Step-by-step derivation
S001 · implies
Before
source roles = [qf-ext18-gcd-common-divisor-audit, Nat_gcd, gcd-common-divisor-audit, no-form-field]
After
derived = [[a, b, d, divides_a, divides_b, equals_gcd, gcd, common_AND_gcd]]
Conditions: none
Check THIS bank interface:BASE equations use form/domain;BASE events/complement have no operation;factorials have no domain.ext18 uses local hyphens,other legacy operations underscores.Do not force every answer into a tuple or fraction.
Step identifiers and dependencies
Rule: R62_EXACT_source_keys_domain_operation_form_AND_output_types · Uses: G001
S002 · calculation
Before
GCD Euclid row0 ORIGINAL quotient times y := mul(1, 30)
After
GCD Euclid row0 ORIGINAL quotient times y = 30
Conditions: none
Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S001
S003 · calculation
Before
GCD Euclid row0 ORIGINAL x minus qy := sub(40, 30)
After
GCD Euclid row0 ORIGINAL x minus qy = 10
Conditions: none
Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S002
S004 · implies
Before
source roles = [40, 30, 1]
After
derived = [30, 10, [40, 30, 10]]
Conditions: none
With y≠0,divide to get x=qy+r and0≤r<y.A divisor common to x,y divides y,r,and the converse follows from x=qy+r,so common divisors are unchanged.Positive remainders strictly decrease,ensuring termination.
Step identifiers and dependencies
Rule: GCD_EUCLID_nonzero_divisor_exact_quotient_remainder_INVARIANT · Uses: G001, S003
S005 · calculation
Before
GCD Euclid row1 ORIGINAL quotient times y := mul(3, 10)
After
GCD Euclid row1 ORIGINAL quotient times y = 30
Conditions: none
Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S004
S006 · calculation
Before
GCD Euclid row1 ORIGINAL x minus qy := sub(30, 30)
After
GCD Euclid row1 ORIGINAL x minus qy = 0
Conditions: none
Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S005
S007 · implies
Before
source roles = [30, 10, 3]
After
derived = [30, 0, [30, 10, 0]]
Conditions: none
With y≠0,divide to get x=qy+r and0≤r<y.A divisor common to x,y divides y,r,and the converse follows from x=qy+r,so common divisors are unchanged.Positive remainders strictly decrease,ensuring termination.
Step identifiers and dependencies
Rule: GCD_EUCLID_nonzero_divisor_exact_quotient_remainder_INVARIANT · Uses: G001, S006
S008 · implies
Before
source roles = [40, 30, [[40, 30, 10], [30, 10, 0]]]
After
derived = [10, 0]
Conditions: none
At second value0 retain the last x.For nonzero x,gcd(x,0)=x;the two-way common-divisor invariant transfers this to the original pair.At(0,0)use the EXPLICIT convention gcd=0,not a claim that0 is the largest positive divisor.
Step identifiers and dependencies
Rule: GCD_TERMINAL_last_x_ZEROsecond_and_explicit_gcd00_convention · Uses: G001, S007
S009 · calculation
Before
GCD candidate divides a ORIGINAL exact multiplier := div(40, 1)
After
GCD candidate divides a ORIGINAL exact multiplier = 40
Conditions: none
Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S008
S010 · implies
Before
source roles = [GCD candidate divides a, 1, 40]
After
derived = [Yes, 40]
Conditions: none
Natural divisibility at d=0 holds ONLY for x=0;choose multiplier0 for0∣0 without claiming uniqueness.For nonzero d,show a multiplier ONLY at remainder0.Two divisibility flags alone do not establish greatestness.
Step identifiers and dependencies
Rule: GCD_DIVISIBILITY_ZERO_guard_before_natural_multiplier · Uses: G001, S009
S011 · calculation
Before
GCD candidate divides b ORIGINAL exact multiplier := div(30, 1)
After
GCD candidate divides b ORIGINAL exact multiplier = 30
Conditions: none
Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S010
S012 · implies
Before
source roles = [GCD candidate divides b, 1, 30]
After
derived = [Yes, 30]
Conditions: none
Natural divisibility at d=0 holds ONLY for x=0;choose multiplier0 for0∣0 without claiming uniqueness.For nonzero d,show a multiplier ONLY at remainder0.Two divisibility flags alone do not establish greatestness.
Step identifiers and dependencies
Rule: GCD_DIVISIBILITY_ZERO_guard_before_natural_multiplier · Uses: G001, S011
S013 · implies
Before
source roles = [40, 30, 1, 10]
After
derived = [Yes, Yes, No, 10, No]
Conditions: none
FIRST compute the actual gcd from ORIGINAL a,b independently of candidate d.A common divisor need not be greatest.Keep d∣a,d∣b,d=gcd,and their conjunction SEPARATE.
Step identifiers and dependencies
Rule: GCD_AUDIT_actual_gcd_INDEPENDENT_of_candidate_common_AND_greatest · Uses: G001, S012