┌ Info: 
└   FUNCTION_NAME = "find_identifiable_functions"
┌ Info: 
└   PROBLEM_NAME = "St"
┌ 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, R, W
[ Info: Parameters: e, rR, dr, d, g, r, a, T, Dd
[ 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 17.258683911 seconds
│   :ioeq_time = :ioeq_time
└   ioeq_time = 17.258683911
[ Info: Computing Wronskians
┌ Info: Computed in 11.560008122 seconds
│   :wrnsk_time = :wrnsk_time
└   wrnsk_time = 11.560008122
[ Info: Dimensions of the Wronskians [34, 2]
┌ Info: Ranks of the Wronskians computed in 0.03312828 seconds
│   :rank_time = :rank_time
└   rank_times = 0.03312828

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[ 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: 7 for num. and 2 for den.
└ Points used: 96.
[ Info: Groebner basis computed in 16.199455292 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 5.025309584 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: 21 for num. and 20 for den.
└ Points used: 640.
[ Info: Groebner basis computed in 13.074567867 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.527874369 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 77 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 7.305299799 seconds. Result: true
[ Info: Out of 39 initial generators there are 7 indepdendent
[ Info: The ranking of the new set of generators is 20503
[ 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: 7 for num. and 4 for den.
└ Points used: 96.
[ Info: Groebner basis computed in 2.018077495 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.226197947 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: 22 for num. and 12 for den.
└ Points used: 640.
[ Info: Groebner basis computed in 8.994702587 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.158968909 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 64 rational functions
┌ Info: Computing normal forms (probabilistic)
│ Variables (12 in total): Nemo.QQMPolyRingElem[e, rR, dr, d, g, r, a, T, Dd, S, R, W]
│ Up to degree: 2
└ Modulo: Finite field of characteristic 1073741827
[ Info: Used specialization points: 10
┌ Info: Final cleaning and simplification of generators. 
└ Out of 19 fractions 15 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 3.205094555 seconds. Result: true
[ Info: Out of 12 initial generators there are 11 indepdendent
[ Info: The ranking of the new set of generators is 1824206
[ Info: The search for identifiable functions concluded in 107.270947627 seconds
[ Info: Processing St
┌ 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 1.628550952 seconds
│   :ioeq_time = :ioeq_time
└   ioeq_time = 1.628550952
[ Info: Computing Wronskians
┌ Info: Computed in 0.085729298 seconds
│   :wrnsk_time = :wrnsk_time
└   wrnsk_time = 0.085729298
[ Info: Dimensions of the Wronskians [34, 2]
┌ Info: Ranks of the Wronskians computed in 8.8085e-5 seconds
│   :rank_time = :rank_time
└   rank_times = 8.8085e-5

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[ 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: 7 for num. and 2 for den.
└ Points used: 96.
[ Info: Groebner basis computed in 2.419476127 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.562057172 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: 21 for num. and 20 for den.
└ Points used: 640.
[ Info: Groebner basis computed in 8.844723147 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.396376502 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 77 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 3.952837116 seconds. Result: true
[ Info: Out of 39 initial generators there are 7 indepdendent
[ Info: The ranking of the new set of generators is 20503
[ 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: 7 for num. and 4 for den.
└ Points used: 96.
[ Info: Groebner basis computed in 1.652280197 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.21800423 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: 22 for num. and 12 for den.
└ Points used: 640.
[ Info: Groebner basis computed in 12.251662553 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.208063728 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 64 rational functions
┌ Info: Computing normal forms (probabilistic)
│ Variables (12 in total): Nemo.QQMPolyRingElem[e, rR, dr, d, g, r, a, T, Dd, S, R, W]
│ Up to degree: 2
└ Modulo: Finite field of characteristic 1073741827
[ Info: Used specialization points: 10
┌ Info: Final cleaning and simplification of generators. 
└ Out of 19 fractions 15 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 3.15107478 seconds. Result: true
[ Info: Out of 12 initial generators there are 11 indepdendent
[ Info: The ranking of the new set of generators is 1824206
[ Info: The search for identifiable functions concluded in 37.541896901 seconds
┌ Info: Result is
│   result =
│    11-element Vector{AbstractAlgebra.Generic.Frac{Nemo.QQMPolyRingElem}}:
│     Dd
│     T
│     S + R
│     rR*R - S*d*T + S*r - dr*R*T
│     ⋮
│     (dr*d + dr*a)//(dr^2 + d^2 + 2*d*a + a^2)
│     (-e*dr^2*g + 2*e*dr*d*g + e*dr*g*a - e*d^2*g - e*d*g*a - rR*dr*g*a + rR*d*g*a + rR*g*a^2 + dr*g^2*a + dr*g*r*a - d*g^2*a - d*g*r*a - g*r*a^2)//(S^2*dr*a - S^2*d*a - S*dr^2*R + 2*S*dr*R*d + S*dr*R*a - S*R*d^2 - S*R*d*a)
└     (-e*S*dr^2 + 2*e*S*dr*d + e*S*dr*a - e*S*d^2 - e*S*d*a - rR*S*dr*a + rR*S*d*a + rR*S*a^2 + 2*S*dr*g*a + S*dr*r*a - 2*S*d*g*a - S*d*r*a - S*r*a^2 - dr^2*R*g + 2*dr*R*d*g + dr*R*g*a - R*d^2*g - R*d*g*a)//(S^2*dr*a - S^2*d*a - S*dr^2*R + 2*S*dr*R*d + S*dr*R*a - S*R*d^2 - S*R*d*a)
