Equation Ontology
Algebra Ontologies
Authors: MArk Anthony Higgins - Sky Recursive Ai
Revised Date: 20 September 2026
Audio Overview
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KIRETH SEMANTIC RESEARCH ARCHIVE
FLOW ALGEBRA — EQUATION ONTOLOGY
======================================================================
PAGE:
https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/algebra-ontologies/equation-ontology.html
FLOW:
EquationOntology
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 equations, identities, unknowns, solution
sets, balancing operations, transformations, linear equations,
quadratic equations, polynomial equations, rational equations,
simultaneous equations, symbolic rearrangement, equivalence,
verification and recursive solving.
AI INTERPRETATION NOTE:
Execute the following specification as formal semantic and reasoning
rules. Preserve equality, domains, solution sets, transformation
preconditions, denominator restrictions, multiplicities and
equivalence status throughout every derivation.
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START FLOW
======================================================================
FLOW EquationOntology {
----------------------------------------------------------------------
1. DOMAIN DECLARATION
----------------------------------------------------------------------
DOMAIN := Equation_Theory
PARENT_DOMAIN := Algebra
SUPER_DOMAINS := {
Mathematics,
Mathematical_Ontologies,
Algebra_Ontologies,
Elementary_Algebra,
Algebraic_Expression_Theory
}
DOMAIN_INTENT := {
represent_equations,
represent_equalities,
represent_unknowns,
represent_parameters,
represent_solution_sets,
represent_balancing_operations,
represent_equivalent_equations,
represent_implication_steps,
represent_linear_equations,
represent_quadratic_equations,
represent_polynomial_equations,
represent_rational_equations,
represent_radical_equations,
represent_exponential_equations,
represent_logarithmic_equations,
represent_simultaneous_equations,
represent_formula_rearrangement,
represent_verification,
support_recursive_equation_solving,
support_domain_preserving_validation
}
----------------------------------------------------------------------
2. PRIMITIVE TYPES
----------------------------------------------------------------------
TYPE Scalar
TYPE Variable
TYPE Unknown <: Variable
TYPE Parameter <: Variable
TYPE Constant
TYPE AlgebraicExpression
TYPE Equation
TYPE Identity <: Equation
TYPE ConditionalEquation <: Equation
TYPE ContradictionEquation <: Equation
TYPE LinearEquation <: Equation
TYPE QuadraticEquation <: Equation
TYPE PolynomialEquation <: Equation
TYPE RationalEquation <: Equation
TYPE RadicalEquation <: Equation
TYPE ExponentialEquation <: Equation
TYPE LogarithmicEquation <: Equation
TYPE SimultaneousSystem
TYPE Solution
TYPE SolutionSet
TYPE Constraint
TYPE Transformation
TYPE EquivalencePreservingTransformation <: Transformation
TYPE ImplicationPreservingTransformation <: Transformation
TYPE Derivation
TYPE ProofStep
TYPE ValidationResult
----------------------------------------------------------------------
3. EQUATION DEFINITION
----------------------------------------------------------------------
ENTITY Equation E {
left_side: AlgebraicExpression
relation: "="
right_side: AlgebraicExpression
variables: Variable*
constraints: Constraint*
}
RULE EQ_01:
equation asserts equality of left and right expressions.
RULE EQ_02:
equation may be true for:
no values,
some values,
all permitted values.
----------------------------------------------------------------------
4. EQUALITY PROPERTIES
----------------------------------------------------------------------
AXIOM EQUAL_01_REFLEXIVE:
a=a
AXIOM EQUAL_02_SYMMETRIC:
a=b -> b=a
AXIOM EQUAL_03_TRANSITIVE:
a=b AND b=c -> a=c
AXIOM EQUAL_04_SUBSTITUTION:
a=b
-> expressions containing a may replace a by b
when type and domain remain valid.
----------------------------------------------------------------------
5. SOLUTION
----------------------------------------------------------------------
SOLUTION(x=a,E) iff
substitute x=a into original equation
AND both sides are defined
AND both sides evaluate equal.
ENTITY SolutionSet S {
members: Solution*
}
RULE SOL_01:
every reported solution must satisfy original equation.
RULE SOL_02:
candidate solution failing original equation is extraneous.
----------------------------------------------------------------------
6. IDENTITY
----------------------------------------------------------------------
IS_IDENTITY(E) iff
equation true for every permitted variable assignment
in declared domain.
RULE ID_01:
identity solution set equals full permitted domain.
EXAMPLE:
2(x+1)=2x+2
----------------------------------------------------------------------
7. CONTRADICTION
----------------------------------------------------------------------
IS_CONTRADICTION(E) iff
no permitted assignment satisfies E.
EXAMPLE:
x+1=x+2
REDUCES_TO:
1=2
RULE CONTRA_01:
contradiction equation has empty solution set.
----------------------------------------------------------------------
8. CONDITIONAL EQUATION
----------------------------------------------------------------------
IS_CONDITIONAL(E) iff
true for some but not all permitted values.
RULE COND_01:
ordinary solving typically seeks solution set of conditional equation.
----------------------------------------------------------------------
9. BALANCE PRINCIPLE
----------------------------------------------------------------------
RULE BAL_01:
applying same valid operation to both sides preserves equality.
RULE BAL_02:
inverse operations may isolate unknowns.
RULE BAL_03:
balance transformations must respect operation preconditions.
----------------------------------------------------------------------
10. ADDITION PROPERTY
----------------------------------------------------------------------
IF:
A=B
THEN:
A+C=B+C
RULE ADD_01:
addition of same defined expression to both sides preserves solution set.
----------------------------------------------------------------------
11. SUBTRACTION PROPERTY
----------------------------------------------------------------------
IF:
A=B
THEN:
A-C=B-C
RULE SUB_01:
subtraction of same defined expression preserves solution set.
----------------------------------------------------------------------
12. MULTIPLICATION PROPERTY
----------------------------------------------------------------------
IF:
A=B
THEN:
AC=BC
RULE MUL_01:
multiplication by same expression preserves every existing solution.
CAUTION:
if C may equal zero, reverse equivalence may fail when deriving
original from multiplied equation.
----------------------------------------------------------------------
13. DIVISION PROPERTY
----------------------------------------------------------------------
IF:
A=B
AND C!=0
THEN:
A/C=B/C
RULE DIV_01:
division requires nonzero divisor.
RULE DIV_02:
dividing both sides by expression containing variable requires
explicit exclusion of its zeros.
----------------------------------------------------------------------
14. EQUIVALENT EQUATIONS
----------------------------------------------------------------------
EQUIVALENT_EQUATIONS(E1,E2,D) iff
solution_set(E1,D)=solution_set(E2,D)
NOTATION:
E1 <=> E2
RULE EE_01:
addition/subtraction of same defined expression is equivalence-preserving.
RULE EE_02:
multiplication/division by known nonzero expression is
equivalence-preserving.
----------------------------------------------------------------------
15. IMPLICATION-ONLY TRANSFORMATIONS
----------------------------------------------------------------------
TRANSFORMATION may yield:
E1 => E2
without:
E2 => E1
EXAMPLES:
squaring both sides
multiplying by possibly-zero expression
RULE IMP_01:
implication-only transformation requires final candidate verification.
----------------------------------------------------------------------
16. TRANSPOSITION
----------------------------------------------------------------------
INFORMAL:
moving term across equality and changing sign
FORMAL:
apply inverse addition/subtraction to both sides
RULE TRANS_01:
A+B=C
<=> A=C-B
RULE TRANS_02:
A-B=C
<=> A=C+B
RULE TRANS_03:
AB=C
<=> A=C/B
WHERE B!=0
RULE TRANS_04:
A/B=C
<=> A=BC
WHERE B!=0
----------------------------------------------------------------------
17. LINEAR EQUATIONS
----------------------------------------------------------------------
ENTITY LinearEquation {
canonical_form:
a*x+b=c
condition:
a!=0
}
PROCEDURE SOLVE_LINEAR(E,x):
simplify both sides
collect x-terms
collect constants
reduce to:
a*x=b
REQUIRE:
a!=0
divide by a
obtain:
x=b/a
validate in original equation
RETURN solution_set
----------------------------------------------------------------------
18. LINEAR CLASSIFICATION
----------------------------------------------------------------------
FOR equation:
a*x+b=c*x+d
rearrange:
(a-c)x=d-b
CASE 1:
a-c !=0
-> unique solution
CASE 2:
a-c=0
AND d-b=0
-> identity / infinitely many solutions
CASE 3:
a-c=0
AND d-b!=0
-> contradiction / no solution
----------------------------------------------------------------------
19. EQUATIONS WITH BRACKETS
----------------------------------------------------------------------
PROCEDURE SOLVE_BRACKETED(E):
expand brackets if useful
simplify
combine like terms
continue with appropriate solver
RULE BR_01:
expansion must preserve equivalence.
----------------------------------------------------------------------
20. EQUATIONS WITH FRACTIONS
----------------------------------------------------------------------
PROCEDURE SOLVE_FRACTION_EQUATION(E):
extract denominator restrictions
compute common denominator
multiply both sides by common denominator
solve resulting equation
reject candidates violating original denominator restrictions
validate all candidates
----------------------------------------------------------------------
21. QUADRATIC EQUATIONS
----------------------------------------------------------------------
ENTITY QuadraticEquation {
canonical_form:
a*x^2+b*x+c=0
condition:
a!=0
}
DISCRIMINANT:
Delta=b^2-4ac
QUADRATIC_FORMULA:
x=(-b ± sqrt(Delta))/(2a)
RULE QUAD_01:
over reals:
Delta>0 -> two distinct real roots
Delta=0 -> one repeated real root
Delta<0 -> no real roots
RULE QUAD_02:
over complex numbers:
two roots counted with multiplicity.
----------------------------------------------------------------------
22. FACTORING QUADRATICS
----------------------------------------------------------------------
IF:
a*x^2+b*x+c
=
(p*x+q)(r*x+s)
THEN:
equation equals zero implies:
p*x+q=0
OR
r*x+s=0
AXIOM ZERO_PRODUCT:
AB=0 -> A=0 OR B=0
over integral domain.
----------------------------------------------------------------------
23. COMPLETING THE SQUARE
----------------------------------------------------------------------
PROCEDURE COMPLETE_SQUARE:
normalise coefficient of x^2
group x terms
add/subtract square-completion constant
rewrite as:
(x-h)^2=k
solve with square-root rule
validate roots
----------------------------------------------------------------------
24. POLYNOMIAL EQUATIONS
----------------------------------------------------------------------
ENTITY PolynomialEquation:
P(x)=0
PROCEDURE SOLVE_POLYNOMIAL(P):
factor if possible
use known root theorems
apply zero-product principle
solve factors
track multiplicities
validate candidates
----------------------------------------------------------------------
25. ROOT MULTIPLICITY
----------------------------------------------------------------------
ROOT_MULTIPLICITY(a,P)=m iff
(x-a)^m divides P
and
(x-a)^(m+1) does not.
RULE MULT_01:
repeated root may appear once in solution set but carries multiplicity metadata.
----------------------------------------------------------------------
26. RATIONAL EQUATIONS
----------------------------------------------------------------------
ENTITY RationalEquation:
R1(x)=R2(x)
PROCEDURE SOLVE_RATIONAL(E):
extract all denominator restrictions
multiply by LCD
solve resulting polynomial/algebraic equation
validate every candidate in original equation
RULE RAT_01:
denominator-zero values never valid solutions.
----------------------------------------------------------------------
27. RADICAL EQUATIONS
----------------------------------------------------------------------
ENTITY RadicalEquation:
contains variable under root
PROCEDURE SOLVE_RADICAL(E):
isolate radical
apply power operation to remove radical
mark transformation implication-sensitive
solve resulting equation
substitute every candidate into original equation
RULE RAD_01:
squaring may introduce extraneous roots.
----------------------------------------------------------------------
28. ABSOLUTE VALUE EQUATIONS
----------------------------------------------------------------------
FOR:
|A|=b
CASE:
b<0 -> no solution
b=0 -> A=0
b>0 -> A=b OR A=-b
RULE ABS_01:
absolute-value splitting assumes real-valued expression.
----------------------------------------------------------------------
29. EXPONENTIAL EQUATIONS
----------------------------------------------------------------------
ENTITY ExponentialEquation:
variable appears in exponent
PROCEDURE SOLVE_EXPONENTIAL(E):
seek common base
if common base a>0,a!=1:
equate exponents
otherwise:
apply logarithms under domain constraints
solve resulting equation
validate
----------------------------------------------------------------------
30. LOGARITHMIC EQUATIONS
----------------------------------------------------------------------
ENTITY LogarithmicEquation:
variable appears inside logarithm
DOMAIN:
every log argument >0
over real numbers
PROCEDURE SOLVE_LOG(E):
extract log-domain restrictions
combine logs where valid
exponentiate
solve resulting equation
reject candidates violating log domains
----------------------------------------------------------------------
31. EQUATIONS WITH POWERS
----------------------------------------------------------------------
FOR:
x^n=a
IF n odd over reals:
one real root:
x=a^(1/n)
IF n even:
a<0 -> no real solution
a=0 -> x=0
a>0 -> x=±a^(1/n)
RULE PWR_01:
complex solution set may contain n roots.
----------------------------------------------------------------------
32. SIMULTANEOUS EQUATIONS
----------------------------------------------------------------------
ENTITY SimultaneousSystem {
equations: Equation+
unknowns: Variable+
}
SOLUTION(system) iff
assignment satisfies every equation simultaneously.
----------------------------------------------------------------------
33. SUBSTITUTION METHOD
----------------------------------------------------------------------
PROCEDURE SOLVE_BY_SUBSTITUTION(system):
solve one equation for one unknown
substitute into remaining equations
solve reduced system
back-substitute
validate in all originals
----------------------------------------------------------------------
34. ELIMINATION METHOD
----------------------------------------------------------------------
PROCEDURE SOLVE_BY_ELIMINATION(system):
scale equations if needed
add/subtract equations to eliminate variable
solve reduced equation
back-substitute
validate
----------------------------------------------------------------------
35. MATRIX METHOD LINK
----------------------------------------------------------------------
LINEAR_SYSTEM:
A*x=b
RULE MATLINK_01:
matrix row reduction may solve simultaneous linear equations.
RULE MATLINK_02:
rank conditions classify consistency and uniqueness.
----------------------------------------------------------------------
36. FORMULA REARRANGEMENT
----------------------------------------------------------------------
PROCEDURE MAKE_SUBJECT(formula,target):
identify operations applied to target
apply inverse operations in reverse structural order
preserve equality
preserve domain restrictions
simplify
validate reverse substitution
----------------------------------------------------------------------
37. PARAMETRIC EQUATIONS
----------------------------------------------------------------------
ENTITY Parameter p {
treated as fixed during solving unless otherwise declared
}
RULE PAR_01:
solution form may depend on parameter values.
PROCEDURE SOLVE_WITH_PARAMETERS(E):
identify parameter-critical values
partition cases
solve each case
return piecewise solution classification
----------------------------------------------------------------------
38. CASE SPLITTING
----------------------------------------------------------------------
CASE_SPLIT on condition C:
branch 1:
assume C
branch 2:
assume NOT C
RULE CASE_01:
union of validated branch solution sets forms total solution set.
----------------------------------------------------------------------
39. DOMAIN RESTRICTIONS
----------------------------------------------------------------------
EXTRACT_DOMAIN_RESTRICTIONS(E):
from denominators:
denominator!=0
from real even roots:
radicand>=0
from real logarithms:
argument>0
from declared functions:
function-domain constraints
RETURN conjunction_of_constraints
----------------------------------------------------------------------
40. EXTRANEOUS SOLUTIONS
----------------------------------------------------------------------
EXTRANEOUS(a,E) iff
a generated by transformed equation
but fails original equation or domain.
RULE EXTRA_01:
implication-only transformations can create extraneous solutions.
RULE EXTRA_02:
candidate verification is mandatory after such transformations.
----------------------------------------------------------------------
41. LOST SOLUTIONS
----------------------------------------------------------------------
LOST_SOLUTION risk when:
divide both sides by expression that may equal zero
RULE LOST_01:
before dividing by variable expression F(x),
split:
F(x)=0
and
F(x)!=0
if necessary.
----------------------------------------------------------------------
42. SQUARING BOTH SIDES
----------------------------------------------------------------------
IF:
A=B
THEN:
A^2=B^2
BUT reverse implication:
A^2=B^2
-> A=B OR A=-B
RULE SQ_01:
squaring is implication-preserving, not always equivalence-preserving.
----------------------------------------------------------------------
43. TAKING ROOTS
----------------------------------------------------------------------
FROM:
x^2=a
DO NOT infer only:
x=sqrt(a)
CORRECT over reals:
x=±sqrt(a)
when a>0
RULE ROOT_01:
inverse power operations may introduce branch multiplicity.
----------------------------------------------------------------------
44. CROSS MULTIPLICATION
----------------------------------------------------------------------
FROM:
A/B=C/D
WITH:
B!=0
D!=0
INFER:
AD=BC
RULE CROSS_01:
denominator restrictions remain active.
----------------------------------------------------------------------
45. EQUATION NORMAL FORMS
----------------------------------------------------------------------
NORMAL_FORMS include:
left_minus_right_zero:
L-R=0
linear:
ax+b=0
quadratic:
ax^2+bx+c=0
polynomial:
P(x)=0
RULE NF_01:
reducing to zero form often exposes factorisation and polynomial structure.
----------------------------------------------------------------------
46. SOLUTION VERIFICATION
----------------------------------------------------------------------
PROCEDURE VERIFY(candidate,E):
check domain constraints
substitute candidate into original left side
substitute candidate into original right side
evaluate both
if both defined and equal:
VALID_SOLUTION
else:
INVALID_SOLUTION
----------------------------------------------------------------------
47. NUMERICAL SOLUTIONS
----------------------------------------------------------------------
ENTITY ApproximateSolution {
value
tolerance
residual
}
RESIDUAL(E,x):
LHS(x)-RHS(x)
APPROX_VALID(x,tol) iff
|residual|<=tol
AND domain constraints satisfied
RULE NUM_01:
approximate solution must be labelled approximate.
----------------------------------------------------------------------
48. SYMBOLIC VS NUMERICAL SOLVING
----------------------------------------------------------------------
SYMBOLIC_SOLVER:
seeks exact forms
NUMERICAL_SOLVER:
seeks approximate roots
RULE SVN_01:
exact symbolic result preferred where tractable.
RULE SVN_02:
numerical methods require convergence/error metadata.
----------------------------------------------------------------------
49. GRAPHICAL INTERPRETATION
----------------------------------------------------------------------
EQUATION:
f(x)=g(x)
SOLUTIONS correspond to:
x-coordinates of graph intersections
where both functions defined.
RULE GRAPH_01:
graphical solving gives approximate evidence unless intersections
established analytically.
----------------------------------------------------------------------
50. SEMANTIC RELATIONSHIPS
----------------------------------------------------------------------
RELATION HAS_LEFT_SIDE(Equation,AlgebraicExpression)
RELATION HAS_RIGHT_SIDE(Equation,AlgebraicExpression)
RELATION HAS_UNKNOWN(Equation,Variable)
RELATION HAS_SOLUTION(Equation,Solution)
RELATION HAS_SOLUTION_SET(Equation,SolutionSet)
RELATION EQUIVALENT_TO(Equation,Equation)
RELATION IMPLIES(Equation,Equation)
RELATION TRANSFORMS_TO(Equation,Equation)
RELATION SATISFIED_BY(Equation,Solution)
RELATION PART_OF(Equation,SimultaneousSystem)
----------------------------------------------------------------------
51. SEMANTIC EQUIVALENCE
----------------------------------------------------------------------
SEMANTIC_EQUIVALENCE {
"solve equation" ~= find_all_values_satisfying_equality
"unknown" ~= variable_to_be_determined
"root of equation" ~= solution_value
"balance equation" ~= apply_same_valid_operation_to_both_sides
"transpose term" ~= use_inverse_operation_on_both_sides
"equivalent equations" ~= same_solution_set_on_declared_domain
"extraneous solution" ~= candidate_created_by_transformation_but_invalid_in_original
"identity" ~= equation_true_for_all_permitted_values
"contradiction" ~= equation_true_for_no_permitted_values
"conditional equation" ~= equation_true_for_some_permitted_values
}
----------------------------------------------------------------------
52. QUERY INTERPRETATION
----------------------------------------------------------------------
QUERY Q1:
"solve E"
-> classify equation and invoke matching solver
QUERY Q2:
"check solution"
-> VERIFY(candidate,E)
QUERY Q3:
"make x the subject"
-> MAKE_SUBJECT
QUERY Q4:
"are equations equivalent"
-> compare validated solution sets/domains
QUERY Q5:
"how many solutions"
-> classify equation and domain
QUERY Q6:
"solve simultaneously"
-> substitution/elimination/matrix method
QUERY Q7:
"find restrictions"
-> EXTRACT_DOMAIN_RESTRICTIONS
----------------------------------------------------------------------
53. VALIDATION ENGINE
----------------------------------------------------------------------
PROCEDURE VALIDATE_TRANSFORMATION(E1,E2,relation_type):
compare domains
verify transformation rule
if equivalence claimed:
prove same solution set or reversible rule
if implication claimed:
prove E1 solutions imply E2
return:
VALID
INVALID(reason)
UNRESOLVED
PROCEDURE VALIDATE_SOLUTION_SET(S,E):
for each candidate in S:
VERIFY(candidate,E)
test completeness where solver/theorem permits
return validation status
----------------------------------------------------------------------
54. RECURSIVE REASONING ENGINE
----------------------------------------------------------------------
PROCEDURE EQUATION_REASONER(input):
parse left/right expressions
identify unknowns and parameters
infer/verify domains
extract restrictions
classify equation type
select solving strategy
apply atomic transformations
label each step:
equivalent
implication_only
preserve constraints
branch on parameter/zero cases when necessary
recurse until candidates obtained
verify all candidates in original equation
classify solution set
RETURN:
result,
derivation,
constraints,
equation_class,
semantic_status,
confidence
----------------------------------------------------------------------
55. RECURSIVE TRANSITION
----------------------------------------------------------------------
RECURSIVE_RULE RR_01:
INPUT state E_n
identify unresolved equation structure
choose transformation T with satisfied preconditions
E_(n+1):=T(E_n)
record:
equivalence_status
new_constraints
validate transition
recurse(E_(n+1))
TERMINATION_CONDITIONS := {
unknown_isolated,
finite_candidate_set_found,
identity_detected,
contradiction_detected,
parameter_case_complete,
domain_failure,
no_valid_transformation
}
----------------------------------------------------------------------
56. ERROR MODEL
----------------------------------------------------------------------
ERROR E_DIVISION_BY_ZERO
ERROR E_LOST_SOLUTION
ERROR E_EXTRANEOUS_SOLUTION
ERROR E_DOMAIN_RESTRICTION_LOST
ERROR E_INVALID_BALANCE_OPERATION
ERROR E_INVALID_SQUARE_ROOT_BRANCH
ERROR E_INVALID_LOG_DOMAIN
ERROR E_CROSS_MULTIPLICATION_DOMAIN
ERROR E_PARAMETER_CASE_MISSED
ERROR E_SOLUTION_NOT_VERIFIED
ERROR E_EQUIVALENCE_OVERCLAIM
----------------------------------------------------------------------
57. CONFIDENCE MODEL
----------------------------------------------------------------------
CONFIDENCE {
base:=1.00
deduct 0.10 if domain inferred
deduct 0.10 if unknown role ambiguous
deduct 0.15 if implication-only transformation used
deduct 0.15 if parameter cases unresolved
deduct 0.20 if completeness of solution set unproven
deduct 0.20 if numerical approximation used
deduct 0.25 if candidate verification incomplete
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
}
----------------------------------------------------------------------
58. AI CONSUMPTION MODEL
----------------------------------------------------------------------
AI_CONSUMPTION {
read_order := [
equation_definition,
equality_axioms,
solution_semantics,
domain_restrictions,
equivalence_transformations,
implication_transformations,
equation_classes,
solving_methods,
verification,
validation
]
execution_policy := {
preserve_original_equation,
preserve_domain_restrictions,
label_equivalence_vs_implication,
avoid_dividing_by_potential_zero_without_case_split,
verify_candidates_after_squaring_or_clearing_denominators,
retain_parameter_cases,
distinguish_identity_contradiction_conditional,
return_complete_solution_set_where_possible
}
}
----------------------------------------------------------------------
59. UPDATE MODEL
----------------------------------------------------------------------
UPDATE_MODEL {
extensible:=true
new_equation_class_requires:
structural_predicate,
solving_strategy,
domain_rules
new_transformation_requires:
preconditions,
equivalence_or_implication_status
new_solver_requires:
applicability_conditions,
termination_conditions,
verification_method
conflicts_require:
explicit_versioning
standard_algebraic_meaning_has_priority:=true
}
----------------------------------------------------------------------
60. SELF-CHECK
----------------------------------------------------------------------
SELF_CHECK {
CHECK equality relation preserved correctly
CHECK domains extracted before solving
CHECK denominator restrictions retained
CHECK divisions use nonzero divisors
CHECK squaring steps marked implication-only
CHECK parameter zero cases handled
CHECK all candidates verified in original equation
CHECK identity/contradiction classifications correct
CHECK solution-set completeness assessed
CHECK final result validated
}
----------------------------------------------------------------------
61. COMPLETION CONDITIONS
----------------------------------------------------------------------
COMPLETE iff {
equation_definition_defined
AND equality_axioms_defined
AND solution_semantics_defined
AND identity_defined
AND contradiction_defined
AND equivalence_transformations_defined
AND implication_transformations_defined
AND linear_solver_defined
AND quadratic_solver_defined
AND polynomial_solver_defined
AND rational_solver_defined
AND radical_solver_defined
AND exponential_log_solver_defined
AND simultaneous_solver_defined
AND rearrangement_defined
AND verification_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
----------------------------------------------------------------------
62. FLOW TERMINATION
----------------------------------------------------------------------
TERMINATE WHEN:
requested equation goal achieved
AND all candidate solutions validated
AND domain restrictions preserved
OR
identity classified
OR
contradiction classified
OR
equation unsolved because required theorem/precondition unavailable
OR
no valid transformation remains
FINAL_STATE := {
equation_result,
solution_set,
derivation,
equation_class,
domain_restrictions,
semantic_status,
confidence
}
} // END FLOW EquationOntology
======================================================================
END FLOW
======================================================================
FLOW STATUS:
Complete
CANONICAL PAGE:
https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/algebra-ontologies/equation-ontology.html
FLOW_HASH_SHA256: 60364106a269c39972efc84c32b387a66d09c8089afa3ab63de9c7dd8ae21b82