┌ Info: 
└   FUNCTION_NAME = "find_identifiable_functions"
┌ Info: 
└   PROBLEM_NAME = "SIR 6"
┌ Info: 
└   KWARGS = (with_states = true, strategy = (:normalforms, 2))
┌ Info: 
└   GLOBAL_ID = Symbol("(:normalforms, 2)_with_states")
[ Info: Summary of the model:
[ Info: State variables: N, S, I, R
[ Info: Parameters: beta, gamma, K
[ 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 14.894061447 seconds
│   :ioeq_time = :ioeq_time
└   ioeq_time = 14.894061447
[ Info: Computing Wronskians
┌ Info: Computed in 11.582395517 seconds
│   :wrnsk_time = :wrnsk_time
└   wrnsk_time = 11.582395517
[ Info: Dimensions of the Wronskians [3, 1]
┌ Info: Ranks of the Wronskians computed in 0.031683844 seconds
│   :rank_time = :rank_time
└   rank_times = 0.031683844

⌜ # Computing specializations..  	 Time: 0:00:10[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: 1 for num. and 1 for den.
│ Maximal number of interpolated terms are: 1 for num. and 1 for den.
└ Points used: 8.
[ Info: Groebner basis computed in 13.081403646 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 4.328903063 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 4 rational functions
┌ Info: Final cleaning and simplification of generators. 
└ Out of 2 fractions 2 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 5.033323009 seconds. Result: true
[ Info: Out of 3 initial generators there are 2 indepdendent
[ Info: The ranking of the new set of generators is 30
[ 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 2 for den.
│ Maximal number of interpolated terms are: 1 for num. and 1 for den.
└ Points used: 10.
[ Info: Groebner basis computed in 4.869793422 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 1.460323289 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 0.010303523 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.038785817 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 7 rational functions
┌ Info: Computing normal forms (probabilistic)
│ Variables (7 in total): Nemo.QQMPolyRingElem[beta, gamma, K, N, 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 12 fractions 10 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 2.272401785 seconds. Result: true
[ Info: Out of 6 initial generators there are 5 indepdendent
[ Info: The ranking of the new set of generators is 21
[ Info: The search for identifiable functions concluded in 79.494977939 seconds
[ Info: Processing SIR 6
┌ 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 0.013263428 seconds
│   :ioeq_time = :ioeq_time
└   ioeq_time = 0.013263428
[ Info: Computing Wronskians
┌ Info: Computed in 0.005970669 seconds
│   :wrnsk_time = :wrnsk_time
└   wrnsk_time = 0.005970669
[ Info: Dimensions of the Wronskians [3, 1]
┌ Info: Ranks of the Wronskians computed in 2.2861e-5 seconds
│   :rank_time = :rank_time
└   rank_times = 2.2861e-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: 1 for num. and 1 for den.
│ Maximal number of interpolated terms are: 1 for num. and 1 for den.
└ Points used: 8.
[ Info: Groebner basis computed in 0.004213319 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.000975665 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 4 rational functions
┌ Info: Final cleaning and simplification of generators. 
└ Out of 2 fractions 2 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 0.000827314 seconds. Result: true
[ Info: Out of 3 initial generators there are 2 indepdendent
[ Info: The ranking of the new set of generators is 30
[ 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 2 for den.
│ Maximal number of interpolated terms are: 1 for num. and 1 for den.
└ Points used: 10.
[ Info: Groebner basis computed in 0.063941383 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.001725929 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 0.010994837 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.002350587 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 7 rational functions
┌ Info: Computing normal forms (probabilistic)
│ Variables (7 in total): Nemo.QQMPolyRingElem[beta, gamma, K, N, 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 12 fractions 10 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 0.002637676 seconds. Result: true
[ Info: Out of 6 initial generators there are 5 indepdendent
[ Info: The ranking of the new set of generators is 21
[ Info: The search for identifiable functions concluded in 0.201855831 seconds
┌ Info: Result is
│   result =
│    5-element Vector{AbstractAlgebra.Generic.Frac{Nemo.QQMPolyRingElem}}:
│     N
│     gamma
│     I*K
│     K*S
└     beta*I
