┌ Info: 
└   FUNCTION_NAME = "find_identifiable_functions"
┌ Info: 
└   PROBLEM_NAME = "HIV2_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: x1, x2, x3, x4
[ Info: Parameters: b, c, q1, w1, k2, d, s, k1, w2, q2
[ 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 14.39345378 seconds
│   :ioeq_time = :ioeq_time
└   ioeq_time = 14.39345378
[ Info: Computing Wronskians
┌ Info: Computed in 11.745306305 seconds
│   :wrnsk_time = :wrnsk_time
└   wrnsk_time = 11.745306305
[ Info: Dimensions of the Wronskians [4, 7]
┌ Info: Ranks of the Wronskians computed in 0.036473836 seconds
│   :rank_time = :rank_time
└   rank_times = 0.036473836

⌜ # Computing specializations..  	 Time: 0:00:09[K
✓ # Computing specializations..  	 Time: 0:00:10[K

⌜ # Computing specializations.. 	 Time: 0:00:00[K
  Points:  191[K
[K[A
✓ # Computing specializations.. 	 Time: 0:00:00[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 1 for den.
│ Maximal number of interpolated terms are: 5 for num. and 1 for den.
└ Points used: 80.
[ Info: Groebner basis computed in 12.671052123 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 4.308751347 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: 3 for num. and 1 for den.
│ Maximal number of interpolated terms are: 5 for num. and 1 for den.
└ Points used: 96.
[ Info: Groebner basis computed in 0.049694168 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.003686827 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 4.840188122 seconds. Result: true
[ Info: Out of 9 initial generators there are 8 indepdendent
[ Info: The ranking of the new set of generators is 120
[ 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: 5 for num. and 1 for den.
└ Points used: 96.
[ Info: Groebner basis computed in 0.869296378 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.006902103 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 4 for den.
│ Maximal number of interpolated terms are: 11 for num. and 8 for den.
└ Points used: 320.
[ Info: Groebner basis computed in 0.655256223 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.08457175 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 28 rational functions
┌ Info: Computing normal forms (probabilistic)
│ Variables (14 in total): Nemo.QQMPolyRingElem[b, c, q1, w1, k2, d, s, k1, w2, q2, x1, x2, x3, x4]
│ Up to degree: 2
└ Modulo: Finite field of characteristic 1073741827
[ Info: Used specialization points: 10
┌ Info: Final cleaning and simplification of generators. 
└ Out of 27 fractions 19 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 0.532104011 seconds. Result: true
[ Info: Out of 14 initial generators there are 12 indepdendent
[ Info: The ranking of the new set of generators is 866
[ Info: The search for identifiable functions concluded in 72.343451044 seconds
[ Info: Processing HIV2_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 0.013802676 seconds
│   :ioeq_time = :ioeq_time
└   ioeq_time = 0.013802676
[ Info: Computing Wronskians
┌ Info: Computed in 0.009375598 seconds
│   :wrnsk_time = :wrnsk_time
└   wrnsk_time = 0.009375598
[ Info: Dimensions of the Wronskians [4, 7]
┌ Info: Ranks of the Wronskians computed in 2.7021e-5 seconds
│   :rank_time = :rank_time
└   rank_times = 2.7021e-5

⌜ # Computing specializations.. 	 Time: 0:00:00[K
  Points:  214[K
[K[A
✓ # Computing specializations.. 	 Time: 0:00:00[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 1 for den.
│ Maximal number of interpolated terms are: 5 for num. and 1 for den.
└ Points used: 80.
[ Info: Groebner basis computed in 0.044873282 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.002400431 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: 3 for num. and 1 for den.
│ Maximal number of interpolated terms are: 5 for num. and 1 for den.
└ Points used: 96.
[ Info: Groebner basis computed in 0.048598179 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.071751119 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 0.004604849 seconds. Result: true
[ Info: Out of 9 initial generators there are 8 indepdendent
[ Info: The ranking of the new set of generators is 120
[ 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: 5 for num. and 1 for den.
└ Points used: 96.
[ Info: Groebner basis computed in 0.230283908 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.006253371 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 4 for den.
│ Maximal number of interpolated terms are: 11 for num. and 8 for den.
└ Points used: 320.
[ Info: Groebner basis computed in 0.571994021 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.011465661 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 28 rational functions
┌ Info: Computing normal forms (probabilistic)
│ Variables (14 in total): Nemo.QQMPolyRingElem[b, c, q1, w1, k2, d, s, k1, w2, q2, x1, x2, x3, x4]
│ Up to degree: 2
└ Modulo: Finite field of characteristic 1073741827
[ Info: Used specialization points: 10
┌ Info: Final cleaning and simplification of generators. 
└ Out of 27 fractions 19 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 0.515271673 seconds. Result: true
[ Info: Out of 14 initial generators there are 12 indepdendent
[ Info: The ranking of the new set of generators is 866
[ Info: The search for identifiable functions concluded in 1.994894319 seconds
┌ Info: Result is
│   result =
│    12-element Vector{AbstractAlgebra.Generic.Frac{Nemo.QQMPolyRingElem}}:
│     x4
│     x1
│     s
│     d
│     ⋮
│     c*w1 + c*k1 + c*w2 + w1*w2 + k1*w2
│     (q1*k1*x4 - k2*x3*q2 - w2*q2*x4)//(q2*x4)
└     (c*x3*x4 - k2*x3^2 - x3*w2*x4 + k1*x2*x4)//(q2*x4)
