┌ Info: 
└   FUNCTION_NAME = "find_identifiable_functions"
┌ Info: 
└   PROBLEM_NAME = "PK1"
┌ 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: k5, k3, s3, k4, k2, s2, k6, k1, k7
[ Info: Inputs: u1
[ 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 14.813380484 seconds
│   :ioeq_time = :ioeq_time
└   ioeq_time = 14.813380484
[ Info: Computing Wronskians
┌ Info: Computed in 10.922344015 seconds
│   :wrnsk_time = :wrnsk_time
└   wrnsk_time = 10.922344015
[ Info: Dimensions of the Wronskians [6, 6]
┌ Info: Ranks of the Wronskians computed in 0.032538919 seconds
│   :rank_time = :rank_time
└   rank_times = 0.032538919

⌜ # Computing specializations..  	 Time: 0:00:10[K
✓ # Computing specializations..  	 Time: 0:00:11[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: 3 for num. and 2 for den.
└ Points used: 48.
[ Info: Groebner basis computed in 13.50388363 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 4.367535206 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 10 rational functions
┌ Info: Final cleaning and simplification of generators. 
└ Out of 8 fractions 8 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 4.855497461 seconds. Result: true
[ Info: Out of 10 initial generators there are 8 indepdendent
[ Info: The ranking of the new set of generators is 2211
[ 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: 3 for num. and 2 for den.
└ Points used: 48.
[ Info: Groebner basis computed in 0.63031742 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.055419248 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 17 rational functions
┌ Info: Computing normal forms (probabilistic)
│ Variables (13 in total): Nemo.QQMPolyRingElem[k5, k3, s3, k4, k2, s2, k6, k1, k7, x1, x2, x3, x4]
│ Up to degree: 2
└ Modulo: Finite field of characteristic 1073741827
[ Info: Used specialization points: 32
┌ Info: Final cleaning and simplification of generators. 
└ Out of 34 fractions 28 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 0.00953511 seconds. Result: true
[ Info: Out of 16 initial generators there are 12 indepdendent
[ Info: The ranking of the new set of generators is 1842
[ Info: The search for identifiable functions concluded in 71.208893628 seconds
[ Info: Processing PK1
┌ 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 0.019911708 seconds
│   :ioeq_time = :ioeq_time
└   ioeq_time = 0.019911708
[ Info: Computing Wronskians
┌ Info: Computed in 0.010543607 seconds
│   :wrnsk_time = :wrnsk_time
└   wrnsk_time = 0.010543607
[ Info: Dimensions of the Wronskians [6, 6]
┌ Info: Ranks of the Wronskians computed in 2.6044e-5 seconds
│   :rank_time = :rank_time
└   rank_times = 2.6044e-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: 2 for num. and 2 for den.
│ Maximal number of interpolated terms are: 3 for num. and 2 for den.
└ Points used: 48.
[ Info: Groebner basis computed in 0.04151999 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.081180041 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 10 rational functions
┌ Info: Final cleaning and simplification of generators. 
└ Out of 8 fractions 8 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 0.006213159 seconds. Result: true
[ Info: Out of 10 initial generators there are 8 indepdendent
[ Info: The ranking of the new set of generators is 2211
[ 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: 3 for num. and 2 for den.
└ Points used: 48.
[ Info: Groebner basis computed in 0.047297256 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.044527799 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 17 rational functions
┌ Info: Computing normal forms (probabilistic)
│ Variables (13 in total): Nemo.QQMPolyRingElem[k5, k3, s3, k4, k2, s2, k6, k1, k7, x1, x2, x3, x4]
│ Up to degree: 2
└ Modulo: Finite field of characteristic 1073741827
[ Info: Used specialization points: 32
┌ Info: Final cleaning and simplification of generators. 
└ Out of 34 fractions 28 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 0.009709858 seconds. Result: true
[ Info: Out of 16 initial generators there are 12 indepdendent
[ Info: The ranking of the new set of generators is 1842
[ Info: The search for identifiable functions concluded in 0.735015714 seconds
┌ Info: Result is
│   result =
│    12-element Vector{AbstractAlgebra.Generic.Frac{Nemo.QQMPolyRingElem}}:
│     x1
│     k6
│     k4
│     k5
│     ⋮
│     k2 + k1
│     k3 + k7
└     (k2*s2)//(k3*k2 + k3*k1)
