← Question derivation index · Original question batch
R0054-27795 · q-20261015-advance-20261118-r04-wilson-theorem-0045
Use prime modulus p=71. Give (p, all_inverse_pairs, self_inverse_residues, distinct_cancelling_pairs, self_inverse_scope, factorial_residue, Wilson_congruence).
G001 — Use prime modulus p=71. Give (p, all_inverse_pairs, self_inverse_residues, distinct_cancelling_pairs, self_inverse_scope, factorial_residue, Wilson_congruence).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.
Before
source roles = [wilson-prime-inverse-pairing]
After
derived = [[p, all_inverse_pairs, self_inverse_residues, distinct_cancelling_pairs, self_inverse_scope, factorial_residue, Wilson_congruence]]
Conditions: none
List the complete operation/mode field order without JavaScript or answer-shape inference;Boolean,string,null and an unclaimed result remain distinct.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
source roles = [71, 1]
After
derived = [1, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S001
Before
source roles = [71, 2]
After
derived = [36, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S002
Before
source roles = [71, 3]
After
derived = [24, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S003
Before
source roles = [71, 4]
After
derived = [18, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S004
Before
source roles = [71, 5]
After
derived = [57, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S005
Before
source roles = [71, 6]
After
derived = [12, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S006
Before
source roles = [71, 7]
After
derived = [61, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S007
Before
source roles = [71, 8]
After
derived = [9, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S008
Before
source roles = [71, 9]
After
derived = [8, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S009
Before
source roles = [71, 10]
After
derived = [64, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S010
Before
source roles = [71, 11]
After
derived = [13, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S011
Before
source roles = [71, 12]
After
derived = [6, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S012
Before
source roles = [71, 13]
After
derived = [11, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S013
Before
source roles = [71, 14]
After
derived = [66, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S014
Before
source roles = [71, 15]
After
derived = [19, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S015
Before
source roles = [71, 16]
After
derived = [40, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S016
Before
source roles = [71, 17]
After
derived = [46, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S017
Before
source roles = [71, 18]
After
derived = [4, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S018
Before
source roles = [71, 19]
After
derived = [15, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S019
Before
source roles = [71, 20]
After
derived = [32, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S020
Before
source roles = [71, 21]
After
derived = [44, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S021
Before
source roles = [71, 22]
After
derived = [42, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S022
Before
source roles = [71, 23]
After
derived = [34, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S023
Before
source roles = [71, 24]
After
derived = [3, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S024
Before
source roles = [71, 25]
After
derived = [54, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S025
Before
source roles = [71, 26]
After
derived = [41, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S026
Before
source roles = [71, 27]
After
derived = [50, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S027
Before
source roles = [71, 28]
After
derived = [33, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S028
Before
source roles = [71, 29]
After
derived = [49, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S029
Before
source roles = [71, 30]
After
derived = [45, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S030
Before
source roles = [71, 31]
After
derived = [55, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S031
Before
source roles = [71, 32]
After
derived = [20, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S032
Before
source roles = [71, 33]
After
derived = [28, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S033
Before
source roles = [71, 34]
After
derived = [23, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S034
Before
source roles = [71, 35]
After
derived = [69, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S035
Before
source roles = [71, 36]
After
derived = [2, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S036
Before
source roles = [71, 37]
After
derived = [48, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S037
Before
source roles = [71, 38]
After
derived = [43, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S038
Before
source roles = [71, 39]
After
derived = [51, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S039
Before
source roles = [71, 40]
After
derived = [16, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S040
Before
source roles = [71, 41]
After
derived = [26, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S041
Before
source roles = [71, 42]
After
derived = [22, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S042
Before
source roles = [71, 43]
After
derived = [38, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S043
Before
source roles = [71, 44]
After
derived = [21, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S044
Before
source roles = [71, 45]
After
derived = [30, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S045
Before
source roles = [71, 46]
After
derived = [17, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S046
Before
source roles = [71, 47]
After
derived = [68, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S047
Before
source roles = [71, 48]
After
derived = [37, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S048
Before
source roles = [71, 49]
After
derived = [29, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S049
Before
source roles = [71, 50]
After
derived = [27, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S050
Before
source roles = [71, 51]
After
derived = [39, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S051
Before
source roles = [71, 52]
After
derived = [56, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S052
Before
source roles = [71, 53]
After
derived = [67, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S053
Before
source roles = [71, 54]
After
derived = [25, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S054
Before
source roles = [71, 55]
After
derived = [31, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S055
Before
source roles = [71, 56]
After
derived = [52, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S056
Before
source roles = [71, 57]
After
derived = [5, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S057
Before
source roles = [71, 58]
After
derived = [60, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S058
Before
source roles = [71, 59]
After
derived = [65, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S059
Before
source roles = [71, 60]
After
derived = [58, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S060
Before
source roles = [71, 61]
After
derived = [7, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S061
Before
source roles = [71, 62]
After
derived = [63, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S062
Before
source roles = [71, 63]
After
derived = [62, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S063
Before
source roles = [71, 64]
After
derived = [10, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S064
Before
source roles = [71, 65]
After
derived = [59, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S065
Before
source roles = [71, 66]
After
derived = [14, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S066
Before
source roles = [71, 67]
After
derived = [53, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S067
Before
source roles = [71, 68]
After
derived = [47, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S068
Before
source roles = [71, 69]
After
derived = [35, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S069
Before
source roles = [71, 70]
After
derived = [70, 1]
Conditions: none
Every nonzero residue at a prime modulus is a unit.For each a find the first source inverse;two inverses coincide by cancellation of a.
Rule: Wilson_EVERY_nonzero_prime_residue_unique_inverse · Uses: G001, S070
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product1 := mul(1, 1)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product1 = 1
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S071
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 1, 1, 1]
After
derived = [0, 1, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S072
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product2 := mul(1, 2)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product2 = 2
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S073
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 2, 1, 2]
After
derived = [0, 2, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S074
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product3 := mul(2, 3)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product3 = 6
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S075
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 3, 2, 6]
After
derived = [0, 6, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S076
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product4 := mul(6, 4)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product4 = 24
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S077
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 4, 6, 24]
After
derived = [0, 24, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S078
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product5 := mul(24, 5)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product5 = 120
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S079
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 5, 24, 120]
After
derived = [1, 49, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S080
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 := mul(49, 6)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 = 294
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S081
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 6, 49, 294]
After
derived = [4, 10, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S082
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product7 := mul(10, 7)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product7 = 70
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S083
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 7, 10, 70]
After
derived = [0, 70, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S084
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 := mul(70, 8)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 = 560
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S085
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 8, 70, 560]
After
derived = [7, 63, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S086
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 := mul(63, 9)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 = 567
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S087
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 9, 63, 567]
After
derived = [7, 70, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S088
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 := mul(70, 10)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 = 700
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S089
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 10, 70, 700]
After
derived = [9, 61, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S090
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product11 := mul(61, 11)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product11 = 671
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S091
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 11, 61, 671]
After
derived = [9, 32, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S092
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product12 := mul(32, 12)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product12 = 384
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S093
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 12, 32, 384]
After
derived = [5, 29, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S094
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product13 := mul(29, 13)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product13 = 377
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S095
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 13, 29, 377]
After
derived = [5, 22, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S096
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product14 := mul(22, 14)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product14 = 308
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S097
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 14, 22, 308]
After
derived = [4, 24, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S098
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product15 := mul(24, 15)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product15 = 360
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S099
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 15, 24, 360]
After
derived = [5, 5, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S100
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product16 := mul(5, 16)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product16 = 80
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S101
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 16, 5, 80]
After
derived = [1, 9, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S102
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product17 := mul(9, 17)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product17 = 153
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S103
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 17, 9, 153]
After
derived = [2, 11, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S104
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product18 := mul(11, 18)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product18 = 198
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S105
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 18, 11, 198]
After
derived = [2, 56, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S106
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product19 := mul(56, 19)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product19 = 1064
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S107
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 19, 56, 1064]
After
derived = [14, 70, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S108
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product20 := mul(70, 20)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product20 = 1400
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S109
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 20, 70, 1400]
After
derived = [19, 51, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S110
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product21 := mul(51, 21)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product21 = 1071
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S111
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 21, 51, 1071]
After
derived = [15, 6, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S112
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product22 := mul(6, 22)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product22 = 132
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S113
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 22, 6, 132]
After
derived = [1, 61, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S114
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product23 := mul(61, 23)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product23 = 1403
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S115
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 23, 61, 1403]
After
derived = [19, 54, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S116
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product24 := mul(54, 24)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product24 = 1296
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S117
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 24, 54, 1296]
After
derived = [18, 18, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S118
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product25 := mul(18, 25)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product25 = 450
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S119
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 25, 18, 450]
After
derived = [6, 24, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S120
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product26 := mul(24, 26)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product26 = 624
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S121
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 26, 24, 624]
After
derived = [8, 56, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S122
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product27 := mul(56, 27)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product27 = 1512
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S123
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 27, 56, 1512]
After
derived = [21, 21, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S124
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product28 := mul(21, 28)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product28 = 588
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S125
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 28, 21, 588]
After
derived = [8, 20, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S126
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product29 := mul(20, 29)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product29 = 580
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S127
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 29, 20, 580]
After
derived = [8, 12, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S128
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product30 := mul(12, 30)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product30 = 360
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S129
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 30, 12, 360]
After
derived = [5, 5, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S130
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product31 := mul(5, 31)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product31 = 155
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S131
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 31, 5, 155]
After
derived = [2, 13, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S132
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product32 := mul(13, 32)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product32 = 416
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S133
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 32, 13, 416]
After
derived = [5, 61, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S134
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product33 := mul(61, 33)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product33 = 2013
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S135
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 33, 61, 2013]
After
derived = [28, 25, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S136
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product34 := mul(25, 34)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product34 = 850
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S137
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 34, 25, 850]
After
derived = [11, 69, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S138
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product35 := mul(69, 35)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product35 = 2415
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S139
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 35, 69, 2415]
After
derived = [34, 1, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S140
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product36 := mul(1, 36)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product36 = 36
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S141
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 36, 1, 36]
After
derived = [0, 36, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S142
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product37 := mul(36, 37)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product37 = 1332
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S143
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 37, 36, 1332]
After
derived = [18, 54, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S144
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product38 := mul(54, 38)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product38 = 2052
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S145
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 38, 54, 2052]
After
derived = [28, 64, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S146
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product39 := mul(64, 39)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product39 = 2496
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S147
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 39, 64, 2496]
After
derived = [35, 11, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S148
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product40 := mul(11, 40)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product40 = 440
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S149
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 40, 11, 440]
After
derived = [6, 14, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S150
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product41 := mul(14, 41)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product41 = 574
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S151
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 41, 14, 574]
After
derived = [8, 6, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S152
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product42 := mul(6, 42)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product42 = 252
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S153
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 42, 6, 252]
After
derived = [3, 39, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S154
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product43 := mul(39, 43)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product43 = 1677
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S155
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 43, 39, 1677]
After
derived = [23, 44, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S156
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product44 := mul(44, 44)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product44 = 1936
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S157
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 44, 44, 1936]
After
derived = [27, 19, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S158
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product45 := mul(19, 45)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product45 = 855
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S159
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 45, 19, 855]
After
derived = [12, 3, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S160
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product46 := mul(3, 46)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product46 = 138
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S161
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 46, 3, 138]
After
derived = [1, 67, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S162
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product47 := mul(67, 47)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product47 = 3149
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S163
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 47, 67, 3149]
After
derived = [44, 25, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S164
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product48 := mul(25, 48)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product48 = 1200
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S165
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 48, 25, 1200]
After
derived = [16, 64, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S166
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product49 := mul(64, 49)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product49 = 3136
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S167
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 49, 64, 3136]
After
derived = [44, 12, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S168
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product50 := mul(12, 50)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product50 = 600
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S169
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 50, 12, 600]
After
derived = [8, 32, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S170
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product51 := mul(32, 51)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product51 = 1632
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S171
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 51, 32, 1632]
After
derived = [22, 70, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S172
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product52 := mul(70, 52)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product52 = 3640
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S173
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 52, 70, 3640]
After
derived = [51, 19, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S174
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product53 := mul(19, 53)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product53 = 1007
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S175
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 53, 19, 1007]
After
derived = [14, 13, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S176
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product54 := mul(13, 54)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product54 = 702
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S177
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 54, 13, 702]
After
derived = [9, 63, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S178
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product55 := mul(63, 55)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product55 = 3465
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S179
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 55, 63, 3465]
After
derived = [48, 57, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S180
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product56 := mul(57, 56)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product56 = 3192
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S181
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 56, 57, 3192]
After
derived = [44, 68, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S182
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product57 := mul(68, 57)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product57 = 3876
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S183
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 57, 68, 3876]
After
derived = [54, 42, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S184
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product58 := mul(42, 58)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product58 = 2436
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S185
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 58, 42, 2436]
After
derived = [34, 22, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S186
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product59 := mul(22, 59)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product59 = 1298
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S187
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 59, 22, 1298]
After
derived = [18, 20, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S188
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product60 := mul(20, 60)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product60 = 1200
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S189
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 60, 20, 1200]
After
derived = [16, 64, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S190
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product61 := mul(64, 61)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product61 = 3904
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S191
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 61, 64, 3904]
After
derived = [54, 70, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S192
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product62 := mul(70, 62)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product62 = 4340
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S193
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 62, 70, 4340]
After
derived = [61, 9, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S194
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product63 := mul(9, 63)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product63 = 567
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S195
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 63, 9, 567]
After
derived = [7, 70, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S196
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product64 := mul(70, 64)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product64 = 4480
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S197
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 64, 70, 4480]
After
derived = [63, 7, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S198
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product65 := mul(7, 65)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product65 = 455
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S199
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 65, 7, 455]
After
derived = [6, 29, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S200
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product66 := mul(29, 66)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product66 = 1914
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S201
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 66, 29, 1914]
After
derived = [26, 68, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S202
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product67 := mul(68, 67)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product67 = 4556
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S203
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 67, 68, 4556]
After
derived = [64, 12, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S204
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product68 := mul(12, 68)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product68 = 816
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S205
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 68, 12, 816]
After
derived = [11, 35, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S206
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product69 := mul(35, 69)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product69 = 2415
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S207
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 69, 35, 2415]
After
derived = [34, 1, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S208
Before
Wilson ORIGINAL prime factorial ORIGINAL running factorial product70 := mul(1, 70)
After
Wilson ORIGINAL prime factorial ORIGINAL running factorial product70 = 70
Conditions: none
Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.
Rule: exact_rational_arithmetic · Uses: G001, S209
Before
source roles = [Wilson ORIGINAL prime factorial, 71, 70, 1, 70]
After
derived = [0, 70, Yes, Yes]
Conditions: none
Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.
Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S210
Before
source roles = [71, [[1, 1], [2, 36], [3, 24], [4, 18], [5, 57], [6, 12], [7, 61], [8, 9], [9, 8], [10, 64], [11, 13], [12, 6], [13, 11], [14, 66], [15, 19], [16, 40], [17, 46], [18, 4], [19, 15], [20, 32], [21, 44], [22, 42], [23, 34], [24, 3], [25, 54], [26, 41], [27, 50], [28, 33], [29, 49], [30, 45], [31, 55], [32, 20], [33, 28], [34, 23], [35, 69], [36, 2], [37, 48], [38, 43], [39, 51], [40, 16], [41, 26], [42, 22], [43, 38], [44, 21], [45, 30], [46, 17], [47, 68], [48, 37], [49, 29], [50, 27], [51, 39], [52, 56], [53, 67], [54, 25], [55, 31], [56, 52], [57, 5], [58, 60], [59, 65], [60, 58], [61, 7], [62, 63], [63, 62], [64, 10], [65, 59], [66, 14], [67, 53], [68, 47], [69, 35], [70, 70]]]
After
derived = [[1, 70], [[2, 36, 1], [3, 24, 1], [4, 18, 1], [5, 57, 1], [6, 12, 1], [7, 61, 1], [8, 9, 1], [10, 64, 1], [11, 13, 1], [14, 66, 1], [15, 19, 1], [16, 40, 1], [17, 46, 1], [20, 32, 1], [21, 44, 1], [22, 42, 1], [23, 34, 1], [25, 54, 1], [26, 41, 1], [27, 50, 1], [28, 33, 1], [29, 49, 1], [30, 45, 1], [31, 55, 1], [35, 69, 1], [37, 48, 1], [38, 43, 1], [39, 51, 1], [47, 68, 1], [52, 56, 1], [53, 67, 1], [58, 60, 1], [59, 65, 1], [62, 63, 1]], 70, Yes]
Conditions: none
Nonself inverse pairs multiply to1.A self-inverse satisfies(r-1)(r+1)=0 modulo a prime,so is±1.For p>2 their product is-1;at p2 only the single1=-1 remains.This general teaching argument does not turn the finite table/program into a Lean proof.
Rule: Wilson_GENERAL_inverse_pairing_and_p2_exception · Uses: G001, S211
S212 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S211 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S210 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product70 with exact rational arithmetic, not rounded decimals.S209 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S208 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product69 with exact rational arithmetic, not rounded decimals.S207 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S206 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product68 with exact rational arithmetic, not rounded decimals.S205 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S204 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product67 with exact rational arithmetic, not rounded decimals.S203 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S202 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product66 with exact rational arithmetic, not rounded decimals.S201 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S200 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product65 with exact rational arithmetic, not rounded decimals.S199 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S198 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product64 with exact rational arithmetic, not rounded decimals.S197 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S196 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product63 with exact rational arithmetic, not rounded decimals.S195 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S194 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product62 with exact rational arithmetic, not rounded decimals.S193 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S192 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product61 with exact rational arithmetic, not rounded decimals.S191 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S190 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product60 with exact rational arithmetic, not rounded decimals.S189 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S188 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product59 with exact rational arithmetic, not rounded decimals.S187 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S186 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product58 with exact rational arithmetic, not rounded decimals.S185 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S184 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product57 with exact rational arithmetic, not rounded decimals.S183 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S182 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product56 with exact rational arithmetic, not rounded decimals.S181 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S180 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product55 with exact rational arithmetic, not rounded decimals.S179 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S178 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product54 with exact rational arithmetic, not rounded decimals.S177 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S176 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product53 with exact rational arithmetic, not rounded decimals.S175 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S174 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product52 with exact rational arithmetic, not rounded decimals.S173 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S172 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product51 with exact rational arithmetic, not rounded decimals.S171 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S170 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product50 with exact rational arithmetic, not rounded decimals.S169 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S168 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product49 with exact rational arithmetic, not rounded decimals.S167 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S166 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product48 with exact rational arithmetic, not rounded decimals.S165 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S164 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product47 with exact rational arithmetic, not rounded decimals.S163 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S162 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product46 with exact rational arithmetic, not rounded decimals.S161 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S160 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product45 with exact rational arithmetic, not rounded decimals.S159 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S158 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product44 with exact rational arithmetic, not rounded decimals.S157 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S156 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product43 with exact rational arithmetic, not rounded decimals.S155 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S154 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product42 with exact rational arithmetic, not rounded decimals.S153 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S152 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product41 with exact rational arithmetic, not rounded decimals.S151 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S150 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product40 with exact rational arithmetic, not rounded decimals.S149 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S148 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product39 with exact rational arithmetic, not rounded decimals.S147 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S146 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product38 with exact rational arithmetic, not rounded decimals.S145 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S144 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product37 with exact rational arithmetic, not rounded decimals.S143 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S142 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product36 with exact rational arithmetic, not rounded decimals.S141 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S140 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product35 with exact rational arithmetic, not rounded decimals.S139 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S138 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product34 with exact rational arithmetic, not rounded decimals.S137 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S136 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product33 with exact rational arithmetic, not rounded decimals.S135 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S134 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product32 with exact rational arithmetic, not rounded decimals.S133 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S132 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product31 with exact rational arithmetic, not rounded decimals.S131 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S130 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product30 with exact rational arithmetic, not rounded decimals.S129 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S128 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product29 with exact rational arithmetic, not rounded decimals.S127 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S126 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product28 with exact rational arithmetic, not rounded decimals.S125 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S124 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product27 with exact rational arithmetic, not rounded decimals.S123 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S122 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product26 with exact rational arithmetic, not rounded decimals.S121 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S120 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product25 with exact rational arithmetic, not rounded decimals.S119 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S118 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product24 with exact rational arithmetic, not rounded decimals.S117 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S116 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product23 with exact rational arithmetic, not rounded decimals.S115 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S114 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product22 with exact rational arithmetic, not rounded decimals.S113 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S112 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product21 with exact rational arithmetic, not rounded decimals.S111 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S110 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product20 with exact rational arithmetic, not rounded decimals.S109 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S108 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product19 with exact rational arithmetic, not rounded decimals.S107 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S106 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product18 with exact rational arithmetic, not rounded decimals.S105 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S104 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product17 with exact rational arithmetic, not rounded decimals.S103 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S102 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product16 with exact rational arithmetic, not rounded decimals.S101 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S100 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product15 with exact rational arithmetic, not rounded decimals.S099 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S098 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product14 with exact rational arithmetic, not rounded decimals.S097 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S096 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product13 with exact rational arithmetic, not rounded decimals.S095 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S094 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product12 with exact rational arithmetic, not rounded decimals.S093 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S092 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product11 with exact rational arithmetic, not rounded decimals.S091 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S090 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 with exact rational arithmetic, not rounded decimals.S089 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S088 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 with exact rational arithmetic, not rounded decimals.S087 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S086 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 with exact rational arithmetic, not rounded decimals.S085 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S084 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product7 with exact rational arithmetic, not rounded decimals.S083 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S082 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 with exact rational arithmetic, not rounded decimals.S081 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S080 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product5 with exact rational arithmetic, not rounded decimals.S079 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S078 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product4 with exact rational arithmetic, not rounded decimals.S077 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S076 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product3 with exact rational arithmetic, not rounded decimals.S075 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S074 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product2 with exact rational arithmetic, not rounded decimals.S073 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S072 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product1 with exact rational arithmetic, not rounded decimals.S071 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S070 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S069 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S068 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S067 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S066 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S065 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S064 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S063 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S062 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S061 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S060 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S059 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S058 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S057 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S056 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S055 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S054 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S053 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S052 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S051 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S050 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S049 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S048 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S047 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S046 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S045 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S044 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S043 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S042 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S041 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S040 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S039 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S038 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S037 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S036 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S035 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S034 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S033 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S032 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S031 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S030 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S029 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S028 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S027 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S026 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S025 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S024 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S023 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S022 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S021 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S020 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S019 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S018 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S017 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S016 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S015 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S014 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S013 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S012 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S011 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S010 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S009 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S008 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S007 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S006 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S005 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S004 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S003 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S002 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S001 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.(71, [[1, 1], [2, 36], [3, 24], [4, 18], [5, 57], [6, 12], [7, 61], [8, 9], [9, 8], [10, 64], [11, 13], [12, 6], [13, 11], [14, 66], [15, 19], [16, 40], [17, 46], [18, 4], [19, 15], [20, 32], [21, 44], [22, 42], [23, 34], [24, 3], [25, 54], [26, 41], [27, 50], [28, 33], [29, 49], [30, 45], [31, 55], [32, 20], [33, 28], [34, 23], [35, 69], [36, 2], [37, 48], [38, 43], [39, 51], [40, 16], [41, 26], [42, 22], [43, 38], [44, 21], [45, 30], [46, 17], [47, 68], [48, 37], [49, 29], [50, 27], [51, 39], [52, 56], [53, 67], [54, 25], [55, 31], [56, 52], [57, 5], [58, 60], [59, 65], [60, 58], [61, 7], [62, 63], [63, 62], [64, 10], [65, 59], [66, 14], [67, 53], [68, 47], [69, 35], [70, 70]], [1, 70], [[2, 36, 1], [3, 24, 1], [4, 18, 1], [5, 57, 1], [6, 12, 1], [7, 61, 1], [8, 9, 1], [10, 64, 1], [11, 13, 1], [14, 66, 1], [15, 19, 1], [16, 40, 1], [17, 46, 1], [20, 32, 1], [21, 44, 1], [22, 42, 1], [23, 34, 1], [25, 54, 1], [26, 41, 1], [27, 50, 1], [28, 33, 1], [29, 49, 1], [30, 45, 1], [31, 55, 1], [35, 69, 1], [37, 48, 1], [38, 43, 1], [39, 51, 1], [47, 68, 1], [52, 56, 1], [53, 67, 1], [58, 60, 1], [59, 65, 1], [62, 63, 1]], self_inverse_residues_1_and_p_minus_1, 70, Yes)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 6c46df0f139a763f95e6e918574e61ecb258e6b890066591e9ff75ebcbb92eadcandidate record SHA-256: e1be73cffc207f1417187bd984a602aed580abccfecf0cdf2a1963b21a96166dsemantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437Lean: NOT_RUN · human review: NOT_PERFORMED