┌ Info: 
└   FUNCTION_NAME = "find_identifiable_functions"
┌ Info: 
└   PROBLEM_NAME = "KD1999"
┌ Info: 
└   KWARGS = (with_states = true, strategy = (:normalforms, 2))
┌ Info: 
└   GLOBAL_ID = Symbol("(:normalforms, 2)_with_states")
[ Info: Summary of the model:
[ Info: State variables: Ca, Cb, T, Tj, Arr
[ Info: Parameters: Th, cph, V, Vh, UA, roh, k0, ro, E, R, Ca0, Ta, cp, DH
[ Info: Inputs: u1, u2
[ 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 14.859055474 seconds
│   :ioeq_time = :ioeq_time
└   ioeq_time = 14.859055474
[ Info: Computing Wronskians
┌ Info: Computed in 11.368783578 seconds
│   :wrnsk_time = :wrnsk_time
└   wrnsk_time = 11.368783578
[ Info: Dimensions of the Wronskians [17, 76]
┌ Info: Ranks of the Wronskians computed in 0.033745197 seconds
│   :rank_time = :rank_time
└   rank_times = 0.033745197

⌜ # Computing specializations..  	 Time: 0:00:10[K
✓ # Computing specializations..  	 Time: 0:00:11[K

⌜ # Computing specializations..  	 Time: 0:00:04[K
✓ # Computing specializations..  	 Time: 0:00:04[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: 1 for num. and 1 for den.
└ Points used: 12.
[ Info: Groebner basis computed in 13.539701276 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 4.269553586 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 11 rational functions
┌ Info: Final cleaning and simplification of generators. 
└ Out of 9 fractions 9 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 4.966288486 seconds. Result: true
[ Info: Out of 129 initial generators there are 9 indepdendent
[ Info: The ranking of the new set of generators is 318
[ 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: 1 for num. and 1 for den.
└ Points used: 10.
[ Info: Groebner basis computed in 4.990149965 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 1.95632299 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 15 rational functions
┌ Info: Computing normal forms (probabilistic)
│ Variables (19 in total): Nemo.QQMPolyRingElem[Th, cph, V, Vh, UA, roh, k0, ro, E, R, Ca0, Ta, cp, DH, Ca, Cb, T, Tj, Arr]
│ Up to degree: 2
└ Modulo: Finite field of characteristic 1073741827
[ Info: Used specialization points: 23
┌ Info: Final cleaning and simplification of generators. 
└ Out of 24 fractions 14 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 2.329164488 seconds. Result: true
[ Info: Out of 33 initial generators there are 14 indepdendent
[ Info: The ranking of the new set of generators is 383
[ Info: The search for identifiable functions concluded in 79.801675661 seconds
[ Info: Processing KD1999
┌ 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 0.121730564 seconds
│   :ioeq_time = :ioeq_time
└   ioeq_time = 0.121730564
[ Info: Computing Wronskians
┌ Info: Computed in 0.255074091 seconds
│   :wrnsk_time = :wrnsk_time
└   wrnsk_time = 0.255074091
[ Info: Dimensions of the Wronskians [17, 76]
┌ Info: Ranks of the Wronskians computed in 0.000493863 seconds
│   :rank_time = :rank_time
└   rank_times = 0.000493863
[ 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: 1 for num. and 1 for den.
└ Points used: 12.
[ Info: Groebner basis computed in 0.035301251 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.078427221 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 11 rational functions
┌ Info: Final cleaning and simplification of generators. 
└ Out of 9 fractions 9 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 0.010188161 seconds. Result: true
[ Info: Out of 129 initial generators there are 9 indepdendent
[ Info: The ranking of the new set of generators is 318
[ 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: 1 for num. and 1 for den.
└ Points used: 10.
[ Info: Groebner basis computed in 0.042775762 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.006411253 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 15 rational functions
┌ Info: Computing normal forms (probabilistic)
│ Variables (19 in total): Nemo.QQMPolyRingElem[Th, cph, V, Vh, UA, roh, k0, ro, E, R, Ca0, Ta, cp, DH, Ca, Cb, T, Tj, Arr]
│ Up to degree: 2
└ Modulo: Finite field of characteristic 1073741827
[ Info: Used specialization points: 23
┌ Info: Final cleaning and simplification of generators. 
└ Out of 24 fractions 14 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 0.012873682 seconds. Result: true
[ Info: Out of 33 initial generators there are 14 indepdendent
[ Info: The ranking of the new set of generators is 383
[ Info: The search for identifiable functions concluded in 0.972061858 seconds
┌ Info: Result is
│   result =
│    14-element Vector{AbstractAlgebra.Generic.Frac{Nemo.QQMPolyRingElem}}:
│     Tj
│     T
│     Cb
│     Ca
│     ⋮
│     UA//DH
│     (ro*cp)//DH
└     (cph*roh)//DH
