┌ Info: 
└   FUNCTION_NAME = "find_identifiable_functions"
┌ Info: 
└   PROBLEM_NAME = "HIV"
┌ Info: 
└   KWARGS = (with_states = true, strategy = (:normalforms, 2))
┌ Info: 
└   GLOBAL_ID = Symbol("(:normalforms, 2)_with_states")
[ Info: Summary of the model:
[ Info: State variables: x, y, v, w, z
[ Info: Parameters: b, c, h, lm, d, k, u, q, a, beta
[ 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 15.121598103 seconds
│   :ioeq_time = :ioeq_time
└   ioeq_time = 15.121598103
[ Info: Computing Wronskians
┌ Info: Computed in 11.600649557 seconds
│   :wrnsk_time = :wrnsk_time
└   wrnsk_time = 11.600649557
[ Info: Dimensions of the Wronskians [18, 40]
┌ Info: Ranks of the Wronskians computed in 0.03182239 seconds
│   :rank_time = :rank_time
└   rank_times = 0.03182239

⌜ # 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 13.628719271 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 4.469603793 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: 4 for num. and 2 for den.
│ Maximal number of interpolated terms are: 1 for num. and 1 for den.
└ Points used: 16.
[ Info: Groebner basis computed in 0.031881999 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.006116587 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 8 fractions 8 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 4.949745412 seconds. Result: true
[ Info: Out of 81 initial generators there are 8 indepdendent
[ Info: The ranking of the new set of generators is 313
[ 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 5.329090161 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 1.430969866 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 14 rational functions
┌ Info: Computing normal forms (probabilistic)
│ Variables (15 in total): Nemo.QQMPolyRingElem[b, c, h, lm, d, k, u, q, a, beta, x, y, v, w, z]
│ Up to degree: 2
└ Modulo: Finite field of characteristic 1073741827
[ Info: Used specialization points: 33
┌ Info: Final cleaning and simplification of generators. 
└ Out of 21 fractions 13 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 2.288807222 seconds. Result: true
[ Info: Out of 15 initial generators there are 13 indepdendent
[ Info: The ranking of the new set of generators is 694
[ Info: The search for identifiable functions concluded in 81.266798602 seconds
[ Info: Processing HIV
┌ 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 0.041531383 seconds
│   :ioeq_time = :ioeq_time
└   ioeq_time = 0.041531383
[ Info: Computing Wronskians
┌ Info: Computed in 0.039119538 seconds
│   :wrnsk_time = :wrnsk_time
└   wrnsk_time = 0.039119538
[ Info: Dimensions of the Wronskians [18, 40]
┌ Info: Ranks of the Wronskians computed in 0.000163735 seconds
│   :rank_time = :rank_time
└   rank_times = 0.000163735
[ 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 0.108065895 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.005199491 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: 4 for num. and 2 for den.
│ Maximal number of interpolated terms are: 1 for num. and 1 for den.
└ Points used: 16.
[ Info: Groebner basis computed in 0.030773286 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.004982757 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 8 fractions 8 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 0.008484532 seconds. Result: true
[ Info: Out of 81 initial generators there are 8 indepdendent
[ Info: The ranking of the new set of generators is 313
[ 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 0.020476693 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.003751261 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 14 rational functions
┌ Info: Computing normal forms (probabilistic)
│ Variables (15 in total): Nemo.QQMPolyRingElem[b, c, h, lm, d, k, u, q, a, beta, x, y, v, w, z]
│ Up to degree: 2
└ Modulo: Finite field of characteristic 1073741827
[ Info: Used specialization points: 33
┌ Info: Final cleaning and simplification of generators. 
└ Out of 21 fractions 13 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 0.031848171 seconds. Result: true
[ Info: Out of 15 initial generators there are 13 indepdendent
[ Info: The ranking of the new set of generators is 694
[ Info: The search for identifiable functions concluded in 0.78024452 seconds
┌ Info: Result is
│   result =
│    13-element Vector{AbstractAlgebra.Generic.Frac{Nemo.QQMPolyRingElem}}:
│     z
│     w
│     a
│     u
│     ⋮
│     lm//y
│     (k*y)//v
└     (k*beta)//(c*y)
