┌ Info: 
└   FUNCTION_NAME = "find_identifiable_functions"
┌ Info: 
└   PROBLEM_NAME = "SIR 24"
┌ Info: 
└   KWARGS = (with_states = true, strategy = (:normalforms, 2))
┌ Info: 
└   GLOBAL_ID = Symbol("(:normalforms, 2)_with_states")
[ Info: Summary of the model:
[ Info: State variables: A, S, I, R
[ Info: Parameters: gamma, c, K, phi, mu
[ Info: Inputs: u1
[ Info: Outputs: y1
[ 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 10.757810632 seconds
│   :ioeq_time = :ioeq_time
└   ioeq_time = 10.757810632
[ Info: Computing Wronskians
┌ Info: Computed in 8.865002047 seconds
│   :wrnsk_time = :wrnsk_time
└   wrnsk_time = 8.865002047
[ Info: Dimensions of the Wronskians [27]
┌ Info: Ranks of the Wronskians computed in 0.021543348 seconds
│   :rank_time = :rank_time
└   rank_times = 0.021543348

⌜ # Computing specializations..  	 Time: 0:00:09[K
✓ # Computing specializations..  	 Time: 0:00:10[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: 2 for num. and 1 for den.
└ Points used: 24.
[ Info: Groebner basis computed in 12.851896459 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 4.680837467 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 5 rational functions
┌ Info: Final cleaning and simplification of generators. 
└ Out of 4 fractions 4 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 5.786824917 seconds. Result: true
[ Info: Out of 68 initial generators there are 3 indepdendent
[ Info: The ranking of the new set of generators is 9
[ 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 0 for den.
│ Maximal number of interpolated terms are: 2 for num. and 1 for den.
└ Points used: 16.
[ Info: Groebner basis computed in 5.304351725 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 1.492009406 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 8 rational functions
┌ Info: Computing normal forms (probabilistic)
│ Variables (9 in total): Nemo.QQMPolyRingElem[gamma, c, K, phi, mu, A, S, I, R]
│ Up to degree: 2
└ Modulo: Finite field of characteristic 1073741827
[ Info: Used specialization points: 7
┌ Info: Final cleaning and simplification of generators. 
└ Out of 21 fractions 16 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 2.384757791 seconds. Result: true
[ Info: Out of 18 initial generators there are 6 indepdendent
[ Info: The ranking of the new set of generators is 29
[ Info: The search for identifiable functions concluded in 68.019583352 seconds
[ Info: Processing SIR 24
┌ 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 0.07451518 seconds
│   :ioeq_time = :ioeq_time
└   ioeq_time = 0.07451518
[ Info: Computing Wronskians
┌ Info: Computed in 0.023554032 seconds
│   :wrnsk_time = :wrnsk_time
└   wrnsk_time = 0.023554032
[ Info: Dimensions of the Wronskians [27]
┌ Info: Ranks of the Wronskians computed in 6.2525e-5 seconds
│   :rank_time = :rank_time
└   rank_times = 6.2525e-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 0.023634679 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.004244255 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 5 rational functions
┌ Info: Final cleaning and simplification of generators. 
└ Out of 4 fractions 4 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 0.008183417 seconds. Result: true
[ Info: Out of 68 initial generators there are 3 indepdendent
[ Info: The ranking of the new set of generators is 9
[ 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 0 for den.
│ Maximal number of interpolated terms are: 2 for num. and 1 for den.
└ Points used: 16.
[ Info: Groebner basis computed in 0.080243513 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.004594389 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 8 rational functions
┌ Info: Computing normal forms (probabilistic)
│ Variables (9 in total): Nemo.QQMPolyRingElem[gamma, c, K, phi, mu, A, S, I, R]
│ Up to degree: 2
└ Modulo: Finite field of characteristic 1073741827
[ Info: Used specialization points: 7
┌ Info: Final cleaning and simplification of generators. 
└ Out of 21 fractions 16 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 0.015280007 seconds. Result: true
[ Info: Out of 18 initial generators there are 6 indepdendent
[ Info: The ranking of the new set of generators is 29
[ Info: The search for identifiable functions concluded in 0.359294394 seconds
┌ Info: Result is
│   result =
│    6-element Vector{AbstractAlgebra.Generic.Frac{Nemo.QQMPolyRingElem}}:
│     I
│     S
│     K
│     c*phi
│     -A + mu
└     gamma + A
