┌ Info: 
└   FUNCTION_NAME = "find_identifiable_functions"
┌ Info: 
└   PROBLEM_NAME = "LLW1987_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
[ Info: Parameters: p2, p3, p4, p1
[ Info: Inputs: u
[ 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 11.187324844 seconds
│   :ioeq_time = :ioeq_time
└   ioeq_time = 11.187324844
[ Info: Computing Wronskians
┌ Info: Computed in 11.690890688 seconds
│   :wrnsk_time = :wrnsk_time
└   wrnsk_time = 11.690890688
[ Info: Dimensions of the Wronskians [7]
┌ Info: Ranks of the Wronskians computed in 0.032717557 seconds
│   :rank_time = :rank_time
└   rank_times = 0.032717557

⌜ # 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: 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 13.300662106 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 4.500891006 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 3 fractions 3 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 5.500234706 seconds. Result: true
[ Info: Out of 12 initial generators there are 3 indepdendent
[ Info: The ranking of the new set of generators is 12
[ 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 2 for den.
└ Points used: 24.
[ Info: Groebner basis computed in 4.9596624 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 1.43032905 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 11 rational functions
┌ Info: Computing normal forms (probabilistic)
│ Variables (7 in total): Nemo.QQMPolyRingElem[p2, p3, p4, p1, x1, x2, x3]
│ Up to degree: 2
└ Modulo: Finite field of characteristic 1073741827
[ Info: Used specialization points: 5
┌ Info: Final cleaning and simplification of generators. 
└ Out of 15 fractions 10 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 2.330507441 seconds. Result: true
[ Info: Out of 16 initial generators there are 7 indepdendent
[ Info: The ranking of the new set of generators is 273
[ Info: The search for identifiable functions concluded in 76.083377872 seconds
[ Info: Processing LLW1987_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 0.008702005 seconds
│   :ioeq_time = :ioeq_time
└   ioeq_time = 0.008702005
[ Info: Computing Wronskians
┌ Info: Computed in 0.005521375 seconds
│   :wrnsk_time = :wrnsk_time
└   wrnsk_time = 0.005521375
[ Info: Dimensions of the Wronskians [7]
┌ Info: Ranks of the Wronskians computed in 2.2835e-5 seconds
│   :rank_time = :rank_time
└   rank_times = 2.2835e-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 0 for den.
│ Maximal number of interpolated terms are: 2 for num. and 1 for den.
└ Points used: 16.
[ Info: Groebner basis computed in 0.012522053 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.001524342 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 3 fractions 3 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 0.07354508 seconds. Result: true
[ Info: Out of 12 initial generators there are 3 indepdendent
[ Info: The ranking of the new set of generators is 12
[ 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 2 for den.
└ Points used: 24.
[ Info: Groebner basis computed in 0.021807834 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.003364582 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 11 rational functions
┌ Info: Computing normal forms (probabilistic)
│ Variables (7 in total): Nemo.QQMPolyRingElem[p2, p3, p4, p1, x1, x2, x3]
│ Up to degree: 2
└ Modulo: Finite field of characteristic 1073741827
[ Info: Used specialization points: 5
┌ Info: Final cleaning and simplification of generators. 
└ Out of 15 fractions 10 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 0.006264301 seconds. Result: true
[ Info: Out of 16 initial generators there are 7 indepdendent
[ Info: The ranking of the new set of generators is 273
[ Info: The search for identifiable functions concluded in 0.252203768 seconds
┌ Info: Result is
│   result =
│    7-element Vector{AbstractAlgebra.Generic.Frac{Nemo.QQMPolyRingElem}}:
│     x3
│     x2*x1
│     p3*p1
│     p2*p4
│     p3 + p1
│     p2*x2 + p4*x1
└     (p2*x2 - p4*x1)//(p3 - p1)
