┌ Info: 
└   FUNCTION_NAME = "find_identifiable_functions"
┌ Info: 
└   PROBLEM_NAME = "QY"
┌ Info: 
└   KWARGS = (with_states = true, strategy = (:normalforms, 2))
┌ Info: 
└   GLOBAL_ID = Symbol("(:normalforms, 2)_with_states")
[ Info: Summary of the model:
[ Info: State variables: P0, P1, P2, P3, P4, P5
[ Info: Parameters: siga1, beta_SI, phi, alpa, M, Mar, Ks, beta, siga2, beta_SA
[ Info: Inputs: 
[ Info: Outputs: y
[ 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 10.307223936 seconds
│   :ioeq_time = :ioeq_time
└   ioeq_time = 10.307223936
[ Info: Computing Wronskians
┌ Info: Computed in 7.663582183 seconds
│   :wrnsk_time = :wrnsk_time
└   wrnsk_time = 7.663582183
[ Info: Dimensions of the Wronskians [64]
┌ Info: Ranks of the Wronskians computed in 0.034241084 seconds
│   :rank_time = :rank_time
└   rank_times = 0.034241084

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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 1 for den.
│ Maximal number of interpolated terms are: 3 for num. and 1 for den.
└ Points used: 40.
[ Info: Groebner basis computed in 13.026478741 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 4.615992501 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 3 for den.
│ Maximal number of interpolated terms are: 14 for num. and 3 for den.
└ Points used: 288.
[ Info: Groebner basis computed in 1.402870684 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.041799746 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 9 fractions 9 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 12.799673206 seconds. Result: true
[ Info: Out of 64 initial generators there are 9 indepdendent
[ Info: The ranking of the new set of generators is 106246
[ 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 5.651133836 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 1.622430914 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 3 for den.
│ Maximal number of interpolated terms are: 11 for num. and 3 for den.
└ Points used: 288.
[ Info: Groebner basis computed in 1.62649271 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.116181288 seconds. Result: false
┌ Info: Computing parametric Groebner basis up to degrees (8, 8)
│ 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: 8 for num. and 7 for den.
│ Maximal number of interpolated terms are: 63 for num. and 8 for den.
└ Points used: 2176.
[ Info: Groebner basis computed in 10.268999723 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.19508961 seconds. Result: false
┌ Info: Computing parametric Groebner basis up to degrees (16, 16)
│ 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: 9 for num. and 8 for den.
│ Maximal number of interpolated terms are: 98 for num. and 8 for den.
└ Points used: 4864.
[ Info: Groebner basis computed in 23.974627868 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.129613763 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 17 rational functions
┌ Info: Computing normal forms (probabilistic)
│ Variables (16 in total): Nemo.QQMPolyRingElem[siga1, beta_SI, phi, alpa, M, Mar, Ks, beta, siga2, beta_SA, P0, P1, P2, P3, P4, P5]
│ Up to degree: 2
└ Modulo: Finite field of characteristic 1073741827
[ Info: Used specialization points: 22
┌ Info: Final cleaning and simplification of generators. 
└ Out of 35 fractions 26 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 5.44060277 seconds. Result: true
[ Info: Out of 16 initial generators there are 15 indepdendent
[ Info: The ranking of the new set of generators is 7310506
[ Info: The search for identifiable functions concluded in 115.613875544 seconds
[ Info: Processing QY
┌ 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 0.093080173 seconds
│   :ioeq_time = :ioeq_time
└   ioeq_time = 0.093080173
[ Info: Computing Wronskians
┌ Info: Computed in 0.053931103 seconds
│   :wrnsk_time = :wrnsk_time
└   wrnsk_time = 0.053931103
[ Info: Dimensions of the Wronskians [64]
┌ Info: Ranks of the Wronskians computed in 0.000197964 seconds
│   :rank_time = :rank_time
└   rank_times = 0.000197964

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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 1 for den.
│ Maximal number of interpolated terms are: 3 for num. and 1 for den.
└ Points used: 40.
[ Info: Groebner basis computed in 0.182431751 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.039668511 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 3 for den.
│ Maximal number of interpolated terms are: 14 for num. and 3 for den.
└ Points used: 288.
[ Info: Groebner basis computed in 0.865865358 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.025456972 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 9 fractions 9 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 5.939944407 seconds. Result: true
[ Info: Out of 64 initial generators there are 9 indepdendent
[ Info: The ranking of the new set of generators is 106246
[ 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 0.196585866 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.131901927 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 3 for den.
│ Maximal number of interpolated terms are: 11 for num. and 3 for den.
└ Points used: 288.
[ Info: Groebner basis computed in 1.515592485 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.118398496 seconds. Result: false
┌ Info: Computing parametric Groebner basis up to degrees (8, 8)
│ 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: 8 for num. and 7 for den.
│ Maximal number of interpolated terms are: 63 for num. and 8 for den.
└ Points used: 2176.
[ Info: Groebner basis computed in 9.527328611 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.197369219 seconds. Result: false
┌ Info: Computing parametric Groebner basis up to degrees (16, 16)
│ 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: 9 for num. and 8 for den.
│ Maximal number of interpolated terms are: 98 for num. and 8 for den.
└ Points used: 4864.
[ Info: Groebner basis computed in 22.71748754 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.192174759 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 17 rational functions
┌ Info: Computing normal forms (probabilistic)
│ Variables (16 in total): Nemo.QQMPolyRingElem[siga1, beta_SI, phi, alpa, M, Mar, Ks, beta, siga2, beta_SA, P0, P1, P2, P3, P4, P5]
│ Up to degree: 2
└ Modulo: Finite field of characteristic 1073741827
[ Info: Used specialization points: 22
┌ Info: Final cleaning and simplification of generators. 
└ Out of 35 fractions 26 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 6.200444059 seconds. Result: true
[ Info: Out of 16 initial generators there are 15 indepdendent
[ Info: The ranking of the new set of generators is 7310506
[ Info: The search for identifiable functions concluded in 49.212118213 seconds
┌ Info: Result is
│   result =
│    15-element Vector{AbstractAlgebra.Generic.Frac{Nemo.QQMPolyRingElem}}:
│     P4
│     P3
│     P2
│     P1
│     ⋮
│     (siga1*beta_SI*phi*siga2 - siga1*beta_SI*siga2 - siga1*phi*siga2*beta_SA - siga1*M*beta_SA - beta_SI*phi*Mar*siga2 + beta_SI*Mar*siga2 - phi*M*Mar*beta_SA + M*Mar*beta_SA)//(siga1*beta_SA + phi*Mar*beta_SA - Mar*beta_SA)
│     (siga1^2*beta_SI*beta_SA - siga1^2*beta_SA^2 + siga1*beta_SI^2*phi*siga2 - siga1*beta_SI^2*siga2 + siga1*beta_SI*phi*Mar*beta_SA - 2*siga1*beta_SI*phi*siga2*beta_SA - siga1*beta_SI*M*beta_SA - siga1*beta_SI*Mar*beta_SA + siga1*beta_SI*siga2*beta_SA - siga1*phi*Mar*beta_SA^2 + siga1*phi*siga2*beta_SA^2 + siga1*Mar*beta_SA^2 - beta_SI^2*phi*Mar*siga2 + beta_SI^2*Mar*siga2 - beta_SI*phi*M*Mar*beta_SA + beta_SI*phi*Mar*siga2*beta_SA + beta_SI*M*Mar*beta_SA - beta_SI*Mar*siga2*beta_SA)//(siga1*M*beta_SA + phi*M*Mar*beta_SA - M*Mar*beta_SA)
└     (siga1^4*beta_SI*beta_SA^2 - siga1^4*beta_SA^3 + 2*siga1^3*beta_SI^2*phi*siga2*beta_SA - 2*siga1^3*beta_SI^2*siga2*beta_SA + 2*siga1^3*beta_SI*phi*Mar*beta_SA^2 - 4*siga1^3*beta_SI*phi*siga2*beta_SA^2 - siga1^3*beta_SI*M*beta_SA^2 - 2*siga1^3*beta_SI*Mar*beta_SA^2 + 3*siga1^3*beta_SI*siga2*beta_SA^2 - 2*siga1^3*phi*Mar*beta_SA^3 + 2*siga1^3*phi*siga2*beta_SA^3 + siga1^3*M*beta_SA^3 + 2*siga1^3*Mar*beta_SA^3 - siga1^3*siga2*beta_SA^3 + siga1^2*beta_SI^3*phi^2*siga2^2 - 2*siga1^2*beta_SI^3*phi*siga2^2 + siga1^2*beta_SI^3*siga2^2 + 2*siga1^2*beta_SI^2*phi^2*Mar*siga2*beta_SA - 3*siga1^2*beta_SI^2*phi^2*siga2^2*beta_SA - siga1^2*beta_SI^2*phi*M*siga2*beta_SA - 6*siga1^2*beta_SI^2*phi*Mar*siga2*beta_SA + 5*siga1^2*beta_SI^2*phi*siga2^2*beta_SA + siga1^2*beta_SI^2*M*siga2*beta_SA + 4*siga1^2*beta_SI^2*Mar*siga2*beta_SA - 2*siga1^2*beta_SI^2*siga2^2*beta_SA + siga1^2*beta_SI*phi^2*Mar^2*beta_SA^2 - 4*siga1^2*beta_SI*phi^2*Mar*siga2*beta_SA^2 + 3*siga1^2*beta_SI*phi^2*siga2^2*beta_SA^2 - 2*siga1^2*beta_SI*phi*M*Mar*beta_SA^2 + 2*siga1^2*beta_SI*phi*M*siga2*beta_SA^2 - 2*siga1^2*beta_SI*phi*Mar^2*beta_SA^2 + 10*siga1^2*beta_SI*phi*Mar*siga2*beta_SA^2 - 4*siga1^2*beta_SI*phi*siga2^2*beta_SA^2 + 2*siga1^2*beta_SI*M*Mar*beta_SA^2 - siga1^2*beta_SI*M*siga2*beta_SA^2 + siga1^2*beta_SI*Mar^2*beta_SA^2 - 6*siga1^2*beta_SI*Mar*siga2*beta_SA^2 + siga1^2*beta_SI*siga2^2*beta_SA^2 - siga1^2*phi^2*Mar^2*beta_SA^3 + 2*siga1^2*phi^2*Mar*siga2*beta_SA^3 - siga1^2*phi^2*siga2^2*beta_SA^3 + 2*siga1^2*phi*M*Mar*beta_SA^3 - siga1^2*phi*M*siga2*beta_SA^3 + 2*siga1^2*phi*Mar^2*beta_SA^3 - 4*siga1^2*phi*Mar*siga2*beta_SA^3 + siga1^2*phi*siga2^2*beta_SA^3 - 2*siga1^2*M*Mar*beta_SA^3 - siga1^2*Mar^2*beta_SA^3 + 2*siga1^2*Mar*siga2*beta_SA^3 - 2*siga1*beta_SI^3*phi^2*Mar*siga2^2 + 4*siga1*beta_SI^3*phi*Mar*siga2^2 - 2*siga1*beta_SI^3*Mar*siga2^2 - siga1*beta_SI^2*phi^2*M*Mar*siga2*beta_SA - 2*siga1*beta_SI^2*phi^2*Mar^2*siga2*beta_SA + 5*siga1*beta_SI^2*phi^2*Mar*siga2^2*beta_SA + 3*siga1*beta_SI^2*phi*M*Mar*siga2*beta_SA + 4*siga1*beta_SI^2*phi*Mar^2*siga2*beta_SA - 9*siga1*beta_SI^2*phi*Mar*siga2^2*beta_SA - 2*siga1*beta_SI^2*M*Mar*siga2*beta_SA - 2*siga1*beta_SI^2*Mar^2*siga2*beta_SA + 4*siga1*beta_SI^2*Mar*siga2^2*beta_SA - siga1*beta_SI*phi^2*M*Mar^2*beta_SA^2 + 2*siga1*beta_SI*phi^2*M*Mar*siga2*beta_SA^2 + 3*siga1*beta_SI*phi^2*Mar^2*siga2*beta_SA^2 - 4*siga1*beta_SI*phi^2*Mar*siga2^2*beta_SA^2 + 2*siga1*beta_SI*phi*M*Mar^2*beta_SA^2 - 4*siga1*beta_SI*phi*M*Mar*siga2*beta_SA^2 - 6*siga1*beta_SI*phi*Mar^2*siga2*beta_SA^2 + 6*siga1*beta_SI*phi*Mar*siga2^2*beta_SA^2 - siga1*beta_SI*M*Mar^2*beta_SA^2 + 2*siga1*beta_SI*M*Mar*siga2*beta_SA^2 + 3*siga1*beta_SI*Mar^2*siga2*beta_SA^2 - 2*siga1*beta_SI*Mar*siga2^2*beta_SA^2 + siga1*phi^2*M*Mar^2*beta_SA^3 - siga1*phi^2*M*Mar*siga2*beta_SA^3 - siga1*phi^2*Mar^2*siga2*beta_SA^3 + siga1*phi^2*Mar*siga2^2*beta_SA^3 - 2*siga1*phi*M*Mar^2*beta_SA^3 + siga1*phi*M*Mar*siga2*beta_SA^3 + 2*siga1*phi*Mar^2*siga2*beta_SA^3 - siga1*phi*Mar*siga2^2*beta_SA^3 + siga1*M*Mar^2*beta_SA^3 - siga1*Mar^2*siga2*beta_SA^3 + beta_SI^3*phi^2*Mar^2*siga2^2 - 2*beta_SI^3*phi*Mar^2*siga2^2 + beta_SI^3*Mar^2*siga2^2 + beta_SI^2*phi^2*M*Mar^2*siga2*beta_SA - 2*beta_SI^2*phi^2*Mar^2*siga2^2*beta_SA - 2*beta_SI^2*phi*M*Mar^2*siga2*beta_SA + 4*beta_SI^2*phi*Mar^2*siga2^2*beta_SA + beta_SI^2*M*Mar^2*siga2*beta_SA - 2*beta_SI^2*Mar^2*siga2^2*beta_SA - beta_SI*phi^2*M*Mar^2*siga2*beta_SA^2 + beta_SI*phi^2*Mar^2*siga2^2*beta_SA^2 + 2*beta_SI*phi*M*Mar^2*siga2*beta_SA^2 - 2*beta_SI*phi*Mar^2*siga2^2*beta_SA^2 - beta_SI*M*Mar^2*siga2*beta_SA^2 + beta_SI*Mar^2*siga2^2*beta_SA^2)//(siga1^2*phi*M^2*siga2*beta_SA - siga1^2*M^2*siga2*beta_SA + siga1*phi^2*M^2*Mar*siga2*beta_SA - 3*siga1*phi*M^2*Mar*siga2*beta_SA + 2*siga1*M^2*Mar*siga2*beta_SA - phi^2*M^2*Mar^2*siga2*beta_SA + 2*phi*M^2*Mar^2*siga2*beta_SA - M^2*Mar^2*siga2*beta_SA)
