┌ Info: 
└   FUNCTION_NAME = "find_identifiable_functions"
┌ Info: 
└   PROBLEM_NAME = "Biohydrogenation_io"
┌ Info: 
└   KWARGS = (with_states = true, strategy = (:normalforms, 2))
┌ Info: 
└   GLOBAL_ID = Symbol("(:normalforms, 2)_with_states")
[ Info: Summary of the model:
[ Info: State variables: x4, x5, x6, x7
[ Info: Parameters: k5, k8, k9, k6, k10, k7
[ 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 9.76925305 seconds
│   :ioeq_time = :ioeq_time
└   ioeq_time = 9.76925305
[ Info: Computing Wronskians
┌ Info: Computed in 7.430508235 seconds
│   :wrnsk_time = :wrnsk_time
└   wrnsk_time = 7.430508235
[ Info: Dimensions of the Wronskians [3, 36]
┌ Info: Ranks of the Wronskians computed in 0.021579832 seconds
│   :rank_time = :rank_time
└   rank_times = 0.021579832

⌜ # Computing specializations..  	 Time: 0:00:06[K
✓ # Computing specializations..  	 Time: 0:00:07[K

⌜ # Computing specializations..  	 Time: 0:00:02[K
✓ # Computing specializations..  	 Time: 0:00:02[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 8.739467815 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 2.917315027 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 9 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 3.175144434 seconds. Result: true
[ Info: Out of 37 initial generators there are 6 indepdendent
[ Info: The ranking of the new set of generators is 28
[ 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: 2 for num. and 1 for den.
└ Points used: 12.
[ Info: Groebner basis computed in 3.330130616 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.94422884 seconds. Result: false
┌ Info: Computing parametric Groebner basis up to degrees (4, 4)
│ 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: 2 for num. and 1 for den.
└ Points used: 12.
[ Info: Groebner basis computed in 0.010456122 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.002751883 seconds. Result: false
┌ Info: Computing parametric Groebner basis up to degrees (8, 8)
│ 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: 7 for num. and 0 for den.
│ Maximal number of interpolated terms are: 24 for num. and 1 for den.
└ Points used: 576.
[ Info: Groebner basis computed in 0.183661122 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.036062021 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 12 rational functions
┌ Info: Computing normal forms (probabilistic)
│ Variables (10 in total): Nemo.QQMPolyRingElem[k5, k8, k9, k6, k10, k7, x4, x5, x6, x7]
│ Up to degree: 2
└ Modulo: Finite field of characteristic 1073741827
[ Info: Used specialization points: 17
┌ Info: Final cleaning and simplification of generators. 
└ Out of 32 fractions 26 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 1.405326039 seconds. Result: true
[ Info: Out of 11 initial generators there are 9 indepdendent
[ Info: The ranking of the new set of generators is 57
[ Info: The search for identifiable functions concluded in 51.897747476 seconds
[ Info: Processing Biohydrogenation_io
┌ 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 0.012523842 seconds
│   :ioeq_time = :ioeq_time
└   ioeq_time = 0.012523842
[ Info: Computing Wronskians
┌ Info: Computed in 0.010737529 seconds
│   :wrnsk_time = :wrnsk_time
└   wrnsk_time = 0.010737529
[ Info: Dimensions of the Wronskians [3, 36]
┌ Info: Ranks of the Wronskians computed in 6.3881e-5 seconds
│   :rank_time = :rank_time
└   rank_times = 6.3881e-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: 2 for num. and 1 for den.
└ Points used: 24.
[ Info: Groebner basis computed in 0.01186811 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.001831845 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 9 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 0.00270046 seconds. Result: true
[ Info: Out of 37 initial generators there are 6 indepdendent
[ Info: The ranking of the new set of generators is 28
[ 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: 2 for num. and 1 for den.
└ Points used: 12.
[ Info: Groebner basis computed in 0.008367528 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.001915607 seconds. Result: false
┌ Info: Computing parametric Groebner basis up to degrees (4, 4)
│ 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: 2 for num. and 1 for den.
└ Points used: 12.
[ Info: Groebner basis computed in 0.009766999 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.001699177 seconds. Result: false
┌ Info: Computing parametric Groebner basis up to degrees (8, 8)
│ 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: 7 for num. and 0 for den.
│ Maximal number of interpolated terms are: 24 for num. and 1 for den.
└ Points used: 576.
[ Info: Groebner basis computed in 0.178461306 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.027700632 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 12 rational functions
┌ Info: Computing normal forms (probabilistic)
│ Variables (10 in total): Nemo.QQMPolyRingElem[k5, k8, k9, k6, k10, k7, x4, x5, x6, x7]
│ Up to degree: 2
└ Modulo: Finite field of characteristic 1073741827
[ Info: Used specialization points: 17
┌ Info: Final cleaning and simplification of generators. 
└ Out of 32 fractions 26 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 0.003331788 seconds. Result: true
[ Info: Out of 11 initial generators there are 9 indepdendent
[ Info: The ranking of the new set of generators is 57
[ Info: The search for identifiable functions concluded in 0.436713338 seconds
┌ Info: Result is
│   result =
│    9-element Vector{AbstractAlgebra.Generic.Frac{Nemo.QQMPolyRingElem}}:
│     x5
│     x4
│     k7
│     k6
│     ⋮
│     k9*k10
│     k10 - 2*x6
└     k8 + x6
