┌ Info: 
└   FUNCTION_NAME = "find_identifiable_functions"
┌ Info: 
└   PROBLEM_NAME = "SIWR with extra output"
┌ Info: 
└   KWARGS = (with_states = true, strategy = (:normalforms, 2))
┌ Info: 
└   GLOBAL_ID = Symbol("(:normalforms, 2)_with_states")
[ Info: Summary of the model:
[ Info: State variables: S, I, W, R
[ Info: Parameters: bi, gam, mu, bw, k, xi, a
[ Info: Inputs: 
[ Info: Outputs: y, 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 15.762548034 seconds
│   :ioeq_time = :ioeq_time
└   ioeq_time = 15.762548034
[ Info: Computing Wronskians
┌ Info: Computed in 11.46851165 seconds
│   :wrnsk_time = :wrnsk_time
└   wrnsk_time = 11.46851165
[ Info: Dimensions of the Wronskians [2, 123]
┌ Info: Ranks of the Wronskians computed in 0.034107292 seconds
│   :rank_time = :rank_time
└   rank_times = 0.034107292

⌜ # Computing specializations..  	 Time: 0:00:10[K
✓ # Computing specializations..  	 Time: 0:00:11[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 0 for den.
│ Maximal number of interpolated terms are: 1 for num. and 1 for den.
└ Points used: 6.
[ Info: Groebner basis computed in 13.626079677 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 4.503365033 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 8 rational functions
┌ Info: Final cleaning and simplification of generators. 
└ Out of 7 fractions 7 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 4.995673876 seconds. Result: true
[ Info: Out of 134 initial generators there are 7 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: 1 for num. and 1 for den.
└ Points used: 6.
[ Info: Groebner basis computed in 0.599177174 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.004197108 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 13 rational functions
┌ Info: Computing normal forms (probabilistic)
│ Variables (11 in total): Nemo.QQMPolyRingElem[bi, gam, mu, bw, k, xi, a, S, I, W, R]
│ Up to degree: 2
└ Modulo: Finite field of characteristic 1073741827
[ Info: Used specialization points: 1
┌ Info: Final cleaning and simplification of generators. 
└ Out of 22 fractions 11 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 0.004610447 seconds. Result: true
[ Info: Out of 13 initial generators there are 11 indepdendent
[ Info: The ranking of the new set of generators is 66
[ Info: The search for identifiable functions concluded in 70.932284768 seconds
[ Info: Processing SIWR with extra output
┌ 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 0.069094782 seconds
│   :ioeq_time = :ioeq_time
└   ioeq_time = 0.069094782
[ Info: Computing Wronskians
┌ Info: Computed in 0.111163694 seconds
│   :wrnsk_time = :wrnsk_time
└   wrnsk_time = 0.111163694
[ Info: Dimensions of the Wronskians [2, 123]
┌ Info: Ranks of the Wronskians computed in 0.001259917 seconds
│   :rank_time = :rank_time
└   rank_times = 0.001259917
[ 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: 1 for num. and 1 for den.
└ Points used: 6.
[ Info: Groebner basis computed in 0.129467991 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.035984269 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 8 rational functions
┌ Info: Final cleaning and simplification of generators. 
└ Out of 7 fractions 7 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 0.102916353 seconds. Result: true
[ Info: Out of 134 initial generators there are 7 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: 1 for num. and 1 for den.
└ Points used: 6.
[ Info: Groebner basis computed in 0.017762352 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.060670404 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 13 rational functions
┌ Info: Computing normal forms (probabilistic)
│ Variables (11 in total): Nemo.QQMPolyRingElem[bi, gam, mu, bw, k, xi, a, S, I, W, R]
│ Up to degree: 2
└ Modulo: Finite field of characteristic 1073741827
[ Info: Used specialization points: 1
┌ Info: Final cleaning and simplification of generators. 
└ Out of 22 fractions 11 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 0.011634936 seconds. Result: true
[ Info: Out of 13 initial generators there are 11 indepdendent
[ Info: The ranking of the new set of generators is 66
[ Info: The search for identifiable functions concluded in 0.671278301 seconds
┌ Info: Result is
│   result =
│    11-element Vector{AbstractAlgebra.Generic.Frac{Nemo.QQMPolyRingElem}}:
│     R
│     W
│     I
│     S
│     ⋮
│     mu
│     gam
└     bi
