┌ Info: 
└   FUNCTION_NAME = "find_identifiable_functions"
┌ Info: 
└   PROBLEM_NAME = "Goodwin oscillator"
┌ Info: 
└   KWARGS = (with_states = true, strategy = (:normalforms, 2))
┌ Info: 
└   GLOBAL_ID = Symbol("(:normalforms, 2)_with_states")
[ Info: Summary of the model:
[ Info: State variables: x1, x2, x3, x4
[ Info: Parameters: b, alpha, c, gama, delta, sigma, beta
[ Info: Inputs: 
[ Info: Outputs: y
[ Info: Summary of the model:
[ Info: State variables: x1, x2
[ Info: Parameters: a, b, d, c
[ Info: Inputs: 
[ Info: Outputs: y
[ Info: Computing IO-equations
┌ Info: Computed in 7.078050781 seconds
│   :ioeq_time = :ioeq_time
└   ioeq_time = 7.078050781
[ Info: Computing Wronskians
┌ Info: Computed in 7.602338717 seconds
│   :wrnsk_time = :wrnsk_time
└   wrnsk_time = 7.602338717
[ Info: Dimensions of the Wronskians [69]
┌ Info: Ranks of the Wronskians computed in 0.023306329 seconds
│   :rank_time = :rank_time
└   rank_times = 0.023306329

⌜ # Computing specializations..  	 Time: 0:00:06[K
✓ # Computing specializations..  	 Time: 0:00:07[K
[ Info: Simplifying identifiable functions
┌ Info: Computing parametric Groebner basis up to degrees (2, 2)
│ Ordering, input / target: degrevlex / InputOrdering
│ Rational interpolator: VanDerHoevenLecerf
│ Polynomial interpolator: PrimesBenOrTiwari
│ Estimate degrees: true
└ Assess correctness: false
┌ Info: Basis interpolated exponents summary:
│ Maximal interpolated degrees are: 2 for num. and 2 for den.
│ Maximal number of interpolated terms are: 2 for num. and 1 for den.
└ Points used: 24.
[ Info: Groebner basis computed in 8.87300459 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 2.983427603 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 7 rational functions
┌ Info: Final cleaning and simplification of generators. 
└ Out of 6 fractions 6 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 3.619306041 seconds. Result: true
[ Info: Out of 90 initial generators there are 5 indepdendent
[ Info: The ranking of the new set of generators is 18
[ Info: Simplifying identifiable functions
┌ Info: Computing parametric Groebner basis up to degrees (2, 2)
│ Ordering, input / target: degrevlex / InputOrdering
│ Rational interpolator: VanDerHoevenLecerf
│ Polynomial interpolator: PrimesBenOrTiwari
│ Estimate degrees: true
└ Assess correctness: false
┌ Info: Basis interpolated exponents summary:
│ Maximal interpolated degrees are: 2 for num. and 1 for den.
│ Maximal number of interpolated terms are: 2 for num. and 1 for den.
└ Points used: 20.
[ Info: Groebner basis computed in 0.377815049 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.003870143 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 12 rational functions
┌ Info: Computing normal forms (probabilistic)
│ Variables (11 in total): Nemo.QQMPolyRingElem[b, alpha, c, gama, delta, sigma, beta, x1, x2, x3, x4]
│ Up to degree: 2
└ Modulo: Finite field of characteristic 1073741827
[ Info: Used specialization points: 10
┌ Info: Final cleaning and simplification of generators. 
└ Out of 22 fractions 15 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 0.008533199 seconds. Result: true
[ Info: Out of 10 initial generators there are 9 indepdendent
[ Info: The ranking of the new set of generators is 138
[ Info: The search for identifiable functions concluded in 44.482149407 seconds
[ Info: Processing Goodwin oscillator
┌ Info: Averaging over 1 runs.
│ Using keyword arguments:
│ NamedTuple{(:with_states, :strategy), Tuple{Bool, Tuple{Symbol, Int64}}}
│ (with_states = true, strategy = (:normalforms, 2))
└ ID: (:normalforms, 2)_with_states
[ Info: Computing IO-equations
┌ Info: Computed in 0.011569849 seconds
│   :ioeq_time = :ioeq_time
└   ioeq_time = 0.011569849
[ Info: Computing Wronskians
┌ Info: Computed in 0.068262591 seconds
│   :wrnsk_time = :wrnsk_time
└   wrnsk_time = 0.068262591
[ Info: Dimensions of the Wronskians [69]
┌ Info: Ranks of the Wronskians computed in 0.000247482 seconds
│   :rank_time = :rank_time
└   rank_times = 0.000247482
[ Info: Simplifying identifiable functions
┌ Info: Computing parametric Groebner basis up to degrees (2, 2)
│ Ordering, input / target: degrevlex / InputOrdering
│ Rational interpolator: VanDerHoevenLecerf
│ Polynomial interpolator: PrimesBenOrTiwari
│ Estimate degrees: true
└ Assess correctness: false
┌ Info: Basis interpolated exponents summary:
│ Maximal interpolated degrees are: 2 for num. and 2 for den.
│ Maximal number of interpolated terms are: 2 for num. and 1 for den.
└ Points used: 24.
[ Info: Groebner basis computed in 0.018145012 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.003494806 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 7 rational functions
┌ Info: Final cleaning and simplification of generators. 
└ Out of 6 fractions 6 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 0.006524787 seconds. Result: true
[ Info: Out of 90 initial generators there are 5 indepdendent
[ Info: The ranking of the new set of generators is 18
[ Info: Simplifying identifiable functions
┌ Info: Computing parametric Groebner basis up to degrees (2, 2)
│ Ordering, input / target: degrevlex / InputOrdering
│ Rational interpolator: VanDerHoevenLecerf
│ Polynomial interpolator: PrimesBenOrTiwari
│ Estimate degrees: true
└ Assess correctness: false
┌ Info: Basis interpolated exponents summary:
│ Maximal interpolated degrees are: 2 for num. and 1 for den.
│ Maximal number of interpolated terms are: 2 for num. and 1 for den.
└ Points used: 20.
[ Info: Groebner basis computed in 0.041792148 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.00367098 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 12 rational functions
┌ Info: Computing normal forms (probabilistic)
│ Variables (11 in total): Nemo.QQMPolyRingElem[b, alpha, c, gama, delta, sigma, beta, x1, x2, x3, x4]
│ Up to degree: 2
└ Modulo: Finite field of characteristic 1073741827
[ Info: Used specialization points: 10
┌ Info: Final cleaning and simplification of generators. 
└ Out of 22 fractions 15 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 0.008341455 seconds. Result: true
[ Info: Out of 10 initial generators there are 9 indepdendent
[ Info: The ranking of the new set of generators is 138
[ Info: The search for identifiable functions concluded in 0.287172966 seconds
┌ Info: Result is
│   result =
│    9-element Vector{AbstractAlgebra.Generic.Frac{Nemo.QQMPolyRingElem}}:
│     x4
│     x1
│     sigma
│     c
│     ⋮
│     delta + beta
│     (alpha*gama)//x3
└     (gama*x2 + x3*beta)//x3
