Step-by-step derivation
S001 · implies
Before
source operation=eulerthm-unit-permutation;original snapshot and mathematics retained
After
explicit ordered source output fields=[units, sorted_mapped_units, permutation, certificate]
Conditions: none
Retain the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field labels are not additional mathematical evidence,and No retains the source claim scope.
Step identifiers and dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · calculation
Before
Euler ORIGINAL signed base normalization exact quotient := div(25, 27)
After
Euler ORIGINAL signed base normalization exact quotient = 25/27
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S001
S003 · calculation
Before
Euler ORIGINAL signed base normalization floor quotient multiple := mul(0, 27)
After
Euler ORIGINAL signed base normalization floor quotient multiple = 0
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S002
S004 · calculation
Before
Euler ORIGINAL signed base normalization normalized remainder := sub(25, 0)
After
Euler ORIGINAL signed base normalization normalized remainder = 25
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S003
S005 · implies
Before
Euler ORIGINAL signed base normalization:ORIGINAL signed integer=25;modulus=27;exact ratio=25/27;floor quotient=0
After
value=q*n+remainder;normalized residue=25 in[0,27);same congruence class,NOT absolute-value replacement
Conditions: none
Use the floor quotient for a positive modulus,retaining the congruence class of a negative base rather than replacing it by abs(a).Equal residues mean congruence modulo n,not equality of original integers.
Step identifiers and dependencies
Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S004
S006 · implies
Before
Euler ORIGINAL base applicability gcd:ORIGINAL gcd inputs=[25, 27];nonnegative initial pair=[25, 27]
After
ALL exact Euclidean rows[left,right,q,r]=[[25, 27, 0, 25], [27, 25, 1, 2], [25, 2, 12, 1], [2, 1, 2, 0]];terminal gcd=1
Conditions: none
Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.
Step identifiers and dependencies
Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S005
S007 · implies
Before
Euler residue representative0:ORIGINAL gcd inputs=[0, 27];nonnegative initial pair=[0, 27]
After
ALL exact Euclidean rows[left,right,q,r]=[[0, 27, 0, 0]];terminal gcd=27
Conditions: none
Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.
Step identifiers and dependencies
Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S006
S008 · implies
Before
Euler residue representative1:ORIGINAL gcd inputs=[1, 27];nonnegative initial pair=[1, 27]
After
ALL exact Euclidean rows[left,right,q,r]=[[1, 27, 0, 1], [27, 1, 27, 0]];terminal gcd=1
Conditions: none
Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.
Step identifiers and dependencies
Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S007
S009 · implies
Before
Euler residue representative2:ORIGINAL gcd inputs=[2, 27];nonnegative initial pair=[2, 27]
After
ALL exact Euclidean rows[left,right,q,r]=[[2, 27, 0, 2], [27, 2, 13, 1], [2, 1, 2, 0]];terminal gcd=1
Conditions: none
Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.
Step identifiers and dependencies
Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S008
S010 · implies
Before
Euler residue representative3:ORIGINAL gcd inputs=[3, 27];nonnegative initial pair=[3, 27]
After
ALL exact Euclidean rows[left,right,q,r]=[[3, 27, 0, 3], [27, 3, 9, 0]];terminal gcd=3
Conditions: none
Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.
Step identifiers and dependencies
Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S009
S011 · implies
Before
Euler residue representative4:ORIGINAL gcd inputs=[4, 27];nonnegative initial pair=[4, 27]
After
ALL exact Euclidean rows[left,right,q,r]=[[4, 27, 0, 4], [27, 4, 6, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal gcd=1
Conditions: none
Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.
Step identifiers and dependencies
Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S010
S012 · implies
Before
Euler residue representative5:ORIGINAL gcd inputs=[5, 27];nonnegative initial pair=[5, 27]
After
ALL exact Euclidean rows[left,right,q,r]=[[5, 27, 0, 5], [27, 5, 5, 2], [5, 2, 2, 1], [2, 1, 2, 0]];terminal gcd=1
Conditions: none
Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.
Step identifiers and dependencies
Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S011
S013 · implies
Before
Euler residue representative6:ORIGINAL gcd inputs=[6, 27];nonnegative initial pair=[6, 27]
After
ALL exact Euclidean rows[left,right,q,r]=[[6, 27, 0, 6], [27, 6, 4, 3], [6, 3, 2, 0]];terminal gcd=3
Conditions: none
Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.
Step identifiers and dependencies
Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S012
S014 · implies
Before
Euler residue representative7:ORIGINAL gcd inputs=[7, 27];nonnegative initial pair=[7, 27]
After
ALL exact Euclidean rows[left,right,q,r]=[[7, 27, 0, 7], [27, 7, 3, 6], [7, 6, 1, 1], [6, 1, 6, 0]];terminal gcd=1
Conditions: none
Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.
Step identifiers and dependencies
Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S013
S015 · implies
Before
Euler residue representative8:ORIGINAL gcd inputs=[8, 27];nonnegative initial pair=[8, 27]
After
ALL exact Euclidean rows[left,right,q,r]=[[8, 27, 0, 8], [27, 8, 3, 3], [8, 3, 2, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1
Conditions: none
Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.
Step identifiers and dependencies
Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S014
S016 · implies
Before
Euler residue representative9:ORIGINAL gcd inputs=[9, 27];nonnegative initial pair=[9, 27]
After
ALL exact Euclidean rows[left,right,q,r]=[[9, 27, 0, 9], [27, 9, 3, 0]];terminal gcd=9
Conditions: none
Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.
Step identifiers and dependencies
Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S015
S017 · implies
Before
Euler residue representative10:ORIGINAL gcd inputs=[10, 27];nonnegative initial pair=[10, 27]
After
ALL exact Euclidean rows[left,right,q,r]=[[10, 27, 0, 10], [27, 10, 2, 7], [10, 7, 1, 3], [7, 3, 2, 1], [3, 1, 3, 0]];terminal gcd=1
Conditions: none
Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.
Step identifiers and dependencies
Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S016
S018 · implies
Before
Euler residue representative11:ORIGINAL gcd inputs=[11, 27];nonnegative initial pair=[11, 27]
After
ALL exact Euclidean rows[left,right,q,r]=[[11, 27, 0, 11], [27, 11, 2, 5], [11, 5, 2, 1], [5, 1, 5, 0]];terminal gcd=1
Conditions: none
Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.
Step identifiers and dependencies
Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S017
S019 · implies
Before
Euler residue representative12:ORIGINAL gcd inputs=[12, 27];nonnegative initial pair=[12, 27]
After
ALL exact Euclidean rows[left,right,q,r]=[[12, 27, 0, 12], [27, 12, 2, 3], [12, 3, 4, 0]];terminal gcd=3
Conditions: none
Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.
Step identifiers and dependencies
Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S018
S020 · implies
Before
Euler residue representative13:ORIGINAL gcd inputs=[13, 27];nonnegative initial pair=[13, 27]
After
ALL exact Euclidean rows[left,right,q,r]=[[13, 27, 0, 13], [27, 13, 2, 1], [13, 1, 13, 0]];terminal gcd=1
Conditions: none
Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.
Step identifiers and dependencies
Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S019
S021 · implies
Before
Euler residue representative14:ORIGINAL gcd inputs=[14, 27];nonnegative initial pair=[14, 27]
After
ALL exact Euclidean rows[left,right,q,r]=[[14, 27, 0, 14], [27, 14, 1, 13], [14, 13, 1, 1], [13, 1, 13, 0]];terminal gcd=1
Conditions: none
Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.
Step identifiers and dependencies
Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S020
S022 · implies
Before
Euler residue representative15:ORIGINAL gcd inputs=[15, 27];nonnegative initial pair=[15, 27]
After
ALL exact Euclidean rows[left,right,q,r]=[[15, 27, 0, 15], [27, 15, 1, 12], [15, 12, 1, 3], [12, 3, 4, 0]];terminal gcd=3
Conditions: none
Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.
Step identifiers and dependencies
Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S021
S023 · implies
Before
Euler residue representative16:ORIGINAL gcd inputs=[16, 27];nonnegative initial pair=[16, 27]
After
ALL exact Euclidean rows[left,right,q,r]=[[16, 27, 0, 16], [27, 16, 1, 11], [16, 11, 1, 5], [11, 5, 2, 1], [5, 1, 5, 0]];terminal gcd=1
Conditions: none
Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.
Step identifiers and dependencies
Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S022
S024 · implies
Before
Euler residue representative17:ORIGINAL gcd inputs=[17, 27];nonnegative initial pair=[17, 27]
After
ALL exact Euclidean rows[left,right,q,r]=[[17, 27, 0, 17], [27, 17, 1, 10], [17, 10, 1, 7], [10, 7, 1, 3], [7, 3, 2, 1], [3, 1, 3, 0]];terminal gcd=1
Conditions: none
Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.
Step identifiers and dependencies
Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S023
S025 · implies
Before
Euler residue representative18:ORIGINAL gcd inputs=[18, 27];nonnegative initial pair=[18, 27]
After
ALL exact Euclidean rows[left,right,q,r]=[[18, 27, 0, 18], [27, 18, 1, 9], [18, 9, 2, 0]];terminal gcd=9
Conditions: none
Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.
Step identifiers and dependencies
Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S024
S026 · implies
Before
Euler residue representative19:ORIGINAL gcd inputs=[19, 27];nonnegative initial pair=[19, 27]
After
ALL exact Euclidean rows[left,right,q,r]=[[19, 27, 0, 19], [27, 19, 1, 8], [19, 8, 2, 3], [8, 3, 2, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1
Conditions: none
Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.
Step identifiers and dependencies
Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S025
S027 · implies
Before
Euler residue representative20:ORIGINAL gcd inputs=[20, 27];nonnegative initial pair=[20, 27]
After
ALL exact Euclidean rows[left,right,q,r]=[[20, 27, 0, 20], [27, 20, 1, 7], [20, 7, 2, 6], [7, 6, 1, 1], [6, 1, 6, 0]];terminal gcd=1
Conditions: none
Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.
Step identifiers and dependencies
Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S026
S028 · implies
Before
Euler residue representative21:ORIGINAL gcd inputs=[21, 27];nonnegative initial pair=[21, 27]
After
ALL exact Euclidean rows[left,right,q,r]=[[21, 27, 0, 21], [27, 21, 1, 6], [21, 6, 3, 3], [6, 3, 2, 0]];terminal gcd=3
Conditions: none
Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.
Step identifiers and dependencies
Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S027
S029 · implies
Before
Euler residue representative22:ORIGINAL gcd inputs=[22, 27];nonnegative initial pair=[22, 27]
After
ALL exact Euclidean rows[left,right,q,r]=[[22, 27, 0, 22], [27, 22, 1, 5], [22, 5, 4, 2], [5, 2, 2, 1], [2, 1, 2, 0]];terminal gcd=1
Conditions: none
Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.
Step identifiers and dependencies
Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S028
S030 · implies
Before
Euler residue representative23:ORIGINAL gcd inputs=[23, 27];nonnegative initial pair=[23, 27]
After
ALL exact Euclidean rows[left,right,q,r]=[[23, 27, 0, 23], [27, 23, 1, 4], [23, 4, 5, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal gcd=1
Conditions: none
Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.
Step identifiers and dependencies
Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S029
S031 · implies
Before
Euler residue representative24:ORIGINAL gcd inputs=[24, 27];nonnegative initial pair=[24, 27]
After
ALL exact Euclidean rows[left,right,q,r]=[[24, 27, 0, 24], [27, 24, 1, 3], [24, 3, 8, 0]];terminal gcd=3
Conditions: none
Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.
Step identifiers and dependencies
Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S030
S032 · implies
Before
Euler residue representative25:ORIGINAL gcd inputs=[25, 27];nonnegative initial pair=[25, 27]
After
ALL exact Euclidean rows[left,right,q,r]=[[25, 27, 0, 25], [27, 25, 1, 2], [25, 2, 12, 1], [2, 1, 2, 0]];terminal gcd=1
Conditions: none
Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.
Step identifiers and dependencies
Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S031
S033 · implies
Before
Euler residue representative26:ORIGINAL gcd inputs=[26, 27];nonnegative initial pair=[26, 27]
After
ALL exact Euclidean rows[left,right,q,r]=[[26, 27, 0, 26], [27, 26, 1, 1], [26, 1, 26, 0]];terminal gcd=1
Conditions: none
Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0.
Step identifiers and dependencies
Rule: euler_gcd_full_remainder_chain_unit_test · Uses: G001, S032
S034 · calculation
Before
Euler complete reduced residue count phi := sum(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)
After
Euler complete reduced residue count phi = 18
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S033
S035 · implies
Before
COMPLETE original residue carrier0..26;ALL [representative,gcd,unit] rows=[[0, 27, No], [1, 1, Yes], [2, 1, Yes], [3, 3, No], [4, 1, Yes], [5, 1, Yes], [6, 3, No], [7, 1, Yes], [8, 1, Yes], [9, 9, No], [10, 1, Yes], [11, 1, Yes], [12, 3, No], [13, 1, Yes], [14, 1, Yes], [15, 3, No], [16, 1, Yes], [17, 1, Yes], [18, 9, No], [19, 1, Yes], [20, 1, Yes], [21, 3, No], [22, 1, Yes], [23, 1, Yes], [24, 3, No], [25, 1, Yes], [26, 1, Yes]]
After
increasing COMPLETE U(n)=[1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26];phi(n)=18;source count over1..n agrees because0 and n are BOTH nonunits for n>=2
Conditions: none
Test gcd1 for every representative to obtain the complete unit set,which contains1.The source totient count uses1..n while the unit list uses0..n−1.For n≥2 both swapped endpoints0 and n are nonunits,so the counts agree without assuming a prime modulus.
Step identifiers and dependencies
Rule: euler_full_units_totient_count_convention_not_prime_only · Uses: G001, S034
S036 · calculation
Before
Euler ACTUAL base-unit inverse product := mul(25, 13)
After
Euler ACTUAL base-unit inverse product = 325
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S035
S037 · calculation
Before
Euler ACTUAL inverse check exact quotient := div(325, 27)
After
Euler ACTUAL inverse check exact quotient = 325/27
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S036
S038 · calculation
Before
Euler ACTUAL inverse check floor quotient multiple := mul(12, 27)
After
Euler ACTUAL inverse check floor quotient multiple = 324
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S037
S039 · calculation
Before
Euler ACTUAL inverse check normalized remainder := sub(325, 324)
After
Euler ACTUAL inverse check normalized remainder = 1
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S038
S040 · implies
Before
Euler ACTUAL inverse check:ORIGINAL signed integer=325;modulus=27;exact ratio=325/27;floor quotient=12
After
value=q*n+remainder;normalized residue=1 in[0,27);same congruence class,NOT absolute-value replacement
Conditions: none
Use the floor quotient for a positive modulus,retaining the congruence class of a negative base rather than replacing it by abs(a).Equal residues mean congruence modulo n,not equality of original integers.
Step identifiers and dependencies
Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S039
S041 · implies
Before
ACTUAL modulus n=27;gcd(original a=25,n)=1;normalized base=25;ACTUAL inverse=13;COMPLETE U(n)=[1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26]
After
for ANY r in U(n),a*r remains a unit;actual inverse makes multiplication injective;finite self-injection on U(n) is surjective,so a*U(n)=U(n)
Conditions: none
The reverse product of inverses inverts a product of units,so multiplication by a stays in U(n).Its actual inverse proves injectivity and a finite self-injection is surjective.The modulus may be composite;not every nonzero residue must be a unit.
Step identifiers and dependencies
Rule: euler_all_unit_residue_permutation_composite_allowed · Uses: G001, S040
S042 · calculation
Before
Euler ORIGINAL a times unit representative1 := mul(25, 1)
After
Euler ORIGINAL a times unit representative1 = 25
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S041
S043 · calculation
Before
Euler unit representative1 raw image exact quotient := div(25, 27)
After
Euler unit representative1 raw image exact quotient = 25/27
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S042
S044 · calculation
Before
Euler unit representative1 raw image floor quotient multiple := mul(0, 27)
After
Euler unit representative1 raw image floor quotient multiple = 0
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S043
S045 · calculation
Before
Euler unit representative1 raw image normalized remainder := sub(25, 0)
After
Euler unit representative1 raw image normalized remainder = 25
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S044
S046 · implies
Before
Euler unit representative1 raw image:ORIGINAL signed integer=25;modulus=27;exact ratio=25/27;floor quotient=0
After
value=q*n+remainder;normalized residue=25 in[0,27);same congruence class,NOT absolute-value replacement
Conditions: none
Use the floor quotient for a positive modulus,retaining the congruence class of a negative base rather than replacing it by abs(a).Equal residues mean congruence modulo n,not equality of original integers.
Step identifiers and dependencies
Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S045
S047 · calculation
Before
Euler ORIGINAL a times unit representative2 := mul(25, 2)
After
Euler ORIGINAL a times unit representative2 = 50
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S046
S048 · calculation
Before
Euler unit representative2 raw image exact quotient := div(50, 27)
After
Euler unit representative2 raw image exact quotient = 50/27
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S047
S049 · calculation
Before
Euler unit representative2 raw image floor quotient multiple := mul(1, 27)
After
Euler unit representative2 raw image floor quotient multiple = 27
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S048
S050 · calculation
Before
Euler unit representative2 raw image normalized remainder := sub(50, 27)
After
Euler unit representative2 raw image normalized remainder = 23
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S049
S051 · implies
Before
Euler unit representative2 raw image:ORIGINAL signed integer=50;modulus=27;exact ratio=50/27;floor quotient=1
After
value=q*n+remainder;normalized residue=23 in[0,27);same congruence class,NOT absolute-value replacement
Conditions: none
Use the floor quotient for a positive modulus,retaining the congruence class of a negative base rather than replacing it by abs(a).Equal residues mean congruence modulo n,not equality of original integers.
Step identifiers and dependencies
Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S050
S052 · calculation
Before
Euler ORIGINAL a times unit representative4 := mul(25, 4)
After
Euler ORIGINAL a times unit representative4 = 100
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S051
S053 · calculation
Before
Euler unit representative4 raw image exact quotient := div(100, 27)
After
Euler unit representative4 raw image exact quotient = 100/27
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S052
S054 · calculation
Before
Euler unit representative4 raw image floor quotient multiple := mul(3, 27)
After
Euler unit representative4 raw image floor quotient multiple = 81
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S053
S055 · calculation
Before
Euler unit representative4 raw image normalized remainder := sub(100, 81)
After
Euler unit representative4 raw image normalized remainder = 19
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S054
S056 · implies
Before
Euler unit representative4 raw image:ORIGINAL signed integer=100;modulus=27;exact ratio=100/27;floor quotient=3
After
value=q*n+remainder;normalized residue=19 in[0,27);same congruence class,NOT absolute-value replacement
Conditions: none
Use the floor quotient for a positive modulus,retaining the congruence class of a negative base rather than replacing it by abs(a).Equal residues mean congruence modulo n,not equality of original integers.
Step identifiers and dependencies
Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S055
S057 · calculation
Before
Euler ORIGINAL a times unit representative5 := mul(25, 5)
After
Euler ORIGINAL a times unit representative5 = 125
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S056
S058 · calculation
Before
Euler unit representative5 raw image exact quotient := div(125, 27)
After
Euler unit representative5 raw image exact quotient = 125/27
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S057
S059 · calculation
Before
Euler unit representative5 raw image floor quotient multiple := mul(4, 27)
After
Euler unit representative5 raw image floor quotient multiple = 108
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S058
S060 · calculation
Before
Euler unit representative5 raw image normalized remainder := sub(125, 108)
After
Euler unit representative5 raw image normalized remainder = 17
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S059
S061 · implies
Before
Euler unit representative5 raw image:ORIGINAL signed integer=125;modulus=27;exact ratio=125/27;floor quotient=4
After
value=q*n+remainder;normalized residue=17 in[0,27);same congruence class,NOT absolute-value replacement
Conditions: none
Use the floor quotient for a positive modulus,retaining the congruence class of a negative base rather than replacing it by abs(a).Equal residues mean congruence modulo n,not equality of original integers.
Step identifiers and dependencies
Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S060
S062 · calculation
Before
Euler ORIGINAL a times unit representative7 := mul(25, 7)
After
Euler ORIGINAL a times unit representative7 = 175
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S061
S063 · calculation
Before
Euler unit representative7 raw image exact quotient := div(175, 27)
After
Euler unit representative7 raw image exact quotient = 175/27
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S062
S064 · calculation
Before
Euler unit representative7 raw image floor quotient multiple := mul(6, 27)
After
Euler unit representative7 raw image floor quotient multiple = 162
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S063
S065 · calculation
Before
Euler unit representative7 raw image normalized remainder := sub(175, 162)
After
Euler unit representative7 raw image normalized remainder = 13
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S064
S066 · implies
Before
Euler unit representative7 raw image:ORIGINAL signed integer=175;modulus=27;exact ratio=175/27;floor quotient=6
After
value=q*n+remainder;normalized residue=13 in[0,27);same congruence class,NOT absolute-value replacement
Conditions: none
Use the floor quotient for a positive modulus,retaining the congruence class of a negative base rather than replacing it by abs(a).Equal residues mean congruence modulo n,not equality of original integers.
Step identifiers and dependencies
Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S065
S067 · calculation
Before
Euler ORIGINAL a times unit representative8 := mul(25, 8)
After
Euler ORIGINAL a times unit representative8 = 200
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S066
S068 · calculation
Before
Euler unit representative8 raw image exact quotient := div(200, 27)
After
Euler unit representative8 raw image exact quotient = 200/27
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S067
S069 · calculation
Before
Euler unit representative8 raw image floor quotient multiple := mul(7, 27)
After
Euler unit representative8 raw image floor quotient multiple = 189
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S068
S070 · calculation
Before
Euler unit representative8 raw image normalized remainder := sub(200, 189)
After
Euler unit representative8 raw image normalized remainder = 11
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S069
S071 · implies
Before
Euler unit representative8 raw image:ORIGINAL signed integer=200;modulus=27;exact ratio=200/27;floor quotient=7
After
value=q*n+remainder;normalized residue=11 in[0,27);same congruence class,NOT absolute-value replacement
Conditions: none
Use the floor quotient for a positive modulus,retaining the congruence class of a negative base rather than replacing it by abs(a).Equal residues mean congruence modulo n,not equality of original integers.
Step identifiers and dependencies
Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S070
S072 · calculation
Before
Euler ORIGINAL a times unit representative10 := mul(25, 10)
After
Euler ORIGINAL a times unit representative10 = 250
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S071
S073 · calculation
Before
Euler unit representative10 raw image exact quotient := div(250, 27)
After
Euler unit representative10 raw image exact quotient = 250/27
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S072
S074 · calculation
Before
Euler unit representative10 raw image floor quotient multiple := mul(9, 27)
After
Euler unit representative10 raw image floor quotient multiple = 243
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S073
S075 · calculation
Before
Euler unit representative10 raw image normalized remainder := sub(250, 243)
After
Euler unit representative10 raw image normalized remainder = 7
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S074
S076 · implies
Before
Euler unit representative10 raw image:ORIGINAL signed integer=250;modulus=27;exact ratio=250/27;floor quotient=9
After
value=q*n+remainder;normalized residue=7 in[0,27);same congruence class,NOT absolute-value replacement
Conditions: none
Use the floor quotient for a positive modulus,retaining the congruence class of a negative base rather than replacing it by abs(a).Equal residues mean congruence modulo n,not equality of original integers.
Step identifiers and dependencies
Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S075
S077 · calculation
Before
Euler ORIGINAL a times unit representative11 := mul(25, 11)
After
Euler ORIGINAL a times unit representative11 = 275
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S076
S078 · calculation
Before
Euler unit representative11 raw image exact quotient := div(275, 27)
After
Euler unit representative11 raw image exact quotient = 275/27
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S077
S079 · calculation
Before
Euler unit representative11 raw image floor quotient multiple := mul(10, 27)
After
Euler unit representative11 raw image floor quotient multiple = 270
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S078
S080 · calculation
Before
Euler unit representative11 raw image normalized remainder := sub(275, 270)
After
Euler unit representative11 raw image normalized remainder = 5
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S079
S081 · implies
Before
Euler unit representative11 raw image:ORIGINAL signed integer=275;modulus=27;exact ratio=275/27;floor quotient=10
After
value=q*n+remainder;normalized residue=5 in[0,27);same congruence class,NOT absolute-value replacement
Conditions: none
Use the floor quotient for a positive modulus,retaining the congruence class of a negative base rather than replacing it by abs(a).Equal residues mean congruence modulo n,not equality of original integers.
Step identifiers and dependencies
Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S080
S082 · calculation
Before
Euler ORIGINAL a times unit representative13 := mul(25, 13)
After
Euler ORIGINAL a times unit representative13 = 325
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S081
S083 · calculation
Before
Euler unit representative13 raw image exact quotient := div(325, 27)
After
Euler unit representative13 raw image exact quotient = 325/27
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S082
S084 · calculation
Before
Euler unit representative13 raw image floor quotient multiple := mul(12, 27)
After
Euler unit representative13 raw image floor quotient multiple = 324
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S083
S085 · calculation
Before
Euler unit representative13 raw image normalized remainder := sub(325, 324)
After
Euler unit representative13 raw image normalized remainder = 1
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S084
S086 · implies
Before
Euler unit representative13 raw image:ORIGINAL signed integer=325;modulus=27;exact ratio=325/27;floor quotient=12
After
value=q*n+remainder;normalized residue=1 in[0,27);same congruence class,NOT absolute-value replacement
Conditions: none
Use the floor quotient for a positive modulus,retaining the congruence class of a negative base rather than replacing it by abs(a).Equal residues mean congruence modulo n,not equality of original integers.
Step identifiers and dependencies
Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S085
S087 · calculation
Before
Euler ORIGINAL a times unit representative14 := mul(25, 14)
After
Euler ORIGINAL a times unit representative14 = 350
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S086
S088 · calculation
Before
Euler unit representative14 raw image exact quotient := div(350, 27)
After
Euler unit representative14 raw image exact quotient = 350/27
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S087
S089 · calculation
Before
Euler unit representative14 raw image floor quotient multiple := mul(12, 27)
After
Euler unit representative14 raw image floor quotient multiple = 324
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S088
S090 · calculation
Before
Euler unit representative14 raw image normalized remainder := sub(350, 324)
After
Euler unit representative14 raw image normalized remainder = 26
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S089
S091 · implies
Before
Euler unit representative14 raw image:ORIGINAL signed integer=350;modulus=27;exact ratio=350/27;floor quotient=12
After
value=q*n+remainder;normalized residue=26 in[0,27);same congruence class,NOT absolute-value replacement
Conditions: none
Use the floor quotient for a positive modulus,retaining the congruence class of a negative base rather than replacing it by abs(a).Equal residues mean congruence modulo n,not equality of original integers.
Step identifiers and dependencies
Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S090
S092 · calculation
Before
Euler ORIGINAL a times unit representative16 := mul(25, 16)
After
Euler ORIGINAL a times unit representative16 = 400
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S091
S093 · calculation
Before
Euler unit representative16 raw image exact quotient := div(400, 27)
After
Euler unit representative16 raw image exact quotient = 400/27
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S092
S094 · calculation
Before
Euler unit representative16 raw image floor quotient multiple := mul(14, 27)
After
Euler unit representative16 raw image floor quotient multiple = 378
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S093
S095 · calculation
Before
Euler unit representative16 raw image normalized remainder := sub(400, 378)
After
Euler unit representative16 raw image normalized remainder = 22
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S094
S096 · implies
Before
Euler unit representative16 raw image:ORIGINAL signed integer=400;modulus=27;exact ratio=400/27;floor quotient=14
After
value=q*n+remainder;normalized residue=22 in[0,27);same congruence class,NOT absolute-value replacement
Conditions: none
Use the floor quotient for a positive modulus,retaining the congruence class of a negative base rather than replacing it by abs(a).Equal residues mean congruence modulo n,not equality of original integers.
Step identifiers and dependencies
Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S095
S097 · calculation
Before
Euler ORIGINAL a times unit representative17 := mul(25, 17)
After
Euler ORIGINAL a times unit representative17 = 425
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S096
S098 · calculation
Before
Euler unit representative17 raw image exact quotient := div(425, 27)
After
Euler unit representative17 raw image exact quotient = 425/27
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S097
S099 · calculation
Before
Euler unit representative17 raw image floor quotient multiple := mul(15, 27)
After
Euler unit representative17 raw image floor quotient multiple = 405
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S098
S100 · calculation
Before
Euler unit representative17 raw image normalized remainder := sub(425, 405)
After
Euler unit representative17 raw image normalized remainder = 20
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S099
S101 · implies
Before
Euler unit representative17 raw image:ORIGINAL signed integer=425;modulus=27;exact ratio=425/27;floor quotient=15
After
value=q*n+remainder;normalized residue=20 in[0,27);same congruence class,NOT absolute-value replacement
Conditions: none
Use the floor quotient for a positive modulus,retaining the congruence class of a negative base rather than replacing it by abs(a).Equal residues mean congruence modulo n,not equality of original integers.
Step identifiers and dependencies
Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S100
S102 · calculation
Before
Euler ORIGINAL a times unit representative19 := mul(25, 19)
After
Euler ORIGINAL a times unit representative19 = 475
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S101
S103 · calculation
Before
Euler unit representative19 raw image exact quotient := div(475, 27)
After
Euler unit representative19 raw image exact quotient = 475/27
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S102
S104 · calculation
Before
Euler unit representative19 raw image floor quotient multiple := mul(17, 27)
After
Euler unit representative19 raw image floor quotient multiple = 459
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S103
S105 · calculation
Before
Euler unit representative19 raw image normalized remainder := sub(475, 459)
After
Euler unit representative19 raw image normalized remainder = 16
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S104
S106 · implies
Before
Euler unit representative19 raw image:ORIGINAL signed integer=475;modulus=27;exact ratio=475/27;floor quotient=17
After
value=q*n+remainder;normalized residue=16 in[0,27);same congruence class,NOT absolute-value replacement
Conditions: none
Use the floor quotient for a positive modulus,retaining the congruence class of a negative base rather than replacing it by abs(a).Equal residues mean congruence modulo n,not equality of original integers.
Step identifiers and dependencies
Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S105
S107 · calculation
Before
Euler ORIGINAL a times unit representative20 := mul(25, 20)
After
Euler ORIGINAL a times unit representative20 = 500
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S106
S108 · calculation
Before
Euler unit representative20 raw image exact quotient := div(500, 27)
After
Euler unit representative20 raw image exact quotient = 500/27
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S107
S109 · calculation
Before
Euler unit representative20 raw image floor quotient multiple := mul(18, 27)
After
Euler unit representative20 raw image floor quotient multiple = 486
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S108
S110 · calculation
Before
Euler unit representative20 raw image normalized remainder := sub(500, 486)
After
Euler unit representative20 raw image normalized remainder = 14
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S109
S111 · implies
Before
Euler unit representative20 raw image:ORIGINAL signed integer=500;modulus=27;exact ratio=500/27;floor quotient=18
After
value=q*n+remainder;normalized residue=14 in[0,27);same congruence class,NOT absolute-value replacement
Conditions: none
Use the floor quotient for a positive modulus,retaining the congruence class of a negative base rather than replacing it by abs(a).Equal residues mean congruence modulo n,not equality of original integers.
Step identifiers and dependencies
Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S110
S112 · calculation
Before
Euler ORIGINAL a times unit representative22 := mul(25, 22)
After
Euler ORIGINAL a times unit representative22 = 550
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S111
S113 · calculation
Before
Euler unit representative22 raw image exact quotient := div(550, 27)
After
Euler unit representative22 raw image exact quotient = 550/27
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S112
S114 · calculation
Before
Euler unit representative22 raw image floor quotient multiple := mul(20, 27)
After
Euler unit representative22 raw image floor quotient multiple = 540
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S113
S115 · calculation
Before
Euler unit representative22 raw image normalized remainder := sub(550, 540)
After
Euler unit representative22 raw image normalized remainder = 10
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S114
S116 · implies
Before
Euler unit representative22 raw image:ORIGINAL signed integer=550;modulus=27;exact ratio=550/27;floor quotient=20
After
value=q*n+remainder;normalized residue=10 in[0,27);same congruence class,NOT absolute-value replacement
Conditions: none
Use the floor quotient for a positive modulus,retaining the congruence class of a negative base rather than replacing it by abs(a).Equal residues mean congruence modulo n,not equality of original integers.
Step identifiers and dependencies
Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S115
S117 · calculation
Before
Euler ORIGINAL a times unit representative23 := mul(25, 23)
After
Euler ORIGINAL a times unit representative23 = 575
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S116
S118 · calculation
Before
Euler unit representative23 raw image exact quotient := div(575, 27)
After
Euler unit representative23 raw image exact quotient = 575/27
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S117
S119 · calculation
Before
Euler unit representative23 raw image floor quotient multiple := mul(21, 27)
After
Euler unit representative23 raw image floor quotient multiple = 567
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S118
S120 · calculation
Before
Euler unit representative23 raw image normalized remainder := sub(575, 567)
After
Euler unit representative23 raw image normalized remainder = 8
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S119
S121 · implies
Before
Euler unit representative23 raw image:ORIGINAL signed integer=575;modulus=27;exact ratio=575/27;floor quotient=21
After
value=q*n+remainder;normalized residue=8 in[0,27);same congruence class,NOT absolute-value replacement
Conditions: none
Use the floor quotient for a positive modulus,retaining the congruence class of a negative base rather than replacing it by abs(a).Equal residues mean congruence modulo n,not equality of original integers.
Step identifiers and dependencies
Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S120
S122 · calculation
Before
Euler ORIGINAL a times unit representative25 := mul(25, 25)
After
Euler ORIGINAL a times unit representative25 = 625
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S121
S123 · calculation
Before
Euler unit representative25 raw image exact quotient := div(625, 27)
After
Euler unit representative25 raw image exact quotient = 625/27
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S122
S124 · calculation
Before
Euler unit representative25 raw image floor quotient multiple := mul(23, 27)
After
Euler unit representative25 raw image floor quotient multiple = 621
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S123
S125 · calculation
Before
Euler unit representative25 raw image normalized remainder := sub(625, 621)
After
Euler unit representative25 raw image normalized remainder = 4
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S124
S126 · implies
Before
Euler unit representative25 raw image:ORIGINAL signed integer=625;modulus=27;exact ratio=625/27;floor quotient=23
After
value=q*n+remainder;normalized residue=4 in[0,27);same congruence class,NOT absolute-value replacement
Conditions: none
Use the floor quotient for a positive modulus,retaining the congruence class of a negative base rather than replacing it by abs(a).Equal residues mean congruence modulo n,not equality of original integers.
Step identifiers and dependencies
Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S125
S127 · calculation
Before
Euler ORIGINAL a times unit representative26 := mul(25, 26)
After
Euler ORIGINAL a times unit representative26 = 650
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S126
S128 · calculation
Before
Euler unit representative26 raw image exact quotient := div(650, 27)
After
Euler unit representative26 raw image exact quotient = 650/27
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S127
S129 · calculation
Before
Euler unit representative26 raw image floor quotient multiple := mul(24, 27)
After
Euler unit representative26 raw image floor quotient multiple = 648
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S128
S130 · calculation
Before
Euler unit representative26 raw image normalized remainder := sub(650, 648)
After
Euler unit representative26 raw image normalized remainder = 2
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S129
S131 · implies
Before
Euler unit representative26 raw image:ORIGINAL signed integer=650;modulus=27;exact ratio=650/27;floor quotient=24
After
value=q*n+remainder;normalized residue=2 in[0,27);same congruence class,NOT absolute-value replacement
Conditions: none
Use the floor quotient for a positive modulus,retaining the congruence class of a negative base rather than replacing it by abs(a).Equal residues mean congruence modulo n,not equality of original integers.
Step identifiers and dependencies
Rule: euler_exact_signed_remainder_not_abs_or_integer_equality · Uses: G001, S130
S132 · implies
Before
COMPLETE original increasing U(n)=[1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26];actual images in ORIGINAL unit order=[25, 23, 19, 17, 13, 11, 7, 5, 1, 26, 22, 20, 16, 14, 10, 8, 4, 2]
After
SOURCE output sorts the mapped list numerically=[1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26];full-list equality=Yes;not a claim each point is fixed
Conditions: none
The source's second field is the sorted complete image list,not images in original unit order.Equal lists certify a permutation,not that multiplication by a fixes every point.The raw images are listed separately here.
Step identifiers and dependencies
Rule: euler_sorted_mapped_units_not_original_image_order_or_identity · Uses: G001, S131