Rational Expression Ontology
Algebra Ontologies
Authors: MArk Anthony Higgins - Sky Recursive Ai
Revised Date: 11 September 2026
Audio Overview
======================================================================
KIRETH SEMANTIC RESEARCH ARCHIVE
FLOW ALGEBRA — RATIONAL EXPRESSION ONTOLOGY
======================================================================
PAGE:
https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/algebra-ontologies/rational-expression-ontology.html
FLOW:
RationalExpressionOntology
VERSION:
1.0
LANGUAGE:
en-GB
AUTHORS:
Mark Anthony Higgins — Sky Recursive AI
PURPOSE:
To provide a machine-readable, recursively executable semantic and
reasoning specification for rational expressions, rational functions,
numerators, denominators, domain restrictions, factor cancellation,
common denominators, arithmetic operations, simplification,
equivalence, asymptotic structure and validation.
AI INTERPRETATION NOTE:
Execute the following specification as formal semantic and reasoning
rules. Preserve denominator nonzero constraints, coefficient domains,
factor structure, excluded values, equivalence domains and operation
preconditions throughout every transformation.
======================================================================
START FLOW
======================================================================
FLOW RationalExpressionOntology {
----------------------------------------------------------------------
1. DOMAIN DECLARATION
----------------------------------------------------------------------
DOMAIN := Rational_Expression_Theory
PARENT_DOMAIN := Algebra
SUPER_DOMAINS := {
Mathematics,
Mathematical_Ontologies,
Algebra_Ontologies,
Elementary_Algebra,
Polynomial_Theory
}
DOMAIN_INTENT := {
represent_rational_expressions,
represent_rational_functions,
represent_numerators,
represent_denominators,
represent_domain_restrictions,
represent_excluded_values,
represent_factor_cancellation,
represent_common_denominators,
represent_addition,
represent_subtraction,
represent_multiplication,
represent_division,
represent_simplification,
represent_equivalence,
represent_zeroes,
represent_poles,
represent_holes,
represent_asymptotic_structure,
support_recursive_rational_reasoning,
support_domain_preserving_validation
}
DEFAULT_COEFFICIENT_DOMAIN := Integer | Rational | Real | Complex | Field
----------------------------------------------------------------------
2. PRIMITIVE TYPES
----------------------------------------------------------------------
TYPE Scalar
TYPE Variable
TYPE Polynomial
TYPE Numerator <: Polynomial
TYPE Denominator <: Polynomial
TYPE RationalExpression
TYPE RationalFunction <: RationalExpression
TYPE Factor
TYPE CommonDenominator
TYPE DomainRestriction
TYPE ExcludedValue
TYPE Zero
TYPE Pole
TYPE RemovableDiscontinuity
TYPE VerticalAsymptote
TYPE HorizontalAsymptote
TYPE ObliqueAsymptote
TYPE Quotient
TYPE Remainder
TYPE Constraint
TYPE Derivation
TYPE ValidationResult
----------------------------------------------------------------------
3. RATIONAL EXPRESSION DEFINITION
----------------------------------------------------------------------
ENTITY RationalExpression R {
numerator: P
denominator: Q
coefficient_domain: F
}
FORM:
R(x)=P(x)/Q(x)
CONSTRAINT RE_01:
Q is not the zero polynomial.
CONSTRAINT RE_02:
Q(x)!=0 for every permitted evaluation point.
RULE RE_03:
numerator and denominator are polynomials over compatible coefficient domains.
RULE RE_04:
rational expression is undefined at values making denominator zero,
even when a common factor later cancels.
----------------------------------------------------------------------
4. DOMAIN OF A RATIONAL EXPRESSION
----------------------------------------------------------------------
DOMAIN(R):
declared scalar domain
minus
{x | Q(x)=0}
RULE DOM_01:
original denominator determines excluded values.
RULE DOM_02:
simplification must preserve excluded-value metadata.
RULE DOM_03:
cancelled denominator factors may produce removable discontinuities,
not restoration of original domain points.
----------------------------------------------------------------------
5. NUMERATOR AND DENOMINATOR
----------------------------------------------------------------------
NUMERATOR(R):=P
DENOMINATOR(R):=Q
RULE ND_01:
denominator must not be zero polynomial.
RULE ND_02:
denominator evaluation must be nonzero before numerical evaluation.
----------------------------------------------------------------------
6. RATIONAL FUNCTION
----------------------------------------------------------------------
ENTITY RationalFunction f {
algebraic_form: P/Q
domain: scalar_domain minus zeros(Q)
codomain: declared scalar domain
}
RULE RF_01:
two algebraically simplified forms may define different partial
functions if original excluded values are ignored.
----------------------------------------------------------------------
7. FACTORISED FORM
----------------------------------------------------------------------
FACTORISED(R):
factor numerator P
factor denominator Q
RETURN:
product_i A_i / product_j B_j
RULE FAC_01:
factorisation domain matters.
RULE FAC_02:
only common nonzero factors may be cancelled.
----------------------------------------------------------------------
8. COMMON FACTORS
----------------------------------------------------------------------
COMMON_FACTOR(F,P,Q) iff
F divides P
AND F divides Q
RULE CF_01:
if F common factor:
P/Q
=
(P/F)/(Q/F)
only on domain where F!=0 and Q!=0.
RULE CF_02:
cancellation changes representation, not original domain restrictions.
----------------------------------------------------------------------
9. SIMPLIFICATION
----------------------------------------------------------------------
PROCEDURE SIMPLIFY_RATIONAL(R):
record original_domain_restrictions
factor numerator
factor denominator
cancel common factors symbolically
reduce coefficient fractions
normalise signs
return simplified_form
plus preserved_excluded_values
RULE SIMP_01:
simplified expression must remain equivalent on original domain.
----------------------------------------------------------------------
10. EQUIVALENCE
----------------------------------------------------------------------
RATIONAL_EQUIVALENT(R1,R2,D) iff
forall x in D:
both defined
AND R1(x)=R2(x)
RULE EQ_01:
cross multiplication may establish equivalence where denominators are nonzero:
P1/Q1 = P2/Q2
iff
P1*Q2=P2*Q1
on common valid domain.
RULE EQ_02:
algebraic equality alone is insufficient to erase domain exclusions.
----------------------------------------------------------------------
11. MULTIPLICATION
----------------------------------------------------------------------
MULTIPLY(P1/Q1,P2/Q2):
REQUIRE:
Q1!=0 polynomial
Q2!=0 polynomial
RESULT:
(P1*P2)/(Q1*Q2)
domain restrictions:
Q1(x)!=0
Q2(x)!=0
simplify with preserved restrictions.
----------------------------------------------------------------------
12. DIVISION
----------------------------------------------------------------------
DIVIDE(P1/Q1,P2/Q2):
REQUIRE:
Q1(x)!=0
Q2(x)!=0
P2(x)!=0
RESULT:
(P1*Q2)/(Q1*P2)
RULE DIV_01:
divisor rational expression itself must be nonzero.
RULE DIV_02:
reciprocal introduces numerator-of-divisor nonzero restriction.
----------------------------------------------------------------------
13. ADDITION
----------------------------------------------------------------------
ADD(P1/Q1,P2/Q2):
common denominator:
Q1*Q2
or least common denominator
RESULT:
(P1*Q2 + P2*Q1)/(Q1*Q2)
preserve:
Q1!=0
Q2!=0
----------------------------------------------------------------------
14. SUBTRACTION
----------------------------------------------------------------------
SUBTRACT(P1/Q1,P2/Q2):
RESULT:
(P1*Q2 - P2*Q1)/(Q1*Q2)
preserve:
Q1!=0
Q2!=0
----------------------------------------------------------------------
15. LEAST COMMON DENOMINATOR
----------------------------------------------------------------------
LCD(Q1,...,Qn):
factor denominators
include each irreducible factor
to maximum exponent appearing
multiply coefficient LCM under declared domain convention
RULE LCD_01:
every denominator divides LCD.
----------------------------------------------------------------------
16. COMPLEX FRACTIONS
----------------------------------------------------------------------
ENTITY ComplexRationalExpression {
numerator_or_denominator_contains_rational_expression
}
PROCEDURE SIMPLIFY_COMPLEX_FRACTION(R):
identify all inner denominators
compute common denominator
multiply numerator and denominator by that common denominator
preserve all original restrictions
simplify resulting rational expression
----------------------------------------------------------------------
17. SIGN NORMALISATION
----------------------------------------------------------------------
RULE SIGN_01:
P/(-Q)=(-P)/Q
RULE SIGN_02:
(-P)/(-Q)=P/Q
RULE SIGN_03:
prefer canonical denominator sign convention where applicable.
----------------------------------------------------------------------
18. EVALUATION
----------------------------------------------------------------------
EVALUATE(R,x=a):
compute Q(a)
if Q(a)=0:
return UNDEFINED
else:
return P(a)/Q(a)
RULE EVAL_01:
domain check precedes cancellation-based evaluation.
----------------------------------------------------------------------
19. ZEROES
----------------------------------------------------------------------
ZERO_OF_RATIONAL(R,a) iff
P(a)=0
AND Q(a)!=0
RULE ZERO_01:
cancelled numerator factors may correspond to holes rather than zeros
if denominator also vanishes there in original expression.
----------------------------------------------------------------------
20. POLES
----------------------------------------------------------------------
POLE_CANDIDATE(R,a) iff
Q(a)=0
AFTER_COMMON_FACTOR_ANALYSIS:
if denominator zero remains after reduction:
a may be pole
if corresponding common factor cancels:
a is removable discontinuity under analytic interpretation
RULE POLE_01:
pole order determined by uncancelled denominator factor multiplicity.
----------------------------------------------------------------------
21. REMOVABLE DISCONTINUITIES
----------------------------------------------------------------------
REMOVABLE_DISCONTINUITY(R,a) iff
original P(a)=0
AND original Q(a)=0
AND common factor cancellation removes zero denominator factor
AND reduced denominator nonzero at a
RULE HOLE_01:
original function remains undefined at a.
----------------------------------------------------------------------
22. VERTICAL ASYMPTOTES
----------------------------------------------------------------------
VERTICAL_ASYMPTOTE(x=a) candidate iff
reduced denominator vanishes at a
AND numerator does not cancel that factor.
RULE VA_01:
exact asymptotic behaviour may depend on multiplicity and side limits.
----------------------------------------------------------------------
23. HORIZONTAL ASYMPTOTES
----------------------------------------------------------------------
FOR R=P/Q over real variable x:
let:
m=deg(P)
n=deg(Q)
RULE HA_01:
if m<n:
horizontal asymptote y=0
RULE HA_02:
if m=n:
horizontal asymptote
y=LC(P)/LC(Q)
RULE HA_03:
if m>n:
no horizontal asymptote from this degree rule.
----------------------------------------------------------------------
24. OBLIQUE / SLANT ASYMPTOTES
----------------------------------------------------------------------
IF:
deg(P)=deg(Q)+1
THEN:
polynomial division:
P/Q = S + Remainder/Q
linear quotient S gives oblique asymptote.
RULE OA_01:
more generally polynomial asymptote arises from quotient when
degree numerator >= degree denominator.
----------------------------------------------------------------------
25. POLYNOMIAL DIVISION FORM
----------------------------------------------------------------------
DIVISION_DECOMPOSITION(P/Q):
P=DQ+R
with deg(R)<deg(Q)
then:
P/Q=D+R/Q
RULE PDF_01:
useful for asymptotic structure and improper rational expressions.
----------------------------------------------------------------------
26. PROPER AND IMPROPER RATIONAL EXPRESSIONS
----------------------------------------------------------------------
PROPER(R) iff
deg(P)<deg(Q)
IMPROPER(R) iff
deg(P)>=deg(Q)
RULE PI_01:
improper rational expression may be decomposed by polynomial division.
----------------------------------------------------------------------
27. DOMAIN RESTRICTION EXTRACTION
----------------------------------------------------------------------
EXTRACT_RESTRICTIONS(R):
factor or solve:
Q(x)=0
create constraints:
x != each denominator root
over declared domain
RULE DR_01:
unresolved denominator roots remain symbolic restrictions.
----------------------------------------------------------------------
28. CROSS MULTIPLICATION
----------------------------------------------------------------------
CROSS_MULTIPLY:
P1/Q1 = P2/Q2
-> P1 Q2 = P2 Q1
PRECONDITIONS:
Q1!=0
Q2!=0
RULE CM_01:
solutions of cross-multiplied equation must be checked against original
denominator restrictions.
----------------------------------------------------------------------
29. RATIONAL EQUATIONS
----------------------------------------------------------------------
PROCEDURE SOLVE_RATIONAL_EQUATION(E):
extract every denominator restriction
compute LCD
multiply equation by LCD
solve resulting polynomial/algebraic equation
test every candidate in original equation
reject excluded or extraneous values
return validated_solution_set
----------------------------------------------------------------------
30. RATIONAL INEQUALITIES
----------------------------------------------------------------------
PROCEDURE SOLVE_RATIONAL_INEQUALITY(P/Q relation 0):
factor numerator and denominator
identify critical points:
zeros(P)
zeros(Q)
create sign intervals
test sign on each interval
include/exclude endpoints according to:
relation
denominator restrictions
return interval_solution_set
RULE RI_01:
denominator-zero points are always excluded.
----------------------------------------------------------------------
31. RECIPROCAL
----------------------------------------------------------------------
RECIPROCAL(P/Q):
Q/P
REQUIRE:
P!=0 at evaluated point
Q!=0 at evaluated point
RULE REC_01:
reciprocal swaps numerator and denominator.
RULE REC_02:
zeros of original become excluded/pole candidates of reciprocal.
----------------------------------------------------------------------
32. POWERS
----------------------------------------------------------------------
POWER(P/Q,n):
FOR integer n>0:
P^n/Q^n
FOR n=0:
1
where original expression defined and nonzero convention handled
FOR n<0:
Q^(-n)/P^(-n)
introducing P!=0
RULE POW_01:
negative powers introduce numerator nonzero restrictions.
----------------------------------------------------------------------
33. DERIVATIVE LINK
----------------------------------------------------------------------
FOR differentiable polynomial P,Q:
d/dx(P/Q)
=
(P'Q-PQ')/Q^2
REQUIRE:
Q!=0
NOTE:
calculus execution belongs to analysis ontology;
quotient-rule structural link declared here.
----------------------------------------------------------------------
34. PARTIAL FRACTIONS LINK
----------------------------------------------------------------------
PROCEDURE PARTIAL_FRACTION_SETUP(R):
REQUIRE:
proper rational expression
or first perform polynomial division
factor denominator over selected field
generate decomposition template based on:
distinct linear factors
repeated linear factors
irreducible quadratic factors
NOTE:
detailed coefficient solving may delegate to algebra solver.
----------------------------------------------------------------------
35. RATIONAL FUNCTION FIELD
----------------------------------------------------------------------
ENTITY RationalFunctionField F(x) {
elements:
equivalence classes of P/Q
with P,Q in F[x], Q!=0
}
RULE RFF_01:
F(x) is a field when F is a field.
RULE RFF_02:
equality in F(x) determined by:
P1 Q2 = P2 Q1.
----------------------------------------------------------------------
36. MULTIVARIATE RATIONAL EXPRESSIONS
----------------------------------------------------------------------
FORM:
P(x1,...,xn)/Q(x1,...,xn)
DOMAIN:
all points where Q!=0
RULE MRE_01:
cancellation preserves original algebraic-domain exclusions.
RULE MRE_02:
solving Q=0 may define algebraic hypersurface rather than isolated values.
----------------------------------------------------------------------
37. FACTOR MULTIPLICITY
----------------------------------------------------------------------
IF:
P contains factor F^m
Q contains factor F^n
THEN after cancellation:
net exponent:
numerator if m>n: F^(m-n)
denominator if n>m: F^(n-m)
none if m=n
ORIGINAL RESTRICTION:
F!=0 remains.
----------------------------------------------------------------------
38. NORMAL FORM
----------------------------------------------------------------------
NORMAL_FORM(R):
numerator and denominator coprime
up to units
denominator canonicalised
coefficient common factors removed
original excluded-value metadata stored separately
RULE NF_01:
reduced algebraic fraction plus restriction set forms complete
semantic representation.
----------------------------------------------------------------------
39. COPRIME NUMERATOR AND DENOMINATOR
----------------------------------------------------------------------
REDUCED(R) iff
gcd(P,Q)=1
up to units
RULE RED_01:
reduced form unique up to multiplication by common unit,
under suitable UFD/field conventions.
----------------------------------------------------------------------
40. GCD-BASED SIMPLIFICATION
----------------------------------------------------------------------
SIMPLIFY_BY_GCD(P/Q):
G=gcd(P,Q)
P1=P/G
Q1=Q/G
return P1/Q1
with original restrictions retained.
----------------------------------------------------------------------
41. RATIONAL IDENTITY
----------------------------------------------------------------------
IDENTITY(R1,R2) iff
cross-products equal as polynomial identity
and semantic domain comparison explicitly handled.
RULE RID_01:
same reduced fraction does not imply same original partial-function
domain if restrictions differ.
----------------------------------------------------------------------
42. SPECIAL VALUES
----------------------------------------------------------------------
RULE SV_01:
numerator zero and denominator nonzero -> value 0
RULE SV_02:
denominator zero -> undefined
RULE SV_03:
0/0 -> indeterminate/undefined, not automatic cancellation result
RULE SV_04:
nonzero/0 -> undefined in ordinary field arithmetic
----------------------------------------------------------------------
43. APPROXIMATE EVALUATION
----------------------------------------------------------------------
APPROX_EVAL(R,a,tolerance):
if |Q(a)| near zero:
flag numerical_instability
else:
evaluate with declared numeric precision
RULE AE_01:
near-pole numerical evaluation requires caution.
----------------------------------------------------------------------
44. SIGN OF RATIONAL EXPRESSION
----------------------------------------------------------------------
SIGN(R,x):
REQUIRE:
Q(x)!=0
sign(R)=sign(P)*sign(Q)
RULE SIGN_01:
factor multiplicities determine sign changes across real roots.
RULE SIGN_02:
even multiplicity factor does not change sign crossing root.
RULE SIGN_03:
odd multiplicity factor changes sign crossing root.
----------------------------------------------------------------------
45. GRAPH-STRUCTURAL LINK
----------------------------------------------------------------------
RATIONAL_GRAPH_FEATURES may include:
x_intercepts from valid numerator zeros
y_intercept if x=0 in domain
holes from cancelled denominator factors
vertical_asymptotes from uncancelled denominator roots
horizontal_or_polynomial_asymptotes from degree comparison/division
NOTE:
graphing belongs to function/geometry ontology;
structural relationships declared here.
----------------------------------------------------------------------
46. SEMANTIC RELATIONSHIPS
----------------------------------------------------------------------
RELATION HAS_NUMERATOR(RationalExpression,Polynomial)
RELATION HAS_DENOMINATOR(RationalExpression,Polynomial)
RELATION HAS_DOMAIN_RESTRICTION(RationalExpression,Constraint)
RELATION SIMPLIFIES_TO(RationalExpression,RationalExpression)
RELATION EQUIVALENT_TO(RationalExpression,RationalExpression)
RELATION HAS_ZERO(RationalExpression,Zero)
RELATION HAS_POLE(RationalExpression,Pole)
RELATION HAS_HOLE(RationalExpression,RemovableDiscontinuity)
RELATION HAS_ASYMPTOTE(RationalExpression,Asymptote)
----------------------------------------------------------------------
47. SEMANTIC EQUIVALENCE
----------------------------------------------------------------------
SEMANTIC_EQUIVALENCE {
"algebraic fraction" ~= RationalExpression
"rational expression" ~= quotient_of_polynomials
"rational function" ~= rational_expression_with_declared_domain
"excluded value" ~= denominator_zero_domain_restriction
"cancel common factor" ~= divide numerator_and_denominator_by_same_nonzero_factor
"hole" ~= removable_discontinuity
"vertical asymptote" ~= uncancelled denominator-zero asymptotic location
"proper rational expression" ~= numerator_degree_less_than_denominator_degree
"improper rational expression" ~= numerator_degree_at_least_denominator_degree
"LCD" ~= least_common_denominator
}
----------------------------------------------------------------------
48. QUERY INTERPRETATION
----------------------------------------------------------------------
QUERY Q1:
"simplify rational expression"
-> SIMPLIFY_RATIONAL
QUERY Q2:
"find restrictions"
-> EXTRACT_RESTRICTIONS
QUERY Q3:
"add rational expressions"
-> ADD
QUERY Q4:
"multiply rational expressions"
-> MULTIPLY
QUERY Q5:
"divide rational expressions"
-> DIVIDE
QUERY Q6:
"solve rational equation"
-> SOLVE_RATIONAL_EQUATION
QUERY Q7:
"find holes/asymptotes"
-> factor, reduce, compare degree and restrictions
QUERY Q8:
"are these equivalent"
-> cross-product identity plus domain comparison
----------------------------------------------------------------------
49. VALIDATION ENGINE
----------------------------------------------------------------------
PROCEDURE VALIDATE_RATIONAL_OPERATION(operation,inputs,result):
verify each denominator nonzero polynomial
collect all original restrictions
verify algebraic transformation
verify cancelled factors recorded as restrictions
verify resulting numerator/denominator domains
for equivalence:
compare on common valid domain
for equation solving:
substitute candidates into original equation
return:
VALID
INVALID(reason)
UNRESOLVED
----------------------------------------------------------------------
50. RECURSIVE REASONING ENGINE
----------------------------------------------------------------------
PROCEDURE RATIONAL_REASONER(input):
parse numerators
parse denominators
infer coefficient domain
extract denominator restrictions
classify requested operation
factor when useful
select valid rational rules
execute atomic transformations
preserve exclusion metadata
validate intermediate states
recurse until goal achieved
validate final result
RETURN:
result,
derivation,
domain_restrictions,
coefficient_domain,
semantic_status,
confidence
----------------------------------------------------------------------
51. RECURSIVE TRANSITION
----------------------------------------------------------------------
RECURSIVE_RULE RR_01:
INPUT state S_n
identify unresolved rational-expression property
choose rule R such that:
denominator_preconditions=true
coefficient_domain_preconditions=true
factor_preconditions=true
S_(n+1):=apply(R,S_n)
preserve original restriction set
validate transition
recurse(S_(n+1))
TERMINATION_CONDITIONS := {
rational_goal_achieved,
reduced_form_reached,
invalid_denominator,
excluded_candidate,
coefficient_domain_failure,
no_valid_rule
}
----------------------------------------------------------------------
52. ERROR MODEL
----------------------------------------------------------------------
ERROR E_ZERO_DENOMINATOR_POLYNOMIAL
ERROR E_DENOMINATOR_ZERO_AT_VALUE
ERROR E_LOST_DOMAIN_RESTRICTION
ERROR E_INVALID_CANCELLATION
ERROR E_DIVISION_BY_ZERO_RATIONAL
ERROR E_LCD_ERROR
ERROR E_CROSS_MULTIPLICATION_EXTRANEOUS
ERROR E_EQUIVALENCE_DOMAIN_MISMATCH
ERROR E_ASYMPTOTE_MISCLASSIFICATION
ERROR E_HOLE_MISCLASSIFICATION
ERROR E_COEFFICIENT_DOMAIN_MISMATCH
----------------------------------------------------------------------
53. CONFIDENCE MODEL
----------------------------------------------------------------------
CONFIDENCE {
base:=1.00
deduct 0.10 if coefficient domain inferred
deduct 0.10 if factorisation domain inferred
deduct 0.15 if denominator roots unresolved
deduct 0.15 if cancellation restrictions implicit
deduct 0.20 if asymptotic classification incomplete
deduct 0.20 if equivalence domains unclear
deduct 0.25 if original expression unavailable after simplification
clamp_to_[0,1]
classification:
[0.95,1.00] -> very_high
[0.85,0.95) -> high
[0.70,0.85) -> moderate
[0.50,0.70) -> low
[0.00,0.50) -> insufficient
}
----------------------------------------------------------------------
54. AI CONSUMPTION MODEL
----------------------------------------------------------------------
AI_CONSUMPTION {
read_order := [
rational_definition,
domain_restrictions,
factorisation,
simplification,
arithmetic,
equivalence,
roots_and_poles,
asymptotes,
rational_equations,
validation
]
execution_policy := {
preserve_original_denominator_restrictions,
never_cancel_without_recording_nonzero_condition,
distinguish_algebraic_simplification_from_domain_equivalence,
validate_cross_multiplication_candidates,
distinguish_holes_from_vertical_asymptotes,
preserve_coefficient_domain,
prefer_factored_form_for_structure_analysis
}
}
----------------------------------------------------------------------
55. UPDATE MODEL
----------------------------------------------------------------------
UPDATE_MODEL {
extensible:=true
new_rational_operation_requires:
numerator_rule,
denominator_rule,
domain_restrictions
new_simplification_rule_requires:
equivalence_condition,
restriction_preservation
new_asymptotic_rule_requires:
degree_or_limit_conditions
conflicts_require:
explicit_versioning
standard_algebraic_meaning_has_priority:=true
}
----------------------------------------------------------------------
56. SELF-CHECK
----------------------------------------------------------------------
SELF_CHECK {
CHECK denominator is not zero polynomial
CHECK all denominator-zero points excluded
CHECK cancellation factors valid
CHECK cancelled zeros retained as restrictions
CHECK addition/subtraction use valid common denominator
CHECK division adds divisor-numerator nonzero restriction
CHECK reduced numerator and denominator coprime where claimed
CHECK rational equation candidates checked in original equation
CHECK holes distinguished from poles
CHECK asymptote classification consistent with reduced form
CHECK final result validated
}
----------------------------------------------------------------------
57. COMPLETION CONDITIONS
----------------------------------------------------------------------
COMPLETE iff {
rational_expression_defined
AND rational_function_defined
AND domain_restrictions_defined
AND simplification_defined
AND arithmetic_defined
AND LCD_defined
AND equivalence_defined
AND zeroes_defined
AND poles_defined
AND holes_defined
AND asymptotes_defined
AND rational_equation_solver_defined
AND rational_inequality_solver_defined
AND rational_function_field_defined
AND semantic_equivalence_defined
AND validation_defined
AND recursive_reasoning_defined
AND confidence_defined
AND ai_consumption_defined
AND update_model_defined
}
IF NOT COMPLETE:
semantic_status:=incomplete_flow
----------------------------------------------------------------------
58. FLOW TERMINATION
----------------------------------------------------------------------
TERMINATE WHEN:
requested rational-expression goal achieved
AND algebraic result validated
AND original domain restrictions preserved
OR
denominator condition makes requested evaluation undefined
OR
candidate solution violates original restrictions
OR
coefficient-domain conditions prevent operation
OR
no valid rational transformation remains
FINAL_STATE := {
rational_result,
derivation,
reduced_form,
original_domain_restrictions,
coefficient_domain,
semantic_status,
confidence
}
} // END FLOW RationalExpressionOntology
======================================================================
END FLOW
======================================================================
FLOW STATUS:
Complete
CANONICAL PAGE:
https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/algebra-ontologies/rational-expression-ontology.html
FLOW_HASH_SHA256: 3f1f4368a8c66ceca157909d920d034f333308b2c68f77aff729d1de79a1d634