After Exp term25 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S377 S379 · calculationBefore Exp term25 factorial step2 := mul(1, 2) After Exp term25 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S378 S380 · calculationBefore Exp term25 factorial step3 := mul(2, 3) After Exp term25 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S379 S381 · calculationBefore Exp term25 factorial step4 := mul(6, 4) After Exp term25 factorial step4 = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S380 S382 · calculationBefore Exp term25 factorial step5 := mul(24, 5) After Exp term25 factorial step5 = 120 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S381 S383 · calculationBefore Exp term25 factorial step6 := mul(120, 6) After Exp term25 factorial step6 = 720 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S382 S384 · calculationBefore Exp term25 factorial step7 := mul(720, 7) After Exp term25 factorial step7 = 5040 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S383 S385 · calculationBefore Exp term25 factorial step8 := mul(5040, 8) After Exp term25 factorial step8 = 40320 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S384 S386 · calculationBefore Exp term25 factorial step9 := mul(40320, 9) After Exp term25 factorial step9 = 362880 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S385 S387 · calculationBefore Exp term25 factorial step10 := mul(362880, 10) After Exp term25 factorial step10 = 3628800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S386 S388 · calculationBefore Exp term25 factorial step11 := mul(3628800, 11) After Exp term25 factorial step11 = 39916800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S387 S389 · calculationBefore Exp term25 factorial step12 := mul(39916800, 12) After Exp term25 factorial step12 = 479001600 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S388 S390 · calculationBefore Exp term25 factorial step13 := mul(479001600, 13) After Exp term25 factorial step13 = 6227020800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S389 S391 · calculationBefore Exp term25 factorial step14 := mul(6227020800, 14) After Exp term25 factorial step14 = 87178291200 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S390 S392 · calculationBefore Exp term25 factorial step15 := mul(87178291200, 15) After Exp term25 factorial step15 = 1307674368000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S391 S393 · calculationBefore Exp term25 factorial step16 := mul(1307674368000, 16) After Exp term25 factorial step16 = 20922789888000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S392 S394 · calculationBefore Exp term25 factorial step17 := mul(20922789888000, 17) After Exp term25 factorial step17 = 355687428096000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S393 S395 · calculationBefore Exp term25 factorial step18 := mul(355687428096000, 18) After Exp term25 factorial step18 = 6402373705728000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S394 S396 · calculationBefore Exp term25 factorial step19 := mul(6402373705728000, 19) After Exp term25 factorial step19 = 121645100408832000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S395 S397 · calculationBefore Exp term25 factorial step20 := mul(121645100408832000, 20) After Exp term25 factorial step20 = 2432902008176640000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S396 S398 · calculationBefore Exp term25 factorial step21 := mul(2432902008176640000, 21) After Exp term25 factorial step21 = 51090942171709440000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S397 S399 · calculationBefore Exp term25 factorial step22 := mul(51090942171709440000, 22) After Exp term25 factorial step22 = 1124000727777607680000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S398 S400 · calculationBefore Exp term25 factorial step23 := mul(1124000727777607680000, 23) After Exp term25 factorial step23 = 25852016738884976640000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S399 S401 · calculationBefore Exp term25 factorial step24 := mul(25852016738884976640000, 24) After Exp term25 factorial step24 = 620448401733239439360000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S400 S402 · calculationBefore Exp term25 factorial step25 := mul(620448401733239439360000, 25) After Exp term25 factorial step25 = 15511210043330985984000000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S401 S403 · impliesBefore (22/23)^25 After Exp ORIGINAL x power25 = 3635524038174209847159494312722432/11045767571919545466173812409689943 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S402 S404 · calculationBefore Exp term25 power over factorial := div(3635524038174209847159494312722432/11045767571919545466173812409689943, 15511210043330985984000000) After Exp term25 power over factorial = 7163441946041898977972248/337595232060838677019594991222363764378946055890625 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S403 S405 · calculationBefore Exp cumulative sum through25 := sum(1528060582074088557918598825298750502144043043041/587122142714502046990599984734545677180775749375, 7163441946041898977972248/337595232060838677019594991222363764378946055890625) After Exp cumulative sum through25 = 32542030914540774844562753026304573510174952878549/12503527113364395445170184860087546828849853921875 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S404 S406 · calculationBefore Exp term26 factorial step1 := mul(1, 1) After Exp term26 factorial step1 = 1 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S405 S407 · calculationBefore Exp term26 factorial step2 := mul(1, 2) After Exp term26 factorial step2 = 2 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S406 S408 · calculationBefore Exp term26 factorial step3 := mul(2, 3) After Exp term26 factorial step3 = 6 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S407 S409 · calculationBefore Exp term26 factorial step4 := mul(6, 4) After Exp term26 factorial step4 = 24 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S408 S410 · calculationBefore Exp term26 factorial step5 := mul(24, 5) After Exp term26 factorial step5 = 120 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S409 S411 · calculationBefore Exp term26 factorial step6 := mul(120, 6) After Exp term26 factorial step6 = 720 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S410 S412 · calculationBefore Exp term26 factorial step7 := mul(720, 7) After Exp term26 factorial step7 = 5040 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S411 S413 · calculationBefore Exp term26 factorial step8 := mul(5040, 8) After Exp term26 factorial step8 = 40320 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S412 S414 · calculationBefore Exp term26 factorial step9 := mul(40320, 9) After Exp term26 factorial step9 = 362880 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S413 S415 · calculationBefore Exp term26 factorial step10 := mul(362880, 10) After Exp term26 factorial step10 = 3628800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S414 S416 · calculationBefore Exp term26 factorial step11 := mul(3628800, 11) After Exp term26 factorial step11 = 39916800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S415 S417 · calculationBefore Exp term26 factorial step12 := mul(39916800, 12) After Exp term26 factorial step12 = 479001600 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S416 S418 · calculationBefore Exp term26 factorial step13 := mul(479001600, 13) After Exp term26 factorial step13 = 6227020800 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S417 S419 · calculationBefore Exp term26 factorial step14 := mul(6227020800, 14) After Exp term26 factorial step14 = 87178291200 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S418 S420 · calculationBefore Exp term26 factorial step15 := mul(87178291200, 15) After Exp term26 factorial step15 = 1307674368000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S419 S421 · calculationBefore Exp term26 factorial step16 := mul(1307674368000, 16) After Exp term26 factorial step16 = 20922789888000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S420 S422 · calculationBefore Exp term26 factorial step17 := mul(20922789888000, 17) After Exp term26 factorial step17 = 355687428096000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S421 S423 · calculationBefore Exp term26 factorial step18 := mul(355687428096000, 18) After Exp term26 factorial step18 = 6402373705728000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S422 S424 · calculationBefore Exp term26 factorial step19 := mul(6402373705728000, 19) After Exp term26 factorial step19 = 121645100408832000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S423 S425 · calculationBefore Exp term26 factorial step20 := mul(121645100408832000, 20) After Exp term26 factorial step20 = 2432902008176640000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S424 S426 · calculationBefore Exp term26 factorial step21 := mul(2432902008176640000, 21) After Exp term26 factorial step21 = 51090942171709440000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S425 S427 · calculationBefore Exp term26 factorial step22 := mul(51090942171709440000, 22) After Exp term26 factorial step22 = 1124000727777607680000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S426 S428 · calculationBefore Exp term26 factorial step23 := mul(1124000727777607680000, 23) After Exp term26 factorial step23 = 25852016738884976640000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S427 S429 · calculationBefore Exp term26 factorial step24 := mul(25852016738884976640000, 24) After Exp term26 factorial step24 = 620448401733239439360000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S428 S430 · calculationBefore Exp term26 factorial step25 := mul(620448401733239439360000, 25) After Exp term26 factorial step25 = 15511210043330985984000000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S429 S431 · calculationBefore Exp term26 factorial step26 := mul(15511210043330985984000000, 26) After Exp term26 factorial step26 = 403291461126605635584000000 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S430 S432 · impliesBefore (22/23)^26 After Exp ORIGINAL x power26 = 79981528839832616637508874879893504/254052654154149545721997685422868689 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S431 S433 · calculationBefore Exp term26 power over factorial := div(79981528839832616637508874879893504/254052654154149545721997685422868689, 403291461126605635584000000) After Exp term26 power over factorial = 78797861406460888757694728/100940974386190764428858902375486765549304870711296875 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S432 S434 · calculationBefore Exp cumulative sum through26 := sum(32542030914540774844562753026304573510174952878549/12503527113364395445170184860087546828849853921875, 78797861406460888757694728/100940974386190764428858902375486765549304870711296875) After Exp cumulative sum through26 = 52542363114617535064031021052030936670820656669244161/20188194877238152885771780475097353109860974142259375 Conditions: none Substitute original roles and established intermediates exactly;check nonzero denominators and do not round. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S433 S435 · impliesBefore source roles = [22/23, 26, [1, 22/23, 242/529, 5324/36501, 29282/839523, 644204/96545145, 7086244/6661615005, 155897368/1072520015805, 428717762/24667960363515, 9431790764/5106267795247605, 103749698404/587220796453474575, 207499396808/13506078318429915225, 1141246682444/931919403971664150525, 25107427013768/278643901787527581006975, 276181697151448/44861668187791940542122975, 6075997337331856/15477275524788219487032426375, 8354496338831302/355977337070129048201745806625, 183798919454288644/139187138794420457846882610390375, 2021788113997175084/28811737730445034774304700350807625, 44479338507937851848/12590729388204480196371154053302932125, 244636361793658185164/1447933879643515222582682716129837194375, 5381999959460480073608/699352063867817852507435751890711364883125, 5381999959460480073608/16085097468959810607671022293486361392311875, 118403999108130561619376/8509016561079739811457970793254285176532981875, 325610997547359044453284/587122142714502046990599984734545677180775749375, 7163441946041898977972248/337595232060838677019594991222363764378946055890625, 78797861406460888757694728/100940974386190764428858902375486765549304870711296875], [1, 45/23, 1277/529, 93437/36501, 726111/279841, 251152499/96545145, 3467321735/1332323001, 310149988227/119168890645, 64201476280751/24667960363515, 13289715021906221/5106267795247605, 169813036807657091/65246755161497175, 7030259765336882929/2701215663685983045, 2425439620182471292949/931919403971664150525, 26859498017024679392797/10320144510649169666925, 116758237880282463017640007/44861668187791940542122975, 40281592068703525738423134271/15477275524788219487032426375, 6862789759853255159111636441/2636869163482437394087005975, 362252357473854257372627183826829/139187138794420457846882610390375, 5768172153622141022917072388409899/2216287517726541136484976950062125, 3640998444969709243963469638499387563/1398969932022720021819017117033659125, 3768433390543649067746827437640524312869/1447933879643515222582682716129837194375, 364030665526516499945419930467966744637867/139870412773563570501487150378142272976625, 1550500982798125833101062000139838374571419/595744350702215207691519344203198570085625, 22145805537305631274182586951996419634565196953/8509016561079739811457970793254285176532981875, 1528060582074088557918598825298750502144043043041/587122142714502046990599984734545677180775749375, 32542030914540774844562753026304573510174952878549/12503527113364395445170184860087546828849853921875, 52542363114617535064031021052030936670820656669244161/20188194877238152885771780475097353109860974142259375]] After derived = [52542363114617535064031021052030936670820656669244161/20188194877238152885771780475097353109860974142259375, No, finite sum is not a general infinite-value certificate] Conditions: none This calculation includes degree0 through the cutoff.No withholds a general exact infinite-value certificate.At x=0 the finite sum can equal E(0)=1,so do not assert universal inequality. Step identifiers and dependenciesRule: Exp_finite_sum_No_is_not_strict_inequality_at_every_x · Uses: G001, S434 ## Reverse checks - S435 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S434 · premise_and_result_audit — Recompute Exp cumulative sum through26 with exact rational arithmetic, not rounded decimals. - S433 · premise_and_result_audit — Recompute Exp term26 power over factorial with exact rational arithmetic, not rounded decimals. - S432 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S431 · premise_and_result_audit — Recompute Exp term26 factorial step26 with exact rational arithmetic, not rounded decimals. - S430 · premise_and_result_audit — Recompute Exp term26 factorial step25 with exact rational arithmetic, not rounded decimals. - S429 · premise_and_result_audit — Recompute Exp term26 factorial step24 with exact rational arithmetic, not rounded decimals. - S428 · premise_and_result_audit — Recompute Exp term26 factorial step23 with exact rational arithmetic, not rounded decimals. - S427 · premise_and_result_audit — Recompute Exp term26 factorial step22 with exact rational arithmetic, not rounded decimals. - S426 · premise_and_result_audit — Recompute Exp term26 factorial step21 with exact rational arithmetic, not rounded decimals. - S425 · premise_and_result_audit — Recompute Exp term26 factorial step20 with exact rational arithmetic, not rounded decimals. - S424 · premise_and_result_audit — Recompute Exp term26 factorial step19 with exact rational arithmetic, not rounded decimals. - S423 · premise_and_result_audit — Recompute Exp term26 factorial step18 with exact rational arithmetic, not rounded decimals. - S422 · premise_and_result_audit — Recompute Exp term26 factorial step17 with exact rational arithmetic, not rounded decimals. - S421 · premise_and_result_audit — Recompute Exp term26 factorial step16 with exact rational arithmetic, not rounded decimals. - S420 · premise_and_result_audit — Recompute Exp term26 factorial step15 with exact rational arithmetic, not rounded decimals. - S419 · premise_and_result_audit — Recompute Exp term26 factorial step14 with exact rational arithmetic, not rounded decimals. - S418 · premise_and_result_audit — Recompute Exp term26 factorial step13 with exact rational arithmetic, not rounded decimals. - S417 · premise_and_result_audit — Recompute Exp term26 factorial step12 with exact rational arithmetic, not rounded decimals. - S416 · premise_and_result_audit — Recompute Exp term26 factorial step11 with exact rational arithmetic, not rounded decimals. - S415 · premise_and_result_audit — Recompute Exp term26 factorial step10 with exact rational arithmetic, not rounded decimals. - S414 · premise_and_result_audit — Recompute Exp term26 factorial step9 with exact rational arithmetic, not rounded decimals. - S413 · premise_and_result_audit — Recompute Exp term26 factorial step8 with exact rational arithmetic, not rounded decimals. - S412 · premise_and_result_audit — Recompute Exp term26 factorial step7 with exact rational arithmetic, not rounded decimals. - S411 · premise_and_result_audit — Recompute Exp term26 factorial step6 with exact rational arithmetic, not rounded decimals. - S410 · premise_and_result_audit — Recompute Exp term26 factorial step5 with exact rational arithmetic, not rounded decimals. - S409 · premise_and_result_audit — Recompute Exp term26 factorial step4 with exact rational arithmetic, not rounded decimals. - S408 · premise_and_result_audit — Recompute Exp term26 factorial step3 with exact rational arithmetic, not rounded decimals. - S407 · premise_and_result_audit — Recompute Exp term26 factorial step2 with exact rational arithmetic, not rounded decimals. - S406 · premise_and_result_audit — Recompute Exp term26 factorial step1 with exact rational arithmetic, not rounded decimals. - S405 · premise_and_result_audit — Recompute Exp cumulative sum through25 with exact rational arithmetic, not rounded decimals. - S404 · premise_and_result_audit — Recompute Exp term25 power over factorial with exact rational arithmetic, not rounded decimals. - S403 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S402 · premise_and_result_audit — Recompute Exp term25 factorial step25 with exact rational arithmetic, not rounded decimals. - S401 · premise_and_result_audit — Recompute Exp term25 factorial step24 with exact rational arithmetic, not rounded decimals. - S400 · premise_and_result_audit — Recompute Exp term25 factorial step23 with exact rational arithmetic, not rounded decimals. - S399 · premise_and_result_audit — Recompute Exp term25 factorial step22 with exact rational arithmetic, not rounded decimals. - S398 · premise_and_result_audit — Recompute Exp term25 factorial step21 with exact rational arithmetic, not rounded decimals. - S397 · premise_and_result_audit — Recompute Exp term25 factorial step20 with exact rational arithmetic, not rounded decimals. - S396 · premise_and_result_audit — Recompute Exp term25 factorial step19 with exact rational arithmetic, not rounded decimals. - S395 · premise_and_result_audit — Recompute Exp term25 factorial step18 with exact rational arithmetic, not rounded decimals. - S394 · premise_and_result_audit — Recompute Exp term25 factorial step17 with exact rational arithmetic, not rounded decimals. - S393 · premise_and_result_audit — Recompute Exp term25 factorial step16 with exact rational arithmetic, not rounded decimals. - S392 · premise_and_result_audit — Recompute Exp term25 factorial step15 with exact rational arithmetic, not rounded decimals. - S391 · premise_and_result_audit — Recompute Exp term25 factorial step14 with exact rational arithmetic, not rounded decimals. - S390 · premise_and_result_audit — Recompute Exp term25 factorial step13 with exact rational arithmetic, not rounded decimals. - S389 · premise_and_result_audit — Recompute Exp term25 factorial step12 with exact rational arithmetic, not rounded decimals. - S388 · premise_and_result_audit — Recompute Exp term25 factorial step11 with exact rational arithmetic, not rounded decimals. - S387 · premise_and_result_audit — Recompute Exp term25 factorial step10 with exact rational arithmetic, not rounded decimals. - S386 · premise_and_result_audit — Recompute Exp term25 factorial step9 with exact rational arithmetic, not rounded decimals. - S385 · premise_and_result_audit — Recompute Exp term25 factorial step8 with exact rational arithmetic, not rounded decimals. - S384 · premise_and_result_audit — Recompute Exp term25 factorial step7 with exact rational arithmetic, not rounded decimals. - S383 · premise_and_result_audit — Recompute Exp term25 factorial step6 with exact rational arithmetic, not rounded decimals. - S382 · premise_and_result_audit — Recompute Exp term25 factorial step5 with exact rational arithmetic, not rounded decimals. - S381 · premise_and_result_audit — Recompute Exp term25 factorial step4 with exact rational arithmetic, not rounded decimals. - S380 · premise_and_result_audit — Recompute Exp term25 factorial step3 with exact rational arithmetic, not rounded decimals. - S379 · premise_and_result_audit — Recompute Exp term25 factorial step2 with exact rational arithmetic, not rounded decimals. - S378 · premise_and_result_audit — Recompute Exp term25 factorial step1 with exact rational arithmetic, not rounded decimals. - S377 · premise_and_result_audit — Recompute Exp cumulative sum through24 with exact rational arithmetic, not rounded decimals. - S376 · premise_and_result_audit — Recompute Exp term24 power over factorial with exact rational arithmetic, not rounded decimals. - S375 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S374 · premise_and_result_audit — Recompute Exp term24 factorial step24 with exact rational arithmetic, not rounded decimals. - S373 · premise_and_result_audit — Recompute Exp term24 factorial step23 with exact rational arithmetic, not rounded decimals. - S372 · premise_and_result_audit — Recompute Exp term24 factorial step22 with exact rational arithmetic, not rounded decimals. - S371 · premise_and_result_audit — Recompute Exp term24 factorial step21 with exact rational arithmetic, not rounded decimals. - S370 · premise_and_result_audit — Recompute Exp term24 factorial step20 with exact rational arithmetic, not rounded decimals. - S369 · premise_and_result_audit — Recompute Exp term24 factorial step19 with exact rational arithmetic, not rounded decimals. - S368 · premise_and_result_audit — Recompute Exp term24 factorial step18 with exact rational arithmetic, not rounded decimals. - S367 · premise_and_result_audit — Recompute Exp term24 factorial step17 with exact rational arithmetic, not rounded decimals. - S366 · premise_and_result_audit — Recompute Exp term24 factorial step16 with exact rational arithmetic, not rounded decimals. - S365 · premise_and_result_audit — Recompute Exp term24 factorial step15 with exact rational arithmetic, not rounded decimals. - S364 · premise_and_result_audit — Recompute Exp term24 factorial step14 with exact rational arithmetic, not rounded decimals. - S363 · premise_and_result_audit — Recompute Exp term24 factorial step13 with exact rational arithmetic, not rounded decimals. - S362 · premise_and_result_audit — Recompute Exp term24 factorial step12 with exact rational arithmetic, not rounded decimals. - S361 · premise_and_result_audit — Recompute Exp term24 factorial step11 with exact rational arithmetic, not rounded decimals. - S360 · premise_and_result_audit — Recompute Exp term24 factorial step10 with exact rational arithmetic, not rounded decimals. - S359 · premise_and_result_audit — Recompute Exp term24 factorial step9 with exact rational arithmetic, not rounded decimals. - S358 · premise_and_result_audit — Recompute Exp term24 factorial step8 with exact rational arithmetic, not rounded decimals. - S357 · premise_and_result_audit — Recompute Exp term24 factorial step7 with exact rational arithmetic, not rounded decimals. - S356 · premise_and_result_audit — Recompute Exp term24 factorial step6 with exact rational arithmetic, not rounded decimals. - S355 · premise_and_result_audit — Recompute Exp term24 factorial step5 with exact rational arithmetic, not rounded decimals. - S354 · premise_and_result_audit — Recompute Exp term24 factorial step4 with exact rational arithmetic, not rounded decimals. - S353 · premise_and_result_audit — Recompute Exp term24 factorial step3 with exact rational arithmetic, not rounded decimals. - S352 · premise_and_result_audit — Recompute Exp term24 factorial step2 with exact rational arithmetic, not rounded decimals. - S351 · premise_and_result_audit — Recompute Exp term24 factorial step1 with exact rational arithmetic, not rounded decimals. - S350 · premise_and_result_audit — Recompute Exp cumulative sum through23 with exact rational arithmetic, not rounded decimals. - S349 · premise_and_result_audit — Recompute Exp term23 power over factorial with exact rational arithmetic, not rounded decimals. - S348 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S347 · premise_and_result_audit — Recompute Exp term23 factorial step23 with exact rational arithmetic, not rounded decimals. - S346 · premise_and_result_audit — Recompute Exp term23 factorial step22 with exact rational arithmetic, not rounded decimals. - S345 · premise_and_result_audit — Recompute Exp term23 factorial step21 with exact rational arithmetic, not rounded decimals. - S344 · premise_and_result_audit — Recompute Exp term23 factorial step20 with exact rational arithmetic, not rounded decimals. - S343 · premise_and_result_audit — Recompute Exp term23 factorial step19 with exact rational arithmetic, not rounded decimals. - S342 · premise_and_result_audit — Recompute Exp term23 factorial step18 with exact rational arithmetic, not rounded decimals. - S341 · premise_and_result_audit — Recompute Exp term23 factorial step17 with exact rational arithmetic, not rounded decimals. - S340 · premise_and_result_audit — Recompute Exp term23 factorial step16 with exact rational arithmetic, not rounded decimals. - S339 · premise_and_result_audit — Recompute Exp term23 factorial step15 with exact rational arithmetic, not rounded decimals. - S338 · premise_and_result_audit — Recompute Exp term23 factorial step14 with exact rational arithmetic, not rounded decimals. - S337 · premise_and_result_audit — Recompute Exp term23 factorial step13 with exact rational arithmetic, not rounded decimals. - S336 · premise_and_result_audit — Recompute Exp term23 factorial step12 with exact rational arithmetic, not rounded decimals. - S335 · premise_and_result_audit — Recompute Exp term23 factorial step11 with exact rational arithmetic, not rounded decimals. - S334 · premise_and_result_audit — Recompute Exp term23 factorial step10 with exact rational arithmetic, not rounded decimals. - S333 · premise_and_result_audit — Recompute Exp term23 factorial step9 with exact rational arithmetic, not rounded decimals. - S332 · premise_and_result_audit — Recompute Exp term23 factorial step8 with exact rational arithmetic, not rounded decimals. - S331 · premise_and_result_audit — Recompute Exp term23 factorial step7 with exact rational arithmetic, not rounded decimals. - S330 · premise_and_result_audit — Recompute Exp term23 factorial step6 with exact rational arithmetic, not rounded decimals. - S329 · premise_and_result_audit — Recompute Exp term23 factorial step5 with exact rational arithmetic, not rounded decimals. - S328 · premise_and_result_audit — Recompute Exp term23 factorial step4 with exact rational arithmetic, not rounded decimals. - S327 · premise_and_result_audit — Recompute Exp term23 factorial step3 with exact rational arithmetic, not rounded decimals. - S326 · premise_and_result_audit — Recompute Exp term23 factorial step2 with exact rational arithmetic, not rounded decimals. - S325 · premise_and_result_audit — Recompute Exp term23 factorial step1 with exact rational arithmetic, not rounded decimals. - S324 · premise_and_result_audit — Recompute Exp cumulative sum through22 with exact rational arithmetic, not rounded decimals. - S323 · premise_and_result_audit — Recompute Exp term22 power over factorial with exact rational arithmetic, not rounded decimals. - S322 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S321 · premise_and_result_audit — Recompute Exp term22 factorial step22 with exact rational arithmetic, not rounded decimals. - S320 · premise_and_result_audit — Recompute Exp term22 factorial step21 with exact rational arithmetic, not rounded decimals. - S319 · premise_and_result_audit — Recompute Exp term22 factorial step20 with exact rational arithmetic, not rounded decimals. - S318 · premise_and_result_audit — Recompute Exp term22 factorial step19 with exact rational arithmetic, not rounded decimals. - S317 · premise_and_result_audit — Recompute Exp term22 factorial step18 with exact rational arithmetic, not rounded decimals. - S316 · premise_and_result_audit — Recompute Exp term22 factorial step17 with exact rational arithmetic, not rounded decimals. - S315 · premise_and_result_audit — Recompute Exp term22 factorial step16 with exact rational arithmetic, not rounded decimals. - S314 · premise_and_result_audit — Recompute Exp term22 factorial step15 with exact rational arithmetic, not rounded decimals. - S313 · premise_and_result_audit — Recompute Exp term22 factorial step14 with exact rational arithmetic, not rounded decimals. - S312 · premise_and_result_audit — Recompute Exp term22 factorial step13 with exact rational arithmetic, not rounded decimals. - S311 · premise_and_result_audit — Recompute Exp term22 factorial step12 with exact rational arithmetic, not rounded decimals. - S310 · premise_and_result_audit — Recompute Exp term22 factorial step11 with exact rational arithmetic, not rounded decimals. - S309 · premise_and_result_audit — Recompute Exp term22 factorial step10 with exact rational arithmetic, not rounded decimals. - S308 · premise_and_result_audit — Recompute Exp term22 factorial step9 with exact rational arithmetic, not rounded decimals. - S307 · premise_and_result_audit — Recompute Exp term22 factorial step8 with exact rational arithmetic, not rounded decimals. - S306 · premise_and_result_audit — Recompute Exp term22 factorial step7 with exact rational arithmetic, not rounded decimals. - S305 · premise_and_result_audit — Recompute Exp term22 factorial step6 with exact rational arithmetic, not rounded decimals. - S304 · premise_and_result_audit — Recompute Exp term22 factorial step5 with exact rational arithmetic, not rounded decimals. - S303 · premise_and_result_audit — Recompute Exp term22 factorial step4 with exact rational arithmetic, not rounded decimals. - S302 · premise_and_result_audit — Recompute Exp term22 factorial step3 with exact rational arithmetic, not rounded decimals. - S301 · premise_and_result_audit — Recompute Exp term22 factorial step2 with exact rational arithmetic, not rounded decimals. - S300 · premise_and_result_audit — Recompute Exp term22 factorial step1 with exact rational arithmetic, not rounded decimals. - S299 · premise_and_result_audit — Recompute Exp cumulative sum through21 with exact rational arithmetic, not rounded decimals. - S298 · premise_and_result_audit — Recompute Exp term21 power over factorial with exact rational arithmetic, not rounded decimals. - S297 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S296 · premise_and_result_audit — Recompute Exp term21 factorial step21 with exact rational arithmetic, not rounded decimals. - S295 · premise_and_result_audit — Recompute Exp term21 factorial step20 with exact rational arithmetic, not rounded decimals. - S294 · premise_and_result_audit — Recompute Exp term21 factorial step19 with exact rational arithmetic, not rounded decimals. - S293 · premise_and_result_audit — Recompute Exp term21 factorial step18 with exact rational arithmetic, not rounded decimals. - S292 · premise_and_result_audit — Recompute Exp term21 factorial step17 with exact rational arithmetic, not rounded decimals. - S291 · premise_and_result_audit — Recompute Exp term21 factorial step16 with exact rational arithmetic, not rounded decimals. - S290 · premise_and_result_audit — Recompute Exp term21 factorial step15 with exact rational arithmetic, not rounded decimals. - S289 · premise_and_result_audit — Recompute Exp term21 factorial step14 with exact rational arithmetic, not rounded decimals. - S288 · premise_and_result_audit — Recompute Exp term21 factorial step13 with exact rational arithmetic, not rounded decimals. - S287 · premise_and_result_audit — Recompute Exp term21 factorial step12 with exact rational arithmetic, not rounded decimals. - S286 · premise_and_result_audit — Recompute Exp term21 factorial step11 with exact rational arithmetic, not rounded decimals. - S285 · premise_and_result_audit — Recompute Exp term21 factorial step10 with exact rational arithmetic, not rounded decimals. - S284 · premise_and_result_audit — Recompute Exp term21 factorial step9 with exact rational arithmetic, not rounded decimals. - S283 · premise_and_result_audit — Recompute Exp term21 factorial step8 with exact rational arithmetic, not rounded decimals. - S282 · premise_and_result_audit — Recompute Exp term21 factorial step7 with exact rational arithmetic, not rounded decimals. - S281 · premise_and_result_audit — Recompute Exp term21 factorial step6 with exact rational arithmetic, not rounded decimals. - S280 · premise_and_result_audit — Recompute Exp term21 factorial step5 with exact rational arithmetic, not rounded decimals. - S279 · premise_and_result_audit — Recompute Exp term21 factorial step4 with exact rational arithmetic, not rounded decimals. - S278 · premise_and_result_audit — Recompute Exp term21 factorial step3 with exact rational arithmetic, not rounded decimals. - S277 · premise_and_result_audit — Recompute Exp term21 factorial step2 with exact rational arithmetic, not rounded decimals. - S276 · premise_and_result_audit — Recompute Exp term21 factorial step1 with exact rational arithmetic, not rounded decimals. - S275 · premise_and_result_audit — Recompute Exp cumulative sum through20 with exact rational arithmetic, not rounded decimals. - S274 · premise_and_result_audit — Recompute Exp term20 power over factorial with exact rational arithmetic, not rounded decimals. - S273 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S272 · premise_and_result_audit — Recompute Exp term20 factorial step20 with exact rational arithmetic, not rounded decimals. - S271 · premise_and_result_audit — Recompute Exp term20 factorial step19 with exact rational arithmetic, not rounded decimals. - S270 · premise_and_result_audit — Recompute Exp term20 factorial step18 with exact rational arithmetic, not rounded decimals. - S269 · premise_and_result_audit — Recompute Exp term20 factorial step17 with exact rational arithmetic, not rounded decimals. - S268 · premise_and_result_audit — Recompute Exp term20 factorial step16 with exact rational arithmetic, not rounded decimals. - S267 · premise_and_result_audit — Recompute Exp term20 factorial step15 with exact rational arithmetic, not rounded decimals. - S266 · premise_and_result_audit — Recompute Exp term20 factorial step14 with exact rational arithmetic, not rounded decimals. - S265 · premise_and_result_audit — Recompute Exp term20 factorial step13 with exact rational arithmetic, not rounded decimals. - S264 · premise_and_result_audit — Recompute Exp term20 factorial step12 with exact rational arithmetic, not rounded decimals. - S263 · premise_and_result_audit — Recompute Exp term20 factorial step11 with exact rational arithmetic, not rounded decimals. - S262 · premise_and_result_audit — Recompute Exp term20 factorial step10 with exact rational arithmetic, not rounded decimals. - S261 · premise_and_result_audit — Recompute Exp term20 factorial step9 with exact rational arithmetic, not rounded decimals. - S260 · premise_and_result_audit — Recompute Exp term20 factorial step8 with exact rational arithmetic, not rounded decimals. - S259 · premise_and_result_audit — Recompute Exp term20 factorial step7 with exact rational arithmetic, not rounded decimals. - S258 · premise_and_result_audit — Recompute Exp term20 factorial step6 with exact rational arithmetic, not rounded decimals. - S257 · premise_and_result_audit — Recompute Exp term20 factorial step5 with exact rational arithmetic, not rounded decimals. - S256 · premise_and_result_audit — Recompute Exp term20 factorial step4 with exact rational arithmetic, not rounded decimals. - S255 · premise_and_result_audit — Recompute Exp term20 factorial step3 with exact rational arithmetic, not rounded decimals. - S254 · premise_and_result_audit — Recompute Exp term20 factorial step2 with exact rational arithmetic, not rounded decimals. - S253 · premise_and_result_audit — Recompute Exp term20 factorial step1 with exact rational arithmetic, not rounded decimals. - S252 · premise_and_result_audit — Recompute Exp cumulative sum through19 with exact rational arithmetic, not rounded decimals. - S251 · premise_and_result_audit — Recompute Exp term19 power over factorial with exact rational arithmetic, not rounded decimals. - S250 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S249 · premise_and_result_audit — Recompute Exp term19 factorial step19 with exact rational arithmetic, not rounded decimals. - S248 · premise_and_result_audit — Recompute Exp term19 factorial step18 with exact rational arithmetic, not rounded decimals. - S247 · premise_and_result_audit — Recompute Exp term19 factorial step17 with exact rational arithmetic, not rounded decimals. - S246 · premise_and_result_audit — Recompute Exp term19 factorial step16 with exact rational arithmetic, not rounded decimals. - S245 · premise_and_result_audit — Recompute Exp term19 factorial step15 with exact rational arithmetic, not rounded decimals. - S244 · premise_and_result_audit — Recompute Exp term19 factorial step14 with exact rational arithmetic, not rounded decimals. - S243 · premise_and_result_audit — Recompute Exp term19 factorial step13 with exact rational arithmetic, not rounded decimals. - S242 · premise_and_result_audit — Recompute Exp term19 factorial step12 with exact rational arithmetic, not rounded decimals. - S241 · premise_and_result_audit — Recompute Exp term19 factorial step11 with exact rational arithmetic, not rounded decimals. - S240 · premise_and_result_audit — Recompute Exp term19 factorial step10 with exact rational arithmetic, not rounded decimals. - S239 · premise_and_result_audit — Recompute Exp term19 factorial step9 with exact rational arithmetic, not rounded decimals. - S238 · premise_and_result_audit — Recompute Exp term19 factorial step8 with exact rational arithmetic, not rounded decimals. - S237 · premise_and_result_audit — Recompute Exp term19 factorial step7 with exact rational arithmetic, not rounded decimals. - S236 · premise_and_result_audit — Recompute Exp term19 factorial step6 with exact rational arithmetic, not rounded decimals. - S235 · premise_and_result_audit — Recompute Exp term19 factorial step5 with exact rational arithmetic, not rounded decimals. - S234 · premise_and_result_audit — Recompute Exp term19 factorial step4 with exact rational arithmetic, not rounded decimals. - S233 · premise_and_result_audit — Recompute Exp term19 factorial step3 with exact rational arithmetic, not rounded decimals. - S232 · premise_and_result_audit — Recompute Exp term19 factorial step2 with exact rational arithmetic, not rounded decimals. - S231 · premise_and_result_audit — Recompute Exp term19 factorial step1 with exact rational arithmetic, not rounded decimals. - S230 · premise_and_result_audit — Recompute Exp cumulative sum through18 with exact rational arithmetic, not rounded decimals. - S229 · premise_and_result_audit — Recompute Exp term18 power over factorial with exact rational arithmetic, not rounded decimals. - S228 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S227 · premise_and_result_audit — Recompute Exp term18 factorial step18 with exact rational arithmetic, not rounded decimals. - S226 · premise_and_result_audit — Recompute Exp term18 factorial step17 with exact rational arithmetic, not rounded decimals. - S225 · premise_and_result_audit — Recompute Exp term18 factorial step16 with exact rational arithmetic, not rounded decimals. - S224 · premise_and_result_audit — Recompute Exp term18 factorial step15 with exact rational arithmetic, not rounded decimals. - S223 · premise_and_result_audit — Recompute Exp term18 factorial step14 with exact rational arithmetic, not rounded decimals. - S222 · premise_and_result_audit — Recompute Exp term18 factorial step13 with exact rational arithmetic, not rounded decimals. - S221 · premise_and_result_audit — Recompute Exp term18 factorial step12 with exact rational arithmetic, not rounded decimals. - S220 · premise_and_result_audit — Recompute Exp term18 factorial step11 with exact rational arithmetic, not rounded decimals. - S219 · premise_and_result_audit — Recompute Exp term18 factorial step10 with exact rational arithmetic, not rounded decimals. - S218 · premise_and_result_audit — Recompute Exp term18 factorial step9 with exact rational arithmetic, not rounded decimals. - S217 · premise_and_result_audit — Recompute Exp term18 factorial step8 with exact rational arithmetic, not rounded decimals. - S216 · premise_and_result_audit — Recompute Exp term18 factorial step7 with exact rational arithmetic, not rounded decimals. - S215 · premise_and_result_audit — Recompute Exp term18 factorial step6 with exact rational arithmetic, not rounded decimals. - S214 · premise_and_result_audit — Recompute Exp term18 factorial step5 with exact rational arithmetic, not rounded decimals. - S213 · premise_and_result_audit — Recompute Exp term18 factorial step4 with exact rational arithmetic, not rounded decimals. - S212 · premise_and_result_audit — Recompute Exp term18 factorial step3 with exact rational arithmetic, not rounded decimals. - S211 · premise_and_result_audit — Recompute Exp term18 factorial step2 with exact rational arithmetic, not rounded decimals. - S210 · premise_and_result_audit — Recompute Exp term18 factorial step1 with exact rational arithmetic, not rounded decimals. - S209 · premise_and_result_audit — Recompute Exp cumulative sum through17 with exact rational arithmetic, not rounded decimals. - S208 · premise_and_result_audit — Recompute Exp term17 power over factorial with exact rational arithmetic, not rounded decimals. - S207 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S206 · premise_and_result_audit — Recompute Exp term17 factorial step17 with exact rational arithmetic, not rounded decimals. - S205 · premise_and_result_audit — Recompute Exp term17 factorial step16 with exact rational arithmetic, not rounded decimals. - S204 · premise_and_result_audit — Recompute Exp term17 factorial step15 with exact rational arithmetic, not rounded decimals. - S203 · premise_and_result_audit — Recompute Exp term17 factorial step14 with exact rational arithmetic, not rounded decimals. - S202 · premise_and_result_audit — Recompute Exp term17 factorial step13 with exact rational arithmetic, not rounded decimals. - S201 · premise_and_result_audit — Recompute Exp term17 factorial step12 with exact rational arithmetic, not rounded decimals. - S200 · premise_and_result_audit — Recompute Exp term17 factorial step11 with exact rational arithmetic, not rounded decimals. - S199 · premise_and_result_audit — Recompute Exp term17 factorial step10 with exact rational arithmetic, not rounded decimals. - S198 · premise_and_result_audit — Recompute Exp term17 factorial step9 with exact rational arithmetic, not rounded decimals. - S197 · premise_and_result_audit — Recompute Exp term17 factorial step8 with exact rational arithmetic, not rounded decimals. - S196 · premise_and_result_audit — Recompute Exp term17 factorial step7 with exact rational arithmetic, not rounded decimals. - S195 · premise_and_result_audit — Recompute Exp term17 factorial step6 with exact rational arithmetic, not rounded decimals. - S194 · premise_and_result_audit — Recompute Exp term17 factorial step5 with exact rational arithmetic, not rounded decimals. - S193 · premise_and_result_audit — Recompute Exp term17 factorial step4 with exact rational arithmetic, not rounded decimals. - S192 · premise_and_result_audit — Recompute Exp term17 factorial step3 with exact rational arithmetic, not rounded decimals. - S191 · premise_and_result_audit — Recompute Exp term17 factorial step2 with exact rational arithmetic, not rounded decimals. - S190 · premise_and_result_audit — Recompute Exp term17 factorial step1 with exact rational arithmetic, not rounded decimals. - S189 · premise_and_result_audit — Recompute Exp cumulative sum through16 with exact rational arithmetic, not rounded decimals. - S188 · premise_and_result_audit — Recompute Exp term16 power over factorial with exact rational arithmetic, not rounded decimals. - S187 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S186 · premise_and_result_audit — Recompute Exp term16 factorial step16 with exact rational arithmetic, not rounded decimals. - S185 · premise_and_result_audit — Recompute Exp term16 factorial step15 with exact rational arithmetic, not rounded decimals. - S184 · premise_and_result_audit — Recompute Exp term16 factorial step14 with exact rational arithmetic, not rounded decimals. - S183 · premise_and_result_audit — Recompute Exp term16 factorial step13 with exact rational arithmetic, not rounded decimals. - S182 · premise_and_result_audit — Recompute Exp term16 factorial step12 with exact rational arithmetic, not rounded decimals. - S181 · premise_and_result_audit — Recompute Exp term16 factorial step11 with exact rational arithmetic, not rounded decimals. - S180 · premise_and_result_audit — Recompute Exp term16 factorial step10 with exact rational arithmetic, not rounded decimals. - S179 · premise_and_result_audit — Recompute Exp term16 factorial step9 with exact rational arithmetic, not rounded decimals. - S178 · premise_and_result_audit — Recompute Exp term16 factorial step8 with exact rational arithmetic, not rounded decimals. - S177 · premise_and_result_audit — Recompute Exp term16 factorial step7 with exact rational arithmetic, not rounded decimals. - S176 · premise_and_result_audit — Recompute Exp term16 factorial step6 with exact rational arithmetic, not rounded decimals. - S175 · premise_and_result_audit — Recompute Exp term16 factorial step5 with exact rational arithmetic, not rounded decimals. - S174 · premise_and_result_audit — Recompute Exp term16 factorial step4 with exact rational arithmetic, not rounded decimals. - S173 · premise_and_result_audit — Recompute Exp term16 factorial step3 with exact rational arithmetic, not rounded decimals. - S172 · premise_and_result_audit — Recompute Exp term16 factorial step2 with exact rational arithmetic, not rounded decimals. - S171 · premise_and_result_audit — Recompute Exp term16 factorial step1 with exact rational arithmetic, not rounded decimals. - S170 · premise_and_result_audit — Recompute Exp cumulative sum through15 with exact rational arithmetic, not rounded decimals. - S169 · premise_and_result_audit — Recompute Exp term15 power over factorial with exact rational arithmetic, not rounded decimals. - S168 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S167 · premise_and_result_audit — Recompute Exp term15 factorial step15 with exact rational arithmetic, not rounded decimals. - S166 · premise_and_result_audit — Recompute Exp term15 factorial step14 with exact rational arithmetic, not rounded decimals. - S165 · premise_and_result_audit — Recompute Exp term15 factorial step13 with exact rational arithmetic, not rounded decimals. - S164 · premise_and_result_audit — Recompute Exp term15 factorial step12 with exact rational arithmetic, not rounded decimals. - S163 · premise_and_result_audit — Recompute Exp term15 factorial step11 with exact rational arithmetic, not rounded decimals. - S162 · premise_and_result_audit — Recompute Exp term15 factorial step10 with exact rational arithmetic, not rounded decimals. - S161 · premise_and_result_audit — Recompute Exp term15 factorial step9 with exact rational arithmetic, not rounded decimals. - S160 · premise_and_result_audit — Recompute Exp term15 factorial step8 with exact rational arithmetic, not rounded decimals. - S159 · premise_and_result_audit — Recompute Exp term15 factorial step7 with exact rational arithmetic, not rounded decimals. - S158 · premise_and_result_audit — Recompute Exp term15 factorial step6 with exact rational arithmetic, not rounded decimals. - S157 · premise_and_result_audit — Recompute Exp term15 factorial step5 with exact rational arithmetic, not rounded decimals. - S156 · premise_and_result_audit — Recompute Exp term15 factorial step4 with exact rational arithmetic, not rounded decimals. - S155 · premise_and_result_audit — Recompute Exp term15 factorial step3 with exact rational arithmetic, not rounded decimals. - S154 · premise_and_result_audit — Recompute Exp term15 factorial step2 with exact rational arithmetic, not rounded decimals. - S153 · premise_and_result_audit — Recompute Exp term15 factorial step1 with exact rational arithmetic, not rounded decimals. - S152 · premise_and_result_audit — Recompute Exp cumulative sum through14 with exact rational arithmetic, not rounded decimals. - S151 · premise_and_result_audit — Recompute Exp term14 power over factorial with exact rational arithmetic, not rounded decimals. - S150 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S149 · premise_and_result_audit — Recompute Exp term14 factorial step14 with exact rational arithmetic, not rounded decimals. - S148 · premise_and_result_audit — Recompute Exp term14 factorial step13 with exact rational arithmetic, not rounded decimals. - S147 · premise_and_result_audit — Recompute Exp term14 factorial step12 with exact rational arithmetic, not rounded decimals. - S146 · premise_and_result_audit — Recompute Exp term14 factorial step11 with exact rational arithmetic, not rounded decimals. - S145 · premise_and_result_audit — Recompute Exp term14 factorial step10 with exact rational arithmetic, not rounded decimals. - S144 · premise_and_result_audit — Recompute Exp term14 factorial step9 with exact rational arithmetic, not rounded decimals. - S143 · premise_and_result_audit — Recompute Exp term14 factorial step8 with exact rational arithmetic, not rounded decimals. - S142 · premise_and_result_audit — Recompute Exp term14 factorial step7 with exact rational arithmetic, not rounded decimals. - S141 · premise_and_result_audit — Recompute Exp term14 factorial step6 with exact rational arithmetic, not rounded decimals. - S140 · premise_and_result_audit — Recompute Exp term14 factorial step5 with exact rational arithmetic, not rounded decimals. - S139 · premise_and_result_audit — Recompute Exp term14 factorial step4 with exact rational arithmetic, not rounded decimals. - S138 · premise_and_result_audit — Recompute Exp term14 factorial step3 with exact rational arithmetic, not rounded decimals. - S137 · premise_and_result_audit — Recompute Exp term14 factorial step2 with exact rational arithmetic, not rounded decimals. - S136 · premise_and_result_audit — Recompute Exp term14 factorial step1 with exact rational arithmetic, not rounded decimals. - S135 · premise_and_result_audit — Recompute Exp cumulative sum through13 with exact rational arithmetic, not rounded decimals. - S134 · premise_and_result_audit — Recompute Exp term13 power over factorial with exact rational arithmetic, not rounded decimals. - S133 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S132 · premise_and_result_audit — Recompute Exp term13 factorial step13 with exact rational arithmetic, not rounded decimals. - S131 · premise_and_result_audit — Recompute Exp term13 factorial step12 with exact rational arithmetic, not rounded decimals. - S130 · premise_and_result_audit — Recompute Exp term13 factorial step11 with exact rational arithmetic, not rounded decimals. - S129 · premise_and_result_audit — Recompute Exp term13 factorial step10 with exact rational arithmetic, not rounded decimals. - S128 · premise_and_result_audit — Recompute Exp term13 factorial step9 with exact rational arithmetic, not rounded decimals. - S127 · premise_and_result_audit — Recompute Exp term13 factorial step8 with exact rational arithmetic, not rounded decimals. - S126 · premise_and_result_audit — Recompute Exp term13 factorial step7 with exact rational arithmetic, not rounded decimals. - S125 · premise_and_result_audit — Recompute Exp term13 factorial step6 with exact rational arithmetic, not rounded decimals. - S124 · premise_and_result_audit — Recompute Exp term13 factorial step5 with exact rational arithmetic, not rounded decimals. - S123 · premise_and_result_audit — Recompute Exp term13 factorial step4 with exact rational arithmetic, not rounded decimals. - S122 · premise_and_result_audit — Recompute Exp term13 factorial step3 with exact rational arithmetic, not rounded decimals. - S121 · premise_and_result_audit — Recompute Exp term13 factorial step2 with exact rational arithmetic, not rounded decimals. - S120 · premise_and_result_audit — Recompute Exp term13 factorial step1 with exact rational arithmetic, not rounded decimals. - S119 · premise_and_result_audit — Recompute Exp cumulative sum through12 with exact rational arithmetic, not rounded decimals. - S118 · premise_and_result_audit — Recompute Exp term12 power over factorial with exact rational arithmetic, not rounded decimals. - S117 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S116 · premise_and_result_audit — Recompute Exp term12 factorial step12 with exact rational arithmetic, not rounded decimals. - S115 · premise_and_result_audit — Recompute Exp term12 factorial step11 with exact rational arithmetic, not rounded decimals. - S114 · premise_and_result_audit — Recompute Exp term12 factorial step10 with exact rational arithmetic, not rounded decimals. - S113 · premise_and_result_audit — Recompute Exp term12 factorial step9 with exact rational arithmetic, not rounded decimals. - S112 · premise_and_result_audit — Recompute Exp term12 factorial step8 with exact rational arithmetic, not rounded decimals. - S111 · premise_and_result_audit — Recompute Exp term12 factorial step7 with exact rational arithmetic, not rounded decimals. - S110 · premise_and_result_audit — Recompute Exp term12 factorial step6 with exact rational arithmetic, not rounded decimals. - S109 · premise_and_result_audit — Recompute Exp term12 factorial step5 with exact rational arithmetic, not rounded decimals. - S108 · premise_and_result_audit — Recompute Exp term12 factorial step4 with exact rational arithmetic, not rounded decimals. - S107 · premise_and_result_audit — Recompute Exp term12 factorial step3 with exact rational arithmetic, not rounded decimals. - S106 · premise_and_result_audit — Recompute Exp term12 factorial step2 with exact rational arithmetic, not rounded decimals. - S105 · premise_and_result_audit — Recompute Exp term12 factorial step1 with exact rational arithmetic, not rounded decimals. - S104 · premise_and_result_audit — Recompute Exp cumulative sum through11 with exact rational arithmetic, not rounded decimals. - S103 · premise_and_result_audit — Recompute Exp term11 power over factorial with exact rational arithmetic, not rounded decimals. - S102 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S101 · premise_and_result_audit — Recompute Exp term11 factorial step11 with exact rational arithmetic, not rounded decimals. - S100 · premise_and_result_audit — Recompute Exp term11 factorial step10 with exact rational arithmetic, not rounded decimals. - S099 · premise_and_result_audit — Recompute Exp term11 factorial step9 with exact rational arithmetic, not rounded decimals. - S098 · premise_and_result_audit — Recompute Exp term11 factorial step8 with exact rational arithmetic, not rounded decimals. - S097 · premise_and_result_audit — Recompute Exp term11 factorial step7 with exact rational arithmetic, not rounded decimals. - S096 · premise_and_result_audit — Recompute Exp term11 factorial step6 with exact rational arithmetic, not rounded decimals. - S095 · premise_and_result_audit — Recompute Exp term11 factorial step5 with exact rational arithmetic, not rounded decimals. - S094 · premise_and_result_audit — Recompute Exp term11 factorial step4 with exact rational arithmetic, not rounded decimals. - S093 · premise_and_result_audit — Recompute Exp term11 factorial step3 with exact rational arithmetic, not rounded decimals. - S092 · premise_and_result_audit — Recompute Exp term11 factorial step2 with exact rational arithmetic, not rounded decimals. - S091 · premise_and_result_audit — Recompute Exp term11 factorial step1 with exact rational arithmetic, not rounded decimals. - S090 · premise_and_result_audit — Recompute Exp cumulative sum through10 with exact rational arithmetic, not rounded decimals. - S089 · premise_and_result_audit — Recompute Exp term10 power over factorial with exact rational arithmetic, not rounded decimals. - S088 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S087 · premise_and_result_audit — Recompute Exp term10 factorial step10 with exact rational arithmetic, not rounded decimals. - S086 · premise_and_result_audit — Recompute Exp term10 factorial step9 with exact rational arithmetic, not rounded decimals. - S085 · premise_and_result_audit — Recompute Exp term10 factorial step8 with exact rational arithmetic, not rounded decimals. - S084 · premise_and_result_audit — Recompute Exp term10 factorial step7 with exact rational arithmetic, not rounded decimals. - S083 · premise_and_result_audit — Recompute Exp term10 factorial step6 with exact rational arithmetic, not rounded decimals. - S082 · premise_and_result_audit — Recompute Exp term10 factorial step5 with exact rational arithmetic, not rounded decimals. - S081 · premise_and_result_audit — Recompute Exp term10 factorial step4 with exact rational arithmetic, not rounded decimals. - S080 · premise_and_result_audit — Recompute Exp term10 factorial step3 with exact rational arithmetic, not rounded decimals. - S079 · premise_and_result_audit — Recompute Exp term10 factorial step2 with exact rational arithmetic, not rounded decimals. - S078 · premise_and_result_audit — Recompute Exp term10 factorial step1 with exact rational arithmetic, not rounded decimals. - S077 · premise_and_result_audit — Recompute Exp cumulative sum through9 with exact rational arithmetic, not rounded decimals. - S076 · premise_and_result_audit — Recompute Exp term9 power over factorial with exact rational arithmetic, not rounded decimals. - S075 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S074 · premise_and_result_audit — Recompute Exp term9 factorial step9 with exact rational arithmetic, not rounded decimals. - S073 · premise_and_result_audit — Recompute Exp term9 factorial step8 with exact rational arithmetic, not rounded decimals. - S072 · premise_and_result_audit — Recompute Exp term9 factorial step7 with exact rational arithmetic, not rounded decimals. - S071 · premise_and_result_audit — Recompute Exp term9 factorial step6 with exact rational arithmetic, not rounded decimals. - S070 · premise_and_result_audit — Recompute Exp term9 factorial step5 with exact rational arithmetic, not rounded decimals. - S069 · premise_and_result_audit — Recompute Exp term9 factorial step4 with exact rational arithmetic, not rounded decimals. - S068 · premise_and_result_audit — Recompute Exp term9 factorial step3 with exact rational arithmetic, not rounded decimals. - S067 · premise_and_result_audit — Recompute Exp term9 factorial step2 with exact rational arithmetic, not rounded decimals. - S066 · premise_and_result_audit — Recompute Exp term9 factorial step1 with exact rational arithmetic, not rounded decimals. - S065 · premise_and_result_audit — Recompute Exp cumulative sum through8 with exact rational arithmetic, not rounded decimals. - S064 · premise_and_result_audit — Recompute Exp term8 power over factorial with exact rational arithmetic, not rounded decimals. - S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S062 · premise_and_result_audit — Recompute Exp term8 factorial step8 with exact rational arithmetic, not rounded decimals. - S061 · premise_and_result_audit — Recompute Exp term8 factorial step7 with exact rational arithmetic, not rounded decimals. - S060 · premise_and_result_audit — Recompute Exp term8 factorial step6 with exact rational arithmetic, not rounded decimals. - S059 · premise_and_result_audit — Recompute Exp term8 factorial step5 with exact rational arithmetic, not rounded decimals. - S058 · premise_and_result_audit — Recompute Exp term8 factorial step4 with exact rational arithmetic, not rounded decimals. - S057 · premise_and_result_audit — Recompute Exp term8 factorial step3 with exact rational arithmetic, not rounded decimals. - S056 · premise_and_result_audit — Recompute Exp term8 factorial step2 with exact rational arithmetic, not rounded decimals. - S055 · premise_and_result_audit — Recompute Exp term8 factorial step1 with exact rational arithmetic, not rounded decimals. - S054 · premise_and_result_audit — Recompute Exp cumulative sum through7 with exact rational arithmetic, not rounded decimals. - S053 · premise_and_result_audit — Recompute Exp term7 power over factorial with exact rational arithmetic, not rounded decimals. - S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S051 · premise_and_result_audit — Recompute Exp term7 factorial step7 with exact rational arithmetic, not rounded decimals. - S050 · premise_and_result_audit — Recompute Exp term7 factorial step6 with exact rational arithmetic, not rounded decimals. - S049 · premise_and_result_audit — Recompute Exp term7 factorial step5 with exact rational arithmetic, not rounded decimals. - S048 · premise_and_result_audit — Recompute Exp term7 factorial step4 with exact rational arithmetic, not rounded decimals. - S047 · premise_and_result_audit — Recompute Exp term7 factorial step3 with exact rational arithmetic, not rounded decimals. - S046 · premise_and_result_audit — Recompute Exp term7 factorial step2 with exact rational arithmetic, not rounded decimals. - S045 · premise_and_result_audit — Recompute Exp term7 factorial step1 with exact rational arithmetic, not rounded decimals. - S044 · premise_and_result_audit — Recompute Exp cumulative sum through6 with exact rational arithmetic, not rounded decimals. - S043 · premise_and_result_audit — Recompute Exp term6 power over factorial with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S041 · premise_and_result_audit — Recompute Exp term6 factorial step6 with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute Exp term6 factorial step5 with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Recompute Exp term6 factorial step4 with exact rational arithmetic, not rounded decimals. - S038 · premise_and_result_audit — Recompute Exp term6 factorial step3 with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Recompute Exp term6 factorial step2 with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute Exp term6 factorial step1 with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute Exp cumulative sum through5 with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute Exp term5 power over factorial with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S032 · premise_and_result_audit — Recompute Exp term5 factorial step5 with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute Exp term5 factorial step4 with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute Exp term5 factorial step3 with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute Exp term5 factorial step2 with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute Exp term5 factorial step1 with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute Exp cumulative sum through4 with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute Exp term4 power over factorial with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S024 · premise_and_result_audit — Recompute Exp term4 factorial step4 with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute Exp term4 factorial step3 with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute Exp term4 factorial step2 with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute Exp term4 factorial step1 with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute Exp cumulative sum through3 with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute Exp term3 power over factorial with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S017 · premise_and_result_audit — Recompute Exp term3 factorial step3 with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute Exp term3 factorial step2 with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute Exp term3 factorial step1 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute Exp cumulative sum through2 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Exp term2 power over factorial with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Recompute Exp term2 factorial step2 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Exp term2 factorial step1 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Exp cumulative sum through1 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Exp term1 power over factorial with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Recompute Exp term1 factorial step1 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Exp cumulative sum through0 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Exp term0 power over factorial with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result (22/23, 26, [1, 22/23, 242/529, 5324/36501, 29282/839523, 644204/96545145, 7086244/6661615005, 155897368/1072520015805, 428717762/24667960363515, 9431790764/5106267795247605, 103749698404/587220796453474575, 207499396808/13506078318429915225, 1141246682444/931919403971664150525, 25107427013768/278643901787527581006975, 276181697151448/44861668187791940542122975, 6075997337331856/15477275524788219487032426375, 8354496338831302/355977337070129048201745806625, 183798919454288644/139187138794420457846882610390375, 2021788113997175084/28811737730445034774304700350807625, 44479338507937851848/12590729388204480196371154053302932125, 244636361793658185164/1447933879643515222582682716129837194375, 5381999959460480073608/699352063867817852507435751890711364883125, 5381999959460480073608/16085097468959810607671022293486361392311875, 118403999108130561619376/8509016561079739811457970793254285176532981875, 325610997547359044453284/587122142714502046990599984734545677180775749375, 7163441946041898977972248/337595232060838677019594991222363764378946055890625, 78797861406460888757694728/100940974386190764428858902375486765549304870711296875], [1, 45/23, 1277/529, 93437/36501, 726111/279841, 251152499/96545145, 3467321735/1332323001, 310149988227/119168890645, 64201476280751/24667960363515, 13289715021906221/5106267795247605, 169813036807657091/65246755161497175, 7030259765336882929/2701215663685983045, 2425439620182471292949/931919403971664150525, 26859498017024679392797/10320144510649169666925, 116758237880282463017640007/44861668187791940542122975, 40281592068703525738423134271/15477275524788219487032426375, 6862789759853255159111636441/2636869163482437394087005975, 362252357473854257372627183826829/139187138794420457846882610390375, 5768172153622141022917072388409899/2216287517726541136484976950062125, 3640998444969709243963469638499387563/1398969932022720021819017117033659125, 3768433390543649067746827437640524312869/1447933879643515222582682716129837194375, 364030665526516499945419930467966744637867/139870412773563570501487150378142272976625, 1550500982798125833101062000139838374571419/595744350702215207691519344203198570085625, 22145805537305631274182586951996419634565196953/8509016561079739811457970793254285176532981875, 1528060582074088557918598825298750502144043043041/587122142714502046990599984734545677180775749375, 32542030914540774844562753026304573510174952878549/12503527113364395445170184860087546828849853921875, 52542363114617535064031021052030936670820656669244161/20188194877238152885771780475097353109860974142259375], 52542363114617535064031021052030936670820656669244161/20188194877238152885771780475097353109860974142259375, No, finite_partial_sum_is_not_the_infinite_exponential_value) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 1e8e154bfa1ee56b939ad6f5f524d5317def42d24a96c107b2757d3a187901d4 - candidate record SHA-256: c182de2625f129557bf8d34f44aee4bdb2e2021dabe7024bb182de547af36f70 - semantic review receipt SHA-256: 9b43b9823e9bd2478d6caa5b57ffed161c2c8956af86974e901d5662ad994386 - question-first witness SHA-256: eb97596068e2d5648033bab2b8f6ca9b653bbe8cb77c0f8e1d16d0c40e46f50b - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261113-r02-real-exponential-function-0022.json)