┌ Info: 
└   FUNCTION_NAME = "find_identifiable_functions"
┌ Info: 
└   PROBLEM_NAME = "generalizedLoktaVolterra (2o)"
┌ 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
[ Info: Parameters: beta22, beta21, r1, beta11, beta12, r2
[ Info: Inputs: 
[ Info: Outputs: y1, y2
[ 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 8.918950837 seconds
│   :ioeq_time = :ioeq_time
└   ioeq_time = 8.918950837
[ Info: Computing Wronskians
┌ Info: Computed in 7.515286501 seconds
│   :wrnsk_time = :wrnsk_time
└   wrnsk_time = 7.515286501
[ Info: Dimensions of the Wronskians [4, 4]
┌ Info: Ranks of the Wronskians computed in 0.022594012 seconds
│   :rank_time = :rank_time
└   rank_times = 0.022594012

⌜ # Computing specializations..  	 Time: 0:00:08[K
✓ # Computing specializations..  	 Time: 0:00:08[K

⌜ # Computing specializations..  	 Time: 0:00:03[K
✓ # Computing specializations..  	 Time: 0:00:03[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: 1 for num. and 0 for den.
│ Maximal number of interpolated terms are: 1 for num. and 1 for den.
└ Points used: 6.
[ Info: Groebner basis computed in 10.530526848 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 2.935004419 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 8 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 3.048521898 seconds. Result: true
[ Info: Out of 6 initial generators there are 6 indepdendent
[ Info: The ranking of the new set of generators is 21
[ 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: 1 for num. and 0 for den.
│ Maximal number of interpolated terms are: 1 for num. and 1 for den.
└ Points used: 6.
[ Info: Groebner basis computed in 4.081021784 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.959361102 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 10 rational functions
┌ Info: Computing normal forms (probabilistic)
│ Variables (8 in total): Nemo.QQMPolyRingElem[beta22, beta21, r1, beta11, beta12, r2, x1, x2]
│ Up to degree: 2
└ Modulo: Finite field of characteristic 1073741827
[ Info: Used specialization points: 1
┌ Info: Final cleaning and simplification of generators. 
└ Out of 16 fractions 8 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 1.483561774 seconds. Result: true
[ Info: Out of 10 initial generators there are 8 indepdendent
[ Info: The ranking of the new set of generators is 36
[ Info: The search for identifiable functions concluded in 53.059704748 seconds
[ Info: Processing generalizedLoktaVolterra (2o)
┌ 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 0.003457166 seconds
│   :ioeq_time = :ioeq_time
└   ioeq_time = 0.003457166
[ Info: Computing Wronskians
┌ Info: Computed in 0.002119997 seconds
│   :wrnsk_time = :wrnsk_time
└   wrnsk_time = 0.002119997
[ Info: Dimensions of the Wronskians [4, 4]
┌ Info: Ranks of the Wronskians computed in 1.431e-5 seconds
│   :rank_time = :rank_time
└   rank_times = 1.431e-5
[ 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: 1 for num. and 0 for den.
│ Maximal number of interpolated terms are: 1 for num. and 1 for den.
└ Points used: 6.
[ Info: Groebner basis computed in 0.003220748 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.000792972 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 8 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 0.000707984 seconds. Result: true
[ Info: Out of 6 initial generators there are 6 indepdendent
[ Info: The ranking of the new set of generators is 21
[ 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: 1 for num. and 0 for den.
│ Maximal number of interpolated terms are: 1 for num. and 1 for den.
└ Points used: 6.
[ Info: Groebner basis computed in 0.004382842 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.00154479 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 10 rational functions
┌ Info: Computing normal forms (probabilistic)
│ Variables (8 in total): Nemo.QQMPolyRingElem[beta22, beta21, r1, beta11, beta12, r2, x1, x2]
│ Up to degree: 2
└ Modulo: Finite field of characteristic 1073741827
[ Info: Used specialization points: 1
┌ Info: Final cleaning and simplification of generators. 
└ Out of 16 fractions 8 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 0.001484219 seconds. Result: true
[ Info: Out of 10 initial generators there are 8 indepdendent
[ Info: The ranking of the new set of generators is 36
[ Info: The search for identifiable functions concluded in 0.087953814 seconds
┌ Info: Result is
│   result =
│    8-element Vector{AbstractAlgebra.Generic.Frac{Nemo.QQMPolyRingElem}}:
│     x2
│     x1
│     r2
│     beta12
│     beta11
│     r1
│     beta21
└     beta22
