┌ Warning: Module Groebner with build ID fafbfcfd-23ce-df9e-0000-0c127c6f1c94 is missing from the cache.
│ This may mean Groebner [0b43b601-686d-58a3-8a1c-6623616c7cd4] does not support precompilation but is imported by a module that does.
└ @ Base loading.jl:1793
┌ Warning: Module Groebner with build ID fafbfcfd-e38f-37d1-0000-0c1083141a1f is missing from the cache.
│ This may mean Groebner [0b43b601-686d-58a3-8a1c-6623616c7cd4] does not support precompilation but is imported by a module that does.
└ @ Base loading.jl:1793
┌ Warning: Module Groebner with build ID fafbfcfd-e38f-37d1-0000-0c1083141a1f is missing from the cache.
│ This may mean Groebner [0b43b601-686d-58a3-8a1c-6623616c7cd4] does not support precompilation but is imported by a module that does.
└ @ Base loading.jl:1793
┌ Warning: Module Symbolics with build ID ffffffff-ffff-ffff-0000-0c2883439594 is missing from the cache.
│ This may mean Symbolics [0c5d862f-8b57-4792-8d23-62f2024744c7] does not support precompilation but is imported by a module that does.
└ @ Base loading.jl:1793
WARNING: using OrdinaryDiffEq.islinear in module ModelingToolkit conflicts with an existing identifier.
WARNING: using OrdinaryDiffEq.isconstant in module ModelingToolkit conflicts with an existing identifier.
┌ Info: 
└   FUNCTION_NAME = "find_identifiable_functions"
┌ Info: 
└   PROBLEM_NAME = "SEIRT"
┌ 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, E, I, R, N
[ Info: Parameters: lambda, alpha, beta
[ 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 14.963437926 seconds
│   :ioeq_time = :ioeq_time
└   ioeq_time = 14.963437926
[ Info: Computing Wronskians
┌ Info: Computed in 11.513246231 seconds
│   :wrnsk_time = :wrnsk_time
└   wrnsk_time = 11.513246231
[ Info: Dimensions of the Wronskians [1, 5]
┌ Info: Ranks of the Wronskians computed in 0.033544397 seconds
│   :rank_time = :rank_time
└   rank_times = 0.033544397

⌜ # Computing specializations..  	 Time: 0:00:10[K
✓ # Computing specializations..  	 Time: 0:00:11[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 13.326565836 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 4.580654287 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 3 fractions 3 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 5.111312332 seconds. Result: true
[ Info: Out of 6 initial generators there are 3 indepdendent
[ Info: The ranking of the new set of generators is 9
[ 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: 4 for num. and 2 for den.
└ Points used: 48.
[ Info: Groebner basis computed in 5.30852175 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 1.441456615 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: 3 for num. and 2 for den.
│ Maximal number of interpolated terms are: 4 for num. and 2 for den.
└ Points used: 56.
[ Info: Groebner basis computed in 0.027933207 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.003681361 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 13 rational functions
┌ Info: Computing normal forms (probabilistic)
│ Variables (8 in total): Nemo.QQMPolyRingElem[lambda, alpha, beta, S, E, I, R, N]
│ Up to degree: 2
└ Modulo: Finite field of characteristic 1073741827
[ Info: Used specialization points: 19
┌ Info: Final cleaning and simplification of generators. 
└ Out of 19 fractions 16 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 2.278346117 seconds. Result: true
[ Info: Out of 8 initial generators there are 7 indepdendent
[ Info: The ranking of the new set of generators is 40
[ Info: The search for identifiable functions concluded in 80.526661083 seconds
[ Info: Processing SEIRT
┌ 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 0.017409073 seconds
│   :ioeq_time = :ioeq_time
└   ioeq_time = 0.017409073
[ Info: Computing Wronskians
┌ Info: Computed in 0.064441137 seconds
│   :wrnsk_time = :wrnsk_time
└   wrnsk_time = 0.064441137
[ Info: Dimensions of the Wronskians [1, 5]
┌ Info: Ranks of the Wronskians computed in 2.53e-5 seconds
│   :rank_time = :rank_time
└   rank_times = 2.53e-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 0.007252134 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.001168218 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 3 fractions 3 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 0.001216442 seconds. Result: true
[ Info: Out of 6 initial generators there are 3 indepdendent
[ Info: The ranking of the new set of generators is 9
[ 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: 4 for num. and 2 for den.
└ Points used: 48.
[ Info: Groebner basis computed in 0.025624842 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.002566499 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: 3 for num. and 2 for den.
│ Maximal number of interpolated terms are: 4 for num. and 2 for den.
└ Points used: 56.
[ Info: Groebner basis computed in 0.030540385 seconds
[ Info: Checking two-sided inclusion modulo a prime
[ Info: Inclusion checked in 0.058848349 seconds. Result: true
[ Info: The coefficients of the Groebner basis are presented by 13 rational functions
┌ Info: Computing normal forms (probabilistic)
│ Variables (8 in total): Nemo.QQMPolyRingElem[lambda, alpha, beta, S, E, I, R, N]
│ Up to degree: 2
└ Modulo: Finite field of characteristic 1073741827
[ Info: Used specialization points: 19
┌ Info: Final cleaning and simplification of generators. 
└ Out of 19 fractions 16 are syntactically unique.
[ Info: Checking inclusion with probability 0.995
[ Info: Inclusion checked in 0.004356683 seconds. Result: true
[ Info: Out of 8 initial generators there are 7 indepdendent
[ Info: The ranking of the new set of generators is 40
[ Info: The search for identifiable functions concluded in 0.354843428 seconds
┌ Info: Result is
│   result =
│    7-element Vector{AbstractAlgebra.Generic.Frac{Nemo.QQMPolyRingElem}}:
│     N
│     I
│     beta
│     alpha*S
│     lambda*alpha
│     lambda + alpha
└     alpha*I + alpha*E
