Step-by-step derivation
S001 · implies
Before
source roles = [qf-function-limit-two-sided-affine, two_sided_affine_limit, punctured_reals, []]
After
derived = [[integer_approach_value_OR_source_nonexistence_STRING]]
Conditions: none
BASE sources do not acquire a generic legacy operation/domain or unknown-metadata ban.Retain metadata WITHOUT using it as an answer oracle.PULSE/legacy enforce their exact source keys,operation/domain.Do not impose universal tuples,/1 or input bounds.
Step identifiers and dependencies
Rule: R63_SOURCE_bank_keys_operation_domain_output_roles_metadata_not_oracle · Uses: G001
S002 · calculation
Before
LEFT affine approach ORIGINAL slope times approach center := mul(4, -9)
After
LEFT affine approach ORIGINAL slope times approach center = -36
Conditions: none
Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S001
S003 · calculation
Before
LEFT affine approach ORIGINAL approach affine constant sum := sum(-36, 27)
After
LEFT affine approach ORIGINAL approach affine constant sum = -9
Conditions: none
Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S002
S004 · calculation
Before
LEFT affine approach ORIGINAL positive slope bound one plus abs m := sum(1, 4)
After
LEFT affine approach ORIGINAL positive slope bound one plus abs m = 5
Conditions: none
Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S003
S005 · implies
Before
source roles = [LEFT affine approach, 4, 27, -9]
After
derived = [-9, 5, error = m*(x-a), delta = epsilon/5]
Conditions: none
For the WHOLE GIVEN affine law,f(x)−L=m(x−a).For ANY ε>0 choose δ=ε/(1+|m|)>0;on the declared side/punctured domain,0<|x−a|<δ gives error<ε.At m0 the error is0.This is an APPROACH value,not assignment of original f(a)or a sample-based proof.
Step identifiers and dependencies
Rule: Limit_AFFINE_whole_law_error_epsilon_delta_NOT_center_assignment · Uses: G001, S004
S006 · calculation
Before
RIGHT affine approach ORIGINAL slope times approach center := mul(-1, -9)
After
RIGHT affine approach ORIGINAL slope times approach center = 9
Conditions: none
Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S005
S007 · calculation
Before
RIGHT affine approach ORIGINAL approach affine constant sum := sum(9, -18)
After
RIGHT affine approach ORIGINAL approach affine constant sum = -9
Conditions: none
Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S006
S008 · calculation
Before
RIGHT affine approach ORIGINAL positive slope bound one plus abs m := sum(1, 1)
After
RIGHT affine approach ORIGINAL positive slope bound one plus abs m = 2
Conditions: none
Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S007
S009 · implies
Before
source roles = [RIGHT affine approach, -1, -18, -9]
After
derived = [-9, 2, error = m*(x-a), delta = epsilon/2]
Conditions: none
For the WHOLE GIVEN affine law,f(x)−L=m(x−a).For ANY ε>0 choose δ=ε/(1+|m|)>0;on the declared side/punctured domain,0<|x−a|<δ gives error<ε.At m0 the error is0.This is an APPROACH value,not assignment of original f(a)or a sample-based proof.
Step identifiers and dependencies
Rule: Limit_AFFINE_whole_law_error_epsilon_delta_NOT_center_assignment · Uses: G001, S008
S010 · calculation
Before
Two-sided ORIGINAL positive joint slope bound := sum(1, 4)
After
Two-sided ORIGINAL positive joint slope bound = 5
Conditions: none
Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S009
S011 · implies
Before
source roles = [-9, 4, 27, -1, -18]
After
derived = [-9, -9, 5, delta = epsilon/5]
Conditions: none
When both approach values AGREE,δ=ε/(1+max(|lm|,|rm|))controls BOTH sides for ANY ε>0,so their common value is the two-sided limit,including both slopes0.No original f(a)is assigned.
Step identifiers and dependencies
Rule: Limit_TWO_sides_AGREE_shared_delta_whole_punctured_law · Uses: G001, S010